Proceedings of the First NCTS Workshoi

Astroparticle Physics

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World Scientific

Astroparticle Physics

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Proceedings of the First NCTS Workshop

Astroparticle Physics Kenting, Taiwan 6 - 8 December 2001

Editors

Husain Athar National Center for Theoretical Sciences, Taiwan

Guey-Lin Lin National Chiao-Tung University, Taiwan

Kin-Wang Ng Academia Sinica, Taiwan

V|S* World Scientific « • NewJersey• London • Sine New Jersey • London • Singapore • Hong Kong

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ASTROPARTICLE PHYSICS Proceedings of the First NCTS Workshop

Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.

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PREFACE

The first NCTS Workshop on Astroparticle Physics was held at Renting National Park, Taiwan, from December 6l to December 8th, 2001. Approximately 45 participants attended the workshop.

At this meeting there were two pedagogical lectures on the ultrahigh energy neutrino physics and the particle physics in the early universe respectively. Besides these lectures, we arranged 12 invited talks addressing recent theoretical and experimental progress in neutrino astrophysics, cosmic-ray physics, and cosmology.

This workshop was supported by Taiwan's National Center for Theoretical Sciences.

Husain Athar Guey-Lin Lin Kin-Wang Ng

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CONTENTS

Preface V

Lectures

The Highest Energy Cosmic Rays, Gamma-Rays and Neutrinos: Facts, Fancy and Resolution 3

F. Halzen

Big Bang Nucleosynthesis, Implications of Recent CMB Data and Supersymmetric Dark Matter 23

K. Olive

Invited Talks

Research Program of the TEXENO Collaboration: Status and Highlights 65

H. T.-K. Wong and J. Li

New Results from AMS Cosmic Ray Measurements 77 M. A. Huang

Measurement of Attenuation Length in Rock Salt and Limestone in Radio Wave for Ultra-High Energy Neutrino Detector 90

M. Chiba, M. Kawaki, M. Inuzuka, T. Kamijo andH. Athar

Expected Performance of a Neutrino Telescope for Seeing AGN/GC Behind a Mountain 105

George W. S. Hou andM. A. Huang

Galactic High-Energy Cosmic-Ray Tau Neutrino Flux 117 K. Cheung, H. Athar, G.-L. Lin and J.-J. Tseng

viii

On Non Hadronic Origin of High Energy Neutrinos 127 H. Athar and G.-L. Lin

Questions in Cosmology and Particle Astrophysics 136 W.-Y. P. Hwang

Noncommutative Early Universe 147 P.-M.Ho

Cosmological Constant, Quintessence and Mini-Universes 156 X.-G. He

Stability of the Anisotropic Brane Cosmology 165 W.F.Kao

Dark Energy, Primordial Magnetic Fields, and Time-Varying Fine-Structure Constant 175

K.-W.Ng

List of Participants 183

Lectures

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T H E H I G H E S T E N E R G Y C O S M I C R A Y S , G A M M A - R A Y S A N D N E U T R I N O S : F A C T S , F A N C Y A N D R E S O L U T I O N

F R A N C I S H A L Z E N

Department of Physics, University of Wisconsin, Madison, WI 5S706, USA

Although cosmic rays were discovered 90 years ago, we do not know how and where they are accelerated. There is compelling evidence that the highest energy cosmic rays are extra-galactic — they cannot be contained by our galaxy's magnetic field anyway because their gyroradius exceeds its dimensions. Elementary elementary-particle physics dictates a universal upper limit on their energy of 5 X 101 9eV, the so-called Greisen-Kuzmin-Zatsepin cutoff; however, particles in excess of this energy have been observed, adding one more puzzle to the cosmic ray mystery. Mystery is nonetheless fertile ground for progress: we will review the facts and mention some very speculative interpretations. There is indeed a realistic hope tha t the oldest problem in astronomy will be resolved soon by ambitious experi­mentation: air shower arrays of 104 km 2 area, arrays of air Cerenkov detectors and kilometer-scale neutrino observatories.

1 The New Astronomy

Conventional astronomy spans 60 octaves in photon frequency, from 104cm radio-waves to 10~14 cm photons of GeV energy; see Fig. 1. This is an amazing expansion of the power of our eyes which scan the sky over less than a single octave just above 10~5 cm wavelength. The new astronomy, discussed in this talk, probes the Universe with new wavelengths, smaller than 10_ 1 4cm, or photon energies larger than 10 GeV. Besides gamma rays, gravitational waves and neutrinos as well as very high energy protons that are only weakly de­flected by the magnetic field of our galaxy, become astronomical messengers from the Universe. As exemplified time and again, the development of novel ways of looking into space invariably results in the discovery of unanticipated phenomena. As is the case with new accelerators, observing the predicted will be slightly disappointing.

Why do high energy astronomy with neutrinos or protons despite the considerable instrumental challenges which we will discuss further on? A mundane reason is that the Universe is not transparent to photons of TeV energy and above (units are: GeV/TeV/PeV/EeV/ZeV in ascending factors of 103). For instance, a PeV energy photon 7 cannot reach us from a source at the edge of our own galaxy because it will annihilate into an electron pair in an encounter with a 2.7 degree Kelvin microwave photon 7CMB before reaching our telescope. Energetic photons are absorbed on background light by pair

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Figure 1. The diffuse flux of photons in the Universe, from radio waves to GeV-photons. Above tens of GeV only limits are reported although individual sources emitting TeV gamma-rays have been identified. Above GeV energy cosmic rays dominate the spectrum.

production 7 + 7bkgnd —* e + + e of electrons above a threshold E given by

4J5e ~ (2me)2 , (1)

where E and e are the energy of the high-energy and background photon, respectively. Eq. (1) implies that TeV-photons are absorbed on infrared light, PeV photons on the cosmic microwave background and EeV photons on radio-waves. Only neutrinos can reach us without attenuation from the edge of the Universe.

At EeV energies proton astronomy may be possible. Near 50 EeV and above, the arrival directions of electrically charged cosmic rays are no longer scrambled by the ambient magnetic field of our own galaxy. They point back to their sources with an accuracy determined by their gyroradius in the inter galactic magnetic field B:

5

where d is the distance to the source. Scaled to units relevant to the problem,

0.1° ~ ( E \ • (6)

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Speculations on the strength of the inter-galactic magnetic field range from 10~7 to 10 - 1 2 Gauss. For a distance of 100 Mpc, the resolution may therefore be anywhere from sub-degree to nonexistent. It is still reasonable to expect that the arrival directions of the highest energy cosmic rays provide informa­tion on the location of their sources. Proton astronomy should be possible; it may also provide indirect information on intergalactic magnetic fields. Deter­mining their strength by conventional astronomical means has turned out to be challenging.

2 The Highest Energy Cosmic Rays: Facts

In October 1991, the Fly's Eye cosmic ray detector recorded an event of energy 3.0 ±o!54 x 1020 eV.1 This event, together with an event recorded by the Yakutsk air shower array in May 1989,2 of estimated energy ~ 2 x 1020 eV, constituted at the time the two highest energy cosmic rays ever seen. Their energy corresponds to a center of mass energy of the order of 700 TeV or ~ 50 Joules, almost 50 times LHC energy. In fact, all experiments3 have detected cosmic rays in the vicinity of 100 EeV since their discovery by the Haverah Park air shower array.4 The AGASA air shower array in Japan5 has by now accumulated an impressive 10 events with energy in excess of 102° eV.6

How well experiments can determine the energy of these events is a crit­ical issue. With a particle flux of order 1 event per km2 per century, these events can only be studied by using the earth's atmosphere as a particle de­tector. The experimental signatures of a shower initiated by a cosmic particle are illustrated in the cartoon shown in Fig. 2. The primary particle creates an electromagnetic and hadronic cascade. The electromagnetic shower grows to a shower maximum, and is subsequently absorbed by the atmosphere. This leads to the characteristic shower profile shown on the right hand side of the figure. The shower can be observed by: i) sampling the electromagnetic and hadronic components when they reach the ground with an array of particle detectors such as scintillators, ii) detecting the fluorescent light emitted by atmospheric nitrogen excited by the passage of the shower particles, iii) de­tecting the Cerenkov light emitted by the large number of particles at shower maximum, and iv) detecting muons and neutrinos underground. Fluorescent and Cerenkov light is collected by large mirrors and recorded by arrays of

6

Figure 2. Particles interacting near the top of the atmosphere initiate an electromagnetic and hadronlc particle cascade. Its profile is shown on the right. The different detection methods are illustrated. Mirrors collect the Cerenkov and nitrogen fluorescent light, arrays of detectors sample the shower reaching the ground, and underground detectors identify the muon component of the shower.

photomultipliers in their focus. The bottom line on energy measurement is that, at this time, several experiments using the first two techniques agree on the energy of EeV-showers within a typical resolution of 25%. Addition-

7

Conditions with E ~ 10 EeV • quasars • blasars • neutron stars

black holes

«grb r > 102 B g 1012 G M s Ms

ally, there is a systematic error of order 10% associated with the modeling of the showers. All techniques are indeed subject to the ambiguity of particle simulations that involve physics beyond LHC. If the final outcome turns out to be erroneous inference of the energy of the shower because of new physics associated with particle interactions, we will be happy to contemplate this discovery instead.

Whether the error in the energy measurement could be significantly larger is a key question to which the answer is almost certainly negative. A variety of techniques have been developed to overcome the fact that conventional air shower arrays do calorimetry by sampling at a single depth. They give results within the range already mentioned. So do the fluorescence experiments that embody continuous sampling calorimetry. The latter are subject to understanding the transmission of fluorescent light in the dark night atmosphere — a challenging problem given its variation with weather. Stereo fluorescence detectors will eliminate this last hurdle by doing two redundant measurements of the same shower from different locations. The HiRes collaborators have one year of data on tape which should allow them to settle any doubts as to energy calibration once and for all.

The premier experiments, HiRes and AGASA, agree that cosmic rays with energy in excess of 10 EeV are not a feature of our galaxy and that their spectrum extends beyond 100 EeV. They disagree on almost everything else. The AGASA experiment claims evidence that they come from point sources, and that they are mostly heavy nuclei. The HiRes data do not support this. Because of statistics, interpreting the measured fluxes as a function of energy is like reading tea leaves; one cannot help however reading different messages in the spectra (see Fig. 3). More about that later.

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Figure 3. The cosmic ray spectrum peaks in the vicinity of 1 GeV and has features near 1015 and 101 9eV. They are referred to as the "knee" and "ankle" in the spectrum. Shown is the flux of the highest energy cosmic rays near and beyond the ankle measured by the AGASA and HiRes experiments.

3 The Highest Energy Cosmic Rays: Fancy

3.1 Acceleration to > 100 EeV?

It is sensible to assume that, in order to accelerate a proton to energy E in a magnetic field B, the size R of the accelerator must be larger than the

9

gyroradius of the particle:

R > flgyro = | . (4)

I.e. the accelerating magnetic field must contain the particle orbit. This condition yields a maximum energy

E = TBR (5)

by dimensional analysis and nothing more. The T-factor has been included to allow for the possibility that we may not be at rest in the frame of the cosmic accelerator resulting in the observation of boosted particle energies. Theorists' imagination regarding the accelerators is limited to dense regions where ex­ceptional gravitational forces create relativistic particle flows: the dense cores of exploding stars, inflows on supermassive black holes at the centers of active galaxies, annihilating black holes or neutron stars? All speculations involve collapsed objects and we can therefore replace R by the Schwartzschild radius

R ~ GM/c2 (6)

to obtain

E ~ TBM. (7)

Given the microgauss magnetic field of our galaxy, no structures are large or massive enough to reach the energies of the highest energy cosmic rays. Dimensional analysis therefore limits their sources to extragalactic objects; a few common speculations are listed in Table 1. Nearby active galactic nu­clei distant by ~ 100 Mpc and powered by a billion solar mass black holes are candidates. With kilo-Gauss fields we reach 100 EeV. The jets (blazars) emitted by the central black hole could reach similar energies in accelerating substructures boosted in our direction by a T-factor of 10, possibly higher. The neutron star or black hole remnant of a collapsing supermassive star could support magnetic fields of 1012 Gauss, possibly larger. Shocks with T > 102

emanating from the collapsed black hole could be the origin of gamma ray bursts and, possibly, the source of the highest energy cosmic rays.

The above speculations are reinforced by the fact that the sources listed happen to also be the sources of the highest energy gamma rays observed. At this point however a reality check is in order. Let me first point out that the above dimensional analysis applies to the Fermilab accelerator: 10 kGauss fields over several kilometers yield 1 TeV. The argument holds because, with optimized design and perfect alignment of magnets, the accelerator reaches efficiencies matching the dimensional limit. It is highly questionable that

10

Nature can achieve this feat. Theorists can imagine acceleration in shocks with efficiency of perhaps 10%.

The astrophysics problem is so daunting that many believe that cosmic rays are not the beam of cosmic accelerators but the decay products of rem­nants from the early Universe, for instance topological defects associated with a grand unified GUT phase transition. A topological defect will suffer a chain decay into GUT particles X,Y, that subsequently decay to familiar weak bosons, leptons and quark- or gluon jets. Cosmic rays are the fragmentation products of these jets. We know from accelerator studies that, among the frag­mentation products of jets, neutral pions (decaying into photons) dominate protons by two orders of magnitude. Therefore, if the decay of topological defects is the source of the highest energy cosmic rays, they must be photons. This is a problem because the highest energy event observed by the Fly's Eye is not likely to be a photon.7 A photon of 300 EeV will interact with the magnetic field of the earth far above the atmosphere and disintegrate into lower energy cascades — roughly ten at this particular energy. The measured shower profile of the event does not support this assumption; see Fig. 4. One can live and die by a single event!

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Figure 4. The composite atmospheric shower profile of a 3 x 102 0 eV 7-ray shower calculated with Landau-Pomeranchuk-Migdal (solid) and Bethe-Heitler (dashed) electromagnetic cross sections. The central line shows the average shower profile and the upper and lower lines show 1 a deviations — not visible for the BH case, where lines overlap. The experimental shower profile is shown along with the data points. It does not fit the profile of a photon shower.

3.10™ eV y ray

11

3.2 Are Cosmic Rays Really Protons: the GZK Cutoff?

All experimental signatures agree on the particle nature of the cosmic rays — they look like protons, or, possibly, nuclei. We mentioned at the beginning of this article that the Universe is opaque to photons with energy in excess of tens of TeV because they annihilate into electron pairs in interactions with background light. Also protons interact with background light, predominantly by photoproduction of the A-resonance, i.e. P + JCMB —* A —* ir + p above a threshold energy Ep of about 50 EeV given by:

2Epe > (m\ - m2) . (8)

The major source of proton energy loss is photoproduction of pions on a target of cosmic microwave photons of energy e. The Universe is therefore also opaque to the highest energy cosmic rays, with an absorption length:

\ p = ( « C M B 0 - P + 7 C M B ) _ 1 (9)

= lOMpc, (10)

or only tens of megaparsecs when their energy exceeds 50 EeV. This so-called GZK cutoff establishes a universal upper limit on the energy of the cosmic rays. The cutoff is robust, depending only on two known numbers: ncMB =

400cm - 3 and 0-p+7cMB = 10_ 2 8cm2 . Protons with energy in excess of 100 EeV, emitted in distant quasars and

gamma ray bursts, will have lost their energy to pions before reaching our detectors. They have, nevertheless, been observed, as we have previously dis­cussed. They do not point to any sources within the GZK-horizon however, i.e. to sources in our local cluster of galaxies. There are three possible reso­lutions: i) the protons are accelerated in nearby sources, ii) they do reach us from distant sources which accelerate them to much higher energies than we observe, thus exacerbating the acceleration problem, or iii) the highest energy cosmic rays are not protons.

The first possibility raises the challenge of finding an appropriate accel­erator by confining these already unimaginable sources to our local galaxy cluster. It is not impossible that all cosmic rays are produced by the active galaxy M87, or by a nearby gamma ray burst which exploded a few -hun­dred years ago. The sources identified by the AGASA array do not correlate however with any such candidates.

Stecker8 has speculated that the highest energy cosmic rays are Fe nuclei with a delayed GZK cutoff. The details are compicated but the relevant quantity in the problem is 7 = E/AM, where A is the atomic number and M the nucleon mass. For a fixed observed energy, the smallest boost above GZK threshold is associated with the largest atomic mass, i.e. Fe.

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3.3 Could Cosmic Rays be Photons or Neutrinos?

When discussing topological defects, I already challenged the possibility that the original Fly's Eye event is a photon. The detector collects light produced by the fluorescence of atmospheric nitrogen along the path of the high-energy shower traversing the atmosphere. The anticipated shower profile of a 300 EeV photon is shown in Fig. 4. It disagrees with the data.

The observed shower profile roughly fits that of a primary proton, or, possibly, that of a nucleus. The shower profile information is however sufficient to conclude that the event is unlikely to be of photon origin. The same conclusion is reached for the Yakutsk event that is characterized by a huge number of secondary muons, inconsistent with an electromagnetic cascade initiated by a gamma-ray. Finally, the AGASA collaboration claims evidence for "point" sources above 10 EeV. The arrival directions are however smeared out in a way consistent with primaries deflected by the galactic magnetic field. Again, this indicates charged primaries and excludes photons.

Neutrino primaries are definitely ruled out. Standard model neutrino physics is understood, even for EeV energy. The average x of the parton mediating the neutrino interaction is of order x ~ y/M^,/s ~ 10~6 so that the perturbative result for the neutrino-nucleus cross section is calculable from measured HERA structure functions. Even at 100 EeV a reliable value of the cross section can be obtained based on QCD-inspired extrapolations of the structure function. The neutrino cross section is known to better than an order of magnitude. It falls 5 orders of magnitude short of the strong cross sections required to make a neutrino interact in the upper atmosphere to create an air shower.

Could EeV neutrinos be strongly interacting because of new physics? In theories with TeV-scale gravity one can imagine that graviton exchange dominates all interactions and thus erases the difference between quarks and neutrinos at the energies under consideration. Notice however that the actual models performing this feat require a fast turn-on of the cross section with energy that violates S-wave unitarity.9

We thus exhausted the possibilities: neutrons, muons and other candidate primaries one may think of are unstable. EeV neutrons barely live long enough to reach us from sources at the edge of our galaxy.

4 A Three Prong Assault on the Cosmic Ray Puzzle

We conclude that, where the highest energy cosmic rays are concerned, both the accelerator mechanism and the particle physics are totally enigmatic. The

13

mystery has inspired a worldwide effort to tackle the problem with novel ex­perimentation in three complementary areas of research: air shower detection, atmospheric Cerenkov astronomy and underground neutrino physics. While some of the future instruments have other missions, all are likely to have a major impact on cosmic ray physics.

4-1 Giant Cosmic Ray Detectors

With super-GZK fluxes of the order of a single event per kilometer-squared per century, the outstanding problem is the lack of statistics; see Fig. 3. In the next five years, a qualitative improvement can be expected from the operation of the HiRes fluorescence detector in Utah. With improved instrumentation yielding high quality data from 2 detectors operated in coincidence, the in­terplay between sky transparency and energy measurement can be studied in detail. We can safely anticipate that the existence of super-Greisen energies will be conclusively demonstrated by using the instrument's calorimetric mea­surements. A mostly Japanese collaboration has proposed a next-generation fluorescence detector, the Telescope Array.

The Auger air shower array is tackling the low rate problem with a huge collection area covering 3000 square kilometers on an elevated plain in Western Argentina. The instrumentation consists of 1600 water Cerenkov detectors spaced by 1.5 km. For calibration, about 15 percent of the showers occurring at night will be viewed by 3 HiRes-style fluorescence detectors. The detector will observe several thousand events per year above lOEeV and tens above 100 EeV, with the exact numbers depending on the detailed shape of the observed spectrum which is at present a matter of speculation; see Fig. 3.

4-2 Gamma-Rays from Cosmic Accelerators

An alternative way to identify the sources of the cosmic rays is illustrated in Fig. 5. The cartoon draws our attention to the fact that cosmic accelerators are also cosmic beam dumps producing secondary photon and neutrino beams. Accelerating particles to TeV energy and above requires high-speed, massive bulk flows. These are likely to have their origin in exceptional gravitational forces associated with dense cores of exploding stars, inflows onto supermas-sive black holes at the centers of active galaxies, annihilating black holes or neutron stars. In such situations, accelerated particles are likely to pass through intense radiation fields or dense clouds of gas leading to production of secondary photons and neutrinos that accompany the primary cosmic-ray beam. An example of an electromagnetic beam dump is the X-ray radiation

14

NEUTRINO BEAMS: HEAVEN & EARTH

; directional beam

black hole

radiation enveloping black hole

Figure 5.

fields surrounding the central black holes of active galaxies. The target mate­rial, whether a gas or particles or of photons, is likely to be sufficiently tenuous so that the primary beam and the photon beam are only partially attenuated. However, it is also a real possibility that one could have a shrouded source from which only the neutrinos can emerge, as in terrestrial beam dumps at CERN and Fermilab.

The astronomy event of the 21st century could be the simultaneous ob­servation of TeV-gamma rays, neutrinos and gravitational waves from cata­clysmic events associated with the source of the cosmic rays.

We first concentrate on the possibility of detecting high-energy photon beams. After two decades, ground-based gamma ray astronomy has become a mature science.10 A large mirror, viewed by an array of photomultipliers, collects the Cerenkov light emitted by air showers and images the showers in order to determine the arrival direction as well as the nature of the pri­mary particle; see Fig. 2. These experiments have opened a new window

15

in astronomy by extending the photon spectrum to 20 TeV, possibly beyond. Observations have revealed spectacular TeV-emission from galactic supernova remnants and nearby quasars, some of which emit most of their energy in very short burst of TeV-photons.

But there is the dog that didn't bark. No evidence has emerged for 7r° origin of the TeV radiation and, therefore, no cosmic ray sources have yet been identified. Dedicated searches for photon beams from suspected cosmic ray sources, such as the supernova remnants IC433 and 7-Cygni, came up empty handed. While not relevant to the topic covered by this talk, supernova remnants are theorized to be the sources of the bulk of the cosmic rays that are of galactic origin. The evidence is still circumstantial.

The field of gamma ray astronomy is buzzing with activity to construct second-generation instruments. Space-based detectors are extending their reach from GeV to TeV energy with AMS and, especially, GLAST, while the ground-based Cerenkov collaborations are designing instruments with lower thresholds. In the not so far future both techniques should generate over­lapping measurements in the 10~102 GeV energy range. All ground-based air Cerenkov experiments aim at lower threshold, better angular- and energy-resolution, and a longer duty cycle. One can however identify three pathways to reach these goals:

1. larger mirror area, exploiting the parasitic use of solar collectors during nighttime (CELESTE, STACEY and SOLARII),11

2. better, or rather, ultimate imaging with the 17 m MAGIC mirror,12

3. larger field of view using multiple telescopes (VERITAS, HEGRA and HESS).

The Whipple telescope pioneered the atmospheric Cerenkov technique. VERITAS13 is an array of 9 upgraded Whipple telescopes, each with a field of view of 6 degrees. These can be operated in coincidence for improved angular resolution, or be pointed at 9 different 6 degree bins in the night sky, thus achieving a large field of view. The HEGRA collaboration14 is already operating four telescopes in coincidence and is building an upgraded facility with excellent viewing and optimal location near the equator in Namibia.

There is a dark horse in this race: Milagro.15 The Milagro idea is to lower the threshold of conventional air shower arrays to 100 GeV by instrumenting a pond of five million gallons of ultra-pure water with photomultipliers. For time-varying signals, such as bursts, the threshold may be lower.

16

4-3 High Energy Neutrino Telescopes

Although neutrino telescopes have multiple interdisciplinary science missions, the search for the sources of the highest-energy cosmic rays stands out because it clearly identifies the size of the detector required to do the science.16 For guidance in estimating expected signals, one makes use of data covering the highest-energy cosmic rays in Fig. 3 as well as known sources of non-thermal, high-energy gamma rays. Accelerating particles to TeV energy and above involves neutron stars or black holes. As already explained in the context of Fig. 5, some fraction of them will interact in the radiation fields surrounding the source, whatever it may be, to produce pions. These interactions may also be hadronic collisions with ambient gas. In either case, the neutral pions decay to photons while charged pions include neutrinos among their decay products with spectra related to the observed gamma-ray spectra. Estimates based on this relationship show that a kilometer-scale detector is needed to see neutrino signals.

The same conclusion is reached in specific models. Assuming, for instance, that gamma ray bursts are the cosmic accelerators of the highest-energy cos­mic rays, one can calculate from textbook particle physics how many neutrinos are produced when the particle beam coexists with the observed MeV energy photons in the original fireball. We thus predict the observation of 10-100 neu­trinos of PeV energy per year in a detector with a kilometer-square effective area. In general, the potential scientific payoff of doing neutrino astronomy arises from the great penetrating power of neutrinos, which allows them to emerge from dense inner regions of energetic sources.

Whereas the science is compelling, the real challenge has been to develop a reliable, expandable and affordable detector technology. Suggestions to use a large volume of deep ocean water for high-energy neutrino astronomy were made as early as the 1960s. In the case of the muon neutrino, for instance, the neutrino (i/M) interacts with a hydrogen or oxygen nucleus in the water and produces a muon travelling in nearly the same direction as the neutrino. The blue Cerenkov light emitted along the muon's ~kilometer-long trajectory is detected by strings of photomultiplier tubes deployed deep below the surface. With the first observation of neutrinos in the Lake Baikal and the (under-ice) South Pole neutrino telescopes, there is optimism that the technological challenges to build neutrino telescopes have been met.

The first generation of neutrino telescopes, launched by the bold decision of the DUMAND collaboration to construct such an instrument, are designed to reach a large telescope area and detection volume for a neutrino threshold of order 10 GeV. The optical requirements of the detector medium are severe.

17

A large absorption length is required because it determines the spacings of the optical sensors and, to a significant extent, the cost of the detector. A long scattering length is needed to preserve the geometry of the Cerenkov pattern. Nature has been kind and offered ice and water as adequate natural Cerenkov media. Their optical properties are, in fact, complementary. Water and ice have similar attenuation length, with the role of scattering and absorption reversed. Optics seems, at present, to drive the evolution of ice and water detectors in predictable directions: towards very large telescope area in ice exploiting the long absorption length, and towards lower threshold and good muon track reconstruction in water exploiting the long scattering length.

DUMAND, the pioneering project located off the coast of Hawaii, demon­strated that muons could be detected by this technique, but the planned detector was never realized. A detector composed of 96 photomultiplier tubes located deep in Lake Baikal was the first to demonstrate the detec­tion of neutrino-induced muons in natural water.17 In the following years, NT-200 will be operated as a neutrino telescope with an effective area be­tween 103~5 x 103 m2, depending on energy. Presumably too small to detect neutrinos from extraterrestrial sources, NT-200 will serve as the prototype for a larger telescope. For instance, with 2000 OMs, a threshold of 10~20 GeV and an effective area of 5 x 104~105 m2, an expanded Baikal telescope would fill the gap between present detectors and planned high-threshold detectors of cubic kilometer size. Its key advantage would be low threshold.

The Baikal experiment represents a proof of concept for deep ocean projects. These do however have the advantage of larger depth and optically superior water. Their challenge is to find reliable and affordable solutions to a variety of technological challenges for deploying a deep underwater detector. The European collaborations ANTARES18 and NESTOR19 plan to deploy large-area detectors in the Mediterranean Sea within the next year. The NEMO Collaboration is conducting a site study for a future kilometer-scale detector in the Mediterranean.20

The AMANDA collaboration, situated at the U.S. Amundsen-Scott South Pole Station, has demonstrated the merits of natural ice as a Cerenkov detec­tor medium.21 In 1996, AMANDA was able to observe atmospheric neutrino candidates using only 80 eight-inch photomultiplier tubes.21

With 302 optical modules instrumenting approximately 6000 tons of ice, AMANDA extracted several hundred atmospheric neutrino events from its first 130 days of data. AMANDA was thus the first first-generation neutrino telescope with an effective area in excess of 10,000 square meters for TeV muons.22 In rate and all characteristics the events are consistent with atmo­spheric neutrino origin. Their energies are in the 0.1-1 TeV range. The shape

18

Figure 6. Reconstructed zenith angle distribution. The points mark the data and the shaded boxes a simulation of atmospheric neutrino events, the widths of the boxes indicating the error bars.

Figure 7. Distribution in declination and right ascension of the up-going events on the sky.

of the zenith angle distribution is compared to a simulation of the atmospheric neutrino signal in Fig. 6. The variation of the measured rate with zenith an­gle is reproduced by the simulation to within the statistical uncertainty. Note that the tall geometry of the detector strongly influences the dependence on zenith angle in favor of more vertical muons.

The arrival directions of the neutrinos are shown in Fig. 7. A statistical analysis indicates no evidence for point sources in this sample. An estimate

19

of the energies of the up-going muons (based on simulations of the number of reporting optical modules) indicates that all events have energies consis­tent with an atmospheric neutrino origin. This enables AMANDA to reach a level of sensitivity to a diffuse flux of high energy extra-terrestrial neutri­nos of order22 dN/dEv — 10~ 6 E t 7

2 cm - 2 s~ 1 s r _ 1 GeV - 1 , assuming an E"2

spectrum. At this level they exclude a variety of theoretical models which as­sume the hadronic origin of TeV photons from active galaxies and blazars.23

Searches for neutrinos from gamma-ray bursts, for magnetic monopoles, and for a cold dark matter signal from the center of the Earth are also in progress and, with only 138 days of data, yield limits comparable to or better than those from smaller underground neutrino detectors that have operated for a much longer period.

In January 2000, AMANDA-II was completed. It consists of 19 strings with a total of 677 OMs arranged in concentric circles, with the ten strings from AMANDA forming the central core of the new detector. First data with the expanded detector indicate an atmospheric neutrino rate increased by a factor of three, to 4-5 events per day. AMANDA-II has met the key challenge of neutrino astronomy: it has developed a reliable, expandable, and affordable technology for deploying a kilometer-scale neutrino detector named IceCube.

Neutrino flavor

ve

6 9 12 15 18 21 Log(energy/eV)

Figure 8. Although IceCube detects neutrinos of any flavor above a threshold of ~ 0.1 TeV, it can identify their flavor and measure their energy in the ranges shown. Filled areas: particle identification, energy, and angle. Shaded areas: energy and angle.

IceCube is an instrument optimised to detect and characterize sub-TeV to multi-PeV neutrinos of all flavors (see Fig. 8) from extraterrestrial sources.

i m m ve

j i i

20

Figure 9. Simulation of a ultra-high energy tau-]epton by the interaction of a 10 million GeV tau-neutrino, followed by the decay of the secondary tau-lepton. The color represents the t ime sequence of the hits (red-orange-yel)ow-green-blue). The size of the dots corresponds to the number of photons detected by the individual photomultipliers.

It will consist of 80 strings, each with 60 10-inch photomultipliers spaced 17 m apart. The deepest module is 2.4 km below the surface. The strings are arranged at the apexes of equilateral triangles 125 m on a side. The effec­tive detector volume is about a cubic kilometer, its precise value depending on the characteristics of the signal. IceCube will offer great advantages over AMANDA II beyond its larger size: it will have a much higher efficiency to re­construct tracks, map showers from electron- and tau-neutrinos (events where both the production and decay of a T produced by a vr can be identified; see Fig. 9) and, most importantly, measure neutrino energy. Simulations indicate that the direction of muons can be determined with sub-degree accuracy and their energy measured to better than 30% in the logarithm of the energy. Even the direction of showers can be reconstructed to better than 10° in both

21

6, 4> above 10 TeV. Simulations predict a linear response in energy of better than 20%. This has to be contrasted with the logarithmic energy resolution of first-generation detectors. Energy resolution is critical because, once one establishes that the energy exceeds 100 TeV, there is no atmospheric neutrino background in a kilometer-square detector.

At this point in time, several of the new instruments, such as the partially deployed Auger array and HiRes to Magic to Milagro and AMANDA II, are less than one year from delivering results. With rapidly growing observational capabilities, one can express the realistic hope that the cosmic ray puzzle will be solved soon. The solution will almost certainly reveal unexpected astrophysics, if not particle physics.

Acknowledgements

I thank Concha Gonzalez-Garcia and Vernon Barger for comments on the manuscript. This research was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-95ER40896 and in part by the Univer­sity of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation.

References

1. D. J. Bird et al., Phys. Rev. Lett. 71, 3401 (1993). 2. N. N. Efimov et al, ICRR Symposium on Astrophysical Aspects of the

Most Energetic Cosmic Rays, ed. M. Nagano and F. Takahara (World Scientific, 1991).

3. http://www.hep.net/experiments/all_sites.html, provides information on experiments discussed in this review. For a few exceptions, I will give separate references to articles or websites.

4. M. Ave et al, Phys. Rev. Lett. 85, 2244 (2000). 5. http://www-akeno.icrr.u-tokyo.ac.jp/AGASA/ 6. Proceedings of the International Cosmic Ray Conference, Hamburg, Ger­

many, August 2001. Some of the results described here can be found in the rapporteur's talks of this meeting which was held two weeks after this conference.

7. R. A. Vazquez et al, Astroparticle Physics 3, 151 (1995). 8. F. W. Stecker and M. H. Salamon, astro-ph/9808110 and references

therein. 9. J. Alvarez-Muniz et al, hep-ph/0107057; R. Emparan et al, hep-

ph/0109287 and references therein.

22

10. T. C. Weekes, Status of VHE Astronomy c.2000, Proceedings of the In­ternational Symposium on High Energy Gamma-Ray Astronomy, Heidel­berg, June 2000, astro-ph/0010431; R. A. Ong, XIX International Sym­posium on Lepton and Photon Interactions at High Energies, Stanford, August 1999, hep-ex/0003014.

11. E. Pare et al, astro-ph/0107301. 12. J. Cortina for the MAGIC collaboration, Proceedings of the Very High

Energy Phenomena in the Universe, Les Arcs, France, January 20-27, 2001, astro-ph/0103393.

13. http://veritas.sao.arizona.edu/ 14. http://hegral.mppmu.mpg.de 15. http://www.igpp.lanl.gov/ASTmilagro.html 16. For reviews, see T.K. Gaisser, F. Halzen and T. Stanev, Phys. Rep.

258(3), 173 (1995); J.G. Learned and K. Mannheim, Ann. Rev. Nucl. Part. Science 50, 679 (2000); R. Ghandi, E. Waxman and T. Weiler, review talks at Neutrino 2000, Sudbury, Canada (2000).

17. I. A. Belolaptikov et al, Astroparticle Physics 7, 263 (1997). 18. E. Aslanides et al, astro-ph/9907432 (1999). 19. L. Trascatti, in Procs. of the 5th International Workshop on "Topics in

Astroparticle and Underground Physics (TAUP97), Gran Sasso, Italy, 1997, ed. by A. Bottino, A. diCredico, and P. Monacelli, Nucl. Phys. B70 (Proc. Suppl.), p.442 (1998).

20. Talk given at the International Workshop on Next Generation Nucleon Decay and Neutrino Detector (NNN99), Stony Brook, 1999, Proceedings to be published by AIP.

21. The AMANDA collaboration, Astroparticle Physics, 13, 1 (2000). 22. E. Andres et al., Nature 410, 441 (2001). 23. F. Stecker, C. Done, M. Salamon, and P. Sommers, Phys. Rev. Lett. 66,

2697 (1991); erratum Phys. Rev. Lett. 69, 2738 (1992).

BIG B A N G NUCLEOSYNTHESIS, IMPLICATIONS OF R E C E N T CMB DATA A N D SUPERSYMMETRIC D A R K

MATTER

KEITH A. OLIVE

Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

E-mail: olive@umn.edu

The BBN predictions for the abundances of the light element isotopes is reviewed and compared with recent observational data. The single unknown parameter of standard BBN is the baryon-to-photon ratio, n, and can be determined by the con­cordance between theory and observation. Recent CMB anisotropy measurements also lead to a determination of n and these results are contrasted with those from BBN. In addition, the CMB data indicate that the Universe is spatially flat. Thus it is clear that some form of non-baryonic dark matter or dark energy is necessary. Here I will also review the current expectations for cold dark matter from minimal supersymmetric models. The viability of detecting supersymmetric dark matter will also be discussed.

1 I n t r o d u c t i o n

The cornerstones of the Big Bang theory are the cosmic microwave background radiat ion (CMB) and big bang nucleosynthesis (BBN) (one could make an argument to include inflation as well). The existence of the former and the success of the latter point unequivocally to a hot and dense origin to the Universe. Indeed, these two aspects of the theory are intimately linked as early work by Alpher and Herman1 on BBN led to their prediction of the CMB with a temperature of order 10K.

These lectures will focus on recent developments in BBN theory and the related observations which test for concordance. Recent measurements of the CMB power spectrum have allowed an unprecedented level of accuracy in the determination of cosmological parameters including the baryon density, which is the key parameter for BBN. The concordance between BBN and the recent CMB measurements will be addressed. It is also becoming clear that while the total density of the Universe is near critical, i.e., we live in a spatially flat Universe, the baryon density is only a small fraction of the total energy density. Something is missing. While much of missing energy appears to be best fit by a smooth component such as a cosmological constant, a sizable

23

24

fraction must lie in the form of non-baryonic dark matter. The second half of these lectures will focus on the specific possibility of supersymmetric dark matter.

2 Big Bang Nucleosynthesis

The standard model2 of big bang nucleosynthesis (BBN) is based on the rel­atively simple idea of including an extended nuclear network into a homoge­neous and isotropic cosmology. Apart from the input nuclear cross sections, the theory contains only a single parameter, namely the baryon-to-photon ratio, T). Other factors, such as the uncertainties in reaction rates, and the neutron mean-life can be treated by standard statistical and Monte Carlo techniques3. The theory then allows one to make predictions (with well-defined uncertainties) of the abundances of the light elements, D, 3He, 4He, and 7Li.

2.1 Theory

Conditions for the synthesis of the light elements were attained in the early Universe at temperatures T ^ 1 MeV. In the early Universe, the energy density was dominated by radiation with

from the contributions of photons, electrons and positrons, and Nv neutrino flavors (at higher temperatures, other particle degrees of freedom should be included as well). At these temperatures, weak interaction rates were in equi­librium. In particular, the processes

n + e+ «-»• p + i>e

n + ve <•+ p + e~

n <-» p + e~ + ve (2)

fix the ratio of number densities of neutrons to protons. At T > 1 MeV, (n/p) ~ 1.

The weak interactions do not remain in equilibrium at lower temper­atures. Freeze-out occurs when the weak interaction rate, Twk ~ G2

FTh

falls below the expansion rate which is given by the Hubble parameter, H ~ y/U^p ~ T2/MP, where MP = l /V^Jv =s 1.2 x 1019 GeV. The /3-interactions in eq. (2) freeze-out at about 0.8 MeV. As the temperature falls and approaches the point where the weak interaction rates are no longer fast

25

enough to maintain equilibrium, the neutron to proton ratio is given approx­imately by the Boltzmann factor, (n/p) ~ e

- A m / T ~ 1/6, where Am is the neutron-proton mass difference. After freeze-out, free neutron decays drop the ratio slightly to about 1/7 before nucleosynthesis begins.

The nucleosynthesis chain begins with the formation of deuterium by the process, p + n —¥ D + 7. However, because of the large number of photons relative to nucleons, r?-1 = n7/nj3 ~ 1010, deuterium production is delayed past the point where the temperature has fallen below the deuterium binding energy, EB = 2.2 MeV (the average photon energy in a blackbody is E~/ ~ 2.7T). This is because there are many photons in the exponential tail of the photon energy distribution with energies E > EB despite the fact that the temperature or E1 is less than EB- The degree to which deuterium production is delayed can be found by comparing the qualitative expressions for the deuterium production and destruction rates,

Tp fa riBcrv (3)

When the quantity ^ _ 1exp(—EB/T) ~ 1, the rate for deuterium destruction (D + 7 —> p + n) finally falls below the deuterium production rate and the nuclear chain begins at a temperature T ~ 0.1 MeV.

In addition to the p (n,~/) D reaction, the other major reactions leading to the production of the light elements are:

D (D, p)T D (n,7) T 3He (n, p) T

D (D, n) 3He D (p, 7) 3He

Followed by the reactions producing 4He:

D (D, 7) 4He 3He (3He, 2p) 4He

D( 3He, p) 4 He T ( p , 7 ) 4 H e

T (D, n) 4He 3He (rc,7) 4He

The gap at A — 5 is overcome and the production of 7Li proceeds through:

3He (4He,7) 7Be

-> 7Li + e+ + ve

T (4He,7) 7Li

26

Figure 1. The nuclear network used in BBN calculations.

The gap at A = 8 prevents the production of other isotopes in any significant quantity. The nuclear chain in BBN calculations was extended4 and is shown in Figure 1.

The dominant product of big bang nucleosynthesis is 4He and its abun-

27

dance is very sensitive to the (n/p) ratio

2(n/p) Y„

[1 + (n/p)} 0.25 (4)

i.e., an abundance of close to 25% by mass. Lesser amounts of the other light elements are produced: D and 3He at the level of about 10 - 5 by number, and 7Li at the level of 10"*10 by number.

Historically, BBN as a theory explaining the observed element abun­dances was nearly abandoned due its inability to explain all element abun­dances. Subsequently, stellar nucleosynthesis became the leading theory for element production5. However, two key questions persisted. 1) The abun­dance of 4He as a function of metallicity is nearly flat and no abundances are observed to be below about 23% as exaggerated in Fig. 2. In partic­ular, even in systems in which an element such as Oxygen, which traces stellar activity, is observed at extremely low values (compared with the so­lar value of O/H & 8.5 x 10 - 4) , the 4He abundance is nearly constant. This is very different from all other element abundances (with the excep­tion of 7Li as we will see below). For example, in Figure 3, the N/H vs. O/H correlation is shown. As one can clearly see, the abundance of N/H goes to 0, as O/H goes to 0, indicating a stellar source for Nitrogen. 2) Stellar sources can not produce the observed abundance of D/H. Indeed, stars destroy deuterium and no astrophysical site is known for the production of significant amounts of deuterium6. Thus we are led back to BBN for the origins of D, 3He, 4He, and 7Li.

MO

o.«

».*>

0.28

f^lte Is Primordial! j

•

•

10' O/H

Figure 2, The 4He mass fraction as determined in extragalactic H II regions as a function of O/H.

28

130

100

80 a z 60

40

20

0 50 100 150 10« 0 / H

Figure 3. The Nitrogen and Oxygen abundances in the same extragalactic HII regions with observed 4He shown in Figure 2.

The resulting abundances of the light elements are shown in Figure 4, over the range in 7710 = 101077 between 1 and 10. The left plot shows the abundance of 4He by mass, Y, and the abundances of the other three isotopes by number. The curves indicate the central predictions from BBN, while the bands correspond to the uncertainty in the predicted abundances based pri­marily the uncertainty in the input nuclear reactions as computed by Monte Carlo in ref. 7. This theoretical uncertainty is shown explicitly in the right panel as a function of 7710- The dark shaded boxes correspond to the observed abundances of 4He and 7Li and will be discussed below. The dashed boxes correspond to the ranges of the elements consistent with the systematic un­certainties in the observations. The broad band shows a liberal range for 7710 consistent with the observations.

At present, there is a general concordance between the theoretical pre­dictions and the observational data, particularly, for 4He and 7Li. These two elements indicate that 7? lies in the range 1.7 < rj < 4.7, corresponding to a range in f2#/i2 = 0.006 — 0.017a. There is limited agreement for D/H as well, as will be discussed below. D/H is compatible with 4He and 7Li at the 2c level in the range 4.7 < 77 < 6.2 {SlBh2 = 0.017 - 0.023).

a f i is the total density of matter relative to the critical density and fifl is the fraction of critical density in baryons. h is Hubble parameter scaled to 100 km M p c - 1 s _ 1 .

_

':

— ~

'• ^ j L L ^ A 7 ^3^ \*t

j[

fK

s

•t ~ll

*

^"'-. ^ -

-

29

10-10 1 0 - » 1 Q - 1 0 1 Q - 9

Figure 4. The light element abundances from big bang nucleosynthesis as a function of rji0.

2,2 Data-4He

The primordial 4He abundance is best determined from observations of Hell -»• Hel recombination lines in extragalactic HII (ionized hydrogen) regions. There is a good collection of abundance information on the 4He mass fraction, Y, O/H, and N/H in over 70 such regions8'9. Since 4He is produced in stars along with heavier elements such as Oxygen, it is then expected that the primordial abundance of 4He can be determined from the intercept of the correlation between Y and O/H, namely Yp = Y(0/E -> 0). A detailed analysis10 of the data found

Yp = 0.238 ± 0.002 ± 0.005 (5)

The first uncertainty is purely statistical and the second uncertainty is an estimate of the systematic uncertainty in the primordial abundance determi-

30

0.30

0.28

0.26

y 0.24

0.22

0.20

0.18

0 SO 100 ISO 200

106O/H

Figure 5. The Helium (Y) and Oxygen (O/H) abundances in extragalactic HII regions, from refs. 8 and 9 . Lines connect the same regions observed by different groups. The regression shown leads to the primordial 4He abundance given in Eq. (5).

nation. The solid box for 4He in Figure 4 represents the range (at 2<rstat) from (5). The dashed box extends this by including the systematic uncer­tainty. The 4He data is shown in Figure 5. Here one sees the correlation of 4He with O/H and the linear regression which leads to primordial abundance given in Eq. (5).

The helium abundance used to derive (5) was determined using assumed electron densities n in the HII regions obtained from SII data. Izotov, Thuan, & Lipovetsky9 proposed a method based on several He emission lines to "self-consistently" determine the electron density. Their data using this method yields a higher primordial value

Yp = 0.244 ±0.002 ±0.005 (6)

One should also note that a recent determination11 of the 4He abundance in a single object (the SMC) also using the self consistent method gives a primordial abundance of 0.234 ± 0.003 (actually, they observe Y — 0.240 ± 0.002 at [O/H] = -0.8, where the abundance [O/H] is defined as the log of the abundance relative to the solar abundance, [X/H] = log([X/H]/[X/H]0)). Indeed, this work was extended in a reanalysis12 of selected regions from ref. 9 and argue for corrections which lower the abundance to 0.239 ±0.002 based on a fit to 5 regions.

As one can see, the resulting primordial 4He abundance shows significant sensitivity to the method of abundance determination, leading one to con­clude that the systematic uncertainty (which is already dominant) may be underestimated13.

31

2.3 Data-7 Li

The abundance of 7Li has been determined by observations of over 100 hot, population-II stars, and is found to have a very nearly uniform abundance 1 4 . For stars with a surface temperature T > 5500 K and a metallicity less than about l / 20 th solar (so that effects such as stellar convection may not be impor tant ) , the abundances show little or no dispersion beyond t ha t which is consistent with the errors of individual measurements. There is, however, an important source of systematic error due to the possibility tha t Li has been depleted in these stars, though the lack of dispersion in the Li da ta limits the amount of depletion. In fact, a small observed slope in Li vs Fe and the tiny dispersion about tha t correlation indicates tha t depletion is negligible in these stars 1 5 . Furthermore, the slope may indicate a lower abundance of Li than that in (6). The observation1 6 of the fragile isotope 6Li is another good indication tha t 7Li has not been destroyed in these s ta rs 1 7 .

The weighted mean of the 7Li abundance in the sample of ref. 1 5 is [Li] = 2.12 ([Li] = log 7 L i / H + 12). It is common to test for the presence of as lope in the Li da ta by fitting a regression of the form [Li] = a + j3 [Fe/H]. These da ta indicate a rather large slope, ft = 0.07 — 0.16 and hence a downward shift in the "primordial" l i thium abundance A [Li] = - 0 . 2 0 0.09. Models of galactic evolution which predict a small slope for [Li] vs. [Fe/H], can produce a value for (3 in the range 0.04 - 0.07 1 8 . Overall, when the regression based on the da ta and other systematic effects are taken into account a best value for Li /H was found to be 1 8

Li/H = 1.23 ± 0.1 x 10~1 0 (7)

with a plausible range between 0.9 - 1.9 x l O - 1 0 . The dashed box in Figure 4 corresponds to this range in Li /H.

Figure 6 shows the different Li components for a model with ( 7 Li /H) p = 1.23 x 1 0 - 1 0 . The linear slope produced by the model is independent of the input primordial value (unlike the log slope given above). The model of ref. 19

includes in addition to primordial 7Li, l i thium produced in galactic cosmic ray nucleosynthesis (primarily a + a fusion), and 7Li produced by the jz-process during type II supernovae. As one can see, these processes are not sufficient to reproduce the population I abundance of 7Li, and additional production sources are needed.

2-4 Likelihood Analyses

At this point, having established the primordial abundance of at least two of the light elements, 4He and 7Li, with reasonable certainty, it is possible

32

^ 3

1 0 - "

10"

Li total '•a

~Li primordial 7] i v proofs

- 2 - 1 [ Fe/H ]

Figure 6. Contributions to the total predicted lithium abundance from the adopted GCE model of ref. 1 9 , compared with low metallicity stars (from ref. 1 5) and a sample of high metallicity stars. The solid curve is the sum of all components.

to test the concordance of BBN theory with observations. Two elements are sufficient for not only constraining the one parameter theory of BBN, but also for testing for consistency20. A theoretical likelihood function for 4He can be defined as

LBBN(Y,YBm) = e-(Y-Y™W2^ (8)

where YBBN(»7) ls the central value for the 4He mass fraction produced in

the big bang as predicted by the theory at a given value of r\. <j\ is the uncertainty in that value derived from the Monte Carlo calculations 3 ,T and is a measure of the theoretical uncertainty in the BBN calculation. Similarly one can write down an expression for the observational likelihood function. Assuming Gaussian errors, the likelihood function for the observations would take a form similar to that in (8).

A total likelihood function for each value of rj is derived by convolving

33

the theoretical and observational distributions, which for 4He is given by

£4Hetotai(r?) = JdYLBBN (Y, YBBN (r,)) LQ(Y, YQ) (9)

An analogous calculation is performed20 for 7Li. The resulting likelihood functions from the observed abundances given in Eqs. (5) and (7) is shown in Figure 7. As one can see there is very good agreement between 4He and 7Li in the range of r/io ~ 1.5 - 5.0.

Figure 7. Likelihood distribution for each of 4He and 7Li, shown as a function of t) in the upper panel. The lower panel shows the combined likelihood function.

The combined likelihood, for fitting both elements simultaneously, is given by the product of the two functions in the upper panel of Figure 7 and is shown in the lower panel. The 95% CL region covers the range 1.7 < TJIQ < 4.7, with the peak value occurring at rji0 = 2.4. This range corresponds to values of Qjg/i2 between

0.006 < QBh2 < 0.017 (10)

34

with a central value of Qsh2 = 0.009. Using the higher value for 4He (Yp = 0.244) would result in an upward shift in rjio by about 0.1 - 0.2.

2.5 Data-D/H

The remaining two light elements produced by BBN are D and 3 He. For the most part , the abundances of these elements are determined either in the local interstellar medium or in our own solar system. As such, one needs a model of galactic chemical evolution to tie them to the BBN abundances. Deuterium is predicted to be a monotonically decreasing function of t ime. The degree to which D is destroyed, however, is a model dependent question and related to the production of 3 He. Stellar models predict tha t substantial amounts of 3He are produced in stars between 1 and 3 M©2 1 . It should be emphasized tha t this prediction is in fact consistent with the observation of high 3 H e / H in planetary nebulae2 2 . However, the implementat ion of s tandard model 3He yields in chemical evolution models leads to an overproduction of 3 H e / H particularly at the solar epoch2 3 . See ref. 2 4 for a t t empts to resolve this problem. Because of this model dependence, I will not consider 3He further here.

Despite the problem of relating many of the locally observed D / H mea­surements to BBN, we can use the ISM values to set a firm lower limit on primordial D / H , and hence an upper limit to r) because of the monotoni­cally decreasing history of D / H in the Galaxy. This is shown as the dashed (half)-box in Fig. 4.

There have been several reported measurements of D / H in high redshift quasar absorption systems. Such measurements are in principle capable of determining the primordial value for D / H and hence r), because of the strong and monotonic dependence of D / H on r\. However, at present, detections of D / H using quasar absorption systems do not yield a conclusive value for D / H . In addition to the earlier determinat ions2 5 in two Lyman limit systems of 4.0 ± .65 and 3.25 ± 0.3 x 10~5 and the upper limit in a third Lyman limit system2 6 of 6.8 x l 0 ~ 5 , there have been three new measurements of D / H in Damped Lyman a systems: 2.5 ± 0.25 2 7 ; 2.25 ± 0.65 2 8 ; and 1.65 ± 0.35 2 9 ; all t imes 1 0 - 5 . This da ta is shown in Fig. 8. Also shown in Fig. 8, are the adjusted D / H abundances when more complex velocity distributions are included as in refs. 3 0 .

Is there a real dispersion in D / H in these high redshift systems? There are two possible trends tha t go in a similar direction. The da t a may show an inverse correlation of D / H abundance with Si 2 7 , 2 9 . This may be an artifact of poorly determined Si abundances, or (as yet unknown) systematics affecting

35

& Q0130-4021

>'

m 10JQ9 + 2956T 9-S< 1 Q1937-l|)09

IHSO I .

I S Q2206--

105+1619

* Q0347-383

L99

.....i. . i ., 1 . , , , 1 , i

i . |

I ISM

l i

- 3 - 2 - 1 0

[Si/H]

Figure 8. The D /H da ta as a function of metallicity given by [Si/H].

the D/H determination in high-column density (damped Lyman-a, hereafter DLA) or low-column density (Lyman limit systems) absorbers. On the other hand, if the correlation is real it would indicate that chemical evolution pro­cesses have occurred in these systems. The second trend is that the data may show an inverse correlation of D/H abundance with HI column density. If real, this would suggest that in the high column density DLA systems, which are most likely to have undergone some star formation, some processing of D/H must similarly have occurred at high redshift. One can only conclude: if the dispersion in D/H is real, it has profound consequences, as it indicates that some processing of D/H must have occurred even at high redshift.

It is interesting to speculate31 that the possible high redshift destruction of D/H is related to recent observations which suggest the existence of a white dwarf population in the Galactic halo32. These observations could be signa­tures of an early population of intermediate-mass stars. Such a population

36

requires a Population III initial mass function different from tha t of the solar neighborhood. Also, to avoid overproduction of C and N, it is required tha t the Z = 0 yields of these stars have low (~ 10~3 solar) abundances as sug­gested by some recent calculations. Under these assumptions, it is possible to model the observed D vs Si t rend 3 1 . Such a scenario predicts a high cos­mic Type la supernova rate, while producing a white dwarf population tha t accounts for only ~ 1.5% of the dark halo.

It is clear that a simple average of D / H abundance determinations does not make sense, at least without a proper enlargement of the error in the mean due to the poor x2 t n a t such a mean would produce. Moreover, if deuter ium destruction has occurred, we must question the extent to which any of these systems determine the value of fls. It is important to note however, tha t for the upper end of the range (~ 5 x 10~5) shown in Fig. 8, all of the element abundances are consistent as will be discussed below.

2.6 More Analysis

It is interesting to compare the results from the likelihood functions of 4He and 7Li with tha t of D / H . To include D / H , one would proceed in much the same way as with the other two light elements. We compute likelihood functions for the BBN predictions as in Eq. (8) and the likelihood function for the observations. These are then convolved as in Eq. (9).

Using D / H = (3.0 ±0 .3 ) x 1 0 - 5 as the primordial abundance, one obtains the likelihood function shown (shaded) in the upper panel of Fig. 9. The 95% CL region covers the range 5.0 < r/io < 7.4, with the peak value occurring at •q10 = 5.8. This range corresponds to values of QB between

0.018 < Q.Bh2 < 0.027 (11)

with a central value of Clsh2 = 0.021. The combined likelihood, for fitting all elements (D, 4 He, and 7Li) simul­

taneously, is shown by the shaded curve in the lower panel of Figure 9. Note tha t it has been scaled upwards by a factor of 25. In this case, the 95% CL region covers the range 4.7 < 7710 < 6.2, with the peak value occurring at r]io = 5.3. This range corresponds to values of QB between

0.017 < ClBh2 < 0.023 (12)

with a central value of Qsh2 = 0.019. It is impor tant to recall however, tha t the true uncertainty in the low D / H

systems might be somewhat larger. If we allow D / H to be as large as 5 x 10~5 , the peak of the D / H likelihood function shifts down to 7710 ~ 4. In this case,

37

Figure 9. Likelihood distribution for D/H, shown (shaded) as a function of f] in the upper panel. Also shown are the 4 He and 7Li likelihoods from Fig. 7. The lower panel shows the combined likelihood function (shaded) compared to the previous case neglecting D/H.

there would be a near perfect overlap with the high t] 7Li peak and since the 4He distribution function is very broad, this would be a highly compatible solution. Given our discussion in the previous section concerning the current status of the D/H data, it is premature to claim a lack of concordance between BBN theory and observations.

3 T h e C M B - B B N connect ion

It is interesting to note the role of BBN in the prediction of the microwave background1. The argument is rather simple. BBN requires tempera­tures greater than 100 keV, which according to the standard model time-temperature relation, tsT^eV = 2A/y/N, where N is the number of relativistic degrees of freedom at temperature T, and corresponds to timescales less than about 200 s. The typical cross section for the first link in the nucleosynthetic

38

chain is

<rv(p + n - ) - D + 7 ) ~ 5 x l(T2 0cm3 /s

This implies that it was necessary to achieve a density

n ~ 1017cnT3

crvt

(13)

(14)

The density in baryons today is known approximately from the density of visible matter to be njg0 ~ 10~7 cm~3 and since we know that that the density n scales as R~z ~ T3, the temperature today must be

To = {nBo/n)1/3TBBN ~ 10K (15)

thus linking two of the most important tests of the Big Bang theory.

> V.

\ ^

/ /

• •—-•• • • •

i i

i. / •j

•

•

,::U-1

40 50 60 70 80 0.6 O.B 1.0 1.2 1.4 0.00 0.05 0.10 0.15 H„ n Q„

> ;»c /

4

1 '

^ / •

V J U - J I - ^ Z .

••«- .,..

0

0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.8 0.8 0.0 0.2 0.4 0.6 0.8 1.0 Qy nA

Figure 10. Ax 2 calculated with the MAXIMA-1 and COBE data as a function of parameter value. Solid blue circles show grid points in parameter space, and the green lines were obtained by interpolating between grid points. The parameter values where the green line intercepts the red dashed (dotted) line corresponds to the 68% (95%) frequentist confidence region .

39

Microwave background anisotropy measurements have made tremendous advances in the last few years. The power spectrum33 '34,35 has been measured relatively accurately out to multipole moments corresponding to I ~ 1000. The details of this spectrum enable one to make accurate predictions of a large number of fundamental cosmological parameters34,36,37 '38. An example of these results as found by a recent frequentist analysis39 is shown in Fig. 10.

The CMB anisotropics thus independently test the BBN prediction of Qsh2. At present, the predicted BBN baryon densities from D/H agree to an uncanny level with the most recent CMB results34 '37. The recent result from DASI37 indicates that QBh2 = 0.022±g 0g|, while that of BOOMERanG-98 34, QBh2 = 0.02lto"ool (using ler errors) which should be compared to the BBN prediction given in eq. 11. These determinations are lower than value found by MAXIMA-1 36 which yields QBh2 = 0.0325 ± 0.006. Given the current uncertainties, these results are consistent as can be seen in Fig. 11 based on the recent frequentist analysis39 which found ClBh2 = 0.026i0'ooe In addition, the BOOMERanG and DASI determinations are higher than the value Qsh2 = 0.009 based on 4He and 7Li. However, the measurements of the Cosmic Background Imager38 at smaller angular scales (higher multipoles) agree with lower BBN predictions and claims a maximum likelihood value for QBh2 = 0.009 (albeit with a large uncertainty).

90

80

70 o

X 60

50

40 0.

Figure 11. Two-dimensional frequentist confidence regions in the (H0,aB) plane3 9 . The red, orange and yellow regions correspond to the 68%, 95%, and 99% confidence regions respectively. Standard calculations from big bang nucleosynthesis and observations of &

predict a 95% confidence region of QBh2 = 0 .021+°°° | ' i n d i c a t e d b y t h e shaded region.

0.15

40

4 Constraints from B B N

Limits on particle physics beyond the standard model are mostly sensitive to the bounds imposed on the 4He abundance. As discussed earlier, the neutron-to-proton ratio is fixed by its equilibrium value at the freeze-out of the weak interaction rates at a temperature Ty ~ 1 MeV modulo the occasional free neutron decay. Furthermore, freeze-out is determined by the competition between the weak interaction rates and the expansion rate of the Universe

GF2Tj5 ~ r w k ( 7 » =H(Tf)~ jG^NTj2 (16)

where N counts the total (equivalent) number of relativistic particle species. At T ~ 1 MeV, N = 43/4. The presence of additional neutrino flavors (or any other relativistic species) at the time of nucleosynthesis increases the overall energy density of the Universe and hence the expansion rate leading to a larger value of Tj, (n/p), and ultimately Yp. Because of the form of Eq. (16) it is clear that just as one can place limits40 on N, any changes in the weak or gravitational coupling constants can be similarly constrained (for a discussion see ref. 4 1 ) .

Changes in Nu actually affect not only 4He, but also the abundances of the other light elements as seen in Fig. 12 42. We see the typical large dependence on Nu in 4He, but also note the shifts in the other elements, particularly D, and also Li over some ranges in 77. Because of these variations, one is not restricted to only 4He in testing N„ and particle physics.

Just as 4He and 7Li were sufficient to determine a value for 77, a limit on N„ can be obtained as well20,43,44. The likelihood approach utilized above can be extended to include Nu as a free parameter. Since the light element abundances can be computed as functions of both 77 and Nu, the likelihood function can be defined by43 replacing the quantity YBBN (V)

m ecl- (8) with YBBN (T}I Nv) to obtain L Hetotai(??, N„). Again, similar expressions are needed for 7Li and D.

The likelihood distribution derived from the analysis of ref. 42 is shown in Fig. 13 where iso-likelihood contours representing 68, 95, and 99 % CL are projected onto the 7710 — Nu plane. As one can see, when only 4He and 7Li are used, the allowed range in 7710 is rather broad and the upper limit to Nv < 4.2 at relatively low 77. At present, one can not use D/H to fix Nv, unless one has additional information on the value of 77, e.g., from the CMB as is demonstrated in Fig. 14. On the other hand when all three light elements are used, the narrow range in 77 at relatively high values, allows for a tight constraint on N„ < at 95 % CL. Furthermore, it is clear that the most realistic value, Nv — 3 is in fact consistent with all of the observed abundances at the

41

10-s

Figure 12. BBN abundance predictions42 as a function of the baryon-to-photon ratio »j, for Nv = 2 to 7. The bands show the lcr error bars. Note that for the isotopes other than Li, the error bands are comparable in width to the thickness of the abundance curve shown. All bands are centered on Nv = 3.

2 <r level. Of course as discussed above, the CMB provides an independent deter­

mination of T}, we can use that to fix TJ, and test the available constraints on N„ 4 2 . For example, we can fix rj = 5.8 x 10 - 1 0 which corresponds to the BOOMERanG and DASI determinations, and use the current 4He and 7Li data to compute a BBN likelihood function for Nv. This is shown in the first panel of Fig. 14. Assuming a 10% uncertainty in rj, the 95 % CL range on

42

68% CL

95% CL

99% CL

68% CL

9S% CL

99% CL

1-147

Figure 13. (a) Likelihood contours representing 68, 95, and 99 % CL projected onto the Tjio — Nv plane. In a) only 4He and 7Li are used. In b) only D is used, and in c) all three light elements are used. The crosses denote the position of the peak of the likelihood distribution.

N„ is 1.8 - 3.3, whereas the 1 sided upper limit with a Nu > 3 prior45 is 3.5. Using instead the D/H data, at the same baryon density, one finds an upper limit Nu < 6.3. This likelihood is shown in panel b) of Fig. 14. Alternatively, we could choose a lower value of rj = 2.4 x 10 - 5 , and using 4He and 7Li we find that the 95 % CL range is 2.2 - 3.9. This case is exemplified in panel c).

5 Something ' s Missing

There is considerable evidence for dark matter in the Universe46. The best observational evidence is found on the scale of galactic halos and comes from the observed flat rotation curves of galaxies. There is also good evidence for dark matter in elliptical galaxies, as well as clusters of galaxies coming from X-ray observations of these objects. In theory, we expect dark matter because 1)

43

Figure 14. (a) The distribution in Nv assuming a value of r) = 5.8 X 10—10 and primordial 4 He & 7Li abundances as in eqs. 5 and 7. (b) As in (a) assuming a D measurement at the current precision (taken to be 3.0 ± 0.3 X 10~ s ) . (c) As in (a), but with a value for ri = 2.4 X 10~1 0 .

inflation predicts Cl = 1, and the upper limit on the baryon (visible) density of the Universe from big bang nucleosynthesis is Q,B < 0.1; 2) Even in the absence of inflation (which does not distinguish between matter and a cosmological constant), the large scale power spectrum is consistent with a cosmological matter density of ft ~ 0.3, still far above the limit from nucleosynthesis; and 3) our current understanding of galaxy formation is inconsistent with observations if the Universe is dominated by baryons.

Indeed, we now have direct evidence from CMB anisotropy measurements

44

1.0

0.8

0.6

0.4

0.2

0.0

• « . ' ' • '

11 -S&Ste.

i . i .

. , . , . ,

..^JsH

• • • .

•

•

"

-^

h i Hgii^-j 0.0 0.2 0.4 0.6 0.8 1.0

Figure 15. Two-dimensional frequentist confidence regions in the (QM>£2A) plane. The red, orange and yellow regions correspond to the 68%, 95%, and 99% confidence regions respectively. The dashed black line corresponds to a flat universe, fl = O M + &A = !•

that Qtot = 1 to with in about 20 %. The CMB anisotropy allows for a determination of the curvature of the Universe. Therefore, while one can not determine unambiguously the value of Omatter, one can fix the sum of the matter and dark energy (cosmological constant) contributions. This is shown in Fig. 15 from the frequentist analysis in39. When combined with high redshift supernovae measurements indicating that the Universe is currently accelerating47, one would conclude that the dark energy constitutes about 65 % of the closure density leaving the remaining 35 % for matter. Given that the baryon density is less than 5 %, this leaves us with about 30 % of closure density for non-baryonic dark matter. Other dynamical arguments estimating the density of matter are in agreement with this allotment (see ref. 4 8 for a recent review).

In fact, there are many reasons why most of the dark matter must be non-baryonic. In addition to the problems with baryonic dark matter associated with nucleosynthesis or the growth of density perturbations, it is very difficult to hide baryons. There are indeed very good constraints on the possible forms for baryonic dark matter in our galaxy. Strong cases can be made against hot gas, dust, jupiter size objects, and stellar remnants such as white dwarfs and neutron stars49.

In what follows, I will focus on supersymmetric candidates in which the relic abundance of dark matter contributes a significant though not excessive amount to the overall energy density. Denoting by Qx the fraction of the critical energy density provided by the dark matter, the density of interest

45

falls in the range

0.1 < Q,xh2 < 0.3 (17)

The lower limit in eq.(17) is motivated by astrophysical relevance. For lower values of Clxh

2, there is not enough dark matter to play a significant role in structure formation, or constitute a large fraction of the critical density. The upper bound in (17), on the other hand, is an absolute constraint, derivable from the age of the Universe, which can be expressed as

H0t0= J dx(l-Q-QA + QAx2 +Q/x)~1/2 (18) Jo

Given a lower bound on the age of the Universe, one can establish an upper bound on fi/i2 from eq.(18). The limit t0 J> 12 Gyr translates into the upper bound given in (17). Adding a cosmological constant does not relax the upper bound on Q,h?, so long as fi + QA < 1- If indeed, the indications for a cosmological constant from recent supernovae observations47 turn out to be correct, the density of dark matter will be constrained to the lower end of the range in (17). Indeed for h2 ~ 1/2, we expect Qxh

2 ~ 0.15.

6 Supersymmetric Dark Matter

Although there are many reasons for considering supersymmetry as a can­didate extension to the standard model of strong, weak and electromagnetic interactions50, one of the most compelling is its role in understanding the hi­erarchy problem51 namely, why/how is mw <S Mp. One might think naively that it would be sufficient to set mw <S Mp by hand. However, radiative corrections tend to destroy this hierarchy. For example, one-loop diagrams generate

Sm2w = O ( £ ) A2 » m2

w (19)

where A is a cut-off representing the appearance of new physics, and the in­equality in (19) applies if A ~ 103 TeV, and even more so if A ~ maur ~ 1016

GeV or ~ Mp ~ 1019 GeV. If the radiative corrections to a physical quan­tity are much larger than its measured values, obtaining the latter requires strong cancellations, which in general require fine tuning of the bare input parameters. However, the necessary cancellations are natural in supersym­metry, where one has equal numbers of bosons B and fermions F with equal couplings, so that (19) is replaced by

&m2w=o(pj \m2

B-m2F\ . (20)

46

The residual radiative correction is naturally small if \m2B — mp\ £ 1 TeV .

In order to justify the absence of superpotential terms which can be re­sponsible for extremely rapid proton decay, it is common in the minimal super-symmetric standard model (MSSM) to assume the conservation of R-parity. If R-parity, which distinguishes between "normal" matter and the supersym-metric partners and can be defined in terms of baryon, lepton and spin as R = (_ i ) 3 B + L + 2 S

) is unbroken, there is at least one supersymmetric particle (the lightest supersymmetric particle or LSP) which must be stable.

The stability of the LSP almost certainly renders it a neutral weakly in­teracting particle52. Strong and electromagnetically interacting LSPs would become bound with normal matter forming anomalously heavy isotopes. In­deed, there are very strong upper limits on the abundances, relative to hy­drogen, of nuclear isotopes53, n/nn ^ 10~15 to 10 - 2 9 for 1 GeV ^ m < 1 TeV. A strongly interacting stable relics is expected to have an abundance n/riH ^ 10 - 1 0 with a higher abundance for charged particles.

There are relatively few supersymmetric candidates which are not col­ored and are electrically neutral. The sneutrino54 is one possibility, but in the MSSM, it has been excluded as a dark matter candidate by direct55 and indirect56 searches. In fact, one can set an accelerator based limit on the sneu­trino mass from neutrino counting, mp <; 43 GeV 57. In this case, the direct relic searches in underground low-background experiments require mp J> 1 TeV 58. Another possibility is the gravitino which is probably the most diffi­cult to exclude. I will concentrate on the remaining possibility in the MSSM, namely the neutralinos.

6.1 Relic Densities

There are four neutralinos, each of which is a linear combination of the R = — 1 neutral fermions,52: the wino W3, the partner of the 3rd component of the SU(2)L gauge boson; the bino, B, the partner of the U(1)Y gauge boson; and the two neutral Higgsinos, H\ and R.2- Assuming gaugino mass universality at the GUT scale, the identity and mass of the LSP are determined by the gaugino mass mj/2 (or equivalently by the SU(2) gaugino mass M2 at the weak scale - at the GUT scale, M2 — mi/2), the Higgs mixing mass \i, and the ratio of Higgs vevs, tan/3. In general, neutralinos can be expressed as a linear combination

X = aB + pW3 + ~/Hi + SH2 (21)

47

The solution for the coefficients a, /3, 7 and 6 for neutralinos that make up the LSP can be found by diagonalizing the mass matrix

(W3,B,H°,H°)

( M2

0

- 9 2 f 1

%/2

0 Mi g i f 1

- g i f 2

V2

0

- 3 1 ^ 2

0 /

/w3\ B Hi

(22)

where Mi is a soft supersymmetry breaking term giving mass to the U(l) gaugino. In a unified theory Mi = Mi at the unification scale (at the weak scale, Mi = | ^ L M2). As one can see, the coefficients a,/3,7, and S depend only on mi/2, fJ>, and tan/3.

In Figure 16 59, regions in the M2,\i plane with tan/3 = 2 are shown in which the LSP is one of several nearly pure states, the photino, 7, the bino, B, a symmetric combination of the Higgsinos, #(12) > or the Higgsino, S — sin0H\ + cos(iH2- The dashed lines show the LSP mass contours. The cross hatched regions correspond to parameters giving a chargino (W^, H^) state with mass m^ < 45GeV and as such are excluded by LEP60. This constraint has been extended by LEP61 and is shown by the light shaded region and corresponds to regions where the chargino mass is < 104 GeV. The newer limit does not extend deep into the Higgsino region because of the degeneracy between the chargino and neutralino. Notice that the parameter space is dominated by the B or H12 pure states and that the photino only occupies a small fraction of the parameter space, as does the Higgsino combination S. Both of these light states are now experimentally excluded.

The relic abundance of LSP's is determined by solving the Boltzmann equation for the LSP number density in an expanding Universe. The technique62 used is similar to that for computing the relic abundance of mas­sive neutrinos63. The relic density depends on additional parameters in the MSSM beyond M2,//, and tan/3. These include the sfermion masses, mj, the Higgs pseudo-scalar mass, TUA, and the tri-linear masses A as well as two phases 9^ and 6A • To determine, the relic density it is necessary to obtain the general annihilation cross-section for neutralinos. In much of the param­eter space of interest, the LSP is a bino and the annihilation proceeds mainly through sfermion exchange as shown in Figure 17. For binos, it is possible to adjust the sfermion masses to obtain closure density in a wide mass range. Ad­justing the sfermion mixing parameters64 or CP violating phases65,66 allows even greater freedom.

Because of the p-wave suppression associated with Majorana fermions, the s-wave part of the annihilation cross-section is suppressed by the outgoing

48

10000

10 30 100 300 1000 3000 10000

M2 (GeV)

Figure 16. Mass contours and composition of nearly pure LSP states in the MSSM 5 9 .

f X ' ^ X

Figure 17. Typical annihilation diagram for neutralinos through sfermion exchange.

fermion masses. This means that it is necessary to expand the cross-section to include p-wave corrections which can be expressed as a term proportional to the temperature if neutralinos are in equilibrium. Unless the neutralino mass happens to lie near near a pole, such as mx ~ mz/2 or ra/j/2, in which case there are large contributions to the annihilation through direct s-channel

49

resonance exchange, the dominant contribution to the BB annihilation cross section comes from crossed ^-channel sfermion exchange. In the absence of such a resonance, the thermally-averaged cross section for BB —> / / takes the generic form

(-) = (I-5-)1 /2TI m2,

m\' 128TT

+ (Y? + Y*)(^)(l + ...)x

= a + bx (23)

where Y^B.) is the hypercharge of //,(#), A/ = m2- + m2- — nil, and we have shown only the leading P-wave contribution proportional to x = T/mg. Such an expansion yields very accurate results unless the LSP is near a threshold, near a resonance, or is nearly degenerate in mass with another SUSY particle in which cases a more accurate treatment is necessary67. Annihilations in the early Universe continue until the annihilation rate T ~ (?vnx drops below the expansion rate, H, For particles which annihilate through approximate weak scale interactions, this occurs when T ~ m x /20. Subsequently, the relic density of neutralinos is fixed relative to the number of relativistic particles.

As noted above, the number density of neutralinos is tracked by a Boltzmann-like equation,

f = -3!"-<™X"2-"o) (24) where no is the equilibrium number density of neutralinos. By defining the quantity / = n/T3, we can rewrite this equation in terms of x, as

df (\ N 1 / 2

dx = mx ( ^ * V t f ) (/2 - f0) (25)

The solution to this equation at late times (small x) yields a constant value of / , so that n ocT 3 . The final relic density expressed as a fraction of the critical energy density can be written as52

,2 _ T o „ m - i i l'Tx\ »ri/2 / GeV O ^ ^ x l O - ^ ^ j ^ S ? ; (26)

where (Tx/Ty)3 accounts for the subsequent reheating of the photon tem­

perature with respect to x, due to the annihilations of particles with mass

50

m < xjrnx 68. The subscript / refers to values at freeze-out, i.e., when

annihilations cease. In Figure 18 69, regions in the Mi — \i plane (rotated with respect to

Figure 16) with tan/3 = 2, and with a relic abundance 0.1 < Qh2 < 0.3 are shaded. In Figure 18, the sfermion masses have been fixed such that mo = 100 GeV (the dashed curves border the region when mo = 1000 GeV). Clearly the MSSM offers sufficient room to solve the dark matter problem. In the higgsino sector H^, additional types of annihilation processes known as coannihilations 67>70>71 between H(\2) and the next lightest neutralino (i?[i2]) must be included. These tend to significantly lower the relic abundance in much of this sector and as one can see there is little room left for Higgsino dark matter69.

1000

100

(a) ^\

•

.Qh2=0 3

m =100 h

\

"-

p=0.9

a h W ^ ' V °-5

"'(U

-1000 p.

-100

Figure 18. Regions in the M2-(i plane where 0.1 < €lh2 < 0.3 6 9 . Also shown are the Higgsino purity contours (labeled 0.1, 0.5, and 0.9). As one can see, the shaded region is mostly gaugino (low Higgsino purity). Masses are in GeV.

51

As should be clear from Figures 16 and 18, binos are a good and likely choice for dark mat te r in the MSSM. For fixed mi, Q,h2 J> 0.1, for all m g = 20 — 250 GeV largely independent of tan /? and the sign of /i. In addition, the requirement tha t m: > m^ translates into an upper bound of about 250 GeV on the bino mass 5 9 ' 7 2 . By further adjusting the trilinear A and accounting for sfermion mixing this upper bound can be relaxed6 4 and by allowing for non-zero phases in the MSSM, the upper limit can be extended to about 600 GeV 6 5 . For fixed Qh2 = 1/4, we would require sfermion masses of order 120 - 250 GeV for binos with masses in the range 20 - 250 GeV. The Higgsino relic density, on the other hand, is largely independent of m*. For large /i, annihilations into W and Z pairs dominate, while for lower n, it is the annihilations via Higgs scalars which dominate. Aside from a narrow region with rrijj < my/ and very massive Higgsinos with rrijj > 500 GeV, the relic density of H\i is very low. Above about 1 TeV, these Higgsinos are also excluded.

As discussed in section 4, one can make a further reduction in the number of parameters by setting all of the soft scalar masses equal at the G U T scale (similarly for the A parameters as well). In this case the free parameters are

m 0 i m i / 2 i j 4 . and tan/3 , (27)

with fj, and m^ being determined by the electroweak vacuum conditions, the former up to a sign. We refer to this scenario as the constrained MSSM (CMSSM). In Figure 19 7 3 , this parameter space is shown for tan/3 = 10. The light shaded region corresponds to the portion of parameter space where the relic density fix/i

2 is between 0.1 and 0.3. The dark shaded region (in the lower right) corresponds to the parameters where the LSP is not a neutralino but rather a TR. The cosmologically interesting region at the left of the figure is due to the appearance of pole effects. There, the LSP can annihilate through s-channel Z and h (the light Higgs) exchange, thereby allowing a very large value of mo- However, this region is excluded by the phenomenological constraints described below. The cosmological region extends toward large values of m j / j , due to coannihilation effects between the lightest neutralino and the TR 78. For non-zero values of A$, there are new regions for which X — t are impor tant 7 9 .

Figure 19 also shows the current experimental constraints on the CMSSM parameter space provided by direct searches at LEP. One of these is the limit m x ± J> 103.5 GeV provided by chargino searches at LEP 6 1 . LEP has also provided lower limits on slepton masses, of which the strongest is rag ^ 99 GeV 7 5 . This is shown by dot-dashed curve in the lower left corner of Fig. 19.

Another impor tant constraint is provided by the LEP lower limit on the

52

800-tanp = 10, ^i>0

T^^twyil'fiU'JU'fmM^Wjjry^Wl'y

mh * 114 GeV

100 200 300 400 500 600 700 800 900 1000

m1/2 (GeV)

Figure 19. Compilation of phenomenological constraints on the CMSSM for tan 0 = 10, fi > 0, assuming A0 = 0,mt = 175 GeV and m t ( m j , ) ^ | = 4.25 GeV 7 S . The near-vertical lines are the LEP limits m x ± = 103.5 GeV (dashed and black)6 1 , and roj, = 114.1 GeV (dotted and red) 7 4 . Also, in the lower left corner we show th mg = 99 GeV contour7 5 . In the dark (brick red) shaded regions, the LSP is the charged f i , so this region is ex­cluded. The light(turquoise) shaded areas are the cosmologically preferred regions with 0.1 < Qh2 < 0.3 7 e . The medium (dark green) shaded regions are excluded by 6 - • s-y 7 7 . The shaded (pink) regions in the upper right regions delineate the 2<7 range of sM - 2. The ± 1 a contours are also shown as solid black lines inside the shaded region.

Higgs mass: mu > 114.1 GeV 74. This holds in the Standard Model, for the lightest Higgs boson h in the general MSSM for tan/3 £ 8, and almost always in the CMSSM for all tan [3. Since m/, is sensitive to sparticle masses, particularly rat-, via loop corrections, the Higgs limit also imposes important constraints on the CMSSM parameters, principally m ^ as seen in Fig. 19.

53

The constraints are evaluated using FeynHiggs80, which is estimated to have a residual uncertainty of a couple of GeV in m/,.

Also shown in Fig. 19 is the constraint imposed by measurements of b —>• sf 77. These agree with the Standard Model, and therefore provide bounds on MSSM particles, such as the chargino and charged Higgs masses, in particular. Typically, the b —> s~f constraint is more important for fx < 0, but it is also relevant for fi > 0, particularly when tan /3 is large. The region excluded by the b —y 57 constraint is the dark shaded (green) region to the left of the plot.

The final experimental constraint considered is that due to the measure­ment of the anomalous magnetic moment of the muon. The BNL E821 exper­iment reported last year a new measurement of a^ = ^{dv ~2) which deviated by 2.6 standard deviations from the best Standard Model prediction available at that time 81. The largest contribution to the errors in the comparison with theory was thought to be the statistical error of the experiment, which will soon be significantly reduced, as many more data have already been recorded. However, it has recently been realized that the sign of the most important pseudoscalar-meson pole part of the light-by-light scattering contribution82

to the Standard Model prediction should be reversed, which reduces the ap­parent experimental discrepancy to about 1.6 standard deviations.

As many authors have pointed out83, a discrepancy between theory and the BNL experiment could well be explained by supersymmetry. As seen in Fig. 19, this is particularly easy if /i > 0. The medium (pink) shaded region in the figure is the new 2 a allowed region: —6 < <5aM x 1010 < 58. With the change in sign of the meson-pole contributions to light-by-light scattering, good consistency is also possible for n < 0 so long as either mj/2 or mo are taken sufficiently large.

In addition to the coannihilation region discussed above, another mech­anism for extending the allowed CMSSM region to large mx is rapid anni­hilation via a direct-channel pole when mx ~ hn^A 84, . This may yield a 'funnel' extending to large mj/2 and mo at large tan/3, as seen in Fig. 20 73.

6.2 Detection

As an aid to the assessment of the prospects for detecting sparticles at dif­ferent accelerators, benchmark sets of supersymmetric parameters have often been found useful, since they provide a focus for concentrated discussion. A set of proposed post-LEP benchmark scenarios85 in the CMSSM are illus­trated schematically in Fig. 21. They take into account the direct searches for sparticles and Higgs bosons, 6 —> S7 and the preferred cosmological density range. All but one of the benchmark points are consistent with g^ — 2 at the

54

o s

15001

1000-

0-

r \

\

s

>

\

- ii4c;c\

-

<. \

i

tan (3 = 50, \L > 0

- ° ; \

s s

s

^ S 1

" s s s s s s s **

x ; s s s s

^ - - ' ]

s * 1

s "s

•

•

100 1000 2000

m1/2 (GeV) 3000

Figure 20. As in Fig. 19 for tan/8 = 50 '

2tr level. The proposed points were chosen not to provide an 'unbiased' statistical

sampling of the CMSSM parameter space, but rather are intended to illustrate the different possibilities that are still allowed by the present constraints85. Five of the chosen points are in the 'bulk' region at small mi/j and m0 , four are spread along the coannihilation 'tail' at larger mx^ for various values of tan/?, two are in rapid-annihilation 'funnels' at large roi/2 and ra0. At large values of TO0, (larger than that shown in Figs. 19 and 20, there is another region where the cosmological range is satisfied, namely in the 'focus-point' region86 along the boundary where electroweak symmetry no longer occurs (shown in Fig. 21 as the shaded region in the upper left corner.) Two points were chosen in the 'focus-point' region at large ra0. The proposed points range

55

>rwtf1 fW^TWW^WjpfW'.'ftHW'l'j? ilWlVWtf ivwvriirpTli"ij'i"i''rf<f'ii in J;i u t rfwi| |"|i1i"»'rp u M ; 11 m

m l / 2

Figure 21. Schematic overview of the CMSSM benchmark points proposed in 8 S . The points are intended to illustrate the range of available possibilities. The labels correspond to the approximate positions of the benchmark points in the (m 1 y 2 , " io) plane. They also span values of tan /? from 5 to 50 and include points with n < 0.

over the allowed values of tan /? between 5 and 50. Prospects for the detection of supersymmetry at future colliders was studied extensively in 85 for these points in particular.

Because, the LSP as dark matter is present locally, there are many avenues for pursuing dark matter detection. Here I conclude by showing the prospects for direct detection for the benchmark points discussed above87. Fig. 22 shows rates for the elastic spin-independent scattering of supersymmetric relics87, including the projected sensitivities for CDMS II 8 8 and CRESST89 (solid) and GENIUS90 (dashed). Also shown are the cross sections calculated in the proposed benchmark scenarios discussed in the previous section, which are

56

' " " i - r 'T; ; . , - " , r-T", — — 1 I r " l " T

m j ; (GeV) m j ; (GeV)

Figure 22. Elastic spin-independent scattering of supersymmetric relics on (a) protons and (b) neutrons calculated in benchmark scenarios87, compared with the projected sensitivities for CDMS II 8 8 and CRESST8 9 (solid) and GENIUS9 0 (dashed). The predictions of our code (blue crosses) and Heutdr iver 9 2 (red circles) for neutralino-nucleon scattering are compared. The labels A, B, ...,L correspond to the benchmark points as shown in Fig. 21.

considerably below the DAM A91 range (10 - 5 — 10 - 6 pb), but may be within reach of future projects. Indirect searches for supersymmetric dark matter via the products of annihilations in the galactic halo or inside the Sun also have prospects in some of the benchmark scenarios87.

Acknowledgments

This work was supported in part by DOE grant DE-FG02-94ER40823 at Minnesota.

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Invited Talks

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RESEARCH P R O G R A M OF THE TEXONO COLLABORATION: STATUS A N D HIGHLIGHTS

HENRY TSZ-KING WONG

Institute of Physics, Academia Sinica, Nankang 11529, Taipei, Taiwan. E-mail: htwong@phys.sinica.edu.tw

JIN LI

Institute of High Energy Physics, Beijing 100039, China. Email: lij@mail.ihep.ac.cn

This article reviews the research program and efforts for the TEXONO Collab­oration among scientists from Taiwan and China. These include reactor-based neutrino physics at the Kuo-Sheng Power Plant in Taiwan as well as various R&D efforts related to the experimental techniques in neutrino and astro-particle physics.

1 Introduction and History

The TEXONO" Collaboration1 has been built up since 1997 to initiate and pursue an experimental program in Neutrino and Astroparticle Physics2. By the end of 2001, the Collaboration comprises more than 40 research scien­tists from major institutes/universities in Taiwan (Academia Sinica^, Chung-Kuo Institute of Technology, Institute of Nuclear Energy Research, National Taiwan University, National Tsing Hua University, and Kuo-Sheng Nuclear Power Station), China (Institute of High Energy Physics*, Institute of Atomic Energy*, Institute of Radiation Protection, Nanjing University, Tsing Hua University) and the United States (University of Maryland), with AS, IHEP and IAE (with +) being the leading groups. It is the first research collabo­ration of this size and magnitude, among Taiwanese and Chinese scientists from major research institutes.

The research program3 is based on the the unexplored and unexploited theme of adopting detectors with high-Z nuclei, such as solid state device and scintillating crystals, for low-energy low-background experiments in Neutrino and Astroparticle Physics4. The "Flagship" program5 is a reactor neutrino experiment to study low energy neutrino properties and interactions. It is the first particle physics experiment performed in Taiwan where local scientists are taking up major roles and responsibilities in all aspects of its operation: conception, formulation, design, prototype studies, construction, commission­ing, as well as data taking and analysis.

"Taiwan Experiment On NeutrinO

65

66

Kuo-sheng Nuclear Power Station : Reactor Building

Figure 1. Schematic side view, not drawn to scale, of the Kuo-sheng Nuclear Power Station Reactor Building, indicating the experimental site. The reactor core-detector distance is about 28 m.

In parallel to the reactor experiment, various R&D efforts coherent with the theme are initiated and pursued. Subsequent sections give the details and status of the program.

2 Kuo-Sheng Neutrino Laboratory

The "Kuo-Sheng Neutrino Laboratory" is located at a distance of 28 m from the core # 1 of the Kuo-Sheng Nuclear Power Station (KSNPS) at the northern shore of Taiwan5. A schematic view is depicted in Figure 1.

A multi-purpose "inner target" detector space of 100 cmx80 cmx75 cm is enclosed by 4n passive shielding materials cosmic-ray veto scintillator panels, the schematic layout of which is shown in Figure 2 The shieldings provide attenuation to the ambient neutron and gamma background, and are made up of, inside out, 5 cm of OFHC copper, 25 cm of boron-loaded polyethylene, 5 cm of steel and 15 cm of lead.

Different detectors can be placed in the inner space for the different scien­tific goals. The detectors will be read out by a versatile electronics and data acquisition systems6 based on a 16-channel, 20 MHz, 8-bit Flash Analog-to-Digital-Convertor (FADC) module. The readout allows full recording of all the relevant pulse shape and timing information for as long as several ms af-

67

Shielding Design [Only One out of Six Sides Shown]

I 'IU - ' »x75(H)c

Copper :5cm

Boras-loaded Polyethylene: 25 cm

Staialess Steel Frame: 5 cm

Lead: IS era

Veto Plastic Scintillator: 3 cm

Figure 2. Schematic layout of the inner target space, passive shieldings and cosmic-ray veto panels. The coverage is 4JT but only one face is shown.

ter the initial trigger. The reactor laboratory is connected via telephone line to the home-base laboratory at AS, where remote access and monitoring are performed regularly. Data are stored and accessed in a multi-disks array with a total of 600 Gbyte memory via IDE-bus in PCs.

It is recognized recently8 that due to the uncertainties in the modeling of the low energy part of the reactor neutrino spectra, experiments to mea­sure Standard Model neutrino-electron cross sections with reactor neutrinos should focus on higher electron recoil energies (T>1.5 MeV), as with (b), while neutrino magnetic moment searches should base on measurements with T<100 keV.

Accordingly, data taking for Period I Reactor ON/OFF has started in July 2001 and will continue till March 2002. Two detector systems are running in parallel using the same data acquisition system but independent triggers: (a) an Ultra Low Background High Purity Germanium (ULB-HPGe), with a fiducial mass of 1.06 kg, and (b) 46 kg of CsI(Tl) crystal scintillators. The target detectors are housed in a nitrogen environment to prevent background events due to the diffusion of the radioactive radon gas.

68

Kuo-Sheng Experiment: HPGe Detector

/?/ Radon Purge Plastic Bag

Figure 3. Schematic drawings of the ULB-HPGe detector with its anti-Compton scintilla­tors and passive shieldings.

2.1 Germanium Detector

As depicted in Figure 3, the ULB-HPGe is surrounded by Nal(Tl) and CsI(Tl) crystal scintillators as anti-Compton detectors, and the whole set-up is further enclosed by another 3.5 cm of OFHC copper and lead blocks.

The measured spectrum, after cuts of cosmic and anti-Compton vetos, during 12.2 days of reactor ON data taking is displayed in Figure 4. Back­ground (order of 1 keV - 1 kg _ 1 day - 1 ) and threshold (5 keV) levels comparable to underground Dark Matter experiment has been achieved on site. Additional cuts based on pulse shape and timing information are expected to further re­duce the background level at low energy. It is expected the data taken in Period I would allow us to achieve world level sensitivities in i7e magnetic moments (£t„)9, and therefore indirectly, radiative lifetimes (r„) 1 0 . These are the lowest threshold data so far for reactor neutrino experiments, and therefore allow the studies of more speculative topics, like y,v and Tv for ve

from reactors, possible nuclear cross-sections, as well as anomalous neutrino interactions.

69

I I I | I I I I [ I M I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I

10 20 30 40 50 60 70 80 90 100

Reactor ON Spectra (keV)

Figure 4. The measured spectrum from the ULB-HPGe, after cuts of cosmic and anti-Compton vetos, during 12.2 days of reactor ON data taking.

2.2 Scintillating CsI(Tl) Crystals

The potential merits of crystal scintillators for low-background low-energy experiments were recently discussed4.

The CsI(Tl) detector system is displayed in Figure 5. Each crystal mod­ule is 2 kg in mass and consists of a hexagonal-shaped cross-section with 2 cm side and a length 40 cm. The first sample is with two 20 cm crystals glued optically at one end to form a module (L20+20). Techniques to grow CsI(Tl) mono-crystal of length 40 cm (L40), the longest in the world for commer­cial production, have been developed and are deployed in the production for subsequent batches. The light output are read out at both ends by custom-designed 29 mm diameter photo-multipliers (PMTs) with low-activity glass. The sum and difference of the PMT signals gives information on the energy and the longitudinal position of the events, respectively.

Extensive measurements on the crystal prototype modules have been performed11. The energy and spatial resolutions as functions of energy are depicted in Figure 6. The energy is defined by the total light collection Qi + Q2. It can be seen that a ~10% FWHM energy resolution is achieved at 660 keV. The detection threshold (where signals are measured at both

70

Csl(TI) Longitudinal View Cross-Sectional View

Figure 5. Schematic drawings of the CsTJTl) target configuration. A total of 23 modules (46 kg) is installed for Period I.

PMTs) is <20 keV. The longitudinal position can be obtained by considering the variation of the ratio R = (Qi - Q2)/(Qi + Q2) along the crystal. Res­olutions of ~2 cm and ~3.5 cm at 660 keV and 200 keV, respectively, have been demonstrated.

In addition, CsI(Tl) provides powerful pulse shape discrimination capabil­ities to differentiate 7/e from a events, with an excellent separation of >99% above 500 keV. The light output for a's in CsI(Tl) is quenched less than that in liquid scintillators. The absence of multiple a-peaks above 3 MeV 12 in the prototype measurements suggests that a 238U and 2 3 2Th concentration (assuming equilibrium) of < 10 - 1 2 g/g can be achieved.

The data taken from the CsI(Tl) detector for Period I would be used for further optimization of the operation parameters as well as for studying the background. A cosmic muon event is shown in Figure 7. A >150 kg system will be installed for Period II. The physics goals include studies of neutrino-electron and neutrino-nuclei scattering cross sections.

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3 R&D Program

Various projects with stronger R&D flavors are proceeding in parallel to the reactor experiment. The highlights are :

3.1 Low Energy Neutrino Detection

It is recognized recently that 176Yb and 160Gd are good candidate targets in the detection of solar neutrino (i/e) by providing a flavor-specific time-delayed

72

EVENT

Z ± L

TOP

LIZZI

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Figure 7. Two typical cosmic ray events taken on site at KS Lab with the CsI(Tl) detector system.

tag13. Our work on the Gd-loaded scintillating crystal GSO14 indicated ma­jor background issues to be addressed. We are exploring the possibilities of developing Yb-based scintillating crystals, like doping the known crystals YbAl15012(YbAG) and YbA103(YbAP) with scintillators.

In addition, we have completed a feasibility study on boron-loaded liquid scintillator for the detector of i7e

15. The case of "Ultra Low-Energy" HPGe detectors, with the potential applications of Dark Matter searches neutrino-nuclei coherent scatterings, are now being investigated.

3.2 Dark Matter Searches with CsI(Tl)

Experiments based on the mass range of 100 kg of Nal(Tl) are producing some of the most sensitive results in Dark Matter "WIMP" searches16. The feasibilities and technical details of adapting CsI(Tl) or other good candi­date crystal like CaF2(Eu) for WIMP Searches have been studied. A neutron test beam measurement was successfully performed at IAE 13 MV Tandem accelerator17. We have collected the world-lowest threshold data for nuclear recoils in Csl, enabling us to derive the quenching factors, displayed in Fig­ure 8, as well as to study the pulse shape discrimination techniques at the

EVENT 55

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V—

73

120 140 Recoil energy (keV)

Figure 8. The quenching factors, shown as black circles, measured at IAE Tandem, as compared to Open triangles and open squares from previous work.

realistically low light output regime. The KIMS Collaboration will pursue such an experiment in South Korea18.

3.3 Radio-purity Measurements with Accelerator Mass Spectrometry

Measuring the radio-purity of detector target materials as well as other labo­ratory components are crucial to the success of low-background experiments. The typical methods are direct photon counting with high-purity germanium detectors, a-counting with silicon detectors or the neutron activation tech­niques. We are exploring the capabilities of radio-purity measurements fur­ther with the new Accelerator Mass Spectroscopy (AMS) techniques19. This approach may be complementary to existing methods since it is in principle a superior and more versatile method as demonstrated in the 13C system, and it is sensitive to radioactive isotopes that do not emit 7-rays (like single beta-

74

decays from 87Rb and 129I) or where 7 emissions are suppressed (for instance, measuring 39K provides a gain of 105 in sensitivity relative to detecting 7's from 40K). A pilot measurement of the 1 2 9I/1 2 7I ratio (< 10"12) in Csl was successfully performed demonstrating the capabilities of the Collaboration. Further beam time is scheduled at the IAE AMS facilities20 to devise mea­suring schemes for the other other candidate isotopes like 238U, 232Th, 87Rb, 40 K in liquid and crystal scintillators beyond the present capabilities by the other techniques.

3.4 Upgrade of FADC for LEPS Experiment

Following the success in the design and operation of the FADCs at the KS Lab, we will develop new FADCs for the Time Projection Chamber (TPC) constructed as a sub-detector for the LEPS experiment at the SPring-8 Syn­chrotron Facilities in Japan21. The current FADCs are being used to provide readout to test the prototype TPC, an event of which is depicted in Figure 9. The upgraded FADCs will have 40 MHz sampling rate, 10-bit dynamic range and be equipped with Field Programmable Gate Array (FPGA) capabilities for real time data processing. This new system is expected to be commissioned in Fall 2002.

In addition, the Collaboration is participating in the discussions on the scientific program and technical feasibilities of (a) the "H2B" project22: a 2000 km Very Long Baseline High Energy Neutrino Experiment at Beijing to receive a neutrino beam from the HIPA Facilities in Tokyo due to be commis­sioned by 2006 in Japan, and (b) the detection scheme of very high energy tau-neutrinos using mountain ranges as target and air as the subsequent show­ering volume23.

4 Outlook

With the strong evidence d neutrino oscillations from atmospheric and solar neutrino experiments7, there are intense world-wide efforts to pursue the next-generation of neutrino projects. Neutrino physics and astrophysics will remain a central subject in experimental particle physics in the coming decade and beyond. There are room for ground-breaking technical innovations - as well as potentials for surprises in the scientific results.

A Taiwan, China and U.S.A. collaboration has been built up with the goal of establishing a qualified experimental program in neutrino and astro-particle physics. It is the first generation collaborative efforts in large-scale basic research between scientists from Taiwan and Mainland China. The tech-

75

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nical strength and scientific connections of the Collaboration are expanding and consolidating. The flagship experiment is to perform the first-ever par­ticle physics experiment in Taiwan at the Kuo-Sheng Reactor Plant. Prom the Period I data taking, we expect to achieve world-level sensitivities and neutrino magnetic moments and radiative lifetime studies. A wide spectrum of R&D projects are being pursued. New ideas are being explored within a bigger framework.

The importance of the implications and outcomes of the experiment and experience will lie besides, if not beyond, neutrino physics.

5 Acknowledgments

The authors are grateful to the scientific members, technical staff and indus­trial partners of TEXONO Collaboration, as well as the concerned colleagues for the many contributions which "make things happen" in such a short pe­riod of time. Funding are provided by the National Science Council, Taiwan and the National Science Foundation, China, as well as from the operational funds of the collaborating institutes.

76

References

1. Home Page at http://hepmail.phys.sinica.edu.tw/~texono/ 2. C.Y. Chang, S.C. Lee and H.T. Wong, Nucl. Phys. B (Procs. Suppl.)

66, 419 (1998). 3. H.T. Wong and J. Li, Mod. Phys. Lett. A 15, 2011 (2000). 4. H.T. Wong et al., Astropart. Phys. 14, 141 (2000). 5. H.B. Li etal. , TEXONO Coll., hep-ex/0001001, Nucl. Instrum. Methods

A, Nucl. Instrum. Methods A 459, 93 (2001). 6. W.P. Lai et al., TEXONO Coll., hep-ex/0010021, Nucl. Instrum. Meth­

ods A 465, 550 (2001). 7. For the overview of present status, see, for example, "Neutrino 2000 Conf.

Proa" , ed. J. Law, R.W. Ollerhead and J.J. Simpson, Nucl. Phys. B (Procs. Suppl.) 91 (2001), and references therein.

8. H. T. Wong and H.B. Li, hep-ex/0111002 (2001). 9. P.Vogel and J.Engel, Phys. Rev. D 39, 3378 (1989).

10. G.G. Raffelt, Phys. Rev. D 39, 2066 (1989). 11. Y. Liu et al., TEXONO Coll., hep-ex/0105006, in press, Nucl. Instrum.

Methods (2002). 12. U. Kilgus, R. Kotthaus, and E. Lange, Nucl. Instrum. Methods A 297,

425, (1990); R. Kotthaus, Nucl. Instrum. Methods A 329, 433 (1993).

13. R.S. Raghavan, Phys. Rev. Lett. 78, 3618 (1997). 14. S.C. Wang, H.T. Wong, and M. Fujiwara, hep-ex/0009014, Nucl. In­

strum. Methods A, in press (2000). 15. S.C. Wang et a l , Nucl. Instrum. Methods A 432, 111 (1999). 16. R. Bernabei et al., Phys. Lett. B 389, 757 (1996);

R. Bernabei et al., Phys. Lett. B 450, 448 (1999). 17. M.Z. Wang et al., nucl-ex/0110003, submitted to Phys. Rev. C (2001). 18. H.J. Ahn et al., KIMS Coll., Techical Design Report, (2001). 19. D. Elmore and F.M. Phillips, Science 346, 543 (1987). 20. S. Jiang et al., Nucl. Instrum. Methods B 52, 285 (1990);

S. Jiang et al., Nucl. Instrum. Methods B 92, 61 (1994). 21. T. Nakano, LEPS Coll., Nucl. Phys. A 684, 71c (2001). 22. H.S. Chen et al., hep-ph/0104266 (2001). 23. G. Hou, these proceedings.

N E W R E S U L T S F R O M A M S C O S M I C R A Y M E A S U R E M E N T S

M. A. H U A N G

Institute of Physics. Academia Sinica, Taipei, 11529. Taiwan, R.O.C.*

The Alpha Magnetic Spectrometer (AMS) is a detector designed to search for antimatter in the cosmic rays. The physics results from the test flight in June 1998 are analyzed and published. This paper reviews the results in the five published papers of the AMS collaboration, updates the current understanding of two puzzles, albedo e+ /e — and albedo 3He, and disscusses the influence of albedo particles.

1 Introduction

The Alpha Magnetic Spectrometer, AMS, is a space borne charged particle detector 1 that will be installed on the International Space Station in 2005 for three years. The primary goals of AMS are to detect antimatter and dark matter, as well as to perform precision measurement of primary cosmic rays. These goals are closely related to astroparticle physics, especially in the matter-antimatter asymmetry, dark matter candidates, and atmospheric neutrino. This review begins with a short introduction to the AMS. Section 2 summarizes the physics results from the June 1998 shuttle flight. Section 3 reviews the updated information about the two puzzles from AMS measure­ments. Section 4 discusses the influence of albedo particles on atmospheric neutrino and space science.

In June 1998, a prototype detector, called AMS01, was flown in space shuttle Discovery on flight STS-91. The major components of AMS01 iclude a permanent magnet, time of flight detectors, and silicone trackers to provide measurements of charge, velocity and curvature. Thus it can identify particles and antiparticles. In addition, there is an aerogel Cerenkov threshold detector, which helps distinguish leptons from hadrons at high energy.

The AMS collaboration is constructing a new detector called AMS02. AMS02 is upgraded with the following features.

• A super-conducting magnet replaces the permanent magnet. This would increase the magnetic field strength and maximum detectable rigidity by 10 times.

• A 8 layered silicone tracker replaces the 6 layered one. This would im­prove rigidity resolution.

• Current affiliation: Department of Physics, National Taiwan University, Taipei, Taiwan, R.O.C., Email: huangmh@phys.ntu.edu.tw

77

78

• A ring image Cerenkov counter (RICH) replaces the aerogel Cerenkov counter.

• An electromagnetic calorimeter and a transition radiation detector are added.

• A synchrotron radiation detector is being tested and will be added if it performs well.

2 AMS physics results

During the 1998 test flight, AMS01 recorded approximately 108 events. The physics results had been published in five journal papers 1>2'3-4'5. One recent result of deuteron is presented at the 27th International Cosmic Ray Con­ference 6. The results related to cosmic rays are reviewed in this section. Detailed information about data selection and background elimination can be found in original articles or in the review paper 7.

2.1 Search for anti-helium

One of the major problems in the evolution of early universe is the disappear­ance of antimatter. The four requirements for matter-antimatter asymmetry are not fully complied. There are no positive evidences supporting the ex­istence or absence of antimatter. A direct detection of anti-nuclei such as anti-helium or anti-carbon could signal the existence of antimatter.

Under strict selection criteria, no anti-helium was found and 2.86 x 106

helium with rigidity of 1 to 140GV survives 1. The antimatter limit at 95% confidence level is then estimated by assuming that anti-helium has the same spectrum as helium. The anti-helium limit is He/He = 1.1 x 10~6 in the rigidity range of 1 to 140 GV. This result and some previous limits 8 are plotted in Fig. 1. With the upgrade in magnet and longer operation time, the AMS02 could reach the anti-helium limit to 10~9, three order of magnitude lower than that of AMS01.

2.2 Search for dark matter

Recent observation of CMB anisotropy measurements confirmed, once again, the existence of a large amount of dark matter. One of the candidates of dark matter, weakly interacting massive particle (WIMP), could annihilate in the halo of galaxy and produce an excess of positrons. The AMS can make indirect search of WIMP through the detection of positrons.

Cosmic positrons come mainly from the decay of charged pions, which are

79

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produced by interaction of cosmic rays with interstellar medium. Some of the early measurements of cosmic positron fraction show a suspicious peak above the expected flux of secondary origin at above 10 GeV 9 . However, one recent high statistics experiment 10 fail to reproduce previous results.

For AMS01 data, the separation of positrons from large background of protons is limited by the poor performance of the aerogel Cerenkov counter. The energy range is only up-to 3 GeV. The AMS fluxes of positron and elec­tron 3 and the positron fraction, e+/(e + + e - ) are consistent with most pre­vious measurements 11. Fig. 2 shows the cosmic positron fraction of AMS01 and several previous measurements 9>10.11.12.

The AMS01 cannot identify the possible positron signal from annihilation of WIMP at higher energy. The new AMS02 detector will add a ring imaging Cerenkov detector and a calorimeter to enhance the chance of detecting this dark matter signal.

2.3 Cosmic rays spectra

Atmospheric neutrinos come from the interaction of cosmic rays with the atmosphere. The large acceptance and multiple sub-detectors of AMS can

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make precise measurements of cosmic rays flux and composition. Although the primary cosmic ray flux has been measured many times, the AMS is the first instrument that measures cosmic rays globally. This information is essential to the calculation of atmospheric neutrino. Proton spectrum The first study of protons 2 use data from two periods, one with the detector facing space (downward events) and one with the detector facing the Earth (upward events). The data are separated into 10 latitude bins, shown in Fig. 3. For each bin, the spectrum is a mixture of two spectra, a cosmic ray and a sub-cutoff component. Section 3 will discuss the sub-cutoff components in detail. Cosmic proton spectrum All the available data are used in a separate study 4

on primary cosmic ray proton. The rigidity is selected with

R>(l + 2aRc)xRc

where Rc is the maximum of rigidity cutoff in the corresponding geomag­netic latitude, and the O-RC is the relative rigidity resolution at Rc. The final spectrum, shown in Fig. 4, is fitted to the power law spectrum at rigidity 10 < R < 100 GV.

d(j>

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is 2.78 ± 0.009(fit) ± 0.019(sys) and 17.1 ± 0.15(fit) ± 1.3(sys) ± 1.5(7)

Energy (GeV)

Figure 2. The positron fraction of primary cosmic rays measured by the AMS and some previous measurements. At energy < 3 GeV, the AMS and CAPRICE show consistent results.

81

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Figure 3. The upper figure shows the downward proton fluxes in 10 geomagnetic latitude intervals. The solid line is the cosmic proton. For each latitude interval, the proton fluxes have a dip due to the rigidity cutoff; below this cutoff, there is a second spectrum. The lower figure shows the upward proton fluxes. Cosmic rays do not exist in these upward events.

Cosmic helium flux For the study on cosmic helium flux 5, 79 hours of data taken before shuttle landing were used. Helium samples were selected with charged number \Z\ = 2. The major contamination comes from protons with mis-reconstructed charge, and it is estimated to be less than 10~4. The ac­ceptance of detector was determined to be 0.1 m2 sr at rigidity > 20 GV and increased to 0.16 m2 sr at lower rigidity. The overall uncertainty of acceptance is 6%, which include uncertainties from trigger condition (4%), track recon­struction (3%), and particle interactions combined with event selections (2%). The cosmic ray events are selected when the geomagnetic rigidity cutoff of the z of AMS01 detector is less than 12 GV. The differential flux, shown in Fig. 4, was fitted to a power law spectrum at rigidity 20 GV to 200 GV. The differen­tial spectrum index 7 is 2.740 ±0.010(stat)±0.016(sys) and the normalization

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& eessa8.aj,tf.>!*/oooM8i n IMAX 9-), ApJ 533:231, (200C) 0 CAPRICE 94, ApJ 518:457, (1999)

HKKM 95, PRD 52:4985, (1995)

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Figure 4. The cosmic proton and helium fluxes measured by the AMS are plotted with some previous measurements. The solid lines are the primary proton and helium fluxes used in the HKKM-95 atmospheric neutrino model.

constant 0O is 2.52±0.09(stat) ±0.13(sys) ± 0.14(7) GV 2 7 4 / (m 2 s sr MV). Influence of AMS cosmic ray measurement Fig. 4 shows the cosmic proton and helium spectra of AMS, several recent measurements, and spectrum used in the atmospheric neutrino calculation model. The AMS spectra are consis­tent with those of previous measurements; however, the HKKM-95 model 13

seems to have higher flux at energy above 20 GeV. Since the cosmic ray flux is the main input parameter of atmospheric neutrino simulations, it is difficult to compare the difference between results from groups using different models. Inspired by the consistency between recent high statistic measurements from AMS 4, BESS 14, and CAPRICE 15, some groups proposed to use an unify spectrum 16.

2.4 Cosmic ray light isotopes abundance

Cosmic light isotopes such as 2D and 3He play an important role in determin­ing the mean amount of matter traversed by cosmic rays inside the galaxy. The excess of 2D and 3He comes from the spallation of heavy cosmic rays, 4He or CNO, interacting with interstellar medium (ISM). 4He loses one nucleon

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and becomes 3 T or 3He. At AMS observed rigidity R/n <; IGV/n, the 3 T half life is only 7 X 12.26 <; 20 years, which is much smaller than 107 years, the typical residence time of cosmic rays. So 3 T decays almost completely to 3He. 4He also breaks into two 2D. 2D and 4He have the same rigidity for the same energy/nucleon, therefore, they suffer same solar modulation effect. The 2D/4He is an important indicators for studying cosmic ray transportation in galaxy and solar system.

Cosmic 3He From the cosmic helium samples, the helium mass histogram is fitted with two components, 3He and 4He. The result shows that 11.5% of helium is 3He 5. The cosmic ray flux ratio 3He/4He is approximately 13%, much higher than the primordial abundance 3He/4He ~ 10~4. Cosmic 2D The deuteron samples are selected with charge +1 and mass com­patible with that of deuteron 6. The 1/P histograms for several velocity bands for p from 0.4 to 0.85 are fitted with those of proton and deuteron. Approx­imately 10% of deuteron samples are proton with wrongly reconstructed ve­locity. After deducting this tail, the chance of contamination from residual background of proton in the accepted deuteron samples is less than 1%.

Approximately 104 cosmic deuteron samples are selected from geomag­netic latitude Am > 0.9 rad. The deuteron flux is fitted to the solar modula­tion model 17. The best fit of the data is the Local Interstellar Space (LIS) spectrum index 2.75 and modulation parameter <j> = 650 ± 40 MV, consis­tent with the solar condition before the solar maximum in 20001. The flux ratio 2D/(3He + 4 He) is employed to evaluate the effect of cosmic ray trans­portation effect. The AMS measurement is consistent with the prediction of Stephens 18, who used the standard leaky box model, and was not in favor of some non-standard models such as re-acceleration theory 19.

3 Atmospheric albedo particles

3.1 Particles trapped inside geomagnetic field

Charged particles having rigidity below geomagnetic cutoff can be trapped inside the geomagnetic field. Some have energy low enough that their lowest altitudes are well above the atmosphere and they stay trapped for a long time. These particles form the radiation belts and have been studied quite thoroughly in the early years of space age.

AMS flew at an altitude approximately 380 km, well below the radiation belts. Surprisingly, AMS still observed many particles with rigidity below cutoff. For all the particles we studied so far, (including p, 2D, He, e~, e+) , all their spectra contain two components, cosmic rays and sub-cutoff parti-

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cles. Unlike the trapped particles in radiation belts, these sub-cutoff particles originate from and return to atmosphere in a very short time, less than 20 seconds 20. They also have some interesting features 7 , 2° . Two puzzling phe­nomena left unanswered in the AMS publications 3 '5.

The secondary albedo particles are produced in air shower. The decay chain of 7r —• fj, —* e, produce positrons, electrons, muons, and atmospheric neutrinos. The positron electron asymmetry also creates great interest among physicists working on atmospheric neutrino simulation. Several groups had developed Monte-Carlo simulation to study the production and transportation of albedo particles.

3.2 Albedo positron electron ratio

The first puzzle and the most surprising result from AMSOl is the albedo positron electron ratio 3. The flux ratio e + / e~ varies with magnetic latitude and can be as large as 4 near the magnetic equator. Some balloon experi­ments, operated in high latitude regions, obtained a ratio of approximately 1. Most of the radiation belts experiments in the 60s and 70s could not distin­guish between electrons from positrons. However, the presence of positrons in the radiation belts had been reported as early as 1983 21-22. AMS measures at higher energy (~ GeV), almost one order of magnitude higher than that in previous radiation belts experiments. The excess of antimatter raises ques­tions concerning their origin. At high latitude, these albedo positrons could have rigidity higher than the cutoff and be mistaken as cosmic rays.

Huang (1998) 23 proposed the first quantitative model of the positron electron ratio. For positive charged cosmic rays, the rigidity cutoff are lower from the west than that from the east. Therefore, more cosmic rays coming from the west than from the east. Because of the geomagnetic field, e+ com­ing from the west and e~ coming from the east have better chance to move upward. The combination of higher (lower) fluxes from the west (east) and secondary e+ (e~) moving upward produce the positron electron asymmetry. The difference in rigidity cutoff decreases with increasing magnetic latitude so does the flux ratio. Using this model and the observed e + / e~ , Huang (2001) 24 derived the arrival direction of primary cosmic rays must be near west (for albedo e+) and east (for albedo e~), respectively.

Monte-Carlo simulations 25 ,26 had reproduced the e + and e~ fluxes and the e + / e~ as a function of magnetic latitude. They provide information about the arrival direction of primary cosmic rays. Those results confirmed the theoretical prediction from Huang (2001) 24.

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20 30 40 50 60 magnetic latitude |A,AMS|

Figure 5. The AMS long flight-time e + / e ~ can be explained by the east-west effect. The only free parameter, zenith angle 8, in this fitting is adjusted according to the atitude of AMS detector.

3.3 Albedo 3He

The second puzzle is tha t 90% of albedo Helium is 3 He 5 . 3 He had been observed in radiation belts by ONR 2T at kinetic energy 40 - 100 M e V / n and SAMPEX 2 8 at kinetic energy 10-20 MeV/n . Both experiments observed low energy particles t rapped inside radiation belts. The AMS observation is in the high energy region and particles stay in space for a very short t ime, <; 20 seconds, compared with the life time of t rapped particles, which is much longer than days. Spallation of cosmic helium Cosmic ray 4He nuclei interact with air nuclei, they break up into 3 He, whose rigidity would be % times tha t of 4 He. When the incoming 4 He has rigidity less than about 4 / 3 t imes the cut-off rigidity, the 3 He fragment, having rigidity smaller than the cut-off, turns into an albedo 2 9 .

Huang and Stephens 3 0 simulate the interaction of cosmic helium with atmosphere. The result shows tha t 3 He produced by spallation exists only

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in specific phase space of rigidity and magnetic latitude. Only few events near -0.6 < Am < -0 .5 and energy >1 GeV/n can be explained by this mechanism. Pick-up Reaction: p ( 4He, d)3He Selesnick and Mewalt31 proposed that pro­tons in radiation belts picking up one neutron from helium in the upper atmo­sphere may be able to explain the light isotopes in radiation belts. However, the radiation belt protons are not energetic enough to produce the 3He as observed by AMS. Huang and Stephens 30 modified this model using cosmic ray protons. Although 3He could be produced in low latitude and rigidity ranges similar to the AMS measurements, there are some serious difficulties. First, the ambient helium density is too low to produce 3He flux comparable with that measured by the AMS. Second, the spectrum is too steep at energy higher than 0.5 GeV/n. Monte-Carlo simulation of light isotopes Derome and Buenerd 32 also ana­lyzed the light isotopes, such as deuteron, 3He, 3T, from their simulations. The coalescence model is used to explain the production of 3He. The incident proton gets absorbed in the atmospheric nitrogen or oxygen to form a com­pound nucleus, which decays to various light nuclei. The individual nucleons must be close to each other in order to form a nuclei. Therefore, the probabil­ity of forming heavier nuclei is much less then that of forming lighter nuclei. This model explains successfully the existence of albedo proton, 2D, 3He, and 4He. Pugacheva et al. 3 3 also had a similar explaination of hydrogen and he­lium isotopes in radiation belts. Their model can be employed to understand the 3He observed by AMS.

4 Influence of albedo particles

4-1 Influence on atmospheric neutrino simulation

The albedo proton, as shown in the bottom of Fig. 3, could also produce air shower and contribute to the atmospheric neutrino flux. Since the albedo flux is approximately 1% of the primary cosmic ray flux, the contribution should be in the same order of magnitude or less 26>29. A good simulation should be able to explain all the features of albedo particles, including particle types, spectra, spatial and temporal distributions.

However, Plyaskin 34 claimed that his simulation reproduced the albedo positron electron ratio and up-down asymmetry of atmospheric neutrino with­out neutrino oscillation. This simulation uses an approach quite different from the others. Plyaskin used the GEANT simulation code and reduced the size of the Earth and the atmosphere into something similar to the size of an detector

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in accelerator experiment. He reported a large excess of contribution coming from the scattered protons which enter the atmosphere from the forbidden cone, where belong to t rapped particles. These scattered particles are not counted in traditional simulation algorithms. It is suspicious t ha t the large contribution from scattered protons may be the cause of his condensation of atmosphere and amy disappear in realistic case.

Although the current simulations reproduce most of the features of albedo particles. There remain some questions. The main argument is whether AMS or Monte-Carlo simulations over-counted the long flight t ime (LFT) events. While the L F T events passed through the AMS alti tude many times, they were detected in one position only. Some simulations counted the flux as from one single particle, others simulations counted each passage through AMS alti tude. The drift period of albedo particle in GeV range is approximately 10 second, much shorter than the 90-minute period of the space shutt le. AMS could not detect the same particle twice, therefore, AMS did not over-count the L F T particles.

Another question is the precision of simulation. The differences between MC simulation and experimental da ta are much larger than the error bars. Discrepency is the greatest near the cutoff and high lat i tude regions. I t simply shows that much remains to be learnt concering the penumbra region (an intermittent transit ion zone from t rapped particles to cosmic rays) 7 . There is certainly much room for improvement in the Monte-Carlo simulations.

4-2 Influence to space science

Space engineers must design proper shielding for satellites or as t ronauts to re­duce the ionization radiation caused by the charged particles in space. There were models of proton and electron fluxes in radiation belts, such as A P 8 and AE8 3 5 . The albedo particles have energy close to the minimum of ioniza­tion energy loss and could penetrate deep inside the protection layer. This effect had not been considered yet. In recent years, space physicists have re­gained interests in the high energy components in radiation belts. Wi th the large acceptance and particle identification capability, AMS could be the most powerful detector compared with other radiation belt experiments.

The AMS provide a global measurement of high energy particles, the measurements could be used to reconstruct a useful model. However, the current da ta are not very helpful. Although the energy loss in detector mate­rials had been restored to the "original energy" by deconvolution using Bayes theorem 2, this is only a statistical correction. Particles entering the detector suffer different amount of energy loss. Also the effect of residual magnetic field

88

is not corrected. Those two corrections change the pitch angle and energy dis­tributions. So the current flux of albedo particles could have large systematic error and can only be used for crude estimation. Some of the mismatches in AMS measurements and MC simulations might come from the simplification of deconvolution.

Acknowledgments

The author wishes to thank the organizer of this workshop, Professor G.L. Lin for his kind invitation. The author was supported by the topical program "Detecting cosmic rays with a precise space spectrometer" from Academia Sinica, Taiwan, R.O.C..

References

1. AMS collaboration, Alcaraz, J. et al., Phys. Lett. B, 461, 387, (1999) 2. AMS collaboration, Alcaraz, J. et al., Phys. Lett. B, 472, 215, (2000a) 3. AMS collaboration, Alcaraz, J. et al., Phys. Lett. B, 484, 10, (2000b) 4. AMS collaboration, Alcaraz, J. et al., Phys. Lett. B, 490, 27, (2000c) 5. AMS collaboration, Alcaraz, J. et al., Phys. Lett. B, 494, 1, (2000d) 6. Lamanna G., 2000, Ph.D. thesis, Perugia; Proc. 27th Int. Cosmic Ray

Conf. (Hamburg), pl614, (2001) 7. Huang, M.A., The 7th Taipei Astrophysics Workshop on the Cosmic

Rays in the Universe, ed. CM. Ko, Astro. Soc. of the Pacific Conf. Ser. 241, 197, (2001); astrop-ph/0104229

8. Aizu, H. et a l , Phys. Rev., 121, 1206, (1961); Evenson, P., Astro-phy. J., 176, 797, (1972); Smoot, G.F., Buffington, A.,& Orth, CD. , Phys.Rev.Lett., 35, 258, (1975); Badhwar, G.D. et al., Nature, 274, 137, (1978); Golden, R.L. et al., Astrophy. J., 479, 992, (1997); Buffington, A. et al., Astrophy. J., 248, 1179, (1981); Saeki, T. et al., Phys. Lett. B, 422, 319, (1998)

9. Fanselow, J.L. et al., Astrophy. J., 158, 771, (1969); Agrinier, B. et al., Lett. Nuovo Cimento, 1, 153, (1969); Buffington , A. et al., Astrophy. J., 199, 669, (1975); Golden, R.L. et al., Astron. k. Astrophy. 188, 145, (1987); Muller D. & Tang, K.K., Astrophy. J., 312, 183, (1987); Golden, R.L. et al., Astrophy. J., 436, 769, (1994)

10. Barwick, S.W. et al., Phys. Rev. Lett., 75, 390, (1995) 11. Barbiellini, G. et al., Astron. & Astrophy., 309, L15, (1996) 12. Daugherty, J. K. et al., Astrophy. J , 198, 493, (1975) 13. Honda, M. et al., Phys. Rev. D, 52, 4985, (1995)

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14. Sanuki, T. et al., Astroph. J., 545 , 1135, (2000) 15. Boezio, M. et al., Astroph. J., 518 , 457, (1999) & 5 6 1 , 787, (2001) 16. Gaisser, T.K. et a l , Proc. 27th Int. Cosmic Ray Conf., (Hamburg),

1643, (2001)

17. Gleeson L.J. & Axford W.I., Astroph. J., 154, 1011, (1968) 18. Stephens, S.A., Adv. Space Res. 9, 145, (1989) 19. Seo E.S. et al., Astroph. J., 4 3 1 , 705, & 432 , 656, (1994) 20. Huang, M.A., et al., Chinese Journal of Physics, 39, 1, (2001) 21. Just , L. et al., J. Geophys. 52 , 247, (1983) 22. Galper, A.M. et al., Proc. 18th Int. Cosmic Ray Conf., (Bangalore),

M G 1 0 - 3 3 , 497, (1983) 23. Huang, M.A., Proc. of the 8th Asia Pacific Physics Conference

(Taipei), ed. Y.D. Yao, H.Y. Cheng, C.S. Chang, and S.F. Lee, World Scientific, Singapore, p l 6 1 , (2001); a s t r o - p h / 0 0 0 9 1 0 6

24. Huang, M.A., Proc. 27th Int . Cosmic Ray Conf., (Hamburg) , 1733, (2001)

25. Derome L. & M. Buenrd, Phys. Lett . B, 515 , 1, (2001) 26. Honda M., M.A. Huang, __ K. Kasahara, Proc. 27th Int. Cosmic Ray

Conf., (Hamburg), 4120, (2001) 27. Chen, J., et al., Geophy. Res. J. Lett . , 21 , 1583, (1994); Astrophys. J.

442 , 875, (1995); Wefel, J .P. et al., Proc. 24th ICRC (Rome), SH 8.1.11, (1995)

28. Cumming, J.R., et al., Eos Trans. 76 AGU, Fall Meet. SuppL, F501, (1995)

29. Lipari, P., Astrpart . Phys., 16, 295, (2002) 30. Huang, M.A. & S.A. Stephens, Proc. 27th Int. Cosmic Ray Conf.,

(Hamburg), 4208, (2001) 31. Selesnick, R.S. and R.A. Mewaldt, J . Geophys. R e s , 101 , 19745,

(1996) 32. Derome L. & M. Buenrd, Phys. Lett. B, 5 2 1 , 133, (2001) 33. Pugacheva, G.I. et a l , Annales Geophysicae, 16, 931, (1998) 34. Plyaskin, P , Phys. Lett . B, 516, 213, (2001) 35. ht tp: / /nssdc.gsfc.nasa.gov/space/model/models_home.html

MEASUREMENT OF ATTENUATION LENGTH IN ROCK SALT AND LIMESTONE IN RADIO WAVE FOR ULTRA-HIGH ENERGY

NEUTRINO DETECTOR

MASAMICHIBA, MHO KAWAKI AND MASAHIDEINUZUKA

Department of Physics, Tokyo Metropolitan University 1-1 Minami-Ohsawa Hachioji-shi, Tokyo, 192-0397, Japan

E-mail: chiba-masami@cmetro-u.ac.jp

TOSfflO KAMTJO

Department of Electrical Engineering, Tokyo Metropolitan University 1-1 Minami-Ohsawa Hachioji-shi, Tokyo, 192-0397, Japan

E-mail: kamijo@eei.metro-u.ac.jp

H. ATHAR

Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan and Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan

E-mail: athar@phys.cts.nthu.edu.tw

Rock salt and limestone are studied to determine their suitability for use as a radio wave transmission medium in an ultra high energy (UHE) cosmic neutrino detector. The detector would detect radio wave generated by the Askar'yan effect (coherent Cherenkov from negative excess charges in an electromagnetic shower) in the interaction of the UHE neutrinos widi the high-density medium. We have measured the radio wave attenuation lengths of the rock salt samples from the Asse mine in Germany at 9.4 GHz and found it to be longer than 3.3 m and then whereas under the assumption of constant tan6" with respect to frequency, we estimate it by extrapolation to be longer than 330 m at 94 MHz.

1 Introduction

Several cosmologically distant astrophysical systems e.g. active galactic nuclei produce ultra-high-energy (UHE) cosmic neutrinos [1] of energies over the 1015 eV (PeV) whose flux, though very low, probably exceeds that of atmospheric neutrinos [2]. Therefore, despite the low flux and the low cross section [3], we can detect extraterrestrial neutrinos coming from far distance over 100 Mpc (300 million light years) without an atmospheric neutrino background. On the other hand, in spite of GZK (Greisen, Zatsepin and Kuz'min) cutoff [4], protons over 1020eV arrive to the earth [5]. Due to this proton flux, we could expect a neutrino flux to be produced by a process between the high-energy protons and the relic cosmic microwave background photons.

90

91

The aim of UHE-neutrino detection is to study (1) the UHE neutrino interaction, which is not afforded by artificial neutrino beams generated in accelerator [6], (2) the neutrino mass problem through the neutrino oscillation effect after a long flight distance [7] and (3) the UHE accelerating mechanism of the protons existing in the universe.

In order to detect UHE neutrinos, we need a detector with a huge mass (at least 109 tons) since neutrinos interact in the detector volume only via weak interactions and the flux of UHE neutrinos is very low. However, from a practical point of view, it is difficult to construct such a huge detector. We are therefore interested in the possibility of using a natural rock salt mine as a UHE neutrino detector, a Salt Neutrino Detector (SND) [8]. Rock salt deposits are distributed world wide, so there are many candidates for suitable sites [9]. Rock salt deposit does not allow water penetration, which hinders radio wave transmission. Figure 1 shows a scheme for an SND using a large volume of salt dome (1km x 1km x 1km).

1000m

Neutrino

1000m

Salt Neutrino Detector (SND)

Figure 1. Underground Salt Neutrino Detector. Excess electrons in the shower from the UHE neutrino interaction generate a coherent Cherenkov radiation with an emission angle of 66°.

In order to measure UHE neutrinos effectively, the mass of a detector should be considerably greater than that of the existing large neutrino detector, Super Kamiokande (S-K), Kamioka, Gifu Japan, which consists of 5 x 104 tons of pure water [10]. An SND is 4 xlO4 times more massive than S-K. The S-K detects visible light generated by the Cherenkov effect in pure water whose transparency is 100m at most. Rock salt, on the other hand, is one of the most transparent materials for radio waves [8,11,12, 13] as well as ice [14]. Therefore a moderate number of radio wave sensors could detect neutrino interactions in the massive rock salt. G. A. Askar'yan proposed detecting radio emissions with coherent amplification produced by the excess negative charges of electron-photon showers in dense materials, Askar'yan effect [15], which could be used to detect the

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interaction of UHE neutrinos with high-density media. While for the radio wave emission from the air shower, a thin material, the same effect was calculated independently by M. Fujii and J. Nishimura [16]. Recently Askar'yan effect was confirmed by a bunched electron beam supplied by an accelerator [17].

Unfortunately, no natural rock salt deposits are located in Japan. At the age of rock salt formation, Japan was under the sea and hence there was no possibility for deposits to form [9]. To find possible locations for constructing an SND we have visited five rock salt mines outside Japan and taken samples of the rock salt to do. We have also considered the possibility of using limestone as a detector mass. In Japan, limestone mines are abundant and the proportion of CaC03 in Japanese limestone is over 95%, compared with around 80% in limestone from overseas sources. We have taken samples of limestone from a mine at Kamaishi to examine its viability as a detector.

We have previously reported measurements of the complex permittivities of rock salt samples by a free space method [18]. In the present report we present results of the measurements of rock salt and limestone samples by a perturbation method using cavity resonator in which the precision of the imaginary part (the attenuation in a medium) of the permittivities is improved.

2 Consideration on the medium for Cherenkov effect

We compare density, radiation length, refractive index, Chrenkov angle and Cherenkov threshold kinetic energy of electrons for air, ice, rock salt and limestone in Table 1.

Table 1. Comparison among air, ice, rock salt and limestone in density, radiation length X0, refractive index, Chrenkov angle 9C and Cherenkov threshold kinetic energy for electrons To,.

Media

XirlSTP) Ice (H20)

Rock salt (NaCl)

Limestone (CaC03)

Density (g/cm3) . 6.0012 0.924 2.22

2.7

X0

.........(cm)

30420 39

10.1

9.0

Refractive Index

™T66o29l 1.78 2.43

2.9

ec _.(deg)....

1.387 55.8 65.7

69.8

T,h

(keV) 2060407

107 50

33

The density of air is 1/833 times less than rock salt and the radiation length is 3042 times larger than that even on the earth surface. Then the shower length and the diameter are much larger than ice, rock salt and limestone. The detection is not easy for such a horizontal extended air shower produced by neutrinos. In

93

addition Cherenkov or fluorescent light detection in air shower depend on weather condition severely as well as the Sunlight, the Moonlight, lightning and artificial light. Underground detectors neither depend on the weather nor the Sunlight and the Moonlight. In principle they could furnish the same detection ability for all the time and the directions of the neutrino incidence.

In comparison with ice, rock salt has 2.4 times higher density, shorter radiation length (-1/4) and 1.4 times larger refractive index. Further limestone has 20% larger density and 10% shorter radiation length than rock salt. Then the high-energy particle shower size is small owing to the high density. For electrons, the Cherenkov angle is large and the threshold energy is small due to the high refractive index. Consequently, rock salt is adequate medium to get larger Cherenkov radiation power. The frequency at the maximum electric field strength at 9C is -2 GHz without the absorption in ice [14]. Whereas it would increase up to -5.6 GHz taking into account the short radiation length and the wavelength contraction due to the higher refractive index. For lime stone 20% larger refractive index yields larger Cherenkov angle and the lower threshold electron energy as low as 33 keV.

When a UHE proton hits the atmosphere, it could not put the large energy deposit underground in a narrow region. Radio-wave neutrino detector could sense only from many tracks of excess electrons inside the concentrated region with the help of strengthening by the coherence effect. For high sensitivity detector e.g. optical Cherenkov detectors, which could detect single muons, a great number of downward muons become large backgrounds. On the contrary the radio Cherenkov detector is immune from the muons. Therefore, practically only UHE neutrinos could induce the electric field on the radio antennas underground. However it could be a shortcoming when we intend to detect abundant lower-energy neutrinos in high statistics.

Normally, rock salt is covered by thick soil, which absorbs electromagnetic wave completely. Then SND is background free from natural or artificial radio wave coming from the surface on the earth. As a result background is only blackbody radiation corresponding to the temperature of the surrounding rocks. As the remaining potential background, radiation may come from a seismic movement of the surrounding rocks, which may generate radio wave due to the piezoelectric effect by the stress in the rocks. If such a radiation could be detected, the observation contributes to the seismology.

3 Measurement of microwave attenuation length by the free space measurement method

The free space measurement method has been used in a non-destructive manner for the assessment of microwave absorbers [18]. The method involves measuring

94

the amplitude and phase of scattered radiation with and without a metal-plate reflector on the sample. We use a vector network analyzer HP85107A to make the measurements. An important feature of the method is the ability to subtract extraneous scattering from the scattering of interest.

We restate our results of the free space measurement method on rock salt (c.f. Ref. [8]). Three rock salt samples were used for the measurement of the complex permittivity by the free space measurement method. Two were taken from the Hallstadt salt mine in Austria and the third from the Asse salt mine in Germany. Each of the samples was prepared into sheets 200 mm x 200 mm square. The samples from the Hallstadt mine were 30 mm and 11 mm thick, and the sample from the Asse mine was 99 mm thick. The samples from the Hallstadt salt mine are brown in color and have a striped pattern to their coloration. The sample from the Asse salt mine is white and has no striped pattern. The rock salt appears to consist of small single crystals.

The measured real part of the complex permittivities of the three samples are tabulated in Table 2. The polarization A and B indicate that the values are derived from the linear polarization of parallel and perpendicular with the scattering plane, respectively. The estimated uncertainty in each of the real values is ± 0.2. The values of the real part are consistent each other and with the value of 5.9 in the reference material [11].

Table 2. Real part of complex permittivities in rock salts.

Thickness / Polarization A B (a) Hallstadt 1L 1mm 5"? + 65. £ b T a 2 (b) Hallstadt 30.1mm 5.9 ± 0.2 6.0 + 0.2

(c) Asse Mine 99.0mm 5.9 ± 0.2 5.9 ± 0.2

We can calculate the attenuation coefficient a for the case of a low loss material from the equation

co /-7tan5 .,, a = —y/e , (!)

c 2 where the complex permittivity e is

£ = £'-js" = e'(l-jtaxi5), (2)

and the loss angle in the permittivity tan<5is

t a n S = — (3)

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From e and tan 8 we can calculate the complex refractive index n,

n= Ve=Ve' Vl-7 tan 8 (4)

After traveling a distance z through a material, the complex electric field of the electromagnetic wave EQ becomes E,

E = E0ejoi-(a+mz = V ' V M ) (5)

where the propagation constant r= a+jp can be expressed as

a + jp = jcoje^je'-je". (6)

The scalar electric field decreases as

Therefore the electric field attenuation length La where the field strength decreases by a factor of 1/e is

The measurement accuracy was insufficient to allow the imaginary part of the permittivity to be determined. We found that the imaginary permittivity had different values for the samples (a) and (b), even though both were cut from the same block. We were able to estimate that the imaginary part of the permittivity is less than 0.1, or tan<5 is less than 0.017 at 9.4 GHz. We calculate a lower limit on a of 4.1 m"1 at 9.4 GHz. Hence the attenuation length is greater than 0.24 m at 9.4 GHz. Assuming that ian.8 is constant with respect to frequency, the attenuation length is larger than 24 m at 94 MHz.

Unfortunately, the accuracy of this method as used in this experiment is too poor for these low tan<5 samples. A more accurate value for tan<5 could be achieved if we used a sample with a larger area and thickness, and used the transmission configuration. This method also has the advantage in that it can be modified to make in situ measurements. However, in order to improve our results we have chosen to measure the permittivities of rock salt and limestone using the cavity perturbation method.

96

4 Measurement of microwave attenuation length by the cavity perturbation method

We have measured natural rock salt samples by the perturbed cavity resonator method at 9.4 GHz [19]. Note that there are no sample insertion holes in the cavity.

The Q value (ratio of resonance frequency to the resonance width) of this system is around 4000. For this perturbation method small samples, such as 1 mm x 1 mm X 10.2 mm, should be used in order to avoid changing the resonance behavior significantly, e.g. inducing only a small shift in the resonance frequency and resonance width. In addition, the electric field strength should be uniform over a cross section of the sample. It is difficult to cut fragile samples to this size. Mechanical cutting using a milling machine was unsatisfactory for our natural rock salt samples. Synthesized rock salt in single crystals could be cleaved to the size, but it was difficult to cleave natural rock salt and slightly thicker samples of natural rock salt had to be used. Limestone is strong enough and we could cut it with a milling machine.

Using this apparatus the complex permittivity e could be measured. The principle of the measurement is to derive the real and imaginary parts of the complex permittivity, e' and e", from the changes in the center frequency and the width of the resonance, respectively. Measurements were made both with and without the insertion of the sample in the cavity.

The real e' is found from the change in resonance frequency when the sample is placed in the cavity,

u J=ae(e - 1 ) — , (9) /o V

where / and f0 are the resonance frequencies with and without the sample in the cavity, dV and V are the volumes of the sample and the cavity resonator, and a£ is a constant determined by the TEi0n mode and the sample position relative to the electric field maximum—equal to 2 in this case of TE107. Note that the appearance of dV/V indicates that the size of the sample impacts on the perturbation of the resonance behavior and hence a small stick-shaped sample should be used.

The imaginary e" depends on the change in the Q factor, l r . 1 . . 1 ... „dV - ( — ) - ( — ) ] = cc,e — , (10) 2 Q Qo E V

where Q and Q0 are f/df and f(/df0, respectively, and df (~2.7MHz) and df0

(~2.5MHz) are the resonance widths measured at a height of half the peak height. The inverse Q difference (1/Q - 1/Q0) is defined as 1/Q,. The measured Q0 was found to be around 4000. A radio frequency signal was supplied to the cavity

97

resonator by a synthesized CW generator (Anritsu 68047C) and Q was measured by a HP8755B swept amplitude analyzer. The absolute uncertainty of the frequency measurements was well under lxlO"5.

We get the real e'and the imaginary s" of the complex permittivity solving eqs. (9) and (10), respectively.

In order to know the errors by error propagations from the ingredients of the equation, we used the error equations (11) and (12), respectively as:

2 -2 -2 -2 , = a fQ (dV) <

J \ 7/o

I_ 2 V + (/o-/)2 (11)

where of and of0 are the measuring errors (~10kHz) of the resonance peak

frequencies. The av and Z)TO are the measuring errors (~0.1mm)3 of the volumes of the cavity and the sample. The f0-f is the frequency difference (~40MHz) without and with a sample. The (dV/V) is 2.8X10"4 for the sample of 1.0irimxl.0mmxl0.2mm. Then the largest contribution to determine e'is from the measurement of dV. The sample volume is measured by a microscope furnished with a caliper and a micrometer. For the confirmation we also measured the weight and used the density to get the volume. After the calculation of eq. (11), we get ac- =0.144 which is 2.4% of E' (=6). Therefore we could get e' to the 3% accuracy. The magnitude of error in e' is consistent with the results shown later comparing with the value got by the free space measurement. The error propagations to the e value is as follows:

w l - 2 / 2 dj

*4f

( df

/o

2 -2 TV (V)

2 -2 (12)

where ad! and adp are the measuring error (~100kHz) of the resonance width with and without the sample. The largest contribution to the error comes from adj and adp. After the calculation of eq. (12), the oB- becomes 2.5 x 10"4. But the measured values scatter more than that even the samples cut from the same block. In the real measuring condition the difference may comes from where the sample is cut in the block, the impurity might be different even in the same block. The surface condition of the smoothness, the stain and the moisture give the difference also. In addition to them, there may be the causes in the apparatus itself. Multiple reflections of the radio wave in the input and output wave guide between the cavity and the RF generator, and the cavity and the detector might changes output amplitude with respect to the frequency span.

We were able to cleave four synthetic rock salt samples into single crystals of cross sections ranging from 1.0 mm x 1.1 mm to 1.0 x 1.6 mm with filling factors

98

dVIV of 3.2 x 10"* to 4.6 x 10"4. For filling factors this small e' and e" are not affected by perturbations to the resonance due to the presence of the sample. For synthetic rock salt, we found an average e = (5.8 ± 0.2) - j (3.2 ± 0.3) X 103 or tan<5 = (5.5 ± 0.5) x 10"4, which is consistent with the values given in Ref. [11]. We have succeeded in cleaving or milling samples of natural rock salt from the Asse salt mine in Germany and the Hallstadt salt mine in Austria, and limestone from the Kamaishijimestone mine. The samples were formed into small sticks of length 10.2 mm, equal to the height of the cavity resonator.

The results of the measurements of e' and e" are listed in Table 3. We find that for the real part of the permittivity e' the size of the sample is sufficiently thin and the values for the Asse and Hallstadt samples at 5.8 ± 0.2 are consistent with the values for the synthetic rock salt. These values are also consistent with those obtained from the free space measurement, tabulated in Table 2. For the imaginary part of the permittivity e", it seems that the samples were not sufficiently thin, in contrast to the synthetic rock salt samples. The results show that the thinner samples have smaller s" values. Therefore we are only able to estimate an upper limit of e", even for the smallest samples. We have calculated the attenuation coefficient a by eq. (1). If there is no frequency dependence in tand, then since there is no orientation polarization in rock salt and limestone, the attenuation lengths La, calculated at 9.4 GHz, 7.7 m for the synthetic salt, can be extrapolated to other frequencies e.g. at 94 MHz the attenuation length in a pure rock salt crystal is 770 m. Our results show that at lower frequencies, the attenuation length may be sufficiently long for use in a salt neutrino detector. Although there was a large uncertainty in e" we were able to obtain lower limits of the attenuation length.

Table 3. Comparison among single crystal, Asse rock salt, Hallstadt rock salt and limestone in e ' s", tan&=s"/e'a at 9.4GHz and La at 9.4GHz.

Sample e' e"xl0"3 tanSxlO" a at L^Va 4 (m1) (m)

Single crystal (NaCl) 5.8 ± 6.2 3.2 ±0.3 5.5 ±6.5 6.13 ± .01 7.7±6.7 Asse 5.8 ±0.2 <78 <13 <0.31 >3.3

Hallstadt 5.8 ±0.2 <440 <76 <1.79 >0.56

Limestone (CaC03) 8.3 ±0.2 <160 <19 <0.54 >1.9

If the approximation of the perturbation holds, i.e. the resonance is not disturbed by the presence of the sample, -(J-foYfo and 1/Q, are proportional to dV/V. The linearity of -(f-foYfo holds up to a filling factor of about 1.7 x 103, for the Asse samples, 2.5 x 10"3 for the Hallstadt samples and l . lx 10"3 for the limestone

99

samples, on the other hand, the linearity of 1/Q, does not hold for any of the samples. Table 4 summarizes the findings of the analysis of the linearity.

Table 4. Comparison among single crystal, Asse rock salt, Hallstadt rock salt and limestone of sample cross section, dV/V, linearity of -(f-f0)/fo and 1/Q, = 1/Q-l/Qo in dV/V

Sample Cross section dV/V Linearity of (mm) -if-foYfo

rndV/V singie crystal 1.6x1.1 - i.bxi.6 i S x l O ^ i s x i c P -

Asse 1.7x1.8-3.0x3.1 S ^ x l O ^ J x l O " 3 1.7xl0"3

Hallstadt 2.1x1.8-3.0x3.0 1.0xl0'3-2.5xl0-3 2.5xl0"3

Limestone 1.0x1.0-3.0x2.9 2.7xl0"4-2.4xl0-3 l.lxlO"3

In order to improve the accuracy of the measurements of e" we need thinner samples. Alternatively we could use a super conducting cavity resonator, which would have larger values of Q0, e.g. 20000. In this case \IQ0 in eq. (10) becomes smaller and the uncertainty decreases.

5 Discussion

Minimum sized SND is envisaged as shown in Fig. 1. Radio wave sensors of 216 are arrayed regularly in 200m repetitions inside a 1km x 1km x 1km rock salt. The attenuation length becomes longer when we use the lower frequency to be detected. However the detection threshold energy of UHE neutrinos increases at the lower frequency. The optimization should be done to select the frequency. If the attenuation length is 400 m, the sensor separation distance less than 200 m is needed to detect the Cherenkov ring shape as well as the energy. Six radio sensors are hung on a string. The string is lowered in a well with the depth over 1km. At the total 36 wells should be bored.

There is a possibility to detect radio wave from the neutrino reaction as low as ITeV if we could set the antennas nearby the shower [14]. According to the proposal for neutrino physics at LHC constructing at CERN [20], it could supply neutrinos over ITeV from the collision points. The neutrinos are coming from charm and anti-charm quark decays since there is no decay region for 7t's and K's

100

decay. Then the beam would contain an equal number of ve's and v / s unlike conventional beams. The beam would also contain an appreciable number of v*'s resulting from Ds decay. If we could use a space near the LHC tunnel e.g. the transfer line where it is located 500 m from the collision point 1, ATLAS experiment, it is enough to install smaller media e.g. rock salt or limestone. It will become a good calibration test for the neutrino detection as well as the highest energy neutrino detection generated artificially. Due to their estimation, with the luminosity of 4xl034 cirfV1, running time of 107 s and the angular coverage of ±2.5mrad, a 15 m long detector with a density of 7.86 gem"3 and a total weight of 24 tons, 58,000 interactions are expected over 500 GeV.

Limestone is more common in the world than rock salt. Especially Japan has abundant in high purity limestone. Unfortunately, limestone tends to crack to be penetrated by water. Weak acid water melts limestone and makes a limestone cave. Therefore we could not expect a large amount of limestone without including water. However we could expect a relatively small amount of the limestone without water. Such a smaller size could be used to detect stronger neutrino beam emitted by LHC collision points. Mt. Jura is known to be made up of limestone, which is situated within ~10km west from LHC collision point. The LHC ring located 100m underground. The north and the south collision points are used for CMS (point 5) and ATLAS (point 1) experiments, respectively. They are located ~400m above sea level. The neutrinos generated at the collision points are emitted toward east and west directions. The LHC ring is inclined 1.4% upward in the direction of west to fit the surface inclination toward Mt. Jura. The nearest foot of the mountain is -10 km west of the Atlas collision point. The surface is above -600 m of the sea level. At the point the neutrino beam passes 540m from the sea level. Consequently, we need boreholes as deep as ~60m to reach the neutrino beam. The beam size is as small as 10m at 10km since the beam dispersion is 1 mrad at the high intensity portion. Whereas the Cherenkov angle is as large as 70° then tan70° becomes 2.7. It means if we aim to use the target thickness of 100m and 10m, the antenna array with a radius of ~270m and ~27m is needed, respectively. Beyond the peak (le Reculet, 1717m) of the Mt. Jura relatively higher land ~1000m above the sea level continues west, then the neutrino beam does not come up soon. The neutrino beam passes 700m from sea level at 20km from the collision point; up to there the lowest location is 800m from the sea level. If it comes out at the steep cliff, we could deploy the antennas easily without boreholes.

6 Conclusion

Rock salt has been studied as a radio wave transmission medium in a UHE cosmic neutrino detector. The radio waves to be detected are those generated by the

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Askar'yan effect (coherent Chrenkov radiation in solid matter from negative excess charges of an electromagnetic shower) for the interaction of UHE neutrinos in the rock salt. Samples from two rock salt mines were investigated to determine whether they are viable sites for an SND. They were the Asse mine in Germany and the Hallstadt mine in Austria. Furthermore, we took limestone samples from a mine in Kamaishi, Japan. We found that they were all potential sites for an SND.

The attenuation lengths of the samples were determined by the free space and cavity perturbation methods. The attenuation lengths of radio wave were found to be 7.7 m, >3.3 m, >0.56 m and >1.9 m at 9.4 GHz for synthetic single crystals, the Asse rock salt, the Hallstadt rock salt and the Kamaishi limestone. A more definite estimate of the attenuation length will require thinner samples. For rock salt the radiation produced by the Askar'yan effect is strongest at about 6 GHz, at the Cherenkov angle of 66°, estimated by the density and the radiation length. At a frequency of 94 MHz the attenuation length is long enough to make a neutrino detector, although the radiation power is compromised and the threshold energy for the detection of neutrinos becomes higher. Recently, P. Gorham et al. [12] have measured attenuation length at the Waste Isolation Pilot Plant (WIPP), located in an evaporite salt bed in Carlsbad, New Mexico and found short attenuation lengths of 3-7m for frequencies of 150-300 MHz. However, measurements at United Salt's Hockley mine, located in a salt dome near Houston, Texas yielded attenuation lengths in excess of 250 m at similar frequencies. Their results are consistent with our result for the Asse sample.

The preliminary results of radio wave attenuation length in rock salt show that it is a possible medium for a UHE neutrino detector if we select a rock salt mine with a high transparency. However, we need to make perturbed cavity resonator measurements at lower frequencies and with more samples in order to make a concrete conclusion. Before the SND site is decided it is important to measure the attenuation length in situ, as there may be defects and impurities in the salt at the site, as well as intrusions by minerals other than rock salt. For such a study a ground penetrating radar would be useful, a well-explored technique.

The frequency to be detected should be decided upon taking into account the detection energy threshold of the UHE neutrinos and the attenuation length at that frequency. In addition, in order to calibrate the energy of the initial electromagnetic shower produced in the interaction of the neutrinos with the rock salt and the distribution of radiation power, which depends on the degree of coherency, an important study is that of the basic processes of coherent Cherenkov radiation due to a pulsed electron and a neutrino beam in an accelerator [17,20]. The angle resolution of the neutrino incidence and the position resolution of the interaction point are the important issues to be studied.

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7 Acknowledgements

This work was supported partly by Funds for Special Research Project at Tokyo Metropolitan University, Fiscal Year 1999 and Agilent Technologies University Relations Philanthropy Grants Program Fiscal Year 2001. We should appreciate Ms. M. Ikeda, Dr. O.Yasuda, Profs. K. Minakata, T. Kikuchi (TMU) and M. Kobayashi (KEK) to be involved or to support this project. We express our gratitude to M.E. Ryouichi Ueno who discussed with and advised us about the microwave techniques. He was indispensable to carry out this study. The research could not be possible without the assistances from and discussions with many persons because parts of the field researched were far from our specialized field.

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20. Camilleri L. "Neutrino physics at LHC", Large Hadron Collider Workshop, held at Aachen, 4-9 October 1990, Proceedings Vol.III, Editors: G. Jalskog and D. Rein, CERN 90-10 ECFA 90-133 Volume III 3 December 1990.

EXPECTED PERFORMANCE OF A NEUTRINO TELESCOPE FOR SEEING A G N / G C BEHIND A M O U N T A I N

GEORGE W.S. HOU AND M.A. HUANG Department of Physics, National Taiwan University, Taipei, Taiwan, R.O.C.

E-mail: wshou@phys.ntu.edu.tw, huangmh@phys.ntu.edu.tw

We study the expected performance of building a neutrino telescope, which targets at energy greater than 1014 eV utilizing a mountain to interact with neutrinos. The telescope's efficiency in converting neutrinos into leptons is first examined. Then using a potential site on the Big Island of Hawaii, we estimate the acceptance of the proposed detector. The neutrino flux limit at event rate 0.3/year/half decade of energy is estimated to be comparable to that of AMANDA neutrino flux limit at above 1016 eV.

1 Neutrino Astronomy

Neutrino astronomy is still in its infancy. Although neutrinos are abundantly produced in stars, as they live and when they die, one suffers from an ex­tremely low cross section for detection on Earth. Still, it is rather impressive that we already have "neutrino images" of the Sun, as well as neutrino blips of the cataclysmic SN1987A event. At the start of a new century/millennium, we yearn to reach beyond the stars and observe cosmological neutrino sources. Large "km3" ice/water or air shower neutrino "telescopes" are being built, and "the sky is the limit".

Neutrinos could play an important role in connecting several branches of particle astrophysics. The origin of ultra-high energy cosmic rays (UHECR) is still a great puzzle l. Bottom-up theories propose that they originate from energetic processes such as Active Galactic Nuclei (AGN) or Gamma Ray Bursts (GRB). The energetic hadron component could interact with accreting materials near the central black hole and produce neutrinos through the decay of charged pions. On the other hand, top-down theories suggest that UHECR are decay products of topological defects or heavy relic particles. According to these theories, there are more neutrinos than gamma rays and protons 2 . Measurement of the neutrino flux at and above the "knee" region (i.e. ^ 1015

eV) provides a good discriminator to distinguish between the two scenarios. Cosmic gamma rays are attenuated by the infrared, microwave and ra­

dio background photons 3. The recent observation of TeV gamma rays from extragalactic sources such as Mkn421 4 and Mkn501 5 , however, has aroused some concern. In order to reach the Earth from extragalactic distances, these 7 sources must have either a much harder spectrum or more powerful mech-

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anisms, e.g. electromagnetic (EM) processes such as inverse Compton scat­tering, or hadronic processes such asp + X-¥n° + ...-t'y + ... The for­mer produces few neutrinos, while the latter produces comparable amounts of both neutrinos and photons. Neutrinos therefore provide direct probes of the production mechanism of TeV 7 rays from extragalactic sources such as AGN/GRB.

Recent results on atmospheric neutrinos add an interesting twist to cos-mological neutrino detection. Super-Kamiokande (SK) and Sudbury Neutrino Observatory (SNO) data strongly suggest that muon neutrinos oscillate into tau neutrinos. Below 1012 eV, the tau decay length is less than 5 mm, and SK and SNO have difficulty distinguishing between showers initiated by electrons and those by taus. Above 1015 eV, the tau decay length becomes 50 m or more, distinctive enough for identifying the taus. Since cosmic neutrinos are produced via 7r+ decay predominately, one does not expect much directly pro­duced cosmic vT flux 6. Detecting a tau decendent on Earth would not only probe AGN/GRB mechanisms, but would also constitute a tau-appearance experiment.

2 A Genuine Neutrino Telescope

Because of the low interaction cross-section, all neutrino experiments resort to a huge target volume. The target volume is usually surrounded by the detec­tion devices in order to maximize detection efficiency. Thus, the target volume is approximately equal to the detection volume. In other words, the cost of building a detector cost varies in propotional to the target/detection volume. Furthermore, to shield against cosmic rays or even high energy atmospheric neutrinos, these detectors often have to be deep underground. For instance, the km3 size ICECUBE project 7 at the South Pole has a price tag of $100M, aims to look for upward going events, and takes years to build. Variants such as sea/ocean or air watch experiments are similarly large and costly. These "telescopes" tend to bear litte resemblance to their EM counterparts.

Some alternative approaches have been proposed, such as using the Earth 8 or a mountain 9,1° as the target into convert neutrinos to leptons, which will then initiate air showers in the atmosphere. Observing the air showers from a region obscured by a mountain or the Earth can eliminate the contamination of cosmic ray showers. The main difference between this approach and the conventional experiments is that the target volume and the detection volume are different. Moreover, materials in the target volume (mountain, Earth) and the detection volume (atmosphere) are readily avail­able at almost no cost, thus the overall cost (and perhaps schedule) of the

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E * V .

03 3 - _:_t

1 t t i i t i 1 i » i . i i M . i i l i . . i . . . i . i f i . j . i i I I •10 0 10 20 30 40 50 60

Horizontal distance (km)

Figure 1. Illustration of a neutrino convert to tau inside a mountain.

experiment could be reduced dramatically. This makes the approach worthy of further exploration.

Using an approach similar to that of Vannucci 10, a Cerenkov telescope sits on one side of a valley opposite a mountain. Energetic cosmic neutrinos, while passing through the atmosphere with ease, interact inside the mountain and produce leptons. Electrons will shower quickly and have little chance of escaping from the mountain. For muons, the decay/interaction lengths are too large to initiate showers inside the valley. The taus have suitable decay length to escape from the mountain and initiate showers inside the valley upon decay. This process is illustrated in Fig. 1. With this design, the telescope is not only a detector for astrophysical and cosmological neutrinos, but also serves as a tau-appearance experiment.

It is interesting to note that this telescope resembles closely usual EM telescopes and a typical particle experiment. The field piece is the mountain, which functions as both a target and a shield, and the subsequent valley is the shower volume. The actual Cerenkov telescope functions as an "eye piece" that focuses the Cerenkov light from a shower emerging from the mountain onto a sensor plane. The sensor could be a MAPMT array, where fast elec­tronics matches the 10 ns Cerenkov pulse and helps discriminate against other background sources. Using two telescopes in coincidence would be produce better results. The only drawback, in comparison to a regular EM telescope, is that we cannot move the mountain and would have to rely on the Earth's rotation to move the telescope.

Besides cost, expected to be far less than ICECUBE or Auger, the most critical issue is the expected count rate. In the following, we choose a potential site (Hawaii Big Island), examine the neutrino conversion efficiency, and then derive the flux limit and sky coverage of the proposed detector.

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3 Potential Site

The criteria for choosing a potential site are as follows:

• Reduced artificial lights, dry air and cloudless sky, much like usual optical telescopes.

• Target mountain broad enough for the sake of acceptance.

• Valley wide enough for taus to decay and air showers to develop.

In the energy range of 1014 - 1018 eV, the depth of shower maximum ~ 500 - 800 gm/cm2. At altitude around 2 km, this depth corresponds to a horizontal distance of 4.5 to 7.8 km. Therefore, the width of the valley must be larger than 5 km, but less than the attenuation length of light ~ 50 km.

• Good exposure to the Galaxy Center (GC).

The nearest massive black hole — what may be behind astrophysical neutrinos — is our Galaxy Center.

Hawaii Big Island, with its perfect weather conditions, has been a favor­able site for astronomical (EM) telescopes. The Big Island also has a rather unique configuration. Besides the more sought after Mauna Kea, the other 4 km high mountain, Mauna Loa, has a breadth of approximately 90 km. Across from Mauna Loa to the northwest, Mount Hualalai is ~ 20 km away and 2.3 km in altitude. This makes Mauna Loa a good candidate for the tar­get mountain with the detector installed on top of Hualalai. In the following study we assume this configuration.

4 Neutrino conversion efficiency

In this study, the mountain is simplified as a block of thickness L. Neutrinos enter the mountain, pass through distance x, interact in a; to x + dx, produce taus, which then survive through the rest of the mountain without decay.

The probability for neutrinos to survive the atmosphere (Pi) is taken as 1, which is very close to the actual case. The probability for neutrinos to survive distance x inside the mountain is Pi{X) = exp(-x/Av) , where A„ = 1/(NACTP), a is the charged current interaction cross-section u , NA is the Avogadro number, and p is the mean density of the mountain. The chance of neutrino interaction in x to a; + dx is dx/\v. The energy of tau is approximated as ET = (1 — y)E„, where y is the fraction of energy carried by

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the recoiling (shattered) nuclei or electron, which is in the range of 0.2 to 0.5 with mean ~ 0.25, we therefore use ET = 0.75EV.

The probability for taus to survive through the rest of the mountain of distance L - x is Ps{X) = e x p ( - ( i - x ) / \ T ) , where Ar is the decay length of tau and equals (ET/PeV) 48.91 m.

The neutrino conversion efficiency is

e= [ e _ x / A " e _ ( L ~ : r ) / A T — = ^T (Q-L/X" -.Q-L/*A n\

Jo Xv Xv — XT \ J where the integration was done by neglecting the energy loss of tau. The maximum efficiency occurs at ^|L=Lmoa, = 0, i.e.

_ ln(A„/Ar) •L^max — i i \^)

A T AU

The conversion efficiencies at five energies are shown in Fig. 2. The efficiency plateaus above L > Lmax/2. This maximal efficiency scales roughly as a power law in energy.

The mean distance traveled by taus inside the mountain is

. /^(L-arje-^-'e-^-*)/^^ T J O v I A„ ' • ! / . , „ . ,

Lr - „ r , , _ , r _ w . . - T T - A T . \6)

_ / ^ ( L - a r J e - ^ - ' e - ^ - * ) / ^ ^ _ A,, jL Q-X/\„Q-(L-X)/XT dx. ~ \ v - \T '

Because Xv >• Ar, LT ^ AT, the mean production point of tau is approximately one decay length inside the mountain. As long as the thickness of mountain is larger than Ar, LT remains unchanged.

5 Acceptance of flux limit

Fig. 3 shows the panoramic view from the top of Mt. Hualalai towards Mauna Loa. The field of view of the detector is the shaded mountain region inside the box. The azimuth angle extends from south to east. The minimum zenith angle of 86.9° is set by the line from the summit of Hualalai to that of Mauna Loa. The maximum zenith angle of 91.5° is set by the line from the summit of Hualalai to the horizon at the base of Mauna Loa. A cross-section of the Big Island along the line from Hualalai to Mauna Loa is shown in Fig. 4.

The acceptance is defined by the effective area multiplied by the effective solid angle. Owing to lateral distribution of air shower, the Cerenkov light

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^ 10 CD l u

c O 10 w 1 _

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-J

-4

-5

-6

-/

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itf'eV

I I i i i mill ' '

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Thickness of mountain (km)

Figure 2. Neutrino conversion efficiency vs mountain thickness (in kilometers) for five en­ergies. The maximum efficiencies are marked with arrows.

CD o> c e o CO > <D LU

6 4 2 0

-2 (i

-6 -8 10

Sea

Mauna Kea Mauna Loa

Hawai Bg island

-180 -150 -120 -90 -60 -30 0

S W N 30 60 90 120 150 180

E S

Figure 3. The panoramic view from the top of Hualalai. The dash line is the horizon and the shaded region is the field of view obstructed by the terrain of Hawaii Big Island. The region between the horizon and the terrain is the sea to the west of the Big Island.

cone of shower is approximately 5° to 6° 12. The effective solid angle can be determined by the sensitivity of PMT, the distance from the detector to the

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i I t i > i t j i i t t t i t i T" i i r -20 0 20 40 60

Horizontal distance (km)

Figure 4. A cross-section of Big Island along the line from Hualalai to Mauna Loa.

shower maximum, and the Cerenkov light yield of air shower. The number of Cerenkov photons is proportional to the number of secondary particles in the air showers, which is proportional to the tau energy. Also, the lower energy taus decay closer to the mountain, thus farther away from the detector and the Cerenkov light suffers more atmospheric scattering. These two effects reduce the effective solid angle at lower energy. The extend of the effect can be obtained by detailed simulation. To simplify the calculation, we use a constant value of 5 °, which yields the effective solid angle

Q.= sin 9d0d<f> = 2TT (1 - cos 0C) = 0.024 sr Jo

The effective area is the cross-section area where tau decays. The mean distance of decay after taus escape from the mountain is still AT. So the effective area is

aeff(E)= f (r(w) - XT(E))2duj JFOV

where u> is the solid angle of each pixel, FOV is the field of view, and r is the distance from the detector to the mountain surface viewed by that pixel. The total acceptance A{E) is aefi(E) x O, as shown in Fig. 5. Below 1017 eV, the acceptance is approximately 1 km2 sr, similar to ICECUBE. The sharp decrease in acceptance at E > 2 x 1017 eV is due to the increase in decay length of tau beyond 10 km. The valley ~ 20 km is not wide enough to contain these high energy taus.

The target volume is defined as the volume inside the mountain where taus are produced,

V= f f ' r{ufdrdu, JFOV Jiii

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where Ri is the distance from the detector to the mountain surface, and

Rf = (Ri + W_i

\ Ri + LT i if W <L^ if W > LT

where W is the width of mountain in the field of view w. The target volume is then transferred to the water-equivalent target volume by multiplying the den­sity of rock, 2.65g/cm3. LT ~ AT increases almost linearly with energy, so does the target volume. For conventional neutrino telescopes, such as SK or ICE­CUBE, the target volume is identical to the detection volume, therefore the acceptance is propotional to the detection volume. For the Earth-skimming or mountain-valley type neutrino telescopes, the target volume and the de­tection volume are different. Thus, the acceptance and the target volume do not have any direct relation.

1 10 10 10

Neutrino energy (PeV) 1 10 10 10

Neutrino energy (PeV)

Figure 5. Acceptance and water equivalent target volume of the potential site in Hualalai.

This study does not consider the effect of energy loss of taus inside the mountain. The effect becomes more serious for energies > 1017 eV, where tau energy loss leads to a decrease in decay length of taus, thus increasing accep­tance at high energy. When the energy loss of tau is taken into consideration, Eq. (1) cannot be integrated in closed form. At the present stage, we have ignored the energy loss effect for simplicity and treat the results as lower limit of acceptance and upper limit of sensitivity.

Because of the lower light yield and more scattering at lower energy, the acceptance should be lower at lower energy. In view of the two factors

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above, the best energy range for this type of detector is approximately in 1015 < E < 1018 eV.

The flux limit is estimated by

{ ' dTdE A(E)s{E)

where N is the number of events, T is the exposure time, dN(E)/dT is the event rate, dE is the bin width of energy which is approximately equal to the energy resolution of detector. The conversion efficiency e{E) is calculated by similar process as Eq. (1). The exact zenith angle, the atmospheric pres­sure, the mountain width, and the curvature of the Earth are all taken into consideration.

In the conversion from vT to r , some fraction of energy (yE) are brough out by interacting nuclei. Because this interaction take place inside the mountain, this energy can not be measured. ay ~ 0.18 is the largest source of systematic error in energy. With some uncertainties from detection and reconstruction, a simplified value of half a decade 10~ 0 5 = 0.31 is assumed as the energy resolution dE.

The detector sensitivity is defined as the flux when the event rate is 0.3 event in one year. Based on the acceptance of Hualalai site, the sensitivity of the proposed detector and the recent AMANDA B-10 neutrino limit are shown in Fig. 6. Note that the AMANDA B-10 limit is the integral flux limit from null observation of neutrino in the energy range of 1012 to 1015 eV. The null observation in one year of operation of the proposed detector could set an upper limit similar to that of AMANDA B-10 13, but at 1015 < E < 1018

eV.

6 Sky coverage

The detector is operated at moonless and cloudless nights. We simulate the operation from December 2003 to December 2006. The detector operates when the total time of moonless night is longer than one hour. The total exposure time in three years is 5200 hours, corresponding to a duty cycle of ~ 20%. In reality, some cloudy nights have to be excluded.

According to the field of view specified above, the sky covered by the detector can be calculated. The total exposure hours in 1 ° x 1 ° of galactic coordinates are shown in Fig. 7. The galactic center is visible for approxi­mately 70 hours.

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E I O 5

o >

CVJ 1 Q 4 LU

LU

* 10 3

% Mauna Loa + Mauna

-4— Mauna Lea

T Mauna Kea

»»i AMANDA B-10 limrt

Kea

10 10 10 10 Ev(PeV)

Figure 6. The sensitivity of the proposed detector if the event rate is 0.3 event/year/half decade of energy.

7 Discussion

Although the acceptance reaches 1 km2 sr, the optical detection suffers 10% operation time in each calandar year. There are several ways to improve the acceptance.

• Extending the zenith angle coverage to below the horizon can include the Earth-skimming events, which are not studied in this report. This extension could double the acceptance at E < 1016 eV. At higher energy, acceptance does not increase much because of lack of space for taus to

115

„ 60

T3

o CC-30 CO

CD -60 -

f

M M Exposure time (hr) i i r i i i | i i i mm 120

I I I I , I I , I I M M -180 -150 -120 -90 -60 -30 0 30 60

Galactic Longitude

WiMMm®

120 150 180

Figure 7. The exposure time in galactic coordinates for the three-year operation from De­cember 2003 to December 2006.

decay.

• If the detector could also detect the fluorescent light from air showers, the current field of view could be triggered by showers initiated by the taus escaping from Mauna Kea and by Earth-skimming from south-west of Mauna Loa. The large increase in solid angle could increase acceptance by a factor of 3 to 10. This is most effective at energy higher than 1017

eV.

The above improvements can increase the acceptance to 20 km2 sr. The azimuth angle can also be extended to the west side of Hualalai so that the sea-skimming events can be used as well. However, the reflection from waves may create more noise.

The detector should have some coverage of the sky and record cosmic ray events. This can help monitor detector performance, and the cosmic ray flux can be used to cross-calibrate the energy scale with other cosmic ray experiments.

8 Summary

Taking Hawaii Big Island as a potential neutrino telescope site, we calculate the neutrino conversion efficiency. The detector acceptance is approximately

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1.4 km2 sr. The sensitivity of the proposed detector is close to the AMANDA B-10 limit. The exposure time of the galactic center, where the nearest black hole is located, is approximately 70 hours in three years of operation.

This study shows that a compact neutrino telescope utilizing the moun­tain for neutrino conversion is capable of achieving a sensitivity similar to that of a big detector. In addition, the cost and construction time is greatly reduced. This type of detector at 1015 < E < 1018 eV could complement con­ventional neutrino telescopes such as AMANDA aiming at energies E < 1016

eV, and cosmic ray experiments such as Auger aiming at E ;> 1018 eV.

Acknowledgments

The authors would like to thank the HiRes group for providing the source code for moonless nights in Julian time.

References

1. For short review and a comprehensive list of references in cosmic rays and neutrinos: T.J. Weiler, talks at Neutrino-2000, Sudbury, Canada, June 2000, hep-ph/0103023.

2. G. Sigl, lectures given at summer schools in Kopenhagen and Parma (2001), hep-ph/0109202.

3. F.W. Stecker, Astropart. Phys. 11, 83-91, (1999). 4. M. Punch et al., Nature, 358, 477, (1992). 5. J. Quinn et al., Astrophy. J., 456, L83, (1996). 6. Kingman Cheung, this proceedings. 7. F. Halzen, this proceedings. 8. G. Domokos and S. Kovesi-Domokos, proceedings of the workshop: Ob-

serving Giant Cosmic Ray Air Showers for > 1020 eV Particles from Space U. of Maryland, Nov. 1997, hep-ph/9801362.

9. D. Fargion, et al., Proc. 26th ICRC, HE 6.1.09, pp. 396-398 (1999), astro-ph/9906450.

10. F. Vannucci, NATO Advanced Research Workshop, Oujda, Marocco, (2000), hep-ph/0106061.

11. R. Gandhi et al., Phys. Rev. D 58, 093009, (1999). 12. P. Sokolsky, Introduction to ultra-high energy cosmic ray physics, Red­

wood City, CA: Addison-Wesley Pub. Co., pp. 50-64, (1989). 13. G.C. Hill, AMANDA Collaboration, Proceedings of the XXXVIth Recon-

tres de Moriond, Electroweak Interactions and Unified Theories, March 2001, astro-ph/0106064.

GALACTIC HIGH-ENERGY COSMIC-RAY TAU N E U T R I N O FLUX

H. ATHAR1 '2, KINGMAN CHEUNG1, GUEY-LIN LIN2, AND JIE-JUN TSENG2

Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan 2 Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan

In this talk, we summarize a recent calculation on the high-energy tau neutrino flux (103 > E > 1011 GeV) originating from the interactions of high-energy cosmic-rays with the matter present in our galaxy. The main source of this flux is the production and decay of Ds for E < 109 GeV. For 109 < E/GeV < 1 0 u , the tau neutrino flux from other heavier quark decays as well as from direct production is comparable to that from Ds.

1 Introduction

This talk summarize the study performed in Ref. 1. Searching for high-energy neutrinos will yield useful informations about the highest energy phenomenon occurring in the universe 2. In particular, the pp interactions taking place in cosmos may play a decisive role in identifying the astrophysical sources for high-energy neutrinos. The pp interactions produce unstable hadrons that decay into neutrinos. Given the current upper bounds on cosmic neutrino flux from high-energy neutrino telescopes, the role of pp interactions in rela­tively nearby and better known astrophysical sites like our galaxy (the Milky Way) becomes relevant. First of all, the pp interactions in our galaxy forms an irreducible background for extra-galactic high-energy neutrinos. Secondly, such interactions could be the only source of high-energy astrophysical neutri­nos other than the several proposed cosmologically distant sources like AGNs, GRBs, as well as groups and clusters of galaxies, should the search for neu­trinos originating from these sources turns out to be negative. Therefore, it is important to compute the neutrino flux expected from our galaxy.

The origin of high-energy cosmic-rays remains unclear so far. Neverthe­less, they interact with the matter present inside our galaxy provided they are dominantly the extra-galactic protons. Presently, there exists no estimate for high-energy cosmic-ray tau neutrino flux originating from our galaxy in pp interactions. It is interesting to know to what extent the tau neutrino flux can be produced in the context of the standard model physics, particularly, in light of recent growing interests to identify the neutrino flavor in the high-energy cosmic-ray neutrino flux. In Ref. *, we calculated the tau neutrino flux from our galaxy, taking into account all major tau neutrino production

117

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channels up to E < 1011 GeV. The calculation of the tau neutrino flux has less theoretical uncertainties than that of computing the electron and muon neutrino flux, because the perturbative approach can be applied reliably to calculate the production of Ds mesons in pp interactions, which is the main source of high-energy tau neutrinos.

One may argue that the interaction of the high-energy cosmic-rays with the ubiquitous cosmic microwave background (CMB) photons could also lead to high-energy astrophysical tau neutrinos. However, the center-of-mass en­ergy (>/*) needed to produce a r lepton and a vT is at least ~ 1.8 GeV. In a collision between a proton of an energy Ep and a CMB photon of an energy £7 C M B , the invariant mass squared of the system is m^ < s < iEPE-rcMB +mp. Since the peak of the CMB photon spectrum with a temperature ~ 2.7 K is at about 2.3 • 10 - 4 eV, it requires a very energetic proton with Ep > 2.5 • 1012

GeV in order to produce a TVT pair. Thus, the contribution of the direct tau neutrino flux from the interaction between the cosmic proton and the CMB photon is negligible.

2 High-energy cosmic-ray tau neutrino flux

2.1 Formula for tau neutrino flux and the model of galaxy

We use the following formula for computing the tau neutrino flux

g ^ = j T dEp <t>p{Ep) f(Ep) dnpP

d^+X. (1)

with 4>p(Ep) given by 3

J 1.7 (£p/GeV)"2-7 for Ep < E0, <PP(UP) - 1 1 7 4 ( £ p / G e V ) - 3 for Ep > £Q) W

in units of c m - 2 s _ 1 s r _ 1 GeV - 1 and EQ — 5-106 GeV. We assume directional isotropy in <j>p(Ep) for the above energy range. The function f(Ep) is equal to R/\PP(EP), where \PP{EP) = ( c r ^ ' np ) - 1 is the pp interaction length and R is a typical distance in our galaxy. The target is taken to be proton with a constant number density of 1 c m - 3 and R to be ~ 10 kpc. The cr^ci is the total inelastic pp cross section. Since the high-energy protons only traverse a distance R much shorter than Xpp, the proton flux (t>p{Ep) is assumed to be constant over the distance R. Furthermore, we calculate the tau neutrino flux along the galactic plane only to obtain the maximal tau neutrino flux expected. The matter density decreases exponentially in the direction orthogonal to the galactic plane, therefore the amount of absolute neutrino flux decreases by

119

approximately two orders of magnitude for the energy range of our interest. The dn/dEVr is the differential cross section normalized by a™1', i.e.,

dupp^^+x _ 1 dapp-+„T+x ,„.. dEVr a%* dEVr ' l '

which gives the fraction of inelastic pp interactions that goes into vT's. We can simplify Eq. (1) to

dNv- = Rnp r dEp MEP) d a ^ + X • (4) poo

, / dEp JE„T

dE„T J EUT dEv.

The remaining task is to compute the differential cross section da/dE„T in pp interactions. We included all major production channels of tau neutrinos, namely, via the Ds meson, 6-hadron, it, W*, and Z*. We count both vT

and PT using the symbol for vT. We note that the heavy intermediate states such as the Ds meson, 6-hadrons as well as other heavier states decay (into vT) before they can interact with other particles. This is due to the rather small matter density of the medium and the large distance between the proton source and the earth.

2.2 Tau neutrino production

Via Ds mesons. The lightest meson that can decay into a T-VT pair is the Ds

meson. It was pointed out that the production and the decay of Ds meson is the major production channel for tau neutrinos in the AGN 4 . The Ds meson decays into a charged r lepton and a vT. The charged T lepton subsequently decays into the second vT plus other particles. For simplicity we model, using Monte Carlo techniques, the kinematics of the r lepton decay by assuming that the r-lepton decays into a vT and a particle X, which has a mass mx satisfying 0.1 GeV < mx < raT — 0.1 GeV. Consequently, the second vT is much more energetic than the first one because the Ds mass is only slightly larger than m r . We take the branching ratio B(DS -> r+vT) as ~ 0.074 5 . We employed two approaches to calculate the production of D8 mesons: (i) the perturbative QCD (PQCD) and (ii) the quark-gluon string model (QGSM). In the PQCD approach, we use the leading-order result for pp —t cc:

a{pp -+cc) = Yl / / da ;ida ;2 fi/P{x1)fj/p{x2) a (ij ->• cc) , (5) ij J J

where fi/p(x) are the parton distribution functions (we use the CTEQv5 6 ) , while the parton subprocesses are qq, gg —} cc. We use a K factor, K = 2, to account for the NLO corrections. The c or c then undergoes fragmentation

120

into the D8 meson, which we model by the Peterson fragmentation function 7

with e « 0.029 8. The probability fc^D, of a charm quark fragmenting into a Ds is 0.19 8 (we have added the /c->r>, and /<;_•£>•).

The QGSM approach is based on the string fragmentation. It contains a number of parameters determined by experiments 9 . The production cross section of the Ds meson is given by the sum of n-pomeron terms

where x = 2p||/v/s and x± = 2«/(m|,o +p2

±)/s. The functions a£p(s) and

<f>^' (s, x) are given in Appendix B of Ref. 1 . A comparison of these two approaches for Ds meson is shown in Fig. 1.

The vT spectra calculated by these two approaches agree well with each other for E„r < 106 GeV. Beyond this energy, the QGSM approach gives a relatively harder spectrum. Nevertheless, in the region where the two approaches differ, the tau neutrino flux is already small.

An important quantity in the neutrino flux calculation is the average fraction of the injected proton energy being transferred to the neutrino, i.e., the ratio y = EVT /EP. The average value of y is given either by the mean or by the value of y at which the distribution da/dEVT attains the peak. We found that both averages of y are very close to each other. The (y) for Ep

between 103 GeV and 1011 GeV is shown in Fig. 2 for the production channel pp -> cc ->• Da+X ->• vT+X, using the PQCD approach. The (y) ranges from 5 • 1 0 - 3 to 5 • l O - 7 for Ep from 103 to 1011 GeV. The higher the injected proton energy, the smaller the fraction of the incident Ep that goes into hadrons is.

Via b hadrons. The production of bb in pp interactions can be calculated reliably by the PQCD approach, similar to the calculation of cc. The 6 quark so produced will hadronize into a b hadron (either a meson or a baryon). We use the Peterson fragmentation function with e = 0.0047 to describe the hadronization process 5. The particle data group published a branching ratio of 0.026 for a b hadron decaying into TVT + X 5 . The subsequent decay of the T lepton is similar to that of the last subsection.

Via it production. The production of it in pp interactions is calculated by the PQCD approach, similar to the calculation of cc. The t quark so produced will decay promptly before any appreciable hadronization. The top quark decays 100% into a b and a W. The W boson then decays 1/9 of the time into rvT. The r lepton undergoes a subsequent decay into another vT.

Via W* and Z*. The subprocesses are qq' -¥ W* ->• r^v? and qq -> Z* -s-vTvT. The spin- and color-averaged amplitude squared for the subprocesses

121

10"

10"

10"

^oKT

#

10"

10"

-s: 1 1 V.

"N.

N

" "^•^ Injected proton

^ - f c H _ ,

pp->D8->v,+X (PQCD)

pp->Ds->v,+X (QGSM)

^spectrum

-**

L T I "~V

11 Log10(E/GeV)

Figure 1. A comparison between the PQCD and QGSM approach to the spectrum of the vT flux coming from Da mesons.

are given by

Y2\M(du->w*^T-pT)\ = gr2{g_m2J+T2wm2w-(u-m% 4 1

(7)

where #{ = T3 / - Q/ sin2 0W. We then calculated the tau neutrino flux resulting from the pp interactions

using Eq. (4) by taking into account the intermediate states and channels that we have discussed above. The results are shown in Fig. 3. A few observations can be drawn from the figure, (i) The production via Ds mesons dominates for EVr < 109 GeV, followed by k-hadrons, W*,Z*, and tl (ii) For EVr > 109

122

r2

«

S 88 ^

m » r

Eg -

» SB

103 105 107 10* 10" Injected proton energy Ep (GeV)

Figure 2. The average fraction (y) of the injected proton energy being transferred to i/T) calculated in pp —» cc + X -> Da + X -»• i T + X by the PQCD approach.

GeV all these production channels become comparable and a realistic estimate of vT flux has to include all these channels, (iii) The tau neutrino flux is about 10 - 12 orders of magnitude smaller than the injected proton flux.

3 P rospec t s for observations

A search for high-energy cosmic-ray tau neutrinos can be done by utilizing the characteristic tau range in deep inelastic (charged current (CC)) tau-neutrino-nucleon (i/-N) scattering, in addition to the associated showers. For E close to 6 • 106 GeV, the neutrino-electron resonant scattering channel is also available to search for high-energy cosmic-ray tau neutrinos 10. The main advantages of using the latter channel are that the neutrino flavor in the initial state is least affected by neutrino flavor oscillations and that this cross section is free from theoretical uncertainties u .

For downward going or near horizontal high-energy cosmic-ray tau neutri­nos, the deep inelastic i/-N scattering, occurring near or inside the detector, produces two (hadronic) showers 12. The first shower is due to a CC i/-N

I u

1 er­

ic-

10"'

10-

m~'

123

10"

10"

;»io-15

E o

1 10" 2 5

10"

~ PP->D,->v,+X • pp->b„->v,+X - pp->W*->v,+X

pp->Z*->v,\ pp->tf->v,+X

Log,0(E/GeV)

Figure 3. Tau neutrino flux calculated via various intermediate states and channels: via Da, &-hadron, W*, Z*, and it. The injected proton flux spectrum is also shown.

deep inelastic scattering whereas the second shower is due to the decay of the associated tau lepton produced in the first shower. It might be possible for the proposed large neutrino telescopes such as ICECUBE to constrain the two showers simultaneously typically for 106 < EVT/GeV < 107, depending on the achievable shower separation capabilities 13 (see, also, 2) . The two showers develop mainly in ice. Using the same shower separation criteria as given in 13, the proposed neutrino detector, the megaton detector 14, may constrain the two showers separated by > 10 m, typically for 5 • 105 < E„T/GeV < 106. The two showers may also be contained in a large surface area detector ar­ray like Pierre Auger, typically for 5 • 108 < EVr/GeV < 109 15. In contrast to previous situations, here the two showers develop mainly in air. Several different suggestions have recently been made to measure only one shower, which is due to the tau lepton decay, typically for 108 < E„T/GeV < 1010, while the first shower is considered to be mainly absorbed in the earth 16. The upward going high-energy cosmic-ray tau neutrinos, on the other hand, for EVT > 104 GeV may avoid earth shadowing to a certain extent because of the characteristic tau lepton range, unlike the upward going electron and

124

1CT

10"

(A

'E

.o

10"

10-

^

•

'

"

- l -

—

.....

%, ' 1 • 1 • 1 •

v. N

-s.

•>*

" >. "*" ^ Injected proton

• - Galactic v, flux (from v

•™ Galactic v, flux (from vM

— Total Intrinsic Galactic v

* ^

h...

osc.)

,flux

~- ^.spectrum

~"\

extrapolated) L"\..,

•

^ \

l--%_

11 Log10(E/GeV)

Figure 4. A comparison between the total intrinsic galactic tau neutrino flux along the galactic plane (this work, solid line), and that due to neutrino flavor oscillations from the same source (taken from Ingelman and Thunman 18: dot-dashed line).

muon neutrinos, and may appear as a small pile up of v\ (I = e,/i, andr) for EVr ~ 103 GeV 17 However, the empirical determination of incident tau neutrino energy seems rather challenging here.

The above studies indicate that for a rather large range of high-energy cosmic-ray tau neutrino energy, a prospective search may be carried out. The event rate in each experimental configuration is directly proportional to the incident tau neutrino flux and the effective area of the detector concerned. Presently, no direct empirical upper bounds (or observations) for high-energy cosmic-ray tau neutrinos exist.

The neutrino flavor oscillation length for v^ ->• uT is Z0sc ~ (E/Sm2) 19. For 103 < EvJGeV < 1011 and with 5m2 ~ l ( r 3 eV2, we obtain lO"8 < 'osc/pc < 1, so that losc <£ / where I ~ 5 kpc is the typical average distance the high-energy cosmic-ray muon neutrinos traverse after being produced in our galaxy. This implies that on average half of the muon neutrino flux will be converted into tau neutrino flux, reducing its absolute level by one half (assuming maximal flavor mixing between v^ and vT). This is clear from the

125

fact that the total intrinsic tau neutrino flux, as indicated in Fig. 4, is four to five orders of magnitude smaller in its absolute value than the muon neutrino flux of the same origin.

To have an idea of event rates one considers the downward going high-energy cosmic-ray tau neutrinos originating from the galactic plane due to neutrino flavor oscillations. Following Ref. 13, we note that these galactic tau neutrinos give a representative event rate of < 1 per year per steradian for two separable and contained showers with EVT ~ 106 GeV in a km3 volume ice neutrino telescope such as the proposed ICECUBE.

4 results and discussion

The calculation for the galactic tau neutrino flux indicates that the main background for high-energy cosmic-ray tau neutrino search from extra-galactic sources will be due to the muon neutrinos produced in the galactic plane after being oscillated into tau neutrinos for 103 < E ^ / G e V < 1011. Conversely, searching for extra-galactic tau neutrinos orthogonal to the galactic plane is more prospective. The non oscillated tau neutrino flux originating from pp interactions taking place in our galaxy turns out to be four to five orders of magnitude smaller than the muon neutrino flux for the entire neutrino energy range (103 < _EVT/GeV < 1011) considered. In the calculation, we have included the bb, tt as well as W* and Z* channels in addition to the more conventional Ds channel for tau neutrino production. The contributions from bb, tt, W* and Z* channels are comparable to Ds for EVT > 109 GeV.

There are a few sources of uncertainties in our PQCD calculations. The next-to-leading order correction is a source of uncertainty for which we merely use a K(=2) factor to account for it. The value of the K factor is available for energy up to about ~ 103 GeV, above which we still use the same K factor as an approximation. Another source of uncertainty comes from the fact that parton distribution functions are only measured up to Q2 ~ 3 • 104 GeV2 and x down to about 5 • 10 - 5 20. Therefore, the parton distribution functions above this Q2 value or below this x value depend on the extrapolation pro­cedure, which can introduce some uncertainties into the calculation. Other uncertainties include the choice for the parameters used in the calculations, such as mc, mi,, and rtit- Remaining uncertainties are of astrophysical origin, like the injected proton flux spectrum index, the distance R (the size of our galaxy), and the particle number density np, which we simply take as a con­stant. Overall, we estimate that our galactic tau neutrino flux is reliable up to a factor of a few.

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Acknowledgments

The work was supported in part by the NCTS under a grant from the NSC and in part by the NSC of Taiwan R.O.C. under the grant number NSC90-2112-M009-023.

References

1. H. Athar, K. Cheung, G.-L. Lin, and J.-J. Tseng, hep-ph/0112222. 2. F. Halzen, astro-ph/0111059, and references cited therein. 3. T. H. Burnett et al. [JACEE Collaboration], Astrophys. J. Lett. 349,

L25 (1990). 4. H. Athar, Nucl. Phys. Proc. Suppl. 76, 419 (1999). 5. D. E. Groom et al, Eur. Phys. J. C 15, 1 (2000). 6. H. L. Lai et al. [CTEQ Collaboration], Eur. Phys. J. C 12, 375 (2000). 7. C. Peterson, et al, Phys. Rev. D 27, 105 (1983). 8. R. Barate et al [ALEPH Collaboration], Eur. Phys. J. C 16, 597 (2000). 9. G. H. Arakelian and S. S. Eremian, Phys. Atom. Nucl. 62, 1724 (1999)

and references therein. 10. D. Fargion, arXiv:astro-ph/9704205. 11. H. Athar and G. L. Lin, arXiv:hep-ph/0108204. 12. J. G. Learned and S. Pakvasa, Astropart. Phys. 3, 267 (1995). 13. H. Athar, Astropart. Phys. 14, 217 (2000); J. Alvarez-Muniz, F. Halzen

and D. W. Hooper, Phys. Rev. D 62, 093015 (2000); H. Athar, G. Parente and E. Zas, Phys. Rev. D 62, 093010 (2000).

14. H. Chen et al, arXiv:hep-ph/0104266. 15. H. Athar, arXiv:hep-ph/0004083. 16. D. Fargion, astro-ph/0002453; X. Bertou et al, astro-ph/0104452;

J. L. Feng et al, hep-ph/0105067; A. Kusenko and T. Weiler, hep-ph/0106071.

17. F. Halzen and D. Saltzberg, Phys. Rev. Lett. 81,4305 (1998); F. Becattini and S. Bottai, Astropart. Phys. 15, 323 (2001); S. Iyer Dutta, M. H. Reno and I. Sarcevic, arXiv:hep-ph/0110245; J. F. Beacom, P. Crotty and E. W. Kolb, arXiv:astro-ph/0111482.

18. G. Ingelman and M. Thunman, arXiv:hep-ph/9604286. 19. See, for instance, H. Athar, M. Jezabek and O. Yasuda, Phys. Rev. D

62, 103007 (2000), and references cited therein. 20. S. Chekanov et al [ZEUS Collaboration], Eur. Phys. J. C 21 , 443 (2001).

O N N O N H A D R O N I C O R I G I N O F H I G H E N E R G Y N E U T R I N O S

H. A T H A R

Physics Division. National Center for Theoretical Sciences. Hsinchu 300, Taiwan

and Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan

E-mail: atharQphys.cts.nthu.edu.tw

G.-L. LIN

Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan

E-mail: glin@cc.nctu. edu. tw

Some of the non hadronic interactions, such as the 77 resonance formation in the 77 interactions and the muon pair production in the e^/ interactions, are identified as possible source interactions for generating high energy neutrinos in the cosmos.

1 Introduction

At present, a main motivation for high energy neutrino astronomy (Eu > 103

GeV) is that it may identify the role of hadronic interactions taking place in cosmos 1. The hadronic interactions mainly include the P7 and pp interac­tions. These interactions produce unstable hadrons that decay into neutrinos of all three flavors. There is a formation of A resonance in jry interactions, at center-of-mass energy, yfs ~ ra^, that mainly decay into electron and muon neutrinos 2. In an astrophysical site for these interactions, the protons are considered to be accelerated up to a certain maximum energy and then inter­act with the photons and other protons present in the vicinity of the source and those present in the interstellar medium. Our galaxy and the earth at­mosphere are two examples of such astrophysical sites .

Currently, the detectors taking data in the context of high energy neutri­nos are Antarctic Muon and Neutrino Detector Array (AMANDA) at south pole x and the lake Baikal array in Russia 4 . These detectors are primarily based on muon detection and are commonly referred to as high energy neu­trino telescopes. The other high energy neutrino telescope under construction is the Astronomy with a Neutrino Telescope and Abyss environmental RE-Search (ANTARES) project 5 . These high energy neutrino telescopes (envis­age to) measure the showers and the charged leptons produced mainly in the deep-inelastic neutrino-nucleon and resonant (anti electron) neutrino-electron scatterings occurring near or inside the high energy neutrino telescopes 6. The later interaction can be used to calibrate the incident neutrino energy

127

128

in future high energy neutrino telescopes. The Monopole, Astrophysics and Cosmic Ray Observatory (MACRO) in Gran Sasso laboratory, Italy has also recently reported its results for the high energy neutrino searches 7. Given the present upper bounds on the high energy neutrino flux from AMANDA (B 10) and Baikal detector, the role of semi and non hadronic interactions becomes relevant. We shall call the later interactions as purely electromagnetic ones. Examples of these include ep and 77 interactions respectively.

Upper bound from the AMANDA detector rule out some of the high energy neutrino flux models based on hadronic interactions only. However, several variants of these models can still possibly be compatible with the high energy neutrino non observations. These include, for instance, the direct pion production off the A resonance in fry interactions.

The absolute high energy neutrino flux originating from the non hadronic interactions, though expected to be small relative to that from hadronic inter­actions, can be a good scale for future large high energy neutrino telescopes such as IceCube. This will be a guaranteed level of the high energy neutrino flux should the conventional astrophysics explanation for observed high en­ergy photon emission from extra galactic astrophysical sources such as AGNs is correct 8 . Here, only electromagnetic interactions are taken into account for explaining the observations. Thus, the implicit assumption of proton ac­celeration can be avoided. The discussion that follows is also relevant in cases where the highest energy cosmic rays, considered to be mainly protons, may not originate from the GRBs which are the likely sources of high energy gamma rays 9 .

This contribution is organized as follows. In Section 2, we briefly discuss some essentials of purely electromagnetic interactions possibly taking place in astrophysical and cosmological sites. In Section 3, we summarize the main points.

2 Purely electromagnetic interactions

The non hadronic interactions are defined to have e^ and 7 in the initial state, rather than p and 7. Therefore, the possible interactions that may generate high energy neutrinos include

77 —> (Tfi , e7 —• 7 1/(1/;, eTe —> v\vi.

For comparison, note that for yfs ~ TTXA, the cross sections for these interac­tions are typically <§; /xb. For y/s > mx±, other channels such as er) —> e7r+7r~ and 77 —• TT+TV~ also become available for high energy neutrino generation.

129

The non hadronic interactions also include the magnetic field induced interac­tions such as 77 —> i/;P;, which will be briefly commented later in this Section.

A yet another possibility to generate high energy neutrinos in purely electroinagnetic interactions is through the formation and decay of 77 resonance into (charged) pions in 77 interactions (77 —»77 —> 7r+7r~7r°). Let us consider in some detail a simple implication of this purely electromagnetic interaction in the context of high energy photon propagation 10.11,12,13,14 a n ( j consequent high energy neutrino generation. The cross section for this interaction is given by

(1)

10"6 . The peak cross section is aTes(s = ml) < 3 mb. Let us remark that <rres(s = m2^) < 0.5 mb, however here T&/m& ~ 10~3. The branching ratio 15 for the light unflavored 77 meson to decay into 7r+, 7r~ and 7r° is ~ 23%.

A useful quantity in the context of high energy photon propagation is the average interaction length defined by

where T^

< r ( 7 7 - * 77-> 7T+7T 7T0, S) = „ / L r 2 , (s - m 2 ) 2 + r 2 m 2

~ 1.18 KeV and mv ~ 547 MeV, so that T^/m,, ~ :

1(E)-1 = f°° nh(e)de f ' (?—±) <r(s)6», Jm^/iE 7+1 \ ^ J

(2)

with s = 2eE(l — /x) and fx = cos 6. The lower limit of integration for e corresponds to a head on collision so that /J, = — 1. Note that for a high energy photon with energy E ~ GeV, the background photon energy is e ~ GeV to form the 77 resonance. For definiteness, let us consider the interaction between high energy photons and the cosmic microwave background (CMB) radiation which has the following number density

1 e2

7r2 exp(e/Tb(0)) - 1

Here Tb(0) ~ 2.74 K. The other ubiquitous photon backgrounds include the infrared and ultraviolet radiation, considered to exist in the present universe, particularly after galaxy formation epoch. For simplicity, we ignore their effects as well as the effects of a possible magnetic field present in the cosmos. The high energy photons are considered to originate from an astrophysical or cosmological site within the present horizon length cff - 1(0) ~ 3 • 103 Mpc (where 1 pc ~ 3 • 1018 cm), as we take H(0) ~ 65 km s _ 1 Mpc - 1 .

Substituting Eq. (1) and Eq. (3) into Eq. (2), the two integrations can be carried out easily under the narrow width assumption, such that for ^/s a ra,,,

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o Q_

r 10' -

1E10 1E11 1E12 1E13

E (GeV)

Figure 1. The average interaction length using Eq. (2) for high energy photons propagating in cosmic microwave background photon flux as a function of incident photon energy.

we obtain

1(E) 8nE2

5.2TnmvT In

exp(m2/4£T)

exp(m2/4ET - (4)

For 4ET ~ m*, i.e., E ~ 3 • 1011 GeV, we note that I ~ 104Mpc > cH'1^). A such single interaction give a total of 6 neutrinos. For comparison, we display in Fig. 1, the lv = ^(77 —> 77 —> 7r+7r_7r°) along with the more familiar relevant /, namely for 77 —• n+^~- From the figure, we note that the Eq. (4) is a quite good approximation to obtain I in resonance and that Iri ^ 'M+M~ f° r s a m e E. In general, this observation may also have some relevance for high energy photon propagation in a dense photon background with relatively narrow background photon flux spectrum such as those arising in some astrophysical sites in the context of high energy neutrino generation.

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In the limit AET < m^, we obtain

urn 8nE2 I m*\ l{E)^K2T^feXp{AEf)^

whereas, in the opposite limit, namely when AET > m2 , we obtain

u m 8TVE2 / / 4 E T \ \ - :

In the two limiting cases, 1(E) > CHQ1. Let us further remark that although <rres(s = m2) > CTres(s = m\), however 1™S > I™ because of rather narrow rj width.

In the presence of an external magnetic field, we note that the cross section for 77 —» vv is significantly enhanced 16 with respect to its value in the vacuum. However, such an enhancement is still insufficient for this process to be presently relevant for high energy neutrino generation. For comparison with 77 -> n+n~, it is found that 1 7 , for B = 1012 G, (7(77 -» vv) sa 10_49cm2

for s > 4 m2,. This cross section scales as B2 for B < Br RS 4 • 1013 G.

5.i Astrophysical sites

Presently, there exists no model to estimate the high energy neutrino flux in purely electromagnetic interactions taking place in sources of highest energy gamma rays such as the AGNs and the GRBs. To make an order of magnitude estimate, we assume that the above astrophysical sites can accelerate electrons to energies greater than the observed gamma ray energies. As these electrons undergo inverse Compton scattering, the up-scattered high energy photons are produced. The scatterings of high energy photons over the ambient photon fields present in the vicinity of the AGNs or GRBs may lead to the fi+ fi~ final state or three-pion final state through the 77 resonance.

Phenomenologically speaking, the resulting (relative) high energy neu­trino flux can be parameterized as 4>f(Ev) ~ P0£7(i?„), where the proba­bility function P depends on the ratio of high energy photon/electron flux associated with a specific astrophysical site to the corresponding high energy proton flux on the same site. The function P certainly also depends on the ratio of neutrino production cross sections between two mechanisms. Finally it also depends on the magnetic field strength on the site, which are relevant for the acceleration of charged particles. A diffuse non hadronic high en­ergy neutrino flux with a representative P ~ 10~4 — 10~5 can in principle be measurable by future large high energy neutrino telescopes such as IceCube.

(5)

(6)

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2.2 Cosmological sites

Topological defects formed in the early epochs may play some role in the latter epochs of the expanding universe. The cosmological and astrophysical aspects of topological defects are the density or metric perturbations that they may generate, particularly in the epoch of large scale structure formation in the expanding universe 18.

The associated particle physics aspect is the possible release of large amount of energy trapped inside these topological defects in the form of gauge bosons. These gauge bosons subsequently decay into known hadrons and lep-tons. Assuming that (some fraction of) the topological defects are formed in the early epochs of the expanding universe and thus contain a large amount of energy, it becomes possible to explain at least some features of the observed highest energy cosmic rays. For this scenario to work, the observed highest energy cosmic rays have to be dominantly the photons. In this (conventional) scenario, the high energy neutrino flux is generated from the decay of charged

19

pions . Here, we discuss a class of topological defects in which high energy neu­

trino flux generation was postulated to originate in the electromagnetic cas­cade rather than in charged-pion decays which result from the hadronizations of initial jets produced in the decays of GUT-scale heavy bosons 20. This class of sources for ultrahigh energy photons is assumed to be active before the galaxy formation epoch. This corresponds 21 to a red shift, z > 5. Thus, for 2 > 5, the effects of galactic magnetic field as well as the infrared and ultraviolet photon backgrounds can be neglected. Consequently, CMB photon flux is the only important photon background. A search for high energy neu­trinos can provide some useful information about the existence of this class of topological defects in the expanding universe. At high red shift, the 77 in­teractions between the energetic and background photons can produce muons (and charged pions) whose decay generate high energy neutrinos. Note that, at high red shift, Th(z) = (1 + z)Tb(0), whereas nh(z) = (1 + z)3nb(0). For E > Exh{z), where E^ = 10 n GeV/( l + z), the 77b —• H+f-t~ is most relevant for high energy neutrino generation. The A = A(77b —» M + A O obtainable using Eq. (2) is less than the horizon length, cH(z)~l for 5 < 2 < 10. With an invariant mass just above the threshold, the purely electromagnetic inter­action 77b —> /i+/i~ also has a shorter interaction length than the energy attenuation length in the electromagnetic cascade dictated by 77b —* e+e~.

Under the assumption that muons decay before interacting, the high en-

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ergy neutrino flux can be calculated as

4>V(Ev)= / dz6E4h(z,E)Hz,E)anV-;v+x. (7)

Here <j>7(z, E) parameterizes the high energy photon flux from the topological defect. Typically, it is normalized by assuming that the high energy photons produced by topological defects at the high red shift are dominantly respon­sible for the observed high energy photon flux and/or the observed highest energy cosmic rays. The function f(z,E) = ciJ(z)_ 1/A gives the number of 77 interactions within the horizon length. The dn/dE ~ a~1da/dE is the neutrino-energy distribution in 77 interactions. The integration limits follow from the above discussion. The <\>p peaks at Ev ~ E^/3 ~ Et^/3(1 + z) ~ 1011 GeV/3(l + z)2 ~ 108 GeV. The 77 resonance formation can also con­tribute to cjff1. It is a possibility to produce high energy neutrinos through non hadronic interactions in a cosmological setting.

The electromagnetic cascade that generate high energy neutrinos from muon decays in 77 interactions contains roughly equal number of photons and electrons. In Ref 22, it was suggested that, for this class of topological defects that produce ultrahigh energy photons at the high z, the muon pair production (MPP) in e~7 —> e~fj,+fx~ dominates over the triplet pair production (TPP) in e^7 —> e~e+e" for 5 m2 < s < 20 m2 in the electromagnetic cascade, thus enabling the MPP process to be an efficient mechanism for generating high energy neutrinos at the high z. The electrons in the final state of the above processes are considered as originating from the electromagnetic cascade generated by the ultrahigh energy photons scattering over the CMB photons present at the high red shift. This conclusion was based upon the value of the ratio R defined as R ~ (JMPPMTPP^TPP, where TJTPP is the inelasticity for the TPP process. The TJTPP is basically the average fraction of the incident energy carried by the final state positron. The original estimate of Ref. 22

gives R ~ 102, which favors the MPP process as the dominating high energy neutrino generating process. Namely, the electron energy attenuation length due to the TPP process is much longer than the interaction length of the MPP process because OMPP — (0.1 — 1) mb. However, by an explicit calculation 23, instead it was found that <JMPP < 1M^ for s > 5 m2 , thus yielding R < 1. In particular,

. . J 4 • HT3/xb for s = 4m2 . . a M P p ( s ) = \ l - 1 0 - V b f o r S = 20m2. ( 8 )

Therefore, MPP can not be a dominating process for generating high energy neutrinos. We note that the equivalent photon approximation was used in

134

this work to calculate the leading-order contribution to (TMPP(S).

In summary, in an electromagnetic cascade generated by ultrahigh en­ergy photons scattering over the CMB photons at the high red shift, the 77 —> n+n~ can in principle produce high energy neutrinos, typically for 5 < z < 10, through the muon decays. On the other hand, the process e7 —> e~/x+n~~ occurring as the next round of interactions in the same elec­tromagnetic cascade can not produce the high energy neutrinos.

3 Conclusions

Possibilities of high energy neutrino generations in two of the non hadronic interactions, namely 77 and ej reactions are briefly discussed. In the first interaction, the formation and decay of the 77 resonance in addition to the muon pair production may have some implications for high energy neutrino generation. Model dependent analysis is needed to further quantify the high energy neutrino generation in non hadronic interactions.

Acknowledgment s

HA thanks Physics Division of National Center for Theoretical Sciences for fi­nancial support. GLL is supported by the National Science Council of Taiwan under the grant number NSC90-2112-M009-023.

References

1. F. Halzen, these proceedings. 2. K. Greisen, Phys. Rev. Lett. 16, 748 (1966); G. T. Zatsepin and

V. A. Kuzmin, JETP Lett. 4, 78 (1966) [Pisma Zh. Eksp. Teor. Fiz. 4, 114 (1966)].

3. H. Athar, K. Cheung, G.-L. Lin and J.-J. Tseng, arXiv:hep-ph/0112222. 4. G. Domogatsky, arXiv:astro-ph/0112446. 5. T. Montaruli [ANTARES Collaboration], arXiv:hep-ex/0201009. 6. I. F. Albuquerque, J. Lamoureux and G. F. Smoot, arXiv:hep-

ph/0109177; H. Athar and G.-L. Lin, arXiv:hep-ph/0201026 and refer­ences therein.

7. M. Ambrosio [MACRO Collaboration], arXiv:astro-ph/0203181. 8. C. D. Dermer and R. Schlickeiser, Science 257, 1642 (1992). 9. See, for instance, S. T. Scully and F. W. Stecker, Astropart. Phys. 16,

271 (2002). 10. R. J. Gould and G. Schreder, Phys. Rev. Lett. 16, 252 (1966).

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11. A. Zdziarski, Ap. J. 335, 786 (1988). 12. R. J. Protheroe and P. A. Johnson, Astropart. Phys. 4, 253 (1996) [

erratum-ibid., 5, 215 (1996)]. 13. S. Lee, Phys. Rev. D 58, 043004 (1998) and references therein. 14. M. Poppe, Int. J. Mod. Phys. A 1, 545 (1986); X. Bertou, P. Billoir, and

S. Dagoret-Campagne, Astropart. Phys. 14, 121 (2000). 15. D. E. Groom et al. [Particle Data Group Collaboration], Eur. Phys. J.

C 15, 1 (2000). 16. R. Shaisultanov, Phys. Rev. Lett. 80, 1586 (1998). 17. T. K. Chyi, C. W. Hwang, W. F. Kao, G. L. Lin, K. W. Ng and

J. J. Tseng, Phys. Lett. B 466, 274 (1999). 18. For a review, see, R. Durrer, M. Kunz and A. Melchiorri, arXiv:astro-

ph/0110348 and references therein. 19. See, for instance, G. Sigl, arXiv:hep-ph/0109202; F. Halzen and

D. Hooper, arXiv:hep-ph/0110201. 20. A. Kusenko, arXiv:astro-ph/0008369. 21. P. J. Peebles, Principles Of Physical Cosmology (Princeton University

Press, USA, 1993). 22. A. Kusenko and M. Postma, Phys. Rev. Lett. 86, 1430 (2001). 23. H. Athar, G.-L. Lin and J.-J. Tseng, Phys. Rev. D 64, 071302 (2001).

Q U E S T I O N S I N C O S M O L O G Y A N D P A R T I C L E A S T R O P H Y S I C S

W - Y . P A U C H Y HWANG

Center for Academic Excellence on Cosmology and Particle Astrophysics

Department of Physics, National Taiwan University. Taipei, Taiwan, R. 0. C. E-mail: wyhwang@phys.ntu.edu.tw

In this brief review, I wish to first flash some key elements of the standard hot big bang model as the basic language, then move on to report on some of the activities and progresses associated with the subproject on the theoretical studies on cosmology and particle astrophysics, and finally t ry to conclude by illustrating, as an example, the problem of phase transitions in the early universe.

1 The Background: The Homogeneous and Isotropic Universe

In what follows, I shall first review briefly the standard model of a homo­geneous early universe, bearing in mind that such picture work well up to one part in 100,000, i.e., up to the level of the observed magnitudes of CMB fluctuations. We do have to introduce inhomogeneities into our picture of the early universe, if we can ever have a complete understanding of what the CMB fluctuations and polarizations are all about.

Based upon the cosmological principle which state that our universe is homogeneous and isotropic, we use the Robertson-Walker metric to describe our universe.1

fir2

ds2 = dt2 - R2(t){^^ + r2dd2 + r2sin2ed4>2}. (1)

Here the parameter k describes the spatial curvature with k = + 1 , — 1, and 0 referring to an open, closed, and flat universe, respectively. The scale factor R(t) describes the size of the universe at time t.

To a reasonable first approximation, the universe can be described by a perfect fluid, i.e., a fluid with the energy-momentum tensor TM

v — diag(p, , —p, —p, —p) where p is the energy density and p the pressure. Thus, the Einstein equation, C v = 8TTGNT'1 V + A<?M „, gives rise to only two independent equations, i.e., from (/i, v) = (0, 0) and (i, i) components,

R2 k 8nGN A & + &=^-p+3- ( 2 )

136

137

R R? k 2R + V + 1P=-8*GNP + A- ^

Combining with the equation of state (EOS), i.e. the relation between the pressure p and the energy density p, we can solve the three functions R(t), p(t), and p(t) from the three equations. Further, the above two equations yields

^ = - ^ - ( p + 3 P ) + 3 , (4)

showing either that there is a positive cosmological constant or that p+3p must be somehow negative, if the major conclusion of the Supernovae Cosmology Project is correct, i.e. the expansion of our universe still accelerating (J| > 0).

Assuming a simple equation of state, p = wp, we obtain, from Eqs. (2) and (3),

R R2 h 2 - + (1 + 3w)(w + —2) - (1 + «,)A = 0, (5)

so that, with p= —p and k = 0, we find

R - ^ = o, (6)

which has an exponentially growing, or decaying, solution R oc e ± a t , compat­ible with the so-called "inflation" or "big inflation". In fact, considering the simplest case of a real scalar field (j>(t), we have

p = \ & + n<t>), P = IJ>2- VW> (?) so that, when the "kinetic" term \<j>2 is negligible, we have an equation of state, p ~ —p. In addition to its possible role as the "inflaton" responsible for inflation, such field has also been invoked to explain the accelerating expansion of the present universe, as dubbed as " quintessence".

Another simple consequence of the homogeneous model is to derive the continuity equation from Eqs. (2) and (3):

d(pR:i)+pd(R3)=0. (8)

Accordingly, we have p oc R~4 for a radiation-dominated universe (p = p/3) while p oc R~3 for a matter-dominated universe (p « p). The present universe has a matter content between 2 x I0~31g/cm3 and 2 x 10_ 2 95/cm3 ,

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much bigger than its radiation content 5 x 10~35g/cm3, as estimated from the 3° black-body radiation. However, as t —» 0, we anticipate R —> 0, extrapolated back to a very small universe as compared to the present one. Therefore, the universe is necessarily dominated by the radiation during its early enough epochs.

For the radiation-dominated early epochs of the universe with k = 0 and A = 0 (for the sake of simple arguments), we could deduce, also from Eqs. (2) and (3),

"-a^k'"2' T=<3^>V i a i 0 , O r l / 2™- <9> These equations tell us a few important times in the early universe, such as 1 0 " n sec when the temperature T is around 300 GeV during which the elec-troweak (EW) phase transition is expected to occur, or somewhere between 10~5sec (= 300 MeV) and 10_4sec (= 100 MeV) during which quarks and gluons undergo the QCD confinement phase transition.

2 CosPA-3: Theoretical Studies on Cosmology and Particle Astrophysics

The science goal of our theoretical studies of Cosmology and Particle Astro­physics (CosPA-3)2 is to make significant progresses, hopefully some major breakthroughs, in the prime area of cosmology, i.e. the physics of the early universe J . While an important emphasis will be placed mainly on the physics of the cosmic microwave background (CMB)3'4, the primary tracer of the hot big bang, a broad and balanced CosPA theory program is essential for a healthy long-term future development.

2.1 Manpower in CosPA-3

The past year has been a growing and fruitful year for us in CosPA-3. The number of participating faculty members has increased, including Guey-Lin Lin and Win-Fun Kao from National Chiao-Tung University and Darwin Chang from National Tsing-Hua University. The Project has offered attractive opportunities for graduate students at Taida. Currently at National Taiwan University(NTU), we are having about a dozen Ph.D. students supervised di­rectly by CosPA-3 faculty members while there are about the same number of M.S. students associated with CosPA-3. As the primary P.I. of CosPA-3, I fully understand that it is the quality of the research that really counts, not the quantity or the number of the published papers. However, the impact

139

of an important paper, if ever important, is often not felt by the community until years have passed and enough has been said.

What I may try to do is to cite a friendly, but official, remark by Marc Henneaux, a well known string theorist who was invited recently by the Na­tional Science Council to visit Taiwan, "I have been truly impressed by the excellence of the 'Center for Academic Excellence on Cosmology and Particle Astrophysics' which, by combining expertise over a wide range of knowledge, is, in my opinion, one of the leading places in the world where research in this area is conducted." It was indeed in our plan to combine expertise over a wide range of knowledge in order to stimulate significant progresses in the field.

Over the last year, we have also enjoyed the visits and the lectures by several distinguished visitors, such as David Gross, Anthony Zee, Robert Brandenberger, and Andrew Liddle, who in a way introduced very original ideas to the CosPA-3 audience at the lectures. These visitors entertained in­teractions from the audience, enjoyed their stays in Taipei, and, if schedules permit, would wish to come back for another visits. This fact alone may be considered as a gauge of the overall quality of the host institution.

2.2 Some Topics Pursued by the CosPA Theorists

Observationally, we expect, over the next few years, to have more accurate sky maps of CMB anisotropics, in much better angular and temperature res­olutions. Some CMB astronomers are also rushing to become the first one to detect CMB polarizations. In addition, surveys of high-z, or far away, clusters or large-scale structures through the SZ effect in CMB measurements, OIR, supernovae, gravitational lensing, etc. have all become possible and will result in many definitive results, much more quantitative than ever and thus much more useful for pinning down the basic parameters in cosmology. Indeed, cosmology is transforming itself into an experimental science.

On CMB anisotropics, Proty J.H. Wu has been made some good marks in the science team for U.C. Berkeley's MAXIMA 5 . Proty has recently joined the faculty at NTU and thus will be an active member of CosPA-3. K.-W. Ng has invested a significant amount of effort, over the last two years (including his sabbatical year), working on aspects related to CMB polarizations 6. I am pretty sure that Chiueh, Ng, and Wu could work together as a team capable of setting up the data-analysis pipeline for the AMiBA data acquisition in the year of 2004 and beyond.

It has evolved into an exciting area of theoretical speculations in terms of understanding the implication from the Supernovae Cosmology Project 7

that the expansion of our universe is still accelerating. Gu and Hwang 8 were

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the first ones to address the question if the quintessence can be a complex scalar field, an option being dubbed " spintessence" due to the new degree of freedom in the internal complex space. Xiao-Gang He work on the im­plication accelerating universe, "Accelerating Universe and Event Horizon" [astro-ph/0105005], led to an United Press International (UPI) interview of him on the idea. Chiueh and He have also considered the possibility of "Fu­ture Island Universes in a Background Universe Accelerated by Cosmological Constant and by Quintessence".

On possible directions of the inflation models offered by stringy spacetime thinkings, I admire Pei-Ming Ho, Miao Li, several postdocs and students for their productivity in the stringy environment of CosPA-3. There might not be much need for me to cite their papers since many of the papers are well cited and some even top-cited.

There are other ongoing research problems which may in fact lead to sig­nificant contributions in cosmology, including (1) the roles of the electroweak and QCD phase transitions in the early universe, as phase transitions may easily amplify the size of inhomogeneities and may also produce new types of inhomogeneities during the transition, (2) simulations of large-scale structures (LSS), as high-z LSS's serve as another important information to narrow down the range of the basic parameters in cosmology, and recently (3) the physics of ultrahigh energy cosmic rays in the microwave and neutrino cosmic back­ground. It is anticipated that important papers will appear over the next couple of years.

2.3 Remarks

The annual CosPA-3 budget is small, usually over-spent but enough to at­tract and glue many people together (faculty members, visitors, postdocs, students) so as to strive for academic excellence. Significant impacts have al­ready been made on the local community in the area of cosmology and particle astrophysics.

In the immediate future, we would like to see or implement more integra­tions, especially with the other projects, such as to integrate efforts to better AMiBA sciences, to integrate efforts on neutrino and gamma-ray cosmology, as well as to set up opportunities for students to enter the OIR research.

In order to strive for better, internationally, we plan (1) to lift the work­shop to an international symposium, (2) to invite more internationally well-known scholars for visits, (3) to send people to attend and make oral presen­tations at important international meetings, (4) to perform annual evaluation of postdoctoral fellows, and (5) to engage the graduate students in active

141

modes. We are confident that the research efforts of CosPA-3 will become an important treasure to the entire CosPA project.

3 Phase Transitions in the Early Universe

With the qualitative description of the research activities associated with CosPA-3, I wish to close this talk by describing to you the outstanding ques­tion of phase transitions in the early universe. The purpose is to provide an illustrative example of what we are doing, but by no means to dismiss other ongoing research topics as being of lesser importance.

As we know, electroweak (EW) phase transition, which endows masses to the various particles, and QCD phase transition, which gives rise to con­finement of quarks and gluons within hadrons in the true QCD vacuum, are two well-established phenomena in the standard model of particle physics. Presumably, the EW and QCD phase transitions would have taken place in the early universe, respectively, at around 1 0 ~ n sec and at a time between 10~5 sec and 10~4 sec, or at the temperature of about 300 GeV and of about 150 MeV, respectively. Formulation of EW and QCD phase transitions has become one of the most challenging problems in the physics of the early uni­verse.

After having worked on this problem for a couple of years, we have come to the following anatomy of the problem by dividing it into four categories of questions, viz.: (1) how a bubble of different vacuum grows or shrinks; (2) how two growing bubbles collide or squeeze (and merging with) each other; (3) how the various bubbles of lower temperature nucleate or grow out of the vacuum as the temperature lowers (due to the expansion of the universe); and (4) how specific objects, such as back holes or magnetic strings, get produced during the specific phase transition. We give solutions to some of these problems.

3.1 Exploding or Imploding Solitons

Consider a spherical wall of radius R and thickness A separating the true vacuum inside from the false vacuum outside. The energy density difference of the vacua is B, the bag constant in the most simplified situation, and the energy r per unit area associated with the surface tension on the separating wall is a quantity to be calculated but nevertheless is small compared to the latent heat. If the wall expands outward for a distance 6R, then the energy budget arising from the vacuum change is

ATTR2 • 6R - TATT{(R + 5Rf - R2} = -pSV, (10)

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where p is the pressure and is so denned that a negative pressure would push the wall outward.

When the surface tension energy required for making the wall bigger is much less than the latent heat required from the expansion of the bubble, the bubble of the stable vacuum inside will grow in an accelerating way, possibly resulting in explosive growth of the bubble. The scenario may be as follows: When the universe expands and cools, to a temperature slightly above the critical temperature Tc, bubbles of lower vacua will nucleate at the spots where either the temperature is lower, and lower than Tc, or the density is higher, and higher than the critical density pc. As the universe continues to expand and cool further, most places in the universe have the temperature slightly below Tc; that is, the destiny arising from eternal expansion of the universe is driving the average temperature of the entire universe toward below the critical temperature. The universe must find a way to convert itself entirely into another vacuum, the true vacuum at the lower temperature.

Therefore, we have a situation in which bubbles of true vacua pop up (nu­cleate) here and there, now and then, and each of them may grow explosively in the environment made of the false vacuum for now, but previously the true vacuum when the temperature was still well above the critical temperature. The problem can be modelled, in the simplest way, by characterizing the vac­uum structure by a scalar field, complex or real, interacting via the potential V(4>):

V{<P) = ^4>*4>+\{<f<t>)2, M 2 < 0 , A > 0 . (11)

Note that, in the complex scalar field description, the true vacua have de­generacy described by a continuous real parameter 6. 4> = 0 everywhere in the spacetime describes the false vacuum for the universe at a temperature below the critical temperature Tc. Consider the solution for a bubble of true vacuum in this environment. It is required that the field <p must satisfy the field equation everywhere in spacetime, including crossing the wall of thick­ness A to connect smoothly the true vacuum inside and the false vacuum outside. This is why we may call the bubble solution " a soliton", in the sense of a nontopological soliton of T.D. Lee's. However, the soliton grows in an accelerating way, or the name "exploding soliton".

The situation must have changed so explosively that at a very short in­stant later the universe expands even further and cools to even a little more farther away from Tc and most places in the universe must be in the true vacuum, making the previously false vacuum shrink and fractured into small regions of false vacua, presumably dominantly in spherical shape, which is

143

shrinking in an accelerating way, or " implosively". Using again the complex scalar field as our language, we then have "imploding solitons".

In what follows, we attempt to solve the problem of an exploding soliton, assuming that the values of both the potential parameters /J? and A are fairly stable during the period of the soliton expansion. The scalar field must satisfy:

7^r^)-W = v{(t')- (12)

The radius of the soliton is R(t) while the thickness of the wall is A:

A (j> = <t>o, for r < Ro+vt- —,

A = 0, for r> Ro + vt+—, (13)

with R(t) = Ro + vt and v the radial expansion velocity of the soliton. We may write

<f> = f(r + vt); w=(l-v2)r, (14)

so that the field equation becomes

We will be looking for a solution of / across the wall so that it connects smoothly the true-vacuum solution inside and the false vacuum solution out­side.

Introducing g = wf(w), we find

^ ' ^ ( l - ^ - ^ ^ l ^ - l 2 - ^ } , (16) w

an equation which we may solve in exactly the same manner as the colliding-wall problem to be elucidated in the next section.

3.2 Colliding Walls

When bubbles of true vacua grow explosively, the nearby pair of bubbles will soon squeeze or collide with each other, resulting in merging of the two bubbles while producing cosmological objects that have specific coupling to the system. The situation is again extremely complicated. We try to disentangle the complexities by looking at between the two bubble walls that are almost ready to touch and for the initial attempt neglecting the coupling of the vacuum dynamics to the matter content. Between the two bubble walls, especially

144

between the centers of the two bubbles, it looks like a problem of plane walls in collision - and this is where we try to solve the problem to begin with.

Consider the problem of two walls approaching each other. The wall, each of thickness A, separates the true vacuum on one side from the false vacuum on the other side of the wall. For the sake of simplicity, both walls are assumed parallel to the (xy)— plane and are infinite in both the x and y directions. In addition, the wall at z = R is moving to the left with the velocity v while that at z = —R moving to the right with +v. For z > R + y and all x and y, the complex scalar field <j> assumes <j>0, a value of the true vacuum (the ground state). On the other hand, for z < —R — y and all x and y, the complex scalar field 4> assumes 4>'0, another value of the true vacuum which differs in general from <J>Q. In between the two walls, i.e., —R + | < z < R — y , we have 0 = 0, the false vacuum. As indicated earlier, the field 4> must satisfy the field equation everywhere in spacetime:

d2<t> ^ vw\ nn

We may write the wall on the right hand side but moving toward the left with the velocity v.

4>= f(z- vt), for z-vt>0, t< R/v. (18)

so that

( 1 - « Y = A / ( | / | 2 - A <r=|<M>0. (19)

For a real scalar field / , this equation can be integrated by multiplying both sides by / ' :

( l - 2 ) ^ = | / 4 - ^ / 2 + Co, (20)

with CQ the integration constant. This in turn gives rise to

df I y7 4 - 2<T2/2 + *f ±a£ + C l , (21)

with £ = = z — vt — R+-2, a= J2(i-v2) an<^ Cl a n ° t h e r integration constant.

What remains to be done is to choose CQ and c\ so that / connects smoothly to the solutions on both sides of the wall.

The function in Eq. (19) may also be complex:

/ = ueie, (22)

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so that, with A = A/(l - v2),

u "-u(0')2 = \u(u2-a2), (23)

2u'e' + u0" = 0. (24)

Integrating the second equation, we find

u29' = K, (25)

with K an integation constant. The second equation can be integrated out in the same way as in the real case. We record the result as follows:

f = - f dV (26) 2 Jo J-K + ay-2py2+(3y^

with

2 Jo J^K

dy

^~K + ay-2f3y2+py3 (27)

Here j3 = ^<r6, and if and a parameters related to the integration constants. In closing, we note that the equation for the exploding or imploding soli-

ton, Eq. (16), may be integrated and solved in an identical manner.

Acknowledgments

This work is supported through the Taiwan CosPA Project as funded by the Ministry of Education of R.O.C. (89-N-FA01-1-0 & 89-N-FA01-1-3).

References

1. E.W. Kolb and M.S. Turner, The Early Universe, Addison-Wesley Pub­lishing Co. (1994).

2. CosPA-3 is the primary theory component of the Taiwan CosPA project as funded by the Ministry of Education (89-N-FA01-1-0 up to 89-N-FA01-1-5) during the 4-year period of 2000.1-2004.3.

3. G. Smoot et al., Astrophys. J. 396, LI (1992); C. Bennett et al., At-rophys. J. 396, L7 (1992); E. Wright et al. Atrophys. J. 396, L l l (1992).

4. C. L. Bennett, M. S. Turner, and M. White, Physics Today, November 1997, p. 32, for an early general review.

146

5. Cosmological Implications of the MAXIMA-I High Resolution Cosmic Microwave Background Anisotropy Measurement, R. Stompor et al. Ap. J. Lett. 561, L7 (2001); A High Spatial Resolution Analysis of the MAXIMA-1 Cosmic Microwave Background Anisotropy Data, A.T. Lee et al., Ap. J. Lett. 561, LI (2001); Tests for Gaussianity of the MAXIMA-1 CMB Map, J.H.P. Wu et al., Phys. Rev. Lett. 87, 251303 (2001).

6. Complex Visibilities of Cosmic Microwave Background Anisotropics, K.-W. Ng, Phys. Rev. D63, 123001 (2001).

7. S. Perhnutter et al., Astrophys. J. 517, 565 (1999) [arXiv:astro-ph/9812133].

8. Je-An Gu and W-Y. P. Hwang, Phys. Lett. B517, 1 (2001).

NONCOMMUTATIVE EARLY UNIVERSE

P E I - M I N G HO

Department of Physics. National Taiwan University. Taipei 106. Taiwan. R. O. G.

pmho @phys.ntu. edu. tw

We study the implication of the stringy spacetime uncertainty relation on the spectrum of metric perturbations. Assuming that the metric perturbations orig­inate from quantum fluctuations of a scalar field, we find that the initial state is uniquely determined by the uncertainty relation, and the spectrum is modified in the infrared.

1 Introduction

Cosmology needs string theory because classical gravity and the standard model of particle physics are expected to break down at very early stages of the universe. In this paper I would like to discuss possible implications of the stringy spacetime uncertainty relation (SSUR) on cosmology. SSUR was first proposed by Yoneya 2 to be a universal property of string theory. It says that the physical time and space coordinates satisfy

AtpAxp > l2s, (1)

where ls is the string length scale. In comparison, another well known stringy uncertainty relation 2

AxpAp > 1 + l2sAp2, (2)

which implies a minimal length scale

Axp > ls (3)

can be violated by D-branes. In a previous work 3, we suggested that SSUR can be used to solve the

flatness problem and the horizon problem of cosmology, even without inflation. Inflation 4 is successful in solving many cosmological problems, such as the flatness problem, the horizon problem, the monopole problem, etc., and in explaining the observed mass perturbations and cosmic microwave background radiation. However, inflationary cosmology also suffers several conceptual problems 5 . Assume that the universe is created at the time t = 0. Then the temporal uncertainty is bounded by

At < t. (4)

147

148

Combined with SSUR, this implies that

I2

Ax > - i . (5)

Suppose that the earliest moment when the time coordinate makes sense is at t = £*, then the radius of spatial curvature, or any other physical length scale, has to be greater than l2/t* according to (5). For a small £* and a relatively larger ls, the radius of curvature at the Planck time can be much greater than the Planck scale and solve the flatness problem. Similarly, the horizon problem is solved because the correlation length of any physical degree of freedom is bounded from below by (5). Explicit calculation shows that if we choose i* = lp ~ (1019GeV)_1 and ls ~ (TeV) - 1 , then both problems can be solved without inflation.

Now we would like to study the effect of SSUR on primordial metric per­turbations, which are the origin of the observed large-scale structure and of cosmic microwave background anisotropics. As a first step, we consider a sim­ple model with a single scalar field in a homogeneous, isotropic background. Quantum fluctuations of the scalar field are responsible for creating the met­ric perturbations. As such the cosmological perturbations are automatically Gaussian.

The assumed flat, homogeneous isotropic background, can be described by the Friedmann-Robertson-Walker (FRW) metric

ds2 = dt2 -a2(t)dx2

= a2{ri){dr}2 - dx2) (6)

= a-2(r)dT2 - a2(T)dx2,

where dt = a(r])dr] = a~1(r)dT. The coordinate r is useful because it can be used to write SSUR in a time-independent way

AT Ax > l2s. (7)

If we use T], for instance, naively SSUR should look like

a2(?7)A?7A:r > I2. (8)

However, more precisely the factor a2 should be smeared over a length of time depending on A77, and this smearing needs to be further specified.

Since the scale factor has no spatial dependence, SSUR imposes no re­striction on the Hubble constant H = a/a, or H, etc. On the other hand, had we imposed the constraint (3), we would have H < Ms, and H < M2, etc., where Ms = Z"1 is the string energy scale.

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The evolution of the scale factor should in principle be determined by string theory. However, since we do not know how to construct a gravitational theory incorporating SSUR, we will first try to understand the problem for a generic a, and then consider some examples.

2 Power Spectrum

In this section we would like to study the power spectrum of metric pertur­bations in a slightly more general setting than the focus of this work. We will consider a generic field theory for a free scalar field <f> whose Fourier modes are decoupled. The most general form of the action is

S~V [ dTd3k\ (f3+(T)dT4>-kdr<Pk - k2^{T)^k4>k) . (9) JkSKa

Z

To simply our notation, we have denoted 0g as <f>k, a n d \k\ as k. For later

consideration of SSUR, we also allow the possibility that 0^ evolves nontriv-ially with time according to this action only for k within a range K0. Similar restrictions also appeared in field theories obeying (2) 6.

By a change of the time coordinate T —* fj, this action can be further simplified as

S~V [ dr,d?k\zl{fj) {<f/_k4>'k ~ k24>-k<t>k), (10) JkeKo z

where / ' is the derivative of / with respect to fj. The new time coordinate fj is defined by

dfj 1/2

(11)

and

** = (tt+) 1 / 4- (12) For many models, the problem of finding metric perturbations can be

reduced to solving the equation of motion for an action of the form (10). For an ordinary inflationary model with a scalar field, fj = r\ is the conformal time, and

where 00 We have

is the

Vk = z

zero mode of </> and Ji -

Vk

00

n = a /a,

— a

1

for scalar metric

(13)

perturbations.

(14)

150

for tensor metric perturbations. Detailed review of the calculation of metric perturbations can be found in 7.

Assume that we are dealing with metric perturbations which are described by the action (10). By a change of variable

<t>k = ~, (15) Zk

the equation of motion is simplified to be

v'i + (k2 ~f-)vk = 0. (16)

From the action (10), the momentum canonically conjugate to v is

nk = v'_k-£.v-k, (17) Zk

and this leads to the Hamiltonian i z

H Id'k 7 t ' Zk I t t \ ka\ak + - — (akaLk - OfcO-fe)

^ Zk

(18)

where a'k(fj) and ak(fj) are the creation and annihilation operators at time fj. They are related to vk and 11^ by

Vk = -^=(ak+aik), (19)

nk = -i\j^(a-k-al). (20)

In the Heisenberg picture, the creation and annihilation operators evolve with time, and the state does not. Assume that the Universe is in the state |0) defined by

a,(77°)|0) = 0 , (21)

which is the vacuum for mode k at some intial time rfc. It is in general not the vacuum at later times.

The Bogoliubov transformation relates the creation and annihilation op­erators at fjk with the corresponding operators at the time 77:

ak(fj) = ak{r,)ak{r,l) + (3k{ri)alk(f&), (22)

4(V) = Pk(vh-k(f}°k) + ak(f,)al(f,°k), (23) where ak and j3k satisfy

akak -/3kpk = 1, Vfc. (24)

151

The Hamilton equations [H, ak] = iak and its Hermitian conjugate are

tf+(fc2-^)&=0, (26)

in terms of the new variables

Cfc = <**-&, £k = ak+pk. (27)

The larger the value of \Pk\2(fj), the larger the number of particles at time fj created out of the vacuum state |0).

Let us denote the factors in (25), (26) by

z" (z-1)" M = k 2 - ^ , N = k2~^~. (28)

zk (zk*) For small fj (scales smaller than the Hubble radius), both M and N are positive and approximately equal to k2. Thus, £k and £t oscillate. If the initial state is taken to be the local vacuum state,

ak(fj°k) = l, /?*($) = 0 , (29)

then the magnitudes of £ and f are of order 1 until the size of the fluctuation grows outside the Hubble radius. This represents the oscillation of quantum vacuum fluctuations. On scales larger than the Hubble radius (at later times), M and N are dominated by the second (negative) term. In this period, C a n d £ correspond to frozen fluctuations which are undergoing quantum squeezing and which scale like zk and z^1, respectively. If zk increases with time, by (27) we can approximate vk by Ck/V2k for sufficiently late times. When k2

is much smaller than both z'£/zk and {zk1)"/(zk~

1), and if zk is an increasing function, the dominant solutions are simply

Cfc ^ Ckzk, & ~ - J — (30) <^kZk

for real uk, vk. Note that the condition (24) is satisfied for this solution, but not for other solutions of the second order differential equations. Suppose that for sufficiently late times, zk ]» 1 (zk -C 1) then vk ~ ^Qk/y/2k (vk ~ l£k/\/2k~). Since the initial condition is Ck(fjo) = €k(fJo) = 1, it is equivalent to say that for zk ~S> 1, for sufficiently large fj, vk(fj) is given by the solution to the differential equation (25) with the initial condition v(fjo) = l/v2fc. For zk <C 1, we replace zk by z^1. In general we want to find the larger of the two functions Qk and £k. This then determines via the relations (27) the Bogoliubov coefficient /3k.

152

The power spectrum for the state |0) is defined by

mp zk

where Mp is the Planck mass. The normal ordering is defined for the operators a(rjl), a t ( ^ ) at 7$, so that V(k) = 0 at fj%. (Certainly we must have V{k) = 0 for 77 < 77°, before the mode k is allowed to appear.) For fj > 77°, we have

V^^W^f- (32) MP zk

According to the analysis above, for expanding zk, the spectrum is 1 k2

T{k) ~ a a (33) Mpz

2k(rik)

where fjk is the time when the fluctuation mode k crosses the Hubble radius (M = 0). In this paper we also consider the case of fluctuations outside the Hubble radius starting at the vacuum. For them 77* should be taken to be the time when they start at the vacuum. In general, fjk is the time when the mode k first appears outside the Hubble radius.

3 SSUR and Spectrum

Eq. (33) for the power spectrum is very convenient. In the usual expand­ing cosmological models, we have zk oc a for both scalar and tensor metric perturbations for scale factors of the form

a(t) = a0tn = OOT^ , (34)

where a0 = ((n + l ) n a 0 ) "+1. This occurs for constant equation of state w — p/p, and

n = WT^y (35)

For classical cases, we replacing zk by a and fj by 77 in the previous section. For n > 1 we have 77 oc A;-1, and thus Zk{rfk) ex k1^. Therefore, from (33) we can easily find the spectrum for metric perturbations

Pk~ck^, (36)

where c = ((2n2 - n)na;j) " _ 1 /MP. For the spectrum to be exactly scale in­variant, we need n = 00, that is, exponential inflation, for which the spectrum

153

H2

For the more general case when the scalar field is described by the action (10), the major difference from the classical case is that we have an effective scale factor Zk for each mode k, replacing a. The reason why SSUR will have such an effect is easy to understand at least intuitively. To describe a mode k, the spatial uncertainty can not be bigger than its wavelength. SSUR thus implies a lower bound on the uncertainty in time. Therefore, when this mode interacts with the scale factor a, or any other physical quantity, the quantity is smeared over some range of time.

Suppose that the replacement of a by Zk everywhere is the only modifi­cation due to SSUR. The physical momentum for the mode k would be k/z}. instead of k/a. Being a massless scalar field, its energy is also k/zk- Therefore, the mode fc has

Axp < zk/k, Atp < zk/k, (38)

and SSUR implies that

Ms > - ^ - = Ek(fj), (39)

which is the energy for a particle in the mode k. For an expanding universe, o is monotonically increasing, and so is its smeared version z^. This bound implies that the mode A; can not exist before a certain time fj° when

*s4= a l* ) - (40)

The mode k first appears at time 77°. The range of k whose dynamics is described by the action (10) is thus

K0(fj) = {k:fj> fj°k}. (41)

Remarkably, this constraint also uniquely determines the initial state of the universe. Before fjl, the fluctuation mode k does not exist. For a smooth transition to its existence after fj%, the only possibility is for it to start at the vacuum at 77°. This is precisely what we assumed to derive (33) for the power spectrum of metric perturbations.

Interestingly, the power spectrum (33) can also be written as

Hk) * ^ - (42)

154

Hence for all modes k which first appears outside the Hubble radius, their spectrum is scale invariant and is given by

M 2

"«=4- (43)

This is independent of the precise form of z^\ Note that the modes whose wavelengths are larger than the Hubble radius are in the infra-red.

For modes which first appears deep inside the Hubble radius, that is, the ultra-violet modes, the effective scale factor zj. is roughly the same as a when it crosses the Hubble radius at rjk = fjk — l/^> a n d their spectrum is given by the old'result (36). The whole spectrum, therefore, interpolates between a scale invariant one in the IR and the classical result in the UV.

The fact that SSUR can change the power spectrum significantly seems to contradict which claims that the effect of new physics is hard to be observed 8. Although our effective action is of the same kind considered there, the major difference is that our action only describes those modes k below a cutoff, and this cutoff determines our initial state, which is very different from the usual choice.

Acknowledgments

The author thanks Tzihong Chiueh, Je-An Gu, Hsien-chung Kao, Feng-Li Lin, Sanjaye Ramgoolam, Hyun-Seok Yang for helpful discussions. This work is supported in part by the National Science Council, the CosPA project of the Ministry of Education, the National Center for Theoretical Sciences, Taiwan, R.O.C. and the Center for Theoretical Physics at National Taiwan University.

References

1. T. Yoneya, in "Wandering in the Fields", eds. K. Kawarabayashi, A. Ukawa (World Scientific, 1987), p. 419; M. Li and T. Yoneya, "D-particle dynamics and the space-time uncertainty relation," Phys. Rev. Lett. 78, 1219 (1997) [arXiv:hep-th/9611072]; T. Yoneya, "String the­ory and space-time uncertainty principle," Prog. Theor. Phys. 103, 1081 (2000) [arXiv:hep-th/0004074].

2. G. Veneziano, Europhys. Lett. 2, 199 (1986); D. J. Gross and P. F. Mende, Nucl. Phys. B 303, 407 (1988); D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216, 41 (1989); R. Guida, K. Konishi and P. Provero, Mod. Phys. Lett. A 6, 1487 (1991).

3. J.-A. Gu, P.-M. Ho and S. Ramgoolam, [hep-th/0101058].

155

4. A. Guth, "The Inflationary Universe: A Possible Solution To The Horizon And Flatness Problems," Phys. Rev. D 23, 347 (1981). A. D. Linde,. "Chaotic Inflation," Phys. Lett. B 129, 177 (1983).

5. R. H. Brandenberger, "Inflationary cosmology: Progress and problems," [hep-ph/9910410].

6. A. Kempf and J. C. Niemeyer, "Perturbation spectrum in inflation with cutoff," Phys. Rev. D 64, 103501 (2001) [astro-ph/0103225].

7. V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, "Theory of Cosmological Perturbations." Phys. Rept. 215, 203 (1992). A. Liddle and D. Lyth, 'Cosmological inflation and large-scale structure' (Cam­bridge Univ. Press, Cambridge, 2000).

8. N. Kaloper, M. Kleban, A. Lawrence and S. Shenker, "Signatures of short distance physics in the cosmic microwave background," arXiv:hep-th/0201158.

COSMOLOGICAL CONSTANT, QUINTESSENCE A N D MINI-UNIVERSES

XIAO-GANG H E *

Department of Physics. National Taiwan University. Taipei, Taiwan 10764, R-O.C.

In this talk, I report my recent work with Tzihong Chiueh (astro-ph/0107453) on bound object formation in a background universe accelerated by cosmological constant and by quintessence. If the acceleration lasts forever, due to the existence of event horizon, one would have naively expected the universe to approach a state of cold death. However, we find that many local regions in the universe can in fact be protected by their own gravity to form mini-universes, if their present matter densities exceed certain critical value. Within these mini-universes, there can be no event horizon although all of them will eventually fall onto each other's horizon and become isolated island universes. In the case with cosmological constant the condition of forming mini-universe is that the ratio of present density parameters &i / ^A ought to be larger than a critical value 3.63. In the case with quintessence, the condition is that the final ratio Cli/Clq of a mini-universe must be larger than Wq — Wq, where w<j is the equation of state parameter.

1 Introduction

Recently direct evidences from the studies of Hubble diagram for Type la supernovae 1 indicate that our universe is expanding with an increasing rate, i.e., accelerating expansion. Accelerating expansion implies that the deceler­ation parameter q — (fim + (3wq + l)Clq)/2 to be negative, where wq is given by the cosmic equation of state, pq = wqpq, of the quintessence Q. As long as wq < —1/3, the corresponding dark energy provides a negative contribution to q. The cosmological constant A has wq — — 1.

The acceleration may stop in the future, i.e., a transient phenomenon with a time-dependent wq, or may last forever, depending on the nature of the dark energy 2. A positive cosmological constant, consistent with the supernovae data at z = 1.7 3, leads to a forever acceleration. Quintessence scenario with a constant wq, not ruled out by data, can also lead to forever acceleration. There are profound implications for a forever accelerating universe 4 , s . The universe will exhibit an event horizon; that is, there exist regions of the uni­verse inaccessible to light probes. It has been argued that such a universe presents a challenge for string theories or any of its alternatives 4.

*B-MAIL: HEXG@PHYS.NTU.EDU.TW

156

157

Naively speaking, in a forever accelerating universe, two points at different spatial locations will, as measured at any one of the two points, eventually be pulled apart at the speed of light to approach the future horizon. Hence existing structures will be eternally frozen and the universe will approach a state of cold death. Here we show that a local region with strong enough slef-gravity of matter can in fact protect the region from being torn up. A large number of mini-universes, including our own Local Group, will become self-bound and survive the destruction of cosmic repulsive force due to the quintessence. They can naturally arise despite that these mini-universes will still be falling into each other's horizon at a sufficiently late time and become isolated island universes 6.

2 Cosmological Constant and Mini-Universe

We now study the conditions for bound object formation in the presence of cosmological constant.

We adopt a toy evolutionary model for the sutdy of bound object forma­tion which assumes the universe to consist of the background and an isolated cold dark matter fluctuation that grew from a small-amplitude since the early epoch. The localized perturbation possesses spherical symmetry and consists of two concentric solid spheres of different densities. The inner sphere has a radius r,- with a uniform dark-matter density pt, which is greater than the background matter density pm. The outer sphere has a radius rout and a uniform compensating under-density, such that the averaged matter density within rout equals pm. Any particle outside rout feels no extra gravity result­ing from the over-density and expands as it would have been in a homogeneous universe. The inner sphere can be regarded as a closed Friedmann universe (the mini-universe) which has a Hubble parameter, Hi, given by

*-(*)'-¥<«+<*>-*• (1)

where p\ = (l/8irG)A is a constant and Kj is the curvature of the mini-universe.

A test particle at Ti feels an effective potential V(ri),

GMi ATTG 2 V(ri) = 5-PA^i • (2)

T% o

Conservation of matter yields a constant mass Mi within Ti with Mi = 47rpi(rj)r?/3. The potential V(r) has a maximum at rmax determined by V'(rmax) = 0 with V{rmax) = -AitGpKr2

max and rmax = (p^/2pA))1/3r0,

158

where the density at present has been denoted as p° = pi(ro) with ro being the present radius of the over-dense sphere. When the test particle reaches r-max with a zero velocity, it will be marginally bound. The density pi corre­sponding to this situation is called the critical binding density, denoted as p\c

at present, and any local region with density larger than p°c will eventually turn around and collapse.

The curvature of the mini-universe can be conveniently obtained at the turn-around, after which gravitational collapse ensues, by setting r j 2 = 0 and equating —Kj/2 to the potential at the turn-around. That is,

Ki = ^ ( ^ n [ + 1)rL (3) •* PA "ta

where rta is the turn-around radius. In particular for a critically bound mini-universe, we have rta = rmax and Kic = Ki(rmax) = d>-KGpfj-^nax.

The time span needed for the mini-universe to evolve to the present epoch is given by

t = fr° _dri_= fro dri

* ~ Jo uHi ~ 70 riy/{SnG/3)(pi + p A ) - (/Ci/r2)

= 1 t dJL ( 4)

When U is known, Eq.(4) can be inverted to solve for the present local density p°. Thus one needs to obtain information about tj. To this end we note that the mini-universe should have started to grow from a small amplitude perturbation at the beginning of matter domination around z = 3 x 103. Therefore to a good approximation, the evolution time ij is equal to the background evolution time tB. The time ijg for the background Friedmann universe to reach the present ratio 0,^/Q,^ is

r00 da 1 f1 dy tH = I

Jo V^Gp-ZJo V ^ / " A ) + 2 / 2

= _ ^ = B i n h - 1 ( « / ^ ) . (5)

Setting ti —tB, one can determine the present density ratio p°/pA for a given

We consider three cases of interest for the discussions of the density ratio Pi/pA- i) the critical mini-universe, where the test particle reaches rmax with zero velocity and is therefore marginally bound; ii) the case where the over-density is larger than that of case i) and the test particle has a zero velocity

159

and turns around at present; iii) the over-density is even larger so that the turn around occurred at the time when the background matter density p%% equaled p\, a particular epoch prior to which the matter in-fall had been vigorous.

For case i), we have Ki = K,C = SirGpAr^ax- Given the background density parameters determined from various experimental data 7 £1^ = 0.35 and £lA = 0.65, Eq.(4) along with Eq.(5) yield the present critical density ratio P°JPA = 3.63 (Q.ic(r0) — 2.36) and r0 = rmax/1.22.

We note that over-dense regions with the present density parameter larger than the critical value fiiC(ro) are abundant in the universe. In fact nearly all future bound objects will never reach rmax with a turn-around radius rta

smaller than rmax. Inserting nt of Eq.(3) into Eq.(4), one obtains their present densities p«(ro) for a given ratio rta/ro. For case ii) the mini-universe that is presently turning around, we have rta — ro, and it is found that p°/pA = 5.90 with Q.ito = 3.83. Any over-dense region with a density greater than this value has already undergone collapse. For case iii), it is found that p^q / pA ~ 8.55 with £li<eq = 5.52.

Although the toy evolutionary model discussed may seem somewhat sim­plistic, the model nevertheless illustrates the essential physics 8 . From the above discussions, we clearly see that local bound objects can form against the cosmic repulsive force, as long as the local density parameter fij(ro) at present is greater than 2.36. This value is at least a factor of several less than the mean density parameter of the Local Group, which includes the Milky Way, the Andromeda galaxy and more than a dozen of smaller galaxies such as the Magellanic clouds. It suggests that the Local Group has defied the cosmic repulsive force and gravitational collapse is underway.

All the bound objects will virialize to form more compact objects after recollapse. The virialized kinetic energy of a test particle is given by, < T >= — < F • r > /2 . For the simplified model under consideration, < T > is given by < T > = (GMi/2rv) - {AnrGp\rl/'S), where rv is the virialized radius. This leads to

- - l = < T > + < ^ > = - ( — i + —-pArl). (6) 2 Zrv 6

From both energy and mass conservation together with the expression of fcj at the turn-around, eq.(3), with the subscripf'O" replaced by "ia", one finds 9

2

£ t * £ ( 2 - ! ^ ) + 2 ( l -2 -^ - ) = 0. (7) Ph rv rta

Regions with a space curvature Kj infinitesimally greater than Kic of case i) will finally collapse in the infinite future, and we shall also regard this

160

situation as case i) in a broad sense. It is straightforward to obtain that Tv/vta = 1/2.73 for case i), since pitta/pA = 2 and the corresponding rv/r0

is 1/2.24. The ratio of the virialized density to the A energy density is then found to be pv/p\ — 41. For cases ii), we find rv/r0 = 1/2.20 and pv/pA = 62. For case iii), rv/rta = 1/2.13 and pv/pA = 83 6 .

3 Quintessence and Mini-Universe

In this section we study the condition for bound object formation under the repulsive force of the quintessence with a constant wq. At present experimen­tal data allow for the possibility that the accelerating cosmic expansion be due to quintessence with a relatively wide range of constant wq(> —1) and energy densities 10.

In order to have an accelerating universe at present, the equation-of-state parameter wq should satisfy

»,<-(! + §)• W Unfortunately, the value of 0,^/0,° is still rather uncertain, and its plausible value deduced from data correlates with wq, with a larger f2^/fi° for a smaller wq in a flat universe. For example, when £2^ = 35% and 0° = 65% to make up the energy budget for a flat universe, eq.(8) implies wq < —0.5, but the existing data pushes the plausible wq to be near —1 10. Nevertheless, to have an idea as to how the results vary with wq, we shall consider three representative values, wq = —0.5, —0.75 and —1, in our later discussions.

The quintessence energy density varies with the scaling factor as pq = p(

q,(a/a0)~

3(1+wi\ The reference background cosmological time t& in this case is given by

ts = -rL= f , "" (9) ITTGO? JO ' ' '

dy

yjteGfi Jo y^VQO + y - ^

For the study of bound mini-universes, the situation becomes more com­plicated than that with a cosmological constant because the space curvature Ki within the collapse region is time-dependent, and eq.(4) is no longer valid. (The analysis presented in Ref. 11, which assumes a constant Ki, is hence in­valid.) This problem should be analyzed using the momentum component of the Einstein equation, which describes the force balance of inertia with gravity and quintessence force. It can be understood as follows. Since the self-gravity of quintessence is negligible, the repulsive force of quintessence at a radius r

161

varies with time as (a0/a(t))3(1+w"\ This is a time-dependent force, thereby leading to a time-dependent space curvature.

The momentum component of the Einstein equation is a second-order differential equation:

r 4nG - = — g - [ f t + (l + 3«;,)p,]- (10)

Defining x = (a/ao), y = (r/ro), one has

± = J-^-p0^(n°m/n°q)(i/x) + x-^

a = -^f-p°*yl§y~3 + a + a*;,)*-3*1^]. (ii)

Combining these two equations one arrives at

2y"x(l + x-3w") = y'(l + (1 + 3wq)x~!iw-)

- ( ^ + (l + 3wq)^x-3m"), (12) y x

where x = x(n°m/^Y/3a^ y = y(£l0m/n0

qY/3w<(Sl0

m/n?) and y' = dy/dx. We may solve eq.(12) to obtain the conditions for bound object formation

at any background cosmic time ts, c.f., eq.(9). However, due to the wide range of possible parameters for the quintessence cosmology, we shall instead concentrate on the critically bound mini-universe. The asymptotic state of the critical mini-universe can be found by letting both y —> oo and x —> oo in the infinite future. The existence of such a state also requires that the background expansion be accelerating, i.e., wq < —1/3. It then follows that eq.(12) has a scaling solution, y = (£licc/£lqooY^3x^1+Wq\ where

-— =wq- wq, (13) Mqoo

by matching the coefficients in eq.(12). The subscript "oo" refers to evaluation of a quantity at a reference epoch in the infinite future.

The space curvature Kj for this mini-universe defined as minus of twice the sum of kinetic energy and potential energy (of the force used in eq.(ll)) is given, in the asymptotic limit, by

«i = - ( — ) - ( 1 + ^ ( l + 3 ^ ) 4 7 ^ 0 0 7 ^ , , (14)

where eq.(13) has been used. This confirms the statement made earlier that with quintessence the curvature is time dependent.

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The space curvature decreases with time as a~^1+w''' but its magnitude cannot be determined from the scaling solution; it can be fixed only by in­tegrating eq.(12) to obtain the entire solution. In this asymptotic state, the absolute values of kinetic, potential and total energies all have the same time dependence with comparable magnitudes. For wq = —1, we recover the cos-mological constant case with pioo/pqoo = 2 and Kj given in eq.(3).

The above critical mini-universe is very different from that of the case with a cosmological constant, in that its size grows indefinitely since r grows as a^1+w,'\ However, within it there is no event horizon, a fact that can be seen from the expanding velocity f/ro which approaches zero as (1 + wq)(a/ao)~(l+wi\ Thus, the radius of the mini-universe can never grow faster than light travels.

One may further extrapolate the ratio pioo/pqoo back to the present by integrating eq.(12) numerically. The results are shown in Fig. 1. In Fig. 1 we show log(fij/Qq) as a function of log(Og/fim) for wq — —0.5, —0.75 and —1. From Fig.l, one can read off the densities of the critically bound mini-universe on the vertical axis for various background densities on the horizontal axis. (The horizontal axis can be regarded as another way for expressing the background cosmic time, c.f., eq.(9).) Regions whose present local densities exceed by a finite amount above a given curve shown in Fig. 1 will turn around within a finite time and become virialized in a manner similar to that discussed in the previous section, though the details differ 12.

However, it is interesting to note that there are differences for the viri­alized objects between the general quintessence cosmology and the ACDM cosmology. For the case with quintessence, a local region with an energy in-finitesimally smaller than the critically-bound condition (local matter density larger than the critical density by an infinitesimal amount), the region will undergo collapse in the infinite future. The bound object has a nearly zero binding energy, as can be seen from eq. (14), and hence after virialization its virial radius rv discussed in Sec.(2) will be infinitely large, a great contrast to the wq = —1 case where the just bound objects have finite sizes.

As has been mentioned earlier in this section, the precise value of £1^/0° is yet to be determined and the plausible value of fi^/fi° in fact anti-correlates with the value of wq. Though for wq = - 1 the most plausible $1^/0,° — 0.35/0.65, for wq = —0.75 and —0.5 the most plausible £2^/f}° decreases to approximately 0.1/0.9 and (RS)0/1, respectively 10. Precise knowledge for the content of energy forms, which can be acquired from more accurate calibration of Type la supernovae and measurements CMB radiation as well as other observations 10>13, are crucial for making good use of Fig.l.

163

{

\ \

\ \ \ \ Y\\ \v*. \ \ \ ' s

W--0.5 w=.-0.75 w—1

'-_ --.

v

- 4 - 2 0 2 4 6 iog(iyn,.>

Figure 1. The barely bound mini-universe critical density as a function of time. The vertical axis is log(Qi/Clq) and the horizontal axis is log(Clq/Clm).

4 Conclusions

In this work, we have quantitatively shown that mini-universes can form against the repulsive force of cosmological constant or quintessence with a constant equation-of-state parameter wq, as long as they have sufficiently high densities at present. In the case with cosmological constant the condition of forming mini-universe is that the ratio of present density parameters 0°/fiA ought to be larger than a critical value 3.63. In the case with quintessence, the final ratio £li/Slq of a mini-universe is found to be always larger than Wq~wq. The corresponding densities at present time for a given wq and given values of fim,g can be read off from Fig. 1.

The detailed properties of the mini-universes, depend on whether the accelerating expansion is due to the cosmological constant or quintessence. New experimental data from various observations in the coming decades are expected to reveal more information. By then we will have a better under-

164

standing of the properties of mini-universes. If the acceleration of our universe at present is indeed due to cosmological constant or quintessence with con­stant wq, our universe will enter a new era, the era where all island universes are falling onto each other's horizon and appear to fade away. Within each of which most physics remains the same as is today 14.

I thank Professer Tzihong Chiueh for collaboration on the work presented here. This work is supported in part by National Science Council under grants NSC 89-2112-M-002-058 and NSC 89-2112-M-002-065, and in part by the Ministry of Education Academic Excellence Project 89-N-FA01-1-4-3.

References

1. S. Perlmutter et al., Nature 391, 51(1998); Astrophys. J. 517, 565(1999); A.G. Riess et al., Astron. J. 116, 1009(1998).

2. J. Barrow, R. Bean and J. Magueijo, Mon. Not. Astron. Soc. 316, L41-4(2000); A. Albrecht and C. Skordis, Phys. Rev. Lett, bf 84, 2076(2000); S.M. Carroll, Phys. Rev. Lett. 81 , 3067(1998); E. Halyo e-print hep-ph/0105216; J. Cline, e-print hep-ph/0105251.

3. A.G. Riess et al., e-print astro-ph/0104455. 4. S. Hellerman, N. Kaloper and L. Susskind, e-print hep-th/0104180; W.

Fischler et al., e-print hep-th/0104181. 5. X.-G. He, e-print astro-ph/0105005; J. Moffat, e-print hep-th/0105017;

J.-A. Gu and W.-Y, Hwang, e-print astro-ph/0106387. 6. Tzihong Chiueh and Xiao-Gang He, e-print astro-ph/0107453. 7. P. de Bernardis et al., Boomerang Coll., Nature 404, 955(2000); S.

Hanany et al., Maxima Coll., Astrophys. J. 545, L1-L4 (2000). 8. J.A. Peacock, "Cosmological Physics", Cambridge Univ. Press (1999);

A.R. Liddle and D. Lyth, "Cosmological Inflation and Large-Scale Struc­tures", Cambridge Univ. Press (2000).

9. O. Lahav, P.B. Lilje, J.R. Primack and M.J. Rees, Mon. Not. Roy. Astro. Soc. 251, 128 (1991).

10. See for example M. Signore and D. Puy, e-print astro-ph/0108515; P. Binetray, e-print hep-ph/0005037.

11. E. Lokas and Y. Hoffman, e-print astro-ph/0108283. 12. L. Wang and P. Steinhardt, e-print astro-ph/9804015, Astrophys. J. 508,

483(1998). 13. P. Nugent, SNAP collaboration, in San Juan 2000, Particle Physics and

Cosmology, pp263 (2000). 14. F. Adams and G. Laughlin, Rev. Mod. Phys. 69, 337(1997).

STABILITY OF THE ANISOTROPIC B R A N E COSMOLOGY

W. F. KAO Institute of Physics, Chiao Tung University, Hsin Chu, Taiwan

email :wfgore@cc. nctu. edu. tui

The stability of the Bianchi type I anisotropic brane cosmology will be analyzed in this talk. Analysis is presented for the model with a perfect fluid. It is shown that the anisotropic expansion is smeared out dynamically in the large time limit. It is shown that the initial state is highly isotropic for the brane universe except for a very particular case. Moreover, it is also shown that the Bianchi type I anisotropic cosmology is stable against any anisotropic perturbation for the brane model in the large time limit. This talk is based on a paper in collaboration with Chiang-Mei Chen l

Introduction The observation of the cosmic microwave background (CMB) radiation 2 '3 indicates that our Universe is globally isotropic to a very high degree of precision. Therefore, our Universe is usually assumed to be described by the Friedmann-Robertson-Walker (FRW) metric in most of the literatures. The origin of the isotropic universe has also become an interesting research topic ever since.

On the other hand, it is known that there is a small large-angle CMB anisotropics, AT/T ~ 10~5, under the CMB background 2 '3. The isotropy of our Universe may have to do with the choice of initial conditions and the stability of the evolutionary solutions. Therefore, the present isotropic phase could be a dynamic result of the evolution of our Universe, no matter what the initial state started out. In this talk, we will address on this issue in both the standard Einstein's theory and the newly developed brane world scenario 4>5. In particular, we will consider the evolution of an anisotropic cosmology described by the Bianchi type I models 6-13 for with a perfect fluid as the matter source. Our result shows that the anisotropic model considered evolves dynamically from the anisotropic Bianchi type I universe into the isotropic FRW space in the large time limit. This property reveals that the isotropy of the cosmological principle may be justified and made consistent with our current observational data.

For the conventional Einstein's theory (CET), the anisotropic expansion tends to be large in the very early stage. In another words, the universe has to begin from a highly anisotropic initial expansion and then decays to zero as the time increases. On the other hand, in the brane world scenario 14-17

! due to the quadratic correction which significantly changes the early time behavior of the Universe. As a result, any initially non-vanishing anisotropy parameter

165

166

tends to vanish in the very early period. There is a characteristic time, tc, that divides the evolution of mean anisotropy parameter into two different stages. The mean anisotropy parameter is increasing when t < tc and reaches its maximal value at t = tc. After that, mean anisotropy parameter starts to decay. This result remains true for both the model with a perfect fluid and the model with a scalar field. And this appears to be a general feature independent of the types of matter considered. We will also consider the stability of this solution 18-27.

The Brane World The brane world scenario assumes that our Uni­verse is a four-dimensional space-time, a 3-brane, embedded in the 5D bulk space-time. All the matter fields and the gauge fields except the graviton are confined on the the 3-brane as a prior requirement in order to avoid any violations with the empirical results. Moreover, inspired by the string theory/M-theory, the ^-symmetry with respect to the brane is imposed 28. A formal realization of the brane world scenario, which recovers the Newton gravity in the linear theory, is the Randall-Sundrum model 4 , s in which the 4D flat brane(s) is embedded in the 5D anti-de Sitter (AdS) space-time. Later on, a covariant formulation of the effective gravitational field equations on the 3-brane has been obtained via a geometric approach by Shiromizu, Maeda and Sasaki 14>16. It is shown that the effective four-dimensional gravitational field equations on the brane take the the following form

where G^ and T^„ are the Einstein and energy-momentum tensors. «.?pi/ i s a

quadratic contribution of T^ defined as

S^v = ~^2TTv-v ~ 4TMaT"a + -^9iu> (3Ta/3TQ/3 - T 2 ) . (2)

The effective 4D parameters, e.g., the cosmological constant A and gravi­tational coupling &4, are determined by the 5D bulk cosmological constant A5, the 5D gravitational coupling £5 and the tension of the brane A via the following relations

Here g^ is the metric tensor on the brane. In addition, the quantity E^ is a pure bulk effect defined by the bulk Weyl tensor 14.

iFrom the Eq. (1), it is easy to realize that the brane world scenario is different from the CET by two parts: (a) the matter fields contribute local "quadratic" energy-momentum correction via the tensor <SM„, and (b) there are "nonlocal" effects from the free field in the bulk, transmitted via the

167

projection of the bulk Weyl tensor Eav. Therefore, the CET can be treated as a limit of the brane theory by taking Eav = 0 and k^ —» 0 with properly adjusted values of the constants k± and A.

In addition to the generalized Einstein equations (1), the energy-momentum tensor also satisfies the conservation law V /2T' i l / = 0. Therefore, there is a constraint on the tensor E^, V^-E^" = /c|VM5M,/, due to the Bianchi identity on the brane. Here, the operator V is the covariant derivative with re­spect to the metric gal/ on the brane. One should point out here that the field equations on the brane, namely the generalized Einstein equations, the con­servation of energy-momentum and the constraint on E^ are, in general, not a closed system in the 4D brane since the quantity E^ is five-dimensional. It can only be evaluated by solving the field equations in the bulk. In this paper, we will, however, only consider the quadratic effect on the brane world in the anisotropic background. Therefore, we will set E^ = 0 which is equivalent to embedding the 3-brane in the pure AdS bulk space-time.

Bianchi type I metric The line element of the Bianchi type I space, an anisotropic generalization of the fiat FRW geometry, can be written as

ds2 = -dt2 + a((t)dx2 + a?,(t)dy2 + al(t)dz2, (4)

with a,i(t), i = 1,2,3 the expansion factors on each different spatial directions. For later convenience, we will introduce the following variables V = Yli=i ai> as the volume scale factor; H = ^-, for i — 1,2, 3 as the directional Hubble

factors; and H = | J2i=i Hi = W> a s t n e m e a n Hubble factor. In addition, we will also introduce two basic physical observational quantities in cosmology: A = X)i=i 3^2 as the mean anisotropy parameter, and q = ^{H"1) — 1 as the deceleration parameter.

Note that A ~ 0 for an isotropic expansion. Moreover, the sign of the deceleration parameter indicates how the Universe expands. Indeed, a positive sign corresponds to "standard" decelerating models whereas a negative sign indicates an accelerating expansion.

In this section, we will consider the case that the matter energy-momentum tensor, T^, is a perfect fluid whose components are given by

T° 0 = - p , T\=T22=T3

3=p. (5)

Here the energy density p and the pressure p of the cosmological fluid obey a linear barotropic equation of state of the form p = (7 — \)p with 7 a constant in the range 1 < 7 < 2.

The gravitational field equations and the conservation law on the brane

168

take the form, 3 3 7 - 2 , 2 3 7 - l

ZH + £ f l ? = A - ^L-t klP - ± L ^ kip', (6)

±±{VHi) = A-1-^klp-1-^-kip\ i = l ,2 ,3 , (7)

p + 3~/Hp = 0. (8)

The general solution of the above system was obtained in the exact para­metric form in 15. We are able to present all variables, including time, in terms of volume scale factor, V, with V > 0. For instance, the time variable can be expressed as

t-to = JFivy^dV, (9)

where F(V) is denned as

F(V) = 3AF2 + 3klPoV2-> + ifclpgV2"27 + C, (10)

with po and C the constants of integration. The other variables are

Ki fv^Fivy^'dv

A = 3K2F(V)~\ (13)

- 1, (14)

Oi — a0iV1/3exp

37(6fc|poV2-^ + kjplV2-^) + 12C

1 = 1,2,3, (12)

4F(V)

where aoi, i^j, i = 1, 2, 3 are constants of integration and K2 = ^ i = i ^t2- I n

addition, the arbitrary integration constants Ki and C must satisfy the same consistency condition K2 = 2C/3.

One can immediately show that the effect of the energy density quadratic term becomes significant at the high energy epoch, or in another words, at the early stages of the Universe by looking at the Eq. (10). Indeed, F(V) oc V2~2ry

at the limit t —• 0 when V is extremely small. As a result, the F diverges as t —• 0. Therefore, the mean anisotropy parameter A —* 0 at the early universe. On the other hand, in the large time limit, the properties of the Universe should be more or less the same as the case for the CET by looking at the same Eq. (10). Hence the early evolution of the anisotropic Bianchi type I brane Universe is dramatically changed due to the presence of the brane

169

correction terms proportional to the square of the energy density. The time variation of the mean anisotropy parameter of the Bianchi type I space-time is presented, for different values of 7, in Fig.l. From the Fig.l, it is clear that at high energy density the evolution of the brane Universe always starts out from an isotropic state with A —> 0. The mean anisotropy parameter increases and reaches a maximum value after a finite time interval tc. One can show that, when t > tc, the mean anisotropy parameter is a monotonically decreasing function approaching zero in the large time limit. This behavior is in sharp contrast to the usual evolution in the CET, as shown in Fig.2, in which the Universe has to kick off from a state of maximum anisotropy due to the constraint from the field equation.

In addition, the early time evolution of the brane universe is normally not in an inflationary phase. On the other hand, the brane Universe always ends up in an inflationary phase in the large time limit in the presence of a nonvanishing cosmological constant. These are generic features of the brane Universe due to the constraint of the field equations on the brane cosmology. Indeed, a more detailed information can be extracted from the Eq. (9) in the limit t —» 0, or equivalently, the case with a small V. For simplicity, one will take C = 0 again. Indeed, we can show that V oc i 1 ' 7 as t —* 0. Hence, the expansion of the early universe is of the form of power law expansion. In addition, in the early stages of evolution of the brane Universe the mean anisotropy parameter varies as A oc i 2 - 2 / 7 approaching zero as t ~ 0. Moreover, the deceleration parameter is given by q = 37 — 1 which is always positive for all possible values of 7 for the case C •= 0.

Stability Analysis The general perturbations for the FRW background with perfect fluid can be found in Ref. 17. The same consideration is, however, more complicate for the anisotropic background. For the primary effect, we will only consider the scalar mode and neglect the vector and tensor modes 19,20 -pn e m e t r i c perturbation considered here is

O-i —• O-Bi + ?>ai = a B i ( l + $h), (15)

and the perturbations with respect to the perfect fluid considered in this paper is

p -> PB + Sp, p->PB + (l~ l)fy>- (16)

Here the variables with subscript B are the exact solutions presented in the previous sections. For technical convenience, we will use variables Sbi instead of 6ai in our analysis.

The perturbation equations for various quantities, can be obtained by sub­stituting the perturbations (15, 16) into the field equations (6, 7, 8). Leading

170

Figure 1. Mean anisotropy parameter of the Bianchi type I brane Universe with a per­fect fluid: 7 = 2 (solid curve), 7 = 1.5 (dotted curve) and 7 = 1 (dashed curve). The normalization of the parameters is chosen as 3A = 1,3fc|po = 2, k5p0 = 4, and C = 1.

Figure 2. Mean anisotropy parameter of the Bianchi type I conventional Einstein's theory with a perfect fluid: 7 = 2 (solid curve), 7 = 1.5 (dotted curve) and 7 = 1 (dashed curve). The normalization of the parameters is chosen as 3A = 1,3fc|/3o = 2, and C = 1.

terms will reproduce the field equations. Therefore, one has the following perturbation equations from the first order terms 0(5bi, Sp),

J2Sbi + 21£HBiSbi = - ^ 2'-2 kl6p-?l—LktpB5p, (17)

171

Sh + ^8bi + HBi J2 Sbj = -1—^L k\ 6p - 1—^ k%PB 6p, Vi, (18)

3

6p + 3jHB 6p + j^26bipB = 0. (19) t = i

In order to solve the above system of differential equations, we need an in­spiration from the process of constructing the exact solutions. Their dynamics on the perturbation variables follow the constraints

3

6p=-7PBJ26bi> ( 2 ° ) 3 3

j=l B j=l

By summing the Eqs. (18) and then using the result (20), we end up with a

second order differential equation for a variable of the combination X)«=i &»

3 3 1 3

J2 6k + 6HB ^ Sbt - -1PB [3(7 - 2)k2A + ( 7 - 1)A*pB] £ Sbt = 0. (22)

The task is to solve ^ <56j from the above equation and then to construct Sbi and Sp from the Eqs. (21) and (20). For the purpose of the stability analysis in the final stage, it is sufficient to consider the large time limit behaviors of the perturbation variables Sbi and 5p. From the discussion on the asymptotic behavior of the exact solutions, qualitative outcome in both the CET and the brane theory, we should divide our analysis into two different cases in the presence of absence of the cosmological constant.

For the case with a positive cosmological constant, one can extract the asymptotic forms of the background variables from the exact solutions which give

HB -» v/A/3, VB OC exp(V3At), pB ex exp(-V3A-yt). (23)

The asymptotic expression of pB indicates that the third term in Eq. (22) can be neglected in the large time limit. As a result, we have the following equation for the asymptotic 5Z i=1 Sbi which remains valid in both the CET and the brane world

3 3

52 6h + vi2A J2 Sbi = °- (24) i=l 4=1

172

This in term leads to the final result 3

Y, Sbi oc exp(-VT2At). (25) i=l

Therefore, from Eqs. (20, 21) we can obtain the asymptotic expressions of the following perturbation variables

6bi oc exp(-v /12Ai), 8p oc exp[->/3A(7 + 2)t]. (26)

This indicates that that the background solutions are stable. Similarly, for the case with A = 0, we can also show that this model is

stable against the metric perturbations. Indeed, the asymptotic behavior of perturbation variables Sbi and Sp can be shown to be

6bi(xta, 6p<xta-2, (27)

which shows that the background solutions are always stable even when the cosmological constant is vanishing.

Conclusion We have discussed the anisotropic property of cosmological models in the brane theory. Comparison between the models with and without brane corrections indicates that the initial expansions behave very different. A realistic model, being consistent with the current observations, should produce a small value of anisotropic parameter at the later stage of the evolution of our Universe near the last scattering surface. By assuming the Bianchi type I space-time for the evolution of our Universe, we found that the final state of the evolving Universe always approaches the phase of isotropic expansion in both theories.

These two different theories give completely different initial anisotropy at the very early stage of evolution. Indeed, for the CET, the anisotropy tends to be large in the very early stage. In another words, the universe tends to begin from a highly anisotropic initial state. The mean anisotropy parameter A will then decay to zero as the time increases. On the other hand, the early time behavior of the Universe in the brane world scenario changes significantly due to the quadratic correction on the brane. As a result, any non-vanishing mean anisotropy parameter, A(t), tends to vanish in the very early period. There is a characteristic time, tc, that divides the evolution of A(t) into two different stages. The mean anisotropy parameter is increasing when t < tc

and reaches its maximal value at t = tc. After that, A(t) starts to decay. This kind of behavior is clearly shown in the Fig.l and Fig.2. This result remains true for both the model with a perfect fluid and the model with a scalar field. And this appears to be a general feature independent of the types of matter considered.

173

It is worth noting that the only exception is the model with p = 0 (i.e. 7 = 1) of the perfect fluid model. The mean anisotropy of this model behaves similar to the models in the CET where mean anisotropy parameter is large in the very early time. Moreover, we also analyzed the stability problem for those exactly solved anisotropic models shown in this paper. The result indicates that all of the solution known to us are stable in the large time limit. Therefore, the evolution of the Universe in the CET starts with highly initial anisotropic expansion. The dynamic of the system will take the Universe to the phase of isotropic expansion in the large time limit. We also show that the final isotropic expansion will remain stable in the large time limit. In addition, the mean anisotropy parameter will keep decreasing as time increases. The model provided here is a useful and explicit model that is capable of providing us with a Universe that has a tiny anisotropy left over near the last scattering surface.

Acknowledgments This work is supported in part by the National Sci­ence Council under the grant numbers NSC90-2112-M009-021.

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DARK ENERGY, PRIMORDIAL MAGNETIC FIELDS, A N D TIME-VARYING FINE-STRUCTURE CONSTANT

KIN-WANG NG

Institute of Physics, Academia Sinica, Taipei, Taiwan E-mail: nkwQphys.sinica.edu.tw

Evidences indicate that the dark energy constitutes about two thirds of the critical density of the universe. We discuss the cosmological implications for the dark energy being an evolving scalar field that couples to electromagnetism.

1 Introduction

Recent astrophysical and cosmological observations such as dynamical mass, Type la supernovae, gravitational lensing, and cosmic microwave background (CMB) anisotropics, concordantly prevail a spatially flat universe containing a mixture of matter and a dominant smooth component with effective nega­tive pressure l . The simplest possibility for this component is a cosmological constant. A dynamical variation calls for the existence of dark energy whose equation of state approaches that of the cosmological constant at recent epochs. Many possibilities have been proposed to explain for the dark energy. Most of the dark energy models involve the dynamical evolution of classical scalar fields or quintessence (Q). So far, many different kinds of scalar field potentials have been proposed. They include pseudo Nambu-Goldstone boson, inverse power law, exponential, tracking oscillating, and others2 .

Although the physical state of the dark energy can be probed through its gravitational effects on the cosmological evolution, it is important in funda­mental physics to understand whether the quintessence is a nearly massless, slowing rolling scalar field. As long as an ultralight <j> field couples to photon where for a slow-roll condition, the mass m^ is comparable to the Hubble con­stant H0, it is conceivable to have very long-wavelength electromagnetic fields generated via spinodal instabilities from the dynamics of 0 as a possible source of seed magnetic fields for the galactic dynamo 3. Furthermore, the coupling may lead to an effective time-varying fine structure constant. Here we will investigate the implication of the electromagnetic coupling of the evolving 4> field to the origin of the primordial magnetic field (PMF) and the time-varying fine structure constant.

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2 Primordial Magnetic Fields

As we know, the issue of the origin of the observed cluster and galactic magnetic fields of about a few p,G 4 remains a puzzle 5. These magnetic fields could have been resulted from the amplification of a seed field of Bseed ~ 10_23G on a comoving scale larger than Mpc via the so-called galactic dynamo effect. A number of scenarios have been proposed for generating seed fields in the early universe, mainly relying on non-equilibrium conditions such as inflation, the electroweak and the QCD phase transitions. After the phase transitions, the universe became a highly conducting plasma so that the magnetic flux which existed is frozen in, and the ratio of the magnetic energy density and the thermal background, PB/PJ, remains constant thereafter. The required Bseea translates into ps — 10_34/97. However, it turns out that the generated fields in these models are too small, except in somewhat contrived cases, to be of cosmological interest. In an equilibrium condition, a large damping term induced by the high plasma conductivity suppresses significantly any electromagnetic field fluctuations6'7.

3 Time-varying Fine Structure Constant

The idea of time-changing fundamental physical constants such as gauge cou­pling constants, the gravitational constant, and the speed of light has a long history8. Recently, it was claimed that the results of a search for time variabil­ity of the fine structure constant, a, using absorption systems in the spectra of distant quasars yield a smaller a in the past and the optical sample shows a 4-CT deviation: Aa/a = -0.72±0.18 x 10~5 over the redshift range 0.5 < z < 3.59. An analysis of the isotropic abundances in the Oklo natural uranium fission re­actor, active about 1.8 x 109 years ago (corresponding to a redshift of z ~ 0.1), suggests Aa/a = -0 .4 ± 1.4 x 10 - 8 10. The most recent CMB data are con­sistent with a being smaller by a few percent at the time of recombination (z ~ 1000) in a ACDM model11, but the bounds imposed on the variation of a can be significantly ralaxed in models with a change in the equation of state of the dark energy12. Many models have been constructed to accommodate a changing a. Bekenstein formulated a dynamical model which consists of a massless scalar field coupled to electromagnetism13. This model was recently generalized by allowing an additional coupling of the scalar field to the super-symmetric dark matter1 4 . Interestingly, it was proposed that this scalar field can be just the quintessence 1 5 , i e , but so far this idea has not been further pursuited.

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4 Q-photon Coupling in Expanding Universe

Since the scalar potential V{<j>) of the Q field is scarcely known, it is conve­nient to discuss the evolution of <f> through the equation of state, p<p =w<pp()). Physically, 1 > w^ > —1, where the latter equality holds for a pure vacuum state.

Here we consider the ^-photon coupling,

L^ = -f<f> e^F^F^ + -f<f> g^g^F^F^, (1)

where FM„ = d^A,, — dvA^, / is a mass scale, and c and c are coupling constants which we treat as free parameters. For the present consideration, we pick / equal to the reduced Planck mass Mp = (87rG) -1/2. The most stringent limit on the 0-photon coupling c comes from the cooling via the Primakoff conversion of horizontal branch (HB) stars in globular clusters17, c/f < 1.5xlO_11GeV_1 , which gives c < 3.7 x 107. The fifth-force experiments limit c < 10~3 (see e.g., Olive and Pospelov14). We are thus led to study the cosmological evolution of the 0-photon system in a flat universe. The effective action is

S= SM + jd^Xy/g -J^-lg^g^F^F^

-lgf"/dli<t>dv<t>-V(4>) + ~L<h , (2)

where the signature is (—h + + ) and SM is the classical action for matter. We assume that the universe today has matter f i ^ = 0.3 and quintessence fl^ = 0.7, and define an ^-weighted average1

( w 0 ) = / %{ii)w4>{'n)dri x ( / n^(ri)dr]) , (3)

where rjo and TJU are respectively the conformal time today and at the last scattering, defined byr] = Hof dto_ 1(t) with the scale factor a and the Hubble constant HQ. Assuming a spatially homogeneous cj> field, the evolution of the cosmic background is governed by

^ = -3aMl + «v)p«, (4)

dh 3 , o 1 /c\ - = --ah --awm, (5)

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where h = H/H0, p^ = ptj>/(MpHo)2, and we have used 4>2 — (1 + w<t>}p<j> and V{<j>) = (1 — w^,)p<i>/2. We have numerically solved the background equations by proposing a simple square-wave form for w^, as shown in Fig. 1. In order to satisfy the existing observational constraints on w<p, we have chosen w^ = — 1 for z < 2, and a width of the square-wave such that (w^) ~ -0 .7 3. Note that at the present time fi<j, = /0^/(3/i2) = 0.7 and H0t = 0.95. We have also plotted dS/drf = a i / ( l -I- w^p^, where 6 = (j>/Mv.

5 Genera t ion of P M F

From the action (2) with c = 0, the comoving magnetic field in the comoving coordinates (r, x) with r = 7}/HQ satisfies

^ - V x ( v x B ) + 4 c ? V x B , (6) CLT

where V = d/dx, a is the plasma conductivity, and v = dx/dr is the peculiar plasma velocity. After hydrogen recombination, the residual ionization keeps the conductivity high, a/H ~ 102 2(T/eV) - 3 /2 > 1. As a result, the a term under the assumption that v = 0 damps out any growth of the B field on scales above ~ A.U. 6. But this assumption may not hold when the universe has entered the non-linear regime. It could be understood from the recent study by Kulsrud et al.18 where the battery mechanism has been proposed as a source for generating a small initial magnetic field during the large-scale structure (LSS) formation. There they showed that magnetic fields are built up in regions about to collapse into galaxies where velocity flows grow rapidly after extreme nonlinearities develop in the cosmic fluid. As such, the a term would be significantly reduced and the high plasma conductivity is no longer a hindrance to the growth of large-scale magnetic fields. Here we will show that the temporal variation of <f> can also be a generating source for large-scale magnetic fields. Here we will simply omit the a term and solve for the photon equation self-consistently.

Now we write B = V x A T and decompose the transverse field A T ( T , X) into Fourier modes,

• /

d3k

%/2(27r)3fc

(7) e i k - x $ > A k ^ A k ( T ) e A k + h . c . A=±

where 6±k are destruction operators, and e±k are circular polarization unit vectors. Then, defining q = k/H0, the mode equations are

d V±q+(q2T4cqf-)V±q = 0, (8)

dr)2 V dr]

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with initial conditions at early epoch given by

r*-1- i s } 1 " * (9)

Hence, the comoving energy density of the magnetic field is given by pB = (B 2) /8TT = Jdq(dpB/dq) with

dpB T0H$ 2 ^ 2

where To is the current temperature of the CMB whose energy density is Pl = TT2T0

4/15.

We have numerically solved the mode equations (8) using c = 130 and the background solution as shown in Fig. 1, and plotted the ratio (dpB/dq)/p-, in Fig. 2. Although photons are being produced as the scalar field starts rolling at z ~ 60, we have counted the photons produced only after z = 10 when the universe has presumably entered the non-linear regime. The result shows that a sufficiently large seed magnetic field of 10 Mpc scale has been produced before z ~ 4. Moreover, we notice that when c ~ 130 the spinoidal instability19

gives rise to magnetic seed fields of the right magnitude and length scale. The self-consistency condition to reduce significantly this high conductivity effect can be justified in the sense that for the exponentially growing modes that we are interested in, dB/dr ~ amaxHGB ~ (10 - 1 6s~1)B with the coupling c ~ 100 and dB/dr] ~ 1, which is of the same order of the twisting term |V x (v x B) | ~ BV• v during the LSS formation obtained from the numerical simulations18.

Although the value of c = 130 is well below the HB limit, undoubtedly it is much larger than the theoretical expectation. However, as suggested by Carroll15, an unsuppressed (^photon coupling may arise in higher dimensional theories. One possible way to reduce the value of the coupling and at the same time to produce the sufficiently large PMF on large correlation length scales during the LSS formation is to combine the mechanism we proposed here with the battery source proposed by Kulsrud et al.18. Also, note that the growth rate is controlled by the exponent a ~ 2c(dd/drj). As such, it may be possible to reduce the value of c by having a large d9/drj. In the case of scalar quintessence, we have seen in Fig. 1 that dO/dr) < 0.8 for z < 10. We cannot further increase dB/dr] since w^ has already reached the maximum value. Perhaps, in the fc-essence models20, in which the kinetic term is modified to K(<j))<j)2/2 and we have ft — (1 + w^p^/Kicf)), we may make dd/dr] much larger than one by tuning the prefactor K(<j>). This possibility is under investigation.

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Figure 1: The quantities dO/dq, w^, Hot, and fi^ as a function of redshift. Note that dO/dr) is drawn 4 times smaller.

Figure 2: Ratios of the spectral magnetic energy density to the present CMB energy density at various redshifts. The present wavelength of the magnetic field is given by 2-w/(qHo).

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6 Variation of Fine Structure Constant

From the action (2) with c = 0, we have readily the change of a from the present time to a time tfi,

Aa mo = -107TC

a

We have used the dO/dt] in Fig. 1 and c = 10 - 3 to calculate the change of a at the recombination time, Aa/a ~ —10~3. Given the equation of state w^ in Fig. 1, A a / a ~ - 1 0 - 3 at the time of the big bang nucleosynthesis. Although it is fine-tuned, we found that it is easy to modify the equation of state w^ at low redshifts to make the change of a consistent with the observations of quasar absorption spectra and the Oklo limits. Note that we have assumed no interaction between the quintessence and the dark matter. It is interesting to combine the present consideration with the supersymmetric generalization of the Bekenstein model14.

7 Conclusions

In conclusion, we have made an effort to link the dark-energy problem to a solution to another important problem in cosmology, namely, the generation of primordial magnetic fields. The high conductivity effect due to residual ion­ization after hydrogen recombination can be argued to be significantly reduced as a result of the existence of the cosmic flow with nonlinear, twisting peculiar velocity to avoid a hindrance to the growth of the magnetic fields. We have shown that the nonlinear instability that drives the rapid growth of magnetic fields is of spinodal instability where the long-wavelength modes about the Mpc scales evolve as being the inverted harmonic oscillators and the ampli­tude fluctuations begin to grow up to a time at which the scalar field velocity approaches zero at redshift z ~ 4. We have also given a simple quintessence-photon model that does not stick to a particular quintessential potential but satisfies all observational constraints to explain the observational data on the time-variation of the fine structure constant.

Acknowledgments

This work was in collaboration with Da-Shin Lee and Wolung Lee, and was supported in part by the National Science Council of the R.O.C. under the grant number NSC90-2112-M-001-028.

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List of Participants

Athar Husain Bor-Luen Huang Chiang-Mei Chen Chih-Yuan Liu Chi-Wei Wang Chun-Yi Lee Chung-Hsien Chou Chung-Ting Li Feng-Yin Chang Francis Halzen George W. S. Hou Guey-Lin Lin Hsueh-wei Huang Hsungrow Chan Henry Wong I-chin Wang Je An Gu Jie-Jun Tseng Jin Li

Kwang-Chang Lai Kingman Cheung K.-W. Ng K. Olive M. Chiba Ming-Huey Huang Pei-Ming Ho Ping-Hung Kuo Qian Yue

NCTS DeptofPhys,NTHU

DeptofPhys,NTU Inst. Of Astronomy, NTHU

DeptofPhys,NTU Electro-optical Eng, NCTU

InstofPhys,AS Electro-optical Eng, NCTU

Inst of Phys, NCTU Univ of Wisconsin

DeptofPhys,NTU Inst of Phys, NCTU

Dept of Phys, NTHU Ping-Tung Normal College

Inst of Phys, AS Inst of Phys, NCKU Dept of Phys, NTU Inst of Phys, NCTU

Inst of Phys, AS Dept of Phys, NTU

NCTS Inst of Phys, AS

Univ of Minnesota TMU, Japan

Inst of Phys, AS Dept of Phys, NTU

Inst of Astronomy, NCU Inst of Phys, AS

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Ru-feng Liu Shiau-jing Wu Shih-Hung Chen Shih-Yuin Lin Sine-Nan Chiue Shu-Wei Lai Tzu-Chen Peng Venktesh Singh W. F. Kao W.-Y. Pauchy Hwang Wei-Chien Lai Wen-long Lin Wolung Lee Xiao-Gang He Xianwen Wang Y.-H. Chang Ya-Wen Tang Yu-chieh Chung Yu-Chung Chen

DeptofPhys,NCKU DeptofPhys,NCKU DeptofPhys,NTHU

InstofPhys,AS Electro-optical Eng, NCTU

Inst of Astronomy, NTHU DeptofPhys,NTHU

InstofPhys,AS DeptofPhys,NCTU

InstofPhys,NTU DeptofPhys,NCKU DeptofPhys,NTNU

InstofPhys,AS Dept of Phys, NTU

InstofPhys,AS Inst of Phys, NCU

Inst of Astronomy, NTHU Dept of Phys, NCKU

Dept of Phys, NTU