Attraction to a radiation-like era in early superstring cosmologies
CCPHT–RR089.1208, LPTENS–08/64, February 2009
Attraction to a radiation-like erain early superstring cosmologies ∗ Fran¸cois Bourliot , Costas Kounnas and Herv´e Partouche Centre de Physique Th´eorique, Ecole Polytechnique, † F–91128 Palaiseau cedex, France
[email protected]@cpht.polytechnique.fr Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure, ‡
24 rue Lhomond, F–75231 Paris cedex 05, France
Abstract
Starting from an initial classical four dimensional flat background of the heterotic or type IIsuperstrings, we are able to determine at the string one-loop level the quantum correctionsto the effective potential due to the spontaneous breaking of supersymmetry by “geometricalfluxes”. Furthermore, considering a gas of strings at finite temperature, the full “effectivethermal potential” is determined, giving rise to an effective non-trivial pressure. The back-reaction of the quantum and thermal corrections to the space-time metric as well as to themoduli fields induces a cosmological evolution that depends on the early time initial condi-tions and the number of spontaneously broken supersymmetries. We show that for a wholeset of initial conditions, the cosmological solutions converge at late times to two qualita-tively different trajectories: They are either attracted to (i) a thermal evolution similar toa radiation dominated cosmology, implemented by a coherent motion of some moduli fields,or to (ii) a “Big Crunch” non-thermal cosmological evolution dominated by the non-thermalpart of the effective potential or the moduli kinetic energy. During the attraction to theradiation-like era, periods of accelerated cosmology can occur. However, they do not giverise to enough inflation ( e -fold (cid:39) .
2) for the models we consider, where
N ≥ N = 0. ∗ Research partially supported by the European ERC Advanced Grant 226371, ANR (CNRS-USAR) contract 05-BLAN-0079-02, and CNRS PICS contracts 3747 and 4172. † Unit´e mixte du CNRS et de l’Ecole Polytechnique, UMR 7644. ‡ Unit´e mixte du CNRS et de l’Ecole Normale Sup´erieure associ´ee `a l’Universit´e Pierre etMarie Curie (Paris 6), UMR 8549. a r X i v : . [ h e p - t h ] M a r Introduction
A sensible theoretical description of cosmology has certainly to take into account both thequantum and thermal effects. It is naturally the case for an early phase of the history,close to the Planck time, where quantum fluctuations are of same order of magnitude as thewhole size of the universe and where the temperature is extremely high. It has to be thecase as well at very late times, where the description of the cosmology should involve theelectroweak phase transition and provide the missing link to the quantum particle physicsof the standard model (or its supersymmetric stringy extension), account for realistic largescale structure formations, as well as describe correct temperature properties of the cosmicmicrowave background.In the framework of strings, the theory of quantum gravity is well defined [1] (at leastin certain cases) and can be considered at finite temperature as well [2–4]. Thus, in thestringy framework it is natural to attempt to describe the cosmological evolution of our uni-verse. In the very early times and when the temperature approaches the so called Hagedorntemperature [2–5], we are faced to non-trivial stringy singularities indicating a high temper-ature non-trivial phase transition. In the literature, there are many speculative proposalsconcerning the nature of this transition [2, 4, 6–8].Conceptually, it is quite exotic to understand the early stringy phase utilizing the stan-dard field theoretical geometrical notions [8–11]. Indeed, following for instance Refs [9, 11],the notions of geometry and even topology break down in this very early-time era. Onlyby considering stringy approaches based on 2-dimensional conformal field theory techniques,there is a hope to analyze further this phase. Actually, some conceptual progress in thisdirection has been achieved recently [11], but we are not yet in a position to give quantita-tive descriptions of our stringy universe at these very early times. This stringy-conceptualobstruction however is not an obstacle to the study of the cosmological evolution of ouruniverse for much later times than the Hagedorn transition era, namely for late times suchthat t (cid:29) t H , [12–14].A way to bypass the Hagedorn transition ambiguities consists in assuming the emergenceof three large space-like directions for t (cid:29) t H , describing the three dimensional space of ouruniverse, and possibly some internal space directions of an intermediate size, characterizing1he scale of the spontaneous breaking of supersymmetry, via geometrical fluxes, or even other(gauge or else) symmetry breaking scales [15]. Within this assumptions, the ambiguities ofthe “Hagedorn transition exit at t E ” can be parametrized, for t ≥ t E , in terms of initialboundary condition data at t E . This is precisely the scope of the present work, were wefocus on the intermediate cosmological era: t E ≤ t ≤ t w , namely, after the “Hagedorntransition exit” and before the electroweak symmetry breaking phase transition at t w .Modulo the initial boundary condition data at t E , the stringy description of the in-termediate cosmological era, turns out to be under control, at least for the string back-grounds where supersymmetry is spontaneously broken by “geometrical fluxes” [16, 17].These fluxes are implemented by a “stringy Scherk-Schwarz mechanism” involving n -internalcompactified dimensions which are coupled non-trivially to some of the R-(super-)symmetriccharges [18–20] [8, 13, 14]. Furthermore, the finite temperature effects are also implementedby a stringy Scherk-Schwarz compactification attached to the Euclidean time circle of radius β = 1 /T [2–4, 8, 13, 14]. Here, the R-symmetry charge is just the helicity, so that space-timebosons are periodic while fermions are anti-periodic along the S Euclidean time circle.The breaking of supersymmetry and the quantum canonical ensemble at temperature T give rise to a non-trivial free energy at the one-loop string level. The latter is evaluated ina large regularized 3-space volume V . This amounts to considering the free energy of a gasof approximately free strings, F = − ln Zβ (cid:39) − Z (string)1-loop β , (1.1)where Z = Tr e − βH is the partition function at finite temperature and Z (string)1-loop is the one-loop connected graph i.e. genus one vacuum to vacuum string amplitude computed in theEuclidean background. In these equalities, we use the fact that the free energy of an infinitenumber of degrees of freedom can be found by a single string loop computation [21] [12–14].The pressure derived from the free energy, P = − ∂F/∂V , has been computed for heteroticand type II models . Supersymmetry is spontaneously broken by introducing geometricalfluxes `a la Scherk-Schwarz involving n internal directions and by finite temperature effects.The pressure acts as a source occurring at 1-loop that backreacts on the original classicalspace-time metric and moduli fields background. In some cases, the perturbed equations of Type I models realized as orientifolds of type IIB can also be considered. They can be dual to heteroticones with an initially N = 4, 2 or even 1 supersymmetry [22]. T ( t ), the supersymmetry breaking scale M ( t ) and the inverse scalefactor 1 /a ( t ) remain proportional [12–14, 23]: Radiation Dominated Solution : T ( t ) ∝ M ( t ) ∝ /a ( t ) and H ≡ (cid:18) ˙ aa (cid:19) ∝ a . (1.2)Very different time evolutions also exist when temperature effects are neglected [14]. In thiscase, the supersymmetry breaking scale evolves like: Moduli Dominated Solution : M ( t ) ∝ /a ( t ) /n with T ( t ) (cid:28) M ( t ) and H ∝ a n +2) /n . (1.3)In this work we would like to examine the late time evolution of the universe in theintermediate cosmological era, t E ≤ t ≤ t w , in terms of initial boundary conditions (IBC)after the “Hagedorn transition exit at t E ”. We show that the Radiation Dominated Solution(RDS) and the Moduli Dominated Solution (MDS) obtained for particular IBC are actuallygeneric. For instance, for n = 1, there is a whole basin of IBC such that the resultingcosmological evolutions converge – depending on the model – either to the RDS or to theMDS. The first attractor corresponds to an expanding evolution controlled by a Friedmannequation H ∝ /a , where H is the Hubble parameter. The second one describes a BigCrunch behavior dominated by the non-thermal radiative corrections. In addition, all modelsadmit a second basin of IBC that yields to another Big Crunch behavior, dominated by themoduli kinetic energy i.e. where H ∝ /a . Within the scope of our present work, theBig Crunch evolutions can be trusted as long as Hagedorn like singularities are not reached.Our analysis is also limited to the cases where the initial supersymmetry is higher or equalto two, N ≥
2. Although the N = 1 case is quite similar in some respects, it is much moreinvolved in issues like the appearance of an effective cosmological term. It will be analyzedelsewhere.Restricting to cases with N ≥ e -fold number of order 0 .
2, which is too low to account for realisticastrophysical observations. It would be very interesting to extend our analysis to models3ith N = 1 initial supersymmetry and see if the e -fold number is larger enough for aninflation era to occur.In Section 2, the case of a supersymmetry breaking using a single internal direction ( i.e. n = 1) is studied in details. We classify the models and find local basins of attraction.This analysis is completed by a more general exploration of the set of IBC by numericalsimulations. In Section 3, the attraction mechanism to the RDS is extended to modelswhere n = 2 internal directions participate in the supersymmetry breaking. We determinein Section 4 the e -fold number of the periods of acceleration. Our results, their limitationsand our perspectives are summarized in Section 5. n = 1 internal dimension In this section, we summarize first the results and notations introduced in Ref. [13] andthen we analyze the attractor mechanism. The starting background can be the heterotic,type I or type II superstring compactified on a 6-dimensional space M that can be either S ( R ) × T or S ( R ) × S × K
3. The sizes of T and S × K . The supersymmetry breaking is introduced by “geometrical fluxes” via thecoupling of the momentum and winding numbers of the Scherk-Schwarz circle S ( R ) to anR-symmetry charge [3, 13, 14,18–20]. The radius of the circle R controls the supersymmetrybreaking scale M . In all applications, we will replace K K ∼ T / Z .In the heterotic and type I cases initially N = 4 or N = 2, the geometrical fluxesbreak spontaneously all the supersymmetries, N →
0. In type II, when the S ( R ) couplinginvolves non-trivially left- and right- moving R-symmetry charges, all initial N = 8 , N = 4 , or N = 2 supersymmetries are spontaneously broken. When the S ( R ) coupling is left-right- “asymmetric” i.e. involves non-trivially a left- (or right-) moving R-symmetry chargeonly, half of the supersymmetries still survive: N → N / We relax this hypothesis in Ref. [15]. .1 General setup, gravity-moduli equations In all cases the computation of the free energy involves for the heterotic or type II superstringthe background: S ( R ) × T × S ( R ) × M with either M = T or M = S × T / Z . (2.1)The T factor stands for the external space with regularized volume V sent to infinity oncethe thermodynamical limit is taken. S ( R ) is the Euclidean time circle of period β = 2 πR that defines the inverse temperature (in the string frame). In this direction, the Scherk-Schwarz circle amounts to coupling the momentum and winding numbers of the Euclideantime circle to the space-time fermion number [2–4, 8, 13, 14]. The computation of the freeenergy density is well defined if the moduli R and R responsible for the spontaneoussupersymmetry breaking are larger than the Hagedorn radius R H , which is of order 1 instring units. At the Hagedorn radius, some winding modes become tachyonic so that thefree energy develops a severe IR divergence [2–4, 8, 13, 14]. If our goal is not to describethe phase transition that is occurring at the Hagedorn radius, we can suppose that in theintermediate cosmological era, R and R are large. In Refs [13, 14], it is shown that up toexponentially small terms of order e − πR or e − πR , the pressure derived from the free energydensity takes the following form [12–14, 23], P = T p ( z ) where e z = MT , (2.2)with the temperature T and supersymmetry breaking scale M defined as, T = 12 πR √ Re S , M = 12 πR √ Re S . (2.3)In these expressions, the factors involving the dilaton in 4 dimensions, Re S = e − φ D , arisewhen we work in Einstein frame. In (2.2), p is a linear sum of functions of z with integercoefficients n T and n V [13, 14], p ( z ) = n T f ( z ) + n V ˜ f ( z ) , (2.4)where f ( z ) = Γ(5 / π / (cid:88) m ,m e z [(2 m + 1) e z + (2 m ) ] / , ˜ f ( z ) = e z f ( − z ) , (2.5)5nd the sums are over the integers in Z . n T is the number of massless states of the model. Ifthe Scherk-Schwarz circle S ( R ) acts non-trivially on the left- and right- moving R-charges, n V satisfies the inequality: − ≤ n V n T ≤ , (2.6)and depends on the choice of R-symmetries used to spontaneously break all the supersymme-tries [13, 14]. For an asymmetric S ( R ) coupling i.e. trivial on the left- (or right-) movingR-symmetry charges, the supersymmetry breaking is partial and thus, there is no contribu-tion ˜ f to p . In that case, M is the mass of the N / n V = 0 for asymmetric R-symmetrycoupling to the momenta and windings of the 4 th internal direction.From a low energy effective field theory point of view, the pressure P is minus the effectivepotential at finite temperature. It is a source for the quantum fields, namely the dilaton φ D and the modulus R , coupled to gravity [12–14, 23]. It also introduces the “thermal field” R to which there is no associated quantum fluctuation, as can be seen from the string modelin light-cone gauge where oscillator modes in the direction 0 are gauged away. The source P is a 1-loop correction that backreacts on the classical 4-dimensional Lorentzian background,via the effective field theory action, S = (cid:90) d x √− g (cid:20) R − g µν (cid:18) ∂ µ φ D ∂ ν φ D + 12 ∂ µ R ∂ ν R R (cid:19) + P (cid:21) , (2.7)where we write explicitly the fields with non-trivial backgrounds only. Note that computingthe 1-loop correction to the kinetic terms is not needed. Indeed, the wave function renor-malization of the moduli, (to recast the kinetic terms at 1-loop in a canonical form), wouldintroduce an additional correction to the pressure at second order only. Thus, the full lowenergy dynamics at first order in string perturbation theory is described by the above action,(2.7).Before writing the equations of motion, it is useful to redefine the fields, φ := (cid:114)
23 ( φ D − ln R ) , φ ⊥ := 1 √ φ D + ln R ) , (2.8)so that the action becomes: S = (cid:90) d x √− g (cid:20) R − (cid:0) ( ∂φ ) + ( ∂φ ⊥ ) (cid:1) + P (cid:21) , (2.9)6here the pressure and the supersymmetry breaking scale M take the form, P = M e − z p ( z ) , e z = MT , M = e αφ π with α = (cid:114) . (2.10)This shows that in the 2-dimensional moduli space ( φ D , R ), the effective potential liftsthe “no-scale modulus” φ , while keeping flat the orthogonal direction φ ⊥ . Assuming anhomogeneous and isotropic cosmological metric with a flat 3-dimensional subspace, ds = − N ( t ) dt + a ( t ) (cid:0) dx + dx + dx (cid:1) , (2.11)the Friedmann equation in the N ( t ) = 1 gauge takes the form,3 H = 12 ˙ φ + 12 ˙ φ ⊥ + ρ , (2.12)where the energy density ρ is defined by the identity, ρ + P = T ∂P∂T , (2.13)and P is understood as a function of the independent variables T and M . As shown inRef. [14], the expression of ρ follows from the variational principle applied to a gauge choice ofthe function N = 1 /T . It is fundamental to stress here that the energy density defined by thevariational principle is identical to the one derived from the second law of thermodynamics.From Eqs (2.2) and (2.13), it follows that: ρ = T r ( z ) where r = 3 p − p z , (2.14)where z -indices stand for derivatives with respect to z = ln( M/T ). Varying the action withrespect to the scale factor a or the scalar fields φ and φ ⊥ , we obtain:2 ˙ H + 3 H = −
12 ˙ φ −
12 ˙ φ ⊥ − P , (2.15)¨ φ + 3 H ˙ φ = ∂P∂φ ≡ α (3 P − ρ ) , (2.16)¨ φ ⊥ + 3 H ˙ φ ⊥ = 0 = ⇒ ˙ φ ⊥ = √ c ⊥ a , (2.17)where c ⊥ is an arbitrary integration constant. Note that the last identity in Eq. (2.16) is aconsequence of the fact that P is a dimension four object, ( M ∂ M + T ∂ T ) P ≡ P , and fromthe definitions (2.10) and (2.13). The conservation of the total energy-momentum tensorfollows from the Einstein equations (2.12) and (2.15): ddt (cid:18)
12 ˙ φ + 12 ˙ φ ⊥ + ρ (cid:19) + 3 H (cid:16) ˙ φ + ˙ φ ⊥ + ρ + P (cid:17) = 0 . (2.18)7sing further the equations of the fields φ and φ ⊥ , we obtain from Eq. (2.18) that the { ρ, P } -system is not isolated but is coupled non-trivially to the “no-scale modulus field”, φ [24, 25]: ˙ ρ + 3 H ( ρ + P ) + α ˙ φ (3 P − ρ ) = 0 . (2.19) Our aim is to analyze the time behavior of the system described above. For this purpose,one can choose four independent non-linear equations, for instance: Eqs (2.12), (2.16), (2.17)and (2.19), any other choice being equivalent. Furthermore, we find convenient to expressthe dependence in time in terms of a dependence in the logarithm of scale factor a ( t ): λ := ln a ( t ) = ⇒ ˙ y ≡ dydt = H dydλ ≡ H ◦ y , (2.20)for any function y . In particular, the definition z = αφ − ln(2 πT ) implies ˙ T /T = H ( α ◦ φ − ◦ z )that can be used to write ˙ ρ = HT (cid:18) rα ◦ φ + ( r z − r ) ◦ z (cid:19) . (2.21)This equality is useful to express the conservation of energy-momentum (2.19) into a relationgiving ◦ φ in terms of z and ◦ z , α ◦ φ = A z ( z ) ◦ z − A z ( z ) = 4 r − r z r + p ) . (2.22)Integrating, one obtains M = e A ( z ) a , (2.23)where A ( z ) involves an integration constant. Using the above relations, the Friedmann Eq.(2.12) becomes 3 H = 38 H (cid:16) ( A z ( z ) ◦ z − − (cid:17) + 98 e A ( z ) − z ) a r ( z ) + 98 c ⊥ a , (2.24)or, H = T r ( z )3 − K ( z, ◦ z, ◦ φ ⊥ ) with K = 12 α (cid:16) A z ( z ) ◦ z − (cid:17) + 12 ◦ φ ⊥ . (2.25)8he equations for φ and φ ⊥ , (2.16) and (2.17), take the form, r ( z )3 − K ( z, ◦ z, ◦ φ ⊥ ) (cid:16) A z ( z ) ◦◦ z + A zz ( z ) ◦ z (cid:17) + r ( z ) − p ( z )2 A z ( z ) ◦ z + V z ( z ) = 0 r ( z )3 − K ( z, ◦ z, ◦ φ ⊥ ) ◦◦ φ ⊥ + r ( z ) − p ( z )2 ◦ φ ⊥ = 0 (2.26)For the derivation of the above equations, we used the relation ¨ y = ˙ H ◦ y + H ◦◦ y for y = φ or φ ⊥ as well as the Eqs (2.22), (2.25) and the linear sum of (2.12) and (2.15). As in Ref. [23],we have introduced the notion of an effective potential V ( z ), defined by its derivative (andusing α = 3 / V z ( z ) = r ( z ) − p ( z ) . (2.27)We are going to see in the following section that any solution of the system with constant z , i.e. z ≡ z c where z c is an extremum of V ( z ), is giving rise to a particular RDS. Theexistence of such a critical point depends drastically on the shape of the potential V ( z ),which is determined by its asymptotic behaviors for z → ±∞ . It is obvious from Eq. (2.4)and the definition (2.13) of ρ that V ( z ) depends only on two parameters, namely, n T and n V . The asymptotic behavior z → ±∞ can be easily derived: • ( n V / n T ) (cid:54) = V ( z ) ∼ z → + ∞ − e z (cid:18) n V n T (cid:19) (cid:18) n T c o (cid:19) , with c o = Γ(5 / π / (cid:88) m | m + 1 | , (2.28) • ( n V / n T ) (cid:54) = -1/15 ≡ − (cid:16)(cid:80) (cid:48) m m ) (cid:17) (cid:46)(cid:16)(cid:80) m m +1) (cid:17) , V ( z ) ∼ z →−∞ − e z (cid:18) n V n T + 115 (cid:19) (cid:18) n T c o (cid:19) , with c o = Γ(5 / π / (cid:88) m m + 1) . (2.29)According to the ratio n V /n T , three distinct cases can actually arise, (see Fig. 1), • Case (a) : -1 ≤ ( n V / n T ) ≤ -1/15 V ( z ) increases monotonically from 0 to + ∞ . • Case (b) : -1/15 < ( n V / n T ) < V ( z ) admits a unique minimum z c , with p ( z c ) positive. When ( n V /n T ) → − thecritical value z c goes to + ∞ ; when ( n V /n T ) → ( − / + the critical value z c → −∞ .9 Case (c) : ≤ ( n V / n T ) ≤ V ( z ) decreases monotonically from 0 to −∞ . z V Case (a) z V Case (b) c z z V Case (c)
Figure 1:
The shape of the potential V ( z ) in Eq. (2.26) depends on the ratio n V /n T . The Cases (a), (b)and (c) correspond to − ≤ n V /n T ≤ − / , − / < n V /n T < and ≤ n V /n T ≤ , respectively. Qualitatively, one can expect that the system represented by z can slide along its potential.It may run away in Case (a) , z ( t ) → −∞ , be stabilized in Case (b) , z ( t ) → z c , and runaway in Case (c) , z ( t ) → + ∞ . Moreover, these behaviors should imply new ones by timereversal. In particular, other run away behaviors are expected: z ( t ) → + ∞ in Case (a) , z ( t ) → ±∞ in Case (b) , and z ( t ) → −∞ in Case (c) . However, the study of the run away z → −∞ is out of the scope of the present work. This is due to the fact that this limitamounts to R (cid:29) R >
1, so that the thermal system should be studied in 5 dimensions. Inthe following, we will only consider the other behaviors, namely the stabilization of z andthe run away z → + ∞ . Since the latter amounts to R (cid:29) R >
1, it is consistent with ouranalysis in four dimensions but with negligible thermal effects.
Clearly, the first equation in (2.26) admits a constant z solution whenever V has an ex-tremum, i.e. in Case (b) . It follows from Eq. (2.23) that M , T and 1 /a are propotionnalduring the evolution, M ( t ) = e z c T ( t ) = e A ( z c ) a ( t ) . (2.30)The time dependence of the scale factor is dictated by the Friedmann Eq. (2.24) thatsimplifies to:3 H = c r a + c m a where c r = 92 e A ( z c ) − z c ) p ( z c ) , c m = 98 c ⊥ . (2.31)10ntegrating, t can be expressed in terms of the scale factor, t ( a ) = t (cid:90) a/a x dx √ x , t = √ c m | c r | , a = (cid:12)(cid:12)(cid:12)(cid:12) c m c r (cid:12)(cid:12)(cid:12)(cid:12) , ∀ a ≥ , (2.32)up to an overall sign. There are thus two monotonic solutions mapped to one another bytime reversal. For the expanding one, the universe life starts in a “moduli kinetic energydominated era” characterized by a total energy density (cid:39) c m /a , followed by a “radiationdominated era” with total energy density (cid:39) c r /a . For large enough times, the evolutionalways enters in the radiation era: It is asymptotic to the critical solution z ≡ z c , c m = 0 i.e. (2.30) where a ( t ) = √ t × (cid:18) c r (cid:19) / , ˙ φ ⊥ = 0 . (2.33)To study the stability of this solution, we analyze its behavior under small perturbations.In terms of the variable λ = ln a , we define, z ( λ ) = z c + ε ( λ ) , ◦ φ ⊥ ( λ ) = ◦ ε ⊥ ( λ ) , (2.34)for small ε and ◦ ε ⊥ . At first order, the system (2.26) becomes ◦◦ ε + ◦ ε + ξ ε = 0 where ξ = 10 p − p zz p + p zz (cid:12)(cid:12)(cid:12)(cid:12) z c , ◦◦ ε ⊥ + ◦ ε ⊥ = 0 . (2.35)The coefficient ξ is actually a function of the model dependent parameter n V /n T , since thelatter appears explicitly in the definition of p and implicitly via z c . Numerically, one findsthat ξ is always positive. It increases from 0 to + ∞ when n V /n T varies from − /
15 to 0.The solution of the system (2.35) is ◦ ε ⊥ ( λ ) = c e − λ , ε ( λ ) = c e − λ (1+ √ − ξ ) / + c e − λ (1 −√ − ξ ) / if 0 < ξ < / e − λ/ ( c + c λ ) if ξ = 1 / e − λ/ (cid:16) c cos (cid:16) λ (cid:112) ξ − / (cid:17) + c sin (cid:16) λ (cid:112) ξ − / (cid:17)(cid:17) if ξ > / c , , are integration constants determined by the IBC. Since ε ( λ ) and ◦ ε ⊥ ( λ ) convergeto 0 for large λ in all cases, the critical solution is stable under small perturbations. Inother words, the solution arising for arbitrary IBC such that ε (0), ◦ ε (0) and ◦ ε ⊥ (0) are smallis attracted toward the critical one, z ≡ z c , ˙ φ ⊥ ≡
0. The behavior of ε ( λ ) is oscillating withdamping when 0 < ξ < /
4, and exponentially convergent when ξ ≥ /
4. The special value ξ = 1 / n V /n T (cid:39) − . .4 Global attraction to the RDS Since we have shown the existence of a local basin of attraction around the critical solutiongiven in equations (2.30) and (2.33), we want to know if this property is valid for moregeneric IBC. Actually, any initial values ( z , ◦ z , ◦ φ ⊥ ) at λ = 0 are allowed, as long as thepositivity of the right hand side of the Friedmann Eq. (2.25) is guaranted , i.e. r ( z )3 − K ( z , ◦ z , ◦ φ ⊥ ) ≥ . (2.37)We have studied numerically the system (2.26) for generic IBC satisfying the previous bound.Implementing in the code a positive increment for λ = ln a simulates the phases of theevolutions where the scale factor increases. We observe that the right hand side of (2.25)never changes of sign. This means that the scale factor a ( t ) is monotonic . We always findthat the solutions satisfy z ( λ ) → z c and ◦ φ ⊥ ( λ ) → λ → + ∞ , after possible oscillations.To illustrate the convergent behavior for large oscillations, we present a numerical exampleon Fig. 2. It is obtained for n V /n T = − .
02 that corresponds to z c (cid:39) . z , ◦ z , ◦ φ ⊥ ) = (0 . , . , . Examples of damping oscillations of z ( λ ) (solid curve) and convergence to zero of ◦ φ ⊥ ( λ ) (dottedcurve) illustrating the dynamical attraction toward the critical solution z ≡ z c (cid:39) . , ˙ φ ⊥ ≡ , obtained for n V /n T = − . . The initial conditions are ( z , ◦ z , ◦ φ ⊥ ) = (0 . , . , . . are attracted by the RDS that satisfies z ≡ z c , ˙ φ ⊥ ≡ i.e. H = c r /a in Eq. (2.30). IBC such that the right hand side of (2.25) is negative correspond to solutions in Euclidean time. In general, a change of sign in the Friedmann equation during a simulation would be a numerical artifactmeaning that H actually vanishes when the scale factor reaches an extremum. z ≡ z c , ˙ φ ⊥ ≡ λ decreases i.e. for evolutions where the scale factor is contracting. The Big Crunch solutionobtained by time reversal on (2.33) is thus only formal, since it is highly unstable under smallfluctuations. What we observe by numerical simulations is that for generic IBC, when wechoose a negative increment for λ , then either z → + ∞ , or z oscillates with a greater andgreater amplitude. To understand better the dynamics of the contracting evolutions, we arethus led to analyze carefully the system in the regime z (cid:29) We have seen that when the potential (2.27) admits a minimum z c ( Case (b) ), the expandingcosmological evolutions for arbitrary IBC satisfy z ( t ) → z c . The shape of the potential in Case (c) of Fig. 1 suggests that when n V ≥
0, the dynamics could admit a run awaybehavior z ( t ) → + ∞ . We also noticed at the end of the previous subsection that thecontracting evolutions in Case (b) involve the large and positive value regime of z . For thesereasons, we need to study the system for z (cid:29)
1. In this limit the thermal effects O ( T )are sub-dominant compare to the supersymmetry breaking effects O ( M ). The expansionof p ( z ) contains two monomials in e z , p ( z ) = n V e z c o + ( n T c o + n V c e ) + · · · where c e = Γ(5 / π / (cid:88) m (cid:48) m ) , (2.38)and c o , c o are defined in Eqs (2.28) and (2.29). The dots stand for exponentially suppressedterms Ø( e − e z ). It follows that A z ( z ) (cid:39) ◦ z = ◦ M /M + 1 = ⇒ ˙ TT = − H i.e. aT = a T , (2.39)where a T is a positive constant depending on the IBC. From the expansion of p ( z ) forlarge z in Eq. (2.38), it is clear that the behavior of the system is drastically different when n V (cid:54) = 0 and when n V = 0. • n V (cid:54) = z (cid:29) We have ρ (cid:39) − P (cid:39) − M n V c o . The Friedmann Eq. (2.12) and the linear sum of Eqs (2.12)13nd (2.15), together with the matter field Eqs (2.16) and (2.17), become3 H = 12 ˙ φ + 12 ˙ φ ⊥ − M n V c o , (2.40)˙ H + 3 H = − M n V c o , (2.41) α ¨ φ + 3 Hα ˙ φ = 4 α M n V c o , ¨ φ ⊥ + 3 H ˙ φ ⊥ = 0 . (2.42)The above equations would also arise if we had considered the system at zero temperature.This is due to the fact that when temperature effects are not taken into account, the sources ofEinstein gravity always satisfy ρ = − P equal to minus the effective potential (see Eq. (2.13)).However, this does not mean that in our case we shall find that T ( t ) →
0. Actually, theregime z (cid:29) i.e. M (cid:29) T corresponds to thermal effects screened by radiative corrections,even if the temperature can be large. Combining Eq. (2.41) and Eq. (2.42) one determinesthe scalar fields in terms of the scale factor, (with α = (cid:112) / α ˙ φ = − α H + c φ a = ⇒ M = e c φ R t dt (cid:48) a ( t (cid:48) )3 a and ˙ φ ⊥ = √ c ⊥ a , (2.43)where c φ and c ⊥ are constants depending on the IBC. The Friedmann Eq. (2.40) takes thenthe form 3 H = 13 (cid:16) H − c φ a (cid:17) + c ⊥ a − M n V c o . (2.44)We will consider separately the cases c φ = 0 and c φ (cid:54) = 0. (i) c φ = M = M (cid:16) a a (cid:17) , H = − c m a + c M a , with c m = c ⊥ , c M = n V M c o a , (2.45)where M a is a positive constant. This solution exists only for n V > c ⊥ = 0, up to time reversal, the evolution describes a Big Crunch occurring at a finite time t BC , a ( t ) = a ( t BC − t ) / × (16 M n V c o ) / , t < t BC . (2.46)Since a ∼ /T , it follows that e z ∝ aM ∝ /a , which is large if t < ∼ t BC and implies a runaway of z , consistently with our hypothesis z (cid:29)
1. However, close to the Big Crunch, thescale M (as well as T , even if T (cid:28) M ) is formally diverging. This implies that the presentanalysis has to be restricted to the regime where M < M H i.e. z < (5 /
6) ln M H , where M H is the supersymmetry breaking scale above which a Hagedorn-like transition occurs. Tobypass this limit, one could try to extend our study to Hagedorn singularity free models [8].14or c ⊥ (cid:54) = 0, the time can be expressed as a function of the scale factor, t ( a ) = ± t BC (cid:90) a/a max x dx √ − x where t BC = √ | c M | a max , ∀ a ≤ a max = (cid:12)(cid:12)(cid:12)(cid:12) c M c m (cid:12)(cid:12)(cid:12)(cid:12) / . (2.47)Formally, this solution describes a Big Bang at t = − t BC that initiates an expansion era.The latter stops when the scale factor reaches its maximum a max at t = 0. The universe thencontracts till t = t BC , where a Big Crunch occurs. As before, the last era of this evolutioncan be trusted for t < ∼ t BC , where the solution (2.47) is asymptotic to (2.46). Thus, we haveshown that for IBC such that z (cid:29) c φ = 0, the dynamics is attracted to the MDSdescribed by the Big Crunch solution of Eq. (2.46) with c φ = c ⊥ = 0 and 3 H = c M /a . (ii) c φ (cid:54) = We consider IBC such that z (cid:29)
1, with generic c φ (cid:54) = 0. The sign of n V (cid:54) = 0 is arbitrary andwe want again to find the basins of attraction of the dynamics. Instead of Eq. (2.40), it ismore convenient to use the combination of Eq. (2.40) and Eq. (2.41),˙ H = −
12 ˙ φ −
12 ˙ φ ⊥ , (2.48)that becomes, utilizing the expressions (2.43) for ˙ φ and ˙ φ ⊥ :˙ H = − (cid:16) H − c φ a (cid:17) − c ⊥ a . (2.49)Introducing the quantity σ , σ = 94 c φ a ˙ a = 34 c φ ( a ) · = 94 c φ a H , (2.50)Eq. (2.49) takes the form: σ dσσ − σ + (1 + 3( c ⊥ /c φ ) ) = − daa , (2.51)valid in the z (cid:29) ( ◦ M /M ) + ◦ φ ⊥ − ◦ M /M ) ◦ − ◦ M /M − ( ◦ M /M ) + ◦ φ ⊥ − ◦◦ φ ⊥ − ◦ φ ⊥ = 0 (2.52)where the signs of n V and the fractions in front of the second derivatives must be the same,so that the right hand side of Eq. (2.25) is positive. This condition is required to study the15volutions in real time (and not Euclidean). This differential system is easily shown to beequivalent to ◦ M /M ) ◦ = ( ◦ M /M + 6) P ( ◦ M /M ) ◦ φ ⊥ = √ c ⊥ c φ ( ◦ M /M + 6) (2.53)where c ⊥ /c φ is an arbitrary constant denoted this way to make contact with the conventionsof Eq. (2.43), and the polynomial P ( x ) = [1 + 3( c ⊥ /c φ ) ] x + 36 ( c ⊥ /c φ ) x + 9 [12( c ⊥ /c φ ) −
1] is of the sign of n V . (2.54)Using the first equation in (2.43), one obtains σ = 94 1 ◦ M /M + 6 , (2.55)that can be used to identify consistently (2.51) with the first equation of (2.52). By the way,the real time condition translates to the condition that n V and the denominator of the lefthand side of (2.51) have a common sign. We are going to solve this equation separately for n V > n V < - For n V > : i) When ( c ⊥ /c φ ) > /
9, the denominator of (2.51) has no real root. Integrating, oneobtains aa = exp (cid:18) − √ c ⊥ /c φ ) − tan − (cid:18) σ − √ c ⊥ /c φ ) − (cid:19)(cid:19)(cid:0) σ − σ + (1 + 3( c ⊥ /c φ ) ) (cid:1) / , (2.56)that is used to draw σ versus a on Fig. 3 a . Suppose c φ >
0. If at some time σ > σ in Eq. (2.50) implies that a increases. Thus, the scale factor reachesa maximum a max where σ vanishes. We then pass into the σ < a decreases and converges to 0. Since σ → −∞ and a →
0, one has ˙ a → −∞ , which isinterpreted as a Big Crunch occurring at a finite time t BC . Similar arguments apply when c φ < σ <
0, the scale factor increases up to a max before converging to 0. Thiscorresponds to a Big Crunch occurring at some finite time t BC . To summarize, when theIBC are such that c φ > c φ < a from up to down(down to up). The universe starts from a Big Bang, reaches a maximum size, and ends witha Big Crunch. Of course, only the parts of this evolution consistent with the hypothesis16 ) a s max a b) s - s + s aa max Figure 3: σ , defined in Eq. (2.50) versus the scale factor a . Fig. a) corresponds to ( c ⊥ /c φ ) > / andFig. b) corresponds to ( c ⊥ /c φ ) ≤ / . The solid lines are associated to the case n V > , while the dashedone is associated to the case n V < . z (cid:29) t BC , one has σ → − s ∞ , where s = sign( c φ ), and Eq. (2.56) implies, a ( t ) (cid:39) ( t BC − t ) / × ((16 / | c φ | a e s π/ ) / , t < ∼ t BC . (2.57)This result can be used to show that the integral in M in Eqs (2.43) is bounded when t → t BC so that a consistent run away e z ∝ /a → + ∞ is found. An important remark then follows.The behavior (2.57) implies that c φ /a is dominated by H in Eq. (2.43), showing that thedynamics for ( c ⊥ /c φ ) > / c φ = c ⊥ = 0. ii) When ( c ⊥ /c φ ) ≤ /
9, the left hand side of Eq. (2.51) has two real roots. Integrating,one hasfor (cid:18) c ⊥ c φ (cid:19) <
19 : aa = | σ − σ − | σ − σ + − σ − ) | σ − σ + | σ +9( σ + − σ − ) , σ ± = 12 ± (cid:113) − c ⊥ /c φ ) , (2.58)for (cid:18) c ⊥ c φ (cid:19) = 19 : aa = e σ − / | σ − | / , σ ± = 12 , (2.59)where σ − < σ < σ + . Fig. 3 b shows σ as a function of a . When c φ >
0, the arguments usedin case i) apply on the branch σ < σ − , where the system evolves from up to down. Thescale factor grows up to a max before converging to 0. Similarly, when c φ < σ > σ + , the scale factor converges to 0. In both cases, a Big Crunch To be precise, the slope at the point ( a, σ ) = (0 , σ − ) is infinite when (1 − / / < ( c ⊥ /c φ ) ≤ / c ⊥ /c φ ) = (1 − / / c ⊥ /c φ ) < (1 − / / t BC and the behavior (2.57), with s = 0, is valid. The dynamics is again attractedby the solution associated to c φ = c ⊥ = 0.Some new features occur for c φ <
0, when σ < σ − , (see Fig. 3 b ). The scale factorincreases up to its maximum a max before converging to 0, while σ → σ − , implying ˙ a → −∞ .A Big Crunch at a finite time t BC occurs but since σ → σ − , one has a ( t ) (cid:39) ( t BC − t ) / × ((4 / | c φ | σ − ) / , t < ∼ t BC . (2.60)However, this behavior can be trusted as long as e z ∝ aM ∝ | t BC − t | σ − − → + ∞ . Thiscondition implies that the scaling ( t BC − t ) / exists when c φ < − / < (cid:18) c ⊥ c φ (cid:19) ≤ . (2.61)In this case, the evolution is not attracted by the solution c φ = c ⊥ = 0. Actually, Eq. (2.43)can be rewritten as αφ (cid:39) − (cid:18) − σ − (cid:19) ln | t BC − t | + c → + ∞ when t → t BC , (2.62)so that the integration constant c is negligible. There is thus a new basin of attraction, tothe solution (2.60) with c = 0 i.e. such that3 H (cid:39) c m a where c m = 127 c φ σ − , c = 0 . (2.63)As for the previous Big Crunch behavior, the divergence of M (and T ) is only formal sincethe present analysis supposes M < M H , where a Hagedorn-like transition occurs. On thecontrary, for ( c ⊥ /c φ ) smaller than the lower bound of the range (2.61), we formally have e z ∝ | t BC − t | σ − − →
0, which shows that the system actually exits from the regime z (cid:29) c φ > σ > σ + of Fig.3 b , one has a → + ∞ and σ → σ + . This implies a (cid:39) t / × ((4 / c φ σ + ) / and t → + ∞ .However, for large positive z , this implies e z ∝ aM ∝ t σ + − →
0, which shows that thesystem actually exits in the regime z (cid:29) Summary for n V > : We have shown analytically that when z (cid:29)
1, either z → + ∞ or z quits the regime z (cid:29) z , we use again a code that implements Eqs (2.26). Thephases of expansion are simulated when a positive increment for λ is chosen. For generic18BC, we observe that the right hand side of the Friedmann equation (2.25) changes sign atsome finite a = a max . This actually means that the scale factor has reached a maximum, (theHubble parameter H vanishes), and that the evolution enters into a phase of contraction.Such eras are simulated by choosing a negative increment for λ and we observe that forgeneric IBC, z ends by running away. As a conclusion, the generic IBC admit two basins ofattraction to MDS. The evolutions converge either to the solution where c φ = c ⊥ = 0 i.e. such that 3 H = c M /a , or to the solution with c = 0 and 3 H = c m /a . - For n V < : For IBC such that z (cid:29)
1, the real time condition translates to σ − < σ < σ + , (see Fig. 3 b ).If c φ >
0, the definition of σ in Eq. (2.51) reaches ˙ a >
0. It follows that a → + ∞ and σ → σ + . Thus, a (cid:39) t / × ((4 / c φ σ + ) / where t → + ∞ , and e z ∝ aM ∝ t σ + − → z (cid:29)
1. When − / < n V /n T < Case(b) ), this result is compatible with Sect. 2.3, where we found that the expanding evolutionssatisfy z → z c .If on the contrary c φ <
0, one has ˙ a <
0. It follows from Fig. 3 b that a → σ → σ − ,so that ˙ a → −∞ . The dynamics is attracted by the Big Crunch solution (2.60) i.e. (2.63)that can be trusted when the conditions (2.61) are satisfied. For smaller values of ( c ⊥ /c φ ) ,the system exits the regime z (cid:29)
1. These conclusions are compatible with the numericalsimulations of the contracting evolutions described at the end of Sect. 2.4, for generic IBCin
Case (b) . We found there that either z → + ∞ , as expected from the existence of theBig Crunch solution (2.60), or z oscillates with a larger and larger amplitude, implying thatwhen z is large and positive, it can exit the regime z (cid:29) • n V = z (cid:29) We argued in Section 2.3 that when − / < n V /n T <
0, the expanding evolutions are suchthat z is attracted by a critical value z c . Since z c → + ∞ when n V /n T → − , we expectthe expanding solutions for n V = 0 to satisfy z ( t ) → + ∞ . Actually, this run away may benatural since the potential in Eq. (2.27) decreases linearly for large positive z , V ∼ − zn T c o .In fact, a numerical study of the differential system (2.26) with n V = 0 confirms that forgeneric IBC, z enters the regime z (cid:29) z → + ∞ or z oscillates with a larger and larger amplitude. In all these cases, we are thus invited to19onsider analytically the system in the regime z (cid:29) φ in Eq. (2.16) vanishes (up to exponen-tially small terms in z ). Both φ and φ ⊥ are flat directions, and their kinetic energies involvetwo constants c φ , c ⊥ , determined by the IBC, α ˙ φ = c φ a , ˙ φ ⊥ = √ c ⊥ a . (2.64)The energy density in the Friedmann Eq. (2.12) is ρ (cid:39) T n T c o and, using (2.39), one has3 H = c r a + c m a where c r = 3 n T c o ( a T ) , c m = c ⊥ + c φ . (2.65)This equation admits two monotonic solutions (see Eq. (2.32)), mapped to one anotherunder time reversal. However, one can only trust them as long as z (cid:29) a ( t ) → + ∞ when t → + ∞ , Eq. (2.64) impliesthat M ( t ) converges to a constant M . It follows that e z = M/T ∝ a → + ∞ , showingthe validity of the solution for large enough time. We conclude that for generic IBC, theexpanding evolutions are attracted to the RDS, Eq. (2.33), with M = M .- For the contracting evolutions, depending on c φ and c ⊥ , we have a ( t ) (cid:39) ( t BC − t ) / × (3 c m ) / , M (cid:39) M (cid:18) − tt BC (cid:19) − c ⊥ /cφ )2]1 / , t < ∼ t BC , (2.66)if the constraints c φ > (cid:18) c ⊥ c φ (cid:19) <
83 (2.67)are satisfied, (so that z → + ∞ when t → t BC ). Alternatively, if c φ = c ⊥ = 0, the solution a ( t ) (cid:39) √ t BC − t × (cid:18) c r (cid:19) / , M (cid:39) M (2.68)is also consistent with a run away of z . For all other values of c φ and c ⊥ , z exits theregime z (cid:29)
1. Our conclusions for the contracting evolutions are that there are two setsof generic IBC. The first gives rise to a run away of z that corresponds to the Big Crunchsolutions (2.66) or (2.68). The second yields an oscillation regime in z , with larger and largeramplitude. 20 Susy breaking involving n = 2 internal directions In this section we extend our analysis to models where the supersymmetry breaking is in-duced by n = 2 Scherk-Schwarz circles S ( R ) × S ( R ) in the 4 th and 5 th internal dimensionsas in Ref. [14]. Thus, we are considering the heterotic or type II superstrings on S ( R ) × T × S ( R ) × S ( R ) × M , (3.1)where M is either T or K ∼ T / Z . As shown in [14], such models allow cosmologicalevolutions that describe radiation dominated eras, with stabilized complex structure modulus R /R [14]. The present analysis could certainly be extended to the other classes of modelsdescribed in [14], where asymmetric Scherk-Schwarz compactifications in the directions 4and/or 5 are considered. These models would involve run away solutions in the spirit of the Cases (c) and (a) for n = 1. R , R , R are restricted to be large enough in order to avoid Hagedorn-like phasetransitions. In this regime, the free energy and pressure are computed in Ref. [14] : P = T p ( z, Z ) where e z = MT , T = 12 πR √ Re S , M = 12 π √ R R √ Re S , e Z = R R . (3.2) T is the temperature in the Einstein frame, M is the supersymmetry breaking scale whichis given in terms of the geometric mean of the moduli R and R , and Z parametrizes thecomplex structure modulus R /R . The quantity p ( z, Z ) is a linear sum of functions, withmodel-dependent integer coefficients, p ( z, Z ) = n T p ( z, Z ) + n p ( z, Z ) + n p ( z, Z ) + n p ( z, Z ) , (3.3)where n T is again the number of massless states and p ˜ g ˜ g ˜ g = 2 π (cid:88) ˜ m , ˜ m , ˜ m e z [(2 ˜ m + ˜ g ) e z + (2 ˜ m + ˜ g ) e − Z + (2 ˜ m + ˜ g ) e Z ] . (3.4) The low energy effective action (2.7) with the kinetic term of R taken into account can bewritten in terms of redefined fields as : S = (cid:90) d x √− g (cid:20) R − (cid:18) ( ∂φ ) + ( ∂φ ⊥ ) + 12 ( ∂Z ) (cid:19) + P (cid:21) , (3.5)21here φ := φ D − ln (cid:112) R R , φ ⊥ := φ D + ln (cid:112) R R , P = M e − z p ( z, Z ) , M = e φ π . (3.6)For evolutions satisfying the metric ansatz (2.11), the Friedmann equation is3 H = 12 ˙ φ + 14 ˙ Z + 12 ˙ φ ⊥ + T r where r ( z, Z ) = 3 p − p z , (3.7)while the equation for the scale factor,2 ˙ H + 3 H = −
12 ˙ φ −
14 ˙ Z −
12 ˙ φ ⊥ − T p , (3.8)can be replaced by the conservation of the energy-momentum in a form similar to Eq. (2.19),˙ ρ + 3 H ( ρ + P ) + ˙ φ T (3 p − r ) + ˙ Z T p Z = 0 . (3.9)For the scalar fields, one has¨ φ + 3 H ˙ φ = T (3 p − r ) , (3.10)¨ Z + 3 H ˙ Z = T p Z , (3.11)¨ φ ⊥ + 3 H ˙ φ ⊥ = 0 = ⇒ ˙ φ ⊥ = √ c ⊥ a . (3.12)We proceed by introducing the variable λ of (2.20) as in Section 2.2 to express ◦ φ = A ( z, Z ) ◦ z + B ( z, Z ) ◦ Z − A ( z, Z ) = 4 r − r z r + p ) , B ( z, Z ) = − r Z + p Z r + p ) , (3.13)that reaches, by integration, M = e F (ln a ) a where ◦ F ( λ ) := A ( z, Z ) ◦ z + B ( z, Z ) ◦ Z . (3.14)This result can be used to rewrite the Friedmann Eq. (3.7) in either of the following forms,3 H = 35 H (cid:18) ( A ◦ z + B ◦ Z − + 12 ◦ Z − (cid:19) + 65 e F (ln a ) − z ] a r ( z, Z ) + 65 c ⊥ a , (3.15)or H = T r ( z, Z )3 − K ( z, ◦ z, Z, ◦ Z, ◦ φ ⊥ ) where K = 12 (cid:18) A ( z, Z ) ◦ z + B ( z, Z ) ◦ Z − (cid:19) + 14 ◦ Z + 12 ◦ φ ⊥ , (3.16)22hile the equations for the scalars become r − K (cid:18) A ◦◦ z + B ◦◦ Z + A z ◦ z + ( A Z + B z ) ◦ z ◦ Z + B Z ◦ Z (cid:19) + r − p A ◦ z + B ◦ Z ) + V ( z ) z = 0 r − K ◦◦ Z + r − p ◦ Z + V ( Z ) Z = 0 r − K ◦◦ φ ⊥ + r − p ◦ φ ⊥ = 0 (3.17)where we have defined V ( z ) z ( z, Z ) = r − p V ( Z ) Z ( z, Z ) = − p Z . (3.18) A wide range of behaviors can emerge from this system of differential equations. However, inthe spirit of Section 2.3, we choose to look for cosmological solutions with stabilized complexstructures e z = R / √ R R and e Z = R /R i.e. satisfying ( z, Z ) ≡ ( z c , Z c ). From thesystem (3.17), such solutions exist if the following conditions are simultaneously satisfied, V ( z ) z ( z c , Z c ) = 0 i.e. p ( z c , Z c ) + p z ( z c , Z c ) = 0 , (3.19) V ( Z ) Z ( z c , Z c ) = 0 i.e. p Z ( z c , Z c ) = 0 . (3.20)These equations define two sets of curves in the ( z, Z )-plane, whose intersections determinethe critical points ( z c , Z c ). Supposing such a solution exists, the corresponding cosmologicalevolution is characterized by ◦ F = 0 in Eq. (3.14) and thus M ( t ) = e z c T ( t ) = e F a ( t ) where F = cst. , R ( t ) = e Z c R ( t ) , (3.21)together with the Friedmann Eq. (3.15),3 H = c r a + c m a where c r = 6 e F− z c ) p ( z c , Z c ) , c m = 65 c ⊥ . (3.22)The latter is qualitatively identical to the one found for n = 1 in Case (b) . The evolutionwith expanding scale factor is asymptotic to the critical solution ( z, Z ) ≡ ( z c , Z c ), c m = 0,where a ( t ) is given in (2.33). 23o study the local stability around a critical solution with c m = 0, we define smallperturbations around it, z ( λ ) = z c + ε ( λ ) , Z ( λ ) = Z c + E ( λ ) , ◦ φ ⊥ ( λ ) = ◦ ε ⊥ ( λ ) , (3.23)and consider the system (3.17) at first order in ε , E and ◦ ε ⊥ . The latter can be brought intothe form ◦◦ ε + ◦ ε + ξ ε + ω E = 0 where ξ = p (4 p − p zz )+2 p zZ p (26 p + p zz ) (cid:12)(cid:12)(cid:12) ( z c ,Z c ) , ω = p zZ (2 p ZZ − p )2 p (26 p + p zz ) (cid:12)(cid:12)(cid:12) ( z c ,Z c ) , ◦◦ E + ◦ E + Ξ E + Ω ε = 0 where Ξ = − p ZZ p (cid:12)(cid:12)(cid:12) ( z c ,Z c ) , Ω = − p zZ p (cid:12)(cid:12)(cid:12) ( z c ,Z c ) , ◦◦ ε ⊥ + ◦ ε ⊥ = 0 . (3.24)To proceed, we specialize on the set of models whose R-symmetry charges coupled to thewinding and momentum numbers in the directions 4 and 5 are identical. For this set, it isshown in [14] that the function p in Eq. (3.3) takes the form, p ( z, Z ) = n T [ p + p ] + n V [ p + p ] , − ≤ n V n T ≤ , (3.25)and that the range of n V /n T can be divided in four phases whose frontiers are given by {− . , − / , } [14]. • For n V /n T < ∼ − . • For − . < ∼ n V /n T < − /
31, the constraint (3.19) defines a non trivial curve, whilethe condition (3.20) is satisfied on the axis Z = 0. The two loci intersect at a point( z c , • For − / < n V /n T <
0, the constraint (3.19) defines a non trivial curve, while thecondition (3.20) corresponds to 3 curves, one of which being the axis Z = 0. There isa unique intersection point ( z c , • For 0 < n V /n T , the constraint (3.19) has no solution.Thus, a solution to the equations of motion with constant complex structures ( z c , Z c ) existswhen − . < ∼ n V /n T <
0. 24he class of models we consider have a symmetry R ↔ R that translates to Z ↔ − Z .This implies p zZ ( z c ,
0) = 0 and ξ = 92 4 p − p zz p + p zz (cid:12)(cid:12)(cid:12)(cid:12) ( z c ,Z c ) , ω = 0 , Ξ = − p ZZ p (cid:12)(cid:12)(cid:12)(cid:12) ( z c ,Z c ) , Ω = 0 . (3.26)As in the n = 1 case, one finds numerically that ξ and Ξ are always positive. They in-crease from 0 to + ∞ when n T /n V varies from (cid:39) − .
215 to 0, which shows that ε ( λ ), E ( λ )(and ◦ ε ⊥ ( λ )) are always converging to zero. This means that in the class of models underinvestigation, when a solution ( z c , Z c ) exists, it is stable under small perturbations.A numerical study of the exact system (3.17) shows that for generic IBC ( z , ◦ z , Z , ◦ Z , ◦ φ ⊥ )such that the right hand side of Eq. (3.16) is positive, the expanding cosmological solutionssatisfy ( z ( λ ) , Z ( λ )) → ( z c , Z c ) when λ → + ∞ . We conclude that the dynamics of the ex-panding evolutions is attracted by the RDS described by the critical solution ( z, Z ) ≡ ( z c , Z c ),˙ φ ⊥ ≡ i.e. H = c r /a (Eq. (3.21)). We have described how cosmological evolutions can be determined by the R-charges of themassless spectrum. In particular, we found that the dynamics can be attracted to a radiationera, with stabilized complex structures. In standard cosmology, such an era follows a periodof inflation introduced to explain the observed flatness and homogeneity of our universe,or solve the topological defect problem. It is thus important to analyze if our attractionmechanism admits periods of accelerated cosmology.Using the Einstein equations, the inflation condition ¨ a > H + H > ρ , P and the scalar kinetic energies(take ˙ Z = 0 for n = 1), 12 (cid:18) P + 13 ρ (cid:19) < − (cid:18) ˙ φ + 12 ˙ Z + ˙ φ ⊥ (cid:19) . (4.1)Using the variable λ and Eq. (2.25) (or (3.16)), we obtain p − p z < − (cid:18) p − p z (cid:19) K − ≤ , (4.2)where there are two conditions in one: First, p − p z / n = 1 (see Eq. (2.4)) and n = 2 (see Eq.(3.25)), one can show that a necessary condition to have p − p z / < n V <
0. We thuscarry on our discussion in this case. It then follows that the quantity p − p z / n V < K < I where I = − p − p z p − p z . (4.3)Actually, when deriving (4.3) from (4.2), one also finds that I ≤
3. However, this is anempty constraint since the function I satisfies I ≤ For a susy breaking with n = 1
From (4.3), in order to have solutions, we need I ( n V /n T , z ) >
0. This condition defines adomain in the ( n V /n T , z )-plan where, by definition, I = 0 on the boundary. However, ithappens that I < ∼ a . We can thus estimate the a) b) Figure 4:
For one modulus z (Fig. a)): In the domain I > of the ( n V /n T , z ) -plane ( − / < n V /n T < ), I can be approximated by 1, its value for z (cid:29) . For two moduli ( z, Z ) (Fig. b)): For any fixed − . < ∼ n V /n T < , in the domain I > of the ( z, Z ) -plane, I can be approximated by 1, its asymptoticvalue in the sectors II , III , IV , V defined in Eq. (4.9). Fig. b), is drawn for n V /n T = − / . e -fold number by considering z large and positive. In particular, when − / < n V /n T < z ’s allowing periods of accelerated cosmology are always larger than z c . When z (cid:29) K (cid:39)
13 ( ◦ M /M ) + ( c ⊥ /c φ ) ( ◦ M /M + 6) < ∼ , (4.4)where we have used the second equation of (2.53). An era of acceleration exists if ( c ⊥ /c φ ) < /
33. It begins and ends at most when ◦ M /M saturates the inequality (4.4) i.e. equals l b,e = − c ⊥ /c φ ) ± (cid:112) − c ⊥ /c φ ) ]1 + 3( c ⊥ /c φ ) . (4.5)To find the scale factors a b and a e when this arises, we integrate the first equation of (2.53),ln (cid:18) aa (cid:19) ≡ λ − λ = γ ln | ◦ M /M + 6 | + γ + ln | ◦ M /M − l + | + γ − ln | ◦ M /M − l − | , (4.6)where l ± = − c ⊥ /c φ ) ± (cid:112) − c ⊥ /c φ ) c ⊥ /c φ ) ,γ = 31 + 3( c ⊥ /c φ ) l + + 6)( l − + 6) , γ ± = ±
31 + 3( c ⊥ /c φ ) l ± + 6)( l + − l − ) . (4.7)Inserting the particular values (4.5) in (4.6), the maximum e -fold number is found to be e ( c ⊥ /c φ ) = ln a e a b = γ ln (cid:12)(cid:12)(cid:12)(cid:12) l e + 6 l b + 6 (cid:12)(cid:12)(cid:12)(cid:12) + γ + ln (cid:12)(cid:12)(cid:12)(cid:12) l e − l + l b − l + (cid:12)(cid:12)(cid:12)(cid:12) + γ − ln (cid:12)(cid:12)(cid:12)(cid:12) l e − l − l b − l − (cid:12)(cid:12)(cid:12)(cid:12) , (4.8)that satisfies e ( c ⊥ /c φ ) ≤ e (0) (cid:39) . For a susy breaking with n = 2
For any fixed n V /n T <
0, the condition I ( n V /n T , z, Z ) > z, Z )-plan, as shown on Fig. 4 b . Using the notations of [14], this zone spans four asymptoticsectors defined as follows, II : Z → −∞ , z ∼ ηZ, | η | < , III : Z ∼ ηz → −∞ , − < η < ,IV : Z ∼ ηz → + ∞ , < η < , V : Z → + ∞ , z ∼ ηZ, | η | < , (4.9)inside of which I < ∼
1. We will evaluate the e -fold number by approximating p ( z, Z ) by itsasymptotic expansions, which are well defined in each sectors [14]: p II = e z e − Z n V S o + e − z e − Z/ ( n T S o + n V S e ) + e z e Z/ ( n T + n V )( S o + S e ) + · · · p III = e z e − Z n V S o + e z e Z n V ( S o + S e ) + 2( n T S o + n V S e ) + · · · p IV = e z e Z n V S o + e z e − Z n V ( S o + S e ) + 2( n T S o + n V S e ) + · · · p V = e z e Z n V S o + e − z e Z/ ( n T S o + n V S e ) + e z e − Z/ ( n T + n V )( S o + S e ) + · · · (4.10)where the dots stand for exponentially suppressed terms and the coefficients S o,e , , are con-stants . Their precise values are not needed in the following but can be found in the Appendix of [14].
27t is interesting to note that the behavior of the temperature depends drastically on thelocation of the representative point of the system in the ( z, Z )-plan. In sectors
III and IV , one finds A (cid:39) B (cid:39) II (or V ), one has A (cid:39) / B (cid:39) / − / II : M e Z a T = cst. , III ∪ IV : a T = cst. , V : M e − Z a T = cst. . (4.11)However, the Eqs of motion for the scalar fields are identical in all these asymptotic sectors, ( ◦ M /M ) + ˙ Z + ◦ φ ⊥ − ◦ M /M ) ◦ − ◦ M /M − ( ◦ M /M ) + ˙ Z + ◦ φ ⊥ − ◦◦ Z − ◦ Z + 6 = 01 ( ◦ M /M ) + ˙ Z + ◦ φ ⊥ − ◦◦ φ ⊥ − ◦ φ ⊥ = 0 (4.12)where the fractions in front of the second derivatives must have the sign of n V i.e. negative.When ◦ M /M + 4 (cid:54) = 0, it is easy to show that there exist constants c φ (cid:54) = 0, c Z , c ⊥ such that ◦ M /M ) ◦ = ( ◦ M /M + 4) P ( ◦ M /M ) ◦ Z − c Z c φ ( ◦ M /M + 4) ◦ φ ⊥ = √ c ⊥ c φ ( ◦ M /M + 4) (4.13)where P ( x ) = [1 + 2( c Z /c φ ) + 2( c ⊥ /c φ ) ] x + 4[4( c Z /c φ ) + 3 c Z /c φ + 4( c ⊥ /c φ ) ] x +4[8( c Z /c φ ) + 12 c Z /c φ + 8( c ⊥ /c φ ) + 3] is of the sign of n V . (4.14)The condition (4.3) is also identical in all asymptotic sectors II ,..., V , K (cid:39)
12 ( ◦ M /M ) + [( c Z /c φ )( ◦ M /M ) + 3] + ( c ⊥ /c φ ) ( ◦ M /M + 4) < ∼ , (4.15)where we have used the second and third equations of (4.13). A period of acceleratedcosmology exists if( c ⊥ /c φ ) < , c − < c Z /c φ < c + where c ± = − ± (cid:112) − c ⊥ /c φ ) ]7 . (4.16)28t begins and ends at most when ◦ M /M saturates the inequality (4.15) i.e. equals l b,e = − c Z /c φ ) + 3 c Z /c φ + 4( c ⊥ /c φ ) ] ± (cid:112) c + − c Z /c φ )( c Z /c φ − c − )1 + 2( c Z /c φ ) + 2( c ⊥ /c φ ) . (4.17)To find the corresponding scale factors a b , a e , we integrate the first equation of (4.13),ln (cid:18) aa (cid:19) ≡ λ − λ = γ ln | ◦ M /M + 4 | + γ + ln | ◦ M /M − l + | + γ − ln | ◦ M /M − l − | , (4.18)where l ± = − c Z /c φ ) + 3 c Z /c φ + 4( c ⊥ /c φ ) ] ± (cid:112) C + − c Z /c φ )( c Z /c φ − C − )1 + 2( c Z /c φ ) + 2( c ⊥ /c φ ) , with C ± = − ± (cid:112) − c ⊥ /c φ ) ]5 ,γ = 21 + 2( c Z /c φ ) + 2( c ⊥ /c φ ) l + + 4)( l − + 4) ,γ ± = ±
21 + 2( c Z /c φ ) + 2( c ⊥ /c φ ) l ± + 4)( l + − l − ) . (4.19)Inserting (4.17) in (4.18), the maximum e -fold number is found to be e ( c Z /c φ , c ⊥ /c φ ) = ln a e a b = γ ln (cid:12)(cid:12)(cid:12)(cid:12) l e + 4 l b + 4 (cid:12)(cid:12)(cid:12)(cid:12) + γ + ln (cid:12)(cid:12)(cid:12)(cid:12) l e − l + l b − l + (cid:12)(cid:12)(cid:12)(cid:12) + γ − ln (cid:12)(cid:12)(cid:12)(cid:12) l e − l − l b − l − (cid:12)(cid:12)(cid:12)(cid:12) , (4.20)that satisfies e ( c Z /c φ , c ⊥ /c φ ) ≤ e ( − / , (cid:39) . e -foldnumber for n = 1 and n = 2 is thus the same. They cannot account for the astrophysicalobservations. The present work focuses on an intermediate cosmological era, t E ≤ t ≤ t w . The time t E corresponds to the exit of an Hagedorn phase i.e. after the notion of topology and geometryemerge, with 3 large space-like dimensions and possibly internal directions of intermediatescales. The characteristic time t w is associated to the electroweak phase transition, occurringbefore large scale structure formations in the universe.To describe the intermediate era, we consider flat classical 4-dimensional space-timeswithin the context of superstring theory, where N = 8 , , T ,- radiative corrections of magnitude M , the spontaneous supersymmetry breaking scale,- the presence of moduli fields, whose kinetic energies scale as 1 /a .We determine the low energy effective action at the one-loop level and parametrize data atthe exit t E of the Hagedorn phase by a set of IBC.To be specific, when n = 1 internal radius is involved in the supersymmetry breaking, weshow that, depending on the R-symmetry charges of the massless spectra, the models canbe grouped in different Cases (a) , (b) or (c) . For each, there exists a partition of the set ofIBC, whose classes yield cosmological evolutions that share a common behavior. • In Case (a) , one class of IBC seems to induce a spontaneous decompactification of theinternal direction involved in the breaking of supersymmetry. It is out of the scope ofthe present paper. • In Case (b) , the evolutions of the scale factor are monotonic. The expanding solutionsbelong to the same class and are attracted to a radiation dominated era. Asymptoti-cally, one has 3 H ∝ /a ∝ T ∝ M , where a ( t ) ∝ √ t . • In Case (c) , a class of IBC reaches solutions attracted to a Big Crunch era dominatedby the radiative corrections to the initially vanishing vacuum energy. Asymptotically,one has 3 H ∝ M ∝ /a ∝ T , where a ( t ) ∝ ( t BC − t ) / . • In all
Cases , a class of IBC reaches solutions attracted to a Big Crunch era dominatedby the classical moduli kinetic energy. Asymptotically, one has 3 H ∝ /a ∝ T ∝ M / (1 − / (8 σ − )) , where σ − is given in Eq. (2.57) and a ( t ) ∝ ( t BC − t ) / .Our analysis has some limitations. First of all, we have supposed that the radii involvedin the spontaneous breaking of supersymmetry and the radius of the Euclidean time are largeenough to avoid Hagedorn like singularities in the partition functions. Since we found BigCrunch behaviors where T (cid:28) M are formally diverging, these evolutions must be restrictedto M lower than a maximum Hagedorn bound. It would be interesting to relax this constraintby considering models free of such singularities [8].Also, we have studied the thermal and quantum corrections at the one-loop level in flatbackgrounds where moduli have some kinetic energy. This means that the induced sources30 , ρ and their backreactions on the “cold” classical backgrounds we start with must besmall. Thus, we need afterwords to check the consistency of our approach.- For the RDS, we have P ( t ) ∝ ρ ( t ) ∝ a ( t ) ∝ t −→ , (5.1)which shows its validity.- For the Big Crunch attractor a ( t ) ∝ ( t BC − t ) / of the MDS, both the thermal/quantumsources and the classical moduli kinetic energy are diverging. Formally, one has P ( t ) = − ρ ( t ) ∝ M ∝ a ( t ) ∝ t BC − t ) (cid:29) a ∝ √ t BC − t , (5.2)which shows that this solution has to be restricted to times where P and ρ do not exceedtoo much the classical kinetic energy.- This limitation does not exist for the Big Crunch behavior a ( t ) ∝ ( t BC − t ) / . Even ifthe thermal/quantum sources and the classical moduli kinetic energy are diverging, one has P ( t ) = − ρ ( t ) ∝ M ∝ a ( t ) − /σ − ∝ t BC − t ) − /σ − (cid:28) a ∝ t BC − t ) , (5.3)since σ − < /
2. Thus, the backreaction on the classical background is always small. Thedynamical attraction to this solution is only limited by the Hagedorn transitions mentionedpreviously.To go beyond the constraint that the thermal/quantum corrections must be small, onecan try to compute the higher string loop corrections and, if possible, use string-stringdualities to find the non-perturbative corrections to the effective action. For the modelsunder consideration in this work, the extended supersymmetry is only broken spontaneously,at the scale M and/or by thermal effects at the scale T . This means that the UV behavior ofthe theories is not affected by these soft breaking and that the renormalization theorems forextended supersymmetries should persist. To find the cosmological evolutions beyond thefirst order in perturbation theory, both the thermal effective potential and the correctionsto the kinetic terms have to be computed.The attraction phenomena we found can be generalized to models where n internal radiiare involved in the spontaneous breaking of supersymmetry. In particular, we have shownthat for n = 2, the attraction to the radiation era persists. In the process, the complex31tructure modulus associated to the ratio of the two radii converges to a constant i.e. isdynamically stabilized.During the convergence to the radiation dominated era, the ratio M/T reaches its criticalvalue exponentially or with damping oscillations. Periods of accelerated expansions are alsoallowed. However, the e -fold number seems to depend weakly on n and, at least for n = 1 , e -fold ≤ . N = 8 , , i.e. before spontaneous breaking), it wouldbe interesting to extend our work to N = 1 models and see if the order of magnitude of the e -fold is higher. Acknowledgements
We are grateful to I. Antoniadis, J. Estes and N. Toumbas for useful discussions. H.P. thanksthe Ecole Normale Superieure for hospitality. F.B. thanks M. Petropoulos and I. Florakisfor fruitful and stimulating discussions.This work is partially supported by the ANR (CNRS-USAR) contract 05-BLAN-0079-02.The work of F.B and H.P is also supported by the European ERC Advanced Grant 226371,and CNRS PICS contracts 3747 and 4172.
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