Metric gravity theories and cosmology: I. Physical interpretation and viability

Abstract

We critically review some concepts underlying current applications of gravity theories with Lagrangians L = f(gμν, Rαβμν) to cosmology to account for the accelerated expansion of the universe. We argue that one cannot reconstruct the function f from astronomical observations either in cosmology or in the solar system. The Robertson–Walker spacetime is so simple and ‘flexible’ that any cosmic evolution may be fitted by an infinite number of various Lagrangians. We show in the example of Newton's gravity that one cannot recover the correct equation of motion from its approximate solution. Any gravity theory different from general relativity generates a new cosmological theory and all the successes of the standard cosmological model are lost even if a single solution of the theory well fits the observations. Prior to application of a given gravity theory to cosmology or elsewhere it is necessary to establish its physical contents and viability. This study may be performed by a universal method of Legendre transforming the initial Lagrangian in a Helmholtz Lagrangian. In this formalism, Lagrange equations of motion are of second order and are the Einstein field equations with additional massive spin-0 and spin-2 fields. All the gravity theories differ only by a form of interaction terms of the two fields and the metric. Initial conditions for the two fields in the gravitational triplet depend on which frame (i.e., the set of dynamical variables) is physical (i.e. matter is minimally coupled in it). This fact and the multiplicity of possible frames obstruct confrontation of solutions to equations of motion with the observational data. A fundamental and easily applicable criterion of viability of any gravity theory is 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.