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Extended Inflation from Strings ∗

arXiv:hep-th/9109049v1 25 Sep 1991Extended Inflation from Strings ∗J. Garc´ıa–Bellido† and M. Quir´os‡Instituto de Estructura de la MateriaSerrano, 123E–28006 Madrid.

SpainAbstractWe study the possibility of extended inflation in the effective theoryof gravity from strings compactified to four dimensions and find thatit strongly depends on the mechanism of supersymmetry breaking.We consider a general class of string–inspired models which are goodcandidates for successful extended inflation.In particular, the ω–problem of ordinary extended inflation is automatically solved by theproduction of only very small bubbles until the end of inflation. Wefind that the inflaton field could belong either to the untwisted or tothe twisted massless sectors of the string spectrum, depending on thesupersymmetry breaking superpotential.IEM–FT–37/91April 1991∗Work partially supported by CICYT under contract AEN90–0139.†Supported by FPI Grant.

e–mail: bellido@iem.csic.es‡e–mail: imtma27@cc.csic.es and quiros@cernvm.0

It is nowadays commonly believed by most cosmologists that the infla-tionary paradigm may solve most of the problems of the standard cosmo-logical model. However, it is not yet clear how the inflationary scenario canbe successfully implemented.

In fact, the first proposed inflationary model(known as ’old’ inflation) [1], based on a first order phase transition, couldnot provide a satisfactory explanation of how to get out from the inflationaryphase without disturbing the good properties of the standard cosmologicalmodel [1, 2]. The first models proposed to solve this ’graceful exit’ prob-lem, known as ’new’ inflation [3], with a second order phase transition, wereplagued with severe fine–tunings.

Soon after, a different solution withoutphase transition, known as ’chaotic’ inflation [4], was proposed. (Chaotic in-flation has been recently shown to considerably soften the fine–tuning prob-lems of new inflation [5].

)Recently, La and Steinhardt [6] proposed an inflationary model, known as’extended’ inflation, based again on a first order phase transition, where thegraceful exit problem was solved by using a Jordan–Brans–Dicke [7] theoryof gravity with a scalar field, instead of General Relativity [8]. It was soonrealized that the anisotropy at decoupling produced by large bubbles [9]made extended inflation incompatible with the post–Newtonian bounds [10]of General Relativity.

This desease could be cured either by using a moregeneral scalar–tensor theory of gravity [11] or by means of a cosmologicalconstant with a runaway dependence on the scalar field [12].Most particle physicists believe that the theory of gravity at low ener-gies (General Relativity, scalar–tensor theories or whatever) is an effectiveapproximation of some fundamental theory of quantum gravity at scales be-yond the Planck mass (MP). The only reliable candidates for such a funda-mental theory are superstrings [13].

They are known to describe gravity asa low energy effective theory. It is therefore of interest to know whether ornot strings could lead to any kind of cosmological inflation in the low energyeffective theory.The effective theory of superstrings exhibits three properties that makesit a good candidate for some kind of extended inflation.

First of all, the scalarfields of the gravitational multiplet (the dilaton and the moduli) are alwayscoupled to the curvature scalar of the four–dimensional metric, in the sameway as the scalar field in scalar–tensor theories of gravity. Second, the dilatonand moduli are also coupled to the matter sector giving rise to a scalar fielddependent potential.

Finally, the existence of flat directions in the potential1

follows from very general principles [13]. In the presence of supersymmetrybreaking terms and a positive vacuum energy, flat directions become runawayfields, and so are good candidates for solving the anisotropy/post–Newtonianbounds conundrum of extended inflation.In a previous paper [14] we studied the conformal properties and cos-mological solutions in the low energy effective theory of gravity from closedstrings compactified to four dimensions, for different supersymmetric andnon–supersymmetric string scenarios, during the radiation and matter dom-inated eras.

In this paper we study the possibility of extended inflation instring scenarios with spontaneously broken supersymmetry. This problemhas been recently studied under some assumptions (in particular, constantvalues for the moduli) with negative results [15].

However, we will show thatthe possibility of extended inflation strongly relies on the mechanism of su-persymmetry breaking and find the conditions under which it could happen.We will argue that a general necessary condition is the existence of a positive’metastable’ minimum with some runaway direction along the scalar fields.This runaway direction should become flat at the true minimum in order tosolve the cosmological constant problem. In this case, the same symmetryprinciple (if any) that could help solving the cosmological constant problem,could also help extended inflation.

A non–constant value of the moduli alongthe runaway direction will help overcoming the problems found in Ref. [15].At string tree level, and keeping only linear terms in the string tensionα′, the effective Lagrangian for the dilaton (φ), the modulus (σ) 1 and thematter fields (Cn), can be written in the Einstein frame as 2 [16, 17]Leff =q−˜g"˜R −12(∂µφ)2 −6(∂µσ)2 −3Xn=1αn2n e−n(σ+ 12φ) | DµCn |2 −V (φ, σ, Cn, C∗n)#(1)where C1 are untwisted matter fields, C2 twisted moduli (blowing up modes)and C3 twisted matter fields, and αn are some positive constants (α1 = 3,α2 = α3 = 1).For the purpose of this paper, in order to establish the necessary condi-1We take the diagonal direction in the space of moduli fields.2We will work hereafter, unless explicit mention, in units in which MP ≡1.2

tions for extended inflation, it will be enough to expand the potential V in(1) asV (φ, σ, Cn, C∗n) = Vo(φ, σ) +3Xn=1Vn(φ, σ) | Cn |2 + ...(2)In fact, the Lagrangian (1) and the potential (2) can be put in the standardsupergravity form [18] by means of the K¨ahler potential [16, 17]K = −ln(S + S∗) −3 ln(T + T ∗) +3Xn=1αn(T + T ∗)−n | Cn |2 + ...(3)and the superpotentialW(S, T, Cn) = Wo(S, T) +3Xn=1Wn(S, T) C3n + ...(4)whereReS = e3σ−12 φReT = eσ+ 12φ(5)It is important to stress that a superpotential Wo different from zero, nec-essary for supersymmetry breaking, and a non–constant superpotential Wncould be generated by string non–perturbative effects. Also note that we areconsistently studying the Lagrangian along the (strong CP–conserving) realdirections 3 (ImS = ImT = 0).The properties of the potential (2), and in particular its ability to produceextended inflation, will depend in general on the form of the superpotential(4).

We will first give some (by no means sufficient) conditions on the poten-tial (2), and their implications on the superpotential Wo, in order to produceextended inflation:a) We assume that the potential Vo has a minimum along some fielddirection, e.g.σ = −bφ + d(6)3Notice that a minimum along a different direction would just amount to a field redef-inition and so the general results of this paper should remain valid.3

(with b and d some real parameters) and a runaway direction along theorthogonal field4. This condition can be fulfilled depending on the functionalform of the superpotential.

For instance, if Wo = Wo(X) with X = SαT 3β(α and β real), then 5b = 3β −α6(α + β)(7)and the potential V takes the form [20]Vo =116e−6σ−φ vo(σ + bφ)Vn =αn16·2ne−(6+n)σ−(1−n2 )φ vn(σ + bφ)(8)wherevo(σ + bφ) = f 2α + 3f 2β −3f 2ovn(σ + bφ) = f 2α + (3 −n)f 2β −2f 2o(9)withfλ(σ + bφ) ≡Wo −2λX ∂Wo∂X . (10)The minimization of vo in (8) should provide the condition (6).Notice that condition (6) is not essential for extended inflation.It isjust a simplifying assumption where one direction in the (σ, φ) configurationspace is fixed to its vacuum expectation value and so the remaining theory ofgravity has only one scalar field.

However, more general situations suitablefor extended inflation are thinkable. For instance, the case where both σand φ are runaway directions (no field is fixed to its vacuum expectationvalue) can be easily realized in many models.

In particular, in the simple4Otherwise φ and σ would be fixed to their vacuum expectation values and no extendedinflation could be present. Since we are concerned in this paper with extended inflationfrom strings, we will not consider the latter case.

On the other hand, the possibility of newinflation from strings was studied some years ago and shown to require a huge amount offine–tuning [19]. Although these negative results are not general enough to exclude otherkinds of inflation based on General Relativity (e.g.

chaotic) which could arise from stringtheories, they make us search for inflation in more general theories of gravity.5The case α = 0, β = 1/3, giving b = 1/2, has recently been considered [20] and shownto be consistent with target space modular invariance. However, we will consider a moregeneral case since non–perturbative effects could break modular invariance [21].4

case where Wo = constant. (A constant superpotential can be triggered bythe vacuum expectation value of some field.) In this case, vo = | Wo |2 andvn = (2 −n)| Wo |2.b) There should be a positive cosmological constant at the minimum (6),i.e.vo(d) > 0 .

(11)In particular, this implies that supersymmetry is broken at the minimum (6)in such a way thatf 2α(d) + 3f 2β(d) > 3f 2o (d) . (12)In the case Wo = constant, condition (11) is automatically satisfied.c) The last condition is that the minimum (6) is required to be stablealong the inflaton field direction Cn, i.e.vn(d) > 0(13)orf 2α(d) + (3 −n)f 2β(d) > 2f 2o (d)(14)where n is the sector to which the inflaton belongs.

In this way, the inflatonpotential can trigger a first order phase transition from the false vacuumat Cn = 0 to the true physical vacuum at Cn ̸= 0, which we assume tocorrespond to a zero cosmological constant 6.In the simple case of Wo = constant, condition (13) is always satisfiedfor n = 1 (untwisted matter sector) but never satisfied for n = 3 (twistedmatter sector). For n = 2 (blowing–up modes) v2 ≡0 and so the stabilityalong the inflaton direction C2 would rely upon higher order derivatives ofthe potential and therefore upon the superpotential W2.In what follows we will assume that conditions (6), (11) and (13) holdand therefore will write the Lagrangian (1) for φ and the inflaton field Cn asLeff =q−˜g˜R −(6b2 + 12)(∂µφ)2 −e−n( 12−b)φ | DµCn |2 −e−(1−6b)φρo + ...(15)6Of course this would impose extra conditions on the total superpotential W, whichwe will not study here.5

where ρo is a constant energy density, we have used Eq. (6) and absorbed allconstant coefficients in the definition of Cn.

Notice that the energy densityρ(φ) in (15) is proportional to m23/2, the scale of supersymmetry breaking(the gravitino mass), as expected,m23/2 ∝e−(1−6b)φ | Wo |2 . (16)The mass of the observable fields at the true vacuum depends on the globalstructure of the potential V in (1), which is very poorly known in mostcases.

In fact, it depends on the total structure of the K¨ahler potential (3)and the superpotential (4), which could in turn depend on non–perturbativeeffects at high energy scales (string effects) and/or at low energy scales (QCDcondensates, ...). We will assume for the masses a simple dependence˜m2 ∼e−aφ m2o(17)where mo is a constant mass and a is a real coefficient parametrizing ourignorance on the details of supersymmetry breaking in string theory and thelow energy non–perturbative effects.

The case of constant masses (a = 0),considered in the analysis of Ref. [15], is particularly interesting and will becommented later on.Under a conformal redefinition [22, 23, 15, 14] of the metric˜gµν = ecφ gµν(18)˜R = e−cφR −c(D −1)D2φ −14c2(D −1)(D −2)gµν∂µφ∂νφ,(19)the masses transform asm2 = ecφ ˜m2 .

(20)It is therefore convenient to make the conformal redefinition of gµν (18,19) with parameter c = a such that the mass of the observable fields, seeEqs. (17, 20), become φ–independent [14].

Then (15) can be written as 7L = √−gΦR −ωΦ(∂µΦ)2 −12Φ1−β′(∂µCn)2 −Φ2(1−β)ρo(21)7Recall that under a conformal redefinition of the Robertson–Walker metric, thescale factor and the time variable transform as ˜a(˜t) = Φ(t)1/2a(t) and d˜t = Φ(t)1/2dtrespectively.6

whereΦ = eaφ(22)and the parameters ω, β and β′ in (21) are defined as functions of a and b as2ω + 3 = 1 + 12b2a2(23)β = 1 −6b2a(24)β′ = n 1 −2b2a!. (25)Written in terms of a Robertson–Walker metric, Φ(t) is a dimensionlessscalar related to the variation of the Plank massΦ(t) = M2P(t)M2P(26)where M2P stands for M2P(to) ≡1/GN (to is the present age of the universe),and we assume Φ(te) ≃1 at the end of inflation.

We can also define thescales M and mP throughρ(0) = Φ(0)2(1−β)ρo ≡M4(27)Φ(0) ≡m2PM2P. (28)The equations of motion then read ˙aa2+ ka2 = ρo6 Φ1−2β + ω6 ˙ΦΦ!2−˙aa˙ΦΦ¨Φ + 3 ˙aa˙Φ =2β2ω + 3ρoΦ2(1−β)(29)with solutions for k = 0 [12]a(t) = a(0)(1 + Bt)p,p = 2ω + 3 −2β2β(2β −1)Φ(t) = Φ(0)(1 + Bt)q,q =22β −1(30)7

whereB2 = 2β2(2β −1)2ρoΦ(0)1−2β(2ω + 3)(6ω + 9 −4β) =2β2(2β −1)2 M2P(2ω + 3)(6ω + 9 −4β) MMP4 MPmP2. (31)We now raise the question of the sufficient conditions for extended infla-tion and whether or not a ’gracefull exit’ can be achieved.First of all, we require that quantum gravity effects be negligible.Inother words, that the kinetic energy due to de Sitter fluctuations (maxi-mal at beginning of inflation [24, 6]) be less than ρ, see Eqs.

(27–29), i.e.H4(0) ≈M8MPmP4< ρ(0). This gives the constraintmP > M .

(32)We are assuming that the universe at Tc goes through a first order phasetransition in which the high-temperature phase remains metastable down toT = 0 [2], where bubble nucleation is dominated by quantum mechanicaltunneling [25]. Bubbles are assumed to be formed with zero radius at tB andthen expand at the speed of light.

A bubble radius at a later time t > tB isgiven byr(t, tB) =Z ttBdt′a(t′) . (33)We now define the asymptotic radius of a bubble nucleated at t asras(t) =Z ∞tdt′a(t′) =pp −11a(t)H(t)(34)where H(t) is the Hubble expansion factorH(t) = pB Φ(0)Φ(t)!β−1/2= pB a(0)a(t)!1/p.

(35)The end of inflation is determined by the competition between the bubblenucleation rate and the cosmic expansion rate. The dimensionless parameterwhich controls the percolation of the phase transition can be computed as[2]ǫ(t) =Z ttBdt′λ(t′)a3(t′)4π3 r3(t, t′) ≃λ(t)H4(t)(p ≫1)(36)8

where λ(t) is the bubble nucleation rate per unit volume.In our model,λ(t) is time dependent since the energy density which drives inflation is itselftime dependent through Φ(t), see Eq.(21). Holman et al.

[26] compute thisdependence to beλ(t) = λoΦ(t)2(1−β′)e−BohΦ(t)2(β−β′)−1i(37)where 8 λo = Ae−Bo. Bo is the Euclidean bounce action [25, 2], which dependson the inflaton potential and can acquire very big values O(102), while theprefactor A is of order one and has dimensions of T 4c , where Tc ∼M is themass scale of the phase transition.The epsilon parameter can then be written asǫ(t)= ǫo Φ(0)2(1−2β)Φ(t)2(2β−β′)e−BohΦ(t)2(β−β′)−1i= ǫ(te)Φ(t)2(2β−β′)e−BohΦ(t)2(β−β′)−1i(38)where ǫo ≡λoH4(0) is the usual parameter considered in the literature.A measure of the progress of the transition is the fraction of space whichremains in the high temperature phase (’false vacuum’), p(t) = e−ǫ(t).

Weneed a very small epsilon parameter at the beginning of inflation which in-creases very quickly to above a critical value, thus allowing for percolationof the low temperature phase (’true vacuum’). Therefore we requireǫ(te) = ǫo Φ(0)2(1−2β) > ǫcr(39)where 1.1 × 10−6 < ǫcr < 34π was computed in Ref.

[2] for solving the ’grace-full exit’ problem. Thusǫo > MMP4(2β−1)ǫcr(40)which gives ample room for very small values of ǫo, provided that 2β >1 (which is anyhow necessary for an increasing Φ(t)).We must be sure,8In ordinary JBD theories and GR, this rate is essentially time independent and givenby λo.9

however, that ǫ(t) is increasing, that is˙ǫ(t)ǫ(t) = 2(β′ −β)˙Φ(t)Φ(t)"BoΦ(t)−2(β′−β) −β′ −2ββ′ −β#> 0(41)which is satisfied forβ′ > β(42)andBoΦ(t)−2(β′−β) > β′ −2ββ′ −β . (43)This condition is very easily satisfied as we will see.We are now ready to analyze our string model for inflation, see Eq.

(21).The peculiarity of this model is the fact that ω, β and β′ are not indepen-dent but determined by the string effective action, see Eqs.(23–25). Thisdependence corresponds to the conformal redefinition of the metric tensorfor which observable matter have constant masses, as discussed above.

Wewill now impose further constraints on our model.A necessary condition for inflation is that ¨aa > 0, or p > 1, which becomes0 < b < 12 . (44)We must also impose that Φ(t) increases, which gives the conditiona < 1 −6b .

(45)The condition that ǫ(t) increases, see Eqs. (42, 43), then becomesa ≥0(46)Bo > 1(47)which are both sufficient conditions for all values of n, see Eq.

(25).Assuming N orders of magnitude increase in the scale factor,10N = a(te)a(0) = Φ(te)Φ(0)!p/q

imposes the constrainta < 1 + 12b21 −6b!logMPMN + logMPM . (49)Furthermore, in order to solve the horizon problem we need sufficientinflation such that 9 dHo < dH(0)aoa(0).However, since H(t) ∼t−1 ∼T 2and aT ∼constant during the post–inflationary period, and assuming ’goodreheating’ for recovering all the latent energy density of the phase transition(Te ≡T(te) ∼Tc ∼M), we obtain the conditionN >pp −1 logMTo.

(50)Therefore, the required number of orders of magnitude of inflation dependscrucially on the energy scale of the phase transition M.Inflation must occur after the production of monopoles or any other topo-logical defects whose densities might affect cosmology. For the same reason,the universe must also reheat before baryogenesis.

These conditions placethe constraint 102 GeV < M < 1018 GeV [6].However, solving the horizon and monopole problems is not enough. Wemust be sure that the phase transition ends and that all the bubbles per-colate without disturbing too much the isotropy and homogeneity of thecosmic background radiation.

Therefore, we expect that the volume frac-tion contained in bubbles with radius greater than a given comoving radiusr = r(te, t) at the end of inflation be less than 10−n at a temperature T:V>(r, te) = 1 −p(t) ≃ǫ(t) = ǫ(te) TMδe−Boh( MT )δ′−1i< 10−n(51)where we have usedΦ(t)2(2β−β′) =rorδ≃ TMδ(52)where ro ≡ras(te) is the asymptotic radius of a bubble nucleated at the endof inflation, δ ≡8β(2β −β′)2ω + 3 −4β2and δ′ ≡8β(β′ −β)2ω + 3 −4β2 > 0 . In particular,9We use here the notation Ho ≡H(to), ao ≡a(to) and To ≡T (to).11

for the cosmic background radiation, we require that [9] n ≃5 at T ≃1 eVin (51). This condition is trivially satisfied thanks to the exponential, usingMT > 1011 and condition (47).

In this way, the extended inflation problem ofanisotropy at decoupling produced by large bubbles is successfully solved inthis kind of models 10.We still have to be sure of reestablishing a common Robertson–Walkerframe in all the bubble clusters that will coalesce to form our universe. Theremust be some way to remember the original (pre–bubble–nucleation) coor-dinates; such a record can be found in the evolution of a(t) or Φ(t).

Sinceconstant H(t) corresponds in General Relativity to a de Sitter universe withno distinguished frame, we must require sufficient variations of this quantity,e.g. m orders of magnitude in H(t) [9, 12].

In particular, we expect that ho-mogeneity and isotropy must hold by the time of nucleosynthesis (TNS ≃1MeV, m ≃1), thusH(t)H(te) =r + roro1p−1 ≃MT1p−1> 10m(53)corresponding top < 1 + log MTNS≡po(54)which is an explicit bound on the power of the scale factor and gives an extracondition on our parametersa

[15].12

where k is the dimensionless physical scale of reentering perturbations and ˜p isthe power of the scale factor in the Einstein frame ˜a(˜t) ∼˜t ˜p,˜p = 2ω + 34β2!.This imposes a very mild constraint on MM < 1 −6b√1 + 12b2!10−2MP < 10−2MP . (58)Finally, the most stringent bounds will come from the post–Newtonianexperiments of time delay [28, 10] and the nucleosynthesis bound [29] on theω parameter2ω + 3 > 500(2σ)(59)which gives a very strong constraint on our parametersa

(60)It is interesting to notice that the anisotropy of the cosmic background ra-diation, which was the main problem for extended inflation, does not imposeany significant bound on our model, see Eq.(51). The most stringent boundcomes from the post–Newtonian experiments and nucleosynthesis bound, seeEq.

(60), which constraint the parameter a. On the other hand, the strongestconstraint on b comes from the isotropy and homogeneity at nucleosynthesis,see Eqs.

(54, 55).Most of the previous bounds depend on the energy scale M of the phasetransition.We have studied those bounds for two typical values of M.For a phase transition driven by phenomenological supersymmetry break-ing (m3/2 ≃1 TeV) we have M = (m3/2MP)1/2 ≃1011 GeV, while for theusual grand unified theory we take M = 1016 GeV. The inflationary scenariois characterized by two parameters, the power p of the scale factor and thenumber N of orders of magnitude increase during inflation.

Both parame-ters depend on the energy scale of the phase transition, see Eqs. (50, 54).

ForM = 1011 GeV we have p < 16 and N > 26, while for M = 1016 GeV, p < 21and N > 31 to solve the horizon problem without disturbing the isotropyand homogeneity at nucleosunthesis. The actual value of N depends on theparameters of the theory.

Using the bounds (55), (58) and (60) we obtainN > 45, which widely solves the flatness problem.13

In Fig.1 we show the region in parameter space (a, b) allowed by all theinflationary conditions, for M = 1011 GeV and M = 1016 GeV (dashed anddotted curves respectively). The allowed region is the one below the curves.The condition associated with neglecting the quantum gravity effects (32, 49)strongly depends on the energy scale of the phase transition, as expected, andas we can see from the lines labelled QG.

Other conditions depend slightly onM, like those associated with reestablishing the isotropic and homogeneousRobertson–Walker frame (53, 55), and labelled RW in Fig.1 . Finally, thereare those conditions which do not depend at all on the energy scale of thephase transition, like the post–Newtonian bounds (59, 60) and the condition(45) that Φ increases from mP to MP, labelled pN–NS and Φ respectively.However, as we can see from Fig.1, the final allowed region in parameterspace does not depend much on the scale M since it is bounded by the post–Newtonian and nucleosynthesis bound and the isotropy and homogeneitycondition at nucleosynthesis.As we can see from Fig.1, the case of constant observable masses (a = 0)is consistent with all inflationary and post–Newtonian bounds.

This can beeasily obtained by taking the limit a →0 in our explicit solution (30), whichcorresponds toa(t) ∼t1+12b2(1−6b)2Φ(t) ∼1(61)On the other hand the direction b = 0, see Eq. (6), is incompatible withthe necessary condition for inflation, Eq.

(44), and corresponds to the case ofconstant moduli. We thus agree with the negative results found in Ref.

[15].In conclusion, we have studied in this paper the general conditions un-der which the effective theory of gravity from strings compactified to fourdimensions could lead to extended inflation. We have found that a neces-sary condition is the existence of runaway directions in the space of fieldscoupled to the curvature scalar (dilaton and moduli fields).

However, theexistence of runaway directions is a usual feature of the effective theory ofstrings (through classical invariance arguments [13]). It is satisfied for manysupersymmetry breaking potentials.

In the simplest case of supersymmetrybreaking, a constant Wo superpotential in (4), all moduli and the dilatonare runaway fields with a positive potential (∼| Wo |2) and extended infla-tion may follow. However, to simplify the study of the equations of motion,we have assumed just one runaway direction and parametrized it by a real14

parameter b. This is just a simplifying hypothesis since extended inflationmight occur under much more general circumstances.A second necessary condition for extended inflation is the existence of ametastable minimum along some (matter) inflaton field.

This condition isnecessary to enforce a first order phase transition. It also depends on theparticular structure of the supersymmetry breaking superpotential, but thiscondition (see Eq.

(13)) is easily satisfied in many models. For instance, inthe simple case Wo = constant, it holds when the inflaton belongs to theuntwisted sector (n = 1), and does not hold if it belongs to the twistedsector (n = 3).

The case of the inflaton as a blowing–up mode (n = 2) wouldrequire the precise knowledge of the total superpotential.Third, we assume a simple behaviour of the mass of observable fields onthe runaway direction, and parametrize this behaviour with a real parametera. (The case of constant masses corresponds to a = 0.) We make a conformalredefinition of the metric in order to go into the ’physical’ frame, where themasses of the observable fields are constant.Of course, if the functionaldependence of masses were more complicated, we would have needed a moregeneral conformal transformation and the theory would look different, inparticular it would have a non–constant ω parameter.

However, we shouldstress here that a conformal redefinition is not essential since Physics cannotdepend on it. In other words, we could redefine the physical scale factor bytaking its ratio with respect to the Compton wavelength [30] which is thenmanifestly independent of the conformal transformation [15, 14].Finally, we have imposed all the conditions for successful extended infla-tion on the solution of our model and found an allowed region in parameterspace (a, b).

Our results are summarized in Fig.1. The direction b = 0 (theregion of constant moduli) is excluded from the allowed region, while a = 0(the case of constant masses) is inside the permitted region and thereforeconsistent with all experimental bounds.

Notice that our model successfullysolves the ω–problem of extended inflation (namely, that the condition ofisotropy of the cosmic background radiation at decoupling is in conflict withthe post–Newtonian bounds on ω) by producing very small bubbles until theend of inflation when the epsilon parameter increases exponentially up to thecritical value.15

AcknowledgementsOne of us (J.G.–B.) would like to thank M. Cvetiˇc, B. Ovrut and P. Stein-hardt for valuable discussions, and the Theoretical Physics Department ofPennsylvania University for the hospitality given to him.

The other (M.Q. )thanks F. Quevedo for discussions and the T–8 Division of Los Alamos Na-tional Laboratory for its warm hospitality.16

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Figure CaptionsFig.1 Plot of the region in parameter space (a, b) allowed by the inflationaryconditions in the text. The solid lines represent those bounds which donot depend on the scale M of the phase transition.

The dashed curvecorrespond to the bounds associated with the scale M = 1011 GeV andthe dotted curve to those related to M = 1016 GeV. The allowed regionis the one below the curves.

The border b = 0 is excluded from it. Wehave labelled the curves as follows: QG corresponds to the conditionassociated to neglecting quantum gravity effects, Φ corresponds to thecondition for an increasing scalar field, RW corresponds to the boundon isotropy and homogeneity at the time of nucleosynthesis and pN–NScorresponds to the bounds from post–Newtonian experiments and thenucleosynthesis bound on ω.19

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