Analytical approach to core-halo structure of fuzzy dark matter

Ultralight bosonic dark matter called fuzzy dark matter (FDM) has attracted much attention as an alternative to the cold dark matter. An intriguing feature of the FDM model is the presence of a soliton core, a stable dense core formed at the center of halos. In this paper, we analytically study the dependence of the soliton core properties on the halo characteristics by solving approximately the Schr\"odinger-Poisson equation. Focusing on the ground-state eigenfunction, we derive a key expression for the soliton core radius, from which we obtain the core-halo mass relations similar to those found in the early numerical work, but involving the factor dependent crucially on the halo concentration and cosmological parameters. Based on the new relations, we find that for a given cosmology, (i) there exist a theoretical bound on the radius and mass of soliton core for each halo mass (ii) incorporating the concentration-halo mass (C-M) relation into the predictions, the core-halo relations generally exhibit a non power-law behavior, and with the C-M relation suppressed at the low-mass scales, relevant to the FDM model, predictions tend to match the simulations well (iii) the scatter in the C-M relation produces a sizable dispersion in the core-halo relations, and can explain the results obtained from cosmological simulations. Finally, the validity of our analytical treatment are critically examined. A perturbative estimation suggests that the prediction of the core-halo relations is valid over a wide range of parameter space, and the impact of the approximation invoked in the analytical calculations is small, although it is not entirely negligible.