Targeted Continuous-Wave Searches

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A targeted continuous-wave search uses an independently known sky position and timing solution to track a nearly periodic gravitational-wave phase for months or years. The gain comes from coherent integration: signal amplitude accumulates linearly while stationary noise grows only as the square root of observing time. The price is exact bookkeeping of detector motion, spin evolution, binary delays, clock standards, timing noise, and glitches.

Helpful background. Continuous-Wave Phase Models develops the radiation-side phase conventions used here. Antenna Patterns and Polarization Response supplies the time-dependent projection into each interferometer.

From electromagnetic timing to detector phase

Let $t$ be detector time and $\boldsymbol n$ the unit vector from the Solar System barycenter toward the source. A simplified barycentric arrival time is

t_{\rm b} = t+\frac{\boldsymbol r(t)\cdot\boldsymbol n}{c} +\Delta_{\rm E}(t)+\Delta_{\rm S}(t)+\Delta_{\rm bin}(t),

where the terms denote the Roemer, Einstein, Shapiro, and—when relevant—binary delays in a stated time standard and ephemeris. The phase of one emission harmonic is expanded about $t_{\rm ref}$ :

\Phi(t) = \phi_0 +2\pi \sum_{k=0}^{s} \frac{f^{(k)}_{\rm ref}}{(k+1)!} \left[t_{\rm b}(t)-t_{\rm ref}\right]^{k+1}.

For the conventional quadrupolar mountain model, the dominant gravitational-wave frequency is near twice the stellar rotation frequency, but other mechanisms can radiate at different harmonics. An electromagnetic ephemeris therefore supplies a phase model only after the emission hypothesis and any allowed phase offset are declared.

The strain in detector $I$ can be written

h_I(t) = h_0\!\left[ F_I^+(t;\psi)\frac{1+\cos^2\iota}{2}\cos\Phi(t) +F_I^\times(t;\psi)\cos\iota\sin\Phi(t) \right].

Daily antenna modulation and annual Doppler modulation help distinguish a sky-coherent signal from a stationary instrumental line. They also correlate $h_0$ , inclination $\iota$ , and polarization angle $\psi$ , so an upper limit depends on the adopted orientation prior.

Coherent statistics

With Gaussian noise and phase parameters fixed, write the signal as $h=\sum_{\mu=1}^4\mathcal A^\mu h_\mu$ , using sine and cosine quadratures of the two antenna responses. Define $x_\mu=(d|h_\mu)$ and $\mathcal M_{\mu\nu}=(h_\mu|h_\nu)$ . Maximizing the quadratic log likelihood over the four real coefficients gives

\widehat{\mathcal A}^{\mu} =(\mathcal M^{-1})^{\mu\nu}x_\nu, \qquad 2\mathcal F=x_\mu(\mathcal M^{-1})^{\mu\nu}x_\nu.

For a full-rank amplitude basis and known Gaussian-noise covariance, $2\mathcal F$ at one fixed phase template has a central $\chi^2_4$ distribution in noise and a noncentral $\chi^2_4(\rho^2)$ distribution in a matched signal. Maximizing over many phase templates changes the false-alarm distribution Jaranowski, Królak, and Schutz 1998, §§3–4. Heterodyne pipelines instead remove a trial phase,

B(t)=d(t)e^{-i\Phi_{\rm trial}(t)},

then low-pass filter and analyze the slowly varying complex amplitude. This isolates one quadrature of the real waveform; its factor-of-two amplitude normalization must be retained. If antenna and noise weights are approximately constant and the phase error is $\delta\Phi(t)$ , the fractional loss of squared coherent signal-to-noise ratio is

\mu \equiv 1-\left\lvert \frac{1}{T}\int_0^T e^{i\delta\Phi(t)}dt \right\rvert^2 .

For a constant frequency error $\delta f$ ,

1-\mu = \operatorname{sinc}^2(\pi\delta fT),

so the tolerable frequency spacing scales as $T^{-1}$ . A glitch or unmodeled timing-noise excursion can destroy coherence even when the average frequency remains accurate.

Ephemeris uncertainty and model extensions

A “targeted” search need not be one template. Radio or gamma-ray timing supplies a covariance for sky position, frequency derivatives, orbital parameters, and glitch epochs. If the gravitational-wave phase may wander relative to the electromagnetic pulse, one can marginalize a phase offset, divide the observation into coherent segments, or use a hidden-state model. Each relaxation reduces sensitivity and enlarges trials, but is preferable to claiming full coherence that the ephemeris does not support.

Binary sources require the correct orbital convention, eccentricity terms, and time system. Proper motion and parallax matter over long baselines. The reference epoch, observatory clock corrections, Solar System ephemeris, and data gaps must be versioned so another implementation can reproduce the phase to much better than one radian.

Computational and statistical checks

A reproducible targeted analysis should publish or record:

the observing spans, segment list, detector duty factors, calibration, and data-quality vetoes;
the ephemeris, covariance, time standard, Solar System ephemeris, binary model, and glitch treatment;
the statistic, amplitude/orientation priors, line model, and trials;
software version, floating-point precision, frequency grid, and stopping tolerances;
hardware or software injection recovery and off-source or sky-shift null tests.

Convergence is tested by halving the sampling interval or frequency spacing, extending the ephemeris uncertainty grid, and changing coherent segmentation until the statistic and reported upper limit move by less than a declared tolerance. A synthetic exact-phase tone should reproduce the expected $\sqrt{T}$ signal-to-noise scaling; a deliberately offset tone should follow the sinc mismatch above. Lines that do not share the predicted Doppler and antenna modulation should fail detector-coherence and sky-shift tests.

Observational status through 2026-08-18

The official O4a search for known pulsars reports no credible continuous-wave detection and provides source- and model-dependent strain or ellipticity upper limits LVK 2025, O4a known-pulsar search. Their interpretation depends on construction: a Bayesian 95% upper limit encloses 95% of the posterior amplitude probability under the stated priors; a frequentist confidence procedure is defined by repeated-experiment coverage; an injection-based sensitivity reports a recovery fraction for the specified signal population and threshold. These quantities need not coincide. None establishes a subthreshold signal or constrains every emission mechanism.

The result should be refreshed when the official observing-run data set, calibration, timing ephemerides, or search release changes. Mutable headline numbers are deliberately not embedded here; the cited release is the authoritative snapshot.

Exercises

A phase model has a constant frequency error $\delta f=10^{-8}\ {\rm Hz}$ over $T=100$ days. Find the accumulated phase error at the end of the run.

Solution

With $T=8.64\times10^6\ {\rm s}$ ,

\delta\Phi(T) = 2\pi\delta fT = 2\pi(10^{-8})(8.64\times10^6) \simeq0.543\ {\rm rad}.

The mismatch depends on the whole phase trajectory, not just its final value, but this already shows why extremely small frequency errors matter over long baselines.

A sky-position error $\delta\theta$ gives a maximum barycentric timing error of roughly $(1\,{\rm AU}/c)\delta\theta$ . Estimate the phase amplitude for $f=500\ {\rm Hz}$ and $\delta\theta=1$ milliarcsecond.

Solution

Use $1\,{\rm AU}/c\simeq499\ {\rm s}$ and $1$ milliarcsecond $=4.848\times10^{-9}$ rad:

\delta t_{\max} \simeq 499(4.848\times10^{-9})\ {\rm s} \simeq2.42\times10^{-6}\ {\rm s}.

Thus

\delta\Phi_{\max} \simeq 2\pi f\delta t_{\max} \simeq7.6\times10^{-3}\ {\rm rad}.

This isolated error is small, but proper-motion growth and correlated timing uncertainties still require propagation over a multi-year search.

References

Jaranowski, P., Królak, A., and Schutz, B. F. (1998). “Data analysis of gravitational-wave signals from spinning neutron stars: The signal and its detection.” Physical Review D 58, 063001. DOI.
LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2025). “Search for continuous gravitational waves from known pulsars in the first part of the fourth LIGO–Virgo–KAGRA observing run.” arXiv:2501.01495.