Compact-Binary Parameter Estimation

Compact-binary parameter estimation is the inverse problem between calibrated network strain and a physical source model. Its output is never “the binary parameters” without qualification: it is a likelihood or posterior under a stated waveform family, detector and source-frame map, calibration and noise model, cosmology, prior measure, and numerical procedure. This chapter develops that complete conditioning chain while leaving source-independent waveform construction and population-level inference to their dedicated treatments.

Helpful background. The Gravitational-Wave Likelihood supplies the statistical foundation, while Compact-Binary Waveforms supplies the source-model interface.

Enter the inverse problem

For detector data $d_I$ and waveform response $h_I(\theta,\lambda)$, the central object is

$$p(\theta\mid d) \propto \pi(\theta) \int d\lambda\, \mathcal L(d\mid\theta,\lambda)\pi(\lambda\mid\theta),$$

where $\theta$ denotes source parameters and $\lambda$ includes nuisance quantities such as calibration and spectral corrections. Integrating over $\lambda$ averages the likelihood over the allowed detector uncertainties. When source and nuisance priors are independent, $\pi(\lambda\mid\theta)=\pi(\lambda)$; the general conditional form also allows prior correlations.

This likelihood–prior–nuisance architecture and its sampler validation are developed in Veitch et al. 2015, §§ II–V, while Thrane and Talbot 2019, §§ 2–5 make the event-to-population probability measures explicit. Four distinctions organize the chapter:

• detector frame versus source frame: dimensional masses are redshifted;
• waveform coordinate versus identifiable combination: chirp mass, effective spin, and amplitude combinations are often closer to what the data constrain;
• statistical width versus systematic displacement: repeated noise and model misspecification require different diagnostics; and
• likelihood information versus prior structure: posterior shape alone does not reveal which supplied it.

The recurring controlled example is a compact-binary injection recovered under changed network, waveform, calibration, noise, and prior assumptions. It teaches causation more cleanly than treating one catalog event as universal.

Two routes through the chapter

Physical interpretation route. Begin with Binary Parameters and Observable Combinations, continue to Mass, Spin, Distance, and Inclination Degeneracies, and then use Astrophysical and Analysis Priors to decide which posterior features are data-driven.

Validation route. Start with Waveform Marginalization and Model Uncertainty, add Calibration and Noise Marginalization, and finish with Systematic Bias and Injection Recovery. This route is designed for readers evaluating an event-analysis result or building a reproducibility test.

The routes rejoin because a degeneracy determines which nuisance or prior choice can matter most. Distance is sensitive to amplitude calibration and orientation priors; mass ratio and spin are sensitive to waveform phasing and model support; sky position depends on timing, detector geometry, and relative calibration.

Chapter map

1. Binary Parameters and Observable Combinations maps component properties and extrinsic geometry into phase, amplitude, timing, and polarization information.
2. Mass, Spin, Distance, and Inclination Degeneracies explains broad waveform-equivalent directions and the features that can break them.
3. Waveform Marginalization and Model Uncertainty compares model averaging, discrepancy marginalization, and injection-calibrated bounds.
4. Calibration and Noise Marginalization propagates detector-response and spectral uncertainty through the source likelihood.
5. Astrophysical and Analysis Priors distinguishes sampling convenience, physical population assumptions, Jacobians, and support.
6. Systematic Bias and Injection Recovery designs repeated, paired tests that separate deterministic bias from noise fluctuation.

Together these pages answer a practical question: which aspects of an inference would remain if the waveform family, nuisance treatment, and prior were changed within their defensible ranges?

Convention-sensitive points

This chapter uses one-sided detector PSDs and the site time dependence $e^{-i\omega t}$. Masses appearing directly in waveform timescales are detector-frame masses unless marked otherwise:

$$m_{\mathrm{det}}=(1+z)m_{\mathrm{src}}.$$

Credible regions carry posterior probability under a stated prior; they are not automatically confidence regions with repeated-sampling coverage. “Uniform prior” is incomplete without the variable, measure, and bounds. “Marginalized uncertainty” is incomplete without the nuisance model and its prior. “Waveform agreement” is incomplete without detector weighting, frequency band, parameter support, and a held-out validation target.

Empirical examples are frozen at 2026-08-18 and use the versioned GWTC-5.0 release as their current observational interface. Stable derivations remain valid independently of future catalog updates; catalog-specific settings and conclusions do not.

Review the chapter

1. Diagnose a mass statement. An analysis reports a narrow detector-frame chirp-mass posterior but no redshift information. Which mass claim is justified?

Solution

The data can support a narrow statement about $\mathcal M_{\mathrm{det}}$ under the waveform, noise, calibration, and prior choices. They do not by themselves determine

$$\mathcal M_{\mathrm{src}} =\frac{\mathcal M_{\mathrm{det}}}{1+z}.$$

A source-frame claim requires a redshift measurement or a declared cosmology- and population-conditioned distance–redshift inference.

2. Design a discriminating check. A spin posterior shifts when a second waveform family is used. Give a test that distinguishes sampler noise from waveform-induced bias.

Solution

Use paired injections with known spin and identical noise realizations. Run each recovery family with multiple seeds and converged settings. If within-family seed variation is small but the between-family shift repeats across noise realizations and appears in zero noise, sampler noise is disfavored and model mismatch is implicated. A held-out injection family and coherent-residual check are needed to determine which recovery is more accurate.

3. Transfer task. Explain why widening only the distance prior is not a principled treatment of amplitude-calibration uncertainty.

Solution

Calibration error is detector- and frequency-dependent and can change both amplitude and phase. It also correlates with orientation, polarization, timing, masses, and spins. Widening a distance prior changes the source measure but does not represent those correlations or the calibration measurement. The correct treatment introduces calibration nuisance variables with an epoch-specific prior and marginalizes them jointly with source parameters.

After this chapter, continue to Catalog Construction and Selection Effects to learn why event posteriors cannot be assembled into population statements without search provenance and detection efficiency.

References

• Thrane, E., and Talbot, C. (2019). “An Introduction to Bayesian Inference in Gravitational-Wave Astronomy: Parameter Estimation, Model Selection, and Hierarchical Models.” Publications of the Astronomical Society of Australia 36, e010. DOI.
• Veitch, J., et al. (2015). “Parameter Estimation for Compact Binaries with Ground-Based Gravitational-Wave Observations Using the LALInference Software Library.” Physical Review D 91, 042003. DOI.

Further reading

• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). “GWTC-5.0: Methods for Identifying and Characterizing Gravitational-Wave Transients.” arXiv:2605.27224. Open PDF.