*Cosmo Comments*. The following comments can also be viewed as annotations on the paper via Hypothesis.

This paper looks at the possibility of addressing the $H_0$ tension by having recombination occur earlier, which in practice is achieved by varying the electron mass $m_e$ within a curved Universe ($\Omega_k\Lambda$CDM). Combining CMB, BAO, and uncalibrated SNeIa measurements, within an ($\Omega_k\Lambda$CDM+$m_e$ Universe, the authors find $H_0=72.3\pm2.8$, which is consistent with the local measurement of Riess et al. (R19). To the best of my knowledge, this is one of the highest values of $H_0$ ever obtained from the CMB+BAO+SNeIa combination (if not the highest altogether). Therefore, if correct, these results would be extremely interesting as they would clearly represent a very compelling solution to the H0 tension.

Nonetheless, I have a few concerns regarding the accuracy of the results, and in particular whether the ingredients required for the authors' solution to work can arise from a realistic theory. I would love to hear the authors' thoughts on this.

- My main concern regards $m_e$. First of all, it is unclear to me how the authors have treated $m_e$ in the MCMC. They say that they use CosmoMC "modified to incorporate varying $m_e$". Have they actually varied $m_e$, some function thereof, or something else altogether? And, most importantly, what is the value of $m_e$ they recover from the MCMC? Unless I have missed something obvious, I have not seen this value quoted anywhere, neither in the main text nor in the table. It would be very instructive to quote this value. From the discussion I'm guessing the authors recover a value of $m_e$ higher by $\sim 4-5\%$ compared to the standard value of 0.511 MeV (so probably about 0.535 MeV). If my understanding of the text is correct this raises the question whether such a value of $m_e$ is allowed by any experiments? I have not checked this in detail but I am guessing we have *very precise and accurate* lab constraints on the electron mass? Could the authors therefore please clarify the following:
- what value of $m_e$ they obtain from the MCMC,
- what are current external constraints on $m_e$,
- whether external constraints are compatible with the constraints the authors find are required to solve the $H_0$ tension, and if they aren't consistent, how big of a concern this is?

- From Fig. 3, it looks to me as if $m_e$ is actually not helping much in terms of addressing the $H_0$ tension, but most of the work is being done by curvature (compare dark blue against orange curves, or simply the various columns in table I). Could the authors please comment on this?

Also, it would have been nice to see the 1D and 2D posteriors on $m_e$ and $\Omega_k$, or at least quote the values of $m_e$ and $\Omega_k$ obtained from the MCMC in the main text and in the table. The table only contains the values of $H_0$ and $\Delta \chi^2_\mathrm{eff}$, but I think it would be instructive to include $m_e$ and $\Omega_k$ as well. - Finally, in the Hubble hunter's guide, https://arxiv.org/abs/1908.03663, in Section VC, the authors discuss the fact that high-temperature (which presumably is the same scenarios considered in this paper) is an unlikely solution to the $H_0$ tension. Admittedly, the discussion in this Section is rather qualitative and mostly focused on the fine structure constant, $\alpha$, rather than $m_e$, although I tend to agree with the qualitative conclusion that:

"It seems highly unlikely that new physics alters $r_s$ by changing recombination, while having an acceptably small impact on the shape of the CMB damping tail. The unlikeliness is also underscored by the fact that recombination occurs out of chemical equilibrium, therefore the relevant atomic per-particle reaction rates are not much faster than the Hubble rate. The particular details of the ionization history resulting from this out-of-equilibrium recombination are marvelously consistent with the shape of the damping tail. Thus the task is more challenging than simply reproducing a generic equilibrium ionization history at a higher temperature."

Could the authors please comment on this earlier statement? It is not clear to me whether the authors have only focused on equilibrium processes at recombination? Certainly some of their statements around Eq.(9), in particular the reference to [10] (the 2015 Planck paper on variations of the fundamental constants https://arxiv.org/abs/1406.7482) appear to suggest so. For the record, this reference obtains constraints on variations of the electron mass of order $10^{-3}$, which is way less than the $5\%$ the authors quote to solve the $H_0$ tension.

*[These comments were shared with us by a member of the cosmology community. They do not necessarily reflect the opinion of the Cosmo Comments team.]*