University of Amsterdam
MSc Astronomy and Astrophysics
Gravitation and AstroParticle Physics
Master Thesis
Baryons in dwarf galaxies
systematic uncertainties on indirect dark matter searches
by
Monica van Santbrink
10202889
July 2016
60 ECTS
Carried out between September 2015 and July 2016
Supervisors:
Dr. Gianfranco Bertone
Second Examiner:
Dr. Christoph Weniger
Dr. Shin’ichiro Ando
Dr. Jennifer Gaskins
Institute of Physics
i
Abstract
Dwarf spheroidal galaxies (dSphs) are promising objects for indirect dark matter detection.
Their expected gamma-ray flux depends on the J-factor: the square of the dark matter density
profile integrated along the line of sight. In this research a Jeans analysis is performed on seven
classical dSphs for reconstructing their J-factors. The effect of baryons is taken into account by
assuming a density profile that is recently proposed by Di Cintio et al.
The J-factors we obtain are in general somewhat lower than those derived by the Fermi-LAT
collaboration and much lower for the two dSphs with the largest ratio between stellar and halo
mass. We discuss the origin of this discrepancy and the implications for indirect dark matter
searches.
ii
Popular scientific abstract
In 1933 the Swiss astrophysicist Fritz Zwicky studied the motions of galaxies in the so-called
Coma cluster and noticed he could not explain their behavior with the amount of mass from
all stars and gas. There was more mass needed to solve this discrepancy. He referred to this
unseen matter as ‘dunkle Materie’, dark matter. Nowadays the evidence for dark matter is
overwhelming, but its nature is still unknown. One thing is for sure, dark matter is not made
up of atoms that makeup everything visible in the entire Universe. What’s more, only five per
cent of the Universe is visible, the rest is dark.
There are several methods that are currently used for detecting dark matter. One of them is
called indirect detection. This method aims for measuring the ordinary particles that are produced when dark matter particles annihilate or decay. Promising targets for this type of search
are dwarf spheroidal galaxies.
In many indirect dark matter searches of these dwarf galaxies it is assumed that the dark matter
is distributed through a profile that is steep, or cuspy, towards the center. However it has been
shown that ordinary atoms can alter this distribution resulting in a more flattened, or cored,
distribution. Therefore a dark matter mass profile has been proposed recently by Di Cintio et
al. that takes into account these ordinary particles.
In this research we use the profile of Di Cintio et al. to calculate the J-factor, a factor indicating
the strength of the observed annihilation signal, for seven dwarf spheroidal galaxies. We find
that our J-factors are similar to those derived by the Fermi-LAT collaboration, where a cuspy
profile was used.
Contents
Abstract
i
Contents
ii
1 Introduction
1
2 Standard Model of Cosmology
2.1 Overview . . . . . . . . . . . . . . . .
2.2 Evidence Dark Matter . . . . . . . . .
2.2.1 Rotation curves . . . . . . . . .
2.2.2 Gravitational lensing . . . . . .
2.2.3 Cosmic Microwave Background
2.3 Candidates Particle Dark Matter . . .
2.3.1 Neutrinos . . . . . . . . . . . .
2.3.2 Axions . . . . . . . . . . . . . .
2.3.3 WIMPs . . . . . . . . . . . . .
2.4 Thermal production of WIMPs . . . .
2.5 Density profiles . . . . . . . . . . . . .
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2
2
3
4
5
6
7
7
8
9
10
13
3 Indirect dark matter detection
15
3.1 Detection methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Dark matter from gamma-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Dwarf Spheroidal Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Impact of baryons
4.1 Problems with the ΛCDM model
4.2 A baryonic solution . . . . . . . .
4.3 Density profile with baryons . . .
4.4 DC14 for dwarf galaxies . . . . .
5 Analysis
5.1 Jeans Analysis . . . . . . . .
5.1.1 Used profiles . . . . .
5.2 MCMC . . . . . . . . . . . .
5.3 J-factor . . . . . . . . . . . .
5.4 Data . . . . . . . . . . . . . .
5.4.1 Stellar kinematic data
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19
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22
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24
24
25
26
27
28
28
iv
5.4.2
5.4.3
Velocity dispersion profiles . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface brightness data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
30
6 Results
31
7 Discussion and conclusion
7.1 Derived halo masses . . . . . . . . . . . . .
7.2 Derived J-factors . . . . . . . . . . . . . . .
7.3 Impact of parametrizations . . . . . . . . .
7.3.1 Impact of velocity anisotropy profile
7.3.2 Impact of the light profile . . . . . .
7.4 Effect of binned analysis . . . . . . . . . . .
7.5 Assessing the DC14 fit . . . . . . . . . . . .
7.6 Future work . . . . . . . . . . . . . . . . . .
7.7 Conclusion . . . . . . . . . . . . . . . . . .
37
37
38
39
39
39
40
40
41
41
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Chapter 1
Introduction
The ΛCDM (Lambda Cold Dark Matter) model, often referred to as the standard model of
cosmology, describes the evolution of the Universe since the Big Bang. The Universe consist
of ∼ 90% of dark energy (DE), associated with the cosmological constant Λ and dark matter
(DM) [1]. Leaving a mere five per cent for all baryons i.e. Standard Model of particle physics
(SM) particles. The nature of DE and DM is not well understood, the latter is the topic of
interest in this research.
A common view is that DM is made up of a new type of particles, for example WIMPs (Weakly
Interacting Massive Particles). WIMPs can annihilate into SM particles. The study of DM
annihilation products is called indirect detection. Since annihilation is a two body process, the
detected signal depends on an integral over the density squared: the J-factor. Dwarf spheroidal
galaxies are promising objects for indirect detection as they are proximate objects and have low
backgrounds [2]. The Fermi-LAT collaboration has calculated J-factors for the dwarf galaxies
assuming that the density profile is cuspy towards the center. However observations favor a
cored profile [3]. Besides, it has been shown that baryons can alter the density profile creating
a cored profile. Therefore a new density profile has been proposed recently by Di Cintio et al.
(2014) (DC14) that takes into account baryons.
The aim of this research is to recalculate the J-factors assuming the DC14 profile and compare
them to those of the Fermi-LAT collaboration. The structure of this thesis is as follows. First
the ΛCDM model is explained in more detail in Chapter 2, with a strong focus on particle
DM. Then in Chapter 3 indirect detection, in particular for dwarf galaxies, will be discussed.
The last theory section is Chapter 4 which addresses the impact of baryons. In Chapter 5 the
analysis that has been carried out is explained and the used data is described. Then Chapter
6 presents the results of this research. Finally in Chapter 7 the discussion and conclusion are
given.
1
Chapter 2
Standard Model of Cosmology
2.1
Overview
The standard model of cosmology states that the Universe, approximately 13 billion years ago,
began with a hot, dense state which started to cool and eventually reached the cold, sparse
state that is seen today [4]. This model assumes that general relativity is the correct way to
describe gravity on large, i.e. cosmological, scales. This model will be discussed briefly.
About hundred years ago, in 1915 Einstein published his theory of general relativity. In his
work he presented the Einstein field equations, or simply the Einstein equations:
8πG
1
Rµν − Rgµν + Λgµν = 4 Tµν ,
2
c
(2.1)
where Rµν and R are the Ricci curvature tensor and scalar respectively, gµν is the metric tensor,
Λ the cosmological constant, G Newton’s gravitational constant, c the speed of light in vacuum
and Tµν the stress-energy tensor. Setting the cosmological constant Λ to zero, the Einstein
equations state that spacetime is curved by matter and energy. The constant that couples these
quantities is, in SI units, of the order 10−43 , implying that a large mass is needed for a significant
curvature.
The cosmological constant was added by Einstein so that the solution of Equation 1 for the
Universe was stationary. However, as was pointed out first by Lemaı̂tre in 1927, Hubble’s observations showed that the Universe is expanding so that the Lambda-term became redundant.
Despite this, keeping the term doesn’t yield mathematical inconsistencies and therefore it was
kept and thought to be zero.
In 1998 with observations of distant supernovae it was found that the expansion of the Universe
is accelerating, claiming a positive value for Λ. This accelerated expansion is thought to be
2
Chapter 2
Standard Model of Cosmology
3
caused by Dark Energy (DE), whose nature is still unknown.
The standard model of cosmology includes this positive Λ associated with DE. Besides, it contains cold Dark Matter (CDM). This is hypothesized, non-baryonic1 matter that is nonrelativistic and only interacts gravitationally with baryonic matter. The composition of the Universe is
estimated to be ∼ 70% DE, ∼ 25% DM and ∼ 5% baryonic matter [1]. The standard model is
often referred to as the ΛCDM model.
For parametrizing the ΛCDM model, one needs to solve the Einstein equations. This can be
done by introducing isotropy and homogeneity. The former states that space looks the same in
all directions, the latter states that space is the same at all locations. Using these symmetries
one obtains for one component of the Einstein equations, the so-called Friedmann equation:
2
8πG
κc2
ȧ
=
ρtot − 2 ,
a
3
a
(2.2)
where a is the scale factor that parametrizes the expansion of the Universe and ȧ its first time
derivative, G the gravitational constant, ρtot is the total energy density and the constant κ
describes the curvature of the Universe.
At this point it’s common to introduce the Hubble parameter H defined by
H=
ȧ
.
a
(2.3)
The value of H is determined empirically and its current value is set to H0 = 67km s−1 Mpc−1
[1]. Substituting Equation 2.3 into Equation 2.2 and setting κ = 0 in the latter, meaning that
the Universe is flat, one easily obtains the critical density ρcrit
ρcrit =
3H 2
.
8πG
(2.4)
The abundances of matter, energy and vacuum are often expressed in units of the critical density,
Ωi ≡ ρi /ρcrit . Measurements show that the Universe is nearly flat [1], this implies that the total
density is almost equal to the critical density and thus
ΩΛ + Ωmatter = 1 .
2.2
(2.5)
Evidence Dark Matter
In 1922 the Dutch astronomer Kapteyn suggested that the Milky Way contains dark matter
and proposed a method to estimate its mass [5]. Fellow Dutchman Jan Oort found in 1933 a
1
It is conventional to refer to all baryons and leptons from the Standard Model of particle physics as baryons
Chapter 2
Standard Model of Cosmology
4
discrepancy between the observed mass and the mass needed to explain the observed motions
of nearby stars in the Milky Way [6]. Oort used the term dark matter for this missing mass,
but reckoned that the mismatch could be explained by baryonic matter: a significant part of
the stars is either too dim to be observed or obscured by absorbing matter.
Around the same time of Jan Oort, Frits Zwicky studied the Coma cluster. By applying the
virial theorem he calculated the mass of the cluster and found that this was significant greater
than the mass inferred from its luminosity [7].
Since these first suggestions and calculations the evidence for dark matter has increased by
several independent and different studies. In the next sections three of these will be discussed.
2.2.1
Rotation curves
The rotation curves of galaxies are considered as the most convincing and direct evidence for
dark matter on galactic scales. In the 1970s it was found that the circular velocities of stars and
gas in galaxies have roughly the same value and are therefore independent of the distance to the
Galactic Center. This results in a flat rotation curve. However, using Newtonian physics, the
circular velocity is found by equating the centripetal force to the gravitational force resulting
in:
r
vc (r) =
GM (r)
,
r
(2.6)
ρ(r)r2 dr and ρ(r) is the mass density profile. From Equation 2.6 one would
√
expect the circular velocity to scale as vc (r) ∝ 1/ r. However, given the the approximately flat
where M (r) = 4π
R
rotation curve it is needed that M (r) ∝ r and ρ ∝ 1/r2 . This can be explained by introducing
a dark matter halo: an approximately spherical halo that contains non-luminous matter and is
extending far beyond the visible size of the galaxy.
In Figure 2.1 the rotation curve of NGC 3198 is shown. It can be seen that the circular velocity
becomes approximately constant at large distances. Moreover the rotation curves of the disk,
gas and dark matter are shown as dashed, dotted and dash-dotted respectively.
Chapter 2
Standard Model of Cosmology
5
Figure 2.1: Rotation curve of NGC 3198 including the curves of the individual components.
Dashed, dotted and dash-dotted represent the disk, gas and dark matter respecitvely. Taken
from [8]
2.2.2
Gravitational lensing
As discussed in Section 2.1 Einsteins theory of general relativity states that mass curves spacetime and therefore light is bent when it passes a massive object. The bending of light is a purely
geometrical effect and is independent of the energy of the photons. This effect is referred to
as gravitational lensing and can be used to estimate the mass of e.g. galaxies and galaxy clusters.
Gravitational lensing can be divided into two regimes: strong and weak lensing. Studies of
strong lensing showed that galaxy clusters have a mass-to-light ratio of about 300 which is a
direct demonstration of the presence of dark matter [9].
For the majority of sources weak gravitational lensing is the method of interest [10]. The path
of light is still deflected by gravitational fields, but the amplitude is small and it is undetectable
on individual galaxies.
The results of the galaxy cluster 1E0657-56, referred to as the Bullet cluster, are noteworthy.
This cluster is formed by the collision of a smaller cluster with a larger one. While colliding most
mass passed right through. Using weak gravitational lensing the center of the total mass was
determined. On the other hand, the X-ray telescope Chandra determined the center of the mass
in this waveband. By comparing these results it can be concluded that there is a spatial offset
Chapter 2
Standard Model of Cosmology
6
between the total and baryonic mass peaks. Therefore the observations of this cluster support
the existence of dark matter and allow for the conclusion that it is collisionless, otherwise the
particles should have experienced ram pressure and thus slowed down like the luminous gas [10].
Moreover the Bullet cluster disfavors other alternatives such as modified Newtonian dynamics
(MOND) where the mass distribution should coincide with the baryon distribution.
In Figure 2.2 an image of the Bullet cluster is presented. The pink regions represent the X-ray
emission from the hot gas, the blue region is a reconstruction of the total mass determined by
gravitational lensing. The spatial offset can be seen directly.
Figure 2.2: Bullet cluster 1E0657-56. The pink region indicate the X-ray emission from the
hot gas, the blue region is a reconstruction of the total mass determined by gravitational lensing.
Taken from [11]
2.2.3
Cosmic Microwave Background
The Cosmic Microwave Background (CMB) is radiation originating from the propagation of
photons in the early Universe and serves as another piece of evidence for the existence of Dark
Matter.
Predicted in 1948 by Gamow, the CMB was detected in 1965 by the radio astronomers Penzias
and Wilson who had an unexplainable, isotropic noise2 in their measurements.
Today it is known that the CMB has an almost perfect blackbody spectrum of temperature
T = 2.726K [12] and is isotropic at the 10−5 level in temperature. At higher levels anisotropies
2
At present recognized as a signal
Chapter 2
Standard Model of Cosmology
7
are observed which are commonly expanded in spherical harmonics Ylm (θ, φ)
∞
+l
XX
δT
(θ, φ) =
alm Ylm (θ, φ) .
T
(2.7)
2m −l
From a measured set of alm the variance Cl can be obtained by
l
Cl = h|alm |2 i =
1 X
|alm |2 .
2l + 1
(2.8)
−l
By comparing the set of observed Cl ’s with the theoretical ones that are obtained by varying
the parameters of a cosmological model, a very good accuracy can be acquired on the parameters. Both the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellite put
constraints on the abundance of baryons and matter. The latter has obtained the most recent
values and are given by
Ωb = 0.0490 ± 0.0007
Ωm = 0.314 ± 0.020 [1].
(2.9)
As can be seen from Equation 2.9 the CMB allows not only to conclude the presence of dark
matter but also to determine the total amount of dark matter in the Universe. The DM-tobaryon ratio in the Universe, using the Planck estimates, is Ωm /Ωb ≈ 6.
2.3
Candidates Particle Dark Matter
There are numerous dark matter candidates and only some will be discussed in this thesis. A
particle that makes a strong candidate meets several criteria, such as that it should be a cold,
neutral particle that has the appropriate relic density and it needs to be in agreement with
current dark matter searches and with stellar structure and evolution [13].
2.3.1
Neutrinos
The Standard Model (SM) of particle physics classifies the fundamental particles and describes
the electromagnetic, weak and strong interactions between them. The elementary particles that
make up matter are quarks and leptons, these are fermions i.e. particles with half-integer spin.
The gauge bosons, particles with integer spin, mediate the interactions. The leptons can be divided into two main classes: charged leptons and neutral leptons. The latter are better known
as neutrinos.
Chapter 2
Standard Model of Cosmology
8
Neutrinos are stable particles and only interact through the weak nuclear force. Therefore they
have been considered as DM candidates. Since oscillation experiments proved that neutrinos
have nonzero rest mass [4], it is possible to evaluate their total relic density. As there are three
generations of neutrinos νi we sum over their masses mi to obtain the density
Ων h2 =
3
X
mi
.
93 eV
(2.10)
i=1
From tritium β-decay experiments the 95% C.L. upper limit on the neutrino mass is found to
be [14]
mν = 2.05 eV .
(2.11)
This yields the following upper limit on the neutrino density
Ων h2 . 0.07 .
(2.12)
This implies that there are not enough neutrinos to account for all the dark matter. Another
constraint rises from the CMB anisotropies combined with large scale structure surveys suggesting an even lower relic density of Ων h2 = 0.0067 [15].
Besides, neutrinos are disfavored because of their relativistic nature which results in a characteristic length scale or free steaming length that is on the order of ∼ 40Mpc mν /30 eV [16].
Regions separated by a distance larger than this free steaming length will prevail, however all
fluctuations lower this scale will be erased. As a consequence the Universe would have a topdown formation history i.e. large structures are formed first, followed by smaller structures.
However our galaxy appears older than the Local Group [17] and thus contradicts the top-down
formation history.
The mass of a neutrino is given in Equation 2.11, however the SM states that there are only
left handed neutrinos, implying they should be massless. This requires physics beyond the SM.
The solution is given by adding right handed neutrinos to the SM. In this way neutrinos obtain
their mass through the same mechanism as quarks and leptons. For the right handed neutrinos
to be allowed they must have no SM gauge interactions [18] and are therefore referred to as
sterile neutrinos. The mass scale of these particles is not predicted a priori. Sterile neutrinos
are DM candidates and their constraints come from their cosmological abundance and, since a
sterile neutrino can decay into a photon and a neutrino, their decay products [12].
2.3.2
Axions
The charge parity symmetry (CP symmetry) combines the symmetry between positive and
negative charges (C) and the symmetry of spatial coordinates (P). However this symmetry can
Chapter 2
Standard Model of Cosmology
9
be violated.
Although the SM has proven to be successful, there are several deficiencies. Next to the neutrino
mass, there is the strong CP problem. The quantum chromodynamics Lagrangian L contains a
CP violating term θ̄. If θ̄ 6= 0 then QCD violates the CP symmetry. Experiments on the neutron
electric dipole moment found an upper limit of θ̄ < 10−9 [19]. Giving rise to the question: why
is θ̄ so small? As a solution a new particle, the axion3 , was introduced.
After being introduced to solve the CP problem, the axion turned out to be a serious DM
candidate. Firstly they couple weakly to SM particles, secondly they are effectively stable and
finally they have a proper density to be a substantial fraction of dark matter [20].
2.3.3
WIMPs
The most appealing class of candidates are Weakly Interacting Massive Particles (WIMPs).
These particles are electrically neutral and interact through both gravity and the weak nuclear
force. Their generic production mechanism resulting in the correct relic density, which will be
discussed in Section 2.4, combined with their detection possibilities makes them strong DM
candidates [10].
Different extensions of the SM such as supersymmetry (SUSY) propose WIMPs. In the remainder of this section will be explained why SUSY was introduced and how this results in DM
candidates.
SUSY was introduced for solving the hierarchy problem [21]. In the SM particles acquire their
mass through interacting with the Higgs field. The mass of the Higgs boson is on the order
of mh ∼ 100 GeV [18]. However, given the three fundamental constants: the speed of light
c, Planck’s constant h and the gravitational constant Gn , a natural value for mh would be a
p
combination of these. Such a combination is the Planck mass Mpl = hc/G ≈ 1.2 · 1019 GeV.
The hierarchy problem addresses the question of why mh Mpl .
Next to SUSY there are several other solutions proposed for this problem e.g. the Higgs boson
being a composite particle [22] and the existence of large extra spatial dimensions [23].
Supersymmetry relates boson and fermions: each known SM particle has a super partner with
the same mass but which differs by half a unit of spin [24]. Fermions have spin-0 particles as
superpartner (sfermions), the gauge bosons have spin-1/2 partners (gauginos) and by introducing an additional Higgs boson, these bosons are linked to a spin-1/2 super partners (Higgsinos).
This is the minimal supersymmetric extension of the SM (MSSM).
This MSSM solves the hierarchy problem, but another symmetry needs to be introduced. In
3
This name was chosen because it cleans up the CP problem
Chapter 2
Standard Model of Cosmology
10
the SM the decay of protons is prevented by gauge symmetries, however this is not the case in
the MSSM. Therefore R-parity is introduced which is defined as
R = (−1)3B+L+2s ,
(2.13)
where B, L and s are the baryon number, lepton number and spin respectively. All known SM
particles have R = +1 and all superpartners have R = −1. Therefore supersymmetric particles
can only decay into an odd number of lighter supersymmetric particles. As a consequence the
lightest supersymmetric particle (LSP) cannot decay, hence it is stable. A direct consequence
is the LSP being an excellent DM candidate.
There are several LSP dark matter candidates: the sneutrino with spin-0, however experimental
results exclude this particle [25], the gravitino with spin-3/2, this particle is very difficult to
detect [26], and the neutralino with spin-1/2. The lightest neutralino is the most studied DM
candidate within the MSSM.
2.4
Thermal production of WIMPs
In general it is assumed that WIMP pairs were produced and annihilated in the early Universe
in particle-antiparticle collisions such as
χχ̄ ↔ e+ e− , µ+ µ− , W + W − , ZZ ,
(2.14)
where on the left-hand side χ represents the WIMP and on the right-hand side are the standard model particles, the electron e, the muon µ, the W-boson and the Z-boson. Initially the
temperature was much higher than the WIMP mass resulting in an equilibrium between the
producing and annihilating processes, the rate Γ given by
Γ = hσann vineq ,
(2.15)
here σann is the annihilation cross section, v the relative velocity of the annihilating particles,
the brackets indicate that the argument is thermally averaged and neq the equilibrium number
density.
As a result of the expansion of the Universe the temperature decreased and eventually became
smaller than the WIMP mass. This led to a smaller number of WIMPs produced, while the
producing and annihilation rates remained in equilibrium. Moreover, as a consequence of the expansion, the number density decreased which resulted in a smaller production and annihilation
Chapter 2
Standard Model of Cosmology
11
rate Γ as can be directly seen from Equation 2.15. At some point the rate became smaller than
the expansion rate of the Universe H, see Equation 2.3. Accordingly the WIMPs chemically
decoupled and the production ceased. Therefore the WIMP density, the number of WIMPs in
a comoving volume, remained approximately constant and is often referred to as the thermal
relic density.
Closely following [10] the value of the thermal relic density can be computed using the Boltzmann
equation
dn
= −3Hn − hσann vi(n2 − n2eq ) ,
dt
(2.16)
which takes into account the number density n, its first time derivative and other quantities
as defined before in this section. The first term on the right-hand side takes the expansion of
the Universe into account, whereas the second term the change in number density. Besides this
equation the other ingredient is the law of entropy conservation
ds
= −3Hs ,
dt
(2.17)
here s is the entropy density and given by s = 2π 2 g∗ T 3 /45 where g∗ is the number of relativistic
degrees of freedom. By introducing the variables
Y ≡
n
,
s
Y eq ≡
neq
s
,
x≡
m
,
T
(2.18)
where T is the photon temperature, Equations 2.16 and 2.17 can be combined and rewritten as
1 ds
dY
=
hσann vi(Y 2 − Yeq2 ) .
dx
3H dx
(2.19)
This is a Riccati equation i.e. a first-order differential equation that is quadratic in the unknown
function and can be solved numerically. For doing this the boundary condition Y = Yeq at x ' 1
should be used. This condition states that at high temperatures the particles were in thermal
equilibrium as explained at the beginning of this section. Finally an expression for the Hubble
parameter is needed for solving Equation 2.19. This can be obtained from, a modified version
of, Equation 2.4
H2 =
8π
2ρ ,
3Mpl
(2.20)
where Mpl = 1.2 · 1019 GeV is the Planck mass. As the ultra-relativistic species dominate in
the early Universe, the mass-energy density ρ is given by ρU R =
π2
4
30 g∗ T
resulting in an Hubble
expansion rate that scales with temperature squared
q
2
2 ≈ 1.66g∗ T .
H = 8π 3 g∗ T 4 /90Mpl
Mpl
(2.21)
Chapter 2
Standard Model of Cosmology
12
Solving the differential equation, expressing the result in units of critical density Ωi ≡ ρi /ρcrit
and performing an order of magnitude estimate results in
ΩX h2 =
3 · 10−27 cm3 s−1
,
hσann vi
(2.22)
where the subscript X indicates that ΩX is the density of a generic relic and h = H0 /100 km
s−1 Mpc−1 the Hubble parameter.
In Figure 2.3 the numerical solution to Equation 2.19 is shown i.e. the evolution in time of
the WIMP density in the early Universe. As expected, given the explanation at the beginning
of this section, can be seen from the figure that the density Y is close to its equilibrium value
Yeq at high temperatures. As the temperature decreases Yeq becomes suppressed, therefore the
equilibrium ceases. It is at the freeze-out temperature that the WIMP number density becomes
approximately constant. An important feature of the thermal relic density as illustrated by
Figure 2.3 is that the annihilation cross section and the relic density are inversely proportional.
As WIMPs with stronger interactions stay longer in equilibrium, thus decoupling at a lower
freeze-out temperature, their densities are further suppressed.
Figure 2.3: Evolution of WIMP density in the early Universe. Taken from [27]
Chapter 2
Standard Model of Cosmology
13
From a particle physics point of view: new particles interacting through the weak force will
have a mass around mweak ∼ 100 GeV [28], the origin of this mass scale is not well understood.
Using this mass scale an order of magnitude estimation can be performed: hσvi ∼ α2 (100
GeV)−2 ∼ 10−25 cm3 s−1 , where α is the fine-structure constant. A priori there is no reason
that weak force interactions are connected to cosmological quantities. However, this value is
remarkably close to the value given in Equation 2.22 for the relic density of WIMPs. This
coincidence is called the WIMP miracle.
2.5
Density profiles
As discussed in Section 2.2.1 DM halos should scale with ρ ∝ 1/r2 at large radii in order to
recover the approximately flat rotation curves. However the proportionality of ρ(r) at the center
has not been determined yet. The density profiles can be roughly classified into cored models,
preferred by observations [3], and cuspy models, preferred by simulations [29]. In this section
some of the proposed density profiles will be discussed.
One generic dark matter density profile is the (α, β, γ) double power-law model:
ρDM (r) =
(r/rs
)γ
ρs
,
· [1 + (r/rs)α ](β−γ)/α
(2.23)
where ρs is the scale density and rs the scale radius. The values for α, β and γ determine the
transition, outer and inner slope respectively.
One parametrization that is often used is the Navarro–Frenk–White (NFW) profile [30] and is
given by:
FW
ρN
DM (r) =
ρs rs3
.
r(rs + r)2
(2.24)
This is a specific form of Equation 2.23 having (α, β, γ) = (1, 3, 1). Other profiles of this
family are the isothermal profile having (α, β, γ) = (2, 2, 0) [31] and the Moore profile with
(α, β, γ) = (3/2, 3, 3/2) [32]
ρIsothermal
(r) =
DM
oore
ρM
DM (r) =
ρs
,
1 + (r/rs )2
ρs
.
(r/rs )3/2 (1 + (r/rs )3/2 )
(2.25)
(2.26)
An modified version of the isothermal profile, fitted to observations, is the Burkert profile given
by [33]
ρBurkert
(r) =
DM
ρs r3
.
(r + rs )(r2 + rs2 )
(2.27)
Chapter 2
Standard Model of Cosmology
14
A different parametrization is for example the Einasto profile which reads [34]:
ρEinasto
(r)
DM
2
= ρ−2 exp −
α
r
α
r−2
−1
.
(2.28)
In this expression ρ−2 and r−2 are the density and radius for which the logarithmic slope equals
−2 i.e. at which ρ(r) ∝ r−2 .
To compare these different profiles by eye and visualize the classification of cuspy versus cored
models, they are shown in Figure 2.4, having all constants set to unity. The horizontal axis
represents the radius R and the vertical axis the density ρ.
Figure 2.4: Five dark matter density profiles: NFW, isothermal, Moore, Burkert and Einasto.
Units are arbitrary
In this chapter the Standard Model of Cosmology is introduced and evidence for Dark Matter is
discussed together with several candidates. In the next chapter will be discussed how DM can
be detected and specifically what the current results are of the astrophysical objects of interest
in this research: dwarf spheroidal galaxies.
Chapter 3
Indirect dark matter detection
3.1
Detection methods
The WIMP scenario reasons that dark matter can interact with SM particles through collisions
such as χχ ↔ f f¯, W + W − , ZZ. Here χ represents the DM particle and since it needs to be
electrically neutral it is equal to its antiparticle. On the right-hand side are SM particles, a
fermion f with its antifermion f¯, the W-boson and the Z-boson.
This DM-SM particle interaction allows for three different options to study DM: indirect detection, direct detection and collider searches. These three methods are illustrated in Figure 3.1.
Figure 3.1: Three detection methods for dark matter: indirect detection, direct detection and
collider searches
15
Chapter 3, Indirect dark matter detection
16
On the left-hand side in Figure 3.1 there are two DM particles shown and on the right-hand
side two SM particles, the nature of the interaction between the two is unknown. For now will
be assumed that interactions are possible without being able to explain how and why.
As Figure 3.1 is a Feynman diagram one can choose how time flows. First, let time flow from
left to right. One starts at the left-hand side with two DM particles these annihilate or decay
into SM particles. The study of the final stable products is called indirect detection. Second,
let time flow in the upward, vertical direction. In this case the aim is to analyze scatterings
of DM particles off of SM particles in a detector. This type of search is referred to as direct
detection. Lastly, let time flow from right to left. Starting with high energy SM particles, the
objective is to search for the DM production process by letting the SM particles collide. This
type of dark matter detection is called collider searches.
This thesis focuses on indirect detection for which there are several messengers and wavelengths
such as gamma-rays, charged cosmic rays, neutrinos and X-rays [10]. More specifically, this
thesis focuses on indirect detection from gamma-rays and therefore this will be discussed more
thoroughly in the next section.
3.2
Dark matter from gamma-rays
Among the DM annihilation messengers, gamma-rays are interesting since they travel in straight
lines and are not absorbed in the local Universe [10]. The detected gamma-rays per square
centimeter per second is the detected flux φs . The expected signal from dark matter annihilation
for a solid angle ∆Ω is given by:
1 hσνi
φs (∆Ω) =
4π 2m2DM
Z
Emax
Emin
dNγ
dEγ
dEγ
Z
∆Ω
Z
ρ2DM (r)dldΩ0 ,
(3.1)
l.o.s.
where hσνi is the thermally-averaged annihilation cross section, mDM the particle mass and
dNγ /dEγ the number of photons per energy interval produced in the annihilation process and
ρDM (r) is the dark matter mass density distribution. The first part of the expression i.e.
R Emax dNγ
1 hσνi
4π 2m2
Emin dEγ dEγ contains the spectral information and is dependent on particle physics
DM
properties. Consequently this term is dubbed the particle physics factor. The second term,
R R
2
0
∆Ω l.o.s. ρDM (r)dldΩ , contains the angular information and is referred to as the astrophysical
factor or the J-factor.
The gamma-ray instruments can be classified in to two groups: the space-based telescopes
(aboard satellites) and ground-based telescopes. The former (e.g. Fermi-LAT) are pair-conversion
telescopes, the latter (e.g. H.E.S.S., MAGIC and VERITAS) are Cherenkov telescopes. There
are several DM search targets such as the Galactic center, the Sun, galaxy clusters, Milky Way
Chapter 3, Indirect dark matter detection
17
halo and dwarf spheroidal galaxies. As this research aims to analyze dwarf spheroidal galaxies,
these objects will be discussed in the next section.
3.3
Dwarf Spheroidal Galaxies
Dwarf spheroidal galaxies (dSphs) are highly DM dominated as suggested by kinematic data
and have low astrophysical gamma ray backgrounds [35]. Moreover they are proximate objects
with known locations [36]. Therefore dSphs are promising targets for indirect DM gamma-rays
searches.
There has not been found any significant excess of gamma rays from dSphs [37]. These nondetections, together with an assumption for the DM profile, lead to constraints on the cross
section hσνi which are on the order of the value for a thermal relic: 3 · 10−26 cm3 s−1 of
mass . 100 GeV [38], see Section 2.4.
The Fermi-LAT collaboration used six years of data and have recently published their upper
limits on the cross section from a combined analysis of 15 dSphs [36]. In Figure 3.2 these results
are shown in black for the bb̄ and τ + τ − channels as a function of mass. For comparison other
results are included in the figure as well. In solid gray the constraints found by the Fermi-LAT
collaboration from an analysis of the Milky Way halo are shown. The results from H.E.S.S.
obtained from 112 hours of observations of the Galactic Center are presented in dashed red. In
solid orange the constraints derived from 157.9 hours of observations of the dwarf galaxy Segue
1 with MAGIC are plotted. The closed contours and the marker with error bars represent the
best-fit cross section and mass from different analyses of the Galactic Center excess. The dashed
gray curve shows the thermal relic cross section.
Figure 3.2: Constraints on the DM annihilation cross section as function of DM mass. The
left panel shows the results for the bb̄ channel, right for the τ + τ − channel. Taken from [36]
Chapter 3, Indirect dark matter detection
18
From Figure 3.2 can be seen that for the bb̄ channel the best-fit contours are in a parameter
space somewhat above the most recent limits derived by the Fermi-LAT collaboration. However,
as argued in [36], due to uncertainties in the structure of the Galactic dark matter distribution
these best-fit contours can significantly enlarge.
In this chapter both the detection method as the astrophysical objects of interest are discussed.
In the next chapter will be examined how baryons can alter the picture.
Chapter 4
Impact of baryons
4.1
Problems with the ΛCDM model
Whereas the ΛCDM is successful on cosmological scales, explaining the CMB and galaxy clustering, it has problems on galactic scales. These problems are the missing satellite problem, the
too-big-to-fail problem and the cusp-core discrepancy.
Starting with the missing satellite problem. It was found that there is a discrepancy between
the number of observed satellites and the number of predicted dark matter halos, the latter
being larger than the former [39]. This mismatch is referred to as the missing satellite problem.
As the Sloan Digital Sky Survey (SDSS) discovered new, very faint dwarf galaxies and nearly
doubled the amount of known satellites, the problem was somewhat alleviated [40]. A possible
solution is that the lowest DM halos scarcely have star formation, due to early reionization of
the intergalactic medium [41].
The too-big-to-fail problem is connected to the missing satellite problem. In the high mass
range, where halos are too massive to have suppressed star formation due to reionization processes i.e. halos that are too big to fail, there is a discrepancy between the predicted and
observed kinematics of galaxies [42]. The expected velocities of stars in the satellites are higher
than observed.
Finally there is the cusp-core discrepancy. As already discussed in Section 2.5, dark matter
density profiles that are inferred from collisionless N-body simulations are cuspy, meaning that
they are steep towards the halo center [29]. However, observational evidence suggests that the
inner slope of the DM distribution is flat or cored [3].
19
Chapter 4, Impact of baryons
4.2
20
A baryonic solution
Different studies have shown that baryons can affect the dark matter density profile [43], since
it can either contract or expand a halo. The former is the result of gas cooling to the center of a
halo which leads to a strengthened cuspy profile and therefore increases the mismatch between
the theoretical models and observations. On the other hand, expanded halos can alleviate this
mismatch. There are two mechanisms through which baryons can expand halos: outflow driven
by stellar feedback and dynamical friction. The former is effective at expanding low mass halos
and the latter is effective at expanding massive halos [44]. Since the focus of this thesis is on
dwarf galaxies, stellar feedback is the mechanism of interest here.
Stars are formed in the center of the halo as a result of gas cooling. These stars cause repeated
energetic outflows in the form of supernovae feedback. As has been shown in e.g. [43],[45] this
feedback flattens the central dark matter density profile creating a core.
Therefore it can be concluded that a dark matter density profile that takes into account baryons
is a well motivated choice. Such a profile has been proposed by Di Cintio et al. [46] (hereafter
DC14) and will be discussed in Section 4.3.
4.3
Density profile with baryons
Di Cintio et al. [46] proposed a density profile that depends on the stellar-to-halo mass ratio
M? /Mhalo , i.e. the star formation efficiency. They argue that this ratio controls the extent of
the baryonic impact.
Di Cintio et al. analyzed smoothed-particle hydrodynamics (SPH) simulated galaxies taken
from the MaGICC project [47], the initial conditions are taken from MUGS [48]. In these simulated galaxies stellar feedback by supernovae, stellar winds and energy from young, massive
stars were implemented. The used sample comprised ten galaxies with five different initial conditions, covering the mass range 9 · 109 M − 7 · 1011 M .
The dark matter profiles of these SPH simulated galaxies were analyzed using a generic form of
the Hernquist-Zhao profile
ρDM (r) =
(r/rs
)γ
ρs
.
· [1 + (r/rs)α ](β−γ)/α
(4.1)
Chapter 4, Impact of baryons
21
It was found that dark matter halos having a M? /Mhalo ratio below 0.01 per cent maintain the
cuspy NFW profile. The amount of stellar feedback is too small to alter the profile. As the ratio
increases, the stellar feedback becomes strong enough to expand the halo and this results in the
profile becoming progressively flatter. The most cored profiles are found at M? /Mhalo ≈ 0.4 per
cent. For halos having a larger ratio return again to the cuspy NFW profile. The explanation
for this is that in the high mass range more mass collapses at the center which opposes the
expansion process.
In Figure 4.1 is shown how the transition slope α (black), the outer slope β (red) and the inner
slope γ (green) vary as function of the stellar-to-halo mass ratio. The symbols represent the
different initial conditions and their sizes indicate the mass of the halo, a larger symbol implies
a larger halo mass.
Figure 4.1: Transition slope α (black), outer slope β (red) and inner slope γ (green) of the
DC14 density profile. Taken from [46]
The correlations between α, γ and M? /Mhalo were fitted with a four parameter, double power
law function. Whereas for β a parabola was used. The resulting expressions are given by
α = 2.94 − log10 [(10X+2.33 )−1.08 + (10X+2.33 )2.29 ] ,
β = 4.23 + 1.34X + 0.26X 2 ,
γ = −0.06 + log10 [(10X+2.56 )−0.68 + (10X+2.56 )] ,
(4.2)
Chapter 4, Impact of baryons
22
with X = log10 (M? /Mhalo ).
Therefore the DC14 profile is a specific form of Equation 4.1, where the slopes are given by 4.2.
The scale density ρs can be obtained by normalizing the mass integral i.e. by stating that
Z
Rvir
ρs = Mhalo / 4π
0
r2
dr
( rrs )γ [1 + ( rrs )α ](β−γ)/α
!
.
(4.3)
The new quantity appearing in Equation 4.3 is the viral radius Rvir which is defined as the
radius at which the mass of a sphere containing ∆ times the critical density ρcrit at z = 0 is
equal to the mass of the halo, i.e.
4 3
Mhalo = πRvir
∆ρcrit .
3
(4.4)
The remaining parameter in Equation 4.1 is the scale radius rs which should be left as a free
parameter.
4.4
DC14 for dwarf galaxies
The too-big-to-fail problem and the cusp-core problem can be cast as a single problem: the
former being an effect of the latter i.e. the mismatch between the predicted and observed kinematics is due to the existence of a cored density profile. Therefore in [41] the kinematical data
of 40 Local Group (LG) galaxies have been used, both isolated and satellite, to examine whether
the too-big-to-fail problem might be alleviated using the mass dependent DC14 profile. This
was done by finding the halo mass that provided the best fit for each galaxy for both a cuspy
NFW profile as the cored DC14 profile. The approach taken by the authors was to compare
the effect of the two different density profiles on the LG members as a whole, this differs from
the method that will be used in this thesis: a Jeans analysis of the velocity dispersions. The
advantage of the method chosen by [41] is that the mass - anisotropy degeneracy is avoided.
Jeans analysis will be discussed in Section 5.1.
Having halo masses the abundance matching technique can be applied. This method constraints
the relation between the stellar mass and the halo mass of galaxies: the heaviest galaxies are
matched with the heaviest halos and one continues doing this with the less massive galaxies and
halos up to the point that all observed galaxies are assigned to a dark matter halo.
In [41] abundance matching is performed for the found halo masses. With this result they argue
that the DC14 profile rather than the NFW profile alleviates the too-big-to-fail problem since
Chapter 4, Impact of baryons
23
the dwarf galaxies are assigned to more massive halos having a cored density distribution.
In this chapter is discussed how baryons can have an impact on DM. This combined with all
the previous chapters results in a well-motivated decision for examining the J-factors of dSphs
for a density profile that takes baryons into account.
Chapter 5
Analysis
5.1
Jeans Analysis
Closely following [49] for performing a Jeans analysis we assumed that dSphs are steady-state
collisionless systems that can be described by their stellar phase-space density f (x, v, t) =
d3 xd3 v. To the latter we applied the collisionless Boltzmann equation and thus obtained:
v · ∇f − ∇Φ ∂f
∂v
∂f
∂t
+
where Φ is a smooth gravitational potential. Then, assuming spherical symmetry
and negligible rotational support, the second-order Jeans equation was found by integrating
moments of the former expression:
dΦ
ν ¯2
d
(ν v¯r2 ) +
2vr − (v¯θ2 + v¯φ2 ) = −ν
,
dr
r
dr
(5.1)
where ν is the stellar number density and v¯x2 the velocity dispersion in the x ∈ {r, θ, φ} direction.
This can be rewritten by introducing an expression for the velocity anisotropy βani ≡ 1 −
v¯θ2
2.
v¯φ
Implementing this and taking the derivative of the gravitational potential yields:
1 d βani (r)v¯r2 (r)
GM (r)
ν(r)v¯r2 (r) + 2
=−
.
ν(r) dr
r
r2
(5.2)
The enclosed mass at radius r, M (r), of the dSphs has contributions from both the stars and
the dark matter. The former can be neglected since the objects are highly DM dominated.
Therefore we have
r
Z
ρDM (s)s2 ds ,
M (r) = 4π
0
here ρDM is the dark matter mass density profile.
24
(5.3)
Chapter 5, Analysis
25
The generic solution of the Jeans equation relates M (r) to ν(r)v¯r2 (r) and is given by:
ν(r)v¯r2 (r) =
1
f (r)
∞
Z
f (s)ν(s)
r
where
Z
r
f (r) = fr1 exp
r1
GM (s)
ds ,
s2
2
βani (t)dt
t
(5.4)
.
(5.5)
The last expression contains a new variable r1 which after integration gives a normalization
factor that cancels out in the generic solution of the Jeans equation.
However the proper motions of the tracer stars cannot be resolved, but the quantities projected
along the line-of-sight can. Therefore the Jeans solution, Equation 5.4, needs to be projected.
In general, for a spherically symmetric system, the quantity f (r) can be projected into F (R)
by applying an Abel transformation (and de-projected by an inverse Abel transformation):
Z
∞
F (R) = 2
R
f (r)rdr
√
, f (r) =
r 2 − R2
Z
r
∞
dF
−dR
√
.
dR π R2 − r2
(5.6)
By applying this to the Jeans solution, we obtained the following expression:
σp2 (R) =
2
Σ(R)
Z
∞
1 − βani
R
R2
r2
ν(r)v¯r2 (r)r
√
dr ,
r 2 − R2
(5.7)
with R the projected radius, σp2 the projected stellar velocity dispersion and Σ(R) the projected
light profile of the surface brightness. The latter is given by
Z
∞
Σ(R) = 2
R
ν(r)rdr
√
.
r 2 − R2
(5.8)
The quantities σp (R) and Σ(R) can be measured directly. We used parametric models for the
dark matter density profile ρDM (r), the light profile ν(r) and the anisotropy profile βani (r) in
order to fit the velocity dispersion σp (R) to the data.
5.1.1
Used profiles
For the dark matter density profile we used the DC14 profile as introduced in Section 4.3. We
used a Planck cosmology and therefore set ∆ = 104.2 [50] in Equation 4.4. Moreover we set
the velocity anisotropy βani to zero. The impact of this choice will be discussed in Section
7.3.1. For the light profile we used the Hernquist-Zhao profile, given in Equation 2.23. The
Hernquist-Zhao profile is analytical for the density profile ν(r). Since the observed quantity is
the projected light profile Σ(R), the profile ν(r) has to be projected. We did this numerically
Chapter 5, Analysis
26
by using the Abel transform given in Equation 5.6.
From Equations 5.4, 5.5 and 5.7 can be seen that three successive integrations are required.
However it is shown in [51] that for several velocity anisotropy profiles the expression can be
rewritten as a single integration, which enormously speeds up the calculation. For the case of
βani = 0 the expression becomes
σp2 (R)
2G
=
Σ(R)
Z
∞
R
R
with
r
K(u) =
5.2
r
K
1−
ν(r)M (r)
dr
,
r
1
.
u2
(5.9)
(5.10)
MCMC
In order to explore the parameter space efficiently a Markov chain Monte Carlo (MCMC) technique is used. Therefore this method will be discussed in this section.
A sequence of random elements X1 , X2 , . . . is a Markov chain provided that Xn+1 only depends
on Xn [52]. The state space of the chain is the set in which the Xi take values. Changes
xi → xi+1 are called transitions and the corresponding transition probabilities are given by the
transition matrix Wxy . Since the sum of probabilities of going from an state Xi to any other
state is equal to one, the columns of Wxy are normalized:
X
Wxy = 1 .
(5.11)
y
Therefore the Markov process conserves the total probability.
For an MCMC simulation a suitable Markov chain is constructed such that its equilibrium
distribution is the desired target distribution. When such a chain is constructed, one can start
from an arbitrary point and iterate the chain as many times as wanted. Eventually the generated
draws would appear as if they were drawn from the target distribution.
One way to construct a appropriate Markov chain is by using the Metropolis-Hastings algorithm
which works as follows [53]:
1. Start with xi , choose randomly one out of a fixed number N changes ∆x, such that the
new state is x0 = xi + ∆x
2. Calculate the ratio of probabilities P = px0 /pxi
Chapter 5, Analysis
27
3. Draw a random number R ∈ [0, 1]
4. If R < P accept the step, add the new values to the chain. Else reject the step, re-add
the old values to the chain
For this analysis one has to choose prior ranges and a likelihood function. Starting with the
former. In order to not favor the values at the high end, we chose the ranges in log space:
log10 (Mhalo /M ) ∈ [7, 12] and log10 (rs /kpc) ∈ [−1, 2]. The range for the scale radius was based
on [54]. As the MCMC returns samples from the models posterior probability distribution
function (PDF), the credibility intervals (CIs) were computed by drawing random samples from
the distribution.
We used the likelihood function for a binned analysis from [55] which is given by
Lbin =
NY
bins
i=1
"
#
1 σobs (Ri ) − σp (Ri ) 2
(2π)−1/2
,
exp −
∆σi (Ri )
2
∆σi (Ri )
where
2
2
1
[σp (Ri + ∆Ri ) − σp (Ri − ∆Ri )]
.
2
2
∆ σi = ∆ σobs (Ri ) +
(5.12)
(5.13)
In this expression ∆σp is the model value for the velocity dispersion calculated with Equation
5.7, ∆σi is the error on the observed velocity dispersion, Ri is the mean radius of the i -th bin
and ∆Ri the standard deviation of the radii distribution in this bin.
5.3
J-factor
As discussed in Section 3.2 the J-factor is the dark matter density profile squared integrated
along the line-of-sight l and over a solid angle Ω:
Z
Z
J(Ω) =
∆Ω
ρ2DM (r)dldΩ0 .
(5.14)
l.o.s.
In order to integrate ρ(r) along the line-of-sight it has to be slightly rewritten. Following [38],
let z denote the shortest distance between the line-of-sight and the center of the dwarf galaxy
and define the impact parameter b = Dsin(θ) where D is the distance between the observer and
the center of the dwarf. This situation is displayed in Figure 5.1.
Chapter 5, Analysis
28
Figure 5.1: Sketch of the situation for calculating the J-factor. D is the distance between
the observer and the dwarf galaxy, l is the line-of-sight and z the shortest distance between the
dwarf and the line-of-sight
For small angles the impact parameter can be approximated by b = Dθ. Then dl = dz and
√
√
r = b2 + z 2 such that ρ(r) = ρ( b2 + z 2 ). The integration limits become z = −∞ and
z = +∞.
In the Fermi-LAT analysis [36] the J-factors are calculated using a solid angle of ∆Ω ∼ 2.4 · 10−4
sr i.e. an angular radius of 0.5◦ . In this analysis we set the angle to 0.5 degrees in order to be
able to compare the results to those of the Fermi-LAT collaboration.
5.4
Data
In this research we used the classical dwarf galaxies Fornax, Carina, Draco, Leo I, Leo II,
Sculptor and Sextans. We took the stellar masses for all dSphs from [41]. These values combined
with their Galactic longitude and latitude, l and b respectively, and their distance all taken from
[36] are shown in Table 5.1.
Name galaxy
Fornax
Carina
Draco
Leo I
Leo II
Sculptor
Sextans
l
(deg)
237.1
260.1
86.4
226.0
220.2
287.5
243.5
b
(deg)
-65.7
-22.2
34.7
49.1
67.2
-83.2
42.3
Distance
(kpc)
147
105
76
254
233
86
86
Mstar
(106 M )
24.5
0.513
0.912
4.90
1.17
3.89
0.851
Table 5.1: Properties of the used dSphs
5.4.1
Stellar kinematic data
We used the stellar kinematic data as presented in [55] for all but one dwarf in the form of projected positions and line-of-sight velocities for individual stars. For Leo II we used a different
Chapter 5, Analysis
29
set, as theirs is in preparation and has not been made public yet.
For each star the probability for its membership of the dSph Pi has to be taken into account.
For clarity the direct references will now be given as well as a brief explanation of how the
membership had to be determined.
The data for Leo I did we obtain from [56]. A star is defined to be a member if the heliocentric
velocity is in the range +250 to +320 km s−1 .
The data for Carina, Fornax, Sculptor and Sextans originates from [57]. For all individual stars
the probability Pi was provided. This value is estimated using an expectation-maximization
algorithm. Following [55] all stars having a probability lower than 0.95 were discarded.
The data set for Draco we used is presented in [58]. The members are the stars having a lineof-sight velocity vlos ∼ −290km s−1 and a relatively low log10 g and [F e/H].
We used the stellar kinematic data for Leo II from [59]. The membership is defined by having
a velocity ∼ 76.0 km s−1 .
All data sets originate from a similar period, i.e. ∼ 2008.
√
For each dwarf we binned the data into N bins where N is defined as the sum of the memberP
ship probabilities i.e. N = i=1 Pi . To estimate the velocity dispersion for each bin we used
the maximum-likelihood algorithm, this will be discussed in Section 5.4.2.
The number of stars we used for each dwarf galaxy is given in Table 5.2.
Name galaxy
Fornax
Carina
Draco
Leo I
Leo II
Sculptor
Sextans
Number of stars used
2273
746
453
328
172
1349
395
Table 5.2: Number of used stars per dwarf galaxy
5.4.2
Velocity dispersion profiles
For all the dwarfs we had to calculate the velocity dispersion from their measured radial velocities. This we did for each bin by using the maximum-likelihood procedure described in [60]
which will be explained in the next paragraph.
Chapter 5, Analysis
30
For each bin consisting of N member stars, we let vi be the observed radial velocity for the i -th
star. Assuming that these values follow a Gaussian distribution centered on the mean velocity
hui, the probability function is given by
N
Y
"
1 (vi − hui)2
q
p=
exp −
2 (σi2 + σp2 )
2π(σi2 + σp2 )
i=1
1
#
,
(5.15)
where σi is the internal measure uncertainty and σp the intrinsic radial velocity dispersion which
is the quantity of interest. The values for hui and σp we numerically obtained by maximizing
the logarithm of Equation 5.15. We could determine the confidence intervals from the 2x2
covariance matrix A given by
A=
a c
!
c b
.
(5.16)
The elements a and b are the variances of hui and σp respectively. The numerical expressions for
the diagonal elements can be calculated by using the inverse covariance matrix A−1 evaluated
at the estimates of hui and σp denoted by hûi and σ̂p

A−1 =
∂ 2 log(p)
 2∂hui2
∂ log(p)
∂hui∂σp

∂ 2 log(p) ∂σp ∂hui 
∂ 2 log(p) ∂σp2
(hûi,σ̂p )
.
Having the variances a and b, the 68% confidence intervals are given by hûi ±
5.4.3
(5.17)
√
a and σ̂p ±
√
b.
Surface brightness data
The surface brightness data sets we used for all the dSphs are taken from [61]. Here the stellar
surface density profiles are listed as the number of stars counted in elliptical annuli. Each ellipse
has its semi-major and semi-minor axis, a and b respectively, equal to the estimated axis of the
dSph as a whole. Since this research uses spherical symmetry for the Jeans analysis, the ellipti√
cal annuli had to be converted into circular annuli. We did this by setting the radius R ≡ ab
[55].
Chapter 6
Results
The MCMC simulation returns posterior probability distributions for the two free parameters.
These we plotted in a corner plot which also shows the covariance between the parameters. In
Figure 6.1 the corner plot for each dwarf consists of three frames. The upper left frame shows
the posterior probability distribution for the halo mass Mhalo in units solar mass. The lower
right frame presents this distribution for the scale radius rs in kpc. The third frame, the lower
left frame, shows the two dimensional posterior distribution. For both parameters the values
are in Log10 and for both the solid blue line represents the median value.
(a) Fornax
(b) Carina
31
Chapter 6, Results
32
(c) Draco
(d) Leo I
(e) Leo II
(f) Sculptor
Chapter 6, Results
33
(g) Sextans
Figure 6.1: One and two dimensional posterior distributions for the halo mass Mhalo and the
scale radius rs . The solid blue line represents the median value.
In Figure 6.2 the best fit result and the 95% confidence intervals (CIs) to the velocity dispersion
profiles σp for each dwarf are presented. The horizontal axis represents the radius in kpc, the
vertical axis the velocity dispersion in km/s. The CIs were chosen such that one can compare
them by eye to the fit results of [55] which are shown in their Figure 1.
(a) Fornax
(b) Carina
Chapter 6, Results
34
(c) Draco
(d) Leo I
(e) Leo II
(f) Sculptor
(g) Sextans
Figure 6.2: Velocity dispersion profiles for seven classical dwarfs. The black dots represent
the data, the blue solid line the median and the dashed the 95% CIs
Chapter 6, Results
35
The J-factors we obtained by applying the DC14 prescription are shown in green in Figure 6.3.
In order to compare them to those of the Fermi-LAT collaboration published in [36], these are
plotted in blue. The J-factors can be expressed in particle physics units GeV2 /cm5 (see e.g.
[36], [38]) or in astrophysical units M2 /kpc5 (see e.g. [2], [54]). For this plot both units are
used, the particle physics units on the left side and the astrophysical units on the right side.
Figure 6.3: J-factors for seven dwarf galaxies. In green are the J-factors obtained from this
research, in blue are those of the Fermi-LAT collaboration
The errors on the J-factors are derived by propagating the uncertainties on the free parameters
Mhalo and rs , the uncertainties on the light profile are not taken into account. The effect of this
will be discussed in Section 7.3.2.
Chapter 6, Results
36
In Figure 6.4 the derived J-factors by using the DC14 profile are presented as a function of the
log stellar-to-halo mass ratio Mstar /Mhalo . For each dwarf the corresponding J-factor from the
Fermi-LAT collaboration is plotted at the same mass ratio. Similar to Figure 6.3 both particle
physics as astrophysics units are used.
Figure 6.4: Jfactors msmh
The dwarf galaxies on the left side are respectively Leo I, Leo II, Sculptor, Carina and Draco.
The two dwarfs on the right side are respectively Sextans and Fornax.
Chapter 7
Discussion and conclusion
7.1
Derived halo masses
The derived halo masses can be read off of Figure 6.1 and are on the order of 109 M . In the
literature values are found, when assuming an NFW profile, on the order of 107 and 108 M [62]
and [63]. Hence, the DC14 masses are larger.
As explained in Section 4.4 the DC14 profile was applied to several dwarf galaxies in the Local
Group in [41]. In Figure 1 of this reference the fits to the circular velocities V (r1/2 ), r1/2 being
the deprojected half-light radii, are presented. The circular velocity is related to the velocity
√
dispersion σ used in this work through V (r1/2 ) = 3σ 2 . Moreover, the obtained halo masses
are listed in their Table 2.
Table 7.1 compares the masses for the DC14 profile we obtained in this research to those
presented by Di Cintio et al. in [41].
Name of galaxy
Fornax
Carina
Draco
Leo I
Leo II
Sculptor
Sextans
Mhalo (108 M )
This research Di Cintio
25
368
21
20
37
88
577
289
91
70
294
152
2
7
Table 7.1: Halo masses for seven dwarf galaxies derived in this research and By Di Cintio [41]
As the authors used a different approach, their results serve as an independent test. From Table
7.1 can be seen that the obtained values for the masses are similar, except for Fornax where
37
Chapter 7, Discussion and conclusion
38
there is a difference of factor 10. To see how this affects the density profile one has to turn to
Figure 4.1 where the evolution of the inner slope γ of the DC14 profile is depicted. The curve
for the inner slope follows a parabolic shape and due to symmetry the value for the inner slope
is in both cases γ ∼ 0.6. Therefore can be concluded that in both studies similar behavior for
the inner region of Fornax is found. This agrees with observations of Fornax for which a non
cuspy density profile has been measured [64].
7.2
Derived J-factors
As can be seen from Figure 6.3 the derived J-factors are for most dwarfs similar to those of the
Fermi-LAT collaboration. Except for Fornax and Sextans where the J-factor is lower when the
DC14 density profile is used.
This might be explained by Figure 6.4 where the J-factors are shown as a function of their
stellar-to-halo mass ratio. From this figure can be seen that for most of the dwarfs the ratio is
in the range Log10 Mstar /Mhalo ∈ [−3.5, −4.0], which is where the inner slope γ of the DC14
profile returns to the cuspy value of γ = 1. The same value the inner slope of the NFW profile
has. However for Fornax and Sextans the mass ratio is respectively −2.0 and −2.4, indicating
a more cored profile.
In [41] it is argued that environmental effects might have occurred in Sextans which results in an
incorrect halo mass and therefore the J-factor might not be correct. Other papers that derive Jfactors for several dwarf galaxies (e.g. [38], [55], [54]) however do not mention anomaly behavior.
Most of the errors we obtained on the J-factors are on the same order as those of the Fermi-LAT
collaboration except for Fornax and Carina. In the case of the former the error is very small.
An explanation for this is that the simulation got stuck in a local maximum. We tried to solve
this by repeating the MCMC analysis with different prior ranges, however these attempts were
unsuccessful. For Carina the corner plot in Figure 6.1 showed a clear bimodal distribution which
causes a somewhat larger error.
As we identified a potential problem with the J-factor of Fornax, further investigation is currently
in progress to identify the origin of this discrepancy.
Chapter 7, Discussion and conclusion
7.3
7.3.1
39
Impact of parametrizations
Impact of velocity anisotropy profile
The velocity anisotropy profile βani is a key ingredient of the Jeans analysis. It is degenerate with
the mass profile and cannot be measured directly. In [2] the impact of different parametrizations
is studied by comparing the results of a constant velocity anisotropy profile, the Osipkov-Merrit
profile [65] and the Baes and van Hese profile [66]. It is concluded that the Baes and van Hese
profile alleviates some of the biases in the Jeans analysis as it has the most free parameters.
Therefore the authors recommend this profile.
As pointed out in [67] this method of examining the impact of the velocity anisotropy profile
relies on marginalization over a parameter space and integration measure which are arbitrary
choices. Consequently it is not transparent what the impact is of selecting a given parametric
form, priors and integration measures. Therefore the authors argue a different method of assessing the impact. They suggest that the Jeans equation can be rewritten such that the dependence
of the anisotropy profile in the dark matter mass profile becomes explicit. By doing this they
show that for minimizing the J-factor, the results do not alter significantly by introducing a
radial dependence in the anisotropy profile (e.g. Osipkov-Merrit profile and Baes and van Hese
profile) compared to a constant profile as long as the projected stellar velocity dispersion σp (R)
is mildly varying in R.
The effect of setting the velocity anisotropy profile to zero is that the corresponding J-factor
is the minimum value. By setting the velocity anisotropy to a non-zero constant, the J-value
increases [67]. In this research we set the velocity anisotropy profile to zero and therefore the
obtained J-factors are the minimum values.
7.3.2
Impact of the light profile
Another important ingredient of the Jeans analysis is the light profile which appears both
in its projected form Σ(R) and deprojected form ν(r) in the projected solution of the Jeans
equation, see Equation 5.7. In [2] the impact of the light profile has been examined by fitting
simulated data with five different profiles: Plummer [68], exponential [69], Sérsic [70], King [71]
and Hernquist-Zhao which is given in Equation 4.1. It is concluded that the Hernquist-Zhao
parametrization provides the best possible fits to the data and reduces biases in the calculated
J-factors, since this profile is the most flexible i.e. has the most degrees of freedom. Therefore
can be concluded that the choice for using the Hernquist-Zhao profile in this research is wellmotivated.
Chapter 7, Discussion and conclusion
40
As mentioned in Chapter 6 the errors on the light profile have not been propagated to the
J-factor. The effect of this is examined in [2] by using three sets of mock data that differ in
sample size. It is found that for any sample size the obtained J-factor is only weakly affected
by including the uncertainties on the light profile when using the Hernquist-Zhao profile.
7.4
Effect of binned analysis
In this research we binned the data as is explained in Section 5.4.1. The effect of binned analysis
versus unbinned analysis is studied in [2]. The authors conclude that the results obtained by
both methods are very similar. The unbinned analysis reduces the statistical uncertainties on
the J-factors, namely on the ultrafaint dwarf galaxies. In this research we only used classical
dSphs and therefore the derived values for the J-factors will not change significantly when using
unbinned data.
7.5
Assessing the DC14 fit
In order to be able to asses the DC14 fit we calculated the reduced χ2 values for each dwarf.
These values allows one the describe the goodness of fit by comparing observed values to values
of the model in question. For comparing how well the DC14 profile fits the data relative to
other profiles, we repeated the procedure of fitting the data but now using a NFW profile. The
reduced χ2 values for both density profiles are presented in Table 7.2.
Name of galaxy
Fornax
Carina
Draco
Leo I
Leo II
Sculptor
Sextans
DC14
0.9
2.4
1.0
0.6
1.3
1.2
1.3
NFW
1.2
2.4
1.2
0.6
1.3
1.2
1.3
Table 7.2: Reduced χ2 for fits with DC14 and NFW profile
As can be seen from this table the reduced χ2 for each dwarf are very similar, indicating an
equally goodness of fit for the two profiles. Therefore this does not advocate the DC14 profile
over the NFW profile.
Chapter 7, Discussion and conclusion
7.6
41
Future work
At present there has not been detected an excess of gamma-rays from dwarf spheroidal galaxies
resulting in upper limits on the cross section, see Section 3.3. Now having the J-factors for a
profile that takes into account baryons would allow one to put constraints on the dark matter
annihilation cross section that encapsulates this. This is a suggestion for future work.
Another suggestion for further research would be to examine the effect of introducing a non-zero
constant velocity profile and see how the J-factors change. Moreover, as this parametrization is
used in other papers e.g. [38] and [54], the results can be compared more straightforwardly to
the values they find.
For improving the derived results it would be interesting to see what the J-factors are when an
unbinned analysis is performed and the uncertainties of the light profile are propagated to the
J-factor.
7.7
Conclusion
In this research the J-factors for seven classical dwarf galaxies have been calculated by fitting
their velocity dispersion with the DC14 density profile, the velocity anisotropy being set to zero
and a power law for the light profile. The J-factors we obtained are in general somewhat lower
than those derived by the Fermi-LAT collaboration and much lower for the two dSphs with the
largest ratio between stellar and halo mass. Putting constraints on the dark matter annihilation
cross section with these J-factors is for future work.
Chapter 7, Discussion and conclusion
42
Acknowledgments
I would like to express my gratitude to my supervisors Gianfranco, Christoph and Jennifer for
their guidance, useful conversations and patience. Moreover I would like to thank Shin’ichiro
Ando for being my second examiner. For their useful conversations and suggestions I would
like to thank Djoeke and Arianna di Cintio. Besides I thank Patrick Decowski for helping me
throughout my master with sorting out all UvA rules and regulations. Finally, I am thankful
to my friends, my mother and my sister for their unfailing support.
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