Equation of state, universal profiles, scaling and macroscopic quantum effects in Warm Dark Matter galaxies

The Thomas-Fermi approach to galaxy structure determines selfconsistently and nonlinearly the gravitational potential of the fermionic WDM particles given their quantum distribution function f(E). Galaxy magnitudes as the halo radius r_h, mass M_h, velocity dispersion and phase space density are obtained. We derive the general equation of state for galaxies (relation between the pressure and the density), and provide an analytic expression. This clearly exhibits two regimes: (i) Large diluted galaxies for M_h>2.3 10^6 Msun corresponding to temperatures T_0>0.017 K, described by the classical self gravitating WDM Boltzman regime and (ii) Compact dwarf galaxies for 1.6 10^6 Msun>M_h>M_{h,min}=30000 (2keV/m)^{16/5} Msun, T_0<0.011 K described by the quantum fermionic WDM regime. The T_0=0 degenerate quantum limit predicts the most compact and smallest galaxy (minimal radius and mass M_{h,min}). All magnitudes in the diluted regime exhibit square root of M_h scaling laws and are universal functions of r/r_h when normalized to their values at the origin or at r_h. We find that universality in galaxies (for M_h>10^6 Msun) reflects the WDM perfect gas behaviour. These theoretical results contrasted to robust and independent sets of galaxy data remarkably reproduce the observations. For the small galaxies, 10^6>M_h>M_{h,min} corresponding to effective temperatures T_0<0.017 K, the equation of state is galaxy dependent and the profiles are no more universal. These non-universal properties in small galaxies account to the quantum physics of the WDM fermions in the compact regime. Our results are independent of any WDM particle physics model, they only follow from the gravitational interaction of the WDM particles and their fermionic quantum nature.


I. 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.
Warm Dark Matter (WDM), that is dark matter formed by particles with masses in the keV scale receives increasing attention today ( [6,16,17] and references therein).
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. [1,4,15,20,21,27,33,35,38]. 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 [29,30,39]. On the other hand, N -body CDM simulations exhibit cusped density profiles with a typical 1/r behaviour 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 [2,5,22,36]. 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 [7,8]. 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 [7,8]. 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 [19]. 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 obeys the self-consistent and nonlinear Poisson equation 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 selfconsistently the gravitational potential of the quantum fermionic WDM given its distribution function f (E).
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 nonlinear differential equation that determines self-consistently the chemical potential µ(r) and constitutes the Thomas-Fermi approach [7,8]. 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 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 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 [14,34]. It is therefore a useful characteristic scale to express galaxy magnitudes.
Our theoretical results follow by solving the self-consistent and nonlinear Poisson equation eq.(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 appropriated to confront to observations over the whole range of galaxies.
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.
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 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 being 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 [13]. This indicates that WDM is thermalized in the internal regions r r h of galaxies.
We plot in fig. 9 the exact equation of state obtained by solving the Thomas-Fermi equation and the empirical equation of state eq.(1.4).
We find that the presence of universal profiles in galaxies reflect the perfect gas behaviour of the the WDM galaxy equation of state in the diluted regime which is identical to the self-gravitating 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 Section 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 Σ 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 Section 3 we derive the equation of state of galaxies and analyze their main regimes: classical regime which is the perfect inhomogenous equation of state, identical to the WDM selfgravitating gas equation of state, and the quantum regime, which exhibits a steeper equation of state, non universal, 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.

II. 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 time independent 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 [7,8] 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.
Eq. (2.3) provides an ordinary nonlinear differential equation that determines self-consistently the chemical potential µ(r) and constitutes the Thomas-Fermi approach [7,8] (see also ref. [9]). This is a semi-classical 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 .
It must be stressed that the Thomas-Fermi equation (2.3) determine σ 2 (r) in terms of ρ(r) through eq.(2.5). Therefore, the Thomas-Fermi equation determines the equation of state through eq.(2.6). Contrary to the usual situation [3], we do not assume the equation of state, but we derive it from the Thomas-Fermi equation.
The fermionic DM mass density ρ is bounded at the origin due to the Pauli principle [7] which implies the bounded boundary condition at the origin We see that µ(r) fully characterizes the DM halo structure in this Thomas-Fermi framework. The chemical potential is monotonically decreasing in r since eq.(2.3) implies From eq.(2.4) and (2.5) we derive the hydrostatic equilibrium equation Eliminating P (r) between eqs.(2.6) and (2.9) and integrating on r gives (2.10) Inserting this expression in the Poisson's equation yields This nonlinear 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 baryotropic equation.
In this semi-classical 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 non-zero 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 give similar results than the out of equilibrium distribution functions [8].
The choice of Ψ 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 that 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 equation (2.3)-(2.7) become the equations for a self-gravitating 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): 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. One can easily translate from Dirac to Majorana fermions changing the WDM fermion mass as: Then, in dimensionless variables, the self-consistent Thomas-Fermi equation (2.3) for the chemical potential ν(ξ) takes the form 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 as: As a consequence, from eqs.(2.8), (2.13), (2.14), (2.17) and (2.19) the total mass M (r) enclosed in a sphere of radius r and the phase space density Q(r) turn to be .
( 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 is found 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. More precisely, all galaxies seem to have the same value for Σ 0 , namely Σ 0 ≃ 120 M ⊙ /pc 2 up to 10% − 20% [14,18,34]. 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 Σ 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 Σ 0 is the only dimensionful quantity which is constant among the different galaxies.
For a fixed value of the surface density Σ 0 , the solutions of the Thomas-Fermi eqs.(2.16) are parametrized by a single parameter: the dimensionless chemical potential at the origin ν 0 . The value of ν 0 at fixed Σ 0 can be determined by the value of the halo galaxy mass M (r h ) obtained from eq.(2.29) at r = r h .
Also, at fixed surface density Σ 0 , the effective temperature T 0 is only a function of ν 0 .
It is useful to introduce the rescaled dimensionless variableŝ We display in Table I 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 .  We see thatT0 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 forM h < 3 10 5 M⊙, ν0 > −1.8,T0 < 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 ν0 vs. the ordinary logarithm ofM h . We see that ν0 follows with precision the (5/4) logM h law as in the diluted regime eq.(2.37) of the Thomas-Fermi equations. except near the degenerate regime forM h < 3 10 5 M⊙, ν0 > −1.8,T0 < 0.005 K corresponding to compact dwarf galaxies.

A. 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:  Table 1 in [8] based on refs. [23], [28], [29], [30], [31]. 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 %.
M hT0 ν0 = µ(0) T0  , the effective temperatureT0 and the chemical potential at the origin ν0 for WDM galaxies covering the whole range from large diluted galaxies till small ultracompact galaxies.
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 temperatureT 0 , the chemical potential at the originν 0 and the normalized halo radiusr h as functions of the halo massM h as defined by eqs. (2.33). We also depict in fig. 2 the galaxy observations from different sets of data from refs. [23], [28], [29], [30], [31]. 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  and eq.(2.43) following with good precision the square root ofM h eq.(2.37).
We see that the whole set of scaling behaviours 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.
It must be stressed that the scaling relations eqs.(2.37)-(2.47) are a consequence solely of the self-gravitating interaction 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 (2.51) From the minimal known halo mass M h = 3.9 10 4 M ⊙ for Willman I (see Table 1 in [8]) we obtain the lower bound

III. 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 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 [13]. 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 behaviour We plot in fig. 5 F (x) and its asymptotic behaviour 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 [13].
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 fastly when the galaxy mass decreases only by a factor seven fromM h = 1.4 10 5 M ⊙ to the minimal galaxy massM 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 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) = Ψ out (E/T 0 ). For the relevant galaxy physical magnitudes, the Fermi-Dirac distribution gives similar results than the out of equilibrium distribution functions [8].
In the two WDM galaxy regimes, classical diluted regime, and degenerate quantum regime, we can eliminate ν in eqs.(4.1) and obtain P as a function of ρ in close form. Let us take the ratios P/ρ and P/ρ . (4.2) In the diluted limit ν ≪ −1 we have that and therefore we obtain for WDM in the diluted limit the local perfect gas equation of state: The local perfect WDM gas equation of state eq.(4.3) is precisely the equation of state of the Boltzmann self-gravitating gas [10].
In the degenerate limit ν ≫ 1 we have that  For small density and growing T0, the self-gravitating ideal WDM gas behaviour is obtained exhibiting straight lines with unit slope; this describes the physical state of large diluted galaxieŝ M h > 2.3 10 6 M⊙, ν0 < −5, T0 > 0.017 K. For large density and decreasing temperature the fermionic quantum behaviour close to the degenerate state eq.(4.4) shows up as the steeper straight lines with slope approaching 5/3. In particular, the degenerate T0 = 0 state exhibits the slope 5/3 for all densities. The diluted classical regime and the degenerate regime are interpolated smoothly by the quantum behaviour corresponding to compact dwarf galaxies with 1.6 10 6 M⊙ >M h ≥M h,min = 3.10 10 4 M⊙, ν0 > −4, T0 < 0.011 K. For increasing T0 the curves move up. The larger is T0, the larger is the value of the densityρ where the quantum behaviour is attained.
Being the galaxies a nonrelativistic system, P turns 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 ρ ≡ 2 keV m (4.6) We plot in fig. 7 the ordinary logarithm ofP vs. the ordinary logarithm ofρ for different values of T 0 . For small density and for growing effective temperature, the self-gravitating ideal WDM gas behaviour eq.(4.3) of the diluted regime is obtained. On the contrary, for large density and for decreasing temperature the fermionic quantum behaviour close to the degenerate state eq.(4.4) shows up. That is, the straight lines with unit slope in fig. 7 describe the perfect WDM gas behaviour eq.(4.3), while the steeper straight lines with slope 5/3 describe the degenerate quantum behaviour eq.(4.4). We see that the diluted classical and degenerate regimes are interpolated smoothly by the quantum behaviour. For increasing T 0 the curves in fig. 7 move up. The larger is T 0 , the larger is the value of the densityρ where the quantum behaviour 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 eqs.(4.1): The diluted and degenerate gas behaviours eq.(4.3) and (4.4) of WDM galaxies are explicitly seen in fig. 8 FIG. 8: The galaxy pressure P/P0 vs. the density ρ/ρ0, where P0 and ρ0 are the pressure and the density at the origin, respectively defined by eq.(4.7). We see the WDM ideal gas behaviour (unit slope) in the diluted regime, that is for galaxy masseŝ M h > 2.3 10 6 M⊙, ν0 < −5, T0 > 0.017 K. For smaller galaxy masses, 1.6 10 6 M⊙ >M h ≥M h,min = 3.10 10 4 M⊙, ν0 > −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 ν0 = ∞ we obtain a 5/3 slope straight line. We see that the diluted and degenerate regimes are interpolated smoothly by the quantum behaviour. diluted perfect gas behaviour appears for galaxy massesM h > 2.3 10 6 M ⊙ , ν 0 < −5, T 0 > 0.017 K. The degenerate gas behaviour shows up for the minimal mas 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 behaviour 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 [13].
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 to the quantum physics of the self-gravitating WDM fermions in the compact case close to the degenerate state.
Indeed, it will be extremely interesting to dispose of observations which could check these quantum effects in dwarf galaxies. 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 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, Indeed, this invariance is restricted to the diluted regime (M h 10 6 M ⊙ ).
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 [8]. Moreover, for m > 2 keV, an overabundance of small structures appear as solution of the Thomas-Fermi equations, which do not have observed counterpart. Therefore, m between 2 keV and 3 keV is singled out as the most plausible value [8].
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.