Download Generation of Cosmological Perturbations by a First

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Progress of Theoretical Physics, Vol. 68, No.5, November 1982
Generation of Cosmological Perturbations
by a First-Order Phase Transition
Misao SASAKI, Hideo KODAMA * and Katsuhiko SATO**
Research Institute for Fundamental Physics, Kyoto University
Kyoto 606
* Department of Physics, Osaka University, Toyonaka 560
and
Department of Physics, Kyoto University, Kyoto 606
** Department of Physics, Kyoto University, Kyoto 606
(Received May 8, 1982)
Origin and growth of the large scale inhomogeneities of the universe are discussed in a
cosmological model based on grand unified theories in which the unified symmetry breaks down
by a first-order phase transition. It is shown that a fine tuning on the nucleation rate of vacuum
bubbles is necessary for explanation of the observed large scale inhomogeneity. This fine
tuning, however, is shown to be consistent with the suppression of monopoles.
§ 1.
Introduction
It was argued that grand unified theories (GUTs) can provide the mechanism
of generating the net baryon number of the universe.!) Further, it was suggested
that not only the horizon problem can be resolved 2 )-4) but also the overproduction
of monopoles may be avoided,5)-S) if the transition of a grand unified symmetric
phase to a broken phase is of strongly first-order. However a very important
issue of cosmology has been left out of arguments: One has to explain how and
why the universe has attained the large scale inhomogeneities of matter distribution such as superclusters of galaxies. 9 ) This problem must be settled in order to
complete a cosmological scenario based on GUTs. 10 )
A clue to approach this problem was obtained from the investigation into the
space-time structure before and after the cosmological first-order phase transition. It was revealed that not all of the vacuum energy is released as radiation*)
by the nucleation of broken phase bubbles, but some portion remains unreleased
inside domains surrounded by bubbles and is eventually transformed into black
holes and wormholes (BWHs).1l)-13) Due to the stochastic nature of bubble
nucleation, the distribution of these BWHs statistically fluctuates. As a consequence, small inhomogeneities on large scales as well as large inhomogeneities on
small scales are produced. Since the mass of BWHs is very small, they evapo*)
We mean by radiation all effectively massless particles.
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M. Sasaki, H. Kodama and K. Sato
rate rapidly by the Hawking process. Hence the small scale irregularities are
expected to disappear within a short time. On the contrary, inhomogeneities of
BWH distribution on sufficiently large scales may well induce growing modes of
density fluctuation. 9 )
In this paper, we study the evolution of such small fluctuations and examine
whether they are responsible for the large scale inhomogeneities of the present
universe. In § 2 we briefly describe a cosmological scenario deduced from GUTs
with a first-order phase transition. Two qualitatively different evolutional
pictures are possible depending on the mass and the number density of BWHs; in
one case BWHs once become dominant energetically and then evaporate, while in
the other case BWHs evaporate before they could dominate the evolution of the
universe. For each of these cases we discuss, in § 3, the evolution of the small
inhomogeneities on large scales associated with the BWH distribution. In § 4, by
assuming a simple form for the nucleation rate of vacuum bubbles, constraints on
it are derived which are to be satisfied in order to explain the observed inhomogeneities on scales of super cluster size. It is found that these constraints
are rather severe. In § 5 the results are discussed in connection with the
monopole problem and some other problems.
Throughout the paper the absolute units c = Ii = G = 1 are used. Some
quantities are given in the conventional units if necessary.
§ 2.
Scenario
As has been discussed in literature, the universe turns into a supercooling
state and begins to expand exponentially after the temperature becomes lower
than a critical value (:<; 10 15 Ge V) if the phase transition associated with the
breakdown of a grand unified symmetry is of strongly first-order. Then the
transition proceeds through random nucleation of broken phase bubbles (true
vacuum bubbles) in the background of metastable symmetric phase (false
vacuum). If these bubbles could fill up the whole universe, the phase transition
would finish safely and the universe would be described by a radiation-dominated
Friedmann model again.
However it can be easily shown that the phase transition never finishes in the
above sense: Let the universe before the nucleation of bubbles be described by
a de Sitter space,
(2·1)
where
R( r)= I exp( rll).
Generation of Cosmological Perturbations
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The characteristic expansion time scale I is related to the false vacuum energy
density Pv as
3
pv= 87[12 .
(2·2)
Assuming that the critical size of a bubble is smaller than I and bubbles expand
at the speed of light, the volume fraction of false vacuum u( r) is given by 4),7)
(2·3)
where
and PN( r) is the nucleation rate of bubbles. Thus unless PN( r) becomes infinite,
u( r) never vanishes within a finite time.
Recently, Sato, Sasaki, Kodama and Maedal!)-13) have given a resolution to
this problem. According to percolation theories/ 4 ) when u( r) reaches a critical
value uc (~O.3, see, however, the discussion in § 5)15) the false vacuum ceases to
have infinite-size networks and is divided into regions of finite volume surrounded
by bubbles. These false vacuum domains have a peculiar character. Since
bubbles are expanding at light velocity, the wall of a false vacuum domain is
pushed inward. Nevertheless it is possible that the area of the wall increases
forever, since the region inside the wall is unaffected by bubbles from causality
and maintains the de Sitter structure. This seeming paradox was solved in Ref.
n ). It was shown there that these trapped false vacuum (TFV) domains become
either black holes or wormholes and as a consequence they become causally
disconnected from the outer region. Thus the phase transition practically
finishes at the time when u = Uc and the fraction uc left over eventually turns into
BWHs.
The mass M and the number density nH of BWHs depend on the nucleation
rate PN( r). For convenience we parametrize them as
M=.!!..l
2
(2·4 )
and
(2·5)
where r c denotes the time when u = uc. The parameters a and /3 are functionals
of PN( d in general and their dependence will be discussed in § 4 by assuming a
simple form of PN( d. From Eqs. (2·2), (2·4) and (2·5) the energy density of
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M. Sasaki, H. Kodama and K. Sato
BWHs at r = r c is given by
PH( rC)=MnH( rc)
= (47[/ 3)a{Jpv.
(2·6)
We presume that the most of the vacuum energy is converted into radiation.
This corresponds to the assumption that a{J~l. Thus the universe is dominated
by radiation with the density Pr~Pv at r= rc and begins to evolve as a radiation·
dominated Friedmann universe. The time tc in the Friedmann model correspon·
ding to the effective completion time of the transition rc is then given by
3
tc= ( 327[Pv
)112
1
=2·
(2·7)
As the universe cools, PrcxR(t )-4CX r2 while PHcxR(t )-3 CX r3/2. Thus PH
eventually dominates over Pro The time tm at which the universe becomes BWH·
dominated is given by
(2·8)
However, since BWHs evaporate due to the Hawking process, two different
evolutional paths of the universe are possible. The lifetime of BWHs is given by
4
_ 107[8
_
10 16 (Nev)-I( 1
t ev - 27Nev M3 - 6 x
W
108
)2 a 31 ,
(2·9)
where N ev is the number of effectively massless spin degrees of freedom which
participate in the evaporation. Comparing Eq. (2·8) with Eq. (2·9), we find
(2·10)
where we have assumed Nev=102. In the case tev>tm, the universe becomes
BWH-dominated at t = tm and becomes radiation-dominated again at t = tev. In
the case tev< t m, on the other hand, the universe is always radiation-dominated.
In both cases the evolution after t = tev is the same as the standard big-bang model
and the universe reaches the matter-dominated stage at t = teq~3 x 10 12 sec,
provided that an appropriate number of baryons are present before the era of
nucleosynthesis.
§ 3.
Origin and growth of perturbation
As noted in § 1, because of randomness intrinsic to the nucleation of bubbles,
the number of TFV domains within a volume V naturally acquires statistical
Generation of Cosmological Perturbations
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fluctuations proportional to N 1I2 , where N is the mean number of TFV domains
in V. Therefore, although the total energy density remains homogeneous on a
scale Lunder consideration,*) at the time rc, PH is associated with a fluctuation
OPH given by
(3'1)
Since OPH+OPr=O and PH4::..Pr, we have Or=-OPr/Pr4::..oH at t=t c. Thus the
perturbations induced by the first-order phase transition are isothermal modes.
As the time proceeds, BWHs evaporate and OH changes into Or. It can be shown
that an appreciable portion of Or thus produced is in the growing mode (Orcxt).
Details on this point will be given elsewhere. 16) Now, assuming a sufficient
number of baryons are present and their distribution is homogeneous before the
BWH evaporation time tev, the baryon-to-entropy ratio is perturbed by a factor
Or at t = tev. Since the universe evolves adiabatically after t = tev, the perturbation of baryon number distribution grows as 0 r does until baryons decouple from
radiation.
Thus the large scale inhomogeneities of the present universe may be due to
oH generated at t = tc. In order to judge this possibility we now investigate the
growth of such perturbations. Since two different evolutional behaviors of the
universe are possible depending on the parameters a and (3 (see Eq. (2-10)), we
consider each case separately.
Case I:
a 5(32)5X 10- 19 (
1~8
r
2
~ tev> tm
Let us consider a fluctuation on a scale Lo at t = tc which becomes the size of
horizon cteq at t=teq. Then from Eqs. (2·7)~(2·9)
Lo
=
(1£)
112(~)213(~) 112
t
t
t
teq
m
ev
eq
(3·2)
where we have assumed that
12
teq = 3 X 10 sec
=6X1047(t/108)-ll.
Thus from Eq. (2'5), OHUC) is given by
OH( tc)= (nHL o3 )-112
*)
We do not consider the energy·momentum transfer across the scale L.
(3-3)
M. Sasaki, H Kodama and K. Sato
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(3·4)
This perturbation remains constant until t= tm and grows as t 2/3 for t m< t< tev
and as t for tev < t < teq. Therefore the density contrast at t = teq is given by
(3·5)
Since the Jeans length AJ = e s (7[/ Gp )1/2 is comparable to the horizon eteq at t = teq
and teq is comparable to tdec, the matter-radiation decoupling time, we can
approximately consider that this density contract o(teq) begins to grow rapidly at
t == teq. The perturbation on scales larger than Lo at t = tc evolves in the same
manner as the one for Lo and the density contrast at t = teq is given by
L"2.Lo.
(3·6)
For L < Lo at t = t c, the epoch th at which the size of perturbation becomes equal
to that of the horizon eth is given by
(3.7)
After this epoch the perturbation amplitude oscillates*l until t = teq "'" tdec.
fore
L<L o .
There(3'8)
Thus the spectrum of the density contrast has a peak at L = Lo.
Case II.'
a 5(32 < 5X 10- 19 ([/ 10 8 )-2
~
tev< tm
Consider a fluctuation on the scale Lo, the meaning of which is the same as
in Case I. The length Lo in this case is
_( tc
Lo-
t;;; )1/2 teq
(3'9)
Thus the initial fluctuation is given by
OH(tC)= (nHL o3)-1/2
= 3 X 10- 36(3-1/2([/10 8 )314
•
(3·10)
This time 0 H remains constant for t < tev and is converted into radiation density
constant 0 r at t = tev. By a detailed analysis we can show that the portion of 0 r
*1 For L smaller than the size corresponding to a mass <: 10 '2 Mo, the perturbation decays rapidly
due to photon viscosity.
Generation of Cosmological Perturbations
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generated as the most rapidly growing mode (ext) is given by lJrc:::=.(tev!tm)lJH at
t = tev,16) and it grows as t until t = teq. Therefore at t = teq we have
(3·11)
The fluctuations on scales other than Lo can be considered in the same way as in
Case I and we obtain the same results,
L<L o ,
L "?Lo.
(3·12)
If these perturbations at t = teq should explain the observed large scale
inhomogeneities of the universe without conflicting the homogeneity of the
microwave background, lJ(teq) must be about 10- 4 more or less. Assuming that
10- 5 < lJ(teq)< 10- 3 , we obtain the possible range in which the parameters a and (3
should lie. Figure 1 illustrates this constraint, in which the shaded and the dotted
regions correspond to the cases 1=10 8 and 1=10 10 , respectively.
2
-1
-2
-16
-15
-.14
-13
-12
-11
-10
-7
Fig. 1. The constraint on a and /3 is shown. The regions satisfying the condition 10- 5 <3(teq)
< 10- 3 are denoted by the shaded area for 1=10 8 and by the dotted area for 1=10 10 • The solid
and the dashed lines crossing the regions represent tev = tm for 1=108 and l = 10 10 , respectively.
In the region above (below) each of these lines, we have tev> tm Uev< t m).
§ 4.
Constraints on the nucleation rate of bubbles
The nucleation of bubbles can occur either by thermal fluctuations at finite
M. Sasaki, H. Kodama and K. Sato
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temperatures or by quantum tunneling which exists even if the temperature is
zero, !7).18) and in general the nucleation rate has a sharp peak at some temperature
due to the former process and becomes small but almost constant for other
temperatures due to the latter. Thus it is convenient to approximate PN( r) by
PN( r)= {
~r o( r- rT)+ ~: e( rT- r)+ ~~ e( r -
rT )}e( r),
(4-1)
where VT, v' and VQ are dimensionless constants and we assume that VT~VQ, v'.
Furthermore we assume v' = 0, Le., the nucleation at 0 < r < rT is negligible. This
assumption is valid as long as rT4:..(v')-11. Now let ~T be the radius of a bubble
at r= re in units of I nucleated at r= rT. Then
(4-2)
Using Eq. (2-3) we then obtain
_
[ _ ( ~T
ue-exp - VT 1 +~T
)3 - VQ_(re-rT
I
(4-3)
where v;= (4Jl"/ 3)v; (i = T, Q) and we have assumed (re- rT)/ l~ 1. Introducing
Ve defined by
ue=exp[- Ve],
(Ve=1.2 for ue=0.3)
(4-4)
we can rewrite Eq. (4-3) as
_ ( ~T)3
( re- rT
Ve - VT 1 + ~T + VQ
I
11)
6 .
(4-5)
Although we have assumed (re- rT)/ l~ 1, since the typical size of bubbles must
be small enough to maintain the global homogeneity of the universe, we must
require vQ(re-rT)/I4:..1. Hence we only consider the case IVT-Vel4:..1. Therefore defining 6 and P as VT = (1 +36 )Ve and P = vQ/ 3ve respectively and using Eq.
(4-2) we obtain an approximate equation to deter~ine ~T (Le., the time re),
(4-6)
This equation has the following limiting solutions:
(4-7a)
(4- 7b)
(4-7c)
Generation of Cosmological Perturbations
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Having obtained the time fc we now estimate the number density of TFV
domains. The number density of bubbles at f = f e with radius ~ ~ ~ + d~ in units
of I is given by
nB(fe;
~)d~= 13(1~~)4{VT(1+nO'(~-~T)+VQUe(1+~YQe(~T-~)}d~.
(4-8)
Then integrating Eq. (4-8) we obtain
1
00
nB( fc)=
-
-(
nB( fe;
~ )d~
",-3+ VQ ~)1-3
VT<;;T
3
(4-9)
From this equation we find that in case of Eq. (4- 7a) thermally nucleated bubbles
dominate nB(fe), while in cases of Eqs. (4-7b) and (4-7c) quantumly nucleated
bubbles dominate nBUe). This fact suggests that the percolation is due to
thermally nucleated bubbles if E> VQ and due to quantumly nucleated bubbles if
E> VQ. Since it can be shown by using Eq. (4 -8) that the overlapping of thermally
nucleated bubbles is large (~60%), we expect the number density of TFV
domains is approximately equal to nB(fe) in case of Eq. (4-7a). On the other
hand, in the other cases the percolation occurs in the region which would have
been totally in the false vacuum state if the quantum tunneling was absent. A
detailed analysis for these cases will be published elsewhere 19 ) but here we only
quote the result. It can be shown that TFV domains have a disk-like shape with
diameter ~ vQ l[ and thickness ~ VQ 1/3[. Therefore the number density nH( f e) is
proportional to V~/3. Then, also taking account of the fact that TFV domains
occupy the fraction Ue of the total volume, the parameter fJ in Eq. (2-5) is given
by
(4-10)
For the mass of BWHs there exists no presice argument to determine the value.
However from the result obtained in Ref. 11), we expect that M = O( t) which
means that the parameter a is
a= 0(1).
(4-11)
From Eqs. (4 -10), (4 -11) and Fig. 1 we now obtain the constraint which we
summarize in Table I. In addition to Table I, we note an important constraint
on ~T. As noted below Eq. (4-5), ~T should be much less than the length Lo.
M. Sasaki, H. Kodama and K. Sato
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Table I. The possible values for E and IIQ are listed under the assumption
that a is more or less close to unity and uc=O.1. For uc*O.l, the listed
values must be multiplied by (Uc/ 0.1)-"3.
E< IIQ
E>IIQ
1= 10'·
1=1OB
E
5x
10-5~
10-
3
1=10 B
1=10'·
1O-4~1O-3
1O-6~
IIQ
10-'
1O-5~
10- 3
From Eqs. (3-2), (3-9) and (4-7c) we therefore require
if
£<0.
(4-12)
From Table I and Eq. (4-12) we find that a fine tuning for ))T seems inevitable in
order to attribute the origin of the large scale inhomogeneities to the occurrence
of a cosmological first-order phase transition.
§ 5_
Discussion
It has been argued that when the phase transition finishes the number density
of generated monopoles nM is proportional to that of nucleated bubbles nB. 5 ),7)
However we can argue that it is more reasonable to consider nMex. nH. As
discussed in § 4, if thermally nucleated bubbles dominate nB we have nH::>e nB. On
the other hand, if quantumly nucleated bubbles dominate nB, the bubbles nucleated near the time fc are the ones which dominate nB as seen from Eq. (4-8).
Although these bubbles do not form TFV domains by themselves, they are the
very ones that are responsible for the percolation. This means that nM is
proportional to the number density of TFV domains left over at f = f c. Therefore
(5-1)
where f is a factor smaller than unity.
temperature Tv defined as
As usually done, by introducing a
(5-2)
the entropy density at t = tc is given by
s(tc)=
24~2 N l/4 Tv 3 ,
(5-3)
where N is the number of effectively massless spin degrees of freedom at the
epoch t = tc. From Eqs. (5 -1) and (5 -3) the monopole-to-entropy ratio at t = tc is
given by
Generation
0/ Cosmological Perturbations
12 N
= 1.5 X 10- ( 102
)-1/4( lOBI )-3/2//3 .
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(5'4)
It is known that the most severe constraint comes from the consideration on the
energy density of the present universe and it requires that (nMI s ) ~ 10- 25 .20 ) This
condition places the upper bound on /3 as
(5'5)
By comparing this with Fig. 1 we find the production of monopoles is sufficiently
suppressed for a wide range of values of /3. Thus although we need a fine tuning
on the nucleation rate of bubbles we can construct a consistent model in which
both problems of the monopole and the large scale inhomogeneity are
simultaneously solved.
Concerning the baryon number problem, we have to require tev~ tm if the net
baryons are generated at t = tc as in a conventional model of GUTs. Now from
Fig. 1 this is also guaranteed if a= 0(1) and we need no new mechanism for the
generation of baryons. The horizon problem seems a bit more difficult to solve.
If we require that the universe should expand by a factor more than 10 26
(II lOB t1/2 (?:- 10 2(Lol I); see Eq. (3·9)) which is needed to solve the horizon
problem, we obtain
(5'6)
If the radius ~T is as large as ~T ~ 5 x 10 2\ which is possible if, e.g., 1=10 10 ,
X 10- 3 and c= -0.1, we have
e CTi1 ?:-2x 10 4 (11 10 Bt 1/2
l/Q
=2
(5'7)
.*)
Equation (5'7) means that the temperature at r=rT is
T(rT)~5X10-5(lllOB)1/2Tv~5xlO-9
.
(5'8)
For I ~ 10 10 this temperature is close to the "temperature" of de Sitter universe ( ~
1-1) and we may conjecture that there could be a theory in which the desired
values for nucleation rate should be obtained by taking the gravitational effects
into account.21)-26) However, since there exists no reliable principle for calculating PN( r) when gravity can playa significant role, we cannot prove this conjecture at present.
Recently Linde 27 ) has proposed a modified supercooling mechanism which
*) Note that this implies rT < lOt and does not violate our assumption that the nucleation of bubbles
at r< rT is negligible provided l/ <{O.l.
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M. Sasaki, H. Kodama and K. Sato
declares that the present universe is inside one bubble which expanded by an
extremely large factor. However if this was so, the universe would be completely homogeneous and there is no possibility to generate the large scale inhomogeneities after the epoch of the phase transition. We therefore conclude
that if GUTs either provide no reasonable mechanism to allow the fine tuning of
parameters or lead to Linde's result, we must abandon GUTs.
Finally we comment on one delicate problem concerning the percolation by
quantum nucleated bubbles. Using the more precise version of Eq. (4°8), we can
show that the overlapping volume fraction of quantumly nu>cleated bubbles with
themselves and thermally nucleated bubbles always remains O( vQ) even when the
number of bubbles increases. 19 ) This reflects the fact that the later the bubbles
are nucleated, the smaller their maximum coordinate radius becomes in inverse
proportion to the cosmic scale factor. Thus the configuration of quantumly
nucleated bubbles changes similarly with the cosmic expansion. These considerations suggest that uc might be very small if vQ::S 10- 2 and further the percolation
by quantum bubbles might not occur if VQ is too small. 15 ) This problem brings
about another serious difficulty for cosmological models based on GUTs.
Acknowledgements
The authors thank Professor C. Hayashi and Professor H. Sato for continuous encouragement and stimulating discussions. Sasaki is indepted to the Japan
Society for the Promotion of Science and Kodama to the Soryushi Shogakukai for
financial aids. This work was supported in part by the Grand-in-Aid for
Scientific Research Fund from the Ministry of Education, Science and Culture
(56340021 ).
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