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 cold-dark matter(CDM)-like envelopes. Simulations without self-interaction (free-field case) have reported core-halo mass relation of the form ${M}_{c}\ensuremath{\propto}{M}_{h}^{\ensuremath{\beta}}$, either $\ensuremath{\beta}=1/3$ or $\ensuremath{\beta}=5/9$. These results can be understood if core halo follow some special energy velocity scaling relations. We extend these relations here to include case SFDM self-interaction, repulsive attractive, investigate its implications for possible gravitational instability collapse solitonic cores, leading formation supermassive black holes (SMBHs). Core sizes are set by larger two length scales, de Broglie wavelength (in free-field limit) radius ${R}_{\mathrm{TF}}$ ($n=1$)-polytrope Thomas-Fermi regime), depending upon particle $m$ interaction strength $\ensuremath{\lambda}$. For parameters selected previous literature make approximately Kpc-sized CDM-like on large scales but suppressed small we find that stable all galactic interest, from limit. attractive in this regime, however, ${M}_{h}\ensuremath{\sim}{10}^{10}--{10}^{12}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ seed SMBHs ${M}_{\mathrm{SMBH}}\ensuremath{\sim}{10}^{6}--{10}^{8}\text{ }{M}_{\ensuremath{\bigodot}}$, as observations seem require, while smaller-mass masses $m=2.14\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}22}--9.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ }\mathrm{eV}/{c}^{2}$, limit has $\ensuremath{\beta}=1/3$, $m=2.23\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}21}--1.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}\text{ also place bounds $\ensuremath{\lambda}$ case. cases, constraints relaxed allow much smaller (subgalactic scale) then SMBHs, same range hole masses, long $\ensuremath{\beta}=5/9$ is correct In case, would largely indistinguishable CDM. As such, models might not help resolve small-scale problems CDM, they explain quite naturally, which otherwise direct feature Since itself, yet been ruled out, such must viable.