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Palatini $f(R)$ gravity is probably the simplest extension of generalrelativity (GR) and the simplest realization of a metric-affine theory. It hasthe same number of degrees of freedom as GR and, in vacuum, it isstraightforwardly mapped into GR with a cosmological constant. The mappingbetween GR and Palatini $f(R)$ inside matter is possible but at the expense ofreinterpreting the meaning of the matter fields. The physical meaning andconsequences of such mapping will depend on the physical context. Here weconsider three such cases within the weak field limit: Solar System dynamics,planetary internal dynamics (seismology), and galaxies. After revising ourprevious results on the Solar System and Earth's seismology, we consider herethe possibility of $f(R)$ Palatini as a dark matter candidate. For any $f(R)$that admits a polynomial approximation in the weak field limit, we show here,using SPARC data and a recent method that we proposed, that the theory cannotbe used to replace dark matter in galaxies. We also show that the same resultapplies to the Eddington-inspired Born-Infeld gravity. Differently from themetric $f(R)$ case, the rotation curve data are sufficient for this conclusion.This result does not exclude a combination of modified gravity and dark matter.

Palatini $f(R)$ gravity tests in the weak field limit: Solar System, seismology and galaxies