Search templates for gravitational waves from inspiraling binaries: Choice of template spacing

Benjamin J. Owen (Caltech)

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Search templates for gravitational waves from inspiraling binaries: Choice of template spacing

Benjamin J. Owen (Caltech)

General Relativity and Quantum Cosmology · Submitted 10 Nov 1995 · 27 references

Pith's verdict: A clean geometric reformulation of template placement that still holds up; the weak link is the match-to-event-rate conversion, which the author flags and hands to an unpublished companion paper.

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CitationBenjamin J. Owen (Caltech). "Search templates for gravitational waves from inspiraling binaries: Choice of template spacing." arXiv:gr-qc/9511032 (1995). https://pith.science/paper/4RCBARZY

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Owen takes the Sathyaprakash–Dhurandhar template-spacing machinery and rewrites it as a metric on the intrinsic parameter space, then generalizes it to any number of parameters and any noise spectrum. The genuinely new piece is Eqs. (2.27)–(2.28): he builds the metric from the ambiguity function after maximizing over t0 and Phi0, then projects out t0. That is the right move — it makes the metric a search metric rather than a parameter-estimation Fisher matrix — and footnote [20] is explicit about how it differs from the information matrix used earlier. The moment integrals are tabulated, the fc-to-infinity approximation is bounded at roughly ten percent for initial LIGO and one percent for advanced, and the parameter-space volume is a Monte Carlo. The numbers N ~ 2.4e5 and P ~ 20 Gflops (initial), N ~ 7.6e6 and P ~ 270 Gflops (advanced) are forward computations from stated inputs, not fits to an answer. That is the paper's real contribution and it is solid. The soft spots, in order of weight. First, Eq. (2.21), event rate proportional to (MM/rho0)^3, is the only bridge from match to event loss, and it ignores that the statistic is a maximum over N templates, so rho0 itself moves with N. He says so in Sec. II.B and defers to Ref. [19], which is listed as in preparation. That is the one place I would push: a load-bearing step handed to an unpublished companion. Still, the correction shifts the L-to-MM conversion, not the metric or the N(MM) scaling, and neighboring templates are strongly correlated, so N_eff is well below N. I would expect a factor-of-order-unity effect on the quoted L, not a structural problem; the stress-test figure of 20–50% is plausible but not established without an N_eff calculation. Second, Eq. (3.21) prints F ~ 214, while direct evaluation gives about 1e6 for initial and 1.4e7 for advanced LIGO. That changes (16 + 3 log2 F) by roughly a factor of two, so P roughly doubles. Still several hundred Gflops, and he already warns against trusting P beyond order unity, but the arithmetic should be fixed. Third, the abstract's claim of validity for arbitrary noise spectra and template parameterizations overstates the body, which requires fine spacing and fc much greater than f0; the introduction says so plainly. The Mmin = 0.2 solar mass choice drives the Mmin^-2.7 scaling and is the biggest lever on N, but he argues for it against Sathyaprakash's 1 solar mass and it is disclosed, so it is a modeling decision rather than a hidden one. This is for anyone doing template placement for inspiral searches; it is the standard reference for the metric. It deserves a serious referee — the method is sound and the corrections are quantitative.

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Major objections (2)

[Sec. II.B, Eq. (2.21)] The bridge from minimal match to fractional event loss, event rate ∝ (MM/ρ0)^3, treats a reduced match as equivalent to raising the threshold by 1/MM. The author acknowledges in Sec. II.B that the detection statistic is a maximum over N templates, so the false-alarm threshold itself shifts with N and with the covariance of the templates, but justifies ignoring this only qualitatively ('these effects are fairly small at high signal-to-noise ratios') and defers the problem to Ref. [19], which is listed as 'in preparation.' A simple estimate for Gaussian noise, ρ0^2 → ρ0^2 + 2 ln N_eff, suggests that for N_eff ~ 10^2–10^3 and ρ0 ~ 5–8 the threshold shift could cost tens of percent of event rate, comparable to the 10% target L. Because the paper's headline N(L) and P(L) depend on this bridge, the caveat should be stated more prominently and, if possible, bounded roughly; at minimum, the paper should estimate N_eff for the correlated template bank and state whether the feedback (larger N → higher ρ0 → larger L for fixed MM) is negligible or not.
[Sec. III.D, Eq. (3.21)] Eq. (3.21) as printed gives F ≃ 2^14 (Mmin/0.2 Msun)^-5/3 (fs/10 Hz)^-8/3 (f0/70 Hz), but direct evaluation of the defining expression F = 2 fu τ1 (f0/fs)^(8/3) with fu = 4f0, Mmin = 0.2 Msun, f0 = 70 Hz, and fs = 10 Hz gives F ~ 1.4e7 ~ 2^24, a factor of roughly 10^3 larger. This propagates to P through the log2 F term in Eq. (3.22): with log2 F = 24 rather than 14, P rises by about 50%, from ~270 Gflops to ~400 Gflops. The qualitative conclusion ('several hundred Gigaflops') survives, but the quoted number and the internal consistency between Eqs. (3.21) and (3.24) should be corrected.

Minor objections (5)

[Sec. III.B, Eqs. (3.10)–(3.11)] The scaling factor (MM/0.03)^-1 appears to be a typo for ((1-MM)/0.03)^-1. With MM = 0.97, the printed form gives (0.97/0.03)^-1 ≈ 0.031, which would reduce N by a factor of ~32 relative to the stated value of 2.7e5; the intended form gives unity at MM = 0.97 and is consistent with the quoted numbers.
[Sec. III.C, Eq. (3.15)] The first eigenvector for advanced LIGO is printed as 'ex1 = 0.899ex1 + 0.437eτ2'; the first term should presumably be 0.899eτ1, by analogy with Eq. (3.14).
[Fig. 2 caption] The caption states that the region is for binaries 'composed of objects more massive than 1 M⊙,' but the text uses Mmin = 0.2 M⊙ for the numerical estimates. The caption should clarify whether the figure is an illustration at 1 M⊙ or whether it corresponds to the Mmin = 0.2 M⊙ calculation.
[Abstract] The abstract says 'of order several hundred Gigaflops,' while the body gives 20 Gflops for initial LIGO and 270 Gflops for advanced LIGO. 'Several hundred' is a slight overstatement for 270; consider 'a few hundred' or specifying that the figure refers to advanced LIGO.
[Sec. II.B, Ref. [19]] The threshold-selection issue is deferred to Ref. [19], which is listed as 'in preparation.' Since this is the paper's main acknowledged caveat, a sentence in the body summarizing the expected size of the effect—even an order-of-magnitude estimate—would make the paper self-contained.

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The paper

The paper claims

A metric on the space of binary inspiral waveforms yields the number of search templates and the computing power needed for LIGO's matched-filter search, as a function of acceptable event-rate loss.

The reading

LIGO looks for the chirp of two compact stars spiraling together by cross-correlating its noisy data with theoretical waveform templates. The templates form a continuous two-parameter family (the two masses), but only a finite discrete grid can be computed. Too coarse a grid misses signals; too fine a grid costs too much computing. This paper turns that trade-off into geometry. It defines a distance between two nearby templates from how much their overlap (the match) drops when their parameters differ, giving a metric on the mass parameter space. The worst case is a signal sitting exactly between grid points, and the minimal match is the overlap retained there. From the metric one reads off the grid spacing, the number of templates, and the computing power, all as functions of the fractional event-rate loss L. For a post-1-Newtonian two-parameter template family, the author finds roughly 2.4e5 templates and about 20 Gflops for the first LIGO interferometers, and roughly 7.6e6 templates and about 270 Gflops for advanced LIGO, at 10 percent event loss and a minimum mass of 0.2 solar masses. The numbers scale steeply with the minimum mass, as Mmin to the minus 2.7, so the choice of 0.2 rather than 1.0 solar masses is what drives the count up relative to earlier estimates. The author argues the cost is feasible for LIGO-era computers but that a hierarchical search, coarse pass then fine pass, may still be worth pursuing.

Extended reading notes

Core claim

The number of templates and the computing power for a single-pass matched-filter search follow from a metric on template parameter space: N ~ 2.4e5 (L/0.1)^-1 (Mmin/0.2 Msun)^-2.7 and P ~ 20 Gflops for the first LIGO interferometers, and N ~ 7.6e6 and P ~ 270 Gflops for advanced LIGO (Eqs. 3.12, 3.13, 3.23, 3.24). If correct, the most demanding reasonable search is within reach of LIGO-era computers, and the template spacing is fixed by the acceptable fractional event loss L.

Load-bearing premise

The bridge from minimal match to fractional event loss is Eq. (2.21), event rate proportional to (MM/rho0)^3, which treats a reduced match as equivalent to raising the threshold by 1/MM. This ignores that the detection statistic is the maximum over N templates, so the false-alarm threshold itself shifts with N and with the covariance of the templates. The author acknowledges this in Sec. II.B ('In real life, rho0 is affected by the total number of discrete templates...') but justifies it only qualitatively ('these effects are fairly small at high signal-to-noise ratios') without a quantitative bound. Since the whole point of the paper is to choose spacing as a function of L, this unquantified step is load-bearing.

Original abstract

It has long been known that the search for gravitational waves from inspiraling binaries must be aided by signal processing methods, and that the technique of matched filtering is the most likely to be used for the detection of inspiral ``chirps''. This means that the output of an interferometer must be cross- correlated with many waveform templates to dig out faint signals buried in the noise. The templates are characterized by several parameters which vary continuously over some finite range; but because the amount of computing power available to perform the cross-correlations of the search templates with the output is finite, the actual templates used must be picked with certain discrete values of these parameters, which will have some finite spacing. If the spacing is too small, the number of templates (and therefore the computing power) needed to perform an on-line search becomes prohibitive; if the spacing is too large, many chirps will not be detected because the values of their parameters lie too far from those of the nearest template. In this paper I use differential geometry to extend the earlier formalism of Sathyaprakash and Dhurandhar to estimate the template spacing, number of templates, and computing power required for a single-pass, on-line search of the output of a single interferometer in terms of the fraction of events lost to parameter discretization. My formalism obtains these results with little numerical computation, and is valid for arbitrary noise spectra and template parameterizations. I find that the computing power needed for the most computationally intensive, reasonable search is of order several hundred Gigaflops. This will be feasible by the time LIGO is operational, but it is worth pursuing methods of reducing this number.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.67 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.37 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

It has long been known that the search for gravitational waves from inspiraling binaries must be aided by signal processing methods, and that the technique of matched filtering is the most likely to be used for the detection of inspiral ``chirps''. This means that the output of an interferometer must be cross- correlated with many waveform templates to dig out faint signals buried in the noise. The templates are characterized by several parameters which vary continuously over some finite range; but because the amount of computing power available to perform the cross-correlations of the search templates with the output is finite, the actual templates used must be picked with certain discrete values of these parameters, which will have some finite spacing. If the spacing is too small, the number of templates (and therefore the computing power) needed to perform an on-line search becomes prohibitive; if the spacing is too large, many chirps will not be detected because the values of their parameters lie too far from those of the nearest template. In this paper I use differential geometry to extend the earlier formalism of Sathyaprakash and Dhurandhar to estimate the template spacing, number of templates, and computing power required for a single-pass, on-line search of the output of a single interferometer in terms of the fraction of events lost to parameter discretization. My formalism obtains these results with little numerical computation, and is valid for arbitrary noise spectra and template parameterizations. I find that the computing power needed for the most computationally intensive, reasonable search is of order several hundred Gigaflops. This will be feasible by the time LIGO is operational, but it is worth pursuing methods of reducing this number.

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	It has long been known that the search for gravitational waves from inspiraling binaries must be aided by signal processing methods, and that the technique of matched filtering is the most likely to be used for the detection of inspiral ``chirps''.	background 1.00	0.03	0.02	hedged possibility 0.54
s1	This means that the output of an interferometer must be cross- correlated with many waveform templates to dig out faint signals buried in the noise.	background 0.94	0.05	0.02	asserted 0.99
s2	The templates are characterized by several parameters which vary continuously over some finite range; but because the amount of computing power available to perform the cross-correlations of the search templates with the output is finite, the actual templates used must be picked with certain discrete values of these parameters, which will have some finite spacing.	background 0.99	0.06	0.03	asserted 1.00
s3	If the spacing is too small, the number of templates (and therefore the computing power) needed to perform an on-line search becomes prohibitive; if the spacing is too large, many chirps will not be detected because the values of their parameters lie too far from those of the nearest template.	background 0.99	0.06	0.04	asserted 0.95
s4	In this paper I use differential geometry to extend the earlier formalism of Sathyaprakash and Dhurandhar to estimate the template spacing, number of templates, and computing power required for a single-pass, on-line search of the output of a single interferometer in terms of the fraction of events lost to parameter discretization.	contribution 0.79	0.95	0.11	asserted 1.00
s5	My formalism obtains these results with little numerical computation, and is valid for arbitrary noise spectra and template parameterizations.	method 0.69	0.90	0.07	asserted 1.00
s6	I find that the computing power needed for the most computationally intensive, reasonable search is of order several hundred Gigaflops.	result 1.00	0.96	0.40	asserted 1.00
s7	This will be feasible by the time LIGO is operational, but it is worth pursuing methods of reducing this number.	future work 0.69	0.34	0.16	asserted 0.51

What kind of paper the classifier reports this is: theoretical result 0.80 · evidence from proof or derivation 0.99.

What was examined, and what was not

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arXiv listing abstract v1

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full text
figures and tables
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S sections
T body sentences
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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. A schematic depiction of the manifold formed by the co

**Figure 2.** FIG. 2. The two-dimensional region of parameter space inhab

**Figure 3.** FIG. 3. Locations of various discrete templates for the first

Figures reproduced from arXiv:gr-qc/9511032 by the authors.

References

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Note that the Φ 0 used here is the coalescence phase of the waveform. In the restricted post-Newtonian approximation, this is twice the coalescen ce phase of the orbitφ0, which is used by Cutler and Flanagan in Ref. [15]

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P. C. Peters, Phys. Rev. 136, B1224 (1964)

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The ψj given here are diﬀerent from those given in Eq. (13a-3) of Ref . [14] because Sathyaprakash does not use t0 and Φ 0 as kinematical parameters, but rather the time and phase at which the quadrupole part of the waveform reache s a frequency of f0

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S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (Wiley, New York, 1983). 30

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Numerical integrations were performed with the aid of M athematica [S. Wolfram, Math- ematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, California, 1988)] and with a Monte Carlo code based on Numerical Recipes [W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanner y, Numerical Recipes: The Art of Scientiﬁc Comp...

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[26]

The storage size of the template is equal to the eﬀective sampling frequency 2fu times the duration of the template waveform. In the restricted pos t-Newtonian approximation, the duration of a binary chirp waveform template is equal to t he time it takes for the gravitational wave frequency to sweep from the seismic freq uency fs up to inﬁnity. Since τ1 ∝...

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R. Balasubramanian, B. S. Sathyaprakash, and S. V. Dhur andhar, Gravitational waves from coalescing binaries: detection strategies and Monte C arlo estimation of parameters , submitted to Phys. Rev. D (Los Alamos xxx.lanl.gov archive p reprint gr-qc/9508011). 31 FIGURES FIG. 1. A schematic depiction of the manifold formed by the co ntinuous template famil...

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Pith read 18 of the 236 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

7 later papers by other authors are listed below: 7 used its method or tool. 2 more papers by the same authors are listed: 1 built on it and 1 used its method or tool.

Builds on it 1 of 2 papers with this label · 1 by the same authors

By the same authors

Extended the non-spinning inspiral metric of this paper and another work to 3.5 PN order in phase.

Parameter space metric for 3.5 post-Newtonian gravitational-waves from compact binary inspirals (2013) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 8 of 39 papers with this label · 1 by the same authors

Used this paper's approximate expression for template count to estimate a good central point.

Metric Assisted Stochastic Sampling (MASS) search for gravitational waves from binary black hole mergers (2021) · 0.94 · paraphrased: the licence does not allow quoting
Used this paper's definition of the metric components in terms of the mismatch.

A binary tree approach to template placement for searches for gravitational waves from compact binary mergers (2022) · 1.00 · paraphrased: the licence does not allow quoting
Followed this paper to Taylor expand the match between nearby waveforms and express the mismatch in terms of the parameter space metric.

Template bank to search for exotic gravitational wave signals from astrophysical compact binaries (2023) · 0.94 · paraphrased: the licence does not allow quoting
This is given by the parameter space metric [ref] defined in the next section.

Coherent Search for Binary Pulsars across all Five Keplerian Parameters in Radio Observations using the template-bank algorithm (2021) · Signal Model and Detection Statistic · 0.90
Followed this paper's simple procedure to project the time and phase shifts out of the Fisher matrix.

Statistical and systematic errors for gravitational-wave inspiral signals: A principal component analysis (2013) · 1.00 · paraphrased: the licence does not allow quoting
Obtained the match from the Fisher matrix by projecting out the phase and time-shift coordinates of the metric as described in this paper.

The failure of the Fisher Matrix when including tidal terms: Considering construction of template banks of tidally deformed binary neutron stars (2021) · 0.99 · paraphrased: the licence does not allow quoting
Approximated its signal population's distinguishability by the filter banks described in this paper.

Composite gravitational-wave detection of compact binary coalescence (2011) · 0.91 · paraphrased: the licence does not allow quoting

By the same authors

Divided by the proper volume per template from this paper to estimate the number of templates.

How to search for gravitational waves from $r$-modes of known pulsars (2019) (same authors) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

98 arXiv papers cite this in 169 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 · 88 sentences without a confident label

Contests it 1 paper

However there is one significant error in the description in [ref] : the limit approached is that of filtering with an unknown overall phase constant, which is commonly handled by summing the squares of two matched filters a quarter cycle out of phase with each other—e.g., [ref] .

Cross-correlation method for intermediate-duration gravitational wave searches associated with gamma-ray bursts · Matched filter limit (all pairs) · contest 0.93

Builds on it 2 papers

We present and employ a geometric method to construct a bank of eccentric waveforms for matched-filtering spanning the intrinsic parameter space $(m^_1, m^_2, s^_1z, s^_2z, e^_0)$, building on the works of [ref] .

A geometric template bank for the detection of spinning low-mass compact binaries with moderate orbital eccentricity · Geometric Approach · 0.89
Hence in this work we extend the non-spinning inspiral metric of [ref] and [ref] to 3.5 PN order in phase.

Parameter space metric for 3.5 post-Newtonian gravitational-waves from compact binary inspirals · Introduction · 1.00

Uses its method or tool 30 papers

This gives a measure of mismatches between neighbouring points in the manifold of the mass-parameters, and hence a hexagonal 2D lattice placement can be used in the manifold of the mass parameters [ref] (and references therein), to construct a geometric lattice based template bank [ref] .

Template banks to search for low-mass binary black holes in advanced gravitational-wave detectors · Bank Placement metric · 0.86
[ref] [equation] We have implemented two numerical schemes for estimating the metric component values that we call the iterative and deterministic methods.

A binary tree approach to template placement for searches for gravitational waves from compact binary mergers · Methods · 1.00
We first generate a template bank for binaries with zero eccentricity and component spin using the standard hexagonal lattice method of template placement [ref] .

Eccentric Binary Neutron Star Search Prospects for Cosmic Explorer · Binary Neutron Star Searches in Cosmic Explorer · 1.00

Uses its data or values 1 paper

And, the typical desirable FFs for initial LIGO are $≈0.97$ [ref] .

Gravitational waves from compact binaries inspiralling along post-Newtonian accurate eccentric orbits: Data analysis implications · Nonoptimal Matched Filter Searches for GW signals having Some Residual Eccentricities · 0.83

Compares with it 2 papers

For the case of the signal given by Equation ([ref]) our formula for number of templates is equivalent to the original formula derived by Owen [ref] .

Gravitational-Wave Data Analysis. Formalism and Sample Applications: The Gaussian Case · Number of templates · 1.00
In contrast, the number of metric-based templates is proportional to $(1-)^-(3p-1)/2$ [ref] and increases dramatically with $p$ for any fixed $$ (see, for example, Tables [ref] and [ref]).

Reduced Basis representations of multi-mode black hole ringdown gravitational waves · Unconstrained multi-mode Reduced Basis · 0.95

Cites it as support 1 paper

Additionally, the computational cost of these methods do not increase steeply as in matched filtering when the minimum searchable frequency decreases [ref] .

Enabling multi-messenger astronomy with continuous gravitational waves: early warning and sky localization of binary neutron stars in Einstein Telescope · Introduction · 0.97

Cites it as background 29 papers

The two-dimensional metric for the inspiral portion of the signal associated with GW from BNS was originally computed at 1.0 PN order [ref] and then extended to 2.0 PN [ref] .

Parameter space metric for 3.5 post-Newtonian gravitational-waves from compact binary inspirals · Introduction · 0.93
The quantity $g_ab$ is called the parameter-space metric [ref] ; for nearby templates, $g_ab dλ^a dλ^b$ measures the squared fractional deviation or squared dimensionless “interval” between the templates.

Spherical ansatz for parameter-space metrics · The mismatch and the metric approximation to the mismatch · 0.87
Optimal template placement requires a metric for creating a grid of waveforms in the desired region of parameter space [ref] , however no analytic metric exists for the EOBNRv2 waveform.

Template banks to search for low-mass binary black holes in advanced gravitational-wave detectors · Introduction · 0.80

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] [20]

This (N + 1)-dimensional metric γαβ is related to the Fisher information matrix, which was used by Dhurandhar and Sathyaprakash in the appendix of R ef. [13] to compute the template spacing analytically in the small-spacing limit. However, in taking the absolute value to maximize over ∆Φ 0, we have lost some dependence on the dynamical (intrinsic) templat...

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[2] [19]

S. V. Dhurandhar and S. D. Mohanty, in preparation. Also L. S. Finn, in preparation

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[3] [1]

K. S. Thorne, in 300 Years of Gravitation, ed. S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1987), p. 330

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[4] [2]

E. S. Phinney, Astrophys. J. 380, L17 (1991)

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[5] [3]

R. Narayan, T. Piran, and A. Shemi, Astrophys. J. 379, L17 (1991)

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[6] [4]

For example, A. V. Tutukov and L. R. Yungelson, Mon. Not. R oy. Astron. Soc. 260, 675 (1993)

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A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. G¨ ursel , S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt , R. Weiss, S. E. Whitcomb, and M. E. Zucker, Science 256, 325 (1992)

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C. Bradaschia, E. Calloni, M. Cobal, R. Del Fasbro, A. Di V irgilio, A. Giazotto, L. E. Holloway, H. Kautzky, B. Michelozzi, V. Montelatici, D. P ascuello, and W. Velloso, in Gravitation 1990 , Proceedings of the Banﬀ Summer Institute, ed. R. Mann and P. Wesson (World Scientiﬁc, Singapore, 1991)

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[9] [7]

L. Blanchet and T. Damour, Phys. Rev. D 51, 5360 (1995)

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[10] [8]

C. M. Will and A. G. Wiseman, in preparation

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[11] [9]

L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. G. Wis eman, Phys. Rev. Lett. 74, 3515 (1995)

View on Pith 1995

[12] [10]

See, e.g., N. Wiener, The Extrapolation, Interpolation and Smoothing of Station ary Time Series with Engineering Applications (Wiley, New York, 1949) or L. A. Wainstein and V. D. Zubakov, Extraction of Signals from Noise (Prentice-Hall, London, 1962)

View on Pith 1949

[13] [11]

T. A. Apostolatos, Phys. Rev. D 52, 605 (1995)

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[14] [12]

B. S. Sathyaprakash and S. V. Dhurandhar, Phys. Rev. D 44, 3819 (1991). 29

View on Pith 1991

[15] [13]

S. V. Dhurandhar and B. S. Sathyaprakash, Phys. Rev. D 49, 1707 (1994)

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[16] [14]

B. S. Sathyaprakash, Phys. Rev. D 50, R7111 (1994)

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[17] [15]

C. Cutler and E. E. Flanagan, Phys. Rev. D. 49, 2658 (1994)

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B. F. Schutz, in The Detection of Gravitational Radiation , ed. D. Blair (Cambridge University Press, Cambridge, England,1989)

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[19] [17]

E. Poisson and C. M. Will, Phys. Rev. D 52, 848 (1995)

View on Pith 1995

[20] [18]

Note that the Φ 0 used here is the coalescence phase of the waveform. In the restricted post-Newtonian approximation, this is twice the coalescen ce phase of the orbitφ0, which is used by Cutler and Flanagan in Ref. [15]

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[21] [21]

P. C. Peters, Phys. Rev. 136, B1224 (1964)

View on Pith 1964

[22] [22]

The ψj given here are diﬀerent from those given in Eq. (13a-3) of Ref . [14] because Sathyaprakash does not use t0 and Φ 0 as kinematical parameters, but rather the time and phase at which the quadrupole part of the waveform reache s a frequency of f0

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[23] [23]

S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (Wiley, New York, 1983). 30

View on Pith 1983

[24] [24]

J. H. Taylor, Rev. Mod. Phys. 66, 711 (1994)

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[25] [25]

Numerical integrations were performed with the aid of M athematica [S. Wolfram, Math- ematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, California, 1988)] and with a Monte Carlo code based on Numerical Recipes [W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanner y, Numerical Recipes: The Art of Scientiﬁc Comp...

View on Pith 1988

[26] [26]

The storage size of the template is equal to the eﬀective sampling frequency 2fu times the duration of the template waveform. In the restricted pos t-Newtonian approximation, the duration of a binary chirp waveform template is equal to t he time it takes for the gravitational wave frequency to sweep from the seismic freq uency fs up to inﬁnity. Since τ1 ∝...

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[27] [27]

R. Balasubramanian, B. S. Sathyaprakash, and S. V. Dhur andhar, Gravitational waves from coalescing binaries: detection strategies and Monte C arlo estimation of parameters , submitted to Phys. Rev. D (Los Alamos xxx.lanl.gov archive p reprint gr-qc/9508011). 31 FIGURES FIG. 1. A schematic depiction of the manifold formed by the co ntinuous template famil...

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