Stochastic fluctuations of bosonic dark matter

Numerous theories extending beyond the standard model of particle physics predict the existence of bosons that could constitute the dark matter (DM) permeating the universe. In the standard halo model (SHM) of galactic dark matter the velocity distribution of the bosonic DM field defines a characteristic coherence time $\tau_c$. Until recently, laboratory experiments searching for bosonic DM fields have been in the regime where the measurement time $T$ significantly exceeds $\tau_c$, so null results have been interpreted as constraints on the coupling of bosonic DM to standard model particles with a bosonic DM field amplitude $\Phi_0$ fixed by the average local DM density. However, motivated by new theoretical developments, a number of recent searches probe the regime where $T\ll\tau_c$. Here we show that experiments operating in this regime do not sample the full distribution of bosonic DM field amplitudes and therefore it is incorrect to assume a fixed value of $\Phi_0$ when inferring constraints on the coupling strength of bosonic DM to standard model particles. Instead, in order to interpret laboratory measurements (even in the event of a discovery), it is necessary to account for the stochastic nature of such a virialized ultralight field (VULF). The constraints inferred from several previous null experiments searching for ultralight bosonic DM were overestimated by factors ranging from 3 to 10 depending on experimental details, model assumptions, and choice of inference framework.