Implications of $f(R)$ gravity on late-time cosmic structure growth through a complete description of density perturbations

TL;DR

This work develops and applies a complete linear perturbation framework for general $f(R)$ gravity, focusing on the Hu-Sawicki model with $n=2$. By deriving the full density-perturbation and metric-potential equations beyond the quasistatic approximation, the authors show that background evolution can closely mimic ΛCDM, while perturbations introduce scale-dependent effects encoded in $G_{ m eff}(z,k)$ and nontrivial relations between $ ext{Φ}$ and $ ext{Ψ}$. Fitting to late-time growth data, including $f_g \,\sigma_8$ measurements and Planck priors, yields very tight constraints $|f_{R_0}|\lesssim 10^{-6}-10^{-5}$ and $d_{ m HS}\gtrsim 10^4-10^{5}$, driving the model toward effectively ΛCDM behavior with no significant perturbative signatures. The results reinforce the persistence of the $\sigma_8$ tension and demonstrate that the full perturbative treatment is essential to isolate genuine higher-order signatures in more complex $f(R)$ theories. The methodology can be extended to other $f(R)$ constructions to identify observable deviations beyond the QS regime.

Abstract

We provide insight about the full form of the equations for matter density perturbations and the scalar Bardeen metric potentials in general $f(R)$ theories of gravity. When considering viable modifications to the standard $Λ$CDM background, the full scale-dependent equations for the metric perturbations are provided and are shown to match the ones obtained with the quasistatic approximation. We investigate the impact of the $n=2$ Hu-Sawicki model on the late-time growth of structures. We find that updated late-time growth of structure data imposes $|f_{R_0}|\lesssim10^{-6}-10^{-5}$ and thus conclude that the Hu-Sawicki $f(R)$ model contributes no significant phenomenology at both background and perturbative level beyond the effective cosmological constant encompassed in its definition. This conclusion points to the survival of the present tension between early and late measurements of $σ_8$, as the Hu-Sawicki model can only worsen this issue or at best reproduce the results from the current concordance cosmological model. The generalized perturbative method showcased in this work can be applied to more elaborate $f(R)$ models to isolate genuine higher-order signatures beyond the quasistatic approximation.

Figures (8)

• Figure 1: Posteriors for the parameters in the $n=2$ and $n=3$ Hu-Sawicki $f(R)$ models from fitting to SNIa distance moduli data from the DES collaboration. There is a clear linear correlation compatible with the presence of a cosmological constant in the modified gravitational action, with little effect from the choice of exponent $n$ in the model.
• Figure 2: Comparison of the quasistatic and full equations for both the the effective gravitational constant in the differential equation for $\delta$ (left) and the metric potentials (right). We consider the reference values of $k=0.1 \ \text{Mpc}^{-1}$ and $d_{\rm HS}=200$. The metric perturbations $\Phi$ and $\Psi$ differ from each other due to the non-linear nature of the gravitational action, as shown in Eq. \ref{['eq:ijEq']}.
• Figure 3: The modified matter power spectrum in the Hu-Sawicki $f(R)$ model for $z=0$. Fiducial values were used for the cosmological parameters, namely $\Omega_m=0.3$ and $H_0=70 \text{ km/s/Mpc}$. The quasistatic and full equations lead to indistinguishable predictions.
• Figure 4: The distortion parameter $\beta_d(z,k)$ in GR (black) and HS $f(R)$ gravity (color). The results from the quasistatic expressions (dashed) precisely match the ones from the full expressions (full).
• Figure 5: The transverse (top) and parallel (bottom) velocity correlation tensor components at $z=0$ in the HS $f(R)$ model and $\Lambda$CDM. The full and quasistatic $f(R)$ equations yield the same results, while both deviate from the $\Lambda$CDM prediction for small correlation distances.