9.4 Shortcomings of Palatini f (R) gravity

In addition to the fact that Palatini f (R) dark energy models are hardly distinguished from the ΛCDM model from observations of large-scale structure, there are a number of problems in Palatini f (R) gravity associated with non-dynamical nature of the scalar-field degree of freedom.

The dark energy model f = R − μ4∕R based on the Palatini formalism was shown to be in conflict with the Standard Model of particle physics [261Jump To The Next Citation Point262Jump To The Next Citation Point260Jump To The Next Citation Point318Jump To The Next Citation Point55Jump To The Next Citation Point] because of large non-perturbative corrections to the matter Lagrangian [here we use R for the meaning of R (T)]. Let us consider this issue for a more general model 2(n+1) n f = R − μ ∕R. From the definition of φ in Eq. (9.6View Equation) the field potential U(φ ) is given by

--n-+-1--μ2- n∕(n+1) U (φ) = 2nn ∕(n+1)κ2 (φ − 1) , (9.42 )
where 2(n+1) −n−1 φ = 1 + nμ R. Using Eq. (9.7View Equation) for the vacuum (T = 0), we obtain the solution
2(n +-1)- φ (T = 0 ) = n + 2 . (9.43 )

In the presence of matter we expand the field φ as φ = φ (T = 0 ) + δφ. Substituting this into Eq. (9.7View Equation), we obtain

-----n-----κ2T- δ φ ≃ nn++21 μ2 . (9.44 ) (n + 2)
For n = 𝒪(1) we have δφ ≈ κ2T ∕μ2 = T ∕(μ2M p2l) with φ(T = 0) ≈ 1. Let us consider a matter action of a Higgs scalar field ϕ with mass m ϕ:
∫ [ ] S = d4x√ −-g − 1gμν∂ ϕ ∂ ϕ − 1-m2 ϕ2 . (9.45 ) M 2 μ ν 2 ϕ
Since 2 2 T ≈ m ϕδϕ it follows that 2 2 2 2 δφ ≈ m ϕδϕ ∕(μ M pl). Perturbing the Jordan-frame action (9.8View Equation) [which is equivalent to the action in Palatini f (R) gravity] to second-order and using the solution φ ≈ 1 + m2ϕδϕ2∕(μ2M p2l), we find that the effective action of the Higgs field ϕ for an energy scale E much lower than m ϕ (= 100 – 1000 GeV) is given by [55Jump To The Next Citation Point]
( ) ∫ √ ---[ 1 1 ] m2 δϕ2 δSM ≃ d4x − g − --gμν∂μδϕ ∂νδϕ − -m2ϕδ ϕ2 1 + --ϕ--2-+ ⋅⋅⋅ . (9.46 ) 2 2 μ2M pl
Since δϕ ≈ m ϕ for E ≪ m ϕ, the correction term can be estimated as
2 2 ( )2 ( )2 m-ϕδϕ-- m-ϕ- m-ϕ- δφ ≈ μ2M 2pl ≈ μ Mpl . (9.47 )
In order to give rise to the late-time acceleration we require that μ ≈ H0 ≈ 10 −42 GeV. For the Higgs mass m ϕ = 100 GeV it follows that δφ ≈ 1056 ≫ 1. This correction is too large to be compatible with the Standard Model of particle physics.

The above result is based on the models 2(n+1) n f(R) = R − μ ∕R with n = 𝒪 (1 ). Having a look at Eq. (9.44View Equation), the only way to make the perturbation δ φ small is to choose n very close to 0. This means that the deviation from the ΛCDM model is extremely small (see [388Jump To The Next Citation Point] 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 [55Jump To The Next Citation Point], the same result was also obtained by the analysis in the Einstein frame [261262Jump To The Next Citation Point260318].

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 P = cρΓ 0 with 3∕2 < Γ < 2 (where P is the pressure and ρ 0 is the rest-mass density) [56Jump To The Next Citation Point55] (see also [107331]). Again this problem is intimately related with the particular algebraic dependence (9.2View Equation) 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 2 2 f (R) = R + R ∕(6M ), where M 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 [38438795236388509476]. 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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