Predictive model of BEC dark matter halos with a solitonic core and an isothermal atmosphere

We develop a model of Bose-Einstein condensate dark matter halos with a solitonic core and an isothermal atmosphere based on a generalized Gross-Pitaevskii equation [P. H. Chavanis, Eur. Phys. J. Plus 132, 248 (2017)]. This equation provides a heuristic coarse-grained parametrization of the ordinary GrossPitaevskii equation accounting for violent relaxation and gravitational cooling. It involves a cubic nonlinearity taking into account the self-interaction of the bosons, a logarithmic nonlinearity associated with an effective temperature, and a source of dissipation. It leads to superfluid dark matter halos with a core-halo structure. The quantum potential or the self-interaction of the bosons generates a solitonic core that solves the cusp problem of the cold dark matter model. The logarithmic nonlinearity generates an isothermal atmosphere accounting for the flat rotation curves of the galaxies. The dissipation ensures that the system relaxes towards an equilibrium configuration. In the Thomas-Fermi approximation, a dark matter halo is equivalent to a barotropic gas with an equation of state P = 2 pi a(s)h(2)/rho(2)m(3)+ pk(B)T/m, where a(s) is the scattering length of the bosons and m is their individual mass. We numerically solve the equation of hydrostatic equilibrium and determine the density profiles and rotation curves of dark matter halos. We impose that the surface density of the halos has the universal value Sigma(0) =rho(0)r(h) = 141 M-circle dot/pc(2) obtained from the observations. For a boson with ratio a(s)/m(3) = 3.28 x 10(3) fm/(eV/c(2))(3), we find a minimum halo mass (M-h)(min) = 1.86 x 10(8) M-circle dot and a minimum halo radius (r(h))(min) = 788 pc. This ultracompact halo corresponds to a pure soliton which is the ground state of the Gross-Pitaevskii-Poisson equation. For (M-h)(min) h, < (M-h), = 3.30 x 10(9) M circle dot the soliton is surrounded by a tenuous isothermal atmosphere. For M-h > (M-h)(c) we find two branches of solutions corresponding to (i) purely isothermal halos without soliton and (ii) isothermal halos harboring a central soliton. The purely isothermal halos (gaseous phase) are stable. For M-h> (M-h)(c) = 6.86 x 10(10) M-circle dot they arc indistinguishable from the observational Burkert profile. For (M-h) * < M-h < (M-h)(c), the deviation from the isothermal law (most probable state) may be explained by incomplete violent relaxation, tidal effects, or stochastic forcing. The isothermal halos harboring a central soliton (core-halo phase) are canonically unstable (having a negative specific heat) but they are microcanonically stable so they are long-lived. By extremizing the free energy with respect to the core mass, we find that the core mass scales as M-c/(M-h)(min) = 0.626(M-h/(M-h)(min))(1/2) In(M-h/(M-h)(min)). For a halo of mass M-h= 10(12) M-circle dot, similar to the mass of the dark matter halo that surrounds our Galaxy, the solitonic core has a mass M-c= 6.39 x 10(10) M-circle dot and a radius R-c = 1 kpc. The solitonic core cannot mimic by itself a supermassive black hole at the center of the Galaxy but it may represent a large bulge which is either present now or may have, in the past, triggered the collapse of the surrounding gas, leading to a supermassive black hole and a quasiu. On the other hand, we argue that large halos with a mass M-h > 10(12) M-circle dot may undergo a gravothermal catastrophe leading ultimately to the formation of a supermassive black hole (for smaller halos, the gravothermal catastrophe is inhibited by quantum effects). We relate the bifurcation point and the point above which supermassive black holes may form to the canonical and microcanonical critical points (M-h)(CCP) = 3.27 x 10(9) M-circle dot and (M-h)(MCP) similar to 2 x 10(12) M-circle dot of the thermal self-gravitating bosonic gas. Our model has no free parameter so it is completely predictive. Extension of this model to noninteracting bosons and fermions will be presented in forthcoming papers.