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Elements of String Cosmology

arXiv:hep-th/9109048v1 25 Sep 1991HUTP-91/A049JHU-TIPAC-910028Elements of String CosmologyA. A. Tseytlin†Department of Physics and AstronomyThe Johns Hopkins UniversityBaltimore, Maryland 21218, USAC.

VafaLyman Laboratory of PhysicsHarvard UniversityCambrdige, MA 02138, USAAspects of string cosmology for critical and non-critical strings are discussed empha-sizing the necessity to account for the dilaton dynamics for a proper incorporation of “large- small” duality. This drastically modifies the intuition one has with Einstein’s gravity.For example winding modes, even though contribute to energy density, oppose expansionand if not annihilated will stop the expansion.

Moreover we find that the radiation dom-inated era of the standard cosmology emerges quite naturally in string cosmology. Ouranalysis of non-critical string cosmology provides a reinterpretation of the (universal coverof the) recently studied two dimensional black hole solution as a conformal realization ofcosmological solutions found previously by Mueller.9/91† On leave of absence from the Department of Theoretical Physics, P. N. Lebedev Physics Insti-tute, Moscow 117924, USSR.

Address from October 1, 1991 : DAMTP, University of Cambridge,Cambridge CB3 9EW and Trinity College, Cambridge, CB2 1TQ, Cambridge, United Kingdom.

1.IntroductionDespite tremendous progress in our understanding of fundamental strings in the pastdecade, we are still very far from a single quantitative prediction to be observed in exper-iments! The main reason for this unsatisfactory state of affairs is that the natural scalein which the stringy effects become important (Planck scale) is much smaller than thescale we can probe in high energy scattering experiments.

However, it has been known fora while that cosmology provides a window to test fundamental physics, since the scalesof particle physics theories become relevant as the universe evolves in its early history.It is our belief that the most likely area for a confrontation between string theory andexperiments is through extracting cosmological consequences of string theory which maybe measurable today. Many of the riddles of astrophysics, for example ‘dark matter’ andlarge scale structure of the universe may turn out to be related to fundamental strings.

Itis with such a program in mind that we study string cosmology in this paper.To see the effects of strings we have to trace back the universe to its earliest times.Here already string theory provides us with a surprise: strings in very small spaces behavethe same way as strings in very large spaces [1,2]. This is true at least in the simplestknown cases (toroidal backgrounds) and is suspected to be a general principle of stringtheory, because strings should be ‘blind’ to scales smaller than their own natural scale.Thus the universe does not start at a zero size, as that would be equivalent to infinitelylarge universe, rather we should start with a universe of the size of the string scale, i.e., thePlanck length.

This a →1/a duality (in Planck units) already hints of a major modificationto Einstein equations at early times, as ordinary theory of general relativity is not invariantunder such a transformation. It has been suggested that, roughly speaking, string theorynecessitates the existence of two kinds of matter fields, “dual” to each other.

For large radii,one type becomes massless and dominates, while for small radii the other type becomesmassless and dominates. This suggests an effective “doubling” of the number of spacedimensions (see e.g.

[2,3]). Needless to say to have an exact treatment of dynamics in1

such a situation is beyond our scope at the present time. This was one of the main reasonswhy the cosmological ideas in [2] were rather difficult to test, as the dynamical equationsgoverning string cosmology were not properly understood.It turns out that it is possible to introduce a consistent dynamics which describesslowly evolving states of strings in a duality invariant manner.

This involves, in particular,taking into account the dilaton field which transforms under the duality [4-7].It is a fundamental consequence of string theory that the gravitational interactionis effectively described not by the metric alone but by the coupled system of the metricand the dilaton.Dilaton of course should get a mass to avoid a scalar component ofgravitational interaction of massive bodies but this probably happens at a later stage ofevolution of the Universe. That implies that one cannot not ignore the dilaton dynamicsat early “stringy” phases of cosmological evolution.

As we shall see, the assumption thatdilaton is constant is in general inconsistent with the cosmological equations describingevolution of matter with the “stringy” equation of state1.The main body of this paper is devoted to the discussion of the duality invariant cos-mological equations describing the evolution of the “scales” of the metric and the dilaton.We shall study their solutions emphasizing departures from Einstein’s gravity theory. Forexample, we will find that a constant energy density not only does not lead to inflation,but it rather slows and eventually stops expansion2.

Similarly, we will discover that if1Dilaton dynamics was ignored in most previous discussions of string cosmology at finitetemperature (see e.g. [8] and refs.

there). Dilaton was not set equal to constant in ref.

[9] where,however, the space was not taken to be compact and hence many of the stringy effects, and inparticular duality invariance was not discussed.2We shall describe the expansion/contraction in terms of the original unrescaled ( “σ -model”) metric. Though the effective gravitational constant is then time dependent , this is the metricstring directly interacts with and hence the one measured by stringy “rods”.

Similar point ofview was expressed in [10]. The use of this metric is appropriate at early phase of cosmologicalevolution when dilaton is still massless (for a discussion of a possible inflation in string cosmologydescribed in terms of the rescaled metric after the dilaton already got mass see [11]).

Let us notealso that the use of the “unrescaled” metric is particularly natural in the duality symmetric casesince it has a simple transformation law under the duality.2

the winding modes (by “winding modes” we mean a more general notion of stringy stateswhose masses grow with a growing scale of the universe) are not annihilated (which mayhappen if they are not in thermal equilibrium) they halt the expansion. Using our equa-tions we can study some simple aspects of the cosmological scenario proposed in [2] inwhich the three dimensionality of extended space was suggested to be a consequence ofstring cosmology.The equations we consider apply to both critical and non-critical strings and we studythem both in the presence and in the absence of matter.

The simplest solution correspond-ing to non-critical strings was first proposed by Myers [12] as a way to change the dimensionof target space in string theory and was then used for construction of cosmological solutionsin [13]. These solutions were further extended by Mueller [14] who found non-trivial cos-mological solutions for which the radii of the toroidal universe change with time.

We willshow that specializing his solution to d=2 one recovers (a region of) the two dimensionalblack-hole solution, which has been found to be related (in the leading order approxima-tion) to a conformal theory [15]3. We find that the more natural setup to discuss d=2cosmology is to go to an infinite fold cover of SL(2, R)/U(1).

This will avoid getting time-like loops and give us infinitely many universes parametrized by an integer n, such thatthe future of particles in the universe n could be in any universe m ≥n. The exact metricsuggested for the d=2 Euclidean black hole [17] which leads to a puzzle within a black-holeinterpretation seems perfectly consistent once we use such a cosmological interpretation(corresponding to the universal cover of SL(2, R)/U(1)).The organization of this paper is as follows.

In section 2 we derive the basic equationsof string cosmology in the adiabatic approximation and discuss their duality invarianceproperty. In section 3 we specialize to critical strings and consider some basic elements ofstring cosmology by studying the solutions of these equations.

We find a simple intuitive3As was already pointed out in [5], the euclidean continuation of the d=2 Myers solutioncoincides with the euclidean solution independently discovered in [16].3

way to think about the solutions based on the analogy with a mechanical model of a particlerolling on a potential under the influence of a damping force. In section 4 we concentrateon some aspects of non-critical string cosmology and its relation to the d=2 black hole.

Wealso briefly discuss how string one loop effects might modify the non-critical cosmologicalsolutions. In section 5 we present our conclusions.

In appendix A some aspects of thesolutions to our cosmological equations are worked out.2. General Structure of Cosmological EquationsLet us consider strings propagating in a large universe of spatial dimension N. If N isnot the critical spatial dimension of strings (i.e., if N < 25 (9) for (super)strings), we couldhave one of two possibilities: either we have some Nc of compact internal dimensions whichmake up the rest of the spatial dimensions which we take to be static and to correspondto conformal theories on the worldsheet, or we are dealing with a “non-critical” string.

Inother words, if we setc = 23(dc −N −Nc)where dc is the critical spatial dimension, in the first case c = 0, and in the second casec ̸= 0. For simplicity we take the N−dimensional space to be a periodic box (torus) oflengths ai = exp λi ( i runs from 1 to N).

However, many of the ideas developed in thispaper can easily be extended to more general cases.It is a well known principle in string theory that as long as we concentrate on slowlyvarying fields, we can hope to describe the global aspects of the dynamics of the the-ory reasonably well by concentrating on massless modes, and keeping only the leadingderivative terms (‘adiabatic’ approximation). The types of massless modes depend on aparticular choice of vacuum backgrounds strings are propagating in, and on whether weare considering bosonic strings, superstrings, or heterotic strings.

But they always includea gravitational field and a dilaton (for simplicity we shall ignore the antisymmetric tensor4

field). If the radii are large compared to the string scale the leading order terms in the lowenergy expansion of the tree level gravitational-dilaton effective action of the closed stringtheory are [18] (we shall absorb the gravitational coupling constant into φ and set α′=1)4S0 = −ZdN+1x√−G e−2φ [ c + R + 4(Dφ)2 ] .

(2.1)Note that eφ plays the role of coupling ‘constant’ in string theory. The first question wewould like to address is whether we should trust this action at all for Planckian size spatialdimensions, i.e., when λi ∼0.

The answer is no, if we wish to take all the modes in theabove action seriously. The above action manifestly breaks the duality symmetry of stringtheory.

The reason is that when we approach Planckian size universes the momentummodes (which correspond to inhomogeneity in the metric and dilaton fields) become mas-sive, and the winding modes which wrap around the box and were ignored previously willhave to be taken into account, as they have mass comparable to that of the momentummodes. This ‘doubling of the space’ seems very difficult to deal with, and on top of thatwe have to remember that the oscillator (higher level) modes are also going to becomeimportant since they will be of the same energy as the momentum modes.However, the following observation allows us to still take part of the above action se-riously.

The point is that the space independent parts of the above fields have zero energyindependent of the size of the universe, and thus we can naturally concentrate on them,setting the other Planckian modes to zero. Note in particular that these spatially indepen-dent modes are common to both the momentum and winding excitations, so truncating tothese modes alone will be consistent with duality.

Therefore we should expect that eventhough the above action naively seems to pick momentum modes over winding modes,i.e., the space over the dual space, concentrating on spatially constant fields restores theduality symmetry. Thus we should trust the above action as far as the overall scales of4 The volume of the internal space, if there is any, can be gotten rid of by a shift in φ.5

the universe are concerned and not so much as a description of inhomogeneities of theuniverse. And in addition we should assume that the fields are varying slowly with time.Let us see how the duality invariance ai →1/ai is restored in the above action oncewe concentrate on zero modes.

Let us therefore consider the metric and the dilaton fieldof the formds2 = −dt2 +NXi=1a2i (t)dx2i,ai = eλi(t) ,φ = φ(t) .It is useful to introduce the “shifted” (and rescaled by 2) dilaton field ϕ which absorbs thespace volume factorϕ ≡2φ −NXi=1λi ,√−G e−2φ = e−ϕ .Then the action becomesS0 = −Zdt e−ϕp−G00 [ c −G00NXi=1˙λ2i + G00 ˙ϕ2 ] .We have kept G00 as we need to vary the action with respect to it in order to get the fullset of equations of motion ( in the equations of motion G00 is set to −1). Dot denotes thetime derivative.

The above action is manifestly invariant under the duality transformation[4-6]λi →−λi ,φ →φ −λi ,ϕ →ϕ(2.2)(here i is any number among 1,..,N ; more general duality transformations are obtainedby combining these basic ones).The above action is written in the absence of stringy “matter”. To discuss cosmologyit is important to incorporate matter into the system.

Assuming that the effective stringcoupling is small let us take the matter to be a gas of (free) string modes in a thermalequilibrium at the temperature β−1. Then matter contribution to the action is representedbySm =Zdtp−G00 F(λi, βp−G00).6

Here F is the (one loop) free energy and can be represented in terms of the one loop stringpartition function in a torus of radii eλi and periodic Euclidean time of perimeter β√−G00, Z = −βF. The adiabaticity assumption implies that we can replace constant radii andβ by functions of time.

Taking the full actionS = S0 + Sm = −Zdtp−G00 [ e−ϕ( c −G00NXi=1˙λ2i + G00 ˙ϕ2 ) −F(λi, βp−G00)]and varying it with respect to λi, φ and G00 and we find the following equations5c −NXi=1˙λ2i + ˙ϕ2 = eϕE ,(2.3)¨λi −˙ϕ˙λi = 12eϕPi ,(2.4)¨ϕ −NXi=1˙λ2i = 12eϕE ,(2.5)whereE = −2 δSmδG00= F + β ∂F∂β,Pi = −δSmδλi= −∂F∂λi.E is the total energy of the matter and Pi is the pressure in the i-th direction times thevolume. These equations imply a modified conservation law for the energy (following fromthe invariance under reparametrizations of time )˙E +NXi=1˙λiPi = 0 .Since F = F(λ(t), β(t)) this is equivalent to the conservation of the entropy S = β2∂F/∂β,i.e.

our matter is indeed evolving adiabatically. This means that for a given radius deter-mined by λi the temperature β−1 adjusts itself so that S remains constant.

Solving the5 Similar equations were discussed in [9] and also in [6] where the matter was represented by theenergy-momentum tensor of a macroscopic string. Even though we derived the above equationsusing the canonical ensemble, they are valid even when the canonical ensemble description breaksdown (as is the case near Hagedorn temperature) and we have to use the more fundamentalmicrocanonical ensemble.7

adiabaticity condition we can express β in terms of λi and hence represent E as a functionof λi alone,E(λ) = E(λ , β(λ)) .In what follows we shall consider only this new function E(λ). Since entropy is not changingwe findPi = −∂E(λ)∂λi.

(2.6)Note that the time derivative of (2.3) vanishes once we use (2.4) and (2.5), and so thenon-trivial content of equation (2.3) is that it fixes the constant of integration in terms ofc. The above equations are duality invariant since F is invariant under λ →−λ for a fixedtemperature (E(λ) and β(λ) are also invariant and Pi changes sign under the duality).Rewritten in terms of the original dilaton field φ equations (2.3)–(2.5) take the formc −NXi=1˙λ2i + (2 ˙φ −NXi=1˙λi)2 = e2φ ρ ,(2.7)¨λj −(2 ˙φ −NXi=1˙λi) ˙λj = 12e2φ pj ,(2.8)2¨φ −NXi=1¨λi −NXi=1˙λ2i = 12e2φ ρ(2.9)where ρ = E/V , pi = Pi/V and V = exp P λi is the space volume.

To compare theseequations with standard cosmological equations in which the dilaton is ignored considersetting φ (which is related to the coupling constant) to be constant. Then the equation(2.7) takes the well known form (with flat space),( ˙aa)2 = ˙λ2 =−cN 2 −N + Gρ(2.10)for some constant G, where −c/(N 2 −N) plays the role of the cosmological constant6.However, it is easy to see that if c = 0 a solution with φ = const is consistent with all the6 It is amusing to note that in this setup the problem of the vanishing cosmological constantseems to be related to the question of being on or off-criticality in string theory.

This connection(which was already mentioned in the past) seems worth further study.8

three equations only ifNXi=1pi = ρ ,i.e., for a matter with the vanishing energy momentum trace, as is effectively the casefor a gas of massless particles in a thermodynamical equilibrium. This “radiation -type”condition is not satisfied in a high temperature phase of string thermodynamics and is thereason for stringy departure from Einstein’s gravity.Let us discuss some general aspects of equations (2.3), (2.4) and (2.5).

It is importantto notice that the equations for λi (2.4) are the same as that of a point particle in thepresence of a time dependent potential 12eϕE(λ) and damped or boosted (depending onwhether ˙ϕ is negative or positive) by a dilaton “friction” force.Note that within thismechanical interpretation the “integration constant” c in (2.3) plays the role of a fixedenergy of the system.Also, the equation for ϕ can be summarized by subtracting equation (2.5) from (2.3)and defining y = e−ϕ, which leads to¨y + cy = 12E . (2.11)This equation for y can be interpreted as that for the endpoint of a spring, with springconstant c, and with the external force 12E(λ).

These interpretations allow us to developan intuitive picture for the nature of the solutions to our equations. As it turns out, thebehavior of solutions will be rather different depending on whether or not we are describingcritical strings, i.e., whether or not c = 0.3.

Critical String CosmologyThis case corresponds to setting c = 0.The important qualitative simplificationthat happens in this case is that since E is positive (assuming no Casimir - like negativecontribution to energy) from equation (2.3) we conclude that ˙ϕ can never become zero. In9

other words, ˙ϕ never changes sign, and thus it provides a damping or a boosting effect inequation (2.4) for all time. We will consider the damping case ˙ϕ < 0; the boosting casecan be obtained by time reversal.

A strong reason to consider ˙ϕ < 0 is that otherwisethe boosting effect of dilaton on λ will invalidate the adiabatic approximation used in thederivation of the above equations. Also, growing ϕ together with expanding λ implies thegrowth of the effective coupling exp φ in contradiction with a weak coupling assumption.From equation (2.5) it follows that ¨ϕ > 0 and thus ˙ϕ is growing in time.

Since ˙ϕis negative and it will never cross zero we conclude that (as long as E ̸= 0) ˙ϕ continuesgrowing towards 0 and approaches 0 as t →∞.For simplicity let us consider the isotropic case , i.e. assume that all λi are equal toeach other and denoted by λ so that a = exp λ is the cosmological scale.

The equationswe get can be written asc −N ˙λ2 + ˙ϕ2 = eϕE ,(3.1)¨λ −˙ϕ ˙λ = 12eϕP,(3.2)¨ϕ −N ˙λ2 = 12eϕE ,(3.3)where P = −N −1∂E/∂λ and for critical strings we simply set c = 0 in (3.1).In order to solve this system of equations we will need to specify E(λ) as well asprovide the initial conditions for λ, ϕ and ˙ϕ. E(λ) encodes string thermodynamics byspecifying how the total energy in the box has to change as a function of λ in order to keepthe entropy constant.

Motivated by string cosmology, the function E(λ) was introducedand studied using string thermodynamics in [2] ( some properties of E(λ) were furtherclarified in [19], see also [20] ). Its basic structure is indicated in fig.

1.This function is duality symmetric under λ →−λ. Its structure is well understoodnear λ ∼0 and for very large λ.In the first region near λ ∼0, which we call theHagedorn region, the temperature of the strings is very close to the Hagedorn temperatureand the energy is almost independent of λ.

It is given by E = THS where TH is the10

Hagedorn temperature (fixed for a given string theory) and S is the total entropy. For largeenough λ temperature drops significantly compared to the Hagedorn temperature, and themassive modes of string go out of equilibrium so that we are left with the massless modes.In this ‘radiation dominated’ region we thus have the usual relation of how E dependson the radius of the universe for a gas of massless particles.In fact, the temperaturedrops as 1/a = exp(−λ) and E ∝T N+1aN ∝a−1.

In other words E = Ce−λ whereC is determined by S and the number of massless modes. The behaviour of E in theintermediate region between the Hagedorn region and the radiation dominated region isonly partially understood but it has been shown that E is monotonically decreasing (inthe Hagedorn regime) with increasing |λ| [19], and it is natural to believe that this istrue for all λ.

So if we start with the initial conditions that ˙ϕ < 0 the decreasing of thepotential E with increasing |λ| will force λ to slide down the potential towards increasingvalues of |λ|, until we emerge at the radiation dominated era. The universe will continueits expansion, and in fact it will approach the radiation dominated era of the standardcosmology.

In other words, independently of the details of E, or the initial conditions, onefinds that 2φ = ϕ+Nλ, which determines the string coupling constant, rapidly approachesa constant and the radius ultimately grows as a ∼A t2N+1 for large t, as is the case forstandard radiation dominated cosmology. This is discussed in appendix A.What determines N, the number of extended spatial dimensions?

One idea to explainwhy N ≤3 was suggested in [2]. It was based on the possibility that if the space expandsin more than three dimensions, the assumption of maintaining thermal equilibrium maynot be a good one, as for example the winding modes of strings will have a hard timefinding and annihilating one another.

Thus they would fall out of equilibrium and will bearound. It was suggested there that if winding modes are around this may stop expansion.The universe would not expand until it learns the lesson that it is only possible to expandin three or smaller number of dimensions.Let us note that Einstein’s theory of gravity leaves us uncomfortable with the abovesuggestion that winding modes may prevent expansion.

Recall that cosmological expansion11

rate for a flat universe with vanishing cosmological constant satisfies ˙λ2 = Gρ in FRWcosmologies. In particular any form of matter, since it contributes to ρ, helps accelerateexpansion.This in particular implies that the winding modes accelerate expansion instandard gravity theory, rather than prevent expansion.

On the other hand the idea in [2]was based on duality which states that what usual matter does for the size of the universe(i.e. accelerates expansion) the dual matter (the winding states) should do for the dualsize of the universe (i.e., accelerate contraction).

This was a puzzle which was not fullyresolved in [2] . We will now see that our equations resolve this paradox by showing thatthey drastically differ from Einstein’s equations in such cases and indeed are consistentwith the idea that winding modes slow down and ultimately stop the expansion.If the winding modes are around the energy E is going to grow with λ, as larger λmean bigger boxes and thus more stretched winding strings with higher energies.

The massof the winding states will increase with λ as expλ. E growing with λ means that ¨λ < 0 asλ will have to climb the potential E. So this slows the expansion and in fact ultimatelystops it as it is shown in appendix A.

More generally, it is shown there that if we take forlarge λE ∝exp(αλ)then for α positive the universe reaches a maximum size and then contracts back. Con-stant matter energy density corresponds to α = N and it prevents expansion even morestrongly!

So we conclude that dilaton suppresses the inflationary mechanism based on con-stant energy density7. If α is negative the universe expands forever (as is the case for theradiation dominated era where α = −1).The duality symmetry of E implies that while the winding modes (providing thegrowth of E at large λ) oppose expansion, the momentum modes (providing the growth of7As was already noted in the Introduction, this does not rule out the possibility of inflationat a later stage of evolution when dilaton gets a mass and it is appropriate to use the redefinedmetric.12

E at large negative λ ) oppose contraction. As a result, in the absence of equilibrium theuniverse sees the effective E represented as a dashed curve in fig.

1 and the radius of theUniverse will be oscillating between maximal and minimal values.So the basic picture one is led to is a universe which oscillates in many directionsfor a while around the Planck scale (maybe within a few orders of magnitude), until bycoincidence it starts expanding in smaller number of directions (the most likely one beingthe largest possible dimension consistent with maintaining equilibrium, namely 3). Thenit expands forever and we find ourselves in the radiation dominated era of the standardcosmology.

This may also explain the large entropy problem. The entropy may be largebecause during this “trial and error” period of the early universe, as the strings were outof equilibrium we would generate ( if this process takes long ) a lot of entropy.

In otherwords, the large entropy problem may be related to having an ‘old’ universe.An important issue to resolve in string cosmology is to explain the absence of masslessdilatons at the present time. One should expect that somehow a potential is generated forthe dilaton and it will have to sit at its minimum picking up a mass.

The mechanism ofhow this precisely should happen in string theory is not known yet and is the biggest gapin connecting string cosmology to observable cosmology. This is presumably related to thefundamental question of how supersymmetry breaking takes place in string theory whilemaintaining vanishing cosmological constant.4.

Non-critical String CosmologyThe case of non-critical string cosmology is an analog of non-vanishing cosmologicalconstant case of cosmology in standard theory of gravity. We shall consider the case ofc > 0.

Cosmological solutions for the toroidal space and non-zero c in the absence ofmatter ( E = 0 ) has been previously studied in [14]. One obtains from (2.11)( settingE = 0 )y = e−ϕ = A sin 2bt ,b2 = c/4 .13

From (2.4) ( with Pi = 0) one learns that˙λi = ki eϕ ,ki = const ,which can be easily integrated to yieldλi = λi0 + qi ln tan bt ,subject to the condition resulting from equation (2.3)NXi=1q2i = 1 .These solutions can also be extended to c ≤0 [14] . Note that for c > 0 these solutionsare singular, i.e., the fields blow up at finite time.

In fact, as it is easy to see from (2.11),the oscillatory nature of solution forces y, which should be positive, to cross zero in finitetime ( so that ϕ blows up in finite time). This could in principle be balanced by a positiveenergy density appearing on the right hand side of (2.11).For the above singular solutions the adiabatic approximation which went into theirderivation breaks down, and so we should not trust them near the singularity.

Even so,one would expect that there should be a solution for any reasonable initial condition, ifwe believe in completeness of string theory. This leads us to expect that there should bean exact conformal theory which asymptotically reproduces Mueller’s solutions.

For thecase N = 1 we will now see directly that this is indeed the case. In fact the correspondingconformal theory turns out to be the SL(2, R)/U(1) coset model which was recently linkedto the black hole geometry in [15].For N = 1 we have q1 = ±1.

Let us take the plus sign for q as the other sign (givingthe dual solution [5]) corresponds simply to shifting of time. We thus haveϕ = ϕ0 −ln sin 2bt ,λ = λ0 + ln tan bt ,b2 = 4 .14

This background describes a universe with one spatial dimension with the metric given by(we fix a particular value for λ0 and assume that x is periodic with period r , e.g. 2π )ds2 = −dt2 + e2λdx2 = −dt2 + b−2tan2bt dx2.The universe starts at t = 0 with zero size and grows to infinite size at t = π/2b.

Let uschange our coordinates. Letu = sin bt ex ,v = sin bt e−x .

(4.1)We find thatds2 = −b−2 dudv1 −uv,and that the unshifted dilaton field φ is given by2φ = ϕ + λ = 2φ0 −2ln cos bt = 2φ0 −ln(1 −uv) .This is the same metric and dilaton background found in [15] in the region 0 ≤uv ≤1 (hereuv = sin2bt), i.e. the region between the horizon and singularity.

There is one differencethough: here we are working on a periodic space, so x is identified with x + r. This meansthat we have to identify(u, v) ∼(αu, α−1v)(4.2)i.e. we get a wedge in the region between horizon and singularity (see fig.

2). Of coursethe system of equation we looked at is also valid for infinite radius r, which implies that wecan in fact reproduce the full region between horizon and singularity with no identification.In this case in order to complete the metric we will have to add regions I and III as well,and we end up getting the full black hole solution.

It is amusing that the transformation(4.2) is an exact symmetry of the conformal theory (in an appropriate basis it correspondsto conjugating SL(2, R) group element by the diagonal matrix (α1/2, α−1/2)). So we canconsider an infinite orbifold of SL(2, R)/U(1) by the group generated by this symmetry15

and obtain the compact universe solution. One should note that since u = v = 0 is a fixedpoint of this transformation we end up getting a new singularity at u = v = 0, which inour cosmological interpretation corresponds to the initial time t = 0, where the universehas zero radius.

There is another wedge branch of the solution in regions I and III whichhas periodic time which touches the cosmological region at the singular point u = v = 0.Note that in this language the duality of conformal theory [17, 21] corresponds (forappropriate choice of radius) to the standard a →1/a duality of our one dimensionaluniverse, i.e. bt →π2 −bt changes uv →1 −uv and λ →−λ.Another aspect of Mueller’s solution is that we can continue time past the singularityat t = π/2b.

In particular, if we introduceu1 = −sin bt ex ,v1 = −sin bt e−xwe get an identical copy of the interior region of black hole (on the overlap u1 = −u , v1 =−v) where the universe now starts at the lower singularity, and evolves till t = π/b at whichit reaches zero size, it grows as t increases until it becomes infinite in size again at t = 3π/2b.Then again we use the coordinate transformationu2 = sin bt ex ,v2 = sin bt e−xwhere the radius shrinks to zero size at t = 2π/b. By now the universe has undergone twooscillations, but we have covered SL(2, R) only once.

For example, if we take x = 0, thecorresponding element of SL(2, R) , as we evolve in time, is represented by the matrixcos btsin bt−sin btcos bt(it is known that the SL(2, R)/U(1) has two copies of the interior region of black hole–toget only one we could have used PSL(2, R)/U(1)). But now we will not identify (u2, v2)with (u, v), as that would have given us periodic time.

Instead we continue forever. Thismeans that in the above matrix realization of SL(2, R) we are not identifying t →t+2π/b.16

In other words the cosmological interpretation suggests that we go to an infinite fold coverof SL(2, R). In this way we end up with a universe which had no beginning and no end,and it undergoes infinitely many oscillations.

Again in this picture we have the option ofchoosing a finite radius or infinite radius for our space. If we have a finite radius, thenwe will have to restrict to the wedge region between the horizon and singularity shown infig.2, otherwise we will have to add the regions I and III to complete the metric.

In thislatter interpretation an observer in region I or III of one of these universes, may decideto enter the next universe. The Penrose diagram for this series of universes is shown infig.3.

It is amusing to note that in the Euclidean version of each of these universes we havea periodic coordinate which suggests a thermal bath interpretation. This suggests, evenin the cosmological picture, we get radiation from the horizons.

It would be interesting tostudy this further.Now we ask the question if this one dimensional cosmological solution should be takenseriously. Naively the answer would be no, because we have violated the assumption of adi-abaticity of the fields near the singularity, when the size of the universe is infinitely large.However, since we know there is an exact conformal theory with the correct asymptoticbehavior, we are led to expect that there exist a corrected metric which gives the exactanswer.

Based on comparison with the coset model an exact metric for the Euclidean blackhole was suggested in [17]. As was checked in [22] the corresponding metric - dilaton back-ground solves the σ model conformal invariance conditions in the 3-loop approximation.When one analytically continues the conjectured metric to the region between horizon andsingularity it readsds2 = −dt2 + b−2tan2bt1 + qtan2btdx2 ,q = 2/k = 8/9 .If this metric is indeed exact, it suggests that the space-time is actually nonsingular att = π/2b so that we see no singularity in the metric!

In the finite radius scenario (wherewe introduce an additional singularity at u = v = 0) we have a universe starting at zero17

radius, reaching a finite maximum radius at finite time and then contracting back to zeroradius and continuing this indefinitely. The conjectured form for dilaton ϕ is unchangedfrom the leading approximation and is thus still singular at t = π/2b.

The cosmologicalinterpretation of the black hole gets rid of a puzzling consequence of this conjectured exactmetric: if analytically continued to the region behind black hole singularity the metricbecomes Euclidean between 1 ≤uv ≤9 with a singularity at uv = 9. This is avoided inour interpretation because the region beyond the singularities are never reached (they areconnected to universes labeled by ±∞).It would be interesting to find exact conformal field theory generalisations of higherdimensional Mueller–type cosmological solutions8.

The coset theories like SO(d, 1)/SO(d−1, 1) which should be analogs of the (anti) DeSitter backgrounds (see Bars [21] and refs.there) does not, however, correspond to a flat N-space. The generalization of the leadingorder conformal invariance equations to the case of isotropic homogeneous metric with acurved space (with positive, zero or negative curvature k ) is given by (cf.

(3.1)–(3.3) )c −N ˙λ2 + ˙ϕ2 = −kN(N −1)e−2λ ,¨λ −˙ϕ˙λ = −k(N −1)e−2λ ,¨ϕ −N ˙λ2 = 0(the absence of the correction in the third equation is due to the fact that the “potential”in the present case is “classical”, i.e. does not depend on the dilaton).Let us assume thatc < 0, k > 0.

Then at large negative λ (small times) the solution is approximately givenby the DeSitter metric and a constant dilaton φ = 12(ϕ + Nλ). In this limit, however, the8In such a case it would be natural to allow the full moduli of toroidal compactifications tobecome involved.

The picture is a more or less straight-forward generalization of the equationswe have considered, and we would simply have to study geodesic equations on the correspondingmoduli spaces of toroidal compactifications.18

adiabatic approximation is not fully reliable. At large positive λ the effect of the spacecurvature becomes irrelevant and the solution approaches the Mueller’s isotropic solutionλ = λ0 + N −12 ln tanh bt →const , ϕ →−bt .So the universe is born at t = 0 as a d = N +1 DeSitter space and evolves into the productof a time line and an N–sphere at large t. This appears to be a direct higher dimensional(Minkowski signature) analog of the d = 2 (euclidean) “cigar” metric.So far we have discussed non-critical string cosmology in absence of matter.

It isnatural to ask how can one introduce ‘thermal’ non-critical matter, in order to have aricher non-critical string cosmology. It turns out that this can be done at least formally.The usual way we consider thermal ensemble in field theory is by making the time Euclideanwith period β.

For non-critical strings the Liouville field plays the role of time and thissuggests making it Euclidean and periodic with period β. The computation of thermalfree energy, which is a one loop computation generalizing that of [23] can be carried outexplicitly for the non-critical strings cm ≤1.

For cm = 1 noncritical strings at radius a,for example, one finds that the necessary modular integral has been evaluated in [24] in acompletely different context. This is related to the S1 × S1 partition function integratedover the fundamental domain and turns out to be the same as that for the critical N = 2strings partition function [25] ( or, equivalently, for cm = 1 non-critical N = 2 stringpartition function).

One finds for the free energyβF = ln[ τ2η2(τ)¯η2(¯τ) ] + (τ →ρ)where τ = iτ2 = iβa/8π, ρ = iβ/8πa and η is the Dedekind eta function. Solving theadiabaticity condition one can obtain the energy E(a) from the above expression for F.For large entropy and small a we haveE(a) =Aa + a−119

where A depends on the entropy. The profile for E(a) is very similar to that of criticalstrings (fig.1) except that there is no flat Hagedorn region and we only get a maximumat λ = 0.

This is to be expected since at cm = 1 there is no exponential degeneracy ofstates, as there is only one massless scalar (and some additional discrete states [26]) andso there is no limiting temperature. In fact it has been suggested [27] that introductionof temperature for non-critical strings might lead to a more clear phase diagram picturefor non-critical strings.

Note that for large a, E(a) behaves like the energy for that of amassless particle (E ∝1/a). For yet larger radii, the temperature drops to zero, but Edoes not go to zero.

It becomes negative and behaves asE = −124(a + 1a) .This is a Casimir - type contribution which is independent of the temperature. It is inaccord with the result of [23] which corresponds to computing the free energy at zerotemperature.Note that this Casimir effect is absent in critical (super)-strings, as thecosmological constant vanishes at one loop.

Putting the potential E(a) on the right handside of our equations will modify Muellers solution. The intuition based on the λ rollingon the potential hill E(a) implies that the solutions are still singular (and blow up in finitetime).These ideas can be applied to cm < 1 as well.

In this case we have a one dimensionalfield theory (corresponding to the Liouville field), which we can identify with a spacecoordinate x. The relevant equation to solve for the dilaton at the tree level is c−ϕ′2 = 0which gives rise to the familiar linear dependence of the dilaton: ϕ = −√c x. However,the one loop partition function [23] modifies the above equation toc −ϕ′2 = −heϕwhere h is a positive constant depending on which minimal model we are dealing with.Solving the above equation we findϕ = ϕ0 −ln sinh2√c x2.This suggests a truncation of the Liouville space to a half-line.

Again we are going be-yond the validity of adiabatic approximations but we are getting the hint that the lineardependence of dilaton is modified at one loop.20

5. ConclusionsWe have discussed some aspects of string cosmology taking into account the dynamicsof the dilaton field which is needed to restore the duality symmetry in string theory.

Wehave considered both critical and non-critical strings, the non-critical strings being theanalog of having a non-vanishing cosmological constant in the Einstein theory.For critical strings we find a major modification of the description of the early universe.In particular, winding string states, if not annihilated, can halt the expansion of theuniverse. This is against the intuition based on standard gravitational theory where aconstant energy density leads to inflation.

We have seen that string dynamics is consistentwith the following picture of cosmology which needs to be verified. A nine dimensionalspatial universe of Planck size which we take to be a periodic “box” , at the Hagedorntemperature, expands in all directions.

Not being able to get rid of winding modes theexpansion stops and the universe contracts to even smaller than the Planck scale. Now themomentum modes cannot annihilate one another and the contraction stops.

The universestarts to grow once again. The universe oscillates for a while in this fashion.

In this wayone can in principle generate a lot of entropy. Fluctuations cause different directions toexpand at different times.

One of these fluctuations leads to three dimensions expanding,and now the winding modes can get annihilated and the universe expands, getting rid ofthe winding modes. As the temperature drops, the Planckian string states get suppressedby the Boltzman factor and we are left with the massless modes in a three dimensionalexpanding universe, which looks very much like the radiation dominated era of the standardcosmology.

It is remarkable that the radiation dominated era of the standard cosmologyemerges with no fine tuning.How can one test this picture? One idea is to treat strings classically as one does forcosmic strings.

So we can study the evolution of a universe filled with these strings whichinteract with each other by cutting and rejoining. The main difference with the cosmicstrings is that we have a different evolution equation for the universe, and the coupling21

constant for the cutting and rejoining which is related to dilaton field is also evolving. Thisapproach will presumably answer questions of entropy production and homogeneity of theearly universe in the cosmological scenario described above.We have also studied non-critical string cosmologies, which seem qualitatively ratherdifferent from the critical string case.

In particular we have related the empty universecosmology in d = 2 to the two dimensional ‘black hole’ solution (more precisely to itsuniversal cover). Could it be that stringy black-holes in 4d behave in a similar fashion?

Inparticular, as we cross the horizon we enter another universe instead of encountering thesingularity? It would be remarkable if the “cosmology–black hole” connection discoveredin two dimensions has higher dimensional analogs.

Indeed, the fact that the singularity ofstandard 4d black hole solution is also spacelike, i.e. appears at a given time rather thanspace, suggests a cosmological interpretation.To discuss non-critical string cosmologies with matter we have to introduce the notionof temperature for non-critical strings.

We have made a suggestion of how this may bedone (by a formal wick rotation of Liouville field and making it periodic with period βas one does for cm = 25 in order to define critical string thermodynamics )9. Using thisdefinition we have computed the thermal properties of d = 1 matter (which can be easilygeneralized to d < 1 as in [23]).

It is an interesting question to see if there exists a matrixmodel analog of this finite temperature interpretation (for example are unitary matrixmodels related to thermalizations of standard matrix theories? ).We would like to thank M. Bershadsky, A. Guth, S. Jain, J. Polchinski, M. Tsypin,G.

Veneziano and E. Verlinde for interesting discussions. A.T. would like to acknowledgeJ.Bagger for hospitality at Johns Hopkins University and various kinds of help.

The re-search of A.T. was supported by NSF grant PHY-90-96198 and that of C.V. was supportedin part by Packard Foundation and NSF grants PHY-89-57162 and PHY-87-14654.9 At higher genus due to the coupling of Liouville field to the curvature, this may be consistentonly if β is quantized.22

Appendix A.In this appendix we discuss the solution to the equations (3.1),(3.2),(3.3) with c = 0andE ∝exp(αλ)(A.1)starting with initial conditions ˙ϕ < 0, ˙λ > 0. In particular we show that if the total energygrows with volume, i.e., if α > 0 the universe stops expanding in finite time and startscontracting, i.e., we reach ˙λ = 0 in finite time as λ is decelerating.

If α < 0 the universeexpands forever and if α = 0 the universe comes to a halt, without turning back. We willshow in addition that if α = −1, as would be the case for radiation dominated era, theradius eventually grows as in standard Einstein’s gravity in the radiation dominated era.One can reach the above conclusions more or less immediately by using the ideasdeveloped in section 3.

As discussed there starting with ˙ϕ < 0, we conclude that ˙ϕ cannotbecome zero, and thus it will always remain negative. Since ¨ϕ > 0 from (3.3) we concludethat ˙ϕ approaches zero and gets there at t = ∞.

Equation (3.2) implies that we can treat λas the position of a particle rolling on a potential described by eϕE(λ)/2N with a dampingterm (because of ˙ϕ terms in (3.2)). The potential is getting weaker as its overall size ismodulated by eϕ which is decreasing because ˙ϕ < 0 and the damping term gets weakeras ˙ϕ is approaching zero.

For α < 0 it is clear that since E is decreasing with λ, λ willcontinue to increase indefinitely. For α > 0, since the energy increases with λ, the universecannot grow indefinitely, and will reach a maximum and start contracting.

This is almostobvious, were it not for the fact that the energy is modulated with eϕ which is gettingsmaller. So the universe could conceivably continue expanding at ever slower rates.

Thiswe will show is not the case. We shall find by a careful analysis of these equations thatthe universe reaches a maximum size and starts contracting after that.

The fact that forα = 0 (i.e., constant energy), the universe reaches a maximum size and stops is also clearfrom (3.2) as the potential is flat in this case, and the damping term will eventually stopλ.23

First we note that P = −N −1∂E/∂λ = −αE/N. Then we can eliminate eϕE fromequations (3.1),(3.2), (3.3) and obtain¨λ −˙ϕ˙λ = −α2N (−N ˙λ2 + ˙ϕ2) ,¨ϕ −N ˙λ2 = 12(−N ˙λ2 + ˙ϕ2) .Now we can represent these equations in the first order form by definingl = ˙λ ,f = ˙ϕto obtain˙l = αl22 + lf −αf 22N,˙f = Nl22+ f 22.

(A.2)The initial conditions are f < 0 and l > 0. Actually we have another restriction, whichcomes from positivity of E in (3.1)|f| ≥|√Nl| .The important point to note in solving (A.2) is that these equations are homogeneous.

Inother words, if we rescale (l, f) →r(l, f) and rescale t →r−1t we get a new solution. Thismeans, in particular, that if we look for solutions withdfdl = fl(A.3)we get straight lines in the (f, l) plane which pass through (0, 0).

Studying these particularsolutions will give us a handle to study a qualitative behavior of all solutions. These straightline solutions can be easily found by noting that df/dl = ˙f/˙l.

Solving (A.3) we find threesolutionsfl = ±√N ,Nα.24

By the rescaling argument, f and l can approach zero only along the above lines. So inthe region α > 0, if we start with the initial conditions l > 0 , f < 0 , |f| >√Nl , theonly possibility for not crossing the l = 0 line is for the flow to approach zero through theline f/l = −√N.

But a simple check of the equation shows that this line is repulsive, i.e.,the flows are driven away from it. They will be attracted to the next fixed line which isin the l < 0 region.

They will cross the l = 0 line which means that at finite time ˙λ = 0.According to (3.2) ¨λ is not zero so the universe bounces back and λ starts to decrease.For α = 0, the solution with l = 0 is an attractive solution and all solutions willapproach it. This simply means that if we start with ˙λ > 0 the expansion will eventuallystop because of the damping term.

In fact, it is easy to write the general solution in thiscase: using (2.11) and (3.2)( with P = 0 ) we gete−ϕ = E0t24−NA2E0, λ = λ0 +1√Nlnt −2√NA/E0t + 2√NA/E0.This solution approximately describes the evolution in the Hagedorn region (in fact, E isnot quite constant and decreases with increasing λ so the expansion never quite stops).If −√N < α < 0, the line f/l = N/α is an attractive solution and all solutions willapproach it. Note that for α = −1 which corresponds to the case of the radiation dominatedera, the fixed line is f/l = −N.

For the fixed line solution 2 ˙φ = ˙ϕ+N ˙λ = f +N l = 0, i.e.the original dilaton φ is constant and we get the standard (flat) FRW - type cosmology inthe radiation dominated era, a ∼t2/(N+1). An important point is that this is an attractivesolution, i.e.

all solutions (with initial ˙ϕ < 0) will eventually approach it ! So at latetimes we get the standard radiation dominated era of cosmology without any fine tuning.If α < −√N the attractive solution will be f/l = −√N.25

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Figure CaptionsFig. 1.The solid curve represents the adiabatic variation of energy E as a function ofscale λ.

Note the duality symmetry λ →−λ. Near the Planck scale (λ ∼0) theenergy is more or less independent of scale, as we are in the Hagedorn regimeof string thermodynamics.

For large radii, we enter the radiation dominated eraand E ∝exp−λ. The dashed curve represents the effective energy if the windingmodes (and momentum modes for λ << 0) fail to annihilate.Fig.

2.The 2d black hole geometry is represented here. The cosmological solution cor-respond to regions II and IV .

If we wish to deal with a compact universe, weare limited to the wedge drawn here. In this case the curve drawn in region IIinside the wedge represents the space, with its endpoints identified.Fig.

3.The ‘accordion-like’ Penrose diagram of 2d cosmology. An observer in universe ncan end up in any universe m ≥n.

The regions inside the curves represent theuniverse if we wish to consider compact space.28

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