Equations

Jaxion solves the following equations:

Fuzzy Dark Matter

Fuzzy dark matter is represented by the wave function, \(\psi\), normalized such that its density is \(\rho_{\rm dm}=|\psi|^2\). The field has a boson mass of \(m\) and is evolved according to the Schrödinger–Poisson equations:

\[i\,\frac{\partial \psi}{\partial t} = -\frac{\hbar}{2m}\,\nabla^2 \psi + \frac{m}{\hbar}\,V \psi\]

\[\nabla^2 V = 4\pi G \left(\rho_{\rm tot} - \overline{\rho}_{\rm tot}\right)\]

where the gravitational potential \(V\) is sourced by the total density of matter: \(\rho_{\rm tot}=\rho_{\rm dm}+\rho_{\rm gas}+\rho_{\rm stars}\), and \(\overline{\rho}_{\rm tot}\) is the mean density in the periodic box.

For cosmological simulations, one can define:

\[\tilde{\mathbf{r}} = a^{-1}\mathbf{r}, \qquad d\tilde{t} = a^{-2} dt, \qquad \tilde{\psi} = a^{3/2}e^{-imHr^2/(2\hbar)}\psi, \qquad \tilde{V} = a^2 V\]

to rewrite the equations as:

\[i\,\frac{\partial \tilde{\psi}}{\partial \tilde{t}} = -\frac{\hbar}{2m}\,\tilde{\nabla}^2 \tilde{\psi} + \frac{m}{\hbar}\,\tilde{V} \tilde{\psi}\]

\[\tilde{\nabla}^2 \tilde{V} = 4\pi G a \left(\tilde{\rho}_{\rm tot} - \overline{\tilde{\rho}}_{\rm tot}\right)\]

where \(a\) is the cosmological scale factor, \(H\equiv\dot{a}/a\) is the Hubble parameter, and \(\tilde{t}\) is super-comoving time.

Gas

For the gas, we consider the compressible isothermal Euler equations, with density \(\rho_{\rm gas}\equiv\rho\), velocity \(\mathbf{v}\), and (fixed) sound-speed \(c_s\):

\[\begin{split}\frac{\partial}{\partial t} \begin{pmatrix} \rho \\ \rho \mathbf{v} \end{pmatrix} + \nabla \cdot \begin{pmatrix} \rho \mathbf{v} \\ \rho \mathbf{v}\mathbf{v}^T + \rho c_s^2 \end{pmatrix} = \begin{pmatrix} 0 \\ -\rho \nabla V \end{pmatrix}\end{split}\]

(where the gas pressure is \(P=\rho c_s^2\)).

Stars

Finally, star particles (position \(\mathbf{x}_s\), velocity \(\mathbf{v}_s\), mass \(m_s\)) evolve according to the collisionless Boltzmann equation:

\[\frac{d\mathbf{x}_s}{dt} = \mathbf{v}_s, \qquad \frac{d\mathbf{v}_s}{dt} = -\nabla V\]