Bose–Einstein condensates & the Gross–Pitaevskii equation

Fuzzy dark matter is a cosmic Bose–Einstein condensate: an astronomical number of ultralight bosons sharing a single quantum state, described by one classical wavefunction. This article introduces the condensate and its governing equation, and shows why the occupation numbers are so vast that the wave is effectively classical.

A macroscopic quantum state

At low temperature and high density, identical bosons pile into the same lowest-energy state — a Bose–Einstein condensate — whose entire many-body physics collapses onto a single complex field $\psi(\mathbf x,t)$, the condensate wavefunction. Ultralight dark matter is the extreme case: bosons of mass $\sim10^{-22}$ eV at cosmic densities have enormous phase-space occupancy, forming a condensate that spans the Universe.

The Gross–Pitaevskii equation

A condensate evolves by the Gross–Pitaevskii (GP) equation — the Schrödinger equation plus a self-interaction term:

$$i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi+g|\psi|^2\psi.$$

In a laboratory BEC $V$ is the trap and $g$ encodes short-range contact interactions. For the simplest fuzzy dark matter the contact term is negligible ($g\approx0$) and the dominant "interaction" is gravity: $V$ is the potential the condensate itself sources through Poisson's equation. Dropping $g$ and closing the loop with gravity turns GP into the Schrödinger–Poisson system (Topic 4) — the equations all our wave codes solve.

Why FDM is a classical field

Although born of quantum mechanics, the FDM field behaves classically, because its mode occupation numbers are astronomical.

Worked example — the occupation number

The number of bosons within a de Broglie volume is $N\sim(\rho/m)\,\lambda_{\rm dB}^3$. At a galactic density $\rho\sim0.1\,M_\odot\,{\rm pc^{-3}}\approx7\times10^{-21}\,{\rm kg\,m^{-3}}$, with $m=1.8\times10^{-58}$ kg and $\lambda_{\rm dB}\sim1$ kpc:

$$N\sim\frac{7\times10^{-21}}{1.8\times10^{-58}}\times(3.1\times10^{19})^3\approx10^{96}.$$

With $\sim10^{96}$ quanta per mode, quantum fluctuations are utterly negligible and $\psi$ is a deterministic classical field — which is precisely why we can lay it on a grid and integrate it.

Derivation — why misalignment gives $\rho\propto a^{-3}$

Track the energy of the oscillating field after release ($H

The field obeys $\ddot\phi+3H\dot\phi+m^2\phi=0$. For $H\ll m$ this is a damped oscillator, $\phi\approx\phi_a(t)\cos(mt)$ with slowly-varying amplitude $\phi_a$.
The energy density is $\rho_\phi=\tfrac12\dot\phi^2+\tfrac12 m^2\phi^2$; averaging over a fast oscillation, $\langle\rho_\phi\rangle=\tfrac12 m^2\phi_a^2$.
Adiabatic invariance of the oscillator gives $\rho_\phi/m\propto$ (number of quanta) $\propto a^{-3}$ (quanta conserved, volume grows as $a^3$).
Since $m$ is constant, $\rho_\phi\propto a^{-3}$ — identical to pressureless matter.

So an oscillating scalar field is cold dark matter cosmologically, even though it is a wave: the fast oscillation averages to zero pressure, leaving only the $a^{-3}$ dilution of matter.

Solitons in the GP/SP system

Every self-bound BEC has a ground state — a soliton (figure) — where the kinetic (quantum-pressure) energy balances the confining potential. In the laboratory that balance is contact repulsion vs the trap; in fuzzy dark matter it is quantum pressure vs self-gravity. The FDM soliton (Topic 5) is nothing but the gravitational GP ground state.

The soliton is the ground state of the gravitational Gross–Pitaevskii (Schrödinger–Poisson) system — a self-bound condensate with a flat core, shown here in scaled units.

Bosons per de Broglie volume versus density — at galactic densities $N\sim10^{96}$, so $\psi$ behaves as a classical field.

In our research

This is the conceptual license for the whole program: fuzzy dark matter is a self-gravitating BEC, so we evolve the Gross–Pitaevskii/Schrödinger–Poisson field (JAXiON, GAMER) rather than particles. The $N\sim10^{96}$ occupancy is why treating $\psi$ as a classical field is exact for our purposes.

Key references

Pethick & Smith, Bose–Einstein Condensation in Dilute Gases — the GP equation.
Hui, Ostriker, Tremaine & Witten (2017), Phys. Rev. D 95, 043541 (arXiv:1610.08297).
Mocz et al. (2017), MNRAS 471, 4559 (arXiv:1705.05845).

PreviousThe uncertainty principle and quantum pressure NextThe Madelung (hydrodynamic) formulation