Inflation Phy Rev Guth

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PHYSICAL REVIEW D VOLUME 23, NUMBER 2 15 JANUARY 1981

In flation ary u niverse: A p ossib le solu tion to th e h orizon an d flatn ess p ro blem s

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

(Received 11August 1980)

The st andard model of hot big-bang cosmology requi res ini tial conditi ons which are problemati c in two ways: (I)The early universe is assumed to be highly homogeneous, in spite of the fact that separated regions were causallydisconnect ed (hori zon problem); and (2) the ini tial value of the Hubble constant must be fine tuned to ext raordinaryaccuracy 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, i n i ts early hist ory, the universe supercool ed to temperatur es 28 or more ordersof magnit ude below the crit ical temperature for some phase t ransiti on. A huge expans ion factor would then resultfrom a period of exponential growth, and the entropy of the universe would be multiplied by a huge factor when thelat ent heat is released. Such a scenari o is complet el y natural in t he context of grand uni fied model s of elementary"parti cl e i nteractions. In such model s, the super cooli ng i s al so rel evant to the problem of monopole suppres sion.Unfor tunate ly, the scenario seems to lead to some unacceptable consequences, so modifica tions 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 adiabatically

expanding radiation-dominated universe describedby a Robertson-Walker 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 = O. As t - 0, the

temperature T -00.

Thus, no initial-value prob-lem can be defined at t = O. However, when T isof the order of the Planck mass (M p'" 1/Vc = 1.22x 10 19 GeV)1 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 ourknowledge, it is sensible to begin the hot big-bang

scenario at some temperature To which is com-fortably below Mp; let us say To=101! 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 is

taken to be homogeneous and isotropic, and filled

with a gas of effectively massless particles inthermaL equilibrium at temperature To. The ini-tial value 'of the Hubble expansion "constant" H istaken to be H o , and the model universe is then

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

well-known horizon problem. 2-4 The initial uni-verse is assumed to be homogeneous, yet it con-sists of at least -10 83 separate regions which arecausally disconnected (i. e., these regions havenot yet had time to communicate with each othervia light signals). 5 (The precise assumptionswhich lead to these numbers will be spelled out in

Sec. II.) Thus, one must assume that the forceswhich created these initial condi tions were capable

of violating causality.The second puzzle is the flatness problem. This

puzzle seems to be much less celebrated than thefirst, but it has been stressed by Dicke and Pee-blas." I feel that it is of comparable importance

to the first.It

is known that the energy densityP

of the universe today is near the critical value Per(corresponding to the borderline between an openand closed universe). One can safely assume that!

0.01 < Q p < 10,

where

(1. 1)

(1. 2)

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 that

the condition Q"" 1 is unstable. Furthermore, theonly time scale which appears in the equations for

a radiation-dominated universe is the Planck time,1 IMp = 5. 4 x 10-.14 sec. A typical closed universewill reach its maximum size on the order of thistime scale, while a typical open universe will

dwindle to a value of P much less than Per. A uni-verse can survive -10 10 years only by extreme finetuning of the initial values of P and H, so that P isvery near P cr ' For the initial conditions taken at

23 347


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34 8 ALAN H. GUTH 23

To = 10 17 GeV, the value of Ho must be fine tunedto an accuracy of one part in 10 55 In the standardmodel this incredibly precise initial relationshipmust be assumed without explanation. (For any

reader 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 To. If I had chosen a modestvalue such as To=l MeV, I would still have con-cluded that the "initial" universe consisted of atleast -10 22 causally disconnected regions, and thatthe initial value of Ho was fine tuned to one part in10 15 These numbers are much smaller than theprevious set, but they are still very impressive.Of course, any problem involving the initial

conditions can always be put off until we under-

stand the physics of T?; M r - However, it is thepurpose of this paper to show that these puzzlesmight be obviated by a scenario for the behaviorof the universe at temperatures well below Mp.The paper is organized as follows. The assump-

tions and basic equations of the standard model aresummarized in Sec. II. In Sec. III, 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. '

II. THE STANDARD MODEL OF THE VERY EARLYUNIVERSE

In this section I will summarize the basic equa-tions of the standard model, and I will spell out theassumptions which lead to the statements made inthe Introduction.The universe is assumed to be homogeneous and

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

dr =dt2-R2(t)[1 ~~YJ +YJ(de2 + sin 2ed2)] ,

(2.1)where k= + 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 rand R(t) are rescaled so that k takes onOneof the three discrete values. The evolution ofR(t) is governed by the Einstein equations

. . 4 1 TR = -3 C(p +3p)R, (2.2a)

k 81 TH2 + R2 = -3 Cp ,

where H o= RIR is the Hubble"constant" (the dot de-notes the derivative with respect to t). Conserva-

tion of energy is expressed by

(2.2b)

d 3 d ( 3)dt (pR ) = - P dt R , (2.3)

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

(2.4)

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 Nb(T) de-note the number of bosonic spin degrees of free-domwhich are effectively massless at temperatureT (e. g., the photon contributes two units to N b);and let Nf(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

1T 2(2.5)= 3p = -;n(T)r4,

30

= 2 1 7 ' 2;n(T)T 3 (2..6)s 45 . ,

n = b(~);n'(T)T3, (2.7)1 7 '

where

;n(T) =Nb(T) + tN,(T), ( 2 . 8 )

;n'(T) =Nb(T) + ~ Nj(T) . (2.9)

Here n denotes the particle number density, andb(3) = 1. 20206 ... 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

(1 ')2 4 1T 3T + E(T)T 2 = 45 C;n(T)r4 , (2.10)

where

k r 2 1 T2 ;n(T)] 21 3E(T) = R2T2 = kL45S- ,

where So=R3S denotes the total entropy in a volume

(2.11)


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23 INFLATIONARY UNIVERSE: A POSSIBLE SOLUTION TO ... 34 9

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) by

current observations. Taking p < 10Per today, itfollows that today

I;2 \< 9H2 . (2.12)From now on I will take k= 1; the special case k

= 0 is still included as the limit R - 00. Then todayR > t H . . -1 - 3 X 10 9 years. Taking the present photontemperature T, as 2. 7 "K, one then finds that thephoton contribution to S is bounded by

(2.13)

Assuming that there are three species of masslessneutrinos (e, 11, and T), all of which decouple at a .time when the other effectively massless particlesare the electrons and photons, then S; = 21/22S,.Thus,

(2.14

and

lE I < 1 0-58~ 2/3

But then

IP - p P L \=_i~ M i lE I


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3 50 ALAN H. GUrH 23

continue to cool as it expands, and it will then su-

percool in the high-temperature phase. Suppose

that this supercooling continues down to some tem-

perature T., many orders of magnitude below Te.When the phase transition finally takes place at

temperature T., the latent heat is released. How-ever, this latent heat is characteristic of the ener-

gy scale Te, which is huge relative to Ts. The uni-verse is then reheated to some temperature T';

which is comparable to Te. The entropy density isthen increased by a factor of roughly (TrITs)3 (as-suming that the number ~ of degrees of freedomfor the two phases are comparable), while the val-

ue of R remains unchanged. Thus,

(3.4)

If the universe supercools by 28 or more orders of

magnitude below some critical temperature, the

horizon and flatness problems disappear.

In order for this scenario to work, it is neces-

sary for the universe to be essentially devoid of

any strictly conserved quantities. Let n denote thedensity of some strictly conserved quantity, and

let r = =nl s denote the ratio of this conserved quan-tity to entropy. Then rp = Z-3ro < 10- S4ro. Thus,only an absurdly large value for the initial ratio

would lead to a measurable value for the present

ratio. 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 net

baryon number of the universe is believed to be

created by CP-violating interactions at a tempera-

ture of 1013 _1014 GeY. 9 Thus, provided that Te liesin this range or higher, there is no problem. The

baryon production would take place after the re-heating. (However, strong constraints are im-

posed on the entropy which can be generated in any

phase transition with Te 1014 GeY, in particular,the Weinberg-Salam phase transition. 36)

Let us examine the properties of the supercooling

universe 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 ~ 0, the system iscooling not toward the true vacuum, but rather to-

ward some metastable false vacuum with an energy

density Po which is necessarily higher than that of

the true vacuum. Thus, to a good approximation(ignoring mass thresholds)

1 T2

p(T) = 30~(T)y4 + Po . (3.5)

Perhaps a few words should be said concerningthe zero point of energy. Classical general rela-tivity couples to an energy-momentum tensor of

matter, T ILV' which is necessarily (covartantly)conserved. When matter is described by a field

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

T ILV - T"v + Ag"v, (3.6)

for any constant A. (A cannot depend on the valuesof the fields, nor can it depend on the temperature

or 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's

equations without an explicit cosmological term;

the cosmological constant A is then defined by

(3.7)

where 10 ) 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 1< 10-.16Gey4)10 so I will take its value to be zero. 11 The

value of Po is then necessarily positive and is de-termined by the particle theory. 12 It is typically

of O(T/).Using (3.5), Eq. (2.10) becomes

(T 'f 47T3 4 2 87TT J =45G~(T)T - E(T)T +3 Gpo. (3.8)

This equation has two types of solutions, depending

on the parameters. If E> EO, where

8 7T2,j 30EO =~G,j~po, (3.9)

then the expansion of the universe is halted at a

temperature T min given by

4 30p {E [( E )2]U2}2Tmin =~ -+ - -17T2

EO EO '(3.10)

and then the universe contracts again. Note that

Tmin is of O(T e), so this is not the desired scenar-

io. The case of interest is E< EO, in which case theexpansion of the universe is unchecked. lNote thatEo - ,ffiTc 2 IMp 2 is presumably a very 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 Po termto dominate over the other two terms on the RHSof (3.8), one has '

T(t) '" const x e -x t ,

where

(3.H)

2 87T GX = 3 Po (3.12)

Since RT = const, one has 13

R(t)=constxe xt (3.13)

The universe is expanding exponentially, in a false


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23 INFLATIONARY UNIVERSE: A POSSIBLE SOLUTION TO .. 3 51

vacuum state of energy density Po. The Hubbleconstant is given by H = R /R = X. (More precisely,H approaches X monotonically.from above. This

behavior differs markedly from the standard mod-el, in which H falls as ["1. )

The false vacuum s tate is LOrentz invariant, so

T uv = P~uv' It follows that p =- Po, 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 force

behind the exponential expansion.The Lorentz invar iance of the false vacuum has

one other consequence: The metric described by(3.13) (with k= 0) does not single out a comovingframe. The metric is invariant under an 0(4, 1)

group of transformations, in contrast to the usualRobertson-Walker invariance of 0(4). 14 It is known

as the de Sitter metric, and it is discussed in thestandard literature. 15

Now consider the process of bubble formation ina Robertson-Walker universe. The bubbles formrandomly, so there is a certain nucleation rateA(t), which is the probability per (physical) volumeper time that a bubble will form in any regionwhich is still in the high- temperature phase. I

will idealize the situation slightly and assume that

the bubbles start at a point and expand at the speed

of light. Furthermore, I neglect k in the metric,so dr =dt 2 - R 2(t)d X2

I want to calculate pe t ) , the probability that anygiven point remains in the high-temperature phase

at 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 which

form inside the real bubbles with the same nuclea-tion rate A(t). With all bubbles expanding at thespeed of light, the fictitious bubbles will be forever

inside the real bubbles and will have no effect onpe t ) . The distribution of all bubbles, real and fic-titious, is then totally uncorrelated.pe t ) is the probability that there are no bubbles

which engulf a given point in space. But the num-ber of bubbles which engulf a given point is a Pois-son-distributed variable, sop(t)=exp[-N(t)],where N(t) is the expectation value of the numberof bubbles engulfing the point. ThuS 16

(3.14)

where

Ve t ,t1) = 4 ; [ 1 t: fh ]3tl 2

is the coordinate volume at time t of a bubble which

(3.15)

formed at time t1I will now assume that the nucleation rate is suf-

ficiently slow so that no Significant nucleation takes

place until T T e, when exponential growth has set

in. I will further assume that by this time A(t) isgiven approximately by the zero- temperature nu-cleation rate Ao. One then has

(3.16)

where

(3.17)

and 0(1) refers to terms which approach a constantas xt - 00. During one of these time constants, theuniverse will expand by a factor

(

3X4 )

Z, = exp(XT) = exp 47TAo.(3.18)

If the phase transition is associated with the ex-

pectation value of a Higgs field, then Aocan be cal-culated using the method of Coleman and Callan. 17The key point is that nucleation is a tunneling pro-

cess, so that Aois typically very small. TheColeman-Callan method gives an answer of theform

(3.19)

where B is a barrier penetration term and A is adimensionless coefficient of order unity. Since Z,

is then an exponential of an exponential, one canvery easily 18,19, 36 obtain values as large as 10gioZ

'"28, or even 10gloZ'" 10 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 early universe canbe described from the beginning by a Robertson-Walker metric. If this assumption were really

necessary, 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, was rather chaotic. Onewould then expect that statistical effects would tendto thermalize the particle distribution on a localscale. 2o It has also been shown (in idealized cir-cumstances) that anisotropies in the metric aredamped out on the time scale of -10 3 Plancktimes. 21 The damping of inhomogeneities in the

metric has also been studied,22 and it is reasonable

to expect such damping to occur. Thus, assumingthat at least some region of the universe started at


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35 2 ALAN H. GUTH 23

temperatures high compared to Te, one would ex-

pect that, by the time the temperature in one of

these regions falls to Te, it will be locally homoge-neous, isotropic, and in thermal equilibrium. By

locally, I am talking about a length scale ~whichis of course less than the horizon distance. It will

then 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 Te, the inflationary scenario will take place.

The end result will be a huge region of space whichis homogeneous, isotropic, and of nearly criticalmass density. If Z is sufficiently large, this re-gion can be bigger than (or much bigger than) our

observed 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-particle interactions. 23,24

A grand unified model begins with a Simple gaugegroup G which is a valid symmetry at the highestenergies. As the energy is lowered, the theory

undergoes a hierarchy of spontaneous symmetry

breaking into successive subgroups: G ~ H;- . - H o, where H1 = SU 3XSU z XU 1 [QeD (quantum

chromodynamics) x Weinberg-Salam] and Ho = SU 3XUfM. In the Georgi-Glashow model,23 which is

the simplest model of this type, G = SU 5 and n = 1.The symmetry breaking of SU, -SU 3XSU 2XU 1 oc-curs at an energy scale M x -10

14 GeV.

At high temperatures, it was suggested by Kirzh-nits and Linde 25 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 the

tree potential is valid, the phase structure of theSU 5 model was analyzed by Tye and me. 16,27 We

found that the SU 5 symmetry is restored at T> -IOl4 GeV and that for most values of the param-eters there is an intermediate-temperature phasewith gauge symmetry SU 4 x U 1, which disappears atT-10 13 GeV. Thus, grand unified models tend toprovide phase transitions which could lead to an in-flationary scenario 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.i'' and will be

present in any model satisfying the above descrip-tion. 29 These monopoles typically have masses of

order MX/CI-10 16 GeV, where CI=i/41T is thegrand unified fine structure constant. Since the

monopoles are really topologically stable knots inthe Higgs field expectation value, they do not existin the high-temperature phase of the theory. They

therefore come into existence during the course ofa phase transrtion, 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 Zeldovich and Khlopov''" andby Preskill. 31 In Pre skill' s analysts, which was

more specifically geared toward grand unified

models, it was found that relic monopoles wouldexceed present bounds by roughly 14 orders of

magnitude. Since it seems difficult to modify theestimated annihilation rate, one must find a scen-

ario which suppresses the production of thesemonopoles.

Kibble 32 has pointed out that monopoles are pro-duced in the course of the phase transition by the

process 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 will

be impossible for the uncorrelated Higgs fields toalign uniformly. One expects to find topologicalknots, and these knots are the monopoles. The

number 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. 33

At the time of bubble coalescence tcoal the size l ofthe bubbles cannot exceed the horizon distance atthat time. So

l 2 ~< tcoal = 2 'Y T coal

(4.1)

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

3T 6n


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23 INFLATIONARY UNIVERSE: A POSSIBLE SOLUTION TO ... 353

(4.4)

If T; - 10 14 GeV, this bound implies that the uni-verse must supercool by at least about four orders

of 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. 16,34 We

showed how to explicitly calculate the bubble den-

sity i.n terms of the nucleation rate, and we con-

sidered the effects of the latent heat released in

the phase transition. Our conclusion was that

(4.4) should be replaced by

Tcoal < 2x10 11 GeV, (4.5)

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

Tye and I omitted the crucial effects of the mass

density P o of the false vacuum. However, our work

has one clear implication: If the nucleation rate is

sufficiently large to avoid exponential growth, then

far too many monopoles would be produced. Thus,the monopole problem seems to also force one into

the inflationary scenario. 35

In the simplest SU 5 model, the nucleation rates

have been calculated (approximately) by Weinberg

and me. 19 The model contains unknown parame-

ters, so no definitive answer is possible. We do

find, however, that there is a sizable range of pa-

rameters which lead to the inflationary scenario. 36

V. PROBLEMS OF THE INFLATIONARY SCENARI037

As I mentioned earlier, the inflationary scenario

seems to lead to some unacceptable consequences.It is hoped that some variation can be found which

avoids these undesirable features but maintains the

desirable ones. The problems of the model will be

discussed in more detail etsewhere;" but for com-pleteness I will give a brief description here.

The central problem is the difficulty in finding a

smooth ending to the period of exponential expan-

sion. Let us assume that A(t) approaches a con-stant as t - 00 and T - O. To achieve the desiredexpansion factor Z > 10 28, one needs Ao / x 4< 10- 2

[see (3.18)], which means that the nucleation rate

is slow compared to the expansion rate of the uni-

verse. (Explicit calculations show that Ao / x 4 istypically much smaller than this value. 18,19,36) The

randomness of the bubble formation process then

leads 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

bubble. 17 This energy can be thermalized only

when the bubble walls undergo many collisions.

(n) The de Sitter metric does not single out a co-

moving frame. The 0(4, 1) invariance of the de

Sitter metric is maintained even after the forma-tion of one bubble. The memory of the original

Robertson-Walker comoving frame is maintained

by the probability distribution of bubbles, but the

local comoving frame can be reestablished only af-

ter enough bubbles have collided.

'(iii) The size of the largest bubbles will exceed

that of the smallest bubbles by roughly a factor of

Z; the range of bubble sizes is immense. Thesurface energy density grows with the size of the

bubble, so the energy in the walls of the largest

bubbles can be thermalized only by colliding with

other 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 subtle

question about the region of space which is in the

new phase: Is the region composed of finite sepa-

rated clusters, or do these clusters join together

to form an infinite region? The latter poasibi.li.ty

is called "percolation." It can be showrr" that thesystem percolates for large values of Ao / x 4 , butthat for sufficiently small values it does not. Thecritical value of Aoli has not been determined,but presumably an inflationary universe would have

a value of Ao / x 4 below critical. Thus, no matterhow long one waits, the region of space in the new

phase will consist of finite clusters, each totally

surrounded 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 the

scenario is impossible, but these consequences

are at best very unattractive. Thus, it seems that

the 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 trapped

in a false vacuum state, unless A o / X 4 2 :1. Such alarge value of Ao / x 4 does not seem likely, but it

is possible. 19

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:

(t) The horizon problem. Causally disconnectedregions are assumed to be neatly identical; in par-


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354 ALAN H. GUTH 23

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

(n) 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 the

universe 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 if

one could avoid the assumption that the nucleationrate i\(t) approaches a small constant i\o as the

temperature T - O. 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 ifTj ~ 1O- 28Tc'Recently Witten 39 has suggested that the above

chain of events may in fact occur if the parametersof the SU 5 Higgs field potential are chosen to obeythe Coleman-Weinberg conditton'" (i, e., that a2 VIacp2=0at cp=O) . Witten 4j has studied this possi-bility in detail for the case of the Weinberg-Salamphase transition. Here he finds that thermal tun-neling is totally ineffective, but instead the phasetransition is driven when the temperature of theQCD chiral-symmetry-breaking phase transitionis reached. For the SU 5 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 left

out of the analysis is the production of particlesdue to the changing gravitational metric. 42 Thiseffect may become important in an exponentiallyexpanding universe at low temperatures.In conclusion, the inflationary scenario seems

like 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 wouldlike 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, andEdward Witten. 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 some

skeptics 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 = O. Then if onefixes the value of H corresponding to some arbi-trary temperature Ta, one always finds that whenthe equations are extrapolated backward in time,n -1 as t - O. Thus, they would argue, it is na-tural for n 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 Ho and To have no significance, themodel universe must be specified by its conservedquantities. In fact, the model universe is com-

pletely specified by the dimensionless constant E=" k1R2T2, where k and R are parameters of theRobertson-Walker metric, Eq. (2.1). For ouruniverse, one must take IE 1< 3 X 10-57. The prob-lem then is the to explain why IE I should have sucha startlingly small value.Some physicists also take the point of view that

E="O is plausible enough, so to them there is noproblem. 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.


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23 INFLATIONARY UNIVERSE: A POSSIBLE SOLUTION TO ... 355

*Present address: Center for Theoretical Physics,.Massachusetts Institute of Technology, Cambridge,

Massachusetts 02139.II use units for which 'If= c =k (Boltzmann constant)= 1. Then 1 m= 5.068x 10 15 GeV- 1 , 1 kg= 5.610x 10 26

GeV, 1 sec=1.519x10 24 Gev- 1 , and1K=8.617x10- 14GeV.

2W.R indler, Mon. Nor R. Astron. Soc. 116, 663 (1956).See also Ref. 3, pp. 489-490, 525-526; and Ref. 4, pp.740 and 815.

3S. Weinberg, Gravitation and Cosmology (Wiley, NewYork, 1972).

4C. W. Misner, K. S. Thorne, and J. A. Wheeler, Grav-itation (Freeman, San Francisco, 1973).

5Ino rder to calculate the horizon distance, one must ofcourse follow the light trajectories back to t = O. Thisviolates my contention that the equations are to betrusted only for T -;:;,o. Thus, the horizon problemcould be obviated if the full quantum gravitationaltheory had a radically different behavior from thenaive extrapolation. Indeed, solutions of this sort

have been proposed by A. Zee, Phys. Rev. Lett. 44,703 (1980) and by F. W. Stecker, Astrophys. J. 235,L1 (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 To.

6R. 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, Lon-

don, 1979).7See Ref. 3, pp. 475-481; and Ref. 4, pp. 796-797.BFor example, see Ref. 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-:-R"ev. 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. Rev. D 19, 1036 (1979);S. Weinberg, Phys. Rev. Lett. 42, 850 (1979); D. V.Nanopoulos and S. Weinberg, Phys. Rev. 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) [Sov. Phys.-JETP

49, 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) ; Nuc!. 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 andF. Wilczek, Phys. Lett. 95B, 222 (1980) ; G. Senjanovicand F. W. Stecker, Phys~tt. B (to be published).

llThe 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.

12S.A. Bludman and M. A. Ruderman, Phys. Rev. Lett.

38, 255 (1977).13The 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 TECHRepor t No. 79-0984 (unpublished), and by S. A. Blud-

man, University of Pennsylvania Report No. UPR-0143T, 1979 (unpublished).

14More precisely, the usual invariance is 0(4) ifk =1,0(3,1) ifk = -1, and the group of rotations and trans-lations in three dimensions ifk = O.

15Seefor example, Ref. 3, pp. 385-392.16A.H. Guth and S. -H. Tye, Phys. Rev. Lett. 44, 631(1980); 44, 963 (1980).

17S.Coleman, Phys. Rev. D 15, 2929 (1977); 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. 1979),

18A.H. Guth and E. J. Weinberg, Phys. Rev. Lett. 45,1131 (1980).

19E. J. Weinberg and I are preparing a manuscript onthe possible cosmological implications of the phasetransitions in the SU 5 grand unified model.

20J. Ellis and G. Steigman, Phys. Lett. 89B, 186 (1980);J. Ellis, M. K. Gaillard, and D. V. Nanopoulos, ibid.90B, 253 (1980).

21B."L. Hu and L. Parker, Phys. Rev. D 17, 933 (1978).22SeeRef. 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. Weinberg, ibid. 33, 451 (1974); and A. J. Buras,J. Ellis, M. K. Gaillard, and D. V. Nanopoulos, Nucl,Phys. B135, 66 (1978).

240ther grand unified models include the SO(10) model:

H. Georgi, in Particles and Fields-1975, proceedings

of the 1975 meeting of the Division of Particles andFields of the American Physical Society, edited byCarl Carlson (AlP, New York, 1975); 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. Giirsey, P. Ramond, and P. Sikivie, Phys.Lett. 60B, 177 (1975); F. Giirsey and M. Serdaroglu,Lett. Nuovo Cimento 21, 28 (1978). The E(7) model:F. Giirsey and P, Sikivie, Phys. Rev. Lett. 36, 775(1976), and Phys. Rev. D 16, 816 (1977); P. Ramond,Nucl. Phys. BllO, 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).

25D.A. Kirzhnits and A. D. Linde, Phys. Lett. 42B,471 (1972).

26S.Weinberg, Phys. Rev. D~, 3357 (1974); L. Dolanand R. Jackiw, ibid.~, 3320 (1974); see also D. A.Kirzhnits and A. D. Linde, Ann. Phys. (N.Y.) 101, 195(1976); A. D. Linde, Rep. Prog. Phys. 42, 389 (1979).c-expanslon techniques are employed by P. Ginsparg,Nucl, Phys. B (to be published).

27Inthe case that the Higgs quartic cotiplingsare com-parable tog or smaller Ig= gauge coupling), thephase structure has been studied by M. Daniel andC. E. Vayonakis, CERN Repor t No. TH.2860 1980


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356 ALAN H. GUTH 23

(unpublished); and by P. Suranyi, University of Cin-cinnati Report No. 80-0506 (unpublished).

28G.'t Hooft, Nucl. Phys. B79, 276 (1974); A. M. Poly-akov, Pis 'ma Zh. Eksp. Teor. Fiz. 20, 430 (1974)[JETP Lett. 20, 194 (1974)]. For a review, see

P. Goddard and D. 1. Olive, Rep. Prog. Phys. 41, 1357(1978).29lfTIt

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