Models

The Fit object interfaces with a of numpyro model for sampling. This model defines the likelihood, prior and signal templates and is constructed according to a number of options per the Module Index. The model can run in three broad configurations.

Generic damped sinusoids

Fit an arbitrary number of damped sinusoids parameterized in terms of their amplitudes \(A_n\), phases \(\phi_n\), frequencies \(f_n\) and damping rates \(\gamma_n = 1/\tau_n\), where \(\tau_n\) is the damping time. The signal model is thus

\[h(t) = \sum_n A_n \cos(2\pi f_n t - \phi_n) \exp(-\gamma_n t)\]

although this is internally parameterized in terms of cosine and sine quadratures, \(A_x = A \cos \phi\) and \(A_y = A \sin \phi\), for efficiency. Importantly, label switching problems are avoided by defining \(f_n < f_{n+1}\) or \(\gamma_n < \gamma_{n+1}\).

Priors are flat in \(A_n, \phi_n, f_n, \gamma_n\), modulo the \(f_n < f_{n+1}\) or \(\gamma_n < \gamma_{n+1}\) restriction.

👉 See this model in action! Analyzing a single damped sinusoid.

Kerr ringdowns

Fit an arbitrary number of Kerr ringdown modes with arbitrary elliptical polarization as in Isi & Farr (2021). Modes are specified by their \((p, s, \ell, |m|, n)\) numbers: prograde vs retrograde (\(p = \pm 1\)), spin-weight (\(s = -2\) for GWs), angular harmonic numbers (\(\ell\) and \(0 \leq |m| \leq \ell\)), and overtone number (\(n\)).

The waveform is such that the two GW polarizations for each \(j \equiv (+1, -2, \ell, |m|, n)\) mode are given by six parameters \(A_j, \epsilon_j, \theta_j,\phi_j\) following

\[\begin{split}\begin{eqnarray} h^{(+)}_{j} &= h^c_{j}\, \cos \theta_{j} - \epsilon_{j} h^s_{j}\, \sin\theta_{j}\, , \\ h^{(\times)}_j &= h^c_{j}\, \sin \theta_j + \epsilon_j h^s_{j}\, \cos\theta_j\, , \end{eqnarray}\end{split}\]

for cosine and sine quadratures

\[\begin{split}\begin{eqnarray} h^c_j &\equiv A_j\, e^{-t/\tau_j} \cos(\omega_j t - \phi_j) \, , \\ h^s_j &\equiv A_j\, e^{-t/\tau_j} \sin(\omega_j t - \phi_j) \, . \end{eqnarray}\end{split}\]

For each detector the template will thus be

\[h = \sum_j \left( F_+ h^{(+)}_{j} + F_\times h^{(\times)}_{j} \right)\, ,\]

summing over the requested mode indices \(j\). The antenna patterns \((F_+, F_\times)\) are determined automatically by the Fit object based on the target sky location and polarization angle; these are currently fixed, and their only effect is to scale the relative amplitudes at different detectors (otherwise, they are degenerate with the mode amplitudes and phases).

In the mchi model, the mode frequencies and damping rates are parameterized by two parameters: the Kerr black-hole mass \(M\) and dimensionless spin magnitude \(\chi\). To replicate this functional dependence efficiently, the model makes use of fitting coefficients precomputed through the qnm package.

The priors are uniform in \(M\) and \(\chi\). The priors can also be made uniform on \(A_j\) and \(\epsilon_j\) using the flat_A and flat_A_ellip options (see Fit.update_prior); by default, however, they correspond to Gaussian priors on the cosine and sine quadratures of each polarization (see Appendix of Isi & Farr (2021)).

This model supports deviations from the Kerr spectrum, which can be turned on via the df and dg options. This activates deviation parameters \(\delta f_j\) and \(\delta\gamma_j\) that modify the frequencies and damping times such that

\[\begin{split}\begin{eqnarray} f_j &= f_j(M,\chi) \exp(\delta f_j) \, , \\ \gamma_j &= \gamma_j(M,\chi) \exp(\delta \gamma_j) \, . \end{eqnarray}\end{split}\]

👉 See this model in action! Fitting quasi-normal modes in GW190521.

Kerr ringdowns with restricted polarizations

Fit an arbitrary number of Kerr overtones with polarizations parameterized by a single “inclination” parameter \(\cos\iota\), controlling the degree of circular polarization for all modes. This is equivalent to assuming all \(m=+2\) and \(m=-2\) components are equally excited, so that the ellipticity of the observed signal is only a function of the viewing angle (see appendix in Isi & Farr (2021)); we might expect this in the case of nonprecessing systems, which possess equatorial reflection symmetry (hence the naming aligned).

In this more restricted version of the mass-spin model above, the polarizations are given by

\[\begin{split}\begin{eqnarray} h^{(+)} &= \sum_{\ell |m| n} Y^+_{\ell |m| n}(\cos\iota) A_{\ell |m| n} \cos(\omega_{\ell |m| n} t - \phi_{\ell |m| n})\exp(-\gamma_{\ell |m| n} t)\, , \\ h^{(\times)} &= \sum_{\ell |m| n} Y^\times_{\ell |m| n}(\cos\iota) A_{\ell |m| n} \sin(\omega_{\ell |m| n} t - \phi_{\ell |m| n}) \exp(-\gamma_{\ell |m| n} t)\, . \end{eqnarray}\end{split}\]

where \(Y^{+/\times}_{\ell |m| n}(\cos\iota)\) are given in Eq. (31) of Isi (2022). Other options for this model are analogous to those in the generic mass-spin configuration.

👉 See this model in action! Fitting quasi-normal modes in GW150914.