Self-similar solutions for fuzzy dark matter

Fuzzy dark matter (FDM) models admit self-similar solutions which are very different from the standard cold dark matter (CDM) self-similar solutions and do not converge to the latter in the semiclassical limit. In contrast with the familiar CDM hierarchical collapse, they correspond to an inverse-hierarchy blowup. Constant-mass shells start in the nonlinear regime, at early times, with small radii and high densities, and expand to reach at late times the Hubble flow, up to small linear perturbations. Thus, larger masses become linear first. This blowup approximately follows the Hubble expansion, so that the central density contrast remains constant with time, although the width of the self-similar profile shrinks in comoving coordinates. As in a gravitational cooling process, matter is ejected from the central peaks through successive clumps. As in wave systems, the velocities of the geometrical structures and of the matter do not coincide, and matter slowly moves from one clump to the next, with intermittent velocity bursts at the transitions. These features are best observed using the density-velocity representation of the nonrelativistic scalar field, or the mass-shell trajectories, than with the Husimi phase-space distribution, where an analog of the Heisenberg uncertainty principle blurs the resolution in the position or velocity direction. These behaviors are due to the quantum pressure and the wavelike properties of the Schr??dinger equation. Although the latter has been used as an alternative to N-body simulations for CDM, these self-similar solutions show that the semiclassical limit needs to be handled with care.

Automated summary

Fuzzy dark matter (FDM) models have self-similar solutions that differ greatly from the self-similar solutions of standard cold dark matter (CDM) models and do not converge to the latter in the semiclassical limit. These self-similar solutions in FDM models exhibit an inverse-hierarchy blowup, where larger masses become linear first, in contrast to the familiar CDM hierarchical collapse. This blowup process roughly follows the Hubble expansion and maintains a constant central density contrast over time, although the width of the self-similar profile shrinks in comoving coordinates.