Accurate characterization of the stochastic gravitational-wave background with pulsar timing arrays by likelihood reweighting

An isotropic stochastic background of nanohertz gravitational waves creates excess residual power in pulsar-timing-array datasets, with characteristic inter-pulsar correlations described by the Hellings--Downs function. These correlations appear as nondiagonal terms in the noise covariance matrix, which must be inverted to obtain the pulsar-timing-array likelihood. Searches for the stochastic background, which require many likelihood evaluations, are therefore quite computationally expensive. We propose a much more efficient method: we first compute approximate posteriors by ignoring cross correlations, and then reweight them to exact posteriors via importance sampling. We show that this technique results in accurate posteriors and marginal likelihoods, because the approximate and exact posteriors are similar, which makes reweighting especially accurate. The Bayes ratio between the approximate and exact marginal likelihoods is commonly used as a detection statistic, and it is also estimated reliably by our method, up to ratios of at least $10^6$.