In this section we discuss f (R) theory in the Palatini formalism [481]. In this approach the action (2.1) is varied with respect to both the metric and the connection . Unlike the metric approach, and are treated as independent variables. Variations using the Palatini approach [256, 607, 608, 261, 262, 260] lead to second-order field equations which are free from the instability associated with negative signs of [422, 423]. We note that even in the 1930s Lanczos [378] proposed a specific combination of curvature-squared terms that lead to a second-order and divergence-free modified Einstein equation.

The background cosmological dynamics of Palatini f (R) gravity has been investigated in [550, 553, 21, 253, 495], which shows that the sequence of radiation, matter, and accelerated epochs can be realized even for the model with (see also [424, 457, 495]). The equations for matter density perturbations were derived in [359]. Because of a large coupling between dark energy and non-relativistic matter dark energy models based on Palatini f (R) gravity are not compatible with the observations of large-scale structure, unless the deviation from the CDM model is very small [356, 386, 385, 597]. Such a large coupling also gives rise to non-perturbative corrections to the matter action, which leads to a conflict with the Standard Model of particle physics [261, 262, 260] (see also [318, 472, 473, 475, 55]).

There are also a number of works [470, 471, 216, 552] about the Newtonian limit in the Palatini formalism (see also [18, 19, 107, 331, 511, 510]). In particular it was shown in [55, 56] that the non-dynamical nature of the scalar-field degree of freedom can lead to a divergence of non-vacuum static spherically symmetric solutions at the surface of a compact object for commonly-used polytropic equations of state. Hence Palatini f (R) theory is difficult to be compatible with a number of observations and experiments, as long as the models are constructed to explain the late-time cosmic acceleration. Moreover it is also known that in Palatini gravity the Cauchy problem [609] is not well-formulated due to the presence of higher derivatives of matter fields in field equations [377] (see also [520, 135] for related works). We also note that the matter Lagrangian (such as the Lagrangian of Dirac particles) cannot be simply assumed to be independent of connections. Even in the presence of above mentioned problems it will be useful to review this theory because we can learn the way of modifications of gravity from GR to be consistent with observations and experiments.

9.1 Field equations

9.2 Background cosmological dynamics

9.3 Matter perturbations

9.4 Shortcomings of Palatini f (R) gravity

9.2 Background cosmological dynamics

9.3 Matter perturbations

9.4 Shortcomings of Palatini f (R) gravity

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