Dark matter self-annihilation signals from Galactic dwarf spheroidals

44 Pages·2014·0.32 MB·English

University of Amsterdam
Dark matter self-annihilation signals
from Galactic dwarf spheroidals;
intermediate mass black holes and
mini-spikes
Mark Wanders
Supervisor: Gianfranco Bertone
GRAPPA Institute, University of Amsterdam
June 17, 2014
Abstract.Theformationofanintermediatemassblackholeinevitablyaffectsthedistri-
bution of dark and baryonic matter in its vicinity. Under the assumption that the black
hole grows adiabatically, large overdensities in the halo’s dark matter density, called
spikes, can be achieved, resulting in signi(cid:12)cant ampli(cid:12)cations to the gamma-ray signal
originating from self-annihilation of WIMP dark matter. Galactic dwarf spheroidals in
particular are expected to contain and maintain mini-spikes, and we use (cid:12)ve years of
Fermi-LAT gamma-ray data to derive upper limits on the mass of an intermediate mass
black hole in the center of the subhalo. Our limits are more constraining than those
previously calculated using other methods.
Contents
1 Dark matter 2
1.1 Evidence 2
1.2 Candidates 5
1.3 Origin of WIMP dark matter and relic density 7
1.4 Density pro(cid:12)le 10
1.5 Gamma-rays from dark matter annihilation 10
2 Mini-spikes 14
2.1 Dwarf spheroidals 14
2.2 Intermediate mass black holes 15
2.2.1 Motivation 15
2.2.2 Formation scenarios 16
2.2.3 Dark matter mini-spikes 18
3 Analysis 23
3.1 Fermi Large Area Telescope 23
3.2 Data acquisition and preparation 23
3.3 Maximum likelihood analysis 24
3.4 Error propagation 26
4 Results 28
4.1 Flux and cross-section constraints 28
4.2 Black hole mass upper limit 29
4.3 Age of the black hole 30
4.4 Other dwarf spheroidals 30
5 Discussion and conclusion 33
5.1 Persistence of the mini-spike 33
5.2 Other limits on black hole mass 34
5.3 Conclusion 35
{ 1 {
1 Dark matter
Current astrophysical evidence suggests non-baryonic dark matter accounts for roughly
27% of the energy content of the present-day universe [1]. The nature of dark matter
(DM) beyond its gravitational in(cid:13)uence is largely a mystery, but a weakly interacting
massive particle (WIMP) is a popular candidate [2]. A principle example of such a
particle is the lightest neutralino, a stable mixture of supersymmetric partners of the Z
boson, photon and Higgs boson, as predicted in the Minimal Supersymmetric Standard
Model (MSSM). Since constraining the properties of such a weakly-interacting particle
through direct detection (i.e., looking for nuclear recoils from scattering of dark matter
particles off nucleons) appears difficult, indirect detection (i.e. looking for gamma-rays
resulting from dark matter self-annihilation in astrophysical objects) could prove to be
a useful alternative.
Nowadays, evidence for the existence of stellar mass black holes and supermassive black
holes (SMBHs, M(cid:15) (cid:24) 106 (cid:0) 108M⊙), is compelling. Recently, evidence has emerged
suggesting the existence of intermediate mass black holes (IMBHs), which could have
masses from 102M⊙ (cid:0)106M⊙, bridging the mass gap between stellar mass black holes
and SMBHs. The formation of a massive black hole inevitably affects the distribution
of (both baryonic and non-baryonic) matter around it, and can lead to large overdensi-
ties called spikes. As the DM annihilation rate scales with density squared, this could
improve prospects for indirect detection. However, mergers, off-center formation of the
seed BH and gravitational scattering off of stars are likely to disrupt and reduce such
spikes [3]. These overdensities could persist in a milder manifestation, called a mini-
spike, in the case of an IMBH, as they are unlikely to undergo the processes that disrupt
the overdensity. This is especially true in the case of an IMBH in the center of a dwarf
spheroidal (dSph); objects orbiting the Milky Way with low central stellar densities and
a relatively high DM content.
In this work, we shall examine the consequences of mini-spikes in the dark matter dis-
tribution caused by the adiabatic growth of IMBHs on indirect detection of WIMP dark
matter using data from the Fermi-LAT space telescope. This work is organized as fol-
lows: in section 1 we will give an introduction to particle dark matter, the motivation
andevidence1.1, possiblecandidates1.2, WIMPrelicdensity1.3anditsdistribution1.4
before moving on to the theory of mini-spikes, including dwarf spheroidals 2.1 and in-
termediate mass black holes 2.2, evidence for their existence 2.2.1, possible formation
scenarios 2.2.2 and mini-spikes themselves 2.2.3, (cid:12)nishing with a description of our data
analysis 3.1{3.3. In section 4 we will cover our results, before moving on to the discus-
sion 5 and ending with our conclusions 5.3.
1.1 Evidence
Galactic rotational velocity curves
The(cid:12)rstsignsofoverlookedmassintheuniversecamefromobservationsofstarsandgas
{ 2 {
ingalaxiesandcomparingtheirobservedvelocitiestothoseexpectedfrommovinginthe
gravitational potential of the visible, luminous matter. Speci(cid:12)cally, the rotation curve
of a galaxy is usually obtained through observations of the 21cm line, i.e. radio-emission
fromneutralatomichydrogen(HI)intheinterstellargas, andshowsthecircularvelocity
of stars and gas as a function of their distance to the center of the galaxy. If we consider
simple Newtonian dynamics, we would expect the circular velocity at a radius r to be
√
GM(r)
v(r) = ; (1.1)
r
∫
where the enclosed mass is M(r) = 4(cid:25)r2(cid:26)(r)dr and (cid:26)(r) is the density pro(cid:12)le. We
p
would then, from observed stellar densities, naively expect v(r) / 1= r, i.e. Keplerian
behavior. However, observations revealed that the velocity curve remained (cid:13)at up to
large radii (see e.g. [4]). From equation 1.1 we see that a (cid:13)at rotation curve requires
M(r) / r, or(cid:26) / r(cid:0)2. Toexplainthe(cid:13)atbehavioroftherotationalvelocitycurveoutto
largeradii,theconceptofthedarkmatterhalowasintroduced; alarge,roughlyspherical
halo of non-luminous matter extending far beyond the visible size of the galactic disk.
Gravitational lensing
Another way to probe the amount of dark matter in a galaxy, given to us by Einstein’s
Theory of General Relativity, is through gravitational lensing. General Relativity pos-
tulates that space-time can be bent and stretched by massive objects, which in turn
affects the motion of bodies, thus causing what we perceive as gravity. This space-time
curvature not only affects massive bodies such as stars, which now follow geodesics in-
stead of simple straight lines, the path of light traveling through curved space-time is
bent as well; massive objects effectively act as a lens. In the case where a very massive
object (such as a cluster of galaxies) is directly in front of a very bright object (such
as a galaxy), one would see an \Einstein ring"; the closer, central object surrounded
by a distorted ring-shaped image of the more distant object. However, such a perfect
alignment is very rare, and so we generally observe only partial Einstein rings, called
arclets. The Einstein radius (cid:18) , the radius of such an arclet in radians, gives us a way
E
to estimate the mass within a lensing cluster [5]:
√
4GM d
LS
(cid:18) = ; (1.2)
E c2 d d
L S
where M is the mass of the lens and d , d and d are the (angular-diameter) distance
LS L S
between the lens and the source, distance to the lens and distance to the source, respec-
tively. Deriving the mass of the cluster this way and comparing it to the luminous mass
again con(cid:12)rms that a large portion of the mass must be non-luminous. For example, an
analysis of the cluster Abell 370 resulted in a M=L ratio of 102{103M⊙=L⊙ [6].
Gravitational lensing has another use in the search for dark matter, through mi-
crolensing, i.e. the change in brightness of a distant object due to the lensing effect of
the passing of a massive object in front of it. The (cid:12)rst attempt to explain the miss-
ing mass in galaxies was to turn to massive objects already known to exist in galaxies
and comprised of normal, baryonic matter. These objects would need to be \dark",
{ 3 {
and thus possible candidates are brown dwarfs, neutron stars, black holes and unassoci-
ated planets. Together, these objects are referred to as Massive Compact Halo Objects
(MACHOs).
Searches for these microlensing events have been conducted by several collabora-
tions, e.g. the MACHO Collaboration, the EROS-2 Survey, and their results are conclu-
sive. The MACHO Collaboration analyzed 5.7 years of photometry on 11.9 million stars
in the Large Magellanic Cloud, (cid:12)nding only 13 { 17 possible microlensing events from
MACHOs in the Milky Way halo [7]. Other collaborations found similar or even smaller
numbers, leading to the conclusion that MACHOs can not account for a signi(cid:12)cant part
of the mass in DM.
Cosmic microwave background anisotropies
StrongevidencethatDMexistsinsteadintheformofparticlescomestousfromcosmol-
ogy. During the (cid:12)rst seconds to minutes after the Big Bang, free neutrons and protons
fused together to form light elements such as deuterium, helium and trace amounts of
lithium and several others. As it turns out, this Big Bang Nucleosynthesis (BBN) is in
fact the primary source of deuterium in the universe, as deuterium produced by stars
is almost immediately lost in fusion to produce 4He. By treating the current amount
of deuterium in the universe as a lower limit on the amount produced during BBN, it
is possible to estimate the primordial deuterium to hydrogen ratio by determining the
D=H abundance of regions with low levels of elements heavier than lithium. A major
success of the BBN model is that theoretical abundances match up very well with obser-
vational ranges, and one of the results of BBN modeling is that the D=H ratio depends
strongly on the baryon density. The baryon density is usually written as Ω h2, where Ω
b b
is the baryon density relative to the critical density (cid:26) and h = H=100 km s(cid:0)1 Mpc(cid:0)1
c
is the reduced Hubble constant. Depending on which deuterium measurement is used,
calculations results in two numbers for the baryon density[8]: Ω h2 = 0:0229(cid:6)0:00013
b
or Ω h2 = 0:0216+0:0020. For an alternative method of determining the baryon density,
b (cid:0)0:0021
we now turn to the cosmic microwave background (CMB).
TheCMBisa(nearly)isotropicbackgroundofphotonswithatemperatureofabout
2.73 K, (cid:12)rst discovered in the ’60s by Penzias and Wilson. The early universe was (cid:12)lled
with a hot, dense plasma of charged particles and photons that cooled as the universe
expanded. After 380,000 years or so, the universe had cooled sufficiently to allow for
recombination; protons and electrons formed the (cid:12)rst neutral atoms. Photons, which
previouslyhadbeenlockedinendlessscatteringwithprotonsandelectrons,nowtraveled
freelythroughthenewlytransparentuniverse. Thelightfromthislastscatteringpersists
and permeates the entire universe in the form of the CMB, now signi(cid:12)cantly redshifted
to a very low temperature due to the expansion of the universe.
Several space-based missions have been launched to map out the full-sky CMB,
startingwith COBE in1989, WMAP in2001 and mostrecentlyESA’s Planck, launched
in May 2009. As it turns out, the CMB (after subtraction of foregrounds and the
dipole caused by the movement of the telescope with respect to the CMB frame) is
not entirely uniform; its temperature (cid:13)uctuates on the order of one part per million.
These tiny anisotropies can be attributed to two primary causes. Firstly, large scale
(cid:13)uctuations are caused by the Sachs-Wolfe effect; photons lose energy traveling out of a
{ 4 {
gravity well, and so we observe more low energy photons from parts of the universe that
were more dense at last scattering. Secondly and for our argument, more importantly,
small scale (cid:13)uctuations are attributed to something called acoustic oscillations. These
oscillations emerge as the photon-baryon (cid:13)uid (cid:12)lling the early universe goes through
repeated contractions and expansions, as it compresses under gravity and expands due
to radiation pressure. This creates acoustic waves, whose propagation is abruptly cut
off when recombination severely decreases the speed of sound in the universe.
Clearly, these (cid:13)uctuations are dependent on both the baryonic mass (as it is the
interaction between baryons and photons that causes the expansion) and the total mass
(thegravityofwhichcausesthecontraction). Oneofthemostinterestingresultsofstud-
iesoftheCMBisthatthesenumbersarenotthesame. Infact,theydiffersigni(cid:12)cantly[1]:
Ω h2 = 0:14300(cid:6)0:0029; Ω h2 = 0:02207(cid:6)0:00033; (1.3)
m b
leadingtotheconclusionthatover85%oftheuniverse’smass-energyexistsintheformof
matter that interacts gravitationally but not electromagnetically; particle dark matter.
1.2 Candidates
Standard model and sterile neutrinos
It is prudent to begin our search for candidates of particle dark matter amongst those
particles we already know to exist, in other words, the Standard Model (SM). The
standard model of particle physics is the quantum (cid:12)eld theory that describes three out
of the four fundamental forces; electromagnetism, the weak force and the strong force,
and their interactions with the constituents of matter, fermions, through force carriers
called bosons. The model contains 17 con(cid:12)rmed particles, of which 8 were predicted
by the model before being con(cid:12)rmed experimentally, the (cid:12)nal one being the Higgs,
discovered in July 2012, which completed the model. The fermions are divided into 6
quarks; up (with symbol u), down (d), top (t), bottom (b), charm (c) and strange (s),
and 6 leptons; the electron (e), muon ((cid:22)) and tau lepton ((cid:28)) and their corresponding
neutrinos ((cid:23) ). Unlike leptons, quarks are not found individually in nature, instead they
l
are the building blocks for hadrons, which each contain 3 quarks (like the proton; uud),
and mesons, each containing 2 quarks (e.g. the pion; ud(cid:22)).
The (cid:12)ve bosons of the SM are the photon ((cid:13)), responsible for electromagnetism,
(cid:6)
the gluons (g), force carriers of the strong force, W and Z bosons, mediating the
weak force and the Higgs (H), which gives mass to the other SM particles through the
Higgs (cid:12)eld. Particles in the SM are de(cid:12)ned by their quantum numbers (i.e. charge,
spin etc.), for instance, all fermions have half-integer spin, whereas bosons have integer
spin. Additionally, each particle has an anti-particle, with the same mass but opposite
quantum numbers, usually denoted with a bar (e.g. a quark q and an anti-quark q).
When a particle collides with its anti-particle they annihilate, converting their mass-
energy into photons (favored in the low-energy limit) or other particles (favored in the
high-energy limit). Some particles, such as the photon and the Z boson, are their own
anti-particle, thus they self-annihilate. Interactions of SM particles obey the standard
laws of conservation of energy and momentum as well as internal gauge symmetries, i.e.
conservation of the respective quantum numbers.
{ 5 {
Charge Spin 1=2 Spin 0 Charge
2=3 u c t g 0
Quarks
1=3 d s b (cid:13) 0 B
o
(cid:0)1 e (cid:22) (cid:28) Z 0 s
o
Leptons n
0 (cid:23) (cid:23) (cid:23) W (cid:6)1 s
e (cid:22) (cid:28)
Table 1: Particles of the standard model.
The only particle in the SM that, at (cid:12)rst glance, ful(cid:12)lls all the criteria for a DM particle
(electrically neutral, stable, massive and weakly-interacting) is the neutrino. However,
neutrinos are a form of \hot" dark matter; meaning their velocities are relativistic,
causing them to erase density (cid:13)uctuations below the scale of their free-streaming length,
(cid:24) 40Mpc. This implies structure formation in the early universe followed a top-down
process, whereby large-scale structures form before small-scale ones, something which
seems excluded from the observations of galaxies, considered small-scale in the context
of the structure of the universe, already having formed at high z.
More convincing, perhaps, are the constraints on neutrinos as DM coming from
their expected relic abundance. Considering that there are three neutrino eigenstates,
their total relic density will be [2]:
∑3
m
Ω h2 = i : (1.4)
(cid:23)
93 eV
i=1
Given the upper limits on total neutrino mass arising from cosmology [1], we (cid:12)nd an
upper limit on the neutrino relic density of
Ω h2 (cid:20) 0:002; (1.5)
(cid:23)
showing us that, while neutrinos constitute a small part of the DM in the universe, they
can not comprise the majority of it.
As the Standard Model does not contain a proper candidate, we will now examine
extensions of the SM. The sterile neutrino is a hypothetical, right-chiral counterpart of
the left-chiral SM neutrino and receives its name from the fact that it does not interact
through the electromagnetic, the strong or the weak interaction. The neutrino sector is
a good candidate for physics beyond the SM, as, in theory, the SM neutrino is massless,
which con(cid:13)icts with observations that neutrinos in fact do have mass, and the intro-
duction of a sterile neutrino would provide an explanation for neutrino (cid:13)avor oscillation
anomalies. The sterile neutrino was (cid:12)rst considered as a DM candidate in 1993 [9] and,
depending on its mass, can function as either cold or warm dark matter. However,
current cosmological searches have as of yet found no evidence of sterile neutrinos [1].
Supersymmetry and WIMPs
Another possible extension of the SM is supersymmetry (SUSY). SUSY, in essence, pro-
poses an additional symmetry beyond the Lorentz invariance and gauge symmetries of
{ 6 {
the SM, namely a symmetry between fermions and bosons. In the SM, such a symmetry
is forbidden by the Coleman-Mandula theorem, which states that space-time and inter-
nal symmetries can not be combined in a non-trivial way [10]. This explicitly forbids
particlesfromchangingspin,i.e. fermionsturningintobosonsorvice-versa. Theallowed
symmetries in SM have generators belonging to Lie algebras, and SUSY circumvents the
restriction by introducing graded Lie algebras, whose operators anti-commute. This
new symmetry means that every boson in the SM now has a fermionic superpartner and
every fermion has a bosonic superpartner, effectively doubling the number of particles
in the model. This might seem like an unnecessary complication, but there are several
theoretical motivations for introducing SUSY, such as its role in solving the problem of
the low mass of the Higgs boson [2].
Speci(cid:12)cally, we will focus on one incarnation of SUSY, the Minimally Supersym-
metricStandardModel(MSSM),containingthesmallestpossible(cid:12)eldcontentnecessary
to reproduce the SM (cid:12)elds. This is done by introducing fermionic superpartners for all
gauge (cid:12)elds, called gauginos, introducing squarks and sleptons as bosonic superpartners
to the fermion sector and adding one additional Higgs (cid:12)eld with one Higgsino to each of
the now 5 Higgs bosons. Additionally, the MSSM assumes R-parity (where SM particles
have +1 R-parity and SUSY partners have -1 R parity) is conserved, one consequence
of which being that the lightest supersymmetric particle is cannot decay and is therefor
stable.
The MSSM in principle contains three candidates for WIMP dark matter; the sneu-
e
trino ((cid:23)~, superpartner of the neutrino), the gravitino (G, superpartner of the theoretical
graviton) and the neutralino ((cid:31), actually the lightest of four mass eigenstates of a mix of
the bino, wino and two neutral higgsinos) [5]. However, sneutrinos annihilate so rapidly
in the early universethat their relic densityis far too lowto be the WIMP[11], and grav-
itinos act like hot dark matter [12], something already ruled out (as discussed earlier).
We are then left with one WIMP candidate from MSSM; the lightest neutralino.
1.3 Origin of WIMP dark matter and relic density
The neutralino, and WIMPs in general, are expected to have masses at the weak scale,
m (cid:24) 100 GeV { 1 TeV. The (cid:3)-Cold Dark Matter ((cid:3)CDM) model predicts that, in
(cid:31)
the very early universe, the temperature of the universe was high enough that such very
massiveparticleswerecreatedand,throughprocesseslikepair-productionandcollisions,
existed in thermal equilibrium with SM particles:
(cid:31)+(cid:31) ⇋ SM+SM: (1.6)
However, the expansion and subsequent cooling of the universe contributed to the end
of this equilibrium in two ways; (cid:12)rstly, as the temperature decreased, lighter particles no
longerhadthekineticenergynecessarytoproducemassiveparticlesthroughinteractions,
and secondly, the expansion of the universe itself diluted the number density of the
particles, thus decreasing the interaction rate. When the interaction rate (cid:0) equals the
expansion rate i.e. the Hubble constant H, (cid:0) (cid:17) n⟨(cid:27)v⟩ = H (cid:17) a_=a, massive particles
will ‘freeze-out’ and their comoving number density remains constant. This freeze-out
density is called the relic density.
{ 7 {
WecanusetheBoltzmannequationtoderiveanapproximationto⟨(cid:27)v⟩,thethermal
average of the annihilation cross section (cid:27) times the relative velocity v of the particles,
whichinthefollowingwewillsimplycallthecrosssection. Forsupersymmetricparticles,
the Boltzmann equation will have four terms, due to: 1) the expansion of the universe,
2) coannihilation of two SUSY particles into a standard model particle, 3) particle decay
and4)scatteringoffthethermalbackground. WecanthuswritetheBoltzmannequation
for N SUSY particles as:
∑N ( ) ∑[ ( )]
dn
i =(cid:0)3Hn (cid:0) ⟨(cid:27) v ⟩ n n (cid:0)neqneq (cid:0) (cid:0) (n (cid:0)neq)(cid:0)(cid:0) n (cid:0)neq
dt i ij ij i j i j ij i i ji j j
j=1 j̸=i
∑N [ ( ) ( )]
(cid:0) ⟨(cid:27)′ v ⟩ n n (cid:0)neqneq (cid:0)⟨(cid:27)′ v ⟩ n n (cid:0)neqneq :
Xij ij i X i X Xji ji j X j X
j̸=i
(1.7)
We can simplify this equation by assuming R-parity (already an inherent feature of the
MSSM) and by assuming the decay rate of SUSY particles is very high compared to the
age of the universe, meaning that all SUSY particles present at the early universe have
since decayed into the lightest stable particle, the neutralino (i.e. n = n +:::+n ).
i N
This means the third and fourth terms in equation 1.7 cancel. If we now additionally
simplify the second term by excluding coannihilations between SUSY particles, we are
left with
( )
dn
= (cid:0)3Hn(cid:0)⟨(cid:27)v⟩ n2(cid:0)n2 : (1.8)
dt eq
In the non-relativistic limit of the Maxwell-Boltzmann distribution, we can express the
number density of particles in thermal equilibrium as:
( )
mT 3=2
n = g e(cid:0)m=T; (1.9)
eq
2(cid:25)
where T is the temperature and m the particle mass. If we now introduce the new
variable
n
Y (cid:17) ; (1.10)
s
where
2(cid:25)2g(cid:3)T3
s = (1.11)
45
is the entropy density of the universe (and g(cid:3) denotes the number of relativistic degrees
of freedom), we can, under the assumption of conservation of entropy per comoving
volume, write
dY dn nds dn
s = (cid:0) = +3Hn: (1.12)
dt dt s dt dt
Substituting this into equation 1.8, we get
( )
dY
s = (cid:0)s2⟨(cid:27)v⟩ Y2(cid:0)Y2 ; (1.13)
dt eq
{ 8 {
We will now de(cid:12)ne a new variable
m
x (cid:17) ; (1.14)
T
and, since T / a(cid:0)1, we can write dx = xH. Using this, we can rewrite equation 1.13 as:
dt
dY dt dY (cid:0)s⟨(cid:27)v⟩ ( )
= = Y2(cid:0)Y2 : (1.15)
dt dx dx Hx eq
For heavy states, ⟨(cid:27)v⟩ can be approximated with the non-relativistic expansion in v2 [2]:
( )
⟨(cid:27)v⟩ = a+b⟨v2⟩+O ⟨v4⟩ (cid:25) a+6b=x: (1.16)
Introducing a (cid:12)nal new variable ∆ = Y (cid:0)Y , leads to [2]
eq
( )
d∆ dY s
∆′ = = (cid:0) eq (cid:0)(a+6b=x) ∆ ∆+2Y2
dx dx √Hx eq
(1.17)
( )
= (cid:0)Y′ (cid:0)(a+6b=x) (cid:25)g(cid:3)mMPl∆ ∆+2Y2 ;
eq 45 x2 eq
where M is the Planck mass.
Pl
Wecansolvethisequationanalyticallyinthecaselongafterfreeze-out, x ≫ x (i.e.
F
the situation today), since in that case Y ≫ Yeq and thus√also ∆ = Y (cid:0)Yeq (cid:25) Y ≫ Yeq
andequation1.17canbeapproximatedas∆′ = (a+6b=x) (cid:25)g(cid:3)mMPl∆2. Ifweintegrate
45 x2
this last equation between xF and 1 and simplify using ∆1 ≪ ∆x we arrive at:
p f
Y(cid:0)1 = 1 (cid:25)g(cid:3)mMPl; (1.18)
1
(a+6b=x) 45 x
F
and, since the present density of a relic (cid:31) is given by (cid:26)(cid:31) = m(cid:31)n(cid:31) = m(cid:31)s0Y1, we can
write for the relic density [2]:
1:07(cid:2)109 GeV(cid:0)1 x 1
Ω h2 = (cid:26) =(cid:26) (cid:25) pF
(cid:31) (cid:31) c
MPl g(cid:3)a+3b=xF
(1.19)
3(cid:2)10(cid:0)27cm3s(cid:0)1
(cid:25)
⟨(cid:27)v⟩
where in the last step we employed an order of magnitude approximation [13].
Equation 1.19, with the current value for the relic density Ω h2 (cid:25) 0:1 [1] means
X
that ⟨(cid:27)v⟩ (cid:25) 3(cid:2)10(cid:0)26cm3s(cid:0)1. This result is a strong argument in favor for the WIMP as
a DM candidate. For a new particle with a weak-scale interaction, we can estimate its
annihilation cross-section to be [13] ⟨(cid:27)v⟩ (cid:24) (cid:11)2(100 GeV)(cid:0)2 (cid:24) 10(cid:0)25cm3s(cid:0)1, where the
(cid:12)ne-structure constant (cid:11) (cid:24) 10(cid:0)2. This is remarkably close to the cross-section derived
from the relic abundance of DM if comprised of neutralinos.
Itshouldbenotedthatinequation1.8,wehaveignoredtheeffectsofcoannihilations
between neutralinos and heavier SUSY particles. Including such reactions yields instead
∑N ( )
dn
= (cid:0)3Hn(cid:0) ⟨(cid:27) v ⟩ n n (cid:0)neqneq ; (1.20)
dt ij ij i j i j
i;j=1
which, demanding that the current relic density of neutralinos be the same as in the case
without coannihilations, requires a much smaller ⟨(cid:27)v⟩ [5].
{ 9 {