Dwarf spheroidal galaxies as degenerate gas of free fermions

In this paper we analyze a simple scenario in which Dark Matter (DM) consists of free fermions with mass $m_f$. We assume that on galactic scales these fermions are capable of forming a degenerate Fermi gas, in which stability against gravitational collapse is ensured by the Pauli exclusion principle. The mass density of the resulting configuration is governed by a non-relativistic Lane-Emden equation, thus leading to a universal cored profile that depends only on one free parameter in addition to $m_f$. After reviewing the basic formalism, we test this scenario against experimental data describing the velocity dispersion of the eight classical dwarf spheroidal galaxies of the Milky Way. We find that, despite its extreme simplicity, the model exhibits a good fit to the data and realistic predictions for the size of DM halos providing that $m_f\simeq 200$ eV. Furthermore, we show that in this setup larger galaxies correspond to the non-degenerate limit of the gas. We propose a concrete realization of this model in which DM is produced non-thermally via inflaton decay. We show that imposing the correct relic abundance and the bound on the free-streaming length constrains the inflation model in terms of inflaton mass, its branching ratio into DM and the reheating temperature.


I. INTRODUCTION
The nature of Dark Matter (DM) is one of the most important problems yet unsolved in physics. The most popular candidate for DM is a WIMP, a weakly interacting particle with a mass in the GeV-TeV range that freezes out from thermal equilibrium in the early Universe. As a consequence of this decoupling, the WIMPs cool off rapidly as the Universe expands. Therefore, in the WIMP scenario the DM that we observe today is made of slowly-moving, non-relativistic cold particles.
The cold DM scenario agrees astonishingly well with observations on cosmological scales [1]. However, cold DM simulations do not match observations at small scales (∼ kpc, the typical galactic scale). They incorrectly predict: i ) too many galactic bulges, ii ) steep density profiles in dwarf galaxies ("cusp/core" problem), iii ) too dense subhalos/satellites ("too big to fail" problem), iv ) too many subhalos/satellites ("missing satellites" problem). These problems drew an increasing level of attention, and the expression "small-scale crisis of cold DM" has been coined [2][3][4][5].
In order to solve these problems, two different approaches are commonly pursued. 1) Introduction of baryons [6,7]. Including models for baryons in the Universe can significantly alter the results from structure formation simulations. 2) Alternative DM paradigm. Along this line, e.g., models of fermionic warm DM (WDM) [8,9] and self-interacting DM [10] have been studied. Recently, the possibility that DM is made of ultra-light bosons able to form a Bose-Einstein condensate (BEC) on galactic scales has been proposed [11][12][13][14][15][16][17][18][19][20][21][22] 26] for the special case in which the BEC is made of QCD axions, and ref. [27] for a more general scenario dubbed fuzzy DM). Neglecting for simplicity self-interactions, in a BEC the pressure supporting the system from gravitational collapse is provided by the Heisenberg uncertainty principle. However, this mechanism alone is capable to sustain a galactic structure only if the boson is ultra-light, i.e. with a mass of order O(10 −25 eV).
The situation is completely different if we consider a fermion rather than a boson, since the pressure arising from the Heisenberg uncertainty principle is enforced by the Pauli exclusion principle. As a consequence, galactic structures can be protect from gravitational collapse even for larger values of the DM mass.
In refs. [28][29][30] the observed properties of galactic structures have been studied in the context of WDM. The starting assumption is that a galactic halo can be described by a self-gravitating Fermi gas of DM particles. The outcome of the analysis outlines an extremely interesting scenario. In particular, it is argued that compact dwarf galaxies correspond to the quantum degenerate limit of the Fermi gas, while larger galaxies correspond to the classical Maxwell-Boltzmann limit.
In this paper we are interested in the degenerate Fermi limit, and our aim is to test this scenario against the experimental data describing the kinematic of the Milky Way's dwarf spheroidal galaxies. Since these galaxies are completely dominated by DM, it is reasonable to expect that all the observed kinematic data are tightly linked to the fundamental properties of their DM content. Moreover, since the possibility to form a degenerate Fermi configuration only relies on the fermionic nature of DM particles, we are not forced to have in mind any specific WDM candidate, like for instance sterile neutrinos. On the contrary, our analysis assumes the simplest DM candidate one can imagine: a free fermion with mass m f .
In more detail, this paper is organized as follows. In section II we sketch, using qualitative arguments, the role of the quantum pressure. In section III we review the basic properties of a degenerate Fermi gas model. In section IV we fit the model against the data describing the velocity dispersion of the eight classical dwarf spheroidal galaxies of the Milky Way. In section V we discuss the results in the context of an explicit realization of the model in which DM is produced non-thermally via inflaton decay. In this section we also test the hypothesis that spiral galaxies can be described by non-degenerate configurations of the same DM particle. Finally, we conclude in section VI. In appendix A we collect the details of the fit performed in section IV. In appendix B we shortly review the statistical mechanics of self-gravitating fermions.

II. QUANTUM PRESSURE VERSUS GRAVITY
Galaxy formation is driven by classical gravitational physics. However, in order to protect a galaxy from gravitational collapse, one needs to counterbalance gravity with a supporting pressure, ordinarily given by thermal pressure.
Here, we want to explore the possibility that the equilibrium is sustained by quantum pressure. Quantum pressure arises from the two fundamental principles of quantum theory: the Heisenberg uncertainty principle and the Pauli exclusion principle. The quantum pressure is present even when the temperature of the gas is so low that the ordinary pressure does not hold it up. This is exactly the degenerate limit we are interested in. For bosons, the degenerate configuration is the BEC; for fermions, it corresponds to the degenerate Fermi gas. In the following, we will discuss both these situations. The aim of this section is to provide an intuitive picture of the physics involved. The system we are interested in is a gas of N fermionic (bosonic) DM particles with mass m f (m b ), confined in a volume V , with number density n = N/V , total mass M , radius R, and mass density ρ = M/V .

A. Ultra-light bosons and the BEC
First, let us illustrate the bosonic case. We start our discussion by computing the least momentum allowed by the Heisenberg uncertainty principle. The minimum momentum -minimum since we are interested in the ground state of the system -is associated with the maximum uncertainty in the position, i.e. ∆x ∼ R. Therefore, the minimum momentum is given by p ∼ h/R, where h is the Planck constant. The quantum pressure P Q , i.e. the flux of Heisenberg momentum, is given by P Q ∼ nvp ∼ h 2 ρ/m 2 b R 2 , where we used n = ρ/m b , and the typical momentum-velocity relation for non-relativistic particles. Remarkably, this qualitative result can be obtained in a more formal way using the Gross-Pitaevskii-Poisson description of a self-gravitating BEC [13][14][15][16][17][18]. The pressure from gravitational attraction, i.e. the gravitational force per unit of area, is given by P G ∼ GM 2 /R 4 , where G is the Newton constant. Therefore, the condition for equilibrium is given by Using the dimensional estimate ρ ∼ M/R 3 we extract the typical size of a self-gravitating BEC with mass M ( Considering for concreteness a typical size R ∼ 100 kpc, M = 10 12 M (with solar mass M = 1.98 × 10 30 kg), one obtains m b ∼ 10 −25 eV.

B. Degenerate Fermi gas
According to the Pauli exclusion principle if there are N fermions in the system they can not occupy the same state with minimum momentum -as they would if they were bosons -but they must form pairs with increasing value of momentum separated by at least p ∼ h/R. The minimum momentum must therefore be p ∼ N 1/3 h/R. Therefore, if N 1 the quantum pressure for a system of fermions is much bigger than in the bosonic case. The related quantum pressure is given by Using again the dimensional estimate ρ ∼ M/R 3 we extract the typical size of a self-gravitating degenerate Fermi gas with mass M Considering for concreteness the typical size R ∼ 100 kpc, M = 10 12 M , one obtains m f ∼ 20 eV. Therefore, a degenerate Fermi gas may sustain a galactic structure with m f m b . Motivated by these results, in the next section we will study the details of the degenerate Fermi gas configuration.

III. DEGENERATE FERMI GAS: QUANTITATIVE ANALYSIS
The assumption underlying this paper is that in the present Universe a galactic structure can be described by a self-gravitating Fermi gas of DM particles in a state of statistical equilibrium. More formally, this equilibrium state is represented by a Fermi-Dirac distribution.
Throughout our analysis, we neglect the role of baryons. This assumption is perfectly justified even in the case of large galaxies, where the baryonic component covers at most a few percent of the total mass [31]. Following refs. [28][29][30], we assume the same distinction between small and large galaxies: the former are close to the degenerate limit of the Fermi-Dirac distribution, the latter are described by the opposite Maxwell-Boltzmann regime. As mentioned in the introduction, in this paper we are interested in the degenerate limit of the Fermi gas, and we aim to test it against the kinematic data describing the classical dwarf spheroidal galaxies of the Milky Way. In the following, we describe quantitatively the degenerate limit (see appendix B for a more detailed discussion); in particular, we provide all the formulas that are relevant for the phenomenological analysis that will be performed in section IV. Most of the arguments faced in this section are known from standard textbooks. We refer to ref. [32] for a general introduction.

A. General overview
At T = 0 the Fermi-Dirac distribution is a step function where p F is the Fermi momentum. Since the temperature is zero, particles have zero kinetic energy. If they were bosons, they would occupy the lowest energy level; fermions, on the contrary, are subject to the Pauli exclusion principles, and they will fill all states with momentum lower than p F . It is possible to associate to the Fermi momentum a Fermi velocity v F = p F /m f . The number of levels with momentum between p and p + dp is given by dχ = (4π/h 3 )p 2 dp; since there are two particles for each level, the number of fermions per unit of volume 1 is n = pF 0 2dχ = 8πp 3 F /3h 3 . The mass density is ρ = mn = 8πmp 3 F /3h 3 , from which we get p F = (3h 3 ρ/8πm) 1/3 . Knowing p F , we can compute the pressure via the usual integral P = (8π/3h 3 ) pF 0 (p 4 / p 2 + m 2 f )dp. In the nonrelativistic case we obtain while in the ultra-relativistic case A given configuration of matter will be in equilibrium if the gradient of the pressure is balanced by the gravitational attractive force per unit of volume. More formally, the equation for the hydrostatic equilibrium is given by Throughout this paper we assume spherical symmetry; in eq. (8) M (r) is the mass within the radius r, and the same radial dependence is explicitly written also for the pressure and the mass density. A simple order-ofmagnitude estimate reveals that in the non-relativistic case dP/dr ∼ M 5/3 /R 6 , while in the ultra-relativistic case dP/dr ∼ M 4/3 /R 5 , where M and R are, respectively, the characteristic mass and length scale of the configuration. On the other hand, from eq. (8), the gravitational force per unit of volume scales according to the ratio M 2 /R 5 . In the non-relativistic case the Pauli pressure and the gravitational force depend on the radius with a different power; it means that -for a given value of mass M -the gas can always adjust the radius until the two forces are equal. In the ultra-relativistic case the Pauli pressure and the gravitational force depend on the radius in the same way; therefore, equilibrium is possible only for one value of the mass. As a consequence, it is important to understand under which conditions either of the two limits is realized. To this purpose, it is useful to define the critical density ρ crit as the density at which the Fermi momentum becomes equal to the fermion mass, i.e. m f = (3h 3 ρ crit /8πm f ) 1/3 , from which we get If ρ ρ crit (ρ ρ crit ), the gas is non-relativistic (ultrarelativistic) since p F m f (p F m f ). As we shall see later, for m f > 1 eV the critical value in eq. (9) is well above the typical mass density characterizing DM halos of galactic size. In the analysis of the dispersion velocities of the dwarf spheroidal galaxies, therefore, we will always use the non-relativistic limit. Nevertheless, it is instructive to keep both limits for the rest of this discussion, writing in full generality the equation of state in the form P = Kρ γ , with K ≡ h 2 /5m 4/3 f (3/π) 1/3 ) and polytropic index γ = 5/3 (γ = 4/3) in the non-relativistic (ultra-relativistic) limit. Coupling eq. (8) with the continuity equation dM/dr = 4πr 2 ρ(r), we obtain Using γ = 1 + 1/n -with n = 3/2 (n = 3) in the nonrelativistic (ultra-relativistic) limit -and rescaling the radial coordinate according to ξ = r/α, with α ≡ (n + 1)Kρ it is straightforward to show that eq. (10) is equivalent to the Lane-Emden equation where θ is related to the density via ρ = ρ 0 θ n (ξ), for central density ρ 0 . The Lane-Emden equation (12) can be solved numerically for the values of n that are relevant in the present analysis, using the boundary conditions θ(0) = 1, θ (0) = 0. The first zero of the solution, θ(ξ 1 ) = 0, defines the radius of the configuration R = ξ 1 α while the total mass is given by M = 4π R 0 r 2 ρ(r)dr. We find Combining eq. (13) and eq. (14), we find the mass-radius relation For definiteness, we find the numerical approximations ξ 1 = 3.65, ξ 2 1 θ (ξ 1 ) = −2.714 for γ = 5/3 (ξ 1 = 6.89, ξ 2 1 θ (ξ 1 ) = −2.018 for γ = 4/3). Besides these analytical results, it is important to keep in mind -using more qualitative arguments -the general features of the model. The crucial physical properties can be understood looking again at the dimensional analysis of the equilibrium condition in eq. (8), that we rewrite here for convenience in the non-relativistic limit we are interested in Suppose now to have an equilibrium condition for a given mass and radius M , R. If we increase the mass, the gravitational force per unit of volume grows faster than the repulsive force induced by the Pauli pressure. To maintain the equilibrium, the system decreases its radius, since the Pauli pressure increases faster than the gravitational force going towards smaller distances. As a consequence, the mean density of the configuration, ρ = 3M/4πR 3 , increases. In a degenerate Fermi configuration, larger values of mass correspond to more compact objects. We plot the mass-radius relation, eq. (15), in fig. 1 for the non-relativistic case. From this plot, it is clear that large galactic structures (i.e. galaxies with representative values of total mass and radius equal to M ∼ 10 10 -10 12 M , R ∼ 10 -100 kpc) can be reproduced only considering small values of m f , i.e. m f 10 eV, while, on the contrary, dwarf galaxies (i.e. galaxies with representative values of total mass and radius equal to M ∼ 10 7 -10 9 M , R ∼ few kpc) require larger values, i.e. m f 100 eV. In the remainder of this paper, we will mainly focus on the classical dwarf spheroidal galaxies of the Milky Way, and in section IV we will test the degenerate Fermi gas model against experimental data describing their velocity dispersion. Dwarf galaxies, in fact, are astrophysical objects largely dominated by their DM component -as inferred from the analysis of the stellar-to-halo mass ratio [33,34]. The velocity dispersion of the dwarf spheroidal galaxies remains approximately flat with radius, thus suggesting a core profile for the DM mass density and a mass M (r) linearly increasing with the radial distance.
In the left panel of fig. 2, we show the mass density ρ(r), solution of eq. (10), as a function of the radius for a fixed value of mass, m f = 200 eV, and different values of the central density ρ 0 . In the right panel of fig. 2, we show the mass M (r) of the configuration as a function of the radius. On the qualitative level, both the mass density and the mass profile seem to possess the right prerogatives to fit the observed galactic rotation curves: the former clearly exhibits a core profile, the latter a linear increase with r. It is striking to observe how these two properties are directly connected to the physical assumptions underlying the degenerate Fermi gas model, instead of be the outcome of a complicated numerical simulation. We postpone to section IV a careful phenomenological analysis.
Finally, notice that for n = 3, corresponding to the ultra-relativistic limit, the radius of the configuration, R, disappears from eq. (15) thus defining -for a fixed value of m f -a unique value of total mass, M Ch (the analogue of the Chandraseckhar limit). We find In ref. [35], the possibility to extract lower bounds on the DM mass from the analysis of DM phase-space distribution in dwarf spheroidal galaxies has been discussed, referred to as the Tremaine-Gunn bound (see also refs. [36][37][38]). In ref. [39], the special case of a degenerate fermionic self-gravitating gas has been explicitly discussed. To be more concrete, the strategy adopted in ref. [39] is the following. For a spherically symmetric DM-dominated object with the mass M within the region R, it is possible to obtain a lower bound on the DM mass by requiring that the Fermi velocity v F of the degenerate gas of mass M in the volume V = 4πR 3 /3 does not exceed the escape velocity v ∞ = (2GM/R) 1/2 . More formally, this condition amounts to imposing the following inequality where we used the mean density ρ = 3M/4πR 3 . Inverting for m f , one obtains the bound [39] Finally, using specific values for R and M extracted from observations, 2 it is possible -for each one of the dwarf galaxy analyzed -to convert eq. (19) into a numerical bound on m f . To give a taste, in ref. [39] the lower bound obtained, e.g., using the Carina dwarf spheroidal galaxy is m f > 215 eV. 3 The validity of this method clearly depends on the accuracy of the mass profile estimate. In the case of a degenerate non-relativistic Fermi gas, moreover, we argue that the inversion of eq. (18) with v F < v ∞ . As a consequence, in eq. (18) both the m f -and ρ 0 -dependence disappear, making eq. (19) useless if applied to the relevant parameter space of the model, i.e. (m f , ρ 0 ). In this paper, instead of using eq. (18), we extract a bound on the parameter space imposing the condition where the Fermi velocity, as discussed above, depends on the parameters m f , ρ 0 while v obs ∞ is related to the veloc-3 Stronger bounds can be derived assuming an initial thermal distribution of the DM particles [39]. As we will see in sec. V, this is not the case we will be interested in here and these bounds hence do not apply. 4 These equation are written considering the whole degenerate configuration, i.e. for ξ = ξ 1 . However, they still hold using a generic value of ξ, with r = ξα and M (r) = 4π r 0 s 2 ρ(s)ds.
ity dispersion directly measured in astrophysical observations via v obs ∞ √ 6σ, 5 where σ is the one-dimensional velocity dispersion. We postpone to the next section the exact definition of these quantities.

IV. VELOCITY DISPERSION OF THE MILKY WAY'S CLASSICAL DWARF SPHEROIDAL GALAXIES
The Milky Way's dwarf spheroidal galaxies are low luminosity, low surface-brightness satellite galaxies characterized by no net rotation, a very large dynamical massto-light ratio and a small baryonic component. Eight of them, dubbed classical, are characterized by high-quality data sets describing their stellar kinematic. In this section, we aim to use these data in order to test the degenerate Fermi gas model introduced in section III.
In the analysis of the classical dwarf spheroidal galaxies only two quantities are directly observed: i ) the line-ofsight velocity dispersion as a function of the projected radius, and ii ) the surface brightness profile as a function of the projected radius. We refer the interested reader to ref. [40] for a proper definition of these quantities. In the following we summarize, for the sake of clarity, the relevant formulas used in the fit.
The square of the projected velocity dispersion along the line-of-sight is We adopt the Plummer profile for the projected stellar density where L is the total luminosity and r half the half-light radius. We take the corresponding values from ref. [40]. 5 The quantity directly observed is the projection of the velocity of the stars along the line-of-sight, and it is a function of the projected radius (the so-called velocity dispersion, see eq. (23) below). For DM-dominated objects -like the dwarf spheroidal galaxies under scrutiny in this paper -rotation curves flatten, and the projected velocity can be characterized by a single, constant value σ. We take the corresponding values from ref. [40]. The relation v obs ∞ √ 6σ holds under the assumption of isotropic velocity distributions. This approximation seems to be reasonable for DM particles [39], for which numerical simulations usually predict a value of velocity anisotropy close to zero. As far as luminous stars are concerned, on the contrary, large values of velocity anisotropy are expected. We will discuss this issue in the next section.
The 3-dimensional stellar density, assuming spherical symmetry, is given by Using in eq. (23) the mass distribution obtained in section III A, we perform a χ 2 fit of the degenerate Fermi gas model against the velocity dispersion of the eight classical dwarf spheroidal galaxies of the Milky Way. We take the corresponding values from ref. [40].  15). Equipped with these results, we can compute for each dwarf galaxy the value of M (r half ), i.e. the mass enclosed in a sphere of radius r half , predicted by the degenerate Fermi gas models. These values have been extracted in ref. [40] by means of a numerical Markov-chain Monte Carlo analysis. We compare the result of ref. [40] with our predictions in table II. Even if this comparison is not completely meaningful -in ref. [40] M (r half ) is extracted assuming a specific halo profile (the generalized Hernquist profile, see also ref. [41]) -it is nevertheless interesting to notice that we find a fairly good agreement. Let us now briefly comment about the values of the stellar anisotropy β that we obtain from the fit. On a general ground, it is possible to establish the following correspondences (see, e.g., ref. [20]). If all the stellar orbits are circular, β = ∞; if they are isotropic, β = 0; if they are perfectly radial, β = 1; finally, tangentially biased systems correspond to β < 0. In principle, there is no a priori preference for anyone of these values; however, values β 1 seem to be disfavored by the peculiar condition that they would require [20]. In our analysis we do not find any particular preference for β; it ranges from β = 1 for Leo II if m f = 150 eV, to β = −1.3 for Fornax if m f = 200 eV.
In fig. 6, we compare the results of our analysis against the bound in eq. (22). We plot the best-fit value for the central density ρ 0 as a function of the mass m f , obtained marginalizing over the orbital anisotropy β. The value m f = 150 eV is consistent with the bound v F v obs ∞ for all the analyzed dwarf spheroidal galaxy but Leo II. Note  table I and table II. that this result has been obtained using a fixed profile for the stellar density, i.e. the Plummer profile in eq. (25). This profile is completely fixed once the value of r half is specified. Different choices can be made (e.g. the King profile in ref. [42]), thus introducing extra free parameters that may change the result of the fit. The value m f = 200 eV, on the contrary, is compatible with the bound in eq. (22). Since in this paper we are not interested in exploring more complicated setups for the stellar density, we identify the value m f 200 eV as the best outcome of our phenomenological analysis. On a qualitative level, moreover, an upper bound on m f follows from the fact that larger values of m f imply smaller size for the DM halos, thus leading to unrealistic results. To be more concrete, we find that even a value as large as m f 250 eV leads to a significant worsening of the fit.
In summary, we find that the degenerate Fermi gas model can reproduce in a realistic way the kinematic data describing the velocity dispersion of the eight classical dwarf spheroidal galaxies of the Milky Way if m f 200 eV. Let us stress once again that this is a remarkable result given the simplicity of the model: it provides a universal halo profile able to adapt -for a nearly unique value of m f and with the only freedom of the structural parameter ρ 0 -to all the observed kinematic data. Furthermore, it has been obtained under the simplifying assumption that the whole system is in a perfect degenerate limit. The reader should keep in mind, in fact, that in a more realistic situation this is probably true only for a fraction of the DM particles. DM particles in the mass range identified in the last section, m f 200 eV, are most likely produced as ultrarelativistic particles in the early Universe. Due to absence of interactions (apart from gravity) with the Standard Model (SM), they decouple immediately and are hence classified as WDM. As such their free-streaming length λ FS -measuring the length-scale below which structures are erased -must be checked against observations of the Lyman-α forest, of high-and low-z galaxies as well as high-z gamma ray bursts (see e.g. ref [43] and references therein for a recent analysis). To give a rough estimate, a free-streaming length above λ 0.5 Mpc is excluded, 0.3 Mpc λ FS 0.5 Mpc is an interesting window in which one might account for the deficit of structure on small scales compared to N-body simulations, and finally for λ FS < 0.1 Mpc the particles behave as cold DM from the point of view of structure formation [44]. In the simplest situation of a thermal relic, there is a one-to-one correspondence between λ FS and the DM mass, leading to the exclusion of WDM with a mass ∼ 1 ÷ 2 keV.
To retain our window of m f ∼ 0.1 ÷ 0.3 keV, we will in the following assume that the DM particles are produced non-thermally in the decay of the inflaton parti- cle. 6 In this case, their initial momentum distribution is peaked a specific momentum p i = m φ /2, where m φ is the inflaton mass. Subsequently, the distribution is redshifted as p(t)/p i = a i /a(t), retaining the approximately monochromatic distribution. The free-streaming length can be calculated as where v(t) = p(t)/E(t) is the velocity of the DM particle, x ≡ 4 √ t eq m f /( √ t p m φ ), H 0 = 67.11 km/(s Mpc) is the Hubble constant today, z eq = 3402 is the redshift at matter-radiation equality and t p , t eq = 0.695 Myr and t 0 = 13.819 Gyr denote the time of DM production, of matter-radiation equality and today, respectively. Assuming a small branching ratio of the inflaton into DM particles, Br = Γ φ→DM /Γ φ 1, the production time t p is determined by the reheating temperature T RH t p = 1 Br 45 π 2 g γ,RH Here M P = 2.4 × 10 18 GeV is the reduced Planck mass and g γ denotes the relativistic degrees of freedom contributing to the entropy of the photon bath, for the SM: g γ,0 = 2, g γ,RH = 427/4, where the indices 'RH' and '0' stand for the time of reheating and today, respectively. To obtain eq. (28) we have exploited that the reheating temperature is determined by the decay rate into SM particles Γ φ→SM Γ φ , T RH = (45/(π 2 g γ,RH )) 1/4 Γ φ→SM M P . In summary, this yields the free-streaming length as a function of the inflaton mass, the reheating temperature and the branching ratio, cf. gray shaded regions in fig. 4.
From the reheating temperature and the branching ratio we can calculate the DM abundance where we have used ρ φ = m φ n φ π 2 /30 g γ,RH T 4 RH . The observed abundance today [1] implies Together with the constraint from the free-streaming length, λ FS < 0.5, this implies for our reference value of m f = 200 eV. The viable parameter space of our model covers both small values of λ FS in which structure forms as in the cold DM paradigm, as well as larger values (close to the bounds in eq. (31)), in which the observed deficit of structure on small scales might be explained. Moreover, it is indeed remarkable that the hypothesis of a degenerate fermion gas as main component of dwarf galaxies has lead us to conclusions on the preceding inflation model, even more so as these constraints can indeed be fulfilled in some typical inflation models.

B. A criterium for degeneracy
In the analysis of section IV we have assumed that dwarf galaxies have reached a degenerate configuration. Given the production of the DM particles in the early Universe at high energies, this implies the occurrence of a phase transition. A comprehensive study of this problem is of course beyond the purposes of this paper; nevertheless it is interesting to provide at least some simple arguments in favor of this possibility. 7 For a gas of fermions, 7 See ref. [46] for a general overview of the problem. the degeneracy temperature T DEG is defined as follows 8 where k B is the Boltzmann constant. To good approximation, this corresponds to the Fermi temperature of a gas of non-relativistic particles: If the present temperature of the fermionic gas is much lower than this degeneracy temperature, then a necessary condition to have a degenerate configuration is satisfied.
To address the question of consistency of such a degenerate configuration today, let us have a brief look at its cosmological history. Let us consider a particle of mass m f 200 eV with no sizable couplings to the SM, produced relativistically in the early Universe in the 2body decay of e.g. the inflaton particle. As described above, the resulting momentum distribution is strongly peaked. This changes only when structure formation sets in at a redshift of z = O(1 − 10). 9 Due to their gravitational interaction, local overabundances of DM density (seeded by primordial fluctuations) form virialized DM halos. The time-dependent gravitational forces lead to violent relaxation, converting the non-thermal peaked distribution into an equilibrium Fermi-Dirac distribution within a few dynamical times [32,[46][47][48][49]. The corresponding temperature can be estimated using the virial theorem equating the total kinetic and potential energies, For typical values for the mass and radius of dwarf galaxies, cf. fig. 1, this estimate yields a value close to the degeneracy temperature. On the other hand, for large galaxies we find T DEG 10 −4 K T DM,0 ∼ 10 −1 K, where we have used m f = 200 eV, M = 10 11 M , R = 50 kpc, ρ ∼ M/( 4 3 πR 3 ) as a reference. This estimate shows that temperatures below the degeneracy limit are indeed within reach for dwarf galaxies, while larger galaxies might be described by a thermal distribution of the same DM particle, which is however not in the degenerate limit. This simple argument thus supports the picture sketched in refs. [28][29][30]. In the next section, we will discuss this point in more details.
Finally, there is one more consistency check we can perform on the cosmological history of our DM particle. Ignoring structure formation, we can calculate the 'would-be' temperature T DM of the DM particles today, based solely on the redshift of their peaked distribution. Since the gravitational collapse during structure formation heats up the DM gas, a necessary condition is where E k is the kinetic energy. Based on eq. (34) we can estimate the would-be temperature of the DM sector today as where the subscripts 0 and i stand for today and an early initial time, respectively. T γ denotes the temperature of the photon bath with T γ,0 = 2.7 K. For m f = 200 eV, the condition T DM < T DM,0 is fulfilled as long as m φ 10 2 T RH . From fig. 4 we see that for the values of m φ and T RH in accordance with bounds from structure formation, this is easily fulfilled: indeed m φ 10 −4 T RH .

C. Larger galaxies and the non-degenerate configuration
We have mentioned the possibility that, within the framework studied in this paper, larger galaxies correspond to non-degenerate configurations of the gas. This picture has been studied in the context of WDM in refs. [28][29][30]. However, since we are considering a significantly different value of the DM mass, their conclusions do not apply straightforwardly to our case. The aim of this section is to provide a simple quantitative analysis able to attest the validity of the aforementioned hypothesis.
The statistical analysis of a self-gravitating Fermi gas at non-zero temperature can be carried out in analogy to what was discussed in section III A for the special case of the degenerate configuration. We review the basic formalism in appendix B. In a nutshell, it is possible to retrace the same discussion outlined in section III A using the Fermi-Dirac distribution at finite temperature instead of the degenerate limit in eq. (5). It turns out that in a non-degenerate configuration the gas is characterized by the following equation This equation relates the temperature T , the central density ρ 0 , and the dimensionless parameter k which controls the degree of degeneracy of the gas. 10 The limit k → ∞ corresponds to the classical limit of an isothermal gas described by the Maxwell-Boltzmann statistic, while the limit k → 0 corresponds to the Fermi degenerate gas (see appendix B 1). For a given value of k the mass density ρ(r) of the system can be obtained by numerically solving a generalized Lane-Emden equation (see eq. (B8)), and the central value ρ 0 -or, equivalently, the temperature via eq. (36) -is a free parameter that we need to extract from observations. Our approach is the following. As customary in the analysis involving the Burkert DM profile [50] we define the core radius R H as the radius where the DM density equals one fourth of its central value.
The corresponding core mass is M H = RH 0 4πr 2 ρ(r)dr. We compute both these quantities for different values of k and ρ 0 , while we keep the DM mass fixed at m f = 200 eV. In fig. 5 we compare our results against the charac- Core mass-radius relation for the non-degenerate Fermi gas against data describing small, medium and large spiral galaxies. The DM mass is fixed to m f = 200 eV. We plot the theoretical prediction of the model for four different values of the central density ρ0, while on each curve the gradient color marks different values of k. The limit k → ∞ correspond to the Maxwell-Boltzmann regime.
teristic values of core mass and radius for small spiral galaxies (SSG), medium spiral galaxies (MSG) and large spiral galaxies (LSG). These characteristic data are taken from ref. [28]. We plot the theoretical prediction for the core mass-radius relation of the non-degenerate Fermi gas considering four different values of central density, from ρ 0 = 10 −28 kg/cm 3 to ρ 0 = 10 −26.5 kg/cm 3 , while the color gradient spans the range k = 1÷10 7 . The model can easily accomodate large galactic structures for increasing value of k, thus confirming the general picture in which large galaxies are described by the Maxwell-Boltzmann limit of the theory. To be more concrete, we find that SSG correspond to k ∼ O(10) and T ∼ 0.1 K, MSG to k ∼ O(10 3 ) and T ∼ 0.5 K, and LSG to k ∼ O(10 6 ) and T ∼ 1 K.
Before concluding, let us discuss one more interesting point. In ref. [51] it has been noticed that the observed surface density Σ 0 ≡ R H ρ 0 is nearly constant for the observed galaxies and does not depend on the galaxy luminosity. This is a remarkable result, in particular since it has been obtained analyzing different galactic systems in a range that covers over 14 magnitudes in luminosity. The best-fit value of the surface density obtained in ref. [51] is Σ 0 = 141 +81 −52 M /pc 2 . We can compare the prediction of the non-degenerate Fermi model extracted from our fig. 5 against this value. We find Σ 0 238 M /pc 2 (SSG), Σ 0 280 M /pc 2 (MSG), Σ 0 275 M /pc 2 (LSG). Therefore, the value of surface density predicted by the model is nearly constant, even if slightly larger w.r.t. the one observed in ref. [51] (but still compatible within the errors). The reader should keep in mind, moreover, that our analysis is not based on real data but only on an order-of-magnitude estimate for the core mass and radius of spiral galaxies. A more detailed analysis is mandatory but it is beyond the purpose of this paper.
In conclusion, we found that the picture according to which large galaxies correspond to the non-degenerate limit of the Fermi gas is consistent in our model with m f = 200 eV.

VI. CONCLUSIONS
Numerical simulations with cold DM deviate from observations at galactic scales. Two of the most problematic aspects are related to the prediction of cuspy (rather than cored) density profiles for dwarf galaxies, and to an overwhelming abundance of small structures that are not observed. Motivated by these problems, in this paper we proposed a simple alternative paradigm in which DM is made of free fermions with mass m f . We can summarize the most tantalizing consequences of this assumption as follows.
In the first part of the paper, we started our analysis from the Universe that we observe today.
• Describing a galactic structure as a self-gravitating Fermi gas of DM particles, we focused our attention on the case of dwarf spheroidal galaxies. In particular, we assumed that a dwarf spheroidal galaxy corresponds to the degenerate limit of the gas, where the attractive force of gravity is entirely balanced by the quantum pressure arising from the Pauli exclusion principle. In this picture, therefore, dwarf spheroidal galaxies are quantum astrophysical objects.
• In the degenerate configuration DM halos are described by a cored mass density profile. We tested the model against the kinematic data describing the velocity dispersion of the eight classical dwarf spheroidal galaxies of the Milky Way. We found a good agreement with data providing that m f 200 eV. We pointed out why this value is not in violation of the Tremaine-Gunn bound.
• Larger galaxies correspond to non-degenerate configurations of the Fermi gas. We tested this picture for the value m f 200 eV. We showed that large spiral galaxies correspond to the classical Maxwell-Boltzmann limit, while small spiral galaxies are closer to the degenerate configuration.
Going back in time, we analyzed the implications of the value m f 200 eV throughout the history of the Universe.
• We discussed a concrete realization in which DM is produced non-thermally by the decay of the inflaton during reheating. This mechanism offers a remarkable connection between the primordial Universe and the Universe observed today. DM particles are ultra-relativistic at the time of their production but they have to be non-relativistic at the time of matter-radiation equality, since otherwise the structures that we observe today would be erased.
• By computing the free-streaming length, imposing the bound from the Lyman-α forest, and requiring to reproduce the correct value of DM relic density, we obtained a consistent picture only for specific values of the inflaton mass, reheating temperature and branching ratio for the decay of the inflaton into DM. Moreover, we also noticed that in a region of the allowed parameter space the model can easily explain the deficit of small structures observed today.
To sum up, starting from the assumption that dwarf spheroidal galaxies are quantum astrophysical objects we presented a simple model of fermionic DM consistent with observations. Most importantly, the model features a remarkable connection between the early and the present Universe that can be tested by constraining the parameters of the early Universe (e.g. by a future measurement of the stochastic gravitational wave background of inflation or of the amplitude of the primordial B-modes in the CMB) or by improving the current kinematic description of Milky Way's dwarf spheroidal galaxies. In particular, it would be interesting to perform the same analysis outlined in this paper for the faintest dwarf spheroidal galaxies for which velocity dispersion profiles are not yet available.
Open questions remain. In particular, it would be interesting to extend our qualitative arguments for larger galaxies by a more quantitative study, focusing on the value of m f found here. Furthermore, the impact of the constraints found here on inflation model building remain to be analyzed in more detail. We leave these questions to future work.
where g = 2s + 1 is the number of internal (spin) degrees of freedom. In this paper we take g = 2. The function f ( x, p, τ ) represents, by definition, the phase-space density. For a given galaxy, it is reasonable to assume that at the present time the phase-space density reaches a timeindependent equilibrium form. The dynamics leading to this equilibrium configuration is governed by the Liouville theorem which asserts that for a dissipationless and collisionless system the phase-space density is constant, df /dτ = 0. Writing explicitly the derivatives, one obtains the Vlasov equation where we used d p/dτ = −am f ∇Φ, with Φ the gravitational potential. Starting from an arbitrary initial condition far from equilibrium, the Vlasov equation develops a complicated mixing process in phase-space, known as phase-space mixing. This process begins at matter-radiation equality, when density perturbations start growing shaped by the action of gravity. The only practical way of integrating the Vlasov equation is by Nbody simulations; however, we are interested in a probabilistic description, i.e. we want to know the most probable distribution of self-gravitating fermions at statistical equilibrium. To determine the equilibrium distribution of the system, it is possible to introduce an entropy functional like in ordinary statistical mechanics. The statistical equilibrium state is obtained by maximizing the entropy. As shown in refs. [46][47][48][49], critical points of the entropy correspond to the Fermi-Dirac distribution. Therefore, regardless of the dynamics of the system, we can directly focus on the Fermi-Dirac distribution since we are only interested in the final equilibrium state. 12 In the following, we take a closer look the the Fermi-Dirac distribution describing a system of self-gravitating fermions. Even if in this paper we are only interested in the degenerate limit, we discuss the general situation at a given non-zero temperature T . The aim of this approach is to properly define the degeneracy temperature used in section V.  a In ref. [40], M (r half ) is estimated in two ways: using a numerical Markov chain Monte Carlo method and by means of a simple analytic model based on the Jeans equation. Since the latter assumes, contrary to our analysis, a stellar velocity distribution that is isotropic β = 0, we compare the prediction of the degenerate Fermi model with the value of M (r half ) extracted in ref. [40] from the full numerical analysis.  in table I and table II, and the values of M (r half ) obtained in ref. [40].

Fermi-Dirac distribution at temperature T
The non-relativistic Fermi-Dirac distribution describing a system of self-gravitating fermions is given by where Φ is the gravitational potential and µ the chemical potential. In order to provide a closed description of the system, we need to couple eq. (B3) with the Poisson equation Φ = 4πGρ. To achieve this goal, we need an explicit expression for the mass density ρ; this is given by ρ = nm f , where n is the number density of the system (B4) Introducing the Fermi integral I n (t) and introducing the variable ψ ≡ m f (Φ − Φ 0 )/k B T , with Φ(0) ≡ Φ 0 . We find 1 ξ 2 d dξ ξ 2 dψ dξ = I 1/2 (ke ψ ) , with k ≡ λe m f Φ0/kBT and boundary conditions ψ(0) = ψ (0) = 0. The solution of eq. (B8) -ψ k (ξ) in the following -depends on the value of k, that controls the degree of degeneracy of the gas; as we shall see in appendix B 2, in fact, the limit k → ∞ corresponds to the classical limit of an isothermal gas described by the Maxwell-Boltzmann statistic, while the limit k → 0 corresponds to the Fermi degenerate gas. Notice that, using the previous definition of k, eq. (B5) can be rewritten as Eq. (B8) has to be integrated from ξ = 0 till some value ξ 1 that defines, via eq. (B7), the total radius of the configuration R. 13 Using the Gauss's theorem at the boundary r = R it is possible to obtain the condition (B10) where we have introduced the normalized temperature η and where M = M (R) is the total mass of the configuration. Using eq. (B7), simple algebra allows to show that is called degeneracy parameter. It follows that A given configuration, with a total mass M and radius R, is characterize by a specific value of the degeneracy parameter µ D ; the integration of eq. (B8), therefore, can be performed till the value ξ 1 that satisfies the condition given by eq. (B12). In order to proceed in the thermodynamic description of the system, we need to compute the pressure and the total energy; the former, in the non-relativistic limit, is given by 13 In the degenerate limit of the self-gravitating Fermi gas, it is obvious to identify ξ 1 with the first zero of the mass density ρ(r) (see eq. (13) and appendix B 2 below). However, in a generic non-degenerate configuration at a given temperature T , the mass density ρ(r) goes to zero only asymptotically, thus requiring the introduction of an empirical cut-off. In ref. [28] , for instance, the integration is performed from zero till the boundary R 200 , defined as the radius where the mass density equals 200 times the mean DM density.
while the latter is the sum of kinetic and potential energy, E = E k + W . From eq. (B13), using the identity dI n (t)/dt = −(n/t)I n−1 (t) and eq. (B9), it is straightforward to show that dP (r) dr = − 8π √ 2m 5/2 f (k B T ) 3/2 h 3 I 1/2 (ke ψ k (ξ) ) dΦ dr that is nothing but the condition for hydrostatic equilibrium. The kinetic energy is given by the following integral E k = (3/2) P d x, from which we get The potential energy can be obtained from the virial theorem 2E k + W = 3V P (R), with V = 4πR 3 /3 and P (R) given by eq. (B13) with ξ = ξ 1 . We find The normalized energy is therefore given by The pressure in eq. (B13) reduces to P = 2(2π) 3/2 m