In this paper, we introduce a scale-independent energy-momentum squared gravity (EMSG) that allows different gravitational couplings for different types of sources, which may lead to scenarios with many interesting applications/implications in cosmology. In the present study, to begin with, we study 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 other relativistic relics) couple to the spacetime in accordance with EMSG. This scenario induces pseudo nonminimal interactions on these components, leading to modification at both the background and perturbative levels. A consequence of this scenario is that the dimensionless free parameter of the theory may induce direct changes on the effective number of the relativistic species, without the need to introduce new extra species. In order to quantify the observational consequences of the cosmological scenario, we use the cosmic microwave background Planck data (temperature, polarization, and lensing power spectrum) and baryonic acoustic oscillations data. We find that the free model parameter is too small to induce statistically significant corrections on the $Lambda$CDM model due to EMSG. We deduce that the model presented here is quite rich with promising cosmological applications/implications that deserve further investigations.
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