The dynamical viability of scalar-tensor gravity theories

TL;DR

This work develops a comprehensive dynamical-systems framework for scalar-tensor gravity as dark energy, analyzing background evolution in both Einstein and Jordan frames and showing the frame-independent nature of attractor behavior. By introducing the variables m ≡ d ln F/d ln Φ and r ≡ −ΦF/f, and mapping between frames, the authors classify evolution into five classes, including a novel regime for |β|<√3/4 not present in f(R) theories. They identify seven critical points (P1–P7) with explicit stability conditions and connect these to observationally relevant quantities such as Ωm and w_eff in the Jordan frame. The study further reveals that, beyond the standard f(R) correspondence (β = 1/2), there exist viable saddle-acceleration trajectories and a Class V of models that can realize transient acceleration, broadening the space of dynamically admissible scalar-tensor theories.

Abstract

We establish the dynamical attractor behavior in scalar-tensor theories of dark energy, providing a powerful framework to analyze classes of theories, predicting common evolutionary characteristics that can be compared against cosmological constraints. In the Jordan frame the theories are viewed as a coupling between a scalar field, Φ, and the Ricci scalar, R, F(Φ)R. The Jordan frame evolution is described in terms of dynamical variables m \equiv d\ln F/d\ln Φand r \equiv -ΦF/f, where F(Φ) = d f(Φ)/dΦ. The evolution can be alternatively viewed in the Einstein frame as a general coupling between scalar dark energy and matter, β. We present a complete, consistent picture of evolution in the Einstein and Jordan frames and consider the conditions on the form of the coupling F and βrequired to give the observed cold dark matter (CDM) dominated era that transitions into a late time accelerative phase, including transitory accelerative eras that have not previously been investigated. We find five classes of evolutionary behavior of which four are qualitatively similar to those for f(R) theories (which have β=1/2). The fifth class exists only for |β| < \sqrt{3}/4, i.e. not for f(R) theories. In models giving transitory late time acceleration, we find a viable accelerative region of the (r,m) plane accessible to scalar-tensor theories with any coupling, β(at least in the range |β| \leq 1/2, which we study in detail), and an additional region open only to theories with |β| < \sqrt{3}/4.

Figures (3)

• Figure 1: The effective equation of state $w_{eff}$ as a function of $\beta$ for the point $\mathbbm{P}4$.
• Figure 2: Class V models of scalar-tensor theories on the $(r,m)$ plane. The solid line with a slope of -1 is the critical line $m=-r-1$ and the dashed curves are possible trajectories for a Class V model. The point $\mathbbm{P}5^{(0)}$$(-1,0)$ is the starting saddle matter dominated point, with the triangles around it representing the forbidden direction regions. The $m(r)$ curve for a Class V model intersects either the $m$-axis between 0 and -1 corresponding to the point $\mathbbm{P}4$, or the critical line in region (E) of point $\mathbbm{P}6$, to get a stable accelerated epoch.
• Figure 3: Scalar-tensor theories which lead to a saddle accelerated expansion of the universe. The solid line with a slope of -1 is the critical line $m=-r-1$ and the dashed curves are possible trajectories for a scalar-tensor theory. The point $\mathbbm{P}5^{(0)}$$(-1,0)$ is the starting saddle matter dominated point, with the triangles around it representing the forbidden direction regions. In the first graph the $m(r)$ curve connects the matter dominated point to one of the three accelerated regions (A, B or C) of the point $\mathbbm{P}6$. The accelerated expansion is stable on the subspace $\tilde{z}=r\tilde{y}$. In the second graph the $m(r)$ curve connects the matter dominated point to the point $\mathbbm{P}4$ on $m(r=0)>0$ which gives a period of, potentially transient, acceleration.