Generalized Galileons: instabilities of bouncing and Genesis cosmologies and modified Genesis

We study spatially flat bouncing cosmologies and models with the early-time Genesis epoch in a popular class of generalized Galileon theories. We ask whether there exist solutions of these types which are free of gradient and ghost instabilities. We find that irrespectively of the forms of the Lagrangian functions, the bouncing models either are plagued with these instabilities or have singularities. The same result holds for the original Genesis model and its variants in which the scale factor tends to a constant as $t\to -\infty$. The result remains valid in theories with additional matter that obeys the Null Energy Condition and interacts with the Galileon only gravitationally. We propose a modified Genesis model which evades our no-go argument and give an explicit example of healthy cosmology that connects the modified Genesis epoch with kination (the epoch still driven by the Galileon field, which is a conventional massless scalar field at that stage).

The situation is not so bright in more complete cosmological models. Known models of the bouncing Universe, employing generalized Galileons, are in fact plagued by the gradient instabilities, provided one follows the evolution for long enough time [24][25][26][27][28]. Gradient instabilities occur also in the known Genesis models, once one requires that the early Genesis regime turns into more conventional expansion (inflationary or not) at later times [16,30,31]. An intriguing exception is the model [16] in which Genesis-like super-accelerated expansion starts from the de Sitter, rather than Minkowski, epoch. We comment on this model in Section 3.
One way to get around the gradient instability problem is to arrange the model in such a way that the quadratic in spatial gradients, wrong sign term in the action is small, and higher derivative terms restore stability at sufficiently high spatial momenta [16,29]. There is also a possibility that the strong coupling momentum scale is low enough [28]. In both cases the exponential growth of trustworthy perturbations does not have catastrophic consequences, provided that the time interval at which the instability operates is short enough. Another option is to introduce extra terms in the action which are not invariant under general coordinate transformations [27,30].
Clearly, it is of interest to understand whether gradient or ghost instabilities are inherent in all "complete" bouncing models and Genesis models with initial Minkowski space, which are based on classical generalized Galileons and General Relativity, or these instabilities are merely drawbacks of concrete models constructed so far. In the latter case, it is worth designing examples in which the gradient and ghost instabilities are absent.
It is this set of issues we address in this paper. We consider the simplest and best studied generalized Galileon theory interacting with gravity. The Lagrangian is (mostly negative signature; κ = 8πG) where π is the Galileon field, F and K are smooth Lagrangian functions, and We also allow for other types of matter, assuming that they interact with the Galileon only gravitationally and obey the NEC: To see that our observations are valid in any dimensions, we study this theory in (d + 1) space-time dimensions with d ≥ 3; the case of interest is of course d = 3. We consider spatially flat FLRW Universe with the scale factor a(t) where t is the cosmic time, and study spatially homogeneous backgrounds π(t).
Our framework is quite general. In the Genesis case we require that neither a(t) nor π(t) has future singularity (i.e., a(t), π(t) and their derivatives are finite for all −∞ < t < +∞). Our definition of the bouncing Universe is that the scale factor a(t) either is constant in the past and future, a(t) → a ∓ as t → ∓∞, or diverges in one or both of the asymptotics (i.e., a − = ∞ or/and a + = ∞), and that there is no singularity in between.
Somewhat surprisingly, our results for the bouncing and Genesis scenarios are quite different. In the bouncing Universe case, we show that the gradient (or ghost) instability is inevitable. This result is a cosmological counterpart of the observation that a static, spherically symmetric Lorentzian wormhole supported by the generalized Galileon always has the ghost or gradient instability [32] (see also Ref. [33]); the technicalities involved are also similar.
Analogous no-go theorem does not hold in the Genesis case. Yet the requirement of the absence of the gradient and ghost instabilities strongly constrains the Galileon theories (i.e., Lagrangian functions F and K). In particular, the gradient or ghost instability (or future singularity) does exist, if the initial stage is the original Genesis [11] or its versions in which a(t) → const as t → −∞, which is the case, e.g., in the subluminal Genesis [12] as well as in the DBI [13] and generalized Genesis [17] models in which the Lagrangians have the general form 2 (1.1) (in the language of Ref. [13], the Lagrangians from this subclass do not contain terms L 4 and L 5 ).
Equipped with better understanding of the instabilities in the Genesis models with generalized Galileons, we propose a modified Genesis behavior in which the space-time curvature, energy and pressure vanish as t → −∞ and which is not inconsistent with the absence of the gradient and ghost instabilities and the absence of future singularity. The pertinent Galileon Lagrangian is similar to ones considered in Refs. [15,17]; in particular, the action is not scale-invariant. Starting from this Lagrangian, we give an example of a "complete" model, with Genesis at the initial stage and kination (the epoch still driven by the Galileon field which, however, is a conventional massless scalar field at that stage) at later times. This model is free of the gradient instabilities, ghosts and superluminal propagation about the homogeneous solution, while the kination stage may possibly be connected to the radiation domination epoch via, e.g., gravitational particle creation, cf. Ref. [34]. This paper is organized as follows. In Section 2 we discuss, in general terms, the conditions for the absence of the gradient and ghost instabilities in the cosmological setting. We show in Section 3 that irrespectively of the forms of the Lagrangian functions F (π, X) and K(π, X), these conditions cannot be satisfied in the bouncing Universe scenario as well as in the Genesis models with time-independent past asymptotics of the scale factor. We propose a modified Genesis model in Section 4, where we first study general properties and concrete example of early Genesis-like epoch which evades the no-go argument of Section 3, then give an explicit example of healthy model connecting Genesis and kination and, finally, briefly discuss a spectator field whose perturbations may serve as seeds of the adiabatic perturbations. We conclude in Section 5. For completeness, the general expressions for the Galileon energy-momentum tensor and quadratic Lagrangian of the Galileon perturbations are given in Appendix.

Generalities
The general expression for the Galileon energy-momentum tensor is given in Appendix, eq. (A.1). In the cosmological context the energy density and pressure are where H is the Hubble parameter and X =π 2 .
Our main concern is the Galileon perturbations χ of high momentum and frequency. The general expression for the effective quadratic Lagrangian for perturbations is again given in Appendix, eq. (A.3). For homogeneous background we obtain and terms omitted in (2.2) do not contain second derivatives of χ. These terms are irrelevant for high momentum modes. The absence of ghosts and gradient instabilities requires A > 0, B ≥ 0.
In particular, if B < 0, there are ghosts (for A < 0) or gradient instability (for A > 0). Our focus is on the coefficient B. Despite appearance, it can be cast in a simple form. To this end we make use of the Friedmann and covariant conservation equations Here ρ and p are Galileon energy density and pressure, while ρ M and p M are energy and pressure of conventional matter, if any. The latter obey the NEC, eq. (1.2). We recall that we assume that conventional matter does not interact with the Galileon directly, so the covariant conservation equations (2.4b) and (2.4c) have to be satisfied separately. Equations (2.1), (2.3b) and (2.5) lead to a remarkable relation It is natural to introduce a combination and write Another representation is in terms of the function Since we assume that the conventional matter, if any, obeys the NEC, the positivity of B requireṡ As we now see, these requirements are prohibitively restrictive in the bouncing Universe case and place strong constraints on the Genesis models.

Bouncing Universe and original Genesis: no-go
We now show that the inequality (2.9) cannot be satisfied in the bouncing Universe scenario. We write it as follows,Ṙ and integrate from t i to t f > t i : SinceṘ > 0 in view of (2.9), R increases in time and remains positive. We have 1 Since a(t) is either a constant or growing function of t at large t, the right hand side of the latter inequality eventually becomes negative at large t f . Thus R −1 (t f ) as function of t f starts positive (at t f = t i ) and necessarily crosses zero. At that time R −1 = 0, and R = ∞, which means a singularity.
A remaining possibility is that R(t) is negative at all times. In particular, R(t f ) < 0. In that case a useful form of the inequality (3.1) is Now, a(t) is either a constant or tends to infinity as t → −∞, so the right hand side is positive at large negative t i . Hence, there is again a singularity R = ∞ at t i < t < t f . This completes the argument. The same argument applies to the original Genesis model [11] and many of its versions, like subluminal Genesis [12] and the DBI Genesis [13], provided the Lagrangian has the general form (1.1). In these versions, the scale factor tends to a constant as t → −∞ and, assuming that the Universe ends up in the conventional expansion regime, the scale factor grows at large times. The integral in eq. (3.1) blows up at large t f or large negative t i , so the inequalities (3.2), (3.3) are impossible to satisfy without hitting the singularity 3 R = ∞. In fact, in the models of Refs. [11][12][13], one has Q > 0, which is consistent with healthy behavior at early times but implies either gradient (or ghost) instabitity or singularity in future.
At this point let us make contact with the model of Ref. [16] in which the Genesis-like superaccelerated expansion starts from the de Sitter rather than Minkowski epoch, d = 3, a ∝ e λt . In that case the integral in (3.3) is convergent as t i → −∞. Hence, our argument does not work: one can have R < 0 at all times, leaving a room for the stable evolution. In fact, in the model of Ref. [16], our parameter Q defined in (2.6) is constant in time and negative, while B > 0 in full accordance with (2.7). We generalize this construction in Section 4.

Early-time evolution
In this Section we construct a model which interpolates between a stage similar to Genesis (in the sense that space-time curvature, energy and pressure vanish as t → −∞) and kination epoch at with the Galileon behaves as a conventional massless scalar field. The model is purely classical and does not have gradient or ghost instability at any time. We begin with the early Genesis-like stage, having in mind the observations made in Section 3.
Since we would like the scale factor to increase at late times, and in view of the inequality (3.2), we require that at the Genesis-like stage R < 0 and hence Furthermore, the second term in the right hand side of eq. (2.8) must be larger than |Q| at the Genesis-like epoch: since H increases at that epoch from originally zero value, so does |Q| (barring cancelations), and we haveQ < 0. Thus, besides the inequality (4.1) we require Assuming power law behavior of Q, we see that the simplest option is that at large negative times From eq. (4.1) we deduce that H cannot rapidly tend to zero as t → −∞; we can only have in contrast to the conventional Genesis, in which H ∝ (−t) −3 . Thus both energy density and pressure should behave like t −2 as t → −∞ (while in the originl Genesis one has p ∝ t −4 , ρ ∝ t −6 ). Now, the no-go argument based on (3.3) is not valid provided that the integral in the right hand side is convergent at the lower limit of integration 4 , t −∞ dt a d−2 < ∞ . (4.5) Thus, we require that These are the general properties of the Genesis-like stage which is potentially consistent with the overall healthy dynamics. Note that unlike in the original Genesis scenario, the scale factor does not tend to a constant as t → −∞. Yet the geometry tends to Minkowski at large negative times in the sense that the space-time curvature tends to zero, and so do the energy density and pressure. One way to realize this scenario is to choose the Lagrangian functions in the following form The resulting Lagrangian is similar to those introduced in Ref. [17], although the particular exponential dependence that we have in (4.7) was not considered there. There is a solution with time-independent H * and h; we relate them to the parameters α 0 , β 0 and f shortly. For this solution Thus, one has Q < 0 and no gradient instability at early times (B > 0), provided that Note that with this choice of parameters, the inequality (4.6) is satisfied, as it should.
The parameters H * and h are related to the parameters of the Lagrangian via the field equations (2.4). Two independent combinations of these equations are Using these relations and inequalities (4.9) one can check that f 2 > 0 and, importantly, the coefficient of the kinetic term in the Lagrangian for perturbations is positive, A > 0. The modified Genesis regime is stable.

From Genesis to kination: an example
A scenario for further evolution is as follows. The function π(t) is monotonous,π > 0. However, the Lagrangian functions F and K depend on π and hence on time in a non-trivial way. The variable Q remains negative at all times, but eventually (at t = t c )Q changes sign and Q starts to increase towards zero. Choosing K X > 0, we find that at t > t c the stability condition B > 0 is satisfied trivially, One should make sure, however, that at t < t c the inequality (4.2) is always satisfied. Another point to check is that the coefficient of the kinetic term in the Lagrangian for perturbations is positive, A > 0, at all times.
To cook up a concrete example of a model in which the Genesis regime is smoothly connected to kination (the regime at which Galileon is a conventional massless scalar field dominating the cosmological expansion), the simplest way is to introduce a field in such a way that the solution is φ = t . (4.11) The general formulas of Sections 2, 3 remain valid, with understanding that the Lagrangian functions are now functions of φ and X = (∂φ) 2 ; in particular, the combination Q is the same as in (2.6) with φ substituted for π. We choose the Lagrangian functions in the following form: On the solution (4.11) one has One way to proceed is to postulate suitable forms of Q(t) < 0 and β(t), such that the inequality (4.2) is satisfied, evaluate H = κ d−1 (2β − Q), reconstruct v(t) and α(t) from the field equations and then check that A > 0 at all times. With the convention (4.11), once v(t), α(t) and β(t) are known, the Lagrangian functions are also known, v(φ) = v(t = φ), etc. As we anticipated in eqs. (4.3), (4.4), the initial behavior is with time-independentq andβ. To satisfy the inequality (2.8), we impose the condition .
We would like the Galileon to become a conventional massless scalar field at large positive t, whose equation of state is p = ρ, and require that at large t the function β(t) rapidly vanishes, while H = (d · t) −1 , and hence It is convenient to introduce rescaled variables where P is positive. In terms of these variables the Hubble parameter is The combinations of the field equations, analogous to eqs. (4.10), give Finally, the coefficients in the Lagrangian for perturbations are The asymptotics of the solution should be t → −∞ : As a cross check, the late-time asymptotics (4.13b) imply α = 0 and which corresponds to the Lagrangian of free massless scalar field, albeit written in somewhat unconventional form, (4.14) On the other hand, the early-time asymptotics (4.13a), according to eq. (4.12), give so that the Lagrangian at early times (φ → −∞) reads Upon introducing new field at early times one writes the Lagrangian in the following form This is precisely the form (4.7). One more point to note is that if the parametric form of b(t) and P (t) is withb andP of order 1 (which is consistent with the asymptotics (4.13)), then for large τ the dynamics is sub-Planckian during entire evolution: in that case one has sub-Planckian H ∼ τ −1 , α ∼ κ −1 τ −2 , etc. A random example is The fact that these functions tend to zero as t → +∞ does not indicate the onset of strong coupling; this behavior rather has to do with the field redefinition from the canonical massless scalar field to the field φ described by the Lagrangian (4.14). The late-time theory is the theory of free massless scalar field, with no strong coupling or instabilities.

Curvaton
It is unlikely that the Galileon perturbations would produce adiabatic perturbations with nearly flat power spectrum. Like in Ref. [11], the adiabatic perturbations may originate from perturbations of an additional scalar field, "curvaton". Let us consider this point, specifying to 4-dimensional space-time, d = 3.   We are interested in the early modified Genesis stage when the Galileon Lagrangian functions have the form (4.7). The modified Galileon action is not invariant under the scale transformations π(x) →π(x) = π(λx) + ln λ , g µν (x) →ĝ µν = g µν (λx) .
One has instead S(π,ĝ µν ) = λ −2 S(π, g µν ), just like for the Einstein-Hilbert action. Let us introduce a spectator curvaton field θ which is invariant under the scale transformations, θ(x) →θ(x) = θ(λx), and require that its action be scale-invariant. This requirement gives In the backround (4.8) this action reads where the scale factor is given by (4.4). Let us introduce conformal time Then the action for θ has the form where a ef f (η) = − 1 H * (h + 1)η , which is precisely the action of the massless scalar field in de Sitter space-time. We immediately deduce that the power spectrum of perturbations δθ generated at the modfied Genesis epoch is flat. This is a pre-requisite for the nearly flat power spectrum of adiabatic perturbations, which may be generated from the curvaton perturbations at later stage.

Conclusion
We have seen in this paper that with generalized Galileons, it is possible to construct healthy Genesis-like cosmologies, albeit with somewhat different properties as compared to the original Genesis scenario. On the other hand, bouncing cosmologies with generalized Galileons are plagued by the gradient (or ghost) instability, at least at the level of second derivative Lagrangians of the form (1.1). This is not particularly surprising. The theory appears to protect itself [32] from having stable wormhole solutions which can be converted into time machines [35]. Technically the same protection mechanism, with radial coordinate and time interchanged, forbids the existence of spatially flat bouncing cosmologies. It would be interesting to understand how general are these features.

(A.3)
We specify to spatially homogeneous Galileon background in Section 2.