Galaxy formation with BECDM - I. Turbulence and relaxation of idealized haloes

We present a theoretical analysis of some unexplored aspects of relaxed Bose-Einstein condensate dark matter (BECDM) haloes. This type of ultralight bosonic scalar field dark matter is a viable alternative to the standard cold dark matter (CDM) paradigm, as it makes the same large-scale predictions as CDM and potentially overcomes CDM's small-scale problems via a galaxy-scale de Broglie wavelength. We simulate BECDM halo formation through mergers, evolved under the Schrödinger-Poisson equations. The formed haloes consist of a soliton core supported against gravitational collapse by the quantum pressure tensor and an asymptotic r^-3 NFW-like profile. We find a fundamental relation of the core-to-halo mass with the dimensionless invariant Ξ ≡ |E|/M³/(Gm/ℏ)² or M_c/M ≃ 2.6Ξ^1/3, linking the soliton to global halo properties. For r ≥ 3.5 r_c core radii, we find equipartition between potential, classical kinetic and quantum gradient energies. The haloes also exhibit a conspicuous turbulent behaviour driven by the continuous reconnection of vortex lines due to wave interference. We analyse the turbulence 1D velocity power spectrum and find a k^-1.1 power law. This suggests that the vorticity in BECDM haloes is homogeneous, similar to thermally-driven counterflow BEC systems from condensed matter physics, in contrast to a k^-5/3 Kolmogorov power law seen in mechanically-driven quantum systems. The mode where the power spectrum peaks is approximately the soliton width, implying that the soliton-sized granules carry most of the turbulent energy in BECDM haloes.