Metric gravity theories and cosmology: II. Stability of a ground state in f(R) theories

Abstract

In the second part of the investigation of metric nonlinear gravity theories, we study a fundamental criterion of viability of any gravity theory: the existence of a stable ground-state solution being either Minkowski, de Sitter or anti-de Sitter space. Stability of the ground state is independent of which frame is physical. In general, a given theory has multiple ground states and splits into independent physical sectors. The fact that all L = f(gαβ, Rμν) gravity theories (except some singular cases) are dynamically equivalent to Einstein gravity plus a massive spin-2 and a massive scalar field allows us to investigate the stability problem using methods developed in general relativity. These methods can be directly applied to L = f(R) theories wherein the spin-2 field is absent. Furthermore for these theories which have anti-de Sitter space as the ground state we prove a positive-energy theorem allowing to define the notion of conserved total gravitational energy in the Jordan frame (i.e., for the fourth-order equations of motion). As is shown in 13 examples of specific Lagrangians the stability criterion works effectively without long computations whenever the curvature of the ground state is determined. An infinite number of gravity theories have a stable ground state and further viability criteria are necessary.