*8.1. An Example for Low Redshifts*

As a viable example for the Dark Energy dominated Universe (slightly different from the traced above), we consider a potential term (and the associated slow-rolling phase) similar to the one adopted in the so-called chaotic inflation [389,390], i.e.,:

$$V(\phi) = \delta + 6H\_0^2 \phi^2 \,, \tag{48}$$

where *<sup>δ</sup>* is a positive constant, such that *<sup>δ</sup>* <sup>6</sup>*H*<sup>2</sup> 0 . From Equation (45), we immediately get

$$H^2 \simeq H\_0^2 \Phi \sim H\_0^2 \,\,\,\,\,\,\,\tag{49}$$

where, we recall that we are considering the slow-rolling phase near *φ* → 1. Analogously, from Equation (47), we immediately get:

$$
\eta \sim -\frac{\delta}{9H\_0^2}.\tag{50}
$$

The negative value of *η* is coherent with the behavior *H*<sup>2</sup> ∝ *φ*. Hence, we can reproduce the requested behavior of *φ*(*z*) by properly fixing the value of *δ* to get *η* as it comes out from the data analysis of Section 4. Specifically, we get *<sup>δ</sup>* <sup>∼</sup> <sup>10</sup>−3*H*<sup>2</sup> 0 to have *<sup>η</sup>* <sup>∼</sup> <sup>10</sup>−<sup>2</sup> . Furthermore, it is easy to check that, for *φ* → 1, Equations (44) and (49), we find the relation

$$
\Phi \sim |\eta| \,\,\, \mathcal{H}\_0 \Phi \ll \mathfrak{B} \mathcal{H}\_0 \Phi \,\, \,\, \tag{51}
$$

which ensures that we are dealing with a real slow-rolling phase.

Finally, we compute the *f*(*R*) function corresponding to the potential in Equation (48), recalling the relation (28):

$$f(R) = \frac{R^2}{24H\_0^2} - \delta\tag{52}$$

We conclude by observing that this specific model is reliable only as far the universe matter component is negligible, *z* < 0.3. The Einstein constant in front of the matterenergy density *ρ* would run as (1 + *z*) 2*η* . The example above confirms that the *f*(*R*) gravity in the JF is a possible candidate to account for the observed effect of *H*0(*z*), but the accomplishment of a satisfactory model for the whole ΛCDM phase requires a significant effort in further investigation, especially accounting for the constraints that observations in the local universe provided for modified gravity.
