Physics Letters B 271 ( 1991 ) 52-60 North-Holland PHYSICS LETTERS B

Inflationary models with g? # 1

G . F . R . Ell is a,b, D . H . Ly th c a n d M.B. Mi j id a.d

" SISSA, Miramare, 1-34014 Trieste, Italy u Applied Mathematics Department, University of Cape Town, Z4- 7700 Rondebosch, RSA ~ Department of Physics, University of Lancaster, Lancaster LA I 4 YB, UK

Department of Physics and Astronomy, California State University, Los Angeles, CA 90032, USA

Received 6 April 1990; revised manuscript received 17 June 1991

We describe the general features and a few specific models of inflationary cosmology that lead to -(2¢ 1 today, solve all standard problems of classical cosmology, and generate density perturbations that may be responsible for the formation of structures in the universe. Further development along these lines may require a more elaborate analysis of the initial conditions, both for the classical evolution and for the density perturbations.

1. Introduction

Inf lat ionary cosmology [1,2] provides a unif ied scenario that may explain several outs tanding and very different problems of classical cosmology. How- ever, it is usually held responsible for the " theoret ica l pre judice" that the densi ty of the universe must be equal to the critical density, to a very high accuracy. Present observat ion says only that the densi ty pa- rameter £20 is roughly in the range 0.1 < ~2o < 10, with the upper decade excluded if the energy density comes mostly from mat ter or radia t ion as opposed to a cos- mological constant [ 3 ].

The purpose of this letter is to demons t ra te the ex- istence of scalar field driven inf lat ionary models that solve all the usual problems of s tandard cosmology, including the generation of density perturbations, but which also lead to ~ different from unity today. All the models are buil t from solutions of s tandard gen- eral relativity, with a homogeneous scalar field as the source of the energy density and with vanishing cos- mological constant today. The significance of these models may be challenged from the point of view of initial condit ions, or the restr ict ions on the scalar po- tentials. The point is, however, that the very essence of a successful inf lat ionary scenario need not be in conflict with the present observed value of the den- sity of the universe, and, as our discussion will show,

that investigation of inf lat ionary models may be car- ried out in a wider f ramework than usually done. The last point may be useful in the light of our uncer- tainty about initial condit ions, and persistent frustra- t ion in the construct ion of a fully successful inflation- ary model within the given fundamenta l theory.

We will call the expansion of the universe in f la t ion-

ary when

/ /= al l= ( 1 + [ I / H 2 ) > 0

(a is the scale factor, H - ( z / a is the Hubble parame- ter ) . This requires e i t he r / : /> 0, so-called super in f la-

t ion, or k / < 0 with I[II < H 2. When the lat ter in- equali ty becomes strong we have exponent ial expansion. During an inflat ionary phase, the densi ty paramete r

n = 8~rGp 1+ K (1) 3 H 2 -- a 2 H 2

(K is posi t ive for closed models, zero for the flat one, negative for open models other than the flat one) , is a t t racted towards unity, and during a F r i e d m a n n - L e m a i t r e - R o b e r t s o n - W a l k e r ( F L R W ) phase moves away from it [4,5 ]. It is usually considered that min- imal requirements for solving the well-known prob- lems of s tandard cosmology make the density param- eter at the end of inflation so close to unity that the subsequent FLRW phase, up to the present epoch,

52 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

Volume 271, number 1,2 PHYSICS LETTERS B 14 November 1991

does not make a not iceable difference to its value. Here we construct models that avoid this conclusion, our t rea tment differing from the s tandard one in two main respects. First, we keep ,O in the equat ions from the beginning, instead of assuming it takes the criti- cal value 1. The resulting solutions are then some- what different from the usual ones. Second, we do not restrict ourselves to the usual slow-rolling regime: in- deed, we show it is precisely the inf lat ionary phase away from that regime which allows us to avoid con- straints on the usual models implying a densi ty near the critical density. In addi t ion, for some models we use the fact that the horizon problem may be under- stood in such a way that it is solved without a long period of inflat ion [4,6]. We now describe these ele- ments in turn.

2. More general models

In most works on inf la t ionary cosmology, it is usu- ally assumed right from the beginning that the den- sity has the critical value. The equations are then eas- ier to hand le but there is no need to make this assumption.

2. 1. Asymptotically de Sitter models

Just as all the s tandard models were inspired by the flat de Sitter solution, we may use exact inf lat ionary solutions for FLRW universes, with a classical scalar field as the e n e r g y - m o m e n t u m source, that are asymptot ic to open or closed de Sitter universes, but have non-negligible curvature and do not make the slow-rolling assumpt ion [ 7 ]. (A de Sitter universe is one in which p = - p is t ime independent , where p is the energy densi ty and p is the pressure [8 ]. )

For example, consider the usual exponent ial ex- pansion, a( t )=A e x p ( H d ) . When driven by a con- stant potent ial I<] =- 3H~/8~rG this is the flat de Sitter universe. This is not possible for an open universe, but for a closed universe it can occur for a scalar field 0 with the potential [7]

)7( 0 ) = ~71 -~ t I ] ( O,~ - __~))2. (2)

The field evolves as

B 0=0<~ + . ] ~ [ e x p ( - H , t ) , (3)

where B e - K / 4 ~ G is a constant, and K is the (posi- t ive) spatial curvature. Here and elsewhere, the' two signs correspond to rolling from the left or from the right. The density parameter

t2( t ) = 1 + [ - (2 (0 ) - 1 ] exp( - 2 H , t ) , (4)

descends from infinity to un i ty . .Q(0) = 1 +K/A2H~ is its value at some convenient moment , chosen to be l = 0 .

Next consider the behaviour a (t) = H ~ 1 sinh (Hit) . In the simplest case it is dr iven by the constant poten- tial lq, and the density parameter makes a smooth t ransi t ion from 0 to l : - Q = t a n h 2 ( H l t ) . This is the open de Sitter universe [8] . However, for a specific class of scalar potentials [ 7 ]

V(0) = tJ'~ + B 2 sinh 2 2 ~ - ( 0 - 0 2 ) , (5)

with

a solution a=A s inh(Hj t ) is possible for an initially open, closed or flat universe. The field evolves as

B 1 { e x p ( H , t ) - ~ ) 0=0~+ ~, o g ~ l ) + , (7)

and the density parameter as

4TtGB2 "~ 2 f 2 ( t ) = 1+ H : ~ , t ) - ) t a n h (Hzt) , (8)

from an arbi t rary initial value ,Q(0 ) = 1 + K/H~A 2 to unity.

There is also the closed de Sitter universe with a(t) = I t j ~ cosh(Hl / ) and with the density parame- ter descending to unity as -Q=cotanh2(Htt) [8 ]. This solution may be dr iven by a constant potent ial as be- fore, or by a scalar field with the potent ial worked out in ref. [7] .

These examples show some of the variety of possi- ble inflat ionary models. As usual, the constant poten- tial I} in these solutions should be considered as an approx imat ion for some nearly flat potential , varia- tion in this quant i ty eventually (when the field

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Volume 271 number 1.2 PHYSICS LETTERS B 14 November 1991

reaches the non-flat part of the potential) allowing an exit f rom inflation. We will return to this later.

2.2. Coasting models

Quite different is the coasting solution, a ~ t. (The standard case of a coasting solution is the so-called Milne universe, with p = p = 0 [8] . ) The potential is [7]

V(0) = VI exp(2 -0B) , (9)

/dec f at Hpph~- a( t ) (12) ti

(where ti refers to whatever is considered as the be- ginning of the universe in the given cosmological model) , larger than the visual horizon at present [4,111,

dt to Uvh- a ~ ~3--ao = 2 . (13)

/dec

and the field evolves as

where B is a constant and tc = B , ~ Vj is the duration of the coasting phase, f rom some arbitrary ~ > 0 to 0 = 0 . During this phase a H = c i = c o n s t . , the slow- rolling approximat ion is never satisfied, and the den- sity parameter does not change. The constant B may be expressed as

£2 B 2 - (11)

47rG"

This solution can play a very useful role as a pre-in- flationary phase in inflationary scenarios. Although the potential for the coasting solution needs to be rather specific, it does appear in the context of super- strings [9]. Also, the coasting solution need not be driven by a scalar field, but may have as a source any fluid with the equation o f s t a t e p + 3p=0 , such as uni- formly expanding network of strings, or cosmic tex- tures [10].

3. Solving the horizon and flatness problems

We now recall that there are several different re- quirements that are imposed on the duration of infla- tion. The first one is to solve the horizon problem, which is that without inflation the practically iso- tropic microwave background originates from causally disconnected regions. For this, all that one needs is to make the pr imordial particle horizon at the decoupling time,

(This last equality holding only in open or closed universes.) Of course, the requirement that the mi- crowave background is highly isotropic is part of the more general requirement that the universe is very homogeneous on large scales. As we discuss in see- lion 4.1, if the departures from the homogeneity are ascribed to vacuum fluctuations during inflation, they can be made sufficiently small for a suitable infla- t ionary potential without explicitly referring to the horizon problem.

The second requirement is to solve the flatness problem, which is usually understood as being that without inflation the initial value of I 1 -£21 must be tiny. In order that the value of I 1 -£21 should recover its present value N e-folds before the end of an era of almost exponential inflation, it is necessary that [ 12 ]

[ ( I / 4 ) [~114 \ w log( J N = 2 3 + 2 . 3 log ~ ~ ,4,3 /

+ 2 l o g ( ~ 7 ~ ) ] ' pl/4 (14)

where p~, ,o 2 and/93 are respectively the energy densi- ties at the beginning of the almost exponential infla- tion, at the end of this era, and at the beginning of radiation domination. With the usual assumption that radiation dominat ion persists back to a temperature only a few orders of magnitude below the Planck scale, this requires 60 or so e-foldings of exponential infla- tion. (See also fig. 1. ) This generates a huge particle horizon, automatically solving the horizon problem [ 1,2 ]. Indeed for a de Sitter phase with Hubble pa- rameter HI, starting at t ime t~ with the radius a,, and ending at t ime tr~h, we have

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U2 a H = I_O_- 11

I I [ I ~> t. t, t treh t0 [

Fig. I. The evolution of scales in inflationary, cosmology. Infla- tion starts at li and ends at/reh- At the time t. the density param- eter had the same value as at the present time to. If we need infla- tion just for the generation of density perturbations, we may have t.<t~. At any time t<~t. the preceding phase may be also non- inflationaw (NI), coasting (C), or non-standard inflation ( NSI ), i.e., not slow-rolling or not de Sitter-like. For a given £2o today, the behaviour of the density parameter and the relation to the initial values will be very different in each case.

1 1 /,t pph - [ 1--e{ ' ( t i - - t reh) ]

u,h aiHl

Ho = 2H~ (El - - 1 )ERDEMD ' (15)

where El = areh/ ai, ERD--= adec/ arch, and EMD -- ao/ ad~,, are, respectively, expansion factors in the inflation- ary, radiation dominated and matter dominated phases, and Ho~ 10-6°ram is the present value of the Hubble parameter. I f inflation lasts at least 60 e-fold- ings, those three phases are sufficient to make the

particle horizon greater than the visual horizon. However, the p r imord ia l part icle horizon grows

also in a pre- inf la t ionary phase. During such a phase,

by defini t ion, the densi ty paramete r is not dr iven to

unity. Models in which the pre- inf la t ionary phase is

radia t ion domina ted have been discussed in refs. [4,6]. This is a natural assumpt ion for inflat ion ac- companied by a phase t ransi t ion. Other possibil i t ies, including anisot ropic and inhomogeneous solutions, might he more appropr ia te in chaot ic inflation, but the most d ramat ic example is the coasting solution,

discussed above. In these models, the particle hori- zon grows as

f dt 1 , { r ( O i ) " ~ Uvph= - J o g [ ~ j . (16) li

Thus, we may build a large particle horizon and have a large expansion, without any change in £2, which itself is assigned freely. The duration of the subse- quent inflat ionary phase is then de te rmined by re- quirements other than solving the horizon problem.

4. R e q u i r e m e n t s o n an i n f l a t i o n a r y s c e n a r i o

Now we can see what the general requirements are on an inf lat ionary scenario, while keeping track of£2. During the FLRW phase the spatial curvature is cer- tainly less impor tan t at earl ier epochs, but it does af- fect the evolut ion of the inflaton field. The simplest argument that such a scenario will differ from the usual one is as follows. From the F r i edmann con- straint equation and the Raychaudhur i equation we find that [ 7 ]

1

Using the defini t ion of the densi ty parameter £2 and demanding a posit ive kinetic term, 0 2 > 0, we find

/:/ g2>~ l + t t ~ _ . (18)

It follows that superinflat ion is possible only in a closed universe, and that there is no exponential in- flation for a very underdense universe.

Various possibil i t ies may be exhibi ted as follows. From the defini t ion of the density parameter , we find that var ia t ions in one Hubble t ime are related as

~2 = H ' p-" - 2 / / H ' ~ p /~5. (19)

In the spatial ly flat case £2 is constant, and an expo- nential expansion implies a slowly varying energy density, IPl << Hp, and (for scalar field driven infla- t ion ) that the potent ial energy dominates over the ki- netic one, 302/p<< I ~ 5 T < < V(0). The slow-rolling condi t ion is that their increments also obey the same relation, 5 aT<< 8V, and this need not be obeyed in

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general. However, there is a wide class of potentials and initial conditions for which this is the case, and this is normally the only inflationary regime that is considered [2]. In this case the equation of state is p ~ - p .

However, if ~2 is different from unity it generally varies rapidly, because [ 5 ]

~ = ( 2 - 3 V ) £ 2 ( 1 - ~ ) H , (20)

where y - 1 +p/p. If the energy density is slowly vary- ing, 1/51 << lip, the equation of state is again approxi- mately p = - p . From the last two equations we find that this is possible only when the Hubble parameter varies as I H I / H 2 ~ ½ I 1 - g21 . This is not exponential inflation in general, except when the density param- eter is close to unity. The complementary possibility is exponential inflation, but with rapidly varying en- ergy density 1151 ~Hp. During such an inflationary phase the kinetic energy is comparable with the po- tential energy, the slow-rolling condition cannot be obeyed, and the equation of state is different from p = - p . A marginal case is when 7 is just a little bit less than 3, which slows down the variation in ~, and makes every term in eq. (20) just somewhat less than unity. In this case we have power law inflation, with the exponent very close to unity. This is what might happen in some superstring inspired models [9].

4. I. The homogeneity problem and the magnitude q[ perturbations

For any of these inflationary solutions, the evolu- tion of ,(2 may be read from the standard graph that shows the evolution of the expanding scales with re- spect to the Hubble radius, fig. 1. Using the expres- sion for ~, we obtain for any comoving distance scale a/k

k 2 k 2

where ao, f2o, and to refer to the present time and k is the comoving wave number. At the moment t, dur- ing the inflationary phase when ( i f ) the current Hubble scale was crossing the Hubble radius, the density parameter had the same value as today (fig. 1).

In order to solve the flatness problem, we require

that I .Q- 1 I is not tiny at the beginning of inflation. In the usual scenario where I g~o- 1 I is supposed to be tiny, this requires that the observable universe leaves the Hubble distance long after inflation be- gins. If in contrast I ~2o- 1 [ ~ 1, the flatness problem can be solved even if the observable universe starts off somewhat outside the Hubble distance.

Now let us consider the question of perturbations away from homogeneity. In the standard case ]~2o- 1] << 1, the inhomogeneity during inflation is supposed to be a fluctuation in the conformal (Bunch-Davies) vacuum. In other words, the non- zero modes of the inflaton field and gravitational fields are supposed to be in the conformal vacuum. With this assumption the expectation value of the square of each mode amplitude can be calculated in terms of the inflaton potential, and can be made as small as observation requires by taking the potential to be small enough. In this way one finds [ 13 ] that

~1/4 l~ . < 1 X 10 3/npl = 1016 GeV, (22)

where the subscript * indicates the epoch at which the observable universe leaves the horizon.

Since observations suggest that £2o is not too far from unity, we conclude that if there was inflation, all scales well within the present Hubble distance leave and enter the Hubble radius at nearly the critical den- sity. For them the magnitude and the spectrum of the perturbations is therefore obviously the same as in the standard case. Recently, by explicit computat ion [ 14], it has been shown that this conclusion remains true to a good approximation even on larger scales, over the relevant range 0.1 <£2o< 1. Thus eq. (22) holds over this range. There are, however, two differ- ences regarding the status of the assumptions needed to obtain it: - The conformal vacuum hypothesis is needed on scales of order the curvature scale a/IKI , whereas in the standard case it is only needed on much smaller scales. - In order for eq. (22) to be meaningful, it is neces- sary that the observable universe starts off inside the horizon, which we noted earlier is not required by any other consideration. This is, however, a technical condition, in that the spectrum could still be calcu- lated without it for a given potential.

We note that the conformal vacuum hypothesis is normally considered [2] as the most natural for all

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observable scales and beyond. This has also been jus- tified in quantum cosmology [ 15 ], in the case o f a flat or closed universe, adopting some form of the regularity condition on the wave function of the uni- verse (which really is not that much different from the conformal vacuum hypothesis itself).

5. Discussion

Let us summarise the main features of inflationary scenarios with £2o# 1. In many aspects they are the same as in the flat case. The horizon problem is solved, as the particle horizon exceeds the visual ho- rizon. The flatness problem is "solved" in the sense that it has been referred to an earlier epoch, and by construction of models that allow many different ini- tial values for the spatial curvature for one and the same value for £20. The monopole problem is trivially solved, in the manner of chaotic inflation, as the den- sity perturbation requirement makes the reheating temperature too low for the typical monopole-pro- ducing phase transition. The density perturbations on observable scales are generated as usual, essentially because we know from observations that £20 is not too different from unity. Recently this has been explicitly checked for the case £2o< 1 [ 14]. Finally, the homo- geneity o f the observed universe is explained by a sufficiently large expansion, as usual (with some re- strictions in case of a £2<< 1 universe, as we will ex- plain below). This does not lead to £2o = 1, due to the different behaviour of the solutions at early epochs.

Even if our models are to be considered as ad hoc choices, their existence is sufficient to show the falsi- ty of some commonly heard arguments that inflation necessarily implies £20 = 1. It is not true that this is necessarily true due to a huge expansion of 60-fold- ings, as one glance at fig. 1 shows. It is also not true that £2# 1 may be achieved only in models with a true cosmological constant, and that this would be de- stroyed once the dynamics of a scalar field is taken into account. Neither is true that £2# 1 is incompati- ble with a scalar field driven inflation properly con- strained by the demand for perturbations of the de-

sired magnitude ~. Elsewhere [ 14,17,18 ] it is shown how £2< 1 inflationary models avoid the homogene- ity problem raised in ref. [ 19]. For explicit discus- sion of some models see refs. [ 18,20].

There is no conflict between the essential dynam- ics of the inflationary phase and £2o#I as an out- come. In fact, accounting for finite spatial curvature and allowing non-slow-rolling solutions introduces a whole new class of inflationary models. Among them, models that include a coasting phase seem to stand out. During such a phase the particle horizon grows while the density parameter stays constant. The sub- sequent inflationary phase may start with any value for £2, and may be of any duration. The present value of the density parameter is then determined essen- tially by the initial conditions.

The existence of our models should demonstrate that, together with the choice of a definite fundamen- tal theory, the real issue is the choice of initial conditions.

5.1. Setting initial conditions

There have been many investigations of the initial conditions in inflationary cosmology, e.g. refs. [21- 26 ], and it is interesting to see if and how our models avoid their conclusions. We will make just a few re- marks here, leaving a more complete discussion for elsewhere. First, as may be seen from our exact solu- tions, throughout the non-slowly-rolling phases of our solutions, the kinetic and potential energy of the sca- lar field decay at the same rate. This is in contrast to the sequence of kinetic-term dominated and poten- tial-term dominated phases in ref. [21 ] that would quickly lead to the usual slowly rolling solution. Sec- ond, exponential attraction towards an inflationary phase, observed in ref. [22] , is strictly true only if one has a true cosmological constant. Whenever there is a true potential, an additional check needs to be performed to assure that potential energy indeed de- cays at the slower rate. This has been done in some cases, see e.g. ref. [23], and an affirmative answer

The recent negative conclusion in ref. [ 16 ] does not contra- dict our results, both because of the restricted form of the po- tentials used there, and because of the restriction lhere to the slow-rolling regime. The latter property is incompatible with the standard de Sitter-like expansion, [ t / t t 2 << 1, also used there, except when 17 is close to one.

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has been found, but clearly this behaviour is model dependent. Recent discussion of this issue in ref. [24] should highlight both the restricted range of assump- tions about the dynamics of a scalar field and the va- riety in the use of a concept of measure. It is similar with the thorough investigation of the phase space trajectories performed in ref. [25], for the case of a massive scalar field ,2: one may hope to find counter- examples, with different potentials, to this suppos- edly generic behaviour. Our inflationary solutions are not attractors indeed, but it is an open issue for the theory of initial conditions to show the existence or non-existence of initial distributions which would sufficiently favour these solutions.

A more difficult subject is the proposal of "natu- ral" or "typical" initial conditions (e.g. ref. [21 ] ), which may favour the standard inflationary scenario. This approach is based on analysis of internal fea- tures of the inflationary models, and therefore the conclusions may be different for some of the cases introduced here. This way of thinking is complemen- tary to the approach of quantum cosmology, e.g. ref. [26], where definite boundary conditions are im- posed on the wave function of the universe and their consequences for an inflationary phase worked out. In this case too, we may expect the conclusions to be somewhat model dependent.

5.2. Stochastic inflation

There is, however, one important aspect of infla- tionary dynamics that has not been considered in this work. It has been shown that quantum fluctuations during an inflationary phase significantly affect the evolution of the inflaton, making it stochastic [27], and typically leading to the scenario of eternal infla- tion [28] where most of the volume in the universe is occupied by Planck density self-reproducing inflat- ing domains. At a first glance, these quantum effects would destroy any reasonable likelihood of £2¢ 1, as most of the volume in a post-inflationary phase would come from domains that have been inflating indefi- nitely long, thereby reducing the local value of the scalar curvature below any measurable level. We may

~2 Which do not show that curvature may be ignored in almost all cases.

offer just a few speculations that may counter this expectation.

Firstly, we may look for models where the stochas- tic force is made deliberately small compared to the classical force. This may prevent self-reproduction of inflationary domains, or postpone their dominance to exponentially late times. Secondly, the whole sto- chastic description has been derived under the as- sumption of a flat, de Sitter-like background, with a slowly rolling scalar field. These conditions are not realised in early phases of our models, and we would not expect them to hold widely in a truly stochastic situation (which essentially demands consideration of all feasible initial conditions). (Stochastic infla- tion with a non-slowly rolling field in the fiat uni- verse case has been considered in ref. [29].) Fur- thermore, only partial results are known about the behaviour of the metric, and a complete treatment of stochastic effects in the gravitational sector is neces- sary for a definite conclusion. Finally, stochastic ef- fects may actually be helpful in some of our models. For example, they would lead to the spread of the field q~ on the plateau, and consequently to an ensemble with different values for £20. The multicomponent cases may even be more promising.

With all these hopeful speculations in mind, we stress that our present analysis strictly applies only to the classical inflationary phase, and should be ade- quate to demonstrate the possibility of solving the £2 problem within classical inflation, contrary to the usual opinion.

5.3. Low and high density models

Granted favourable initial conditions, both low density and high density universes are compatible with inflation and with the formation of large scale structures, but we wish to comment about some dif- ferences between them in this context. First, as seen from eq. (21), when £2o is very near 1 the presently observable universe can be initially well inside the Hubble distance. But this is impossible if g2 o is much less than 1, because the minimum fraction of the Hubble distance ever occupied by the present uni- verse is ~ . This would mean that the homoge- neity of the universe can no longer be explained by postulating that the universe started out by being very homogeneous on scales much less than the initial

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Hubble distance. Rather, if observation show that £2o < 1, we will know that the universe started out very homogeneous on scales of order the initial Hubble distance, and we will have to explain why. As noted in section 4.1, the usual explanation in terms of vac- uum fluctuations works just as well for .O0=0.1 as it does for £2o= 1. A somewhat secondary problem is that, if the universe is rather underdense, in order to produce galaxies we may have to increase the nor- malisation of the perturbations so much that we may have a conflict with the observed absence o f a cosmic blackbody radiation quadrupole anisotropy. In this case an open, but not too underdense, universe might be preferred, or an overdense universe with positive cosmological constant to avoid conflict with the ages of globular clusters. Both these options could avoid the quadrupole problem. Such models with k = + 1 have the additional advantage that horizons are fully broken early on, in inflation with a prior radiation- dominated era or prior contraction, in the sense that once j-',, dt'/a(t') > ~r, all the matter in the universe is in causal contact with all the other matter [ 11 ]. For more detailed discussions of the possibility of having -O¢ 1 in the post-inflationary universe see ref. [30].

utility of inflationary solutions that are not slow-roll- ing or de Sitter-like, and presented models that in principle lead to -O4= 1 for more than just an exponen- tially small fraction of all initial conditions. In partic- ular, models involving a coasting solution seem to stand out as promising. We should also remember that most proposals for initial conditions assert anyway that the initial state was in some sense special. The possibilities discussed here should perhaps be kept in mind in more detailed model building, and in the further study of initial conditions, which have yet to provide the final answer.

Acknowledgement

We acknowledge useful discussions with, or com- ments from M. Bruni, A. Linde and S. Mollerach. This work has been supported by the Italian Ministero per la Ricerca Scientifica.

References

6. Conclusion

Of course, the outstanding task in cosmology has always been not only to obtain the observed universe as a solution of the fundamental theory, but also to understand if this is perhaps the only possible solu- tion, and why. The scenarios described here do not seem to point to any unique solution, but rather to many possible Friedmann universes, in particular with many possible values for the density parameter.

Our conclusion is that a scenario as attractive as inflationary cosmology need not be viewed with sus- picion if the observed density of the universe turns out to be different from the critical value. Nor should it constrain our ideas about structure formation, the CMB anisotropy, etc., entirely within the framework of the flat Friedmann model. Inflation apparently may allow for the present density parameter to be notice- ably different from unity. We have illustrated here what the price for this is, in terms of constraints on the initial conditions and general requirements on the appropriate inflationary phase. We have shown the

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