University of Amsterdam
MSc Physics
Gravitational and Astroparticle Physics
Master Thesis
Gamma rays from dark matter subhalos in the Milky Way
Predictions for detectability with Fermi-LAT
by
Djoeke Schoonenberg
10610790
July 2015
60 ECTS
Research carried out between September 2014 and June 2015
Supervisors:
Dr. Gianfranco Bertone
Dr. Jennifer Gaskins
Second Examiner:
Dr. Shin’ichiro Ando
Institute of Physics
Acknowledgments
First of all, I would like to thank my supervisors Jennifer Gaskins and Gianfranco Bertone.
You were the best supervisors I could have wished for. I am also thankful to Christoph
Weniger, Mark Lovell, Arianna di Cintio and Jürg Diemand for useful discussions. I would
also like to thank Shin’ichiro Ando for being my second reader, Michael Feyereisen for
proof-reading my manuscript, and Patrick Decowksi for helping me with the bureaucratic
parts of the Master’s program.
Also many thanks to my friends on ‘the fourth floor’; you always provided the necessary
distraction. Without you, last year at Science Park would have been a lot less fun!
I would also like to thank my friends in Utrecht, who are always there for me.
Lastly, I am very grateful to my parents, my brother, and my sister, for supporting and
loving me throughout my life.
2
Abstract
In the case that dark matter consists of weakly interacting massive particles (WIMPs), dark
matter subhalos in the Milky Way could be detectable as gamma-ray point sources due
to WIMP annihilations. In this thesis, we study the detectability of dark matter subhalos
with Fermi-LAT. We improve on previous work in two ways. Firstly, we take into account
the effects of baryons on the density profile within subhalos by adopting a profile recently
proposed by Di Cintio et al. Secondly, we use the results of the Via Lactea II simulation
— scaled to the latest cosmological parameters — to predict the local dark matter subhalo
distribution, instead of making assumptions on the effects of tidal stripping.
We find that baryons have a negligible effect on the detectability of point-like subhalos,
and we predict that about 3 subhalos are present in the Fermi-LAT point-source catalog
3FGL in the case of a 40 GeV WIMP annihilating to bb at a thermal cross-section. This
result is in conflict with the result recently found by Bertoni et al. due to their optimistic
treatment of tidal effects. Along with our predictions for the detectability of subhalos,
we use the number of subhalo candidate sources in 3FGL based on a spectral analysis
presented by Bertoni et al. to place upper limits on the WIMP annihilation cross-section.
In case there would be no candidate sources in 3FGL, our constraints are competitive with
those found by other studies. Using an optimistic flux threshold, we also predict that about
5 subhalos would be present in 3FGL in the case of a 100 GeV WIMP annihilating to bb at
a thermal cross-section — a scenario that has not yet been excluded by current constraints.
In addition, adopting the profile proposed by Di Cintio et al., we calculate the J-factor
(which parameterises the expected gamma-ray flux resulting from WIMP annihilations in
an astrophysical object independently of the exact choice of WIMP particle model) for some
of the dwarf galaxies used in the combined dwarf analysis of the Fermi-LAT collaboration.
We find that for two of them, the J-factors exceed the error margin quoted in the FermiLAT analysis. This suggests that the upper limits on the annihilation cross-section placed
in this analysis are not very robust against the choice of density profile.
4
Contents
1 Introduction
7
2 Particle Dark Matter
2.1 Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Rotation curves . . . . . . . . . . . . . . . . . . . . .
2.1.2 Gravitational lensing . . . . . . . . . . . . . . . . . .
2.1.3 The cosmic microwave background . . . . . . . . . .
2.2 Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Standard Model neutrinos . . . . . . . . . . . . . . .
2.2.2 Sterile neutrinos . . . . . . . . . . . . . . . . . . . .
2.2.3 WIMPs . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Thermal production of WIMPs in the early Universe . . . .
2.4 Dark matter structure . . . . . . . . . . . . . . . . . . . . .
2.4.1 Hierarchical structure formation . . . . . . . . . . .
2.4.2 Cosmological numerical simulations: Via Lactea II &
2.4.3 Density profiles . . . . . . . . . . . . . . . . . . . . .
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4 The Effect of Baryons on Dark Matter Halos
4.1 Discrepancies between observations and simulations . . . . . . . . . . . . .
4.1.1 Missing satellite problem . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Too-big-to-fail problem . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Cusp/core problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 DC14: a mass-dependent halo profile taking into account galaxy formation
4.2.1 Reionisation: only a fraction of halos form galaxies . . . . . . . . . .
4.3 DC14 for dwarf galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Methods
5.1 Calculating the annihilation flux from NFW halos in VL-II
5.2 Implementing DC14 profiles in VL-II . . . . . . . . . . . . .
5.3 Dark matter subhalo candidate sources in 3FGL . . . . . .
5.4 Calculating J-factors of dwarf galaxies with DC14 profiles .
45
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3 Indirect Detection of Dark Matter
3.1 Gamma rays from WIMP annihilations .
3.2 The Fermi Large Area Telescope . . . . .
3.3 The 3FGL point source catalog . . . . . .
3.4 The GeV excess from the Galactic Center
3.5 Null detection from dwarf galaxies . . . .
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Aquarius
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6 Results — Subhalos in VL-II
6.1 Detectability of subhalos . . . . . . . . . . . . . . . .
6.1.1 J-factors . . . . . . . . . . . . . . . . . . . .
6.1.2 Ndet against Mhalo . . . . . . . . . . . . . . .
6.1.3 Ndet against hσvi . . . . . . . . . . . . . . . .
6.1.4 Upper limits on the annihilation cross-section
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7 Results — J-factors of Dwarf Galaxies with DC14 Profiles
69
8 Discussion
8.1 Detectability of subhalos in VL-II . . . . . . . . . . . . . . . . . . . . . . . .
8.2 J-factors of dwarf galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Conclusions
75
10 Appendix
10.1 Line-of-sight integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Luminosity of a point-like dark matter halo with a NFW density profile . .
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11 Lay-man summary
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12 Samenvatting voor leken
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6
1
Introduction
About 85 % of all matter in the Universe is dark. One proposed class of candidates that
could make up the dark matter (DM) are WIMPS: weakly interacting massive particles.
WIMPs are a viable DM candidate for several reasons; such as that they naturally arise
from theories that were invented to solve problems other than the DM problem, and that
thermal production of WIMPs in the early Universe leads to the correct relic DM density
we observe today. See section ‘Particle Dark Matter’.
In the current model of cosmology, the DM halo hosting our Galaxy is predicted to
contain numerous substructures [1] [2] [3]. The most massive of these subhalos host the
dwarf galaxies orbiting the Milky Way, whereas smaller subhalos are expected to contain
no baryons. However, if DM consists of weakly interacting massive particles (WIMPs),
such subhalos could be detectable in gamma rays due to WIMP annihilations [4] [5].
The idea of this study is to update the work on the detectability of point-like DM
subhalos in gamma rays in the literature. No previous works addressing this topic have
taken into account the effects of baryons on the density profiles of DM subhalos (e.g.,
[6], [7], [8], [9]). Moreover, the most recent of these studies have either included a spatial
extension test — requiring DM subhalo candidate sources to be spatially extended — (e.g.,
[7]), or included only subhalos with masses up to 107 M [9].
In this thesis, we will consider the scenario where baryons flatten the profiles of relatively
massive DM subhalos, in line with recent findings (e.g., [10], [11], [12]). We will take this
effect into account by describing subhalos in the Milky Way with the mass-dependent
density profile proposed by Di Cintio et al. (hereafter ‘DC14’), which depends on the
stellar-to-total mass ratio of the halo [13].
As opposed to [9], who used a mass function and radial distribution of subhalos in the
Milky Way as determined from numerical simulations and extrapolated to masses below
the resolution limits of these simulations, we will consider the actual subhalos in the Via
Lactea II (VL-II) simulation with halo concentrations scaled to the latest cosmological
parameters as measured by Planck [14]. We consider all subhalos in VL-II likely bound
to the host halo, covering a mass range of ∼ 105 − 1011 M . We focus on subhalos that
would appear point-like to Fermi-LAT, but comment on the presence of spatially extended
subhalos.
Taking the VL-II halo as a proxy for the DM halo hosting the Milky Way, we will then
calculate the number of DM subhalos one would expect to show up as unidentified sources
in the recently released Fermi-LAT third point-source catalog 3FGL ([15]) for different
benchmark DM particle models. We will compare our predicted number of detectable
subhalos to the number of DM subhalo candidate sources in 3FGL recently found by [9] to
place upper limits on the DM annihilation cross-section.
In calculating the gamma-ray flux resulting from WIMP annihilations, the so-called Jfactor is often introduced. The J-factor accounts for the amount of dark matter annihilation
one is looking at, and does not depend on the exact choice of the dark matter particle model.
7
It is therefore called the ‘astrophysical term’ in the formula for the annihilation flux (see
Eq. 34 and the Appendix). We will calculate J-factors of dwarf galaxies in the Milky Way
with DC14 density profiles and compare our results to the J-factors adopted by Fermi-LAT
in their combined likelihood analysis of dwarf galaxies [16].
8
2
Particle Dark Matter
The Standard Model of cosmology is the ΛCDM model, which states that the Universe
started as a hot and dense environment some 13.8 billion years ago ([14]), after which it
expanded and cooled down to the Universe we live in today. According to this model,
the Universe today contains a positive cosmological constant Λ1 , that accounts for the
accelerated expansion of the Universe that was discovered in 1998 observing type Ia supernovae [18] [19]. Besides the presence of a positive cosmological constant, the ΛCDM
model includes a Cold Dark Matter (CDM) component that contributes about 6 times
as much to the total matter density of the Universe as baryonic matter does. CDM is
a hypothesised form of matter that is non-baryonic and interacts gravitationally, but not
electromagnetically, with particles of the Standard Model of particle physics.
The matter density of the Universe is commonly parametrised as the dimensionless
parameter Ωmatter [17]:
ρmatter
ρcrit
Ωmatter =
(1)
where ρcrit is the critical density of the Universe, that is, the density that makes the spatial
geometry of the Universe Euclidean (flat). It is given by:
ρcrit, 0 =
3H02
8πG
(2)
where the subscript 0 denotes the value today and where H0 is the current value of the Hubble constant. Plugging in the measured value H0 = 67.8 km s−1 Mpc−1 [14] and converting
the resulting value for ρcrit, 0 to units of GeV kpc−3 , we find ρcrit, 0 = 127.5 GeV kpc−3 .
Similarly, the energy density associated with the cosmological constant can be parametrised as [17]:
ΩΛ =
Λ
3H02
(3)
1
The cosmological constant was included in the field equation of general relativity by Einstein, as a
reaction to the work of Friedmann in 1922 that the equations of general relativity in an isotropic and homogeneous Universe are not static and that therefore the Universe should be either expanding or contracting
[17]. This prediction by Friedmann was confirmed by Hubble in 1922, who showed that galaxies are moving
away from us with velocities proportional to their distance: v = H0 ·d, where H0 is the value of the so-called
Hubble constant today. Following this proof of the expansion of the Universe, Einstein is said to have called
his invention of Λ the ‘biggest blunder of his life’. However, it was common among physicists to keep the
factor in the equation, since nothing really forbids it to be there. In fact, in 1999 it was discovered that
the expansion of the Universe is accelerating, an observation favouring a non-zero value of Λ. Although the
value of Λ is being constrained by cosmological observations nowadays, it is still unknown what makes up
this ‘dark energy’.
9
The Universe is measured to be nearly perfectly flat on large scales [14], meaning that the
total energy density, consisting of the matter component ρmatter and the energy component
Λ, has a value very close to the critical energy density. Therefore:
Ωmatter + ΩΛ = 1
(4)
Evidence for a DM contribution to Ωmatter comes from several independent observations
on different length scales that will be discussed in more detail in the section ‘Evidence’.
On large length scales, the ΛCDM model has been very successful at explaining observations [1]. Besides the observation that the expansion of the Universe is accelerating, two
other key predictions of ΛCDM that match observations are the existence of the cosmic microwave background radiation (CMB), and the abundances of the light elements hydrogen,
deuterium, and helium.
In spite of its successes on cosmological scales, the ΛCDM model is not perfect. Although the cosmological constant (also called ‘dark energy’ (DE)) and CDM are vital
ingredients to the theory, it is still a mystery what they exactly are. The value of the cosmological constant predicted by quantum field theories is 123 orders of magnitude larger
than the measured value of Λ ([20])2 , and though the gravitational effects of DM are clearly
observed, its particle nature is still unknown, despite great efforts worldwide. Furthermore,
there seems to be some tension between observations of DM halos and ΛCDM predictions,
although these arise from comparing DM-only simulations with observations and might be
solved by taking into account baryonic effects (e.g., [10], [21], [11], [12], [13]). This will be
discussed in more detail in the section ‘The effect of baryons on dark matter subhalos’.
The nature of CDM is central to this study and we will therefore leave DE for whatever
it might be, and focus in the remainder of this section on DM: the evidence for its existence,
some proposed candidates for DM particles and their production in the early Universe, and
the distribution of DM throughout the Universe. It is impossible to provide a complete
overview of the evidence for the existence of DM and the enormous number of proposed
DM particle candidates. In this thesis, we will focus on a few compelling pieces of evidence,
and treat only those DM particle candidates that are relevant for this study. For a more
complete overview we refer the reader to [1] or [22].
2
This discrepancy is called the ‘cosmological constant problem’, and is one of the greatest (fine-tuning)
problems in theoretical physics.
10
2.1
2.1.1
Evidence
Rotation curves
The earliest evidence for the existence of missing matter on galactic and galaxy cluster
scales came from the observations of the motions of stars. In 1932, Jan Oort studied the
motions of nearby stars in the Milky Way and discovered that the visible mass alone could
not account for these motions, but that there should be more mass in our Galaxy than
only the luminous matter [23].
Around the same time, Fritz Zwicky discovered that the radial velocities of galaxies
in the Coma cluster could not be explained by gravitational effects of only the luminous
matter. Instead, his observations implied the presence of an amount of DM much greater
than the amount of observed luminous matter [24].
Since these pioneering discoveries, the evidence for the existence of a large amount of
DM in galaxies and galaxy clusters has accumulated. The most convincing of these are
galactic rotation curves — graphs of the circular velocities of stars and gas as a function of
their distance from the galactic centre [22]. In 1980, Vera Rubin noticed that the rotation
curves of spiral galaxies did not agree with what was expected from the gravitational effects
of the visible matter in those galaxies [25].
In Newtonian mechanics, Kepler’s third law gives a circular velocity vcirc expressed as:
r
GM (r)
vcirc =
(5)
r
where M (r) is the mass enclosed within radius r. From this relation one would expect for
a galaxy where most of the mass is concentrated in the centre that the orbital velocities
of stars decrease with the distance from the galaxy core r as vorb ∝ r−1/2 . This should be
the case for stars beyond the visible galaxy disk if there were no mass beyond the visible
galaxy disk.
However, rotation curves of galaxies were discovered to exhibit a flat behaviour at
distances beyond the visible disks [25]. A typical example is shown in Fig. 1. The fact
that the rotation curves of galaxies are nearly constant at large radii
R implies that the mass
at those radii should be proportional to r. Because M (r) ≡ 4π ρ(r)r2 dr, where ρ(r) is
the mass density profile, the flat rotation curves provide evidence for the existence of a
dark halo with ρ(r) ∝ 1/r2 .
Recently, Iocco et al. compiled many rotation curve measurements of the Milky Way
and compared the result to baryonic mass distribution models [26]. They concluded that
rotation curve measurements provide evidence that DM must be present even within our
solar circle, as well as in the centre of our Milky Way — even without making assumptions
about its distribution.
11
Figure 1: The measured rotation curve of the galaxy NGC 2403. The dotted, dashed, and
dash-dotted line correspond to the contributions of the gas, the disk, and dark matter,
respectively. Picture adapted from [27].
2.1.2
Gravitational lensing
Bullet clusters are providing additional evidence for the existence of a non-baryonic contribution to the mass of galaxy clusters. In a Bullet cluster scenario, two clusters of galaxies
have collided. The location of the gas, which constitutes most of the baryonic matter
in galaxy clusters, can be determined from its X-ray radiation. The centre of the total
mass can be determined from gravitational lensing of background galaxies by the clusters:
according to Einstein’s theory of general relativity, mass bends space-time and therefore
the path of light [28]. In Fig. 2, a typical example of a Bullet Cluster is depicted. In
the left panel, the blue colour corresponds to the centre of the total mass, obtained from
gravitational lensing. The red colour corresponds to X-ray data and coincides with the intergalactic gas. In the right panel, mass density contours obtained from weak gravitational
lensing are superimposed over a picture from the Hubble Space Telescope. Bullet clusters
have been interpreted as follows: the populations of hydrogen gas from both colliding clusters are still interacting in the middle of the scene after the collision, whilst most of the
12
mass from both clusters has moved through each other without colliding. Therefore, Bullet clusters provide evidence for a massive DM particle that self-interacts only weakly [29].
While the need for DM from the observations of rotation curves might be circumvented by
modifying the theory of gravity on large scales, the observations of Bullet clusters are very
hard to explain without invoking the presence of CDM [1].
Figure 2: The Bullet cluster 1E 0657-56. Left panel: X-ray images from Chandra (red)
together with gravitational lensing results (blue). Right panel: mass density contours
(green) from weak gravitational lensing, superimposed over a picture from Hubble Space
Telescope. Picture taken from [28].
Even for non-colliding clusters, we know that there must be more matter present than
visible matter: gravitational lensing studies show that galaxy clusters and galaxies must
contain a lot more mass than the amount of mass they contain according to the massto-brightness relation, to account for the gravitational lensing they perform. Clusters of
galaxies are highly DM dominated, having mass-to-light ratios of about 100 to 300, while
galaxies have mass-to-light ratios of about 10 to 20 [1].
2.1.3
The cosmic microwave background
In 1965, Penzias and Wilkinson published the discovery of the cosmic microwave background radiation (CMB). They were radio astronomers at the Bell Telephone Laboratories
and could not get rid of an excess noise with a frequency of 4080 MHz that came from all
directions. It was estimated that the noise corresponded to a black body with a temperature of 3.5 ± 1.0 Kelvin. Today, the temperature of this isotropic background radiation
has been measured to be 2.725 ± 0.0004 K [17]. The ‘noise’ turned out to be the CMB: a
revolutionary discovery providing great support for the Big Bang theory. The CMB is a
residual from the very early Universe: at the time, photons were in thermal equilibrium
with electrons. A few hundred thousand years after the Big Bang, electrons got bound
13
to protons to form hydrogen atoms, and the photons decoupled from the thermal equilibrium. From that moment onward, the photons propagated mostly unscattered through
the Universe for about 13 billion years before being detected today. Due to the expansion
of the Universe, the wavelengths of the photons got red-shifted. Since the wavelength of
radiation is inversely proportional to the energy, the CMB photons are 1000 times less
energetic today than when they were produced [17].
The CMB is isotropic in temperature apart from small angular fluctuations: ∆T /T ∼
−5
10 [22]. Small differences in CMB temperature between different locations in the sky are
generated by regions of density enhancements at the time of photon decoupling: photons in
a region of higher density got red-shifted more because they had to climb out of the gravitational potential well generated by the density enhancement. Therefore, the temperature
fluctuations in the CMB are related to small density perturbations in the primordial Universe, which eventually turned into the stars and galaxies of the inhomogeneous Universe
we live in today [17].
The temperature anisotropies in the CMB are usually expanded into spherical harmonics Ylm (θ, φ):
∞ X
+l
X
δT
(θ, φ) =
alm Ylm (θ, φ)
T
(6)
l=2 m=−l
The coefficients alm have been measured by satellite experiments such as WMAP [30]
and its successor Planck [14]. In Fig. 3, the quantity l(l + 1)Cl /2π is plotted against
the multipole l, as measured by WMAP. The best-fit cosmological parameters can be
determined from the data and provide an independent measurement of the abundance of
baryons and matter in the Universe. The WMAP estimates of these abundances from
fitting the ΛCDM model to the CMB data are [17]:
Ωmatter = 0.27 ± 0.04, Ωb = 0.044 ± 0.004
which agree with the predictions of ΛCDM concerning the abundances of the light elements
hydrogen, deuterium, and helium [22], and provide an independent measurement of the
DM-to-baryonic density ratio in the Universe: Ωmatter /Ωb ∼ 6.
2.2
2.2.1
Candidates
Standard Model neutrinos
Before invoking theories beyond the Standard Model of particle physics (SM), let us begin
our discussion of candidates for DM particles in the context of the SM. Until recently, SM
neutrinos were considered excellent candidates for DM [22]: they are stable, do not couple to
the electromagnetic force and interact only via the weak nuclear force. In the SM, neutrinos
14
Figure 3: The angular power spectrum of the temperature anisotropies in the CMB as
measured by WMAP. The solid line corresponds to the prediction of the ΛCDM model
with the best-fit parameters. Figure taken from [30].
are massless. However, the Super-Kamiokande experiment discovered atmospheric neutrino
oscillations, which implies that neutrinos in fact do have mass [31], another requirement
for being a good DM candidate. However, their total relic density is predicted to be [22]:
Ων h2 =
3
X
mi
93 eV
(7)
i=1
and the latest upper limit on the sum of the masses of the three neutrino species is [14]:
3
X
mi < 0.23 eV
(8)
i
which means that their total relic density cannot exceed the following upper bound:
Ων h2 < 0.0025
(9)
Hence, neutrinos cannot account for the total DM relic density. Another argument against
neutrinos making up all of the DM is that their relativistic nature makes them a form of ‘hot
15
dark matter’, that is, they would have erased density fluctuations below their free-streaming
length — around a scale of ∼ 40 Mpc [32]. This would mean that structure formation
happened through a top-down process — large-scale structures forming before small-scale
ones. This seems to be excluded by observations of galaxies (small-scale structures) at high
red-shifts, which favour a bottom-up scenario of structure formation instead [22].
2.2.2
Sterile neutrinos
Sterile neutrinos are hypothetical particles that are the right-handed counterparts to the
left-handed (active) SM neutrinos. They do not couple to SM particles, except through
mixing with the left-handed SM neutrinos [22]. They were proposed as viable DM candidates in 1993 [33].
2.2.3
WIMPs
The most popular class of DM candidates are WIMPs: Weakly Interacting Massive Particles. WIMPs do not only interact gravitationally, but also couple to the weak nuclear
force. Many theories that extend the Standard Model of particle physics naturally propose WIMPs. One such theory is Supersymmetry (SUSY), which was invented to solve the
hierarchy and unification problems of the SM [34] [22]. In SUSY, all fermions have a bosonic
superpartner, and vice versa. Hence, SUSY doubles the number of particles in the SM. To
ensure the stability of the proton, an additional symmetry called R-parity is introduced. If
R-parity is conserved, the lightest supersymmetric particle (LSP) is stable — which makes
it an ideal WIMP candidate. Of the SUSY particle spectrum, a neutralino is typically the
LSP, and also fulfils another requirement for a DM particle by being electrically neutral.
Typical WIMP masses are in the range 1 GeV - 1 TeV [1].
2.3
Thermal production of WIMPs in the early Universe
In the standard scenario of dark matter production in the early Universe, collisions between
particles and antiparticles produced DM particle pairs, which could again annihilate into
SM particle-antiparticle pairs [1] via:
χ + χ SM + SM
(10)
where χ represents a DM particle and SM a standard model particle. At temperatures
higher than the WIMP mass, that is, early enough in the Universe, production and annihilation reactions of WIMPs were in thermal equilibrium. As the Universe expanded, it
cooled down. When the temperature fell below the dark matter particle mass, the number
of produced WIMPs decreased with e−mχ /T (the Boltzmann factor), since only particles
with kinetic energies in the tail of the Boltzmann distribution were still energetic enough
16
to produce WIMPs. The annihilation and production rate both decreased due to the expansion of the Universe, since these rates are proportional to the square of the number
density of particles (i.e., the chance of particles colliding), which is inversely proportional
to the volume of the Universe. When the expansion rate exceeded the annihilation and production rates, WIMP ‘freeze-out’ happened, and the comoving number density of WIMPs
became constant. Assuming this process happened, let us calculate the number density of
DM particles today, using a manipulated form of the Boltzmann equation [22]:
dn
= −3Hn − hσann vi(n2 − n2eq )
dt
(11)
where t is time, H is the Hubble parameter, neq is the WIMP number density in equilibrium,
and hσann vi is the velocity-averaged WIMP annihilation cross-section. In Eq. 11, the lefthand-side corresponds to the change in n with time, which equals the decrease in n due
to the expansion of the Universe (first term on the right-hand-side), plus the decrease in
n due to annihilations plus the increase in n due to pair-production (second term on the
right-hand-side).
Because the early Universe is radiation dominated, the expansion rate parameter H
falls with the temperature as [35]:
√
H(T ) = 1.66 g∗ T 2 m−1
pl
(12)
where g∗ is the effective number of relativistic degrees of freedom (that decreases slowly in
time as more particle species decouple from thermal equilibrium) and mpl ∼ 1019 GeV is
the Planck mass. Assuming that entropy is conserved, the entropy per comoving volume
is constant, which means that nχ /s is constant, where s ∼ 0.4g∗ T 3 is the entropy density
of the Universe. When the expansion rate H falls below the WIMP annihilation rate
Γ = nhσann vi, WIMP freeze-out happens and from that moment onwards the comoving
WIMP density remains constant. For typical weak-scale annihilations, the DM freeze-out
temperature is Tf ∼ mχ /20 [35]. Using the above relations, we can now write:
n χ
s
∼
0
√
1.66 g∗ Tf2
shσann vimpl
√
√
1.66 g∗
1.66 · 20 g∗
∼
∼
0.49Tf hσann vimpl
0.49hσann vimχ mpl
(13)
where the subscript 0 denotes the value today. Plugging in approximate values for the
Planck mass mpl ∼ 1019 GeV, the number of relativistic degrees of freedom g∗ at the
freeze-out temperature of a 100 GeV WIMP, the current entropy density s0 ∼ 4000 cm−3
and the critical density today ρc, 0 ∼ 10−5 h2 GeVcm−3 , where h is the dimensionless Hubble
constant that defines H0 through H0 = h · 100 km/s/Mpc, we find for the relic DM density
today [35]:
17
Ω χ h2 =
mχ nχ
3 · 10−27 cm3 s−1
≈
ρc, 0
hσann vi
(14)
where the WIMP mass mχ dropped out of the equation (but did have an influence on
the value of g∗ ). The result that the DM relic density is inversely proportional to the
annihilation cross-section makes sense: the larger the annihilation cross-section, the longer
the particles stay in equilibrium, hence the colder the Universe is when they decouple, which
means their density is further suppressed by a smaller Boltzmann factor. The dependence
of the relic WIMP density on the annihilation cross-section is shown in Fig. 4.
Plugging in the current value of Ωχ h2 ≈ 0.14 [14] into Eq. 14, we find:
hσann vi ≈ 2.1 · 10−26 cm3 s−1
(15)
which is remarkably close to the annihilation cross-section of a weakly interacting particle
with mass of order 100 GeV: hσann vi ∼ α2 (100 GeV)−2 ∼ 10−25 cm3 s−1 , where α ∼ 10−2
is the fine-structure constant [35].
This coincidence is called the ‘WIMP miracle’, and has sparked a lot of interest in the
WIMP scenario for dark matter. Using the less recent measurement of Ωχ h2 ≈ 0.11 by
WMAP3 [17], we find hσann vi ≈ 3 · 10−26 cm3 s−1 . Since this value is often used as a
benchmark in the literature, we will refer to this value of the annihilation cross-section as
the thermal cross-section.
The required annihilation cross-section to arrive at the correct relic density of dark
matter might be different from the value hσann vi ≈ 2.1 · 10−26 cm3 s−1 that resulted from
the rough calculation above, if there exists a particle only slightly heavier than the WIMP.
In this case, coannihilations occur between the WIMPs and the heavier particles, which
change the dark matter freeze-out time and therefore the relic abundance [35].
18
Figure 4: The comoving number density of WIMPs in the early Universe. The solid
line corresponds to the WIMP abundance in equilibrium; the dashed line is the actual
abundance after dark matter freeze-out. For a larger annihilation cross-section, the WIMPs
stayed in thermal equilibrium for a longer time, which results in a lower relic density. Figure
taken from [35].
19
2.4
2.4.1
Dark matter structure
Hierarchical structure formation
In ΛCDM, the Universe started out as smooth, with small structures collapsing under
their self-gravity first and merging into ever bigger structures as the Universe grew into
the lumpy Universe with galaxy clusters we observe today [2].
This theory of structure formation is called hierarchical structure formation, and dark
matter plays a crucial role in this theory. Baryons alone cannot explain structure formation
since they decouple too late from the photons, such that gravitational overdensities would
not have had enough time to grow [22].
Hierarchical structure formation predicts that the DM halos that host galaxies like the
Milky Way contain lots of smaller substructures that have fallen into it, and that are either
gravitationally bound (subhalos) or unbound (streams). The mass of the smallest subhalos
m0 depends on the particle nature of dark matter, since it is set by collisional damping
and free-streaming in the early Universe [36] [37]. For WIMP masses in the range of a few
GeV to a few TeV, the cutoff is in the range m0 = 10−12 - 10−4 M [38].
When halos fall into the host halo, their total mass gets altered due to tidal effects. Tidal
stripping removes mass from the outer parts of subhalos. This effect can be approximated
by removing the mass beyond the tidal radius, where the tidal radius is defined as the
radius at which the subhalo density is equal to the host density. Since the tidal radius is
often larger than the scale radius of an infalling halo, the matter within the scale radius
of a halo is mostly preserved [5]. The concept of scale radius will be defined in the section
‘Density profiles’.
2.4.2
Cosmological numerical simulations: Via Lactea II & Aquarius
Because structure formation is a highly non-linear process [17], and galaxies are constituted
of a great many particles, it is practically impossible to analytically calculate the evolution
of the dark matter distribution from the initial conditions in the early Universe. Therefore,
cosmological numerical simulations are performed to deduce the dark matter distribution
in a galaxy such as the Milky Way. In the past years, several such simulations have been
run on the most powerful supercomputers. Two of these were the Via Lactea II (VL-II)
project [3] and the Aquarius project [39]. Both of these simulations include only dark
matter particles and do not take into account baryons. Aquarius contains a sample of six
simulated Milky Way-like dark matter halos, whereas VL-II has one. One ‘particle’ in the
simulations corresponds to a few thousand solar masses, and in total one billion (VL-II)
or two hundred million (Aquarius) particles were included. With the linear power spectrum derived from the cosmic microwave background and other cosmological parameters
as input parameters, the particles were evolved from the early Universe until presentday. Aquarius used the WMAP1 values for the cosmological parameters, VL-II used the
WMAP3 values. This difference resulted in slightly different concentrations of subhalos
20
between the two simulations (the concept of concentration will be defined in the section
‘Density Profiles’).
The simulated halos contain numerous smaller substructures, which is predicted by the
ΛCDM theory of hierarchical structure formation. The smallest subhalos resolved in VL-II
are ∼ 105 M . The mass function of subhalos in the simulated halos can be approximated
by:
F (µ, Msub ) = F0
Msub
M
−µ
(16)
where the logarithmic slope parameter µ equals -2 for VL-II ([40]) and -1.9 for Aquarius
([39]) and where F0 is a normalisation factor. The mass function can be thought of as
a probability distribution function where F (µ, Msub ) gives the normalised probability of
finding a subhalo of mass M < Msub in the host halo.
The mass density profiles of VL-II and Aquarius, extrapolated to small radii, are plotted
in Fig. 5.
Figure 5: The mass density profiles of the halos in the VL-II and Aquarius simulations,
extrapolated to small radii. Figure taken from [6].
21
In this thesis, we will use the results of the VL-II simulation to calculate the number of
subhalos that could be detectable in gamma rays. We will scale the concentrations of the
subhalos in VL-II to match the latest cosmological parameters.
2.4.3
Density profiles
In 1997, Navarro, Frenk and White proposed a density profile — thereafter dubbed the
NFW profile — to describe the dark matter distribution within CDM halos in simulations
[41]:
ρs
ρ(r) = h
i2
r
r
1
+
rs
rs
(17)
where ρs and rs , the scale density and scale radius, respectively, are characteristics of
individual halos. The NFW profile falls off as r−1 in the inner region (r < rs ) and continues
to fall off more rapidly, as r−3 , in the outer parts. Formally, the density is infinite at r = 0.
Therefore, it is called a ‘cuspy’ profile. The integral of the NFW profile over an infinite
volume is divergent, which means that the halo mass would be infinitely large if one goes
to infinitely large radii. This is of course non-physical, and to circumvent this problem one
can integrate ρ(r) up to a certain radius. A common definition of halo mass is M200 , which
is defined as the mass enclosed within a sphere that has an average density of 200 times
the cosmological critical matter density. The radius r200 is often called the ‘virial radius’.
The dimensionless concentration parameter c of a halo is defined as its virial radius divided
by its scale radius: c ≡ r200 /rs .
Another profile that is used to describe dark matter density distributions is the Einasto
profile, which was originally introduced in a two-dimensional form in the 60s to model
the surface brightness distributions of galaxies ([42]) and found its three-dimensional dark
matter applications in the 80s [43]. In the Einasto profile the logarithmic density is given
by [44]:
d ln ρ
r α
∝
(18)
d ln r
r−2
such that
ρ(r) ∝ exp(−Ara )
(19)
where A and a are constants to be determined for each halo. The Einasto profile seems to
provide good fits to simulated dark matter halos, indeed, often better than NFW profiles
(e.g., [45]). The value of A is related to the concentration of the halo and the value of the
shape parameter a is thought to depend on the accretion history of the halo [44].
22
ΡHrL
104
NFW
Einasto
Isothermal
100
Burkert
1
0.01
10-4
10-5
r
0.001
0.1
10
Figure 6: A plot of the dark matter density as a function of radius for the four different
density profiles discussed in the text. All constants were set to 1, and the axes have
arbitrary units.
Other profiles sometimes discussed in literature are the isothermal sphere [4] and the Burkert profile [46], characterised by:
ρ0
,
1 + (r/r0 )2
(20)
ρ0
,
(1 + r/r0 )(1 + (r/r0 )2 )
(21)
ρ(r) =
and
ρ(r) =
respectively. The NFW, Burkert and isothermal profiles can be generalised to a fiveparameter double power-law [47]:
ρ(r) =
ρ0
(r/r0 )γ (1 + (r/r0 )α )(β−γ)/α
23
(22)
where (α, β, γ) are (1, 3, 1) for the NFW profile, (1, 3, 2) for the Burkert profile, and
(2, 2, 0) for the isothermal profile. The four profiles discussed above are illustrated in Fig.
6.
The NFW and Einasto profile are successful at describing the density distributions of
CDM halos in simulations, however, they seem to be in conflict with certain observations.
The problems that arise when matching these profiles to observations are discussed in the
section ‘Discrepancies between observations and simulations’. It has been proposed that
including baryonic effects in numerical cosmological simulations can resolve these issues
(e.g., [10], [12]). The effects of baryons on dark matter distributions will be discussed
together with a modified density profile introduced by Di Cintio et al. [13], which is
based on the five-parameter profile (Eq. 22), where the slope parameters α, β, and γ are
dependent on the baryonic content of the halo. However, we will first take the NFW profile
as our starting point for the discussion of the subhalos in VL-II.
From rV max and Vmax to NFW profile parameters
Two observable quantities of subhalos in the VL-II simulation are the maximum circular
velocity, Vmax , and the radius at which this velocity is reached, rV max . For a NFW density
profile, the following equations hold, where rs and ρs are the scale radius and scale density,
respectively (Eq. 7 through 11 in [48]):
rs = rV max /2.163
Vc2 (r) = 4πGρs rs3
(23)
f (r)
,
r
(24)
r/rs
.
1 + r/rs
(25)
where
r
f (r) = ln 1 +
rs
−
Plugging in Vmax for Vc (r) in Eq. 24, we find:
2
Vmax
= 4πGρs rs3
1
2.163
ln(3.163) −
,
2.163rs
3.163
(26)
such that we can write ρs in terms of rs and Vmax :
ρs =
2
2.163 · Vmax
4πGrs2 [ln(3.163) − 2.163/3.163]
24
(27)
Using Eqs. 23 and 27, we can fully determine a NFW-profile for each halo in the simulation.
The virial radius r200 , which is defined as the radius at which the included mass equals 200
times the critical density of the Universe, can be found by requiring:

Z
r200

r 2 ρs

h
i2 dr /
r
1 + rrs
rs

200ρcrit = 4π
0
4 3
πr200 ,
3
(28)
i.e., by requiring that 200 times the critical density (left-hand-side) equals the halo mass
integrated out to the virial radius divided by the volume enclosed by that radius (righthand-side), i.e., the average density within that volume — and that must be true by
definition of r200 .
Substituting u = 1 + rrs in the integral:
Z
4π
0
Z
r200
r2 ρs
h
i2 dr =
r
r
1
+
rs
rs
4πρs rs3
=
4πρs rs3
1+r200 /rs
du
1
1
− 2
u u
1
= 4πρs rs3
= 4πρs rs3
r200
1 1+ rs
ln(u) +
u 1
!
r200
1
ln 1 +
+
−1
rs
1 + rr200
s
r200 + rs
ln
+
rs
1
r200 +rs
rs
r200 + rs
−
r200 + rs
r200 + rs
r200
= 4πρs rs3 ln
−
rs
r200 + rs
we can determine r200 by solving the following equation for r200 :
rs + r200
r200
3
3
200r200 ρcrit = 3ρs rs ln
−
rs
rs + r200
!
(29)
(30)
Subsequently, the virial mass M200 can be calculated using:
4 3
· 200 · ρcrit .
M200 = πr200
3
25
(31)
Scaling rVmax to Planck15 cosmological parameters
The concentrations and velocity profiles of subhalos in CDM simulations such as VL-II
are dependent on the adopted cosmological parameters [49]. As predicted by the theory
of hierarchical structure formation, the later small-mass halos (which could eventually become subhalos in a Milky Way-like host halo) form in the history of the Universe, the less
concentrated they are, reflecting the lower density of the Universe at later times. Therefore, adopting cosmological values that shift the small-mass halo formation to later epochs
results in less concentrated subhalos. The concentration of a halo is related to its radius
of maximum circular velocity rVmax and is an observable quantity in simulations. In [49],
Polisensky and Ricotti show how rVmax in CDM simulations scales with the cosmological parameters σ8 and ns at fixed Vmax (they found no dependence of rVmax on other
cosmological parameters). This scaling is given by:
rVmax ∝ (σ8 5.5ns )−1.5
(32)
The VL-II simulation is based on the WMAP3 cosmology and has σ8 = 0.74 and ns =
0.951. The latest Planck results provide σ8 = 0.82 and ns = 0.9667 [14], such that the
scaling becomes:
rVmax, VLII
rVmax, Planck15
=
(0.74 · 5.50.951 )−1.5
(0.82 · 5.50.9667 )−1.5
rVmax, Planck15 =
rVmax, VLII
1.21
(33)
For our analysis, we have scaled the rVmax -values of the halos in VL-II according to Eq. 33.
For a given maximum circular velocity, the radius of maximum circular velocity becomes
smaller, which means that the subhalos in VL-II become more concentrated after applying
Eq. 33.
26
3
Indirect Detection of Dark Matter
Figure 7: Three different ways of searching for Dark Matter: indirect, direct and with
colliders. Figure from [50].
There are different ways in which one can search for DM. These are schematised in Fig. 7,
which shows a Feynman diagram of two initial particles and two final particles, where the
interaction concerns New Physics and will for convenience be considered a black box. There
are three options. If time flows from right to left in the diagram, two Standard Model (SM)
particles interact to form two DM final particles. Detecting DM via such a process could
happen in a collider such as the LHC, where protons (SM particles) are collided with each
other at high energies, in the hope that new particles (possibly DM) are produced. If time
flows from top to bottom or from bottom to top, a DM particle scatters off a SM particle.
Experiments dedicated to this process are called direct detection experiments — they are
looking for signatures of DM particles scattering off of atomic nuclei3 .
3
The cross-section of WIMPs scattering on other particles is related to the WIMP annihilation crosssection, but is model-dependent [51].
27
The third option is if time flows from left to right in the diagram. In that case, two DM
particles annihilate to form two SM particles in the final state. Detecting DM through its
annihilation products is called indirect detection and is the method we are concerned with
in this thesis.
Among possible SM final particles in the annihilation process of DM, photons are
especially interesting, because they do not carry charge and are therefore not deflected by
magnetic fields. This means that photons arriving at Earth point directly back to their
production site. WIMP annihilations typically produce photons with energies around the
GeV scale [1]. Such high-energy photons are called gamma rays and can be measured with
Cherenkov telescopes on Earth or pair-conversion telescopes in space — such as the Fermi
Large Area Telescope, which will be considered in this thesis. See section ‘The Fermi Large
Area Telescope’.
3.1
Gamma rays from WIMP annihilations
The annihilation of WIMP pairs into one photon plus one other neutral particle (another
photon, a Z-boson or a Higgs boson), would lead to monochromatic gamma rays of a
given energy. The observation of such a gamma-ray line would provide a smoking-gun
signal of DM annihilations. Unfortunately, however, direct annihilation to photons is loopsuppressed and subdominant to the continuous γ-ray spectrum that is produced in the
cascades following DM annihilations to lepton-, W boson-, Z boson-, or quark-pairs [52].
In this thesis, we focus on the continuous γ-ray spectrum produced in DM annihilations.
The annihilation flux in gamma rays per gamma-ray energy from a DM halo is given
by the following formula:
dΨ
dN hσvi
=
· J,
dEγ
dEγ 8πm2χ
(34)
dN
where dE
is the number of photons per photon energy produced in the annihilation process,
γ
hσvi is the velocity-averaged annihilation cross-section, mχ is the DM particle mass, and
J — the astrophysical term — is defined as:
Z
J=
ρ2χ (l, Ω)dl
(35)
l.o.s.
where ρχ is the DM mass density. Hence, J is the integral over the DM mass density squared
along the line-of-sight, and quantifies ‘how much dark matter annihilation one is looking at’.
See section ‘Line-of-sight integral’ in the Appendix for more details. Eqs. 34 and 35 show
that the DM annihilation flux depends on the DM density squared. Therefore, natural
places to search for DM annihilation fluxes are regions with high densities. Interesting
regions for indirect detection of DM are hence the Galactic Center (GC), where the DM
28
density profile is proportional to r−1 , with r the distance to the GC4 . On top of this, DM
might be adiabatically contracted onto the Supermassive Black Hole that lives in the centre
of the Galaxy, resulting in an even higher DM density at the GC [53]. Other interesting
objects to look at when searching for annihilation fluxes are DM substructures in the Milky
Way (see section ‘Dark matter structure’). In this thesis, we are interested in searching for
gamma rays produced in DM annihilations in such DM subhalos of the Milky Way.
The annihilation flux in a certain gamma-ray energy range [E1 − E2 ] is obtained by integrating Eq. 34:
hσvi
·J ·
8πm2χ
Ψ[E1 −E2 ] =
Z
E2
E1
dN
dEγ
dEγ
(36)
To calculate the last term in Eq. 36, we use the Mathematica notebook from “A Poor
Particle Physicist Cookbook for Dark Matter Indirect Detection” (PPPC) ([54]) which
provides a function for the gamma-ray spectra resulting from dark matter annihilation in
the following form:
dN
log10
(37)
d log10 X
where X equals Eγ /mχ with Eγ the photon energy and mχ the DM particle mass. In the
following, we will write E instead of Eγ for simplicity. The spectra are normalised per one
DM annihilation, and one has to specify the primary annihilation product, the secondary
product and the dark matter particle mass mχ .
To obtain the total N in a certain energy range [E1 − E2 ], we integrate:
Z
X2
log10
10
h
dN
d log10 X
i
d log10 X
(38)
X1
where X1 and X2 are E1 /mχ and E2 /mχ , respectively. Plugging in Eq. 38 for the last
term in Eq. 36 gives us the number flux: the total number of photons in the energy range
[E1 − E2 ], with units cm−2 s−1 . However, it will be useful to calculate the energy flux of
gamma rays with energies that can be detected by Fermi-LAT, since we will define the
detection threshold for 3FGL in terms of an energy flux rather than a number flux (see
section ‘Methods’). Therefore, rather than integrating dN/dE in Eq. 36, we integrate
E dN/dE in order to obtain an energy flux. Using Eq. 38, we can write:
Z
X2
log10
10
h
dN
d log10 X
i
· E d log10 X
X1
4
This is true in the case of a NFW density profile. See Fig. 6.
29
(39)
E dNdE
100
30 GeV
10
40 GeV
50 GeV
1
100 GeV
0.1
200 GeV
0.01
0.5
1.0
5.0
10.0
E @GeVD
100.0
50.0
Figure 8: The energy spectrum of photons produced per process of two DM particles
annihilating to bb, for different DM masses mχ .
Using that:
dE
,
E · ln 10
d log10 X = d log10 [E/mχ ] =
(40)
we find:
Z
X2
10
log10
h
dN
d log10 X
i
Z
E d log10 X
E2
=
log10
10
h
dN
d log10 [E/mχ ]
E1
X1
Z
E2
=
log10
10
E1
h
i
·E
dE
E · ln 10
dN
d log10 [E/mχ ]
i
·
dE
ln 10
(41)
In Table 1, the resulting gamma-ray values for
Z
E2
NE =
E
E1
dN
dE
dE
are listed for different commonly considered values of mχ , in the case of 100 % annihilation
to bb-quarks, and for [E1 , E2 ] = [0.1GeV, 100GeV] as well as for [E1 , E2 ] = [1GeV, 100GeV].
30
E dNdE
100
30 GeV
10
40 GeV
50 GeV
1
100 GeV
0.1
200 GeV
0.01
0.5
1.0
5.0
10.0
50.0
E @GeVD
100.0
Figure 9: The energy spectrum of photons produced per process of two DM particles
annihilating to τ + τ − , for different DM masses mχ .
Table 1: Results NE of integrating E dN/dE for 100 % annihilation to bb and gamma rays
as secondary products, for different DM masses mχ and for two different energy ranges.
mχ [GeV]
NE [GeV] (0.1 - 100 GeV)
NE [GeV] (1 - 100 GeV)
30
15.99
9.653
40
21.42
14.52
50
26.87
19.56
100
54.06
45.76
200
108.2
99.03
1000
474.1
461.6
3.2
The Fermi Large Area Telescope
Gamma rays are photons in the energy band ranging from 100 KeV up to 100s of TeV. Since
the Earth’s atmosphere is opaque to these highly energetic photons, gamma-ray telescopes
either have to be located above it, or should be dedicated to measuring the interactions
of gamma rays with Earth’s atmosphere. Ground-based telescopes measuring very highenergy (VHE) gamma rays through their interactions in the atmosphere are Cerenkov-light
telescopes such as H.E.S.S., MAGIC and VERITAS. In this work, we consider the Fermi
31
Large Area Telescope (Fermi-LAT), the most sensitive space telescope to detect high-energy
(HE) gamma rays in the energy range 0.1 - 100 GeV [55].
The Fermi-LAT was launched in June 2008 and has an orbit of about 95 minutes. It
has been taking data since August 2008. Most of the time the telescope is in sky survey
mode, but if a signal consistent with a gamma-ray burst is measured, it can point itself to
the direction of the burst to take more data from that direction for a while [56].
The way detection of gamma rays works in the Fermi-LAT is through the process of
pair-production: at energies above 1 MeV (twice the electron mass), it becomes possible for
an incoming photon to produce an electron-positron pair in the Coulomb field of a nucleus of
the detector material. Above 100 MeV, this pair-production process completely dominates
over other interaction processes of the incoming photon with the detector material. The
produced electron-positron pair in turn produces high-energy photons, which again form
electron-positron pairs, until the energy of the produced photons falls below the energy
equivalent to twice the electron mass, and the cascade of electrons and photons stops. The
energy of the initial incoming photon (gamma ray) is determined by means of a calorimeter,
which measures the energy of the electromagnetic shower initiated by the incoming photon.
A combination tracker measures the positions of the electron-positron pairs to infer an
estimate of the arrival direction of the incoming gamma ray [55] [56].
Electrons from cosmic rays can also generate electromagnetic cascades in the detector,
which are difficult to distinguish from those generated by the photons we are interested in.
This background is rejected by means of an anticoincidence shield. Another background
component the Fermi-LAT is subject to is the gamma-ray background from the Earth. The
Fermi-LAT eliminates this background by rejecting upward-going gamma rays [56].
In Fig. 10, the sky in gamma rays of energies greater than 1 GeV as measured by FermiLAT is depicted. The Galactic Plane, and the Galactic Center in particular, is clearly a
bright gamma-ray source. Multiple point sources can also be seen as bright spots on the
sky-map.
Some of the sources emitting gamma rays are:
1. Cosmic rays from the Milky Way. Cosmic rays produce a diffuse gamma-ray background due to interactions between cosmic-ray electrons and cosmic-ray protons and
the interstellar gas and starlight. Cosmic-ray electrons emit gamma rays due to inverse Compton scattering on starlight and the cosmic microwave background, and
due to Bremsstrahlung when travelling through interstellar gas. Cosmic-ray protons
inelastically scatter with interstellar gas producing pions, which subsequently decay
to gamma rays. These effects cause the Milky Way to appear as a diffuse Galactic
γ-ray background due to the cosmic-ray production in the Galactic plane [57] [55], as
can be appreciated from Fig. 10. Due to the intense γ-ray emission from the Galactic
Plane, point sources in the plane are more difficult to detect than point sources at
higher latitude [15].
32
Figure 10: The sky in gamma rays greater than 1 GeV, based on five years of data from
the Fermi-LAT (Picture from NASA/DOE/Fermi-LAT Collaboration).
2. Cosmic rays from starburst galaxies. Cosmic rays are accelerated by supernovae
shocks. Since starburst galaxies have a high rate of star-formation and therefore a
high rate of supernova explosions, starburst galaxies can be detected by Fermi-LAT
[58].
3. Cosmic rays from the Earth’s atmosphere. Because it is so close-by, the limb of the
Earth is the brightest source of γ-ray emission measured by Fermi-LAT [59]. In the
analysis of Fermi-LAT data, the measurements due to the Earth’s limb are always
carefully eliminated [55].
4. Pulsars. Pulsars are rapidly rotating neutron stars which accelerate charged particles to high energies. The charged particles subsequently produce γ-rays due to
synchrotron and inverse Compton radiation. A population of milli-second pulsars
has been observed by Fermi-LAT [60], as well as a few pulsars that do not emit at
radio wavelengths [61] — through which most pulsars have been detected [55].
5. Active Galactic Nuclei. If the relativistic jets of super massive black holes at the
centres of galaxies happen to point in our direction, these Active Galactic Nuclei are
called blazars and are detectable in gamma rays, due to inverse Compton scattering
of electrons of the jet on surrounding photons [55].
33
Dark matter annihilations might also be included in the above list. Dark matter subhalos
could be detected as faint gamma-ray sources by Fermi-LAT, and the Galactic Center too
might produce a gamma-ray excess over the expected Galactic background. In this thesis,
we are interested in the subhalos that could appear as unidentified sources in the latest
Fermi-LAT point source catalog. We will also briefly discuss the measured gamma-ray
excess from the Galactic Center and its interpretations.
3.3
The 3FGL point source catalog
Recently, the Fermi-LAT collaboration has released its third point source catalog 3FGL [15],
in which all point sources that were detected at 5σ with 6 years of data are listed. In total,
this catalog contains 3034 sources, many of which are associated to Active Galactic Nuclei
or pulsars thanks to astronomical studies at other wavelengths. 992 sources, however, have
not been associated with emission at other wavelengths. A subset of these are expected to
be pulsars and AGNs that have not yet been associated — sources that were not associated
in earlier point source catalogs of Fermi-LAT have been listed as pulsars or AGNs in 3FGL,
and the same is expected to happen for many of the unassociated sources in 3FGL with
more multi-wavelength data. However, a subset of the unassociated sources might be dark
matter subhalos. In Fig. 11 we have plotted the positions on the sky of the unidentified
sources in 3FGL. Here we have selected the 934 unidentified sources that are non-variable,
since the gamma-ray flux resulting from DM annihilation is expected to be constant in
time. We used a variability index cut recommended by [15].
3.4
The GeV excess from the Galactic Center
Recently, several groups have reported the detection of an excess over the expected background of gamma rays from the region of the Galactic Center (e.g., [62], [63]). The
spectrum and angular distribution of the signal is claimed to be compatible with that
predicted from 30-40 GeV DM particles annihilating into quarks with a cross-section of
hσvi ∼ 10−26 cm3 s−1 [64].
However, the DM interpretation of the GeV Excess is still heavily debated. Recently,
the presence of a population of millisecond pulsars near the Galactic Center has been
invoked to explain the spectrum and morphology of the excess [65]. Another proposed
origin of the GeV excess are injections of charged leptons into the Galactic Center region
by supernova outbursts around a megayear ago. These injected charged leptons might be
able to produce the same spectrum of gamma rays through processes of inverse Compton
scattering and Bremsstrahlung [66].
The region in the DM mass - annihilation cross-section parameter space that fits well
the observed GeV excess from the Galactic Center lies just below the exclusion limits from
the combined analysis of dwarf galaxies [64] [16]. See Fig. 12 in section ‘Null detection from
dwarf galaxies’ below. If one improves the upper limits on the cross-section, one might well
34
Figure 11: Aitoff projection of the positions on the sky of the 934 unassociated, non-variable
point sources in 3FGL.
exclude the dark matter interpretation of the GeV excess from the GC. Therefore, this is an
extremely interesting parameter region to study. This is why we will revisit the calculation
of J-factors of dwarf galaxies, taking into account the effects of baryons on the DM density
profiles of the dwarfs.
Even though the DM explanation of the GeV excess is still under debate, we will focus
on the range of DM masses that can account for the signal, and use it as a benchmark
model in our analysis of the detectability of subhalos with Fermi-LAT.
3.5
Null detection from dwarf galaxies
Dwarf galaxies are heavily dark matter dominated, with mass-to-light ratios up to 1000
[67]. This makes dwarf galaxies interesting objects for indirect detection of DM: they
have relatively high J-factors due to their high DM content and their relative proximity,
35
compared to for example galaxy clusters [1]. Moreover, the astrophysical background is
relatively low for high-latitude dwarf galaxies (see Fig. 11) which would make it relatively
easy to identify a possible signal with annihilating DM. Since we know where the dwarf
galaxies are (thanks to their stellar component), it is possible to perform dedicated searches
for DM annihilation signals by analysing the Fermi-LAT gamma-ray data from regions
centred around these galaxies.
Up to now, no significant gamma-ray emission from dwarf galaxies has been detected
[16]. This has enabled the Fermi-LAT collaboration to place upper limits on the annihilation cross-section of DM (assuming a model for the distribution of DM in the dwarfs):
if the annihilation cross-section would be larger than those limits, Fermi-LAT should have
seen a gamma-ray signal from the dwarfs.
Recently, the Fermi-LAT collaboration presented upper limits on the DM annihilation
cross-section from a combined analysis of 15 dwarf galaxies using 6 years of data [16]. In
Fig. 12, these upper limits are shown, together with limits from other analyses.
Figure 12: Upper limits on the DM annihilation cross-section in the case of annihilations to
bb (left) and τ + τ − (right). The black line corresponds to the upper limit from the combined
dwarf analysis of the Fermi-LAT collaboration [16]. The pink contour corresponds to the
region in parameter space that can account for the GeV excess, according to the analysis
of [68]. Figure taken from [16].
36
4
The Effect of Baryons on Dark Matter Halos
4.1
Discrepancies between observations and simulations
Although the ΛCDM model of cosmology has been successful on large scales (e.g., accounting for structure formation, being compatible with the acoustic peaks in the cosmic
microwave background and predicting the right abundances of light elements [22]), there
have been problems reconciling ΛCDM with small-scale observations [69]. Three of these
problems have come to be known as the ‘missing satellite problem’, the ‘too-big-to-fail’
problem, and the ‘cusp/core’ problem, which may be interrelated. These problems will
each be briefly discussed below. Since we are interested in the effects of baryons on the
DM distribution, we will only discuss the proposed solutions involving baryonic effects.
4.1.1
Missing satellite problem
Moore et al. [70] pointed out that the number of observed subhalos in the Milky Way
was much smaller than the number predicted by N-body simulations. Dark matter-only
(DMO) simulations such as VL-II predict an order of magnitude more dwarf-sized subhalos
in a Milky Way-like halo than the number of observed dwarf galaxies [40]. The problem
was slightly mitigated when recently a number of ultra-faint Milky Way satellites were
discovered with the Dark Energy Survey ([71], [72]), but the discrepancy can only be really
solved once one takes into account that not all halos form galaxies due to reionisation [73].
Indeed, for halos with masses between 108 and 109 M , the luminous fraction at z = 0
is expected to be around 10 percent if reionisation happened at around z = 10 [73]. For
comparison, the masses of dwarf galaxies and low surface brightness galaxies are in the
range ∼ 108 − 1010 M [74] [75]. Thus, the subhalos are out there — we just do not see
them because they do not contain any stars. We will discuss the effect of reionisation in
more detail in the section ‘Reionisation: Only a fraction of halos form galaxies’.
Other effects that reduce the discrepancy include stellar and supernova feedback and
dynamical stripping, which lead to cored dark matter halos. Since cored halos are more
prone to tidal stripping than cuspy ones, the number of predicted satellites decreases [28].
4.1.2
Too-big-to-fail problem
The too-big-to-fail problem reflects the mismatch between observations and simulations
in the sense that simulations predict ∼ 10 subhalos with larger Vmax (i.e., that are more
massive) than the heaviest companions of the Milky Way [76] [77]. The most massive dwarf
spheroidals in the Milky Way have 12 < Vmax [km/s] < 25, whereas VL-II contains a few
tens of halos with Vmax > 25 km/s (see Fig. 13). The problem is called too-big-to-fail
because one does not expect such massive subhalos not to have formed galaxies; so why
have we not observed them? This problem is claimed to be solved if one models subhalo
density distributions as cored profiles instead of cuspy NFW profiles (e.g., [78], [74]). Cored
37
profiles can be obtained by introducing baryonic effects such as supernovae feedback (e.g.,
[10], [12]). A recently proposed density profile taking into account baryonic effects is the
DC14 profile [13], in which the slope parameters depend on the baryonic content of the
halo. For DC14 profiles, the fitted halo masses of observed dwarf galaxies are higher than
their corresponding NFW masses, which matches them to the heaviest halos in simulations
[74]. We will discuss the DC14 profile extensively below.
Figure 13: The maximum circular velocities of the subhalos in VL-II plotted against their
NFW (M200 ) masses. The horizontal line corresponds to the largest measured maximum
circular velocity of dwarf spheroidal galaxies of the Milky Way [76]. The fact that there
are a handful of VL-II halos above this line represents the too-big-to-fail problem.
4.1.3
Cusp/core problem
The universal NFW profile that is inferred from dark matter-only simulations is cuspy.
That is, the dark matter density increases as ρ ∝ r−1 from the scale radius toward the
halo centre (Eq. 17; Fig. 6). However, observations of rotation curves of field and dwarf
38
galaxies suggest flatter, or ‘cored’, inner slopes (e.g., [79]). Einasto profiles (Eq. 19) provide
a slightly better fit but can still not solve the discrepancy [80].
We can conclude that having a — physically motivated — cored dark matter density profile
in combination with the effects of reionisation solves the dark matter-related small-scale
problems of ΛCDM discussed above, without having to resort to more exotic forms of dark
matter. Let us focus on one such density profile, in which cores are effectuated by baryonic
feedback, introduced by Di Cintio et al. in 2014 [13].
4.2
DC14: a mass-dependent halo profile taking into account galaxy
formation
Several groups have found that implementing stellar feedback — such as blastwave supernova feedback [81] and energy input into the interstellar medium by massive stars prior to
their explosions [82] — results in expansion of the halo and flattening of the inner density
slope (e.g. [12], [21], [11]).
Di Cintio et al. argue that the stellar-to-total mass ratio is the quantity that determines
the strength of the baryonic effects on the density profile of a DM halo [12]. In halos with
low stellar-to-total mass ratios, baryons do not have a large effect: the stellar contribution is
simply too small to make a difference, and the halo retains a cuspy NFW profile. For halos
with stellar-to-total mass ratios of around 0.5 per cent, baryonic feedback is strong enough
to achieve a cored profile. For even higher stellar-to-total mass ratios, the gravitational
potential generated by baryons in the centre of the halo is so high, that the outflow of gas
from the inner region of the halo due to supernovae explosions does not have a significant
effect, and a NFW-like cusp is formed in these halos [12].
The stellar-to-total-mass ratio is a function of the total halo mass:
Mstar
Mstar
=
(Mhalo )
Mhalo
Mhalo
(42)
and this function can be determined through abundance matching. In abundance matching,
the heaviest observed galaxies (with measured stellar content) are matched to the heaviest
DM halos from a simulation, and one continues doing this for ever less massive observed
halos until all observed halos are matched to DM halos. This way, the stellar-to-total mass
relation can be determined. In Fig. 14, the abundance matching relation from Moster et
al. [83] is plotted. The colour coding corresponds to the value of the inner slope of the
density profiles, where -1 corresponds to a NFW profile and 0 corresponds to a maximally
cored profile. The flattest density profiles occur at halo masses of around 3 · 1010 M [12].
In [13], Di Cintio et al. not only look at the central parts of dark matter halos, but model
the complete density profiles of a suite of galaxies from the hydrodynamical MaGICC [82]
and MUGS [84] simulations. They find, in accordance with their previous work, that the
39
Figure 14: The dependence of the inner slope α on the stellar-to-total-mass ratio colour
coded on the abundance matching relation from Moster et al. [83]. In this plot, the
halo mass Mhalo is defined as the mass within a sphere containing 360 times the cosmic
background matter density at z = 0. Figure taken from [12].
stellar-to-total mass ratio is the determining factor that constrains both the inner and
outer slopes of the density distribution. They use the five-parameter profile function (Eq.
22) and express the slope parameters α, β, and γ as functions of the stellar-to-total-mass
ratio at zero redshift. The DC14 profile then is [13]:
ρs
ρDC14 (r) = γ h
α i(β−γ)/α
r
1 + rrs
rs
(43)
where ρs and rs are the scale density and scale radius, respectively. For a NFW profile
(with (α, β, γ) = (1, 3, 1)), the scale radius equals the radius at which the logarithmic
slope equals -2: r−2 = rs . For a profile with a general (α, β, γ) we have:
40
r−2 =
2−γ
β−2
1/α
rs
(44)
The functional forms of the slope parameters are:
α = 2.94 − log10 [(10X+2.33 )−1.08 + (10X+2.33 )2.29 ]
β = 4.23 − 1.34X + 0.26X 2
(45)
γ = −0.06 + log10 [(10X+2.56 )−0.68 + (10X+2.56 )]
where X = log10 (Mstar /Mhalo ) and should be in the range −4.1 < X < −1.3. At lower
values of X the profile returns to NFW.
The DC14 concentration cDC14 is defined as cDC14 ≡ rvir /r−2 and is related to the NFW
concentration cNFW via:
cDC14 = (1.0 + 0.00003e3.4[X+4.5] ) · cNFW
(46)
where X = log10 (Mstar /Mhalo ) as before. The NFW concentration can be obtained from a
concentration-mass relation such as the one by Correa et al. [85] or by Dutton&Macciò [86].
Following [74], this is how we will determine the concentrations of the dwarf spheroidal
galaxies of the Milky Way. More details are provided in the ‘Methods’ section. Alternatively, one can determine the NFW concentration of a simulated halo from its observed
rVmax - and VMax -values using Eqs. 23 and 27. This is how we will determine the concentrations of the halos in VL-II, in order not to have to rely on a concentration-mass relation.
Again, more details are provided in the ‘Methods’ section.
Recall that the halo mass is defined as the mass contained within a sphere of radius
rvir containing ∆ (common choice: ∆ = 200) times the critical density of the Universe
ρcrit = 3H 2 /8πG. Therefore, the halo mass defines the value of rvir . Thus, once one knows
cDC14 , one also knows r−2 . The corresponding DC14 scale radius can then be determined
using Eq. 44.
At this point, we know how to use an abundance matching relation to get X — which
gives us α, β, and γ — and a concentration-mass relation to find rs for a given halo mass.
The only DC14 parameter to be determined is ρs , and we find it by requiring:
41
Z
Mhalo
rvir
=
0
4πr2 ρs
h
α i(β−γ)/α dr
r
r
γ
1
+
rs
rs

→ ρs
Z

= Mhalo / 4π
(47)

rvir
γ h
r
rs
0
1+
r2

α i(β−γ)/α dr
(48)
r
rs
For a given halo mass, now, one can fully determine the DC14 profile following the steps
above. We compare the NFW profile of a halo of mass M200 = 3 · 1010 M with its DC14
counterpart in Fig. 15.
ΡHrL @MSun ^2kpc^5D
DC14
106
NFW
104
100
1
5
10
50
100
500
1000
r @kpcD
Figure 15: NFW and DC14 density profiles for a field halo with mass M200 = 3 · 1010 M .
The concentration has been calculated using the c(M )-relation from Correa et al. [85],
and the stellar-to-total mass ratio was determined from the abundance matching relation
of Moster et al. [83].
42
4.2.1
Reionisation: only a fraction of halos form galaxies
Obviously, baryonic effects need to be taken into account only if the halo of interest contains
baryons. Astrophysical processes such as reionisation can suppress star formation in lowmass halos (e.g., [87]). Reionisation helps resolving the ‘missing satellite problem’, as was
described above.
In [73], Sawala et al. simulated Milky Way-Andromeda pairs based on the WMAP7
cosmology. They implemented instantaneous hydrogen reionisation at z = 11.5 and the
redshift of helium reionisation was modelled as a gaussian around z = 2.5 with σ(z) = 0.5.
Besides reionisation, stellar formation and evolution, as well as black hole growth and AGN
feedback were included. They ran their simulations three times; once with only dark matter,
once with dark matter plus baryons but without reionisation, and once with all effects
included. They found that when reionisation was included, the fraction of luminous halos
— halos that had formed a galaxy — was much reduced compared to when reionisation
was not taken into account. All halos with a mass of around 1010 M and higher at z = 0
contained a galaxy, but for halos of 109 M the luminous fraction was reduced to about
20 %. Among halos with M < 108 M , none were luminous. For comparison, the masses
of dwarf galaxies and low surface brightness galaxies are in the range ∼ 108 − 1010 M [74]
[75].
As for the halos in VL-II, we will take the effect of reionisation into account by assigning
a chance of forming a galaxy (i.e., a chance of getting assigned a DC14 profile instead of a
NFW profile) to each halo based on the results of [73].
4.3
DC14 for dwarf galaxies
As explained before, the ‘too-big-to-fail’ problem represents the lack of observed high-mass
halos — which are too massive to not have formed any galaxy — in the Milky Way, but
which are present in simulations [76]. The problem could be alleviated once the DM halos
are modelled with cored profiles ([78], [74]), because using a cored profile, the dwarf galaxies
are assigned to more massive halos. Moreover, the cusp/core problem is simultaneously
resolved [74].
In [74], kinematical data of isolated and satellite galaxies in the Local Group have
been used to derive the best-fit mass of the DM halos hosting these observed galaxies.
This is done for two different density profiles; the cuspy NFW profile (Eq. 17) and the
mass-dependent DC14 profile (Eq. 43).
To constrain the halo mass, a concentration-mass relation has to be adopted. The
concentration-mass relation for the NFW profiles is taken from [86], which is based on
a Planck cosmology. The DC14 concentrations are subsequently obtained from Eq. 46.
Given a density profile and a concentration, the halo mass is now the only free parameter
left in both the NFW and the DC14 case, since α, β, and γ depend on Mstar /Mhalo only,
and Mstar is an observable quantity.
43
In Table 2 of [74], the best-fit halo mass (NFW and DC14) and corresponding α, β, and
γ values for the forty isolated and dwarf galaxies that were studied are given. Part of this
information is contained in Table 2 of this thesis.
Then, abundance matching in the Local Group is performed by Di Cintio et al. ([74]):
assigning the observed halos derived from the kinematical data to DM halos present in
a numerical simulation of the Local Group. According to [74], adopting DC14 profiles
rather than NFW profiles solves the too-big-to-fail problem since the observed galaxies are
assigned to more massive halos in the simulated Local Group in the DC14 case than in the
NFW case.
44
5
Methods
The aim of this project is two-fold. Firstly, we will update the predictions of the number
of DM subhalos that might be present as unidentified sources in the 3FGL catalog ([15]).
We will consider the effects of baryons on the DM density profiles of DM subhalos and
examine if these effects change the detectability of subhalos. We will use the measured
characteristics of the subhalos of the VL-II halo as a proxy for the subhalos in the Milky
Way, which means we do not have to make additional assumptions on the amount of tidal
stripping, as opposed to [9].
Secondly, we will recalculate the J-factors of the known dwarf galaxies of the Milky
Way in the case of the mass-dependent DC14 density profile. Recently, the Fermi-LAT
collaboration has used six years of gamma-ray data to constrain the DM annihilation crosssection from the null detection of the Milky Way’s dwarf spheroidal galaxies in gamma rays
[16]. In their analysis, they took the J-factors from [88], which were calculated assuming
NFW profiles. In light of the work by Di Cintio et al. that proposes the mass-dependent
DC14 profile which takes into account baryonic effects ([13]) — in contrast with the NFW
profile which is based on DMO simulations — the limits on the DM annihilation crosssection that were found by [16] might change when one adopts the DC14 profile instead of
the NFW profile for dwarf galaxies. Since these upper limits probe an extremely interesting
region in parameter space for reasons explained before (see Fig. 12), it is worthwhile to
check their robustness.
5.1
Calculating the annihilation flux from NFW halos in VL-II
We will make use of the publicly available data of the VL-II project [3] to calculate the
predicted DM annihilation flux from subhalos. The data we will use consists of the positions
of subhalos with respect to the host halo centre (GC), their maximum circular velocity
Vmax , and their radius of maximum circular velocity rVmax . We discard all subhalos at
a distance greater than 400 kpc (approximately the virial radius of the Milky Way) from
the GC, since we suspect them to be unbound. We include all subhalo masses resolved in
VL-II (∼ 105 − 1011 M ).
NFW profile parameters
As explained in the section ‘Density profiles’, the observable quantities Vmax and rVmax , the
maximum circular velocity and the radius of maximum circular velocity, respectively, can
be related to the NFW profile parameters. This way, we calculate the concentrations of all
subhalos in the VL-II simulation, rescaling the rVmax -values to the most recent cosmological
data as measured by Planck [14].
Fig. 16 shows the concentrations of all subhalos in VL-II that are closer than 400 kpc
to the GC and are therefore probably gravitationally bound to the Milky Way halo. The
blue dots correspond to the concentrations of individual halos in VL-II, calculated from
45
rV max and Vmax . The red line corresponds to the concentration-mass relation from [85].
The fact that the concentrations of halos in VL-II are systematically higher than the
concentration-mass relation prescribes is most probably due to the effect of tidal stripping:
[85] considered isolated field halos, whereas the halos in VL-II are subhalos that have been
tidally stripped ([40]), which resulted in higher concentrations.
To check our values for the concentration, we also plotted the concentrations of a subset
of halos in VL-II with distances to the Galactic Center in a range around 8 kpc. The result
is shown in Fig. 17, in which the blue dots are calculated using rV max and Vmax , and the
red dots are calculated using the relation in Eq. 11 in [6] at RGC = 8 kpc. This relation is
derived from the concentration of halos in VL-II and should therefore be compatible with
our results, which it indeed is, as will be shown below.
In Eq. 11 from [6], the dependence of the concentration on the subhalo mass and the
distance from the GC is parameterised. The first term of this equation reads
R
Rvir
−αR
,
where Rvir = 402 kpc is the virial radius of the MW halo, and the best-fit value for αR
is 0.286. This term equals 4.64 for R = 1.87 kpc (the smallest rGC value included in Fig.
17), 2.36 for R = 20.0 kpc (the largest rGC value included), and 3.07 for R = 8.00 kpc.
Hence, we would expect to find, at a given halo mass, a difference of a factor 4.64/2.36 ≈ 2
between the highest and lowest concentration values.
Therefore, the scatter in the blue dots can be explained by the fact that the halos we
considered are not all at a distance of exactly 8 kpc from the GC, but in a wider range of
distances.
46
Figure 16: The concentration parameter of the 9369 halos in VL-II that are closer than
400 kpc to the GC. The blue dots correspond to the concentrations calculated from rV max
(rescaled to the Planck 2015 cosmological parameters) and Vmax ; the red dots correspond
to the concentrations calculated using Eq. 21 in Correa et al. [85].
47
Figure 17: The concentration parameters of the 144 halos in VL-II with rGC between 0 and
20 kpc. The blue dots correspond to the concentrations calculated from rV max (rescaled
to the Planck 2015 cosmological parameters) and Vmax ; the red dots correspond to the
concentrations calculated from Eq. 11 in Pieri et al. ([6]) at R = 8 kpc. The scatter in the
blue dots can be explained by the fact that these values correspond to subhalos that are
not all at exactly 8 kpc from the GC, but between 0 and 20 kpc, as explained in the text.
The values of rV max and Vmax define the NFW profile of each subhalo. We subsequently
define the halo mass as:
4 3
Mhalo ≡ M200 = πr200
· 200ρcrit
3
where r200 is the virial radius given by rs = c · rs with c the concentration.
48
(49)
Point-like approximation
If one considers the object of interest (e.g., a DM subhalo) to be at a distance great enough
for it to be treated as a point source, one can write:
Z rs
1
4πr2 ρ(r)2 dr
(50)
Jpointlike = 2
d 0
where we choose to integrate out to the scale radius instead of the virial radius, since ∼ 90 %
of the luminosity is generated within the scale radius (see Appendix).
We are interested in subhalos that would appear point-like to Fermi-LAT, since we are
going to compare our predicted number of detectable sources to the number of candidate
sources in the 3FGL point-source catalog.
For this reason, we declare a subhalo to be spatially extended if its angular distance
on the sky is larger than the containment angle of Fermi-LAT at 1 GeV, since we expect
most photons produced in DM annihilations to be of such energies (see section ‘Gamma
rays from DM annihilations’).
The 68 % containment angle at 1 GeV of Fermi-LAT is ∼ 0.8 degrees (see Fig. 18),
thus, our spatial extension threshold for defining a source as point-like is:
ext =
180
arctan(rs /d) < 0.8 degrees
π
(51)
where rs is the scale radius and d is the distance between the observer and the subhalo.
The J-factors of point-like subhalos will be calculated using Eq. 50. For spatially extended
subhalos, however, we will correct the J-factors for spatial extension by integrating the
luminosity out to the containment angle rather than the scale radius. This is not as correct
as performing a line-of-sight integral (see Appendix), but it is a better approximation than
treating them the same way as point-like sources (i.e., integrating the luminosity out to
the scale radius).
We will calculate the J-factors for the case of NFW profiles and for the case of DC14
profiles. We determine the NFW-profile parameters from rVmax and Vmax , where we scale
the values of rVmax to match the latest values of the cosmological parameters as measured
by Planck [14]. The DC14 profile parameters will be determined from the NFW-profile
parameters, see section ‘Implementing DC14 profiles in VL-II’.
Annihilation boost
We take a conservative approach by not including possible annihilation boosts by the
presence of substructure within subhalos (e.g., [89]).
49
Figure 18: The containment angle of Fermi-LAT. Image taken from http://www.slac.
stanford.edu/exp/glast/groups/canda/lat_Performance.htm.
Rotating the observer around the GC
The VL-II halo can be regarded as a model of the Milky Way, but it is of course not clear
where the Earth would be in the simulated halo. However, the VL-II halo is triaxial, and
one could argue that the Galactic Disk should be coinciding with the plane perpendicular
to the major axis. Because the z-direction in VL-II is roughly coincident with the major axis of the halo, we assume the Galactic Plane to be corresponding to z = 05 . To
generate multiple realisations of the DM subhalo population around Earth, we place the
observer at 8.5 kpc (roughly the distance from the GC to Earth) from the centre of the halo
(0,0,0) and rotate it around this point in the x, y-plane, keeping the distance to the GC
constant.
5
Jürg Diemand, private communication.
50
Particle physics
The terms in the formula for the DM annihilation flux (Eq. 34) that concern the particle
physics parameters will be calculated for different DM masses, and for annihilations into
bb and into τ + τ − . The photon spectra dN/dE will be determined using [54], see section
‘Gamma rays from DM annihilations’.
Flux threshold
Our aim is to predict the number of DM subhalos that might be present as unidentified
point sources in the 3FGL catalog. Therefore, we want to define the detectability of a
source according to the threshold for inclusion in 3FGL (which corresponds to a 5-sigma
detection [15]).
In Fig. 19, the energy flux distributions of sources in 3FGL are shown for the photon
energy range 100 MeV - 100 GeV and for the photon energy range 1 GeV - 100 GeV.
The peaks of the distribution are at ∼ 4.0 · 10−12 erg/cm2 /s and ∼ 1.35 · 10−12 erg/cm2 /s,
respectively. The figure also shows that the energy flux above 1 GeV of the faintest source
in 3FGL is ∼ 4.0 · 10−13 erg/cm2 /s.
We expect DM annihilation spectra to be ‘harder’ than the gamma-ray spectra produced by pulsars and blazars, that is, we expect gamma rays from DM annihilations
to have in general higher energies than gamma rays from blazars or pulsars [15]. Because a lot of unidentified sources are expected to be blazars or pulsars, we choose not
to use the detectability threshold inferred from the energy flux distribution in the range
100 MeV - 100 GeV, but rather that inferred from the energy flux distribution in the range
1 GeV - 100 GeV.
As a conservative detectability threshold, we will take the energy flux above 1 GeV
corresponding to the peak of the distribution: F>1 GeV > 1.35 · 10−12 erg/cm2 /s. We expect
the catalog to be complete down to the peak of the distribution, i.e., all sources with fluxes
above the peak value have been detected regardless of their spectra. Therefore, we are
confident this is a conservative detectability threshold. As a (very) optimistic detectability
threshold, we will take the energy flux above 1 GeV corresponding to the energy flux of
the faintest source in 3FGL: F>1 GeV > 4.0 · 10−13 erg/cm2 /s.
Because the γ-ray spectrum resulting from DM annihilations to τ + τ − is harder than
that produced in annihilations to bb (See Figs. 8 and 9), the energy flux in the range
10 - 100 GeV is higher in the case of annihilations to τ + τ − than in the case of annihilations to bb. Therefore, we expect more detectable sources for DM annihilating to τ + τ −
than for annihilations to bb if we use a detectability threshold inferred from the energy
flux distribution of sources in 3FGL in the energy range 10 - 100 GeV. The peak of the
energy flux distribution of 3FGL sources in this energy range is 4.43 · 10−13 erg/cm2 /s
(Fig. 20), and we will use this detectability threshold in the case of annihilations to
τ +τ −.
51
Figure 19: Energy flux distribution of sources at high galactic latitude (|b| > 10◦ ) in the
Fermi-LAT 3FGL point-source catalog, in the photon energy ranges 100 MeV to 100 GeV
(blue) and 1 GeV to 100 GeV (green).
52
Figure 20: Energy flux distribution of sources at high galactic latitude (|b| > 10◦ ) in the
Fermi-LAT 3FGL point-source catalog, in the photon energy range 10 GeV to 100 GeV.
We exclude all subhalos in VL-II with latitudes smaller than 10 degrees, because a point
source at lower latitude — that is, closer to the Galactic Plane — would be much harder
to detect due to the strong Galactic background (see Fig. 10) and would therefore have to
have a higher energy flux than the threshold we inferred from the energy flux distribution
of sources in 3FGL at latitudes above 10 degrees.
5.2
Implementing DC14 profiles in VL-II
The slope parameters in the DC14 profile depend on the stellar-to-total mass fraction of
the halo. We will use the abundance matching relation of Brook et al. [90] to assign
a stellar-to-total mass ratio to each halo mass. As opposed to the abundance matching
relation of Sawala et al. [91], the stellar-to-total mass relation provided in [90] does not
53
have an upturn at low masses, as expected on physical grounds6 . This is why we adopt the
abundance matching relation of [90] rather than that of [91]. Following [83], we model the
stellar-to-total mass relation as a log-normal distribution centred around the relation of
[90] with an intrinsic scatter of σ = 0.15. We use the results of [73] in order to account for
reionisation, by assigning a chance of forming a galaxy (and therefore a chance of getting
assigned a DC14 profile rather than a NFW profile) to each halo, based on its mass. We
run our analysis multiple times to take into account the scatter resulting from the intrinsic
scatter in the stellar-to-total mass relation and the chance for halos to become luminous.
5.3
Dark matter subhalo candidate sources in 3FGL
We will use the results of the spectral analysis of unidentified sources in 3FGL by Bertoni
et al. [9]. For unidentified sources they identified as compatible with DM subhalos, they
have provided DM mass ranges for which the measured spectra provide good fits to the
data in the case of annihilations to bb, based on a chi-squared analysis.
For each DM mass, then, we take the number of compatible unidentified sources (see
Fig. 21) and compare this number to our predicted number of detectable subhalos to place
upper limits on the annihilation cross-section in the case of annihilations to bb.
5.4
Calculating J-factors of dwarf galaxies with DC14 profiles
In this subsection, we will provide the method for calculating the J-factor of a dwarf galaxy
with a NFW profile and a DC14 profile, using the results from Brook et al. as presented
in [74]. Let us start with JNFW . From Table 2 in [74] we take Mhalo, NFW . This is the
best-fit halo mass for the observed satellites assuming a NFW profile. In the process of
fitting, Brook et al. have assumed that the inner regions of the galaxies, at their half-light
radii, have density profiles related to the mass of the halo prior to infall — i.e., they did
not take into account tidal effects. For this reason, in our analysis we will not include any
satellites that show signs of tidal disruption according to [74].
Mhalo is defined as the mass of a sphere with virial radius rvir that contains ∆vir times
the critical matter density of the Universe ρcrit = 3H 2 /8πG at z = 0, where ∆vir =
18π 2 + 82x − 39x2 and x = Ωm − 1. In [74], Brook et al. use Ωm = 0.308 corresponding to
the Planck cosmology defined in [86]. With this definition of Mhalo 7 we can write rvir as:
rvir =
3Mhalo [M ]
4π · 1.32 · 104
6
1/3
kpc
(52)
Arianna di Cintio, private communication.
Note that this definition of Mhalo differs from the definition Mhalo = M200 we have been using before.
Since we are using the results of [74] in our analysis of observed Milky Way dwarf galaxies, we follow [74]
in their definition of Mhalo in this part of the analysis instead of using M200 .
7
54
Figure 21: The number of candidate sources in 3FGL as a function of DM mass in the case
of annihilations to bb. Candidate sources are compatible with a DM annihilation spectrum
at 95 % confidence level. Figure according to results of [9].
The concentration-mass relation at redshift zero is taken from [86] and is given by:
log10 cvir = 1.025 − 0.097 log10 (Mvir /[1012 h−1 M ])
(53)
where h is the dimensionless Hubble constant and Mvir = Mhalo mentioned above. The
concentration calculated using Eq. 53 is the concentration corresponding to a NFW profile,
defined as c = rvir /rs with rs the scale radius. Hence, the concentration-mass relation also
defines a NFW scale radius for a given mass.
The concentration and halo mass in these relations are the virial concentration and
mass, i.e., valid for isolated halos. Dwarf galaxies live in satellite halos, which are subhalos
of the Milky Way host halo, and these halos are therefore expected to be (on average) more
55
concentrated than isolated ones due to the effects of tidal stripping [40]. According to [92],
this effect is small and according to [74] the effect does not influence the results of their
study. We will ignore this effect as well and directly take the results of Eq. 53 as the NFW
concentration values for the satellite galaxies.
Once the NFW concentration of a dwarf galaxy is set, we determine the appropriate
DC14 concentration through Eq. 46, taking the values for Mstar and Mhalo, DC14 from Table
2 in [74].
Now we have fully specified the NFW and DC14 profiles (Eq. 17 and Eq. 43). Instead
of assuming point-like subhalos when calculating J-factors, as we do for the subhalos in
VL-II, for the J-factors of the observed dwarf galaxies we perform the line-of-sight integral:
Z
∆Ω
Z
lmax
dΩ
J(∆Ω) =
ρ2 [r(l, Ω)]dl
(54)
lmin
0
i.e., the integral of the density profile squared over l which runs along the line-of-sight and
is bounded by lmin and lmax , integrated over a certain solid angle ∆Ω. We follow [16] in
integrating over a cone with a solid angle of 0.5 degrees. See the section ‘Line-of-sight
integral’ in the Appendix for more details.
The coordinates and distances of the galaxies will be taken from the Fermi-LAT dwarf
analysis paper (Table 1 in [16], partially reproduced in Table 2 in this thesis).
56
6
Results — Subhalos in VL-II
Luminosities: NFW vs. DC14
Following [12], we assigned DC14 profiles to subhalos with −4.1 < log10 (Mstar /Mhalo ) <
−1.3, where we took Mstar /Mhalo from the abundance matching relation of Brook et al.
[90]. For the halos with DC14 profiles, we calculated the luminosities (defined as L = J ·d2 )
by integrating Eq. 50. For NFW profiles, it is straightforward to analytically solve the
integral (see Appendix):
Z
0
rs
7 3 2
r2 ρ2s
i4 dr = rs ρs
2 h
24
r
1 + rrs
rs
(55)
In Fig. 22, we plotted the luminosities of all subhalos in VL-II with rGC < 400 kpc.
We assigned DC14 profiles when appropriate. From this figure, it becomes clear that
implementing DC14 profiles for halos in VL-II is irrelevant if one adopts the abundance
matching relation of Brook et al. ([90]), since only one halo gets assigned a DC14 profile in
this case: only one halo in VL-II has a mass larger than 5 · 109 M ; the other VL-II halos
are not massive enough to be assigned a DC14 profile (see Fig. 14).
6.1
Detectability of subhalos
To arrive at the results presented in this subsection, we have not implemented DC14 profiles
in the calculations. Rather, we assigned NFW profiles to all halos in VL-II, recalling that
merely one subhalo was eligible to be assigned a DC14 profile. The DC14 profiles will
return in our discussion of the observed dwarf satellites of the Milky Way. In that part
of our analysis, we use the results of [74] concerning the density profiles of the observed
MW dwarfs. According to [74], the total masses of the MW dwarf galaxies are larger
when adopting DC14 profiles (∼ 1010 M ) than when adopting NFW profiles (∼ 108 M ).
Therefore, baryonic effects in observed MW dwarf galaxies should be taken into account
when considering the DC14 framework.
6.1.1
J-factors
In Fig. 23, the J-factors for all subhalos in VL-II with rGC < 400 kpc are plotted, for
six observer positions. Therefore, each halo corresponds to six points in the plot. The
upper and lower horizontal lines correspond to the J-factor of Draco and that of LeoII,
respectively. This figure illustrates that even relatively small halos can have J-factors
higher than that of the dwarf galaxies, if they happen to be nearby enough.
57
Figure 22: Luminosities (J · d2 ) of all halos with rGC < 400 kpc in VL-II. Halos with
Mhalo > 107 M and −4.1 < Mstar /Mhalo < −1.3 were assigned DC14 profiles (red dots; in
this case only one); the others were assigned NFW profiles (blue dots). Mstar /Mhalo -values
were calculated from the abundance matching relation of Brook et al. [90]. The luminosities
of spatially extended sources were integrated out to the Fermi-LAT containment angle at
Eγ = 1 GeV; the luminosities of point-like sources were integrated out to their scale radii.
58
Figure 23: Masses plotted against J-factors for all subhalos with rGC < 400 kpc in VL-II.
Each subhalo corresponds to 6 points in the plot, corresponding to 6 different observer
positions (with the distance between observer and GC equal at 8.5 at all times). Blue dots
correspond to point-like sources; red dots correspond to spatially extended sources. The
J-factors of spatially extended sources were calculated by integrating out to to the containment angle of Fermi-LAT at Eγ = 1 GeV. The upper and lower black lines correspond
to the J-factor of Draco and the J-factor of LeoII, respectively, as provided by [16].
59
6.1.2
Ndet against Mhalo
The following two plots show the mass histograms of detectable subhalos, in the case of
annihilations to bb and τ + τ − , respectively. In both plots, a relatively high annihilation
cross-section of 3 · 10−25 cm3 s−1 is assumed, to illustrate the typical masses of detectable
subhalos. In the case of annihilations to bb, an energy flux threshold above 1 GeV of
1.35 · 10−12 erg s−1 cm−2 is used, corresponding to the peak of the energy flux distribution
of sources in 3FGL above 1 GeV (see Fig. 19). In the case of annihilations to τ + τ − , an
energy flux threshold above 10 GeV of 4.43 · 10−13 erg s−1 cm−2 is used, corresponding to
the peak of the energy flux distribution of sources in 3FGL above 10 GeV (see Fig. 20).
Figure 24: Mass histogram of detectable subhalos in VL-II in the case of a 40 GeV dark
matter particle annihilating to bb at a cross-section of hσvi = 3 · 10−25 cm3 s−1 (ten times
larger than the thermal cross-section). Subhalos are considered detectable if their gammaray energy flux above 1 GeV exceeds 1.35 · 10−12 erg s−1 cm−2 , corresponding to the peak
of the distribution of the energy flux above 1 GeV of sources in 3FGL. The error bars
correspond to 1 sigma due to rotating the observer position around the Galactic Center.
The J-factors of spatially extended sources were calculated by integrating the luminosity
out to to the containment angle of Fermi-LAT at Eγ = 1 GeV.
60
Figure 25: The same as Fig. 24, but for annihilations to τ + τ − instead of bb. The energy
flux threshold used in this plot is 4.43 · 10−13 erg s−1 cm−2 above 10 GeV, corresponding
to the peak of the distribution of the energy flux above 10 GeV of sources in 3FGL.
61
6.1.3
Ndet against hσvi
In the following figures, the number of detectable subhalos is plotted against the annihilation cross-section for several choices of WIMP masses, annihilation channels and choices
of the energy flux threshold.
Figure 26: Number of detectable point-like (blue) and spatially extended (green) subhalos
plotted against the annihilation cross-section in the case of a 40 GeV dark matter particle
annihilating to bb. Subhalos are considered detectable if their gamma-ray energy flux above
1 GeV exceeds 1.35 · 10−12 erg s−1 cm−2 , corresponding to the peak of the distribution of
the energy flux above 1 GeV of sources in 3FGL. The error bars correspond to 1 sigma due
to rotating the observer position around the Galactic Center. The J-factors of spatially
extended sources were calculated by integrating the luminosity out to to the containment
angle of Fermi-LAT at Eγ = 1 GeV. For a thermal cross-section, we predict about 3
detectable point-like subhalos.
62
Figure 27: The same as Fig. 26, but for an energy flux threshold above 1 GeV
of 4.0 · 10−13 erg s−1 cm−2 (corresponding to the faintest source in 3FGL) instead of
1.35 · 10−12 erg s−1 cm−2 (corresponding to the peak of the energy flux distribution of
sources in 3FGL). For a thermal cross-section, we predict about 10 detectable point-like
subhalos.
63
Figure 28: The same as Fig. 26, but for annihilations to τ + τ − instead of bb. The energy
flux threshold used in this plot is 4.43 · 10−13 erg s−1 cm−2 above 10 GeV, corresponding
to the peak of the distribution of the energy flux above 10 GeV of sources in 3FGL. For a
thermal cross-section, we predict about 4 detectable point-like subhalos.
64
Figure 29: The same as Fig. 27, but for a 100 GeV dark matter particle instead of a 40 GeV
particle. For a thermal cross-section, we predict about 5 detectable point-like subhalos.
65
Figure 30: Number of detectable point-like subhalos plotted against the annihilation crosssection for different values of the dark matter particle mass mχ , in the case of 100%
annihilation into bb. Subhalos are considered detectable if their gamma-ray energy flux
above 1 GeV exceeds 1.35·10−12 ergs−1 cm−2 , corresponding to the peak of the distribution
of the energy flux above 1 GeV of sources in 3FGL. The number of detectable subhalos
per value of the annihilation cross-section was averaged over 24 observer positions.
66
Figure 31: The same as Fig. 30, but for annihilations to τ + τ − instead of bb. The energy
flux threshold used in this plot is 4.43 · 10−13 erg s−1 cm−2 above 10 GeV, corresponding
to the peak of the distribution of the energy flux above 10 GeV of sources in 3FGL.
6.1.4
Upper limits on the annihilation cross-section
In Fig. 32 we present our upper limits on the DM annihilation cross-section as function of
the DM particle mass mχ in the case of annihilations to bb. The solid line was obtained
by calculating the value of the annihilation cross-section for which the predicted number
of detectable subhalos equals zero. Thus, the solid line corresponds to our constraints in
the case of no candidate DM subhalos in 3FGL. The dashed line was obtained in the same
way as the solid line, but by requiring that the number of predicted detectable subhalos
for a given value of mχ equals the number of candidate subhalos in 3FGL compatible with
that value of mχ , as presented by [9] (see Fig. 21).
67
Figure 32: Upper limits on the DM annihilation cross-section as function of the DM particle
mass mχ , in the case of 100% annihilation into bb. The solid line represents the constraint
that would have been obtained if there were no DM subhalo candidate sources in 3FGL.
The dashed line represents the constraint taking into account the population of candidate sources as provided by [9]. Candidate sources are compatible with DM annihilation
spectra at 95 % confidence level. The dotted line corresponds to a thermal cross-section.
Subhalos were considered detectable if their gamma-ray energy flux above 1 GeV exceeds
1.35 · 10−12 erg s−1 cm−2 , corresponding to the peak of the distribution of the energy flux
above 1 GeV of sources in 3FGL. The number of detectable subhalos per value of the
annihilation cross-section was averaged over 24 observer positions.
68
7
Results — J-factors of Dwarf Galaxies with DC14 Profiles
We calculated the J-factors of six dwarf galaxies with NFW profiles and DC14 profiles.
As described in the ‘Methods’ section, we followed [16] by integrating over a cone with a
radius of 0.5 degrees and along the line-of-sight connecting the observer with the centre
of the dwarf galaxy. We took the values for the distance, latitude and longitude of the
dwarfs from Table 1 in [16] and the parameters that define the NFW and DC14 profiles
from Table 2 in [74]. All parameters we used are shown in Table 2.
The results are shown in Table 3. All JNFW -values are compatible with the values
quoted by the Fermi-LAT collaboration in their combined dwarf galaxies analysis ([16]),
as expected. For four dwarf galaxies, the JDC14 -values are compatible with the values of
[16], but for LeoI and LeoII we find significantly higher J-values using the DC14 profile
compared with the values of [16]. Using the Planck 2015 cosmological parameters rather
than the Planck 2013 values as quoted in [86] does not alter the results in a qualitative
way.
Table 2: Values of parameters for NFW and DC14 profiles of the six dwarfs that were
studied in both [74] and [16].
Mstar
Mhalo, NFW
Mhalo, DC14
α
β
γ
l (◦ )
b (◦ )
d (kpc)
Fornax
2.45 · 107
2.00 · 109
3.68 · 1010
2.02
2.60
0.393
237.1
-65.7
147
LeoI
4.90 · 106
1.99 · 109
2.89 · 1010
1.38
2.87
0.761
226.0
49.1
254
LeoII
1.17 · 106
8.01 · 108
7.03 · 109
1.38
2.88
0.765
220.2
67.2
233
Draco
9.12 · 105
5.02 · 109
8.82 · 109
1.15
3.02
0.904
86.4
34.7
76
Sculptor
3.89 · 106
2.02 · 109
1.52 · 1010
1.58
2.77
0.643
287.5
-83.2
86
Carina
5.13 · 105
3.99 · 108
2.00 · 109
1.58
2.77
0.644
260.1
-22.2
105
69
Table 3: Values of J-factors for NFW and DC14 profiles of the six dwarfs that were studied
in both [74] and [16]. In calculating these values we used the Planck cosmology as in [86]
in the definition of Mvir and in the c-M -relation of [86]. This cosmology was also used in
the analysis of [74]. The J-factors are integrated over a cone with radius of 0.5 degrees.
All JNFW -values are within the error bars of the J-values used in the Fermi dwarf analysis
[16]. The JDC -values of Fornax, Draco, Sculptor and Carina are within Fermi’s error bars,
but those of LeoI and LeoII are significantly higher.
JNFW [GeV2 cm−5 ]
log10 JNFW
JDC14 [GeV2 cm−5 ]
log10 JDC
log10 JNFW, Fermi
Fornax
2.06 · 1018
18.3
2.45 · 1018
18.4
18.2 ± 0.21
LeoI
7.20 · 1017
17.9
2.75 · 1018
18.4
17.7 ± 0.18
LeoII
4.16 · 1017
17.6
1.08 · 1018
18.0
17.6 ± 0.18
Draco
8.85 · 1018
18.9
6.64 · 1018
18.8
18.8 ± 0.16
Sculptor
4.76 · 1018
18.7
4.74 · 1018
18.7
18.6 ± 0.18
Carina
1.18 · 1018
18.1
1.23 · 1018
18.1
18.1 ± 0.23
70
8
8.1
Discussion
Detectability of subhalos in VL-II
The effect of baryons on the detectability of DM subhalos is negligible if one uses the
DC14 density profile along with the abundance matching relation of Brook et al. [90]
and the effect of reionisation. There are simply not enough high-mass halos in VL-II for
the baryons to have an impact. Therefore, we abandoned the inclusion of DC14 profiles
in our analysis of the subhalos in VL-II. In our results we have also shown the number of spatially extended sources that would be detectable. However, this result should
be critically assessed: in calculating this number, we have not performed a line-of-sight
integral of the DM density squared, rather, we integrated the density squared out to
the 68 % containment angle at 1 GeV of Fermi-LAT. Moreover, we used a detectability threshold corresponding to the threshold for the inclusion of point sources in 3FGL,
whereas spatially extended sources are less detectable than point sources due to background
modelling [15].
Adopting the peak of the energy flux distribution above 1 GeV of sources in 3FGL as
the energy flux threshold for inclusion in 3FGL, we have found that about 3 subhalos in a
Milky Way-like halo would be detectable with Fermi-LAT as a point source at 5-sigma level
in the case of a 40 GeV DM particle annihilating to bb-quarks at a thermal cross-section.
This is compatible with the findings of Pieri et al. [6]. If we instead take an optimistic
approach by taking the energy flux above 1 GeV of the faintest source included in 3FGL
as the threshold for inclusion in 3FGL, we found ∼ 10 detectable subhalos.
For a 100 GeV DM particle annihilating to bb at a thermal cross-section, a scenario
which has not yet been excluded by the most stringent upper limits on the annihilation
cross-section set by Fermi-LAT in their combined dwarf analysis ([16], see Fig. 12), we
predict ∼ 5 detectable point-like subhalos using the energy flux above 1 GeV of the faintest
source in 3FGL as the threshold for detectability. Although this choice of threshold is
probably too optimistic, this result does indicate that even though no significant gammaray emission was observed from dwarf galaxies, we might still expect to find DM subhalos
among the unidentified sources of Fermi-LAT.
Our results are slightly in tension with the results of Bertoni et al. [9]. Whereas we
predicted ∼ 3 detectable subhalos for a 40 GeV DM particle annihilating to bb, Bertoni
et al. predicted ∼ 10 detectable subhalos for the same DM particle parameters and a
slightly more conservative detectability threshold. For this reason, our upper limits on the
annihilation cross-section using the DM subhalo candidate population in 3FGL from [9]
are slightly weaker than those presented in [9]. The discrepancy is due to their optimistic
choice of DM density profile. The details of the analysis of [9] are found in [8]. From
this paper we learn that Bertoni et al. describe their halos with an Einasto profile, and
by comparing the mass fraction in subhalos in the local volume with the mass fraction in
subhalos at the virial radius of a halo in the Aquarius simulation, they let the halos lose
71
99.5 % of their initial mass due to tidal stripping. This means that a 105 M stripped halo
has a scale radius corresponding to a 2 · 107 M halo before tidal stripping. In Fig. 34 we
compare the density profile of a 107 M halo used in our analysis with the corresponding
profile used in [9].
To examine the difference between their and our approach, we have reproduced the left
panel of Fig. 1 in [8], in which contours of constant gamma-ray flux from DM subhalos are
plotted in the mass-distance plane. The result using our approach is presented in Fig. 33.
Comparing this figure to Fig. 1 in [8], we indeed see that our predicted annihilation fluxes
are systematically lower than those predicted by [8] for the same values of halo mass and
distance.
If we consider a 105 subhalo after tidal stripping, in the analysis of Bertoni et al. this
subhalo had a mass of 2 · 107 before infall into the host halo. Following Bertoni et al. in
adopting the concentration-mass relation presented in [93], a 2 · 107 halo with a density
distribution described by the Einasto profile used by Bertoni et al. has a scale radius (the
radius at which the logarithmic slope of the profile equals -2) of 0.42 kpc (using a NFW
profile, the halo would have a scale radius of 0.33 kpc instead). The radius within which
105 M is contained is 0.056 kpc. Thus, Bertoni et al. assume that even the parts within
the scale radius of a halo get destroyed due to tidal effects. However, the tidal radii —
defined as the radius at which the density of the subhalo is equal to the density of the
host halo [3] — of the subhalos in VL-II are larger than their scale radii. This means that
the matter within the scale radii of the subhalos in VL-II have survived tidal stripping, in
contradiction with the assumption of Bertoni et al.
In conclusion, we have treated the effect of tidal stripping on halos in a careful way by
directly taking the results from the VL-II simulation, as opposed to Bertoni et al., who
assumed an amount of tidal mass loss that is not supported by the results of VL-II.
When this project was finished, we came across a paper by Zhu et al., in which they
argue that the stellar disk of a Milky Way-size galaxy depletes subhalos near the central
region [94]. The total number of low-mass subhalos in their hydrodynamic simulation is
nearly twice as low as that in a DM-only simulation. This would change the predictions
regarding the detectability of DM subhalos, but was not taken into account in this study.
72
Figure 33: Contours of constant gamma-ray flux in the energy range 1 - 100 GeV from
DM subhalos in the mass-distance plane. J-factors of subhalos were calculated assuming
NFW-profiles with concentrations given by Eq. 11 in [6] at RGC = 8 kpc, where we
integrated the luminosity out to the scale radius. The corresponding fluxes were calculated
for the case of a 100 GeV dark matter particle annihilating to bb at a thermal cross-section
hσvi = 3 · 10−26 cm3 s−1 . Going one contour to the right in the figure means going one
order in magnitude higher in flux. The dashed line corresponds to a flux of 10−9 cm−2 s−1 .
73
ΡHrL @MŸ kpc-3 D
1010
NFW, 1e7
108
Einasto, 2e9
106
104
100
1
0.01
0.01
0.1
1
10
100
r @kpcD
Figure 34: Comparison of the NFW profile we assigned to a 107 M halo in VL-II with the
Einasto profile Bertoni et al. assigned to the same halo before tidal stripping [9].
8.2
J-factors of dwarf galaxies
The result that the JDC -values of LeoI and LeoII are significantly higher than their NFW
counterparts was not expected at first glance: DC14 profiles are more cored than NFW
profiles, so one would naively expect the JDC14 -values to be smaller than the JNFW -values.
However, there are two effects in play that are countering each other. The first one is
the flatness of the DC14-profiles, which reduces the annihilation luminosity from the inner
region of the halo. The second effect, which is due to the coredness of DC14 profiles, is that
the fitted total halo mass is larger for a DC14 profile than for a NFW profile in order to
account for the kinematic star data. LeoI and LeoII with DC14 profiles are approximately
10 times heavier than LeoI and LeoII with NFW profiles. Their inner regions are apparently
not cored enough to counterbalance the effect of the larger mass, which causes their JDC14 values to be larger than their JNFW -values.
These results indicate that the J-values used to place some of the most stringent constraints on the dark matter annihilation cross-section are off when one considers a density
profile of dwarf galaxies that accounts for baryonic effects. It would be interesting to calculate JDC14 for the other dwarfs used in the combined analysis of [16] as well, such that
we can redo the combined analysis with the new J-values and place even more stringent
constraints on the dark matter annihilation cross-section.
74
9
Conclusions
The main conclusions from our analysis can be summarised as follows:
1. In the case of a 40 GeV WIMP annihilating to bb at a thermal cross-section, we
predict between 3 and 10 (depending on the choice of detectability threshold) DM
subhalos among the unidentified sources of 3FGL.
2. For a 100 GeV WIMP annihilating to bb at a thermal cross-section, a scenario that
is not excluded by current constraints, we predict about 5 detectable subhalos using
an optimistic detection threshold.
3. We placed upper limits on the DM annihilation cross-section using our results obtained for a conservative detectability threshold. These limits are competitive with
constraints by other studies in case there are no DM subhalo candidates in 3FGL.
4. Our results are in tension with those of Bertoni et al. ([9]), due to their too optimistic
assumption concerning the amount of tidal stripping.
5. Adopting a DC14 profile with parameters for individual dwarf galaxies provided by
[74], rather than the commonly used NFW profile, we found that the J-factors of
LeoI and LeoII exceed the error margins of the J-factors of these dwarf galaxies used
in the combined dwarf analysis of Fermi-LAT [16].
75
10
10.1
Appendix
Line-of-sight integral
The J-factor is the astrophysical term in the formula for the dark matter annihilation flux
(Eq. 34); it quantifies how much dark matter annihilation the observer is looking at. It is
given by:
Z
∆Ω
Z
lmax
dΩ
J(∆Ω) =
ρ2 [r(l, Ω)]dl,
(56)
lmin
0
i.e., the integral of the the dark matter density profile squared along the line-of-sight
(bounded by lmin and lmax ) and over a solid angle ∆Ω. Fig. 35 shows a sketch of the
situation. The observer is located at O and is looking along the line-of-sight l that is
separated from the line d connecting the observer and the centre of the halo by an angle ψ.
The parameter r connects the centre of the halo with the line-of-sight, and is the parameter
the dark matter density ρ depends on. In principle, we would like to integrate ρ(r)2 over
l from 0 to ∞, to include all dark matter that we observe when looking in that direction.
However, this integral is not bounded, so we have to make a cut and integrate from lmin
to lmax . A sensible choice for lmin and lmax are values that correspond to the minimum
value of r possible under the angle ψ and rs , respectively, such that we integrate ρ2 up to
the scale radius, which contains ∼ 90 % of the luminosity. The behaviour of r will become
more clear later on.
Figure 35: A sketch of the situation. O corresponds to the observer (Fermi-LAT), d is the
distance to the halo, l is the line-of-sight, which is separated from the vector d by an angle
ψ. r is the parameter that connects the centre of the halo to the line-of-sight l.
Let us parametrize r in terms of l and ψ. Using the cosine rule, we find:
r=
p
l2 + d2 − 2ld cos ψ
76
(57)
and
l± = d cos ψ ±
q
rs2 − d2 sin2 ψ.
(58)
Fig. 36 shows the behaviour of r when l runs from 0 to ∞. For this plot, a distance d
of 147 kpc (the distance to the Fornax dwarf galaxy) and an angle ψ of 0.5 degrees were
chosen. As expected, at l = 0, r equals d. With increasing l, r decreases up to its smallest
value, after which it increases again to infinity.
r @kpcD
140
120
100
80
60
40
20
0
0
50
100
150
200
250
l @kpcD
Figure 36: r as a function of l, for d = 147 kpc and ψ = 0.5 degrees.
Since the angle ψ is non-zero, the parameter r does not become zero. This is more clearly
depicted in Fig. 37, which shows a close-up of Fig. 36.
Because a telescope has a certain angular resolution, we can think of looking to the sky
in the direction of l through a cone of a certain width (the angular resolution). In the
analysis of dwarf galaxies by the Fermi-LAT collaboration, J-factors are integrated over a
cone with a radius of 0.5 degrees [16]. Since
∆Ω = 2π(1 − cos ψ),
(59)
dΩ = 2π sin ψdψ,
(60)
77
r @kpcD
20
15
10
5
0
135
140
145
150
155
160
l @kpcD
Figure 37: r as a function of l, for d = 147 kpc and ψ = 0.5 degrees.
Eq. 56 can be written as:
Z
J0.5◦ = 2π
0.5◦
Z
lmax
sin ψdψ
0
lmin
where r(l, ψ) is given by Eq. 57.
78
ρ2 [r(l, ψ)]dl.
(61)
10.2
Luminosity of a point-like dark matter halo with a NFW density
profile
The luminosity of a point-like dark matter halo with a NFW density profile can be calculated by integrating the NFW profile squared out to the scale radius (which contains
∼ 90 % of the luminosity):
Z
Z
R
r2 ρ2s
2 2
2 h
i4 dr = ρs rs
r
r
1 + rs
rs
0
0
R
1
h
i4 dr
1 + rrs
(62)
Substituting u = 1 + r/rs :
Z
ρ2s rs2
0
Z
R
1
h
i4 dr
1 + rrs
1+ rR
s
1
du
u4
= rs3 ρ2s
1
=
rs3 ρ2s
1
− u−3
3
1+ R
rs
1

1

= rs3 ρ2s − 3 1+

1
3 + 
3
R
rs
1
r 3 ρ2
= rs3 ρ2s − s s 3
3
3 1 + rRs
Integrating out to R = rs , we find:
Z
0
rs
r2 ρ2s
7 3 2
2 h
i4 dr = rs ρs
24
r
1 + rrs
rs
79
(63)
11
Lay-man summary
In the 1930s, Jan Oort and Fritz Zwicky discovered a discrepancy between the amount of
visible matter in the galaxy and in galaxy clusters and the amount of matter that should
be there to account for the motions of stars. The missing matter was called ‘dark matter’,
and since then the evidence for its existence has accumulated. Although the gravitational
effect of dark matter can be measured, little is known about its nature, except that it
cannot be made up of already known particles — dubbed ‘baryons’. Many theories that
extend the Standard Model of Particle Physics predict the existence of (a) new type(s) of
particle(s), which could in principle be dark matter.
Although there is little restriction on the dark matter particle mass and it is not clear
if dark matter is made up of one particle species or several, great effort is being made
to unravel the nature of particle dark matter. Theorists are building models of particle
dark matter and their interactions; the LHC at CERN is colliding beams of protons at
ever higher energies in the hope to produce particles that have not been seen before; giant
experiments such as LUX are trying to detect dark matter particles scattering off of atomic
nuclei. Besides collider and direct searches for dark matter, scientists are searching for the
annihilation products of dark matter, a method called indirect detection.
The standard model particles that could be produced in dark matter annihilation processes could subsequently decay into photons, which travel freely from their site of production to Earth, not being affected by magnetic fields. The Fermi Large Area Telescope
(Fermi-LAT), a satellite orbiting the Earth, measures these photons with energies in the
range 20 MeV to > 1 TeV.
From regions with a higher dark matter density one expects a higher flux of photons,
since the chance that two dark matter particles bump into each other is proportional to
the dark matter density squared. The current model of cosmology predicts that the dark
matter halo in which the Milky Way resides contains a large number of over-dense regions
called dark matter subhalos. Large N-body simulations have been performed to predict the
dark matter subhalo distribution in our Galaxy. Emitting a constant flux of photons from
dark matter annihilations, dark matter subhalos might be present in the Fermi-LAT data
as faint point sources against the Galactic and extragalactic diffuse photon background.
Using the results of the Via Lactea II simulation to predict the dark matter subhalo
distribution around Earth, we have calculated the number of subhalos in the Milky Way
that might have been detected as unidentified point sources by Fermi-LAT. For a dark
matter particle model that has not been excluded by other analyses, we make an optimistic
prediction that about five subhalos would be present in the latest Fermi-LAT point source
catalog 3FGL.
We compared our predictions of the number of subhalos in 3FGL to the number of
sources in this catalog that have spectra compatible with dark matter annihilation spectra.
This comparison allowed us to place upper limits on the annihilation cross-section of dark
matter particles: if the annihilation cross-section would be higher than our upper limit, we
80
would predict more detected dark matter subhalos than the number of candidate subhalos.
In case no candidate dark matter subhalos are present in the catalog — if they all get
associated with other astrophysical objects —, our upper limits are competitive with limits
placed by other studies.
The most stringent upper limits are placed by the Fermi-LAT collaboration in an analysis of dwarf galaxies, from which no significant high-energy photon flux was observed. This
analysis made an assumption on the dark matter density distribution within the dwarf
galaxies. We checked if this assumption was robust by calculating the predicted highenergy photon flux from the dwarf galaxies using a recently proposed dark matter density
profile called ‘DC14’ that takes into account the effect of baryons on the dark matter distribution. We found that for two of the fifteen galaxies that were used in the Fermi-LAT
analysis, the predicted flux using the DC14 profile exceeds the error margin quoted in the
Fermi-LAT analysis.
12
Samenvatting voor leken
In de jaren 30 ontdekten Jan Oort en Fritz Zwicky dat er een verschil was tussen de
hoeveelheid zichtbare materie in ons sterrenstelsel (en in clusters van sterrenstelsels), en
de hoeveelheid materie die nodig is om de bewegingen van sterren te kunnen verklaren.
De missende materie werd ‘donkere materie’ genoemd, en sinds die eerste observaties heeft
het bewijs voor het bestaan van donkere materie zich opgestapeld.
Hoewel de zwaartekrachtseffecten van donkere materie gemeten worden, weten we
weinig tot niets over de deeltjeseigenschappen van donkere materie, behalve dat donkere
materie niet kan bestaan uit deeltjes die we al eens geobserveerd hebben en die gezamenlijk
‘baryonen’ worden genoemd. Veel theorieën die problemen in het standaardmodel van de
deeltjesfysica proberen op te lossen voorspellen het bestaan van (een) nieuw(e) deeltje(s),
die donkere materie zouden kunnen zijn.
Wereldwijd wordt er actief gezocht naar donkere materie. Er worden theoretische modellen bedacht voor donkere materie-deeltjes en hun interacties met andere deeltjes; de
deeltjesversneller in CERN laat protonen op elkaar botsen in de hoop om nieuwe deeltjes
te produceren; grote xenon-experimenten zoals LUX proberen donkere materie-deeltjes te
detecteren middels hun botsingen met atoomkernen. Naast deze zoektochten, wordt er ook
gezocht naar de annihilatie-producten van donkere materie. Deze methode wordt ‘indirecte
detectie’ genoemd.
De deeltjes die worden geproduceerd in annihilatie-processen van donkere materie kunnen op hun beurt vervallen naar lichtdeeltjes met enorme energieën. Deze lichtdeeltjes
reizen rechtstreeks vanwaar ze geproduceerd zijn naar de aarde, omdat ze niet afgebogen
worden door magnetische velden. De Fermi Large Area Telescope (Fermi-LAT), een satelliet die om de aarde cirkelt, meet deze lichtdeeltjes met energieën tussen de 20 MeV en > 1
TeV.
81
Hoe groter de dichtheid van donkere materie in een bepaalde regio, hoe hoger de flux van
lichtdeeltjes zal zijn, omdat de kans dat twee donkere materie-deeltjes op elkaar botsen
groter is naarmate de dichtheid groter is. Het standaard model van kosmologie voorspelt
dat de donkere materie halo waarin de Melkweg leeft, heel erg ‘klonterig’ is. Deze klonten
worden subhalos genoemd. Gigantische computersimulaties bevestigen deze theorie, en
voorspellen de distributie van subhalos in ons sterrenstelsel. Aangezien subhalos een constante flux van lichtdeeltjes uitzenden vanwege de annihilaties van donkere materie-deeltjes,
zouden subhalos gedetecteerd kunnen worden met de Fermi-LAT als zwakke puntbronnen.
In deze scriptie maken we gebruik van de Via Lactea II simulatie om de distributie
van subhalos rondom de aarde te voorspellen. We hebben uitgerekend hoeveel subhalos
in de Melkweg gedetecteerd zouden kunnen zijn als niet-geı̈identificeerde puntbronnen met
de Fermi-LAT. Voor een donkere materie-model dat nog niet uitgesloten is door andere
analyses, doen we de optimistische voorspelling dat ongeveer vijf subhalos aanwezig zijn in
de meest recente bronnencatalogus van Fermi-LAT, 3FGL.
We hebben onze voorspellingen vergeleken met het aantal bronnen in 3FGL dat een
spectrum heeft dat overeenkomt met het spectrum dat verwacht wordt van donkere materieannihilaties. Deze vergelijking stelde ons in staat om bovenlimieten te zetten op de hoeveelheid annihilatie van donkere materie: als deze hoeveelheid groter zou zijn dan onze bovenlimieten zouden we meer gedetecteerde subhalos voorspellen dan er subhalo-kandidaten
gedetecteerd zijn. Als er geen kandidaten voor subhalos zijn in de catalogus, bijvoorbeeld
als alle kandidaten nog worden geı̈dentificeerd met astrofysische objecten, zijn onze bovenlimieten vergelijkbaar met de bovenlimieten die door andere studies zijn gezet.
De sterkste bovenlimieten op de hoeveelheid annihilatie van donkere materie zijn geplaatst
door de Fermi-LAT collaboratie in hun analyse van dwergsterrenstelsels, waarvan geen significante flux van lichtdeeltjes werd geobserveerd. Deze analyse maakte een zekere aanname
wat betreft de verdeling van donkere materie in de dwergsterrenstelsels. We hebben uitgerekend in hoeverre de voorspelde flux verandert als men een andere verdeling van donkere
materie aanneemt, in het bijzonder de verdeling die voorgesteld is door Di Cintio et al.,
waarin de effecten van baryonen op de donkere materie-verdeling zijn meegenomen. We
concluderen dat voor twee van de vijftien dwergsterrenstelsels die geanalyseerd waren, de
voorspelde flux van lichtdeeltjes significant hoger is als men de verdeling van Di Cintio et
al. aanneemt in plaats van de verdeling die gebruikt werd door de Fermi-LAT collaboratie.
82
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