Temperature-redshift relation in energy-momentum-powered gravity models

There has been recent interest in the cosmological consequences of energy-momentum-powered gravity models in which the matter side of Einstein's equations includes a term proportional to some power, $n$, of the energy-momentum tensor, in addition to the canonical linear term. Previous works have suggested that these models can lead to a recent accelerating universe without a cosmological constant, but they can also be seen as phenomenological extensions of the standard $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$, which are observationally constrained to be close to the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ limit. Here we show that these models violate the temperature-redshift relation, and are therefore further constrained by astrophysical measurements of the cosmic microwave background temperature. We provide joint constraints on these models from the combination of astrophysical and background cosmological data, showing that this power is constrained to be about $|n|<0.01$ and $|n|<0.1$, respectively, in models without and with a cosmological constant, and improving previous constraints on this parameter by more than a factor of 3. By breaking degeneracies between this parameter and the matter density, constraints on the latter are also improved by a factor of about 2.