Stochastic effect on ultralight dark matter searches with astronomical polarized sources

Ultralight axion-like particles (ALP) are the well-motivated candidates for the dark matter. In the standard halo model, the ultralight galactic dark matter behaves as a stochastic Gaussian random field, and the polarization angle of the linearly polarized light propagating through the field stochastically oscillates proportional to the field difference between the observer and the source. The statistics of this signal depend on the source size and its distance from the observer relative to the ALP Compton length and coherence length, and therefore on the ALP mass. We present a unified statistical framework for the polarization-oscillation searches with light sources of arbitrary size and distance. We classify the statistics of the stochastic oscillation amplitude into four distinct regimes and provide a seamless description including the transitional regimes between them. Our framework reproduces the signal models in previously known cases including the laboratory-scale searches and searches using pulsars and the cosmic microwave background. We provide a likelihood prescription that incorporates the stochasticity of the signal, and show that uncertainties in source distance must be carefully accounted for, as they can alter the resulting constraints on the ALP-photon coupling by up to a factor of three. Our results provide a consistent basis for interpreting and comparing a broad range of ALP searches with astronomical polarized sources.