Cosmological Structure Growth in Energy-Momentum Squared Gravity (EMSG)

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, growth index. We consider representative modified gravity parameter 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 behaviour at early times, while mild scale-dependent deviations from the ΛCDM 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 α. We further compare the theoretical predictions for f σ8 with current observational data and find that viable parameter choices remain within the observational ±2σ bounds. These results indicate that EMSG gravity can provide a viable description of late-time cosmic acceleration and large-scale structure formation while remaining consistent with current growth observations.