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

TL;DR

The paper tackles the challenge of modeling the dynamic interplay between fuzzy dark matter (FDM) solitons and baryons by constructing an analytical emulator for the baryon EoM inside the soliton. It begins with stationary cylindrical Schrödinger–Poisson solutions in a Milky Way–like baryon background and generates mock data to train a genetic-algorithm–based emulator, expressing the baryon density as a function of the FDM density and total potential. The emulator achieves about $2.5\%$ accuracy on the mock data, and validating it within an enlarged SP system yields fractional differences $\lesssim 0.04$ relative to the fixed-background case, demonstrating reliable performance for perturbations and mild evolutions. The approach provides a practical tool to study FDM soliton evolution in evolving baryon environments, with explicit limitations for highly dynamical events such as strong collisions or tidal disruptions.

Abstract

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 cylinderical symmetric FDM soliton solution about the FDM density and the total potential of FDM and baryon. Then, we build an analytical baryon EoM from the obtained FDM density and total potential by machine learning. Finally, we check that this baryon EoM works as well as an empirical baryon density profile for the FDM soliton formation, with the fractional errors $\lesssim0.04$. It should also work well for some other simple FDM soliton evolutions.

Figures (6)

• Figure 1: FDM soliton density contours in the inner Milky Way, which is consistent with Fig. 2 of Bar:2019bqz. The corresponding dimensionless total baryon mass is $\tilde{M}_{\rm b}=1.03$.
• Figure 2: Total potential contours inside the FDM soliton, where $\tilde{\Phi}(\tilde{R}=0,\tilde{z}=0)=-2.75395$ which is accompanied by $\tilde{\rho}_{\rm b}(\tilde{R}=0,\tilde{z}=0)=21551.2$ and $\tilde{\phi}^2(\tilde{R}=0,\tilde{z}=0)=1$.
• Figure 3: Accuracy of the analytical emulator for baryon EoM as a function of $\theta$.
• Figure 4: Comparison between the analytical emulator for baryon EoM $\tilde{\rho}_{\rm b,EoM}(x,\theta)$ (colored points) and the mock data $\tilde{\rho}_{\rm b}(x,\theta)$ (black points) at four angles $\theta=\{0^{\circ},30^{\circ},60^{\circ},90^{\circ}\}$ respectively. The unphysical jump in $\tilde{\rho}_{\rm b}(x,\theta)$ for larger $\theta$ comes from the cutoff of Eq. (\ref{['eq:emp']}), which decreases the accuracy to some extent.
• Figure 5: Distribution of fractional error of $\delta\rho(R,z)$.