On the Metric Propagator and Affine Modes of Extended Hybrid Metric--Palatini Gravity with Ricci--Squared Invariants

Abstract

We investigate an extended hybrid metric--Palatini theory defined by an arbitrary function $f(R,\mathcal{R},\hat{Q},Q,\mathcal{Q})$, where $R$ and $\mathcal{R}$ are the metric and Palatini scalar curvatures, and $Q=R_{\mu \nu }{R}^{\mu \nu }$, $\mathcal{Q}={\mathcal{R}}_{(\mu \nu )}{\mathcal{R}}^{(\mu \nu )}$, and $\hat{Q}=R_{\mu \nu }{\mathcal{R}}^{(\mu \nu )}$ are the metric, Palatini, and mixed quadratic Ricci invariants, respectively. We derive the field equations and analyze the Minkowski weak--field limit. We obtain the metric propagator and identify the additional spin--$2$ and spin--$0$ poles that couple to the metric source. When these poles are finite, real, and simple, the spin-2 sector of the metric propagator contains two additional poles, at least one of which has a negative residue. Once the spin-2 sector is restricted to contain at most one additional pole, the absence of ghostlike metric-coupled excitations in the generic two-scalar branch requires eliminating the remaining spin--$2$ pole, whereas one healthy metric-coupled spin--$2$ pole may coexist with at most one healthy scalar pole when the scalar denominator becomes linear in $k^2$. We derive the corresponding algebraic conditions on the Minkowski-background derivatives of $f$ and specialize them to hybrid $f(R,\mathcal{R})$, mixed $f(R,\mathcal{Q})$, and purely Palatini $f(\mathcal{R},\mathcal{Q})$ theories, including the metric $f(R)$ and Palatini $f(\mathcal{R})$ limits. We further show that the complete linearized system can admit modes carried entirely by the independent connection that are absent from the metric propagator. These results therefore provide necessary flat-space stability criteria for the metric-coupled sector on the branches analyzed, supplemented by the analysis of purely affine modes.

Substantially revised and expanded. We clarify and extend the metric propagator and stability analysis, including the reduced order scalar branch, and separately analyze the full linearized metric connection system, revealing purely affine modes absent from the metric propagator. Degenerate branches, appendices, title, abstract, discussion, and conclusions were also revised