Cosmological perturbations in the Palatini formulation of modified gravity

TL;DR

The paper addresses cosmological perturbations in generalized gravity within the Palatini formalism, where the metric and independent connection yield second-order field equations. It develops a gauge-ready perturbation framework and derives scalar, vector, and tensor equations around a homogeneous, possibly curved background for broad f(R, φ) theories, with a focus on late-time structure formation. A key finding is that curvature corrections can induce an effective pressure gradient in cold dark matter, posing challenges for the growth of structure, as demonstrated analytically for the case f(R) ~ R^n. The work also proves a precise equivalence to metric scalar-tensor theories upon appropriate rescaling of the scalar kinetic term, clarifying the relation between the two formalisms and guiding extensions to more general theories.

Abstract

Cosmology in extended theories of gravity is considered assuming the Palatini variational principle, for which the metric and connection are independent variables. The field equations are derived to linear order in perturbations about the homogeneous and isotropic but possibly spatially curved background. The results are presented in a unified form applicable to a broad class of gravity theories allowing arbitrary scalar-tensor couplings and nonlinear dependence on the Ricci scalar in the gravitational action. The gauge-ready formalism exploited here makes it possible to obtain the equations immediately in any of the commonly used gauges. Of the three type of perturbations, the main attention is on the scalar modes responsible for the cosmic large-scale structure. Evolution equations are derived for perturbations in a late universe filled with cold dark matter and accelerated by curvature corrections. Such corrections are found to induce effective pressure gradients which are problematical in the formation of large-scale structure. This is demonstrated by analytic solutions in a particular case. A physical equivalence between scalar-tensor theories in metric and in Palatini formalisms is pointed out.