Parameter estimation on gravitational waves from multiple coalescing binaries

Future ground-based and space-borne interferometric gravitational-wave detectors may capture between tens and thousands of binary coalescence events per year. There is a significant and growing body of work on the estimation of astrophysically relevant parameters, such as masses and spins, from the gravitational-wave signature of a single event. This paper introduces a robust Bayesian framework for combining the parameter estimates for multiple events into a parameter distribution of the underlying event population. The framework can be readily deployed as a rapid post-processing tool.

Figure 1

The left panel shows a possible multimodal marginalized PDF for a single parameter of a single detected event. The right panel shows the results of estimating the distribution from individual samples by means of a kernel density estimator. Twenty samples are taken from a bimodal distribution (red solid line); the blue dashed curve shows a KDE where the smoothing parameter (bandwidth) was chosen to be too small, while the black dash-dotted curve shows the effect of a KDE with a large bandwidth parameter that over-smoothes the distribution.

Figure 2

Recovered one-dimensional posterior probability distributions for mean $\mu$, standard deviation $\sigma$, and their joint two-dimensional distribution. The true probability distribution with ${\mu }_{\mathrm{true}}=0.4$ and ${\sigma }_{\mathrm{true}}=0.1$ was sampled with $k=20$ detected events. The PDF obtained for each event is a Gaussian centered on its true value with standard deviation taken uniformly from [0,0.2], sampled with $N=100$ points. The weighted averages of the recovered parameters over the posterior density of the reconstructed population distribution are $\overline{\mu }=0.4055$ and $\overline{\sigma }=0.0958$, respectively.

Figure 3

The left panel shows the error in the recovered mean of the population distribution $\delta \overline{\mu }=\overline{\mu }-{\mu }_{\mathrm{true}}$ (red solid line) and the standard deviation $\delta \overline{\sigma }=\overline{\sigma }-{\sigma }_{\mathrm{true}}$ (blue dashed line) as a function of the number of detected events $k$. The black dash-dotted curve is the approximate analytic fit $\delta \overline{\mu }\propto 1/\sqrt{k}$. The right panel shows the estimated mean $\overline{\mu }$ as a function of the statistical uncertainty in each event $a=b$. The dashed line is the true mean of the population distribution ${\mu }_{\mathrm{true}}=0.4$, while the dotted line is the mean of the prior on $\mu$.