$Q$-balls, neural networks and galaxy rotation curves

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 axially-symmetric structure of disk galaxies. Motivated by this similarity, we investigate whether rotating scalar $Q$-balls can reproduce the observed rotation curves of disk galaxies. 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 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, and a reasonable fit when compared to standard dark matter profiles such as the Navarro-Frenk-White; 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.