An empirical Bayesian analysis applied to the globular cluster pulsar population

Abstract: We describe an empirical Bayesian approach to determine the most likely size of an astronomical population of sources of which only a small subset are observed above some limiting flux density threshold. The method is most naturally applied to astronomical source populations at a common distance (e.g.,stellar populations in globular clusters), and can be applied even to populations where a survey detects no objects. The model allows for the inclusion of physical parameters of the stellar population and the detection process. As an example, we apply this method to the current sample of radio pulsars in Galactic globular clusters. Using the sample of flux density limits on pulsar surveys in 94 globular clusters published by Boyles et al., we examine a large number of population models with different dependencies. We find that models which include the globular cluster two-body encounter rate, $\Gamma$, are strongly favoured over models in which this is not a factor. The optimal model is one in which the mean number of pulsars is proportional to $\exp(1.5 \log \Gamma)$. This model agrees well with earlier work by Hui et al. and provides strong support to the idea that the two-body encounter rate directly impacts the number of neutron stars in a cluster. Our model predicts that the total number of potentially observable globular cluster pulsars in the Boyles et al. sample is 1070$^{+1280}_{-700}$, where the uncertainties signify the 95% confidence interval. Scaling this result to all Galactic globular clusters, and to account for radio pulsar beaming, we estimate the total population to be 2280$^{+2720}_{-1490}$.