In this section we discuss the compatibility of f (R) models with local gravity constraints (see [469, 470, 245, 233, 154, 448, 251] for early works, and [31, 306, 134] for experimental constraints on viable f (R) dark energy models, and [101, 210, 330, 332, 471, 628, 149, 625, 329, 45, 511, 277, 534, 133, 445, 309, 89] for other related works). In an environment of high density such as Earth or Sun, the Ricci scalar is much larger than the background cosmological value . If the outside of a spherically symmetric body is a vacuum, the metric can be described by a Schwarzschild exterior solution with . In the presence of non-relativistic matter with an energy density , this gives rise to a contribution to the Ricci scalar of the order .
If we consider local perturbations on a background characterized by the curvature , the validity of the linear approximation demands the condition . We first derive the solutions of linear perturbations under the approximation that the background metric is described by the Minkowski metric . In the case of Earth and Sun the perturbation is much larger than , which means that the linear theory is no longer valid. In such a non-linear regime the effect of the chameleon mechanism [344, 343] becomes important, so that f (R) models can be consistent with local gravity tests.
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