Constraining $f(R)$ gravity and evolving dark energy via large-scale structure and phase-space trajectories

Abstract

We present a joint observational analysis confronting viable $f(R)$ modified gravity theories, specifically the Hu \& Sawicki and Starobinsky models, with background and large-scale structure (LSS) data. Utilizing Monte-Carlo Markov chain (MCMC) sampling across datasets including baryon acoustic oscillations (BAO), type Ia supernovae (SNeIa), cosmic microwave background (CMB) distance priors, and linear growth measurements ($fσ_8$, $f$, $σ_8$), we place tight constraints on the model parameters governing deviations from General Relativity. For the full dataset combination, we obtain $\log_{10} b_\mathrm{HS} = -6.325_{-1.138}^{+1.216}$ for the Hu \& Sawicki model and $b_\mathrm{S} = (0.8\pm61.0)\times10^{-4}$ for the Starobinsky model. Model comparison based on the Akaike Information Criterion indicates that these $f(R)$ extensions are statistically favored over flat $Λ\text{CDM}$ ($|Δ\text{AIC}| \ge 3.99$) for the combined data. However, when considering the Bayesian Information Criterion, the evidence for support is significantly reduced. Furthermore, we construct two-dimensional phase-space diagrams in the $(μ, γ)$ and $(μ, Σ)$ planes across several redshifts, establishing a novel diagnostic null-test allowing us to probe for deviations from $Λ\text{CDM}$, corresponding to the fixed point $(1,1)$ in both planes, using LSS observables. Should future weak-lensing and galaxy surveys provide data points with $μ-1<0$ and $γ-1>1$ or $Σ-1 < 0 $, then the aforementioned models could be directly ruled out.