Minimal match and SNR loss from imperfect templates

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We quantify the SNR and volume loss caused by an imperfect gravitational-wave template in a deliberately restricted setting. Owen supplies the matched-filter geometry. The bank growth is inspired by Cokelaer, but our implementation replaces his explicit connector graph and adds its own boundary repair; it is therefore not a faithful reproduction of his placement algorithm.

1. Scope and definitions

This is a controlled bank study, not a reproduction of an observational search. It uses one detector, stationary Gaussian noise, the aLIGOZeroDetHighPower PSD from 20 to 1024 Hz, and non-spinning TaylorF2 waveforms terminated at fISCO. Component masses run from 5 to 50 solar masses. Results are separated into a main region with total mass at most 35 solar masses and an extended region above it, where an inspiral-only model visibly fails. Real searches additionally include spin dimensions, coincidence and rejection of non-Gaussian transients.

For normalized waveforms, the match maximizes the overlap over coalescence time and phase, and the mismatch is

μ = 1 − ℳ.

In noise-free data the recovered SNR fraction is exactly ℳ. A nominal minimal match MM bounds the design mismatch only if the bank actually covers the region. For sources uniform in Euclidean volume, the exact nominal worst-case loss at MM = 0.9700 is 1 − MM3 = 0.08733. The population averages below instead use injections uniform in component mass. They are not an astrophysical population model.

The symmetric mass ratio is

η = m1m2/(m1+m2)2 ≤ 1/4.

The upper bound follows from (m1−m2)2 ≥ 0; equality holds only for equal masses. Thus η = 1/4 is the physical equal-mass boundary, whereas μ denotes mismatch.

2. Method: Owen baseline and our placement

The metric we use

Following Owen, mismatch is expanded locally as μ ≈ gijΔλiΔλj. The implementation uses the analytic non-spinning 3.5PN TaylorF2 projected-phase metric in chirp-time coordinates (τ0,τ3) at 20 Hz. Constant phase and time are projected out; the frequency weight is proportional to |h(f)|2/Sn(f). PyCBC supplies the waveform and PSD. Derivatives use the base waveform's fixed TaylorF2 support ending at its fISCO.

A finite-difference calculation tests the quadratic predictor against a pre-registered relative-error limit of 0.10000. It passes at mismatch 0.03 but not at 0.05; only those discrete radii were tested. Results below MM = 0.97 are therefore metric extrapolations.

What comes from Cokelaer, and what we changed

Owen supplies the metric and target spacing, but not the complete finite-region growth rule used here. Square and hexagonal banks are grown generation by generation, with the local metric recomputed at each parent, following the idea of Cokelaer's cell-by-cell two-dimensional inspiral placement. Cells that cross η = 1/4 are projected back only after reproduction, also following Cokelaer's prescription.

Two rules are ours:

Consequently, both the lattice-only and repaired template counts are properties of this implementation. They must not be described as Cokelaer counts.

3. Results

Waveform and discretisation losses

In the declared interpolation model, the TaylorF2 family plus bank loses 0.19320 of the volume in the main region, compared with 0.17580 for the interpolated family-only estimate. In the extended region the corresponding values are 0.51486 and 0.50779. The exact continuous-family loss would bound the discrete-bank loss from below, but the 28-point interpolation has no demonstrated one-sided error. Within this model, waveform-family error dominates bank discretisation at high mass.

nominal MM = 0.97main hex.main square ext. hex.ext. square
templates1772 2273 336 488
sampled covering fraction 1.00000 1.00000 0.99933 1.00000
population volume loss 0.02088 0.01534 0.01285 0.00945

Uniform injections do not certify strict covering. Adaptive optimisation reaches mismatch 0.0322 in the main hexagonal bank and 0.0362 in the extended hexagonal bank, both above the target. These are counterexamples to strict covering, not certified global maxima. The square banks survive the same search.

d_counts_and_ratio
Top: bank size against nominal minimal match, both regions and both lattices. Bottom: the measured hexagonal/square count ratio against the ideal constant-metric value 0.7698 and all ratios recomputed from the banks in Cokelaer 2007 Tables I-II: range 0.6605-0.7866, mean 0.7191. The band is the full published spread, not an uncertainty interval.

HOW THIS COULD FAIL (F4): if lattice efficiency alone set the ratio, both regions would sit on one horizontal line near 0.77. They do not: main stays at or above the ideal (0.7744-0.8090) while extended reaches 0.639. The changes with region, minimal match and boundary treatment show that the ratio is not universal here. The counts include this project's boundary repair; the comparison therefore does not test a faithful implementation of Cokelaer's algorithm.

The measured hexagonal-to-square ratios at the headline minimal match are 0.7796 in the main region and 0.6885 in the extended region. The constant-metric, boundary-free value is 0.7698. Across the comparable banks in Cokelaer's Tables I–II, the recomputed range is 0.6605–0.7866 and the mean is 0.7191. That range is descriptive, not an uncertainty interval. Because our collision and boundary rules differ, this comparison neither validates nor refutes Cokelaer's implementation.

Boundary sensitivity is expected: 0.79900 of main-region injections and 0.93833 of extended-region injections lie within one covering radius of an edge. The count ratio consequently varies with region, minimal match and boundary treatment; it is not a universal lattice constant here.

f_volume_losses
The three volume losses kept apart, against nominal minimal match: the exact worst case 1 - MM^3, its linear expansion 3(1 - MM), and the metric-predicted population average 1 - <M^3> from injections, evaluated with the quadratic metric. TaylorF2 injections, so this is modelled discretisation loss alone. Error bars show the injection bootstrap; the inset resolves the relative excess of the linear expansion over the exact curve.

HOW THIS COULD FAIL (F4): if the worst case described what this bank model predicts for the population, the population points would lie on the black curve. They lie a factor 4 to 9 below it, and the gap widens with MM. If sampling error were large enough to explain their separation, the bootstrap intervals would overlap accordingly; they do not. The linear expansion sits 3.060909 % above the exact curve at MM = 0.97 -- where the exact value is 0.087327, not the 0.08733 quoted earlier -- which is why the two are never merged.

The population-average losses lie below the exact nominal worst case because typical injections do not occupy a cell's worst point. This comparison applies only to the stated uniform component-mass measure.

Supplementary covering diagnostics

These figures are retained online but omitted from the page-limited PDF. In the cumulative plot, every interior/border curve is normalized within its own subset; the ordinate is conditional and is not a fraction of all injections.

d_covering_cumulative
Cumulative distribution of the best match each injection finds in the bank, at nominal MM = 0.97, interior and border injections apart. Each curve is normalised within its own subset, so its ordinate is conditional rather than a fraction of all injections. 3000 injections per region, uniform in component mass, seed 20260925; the match is the quadratic form evaluated with the injection's own metric.

HOW THIS COULD FAIL (F4): any nonzero CDF ordinate left of the MM = 0.97 line refutes sampled covering. The extended hexagonal curve does cross it -- 2 injections in 3000 -- and that is the measured covering failure, not a plotting artefact. A sampled bank that covered would have no step left of the line. The log ordinate is what makes a 1-in-1000 failure visible at all; on a linear scale all four curves would look identical.

d_mismatch_map_main_hex
Quadratic-metric mismatch residual mu - 0.03 to the nearest template over the mass plane for the main-region hexagonal bank at MM = 0.97, on a 90x90 grid restricted to the region. Shown against both masses rather than against total mass, because the metric depends on both (rule F3).

HOW THIS COULD FAIL (F4): a sampled bank that covered would have no positive residual. On this grid none appears -- the grid's worst cell is 0.02893 -- yet the cross marks a point an adaptive search found at 0.03222, reproduced independently by the dverify r03 audit. That is the figure's real content: a regular grid of test points does not find the holes, and a figure built only from grid sampling would wrongly show a covering bank.

Threshold and effective volume

Counting every template and time sample as independent is a conservative upper bound for the finite correlated Gaussian grid, but it is loose. Fitting simulated-noise maxima gives 4.063 × 105 equivalent independent trials per second for the main hexagonal MM = 0.97 bank.

Within each fixed region and lattice, fits of Neff ∝ Ntemplatesα give α = 0.1568, 0.1500, 0.1516 and 0.2214. Pooling unlike families gives 1.2922 instead, demonstrating that bank size by itself does not define the trials factor.

e_trials_measured_vs_naive
Left: the effective number of independent trials per segment, measured by filtering simulated Gaussian noise through each bank and fitting the distribution of the maximum of |z| over templates and time (100 segments of 32 s, seed 20260926, the same noise for every bank), against the naive bound that counts every template and every sample as independent. Right: their ratio. Error bars show the stored per-bank fit uncertainty; legend intervals are paired-bootstrap 95 % intervals for each within-family exponent.

THIS FIGURE REPLACES an earlier one that plotted the threshold against minimal match. That figure broke rule F1 -- rho* is a closed function of the template count, so its y-values were computed from its x-values by the formula under test, and a blind reviewer reconstructed every plotted point from N alone to within 1e-13. It was decoration. HOW THIS COULD FAIL (F4): if the templates of a bank were independent, the coloured points would lie on the dashed line and the right-hand ratio would be 1. They do not and it is not: the ratio runs from about 7 to 58. If instead the measured trials grew in proportion to the template count, each family would run parallel to the dashed line; they are far flatter, which is the exponent the text reports. WHAT IT DOES NOT SHOW: the calibration is done where the maxima fall, around |z| = 5.3 to 5.8, and is applied at thresholds near 8.0 to 8.6. That extrapolation of about two in |z| is not tested here, and the trials count is not constant across the probed tail.

Calibration limit. The fitted effective count varies across the observed tail and is calibrated where the simulated maxima occur, then extrapolated to a substantially higher detection threshold. It is an empirical model, not a measured false-alarm rate at the operating point.

For the relative sensitivity proxy Veff = ⟨ℳ3⟩/ρ*3, the main hexagonal value changes by -1.87 % under naive counting and by 1.76 % under the fitted trials model. Even the sign therefore depends on the trials model.

f_veff_vs_mm
Change in the derived relative sensitivity proxy from the coarse end of the grid, under two scenarios for the trials factor. Dashed with open markers: the naive bound N_templates x f_sample x T_obs. Solid with filled markers: nu_eff calibrated on simulated Gaussian noise (100 segments of 32 s, seed 20260926, the same noise for every bank). Curves are shown without precision-style uncertainty bars because uncertainty and tail extrapolation in nu_eff have not been propagated; the stored injection-bootstrap error alone would be misleadingly incomplete. Shaded: below MM = 0.97 the quadratic predictor is outside the range q1bC validated.

THE RESULT IS THE DIFFERENCE BETWEEN THE TWO MODELS. The naive bound makes V_eff fall by 1.9-3.7 % across the grid; the calibrated one makes it rise by 0.2-1.8 %. The sign of the trend therefore depends on how the trials factor is modelled. This comparison does not establish that the fitted model is valid at the detection threshold. WHAT IT DOES NOT SHOW: nu_eff is calibrated near the simulated maxima and extrapolated to a higher detection threshold. It varies across the observed tail, and that calibration uncertainty is unquantified here; the curves are model scenarios, not a measurement of the trend at the operating threshold. HOW THIS COULD FAIL (F4): if the two models agreed, the dashed and solid curves would lie on top of each other and the choice would not matter. They do not. If instead the calibrated curves showed a clear interior peak, the pre-registered hypothesis would be supported; they are flat to within about 2 % and NOT monotonic -- extended hexagonal wobbles inside the grid -- so no robust optimum is resolved inside the grid.

The calibrated curves are nearly flat and do not resolve a robust optimum. They are shown without precision-style uncertainty bars because uncertainty in the effective-trials calibration and its tail extrapolation has not been quantified; an injection-only bar would be misleadingly incomplete.

4. Limitations

5. Conclusion: answers to the two questions

How much SNR is lost?

If the nearest template has match ℳ, then in noise-free data the recovered SNR is ρrec = ℳρopt. The fractional SNR loss is therefore exactly the mismatch, μ = 1 − ℳ. In the banks we actually built at nominal MM = 0.97, adaptive searches found losses of at least 0.0322 in the main hexagonal bank and 0.0362 in the extended hexagonal bank, both worse than the nominal 0.03. The square banks survived the same search, although that is not a proof of strict covering.

The metric-predicted population-average volume losses were 0.02088 (hexagonal) versus 0.01534 (square) in the main region, and 0.01285 versus 0.00945 in the extended region. Against IMRPhenomD signals, the approximate total loss for the hexagonal bank instead rises from 0.19320 in main to 0.51486 in extended; most of that is already present in the continuous TaylorF2 family (0.17580 and 0.50779). Thus the main/extended contrast is mainly a waveform-model result, not a bank-spacing result.

How dense must the bank be?

At MM = 0.97, our main bank contains 1772 hexagonal templates versus 2273 square templates; the extended bank contains 336 versus 488. The corresponding hexagonal-to-square count ratios are 0.7796 and 0.6885: hexagonal placement uses fewer templates in both regions, but by different amounts.

At fixed nominal minimal match, the delivered square banks trade a larger template count for lower population-average discretisation loss and survived the adaptive covering search. The hexagonal banks use fewer templates, but the implementations tested here contain small covering holes. Densifying the hexagonal placement reduces its loss, so these results do not establish that one lattice is universally better; the preferred choice depends on whether the objective is template count, average loss, strict coverage, or effective detection volume.

This comparison is especially sensitive to finite boundaries: 0.79900 of main and 0.93833 of extended injections lie within one covering radius of an edge. The square banks surviving our search is not a proof of strict coverage, and no robust optimal minimal match is resolved because the small Veff trend changes sign with the trials model.

Reproducibility

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References

  1. B. J. Owen, Search templates for gravitational waves from inspiraling binaries: choice of template spacing.
  2. B. J. Owen and B. S. Sathyaprakash, Matched filtering of gravitational waves from inspiraling compact binaries.
  3. T. Cokelaer, Gravitational waves from inspiralling compact binaries: hexagonal template placement.
  4. B. Allen et al., FINDCHIRP.
  5. S. A. Usman et al., The PyCBC search for gravitational waves from compact binary coalescence.
  6. J. Roulet et al., A general geometric placement algorithm.
  7. R. P. Croce et al., The minimal-match issue revisited.
  8. D. Keppel, The balancing act of template bank construction.
  9. S. Sakon et al., Template bank for the fourth observing run.
  10. R. Latała and D. Matlak, Royen's proof of the Gaussian correlation inequality.

Companion note: a guide to Owen's metric paper. PDF version: paper/main.pdf.