On the maximum compactness of neutron stars

Dear SciPost Team,

 thank you for sending us the report of the Referee, whom we thank for the careful reading of the paper and the recommendation made to publish the work. Our responses to the Referee's comments (indicated with a "> " symbol) are outlined below, while the changes to the manuscript will be highlighted with magenta boldface text in the revised version.

 We hope that the replies clarify the points raised by the Referee.

 Best regards,

 Luciano Rezzolla and Christian Ecker

In this manuscript the authors adopt a statistical approach for a large set of (isotropic) fluid equations of state parameterized by the speed of sound and satisfying several physical and observational constraints. By solving the TOV equations numerically they show that perturbative-QCD effects constraints the maximum compactness to that of the photon sphere. On the way authors also state a general conjecture that the maximum compactness is attained by the solution with the maximum mass and show that the considered set of EOS satisfy the conjecture.

In my opinion the results are interesting and deemed for publication. I just have two minor suggestions.

Firstly, that the authors better explain how the pQCD constraints act on the EOS.

To easiest way to "see" the impact of the pQCD constrain is to look at Fig. 3, which shows the probability distribution of sampled EOSs with the pQCD constraint imposed (blue density map) and compares it with the corresponding distribution obtained when the constraint is not imposed (black contour). The comparison clearly demonstrates the effect of pQCD: with the constraint enforced, the maximum compactness remains always below 1/3, whereas without it the compactness can (slightly) exceed this value.

The easiest way to "explain" the impact of the pQCD constraint is to recall that matching the EOSs at 40 times the saturation density has the ultimate effect of softening of the EOS at high densities, that is, it has the effect of reducing pressure and the sound speed (see, e.g., Fig. 1 of arXiv:2204.11877).

That said, why the bound is 1/3 and not another number is not known to us yet, although we are presently working on exactly this point. We hope to present a discussion on this in forthcoming work.

We have revised the manuscript to reflect this discussion.

Secondly, how can these results fit in a larger set containing anisotropic EOS ?

This is a question that has a very simple answer: the limit on compactness we have derived here (C < 1/3) will apply only to stellar models obeying isotropic EOSs.

The reason for this is simple: EOSs allowing for a degree of anisotropy will lead to compactness that are comparable to those of a black hole (C=1/2), even in the presence of modest amounts of anisotropy. A "classical" example of this statement is given by gravastars, which can reach compactnesses limiting the value 1/2 from below (see, e.g., arXiv:0706.1513), but a larger classes of compact objects can be considered (see, e.g., arXiv:1811.07917).

Finally, we note that it is possible to evade the compactness constraint C < 1/3 even with fully isotropic EOSs and we show this with the solid black contour in Fig. 3. Hence, it's really the pQCD constraint that generates the bound and for the reasons given above.

We have revised the manuscript to reflect this discussion.