Current pulsar timing array (PTA) techniques for characterizing the spectrum
of a nanohertz-frequency stochastic gravitational-wave background (SGWB) begin
at the stage of timing data. This can be slow and memory intensive with
computational scaling that will worsen PTA analysis times as more pulsars and
observations are added. Given recent evidence for a common-spectrum process in
PTA data sets and the need to understand present and future PTA capabilities to
characterize the SGWB through large-scale simulations, we have developed
efficient and rapid approaches that operate on intermediate SGWB analysis
products. These methods refit SGWB spectral models to previously-computed
Bayesian posterior estimations of the timing power spectra. We test our new
methods on simulated PTA data sets and the NANOGrav 12.5-year data set, where
in the latter our refit posterior achieves a Hellinger distance from the
current full production-level pipeline that is ≲0.1. Our methods are
∼102--104 times faster than the production-level likelihood and scale
sub-linearly as a PTA is expanded with new pulsars or observations. Our methods
also demonstrate that SGWB spectral characterization in PTA data sets is driven
by the longest-timed pulsars with the best-measured power spectral densities
which is not necessarily the case for SGWB detection that is predicated on
correlating many pulsars. Indeed, the common-process spectral properties found
in the NANOGrav 12.5-year data set are given by analyzing only the ∼10
longest-timed pulsars out of the full 45 pulsar array, and we find that the
``shallowing'' of the common-process power-law model occurs when
gravitational-wave frequencies higher than ∼50~nanohertz are included.
The implementation of our methods is openly available as a software suite to
allow fast and flexible PTA SGWB spectral characterization and model selection.Comment: 19 pages, 12 figures. Submitting to Physical Review