Bias from small-scale leakage in Pulsar Timing Array maps

Pulsar Timing Array experiments are rapidly approaching the era of gravitational wave background anisotropy detection. The timing residuals of each pulsar are an integrated measure of the gravitational-wave power across all angular scales. However, due to the limited number of monitored pulsars, current analyses are only able to reconstruct the angular structure of the background at large scales. We show analytically that this mismatch between the integrated all-sky signal and the truncated reconstruction introduces a previously unaccounted source of systematic bias in the anisotropic background angular power spectrum. The source of this systematic error, that we call ``small-scale leakage'', is the intrinsic presence of unaccounted gravitational wave power at scales smaller than the reconstructed scales. This unmodeled power leaks into large-scale modes, artificially increasing the recovered value of the inferred angular power spectrum by at least one order of magnitude in a wide range of scales. Importantly, this effect is fundamentally independent of the geometry of the pulsar configuration, the anisotropy reconstruction method, the use of different regularization schemes, and the presence of pulsar noise. As the quality of pulsar timing array experiments improves, a robust understanding of small-scale leakage will become paramount for reliable detection and characterization of the gravitational wave background. Thus, the theoretical formalism developed here will be essential to estimate the magnitude of this systematic uncertainty in anisotropy searches.