Derivation of the core mass -- halo mass relation of fermionic and bosonic dark matter halos from an effective thermodynamical model

We consider the possibility that dark matter halos are made of quantum particles such as fermions or bosons in the form of Bose-Einstein condensates. In that case, they generically have a "core-halo" structure with a quantum core that depends on the type of particle considered and a halo that is relatively independent of the dark matter particle and that is similar to the NFW profile of cold dark matter. We model the halo by an isothermal gas with an effective temperature $T$. We then derive the core mass -- halo mass relation $M_c(M_v)$ of dark matter halos from an effective thermodynamical model by extremizing the free energy $F(M_c)$ with respect to the core mass $M_c$. We obtain a general relation that is equivalent to the "velocity dispersion tracing" relation according to which the velocity dispersion in the core $v_c^2\sim GM_c/R_c$ is of the same order as the velocity dispersion in the halo $v_v^2\sim GM_v/r_v$. We provide therefore a justification of this relation from thermodynamical arguments. In the case of fermions, we obtain a relation $M_c\propto M_v^{1/2}$ that agrees with the relation found numerically by Ruffini {\it et al.}. In the case of noninteracting bosons, we obtain a relation $M_c\propto M_v^{1/3}$ that agrees with the relation found numerically by Schive {\it et al.}. In the case of bosons with a repulsive self-interaction in the Thomas-Fermi limit, we predict a relation $M_c\propto M_v^{2/3}$ that still has to be confirmed numerically. We also obtain a general approximate core mass -- halo mass relation that is valid for bosons with arbitrary repulsive or attractive self-interaction. For an attractive self-interaction, we determine the maximum halo mass that can harbor a stable quantum core (dilute axion "star")

Phase transitions between dilute and dense axion stars

We study the nature of phase transitions between dilute and dense axion stars interpreted as self-gravitating Bose-Einstein condensates. We develop a Newtonian model based on the Gross-Pitaevskii-Poisson equations for a complex scalar field with a self-interaction potential $V(|\psi|^2)$ involving an attractive $|\psi|^4$ term and a repulsive $|\psi|^6$ term. Using a Gaussian ansatz for the wave function, we analytically obtain the mass-radius relation of dilute and dense axion stars for arbitrary values of the self-interaction parameter $\lambda\le 0$. We show the existence of a critical point $|\lambda|_c\sim (m/M_P)^2$ above which a first order phase transition takes place. We qualitatively estimate general relativistic corrections on the mass-radius relation of axion stars. For weak self-interactions $|\lambda|<|\lambda|_c$, a system of self-gravitating axions forms a stable dilute axion star below a general relativistic maximum mass $M_{\rm max,GR}^{\rm dilute}\sim M_P^2/m$ and collapses into a black hole above that mass. For strong self-interactions $|\lambda|>|\lambda|_c$, a system of self-gravitating axions forms a stable dilute axion star below a Newtonian maximum mass $M_{\rm max,N}^{\rm dilute}=5.073 M_P/\sqrt{|\lambda|}$, collapses into a dense axion star above that mass, and collapses into a black hole above a general relativistic maximum mass $M_{\rm max,GR}^{\rm dense}\sim \sqrt{|\lambda|}M_P^3/m^2$. Dense axion stars explode below a Newtonian minimum mass $M_{\rm min,N}^{\rm dense}\sim m/\sqrt{|\lambda|}$ and form dilute axion stars of large size or disperse away. We determine the phase diagram of self-gravitating axions and show the existence of a triple point $(|\lambda|_*,M_*/(M_P^2/m))$ separating dilute axion stars, dense axion stars, and black holes. We make numerical applications for QCD axions and ultralight axions

Statistical mechanics of 2D turbulence with a prior vorticity distribution

We adapt the formalism of the statistical theory of 2D turbulence in the case where the Casimir constraints are replaced by the specification of a prior vorticity distribution. A phenomenological relaxation equation is obtained for the evolution of the coarse-grained vorticity. This equation monotonically increases a generalized entropic functional (determined by the prior) while conserving circulation and energy. It can be used as a thermodynamical parametrization of forced 2D turbulence, or as a numerical algorithm to construct (i) arbitrary statistical equilibrium states in the sense of Ellis-Haven-Turkington (ii) particular statistical equilibrium states in the sense of Miller-Robert-Sommeria (iii) arbitrary stationary solutions of the 2D Euler equation that are formally nonlinearly dynamically stable according to the Ellis-Haven-Turkington stability criterion refining the Arnold theorems.Comment: Proceedings of the international conference "Euler Equations: 250 Years" (Aussois, France 18-23 June 2007). Edited by G. Eyink, U. Frisch, R. Moreau and A. Sobolevsk

Kinetic theory of spatially inhomogeneous stellar systems without collective effects

We review and complete the kinetic theory of spatially inhomogeneous stellar systems when collective effects (dressing of the stars by their polarization cloud) are neglected. We start from the BBGKY hierarchy issued from the Liouville equation and consider an expansion in powers of 1/N in a proper thermodynamic limit. For $N\rightarrow +\infty$, we obtain the Vlasov equation describing the evolution of collisionless stellar systems like elliptical galaxies. At the order 1/N, we obtain a kinetic equation describing the evolution of collisional stellar systems like globular clusters. This equation does not suffer logarithmic divergences at large scales since spatial inhomogeneity is explicitly taken into account. Making a local approximation, and introducing an upper cut-off at the Jeans length, it reduces to the Vlasov-Landau equation which is the standard kinetic equation of stellar systems. Our approach provides a simple and pedagogical derivation of these important equations from the BBGKY hierarchy which is more rigorous for systems with long-range interactions than the two-body encounters theory. Making an adiabatic approximation, we write the generalized Landau equation in angle-action variables and obtain a Landau-type kinetic equation that is valid for fully inhomogeneous stellar systems and is free of divergences at large scales. This equation is less general than the Lenard Balescu-type kinetic equation recently derived by Heyvaerts (2010) since it neglects collective effects, but it is substantially simpler and could be useful as a first step. We discuss the evolution of the system as a whole and the relaxation of a test star in a bath of field stars. We derive the corresponding Fokker-Planck equation in angle-action variables and provide expressions for the diffusion coefficient and friction force

Gravitational phase transitions with an exclusion constraint in position space

We discuss the statistical mechanics of a system of self-gravitating particles with an exclusion constraint in position space in a space of dimension $d$. The exclusion constraint puts an upper bound on the density of the system and can stabilize it against gravitational collapse. We plot the caloric curves giving the temperature as a function of the energy and investigate the nature of phase transitions as a function of the size of the system and of the dimension of space in both microcanonical and canonical ensembles. We consider stable and metastable states and emphasize the importance of the latter for systems with long-range interactions. For $d\le 2$, there is no phase transition. For $d>2$, phase transitions can take place between a "gaseous" phase unaffected by the exclusion constraint and a "condensed" phase dominated by this constraint. The condensed configurations have a core-halo structure made of a "rocky core" surrounded by an "atmosphere", similar to a giant gaseous planet. For large systems there exist microcanonical and canonical first order phase transitions. For intermediate systems, only canonical first order phase transitions are present. For small systems there is no phase transition at all. As a result, the phase diagram exhibits two critical points, one in each ensemble. There also exist a region of negative specific heats and a situation of ensemble inequivalence for sufficiently large systems. By a proper interpretation of the parameters, our results have application for the chemotaxis of bacterial populations in biology described by a generalized Keller-Segel model including an exclusion constraint in position space. They also describe colloids at a fluid interface driven by attractive capillary interactions when there is an excluded volume around the particles. Connexions with two-dimensional turbulence are also mentioned