Pulsar Timing Sensitivity to Dark Matter Substructure in the Presence of a Stochastic Gravitational-Wave Background

Abstract

Pulsar timing arrays (PTAs) can detect dark matter (DM) substructure through the small shifts a transiting DM subhalo imprints on pulse arrival times. Recently found evidence for a stochastic gravitational-wave background (GWB) acts as red noise and competes with the substructure signal. Here we provide an analytic understanding of how this background degrades PTA sensitivity to DM substructure. We derive the full gauge-invariant proper-time observable induced by a transiting DM subhalo and develop a framework for the expected signal-to-noise ratio in the presence of red noise, accounting for the degeneracy with the pulsar timing model. From this we obtain simple scaling relations for the reach across the static, dynamic, and stochastic regimes, and numerically compute the reach for a Square Kilometre Array benchmark at three representative points of the NANOGrav 15-year posterior. The GWB background suppresses the sensitivity to DM substructure by one to three orders of magnitude compared to forecasts in the presence of only white noise, and the suppression depends on the amplitude and spectral index of the background within a factor of three. The dynamic Shapiro signal suffers the smallest suppression and gives the best sensitivity to DM substructure near $10^{-2}\, M_\odot$. Probing the regime where subhalos make up all or part of the DM remains a challenge even for surveys with more pulsars and longer observing time. Despite this, PTA measurements remain a competitive probe of DM substructure, and future surveys will increase in sensitivity by up to two orders of magnitude from existing NANOGrav limits.