© ESO, 2013

1. Introduction

It is accepted that most matter in the universe is dark. This belief comes from the fact that more mass than visible matter is required to explain the observed structure. At the end of the 1980s, it was suggested that baryonic dark matter could also be clumped in dark clusters (for a review, see for example, Carr 1994). The search for baryonic dark objects started at the beginning of the 1990s (see, for example, Moscoso 1993; Kerins & Carr 1994; Wasserman & Salpeter 1994). However, baryonic dark matter does not seem to be sufficient to describe the rotation curves observed in galaxies (see, for example, Caldwell 1995). Moreover, baryon-dominated models are ruled out by the observations of anisotropies of the cosmic microwave background (CMB; see, for example: Bond 1988; Strukov et al. 1987). Indeed, cosmological evidence favors non-baryonic dark matter (e.g., weakly interacting massive particles; see, for example, Trimble 1987), which reproduces rotational curves. The most natural non-baryonic dark matter candidate is the massive neutrino, but there are other alternatives such as the axion and various supersymmetric particles. The main argument in favor of non-baryonic dark matter comes from the primordial or big bang nucleosynthesis (BBN): it estimates a baryonic energy density much lower than present estimates of the energy density of matter. In addition, dominant non-baryonic dark matter models are consistent with other cosmological observations (the angular power spectrum of CMB, the large-scale structure of the universe (LSS), lensing clusters), which also impose constraints on them (see, for example, Peacock 1999). In this work we will deal with fully degenerate fermionic dark matter. Semi-degenerate massive neutrinos, with different degrees of degeneracy, have been extensively addressed; see, for example, Freese et al. (1983); Kang & Steigman (1992); Lesgourgues & Pastor (1999); Lesgourgues & Liddle (2001); Dolgov et al. (2002b); and Dolgov (2002a).

The evolution of the non-baryonic dark matter distribution is collisionless. According to the Liouville theorem, the phase-space density of a collisionless system is preserved in its evolution. Therefore, the maximum phase-space density is conserved. Tremaine & Gunn (1979) applied this result to leptons. They used the condition that the maximum phase-space density must not increase and that the energy density of the cosmological background of leptons must be lower than that of matter. They assumed that initially the distribution function is that of Fermi-Dirac, very far from degeneracy, i.e., neglecting the chemical potential, and they took a Maxwellian velocity dispersion for the final relaxed cluster. Tremaine and Gunn concluded that massive galactic halos cannot be composed of stable neutral leptons of mass ≤1   MeV, which are relativistic when they decouple from radiation. Assuming a typical isothermal massive galactic halo, the phase-space constraint for neutrinos is mc² > 33   eV (see, for example, Peacock 1999), close to the result required for a critical-density universe. Thus, non-degenerate isothermal neutrinos cannot be the components of these dark haloes. However, neutrinos could constitute dark haloes in clusters of galaxies; for them, the constraint gives mc² > 1.5   eV. Leptons with masses >1   MeV are non-relativistic at the decoupling, and they would not be ruled out by the phase-space density arguments.

In any case, the Tremaine and Gunn conclusion cannot be claimed for degenerate fermions such as those treated here. It must be taken into account that in a fully degenerate state, all quantum phase-space cells, from momentum 0 up to the Fermi momentum, are occupied. Therefore, the maximum phase-space density of fully degenerate fermions in bound clusters is equal to g/h³, where g is the spin degeneracy factor. Because in the evolution the maximum phase-space density cannot increase, the maximum phase-space density of the cosmological background of such fermions should also have been g/h³. Hence, no constraint such as that derived by Tremaine and Gunn can be derived for the mass of the fermion. Nevertheless, in this work we look for restrictions on the mass of degenerate fermions by imposing the degeneracy condition on fermions that are relativistic when they decouple and on fermions that are non-relativistic at the decoupling.

There is strong evidence that the expansion of our flat universe (Efstathiou et al. 1999) is speeding up (see, for example: Perlmutter et al. 1997, 1999; Schmidt et al. 1998), which cannot be explained only in terms of matter and radiation. Thus, it is necessary to assume that another component is involved. This new component has been named “dark energy”, and its pressure (from a hydrodynamical point of view) must be a negative function of its density. The first models for dark energy were hydrodynamical (Steinhardt 1996; Turner & White 1997), but dynamical scalar field models were soon proposed (see, for example, Coble et al. 1997; Caldwell et al. 1998). When the dark energy density is assumed to be a constant and pressure is taken as minus density, a cosmological constant model is considered.

Pacheco et al. (1986) found that a cosmological constant imposes a minimum mass for the existence of Newtonian selfgravitating degenerate fermion systems. Though bound states were allowed by the repulsive effect of the cosmological constant, clusters could also suffer gravitational instability due to an excessive mass (see, for example, Shapiro & Teukolski 1983). In this work, we study both the lower bound for the cluster mass imposed by the repulsive effect of the cosmological constant and the upper bound imposed by the gravitational instability.

Finally, we calculate the Jeans mass and Jeans length of a cosmological background of Newtonian degenerate fermions as a function of their redshift of formation.

The paper is organized as follows. In Sect. 2, Einstein equations are written for a fluid composed of a pure degenerate Fermi gas where the cosmological constant is present; the boundary conditions to solve these equations are also shown. In Sect. 3, we assume a weak field approximation to deal with Newtonian clusters. Section 4 shows the process we have followed to impose the stability conditions that limit the existence of these clusters. Numerical results are presented in Sect. 5. Section 6 deals with the degeneracy of a cosmological background of fermions as a function of the fermion mass. The Jeans mass for Newtonian degenerate fermions is calculated in Sect. 7. Finally, Sect. 8 contains our conclusions.

2. Einstein equations

We consider a static spherically symmetric cluster whose components are perfect fluids fulfilling different equations of state. The energy-stress-momentum tensor T^μν is then given by (see, for example, Glendenning 1997) ${\mathit{T}}^{\mathit{\mu \nu }}\mathrm{=}\mathrm{-}\mathit{P}{\mathit{g}}^{\mathit{\mu \nu }}\mathrm{+}\mathrm{(}\mathit{P}\mathrm{+}\mathit{\rho }\mathrm{)}{\mathit{U}}^{\mathit{\mu }}{\mathit{U}}^{\mathit{\nu }}\mathit{.}$(1)In Eq. (1), P and ρ are the pressure and the energy density of the fluid; g_μν is the metric which, using polar coordinates (x⁰ = ct, x¹ = r, x² = θ, x³ = φ), is given by ${\mathit{g}}_{\mathrm{00}}\mathrm{=}\mathit{B}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathit{,}{\mathit{g}}_{\mathrm{11}}\mathrm{=}\mathrm{-}\mathit{A}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathit{,}{\mathit{g}}_{\mathrm{22}}\mathrm{=}\mathrm{-}{\mathit{r}}^{\mathrm{2}}\mathit{,}{\mathit{g}}_{\mathrm{33}}\mathrm{=}\mathrm{-}{\mathit{r}}^{\mathrm{2}}\hspace{0.17em}{\sin }^{\mathrm{2}}\mathit{\theta }$(2)(ds² = B(r)(cdt)² − A(r)(dr)² − r² [(dθ)² + sin²(dφ)²] is therefore the line element); U⁰ = (dx⁰/ds) = B^−1/2 and U = (dx/ds) = U⁰(v/c) = 0 are the components of the four-velocity tensor (they come from assuming a static cluster, so v = (dx/dt) = 0, and from U^μU_μ = 1). B(r), A(r), ρ(r) and P(r) must be solutions of Einstein equations.

We consider a cluster composed of degenerate fermions of mass m in a cosmological background of dark energy treated by the cosmological constant model. Then, the total energy density and pressure are: $\begin{array}{ccc}\mathit{\rho }& \mathrm{=}& {\mathit{\rho }}_{\mathrm{F}}\mathrm{+}{\mathit{\rho }}_{\mathrm{\Lambda }}\mathit{,}\\ \mathit{P}& \mathrm{=}& {\mathit{P}}_{\mathrm{F}}\mathrm{+}{\mathit{P}}_{\mathrm{\Lambda }}\mathit{,}\end{array}$where F and Λ represent fermions and dark energy, respectively.

In a degenerate state, fermions have the lowest energy without violating the Pauli principle. In this state, all quantum phase-space cells, from zero momentum up to Fermi momentum p_F(r) (maximum momentum that a fermion can have at a distance r from the cluster center), are occupied by g fermions, where g is the degeneracy per kinematic state. Hence, the distribution function, f(p), called Fermi distribution function (see, for example, Kippenhahn & Weigert 1994), is $\mathit{f}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{=}\{\begin{array}{c}\\ \mathit{g}\mathit{/}{\mathit{h}}^{\mathrm{3}}\mathit{,}& \mathrm{0}\mathrm{\le }\mathit{p}\mathrm{\le }{\mathit{p}}_{\mathrm{F}}\\ \mathrm{0}\mathit{,}& {\mathit{p}}_{\mathrm{F}}\mathit{<}\mathit{p}\mathit{.}\end{array}$(5)Thus, the number of fermions per unit volume whose absolute value of momentum lies between p and p + dp is 4πf(p)p²dp. Eq. (5) can be compared with the Fermi-Dirac distribution function (see Landau & Lifshitz 2002), ${\mathit{f}}_{\mathrm{FD}}\mathrm{\left(}\mathit{E}\mathrm{\right)}\mathrm{=}\frac{\mathit{g}}{{\mathit{h}}^{\mathrm{3}}}{\left[\exp \left(\frac{\mathit{E}}{{\mathit{K}}_{\mathrm{B}}\mathit{T}}\mathrm{-}\frac{\mathit{\mu }}{{\mathit{K}}_{\mathrm{B}}\mathit{T}}\right)\mathrm{+}\mathrm{1}\right]}^{-1}\mathrm{·}$(6)In Eq. (6), μ and T are the chemical potential and the temperature of fermions, and E is the kinetic energy related with the momentum, p, by $\mathit{E}\mathrm{=}{\mathrm{[}{\left(\mathit{pc}\right)}^{\mathrm{2}}\mathrm{+}\mathrm{(}\mathit{m}{\mathit{c}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{2}}\mathrm{]}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{-}\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathit{.}$(7)(Equation (6) can also be expressed as f_FD = (g/h³) [exp(U/K_BT − μ^∗/K_BT) + 1] ^-1, where U = E + mc² is the total energy and μ^∗ = μ + mc²; see, for example, Raychaudhuri et al. 1992). As can be seen, when μ ≫ K_BT, Eq. (6) coincides with Eq. (5) expressed as a function of the kinetic energy, E, $\mathit{f}\mathrm{\left(}\mathit{E}\mathrm{\right)}\mathrm{=}\{\begin{array}{c}\\ \mathit{g}\mathit{/}{\mathit{h}}^{\mathrm{3}}\mathit{,}& \mathrm{0}\mathrm{\le }\mathit{E}\mathrm{\le }{\mathit{E}}_{\mathrm{F}}\mathrm{=}\mathit{\mu }\\ \mathrm{0}\mathit{,}& \mathit{E}\mathit{>}{\mathit{E}}_{\mathrm{F}}\mathrm{=}\mathit{\mu }\mathit{.}\end{array}$(8)In Eq. (8), E_F is the Fermi kinetic energy, so ${\mathit{E}}_{\mathrm{F}}\mathrm{=}{\left[{\left({\mathit{p}}_{\mathrm{F}}\mathit{c}\right)}^{\mathrm{2}}\mathrm{+}{\mathrm{\left(}\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathrm{\right)}}^{\mathrm{2}}\right]}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{-}\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathit{.}$(9)The number density of particles, n_F, is then ${\mathit{n}}_{\mathrm{F}}\mathrm{=}\mathrm{4}\mathit{\pi }{\mathrm{\int }}_{\mathrm{0}}^{\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{p}\mathrm{\right)}{\mathit{p}}^{\mathrm{2}}\mathrm{d}\mathit{p}\mathrm{=}\frac{\mathrm{4}\mathit{\pi }}{\mathrm{3}}\frac{\mathit{g}}{{\mathit{h}}^{\mathrm{3}}}{\mathit{p}}_{\mathrm{F}}^{\mathrm{3}}\mathrm{;}$(10)and as a function of the dimensionless Fermi momentum, x = p_F/mc, ${\mathit{n}}_{\mathrm{F}}\mathrm{=}\mathit{g}\frac{{\mathit{m}}^{\mathrm{3}}{\mathit{c}}^{\mathrm{3}}{\mathit{x}}^{\mathrm{3}}}{\mathrm{6}{\mathit{\pi }}^{\mathrm{2}}{ħ}^{\mathrm{3}}}$(11)(the end of a gravitationally bound system at r = R is then characterized by the fact that no cell is occupied, so p_F(R) = 0, and therefore n_F(R) = 0). Energy density, ρ_F, and pressure, P_F, of degenerate fermions are given by $\begin{array}{ccc}{\mathit{\rho }}_{\mathrm{F}}& \mathrm{=}& \mathrm{4}\mathit{\pi }{\mathrm{\int }}_{\mathrm{0}}^{\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathit{U}\mathrm{\left(}\mathit{p}\mathrm{\right)}{\mathit{p}}^{\mathrm{2}}\mathrm{d}\mathit{p}\mathrm{=}\frac{\mathit{g}{\mathit{m}}^{\mathrm{4}}{\mathit{c}}^{\mathrm{5}}}{\mathrm{2}{\mathit{\pi }}^{\mathrm{2}}{ħ}^{\mathrm{3}}}\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{,}\\ {\mathit{P}}_{\mathrm{F}}& \mathrm{=}& \frac{\mathrm{4}\mathit{\pi }}{\mathrm{3}}{\mathrm{\int }}_{\mathrm{0}}^{\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathit{pv}\mathrm{\left(}\mathit{p}\mathrm{\right)}{\mathit{p}}^{\mathrm{2}}\mathrm{d}\mathit{p}\mathrm{=}\frac{\mathit{g}{\mathit{m}}^{\mathrm{4}}{\mathit{c}}^{\mathrm{5}}}{\mathrm{2}{\mathit{\pi }}^{\mathrm{2}}{ħ}^{\mathrm{3}}}\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{,}\end{array}$where U(p) = (p²c² + m²c⁴)^1/2 and v(p) = pc/(p² + m²c²)^1/2 are the total energy and velocity of a fermion with momentum p, and $\begin{array}{ccc}\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{8}}\left[\mathit{x}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{(}\mathrm{2}{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathrm{1}}^{\mathrm{)}}\mathrm{-}\mathit{ln}\left(\mathit{x}\mathrm{+}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\right)\right]\mathit{,} \\ \mathit{v}\mathrm{\left(}\mathit{x}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{24}}\left[\mathit{x}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{(}\mathrm{2}{\mathit{x}}^{\mathrm{2}}\mathrm{-}{\mathrm{3}}^{\mathrm{)}}\mathrm{+}\mathrm{3}\mathit{ln}\left(\mathit{x}\mathrm{+}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\right)\right]\mathit{.} \end{array}$For dark energy, we assume a cosmological constant model. Because the metric tensor g^μν fulfills ${\mathit{g}}_{\mathrm{;}\mathit{\nu }}^{\mathit{\mu \nu }}\mathrm{=}\mathrm{0}$, we can write a modified set of field equations, G^μν + Λg^μν = −(8πG/c⁴)T^μν from the Einstein equations (G^μν being Einstein tensor) that are also consistent with the conservation laws, ${\mathit{T}}_{\mathrm{;}\mathit{\nu }}^{\mathit{\mu \nu }}\mathrm{=}\mathrm{0}$. Λ is the cosmological constant and was introduced by Albert Einstein. Therefore, the existence of an energy-momentum tensor of the vacuum can be considered, ${\mathit{T}}_{\mathrm{vac}}^{\mathit{\mu \nu }}\mathrm{=}\frac{\mathrm{\Lambda }{\mathit{c}}^{\mathrm{4}}}{\mathrm{8}\mathit{\pi G}}{\mathit{g}}^{\mathit{\mu \nu }}$. From a hydrodynamical point of view, ${\mathit{T}}_{\mathrm{vac}}^{\mathit{\mu \nu }}$ will agree with Eq. (1) if we take $\begin{array}{ccc}{\mathit{\rho }}_{\mathrm{\Lambda }}& \mathrm{=}& \frac{{\mathit{c}}^{\mathrm{4}}}{\mathrm{8}\mathit{\pi G}}\mathrm{\Lambda }\mathit{,}\\ {\mathit{P}}_{\mathrm{\Lambda }}& \mathrm{=}& \mathrm{-}\frac{{\mathit{c}}^{\mathrm{4}}}{\mathrm{8}\mathit{\pi G}}\mathrm{\Lambda }\mathit{.}\end{array}$Defining Ω_Λ0 = ρ_Λ/ρ_c0, where ${\mathit{\rho }}_{\mathit{c}\mathrm{0}}\mathrm{=}\mathrm{\left(}\mathrm{3}{\mathit{c}}^{\mathrm{2}}{\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}\mathrm{\right)}\mathit{/}\mathrm{\left(}\mathrm{8}\mathit{\pi G}\mathrm{\right)}$ is the critical energy density of the universe at present (see, for example, Padmanabhan, 1995) and H₀ is the Hubble function at present, $\mathrm{\Lambda }\mathrm{=}\mathrm{3}{\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}{\mathrm{\Omega }}_{\mathrm{\Lambda }\mathrm{0}}\mathit{/}{\mathit{c}}^{\mathrm{2}}$; i.e. $\mathrm{\Lambda }\mathrm{=}\mathrm{1.25}\mathrm{\times }{\mathrm{10}}^{-56}\hspace{0.17em}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-2}\mathit{,}$(18)where ${\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}\mathrm{=}{\left(\frac{{\mathit{H}}_{\mathrm{0}}}{\mathrm{70}\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}\hspace{0.17em}\mathrm{Mp}{\mathrm{c}}^{-1}}\right)}^{\mathrm{2}}\left(\frac{{\mathrm{\Omega }}_{\mathrm{\Lambda }\mathrm{0}}}{\mathrm{0.73}}\right)\mathrm{·}$(19)Values for H₀ and Ω_Λ0 can be taken from Spergel et al. (2003).

In the absence of other forces, except gravitation, the nontrivial components of the field equation, ${\mathit{R}}_{\mathit{\mu \nu }}\mathrm{=}\mathrm{-}\mathrm{(}\mathrm{8}\mathit{\pi G}\mathit{/}{\mathit{c}}^{\mathrm{4}}\mathrm{)}\mathrm{[}{\mathit{T}}_{\mathit{\mu \nu }}\mathrm{-}\mathrm{(}\mathrm{1}\mathit{/}\mathrm{2}\mathrm{)}{\mathit{g}}_{\mathit{\mu \nu }}{\mathit{T}}_{\mathit{\alpha }}^{\mathit{\alpha }}$, read as (see, for example, Weinberg 1972)

$\begin{array}{ccc}& & {\mathit{R}}_{\mathit{rr}}\mathrm{=}\frac{{\mathit{B}}^{\mathrm{\prime \prime }}}{\mathrm{2}\mathit{B}}\mathrm{-}\frac{{\mathit{B}}^{\mathrm{\prime }}}{\mathrm{4}\mathit{B}}\left(\frac{{\mathit{A}}^{\mathrm{\prime }}}{\mathit{A}}\mathrm{+}\frac{{\mathit{B}}^{\mathrm{\prime }}}{\mathit{B}}\right)\mathrm{-}\frac{{\mathit{A}}^{\mathrm{\prime }}}{\mathit{rA}}\mathrm{=}\mathrm{-}\frac{\mathrm{4}\mathit{\pi GA}}{{\mathit{c}}^{\mathrm{4}}}\mathrm{(}\mathit{\rho }\mathrm{-}\mathit{P}\mathrm{)}\mathit{,}\\ & & {\mathit{R}}_{\mathit{\theta \theta }}\mathrm{=}\mathrm{-}\mathrm{1}\mathrm{+}\frac{\mathit{r}}{\mathrm{2}\mathit{A}}\left(\mathrm{-}\frac{{\mathit{A}}^{\mathrm{\prime }}}{\mathit{A}}\mathrm{+}\frac{{\mathit{B}}^{\mathrm{\prime }}}{\mathit{B}}\right)\mathrm{+}\frac{\mathrm{1}}{\mathit{A}}\mathrm{=}\mathrm{-}\frac{\mathrm{4}\mathit{\pi G}{\mathit{r}}^{\mathrm{2}}}{{\mathit{c}}^{\mathrm{4}}}\mathrm{(}\mathit{\rho }\mathrm{-}\mathit{P}\mathrm{)}\mathit{,}\\ & & {\mathit{R}}_{\mathit{tt}}\mathrm{=}\mathrm{-}\frac{{\mathit{B}}^{\mathrm{\prime \prime }}}{\mathrm{2}\mathit{A}}\mathrm{+}\frac{{\mathit{B}}^{\mathrm{\prime }}}{\mathrm{4}\mathit{A}}\left(\frac{{\mathit{A}}^{\mathrm{\prime }}}{\mathit{A}}\mathrm{+}\frac{{\mathit{B}}^{\mathrm{\prime }}}{\mathit{B}}\right)\mathrm{-}\frac{{\mathit{B}}^{\mathrm{\prime }}}{\mathit{rA}}\mathrm{=}\mathrm{-}\frac{\mathrm{4}\mathit{\pi GB}}{{\mathit{c}}^{\mathrm{4}}}\mathrm{(}\mathit{\rho }\mathrm{+}\mathrm{3}\mathit{P}\mathrm{)}\mathit{,}\end{array}$where R_μν is the Ricci tensor and T_μν is that of Eq. (1), with ρ and P given by Eqs. (3) and (4). In Eqs. (20)−(22), a prime represents d/dr. A fourth non-independent equation to (20)−(22) comes from the conservation of the energy-momentum tensor, ${\mathit{T}}_{\mathrm{;}\mathit{\nu }}^{\mathit{\mu \nu }}\mathrm{=}\mathrm{0}$; ${\mathit{T}}_{\mathrm{;}\mathit{\nu }}^{\mathrm{0}\mathit{\nu }}$ is identical to null, and ${\mathit{T}}_{\mathrm{;}\mathit{\nu }}^{\mathrm{1}\mathit{\nu }}\mathrm{=}\mathrm{0}$ gives the equation of the hydrostatic equilibrium of the system, i.e., $\frac{{\mathit{B}}^{\mathrm{\prime }}}{\mathit{B}}\mathrm{=}\mathrm{-}\frac{\mathrm{2}{\mathit{P}}^{\mathrm{\prime }}}{\mathit{P}\mathrm{+}\mathit{\rho }}\mathrm{·}$(23)And a fifth equation is also obtained from Eqs. (20)−(22), as follows: $\frac{{\mathit{r}}^{\mathrm{2}}}{\mathrm{2}}\left(\frac{{\mathit{R}}_{\mathit{rr}}}{\mathit{A}}\mathrm{+}\frac{{\mathit{R}}_{\mathit{tt}}}{\mathit{B}}\right)\mathrm{+}{\mathit{R}}_{\mathit{\theta \theta }}\mathrm{=}{\left(\frac{\mathit{r}}{\mathit{A}}\right)}^{\mathrm{\prime }}\mathrm{-}\mathrm{1}\mathrm{=}\mathrm{-}\frac{\mathrm{8}\mathit{\pi G}}{{\mathit{c}}^{\mathrm{4}}}\mathit{\rho }{\mathit{r}}^{\mathrm{2}}\mathrm{·}$(24)Now, let us define a length scale, a, given by $\begin{array}{ccc}\mathit{r}& \mathrm{=}& \mathit{ay,}\\ \mathit{a}& \mathrm{=}& {\left(\frac{\mathrm{3}\mathit{\pi }}{\mathrm{4}\mathit{g}}\right)}^{\mathrm{1}\mathit{/}\mathrm{2}}\frac{{ħ}^{\mathrm{3}\mathit{/}\mathrm{2}}}{{\mathit{G}}^{\mathrm{1}\mathit{/}\mathrm{2}}{\mathit{c}}^{\mathrm{1}\mathit{/}\mathrm{2}}{\mathit{m}}^{\mathrm{2}}}\mathrm{=}\mathrm{1.20}\mathrm{\times }{\mathrm{10}}^{\mathrm{2}}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\mathrm{eV}}\right)}^{-2}\mathrm{kpc}\mathit{.}\end{array}$Inserting (25) and (26) in (20)−(24), and using Eqs. (3), (4), (12), (13), (16) and (17), we have $\begin{array}{ccc}& & \frac{\begin{array}{c}\mathrm{¨}\\ \mathit{B}\end{array}}{\mathrm{2}\mathit{B}}\mathrm{-}\frac{\mathit{Ḃ}}{\mathrm{4}\mathit{B}}\left(\frac{\mathit{Ȧ}}{\mathit{A}}\mathrm{+}\frac{\mathit{Ḃ}}{\mathit{B}}\right)\mathrm{-}\frac{\mathit{Ȧ}}{\mathit{yA}}\mathrm{=}\mathrm{-}\frac{\mathrm{3}}{\mathrm{2}}\mathit{A}\left(\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{-}\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{+}\frac{\mathrm{2}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right)\mathit{,} \\ & & \mathrm{-}\mathrm{1}\mathrm{+}\frac{\mathit{y}}{\mathrm{2}\mathit{A}}\left(\mathrm{-}\frac{\mathit{Ȧ}}{\mathit{A}}\mathrm{+}\frac{\mathit{Ḃ}}{\mathit{B}}\right)\mathrm{+}\frac{\mathrm{1}}{\mathit{A}}\mathrm{=}\mathrm{-}\frac{\mathrm{3}}{\mathrm{2}}{\mathit{y}}^{\mathrm{2}}\left(\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{-}\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{+}\frac{\mathrm{2}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right)\mathit{,} \\ & & \mathrm{-}\frac{\begin{array}{c}\mathrm{¨}\\ \mathit{B}\end{array}}{\mathrm{2}\mathit{A}}\mathrm{+}\frac{\mathit{Ḃ}}{\mathrm{4}\mathit{A}}\left(\frac{\mathit{Ȧ}}{\mathit{A}}\mathrm{+}\frac{\mathit{Ḃ}}{\mathit{B}}\right)\mathrm{-}\frac{\mathit{Ḃ}}{\mathit{yA}}\mathrm{=}\mathrm{-}\frac{\mathrm{3}}{\mathrm{2}}\mathit{B}\left(\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{+}\mathrm{3}\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{-}\frac{\mathrm{2}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right)\mathit{,} \\ & & \frac{\mathit{Ḃ}}{\mathit{B}}\mathrm{=}\mathrm{-}\mathrm{2}\frac{\mathit{ẋ}\mathit{x}}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}\mathit{,}\\ & & \mathrm{-}\frac{\mathit{Ȧ}}{{\mathit{A}}^{\mathrm{2}}}\mathit{y}\mathrm{+}\frac{\mathrm{1}}{\mathit{A}}\mathrm{-}\mathrm{1}\mathrm{=}\mathrm{-}\mathrm{3}\left(\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right){\mathit{y}}^{\mathrm{2}}\mathrm{·}\end{array}$In Eqs. (27)−(31), a dot means d/dy. The value of a given in Eq. (26) was chosen so that the coefficient on the right hand term in Eqs. (27)−(29) is 3/2.

From Eqs. (27)−(31) coupled differential equations for x and A can be obtained. Using Ḃ (from Eq. (30)), Ȧ (from Eq. (31)) and $\begin{array}{c}{\mathrm{¨}}_{}\\ \mathit{B}\end{array}$ (from the differentiation of Eq. (30) with respect to y) in Eq. (27), and after some manipulations, we obtain $\begin{array}{ccc}\begin{array}{c}\mathrm{¨}\\ \mathit{x}\end{array}\mathrm{=}\frac{\mathit{ẋ}\mathrm{2}\mathrm{(}\mathrm{2}{\mathit{x}}^{\mathrm{2}}\mathrm{-}\mathrm{1}\mathrm{)}}{\mathit{x}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}\mathrm{+}\frac{\mathit{Ȧ}}{\mathit{A}}\left(\frac{\mathit{ẋ}}{\mathrm{2}}\mathrm{-}\frac{\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}}{\mathit{yx}}\right)\mathrm{+}\frac{\mathrm{3}}{\mathrm{2}}\mathit{A}\frac{\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}}{\mathit{x}}\left[\mathit{F}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{+}\frac{\mathrm{2}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right]\mathit{,}& & \end{array}$(32)where $\mathit{F}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{12}}\mathit{x}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{(}\mathrm{2}{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathrm{3}}^{\mathrm{)}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{4}}\ln \left(\mathit{x}\mathrm{+}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\right)\mathit{.}$(33)The differentiation of Eq. (31) with respect to the dimensionless radius y gives $\begin{array}{ccc}\begin{array}{c}\mathrm{¨}\\ \mathit{A}\end{array}\mathrm{=}\frac{\mathit{A}}{{\mathit{y}}^{\mathrm{2}}}\mathrm{(}\mathit{A}\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{+}\frac{\mathit{Ȧ}}{\mathit{y}}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{2}\mathit{A}\mathrm{)}\mathrm{+}\mathrm{3}\mathit{A}\mathrm{(}\mathrm{2}\mathit{Ȧ}\mathit{y}\mathrm{+}\mathit{A}\mathrm{)}& & \\ \mathrm{\times }\left[\mathit{G}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right]\mathrm{3}\mathit{y}{\mathit{A}}^{\mathrm{2}}\mathit{ẋ}{\mathit{x}}^{\mathrm{2}}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathit{,}& & \end{array}$(34)where $\mathit{G}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{8}}\left[\mathit{x}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{(}\mathrm{2}{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathrm{1}}^{\mathrm{)}}\mathrm{-}\ln \left(\mathit{x}\mathrm{+}{\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\right)\right]\mathit{.}$(35)If the cosmological constant terms were not considered in Eqs. (32) and (34), we would have the well-known Tolman-Oppenheimer-Volkoff equations and universal solutions for x and A, i.e., solutions that are m independent. However, as cosmological constant terms (a²Λ) depend on the fermion mass, it will be necessary to solve Eqs. (32) and (34) for each m.

From Eqs. (32) and (34), the behavior of x and A for y → 0 can be deduced. Up to second order in y, we obtain $\begin{array}{ccc}\mathit{x}\mathrm{(}\mathit{y}\mathrm{\to }\mathrm{0}\mathrm{)}& \mathrm{=}& {\mathit{x}}_{\mathrm{0}}\mathrm{+}\frac{\mathrm{1}\mathrm{+}{\mathit{x}}_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{2}{\mathit{x}}_{\mathrm{0}}}\left[\mathrm{-}\mathrm{2}\mathit{G}\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}\mathrm{\right)}\mathrm{+}\frac{\mathrm{3}}{\mathrm{2}}\mathit{F}\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right]{\mathit{y}}^{\mathrm{2}}\mathit{,} \\ \mathit{A}\mathrm{(}\mathit{y}\mathrm{\to }\mathrm{0}\mathrm{)}& \mathrm{=}& \mathrm{1}\mathrm{+}\left[\mathit{G}\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right]{\mathit{y}}^{\mathrm{2}}\mathit{.}\end{array}$According to Eqs. (36) and (37), the coupled differential equations (32) and (34) must be integrated with the four boundary conditions: $\mathit{x}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}{\mathit{x}}_{\mathrm{0}}\mathit{,}\mathit{ẋ}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\mathit{A}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{1}\mathit{,}\mathit{Ȧ}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0.}$(38)The mass of the clusters inside a distance r = ay is $\mathit{M}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\frac{\mathrm{4}\mathit{\pi }}{{\mathrm{c}}^{\mathrm{2}}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{r}}{\mathit{\rho }}_{\mathrm{F}}{\mathit{r}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{d}{\mathit{r}}_{\mathrm{1}}\mathrm{=}\frac{\mathrm{3}{\mathrm{c}}^{\mathrm{2}}\mathit{a}}{\mathrm{2}\mathit{G}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{y}}\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}{\mathit{y}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{d}{\mathit{y}}_{\mathrm{1}}\mathit{.}$(39)It can also be calculated by integrating Eq. (31), $\frac{\mathrm{d}}{\mathrm{d}\mathit{y}}\left(\frac{\mathit{y}}{\mathit{A}}\right)\mathrm{=}\mathrm{1}\mathrm{-}\mathrm{3}\left[\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\right]\mathit{,}$(40)from 0 up to some y. Thus, $\mathit{M}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{=}\frac{{\mathit{c}}^{\mathrm{2}}}{\mathrm{2}\mathit{G}}\mathit{ay}\left[\mathrm{1}\mathrm{-}\frac{\mathrm{1}}{\mathit{A}\mathrm{\left(}\mathit{y}\mathrm{\right)}}\mathrm{-}\frac{\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}^{\mathrm{2}}}{\mathrm{3}}\right]\mathrm{·}$(41)When A is close to unity, it is better to use Eq. (39) to calculate M(y). The radius of the cluster will be the distance, R_c = ay_c, that, as has been said above, fulfills x(y_c) = 0; so, the cluster mass will be M_c = M(y_c).

The number of fermions inside a sphere of dimensionless radius y is determined by $\mathit{N}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{=}\frac{\mathrm{4}\mathit{\pi }{\mathit{a}}^{\mathrm{3}}}{{\mathit{c}}^{\mathrm{2}}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{y}}{\mathit{n}}_{\mathrm{F}}\mathrm{\left(}{\mathit{y}}_{\mathrm{1}}\mathrm{\right)}{\mathit{y}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{d}{\mathit{y}}_{\mathrm{1}}\mathit{,}$(42)where n_F is given by Eq. (10).

From Eq. (41), A(y) can be expressed as $\mathit{A}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{=}{\left[\mathrm{1}\mathrm{-}\frac{\mathrm{2}\mathit{GM}\mathrm{\left(}\mathit{y}\mathrm{\right)}}{{\mathit{c}}^{\mathrm{2}}\mathit{ay}}\mathrm{-}\frac{\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}^{\mathrm{2}}}{\mathrm{3}}\right]}^{-1}\mathrm{·}$(43)With respect to B, the integration of Eq. (30), $\frac{\mathrm{d}\ln \mathit{B}}{\mathrm{d}\mathit{x}}\mathrm{=}\ln \mathrm{(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{)}\mathit{,}$(44)from a point y up to y_c (i.e., from B(y) up to B(y_c) and from x(y) up to x(y_c) = 0) leads to $\mathit{B}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{=}\frac{\mathit{B}\mathrm{\left(}{\mathit{y}}_{\mathrm{c}}\mathrm{\right)}}{\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{\left(}\mathit{y}\mathrm{\right)}}\mathrm{·}$(45)B(y_c)can be calculated from Eqs. (27) and (29) for y ≥ y_c where u = v = 0. Adding the product of B and Eq. (27) to the product of A and Eq. (29), we have $\frac{\mathit{Ȧ}}{\mathit{A}}\mathrm{=}\mathrm{-}\frac{\mathit{Ḃ}}{\mathit{B}}\mathit{,}\mathit{y}\mathrm{\ge }{\mathit{y}}_{\mathrm{c}}\mathit{,}$(46)i.e., d(ln(AB))/dy = 0, and therefore, $\mathit{A}\mathrm{(}\mathit{y}\mathrm{\ge }{\mathit{y}}_{\mathrm{c}}\mathrm{)}\mathit{B}\mathrm{(}\mathit{y}\mathrm{\ge }{\mathit{y}}_{\mathrm{c}}\mathrm{)}\mathrm{=}\mathrm{constant}\mathrm{=}\mathit{A}\mathrm{(}\mathit{y}\mathrm{=}\mathrm{\infty }\mathrm{)}\mathit{B}\mathrm{(}\mathit{y}\mathrm{=}\mathrm{\infty }\mathrm{)}\mathit{.}$(47)At y = ∞, the metric given by Eq. (2) has to be the Minkowski metric (that of special relativity), so A(y = ∞) = B(y = ∞) = 1; therefore $\mathit{B}\mathrm{\left(}{\mathit{y}}_{\mathrm{c}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathit{A}\mathrm{\left(}{\mathit{y}}_{\mathrm{c}}\mathrm{\right)}}\mathrm{·}$(48)And from Eq. (43) evaluated at y_c, $\mathit{B}\mathrm{\left(}{\mathit{y}}_{\mathrm{c}}\mathrm{\right)}\mathrm{=}\mathrm{1}\mathrm{-}\frac{\mathrm{2}\mathit{G}{\mathit{M}}_{\mathrm{c}}}{{\mathit{c}}^{\mathrm{2}}\mathit{a}{\mathit{y}}_{\mathrm{c}}}\mathrm{-}\frac{\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}_{\mathrm{c}}^{\mathrm{2}}}{\mathrm{3}}\mathrm{·}$(49)

3. Weak field approximation

When fermions are non-relativistic (x ≪ 1), the weak field approximation can be considered; i.e., $\begin{array}{ccc}\mathit{A}& \mathrm{=}& \mathrm{1}\mathrm{+}\mathit{\gamma ,}\mathrm{\left|}\mathit{\gamma }\mathrm{\right|}\mathrm{\ll }\mathrm{1}\mathit{,}\\ \mathit{B}& \mathrm{=}& \mathrm{1}\mathrm{+}\mathit{\beta ,}\mathrm{\left|}\mathit{\beta }\mathrm{\right|}\mathrm{\ll }\mathrm{1.}\end{array}$So, taking into account Eqs. (43), (45) and (49), the leading terms in γ and β are $\begin{array}{ccc}\mathit{\gamma }\mathrm{\left(}\mathit{y}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{2}\mathit{GM}\mathrm{\left(}\mathit{y}\mathrm{\right)}}{{\mathit{c}}^{\mathrm{2}}\mathit{ay}}\mathrm{+}\frac{\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}^{\mathrm{2}}}{\mathrm{3}}\mathit{,}\\ \mathit{\beta }\mathrm{\left(}\mathit{y}\mathrm{\right)}& \mathrm{=}& \mathrm{-}{\mathit{x}}^{\mathrm{2}}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{-}\frac{\mathrm{2}\mathit{G}{\mathit{M}}_{\mathrm{c}}}{{\mathit{c}}^{\mathrm{2}}\mathit{a}{\mathit{y}}_{\mathrm{c}}}\mathrm{-}\frac{\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}_{\mathrm{c}}^{\mathrm{2}}}{\mathrm{3}}\mathrm{·}\end{array}$Using Eqs. (50) and (51) in Eq. (29), and taking into account that $\begin{array}{ccc}\mathit{u}\mathrm{(}\mathit{x}\mathrm{\ll }\mathrm{1}\mathrm{)}\mathrm{=}\frac{{\mathit{x}}^{\mathrm{3}}}{\mathrm{3}}\mathit{,}& & \\ \mathit{v}\mathrm{(}\mathit{x}\mathrm{\ll }\mathrm{1}\mathrm{)}\mathrm{=}\frac{{\mathit{x}}^{\mathrm{5}}}{\mathrm{15}}\mathit{,}& & \end{array}$we have $\begin{array}{c}\mathrm{¨}\\ \mathit{\beta }\end{array}\mathrm{+}\frac{\mathrm{2}}{\mathit{y}}\mathit{\beta ̇}\mathrm{=}{\mathit{x}}^{\mathrm{3}}\mathrm{-}\mathrm{2}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\mathit{.}$(56)Equation (56) is the classical potential equation of gravity with $\mathit{\beta }\mathrm{=}\mathrm{2}\mathit{\phi }\mathit{/}{\mathit{c}}^{\mathrm{2}}\mathit{,}$(57)φbeing the gravitational potential; i.e., $\begin{array}{c}\mathrm{¨}\\ \mathit{\phi }\end{array}\mathrm{+}\frac{\mathrm{2}\mathit{\phi ̇}}{\mathit{y}}\mathrm{=}\frac{{\mathit{c}}^{\mathrm{2}}{\mathit{x}}^{\mathrm{3}}}{\mathrm{2}}\mathrm{-}{\mathit{c}}^{\mathrm{2}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\mathit{.}$(58)To derive a differential equation for x from Eq. (58), we can replace β by φ (see (57)) in Eq. (53), i.e., $\mathit{\phi }\mathrm{=}\mathrm{-}\frac{{\mathit{c}}^{\mathrm{2}}{\mathit{x}}^{\mathrm{2}}}{\mathrm{2}}\mathrm{-}\frac{\mathit{G}{\mathit{M}}_{\mathrm{c}}}{\mathit{a}{\mathit{y}}_{\mathrm{c}}}\mathrm{-}\frac{{\mathit{c}}^{\mathrm{2}}\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}_{\mathrm{c}}^{\mathrm{2}}}{\mathrm{6}}\mathit{,}$(59)and insert this equation in (58). Thus, we have $\begin{array}{c}\mathrm{¨}\\ \mathit{x}\end{array}\mathrm{=}\mathrm{-}\frac{\mathit{ẋ}\mathrm{2}}{\mathit{x}}\mathrm{-}\frac{\mathrm{2}\mathit{ẋ}}{\mathit{y}}\mathrm{-}\frac{{\mathit{x}}^{\mathrm{2}}}{\mathrm{2}}\mathrm{+}\frac{{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }}{\mathit{x}}\mathrm{·}$(60)From Eq. (60), the behavior of x in the neighborhood of the cluster center, up to second order in y, is given by $\mathit{x}\mathrm{(}\mathit{y}\mathrm{\to }\mathrm{0}\mathrm{)}\mathrm{=}{\mathit{x}}_{\mathrm{0}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{6}}\left[\frac{{\mathit{x}}_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{2}}\mathrm{-}\frac{{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }}{{\mathit{x}}_{\mathrm{0}}}\right]{\mathit{y}}^{\mathrm{2}}\mathrm{·}$(61)Evidently, Eq. (36) converts into Eq. (61) when x₀ ≪ 1 (taking into account that $\mathit{F}\mathrm{(}{\mathit{x}}_{\mathrm{0}}\mathrm{\ll }\mathrm{1}\mathrm{)}\mathrm{=}\mathit{G}\mathrm{(}{\mathit{x}}_{\mathrm{0}}\mathrm{\ll }\mathrm{1}\mathrm{)}\mathrm{=}{\mathit{x}}_{\mathrm{0}}^{\mathrm{3}}\mathit{/}\mathrm{3}$). Thus, from Eq. (61), Eq. (60) must be integrated with the boundary conditions $\mathit{x}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}{\mathit{x}}_{\mathrm{0}}\mathit{,}\mathit{ẋ}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0.}$(62)Using (54) in (39), (39) in (52), and (52) in (50), A can be expressed as $\mathit{A}\mathrm{=}\mathrm{1}\mathrm{+}\frac{\mathrm{1}}{\mathit{y}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{y}}{\mathit{x}}^{\mathrm{3}}{\mathit{y}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{d}{\mathit{y}}_{\mathrm{1}}\mathrm{+}\frac{{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }}{\mathrm{3}}{\mathit{y}}^{\mathrm{2}}\mathrm{;}$(63)so, $\mathit{Ȧ}\mathrm{=}\mathrm{-}\frac{\mathrm{1}}{{\mathit{y}}^{\mathrm{2}}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{y}}{\mathit{x}}^{\mathrm{3}}{\mathit{y}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{d}{\mathit{y}}_{\mathrm{1}}\mathrm{+}{\mathit{x}}^{\mathrm{3}}\mathit{y}\mathrm{+}\frac{\mathrm{2}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }}{\mathrm{3}}\mathit{y}\mathit{.}$(64)The integral in Eqs. (63) and (64) can be expressed as a function of x and y as follows: integrating Eq. (56), $\mathrm{d}\mathrm{\left(}{\mathit{y}}^{\mathrm{2}}\mathit{\beta ̇}\mathrm{\right)}\mathrm{=}\mathrm{(}{\mathit{x}}^{\mathrm{3}}{\mathit{y}}^{\mathrm{2}}\mathrm{-}\mathrm{2}\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}^{\mathrm{2}}\mathrm{)}\hspace{0.17em}\mathrm{d}\mathit{y,}$(65)from 0 up to some y, we have $\mathit{\beta ̇}\mathrm{=}\frac{\mathrm{1}}{{\mathit{y}}^{\mathrm{2}}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{y}}{\mathit{x}}^{\mathrm{3}}{\mathit{y}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{d}{\mathit{y}}_{\mathrm{1}}\mathrm{-}\frac{\mathrm{2}}{\mathrm{3}}\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}\mathit{y}\mathrm{;}$(66)and as from Eq. (53), , $\frac{\mathrm{1}}{\mathit{y}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{y}}{\mathit{x}}^{\mathrm{3}}{\mathit{y}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{d}{\mathit{y}}_{\mathrm{1}}\mathrm{=}\mathrm{-}\mathrm{2}\mathit{x}\mathit{ẋ}\mathit{y}\mathrm{+}\frac{\mathrm{2}}{\mathrm{3}}\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}^{\mathrm{2}}\mathit{.}$(67)Hence $\begin{array}{ccc}\mathit{A}& \mathrm{=}& \mathrm{1}\mathrm{-}\mathrm{2}\mathit{x}\mathit{ẋ}\mathit{y}\mathrm{+}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }{\mathit{y}}^{\mathrm{2}}\mathit{,}\\ \mathit{Ȧ}& \mathrm{=}& \mathrm{2}\mathit{x}\mathit{ẋ}\mathrm{+}{\mathit{x}}^{\mathrm{3}}\mathit{y}\mathit{.}\end{array}$Instead of solving Eq. (60), we can also make the following changes of variable $\begin{array}{ccc}\mathit{\phi }& \mathrm{=}& \mathrm{-}\frac{\mathit{G}{\mathit{M}}_{\mathrm{c}}}{\mathit{r}}\mathit{\chi }\mathrm{-}\frac{\mathit{G}{\mathit{M}}_{\mathrm{c}}}{\mathit{a}{\mathit{y}}_{\mathrm{c}}}\mathrm{-}\frac{{\mathit{c}}^{\mathrm{2}}\mathrm{\Lambda }{\mathit{a}}^{\mathrm{2}}{\mathit{y}}_{\mathrm{c}}^{\mathrm{2}}}{\mathrm{6}}\mathit{,}\\ \mathit{r}& \mathrm{=}& \mathit{bz}\mathit{.}\end{array}$Equating Eqs. (59) and (70), and using (71), we obtain that $\left(\frac{\mathrm{2}\mathit{G}{\mathit{M}}_{\mathrm{c}}}{\mathit{b}{\mathit{c}}^{\mathrm{2}}}\right)\hspace{0.17em}\frac{\mathit{\chi }}{\mathit{z}}\mathrm{=}{\mathit{x}}^{\mathrm{2}}\mathit{.}$(72)And using Eqs. (70)−(72) and (25) in (58), we have $\frac{{\mathrm{d}}^{\mathrm{2}}\mathit{\chi }}{\mathrm{d}{\mathit{z}}^{\mathrm{2}}}\mathrm{=}\mathrm{-}{\left(\frac{\mathrm{2}\mathit{G}{\mathit{M}}_{\mathrm{c}}}{\mathit{b}{\mathit{c}}^{\mathrm{2}}}\right)}^{\mathrm{1}\mathit{/}\mathrm{2}}{\left(\frac{\mathit{b}}{\mathit{a}}\right)}^{\mathrm{2}}\left[\frac{{\mathit{\chi }}^{\mathrm{3}\mathit{/}\mathrm{2}}}{{\mathit{z}}^{\mathrm{1}\mathit{/}\mathrm{2}}}\mathrm{-}\mathrm{2}{\mathit{a}}^{\mathrm{2}}{\left(\frac{\mathit{b}{\mathit{c}}^{\mathrm{2}}}{\mathrm{2}\mathit{G}{\mathit{M}}_{\mathrm{c}}}\right)}^{\mathrm{3}\mathit{/}\mathrm{2}}\mathrm{\Lambda }\mathit{z}\right]\mathrm{·}$(73)Equation (73) depends on two length scales (a, given by Eq. (26), and the free parameter b), on the mass of the cluster (M_c), and on the cosmological constant (Λ). However, by taking $\frac{\mathrm{2}\mathit{G}{\mathit{M}}_{\mathrm{c}}}{\mathit{b}{\mathit{c}}^{\mathrm{2}}}\mathrm{=}{\left(\frac{\mathit{a}}{\mathit{b}}\right)}^{\mathrm{4}}\mathit{,}$(74)we can make Eq. (73) depend on just one parameter. Thus, Eq. (73) reads $\frac{{\mathrm{d}}^{\mathrm{2}}\mathit{\chi }}{\mathrm{d}{\mathit{z}}^{\mathrm{2}}}\mathrm{=}\mathrm{-}\frac{{\mathit{\chi }}^{\mathrm{3}\mathit{/}\mathrm{2}}}{{\mathit{z}}^{\mathrm{1}\mathit{/}\mathrm{2}}}\mathrm{+}\mathit{\alpha z,}$(75)where $\mathit{\alpha }\mathrm{=}\frac{\mathrm{2}{\mathit{b}}^{\mathrm{6}}}{{\mathit{a}}^{\mathrm{4}}}\mathrm{\Lambda }\mathit{.}$(76)Finally, using the value of a, given in Eq. (26), in Eqs. (74) and (76), b and α are $\begin{array}{ccc}\mathit{b}& \mathrm{=}& {\left(\frac{\mathrm{9}{\mathit{\pi }}^{\mathrm{2}}}{\mathrm{32}{\mathit{g}}^{\mathrm{2}}}\right)}^{\mathrm{1}\mathit{/}\mathrm{3}}\frac{{ħ}^{\mathrm{2}}}{\mathit{G}{\mathit{m}}^{\mathrm{8}\mathit{/}\mathrm{3}}{\mathit{M}}_{\mathrm{c}}^{\mathrm{1}\mathit{/}\mathrm{3}}}\mathit{,}\\ \mathit{\alpha }& \mathrm{=}& \frac{\mathrm{9}{\mathit{\pi }}^{\mathrm{2}}}{\mathrm{32}{\mathit{g}}^{\mathrm{2}}}\frac{{ħ}^{\mathrm{6}}{\mathit{c}}^{\mathrm{2}}\mathrm{\Lambda }}{{\mathit{G}}^{\mathrm{4}}{\mathit{m}}^{\mathrm{8}}{\mathit{M}}_{\mathrm{c}}^{\mathrm{2}}}\mathrm{;}\end{array}$i.e., $\begin{array}{ccc}\mathit{b}& \mathrm{=}& \mathrm{9.46}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{20}\hspace{0.17em}\mathrm{eV}}\right)}^{\mathrm{-}\mathrm{8}\mathit{/}\mathrm{3}}{\left(\frac{{\mathit{M}}_{\mathrm{c}}}{{\mathrm{10}}^{\mathrm{11}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}}\right)}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}\mathrm{kpc}\mathit{,}\\ \mathit{\alpha }& \mathrm{=}& \mathrm{2.10}\mathrm{\times }{\mathrm{10}}^{-5}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{20}\hspace{0.17em}\mathrm{eV}}\right)}^{-8}{\left(\frac{{\mathit{M}}_{\mathrm{c}}}{{\mathrm{10}}^{\mathrm{11}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}}\right)}^{-2}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}\mathit{.}\end{array}$In the neighborhood of the cluster center, the behavior of χ up to second order in z is $\mathit{\chi }\mathrm{(}\mathit{z}\mathrm{\to }\mathrm{0}\mathrm{)}\mathrm{=}{\mathit{\chi }}_{\mathrm{1}}\mathit{z}\mathrm{-}\frac{\mathrm{1}}{\mathrm{6}}\mathrm{[}{\mathit{\chi }}_{\mathrm{1}}^{\mathrm{3}\mathit{/}\mathrm{2}}\mathrm{-}{\mathit{\alpha }}^{\mathrm{]}}{\mathit{z}}^{\mathrm{3}}\mathit{.}$(81)Comparing (61) and (81), and taking into account (72) and (74), it can be seen that ${\mathit{x}}_{\mathrm{0}}\mathrm{=}{\left(\frac{\mathit{a}}{\mathit{b}}\right)}^{\mathrm{2}}{\mathit{\chi }}_{\mathrm{1}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathit{.}$(82)Equation (75) has to be integrated for each value of α, imposing the boundary conditions $\mathit{\chi }\mathrm{(}\mathit{z}\mathrm{=}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{0}\mathit{,}\frac{\mathrm{d}\mathit{\chi }}{\mathrm{d}\mathit{z}}\mathrm{(}\mathit{z}\mathrm{=}\mathrm{0}\mathrm{)}\mathrm{=}{\mathit{\chi }}_{\mathrm{1}}\mathit{,}$(83)which derives from Eq. (81). For each α, χ₁ must be that which causes the mass condition to be fulfilled; i.e., ${\mathit{M}}_{\mathrm{c}}\mathrm{=}\mathrm{4}\mathit{\pi }{\mathrm{\int }}_{\mathrm{0}}^{{\mathit{R}}_{\mathrm{c}}}\mathit{m}{\mathit{n}}_{\mathrm{F}}{\mathit{r}}^{\mathrm{2}}\mathrm{d}\mathit{r,}$(84)where R_c = bz_c with χ(z_c) = 0. So using Eqs. (11), (72) and (79) in Eq. (84), $\mathrm{1}\mathrm{=}{\mathrm{\int }}_{\mathrm{0}}^{{\mathit{z}}_{\mathrm{c}}}{\mathit{\chi }}^{\mathrm{3}\mathit{/}\mathrm{2}}{\mathit{z}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{d}\mathit{z}$(85)must be fulfilled.

Finally, when the cosmological constant is neglected, χ₁ = 0.264 and z_c = 5.10, so the cluster radius is given by ${\mathit{R}}_{\mathrm{c}}\mathrm{=}\mathrm{48.1}\hspace{0.17em}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{20}\hspace{0.17em}\mathrm{eV}}\right)}^{\mathrm{-}\mathrm{8}\mathit{/}\mathrm{3}}{\left(\frac{{\mathit{M}}_{\mathrm{c}}}{{\mathrm{10}}^{\mathrm{11}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}}\right)}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}\mathrm{kpc}\mathit{.}$(86)

4. Minimum and maximum mass allowed for degenerate fermion clusters

As pointed out in Sect. 2, because the Λ terms in Eqs. (32) and (34) depend on fermion mass, it is necessary to solve these equations for each m. Thus, the discussion presented in the following paragraph must be made for each fermion mass.

The repulsive effect of the cosmological constant is already present in the neighborhood of the cluster center, as can be seen from Eqs. (36) and (37). According to Eq. (36), models with ${\mathit{x}}_{\mathrm{0}}\mathit{<}{\mathit{x}}_{\mathrm{0}}^{\mathrm{\ast }}$, ${\mathit{x}}_{\mathrm{0}}^{\mathrm{\ast }}$ fulfilling $\mathrm{-}\mathrm{2}\mathit{G}\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}^{\mathrm{\ast }}\mathrm{\right)}\mathrm{+}\frac{\mathrm{3}}{\mathrm{2}}\mathit{F}\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}^{\mathrm{\ast }}\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{3}}{\mathit{a}}^{\mathrm{2}}\mathrm{\Lambda }\mathrm{=}\mathrm{0}\mathit{,}$(87)show dx/dy|_{y    =    0} > 0 and are therefore unstable (see, for example, Jeans 1919; Shapiro & Teukolski 1983). As we take greater x₀’s, with ${\mathit{x}}_{\mathrm{0}}\mathit{>}{\mathit{x}}_{\mathrm{0}}^{\mathrm{\ast }}$, the instability condition dx/dy > 0 appears at greater dimensionless radii y’s (by solving Eqs. (32) and (34)). This will continue to happen until some x₀ = x_0,m, for which the instability condition appears exactly at y = y_c,m with x(y_c,m) = 0. The mass inside the sphere of dimensionless radius y_c,m, calculated from Eq. (39), is denoted by M_m. Clusters with x₀ > x_0,m do not fulfill dx/dy > 0 at any y ≤ y_c, with x(y_c) = 0; therefore, M_m is a lower limit for the mass of bound degenerate fermion clusters. The minimum number of fermions, N_m, that build a bound fermion cluster can then be calculated from Eq. (42) taking y = y_c,m.

Pacheco et al. (1986) found, using Eq. (75) for non-relativistic clusters, that α > α^∗ = 6.29 × 10^-3 does not allow bound states. Indeed, for α > α^∗, dx/dz > 0 at some z, so such a cluster will be unstable. Therefore, for each fermion mass, m, α^∗ fixes the lower mass, M_m, for bound non-relativistic clusters. Taking α = α^∗ in Eq. (75) and integrating that equation, the condition (85) is fulfilled for χ₁ = 0.180 and z_c = 7.71. Thus, minimum mass, ${\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}$ (from Eq. (80), taking α = α^∗ = 6.29 × 10^-3), minimum fermion number, ${\mathit{N}}_{\mathit{m}}^{\mathrm{NR}}\mathrm{=}{\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}\mathit{/}\mathit{m}$, and maximum radius, ${\mathit{R}}_{\mathit{m}}^{\mathrm{NR}}$ (using (79) in (71), with z = z_c = 7.71 and (88)), are $\begin{array}{ccc}{\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}& \mathrm{=}& \mathrm{9.24}\mathrm{\times }{\mathrm{10}}^{\mathrm{14}}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{2}}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{-4}{\mathit{M}}_{\mathrm{\odot }}\mathit{,}\\ {\mathit{N}}_{\mathit{m}}^{\mathrm{NR}}& \mathrm{=}& \mathrm{1.03}\mathrm{\times }{\mathrm{10}}^{\mathrm{81}}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{2}}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{-5}\mathit{,}\\ {\mathit{R}}_{\mathit{m}}^{\mathrm{NR}}& \mathrm{=}& \mathrm{1.01}\mathrm{\times }{\mathrm{10}}^{\mathrm{4}}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{6}}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{\mathrm{-}\mathrm{4}\mathit{/}\mathrm{3}}\mathrm{kpc}\mathit{.}\end{array}$Therefore, ${\mathit{R}}_{\mathit{m}}^{\mathrm{NR}}\mathrm{=}\mathrm{4.83}\mathrm{\times }{\mathrm{10}}^{\mathrm{2}}\hspace{0.17em}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}{\left(\frac{{\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}}{{\mathrm{10}}^{\mathrm{11}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}}\right)}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathrm{kpc}\mathit{.}$(91)Equation (60) could also be used to determine M_m; however, it would be necessary to calculate x_0,m for each fermion mass, so the calculation time would be longer.

Fig. 1 Mass limits, MM (upper continuous line) and Mm (lower continuous lines), versus fermion mass, m, for dark clusters of degenerate fermions. is assumed. For details see the text.

So far, we have dealt with lower bounds in the cluster mass. Now, we study upper bounds caused by the gravitational instability. There are x₀ > x_0,m models that are not stable. An upper limit for x₀, and therefore for the mass of clusters, comes from the condition $\frac{\mathrm{d}{\mathit{M}}_{\mathrm{c}}}{\mathrm{d}{\mathit{x}}_{\mathrm{0}}}{|}_{{\mathit{x}}_{\mathrm{0}\mathit{,M}}}\mathrm{=}\mathrm{0.}$(92)Equation (92) indicates that stability changes at x_0,M. Our x_0,m<x₀ models are stable for x₀ < x_0,M fulfilling dM_c/dx₀ > 0, and unstable for x₀ > x_0,M fulfilling dM_c/dx₀ < 0. Therefore, M_M = M_c(x_0,M) is the Oppenheimer-Volkoff limit for the mass of degenerate fermion clusters. When the repulsive effects of the cosmological constant are negligible, ${\mathit{M}}_{\mathit{M}}^{\mathrm{NCC}}$, ${\mathit{N}}_{\mathit{M}}^{\mathrm{NCC}}$ and ${\mathit{R}}_{\mathit{M}}^{\mathrm{NCC}}$ are given by $\begin{array}{ccc}{\mathit{M}}_{\mathit{M}}^{\mathrm{NCC}}& \mathrm{=}& \mathrm{8.87}\mathrm{\times }{\mathrm{10}}^{\mathrm{17}}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}{\mathit{mc}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{-2}{\mathit{M}}_{\mathrm{\odot }}\mathit{,}\\ {\mathit{N}}_{\mathit{M}}^{\mathrm{NCC}}& \mathrm{=}& \mathrm{9.04}\mathrm{\times }{\mathrm{10}}^{\mathrm{83}}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{-3}\mathit{,}\\ {\mathit{R}}_{\mathit{M}}^{\mathrm{NCC}}& \mathrm{=}& \mathrm{3.72}\mathrm{\times }{\mathrm{10}}^{\mathrm{2}}{\left(\frac{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{-2}\mathrm{kpc}\mathit{.}\end{array}$As a brief summary of this section, we emphasize that Eqs. (88)−(90) are non-relativistic results including repulsion effects, whilst Eqs. (93)−(95) are pure relativistic results that do not take repulsion into account.

A dimensional analysis of the lower bounds ${\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}$ and upper bounds ${\mathit{R}}_{\mathit{m}}^{\mathrm{NR}}$ for fermions and bosons is described in Membrado & Pacheco (2012).

5. Numerical results and discussion

Numerically exact results for M_m and M_M versus fermion mass, m   c², up to 10   g^−1/4   GeV, are shown in the lower and upper curves of Fig. 1.

If relativistic effects are not taken into account in M_m (from Eq. (88), ${\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}\mathrm{\propto }{\mathit{m}}^{-4}$ ) and if repulsive effects in M_M are not considered (from Eq. (93), ${\mathit{M}}_{\mathit{M}}^{\mathrm{NCC}}\mathrm{\propto }{\mathit{m}}^{-2}$), then, ${\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}\mathrm{=}{\mathit{M}}_{\mathit{M}}^{\mathrm{NCC}}\mathrm{=}\mathrm{8.51}\mathrm{\times }{\mathrm{10}}^{\mathrm{20}}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{2}}{\mathit{M}}_{\mathrm{\odot }}$ for $\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathrm{=}\mathrm{3.22}\mathrm{\times }{\mathrm{10}}^{-2}{\mathit{g}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{4}}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{4}}$ eV. However, for this fermion mass, relativistic and repulsive effects can not be neglected. Our exact calculation, for , gives m_∗c² = 2.38 × 10^-2  g^−1/4eV, M_∗ = 1.66 × 10²¹M_⊙ (see the point in Fig. 1 where the two continuous lines intersect) and R_∗ = 1.26 × 10³Mpc. m_∗ is, therefore, the minimim fermion mass that can build bound degenerate fermion clusters.

We can now compare values of M_m obtained by neglecting relativistic effects, ${\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}$ (see Eq. (88)), with those shown in Fig. 1. For fermion masses 1, 0.1, 0.04 and 0.02   g^−1/4   eV, $\mathrm{100}\hspace{0.17em}\mathrm{\times }\mathrm{|}{\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}\mathrm{-}{\mathit{M}}_{\mathit{m}}\mathrm{|}\mathit{/}{\mathit{M}}_{\mathit{m}}$ is about 0.1%, 0.6%, 10% and 72%. Thus, we can say that relativistic effects on M_m appear for mc² < 0.1   g^−1/4  eV. With respect to M_M, the repulsive effects of the cosmological constant begin to be appreciable for mc² < 0.1   g^−1/4  eV; for m_∗c², M_M = M_∗ = 1.66 × 10²¹   M_⊙ and ${\mathit{M}}_{\mathit{M}}^{\mathrm{NCC}}\mathrm{=}\mathrm{1.56}\mathrm{\times }{\mathrm{10}}^{\mathrm{21}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}$, so $\mathrm{100}\hspace{0.17em}\mathrm{\times }\mathrm{|}{\mathit{M}}_{\mathit{M}}^{\mathrm{NCC}}\mathrm{-}{\mathit{M}}_{\mathit{M}}\mathrm{|}\mathit{/}{\mathit{M}}_{\mathit{M}}\mathrm{\approx }\mathrm{6}\mathrm{\%}$.

Fig. 2 Cluster radii, R, versus fermion mass, m, for several cluster masses, M. is assumed. Horizontal lines correspond to 1, 10, 30 kpc and 1 Mpc. For details see the text.

Numerical results for the radii of clusters, R, versus fermion mass, m, for several cluster masses, M, are shown in Fig. 2. In this figure, each line represents radii allowed for degenerate fermion clusters of the same mass, M. The maximum radius shown in each line corresponds to the limit imposed by the cosmological constant. The minimum radius in each line is the Oppenheimer-Volkoff limit.

We consider the mass of a typical galaxy dark halo (G). This mass would be on the order of ~10¹¹   M_⊙. According to Eq. (88) the minimum fermion mass that creates a degenerate cluster with ${\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}\mathrm{=}{\mathrm{10}}^{\mathrm{11}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}$ is ${\mathit{m}}_{\mathrm{m}\mathit{,}\mathrm{G}}{\mathit{c}}^{\mathrm{2}}\mathrm{=}\mathrm{9.80}\hspace{0.17em}{\mathit{g}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{4}}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{8}}\mathrm{eV}$; from Eq. (90) we know that its radius is ${\mathit{R}}_{\mathrm{m}\mathit{,}\mathrm{G}}^{\mathrm{NR}}\mathrm{=}\mathrm{482}\hspace{0.17em}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}\hspace{0.17em}\mathrm{kpc}$. Thus, the contributions to the rotational velocity would be about ${\mathit{v}}_{\mathit{h}}\mathrm{=}\mathrm{(}\mathit{G}{\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}\mathit{/}{\mathit{R}}_{\mathrm{m}\mathit{,}\mathrm{G}}^{\mathrm{NR}}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{=}\mathrm{30}\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}$, too small to reproduce the rotational curve of a typical galaxy. The stability limit for ${\mathit{M}}_{\mathit{M}}^{\mathrm{NCC}}\mathrm{=}{\mathrm{10}}^{\mathrm{11}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}$ is reached with m_M,Gc² = 2.97 × 10³   g^−1/4   eV (see Eq. (93)); in that case, from Eq. (95), ${\mathit{R}}_{\mathrm{M}\mathit{,}\mathrm{G}}^{\mathrm{NCC}}\mathrm{\approx }\mathrm{4}\mathrm{\times }{\mathrm{10}}^{-2}\hspace{0.17em}\mathrm{pc}$, so v_h = 10⁵   km   s^-1. Therefore, that degenerate cluster has too small a radius and too great a v_h to be the dark halo of a typical galaxy. However, as we show below, there are fermion masses between m_m,G and m_M,G that could give rise to dark clusters whose v_h could lead to the observed values of the rotational velocities of typical galaxies.

We now look at the mass of a dark halo of a typical galaxy cluster (GC), i.e., ~10¹⁵   M_⊙. If in Eq. (88) we take ${\mathit{M}}_{\mathit{m}}^{\mathrm{NR}}\mathrm{=}{\mathrm{10}}^{\mathrm{15}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}$, we conclude that the minimum fermion mass that can build a degenerate fermion object with such a mass is ${\mathit{m}}_{\mathrm{m}\mathit{,}\mathrm{GC}}{\mathit{c}}^{\mathrm{2}}\mathrm{=}\mathrm{9.80}\mathrm{\times }{\mathrm{10}}^{-1}\hspace{0.17em}{\mathit{g}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{4}}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{8}}\hspace{0.17em}\mathrm{eV}$; and from Eq. (90), its radius is ${\mathit{R}}_{\mathrm{m}\mathit{,}\mathrm{GC}}^{\mathrm{NR}}\mathrm{=}\mathrm{10}\hspace{0.17em}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}\hspace{0.17em}\mathrm{Mpc}$. v_h is then ≈650   km   s^-1, which is too small to reproduce the velocities of galaxies in clusters. It can also be seen that the maximum fermion mass that creates a 10¹⁵   M_⊙ degenerate cluster is, from Eq. (93), m_M,GCc² = 2.97 × 10¹   g^−1/4   eV; in that case, from Eq. (95), ${\mathit{R}}_{\mathrm{M}\mathit{,}\mathrm{GC}}^{\mathrm{NCC}}\mathrm{=}\mathrm{4.21}\mathrm{\times }{\mathrm{10}}^{-1}\hspace{0.17em}\mathrm{kpc}$, so v_h ≈ 10⁵   km   s^-1. Therefore, the radius is too small and v_h too high with respect to the size of a typical galaxy cluster and the velocities of their galaxies. We can also consider the case of 1 Mpc, which has about the visible size of a typical galaxy cluster. Figure 2 shows that a degenerate cluster of 10¹⁵   M_⊙ and 1 Mpc (so, v_h = 2 × 10³   km   s^-1) can be built using a 2   g^−1/4   eV fermion (see Eq. (86)). However, this particle cannot build clusters with masses smaller than 5.8 × 10¹³   M_⊙ (see Eq. (88)), so it cannot be the component of the dark halos of galaxies.

It can be seen from Eq. (86) that fermions with masses between 15   g^−1/4 and 30   g^−1/4   eV could create bound degenerate dark haloes that would contribute to the rotational velocity with values, v_h = (GM_c/R_c)^1/2, between 150 and 300   km   s^-1; these haloes would have radii between 15 and 50   kpc and masses between 8 × 10¹⁰   M_⊙ and 7 × 10¹¹   M_⊙. However, they could not compose the dark halo in a galaxy cluster, unless they were composed of dark galaxies. We consider some examples. For 20   g^−1/4   eV, M_c = (2−6) × 10¹¹   M_⊙ degenerate dark clusters show R_c = (38−27)   kpc and v_h = (150−313)   km   s^-1; and those with M_c = (10¹²−10¹⁵)   M_⊙ present R_c = (22−2.2)   kpc and v_h = (4.4 × 10²−4.4 × 10⁴)   km   s^-1, and therefore are incompatible with the observations of the largest galaxies and clusters of galaxies. For 25   g^−1/4   eV, M_c = (1−4) × 10¹¹   M_⊙ clusters are built with R_c = (26−17)   kpc, showing v_h = (129−318)   km   s^-1; while for M_c = (10¹²−10¹⁵)   M_⊙, R_c = (12−1.3)   kpc and v_h = (5.9 × 10²−5.9 × 10⁴)   km   s^-1, so they are again incompatible with observations.

For fermion masses smaller than 15   g^−1/4   eV, only clusters with masses greater than 10¹²   M_⊙ are able to give the kind of v_h that is needed to reproduce the values of the rotational curves of typical galaxies; so, they cannot be the dark haloes of galaxies. The same conclusion is obtained for fermions with masses greater than 30   g^−1/4   eV, but now because degenerate clusters show radii smaller than those of typical galaxies.

We now look at the problem of dwarf galaxies. Dwarf galaxies are the most abundant type of galaxy in the Universe. The largest dwarf irregulars have stellar masses of ~10⁹   M_⊙ and are rotationally supported, while dwarf spheroidal galaxies have stellar masses from 10⁷   M_⊙ to below 10⁶   M_⊙ with no rotational support (the lightest known is Segue 1 with a mass of ~6 × 10⁵   M_⊙; see, for example, Simon et al. 2011).

High-resolution simulations of the standard ΛCDM (cosmological constant and cold dark matter) cosmology (see, for example: Klypin et al. 1999; Moore et al. 1999; Diemand et al. 2007; Springel et al. 2008) were applied to the Local Group; they showed a total number of dark matter subhaloes much greater than that of dwarf galaxies predicted using luminosity functions corrected by completeness and bias (Tollerud et al. 2008; Koposov et al. 2008) (one order of magnitud higher than the number of observed dwarf galaxies). It has been attempted to solve this problem found in ΛCDM simulations by suggesting that the supernova feedback could lead to the loss of baryonic components in haloes with masses smaller than a critical value, and that the effect of the photoionisation could prevent star formation in the smallest haloes (see, for example: Larson 1974; Dekel & Silk 1986; Ferrara & Tolstoy 2000; Efstathiou 1992; Somerville 2002; Hoeft et al. 2006; Simon & Geha 2007). These mechanisms would make the dark matter mass-light ratio very high in haloes on the order of or smaller than ~10⁹   M_⊙ (Sawala et al. 2010).

Another problem found in simulations of dark halos using the standard ΛCDM cosmology comes from dark halos of ~10¹⁰   M_⊙. ΛCDM simulations of ~(1−2) × 10¹⁰   M_⊙ dark halos show the formation of dwarf galaxies with stellar masses between 5 × 10⁷   M_⊙ and 2 × 10⁸   M_⊙ (Pelupessy et al. 2004; Stinson et al. 2007; Governato et al. 2010; Sawala et al. 2011). Guo et al. (2010) compared the abundance of dark haloes in the Millennium and Millennium-II Simulations to the observed abundances of galaxies as a function of the stellar mass obtained from SDSS DR-7 by Li & White (2009). They found that the abundance of ~10¹⁰   M_⊙ dark haloes obtained from ΛCDM simulations, and whose stellar mass is ~5 × 10⁷   M_⊙, is about the abundance of observed galaxies with stellar mass of ~10⁶   M_⊙. Therefore, ΛCDM simulations predict galaxies with about fifty times more stellar mass than observed. In any case, if dark halos of 10¹⁰   M_⊙ contain dwarf galaxies with stellar masses of 10⁸   M_⊙, a ΛCDM universe would predict about four times more of those galaxies than observed.

Thus, ΛCDM simulations present serious problems for describing dwarf galaxies. In this respect, it should be said that warm dark matter models (ΛWDM) predict about three times fewer dark haloes of 10¹⁰   M_⊙ than ΛCDM models for m_WDM = 1   keV (Zavala et al. 2009).

We now look at our results for small degenerate dark haloes. We take as an example the case of fermions of 20   g^−1/4   eV. According to Eq. (88), bound dark haloes with masses smaller than $\mathrm{6}\mathrm{\times }{\mathrm{10}}^{\mathrm{9}}{\mathrm{𝒞}}_{\mathrm{\Lambda }\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{2}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}$ cannot exist due to the repulsive effect of the cosmological constant. These small degenerate dark haloes would be very extensive (a degenerate dark halo of M_c = 10¹⁰   M_⊙ would have a radius R_c ≈ 100   kpc). However, these extensive degenerate dark haloes should be truncated by the tidal effects of larger neighboring galaxies (below we show as an example the case of the Large Magellanic Cloud). With respect to their contributions to the rotational velocities, v_h = (GM_c/R_c)^1/2, we see that they are small; thus, for a dark cluster of 10¹⁰   M_⊙, v_h(100   kpc) ≈ 20   km   s^-1. We may also wonder about the mass and velocity contribution at 5 and 10 kpc, which could be the size of dwarf galaxies contained in the dark haloes. For a 10¹⁰   M_⊙ cluster, M(5   kpc) ≈ 6.7 × 10⁶   M_⊙ and v_h(5   kpc) ≈ 2.4   km   s^-1; M(10   kpc) ≈ 5.4 × 10⁷   M_⊙ and v_h(10   kpc) ≈ 4.8   km   s^-1.

We now consider the case of the Large Magellanic Cloud (LMC). This is a nearby barred spiral galaxy, a satellite of the Milky Way (MW). It is located at D ≈ 48.5   kpc from the center of MW. Up to 8.9 kpc its stellar disk mass is ~2.7 × 10⁹   M_⊙ (de Vaucouleurs & Freeman 1972; Bothum & Thompson 1988; Zaritsky 1999; Bell & de Jong 2001) and the neutral gas mass is ~0.5 × 10⁹   M_⊙ (Kim et al. 1998); these masses contribute to the circular velocity of 39.3 km s^-1. The circular velocity derived by Van der Marel et al. (2002) at 8.9 kpc is 64.8 ± 15.9   km   s^-1; consequently, a dark halo must exist in this galaxy.

The LMC suffers the tidal effects exerted by the MW. The gravitational interaction between LMC and MW is a very complex problem. But, for the sake of simplicity and merely to illustrate the problem, we have made very considerable simplifications. Therefore, the results described in the following must be considered as very rough estimates.

For the Milky Way we take, as an example, a degenerate dark halo composed of fermions of 23   g^−1/4   eV with a mass of M_MW = 2.5 × 10¹¹   M_⊙, and therefore a radius (see Eq. (86)) of R_MW = 24.4   kpc. At the Sun position, R_⊙ ≈ 8.5   kpc, the contribution to the rotational velocity is v_h ≈ 158   km   s^-1. Using the galactic bulge-disk model developed by Klypin et al. (2002), and taking for the bulge and disk masses, M_B = 7 × 10⁹   M_⊙ and M_D = 3.5 × 10¹⁰   M_⊙, the rotational velocity at 8.5 kpc is ${\mathit{v}}_{\mathrm{r}}\mathrm{=}\mathrm{(}{\mathit{v}}_{\mathrm{d}}^{\mathrm{2}}\mathrm{+}{\mathit{v}}_{\mathrm{b}}^{\mathrm{2}}\mathrm{+}{\mathit{v}}_{\mathit{h}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{\approx }\mathrm{220}\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}$, v_d and v_b being the contributions of the bulge and the disk (we have taken M_D/M_B = 5, as happens with the luminosity ratio L_D/L_B ≈ 5 (see Kent et al. 1991); the k-band luminosity of the Galaxy measured by Kent et al. (1991) is L_K = 6.7 × 10¹⁰   L_⊙, and measured by Drimmel & Spergel (2001), L_K = 8.9 × 10¹⁰   L_⊙). At 20 kpc, v_h ≈ 225   km   s^-1, which together with the contributions of the disk and bulge make v_r ≈ 250   km   s^-1 (the rotation curve derived by Honma & Sofue (1996) based on the geometry of the HI disk shows v_r ≈ 265 ± 30   km   s^-1).

For the LMC, we could propose a dark halo composed of 23   g^−1/4   eV degenerate fermions which, if isolated, would have M_c = 8 × 10¹⁰   M_⊙ (according to Eq. (86), its radius would be R_c = 35.7   kpc). However, this halo should be truncated by the gravitational effects of the MW. We can derive a rough estimate of a tidal radius, r_T and a mass of the degenerate dark halo, M_LMC ≈ M(r_T), as those that fulfill $\mathit{G}{\mathit{M}}_{\mathrm{MW}}\mathit{/}{\mathit{D}}^{\mathrm{2}}\mathrm{+}\mathit{GM}\mathrm{\left(}{\mathit{r}}_{\mathrm{T}}\mathrm{\right)}\mathit{/}{\mathit{r}}_{\mathrm{T}}^{\mathrm{2}}\mathrm{\approx }\mathit{G}{\mathit{M}}_{\mathrm{MW}}\mathit{/}\mathrm{(}\mathit{D}\mathrm{-}{\mathit{r}}_{\mathrm{T}}{\mathrm{)}}^{\mathrm{2}}$. This equation is the balance of strengths per unit mass in the direction that joins the centers of the LMC and the MW. The first term on the left hand denotes the rotation of the LMC around the MW (${\mathit{v}}_{\mathrm{LMC}}^{\mathrm{2}}\mathit{/}\mathit{D}$) and the other terms are the gravitational strength per unit mass due to the LMC and the MW at a distance r_T from the center of the LMC. For this calculation it is necessary to know M(r). This can be derived by solving Eq. (75), neglecting the cosmological constant, and with the boundary conditions given by (83) and (85). Thus, we have obtained that R_LMC = r_T ≈ 14.5   kpc and M_LMC ≈ 2.3 × 10¹⁰   M_⊙. Assuming that the inner structure of the degenerate dark cluster is not changed by the truncation, the halo mass and the contribution to the circular velocity of LMC at r = 8.9   kpc are M(8.9   kpc) ≈ 6.5 × 10⁹   M_⊙ and v_h(9   kpc) ≈ 56   km   s^-1. Accordingly, the total circular velocity would be v_r = 69   km   s^-1, within the range given by Van der Marel et al. (2002). Smaller masses than M_c lead to smaller r_T, v_h and v_r, while greater masses than M_c lead to greater r_T, v_h and v_r.

With respect to dwarf spheroidal galaxies, we can consider the ultra-faint Milky Way satellite galaxy Segue 1 (Simon et al. 2011; Martinez et al. 2011). This is located at a distance of 23 kpc from the MW center. The stars in this galaxy show a velocity dispersion σ ≈ 3.7   km   s^-1. This velocity dispersion could imply a dark halo mass of M_1/2 ≈ 5.8 × 10⁵   M_⊙ within a sphere with a radius r_1/2 ≈ 36   pc that encloses half the stellar luminosity of the galaxy. As the V-band luminosity of Segue 1 is 240   L_⊙, the V-band mass-to-light ratio within the half-light radius is ≈3400   M_⊙/L_⊙. Assuming for the dark halo of Segue 1 a mass of 6 × 10⁵   M_⊙ and using the same formula as we used above to derive the tidal radius of LMC, we obtain that the tidal radius of Segue 1 is ≈240   pc; this is much greater than r_1/2. This means that Segue 1 should not suffer tidal effects from the MW.

We assume that the dark halo of Segue 1 is composed of degenerate fermions like those proposed here to describe the dark halo of the MW. In that case, the condition that within a sphere of 36 pc there is a mass of 5.8 × 10⁵   M_⊙ is fulfilled by a degenerate dark halo with a total mass of ≈3 × 10¹²   M_⊙ and a radius of ≈10 Kpc. This has no sense because the MW would suffer strong tidal effects. Therefore, the dark halo of Segue 1 cannot be composed of degenerate fermions like those proposed in this paper to describe the dark halo of typical galaxies. However, if the halo were composed of degenerate fermions of ≈1.3   keV, a cluster with total mass M_1/2 and radius r_1/2 could be reproduced (see Eq. (86)). The problem with this assumption is that, as can be seen from Eq. (125) shown below, such fermions are not degenerate; so Eq. (86) cannot be used. Therefore, such a non-degenerate fermion cluster with mass M_1/2 should have a radius greater than r_1/2.

Hence, it is apparent that to adequately model the dark halo of Segue 1 it is necessary to have more compact self-gravitating spheres than those provided by degenerate fermions. In this sense, spheres formed by fermions that are not completely degenerate are of no help because they are obviously less compact than their degenerate counterparts. Self-gravitating boson spheres, however, might be an answer. One kind of these objects is the self-gravitating collisionless Newtonian boson cluster in which all bosons occupy the lowest lying one-particle Hartree orbital (Membrado et al. 1989a,b; Membrado & Aguerri 1996). For these condensates, the radius R_BC, which covers 99% of the assembly, is ${\mathit{R}}_{\mathrm{BC}}\mathrm{=}\mathrm{9.9}\hspace{0.17em}\mathrm{(}{ħ}^{\mathrm{2}}\mathit{/}\mathit{G}{\mathit{M}}_{\mathrm{BC}}{\mathit{m}}_{\mathrm{B}}^{\mathrm{2}}\mathrm{)}\mathrm{=}\mathrm{84.6}\hspace{0.17em}\mathrm{(}{\mathit{M}}_{\mathrm{BC}}\mathit{/}{\mathrm{10}}^{\mathrm{6}}\hspace{0.17em}{\mathit{M}}_{\mathrm{\odot }}{\mathrm{)}}^{-1}\mathrm{(}{\mathit{m}}_{\mathrm{B}}{\mathit{c}}^{\mathrm{2}}\mathit{/}{\mathrm{10}}^{-20}\hspace{0.17em}\mathrm{eV}{\mathrm{)}}^{-2}\hspace{0.17em}\mathrm{pc}$, where M_BC is the mass of the boson cluster and m_B the boson mass. An upper limit for the mass of the boson that could reproduce the dark halo of Segue 1 can be obtained assuming M_BC = M_1/2 and R_BC = r_1/2. Thus, m_Bc² < 2    ×    10^-20   eV. It should be said that invoking the uncertainly principle, it may be deduced that the linear momentum of each boson in the cluster is on the order of p_B ~ ħ/R_BC and hence that the de-Broglie wavelength (non-relativistic particles) for the boson is derived as R_BC. Then, the phase-space condition for degeneracy in this condensate of bosons is trivially fulfilled: within a de-Broglie cell there are M_BC/m_B particles.

Scalar fields fulfilling different potentials have been proposed to model dark matter (see, for example, Lee & Koh 1996; Sahni & Wang 2000; Matos & Ureña-López 2000, 2001). The mass of this scalar dark matter, m_B, ranges from 10^-26 eV up to 10^-23   eV. The dark clusters composed of these fields show a minimal cutoff radius, greater than their Compton legth Λ_C = ħ/m_Bc. Indeed, the enormous Compton length of these particles would prevent structure formation on small subgalactic scales. In this sense, collisionless boson spheres do not have this limitation.

Finally, we look at isolated dwarf galaxies of visible radii of about 10 Kpc. They show rotational velocities ranging between 50 and 150   km   s^-1 (see, for example, Kent 1987). Fermions with masses between 15   g^−1/4 and 30   g^−1/4   eV could build more extensive degenerate structures that at 10 kpc would show masses in the range (0.6−5.2) × 10¹⁰   M_⊙. These would lead to rotational velocity contributions, v_h, similar to those shown by the visible dwarf galaxies. From the calculation of M(r) for degenerate fermion clusters (by integrating Eq. (75) with the boundary conditions Eqs. (83) and (85)), it is deduced that M(r) ∝ r³ for approximately r < 0.4R_c, so v_h(r) ∝ r. Incidentally, this behavior of v_h(r) is what is needed to describe the rotation velocity in some of those galaxies (see, for example, NGC 247, NGC 3109, NGC 2259 from Kent 1987).

6. Dependence on m of the degeneracy of a cosmological background of fermions

In this section we consider that the degenerate fermions decouple from radiation at an expansion parameter a_D. For simplicity, we assume that the decoupling occurs instantaneously. Before the decoupling, the fermions are in equilibrium with the radiation, so their temperatures are equal, i.e., T(a ≤ a_D) = T_γ(a ≤ a_D). We assume that at a_D, such fermions have a kinetic energy E_D, a chemical potential μ_D and a temperature T_D = T(a_D) = T_γ(a_D). Because we are considering degenerate fermions, μ_D ≫ K_BT_D. We also assume that during a_NR < a ≤ a₀ degenerate fermions are non-relativistic, and that their temperature at a_NR is T_NR = T(a_NR).

We distinguish between fermions that are ultra-relativistic at a_D (hot and warm dark matter) and non-relativistic fermions at a_D (cold dark matter). The degeneracy condition μ_D/K_BT_D ≫ 1 together with the condition a_NR > a_D for the former and a_NR < a_D for the latter must lead to restrictions on the mass of fermions. Obtaining these restrictions is the aim of this section.

Fermions that decouple when they are ultra-relativistic could be e, μ and τ neutrinos. Direct observational bounds on neutrino masses, found kinematically, are: m_νe < 10   eV (Weinheimer et al. 1999; Lobashev et al. 1999; Particle Data Group 2000); m_ντ < 170   keV (Assmagan et al. 1996); and m_ντ < 18   MeV (Buskulic et al. 1995; Passalacqua 1997). In the absence of degeneracy, e-neutrinos decouple at K_BT_D(ν_e) ≈ 2.0   MeV, and μ and τ neutrinos at K_BT_D(ν_μ) ≈ K_BT_D(ν_τ) ≈ 3.5   MeV (see, for example, Freese et al. 1983). For semi-degenerate neutrinos, T_D is greater and depends on their degree of degeneration (see, for example, Kang & Steigman, 1992). Heavy neutrinos could be an example of fermions that are non-relativistic when they decouple. For a review of neutrinos in cosmology, see, for example, Dolgov (2002a).

6.1. Degenerate fermions that decouple when they are ultra-relativistic

At a_D, before the decoupling, the degenerate fermions, which are in equilibrium with the radiation, follow the Fermi distribution function given by Eq. (8) ${\mathit{f}}_{\mathrm{equi}}\mathrm{\left(}{\mathit{E}}_{\mathrm{D}}\mathit{,}{\mathit{a}}_{\mathrm{D}}\mathrm{\right)}\mathrm{=}\{\begin{array}{c}\\ \mathit{g}\mathit{/}{\mathit{h}}^{\mathrm{3}}\mathit{,}& \mathrm{0}\mathrm{\le }{\mathit{E}}_{\mathrm{D}}\mathrm{\le }{\mathit{\mu }}_{\mathrm{D}}\\ \mathrm{0}\hspace{0.17em}& {\mathit{E}}_{\mathrm{D}}\mathit{>}{\mathit{\mu }}_{\mathrm{D}}\mathit{,}\end{array}$(96)where E_D and μ_D are related with the momentum and the Fermi momentum at a_D, p_D and P_FD, by $\begin{array}{ccc}{\mathit{E}}_{\mathrm{D}}& \mathrm{=}& {\left[{\left({\mathit{p}}_{\mathrm{D}}\mathit{c}\right)}^{\mathrm{2}}\mathrm{+}{\mathrm{(}\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{2}}\right]}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{-}\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathit{,}\\ {\mathit{\mu }}_{\mathrm{D}}& \mathrm{=}& {\left[{\left({{\mathit{p}}_{\mathrm{F}}}_{\mathrm{D}}\mathit{c}\right)}^{\mathrm{2}}\mathrm{+}{\mathrm{(}\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{2}}\right]}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{-}\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathit{.}\end{array}$If fermions are ultra-relativistic at a_D, p_FDc/mc² ≫ 1, only fermions with p_Dc ≫ mc² contribute appreciably to physical quantities. Therefore, from Eqs. (97) and (98) $\begin{array}{ccc}{\mathit{E}}_{\mathrm{D}}\mathrm{\approx }{\mathit{p}}_{\mathrm{D}}\hspace{0.17em}\mathit{c,}& & \\ {\mathit{\mu }}_{\mathrm{D}}\mathrm{\approx }{{\mathit{p}}_{\mathrm{F}}}_{\mathrm{D}}\hspace{0.17em}\mathit{c,}& & \end{array}$and then Eq. (96) expressed as a function of the momentum, p_D, is given by ${\mathit{f}}_{\mathrm{equi}}^{\mathrm{UR}}\mathrm{\left(}{\mathit{p}}_{\mathrm{D}}\mathit{,}{\mathit{a}}_{\mathrm{D}}\mathrm{\right)}\mathrm{=}\{\begin{array}{c}\\ \mathit{g}\mathit{/}{\mathit{h}}^{\mathrm{3}}\mathit{,}& \mathrm{0}\mathrm{\le }{\mathit{p}}_{\mathrm{D}}\mathrm{\le }{\mathit{\mu }}_{\mathrm{D}}\mathit{/}\mathit{c}\\ \mathrm{0}\mathit{,}& {\mathit{p}}_{\mathrm{D}}\mathit{>}{\mathit{\mu }}_{\mathrm{D}}\mathit{/}\mathit{c}\mathit{.}\end{array}$(101)At a_D, after the decoupling, the fermions travel along geodesics in space-time. During this free propagation, the distribution function of the fermions is conserved (the expansion of the universe causes the proper volume occupied by any number of these fermions to increase as a³ while their volume in the momentum space is redshifted by a^-3). This allows us to obtain their distribution function after the decoupling, ${\mathit{f}}_{\mathrm{dec}}^{\mathrm{UR}}$. Thus, taking into account that as the universe expands, the fermion momentum, p, decreases as $\mathit{p}\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{=}{\mathit{p}}_{\mathrm{D}}\mathrm{(}{\mathit{a}}_{\mathrm{D}}\mathit{/}\mathit{a}\mathrm{)}\mathit{,}$(102)the conservation of the distribution function implies ${\mathit{f}}_{\mathrm{dec}}^{\mathrm{UR}}\mathrm{\left(}\mathit{p,a}\mathrm{\right)}\mathrm{=}{\mathit{f}}_{\mathrm{equi}}^{\mathrm{UR}}\mathrm{(}\mathit{pa}\mathit{/}{\mathit{a}}_{\mathrm{D}}\mathit{,}{\mathit{a}}_{\mathrm{D}}\mathrm{)}$, so ${\mathit{f}}_{\mathrm{dec}}^{\mathrm{UR}}\mathrm{\left(}\mathit{p,a}\mathrm{\right)}\mathrm{=}\{\begin{array}{c}\\ \mathit{g}\mathit{/}{\mathit{h}}^{\mathrm{3}}\mathit{,}& \mathrm{0}\mathrm{\le }\mathit{p}\mathrm{\le }\mathrm{(}{\mathit{\mu }}_{\mathrm{D}}\mathit{/}\mathit{c}\mathrm{)}\mathrm{(}{\mathit{a}}_{\mathrm{D}}\mathit{/}\mathit{a}\mathrm{)}\\ \mathrm{0}\mathit{,}& \mathit{p}\mathit{>}\mathrm{(}{\mathit{\mu }}_{\mathrm{D}}\mathit{/}\mathit{c}\mathrm{)}\mathrm{(}{\mathit{a}}_{\mathrm{D}}\mathit{/}\mathit{a}\mathrm{)}\mathit{.}\end{array}$(103)Finally, as the fermion temperature, T, cools as $\mathit{T}\mathrm{=}{\mathit{T}}_{\mathrm{D}}\mathrm{(}{\mathit{a}}_{\mathrm{D}}\mathit{/}\mathit{a}\mathrm{)}\mathit{,}$(104)even though the fermions are no longer in thermodynamical equilibrium with radiation (see, for example, Padmanabhan 1995), Eq. (103) reads as ${\mathit{f}}_{\mathrm{dec}}^{\mathrm{UR}}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{=}\{\begin{array}{c}\\ \mathrm{(}\mathit{g}\mathit{/}{\mathit{h}}^{\mathrm{3}}\mathrm{)}\mathit{,}& \mathrm{0}\mathrm{\le }\mathit{p}\mathrm{\le }\mathrm{(}{\mathit{\mu }}_{\mathrm{D}}\mathit{/}\mathit{c}\mathrm{)}\mathrm{(}\mathit{T}\mathit{/}{\mathit{T}}_{\mathrm{D}}\mathrm{)}\\ \mathrm{0}\mathit{,}& \mathit{p}\mathit{>}\mathrm{(}{\mathit{\mu }}_{\mathrm{D}}\mathit{/}\mathit{c}\mathrm{)}\mathrm{(}\mathit{T}\mathit{/}{\mathit{T}}_{\mathrm{D}}\mathrm{)}\mathit{.}\end{array}$(105)Equation (105) tells us that the Fermi momentum, p_F, is ${\mathit{p}}_{\mathrm{F}}\mathrm{=}\frac{{\mathit{\mu }}_{\mathrm{D}}}{\mathit{c}}\left(\frac{\mathit{T}}{{\mathit{T}}_{\mathrm{D}}}\right)\mathrm{·}$(106)Hence, from Eq. (106), the fermions will be non-relativistic when p_Fc/mc² = (μ_D/mc²)(T/T_D) ≪ 1, and T_NR can then be estimated taking p_Fc/mc² = 1; thus, ${\mathit{T}}_{\mathrm{NR}}\mathrm{=}\frac{\mathit{m}{\mathit{c}}^{\mathrm{2}}}{{\mathit{\mu }}_{\mathrm{D}}}{\mathit{T}}_{\mathrm{D}}\mathit{.}$(107)Dividing now Eq. (107) by the fermion temperature at present, T₀ = T(a₀), and taking into account that T/T₀ = a₀/a, $\frac{{\mathit{a}}_{\mathrm{NR}}}{{\mathit{a}}_{\mathrm{0}}}\mathrm{=}\left(\frac{{\mathit{\mu }}_{\mathrm{D}}}{\mathit{m}{\mathit{c}}^{\mathrm{2}}}\right)\frac{{\mathit{a}}_{\mathrm{D}}}{{\mathit{a}}_{\mathrm{0}}}\mathrm{·}$(108)For a > a_NR, degenerate fermions are non-relativistic. Then, their number density and energy density, n_F and ρ_F, are given by $\begin{array}{ccc}& & {\mathit{n}}_{\mathrm{F}}\mathrm{=}\mathrm{4}\mathit{\pi }{\mathrm{\int }}_{\mathrm{0}}^{\mathrm{\infty }}{\mathit{f}}_{\mathrm{dec}}^{\mathrm{UR}}\mathrm{\left(}\mathit{p}\mathrm{\right)}{\mathit{p}}^{\mathrm{2}}\mathrm{d}\mathit{p}\mathrm{=}\frac{\mathit{g}}{\mathrm{6}{\mathit{\pi }}^{\mathrm{2}}{ħ}^{\mathrm{3}}{\mathit{c}}^{\mathrm{3}}}{\left[{\mathit{\mu }}_{\mathrm{D}}\left(\frac{\mathit{T}}{{\mathit{T}}_{\mathrm{D}}}\right)\right]}^{\mathrm{3}}\mathit{,}\\ & & {\mathit{\rho }}_{\mathrm{F}}\mathrm{=}\mathit{m}{\mathit{c}}^{\mathrm{2}}{\mathit{n}}_{\mathrm{F}}\mathrm{=}\frac{\mathit{gm}{\mathit{c}}^{\mathrm{2}}}{\mathrm{6}{\mathit{\pi }}^{\mathrm{2}}{ħ}^{\mathrm{3}}{\mathit{c}}^{\mathrm{3}}}{\left[{\mathit{\mu }}_{\mathrm{D}}\left(\frac{\mathit{T}}{{\mathit{T}}_{\mathrm{D}}}\right)\right]}^{\mathrm{3}}\mathit{,}\end{array}$where the distribution function given by Eq. (105) has been used. Assuming a value for Ω_F0 = ρ_F0/ρ_c0, the energy density at present will be ${\mathit{\rho }}_{\mathrm{F}\mathrm{0}}\mathrm{=}\frac{\mathrm{3}{\mathit{c}}^{\mathrm{2}}}{\mathrm{8}\mathit{\pi G}}{\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}{\mathrm{\Omega }}_{\mathrm{F}\mathrm{0}}\mathrm{=}\mathrm{2.23}\mathrm{\times }{\mathrm{10}}^{-9}\hspace{0.17em}{\mathrm{𝒞}}_{\mathrm{F}\mathrm{0}}\hspace{0.17em}\mathrm{erg}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-3}\mathit{,}$(111)with ${\mathrm{𝒞}}_{\mathrm{F}\mathrm{0}}\mathrm{=}{\left(\frac{{\mathit{H}}_{\mathrm{0}}}{\mathrm{70}{\hspace{0.17em}\mathrm{km}\hspace{0.17em}\mathrm{Mpc}}^{-1}\hspace{0.17em}{\mathrm{s}}^{-1}}\right)}^{\mathrm{2}}\left(\frac{{\mathrm{\Omega }}_{\mathrm{F}\mathrm{0}}}{\mathrm{0.27}}\right)\mathrm{;}$(112)so, equating Eqs. (110) at a₀ and (111), ${\mathit{\mu }}_{\mathrm{D}}\left(\frac{{\mathit{T}}_{\mathrm{0}}}{{\mathit{T}}_{\mathrm{D}}}\right)\mathrm{=}{\left(\frac{\mathrm{9}\mathit{\pi }}{\mathrm{4}\mathit{g}}\right)}^{\mathrm{1}\mathit{/}\mathrm{3}}\frac{ħ{\mathit{c}}^{\mathrm{5}\mathit{/}\mathrm{3}}}{{\mathit{G}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{(}\mathit{m}{\mathit{c}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{3}}}{\mathrm{\left(}{\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}{\mathrm{\Omega }}_{\mathrm{F}\mathrm{0}}\mathrm{\right)}}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathit{.}$(113)Finally, using Eq. (113) in (107), we have $\frac{{\mathit{T}}_{\mathrm{NR}}}{{\mathit{T}}_{\mathrm{0}}}\mathrm{=}{\left(\frac{\mathrm{4}\mathit{g}}{\mathrm{9}\mathit{\pi }}\right)}^{\mathrm{1}\mathit{/}\mathrm{3}}\frac{{\mathit{G}}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathrm{(}\mathit{m}{\mathit{c}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{4}\mathit{/}\mathrm{3}}}{ħ{\mathit{c}}^{\mathrm{5}\mathit{/}\mathrm{3}}}{\mathrm{\left(}{\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}{\mathrm{\Omega }}_{\mathrm{F}\mathrm{0}}\mathrm{\right)}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}\mathrm{;}$(114)and, as T/T₀ = a₀/a, $\frac{{\mathit{a}}_{\mathrm{NR}}}{{\mathit{a}}_{\mathrm{0}}}\mathrm{=}{\left(\frac{\mathrm{9}\mathit{\pi }}{\mathrm{4}\mathit{g}}\right)}^{\mathrm{1}\mathit{/}\mathrm{3}}\frac{ħ{\mathit{c}}^{\mathrm{5}\mathit{/}\mathrm{3}}}{{\mathit{G}}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathrm{(}\mathit{m}{\mathit{c}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{4}\mathit{/}\mathrm{3}}}{\mathrm{\left(}{\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}{\mathrm{\Omega }}_{\mathrm{F}\mathrm{0}}\mathrm{\right)}}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathit{,}$(115)i.e., $\frac{{\mathit{a}}_{\mathrm{NR}}}{{\mathit{a}}_{\mathrm{0}}}\mathrm{=}\mathrm{8.61}\mathrm{\times }{\mathrm{10}}^{-4}{\mathit{g}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}{\left(\frac{\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{\mathrm{-}\mathrm{4}\mathit{/}\mathrm{3}}{\mathrm{𝒞}}_{\mathrm{F}\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathit{.}$(116)As an example, at z = 1000 (a relatively short epoch after recombination when the growth of perturbations is fully given by linear theory and late enough for the decaying modes of perturbations to have decayed away), i.e., at a/a₀ = 1/(1 + z) = 9.99 × 10^-4, degenerate fermions are non-relativistic if $\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathit{>}\mathrm{0.90}\hspace{0.17em}{\mathit{g}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{4}}{\mathrm{𝒞}}_{\mathrm{F}\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{4}}\hspace{0.17em}\mathrm{eV}\mathit{.}$(117)At present, the fermion temperature, T₀, differs from that of radiation, T_γ0, due to different particle annihilation and decoupling occurring after a_D. If only e⁺e⁻ annihilate, T₀ = (4/11)^1/3T_γ0 (see, for example, Padmanabhan 1995). Thus, expressing the temperature of fermions at present, T₀, as ${\mathit{T}}_{\mathrm{0}}\mathrm{=}\mathrm{1.96}{\mathit{F}}_{\mathrm{0}}\left(\frac{{\mathit{T}}_{\mathit{\gamma }\mathrm{0}}}{\mathrm{2.75}\hspace{0.17em}\mathrm{K}}\right)\mathit{,}$(118)with ${\mathit{F}}_{\mathrm{0}}\mathrm{=}\left(\frac{{\mathit{T}}_{\mathrm{0}}\mathit{/}{\mathit{T}}_{\mathit{\gamma }\mathrm{0}}}{\mathrm{(}\mathrm{4}\mathit{/}\mathrm{11}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{3}}}\right)\mathit{,}$(119)the expansion factor of the universe at the fermion decoupling, a_D, can be expressed as $\frac{{\mathit{a}}_{\mathrm{D}}}{{\mathit{a}}_{\mathrm{0}}}\mathrm{=}\frac{{\mathit{T}}_{\mathrm{0}}}{{\mathit{T}}_{\mathrm{D}}}\mathrm{=}\mathrm{1.69}\mathrm{\times }{\mathrm{10}}^{-10}{\left(\frac{{\mathit{K}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{D}}}{\mathrm{1}\hspace{0.17em}\mathrm{MeV}}\right)}^{-1}{\mathit{F}}_{\mathrm{0}}\left(\frac{{\mathit{T}}_{\mathit{\gamma }\mathrm{0}}}{\mathrm{2.75}\hspace{0.17em}\mathrm{K}}\right)\mathrm{·}$(120)Thus, as degenerate fermions are relativistic when decoupling if a_D/a₀ < a_NR/a₀, we deduce from Eqs. (116) and (120) that m must fulfill $\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathit{<}\frac{\mathrm{107}}{{\mathit{g}}^{\mathrm{1}\mathit{/}\mathrm{4}}}{\left(\frac{{\mathit{K}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{D}}}{\mathrm{1}\hspace{0.17em}\mathrm{MeV}}\right)}^{\mathrm{3}\mathit{/}\mathrm{4}}{\mathrm{𝒞}}_{{\mathrm{F}}_{\mathrm{0}}}^{\mathrm{1}\mathit{/}\mathrm{4}}{\mathit{F}}_{\mathrm{0}}^{\mathrm{-}\mathrm{3}\mathit{/}\mathrm{4}}{\left(\frac{{\mathit{T}}_{\mathit{\gamma }\mathrm{0}}}{\mathrm{2.75}\hspace{0.17em}\mathrm{K}}\right)}^{\mathrm{-}\mathrm{3}\mathit{/}\mathrm{4}}\hspace{0.17em}\mathrm{keV}\mathit{.}$(121)If the fermions were non-degenerate, we could estimate a_NR/a₀ assuming K_BT_NR ≈ mc²/3. Thus, $\frac{{\mathit{a}}_{\mathrm{NR}}}{{\mathit{a}}_{\mathrm{0}}}\mathrm{=}\frac{{\mathit{T}}_{\mathrm{0}}}{{\mathit{T}}_{\mathrm{NR}}}\mathrm{=}\frac{\mathrm{3}{\mathit{K}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{0}}}{\mathit{m}{\mathit{c}}^{\mathrm{2}}}\mathrm{=}\mathrm{5.08}\mathrm{\times }{\mathrm{10}}^{-4}{\left(\frac{\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{-1}{\mathit{F}}_{\mathrm{0}}\left(\frac{{\mathit{T}}_{\mathit{\gamma }\mathrm{0}}}{\mathrm{2.75}\hspace{0.17em}\mathrm{K}}\right)\mathrm{·}$(122)Therefore, non-degenerate fermions would be relativistic when decoupling (a_NR/a₀ > a_D/a₀) if $\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathit{<}\mathrm{3.01}\left(\frac{{\mathit{K}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{D}}}{\mathrm{1}\hspace{0.17em}\mathrm{MeV}}\right)\hspace{0.17em}\mathrm{MeV}\mathit{.}$(123)The degeneracy parameter ξ_D = μ_D/(K_BT_D) can be estimated from Eqs. (113) and (118). Thus, ${\mathit{\xi }}_{\mathrm{D}}\mathrm{=}\mathrm{5.08}\hspace{0.17em}{\mathit{g}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}{\left(\frac{\mathit{m}{\mathit{c}}^{\mathrm{2}}}{\mathrm{1}\hspace{0.17em}\mathrm{eV}}\right)}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{3}}{\mathrm{𝒞}}_{\mathrm{F}\mathrm{0}}^{\mathrm{1}\mathit{/}\mathrm{3}}{\mathit{F}}_{\mathrm{0}}^{-1}{\left(\frac{{\mathit{T}}_{\mathit{\gamma }\mathrm{0}}}{\mathrm{2.75}\hspace{0.17em}\mathrm{K}}\right)}^{-1}\mathrm{·}$(124)Therefore, only fermions with mass m fulfilling $\mathit{m}{\mathit{c}}^{\mathrm{2}}\mathrm{\ll }\mathrm{131}\hspace{0.17em}{\mathit{g}}^{-1}{\mathrm{𝒞}}_{\mathrm{F}\mathrm{0}}{\mathit{F}}_{\mathrm{0}}^{-3}{\left(\frac{{\mathit{T}}_{\mathit{\gamma }\mathrm{0}}}{\mathrm{2.75}\hspace{0.17em}\mathrm{K}}\right)}^{-3}\hspace{0.17em}\mathrm{eV}$(125)will be degenerate (ξ_D ≫ 1). Evidently, the degeneracy condition (125) is more restrictive for m than the relativistic condition at a_D given in (121).