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

where . Using Eq. (9.7) for the vacuum (), we obtain the solutionIn the presence of matter we expand the field as . Substituting this into Eq. (9.7), we obtain

For we have with . Let us consider a matter action of a Higgs scalar field with mass : Since it follows that . Perturbing the Jordan-frame action (9.8) [which is equivalent to the action in Palatini f (R) gravity] to second-order and using the solution , we find that the effective action of the Higgs field for an energy scale much lower than (= 100 – 1000 GeV) is given by [55] Since for , the correction term can be estimated as In order to give rise to the late-time acceleration we require that . For the Higgs mass it follows that . This correction is too large to be compatible with the Standard Model of particle physics.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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