Decoupling perturbations from background in $f(Q)$ gravity: the square-root correction and the alleviation of the $σ_8$ tension

We investigate a perturbation-level modification of symmetric teleparallel gravity of the form $f(Q)=F(Q)+M\sqrt{Q}$ and assess its ability to ease the $σ_8$ tension. The square-root term leaves the background expansion unchanged while modifying the effective gravitational coupling, providing a pure decoupling between background cosmology and structure-growth evolution. Using the latest redshift-space distortion data, including DESI DR1 Full-Shape measurements, we constrain $M$ and $σ_8$ across three representative backgrounds: $Λ$CDM, an $H_0$-tension-reducing model, and a DESI-motivated dynamical dark energy scenario. In all cases, the square-root correction suppresses growth and can reconcile $σ_8$ with Planck at the $1σ$ level, with the strongest improvement occurring in the $H_0$-tension-oriented background. A residual degeneracy between $M$ and $σ_8$ remains, indicating that future multi-probe analyses combining lensing and full-shape clustering will be required to determine whether the $\sqrt{Q}$ term represents a genuine signal of modified gravity.

💡 Research Summary

In this work the authors explore a novel way to address the long‑standing σ₈ (or S₈) tension within the framework of symmetric teleparallel gravity, also known as f(Q) gravity. The key idea is to add a square‑root term, M √Q, to the Lagrangian function f(Q)=F(Q)+M √Q. Because the non‑metricity scalar Q equals 6 H² in a flat FLRW universe, the √Q contribution cancels exactly from the modified Friedmann equations. Consequently the background expansion history H(z) is governed solely by the chosen function F(Q) and remains identical to that of the underlying cosmology (ΛCDM, an H₀‑tension‑alleviating exponential model, or a DESI‑motivated quintom‑like dynamical dark‑energy model).

At the level of linear perturbations the √Q term does not vanish. In the Newtonian gauge the Poisson equation becomes –k²Ψ = 4πG_N f_Q a²ρ_m δ, where f_Q≡∂f/∂Q = F_Q+M/(2√Q). The effective gravitational coupling is therefore G_eff = G_N f_Q. For M>0 the factor f_Q increases, reducing G_eff and suppressing the growth of matter perturbations. This mechanism directly lowers the predicted fσ₈(z) without altering distances or the Hubble rate, offering a clean way to reconcile low‑redshift large‑scale‑structure measurements with the higher σ₈ inferred from Planck CMB data.

The authors confront this theory with the most up‑to‑date redshift‑space distortion (RSD) data set, comprising 28 fσ₈ measurements, including six recent DESI DR1 full‑shape points. They apply the Alcock–Paczynski geometric correction to each datum, construct a χ² likelihood, and explore the joint posterior of σ₈ and M using flat priors (σ₈∈