True vacuum bubbles and the origin of voids

Physics Letters B 265 ( 1991 ) 232-238 North-Holland

PHYSICS LETTERS B

True vacuum bubbles and the origin of voids Daile La

Centerfor ParticleAstrophysics, Universityof California at Berkeley, Berkeley, CA 94720, USA Received 16 December 1990; revised manuscript received 4 June 1991

Extended inflation theory can provide astrophysically interesting, nearly energy-empty bubbles at the epoch of matter-radiation equality, t = teq. The physical size and number of such bubbles at t= teq within the present physical horizon as a function of the model parameters is given. If non-baryonic cold dark matter (CDM) is not present in the universe, the filling fraction of these bubbles in the physical volume of the present universe becomes less than a few percent, so the bubbles are marginally detectable. If CDM is present in the universe, true vacuum bubbles can grow to occupy a major fraction of the volume of the universe today.

The f o r m a t i o n o f the large-scale structure o f the universe within the context o f the s t a n d a r d big-bang m o d e l ( S B B M ) has been one o f the least u n d e r s t o o d puzzles for decades. The leading p r o b l e m s are twofold. First, the (general relativistic) growing speed of the seed p e r t u r b a t i o n is not fast enough. D u r i n g the matt e r - d o m i n a t e d epoch, the growing rate o f the density p e r t u r b a t i o n is (Ap/p) ~c 12/3. Thus if there were density p e r t u r b a t i o n s (Ap/p)td at decoupling t=td, the fluctuations would have grown to (Ap/p)tp~ (tp/td)2/3(Ap/p),d at the present time t=tp. F o r (Ap/p)t~~10 -5 , a n d td~ 10-S tp, one finds (Ap/p)tp
The currently working model, which is a powerful model in describing the universe at m o d e r a t e scales, is the biased C D M m o d e l [4]. Unfortunately, this model has not fully escaped criticism, mostly stemming from observational aspects. Leading criticisms are: ( a ) t e m p e r a t u r e fluctuations o f the cosmic background radiation ( C B R ) could be much lower than the 10 -5 level, ( b ) quasars with redshift z > 5 m a y exist, and (c) there is no physically compelling evidence supporting the " b i a s e d " distribution o f luminous matter. Other d a t a discrediting the simpler C D M model are: ( d ) the large-scale coherent structures larger than ~ 100 Mpc [5] ~2, (e) the high velocity flow over ~ 80 M p c scale [6], and ( f ) the angular correlation function in the A P M survey [ 7 ]. The A P M data, especially, d e m a n d s the C D M m o d e l to have more power at large scales. This p a p e r is designed to introduce a new scenario for the f o r m a t i o n o f structure that is based on the extended inflation ( E l ) theory. Although it may be prem a t u r e to say that this model presents a viable alternative to pre-existing models, this m o d e l m a y distinguish itself by several new ingredients. First, this model provides the " b u b b l y " isocurvature p e r t u r b a t i o n to the post-inflationary universe. As El inflation ends (or at the beginning o f the post-inflationary r a d i a t i o n e p o c h ) , true v a c u u m bubbles

~ We reserve a note that depending on data, (6P/P)t, could be somewhat less than unity.

.2 For a model with quadrupole CBR anisotropy 5× 10 -5, a structure as large as 200h - t Mpc can be generated.

232

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

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nucleated during inflation can leave energy-empty voids randomly scattered in the sea of radiation [8] ~3, [9]. The key advantage of the bubbly isocurvature perturbations is that without interfering with constraints of the CBR, very large ( ~ 100 Mpc) or frothy structure (similar to the one observed in the CfA survey) [ 10 ] emerges rather easily. We will prove this claim in such a way that inflationary true vacuum bubbles can provide an astrophysically interesting number (and size) of energy-void regions in the present universe. Second, like other conventional inflation models, this model has (nearly) scale-invariant adiabatic perturbations, ,

where Mpi ~ 1019 GeV is the Planck mass, Tcuxis the symmetry breaking temperature of the grand unified theories (GUTs), and the quantity b is the model parameter. The model parameter is restricted to a range O ( 1 ) - O ( 1 0 ) for successful extended inflation [9]. For TauT~< 10165 GeV, we find (Sp/p)ta <~10 -5. Thus depending on Taux and b, we are left with the freedom to conceive a hybrid EI CDM model (i.e., a model with both the isocurvature and adiabatic perturbations) #4

The graceful exit problem. The EI theory is motivated by the "graceful exit" problem of the old inflation theory (OIT). The problem of the OIT is that, once inflation starts, it never ends [12,13]. This problem can be demonstrated in the following way. The success/failure of the bubble nucleation and percolation processes depends entirely on the dimensionless bubble nucleation parameter e. This is a quantity defined as the number of true vacuum bubbles nucleated in the Hubble four-volume H -4. For the OIT, the condition for enough inflation (in order to resolve major cosmological puzzles) is E< 10 -6, while the condition for successful percolation of the universe is E/> O (~o). The problem is that, for expo-

~3 I thank the referee for this information. ~4 In El models, the present density parameter Q(tv) can be less than unity even if the universe is geometricallyflat (i.e., the curvature index k=0); for details, see ref. [ l 1l-

15 August 1991

nential inflation models (like the O I T / N I T ) , the quantity E remains constant. Thus it is impossible to satisfy the two conditions simultaneously. E1 models are equipped with a time-dependent ~. This is a typical characteristic of E1 models based on most non-minimal gravitation theories, where gravitational coupling changes in time. During the inflationary epoch, the scale factor R (t) grows as a power law, which yields the time-decreasing Hubble parameter H4(t) = ( / ~ / R ) 4 ~ l i t 4. Consequently, ~ocH4(t) becomes time-increasing, ~oct 4. Thus in E1 models, even if ~ is small at the beginning of the inflationary epoch, the percolation of the universe is guaranteed as ~(t) grows larger than O ( ~ ) ~5

The "bubbly" isocurvature perturbation. The way these perturbations are generated is as follows. As a true vacuum bubble nucleates, the false vacuum energy (the latent heat of the phase transition) swept by its expanding bubble wall is transformed into the kinetic energy of the bubble wall. As bubbles collide, this kinetic energy is transformed into matter and radiation and flows into the interiors of the bubbles. It has been shown that for bubbles nucleated when E ( t ) < < O ( ~ ) , their mutual collision is quite rare [ 13 ]. Thus follows the failure of thermalization of the kinetic energy in bubble walls. However, for bubbles nucleated when x / / ~ ~ O ( ~ ) , which occurs near the end of the inflation, there are violent collisions between bubbles of comparable sizes. In this case, the thermalization of the kinetic energy deposited in the bubble walls is complete. Thus as extended inflation ends, the beginning of the (post-inflationary) radiation epoch is characterized by the presence of energy-empty voids (the interiors of bubbles nucleated at an early stage of inflation) randomly scattered in the uniform background radiation field. This paper is designed to investigate the astrophysical significance of the bubble perturbation. In the next section, we will investigate the evolution of true vacuum bubbles from the beginning of the radiation ~5 Let us note that the percolation of the universe is not a sulhcient condition for successful inflation. It is merely a necessary condition. One has to derive the bubble size spectrum and checkthe induced perturbations againstthe CBR fluctuation constraint. At present, this condition poses the most stringent constraint on workable E1models. 233

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PHYSICS LETTERS B

epoch to the present time. Derivation and conclusions in this paper are reached via analytical means. Computer simulation details will follow in subsequent publications.

Extended inflationary bubble spectrum. Near the end of (extended) inflation, where tb<~t' <~te, the number of true vacuum bubbles NB ( t ' ) is given by dNB(t' ) = 2 ( ( )R(t' )pf(t', tb) d t ' .

( 1)

Here, the quantity 2 - e(t)H4(t) is the quantum tunneling rate between two phases of the phase transition, R ( t ' ) is the scale-factor of the universe, the quantity

15 August 1991

wide use of the dimensionless quantity H~x(t~): the new unit. The quantity Hc~2×IOII(TGuT/IO 1~ GeV)Z GeV is the Hubble parameter at the end of inflation, and X(te) denotes the present physical scale x (tp), which has been scaled back to the end of inflation. The advantage of this unit is that physical units of the BBM and inflation theory are combined to yield a dimensionless number. The key-standard unit is derived from the present horizon Lh (t o ), whose extrapolated size at the end of inflation is Lh(te)~Lh(tp)[2.7 K/TouT]. Thus in H~X( te) units,

{ TOUT ) H~Lh(t~) ~ 4 . 3 × 102s \101Sh G e V J "

(3)

t'

pf=exp[--fat"

tb

)t(t" ) (ti dT ~3] ~¢, R(~')/] .J

is the (volume) fraction of the false vacuum, and tb and t~ denote the beginning and end of inflation, respectively [12,13]. For a bubble nucleated with a negligible radius at t' and that has grown at the speed of light until t = h, its proper (or physical) size is

[(,y '2

x(r,h)=H-t(t')

~

-1

]

.

From this, one can read off other important physical scales (at the present time) in Hexe units. For example, consider the present scale A(tp) .~ 30 Mpc. Since this quantity"IS 766 J Lh(to), by eq. (3),

HeA(G)=~HeLh(te) .~2.2X1023 ( Tout "~ \10'Sh G e V J "

(4)

Similarly, the size of a clime [ 1 c m ~ 10 -28Lh(tp) ] in H¢c¢ units is ~ 10 -4.

Substituting this into eq. ( 1 ) to eliminate t', we find

dNB 4 (HeLh) 3 d[H~x] ~ b IBEX] 4+4/b"

(2)

Thus one can compute the number of bubbles in a volume [HcLh(t~)]3 at tl---~te. Here, the quantity He is the Hubblc parameter at the end of inflation, and Lh(/¢) is the physical size of the present horizon Lh(to), which has been extrapolated back to l-~l c. Throughout this work, we will set Lh(tp)=-2c X H - ~ (tp) ~ 6000 h -~ Mpc, where c = 1 is the speed of light, and 0.4 ~> 1. This is a good approximation, since we will show shortly that H~x >> 1 for bubbles of astrophysical interest.

New unit for the physical size. Before proceeding with the main issue, let us introduce a new unit of distance. For the rest of this paper, we will make a ,6 Numerical values are from refi [ 141. 234

The wall-to-wall distance between true vacuum bubbles at decoupling. During the radiation epoch, bubbles shrink due to the hydrodynamic inflow of cosmic matter into the empty bubble interiors. This process continues until the matter-radiation equality t= teq. If the CDM is not present in the universe, the photon-baryon plasma flows into the bubble interiors with the speed ~ c/v/3. If the CDM is present in the universe, its flow distance is limited, since the CDM lacks mobility. Thus it is convenient to parametrize the flow distance during t= tc to t = teq as A(teq, ~): A(t~q, if) =,f~3 2H-~ (t~q).

(5)

For the photon-baryon plasma, A ~ ( c / x f 3 ) × 2 H - t (t~q) and A--,0 for the collisionless CDM. The case 0 < i f < 1 includes (a) the presence of the dark matter that decays during the radiation epoch, (b) stable cosmions that are loosely coupled to photons, or (c) possible (minor) corrections due to the expansion of the bubble interior during the radiation

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epoch. In this paper, however, we shall mainly be concerned with two limiting cases: ( ~ 1 and ( ~ 0. Finally, let us note that the quantity A(6q, () represents the wall-to-wall distance between bubbles at decoupling. Now let us determine the physical size ofA(6q, () at the present time, A(tp, (). The quantity A(6, 5) can be computed in the following way. First, note that the present CBR temperature Tv ~ 2.7 K ~ 2.3 × 10- ~3 GeV, and the temperature of the universe at decoupling T(teq)~5.5×lO-9"£2(6)h 2 GeV. For adiabatic expansion of the universe RT=const., which yields g ( t p ) _ Teq ~2.4X R(teq) Tp

104g'~(lo)h

2 .

(6)

Since R(6)/R(teq ) = [H(6q)/H(6 ) ]2/3, H-'(teq) =H-'(6)

(R(teq)~ 3/2 \ R(tp) /

~2.7X 10 -7 H - l (/p) [g2(6)h 2] -3/2

(7)

Therefore, A(6q, 5) = ~

2H-J ( 6 )

~l.6(XlO-7[g2(tp)hZ]-3/22H-J(tp).

15 August 1991

A(te, 5) ~-~ 8 . 2 ~

1011GeV_ l g2_l/2h_ 2 (1015 GeV~

\ yo~ /"

Therefore, HeA (te, 5) ~ 1.65× 1023 ~ - l / 2 h - 2 ( TGUT/ 10 ~5 GeV). We will use this quantity when we compute the filling fraction of bubbles at the present time.

The bubble spectrum at the end of inflation. Let us first find the number of bubbles in a volume L~(6), i.e., the number of bubbles in the present horizon volume scaled back to the end of inflation. Second, we have to sort out bubbles which are larger than A ( teq , 5) at t = 6 but not larger than L~ (te). [ For simplicity, we will denote A ( teq, 5) which has been scaled back to t=6 as A(te, 5).] The presence of the lower limit of the bubble size is apparent, since bubbles smaller than A(te, 5) have been erased by the inflow of cosmic matter during the radiation-dominated epoch. Thus multiplying by the step function 0([HeX] -[H~(6, 5)]) on the right hand side of eq, (2) [0(y) = 1 for y > 0 and zero otherwise] and integrating it, one finds the number of bubbles in a volume [H~Lh(te) ]3 having a size x larger than A(te, if):

(8)

4[H~Lh(6) ] 3 b

NB[te;x>A(6'~)]~ Substituting 2 H - ~(tp) = Lh (tp) ~ 6000h - l Mpc 9.4X 1041h -~ GeV -~ into eq. (8), we find

1.5~X 103s GeV-lg2-3/2(tp)h-4.

f 0

]

d[Hcx]

[Hexp+4/b

(9)

Thus the physical separation between bubbles at the present time is

× O( [Hex] - [He A(6) ] ) 4 b

4[HeLh(6)l- / ~2(Lh(te)~ 4+3b

R(to)

A(6, 5 ) ~ A(teq, 5) R(6q) ~ 23(£2-l/2(tp)h-2 Mpc.

He[Lh(te)+A(te,~)

X

A(teq, 5) ~ 9.6(X lO-4#-3/2(tp)h -4 Mpc

(ll)

k kx(te) ]

3+4/b

-ll J" (12)

(10)

This is the desired formula. For ( ~ O (~o), h ~ 1, and

£2(to) ~ 1, A (to, 5) ofeq. (10) yields a separation scale not much different from those of the CfA voids [ 11 ]. Further, for later convenience, let us compute A (t~, 5). Since A(te, 5) =A(teq, ()R(te)/R(teq), forR(te)/ R(teq)~5.5XlO-24g2(tp)h2(lOlSGeV/TGuT), we find

The formula is valid up to order x(tc)/L~(&). We note that eq. (12) has two zeros: for b-*0 and b--,oo. The first zero corresponds to the case when the frequent collisions with other bubbles prevent bubbles from growing larger than their typical Hubble radius, H g ~, and the second limit occurs when the observable horizon volume is engulfed by a single bubble. Both are expected features.

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The bubble spectrum at matter-radiation equality. The next task is to show how the bubble spectrum of eq. ( 12 ) evolves until t = teq. This job can be accomplished in the following way. First, note that the right hand side ofeq. (12) is independent of He. Thus one can substitute Lh (te)/x (te) on the right hand side of eq. (12) by Lh(teq)/X(teq). Then NB(te) simply becomes NB ( teq ): NB(teq)~ 4[HeLh(tD]-4/a4+3b 2 [L~(teq)~ \ X(teq) .1

_ 1 . (13)

Thus via eq. (13), one can compute the number of bubbles of size x(teq), which is larger than A(teq, ~)/ Lh (teq), but not larger than Lh (teq). This is not enough, though. What we really want is eq. (13) expressed in terms o f x ( t p ) , i.e., x(teq) expressed in terms of its present size X(tp). Since the interiors of the bubbles expand as R(t)oct 213+2/15, which is faster than the global expansion R (t)oc t 2/3 during the matter-dominated epoch [15 ], this correction should be incorporated when expressing x(te) in terms of its present counterpart X(tp). This correction can be achieved in the following way. First, we will express the scale X(tp) as a fraction of the horizon scale Lh(tp), i.e., we define X(tp)--qLH(tp), where r/is a dimensionless number. Then

.

volume. The condition NB(to) < 1 yields 7.5 N2h/(4+3b) qm,x ~ [ 1 + ( 4 + 3b) (4.3/4)4/blol°°/b] b/(n+3b)

(lO~5h G e V ) 4/(4+3/~)

×[O(t~)h2l,/5,

~ 7.5q-1 [g2(tp)h 2] 1/5

~

(16)

/

Now one can compute the size of the largest bubble in the present horizon volume. For b=20, 16, 10, 8, . Q ( t p ) ~ h ~ l and TGUT~1016"5 GeV, one finds Xmax=~maxLh(tp) as ~ 336 Mpc, ~ 142 Mpc, ~ 14.5 Mpc, ~ 3.4 Mpc, respectively. The number of smaller bubbles increases as ~/-(3+4/b); for example, for b= 20 there is one bubble (in the present universe) with a size larger than ~ 336 Mpc, and eight bubbles larger than ~ 168 Mpc, 64 bubbles larger than ~ 84 Mpc, ..., etc.

The fillingfraction of the bubble. The last question is whether those large bubbles are, indeed, observables. To answer this question, it is necessary to compute the filling fraction of bubbles f(tcq, ~). By eqs. (2) and ( 11 ), one can show that the filling fraction takes the form tte[Lh(te)+A(te,~) ] f d[Hex] f(teq, ~ ) - 4zr4 3 b 0

.1/5

Lh(tp)(teq/tp) 2/3 =~_1 (teq~ x(te) - x(tp)(teqltp) 2/3+2115 \Tp)

Lh(t~)

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;'(

{He [ x - A(te, () ] }3 [HeX]4+4/t, O([Hex]-[HeA(te,~)]) 4zc( ~c 3 - [HcA(~, ~)]4i~,

(14)

1

)

[HeLh(te)]411;

Substituting this into NB (/eq ) of eq. ( 13 ), we find

47r_3F{6.3~ 4/b

4 10 -l°°/b NB[tp;q]~ (4.3)4/h 4 + 3 b

X

(15)

Here qmi~
(17)

Here,

TGUT ~--4/t, 10~ShGeV ]

lO_96/t,g22/t,(tp)h8/b

LkT)

b 4 bq{lOI5GeV\4/t' _(23),1,,×10 --104'h'jt, )

( 2 × (7"5)3+4/'' ) q3+4//~ [.Q(tp)h 2 ] 3/5+4/5b__ 1 (

×

-

K-2 ( l-

3

l-V3 b

1 -t213b) .

We neglected terms of order larger than A/Lh. For O( 1 ) < b~ 25, we find O(~o) ~
Volume 265, number 3,4

f(t¢q, ~) ~

PHYSICS LETTERSB

- 4 / b x o ( 10 - 2 )

( 18 )

at matter-radiation equality. For ~ 1, which corresponds to the CDM-free case, we find f(teq, ~ ) ~ l 0 -96/b. In this case, the filling fraction has a rather sensitive dependence on b. For small b, f(tCq) << 1. The filling fraction becomes negligible, a consistent limit. For b near its upper limit b~25, however, f(leq)~ 10-3"9. In this case, let us compute the filling fraction at the present timef(tp). The comoving energy-void regions during the matter-dominated epoch grows a s f ( t > t~q) ~c (t/teq) 2/5. T h u s f ( t p ) ~ 10-39(tp/teq) 2/5~ 10 -2. Therefore, we reach the conclusion that true vacuum bubbles can fill a few percent of the volume of the present universe. Now let us consider when the motionless CDM is present in the universe. In this case, (TGuT~ -4/' 0
/

\2/5

~12.

(20,

Then the filling factor of during t, < t~< tp is

cosmic matterfm(t, t,, ~)

fm(t,t,,~)~exp[-f(t,,~)H(t,)3"~(t,t,)],

(21)

where the quantity ~ ( t , t, ) denotes the net increase in the void volume during t, < t ~
( to /2J,1 \7,j j

~exp{--

lOl2/5[f2(to)h 2 ] 3/sf(tcq, ( ) } .

For b ~ 2 0 and T c u v ~ 10 ~5 GeV,

(23)

f(t~q, ( ) ~ ( - 4 / b

x 10-4-8 '

fm (to, t,, ~ ) ~ e x p ( \

1

~o.2X 1024)['Q(tp)h2] °'6" (24)

For f2(to)h2~ 1 and ~< 10 -12 (this is a condition satisfied by most of the motionless CDM), we find fm(tp, t,, ~') ~ e - i . Thus, if the CDM is present in the universe, true vacuum bubbles can occupy a few tens of percent of the volume of the present universe. Indeed, the CDM E1 model may be a promising candidate for explaining the "bubbly" large-scale structure observed so far.

To summarize. Extended inflation theory can predict an astrophysically interesting size of the (nearly) matter-empty voids at the present time. This claim does not depend on whether the CDM is present in the universe or not. For the CDM-free universe, these bubbles occupy only a minor volume fraction (up to a few percent) of the present universe, hence they become marginally detectable. If the (nearly motionless) CDM is present in the universe, these bubbles occupy a few tens of percent of the volume of the universe. This work has improved by a useful conversation with David Weinberg #7. I am also grateful to Changbom Park for his help on the earlier version of this work and for useful comments on the quantity A(teq, ~) and the CDM model. This work was supported by National Science Foundation Cooperative Agreement AST8809616. ~7 NB(/eq)~c l/[x+A(leq)]3+4/b, where x>0 is an alternative

form of the bubble spectrum at t = teq. I thank D. Weinbergfor pointing this out.

(22)

t,=teq [1/2f(tcq, ~)]5/2, fm~eXp[_f(tcq, ) (to~ teq) Z/S ]. Therefore,

Since

(R(,.> l "lj

fm(t o, t,, ~)-~exp[--f(teq, () \ ~ /

(19)

As in the previous case, the energy-void regions surrounded by the CDM grow as f(t>teq, ~)ocf(t¢q, ~) (t/tcq) 2Is. This is a crude estimate, though, since as f ( t ) ~ ~, overlapping corrections of the voids become important. If we denote this moment as t= t,,

f(t,)=_f(t~q)(t~e,)

15 August 1991

References

[ 1] See J. Silk, Inner space/outer space (University of Chicago Press, Chicago, 1986) p. 143; 237

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W. Saunders, C. Frenk, M. Rowan-Robinson, G. Efstathiou, A. Lawrence, N. Kaiser, R. Ellis, J. Crawford, X. Xia and I. Parry, Nature 349 (1981) 32. [2] A.D. Linde, Phys. Lett. B 108 (1982) 389; B 129 (1983) 177; A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 ( 1982 ) 1220; F. Abbott and M.B. Wise, Nucl. Phys. B 244 (1984) 541; F. Luccin and S. Matarrese, Phys. Rev. D 32 (1985) 1316; Phys. Lett. B 164 (1985) 282; F.S. Accetta, D.J. Zoller and M.S. Turner, Phys. Rev. D 31 (1985) 3046. [3] A. Guth and S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110; S.W. Hawking, Phys. Rev. Lett. 115 (1982) 295; A. Starobinskii, Phys.Rev. Lett. 117 (1982) 175; J.M. Bardeen, P.J. Steinhardt and M.S. Turner, Phys. Rev. D28 (1983) 679. [4] G.R. Blumenthal, S.M. Faber, J.R. Primack and M.J. Rees, Nature 311 (1984) 517; S.D.M. White, C.S. Frenk, M. Davis and G. Efstathiou, Astrophys, J. 313 (1987) 505. [ 5 ] T.J. Broadhurst, R.S. Ellis, D.C. Koo and A.S. Szalay, Nature 343 (1990) 726; C. Park and J.R. Gott, Princeton Observatorium preprint POP-363 (1990).

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[6] D. Lynden-Bell, S. Faber, D. Burstein, R. Davies, A. Dressier, R. Terlevich and G. Wegener, Astrophys. J. 326 (1988) 19. [7 ] S. Maddox, G. Efstathiou, W. Sutherland and J. Loveday, Mon. Not. R. Astron. Soc. 242 (1990) 43. [ 8 ] For earlier related works see D. Seckel and M. Turner, Phys. Rev. D 32 (1985) 3178; L. Kofman and A. Linde, Nucl. Phys. 282 (1987) 555; L. Kofman, A. Linde and J. Einasto, Nature 326 ( 1987 ) 48. [9] D. La and P.J. Steinhardt, Phys. Rev. Lett. 62 (1989) 376; Phys. Lett. B 220 (1989) 375; D. La, P.J. Steinhardt and E. Bertschinger, Phys. Lett. B 231 (1989) 231. [ 10] V. de Lapparent, M.J. Geller and J.P. Huchra, Astrophys. J. Lett. 302 (1986) L1. [ 11 ] P. Steinhardt, Nature 345 (1990) 45; D. La, Center for Particle Astrophysics preprint CfPA-TH90-015 (1990). [ 12] A.H. Guth, Phys. Rev. D 23 ( 1981 ) 347. [13] A.H. Guth and E.J. Weinberg, Nucl. Phys. B 212 (1983) 321. [ 14] E. Kolb and M. Turner, The very early universe (AddisonWesley, Reading, MA, 1990) pp. 76, 504. [15] K. Thompson and E. Vishniac, Astrophys. J 313 (1987) 517; E. Bertschinger, Astrophys. J. Suppl. 58 (1985) 1.