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Deviations from the predictions of general relativity due to energy-momentumsquared gravity (EMSG) are expected to become pronounced in the high densitycores of neutron stars. We derive the hydrostatic equilibrium equations in EMSGand solve them numerically to obtain the neutron star mass-radius relations forfour different realistic equations of state. We use the existing observationalmeasurements of the masses and radii of neutron stars to constrain the freeparameter, $\alpha ,$ that characterizes the coupling between matter andspacetime in EMSG. We show that $-10^{-38}\,\mathrm{cm^{3}/erg}<\alpha<+10^{-37}\,\mathrm{cm^{3}/erg}$. Under this constraint, we discuss whatcontributions EMSG can provide to the physics of neutron stars, in particular,their relevance to the so called \textit{hyperon puzzle} in neutron stars. Wealso discuss how EMSG alters the dynamics of the early universe from thepredictions of the standard cosmological model. We show that EMSG leaves thestandard cosmology safely unaltered back to $t\sim 10^{-4}$ seconds at whichthe energy density of the universe is $\sim 10^{34}\,\mathrm{erg\,cm^{-3}}$.

Constraint on energy-momentum squared gravity from neutron stars and its cosmological implications