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Asymptotically matched quasicircular inspiral and transitiontoplunge in the small mass ratio expansion
by Geoffrey Compère, Lorenzo Küchler
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Authors (as registered SciPost users):  Geoffrey Compère · Lorenzo Küchler 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.02114v1 (pdf) 
Code repository:  https://github.com/gcompere/Asymptoticallymatchedquasicircularinspiralandtransitiontoplungeinthesmallmassratioexpa.git 
Date submitted:  20211216 15:55 
Submitted by:  Küchler, Lorenzo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
In the small mass ratio expansion and on the equatorial plane, the twobody problem for point particles in general relativity admits a quasicircular inspiral motion followed by a transitiontoplunge motion. We first derive the equations governing the quasicircular inspiral in the Kerr background at adiabatic, postadiabatic and postpostadiabatic orders in the slowtimescale expansion in terms of the selfforce and we highlight the structure of the equations of motion at higher subleading orders. We derive in parallel the equations governing the transitiontoplunge motion to any subleading order, and demonstrate that they are governed by sourced linearized Painlev\'e transcendental equations of the first kind. We propose a scheme that matches the slowtimescale expansion of the inspiral with the transitiontoplunge motion to all perturbative orders in the overlapping region around the last stable circular orbit where both expansions are valid. We explicitly verify the validity of the matching conditions for a large set of coefficients involved, on the one hand, in the adiabatic or postadiabatic inspiral and, on the other hand, in the leading, subleading or higher subleading transitiontoplunge motion. This result is instrumental at deriving gravitational waveforms within the selfforce formalism beyond the innermost stable circular orbit.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022323 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.02114v1, delivered 20220322, doi: 10.21468/SciPost.Report.4752
Report
This paper derives the asymptotic matching between a quasicircular, equatorial inspiral around a Kerr black hole and the plunge into the black hole including selfforce 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.
Requested changes
1 The abstract or at least the introduction should highlight that one only needs the secondorder selfforce to obtain quite highorder expressions for the transition in this framework, since this will clarify that the highorder expressions obtained here do not require significant developments in selfforce calculations to obtain the thirdorder selfforce 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 highorder 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/annurevastro081913040031, 10.12942/lrr20142, and 10.1007/9783319194165_7) rather than original papers in the second sentence. Also, the citation of observational results should presumably cite the latest GWTC3 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 GWTC2 catalogue paper is still also cited, the associated testing GR paper should be cited, as well, as well as perhaps the GWTC1 catalogue and testing GR papers, for completeness. (See https://www.ligo.caltech.edu/page/detectioncompanionpapers 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 secondorder 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 CD, 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 massratio waveforms used for this calibration, dois:10.1103/PhysRevD.90.084025 and 10.1088/02649381/31/24/245004, likely instead of most of the older papers. You should probably also cite some of the nonEOB waveform papers that use extreme massratio 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 postpostadiabatic order should be clarified, since \delta_{(1)} is nonzero (and this calculation is in the 1postadiabatic 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. (1445) 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. AmaroSeoane et al., not H. Audley et al.
Report #1 by Anonymous (Referee 1) on 2022221 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.02114v1, delivered 20220221, doi: 10.21468/SciPost.Report.4482
Report
The present paper develops a method to compute the dynamics of an inspiraling compact binary beyond the innermost stable circular orbit in the small massratio approximation (i.e. within the gravitational selfforce formalism). This is important for constructing accurate and complete (inspiralmergerringdown) 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 followup work. I recommend to accept it.
Author: Lorenzo Küchler on 20220304 [id 2268]
(in reply to Report 1 on 20220221)I would like to thank the referee for the careful reading and the positive assessment of our manuscript.
Author: Lorenzo Küchler on 20220503 [id 2434]
(in reply to Report 2 on 20220323)We would like to thank the referee for at least partially reproducing our computations and results and for his/her pertinent comments. We addressed all questions and consequently improved the manuscript. The answers to the individual questions are listed below.
1The abstract has been completed with 2 additional clarifications.
2We mentioned the expected improvement in errors in the Introduction and justify these errors at the end of Section VI: we added (201)(203) and commented on the error in the composite expansion.
3We think that citing the founding papers of the topic is as appropriate as citing relevant reviews. We will therefore cite both. We agree with the other citation comments.
4We agree and added the reference.
5We agree and added the reference.
6.We adapted this convention section indeed totally shared with [35] as already stated and absolutely necessary for this paper as well. Thank you for pointing to the typo.
7We corrected the phrasing.
8We answered this question in the text after Eq. (30).
9We rephrased after Eq. (31).
10The T_i coefficients have been renamed in order of appearance. These coefficients are not specific to any expression: they are auxiliary functions to write more compact expressions for the quantities relevant to the inspiral.
11Indeed, we changed the notation $\kappa$ to be coherent with previous notation $\ell$.
12We will use "reciprocal".
13Indeed, that reference should not be there. We removed it.
14Thank you for pointing this out. We added the comment.
15We added the first coefficients of the expansion that do not vanish when $d=0$.
16This is indeed a check only. We rephrased.
17We updated the references and defined "EOB" in the Introduction section.
18We clarified the discussion in the conclusion and added a remark at the end of Section VD.
19We now use present tense.
20Fixed.
21The referenced equations now appear in order of equation number.
22Fixed.
23We changed the order of appearance of the Appendices.
24Fixed.