_-_r- S

Fermi National Accelerator Laboratory

SUSSEX-AST 90/5-1

FNAL--PUB-90/62-A

(May, 1990)

Baryogenesis in Extended Inflation. I. Baryogenesi s

via Production and Decay of Supermassive Bosons

0

I ,..-

O_ r'r_

Z _O

John D. Barrow, 1 Edmund J. Copela-ud, 1 Edward W. Kolb, u and Andrew R. Liddle _

2Division of Physics and Astronomy

University of Sussez, Brighton BN1 9QH U.K.

_NASA/Fermilab Astrophysics Center

Fermi National Accelerator Laboratory

Batavia, illinois 60510-0500 U.S.A.

and

Department of Astronomy and Astrophysics

Enrico Fermi Institute, The University of Chicago

Chicago, Illinois 60657 U.S.A.

/Z/- 7Z

Abstract

We consider baryogenesis occurring during the thermal.ization stage at

the end oi' extended inflation. In extended inflation, the Universe passes

through a first-order phase transition via bubble nucleation; inflation

comes to a_a end when bubbles collide and their collisions convert energy stored in the bubble walls into particles. This naturally provides

conditions well out of thermal equilibrium in which baryon number violating processes may proceed; we estimate the amount of baryon asymmetry

which may be produced this way. The avoidance of a monopole or domdnwall problem can also be ensured and isothermal density perturbations

may arise as a remnant of spatial variation in the baryon asymmetry.

_. Operated by Universities Research Association Inc. under contract with the United States Department of Energy

I. INTRODUCTION

Recently the spirit of the orginal inflationary cosmology 1 has been revived _n the context of "extended" inflationary models by La and Steinhardt. 2 In such models, the

Universe is trapped in a false vacuum state as it cools from high temperatures; the

energy density of this false vacuum then drives a rapid expansion in the scale factor

of the Universe, solving a variety of cosmological conundrums. Inflation is ended by

the quantum-mechanical process of formation of bubbles of the true vacuum via tunneUlng; bubbles form with a characteristic size determined by microphysics s (provided

gravitational corrections are small). These bubbles then expand at the speed of light,

eventually colliding with adjacent bubbles. The percolation of these bubbles then

brings the inflationary era to an end. The original %ld inflation" scenario of Guth 1

was flawed by what became known as the "graceful exit" problem: regions trapped

in the false vacuum state expand exponentially; the expansion genetically overcomes

the decay to the true vacuum state and percolation of the Universe by true-vacuum

bubbles never occurs. 4 Extended inflation circumvents this obstacle by considering

modified gravitational theories (such as the 3ordan-Brans-Dicke theory) in which

the gravitational constant may vary. In such theories the inflationary expansion is

a rapid power-law rather than exponential, and the exponential bubble nucleation

rate will always eventually overcome the expansion and bring the inflationary era to

a satisfactory end.

As pointed out by Welnberg s and by La, Steinhardt, and Bertschinger, s the original extended inflation model based on a ,Jordan-Brans-Dicke theory fails because

bubbles nucleated early in inflation have time to grow to large sizes. Such bubbles

do not have time to thermalize before radiation decoupling (a lower bound on the

thermalization time being easily obtained simply from causality) and would cause
unacceptablylarge distortions in the microwave background. To resolve this conflict,

severM more involved models have been proposed, 7,s,9,I° with the common theme of

arranging that the production of bubbles early in inflation is suppressed. This appears to be a necessary ingredient for a successful extended inflation model, and here

we sha]] assume, without tying ourselves down to a particular model, that the vast

majority of bubbles are produced in a rapid burst right at the end of inflation. These

bubbles have little time to grow before the inflationary era is brought to an end by

percolation. A detailed examination of the dynamics of extended inflationary models

is given in Ref. Ii. We note also that it is simply the fa_ng Hubble expansion rate

that enables the phase transition to proceed to completion in extended inflationary

models and similar conclusions could be drawn in any power-law inflationary model lu

in which a first order transition occurs. Thus the picture we shall present is more

general than the "extended" inflationary universe model which we use to provide a

context.

In this paper we will not address problems in the dynamics of the bubble nucleation

rate, and only assume that some satisfactory explanation will result in an acceptable

bubble distribution at the end of extended inflation. Rather, we will concentrate

on the inflaton sector of the theory, and investigate whether an acceptable baryon

asymmetry can be produced after extended inflation.

One of the most important results in particle astrophysics is the development of

a framework that provides a dynsJ_icM mechanism for the generation of the baryon

asymmetry. Before reviewing the basic ingredients necessary, it is useful to quantify

exactly what is meant by the baryon asymmetry. The baryon number density is

defined as the number density of baryons, minus the number density of antibaryons:

ns -- nb-n_. Today, nB = ns = 1.13x 10-s(f_Bh _) cm -s, where h is Hubble's constant

in units of 100 km s-a Mpc -1. Of course, the baryon number density changes with

2
expansion, so it is most useful to define a quantity B, called the bcr_jon number of

the universe, which is the ratio of the baryon number density to the entropy density

s. Assuming three species of light neutrinos, the present entropy density is s = 2970

cm -3, and the baryon number is

B = 3.81 x 10-_(f_vh2). (1.1)

Primordial nucleosynthesis provides the constraint 0.010 < fish 2 < 0.017, is which

implies B = (3.81 to 6.48) x 10 -11. So long as baryon number violating processes

are slow compared to the expansion rate and no entropy is crested in the expansion,

B is constant.

A key feature of inflation is the creation of a large amount of entropy in a volume

that was at one point in causal contact. The creation of entropy in inflation would

dilute any pre-existing baryon asymmetry, so it is necessary to create the asymmetry

after, or very near the end of, inflation. In order for the baryon number to arise after

inflation in the usual picture, where CPT invariance and unitarity hold, it is necessary

for three criteria to be satisfied: baryon number (B) violating reactions must occur,

C and CP invariance must be broken, and non-equilibrium conditions must obtain.

There are two standard scenarios for baryogenesis: 14 In the first picture the baryon

asymmetry is produced by the "out of equilibrium" B, C, and CP violating decays

of some massive particle, while the second scenario involves the evaporation of black

holes, is We shall discuss the role of the latter mechanism in a second paper on this

subject.

In the out of equilibrium decay scenario, the most likely candidate for the decaying

particle is a massive boson that arises in Grand Unified Theories (GUTs). In the

simplest models, the degree of C and CP violation is larger for Higgs scalars than for

the gauge vector bosons, so we will assume that the relevant boson is a massive Higgs

3
particle. This Higgs is also taken to be different from the inflaton. The Higgs of GUTs

naturally violate B. The origin of the C and CP violation necessary for baryogenesis

is uncertain. It is practical simply to parameterize the degree of C and CP violation

in the decay of the particle. To illustrate such a parameterization, imagine that

some Higgs scalar H has two possible decay chanels, to final states fl, with baryon

number B1, and f2, with baryon number B2. Consider the initial condition of an

equal number of H and its antiparticle, _r. The H's decay to final states fl and f2

with decay widths F(H -4 fl) and F(H --, f2), while the H's decay to final states

L and L withdecaywidths L) and L). Thedecaysproducea net

baryon asymmetry per H-/_ r given by

= _: B,r(H -_/,)- r(H -_ ],) (1.2)

i=1,2 FH

where FH is the total decay width. Of course e can be calculated if one knows the

masses and couplings of the relevant particles. Reasonable upper bounds for e are in

the range of 10 -2 to 10 -s, but it could be much smaller. For more details, the reader

is referred to Ref. (16).

The non-equilibrium condition is most easily realized if the particle interacts

weakly enough so that by the time it decays when the age of the Universe is equal to

its lifetime, the particle is nonrelativistic. Then the decay products will be rapidly

thermalized, and the "back reactions" that would destroy the baryon asymmetry

produced in the decay will be suppressed.

In most successful models of new inflation the reheat temperature is constrained to

be rather low. This is due to the fact that new inflation requires flat scalar potentials

in order for inflation to occur during the "slow roll" of the scalar field toward its

minimum. In order to maintain the flatness of the potential, the inflaton field must

be very weakly coupled to all fields so that one-loop corrections to the scalar potential
do not interfere with the desired flatness of the potential. The feeble coupling of the

inflaton to other fields means that the process of converting the energy stored in the

scalar field to radiation ("re'heating) is inherently inefficient. Although it is possible

to overcome this difficulty in several ways, it remains a concern for new inflation.

The thermalization process of bubble wall collision at the end of extended inflation

provides a natural arena for baryogenesis in the early Universe, as it automatically

creates conditions far from thermal equilibrium, exactly as required for B, C, and

CP violating GUT processes to produce an asymmetry. The aim of this paper is to

investigate how the baryon asymmetry produced at the end of extended inflation can

be estimated.

In this, the first paper of two, we consider the production in bubble-wall collisions

of supermassive baryon-number violating bosons whose decays generate the baryon

asymmetry. In the second paper we consider the further possibility that the bubblewall collisions may produce a significant density of black holes, which then decay via

the emission of Hawking radiation. These decays may lead to the radiation of more

baryons than antibaryons, providing an alternative mechanism for the generation of

the baryon asymmetry.

In the next section we wiU describe the Universe at the end of extended inflation.

In particular we will derive the physical parameters that describe the true-vacuum

bubbles. In Section III we will discuss baryogenesis from the decay of Higgs particles

produced in the bubble wall collisions. The final section discusses our results.
II. THE END OF EXTENDED INFLATION

Most of the work on extended inflation has concerned the gravitation sector of

the theory, which will not concern us. Our only assumption about the gravitation sector is that the parameter that determines the efficiency of bubble nucleation,

eN(t) = PN(t)/H4(t), where FN is the nucleation rate per volume and H is the expansion rate of the Universe, has a time dependence that suppresses bubble nudeation

early in inflation, then rapidly increases so inflation is brought to a successful conclusion in a burst of bubb]e nucleation. This will be the case so long as H(t) falls as t

increases. Hence we see that it could occur in any power-law inflationary universe lz

driven by an appropriate phase transition. In fact, our experience with "new" and

"chaotic" inflation, as well as inflation driven by higher-order curvature terms in

the gravitational Lagrangian, lr,ls,ls, indicates that we can have "intermediate" and

hyperinflation where the scale factor of the Universe increases as exp(Atn), with A

constant, and n < 1 or n > 1 respectively. When n > 1, as is possible in some

quadratic Lagrangian inflationary scenarios, _° we WIU have//> 0 and a phase transition could not complete even if the effective potential allowed one to occur. However,

when 0 < n < 1, as considered in Ref. 18, the phase transition could proceed to

completion just as in the power-law and extended inflationary models.

Here we shall refer to the extended inflationary model for definiteness and we shall

be concerned with the inflaton sector of the theory. So far the only restriction on the

inflaton sector has been that it must result in a first-order phase transition. Here, we

examine the results of requiring that it must also produce a baryon asymmetry.

In order to keep our discussion as general as possible, we will not specify any

particular inflaton model, but rather describe the salient features of the potential in

terms of a few parameters that can be easily identified with any scalar potential that

6
undergoes spontaneous symmetry breaking. We denote the inflaton field throughout

to be _r, which has a potential of the general form suitable to provide a first order

phase transition necessary for extended inflation. The parameters of the potential

are assumed to be:

1. _ro, the energy scale for SSB, i.e., the VEV of the scalar field.

2. A, a dimensionless coupling constant of the inflaton potential. We will assume

that the potential is proportional to A.

3. _, a dimensionless number that measures the difference between the false and

the true vacuum energy density via pv = _A_r_. _ must be less than unity for

sufficient inflation to occur; this is also precisely the condition that allows the

thin-wall approximation (discussed below) to be made.

From these few parameters we can find all the information we require about the

bubbles formed in the phase transition, s For instance, an important parameter is

the size of bubbles nucleated in the tunnelling to the true vacuum. In the thin-wall

approximation, the size of a nucleated bubble is given by

Rc ~ ( A1/2 o) -1. (2.1)

Bubbles smaller than this critical size will not grow, and it is exponentially unlikely

to nucleate bubbles larger than this critical size. We will assume that all the truevacuum bubbles are initially created with size R = Rc.

Another crucial parameter is the thickness of the bubble wall separating the truevacuum region inside from the false-vacuum region outside the bubble. For the potential described above, the bubble wall thickness is

A _, (A1/2cr0) -1. (2.2)
Note that the ratio of the bubble thickness to its size is A/Rc -,, _; as advertised, if

_ 1, the thin-wall approximation is valid.

Finally, the energy per unit area of the bubble wall is

~ A1/2 o . (2.3)

At the end of extended inflation all of the energy is in these bubbIe walls.

We must have some idea of the size of bubbles at the end of inflation, when

bubbles of true vacuum percolate, collide, and release the energy density tied up in

the bubble walls, so creating the entropy of the Universe. The bubbles of true vacuum

are nucleated with size R - Re. After nucleation the bubble will grow until it collides

with other bubbles. We now show that the size of the bubble at the end of extended

inflation is still approximately Re.

Consider first the growth of the bubble in comoving coordinates. If a bubble

is nucleated with coordinate radius r at time tsuc, then at some later time t the

coordinate radius of the bubble will have grown by an amount At(t, tsuc), given by

At(t. _' dt' (2.4) t ,o) = .,.,o '

where a(t) is the Robertson-Walker scale factor. Typically in extended inflation

a(t) grows as a power-law in time, say a(t) oct v, p >> 1. If this is true, then

Ar(t,tNuc) ~ tNUC 1-j' --t 1-_, which approaches an asymptotic value Ar(oo, tNUC)

tNUC 1-P. Clearly bubbles nucleated at late time (large tNUC) will have little growth in

coordinate radius, and any increase in the physical size of such a bubble is due solely

to the growth in the scale factor between the time the bubble is nucleated and the

end of inflation.

The physical size of a bubble nucleated at time tNuc is related to its coordinate size

by R(tNuc) = r(tNUC)a(tNUC) = P_. If there is negligible growth in the coordinate

8
sizeof the bubble between the tNUC and end of inflation tEND, then at the end of

inflation the bubble will have a physical size

R(tEND) _- _---- 7'(_NUC)g(tEND) = RC[a(tEND)/a(tNUC)]" (2.5)

With the assumption that successful extended inflation will have a burst of bubble nucleation at the end of inflation, and that there is not much growth in the

size of a typical bubble between the time of nucleation and When it collides with

other bubbles at the end of inflation, for a first approximation we will assume that

at the end of extended inflation all bubbles have the same size, R = a_, where

a _ a(tEND)/a(_NUC) _" 0(1). In the concluding section we will discuss possible

implications of relaxing this assumption.

We conclude this section by a description of the Universe at the end of extended

inflation. To a good approximation the Universe is percolated by bubbles of true

vacuum of size ._ = arc, with all the energy density residing in the bubble walls.

We have spoken of the "end of inflation" as if it was a well-deflned time, but in fact it

is not. We simply define the end of inflation to be when the Universe is first percolated

by bubbles of true vacuum. The "fiLling fraction" of the Universe at percolation is

some fraction fv. Based upon Monte-Carlo simulations, it is expected that f_" ,_ 0.3. 21

The final step is the release of this energy into radiation via bubble-wall collisions.
III. BARYOGENESIS BY DIRECT PRODUCTION

OF SUPERMASSIVE BOSONS

Let us concentrate on a single bubble of radius R. The total mass of the bubble is

M = 47r_TR_ .._ 41rA'/2_3oR_. (3.1)

The collisions of the bubble wafts produces some spectrum of particles, which are

subsequently thermafized. The mean energy of a particle produced in the collision

is of order of the thickness of the wall, (E) -._ A-l, and hence the mean number of

particles produced in the collisions is

<N} __ M/(E) .._ 41rAA'/'a'3oR 2. (3.2)

In general, the bubble collisions wiI] produce all species of particles, at least all species

with masses not too large compared to (E). In the following we will assume that

this is the case for the baryon-number violating Higgs particles. If the Higgs mass

exceeds A -1 by a significant amount, we can expect some suppression, presumably

exponential, in the number of Higgs formed. Tiffs possibility will be discussed at

the end of tiffs section. For now, we simply parameterize the fraction of the primary

annihilation products that axe supermassive Higgs by a fraction fs. The typical

number of Higgs particles produced per bubble is

(NR) "_ fR(N) "_ 4_f_A_l/2_,aoR2. (3.3)

We will now assume that the only source of the supermassive Higgs is from the

primary particles produced in the bubble-wall collisions. This will be true if the

reheat temperature, T/Is, is below the Higgs mass. (Note that throughout this paper

we have set the Boltzmann constant equal to 1.) The validity of this approximation

will also be discussed at the end of this section.

10
The Higgs particlesproduced in the wall collisionsdecay, producing a net baryon

asymmetry e per decay, where e isgiven in Eq. (1.2). Hence, the excess of baryons

over antibaryons produced from a singlebubble, Are - Nb --N_, is given by

Nv -- c(NB) ~ 4_refR_r_R a, (3.4)

where we have substituted in for the bubble thickness from Eq. (2.2). This results in

a baryon number density of

nB= fv Ns/(4rrRa/3) = 3e.fvfHo'gR -1. (3.5)

We must now calculate the entropy generated in bubble-wall collisions. As stated

above, the mass of a bubble is M = 41r_Ax/2R 2. Thermalization of the mass in

the bubble walls will redistrlbutcthis energy throughout the bubble, resultingin a

radiation energy density (recallthat at percolation,a fractionfv of the Universe is

taken up by bubbles of radius R)

PR ~ fvg/(47rRa/3) ~ 3fvA'/acr_o/R. (3.6)

We may now express the reheat temperature to the radiation energy density via

g'Tr2T4 (3.7)

PR : "_- RIt,

where 9. is the effective number of degrees of freedom in all the species of paxticles

which may be formed in the thermaiization process. From this we obtain the entropy

density, 8, produced by the thermMization of the debris from bubble-wa_l collisions:

2_'2 --3 ,,,1/4 ¢3/413/S,.eg/4_-3/4

a = -_9.T_z ~ 2.3 (3.8) _° J V "_ "0 "" "

From Eqs. (3.5) and (3.8) we can calculate the bsxyon asymmetry B as

B =_ ~ -'Is (3.9)

11
As stated above, a successful extended inflation model is expected to have the

feature that a typical bubble does not grow significantly between nucleation and

percolation, so we can assume that the bubble size at percolation is R = arc =

a(_,_I/2_o)-I. The baryon asymmetry B is then given by

B = ef_/'fH a-1/4g:l/4_-l/4_ 1/4. (3.10)

Provided the mass of the Higgs is less than T_B, fH is just gH/g., where gH is the

number of Higgs degrees of freedom. Substituting this in gives the final result

B = ef_/igs a-1/4g=S/4)_-1/4_ 1/4. (3.11)

This allows us to make numerical estimates of B based on sample values of these parameters. Notice that the dependence on both A and _, which are the two parameters

on which the inflaton's potential depends, is very weak, as is that on the filling factors

fv and a. The important contributions axe the degree of asymmetry in CP violating

Higgs decays and the number of particle species available for production in the wall

collisions. Numerical estimates for B based upon this expression will be made in the

concluding section.

We now elaborate upon the implications of two assumptions of our scenario. The

first assumption is that the mass of the Higgs is not much larger than the typical

energy of particles produced in bubble wall collisions, i.e., mH < A-1 = _1/2_0" If

we take GUT theories as a guide, the Higgs mass is of order _]{2_r0, where _B is

the coupling constamt of the quartic term in the Higgs potential coupling cr and H.

Clearly _]{2 must not be too much larger than )_1/_, or there will be a large supression

in lB.

The second assumption is that the reheat temperature is less than the mass of

the Higgs, so that thermal production of H is not important. This implies that

rrtH > ._l/4g_l/4f_/4a-1/4_l/40"O. Again assuming that mH = _B _ro, the requirement

12

=
becomes AB > Al12gjl12f_/2a-l12{*l_. If this inequality is not satisfied, then H's

will be copiously produced in the thermalization process and baryogenesis will follow

the standard out of equilibrium decay scenario rather than the mechanism we have

outlined above.

The compatibilityand naturalness of these two requirements willbe discussed in

the concluding section.

IV. DISCUSSION AND CONCLUSIONS

Here we examine some typical numbers for the baryon asymmetry which may be

obtained from Eq. (3.11), in the light of the experimental limits discussed in the

introductory section setting B at around 10 -l°. The two filling factors fv and a both

appear to the quarter power, and hence as they are both of order one, they can be

dropped. The number of Higgs degrees of freedom, gH, is expected also to be of order

one, with simply one degree of freedom for each polarization in the case of a single

Higgs and further degrees in the case of a doublet or more of Higgs particles. The

total number of degrees of freedom g, is expected to be of order 100-800 in a grand

unified theoryJ _ This implies

B ~ i0 (4.1)

The remaining parameters e,A and _ arelesscertain,with some dependence on the

particularunifiedtheory under examination, though itisreassuringthat both A and

also enter only to the quarter power and hence the dependence on these quantities

is weak. This does however have the further implication that e should be very small,

as we shall shortly see. That a suitable baryon asymmetry can be produced with

13
such a small • indicates that the bubble wall collisions are very efficient in producing

a baryon asymmetry. A reasonable estimate of _ is that it may be of order 10 -2

(recalling _ << I is the condition both for sufficient inflation and for the thin wail

approximation tobe valid). The parameter )_ should probably be less than of order 1,

though nothing in principle prevents it from being much smaller; note that a smaller

increases the baryon asymmetry as it leads to a less efficient production of entropy,

though _ must also be sufficently large that Higgs particles can be produced in the

wall collisions. From these arguments, it seems llkely that the ratio _/)_ will be within

a few orders of magnitude of unity, implying that if this mechanism is to generate

the appropriate baryon asymmetry e must be of order 10 -s. This argument will be

made tighter below.

There are constraints that must be satisfied in order for this scenario to work.

As mentioned at the end of the preceding section, the typical energy of particles

produced in wall collisions, A -1, should exceed the Higgs mass. (If this does not

hold, then fH will have an extra suppression. While this may allow a larger e it

will most likely require some fine tuning of the amount of suppression.) A further

constraint is that the reheat temperature be less than the Higgs mass in order to

avoid the Higgs produced in Wall collisions reaching a state of thermal equilibrium.

These two constraints translate into an upper and lower bound for )_x (neglecting the

volume factors and substituting for go as before)

Clearly suitable values of _H are only possible provided

although to aLlow a range of _zr this bound should be stronger. Although this provides

a non-trivial constraint on _ and A that _ < 100A, it is not a particularly strong one

14
and it leads only to a weak lower bound on e of around I0 -s. Hence this baryogenesis

scenario appears satisfactory for a large set of possible model parameters.

We should discuss more general requirements of the model in order for this type

of extended inflation scenario to be considered as s sensible candidate for baryogenesis. One important requirement is that the symmetry breakings do not lead to an

unacceptable density of relic monopoles (monopo]es being the inevitable outcome of

any symmetry breaking from a semi-simple group to the standard model). In several

theories this can be arranged by creating the monopo]es in a pre-inflationary breaking. The monopoles are then subsequently diluted during the inflationary era and

present no further problems. For example, in a specific model proposed by Olive and

Turok, 23,24 S0(10) is broken in a two-step process

so( o) SU(3) _} SU(2)L _ SU(2)R _ U(1)B-L,

-----4 SU(3) ® SU(2)L _ _(1)y _ _2, (4.4)

where the first breaking is through the 45-dimensional representation of SO(10), and

the second breaking is through the 126-dimensional representation. _s In the symmetry

breaking scheme of Eq. (4.4), inonopoles are produced at the first transition, but not

at the second (existing monopoles are converted rather than new ones formed24).

Hence, in such a picture at least two symmetry breakings are needed to reach the

standard model, the latter causing both inflation and the out of equilibrium conditions

required for baryogenesis. It is even possible for this later transition to produce cosmic

strings as defects in the inflaton field. 26

We should also comment on the possible role of the baryon-number violating

anomolous currents in the non-perturbative sector of the electroweak theory. It has

been conjectured _r that this anomolous current may cause a awash-out" of any preexisting baryon asymmetry at temperatures above the e]ectroweak phase transition

15
as the barrier height between sectors of different baryon number may only be around

10 TeV, and baryon number non-conserving interactions could proceed in thermal

equilibrium. If these calculations prove correct, then this may destroy any baryon

asymmetry generated in the wall collisions (this would wash out any primordial baryon

asymmetry of course, and is not a problem specific to the scenario we are proposing

here). It has also been proposed that the electroweak phase transition may actually

create a suitable baryon asymmetry but so far these models have ex_bited only

limited success. The effect of sphalerons may be mitigated if a non-zero value of

B - L is generated, such as possible with the breaking scheme of Eq. (4.4).

We should also comment here on the possibility of isothermal perturbations _ising

from the thermalization process. While we have assumed throughout this paper

that at percolation all the true vacuum bubbles have the same size, the full picture

is somewhat more complicated, as bubbles formed earlier in inflation will grow to

larger sizes than those formed right at the end. While homogeneity of the microwave

background requires large bubbles to be suppressed, _s one would still expect to see

a range of sizes of small bubbles, and hence spatial variations in the ratio of baryon

number density to entropy density from point to point.

In conclusion then, we have seen that the result of the first-order phase transition

bringing extended inflation to an end is an environment well out of thermal equiLibrium. In such conditions baryogenesis via the decay of baryon number violating

Higgs particles can proceed, and we have demonstrated a means by which the baryon

number can be estimated. The mechanism has further been shown to work for a

large range of model parameters and to have the capability of predicting a baryon

asymmetry of the required magnitude. In a second paper, we shall consider a slightly

different mech_sm for haryogenesis in which primordial black holes formed during

the bubble-wall coUisions may play an important role.

16
Acknowledgements

EJC andARL aresupportedby the SERC,and EWK wasan SERC-supported Visiting Research Fellow at the Astronomy Centre, University of Sussex. EWK was also

supported by the DoE and NASA (grant #NAGW-1340) at Fermilab. EWK would

also llke to thank John Barrow for his hospitality at the University of Sussex where

part of this work was done.

References

1. A. H. Guth, Phys. Rev. D 23, 347 (1981).

2. C. Mathiszhagen and V. B. Johri, Clozs. Quant. Gray. 1, L29 (1984)

D. La and P. J. Steinhardt, Phys. Rev. Left. 62, 376 (1989).

3. C. Callan and S. Coleman, Phys. Rev. D 16, 1762 (1977).

4. A. H. Guth and E. J. Weinberg, Nucl. Phys. B212, 321 (1983).

5. E. J. Welnberg, Phys. Rev. D 40, 3950 (1989).

6. D. La, P. J. Steinhardt, and E. W. Bertschinger, Phys. Left. 231B, 231 (1989).

7. P. J. Steinhardt and F. S. Accetta, "Hyperextended Inflation" University of

Pennsylvania preprint 1990, unpublished.

8. F. S. Accetta and J. J. Trester, Phys. Rev. D 39, 2854 (1989).

9. R. Holmsn, E. W. Kolb, S. L. Vadas, and Y. Wang, "Kaluza-Klein Extended

Inflation," Fermilab preprint 1990, unpublished.

10. R. Holman, E. W. Kolb, and Y. Wang, "Gravitational Couplings of the Inflaton

in Extended Inflation," Fermilab preprint 1990, unpublished.

17
11. J. D. Barrow and K. Maeda, to appear, Nucl. Phys. B.

12. L. F. Abbott and M. B. Wise, Nucl. Phys. B244 541 (1984); J. D. Barrow,

Phys. Left. B187, 12 (1987); F. Lucchin and S. Matarrese, Phys. Rev. D 32,

1316 (1985); J. D. Barrow and P. Coles, Mon. Not. Roy. astron. Sac. 244, 188

(1990).

13. See, e.g., K. A. Olive, "Big Bang Nucleosynthesis: Theory and Observations",

Univ. Minnesota preprint (1990); B. E. J. Pagel, in Evolutionary Phenomena in

Ga/a_[e#, eds. J. E. Beckman and B. E. J. Pagel (Cambridge University Press,

Cambridge, 1989).

14. For a review of baryogenesis, see E. W. Kolb and M. S. Turner, The Early

Universe (Addison Wesley, Redwood City, Ca., 1990).

15. J. D. Barrow, Mon. Not. R. astron. Sac. 192,427 (1980); J. D. Barrow and G.

Ross, Nuc/. Phys. B181, 461 (1980).

16. J. A. Harvey, E. W. Kolb, D. B. Reiss, and S. Wolfram, Nucl. Phys. B201, 16

(1982).

17. J. D. Barrow, Quart..ll. Roy. ast_n. Sac. 29, 101 (1988).

18. J. D. Barrow, Phys. Left. B255, 40 (1990).

19. J. D. Barrow and S. Cotsakls, Phys. Left. B214, 515 (1988).

20. J. D. Barrow and A. C. OttewiU, J. Phys A 16, 2757 (1983).

21. See e.g., D. Stauffer, Phys. Rep. 54, 1 (1979). The Monte Carlo results discussed

in this reference assume that all bubbles axe the same size.

18
22. The ideal gas condition requires only that, roughly, 9. < 10aa-z << 10a_-z if

e < a; see J. D. Barrow, Fund. Cosmic. Phys. 8, 83 (1983).

23. D. Olive and N. Turok, Phys. Left. l17B, 193 (1982).

24. N. Turok, NucL Phys. B242, 520 (1984).

25. J. A. Harvey, D. B. Reiss, and P. Ramond, Nucl. Phys. B199, 223 (1982).

26. E.J. Copeland, E. W. Kolb, and A. R. Liddie, "Topological Defects in Extended

Inflation," University of Sussex preprint 1990, unpublished.

27. V. A. Kuzmln, V. A. Rubakov, and M. E. Shaposhnikov, Phys. Left.155B, 36

(1985).

28. P. Coles and J. D. Barrow, to appear, Mon. Not. Roy. astron. Soc..

19