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Nuclear Physics B 795 (2008) 334-360
www.elsevier.com/locate/nuclphysb
Inflationary de Sitter solutions from superstrings *
Costas Kounnasa, Hervé Partoucheb*
a Laboratoire de Physique Théorique, Ecole Normale Supérieure, 1 24 rue Lhomond, F-75231 Paris cedex 05, France b Centre de Physique Théorique, Ecole Polytechnique, 2 F-91128 Palaiseau, France
Received 26 August 2007; accepted 21 November 2007
Available online 28 November 2007
Abstract
In the framework of superstring compactifications with N = 1 supersymmetry spontaneously broken (by either geometrical fluxes, branes or else), we show the existence of new inflationary solutions. The time-trajectory of the scale factor of the metric a, the supersymmetry breaking scale m = m(&) and the temperature T are such that am and aT remain constant. These solutions request the presence of special moduli-fields: (i) The universal "no-scale-modulus" which appears in all N = 1 effective supergravity theories and defines the supersymmetry breaking scale m(&). (ii) The modulus , which appears in a very large class of string compactifications and has a &-dependent kinetic term. During the time evolution, a4 ps remains constant as well (ps being the energy density induced by the motion of ). The cosmological term A(am), the curvature term k(am, aT) and the radiation term cr = a4p are dynamically generated in a controllable way by radiative and temperature corrections; they are effectively constant during the time evolution. Depending on A, k and cr , either a first or second order phase transition can occur in the cosmological scenario. In the first case, an instantonic Euclidean solution exists and connects via tunneling the inflationary evolution to another cosmological branch. The latter starts 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. © 2007 Elsevier B.V. All rights reserved.
* 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, CNRS PICS 2530, 3059 and 3747, and ANR (CNRS-USAR) contract 05-BLAN-0079-01.
* Corresponding author.
E-mail addresses: costas.kounnas@lpt.ens.fr (C. Kounnas), herve.partouche@cpht.polytechnique.fr (H. Partouche).
1 Unité mixte du CNRS et de l'Ecole Normale Supérieure associée à l'Université Pierre et Marie Curie (Paris 6), UMR 8549.
2 Unité mixte du CNRS et de l'Ecole Polytechnique, UMR 7644.
0550-3213/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.nuclphysb.2007.11.020
1. Introduction
In the framework of superstring and M-theory compactifications, there are always moduli fields 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 the effective La-grangian have the generic form
L = yj- det g
1R - g^Kijdrfidvtj - V(ffi,fi)
where Ki j is the metric of the scalar manifold and V is the scalar potential of the N = 1 super-gravity. (We will always work in gravitational mass units, with M = = 2.4 x 1018 GeV.)
What will be crucial in this work is the non-triviality of the scalar kinetic terms Kij in the N = 1 effective supergravity theories that will provide us, in some special cases, accelerating cosmo-logical solutions once the radiative and temperature corrections are taken into account.
Superstring vacua with spontaneously broken supersymmetry [1] that are consistent at the classical level with a flat space-time define a very large class of "no-scale" supergravity models [2]. Those with N = 1 spontaneous breaking deserve more attention. Some of them are candidates for describing (at low energy) the physics of the standard model and extend it up to 0(1) TeV energy scale. This class of models contains an enormous number of consistent string vacua that can be constructed either via freely acting orbifolds [1,3] or "geometrical fluxes" [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 which are described by an effective N = 1 "no-scale supergravity theory". Namely the vacua in which the supersymmetry is spontaneously broken with a vanishing classical potential with undetermined gravitino mass due to at least one flat field direction, the "no-scale" modulus @. At the quantum level a non-trivial effective potential is radiatively generated which may or 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. As we will show in Section 4, the quantum and thermal corrections are under control (thanks to super-symmetry and to the classical structure of the "no-scale models"), showing interesting scaling properties.
In Section 2, we set up our notations and conventions in the effective N = 1 "no-scale" su-pergravities of the type IIB orientifolds with D3-branes and non-trivial NS-NS and RR three form fluxes H3 and F3. We identify the "no-scale" modulus @, namely the scalar superpartner of the Goldstino which has the property to couple to the trace of the energy momentum tensor of a subsector of the theory [6]. More importantly, it defines the field-dependence of the gravitino mass [2]
m($) = ea0. (1.2)
Other extra moduli that we will consider here are those with @-dependent kinetic terms. These moduli appear naturally in all string compactifications [7]. We are in particular interested in scalars (@s) which are leaving on D3-branes and whose kinetic terms scale as the inverse volume of the "no-scale" moduli space.
In Section 3, we display the relevant gravitational, fields and thermal equations of motion in the context of a Friedman-Robertson-Walker (FRW) space-time. We actually generalize the
mini-superspace (MSS) action by including fields with non-trivial kinetic terms and a generic, scale factor dependent, thermal effective potential.
In our analysis we restrict ourselves to the large moduli limit, neglecting non-perturbative terms and world-sheet instanton corrections O(e-S), O(e-Ta). On the other hand we keep the perturbative quantum and thermal corrections.
Although this study looks hopeless and out of any systematic control even at the perturba-tive level, it turns out to be manageable thanks to the initial no-scale structure appearing at the classical level (see Section 4).
In Section 5, we show the existence of a critical solution to the equations of motion that follows from the scaling properties derived in Section 4. We have to stress here that we extremize the effective action by solving the gravitational and moduli equations of motion and do not consider the stationary solutions emerging from a minimization of the effective potential only. We find in particular that a universal solution exists where all scales evolve in time 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 A, curvature term k and radiation term are generated in the MSS action, characterizing the cosmological evolution.
Obviously, the validity of the cosmological solutions based on (supergravity) effective field theories is limited. For instance, in the framework of more fundamental theories such as string theory, there are high temperature instabilities occurring at T ~ TH, where TH is the Hagedorn temperature of order the mass of the first string excited state. To bypass these limitations, one needs to go beyond the effective field theory approach and consider the full string theory (or brane, M-theory,...) description. Thus, the effective solutions presented in this work are not valid anymore and must be corrected for temperatures above TH. Moreover, Hagedorn-like instabilities can also appear in general in other corners of the moduli space of the fundamental theory, when space-time supersymmetry is spontaneously broken.
Regarding the temperature scale as the inverse radius of the compact Euclidean time, one could conclude that all the internal radii of a higher dimensional fundamental theory have to be above the Hagedorn radius. This would mean that the early time cosmology should be dictated by a 10-dimensional picture rather than a 4-dimensional one where the internal radii are of order the string scale. There is however a loophole in this statement. Indeed, no tachyonic instability is showing up in the whole space of the moduli which are not involved in the spontaneous breaking of supersymmetry, as recently shown in explicit examples [8]. This leeds us to the conjecture that the only Hagedorn-like restrictions on the moduli space depend on the supersymmetry breaking. In our cosmological solutions, not only the temperature T scale is varying, but also the super-symmetry breaking scale m, which turns to be a moduli-dependent quantity. Based on the above statements, we expect that in a more accurate stringy description of our analysis, there should be restrictions on the temperature as well as the supersymmetry breaking scale. This has been recently explicitly shown in the stringy examples considered in [8].
In Section 6, our cosmological solutions are generalized by including moduli with other scaling properties of their kinetic terms.
Finally, Section 7 is devoted to our conclusions and perspective for future work.
2. N = 1 no-scale Sugra from type IIB orientifolds
In the presence of branes and fluxes, several moduli can be stabilized. For instance, in "generalized" Calabi-Yau compactifications, either the h1,1 Kahler structure moduli or the h2,1 complex structure moduli can be stabilized according to the brane and flux configuration in
type IIA or type IIB orientifolds [4-6,9]. The (partial) stabilization of the moduli can lead us at the classical level to AdS like solutions, domain wall solutions or "flat no-scale like 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 D3-branes and non-trivial NS-NS and RR three form fluxes H3 and F3. This particular configuration induces a well known superpotential W(S, Ua) that can stabilize all complex structure moduli Ua and the coupling constant modulus S [4,5]. The remaining hi,i moduli Ta "still remain flat directions at the classical level", e.g. neglecting world-sheet instanton corrections 0(e—Ta) and the perturbative and non-perturbative quantum corrections [5].
It is also well known by now that in the large Ta limit the Kahler potential is given by the intersection numbers dabc of the special geometry of the Calabi-Yau manifold and orbifold com-pactifications [10,11]:
K = log dabc(Ta + Ta)(Tb + Tb)(Tc + Tc). (2.1)
Thus, after the S and Ua moduli stabilization, the superpotential W is effectively constant and implies a vanishing potential in all Ta directions. The gravitino mass term is however non-trivial [1,2,4,5,11],
m2 = |W |2eK. (2.2)
This classical property of "no-scale models" emerges from the cubic form of K in the moduli Ta and is generic in all type IIB orientifold compactifications with D3 -branes and three form H3 and F3 fluxes [4,5]. Keeping for simplicity the direction Ta = yaT (for some constants ya) and freezing all other directions, the Kahler potential is taking the well known SU(1,1) structure [2],
K = —3log(T + T). (2.3)
This gives rise to the kinetic term and gravitino mass term,
d„TdvT o v c
g^v 3 —T and m2 = ceK =-(2.4)
(T + T)2 (T + T)3
where c is a constant. Freezing Im T and defining the field ® by
e2a® = m2 =-(2.5)
(T + T)3
one finds a kinetic term
B^TBvT _ a2 (T + T)2 = g"v 3
The choice a2 = 3/2 normalizes canonically the kinetic term of the modulus @.
The other extra moduli that we will consider are those with @-dependent kinetic terms. We are in particular interested to the scalars whose kinetic terms scale as the inverse volume of the T-moduli. For one of them, , one has
Ks = —a2 e2a*g^sdv*s = (2.7)
Moduli with this scaling property appear in a very large class of string compactifications. Some examples are:
g^v3 " v- 2 = g^—d^dv®. (2.6)
(i) All moduli fields leaving in the parallel space of D3-branes [4,5].
(ii) All moduli coming from the twisted sectors of Z3-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, namely those with kinetic terms of the form:
Kw = eV-vWg^d^wdrfw, (2.8)
with weight w = 0, 2 and 6.
3. Gravitational, moduli and thermal equations
In a fundamental theory, the number of degrees of freedom is important (and actually infinite in the context of string or M-theory). However, in an effective field theory, an ultraviolet cut-off set by the underlying theory determines the number of states to be considered. We focus on cases where these states include the scalar moduli fields @ and , with non-trivial kinetic terms given by
L = V- det g
2r - 2+ e2act - /)
+ ■■■. (3.1)
In this Lagrangian, the "■ ■ ■" denote all the other degrees of freedom, while the effective potential V depends on @ and the renormalization scale n-. We are looking for gravitational solutions based on isotropic and homogeneous FRW space-time metrics,
ds2 = -N(t)2 dt2 + a(t)2 dffi2, (3.2)
where £23 is a 3-dimensional compact space with constant curvature k, such as a sphere or an orbifold 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 matter degrees 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 degrees of freedom are parameterized by a pressure P(T,mi) andadensity p(T, mi), where mi are the non-vanishing masses of the theory. Note that P and p have an implicit dependence on @, through the mass m(&) defined in Eq. (1.2) [6]. The presence of the thermal bath modifies the effective MSS action, including the corrections due to the quantum fluctuations of the degrees of freedom whose masses are below the temperature scale T. The result, together with the fields @ and &s, reads
Seff =-kLL [dta3(l(^2 - ^ - 2.*2 - !_e2«*é2 'J \N\aJ a2 2N 2N s
+ NV - —(P + P) + y (P - P)l (3.3)
where a "dot" denotes a time derivation. N(t) is a gauge dependent function that can be arbitrarily chosen by a redefinition of time. We will always use the gauge N = 1, unless it is explicitly specified.
The variation with respect to N gives rise to the Friedman equation,
3H2 = —k + p + 1 <2 + 1 e2a<4> 2 + V, (3.4)
a2 2 2
where H = (a/a).
The other gravitational equation is obtained by varying the action with respect to the scale factor a:
2H + 3H2 = —k — P — 1 <2 — 1 e2a<<2 + V + 1 a — . (3.5)
a2 2 2 s 3 da
In the literature, the last term a(dV/da) is frequently taken to be zero. However, this is not valid due to the dependence of V on /, when this scale is chosen appropriately as will be seen in Section 3. We thus keep this term and will see that it plays a crucial role in the derivation 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 the kinetic terms of < and <s drop out,
2 2k 1 1 dV
H + 3H2 = —^ + - (p — P)+ V + - a — . (3.6)
a2 2 6 da
The other field equations are the moduli ones,
< + 3H< + -¡^(v — P — 1 e2a<< = 0 (3.7)
<s + (3H + 2a< )< s = 0. (3.8) The last equation (3.8) can be solved immediately,
1 e—2a<
Ks = - e2a<<t>j = Cs-6—, (3.9)
where Cs is a positive integration constant. It is important to stress here that we insist to keep in Eq. (3.7) both terms dP/d< and dKs/d< that are however usually omitted in the literature. The first term vanishes only under the assumption that all masses are taken to be < -independent, while the absence of the second term assumes a trivial kinetic term. However, both assumptions are not valid in string effective supergravity theories! (see Section 3).
Finally, we display for completeness the total energy conservation of the system,
dt(p + 2 <2 + Ks + V^ + 3H (p + P + <2 + 2Ks) = 0. (3.10)
Before closing this section, it is useful to derive some extra useful formulas that are associated to the thermal system. The integrability condition of the second law of thermodynamics reaches, for the thermal quantities p and P,
T— = p + P. (3.11)
The fact that these quantities are four-dimensional implies
9 9 \ / 9 9 \
~ ~ -IP=<
mi— + T-T)P = 4P and (m^ + T—)P = 4P. (3.12)
Then, the second Eq. (3.12) together with Eq. (3.11) implies [6]: dP
mi-— = -(p - 3P). (3.13)
Among the non-vanishing mi, let us denote with "hat-indices" the masses m; that are @-independent, and with "tild-indices" the masses m~t that have the following @ -dependence:
{mi } = {mj}U{m-l} where mj = cjea0, (3.14)
for some constants cj. Then, utilizing Eq. (3.13), we obtain a very fundamental equation involving the modulus field @ [6],
--= a(p- 3P), (3.15)
where p and P are the contributions to p and P associated to the states with @ -dependent masses mj. The above equation (3.15) clearly shows that the modulus field @ couples to the (sub-)trace of the energy momentum tensor associated to the thermal system [6] p, P of the states with @-dependent masses defined in Eq. (3.14). We return to this point in the next section.
4. Effective potential and thermal corrections
In order to find solutions to the coupled gravitational and moduli equations discussed in the previous section, it is necessary to analyze the structure of the scalar potential V and the thermal functions p, P. More precisely, we have to specify their dependence on @, T, a and . Although this analysis looks hopeless in a generic field theory, it is perfectly under control in the string effective no-scale supergravity theories.
Classically the potential Vcl is zero along the moduli directions @ and . At the quantum level, it receives radiative and thermal corrections that are given in terms of the effective potential [11], V(mi ,ix), and in terms of the thermal function, -P(T, mi). Let us consider both types of corrections.
4.1. Effective potential
The one loop effective potential has the usual form [11,17],
1 ^2 1
V = Vcl + Strlog 7f + ^ Str-M^o
1 / A//2 \
+ 64^ Str(-4l°g -¿r) ^
where Vcl is the classical part, which vanishes in the string effective "no-scale" supergravity case. An ultraviolet cut-off Aco is introduced and \x stands for the renormalization scale.
Str Mn = J2(-)2J' (2Ji + 1)mnj (4.2)
is a sum over the nth power of the mass eigenvalues. In our notations, the index I is referring to both massless and massive states (with eventually @-dependant masses). The weights account for the numbers of degrees of freedom and the statistics of the spin Ji particles.
The quantum corrections to the vacuum energy with the highest degree of ultraviolet divergence is the A^ term, whose coefficient StrM0 = (nB - nF) is equal to the number of bosonic minus fermionic degrees of freedom. This term is thus always absent in supersymmetric theories since they possess equal numbers of bosonic and fermionic states.
The second most divergent term in Eq. (4.1) is the A;:o contribution proportional to Str M2. In the N = 1 spontaneously broken supersymmetric theories, it is always proportional to the square of the gravitino mass-term m(@)2,
Str M2 = c2m(&)2. (4.3)
The coefficient c2 is a field independent number. It depends only on the geometry of the kinetic terms of the scalar and gauge manifold, and not on the details of the superpotential [11,17]. This property is very crucial in our considerations.
The last term has a logarithmic behavior with respect to the infrared scale \x and is independent of the ultraviolet cut-off Aco. Following the infrared regularization method valid in string theory (and field theory as well) adapted in Ref. [18], the scale \x is proportional to the curvature of the three-dimensional space,
M =—, (4.4)
where y is a numerical coefficient chosen appropriately according to the renormalization group equation arguments. Another physically equivalent choice for m is to be proportional to the temperature scale, m = ZT. The curvature choice (4.4) looks more natural and has the advantage to be valid even in the absence of the thermal bath.
Modulo the logarithmic term, the Str M4 can be expanded in powers of gravitino mass m(&),
1 Str M4 = C4m4 + C2m2 + Co. (4.5)
Including the logarithmic terms and adding the quadratic contribution coming from the Str M2, we obtain the following expression for the effective potential organized in powers of m(&):
V = V4($,a) + V2($,a) + V0($,a), (4.6)
V„(0, a) = mn(<P)(Cn + Qn log(m(&)ya)), (4.7)
for constant coefficients Cn and Qn (n = 4,2,0). These contributions satisfy
dVn(&,a) ) dVn($,a) nn
-—-= a(nVn + mnQn) and a---= mnQn. (4.8)
d0 v da
The logarithmic dependence in the effective potential can be derived in the effective field theory by considering the renormalization group equations (RGE). They involve the gauge couplings, the Yukawa couplings and the soft-breaking terms [11,19]. These soft-breaking terms are usually parameterized by the gaugino mass terms Mi/2, the soft scalar masses m0, the trilinear coupling mass term A and the analytic mass term B [11,19]. However, what will be of main importance in this work is that all soft breaking mass terms are proportional to m(&) [11,17].
4.2. Thermal potential
For bosonic (or fermionic) fluctuating states of masses mb (or mf) in thermal equilibrium at temperature T, the general expressions of the energy density p and pressure P are
p = T4( E p(m) + E P(T ))•
^ boson b fermion f ^ ' '
P = E •f{mf) + E iFf)- (4.9)
boson b fermion f
'm) f d q2E(q-mm) b(f) 1 f d q'/Ejq^) (410)
P \ T ) = I dq E(q.Z) T 1 • 7B = 3j dq eE(q,f) T 1 (4.10)
and E(q,f) = Jq2 + m2.
There are three distinct sub-sectors of states:
(i) The subsector of nB bosonic and n^ fermionic massless states. From Eqs. (4.9) and (4.10), their energy density p0 and pressure P0 satisfy
Po = 3P0 = ^-(nB + 7 n^TA. (4.11)
In particular, we have p0 - 3P0 = 0 and dP0/d< = 0.
(ii) The subsector of states with non-vanishing masses independent of m(<).
• Consider the nB bosons and nF fermions whose masses we denote by mi0 are below T. The energy density P and pressure P associated to them satisfy
P(T, mi0) = P(T, mi0 = 0) + m? dp- = + 8nF V - £^m20t2, (4.12)
m V ' Î0
p(T, mj0) = p(T, mj0 = 0) + m? dP = + 8nF) T4 - £ ^m20T2, (4.13)
m V ' Î0
where the ci0's are non-vanishing positive constants. In particular, one has dP/d< = 0.
• For the masses mi above T, the contributions of the particular degrees of freedom are exponentially suppressed and decouple from the thermal system. We are not including their contribution.
(iii) The subsector with non-vanishing masses proportional to m(<). Its energy density p and pressure P satisfy
-= -a(p- 3P), (4.14)
as was shown at the end of Section 2. This identity is also valid for the massless system we consider in case (i).
According to the scaling behaviors with respect to T and m(<), we can separate
P = P4 + P2, P = P4 + P2, (4.15)
+ Tlyf) (fin>Pn) = n{P",Pn)- (4'16)
p4 and P4 are the sums of the contributions of the massless states (case (i)), the f4 parts of p and P (case (ii)), and p and P (case (iii)),
p4=^(M+nB)+8 n+ iBP{m) + e n
boson b fermion f
(4.17)
P4=T4(n?((nB+nB)+8n+ E iBP(m)+ £
boson b fermion f
(4.18)
while p2 and P2 arise from the T2 parts of p and P (case (ii)):
P2 = P2 = - ££hm20T2 = -M2T2. (4.19)
Com2rr2
5. Critical solution
The fundamental ingredients in our analysis are the scaling properties of the total effective potential at finite temperature,
Vtotal = V - P. (5.1)
Independently of the complication appearing in the radiative and temperature corrected effective potential, the scaling violating terms are under control. Their structure suggests to search for a solution where all the scales of the system, m(0), T and \x = (1/ya), remain proportional during their evolution in time,
ea0 = m(0) = — H = -a0 and fm(0) = T. (5.2)
Our aim is thus to determine the constants y' and f in terms of Cs in Eq. (3.9), y, and the computable quantities Cn, Qn (n = 4,2, 0) in string theory, such that the equations of motion for 0, 0s and the gravity are satisfied. On the trajectory (5.2), the contributions Vn (n = 4, 2, 0) defined in Eq. (4.7) satisfy
Vn = mnC'n where Cn = Cn + Qn log(Yj), (5.3)
dV„ „ , dV„ „
-V = am (nCn + Qn), a-V = mn Qn- (5.4)
Also, the contributions of 0 and 1/a6 in Ks in Eq. (3.9) conspire to give a global 1/a4 dependence,
Ks = CSY—. (5.5)
Finally, the sums over the full towers of states with 0 -dependent masses behave in p4/T4 and P4/T4 as pure constants (see Eqs. (4.17) and (4.18)),
p4 = r4T4 where
^4=^ (n+no) + 8 n+4)) + E P (f) + E iFf. (5.6)
P4 = p4T4 where
P4 = (("B ) + 8 (4+"F)) + E P (f) + E iFf ■ (5.7)
As a consequence, using Eqs. (4.8) and (4.14), the 0-equation (3.7) becomes,
H + 3H2 = a2((4C4 + Q4)m4 + (2C'2 + Q2)m2 + Qo
+ (r4 - 3p4)f 4m4 - 2Csy'6m4). (5.8)
On the other hand, using Eq. (4.8), the gravity equation (3.6) takes the form
H + 3H2 = -Iky'2m2 + 1 (r4 - p4)f 4m4 + (C4m4 + C2m2 + CO)
+ 1 (Q4m4 + Q2 m2 + Qo). (5.9)
The compatibility of the 0 -equation and the gravity equation along the critical trajectory implies an identifica term of Qo
an identification of the coefficients of the monomials in m. The constant terms determine CO in
, 6a2 - 1
C0 = Qo, (5.10)
which amounts to fixing y',
Co 6a2-1
Y ' = yeQo 6 . (5.11)
The quadratic terms determine the parameter k: 1 ( 2a2 - 1 , 6a2 - 1 \
k =-y^( — C2 + Q2) . (5.12)
Finally, the quartic terms relate f to the integration constant Cs appearing in Ks,
1 (4a2 - 1 , 6a2 - 1 2a2 - 1 4 6a2 - 1 4\
C = ^ { — C4 + Q4 + *f 4 - 4) . (5.13)
At this point, our choice of ansatz (5.2) and constants y', f allows to reduce the differential system for 0s, 0 and the gravity to the last equation. We thus concentrate on the Friedman equation (3.4) in the background of the critical trajectory 02 = (H2/a2),
'6a2 - 1 \ 2 3k 1 2a0 ■ 2
3H2 =--2 + p + - e 00 2 + V. (5.14)
6a2 ) a2 2
The dilatation factor in front of 3H2 can be absorbed in the definition of k, k and CR, once we take into account Eqs. (5.10), (5.12) and (5.13),
o 3k Cr
3H2 1 R
3H2 - 3k + -R, (5.15)
3k - a2Qo, (5.16)
k - ^(âO^T^ - C2- 1 Q2\ (5.17)
3 ( 4 , 1
-4 , O w
Cr = (r4 - P4)h4 + 2C4 + 3Q4 )• (5.18)
We note that for Q0 > 0, k is positive. In that case, the constraint (5.13) allows us to choose a lower bound for the arbitrary constant Cs, so that f4 is large enough to have k > 0. This means that the theory is effectively indistinguishable with that of a universe with cosmological constant 3k, uniform space curvature k, and filled with a thermal bath of radiation coupled to gravity. This can be easily seen by considering the Lagrangian
V-det g
R - 3k
(5.19)
and the metric ansatz (3.2), with a 3-space of constant curvature k. In the action, one can take into account a uniform space filling bath of massless particles by adding a Lagrangian density proportional to 1 /a4 (see [14-16]) in the MSS form. One finds
\k\-3 f ,( 3 (a\2 3k CR\
SeffdtNa^-( -a) + 3k - ^ + j), (5.20)
whose variation with respect to N gives (5.15). Actually, the thermal bath interpretation is allowed as long as CR > 0, since the 1/a4 term is an energy density. However, in the case under consideration, the effective CR can be negative due to the m4 contribution of the effective potential. The general solution of the effective MSS action of Eq. (5.20) with k > 0, k > 0 and CR > 0 was recently investigated in Ref. [16]. It amounts to a thermally deformed de Sitter solution, while the pure radiation case where k - k - 0, CR> 0 was studied in Ref. [6]. In the latter case, the time trajectory (5.2) was shown to be an attractor at late times, giving rise to a radiation evolving universe with
4Cr\1/4 1/2 T 1
—R) t1/2, m(0) - - -—. (5.21)
3 ) H Y 'a
Following Ref. [16], the general case with k > 0, k > 0 and CR> 0 gives rise to cosmological scenarios we summarize here. Depending on the quantity
4 = 3 k2 cr, (5.22)
a first or second order phase transition can occur:
52 < 1 ^^ 1st order transition, 8T > 1 ^^ 2d order transition. (5.23)
(i) The case 82 < 1
There are two cosmological evolutions connected by tunnel effect:
c(t) = N^s + cosh2(Vît), t e R, (5.24)
r (t) = N^s - sinh2(Vît), ti < t < -ti,
(5.25)
N=H (1 - ¿2 )1/4,
s = ^ —--1
2 vyl — 52 1 _
ti = —— arcsinh V e. (5.26)
The "cosh"-solution corresponds to a deformation of a standard de Sitter cosmology, with a contracting phase followed at t = 0 by an expanding one. The "sinh"-solution describes a big bang with a growing up space till t = 0, followed by a contraction till a big crunch arises. The two evolutions are connected in Euclidean time by a @-gravitational instanton
aE(T) = e + cos2 (-Xt), ®e(t) = —^log^ WW). (5.27)
The cosmological scenario is thus starting with a big bang at ti = — --X arcsinh^/e and expands up to t = 0, following the "sinh"-evolution. At this time, performing the analytic continuation t = —i(n/2VX + t) reaches (5.27) (where t is chosen in the range3 —n ^ t ^ 0). At t = 0, a different analytic continuation to real time exists, t = it, that gives rise to 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 on its "sinh"-evolution and starts to contract, or a first order transition occurs and the universe enters into the inflationary phase of the "cosh"-evolution. In that case, the scale factor jumps instantaneously from a— to a+ at t = 0,
1 -y/1 - STj a+= y — (1 + V 1 - 8ZT). (5.28)
An estimate of the transition probability is given by
p a e-2SEeff, (5.29)
where SE eff is the Euclidean action computed with the instanton solution (5.27), for t e [-n/2Vk, 0]. Actually, following Refs. [15,16], one has:
eff = -3Î\|
1 +J1 - 5
(e(U) - (1 -sj 1 - ¿2)K(u)j, (5.30)
3 It is also possible to consider the instantons associated to the ranges a/Xt € [—(2n + 1)n/2,0], n € N, see [16].
Fig. 1. A first order phase transition can occur. The two cosmological evolutions as and ac are connected by an instanton aE. The universe starts with a big bang at t = ti and expands till t = 0 along the solution as. Then, the scale factor can either contract, or jump instantaneously and enter into the inflationary phase of ac.
where K and E are the complete elliptic integrals of first and second kind, respectively, and
2(1 - ST)1/4
(5.31)
1 + 1 - S
(ii) The case ST > 1
There is a cosmological solution,
a(t) = 1 + - 1sinh(2VXt), t > ti,
ti = -
arcsinh-
s 2 -1
(5.32)
(5.33)
As in the previous case, it starts with a big bang. However, the behavior evolves toward the inflationary phase in a smooth way (see Fig. 2). The transition can be associated to the inflection point arising at tinf, where a(tinf) = ainf,
tinf =
arcsinh,
I sT - 1 st + 1,
ainf =
Another solution, obtained by time reversal t ending in a big crunch. (iii) The case ST2 = 1
There is a static solution,
(5.34)
-t, describes a contracting universe that is
a(t) = a0 where a0 = ^ —, 2a
(5.35)
Fig. 2. A second order phase transition occurs. The universe starts with a big bang at t = t( and evolves smoothly toward the inflationary phase.
corresponding to an S3 universe of constant radius. This trivial behavior can be reached from the cases (i) and (ii) by taking the limit 8^ ^ 1. Beside it, there are two expanding cosmological evolutions,
a< (t) = a^yj 1 - e—t > 0, (5.36)
a> (t) = ao/^, t e R. (5.37)
The first one starts with a big bang at t = 0, while the second is inflationary. Both are asymptotic to the static one (see Fig. 3). Contracting universes are described by the solutions obtained under the transformation t ^—t.
6. Inclusion of moduli with other scaling properties
We would like to consider generalizations of the previous set up. They are consisting in the inclusion of moduli fields with kinetic terms obeying different scaling properties with respect to 0. Namely, we take into account the effects of the class of moduli with Lagrangian density
V- det g 2 M6 dy06 + e4a03^2 + e6a03^o ), (6.1)
to be added to (3.1). With the metric ansatz (3.2), the MSS action (3.3) is completed by the contributions of the 's (w = 0, 2, 6),
-]*H f dta3(- —tâ - — e4022 - — e6"0tâ). (6.2)
6 J V 2N 2N *2 2N V 7
The equation of motion for 0 has now terms arising from 02 and 0O,
0 + 3H0 + JL(y - P - 1 e*>*0j - 2e4a0(^2 - 1 e6a%2) = 0, (6.3)
Fig. 3. There are two expanding cosmological evolutions. The first one, a<, starts with a big bang and converges quickly to the static solution. The second one, a>, is almost static for negative time and inflationary for positive time.
while <s and 0w satisfy
<PS + (3H + 2a<)<s = 0, 4>w + (3H + (6 - w)a<)0>w = 0. (6.4)
Eqs. (6.4) are trivially solved,
Ks - Ie2a<<>2 = Cs, Kw - 2e(6-w)a*<P2w = , (6.5)
where Cs and the C^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 0w (w = 0, 2, 6). However, these additional terms cancel out from the linear sum of the two equations and thus (3.6) remains invariant.
On the critical trajectory (5.2), one has
Ks = CsOf, Kw = (6.6)
This implies that the new contributions arising in the < equation (6.3) have dimensions two and zero and thus do not spoil the possible identification between (6.3) and the gravity equation (3.6). In particular, the 1/a6 scaling properties of 06 play no role at this stage. The < -equation becomes
H + 3H2 = a2((4C4 + Q4)mA + (2C2 + Q2)m2 + Qo + (n - 3p4)$4m4
- 2CsY'6m4 - 4C02y'6m2 - 6C0oy'6), (6.7)
to be compared with Eq. (5.9). The identification of the constant terms implies
o /A 6a2 — 1
C0 + 6a 2 C^o y'6 = —-—Qo, (6.8)
which is an equation for y'. For Q0 < 0, there is always a unique solution for y'. However, it is interesting to note that for Q0 > 0, there is a range for C00 > 0 where there are always two solutions for y'. This case is thus giving rise to two different critical trajectories. The m2
contributions impose
1 / 2 ,6 2a2 -1 , 6a2 -1 ' - . .J 6
k = —j^yZaCfo / 6--— C'2 + Qij. (6.9)
This fixes the value of k for any arbitrary C^2. Finally, the equation implied by the quartic mass terms is identical to the one of the previous section, and relates f to Cs. We repeat it here for completeness,
1 (4a2 — 1 , 6a2 — 1 2a2 — 1 4 6a2 — 1 A
C = ^ { — C4 + -VOT Q4 + — r4f 4 — — P4f 4) . (6.10)
We would like to stress again that for Cs sufficiently large, f4 can take any value other some bound we may wish.
The Friedman equation in the presence of the extra moduli becomes,
3H2 = — 3k + p + 1 $ 2 + 1 ft + 2 e2a*<i>2 + 1 e4a%2 + 2 e6a%2 + V. (6.11)
a 2 2 2 2 2
On the critical trajectory where <i>2 = (H2/a2), and taking into account Eqs. (6.8), (6.9) and (6.10), one obtains
o 3k Cr Cm
3H2 = 3A — + Cr + CM, (6.12)
a2 a4 a6
3X = a2(Qo — 6Q0 y '6), (6.13)
k = Y^C Y'6 + ^ f2M 2 — C2 — 1Q2), (6.14)
CR = (r4 — P4)f4 + 2C4 + 3 Q^j, (6.15)
Cm = C. (6.16) Some observations are in order:
• C$0 gives rise to a negative contribution to the cosmological term 3A..
• As previously, it is possible to have CR positive if one wishes, by considering large enough values for f4. This condition can always be satisfied due to the freedom on Cs in Eq. (6.10). To reach positive values of k, one can either consider a large enough f4 or utilize C^2 as a parameter.
• CM is always positive and is determined by the modulus only.
• Here also the system can be described by an effective MSS action similar to the one examined in [16],
\k\ — 2 f 3 fa\2 3k CR CM\
Seff =JJTj dtNa {A.a) + 3X — ^ + ^ + -or), (6.17)
whose associated Friedman equation is precisely (6.12). Thus, once taking into account the thermal and quantum corrections as well as the effects of the moduli we are considering here, the system admits solutions that cannot be distinguished from the de Sitter cosmology deformed by
Fig. 4. Phase diagram in the (52 )-plane. Inside the "almost triangular" domain (6.19), a first order transition can connect two cosmological solutions by tunnel effect. Outside the domain, there is a single expanding solution (and one contracting) that describes a second order transition. On the boundary, there are two expanding (and two contracting) solutions, beside a static one.
the presence of thermal radiation and time dependent 06-moduli fields [16]; this interpretation is valid only when CR is positive.
Assuming X, k, CR positive and utilizing the equivalence of the effective MSS action to the thermally and moduli deformed de Sitter action studied in [16], we can immediately derive the 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 82, &2M,
5 2=3 k2Cs'
and define the domain
2 9 X2 5M = 4 k3 Cm>
(6.18)
5 2 < 1 and 5M /1 -T 5 2 + 1
in the (82,&2M)-plane, as shown in Fig. 4. The Friedman equation (6.12) admits solutions that involve a first order phase transition inside this domain and a second order one outside it. It is convenient to express these conditions in terms of k , the only real positive root of the polynomial equation
1 — 52 - -4 T 2
(6.19)
by defining
k 3 + K 2 + — K--5M = 0,
+ + 4 27 M
4k 2 + 4k + 52
(1 + k)2 27 k(1 + k)2' where the second equality in Eq. (6.21) is just a consequence of Eq. (6.20):
(6.20)
(6.21)
1st order transition,
2d order transition.
(6.22)
(i) The case A < 1
There are two cosmological solutions connected via tunneling. The first one takes a simple form in terms of a function t(t),
ac (t(t)) = N^ e + cosh2 (Vkt(t)), (6.23)
and t(t) is found by inverting the definition of t as a function of t,
/1 cosh2 (VXv) + s k
dvj--——- where s =-, . (6.25)
V cosh2(^flv) + s + s (1 + kW 1 - A
In ac, the variables t and t are arbitrary in R. The second cosmological evolution is
as (t(t)) = N^s - sinh2(VXt(t)) where
, I s — sinh2 (VXv) t =— dvj-—-- (6.26)
e + e — sinh2(VXv) '
with the range of time
_ 1 ^ _ _
ti =--= arcsinh Ve < t < —ti, i.e.
. , e — sinh2 (\fXv) ti = — dvj-;-\ v) < t < —ti. (6.27)
f ye + e — sinh2 (+/Xv) ti
As in the previous section, the "cosh"-solution describes a contracting phase followed by an expanding one and approaches a standard de Sitter cosmology for positive or negative large times. The "sinh"-solution starts with a big bang, ends with a big crunch, while the scale factor reaches its maximum at t = 0. The two cosmological solutions are connected by a 0 -gravitational instanton
aE (t(t)) = N^e + cos2 (VxT(t))
(r(T)) = —log(y'ae (t(t))), (6.28)
where T(t) is the inverse function of
, cos2(VIv) + e T = — dvj----^, (6.29)
2(VXv) + e + e '
and the range of Euclidean time is4
< t < 0 i.e. ti < t < 0 where
, cos2 (\/Xv) + £
ti = - dvj „ (1 ) + _. (6.30)
cos2 (V^v) + £ + £
The cosmological scenario starts with an initial singularity at ti and follows the "sinh"-expansion till t = 0. At this time, the solution can be analytically continued to the instantonic one by choosing t = -i(n/2\pk + t) i.e. t = -i(-Ti + t). When the Euclidean time t = 0 is reached, a second analytic continuation to real time, t = it i.e. t = it, gives rise to 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 along the "sinh"-solution i.e. enter in a phase of contraction. Or, a first order phase transition occurs and the trajectory switches to the "cosh"-evolution. In that case, the scale factor jumps instantaneously from a- to a+,
k(1 + -) , ,-\ k(1 + -) , ,-\
a-=i k^-^-Sl-^) a+ = ]j k^-^l + VT-A). (6.31)
The transition probability is controlled by the Euclidean action, p a e-2SEeff, where SE eff has been computed in [16]. For V^t e [-n/2,0], one has
c - tA aS2\1/4 / . / 0 +n
SE eff =-
<4 - 34>,/4,f< 3
/ J4 - 352 cos(3) - 1 + 3&T \
x E(u) ---K(u)\, (6.32)
\ ^3^4 - 352 sin^) /
sin(3 ) /165^+952-8 ^
0 = arccos —M-> . (6.33)
sin^)' V (4 - 352)3/2
We have displayed the scale factors ac and as in terms of and 52M. However, it is interesting to 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 fact that a±T±= %/y', one finds
T±(52 ,52m> = T ^ where Tm = I(6.34)
It is shown in [16] that the cosmological solutions can be written as
c(t(t) = cosh(2^~(t)), t>0 (6.35)
4 Actually, one can also consider the instantons associated to the ranges -TLt e [-(2n + 1)n/2, 0], n e N, see [16].
as {m) = A)]j 1 + cosh(2t < 0, (636)
where we have defined
(T+/T- + T_/T+)2' ' '
We note that there is a temperature duality T+ ^ T_ that switches the two cosmological solutions ac(t(t)) for t > 0 and as(t(t)) for t < 0 into each other:
(T+ ^ T_) ^^ (ac(tit) for t> 0 ^ as(t(t)) for t < 0). (6.38)
Once quantum and thermal corrections are taken into account, the no-scale supergravities we are considering share a common effective behavior with the thermally and moduli deformed de Sitter evolution (see Eq. (6.12)). This means that the temperature Tm can be defined in both contexts. In Eq. (6.34), Tm is expressed in terms of critical trajectory quantities. However, one can consider the effective 1/a4-radiation energy density in Eq. (6.12) to define the "would-be number" of massless bosonic (fermionic) degrees of freedom, nfF), of the equivalent deformed de Sitter point of view:
PR — — J4 (nfff + 8nf T4, (6.39)
where 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 Tm as
Tm = ( ^ n^TT^ . (6.40)
V neff + 8 neff '
(ii) The case A> 1
There is an expanding solution we briefly describe (a contracting one is obtained by time reversal t ^ _t),
(tit)) = J ^ 1 + VA_Tsinh(2VXtit)), (6.41)
/k(1 + k) 2X
where t as a function of t is given by
t = J dv
and we consider
' ' VA _ 1sinh(2+Jkv) + 1
(6.42)
^/A_Tsinh^^/2Xv) + 1 +
11 t > ti —--— arcsinh , i.e.
2*Jk VA _ 1 0
t ^ ti — _ I dv
VA _ 1sinh(2VXv) + 1
V ( ^ (6.43)
VA_1sinh^V2Xv) + 1 +
This cosmological solution describes a smooth evolution from a big bang to an inflationary era (see Fig. 2). It has a single inflection point arising when a = ainf, which is defined by the following condition,
H2nf k 2
> 0 satisfies
x3 - x - — S2M = 0. 4 27 M
(6.44)
(iii) The case A = 1
Beside the following static solution,
k(1 + k) k ( / 3 2 a(t) = ao where ao = J ——— ^ 3X1 1 + V 1 - 452
(6.45)
the two expanding cosmological evolutions we found for vanishing B2M are generalized by,
. (t(t)) = a0^j 1 - e-2^t(t) where t = j dv
1 _ e-2\fkv
1 + 12FF- e-2^kV
(6.46)
for t^ 0 i.e. t > 0, and
, (t(t) = a0^ 1 + e2^t(t) where t = j dv
1 + TFF + e
(6.47)
for arbitrary t and t. They are monotonically increasing as in the pure thermal case: the first one starts with a big bang and the second one is inflationary (see Fig. 3). Two other solutions obtained under t ^—t are decreasing. These time-dependent solutions are asymptotic to the static one.
Before concluding let us signal that the number nB(F) of states with @-independent masses below T(t) is not strictly speaking a constant. It is actually lowered by one unit each time the temperature T(t ) passes below the mass threshold of a boson (fermion). Our critical solutions for the scale factor are thus well defined in any range of time where nfi and nF are constant. The full cosmological scenario is then obtained by gluing one after another these ranges. Each time a mass threshold is passed, the values of nMl2, r4 and p4 decrease (see Eqs. (4.19), (5.6) and (5.7)), and the parameters of the critical trajectory have to be evaluated again. However, the constraint (6.10) implies that (nfi + 8nF)%4 remains constant, due to the fact that Cs (and any other C$w) is constant along the full cosmological evolution. This implies that CR defined in Eq. (6.15) is also invariant. However, the term f 2M2 and thus k can decrease (see Eq. (6.14)). The positivity of k can nevertheless be guaranteed by the modulus term C^2 y6. Actually, this procedure that is consisting in gluing time intervals as soon as an energy threshold is reached is identical to what is assumed in Standard Cosmology. In the latter case, the full time evolution is divided in different phases (e.g. radiation dominated, matter dominated, and so on).
7. String perspectives and conclusions
At the classical string level, it is well known that it is difficult to construct exact cosmological string solutions. It is even more difficult to obtain de Sitter like inflationary evolutions, even in less than four dimensions.
The main difficulty comes from the fluxes and torsion terms which are created via non-trivial field strength (kinetic terms) and have the tendency to provide negative contributions to the cos-mological term, thus anti-de Sitter like vacua. To illustrate a relative issue, consider for instance the Euclidean version of the de Sitter space in three dimensions, dS3, which is nothing but the 3-sphere S3. Although S3 can be represented by an exact conformal field theory based on an SU(2)k WZW model, the latter does not admit any analytic continuation to real time. This is due to the existence of a non-trivial torsion Hlvp that becomes imaginary under an analytic continuation [20-22].
This obstruction in string and M-theory is generic and follows from the kinetic origin of the 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 tension objects such as orientifolds. This idea was explored in the field theory approximation in Ref. [24]. To go further, it is necessary to work with string cosmological backgrounds based on exact conformal field theories. However, the only known exact cosmological solution without the torsion problem described above is that of SL(2, R)/^(1)_|k| x K [25]. Its Euclidean 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 classical superstring background with spontaneously broken N = 1 supersymmetry defined on a flat 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, we have shown that the quantum and thermal corrections create dynamically universal effective potential terms that give rise to non-trivial cosmological accelerating solutions.
The main ingredients we have used are the scaling properties of the effective potential at finite temperature in the "no-scale N = 1 spontaneously broken supergravities", once the backgrounds follow critical trajectories where all fundamental scales have a similar evolution in time. Namely, the supersymmetry breaking scale m(&), the inverse of the scale factor a, the temperature T the infrared scale i remain proportional to each other:
— m(0) = -L = T = Yl.
Y a f Y
The "no-scale modulus 0" is very special in the sense that it is the superpartner of the goldstino and couples to the trace of the energy momentum of a subsector of the theory. It also provides non-trivial dependences in the kinetic terms of other special moduli of the type:
Kw — _1 e(6_w)a*(d<fiw)2 (w = 0, 2,4, 6),
where 04 — 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 Sitter evolutions. The cosmological term 3X(am), the curvature term k(am,aT) and the radiation term CR(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 are valid below Hagedorn-like scales associated to the temperature as well as the supersymmetry breaking scale m, where instabilities would occur in the extension of our work in a stringy framework. These restrictions on m are supported by the analysis of the string theory examples considered in Ref. [8].
When the deformation of the de Sitter evolution is below some critical value, there are two cosmological solutions which are connected by tunnel effect and interchanged under a temperature 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 deformation of a standard de Sitter evolution, with a contracting phase followed at t = 0 by an expanding one. The universe starts on the big bang cosmological branch and expands up to t = 0 along the first solution. At this time, two distinct behaviors can occur. Either the universe starts to contract, or a first order phase transition arises via a 0 -gravitational instanton, toward the inflationary phase of the deformed de Sitter evolution. The transition probability p can be estimated.
If on the contrary the induced cosmology corresponds to a de Sitter-like universe with deformation above the critical value, the previous big bang and inflationary behaviors are smoothly connected via a second order phase transition.
It is of main importance that the field theory approach we developed here can easily be adapted at the string level [8], following the recent progress in understanding the stringy wave-function of the universe [27,28]. This will permit us to go beyond the Hagedorn temperature and understand better the very early "stringy" phase of our universe [29]. At this point, we insist again on a highly interesting question that can be raised concerning the common wisdom which states that all radii-like moduli should be large to avoid Hagedorn-like instabilities. If this statement was true, then the quantum and thermal corrections should be considered in a 10 rather than in a 4-dimensional picture. However, the recent results of Ref. [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 remaining ones are free from any Hagedorn-like instabilities and can take very small values, even of the order of the string scale.
Acknowledgements
We are grateful to Constantin Bachas, Ioannis Bakas, Adel Bilal, Gary Gibbons, John Iliopou-los, 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 supported by the UE contract MRTN-CT-2004-512194, while H.P. is supported by the UE contracts MRTN-CT-2004-503369 and MEXT-CT-2003-509661, INTAS grant 03-51-6346, and CNRS PICS 2530, 3059 and 3747.
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