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

This paper derives the asymptotic matching between a quasicircular, equatorial inspiral around a Kerr black hole and the plunge into the black hole including self-force effects, showing how to carry out this matching to high orders. This is a potentially important advance in modelling this portion of binary merger, though the paper leaves explicit calculations and the generalization to more generic orbits and finite size effects to later work. It thus satisfies expectation 3. It also satisfies all of the acceptance criteria except that one important bit of context (the accuracy of current calculations) is missing in the introduction, and there are some citations that are omitted or not the best ones. I give explicit suggestions in the requested changes. Once these and various other small issues I noticed are addressed, this will be suitable for publication in SciPost Physics.

The lengthy expressions obtained in the derivation are provided in Mathematica notebooks as well as typeset in the appendices of the paper. It is possible some of these expressions could just be given in Mathematica form, shortening the paper, but the expressions in the appendices are typeset well, so it is probably worth keeping them.

1- The abstract or at least the introduction should highlight that one only needs the second-order self-force to obtain quite high-order expressions for the transition in this framework, since this will clarify that the high-order expressions obtained here do not require significant developments in self-force calculations to obtain the third-order self-force in order to be useful.

2- The introduction should, to the extent possible without lengthy additional calculations, give some indication of the size of the errors that are being committed in current calculations of the matching to the plunge compared to the errors expected using the high-order scheme detailed here.

3- In the introduction, it seems preferable to cite recent reviews for NR, PN/PM, and EOB (e.g., dois:10.1146/annurev-astro-081913-040031, 10.12942/lrr-2014-2, and 10.1007/978-3-319-19416-5_7) rather than original papers in the second sentence. Also, the citation of observational results should presumably cite the latest GWTC-3 catalogue paper, arXiv:2111.03606, as well as the associated testing GR paper, arXiv:2112.06861, since the confrontation of models with observations is mentioned. If the GWTC-2 catalogue paper is still also cited, the associated testing GR paper should be cited, as well, as well as perhaps the GWTC-1 catalogue and testing GR papers, for completeness. (See https://www.ligo.caltech.edu/page/detection-companion-papers for the references.) I do not see a reason to cite the GW150914 paper here.

4- It is likely appropriate to cite a paper about EMRI science with LISA in the introduction, in addition to the general introduction to the mission in [11], e.g., doi:10.1103/PhysRevD.95.103012

5- The discussion in the final sentence of the first paragraph of the introduction should also mention the second-order results in arXiv:2112.12265 that appeared after this paper was submitted to the arXiv.

6- The opening text of Sec. II [through Eq. (2)] is taken almost verbatim from the authors’ previous work, Ref. [30], including the typo “Linquist” for “Lindquist.” This copying should at least be noted explicitly, though it would be better to rewrite the material.

7- The mention in the first paragraph of Sec. II that e = -p_t/m is made dimensionless using M is confusing, since it is already dimensionless.

8- The discussion above Eq. (30) is a bit confusing. It appears that what is meant is that D has a single root outside of the horizon and that root occurs at the location of the ISCO. Also, it is not clear why the roots of A, B, and C are not of interest here. Is it just that they all are inside the ISCO?

9- Below Eq. (31), it appears that the statement about even and odd quantities also involves switching the sign of a as well as the sign of \sigma, given the discussion in Appendix A. This should be made clear here, as I was initially confused.

10- In Sec. III C-D, is there a reason not to give T_1, …, T_20 names that reflect the expressions that they enter? The current notation suggests that they are all related, while they appear in very different expressions. They are not even labelled in the order they appear in the paper—T_15 through T_20 appear before T_8 through T_14.

11- It seems that the reason for using \kappa instead of \ell in Eq. (57a) is because \kappa appears in a similar manner to other Greek letters in Sec. IV. This probably deserves some comment.

12- Above Eq. (76), “inverse” -> “reciprocal” or “multiplicative inverse” so it’s clear that this does not refer to the functional inverse.

13- Above Eq. (92), it’s not clear to me how Eq. (82) is used in this calculation.

14- Below Eq. (98), it might be a good idea to recall that \delta_{(0)} = 0, to explain why there is a special case for q’_{(n)}.

15- In Sec. IV C, he discussion of the asymptotic solution is unclear, since the explicit examples of coefficients for the general case all vanish when d = 0, so it’s not clear how exactly one gets the expression in Eq. (142) from Eq. (141). Some more discussion is in order, e.g., mentioning which coefficients are nonzero for d = 0 and giving a few examples of these.

16- Why is it necessary to identify expressions in Eq. (199) instead of showing that they are equal, as in the previous matching calculations? Explicitly, what free parameters are being fixed by this identification?

17- In the conclusions, the discussion of using these results to calibrate EOB should cite the more recent EOB papers that perform this calibration, e.g., dois: 10.1103/PhysRevD.98.084028 and 10.1103/PhysRevD.102.024077 (for the higher modes, and perhaps also dois:10.1103/PhysRevD.95.044028 and 10.1103/PhysRevD.98.104052, which do the calibration for the dominant mode) and as well as the papers describing the extreme mass-ratio waveforms used for this calibration, dois:10.1103/PhysRevD.90.084025 and 10.1088/0264-9381/31/24/245004, likely instead of most of the older papers. You should probably also cite some of the non-EOB waveform papers that use extreme mass-ratio waveforms in the calibration, e.g., doi:10.1103/PhysRevD.102.064002 and arXiv:2012.11923. Also, “EOB” is not defined.

18- Also in the conclusions, the discussion of why the departure from quasicircularity in the inspiral is only at post-post-adiabatic order should be clarified, since \delta_{(1)} is nonzero (and this calculation is in the 1-post-adiabatic inspiral section).

* Minor issues:

19- In the third paragraph of the introduction, I find the use of the future tense when describing the contents of the paper distracting. I am used to the present tense in these sorts of guides to the contents of the paper, since the work is already completed.

20- There is an Eq. (104a) but no Eq. (104b).

21- Below Eq. (123), is there a reason not to list the equations being substituted in order of equation number? This also applies above Eqs. (197) and (198).

22- In Eqs. (144-5) and intervening expressions, as well as before Eq. (146), the “lin” subscript and superscript should always be set in Roman

23- Appendix B is only referred to in Appendices C and D, so perhaps it should go after them.

24- Ref. [11] should be P. Amaro-Seoane et al., not H. Audley et al.

The present paper develops a method to compute the dynamics of an inspiraling compact binary beyond the innermost stable circular orbit in the small mass-ratio approximation (i.e. within the gravitational self-force formalism). This is important for constructing accurate and complete (inspiral-merger-ringdown) waveform models for gravitational wave astronomy. While this problem has been investigated in the past, the present work provides a systematic and rigorous treatment. Hence I think it fulfills the acceptance criteria for SciPost Physics, since it opens a new pathway in an existing research direction, with clear potential for multipronged follow-up work. I recommend to accept it.