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

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by  H. J. de Vega

Mon.Not.R.Astron.Soc.000,1–11(0000) Printed11January2010 (MNLATEXstylefilev2.2)
Model independent analysis of dark matter points to a
particle mass at the keV scale
0
H. J. de Vega1,2⋆, N. G. Sanchez2
1
0 1LPTHE, Universit´e Pierre et Marie Curie (Paris VI†) et DenisDiderot (Paris VII),
2 Laboratoire Associ´e au CNRS UMR 7589, Tour 24, 5`eme. ´etage, Boite 126, 4, Place Jussieu, 75252 Paris, Cedex 05, France
2Observatoire de Paris, LERMA. Laboratoire Associ´e au CNRS UMR 8112.
n
61, Avenue de l’Observatoire, 75014 Paris, France.
a
J

1
1 11January2010

]
O
ABSTRACT
C We present a model independent analysis of dark matter (DM) both decoupling
. ultrarelativistic(UR) andnon-relativistic(NR) basedonthe DMphase-spacedensity
h = ρDM/σD3M. We derive explicit formulas for the DM particle mass m and for
p D
the number of ultrarelativistic degrees of freedom gd at decoupling. We find that
-
o for DM particles decoupling UR both at local thermal equilibrium (LTE) and out
r of LTE, m turns to be at the keV scale. For example, for DM Majorana fermions
t
s decoupling at LTE the mass results m 0.85 keV. For DM particles decoupling NR,
a √mTd results in the keV scale (Td is≃the decoupling temperature) and the m value
[ is consistent with the keV scale. In all cases, DM turns to be cold DM (CDM). Also,

3 lower and upper bounds on the DM annihilation cross-section for NR decoupling are
v derived. We evaluate the free-streaming (Jeans’) wavelength and Jeans’ mass: they
2 result independent of the type of DM except for the DM self-gravity dynamics. The
2 free-streaming wavelength today results in the kpc range. These results are based
9 on our theoretical analysis, astronomical observations of dwarf spheroidal satellite
0 galaxiesintheMilkyWayandN-bodynumericalsimulations.Weanalyzeanddiscuss
.
1 the results on from analytic approximate formulas both for linear fluctuations and
D
0 the (non-linear) spherical model and from N-body simulations results. We obtain in
9 this way upper bounds for the DM particle mass which all result below the 100 keV
0 range.
:
v
Key words: dark matter – keV mass scale.
i
X
r
a
1 THE DARK MATTER PARTICLE MASS and the comoving momentum of the dark matter particles
pc.
Although dark matter was noticed seventy-five years ago
Knowingthedistributionfunction Fd(pc),wecancom-
(Zwicky 1933; Oort 1940) its nature is not yet known.
pute physical magnitudes as the DM velocity fluctuations
Darkmatter(DM)mustbenon-relativisticbythetime
and the DM energy density. For the relevant times t dur-
of structure formation (z < 30) in order to reproduce the
ing structure formation, when the DM particles are non-
observed small structure at 2 3 kpc.
∼ − relativistic, we have
DM particles can decouple being ultrarelativistic (UR)
watherTedm≫ismtheomr absesinogftnhoend-raerlkatmivaitstteicrp(NarRti)claestanTddT≪d thme V~2 (t)= p~p2h (t)= Z d(23πpp)h3 p~mp2h2 Fd[a(t)pph] (1)
decouplingtemperature.Weconsiderinthispaperparticles h i hm2i d3p
ph F [a(t)p ]
thatdecoupleatoroutoflocalthermalequilibrium(LTE). (2π)3 d ph
Z
The DM distribution function F freezes out at decou-
d where we use the physical momentum of the dark matter
pling. Therefore, for all times after decoupling F coincides
d particles pph(t) pc/a(t) as integration variable. The scale
with itsexpression at decoupling. F is a function of T , m ≡
d d factor a(t)is normalized as usual,
1
a(t)= , a(today)=1, (2)
1+z(t)
⋆ [email protected] namely,thephysicalmomentumpph(t)coincidestodaywith
† [email protected] thecomoving momentum pc.
2 H. J. de Vega, N. G. Sanchez
WecanrelatethecovariantdecouplingtemperatureT , whereweconsidertherelevanttimestduringstructurefor-
d
theeffectivenumberofURdegreesoffreedomatdecoupling mation when the DM particles are non-relativistic. (t) is
D
gd and the photon temperature today Tγ by using entropy a constant in absence of self-gravity. In the non-relativistic
conservation(Kolb & Turner1990;B¨orner2003;Yao2006): regime (t)canonlydecreasebycollisionlessphasemixing
D
1 orself-gravitydynamics(Lynden-Bell1967;Tremaine et al.
2 3
Td = Tγ , where Tγ =0.2348meV (3) 1986).
(cid:18)gd(cid:19) We derive a useful expression for the phase-space den-
and 1 meV =10−3 eV. sity from eqs.(5), (10) and (12) with theresult
D
The DM energy density can be written as 5
g I2
d3p = 2 , (13)
ρDM(t)=g (2πp)h3 m2+p2ph Fd[a(t)pph], (4) D 2π2I432
Z q
where g is thenumberof internal degrees of freedom of the Observing dwarf spheroidal satellite galaxies in the Milky
DM particle, typically 16g64. Way (dSphs) yields for the phase-space density today
BythetimewhentheDMparticlesarenon-relativistic, (Wyse& Gilmore 2007; Gilmore et. al. 2007):
theenergy density eq.(4) becomes ρs 5 103 keV/cm3 =(0.18keV)4. (14)
ρDM(t)= 2mπg2 aT3(d3t) I2 ≡mn(t), (5) σs3 ∼ × (km/s)3
The precision of theseresults is about a factor 10.
where Aftertheradiationdominatederathephase-spaceden-
∞ sity reduces by a factor that we call Z
I y2F (y)dy,
2 d
≡Z0 (0)= 1 (z 3200) (15)
n(t) is thenumberof DMparticles perunit volumeand we D Z D ∼
used as integration variable Recall that D(z) = ρDM/(3 √3 m4 σD3M) [according to
y pph(t) = pc . (6) eq.(12)] is independent of z for z & 3200 since density
≡ Td(t) Td fluctuations were . 10−3 before the matter dominated era
(Dodelson 2003).
From eq.(5) at t=0 and from thevalueobserved today for
TherangeofvaluesofZ (whichisnecessarilyZ >1)is
ρDM (Komatsu et al. 2009; Yao 2006),
analyzed in detail in sec. 2.3 below.
ρDM = ΩDM ρc=0.228ρc , We can express the phase-space density today from
eqs.(12) and (14) as
ρc = 3MP2l H02 =(2.518meV)4, (7) (0)= 1 ρs . (16)
and MP2l =1/[8πG], we findthe valueof the DMmass: D 3√3m4 σs3
m=π2 ΩDM Tρc3 ggId =6.986eV ggId , (8) Therefore, eqs.(12), (15) and (16) yield,
γ 2 2
where ρc is the critical density. σρs3 = Z1 σρD3M(z∼3200), (17)
Using as integration variable y [eq.(6)], eq.(1) for the s DM
velocity fluctuations, yields where ρDM/σD3M(z∼3200) follows from eqs.(12) and (13),
hV~2i(t)=(cid:20)mTad(t)(cid:21)2 II24 , (9) σρD3M(z∼3200)= 3√23π2m4 g I2235 . (18)
DM I2
where 4
∞ We can express m from eqs.(14)-(18) in terms of and
I y4F (y)dy. D
4≡ d observable quantitiesas
Z0
Expressing Td in terms of the CMB temperature today ac- m4= Z ρs = 2π2 Z ρs I432 , (19)
pcoerrdsiionng,to eq.(3) gives for theone-dimensional velocity dis- 3√3 Dσs3 3√3 g σs3 I252
1 3
σDM(z)=q13 hV~2i(z)= √2133 1g+d31z Tmγ rII42 = (10) m=0.2504 (cid:18)Zg(cid:19)4 II42885 keV. (20)
1 Combining this with eq.(8) for m we obtain the number of
=0.05124 1+z keV I4 2 km . (11) ultrarelativistic degrees of freedom at decoupling as
1 m I s
g3 (cid:20) 2(cid:21)
Idtensiitsy veirnyvariuansetfulduntodercontshideer utnhieverspehasee-xsppaance- gd = 338241π23 ΩgD34M Tργc3 (cid:18)Zσρs3s(cid:19)14 [I2 I4]38
1 3 3
sion (Boyanovsky,de Vega& Sanchez 2008a; =35.96Z4 g4 [I2 I4]8 . (21)
Hogan & Dalcanton 2000; Madsen 1990,2001)
Ifweassumethatdarkmattertodayisaself-gravitatinggas
(t) n(t) non=−rel 1 ρDM(t) , (12) in thermal equilibrium described by an isothermal sphere
D ≡ hP~p2h(t)i32 3√3m4 σD3M(t) solution of theLane-Emdenequation,therelevant quantity
Model independent analysis of dark matter points to a particle mass at the keV scale 3
characterizing the dynamics is the dimensionless variable We obtain theprimordial DM dispersion velocity σDM
(deVega & S´anchez 2002; Destri & de Vega 2007) from eqs. (5), (7) and (17),
η= GLmT2 N = 32 GL2 σρss2 , (22) r13 hV~2i(0)=σDM =(cid:18)3 MP2l H02ΩDM Z1 σρss3(cid:19)31 (27)
whichisboundtobeη.1.6topreventthegravitationalcol- ThisexpressionisvalidforanykindofDMparticles.Insert-
lapseofthegas(deVega & S´anchez2002;Destri & deVega ingeq.(27)intoeq.(26)yieldsforthephysicalfree-streaming
2007). Here V =L3 stands for the volume occupied by the wavelength
gaansd,NT f=ort3hemnuσm2bisertohfepgaartsictleems,pGerfaotruNree.w(tTohne’slceonngstthanLt λ (z) = 2√2π 3MP2l 13 σs3 13 1 =
is similar t2o the so-called King radius (Binney & Tremaine fs Ω16 (cid:18) H0 (cid:19) (cid:18)Z ρs(cid:19) √1+z
DM
1987). Notice however that the King radius follows from 16.3 1
= kpc. (28)
the singular isothermal sphere solution while L is the Z13 √1+z
characteristic size of a stable isothermal sphere solution
where we used 1 keV =1.5637381029 (kpc)−1.
(deVega & S´anchez 2002; Destri & de Vega 2007).)
Notice that λfs and therefore λJ turn to be indepen-
The compilation of recent photometric and kinematic dentof thenatureof theDMparticle except for thefactor
datafromtenMilkyWaydSphssatellites(Wyse & Gilmore Z.
2007; Gilmore et. al. 2007) yields values for the one dimen- The approximated analytic evaluations in sec.
sional velocity dispersion σs and the radius Lin theranges 2 together with the results of N-body simulations
(Peirani et al. 2006; Hoffman et al. 2007; Lapi & Cavaliere
0.5 kpc6L61.8 kpc , 6.6 km/s6σs 611.1 km/s. 2009; Romano-Diaz et al. 2006, 2007; Vass et al. 2009)
(23)
indicate that for dSphsZ is in the range
Combining eq.(12), eq.(17) and (22) yields the explicit ex-
pression for theDM particle mass, 1<Z <10000.
1
1 Z 4π Z Therefore, 1 < Z3 < 21.5 and the free-streaming wave-
m4 ∼ 2√3G η L2 σs = √3 MP2l η L2 σs length results in the range
D D
=0.527910−4 ηZ 10km/s kpc 2 (keV)4 . (24) 0.757 √11+z kpc<λfs(z)<16.3 √11+z kpc.
D σs (cid:18) L (cid:19)
These values at z = 0 are consistent with the N-body
This formula provides an expression for the DM particle
simulations reported in Gao & Theuns (2007) and are of
mass independent of eq.(19). We shall see below that
the order of the small DM structures observed today
eqs.(19) and (24) yield similar results.
(Wyse& Gilmore 2007; Gilmore et. al. 2007).
We investigate in the subsequent sections the cases
The Jeans’ mass is given by
where DM particles decoupled UR or NR both at LTE and
4
oduiffteorefnLtTcEa.seWs eacccoomrdpinugtettohetrheemgenaenrdalgdforemxpulliacsitleyqsin.(2t0h)e, MJ(t)= 3 πλ3J(t)ρDM(t). (29)
(21) and (24). andprovidesthesmallestunstablemassbygravitationalcol-
lapse Kolb & Turner (1990); B¨orner (2003). Inserting here
eq.(5) for the DM density and eq.(28) for λJ(t) = λfs(t)
1.1 Jeans’ (free-streaming) wavelength and Jeans’ yields
ItisvemryaismsportanttoevaluatetheJeans’lengthandJeans’ MJ(z) = 192√2π4 √ΩDM MP4l H0 Zσρs3s (1+z)23 =
mass in the present context (B¨orner 2003; Gilbert 1968; = 0.4464 107 M⊙ (1+z)23 . (30)
Bond & Szalay1983).TheJeans’lengthisanalogoustothe Z
free-streaming wavelength. The free-streaming wavevector Taking intoaccount theZ-values range yields
iasntdhcehlaarragcetsetriwzaesvetvheecstcoarleexohfisbuiptipnrgesgsriaovnitoafttiohneaDlMinsttraabnislifteyr 0.4464103 M⊙ <MJ(z)(1+z)−32 <0.4464107 M⊙ .
function T(k) (Boyanovsky,de Vega& Sanchez2008b). Thisgivesmassesoftheorderofgalacticmasses 1011 M⊙
∼
The physical free-streaming wavelength can be ex- by the beginning of the MD era z 3200. In addition, the
∼
pressed as (B¨orner 2003; Boyanovsky,de Vega& Sanchez comoving free-streaming wavelength scale by z 3200
∼
2008b)
3200 λ (z 3200) 100kpc,
fs
× ∼ ∼
2π
λfs(t)=λJ(t)= k (t) (25) turnsto beof theorder of the galaxy sizes today.
fs
where kfs(t) = kJ(t) is the physical free-streaming
wavenumbergiven by 2 THE PHASE-SPACE DENSITY FROM
D
ANALYTIC APPROXIMATION METHODS
k2 (t)= 4πGρDM(t) = 3 [1+z(t)] H02 ΩDM . (26) AND FROM N-BODY SIMULATIONS
fs V~2 (t) 2 V~2 (0)
h i h i Weanalyticallyderivehereformulasforthereductionfactor
where we used that ρDM(t)=ρDM(0)(1+z)3 and eq.(7). Z definedbyeq.(15)inthelinearapproximation andin the
4 H. J. de Vega, N. G. Sanchez
spherical model. The results obtained (see Table I) are in R= Ri (1 cosθ) , 2Ri3 =GM ,
fact upper bounds for Z. We then analyze the results on 2δi − 9t2i
D from N-body simulations. R˙ = 2Ri √δi sinθ ,
3ti 1 cosθ
−
2.1 Linear perturbations 4 23 1
TpasehretusrimbaptlieosntscaarlocuunladtitohneohfomDofgoelnloewoussbdyisctornibsuidteiorninρgDliMne(azr) ρ1+=zρD=M((1z+) 29zi)((1θδi−(cid:18)csoi3ns(cid:19)θθ))23 (,θ−sinθ)23 , (38)
−
ρ=ρDM(z)[1+δ(z,k)] , (31) Here,Ri andzi aretheradiusandtheredshiftattheinitial
We have ρDM(z)=ρDM(0)(1+z)3 and in a matter domi- timeCtihoaonsdinθgitshaeninaiutxiailliatrimyetimbyeedqeupielinbdraentitonpawraitmhezteir. 1
nated universe ≫
we haveθi 1 and we find from eqs.(38),
≪
1+z
δ(z,k)∼δi 1+zi . (32) θi=2√δi , R˙(ti)= 23Rtii , ρi=ρDM(zi). (39)
The peculiar velocity in the MD universe behaves as
The spherical shell reaches its maximum radius of expan-
(Dodelson 2003),
sion Rm = Ri/δi at θ = π and then it turns around and
v aH δ 1/√1+z. (33) collapses to a point at θ = 2π. However, well before that,
∼ ∼
theapproximationthatmatteronlymovesradiallyandthat
We can thus relate the phase-space density at redshift z
random particle velocities are small will break down. Actu-
(z) ρ/v3eq.(12)withthephase-spacedensityatredshift
D ∼ ally, the DM relaxes to a virialized configuration where the
zi as, velocityand thevirialradiusfollow from thevirial theorem
9 (Padmanabhan 1999)
1+z 2
D(z)∼D(zi) (cid:18)1+zi(cid:19) . (34) v2 = 6GM , Rv = 1 Rm . (40)
Sincethelinear approximation isvalid for δ2 1, wefind 5Rm 2
fromeq.(32)thateq.(34)appliesinthereds|hi|ft≪range(zi, z) Wecan now computetheinitial phase-spacedensity(at zi)
where and the phase-space density at virialization. We get at zi
from eqs.(12) and (39)
1+z (1+zi)δi. (35)
We can apply eq.(34) to≫relate (z) at equilibration (z = i = ρDM(0) (1+zi)3 3ti 3 , (41)
zi 3200)and (z)atthebeginDningofstructureformation D 3√3m4 (cid:18)2Ri(cid:19)
z ≃30, since δDi 10−3 as a scale average of the density and at virialization for θ=2π from eqs.(12) and (40)
∼ ∼
fl2in0u0ct3htu)eaaltininodenaserqaa.t(p3tp5hr)eoxiesinmsdaatotiisfofitneh:de.RWDedtohmusinoabtteadinerfaro(mDoedqe.(ls3o4n) Dv = 3ρD√M3(m0)4 (1+zi)3 (cid:18)Rtii(cid:19)3 93π22 (cid:18)145 δi(cid:19)23 . (42)
Therefore, the Z factor in the spherical model takes the
(z 3200)
D ≃ 1.3 109 . (36) value
(z 30) ∼ ×
Notice that eq.(35D) do≃es not hold for z 0. Therefore, in Z = Di = 9π2 3 23 = 1.29009 .
order to evaluate Z, we should combine t∼he linear approxi- Dv 32 (cid:18)5δi(cid:19) δi32
mation result eq.(36) for 30 .z .3200 with the results of
N-body simulations for 0.z .30. This is done in sec. 2.4 Setting δi 10−3 as a scale average of the density fluctua-
∼
tions at the end of the RD dominated era (Dodelson 2003)
to obtain upperboundsfor Z.
yields
Z 4.08 104 . (43)
2.2 The spherical model ∼ ×
Thesphericalmodelapproximatestheevolutionasapurely
Let us now consider the spherical model where particles
radial expansion followed by a radial collapse. Since no
only move in the radial direction but where the non-
transverse motion is allowed neither mergers, the spherical
linearevolutionisexactlysolved(Fillmore & Goldrich1984;
modelresultforZ eq.(43)isactuallyanupperboundonZ.
Bertschinger 1985; Peebles 1993; Padmanabhan 1999). The
proper radius of thespherical shell obeys theequation
GM 2.3 The phase-space density from N body
R¨= (37) D
− R2 simulations
whereGisthegravitationalconstantandM the(constant) The phase-space density (z) is invariant under the
D
mass enclosed by the shell. Eq.(37) can be solved in close universe expansion except for the self-gravity dynam-
form with the solution (Bertschinger 1985; Peebles 1993) ics that diminishes (z) in its evolution (Lynden-Bell
D
t= 43δti32 (θ−sinθ), 1th9a6t7; DT(rze)madinecereetasaels. 1s9h8a6r)p.lyNudmureirnicgalpshiamsueslatoiofnsvioshleonwt
i mergers followed by quiescent phases (Peirani et al.
Model independent analysis of dark matter points to a particle mass at the keV scale 5
1
Approximationused UpperlimitonZ Upperlimitonm≃0.5Z4 keV
Linearfluctuations ∼1.3×1011 96keV
−3
SphericalModel ∼1.29×δ 2 ≃4.1×104 7.1keV
i
Table 1.Upperbounds fortheZ-factor[definedbyeq.(15)]andforthemassoftheDMparticleobtained
fortwodifferentapproximationmethods.Noticethatonlythesphericalmodeltakesintoaccountnon-linear
self-gravityeffects. Themassmmildlydepends onZ throughthepower 1/4.Inanycasemresultsinthe
keVrange.
2006; Hoffman et al. 2007; Lapi & Cavaliere 2009; is restricted to a factor of order one at each violent phase
Romano-Diaz et al. 2006, 2007; Vasset al. 2009). (z) (Peirani et al. 2006; Hoffman et al. 2007; Lapi & Cavaliere
D
decreases at these violent phases by a factor of the or- 2009; Romano-Diaz et al. 2006, 2007; Vasset al. 2009).
der & 1. (See fig. 3 in Peirani et al. (2006), fig. 1 in
Insummary,thelinearapproximationsuggestsareduc-
Hoffman et al.(2007),fig.6inLapi & Cavaliere(2009)and
tion of at each violent phase by a factor & 1 while such
fig. 5 in Vass et al. (2009)). These sharp decreasings of D
D approximation is valid. Succesive violent phases can reduce
are in agreement with the linear approximation of sec. 2.1
byafactorupto 10intherange0.z.30asshownin
as we show below. D ∼
thesimulationsPeirani et al.(2006);Hoffman et al. (2007);
A succession of several violent phases happens during
Lapi & Cavaliere (2009); Romano-Diaz et al. (2006, 2007);
thestructureformation stage(z .30). Theircumulatedef-
Vasset al. (2009).
fect together with the evolution of for 3200 & z & 30
D
produces a range of values of the Z factor which we can Combining the approximate decrease of (z) given by
conservatively estimate on the basis of the N-body sim- eq.(36)withanupperboundofadecreasebyaDfactor 100
ulations results (Peirani et al. 2006; Hoffman et al. 2007; fortheinterval0.z.30yieldsinthelinearapproxim∼ation
Lapi & Cavaliere 2009; Romano-Diaz et al. 2006, 2007; theupperbound
Vass et al.2009)andtheapproximationresultseqs.(36)and
(43).This gives a range of values 1<Z <10000 for dSphs. Z <1.3 1011 , (44)
×
Indeed, more accurate analysis of N body simulations
and we have eq.(43) for Z in the spherical model. The fact
should narrow this range for Z which depends on the type
that Z in the spherical model turns to be several orders of
and size of the galaxy considered.
magnitude below the Z value in the linear approximation
The dSphs observations, for which the best observa-
arises from the fact that the spherical model does include
tionaldataareavailable,takemostlyintoaccountthecores
non-lineareffectsanditisthereforesomehowmorereliable
ofthestructuressincethedSphshavebeenstrippedoftheir
than thelinear approximation.
external halos. Hence,the observed values of ρs/σs3 may be
higher than the space-averaged valued represented by the Therange1<Z <10000fordSphsfromN-bodysimu-
righthandsideofeqs.(15)and(17).Highervaluesforρs/σs3 lationscorresponds torealistic initialconditions inthesim-
correspond to lowervalues for Z. ulations.
The approximate formula eq.(34) indicates a sharp The evolutions in the two approximations considered
decrease of the phase-space density with the red- (see Table I) are simple spatially isotropic expansions, fol-
shift. This sharp decreasing is in qualitative agreement lowedbyacollapseinthecaseofthesphericalmodel.There
with the simulations in the violent phases Peirani et al. is no possibility of non-radial motion neither of mergers in
(2006); Hoffman et al. (2007); Lapi & Cavaliere (2009); theseapproximations contrary to thecase in N-bodysimu-
Romano-Diaz et al. (2006, 2007); Vass et al. (2009). lations.Forsuchreasons,theZ-valuesinTableIareupper
bounds to the true values of Z in galaxies. The largest
bound on Z yield DM particle masses below 100 keV.
2.4 Sinthetic discussion on the evaluation of Z ∼
Moreover,themorereliablesphericalmodelyields7.1keV
and its upper bounds.
as upperbound for theDM particle mass.
TheDMparticlemassscaleissetbythephase-spacedensity
Insummary,withrealisticinitialconditions willnot
fordSphseq.(14).Thosegalaxiesareparticularlydenseand D
decrease more than . 10000 and it is therefore fair to as-
exhibitlargervaluesforρs/σs3thanspiralgalaxies.Sincethe sumethat Z <10000 for dSphs.
primordialphase-spacedensityisanuniversalquantityonly
dependingoncosmologicalparameters,theZ-factormustbe
galaxy dependent,larger for spiral galaxies than for dSphs.
3 DM PARTICLES DECOUPLING BEING
Wewant tostressthat thevaluesof therelevant quan-
ULTRARELATIVISTIC
tities m and g are mildly affected by the uncertainity of
d
1
Z through the factor Z4 [see eqs.(20)-(21)]. 3.1 Decoupling at Local Thermal Equilibrium
(LTE)
Eqs.(34) provides an extreme high estimate for the de-
crease of and hencean extremehigh estimate for Z. The If the dark matter particles of mass m decoupled at a tem-
D
N-body simulations show that the violent decrease of perature T m their freezed-out distribution function
d
D ≫
6 H. J. de Vega, N. G. Sanchez
only dependson
20
Fermions
pc = pph(t), where T (t) Td . Bosons
Td Td(t) d ≡ a(t) 18
That is, the distribution function for dark matter particles
that decoupled in thermal equilibrium takes theform 16
Fequil pph(t) =Fequil pc , 14
d T (t) d T
(cid:20) d (cid:21) (cid:20) d(cid:21)
where Fequil is a Bose-Einstein or Fermi-Dirac distribution 12
d
function:
1 10
Fdequil[pc]= exp[ m2+p2/T ] 1 . (45)
c d ±
8
Notice that for eq.(45) in thpisregime:
m2+p2c Td=≫my+ m2 . 6
p Td O(cid:18)Td2(cid:19)
4
where y is defined by eq.(6) and we can use as distribution
functions
2
1
Fequil(y)= . (46)
d ey 1
± 0
Using eqs.(8) and (45), we find then for Fermions and for 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Bosons decoupling at LTE
Figure 1. The free-streaming wavelength today λfs(0) in kpc
m= gd 3.874eV Fermions . (47) vs.theDMparticlemassminkeVtimesg14 forultrarelativistic
g 2.906eV Bosons decouplingatLTEaccordingtoeq.(51).
(cid:26)
We see that for DM that decoupled at the Fermi scale:
T 100GeVandg 100, mresultsinthekeVscaleasal-
d ∼ d ∼ A further estimate for the DM mass m follows by in-
readyremarkedinBond & Szalay(1983);Pagels & Primack
serting eq.(48) for in eq.(24)
(1982); Bond, Szalay & Turner (1982). DM particles may D
decouple earlier with Td > 100 GeV but gd is always in ηZ 10km/s 41 kpc 0.405 Fermions
m keV .
rtehaechhutnhdereGdUs Teveenneirngygrsacnadle.unTihfieerdeftohreeo,reieqs.(w47h)ersetrTodngcalyn ∼(cid:20) g σs (cid:21) r L (cid:26) 0.347 Bosons (50)
suggests that the mass of the DM particles which decou- Taking into account the observed values for σs and L from
pled UR in LTE is in the keV scale. eq.(23) and the fact that η .1.6, g 1 4, 1<Z14 <10,
It should be noticed that the Lee-Weinberg ≃ −
eq.(50) givesagain a mass m in thekeVscale as in eq.(49).
(Lee & Weinberg 1977; Sato & Kobayashi 1977;
Bothequations(49)and(50)yieldamasslargerin17%for
Vysotsky,Dolgov & Zeldovich 1977) lower bound as
thefermion than for the boson.
well as the Cowsik-McClelland (Cowsick & McClelland
1972) upper bound follow from eq.(8) as shown in Wecanexpressthefree-streamingwavelengthasafunc-
Boyanovsky,deVega & Sanchez(2008a). tion of the DM particle mass from eqs.(28) and (49) with
theresult,
Computingtheintegralsineq.(13)withthedistribution
functions eq.(45) yields for DMdecoupling UR in LTE keV 43 kpc 1 7.67 Fermions
λ (z)= .
1 ζ5(3) =1.9625 10−3 Fermions fs (cid:18) m (cid:19) g13 √1+z (cid:26) 6.19 Bosons
=g 4π2 15ζ3(5) × (51)
D  81π2 q3ζζ53((35)) =3.6569×10−3 Bosons Wedisplay in fig. 1 λfs(0) in kpcvs. mg14 in keV.
q (48)
where ζ(3)=1.2020569... and ζ(5)=1.0369278....
3.2 Decoupling out of LTE
Insertingthedistributionfunctioneq.(46)intoeqs.(19)
In general, for DM decoupling out of equilibrium, the DM
and (21) for m and g , respectively, we obtain
d
particle d istribution function takes theform
1
Z 4 0.568 Fermions
m = (cid:18)g(cid:19) keV (cid:26) 0.484 Bosons , Fd(pc)=Fd(cid:18)Tpcd;Tmd;...(cid:19) (52)
3 1 155 Fermions
gd = g4 Z4 180 Bosons . (49) Typically, thermalization is reached by the mixing of
(cid:26) the particle modes and scattering between particles that
Sinceg=1 4, for DMparticle decouplingat LTE, we see redistributes the particles in phase space: the larger
−
fromeq.(49)thatgd >100andthus,theDMparticleshould momentum modes are populated by a cascade whose
1
decouple for Td > 100 GeV. Notice that 1 < Z4 < 10 for front moves towards the ultraviolet akin to a direct
1<Z <10000. cascade in turbulence, leaving in its wake a state
Model independent analysis of dark matter points to a particle mass at the keV scale 7
1 gd ∼g43 Z14 ξ3X±(s) 115850 FBeorsmoniosns . (57)
(cid:26)
0.9 where
X+(s)vs.s
0.8 XW−+((ss))vvss..ss W±(s)≡(cid:20)GG3±3±((s∞))FF±5±(5∞(s))(cid:21)81 , X±(s)≡(cid:20)GG±±(∞(s))FF±±((s∞))(cid:21)83
(58)
0.7
W−(s)vs.s HereF±(s) is definedby eq.(55) and
0.6 s y4dy 45
G±(s)≡ ey 1 , G+(∞)= 2 ζ(5), G−(∞)=24ζ(5).
Z0 ±
0.5 (59)
Forsmall arguments s we have:
0.4 √3 ζ5(3) 18
W (0)= =0.5732982... ,
0.3 + 543 (cid:20)ζ3(5)(cid:21)
0.2 W−(s)s→=0 2s1458 (cid:20)27ζ5ζ(33()5)(cid:21)81 =0.4753169... s14 .
0.1
X (s)s→=0 s3 =0.0529923... s3 ,
0 + 3
0 2 4 6 8 10 12 14 [2025ζ(3)ζ(5)]8
Figure 2. X+(s), X−(s), W+(s) and W−(s) as functions of s 9
accordingtoeq.(58). X−(s)s→=0 s4 3 =0.0398856... s49 .
[4320ζ(3)ζ(5)]8
of nearly LTE but with a lower temperature than Asseen in fig. 2,
that of equilibrium (Boyanovsky,Destri & deVega 2004;
Destri & de Vega 2006). Hence, in the case the dark mat- W±(s)61 and X±(s)61 for s>0 .
ter particles are not yet at thermodynamical equilibrium Weseefromeq.(56)thatforrelicsdecouplingoutofLTE,m
at decoupling, their momentum distribution is expected to isinthekeVrange.Fromeqs.(56)-(57)weseethatbothm
bepeaked atsmaller momentasincetheultraviolet cascade andg for relics decouplingout of LTEaresmallerthanif
d
is not yet completed (Boyanovsky,Destri & de Vega 2004; they would decouple at LTE. In addition, since X±(s) van-
Destri & de Vega2006).Thefreezed-outofequilibriumdis- ishesfors 0,g maybemuchsmallerthanfordecoupling
d
→
tribution function can bethen written as at LTE.
a(t)p (t)
Fdoutof LTE(pc)=F0 Fdequil(cid:20) ξTpdh (cid:21) θ(p0c−pc), toWtheenoouwtgoefnLeTraElizceaseeq.e(q4.8()48f)o.rUthsiengpheaqsse.-(s1p3a)c,e(4d6e)nasintdy
(53) D
(53) we have
where ξ=1 at thermal equilibrium and ξ<1 before ther-
tmioondyfancatmoricaanldeqpu0cilicburtiusmthiessaptteacitnruedm.Fin0 ∼the1UisVanreogrmionalinzoat- D=g WF±40(s) (cid:26) 13..96652659××1100−−33 FBeorsmoniosns (60)
yet reached bythe cascade.
Inserting the out of equilibrium distribution eq.(53) in Inserting now eq.(60) into eq.(24) leads to the out of LTE
the expression for the DM particle mass eq.(8) and using generalization of the estimate for the DM particle mass m
eq.(46), we obtain the generalization of eq.(47) for the out eq.(50)
of LTE case: 1
ηZ 10 km 4 kpc 0.405 Fermions
m= gFg0dξ3 eV ( 32..569935 FFFF+−+−((((∞∞ss)))) FBeorsmoniosns , (54) m∼" g σss # W±(s)r L keV (cid:26) 0.347 Boso(n6s1)
where s=p0/[ξ T ] . Here we used eq.(46) and Taking into account the observed values for σs and L from
c d eq.(23) and the fact that η .1.6, g 1 4, 1<Z14 <10,
s y2 dy 3 eq.(61)givesagainamassminthek≃eVs−caleasineq.(56).
F±(s)≡ ey 1 , F+(∞)= 2 ζ(3) , F−(∞)=2ζ(3) .Bothequations(56)and(61)yieldamasslargerin17%for
Z0 ±
(55) thefermion than for the boson.
Inserting the out of equilibrium distribution eq.(53) into
eqs.(19) and (21) for m and g , respectively, and using
d
eq.(46),we obtain the estimates 3.2.1 An instructive example: Sterile neutrinos decoupling
out of LTE.
1
Z 4 0.568 Fermions
m∼ g W±(s)keV 0.484 Bosons (56) We consider in this subsection a sterile neutrino ν as DM
(cid:18) (cid:19) (cid:26) particle decoupling out of LTE in a specific model where
8 H. J. de Vega, N. G. Sanchez
ν is a singlet Majorana fermion (g = 2) with a Ma- More precisely, for the typical range 0.035 .τ .0.35,
jorana mass mν coupled with a small Yukawa-type cou- from eq.(65) we find
pling (Y 10−8) to a real scalar field χ (Chikashige et al.
1981; Gel∼mini & Roncadelli 1981; Schechter& Valle 1982; 0.56keV.mν Z−14 .1.0keV , 15.gd Z−41 .84,
Shaposhnikov & Tkachev2006;McDonald & Sahu2009).χ
whilefor g=2fermions decouplingin LTE,themass turns
is more strongly coupled to the particles in the Standard to be smaller: mZ−41 =0.48 keV and gd larger: gd Z−14 =
Model plus tothreeright-handed neutrinos.As aresult, all
184 [from eq.(49)].
particles (except ν) remain in LTE well after ν decouples
A further estimate for mν, independent of eq.(65), fol-
from them.
lows by inserting eq.(64) for into eq.(24) valid for a self-
Thedistributionfunctionafterdecouplingofthesterile D
gravitating gas of DM:
neutrinoν isknownforsmallcouplingY tobe(Boyanovsky
2008), ηZ 10km/s 14 kpc
mν 0.3105 keV, (66)
Fν(y)=τ g25(y) where g (y) ∞ e−ny (62) ∼ (cid:20) τ σs (cid:21) r L
d √y 25 ≡n=1 n52 which gives for thetypical τ range,
X
−1
and the coupling τ is in the range 0.035 . τ . 0.35 10km/s 4 L
(Boyanovsky 2008). 0.40keV.mν (cid:20)ηZ σs (cid:21) rkpc .0.72keV
Itisinterestingtocomparethesmall(y 0)andlarge
momenta(y ) behaviourofthisoutofe→quilibrium dis- [Recall that 0.1<Z−41 <1.]
tribution Fν(→y)∞with the Fermi-Dirac equilibrium distribu-
d In summary the results for the sterile neutrino decou-
tion eq.(46). We find
pling out of LTE in the model of Chikashige et al. (1981);
Fdν(y) y=→0 2τ ζ 25 , ζ 5 =1.341... , GShealmpoinshin&ikRovon&caTdkeallcihe(v19(8210)0;6)S;chBeocyhatnerov&skVyal(le200(81)98a2r)e;
Fdequil(y) √y(cid:0) (cid:1) →∞ (cid:18)2(cid:19) qualitatively similar to those for fermions decoupling at
LTE.
Fdν(y) y→=∞ τ 0.
Fequil(y) √y →
d
Therefore,Fν(y)exhibitsanenhancementcomparedwith 4 DM PARTICLES DECOUPLING BEING
d
the Fermi-Dirac equilibrium distribution for small (y 0) NON-RELATIVISTIC
→
and a suppression for large momenta (y ). Quali-
tatively,theout ofequilibrium distribution e→q.(5∞3) exhibits Particlesdecouplingnon-relativisticatatemperatureTd ≪
maredescribedbyafreezed-outMaxwell-Boltzmanndistri-
thesameeffectwhencomparedtotheequilibriumdistribu-
tion Fequil as aconsequenceof theincompleteUVcascade. bution function dependingon
d
p2 a2(t)p2 (t)
Wenowevaluatetherelevantphysicalquantitiesinsert- c = ph =T y2 .
d
ing Fν(y) in the relevant equations of sec. 3.1. We find for Td Td
d
mν from eqs.(8) and (62) That is,
mν =2.34 gτd eV, (63) Fdequil(pc)= 25245π72 gd Y∞ Tmd 23 e−2mp2cTd =
which must be compared with the LTE result for fermions (cid:18) (cid:19)
eq.(4T7)hewpithhasge-=sp2a.cedensity fromeqs.(13)and(62)takes = 22545π27 gd Y∞ Tmd 32 e−a2(2t)mpT2phd(t)
D (cid:18) (cid:19)
thevalue, = 252 π27 gd Y∞ e−2y2x , (67)
= 6τ ζ25(5) =5.627 10−3 τ where ζ(7)=1.0083493... . 45 x23
D [35πζ(7)]32 × wheregdistheeffectivenumberofultrarelativisticdegreesof
(64) freedomatdecoupling,Y(t)=n(t)/s(t),n(t)isthenumber
This result is to be compared with the LTE result for ofDMparticlesperunitvolume,s(t)theirentropyperunit
fermions eq.(48) with g=2. volume, x m/Td and Y∞ follows from the late time limit
≡
Inserting the sterile neutrino distribution function of the Boltzmann equation (Kolb & Turner 1990; B¨orner
eq.(62) into eqs.(19) and (21), that take into account the 2003).
decrease of the phase-space density due to the self-gravity For particles that decoupled NR we obtain inserting
dynamics, we obtain the following mass estimates for the ν eq.(67) into the general formula for the DM particle mass
DM particles that decoupled out of LTE, eq.(8),
Z 41 3 1 m= 45 ΩDM ρc = 0.748 eV. (68)
mν ∼ τ 0.434keV , gd ∼τ4 Z4 185. (65) 4π2 gTγ3Y∞ gY∞
(cid:18) (cid:19)
Again, these formulas must be compared with the LTE re- Solving the Boltzmann equation gives for Y∞
(Kolb & Turner1990; B¨orner 2003),
sultforfermionseqs.(49).g 100correspondstoT 100
d d
∼ ∼
GfoerVTd(sieneBKooylban&ovTskuyrn(e2r0(0189)9.0))whichistheexpectedvalue Y∞ = 4√425π72 ggd xe−x . (69)
Model independent analysis of dark matter points to a particle mass at the keV scale 9
Noticethatx&1sincetheDMparticlesdecoupledNR.Y∞ 4.1 Allowed ranges for m, Td and the annihilation
canalsobeexpressedintermsofσ (thethermallyaveraged cross section σ .
0 0
totalannihilation cross-section timesthevelocity whichap-
Wederive here individual boundson m, T and σ for DM
pears in the Boltzmann equation) as (Kolb & Turner 1990; d 0
particles decoupling NR.
B¨orner 2003),
Using that T < m for DM particles that decoupled NR
d
1 45 x we obtain from eq.(74) a lower bound for m and an upper
Y∞ = π r 8 √gd mσ0MPl (70) boundon Td.Furthermore, takingintoaccount that Td >b
eVwhereb>1orb 1forDMparticlesthatdecoupledin
(We assume for simplicity S-wave annihilation). It follows ≫
theRD era, we obtain an upperbound for m. In summary,
from this relation and eq.(68) that
1 2
0.41410−9 gx Z 3 1.47keV<m< 2.16 MeV Z 3 ,
σ0 = GeV2 √gd (71) (cid:18)gd(cid:19) b (cid:18)gd(cid:19)
1
Z 3
The trascendental equations (68) and (69) fix the values of beV<T < 1.47keV. (78)
d g
m and x.They can be combined as (cid:18) d(cid:19)
ex g2 m Recalling that (Kolb & Turner1990)
=193.5 . (72)
x gd keV gd 3 for 1 eV<Td <100 keV, (79)
≃
This equation has solutions for x>1 provided and that 1<Z <104,we see from eqs.(78) that
m e g g
> d =0.014 d . 482
keV 193.5 g2 g2 1.02keV<m< MeV , Td <10.2keV.
b
For x=m/Td &1 we have theanalytic solution of eq.(72) Notice that b may be much larger than one with b <
1
m g2 m g2 m 1470 (Z/gd)3 < 21960 to ensure Td < m for consistency
=x log 193.5 =5.265+log .
Td ≃ (cid:18) gd keV(cid:19) (cid:18)gd keV(cid:19) in eqs.(78).
We obtain the mass of the DM particle inserting the non- In addition, lower and upper bounds for the cross-
relativisticdistributionfunctioneq.(67)intothegeneralfor- section σ0 can be derived. From eqs.(71), (79) and x > 1
mula eq.(19) for m4 with theresult, a lower bound follows,
m25 T23 = 45 1 Z ρs , (73) σ0>0.23910−9 GeV−2g. (80)
d 2π2 ggd Y∞ σs3 On the other hand, upper bounds for the total DM
Combining eq.(68) for Y∞ with eq.(73) we obtain for the self-interaction cross sections σT have been given by
product mTd comparing X-ray, optical and lensing observations of
1 the merging of galaxy clusters with N-body simulations
Z 3
√mT =1.47 keV NRMaxwell Boltzmann. in Markevitch et al. (2004); Randall et al. (2008); Brada
d
(cid:18)gd(cid:19) − (2008)(seealsoMiralda-Escud´e(2002);Hennawi & Ostriker
(74)
(2002); Arabadjis et al. (2002)):
Typicalwimpsareassumedtohavem=100GeVandT =
d
5GeV(Particle Data Group2009).SuchvalueforTdimplies σT <0.7 cm2 =3200GeV−3 .
gd 80 (Kolb & Turner 1990). Eq.(74) thus requires for m gr
such≃heavy wimps Z 1023 well above the upper bounds
∼ Since the annihilation cross-section must be smaller than
derived in sec. 2 (see Table I). Therefore, wimps in the 100
thetotal cross-section we can write thebound
GeV scale are strongly disfavoured.
σ <3200mGeV−3 . (81)
0
Wefindfromeqs.(13)and(67)thephase-spacedensity
for DM decoupling NR in LTE Usingtheupperboundeq.(78)formyieldstheupperbound
for σ
D=g 1325π√23 gd Y∞ (cid:18)Tmd(cid:19)23 =8.4418×10−2 ggd Y∞x(7−523). 0 σ0 < 3.32bZ23 GeV−2 . (82)
Weobtain using here thevaluefor Y∞ from eq.(68) This result leaves at least fiveorders of magnitudebetween
the lower bound [eq.(80)] and the upper bound for σ . The
D=0.631510−4 gd kmeV52 Td23 . (76) iDnMthnisonco-gnrtaevxitt.ational self-interaction is therefore neg0ligible
Insertingthisexpression for intothegeneral estimate for
theDM mass eq.(24) yields,D Exoticmodelswhereveryheavy( 10TeV)DMparti-
∼
clesareproducedverylateanddecouplenon-relativistically
1 2 1
ηZ 3 kpc 3 10km/s 3 were proposed introducing two new fine tuned param-
√mT 0.942 keV.
d ∼ (cid:18) gd (cid:19) (cid:18) L (cid:19) (cid:18) σs (cid:19) eters: (a) the lifetime of unstable particles (sneutrinos)
(77) that decay into DM (gravitinos) (b) the mass difference
As in eq.(74) but independently from it, we reach a result between the two particles which must be small enough
for √mT in the keV scale assuming the DM is a self- to led to non-relativistic DM (Cembranos et al. 2005;
d
gravitating gas in thermal equilibrium. Strigari et al. 2007). Itis stated in Cembranos et al. (2005)
10 H. J. de Vega, N. G. Sanchez
andStrigari et al.(2007)thatsuchmodelsmaydescribethe eq.(28) an extremely short λ today
fs
observedphase-spacedensity.ItisstatedinBorzumati et al.
λ (0) 3.5110−4 pc=72.4AU.
(2008)thatitisinherentlydifficulttofulfilallobservational fs ∼
constraints in such models. Iftheflybyanomaly wouldbeexplainedbyDM,akeV
scale DMmass is preferred (Adler2009).
Further evidence for the DM particle mass in the keV
scale follows by contrasting the observed value of the con-
5 CONCLUSIONS stant surface density of galaxies to the theoretical cal-
culation from the linearized Boltzmann-Vlasov equation
Our results are independent of the particle model that
(deVega & S´anchez 2009). Independent further evidence
will describe the dark matter. We consider both DM parti-
for the DM particle mass in the keV scale is given by
cles that decouple being NR and UR and both decoupling
Tikhonov et al. (2009). [See also Gilmore et. al. (2007)].
at LTE and out of LTE. Our analysis and results refer to
the mass of the dark matter particle and the number of In summary, our analysis shows that DM particles de-
ultrarelativistic effective degrees of freedom when the DM coupling UR in LTE have a mass m in the keV scale with
particlesdecoupled.Wedonotmakeassumptionsaboutthe g &150 as shown in sec. 3.1. That is, decoupling happens
d
natureoftheDMparticleandweonlyassumethatitsnon- atleastatthe100GeVscale.Thevaluesofmandg maybe
d
gravitational interactions can be neglected in the present smaller for DM decoupling UR out of LTE than for decou-
context (which is consistent with structure formation and plingURinLTE(seesec.3.2).ForDMparticlesdecoupling
observations). NR in LTE (T < m) we find that √mT is in the keV
d d
In case DM particles explain the formation of galactic range. This is consistent with the DM particle mass in the
centerblackholes,DMparticlesmustbefermionswithkeV- keVrange.
scale mass (Munyaneza & Biermann 2006). Notice that the present uncertainity by one order of
ThemassfortheDMparticleinthekeVrangeismuch magnitudeoftheobservedvaluesofthephase-spacedensity
largerthanthetemperatureduringtheMDera,hencedark ρs/σs3 only affects the DM particle mass through a power
matter is cold (CDM). 1/4ofthisuncertainityaccordingtoeqs.(19)-(20).Namely,
A possible CDM candidate in the keV scale is a bya factor 1014 1.8.
sterile neutrino (Dodelson & Widrow 1994; Shi& Fuller We find th≃at the free streaming wavelength (Jeans’
1999; Abazajian, Fuller & Patel 2001; Abazajian 2006; length) is independent of the nature of the DM particle
Munyaneza& Biermann2006;Kusenko2007)producedvia except for the Z factor characterizing the decrease of the
theirmixingandoscillation withanactiveneutrinospecies. phase-space density through self-gravity [sec. 1.1]. The val-
Other putative CDM candidates in the keV scale are the uesfoundfortheJeans’lengthandtheJeans’massformin
gravitino(Gorbunov et al.2008;Steffen2009),thelightneu- the keV scale are consistent with the observed small struc-
tralino (Profumo 2008) and the majoron (Lattanzi & Valle tureand with the masses of thegalaxies, respectively.
2007).
Actually,manymoreextensionsoftheStandardModel
ofParticlePhysicscanbeenvisagedtoincludeaDMparticle
ACKNOWLEDGMENTS
with mass in the keV scale and weakly enough coupled to
theStandard Model particles. We thank C. Alard, D. Boyanovsky, C. Frenk, G. Gilmore,
Lyman-α forest observations provide indirect lower B. Sadoulet, P. Salucci for fruitful discussions.
bounds on the masses of sterile neutrinos (Viel et al. 2005,
2007) while constraints from the diffuse X-ray background
yield upper bounds on the mass of a putative sterile neu- REFERENCES
trino DM particle (Dolgov & Hansen 2002; Watson et al.
C. E. Aalseth et al. Phys.Rev. Lett.101, 251301 (2008).
2006; Boyarsky et al. 2007; Riemer-Sorensen et al. 2006,
K. Abazajian, G. M. Fuller, M. Patel, Phys. Rev. D64,
2007; Loewenstein et al. 2009). All these recent constraints
023501 (2001).
are consistent with DM particle masses at the keVscale.
K. Abazajian, Phys. Rev.D73, 063506 (2006).
The DAMA/LIBRA collaboration has confirmed the
S. L. Adler,Phys. Rev.D79, 023505 (2009).
presence of a signal in the keV range (Bernabei et al.
Z. Ahmed et al., arXiv:0907.1438.
2008a). Whether this signal is due to DM particles in
J. S.Arabadjis et al. Ap.J. 572, 66 (2002).
the keV mass scale is still unclear (Bernabei et al. 2006,
R. Bernabei et al. Int.J. Mod. Phys. A21, 1445 (2006).
2008b; Pospelov et al. 2008). On the other hand, the
R. Bernabei et al. (DAMA/LIBRA coll.) Eur. Phys. J.
DAMA/LIBRAsignals cannot beexplained bya hypothet-
C56:333 (2008a).
icalWIMPparticlewithmass&O(1)GeVsincethiswould
R. Bernabei et al. Mod. Phys.Lett. A23:2125 (2008b).
be in conflict with previous WIMPS direct detection ex-
E. Bertschinger, ApJS, 58, 1 and 39 (1985).
periments (Aalseth et al. 2008; Fairbairn & Schwetz 2009;
J.Binney,S.Tremaine,GalacticDynamics,PrincetonUni-
Hooper et al. 2009; Savage et al. 2008; Ahmed et al. 2009).
versity Press, 1987.
Asdiscussedinsec.4,typicalwimpswithm=100GeV J R Bond, A S Szalay, M S Turner, Phys. Rev. Lett. 48,
and T =5 GeV (Particle Data Group 2009) would require 1636 (1982).
d
a huge Z 1023, well above the upperbounds displayed in J R Bond, A S Szalay, Astrophys.J. 274, 443 (1983).
∼
TableI.Hence,wimpscannotreproducetheobservedgalaxy G. B¨orner, The Early Universe, Springer, 2003.
properties. In addition, recall that Z 1023 produces from F. Borzumati et al., Phys.Rev.D77, 063514 (2008).
∼