grand unification and the big-bang cosmology

Forbchr . Phys . 31 (1983) 11. 569-590

Grand Unification and the Big-Bang Cosmology

VIRENDRA SINGH

Tata Institute of Fundamental Research. Homi Bhabha Road. Bomba.y 400005. India

Contents

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

2 . Standard Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 2.1. Robertson-Walker Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 2.2. Friedman Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 2.3. Early History of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . 673 2.4. Baryon-Photon Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

3 . Baryosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 3.1. The Basic Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 3.2. A Possible Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 3.3. Estimate of AB Using GUT . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 3.4. Galaxy Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

4 . Superheavy Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . . . . 581 4.1. The Monopole Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 4.2. First Order Phase Transition and Supercooling . . . . . . . . . . . . . . . . . . 681 4.3. Magnetic Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 4.4. Spontaneous Breakdown of Electromagnetic Gauge Invariance . . . . . . . . . . . 585 4.5. Second Order Phase Transition with Thermal Fluctuations . . . . . . . . . . . . 586

5 . Further Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

Appendix: Field Theory a t Finite Temperature . . . . . . . . . . . . . . . . . . 588

1 . Introduction

Recently great progress has been made in our understanding of the elementary particles and their interactions [ I ] . The electromagnetic and weak interactions have been unified into electroweak interactions in Salam-Weinberg-Glashow model using S U ( 2 ) x U ( 1) gauge symmetry . This symmetry is spontaneously broken. via Higg’s mechanism. to the observed U ( l)e.m. gauge symmetry of Quantum electrodynamics . As a result weak vector gauges bosons acquire a mass of around 100GeV . The strong interactions are widely believed to be correctly described by quantum chromodynamics based on SU( 3), gauge symmetry . The color gauge symmetry is presumably an exact symmetry as no colored hadrons have been seen . It is natural t o hope for a grand unification of electroweak and strong interaction into electronuclear interaction . Suppose that this electro-

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670 VIRENDRA SINUH

nuclear interaction involves a gauge symmetry 3. Clearly S 3 G E SU(3) , X S U ( 2 ) x U(1). Further the symmetry 3 must be spontaneously broken a t some heavy mass scale M to the lower symmetry C?. The gauge vector boson, corresponding to broken symmetry generators of 3, would acquire a mass of the order of this mass scale M . The estimates of this mass M are around GeV. We must have the general scenario,

The syinnietry 3 could, of coiirse, breakdown to the symmetry G in more than one step also. Generally quarks and leptons toget her transform as an irreducible representat ion in these grand unification symmetry schemes. Consequently, apart from the usual gauge bosons, namely color gluons, electroweak bosons W * , Zo and photon, we also have new lepto-quark gauge bosons which convert a quark into a lepton and vice versa. The exchange of these lepto-quark gauge bosons, in general, leads to a violation of t hc conservation laws of Baryon number B and lepton number L. Where one can study “new physics” corresponding to grand unification? A s thr. cha- racteristic energies corresponding to relevant new interactions are of the order of M i.?. lo1‘ GeV, we should ideally study particle interactions a t these energies. Surely, in conceivable future, we can not hope to study phenomenon at such energies using high energy accelerators. A few TeV seems to he the practical limit for accelerator 1)iiset.l physics. Our only recourse to study such high tmergy interactions in laboratory is to (i) look for unusual signatures of new physics which survive a t low energies (i.e. less than 100 GeV) and (ii) to do extremely high precision experiments sensitive to new physics via virtual processes. Since B and L are exactly conserved in the usual physics, the experinicnts involving a violation of B and L conservation such as proton decay, neutron-mtineii- tron oscillation, neutrino oscillation would constitute examples of the first kind. A n example of the later would be variation of Weinherg weak angle with momentum trans- fer involved. Clearly the scope of studying new physics in the laboratory is rather linii- ted. It is here that the standard cosmology comes to our help [2] . A t present, primarily due to the discovery by PENZIAS and WILSON in 1965 of the universal “3 K” microwave background radiation, t he hot big-bang model of cosmology has come to be generally accepted as the standard cosmology. The universe in its’ early stages is extremely hot and its’ energy density is very high. A s it evolves and cspands the temperature falls and the energy density becomes smaller. In the period when the temperature was within few orders of the heavy mass scale M there was enough energy to create the particles with these energies copiously. If these processes, which could take place at that time, have left any relic then a study of cosmology would illuminate physics a t such high energies. It is an exciting possibility that presently observed baryon- photon ratio and baryon-antibaryon asymmetry may be such a phenomenon. Standard cosmology without grand unification has no real way to account for the observed baryon content and baryon-excess of the universe. These ideas have ramifications for the theory of galaxy formation also. The presently available limits on magnetic nionopole density of the universe also place severe constraints on the grand unified models.

2. Standard Cosmology

We begin by providing a brief introduction to the “geography” and “history” of the universe as conceived in standard cosmology.

Grand Unification 571

2.1. Robertson-Walker Metric

The “geography”of the universe is described by the Robertson-Walker metric given by

(2.1)

where (dQ)2 = (d8)2 + (sin 8 d$)2 and R(t) is the cosmic scale factor. Here k = + 1, 0, - 1 correspond respectively to a closed, parabolic and open universe. These metrices describe a universe obeying the comological principle i.e. “The universe is isotopic and homogeneous a t all times”. The presently observed universe does seem to be quite homogeneous and isotropic provided we look a t it a t the scale in which we average over local clusters of galaxies or distances of the order of 2 10 Mpc. The best evidence for the cosmological principle is however provided by the high isotropy observed for the universal background “3 K” radiation. We can conclude from these observations that cosmological principle has been valid during the period beginning at the epoch of decoupling of matter and radiation at teinperaturc of about 3000 K. This happened around a million years after the big- bang. In our discussion we assume its validity a t all times.

2.2. Friedman Universe

In an isotopic homogeneous universe the energy-momentum tensor TPy must have the “perfect fluid” form

T,” = - P q p + (P + e ) U P ” (2.2) 2

with U p = (1, 0). Here pressure p and energy density e can be functions of time t only. Combining (2.1), (2.2) with the Einstein’s field equation we get the evolution equations,

8nG 3

+ k = - eR2 (2.3)

(2.4)

The models of the universe satisfying (2.3)-(2.4) are known as Friedinan models. In order to close the system of equations (2.3)-(2.4) we also need equation of state for niatter and radiation in the universe specifying the relationship between pressure and energy density. In the early universe, when temperatures T and energy density were very high, the effective equation of state is that of a ideal gas of massless particles as we can neglect particle masses which are much less than kBT (kB = Boltzmann const.). We thus have (in units with ti = c = kB = 1)

where J ( T ) = effective number of massless degrees of freedom a t temperature T and would generally stay constant over various periods of universal history. It is given by

1 *

672 VIRENDRA SINQH

where Nb,,(T) = number of helicity states for effectively massless bosons (fermions). Clearly a particles can be regarded as massless only for temperature such that Ic,T > mass of the particle. For example in SU(5) with its 24 vector bosons, Higgs mesons belonging to 5 and 5 and 24-plet representation, 3 generations of 5 + 10 fermions with spin 112 we have

NB = 2 * 24 + 5 + 5 + 24 = 82

lV, = 3 * 2 . (5 + 10) = 90

N ( T ) = 160.75

a t such high temperatures that masses of all these particle can be neglected. The eqns. (2.3)-(2.6) essentially describe the “history” of the early universe. The entropy density s of the universe is given by

With the above approximations it is easy to see that

a - (sR3) = 0 , dt

i.e. the expansion is adiabatic. We further note that eqn. (2.4) and (2.5) imply

d dt - ($24) = 0

and since e , from Stefans law (2.5) is proportional t.0 T‘, we get

R(t) T( t ) N constant (2.10)

during each period with M ( T ) effectively constant. For very early times, i.e. t -+ 0, we can neglect the term k in eqn. (2.3) as both & and Q R ~ tend to infinity in that limit. Writing the equation for the evolution of the temperature of the universe we obtain

4n3G 45 (9’iT)Z c? - N ( T ) T4 as t -+ 0

(2.11 a ) 4n3N( T)

T(t) Y 1/R where y2 = 45

and Mp the “Planck Mass” = l/@ = 1.22 x lOl0 GeV. If we measure temperature in units of GeV (1 GeV = 1.16 x 1013 K) we obtain

2.4 x piF *

(t/l s) (T/1 III (2.11)


2.3. Early History of the Universe

(i) times earlier than the Planck-time: The Planck time is defined as ( t ~ G / c 6 ) ~ ’ ~ (h 0.5 x 10-43 s) and is the only quantity which we can construct out of h, c, B and having the dimension of time. In this epoch the gravitational interactions between the particles become strong as the particles are highly energetic due to prevalent high temperatures which are greater than 10aaK. Not only that we are faced here with the conceptual problems of quantishg gravity itself. A t these times the horizon and the wavelength of a particle in thermal equilibrium are comparable. Is then the particle concept valid? This is truly the prehistoric period of the universe. After this period unquantised gravity should be adequate. (ii) time M 10-36 s. The temperature a t this epoch is of the order of M i.e. 10l6 CeV i.e. the grand unification mass. This is when the presently observed baryon-antibaryon asymmetry of the universe is generated. We will come back to a discussion of this epoch later. (iii) time m The Salam-Weinberg symmetry SU(2) x V(1) is broken to U(l)e.m.. Previous to this epoch the electroweak bosons were essentially massless and electroweak forces were long-range. The electroweak bosons now acquire their present epoch masses of about 100 GeV. (iv) time rn The quark-gluon soup goes into hadrons. We have

s, T M 100 GeV:

s, T w 1 GeV

nB M njj M my and nB - njj M 10-lOnB.

(v) time = Most of the hadrons and antihadrons have annihilated each other by now leaving a small residue. The neutrino interactions have become so weak by now that they decouple from the rest of the universe. From now on neutrino have thermal equilibrium corresponding to temperature T, given by T, = (4/1l)ll3 T. (vi) time w 3 to 4 minutes: The temperature is now about 109K a t the beginning of this epoch. The “deuterium bottleneck” to nucleosynthesis is operative. The presence of large number of photons as compared to nucleons, leads to deuterium break up and subsequent nuclei do not get built up. A little later the temperature drop a little more an deuterium bottleneck becomes inoperative. Nucleosynthesis of deuterium, He3, tritium and He4 takes place. Indeed a successful prediction of primordial He4 abundance is one of the strong evidences for the big-bang model. Nuclei beyond He4 do not get cooked a t this stage. (vii) time w 0.7 million years: The temperature has now dropped to about 3000 K (i.e. N 0.24 eV). The nuclei and electrons combine to form atomic systems. Universe becomes transpar?nt to photons and the photon decouple from the mat,ter. The microwave universal background radiation of “3 K” is this radiation which has cooled by a factor of a thousand. The era of radiation dominated universe ends to give place to matter dominated era. (viii) time w lo9 years: Galaxy formation takes place (iv) time rn 1O1O years: Present period of the universal history.

s, T m 10 MeV.

574 VIRENDRA SINQH

2.4. Baryon-Photon Ratio

At present our universe contains a very large number of photons compared to the number of t,he baryons. As these photons are mainly in the universal microwave background radiation their present number density ny can calculated by using Planck's law. We have. a t ternnerature T. I

(2.12)

where ((3) w 1.202.. ,. The present measurement of the temperature of the background radiation are consistent with 2.7 K < T < 3 K. We thus have

(n,)o 'v 400 (&Y/(cn~)~ (2.13)

We use a subscript 0 for denoting the value of a quantity a t present epoch. We now come to estimates of present values of density of number of baryons 7tg and the density number of antibaryons n,- in the universe. Note that the Baryon-Number density is given by nB - ng. At present we have no evidence for m y large scale antibaryon concentration in the universe. There is no serious indication for antibaryons in cosmic rays over and above those which can be accounted for by production in usual interaction of high energy hitryons. The ratio of proton to antiprotons is N 3 x while anti-Helium nuclei to Helium nuclei is less than One has also not seen any gamma rays corresponding to Baryon- antibaryon annihilation [31. The direct estimates of the present density of number of baryons can be obtained in the following two ways. (i) From the equations (2.3) and (2.4) one can deduce, without any approximation,

(2.14)

and

SZGQO = 3 (j$ + Ho2) (2.15)

where H ( t ) = RIB is the Hubble parameter and q( t ) =: -RR/(R)2 is the deceleration parameter. In the present era of matter dominating, over radiation, we have eR3 essentially constant i.e.

po w 0. (2.16) Combining (2.14) -(2.15) we obtain

eo R3 %oec (2.17) where

(2.18)

and H o = 100h, km/s Mpc is the present value of Hubble constant and mg is the nu- cleon mass. The present data is consistent with

1 > h0 > 0.5. (2.19)


Note that, as is obvious from eqn. (3.4), that the universe is closed if eo > ec, open if ec > o0 and parabolic if Po = ec. The data on qo are not very good and as 2q0 is consistent with a range of values around unity the universe is either a little closed or open. We further note that this estimate is for total matter density and as such is an upper estimate to nB + nB’ l4-e thus have

(no + ng)o < 2.26 x 10-6(qoho2) c n r 3

and therefore, since ( n ~ ) ~ is negligible,

( nB nE)o 5 0.56 x 10- 7 q,, h 2 r*;oK):. - (2.20)

(ii) Direct estimates of the amount of luminous matter in galaxies can be made by niea- suring Mass to light ratios for various type of objects such as galaxies in binaries and small groups. This method leads to the estimate

0.04 5 eo/ec 5 0.13.

Here the determination refers essentially to the baryon content of the universe. We thus have

0.045 x 10-6h02 C M - ~ 5 nB 5 0.15 x cme3

and thus leading to

note that if both these direct estimates (2.20) and (2.21) are reliable they would imply that a lot of matter in the universe is in non-baryonic form. It could for example be in the form of massive neutrinos. This is the problem of “missing mass”. We thus see that (ne - ng)d/(n,)O is a fairly small quantity which is of the order of 10-D*l. A theoretically more meaningful quantity to consider is

= N B / s (2.22)

where N , =; nB - ng and s is the entropy of t,he universe. A t present s is mostly concentrated in the universal background “3 K” radiation and the cosmic “neutrino background” of three types of neutrinos. At presenq

so % 7(n,)o. (2.23)

We now note that quantity q, whose present value is w (NE/7n,), , is thus w lO-lO*l, is a quantity which has not changed through most of the universal history. The baryon number, though violated by small amounts, is essentially conserved and the Hubble rspansion has been isoentropic for photons and neutrinos. The value of 7 has to be specified from outside, as an input, in the big-bang model of niicleosynthesis for a calculation of primordial deuterium nuclei and Helium nuclei production. The latest calculation [4] of q from Big-bang nucleosynthesis for a primordial 4He abundance of about 0,25, by mass, leads to

7 = 10-10,8+1. (2.24)

Jt looks extremely unaesthetic to specify such a small number as a boundary condition in cosmology. Note that for times less than 8, this value of 7 implies (nB - njj)/nB

576 VIRENDRA SINQH

rn 10-lO i.e. the universe is quite symmetric. There is only one unbalanced baryon for every 1O1O baryons. However completely symmetric cosmologies have repeatedly failed to account for the present epoch's baryon-antibaryon asymmetry and as such there is not much choice. As we shall see'in the next section the Grand Unification ideas offer a possibility of explaining this small number.

3. Baryosynthesis

Suppose the universe initially, i.e. say around Planck-time ( M s), is baryon-antibaryon symmetric i.e. has Baryon number B equal to zero. The B-violating interactions in the subsequent period of very early universe can then dynamically give rise of a tiny but non-zero baryon number. In the epoch when the temperature is around 1 GeV to 20 MeV most of the baryons and antibaryons would annihilate each other but leaving about one unbalanced baryon per 1O1O baryons and antibaryons and t,his would explain the presently observed value of baryon-photon ratio 7. This possibility was raised even before the grand unification ideas suggested the possibility of Baryon-number violating interactions [5] but has become more appealing, as well as more concrete, now as the grand-unification schemes naturally lead to such B-violating interactions [6, 71.

3.1. The Basic Requirements

It is however not enough, though necessary, to have B-violating interactions only in the theory in order to generate the observed Baryon number. We must clearly also have charge conjugation C, and combined-parity CP noninvariance as well. Otherwise equal number of baryons and antibaryons would be produced and if initially we have B = 0, it would remainso. Another important ingredient is the possibility of a departure from thermal equilibiriuni'). Roughly can see it as follows : In thermal equilibrium we have for the number density nj (p) of particle, of type i, with momentum p , and a t temperature T ,

(3.1)

where pi and mi are the chemical potential and mass of the particle of the type i. Here 8, = +1 for a boson and Bi = -1 for a fermion. In thermal equilibrium the reactions like

y + y + baryon + antibaryon

lead to ,&aryo,, f ,&,tibaryon f 0 while the reaction, involving baryon number violation, such as two baryons -+ three baryons lead to PbaryOn = 0. We thus have

p h a r y o n = pan t iba ryon = 0 in equilibrium. Further since mass of a baryon and its' antibaryon are equal from CPT, it follows from (3.1) that we have same number of baryons and antibaryons and thus net B = 0. While the microphysics provides mechanisms for B, C , CP violation the Hubble expansion of the universe provides a mechanism for a departure from thermal equilibrium. There is one more general requirement which is not so obvious. If the temperatures involved is so high that all the particles species involved can be regarded as rnass1e.s.s

l) Incidently the existence of thermal equilibrium, or of Boltzmann's H-theorem, can be proved even with CP nonconservation (i.e. also time reversal noninvariance) interactions if CPT is good.


then no baryon asymmetry would result. In this situat,ion Uehling-Uhlenbeck equation, which is Boltzmann equat,ion generalised to take care of quantum statistics, in an ex- panding universe reads as follows :

PIsPk*PI

x7bi(l?i) ni(pi) ( I - ekn(pk) ) ( I - x I(ipi? ipjl T Ikpk, bl)12 nk(pk) nL(pl) (l - ein(pi)) (l - ejn(pj))]

+ (b(pk, pl ; pi, p i )

+ ..*. (3.3)

where (b is the phase space and statistics factor and T is the scattering T-matrix. We have only displayed collision terms corresponding to two particle reactions. An exact solution of eqn. (3.2) is given by, if all the particles are massless,

= [e+~t/w - 0~1-1 where

T o r R(t) T(t) = const. dT(t) - fi

dt R -- -- (3.4)

This solution can be checked by simply using unitarihy. Clearly number of baryons and antibaryons are equal a t all such times since effectively it is only an adiabatic expansion of a system in equilibrium distribution which is taking place [7].

3.2. A Possible Scenario

We describe here the scenario for Baryoproduction due to W ~ C Z E K and WEINBERG [7], sometimes referred to as the ‘Standard Scenario’. The baryon asymmetry in this scenario conies from ‘out of the equilibrium’ interactions and decays involving superheavy X bosons which have B-violating interactions. The X boson could, for example, be either a superheavy gauge-boson or a superheavy Higgs boson. We shall see later than Higgs boson possibility is preferred. We now estimate the reactions crosssections etc. to find out when t,he interactions of X would go out of thermal equilibrium. Let the mass off the boson be m, and its’ coupling to fermions denoted by ax. We then have the rough estimates

r, fl olxmx2.N/(T2 + mx2)1/2 (3.5) (3.6)

(3.7)

for the decay rate rx, collision rate r, of X and the Hubble parameter. In order that a certain process should go out of equilibrium due to Hubble expansion it is required that we have

Reaction Rate < Expansion Rate.

It is assumed that because of strong gravitational interactions before the Planck-time all particles are in thermal equilibrium. Therefore the baryon asymmetry is zero a t the Planck-time. A t Planck-time both r, and r, < H as nix < mp and aAr2N < 1.

678 VIRENDRA SINQH

After the Planck-time the universe expands and the temperature T falls. Nothing much in the way of baryoproduction can happen as long as T > mx since we can ignore mx and effectively we have the massless situation. The X boson decay becomes interesting, as source of baryon excess, when T 5 n ~ x and further 11 > r D i.e. T 2 TD where

T O cz ~ [ L x ~ . / V - ' / ~ ~ P ~ ~ ~ V L ~ ] . (3.8)

?ndK 2 cuxm,.M~12. (3.9)

In order that rnx > TD we must have

We also note that under these conditions we always have H > r, and we thus can neglect collisions as a source of baryon excess. For gauge bosom ag - a, the gauge collpling which is N 1/40. We therefore can not satisfy the condition for mnx, given by (3.9), as it implies an unacceptably large value of mX which is presumably rn 10la*l GeV. On the other hand if X is asuperheavy Higgs then ax N 10-6*1 and leads to acceptable masses for superheavy Higgs. The production of nonzero baryon number B is thus attributed here to decay processes of Higgs beginning a t temperature T - To. The inverse process is blocked by Boltzmann factor crnxlT. We now proceed to estimate the amount of Baryon-number which is generated by this mechanism. The number density of X at T = T, before they decayed is given by

nx N [(a) T D 3 N , j ~ 2

where Nx is thenumber of helicity states of X and x. On the other hand total entropy S is given by

4 Z 2 T ~ 3 x 45

S -

where JV is total number of effective helicity states defined by (2.6). Here we neglect contribution of the decay to the entropy. We thus have

If LIB is the mean net Baryon number produced in X or x decay we get [a]

The ratio Mx/Jr/- is of the order Grand unification scheme and we shall turn to it next.

For a calculation of AB we have to use specific

3.3. Estimate of /IU Using GUT

We continue with our consideration of X and x boson decays. Suppose, e.g., the X boson has two decay channels XI, X, with branching ratios r and (1 - r ) and respective baryon number B, and B,. The first channel could be, e.g., X + d +l with B, = 113 while the second could be e.g. X +- a + d with B, = -213. Let the corresponding quan- tities for x decay be F with (1 - F ) and, of course, with El = -Bl, B2 = -I&. The


mean baryon-number produced in X and decay is

B, = rB, + (1 - r ) B,

Hence the baryon number produced on the decay of an X and an x is given by

1 1 2 2

AB = -(B-r $- B z ) = - (B, - B,) ( r - P ) .

Clearly r - f = 0 if either C or CP is conserved. Indeed r = F from TCP theorem if only tree diagrams are kept in the calculation of r - F. In general we would have the amplitude for channel 1 (i.e. X -+ x,) given by [9]

and

where ( I , refer to the coupling-constant combinations associated with different Feyn- man diagrams and Yj( +i&) are the corresponding integrals with proper i& prescription.

For a nonzero r - P we must thus have both complex couplings which comes from CP or G violation and complex amplitudes which can come only from the interference between absorptive and dispersive parts of the amplitude. Jn SU(5) with more than one 5-plet of Higgs we find for Higgs decaying intoq channel,

AB a Im Tr ( P t P y q t y p )

where P is coupling constant matrix for trilinear coupling of the p t h Higgs to two decimets of Fermions while yp refer to coupling constant matrix for pth Higgs - 5* fermion - 10 fermion coupling. It arisesfrom the interference between tree diagram (a) and loop diagram (b). Note that the interference between tree diagram (a) and loop diagram (c) is zero and hence does not contribute to AB. It also follows from (3.8) that to this order AB = 0 for the minimal SU(5) scheme. A t three loop level one get a very small AB M lO-l7*9 in minimal model. The detailed estimates of the integrals and coupling involved here show that this mechanism is ca- pable of giving the observed value of r,i since they can get AB w

In XO(10) the basic fermionic multiplet, i.e. 16-plet, is symmetric under C and as such the exact SO( 10) theory is C-symmetric. This the unbroken SO( 10) theory would lead to AB = 0. When one breaks the C symmetry, the L-R symmetry also gets broken i.e. N R , the right hand analog of vL, acquires a large mass M N . Unless MN is not too different from heavier gauge-boson, associated with Baryon nonconservations, mass this leads, in general, t~o a small AB. There are many other scenarios, e.g. those involving heavy fermion decays, etc. which have been investigated in this connection [9]. While here we have here concentrated on only qualitative aspects there has been many detailed numerical calculation, involving a solution of Uehling-Uhlenheck equation, to see how the baryon asymmetry evolves dvnam ically.

\

580 VIRENDRA SINW ----< P

Ul

c c P

A t this stage it is not possible toclainl thatone has explainedthe observed photon-baryon ratio but in principle, the way is clear to anunderstanding of this number from grand unification dynamics. Meanwhile the observed 71 value provides a strong constraint on possible grand unification gauge models. For example minimal SU(5) model with three ferniion generations is ruled out.

3.4. Galaxy Formation

The above scenario for the generation of baryons has implications for theories of galaxy formation. If any inhomogeneities existed in the universe in the baryon number density initially these would have been washed away due to baryon-number nonconserving interactions in the period T > 10l6 GeV. As the universe cools to below T N GeV we would have a uniform generation of baryon-photon ratio all over universe at the same rate i.e. initial perturbations at the time can be only ‘adiabatic’ [ZO]. Thus these consideration do not allow for ‘isothermal’ perturbations which would have spatial variation in baryon-photon ratio but uniform temperatures. In the usual picture of galaxy formation these initial perturbations, from the era of radiation dominance, which are present a t the time of the recombination of electrons and nuclei i.e. around T N 3000 K at t N lo6 yr, evolve in the subsequent matter doniina- ted era of the universe into clusters clusters of galaxies, galaxies, stars etc. In the scenario based on ‘adiabatic perturbations of Zeldovich et a1 these evolve into flat pancake like structures of about 1014 solar masses (superclusters !). The fluctuations corresponding to smaller masses are damped by photon diffusion during recombination. These objects with 1014 M a later fragment in glaxies and stars. Another ingredient in Zel’dovich et a1 initial conditions is (6Too/Too) N 1/12 where the average of the ratio of density fluctuation to itself is taken over a length scale 1. There are some arguments by Press that even this may be compatible with GUTS [ l o ] .


However the ‘adiabatic’ scenarios has some problems as it predict thermal fluctuations on the small angular scales in the universal “3 K” radiation which are larger than the observed one. There is a possibility that ‘massive neutrinos’ are the needed cure.

4. Superheavy Magnetic Monopoles

If a compact symmetry group a breaks down to H x U( 1) a t a mass scale M H then the theory has monopole solutions, corresponding to the U(1) group. The mass M , and magnetic charge gm of these monopoles is given by

M~ - MHIOI and gm - l& (4.1)

where 01 ,- g2/4n and g is the coupling constant associated with U ( 1). For example, in SU(5) , one has

SU(5) M,‘ S U ( W x SU(2)L x U(1)”.

Taking 01 - and Mx - lo1* GeV we obtain M , - l O l a GeV i.e. the theory has superheavy monopoles. Since all grand unified theories eventually break down to S U ( 3 ) , x U ( l)e,,,. the magnetic monopoles are invariably associated with such theories [U].

4.1. The Monopole Problem

In the early universe these superheavy magnetic monopoles must have been produced. Further since a magnetic monopole is stable due to their carrying a topological quantum number, the only way these monopoles can have disappeared would be via M - g annihilation. The present monopole density in the universe would thus be another relic of early universe just like the present baryon number density. The cosmic ray data can not give any evidence about these superheavy M’s since galac- tic magnetic fields are not strong enough to accelerate them to relativistic velocities. They would therefore not be very ionising. A search for them in terrestrial iron deposits has been proposed. Their present mass density should however be such that it does not overwhelm presently known limit on mass density given by our knowledge of deceleration parameter and present Hubble-constant i.e.

mM(nM)o 5 2qoec N 2.26 x 10-6m~(qoho2) i.e.

5 4 x 10-21 with qo < 2, ho < 1 or

r 5 a t T = 3.9 K (4.2)

where r = nM/n, a t the present epoch. Similarly the standard scenario for primordial He4 synthesis demands that the monopole mass density should not overwhelm the universe at that time i.e. when temperatures w 1 MeV.

10-19, (1 MeV)

r(T = 1 MeV) < ~

mM (4.3)

We now proceed to estimate number of M’s which would have survived M - B annihilation until the era of nucleosynthesis. Consider ‘the period in which production rate of M and is small compared with Hubble expansion but the annihilations are taking

682 VIRENDRA SINGH

place. The magnetic Coulomb energy of a M-@ pair would dominate over the thermal energy if their separation d is such that gM2/d > T i.e. d 5 l/ciT = d,. If the iiiean free path 1. of monopoles in less than d, then they will annihilate efficiently while if 2. > d, then annihilation rates would be negligible. Since A - 1/P the annihilations get less frequent. Detailed analysis by Preskill shows that M z annihilation cannot reduce r below 10-10 if r ( T i ) > while if r(T,) 5 10-lO then annihilations basically do not disturb it. Combining this analysis with (4.3) we find we must have [I21

T ( ~ , ) 5 10-19. (4.4)

We have now this number to confront with the estimates of monopole production in the process of symmetry breaking at Ti, and this is rather difficult to do. Let us now, following KIBBLE [13,14], try to visualize nionopole production. In the syiii- metric phase at high temperature the order parameter, i.e. the expectation value of Eliggs field (+), would be zero. As the temperature falls below critical temperature T,, a t which the a new phase results, the order parameter will acquire a nonzero value but in general it will be pointing, in different regions of space, in different direction in the internal symmetry space. In general when these domains come together with differently pointing order parameters there would have to be regions of zeros of Higgs field and these are the monopoles. If 5 he the distance over which Higgs field directions become uncorrelated, then number density of monopoles would be given by

- PIP (4.5)

where p is the probability of a non-trivial winding of vector representing the direction of (4) in group space. I n general p - 10-1 (it is 118 for Georgi-Glashow model).

We now estimate 5 in two ways: (i) Classical relativity tells us the there is a particle horizon at 2ct where t is the epoch. We’can thus take 6 - 4ct and using eqn. (2.11a) we get 6 ,- 2MplyTC2 i.e.

r (T, / Z / M ~ ) ~ 10-12 (4.6)

for T, - 1014 GcV and JV - loz and p - 1/10. Because of homogeneity problems of the universe, which are hard t o understand if particle horizons are seriously taken, this estimate may be suspect. (ii) For a second order phase transition the correlation length E , in equilibrium, is given bY

and is infinite a t T = T,. Because of Hubble expansion there is no time for le to become so large. Indeed we do not expect the dE,/dt to be more than velocit,y of light. We can take df,/dt to achieve a value c and be at that value after that. This leads to

since ,u w T,. In either of the two cases it is clear we monopole production is too copious to be consistent with the estimate (4.4) and threatens to question the compatibility of the grand unified interactions with big-bang cosmology. This is the monopole problem !


How to resolve this problem? There are basically three responses to it :h) The transition was strongly first order and conconimitant supercooling, (ii) The M-M annihilation is much larger than estimated because of a magnetic confinement niechanisin and (iii). The U(1) symmetry appeared in the symmetry breakdown at such a late stage (e.g. at T M 103GeV) that there was no production of superheavy monopoles. On the other hand it has also been argued that a second order transition may resolve the problem if adequate account is taken of the thermal fluctuations. All these scenarios impose serious constraints on the Higgs-sector of the theory for a given GUT. We now turn to a discussion of all these possibilities.

4.2. First Order Phase Transition and Supercooling [ I d , IS]

We first give a 7wiae argunient as to why the first order transition may resolve the proh- lein. If the monopoles were in statistical equilibrium (and this is probably not quite light for the whole space) then their number density would be given by

(4.8)

For a second order transition wzM(T) vanishes like (T - T,)2 around T = T , and as such there is no suppression of monopole density and there is copious amount. In con- trast for a first order transition ni,(T)/T would have a value GZ -1oc mx(T,) /aT, m -n/Au(0)/T, for T a little below T,, and since 01 N and m, w T , we have a large suppression. Consider for illustration SU(5). Let the Higgs potential be given by

1 1 1 1 2 4 2 3 V(@, I’ = 0) = --p2 Tr @2 + --a(Tr 0 2 ) 2 + - b Tr cD4 + - cTr Q3. (4.9)

where O is a 24-plet of Higgs fields. The i~linimuin of V(@, T = 0) would determine the correct phase a t T = 0. A number of possibilities can arise, depending on the relative ralues of the parameter in P, and are given by: (i) Phase I: Exact SU(5) is maintained, (i i) Phase IT: The unbroken symmetry of the phase is XU(4) x U’( 1) and (iii) Phase TII: The unbroken symmetry is S U ( 3 ) xSU(2) x U(1). The parameter are sochosen that the Phase IT1 is the realised one. We now consider this theory a t finite temperature. We have

(4.10) 1 2

T r e f f ( @ , T’) w V(@, I’ = 0 ) + - d ’ 2

where c = ( 1 3 0 ~ + 946 + 75g2)/60.

At very high temperatures we have Phase I. After that it goes into Phase TI and then into Phase I11 or directly into Phase I11 from Phase I depending on the parameter values. The transition from Phase 11 to Phase I11 is first order since there is no continuous way to deform one into the another. The presence of a cubic term is necessary for the first order transition. The approximation (4.10) while good a t T > mx gives qualitatively correct answers a t other temperatures as well. We now have the following scenario using the first order phase transitions. Upto T > TI we have the Phase I. A t T = T, the universe goes in Phase I1 (i.e. S U ( 4 ) x U‘(1) phase). Since this is a continuous transition it takes place quickly. The magnetic monopoles corresponding to U’( 1). The universe now cools in this phase. At the temperature T = T ,

584 VIRENDRA SINGE

the minima in potential is degenerate with the phase 111. This transition however being a first order one a large potential barrier separates the two phase and the universe supercools to the temperature T = T,, where the minima corresponding to Phase I11 disappears, in the “false vacuum” i.e. Phase 11. Between T = Tc and T = T, there is formation of the bubbles of “true vacuum”, i.e. in Phase I11 or S U ( 3 ) x S U ( 2 ) x U(1) symmetric phase. In theprocess of transition from Phase I1 to Phase 111, the monopoles corresponding to U’(1) disappear through annihilation. A t the temperature T = T, when the transition to phase IT1 is complete the latent heat corresponding to this transition is released. This heats the universe to a T = TR. KENNEDY et al. [I61 estimate for M,(T = 0) w 6 x 1014 GeV and OL - 1/45, N w 100, T , M 1015 GeV, T, w 8 x lo1* GeVJ T, 5 x 1013 GeV and TR m 3 x lo1* GeV. The monopoles, corresponding to final U( 1) factor of t,he phase I11 symnietry SU(3) XSU(2) x U(1), can be expected to be given by

and leads to r w 10-36 which is quite compatible with observation. It is not however completely clear that in all these scenarios one does not generate too many monopoles during the reheating and whether the process of supercooling itself really terminates

4.3. Magnetic Confinement [171*

Let us first consider the usual SU(5) symmetry breaking chain

The monopoles produced, when T w M,, would be superheavy and these will carry U( l ) y magnetic charge. Consider the monopole with minimum magnetic charge. It would be an 8 U ( 3 ) , magnetic triplet, XU(2), magnetic doublet and U(1), magnetic charge of ( -1 /2 ) . What happens to these monopoles as temperature drops below T M Mw. Denoting by Q and Z the generators of U( l)e.m. and lJ( 1) generator in the ZO-Boson- direction, the S U ( 2 ) , x U ( l ) Y magnetic field strength of the monopole is given by

B = qQ + zZ

where (q, z ) = (-1/2, 0) or (-l/S, -31’8). The monopole with (q, 2 ) = ( -1 /2 , 0 ) , since it carries no U(1)zo charge becomes a pure U(l)e.m. magnetic monopole and survives. The monopole with (9, z ) = ( - 1/8, -3/8) however carries a U( 1)p charge and becomes strongly bound to an antimonopole by a strong magnetic flux tube of Zo field. In short it annihilates quickly. We thus see that if magnetic monopoles carry apart for U( l)e.m. magnetic charge other U(1), charge as well they will be strongly annihilated [18]. The mechanism does not affect pure U(l)e.m. magnetic monopoles and as such does not directly apply to SU(5), SO(10) and E(6) with their usual scenarios of symmetry break- ings. In SO( 10) theory, if the symmetry breaking pattern is

so(1o) M ~ ~ ~ ~ I s G ~ v ~ ’ slJ(4)c x [au(2 )L XSu(2)RI and

M c . c l ~ 1 4 ~ e ~ ’ S u ( 3 ) C x su(2)L x u ( l ) Y


and further, if the transition to S U ( ~ ) , X ( S U ( ~ ) ~ X S U ( ~ ) R ) is strongly first order, then monopole suppression is almost perfect [ 19). The superheavy monopoles associated with Mar are suppressed by first order transitions and lighter monopoles associated by SU(3) , X S U ( 2 ) L x U(l)y are completely annihilated by magnetic confinement.

4.4. Spontaneous Breakdown of Electromagnetic Gauge Invariance

Imagine the scenario for symmetry variation with temperature T is as follows

Clearly no monopoles would be generated a t the temperature T,. Besides any magnetic monopoles which were produced for T > Tx would annihilate during SU(3) , phase of the universe i.e. (T,r > T > T,) as there is no U(l)e.,,,. symmetry in this phase [20]. Let us discuss this scheme in the framework of the usual x U ( l ) y electroweak model. To implement the idea one needs a t least three coinplex Higgs doublets. Let the potential term be

+ rij(4i+4i)2 + ~;($j+4i)~I-

The parameters occuring here are chosen such that a t T = 0 we have the usual gauge hierarchy i.e.

flu(%, X u(1 )~ u ( 1 ) e . m .

so we can choose parameters to give at, T = 0

Now consider the theory a t temperature T. The effective potential is given by

1 Vetf(T) = + C Ppdit4i

i

where Fi = 1/8 (3g2 + 9'2) + l i + for s U ( 2 ) ~ x U ( l ) y . We further choose parameters such that

[l/30ii + l/6pij], and g, g' are gauge parameters j+i

F, , F2 C 0, F3 > 0 .

In fact the niinimum can be made to occur in such a fashion that, for T 2 Mw,

Note that since (+2+) + 0 we have violation of electric charge conservation. Also the photon becomes massive a t T > Tw. We thus see that the unusual scenario mentioned earlier can be implemented. The embedding of this scheme in a proper GUT scheme remains to be done but can presumably be carried out.

2 Fortschr. Phys., Bd. 31, H. 11

686 VIRENDRA SINGE

This scheme, because of the violation of the electric charge conservation above T > l ’w , implies that the present universe need not be electrically neutral. However the rate T of charge nonconserving processes a t 1’ - M w is of the order a2MW while expansion rate a t this epoch is of the order of Mw2/Mp. The electric charge violating interactions are therefore in equilibrium a t this epoch. We may thus expect the electric charge Q of the universe to be zero, except that due to usual fluctuations a charge Q of the order of vy) where N , is the number of photons in the observable universe i.e. los6, could arise. We thus have

&IN,, - l/my N

i.e. Q/Nn N 10-431V,/Ns N

and this is quite consistent with observational limits

4.5. Second Order Phase-Transit ion with Thermal Fluctuations [21].

For a second order transition the expectation value of the Higgs field (4)T is zrrofor T > T,, the critical tetnperature and goes as

for T , > T. (see Appendix). The same considerations tell us that miniinuni of Veff($, 1’) is given by (1 - T2/T,2)2 x min V,if(4, 5” = 0) . The minirnumis therefore quite shallow for T w T, and thermal fluctuation caneasily drive the system from “true V ~ C I I U I I I ” to “false vacuum”. There exists another temperature T O known as Ginzburg temperature, which is less than T,, below which thermal fluctuation is not sufficient to cause these changes i.e. bubbles of the wrong kind of phase do not form easilyin theright phase of the system (Fig. 1). The Ginzburg temperature can be estimated by requiring the thermal energy which is essentially k B T G should be comparable to difference of the free energyP between the two phases contained in the correlation volume i.e.

TG = t3(1’c) [ ~ q w G ) ) l .

Here E(TG) w l/nL,(TG) where mH(TG) is the mass of Higgs boson a t Tc. Furthrr

It thus follows that

(4.11)

where C is a model-dependent constant. We now come back to an estimate of monopole density. Divide the space into cells of size N t3. Each cell has a possibility of being either innormalor “false” stat.e. If this pro-


c

Fig. 1. Second order phase transition with thermal fluctuation8 (After ELLIS et al. 121)

bability is greater than w 1/4 then analogy to percolation problem tells one that there would be infinite interconnected cluster of normal vacuum. Thus if monopoles may be kept in equilibrium right down to T = TG. We then would have

On using (4.11) we obtain]

r(T,) w ( c - exp (-c 2). rnli 1(4.12)

The quantity C was estimated to be C m 36 in an S U ( 5 ) modelcalculation. UsingmH/mx - 2.5 we obtain r ( T G ) FZ

which is quite alright. However the assumption that the monopoles remain in equilibrium right upto Ginzburg temperature may need further justification.

5. Further Problems

We have seen that GUT leads to the possibility of a dynamical understanding of the baryosynthesis. This probably constitutes the greatest advance in the big-bang cosmology since nucleosynthesis ideas. We have also seen that the superheavy magnetic monopoles constitute something of an embarrasment. GUT also opens the possibility of neutrinos acquiring a small mass and this can signi- ficant cosmological implications.

2*

688 VIRENDRA SINGH

In the present universe the value of the cosinological constant A is awfully small. We have A/8zG 5 10-46 (GeV)4. If the universe has undergone phase transitions of the first order type then there would be epochs were A would be nonzero and since the energy density is of the order TC4 a t temperature T , we would have A/8nG - Tc4. This can be enormous for T, N 1015 GeV (GUT transition) or even for T, - 100 GeV (Electroweak transition). In these epochs A could even doininate the dynamics. It has, for example, been suggested in A. GUTH [ZZ] , in his ‘inflationary Universe’ scenario, that this niay be the mechanism which leads to a homogeneous and isotropic universe. In the standard cosmology it has always been somewhat of a mystery as to how parts of the universe which in early times were not causally connected eventually evolved into similar parts. It is, on the other hand, quite probable that the solution to these mysteries would in- volve quantum gravity [23], or even modifications of gravity [24].

Appendix

Field Theory at Finite Temperature

Consider SU(5) symmetry. The spontaneous symmetry breakdown associated with SU(5) -+ SU(3) , x SU(2) x U ( 1) is associated with a order parameter given by vacuum expectation value of (+21) where Oz4 is a 24-plet Higgs boson field. We have ( 4 2 3 M m,, the heavy gauge boson mass. The SU(2) x U( 1) -+ U ( l)e.m. is similarly associated with a {&). This is the pattern for the cold theory i.e. if the system is a t T = 0. Tf one is discussing the situation in early universe we have the system a t a high temperature and we have to find the correct pattern for a hot theory. A t nonzero temperature T , the vacuum espectation value has to repoaced by the ensemble average t2-51. Thus

At very high temperatures clearly all masses would be negligible i.e. it would effectively be a massless situation. We thus expect complete S U ( 5 ) symmetry t o be good. On the other hand at T = 0 we would expect only SU(3) , x U ( l)e,m4 symmetry to survive. As the universe evolves from very high temperatures to lower temperature the symmetry breaking pattern a t T = 0 in the hierarchy of masses would roughly parallel that of the hot theory as temperature coines down. The pattern however need not be the same. Consider, for illustration, temperature dependent effective potential given by

near the critical temperature T = Yc. This would give

= o for T 2 T,.

We thus see that order parameter is continuous as T goes through T = T , and provides an example of a second order phase transition. In this case the mass of Higgs boson m,(T) M I/;i (4)T and is thus teiiiperature dependent.


An example of a first order transition is provided by the following exaniple of an Abe- lian Higgs model with gauge coupling g,

In this case there is a temperature T, at which an asymmetric inininium at (4) =/= 0 first appears, while another temperature TI at which symmetric minimum a t (4) = 0 disappears and leaving only an asymmetric minimum with (4) =/= 0. I n between, i.e. T2 > T , > Y’,, there is a temperature T, at which the symmetric and asymmetric niiniiiium are degenerate. The value of the order parameter (4) would be discontinuous a t T = T,. (Fig. 2).

1 $ 1 - Fig. 2. First order phase transition

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grand unification and the big-bang cosmology

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