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 H ( z ) is defined as the ratio between the modified Hubble parameter and that of the standard cosmological model (ΛCDM), multiplied by H 0 . H ( z ) serves as a diagnostic tool to capture deviations from the ΛCDM 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 reliably approximated by a second-order Taylor expansion at low redshift. While this more general formulation yields a phenomenological fit that is better than the ΛCDM 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 non-minimally coupled 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 the 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.