Probing the neutrino chemical potential with cosmological observations

The electron neutrino degeneracy parameter, $\xi_{\nu_\mathrm{e}} = \mu_{\nu_\mathrm{e}} / T$, is tightly constrained by Big Bang Nucleosynthesis (BBN), while the degeneracy parameters of the other neutrino species, $\xi_{\nu_\mathrm{x}}$, remain weakly constrained by cosmological observations alone. In this manuscript we shall compute up-to-date bounds on $\xi_{\nu_\mathrm{e}}$ and $\xi_{\nu_\mathrm{x}}$ assuming that either they are constant free-parameters along the cosmic history or that they are redshift dependent quantities. In the latter case we employ a model-independent reconstruction approach based on the Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) formalism with four nodes, located at $z\simeq$ 10, 100, 1000 and $10^8$. We shall also consider two scenarios for neutrinos, specifically three degenerate neutrinos ($\xi_{\nu_\mathrm{e}}$ = $\xi_{\nu_\mathrm{x}}$) and the case in which we actually differentiate between $\xi_{\nu_\mathrm{e}}$ and $\xi_{\nu_\mathrm{x}}$. We perform a cosmological analysis combining CMB data from Planck, SPT, and ACT with BAO measurements from DESI, showing the impact of including BBN observables from either EMPRESS results, which allow for a non-zero chemical potential, or from LBT observations, compatible with the standard $\xi_\nu$ = 0 prediction. We explicitly show that the BBN data, via the change in neutron-to-proton interconversion rates, mostly constrain $\xi_{\nu_\mathrm{e}}$, parameter for which we observe a preferred non-zero positive value at $95\%$ C.L. in the non-degenerate neutrino case at the BBN period. Since the Hubble constant is correlated with $\xi_{\nu}$, through $N_{\rm eff}$, a larger value of $H_0$ is allowed within these models, making them really interesting scenarios where to test non-standard physics models.