Turbulence and large-scale structures in self-gravitating superfluids

Abstract

We study turbulence in self-gravitating superfluids by performing direct numerical simulations of the 3D Gross-Pitaevskii-Poisson (GPP) equation, which is also a model for dark matter haloes around galaxies. In the absence of self-gravity, the spectrally truncated Gross-Pitaevskii (GP) equation shows the emergence of Kolmogorov's $5/3$ scaling in the incompressible kinetic energy spectrum. Introducing self-gravity, we observe the formation of different structures, from sheet-like to spherically collapsed structures, which introduce a minimum in the kinetic energy spectrum that corresponds to the sizes of these structures. The system shows early convergence towards statistically stationary states, which we show by the onset of thermalisation in the compressible kinetic energy spectrum, where $E_{\rm kin}^c \propto k^2$. We also show that the formation of such large-scale structures suggests that the particles (bosons) move from small to large scales through an inverse cascade, supporting a mechanism for the formation of large-scale structures, such as dark matter haloes, around our galaxy Milky Way.

Figures (8)

• Figure 1: Plots of the dispersion relation in Eq. \ref{['eq:dispersion']} showing $\omega^2$ versus the wavenumber $k$ for different values of the ratio $l=\tfrac{\xi}{\lambda_{\rm J}}$ [Eq. \ref{['eq:ratio_l']}]. The wavenumber where $\omega^2$ passes the $k$-axis gives the Jeans wavenumber $k_{\rm J}$.
• Figure 2: One level contour plots of the density $\rho = m |\psi_{\rm AR}|^2$ describing the Taylor-Green vortex flow from Eq. \ref{['eq:TG_velocity']}.
• Figure 3: Contour plots of the density $\rho=m|\psi|^2$ in the stationary states after evolving ARGL equation \ref{['eq:ARGLE']} for four values of the ratio $l=\tfrac{\xi}{\lambda_{\rm J}}$: (a)$l=0.001$, (b)$l=0.136$, (c)$l=0.145$, and (d)$l=0.155$.
• Figure 4: Contour plots of the density $\rho=m|\psi|^2$ from the real-time GPE \ref{['eq:GPE_rescaled']} simulations at three representative times for three different values of the ratio $l=\tfrac{\xi}{\lambda_{\rm J}}$: (a)- (c)$l=0.001$, (e)- (g)$l=0.145$, and (i)- (k)$l=0.155$. Figs. (d), (h), and (l) show the incompressible kinetic energy spectra $E_{\rm kin}^i$ [Eq. \ref{['eq:kinetic_spectra']}].
• Figure 5: Plots of the incompressible [solid curve] and compressible [dashed curve] kinetic energies versus time [in sec] for the two values of $l=0.001$ and $l=0.155$ corresponding to no collapse and spherical collapse, respectively.