The dark energy model based on the Palatini formalism was shown to be in conflict
with the Standard Model of particle physics [261
, 262
, 260
, 318
, 55
] because of large non-perturbative
corrections to the matter Lagrangian [here we use
for the meaning of
]. Let us consider this
issue for a more general model
. From the definition of
in Eq. (9.6
) the field
potential
is given by
In the presence of matter we expand the field as
. Substituting this into
Eq. (9.7
), we obtain
The above result is based on the models with
. Having a look at
Eq. (9.44
), the only way to make the perturbation
small is to choose
very close to 0. This means
that the deviation from the
CDM model is extremely small (see [388
] for a related work).
In fact, this property was already found by the analysis of matter density perturbations in
Section 9.3. While the above analysis is based on the calculation in the Jordan frame in which test
particles follow geodesics [55
], the same result was also obtained by the analysis in the Einstein
frame [261, 262
, 260, 318].
Another unusual property of Palatini f (R) gravity is that a singularity with the divergent Ricci scalar
can appear at the surface of a static spherically symmetric star with a polytropic equation of state
with
(where
is the pressure and
is the rest-mass density) [56
, 55] (see
also [107, 331]). Again this problem is intimately related with the particular algebraic dependence (9.2
) in
Palatini f (R) gravity. In [56] it was claimed that the appearance of the singularity does not very much
depend on the functional forms of f (R) and that the result is not specific to the choice of the polytropic
equation of state.
The Palatini gravity has a close relation with an effective action which reproduces the dynamics of loop
quantum cosmology [477]. [474] showed that the model , where
is of the
order of the Planck mass, is not plagued by a singularity problem mentioned above, while the singularity
typically arises for the f (R) models constructed to explain the late-time cosmic acceleration (see also [504]
for a related work). Since Planck-scale corrected Palatini f (R) models may cure the singularity
problem, it will be of interest to understand the connection with quantum gravity around the
cosmological singularity (or the black hole singularity). In fact, it was shown in [60] that non-singular
bouncing solutions can be obtained for power-law f (R) Lagrangians with a finite number of
terms.
Finally we note that the extension of Palatini f (R) gravity to more general theories including Ricci and Riemann tensors was carried out in [384, 387, 95, 236, 388, 509, 476]. While such theories are more involved than Palatini f (R) gravity, it may be possible to construct viable modified gravity models of inflation or dark energy.
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