Constructing likelihoods for a non-Gaussian gravitational-wave background

Author(s)

Boybeyi, Tore, Mandic, Vuk, Romano, Joseph

Abstract

The stochastic gravitational-wave background from stellar-mass compact binaries is non-Gaussian since it is made up of a finite number of individual mergers. Over the years, several methods have been developed to search for this background beyond the standard cross-correlation statistic. We reformulate the data analysis problem for such astrophysical backgrounds in terms of the characteristic function of the data in multiple detectors, and show that the existing searches arise as different limiting cases. We also show that new searches can be constructed through different approximations of the characteristic function, which gives a concrete way to engineer suitable approximate sufficient statistics. To illustrate this framework in the context of a simple example, we use the formalism proposed here to analyze an exactly solvable toy model.

Figures

Plots of $\exp[\bar N(z-1)]$ as a function of $\eta_1+\eta_2$, along with the approximations used in the mixture members $\mathcal{L}_{M=1}$, $\mathcal{L}_{M=2}$, $\mathcal{L}_{M=3}$, for different values of $\bar N$ at unit per-event power $A_0^2=1$. As the number of Gaussian components grows, the mixture converges to the true characteristic function. Each member of this likelihood family is therefore an approximation to the exact likelihood.
Caption Plots of $\exp[\bar N(z-1)]$ as a function of $\eta_1+\eta_2$, along with the approximations used in the mixture members $\mathcal{L}_{M=1}$, $\mathcal{L}_{M=2}$, $\mathcal{L}_{M=3}$, for different values of $\bar N$ at unit per-event power $A_0^2=1$. As the number of Gaussian components grows, the mixture converges to the true characteristic function. Each member of this likelihood family is therefore an approximation to the exact likelihood.
Detection and recovery for the event-count family of~\eqref{eq:Ktrunc}, for a Gaussian intermittent signal in two co-located and co-aligned detectors. The top-left panel is the smallest detectable cross power $\bar N A_0^2$ versus the rate $\bar N$ for $\mathcal{L}_{K\le1}$, $\mathcal{L}_{K\le2}$, $\mathcal{L}_{K\le3}$ and the exact mixture; this is the detection floor of each likelihood. The callouts give, at the three marked rates, the maximum-likelihood recovery of $(\log_{10}\bar N,\log_{10}A_0^2)$. Each thin curve is the $\Delta\ln\mathcal{L}=-3$ contour of one realization, and the cyan cross marks the true parameter values. Where the background is rare and loud, the one-event truncation already reaches the exact detection floor and centers on the true parameter values, while at higher rates the low truncations stay on the exact floor and bias the estimate. The family closes in on the exact mixture as more event counts are kept.
Caption Detection and recovery for the event-count family of~\eqref{eq:Ktrunc}, for a Gaussian intermittent signal in two co-located and co-aligned detectors. The top-left panel is the smallest detectable cross power $\bar N A_0^2$ versus the rate $\bar N$ for $\mathcal{L}_{K\le1}$, $\mathcal{L}_{K\le2}$, $\mathcal{L}_{K\le3}$ and the exact mixture; this is the detection floor of each likelihood. The callouts give, at the three marked rates, the maximum-likelihood recovery of $(\log_{10}\bar N,\log_{10}A_0^2)$. Each thin curve is the $\Delta\ln\mathcal{L}=-3$ contour of one realization, and the cyan cross marks the true parameter values. Where the background is rare and loud, the one-event truncation already reaches the exact detection floor and centers on the true parameter values, while at higher rates the low truncations stay on the exact floor and bias the estimate. The family closes in on the exact mixture as more event counts are kept.
Detection and recovery for the Gaussian-mixture family of~\eqref{eq:mmansatz}, in the same setup as Fig.~\ref{fig:mixcallout}. The top-left panel is the smallest detectable cross power $\bar N A_0^2$ versus the rate $\bar N$ for $\mathcal{L}_{M=1}$, $\mathcal{L}_{M=2}$, $\mathcal{L}_{M=3}$ and the exact mixture. The callouts give, at the three marked rates, the maximum-likelihood recovery of $(\log_{10}\bar N,\log_{10}A_0^2)$. Each thin curve is the $\Delta\ln\mathcal{L}=-3$ contour of one realization and the cyan cross marks the true parameter values. The posterior from the single-component likelihood $\mathcal{L}_{M=1}$ is informative only on the degenerate combination $\bar N A_0^2$, and its detection floor is above that of the exact likelihood. Each additional component narrows the posterior and lowers the detection floor toward that of the exact mixture.
Caption Detection and recovery for the Gaussian-mixture family of~\eqref{eq:mmansatz}, in the same setup as Fig.~\ref{fig:mixcallout}. The top-left panel is the smallest detectable cross power $\bar N A_0^2$ versus the rate $\bar N$ for $\mathcal{L}_{M=1}$, $\mathcal{L}_{M=2}$, $\mathcal{L}_{M=3}$ and the exact mixture. The callouts give, at the three marked rates, the maximum-likelihood recovery of $(\log_{10}\bar N,\log_{10}A_0^2)$. Each thin curve is the $\Delta\ln\mathcal{L}=-3$ contour of one realization and the cyan cross marks the true parameter values. The posterior from the single-component likelihood $\mathcal{L}_{M=1}$ is informative only on the degenerate combination $\bar N A_0^2$, and its detection floor is above that of the exact likelihood. Each additional component narrows the posterior and lowers the detection floor toward that of the exact mixture.
Detection and recovery for the Taylor family of~\eqref{eq:famthree}, in the same setup as Fig.~\ref{fig:mixcallout}. The top-left panel is the smallest detectable cross power $\bar N A_0^2$ versus the rate $\bar N$ for $\mathcal{L}_2$, $\mathcal{L}_4$, $\mathcal{L}_6$ and the exact mixture. The callouts give, at three benchmark rates, the maximum-likelihood recovery of $(\log_{10}\bar N,\log_{10}A_0^2)$. Each thin curve is the $\Delta\ln\mathcal{L}=-3$ contour of one realization and the cyan cross marks the true parameter values. Each added data power tightens the recovery and lowers the floor toward the exact mixture.
Caption Detection and recovery for the Taylor family of~\eqref{eq:famthree}, in the same setup as Fig.~\ref{fig:mixcallout}. The top-left panel is the smallest detectable cross power $\bar N A_0^2$ versus the rate $\bar N$ for $\mathcal{L}_2$, $\mathcal{L}_4$, $\mathcal{L}_6$ and the exact mixture. The callouts give, at three benchmark rates, the maximum-likelihood recovery of $(\log_{10}\bar N,\log_{10}A_0^2)$. Each thin curve is the $\Delta\ln\mathcal{L}=-3$ contour of one realization and the cyan cross marks the true parameter values. Each added data power tightens the recovery and lowers the floor toward the exact mixture.
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