Schrödinger-Poisson Solitons: Perturbation Theory

Abstract

Self-gravitating quantum matter may exist in a wide range of cosmological and astrophysical settings from the very early universe through to present-day boson stars. Such quantum matter arises in a number of different theories, including the Peccei-Quinn axion and UltraLight (ULDM) or Fuzzy (FDM) dark matter scenarios. We consider the dynamical evolution of perturbations to the spherically symmetric soliton, the ground state solution to the Schrödinger-Poisson system common to all these scenarios. We construct the eigenstates of the Schrödinger equation, holding the gravitational potential fixed to its ground state value. We see that the eigenstates qualitatively capture the properties seen in full ULDM simulations, including the soliton "breathing" mode, the random walk of the soliton center, and quadrupolar distortions of the soliton. We then show that the time-evolution of the gravitational potential and its impact on the perturbations can be well described within the framework of time-dependent perturbation theory. Applying our formalism to a synthetic ULDM halo reveals considerable mixing of eigenstates, even though the overall density profile is relatively stable. Our results provide a new analytic approach to understanding the evolution of these systems as well as possibilities for faster approximate simulations.

Not so fuzzy: excluding FDM with sizes and stellar kinematics of ultra-faint dwarf galaxies

“…This gives time-averaged density profiles that agree reasonably well with the desired input profile. The density fluctuations in the soliton region resemble those found in full Schrödinger-Poisson simulations [26], in agreement with previous work that found that soliton fluctuations can be explained by wave interference between the lowest eigenmodes [22,23]. Our simulations find that including the soliton can enormously enhance heating of stellar orbits, depending on the soliton mass, in agreement with previous work [24].…”

“…1 shows examples. This approach is accurate to first order in perturbation theory and neglects the nonlinear interactions between different modes, which should be sufficient for our calculations, although it may be worthwhile to study whether greater accuracy can be achieved by going to higher order in time-dependent perturbation theory [23].…”

“…At very small radii, the solitons discussed above can produce additional, strong dynamical heating. Fluctuations in the soliton's size, amplitude, and centroid location, due to wave interference effects [22,23], can dominate the heating of central stars compared to the more diffusive effect described above [24]. Stellar self-gravity can suppress this effect [25,26], but in systems where gravity is completely dominated by dark matter, FDM heating can be extremely efficient.…”

Dynamical Friction From Ultralight Dark Matter

“…In a post-merger halo, the central soliton may be far from its ground state, suggesting that these effects might be substantially enhanced in astrophysical settings, resulting in the outward diffusion of light objects residing in the center of the soliton [36]. In addition, the coupling and impulsive heating associated with a single SMBH-soliton interaction could be analyzed in detail using eigenstate expansions of the soliton potential [42], facilitating the semi-analytic treatment of these systems.…”

Construction of Wave Dark Matter Halos: Numerical Algorithm and Analytical Constraints

“…This opens the door to detailed investigations of wave halo substructure. Indeed, there are a few recent papers heading in this direction: Dalal et al [60] employed the method of Widrow and Kaiser to construct wave dark matter halos, for the purpose of studying the scattering of tidal streams by the interference substructure; we in [61] studied soliton random walk and oscillations by decomposing a wave dark matter halo into its eigenmodes; Zagorac et al [62] studied the distortions of solitons using perturbation theory.…”