Burst, Continuous-Wave, and Stochastic Signal Models

Not every gravitational-wave signal is a compact-binary chirp. Short transients may support only a controlled time-frequency morphology; core collapse maps matter stages into simulation-conditional features; continuous waves demand long-term phase bookkeeping; and stochastic radiation is described by spectra, sky covariance, and polarization statistics. This chapter builds those physical signal models while keeping source microphysics, numerical production, detector searches, upper limits, and current detections in their proper neighboring subjects.

Helpful background. Einstein Equations supplies radiative gravity. Waves, Green Functions, and Geometric Optics supplies Fourier localization and random fields. Gravitational Radiation and Waveforms supplies strain, flux, polarization, and frame conventions. Relativistic Matter and Compact Objects supplies transient and rotating-star microphysics. Relativistic Cosmology supplies redshift, transfer, and critical density.

Enter by signal duration and predictability

The first comparison axes are coherence time and model strength. A burst can be localized without a phase-accurate source template. A continuous wave can be weak in amplitude yet demand sub-cycle phase prediction over months or years. A stochastic background abandons realization-by-realization phase and instead predicts a covariance. These are different mathematical objects, not merely different frequency bands.

The second axes are source multiplicity and stationarity. One transient has a waveform; many unresolved transients can approach a stationary Gaussian background when their duty cycle is high. A low-duty-cycle population can instead retain intermittent higher-order statistics even when its two-point spectrum is smooth. Long-lived deterministic sources can overlap into an apparently stochastic field. The signal category must therefore state which information has been averaged away.

Shared Fourier and spectral dictionary

All pages use

$$h_A(t)=\int_{-\infty}^{\infty} \widetilde h_A(f)e^{-2\pi ift}\,df, \qquad h=h_+-ih_\times.$$

For a burst,

$$h_{\rm rss}^2=\int \left(h_+^2+h_\times^2\right)dt.$$

For a stationary field, positive-frequency one-sided spectra satisfy

$$\left\langle \widetilde h_A(f)\widetilde h_B(f')^* \right\rangle =\frac12\delta(f-f')\delta_{AB}S_A(f),$$

with total tensor field spectrum $S_h=S_++S_\times$. In the isotropic unpolarized convention,

$$\Omega_{\rm gw}(f) =\frac{2\pi^2}{3H_0^2}f^3S_h(f), \qquad h_c^2=fS_h.$$

These are field quantities. Antenna averages, overlap-function normalizations, channel noise, and likelihoods are added later. The stochastic convention is traced to Allen and Romano 1999, §§II–IV, PDF and Romano and Cornish 2017, §§2–4, PDF.

For a whole-sky background the polarization powers in this shorthand are integrated over independent propagation directions, with a local screen at each direction. The spectra page restores the sphere delta function and derives the factor five between the total field spectrum and the signal spectrum of an ideal right-angle interferometer. That distinction is essential before comparing any published amplitude curves.

Six pages, three modeling regimes

1. Unmodeled and Weakly Modeled Bursts defines duration, bandwidth, sine-Gaussian quality factor, polarization, $h_{\rm rss}$, and isotropic-equivalent energy without pretending that “unmodeled” means assumption free.
2. Core-Collapse Waveform Morphologies relates bounce, convection, SASI, proto-neutron-star modes, and explosion offsets to conditional simulations. It includes a pinned frequency-recovery benchmark and separates estimator convergence from simulation validity. The physical synthesis follows Abdikamalov, Pagliaroli, and Radice 2022, §§2–5, PDF.
3. Continuous-Wave Phase Models integrates intrinsic frequency derivatives and composes Solar-System and binary delays in declared time standards, following Riles 2023, §§2–3, PDF.
4. Stochastic Background Spectra derives the conversion among $S_h$, $h_c$, and $\Omega_{\rm gw}$ and explains why detector-normalized formulas carry different geometric factors.
5. Anisotropic and Polarized Backgrounds promotes intensity to a sky map, builds the Stokes coherency matrix, and expands linear polarization in spin-four harmonics.
6. Astrophysical and Cosmological Backgrounds compares population and early-Universe mechanisms through a dated comparison, counterexamples, and tests that could change the conclusion rather than a mutable empirical verdict. Primordial production and transfer are reviewed in Caprini and Figueroa 2018, §§2–3, PDF.

What remains robust under weaker modeling

For a burst, central time and frequency, effective duration and bandwidth, polarization covariance, and integrated strain can remain meaningful when exact phase does not. Radiated energy additionally requires distance, angular emission, frame, and frequency weighting. A sine-Gaussian is a benchmark atom, not a source identification.

For a continuous wave, phase derivatives are useful only with their epoch and time coordinate. The chain

$$t_{\rm det}\longrightarrow T_{\rm SSB} \longrightarrow\tau_{\rm source} \longrightarrow\Phi(\tau)$$

must record clock, ephemeris, sky motion, and binary-delay signs. A millisecond timing discrepancy at kilohertz frequency is already many radians.

For stochastic radiation, the two-point covariance survives averaging. Gaussianity, isotropy, stationarity, and absence of polarization are four independent assumptions. A spectrum alone cannot measure intermittency or prove a source origin.

Interpreting core-collapse predictions

Core-collapse morphology is especially vulnerable to overgeneralization. A published product must record progenitor, rotation law, gravity treatment, hydrodynamics, neutrino transport, equation of state, dimension, resolution, extraction, distance, and orientation. The reviewed $12M_\odot$ rotation sequence, for example, has reported bounce peaks near $800$, $750$, and $372\,\mathrm{Hz}$ as centrifugal support changes Abdikamalov, Pagliaroli, and Radice 2022, Fig. 8 and pp. 16–19, Open PDF; it is a conditional model family, not a universal rotation-frequency law.

Convergence should track phase/frequency, amplitude, and radiated energy separately. Turbulent realizations need ensemble or spectral comparisons, while axisymmetric bounce can admit cleaner waveform convergence. Independent codes and extraction methods are stronger evidence than another fit to the same product.

Background-origin logic

An astrophysical background integrates source energy over a redshift-dependent rate. A cosmological background evolves a production spectrum through an expansion transfer function. Within a finite band, both can share amplitude and slope. Distinguishing them can require spectral curvature, anisotropy, non-Gaussianity, polarization, cross-frequency coherence, and cross-correlation with matter—each with its own nuisance model.

The evidence cutoff on the frontier comparison is 2026-08-18. The page should be revisited after new mechanism calculations, materially revised population synthesis, official empirical results, or corrected transfer functions. Such changes update a conditional comparison; they do not relocate search pipelines or current results into this waveform chapter.

Chapter diagnostic

A signal has a narrow feature at source frequency $600\,\mathrm{Hz}$, a duration of $20\,\mathrm{ms}$ in the source frame, and comes from $z=2$. An unresolved population reproduces this feature over a broad redshift distribution. Find the observed central frequency and duration for the $z=2$ member. Explain why the population spectrum broadens, and state whether a smooth power law by itself distinguishes this population from a cosmological mechanism.

Solution

Cosmological redshift gives

$$f_{\rm det}=\frac{600\,\mathrm{Hz}}{1+2} =200\,\mathrm{Hz}, \qquad \Delta t_{\rm det}=(1+2)(20\,\mathrm{ms}) =60\,\mathrm{ms}.$$

Members at different redshifts map the same source-frame feature to different observed frequencies, so integrating the rate kernel smears it. The weights also include source-time dilation, comoving expansion, population evolution, and source energy.

A smooth power law is not a unique origin test. One flank of a cosmological peak or a broad string spectrum can mimic a population slope over a finite band. Spectral curvature over a wider band, angular clustering, duty-cycle-dependent non-Gaussianity, polarization, and cross-correlation with large-scale structure supply additional conditional discriminants.

Connections to other topics

This chapter defines physical signal and spectral models. Source microphysics is treated in Relativistic Matter and Compact Objects; background histories and primordial mechanisms in Relativistic Cosmology; production simulations in Computational Relativity; and searches, calibration, upper limits, and current detections in Gravitational-Wave Astronomy. The pages remain useful precisely because their stable definitions are separated from mutable observational status.

References

• Abdikamalov, Ernazar, Giulia Pagliaroli, and David Radice. “Gravitational Waves from Core-Collapse Supernovae.” In Handbook of Gravitational Wave Astronomy. Singapore: Springer, 2022. DOI. Open PDF.
• Allen, Bruce, and Joseph D. Romano. “Detecting a Stochastic Background of Gravitational Radiation: Signal Processing Strategies and Sensitivities.” Physical Review D 59 (1999): 102001. DOI. Open PDF.
• Caprini, Chiara, and Daniel G. Figueroa. “Cosmological Backgrounds of Gravitational Waves.” Classical and Quantum Gravity 35 (2018): 163001. DOI. Open PDF.
• Riles, Keith. “Searches for Continuous-Wave Gravitational Radiation.” Living Reviews in Relativity 26 (2023): 3. DOI. Open PDF.
• Romano, Joseph D., and Neil J. Cornish. “Detection Methods for Stochastic Gravitational-Wave Backgrounds: A Unified Treatment.” Living Reviews in Relativity 20 (2017): 2. DOI. Open PDF.