Orthonormal bases for reconstructing pairwise interpulsar correlations in pulsar timing arrays

For pulsar timing arrays (PTAs), the telltale signature of an isotropic stochastic background of gravitational waves is a pattern of pairwise interpulsar timing correlations approximately following the Hellings and Downs (HD) curve. Certain systematic errors and new physics processes also lead to interpulsar correlations with different patterns that can be distinguished from the HD curve to varied degrees. As evidence of HD correlations in PTA data mounts in coming years, it is important to develop principled strategies for flexibly and optimally reconstructing the pattern of interpulsar timing correlations, both to test how well the correlations track the HD pattern and to possibly detect additional effects, systematic or otherwise. To this end, we develop orthonormal basis functions that fully capture HD correlations and eliminate covariances between the HD curve and any additional correlated structure. We do this analytically and in a data-adaptive way informed by ``optimal statistic'' analysis techniques widely used by PTA groups. These bases are adaptive in that they will vary from PTA to PTA and from data release to data release as new pulsars of varied timing quality and baseline are added to arrays, as instrumentation advances, and as observations accrue. We discuss how the techniques we introduce can be extended to future multisignal searches by PTAs and to robust assessment of HD detection significance.