Dark matter by design: $Q$-balls, neural networks and galaxy rotation curves

TL;DR

The paper investigates whether rotating scalar Q-balls can account for disk-galaxy dark-matter halos by solving for static and rotating configurations under two self-interactions: a sextic polynomial potential and an axion-like periodic potential. It introduces SpectralPINN, a hybrid solver that embeds a pseudo-spectral basis within a physics-informed neural-network framework to efficiently construct smooth Q-ball solutions, then maps the energy density to galactic rotation curves using Newtonian gravity in the equatorial plane. By fitting to a carefully curated SPARC sample, the study finds that both potentials can reproduce RCs reasonably well, with the axion-like case mildly preferred and particle masses clustered around $m \\sim 10^{-27}$ eV, consistent with galactic-scale solitons. The work highlights rotating Q-balls as viable solitonic DM candidates on galactic scales and showcases SpectralPINN as a powerful forward-modeling tool, while recognizing the simplifying assumptions (neglecting self-gravity) and outlining paths for more complete future analyses.

Abstract

Can a dynamically robust (\textit{aka} stable) $Q$-ball reproduce the rotation curve of a disk galaxy? In an astrophysical environment, $Q$-balls are non-topological solitons that are transparent and only perceived by their gravitational effects. Traditionally, scalar $Q$-balls are modelled with a polynomial potential, but axion-like periodic potentials are also expected to support such solitonic configurations. In the presence of angular momentum, $Q$-balls acquire a toroidal structure with a central density void, qualitatively resembling the structure of disk galaxies. Motivated by this similarity, we investigate whether rotating scalar $Q$-balls can reproduce the observed galactic rotation curves. In this work, we use a recently developed hybrid numerical framework that combines a high-accuracy pseudo-spectral method with a physics-informed neural network approach to construct both static and rotating $Q$-ball solutions. We then assess their ability to act as the dark matter halos in galaxies by fitting the observed rotation curves of a sample of disk galaxies from the SPARC catalogue. Our simplified model provides an overall good agreement with observational data; we have further found an average constraint on the scalar field particle's mass $m\sim 10^{-27}$ eV, in agreement with similar galactic-scale soliton solutions.

Figures (6)

• Figure 1: Schematic representation of the SpectralPINN network architecture, where each neuron corresponds to a basis function, with each layer associated with a coordinate dependence. Scheme inspired by luna2023solving.
• Figure 2: Radial profiles for $\omega/\mu = 0.90$ in code units ($\mu = 1$) of the scalar field $\phi(r)$ (solid blue), the normalized energy density $\rho(r)/\rho_0$ (dashed green), the differential charge density $Q'(r)/Q_{\rm tot}$ (dotted red), and the potential left: $U_P$; right: $U_A$ (dot-dashed yellow). Top: spherical ($m = 0$) solutions. Bottom: spinning ($m = 1$) solutions evaluated at the equator ($\theta = \pi/2$). Here $\rho_0$ denotes the maximum of the energy density $\rho(r)$.
• Figure 3: 2D density profiles $\rho(r, \theta)$ for $\omega/\mu=0.90$ and $m=1$, in code units ($\mu = 1$). Left: $U_P$. Right: $U_A$.
• Figure 4: Domain of existence in terms of the energy, $E(\omega)$ (solid blue), and Noether charge, $Q(\omega)$ (dashed red), for: Top spherical ($m=0$); Bottom: spinning $m=1$$Q$-ball solutions. Left: polynomial potential, $U_P$; Right: axion-like potential, $U_A$. All the quantities are in code units.
• Figure 5: Example of fitted rotation curves for the galaxy NGC 4183 of the SPARC sample. The orange solid lines and the light blue region indicate the best fit model from \ref{['E7.1']} and the 68% confidence region. Blue points with error bars are the observed rotation curves. The green dashed lines represent the best-fit rotation curve contributions of the $Q$-ball, while the yellow lines refer to the combined contribution of the baryonic components. Left: axion-like potential, $U_A$. Right: polynomial potential, $U_P$.