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 a core-halo mass relation of the form M₂M₇^, with either =1/3 or =5/9. These results can be understood if the core and halo follow some special energy or velocity scaling relations. We extend these core-halo mass relations here to include the case of SFDM with self-interaction, either repulsive or attractive, and investigate its implications for the possible gravitational instability and collapse of solitonic cores, leading to the formation of supermassive black holes (SMBHs). Core sizes are set by the larger of two length scales, the de Broglie wavelength (in the free-field limit) or the radius Rₓ₅ of the (n=1) -polytrope for repulsive SFDM (in the Thomas-Fermi regime), depending upon particle mass m and interaction strength. For parameters selected by previous literature to make approximately Kpc-sized cores and CDM-like structure formation on large scales but suppressed on small scales, we find that cores are stable for all galactic halos of interest, from the free-field to the repulsive Thomas-Fermi limit. For attractive self-interaction in this regime, however, halos of mass M₇10^10--10^12 M_ have cores that collapse to form seed SMBHs with Mₒ₌₁₇10^6--10^8 M_, as observations seem to require, while smaller-mass halos have stable cores, for particle masses m=2. 1410^-22--9. 910^-20 eV/c^2, if the free-field limit has =1/3, or m=2. 2310^-21--1. 710^-18 eV/c^2, if =5/9. We also place bounds on for this case. For free-field and repulsive cases, if previous constraints on particle parameters are relaxed to allow much smaller (subgalactic scale) cores, then halos can also form SMBHs, for the same range of halo and black hole masses, as long as =5/9 is correct for the free-field limit. In that case, structure formation in SFDM would be largely indistinguishable from that in CDM. As such, while these SFDM models might not help to resolve the small-scale structure problems of CDM, they would explain the formation of SMBHs quite naturally, which is otherwise not a direct feature of CDM. Since CDM, itself, has not yet been ruled out, such SFDM models must also be viable.