Higher order gravity theories and scalar tensor theories

TL;DR

This paper extends the equivalence between higher order gravity and scalar tensor theories to a broader class involving two scalar fields, analyzing their properties and implications for solar system tests.

Contribution

It introduces a new class of theories with two scalar fields, showing their structure, coupling, and constraints from experimental tests.

Findings

Theories can be expressed as tensor-multi-scalar models with constant negative curvature.

The coupling to matter is universal and independent of the function f.

If both scalar fields are long-ranged, the PPN parameter is 1/2, conflicting with solar system tests.

Abstract

We generalize the known equivalence between higher order gravity theories and scalar tensor theories to a new class of theories. Specifically, in the context of a first order or Palatini variational principle where the metric and connection are treated as independent variables, we consider theories for which the Lagrangian density is a function f of (i) the Ricci scalar computed from the metric, and (ii) a second Ricci scalar computed from the connection. We show that such theories can be written as tensor-multi-scalar theories with two scalar fields with the following features: (i) the two dimensional sigma-model metric that defines the kinetic energy terms for the scalar fields has constant, negative curvature; (ii) the coupling function determining the coupling to matter of the scalar fields is universal, independent of the choice of function f; and (iii) if both mass eigenstates are long ranged, then the Eddington post-Newtonian parameter has value 1/2. Therefore in order to be compatible with solar system experiments at least one of the mass eigenstates must be short ranged.