Self-gravitating Superfluids: The Gross-Pitaevskii-Poisson Framework

TL;DR

The paper surveys the Gross-Pitaevskii-Poisson equation (GPPE) framework as a unifying, nonrelativistic approach to self-gravitating superfluids, applicable from galactic dark-matter halos to the dense interiors of neutron stars. It discusses extensions that couple the condensate to gravity and, in magnetized or superconducting contexts, to Maxwell fields, enabling minimal models of vortices in neutron superfluids and flux tubes in proton superconductors. The authors highlight numerical strategies, including Fourier-truncated GPPE (T-GPPE), stochastic imaginary-time simulations (SGLP), and fully coupled neutron-proton-Maxwell-Poisson formulations, to study equilibrium, finite-temperature effects, rotation, and glitch-like dynamics. The work emphasizes how self-interaction sign and strength, along with gravitational coupling, shape halo density profiles, axion-star stability, vortex formation, and crust–superfluid coupling in pulsars, offering a flexible framework to connect theory with observations across scales.

Abstract

We provide an overview of the Gross-Pitaevskii-Poisson equation (GPPE) that is used to model self-gravitating superfluid systems, which include gravitationally collapsed boson and axion stars and dark-matter haloes. We outline how this framework can be used to develop minimal models for neutron stars and for pulsars and their glitches. We account not only for vortices in the neutron superfluid inside these stars, but also for the flux tubes in the proton-superconductor subsystem, using a coupled model with the neutron superfluid, proton superconductor, the Maxwell equations for the vector potential ${\bf A}$, and the Poisson equation for self-gravity.

Figures (9)

• Figure 1: (a) A schematic diagram of the spherical distribution of the dark-matter halo (say around the Milky Way) composed of ultra-light bosons that exhibit wave-like properties. (b) Dark-matter halo from our direct numerical simulation (DNS) of the Gross-Pitaevskii-Poisson equation \ref{['eq:GP_poisson']}.
• Figure 2: Columns 1–3 show 10-level contour plots of density $|\psi|^2$ at representative times using SGLPE (imaginary-time) in the top row and GPPE (real-time) in the bottom row, respectively. Column 4: Plots of the scaled radius of gyration $R/L$ [\ref{['eq:radius_gyration']}] versus the scaled time [adapted from Ref. AK_Verma_PhysRevResearch.3.L022016 with permission from the APS].
• Figure 3: (a) Log-linear plots of the effective potential \ref{['eq:eff_pot']} versus the axionic condensate’s radius. The minima labeled ${\bf LM}$ and ${\bf GM}$ correspond, respectively, to low- and high-density phases. (b) Plots of Eq. \ref{['eq:eff_pot']} versus the radius of gyration obtained by using the Gaussian Ansatz (red) and our DNS (black) [adapted from Ref. Shukla_PhysRevD.109.063009, with permission from the APS].
• Figure 4: Plots of the scaled radius of gyration $R/L$ versus the gravitational interaction parameter $G$ in the cq-GPPE; blue and green curves show, respectively, curves along which $G$ increases and decreases [adapted from Ref. Shukla_PhysRevD.109.063009, with permission from the APS].
• Figure 5: Contour plots of the density $|\psi|^2$, for a single rotating compact axionic object, which we obtain by solving the cq-SGLPE for (a) $\Omega=3$, (b) $\Omega=4$, and (c) $\Omega=5$. Vortices appear once $\Omega=\Omega_c$, a critical angular speed. The $z$ axis of rotation is indicated by the green arrow. (d) Plot of $\Omega_c$ versus $g_2$ [adapted from Ref. Shukla_PhysRevD.109.063009, with permission from the APS].