Scale-Dependent Structure Growth in Chameleon Scalar-Tensor Gravity: Alleviating the \ (S₈\) Tension

We present the linear cosmological perturbations of the Unified Chameleon Scalar-Tensor Gravity (CSTG) model and investigate their impact on structure formation. The density-dependent effective mass of the scalar field induces a scale- and time-dependent modification of the gravitational coupling, μ (k, a) = Gₑff (k, a) /GN, which tends to unity on small scales (k ≳ 1 h Mpc⁻¹) but remains larger than one on large scales (k ∼ 0. 01 h Mpc⁻¹). Solving the linear growth equation numerically, we compute the evolution of the matter power spectrum and the growth rate fσ₈ (z). Our results show a suppression of structure growth at late times compared to ΛCDM, which brings the predicted fσ₈ values into better agreement with recent DESI and KiDS measurements. This naturally alleviates the S₈ tension without spoiling the excellent fit to CMB data. The scale-dependence of μ (k, a) offers a falsifiable signature of the chameleon mechanism in upcoming large-scale structure surveys.