On the Metric $f(R)$ gravity Viability in Accounting for the Binned Supernovae Data
In this work, two models of metric $f(R)$ gravity in the Jordan frame are investigated as a dynamical description of the late-time cosmic expansion using binned Type Ia Supernovae data. The aim is to provide an explanation for the effective running of the Hubble constant observed in both the binned Pantheon Sample and the Master Sample. To this end, the effective running Hubble constant $\mathcal{H}(z)$ is defined as the ratio between the modified Hubble parameter and that of the $\Lambda$CDM, multiplied by $H_0$. $\mathcal{H}(z)$ serves as a diagnostic tool to capture deviations from the $\Lambda$CDM model. The first model used is a general representation of metric $f(R)$ gravity in which the gravitational Lagrangian is encoded in an effective redshift-dependent function that mimics the evolution of the Hubble parameter. This function can be approximated by a second-order Taylor expansion at low redshift due to the limited redshift range covered by the Supernovae data. While this general formulation yields a phenomenological fit compatible with that of the $\Lambda$CDM model for the binned Pantheon Sample, the model generically leads to the emergence of an unphysical mass of the scalar field. This issue originates from an implicit restriction imposed on the Cauchy problem for the scalar field. To address this limitation, following previous studies, an additional condition on the modified Friedmann equation is introduced, enabling a fully consistent reformulation of the dynamics. It is clarified that this additional condition has a precise dynamical origin, being necessary to restore a consistent Cauchy problem and to ensure a finite, positive scalar field mass. The resulting framework not only preserves the agreement with binned Supernova Ia data, but also provides a physical justification for the additional condition adopted in earlier analyses of late-time cosmological dynamics.