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In this paper we investigate a time-varying neutrino mass model, motivated bythe mild tension between cosmic microwave background (CMB) measurements of thematter fluctuations and those obtained from low-redshift data. We modify theminimal case of the model proposed by Dvali and Funcke (2016) that predictslate neutrino mass generation in a post-recombination cosmic phase transition,by assuming that neutrino asymmetries allow for the presence of relic neutrinosin the late-time Universe. We show that, if the transition is supercooled,current cosmological data (including CMB temperature, polarization and lensing,baryon acoustic oscillations, and Type Ia supernovae) prefer the scale factor$a_s$ of the phase transition to be very large, peaking at $a_s\sim 1$, andtherefore supporting a cosmological scenario in which neutrinos are almostmassless until very recent times. We find that in this scenario thecosmological bound on the total sum of the neutrino masses today issignificantly weakened compared to the standard case of constant-massneutrinos, with $\sum m_\nu<4.8$~eV at 95\% confidence, and in agreement withthe model predictions. The main reason for this weaker bound is a largecorrelation arising between the dark energy and neutrino components in thepresence of false vacuum energy that converts into the non-zero neutrino massesafter the transition. This result provides new targets for the coming KATRINand PTOLEMY experiments. We also show that the time-varying neutrino mass modelconsidered here does not provide a clear explanation to the existingcosmological $\Omega_m$-$\sigma_8$ discrepancies.

Time-varying neutrino mass from a supercooled phase transition: current cosmological constraints and impact on the $\Omega_m$-$\sigma_8$ plane