Inflationary de Sitter solutions from superstrings
aa r X i v : . [ h e p - t h ] J a n LPTENS–07/22, CPHT–RR025.0407June 2007
Inflationary de Sitter Solutionsfrom Superstrings ∗ Costas Kounnas and Herv´e Partouche Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure, †
24 rue Lhomond, F–75231 Paris cedex 05, France
[email protected] Centre de Physique Th´eorique, Ecole Polytechnique, ⋄ F–91128 Palaiseau, France
Abstract
In the framework of superstring compactifications with N = 1 supersymmetry sponta-neously broken, (by either geometrical fluxes, branes or else), we show the existence of newinflationary solutions . The time-trajectory of the scale factor of the metric a , the supersym-metry breaking scale m ≡ m (Φ) and the temperature T are such that a m and a T remainconstant. These solutions request the presence of special moduli-fields: i ) The universal “no-scale-modulus” Φ, which appears in all N = 1 effective supergravitytheories and defines the supersymmetry breaking scale m (Φ). ii ) The modulus Φ s , which appears in a very large class of string compactifications and hasa Φ-dependent kinetic term. During the time evolution, a ρ s remains constant as well, ( ρ s being the energy density induced by the motion of Φ s ).The cosmological term Λ( am ), the curvature term k ( am, aT ) and the radiation term c R = a ρ are dynamically generated in a controllable way by radiative and temperaturecorrections; they are effectively constant during the time evolution.Depending on Λ, k and c R , either a first or second order phase transition can occur inthe cosmological scenario. In the first case, an instantonic Euclidean solution exists andconnects via tunneling the inflationary evolution to another cosmological branch. The latterstarts with a big bang and, in the case the transition does not occur, ends with a big crunch.In the second case, the big bang and the inflationary phase are smoothly connected. ∗ Research partially supported by the EU (under the contracts MRTN-CT-2004-005104, MRTN-CT-2004-512194, MRTN-CT-2004-503369, MEXT-CT-2003-509661), INTAS grant 03-51-6346, CNRSPICS 2530, 3059 and 3747, and ANR (CNRS-USAR) contract 05-BLAN-0079-01. † Unit´e mixte du CNRS et de l’Ecole Normale Sup´erieure associ´ee `a l’Universit´e Pierre et MarieCurie (Paris 6), UMR 8549. ⋄ Unit´e mixte du CNRS et de l’Ecole Polytechnique, UMR 7644.
Introduction
In the framework of superstring and M -theory compactifications, there are always modulifields coupled in a very special way to the gravitational and matter sector of the effective N = 1 four-dimensional supergravity. The gravitational and the scalar field part of theeffective Lagrangian have the generic form L = p − det g (cid:20) R − g µν K i ¯ ∂ µ φ i ∂ ν ¯ φ ¯ − V ( φ i , ¯ φ ¯ ı ) (cid:21) , (1.1)where K i ¯ is the metric of the scalar manifold and V is the scalar potential of the N = 1supergravity. (We will always work in gravitational mass units, with M = √ πG N = 2 . × GeV). What will be crucial in this work is the non-triviality of the scalar kinetic terms K i ¯ in the N = 1 effective supergravity theories that will provide us, in some special cases,accelerating cosmological solutions once the radiative and temperature corrections are takeninto account.Superstring vacua with spontaneously broken supersymmetry [1] that are consistent at theclassical level with a flat space-time define a very large class of “no-scale” supergravitymodels [2]. Those with N = 1 spontaneous breaking deserve more attention. Some of themare candidates for describing (at low energy) the physics of the standard model and extend itup to O (1) TeV energy scale. This class of models contains an enormous number of consistentstring vacua that can be constructed either via freely acting orbifolds [1, 3] or “geometricalfluxes” [4] in heterotic string and type IIA,B orientifolds, or with non-geometrical fluxes [5](e.g. RR-fluxes or else).Despite the plethora of this type of vacua, an interesting class of them are those whichare described by an effective N = 1 “no-scale supergravity theory”. Namely the vacua inwhich the supersymmetry is sponaneously broken with a vanishing classical potential withundetermined gravitino mass due to at least one flat field direction, the “no-scale modulusΦ. At the quantum level a non-trivial effective potential is radiatively generated which mayor may not stabilize the “no-scale” modulus [2].What we will explore in this work are the universal scaling properties of the “thermal”effective potential at finite temperature that emerges at the quantum level of the theory. As1e will show in section 4, the quantum and thermal corrections are under control, (thanks tosupersymmetry and to the classical structure of the “no-scale models”), showing interestingscaling properties.In section 2, we set up our notations and conventions in the effective N = 1 “no-scale”supergravities of the type IIB orientifolds with D -branes and non-trivial NS-NS and RRthree form fluxes H and F . We identify the “no-scale” modulus Φ, namely the scalarsuperpartner of the Goldstino which has the property to couple to the trace of the energymomentum tensor of a sub-sector of the theory [6]. More importantly, it defines the field-dependence of the gravitino mass [2] m (Φ) = e α Φ . (1.2)Other extra moduli that we will consider here are those with Φ-dependent kinetic terms.These moduli appear naturally in all string compactifications [7]. We are in particularinterested in scalars (Φ s ) which are leaving on D -branes and whose kinetic terms scale asthe inverse volume of the “no-scale” moduli space.In section 3, we display the relevant gravitational, fields and thermal equations of motionin the context of a Friedman-Robertson-Walker (FRW) space-time. We actually generalizethe mini-superspace (MSS) action by including fields with non-trivial kinetic terms and ageneric, scale factor dependent, thermal effective potential.In our analysis we restrict ourselves to the large moduli limit, neglecting non-perturbativeterms and world-sheet instanton corrections O ( e − S ), O ( e − T a ). On the other hand we keepthe perturbative quantum and thermal corrections.Although this study looks hopeless and out of any systematic control even at the perturbativelevel, it turns out to be manageable thanks to the initial no-scale structure appearing at theclassical level (see section 4).In section 5, we show the existence of a critical solution to the equations of motion thatfollows from the scaling properties derived in section 4. We have to stress here that weextremize the effective action by solving the gravitational and moduli equations of motion2nd do not consider the stationary solutions emerging from a minimization of the effectivepotential only. We find in particular that a universal solution exists where all scales evolve intime in a similar way, so that their ratios remain constant: m ( t ) /T ( t ) = const., a ( t ) m ( t ) =const.. Along this trajectory, effective time-independent cosmological term Λ, curvatureterm k and radiation term are generated in the MSS action, characterizing the cosmologicalevolution.Obviously, the validity of the cosmological solutions based on (supergravity) effective fieldtheories is limited. For instance, in the framework of more fundamental theories such asstring theory, there are high temperature instabilities occuring at T ≃ T H , where T H is theHagedorn temperature of order the mass of the first string excited state. To bypass theselimitations, one needs to go beyond the effective field theory approach and consider the fullstring theory (or brane, M-theory,...) description. Thus, the effective solutions presented inthis work are not valid anymore and must be corrected for temperatures above T H . Moreover,Hagedorn-like instabilities can also appear in general in other corners of the moduli space ofthe fundamental theory, when space-time supersymmetry is spontaneously broken.Regarding the temperature scale as the inverse radius of the compact Euclidean time, onecould conclude that all the internal radii of a higher dimensional fundamental theory haveto be above the Hagedorn radius. This would mean that the early time cosmology shouldbe dictated by a 10-dimensional picture rather than a 4-dimensional one where the internalradii are of order the string scale. There is however a loophole in this statement. Indeed, notachyonic instability is showing up in the whole space of the moduli which are not involved inthe spontaneous breaking of supersymmetry, as recently shown in explicit examples [8]. Thisleeds us to the conjecture that the only Hagedorn-like restrictions on the moduli space dependon the supersymmetry breaking. In our cosmological solutions, not only the temperature T scale is varying, but also the supersymmetry breaking scale m , which turns to be a moduli-dependent quantity. Based on the above statements, we expect that in a more accuratestringy description of our analysis, there should be restrictions on the temperature as wellas the supersymetry breaking scale. This has been recently explicitly shown in the stringyexamples considered in [8]. 3n section 6, our cosmological solutions are generalized by including moduli with other scalingproperties of their kinetic terms.Finally, section 7 is devoted to our conclusions and perspective for future work. In the presence of branes and fluxes, several moduli can be stabilized. For instance, in“generalized” Calabi-Yau compactifications, either the h , K¨alher structure moduli or the h , complex structure moduli can be stabilized according to the brane and flux configurationin type IIA or type IIB orientifolds [4–6, 9]. The (partial) stabilization of the moduli canlead us at the classical level to AdS like solutions, domain wall solutions or “flat no-scalelike solutions”. Here we will concentrate our attention on the “flat no-scale like solutions”.In order to be more explicit, let us consider as an example the type IIB orientifolds with D -branes and non-trivial NS-NS and RR three form fluxes H and F . This particularconfiguration induces a well known superpotential W ( S, U a ) that can stabilize all complexstructure moduli U a and the coupling constant modulus S [4, 5]. The remaining h , moduli T a “still remain flat directions at the classical level”, e.g. neglecting world-sheet instantoncorrections O ( e − T a ) and the perturbative and non-perturbative quantum corrections [5].It is also well known by now that in the large T a limit the K¨alher potential is given by theintersection numbers d abc of the special geometry of the Calabi-Yau manifold and orbifoldcompactifications [10, 11]: K = − log d abc ( T a + ¯ T a )( T b + ¯ T b )( T c + ¯ T c ) . (2.1)Thus, after the S and U a moduli stabilization, the superpotential W is effectively constantand implies a vanishing potential in all T a directions. The gravitino mass term is howevernon-trivial [1, 2, 4, 5, 11], m = | W | e K . (2.2)This classical property of “no-scale models” emerges from the cubic form of K in the moduli T a and is generic in all type IIB orientifold compactifications with D -branes and three form4 and F fluxes [4, 5]. Keeping for simplicity the direction T a = γ a T (for some constants γ a ) and freezing all other directions, the K¨alher potential is taking the well known SU (1 , K = − T + ¯ T ) . (2.3)This gives rise to the kinetic term and gravitino mass term, g µν ∂ µ T ∂ ν ¯ T ( T + ¯ T ) and m = ce K = c ( T + ¯ T ) , (2.4)where c is a constant. Freezing Im T and defining the field Φ by e α Φ = m = c ( T + ¯ T ) , (2.5)one finds a kinetic term g µν ∂ µ T ∂ ν ¯ T ( T + ¯ T ) = g µν α ∂ µ Φ ∂ ν Φ . (2.6)The choice α = 3 / T -moduli. For one of them, Φ s , one has K s ≡ − α e α Φ g µν ∂ µ Φ s ∂ ν Φ s = − α c g µν ∂ µ Φ s ∂ ν Φ s ( T + ¯ T ) . (2.7)Moduli with this scaling property appear in a very large class of string compactifications .Some examples are: i ) All moduli fields leaving in the parallel space of D -branes [4, 5]. ii ) All moduli coming from the twisted sectors of Z -orbifold compactifications in het-erotic string [7], after non-perturbative stabilization of S by gaugino condensation and flux-corrections [12].Our analysis will also consider other moduli fields with different scaling properties, namelythose with kinetic terms of the form: K w ≡ − e (6 − w ) α Φ g µν ∂ µ φ w ∂ ν φ w , (2.8)with weight w = 0 , . Gravitational, Moduli and Thermal Equations
In a fundamental theory, the number of degrees of freedom is important (and actually infinitein the context of string or M-theory). However, in an effective field theory, an ultravioletcut-off set by the underlying theory determines the number of states to be considered. Wefocus on cases where these states include the scalar moduli fields Φ and Φ s , with non-trivialkinetic terms given by L = p − det g (cid:20) R − g µν (cid:0) ∂ µ Φ ∂ ν Φ + e α Φ ∂ µ Φ s ∂ ν Φ s (cid:1) − V (Φ , µ ) (cid:21) + · · · (3.1)In this Lagrangian, the “ · · · ” denote all the other degrees of freedom, while the effectivepotential V depends on Φ and the renormalization scale µ . We are looking for gravitationalsolutions based on isotropic and homogeneous FRW space-time metrics, ds = − N ( t ) dt + a ( t ) d Ω , (3.2)where Ω is a 3-dimensional compact space with constant curvature k , such as a sphere or anorbifold of hyperbolic space. This defines an effective one dimensional action, the so called“mini-super-space” (MSS) action [13–16].A way to include into the MSS action the quantum fluctuations of the full metric and matterdegrees of freedom (and thus taking into account the back-reaction on the space-time metric),is to switch on a thermal bath at temperature T [14–16]. In this way, the remaining degreesof freedom are parameterized by a pressure P ( T, m i ) and a density ρ ( T, m i ), where m i arethe non-vanishing masses of the theory. Note that P and ρ have an implicit dependence onΦ, through the mass m (Φ) defined in eq. (1.2) [6]. The presence of the thermal bath modifiesthe effective MSS action, including the corrections due to the quantum fluctuations of thedegrees of freedom whose masses are below the temperature scale T . The result, togetherwith the fields Φ and Φ s , reads S eff = − | k | − Z dt a N (cid:18) ˙ aa (cid:19) − kNa − N ˙Φ − N e α Φ ˙Φ s + N V − N ( ρ + P ) + N ρ − P ) ! , (3.3)where a “dot” denotes a time derivation. N ( t ) is a gauge dependent function that can bearbitrarily chosen by a redefinition of time. We will always use the gauge N ≡
1, unless it6s explicitly specified.The variation with respect to N gives rise to the Friedman equation,3 H = − ka + ρ + 12 ˙Φ + 12 e α Φ ˙Φ s + V , (3.4)where H = ( ˙ a/a ).The other gravitational equation is obtained by varying the action with respect to the scalefactor a : 2 ˙ H + 3 H = − ka − P −
12 ˙Φ − e α Φ ˙Φ s + V + 13 a ∂V∂a . (3.5)In the literature, the last term a ( ∂V /∂a ) is frequently taken to be zero. However, this isnot valid due to the dependence of V on µ , when this scale is chosen appropriately as willbe seen in section 3. We thus keep this term and will see that it plays a crucial role in thederivation of the inflationary solutions under investigation.We find useful to replace eq. (3.5) by the linear sum of eqs. (3.4) and (3.5), so that thekinetic terms of Φ and Φ s drop out,˙ H + 3 H = − ka + 12 ( ρ − P ) + V + 16 a ∂V∂a . (3.6)The other field equations are the moduli ones,¨Φ + 3 H ˙Φ + ∂∂ Φ (cid:18) V − P − e α Φ ˙Φ s (cid:19) = 0 (3.7)and ¨Φ s + (3 H + 2 α ˙Φ) ˙Φ s = 0 . (3.8)The last equation (3.8) can be solved immediately, K s ≡ e α Φ ˙Φ s = C s e − α Φ a , (3.9)where C s is a positive integration constant. It is important to stress here that we insist tokeep in eq. (3.7) both terms ∂P/∂ Φ and ∂K s /∂ Φ that are however usually omitted in theliterature. The first term vanishes only under the assumption that all masses are taken to beΦ-independent, while the absence of the second term assumes a trivial kinetic term. However,7 oth assumptions are not valid in string effective supergravity theories ! (See section 3.)Finally, we display for completeness the total energy conservation of the system, ddt (cid:18) ρ + 12 ˙Φ + K s + V (cid:19) + 3 H (cid:16) ρ + P + ˙Φ + 2 K s (cid:17) = 0 . (3.10)Before closing this section, it is useful to derive some extra useful formulas that are associatedto the thermal system. The integrability condition of the second law of thermodynamicsreaches, for the thermal quantities ρ and P , T ∂P∂T = ρ + P . (3.11)The fact that these quantities are four-dimensional implies (cid:18) m i ∂∂m i + T ∂∂T (cid:19) ρ = 4 ρ and (cid:18) m i ∂∂m i + T ∂∂T (cid:19) P = 4 P . (3.12)Then, the second eq. (3.12) together with the eq. (3.11) implies [6]: m i ∂P∂m i = − ( ρ − P ) . (3.13)Among the non-vanishing m i , let us denote with “hat-indices” the masses m ˆ ı that are Φ-independent, and with “tild-indices” the masses m ˜ ı that have the following Φ-dependence: { m i } = { m ˆ ı } ∪ { m ˜ ı } where m ˜ ı = c ˜ ı e α Φ , (3.14)for some constants c ˜ ı . Then, utilizing eq. (3.13), we obtain a very fundamental equationinvolving the modulus field Φ [6], − ∂P∂ Φ = α ( ˜ ρ − P ) , (3.15)where ˜ ρ and ˜ P are the contributions to ρ and P associated to the states with Φ-dependentmasses m ˜ ı . The above equation (3.15) clearly shows that the modulus field Φ couples to the(sub-)trace of the energy momentum tensor associated to the thermal system [6] ˜ ρ , ˜ P of thestates with Φ-dependent masses defined in eq. (3.14). We return to this point in the nextsection. 8 Effective Potential and Thermal Corrections
In order to find solutions to the coupled gravitational and moduli equations discussed inthe previous section, it is necessary to analyze the structure of the scalar potential V andthe thermal functions ρ , P . More precisely, we have to specify their dependence on Φ , T, a and Φ s . Although this analysis looks hopeless in a generic field theory, it is perfectly undercontrol in the string effective no-scale supergravity theories.Classically the potential V cl is zero along the moduli directions Φ and Φ s . At the quantumlevel, it receives radiative and thermal corrections that are given in terms of the effectivepotential [11], V ( m i , µ ), and in terms of the thermal function, − P ( T, m i ). Let us considerboth types of corrections. The one loop effective potential has the usual form [11, 17], V = V cl + 164 π Str M Λ co log Λ co µ + 132 π Str M Λ co + 164 π Str (cid:18) M log M µ (cid:19) + · · · , (4.1)where V cl is the classical part, which vanishes in the string effective “no-scale” supergravitycase. An ultraviolet cut-off Λ co is introduced and µ stands for the renormalization scale.Str M n ≡ X I ( − ) J I (2 J I + 1) m nI (4.2)is a sum over the n -th power of the mass eigenvalues. In our notations, the index I is refer-ring to both massless and massive states (with eventually Φ-dependant masses).The weightsaccount for the numbers of degrees of freedom and the statistics of the spin J I particles.The quantum corrections to the vacuum energy with the highest degree of ultraviolet di-vergence is the Λ co term, whose coefficient Str M = ( n B − n F ) is equal to the number ofbosonic minus fermionic degrees of freedom. This term is thus always absent in supersym-metric theories since they possess equal numbers of bosonic and fermionic states.The second most divergent term in eq. (4.1) is the Λ co contribution proportional to Str M .9n the N = 1 spontaneously broken supersymmetric theories, it is always proportional tothe square of the gravitino mass-term m (Φ) ,Str M = c m (Φ) . (4.3)The coefficient c is a field independent number. It depends only on the geometry of thekinetic terms of the scalar and gauge manifold, and not on the details of the superpoten-tial [11, 17]. This property is very crucial in our considerations.The last term has a logarithmic behavior with respect to the infrared scale µ and is inde-pendent of the ultraviolet cut-off Λ co . Following the infrared regularization method valid instring theory (and field theory as well) adapted in ref. [18], the scale µ is proportional to thecurvature of the three dimensional space, µ = 1 γa , (4.4)where γ is a numerical coefficient chosen appropriately according to the renormalizationgroup equation arguments. Another physically equivalent choice for µ is to be proportionalto the temperature scale, µ = ζ T . The curvature choice (4.4) looks more natural and hasthe advantage to be valid even in the absence of the thermal bath.Modulo the logarithmic term, the Str M can be expanded in powers of gravitino mass m (Φ),164 π Str M = C m + C m + C . (4.5)Including the logarithmic terms and adding the quadratic contribution coming from theStr M , we obtain the following expression for the effective potential organized in powers of m (Φ): V = V (Φ , a ) + V (Φ , a ) + V (Φ , a ) , (4.6)where V n (Φ , a ) = m n (Φ) (cid:16) C n + Q n log ( m (Φ) γa ) (cid:17) , (4.7)for constant coefficients C n and Q n , ( n = 4 , , ∂V n (Φ , a ) ∂ Φ = α ( nV n + m n Q n ) and a ∂V n (Φ , a ) ∂a = m n Q n . (4.8)10he logarithmic dependence in the effective potential can be derived in the effective fieldtheory by considering the Renormalization Group Equations (RGE). They involve the gaugecouplings, the Yukawa couplings and the soft-breaking terms [11, 19]. These soft-breakingterms are usually parameterized by the gaugino mass terms M / , the soft scalar masses m ,the trilinear coupling mass term A and the analytic mass term B , [11, 19]. However, whatwill be of main importance in this work is that all soft breaking mass terms are proportionalto m (Φ) [11, 17]. For bosonic (or fermionic) fluctuating states of masses m b (or m f ) in thermal equilibrium attemperature T , the general expressions of the energy density ρ and pressure P are ρ = T X boson b I Bρ (cid:16) m b T (cid:17) + X fermion f I Fρ (cid:16) m f T (cid:17)! , P = T X boson b I BP (cid:16) m b T (cid:17) + X fermion f I FP (cid:16) m f T (cid:17)! , (4.9)where I B ( F ) ρ (cid:16) mT (cid:17) = Z ∞ dq q E (cid:0) q, mT (cid:1) e E ( q, mT ) ∓ , I B ( F ) P = 13 Z ∞ dq q /E (cid:0) q, mT (cid:1) e E ( q, mT ) ∓ E (cid:0) q, mT (cid:1) = q q + m T .There are three distinct sub-sectors of states: i ) The sub-sector of n B bosonic and n F fermionic massless states. From eqs. (4.9) and(4.10), their energy density ρ and pressure P satisfy ρ = 3 P = π (cid:18) n B + 78 n F (cid:19) T . (4.11)In particular, we have ρ − P = 0 and ∂P /∂ Φ = 0. ii ) The sub-sector of states with non vanishing masses independent of m (Φ). • Consider the ˆ n B bosons and ˆ n F fermions whose masses we denote by m ˆ ı are below T.The energy density ˆ ρ and pressure ˆ P associated to them satisfyˆ ρ ( T, m ˆ ı ) = ˆ ρ ( T, m ˆ ı = 0) + m ı ∂ ˆ ρ∂m ı = π (cid:18) ˆ n B + 78 ˆ n F (cid:19) T − X ˆ ı ˆ c ˆ ı m ı T , (4.12)ˆ P ( T, m ˆ ı ) = ˆ P ( T, m ˆ ı = 0) + m ı ∂ ˆ P∂m ı = π (cid:18) ˆ n B + 78 ˆ n F (cid:19) T − X ˆ ı ˆ c ˆ ı m ı T , (4.13)11here the ˆ c ˆ ı ’s are non-vanishing positive constants. In particular, one has ∂ ˆ P /∂
Φ = 0. • For the masses m ˆ ı above T , the contributions of the particular degrees of freedom areexponentially suppressed and decouple from the thermal system. We are not including theircontribution. iii ) The sub-sector with non vanishing masses proportional to m (Φ). Its energy density ˜ ρ and pressure ˜ P satisfy ∂P∂ Φ = − α ( ˜ ρ − P ) , (4.14)as was shown at the end of section 2. This identity is also valid for the massless system weconsider in case i ).According to the scaling behaviors with respect to T and m (Φ), we can separate ρ = ρ + ρ , P = P + P , (4.15)where (cid:18) m (Φ) ∂∂m (Φ) + T ∂∂T (cid:19) ( ρ n , P n ) = n ( ρ n , P n ) . (4.16) ρ and P are the sums of the contributions of the massless states (case i )), the T parts ofˆ ρ and ˆ P (case ii )), and ˜ ρ and ˜ P (case iii )), ρ = T π (cid:18)(cid:0) n B + ˆ n B (cid:1) + 78 (cid:0) n F + ˆ n F (cid:1)(cid:19) + X boson ˜ b I Bρ (cid:16) m ˜ b T (cid:17) + X fermion ˜ f I Fρ (cid:16) m ˜ f T (cid:17) , (4.17) P = T π (cid:18)(cid:0) n B + ˆ n B (cid:1) + 78 (cid:0) n F + ˆ n F (cid:1)(cid:19) + X boson ˜ b I BP (cid:16) m ˜ b T (cid:17) + X fermion ˜ f I FP (cid:16) m ˜ f T (cid:17) , (4.18)while ρ and P arise from the T parts of ˆ ρ and ˆ P (case ii )): ρ = P = − X ˆ ı ˆ c ˆ ı m ı T ≡ − ˆ M T . (4.19) The fundamental ingredients in our analysis are the scaling properties of the total effectivepotential at finite temperature, V total = V − P . (5.1)12ndependently of the complication appearing in the radiative and temperature correctedeffective potential, the scaling violating terms are under control. Their structure suggeststo search for a solution where all the scales of the system, m (Φ), T and µ = (1 /γa ), remainproportional during their evolution in time, e α Φ ≡ m (Φ) = 1 γ ′ a = ⇒ H = − α ˙Φ and ξ m (Φ) = T . (5.2)Our aim is thus to determine the constants γ ′ and ξ in terms of C s in eq. (3.9), γ , andthe computable quantities C n , Q n , ( n = 4 , ,
0) in string theory, such that the equations ofmotion for Φ, Φ s and the gravity are satisfied. On the trajectory (5.2), the contributions V n ,( n = 4 , ,
0) defined in eq. (4.7) satisfy V n = m n C ′ n where C ′ n = C n + Q n log (cid:18) γγ ′ (cid:19) , (5.3)and ∂V n ∂ Φ = α m n ( nC ′ n + Q n ) , a ∂V n ∂a = m n Q n . (5.4)Also, the contributions of Φ and 1 /a in K s in eq. (3.9) conspire to give a global 1 /a dependence, K s = C s γ ′ a . (5.5)Finally, the sums over the full towers of states with Φ-dependent masses behave in ρ /T and P /T as pure constants, (see eqs. (4.17) and (4.18)), ρ = r T where r = π (cid:18)(cid:0) n B + ˆ n B (cid:1) + 78 (cid:0) n F + ˆ n F (cid:1)(cid:19) + X ˜ b I Bρ (cid:18) ˜ c ˜ b ξ (cid:19) + X ˜ f I Fρ (cid:18) ˜ c ˜ f ξ (cid:19) , (5.6) P = p T where p = π (cid:18)(cid:0) n B + ˆ n B (cid:1) + 78 (cid:0) n F + ˆ n F (cid:1)(cid:19) + X ˜ b I BP (cid:18) ˜ c ˜ b ξ (cid:19) + X ˜ f I FP (cid:18) ˜ c ˜ f ξ (cid:19) . (5.7)As a consequence, using eqs. (4.8) and (4.14), the Φ-equation (3.7) becomes,˙ H + 3 H = α (cid:16) (4 C ′ + Q ) m + (2 C ′ + Q ) m + Q + ( r − p ) ξ m − C s γ ′ m (cid:17) . (5.8)On the other hand, using eq. (4.8), the gravity equation (3.6) takes the form˙ H +3 H = − kγ ′ m + 12 ( r − p ) ξ m +( C ′ m + C ′ m + C ′ )+ 16 ( Q m + Q m + Q ) . (5.9)13he compatibility of the Φ-equation and the gravity equation along the critical trajectoryimplies an identification of the coefficients of the monomials in m . The constant termsdetermine C ′ in term of Q C ′ = 6 α − Q , (5.10)which amounts to fixing γ ′ , γ ′ = γ e C Q − α − . (5.11)The quadratic terms determine the parameter k : k = − γ ′ (cid:18) α − C ′ + 6 α − Q (cid:19) . (5.12)Finally, the quartic terms relate ξ to the integration constant C s appearing in K s , C s = 1 γ ′ (cid:18) α − α C ′ + 6 α − α Q + 2 α − α r ξ − α − α p ξ (cid:19) . (5.13)At this point, our choice of anzats (5.2) and constants γ ′ , ξ allows to reduce the differentialsystem for Φ s , Φ and the gravity to the last equation. We thus concentrate on the Friedmanequation (3.4) in the background of the critical trajectory ˙Φ = ( H /α ), (cid:18) α − α (cid:19) H = − ka + ρ + 12 e α Φ ˙Φ s + V . (5.14)The dilatation factor in front of 3 H can be absorbed in the definition of λ , ˆ k and C R , oncewe take into account eqs. (5.10), (5.12) and (5.13),3 H = 3 λ − ka + C R a , (5.15)where 3 λ = α Q , (5.16)ˆ k = α γ ′ (cid:18) α − ξ ˆ M − C ′ − Q (cid:19) , (5.17)and C R = 32 γ ′ (cid:18) ( r − p ) ξ + 2 C ′ + 13 Q (cid:19) . (5.18)We note that for Q > λ is positive. In that case, the constraint (5.13) allows us tochoose a lower bound for the arbitrary constant C s , so that ξ is large enough to have14 k >
0. This means that the theory is effectively indistinguishable with that of a universewith cosmological constant 3 λ , uniform space curvature ˆ k , and filled with a thermal bath ofradiation coupled to gravity. This can be easily seen by considering the Lagrangian p − det g (cid:20) R − λ (cid:21) (5.19)and the metric anzats (3.2), with a 3-space of constant curvature ˆ k . In the action, one cantake into account a uniform space filling bath of massless particles by adding a Lagrangiandensity proportional to 1 /a (see [14–16]) in the MSS form. One finds S eff = − | ˆ k | − Z dt N a N (cid:18) ˙ aa (cid:19) + 3 λ − ka + C R a ! , (5.20)whose variation with respect to N gives (5.15). Actually, the thermal bath interpretationis allowed as long as C R ≥
0, since the 1 /a term is an energy density. However, in thecase under consideration, the effective C R can be negative due to the m contribution of theeffective potential. The general solution of the effective MSS action of eq. (5.20) with λ > k > C R > λ = ˆ k = 0, C R > a = (cid:18) C R (cid:19) / t / , m (Φ) = Tξ = 1 γ ′ a . (5.21)Following ref. [16], the general case with λ > , ˆ k > C R > δ T = 43 λ ˆ k C R , (5.22)a first or second order phase transition can occur: δ T < ⇐⇒ st order transition , δ T > ⇐⇒ d order transition . (5.23) i) The case δ T < There are two cosmological evolutions connected by tunnel effect: a c ( t ) = N q ε + cosh ( √ λ t ) , t ∈ R , (5.24)15nd a s ( t ) = N q ε − sinh ( √ λ t ) , t i ≤ t ≤ − t i , (5.25)where N = s ˆ kλ (1 − δ T ) / , ε = 12 p − δ T − ! , t i = − √ λ arcsinh √ ε . (5.26)The “cosh”-solution corresponds to a deformation of a standard de Sitter cosmology, witha contracting phase followed at t = 0 by an expanding one. The “sinh”-solution describesa big bang with a growing up space till t = 0, followed by a contraction till a big cruncharises. The two evolutions are connected in Euclidean time by a Φ -Gravitational Instanton a E ( τ ) = N q ε + cos ( √ λ τ ) , Φ E ( τ ) = − α log (cid:16) γ ′ a E ( τ ) (cid:17) . (5.27)The cosmological scenario is thus starting with a big bang at t i = − √ λ arcsinh √ ε andexpands up to t = 0, following the “sinh”-evolution. At this time, performing the analyticcontinuation t = − i ( π/ √ λ + τ ) reaches (5.27), (where τ is chosen in the range − π/ √ λ ≤ τ ≤ τ = 0, a different analytic continuation to real time exists, τ = it , that gives riseto the inflationary phase of the “cosh”-evolution, for t ≥
0, (see fig. 1).There are thus two possible behaviors when t = 0 is reached. Either the universe carries onits “sinh”-evolution and starts to contract, or a first order transition occurs and the universeenters into the inflationary phase of the “cosh”-evolution. In that case, the scale factorjumps instantaneously from a − to a + at t = 0, a − = s ˆ k λ (cid:18) − q − δ T (cid:19) −→ a + = s ˆ k λ (cid:18) q − δ T (cid:19) . (5.28)An estimate of the transition probability is given by p ∝ e − S E eff , (5.29)where S E eff is the Euclidean action computed with the instanton solution (5.27), for τ ∈ [ − π/ √ λ, S E eff = − λ s p − δ T (cid:18) E ( u ) − (cid:18) − q − δ T (cid:19) K ( u ) (cid:19) , (5.30) It is also possible to consider the instantons associated to the ranges √ λτ ∈ [ − (2 n + 1) π/ , n ∈ N ,see [16]. a − i τ a + a s E a a cscalefactor t τ i Figure 1:
A first order phase transition can occur. The two cosmological evolutions a s and a c are connectedby an instanton a E . The universe starts with a big bang at t = t i and expands till t = 0 along the solution a s . Then, the scale factor can either contract, or jump instantaneously and enter into the inflationary phaseof a c . where K and E are the complete elliptic integrals of first and second kind, respectively, and u = 2(1 − δ T ) / q p − δ T . (5.31) ii) The case δ T > There is a cosmological solution, a ( t ) = s ˆ k λ r q δ T − √ λ t ) , t ≥ t i , (5.32)where t i = − √ λ arcsinh 1 p δ T − . (5.33)As in the previous case, it starts with a big bang. However, the behavior evolves towardthe inflationary phase in a smooth way, (see fig. 2). The transition can be associated to theinflection point arising at t inf , where a ( t inf ) = a inf , t inf = 12 √ λ arcsinh r δ T − δ T + 1 , a inf = s ˆ k δ T λ . (5.34)17 scalefactor t i tt infinf a Figure 2:
A second order phase transition occurs. The universe starts with a big bang at t = t i and evolvessmoothly toward the inflationary phase. Another solution, obtained by time reversal t → − t , describes a contracting universe that isending in a big crunch. iii) The case δ T = 1 There is a static solution, a ( t ) ≡ a where a = s ˆ k λ , (5.35)corresponding to an S universe of constant radius. This trivial behavior can be reachedfrom the cases i ) and ii ) by taking the limit δ T →
1. Beside it, there are two expandingcosmological evolutions, a < ( t ) = a p − e − √ λ t , t ≥ , (5.36)and a > ( t ) = a p e √ λ t , t ∈ R . (5.37)The first one starts with a big bang at t = 0, while the second is inflationary. Both areasymtotic to the static one, (see fig. 3). Contracting universes are described by the solutionsobtained under the transformation t → − t . 18 calefactor aa ><0 a t Figure 3:
There are two expanding cosmological evolutions. The first one, a < , starts with a big bangand converges quickly to the static solution. The second one, a > , is almost static for negative time andinflationary for positive time. We would like to consider generalizations of the previous set up. They are consisting in theinclusion of moduli fields with kinetic terms obeying different scaling properties with respectto Φ. Namely, we take into account the effects of the class of moduli with Lagrangian density p − det g g µν (cid:0) ∂ µ φ ∂ ν φ + e α Φ ∂ µ φ ∂ ν φ + e α Φ ∂ µ φ ∂ ν φ (cid:1) , (6.1)to be added to (3.1). With the metric anzats (3.2), the MSS action (3.3) is completed bythe contributions of the φ w ’s, ( w = 0 , , − | k | − Z dt a (cid:18) − N ˙ φ − N e α Φ ˙ φ − N e α Φ ˙ φ (cid:19) . (6.2)The equation of motion for Φ has now terms arising from φ and φ ,¨Φ + 3 H ˙Φ + ∂∂ Φ (cid:18) V − P − e α Φ ˙Φ s − e α Φ ˙ φ − e α Φ ˙ φ (cid:19) = 0 , (6.3)while Φ s and φ w satisfy¨Φ s + (3 H + 2 α ˙Φ) ˙Φ s = 0 , ¨ φ w + (cid:16) H + (6 − w ) α ˙Φ (cid:17) ˙ φ w = 0 . (6.4)19quations (6.4) are trivially solved, K s ≡ e α Φ ˙Φ s = C s e − α Φ a , K w ≡ e (6 − w ) α Φ ˙ φ w = C φ w e − (6 − w ) α Φ a , (6.5)where C s and the C φ w ’s are positive constants. The equivalent equations of motion for N and the scale factor a have new contributions from the kinetic terms of φ w , ( w = 0 , , K s = C s γ ′ a , K w = C φ w γ ′ (6 − w ) a w . (6.6)This implies that the new contributions arising in the Φ equation (6.3) have dimensionstwo and zero and thus do not spoil the possible identification between (6.3) and the gravityequation (3.6). In particular, the 1 /a scaling properties of φ play no role at this stage.The Φ-equation becomes˙ H + 3 H = α (cid:16) (4 C ′ + Q ) m + (2 C ′ + Q ) m + Q + ( r − p ) ξ m − C s γ ′ m − C φ γ ′ m − C φ γ ′ (cid:17) , (6.7)to be compared with eq. (5.9). The identification of the constant terms implies C ′ + 6 α C φ γ ′ = 6 α − Q , (6.8)which is an equation for γ ′ . For Q ≤
0, there is always a unique solution for γ ′ . However,it is interesting to note that for Q >
0, there is a range for C φ > γ ′ . This case is thus giving rise to two different critical trajectories. The m contributions impose k = 1 γ ′ (cid:18) α C φ γ ′ − α − C ′ + 6 α − Q (cid:19) . (6.9)This fixes the value of k for any arbitrary C φ . Finally, the equation implied by the quarticmass terms is identical to the one of the previous section, and relates ξ to C s . We repeat ithere for completeness, C s = 1 γ ′ (cid:18) α − α C ′ + 6 α − α Q + 2 α − α r ξ − α − α p ξ (cid:19) . (6.10)20e would like to stress again that for C s sufficiently large, ξ can take any value other somebound we may wish.The Friedman equation in the presence of the extra moduli becomes,3 H = − ka + ρ + 12 ˙Φ + 12 ˙ φ + 12 e α Φ ˙Φ s + 12 e α Φ ˙ φ + 12 e α Φ ˙ φ + V . (6.11)On the critical trajectory where ˙Φ = ( H /α ), and taking into account eqs. (6.8), (6.9) and(6.10), one obtains 3 H = 3 λ − ka + C R a + C M a , (6.12)where 3 λ = α (cid:0) Q − C φ γ ′ (cid:1) , (6.13)ˆ k = α γ ′ (cid:18) C φ γ ′ + 26 α − ξ ˆ M − C ′ − Q (cid:19) , (6.14) C R = 32 γ ′ (cid:18) ( r − p ) ξ + 2 C ′ + 13 Q (cid:19) , (6.15)and C M = 6 α α − C φ . (6.16)Some observations are in order: • C φ gives rise to a negative contribution to the cosmological term 3 λ . • As previously, it is possible to have C R positive if one wishes, by considering large enoughvalues for ξ . This condition can always be satisfied due to the freedom on C s in eq. (6.10).To reach positive values of ˆ k , one can either consider a large enough ξ or utilize C φ as aparameter. • C M is always positive and is determined by the modulus φ only . • Here also the system can be described by an effective MSS action similar to the oneexamined in [16], S eff = − | ˆ k | − Z dt N a N (cid:18) ˙ aa (cid:19) + 3 λ − ka + C R a + C M a ! , (6.17)whose associated Friedman equation is precisely (6.12). Thus, once taking into account thethermal and quantum corrections as well as the effects of the moduli we are consideringhere, the system admits solutions that cannot be distinguished from the de Sitter cosmology21eformed by the presence of thermal radiation and time dependent φ -moduli fields [16]; thisinterpretation is valid only when C R is positive.Assuming λ , ˆ k , C R positive and utilizing the equivalence of the effective MSS action to thethermally and moduli deformed de Sitter action studied in [16], we can immediately derivethe general solution of the system under investigation. Our results are summarized as follows(more details can be found in [16]).For convenience, we choose rescaled parameters δ T , δ M , δ T = 43 λ ˆ k C R , δ M = 94 λ ˆ k C M , (6.18)and define the domain δ T ≤ δ M ≤ r − δ T + 1 ! r − δ T − ! (6.19)in the ( δ T , δ M )-plane, as shown in fig. 4. The Friedman equation (6.12) admits solutions T δ M Figure 4:
Phase diagram in the ( δ T , δ M ) -plane. Inside the “almost triangular” domain (6.19), a first ordertransition can connect two cosmological solutions by tunnel effect. Outside the domain, there is a singleexpanding solution (and one contracting) that describes a second order transition. On the boundary, thereare two expanding (and two contracting) solutions, beside a static one. that involve a first order phase transition inside this domain and a second order one outside22t. It is convenient to express these conditions in terms of κ , the only real positive root ofthe polynomial equation κ + κ + δ T κ − δ M = 0 , (6.20)by defining ∆ ≡ κ + 4 κ + δ T (1 + κ ) = 1627 δ M κ (1 + κ ) , (6.21)where the second equality in eq. (6.21) is just a consequence of eq. (6.20):∆ < ⇐⇒ st order transition , ∆ > ⇐⇒ d order transition . (6.22) i) The case ∆ < There are two cosmological cosmological solutions connected via tunneling. The first onetakes a simple form in terms of a function ˜ t ( t ), a c (˜ t ( t )) = N q ε + cosh ( √ λ ˜ t ( t )) , (6.23)where N = s ˆ k (1 + κ ) λ (1 − ∆) / , ε = 12 (cid:18) √ − ∆ − (cid:19) , (6.24)and ˜ t ( t ) is found by inverting the definition of t as a function of ˜ t , t = Z ˜ t dv s cosh ( √ λv ) + ε cosh ( √ λv ) + ε + ˜ ε where ˜ ε = κ (1 + κ ) √ − ∆ . (6.25)In a c , the variables t and ˜ t are arbitrary in R . The second cosmological evolution is a s (˜ t ( t )) = N q ε − sinh ( √ λ ˜ t ( t )) where t = − Z t dv s ε − sinh ( √ λv ) ε + ˜ ε − sinh ( √ λv ) , (6.26)with the range of time˜ t i ≡ − √ λ arcsinh √ ε ≤ ˜ t ≤ − ˜ t i i.e. t i = − Z t i dv s ε − sinh ( √ λv ) ε + ˜ ε − sinh ( √ λv ) ≤ t ≤ − t i . (6.27)As in the previous section, the “cosh”-solution describes a contracting phase followed by anexpanding one and approaches a standard de Sitter cosmology for positive or negative largetimes. The “sinh”-solution starts with a big bang, ends with a big crunch, while the scale23actor reaches its maximum at t = 0. The two cosmological solutions are connected by aΦ -Gravitational Instanton a E (˜ τ ( τ )) = N q ε + cos ( √ λ ˜ τ ( τ )) , Φ E (˜ τ ( τ )) = − α log (cid:16) γ ′ a E (˜ τ ( τ )) (cid:17) , (6.28)where ˜ τ ( τ ) is the inverse function of τ = − Z τ dv s cos ( √ λv ) + ε cos ( √ λv ) + ε + ˜ ε , (6.29)and the range of Euclidean time is − π √ λ ≤ ˜ τ ≤ i.e. τ i ≤ τ ≤ τ i = − Z − π √ λ dv s cos ( √ λv ) + ε cos ( √ λv ) + ε + ˜ ε . (6.30)The cosmological scenario starts with an initial singularity at t i and follows the “sinh”-expansion till t = 0. At this time, the solution can be analytically continued to the instan-tonic one by choosing ˜ t = − i ( π/ √ λ + ˜ τ ) i.e. t = − i ( − τ i + τ ). When the Euclidean time τ = 0 is reached, a second analytic continuation to real time, ˜ τ = i ˜ t i.e. τ = it , gives riseto the inflationary phase of the “cosh”-solution, for later times t ≥
0, (see fig. 1). At t = 0,the universe has thus two different possible behaviors. It can carry on its evolution alongthe “sinh”-solution i.e. enter in a phase of contraction. Or, a first order phase transitionoccurs and the trajectory switches to the “cosh”-evolution. In that case, the scale factorjumps instantaneously from a − to a + , a − = s ˆ k (1 + κ )2 λ (cid:16) − √ − ∆ (cid:17) −→ a + = s ˆ k (1 + κ )2 λ (cid:16) √ − ∆ (cid:17) . (6.31)The transition probability is controlled by the Euclidean action, p ∝ e − S E eff , where S E eff hasbeen computed in [16]. For √ λ ˜ τ ∈ [ − π/ , S E eff = − / λ (cid:0) − δ T (cid:1) / s sin (cid:18) θ + π (cid:19) E ( u ) − p − δ T cos (cid:0) θ (cid:1) − δ T √ p − δ T sin (cid:0) θ + π (cid:1) K ( u ) ! , (6.32)where u = s sin (cid:0) θ (cid:1) sin (cid:0) θ + π (cid:1) , θ = arccos δ M + 9 δ T − − δ T ) / ! . (6.33) Actually, one can also consider the instantons associated to the ranges √ λ ˜ τ ∈ [ − (2 n + 1) π/ , n ∈ N ,see [16].
24e have displayed the scale factors a c and a s in terms of δ T and δ M . However, it is interestingto express the solutions with more intuitive quantities, namely the temperatures T + at t = 0 + (along the “cosh”-evolution) and T − at t = 0 − (along the “sinh”-evolution). Using the factthat a ± T ± = ξ/γ ′ , one finds T ± ( δ T , δ M ) = T m √ δ T q (1 + κ ) (cid:0) ± √ − ∆ (cid:1) where T m = ξγ ′ s λ ˆ kδ T . (6.34)It is shown in [16] that the cosmological solutions can be written as a c (˜ t ( t )) = p ˆ k p λ (1 − A ) s T − − T T − + T cosh(2 √ λ ˜ t ( t )) , t > a s (˜ t ( t )) = p ˆ k p λ (1 − A ) s T − T − T + T − cosh(2 √ λ ˜ t ( t )) , t < , (6.36)where we have defined A = 1 − T T − /T m ( T + /T − + T − /T + ) . (6.37)We note that there is a temperature duality T + ↔ T − that switches the two cosmologicalsolutions a c (˜ t ( t )) for t > a s (˜ t ( t )) for t < T + ↔ T − ) ⇐⇒ (cid:16) a c (˜ t ( t )) for t > ↔ a s (˜ t ( t )) for t < (cid:17) . (6.38)Once quantum and thermal corrections are taken into account, the no-scale supergravities weare considering share a common effective behavior with the thermally and moduli deformedde Sitter evolution (see eq. (6.12)). This means that the temperature T m can be definedin both contexts. In eq. (6.34), T m is expressed in terms of critical trajectory quantities.However, one can consider the effective 1 /a -radiation energy density in eq. (6.12) to definethe “would-be number” of massless bosonic (fermionic) degrees of freedom, n B ( F ) eff , of theequivalent deformed de Sitter point of view: ρ R ≡ C R a ≡ π (cid:18) n B eff + 78 n F eff (cid:19) T , (6.39)25here we have applied the relations (4.9) and (4.10) for massless states. Using eq. (6.39)and the fact that aT ≡ a ± T ± , one can rewrite T m as T m = (cid:18) π λn B eff + n F eff (cid:19) / . (6.40) ii) The case ∆ > There is an expanding solution we briefly describe (a contracting one is obtained by timereversal t → − t ), a (˜ t ( t )) = s ˆ k (1 + κ )2 λ q √ ∆ − √ λ ˜ t ( t )) , (6.41)where t as a function of ˜ t is given by t = Z ˜ t dv vuut √ ∆ − √ λv ) + 1 √ ∆ − √ λv ) + 1 + κ κ (6.42)and we consider˜ t ≥ ˜ t i ≡ − √ λ arcsinh 1 √ ∆ − i.e. t ≥ t i ≡ − Z t i dv vuut √ ∆ − √ λv ) + 1 √ ∆ − √ λv ) + 1 + κ κ . (6.43)This cosmological solution describes a smooth evolution from a big bang to an inflationaryera, (see fig. 2). It has a single inflection point arising when a = a inf , which is defined by thefollowing condition, λa inf ˆ k > x − δ T x − δ M = 0 . (6.44) iii) The case ∆ = 1 Beside the following static solution, a ( t ) ≡ a where a = s ˆ k (1 + κ )2 λ ≡ vuut ˆ k λ r − δ T ! , (6.45)the two expanding cosmological evolutions we found for vanishing δ M are generalized by, a < (˜ t ( t )) = a p − e − √ λ ˜ t ( t ) where t = Z ˜ t dv vuut − e − √ λv κ ′ κ ′ − e − √ λv , (6.46)26or ˜ t ≥ i.e. t ≥
0, and a > (˜ t ( t )) = a p e √ λ ˜ t ( t ) where t = Z ˜ t dv vuut e √ λv κ ′ κ ′ + e √ λv , (6.47)for arbitrary ˜ t and t . They are monotonically increasing as in the pure thermal case: the firstone starts with a big bang and the second one is inflationary, (see fig. 3). Two other solutionsobtained under t → − t are decreasing. These time-dependent solutions are asymtptic to thestatic one.Before concluding let us signal that the number ˆ n B ( F )0 of states with Φ-independent massesbelow T ( t ) is not strictly speaking a constant. It is actually lowered by one unit each timethe temperature T ( t ) passes below the mass threshold m ˆ ı of a boson (fermion). Our criticalsolutions for the scale factor are thus well defined in any range of time where ˆ n B and ˆ n F are constant. The full cosmological scenario is then obtained by gluing one after anotherthese ranges. Each time a mass threshold is passed, the values of ˆ n B ( F )0 , ˆ M , r and p decrease (see eqs. (4.19), (5.6) and (5.7)), and the parameters of the critical trajectory haveto be evaluated again. However, the constraint (6.10) implies that (cid:0) ˆ n B + ˆ n F (cid:1) ξ remainsconstant , due to the fact that C s (and any other C φ w ) is constant along the full cosmologicalevolution. This implies that C R defined in eq. (6.15) is also invariant. However, the term ξ ˆ M and thus ˆ k can decrease (see eq. (6.14)). The positivity of ˆ k can nevertheless beguaranteed by the modulus term C φ γ . Actually, this procedure that is consisting in gluingtime intervals as soon as an energy threshold is reached is identical to what is assumed inStandard Cosmology. In the latter case, the full time evolution is divided in different phases(e.g. radiation dominated, matter dominated, and so on). At the classical string level, it is well known that it is difficult to construct exact cosmologicalstring solutions. It is even more difficult to obtain de Sitter like inflationary evolutions, evenin less than four dimensions.The main difficulty comes from the fluxes and torsion terms which are created via non-trivial27eld strength (kinetic terms) and have the tendency to provide negative contributions to thecosmological term, thus anti de Sitter like vacua. To illustrate a relative issue, considerfor instance the Euclidean version of the de Sitter space in three dimensions, dS , which isnothing but the 3-sphere S . Although S can be represented by an exact conformal fieldtheory based on an SU (2) k WZW model, the latter does not admit any analytic continuationto real time. This is due to the existence of a non-trivial torsion H µνρ that becomes imaginaryunder an analytic continuation [20–22].This obstruction in string and M-theory is generic and follows from the kinetic origin ofthe flux terms. A way to bypass this fact is to take into account higher derivative terms[23]. Another strategy is to assume non-trivial effective fluxes coming from negative tensionobjects such as orientifolds. This idea was explored in the field theory approximation inref. [24]. To go further, it is necessary to work with string cosmological backgrounds basedon exact conformal field theories. However, the only known exact cosmological solutionwithout the torsion problem described above is that of SL (2 , R ) /U (1) −| k | × K , [25]. ItsEuclidean version is also well defined by the parafermionic T-fold [26, 27].In this work we have implemented a more revolutionary approach. We start with a classicalsuperstring background with spontaneously broken N = 1 supersymmetry defined on aflat space-time. The effective field theories associated to these cases are nothing but the N = 1 string induced “no-scale supergravity models”. Working at the field theory level, wehave shown that the quantum and thermal corrections create dynamically universal effectivepotential terms that give rise to non-trivial cosmological accelerating solutions.The main ingredients we have used are the scaling properties of the effective potential atfinite temperature in the “no-scale N = 1 spontaneously broken supergravities”, once thebackgrounds follow critical trajectories where all fundamental scales have a similar evolutionin time. Namely, the supersymmetry breaking scale m (Φ), the inverse of the scale factor a ,the temperature T an the infrared scale µ remain proportional to each other: e α Φ ≡ m (Φ) = 1 γ ′ a = Tξ = γγ ′ µ . The “no-scale modulus Φ” is very special in the sense that it is the superpartner of thegoldstino and couples to the trace of the energy momentum of a sub-sector of the theory.28t also provides non-trivial dependences in the kinetic terms of other special moduli of thetype: K w ≡ − e (6 − w ) α Φ ( ∂φ w ) , ( w = 0 , , , , where φ ≡ Φ s in the text. The quantum and thermal corrections, together with the non-trivial motion of the special moduli, allow to find thermally and moduli deformed de Sitterevolutions. The cosmological term 3 λ ( am ), the curvature term ˆ k ( am, aT ) and the radiationterm C R ( am, aT ) (see eqs. (5.15) or (6.12)), are dynamically generated in a controllable way and are effectively constant. Obviously, as stated in the introduction, these solutions arevalid below Hagedorn-like scales associated to the temperature as well as the supersymmetrybreaking scale m , where instabilities would occur in the extension of our work in a stringyframework. These restrictions on m are supported by the analysis of the string theory ex-amples considered in ref. [8].When the deformation of the de Sitter evolution is below some critical value, there are twocosmological solutions which are connected by tunnel effect and interchanged under a tem-perature duality. The first one describes a big bang with a growing up space till t = 0,followed by a contraction that ends with a big crunch. The second corresponds to a defor-mation of a standard de Sitter evolution, with a contracting phase followed at t = 0 by anexpanding one. The universe starts on the big bang cosmological branch and expands upto t = 0 along the first solution. At this time, two distinct behaviors can occur. Eitherthe universe starts to contract, or a first order phase transition arises via a Φ-gravitationalinstanton, toward the inflationary phase of the deformed de Sitter evolution. The transitionprobability p can be estimated.If on the contrary the induced cosmology corresponds to a de Sitter-like universe with de-formation above the critical value, the previous big bang and inflationary behaviors aresmoothly connected via a second order phase transition.It is of main importance that the field theory approach we developed here can easily beadapted at the string level [8], following the recent progress in understanding the stringywave-function of the universe [27, 28]. This will permit us to go beyond the Hagedorn tem-perature and understand better the very early “stringy” phase of our universe [29]. At this29oint, we insist again on a highly interesting question that can be raised concerning the com-mon wisdom which states that all radii-like moduli should be large to avoid Hagedorn-likeinstabilities. If this statement was true, then the quantum and thermal corrections shouldbe considered in a 10 rather than in a 4 dimensional picture. However, the recent results ofref. [8] where explicit string models are considered show that this is only valid for the radii-moduli which are participating to the supersymmetry breaking mechanism. The remainingones are free from any Hagedorn-like instabilities and can take very small values, even of theorder of the string scale. Acknowledgements
We are grateful to Constantin Bachas, Ioannis Bakas, Adel Bilal, Gary Gibbons, John Il-iopoulos, Jan Troost and Nicolas Toumbas for discussions.The work of C.K. and H.P. is partially supported by the EU contract MRTN-CT-2004-005104 and the ANR (CNRS-USAR) contract 05-BLAN-0079-01. C.K. is also supportedby the UE contract MRTN-CT-2004-512194, while H.P. is supported by the UE contractsMRTN-CT-2004-503369 and MEXT-CT-2003-509661, INTAS grant 03-51-6346, and CNRSPICS 2530, 3059 and 3747.
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