Cosmological structure growth in energy-momentum squared gravity

We investigate the cosmological evolution of matter perturbations in the modified gravity model $f(R,T^2)$, where $T^2=T_{\mu\nu}T^{\mu\nu}$ denotes the quadratic contraction of the energy--momentum tensor. Using the gauge-invariant 1+3 covariant formalism, we study the evolution of the matter density contrast and analyze several growth observables, including the growth factor, the growth index, and the weighted growth rate $f\sigma_8$. We consider representative values $n=1/2$ and $n=1/4$, which probe different regimes of the matter--geometry coupling. We show that the growth index decreases with increasing redshift and approaches the standard matter-dominated behavior at early times, while mild scale-dependent deviations from the $\Lambda$CDM model emerge at late times. The model predicts small departures from General Relativity for $n=1/4$, whereas stronger deviations appear for $n=1/2$ and larger values of the coupling parameter $\alpha$. We further compare the theoretical predictions for $f\sigma_8$ with current observational data and find that viable parameter choices remain within the observational $\pm2\sigma$ bounds. These results indicate that $f(R,T^2)$ gravity can provide a viable description of late-time cosmic acceleration and large-scale structure formation while remaining consistent with current growth observations.