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CHALLENGES FOR SUPERSTRING COSMOLOGY

arXiv:hep-th/9212049v1 8 Dec 1992UPR-541TCHALLENGES FOR SUPERSTRING COSMOLOGYRam Brustein and Paul J. SteinhardtDepartment of Physics,University of Pennsylvania, Philadelphia, PA 19104ABSTRACTWe consider whether current notions about superstring theory below thePlanck scale are compatible with cosmology. We find that the anticipatedform for the dilaton interaction creates a serious roadblock for inflation andmakes it unlikely that the universe ever reaches a state with zero cosmologicalconstant and time-independent gravitational constant.

Superstring theories have been extensively studied as models of unifiedtheories [1]. Effective field theories in four dimensions are used to describestring theory below the Planck scale.There are strong prejudices aboutthe form that the four dimensional low energy effective field theory derivedfrom superstrings must take in order for superstrings to be a viable modelof particle interactions.

In this paper, we extract the most generic elementsof the “current superstring lore” (CSSL) and examine their implications forthe early evolution of the universe.Our goal here is to be strict and systematic in bringing CSSL and cosmol-ogy together. Hence, it is necessary to carefully lay down the components ofCSSL and to not deviate from them.

In the past, several authors have en-countered some of the cosmological problems of superstrings outlined belowand have resorted to adding new terms in the effective potential to solve them[2]-[5]. However, these added terms violate one or more key tenets of presentsuperstrings lore; the cosmology problems of CSSL are thereby masked.

Themain components of CSSL are:(1) The effective theory below the Planck scale is described by a weakly-coupled, four-dimensional, N = 1 supergravity field theory. Hence, the effec-tive action is Kahler invariant, described by a real Kahler potential, K(φi),and a holomorphic superpotential W(φi):Leff=Xi,j12Kφi,¯φj∇φi·∇¯φj −eKXi,jKφi ¯φj(DφiW)(D¯φjW)† −3|W|2(1)where the sum is over (complex-valued) chiral superfields fields φi, and Dφi =∂φi −Kφi, Kφi = ∂φiK and Kφi ¯φj = K−1φi ¯φj.

The Kahler potential,K = −ln (S + S∗) −3ln (T + T ∗) + . .

. (2)includes two fields: S (related to the dilaton) and T (related to the breathing1

mode). According to CSSL, the contributions of other moduli (fields similarto T), matter fields and higher order corrections not shown explicitly herecan be expressed as corrections to K and W.(2) The unified gauge coupling constant is fixed at small value gGUT ≈.7to agree with recent hints from LEP suggesting that, within the minimalsupersymmetric standard model, a unification of coupling constants occursat a scale MGUT ≈1016 GeV.

The gauge coupling constant is determined bythe expectation value of the real part of S, < ReS >∼1g2 > 1. (3) The source of dilaton interactions is in the hidden sector of the theoryand appears at some intermediate scale Λc ≈1014 GeV.

The dilaton andmoduli interactions induce supersymmetry breaking in the observed sectorof the theory [6].Supersymmetry breaking in the observed sector occursat approximately the electroweak scale 103 GeV. The hierarchy between thescale Λc of the hidden sector and supersymmetry breaking in the observedsector, is created because of the weakness of the induced dilaton interactions.

(4) The dilaton potential is set by non-perturbative interactions. An impor-tant consequence of Condition (3) above is that the dilaton potential vanishesat temperatures between the Planck scale (1019 GeV) and the intermediatescale, Λc ≈1014 GeV.

Below Λc, the superpotential is of the form:W(S, T) =Xje−αjSfj(S, T, . .

. ),(3)where αj are constants and fj depend on matter fields, and may containpowers of S. The potential is, using Eq.

(1):V (S) = (S + S∗)|∂SW(S) −1S + S∗W(S)|2 −3S + S∗|W(S)|2(4)The dependence on T and other moduli is not shown explicitly here becausetheir expectation values are fixed by modular invariance [7].2

Although some of our arguments depend only on the general, non-perturbativeform of the superpotential, for concreteness we will assume the current lead-ing candidate as a source of this non-perturbative potential: gaugino conden-sation [8],[9]. Above the scale of gaugino condensation, Λc, the superpotentialis identically zero.

Below Λc, the superpotential has the form given in Eq. (3).In this case, αj = 3k/2β, where k is a positive integer, β is determined by theone-loop beta-function of the hidden gauge group, and the fj’s are knownconstants, determined by the structure of the hidden gauge group and thepotential for the T field and matter fields [10].W(S) =Xjaje−αjS(5)If the gauge group is simple there is only one term in the sum.

If the gaugegroup is semi-simple, each additional term corresponds to gaugino condensa-tion in one of the distinct factors of the group. The rank of the gauge groupis restricted to be at most 18.

The sum in Eq. (5) is therefore a small finitesum.Now we wish to show how this prevalent view of superstrings runs intoimmediate problems once cosmology is considered.

First, we shall argue howdilaton interactions create a serious roadblock for inflationary cosmology.Since inflation is not yet a proven idea, this problem is arguably the leastsevere. Then, we shall see how the anticipated dilaton interaction makes ithighly unlikely that the universe ever reaches a state with zero (or small)cosmological constant and a state in which Newton’s constant G is time-independent, ˙G = 0.

Here, the contradictions are with strong observationallimits; e.g., see [11]-[14]. In the end, we do not claim a rigorous theorem.However, we have been unable to identify a viable model, and we seem to bedriven to unattractive extremes (at best).3

Any type of inflationary scenario requires an epoch in which the false vac-uum (potential) energy density dominates the energy density of the universe[15]. The false vacuum energy density must drive the scale factor a(t) to growas a(t) ∝tm where m >∼4 [16] in order to solve the cosmological flatness,horizon, and monopole problems.

The anticipated dilaton interaction canprevent the universe from undergoing superluminal expansion in two ways.Before the dilaton settles into a stable minimum of its potential, the dilatonkinetic energy density can dominate the vacuum energy density. Once thedilaton settles at a minimum and its kinetic energy becomes negligible, theproblem may be that there is not sufficiently large potential energy densityto drive inflation.Let us first consider the problem before the dilaton settles to a minimum.Numerical integration is required to obtain quantitative results.

However,there is a useful analogy to Brans-Dicke theory that allows us to explain theessence of our results in a simple way. Both a Brans-Dicke scalar and thedilaton are non-minimally coupled to the scalar curvature.

Through its non-minimal coupling, the Brans-Dicke field φ is subject to a force proportional tothe false vacuum energy density. Instead of the vacuum energy density beingfocused totally into inflating the scale factor a(t), some of it is funneled offinto driving the φ field [17].

The kinetic energy density of the φ field changesthe overall equation of state, slowing down the expansion rate (i.e., reducingm).If the Brans-Dicke field is Weyl transformed so that the scalar curvatureterm assumes standard Einstein form ((16πM2P l)−1R), φ has canonical kineticenergy density, but the false vacuuum energy density VF is modified by φ-dependent factor, V (φ) = VF exp(−βφ/MP l), where β =q64π/(2ω + 3) andω is the Brans-Dicke parameter that determines the deviation from Einstein4

gravity. The equations of motion for the Weyl-transformed theory are:¨φ + 3H ˙φ = −dV (φ)dφ,(6)where H2 ≡(˙a/a)2 = 8π( 12 ˙φ2+V )/(3M2P l).

The solution is a(t) ∝t(2ω+3)/4 sothat ω ≥5 is required to resolve the horizon and flatness problems. A slightlyweaker bound ω > 1/2 is required to have any superluminal expansion [17].For ω < 1/2, the potential for φ in the Weyl transformed theory is so steepthat the φ kinetic energy density grows to dominate any potential energydensity.In superstrings, the dilaton φ is related to S by Re S = eφ.

The effectivepotential is identically zero until some intermediate scale Λc ≈1014 GeV.Since inflation requires a non-zero vacuum energy density, inflation can onlyoccur after a non-zero potential is generated below Λc. Below Λc, the dilatonpotential assumes the form given in Eq.

(5). Any prospective inflaton mustcouple to S.Under the assumption that the dilaton potential is strictlynon-perturbative, the false vacuum energy must depend on S through one ormore terms in Eq.

(5). Hence, the following argument is independent of thetype of inflationary model (e.g., chaotic, extended, etc.

).In Figure 1, we plot some typical dilaton potentials corresponding to oneand two gaugino condensates. Note that the potential is unbounded belowfor φ < 0; this corresponds to the strong coupling region, where the non-perturbative analysis that led to this potential cannot be trusted.

Hence, wewill restrict ourselves to φ > 0. The striking feature of the potential is itssteepness.

Because the potential is non-perturbative in origin, the potentialdrops as e−2αS ∼e−2αeφ, for large φ, where α is one of the coefficients inEq. (5).5

Figure 1. Non Perturbative Dilaton Potential V (φ)/M4P l. The dashed linecorresponds to one gaugino condensate and the solid line correspondsto two gaugino condensates.

The insert shows V (φ) in the region wherethe potential resulting from two gaugino condensates has a minimum,scaled to show the minimum and the steepness of the potential.The precise value of α depends on the hidden gauge group, but among theknown examples we find α ≥24π2/r, where r is an integer number of orderO(10). Expanded about any φ0 > 0 (S0 ≡eφ0 > 1), the potential can beapproximated by exp(−2αS0(φ−φ0)).

Comparing this with the Brans-Dicketheory, we find a small effective ω ≈4.5 × 10−4(r/S0)2 −1.5. For typicalhidden sector groups, ω < 1/2, a value too small for inflation owing to thesteepness of the potential; for E8, we find the maximal value r = 90 which,combined with S0 = 1 squeaks past superluminal expansion with ω < 2, butstill ω is too small to solve the cosmological horizon and flatness problems.6

Consequently, until φ settles near a minimum of its potential, the dilatonkinetic energy totally dominates the potential energy density, thereby blockinginflation of any kind.1The best-hope for inflation then becomes a scenario in which the dilatonfirst rolls to a minimum, and then inflation is driven by other fields. There aretwo problems with this approach, though.

First, as is apparent from Fig. 1,for φ be trapped in any minimum some very strict constraints on the initialconditions of φ have to be satisfied.

Initially (above Λc) there is no potentialand it seems reasonable to assume that φ is has an arbitrary value that isspatially varying. If φ begins to the right of the “bumps” in the potential, itrolls continuously towards φ →∞.

If φ begins close to φ = 0, the potentialis so steep that φ rolls right over the bumps and continues onto φ →∞.Only for a narrow region about the minimum will φ be properly trapped.However, this condition is not sufficient: the energy density must be positiveto have inflation, whereas the example in Fig. 1 has a minimum with negativevacuum energy density.

The minimum must also be metastable since, afterinflation, the universe needs to find its way down to a lower energy statewith zero cosmological constant. Thus far, we have been unable to constructa superpotential of the form in Eq.

(5) which has the desired properties.Even without inflation, we have the embarrassing problem of understand-ing why φ gets trapped at a minimum rather than rolling past the bumpsand evolving forever. The dilaton must be trapped because, otherwise, itsevolution leads to an unacceptably large time-variation of Newton’s constantG [13],[14].

As we have argued, so long as φ is unpinned, the effective ω is1Strictly speaking, there can be inflation if φ happens to begin very close to the peakof one of the small hills, e.g., see the two-gaugino potential in Fig. 1 .

However, this seemsto be an extraordinarly fine-tuned and unattractive initial condition.7

less than unity, whereas cosmology demands ω to be greater than 50 in orderto meet the constraints of primordial nucleosynthesis and solar system tests(e.g., Viking radar-ranging) push the constraints even higher, ω > 500 [12].At this point, the only means of trapping φ is if its initial value is close to oneof the minima in the first place. It would be nice if some natural dampingmechanism existed which forces the dilaton to settle in its shallow minimum.But is the minimum of a dilaton potential where the universe really oughtto be?

The problem here is that it is very difficult2 avoid a minimum withnegative cosmological constant! This was already a problem in generic su-pergravity theories [19]; the problem is exacerbated when one considers themore special non-perturbative potential in Eq.

(5).A minimum of the potential satisfies ∂SV (S) = 0 and can be super-symmetric in the S sector, DSW = 0, or produce supersymmetry breaking,DSW ̸= 0, where D is the Kahler derivative defined below Eq. (3).

It isstraightforward to show that any extremum of V (S) in the weak couplingregion (namely, ReS > 1, where the desirable minimum should be accordingto condition 2 of the CSSL), with W(S) of the form in Eq. (5) and DSW ̸= 0is a saddle point of V (S), rather than a stable minimum [10].

Hence, thepresent universe could not be in a vacuum state which is supersymmetrybreaking in the S sector.The only remaining possibility is that the present state of the universe cor-responds to a minimum with DSW = 0: either ∂SW(S)|Smin = W(Smin) = 0(a minimum of Type A); or W(Smin) and ∂SW(S)|Smin are both non-zero2 A special mechanism for obtaining zero cosmological constant [11],[18] has been usedin the literature [8],[2] in which a field appears in the Kahler potential but not in thesuperpotential. This approach necessarily leads to a flat direction in the potential anda massless field.

It is unknown if there is a mechanism to give the field a mass and yetmaintain zero cosmological constant.8

(a minimum of Type B). If supersymmetry were unbroken in the whole the-ory then the potential in Eq.

(4) is exactly correct and because DSW = 0but W(Smin) ̸= 0 in Type B, V (Smin) is negative (see Eq. (4)) and Type Bminima would necessarily have an unacceptable negative cosmological con-stant.

However, supersymmetry has to be broken in some sector the theory.This means that the factor 3, in Eq. (4) gets modified to a smaller numbern < 3.

In all presently known examples, supersymmetry breaking occurs inthe moduli sector [7],[10] and n > 1 always. Provided n > 1, the conclusionthat the potential is negative at Type B minima is not modified.The last hope for a realistic vacuum state is a minimum of Type A.However, we shall now argue that even if one is able to arrange a minimumwith zero cosmological constant of Type A, it is almost always accompaniedby a lower energy minimum of Type B with negative cosmological constant inwhich the universe is more likely to be trapped.

This is the situation depictedin Figure 2, it corresponds to an example found in ref.[20]. We ignore herecontributions from supersymmetry breaking in other sectors of the theory.Figure 2.

Dilaton potential corresponding to gaugino condensation inthe hidden group G = SU(2)k=1 ×SU(2)k=24×SU(2)k=34.First, let us consider the case where the coefficients in Eq. (5) are all realand there is a minimum of Type A for some real value of S = SA.

For any9

hidden gauge group, the coefficients αj in Eq. (5) can all be written as nj ˜α,where the nj’s are integers.

Hence, if we set z = e−˜αS, W(S) becomes apolynomial P(z). As S →∞, W(S) →0 exponentially for superpotentialsof non-perturbative form; consequently, P(z = 0) = 0.

If zA = e−˜αSA (whichis real by our assumptions), then, since W(SA) = ∂SW(SA) = 0, we canwrite P(z) = z(z −zA)2 eP(z). There may be many Type A minima along thereal axis, but let us assume that SA is the most positive value.

Now let usconsider P(z) between z = zA and z = 0 (corresponding to SA < S < ∞).Rolle’s theorem says that between every two zeros of a real function lies a zeroof its derivative. Since P(z) has a zeros at z = zA and z = 0, this means thatwhenever we have a minimum of Type A, there will be a point between, call itz′, corresponding to S′ > SA, where P ′(z′) = 0 or ∂S(W) = 0.

If we also hadW = 0 and this were a minimum, then it would be of Type A, contradictingour assumption that SA is the most positive minimum of Type A. Thatleaves two possibilities: (i) W ̸= 0, in which case V (S′) < 0 (see Eq. (4); or(ii) W = 0, and, hence, V (S′) = 0, but S′ is not a local minimum.

In eithercase, we see that there must be a range of the effective potential with negativeV (S), in the neighborhood of S′. This means that the global minimum ofV (S) is negative.

In practice, these are relatively deep minima which theuniverse can avoid only for special initial conditions.If we consider complex coefficients in Eq. (2), there is a useful, albeitsomewhat weaker generalization of Rolle’s theorem that applies specificallyto polynomial P(z) [21].

We begin with the same assumptions that P(z)has a zero at z = 0 (S →∞) with multiplicity k0 ≥1 and a multiplicitykA ≥2 zero (Type A minimum) at some real z = zA. Then, there is atleast one additional z = z′ in the complex plane for which P ′(z′) = 0.

Thepoint z′ lies within a bounded region A about the segment OA which joins10

z0 and zA. The same reasoning as above implies that there is a neighborhoodof z′ where V (S) < 0, a potentially dangerous region of the potential withnegative cosmological constant.

Region A for polynomial P(z) of order nis the union of the two circles for which OA is a chord subtending an angle2π/(n + 1 −k0 −kA). For fixed k0 and kA, A grows as n increases.

Forconcreteness, let us consider the minimal case, k0 = 1 and kA = 2. Then, forn ≤4, A lies totally in the physically allowable region Re S > 0 (|z| < 1); inthis case, the region of V (S) with negative cosmological constant near z′ isa real danger.

However, for sufficiently large n > 4 (depending on the valueof zA), A extends to the unphysical region Re S < 0 (|z| > 1). Hence, forsufficiently large n, it is conceivable that V (S) only has Type A minima inthe physical region.

So, the situation is not completely hopeless, but onemust pay a heavy price: (a) the coefficients aj in Eq. (5) must be complex,or equivalently, < ImS ≯= 0, which may not be possible for realistic CPinvariant theories; and (b) in order to satisfy n > 5 for k0 = 1 and kA = 2(or, more generally, n > 2 + k0 + kA) so that the Type B minima are pushedinto the unphysical region, at least five gaugino condensates are required.This combination of conditions seems unwieldy, unlikely and unattractive.We are, therefore, forced to conclude that it seems difficult to constructa model consistent with current superstring lore which can support inflationor which can lead to a vacuum state with zero cosmological constant andtime-invariant Newton’s constant.

While we have pointed out some exoticloopholes entailing special initial conditions and complex superpotentials,we would recommend against this unattractive approach. Rather, we havepresented this discussion because we believe that superstring theorists shouldbe aware of the impending cosmological disasters and take up our challenge:Is there a plausible modification of superstring lore that will fit better with11

cosmology, and does this modification give us a promising implications forparticle physics?ACKNOWLEDGEMENT: It is a pleasure to thank Charles Epstein, JensErler, Vadim Kaplunovsky, Burt Ovrut and Steven Weinberg for useful dis-cussions and Mirjam Cvetic for collaboration in the early stages of this work.This work was supported in part by the Department of Energy under contractNo. DOE-AC02-76-ERO-3071.References[1] M.B.

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