Some late-time asymptotics of general scalar–tensor cosmologies

Abstract

We study the asymptotic behaviour of isotropic and homogeneous universes in general scalar–tensor gravity theories containing a p = −ρ vacuum fluid stress and other sub-dominant matter stresses. It is shown that in order for there to be an approach to a de Sitter spacetime at large 4-volumes the coupling function, ω(ϕ), which defines the scalar–tensor theory, must diverge faster than |ϕ∞ − ϕ|−1+ϵ for all ϵ > 0 as ϕ → ϕ∞ ≠ 0 for large values of the time. Thus, for a given theory, specified by ω(ϕ), there must exist some ϕ∞ ∊ (0, ∞) such that ω → ∞ and ω′/ω2+ϵ → 0 as ϕ → ϕ∞ in order for cosmological solutions of the theory to approach de Sitter expansion at late times. We also classify the possible asymptotic time variations of the gravitation ‘constant’ G(t) at late times in scalar–tensor theories. We show that (unlike in general relativity) the problem of a profusion of ‘Boltzmann brains’ at late cosmological times can be avoided in scalar–tensor theories, including Brans–Dicke theory, in which ϕ → ∞ and ω ∼ o(ϕ1/2) at asymptotically late times.