Schr?dinger-Poisson solitons: Perturbation theory

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 dark matter scenarios. We consider the dynamical evolution of perturbations to the spherically symmetric soliton, the ground state solution to the Schr??dinger-Poisson system common to all these scenarios. We construct the eigenstates of the Schr??dinger 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. As an illustrative example, we apply our formalism to a synthetic ULDM halo. We find the soliton core accounts for approximately 30% of the halo???s wave function throughout its evolution, with higher modes accounting for the halo???s Navarro-Frenk-White skirt, and relatively little mixing between different l modes. Our results provide a new analytic approach to understanding the evolution of these systems as well as possibilities for faster approximate simulations.

Automated summary

This study investigates the dynamical evolution of perturbations to the spherically symmetric soliton in self-gravitating quantum matter. The study finds that the eigenstates of the Schr??dinger equation qualitatively capture the properties seen in full ultralight dark matter (ULDM) simulations. Additionally, the study shows that the time evolution of the gravitational potential and its impact on the perturbations can be well-described using time-dependent perturbation theory.