Binary Parameters and Observable Combinations

A compact-binary analysis does not measure a list of source labels independently. It measures combinations that control phase evolution, relative mode amplitudes, arrival times, and detector responses. Chirp mass is usually the leading phase coordinate; mass ratio, spins, tides, distance, orientation, and sky position enter through progressively subtler and often correlated features. This page identifies those combinations without confusing detector-frame measurements with source-frame interpretation.

Helpful background. The network probability model is developed in The Gravitational-Wave Likelihood, while Likelihood Ratios and Matched Filtering defines the one-sided-noise inner product used below.

Intrinsic coordinates and redshift

Let the detector-frame component masses satisfy $m_{1,\mathrm{det}}\ge m_{2,\mathrm{det}}$. Useful combinations are

$$M=m_1+m_2, \qquad q=\frac{m_2}{m_1}\le 1, \qquad \eta=\frac{m_1m_2}{M^2}=\frac{q}{(1+q)^2},$$

Here $M$ is total mass, $q$ is the mass ratio, and $\eta$ is the symmetric mass ratio. The chirp mass is

$$\mathcal M=M\eta^{3/5}.$$

The leading inspiral phase depends on the detector-frame chirp mass. Cosmological redshift rescales every dimensional source timescale, so

$$m_{i,\mathrm{det}}=(1+z)m_{i,\mathrm{src}}, \qquad \mathcal M_{\mathrm{det}}=(1+z)\mathcal M_{\mathrm{src}}.$$

Gravitational-wave data alone therefore do not turn a measured detector-frame mass into a source-frame mass without a redshift or a distance–redshift model. Dimensionless quantities such as $q$ and the spin parameters

$$\boldsymbol\chi_i=\frac{\mathbf S_i}{Gm_i^2}$$

are unaffected by the common redshift factor. Here $\mathbf S_i$ is the spin angular momentum of body $i$. For nearly aligned binaries, the inspiral is especially sensitive to the mass-weighted spin projected along the orbital-angular-momentum unit vector $\hat{\mathbf L}$:

$$\chi_{\mathrm{eff}} = \frac{m_1\boldsymbol\chi_1\!\cdot\!\hat{\mathbf L} +m_2\boldsymbol\chi_2\!\cdot\!\hat{\mathbf L}}{M}.$$

Individual spin components are less directly identified unless precession, higher harmonics, or a long, high-information signal supplies additional structure.

What phase and amplitude constrain

The reason chirp mass is special can be seen before doing Bayesian inference. First use source-frame masses, time, and frequency for a slowly evolving, quasicircular binary. The Newtonian binding energy and quadrupole luminosity give

$$\begin{aligned} E&=-\frac12\eta Mv^2, &\mathcal F&=\frac{32}{5G}\eta^2v^{10},\\ v&=(\pi GMf)^{1/3}, &\frac{dE}{dt}&=-\mathcal F. \end{aligned}$$

Here $E$ is orbital binding energy, $\mathcal F$ is energy radiated per unit time, and $v$ is the dimensionless orbital-velocity parameter. The frequency $f$ is the dominant gravitational-wave frequency, twice the orbital frequency. The chain rule gives $\dot f=-\mathcal F/(dE/df)$. Convert to observed time and frequency using $dt_{\rm det}=(1+z)dt_{\rm src}$ and $f_{\rm det}=f_{\rm src}/(1+z)$, so $\dot f_{\rm det}=\dot f_{\rm src}/(1+z)^2$. In terms of detector-frame chirp mass, the result is

$$\dot f=\frac{96}{5}\pi^{8/3} (G\mathcal M_{\mathrm{det}})^{5/3}f^{11/3}.$$

Thus the measured frequency and its rate of increase already constrain one mass combination. Integrating this equation gives the leading time remaining,

$$t_c-t=\frac{5}{256} (G\mathcal M_{\mathrm{det}})^{-5/3}(\pi f)^{-8/3}.$$

Here $t_c$ is the formal coalescence time obtained by extending this leading inspiral model to arbitrarily high frequency. A real merger ends the approximation earlier. These formulas neglect spin, eccentricity, tides, and higher-order relativistic corrections; their value is to explain the information in a chirp, not to replace a complete waveform. They follow the compact-binary waveform construction reviewed in Christensen and Meyer 2022, § V.A.

For the same approximation, the leading stationary-phase amplitude can be written

$$\tilde h_I(f) = \sqrt{\frac{5}{24}}\,\pi^{-2/3} \frac{(G\mathcal M_{\mathrm{det}})^{5/6}}{D_L} f^{-7/6}\,\mathcal Q_I(\alpha,\delta,\psi,\iota) e^{i\Psi(f)},$$

where $I$ labels a detector and $D_L$ is luminosity distance. The dimensionless factor $\mathcal Q_I$ combines the antenna response and orientation: $\alpha$ and $\delta$ are right ascension and declination, $\psi$ specifies the polarization basis, and $\iota$ is the angle between $\hat{\mathbf L}$ and the line of sight. For the transform $\tilde h(f)=\int h(t)e^{+2\pi ift}dt$ and a gravitational-wave coalescence phase $\phi_c$, the leading phase has the structure

$$\Psi(f) = 2\pi f t_c-\phi_c-\frac{\pi}{4} +\frac{3}{128} \left(\pi G\mathcal M_{\mathrm{det}} f\right)^{-5/3} +\cdots .$$

The large accumulated coefficient of the $f^{-5/3}$ term explains why a sufficiently long inspiral can constrain $\mathcal M_{\mathrm{det}}$ sharply. Higher-order phase terms mix $\eta$ with aligned spin; precessional modulations probe spin components transverse to $\hat{\mathbf L}$; merger and ringdown constrain combinations related to the remnant; and tidal terms enter late in a neutron-star inspiral. The complete waveform, rather than any single coefficient, determines the posterior Christensen and Meyer 2022, § V.

For the dominant quadrupole mode, the two polarizations scale approximately as

$$h_+\propto \frac{1+\cos^2\iota}{2D_L}, \qquad h_\times\propto \frac{\cos\iota}{D_L}.$$

A single detector records $F_I^+h_+ + F_I^\times h_\times$, where $F_I^+$ and $F_I^\times$ are its antenna-response factors for the two polarizations. Distance, inclination, polarization, and sky response are therefore entangled. A network adds relative arrival times, phases, and amplitudes. It can constrain sky position and polarization well only when its baselines and antenna patterns provide genuinely independent information.

Parameter or combination	Primary waveform information	Important qualifications
$\mathcal M_{\mathrm{det}}$	Long inspiral phase accumulation	Source-frame value needs $z$ or a cosmology-conditioned distance relation
$q$, $\eta$, $\chi_{\mathrm{eff}}$	Higher-order phase, merger frequency	Strong mass-ratio–spin correlation; waveform-family dependent
Precessing spins	Amplitude and phase modulation	Weak for face-on, short, or low-information signals
$D_L$, $\iota$, $\psi$	Overall and relative detector amplitudes	Distance–inclination degeneracy; calibration amplitude matters
$(\alpha,\delta,t_c)$	Network timing and antenna response	Depends on detector geometry and which instruments are operating
Tidal combinations	High-frequency inspiral phase	Matter model, waveform, calibration, and prior dependent

The coherent inference map

For independent detector noises with one-sided power spectra $S_{n,I}(f)$, define

$$(a\mid b)_I =4\,\mathrm{Re}\int_{f_{\min}}^{f_{\max}} \frac{\tilde a_I(f)\tilde b_I^*(f)}{S_{n,I}(f)}\,df.$$

The coherent likelihood is

$$\ln \mathcal L(d\mid\theta,\lambda) =-\frac12\sum_I \bigl(d_I-h_I(\theta,\lambda)\mid d_I-h_I(\theta,\lambda)\bigr)_I+C(\lambda),$$

where $\theta$ denotes source parameters and $\lambda$ nuisance quantities such as calibration and noise-spectrum corrections. If the noise spectrum varies with $\lambda$, both the inner product and its Gaussian normalization $C(\lambda)$ change; that normalization cannot be discarded. For a fixed spectrum, $C$ is an irrelevant additive constant. Calibration and Noise Marginalization gives the explicit normalization. The posterior is a statement about this likelihood and a declared prior measure:

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

Consequently, “measured” means constrained under a specified waveform family, detector set, frequency band, calibration model, noise model, cosmology, and prior. The standard coherent compact-binary formulation and its validation logic are given in Veitch et al. 2015, §§ II–V.

Dated observational boundary and checks

As of 2026-08-18, the current observational interface used here is the official GWTC-5.0 release, DOI 10.7935/bk00-6a89, with search and event-analysis choices documented in LVK 2026, GWTC-5.0 Methods, §§ 2–5. No numerical event estimate is imported into the stable derivation above.

A defensible injection check fixes the detector network, strain segment or simulated noise, one-sided PSD, calibration response, frequency band, waveform family and modes, cosmology, prior support, sampler settings, and random seed. It should recover $\mathcal M_{\mathrm{det}}$, display the expected distance–inclination or mass-ratio–spin correlation, and repeat the analysis with an alternative waveform or nuisance treatment. Failure to recover the injection can indicate waveform misspecification, calibration or PSD error, sampler nonconvergence, a parameter boundary, or an inconsistent source/detector-frame conversion.

Exercises

1. Redshifted masses. A binary has source-frame component masses $30M_\odot$ and $20M_\odot$ at redshift $z=0.25$. Find its detector-frame total mass, mass ratio, and chirp mass in both frames.

Solution

The mass ratio is dimensionless, so $q=20/30=2/3$ in both frames. The source-frame total mass is $50M_\odot$, and

$$\eta=\frac{q}{(1+q)^2}=0.24, \qquad \mathcal M_{\mathrm{src}} =50M_\odot\,(0.24)^{3/5} \simeq21.24M_\odot.$$

Multiplication by $1+z=1.25$ gives

$$M_{\mathrm{det}}=62.5M_\odot, \qquad \mathcal M_{\mathrm{det}}\simeq26.55M_\odot.$$

The detector measures the redshifted masses through waveform timescales; assigning the smaller source-frame values requires the redshift assumption.

2. Face-on ambiguity. Show why the leading quadrupole amplitude changes only at second order in a small inclination $\iota$ near face-on orientation.

Solution

Using $\cos\iota=1-\iota^2/2+O(\iota^4)$,

$$\frac{1+\cos^2\iota}{2} =1-\frac{\iota^2}{2}+O(\iota^4), \qquad \cos\iota =1-\frac{\iota^2}{2}+O(\iota^4).$$

Both polarizations therefore change quadratically rather than linearly. A small change of $D_L$ can mimic this weak amplitude change, producing a broad distance–inclination direction unless other modes, precession, or external information break it.

References

• Christensen, N., and Meyer, R. (2022). “Parameter Estimation with Gravitational Waves.” Reviews of Modern Physics 94, 025001. DOI. Open PDF.
• 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.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). GWTC-5.0 Data Release. DOI 10.7935/bk00-6a89. Official release.
• 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. Open PDF.