6.2.2. Metric f(R) Gravity in the Jordan Frame

The observed decaying behavior of the Hubble constant *H*<sup>0</sup> with the redshift draws significant attention for an explanation and, if it is not due to selection effects or systematics in the sample data, we need to interpret our results from a physical point of view. As already argued in [51,380], the simplest way to account for this unexpected behavior of *H*0(*z*) is that the Einstein constant *χ* = 8*πG* (where *G* denotes the gravitational constant), mediating the gravity-matter interaction, is subjected itself to a slow decaying profile with the redshift. In this Section, we consider *c* = 1 for the speed of light. More specifically, since the critical energy density *ρc*<sup>0</sup> = 3 *H*<sup>2</sup> 0 /*χ* today must be a constant, we need an evolution for *χ* ∼ (1 + *z*) −2 *η* , considering the function *H*0(*z*) given by Equation (9). The evolution of *χ*(*z*) is not expected within the cosmological Einsteinian gravity, therefore we are led to think of it as a pure dynamical effect, associated with a modified Lagrangian for the gravitational field beyond the ΛCDM cosmological model. Ref. [143] obtained cosmological constraints within the Brans–Dicke theory considering how the evolution of the gravitational constant *G*, contained in *χ*, affects the SNe Ia peak luminosity. The most natural extended framework is the *f*(*R*)-gravity proposal [162,163,167,381] which contains only an additional scalar degree of freedom. For instance, ref. [323] try to alleviate the *H*<sup>0</sup> tension considering exponential and power-law *f*(*R*) models.

The formulation of the *f*(*R*) theories in an equivalent scalar-tensor paradigm turns out to be particularly intriguing for our purposes: the function *f*(*R*) is restated as a real scalar field *φ*, which is non-minimally coupled to the metric in the JF. The information about the function *f* turns into the expression of the scalar field potential *V*(*φ*). The relevance of modified gravity models relies on the possibility that this revised scenario for the gravitational field can account for the physics of the so-called "dark universe"component without the need for a cosmological constant. Indeed, the observed cosmic acceleration in the late universe via the SNe Ia data is a pure dynamical effect, i.e., associated with a modification of the Einsteinian gravity at very large scales (in the order of the present Hubble length).

According to the standard literature on this field (which includes a large number of proposals), three specific *f*(*R*) models, i.e., the Hu–Sawicki [382], the Starobinsky [383], and Tsujikawa models [384,385], successfully describe the Dark Energy component (say an effective parameter for the Dark Energy *w* = *w*(*z*) < −1/3) and overcome all local constraints. The difference in the form of the Lagrangian densities associated with *f*(*R*) models is reflected in the morphology of the potential term governing the dynamics of the scalar field. For instance, the scalar field potential related to the Hu–Sawicki *f*(*R*) proposal, with the power index *n* = 1, in the JF is given by

$$V(\phi) = \frac{m^2}{c\_2} \left[ c\_1 + 1 - \phi - 2\sqrt{c\_1(1-\phi)} \right],\tag{18}$$

where we have two free parameters *c*<sup>1</sup> and *c*2, while *m*<sup>2</sup> = *χ ρ*0*m*/3. The scalar-tensor dynamics in the JF for a flat FLRW metric with a matter component is summarized by

$$H^2 = \frac{\chi \rho}{3\,\phi} - H\frac{\phi}{\phi} + \frac{V(\phi)}{6\,\phi} \tag{19}$$

$$\frac{d}{a} = -\frac{\chi \rho}{3\,\phi} - \frac{V(\phi)}{6\,\phi} + \frac{1}{6}\frac{dV}{d\phi} + \frac{a}{a}\frac{\phi}{\phi} \tag{20}$$

$$2\ddot{\phi} - 2\, V(\phi) + \phi \frac{dV}{d\phi} + 9\, H\phi = \chi\rho\_\prime \tag{21}$$

which are the generalized Friedmann equation, the generalized cosmic acceleration equation and the scalar field equation, respectively [167]. We recall that *φ* = *φ*(*t*) is a function of the time (or the redshift *z*) only for an isotropic universe. Considering the first term on the right-hand side of Equation (19), it is possible to recognize that *φ* mediates the

gravity-matter coupling, and therefore it mimics a space-time varying Einstein constant. Hence, to account for our observed decay of *H*0(*z*), we have to require that the scalar field assumes a specific behavior with the redshift, i.e.,

$$
\phi(z) = (1+z)^{2\eta}.\tag{22}
$$

Moreover, the remaining terms contained in the gravitational field equations must be negligible. This situation is naturally reached when the potential term is sufficiently slowvarying in a given time interval. We see that the hypothesis of a near-frozen scalar field evolution is a possible assumption, as far as the potential term should provide a dynamical impact, sufficiently close to a cosmological constant term. These simple considerations lead us to claim that this scenario is worth to be investigated for the behavior of *H*0(*z*) here observed.

The specific cosmological models affect the expression of the luminosity distance and this should be the starting point of a careful test of a *f*(*R*) theory versus the comprehension of the *H*<sup>0</sup> tension. A new binned analysis of the PS, using the corrected luminosity distance obtained through a reliable *f*(*R*), may in principle shed new light on the observed decaying trend of *H*0(*z*), testing also new physics. This analysis is performed in the next Section.

As a preliminary approach, we try to understand which profile we could expect for the scalar field potential, inferred from the behavior of *H*0(*z*). This is quite different from a standard analysis of *f*(*R*) models. Generally, a specific *f*(*R*) function is defined a priori, and then the dynamical equations are studied to obtain constraints on the free parameters. Here, instead, starting from the observed decreasing trend of *H*0(*z*) and assuming *φ*(*z*) from Equation (22), we wonder what the scalar field potential would be in a scalar-tensor dynamics. Eventually, we should have a scalar field in near-frozen dynamics, i.e., a slowroll of the scalar field potential, mimicking a cosmological constant term (*φ* → 1). To this end, we rewrite the generalized Friedmann Equation (19) and calculate *V*(*φ*):

$$V(\phi) = 6(1 - 2\eta) \left(\frac{dz}{dt}\right)^2 \phi^{1 - 1/\eta} - 6m^2 \phi^{3/2\eta},\tag{23}$$

where we have used the standard definition of redshift and the relation (22) for *φ*(*z*). Moreover, we recall that for a matter component *ρ* ∼ (1 + *z*) 3 . As a final step, we need to calculate the term *dz dt* . Starting again from the redshift definition, it is well known that

$$\frac{dz}{dt} = -(1+z)\,H(z). \tag{24}$$

In principle, we would need to compute the Hubble parameter *H*(*z*) from the field equations, and then replace *H*(*z*) in the term *dz dt* . However, this procedure is not viable, since we need to fix a well-defined *V*(*φ*) to solve the field equations. Moreover, *H*(*z*) appears also in the right-hand-side of Equation (19), because of the non-minimal coupling with the scalar field. Therefore, we can not calculate exactly *dz dt* to get *V*(*φ*) in the JF.

Then, to obtain *V*(*φ*) inferred from the trend of *H*0(*z*), we require that the Hubble function provides the same physical mechanism suggested from our binned analysis in Section 4, i.e., simply replacing *H*<sup>0</sup> with *H*0(*z*) given by Equation (9) in the standard Friedmann equation in the ΛCDM model. With this new definition of H0, we write the following condition on the Hubble function:

$$H(z) = \frac{\mathcal{H}\_0}{(1+z)^{\eta}} \sqrt{\Omega\_{0m} \left(1+z\right)^3 + 1 - \Omega\_{0m}} \,. \tag{25}$$

In doing so, using Equations (23)–(25), we determine the form of the scalar field potential

$$\frac{V(\phi)}{m^2} = 6\left(1 - 2\eta\right) \left(\frac{1 - \Omega\_{0m}}{\Omega\_{0m}}\right) - 12\eta \left.\phi^{3/2\eta}\right|\tag{26}$$

inferred from the decreasing trend of *H*0(*z*). In other words, the potential Equation (26) might provide an effective Hubble constant that evolves with redshift. In the computation, we have used the expression <sup>Ω</sup>0*<sup>m</sup>* <sup>=</sup> *<sup>m</sup>*2/H<sup>2</sup> 0 .

In Figure 3, we plotted this potential profile, observing that, as expected, a flat region consistently appears, validating our guess on the feasibility of *f*(*R*)-gravity in the JF to account for the observed behavior of *H*0(*z*). We set *η* = 0.009 in Figure 3, according to our binned analysis results for three bins (see Table 1). We stress that the flatness of the potential does not emerge throughout the Pantheon sample redshift range, 0 < *z* < 2.3, but it appears only in a narrow region for 0 < *z* . *z* ∗ , where *z* ∗ = 0.3 is the redshift at the Dark Energy and Matter components equivalence of the universe. This form of *V*(*φ*) is reasonable since the Dark Energy contribution, provided by the scalar field in the JF gravity, dominates the matter component only for 0 < *z z* ∗ . It is the weak dependence of *H*<sup>0</sup> on *z* that ensures the existence of a flat region of the potential, according to the theoretical scenario argued above.

**Figure 3.** Profile of the scalar field potential *V*(*φ*) in the JF equivalent scalar-tensor formalism of the f(R) modified gravity. The form of *V*(*φ*)) is inferred from the behavior of *H*0(*z*) (Equation (9)). Note that *V*(*φ*)/*m*<sup>2</sup> is a dimensionless quantity. A flat profile of *V*(*φ*) occurs only at low redshifts, for 0 < *z* . 0.3 or equivalently *φ* . 1.005. Note, also, the non-linearity of the scale for the redshift axis on top, considering the relation (22) between *φ* and *z*. In this plot, *η* = 0.009.

Finally, we can calculate the form of the *f*(*R*) function associated with the potential profile. Recalling the following general relations in the JF [167]:

$$R = \frac{dV}{d\phi'}\tag{27}$$

$$f(R) = R\,\phi(R) - V(\phi(R)),\tag{28}$$

we can obtain:

$$f(\mathbb{R}) = -6 \, m^2 \left[ (1 - 2\eta) \frac{1 - \Omega\_{0m}}{\Omega\_{0m}} + (3 - 2\eta) \left( -\frac{R}{18 \, m^2} \right)^{\frac{3}{3 - 2\eta}} \right]. \tag{29}$$

Note that the formula above provides a generalization of the Einstein theory of gravity, as it should be in the context of a *f*(*R*) model. Indeed, if *η* = 0, then *f*(*R*) ≡ *R* reproduces exactly the Einstein–Hilbert Lagrangian density in GR with a cosmological constant Λ, as soon

as you recognize that Λ = 3*m*<sup>2</sup> (<sup>1</sup> <sup>−</sup> <sup>Ω</sup>0*m*) / <sup>Ω</sup>0*<sup>m</sup>* for a flat geometry, using *<sup>m</sup>*<sup>2</sup> <sup>=</sup> <sup>H</sup><sup>2</sup> <sup>0</sup> Ω0*m*. In particular, expanding the function (29) for *η* ∼ 0, we can see explicitly the deviation from the Einstein–Hilbert term:

$$f(R) \approx \left(R - 6\,m^2 \frac{1 - \Omega\_{0\text{m}}}{\Omega\_{0\text{m}}}\right) + \frac{2}{3}\eta \left[R\,\ln\left(-\frac{R}{m^2}\right) - \left(1 + \ln 18\right)R + 18m^2 \frac{1 - \Omega\_{0\text{m}}}{\Omega\_{0\text{m}}}\right] + O\left(\eta^2\right). \tag{30}$$

The first term at the zero-th order in *η* is exactly the Einstein–Hilbert Lagrangian density, while the linear term in *η* provides the correction to GR. Therefore, *η*, in addition to being the physical parameter that describes the evolution of *H*0(*z*), also denotes the deviation from GR and the standard cosmological model. It is worthwhile to remark that the expression above may not be the final form of the underlying modified theory of gravity, associated with the global universe dynamics, but only its asymptotic form in the late Universe, i.e., as the scalar of curvature approaches the value corresponding to the cosmological constant in the ΛCDM model. In all these computations we do not consider relativistic or radiation components at very high redshifts, but it may be interesting to test this model with other local probes in the late Universe.

In this discussion, we infer that the dependence of *H*<sup>0</sup> on the redshift points out the necessity of new physics in the description of the universe dynamics and that such a new framework may be identified in the modified gravity, related to metric theories.
