Binary black hole detection rates in inspiral gravitational wave searches

Chris Van Den Broeck

REVIEW

Binary black hole detection rates in inspiral gravitational wave searches

Chris Van Den Broeck

General Relativity and Quantum Cosmology · Submitted 7 Apr 2006 · 13 references

Pith's verdict: New numerical result on amplitude-corrected templates that likely cuts predicted LIGO rates by up to 2x, but the analytic proof is deferred and the robustness claim overreaches.

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arXiv IDgr-qc/0604032 (version 2)

Pith IDAXMZVS5V, Pith's permanent record of this paper and its review

CitationChris Van Den Broeck. "Binary black hole detection rates in inspiral gravitational wave searches." arXiv:gr-qc/0604032 (2006). https://pith.science/paper/AXMZVS5V

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Editor's note

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Chris — quick take.

The genuinely new thing here is showing that the 0.5PN SNR boost seen by Sintes and Vecchio reverses by 2.5PN order, so the best available amplitude-corrected waveforms give lower SNR than restricted ones — up to 25% lower, hence roughly a factor of two in detection rates. That is a concrete, useful correction for anyone planning LIGO injections or interpreting rate upper limits. The numerical work is honest and reproducible: the waveforms come from published PN expressions, the SPA treatment is straightforward, and there are no fitted parameters. The mass dependence plots are informative.

The soft spots are exactly where the reader flagged them. The analytic argument for the central inequality (3.1) is deferred to a companion paper, and the key step (3.6) is asserted without proof. On its own the paper is therefore a numerical observation with a plausibility argument, not a fully supported analytical result. The author admits as much by pointing to [10]. The faithfulness comparison in Eq. (1.3) ignores maximization over intrinsic parameters; the author notes this, so it's a stated limitation, but it means the 'up to 25%' for real searches is not yet established. That said, the numerical tables stand on their own and the trend is consistent across mass ratios. Nothing here looks circular or fitted.

Bottom line: this is a good, important-in-subfield paper that deserves refereeing. The referee should push for the proof of (3.6) or a clear statement that it is elementary and supplied in the follow-up, plus an effectualness study for the (1.3) comparison. I'd accept it as a solid contribution to the search-template literature, with the caveats kept explicit.

What would settle it

One could settle the claim by recomputing $\rho_h[h]$ and $\rho_{h_0}[h_0]$ with an independent code that uses direct fast Fourier transforms of time-domain (2.5,3.5)PN waveforms, rather than the stationary phase approximation, over a grid of masses, mass ratios, and inclination angles in the same noise curve, and checking whether the fractional deficit $\delta = (\rho_{h_0}[h_0]-\rho_h[h])/\rho_h[h]$ reaches 20 to 30 percent in the predicted asymmetric-mass region. If the deficit is absent, changes sign, or only appears under SPA, the central claim is refuted; separately, evaluating the coefficients $\mathrm{Re}[C^*_{(0)}C_{(m/2)}]$ symbolically from the published amplitudes would falsify the analytic mechanism if any is positive for allowed parameters.

Referee report

Significance

If the central claim holds, it has direct practical consequences: simulated searches and detection-rate projections based on restricted waveforms would systematically overestimate expected SNRs, and the effect is large enough to change rate predictions by a factor of two. The paper's numerical construction is transparent and reproducible in structure: the SNR ratios are computed directly from published post-Newtonian waveform expressions with no parameters fitted to the outcome, and the main numerical statement is falsifiable. The paper also makes the useful observation that amplitude corrections to the dominant second harmonic alone reproduce the behavior of the full amplitude-corrected waveform for restricted-template searches (Table II). The main weakness is that the analytic robustness argument, which is used to elevate the numerical trend into a parameter-independent claim, is not actually demonstrated in the manuscript.

Major objections (3)

[Sec. III B, Eq. (3.6)] The inequality Re[C*_(0)C_(m/2)] <= 0 for m = 1,...,5 is stated as 'not difficult to show' but no proof is provided. This inequality is the key step in the analytic explanation of Eq. (3.2) and in the claim that the result is insensitive to detection details. Since the coefficients C_(m/2) are complex functions of the angles and mass ratio and are not written out in this paper, the assertion that the inequality holds 'irrespective of parameter values' is not verifiable from the text. Please supply a full derivation, or at least an explicit listing of the coefficients together with the signs of their real parts, and justify the statements that the m = 1/2 and m = 3/2 terms vanish identically while the others are negative definite.
[Sec. III B and Sec. IV] The proof of the central inequality rho_h[h] < rho_h0[h0], Eq. (3.1), is explicitly deferred to reference [10], which is listed as 'in preparation'. This inequality is the basis of the paper's headline conclusion and of the statement that 'no caveats are needed here' beyond the difference between a genuine signal and a (2.5,3.5)PN waveform. Deferring the proof of the main result to an unpublished manuscript leaves the central claim unsupported within the paper. The numerical tables and figures provide empirical evidence, but the claimed robustness depends on an analytic argument that is not present. The proof, or a detailed and checkable sketch of it, should be included in the paper, or the conclusion should be explicitly conditioned on the validity of the unpublished proof.
[Sec. I and Sec. IV] The comparison in Eq. (1.3) and the related discussion hold for fixed intrinsic parameters, while real searches maximize the SNR over intrinsic parameters as well as extrinsic ones. The paper acknowledges this caveat for the restricted-template case, but then presents the rho_h[h] < rho_h0[h0] result as independent of 'any other detection issues' without discussing how parameter maximization affects it. Since the proof of (3.1) is not given, the reader cannot check whether the inequality survives maximization over the template-bank parameters. Please clarify how maximization over intrinsic parameters enters the main claim, or restrict the statement to the fixed-parameter faithfulness comparison that is actually computed.

Minor objections (5)

[Sec. III A] The 'up to 25%' statement is obtained by combining the left and right panels of Fig. 1, which display delta and gamma separately. Please state explicitly that the same points in (eta, M_c) parameter space are used for both quantities, or add a contour plot of the combined ratio (1+gamma)/(1+delta) so that the headline number can be checked directly.
[Sec. II, Eq. (2.5) and Eq. (3.4)] The stationary-phase expressions are written 'leaving out the step function in frequency', but the inner product integral extends to 7 f_LSO and the higher harmonics are cut off. Please specify how the cutoffs are imposed in the analytic integrand that underlies Eq. (3.5), since the step functions affect the integration limits and hence the sign of the comparison (3.7).
[Sec. III A] In the discussion of the right panel of Fig. 1, the phrase 'astrophysically less interesting' is used for large-M_c and small-eta systems. This is a subjective characterization; please give a quantitative selection criterion for the mass and mass-ratio range considered astrophysically relevant.
[References] Reference [10] is described as 'in preparation'. If it remains unpublished at the time of publication, the manuscript should state that the proof is unavailable and should not rely on it for a central claim. Consider citing a publicly available preprint or including the proof in an appendix.
[Throughout] Several inequalities are typeset with what appears to be 'greaterorsimilar' symbols; please use standard notation (for example, rho_h[h] >= rho_h0[h] with a qualification of the typical size of the difference) so that the quantitative comparisons are unambiguous.

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The paper

The paper claims

The paper claims that replacing restricted post-Newtonian waveforms with amplitude-corrected ones as search templates lowers expected inspiral signal-to-noise ratios by up to 25 percent, which translates into detection rates up to a…

The reading

This paper claims that gravitational-wave searches for inspiraling black-hole binaries have been overestimating the signal-to-noise ratios they can expect, because the waveforms used as templates and as simulated signals are usually 'restricted' post-Newtonian waveforms that keep only the dominant harmonic and drop amplitude corrections. Comparing matched-filter SNRs for the best available amplitude-corrected waveforms against restricted ones, the paper finds the amplitude-corrected SNR is up to 25 percent lower for astrophysically relevant binaries in a first-generation ground-based detector noise curve. Since detection rates scale as the cube of SNR, rate predictions are correspondingly too high by up to a factor of two. The paper traces the effect to amplitude corrections of the dominant second harmonic, which systematically reduce the overlap integral rather than adding useful signal power.

What carries the argument

The load-bearing object is the normalized matched-filter SNR $\rho_{h_1}[h_2] = (h_1|h_2)/\sqrt{(h_1|h_1)}$, with $(x|y)$ the noise-weighted inner product over frequency. The full signal model $h$ is a sum of seven harmonics of the orbital phase, built in the stationary phase approximation from post-Newtonian amplitude coefficients; the restricted model $h_0$ is the single second harmonic at leading amplitude. The analytic mechanism is the expansion of the cross-spectrum for the dominant harmonic, $\mathrm{Re}[\tilde h_0^* \tilde h^{(2)}] = C^2 f^{-7/3}\sum_{m=0}^{5}\mathrm{Re}[C^*_{(0)}C_{(m/2)}](2\pi M f)^{m/3}$, combined with the paper's assertion that each correction coefficient satisfies $\mathrm{Re}[C^*_{(0)}C_{(m/2)}]\le 0$ for $m=1,\ldots,5$. That makes the overlap of the restricted template with the amplitude-corrected dominant harmonic strictly smaller than the self-overlap, which is why the SNR deficit is systematic rather than a random fluctuation of waveform differences.

Extended reading notes

Core claim

The paper's central claim is the ordering $\rho_h[h] < \rho_{h_0}[h_0]$: when both the signal and the template are the best available amplitude-corrected post-Newtonian waveform $h$ (amplitude order 2.5PN, phase order 3.5PN), the matched-filter SNR is smaller than when both are the restricted waveform $h_0$, by up to 25 percent for binaries with total mass below about $30\,M_\odot$. Because detection rate scales as $\rho^3$, this means rate estimates from restricted-waveform simulations are high by up to a factor of two. The paper also finds that using restricted templates against amplitude-corrected signals gives an even larger deficit of up to about 30 percent, but that comparison carries caveats about maximization over intrinsic parameters; the main inequality is presented as free of those caveats. Importantly, switching to amplitude-corrected templates recovers only a few percent of the lost SNR, so the deficit is not compensated by better template models.

Load-bearing premise

The load-bearing premise is the asserted but not demonstrated inequality $\mathrm{Re}[C^*_{(0)}C_{(m/2)}]\le 0$ for $m=1,\ldots,5$ on the amplitude-correction coefficients of the dominant harmonic, together with the deferred proof of the main inequality; if any of those coefficients is positive for some masses or orientations, the analytic explanation collapses, even though the numerical trend might remain.

Editorial extensions

If this is right

A search that uses the best available amplitude-corrected (2.5,3.5)PN waveforms for both templates and simulated signals will see SNRs up to 25 percent lower than restricted-waveform simulations for sources with total mass below about 30 solar masses.
Because detection rates scale as $\rho^3$, the rate predictions from restricted-waveform simulations are too high by up to a factor of two.
Switching from restricted to amplitude-corrected templates recovers only a few percent of SNR over most of parameter space, so the deficit is not a template-choice artifact.
For the restricted-template case, the paper finds an even larger SNR overestimate, up to about 30 percent, but subject to caveats about maximizing over intrinsic parameters; the main inequality is claimed to be robust to those caveats.
Amplitude corrections to the dominant second harmonic alone reproduce the behavior of the full multi-harmonic waveform for restricted-template overlaps, so higher harmonics contribute little to this effect.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that detector search summaries and sensitivity statements should quote benchmarks for both restricted and amplitude-corrected waveforms, since the choice changes predicted rates by up to a factor of two.
A testable extension is to use an independent waveform family with amplitude corrections: the same mechanism predicts the largest deficits for asymmetric-mass, higher-total-mass binaries, so a separate implementation reproducing a deficit above 20 percent there would strengthen the conclusion.
Mapping the fractional deficit onto proposed next-generation detector noise curves would be a direct extension, since the frequency weighting in the SNR integral will shift and the up-to-25 percent figure is not expected to be universal.

Original abstract

The signal-to-noise ratios (SNRs) for quasi-circular binary black hole inspirals computed from restricted post-Newtonian waveforms are compared with those attained by more complete post-Newtonian signals, which are superpositions of amplitude-corrected harmonics of the orbital phase. It is shown that if one were to use the best available amplitude-corrected waveforms for detection templates, one should expect SNRs in actual searches to be significantly lower than those suggested by simulations based purely on restricted waveforms.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,420 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.49 · human participants 0.01 · controlled comparison 0.56 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.91 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The signal-to-noise ratios (SNRs) for quasi-circular binary black hole inspirals computed from restricted post-Newtonian waveforms are compared with those attained by more complete post-Newtonian signals, which are superpositions of amplitude-corrected harmonics of the orbital phase. It is shown that if one were to use the best available amplitude-corrected waveforms for detection templates, one should expect SNRs in actual searches to be significantly lower than those suggested by simulations based purely on restricted waveforms.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The signal-to-noise ratios (SNRs) for quasi-circular binary black hole inspirals computed from restricted post-Newtonian waveforms are compared with those attained by more complete post-Newtonian signals, which are superpositions of amplitude-corrected harmonics of the orbital phase.	method 0.73	0.81	0.06	asserted 0.90
s1	It is shown that if one were to use the best available amplitude-corrected waveforms for detection templates, one should expect SNRs in actual searches to be significantly lower than those suggested by simulations based purely on restricted waveforms.	result 0.66	0.79	0.86	asserted 0.81

What kind of paper the classifier reports this is: simulation study 0.86 · evidence from simulation or computation 0.99.

What was examined, and what was not

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not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Plots of δ (left panel) and γ (right) as functions of η and chirp mass Mc = Mη3/5 . In both plots we have imposed that individual component masses be at least 1 M⊙ and that total mass M < 80M⊙; hence the sudden drop on the left in each panel. δ = 0.2, lighter and heavier component masses m1, m2 are constrained to the ranges 1 M⊙ . m1 . 12 M⊙ and 12 M⊙ . m2 . 15 M⊙, while for δ = 0.3 one has 1 M⊙ . m1 . 8 M⊙ and 17 M⊙ . m2 . 24 M⊙. The implied range of binary systems is of clear astrophysical interest. As already suggested by the results of Table I, the overestimation is more…

Figures reproduced from arXiv:gr-qc/0604032 by the authors.

References

Works cited (13)

[10]

T. Damour, B.R. Iyer, and B.S. Sathyaprakash, Phys. Rev. D 57, 885-907 (1998)

View on Pith 1998
[1]

4M⊙ while for the black hole we choose 10 M⊙. Table I displays the variation of ρh0[h0], ρh0[h], and ρh[h] for (p, 3. 5)PN waveformsh with increasing p, where h0 is the corresponding restricted waveform. The most interesting case is the NS–BH syst em. In going from p = 0 top = 0. 5, both ρh0[h] and ρh[h] increase, as has been noted for ρh[h] by Sintes and...

View on Pith
[2]

5 exp [ i ( 2πft c −π/ 4 +kψ ( 1 kf ))] , (2.5) where [. ]2. 5 denotes consistent truncation to 2 . 5th post-Newtonian order (i.e., the “Newto- nian” prefactorf −7/ 6 is taken outside and the remaining expression is expanded in (2πMf )1/ 3 up to ﬁfth power). A dot denotes derivation with respect to time an d tc is the coalescence time. The function t(f ) ...

View on Pith
[3]

L. Blanchet, Liv. Rev. Rel. 5, 3 (2002)

View on Pith 2002
[4]

L. Blanchet, B.R. Iyer, C.M. Will, and A.G. Wiseman, Clas s. Quantum Grav. 13, 575-584 (1996); K.G. Arun, L. Blanchet, B.R. Iyer, and M.S.S. Qusailah, Clas s. Quantum Grav. 21, 3771-3802 (2004); Erratum ibid. 22, 3115 (2005)

View on Pith 1996
[5]

L. Blanchet, G. Faye, B.R. Iyer, and B. Joguet, Phys. Rev. D 65, 061501(R) (2002); Erratum ibid. D 71, 129902 (2005); L. Blanchet, T. Damour, G. Esposito-Far` ese, and B.R. Iyer, Phys. Rev. Lett. 93, 091101 (2004)

View on Pith 2002
[6]

T. Damour, B.R. Iyer, and B.S. Sathyaprakash, Phys. Rev. D 63, 044023 (2001); Erratum ibid. D 72, 029902 (2005)

View on Pith 2001
[7]

A. Buonanno and T. Damour, Phys. Rev. D 59, 084006 (1999); A. Buonanno and T. Damour, Phys. Rev. D 62, 064015 (2000)

View on Pith 1999
[8]

A. Buonanno, Y. Chen and M. Vallisneri, Phys. Rev. D 67, 024016 (2003); A. Buonanno, Y. Chen, Y. Pan, and M. Vallisneri, Phys. Rev. D 70, 104003 (2004)

View on Pith 2003
[9]

T. Damour, B.R. Iyer, and B.S. Sathyaprakash, Phys. Rev. D 62, 084036 (2000)

View on Pith 2000
[11]

S. Droz, D.J. Knapp, E. Poisson, and B.J. Owen, Phys. Rev. D 59, 124016 (1999)

View on Pith 1999
[12]

C. Van Den Broeck and A.S. Sengupta, in preparation (200 6)

View on Pith
[13]

A.M. Sintes and A. Vecchio, in Rencontres de Moriond: Gravitational waves and experiment al gravity, ed. J. Dumarchez (Editions Fronti` eres, Paris, 2000) 9

View on Pith 2000

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [10]

T. Damour, B.R. Iyer, and B.S. Sathyaprakash, Phys. Rev. D 57, 885-907 (1998)

View on Pith 1998

[2] [1]

4M⊙ while for the black hole we choose 10 M⊙. Table I displays the variation of ρh0[h0], ρh0[h], and ρh[h] for (p, 3. 5)PN waveformsh with increasing p, where h0 is the corresponding restricted waveform. The most interesting case is the NS–BH syst em. In going from p = 0 top = 0. 5, both ρh0[h] and ρh[h] increase, as has been noted for ρh[h] by Sintes and...

View on Pith

[3] [2]

5 exp [ i ( 2πft c −π/ 4 +kψ ( 1 kf ))] , (2.5) where [. ]2. 5 denotes consistent truncation to 2 . 5th post-Newtonian order (i.e., the “Newto- nian” prefactorf −7/ 6 is taken outside and the remaining expression is expanded in (2πMf )1/ 3 up to ﬁfth power). A dot denotes derivation with respect to time an d tc is the coalescence time. The function t(f ) ...

View on Pith

[4] [3]

L. Blanchet, Liv. Rev. Rel. 5, 3 (2002)

View on Pith 2002

[5] [4]

L. Blanchet, B.R. Iyer, C.M. Will, and A.G. Wiseman, Clas s. Quantum Grav. 13, 575-584 (1996); K.G. Arun, L. Blanchet, B.R. Iyer, and M.S.S. Qusailah, Clas s. Quantum Grav. 21, 3771-3802 (2004); Erratum ibid. 22, 3115 (2005)

View on Pith 1996

[6] [5]

L. Blanchet, G. Faye, B.R. Iyer, and B. Joguet, Phys. Rev. D 65, 061501(R) (2002); Erratum ibid. D 71, 129902 (2005); L. Blanchet, T. Damour, G. Esposito-Far` ese, and B.R. Iyer, Phys. Rev. Lett. 93, 091101 (2004)

View on Pith 2002

[7] [6]

T. Damour, B.R. Iyer, and B.S. Sathyaprakash, Phys. Rev. D 63, 044023 (2001); Erratum ibid. D 72, 029902 (2005)

View on Pith 2001

[8] [7]

A. Buonanno and T. Damour, Phys. Rev. D 59, 084006 (1999); A. Buonanno and T. Damour, Phys. Rev. D 62, 064015 (2000)

View on Pith 1999

[9] [8]

A. Buonanno, Y. Chen and M. Vallisneri, Phys. Rev. D 67, 024016 (2003); A. Buonanno, Y. Chen, Y. Pan, and M. Vallisneri, Phys. Rev. D 70, 104003 (2004)

View on Pith 2003

[10] [9]

T. Damour, B.R. Iyer, and B.S. Sathyaprakash, Phys. Rev. D 62, 084036 (2000)

View on Pith 2000

[11] [11]

S. Droz, D.J. Knapp, E. Poisson, and B.J. Owen, Phys. Rev. D 59, 124016 (1999)

View on Pith 1999

[12] [12]

C. Van Den Broeck and A.S. Sengupta, in preparation (200 6)

View on Pith

[13] [13]

A.M. Sintes and A. Vecchio, in Rencontres de Moriond: Gravitational waves and experiment al gravity, ed. J. Dumarchez (Editions Fronti` eres, Paris, 2000) 9

View on Pith 2000