Probing ultralight axionlike dark matter: A pulsar timing arrays-pulsar polarization arrays synergy

Ultralight axionlike dark matter (ALDM) is a leading candidate in the dark matter realm, characterized by its prominent wave properties on astronomical scales. Pulsar timing arrays (PTAs) and Pulsar polarization arrays (PPAs) aim to detect this dark matter through timing and polarization measurements, respectively, of pulsars. The PTA relies on gravitational effects, while the PPA detects nongravitational effects. These two methods complement each other, synergistically enhancing the pulsar array's capability to identify the ALDM signals in the data. In this article, we provide a foundational development of this synergy. We begin by revisiting previously derived two-point correlation functions for both PTA and PPA, and extend the analysis to include the leading-order correlation between timing and polarization signals, encoded as a three-point function. We then explore the constructions of likelihood functions for PTA and combined PTA-PPA analyses within a Bayesian framework, aimed at detecting the characteristic correlations of ALDM signals. We emphasize the non-Gaussianity of the ALDM timing signals, which arises from their nonlinear dependence on the field, in contrast to the Gaussian nature of its polarization signals. To address the complexities introduced, we approach this investigation in two ways: one involves a Gaussian approximation with proper justifications, while the other derives the formalism from the generic Gaussian characteristics of the ALDM field. Particularly, for the combined PTA-PPA case, a clear connection between the three-point correlation function and the likelihood is established. We anticipate that these efforts will lead to further developments in PTA and PTA-PPA analysis methods, better accounting for the influence of non-Gaussianity.