We develop the kinetic theory of classical and quantum particles (fermions and bosons) in gravitational interaction. The kinetic theory of quantum particles may have applications in the context of dark matter. For simplicity, we consider an infinite and spatially homogeneous system (or make a local approximation) and neglect collective effects. This leads to the quantum Landau equation derived heuristically in [Chavanis, Physica A 332, 89 (2004)]. We establish its main properties: conservation laws, $H$-theorem, equilibrium state, relaxation time, quantum diffusion and friction coefficients, quantum Rosenbluth potentials, self-consistent evolution, (thermal) bath approximation, quantum Fokker-Planck equation, quantum King model... For bosonic particles, the Landau equation can describe the process of Bose-Einstein condensation. We discuss the relation of our study with the works of [Levkov et al., Phys. Rev. Lett. 121, 151301 (2018); Bar-Or et al., Astrophys. J. 871, 28 (2019)] on fuzzy dark matter halos and the formation of Bose stars and solitons.
The current paradigm for understanding galaxy formation in the universe depends on the existence of self-gravitating collisionless dark matter. Modeling such dark matter systems has been a major focus of astrophysicists, with much of that effort directed at computational techniques. Not surprisingly, a comprehensive understanding of the evolution of these self-gravitating systems still eludes us, since it involves the collective nonlinear dynamics of many-particle systems interacting via long-range forces described by the Vlasov equation. As a step towards developing a clearer picture of colli…
We examine the evolution of non-relativistic cold dark matter gravitationally coupled to baryons with modes deep inside the Hubble radius (sub-horizon regime) using akinetic theory approach within the realm of Newtonian theory. We obtain the general solution for the total density perturbation and we also show that a baryon perturbation catches up with the dark matter perturbation at late times, which in turn makes possible the formation of bound structures. We extend the linear perturbation analysis by considering the turn-around event, the collapse of matter, and its virialization process.
By solving analytically the various types of Lane-Emden equations with rotation, we have discovered two new coupled fundamental properties of rotating, self-gravitating, gaseous disks in equilibrium: Isothermal disks must, on average, exhibit strict power-law density profiles in radius $x$ on their equatorial planes of the form $A x^{k-1}$, where $A$ and $k-1$ are the integration constants, and "flat" rotation curves precisely such as those observed in spiral galaxy disks. Polytropic disks must, on average, exhibit strict density profiles of the form $\left[\ln(A x^k)\right]^n$, where $n$ is t…
We present an innovative approach to the methodology of dynamical modelling, allowing practical reconstruction of the underlying dark matter mass without assuming both the density and anisotropy functions. With this, the mass-anisotropy degeneracy is reduced to simple model inference, incorporating the uncertainties inherent with observational data, statistically circumventing the mass-anisotropy degeneracy in spherical collisionless systems. We also tackle the inadequacy that the Jeans method of moments has on small datasets, with the aid of Generative Adversarial Networks: we leverage the po…
A new family of nonrelativistic, Newtonian, non-quantum equilibrium configurations describing galactic halos is introduced, by considering strange quark matter conglomerates with masses larger than about 8 GeV as new possible components of the dark matter. Originally introduced to explain the state of matter in neutron stars, such conglomerates may also form in the high-density and temperature conditions of the primordial Universe and then decouple from ordinary baryonic matter, providing the fundamental components of dark matter for the formation of pristine gravitational potential wells and …
We study gravitational collapse with anisotropic pressures, whose end stage can mimic space-times that are seeded by galactic dark matter. To this end, we identify a class of space-times (with conical defects) that can arise out of such a collapse process, and admit stable circular orbits at all radial distances. These have a naked singularity at the origin. An example of such a space-time is seen to be the Bertrand space-time discovered by Perlick, that admits closed, stable orbits at all radii. Using relativistic two- fluid models, we show that our galactic space-times might indicate exotic …
We study the Jeans mass-radius relation of Bose-Einstein condensate dark matter in Newtonian gravity. We show at a general level that it is similar to the mass-radius relation of Bose-Einstein condensate dark matter halos [P.H. Chavanis, Phys. Rev. D {\bf 84}, 043531 (2011)]. Bosons with a repulsive self-interaction generically evolve from the Thomas-Fermi regime to the noninteracting regime as the Universe expands. In the Thomas-Fermi regime, the Jeans radius remains approximately constant while the Jeans mass decreases. In the noninteracting regime, the Jeans radius increases while the Jeans…
Self-gravitating Newtonian systems consisting of a very large number of particles have generally defied attempts to describe them using statistical mechanics. This is paradoxical since many astronomical systems, or simulations thereof, appear to have universal, equilibrium structures for which no physical basis exist. A decade ago we showed that extremizing the number of microstates with a given energy per unit mass, under the constraints of conserved total energy and mass, leads to the maximum entropy state, $n(E) \propto \exp (-β(E-Φ_0))-1$, known as DARKexp. This differential energy distrib…
The field which binds a thermal fermionic cloud is defined as a Hartree integral upon its density. In turn, the density results from the field via a Thomas-Fermi occupation of the local phase space. This defines a complete theory of all properties and observables for the cloud. As an application to dark matter halos, comparisons with astronomic data on dwarf spheroidal galaxies are provided and discussed. Estimates of the elementary fermion mass are obtained, serving as a phase-space bound on fermionic dark matter.
General self-consistent expressions for the coefficients of diffusion and dynamical friction in a stable, bound, multicomponent self-gravitating and inhomogeneous system are derived. They account for the detailed dynamics of the colliding particles and their self-consistent dressing by collective gravitational interactions. The associated Fokker-Planck equation is shown to be fully consistent with the corresponding inhomogeneous Balescu-Lenard equation and, in the weak self-gravitating limit, to the inhomogeneous Landau equation. Hence it provides an alternative derivation to both and demonstr…
Observational studies of nearby galaxies have demonstrated correlations between the mass of the central supermassive black holes (BHs) and properties of the host galaxies, notably the stellar bulge mass or central stellar velocity dispersion. Motivated by these correlations, the theoretical paradigm has emerged, in which BHs and bulges co-evolve. However, this picture was challenged by observational and theoretical studies, which hinted that the fundamental connection may be between BHs and dark matter halos, and not necessarily with their host galaxies. Based on a study of 3130 elliptical gal…
The problem of modeling the distribution of dark matter in galaxies in terms of equilibrium configurations of collisionless self-gravitating quantum particles is considered. We first summarize the pioneering model of a Newtonian self-gravitating Fermi gas in thermodynamic equilibrium developed by Ruffini and Stella (1983), which is shown to be the generalization of the King model for fermions. We further review the extension of the former model developed by Gao, Merafina and Ruffini (1990), done for any degree of fermion degeneracy at the center ($θ_0$), within general relativity. Finally, we …
In this Letter, we investigate the stability of the statistical equilibrium of spherically symmetric collisionless self-gravitating systems. By calculating the second variation of the entropy, we find that perturbations of the relevant physical quantities should be classified as long- and short-range perturbations, which correspond to the long- and short-range relaxation mechanisms, respectively. We show that the statistical equilibrium states of self-gravitating systems are neither maximum nor minimum, but complex saddle-point entropy states, and hence differ greatly from the case of ideal ga…
We study the vertical structure of polytropic, $P\propto ρ^Γ$, centrifugally-supported gaseous discs embedded in cold dark matter (CDM) halos. At fixed radius $R$, the shape of the vertical density profile depends only weakly on whether the disc is self-gravitating (SG) or not (NSG). The disc thickness, set by the midplane sound speed and circular velocity, $(c_s/V_c)R$, in the NSG case, and by the sound speed and surface density, $c_s^2/GΣ$, in SG discs, is smaller than either of these scales. SG discs are typically Toomre unstable, NSG discs are stable. Exponential discs in CDM halos with ro…
Essay selected for Honorable mention 2014 by the Gravity Research Foundation. We study an isothermal system of semi-degenerate self-gravitating fermions in General Relativity (GR). The most general solutions present mass density profiles with a central degenerate compact core governed by quantum statistics followed by an extended plateau, and ending in a power law behaviour $r^{-2}$. By fixing the fermion mass $m$ in the keV regime, the different solutions depending on the free parameters of the model: the degeneracy and temperature parameters at the center, are systematically constructed alon…
MOdified Gravity (MOG) is a covariant modification of Einstein's general relativity. This theory is one of the current alternatives to dark matter models. We describe dynamics of collisionless self-gravitating systems in the context of MOG. By studying the weak field approximation of this theory, we derive the equations governing the dynamics of the self-gravitating systems. More specifically, we consider the Jeans instability for self-gravitating fluid and stellar systems, and derive new Jeans mass limit $\tilde{M}_J$ and wave-number $\tilde{k}_J$. Furthermore, considering the gravitational i…
Efficient numerical evolution of self-gravitating systems with dark-matter-baryon scattering
Johns Hopkins University（约翰斯·霍普金斯大学）；University of Southern California（南加州大学）；Institute for Advanced Study（高等研究院）；Flatiron Institute（平顿研究所）；University of British Columbia（不列颠哥伦比亚大学）
The relation between properties of galaxies and dark matter halos they reside in can be valuable for structure formation and evolution. This paper focus on the baryonic-to-halo mass ratio (BHMR) and its evolution. We first review unique properties of self-gravitating collisionless dark matter flow (SG-CFD), followed by their application to derive BHMR. To maximize system entropy, the long-range interaction requires a broad size of halos to be formed. These halos facilitate inverse mass and energy cascade from small to large scales with a constant rate of energy cascade $\varepsilon_u$. In addi…
This paper studies the scale and redshift variation of density and velocity distributions in self-gravitating collisionless dark matter flow by a halo-based non-projection approach. All particles are divided into halo and out-of-halo particles for redshift variation of distributions. Without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales $r$. We demonstrate that: i) Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained…
Isothermal and self-gravitating systems bound by non-conducting and conducting walls are known to be unstable if the density contrast between the center and the boundary exceeds critical values. We investigate the equilibrium and dynamical evolution of isothermal and self-gravitating system embedded in potential well, which can be the situation of many astrophysical objects such as the central parts of the galaxies, or clusters of galaxies with potential dominated by dark matter, but is still limited to the case where the potential well is fixed during the evolution. As the ratio between the d…
The long timescale evolution of a self-gravitating system is generically driven by two-body encounters. In many cases, the motion of the particles is primarily governed by the mean field potential. When this potential is integrable, particles move on nearly fixed orbits, which can be described in terms of angle-action variables. The mean field potential drives fast orbital motions (angles) whose associated orbits (actions) are adiabatically conserved on short dynamical timescales. The long-term stochastic evolution of the actions is driven by the potential fluctuations around the mean field an…
Recent developments in computer hardware and software enable researchers to simulate the self-gravitating evolution of galaxies at a resolution comparable to the actual number of stars. Here we present the results of a series of such simulations. We performed $N$-body simulations of disk galaxies with between 100 and 500 million particles over a wide range of initial conditions. Our calculations include a live bulge, disk, and dark matter halo, each of which is represented by self-gravitating particles in the $N$-body code. The simulations are performed using the gravitational $N$-body tree-co…
Self-gravitating Bose-Einstein condensates (BEC) have been proposed in various astrophysical contexts, including Bose-stars and BEC dark matter halos. These systems are described by a combination of the Gross-Pitaevskii and Poisson equations (the GPP system). In the analysis of these hypothetical objects, the Thomas-Fermi (TF) approximation is widely used. This approximation is based on the assumption that in the presence of a large number of particles, the kinetic term in the Gross-Pitaevskii energy functional can be neglected, yet it is well known that this assumption is violated near the co…
(abbreviated) The statistical mechanics of self-gravitating systems is a long-held puzzle. In this work, we employ a phenomenological entropy form of ideal gas, first proposed by White & Narayan, to revisit this issue. By calculating the first-order variation of the entropy, subject to the mass- and energy-conservation constraints, we obtain an entropy stationary equation. Incorporated with the Jeans equation, and by specifying some functional form for the anisotropy parameter beta, we numerically solve the two equations, and demonstrate that the velocity anisotropy parameter plays an importan…
We study the global stability of a self-gravitating disk in the context of Modified Gravity (MOG) using N-body simulations. This theory is a relativistic scalar-tensor-vector theory of gravity and presented to address the dark matter problem. In the weak field limit MOG possesses two free parameters $α$ and $μ_0$ which have been already determined using rotation curve data of spiral galaxies. The evolution of a stellar self-gravitating disk and more specifically the bar instability in MOG is investigated and compared to a Newtonian case. Our models have exponential and Mestel-like surface dens…
Analytical tools are extremely hard to find for non-linear gravitational collpase. Only a few simple but powerful tools exist so far. Two examples are the spherical collapse model (SCM) and stable clustering hypothesis (SCH). We present a new analytical tool, a two-body collapse model (TBCM), that plays the same fundamental role as harmonic oscillator in dynamics. For convenience, TBCM is formulated for gravity with any potential exponent $n$ in a static background with a fixed damping ($n$=-1 for Newtonian gravity). The competition between gravity, expanding background (or damping), and angul…
Halo-mediated mass and energy cascades are key to understand dark matter flow. Both cascades origin from the mass exchange between halo and out-of-halo sub-systems. Kinetic energy can be from the motion of halos and particle motion in halos. Similarly, potential energy can be due to the inter- and intra-halo interactions. Intra-halo virial equilibrium is established much faster than inter-halo. Change of energy of entire system comes from virilization in halos. At statistically steady state, continuous mass exchange is required to sustain growth of total halo mass $M_h\propto a^{1/2}$ and ener…
Cosmological N-body simulations have revealed many empirical relationships of dark matter halos, yet the physical origin of these halo properties still remains unclear. On the other hand, the attempts to establish the statistical mechanics for self-gravitating systems have encountered many formal difficulties, and little progress has been made for about fifty years. The aim of this work is to strengthen the validity of the statistical-mechanical approach we have proposed previously to explain the dark matter halo properties. By introducing an effective pressure instead of the radial pressure t…
Very long-term relaxation of harmonic 1D self-gravitating systems
Observatoire Astronomique de Strasbourg, CNRS UMR 7550（斯特拉斯堡天文台）；Institut d’Astrophysique de Paris, CNRS and Sorbonne Université, UMR 7095（巴黎天体物理学研究所）；Sorbonne Université（索邦大学）；Kyung Hee University, Dept. of Astronomy & Space Science（庆熙大学天文与空间科学系）
Twenty three years ago, cosmological N-body simulations revealed quasi-universal NFW dark matter halos, whose physical origin is still unclear. This work tries to solve this issue by Landau-Ginzburg (LG) theory in equilibrium statistical mechanics. We replace the order parameter in LG theory by density and consider the dark matter halos as fluctuations from the equilibrium state of the background, which can be the main halos for the subhalos, or the homogeneous and isotropic Universe for the galaxy cluster halos. With the assumption that the fluctuations of Helmholtz free energy depend mainly …
Starting from the weak field limit, we discuss astrophysical applications of Extended Theories of Gravity where higher order curvature invariants and scalar fields are considered by generalizing the Hilbert-Einstein action linear in the Ricci curvature scalar $R$. Results are compared to General Relativity in the hypothesis that Dark Matter contributions to the dynamics can be neglected thanks to modified gravity. In particular, we consider stellar hydrostatic equilibrium, galactic rotation curves, and gravitational lensing. Finally, we discuss the weak field limit in the Jordan and Einstein f…
We completed the development of simulation code that is designed to study the behavior of a conjectured dark matter galactic halo that is in the form of a Bose-Einstein Condensate (BEC). The BEC is described by the Gross-Pitaevskii equation, which can be solved numerically using the Crank-Nicholson method. The gravitational potential, in turn, is described by Poisson's equation, that can be solved using the relaxation method. Our code combines these two methods to study the time evolution of a self-gravitating BEC. The inefficiency of the relaxation method is balanced by the fact that in subse…
We propose an association between the phase-space mixing level of a self-gravitating system and the indistinguishability of its constituents (stars or dark matter particles). This represents a refinement in the study of systems exhibiting incomplete violent relaxation. Within a combinatorial analysis similar to that of Lynden-Bell, we make use of this association to obtain a distribution function that deviates from the Maxwell-Boltzmann distribution, increasing its slope for high energies. Considering the smallness of the occupation numbers for large distances from the center of the system, we…
The statistical mechanics of self-gravitating systems has not been well understood, and still remains an open question so far. In a previous study by Kang & He, we showed that the fluid approximation may give a clue to further investigate this problem. In fact, there are indeed many dynamical similarities between self-gravitating and fluid systems. Based on a fluid-like entropy, that work explained successfully the outer density profiles of dark matter halos, but there left some drawbacks with the calculation concerning extremizing process of the entropy. In the current paper, with the improve…
In this paper, we consider the problem of the evolution of the disk subsystem of galaxies in view of the halo. To this end, we have studied the dependence of the evolution of a non-linearly non-radially disk oscillating in its plane depending on the basic parameters of the dark matter halo numerically. The dark matter halo stabilizes the instabilities in the plane of the disk, but destabilizes its vertical oscillations. The global disk structure is dependent strongly on the mass and shape of the of dark matter halo. The evolutionary dependence of the oscillation process of a self-gravitating d…
In a previous study, we formulated a framework of the entropy-based equilibrium statistical mechanics for self-gravitating systems. This theory is based on the Boltzmann-Gibbs entropy and includes the generalized virial equations as additional constraints. With the truncated distribution function to the lowest order, we derived a set of second-order equations for the equilibrium states of the system. In this work, the numerical solutions of these equations are investigated. It is found that there are three types of solutions for these equations. Both the isothermal and divergent solutions are …
Noble gravitational atoms: Self-gravitating black hole scalar wigs with angular momentum number
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México（墨西哥国立自治大学核科学研究所）；Departamento de Física, División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato（瓜纳华托大学莱昂校区科学与工程分院物理系）；Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学研究所）；Instituto de Física Enrique Gaviola, CONICET（恩里克·加维亚拉物理研究所，阿根廷国家科学研究委员会）；Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo（圣尼古拉斯伊达尔戈米却肯大学物理与数学研究所）
Large self-gravitating stellar systems share with correlated liquids in condensed matter physics a pattern of hierarchical density variations. While it takes the microscopic time resolution to discern the correlated dynamics of the critical opalescence, characteristic astronomical times hide fluctuational dynamics of stellar liquids, where, governed by interstellar correlations, denser clusters of stars assemble and disperse. For a semi-isolated galaxy, these dynamical fluctuations are dense globular clusters. For a galaxy cluster, these dynamical fluctuations are the member galaxies, elliptic…
We analyze theoretically the Schrodinger-Poisson equation in two transverse dimensions in the presence of a Kerr term. The model describes the nonlinear propagation of optical beams in thermooptical media and can be regarded as an analogue system for a self-gravitating self-interacting wave. We compute numerically the family of radially symmetric ground state bright stationary solutions for focusing and defocusing local nonlinearity, keeping in both cases a focusing nonlocal nonlinearity. We also analyze excited states and oscillations induced by fixing the temperature at the borders of the ma…
It has long been known that no static, spherically symmetric, asymptotically flat Klein-Gordon scalar field configuration surrounding a nonrotating black hole can exist in general relativity. In a series of previous papers we proved that, at the effective level, this no-hair theorem can be circumvented by relaxing the staticity assumption: for appropriate model parameters there are quasi-bound scalar field configurations living on a fixed Schwarzschild background which, although not being strictly static, have a larger lifetime than the age of the universe. This situation arises when the mass …
