The Schive soliton profile and its derivation

The ground-state problem

The soliton is the lowest-energy configuration of the wavefunction $\psi$ obeying the Schrödinger–Poisson (SP) equations (Topic 4). We seek a stationary state oscillating only in phase,

which turns SP into a coupled nonlinear eigenvalue problem for the profile $\phi(r)$ and potential $V(r)$:

The nonlinearity — $V$ sourced by $\phi^2$, acting back on $\phi$ — is what makes the soliton self-bound rather than a spreading packet. There is no closed form; one solves it numerically (Topic 5.5).

Scaling symmetry: one solution generates all

SP has a scaling symmetry: if $\{\phi(r),V(r),\gamma\}$ solves it, so does $\phi\to\lambda^2\phi(\lambda r)$, $V\to\lambda^2V(\lambda r)$, $\gamma\to\lambda^2\gamma$ for any $\lambda>0$. A single computed ground state, rescaled, gives the whole one-parameter family. Since $\rho=m\phi^2$ scales as $\lambda^4$ and length as $\lambda^{-1}$,

the origin of the $M^4$ law.

The Schive fitting profile

Schive, Chiueh & Broadhurst (2014) found the numerical ground state is captured to $<1\%$ inside a few core radii by

with $r_c$ the radius where the density falls to half its central value. The high power (8) makes the profile flat-topped in the centre and steep ($\rho\propto r^{-16}$) outside — nothing like the $r^{-1}$ cusp of cold dark matter. The constant $0.091$ is fixed by the ground-state shape; only $\rho_c$ (or $r_c$) varies between solitons.

Mass, radius and boson mass

Integrating gives the core mass $M_c=4\pi\cdot0.9220\,\rho_c r_c^3$, and combining with the scaling symmetry yields the relations used throughout the campaign,

with $m_{22}=m/10^{-22}\,$eV. A heavier boson makes smaller, denser cores; a more massive core is smaller. The figure shows the universal shape in scaled units.

The M⁴ law and the family

The invariant $\rho_c r_c^4=$ const is the soliton's signature: a core four times as massive is sixteen times denser and half the size. It is sharp and falsifiable, and among the cleanest things a wave simulation can test — it needs only the correct ground-state shape.