On the Metric $f(R)$ gravity Viability in Accounting for the Binned Supernovae Data

TL;DR

This work probes the viability of metric $f(R)$ gravity in the Jordan frame to describe late-time cosmic expansion using binned Type Ia supernova data. A phenomenological first model with a redshift-dependent function $f(z)$ can fit the data but generically induces a tachyonic scalar field and diverging mass at $z=0$, rendering it physically untenable. A revised framework imposes a dynamical condition that preserves agreement with observations while maintaining a positive, finite scalar mass, yielding a physically viable theory that remains competitive with $\\Lambda$CDM and is supported by Pantheon data; Master sample results are compatible but less strongly favorable. The analysis demonstrates that a carefully constructed metric $f(R)$ model can reconcile cosmological data with theoretical viability, offering a concrete route beyond $\\Lambda$CDM while acknowledging Solar-System scale constraints via the chameleon mechanism.

Abstract

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 standard cosmological model ($Λ$CDM), multiplied by $H_0$. $\mathcal{H}(z)$ serves as a diagnostic tool to capture deviations from the $Λ$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. Due to the limited redshift range covered by the available Supernovae Data, this function can be reliably approximated by a second-order Taylor expansion. While this more general formulation yields a phenomenological fit that is compatible with that of the $Λ$CDM for the binned Pantheon Sample, the model is not physically viable, as it 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. The resulting framework not only preserves the agreement with Supernova~Ia data, but also provides a physical justification for the additional condition adopted in earlier analyses of late-time cosmological dynamics.

Figures (11)

• Figure 1: Reconstruction of $\mathcal{H}(z)$ from \ref{['tab:bestfit']} compared with the binned Pantheon data.
• Figure 2: Squared mass of the scalar field for the parameters shown in Fig. \ref{['tab:bestfit']}.
• Figure 3: Reconstruction of $\mathcal{H}(z)$ from Tab. \ref{['tab:master_bestfit']} compared with the binned Pantheon data.
• Figure 4: Scalar field mass squared $u^{2}(z)$ obtained from the MCMC analysis of the Master Sample.
• Figure 5: Solution of Eq. \ref{['eq:massa_finitudine']} for $\alpha=1$, expressing the required relation between $\frac{df}{dz}$ and $\frac{d^{2}f}{dz^{2}}$ at $z=0$ for a finite scalar mass. From the initial conditions of the problem, $\phi'(0)=0$, see Eq. \ref{['problemacauchy']}.