Refitting cosmological data with neutrino mass and degeneracy

A simple and natural extension of the standard Lambda cold dark matter ($\Lambda$CDM) model is to allow relic neutrinos to have finite chemical potentials. We confront this $\Lambda$CDM$\xi$ model, a $\Lambda$CDM with neutrino mass $M_\nu$ and degeneracy $\xi_3$ as additional parameters, with various cosmological data sets. We find that the $H_0$ and $S_8$ tensions become significant only in the presence of the cosmic microwave background (CMB) polarization data. Specifically, the global and local measurements agree to within 0.8$\sigma$ and 1.6$\sigma$ for the $H_0$ and $S_8$ tensions, respectively, when the CMB polarization data are not included. Therefore, the $H_0$ and $S_8$ tensions exist between CMB temperature and polarization data, both being global measurements. Fitting the $\Lambda$CDM$\xi$ model to the CMB temperature data, we find 3$\sigma$ evidence for nonzero neutrino mass ($M_\nu=0.57^{+0.17}_{-0.13}\,\mathrm{eV}$) and degeneracy ($\xi_3=1.13^{+0.41}_{-0.19}$), and the O(1) neutrino degeneracy parameter is compatible with Big Bang nucleosynthesis data. The scalar index $n_s$ exceeds 1 slightly, which is compatible with some hybrid inflation models. Furthermore, the recent DESI baryon acoustic oscillation data prefer the $\Lambda$CDM$\xi$ model to the Planck $\Lambda$CDM model. Similar results are obtained when including additional supernova data, while the inclusion of the Atacama Cosmology Telescope $\tau$ prior shifts the preferred $M_\nu$ and $\xi_3$ values closer to zero and brings $n_s$ back to the values favored when the polarization data are included.