The deepest all-sky searches for continuous gravitational waves (CWs) — nearly monochromatic, years-long signals from spinning non-axisymmetric neutron stars — are hierarchical pipelines whose sensitivity is characterized empirically through costly injection-and-recovery Monte Carlo simulations. The most sensitive such search to date, on LIGO O3 public data (Steltner et al. 2023; all-sky, 20-800 Hz), acknowledged three unresolved methodological issues: the parameter-uncertainty regions of its coherent follow-up stages grew unexpectedly (by 10-30%) and the origin is not understood; it could not be ruled out that a different hierarchical setup achieves greater sensitivity at the same computational budget; and no analytical model predicts the pipeline's sensitivity to the few-percent accuracy needed to replace Monte Carlo characterization, chiefly because of nonlinear candidate-selection steps such as clustering. Successor work by the same group (a Bayesian follow-up of the O3 candidates; 'bucket' and high-frequency searches; a deliberately conservative simplified efficiency estimate) has improved follow-up efficiency and explored the depth-versus-breadth trade, but none of the three issues is resolved: the uncertainty anomaly remains unexplained, no optimality statement at fixed budget exists, and the most recent search's estimate still erred by about 16% precisely where clustering matters. This problem asks for a predictive quantitative theory of such pipelines: a few-percent-accurate sensitivity model including clustering, an explanation of the follow-up uncertainty scaling, and thereby a determination of setup optimality at fixed computational budget.

Background

Continuous gravitational waves (CWs) are the years-long, nearly monochromatic gravitational signals expected from rapidly rotating neutron stars with a sustained mass or current quadrupole (parameterized by an intrinsic strain amplitude h0h_0, often translated into an equatorial ellipticity). Because the signal at any instant lies far below the detector noise floor, detectability comes from long integration; the phase evolves mainly as Φ(τ)=Φ0+2π[f(τ−τ0)+12f˙(τ−τ0)2]\Phi(\tau)=\Phi_0+2\pi[f(\tau-\tau_0)+\frac{1}{2}\dot f(\tau-\tau_0)^2] in barycentric time, with the spin-down f˙\dot f slowly decreasing the frequency. Searches that do not know the source position, frequency, or spin-down face a parameter space so vast that fully coherent matched filtering is computationally impossible; the standard remedy is a semicoherent hierarchical search. In a semicoherent stage the data (conventionally stored as short Fourier transforms of the calibrated strain) are divided into NsegN_{\rm seg} segments of coherence time TcohT_{\rm coh}, the F\mathcal{F}-statistic — a matched-filter statistic that analytically maximizes the likelihood ratio over the signal's four amplitude parameters — is computed coherently per segment and averaged, and template grids are spaced so that the average mismatch (the fractional signal-to-noise loss when a signal falls between grid points) stays within a chosen budget. Surviving candidates are then followed up by a hierarchy of stages with longer coherence times and finer grids.

The Einstein@Home (E@H) volunteer-computing project supplies the computing power for the deepest such searches. The search studied here (Steltner et al. 2023) targeted the whole sky, 20-800 Hz, and spin-downs down to −2.6×10−9-2.6\times10^{-9} Hz/s in LIGO O3 public data (the O3a and O3b sub-runs): its first stage alone surveyed about 6.7×10186.7\times10^{18} templates. Results are ranked with the line- and transient-line-robust statistic β^S/GLtL\hat\beta_{S/GLtL}, a likelihood-ratio statistic testing the signal hypothesis against an expanded noise hypothesis (Gaussian noise, stationary spectral lines, or transient lines), and nearby high-statistic results are grouped by a clustering procedure into single candidates, each represented by a seed point and a parameter uncertainty region. Both the uncertainty regions and the detection efficiency are measured empirically: fake signals from a reference population are injected into the data and recovered by the full pipeline (injection-and-recovery Monte Carlo). Sensitivity is quoted as the sensitivity depth D=S(f)/h0\mathcal{D}=\sqrt{S(f)}/h_0 — the detector noise amplitude spectral density divided by the signal amplitude — at which the pipeline achieves 90% detection efficiency; the deep search aimed at D≈56 Hz−1/2\mathcal{D}\approx56~\mathrm{Hz}^{-1/2}, and the leading all-sky methods on the same data differ by only a few percent in achieved depth, which sets the accuracy any predictive model must reach to be useful for setup selection. The search found no astrophysical signal: the six surviving candidates were all hardware injections (fake signals intentionally added to the detector data for validation purposes), and the most stringent upper limit was h0=8.1×10−26h_0=8.1\times10^{-26} (90% confidence) at 203 Hz.

The pipeline's sensitivity could in principle be predicted rather than simulated. The closest existing tool (Dreissigacker, Prix and Wette 2018) estimates the sensitivity of F\mathcal{F}-statistic-based searches — single- and multi-stage semicoherent searches, Hough-transform searches (which count thresholded per-segment statistic peaks across the data), and Bayesian upper limits — and agrees with Monte Carlo to a few percent at high detection thresholds, but only for idealized pipelines in Gaussian noise: it does not describe clustering, thresholds on the line-robust statistic, vetoes, or the non-Gaussian disturbances of real data. This is why the source paper leaves three issues open. First, the coherent follow-up stages on O3a and O3b (Stages 7 and 8) produced uncertainty regions 10-30% larger than the preceding semicoherent Stage 6 — an effect verified in several ways but not understood — which forced the last follow-ups to be centered and sized on the Stage 6 regions. Second, the setup was selected among the configurations that injection-and-recovery Monte Carlo could afford to explore, so it cannot be claimed that no more sensitive setup exists at the same computational budget. Third, a few-percent-accurate analytical sensitivity model of the full pipeline remains an open technical challenge because of the nonlinear candidate-selection steps.

Problem Statement

Replace the empirical, Monte-Carlo-based characterization of deep hierarchical all-sky continuous-wave searches with predictive quantitative understanding, for the pipeline configuration of Steltner et al. (2023) and its successors. With respect to the published O3 search, its reference signal populations, and its detection criteria, this problem asks for:

(1) A sensitivity model: an analytical or semi-analytical model that predicts each stage's detection efficiency as a function of sensitivity depth — including the nonlinear candidate-selection steps (clustering, thresholds on the line-robust statistic, vetoes) — to the few-percent accuracy at which competing all-sky methods on the same data differ, so that search setups could be selected and tuned without exhaustive injection-and-recovery Monte Carlo.

(2) An explanation of the follow-up uncertainty scaling: a quantitative account of why the parameter-uncertainty regions of the coherent Stages 7 and 8 are 10-30% larger than at the preceding Stage 6, consistent with the verification checks already reported in the source paper, and more generally a prediction of how follow-up uncertainty regions scale with coherence time and data span.

(3) A determination of setup optimality: whether, at the computational budget of the published search, a more sensitive hierarchical setup exists over the searched parameter space (all sky, 20-800 Hz, spin-down down to −2.6×10−9-2.6\times10^{-9} Hz/s).

These three results jointly serve one objective: a predictive theory of the pipeline's statistical behavior accurate enough to design, tune, and certify the deepest all-sky continuous-wave searches without brute-force simulation.

References

  1. B. Steltner, M. A. Papa, H.-B. Eggenstein, R. Prix, M. Bensch, B. Allen, B. Machenschalk, 'Deep Einstein@Home All-sky Search for Continuous Gravitational Waves in LIGO O3 Public Data', The Astrophysical Journal 952, 55 (2023), DOI 10.3847/1538-4357/acdad4, arXiv:2303.04109, https://arxiv.org/abs/2303.04109
  2. C. Dreissigacker, R. Prix, K. Wette, 'Fast and accurate sensitivity estimation for continuous-gravitational-wave searches', Physical Review D 98, 084058 (2018), DOI 10.1103/PhysRevD.98.084058, arXiv:1808.02459, https://arxiv.org/abs/1808.02459
  3. J. Martins, M. A. Papa, B. Steltner, R. Prix, P. B. Covas, 'Bayesian framework to follow-up continuous gravitational wave candidates from deep surveys', Physical Review D 113, 023041 (2026), DOI 10.1103/rbyp-157m, arXiv:2508.18204, https://arxiv.org/abs/2508.18204
  4. B. McGloughlin, J. Martins, B. Steltner, M. A. Papa, H.-B. Eggenstein, B. Machenschalk, R. Prix, M. Bensch, 'Einstein@Home All-sky Bucket Search for Continuous Gravitational Waves in LIGO O3a Public Data', The Astrophysical Journal 997, 149 (2026), DOI 10.3847/1538-4357/ae225a, arXiv:2508.16423, https://arxiv.org/abs/2508.16423
  5. B. McGloughlin, B. Steltner, J. Martins, M. A. Papa, H.-B. Eggenstein, J. Ming, B. Machenschalk, R. Prix, M. Bensch, 'High-frequency Continuous Gravitational Wave Search in LIGO O3 Public Data with Einstein@Home', The Astrophysical Journal 998, 205 (2026), DOI 10.3847/1538-4357/ae2ed8, arXiv:2508.20073, https://arxiv.org/abs/2508.20073
  6. L. Mirasola, R. Tenorio, 'Toward a computationally efficient follow-up pipeline for blind continuous gravitational-wave searches', Physical Review D 110, 124049 (2024), DOI 10.1103/PhysRevD.110.124049, arXiv:2405.18934, https://arxiv.org/abs/2405.18934
  7. A. Tripathee, K. Riles, 'Probing More Deeply in an All-Sky Search for Continuous Gravitational Waves in the LIGO O3 Data Set', Physical Review D 109, 043049 (2024), DOI 10.1103/PhysRevD.109.043049, arXiv:2311.04985, https://arxiv.org/abs/2311.04985