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We revisit the joint constraints in the mixed hot dark matter scenario inwhich both thermally produced QCD axions and relic neutrinos are present. Uponrecomputing the cosmological axion abundance via recent advances in theliterature, we improve the state-of-the-art analyses and provide updated boundson axion and neutrino masses. By avoiding approximate methods, such as theinstantaneous decoupling approximation, and limitations due to the limitedvalidity of the perturbative approach in QCD that forced to artificially dividethe constraints from the axion-pion and the axion-gluon production channels, wefind robust and self-consistent limits. We investigate the two most popularaxion frameworks: KSVZ and DFSZ. From Big Bang Nucleosynthesis (BBN) lightelement abundances data we find for the KSVZ axion $\Delta N_{\rm eff}<0.31$and an axion mass bound $m_a < 0.53 $ eV (i.e., a bound on the axion decayconstant $f_a > 1.07 \times 10^7$ GeV) both at $95\%$ CL. These BBN bounds areimproved to $\Delta N_{\rm eff}<0.14$ and $m_a< 0.16$ eV ($f_a > 3.56 \times10^7$ GeV) if a prior on the baryon energy density from Cosmic MicrowaveBackground (CMB) data is assumed. When instead considering cosmologicalobservations from the CMB temperature, polarization and lensing from the Plancksatellite combined with large scale structure data we find $\Delta N_{\rmeff}<0.23$, $m_a< 0.28$ eV ($f_a > 2.02 \times 10^7$ GeV) and $\sum m_\nu <0.16$ eV at $95\%$ CL. This corresponds approximately to a factor of $5$improvement in the axion mass bound with respect to the existing limits. Verysimilar results are obtained for the DFSZ axion. We also forecast upcomingobservations from future CMB and galaxy surveys, showing that they could reachpercent level errors for $m_a\sim 1$ eV.

Cosmological Bound on the QCD Axion Mass, Redux