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We provide insight about the full form of the equations for matter density perturbations and the scalar Bardeen metric potentials in general f (R) theories of gravity. When considering viable modifications to the standard ΛCDM background, the full scale-dependent equations for the metric perturbations are provided and are shown to match the ones obtained with the quasistatic approximation. We investigate the impact of the n=2 Hu-Sawicki model on the late-time growth of structures. We find that updated late-time growth of structure data imposes |fₑ䃐|10^-6-10^-5 and thus conclude that the Hu-Sawicki f (R) model contributes no significant phenomenology at both background and perturbative level beyond the effective cosmological constant encompassed in its definition. This conclusion points to the survival of the present tension between early and late measurements of σ₈, as the Hu-Sawicki model can only worsen this issue or at best reproduce the results from the current concordance cosmological model. The generalized perturbative method showcased in this work can be applied to more elaborate f (R) models to isolate genuine higher-order signatures beyond the quasistatic approximation.
Varela et al. (Mon,) studied this question.