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 nuT 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 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 plus or minus2 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.