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Cosmological perturbations in Palatini modified gravity

Two approaches to the study of cosmological density perturbations in modified theories of Palatini gravity have recently been discussed. These utilise, respectively, a generalisation of Birkhoff's theorem and a direct linearization of the gravitational field equations. In this paper these approaches are compared and contrasted. The general form of the gravitational lagrangian for which the two frameworks yield identical results in the long-wavelength limit is derived. This class of models includes the case where the lagrangian is a power-law of the Ricci curvature scalar. The evolution of density perturbations in theories of the type $f(R)=R-c /R^ b$ is investigated numerically. It is found that the results obtained by the two methods are in good agreement on sufficiently large scales when the values of the parameters (b,c) are consistent with current observational constraints. However, this agreement becomes progressively poorer for models that differ significantly from the standard concordance model and as smaller scales are considered.