Analytical Emulator for the Baryon Density Distribution inside the Fuzzy Dark Matter Soliton from Machine Learning

An empirical baryon density profile can be included in the Schrödinger-Poisson (SP) equations to influence the fuzzy dark matter (FDM) soliton formation. However, to probe the effects of baryon on the other dynamical evolutions of the FDM soliton, its equation of motion (EoM) inside the corresponding FDM soliton is needed. In this paper, given an empirical baryon density profile, we first provide the cylindrical symmetric FDM soliton solution about the FDM density and the total potential of FDM and baryon. Then, instead of EoM, we build an analytical emulator (AE) for the baryon density distribution from the obtained FDM density and total potential by machine learning. Finally, we check that this AE works as well as an empirical baryon density profile for the FDM soliton formation, with the fractional errors $\lesssim 0.04$. It should also work as a kind of baryon EoM for some other simple FDM soliton evolutions.