Modified Gravity from growth data: goodness-of-fit and gravitational coupling

While background expansion data alone cannot discriminate among cosmological models, the growth of large-scale structures directly probes the underlying theory of gravity, offering a path to distinguish General Relativity from its modifications. In this work, we constrain three representative $F(R)$ modified gravity models (Starobinsky, Hu-Sawicki, and $R^2$-corrected Appleby-Battye) using measurements of the growth rate $f(z)$ and the matter fluctuation amplitude $σ_8(z)$. Our analyses combine MCMC parameter estimation, Gaussian Process reconstructions, and goodness-of-fit statistics including Akaike information criterion (AIC) and Bayesian information criterion (BIC). All investigated models provide statistically comparable fits to the current growth data, with information criteria differences too small to establish a preference for any particular scenario. To overcome this degeneracy, we reconstruct the effective gravitational coupling $μ(z)$ from the MCMC posterior samples, providing a physically motivated diagnostic that complements standard goodness-of-fit criteria. The Starobinsky and Hu-Sawicki models predict moderate departures from General Relativity, remaining compatible with constraints on both $μ(z)$ and $S_8$. In contrast, the $R^2$-AB model predicts $μ_0 \sim 3.5$ and $S_8 = 0.638$, a $3.2σ$ tension with Planck 2018, rendering it physically disfavored despite its competitive statistical performance.