Self-similar kinetics for gravitational Bose-Einstein condensation

TL;DR

The study addresses how a highly populated gas of gravitationally interacting bosons around a Bose star evolves toward self-similar kinetic attractors. By employing a scale-invariant Landau-type kinetic equation, the authors demonstrate the existence of a family of self-similar profiles $F_s(\omega_s)$ with a scaling dimension $D$, notably obtaining a stable attractor at $D=5/2$ in the absence of external forcing. When scale-breaking sources are present, the evolution remains close to self-similarity, described by adiabatic drift of $D(t)$, enabling practical predictions for energy growth and Bose-star condensation; the authors develop an adiabatic framework that matches conservation laws to extract the slow evolution. They extend the method to the growth of Bose stars, deriving a growth law that aligns with Schrödinger–Poisson simulations and offering a pathway to connect kinetic theory with cosmological structure formation. Overall, adiabatic self-similarity provides a powerful, generalizable tool for nonthermal gravitational kinetics and the long-time growth of Bose stars in dark-matter scenarios.

Abstract

We study an overpopulated gas of gravitationally interacting bosons surrounding a droplet of Bose-Einstein condensate - Bose star. We argue that kinetic evolution of this gas approaches with time a self-similar attractor solution to the kinetic equation. If the scale symmetry of the equation is broken by external conditions, the attractor solution exists, remains approximately self-similar, but has slowly drifting scaling dimension. The latter new regime of adiabatic self-similarity can determine growth of dark matter Bose stars in cosmological models.

Figures (13)

• Figure 1: (a) A minicluster from the simulation in Ref. Dmitriev:2023ipv. It includes a weakly bound virialized gas (clumpy spherical pattern) and a newborn Bose star (bright dot in the center). Color indicates dark matter mass density $\rho(x,\, y,\, z)$ at ${z=0}$ in dimensionless units$^{\hbox{\scriptsize }\ref{['foot-units']}}$, the inset zooms onto the Bose star. (b) Typical self-similar profile $F_s(\omega_s)$ in Eqs. (\ref{['eq:an']}), (\ref{['eq:ab']}) (log-log scale).
• Figure 2: (a) A diagram for particle condensation onto the Bose star. (b) Bose star gravitational field $U_{bs}(\bm{x})$ and its mass density $\rho_{bs}(\bm{x})$.
• Figure 3: (a) Numerical solution of the kinetic equation (\ref{['eq:13']}) with the sponge (\ref{['eq:17']}), $J_{\mathrm{ext}}=0$, and Gaussian initial data \ref{['eq:16']}. Lines show $\tilde{F}$ as a function of $\tilde{\omega}$ at different $\tau\equiv t/t_{gr}$. (b) The same solution rescaled by $\alpha(\tau)$ and $\beta(\tau)$ from Eq. (\ref{['eq:15']}) with ${D = 5/2}$ and $\tau_i \approx -2.3$. Chain points display self-similar profile $F_s(\omega_s)$ satisfying Eq. \ref{['eq:4']} with $J_{\mathrm{ext},s}=0$.
• Figure 4: (a) Kinetic evolution with the sponge (\ref{['eq:17']}) and a nonzero energy source (\ref{['eq:19']}) starting from the Gaussian initial profile \ref{['eq:16']}. Graphs show the distribution function $\tilde{F}(\tau,\, \tilde{\omega})$ at different $\tau$. (b) The same graphs rescaled with $\alpha(\tau)$ and $\beta(\tau)$ in Eq. (\ref{['eq:15']}), where ${D = 2.8}$ and ${\tau_i = -1.1}$. Chain points give self-similar profile $\tilde{F}_s(\tilde{\omega}_s)$ extracted from Eq. \ref{['eq:4']}.
• Figure 5: Typical Kolmogorov-Zakharov cascade (not to scale).