Sensitive volume estimation

Searches for gravitational-wave transients are limited by the sensitivity of current detectors. For this reason it is necessary to quantify the fraction of sources that are expected to be observed to avoid biases in the population inference.

\[P_{\rm det}(\Lambda) = \int dd \int d\theta p(d, \theta | \lambda) \Theta(\rho(d) - \rho_{\rm th})\]

Here \(d\) is the observed strain data, \(\theta\) are the parameters of individual sources, e.g., masses, spins, redshifts, etc., and \(\Lambda\) are the population parameters. The quantity \(\rho\) is a detection statistic, e.g., the signal-to-noise ratio, and \(\rho_{\rm th}\) is the threshold for detection.

The most common method to estimate this quantity is to simulate a population of sources from some reference distribution \(p(\theta | \varnothing)\) and compute the fraction of sources that are detected. Using a single reference set of such “injections” one can estimate \(P_{\rm det}(\Lambda)\) using Monte Carlo integration.

\[\hat{P}_{\rm det}(\Lambda) = \frac{1}{N} \sum_{i=1}^N \Theta(\rho_i - \rho_{\rm th}) \frac{p(\theta_i | \Lambda)}{p(\theta_i | \varnothing)}\]

Since the detection statistic is independent of the population model, we can remove the \(\theta_i\) that don’t pass the threshold yielding \(M\) detected sources.

\[\hat{P}_{\rm det}(\Lambda) = \frac{1}{N} \sum_{i=1}^M \frac{p(\theta_i | \Lambda)}{p(\theta_i | \varnothing)}\]

This model is implemented in the gwpopulation.vt.ResamplingVT class.

A simpler model is to interpolate some expression for

\[p_{\rm det}(\theta) = \int dd p(d, \theta) \Theta(\rho(d) - \rho_{\rm th})\]

The quantity \(P_{\rm det}(\Lambda)\) can be computed by integrating over the specified \(\theta\). This model is implemented in the gwpopulation.vt.GridVT class. Note that the computational cost of this approach scales exponentially with the number of parameters.