The Status of Neutrino Cosmology: Standard $\Lambda$CDM, Extensions, and Tensions

We examine the performance of the six-parameter $\Lambda$CDM model and its extensions in light of recent cosmological observations, with particular focus on neutrino properties inferred from cosmology. Using a broad suite of nine combinations of datasets, with three separate analyses of the Planck Cosmic Microwave Background (CMB) data, and three separate supernovae (SNe) survey data, plus the recent DESI baryon acoustic oscillation (BAO) scale results, we derive constraints on the sum of neutrino masses ($\Sigma m_\nu$). Our results show upper limits in the range of $\Sigma m_\nu < 76.9\,\mathrm{meV}$ to $\Sigma m_\nu < 108\,\mathrm{meV}$ (95\% CL). The variation in the limits on $\Sigma m_\nu$ arises from the separate analyses of the Planck CMB data and the separate supernovae datasets, as they relate to the inferred matter density and its relation to the sensitivity of the BAO scale and CMB lensing to $\Sigma m_\nu$. In the context of hierarchical mass models in $\Lambda$CDM, we find a $1.47\sigma$ preference for normal ordering (NO) over inverted ordering (IO), with similar values of preference across all datasets. Despite the strong constraints, an inclination towards massless neutrinos over NO remains weak at $1.36\sigma$. We find that a ``negative'' neutrino mass, inferred from the shape of the likelihood in the physical regime, $\Sigma m_\nu > 0$, is only present at less than $2\sigma$. We confirm that models allowing extra relativistic degrees of freedom, with $N_{\rm eff} \approx 3.5$, alleviate the Hubble tension. Significantly, we find a $3.3\sigma$ preference for a 0.1 eV partially thermalized sterile neutrino when the SH0ES $H_0$ measurement is included, a scale of interest in short-baseline oscillation experiment results. [abridged]