Remarks on (super-)accelerating cosmological models in general scalar-tensor gravity

We consider Friedmann-Lemaitre-Robertson-Walker cosmological models in the framework of general scalar-tensor theories of gravity (STG) with arbitrary coupling functions, set in the Jordan frame. First we describe the general properties of the phase space in the case of barotropic matter fluid and scalar field potential for any spatial curvature (flat, spherical, hyperbolic). Then we address the question under which conditions epochs of accelerated and super-accelerated expansion are possible in STG. For flat models filled with dust matter (and vanishing potential) we give a necessary condition on the coupling function of the scalar field which must be satisfied to allow acceleration and super-acceleration. This is illustrated by a specific example.


Introduction
The last decade has produced an abundance of cosmological precision data, leading to surprising results and implications. Approching the statistics of observations by more relaxed priors suggests that the expansion of the Universe as measured by the scale factor a is not only accelerating (ä a > 0), but might be also about to enter into a super-accelerating phase (Ḣ =ä a −˙a 2 a 2 > 0), sometimes dubbed as "crossing the phantom divide" [1]. The latter possibility can not be accommodated in the cosmological Concordance Model based on the Einstein equations with a cosmological constant in the framework of general relativity (GR). If one prefers to play within the traditional GR, then the onset of super-acceleration can be invoked by adding another matter component with unusual "phantom" properties [2]. An alternative explanation would require superseding GR by a more general theory of gravitation, examples of super-accelerating solutions have been studied, for instance, in the context of f(R) [3] and scalar-tensor [4] theories. Scalar-tensor theories of gravitation (STG) employ a scalar field Ψ besides the usual spacetime metric tensor g µν to describe the gravitational interaction. Scalar field is in the role of a variable gravitational "constant", leaving tensorial metric field and its geodesics to act as trajectories of freely falling particles as in GR. In general, STG form a collection of theories which contain two functional degrees of freedom, a coupling function ω(Ψ) and a scalar potential V (Ψ). Each distinct functional form of these two functions gives us a distinct theory of gravitation together with its field equations. It is of considerable interest to determine which members of this family of theories allow solutions (model Universes) exhibiting periods of accelerating and super-accelerating expansion without introducing any unusual matter component.
The study of global properties of solutions can be greatly facilitated by the mathematical methods of dynamical systems and phase space. Several previous detailed studies which have considered STG cosmology as a dynamical system have focused upon examples with specific coupling functions and potentials [5]. The main properties of the corresponding general phase space geometry were outlined by Faraoni [6] and us [7].
The plan of the paper is the following. Section 2 introduces STG field equations for homogeneous and isotropic cosmological models. In section 3 we describe the phase space in the most general case: one barotropic matter fluid component, non-vanishing scalar field potential and arbitrary spatial geometry (flat, spherical, hyperbolic), thus generalizing the results of previous studies [6,7]. In section 4 we investigate the conditions under which accelerated and super-accelerated expansion is possible, and also when do the solutions enter or leave the epoch of accelerated and super-accelerated expansion. In the simplest and phenomenologically most relevant case of dust matter, vanishing potential, and flat spatial geometry (k = 0) we give a necessary condition on the coupling function ω(Ψ) which must be satisfied for acceleration and super-acceleration to be possible at all. These considerations are illustrated by an example of a particular STG where some solutions undergo a phase of super-acceleration while some solutions do not.

The equations of scalar-tensor cosmology
We consider a general scalar-tensor theory in the Jordan frame given by the action functional Here ω(Ψ) is a coupling function and V (Ψ) is a scalar potential, ∇ µ denotes the covariant derivative with respect to the metric g µν , κ 2 is the non-variable part of the gravitational constant, and S m is the matter contribution to the action as all other fields are included in χ m . In order to keep the effective gravitational constant κ 2 Ψ positive we assume that 0 < Ψ < ∞. The field equations for the Friedmann-Lemaître-Robertson-Walker (FLRW) line element with curvature parameter k = 0 (flat), +1 (spherical), −1 (hyperbolic), and perfect barotropic fluid matter, p = wρ, read where H ≡ȧ/a, K = k a 2 . The matter conservation law is the usual and it is reasonable to assume positive matter energy density, ρ ≥ 0.

Phase space
The system (3)- (6) is characterized by five variables {Ψ,Ψ, H, a, ρ}, but one of them is algebraically related to the others via the Friedmann equation (3). Since the scale factor a is not physically observable, it is reasonable to eliminate K by Eq. (3). This leads to a phase space spanned by four variables {Ψ,Ψ, H, ρ}. By defining Ψ ≡ x,Ψ ≡ y the dynamical system corresponding to equations (3)- (6) can be written as follows: The phase space may be imagined as a four dimensional box filled by the spaghetti of one dimensional trajectories (orbits of solutions) which do not intersect with each other except for special points known as fixed (critical, equilibrium) points. As the curvature invariants of FLRW metric are proportional to H andḢ the phase space boundaries |H| → ∞, |ρ| → ∞, and |Ψ| → ∞ generically entail a spacetime singularity. Analogously, the limit Ψ → 0 in general implies diverging |H| or |Ḣ| and poses a spacetime singularity, obstructing the solutions from safely passing from positive to negative values of Ψ (from "attractive" to "repulsive" gravity). The limit Ψ → ∞ does not call forth a spacetime singularity, however, the gravitational "constant" vanishes.
Within this box there could also be singular hypersurfaces perpendicular to the Ψ axis, depending on the form of ω and V . So, in general terms the limit V → ∞ renders the system singular, and also 2ω + 3 → 0 implies |Ḣ| → ∞ with the same conclusion that passing through ω(Ψ) = − 3 2 (corresponding to the change of the sign of the scalar field kinetic term in the Einstein frame action) would entail a space-time singularity and is impossible. The limit 1 2ω+3 → 0 is also marred by a singularity, unless simultaneouslẏ The latter situation is particularly interesting, since in this limit the system coincides with the FLRW equations of general relativity [7].
The trajectories corresponding to the flat FLRW geometry (k = 0) lie on the 3-surface due to the constraint (3). The trajectories corresponding to spherical and hyperbolic models remain on either side of this surface. In principle the geometry of the 3-surface F in the 4-dimensional phase space is rather complicated to visualize, but a few general characteristics can still be given. We may write Eq. (12) as which for fixed ρ and x can be recognized as describing familiar conic sections: 1) for ρ+V > 0, 2ω + 3 > 0 a hyperbola on the (H + y 2x , y) plane, 2) for ρ + V > 0, 2ω + 3 < 0 an ellipse also on the (H + y 2x , y) plane, while 3) for ρ + V < 0, 2ω + 3 > 0 a hyperbola on the (y, H + y 2x ) plane. The case 4) ρ + V < 0, 2ω + 3 < 0 is not realized as real solutions are absent. This result establishes that the intersection of the 3-surface F with the (fixed ρ, fixed x) 2-plane is constituted in either one piece (ellipse) or two pieces (hyperbola). Thus in case 1) the allowed phase space is divided into two separate regions, the "upper" region where H + y 2x > 0 and the "lower" region where H + y 2x < 0, and there is no way the trajectories can travel from one region to another. In case 2) these two regions meet along a 2-surface where H + y 2x = 0, and the trajectories can in principle cross from one region to another. In case 3) there are again two separate parts, now characterized by y > 0 and y < 0, respectively. At first it may be difficult find a direct physical interpretation for the quantity H + y 2x that characterizes the "upper" and "lower" region in cases 1) and 2), but it turns out that this combination is equal to the Hubble parameter in the Einstein frame [8,7], and thus the "upper" region corresponds to universes which expand in the Einstein frame, while the "lower" region has universes which contract in the Einstein frame.
Related information can be also established by another approach. With general k we may solve the Friedmann constraint, Eq. (3), for H and then the condition for all variables to be real valued imposes an inequality (2ω(x) + 3) Allowed range ofΨ Allowed range of H for k = 0, +1 for k = 0, −1 Allowed range ofΨ Allowed range of H for k = 0, +1 for k = 0, +1 In terms of physics this inequality can be interpreted as a restriction on the allowed values of y. Table 1 summarizes the situation. By the allowed range for a given value of k we mean that if y satisfies the inequality listed, then it is possible to find a real-valued a which fits the Friedmann equation. Thus for k = 0 and k = +1 there is no restriction in case 1), while the case 4) is completely ruled out since no real solutions compatible with the Friedmann constraint exist. For k = −1 there are no restrictions. Similarly, solving the Friedmann constraint for y leads to another inequality, which can be interpreted as a restriction on the allowed values of H (given also in Table 1). Analogously, once ω(x) and V (x) are specified, we may get a third inequality from solving the Friedmann constraint for x as well.

Acceleration and super-acceleration
In the four dimensional phase space there could be regions where the trajectories exhibit super-accelerating behavior, marked by the condition and surrounded by regions of accelerated expansion, delineated by For a cursory comparison with general relativity let us recall that in GR super-acceleration requires matter with barotropic index w < −1, while acceleration demands w < − 1 3 . Cosmological constant behaves as a barotropic fluid with w = −1. Eqs. (16), (17) readily reveal that identical conditions are recovered in STG at the GR limit (11), where the fixed value of the potential is read as the cosmological constant, and the Friedmann constraint (3) should be taken into account along with (16).
However, away from the GR limit, new possibilities occur. First, irrespective of the matter content, the scalar field itself may trigger accelerated and super-accelerated expansion, in the domain where ω < 0, or 1 2x(2ω+3) dω dΨ > 0. Second, acceleration and super-acceleration may also occur for matter with w > 1 3 in the domain where 2ω + 3 > 0, or for w < 1 3 in the domain where 2ω + 3 < 0. Third, the overall effect of the potential is considerably more complicated than that of a simple cosmological constant, depending on the derivative dV dΨ as well as the sign of 2ω + 3, and also implying a possibility that a constant negative potential may lead to super-acceleration, provided that 2ω + 3 > 0.
The region of super-acceleration is bounded by the 3-surface S : S(x, y, H, ρ) = 0. The circumstance whether the trajectories enter this region can be read off from the scalar product of the gradient normal to S and the tangent vector of the phase flow T i = (ẋ,ẏ,Ḣ,ρ), namely ∇ i S · T i S > 0. A completely analogous condition arises for the surface A = 0 bounding the region of acceleration.
These results, expressed in full generality, hint ample possibilities for acceleration and super-acceleration, but in order to come up with more exact conditions one has to narrow down the scope a bit. Therefore let us focus upon the physically most interesting case of spatially flat (k = 0) universe filled with dust matter (w = 0) and vanishing potential. In this case Eq. (4) can be written aṡ Here the first line informs that super-acceleration is only possible, if ω < 0, orΨ < 0. From the second line which has taken Eq. (5) into account, we can read off a neccessary condition on the form of the coupling function ω for super-acceleration to be possible assuming 2ω + 3 > 0. The reason is that if ω > 0 the Fridmann constraint imposes κ 2 ω(ρ) 3Ψ(2ω+3) ≤ H 2 2 , and the only positive contribution towards super-acceleration can arise from the third term in (18). The same is true for − 3 2 < ω < 0 since then the second term is negative itself. (It is easier to achieve super-acceleration if 2ω + 3 < 0, but this option is not so lucrative since in the Einstein frame, where the tensor and scalar degrees are not mixed, the kinetic energy of the latter is negative, and thus problematic [9].) Note that Eq. (19) provides only a neccessary, and not sufficient condition for superaccelerating solutions to be present in a model. More exactly, it states that in the domain of Ψ, where (19) holds, there may be solutions which undergo super-accelerated expansion. In the domain of Ψ, where (19) does not hold, super-acceleration is not possible. Therefore, given a zoo of all possible forms of ω, it can be used to filter out and discard from further investigation those forms of ω, which are decidedly infertile with respect to super-acceleration.
Finally, as an illustration, let us consider 2ω + 3 = 1 2(1−Ψ) for example. Here 2ω + 3 > 0 and (19) holds in the domain 0 < Ψ < 1. Inspection of the phase space flow reveals that the trajectories on the "upper sheet" (H +Ψ 2Ψ > 0), where most of the expanding, H > 0, models lie, belong to two typical classes: either exhibiting a super-accelerating phase or not, see Fig.  1. The dynamics has been characterized by the evolution of the effective barotropic index, w eff = −1 − 2Ḣ 3H 2 , defined as an analogy to single component barotropic fluid FLRW models in GR. In particular, w eff = 0 characterizes the decelerating evolution of usual dust matter, w eff < − 1 3 is required for acceleration, while w eff = −1 corresponds to the "phantom divide line" below which super-acceleration occurs.