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

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## Abstract

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 \( r_h \), mass \( M_h \), 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 \( M_h \gtrsim 2.3 \times 10^6 \; M_\odot \) and effective temperatures \( T_0 > 0.017 \) 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 \( 1.6 \times 10^6 \; M_\odot \gtrsim M_h \gtrsim M_{h,\mathrm{min}} \simeq 3.10 \times 10^4 \; (2 \, {\mathrm{keV}}/m)^{\! \! \frac{16}{5}} \; M_\odot , \; T_0 < 0.011 \) K described by the quantum fermionic WDM regime with a steeper equation of state close to the degenerate state. In particular, the \( T_0 = 0 \) degenerate or extreme quantum limit yields the most compact and smallest galaxy. In the diluted regime, the halo radius \( r_h \), the squared velocity \( v^2(r_h) \) and the temperature \( T_0 \) turn to exhibit square-root of \( M_h \) **scaling** laws. The normalized density profiles \( \rho (r)/\rho (0) \) and the normalized velocity profiles \( v^2(r)/ v^2(0) \) are **universal** functions of \( r/r_h \) 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, \( 10^6 \gtrsim M_h \ge M_{h,\mathrm{min}} \), 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.

## 1 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 ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and references therein).

At intermediate scales \({\sim } 100 \) 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 \({\sim } 100\) 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.

**self-consistent**and

**non-linear**Poisson equation,

*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 \( t_{ff} \) for galaxies is much shorter than the age of galaxies. \( t_{ff} \) is at least one or two orders of magnitude smaller than the age of the galaxy.

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 phase-space, 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 \( r \lesssim r_h \), the density is higher than beyond the halo radius. The gravitational interaction in the inner region is strong enough and thermalizes the self-gravitating 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 \( \mu (0) \). Large positive values of \( \mu (0) \) correspond to dwarf compact galaxies (fermions near the quantum degenerate limit), while large negative values of \( \mu (0) \) 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.

**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.

**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 \( M_{h,\mathrm{min}} \) 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.

**stable configuration of maximal**mass value \( M_h^\mathrm{max} \).

**safely**neglected.

*v*, the values for \( M_h \) predicted by this non-relativistic estimation provide minimal values for \( M_h \) while if the WDM particles become ultrarelativistic we obtain the maximum possible value for \( M_h \) Eqs. (1.13), (1.14).

We thus have a whole range of stable WDM configurations ranging from a minimal \( M_h \) for non-relativistic WDM particles as in Eqs. (1.15), (1.16) till the maximal mass \( M_h^\mathrm{max} \) in the ultrarelativistic limit as given by Eqs. (1.13), (1.14).

In fact, observations show that WDM particles are always non-relativistic and therefore galaxies with high masses near \( M_h^\mathrm{max} \sim 10^{12} \; M_\odot \) 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 \( M_h \simeq 10^6 \; M_\odot \) arises as a solution of the Thomas–Fermi approach near the quantum degenerate regime. We obtain for a halo mass \( M_h \simeq 10^6 \; M_\odot \) a halo radius \( r_h \simeq 100 \) pc as one can see from Fig. 2 and a galaxy temperature \( T_0 \simeq 0.01 \) K (see Table 1). As discussed in Sect. 2, a galaxy solution with mass \( M_h \simeq 10^6 \; M_\odot \) 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.

**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

The theoretical rotation curves and density profiles obtained from the Thomas–Fermi equations remarkably agree with observations for \( r \lesssim r_h \), for all galaxies in the diluted regime [40]. This indicates that WDM is thermalized in the internal regions \( r \lesssim r_h \) of galaxies.

*P*and the density \( \rho \) in a parametric way as

*P*as a function of the density \( \rho \) and therefore provide the equation of state. \( I_2(\nu ) \) and \( I_4(\nu ) \) 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 \( \rho \) for different values of the effective temperature \( T_0 \).

*P*as a function \( {\tilde{\rho }} \) is obtained for the values of the parameters

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 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, \( 10^6 \; M_\odot \gtrsim M_h \ge M_{h,\mathrm{min}} \) corresponding to effective temperatures \( T_0 \lesssim 0.017 \) 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 \( \Sigma _0 \). 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, 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.

## 2 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.

*f*(

*E*) , where \( E = p^2/(2m) - \mu \) is the single-particle energy,

*m*is the mass of the DM particle and \( \mu \) is the chemical potential [30, 31] related to the gravitational potential \( \phi (\mathbf {r}) \) by

*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.

*G*is Newton’s constant and \( \rho (r) \) is the DM mass density.

Equation (2.3) provides an ordinary **non-linear** differential equation that determines **self-consistently** the chemical potential \( \mu (r) \) 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*) .

**determines the equation of state**through Eq. (2.6). Contrary to the usual situation [41], we

**do not assume**the equation of state, but we

**derive**it from the Thomas–Fermi equation.

*r*since Eq. (2.3) implies

*P*(

*r*) between Eqs. (2.6) and (2.9) and integrating on

*r*gives

*f*(

*E*) must be given. We consider the Fermi–Dirac distribution,

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 \( \Psi _{\mathrm{FD}} \) 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 \( \mu /T_0 \rightarrow -\infty \) is attained for large distances *r* since Eq. (2.8) indicates that \( \mu (r) \) 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 self-gravitating Boltzmann gas.

*P*(

*r*) , which are all

*r*-dependent:

*M*(

*r*) enclosed in a sphere of radius

*r*and the phase space density

*Q*(

*r*) turn out to be

*Q*(

*r*) turns to be independent of \( T_0 \) and therefore of \( \rho _0 \).

**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 \( \Sigma _0 \), namely \( \Sigma _0 \simeq 120 \; M_\odot /{\mathrm{pc}}^2 \) up to 10–20% [38, 39, 42]. It is remarkable that at the same time other important structural quantities as \( r_h , \; \rho _0 \), the baryon-fraction and the galaxy mass vary orders of magnitude from one galaxy to another.

The constancy of \( \Sigma _0 \) 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 \( \Sigma _0 \) is the only dimensionful quantity which is constant among the different galaxies.

*r*Eq. (2.36) provide expressions for a space-dependent surface density. They are both proportional to \( \Sigma _0 \) and differ from each other by factors of order one. Notice that \( \hbar , \; G \) and

*m*canceled out in these space-dependent surface densities Eq. (2.36).

### 2.1 Galaxy properties in the diluted Boltzmann regime

**all**these galaxy magnitudes

**scale**as functions of each other.

**scaling**behavior for \( r_h, \; T_0, \; Q(0) , \; \sigma ^2(0) \) and \( v_c^2(r_h) \) as functions of \( M_h \).

**inhomogeneous perfect gas**as we will discuss in the next section.

Corresponding values of the halo mass \( {\hat{M}}_h \), the effective temperature \( {\hat{T}_0} \) and the chemical potential at the origin \( \nu _0 \) for WDM galaxies covering the whole range from large diluted galaxies till small ultracompact galaxies

\( \displaystyle {\hat{M}}_h \) | \( \displaystyle {\hat{T}_0} \) | \(\nu _0 = \frac{\mu (0)}{ T_0} \) |
---|---|---|

\( 6.56 \times 10^{12} \; M_\odot \) | 22.4 K | −23 |

\( 6.45 \times 10^{11} \; M_\odot \quad \) | 7.04 K | −20.1 |

\( 6.34 \times 10^{10} \; M_\odot \) | 2.21 K | −17.2 |

\( 4.9 \times 10^9 \; M_\odot \) | 0.613 K | −14 |

\( 2.16 \times 10^8 \; M_\odot \) | 0.129 K | −10.1 |

\( 1.55 \times 10^7\; M_\odot \) | 0.0344 K | −6.8 |

\( 3.67 \times 10^6\; M_\odot \) | 0.0168 K | −5 |

\( 1.66 \times 10^6\; M_\odot \) | 0.0112 K | −4 |

\( 1.21 \times 10^5\; M_\odot \) | 0.00278 K | −0.4 |

\( 9.73 \times 10^4\; M_\odot \) | 0.00241 K | 0 |

\( 6.31 \times 10^4\; M_\odot \) | 0.00173 K | 1 |

\( 4.06 \times 10^4\; M_\odot \) | 0.00101 K | 3 |

\( 3.48 \times 10^4\; M_\odot \) | \( 6.82 \times 10^{-4} \; \) K | 5 |

\( 3.19 \times 10^4\; M_\odot \) | \( 3.63 \times 10^{-4} \; \) K | 10 |

\( 3.12 \times 10^4\; M_\odot \) | \( 1.84 \times 10^{-4} \; \) K | 20 |

\( {\hat{M}}_h^\mathrm{min} = 3.10 \times 10^4\; M_\odot \) | 0 | \( +\infty \) |

The characteristic temperature \( \hat{T}_0 \) monotonically grows with the halo mass \( {\hat{M}}_h \) of the galaxy as shown by Fig. 1 and Eq. (2.43) following with good precision the square-root of \( {\hat{M}}_h \) equation (2.37).

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 masses \( {\hat{M}}_h < 3 \times 10^5 \; M_\odot \). The deviation from the diluted scaling regime for \( {\hat{M}}_h < 3 \times 10^5 \; M_\odot \) 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 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).

**reproduce**the observed DM halo properties with

**good precision**.

**minimum**values

*Q*(

*r*) takes its

**maximum**value

*m*

## 3 Density and velocity dispersion: universal and non-universal profiles

It is illuminating to normalize the density profiles as \( \rho (r)/\rho (0) \) and plot them as functions of \( r/r_h \). We find that these normalized profiles are **universal** functions of \( x \equiv r/r_h \) in the diluted regime as shown in Fig. 4. This universality is valid for **all** galaxy masses \( {\hat{M}}_h > 10^5 \; M_\odot \).

Our theoretical density profiles and rotation curves obtained from the Thomas–Fermi equations remarkably agree with observations for \( r \lesssim r_h \), for all galaxies in the diluted regime [40]. This indicates that WDM is thermalized in the internal regions \( r \lesssim r_h \) of galaxies.

The theoretical profile \( \rho (r)/\rho (0) \) and the precise fit \( F_{\alpha =1.5913}(x) \) cannot be used for \( x \gg 1 \) where they decay as a power \( {\simeq } 3.2 \), which is a too large number to reproduce the observations.

*F*(

*x*) and its asymptotic behavior \( F_\mathrm{asy}(x) \) vs.

*x*. We see that \( F_\mathrm{asy}(x) \) becomes a very good approximation to

*F*(

*x*) for \( x \gtrsim 3 \). When

*F*(

*x*) behaves as \( \sim 1/x^2 \) the circular velocity for these theoretical density profiles becomes constant as shown in [40].

For galaxy masses \( {\hat{M}}_h < 10^5 \; M_\odot \), near the quantum degenerate regime, the normalized density profiles \( \rho (r)/\rho (0) \) 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 from \( {\hat{M}}_h = 1.4 \times 10^5 \; M_\odot \) to the minimal galaxy mass \( {\hat{M}}_{h,\mathrm{min}} = 3.10 \times 10^4 \; M_\odot \). 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.

**universal and constant**, i.e. independent of the galaxy mass in the diluted regime for \( {\hat{M}}_h> 2.3 \times 10^6 \; M_\odot , \; \nu _0 < -5 , \; T_0 > 0.017\) K. The constancy of \( \sigma ^2(r) = \sigma ^2(0) \) in the diluted regime implies that the equation of state is that of a perfect but inhomogeneous WDM gas. Indeed, from Eq. (2.48)

*P*(

*r*) and the density \( \rho (r) \) depend on the coordinates.

For smaller galaxy masses \( 1.6 \times 10^6 \; M_\odot> {\hat{M}}_h > {\hat{M}}_{h,\mathrm{min}} \), the velocity profiles do depend on *r* and yield decreasing velocity dispersions for decreasing galaxy masses. Namely, the deviation from the universal curves appears for \( {\hat{M}}_h < 10^6 \; M_\odot \) and we see that it precisely arises from the quantum fermionic effects which become important in such range of galaxy masses.

## 4 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 \( \rho \).

*P*and \( \rho \) at a point

*r*as

*P*as a function of the density \( \rho \) and therefore provide the WDM galaxy equation of state.

For fermionic WDM in thermal equilibrium \( I_2(\nu ) \) and \( I_4(\nu ) \) 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 \( I_2(\nu ) \) and \( I_4(\nu ) \) 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 \( f_\mathrm{out}(E) = \Psi _\mathrm{out}(E/T_0) \). For the relevant galaxy physical magnitudes, the Fermi–Dirac distribution gives similar results to the out of equilibrium distribution functions [31].

*P*as a function of \( \rho \) in close form. Let us take the ratios \( P/\rho \) and \( P/\rho ^\frac{5}{3} \) in Eq. (4.1):

*P*turns out to be much smaller than \( \rho \) when both are written in the same units where the speed of light is taken to be unit.

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 \( 1.6 \times 10^6 \; M_\odot> {\hat{M}}_h \ge {\hat{M}}_{h,\mathrm{min}} , \; \nu _0 > -4 , \; T_0 < 0.011 \) 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 for \( {\hat{M}}_h> 2.3 \times 10^6 \; M_\odot , \; \nu _0 < -5 , \; T_0 > 0.017 \) K and they are quantum and compact for \( 1.6 \times 10^6 \; M_\odot> {\hat{M}}_h \ge {\hat{M}}_{h,\mathrm{min}} , \; \nu _0 > -4 , \; T_0 < 0.011 \) K. The degenerate limit \( T_0 = 0 \) corresponds to the extreme quantum situation yielding a minimal galaxy size \( {\hat{r}}_{h,\mathrm{min}} \) and mass \( {\hat{M}}_{h,\mathrm{min}} \) 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 (\( {\hat{M}}_h \gtrsim 10^6 \; M_\odot \)): 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 \( r \lesssim r_h \) [40].

For small galaxy masses, \( 10^6 \; M_\odot \gtrsim {\hat{M}}_h \ge {\hat{M}}_{h,\mathrm{min}} = 3.10 \times 10^4 \; M_\odot \), which correspond to chemical potentials at the origin \( \nu _0 \gtrsim -5 \) and effective temperatures \( T_0 \lesssim 0.017 \) 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.

Indeed, it will be extremely interesting to dispose of observations which could check these quantum effects in dwarf galaxies.

## 5 The dependence on the WDM particle mass in the diluted and quantum regimes

*m*of the WDM particle.

*m*, while the other magnitudes as \( \rho (r), \; M(r), \; \sigma ^2(r), \; P(r), \; Q(r) \), and \( \phi (r) \) do not depend on

*m*. This means that a change in

*m*, namely

*m*implies that the temperature \( T_0 \) and the chemical potential \( \mu (r) \) transform as given by Eq. (5.3). These transformations leave the Boltzmann gas equations (5.1) and (5.2) invariant.

*m*the dimensionless variables \( \xi \) and \( \nu (\xi ) \) transform as

Indeed, this invariance is restricted to the diluted regime (\( {\hat{M}}_h \gtrsim 10^6 \; M_\odot \)).

For galaxy masses \( {\hat{M}}_h < 10^5 \; M_\odot \), 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.

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