Rapid inference for individual binaries and a stochastic background with pulsar timing array data

The analysis of pulsar timing array data has provided evidence for a gravitational wave background in the nanohertz band. This raises the question of what is the source of the signal, is it astrophysical or cosmological in origin? If the signal originates from a population of supermassive black hole binaries, as is generally assumed, we can expect both to see evidence for anisotropy and to be able to resolve signals from individual binaries as more data are collected. The anisotropy and resolvable systems are caused by a small number of loud signals that stand out from the crowd. Here we focus on the joint detection of individual signals and a stochastic background. While methods have previously been developed to perform such an analysis, they are currently held back by the cost of computing the joint likelihood function. Each individual source is described by $N=8+2{N}_{p}$ parameters, where ${N}_{p}$ are the number of pulsars in the array. With the latest combined datasets having over one hundred pulsars, the parameter space is very large, and consequently, it takes a large number of likelihood evaluations to explore these models. Here we present a new approach that extends the Fourier basis method, previously introduced to accelerate analyses for stochastic signals, to also include deterministic signals. Key elements of the method are that the likelihood evaluations are per pulsar, avoiding expensive operations on large matrices, and the templates for individual binaries can be computed analytically or using fast Fourier methods on a sparsely sampled grid of time samples. The net result is an analysis that scales better than quadratically with the size of the dataset, while the approach currently being used in most analyses scales quartically or worse with the number of data points. As datasets grow with more observations, this analysis will be orders of magnitude faster than previous approaches.