Equation of state, universal profiles, scaling and macroscopic quantum effects in warm dark matter galaxies

The Thomas–Fermi approach to galaxy structure determines self-consistently and non-linearly the gravitational potential of the fermionic warm dark matter (WDM) particles given their quantum distribution function f(E). This semiclassical framework accounts for the quantum nature and high number of DM particles, properly describing gravitational bounded and quantum macroscopic systems as neutron stars, white dwarfs and WDM galaxies. We express the main galaxy magnitudes as the halo radius rh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r_h $$\end{document}, mass Mh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M_h $$\end{document}, velocity dispersion and phase space density in terms of the surface density which is important to confront to observations. From these expressions we derive the general equation of state for galaxies, i.e., the relation between pressure and density, and provide its analytic expression. Two regimes clearly show up: (1) Large diluted galaxies for Mh≳2.3×106M⊙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M_h \gtrsim 2.3 \times 10^6 \; M_\odot $$\end{document} and effective temperatures T0>0.017\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T_0 > 0.017 $$\end{document} K described by the classical self-gravitating WDM Boltzman gas with a space-dependent perfect gas equation of state, and (2) Compact dwarf galaxies for 1.6×106M⊙≳Mh≳Mh,min≃3.10×104(2keV/m)165M⊙,T0<0.011\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 1.6 \times 10^6 \; M_\odot \gtrsim M_h \gtrsim M_{h,\mathrm{min}} \simeq 3.10 \times 10^4 \; (2 \, {\mathrm{keV}}/m)^{\! \! \frac{16}{5}} \; M_\odot , \; T_0 < 0.011 $$\end{document} K described by the quantum fermionic WDM regime with a steeper equation of state close to the degenerate state. In particular, the T0=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T_0 = 0 $$\end{document} degenerate or extreme quantum limit yields the most compact and smallest galaxy. In the diluted regime, the halo radius rh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r_h $$\end{document}, the squared velocity v2(rh)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ v^2(r_h) $$\end{document} and the temperature T0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T_0 $$\end{document} turn to exhibit square-root of Mh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M_h $$\end{document}scaling laws. The normalized density profiles ρ(r)/ρ(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rho (r)/\rho (0) $$\end{document} and the normalized velocity profiles v2(r)/v2(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ v^2(r)/ v^2(0) $$\end{document} are universal functions of r/rh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r/r_h $$\end{document} reflecting the WDM perfect gas behavior in this regime. These theoretical results contrasted to robust and independent sets of galaxy data remarkably reproduce the observations. For the small galaxies, 106≳Mh≥Mh,min\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 10^6 \gtrsim M_h \ge M_{h,\mathrm{min}} $$\end{document}, the equation of state is galaxy mass dependent and the density and velocity profiles are not anymore universal, accounting to the quantum physics of the self-gravitating WDM fermions in the compact regime (near, but not at, the degenerate state). It would be extremely interesting to dispose of dwarf galaxy observations which could check these quantum effects.


Introduction
Dark matter (DM) is the main component of galaxies: the fraction of DM over the total galaxy mass goes from 95% for large diluted galaxies till 99.99% for dwarf compact galaxies. Therefore, DM alone should explain the main structure of galaxies. Baryons should only give corrections to the pure DM results.
At intermediate scales ∼100 kpc, WDM gives the correct abundance of substructures and therefore WDM solves the cold dark matter (CDM) overabundance of structures at small scales [11][12][13][14][15][16][17][18][19]. For scales larger than 100 kpc, WDM yields the same results than CDM. Hence, WDM agrees with all the observations: small scale as well as large scale structure observations and CMB anisotropy observations. Astronomical observations show that the DM galaxy density profiles are cored till scales below the kpc [20][21][22][23][24][25]. On the other hand, N -body CDM simulations exhibit cusped density profiles with a typical 1/r behavior near the galaxy center r = 0. Inside galaxy cores, below ∼100 pc, N -body classical physics simulations do not provide the correct structures for WDM because quantum effects are important in WDM at these scales. Classical physics N -body WDM simulations exhibit cusps or small cores with sizes smaller than the observed cores [26][27][28][29]. WDM predicts correct structures and cores with the right sizes for small scales (below kpc) when the quantum nature of the WDM particles is taken into account [30,31]. This approach is independent of any WDM particle physics model.
We follow here the Thomas-Fermi approach to galaxy structure for self-gravitating fermionic WDM [30,31]. This approach is especially appropriate to take into account quantum properties of systems with large number of particles. That is, macroscopic quantum systems as neutron stars and white dwarfs [32,33]. In this approach, the central quantity to derive is the DM chemical potential μ(r), which is the free energy per particle. For self-gravitating systems, the potential μ(r) is proportional to the gravitational potential φ(r), μ(r) = μ 0 − m φ(r), μ 0 being a constant, and it obeys the self-consistent and non-linear Poisson equation, ∇ 2 μ(r) = −4 π g G m 2 d 3 p (2 πh) 3 f p 2 2 m − μ(r) . (1.1) Here G is Newton's gravitational constant, g is the number of internal degrees of freedom of the DM particle, p is the DM particle momentum and f (E) is the energy distribution function. This is a semiclassical gravitational approach to determine self-consistently the gravitational potential of the quantum fermionic WDM given its distribution function f (E).
The terminology "Thomas-Fermi approach" is used here by analogy with the effective quantum mechanical treatement implying a quantum statistical distribution function. Notice, however, that the Thomas-Fermi method in atomic physics does not lead to an integro-differential equation but rather to a non-linear differential equation.
In the Thomas-Fermi approach, DM dominated galaxies are considered in a stationary state. This is a realistic situation for the late stages of structure formation since the free-fall (Jeans) time t f f for galaxies is much shorter than the age of galaxies. t f f is at least one or two orders of magnitude smaller than the age of the galaxy.
We consider spherical symmetric configurations where Eq. (1.1) becomes an ordinary non-linear differential equation that determines self-consistently the chemical potential μ(r ) and constitutes the Thomas-Fermi approach [30,31] (see also Refs. [34][35][36]). We choose for the energy distribution function a Fermi-Dirac distribution where T 0 is the characteristic one-particle energy scale. T 0 plays the role of an effective temperature scale and depends on the galaxy mass. The Fermi-Dirac distribution function is justified in the inner regions of the galaxy, inside the halo radius where we find that the Thomas-Fermi density profiles perfectly agree with the observations. The collisionless self-gravitating gas is an isolated system which is not integrable. Therefore, it is an ergodic system that can thermalize [37]. Namely, the particle trajectories explore ergodically the constant energy manifold in phasespace, covering it uniformly according to precisely the microcanonical measure and yielding to a thermal situation [37].
Physically, these phenomena are clearly understood because in the inner halo region r r h , the density is higher than beyond the halo radius. The gravitational interaction in the inner region is strong enough and thermalizes the selfgravitating gas of DM particles while beyond the halo radius the particles are too dilute to thermalize, namely, although they are virialized, they had not enough time to accomplish thermalization. Notice that virialization always starts before than thermalization.
The solutions of the Thomas-Fermi equations (1.1) are characterized by the value of the chemical potential at the origin μ(0). Large positive values of μ(0) correspond to dwarf compact galaxies (fermions near the quantum degenerate limit), while large negative values of μ(0) yield large and diluted galaxies (classical Boltzmann regime).
Approaching the classical diluted limit yields larger and larger halo radii, galaxy masses and velocity dispersions. On the contrary, in the quantum degenerate limit we get solutions of the Thomas-Fermi equations corresponding to the minimal halo radii, galaxy masses and velocity dispersions.
The surface density 0 ≡ r h ρ 0 120 M /pc 2 up to 10 − 20%, (1.2) whete r h stand for the halo radius and ρ 0 for the density at the center has the remarkable property of being nearly constant and independent of luminosity in different galactic systems (spirals, dwarf irregular and spheroidals, elliptics) spanning over 14 magnitudes in luminosity and over different Hubble types [38,39]. It is therefore a useful characteristic scale to express galaxy magnitudes.
Our theoretical results follow by solving the self-consistent and non-linear Poisson equation (1.1) which is solely derived from the purely gravitational interaction of the WDM particles and their fermionic nature.
The main galaxy magnitudes as the halo radius r h , mass M h , velocity dispersion and phase space density are analytically obtained and expressed in terms of the surface density, which is particularly appropriate to confront to observations over the whole range of galaxies.
In this paper we derive and analyze the general equation of state of galaxies which clearly exhibits two regimes: (1) large diluted galaxies for In particular, the T 0 = 0 degenerate or extreme quantum limit yields the most compact and smallest galaxy: with minimal mass M h,min and minimal radius, and maximal phase space density.
In Ref. [30] careful estimates of the halo mass and radius for the degenerate WDM self-gravitating gas were reported. For clarity, we reproduce and update the estimates here.
For an order-of-magnitude estimate, let us consider a halo of mass M h and radius r h of fermionic matter. Each fermion can be considered inside a cell of size x ∼ 1/n 1 3 and therefore has a momentum The associated quantum pressure P q (flux of the momentum) has the value P q = n σ p ∼h σ n where σ is the mean velocity given by The system will be in dynamical equilibrium if this quantum pressure is balanced by the gravitational pressure, We estimate the number density as and we use p = m σ to obtain from Eq. (1.6) .
(1.8) (1.14) The Chandrasekhar mass limit [32,33] for neutron stars formed by a degenerate gas of neutrons is analogous to the maximal mass Eqs. (1.13), (1.14) for self-gravitating degenerate WDM particles. Notice that strong interactions in neutron stars introduce corrections [32,33] to the self-gravitating degenerate gas of neutrons while the particle physics interactions of WDM are so weak that can be safely neglected. It is useful the comparison with the estimations Eqs. (1.9), (1.10) for non-relativistic particles which imply In fact, observations show that WDM particles are always non-relativistic and therefore galaxies with high masses near M max h ∼ 10 12 M in the degenerate ultrarelativistic regime Eqs. (1.13), (1.14) are not observed. On the contrary, the WDM non-relativistic estimates Eqs. (1.9), (1.10) and (1.15), (1.16) are observationally realistic for ultracompact galaxies.
As we see below in Sect. 2, observations show that real galaxies in their whole range of masses, sizes and velocities turn to be non-degenerate (non-zero temperature) solutions of the Thomas-Fermi equations and in particular the ultracompact galaxies are close to the zero temperature degenerate state.
A dwarf galaxy with a halo mass M h 10 6 M arises as a solution of the Thomas-Fermi approach near the quantum degenerate regime. We obtain for a halo mass M h 10 6 M a halo radius r h 100 pc as one can see from Fig. 2 and a galaxy temperature T 0 0.01 K (see Table 1). As discussed in Sect. 2, a galaxy solution with mass M h 10 6 M exhibits quantum properties: it is near the quantum degenerate regime but is not in a zero temperature degenerate state, dwarf galaxies possessing a small but non-zero temperature.
Dwarf galaxies are macroscopic astrophysical quantum objects as white dwarf stars and neutron stars [32,33], but are different from them.
We find that all magnitudes in the diluted regime exhibit square-root of M h scaling laws and are universal functions of r/r h normalized to their values at the origin or at r h . Conversely, the halo mass M h scales as the square of the halo radius r h as M h = 1.75572 0 r 2 h . Moreover, the proportionality factor in this scaling relation is confirmed by the galaxy data (see Fig. 2).
We find that the universal theoretical density profile obtained from the Thomas-Fermi equation (1.1) in the diluted regime (M h 10 6 M ) is accurately reproduced by the simple formula (see Fig. 5) The fit is precise for r < 2 r h . The theoretical rotation curves and density profiles obtained from the Thomas-Fermi equations remarkably agree with observations for r r h , for all galaxies in the diluted regime [40]. This indicates that WDM is thermalized in the internal regions r r h of galaxies.
We find the WDM galaxy equation of state, that is, the functional relation between the pressure P and the density ρ in a parametric way as (1.17) These equations express parametrically, through the parameter ν, the pressure P as a function of the density ρ and therefore provide the equation of state. I 2 (ν) and I 4 (ν) are integrals (2nd and 4th momenta) of the distribution function. At thermal equilibrium they are given by Eq. (2.15). For the main galaxy physical magnitudes, the Fermi-Dirac distribution gives similar results than the out of equilibrium distribution functions [31]. We plot in Figs. 7 and 8 P as a function of ρ for different values of the effective temperature T 0 .
Interestingly enough, we provide a simple formula representing the exact equation of state (1.17) obtained by solving the Thomas-Fermi equation (1.1) 19) and the best fit to the exact values of P as a functionρ is obtained for the values of the parameters The fitting formula Eq. (1.18) exactly fulfills the diluted and degenerate limiting behaviors: WDM degenerate quantum limit.
We plot in Fig We find that the presence of universal profiles in galaxies reflect the perfect gas behavior of the WDM galaxy equation of state in the diluted regime which is identical to the selfgravitating Boltzman WDM gas.
These theoretical results contrasted to robust and independent sets of galaxy data remarkably reproduce the observations.
For the small galaxies, 10 6 M M h ≥ M h,min corresponding to effective temperatures T 0 0.017 K, the equation of state is steeper, dependent on the galaxy mass and the profiles are not anymore universal. These non-universal properties in small galaxies account to the quantum physics of the self-gravitating WDM fermions in the compact regime with high density close to, but not at, the degenerate state.
It would be extremely interesting to dispose of observations which could check these quantum effects in dwarf galaxies.
In summary, the results of this paper show the power and cleanliness of the Thomas-Fermi theory and WDM to properly describe the galaxy structures and the galaxy physical states.
This paper is organized as follows. In Sect. 2 we present the Thomas-Fermi approach to galaxy structure, we express the main galaxy magnitudes in terms of the solution of the Thomas-Fermi equation and the value of the surface density 0 . We analyze the diluted classical galaxy magnitudes, derive their scaling laws and find the universal density and velocity profiles and their agreement with observations. In Sect. 3 we derive the equation of state of galaxies and analyze their main regimes: classical regime which is the perfect inhomogeneous equation of state, identical to the WDM self-gravitating gas equation of state, and the quantum regime, which exhibits a steeper equation of state, nonuniversal, galaxy mass dependent and describes the quantum fermionic compact states (dwarf galaxies), close to the degenerate limit. Finally, the invariance and dependence on the WDM particle mass m in the classical and quantum regimes is discussed.

Galaxy structure in the WDM Thomas-Fermi approach
We consider DM dominated galaxies in their late stages of structure formation when they are relaxing to a stationary situation, at least not too far from the galaxy center. This is a realistic situation since the free-fall (Jeans) time t f f for galaxies is much shorter than the age of galaxies: The observed central densities of galaxies yield free-fall times in the range from 15 million years for ultracompact galaxies till 330 million years for large diluted spiral galaxies. These free-fall (or collapse) times are small compared with the age of galaxies running in billions of years.
Hence, we can consider the DM described by a timeindependent and non-relativistic energy distribution function f (E), where E = p 2 /(2m)−μ is the single-particle energy, m is the mass of the DM particle and μ is the chemical potential [30,31] related to the gravitational potential φ(r) by where μ 0 is a constant.
In the Thomas-Fermi approach, ρ(r) is expressed as a function of μ(r) through the standard integral of the DM phase-space distribution function over the momentum where g is the number of internal degrees of freedom of the DM particle, with g = 1 for Majorana fermions and g = 2 for Dirac fermions.
We will consider spherical symmetric configurations. Then the Poisson equation for φ(r ) takes the self-consistent form where G is Newton's constant and ρ(r ) is the DM mass density. Equation (2.3) provides an ordinary non-linear differential equation that determines self-consistently the chemical potential μ(r ) and constitutes the Thomas-Fermi approach [30,31] (see also Refs. [34][35][36]). This is a semiclassical approach to galaxy structure in which the quantum nature of the DM particles is taken into account through the quantum statistical distribution function f (E).
The DM pressure and the velocity dispersion can also be expressed as integrals over the DM phase-space distribution function as . (2.5) Equations (2.2), (2.4), and (2.5) imply the equation of state Eliminating P(r ) between Eqs. (2.6) and (2.9) and integrating on r gives (2.10) Inserting this expression in the Poisson equation yields (2.11) This non-linear equation for non-constant σ 2 (r ) generalizes the corresponding equation in the self-gravitating Boltzmann gas. For constant σ 2 (r ) Eq. (2.10) reduces to the barotropic equation.
In this semiclassical framework the stationary energy distribution function f (E) must be given. We consider the Fermi-Dirac distribution, where the characteristic one-particle energy scale T 0 in the DM halo plays the role of an effective temperature. The value of T 0 depends on the galaxy mass. In neutron stars, where the neutron mass is about six orders of magnitude larger than the WDM particle mass, the temperature can be approximated by zero. In galaxies, T 0 ∼ m < v 2 > turns to be nonzero but small in the range: 10 −3 K T 0 10 K for halo galaxy masses in the range 10 5 -10 12 M , which reproduce the observed velocity dispersions for m 2 keV. The smaller values of T 0 correspond to compact dwarfs and the larger values of T 0 are for large and diluted galaxies.
Notice that, for the relevant galaxy physical magnitudes, the Fermi-Dirac distribution gives similar results than the out of equilibrium distribution functions [31].
The choice of FD is justified in the inner regions, where relaxation to thermal equilibrium is possible. Far from the origin, however, the Fermi-Dirac distribution as its classical counterpart, the isothermal sphere, produces a mass density tail 1/r 2 , which overestimates the observed tails of the galaxy mass densities. Indeed, the classical regime μ/T 0 → −∞ is attained for large distances r since Eq. (2.8) indicates that μ(r ) is always monotonically decreasing with r .
More precisely, large positive values of the chemical potential at the origin correspond to the degenerate fermions limit which is the extreme quantum case and oppositely, large negative values of the chemical potential at the origin gives the diluted case which is the classical regime. The quantum degenerate regime describes dwarf and compact galaxies while the classical and diluted regime describes large and diluted galaxies. In the classical regime, the Thomas-Fermi equations (2.3)-(2.7) become the equations for a selfgravitating Boltzmann gas.
It is useful to introduce dimensionless variables ξ, ν(ξ ) where l 0 is the characteristic length that emerges from the dynamical equation (2.3): , R 0 = 7.425 pc, (2.14) and where we use the integration variable y ≡ p/ √ 2 m T 0 . For definiteness, we will take g = 2, Dirac fermions in the sequel. We find the main physical galaxy magnitudes, such as the mass density ρ(r ), the velocity dispersion σ 2 (r ) = v 2 (r )/3 and the pressure P(r ), which are all r -dependent: .
(2. 22) In these expressions, we have systematically eliminated the energy scale T 0 in terms of the central density ρ 0 through Eq. (2.17). Notice that Q(r ) turns to be independent of T 0 and therefore of ρ 0 .
We define the core size r h of the halo by analogy with the Burkert density profile as It must be noticed that the surface density 0 ≡ r h ρ 0 (2.24) is found to be nearly constant and independent of the luminosity in different galactic systems (spirals, dwarf irregular and spheroidals, elliptics) spanning over 14 magnitudes in luminosity and over different Hubble types. More precisely, all galaxies seem to have the same value for 0 , namely 0 120 M /pc 2 up to 10-20% [38,39,42]. It is remarkable that at the same time other important structural quantities as r h , ρ 0 , the baryon-fraction and the galaxy mass vary orders of magnitude from one galaxy to another.
The constancy of 0 seems unlikely to be a mere coincidence and probably reflects a physical scaling relation between the mass and halo size of galaxies. It must be stressed that 0 is the only dimensionful quantity which is constant among the different galaxies.
It is then useful to take here the dimensionful quantity 0 as physical scale to express the galaxy magnitudes in the Thomas-Fermi approach. That is, we replace the central density ρ 0 in the above galaxy magnitudes Eqs. (2.14)-(2.21) in terms of 0 Eq. (2.24) with the following results: l 0 = 9 π 2 9 1 5 h 6 G 3 m 8 1 5 ξ h I 2 (ν 0 ) [I 2 (ν 0 )] 3 5 |ν (ξ h )| 2 keV m 16 5 × 0 pc 2 120 M 3 5 M . (2.32) Also, at fixed surface density 0 , the effective temperature T 0 is only a function of ν 0 . It is useful to introduce the rescaled dimensionless variableŝ r h ≡ r h m 2 keV 8 5 0 pc 2 120 M 1 5 , M h ≡ M h m 2 keV 16 5 120 M 0 pc 2 3 5 T 0 ≡ T 0 2 keV m 120 M 0 pc 2 4 5 ν 0 ≡ ν 0 + 4 ln m 2 keV , σ 2 (r ) ≡ σ 2 (r ) m 2 keV 8 5 120 M 0 pc 2 4 5 . (2.33) We display in Table 1 the corresponding values of the halo massM h , the effective temperatureT 0 and the chemical potential at the origin ν 0 in the whole galaxy mass range, from large diluted galaxies till small ultracompact galaxies. The circular velocity v c (r ) is defined through the virial theorem as (2. 36) In particular, it follows that r h ρ(0) = 0 reproducing the surface density as it must. At a generic point r Eq. (2.36) provide expressions for a space-dependent surface density. They are both proportional to 0 and differ from each other by factors of order one. Notice thath, G and m canceled out in these space-dependent surface densities Eq. (2.36).

Galaxy properties in the diluted Boltzmann regime
In the diluted Boltzmann regime, ν 0 −5, the analytic expressions for the main galaxies magnitudes are given by  48) which shows that in the diluted regime the self-gravitating WDM gas behaves as an inhomogeneous perfect gas as we will discuss in the next section. We plot in Figs. 1 and 2, the dimensionless effective tem-peratureT 0 , the chemical potential at the originν 0 and the normalized halo radiusr h as functions of the halo masŝ M h as defined by Eq. (2.33). We also depict in Fig. 2 the galaxy observations from different sets of data from Refs. [20,21,[43][44][45][46][47][48][49][50]. All data are well reproduced by our theoretical Thomas-Fermi results. The errors of the data can be estimated to be about 10-20%.
The characteristic temperatureT 0 monotonically grows with the halo massM h of the galaxy as shown by Fig. 1  We see that the whole set of scaling behaviors of the diluted regime Eqs. (2.37)-(2.43) are very accurate except near the degenerate regime for halo massesM h < 3 × 10 5 M . The deviation from the diluted scaling regime for M h < 3 × 10 5 M accounts for the quantum fermionic effects in the dwarf compact galaxies obtained in our Thomas-Fermi approach (Figs. 1, 2, 3).
It must be stressed that the scaling relations Eqs. (2.37)-(2.47) are a consequence solely of the self-gravitating interac- Fig. 1 Upper panel the common logarithm (base 10) of the effective temperatureT 0 (vertical axis) versus the common logarithm of the halo massM h . We see thatT 0 grows withM h following with precision the square-root ofM h law as in the diluted regime Eq.
(2.37) of the Thomas-Fermi equations, except for M h < 3 × 10 5 M , ν 0 > −1.8,T 0 < 0.005 K, which is near the quantum degenerate regime and corresponds to compact dwarf galaxies. The deviation from the scaling diluted regime is due to the quantum fermionic effects which become important for dwarf compact galaxies. Lower panel the dimensionless chemical potential at the origin ν 0 versus the common (base 10) logarithm ofM h . We see that ν 0 follows with precision the Exact Thomas-Fermi diluted regime tion of the fermionic WDM. Galaxy data verify the exponent and the amplitude factor in these scaling as shown in Fig. 2 for the square-root scaling relation Eq. (2.37). It is highly remarkable that our theoretical results reproduce the observed DM halo properties with good precision.
The opposite limit, ν 0 1, is the quantum regime corresponding to compact WDM fermions. In particular, in the degenerate limit ν 0 → ∞, the galaxy mass and halo radius take their minimum values  Table 1 in [31]) we obtain the lower bound   Table  1 in [31] based on Refs. [20,21,[43][44][45][46][47][48][49][50]. The data are very well reproduced by the theoretical Thomas-Fermi curve. The errors of the data can be estimated to be about 10-20% observed values theory curve Table 1 Corresponding values of the halo massM h , the effective tem-peratureT 0 and the chemical potential at the origin ν 0 for WDM galaxies covering the whole range from large diluted galaxies till small ultracompact galaxieŝ

Density and velocity dispersion: universal and non-universal profiles
It is illuminating to normalize the density profiles as ρ(r )/ρ(0) and plot them as functions of r/r h . We find that these normalized profiles are universal functions of x ≡ r/r h in the diluted regime as shown in Fig. 4. This universality is valid for all galaxy massesM h > 10 5 M . No analytic form is available for the profile ρ(r ) obtained from the resolution of the Thomas-Fermi equations (2.16). The universal profile F(x) = ρ(r )/ρ(0) can be fitted with precision by the simple formula (3.1) The value α = 1.5913 provides the best fit. We plot in Fig. 5 ρ(r )/ρ(0) from the Thomas-Fermi equations (2. 16) and the precise fitting formula F α=1.5913 (x). The fit is particularly precise for r < 2 r h . Our theoretical density profiles and rotation curves obtained from the Thomas-Fermi equations remarkably agree with observations for r r h , for all galaxies in the diluted regime [40]. This indicates that WDM is thermalized in the internal regions r r h of galaxies. The theoretical profile ρ(r )/ρ(0) and the precise fit F α=1.5913 (x) cannot be used for x 1 where they decay as a power 3.2, which is a too large number to reproduce the observations. The universal density profile ρ(r )/ρ(0) is obtained theoretically in the diluted Boltzmann regime. In such regime the density profile decreases for large x 1 as ∼1/x 2 . More precisely, we find the asymptotic behavior We plot in Fig. 5 F(x) and its asymptotic behavior F asy (x) vs.
x. We see that F asy (x) becomes a very good approximation to F(x) for x 3. When F(x) behaves as ∼ 1/x 2 the circular velocity for these theoretical density profiles becomes constant as shown in [40].
For galaxy massesM h < 10 5 M , near the quantum degenerate regime, the normalized density profiles ρ(r )/ρ(0) are not anymore universal and depend on the galaxy mass.
As we can see in Fig. 4 the density profile shape changes fast when the galaxy mass decreases only by a factor seven fromM h = 1.4 × 10 5 M to the minimal galaxy masŝ M h,min = 3.10 × 10 4 M . In this narrow range of galaxy masses the density profiles shrink from the universal profile till the degenerate profile as shown in Fig. 4. Namely, these dwarf galaxies are more compact than the larger diluted galaxies.
We display in Fig. 6 the normalized velocity dispersion profiles σ 2 (r )/σ 2 (0) as functions of x = r/r h . Again, we see that these profiles are universal and constant, i.e. independent of the galaxy mass in the diluted regime for M h > 2.3 × 10 6 M , ν 0 < −5, T 0 > 0.017 K. The constancy of σ 2 (r ) = σ 2 (0) in the diluted regime implies that the equation of state is that of a perfect but inhomogeneous WDM gas. Indeed, from Eq. (2.48) and Eq. (2.6) implies for the WDM diluted galaxies the perfect gas equation of state (4.3) where both the pressure P(r ) and the density ρ(r ) depend on the coordinates. For smaller galaxy masses 1.6 × 10 6 M >M h > M h,min , the velocity profiles do depend on r and yield decreasing velocity dispersions for decreasing galaxy masses. Namely, the deviation from the universal curves appears forM h < 10 6 M and we see that it precisely arises from the quantum fermionic effects which become important in such range of galaxy masses.

The equation of state of WDM galaxies: classical diluted and compact quantum regimes
The WDM galaxy equation of state is by definition the functional relation between the pressure P and the density ρ.  These equations express parametrically, through the parameter ν, the pressure P as a function of the density ρ and therefore provide the WDM galaxy equation of state. For fermionic WDM in thermal equilibrium I 2 (ν) and I 4 (ν) are given as integrals of the Fermi-Dirac distribution function in Eq. (2.15). For WDM out of thermal equilibrium Eq. (4.1) is always valid but I 2 (ν) and I 4 (ν) should be expressed as integrals of the corresponding out of equilibrium distribution function. In the out of equilibrium case T 0 is just the characteristic scale in the out of equilibrium distribution function f out (E) = out (E/T 0 ). For the relevant galaxy physical magnitudes, the Fermi-Dirac distribution gives similar results to the out of equilibrium distribution functions [31].
In the two WDM galaxy regimes, classical diluted regime, and degenerate quantum regime, we can eliminate ν in Eq. (4.1) and obtain P as a function of ρ in close form. Let us take the ratios P/ρ and P/ρ 5 3 in Eq. (4.1): . (4.2) In the diluted limit ν −1 we have and therefore we obtain for WDM in the diluted limit the local perfect gas equation of state:   The galaxies being non-relativistic systems, P turns out to be much smaller than ρ when both are written in the same units where the speed of light is taken to be unit.
It is useful to introduce the rescaled dimensionless variables  We plot in Fig. 7 Fig. 7 move up. The larger T 0 is, the larger is the value of the densityρ where the quantum behavior is attained. We plot in Fig. 8 the pressure normalized to its value at the origin as a function of the density normalized to its value at the origin according to Eq.
The diluted and degenerate gas behaviors Eqs. (4.3) and (4.4) of WDM galaxies are explicitly seen in Fig. 8. The diluted perfect gas behavior appears for galaxy massesM h > 2.3 × 10 6 M , ν 0 < −5, T 0 > 0.017 K. The degenerate gas behavior shows up for the minimal mass galaxyM h,min = 3.10 × 10 4 M , T 0 = 0. Besides the two limiting regimes, diluted and degenerate, we see from Fig. 8 that the equation of state does depend on the galaxy mass for galaxy masses in the range 1.6 × 10 6 M >M h ≥M h,min , ν 0 > −4, T 0 < 0.011 K. This is a quantum regime, close to but not at, the degenerate limit. The equation of state in this quantum regime is steeper than in the degenerate limit.
We find that WDM galaxies exhibit two regimes: classical diluted and quantum compact (close to degenerate). WDM galaxies are diluted forM h > 2.3 × 10 6 M , ν 0 < −5, T 0 > 0.017 K and they are quantum and compact for 1.6 × 10 6 M >M h ≥M h,min , ν 0 > −4, T 0 < 0.011 K. The degenerate limit T 0 = 0 corresponds to the extreme quantum situation yielding a minimal galaxy sizer h,min and massM h,min given by Eq. (2.49). The equation of state covering all regimes is given by Eq. (4.1).
We therefore find an explanation for the universal density profiles and universal velocity profiles in diluted galaxies (M h 10 6 M ): these universal properties can be traced back to the perfect gas behavior of the self-gravitating WDM gas summarized by the WDM equation of state (4.3). Notice that all these universal theoretical profiles well reproduce the observations for r r h [40].
For small galaxy masses, 10 6 M M h ≥M h,min = 3.10 × 10 4 M , which correspond to chemical potentials at the origin ν 0 −5 and effective temperatures T 0 0.017 K, the equation of state is galaxy mass dependent (see Fig.  8) and the profiles are not anymore universal. These properties account for the quantum physics of the self-gravitating WDM fermions in the compact case close to the degenerate state. Fig. 8 The galaxy pressure P/P 0 versus the density ρ/ρ 0 , where P 0 and ρ 0 are the pressure and the density at the origin, respectively defined by Eq. (4.7). We see the WDM ideal gas behavior (unit slope) in the diluted regime, that is, for galaxy masseŝ M h > 2.3 × 10 6 M , ν 0 < −5, T 0 > 0.017 K. For smaller galaxy masses, 1.6 × 10 6 M > M h ≥M h,min = 3.10 × 10 4 M , ν 0 > −4, the equation of state depends on the galaxy mass and becomes steeper corresponding to the quantum fermionic regime of dwarf galaxies. In the degenerate limit ν 0 = ∞ we obtain a 5/3 slope straight line. We see that the diluted and degenerate regimes are interpolated smoothly by the quantum behavior It is instructive to discuss from Eq. (5.1) the dependence on the mass m of the WDM particle.
In the diluted regime, T 0 and μ(r ) depend on m, while the other magnitudes as ρ(r ), M(r ), σ 2 (r ), P(r ), Q(r ), and φ(r ) do not depend on m. This means that a change in m, namely m ⇒ m , must leave Eq. (5.1) invariant, which implies That is, For galaxy massesM h < 10 5 M , namely in the quantum regime of compact dwarf galaxies, all physical quantities do depend on the DM particle mass m as explicitly displayed in Eqs. (2.17)-(2.35). It is precisely this dependence on m that leads to the lower bound m > 1.91 keV from the minimum observed galaxy mass [31]. Moreover, for m > 2 keV, an overabundance of small structures appears as solution of the Thomas-Fermi equations, which do not have an observed counterpart. Therefore, m between 2 and 3 keV is singled out as the most plausible value [31].
In summary, we see the power of the WDM Thomas-Fermi approach to describe the structure and the physical state of galaxies in a clear way and in very good agreement with observations. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP 3 .