Analysis of Bose-Einstein condensation times for self-interacting scalar
  dark matter
  We investigate the condensation time of self-interacting axion-like particles
in a gravitational well, extending the prior work [arXiv:2007.07438] which
showed that the Wigner formalism is a good analytic approach to describe a
condensing scalar field. In the present work, we use this formalism to affirm
that $\phi^4$ self-interactions will take longer than necessary to support the
time scales associated with structure formation, making gravity a necessary
part of the process to bring axion dark matter into a solitonic form. Here we
show that when the axions' virial velocity is taken into account, the time
scale associated with self-interactions will scale as $\lambda^2$. This is
consistent with recent numerical estimates, and it confirms that the Wigner
formalism described in prior work~\cite{Relax} is a helpful analytic framework
to check computational work for potential numerical artifacts.
