Gravitational waves: Generation and Detection

Joseph BayleySupervisor: Prof. Stewart Boogert

November 6, 2015

Abstract

After the first indirect observation of gravitational waves from the Hulse-Taylor binarysystem in 1975, many groups have set out to try and directly detect them. These groups usea range of methods of detection, each of these detectors have advanced substantially sincethe initial work on the experimental detection of gravitational waves, carried out by JosephWeber. One of the leading methods in the detection of gravitational waves is interferometry;the most recent upgrades to the laser interferometry gravitational observatory (LIGO) pro-vides a high probability of detection. The interferometric experiments operate in the highfrequency range where the coalescence of binary neutron stars generate gravitational waves,the most recent upgrades are expected to be able to detect up to 40 of these coalescencesper year. Other experiments such as pulsar timing arrays focus on a lower frequency range,where they hope to detect sources of the gravitational wave background from supermassiveblack holes. Each of the experiments sensitivity is determined by its signal to noise ratio,therefore a lot of the recent developments in the search for gravitational waves have beenin the reduction of noise in the detectors. This review focusses on the generation and de-tection on gravitational waves, particularly on the interferometry experiments methods ofnoise reduction and results.

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Contents

1 Introduction 31.1 Gravitational Waves and their properties . . . . . . . . . . . . . . . . . . . . . . . 3

2 Sources of Gravitational Waves 72.1 Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Binary mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Core collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Chirp signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Continuous signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Stochastic background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Cosmological gravitational wave background . . . . . . . . . . . . . . . . 112.4.2 Astrophysical gravitational wave background . . . . . . . . . . . . . . . . 12

3 Detectors 123.1 Observations of the cosmic microwave background . . . . . . . . . . . . . . . . . 133.2 Pulsar Timing Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Resonant Mass Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.1 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.2 Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.3 Seismic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Future searches for GW 29

5 Conclusion 31

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1 Introduction

Gravitational waves (GW) were first predicted by Einstein as a consequence of his theory ofgeneral relativity and were theorised to transfer energy in the form of gravitational radiation[1]. Since this prediction many groups have attempted to detect GW. Despite the fact that todate no signals have been detected, the recent upgrades to the laser interferometry gravitationalobservatory (LIGO), and advances in other detectors provide a promising outlook for their de-tection. Other than for the discovery itself, there are many motivations for finding GWs. Dueto gravity being such a weak force, the GW are not absorbed by clouds of interstellar dustlike electromagnetic radiation (EMR), this allows an unobstructed view of the universe. Othermotivations for finding GW include the fact that they are emitted by some of the most ener-getic events in the universe. These events include the formation of black holes from compactbinary coalescences and core collapse supernova (CCSN). GW from these sources would allowthe study of black hole progenitors and their formation. As well as the observational benefitsthat GW offer, they also provide a way to test general relativity itself.

After the prediction of GW from Einstein, it was not until 1975 that physical evidence ofGW were found [2]. The first indirect observation of GW was from the famous Hulse-Taylorbinary system, or pulsar B1913+16, which was the first binary pulsar system to be discovered[2]. This binary orbit was studied for 30 years and the radius of its orbit was found to decreasewith time, this means the system must be losing energy and can be accounted for somewhere.The loss in energy from the orbit matched incredibly well with the prediction from generalrelativity, therefore the loss in energy is assumed to be from the emission of GW. Figure 1 fromRef. [2] shows how well the prediction matched with the observation. The result from the studyof the Hulse-Taylor binary system gave a promising outlook for the existence of gravitationalwaves and led to the construction of many of the experiments that exist today.

Although GW have not been directly detected, many groups and experiments are currentlylooking to do this. These experiments have to be incredibly sensitive; for example if two stellarmass black holes were close to coalescence at a distance of 1 Mpc, the LIGO experiment willhave to detect a strain of ∼ 10−13 m across its 4 km arms [3]. There are a number of differentmethods that are used to try to detect the displacement caused by a GW. The first experimentwas a resonant mass bar designed by Joseph Weber [4], this uses a solid aluminium bar witha particular resonant frequency. If a passing GW is of a similar frequency then it will causethe bar to resonate, this resonance can then be measured. Although many of the bars havebeen decommissioned there are more recent resonant mass experiments such as MINIGrail[5] which use spherical masses for detection. However the resonant masses have a limitedsensitivity, therefore a different type of experiment was developed using interferometry. Thelargest interferometric experiment is the laser interferometry gravitational observatory (LIGO)[6], this is expected to be the first to directly detect GW, especially after its most recent upgrade.Other more indirect methods include pulsar timing arrays (PTAs), which exploit the regularityof millisecond pulsars [7], and experiments that look at patterns within the cosmic microwavebackground (CMB). Future detectors that are planned include the laser interferometry spaceantenna (LISA) [8], this uses a similar technique to LIGO however operates over a distance of5×106 km [8] in space. The future detectors therefore hope to be able to reduce their sensitivityto a level that allows for the direct detection of a GW.

1.1 Gravitational Waves and their properties

In Einstein’s theory of general relativity, gravity is caused by distortions in spacetime, whichare created by a mass or energy. The larger the mass of an object is, the more it will distort

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General Relativity prediction

Cum

ula

tive

shift

of p

eria

stro

n t

ime

/s

Year

Figure 1: This plot shows the shift in periastron time, which is the change in time of when the stars arecloset together, plotted against the time it was observed. It shows how the orbit of pulsar B1913+16decayed over the course of 30 years and how well it matched the general relativity prediction [2].

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spacetime. Figure. 2 shows how spacetime is distorted by three different sized masses. Infor-

Figure 2: This is an illustration of how three different masses distort spacetime. The smallest mass onthe left to the largest on the right [9].

mation about the curvature of spacetime that these masses create is contained in a parametercalled the metric tensor, gµν . The relationship between matter and the curvature of spacetimeis contained within Einstein’s field equation, this is written as

Rµν −1

2Rgµν + Λgµν =

8πG

c4Tµν , (1)

where Rµν is the Ricci curvature tensor, R is the scalar curvature, gµν is the metric tensor, Λis the cosmological constant, G is the gravitational constant, c is the speed of light and Tµν isthe stress-energy tensor. By expanding Einstein’s equations around flat space time ηµν , it ispossible to find the wave equation for gravitational waves. For flat space-time

ηµν =

−1 0 0 00 1 0 00 0 1 00 0 0 1

, (2)

where ηµν is the Minkowski metric tensor in the coordinate system of (t, x, y, z). By perturbingthe metric tensor gµν by hµν , it can be written as

gµν = ηµν + hµν . (3)

The expansion of the equations of motion to linear order hµν is then called linearised theory [1].The linearised equations of motion can be written more simply by defining

hµν = hµν +1

2ηµνh, (4)

where h = ηµνhµν [1]. By adopting the Lorentz gauge in which

∂ν hµν = 0, (5)

the linearisation of Einstein’s equation can be written as a set of wave equations

2hµν = −16πG

c4Tµν , (6)

where 2 = −(1/c2)∂2t +52 [1].This shows that the gravitational radiation takes energy away from the source in the form of

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a wave, distorting space time as it travels. Equation 6 shows the result of generating a GW inlinearised theory, however to see how a GW interacts with test masses, its behaviour outside ofthe source needs to be found. In Eq. 6, the stress-energy tensor Tµν is equal to zero outside ofthe source[1], therefore Eq. 6 becomes

2hµν = 52hµν −1

c2hµν = 0. (7)

Any wave in three dimensional cartesian coordinates satisfies the equation(∂2u

∂x2+∂2u

∂y2+∂2u

∂z2

)− ∂2u

∂t2= 52u− 1

v2∂2u

∂t2= 2u = 0, (8)

where v is the velocity of the wave and u(t, x, y, z) is the wavefunction [10]. By comparingthis to the gravitational wave equation Eq. 7, it is clear that gravitational waves propagate atthe speed of light c. To solve Eq. 7 another gauge is adopted called the transverse-traceless(TT) gauge. In this gauge only the spatial components are non zero as hµ0 = 0, and they aretransverse to the direction of propagation [1]. Additionally the spatial components are tracefree, hii = 0, and divergence free, ∂ihij = 0. These conditions lead to

hµν = hµν = hTTµν . (9)

where hTTµν is the transverse-traceless gauge metric. This then leads to a solution to Eq. 7 for

waves travelling in the z direction. The solution is

hµν =

0 0 0 00 hxx hxy 00 hyx hyy 00 0 0 0

ei(kz−ωt), (10)

where k is the wavenumber and ω = kc. Due to the symmetric nature of hµν , there are twopossible polarisations of the wave, either a plus h+, or cross h× polarisation [11]. These are

h+ = hxx = −hyy, (11)

andh× = hxy = hyx. (12)

The amplitude of a GW h, can usually be interpreted as the physical strain in space and canbe written as a combination of the plus as cross polarisations of the GW, where

h = A+h+ +A×h×; (13)

A+ and A× are the amplitudes of the plus and cross polarisations of the GW [12]. GW affectmatter by a tidal force that ’stretches’ and ’squeezes’ spacetime. An illustration of a GW actingon a ring of test masses can be seen in Fig. 3, where the GW is travelling along the z axis.When acting on test masses the amplitude h, from Eq. 13, can be written as

h = 2δl

l, (14)

where δl is the change in separation of two masses with a separation of l [12]. Figure 3 shows aGW with a large strain amplitude, h > 0.1, however the strains that are expected to be detectedare closer to the order of h = 10−16.

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h+

hx

Time

y

x

y

x

Figure 3: This illustration shows how gravitational waves affect matter by causing a strain tidally. TheGW is moving from the left to the right in time and is propagating in the z direction. It distortsspace-time in two different ways depending if it is plus h+ or cross h× polarised.

Sources in a circular orbit, such as inspiralling neutron stars, will have a GW with a constantamplitude. However, their plane of polarisation changes twice every orbit, therefore their strainis also observed as shown in Fig. 3. If there is any ellipticity in the orbit then the amplitude of thewave will also vary with time, Einstein’s quadrupole formula describes this varying amplitude.The quadrupole formula states that the wave amplitude hµν is proportional to the second timederivative of the quadrupole moment,

hµν =2

r

G

c4QTTµν (t− r

c), (15)

where G is the gravitational constant, c is the speed of light, QTTµν is the quadrupole moment

and t − rc is the retarded time [1]. The retarded time is the time at which the field began to

propagate from a point to an observer. Equation. 15 shows how GW follow an inverse squarelaw similar to electromagnetic radiation (EMR), this means that the energy density of a GWfalls as 1/r2 and its amplitude reduces as 1/r.

2 Sources of Gravitational Waves

There are a range of sources of GW, due to gravity being such an extremely weak force anydetectable GW must originate from high mass or energy systems. The sources can be splitinto roughly four types; two short lived signals, which are bursts and chirps, and two longlived signals, which are continuous signals and the stochastic background. The shortest signalsare bursts, these come from systems such as supernova and compact binary mergers. Chirpsare slightly longer signals which originate from the inspiral stage compact binary coalescences(CBC); CBCs are a mixture of neutron stars (NS) and black holes (BH) in binary orbits.Continuous signals are constant signals which can come from certain types of rapidly rotat-ing neutron stars (pulsar). Finally stochastic GWs are the background of gravitational waves

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(GWB), these can come from a range of sources, however are thought to be primarily fromsupermassive BH mergers. Each of these systems operate in different frequency ranges whichsuit different detectors, these are outlined in Table 1 below.

Table 1: This table shows the four frequency bands of GW, the sources that fall into each band and thedetectors that are designed to work in each band. M is the mass of the sun [13].

Band Typical sources Detectors

Extremely lowfrequency

10−18 − 10−15 HzPrimordial stochastic background

Gravitationalwave signaturesin the CMB

Very lowfrequency 1 nHz -

1 mHz

Supermassive black hole binaries(M ∼ 109M); Stochastic background

(supermassive black hole binaries)

Pulsar timing ar-rays

Low frequency 1mHz - 1 Hz

Supermassive black holebinaries(M ∼ 103M − 109 M);

Extreme mass ratio inspirals; Dwarf/white dwarf binaries; Stochastic

background (dwarf binaries, cosmicstrings)

Space based inter-ferometers

High frequency 1Hz - 10 kHz

Neutron star/ black holebinaries(M ∼ 1− 103M); Supernovae;

Pulsars; X-ray binaries; Stochasticbackground (cosmic strings, binary

mergers)

Ground basedinterferometers;Resonant massdetectors

2.1 Bursts

Burst signals are very short signals that are thought to originate from the same sources asgamma ray bursts (GRBs). The GRBs can be split into two groups classified by length andspectral hardness; short hard GRB last for . 2 s, and long soft GRB last for & 2 s [14]. Shorthard GRB are thought to come from the merging of binary star systems containing a mixtureof neutron stars and black holes and long soft GRBs are though to come from the collapseof massive stars with fast rotating cores. These signals are thought to be dominant in thehigher GW frequency range, > 500 Hz [15]. The following section will use the information fromRef. [14], Ref. [15] and Ref. [16].

2.1.1 Binary mergers

Short hard GRB are thought to come from the merger stage of compact binary coalescences(CBC), this stage is also thought to be an emitter of GW. CBC can be split into three stages,the inspiral stage which will be covered in more detail in Sec. 2.2, the merger stage which is whenthe two compact objects collide and join together, and the ring down stage which is caused bydeformations in the formed BH [16]. The merging of two compact objects leads to a BH with anaccretion disk. The accretion disk forms as a significant fraction of the stellar material retainsenough angular momentum so that it does not immediately cross the BH horizon. If dynamicalinstabilities develop in either the rotating core or disk, then the core or disk could potentiallyradiate GW. The instability or deformation in the core or disk can simply be considered as a

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bar. The strain, h, caused by this bar can be written as

h =

√32

45

G

c4mr2ω2

d, (16)

where G is the gravitational constant, m is the mass of the bar which has a length l = 2r, ωis the angular frequency and d is the distance of the source from the detector [16]. After themerging of the compact objects, the initially created black hole is deformed. A deformed blackhole should emit radiation in the form of a GW until it settles to a Kerr geometry, this is calledthe ring down stage of the binary coalescence [16].

2.1.2 Core collapse

Long soft GRBs are thought to come from the core collapse supernova (CCSN) of rapidlyrotating massive stars of mass between 8−100 M; rotational instabilities in the central enginethat drives the GRB are thought to be a source of GWs. The high rotation rate of the core isrequired to form a disk that is around the central black hole, this is what powers the GRB jet.This high rotation rate may also cause the development of bar or fragmentation instabilities inthe collapsing core or disk [16]. The asymmetrically infalling matter from the disk to the BHperturbs the BHs geometry, this causes a ring down phase of the GW emission. The emission ofGW from collapsing massive stars can then be estimated in a similar way to the merger and ringdown stages of binary coalescences. For more information on GW from core collapse supernovasee Ref. [14] and Ref. [16].

2.2 Chirp signals

Chirp signals are short signals of which the frequency and amplitude increase with time. Ra-diation is typically emitted for ∼ 17 minutes at ∼ 10 Hz; for the last 2 s radiation is emittedat ∼ 100 Hz, and for the last few ms radiation is emitted at ∼ 1 kHz [1]. The source of thesechirp signals are the inspiral stage of CBCs mentioned in Sec. 2.1. There are three types ofbinary coalescing systems which will be detectable, these are NS/NS, NS/BH and BH/BH bi-naries [17]. BH/BH binaries have the strongest signal therefore can be detected to the largestdistance, however they are found to be the least likely to be detected as their event rate is∼ 1/250 that of NS/NS systems, this is summarised in Table 2 [19]. The NS/NS and NS/BH(hereafter NS-NS/BH) systems can form in two different ways, as primordial or dynamical bi-naries. Primordial systems must initially have two stars large enough to undergo collapse toa NS or BH, however when the star explodes it must not destroy its companion star, or shedtoo much mass so that it is no longer in a bound orbit [17]. Dynamical binaries form slightlydifferently, these will happen within the cores of globular clusters. These systems form when acompact object (NS or BH) captures a non-degenerate star, this binary system then interactswith another compact object, ejecting the non-degenerate star and leaving a compact binarysystem [17].

The amplitude of a circularly polarised wave, h0(t), from a binary system can be found fromEq. 15, the mathematics is covered in more detail in Ref. [1] and Ref. [18]. The amplitude canbe found to be

h0(t) =1

r

[5G5M5

2c11

] 14 1

τ14

, (17)

where r is the distance from the source to the detector, G is the gravitational constant, M is themass of each component of the binary, which is the same in this case and τ = tcoal − t, where tis the observer time and tcoal is the coalescence time [1]. Equation 17 shows how the amplitude

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of the GW will increase when the mass of the system increases, and also as the time from thecoalescence decreases. Also this equation demonstrates the inverse square law of gravitationalradiation, showing that the amplitude of the wave falls of as 1/r.

There are a few different models which aim to predict the coalescence rate of compactbinaries, in the simplest models they are thought to be proportional to the rate of star birthwithin the galaxy [19]. This can be found by observing the amount of blue light coming from thegalaxy, however this method does not include the currently existing compact binaries. Anothermore accurate way to find the rate is by looking at gamma ray bursts. In Sec. 2.1 the emissionof short hard gamma ray bursts from merging CBC was discussed, this allows a way to measurethe coalescence rate of CBC. By assuming that these short hard bursts come from mainly NS-BH/NS, an estimate can be made on the event rate of CBCs. These even rates are shown inTable 2 below for all CBCs, they are measured by how many coalescence events happen perMilky Way like galaxy per Myr [19]. There are currently many uncertainties on the number orextragalactic binary coalescences, therefore in Table 2, low, realistic and high estimates havebeen made for the event rate. These binary systems operate across a range of frequencies, from

Table 2: Table showing the predicted event rates for all compact binaries within a Milky Way like galaxy.Rlow is the low estimate of the event rates, Rre is the realistic estimate and Rhigh is the high estimate.[19]

Type Rlow/Myr Rre/Myr Rhigh/Myr

NS/NS 1 100 1000NS/BH 0.5 3 100BH/BH 0.01 0.4 30

1 nHz− 10 kHz depending on which compact objects make up the binary; the larger the massof the system the lower the frequency of GW it will emit. Table 1 shows the frequencies thateach of the binary systems operate in. Chirp gravitational wave signals from CBC are likelyto be most easily detected by the LIGO antenna, this is because these systems emit GWs in afrequency range where the detectors sensitivity operates, which is 40− 1000 Hz [20].Other types of events that will emit chirp signals involve extreme mass ratio inspirals, this iswhen a smaller body inspirals into a much heavier body. The mass of the smaller body is usuallya compact object with a mass in the range m = 1− 102 M, and the larger body has a mass inthe range M = 105−107 M [21]. This give a mass ratio of µ = m/M ∼ 10−7−10−3. Extrememass ratio inspirals are long lasting signals that emit in a frequency range between 10−4−1 Hz,this is the operating range of space based interferometers such as LISA [21].

2.3 Continuous signals

Continuous signals are constant signals that last for many years, they are thought to originatefrom certain types of rapidly rotating neutron stars (pulsar). GWs cannot be produced by apulsar if it is spherically symmetric, therefore gravitational waves are only emitted if there is anasymmetry between the pulsar and its rotation axis. This can be due to either a strain in thesolid parts of the star or magnetic stresses, which is where the magnetic field of the pulsar isnot aligned with the rotation axis [22]. The asymmetry can be characterised by the ellipticityof the star. The ellipticity ε can be defined as the relation between two perpendicular momentsof inertia, I1,2 and the principle moment of inertia I3 [1], this is written as

ε =I1 − I2I3

, (18)

where I1,2,3 are the moments of inertia for the three axis if the ellipsoid. This can be simplifiedas approximately the relation between the radius of the star and the size of its deformation,

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this is written as

ε ∼ ∆r

r, (19)

where r is the radius of the star and ∆r is the size of the deformation [23]. These pulsars can belocated by observing them in the electromagnetic spectrum, specifically in radio, gamma andX-rays. The pulsars that are emitting GW can then be observed to have a ’spin down’, whereits rotational speed decreases, therefore it is losing energy. This loss in energy is thought to bedue to the emission of gravitational waves. From observations of pulsars spin down, limits havebeen placed on their ellipticity. The equatorial distortions have a maximum limit at ε = 10−5

[24], which is the largest amount of stress that a crust of the pulsar could support. This allows acorrelation between the gravitational wave frequency and the rotational frequency of the pulsarto be determined; the gravitational frequency is thought to be twice the rotation frequency [1].Therefore a pulsar that is non-axisymmetric emits a GW at a frequency of twice its rotationfrequency. The frequency space that pulsars operate in is the high frequency range between1 − 10 kHz, as Table 1 shows above. This means that ground based interferometers are bestsuited to detect these sources.

2.4 Stochastic background

The stochastic background is a background of gravitational waves within the universe whichcomes from a range of sources. The gravitational wave background (GWB) can be split into twotypes; the cosmological GWB, which involves processes from times near the beginning of theuniverse, and the astrophysical GWB, which comes from astrophysical sources such as compactbinary coalescences which are too distant to observe individually.

2.4.1 Cosmological gravitational wave background

The cosmological GWB is analogous to the cosmic microwave background (CMB), but forgravitational radiation [25]. The CMB originated from the time of last scattering which was∼ 380, 000 years after the big bang. The cosmological GWB, however, can probe to much earliertimes, near the Planck time of ∼ 10−43 s. This is due to the fact that gravity has a much weakercoupling, therefore decoupled with matter at a much earlier times [25]. Most cosmologicaltheories predict a stochastic background of GW, however the strengths vary substantially. Anumber of the models are shown in Fig. 4, such as the inflationary model and GW from cosmicstrings.

The strength of the gravitational wave background can be parametrised in terms of its energydensity parameter, Ωgw(f). This can be written as

Ωgw(f) =1

ρc

dρgw(f)

d ln(f), (20)

where ρgw is the spectral energy density of GW, f is the frequency and ρc is the critical densitywhere ρc = 3H2

0c2/8πG [27]. Inflation is the mechanism that solves many of the problems which

came with the discovery of the CMB, including the horizon, monopole and flatness problems.As well as solving these problems inflation explains primordial density perturbations in theuniverse as quantum fluctuations in the scalar field driving the exponential expansion [28]. Abackground of GWs is thought to have been generated from the quantisation of the gravitationalfield, coupled to the exponential expansion in inflation, for more information see Ref. [28]. TheseGW are analogous to the primordial density perturbations, however are tensor perturbationsrather than scalar ones. If the scale of inflation is large enough then the GW could producemeasurable effects within the CMB [28].

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Figure 4: This plot shows the normalised GWB density parameter ΩGW plotted against the GW fre-quency. The contours are shown for various models of the GWB which are labelled. Also shown is theinitial LIGO and VIRGO limits as well as the predicted limit for advanced detectors [26]. The BBH andBNS on the plot are the stochastic background caused by binary black holes and binary neutron starsrespectively.

2.4.2 Astrophysical gravitational wave background

Astrophysical GW are thought to come from many different sources that are too distant toobserve individually, this is because the signals become indistinguishable over the much largerdistances. The sources of the astrophysical background include the ones mentioned in Sec. 2,however a large contributor is thought to be supermassive black hole coalescences. This typeof stochastic background is investigated by the pulsar timing arrays and will be investigatedfurther in Sec. 3.2.

3 Detectors

There are a number of methods that have been developed to try to detect gravitational wavesdirectly. These can be split into four main types, CMB experiments, pulsar timing arrays(PTAs), resonant mass detectors and interferometers. Each of these operate in one of thefour main bands of the GW frequency spectrum, either the high, low, very low or ultra-lowfrequencies. Table 1 above shows which sources fall into each category and which detectors arebest suited for their detection. Current detectors do not operate in the low frequency range,therefore the future experiments that operate in this band are covered in Sec. 4, where the

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future spaced based interferometers are investigated. Before each of the detectors methods andresults are summarised it is worth mentioning the measured quantities which are equivalent forall the detectors. The output of any GW detector is characterised by a time series s(t), whichis comprised of the GW signal h(t), and a noise signal n(t) [3]. This time series can be writtenas

s(t) = F+(t, θ, φ, ψ)h+(t) + F×(t, θ, φ, ψ)h×(t) + n(t), (21)

where F+(t, θ, φ, ψ) and F×(t, θ, φ, ψ) are the antenna pattern functions that describe the sen-sitivity from the cross and plus polarisations from different directions, h+(t) and h×(t) arethe plus and cross polarisations of h(t) and n(t) is the noise signal in the detector [3]. Thetime series, s(t), is often represented in the frequency domain in terms of the power spectraldensity Sh(f) = s∗(t)s(t) , where s(t) is the Fourier transform of s(t). The time series is thencharacterised by a strain amplitude spectrum h(f), where

h(f) =√Sh(f). (22)

The noise amplitude spectrum is equivalently given by n(f) =√Sn(f), where Sn(f) = n∗(t)n(t),

and these both have dimensions of 1/√

Hz. Another quantity used to measure the sensitivity isthe characteristic strain, hc(f), which is a dimensionless quantity written as

hc(f) =√fh(f) =

√fSh(f), (23)

where f is the frequency [29]. This also has an equivalent characteristic strain for the noise.These quantities are then plotted against the frequency to show the sensitivity of the detectoracross a range of frequencies. The following sections will cover the methods used by eachexperiment to detect GW as well as some of the limits that have been placed on this strainamplitude.

3.1 Observations of the cosmic microwave background

The cosmic microwave background (CMB) can show traces of GW from different sources whichare outlined in Sec. 2.4.1. The experiments that focus on looking for these patterns in theCMB operate in the ultra low frequency band, and comprise mainly of radio telescopes. Someof the experiments currently in operation are the background imaging of cosmic extragalacticpolarisation (BICEP) [30], and the Keck array [30]. The aim of the experiments are to mapthe polarisations of the CMB, specifically the type called B-mode polarisations, which can be asignature for primordial GW.

Perturbations within the early universe caused the CMB to be polarised, this happenedwhen the photons scattered from free electrons. When a photon is scattered from an electron thephoton will be polarised perpendicular to the incident direction [31]. Therefore for polarisationto be seen, the photons must have a polarisation separated by 90 degrees, this is called aquadrupole anisotropy. Within the CMB there can be two kinds of polarisation, E-mode andB-mode [32]. These polarisations can be induced by a number of sources, from intergalacticmagnetic fields, CMB lensing, scalar perturbations of the second order, rotating dust but mostimportantly primordial GW [31]. The density perturbations generate only E-mode polarisationshowever GW have a component of E-mode and B-mode. This can be observed by looking atthe CMB and observing the polarisations.

The BICEP 2 experiment (upgrade from BICEP) was the first to claim direct observationsof the GWB, however these claims were refuted due to an incorrect map of cosmic dust. Oncethe updated version of this map from the Planck satellite was used, the the polarisations couldnot be distinguished from the patterns that the cosmic dust leaves [32]. However the BICEPteam still remain hopeful that they will find the GW pattern within the CMB.

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3.2 Pulsar Timing Arrays

As well as certain types of pulsar emitting their own source of GW, other types can be used todetect them. Pulsar timing arrays (PTAs) are collections of millisecond pulsars (MSPs) thatare evenly distributed across the sky, and are monitored by an array of single dish radio tele-scopes [33]. There are three main PTA groups that collaborate to form the international pulsartiming array (IPTA), these are the North American NanoHertz Observatory (NANOGrav), theEuropean pulsar timing array (EPTA) and the Parkes pulsar timing array (PPTA) located inAustralia [33]. MSPs are neutron stars that have low spin down rates and therefore rotationalstability. This means that they have regular pulse trains, i.e. their beamed radio emission isvery regular. What PTAs look for is an irregularity within the received signal from a MSP,this irregularity may have been caused by a GW. As a GW passes between the detector andMSP the space-time is distorted, this changes the apparent distance to the pulsar and affectsthe received frequency of rotation. The change in the information from the pulsar is known asits time of arrival (TOA). The radio groups can then observe this frequency change and analyseit to find information about the GW [34].

One of the sources that the PTA groups have focussed on is the GWB from supermassiveblack holes. They have not detected any GW however have placed limits of the amplitudes ofthe GWB. The data is initially collected from the network of radio telescopes outlined above,the data is then analysed by following a ’pipeline’, where a series of processes is applied to thedata to improve the signal to noise ratio (SNR), for more detail on the analysis see Ref. [34].The measurements of the GWB are characterised by either the GW energy density parameterΩGW, mentioned in Eq. 20, or the characteristic strain hc which is defined in [34] as

hc = A

(f

yr−1

)α, (24)

where A is the amplitude of the GW, f is the frequency of the gravitational wave and α isthe spectral index of the GWB. The results for the upper limits of the GWB are shown inFig 5, the two contours show the 1σ and 2σ levels which are the 95% and 65% confidence levelcontours respectively. For a spectral index of α = 2/3 which corresponds to a background fromsupermassive black hole binaries, the upper limit was found as hc(1yr) 6 6× 10−15 with a 95%confidence level [34]. The current PTAs aim to reach sensitivities of hc = 10−15 in the futurewhich is over five times as sensitive as the above sensitivity [34].

3.3 Resonant Mass Detectors

There are a number of resonant mass detectors located around the world which use solid, non-free masses as detectors, they look for patterns within the phonons of the solid mass. Thissection will use information from the Ref. [4] and Ref. [11]. Resonant mass (RM) detectors wereinitially proposed by Joseph Weber as a way to detect gravitational waves. His idea was to usethe fact that the atoms will try to follow the geodesic that the GW creates. The electrostaticforces that oppose the atoms movement create a measurable force within the mass, which causesan oscillation. A large part of the energy in this oscillation is coupled to the 1st longitudinalmode of the bar, this means that the oscillation is amplified [4]. Electromagnetic transducers areplaced at the ends of the bar and have magnetic resonance at same frequency as the bar, thesewill pick up the oscillations induced in the bar. The transducers amplitude will increase untilalmost all of the energy from the bar has been transferred to the transducer. The transducerthen turns it into an electrical signal, which is then pre-amplified by a low noise cryogenicamp. This signal is then recorded and analysed. The main problem encountered with resonantmass detectors is the size of the signal to noise ratio. For RM detectors, thermal noise is the

14

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Expected α for GWB from SMBHBs

van Haasteren et al. (2011)

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Joint GWB (α,hc) distribution

Expected α for GWB from SMBHBs

v Haasteren et al. (2011)

Figure 5: This plot shows the log scales of both the characteristic strain hc(yr−1) and the normalisedGW energy density parameter ΩGW(yr−1) potted against the GW spectral index. The contours showthe 1σ (95% confidence) and 2σ (65% confidence) limits that have been places on the GWB by theEPTA joint analysis by Haasteren et al. in [34]. The vertical line on the plot shows α = 2/3 which isthe spectral index for supermassive black hole binaries.

15

dominating source of noise. Weber decided to choose a material with a high quality factor, thisallowed the signal to be more easily distinguished from the background. The quality factor Qis defined by

Q =ω0m

b, (25)

where ω0 = 2πf0, f0 is the resonant frequency of the bar, m is the mass of each test massand b is the dissipation factor [4]. Webers first RM detector was a cylindrical bar made with1.2 tonnes of aluminium, it was 1.5m long and 61cm diameter. This was then suspendedin a vacuum on acoustic filters to try to dampen any seismic backgrounds [4]. Once Weberhad an array of detectors (i.e. > 1), he claimed to see coincidence events between them.However subsequent detectors that were more sensitive could not replicate any of his results,therefore they were assumed to be incorrect. Nevertheless his findings did spark an interest inthe experimental discovery for GW and led the way for many more GW antenna. The nextgeneration of RM detectors, including the groups at Stanford and Louisiana state university(LSU), started to develop cryogenic resonant mass detectors (CRM) [4]. CRM detectors arecooled to temperatures around 100 mK or lower with liquid helium, and have many advantagesto the previous room temperature devices [11]. The main advantage being that the thermalnoise is reduced as well as an increase in the quality factor of the material. This increases thesensitivity of the device, therefore increases the distance it can observe to.

The international gravitational event collaboration (IGEC) have two main sets of data calledIGEC1 and IGEC2, they use a network of five and four detectors respectively. The four in usefor IGEC2 are ALLEGRO, EXPLORER, NAUTILUS and AURIGA [35]. The strain sensitivityof the four detectors are compared in Fig. 6 below. Figure 6 shows how the RM detectors are

Figure 6: This plot shows the strain amplitude spectrum of the four detectors for IGEC2 as marked inthe legend [35].

16

a narrowband search of the sky, this is because their sensitivity has a bandwidth of ∼ 100 Hz.The IGEC can then only search for specific sources within a certain bandwidth, mainly binarycoalescences and supernova. The ICEG2 collaboration combined the four detectors in Fig. 6,this allowed checks to be made of the number of coincidence events between the operatingdetectors, however no significant events have been observed yet.

The latest resonant mass detectors such as MiniGRAIL [36] and Mario Schenberg [5] arespherical masses, which has the added benefit of having the same sensitivity in all directions,plus they will be more sensitive than its bar counterparts. These two detectors are ∼ 1.3 tonspheres made of CuAl alloy (6%) [5]. The detectors weigh 1400 kg and 1159 kg respectivelywith an operation temperature of 5 mK and resonant frequencies at 3 kHz. Figure 7 shows thestrain sensitivity of the MiniGRAIL experiment. The bandwidth of this detector is larger thanthe ICEG counterparts however it does not currently detect to a higher sensitivity due to onlyone transducer readout. Figure 7, however, shows what the current sensitivity is at 50 mK andthe future MINIgrail II experiments proposed sensitivity.

Figure 7: This plot shows the measured strain sensitivity or strain amplitude spectrum of the MiniGRAILexperiment alongside the predicted strain sensitivities of future detectors. The second from bottomdotted line (MiniGRAIL II) shows the sensitivity that is achievable with current technology, and thelowest dotted line shows the sensitivity of a quantum limited detector [37].

Resonant mass detectors operate in a similar frequency band to interferometers, however,despite being cheaper to build, operate at a lower sensitivity with a smaller bandwidth. Forthis reason most efforts are going into upgrading interferometry experiments, as they are morelikely to achieve detection.

17

3.4 Interferometers

Interferometers are the most promising experiments under-way to directly detect GW, thereforewill be investigated in more detail than other GW detectors. There are a number of GW de-tectors which focus on using interferometry to directly detect gravitational waves. The networkof GW interferometers include the laser interferometry gravitational observatory (LIGO) in theUSA, VIRGO in Italy, GEO 600 in Germany and TAMA 300 in Japan [18]. All these experi-ments collaborate with the LIGO scientific collaboration. All of these are based around a similarlaboratory experiment, the Michelson interferometer, however they have many more sophisti-cated components and operate on a much larger scale (∼ 4 km). The Michelson interferometerworks by taking a laser beam and splitting the beam down two perpendicular arms. This lightis then reflected from a mirror at the end of the arm back to where the beam split and the twobeams recombine; a detector can then read any interference between the two beams. Figure8 shows the setup of a basic interferometer. The phase of the light between each beam may

L1

L2Laser

Mirror 1

Mirror 2

Beam Splitter

Screen or detector

Figure 8: This image shows the basic setup of a Michelson interferometer. A laser source is aimed at abeam splitter which sends a beam down each arm. The beams are then reflected back and recombinedfor the detector to read any interference.

change due to a difference in length of the arms or some other effect, this can then be measuredas a phase shift φ. This phase shift affects whether the light constructively or destructively

18

interferes. For an area of constructive interference

Φ = 2πm, (26)

and for an area of destructive interference

Φ = (2m+ 1)π, (27)

where m = ±1,±2,±3, ... [38]. The total optical path difference (OPD) in the interferometer isthen OPD = (2 · L1− 2 · L2) = 2 ·∆L, where L1 and L2 are the length of the respective arms.The phase difference Φ over the two arms can then be written as

Φ =2π

λ2 ·∆L, (28)

where λ is the wavelength of the light and ∆L is the difference in length of the arms [38].This interference pattern can then be observed at the output of the Michelson interferometer.The design of a GW interferometer follows the same principle as the Michelson interferometermentioned above, where the lengths of the arms are thought to be distorted by a passing GW.A GW interferometer however does not look at the whole interference pattern but focusses ona dark fringe. Also a gravitational wave interferometer has many different components.

The LIGO antenna has undergone three major upgrades in its history, starting at initialLIGO, then improving its sensitivity to enhanced LIGO and most recently has begun opera-tion with advanced LIGO. Throughout these upgrades LIGO has housed 3 interferometers, 2 ofwhich have 4 km arms and one has 2 km arms [39]. These are located in Hanford, Washingtonand Livingston, Louisiana, with a separation in distance of 3002 km [40]. As the two locationsare isolated it allows a check for coincidence events, improving the accuracy of a GW detection.As well as this, the two isolated locations means that the antenna has directionality, thereforethe location of the source can be found.

Compared to the basic design the Michelson interferometer, the GW interferometers haveadditional components which allow it to measure to a much higher sensitivity. A diagram of themost recent upgrades for the S6 run can be seen in Fig. 9. Many of the components shown inFig. 9 are designed to reduce the amount of noise within the system, this is because the amountof noise within a detector determines its sensitivity. By increasing the sensitivity of the detectorit is possible for it to see to a greater distance. This is because the gravitational radiation followsthe inverse square law, therefore weaker signals from more distant sources would be detectable.Gravitational waves are measured by a strain over the interferometer arms, the biggest factorthat affects how accurately the antenna can measure this strain, is the noise strain amplitudethat clouds the GW signal. There are a range of sources of noise, Fig. 10 shows some of the noisesources and how they affect different frequency ranges. Figure 10 shows that the dominatingnoise sources at the LIGO, Hanford observatory is from seismic noise at the lower frequencies,thermal noise at the intermediate frequencies and the shot noise at higher frequencies. Belowis a summary of each of these noise sources and the systems that are in place to reduce them.

3.4.1 Shot noise

Shot noise is due to the fact that photons come quantised wavepackets, the uncertainty arisesfrom the statistical fluctuations in the number of photons [1]. Shot noise is dominant at highfrequencies > 100 Hz, as Fig. 10 shows. The photons in the laser follow Poisson statistics,

19

SRM

T=1.4%

ITM

ETM

InputMode

Cleaner

OutputMode

Cleaner

PRM

BS

4 km

T= 3%

Laser

PD

GW readout

ITM ETM125 W

5.2 kW 750 kW

CP

ERM

SR3

SR2

PR2

PR3 ERM

Figure 9: The layout and components of the Advanced LIGO detector, ETM: End test mass, ERM: Endreaction mass, ITM: Input test mass, CP: compensation plate, BS: Beam splitter, PRM: Power recyclingmirror, SRM: signal recycling miror, PD: photodetector. The power of the laser in each section is alsolabelled and can been seen to increase through each section [41].

therefore there is an error in the number of incident photons σN, and an error in the phase ofthe photons σφ. The error in the number of incident photons is

σN =√Nγ , (29)

where Nγ is the number of incident photons [46]. Heisenberg’s uncertainty principle states that

σNσφ = 1, (30)

where σφ is the error in the phase. Therefore by combining Eq. 29 and 30 one can find theuncertainty in the phase is

σφ =1√Nγ

. (31)

The power Pγ of a laser is related to the number of photons by

Pγ = NγEγ , (32)

where Eγ is the energy of a wavepacket [46]. This means that the error in the phase, i.e theshot noise, is

σφ ∝1√Pγ, (33)

therefore by increasing the power of the laser the shot noise is reduced. The laser that is usedis a neodymium-doped yttrium aluminium garnet (Nd:YAG) laser, with wavelength of 1064nm

20

Figure 10: This plot shows the primary noise contributors to the signal at LIGO (H1). The black lineat the top is the measured strain for the detector, the cyan curve is the root square sum of all thecontributors of noise. Each of the individual noise components are marked on the plot. The peaks inthe measured strain are well identified and have been marked as p - power line harmonic, s - suspensionwire vibrational mode and m - mirror test mass vibrational mode [44].

21

and a power of up to 180W [41]. It is then put through an input mode cleaner, this stabilisesthe beam in position and mode content as well as providing a high quality laser frequencyreference [41]. There are a few additional components that are installed in the interferometer toincrease the power and reduce shot noise, these are Fabry-Perot arms, power recycling mirrorsand signal recycling mirrors. Each of the arms of the interferometer are Fabry-Peirot cavitiesand are characterised by a parameter called finesse, LIGO operates at a finesse of 450 [41]. AFabry-Perot cavity consists of two highly reflecting mirrors, between these light is trapped andreflected multiple times. The finesse of a Fabry-Perot cavity is the ratio of the free spectralrange and the full width half maximum of its resonant peaks. The finesse can then be definedas

F =∆λ

δλ, (34)

where F is the finesse, ∆λ is the free spectral range, which is the frequency spacing of itsresonator modes [42] and δλ is the full width half maximum of the resonance [42]. A Fabry-Perot cavity with a high and low finesse is shown in Fig. 11. This high finesse of the cavityallows for a higher accuracy and sensitivity in the measurement.

Wavelength, λ0.0

0.2

0.4

0.6

0.8

1.0

Transmission,T

F=0.5F=2F=10

Figure 11: This image shows the transmission of the mirrors in a Fabry-Perot cavity, plotted against thewavelength for different finesse values [42]. This plot also shows the free spectral range ∆λ and the fullwidth half maximum of the resonances δλ.

The power recycling mirror is located between the laser and the Fabry-Perot MichelsonInterferometer (FPMI) as shown in Fig. 9. A power recycling mirror is a partially reflectivemirror that recycles ’waste’ light back into the interferometer, this increases the effective powerof the laser [47]. By increasing the power of the laser the shot noise is reduced as Eq. 33 shows.The shot noise is then reduced by a factor of

√Gpr, where Gpr is the gain of the power recycling

mirror. The power recycling gain is defined as

Gpr =

(tRM

1− rRMrFPMI

)2

, (35)

22

where rRM is the amplitude reflectivity of the power recycling mirror, rFPMI is the amplitudereflectivity of the FPMI and tRM is the time spent in the power recycling cavity [47]. A signalrecycling mirror is located at the dark port of the interferometer and forms a signal tuned cavity[48]. The signal recycling mirror can be tuned for different types of operation, either signalrecycling or broadband. For broadband operation the cavity is tuned to effectively reduce thefinesse of the arm cavities at the signal side bands, this increases the bandwidth. For signalrecycling the cavity is tuned to a specific frequency of the source that is being searched for, thisallows a narrow band search with a higher sensitivity [48].

3.4.2 Thermal noise

Thermal noise induces vibrations in both the suspensions of the test masses and the test massesthemselves, this noise is dominant in the intermediate frequencies between 50−250 Hz [43]. Fora GW to be detected the test masses need to be free masses, as they are connected through theearth they are not completely free, however by suspending the masses it means that they are freeto move in the plane of the interferometer, therefore can be treated as free masses. Within thesuspensions, thermal noise can affect it in three different ways: pendulum thermal fluctuations,which induce swinging in the suspensions, vertical thermal fluctuations, which cause verticalmovement in the suspensions, and ’violin modes’, which are fluctuations in the normal modesof the suspensions [1]. The suspension thermal noise has been reduced substantially due to theinput of silica fibres as opposed to the steel wires previously used to suspend the test masses.The test mass thermal noise, similarly, can be split into three categories: Brownian motionof the mirrors, thermo-elastic fluctuations and thermo-refractive fluctuations [1]. Brownianmotion of the mirrors is due to the fact that the mirrors have a temperature T , the atoms inthe mirror then have some kinetic energy causing movement within the mirror. Thermo-elasticfluctuations are due to the fact that in a finite volume, V , the temperature fluctuates with avariance

(δT )2 =kBT

2

ρCV V, (36)

where CV is the specific heat, kB is Boltzmann’s constant, T is the temperature and ρ is thedensity of the material [1]. This generates expansion in the mirror and its coatings, causingthermal noise. Finally, thermo-refractive fluctuations are due to the fact that the mirror coat-ings refractive index is temperature dependant, therefore fluctuations in temperature changethe refraction index of the mirrors, causing noise [1]. The thermo-refractive fluctuations canbe reduced as the noise scales inversely with the beam size. Therefore the size of the beamis made as large as practically possible so that the thermal noise is averaged over as much ofthe mirrors surface as possible [45]. For advanced LIGO the mirror has been upgraded to anultra-high purity fused silica mirror with dimensions of 34 cm diameter and 20 cm depth, fromthe 25 cm diameter and 10 cm thick fused silica mirrors of initial LIGO [45]. The beam size hasincreased from 3.7 cm and 4.3 cm (for the input and end mirrors respectively) to 6 cm for bothmirrors [43], this means that the thermo-refractive fluctuations are reduced. The mass of themirror has also increased from 11 kg to 40 kg, which reduces the amount of radiation pressurenoise on the mirror [45].

3.4.3 Seismic noise

Seismic noise is the noise generated by vibrations in the earths surface, and begins to dominateat lower frequencies < 100 Hz [49]. This can be seen in Fig 10, where the seismic noise is shownby the brown line at lower frequencies. The two main sources of seismic noise are microseismic

23

noise and anthropogenic noise. Microseismic noise is dominant between 0.1 − 0.5 Hz, and isdue to vibrations in the earth’s surface caused by waves of the ocean. Anthropogenic noise isdominant between 1− 10 Hz, this is generated primarily by human activity on the ground [50].These vibrations can prevent the length systems from holding the optics accurately enough, thisincreases the noise signal and reduces the probability of detecting a GW. Therefore this needsto be minimised for successful GW detection, for the Advanced LIGO system the masses areaimed to be held to a value of less than 10−14 m [51].

There are two main ways of dealing with seismic noise, passive and active. Passive isola-tion involves using damping materials which operate without any input, and active systemsuse actuators and sensors to actively counteract any vibrational movement. The initial LIGOexperiment used passive stacks to dampen the seismic noise, however to reach the design sensi-tivity needed for Advanced LIGO, a combination of passive and active systems are used. Thereare several stages to the isolation systems at LIGO, an external active hydraulic stage, twotypes of active internal seismic isolation stages and a passive quadruple pendulum that sus-pends the mirror [3]. The initial system is the hydraulic external pre-isolator (HEPI), shown inFig. 12, which is an active isolation platform located outside of the vacuum chamber. Withinthe vacuum chamber there are internal seismic isolators (ISI) located on the HEPI. There aretwo different systems that house the ISI, these are called horizontal access modules or (HAM orHAM-ISI), Fig. 13, and basic symmetric chambers (BSC or BSC-ISI), Fig. 14. The HAM-ISIisolate the auxiliary optics of the interferometer from ground motion and the BSC-ISI isolatethe core opitics [51]. The quadruple pendulums suspend the 40 kg test masses that are used inthe core optics for the interferometer [53].

The actuators for the HEPI system were chosen as they have a bandwidth of 20 Hz, andbetween 0− 10 Hz they operate with a noise of 10−10 m/

√Hz [50], this then provides suitable

isolation in this frequency range. The HEPI system has four sensor-actuator assemblies locatedon each of the four corner support piers. Each sensor-actuator consists of hydraulic actuators,position sensors, geophones and two offload springs. The two offload springs support the weightof the payload and were initially used to position it. The position sensors measure the relativedistance of the payload from the ground, and the geophones measure the payloads inertialvelocity. The hydraulic actuators are controlled by a central pump, this pumps a viscous fluidto the actuators. The pump controls a differential pressure between two terminals which in turncontrols two axially soft bellows, this can then be transferred to the payload [50].

Both the HAM and BSC ISI systems use a similar passive-active system shown in Fig. 13and Fig. 14 respectively, they both use sets of blades, flexure’s and sensor-actuators to isolateany vibrations. The main difference between the BSC and HAM is that the BSC has a twostage isolation system in place as opposesd to the 1 stage that HAM operates with. The bladesand flexure’s are sets of triangular steel blades, which as well as supporting the mass providepassive isolation. The HAM system can be split into two stages, stage 0 which is supportedby the HEPI system, and stage 1 which is supported with sets of blades and flexure’s by stage0, this is shown in Fig. 13. Both seismometers and geophones are used as sensors to detectany movement from the ground; non contact magnetic actuators use the information from thesensors to adjust to any movement, providing active isolation [51]. The BSC system used forthe core optics has another stage which is supported by stage 1, shown in Fig. 14, and followsa similar isolations system.

The quadruple pendulum consists of four layers of suspension which is shown in Fig. 15.The test mass is the bottom layer which is suspended by silica fibres from an intermediate testmass above. Above these there are three stages of cantilever blades which are suspended asshown in Fig. 15, these are made from maraging steel and provide vertical isolation to the test

24

Feedback instruments(Geophones and relative sensors)

Springs

Quiet hydraulic actuatorsBellowsCross BeamGround

GroundInstrument

Instruments and springssupport structure

Vacuum chamber

Payload attachment point Support

Tubes

Figure 12: This diagram shows the layout for the HEPI system, which consists of seismometers, geophonesand springs which are all labelled [51]

mass [53]. The combined isolation systems for advanced LIGO aim to bring the seismic noisecutoff level down from 40 Hz to 10 Hz [52].

3.4.4 Results

The LIGO experiment operates in the high frequency range which is between 40 − 1000 Hz[20], therefore it has the capability to detect many of the sources of gravitational waves. Thesesources include NS-NS/BH binaries, supernovas, pulsars and the stochastic background, asTable 1 shows. This wide range of sources demonstrates how versatile interferometers are asan instrument in the search for gravitational waves. Although GW have not yet been detected,many limits have been placed on their amplitudes. This section will focus on the results fromthe S5 and S6 data runs from enhanced LIGO; the results originate from Ref. [54], Ref. [55],Ref. [20] and Ref. [56].

The target sources for LIGO have different signal lengths with a varying knowledge of theirwaveform. The more that is known about the waveform of the source the easier it is to searchfor, this is because the signal can be compared to an already predicted model. To analyse data,it is first collected from the network of interferometric detectors, then analysis is completed indifferent ways depending on which source is being searched for. If one is searching for CBC thewaveform is well defined, therefore the detectors strain output can be compared to a theoreticalmodel of the waveforms, this technique is called matched filtering. Any match that occurswith a signal to noise ratio (SNR) above a certain threshold is then investigated further. Thebackground rate of coincidence events between two or more detectors is measured by a methodcalled time shifting. Time shifting is when the triggers from different detectors are shiftedin time relative to each other and the analysis is repeated, this will give a different rate ofcoincidence events which is known not to be a signal. By performing many of these time shiftsit is possible to gain a good estimate of the accidental coincidence events that will occur.

25

Flexure

BladeActuator

Geophone Stage 1 Stage 0

Figure 13: This diagram shows the layout for the HAM-ISI system inside the vacuum chamber. Thissystem will be located on the HEPI system in Fig. 12 [51].

Stage 0

Stage 2

Stage 1

Stage 0-1 blade and flexure

Stage 1-2 blade and flexure

Stage 0-1 vertical and horizontal actuators

Stage 1-2 vertical and horizontal actuators

Stage 2 vertical and horizontal geophones (GS13)

Stage 1 vertical and horizontal geophones (L4C)

Stage 1 seismometer

Figure 14: This diagram shows the layout for the BSC-ISI system that will house the core optics insidethe vacuum chamber. This system will be located on the HEPI system in Fig. 12 [51].

The maximum sensitivity of an interferometric detector is characterised by two quantities,the amount of time spent with two or more detectors in operation, and the distance to whicheach of the detectors observe to. The S6 data run for the LIGO detectors started on the 7th ofJuly 2009, and ended on the 20th of October 2010, each detector recorded approximately sevenmonths of data in this period. Each of the data runs are split up into smaller segments labelledA,B,C and D, these did not run continuously for the whole time however, as the instrumentalstability plays a large role. Each segment that runs continuously is called a science segment, thisis when the instruments are stable enough to be able to record data to a significant sensitivity.These data runs are usually ended when the noise is too great for the electronic control systemsto control the interferometer. Figure 16 is a histogram showing the length of time for eachscience run at each of the LIGO sites. The Livingston site can be seen to have a higher eventcount in the 1− 10 s segments, this is due to the lack of stability in the detector in early runs.The mean length of time for a segment at both detectors is approximately 1 hour.

The other factor that effects the maximum sensitivity of the detector is the distance thatthey can detect to, this is called the horizon distance. This horizon distance is different for each

26

Cantilever blades

Metal masses

Intermediate massHeavy glass

Silica fibres

Test mass Sapphire

Front View Side View

Figure 15: This diagram shows the four stages of the quadruple pendulum suspension system, includingthe cantilever blades and silica fibres. This image shows the front and side view of the system [53].

source as it depends on the amplitude of the emitted GW. For CBC the horizon distance as afunction of the total mass of the binary system is shown in Fig. 17. Figure 17 shows how themean inspiral horizon distance increases with the mass of the binary system, this is due to thefact that the higher mass systems will emit gravitational waves with a larger amplitude. Thisfollows from Eq. 17 in Sec. 2.2.

The overall sensitivity of a GW detector is characterised by a strain spectral sensitivityor a characteristic strain as mentioned in Sec. 3, this is determined by a combination of noisecomponents. The strain sensitivity of the LIGO detectors in the S6 data runs is shown in Fig. 18.The black line is the design sensitivity of the the detectors, therefore it can be seen that thesensitivity has surpassed the design at most frequencies. The exception to this is the lowerfrequency range, this is where the seismic noise dominates. The Livingston (L1) observatorycan be seen to have a higher sensitivity in the low frequency range, this is in part due tothe installation of the prototype version of the hydraulic isolation system (HEPI) covered inSec. 3.4.3. This should be reduced further with the most recent instalment of seismic isolationsystems [54].

From the S5 and S6 data runs, upper limits have been placed of the event rates of compactbinary coalescences. Table 3 shows the upper limit rates and realistic rates for CBC with a 90%confidence level, these are for a neutron star mass of 1.35 M and black hole mass of 1.5 M.These upper limit rates can be see in Fig. 19 compared to the astrophysically predicted ratesfor coalescence.

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Figure 16: This histogram shows the length of time spent in each science segment for the Hanford (H1)and Livingston (L1) observatories in their S6 data run [54].

Figure 17: This plot shows the mean horizon distance as a function of mass for the 4 km Hanford and4 km Livingston interferometers (S6 data run), and the 3 km VIRGO interferometer (V2 and 3 datarun) [55]. The error bars are one standard deviation above and below the mean

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Figure 18: This plot shows the strain spectral density against frequency for the S6 run data. L1 isthe Livingston observatory and H1 is the Hanford observatory. The black line is the design sensitivity,therefore the seismic noise can be seen to be a large problem at the lower frequencies [49].

Table 3: This table shows the upper limit rates and realistic rates for different types of CBC [20]. Theneutron stars in this have a mass of 1.35 M and black holes have a mass of 1.5 M, where M is onesolar mass.

System NS/NS NS/BH BH/BH

Component masses (M) 1.35/1.35 1.35/1.5 1.5/1.5

Upper Limit (Mpc−3yr−1) 1.3× 10−4 3.1× 10−5 6.4× 10−6

Realistic rates (Mpc−3yr−1) 1× 10−6 3× 10−8 5× 10−9

4 Future searches for GW

There are a number of GW detectors proposed for the future, most of these detectors useinterferometry as their method of detection. These experiments will be designed to detect toa much higher sensitivity, spanning a larger portion of the frequency range. The most recentdevelopments in the search for GW have been in the upgrade to Advanced LIGO, this detectorhas been mentioned throughout the review, however has not yet reached design sensitivity andmay not for some time. Once Advanced LIGO has reached its design sensitivity it is expectedto be the first to directly detect GW. The predicted number of compact binary coalescencesthat will be detected by Advanced LIGO can be summarised from Ref. [19]; for NS/NS, NS/BHand BH/BH coalescences the detection rates are predicted to be 40 yr−1 , 10 yr−1 and 20 yr−1

respectively [19]. These are large improvements on the 0.02, 0.004 and 0.007 respective yearlylimits from the initial LIGO experiments [19].

The laser interferometry space antenna (LISA), is the closest third generation detector tobegin operation after Advanced LIGO; the pathfinder for this mission is scheduled to be launchedin November 2015. The space based interferometer will consist of 3 separate test masses whichoperate over a distance of ∼ 106 km [8]. As the antenna operates over a large distance, sources

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BNS NSBH BBH

10 10

10 9

10 8

10 7

10 6

10 5

10 4

10 3

RateEstimates

Mpc

3 yr

1

Figure 19: This plot show the event rates of the three types of CBC [20]. The light grey regions showthe event rates predicted in Ref. [20] in the S5 data run, the dark grey regions show previous upper limitrates and the blue regions show the astrophysically predicted rates. The dotted line shows the realisticrates for CBC.

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in the low frequency range of 0.03 mHz − 0.1 Hz can be detected [8]. Also as the antennaoperates in space it is isolated from seismic noise, which is dominant in the lower frequencyrange. This allows observations of many events including the inspirals of massive black holesand extreme mass ratio inspirals.

The big bang observer (BBO) is another spaced based interferometry network that willoperate on a similar setup to LISA, however will operate over a distance of ∼ 50, 000 km.Although the BBO operates over a smaller distance than LISA, it is planned to have 4 separatetriangular interferometers which will orbit the sun [57], it is also designed to operate over afrequency band that spans parts of both LIGO and LISAs spectrum as Fig. 20 shows.

Other planned ground based interferometers are the Einstein telescope (ET) [58] and theKamioka gravitational wave detector (KAGRA) [59], the main advantages of these experimentsare that they use cryogenics to reduce the thermal noise within the optics and detector. All ofthe advanced detectors will reduce the noise in the detector by a significant amount and allowfor better measurements of GW to be made across a wider frequency range. A plot showingthe sensitivities and frequency ranges of most of these experiments can be seen in Fig. 20 [60].

Figure 20: This plot shows the characteristic strain of the labelled detectors which have been mentionedthroughout this paper, including the planned strain sensitivities of future detectors [60]. This plot alsoshows the frequency bands and strains of a number of sources that each experiment should be able todetect. IPTA is the international pulsar timing array, BBO is the big bang observer, LISA is the laserinterferometry gravitational observatory and LIGO and aLIGO are the initial and advanced versions ofthe laser interferometry gravitational observatory.

5 Conclusion

Ever since the first indirect observation of GWs from the Hulse-Taylor binary system, manygroups have set out to try to directly detect gravitational waves. Despite the fact that these

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groups have not yet succeeded in direct detection, they have placed many limits on the gravi-tational waves themselves.

Some of the most promising searches for GW lie in the interferometry experiments such asLIGO and VIRGO, these detectors operate at the highest sensitivity of all GW detectors over abroad frequency band 40−1000 Hz [20]. Within this frequency band lie some interesting sourcessuch as compact binary coalescences, the observation of these will allow detectors to probe intothe formation of black holes. Without the detection of a GW signal, the LIGO collaborationhave placed upper limits on the rates that binary systems coalesce, these upper limits werefound to be 1.3 × 10−4, 3.1 × 10−5 and 6.4 × 10−6 in units of Mpc−3yr−1 for NS/NS, NS/BHand BH/BH systems respectively.

Due to GW leaving such weak signals, one of the most important aspects in their search is thesignal to noise ratio of the detector. The main sources of noise in a ground based interferometricdetector can be split into three sections: seismic, shot and thermal noise. Systems summarisedin Sec. 3.4 have been introduced in the most recent upgrades to the interferometers to reducethis. By reducing this noise the signal to noise ratio increases, therefore the probability ofdetection of a GW increases.

Other experiments such as pulsar timing arrays have also placed limits on the stochasticbackground in lower frequency ranges, these experiments use the regularity pulsars to observedistortions in the space-time between the pulsar and the detector. They have managed to findthe maximum amplitude of the GWB from supermassive black holes to be hc(1yr) 6 6×10−15,with a 95% confidence level [34]. Future experiments, such as LISA, aim to be sensitive in thefrequency range of massive black holes coalescences, therefore hope to be able to place limits onthe rate of relatively nearby coalesces. In the coming years there are hopes for many more GWantenna to be in operation, such as the LISA experiment for which the pathfinder is scheduledto be launched in 2015. These will allow higher sensitivities across a large frequency bandin which many interesting sources lie. With the upgrades and planned experiments it seemsprobable that a direct detection will be made in the near future. The future therefore hopes tosee the experiments move from placing limits on GWs to become an established, operationalfield of astronomy.

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