From new physics to a running power law and back again: Minimal refitting techniques for the reconstruction of the gravitational-wave background signal in pulsar timing array data

Pulsar timing array (PTA) collaborations recently presented evidence for a gravitational-wave background (GWB) signal at nanohertz frequencies. In this paper, we introduce new refitting techniques for PTA data analysis that elevate related techniques in the literature to a more rigorous level, and thus provide the basis for fast and accurate Bayesian inference and physically intuitive model comparisons. The key idea behind our approach is to construct maps Φ from GWB spectral models to a running-power-law (RPL) reference model, such that the pullback Φ * P RPL of the RPL posterior density P RPL induces a likelihood on the GWB model parameter space; in other words, we refit spectral models to the RPL posterior density. In order to construct Φ , we introduce a matched-filtering approach in which Φ follows from a χ 2 minimization that accounts for the frequency dependence of PTA sensitivity curves. We validate and illustrate our techniques by three concrete examples: GWs from stable cosmic strings, GWs from metastable strings, and scalar-induced GWs.