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.
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