Horndeski Genesis: strong coupling and absence thereof

We consider Genesis in the Horndeski theory as an alternative to or completion of the inflationary scenario. One of the options free of instabilities at all cosmological epochs is the one in which the early Genesis is naively plagued with strong coupling. We address this issue to see whether classical field theory description of the background evolution at this early stage is consistent, nevertheless. We argue that, indeed, despite the fact that the effective Plank mass tends to zero at early time asymptotics, the classical analysis is legitimate in a certain range of Lagrangian parameters.

There is an issue in the Horndeski Genesis models, however. In most cases, spatially flat Genesis solutions are plagued with gradient instabilities occurring sooner or later in the cosmological evolution [22][23][24][25]. This property has been formulated as the "no-go theorem". One of the possible ways out is to consider models which are, at least naively, strongly coupled in the asymptotic past (and/or asymptotic future, the case that can be studied along the lines of this work) [23,26,27]. In these models, the coefficients in quadratic action for perturbations about the classical solution tend to zero as t → −∞, which, indeed, implies that the strong coupling energy scale also tends to zero.
In this paper, we point out that this property does not necessarily mean that one cannot use classical field theory for describing the cosmological evolution at early times. Indeed, the time scale of the classical evolution tends to infinity, and hence its inverse, the classical energy scale, tends to zero, as t → −∞. So, to see whether or not the classical field theory treatment is legitimate, one has to figure out the actual strong coupling energy scale and compare it with the inverse time scale of the classical background evolution. The classical analysis of the background is consistent, provided that the former energy scale much exceeds the latter. In this paper, we consider a simple class of Horndeski Genesis models with the strong coupling at early times, and study scalar perturbations in the asymptotics t → −∞. We derive the conditions ensuring that the classical energy scale is much lower than the strong coupling scale in the scalar sector. We find that these conditions can indeed be satisfied in a certain range of parameters in the Lagrangian, i.e., it is possible to avoid strong coupling regime for Genesis stage at least as far as the scalar sector is concerned, in the sense that the classical treatment of the background evolution is consistent at early times. We argue that tensor and tensor-scalar sectors may leave this result unmodified. This paper is organised as follows. In Section 2 we introduce the model and discuss its early-time asymptotics that enables one to avoid the no-go theorem of Ref. [23]. In Section 3 we discuss strong coupling issue in detail and find a region of the parameter space in which the classical description of Genesis is legitimate despite the low strong coupling energy scale. We conclude in Section 4.

The model
If one uses general relativity to describe gravity, then an important characteristic is the null energy condition (NEC) for the matter energy-momentum tensor T µν , which reads T µν k µ k ν ≥ 0 for every null vector k µ . Once the NEC holds in the cosmological context, then (assuming flat spatial sections) it follows from the Einstein equations that dH/dt ≤ 0, where H is the Hubble parameter. This implies that there is a singularity in the past of the expanding universe. Therefore, one either modifies gravity or violates the NEC to build non-singular cosmology.
A candidate for NEC violating theory is the generalised Galileon scalar field coupled to gravity [1][2][3][4][5][6][7]. The most general form of Lagrangian which leads to the second-order field equations was obtained by G. Horndeski in [11]. It is sufficient for our purposes to consider a subclass of Horndeski Lagrangians instead of the full one: where R is a Ricci scalar and φ = g µν ∇ µ ∇ ν φ. The metric signature is (−, +, +, +). Let us rewrite this Lagrangian (1) in terms of ADM variables, to make contact with Ref. [23]: where we use the unitary gauge in which φ = φ(t), and K ij , R ij are the extrinsic curvature and the Ricci tensor of the spatial slices, respectively. There is one-to-one correspondence between the variables φ and X in the covariant Lagrangian and time variable t and lapse function N in the ADM formalism. The following expressions convert one formalism to another [28][29][30]: where F (φ, X) is an auxiliary function, such that and the following gauge is fixed with Y 0 = const:

Avoiding the no-go theorem
A subclass of Lagrangians in which the no-go theorem can be avoided was given in Ref. [23]: where M P l is the Planck mass, α and δ are constant parameters satisfying and f (t) is some function of time, which has the following asymptotics as t → −∞ As a concrete example, we choose The background metric reads where N is the lapse function (the same as in the Lagrangian (1)). One derives the equations of motion for the homogeneous background directly from the variation of the background part of the Lagrangian [31] and obtains where the Hubble parameter is H =ȧ/(Na) and subscript N denotes the derivative upon lapse function N. From these equations we find an asymptotic solution at early times (t → −∞): where χ is the combination of the Lagrangian parameters An important feature of this solution is that and hence On the one hand, these are necessary conditions to avoid both ghost and gradient instabilities during subsequent evolution [23]. On the other hand, Eqs. (20) and (21) signalise that the strong coupling energy scale in this theory tends to zero as t → −∞. The purpose of this paper is to see whether or not the latter feature spoils the classical field theory description of the early time evolution, t → −∞.

Strong coupling scale for perturbations versus classical scale
We now consider the perturbations about the classical solution and, for technical reasons, study scalar perturbations only. We comment on tensor and cross (tensor-tensor-scalar and scalar-scalartensor) sectors later on. The perturbed metric for the scalar sector has the following form where and α, β, ζ are scalar perturbations. Expanding the action up to the second order, one obtains the following expression for the quadratic action in the unitary gauge where ∂ denotes spatial derivatives, and The early-time asymptotics for the background solution (17) and (18) of the latter coefficients are The fields α and β are constraint variables. One finds them by solving the constraint equations and plugs them back into the action (24). In this way one obtains the following expression for the unconstrained quadratic action: where The asymptotic behaviour of the functions ǫ s and c s is found from (26): Since 2α − δ > 1, see (9), the overall coefficient ǫ s tends to zero as t → −∞, signalling the low strong coupling energy scale at early times.
To figure out the strong coupling scale in the scalar sector, we have to go one step further and consider the cubic action. We use the results presented in [32] for cubic action for all of the scalar perturbations α, β and ζ: where λ 1 , λ 2 , λ 3 , λ 4 , λ 5 are the functions of g ζ , c ζ f α , m α , A 2 , A 3 , A 4 , H and we find their asymptotic behaviour as t → ∞ for our model (8): We solve the constraints in terms of α and β and obtain the following expression for unconstrained cubic action: where ψ = ∂ −2 (aζ/N); Λ 1 ...Λ 18 are functions of g ζ , c ζ f α , m α , A 2 , A 3 , A 4 , H, and hence of time t. All of them have power-law behaviour at early times t → −∞, where x i are combinations of α and δ.
For power-counting purposes, every term L i in the cubic Lagrangian (i = 1, 18) can be schematically written as follows where a i and b i are the numbers of temporal and spatial derivatives, respectively.
The fact that the coefficient here tends to zero as t → −∞ is crucial for what follows.
In terms of the canonically normalised field π one rewrites (34) as follows: By naive dimensional analysis (dimension of Λ i is [Λ i ] = 4 − a − b and [ǫ s ] = 2) we immediately find that the strong coupling energy scale associated with the term L i is On the other hand, the inverse time scale of classical evolution is Thus, the condition for legitimacy of the classical treatment of the early evolution, E class ≪ E (i) strong for all i reads Clearly, the most dangerous terms are those with the smallest combination a i + b i − x i . By inspecting the behaviour of Λ i one finds that this combination is the smallest for i = 1 (given the constraints (9)), when (as an example, the next term has Λ 2 ∼ (−t) −2α+2δ , a 2 = 2, b 2 = 0 and a 2 + b 2 − x 2 = 2 + 2α − 2δ; recall that δ > 0). Thus, the strong coupling regime can be avoided for 2α < 2 − 3δ, which together with (9) gives We conclude that the strong coupling regime is avoided (at least as far as the scalar perturbations are concerned), in the sense that the evolution remains classical at early times, provided one chooses the Lagrangian parameters α and δ in the dark grey allowed region shown in Fig. 1.
To get more confidence in the classical field theory treatment avoiding the strong coupling problem, one has to analyse tensor, tensor-tensor-scalar and scalar-scalar-tensor sectors of perturbations. It is likely, though, that they give weaker constraints than those presented above.

Summary
We have studied the non-singular Genesis scenario in the framework of the Horndeski theory, which is capable of avoiding the gradient instability at the expense of potential strong coupling problem.  [23]. The dark grey area shows the allowed range of the Lagrangian parameters satisfying "no strong coupling" criterion.
The model of Ref. [23] has been used as an example that gives explicit asymptotic solutions at early times. We have seen that with an appropriate choice of parameters, these solutions are actually away from the strong coupling regime inferred from the study of scalar perturbations. This opens up a possibility that the Universe starts up with very low quantum gravity energy scale (effective Planck mass asymptotically vanishes as t → −∞), and yet its classical evolution is so slow that the classical field theory description remains valid.
Even though our analysis has given a promising outcome, it is certainly incomplete. First, we still have to study tensor perturbations and their cubic self-interactions and interactions with scalar perturbations. Second, there is no guarantee that the fourth and higher order interactions give strong coupling energy scales higher or equal to the ones we have found by studying the cubic interactions. We hope to turn to these issues in future.