  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.
