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

The paper proposes, implements, and tests a new algorithm to reconstruct the real-space galaxy distribution from the galaxy distribution in redshift space, attempting to revert redshift space distortions (RSD). This can be useful because the real-space galaxy density is easier to model than that in redshift space, potentially allowing more Fourier modes to be used for the analysis. The paper presents a careful study of the algorithm and its performance on mock data. It is well written and I only have some minor comments:

- From the results shown in the paper, is it possible to see whether analysing the reconstructed real-space galaxy density with a real space model (e.g. for the correlation function or power spectrum) gives more cosmological information than analysing the original redshift-space galaxy density and modeling the RSD? For example, is it possible to estimate how much error bars on $f/b$ or $f\sigma_8$, or $f$ and $b$, or other parameters would improve? One potential worry could be that all one is removing by the algorithm is linear (Kaiser) RSD, because in that case one could as well just model the effect of linear RSD on the redshift-space correlation function or power spectrum and ultimately get the same cosmological parameter constraints. (Maybe the fact that the quadrupole is zero after reconstruction is evidence that the algorithm does more than only removing the linear RSD? I guess my question is, do we know that more than linear RSD is removed.)
- Section 2.3: It is worth mentioning here that everything is at z=0.5.
- Is there any reason why the $C_\mathrm{ani}$ parameter has not much effect on the results? This is surprising – as the authors argue it should help to have $C_\mathrm{ani}<1$ because it should suppress the fingers of god.
- It might be useful to the reader to parse some of the plots showing correlation coefficients in plain text. For example, it might be useful to say somewhere that the reconstruction improves the $k_\mathrm{max}$ up to which $r>90\%$ by a factor of 2 or similar. (One could try to translate this into the number of modes gained, which might give a rough estimate for how much cosmological parameter error bars might shrink in principle.)
- While the paper already has lots of useful performance measures, I think one missing performance measure is the broadband power spectrum or correlation function of the reconstructed initial conditions – at least I could not find that (maybe I just missed it?). Does that look reasonable? Maybe it would be worth including a plot of that.
- In the conclusions, it might be useful to also comment on the covariance after reconstruction. One could argue that the reconstruction algorithm makes a complicated transformation of the observed data, which might induce complicated correlations (e.g. how are fiber collisions propagated, or anisotropic mean number density?). Presumably, one could get the covariance with many simulations, but maybe it's tricky to come up with a good likelihood because cosmological parameters enter the summary statistics after reconstruction (e.g. power spectrum) but also the algorithm itself ($b$ and $f$). It might be interesting to discuss or mention some of this.

*[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.]*