Pulsar Timing Array Science and Axion Phenomenology

Pulsar Timing Arrays are remarkable experiments timing a network of millisecond pulsars over decade-long campaigns, with an approximate cadence of one week. While their primary goal is to detect and characterize the presence of a stochastic gravitational wave background permeating the Universe, they also stand out as a powerful tool to probe fundamental physics. This thesis explores how Pulsar Timing Arrays can shed light on the nature of dark matter in our Galaxy, under the assumption that it consists of ultralight particles. First, we consider a scenario in which ultralight particles interact only gravitationally, with no couplings to Standard Model fields. In this case, we constrain their abundance below a few tenths of the total observed dark matter density, within a specific ultralight mass range. Second, we assume that ultralight particles are conformally coupled to the Standard Model, deriving bounds on the relevant couplings constants in two representative scalar-tensor theories. Finally, we investigate a particular aspect of axion-like particle phenomenology: the axion-photon coupling. This interaction is typically described by a pair of coupled Klein-Gordon equations, which, under the assumption that the axion is relativistic, are approximated by two first-order Schr\"odinger-like equations. Here, we discuss how to evaluate the axion-photon system evolution without relying on such relativistic approximation. In particular, we provide an exact analytical solution for the case of constant magnetic field and plasma frequency, and develop a perturbative expansion to account for spatial inhomogeneities.