Analysis of Bose-Einstein condensation times for self-interacting scalar dark matter

We investigate the condensation time of self-interacting axionlike particles in a gravitational well, extending the prior work [K. Kirkpatrick, A. E. Mirasola, and C. Prescod-Weinstein, Phys. Rev. D 102, 103012 (2020)] 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 [K. Kirkpatrick, A. E. Mirasola, and C. Prescod-Weinstein, Phys. Rev. D 102,103012 (2020)] is a helpful analytic framework to check computational work for potential numerical artifacts.