Late-Time Cosmology and Structure Formation in Quadratic $f(Q)$ Gravity

We investigate the cosmological evolution associated with the quadratic symmetric teleparallel gravity framework, $f(Q)=Q+\alpha Q^2+\beta$ where the relation $Q\propto H^2$ generates an additional $H^4$ contribution to the Friedmann equation. Using the exact algebraic solution for $H(z)$, we reconstruct the effective dark-energy sector and compare the background evolution with $\mathrm{\Lambda }$CDM using Type Ia supernovae, BAO, and cosmic-chronometer data. At the perturbative level, the model modifies the Poisson equation through a time-dependent effective gravitational coupling $G_{\text{e}ff}(z)=G[1+\frac{2}{3}A{E}^2(z){]}^{-1}$, where $A=18\alpha H_0^2$. For $\alpha >0$ this produces a weakened gravitational interaction, suppressing the linear growth factor $D(z)$, the growth rate $f(z)$, and the RSD observable $f{\sigma }_8(z)$. In the nonlinear regime, the reduced gravitational strength increases the spherical-collapse threshold and suppresses the halo mass function, leading to a lower predicted value of $S_8={\sigma }_8\sqrt{{\mathrm{\Omega }}_m\mathrm{/}0.3}$. Thus, the quadratic $f(Q)$ extension can reproduce mild deviations from $\mathrm{\Lambda }$CDM at the background level while naturally alleviating the $S_8$ tension, offering a viable modified-gravity explanation for recent observational hints of dynamical dark energy.