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In this paper, we introduce a scale-independent energy-momentum squaredgravity (EMSG) that allows different gravitational couplings for differenttypes of sources, which may lead to scenarios with many interestingapplications/implications in cosmology. In the present study, to begin with, westudy a modification of the $\Lambda$ cold dark matter ($\Lambda$CDM) model,where photons and baryons couple to the spacetime as in general relativity,while the cold dark matter and relativistic relics (neutrinos and any otherrelativistic relics) couple to the spacetime in accordance with EMSG. Thisscenario induces pseudo nonminimal interactions on these components, leading tomodification at both the background and perturbative levels. A consequence ofthis scenario is that the dimensionless free parameter of the theory may inducedirect changes on the effective number of the relativistic species, without theneed to introduce new extra species. In order to quantify the observationalconsequences of the cosmological scenario, we use the cosmic microwavebackground Planck data (temperature, polarization, and lensing power spectrum)and baryonic acoustic oscillations data. We find that the free model parameteris too small to induce statistically significant corrections on the$\Lambda$CDM model due to EMSG. We deduce that the model presented here isquite rich with promising cosmological applications/implications that deservefurther investigations.

physical review. d/physical review. d.

Cosmological implications of scale-independent energy-momentum squared  gravity: Pseudo nonminimal interactions in dark matter and relativistic  relics