Model independent analysis of dark matter points to a particle mass at the keV scale

# Model independent analysis of dark matter points to a particle mass at the keV scale

H. J. de Vega, N. G. Sanchez
LPTHE, Université Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII),
Laboratoire Associé au CNRS UMR 7589, Tour 24, 5ème. étage, Boite 126, 4, Place Jussieu, 75252 Paris, Cedex 05, France
Observatoire de Paris, LERMA. Laboratoire Associé au CNRS UMR 8112.
61, Avenue de l’Observatoire, 75014 Paris, France.
devega@lpthe.jussieu.frNorma.Sanchez@obspm.fr
July 15, 2019
###### Abstract

We present a model independent analysis of dark matter (DM) both decoupling ultrarelativistic (UR) and non-relativistic (NR) based on the DM phase-space density . We derive explicit formulas for the DM particle mass and for the number of ultrarelativistic degrees of freedom at decoupling. We find that for DM particles decoupling UR both at local thermal equilibrium (LTE) and out of LTE, turns to be at the keV scale. For example, for DM Majorana fermions decoupling at LTE the mass results keV. For DM particles decoupling NR, results in the keV scale ( is the decoupling temperature) and the value is consistent with the keV scale. In all cases, DM turns to be cold DM (CDM). Also, lower and upper bounds on the DM annihilation cross-section for NR decoupling are derived. We evaluate the free-streaming (Jeans’) wavelength and Jeans’ mass: they result independent of the type of DM except for the DM self-gravity dynamics. The free-streaming wavelength today results in the kpc range. These results are based on our theoretical analysis, astronomical observations of dwarf spheroidal satellite galaxies in the Milky Way and -body numerical simulations. We analyze and discuss the results on from analytic approximate formulas both for linear fluctuations and the (non-linear) spherical model and from -body simulations results. We obtain in this way upper bounds for the DM particle mass which all result below the 100 keV range.

###### keywords:
dark matter – keV mass scale.
pagerange: Model independent analysis of dark matter points to a particle mass at the keV scaleReferencespubyear: 0000

## 1 The dark matter particle mass

Although dark matter was noticed seventy-five years ago (Zwicky, 1933; Oort, 1940) its nature is not yet known.

Dark matter (DM) must be non-relativistic by the time of structure formation () in order to reproduce the observed small structure at kpc.

DM particles can decouple being ultrarelativistic (UR) at or being non-relativistic (NR) at where is the mass of the dark matter particles and the decoupling temperature. We consider in this paper particles that decouple at or out of local thermal equilibrium (LTE).

The DM distribution function freezes out at decoupling. Therefore, for all times after decoupling coincides with its expression at decoupling. is a function of and the comoving momentum of the dark matter particles .

Knowing the distribution function , we can compute physical magnitudes as the DM velocity fluctuations and the DM energy density. For the relevant times during structure formation, when the DM particles are non-relativistic, we have

 ⟨→V2⟩(t)=⟨→p2phm2⟩(t)=∫d3pph(2π)3→p2phm2Fd[a(t)pph]∫d3pph(2π)3Fd[a(t)pph] (1)

where we use the physical momentum of the dark matter particles as integration variable. The scale factor is normalized as usual,

 a(t)=11+z(t),a(today)=1, (2)

namely, the physical momentum coincides today with the comoving momentum .

We can relate the covariant decoupling temperature , the effective number of UR degrees of freedom at decoupling and the photon temperature today by using entropy conservation (Kolb & Turner, 1990; Börner, 2003; Yao, 2006):

 Td=(2gd)13Tγ,whereTγ=0.2348meV (3)

and meV eV.

The DM energy density can be written as

 ρDM(t)=g∫d3pph(2π)3√m2+p2ph Fd[a(t)pph], (4)

where is the number of internal degrees of freedom of the DM particle, typically .

By the time when the DM particles are non-relativistic, the energy density eq.(4) becomes

 ρDM(t)=m g2π2T3da3(t)I2≡mn(t), (5)

where

 I2≡∫∞0y2Fd(y)dy,

is the number of DM particles per unit volume and we used as integration variable

 y≡pph(t)Td(t)=pcTd. (6)

From eq.(5) at and from the value observed today for (Komatsu et al., 2009; Yao, 2006),

 ρDM = ΩDMρc=0.228ρc, (8) ρc = 3M2PlH20=(2.518meV)4, (9)

and , we find the value of the DM mass:

 m=π2ΩDMρcT3γgdgI2=6.986eVgdgI2, (10)

where is the critical density.

Using as integration variable [eq.(6)], eq.(1) for the velocity fluctuations, yields

 ⟨→V2⟩(t)=[Tdma(t)]2I4I2, (11)

where

 I4≡∫∞0y4Fd(y)dy.

Expressing in terms of the CMB temperature today according to eq.(3) gives for the one-dimensional velocity dispersion,

 σDM(z)=√13⟨→V2⟩(z)=213√31+zg13dTγm√I4I2= (13) =0.051241+zg13dkeVm[I4I2]12kms . (14)

It is very useful to consider the phase-space density invariant under the universe expansion (Boyanovsky, de Vega & Sanchez, 2008a; Hogan & Dalcanton, 2000; Madsen, 1990, 2001)

 D(t)≡n(t)⟨→P2ph(t)⟩32non−rel=13√3m4ρDM(t)σ3DM(t), (15)

where we consider the relevant times during structure formation when the DM particles are non-relativistic. is a constant in absence of self-gravity. In the non-relativistic regime can only decrease by collisionless phase mixing or self-gravity dynamics (Lynden-Bell, 1967; Tremaine et al., 1986).

We derive a useful expression for the phase-space density from eqs.(5), (13) and (15) with the result

 D=g2π2I522I324, (16)

Observing dwarf spheroidal satellite galaxies in the Milky Way (dSphs) yields for the phase-space density today (Wyse & Gilmore, 2007; Gilmore et. al., 2007):

 ρsσ3s∼5×103 keV/cm3(km/s)3=(0.18keV)4. (17)

The precision of these results is about a factor .

After the radiation dominated era the phase-space density reduces by a factor that we call

 D(0)=1ZD(z∼3200) (18)

Recall that [according to eq.(15)] is independent of for since density fluctuations were before the matter dominated era (Dodelson, 2003).

The range of values of (which is necessarily ) is analyzed in detail in sec. 2.3 below.

We can express the phase-space density today from eqs.(15) and (17) as

 D(0)=13√3m4ρsσ3s. (19)

Therefore, eqs.(15), (18) and (19) yield,

 ρsσ3s=1ZρDMσ3DM(z∼3200), (20)

where follows from eqs.(15) and (16),

 ρDMσ3DM(z∼3200)=3√3m42π2gI522I324. (21)

We can express from eqs.(17)-(21) in terms of and observable quantities as

 m4=Z3√3ρsDσ3s=2π23√3Zgρsσ3sI324I522, (22) (23) m=0.2504(Zg)14I384I582keV. (24)

Combining this with eq.(10) for we obtain the number of ultrarelativistic degrees of freedom at decoupling as

 gd=214338π32g34ΩDMT3γρc(Zρsσ3s)14[I2I4]38 (25) (26) =35.96Z14g34[I2I4]38. (27)

If we assume that dark matter today is a self-gravitating gas in thermal equilibrium described by an isothermal sphere solution of the Lane-Emden equation, the relevant quantity characterizing the dynamics is the dimensionless variable (de Vega & Sánchez, 2002; Destri & de Vega, 2007)

 η=Gm2NLT=23GL2ρsσ2s, (28)

which is bound to be to prevent the gravitational collapse of the gas (de Vega & Sánchez, 2002; Destri & de Vega, 2007). Here stands for the volume occupied by the gas, for the number of particles, for Newton’s constant and is the gas temperature. (The length is similar to the so-called King radius (Binney & Tremaine, 1987). Notice however that the King radius follows from the singular isothermal sphere solution while is the characteristic size of a stable isothermal sphere solution (de Vega & Sánchez, 2002; Destri & de Vega, 2007).)

The compilation of recent photometric and kinematic data from ten Milky Way dSphs satellites (Wyse & Gilmore, 2007; Gilmore et. al., 2007) yields values for the one dimensional velocity dispersion and the radius in the ranges

 0.5 kpc≤L≤1.8 kpc,6.6 km/s≤σs≤11.1 km/s. (29)

Combining eq.(15), eq.(20) and (28) yields the explicit expression for the DM particle mass,

 m4∼12√3GηZL2Dσs=4π√3M2PlηZL2Dσs (31) =0.527910−4ηZD10km/sσs(kpcL)2(keV)4 . (32)

This formula provides an expression for the DM particle mass independent of eq.(22). We shall see below that eqs.(22) and (31) yield similar results.

We investigate in the subsequent sections the cases where DM particles decoupled UR or NR both at LTE and out of LTE. We compute there and explicitly in the different cases according to the general formulas eqs.(24), (25) and (31).

### 1.1 Jeans’ (free-streaming) wavelength and Jeans’ mass

It is very important to evaluate the Jeans’ length and Jeans’ mass in the present context (Börner, 2003; Gilbert, 1968; Bond & Szalay, 1983). The Jeans’ length is analogous to the free-streaming wavelength. The free-streaming wavevector is the largest wavevector exhibiting gravitational instability and characterizes the scale of suppression of the DM transfer function (Boyanovsky, de Vega & Sanchez, 2008b).

The physical free-streaming wavelength can be expressed as (Börner, 2003; Boyanovsky, de Vega & Sanchez, 2008b)

 λfs(t)=λJ(t)=2πkfs(t) (33)

where is the physical free-streaming wavenumber given by

 k2fs(t)=4πGρDM(t)⟨→V2⟩(t)=32[1+z(t)]H20ΩDM⟨→V2⟩(0). (34)

where we used that and eq.(8).

We obtain the primordial DM dispersion velocity from eqs. (5), (8) and (20),

 √13⟨→V2⟩(0)=σDM=(3M2PlH20ΩDM1Zσ3sρs)13 (35)

This expression is valid for any kind of DM particles. Inserting eq.(35) into eq.(34) yields for the physical free-streaming wavelength

 λfs(z) = 2√2πΩ16DM(3M2PlH0)13(σ3sZρs)131√1+z= (37) = 16.3Z131√1+zkpc. (38)

where we used keV .

Notice that and therefore turn to be independent of the nature of the DM particle except for the factor .

The approximated analytic evaluations in sec. 2 together with the results of -body simulations (Peirani et al., 2006; Hoffman et al., 2007; Lapi & Cavaliere, 2009; Romano-Diaz et al., 2006, 2007; Vass et al., 2009) indicate that for dSphs is in the range

 1

Therefore, and the free-streaming wavelength results in the range

 0.7571√1+zkpc<λfs(z)<16.31√1+zkpc.

These values at are consistent with the -body simulations reported in Gao & Theuns (2007) and are of the order of the small DM structures observed today (Wyse & Gilmore, 2007; Gilmore et. al., 2007).

The Jeans’ mass is given by

 MJ(t)=43πλ3J(t)ρDM(t). (39)

and provides the smallest unstable mass by gravitational collapse Kolb & Turner (1990); Börner (2003). Inserting here eq.(5) for the DM density and eq.(37) for yields

 MJ(z) = 192√2π4√ΩDMM4PlH0σ3sZρs(1+z)32= (41) = 0.4464Z107M⊙(1+z)32. (42)

Taking into account the -values range yields

 0.4464103M⊙

This gives masses of the order of galactic masses by the beginning of the MD era . In addition, the comoving free-streaming wavelength scale by

 3200×λfs(z∼3200)∼100kpc,

turns to be of the order of the galaxy sizes today.

## 2 The phase-space density D from analytic approximation methods and from N-body simulations

We analytically derive here formulas for the reduction factor defined by eq.(18) in the linear approximation and in the spherical model. The results obtained (see Table I) are in fact upper bounds for . We then analyze the results on from -body simulations.

### 2.1 Linear perturbations

The simplest calculation of follows by considering linear perturbations around the homogeneous distribution as

 ρ=ρDM(z)[1+δ(z,k)], (43)

We have and in a matter dominated universe

 δ(z,k)∼δi1+z1+zi. (44)

The peculiar velocity in the MD universe behaves as (Dodelson, 2003),

 v∼aHδ∼1/√1+z. (45)

We can thus relate the phase-space density at redshift eq.(15) with the phase-space density at redshift as,

 D(z)∼D(zi)(1+z1+zi)92. (46)

Since the linear approximation is valid for , we find from eq.(44) that eq.(46) applies in the redshift range where

 1+z≫(1+zi)δi. (47)

We can apply eq.(46) to relate at equilibration () and at the beginning of structure formation , since as a scale average of the density fluctuations at the end of the RD dominated era (Dodelson, 2003) and eq.(47) is satisfied. We thus obtain from eq.(46) in the linear approximation:

 D(z≃3200)D(z≃30)∼1.3×109. (48)

Notice that eq.(47) does not hold for . Therefore, in order to evaluate , we should combine the linear approximation result eq.(48) for with the results of -body simulations for . This is done in sec. 2.4 to obtain upper bounds for .

### 2.2 The spherical model

Let us now consider the spherical model where particles only move in the radial direction but where the non-linear evolution is exactly solved (Fillmore & Goldrich, 1984; Bertschinger, 1985; Peebles, 1993; Padmanabhan, 1999). The proper radius of the spherical shell obeys the equation

 ¨R=−GMR2 (49)

where is the gravitational constant and the (constant) mass enclosed by the shell. Eq.(49) can be solved in close form with the solution (Bertschinger, 1985; Peebles, 1993)

 t=3ti4δ32i(θ−sinθ), (50) (51) R=Ri2δi(1−cosθ),2R3i9t2i=GM, (52) (53) ˙R=2Ri√δi3tisinθ1−cosθ, (54) (55) 1+z=(1+zi)δi(43)231(θ−sinθ)23, (56) (57) ρ=ρDM(z)92(θ−sinθ)2(1−cosθ)3, (58)

Here, and are the radius and the redshift at the initial time and is an auxiliary time dependent parameter.

Choosing the initial time by equilibration with we have and we find from eqs.(50),

 θi=2√δi,˙R(ti)=2Ri3ti,ρi=ρDM(zi). (59)

The spherical shell reaches its maximum radius of expansion at and then it turns around and collapses to a point at . However, well before that, the approximation that matter only moves radially and that random particle velocities are small will break down. Actually, the DM relaxes to a virialized configuration where the velocity and the virial radius follow from the virial theorem (Padmanabhan, 1999)

 v2=6GM5Rm,Rv=12Rm. (60)

We can now compute the initial phase-space density (at ) and the phase-space density at virialization. We get at from eqs.(15) and (59)

 Di=ρDM(0)3√3m4(1+zi)3(3ti2Ri)3, (61)

and at virialization for from eqs.(15) and (60)

 Dv=ρDM(0)3√3m4(1+zi)3(tiRi)3329π2(154δi)32. (62)

Therefore, the factor in the spherical model takes the value

 Z=DiDv=9π232(35δi)32=1.29009δ32i.

Setting as a scale average of the density fluctuations at the end of the RD dominated era (Dodelson, 2003) yields

 Z∼4.08×104. (63)

The spherical model approximates the evolution as a purely radial expansion followed by a radial collapse. Since no transverse motion is allowed neither mergers, the spherical model result for eq.(63) is actually an upper bound on .

### 2.3 The phase-space density D from N body simulations

The phase-space density is invariant under the universe expansion except for the self-gravity dynamics that diminishes in its evolution (Lynden-Bell, 1967; Tremaine et al., 1986). Numerical simulations show that decreases sharply during phases of violent mergers followed by quiescent phases (Peirani et al., 2006; Hoffman et al., 2007; Lapi & Cavaliere, 2009; Romano-Diaz et al., 2006, 2007; Vass et al., 2009). decreases at these violent phases by a factor of the order . (See fig. 3 in Peirani et al. (2006), fig. 1 in Hoffman et al. (2007), fig. 6 in Lapi & Cavaliere (2009) and fig. 5 in Vass et al. (2009)). These sharp decreasings of are in agreement with the linear approximation of sec. 2.1 as we show below.

A succession of several violent phases happens during the structure formation stage (). Their cumulated effect together with the evolution of for produces a range of values of the factor which we can conservatively estimate on the basis of the -body simulations results (Peirani et al., 2006; Hoffman et al., 2007; Lapi & Cavaliere, 2009; Romano-Diaz et al., 2006, 2007; Vass et al., 2009) and the approximation results eqs.(48) and (63). This gives a range of values for dSphs.

Indeed, more accurate analysis of body simulations should narrow this range for which depends on the type and size of the galaxy considered.

The dSphs observations, for which the best observational data are available, take mostly into account the cores of the structures since the dSphs have been stripped of their external halos. Hence, the observed values of may be higher than the space-averaged valued represented by the right hand side of eqs.(18) and (20). Higher values for correspond to lower values for .

The approximate formula eq.(46) indicates a sharp decrease of the phase-space density with the redshift. This sharp decreasing is in qualitative agreement with the simulations in the violent phases Peirani et al. (2006); Hoffman et al. (2007); Lapi & Cavaliere (2009); Romano-Diaz et al. (2006, 2007); Vass et al. (2009).

### 2.4 Sinthetic discussion on the evaluation of Z and its upper bounds.

The DM particle mass scale is set by the phase-space density for dSphs eq.(17). Those galaxies are particularly dense and exhibit larger values for than spiral galaxies. Since the primordial phase-space density is an universal quantity only depending on cosmological parameters, the -factor must be galaxy dependent, larger for spiral galaxies than for dSphs.

We want to stress that the values of the relevant quantities and are mildly affected by the uncertainity of through the factor [see eqs.(24)-(25)].

Eqs.(46) provides an extreme high estimate for the decrease of and hence an extreme high estimate for . The -body simulations show that the violent decrease of is restricted to a factor of order one at each violent phase (Peirani et al., 2006; Hoffman et al., 2007; Lapi & Cavaliere, 2009; Romano-Diaz et al., 2006, 2007; Vass et al., 2009).

In summary, the linear approximation suggests a reduction of at each violent phase by a factor while such approximation is valid. Succesive violent phases can reduce by a factor up to in the range as shown in the simulations Peirani et al. (2006); Hoffman et al. (2007); Lapi & Cavaliere (2009); Romano-Diaz et al. (2006, 2007); Vass et al. (2009).

Combining the approximate decrease of given by eq.(48) with an upper bound of a decrease by a factor for the interval yields in the linear approximation the upper bound

 Z<1.3×1011, (64)

and we have eq.(63) for in the spherical model. The fact that in the spherical model turns to be several orders of magnitude below the value in the linear approximation arises from the fact that the spherical model does include non-linear effects and it is therefore somehow more reliable than the linear approximation.

The range for dSphs from -body simulations corresponds to realistic initial conditions in the simulations.

The evolutions in the two approximations considered (see Table I) are simple spatially isotropic expansions, followed by a collapse in the case of the spherical model. There is no possibility of non-radial motion neither of mergers in these approximations contrary to the case in -body simulations. For such reasons, the -values in Table I are upper bounds to the true values of in galaxies. The largest bound on yield DM particle masses below keV. Moreover, the more reliable spherical model yields keV as upper bound for the DM particle mass.

In summary, with realistic initial conditions will not decrease more than and it is therefore fair to assume that for dSphs.

## 3 DM particles decoupling being ultrarelativistic

### 3.1 Decoupling at Local Thermal Equilibrium (LTE)

If the dark matter particles of mass decoupled at a temperature their freezed-out distribution function only depends on

 pcTd=pph(t)Td(t),whereTd(t)≡Tda(t).

That is, the distribution function for dark matter particles that decoupled in thermal equilibrium takes the form

 Fequild[pph(t)Td(t)]=Fequild[pcTd],

where is a Bose-Einstein or Fermi-Dirac distribution function:

 Fequild[pc]=1exp[√m2+p2c/Td]±1. (65)

Notice that for eq.(65) in this regime:

 √m2+p2cTdTd≫m=y+O(m2T2d).

where is defined by eq.(6) and we can use as distribution functions

 Fequild(y)=1ey±1. (66)

Using eqs.(10) and (65), we find then for Fermions and for Bosons decoupling at LTE

 m=gdg{3.874eV  Fermions2.906eV  Bosons. (67)

We see that for DM that decoupled at the Fermi scale: GeV and results in the keV scale as already remarked in Bond & Szalay (1983); Pagels & Primack (1982); Bond, Szalay & Turner (1982). DM particles may decouple earlier with GeV but is always in the hundreds even in grand unified theories where can reach the GUT energy scale. Therefore, eq.(67) strongly suggests that the mass of the DM particles which decoupled UR in LTE is in the keV scale.

It should be noticed that the Lee-Weinberg (Lee & Weinberg, 1977; Sato & Kobayashi, 1977; Vysotsky, Dolgov & Zeldovich, 1977) lower bound as well as the Cowsik-McClelland (Cowsick & McClelland, 1972) upper bound follow from eq.(10) as shown in Boyanovsky, de Vega & Sanchez (2008a).

Computing the integrals in eq.(16) with the distribution functions eq.(65) yields for DM decoupling UR in LTE

 D=g⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩14π2√ζ5(3)15ζ3(5)=1.9625×10−3  Fermions18π2√ζ5(3)3ζ3(5)=3.6569×10−3  Bosons (68)

where and .

Inserting the distribution function eq.(66) into eqs.(22) and (25) for and , respectively, we obtain

 m = (Zg)14keV{0.568   Fermions0.484   Bosons, (70) gd = g34Z14{155   Fermions180   Bosons. (71)

Since , for DM particle decoupling at LTE, we see from eq.(70) that and thus, the DM particle should decouple for GeV. Notice that for .

A further estimate for the DM mass follows by inserting eq.(68) for in eq.(31)

 (72)

Taking into account the observed values for and from eq.(29) and the fact that , eq.(72) gives again a mass in the keV scale as in eq.(70). Both equations (70) and (72) yield a mass larger in 17% for the fermion than for the boson.

We can express the free-streaming wavelength as a function of the DM particle mass from eqs.(37) and (70) with the result,

 λfs(z)=(keVm)43kpcg131√1+z{7.67  Fermions6.19  Bosons. (73)

We display in fig. 1 in kpc vs. in keV.

### 3.2 Decoupling out of LTE

In general, for DM decoupling out of equilibrium, the DM particle d istribution function takes the form

 Fd(pc)=Fd(pcTd;mTd;…) (74)

Typically, thermalization is reached by the mixing of the particle modes and scattering between particles that redistributes the particles in phase space: the larger momentum modes are populated by a cascade whose front moves towards the ultraviolet akin to a direct cascade in turbulence, leaving in its wake a state of nearly LTE but with a lower temperature than that of equilibrium (Boyanovsky, Destri & de Vega, 2004; Destri & de Vega, 2006). Hence, in the case the dark matter particles are not yet at thermodynamical equilibrium at decoupling, their momentum distribution is expected to be peaked at smaller momenta since the ultraviolet cascade is not yet completed (Boyanovsky, Destri & de Vega, 2004; Destri & de Vega, 2006). The freezed-out of equilibrium distribution function can be then written as

 Fout of LTEd(pc)=F0Fequild[a(t)pph(t)ξTd]θ(p0c−pc), (75)

where at thermal equilibrium and before thermodynamical equilibrium is attained. is a normalization factor and cuts the spectrum in the UV region not yet reached by the cascade.

Inserting the out of equilibrium distribution eq.(75) in the expression for the DM particle mass eq.(10) and using eq.(66), we obtain the generalization of eq.(67) for the out of LTE case:

 (76)

where . Here we used eq.(66) and

 F±(s)≡∫s0y2dyey±1,F+(∞)=32ζ(3),F−(∞)=2ζ(3). (77)

Inserting the out of equilibrium distribution eq.(75) into eqs.(22) and (25) for and , respectively, and using eq.(66), we obtain the estimates

 (79) gd∼g34Z14ξ3X±(s){155   Fermions180   Bosons . (80)

where

 (81)

Here is defined by eq.(77) and

 G±(s)≡∫s0y4dyey±1,G+(∞)=452ζ(5),G−(∞)=24ζ(5). (82)

For small arguments we have:

 W+(0)=√3534[ζ5(3)ζ3(5)]18=0.5732982…,
 X+(s)s→0=s3[2025ζ(3)ζ(5)]38=0.0529923…s3,
 X−(s)s→0=s94[4320ζ(3)ζ(5)]38=0.0398856…s94.

As seen in fig. 2,

 W±(s)≤1andX±(s)≤1fors≥0.

We see from eq.(79) that for relics decoupling out of LTE, is in the keV range. From eqs.(79)-(80) we see that both and for relics decoupling out of LTE are smaller than if they would decouple at LTE. In addition, since vanishes for , may be much smaller than for decoupling at LTE.

We now generalize eq.(68) for the phase-space density to the out of LTE case eq.(68). Using eqs.(16), (66) and (75) we have

 D=gF0W4±(s){1.9625×10−3  Fermions3.6569×10−3  Bosons (83)

Inserting now eq.(83) into eq.(31) leads to the out of LTE generalization of the estimate for the DM particle mass eq.(72)

 m∼[ηZg10kmsσs]14W±(s)√kpcLkeV{0.405  Fermions0.347  Bosons (84)

Taking into account the observed values for and from eq.(29) and the fact that , eq.(84) gives again a mass in the keV scale as in eq.(79). Both equations (79) and (84) yield a mass larger in 17% for the fermion than for the boson.

#### 3.2.1 An instructive example: Sterile neutrinos decoupling out of LTE.

We consider in this subsection a sterile neutrino as DM particle decoupling out of LTE in a specific model where is a singlet Majorana fermion () with a Majorana mass coupled with a small Yukawa-type coupling () to a real scalar field (Chikashige et al., 1981; Gelmini & Roncadelli, 1981; Schechter & Valle, 1982; Shaposhnikov & Tkachev, 2006; McDonald & Sahu, 2009). is more strongly coupled to the particles in the Standard Model plus to three right-handed neutrinos. As a result, all particles (except ) remain in LTE well after decouples from them.

The distribution function after decoupling of the sterile neutrino is known for small coupling to be (Boyanovsky, 2008),

 Fνd(y)=τg52(y)√ywhereg52(y)≡∞∑n=1e−nyn52 (85)

and the coupling is in the range (Boyanovsky, 2008).

It is interesting to compare the small () and large momenta () behaviour of this out of equilibrium distribution with the Fermi-Dirac equilibrium distribution eq.(66). We find

 Fνd(y)Fequild(y)y→0=2τζ(52)√y→∞,ζ(52)=1.341…,
 Fνd(y)Fequild(y)y→∞=τ√y→0.

Therefore, exhibits an enhancement compared with the Fermi-Dirac equilibrium distribution for small () and a suppression for large momenta (). Qualitatively, the out of equilibrium distribution eq.(75) exhibits the same effect when compared to the equilibrium distribution as a consequence of the incomplete UV cascade.

We now evaluate the relevant physical quantities inserting in the relevant equations of sec. 3.1. We find for from eqs.(10) and (85)

 mν=2.34gdτeV, (86)

which must be compared with the LTE result for fermions eq.(67) with .

The phase-space density from eqs.(16) and (85) takes the value,

 D=6τζ52(5)[35πζ(7)]32=5.627×10−3τwhereζ(7)=1.0083493…. (87)

This result is to be compared with the LTE result for fermions eq.(68) with .

Inserting the sterile neutrino distribution function eq.(85) into eqs.(22) and (25), that take into account the decrease of the phase-space density due to the self-gravity dynamics, we obtain the following mass estimates for the DM particles that decoupled out of LTE,

 mν∼(Zτ)140.434keV,gd∼τ34Z14185. (88)

Again, these formulas must be compared with the LTE result for fermions eqs.(70). corresponds to GeV (see Kolb & Turner (1990)) which is the expected value for in Boyanovsky (2008).

More precisely, for the typical range , from eq.(88) we find

 0.56keV≲mνZ−14≲1.0keV,