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Self-force framework for transition-to-plunge waveforms

by Lorenzo Küchler, Geoffrey Compère, Leanne Durkan, Adam Pound

Submission summary

Authors (as registered SciPost users): Lorenzo Küchler
Submission information
Preprint Link:  (pdf)
Code repository:
Date submitted: 2024-05-06 16:43
Submitted by: Küchler, Lorenzo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
Approach: Theoretical


Compact binaries with asymmetric mass ratios are key expected sources for next-generation gravitational wave detectors. Gravitational self-force theory has been successful in producing post-adiabatic waveforms that describe the quasi-circular inspiral around a non-spinning black hole with sub-radian accuracy, in remarkable agreement with numerical relativity simulations. Current inspiral models, however, break down at the innermost stable circular orbit, missing part of the waveform as the secondary body transitions to a plunge into the black hole. In this work we derive the transition-to-plunge expansion within a multiscale framework and asymptotically match its early-time behaviour with the late inspiral. Our multiscale formulation facilitates rapid generation of waveforms: we build second post-leading transition-to-plunge waveforms, named 2PLT waveforms. Although our numerical results are limited to low perturbative orders, our framework contains the analytic tools for building higher-order waveforms consistent with post-adiabatic inspirals, once all the necessary numerical self-force data becomes available. We validate our framework by comparing against numerical relativity simulations, surrogate models and the effective one-body approach.

Author indications on fulfilling journal expectations

  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
In refereeing

Reports on this Submission

Report 1 by Scott Hughes on 2024-6-30 (Invited Report)


A totally thorough and comprehensive analysis.
Well written, not difficult to follow.
Excellent comparisons with relevant past literature.


None. (I provide a few minor pieces of feedback and suggestions, none of crucial importance.)


The so-called "two-body" problem in general relativity is an extremely challenging problem in general, thanks to the non-linearity of the underlying gravitational field equations, the existence and contribution of radiation, and complicated boundary conditions associated with event horizons. Indeed, when speaking to general audiences, I like to emphasize that (at least for binary black hole systems) the "two-body" problem is really a "one-spacetime" problem, with this single spacetime having two-body properties in certain important limits. Although in principle just about any interesting two-body configuration can now be evolved numerically, in practice numerical relativity remains somewhat limited in scope. Especially with respect to the gravitational-wave applications which drive this field, the role of numerical relativity may be limited by accuracy and precision for a range of important astrophysical sources. Analytic or quasi-analytic methods which can provide highly precise input to this problem are thus of great value, even if they formally only apply to a limited domain of the two-body problem's parameter space.

This paper provides an extremely comprehensive study of an aspect of this problem, the transition from the inspiral (when the binary consists of two unambiguous separated bodies slowly spiraling together driven by gravitational backreaction) to plunge (when the smaller member of the binary falls into the larger and the system becomes a single black hole). Indeed, the manuscript is so comprehensive that it reads almost more like a monograph that a research paper! (Because of this, it took quite a while to find time to go through the draft thoroughly, and I regret the delay providing this report.)

The technique employed by these authors uses a separation of timescales, which follows in turn from a separation in mass scales between the binary's members. Although limited in the category of problems that it focuses upon here (it studies the transition from quasi-circular inspiral of a secondary body into a Schwarzschild black hole), its presentation is sufficiently general that many aspects of what the authors present can be expected to carry over to more generalized versions of this problem. Although I have a few suggestions that I would like the authors to consider, none are of crucial importance for the paper. Modulo their consideration of these points (which I list in the section "Requested changes", though I emphasize that these points can be considered optional), I am very happy to recommend this paper for publication. Their approach is very elegant and complete. This manuscript essentially reads like a textbook to any practitioner interested in understanding this problem.

Requested changes

The listing I provide here is in order of where I encountered the issue or text for which I have a suggestion; this is not a ranking of importance.

1. On page 2 of the draft manuscript, the second complete paragraph begins "Accurately modeling the transition to plunge is expected to improve parameter estimation...". A related point that may be made here is that a proper handling of the transition and plunge significantly improves the utility of inspiral waveforms currently being investigated for the development of LISA data analysis. An inspiral-only waveform abruptly terminates when the secondary reaches the last stable orbit. This termination introduces spurious features into the time-frequency behavior, reducing the value of inspiral-only waveforms for data-analysis studies. One can taper such a waveform to reduce the influence of the late-inspiral behavior and avoid the abrupt termination, but a physically motivated termination is even more valuable. The framework provided here is probably more than is needed to "smooth" the waveforms' behavior for ongoing data-analysis studies, but it is a valuable motivator for the research program as a whole.

2. In the text following Equation (2.18), the authors describe an expansion of $h_{\mu\nu}$ as identical to the expansion of $\bar h_{\mu\nu}$ but "with the $i = 3, 6$ terms flipped". It would be helpful to describe precisely what "flipped" means here -- are these terms of opposite sign? Does the $i = 3$ term in the $\bar h$ expansion somehow change places with the $i = 6$ term in the $h$ expansion? The term "flipped" on its own is a bit ambiguous, but the authors should be able to easily fix this minor bit of terminology.

3. The paragraph which follows Equation (5.10) contains the text "..the composite solution should not be trusted for sufficiently large mass ratios." One should be somewhat cautious here, since in some contexts and in some papers "large" mass ratio means "one body much more massive than the other". Such a term is nothing more than a simple remapping of the "one body much less massive than the other" small mass ratio used in this paper. I am fairly confident that the authors' concern is for the case of mass ratios close to unity, and if correct would suggest rewording this sentence to something like

"In conclusion, the composite solution should not be trusted as the mass ratio approaches unity."

4. It is very salubrious to see the authors explain the Apte-Hughes model in the context of their significantly more complete framework, and in particular to see how that simpler model could be extended based on the framework developed in this paper. One of the motivations of the Apte-Hughes model was to have a method for describing the transition and plunge (if only approximately) for all black hole spins. In this context, it would be useful to describe at least schematically the challenge of extending these results to Kerr. I imagine that many aspects of the two-timescale expansion remain unchanged, but that solving for the self-force corrections may be significantly more complicated. Though beyond the scope of what the authors consider here, this problem is a natural point for additional work, so a brief discussion of these challenges would be appropriate.


Publish (surpasses expectations and criteria for this Journal; among top 10%)

  • validity: top
  • significance: top
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

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