Can Dark Matter explain the Braking Index of Neutron Stars?

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by  Chris Kouvaris

Can Dark Matter explain the Braking Index of Neutron Stars?
Chris Kouvaris1, M. A´ngeles P´erez-Garc´ıa2
1 CP3-Origins, University of Southern Denmark, Campusvej 55, Odense 5230, Denmark
2 Department of Fundamental Physics and IUFFyM,
University of Salamanca, Plaza de la Merced s/n 37008 Salamanca, Spain
(Dated: January 16, 2014)
We explore a new mechanism of slowing down the rotation of neutron stars via accretion of
millicharged dark matter. We find that this mechanism yields pulsar braking indices that can be
4
substantiallysmallerthanthestandardn∼3ofthemagneticdipoleradiationmodelformillicharged
1
darkmatterparticlesthatarenotexcludedbyexistingexperimentalconstraintsthusaccommodating
0
existing observations.
2
Preprint: CP3-Origins-2014-002 DNRF90 & DIAS-2014-2.
n
a
PACSnumbers: 95.35.+d95.30.Cq
J

5
1 Pulsar spin down has been studied since its discovery I = πR2ρ c is the current quoted in the pioneering
GJ C GJ
but yet to date, there is not a definite clear understand- work of Goldreich and Julian [7], representing the emis-
]
R ing for the handful of existing data values for rotation- sion of relativistic charged particles with charge density
S poweredpulsars[1]. Thebrakingindexnasdefinedfrom ρGJ, due to a large difference in the electrostatic poten-
h. aspindownlawΩ˙ ∼Ωn,whereΩistherotationalangu- tial, from the NS capregions of a surface ∼πRC2. ρGJ is
lar velocity, is consistently smaller than the n = 3 value the electron density which shields the electric field and
p
predicted for energy loss via pure dipole radiation. In yields the only stable static solution for a pulsar magne-
-
o orderto explainthis discrepancyothermechanisms have tosphere(excludingcentrifugalandgravitationalforces).
r beeninvokedincludingrelativisticparticleflow,andincli- At the poles of the NS, it was estimated to be [7]
t
s nation angles or reconnection in the magnetosphere (see
a
[ e.g. the discussionin[2]and[3]). Apartfromshortlived ρ 7 10−2e B0 s cm−3, (3)
glitches or eventual transitions to a more exotic quark GJ ≃ × (cid:18)1012G(cid:19)(cid:16)P(cid:17)
1
star[4],the rateofrotationofa neutronstar(NS) drops
v steadily as a function of time. The deceleration of the where P is the period of the pulsar.
4
rotation is due to torques that are applied on the star The field lines that do not close within the so-called
4
6 impeding its motion. There are basically two compo- light cylinder of the NS (defined as a cylinder of radius
3 nents for the torque: an “orthogonal” one, that maxi- Rlc = c/Ω), are the ones starting from the cap regions
. mizes when the magnetic moment of the NS is perpen- with an angle given by θ where θ arcsin(R/R )1/2.
1 0 0 ≃ lc
0 dicular to the rotationaxis of the star, and an “aligned” For a typical NS, RC = Rsinθ0 100m. Magnetic
≃
4 component, maximized when the rotation and the mag- field lines starting outside of the caps return back to the
1 netic axis of the star are aligned. In the “orthogonal” star and therefore particles trapped in these lines do not
: case,angularmomentumislostthroughemissionofmag- contribute to a net current. If we use the definition of
v
i netic dipole radiation. In the “aligned” case, the torque IGJ, RC, and θ0, we obtain [6],
X
is produced by the electric current created by escaping
r chargedparticles(mainlyelectronsandprotons)thatfol- B R3Ω2 B s 2
a I =I = 0 1.4 1030e 0 s−1.
low the open field lines of the NS magnetosphere. The GJ 2c ≃ × (cid:18)1012G(cid:19)(cid:16)P(cid:17)
rotational kinetic energy loss rate for both mechanisms (4)
present is [5] However as we shall argue, if dark matter (DM) is in
the form of millicharged particles (MC) (as for exam-
L=Lorthsin2θ+Laligncos2θ, (1) ple [8], [9]), accretion of DM onto the NS can lead to
an excess of electric charge that due to overall electric
where θ is the angle between the magnetic and the rota-
neutralitymustbeexpelledfromthecaps,providingthis
tionalaxis. IfwetakeM andRasthemassandradiusof
wayanextracomponentofcurrent. Ongenericgrounds,
the NS respectively, the two components can be written
letusassumethatthetotalcurrenthastwocomponents,
as
I = I +I , where I represents the extra current
GJ DM DM
B2Ω4R6 B ΩΩ R3I due to external accretion from the MCDM galactic dis-
Lorth = 04c3 , Lalign = 0 2cF2 , (2) tribution where the NS may be located [10].
At this point let us recall that the rotational kinetic
where B is the magnetic field strength on the polar energy of a nearly sphericalNS (upon assuming uniform
0
caps of the star, and Ω =Ω Ω (Ω being the density for most of the star in the core central region)
F death death
angular velocity below which−the pulsar emission dies). is E 1 Ω2 where at lowest order = 2MR2 is the
≃ 2I I 5
The current I is approximately equal to I [6], where momentofinertiaofthestar[11]. The rotationalkinetic
GJ
2
energy loss rate is E˙ = ΩΩ˙ = 2MR2ΩΩ˙ = L. Using within the light cylinder given in [7] as
I 5 −
the above along with Eqs. (1) and (4), we obtain
1 R3
Ω˙ =−5B02Ω3R4 sin2θ+ 1− Ωdeath 1+ IDM cos2θ . Bp = 2Rlcr2B0χ, (8)
8Mc3 (cid:20) (cid:18) Ω (cid:19)(cid:18) IGJ(cid:19) (cid:21)
(5) where χ (1) is a coefficient. For the sake of the esti-
Asitcanbeseen, Ω˙ dependsonB0. Infacttherateof mate and∼inOwhat follows we will consider the equatorial
decelerationofthe rotationis usuallyusedtodeduce the latitude, γ 0, as an upper limit on the MCDM cap-
≃
magnitude of the magnetic field. Instead, the braking ture rate. The condition for capture is r << R .
L curv
index n = Ω¨Ω/Ω˙2, is independent of the magnetic field Using the explicit value of the magnetic poloidal field
strength. Using the definition of n and Eq. (5) we find strength of Eq. (8), we find that this is satisfied as long
as r << r/2. r depends on the velocity of the MC
L L
2 1 Ωdeath λ Ωdeath (1+λ) particle. Far away from the NS, assuming a thermal
n=3 − Ω − Ω cos2θ, (6)
− sin(cid:0)2θ+ 1 Ω(cid:1)death (1+λ)cos2θ DM distribution as e.g. in the solar neighbourhood, the
− Ω particle’s velocity is v 230km/s, but as the parti-
(cid:0) (cid:1) 0 ∼
cle approaches the star, it is gravitationally boosted to
where λ= IDM. If we assume that the pulsar is younger
IGJ v = v2+2GM/r. Insertingthisexpressionandrequir-
than 10 to 100 million years, we can safely assume 0
Ωdeath <<1. In that case n takes the simple form ing rpL <<r/2 leads to the condition
Ω
GM
n=3 2λcos2θ . (7) r <<r0 = v2 ( 1+∆2−1), (9)
− 1+λcos2θ 0 p
where∆=ǫev ΩR3B χ/(4GMm),M beingthemassof
As we can see, for λ = 0 (no extra current), we recover 0 0
the star.
n=3foranyangleθ,whichisthewellknownprediction
The above condition means that for r <<r one can-
for the aligned rotatoras well as the magnetic dipole ra- 0
not ignore the helical motion trapping effect in the mag-
diation. λ can take positive (negative) values depending
netosphere. One should emphasize that satisfying this
onthe relativechargesignofthe extraI currentwith
DM relation does not necessarily mean that all particles will
respecttoI . However,onecanseethatfor λ (1),
GJ | |∼O follow the magnetic lines and hit the star. Whether a
the braking index can be either smaller(λ>0)or larger
particle can be trapped by following the magnetic lines
(λ < 0) than 3. There are not many stars where the
ornotdepends alsoonthe pitchanglebetweenparticle’s
braking index has been estimated [2], seeming to range
velocityandthemagneticfield. However,ifthecondition
from n 1.4 to 2.91. If I has the same sign with I
∼ DM GJ is satisfied, a substantial fraction of the particles enter-
anditisroughlyofthesameorderofmagnitude,braking
ingtheareawilldefinitelybetrapped. Theexactfraction
indices n 2 or less are possible. To examine this possi-
∼ will depend on the detailed kinematics in the accretion
bility we need to find if indeed accreted DM in the form
process arising from the DM distribution.
of MC particles can accommodate such a current.
A rough estimate of the accreted number of particles
The capture rate of MCDM strongly depends on the
can be obtained by simply finding the flux of particles
the mass m and electric charge (q = ǫe, in units of the
entering through the surface that defines the minimum
electron charge e) of the particle constituent. For suf-
volumebetween the lightcylinder anda sphere ofradius
ficiently small electric charge and/or large mass for the
r . This can be evaluated approximately by considering
0
particle, the flux of DM particles crossing the surface
the light cylinder as a sphere with radius R = c/Ω.
lc
of the star is determined by the gravitational attraction
In this case, the smaller between r and R defines the
0 lc
of the star. However, if the electric charge is sufficiently
sphere which if millicharged particles cross its surface, a
large,thenanenhancednumberofMCDMparticlesmay
significant fraction of them will be captured.
follow the magnetic lines (which are electrically equipo-
Let us calculate the flux of particles entering a sphere
tential) in the magnetosphere within the light cylinder
of radius R∗. Assuming a typical Maxwell-Boltzmann
andcrossthesurfaceofthestar. Ongeneralgrounds,we
distribution for the velocities of DM particles, the flux
will show that the latter case may lead to a much higher
of particles crossing the surface with velocity between v
accretion rate.
and v+dv and with angle with respect to the normal φ
In this case, if the Larmor radius r = mv⊥, is much
L ǫeB and φ+dφ is
smaller than the curvature radius of the star’s magnetic
filcorewoldsssiidnnegfianthheedelicsaatsalrRfaacsunhrdivo(ntrh,tuγhs)eg=meta(cid:12)(cid:12)tgBi1nnegddtBritc(cid:12)(cid:12)r−afi1pe,pldtehdlie.nepOaenrveteincstlheuoafuolllldy- dF =n0(cid:18)2π3v02(cid:19)3/2πv3exp(cid:18)−23vv022(cid:19)dcos2φdv, (10)
keep in mind that R (r,γ) is a function of both the
curv where n is the number density of DM particles, in the
0
radial distance r and the latitude γ = π2 −θ. v⊥ is par- neighborhoodofthe star. Thetotalflux(number ofpar-
ticle’s velocity perpendicular to the B field line. The ticles per time) crossing the surface of a sphere with
B field strength corresponds to the poloidal component radius r can be written in terms of two new variables:
3
(non-relativistic) energy per mass E = 1v2 and angular where N is the number of MC particles trapped in the
2
momentum per mass J =vrsinφ as star, and α is the fine structure constant. For example,
for a proton with mass m , once N &1.46 1021ǫ−1m
2 3 3/2 −3v2 2 2 (m measured in GeV), pthe Coulomb forc×e is alreadyp
dFtotal =4πr dF =n0(cid:18)2πv02(cid:19) exp(cid:18) 2v02 (cid:19)4π dEdJ . largper than the gravitational force, leading to expulsion
(11) of charge from the star. For electrons, this will happen
The perihelion of the orbit of a particle around a mass
roughly three orders of magnitude lower than the pro-
M in Newtonian dynamics given in terms of E and J
tons. Once equilibrium between accretion and expulsion
is r = a/(1+√1+2ab) where a = J2/GM and b =
per is established, then the electric charge expelled by the
E/GM. Wedemandtheperiheliontobesmallerthanthe
star should balance that of the accretion.
radius of the sphere under consideration i.e. rper < R∗. Some comments are in order here. One might naively
From this we determine the range of J2 < 2GMR∗(1+ expect that the accumulating electric charge onto the
ER/GM). Integrating Eq. (11) first over J2 from zero
NS could stop the further accretion of DM particles due
to the maximum value, and then over E from zero to
to Coulomb repulsion. However, one can trivially check
infinity, we get
that in order for the repulsive Coulomb force to com-
pete with the gravitational force in the case of a MC
Ftotal =n08π2 3 3/2GMR∗ 1+ v02 R∗ . (12) particle, the corresponding Eq. (16) should now give
3v0 (cid:18)2π(cid:19) (cid:18) 3 GM(cid:19) N & 1.4 1021ǫ−2m, where m is the mass of the MC
×
particle in GeV. Electrons or protons (depending on the
Let us discuss the differentconditions arising inthe cap-
chargeofDM)aregettingexpelledbythestarbeforeac-
ture process. In the case where r >> R (case I), the
0 cretion can be stopped as long as ǫ(m /m)<<1, which
total flux of particles captured is given approximately p
is satisfied for all cases of our interest.
by the above equation with R∗ min(r0,Rlc). For a
rotation period P = 1 s (P = 0≃.1 s) we found that if Under these conditions, once equilibrium between ac-
ǫ & 10−5.5m (ǫ & 10−7m) with m being measured in cretion and expulsion of electric charge has been estab-
GeV, R is alwaysthe minimum and the flux for a typi- lished, the expelled charge will be in the form of either
lc
cal NS of M =1.4M⊙ and R=10 km is electrons or protons (depending on the charge of the ac-
cretedMC particles)becauseas longasǫ(m /m)<<1
p,e
FI ≃1.0×1029(cid:16)0.3GρeDVMcm−3(cid:17)(cid:18)1GmeV(cid:19)(cid:18)Ps(cid:19) s−1. (13) itsrosnat)i,sfiperodt(omnsp,oerbeeliencgtrtohnesmaarseseoafstiehretporoetxopneolrththaeneMlecC-
particles.
If r <<R (case II) one can completely ignore the mag-
0
Finally we should comment on our approximation on
netic field and the accretion proceeds as in the case of
the addition of I and I components in order to
a neutral particle. The capture rate of such particles is GJ DM
get the total current. In principle, one has to solve the
[12]
problem of the magnetosphere of the neutron star from
F = 8π2ρDM 3 3/2 GMR v2(1 e−3E0/v2)f, scclersatE~ch.B~In=t0h.e Ianbsaedndcietioonf,atchcereptiootnenotfiaclhoafrgthede cpaaprtiis-
II 3 m (cid:18)2πv2(cid:19) 1 2GM − ·
− R also determined by demanding overall neutrality on the
(14)
emittingparticlesfromthesurfaceofthestartothemag-
where E is the maximum kinetic energy per mass that
0 netosphere. In our case, the presence of extra charge on
can lead to capture and f is a capture efficiency factor the surface of the NS will make E~ B~ =0 and definitely
estimated in [13]. The flux for a typical NS is then · 6
there is not an overall neutrality since it must expel the
excessive charge accumulated by the accreted DM. De-
ρ 1GeV
F =4.2 1026 DM f s−1. (15) spite all this, it is a good first approximation to add up
II × (cid:16)0.3GeVcm−3(cid:17)(cid:18) m (cid:19) thecurrents. FirstofalleveninnonaccretingNSs, E~ B~
·
is never zero in reality. In addition one can compare
TheaccretionofMCDMparticlesontotheNSwillleadat
the total charge on the surface of the star to the charge
somepointtoanequilibriumbetweenincomingandout-
we estimated earlierneeded to startexpelling protons or
going electric charge. If particles with electric charge ǫe
electrons from the caps. The charge on the surface of a
are accumulatedin the NS with a rate givenin Eq. (13),
NSis(2/3)BΩR3e2/α 3 1029e(foratypicalB =1012
and due to the fact that the NS is an excellent electric ∼ ×
G and P = 1 s). This number is much larger than the
conductor, the charge will be distributed at the surface.
1.4 1021ǫ−2m we derived earlier for expelling charges
It is easy to see that for an electron or proton even if we ×
from the star. Therefore, the extra charge (coming from
ignorethe factthatthere is amagnetosphere,andthere-
DM)shouldbeasmallperturbationtothesolutionofthe
fore an electric potential on the surface of the star, the
magnetospherewithoutMC particles. However,andthis
gravitational force becomes smaller than the Coulomb
is a crucial point, the current I can be a substantial
repulsive force once DM
fraction of I . It is expected that the excessive charge
GJ
GMm Nǫα~c (positive or negative) will follow exactly the pattern of
< , (16) expulsion from the cap of the star, thus justifying the
R R
4
addition of the two components. thepulsar,dP/dt,andbyapplyingthemainmechanisms
In order to evaluate the two cases i.e. electromagnetic of angular momentum loss. However, for B < 1012 G,
accretion(caseI)andgravitationalaccretion(caseII)we Eqs.(17), (18) are easier satisfied and dP/dt can still be
calculate the conditions such that λ = IDM 1. Using theobservedonedespitethemagneticfieldbeingsmaller.
IGJ ∼
I ǫeF for case I we obtain the condition, The second remark has to do with the sign of accreted
DM I
≃
charge and the change in the braking index of the NS.
ǫ GeV ≃14 0.3GeVcm−3 s 3 B0 , (17) If Ω B > 0 at the poles, accretion of negative (posi-
(cid:18) m (cid:19) (cid:18) ρDM (cid:19)(cid:16)P(cid:17) (cid:18)1012G(cid:19) tive)·MCDM will reduce (increase) the braking index of
the star wth respect to the value three of the “aligned
while for case II with I ǫeF we get
DM II
≃ rotator”. Obviously it will be the opposite in case the
ǫ GeV ≃ 3.3×103 0.3GeVcm−3 s 2 B0 . magnetic field and the rotation axis are antiparallel.
(cid:18) m (cid:19) f (cid:18) ρDM (cid:19)(cid:16)P(cid:17) (cid:18)1012G(cid:19) So far wediscussedthe accretionofDM particles with
(18) definite charge into NSs. If DM consists of MC particles
At this point one should note that MCDM is already and it is overall electrically neutral, there are two pos-
constrainedby experimentaldata[14–22]. Thesebounds sibilities: i) MC particles appear as positive or negative
have been obtained by a variety of different methods: equal charge (as an absolute value) and mass. In this
accelerator and collider experiments, invisible decay of case the scenario we present in this paper is not valid
orth-positronium, MC particle searches, Lamb shift, Big simply because the net accumulated charge is zero. In
Bang Nucleosynthesis, plasmon decay in white dwarfs, principle fluctuations in the accretion can induce a net
dark matter searches and Supernova 1987A. All the current, but we found that unless the star is immersed
above constrain the ǫ m phase space of MC particles. in a DM environment of extremely high density, there
−
The reader can see Fig. 1 of [20] for a comprehensive is no effect. ii) DM is overall electrically neutral but it
explanation of the different bounds. In addition there consists of two components of positive and negative MC
arecosmologicalboundsbasedonCMB,constrainingthe particleswithdifferentchargeand/ormassinsuchaway
MC particle fraction with some underlying theoretical that accretion onto the star is different for the positive
model dependence [21, 24]. It is worth mentioning that and the negative MC particles. This is not difficult to
additional forms of charged DM under the strangelet, achieve since, as we pointed out, accretion rates depend
Q-balls and other phenomenology may also arise in NS strongly on the ǫ/m ratio for the particle species. If for
catastrophic events [25] although at an expected much examplethenegativeMCparticlesareaccumulatedonto
lower flux. the star via type I accretion (Eq. (13)) and the positive
One can check that MCDM with mass m 1 MeV ones via type II (Eq. (15)), the NS will dominantly ac-
(1 GeV), ǫ 10−4 (0.1) and with a density∼ρ of a crete the negative component and therefore the braking
DM
few GeV/c∼m3 satisfies Eq. (17) while the same time index will become smaller than 3 (if Ω B > 0 at the
·
does not violate experimental bounds. It has been sug- poles). In addition, even within the same regime (I or
gested [26] that magnetic galactic fields can prevent MC II), accretionrates canstill be differentbetween positive
particles from the halo to enter the galactic disk and and negative components. For example by inspection of
thoseinitially trappedinthe disccanbeacceleratedand Eq. (14), one can see that different particle masses can
injected out of the disc within 0.1 to 1 billion years, if lead to different accretion rates.
MC particles fall within 100ǫ2 . m . 108ǫ TeV. How- In this work we present the possibility that accretion
ever,evenifthisistrue,thetwoexampleswegaveabove ofatwo-componentmillichargeddarkmatterontoaneu-
(i.e. the 1 MeV and the 1 GeV particles) lie exactly at tronstarevenatrelativelylowdarkmatterdensitiescan
the left margin of the inequality and therefore it is not significantly change the braking index of a pulsar. For
clear if the constraint is applicaple. In addition, there MC particles that are still evasive of experimental con-
is a large number of NS lying out of the galactic plane straints, accretion onto neutron stars can lead to expul-
where the above constraint is irrelevant. sion of extra electric charge from the poles of the star,
There are two remarks we would like to make here. which consequently can impede further the rotation of
The first one is that P > 1 s, DM densities larger than thestaryieldingbrakingindicesconsistentwiththe ones
1 GeV/cm3 and/or smaller values of B (i.e. B <1012 observed.
∼
G) allow more phase space in terms of ǫ and m in order CK would like to thank I. Contopoulos for very useful
to satisfy Eqs. (17), (18). Pulsars can potentially have discussions. C.K. is supported by the Danish National
periods up to a few seconds and DM densities close to Research Foundation, Grant No. DNRF90. M.A.P.G.
the centerofthe Galaxycanbe substantiallylargerthan wouldliketothankthe newCOMPSTAR,andtheMUL-
that around the Earth. As for the magnetic field, we TIDARK projects, as well as CP3-Origins where part of
should emphasize that usually in a NS its value is esti- this work was developed. M.A.P.G. is supported by the
matedindirectlybyobservingthechangeintheperiodof Spanish MICINN project FIS2012-30926.
5
[1] M.A.Livingstoneetal.,AstrophysicsandSpaceSciences [15] S. Dimopoulos, D. Eichler, R. Esmailzadeh and
308 (2007) 317. G. D.Starkman, Phys.Rev.D 41, 2388 (1990).
[2] Y.L.Yue,R.X.Xu,W.W.Zhu,Adv.inSpaceResearch [16] S. Davidson and M. E. Peskin, Phys. Rev. D 49, 2114
40 (2007) 1491. (1994) [hep-ph/9310288].
[3] I.Contopoulos, [astro-ph/0610156]. [17] R.N.MohapatraandI.Z.Rothstein,Phys.Lett.B247,
[4] N. K. Glendenning, N.K., Compact stars, Ed. Springer- 593 (1990).
Verlag, New York ( 2000) [18] R. N.Mohapatra and S. Nussinov,Int.J. Mod. Phys. A
[5] A.K.Harding,I.Contopoulos,andD.Kazanas,ApJ525 7, 3817 (1992).
L125 (1999) [19] A.A. Prinz et al., Phys.Rev.Lett. 81 (1998) 1175.
[6] I. Contopoulos and A. Spitkovsky, ApJ, 643, 1139 [20] S. Davidson, S. Hannestad and G. Raffelt, JHEP 0005,
(2006). 003 (2000) [hep-ph/0001179].
[7] P.Goldreich and W. H. Julian, ApJ 157, 869 (1969). [21] S.L.Dubovsky,D.S.GorbunovandG.I.Rubtsov,JETP
[8] H. Goldberg and L. J. Hall, Phys. Lett. B 174 (1986) Lett. 79, 1 (2004) [Pisma Zh. Eksp. Teor. Fiz. 79, 3
151. (2004)] [hep-ph/0311189].
[9] C. Kouvaris, Phys. Rev. D 88, 015001 (2013) [22] P. Langacker and G. Steigman, Phys. Rev. D 84 (2011)
[arXiv:1304.7476 [hep-ph]]. 065040.
[10] F. J. Sanchez-Salcedo, E. Martinez-Gomez and J. Ma- [23] C. Burrage, J. Jaeckel, J. Redondo and A. Ringwald,
gana, JCAP 1002, 031 (2010) [arXiv:1002.3145 [astro- JCAP0911,002(2009)[arXiv:0909.0649[astro-ph.CO]].
ph.CO]]. [24] A.D. Dolgov et al., Phys.Rev.D 88(2013) 117701.
[11] J. M. Lattimer and B. F. Schutz,ApJ629 (2005) 979 . [25] M. A. Perez-Garcia, F. Daigne, and J. Silk, ApJ 768
[12] C. Kouvaris and P. Tinyakov, Phys. Rev. Lett. 107, (2013)145[arXiv:1303.2697[astro-ph.HE]];M.A.Perez-
(2011) 091301. Garcia, J. Silk and J. R. Stone, Phys. Rev. Lett. 105
[13] C. Kouvaris, Phys. Rev.D 77, (2008) 023006. (2010) 141101 [arXiv:1108.5206v1 [astro-ph.CO]].
[14] J.Jaeckel,M.JankowiakandM.Spannowsky,Phys.Dark [26] L. Chuzhoy and E. W. Kolb, JCAP 0907, 014 (2009)
Univ.2 (2013) 111. [arXiv:0809.0436 [astro-ph]]. [arXiv:1303.2697 [astro-
ph.HE]]