Unmodeled and Weakly Modeled Bursts

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A burst is a transient whose robust structure is weaker than a phase-accurate template. “Unmodeled” therefore means that the analysis does not assume a complete source waveform, not that the signal has no model at all. Duration, bandwidth, polarization, coherence, time-frequency track, and energy scaling can remain controlled. This page defines those field-side descriptions; search pipelines, upper limits, and current detections are not treated.

Helpful background. Einstein Equations supplies the radiative field equation. Waves, Green Functions, and Geometric Optics supplies Fourier localization and time-frequency resolution. Gravitational Radiation and Waveforms supplies strain, polarization, and flux. Relativistic Matter and Compact Objects supplies source microphysics. Relativistic Cosmology supplies the redshift map when the source is cosmological.

A controlled transient description

Write the radiation in a declared source or detector frame as

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

A compact transient is characterized first by a central time $t_0$ , an effective duration $\Delta t$ , a central frequency $f_0$ , and an effective bandwidth $\Delta f$ . Its time-frequency area cannot be made arbitrarily small: changing the window that defines $\Delta t$ and $\Delta f$ changes numerical factors but preserves the Fourier tradeoff. Additional robust descriptors include whether power follows a chirp-like ridge, a nearly horizontal band, a short impulse, repeated pulses, or a broadband stochastic patch.

Polarization is separate from morphology. One may specify two time series, an ellipticity relation, or only a coherency matrix. Assuming $h_\times=0$ because a sketch shows one curve silently imposes linear polarization.

The sine-Gaussian benchmark

A useful localized benchmark is

\begin{aligned} h_+(t)&=h_0e^{-(t-t_0)^2/(2\tau^2)} \cos\!\left[2\pi f_0(t-t_0)+\phi_0\right],\\ h_\times(t)&=\epsilon h_0e^{-(t-t_0)^2/(2\tau^2)} \sin\!\left[2\pi f_0(t-t_0)+\phi_0\right]. \end{aligned}

Here $\epsilon=0$ is linear and $\lvert\epsilon\rvert=1$ is circular in the stated viewing convention. Define

Q=2\pi f_0\tau.

For $Q\gg1$ , the positive-frequency amplitude is a Gaussian centered at $f_0$ with characteristic width $(2\pi\tau)^{-1}=f_0/Q$ . Small $Q$ produces a broad pulse for which a single carrier cycle is not a faithful physical picture. The parameters describe a morphology; they do not identify a unique source.

Norms that survive weak modeling

The root-sum-square strain is

h_{\rm rss}^2=\int_{-\infty}^{\infty} \left(h_+^2+h_\times^2\right)dt =\int_{-\infty}^{\infty} \left(\lvert\widetilde h_+\rvert^2+ \lvert\widetilde h_\times\rvert^2\right)df,

where the second equality is Parseval’s theorem for the displayed two-sided transform. $h_{\rm rss}$ has units $\mathrm{Hz}^{-1/2}$ and does not depend on an arbitrary time origin. For a finite- $Q$ sine–Gaussian, however, exact carrier-phase independence holds only for circular polarization; a linear or elliptical pulse retains a carrier-envelope correction proportional to $(1-\epsilon^2)e^{-Q^2}\cos(2\phi_0)$ , which is exponentially small when $Q\gg1$ . The value also depends on which polarizations and time interval are included.

The exact Gaussian integral makes the normalization explicit:

h_{\rm rss}^2 =\frac{\sqrt\pi\,\tau h_0^2}{2} \left[1+\epsilon^2 +(1-\epsilon^2)e^{-Q^2}\cos(2\phi_0)\right].

Thus even a quoted peak amplitude $h_0$ does not specify integrated strain without duration and polarization. The circular case cancels the carrier oscillations in $h_+^2+h_\times^2$ point by point.

For a far-zone signal with flux

F=\frac{1}{16\pi G} \left(\dot h_+^2+\dot h_\times^2\right),

the source-frame isotropic-equivalent energy for a nearby source at distance $D$ is

E_{\rm iso}=\frac{D^2}{4G} \int\left(\dot h_+^2+\dot h_\times^2\right)dt.

For a narrowband burst this becomes $E_{\rm iso}\simeq\pi^2D^2f_0^2h_{\rm rss}^2/G$ . At cosmological distance, use the observed strain and detector time instead:

E_{\rm iso,src} =\frac{D_L^2}{4G(1+z)} \int dt_{\rm det}\left(\dot h_+^2+\dot h_\times^2\right) \simeq\frac{\pi^2D_L^2 f_{0,\rm det}^2 \bigl(h_{\rm rss}^{\rm det}\bigr)^2}{G(1+z)}.

The factor follows from $F_{\rm det}=L_{\rm src}/(4\pi D_L^2)$ and $dt_{\rm det}=(1+z)dt_{\rm src}$ : integrate the flux and divide the observed duration stretch back out to recover emitted energy. Dots in this expression are detector-time derivatives. The label “isotropic-equivalent” remains essential because real emission has angular structure. In SI units multiply the displayed strain-rate energy expressions by $c^3$ . The flux normalization follows Flanagan and Hughes 2005, §5.3, eqs. (5.38)–(5.41), p. 29, Open PDF.

Families of weak assumptions

Different physical questions justify different model strengths.

Excess-power description: assume only localization in time and frequency.
Coherent ridge: assume a smooth instantaneous-frequency track but not a source-specific phase.
Wavelet sum: assume sparsity in localized atoms and infer their number or scale.
Damped sinusoid: assume one or more transient resonances with fitted frequencies and damping times.
Catalog-conditioned morphology: compare with a declared simulation family while retaining its microphysical and numerical limitations.

Moving downward in this list gains interpretability but risks excluding signals outside the assumed family. A good burst statement names which features are data-independent definitions and which come from a source prior.

A time-frequency cluster can merge two overlapping sources, split one long signal, or mistake a narrow instrumental line for long-lived radiation. Windowing leaks power, gaps create sidelobes, and a detector transfer function changes apparent morphology. Those are reasons to carry duration and bandwidth definitions with the result, not reasons to abandon modeling.

This chapter develops physical waveform descriptors. Search statistics, background estimation, vetoes, calibration, upper limits, and current detections belong to Gravitational-Wave Astronomy. Matter simulations supply conditional source families rather than universal burst templates.

Exercises

1. Time-frequency scaling. A sine-Gaussian has $f_0=200\,\mathrm{Hz}$ and $Q=10$ . Find $\tau$ and its characteristic amplitude bandwidth.

Solution

From $Q=2\pi f_0\tau$ ,

\tau=\frac{10}{2\pi(200\,\mathrm{Hz})} =7.96\,\mathrm{ms}.

The characteristic amplitude width is $(2\pi\tau)^{-1}=f_0/Q=20\,\mathrm{Hz}$ . A different convention based on power variance or full width at half maximum changes the numerical multiplier, so a reported bandwidth must name its definition.

2. Energy scaling. Two narrowband bursts have equal $h_{\rm rss}$ and distance, but the second has twice the central frequency. Compare their isotropic-equivalent energies.