Substructure of fuzzy dark matter haloes

We derive the halo mass function (HMF) for fuzzy dark matter (FDM) by solving the excursion set problem explicitly with a mass-dependent barrier function, which has not been done before. We find that compared to the naive approach of the Sheth--Tormen HMF for FDM, our approach has a higher cut off mass and the cut off mass changes less strongly with redshifts. Using merger trees constructed with a modified version of the Lacey & Cole formalism that accounts for suppressed small scale power and the scale-dependent growth of FDM haloes and the semi-analytic GALACTICUS code, we study the statistics of halo substructure including the effects from dynamical friction and tidal stripping. We find that if the dark matter is a mixture of cold dark matter (CDM) and FDM, there will be a suppression on the halo substructure on small scales which may be able to solve the Missing Satellites Problem faced by the pure CDM model. The suppression becomes stronger with increasing FDM fraction or decreasing FDM mass. Thus, it may be used to constrain the FDM model.