Ann. Phys. (Leipzig) 15, No. 4 ? 5, 291 ? 301 (2006) / DOI 10.1002/andp.200510188 Cosmological scaling solutions (and their applications to string scenarios) Beatriz de Carlos? Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QJ, UK Received 8 November 2005, accepted 16 January 2006 Published online 6 April 2006 Key words Scaling solutions, superstrings, moduli stabilisation. PACS 98.80.Cq, 04.25.-g In this review we will study the phase space of cosmologies consisting of a scalar ?eld with an exponential potential and a barotropic ?uid. This is a very simple system which, however, gives rise to many interesting solutions that can be classi?ed according to the values of the parameters in the game, namely those that de?ne the type of ?uid and the steepness of the potential. We shall then pay particular attention to scaling solutions, where the scalar energy density tracks that of the ?uid. In the second part of this review we shall turn to speci?c models in order to apply the previous results. The goal is to study the dynamics of scalar ?elds in the Early Universe, and the framework to do so is different types of string compacti?cations. The corresponding scalar ?elds are then the so-called moduli for which non perturbative potentials, of an exponential type, usually arise. We shall see how, contrary to previous expectations, the existence of scaling in this context allows these ?elds to dynamically stabilise at their tiny minima for a large fraction of initial conditions. This represents a further step in the quest for ?nding a suitable Particle Physics model compatible with the existing cosmology. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Scalar ?elds play a crucial role in cosmology. Many relevant events in the Early Universe, such as the existence of an in?ationary period, or the recently detected present expansion of the Universe can be suitably explained in terms of scalar ?elds (the in?aton for the former, and the quintessence ?eld for the latter). This becomes particularly interesting if one is trying to use Particle Physics as the framework where to implement these mechanisms. It is well known that, within the Standard Model of Particle Physics the only available scalar ?eld is the Higgs ?eld, and it has a too steep potential to support either slow-roll in?ation or a slow enough evolution as required for quintessence to work. However it is widely believed that the Standard Model is not the ultimate theory of Particle Physics, but a low energy effective one. We have, by now, enough evidence to think that there is a Grand Uni?ed Theory, formulated at high scales in order to incorporate gravity, which gives the Standard Model as its low energy realisation. It is also assumed that Supersymmetry will be a fundamental ingredient of this ultimate theory. Within this framework, the prospects of doing cosmology within Particle Physics become more promising, as Supersymmetry predicts that, for every known particle of the Standard Model, a superpartner should exist, with identical quantum numbers other than the spin. This means that a new plethora of scalar particles should be around, which become potentially interesting in the cosmological context. The goal of this article is to introduce scaling solutions in a cosmological set up and study their main characteristics. We will see how these solutions become particularly relevant in the context of theories beyond the Standard Model, in particular string/M-theory. They will help us understand better the possible ? E-mail: b.de-carlos@sussex.ac.uk c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 292 B. de Carlos: Cosmological scaling solutions roles that stringy moduli ?elds could play in the evolution of the Early Universe, which is a step forward in unifying Particle Physics and cosmology. 2 Scalar solutions in cosmology In this section we want to review the structure of the solutions to the evolution equations of a simple scalar ?eld in an expanding universe. We will start by writing up the simplest system one could imagine, with an exponential potential. Then we will look for solutions and classify them, and we will perform a phase space analysis. Finally we will devote some extra time to discussing the scaling solution in particular. This is mostly based on [1]. 2.1 Set up Our starting point will be a Friedman-Robertson-Walker universe, de?ned by the metric ds2 = ?dt2 + a2 (t)[dr2 + r2 (d?2 + sin2 ?d?2 )] , (1) where a2 (t) is the scale factor of the universe. Our system will include as well a perfect ?uid with barotropic equation of state p? = (? ? 1)?? , (2) where 0 ? ? ? 2, for example ? = 4/3 stands for radiation, ? = 1 for dust, or ? = 0 for cosmological constant. The ?nal element in this system is a scalar ?eld, ? with an exponential potential V (?) = V0 e??? , (3) with ? de?ning the slope of the potential. A useful quantity from now on will be the Hubble parameter, H, given by H? a? . a (4) We are now ready to write down the evolution equations for this system. 2.2 Evolution equations Those are derived from Einstein?s equations plus the equation of motion for the ?eld ?. A useful way of writing this system is ?2 (?? + p? + ??2 ) , 2 ??? = ?3H(?? + p? ) , H? = ? ?? = ?3H ?? ? (5) dV , d? where ?2 ? 8?G. There is a further equation, known as the Friedmann constraint, which for this system is given by ?2 1 ?? + ??2 + V . (6) H2 = 3 2 c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 4 ? 5 (2006) 293 In order to study the solutions to this system, it is convenient to perform a change of variables and rewrite as an autonomous system. De?ning ? ??? ? V x? ? ; y? ? , (7) 6H 3H the system of equations (5) can be rewritten as 3 2 3 x = ?3x + ? y + x[2x2 + ?(1 ? x2 ? y 2 )] , 2 2 3 3 y = ?? xy + y[2x2 + ?(1 ? x2 ? y 2 )] , 2 2 (8) where we have also changed the independent variable from t to N = lna, therefore ? constraint, Eq. (6), reads now ?2 ?? + x2 + y 2 = 1 . 3H 2 d dN = 1 d H dt . The (9) Other useful de?nitions are those of the scalar ?eld?s energy density, ?? = ??2 + V (?) , 2 (10) so that the fraction of the total energy carried by ? is ?? ? ? 2 ?? = x2 + y 2 . 3H 2 (11) Then, for ?? ? 0, we have the inequality 0 ? x2 + y 2 ? 1 . (12) Also, one can check that the system Eqs. (8) is symmetric under the transformations (x, y) ? (x, ?y) and t ? ?t, therefore it is enough to study the half plane y ? 0 in order to obtain all possible solutions. 2.3 Solutions We proceed now to solve the system Eqs. (8). The ?rst thing to do is to look for ?xed (i.e. stationary) points of the system. Those are de?ned by the condition x = y = 0, where ? has an effective equation of state given by ?? ? ?? + p ? 2x2 = 2 . ?? x + y2 (13) At those points the scale factor behaves as a ? tp with p = 2/(3?? ). The complete list of stationary solutions to x and y, together with their domain of existence, fraction of energy stored in ? and effective equation of state (?? ) are given in Table 1. By looking at the parameter space spanned by ? and ? we can distinguish three regions i) ?2 < 3? which presents three type of solutions ? two kinetic-dominated solutions, which are unstable nodes ? one ?uid-dominated solution, which is a saddle point www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 294 B. de Carlos: Cosmological scaling solutions Table 1 Fixed points for the system of equations given by Eq. (8), together with their domain of existence and energy and equation of state for the scalar ?eld ?. x y existence ?? ?? 0 0 all ? and ? 0 unde?ned 1 0 all ? and ? 1 2 all ? and ? 1 2 1 2 -1 ? ?/ 6 3?/(2?) 0 1? 3(2 ? 2 ?2 /6 ? <6 2 ?)?/(2?2 ) ? > 3? 3?/? ? /6 2 ? ? one scalar-?eld dominated solution, which is an attractor This latter solution gives rise to in?ationary expansion if ?2 < 2 (see after Eq. (13)) [2, 3]. ii) 3? < ?2 < 6 ? two kinetic-dominated solutions, which are unstable nodes ? one ?uid-dominated solution, which is a saddle point ? one scalar-?eld dominated solution, which is an attractor ? one scaling solution, which is a stable node or a spiral iii) 6 < ?2 ? two kinetic-dominated solutions, one of which is an unstable node (?x < 0) and the other (?x > 0) a saddle point ? one ?uid-dominated solution, which is a saddle point ? one scaling solution, which is a stable spiral These three regions of parameter space are depicted in Fig. 1. We can now discuss the cosmological 2 1.8 1.6 i)a 1.4 i)b a 1.2 iii) 1 0.8 0.6 ii) 2. 1. 0. 0 2 4 6 8 10 12 Fig. 1 Regions of parameter space, in the ?2 vs ? plane corresponding to the three cases described in the text. 2 h c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 4 ? 5 (2006) 295 Fig. 2 Phase space for the scaling solution corresponding to ? = 3, ? = 1. The late-time attractor is given by x = y = ? 1/ 6 (see Table 1). Figure taken from [1]. consequences of the existence of these solutions. Being such a simple model, it is surprising how many different features it can display. The most relevant ones are ? an in?ationary regime for ?2 < 2: this is unfortunately ruled out as a realistic model of in?ation, as it would predict too much energy density today. ? a scalar ?eld dominated solution for ?2 < 6: this represents an interesting possibility for quintessence. ? a scaling solution for ?2 > 3?: a scalar ?eld with a steep exponential potential would comprise a signi?cant fraction of the energy density of the universe today [4]. 2.4 The scaling solution This last feature is the one we want to study in the remaining of this lecture. The corresponding phase space is shown in Fig. 2. In order to make our discussion more physical, we would like to see explicitly what the scalar ?eld is doing during its evolution. In order to do that we have to perform, following [5], a further integration of the x and y variables, recall Eq. (7), to obtain an expression for ?(N ). The key point here is to realise that there are two main stages in the evolution of the scalar ?eld. I Pre scaling where, given that ? is very large, the potential is negligible and, therefore, y is neglected. Then the equation for x reads x = ?3x + 3 x[2x2 + ?(1 ? x2 )] , 2 (14) which has as solution ?1/2 1 ? x20 3(2??)N e . x= 1+ x0 (15) A further integration gives (with H = H0 e?3?N/2 ) ? x0 x0 2 6 ?1 ?1 3(2??)N/2 sinh ?I (N ) = ?0 + ? sinh . e 3(2 ? ?) 1 ? x20 1 ? x20 (16) II Scaling, which is characterised by x and y reaching their ?xed points, therefore x = y = 0. The solution is then 2?2 V0 3? 1 + N . (17) ?II (N ) = ln ? 9H02 (2 ? ?)? ? www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 296 B. de Carlos: Cosmological scaling solutions 1.2 1.1 q/MPl 1 0.9 0.8 Fig. 3 Evolution of the ? ?eld (in units of MP ) as a function of the number of efolds, N . Here the exponential potential is characterised by ? = 85 and the background by ? = 1. The initial conditions are given by H0 = 1, x0 = 0.2 and ?0 = 0.5. From [5]. 0.7 0.6 0.5 0 5 10 15 20 25 30 N A number of useful comments are in order here. The scaling regime is determined by a constant value of x which, therefore, means that the scalar ?eld is evolving at a constant pace in units of H. By looking at Table 1 we see that, in this case, the ?eld?s energy density is a fraction of that of the ?uid. We recall here that the scalar potential is a pure, steep, exponential. However, contrary to our naive expectations, if the ?eld enters a scaling regime, its evolution will be rather smooth, as it is shown in Fig 3. These results, as we will see in the next section, are very relevant to string theory, where exponential potentials happen quite often. 3 Application to string scenarios We are going to consider now the compacti?cation of D = 10, 11 string theories down to D = 4. Regardless of the speci?c string theory we consider, this process brings a number of common features which are relevant to us. ? A plethora of scalars in the N = 1, D = 4 effective Supergravity (SUGRA) action. These include, among others, the dilaton and geometrical moduli. ? Their dynamics are generated by both ?uxes and (non-perturbative) instanton effects. In either case exponential potentials are generated. ? An important point to take into account is that these scalar have, generically, non canonical kinetic terms. The issue of moduli (we shall use this word to refer to all stringy scalars in general) evolution and eventual stabilisation is a very important, and unsolved, one in the context of string cosmology and phenomenology. The real parts of these (complex) moduli must acquire non trivial vacuum expectation values as those measure physical quantities such as the size of the compacti?ed space. Therefore a good model of particle physics derived from strings should be able to provide vacua for these ?elds. Moreover, Supersymmetry (SUSY) should be broken in the process, as the world we observe certainly is non supersymmetric. Knowing the mechanism that breaks SUSY would allow us to compute the soft breaking terms and fully determine the low energy Lagrangian and particle spectrum. Finally, given the amount of scalar ?elds present in any effective Lagrangian coming from strings, it would be desirable that one of them would be responsible for in?ation in the Early Universe. As we can see the study of the phenomenology and cosmology of string models is a complicated one, given the number of problems that have to be addressed almost simultaneously. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 4 ? 5 (2006) 297 The topic of superstring cosmology has been addressed for a number of years now, and the main challenges were identi?ed in [6] (after earlier work in [7,8]). There, the possibility of the dilaton being the in?aton was addressed in the context of the weakly coupled heterotic string. Two were the main problems associated to this idea: the ?rst one was that, to that date (1993), all phenomenologically working models built had AdS vacua [9], i.e. negative vacuum energy. This made it quite dif?cult to in?ate. The second problem was that, even if the energy density had been positive at the vacuum, this is so tiny that the dilaton would overrun it. This is known as the runaway dilaton problem. 1� 8� -8 -9 6� -9 4� -9 2� -9 1� -33 8� -34 6� -34 4� -34 2� -34 7.5 8 8.5 9 Fig. 4 Plot of the potential V (?) as a function of ?. Example taken from [12]. The region between ? = 7 ad ? = 9 is being blown up in the box. 5 6 7 8 9 The problem of AdS vacua has been addressed in the past years, and the introduction of ?uxes among the dynamical effects one can consider for moduli, has partially solved the problem [10, 11]. It is the problem of the dynamical evolution the one we want to address here. In order to do that, let us illustrate it with a plot, Fig. 4. There we can see how much one has to blow up a certain region of the modulus potential (25 orders of magnitude) in order to be able to see the minimum. Obviously the naive expectation is that, if the dilaton starts evolving to the left of its minimum, it does not matter where, it will always ignore the minimum and evolve towards in?nity, which is totally unphysical. However, we can attempt to do things carefully and solve its equation of motion in the presence of an expanding universe and a background ?uid. In this part of the lecture we shall follow [5] and consider the dilaton (S) ?eld of the weakly coupled heterotic string. Its dynamics are determined by the N = 1, D = 4 Supergravity Lagrangian, L = KS S? ?� S? � S? + V (S) , (18) where K is the Ka?hler potential, a real function of the ?elds which, for the case of the dilaton is given by K = ?ln(S + S?) , (19) and the function KS S? is its second derivative with respect to both S and S?. V (S) is the scalar potential, which is determined by K and a second, holomorphic, function, the superpotential W (S) through the combination V (S) = eK [(Ki W + Wi )(Kij )?1 (K j W? + W? j ) ? 3|W |2 ] . (20) The kind of dynamics that W (S) encodes includes, in any working model giving rise to minima for S, nonperturbative effects. In particular, gaugino condensation [13,14] was used in the early nineties to construct the so-called racetrack models where the superpotential looked like [9] e??i S , (21) W (S) ? i with ?i related to the one-loop beta functions of the corresponding condensing groups. With this form for W (S) and Eq. (19), we obtain a potential which has a suitable minimum for S (characterised by a physically www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 298 B. de Carlos: Cosmological scaling solutions ?2 acceptable value of the string coupling constant, i.e. gstr ? ReS ? 2 in Planck units), although a negative vacuum energy, as mentioned above. Also, it is worth pointing out that, if one were to canonically normalise the Lagrangian Eq. (18), the potential should be written in terms of the ?eld ? where S = e? . That is, we are actually dealing with the exponential of an exponential, which makes the potential look even steeper. 5 10 (a) 0 10 � 10 V/M 4 Pl � 10 � 10 Fig. 5 Plot of the potential V (S) (in logarithmic units) as a function of ReS (solid line), as given in Eq. (20) and of the approximation V (S) = V0 e??ReS (dashed line). The expressions for V0 and ? can be found in [5]. � 10 � 10 � 10 0 0.5 1 1.5 2 2.5 3 Re S In principle it seems dif?cult to try to achieve any scaling in the simple way it was presented in the previous section. However, the key point is to realise that V (S) can be very ef?ciently represented by a single exponential in the region we are interested in studying, as can be seen in Fig. 5. There we can see that, to the left of the minimum the approximated potential given by just one exponential should be suf?cient to describe the evolution of the ?eld ReS. In fact, one can just rewrite the autonomous system in Eq. (8), for ReS = e? , as follows 3 2 3 ? y + x? [2x2? + ?(1 ? x2? ? y?2 )] , x? = ?3x? + ?e 2 ? 2 3 3 ? x? y? + y? [2x2? + ?(1 ? x2? ? y?2 )] , y? = ??e 2 2 3 H = ? [2x2? + ?(1 ? x2? ? y?2 )] , (22) 2 where we simply make the replacement ? ? ?e? (which is, incidentally, (dV /d?)/V ). One can now solve this system, and the numerical result is shown in Fig. 6. We can, as in the single exponential case, divide the evolution of ReS in different stages, namely I Pre scaling which, given that is independent of the shape of the potential V , would be represented by the same equations as in the previous case. II Scaling which is slightly different now, given that the new critical points for the system, in the scaling regime, are given by 3? c xS = , (23) 2? 3(2 ? ?)? 1 c yS = , 2?2 ReS and are, obviously, ?eld-dependent. The expression we obtained in Eq. (17) is now modi?ed due to the presence of this correction, which is nothing that a re?ection of the fact that the kinetic terms for S are not c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 4 ? 5 (2006) 299 3 (b) 2.5 Re S 2 1.5 1 0.5 0 2 4 6 8 10 12 14 Fig. 6 Evolution of the real part of the dilaton, ReS, as a function of the number of efolds N . for the full scalar potential de?ned by Eqs. (19,21) and different initial conditions. The thick solid line corresponds to the analytic result for the approximate potential given by the dashed line in Fig. 5 (from [5]). N canonically normalised. The new expression reads 3? 1 2?2 V0 + ReSII = ln N + (N ) , ? 9H02 (2 ? ?)? ? where 1 2?2 V0 3? 2 ln + N . (N ) = ? ln ? ? 9H02 (2 ? ?)? ? (24) (25) Finally, there is a third stage in the evolution of ReS, given that this potential has a minimum. Eventually, the scaling regime will end when such minimum is reached, i.e. III Minimum is characterised by ReSIII = ReSmin . We can now comment a bit further on Fig. 6. There we show the numerical evolution of the dilaton ?eld as a function of the number of efolds N . We have chosen a sample of initial conditions and we have also plotted the corresponding, analytic result for scaling, given by Eq. (17) in the case of the approximate potential given by one exponential (and represented as a dashed line in Fig. 5). As we can see, in the scaling regime the real trajectories are not straight lines, something which is taken into account by the small correction in Eq. (24). It is now straightforward to determine the region of initial conditions under which the ?eld ReS will end up at its minimum. Essentially, all trajectories reaching scaling before the minimum will end up in it, and this re?ects on conditions on the initial position of the ?eld, ReSII (0) < ReS0 < ReSmin . (26) Also, the initial velocity, x0 must be such that freezing, at the end of Stage I, occurs before the minimum, ? ReSmin 6 (2 ? ?)ln . (27) 0 < x0 < tanh 4 ReS0 In conclusion, a period of scaling before reaching the minimum would be enough for the dilaton to end up in it, despite of the steepness of its potential. This is very good news from the point of view of building a www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 300 B. de Carlos: Cosmological scaling solutions Fig. 7 Contour plot of the real part of the dilaton, ReS, (in terms of the normalised ?eld ?) and its imaginary part, ?i (normalised in units of ?) which shows the area of initial conditions for which the ?eld will end up at its minimum. The background, radiation, energy density is also indicated at the top (from [15]). viable phenomenology and cosmology based on string models. The runaway dilaton problem is therefore solved provided that there is a background ?uid present in the evolution. As it was mentioned before, the steepness of the moduli potential in string theory was one of the generic problems one had to face when trying to embed these ?elds in any cosmological set up. The other one mentioned was the fact that any vacua built in the nineties would have a negative vacuum energy. This latter problem was, as also mentioned above, partially solved when considering ?uxes as part of the moduli dynamics. It is then a combination of these ?uxes and non-perturbative effects what i) creates a minimum; ii) can give rise to a Minkowski or de Sitter vacuum. In the last part of this section we would like to comment, very brie?y, on how ?uxes affect the evolution of moduli and whether they affect them reaching a scaling regime. For more details the reader can consult [12,15]. The essence of the so-called KKLT models [10] is to consider a superpotential for a generic, single, modulus with a constant ?ux and a non-perturbative exponential term. Then a supersymmetry preserving minimum can be found for the scalar potential, which is invariably AdS. Finally, they add a D-term to the potential which will break Supersymmetry and also lift the vacuum energy to positive values. The ?rst thing to take into account is that, now, the presence of the constant (?ux) term in the superpotential forces us to consider as well the evolution of the axion ?eld, or imaginary part of the modulus. This axion was easily rotated away in the case of the weakly coupled heterotic string but now this is not possible anymore and its evolution must be considered. Therefore we will produce now contour plots of the regions of parameter space (ReS, ImS) which correspond to initial conditions for which the ?eld will end up at its minimum. This is what is shown in Fig. 7. The presence of two real ?elds and the more complex for of the potential make it impossible to obtain analytic approximations to their evolution. There are, however, a number of features common to all the cases we have studied, the main one being the presence of a period of scalar ?eld domination which precedes scaling. This essentially means that, in many occasions, the ?eld will reach scaling too late, i.e. beyond the position of its minimum and will simply runaway to in?nity. In that respect the addition of the D-term harms the chances of the ?eld ending up at its minimum, as it makes the potential shallower than a pure exponential one. 4 Conclusions In this lecture we have tried to explain the structure of scaling solutions in cosmology. We have ?rst chosen a very simple model, namely a scalar ?eld with an exponential potential which evolves in an expanding c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 4 ? 5 (2006) 301 universe in the presence of a background ?uid. We have solved the equations of motion and we have classi?ed all solutions as a function of the two relevant parameters in the model, the steepness of the potential and the type of background ?uid involved. Finally, we have elaborated further on the solution of interest to us, the scaling one. In the second part of the lecture we have applied the previous results to string inspired models. Having noted the similarities between the very simple model studied in the ?rst section and a typical modulus potential, we have solved the equations of motion for the latter (again in an expanding Universe and in presence of a perfect ?uid), and we have con?rmed our expectations that a period of scaling is possible within these models. We have, therefore, found an elegant solution to the dilaton (or moduli, in general) runaway problem, which was one of the main obstacles for working our a successful cosmology based on string theory. Acknowledgements I would like to thank my collaborators, Tiago Barreiro, Ed Copeland and Nelson Nunes, with whom it is a pleasure to work, and the organisers of this meeting for the very stimulating working environment they created. This work is supported by PPARC. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] E. J. Copeland, A. R. Liddle, and D. Wands, Phys. Rev. D 57, 4686 (1998) [arXiv:gr-qc/9711068]. B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988). P. G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503 (1998) [arXiv:astro-ph/9711102]. C. Wetterich, Nucl. Phys. B 302 (1988) 645. T. Barreiro, B. de Carlos and E. J. Copeland, Phys. Rev. D 58, 083513 (1998) [arXiv:hep-th/9805005]. R. Brustein and P. J. Steinhardt, Phys. Lett. B 302, 196 (1993) [arXiv:hep-th/9212049]. M. Dine and N. Seiberg, Phys. Lett. B 162, 299 (1985). N. Kaloper and K.A. Olive, Astropart. Phys. 1, 185 (1993). B. de Carlos, J.A. Casas, and C. Munoz, Nucl. Phys. B 399, 623 (1993) [arXiv:hep-th/9204012]. S. Kachru, R. Kallosh, A. Linde, and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240]. C. P. Burgess, R. Kallosh, and F. Quevedo, J. High Energy Phys. 0310, 056 (2003) [arXiv:hep-th/0309187]. R. Brustein, S. P. deAlwis, and P. Martens, Phys. Rev. D 70, 126012 (2004) [arXiv:hep-th/0408160]. J. P. Derendinger, L. E. Ibanez, and H. P. Nilles, Phys. Lett. B 155, 65 (1985) M. Dine, R. Rohm, N. Seiberg, and E. Witten, Phys. Lett. B 156, 55 (1985). T. Barreiro, B. de Carlos, E. Copeland, and N. J. Nunes, arXiv:hep-ph/0506045. www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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