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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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Submission summary
Authors (as registered SciPost users):  Geoffrey Compère · Lorenzo Küchler 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.02114v2 (pdf) 
Code repository:  https://github.com/gcompere/Asymptoticallymatchedquasicircularinspiralandtransitiontoplungeinthesmallmassratioexpa.git 
Date submitted:  20220503 10:34 
Submitted by:  Küchler, Lorenzo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

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. The first ten perturbative orders do not require any further developments in selfforce theory, as they are determined by the secondorder selfforce. We propose a scheme that matches the slowtimescale expansion of the inspiral with the transitiontoplunge motion to all perturbative orders in the overlapping region exterior to the last stable 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.
Author comments upon resubmission
List of changes
1) We added absolute values above Eq. (118).
2) We added Table II (and a remark below Eq. (180)).
3) We added the reference to the published Erratum of the PRL.
4) We made the nomenclature consistent throughout the text: we use Painlevé "transcendent" as a noun referring to the solution to the Painlevé "transcendental" equation.
5) We changed Last Stable Circular Orbit to Last Stable Orbit (LSO).
Current status:
Reports on this Submission
Anonymous Report 1 on 202259 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.02114v2, delivered 20220509, doi: 10.21468/SciPost.Report.5049
Report
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 leadingorder transition matching, and a number of minor issues, all of which I detail in the requested changes.
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 leadingorder 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:10.1088/13616382/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 stateoftheart 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. (2013), 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 frequencydomain model, doi:10.1103/PhysRevD.102.064001 (the calibration of the dominant mode of the timedomain Phenom model using the extreme massratio 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 quasicircularity appears in the inspiral starting from postadiabatic 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]).
Author: Lorenzo Küchler on 20220511 [id 2454]
(in reply to Report 1 on 20220509)We thank the referee for her/his pertinent comments.
1 We thank the referee for finding this typo: we simply forgot to put the $R_{[1]}$ term in Eq. (201). For clarity, we now denote the composite expansion with both inspiral and transition approximations with the label nPA/mPL in Section VI. The scaling of the error in the introduction is correct and refers to the 0PA/0PL composite expansion, while the 0PA/2PL expansion (201) is precise up to $\eta$ corrections both at fixed $s$ and $\tilde\tau$. This is now explained at the end of Section VI.
2 Fixed, thank you.
3 We are not very knowledgeable of the numerical relativity literature and we thank you for your informative comments. We decided to cite the SXS catalog.
4 Fixed.
5 Fixed.
6 Fixed.
7 Fixed.
8 Fixed with the new notation nPA/mPL.
9 Fixed.
10 Fixed, we put this discussion in a new paragraph.
11 Fixed.
12 Fixed.