Stringy bounces and gradient instabilities

Bouncing solutions are obtained from a generally covariant action characterized by a potential which is a nonlocal functional of the dilaton field at two separated space-time points. Gradient instabilities are shown to arise in this context but they are argued to be nongeneric. After performing a gauge-invariant and frame-invariant derivation of the evolution equations of the fluctuations, a heuristic criterion for the avoidance of pathological instabilities is proposed and corroborated by a number of explicit examples that turn out to be compatible with a quasi-flat spectrum of curvature inhomogeneities for typical wavelengths larger than the Hubble radius.


Introduction
The temperature and polarization anisotropies of the Cosmic Microwave Background can be successfully reproduced by assuming that the initial conditions of the Einstein-Boltzmann hierarchy are predominantly adiabatic and Gaussian [1,2]. Inflationary models are consistent with the presence of a dominant adiabatic mode but the smallness of the tensor to scalar ratio calls for a plateau-like potential in the Einstein frame implying a minute energy density of the inflaton in Planck units (typically ρ inf = O(10 −12 ) M 4 P ). A very plausible chain of arguments [3] stipulates that the kinetic energy of the inflaton and the spatial curvature must then be comparable with the inflaton potential if we do not want the kinetic energy and the spatial curvature to dominate even before inflation starts. This occurrence can be quantitatively scrutinized by following the evolution of the spatial gradients [4] during the preinflationary phase. A complete theory of the initial conditions should account for the emergence of the observable Universe from a sufficiently generic set of initial data. It might well be, however, that inflation should be primarily regarded as a model of the spectral indices rather than a theory of the initial data.
Since bouncing models more often than not lead to a small tensor components over large distance scales, they have been intensively investigated in the last twenty years with various independent motivations ranging from string inspired models to different classes of scenarios concocted in the framework of effective theories (see, for instance, [5] for some reviews on the subject). The current versions of bouncing models have many virtues but they also have various well known problems which have been scrutinized in the past and are currently under active consideration [6,7]. Some of the current approaches even renounce general covariance and often impose ad hoc symmetries (like the shift symmetry).
An interesting class of bouncing models can be obtained, in the low-curvature regime, by the contribution of a nonlocal, though generally covariant, dilaton potential [8]. Even if the potential is nonlocal, the corresponding evolution equations of the background and of the corresponding fluctuations are perfectly local in time. Bouncing solutions have been studied in this framework. The production of massless quanta (e.g. gauge bosons) could heat the background, stabilize the dilaton and eventually provide an exit to radiation (see, in particular, first paper of [8]). The idea that radiation can arise from the backreaction of the produced quanta goes back to the seminal contributions of various authors [9] (see, in particular, the analyses of Parker and Ford). Bouncing solutions may also arise in the context of double field theory [10] which was firstly proposed to realize T -duality explicitly at the level of component fields of closed string field theory; earlier contributions along this direction can be found in [11]. Since the nonlocal potential of [8] depends on a T -duality invariant combination, the corresponding solutions can also be interpreted in a double field theory context [12].
Various bouncing models may experience the so called gradient instability stipulating that either the scalar or the tensor modes of the geometry inherit an imaginary sound speed. The corresponding fluctuations are then exponentially unstable for sufficiently small wavelengths. To cure this pathology non-covariant terms are often added to the action of the fluctuations with the purpose of modifying the effective sound speed without disturbing too much the evolution of the background. In the present investigation, after discussing in detail the derivation of the scalar sound speed, it will be demonstrated that gradient instabilities take place also in the context of bouncing models induced by a nonlocal dilaton potential. Unlike other modes, however, these instabilities do not arise ubiquitously for any form of the dilaton potential and are, in this sense, nongeneric. We provide examples of semi-realistic backgrounds where the instabilities are tamed and the spectrum of the scalar modes turns out to be quasi-flat.
We remark that while the scalar modes of the geometry may evolve differently in the Einstein and in the string frames, the curvature perturbations on comoving orthogonal hypersurfaces are both gauge-invariant and frame-invariant. The corresponding evolution equations inherit an effective sound speed. Such a terminology is justified since the pivotal equation derived in this context coincides, formally, with the equation derived long ago by Lukash [14] and describing the excitations of a relativistic and irrotational fluid in a Friedmann-Robertson-Walker background. Note, however that in the case of Ref. [14] the sound speed is the square root of the ratio between the derivatives of the pressure and on the energy density of the relativistic fluid, as it should be by definition of the sound speed; in the present case, however, the effective sound speed will ultimately be a functional of the potential. A valid help for this analysis is represented by the uniform dilaton gauge which has been discussed in various related contexts [15,16]. In this gauge the dilaton remains unperturbed, the curvature perturbations coincide with the longitudinal degrees of freedom of the perturbed metric and the scalar modes are automatically frame-invariant.
The present paper is organized as follows. In section 2 the bouncing models induced by nonlocal dilaton potentials are introduced. In section 3 we derive explicitly the scalar sound speed and discuss the problem of the gradient instability. The genericness of the gradient instability is scrutinized in section 4 with the aim of constructing solutions where this pathology does not arise. A number of potential drawbacks are anyway suggested in the last part of section 4. Finally section 5 contains the concluding remarks. Various results have been presented and summarized in a self-contained perspective in the appendix A.

Nonlocal potentials and bouncing solutions
A smooth bouncing transition at low curvatures can be achieved in the framework of the following generally covariant action in four space-time dimensions [8]: where we have that V = V (e −ϕ ) and For immediate convenience we also define the following pair of integrals: with a prime denoting differentiation with respect to the argument 2 . The present analysis can be generalized to include antisymmetric tensor fields, gauge fields as well as internal (contracting) extra-dimensions [13].These potential contributions will be ignored and we shall focus on the minimal four-dimensional scenario compatible with the presence of adiabatic curvature perturbations. The variation of the action (2.1) with respect to g µν and ϕ leads, respectively, to the following pair of equations: where the shorthand notation g αβ ∂ α ϕ∂ β ϕ = (∂ϕ) 2 has been consistently employed; in Eq. (2.5) G µν denotes the Einstein tensor and ∇ 2 is defined as: Note that, by definition, we also have the following chain of equalities: A detailed derivation of Eqs. (2.4) and (2.5) has been discussed elsewhere [8] and here we shall only focus on those aspects that are germane to the main theme of the analysis. By combining Eqs. (2.4) and (2.5) the Ricci scalar can be eliminated and the following equation is readily obtained: In Eq. (2.5) as well as in Eqs. (2.3) and (2.8) the prime denotes a derivation with respect to the argument of the given functional and not the derivative with respect to the conformal time coordinate. The two notations cannot be confused since the conformal time derivative only appears in connection with the explicit form of the equations discussed in Eqs. (2.11)-(2.13) and in the forthcoming sections.
In the case of a homogeneous dilaton and for a conformally flat metric of Friedmann-Robertson-Walker type Eqs. (2.4) and (2.8) lead, in units 2λ 2 s = 1, to the following system of equations 3 where the overdot denotes a derivation with respect to the cosmic time coordinate t; as usual H =ȧ s /a s and, in the homogeneous limit ϕ = ϕ − 3 ln a s . Equations (2.9)-(2.10) hold in the string frame 4 . For the discussion of the fluctuations it is practical to write Eqs. (2.9)-(2.10) in the conformal time coordinate τ and, eventually, in the Einstein frame. Unlike the cosmic time coordinate the conformal time is frame-invariant (i.e. τ s = τ e = τ ). The equations for the background in the Einstein frame are 5 :  (2.13). Note finally that the energy density and pressure in the Einstein frame description are simply given by: (2.14) In terms of ρ e and p e Eq. (2.13) simply becomes ρ e + 3H e (ρ e + p e ) = 0, as expected.
As discussed in the introduction, even if the potential is nonlocal in field space the evolution equations of the background are local in time [8]. The effect of the nonlocal modification of the action (2.1) is however apparent in the modification of the equations of motion in a way that makes possible the bouncing solution. To appreciate this important 3 The discussion of the gradient instability and of its implications can be easily extended to the case D = d + n + 1 where d and n denote the number of external and internal (i.e. compactified) dimensions [13]. Since this is analysis is not central to the theme of this paper, it will be omitted. 4 The evolution of the background (and of its fluctuations) can be described either in the string frame (where the dilaton and the Ricci scalar are explicitly coupled) or in the Einstein frame. A self-contained discussion of the relation between the two conformally related frames can be found in appendix A. point, let us suppose, for a moment, that the potential term in Eq. (2.1) is just a local function of ϕ, i.e. V = V (ϕ) and not V = V (ϕ), as Eq. (2.2) stipulates. To avoid confusions between the two situations let us write, for notational convenience, that the local potential corresponds to W (ϕ). In this case the evolution equations can be immediately obtained either in the string frame or in the Einstein frame. To make the comparison more clear let us therefore write the analog of Eqs. (2.11)-(2.13) when the action is given by Eq. (2.1) but the potential is W (ϕ)  (2.19) showing that in this case, depending on the sign of the derivative of V with respect to ϕ, bouncing solutions become possible. All in all we can therefore conclude that even if the evolution equations (2.11)-(2.13) are local in time, the nonlocality of the potential which depends on the shifted dilaton ϕ is reflected in a substantially different form of the equations which cannot be mimicked by a potential term depending only on ϕ.
The essential features of Eq. (2.19) are not affected by the isotropy of the metric. Indeed when the potential depends on the shifted dilaton the solutions can be anisotropic both in 3 + 1 dimensions as well as in higher dimensions [8]. In particular 10 dimensional solutions have been used to discuss the evolution of the vector modes of the geometry (see, in this respect, the first paper of Ref. [8]). Clearly depending on the frame where the solutions are described the features of the solutions will be slightly different. In the anisotropic case, however, the solutions will be of Kasner type: some of the dimensions will expand and some will contract.
Let us finally remark, as we close the section, that the potentials depending on the shifted dilaton may be interpreted as string loop corrections preserving the T -duality symmetry [8]. This interpretation is particularly intriguing in the light of the potential interpretation of the solutions in the light of double field theory [10,11,12]. Having said this, not all the potentials depending on ϕ induce bouncing solutions: this method of regularizing the bouncing solutions remains after all heuristic. This observation is of course a source of concern but it should be contrasted with the effective approaches to bouncing solutions [6,7].

Gradient instabilities
In the present framework the gradient instabilities do not affect the evolution of the tensor modes while the scalar modes inherit an effective sound speed which may get imaginary. Denoting, in general terms, by c 2 t and c 2 s the sound speeds of the tensor and of the scalar modes of the geometry we have that a gradient instability is said to arise when either c 2 t < 0 or when c 2 s < 0. The tensor modes of the geometry are both gauge-invariant (i.e. invariant under infinitesimal coordinate transformations) and frame-invariant (i.e. invariant for the transition from the string to the Einstein frame). The scalar modes are not automatically frame-invariant but, as we shall discuss the gauge-invariant curvature perturbations are also frame-invariant (see also appendix A). Finally, as already mentioned, the conformal time coordinate (unlike the cosmic time parametrization) is frame-invariant.
The evolution equations of the fluctuations in the string frame can be obtained by perturbing to first-order Eq. (2.4). Denoting by δg µν the total fluctuation of the metric, the tensor, scalar and vector modes will be: where the subscripts remind, respectively, of the tensor, scalar and vector perturbations. We shall discuss hereunder the tensor and the scalar modes of the bounce. As a consequence of the bounce the vector zero modes may increase, but their fate depends upon the dynamics. This problem has been specifically addressed (see second paper in Ref. [8]) and the result can be summarized by saying that for non-singular bounces in four-dimensions the growing mode customarily present during the contracting phase matches with the decaying mode after the bounce 6 .

Evolution of the tensor modes
In spite of the specific background solution deduced from the action (2.1) the tensor modes do not experience any sort of gradient instability. To demonstrate this statement the tensor fluctuations can be defined, in the string frame, as: where a s denotes the scale factor in the string frame metric and δ t stands for the tensor fluctuation of the corresponding quantity. Let us therefore perturb Eq. (2.4) when the indices are mixed (i.e. one covariant and the other contravariant): All the quantities not preceded by δ t in Eq. (3.9) must be understood as evaluated on the background. Recalling Eq. (3.2) the explicit fluctuations of the Einstein tensor and of the Christoffel connections can be readily computed; the equation for h ij becomes then: where the prime denotes the derivation with respect to the conformal time coordinate; the relation between H s and H s (defined after Eqs. (2.9)-(2.10)) is given as usual by H s = a s H s . The equations for the tensors in the Einstein frame are given by: Equation (3.5) can be obtained from Eq. (3.4) by appreciating that the tensor amplitudes are frame-invariant (see appendix A for further details). All in all Eqs. (3.4) and (3.5) show that there are no problems with the gradient instability in the case of the tensor modes which are automatically gauge-invariant and frame-invariant. This conclusion should be contrasted with the remark mentioned at the beginning of this section where, in general terms, it has been said that the tensor modes may inherit an effective sound speed. This can happen, for instance, if the bounces are described using effective field theory methods [6,7] analog to the one employed in the context of inflationary modes [20].

Evolution of the scalar modes
The scalar fluctuations of the geometry are given by: where δ s denotes the scalar fluctuation of the corresponding quantity. We fix the coordinate system by setting to zero the perturbation of the dilaton field and of the off-diagonal fluctuations of the metric. This is often referred to as the uniform field gauge [15,16] and it is particularly practical in the present context. Since the dilaton and its fluctuation are frameinvariant we can denote by δ s ϕ = χ the common value of the dilaton fluctuation either in the Einstein or in the string frame. The uniform field gauge stipulates that χ = 0 and B = 0. Clearly for an infinitesimal coordinate transformation 7 parametrized as x µ → x µ = x µ + µ the fluctuation of a rank-two tensor in four-dimensions (like δ s g µν ) change according to the Lie derivative in the direction µ . In explicit terms we will have that where we also reported, for immediate convenience, the gauge transformation of χ. If we start from a generic gauge we can arrive at the uniform dilaton gauge by setting 0 = χ/ϕ . Furthermore by settingB = 0 in Eq. (3.7) the value of can be determined and it is: where c 1 is an integration constant which does not depend on the conformal time coordinate. Since gauge freedom is not completely fixed the evolution equations of the scalar modes (see below) depend on E (and not on E) which is related to the gauge-invariant dilaton fluctuation in the uniform field gauge 8 . In the case of the scalar modes of the geometry the perturbed version of Eq. (2.4) can be written in a more explicit form as: (3.9) where, as in the case of Eq. (3.3), all the quantities not preceded by δ s in Eq. (3.9) must be understood as evaluated on the background. Equation (3.9) shall be further simplified by noting that, in the uniform field gauge, the scalar fluctuations of e −ϕ , I 1 and γ ν µ vanish (i.e. δ s (e −ϕ ) = δ s I 1 = δ s γ ν µ = 0). Using the decomposition (3.6) and imposing the uniform dilaton gauge, the (00) and (0i) components of Eq. (3.9) can be respectively written as: Similarly the component (i = j) of Eq. (3.9) becomes: Finally the explicit form of the (i = j) component of Eq. (3.9) is given by: The following notations shall be employed: Indeed the gauge-invariant fluctuation of the dilaton (see also appendix A) is frame-invariant and it is give by The coefficient of φ in Eq. (3.13) can be expressed as (a 2 s /2)(∂V /∂ϕ) − V a 2 s by repeated use of Eqs. (A.9) and (A.10). In the uniform field gauge the fluctuations of the spatial curvature are given solely in terms of ψ: Up to a sign which changes depending on different conventions, the value of ψ in the uniform field gauge coincides with the curvature perturbations customarily indicated by R. Strictly speaking R defined the curvature perturbation on comoving orthogonal hypersurfaces. However, since R is by definition gauge invariant its evolution equations can be derived in any gauge, such as the one employed in the present discussion 9 . The strategy will therefore to derive the equations first in the string frame by combining Eqs. (3.10)-(3.11) and (3.12)- (3.13). From Eq. (3.11) we have φ = 2ψ /(ϕ + H s ); inserting this relation into Eq. (3.10) the following expression can be readily obtained: According to Eq. (3.14) in the uniform dilaton gauge R = −ψ; in the same gauge we can compute the Bardeen potential and the result is Ψ e = ψ+H e E (see also appendix Eq. (A.8)).
We thus obtain from Eq. (3.15) the following simple equation R = −4(H e /ϕ 2 )∇ 2 Ψ e . Let us finally mention that the relation φ = 2ψ /(ϕ + H s ) derived from Eq. (3.11) implies that Eq. (3.13) is identically satisfied and does not imply further conditions. Indeed inserting φ = 2ψ /(ϕ + H s ) into Eq. (3.13) we have: 17) but this is an identity since the expression between square brackets vanishes on the background. In fact, inserting the two relations of Eq. (A.9) into Eq. (A.10) we obtain that the combination appearing in Eq. (3.17) between square brackets is bound to vanish.

Frame-invariance and scalar sound speed
To derive the gauge-invariant and frame-invariant evolution of the curvature perturbations it is appropriate to rewrite the system of scalar perturbations in a more compact form by 9 In what follows we shall deduce the equation for R and we shall demonstrate that R is not only the correct gauge-invariant variable to be used but it is also frame-invariant. See, in this respect, also the discussion of appendix A.
using Eqs. (A.9)-(A.10): 20) where the new background variable y 1 = ϕ + H s has been introduced. It is now evident from Eqs. (3.18), (3.19 and (3.20) that once the second relation of Eq. (3.18) is inserted into (3.19) an identity is swiftly obtained. Using then y 1 φ = 2ψ into Eq. (3.20) the background equations imply: where the background combination y 2 = ϕ + 3H s has been defined. Equation (3.21) has the same dynamical content of Eq. (3.15) but it is more practical. Let us now take the conformal time derivative of both sides of Eq. (3.21) and replace the terms ∇ 2 E and ∇ 2 E by means of Eqs. (3.18) and (3.21). The final result of this lengthy but straightforward procedure is the following decoupled equation for ψ : Recalling now the explicit expressions of y 1 and y 2 we can rewrite some of the background dependent quantities appearing in Eq. (3.22). In particular the following identity is verified Thus from Eq. (3.22) the evolution equation for curvature perturbations R s = −ψ becomes: 3.24) and the sound speed squared is: The second equality in Eq. (3.25) follows directly from the expressions of y 1 and y 2 ; note, in particular, that y 2 = ϕ + 3H s ≡ ϕ and y 1 = y 2 − 2H s . It is furthermore easy to prove that a combination of Eqs. (A.11) and (A.12) implies the following identity When ϕ → ϕ and H s → H s = H e + ϕ /2 (see also Eq. (A.3)) we have also have that z s → z e and c 2 s → c 2 e , as expected 10 . Unlike the Bardeen potentials (which are gauge-invariant but not necessarily frame-invariant), the variable R will then denote the common value of the curvature perturbations either in the string or in the Einstein frame (i.e. R = R e = R s ). This is in full analogy with the case of the tensor modes of the geometry discussed at the beginning of this section.
It is natural, at this point, to identify c 2 s or c 2 e with an effective sound speed. We note that this identification rests on the analogy of Eqs. (3.24) and (3.28) with the equation describing the normal modes of a gravitating, irrotational and relativistic fluid firstly discussed by Lukash [14] R where p t and ρ t denote, in the context of Eq. (3.30) the pressure and the energy density of a perfect and irrotational fluid. The canonical normal mode identified in Ref. [14] is invariant under infinitesimal coordinate transformations as required in the context of the Bardeen formalism [17] (see also [18]). The subsequent analyses of Refs. [19] follow the same logic of [14] but in the case of scalar field matter; the normal modes of Refs. [14,19] coincide with the (rescaled) curvature perturbations on comoving orthogonal hypersurfaces [17,19].
Owing to the analogy of Eq. (3.28) with the Lukash equation (3.30) it would be tempting to carry this analogy even further by recalling that the evolution equations in the Einstein frame can be phrased in terms of an effective energy density ρ e supplemented by an effective pressure (see, in this respect, Eq. (2.14) and discussion therein). This literal correspondence is however misleading since z e = a 2 e √ p e + ρ e /H e as it would follow from a naive combination of Eqs. (2.14) and (3.30). This is why, in the present context, we shall always qualify the sound speed as effective.
The evolution equations of the linearized fluctuations discussed here all local in time however it is true that the presence of the potential induces a sound speed for the scalar modes of the geometry. This is particularly clear if we notice that the sound speed goes to 1 in the limit of vanishing nonlocal potential: if the potential would just be local (i.e. only dependent on ϕ) the sound speed would coincide with the speed of light. It is therefore correct to conclude that the presence of a potential depending on the shifted dilaton ϕ induces a scalar sound speed.

Imaginary sound speed and gradient instability
We are now going to show that a four-dimensional curvature bounce connecting analytically two duality related solutions leads to a gradient instability. For those wavenumbers k exceeding the typical scale of the bounce (of the order of 1/t 0 in the example discussed below), the solutions of the evolution equation for R instead of oscillating are exponentially amplified. To demonstrate this point let us focus on the following well known solution (see [8]): If we now compare Eqs. (3.4) and (3.24) in the light of the solution (3.31) and (3.32) we conclude that the evolution equation for the tensor modes obtained in Eq. (3.4) does not lead to any gradient instability. The effective sound speed c s appearing in Eq. (3.24) can instead become imaginary when |t| < t 0 , i.e. exactly in the regime where the potential modifies the asymptotic "vacuum" solutions: this means that in the limit kt 0 > 1 and for |t| < t 0 the corresponding Fourier modes of R are exponentially amplified. To prove this statement we can compute c 2 s and simply demonstrate that it is not positive semidefinite. Indeed using Eqs. (3.31) and (3.32) inside Eq. (3.27) we obtain where the first equality merely follows from Eq. |t| < t 0 the square of the effective sound speed can be expanded as When kt 0 1 (or for kt 0 ∼ O (1)) values c 2 s < 0 around the origin are never problematic and typically lead to a steeply increasing spectrum of scalar modes [8]. However when kt 0 1 we can have that the Laplacian of Eq. (3.24) becomes −c 2 s (t)∇ 2 R → k 2 c 2 s (t); but since this term gets sharply negative all modes kt 0 are exponentially amplified for |t| < t 0 . Since the whole description of curvature perturbations is frame-invariant, the occurrence of the gradient instability is also frame-invariant If we would assume that gradient instabilities are generically present in bouncing model the problem could be cured (or at least alleviated) by the arbitrary addition of further terms in the action of the scalar modes of perturbations. For instance the evolution equations of curvature perturbations could be written as: where M is a mass scale possibly related to the maximal curvature scale reached at the bounce, i.e. M = O(H max a max ) = O(1/τ max ). In the limit q 1 (τ ) = q 2 (τ ) = 0 the variation of the second-order action of Eq. (3.35) leads immediately to Eq. (3.24). Conversely if these two terms are present the evolution equation of R can be written as: Let us now suppose, for the sake of concreteness, that the two background dependent functions q 1 (τ ) and q 2 (τ ) will be continuous and differentiable everywhere. Going to Fourier space and rearranging the terms of Eq. (3.36) 3.37) where κ = k/(a max H max ). Even if c 2 s becomes negative in Eq. (3.37), c 2 tot may well be positive depending on the sign of q 1 and q 2 . More specifically let us suppose that c 2 s becomes negative in the neighbourhood of the origin (say for |τ | < τ 0 ); even if, in the same region, q 1 (τ ) > 0 we will have that c 2 tot ≥ 0 for κ 1 provided q 2 (τ ) ≥ 0 when |τ | < τ 0 . Note that corrections such as the ones appearing in Eq. (3.35) arise naturally when applying the methods of effective field to the analysis of generic theories of inflation with a single inflaton field [20]. In generic theories of inflation the dependence of the action on the inflaton field is unconstrained and a similar analysis translates to the case of bouncing models even if a shift symmetry may be imposed on the action so that the Lagrangian density will involve only spacetime derivatives rather than the field itself [6,7]. We will now argue that the gradient instability of Eqs. (3.33) and (3.34) is a property of the solution but not necessarily a property of the model. In other words we can expect that by changing the solution also the gradient instability might disappear.

Taming the gradient instability
The gradient instability arising in the bouncing models based on the action (2.1) and scrutinized in section 3 is nongeneric. For this purpose, a purely qualitative analysis shows that that the sign of the effective scalar sound speed, in both frames, is determined by the sign of the derivative of V (ϕ) with respect to ϕ. Let us now consider the form of the potential reported in Eq. (3.31): away from the bounce, the potential and its derivative are both subleading and this is the reason why, incidentally, the solution in the asymptotic regions exactly matches the (duality related) vacuum solutions. Near the bounce the contribution of the potential and of its derivative always enters the effective scalar sound speed with a negative sign: this is ultimately the reason why c 2 s < 0 for |t| < t 0 . To construct examples where the gradient instability is tamed we must therefore consider more seriously those models where ∂V /∂ϕ does not have a definite sign in the bouncing region.

A class of semi-realistic backgrounds
Guided by the logic spelled out in the previous paragraph let us therefore consider, for the sake of concreteness, the following class of potentials whose derivative with respect to ϕ changes sign for ϕ = ϕ 1 . Inserting Eq. (4.1) into Eqs.
(2.11), (2.12) and (2.13) the evolution equations of the background can be solved in explicit terms. Without loss of generality we shall focus on the solution in the Einstein frame and in the conformal time parametrization which is the most convenient for the analysis of the inhomogeneities. Note, in particular, that in the Einstein frame the following useful relation can be obtained after repeated combinations of Eqs. (2.11), (2.12) and (2.13): The bouncing solution corresponding to Eq. (4.1) can then be written as: where τ 1 denotes the typical scale of the bounce and H = a 1 H 1 = 1/τ 1 ; in the rescaled coordinate x the bouncing region corresponds to |x| < 1. To satisfy consistently all the equation the relation between the integration constants H 1 , V 1 and ϕ 1 must be given by The solution (4.3), (4.4) and (4.5) holds in the Einstein frame and (most importantly) in the conformal time parametrization 12 . From Eqs. (4.3) and (4.4) the total sound speed of the scalar fluctuations can be easily computed from the general expression and the result of this manipulation is given by: Note that once expressed in the conformal time coordinate and on the explicit solution the sound speeds in the string and Einstein frames coincide (i.e. c 2 e (τ ) = c 2 s (τ )). From Eq. (4.6) the overall sign of c 2 e is determined by the value of beta. In principle we could concoct a value of beta leading to c 2 s > 0 and simultaneously describing an accelerated contraction before the bounce. However, for the sake of concreteness, it seems more useful to compute the spectrum and select those values of β allowing for a quasi-flat spectrum of curvature perturbations, as we shall show in a moment.
Even if Eq. (4.5) does not allow for an analytic connection between the conformal and the cosmic time coordinate over the whole range of variation of τ (or x) the relation of τ to t can easily be determined piecewise in the asymptotic regions. In particular in the limits τ −τ 1 and τ τ 1 the scale factor evolves, respectively, as , (4.8) where the last two equalities at the right hand side follow from the relation between the cosmic and the conformal time coordinate (i.e. a e (τ ) dτ = dt e ). The solution of Eqs. (4.1) and (4.3)-(4.4) is a special case of a more general set of solutions characterized by where f 1 is a dimensional constant. By making use of Eq. (4.9) we can easily write the explicit form of the sound speed. More specifically thanks to Eqs. (2.11), (2.12) and (2.13) the result can be expressed as c 2 s = 1 − N s /D s where N s and D s can be written, respectively, as: (4.10) Note that H e is dimensionless, i.e. H e = H 1 H e and λ = f 1 /H 1 is also a dimensionless constant. For a e a 1 (i.e. away from the bounce) the terms weighted by λ are always negligible and Eq. (4.10) reduces to (4.11) As far as the power spectra are concerned what matters is not the behaviour in the region |x| < 1 (i.e. |τ | < τ 1 ) but rather the evolution in the asymptotic regions and, in particular, in the pre-bounce stage. In this respect we can mention that there are different solutions with the same asymptotic behaviour of Eq. (4.3) which have however a different analytical structure in the region |x| < 1. A class of models sharing this property is given by . (4.12) where γ = 1, 2, . . . is an integer. Note that the solution of Eq. (4.5) corresponds to γ = 1 in Eq. (4.12). For γ > 1 the analytical behaviour of the solution is different from the ones discussed before. The relevant point, however, is not the specific analytic form of the solution but rather the positivity of c 2 e which can be realized in different ways.

The spectrum of inhomogeneities
The results obtained so far demonstrate that the occurrence of the gradient instability is nongeneric when the bounce is regularized by means of a nonlocal dilaton potential. We now want to fix the value of β in a more realistic realistic way. In what follows we shall analyze specifically the class of models of Eqs. (4.6) and (4.7) and show that the spectrum of curvature perturbations may indeed be flat when β −1/4. From Eq. (4.7) the first and second (cosmic) time derivatives of the scale factor in the asymptotic region t < −t 1 are given, respectively, by: (4.13) Equation (4.13) indicates that the case of accelerated contraction (i.e.ȧ e < 0 andä e < 0) is realized when β < 0. Thus values β < 0 will be regarded as the most physical ones. Different choices are possible but they are not central to the present discussion. The amplified curvature perturbations in the case where the relevant modes exited the Hubble radius for τ < −τ 1 will now be computed. This terminology is inaccurate but often used. What matters here is not the Hubble radius itself but the nature of the pump field governing the evolution of the scalar modes. The exit refers here to the moment where pump field z e /z e equals approximately c 2 e k 2 (see below Eq. (4.19)). At this turning point the solution of the mode functions change behaviour. Since the initial conditions of the curvature perturbations are set by quantum mechanics, it is essential to remind on the canonical structure of the problem. To be specific we could say that the quantization of the fluctuations follows exactly the same steps outlined by Lukash [14] when discussing the scalar modes of an irrotational relativistic fluid: unlike Ref. [14] the sound speed is only effective. The evolution equation (3.28) can be obtained by functional variation from the following action (4.14) which falls into the same equivalence class of Eq. (3.35). The normal modes of Eq. (4.14) are q = z e R; inserting the normal modes in Eq. (4.14) and dropping an irrelevant total time derivative we get: (4.15) so that a convenient form of the canonical Hamiltonian can be obtained (4.16) whereπ = q . We can therefore promote q and π to field operators obeying equal-time commutation relations, i.e. [q( x, τ ),π( y, τ )] = iδ (3) ( x − y) in unitsh = 1. We eventually want to compute the spectrum ofR =q/z e so that we can write the Fourier representation directly forR:R (4.17) where [â k ,â † p ] = δ (3) ( k − p). The evolution of the mode functions F k (τ ) and F * k (τ ) can be immediately deduced from Eqs. (3.28), (4.14) and (4.15). More specifically F k will obey F k + 2 z e z e F k + k 2 c 2 e F k = 0. (4.18) The limit in Eq. (4.21) follows easily from the first term when x < −1 (i.e. τ < −τ 1 ). Consequently the solution of Eq. (4.21) with the correct boundary conditions can be written, for β < 0, as: (4.22) where N = e iα/2 π/2 and α = i(µ + 1/2)π. The two-point function computed from Eq. (4.17) is then given by The expectation value in Eq. (4.23) is performed over the state minimizing the Hamiltonian of the fluctuations. Since in the limit τ → −∞ the space-time is flat the initial state is well defined. The scale-invariant limit for the large-scale modes is therefore realized when µ 3/2 which implies that for β −1/4 the spectrum of curvature perturbations is quasiflat and the scale-invariant limit is reached in the case β → −1/4. In summary we can say that the gradient instability does not generically appear in bouncing cosmologies constructed from the action (2.1). We even presented a series of examples where the bounce is correctly regularized, the gradient instability does not arise and the Universe evolves from a stage of decelerated contraction.

Post-bounce evolution and potential drawbacks
The sound speed obtained within the strategy presented in this paper does not lead to a gradient instability (i.e. c 2 e is correctly positive semidefinite) but the condition of subluminal sound speed (i.e. c 2 e ≤ 1) is not always respected. This is an improvement in comparison with the original instability. It is however clear that this class of models can only be viewed as semi-realistic even if there are other bouncing models with similar drawbacks [6,7]. Another point of concern is that the obtained bouncing backgrounds do not exit to radiation but to an expanding solution. As argued in the past this problem is closely related to the dilaton stabilization. This drawback is common to other scenarios and it is potentially very serious. In this respect the idea that backreaction effects can produce dynamically a radiation-dominated background has been previously discussed. The possibility of a gravitational reheating of the Universe was pointed out by various authors [9] and here we shall follow, in particular, the approaches developed by Parker and Ford. In [8] a model of gravitational heating of the cold bounce has been proposed by considering the effects coming from the production of Abelian gauge bosons which are directly coupled to ϕ and are copiously produced. The frequencies that are maximally amplified are comparable with the typical curvature of the Universe at the bounce and effectively behave like a gas of massless gauge bosons. Their energy density for τ > τ 1 will be 14 ρ r (τ ) = 0 H 4 1 (a 1 /a) 4 with 0 0.2 While fields of different spin will contribute with similar values of 0 , in the post-bounce regime the dynamically produced radiation may quickly dominate and stabilize the evolution of ϕ as argued in the context of specific solutions (see, in particular, the first paper of Ref. [8]).

Concluding remarks
Bouncing models may experience gradient instabilities when the effective sound speed of the fluctuations becomes imaginary for some of the time range where the background solution is defined. While this pathology can happen either in the case case of the tensor or in the case of the scalar modes, the cures vary depending on the specific context. Gradient instabilities arise in a class of bouncing models regularized by the presence of a nonlocal dilaton potential which is a scalar under general coordinate transformations but depends on the values of the dilaton at two different space-time points. Even if the potential is nonlocal in field space the evolution equations of the background and of the fluctuations are perfectly local in time. The potential is invariant under T -duality transformations and may also arise in cosmological models inspired by double field theory. If the dilaton potential is only relevant around the bounce and negligible elsewhere the asymptotic background solutions follow from the corresponding equations with vanishing potential. In this case the effective sound speed of the scalar modes becomes imaginary for typical time scales comparable with the size of the bounce. The tensor modes are automatically gauge-invariant and frame-invariant and do not suffer any instability.
Following a simple heuristic criterion dictated by the properties of the scalar sound speed, different classes of solutions can be found where the instability does not arise and the effective sound speed is never imaginary. The gradient instabilities are tamed for a set of solutions leading, incidentally, to a quasi-flat spectrum of curvature inhomogeneities. We suggest or speculate that the exit to radiation (a long standing problem of bouncing scenarios) can be naturally addressed by considering the backreaction effects on non-conformally coupled species as suggested long ago by Ford and Parker. The current proliferation of bouncing scenarios suggests a number of interesting possibilities which will certainly mature in the years to come. According to some, realistic scenarios competitive with inflation, are already in sight. The viewpoint of this paper is more modest and we just regard the present model as a useful but not yet ultimate theoretical laboratory.

A Gauge-invariance and frame-invariance
A series of technical results that have been employed in the bulk of the paper will now be presented and derived in a self-contained perspective. We shall focus on the case of four space-time dimensions even if most of the obtained results can easily be generalized to higher dimensions [8,13]. The relation between the metric tensors in the Einstein and in the string frames 15 is given by: g (e) µν = e −ϕ g (s) µν , ϕ s = ϕ e = ϕ. (A.1) The metric fluctuations admit a frame-invariant and gauge-invariant description. In the uniform field gauge [15,16] the scalar perturbation variables are unaltered under a frame redefinition and this property makes this choice particularly effective. Conversely the tensor modes of the geometry are automatically frame-invariant and also gauge-invariant. The above statements will now be scrutinized in some detail. where, as in Eq. (3.2), δ t denotes the tensor fluctuation of the corresponding quantity. The fluctuation of ϕ (i.e. δ s ϕ = χ) affects the scalar but not the tensor modes of the geometry. Therefore recalling Eq.

A.2 Frame-invariant variables for the scalar modes
The scalar fluctuations of the geometry in the two frames are in principle different and they are related as: δ s g (e) µν = e −ϕ [−χg (s) µν + δ s g (s) µν ], χ s = χ e = χ, (A.4) 15 As in the bulk of the paper the subscripts e and s will distinguish the quantities evaluated, respectively, in the Einstein and in the string frames. 16 We consider here the case of conformally flat background geometries in each frame i.e. g where χ denotes the common value of the dilaton fluctuation either in the string or in the

A.3 Frame transformations for the background
We shall now apply the transformation (A.1) to the evolution equations of the background in different coordinate systems. Since the conformal time coordinates are frame-invariant, to perform swiftly the correct transition from string to the Einstein frames, it is appropriate to rewrite the system of Eqs. (2.9)-(2.10) in the conformal time coordinate, i.e. a s (τ )dτ = dt where τ e = τ s = τ is the common value of the conformal time coordinate either in the string or in the Einstein frame and t is the cosmic time coordinate in the string frame. After this coordinate change, Eqs.

A.4 Frame transformation for the fluctuations
Equations (3.18), (3.19) and (3.20) account for the evolution of the scalar modes in the string frame in the uniform dilaton gauge. One of the virtues of this coordinate system is that the corresponding Einstein frame equations follow from Eqs. (3.18), (3.19) and (3.20) by transforming the background according to Eq. (A.3). Therefore, using Eq. (A.3) into Eqs. (3.18), (3.19) and (3.20) we obtain without problems the following set of equations E + 2H e E + ψ − φ = 0, (A. 13) ψ + H e φ = 0, (A.14) [ψ + 2H e ψ ] + H e φ + (H e + 2H 2 e )φ = 0, (A.15) −6H e ψ − V a 2 e e ϕ φ + 2∇ 2 [ψ + H e E ] = 0, (A. 16) where, as already mentioned, φ, ψ and E denote the common values of the metric fluctuations either in the string or in the Einstein frame and the conformal time coordinate is unaltered in the transition between the two frames.