Higher order gravity theories and scalar tensor theories

TL;DR

The paper generalizes the equivalence between higher-order gravity and scalar-tensor theories to Palatini (first-order) theories with a Lagrangian $f(\bar{R},\hat{R})$, where $\bar{R}$ is the metric Ricci scalar and $\hat{R}$ is the connection Ricci scalar. By introducing auxiliary fields and performing conformal (Einstein-frame) reductions, it shows that the theory is classically equivalent to a tensor-miscalar theory with two dynamical fields $\Phi$ and $\Psi$, characterized by a sigma-model metric of constant negative curvature, a universal coupling function $\alpha(\Phi,\Psi)$, and a potential $V(\Phi,\Psi)$ expressed in terms of $f$ and its derivatives. Stability and nondegeneracy conditions constrain the signs $\varepsilon_1$, $\varepsilon_2$, $\varepsilon_3$ and require a nonzero determinant $\Delta(\bar{R},\hat{R})$ to ensure a well-posed dynamics and a bijection between domain ${\mathcal D}$ and the field domain ${\mathcal D}'$. The resulting framework demonstrates a ghost-free, two-field scalar-tensor representation for a broad class of Palatini gravity theories, with implications for solar-system tests (via the PPN parameter $\gamma$) and potential cosmological applications such as explaining late-time acceleration without fully abandoning general relativity.

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.