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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">Φ</a:mi> </a:math> from GWB spectral models to a running-power-law (RPL) reference model, such that the pullback <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msup> <d:mi mathvariant="normal">Φ</d:mi> <d:mo>*</d:mo> </d:msup> <d:msub> <d:mi>P</d:mi> <d:mrow> <d:mi>RPL</d:mi> </d:mrow> </d:msub> </d:math> of the RPL posterior density <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>P</g:mi> <g:mrow> <g:mi>RPL</g:mi> </g:mrow> </g:msub> </g:math> 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 <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi mathvariant="normal">Φ</i:mi> </i:math> , we introduce a matched-filtering approach in which <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi mathvariant="normal">Φ</l:mi> </l:math> follows from a <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msup> <o:mi>χ</o:mi> <o:mn>2</o:mn> </o:msup> </o:math> 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.