**7. The Binned Analysis with Modified f(R) Gravity**

To try to explain the observed trend of *H*0(*z*), we focus on *f*(*R*) theories of gravity, and then we perform the same binned analysis, using the correction for the distance luminosity according to the modified gravity. We start from the gravitational field action [167]:

$$S\_{\mathcal{S}} = \frac{1}{2\chi} \int d^4x \sqrt{-g} f(R) \,\tag{31}$$

where *f*(*R*), as a function of the Ricci scalar *R*, is an extra degree of freedom compared to General Relativity. We rewrite *f*(*R*) = *R* + *F*(*R*) to highlight the deviation from the standard gravity. Varying the total action with respect to the metric, we obtain the flat FLRW metric field equations:

$$H^2(1+F\_R) = \frac{\chi\rho}{3} + \left[\frac{R\,F\_R - F}{6} - F^{RR}H\dot{R}\right]\_{\prime} \tag{32}$$

where *F<sup>R</sup>* ≡ *dF*(*R*) *dR* . The Ricci scalar *R* can be cast in the form

$$R = 12H^2 + 6HH',\tag{33}$$

where the Hubble parameter *H* is expressed as a function of *γ* ≡ *ln*(*a*), and the prime indicates the derivative with respect to *γ*.

Now, we introduce two dimensionless variables [382]

$$y\_H = \frac{H^2}{m^2} - \frac{1}{a^{3\prime}} \qquad \qquad \qquad y\_R = \frac{R}{m^2} - \frac{3}{a^{3\prime}} \tag{34}$$

which denote the deviation of *H*<sup>2</sup> and *R* with respect to the matter contribution when compared to the ΛCDM model. We rewrite the modified Friedmann Equation (32) and the Ricci scalar relation (33) in terms of *y<sup>H</sup>* and *yR*. Then, we have a set of coupled ordinary differential equations:

$$y\_H' = \frac{1}{3}y\_R - 4y\_H \tag{35}$$

$$y\_R' = \frac{9}{a^3} - \frac{1}{y\_H + a^{-3}} \frac{1}{m^2 F\_{RR}} \left[ y\_H - F\_R \left( \frac{1}{6} y\_R - y\_H - \frac{a^{-3}}{2} \right) + \frac{1}{6} \frac{F}{m^2} \right]. \tag{36}$$

The solution of this coupled first-order differential equations system above can not be obtained analytically, but can be numerically calculated. We need initial conditions such that this scenario mimics the ΛCDM model in the matter dominated universe at initial redshift *z<sup>i</sup> z* ∗ . Hence, we impose the following conditions for *y<sup>H</sup>* and *y<sup>R</sup>* at the redshift *z<sup>i</sup>* :

$$y\_H(z\_i) = \frac{\Omega\_{0\Lambda}}{\Omega\_{0m}}\tag{37}$$

$$y\_{\mathcal{R}}(z\_i) = 12 \frac{\Omega\_{0\Lambda}}{\Omega\_{0m}}.\tag{38}$$

The standard ΛCDM model is reached for *z* = *z<sup>i</sup>* or asymptotically, and we consider a flat geometry, such that Ω0<sup>Λ</sup> = 1 − Ω0*m*. Finally, the luminosity distance can be written as

$$d\_L(z) = \frac{(1+z)}{H\_0} \int\_0^z \frac{dz'}{\sqrt{\Omega\_{0m} \left( y\_H(z') + (1+z')^3 \right)}} \,\,\,\tag{39}$$

including the solution *yH*(*z*) from Equation (34) [386].
