The classical field approximation of ultra light dark matter: quantum breaktimes, corrections, and decoherence

The classical field approximation is widely used to better understand the predictions of ultra-light dark matter. Here, we use the truncated Wigner approximation method to test the classical field approximation of ultra-light dark matter. This method approximates a quantum state as an ensemble of independently evolving realizations drawn from its Wigner function. The method is highly parallelizable and allows the direct simulation of quantum corrections and decoherence times in systems many times larger than have been previously studied in reference to ultra-light dark matter. Our study involves simulation of systems in 1, 2, and 3 spatial dimensions. We simulate three systems, the condensation of a Gaussian random field in three spatial dimensions, a stable collapsed object in three spatial dimensions, and the merging of two stable objects in two spatial dimensions. We study the quantum corrections to the classical field theory in each case. We find that quantum corrections grow exponentially during nonlinear growth with the timescale being approximately equal to the system dynamical time. In stable systems the corrections grow quadratically. We also find that the primary effect of quantum corrections is to reduce the amplitude of fluctuations on the deBroglie scale in the spatial density. Finally, we find that the timescale associated with decoherence due to gravitational coupling to Baryonic matter is at least as fast as the quantum corrections due to gravitational interactions. These results strongly imply that quantum corrections do not impact the predictions of the classical field theory.