Fermion-Boson Stars as Attractors in Fuzzy Dark Matter and Ideal Gas Dynamics

Abstract

In the context of Fuzzy Dark Matter (FDM) we study the core formation in the presence of an Ideal Gas (IG). Our analysis is based on the solution of the Schrödinger-Poisson-Euler system of equations that drives the evolution of FDM together with a compressible IG, both coupled through the gravitational potential they produce. Starting from random initial conditions for both FDM and IG, we study the evolution of the system until it forms a nearly relaxed, virialized and close to hydrostatic equilibrium core, surrounded by an envelope of the two components. We find that the core corresponds to Newtonian Fermion-Boson Stars (FBS). If the IG is used to model luminous matter, our results indicate that FBS behave as attractor core solutions of structure formation of FDM along with visible matter.

Figures (5)

• Figure 1: Evolution of $\rho_{FDM}$ and $\rho$ densities for the simulations with $MR=1$, described by color maps and isocontours respectively. These plots are centered at the maximum of the FDM density for ease of illustration. Each column presents snapshots at times $t = 0$, 7, 14, 50, and 100, while each row corresponds to simulations with polytropic constants $K = 0.1$, 1.0, and 10. Similar results are found for $MR=0.1$ and 10 and shown in the SM for comparison SupplementaryMaterial.
• Figure 2: Maximum of $\rho_{FDM}$ (left) and $\rho$ (right) as function of time. Top, middle and bottom rows correspond to initial polytropic constants $K = 0.1$, 1, and 10. Blue, orange and green lines indicate the cases with mass ratios $MR = 0.1$, 1 and $10$.
• Figure 3: Angular average of $\rho_{FDM}$ at the left and $\rho$ at the right for the case $MR=1$ and $K=0.1,1,10$ at time $t=100$ when the core has relaxed. These densities in the core are compared with the densities of a Newtonian FBS. Keeping in mind that FBS are constructed using a polytropic EoS Alvarez_Rios_2023, we find that the FBSs that fit these relaxed densities of the FDM-IG core have polytropic constant $K_{fit}\sim 9.25, 11.00, 23.06$ for each value of the initial $K$. These results indicate that the core of FDM-IG core approaches a stable FBS, with radial-hydrostatic equilibrium and entropy nearly conserved, which justifies the attractor nature of FBS. Simulations for other values of $MR$ have similar fits. For this type of fitting we do not use a phenomenological universal formula describing the densities, instead we solve the eigenvalue problem of FBS many times and search the fitting parameters using a Genetic Algorithm.
• Figure 4: Maximum of $\rho_{FDM}$ (left) and $\rho$ (right) as function of time. Blue, orange and green lines indicate the cases with mass ratios $\sigma = 0.5$, $1$, and 10, respectively.
• Figure 5: Time evolution of the FDM and IG properties as function of time for the simulations with $MR=1$. These plots illustrate the relaxation process of the FDM-IG system, that leads to the formation of a Newtonian FBS. For the FDM component, we show the kinetic energy $K_{\text{FDM}}$, gravitational energy $W_{\text{FDM}}$, total energy $E_{\text{FDM}}$, and the virial scalar $Q_{\text{FDM}}$, all normalized by the initial absolute total energy $|E_{\text{FDM}}(0)|$. For the IG component, we display the kinetic energy $K_{\text{IG}}$, gravitational energy $W_{\text{IG}}$, internal energy $U_{\text{IG}}$, total energy $E_{\text{IG}}$, and the virial scalar $Q_{\text{IG}}$, each one normalized by the initial absolute total energy $|E_{\text{IG}}(0)|$. These energy diagnostics emphasize the stabilization of the system into a virialized configuration, with both FDM and IG components approaching stable energy values over time. Finally $Q_{FDM}\sim 0$ and $Q_{IG}\sim 0$ with time, which indicates that the two components evolve near a virialized state separately. Similar results are found for $MR=0.1$ and 10 in the SM SupplementaryMaterial.