Core-Halo Mass Relation in Scalar Field Dark Matter Models and its Consequences for the Formation of Supermassive Black Holes

Scalar-field dark matter (SFDM) halos exhibit a core-envelope structure with soliton-like cores and CDM-like envelopes. Simulations without self-interaction (free-field case) report a core-halo mass relation $M_c\propto M_{h}^{\beta}$, with either $\beta=1/3$ or $\beta=5/9$, which can be understood if core and halo obey certain energy or velocity scalings. We extend the core-halo mass relations to include SFDM with self-interaction (SI), either repulsive or attractive, and investigate its implications for the gravitational instability and collapse of solitonic cores, leading to supermassive black hole (SMBH) formation. For SFDM parameters that make $\sim$ Kpc-sized cores and CDM-like structure formation on large scales but suppressed on small scales, cores are stable for all galactic halos of interest, from the free-field to the repulsive SI limit. For attractive SI, however, halos masses $M_h\sim (10^{10}-10^{12}) M_\odot$ have cores that collapse to SMBHs with $M_{SMBH}\sim 10^{6}-10^8 M_\odot$, as observations seem to require, while smaller-mass halos have stable cores, for particle masses $m\simeq (2.14\times 10^{-22}-9.9\times 10^{-20})\rm{eV}/c^2$, if the free-field has $\beta=1/3$, or $m = 2.23\times 10^{-21}-1.7\times 10^{-18}\rm{eV}/c^2$, if $\beta=5/9$. For free-field and repulsive cases, however, if previous constraints on particle parameters are relaxed to allow much smaller (sub-galactic scale) cores, then halos can also form SMBHs, for the same range of halo and BH masses, as long as $\beta = 5/9$ is correct for the free-field. In that case, structure formation in SFDM would be largely indistinguishable from Cold Dark Matter (CDM). Such SFDM models might not resolve the small-scale structure problems of CDM, but they would explain the formation of SMBHs quite naturally. Since CDM, itself, has not yet been ruled out, such SFDM models must also be viable (Abbreviated).