Asymptotically matched quasi-circular inspiral and transition-to-plunge in the small mass ratio expansion

The authors have addressed most of my comments quite satisfactorily, though there are a few things still to clarify/correct before the paper is accepted. I just have one remaining relatively significant request, for some clarification about the statements about the accuracy of the leading-order transition matching, and a number of minor issues, all of which I detail in the requested changes.

1- Around Eq. (201), it would be good to explain why there is no \eta^{3/5} term at 0PA, even though there is an \eta^{4/5} term and there is an \eta^{3/5} term at 1PA. Also (and likely relatedly), it is necessary to reconcile the statement in the introduction that “the leading-order transition matching admits an error of the order of \eta^{3/5}” with the statement at the end of Sec. VI that “0PA composite expansion (201) leads to an error in the radius of the order of O_s(\eta) + O_\tilde{\tau}(\eta).” Of course, perhaps the statement in the introduction refers to a different expression, but if so, this needs to be made explicit.

2- Final sentence of abstract: “instrumental at” -> “instrumental for” (apologies for not noticing this previously).

3- I am fine with citing foundational papers along with the more recent reviews, in principle. However, just citing the Pretorius paper for the entire enterprise of numerical relativity (even just numerical relativity simulations of binary black holes) seems a bit odd, since even though it is the first BBH breakthrough paper (though commonly cited along with the two papers giving the moving punctures breakthrough), Pretorius’s code is not used to create the “accurate waveform models currently under confrontation with observations of gravitational waves produced from compact binary mergers” which instead are primarily created by the SXS collaboration with SpeC and secondarily by some of the moving punctures codes. (This also leaves aside numerical simulations with matter, which are also important for comparisons with observations but not so relevant to the current paper.) If you want to cite a single paper for numerical relativity along with the review, I would suggest instead the most recent SXS catalogue paper, doi:10.1088/1361-6382/ab34e2, since this describes the waveforms used to calibrate all binary black hole waveform models. If you want to cite the Pretorius paper, then it should be mentioned as purely being for the breakthrough (though I would argue that it is not the most useful citation here). I do not object to the PN/PM and EOB foundational citations, since the basic methods they introduce are still the ones used in current calculations, and the state-of-the-art calculations are still being carried out at least in part by these groups. Apologies for being picky about this, but I feel that it is important to be careful with these sorts of introductory citations.

4- In the final paragraph of the introduction, “expansions complements” -> “expansions complement”

5- Below Eq. (30), “in denominators as D” -> “in the same denominators that D does” or “in the denominators along with D” or something similar

6- Below Eqs. (31), “symmetry exist” -> “symmetry exists” and “when both” -> “when the signs of both”; also, the discussion of the parity property at the beginning of Appendix A should probably also mention the change of the sign of a, to prevent any confusion.

7- The lengthy numbers in s_{8,0} should be typeset with the same spacing between groups of numbers as for the c coefficients above Eq. (145) for consistency, or those coefficients should not have the spaces—I would be fine with either.

8- In Eqs. (201-3), putting the subscripts “0PA,” “1PA,” and “2PA” on the “r”s on the l.h.s. of the expressions would make things clearer.

9- In the conclusions, it’s probably worthwhile separating out the references for the calculations of EMRI waveforms ([21, 43]) from the papers that calibrate waveform models. Additionally, since you are citing the EOB papers that calibrate the dominant mode as well as those that calibrate the higher modes, it’s probably appropriate to cite the Phenom paper that calibrates the dominant mode of the frequency-domain model, doi:10.1103/PhysRevD.102.064001 (the calibration of the dominant mode of the time-domain Phenom model using the extreme mass-ratio waveforms is done in the paper that’s already cited). (I should have mentioned this explicitly in my previous report, as opposed to just giving the papers calibrating the higher modes with “e.g.”) Finally, it makes sense to cite the papers ordered by model, with the paper giving the dominant mode calibration first, i.e., in the order [41, 39, 42, 40] and 10.1103/PhysRevD.102.064001, [44, 45].

10- The sudden transition from the discussion of calibrating waveforms to “The deviation from quasi-circularity appears in the inspiral starting from post-adiabatic order.” is a bit jarring, so I suggest rewriting to let the reader know you’re changing subjects.

11- In the conclusions “without eccentricity nor inclination” -> “without either eccentricity or inclination” or something similar.

12- Bibliography: VIRGO -> Virgo, and it should be “LIGO Scientific, Virgo, and KAGRA Collaborations” (since they’re separate collaborations who author papers together) and sometimes the “D” in “Phys. Rev. D” gets put in the volume number (e.g., [3], presumably due to using old INSPIRE BibTeX entries), while in other cases it is correctly included as part of the journal name (e.g., [12]).