Measuring gravitational waves from binary black hole coalescences: I. Signal to noise for inspiral, merger, and ringdown

Eanna E. Flanagan; Scott A. Hughes

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Measuring gravitational waves from binary black hole coalescences: I. Signal to noise for inspiral, merger, and ringdown

Eanna E. Flanagan, Scott A. Hughes

General Relativity and Quantum Cosmology · Submitted 16 Jan 1997 · 103 references

Pith's verdict: A foundational, largely honest SNR survey whose headline mass thresholds are softer than the confident framing suggests.

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CitationEanna E. Flanagan, Scott A. Hughes. "Measuring gravitational waves from binary black hole coalescences: I. Signal to noise for inspiral, merger, and ringdown." arXiv:gr-qc/9701039 (1997). https://pith.science/paper/AK2LQIOZ

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The thing to know: this is the paper that set the template for how everyone since has thought about binary black hole detection — three phases, three detectors, one SNR formula, and the observation that the optimal search band shifts with total mass. Eq. (2.34) is clean and checkable, the piecewise noise-curve fits are genuinely useful engineering, and the ringdown template count of order 6000 is a number people still quote. Credit where it is due: the inspiral and ringdown spectra rest on published results, and the merger spectrum is explicitly labeled a crude model rather than smuggled in as fact. The classifier record's 'limitations: none found' is simply wrong — the paper is unusually explicit about its guesses, and that candor is a real virtue here. But the reader's rationale that 'the mass-threshold conclusions do not depend on that amplitude' is only half right, and the stress-test note lands. The inspiral shutoff is amplitude-independent; the crossover is not. From Eq. (5.4), merger over inspiral scales as epsilon_m^{1/2} M^{5/3}, so the crossover mass goes as epsilon_m^{-3/10}. At epsilon_m = 0.01 the 26 solar-mass crossover becomes about 52; at 0.001 it reaches about 103, and the merger-dominated window closes for initial LIGO. Sec. III B itself calls 0.1 'perhaps optimistic' and an upper bound, and notes that contemporaneous axisymmetric simulations found lower efficiencies with most energy in ringdown. So this is not hypothetical. Second, the 102 and 410 solar-mass shutoffs in Eqs. (A4) and (A5) are not physics — they are exactly where f_merge = 0.02/M crosses the 40 and 10 Hz seismic walls. Cook's f_LSCO of about 0.055/M would move them to roughly 279 and 1100. The authors flag the f_merge choice as conservative in the template-availability sense and concede it may overestimate merger SNR, but neither sensitivity is quantified anywhere, and Figs. 4 and 5 carry no uncertainty bands. Minor points: the abstract's 500 Mpc sits a factor of two above the body's 250 Mpc and is reconciled only in a footnote; Refs. [36] and [58] are unpublished and carry the Bayesian justification of the noise-monitoring statistic, with Eq. (2.32) only heuristically derived in the text. None of this touches the qualitative conclusion that a chirp-only search silently loses the high-mass population. Anyone doing BBH searches or parameter estimation should read it, and it deserves referees.

Referee report

Significance

This is the reference calculation for the field, and the phase-by-phase decomposition it introduces is the load-bearing contribution: it converts a qualitative statement ('merger waveforms are poorly known') into a quantitative search-strategy prescription with specific mass thresholds and specific template counts. Several parts deserve explicit credit. The derivation of Eq. (2.34) in Sec. II C is clean and I checked it independently: the beam-pattern averages <F_+^2> = <F_x^2> = 1/5, Parseval, and Eq. (2.42) combine correctly to give the 2/(5 pi^2 r^2) coefficient, and the cosmological extension via graviton-number conservation, Eq. (2.44), is an elegant and correct way to handle the redshift factors. I also verified the ringdown algebra: inserting the delta-function spectrum (3.25) into (2.34) reproduces Eq. (A13), and using f_qnr = F(a)/(2 pi M) with the radiated energy 8 eps_r M = A^2 Q M^2 f_qnr reproduces Eq. (A14) exactly. Appendix B is unusually careful: the observation that rho^2 for a ringdown does not accumulate locally in time, that the naive choice h(t) = 0 for t < 0 produces spurious high-frequency power, and that the correct object is a half-line Wiener filter with a

Major objections (4)

[Abstract vs. Sec. I E and Sec. VI] The headline numbers in the abstract do not match the body, and the discrepancies are large enough to matter for a reader who reads only the abstract. (i) The abstract states BBHs are 'visible to much larger distances (up to 500 Mpc by initial interferometers)', whereas Sec. I E and the Conclusions give 'up to ~250 Mpc for M <~ 50 Msun' and 'about 200 Mpc' for M <~ 30 Msun; Eq. (5.4) confirms ~250 Mpc at the SNR ~ 6 threshold. A factor of two is not a rounding difference. (ii) The abstract gives the inspiral-search mass range as 'up to 50 solar masses for initial LIGO interferometers; 100 for advanced; 10^6 for LISA', whereas Sec. I E gives 30 Msun, 80 Msun, and 3 x 10^6 Msun respectively, and Eq. (6.1)-(6.3) give 60 Msun, 200 Msun, and 10^7 Msun for the inspiral/ringdown boundary. (iii) The abstract says merger searches 'could increase the number of detected massive BBHs by a factor of order 10 or more', whereas Sec. VI B concludes 'a factor of up to ~10' and that templates give 'at most 4, and more typically only 2'. These need to be reconciled so that the abstract, Sec. I E, Sec. VI, and Appendix A quote one set of numbers, and the reader should be told whether the abstract distances refer to angle-averaged or optimally oriented sources (the sqrt(5) amplitude difference between the two conventions, discussed after Eq. (2.34), is exactly the size of the 250 vs 500 Mpc gap).
[Sec. V B, Eq. (5.4), Sec. VI] The crossover mass at which merger SNR overtakes inspiral SNR is presented as a fixed number (~30 Msun for initial LIGO, ~60 Msun for advanced), but it is not amplitude-independent. From Eq. (5.4), merger SNR scales as eps_m^{1/2} M^{5/2} while inspiral scales as M^{5/6}, so the crossover scales as eps_m^{-3/10}. At eps_m = 0.01 the crossover moves from roughly 26 Msun to roughly 52 Msun, and at eps_m = 0.001 it reaches roughly 103 Msun, at which point the merger-dominated window for initial LIGO closes against the inspiral shutoff entirely. Since Sec. III B itself states that the 10 percent efficiency 'should perhaps be interpreted as reasonable upper bounds that could be achieved in favorable cases, rather than as best-guess estimates', and notes that axisymmetric simulations available at the time generally found lower efficiencies with most energy in the ringdown, this sensitivity is not hypothetical. The paper should state the eps_m^{-3/10} scaling explicitly and give the crossover mass for at least two or three values of eps_m, so that the search-strategy prescription in Eqs. (6.1)-(6.3) can be read as conditional on the radiation efficiency rather than as a fixed mass boundary.
[Sec. IV, Eq. (4.1); Eqs. (A4) and (A5); Sec. VI] The inspiral 'shutoff' at 102 Msun (initial LIGO) and 410 Msun (advanced LIGO) is not physics but a joint artifact of two modeling choices, and the paper's language does not make this clear. First, Eq. (4.1) sets h_rms(f) = infinity below f_s, so the shutoff occurs exactly where f_merge/(1+z) crosses f_s = 40 Hz and 10 Hz; a finite but steeply rising noise floor would move the boundary and would leave a small nonzero inspiral SNR above it. Second, the boundary is set by the free choice f_merge = 0.02/M, which Sec. III B adopts while citing Cook [9] for f_LSCO ~ 0.055/M and conceding that 0.02/M 'may optimistically overestimate the merger SNR by increasing the number of cycles in what we define as our merger waveform at the expense of the number of cycles in the inspiral'. Raising f_merge to 0.055/M would move the shutoff to roughly 279 and 1100 Msun and would reassign inspiral SNR in the band between the two cutoff frequencies. The paper should either soften the statement that 'the inspiral SNR completely shuts off' to make clear it is a property of the noise model, or add a short sensitivity paragraph giving the shutoff mass for f_merge = 0.055/M and for a finite noise floor below f_s.
[Sec. II B, Eqs. (2.32)-(2.33); Sec. VI B, Eq. (6.12)] The factor-of-ten headline gain from merger searches without templates rests on the noise-monitoring threshold Eq. (2.32) and the shape-counting formula Eq. (2.33), and both are deferred to manuscripts listed as unpublished: Eq. (2.32) is introduced with 'It can be shown [58]', where Ref. [58] is 'in preparation', and Eq. (2.33) is attributed to Ref. [36], also 'in preparation'. Eq. (6.12), R = (rho*/rho_threshold)^3, which converts the threshold ratio into an event-rate loss, is likewise not derived here. Because this is the one place where the paper makes a quantitative claim about how much is lost by not having numerical-relativity templates, and because the answer (a factor of ~2-4) is small enough that a reader will want to check it, the key step connecting Eq. (2.33) to Eq. (2.32) should be sketched in this paper rather than left entirely to a companion manuscript. A short derivation, or at minimum a statement of the assumptions under which the Gaussian counting argument holds, would make the result self-contained.

Minor objections (6)

[Fig. 1 caption vs. Eq. (5.4)] The Fig. 1 caption quotes inspiral/merger/ringdown SNRs of 2.6/2.1/0.1 for the 20 Msun system at 200 Mpc, while Eq. (5.4) evaluated at 20 Msun gives about 3.1 for the inspiral. The difference is presumably because the figures use the full numerical integration of Eq. (3.24) and the exact piecewise noise model while Eq. (5.4) is a fit; a sentence in Sec. V B saying so would prevent readers from treating the two as interchangeable.
[Sec. III E and Sec. VI B] Sec. III E estimates 10 <~ N_bins <~ 30 with a conservative upper bound of 60, and Sec. VI B then adopts N_bins = 60, 'twice the upper limit estimated in Sec. III E'. It would help to state explicitly that this choice is deliberately pessimistic and that it works against the paper's own conclusion (a larger N_bins gives a larger rho* and hence a smaller gain from templates), so that the quoted factor of ~10 is a floor rather than a ceiling.
[Appendix A] The SNR results are spread across nine piecewise formulas, Eqs. (A4), (A5), (A7), (A10)-(A12), and (A16)-(A18), each with different break frequencies and different normalizations. A single summary table listing, for each detector and each phase, the break masses, the normalization, and the power-law index would make the appendix substantially easier to use and would make the abstract/body reconciliation requested above straightforward to check.
[Sec. IV, Eq. (4.2) and footnote [91]] The observation that Fig. 7 of Ref. [3] is a factor of three too small between 10 Hz and 70 Hz is a useful catch and is properly documented. It would strengthen the section to state how much the fitted parameters (f_s, f_m, h_m, alpha) in Eq. (4.2) shift if the corrected curve is used, since the low-frequency end of the initial-LIGO fit is exactly where the inspiral SNR for the highest-mass binaries accumulates.
[After Eq. (2.34)] The paper notes that its rms angle-averaging convention gives SNRs a factor of sqrt(3/2) smaller than the 'cube root of a mean cube' convention of Refs. [1,10]. Since the abstract and Sec. I E quote distances and mass ranges without stating which convention they use, a one-sentence statement in Sec. I D fixing the convention for all quoted numbers would remove an ambiguity that a casual reader will otherwise resolve incorrectly.
[Sec. III B, Ref. [11]] Ref. [11] (Ori and Thorne, 'in preparation') is cited for the transition from radiation-reaction-driven inspiral to free-fall plunge, a step that feeds the choice of f_merge. Since this is a load-bearing physical input, a published or more accessible reference, or a one-sentence summary of the argument, would be preferable.

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The paper

The paper claims

For heavy black holes, the merger outshouts the chirp

The reading

Flanagan and Hughes ask a practical question about detectors that had not yet taken data: for a binary black hole of a given total mass, which portion of the gravitational-wave signal should a search actually be built around? Their answer is that no single answer exists. The adiabatic inspiral chirp, the violent merger, and the ringdown of the final black hole each dominate the signal-to-noise ratio in a different mass band, so a search tuned only to inspiral waves would silently lose the heavy end of the population. The calculation is a matched-filter signal-to-noise ratio averaged over source orientation and sky position, written as an integral of the source's emitted energy spectrum against the detector's noise curve, then evaluated with fitted noise models for initial LIGO, advanced LIGO, and LISA. If the estimates hold, coalescing black holes would be visible much farther out than the neutron-star binaries then treated as the primary target, and ringdown and merger searches become a requirement rather than an optional extra.

What carries the argument

The load-bearing object is the angle-averaged matched-filter signal-to-noise identity, Eq. (2.34): $4$-normalized optimal filtering squared, averaged over source orientation and position, equals a frequency integral of the source's emitted energy spectrum $dE_e/df_e$ evaluated at the redshifted frequency $(1+z)f$, weighted by $1/(f^2 S_h(f))$ and scaled by $(1+z)^2/D(z)^2$. Its virtue is that the answer depends only on the energy spectrum, not on the detailed waveform or the geometry of the source relative to the detector, so the three phases can be compared on a single footing. Feeding in three model spectra — the Newtonian quadrupole inspiral spectrum cut off at $f=0.02/M$, a flat merger spectrum from $0.02/M$ to $0.13/M$ carrying 10 percent of the mass-energy, and a Breit-Wigner ringdown resonance at the $l=m=2$ quasinormal frequency with 3 percent of the mass-energy — against piecewise power-law fits to the detector noise curves produces the mass-dependent signal-to-noise curves that drive every conclusion. A second piece of machinery is the template-counting metric on the two-dimensional ringdown parameter space of frequency and quality factor, which yields the estimate of roughly 6000 templates for a 10 percent event-rate loss.

Extended reading notes

Core claim

The paper establishes a mass-dependent rule for how binary black hole coalescences should be searched for. Using an angle-averaged optimal-filtering signal-to-noise ratio, it shows that the inspiral phase dominates only for relatively light systems, while the merger and ringdown phases take over above roughly $(1+z)M$ of about $30M_*$ for the initial interferometers and about $60M_*$ for the advanced ones, with the inspiral contribution cutting off entirely near $100M_*$ and $410M_*$ respectively once its whole spectrum falls below the seismic wall. The authors then turn that into concrete reach estimates: a matched-filter ringdown search needs only about 6000 templates or fewer and should find equal-mass systems of roughly $100$ to $700M_*$ out to about 200 Mpc with initial interferometers, $200$ to $3000M_*$ out to $z$ near 1 with advanced ones, and $10^6$ to $3$ times $10^8M_*$ out to $z$ beyond 100 with LISA. Finally, they argue that a nonlinear noise-monitoring search for merger waves, which needs no detailed waveform knowledge, could raise the number of detected massive binaries by a factor of order ten, with a complete set of numerical-relativity merger templates adding at most another factor of about four.

Load-bearing premise

The merger signal-to-noise numbers, and the headline factor-of-ten gain from merger searches, rest on a chosen flat merger energy spectrum carrying 10 percent of the total mass-energy, a value the authors themselves describe as a reasonable upper bound achievable in favorable cases rather than a best guess.

Editorial extensions

If this is right

A search pipeline built exclusively around the inspiral chirp will systematically miss equal-mass binaries above roughly 30 solar masses for initial interferometers and 60 for advanced ones, because their signal-to-noise is carried by the merger and ringdown instead.
Ringdown searches are practical now rather than in some future: the template bank is small compared with inspiral banks, and the waveform family is known analytically from Kerr perturbation theory.
Coalescing black holes, not neutron-star binaries, may be the first gravitational-wave sources detected, because their larger mass buys a distance gain that plausibly outweighs a lower event rate.
Even without any merger waveform templates, a nonlinear noise-monitoring statistic can recover a large share of the massive binaries that template searches would miss, so the unknown merger waveform is not a hard barrier to detection.
LISA would measure supermassive black hole coalescences with signal-to-noise of order $10^4$, high enough that the merger and ringdown phases are visible essentially throughout the observable universe.

Reading between the lines

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

The mass-dependent search-phase rule is a statement about where the signal-to-noise lives, not about where the physics is best understood; it implies a trade between detection reach and interpretability that later observing strategy has to make explicitly.
Because the ringdown frequency and damping time map one-to-one onto final mass and spin, a ringdown detection is simultaneously a measurement of the remnant, so the ringdown search doubles as a black-hole census instrument rather than only a trigger.
The paper's own logic suggests the reverse search problem is worth stating: given a detected ringdown with no inspiral counterpart, the absence of inspiral signal above the seismic cutoff itself constrains the total mass, which is a consistency check that needs no merger templates.
The noise-monitoring statistic is essentially an incoherent power sum over frequency bins, so its advantage shrinks as the number of independent bins grows; this predicts that any future waveform family richer than the merger's roughly 10 to 60 bins would erode the factor-of-ten gain claimed here.

Original abstract

We estimate the expected signal-to-noise ratios (SNRs) from the three phases (inspiral,merger,ringdown) of coalescing binary black holes (BBHs) for initial and advanced ground-based interferometers (LIGO/VIRGO) and for space-based interferometers (LISA). LIGO/VIRGO can do moderate SNR (a few tens), moderate accuracy studies of BBH coalescences in the mass range of a few to about 2000 solar masses; LISA can do high SNR (of order 10^4) high accuracy studies in the mass range of about 10^5 to 10^8 solar masses. BBHs might well be the first sources detected by LIGO/VIRGO: they are visible to much larger distances (up to 500 Mpc by initial interferometers) than coalescing neutron star binaries (heretofore regarded as the "bread and butter" workhorse source for LIGO/VIRGO, visible to about 30 Mpc by initial interferometers). Low-mass BBHs (up to 50 solar masses for initial LIGO interferometers; 100 for advanced; 10^6 for LISA) are best searched for via their well-understood inspiral waves; higher mass BBHs must be searched for via their poorly understood merger waves and/or their well-understood ringdown waves. A matched filtering search for massive BBHs based on ringdown waves should be capable of finding BBHs in the mass range of about 100 to 700 solar masses out to 200 Mpc (initial LIGO interferometers), and 200 to 3000 solar masses out to about z=1 (advanced interferometers). The required number of templates is of order 6000 or less. Searches based on merger waves could increase the number of detected massive BBHs by a factor of order 10 or more over those found from inspiral and ringdown waves, without detailed knowledge of the waveform shapes, using a "noise monitoring" search algorithm. A full set of merger templates from numerical relativity could further increase the number of detected BBHs by an additional factor of up to 4.

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.

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What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.53 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.80 · 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.

We estimate the expected signal-to-noise ratios (SNRs) from the three phases (inspiral,merger,ringdown) of coalescing binary black holes (BBHs) for initial and advanced ground-based interferometers (LIGO/VIRGO) and for space-based interferometers (LISA). LIGO/VIRGO can do moderate SNR (a few tens), moderate accuracy studies of BBH coalescences in the mass range of a few to about 2000 solar masses; LISA can do high SNR (of order 10^4) high accuracy studies in the mass range of about 10^5 to 10^8 solar masses. BBHs might well be the first sources detected by LIGO/VIRGO: they are visible to much larger distances (up to 500 Mpc by initial interferometers) than coalescing neutron star binaries (heretofore regarded as the "bread and butter" workhorse source for LIGO/VIRGO, visible to about 30 Mpc by initial interferometers). Low-mass BBHs (up to 50 solar masses for initial LIGO interferometers; 100 for advanced; 10^6 for LISA) are best searched for via their well-understood inspiral waves; higher mass BBHs must be searched for via their poorly understood merger waves and/or their well-understood ringdown waves. A matched filtering search for massive BBHs based on ringdown waves should be capable of finding BBHs in the mass range of about 100 to 700 solar masses out to 200 Mpc (initial LIGO interferometers), and 200 to 3000 solar masses out to about z=1 (advanced interferometers). The required number of templates is of order 6000 or less. Searches based on merger waves could increase the number of detected massive BBHs by a factor of order 10 or more over those found from inspiral and ringdown waves, without detailed knowledge of the waveform shapes, using a "noise monitoring" search algorithm. A full set of merger templates from numerical relativity could further increase the number of detected BBHs by an additional factor of up to 4.

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	We estimate the expected signal-to-noise ratios (SNRs) from the three phases (inspiral,merger,ringdown) of coalescing binary black holes (BBHs) for initial and advanced ground-based interferometers (LIGO/VIRGO) and for space-based interferometers (LISA).	contribution 1.00	0.89	0.13	asserted 0.91
s1	LIGO/VIRGO can do moderate SNR (a few tens), moderate accuracy studies of BBH coalescences in the mass range of a few to about 2000 solar masses; LISA can do high SNR (of order 10^4) high accuracy studies in the mass range of about 10^5 to 10^8 solar masses.	result 0.95	0.72	0.32	asserted 0.88
s2	BBHs might well be the first sources detected by LIGO/VIRGO: they are visible to much larger distances (up to 500 Mpc by initial interferometers) than coalescing neutron star binaries (heretofore regarded as the "bread and butter" workhorse source for LIGO/VIRGO, visible to about 30 Mpc by initial interferometers).	result 0.49	0.28	0.11	hedged possibility 0.49
s3	Low-mass BBHs (up to 50 solar masses for initial LIGO interferometers; 100 for advanced; 10^6 for LISA) are best searched for via their well-understood inspiral waves; higher mass BBHs must be searched for via their poorly understood merger waves and/or their well-understood ringdown waves.	result 0.70	0.54	0.11	asserted 0.95
s4	A matched filtering search for massive BBHs based on ringdown waves should be capable of finding BBHs in the mass range of about 100 to 700 solar masses out to 200 Mpc (initial LIGO interferometers), and 200 to 3000 solar masses out to about z=1 (advanced interferometers).	result 0.99	0.67	0.13	hedged possibility 0.95
s5	The required number of templates is of order 6000 or less.	result 0.96	0.74	0.07	asserted 0.99
s6	Searches based on merger waves could increase the number of detected massive BBHs by a factor of order 10 or more over those found from inspiral and ringdown waves, without detailed knowledge of the waveform shapes, using a "noise monitoring" search algorithm.	result 0.88	0.42	0.15	hedged possibility 0.99
s7	A full set of merger templates from numerical relativity could further increase the number of detected BBHs by an additional factor of up to 4.	result 0.60	0.35	0.10	hedged possibility 0.94

What kind of paper the classifier reports this is: theoretical result 0.76 · evidence from simulation or computation 0.71.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

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

levels run

P abstract

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C citations
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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.** FIG. 1. An illustration of the relative magnitudes of our

**Figure 2.** FIG. 2. A merger of a binary consisting of two 50

**Figure 3.** [10] K. S. Thorne, in Proceedings of the Snowmass 95 Summer Study on Particle and Nuclear Astrophysics, eds. E.W. Kolb and R. Peccei (World Scientific, Singapore, 1995) (gr-qc/9506086). [11] A. Ori and K. S. Thorne, in preparation. [12] D. Lai and A. G. Wiseman, “Innermost Stable Circular Orbit of Inspiraling Neutron-Star Binaries: Tidal Effects, Post-Newtonian Effects and the Neutron-Star Equation of State”, gr-qc/9609014. [13] S. A. Teukolsky and W. H. Press, Astrophys. J. 193, 443 (1974). [14] S. Chandrasekhar and S. L. Detweiler, Proc. R. Soc. London A344, 441 (1975). [15] F.…

**Figure 4.** FIG. 4. The signal to noise ratio (SNR) for equal-mass

**Figure 5.** FIG. 5. The SNR values for advanced LIGO interferome

**Figure 6.** FIG. 6. The signal to noise ratio (SNR) for equal-mass

**Figure 7.** FIG. 7. This plot shows that factor by which the event

Figures reproduced from arXiv:gr-qc/9701039 by the authors.

References

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A. Ori and K. S. Thorne, in preparation

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D. Lai and A. G. Wiseman, “Innermost Stable Circular Orbit of Inspiraling Neutron-Star Binaries: Tidal Eﬀects, Post-Newtonian Eﬀects and the Neutron-Star Equation of State”, gr-qc/9609014

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S. A. Teukolsky and W. H. Press, Astrophys. J. 193, 443 (1974)

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One can convince oneself of the likelihood of this clean - liness by considering various possible sources of pertur- bations to the gravitational-wave signal. Accretion disks will have masses much smaller than the black hole masses (for the equal mass coalescences that we are considering); and moreover for solar mass BBHs any accretion disks from previous...

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The Grand Challenge Alliance is an NSF funded col- laboration among physicists and computer scientists at eight institutions: University of Texas, Austin; NCSA/University of Illinois; University of North Car- olina, Chapel Hill; Cornell University; Syracuse Uni- versity NPAC; University of Pittsburgh; Northwest- ern University; and Penn State University. ...

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Our convention for the normalization of the spectral no ise densitySh(f ) is such that ⟨n(t)2⟩ = ∫ ∞ 0 Sh(f )df, wheren(t) is the detector noise (“One-sided spectral noise density”). This determines the constant 4 appearing in Eq. (2.7). Note that the alternative convention used in Refs. [53,54] results in a factor 2 instead of 4 in Eq. (2.7). In Eq. (29)...

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Note that we distinguish between the number of statisti - cally independent waveform shapes Nshapes in the mani- fold of waveforms that one is trying to detect using the 38 matched ﬁltering process, and the actual number of tem- plates Ntemplates used in the computations. The number Nshapes is the volume of the subset of the manifold of sig- nals with SNR...

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For instance, if the post-Newtonian expansion is taken to order 3/2, we would ﬁnd in the equal mass case (see, e.g., Ref. [61]) dE df = 112π2/3M 5/3f −1/3 [ 1 − 16 (9 + 14 ) (πMf )2/3]. The correction to the Newtonian formula is ∼ 0.24 at f =fmerge; when taken to higher order, there will be ad- ditional corrections. For the purposes of our discussion, the...

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It would be quite simple to improve on this ﬁlter. For ex- ample, one could divide it by the noise spectral density in order to weight the ﬁlter according to the instrumen- tal noise characteristics, i.e., pre-whiten the data stream. The main result of our analysis, the formula (2.11), would also approximately apply to such an improved ﬁltering method

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Actually the QNR metric perturbation will vanish at the event horizon; one could instead use the boundary of the ergo-region. Also the ratio R of the circumferences will actually be gauge dependent. However, if we expand R as R = 1 +αA +O(A2), then the coeﬃcient α will be gauge invariant and we could take A2 = 1/α

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More speciﬁcally, an ellipsoidal solid body of mass M of uniform density whose axes are given by l2,l 2(1 ± (3/5) sin(ωt)) gives rise to a luminosity L = 9M 2l4ω6/55; comparing this to the luminosity A2M 2ω2qnr/(16π) for the waveform (3.21) at t = 0, and identifying ω = ωqnr = 2π 0.13/M and taking l = r+ = M + √ M 2 −a2 for a = 0.97M say yields A = 0.4. T...

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This error can be seen by inserting the parameter values m = 1000 kg, f0 = 1 Hz, and Q0 = 10 9 given in Ref. [3] into the standard equation for suspension thermal noise due to viscous damping, as given in, e.g., Eq. (4.3) of Ref. [92]. The resulting noise level is 3 times larger than the noise level shown in Fig. 7 of Ref. [3] for 10 Hz <∼ f <∼ 70 Hz. Thi...

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The hrms(f ) = const piece in the noise curve happens to have the same frequency dependence that the test masses’ thermal noise would have if this noise were due to struc- tural damping [95]. We emphasize that it is not our aim to alter the standard LIGO advanced interferometer noise spectrum by assuming that structural internal thermal noise is dominant ...

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See Eq. (32) of Ref. [1]. Note also that our hn(f ) is sim- ilar to the quantity h3/yr deﬁned in Ref. [1], which is the same as the quantity denoted hSB(f ) in Refs. [3,10]. These quantities diﬀer from hn(f ) by the factor √ ln N ∼ 5, where N is the number of independent starting times per third of a year that must be tried in the optimal ﬁltering search ...

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We obtained the metric (6.4) in the following way: (i) We took the large Q limit of the matrix C′ −1ij of Ref. [16], which was calculated assuming white noise, (ii) We in- verted this matrix, took a 2 × 2 sub-block and re-inverted, thus eliminating (eﬀectively integrating out) the variabl es V and T . This procedure for eliminating the time-of- arrival va...

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This is Eq. (2.16) with ln( Nshapes) replaced by ln( Nshapes), cf. Ref. [57] above

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In the situation where merger templates are available , the optimum search method would in fact be a combina- tion of inspiral, merger and ringdown searches, in which the entire inspiral–merger–ringdown signal as a whole is searched for using matched ﬁltering

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K. S. Thorne and R. H. Price, private communication. This bet was posed at the Astrophysical Sources of Gravi- tational Waves workshop held at Penn State in July 1995

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The small z, equal-mass limit of Eq. (A14) agrees with the large Q limit of Eq. (4.7) of Ref. [16] when that equation is multiplied by a correction factor of 2

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To see that angle-averaging the approximate formula ( B3) yields the formula (A13) for the ringdown SNR, corre- sponding to the approximate delta function energy spec- trum (3.25), proceed as follows. Starting from Eq. (B14), average over the angles θ,ϕ,ψ,ι and β using Eqs. (2.38) and (3.22). This yields ⟨h0(θ,ϕ,ψ,ι,β )2⟩ = 110π A2M 2r2 . Substitute this ...

View on Pith

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Science with the space-based interferometer eLISA. I: Supermassive black hole binaries
gr-qc 2015-11 conditional novelty 5.0 of 10

Across 24 eLISA design variants and three black hole population models, only six-link configurations with the better low-frequency noise retain the high-redshift detections and sky localizations needed for multi-messe...
Determining the Hubble constant from gravitational wave observations of merging compact binaries
astro-ph.CO 2013-07 conditional novelty 5.0 of 10

Simulated joint gravitational-wave and electromagnetic observations of 15 to 30 neutron-star mergers could measure the Hubble constant to 5 percent, and to better than 1 percent if the mergers are beamed gamma-ray bursts.
Bayesian model selection for testing the no-hair theorem with black hole ringdowns
gr-qc 2011-11 conditional novelty 5.0 of 10

Simulated ringdown signals show that Bayesian model selection with Einstein Telescope or NGO could detect deviations from general relativity at the several-percent level in quasi-normal mode frequencies, while decay t...
Are black holes in alternative theories serious astrophysical candidates? The case for Einstein-Dilaton-Gauss-Bonnet black holes
gr-qc 2009-02 conditional novelty 5.0 of 10

Dilatonic black holes in Einstein-Dilaton-Gauss-Bonnet gravity are stable against axial perturbations, admit slowly rotating solutions, and differ from Kerr black holes in ISCO location and orbital frequency by up to ...
Post-Newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries
gr-qc 2008-11 accept novelty 5.0 of 10

The post-Newtonian corrections to the Christodoulou gravitational-wave memory for quasicircular inspiralling binaries are computed to 3PN order, with explicit expressions for the plus polarization and the m=0 spherica...
Generation and propagation of nonlinear quasi-normal modes of a Schwarzschild black hole
gr-qc 2022-08 conditional novelty 4.0 of 10

Second-order perturbation theory shows that quadratic quasi-normal modes of a Schwarzschild black hole are generated near the potential peak, and in the eikonal limit only waves generated outside the light ring propag...
Silhouette of M87*: A new window to peek into the world of hidden dimensions
gr-qc 2019-09 conditional novelty 4.0 of 10

The M87* shadow diameter, combined with stellar-dynamical or EHT mass estimates and an assumed spin of 0.5, is better fit by a braneworld black hole with negative tidal charge than by a Kerr black hole, though the pre...
Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (II) Ringdown
gr-qc 2018-01 accept novelty 3.0 of 10

Black hole spectroscopy, extracting multiple quasinormal modes from ringdown signals, can test Kerr dynamics, but needs ringdown signal-to-noise ratios near 100, reachable only with third-generation and space-based detectors.
Black Hole Based Tests of General Relativity
gr-qc 2016-02 accept novelty 3.0 of 10

A structured survey of black-hole-based tests of general relativity, covering model-independent parameterizations and four specific alternative theories, with current and projected constraints from gravitational and e...
Testing General Relativity with Low-Frequency, Space-Based Gravitational-Wave Detectors
gr-qc 2012-12 accept novelty 3.0 of 10

A structured survey of the general-relativity tests that low-frequency space-based gravitational-wave detectors could perform, with quantitative projections for each test and explicit caveats about the modeling those ...
Testing the black hole "no-hair" hypothesis
gr-qc 2016-07 conditional novelty 2.0 of 10

The Kerr no-hair hypothesis is a testable null hypothesis, and ringdown, stellar-orbit, accretion-disk and shadow observations can check it at signal-to-noise ratios the review quantifies.
The Evolution of Compact Binary Star Systems
astro-ph.HE 2014-03 accept novelty 2.0 of 10

The review consolidates formation channels, merger rates and open uncertainties for compact binary stars, concluding that Galactic double neutron star mergers occur roughly once per ten thousand years.
Physics, Astrophysics and Cosmology with Gravitational Waves
gr-qc 2009-03 accept novelty 2.0 of 10

The paper reviews the physics, sources, detectors, data analysis and scientific payoff of gravitational wave astronomy, forecasting that upgraded ground-based interferometers should reach regular detections around 2014.

How later papers used it

134 later papers cite it

Evidence: substantial. 133 of the citing papers share no author with it, and 56 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 189 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

8 later papers by other authors are listed below: 8 used its method or tool.

Uses its method or tool 8 of 52 papers with this label

Computed the SNR accumulated over the observational time following this paper's formalism.

Probing $z \gtrsim 6$ massive black holes with gravitational waves (2023) · 0.99 · paraphrased: the licence does not allow quoting
Calculated the characteristic GW strain for a given frequency using this paper.

Gravitational Wave Signatures of Ab Initio Two-Dimensional Core Collapse Supernova Explosion Models for 12-25 Solar Masses Stars (2015) · 0.99 · paraphrased: the licence does not allow quoting
We use [https://legwork.readthedocs.io] [ref] to determine the detectability of our simulated DWD populations for sources with GW frequencies $f_GW>10^-4 Hz$. calculates the position-, orientation-, and angle-averaged signal to noise ratio (SNR) for inspiraling GW sources closely following the derivations of [ref] and using the LISA noise power spectral density (PSD) of [ref] .

Applying the metallicity-dependent binary fraction to double white dwarf formation: Implications for LISA (2021) · LISA detectability · 1.00
Calculated the position-, orientation-, and angle-averaged SNR for a circular binary system following this paper and another work.

Massive Double White Dwarf Binary Mergers from the Moon: Extending the Reach of Multi-messenger Astrophysics (2025) · 0.99 · paraphrased: the licence does not allow quoting
Crudely estimated the signal to noise ratio of this characteristic strain for DECIGO using this paper.

Gravitational wave signals in the deci-Hz range from neutrinos during the proto-neutron star cooling phase (2022) · 0.98 · paraphrased: the licence does not allow quoting
Took the factors within the minimum functions because each signal is only accumulated within its own lifetime or the LISA observation timescale, following this paper.

Gravitational-Wave Signatures from Compact Object Binaries in the Galactic Center (2020) · 0.83 · paraphrased: the licence does not allow quoting
Used the signal-to-noise ratio for a matched filtered signal, citing this paper as an example, to roughly assess detectability of the GW signals.

Gravitational waves from three-dimensional core-collapse supernova models: The impact of moderate progenitor rotation (2018) · 1.00 · paraphrased: the licence does not allow quoting
Calculated the single-detector SNR of the GW waveforms assuming the optimal orientation using this paper's Eq. (1.1).

Gravitational-wave Signature of a First-order Quantum Chromodynamics Phase Transition in Core-Collapse Supernovae (2020) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

77 arXiv papers cite this in 111 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 59 sentences without a confident label

Contests it 1 paper

This is essentially the approach followed by Thorne [ref] , who examined the Christodoulou memory from unmodeled GW bursts from merging BHs.[Note that Thorne's [ref] normalization of the SNR should be corrected by multiplying his Eq. (A1) by $√(2)$ (see footnote 61 of Ref. [ref] ).] However the ZFL is inadequate for computing the memory's SNR for some sources.

Post-Newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries · Detecting memory with gravitational-wave interferometers · contest 0.77

Uses its method or tool 29 papers

In order to determine the detectability of the GWs, we calculate the characteristic GW strain for a given frequency $f$ [ref] using [equation] [equation] where $dE_GW(f)/df$ is the GW energy spectrum and $Ã_20(f)$ denotes the Fourier transform of $A_20(t)$: [equation] The stochastic nature of GW signals from CCSNe prompts the use of short-time Fourier transform (STFT) techniques to determine the frequency of a signal as it changes over time [ref] : [equation] where $H(t - τ)$ is the Hann window function [ref] .

Gravitational Wave Signatures of Ab Initio Two-Dimensional Core Collapse Supernova Explosion Models for 12-25 Solar Masses Stars · GWs Produced by a Time-Dependent, Mass-Quadrupole Moment · 0.99
To understand the impact of such energy loss on GW signals, we add to the quadrupole formula the energy flux [ref], and compute the correction to the GW phase in the stationary approximation [ref] .

Stirred and shaken: dynamical behavior of boson stars and dark matter cores · Introduction · 0.93
For an optimally oriented detector the matched-filtering SNR squared, averaged over all possible source orientations is [ref]

On the dynamical bar-mode instability in spinning bosonic stars · Detectability · 0.89

Uses its data or values 3 papers

As estimated in Ref. [ref] , the maximum value of $𝒜$ is taken to be $𝒜=0.4$.

Measuring the Black Hole Mass Spectrum from Redshifts of aLIGO Binary Merger Events · Detectability based on aLIGO · 0.97
The values can range between $ϵ_rd∼0.8%-3%$ [ref] depending on the binary mass ratio and spins.

Generation and propagation of nonlinear quasi-normal modes of a Schwarzschild black hole · Importance of nonlinearities · 0.95
Flanagan and Hughes [ref] argue that $E ∼0.03 (4ν)^2 M$.

Physics, Astrophysics and Cosmology with Gravitational Waves · Quasi-normal modes of a black hole · 0.93

Cites it as background 12 papers

The QNM amplitudes $A_ℓm n$ and phases $ϕ_ℓm n$ were unknown before the first numerical BH merger simulations, and early work on BH spectroscopy [ref] had to rely on educated guesses [ref] .

Analysis of Ringdown Overtones in GW150914 · Introduction · 0.90
The inspiral waveform measured in a gravitational wave detector is of the form of [ref] h(t) = h_0(t) cosϕ(t), where h_0(t)=4 Q/D_L G Mη Ω^2 r^2 and ϕ(t) = ∫2πf dt, with $D_L$ being the luminosity distance to the source and where $Q$ encodes the detector response as a function of the angular position and orientation of the binary.

Prospects for axion searches with Advanced LIGO through binary mergers · Waveform · 0.84
With the emergence of GW astronomy, we now know that low-mass IMBH can form and grow through mergers with other BH[The term IMBH first began to appear in published papers from the late 1990s that were predicting the type of merging BH that could potentially be observed with future GW missions [ref] .]

Intermediate-Mass Black Holes in Star Clusters and Dwarf Galaxies · Context and definition · 0.87

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.

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Since ǫ is a scalar quantity, it can only depend on scalar products of S1, S2 and ˆL and not on these vectorial quan- tities themselves. Therefore the number of independent variables that ǫ depends on is six

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J. K. Blackburn and S. Detweiler, Phys. Rev. D 46, 2318 (1992); the geometry at which S = 7.5m1 in their nota- tion

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D. M. Eardley, in Gravitational Radiation, ed. N. Deruelle and T. Piran, North-Holland Publishing Co., 1983

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See, for example, K. S. Thorne, Rev. Mod. Phys. 52, 299 (1980), Eqs. (4.16’) and (4.23’)

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The inequality |Sﬁnal| ≤ M 2 used in deriving Eq. (3.10) should really be |Sﬁnal| ≤ (M −Erad)2, since it is the mass of the ﬁnal black hole and not the initial total mass of the system that is relevant. An argument par- allel to that leading to Eq. (3.12) then yields the inequal- ity ǫrad >∼ [1.4 − (1 −ǫrad)2]/4 or ǫrad >∼ 0.18, where ǫrad =Erad/M. Hence,...

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D. M. Eardley and E. W. Hirschmann, Comment on “In- stabilities in Close Neutron Star Binaries” , ITP Santa Barbara preprint NSF-ITP-95-165 (gr-qc/9601019)

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C. W. Misner et al. , Gravitation (Freeman, New York, 1973), Sec. 36.6

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B. J. Owen, Phys. Rev. D 53, 6749 (1996)

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E. W. Leaver, Proc. R. Soc. Lond. A 402, 285 (1985). Leaver’s data on the l =m = 2 mode, which was used by Echeverria, is unpublished. 39

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K. S. Thorne, Ap. J. 191, 507 (1974)

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Actually the QNR metric perturbation will vanish at the event horizon; one could instead use the boundary of the ergo-region. Also the ratio R of the circumferences will actually be gauge dependent. However, if we expand R as R = 1 +αA +O(A2), then the coeﬃcient α will be gauge invariant and we could take A2 = 1/α

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[90] [90]

More speciﬁcally, an ellipsoidal solid body of mass M of uniform density whose axes are given by l2,l 2(1 ± (3/5) sin(ωt)) gives rise to a luminosity L = 9M 2l4ω6/55; comparing this to the luminosity A2M 2ω2qnr/(16π) for the waveform (3.21) at t = 0, and identifying ω = ωqnr = 2π 0.13/M and taking l = r+ = M + √ M 2 −a2 for a = 0.97M say yields A = 0.4. T...

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[91] [91]

This error can be seen by inserting the parameter values m = 1000 kg, f0 = 1 Hz, and Q0 = 10 9 given in Ref. [3] into the standard equation for suspension thermal noise due to viscous damping, as given in, e.g., Eq. (4.3) of Ref. [92]. The resulting noise level is 3 times larger than the noise level shown in Fig. 7 of Ref. [3] for 10 Hz <∼ f <∼ 70 Hz. Thi...

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L. S. Finn and D. F. Chernoﬀ, Phys. Rev. D 47, 2198 (1993)

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R.E. Vogt et. al., ”Proposal to the National Science Fou n- dation for the Construction, Operation, and Support- ing Research and Development of a Laser Interferometer Gravitational-Wave Observatory”, (California Institute of Technology, 1989), unpublished

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The hrms(f ) = const piece in the noise curve happens to have the same frequency dependence that the test masses’ thermal noise would have if this noise were due to struc- tural damping [95]. We emphasize that it is not our aim to alter the standard LIGO advanced interferometer noise spectrum by assuming that structural internal thermal noise is dominant ...

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A. Gillespie and F. Raab, Phys. Lett. A, 190, 213 (1994)

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See Eq. (32) of Ref. [1]. Note also that our hn(f ) is sim- ilar to the quantity h3/yr deﬁned in Ref. [1], which is the same as the quantity denoted hSB(f ) in Refs. [3,10]. These quantities diﬀer from hn(f ) by the factor √ ln N ∼ 5, where N is the number of independent starting times per third of a year that must be tried in the optimal ﬁltering search ...

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K. D. Kokkotas, A. Kr´ olak, and G. Tsegas, Class. Quant. Grav. 11, 1901 (1994)

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We obtained the metric (6.4) in the following way: (i) We took the large Q limit of the matrix C′ −1ij of Ref. [16], which was calculated assuming white noise, (ii) We in- verted this matrix, took a 2 × 2 sub-block and re-inverted, thus eliminating (eﬀectively integrating out) the variabl es V and T . This procedure for eliminating the time-of- arrival va...

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[99] [99]

This is Eq. (2.16) with ln( Nshapes) replaced by ln( Nshapes), cf. Ref. [57] above

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[100] [100]

In the situation where merger templates are available , the optimum search method would in fact be a combina- tion of inspiral, merger and ringdown searches, in which the entire inspiral–merger–ringdown signal as a whole is searched for using matched ﬁltering

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K. S. Thorne and R. H. Price, private communication. This bet was posed at the Astrophysical Sources of Gravi- tational Waves workshop held at Penn State in July 1995

View on Pith 1995

[102] [102]

The small z, equal-mass limit of Eq. (A14) agrees with the large Q limit of Eq. (4.7) of Ref. [16] when that equation is multiplied by a correction factor of 2

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[103] [103]

To see that angle-averaging the approximate formula ( B3) yields the formula (A13) for the ringdown SNR, corre- sponding to the approximate delta function energy spec- trum (3.25), proceed as follows. Starting from Eq. (B14), average over the angles θ,ϕ,ψ,ι and β using Eqs. (2.38) and (3.22). This yields ⟨h0(θ,ϕ,ψ,ι,β )2⟩ = 110π A2M 2r2 . Substitute this ...

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