PHYSICAL REVIEW D VOLUME 23, NUMBER 2 15 JAN UAR Y 1981

Infiationary universe: A possible solution to the horizon and fiatness problems

Alan H. Guth*Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

(Received 11 August 1980)

The standard model of hot big-bang cosmology requires initial conditions which are problematic in two ways: (1)The early universe is assumed to be highly homogeneous, in spite of the fact that separated regions were causallydisconnected (horizon problem); and (2) the initial value of the Hubble constant must be fine tuned to extraordinary

accuracy to produce a universe as flat (i.e., near critical mass density) as the one we see today (flatness problem).These problems would disappear if, in its early history, the universe supercooled to temperatures 28 or more ordersof magnitude below the critical temperature for some phase transition. A huge expansion factor would then resultfrom a period of exponential growth, and the entropy of the universe would be multiplied by a huge factor when thelatent heat is released. Such a scenario is completely natural in the context of grand unified models of elementary-

particle interactions. In such models, the supercooling is also relevant to the problem of monopole suppression.

Unfortunately, the scenario seems to lead to some unacceptable consequences, so modifications must be sought.

I. INTRODUCTION: THE HORIZON AND FLATNESSPROBLEMS

The standard model of hot big-bang cosmologyrelies on the assumption of initial conditions whichare very puzzling in two ways which I will explainbelow. The purpose of this paper is to suggest amodified scenario which avoids both of these puz-zles.

By "standard model, " I refer to an adiabaticallyexpanding radiation- dominated universe describedby a Robertson-%alker metric. Details will begiven in Sec. II.

Before explaining the puzzles, I would first liketo clarify my notion of "initial conditions. " Thestandard model has a singularity which is conven-tionally taken to be at time t =0. As t -0, thetemperature T —~. Thus, no initial-value prob-lem can be defined at t=0. However, when T isof the order of the Planck mass (Mz, —=I/~6=1. 22&&10~~ GeV)' or greater, the equations of the stan-dard model are undoubtedly meaningless, sincequantum gravitational effects are expected to be-come essential. Thus, within the scope of ourknowl, edge, it is sensible to begin the hot big-bangscenario at some temperature To which is com-fortably below Mp, let us say To ——10"GeV. Atthis time one can take the description of the uni-verse as a set of initial conditions, and the equa-tions of motion then describe the subsequent evolu-tion. Of course, the equation of state for matterat these temperatures is not really known, but onecan make various hypotheses and pursue the con-sequences.

In the standard model, the initial universe istaken to be homogeneous and isotropic, and filledwith a gas of effectively massless particles inthermal equilibrium at temperature To. The ini-tial value of the Hubble expansion "constant" H istaken to be Ho, and the model universe is then

completely described.Now I can explain the puzzles. The first is the

well-known horizon problem. 2 The initial uni-verse is assumed to be homogeneous, yet it con-sists of at least -10" separate regions which arecausally disconnected (i. e. , these regions havenot yet had time to communicate with each othervia light signals). ' (The precise assumptionswhich lead to these numbers will be spelled out inSec. II. ) Thus, one must assume that the forceswhich created these initial conditions were capableof violating causality.

The second puzzle is the flatness problem. Thispuzzle seems to be much less celebrated than thefirst, but it has been stressed by Dicke and Pee-bles. I feel that it is of comparable importanceto the first. It is known that the energy density pof the universe today is near the critical value p„(corresponding to the borderline between an openand closed universe). One can safely assume that~

0. 01 & Q& ( 10,

where

0 —= p/p„= (8w/3)Gp/H2,

and the subscript p denotes the value at the presenttime. Although these bounds do not appear at firstsight to be remarkably stringent, they, in fact,have powerful implications. The key point is thatthe condition 0=1 is unstable. Furthermore, theonly time scale which appears in the equations fora radiation-dominated universe is the Planck time,1/I„=5. 4 && 10 sec. A typical closed universewill reach its maximum size on the order of thistime scale, while a typical open universe willdwindle to a value of p much less than p„. A uni-verse can survive -10' years only by extreme finetuning of the initial values of p and H, so that p isvery near p„. For the initial conditions taken at

ALAN H. GUTH

To ——10 GeV, the value of Ho must be fine tunedto an accuracy of one part in 10". In the standardmodel this incredibly precise initial relationshipmust be assumed without explanation. (For anyreader who is not convinced that there is a realproblem here, variations of this argument are giv-en in the Appendix. )

The reader should not assume that these incredi-ble numbers are due merely to the rather largevalue I have taken for 7.'0. If I had chosen a modestvalue such as To ——1 MeV, I mould still have con-cluded that the "initial" universe consisted of atleast -10" causally disconnected regions, and thatthe initial value of Ho was fine tuned to one part in10 . These numbers are much smaller than theprevious set, but they are still very impressive.

Of course, any problem involving the initialconditions can always be put off until we under-stand the physics of T ~M&. However, it is thepurpose of this paper to show that these puzzlesmight be obviated by a scenario for the behavior

- of the universe at temperatures mell below M&.The paper is organized as follows. The assump-

tions and basic equations of the standard model aresummarized in Sec. II. In Sec. ID, I describe theinflationary universe scenario, showing how it caneliminate the horizon and flatness problems. Thescenario is discussed in the context of grand mod-els in Sec. IV, and comments are made concerningmagnetic monopole suppression. In Sec. V I dis-cuss briefly the key undesirable feature of thescenario: the inhomogeneities produced by therandom nucleation of bubbles. Some vague ideaswhich might alleviate these difficulties are men-tioned in Sec. VI. '

k 8mH +—= —GpR2 3

(2. 2b)

where H—=R/R is the Hubble "constant" (the dot de-notes the derivative with respect to f). Conserva-tion of energy is expressed by

(2. 3)

where p denotes the pressure. In the standardmodel one also assumes that the expansion is adi-abatic, in which case

—(sR )=0dt

(2. 4)

7TR

p = 3p = —3t(T)T',30

(2. 6)

(2. 6)

where s is the entropy density.To determine the evolution of the universe, the

above equations must be supplemented by an equa-tion of state for matter. It is now standard to de-scribe matter by means of a field theory, and athigh temperatures this means that the equation ofstate is to a good approximation that of an idealquantum gas of massless particles. Let N, (T) de-note the number of bosonic spin degrees of free-dom which are effectively massless at temperatureT (e. g. , the photon contributes two units to N, );and let N&(T) denote the corresponding number forfermions (e. g. , electrons and positrons togethercontribute four units). Provided that T is not nearany mass thresholds, the thermodynamic functionsare given by

II. THE STANDARD MODEL OF THE VERY EARI YUNIVERSE n =, 3V(T)T' (2. 7)

4nR = ——G(p+ 3p)R, (2. 2a)

In this section I will summarize the basic equa-tions of the standard model, and I will spell out the

assumptions which lead to the statements made inthe Introduc tion.

The universe is assumed to be homogeneous and

isotropic, and is therefore described by the Rob-ertson-Walker metric:

dJ'dv =dF —R~(t) +r (dB +sin 9dg )

1 —0(2. 1)

where 4=+1, —1, or 0 for a closed, open, orflat universe, respectively. It should be empha-sized that any value of k is possible, but by con-vention r and R(t) are rescaled so that k takes on

one of the three discrete values. The evolution ofR(t) is governed by the Einstein equations

where

m{T)=N, (T)+ '.N, (T}, -3t'(T) =N, (T) + ', N, (T) . -

(2. 6)

(2. 9}

Here n denotes the particle number density, and

t(3) = 1.202 06. . . is the Riemann zeta function.The evolution of the universe is then found by

rewriting (2. 2b) solely in terms of the tempera-ture. Againing assuming that T is not near anymass thresholds, one finds

4~'—~

+ e(T)T' = Got(T)T4, (2. 10)

where

u ~ 2v'~&T} "'RT i45 S (2. 11)

where S—=A s denotes the total entropy in a volume

INFLATIONARY UNIVERSE: A POSSIBLE SOLUTION TO. . .

specified by the radius of curvature R.Since S is conserved, its value in the early uni-

verse can be determined (or at least bounded) bycurrent observations. Taking p & 10p„ today' itfollows that today

ratio of volumes, so

l 11 45I

's~ (gs Mp43 4~') L,T„T )

= 4x10-ss&-t s(M~/T}s (2. 20)

kR2 & 9H2. (2. 12)

Taking 'X- 10s and Ts ——10" GeV, one finds ls'/Lo=10 . This is the horizon problem.

From now on I will take k=+ 1; the special case k=0 is still included as the limit R —~. Then todayB+ 3 g - 3 x 2 0 years. Taking the present photontemperature T„as 2. 7'K, one then finds that thephoton contribution to S is bounded by

S„&3 &&1085 (2. ia)

and

S& 1086

0&s3ts' s .

(2. 14

(2. iS)

Assuming that there are three species of masslessneutrinos (e, p, and r), all of which decouple at a '

time mhen the other effectively massless particlesare the electrons and photons, then S„=21/22S„.Thus,

III. THE INFLATIONARY UNIVERSE

In this section I will describe a scenario whichis capable of avoiding the horizon and flatnessproblems.

From Sec. II one can see that both problemscould disappear if the assumption of adiabaticitywere grossly incorrect. Suppose instead that

Sp ——Z So, (3. i)where S~ and So denote the present and initial val-ues of R s, and Z is some large factor.

Let us look first at the flatness problem. Given(3. 1), the right-hand side (RHS) of (2. 16) is multi-plied by a factor of Z . The "initial" value (at Ts=10'~ GeV) of Ip —p„ I/p could be of order unity,and the flatness problem would be obviated, if

But thenZ& 3 X102'. (3. 2)

—~ s~ & 3 x 10 ssX (M /T)

P I

4m' %2

(2. ie)

Taking T= 10' GeV and%-10 (typical of grandunified models), one finds Ip —p„ I/p & 10+s. Thisis the flatness problem.

The sT term can now be deleted from (2. 10),which is then solved (for temperatures higher thanall particle masses) to give

Now consider the horizon problem. The RHS of(2. 19) is multiplied by Z, which means that thelength scale of the early universe, at any giventemperature, was smaller by a factor of Z thanhad been previously thought. If Z is sufficientlylarge, then the initial region which evolved intoour observed region of the universe would havebeen smaller than the horizon distance at that time.To see how large Z must be, note that the RHS of(2. 20) is multiplied by Z . Thus, if

P2yt ' (2. 17) Z& 5&20", (3.3}

where ys = (4v /45)X. (For the minimal SUs grandunified model, N, =82, N&

——90, and y=21. 05. )Conservation of entropy implies RT= constant, soA o-t" 2. A light pulse beginning at t=0 will havetraveled by time t a physical distance

tl(t) =R(t) dt'R (t') = 2t, (2. iS)

0

and this gives the physical horizon distance. Thishorizon distance is to be compared with the radiusL(t) of the region at time t which will evolve intoour currently observed region of the universe. A-gain using conservation of entropy,

(2. i9)

where s~ is the present entropy density and L~-10' years is the radius of the currently observedregion of the universe. One is interested in the

then the horizon problem disappears. (It should benoted that the horizon will still exist; it will simplybe moved out to distances which have not-been ob-served. )

It is not surprising that the RHS's of (3. 2) and(3. 3) are approximately equal, since they bothcorrespond roughly to Sp of order unity.

I will now describe a scenario, which I call theinflationary universe, which is capable of such alarge entropy production.

Suppose the equation of state for matter (with allchemical potentials set equal to zero} exhibits afirst-order phase transition at some critical tem-perature T,. Then as the universe cools throughthe temperature T„one would expect bubbles ofthe low-temperature phase to nucleate and grow.However, suppose the nucleation rate for thisphase transition is rather low. The universe will

ALA% H. GUTH

continue to cool as it expands, and it will then su-percool in the high- temperature phase. Supposethat this supercooling continues down to some tem-perature T„many orders of magnitude below T,.When the phase transition finally takes place attemperature T„ the latent heat is released. How-ever, this latent heat is characteristic of the ener-gy scale T„which is huge relative to T,. The uni-verse is then reheated to some temperature T„which is compa, rable to T,. The entropy density isthen increased by a factor of roughly (T„/T,) (as-suming that the number X of degrees of freedomfor the two phases are comparable), while the val-ue of R remains unchanged. Thus,

Z= T„/T, . (s. 4)

If the universe supercools by 28 or more orders ofmagnitude below some critical temperature, thehorizon and flatness problems disappear.

In order for this scenario to work, it is neces-sary for the universe to be essentially devoid ofany strictly conserved quantities. Let n denote thedensity of some strictly conserved quantity, andlet y= n/s denot—e the ratio of this conserved quan-tity to entropy. Then x~ = Z 3x0 + 10 +0, Thus,only an absurdly large value for the initial ratiowould lead to a measurable value for the presentratio. Thus, if baryon number were exactly con-served, the inflationary model would be untenable.However, in the context of grand unified models,baryon number is not exactly conserved. The netbaryon number of the universe is believed to becreated by CP-violating interactions at a tempera-ture of 10 -10 GeV. Thus, provided that T, liesin this range or higher, there is no problem. Thebaryon production would take place after the re-heating. (However, strong constraints are im-posed on the entropy which ca.n be generated in anyphase transition with T, «10 GeV, in particular,the Weinberg-Salam phase transition. )

Let us examine the properties of the supercoolinguniverse in more detail. Note that the energy den-sity p(T), given in the standard model by (2. 5),must now be modified. As T-O, the system iscooling not toward the true vacuum, but rather to-ward some metastable false vacuum with an energydensity p0 which is necessarily higher than that ofthe true vacuum. Thus, to a good approximation(ignoring mass thresholds)

theory, the form of T~„ is determined by the con-servation requirement up to the possible modifica-tion

Ttt, v Ttt v +

~gatv

s (s. 6)

for any constant X. (X cannot depend on the valuesof the fields, nor can it depend on the temperatureor the phase. ) The freedom to introduce the mod-ification (3. 6) is identical to the freedom to intro-duce a cosmological constant into Einstein's equa-tions. One can always choose to write Einstein'sequations without an explicit cosmological term;the cosmological constant A is then defined by

(0 [ T,„I0) =Ag. „, (3.7)

Gx(r)r -e(r)r +—Gp .T2 4n 8w

T 45 0' (3.8)

This equation has two types of solutions, dependingon the parameters. If e& e0, where

8w2v' 300 45 GvPO ~ (s. 9)

then the expansion of the universe is halted at atemperature T „given by

3Pp ~ ~-(~ 2q

1/2 2

r '=, '- —+ ~—0 g(60

(s. io)

and then the universe contracts again. Note thatT „is of 0(T,), so this is not the desired scenar-io. The case of interest is e& e0, in which case theexpansion of the universe is unchecked. [Note thatco-v9LT, '/M~' ispresumably avery small number.Thus 0( e & eo (a closed universe) seems unlikely,but e & 0 (an open universe) is quite plausible. ]Once the temperature is low enough for the p0 termto dominate over the other two terms on the HHSof (3. 8), one has

T(t) = const&&e "',where

(3. 11)

where Io) denotes the true vacuum. A is identifiedas the energy density of the vacuum, and, in prin-ciple, there is no reason for it to vanish. Empir-ically A is known to be very small ( IA I ( 10+6

GeV ) so I will take its value to be zero. " Thevalue of p0 is then necessarily positive and is de-termined by the particle theory. '2 It is typicallyof 0(r,').

Using (3. 5), Eq. (2. 10) becomes

p(T) = —Ot(r)T' + po. (s. 5) Sm'X' = —G~0 ~

3(3. 12)

Perhaps a few words should be said concerningthe zero point of energy. Classical general rela-tivity couples to an energy-momentum tensor ofmatter, T~„, which is necessarily (covariantly)conserved. When matter is described by a field

Since RT = const, one has"

R(t) = const&& e" '.The universe is expanding exponentially, in a false

IN F LATIONAR 7 UNIVERSE: A POSSIBLE SOLUTION TO. . . 351

p(t) = exp — dtqX(t~)R (tq) V(t, t~)p

where

4m ' dtv(~, t, ) =-s, R(t, )

(3. 14)

(s. 15)

is the coordinate volume at time t of a bubble which

vacuum state of energy density pp. The Hubbleconstant is given by H=R/R=y. (More precisely,H approaches X monotonically. from above. Thisbehavior differs markedly from the standard rnod-el, in which H falls as t"'. )

The false vacuum state is Lorentz invariant, soT& = p'pg+ . I't follows tha't p = pp the pressureis negative. This negative pressure allows for theconservation of energy, Eq. (2. 3). From the sec-ond-order Einstein equation (2. 2a), it can be seenthat the negative pressure is also the driving forcebehind the exponential expansion.

The Lorentz invariance of the false vacuum hasone other consequence: The metric described by(3. 13) (with @=0)does not single out a comovingframe. The metric is invariant under an O(4, 1)group of transformations, in contrast to the usualRobertson-Walker invariance of O(4). It is knownas the de Sitter metric, and it is discussed in thes tandard literature. '

Now consider the process of bubble formation ina Robertson-Walker universe. The bubbles formrandomly, so there is a certain nucleation rateX(t), which is the probability per (physical) volumeper time that a bubble will form in any regionwhich is still in the high-temperature phase. Iwill idealize the situation slightly and assume thatthe bubbles start at a point and expand at the speedof light. Furthermore, I neglect k in the metric,so d7 =dt —R (t)dx .

I want to calculate p(t), the probability that anygiven point remains in the high-temperature phaseat time t. Note that the distribution of bubbles istotally uncorrelated except for the exclusion prin-ciple that bubbles do not form inside of bubbles.This exclusion principle causes no problem be-cause one can imagine fictitious bubbles whichform inside the real bubbles with the same nuclea-tion rate X(t). With all bubbles expanding at the

speed of light, the fictitious bubbles mill be foreverinside the real bubbles and will have no effect on

p(t). The distribution of all bubbles, real and fic-titious, is then totally uncorrelated.

P(t) is the probability that there are no bubbleswhich engulf a given point in space. But the num-ber of bubbles mhich engulf a given point is a Pois-son-distributed variable, so P(t) =exp[-1V(t)],where Z(t) is the expectation value of the numberof bubbles engulfing the point. Thus

formed at time t&.

I will now' assume that the nucleation rate is suf-ficiently slow so that no significant nucleation takesplace until T«T„when exponential growth has setin. I will further assume that by this time &(t) isgiven approximately by the zero-temperature nu-cleation rate Xp. One then has

P(t) = exp ——+0(1) (3. Is)

where

3XT =—

4nzp' (s. iv)

and O(1) refers to terms which approach a constantas Xt —~. During one of these time constants, theuniverse will expand by a factor

3X4 lZ, = exp(}t~) = exp

4mXp)' (s. is}

If the phase transition is associated with the ex-pectation value of a Higgs field, then Xp can be cal-culated using the method of Coleman and Callan. "The key point is that nucleation is a tunneling pro-cess, so that &p is typically very small. TheColeman-Callan method gives an answer of theform

&0 =A po exp(-B), (s. 19}

where B is a barrier penetration term and A is a,

dimensionless coefficient of order unity. Since Z,is then an exponential of an exponential, one canvery easily' ' obtain values as large as log&pZ

=28, or even log&pZ=10Thus, if the universe reaches a state of exponen-

tial growth, it is quite plausible for it to expandand supercool by a huge number of orders of mag-nitude before a significant fraction of the universeundergoes the phase transition.

So far I have assumed that the ea, rly universe canbe described from the beginning by a Robertson-Walker metric. If this assumption were reallynecessary, then it would be senseless to talk about"solving" the horizon problem; perfect homogeneitywas assumed at the outset. Thus, I must now ar-gue that the assumption can probably be dropped.

Suppose instead that the initial metric, and thedistribution of particles, mas rather chaotic. Onewould then expect that statistical effects mould tendto thermalize the particle distribution on a localscale. 2 It has also been shown (in idealized cir-cumstances} that anisotropies in the metric aredamped out on the time scale of -10' Plancktimes. The damping of inhornogeneities in themetric has also been studied, 22 and it is reasonableto expect such damping to occur. Thus, assumingthat at least some region of the universe started at

ALAR H.

temperatures high compared to T„one would ex-pect that, by the time the temperature in one ofthese regions falls to T„ it will be locally homoge-neous, isotropic, and in thermal equilibrium. Bylocally, I am talking about a length scale $ whichis of course less than the horizon distance. It willthen be possible to describe this local region of theuniverse by a Robertson-Walker metric, which willbe accurate at distance scales small compared to

When the temperature of such a region falls be-low T„ the inflationary scenario will take place.The end result will be a, huge region of space whichis homogeneous, isotropie, and of nearly criticalmass density. If Z is sufficiently large, this re-gion can be bigger than (or much bigger than) ourobserved region of the universe.

IV. GRAND UNIFIED MODELS AND MAGNETICMONOPOLE PRODUCTION

In this section I will discuss the inflationarymodel in the context of grand unified models ofelementary-par ticle interac tions. 2 ~

A grand unified model begins with a, simple gaugegroup G which is a valid symmetry at the highestenergies. As the energy is lowered, the theoryundergoes a hierarchy of spontaneous symmetrybreaking into successive subgroups: G -0„—' ' '-Ho, where Hi ——SU3&&SU2&&Uq [@CD (quantumchromodynamics) & Weinberg-Salam] and Ho ——SU3x U& ". In the Georgi-Glashow model, which isthe simplest model of this type, G = SU, and n = 1.The symmetry breaking of SU& —SU3 x SU2 x U$ oc-curs at an energy scale M~- 10 GeV.

At high temperatures, it was suggested by Kirzh-nits and Linde25 that the Higgs fields of any spon-taneously broken gauge theory would lose their ex-pectation values, resulting in a high-temperaturephase in which the full gauge symmetry is re-stored. A formalism for treating such problemswas developed by Weinberg and by Dolan andJackiw. In the range of parameters for which thetree potential is valid, the phase structure of theSU5 model was analyzed by Tye and me. ' ~ ' Wefound that the SU5 symmetry is restored at T& -10' GeV and that for most values of the param-eters there is an intermediate-temperature phasewith gauge symmetry SU4xU&, which disappears atT-10 - GeV. Thus, grand unified models tend toprovide phase transitions which could lead to an in-flationary scenaxio of the universe.

Grand unified models have another feature withimportant cosmological consequences: They con-tain very heavy magnetic monopoles in their parti-cle spectrum. These monopoles are of the typediscovered by 't Hooft and Polyakov, 2 and will bepresent in any model satisfying the above descrip-tion. These monopoles typically have masses of

l & 2tcoal = 27 T coal

(4. 1)

By Kibble's argument, the density n„of monopolesthen obeys

3T 6~ l'~3 + Y co81

NP

(4. 2)

By considering the contribution to the mass densityof the present universe which could come from 10~

GeV monopoles, Preskill ' concludes that

n„/n„& 10 ~4, (4. 3)

where n, is the density of photons. This ratiochanges very little from the time of the phasetransition, so with (2. 7) one concludes

order Mx/n-10~6 GeV, where n =g /4m is thegrand unified fine structure constant. Since themonopoles are really topologically stable knots inthe Higgs field expectation value, they do not existin the high-temperature phase of the theory. Theytherefore come into existence during the course ofa phase transition, and the dynamics of the phasetransition is then intimately related to the mono-pole production rate.

The problem of monopole production and the sub-sequent annihilation of monopoles, in the context ofa second-order or weakly first-order phase transi-tion, was analyzed by Zeldovieh and Khlopov30 and

by Preskill. ' In Preskill's analysis, which wasmore specifically geared toward grand unifiedmodels, it was found that relic monopoles wouldexceed present bounds by roughly 14 orders ofmagnitude. Since it gems difficult to modify theestimated annihilation rate, one must find a scen-ario which suppresses the production of thesemonopole s.

Kibble 2 has pointed out that monopoles are pro-duced in the course of the phase transition by theprocess of bubble coalescence. The orientation ofthe Higgs field inside one bubble will have no cor-relation with that of another bubble not in contact.When the bubbles coalesce to fill the space, it willbe impossible for the uncorrela, ted Higgs fields toalign uniformly. One expects to find topologicalknots, and these knots are the monopoles. Thenumber of monopoles so produced is then compar-able to the number of bubbles, to within a few or-ders of magnitude.

Kibble's production mechanism can be used toset a "horizon bound" on monopole productionwhich is valid if the phase transition does not sig-nificantly disturb the evolution of the universe.At the time of bubble coalescence t«„ the size l ofthe bubbles cannot exceed the horizon distance atthat time. So

28 ' INFLATIONARY UNIVERSE: A POSSIBLE SOLUTION TO. . .

] 0-24/2 qi/3

2$ (3)(4.4)

If T,-10' GeV, this bound implies that the uni-verse must supercool by at least about four ordersof magnitude before the phase transition is com-pleted.

The problem of monopole production in a strongly.first-order phase transition with supercooling wastreated in more detail by Tye and me. 6 Weshowed how to explicitly calculate the bubble den-sity in terms of the nucleation rate, and we con-sidered the effects of the latent heat released inthe phase transition. Our conclusion was that(4. 4) should be replaced by

T,.„&2&10" GeV, (4. 6)

V. PROBLEMS OF THE INFLATIONARY SCENARIO

As I mentioned earlier, the inflationary scenarioseems to lead to some unacceptable consequences.It is hoped that some variation can be found whichavoids these undesirable features but maintains thedesirable ones. The problems of the model will bediscussed in more detail elsewhere, ' but for com-pleteness I will give a brief description here.

The central problem is the difficulty in finding asmooth ending to the period of exponential expan-sion. Let us assume that X(t) approaches a con-stant as t - and T -0. To achieve the desiredexpansion factor Z & 10'8, one needs Xo/y4 & 10 2

[see (3. 18)], which means that the nucleation rateis slow' compared to the expansion rate of the uni-verse. (Explicit calculations show that &0/}t4 istypically much smaller than this value. '8 '9 ~6) Therandomness of the bubble formation process thenleads to gross inhomogeneities.

To understand the effects of this randomness,the reader should bear in mind the following facts.

(i} All of the latent heat released as a bubble ex-pands is transferred initially to the walls of the

where T„„refers to the temperature just beforethe release of the latent heat.

Tye and I omitted the crucial effects of the massdensity po of the false vacuum. However, our workhas one clear implication: If the nucleation rate issufficiently large to avoid exponential growth, thenfar too many monopoles would be produced. Thus,the monopole problem seems to also force one intothe inflationary scenario. 5

In the simplest SU5 model, the nucleation rateshave been calculated (approximately) by Weinbergand me. ' The model contains unknown parame-ters, so no definitive answer is possible. We dofind, however, that there is a sizable range of pa-rameters which lead to the inflationary scenario.

bubble. This energy can be thermalized onlywhen the bubble walls undergo many collisions.

(ii} The de Sitter metric does not single out a co-moving frame. The O(4, 1) invariance of the deSitter metric is maintained even after the forma-tion of one bubble. The memory of the originalRobertson-Walker comoving frame is maintainedby the probability distribution of bubbles, but thelocal comoving frame can be reestablished only af-ter enough bubbles have collided.

(iii) The size of the largest bubbles will exceedthat of the smallest bubbles by roughly a factor ofZ; the range of bubble sizes is immense. Thesurface energy density grows with the size of thebubble, so the energy in the walls of the largestbubbles can be thermalized only by colliding withother large bubbles.

(iv) As time goes on, an arbitrarily large frac-tion of the space will be in the new phase [see(3. 16)). However, one can ask a more subtlequestion about the region of space which is in thenew phase: Is the region composed of finite sepa-rated clusters, or do these clusters join togetherto form an infinite region& The latter possibilityis called "percolation. " It can be shown. that thesystem percolates for large values of &0/Z4, butthat for sufficiently small values it does not. Thecritical value of Ao/}t has not been determined,but presumably an inflationary universe would havea value of Xo/y be'low critical. Thus, no matterhow long one waits, the region of space in the new

phase will consist of finite clusters, each totallysurrounded by a region in the old phase.

(v) Each cluster will contain only a few of the

largest bubbles. Thus, the collisions discussed in(iii) cannot occur.

The above statements do not quite prove that thescenario is impossible, but these consequencesare at best very unattractive. Thus, it seems thatthe scenario will become viable only if some mod-ification can be found which avoids these inhomog-eneities. Some vague possibilities will be men-'

tioned in the next section.Note that the above arguments seem to rule out

the possibility that the universe was ever trappedin a, false vacuum state, unless Xo/y4 ~ 1. Such alarge value of Xo/y does not seem likely, but itis possible. '

VI. CONCLUSION

I have tried to convince the reader that the stan-dard model of the very early. universe requires theassumption of initial conditions which are very im-plausible for two reasons:

(i) The horizon problem. Causally disconnectedregions are assumed to be near'ly identical; in par-

ALAN H.

ticular, they are simultaneously at the same tem-perature.

(ii) The flatness problem. For a fixed initialtemperature, the initial value of the Hubble "con-stant" must be fine tuned to extraordinary accura-cy to produce a universe which is as flat as the onewe observe.

Both of these problems would disappear if theuniverse supercooled by 28 or more orders ofmagnitude below the critical temperature for somephase transition. (Under such circumstances, theuniverse would be growing exponentially in time. )However, the random formation of bubbles of thenew phase seems to lead to a much too inhomoge-neous universe.

The inhomogeneity problem would be solved ifone couM avoid the assumption that the nucleationrate X(t) approaches a small constant Xp as thetemperature T -0. If, instead, the nucleationrate rose sharply at some T&, then bubbles of anapproximately uniform size would suddenly fillspace as T fell to T&. Of course, the full advant-age of the inflationary scenario is achieved only ifT, &10 "T,.

Recently Witten has suggested that the abovechain of events may in fact occur if the parametersof the SU5 Higgs field potential are chosen to obeythe Coleman-Weinberg condition4P (i. e. , that O'V/

8&fP=O at /=0). Witten has studied this possi-bility in detail for the case of the Weinberg-Salamph3se transition. Here he finds that thermal tun-neling is totally ineffective, but instead the phasetransition is driven when the temperature of the@CD chiral-symmetry-breaking phase transitionis reached. For the SU, case, one can hope that amuch larger amount of supercooling will be found;however, it is difficult to see how 28 orders ofmagnitude could arise.

Another physical effect which has so far been leftout of the analysis is the production of particlesdue to the changing gravitational metric. 2 Thiseffect may become important in an exponentiallyexpanding universe at low temperatures.

In conclusion, the inflationary scenario seemslike a natural and simple way to eliminate both thehorizon and the flatness problems. I am publishingthis paper in the hope that it will highlight the ex-istence of these problems and encourage others tofind- some way to avoid the undesirable features ofthe inflationary scenario.

ACKNOWLEDGMENTS

I would like to express my thanks for the adviceand encouragement I received from Sidney Cole-

man and Leonard Susskind, and for the invaluablehelp I received from my collaborators Henry Tyeand Erick Weinberg. I would also like to acknowl-edge very useful conversations with Michael Aizen-man, Beilok Hu, Harry Kesten, Paul Langacker,Gordon Lasher„So- Young Pi, John Preskill, andEdwardWitten. This work was supported by theDepartment of Energy under Contract No. DE-AC03-76SF00515.

APPENDIX: REMARKS ON THE FLATNESS

PROBLEM

This appendix is added in the hope that someskeptics can be convinced that the flatness problemis real. Some physicists would rebut the argumentgiven in Sec. I by insisting that the equations mightmake sense all the way back to t=0. Then if onefixes the value of II corresponding to some arbi-trary temperature T„one always finds that when

the equations are extrapolated backboard in time,Q -1 as t -0. Thus, they would argue, it is na-tural for 0 to be very nearly equal to 1 at earlytimes. For physicists who take this point of view,the flatness problem must be restated in otherterms. Since Hz and Tz have no significance, themodel universe must be specified by its conservedquantities. In fact, the model universe is com-pletely specified by the dimensionless constant &

=—Ip/R2T2, where k and R are parameters of theRobertson-Walker metric, Eq. (2. 1). For ouruniverse, one must take lel &3&10~ . The prob-lem then is the to explain why le l should have sucha startlingly small value.

Some physicists also take the point of view thate=—0 is plausible enough, so to them there is no

problem. To these physicists I point out that theuniverse is certainly not described exactly by aRobertson-Walker metric. Thus it is difficult toimagine any physical principle which would requirea parameter of that metric to be exactly equal tozero.

In the end, I must admit that questions of plausi-bility are not logically determinable and dependsomewhat on intuition. Thus I am sure that somephysicists will remain unconvinced that there real-ly is a flatness problem. However, I am also surethat many physicists agree with me that the flatnessof the universe is a peculiar situation which atsome point will admit a physical explanation.

INFLATIONARY UNIVERSE: A POSSIBLE SOLUTION TO. . .

~Present address: Center for Theoretical Physics,Massachusetts Institute of Technology, Cambridge,Massachusetts 02139.I use units for which @= c =k (Boltzmann cons tant)= 1. Then 1 m= 5.068x 10~~ GeV, 1 kg= 5.610x 10 6

GeV, 1 sec = 1.519x 10+ GeV ~, and 1 'K = 8.617x 10 ~4

GeV.W. Rindler, Mon. No~. . B. Astron. Soc. 116, 663 {1956).See also Ref. 3, pp. 489-490, 525-526; and Ref. 4, pp.740 and 815.

S. Weinberg, Gravitation and Cosmology (Wiley, NewYork, 1972).

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Grav-itation (Freeman, San Francisco, 1973).

5In order to calculate the horizon distance, one must ofcourse follow the light trajectories back to t = 0. Thisviolates my contention that the equations are to betrusted only for T &Tp Thus, the horizon problemcould be obviated if the full quantum gravitationaltheory had a radically different behavior from thenaive extrapolation. Indeed, solutions of this sorthave been proposed by A. Zee, Phys. Bev. Lett. 44,703 (1980) and by F. W. Stecker, Astrophys. J. 235,Ll (1980). However, it is the point of this paper toshow that the horizon problem can also be obviated bymechanisms which are more within our grasp, occur-ring at temperatures below Tp.

R. H. Dicke and P. J.E. Peebles, General Relativity:An Einstein Centenary Survey, edited by S. W. Hawk-ing and W. Israel {Cambridge University Press, K,on-don, 1979).

See Ref. 3, pp. 475-481; and Bef. 4, pp. 796-797.For example, see Bef. 3, Chap. 14.

9M. Yoshimura, Phys. Rev. Lett. 41, 281 {1978);42,746 (E} (1979); Phys. Lett. 88B, 294 (1979); S.Dimopou- .

los and L. Susskind, Phys. Bev. D 18, 4500 {1978);Phys. Lett. 81B, 416 (1979); A. Yu Ignatiev, N. V.Krashikov, V. A. Kuzmin, and A. N. Tavkhelidze,ibid. 76B, 436 (1978); D. Toussaint, S. B. Treiman,F. Wilczek, and A. Zee, Phys. Bev. D 19, 1036 (1979);S. Weinberg, Phys. Rev. Lett. 42, 850 (1979); D. V.Nanopoulos and S. Weinberg, Phys. Bev. D 20, 2484{1979);J. Ellis, M. K. Gaillard, and D. V. Nanopoulos,Phys. Lett. 80B, 360 {1979);82B, 464 (1979); M. Hon-da and M. Yoshimura, Prog. Theor. Phys. 62, 1704(1979); D. Toussaint and F. Wilczek, Phys. Lett. 81B,238 {1979);S. Barr, G. Segre, and H. A. Weldon,Phys. Rev. D 20, 2494 {1979);A. D. Sakharov, Zh.Eksp. Teor. Fiz. 76, 1172 (1979) fSov. Phys. —JETP49, 594 {1979)];A. Yu Ignatiev, N. V. Krashikov,V. A. Kuzmin, and M. E. Shaposhnikov, Phys. Lett.87B, 114 (1979); E. W. Kolb and S. Wolfram, ibid.91B, 217 (1980); Noel. Phys. B172, 224 (1980);J. N.Fry, K. A. Olive, and M. S. Turner, Phys. Rev. D 22,2953 (1980); 22, 2977 (1980); S. B. Treiman and

F. Wilczek, Phys. Lett. 95B, 222 (1980); G. Senjanovicand F. W. Stecker, Phys. Lett. B (to be published).

~~The reason A is so small is of course one of the deepmysteries of physics. The value of A is not determinedby the particle theory alone, but must be fixed bywhatever theory couples particles to quantum gravity.This appears to be a separate problem from the onesdiscussed in this paper, and I merely use the empiri-cal fact that A=O.S. A. Bludman and M. A. Buderman, Phys. Rev. Lett.

38, 255 (1977).~3The effects of a false vacuum energy density on the

evolution of the early universe have also been con-sidered by E. W. Kolb and S. Wolfram, CAL TECHReport No. 79-0984 (unpublished), and by S. A. Blud-rnan, University of Pennsylvania Report No. UPR-0143T, 1979 (unpublished).More precisely, the usual invariance is O(4) if' =1,O(3,1) ifk = -1, and the group of rotations and trans-1.ations in three dimensions ifk = 0.See for example, Bef. 3, pp. 385-392.

6A. H. Guth and S. -H. Tye, Phys. Rev. Lett. 44, 631(1980); 44, 963 {1980).S. Coleman, Phys. Rev. D 15, 2929 (197?); C. G. Cal-lan and S. Coleman, ibid. 16, 1762 (1977); see alsoS. Coleman, in The Whys of Subnuclear Physics,proceedings of the International School of SubnuclearPhysics, Ettore Majorana, Erice, 1977, edited byA, Zichichi plenum. New York. 19?9).

~ A. H. Guth and E.J. Weinberg, Phys. Rev. Lett. 45,1131 (1980).

~~E. J.Weinberg and I are preparing a manuscript onthe possible cosmological implications of the phasetransitions in the SU& grand unified model.

2PJ. Ellis and G. Steigman, Phys. Lett. 89B, 186 (1980);J. Ellis, M. K. Gaillard, and D. V. Nanopoulos, ibid.SOB, 253 {1980).

2~B. L. Hu and L. Parker, Phys. Bev. D 17, 933 (1978).See Ref. 4, Chap. 30.

23The simplest grand unified model is the SU(5) modelof H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32,438 (1974). See also H. Georgi, H. R. Quinn, andS. Weihberg, ibid. 33, 451 {1974); and A. J.Buras,J. Ellis, M. K. Gaillard, and D. V. Nanopoulos, Nucl.Phys. B135, 66 {1978).Other grand unified models include the SO(10) model:H. Georgi, in Particles and Eields —1975, proceedingsof the 1975 meeting of the Division of Particles andFields of the American Physical Society, edited byCarl Carlson (AIP, New- York, 1S75); H. Fritzsch andP; Minkowski, Ann. Phys. , (N.Y.) 93, 193 (1975);H. Georgi and D. V. Nanopoulos, Phys. Lett. 82B,392 (1979) and Nucl. Phys. B155, 52 (1979). The E(6)model: F. Gursey, P. Ramond, and P. Sikivie, Phys.Lett. 60B, 177 (1975); F. Gursey and M. Serdaroglu,Lett. Nuovo Cirnento 21, 28 (1978). The K(7) model:F. Giirsey and P. Sikivie, Phys. Rev. Lett. 36, 775(1976), and Phys. Bev. D 16, 816 (1977); P. Ramond,Nucl. Phys. B110, 214 {1976). For some generalproperties of grand unified models, see M. Gell-Mann,P. Ramond, and R. Slansky, Rev. Mod. Phys. 50, 721(1978). For a review, see P. Langacker, Report No.SLAC-PUB-2544, 1980 (unpublished).D. A. Kirzhnits and A. D. Linde, Phys. Lett. 42B,471 (1972).S. Weinberg, Phys. Bev. D 9, 3357 (1S74); L. Dolanand R. Jackiw, ibid. 9, 3320 (1974); see also D. A.Kirzhnits and A. D. Linde, Ann. Phys. {¹Y.) 101, 195,(1S76); A. D. Linde, Bep. Prog. Phys. 42, 389 (19?9).&-expansion techniques are employed by P. Ginsparg,Nucl. Phys. B (to be published).

27In the case that the Higgs quartic couplings are corn-parable to g or smaller (g = gauge coupling), thephase structure has been studied by M. Daniel andC. E. Vayonakis, CERN Report No. TH.2860 1980

ALAN H. GUTH

(unpublished); and by P. Suranyi, University of Cin-cinnati Report No. SO-0506 (unpublished).

8G. 't Hooft, Nucl. Phys. 879, 276 {1974);A. M. Poly-akov, Pis'ma Zh. Eksp. Teor. Fiz. 20, 430 (1974)[JETP Lett. 20, 194 (1974)]. For a review, seeP. Goddard and D. I. Olive, Rep. Prog. Phys. 41, 1357(1978).If II&(|") and 02(C) are both trivial, then II2(G/Kp)= IIf (0'p). In our case IIf +p) is the group of integers.For a general review of topology written for physicists,see N. D. Mermin, Rev. Mod. Phys. 51, 591 (1979).Y. B.Zeldovich and M. Y. Khlopov, Phys. Lett. 79B,239 (1978).

@J.P. Preskill, Phys. Rev. Lett. 43, 1365 (1979).T. W. B.Kibble, J. Phys. A 9, 1387 (1976).This argument was first shown to me by John Pres-kill. It is also described by Einhorn et al. , Ref. 34,except that they make no distinction between T~~ and

Tc'+The problem of monopole production was also exam-

ined by M. B. Einhorn, D. L. Stein, and D. Toussaint,Phys. Rev. D 21, 3295 (1980), who focused on second-order transitions. The structure of SU(5) monopoleshas been studied by C. P. Dokos and T. N. Tomaras,Phys. Rev. D 21, 2940 (1980); and by M. Daniel,G. Lazarides, and Q. Shafi, Nucl. Phys. B170, 156(1980). The problem of suppression of the cosmologi-cal production of monopoles is discussed by G. Laza-rides and Q. Shafi, Phys. Lett. 948, 149 {19SO), andG. Lazarides, M. Magg, and Q. Shafi, CERN ReportNo. TH.2856, 1980 (unpublished); the suppression dis-cussed here relies on a novel confinement mechanism,and also on the same kind of supercooling as in Ref.16. See also J. N. Fry and D. N. Schramm, Phys. Rev.Lett. 44, 1361 (1980).

35An alternative solution to the monopole problem has

been proposed by P. Langacker and S.-Y. Pi, Phys.Rev. Lett. 45, 1 {1980). By modifying the Higgsstructure, they have constructed a model in which thehigh-temperature SU5 phase undergoes a phase transi-tion to an SU3 phase at T 10~4GeV. Another phasetransition occurs at T -10 GeV, and below thistemperature the symmetry is SU3x U~

+ . Monopolescannot exist until T + 10 GeV, but their production isnegligible at these low temperatures. The suppressionof monopoles due to the breaking of U~ symmetry athigh temperatures was also suggested by S. -H. Tye,talk given at the 1980 Guangzhou Conference on Theo-retical Particle Physics, Canton, 1980 (unpublished).

+The Weinberg- Salam phase transition has also beeninvestigated by a number of authors: E. Witten, Ref.41; M. A. Sher, Phys. Rev. D 22, 2989 (1980); P. J.Steinhardt, Harvard report, 1980 (unpublished); andA. H. Guth and E.J.Weinberg, Ref. 1S.

37This section represents the work of E. J. Weinberg,H. Kesten, and myself. Weinberg and I are preparinga manuscript on this subject.The proof of this statement was outlined by H. Kesten(Dept. of Mathematics, Cornell University), with de-tails completed by me.

39E. Witten, private communication.4 S. Coleman and E.J. Weinberg, Phys. Rev. . D 7, 1888

(1973); see also, J. Ellis, M. K. Gaillard, D. Nano-poulos, and C. Sachrajda, Phys. Lett. 83B, 339 (1979),and J. Ellis, M. K. GaiQard, A. Peterman, andC. Sachrajda, Nucl. Phys. B164, 253 (1980),

4~E. Witten, Nucl. Phys. B (to be published).L. Parker, in Asymptotic Structure of Spacetime,editedby F.Esposito and L.Witten (Plenum, New York,1977); V. N. Lukash, I. D. Novikov, A. A. Starobin-sky, and Ya. B. Zeldovich, Nuovo Cimento 35B, 293(1976).