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

Abstract

We investigate a perturbation-level modification of symmetric teleparallel gravity of the form $f(Q)=F(Q)+M\sqrt{Q}$ and assess its potential to ease the $σ_8$ tension. The square-root term leaves the background expansion unchanged at the level of FLRW cosmology, while modifying the effective gravitational coupling and thereby providing a decoupling between background evolution and structure-growth. Using the latest redshift-space distortion data, including DESI DR1 Full-Shape measurements, we constrain the model through its impact on the growth observable $fσ_8(z)$ 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 the growth of structure and induces a degeneracy with $σ_8$, leading to weaker constraints from current data. This allows for a wider range of $σ_8$ values consistent with Planck, with the effect being most pronounced in the $H_0$-tension-oriented background model. However, because RSD data constrain $fσ_8$ rather than $σ_8$ directly, 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.