MSc Astronomy and Astrophysics
Gravitation and AstroParticle Physics
Master Thesis
Baryons in dwarf galaxies
systematic uncertainties on indirect dark matter searches
by
Monica van Santbrink 10202889
July 2016 60 ECTS
Carried out between September 2015 and July 2016
Supervisors: Dr. Gianfranco Bertone Dr. Christoph Weniger Dr. Jennifer Gaskins Second Examiner: Dr. Shin’ichiro Ando Institute of Physics
Abstract
Dwarf spheroidal galaxies (dSphs) are promising objects for indirect dark matter detection. Their expected gamma-ray flux depends on the J-factor: the square of the dark matter density profile integrated along the line of sight. In this research a Jeans analysis is performed on seven classical dSphs for reconstructing their J-factors. The effect of baryons is taken into account by assuming a density profile that is recently proposed by Di Cintio et al.
The J-factors we obtain are in general somewhat lower than those derived by the Fermi-LAT collaboration and much lower for the two dSphs with the largest ratio between stellar and halo mass. We discuss the origin of this discrepancy and the implications for indirect dark matter searches.
Popular scientific abstract
In 1933 the Swiss astrophysicist Fritz Zwicky studied the motions of galaxies in the so-called Coma cluster and noticed he could not explain their behavior with the amount of mass from all stars and gas. There was more mass needed to solve this discrepancy. He referred to this unseen matter as ‘dunkle Materie’, dark matter. Nowadays the evidence for dark matter is overwhelming, but its nature is still unknown. One thing is for sure, dark matter is not made up of atoms that makeup everything visible in the entire Universe. What’s more, only five per cent of the Universe is visible, the rest is dark.
There are several methods that are currently used for detecting dark matter. One of them is called indirect detection. This method aims for measuring the ordinary particles that are pro-duced when dark matter particles annihilate or decay. Promising targets for this type of search are dwarf spheroidal galaxies.
In many indirect dark matter searches of these dwarf galaxies it is assumed that the dark matter is distributed through a profile that is steep, or cuspy, towards the center. However it has been shown that ordinary atoms can alter this distribution resulting in a more flattened, or cored, distribution. Therefore a dark matter mass profile has been proposed recently by Di Cintio et al. that takes into account these ordinary particles.
In this research we use the profile of Di Cintio et al. to calculate the J-factor, a factor indicating the strength of the observed annihilation signal, for seven dwarf spheroidal galaxies. We find that our J-factors are similar to those derived by the Fermi-LAT collaboration, where a cuspy profile was used.
Abstract i
Contents ii
1 Introduction 1
2 Standard Model of Cosmology 2
2.1 Overview . . . 2
2.2 Evidence Dark Matter . . . 3
2.2.1 Rotation curves . . . 4
2.2.2 Gravitational lensing . . . 5
2.2.3 Cosmic Microwave Background . . . 6
2.3 Candidates Particle Dark Matter . . . 7
2.3.1 Neutrinos . . . 7
2.3.2 Axions . . . 8
2.3.3 WIMPs . . . 9
2.4 Thermal production of WIMPs . . . 10
2.5 Density profiles . . . 13
3 Indirect dark matter detection 15 3.1 Detection methods . . . 15
3.2 Dark matter from gamma-rays . . . 16
3.3 Dwarf Spheroidal Galaxies . . . 17
4 Impact of baryons 19 4.1 Problems with the ΛCDM model . . . 19
4.2 A baryonic solution . . . 20
4.3 Density profile with baryons . . . 20
4.4 DC14 for dwarf galaxies . . . 22
5 Analysis 24 5.1 Jeans Analysis . . . 24 5.1.1 Used profiles . . . 25 5.2 MCMC . . . 26 5.3 J-factor . . . 27 5.4 Data . . . 28
5.4.1 Stellar kinematic data . . . 28
5.4.2 Velocity dispersion profiles . . . 29
5.4.3 Surface brightness data . . . 30
6 Results 31 7 Discussion and conclusion 37 7.1 Derived halo masses . . . 37
7.2 Derived J-factors . . . 38
7.3 Impact of parametrizations . . . 39
7.3.1 Impact of velocity anisotropy profile . . . 39
7.3.2 Impact of the light profile . . . 39
7.4 Effect of binned analysis . . . 40
7.5 Assessing the DC14 fit . . . 40
7.6 Future work . . . 41
Introduction
The ΛCDM (Lambda Cold Dark Matter) model, often referred to as the standard model of cosmology, describes the evolution of the Universe since the Big Bang. The Universe consist of ∼ 90% of dark energy (DE), associated with the cosmological constant Λ and dark matter (DM) [1]. Leaving a mere five per cent for all baryons i.e. Standard Model of particle physics (SM) particles. The nature of DE and DM is not well understood, the latter is the topic of interest in this research.
A common view is that DM is made up of a new type of particles, for example WIMPs (Weakly Interacting Massive Particles). WIMPs can annihilate into SM particles. The study of DM annihilation products is called indirect detection. Since annihilation is a two body process, the detected signal depends on an integral over the density squared: the J-factor. Dwarf spheroidal galaxies are promising objects for indirect detection as they are proximate objects and have low backgrounds [2]. The Fermi-LAT collaboration has calculated J-factors for the dwarf galaxies assuming that the density profile is cuspy towards the center. However observations favor a cored profile [3]. Besides, it has been shown that baryons can alter the density profile creating a cored profile. Therefore a new density profile has been proposed recently by Di Cintio et al. (2014) (DC14) that takes into account baryons.
The aim of this research is to recalculate the J-factors assuming the DC14 profile and compare them to those of the Fermi-LAT collaboration. The structure of this thesis is as follows. First the ΛCDM model is explained in more detail in Chapter 2, with a strong focus on particle DM. Then in Chapter 3 indirect detection, in particular for dwarf galaxies, will be discussed. The last theory section is Chapter 4 which addresses the impact of baryons. In Chapter 5 the analysis that has been carried out is explained and the used data is described. Then Chapter 6 presents the results of this research. Finally in Chapter 7 the discussion and conclusion are given.
Standard Model of Cosmology
2.1
Overview
The standard model of cosmology states that the Universe, approximately 13 billion years ago, began with a hot, dense state which started to cool and eventually reached the cold, sparse state that is seen today [4]. This model assumes that general relativity is the correct way to describe gravity on large, i.e. cosmological, scales. This model will be discussed briefly.
About hundred years ago, in 1915 Einstein published his theory of general relativity. In his work he presented the Einstein field equations, or simply the Einstein equations:
Rµν−
1
2Rgµν+ Λgµν = 8πG
c4 Tµν , (2.1)
where Rµν and R are the Ricci curvature tensor and scalar respectively, gµν is the metric tensor,
Λ the cosmological constant, G Newton’s gravitational constant, c the speed of light in vacuum and Tµν the stress-energy tensor. Setting the cosmological constant Λ to zero, the Einstein
equations state that spacetime is curved by matter and energy. The constant that couples these quantities is, in SI units, of the order 10−43, implying that a large mass is needed for a significant curvature.
The cosmological constant was added by Einstein so that the solution of Equation 1 for the Universe was stationary. However, as was pointed out first by Lemaˆıtre in 1927, Hubble’s ob-servations showed that the Universe is expanding so that the Lambda-term became redundant. Despite this, keeping the term doesn’t yield mathematical inconsistencies and therefore it was kept and thought to be zero.
In 1998 with observations of distant supernovae it was found that the expansion of the Universe is accelerating, claiming a positive value for Λ. This accelerated expansion is thought to be
caused by Dark Energy (DE), whose nature is still unknown.
The standard model of cosmology includes this positive Λ associated with DE. Besides, it con-tains cold Dark Matter (CDM). This is hypothesized, non-baryonic1matter that is nonrelativis-tic and only interacts gravitationally with baryonic matter. The composition of the Universe is estimated to be ∼ 70% DE, ∼ 25% DM and ∼ 5% baryonic matter [1]. The standard model is often referred to as the ΛCDM model.
For parametrizing the ΛCDM model, one needs to solve the Einstein equations. This can be done by introducing isotropy and homogeneity. The former states that space looks the same in all directions, the latter states that space is the same at all locations. Using these symmetries one obtains for one component of the Einstein equations, the so-called Friedmann equation:
˙a a 2 = 8πG 3 ρtot− κc2 a2 , (2.2)
where a is the scale factor that parametrizes the expansion of the Universe and ˙a its first time derivative, G the gravitational constant, ρtot is the total energy density and the constant κ
describes the curvature of the Universe.
At this point it’s common to introduce the Hubble parameter H defined by
H = ˙a
a . (2.3)
The value of H is determined empirically and its current value is set to H0 = 67km s−1Mpc−1
[1]. Substituting Equation 2.3 into Equation 2.2 and setting κ = 0 in the latter, meaning that the Universe is flat, one easily obtains the critical density ρcrit
ρcrit =
3H2
8πG . (2.4)
The abundances of matter, energy and vacuum are often expressed in units of the critical density, Ωi≡ ρi/ρcrit. Measurements show that the Universe is nearly flat [1], this implies that the total
density is almost equal to the critical density and thus
ΩΛ+ Ωmatter= 1 . (2.5)
2.2
Evidence Dark Matter
In 1922 the Dutch astronomer Kapteyn suggested that the Milky Way contains dark matter and proposed a method to estimate its mass [5]. Fellow Dutchman Jan Oort found in 1933 a
1
discrepancy between the observed mass and the mass needed to explain the observed motions of nearby stars in the Milky Way [6]. Oort used the term dark matter for this missing mass, but reckoned that the mismatch could be explained by baryonic matter: a significant part of the stars is either too dim to be observed or obscured by absorbing matter.
Around the same time of Jan Oort, Frits Zwicky studied the Coma cluster. By applying the virial theorem he calculated the mass of the cluster and found that this was significant greater than the mass inferred from its luminosity [7].
Since these first suggestions and calculations the evidence for dark matter has increased by several independent and different studies. In the next sections three of these will be discussed.
2.2.1 Rotation curves
The rotation curves of galaxies are considered as the most convincing and direct evidence for dark matter on galactic scales. In the 1970s it was found that the circular velocities of stars and gas in galaxies have roughly the same value and are therefore independent of the distance to the Galactic Center. This results in a flat rotation curve. However, using Newtonian physics, the circular velocity is found by equating the centripetal force to the gravitational force resulting in:
vc(r) =
r
GM (r)
r , (2.6)
where M (r) = 4πR ρ(r)r2dr and ρ(r) is the mass density profile. From Equation 2.6 one would expect the circular velocity to scale as vc(r) ∝ 1/
√
r. However, given the the approximately flat rotation curve it is needed that M (r) ∝ r and ρ ∝ 1/r2. This can be explained by introducing a dark matter halo: an approximately spherical halo that contains non-luminous matter and is extending far beyond the visible size of the galaxy.
In Figure 2.1 the rotation curve of NGC 3198 is shown. It can be seen that the circular velocity becomes approximately constant at large distances. Moreover the rotation curves of the disk, gas and dark matter are shown as dashed, dotted and dash-dotted respectively.
Figure 2.1: Rotation curve of NGC 3198 including the curves of the individual components. Dashed, dotted and dash-dotted represent the disk, gas and dark matter respecitvely. Taken
from [8]
2.2.2 Gravitational lensing
As discussed in Section 2.1 Einsteins theory of general relativity states that mass curves space-time and therefore light is bent when it passes a massive object. The bending of light is a purely geometrical effect and is independent of the energy of the photons. This effect is referred to as gravitational lensing and can be used to estimate the mass of e.g. galaxies and galaxy clusters.
Gravitational lensing can be divided into two regimes: strong and weak lensing. Studies of strong lensing showed that galaxy clusters have a mass-to-light ratio of about 300 which is a direct demonstration of the presence of dark matter [9].
For the majority of sources weak gravitational lensing is the method of interest [10]. The path of light is still deflected by gravitational fields, but the amplitude is small and it is undetectable on individual galaxies.
The results of the galaxy cluster 1E0657-56, referred to as the Bullet cluster, are noteworthy. This cluster is formed by the collision of a smaller cluster with a larger one. While colliding most mass passed right through. Using weak gravitational lensing the center of the total mass was determined. On the other hand, the X-ray telescope Chandra determined the center of the mass in this waveband. By comparing these results it can be concluded that there is a spatial offset
between the total and baryonic mass peaks. Therefore the observations of this cluster support the existence of dark matter and allow for the conclusion that it is collisionless, otherwise the particles should have experienced ram pressure and thus slowed down like the luminous gas [10]. Moreover the Bullet cluster disfavors other alternatives such as modified Newtonian dynamics (MOND) where the mass distribution should coincide with the baryon distribution.
In Figure 2.2 an image of the Bullet cluster is presented. The pink regions represent the X-ray emission from the hot gas, the blue region is a reconstruction of the total mass determined by gravitational lensing. The spatial offset can be seen directly.
Figure 2.2: Bullet cluster 1E0657-56. The pink region indicate the X-ray emission from the hot gas, the blue region is a reconstruction of the total mass determined by gravitational lensing.
Taken from [11]
2.2.3 Cosmic Microwave Background
The Cosmic Microwave Background (CMB) is radiation originating from the propagation of photons in the early Universe and serves as another piece of evidence for the existence of Dark Matter.
Predicted in 1948 by Gamow, the CMB was detected in 1965 by the radio astronomers Penzias and Wilson who had an unexplainable, isotropic noise2 in their measurements.
Today it is known that the CMB has an almost perfect blackbody spectrum of temperature T = 2.726K [12] and is isotropic at the 10−5 level in temperature. At higher levels anisotropies
are observed which are commonly expanded in spherical harmonics Ylm(θ, φ) δT T (θ, φ) = ∞ X 2m +l X −l almYlm(θ, φ) . (2.7)
From a measured set of alm the variance Cl can be obtained by
Cl = h|alm|2i = 1 2l + 1 l X −l |alm|2 . (2.8)
By comparing the set of observed Cl’s with the theoretical ones that are obtained by varying
the parameters of a cosmological model, a very good accuracy can be acquired on the parame-ters. Both the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellite put constraints on the abundance of baryons and matter. The latter has obtained the most recent values and are given by
Ωb = 0.0490 ± 0.0007 Ωm = 0.314 ± 0.020 [1]. (2.9)
As can be seen from Equation 2.9 the CMB allows not only to conclude the presence of dark matter but also to determine the total amount of dark matter in the Universe. The DM-to-baryon ratio in the Universe, using the Planck estimates, is Ωm/Ωb ≈ 6.
2.3
Candidates Particle Dark Matter
There are numerous dark matter candidates and only some will be discussed in this thesis. A particle that makes a strong candidate meets several criteria, such as that it should be a cold, neutral particle that has the appropriate relic density and it needs to be in agreement with current dark matter searches and with stellar structure and evolution [13].
2.3.1 Neutrinos
The Standard Model (SM) of particle physics classifies the fundamental particles and describes the electromagnetic, weak and strong interactions between them. The elementary particles that make up matter are quarks and leptons, these are fermions i.e. particles with half-integer spin. The gauge bosons, particles with integer spin, mediate the interactions. The leptons can be di-vided into two main classes: charged leptons and neutral leptons. The latter are better known as neutrinos.
Neutrinos are stable particles and only interact through the weak nuclear force. Therefore they have been considered as DM candidates. Since oscillation experiments proved that neutrinos have nonzero rest mass [4], it is possible to evaluate their total relic density. As there are three generations of neutrinos νi we sum over their masses mi to obtain the density
Ωνh2 = 3 X i=1 mi 93 eV . (2.10)
From tritium β-decay experiments the 95% C.L. upper limit on the neutrino mass is found to be [14]
mν = 2.05 eV . (2.11)
This yields the following upper limit on the neutrino density
Ωνh2 . 0.07 . (2.12)
This implies that there are not enough neutrinos to account for all the dark matter. Another constraint rises from the CMB anisotropies combined with large scale structure surveys sug-gesting an even lower relic density of Ωνh2 = 0.0067 [15].
Besides, neutrinos are disfavored because of their relativistic nature which results in a charac-teristic length scale or free steaming length that is on the order of ∼ 40Mpc mν/30 eV [16].
Regions separated by a distance larger than this free steaming length will prevail, however all fluctuations lower this scale will be erased. As a consequence the Universe would have a top-down formation history i.e. large structures are formed first, followed by smaller structures. However our galaxy appears older than the Local Group [17] and thus contradicts the top-down formation history.
The mass of a neutrino is given in Equation 2.11, however the SM states that there are only left handed neutrinos, implying they should be massless. This requires physics beyond the SM. The solution is given by adding right handed neutrinos to the SM. In this way neutrinos obtain their mass through the same mechanism as quarks and leptons. For the right handed neutrinos to be allowed they must have no SM gauge interactions [18] and are therefore referred to as sterile neutrinos. The mass scale of these particles is not predicted a priori. Sterile neutrinos are DM candidates and their constraints come from their cosmological abundance and, since a sterile neutrino can decay into a photon and a neutrino, their decay products [12].
2.3.2 Axions
The charge parity symmetry (CP symmetry) combines the symmetry between positive and negative charges (C) and the symmetry of spatial coordinates (P). However this symmetry can
be violated.
Although the SM has proven to be successful, there are several deficiencies. Next to the neutrino mass, there is the strong CP problem. The quantum chromodynamics Lagrangian L contains a CP violating term ¯θ. If ¯θ 6= 0 then QCD violates the CP symmetry. Experiments on the neutron electric dipole moment found an upper limit of ¯θ < 10−9 [19]. Giving rise to the question: why is ¯θ so small? As a solution a new particle, the axion3, was introduced.
After being introduced to solve the CP problem, the axion turned out to be a serious DM candidate. Firstly they couple weakly to SM particles, secondly they are effectively stable and finally they have a proper density to be a substantial fraction of dark matter [20].
2.3.3 WIMPs
The most appealing class of candidates are Weakly Interacting Massive Particles (WIMPs). These particles are electrically neutral and interact through both gravity and the weak nuclear force. Their generic production mechanism resulting in the correct relic density, which will be discussed in Section 2.4, combined with their detection possibilities makes them strong DM candidates [10].
Different extensions of the SM such as supersymmetry (SUSY) propose WIMPs. In the remain-der of this section will be explained why SUSY was introduced and how this results in DM candidates.
SUSY was introduced for solving the hierarchy problem [21]. In the SM particles acquire their mass through interacting with the Higgs field. The mass of the Higgs boson is on the order of mh ∼ 100 GeV [18]. However, given the three fundamental constants: the speed of light
c, Planck’s constant h and the gravitational constant Gn, a natural value for mh would be a
combination of these. Such a combination is the Planck mass Mpl =phc/G ≈ 1.2 · 1019 GeV.
The hierarchy problem addresses the question of why mh Mpl.
Next to SUSY there are several other solutions proposed for this problem e.g. the Higgs boson being a composite particle [22] and the existence of large extra spatial dimensions [23].
Supersymmetry relates boson and fermions: each known SM particle has a super partner with the same mass but which differs by half a unit of spin [24]. Fermions have spin-0 particles as superpartner (sfermions), the gauge bosons have spin-1/2 partners (gauginos) and by introduc-ing an additional Higgs boson, these bosons are linked to a spin-1/2 super partners (Higgsinos). This is the minimal supersymmetric extension of the SM (MSSM).
This MSSM solves the hierarchy problem, but another symmetry needs to be introduced. In
3
the SM the decay of protons is prevented by gauge symmetries, however this is not the case in the MSSM. Therefore R-parity is introduced which is defined as
R = (−1)3B+L+2s , (2.13)
where B, L and s are the baryon number, lepton number and spin respectively. All known SM particles have R = +1 and all superpartners have R = −1. Therefore supersymmetric particles can only decay into an odd number of lighter supersymmetric particles. As a consequence the lightest supersymmetric particle (LSP) cannot decay, hence it is stable. A direct consequence is the LSP being an excellent DM candidate.
There are several LSP dark matter candidates: the sneutrino with spin-0, however experimental results exclude this particle [25], the gravitino with spin-3/2, this particle is very difficult to detect [26], and the neutralino with spin-1/2. The lightest neutralino is the most studied DM candidate within the MSSM.
2.4
Thermal production of WIMPs
In general it is assumed that WIMP pairs were produced and annihilated in the early Universe in particle-antiparticle collisions such as
χ ¯χ ↔ e+e−, µ+µ−, W+W−, ZZ , (2.14)
where on the left-hand side χ represents the WIMP and on the right-hand side are the stan-dard model particles, the electron e, the muon µ, the W-boson and the Z-boson. Initially the temperature was much higher than the WIMP mass resulting in an equilibrium between the producing and annihilating processes, the rate Γ given by
Γ = hσannvineq , (2.15)
here σann is the annihilation cross section, v the relative velocity of the annihilating particles,
the brackets indicate that the argument is thermally averaged and neq the equilibrium number
density.
As a result of the expansion of the Universe the temperature decreased and eventually became smaller than the WIMP mass. This led to a smaller number of WIMPs produced, while the producing and annihilation rates remained in equilibrium. Moreover, as a consequence of the ex-pansion, the number density decreased which resulted in a smaller production and annihilation
rate Γ as can be directly seen from Equation 2.15. At some point the rate became smaller than the expansion rate of the Universe H, see Equation 2.3. Accordingly the WIMPs chemically decoupled and the production ceased. Therefore the WIMP density, the number of WIMPs in a comoving volume, remained approximately constant and is often referred to as the thermal relic density.
Closely following [10] the value of the thermal relic density can be computed using the Boltzmann equation
dn
dt = −3Hn − hσannvi(n
2− n2
eq) , (2.16)
which takes into account the number density n, its first time derivative and other quantities as defined before in this section. The first term on the right-hand side takes the expansion of the Universe into account, whereas the second term the change in number density. Besides this equation the other ingredient is the law of entropy conservation
ds
dt = −3Hs , (2.17)
here s is the entropy density and given by s = 2π2g∗T3/45 where g∗ is the number of relativistic
degrees of freedom. By introducing the variables
Y ≡ n s , Y eq≡ neq s , x ≡ m T , (2.18)
where T is the photon temperature, Equations 2.16 and 2.17 can be combined and rewritten as dY dx = 1 3H ds dxhσannvi(Y 2− Y2 eq) . (2.19)
This is a Riccati equation i.e. a first-order differential equation that is quadratic in the unknown function and can be solved numerically. For doing this the boundary condition Y = Yeq at x ' 1
should be used. This condition states that at high temperatures the particles were in thermal equilibrium as explained at the beginning of this section. Finally an expression for the Hubble parameter is needed for solving Equation 2.19. This can be obtained from, a modified version of, Equation 2.4
H2 = 8π
3Mpl2ρ , (2.20)
where Mpl = 1.2 · 1019 GeV is the Planck mass. As the ultra-relativistic species dominate in
the early Universe, the mass-energy density ρ is given by ρU R = π
2
30g∗T4 resulting in an Hubble
expansion rate that scales with temperature squared
H = q 8π3g ∗T4/90Mpl2 ≈ 1.66g∗T2 Mpl . (2.21)
Solving the differential equation, expressing the result in units of critical density Ωi ≡ ρi/ρcrit
and performing an order of magnitude estimate results in
ΩXh2=
3 · 10−27cm3s−1
hσannvi , (2.22)
where the subscript X indicates that ΩX is the density of a generic relic and h = H0/100 km
s−1 Mpc−1 the Hubble parameter.
In Figure 2.3 the numerical solution to Equation 2.19 is shown i.e. the evolution in time of the WIMP density in the early Universe. As expected, given the explanation at the beginning of this section, can be seen from the figure that the density Y is close to its equilibrium value Yeq at high temperatures. As the temperature decreases Yeq becomes suppressed, therefore the
equilibrium ceases. It is at the freeze-out temperature that the WIMP number density becomes approximately constant. An important feature of the thermal relic density as illustrated by Figure 2.3 is that the annihilation cross section and the relic density are inversely proportional. As WIMPs with stronger interactions stay longer in equilibrium, thus decoupling at a lower freeze-out temperature, their densities are further suppressed.
From a particle physics point of view: new particles interacting through the weak force will have a mass around mweak∼ 100 GeV [28], the origin of this mass scale is not well understood.
Using this mass scale an order of magnitude estimation can be performed: hσvi ∼ α2(100 GeV)−2 ∼ 10−25 cm3s−1, where α is the fine-structure constant. A priori there is no reason that weak force interactions are connected to cosmological quantities. However, this value is remarkably close to the value given in Equation 2.22 for the relic density of WIMPs. This coincidence is called the WIMP miracle.
2.5
Density profiles
As discussed in Section 2.2.1 DM halos should scale with ρ ∝ 1/r2 at large radii in order to recover the approximately flat rotation curves. However the proportionality of ρ(r) at the center has not been determined yet. The density profiles can be roughly classified into cored models, preferred by observations [3], and cuspy models, preferred by simulations [29]. In this section some of the proposed density profiles will be discussed.
One generic dark matter density profile is the (α, β, γ) double power-law model:
ρDM(r) =
ρs
(r/rs)γ· [1 + (r/rs)α](β−γ)/α
, (2.23)
where ρs is the scale density and rs the scale radius. The values for α, β and γ determine the
transition, outer and inner slope respectively.
One parametrization that is often used is the Navarro–Frenk–White (NFW) profile [30] and is given by: ρN F WDM (r) = ρsr 3 s r(rs+ r)2 . (2.24)
This is a specific form of Equation 2.23 having (α, β, γ) = (1, 3, 1). Other profiles of this family are the isothermal profile having (α, β, γ) = (2, 2, 0) [31] and the Moore profile with (α, β, γ) = (3/2, 3, 3/2) [32] ρIsothermalDM (r) = ρs 1 + (r/rs)2 , (2.25) ρM ooreDM (r) = ρs (r/rs)3/2(1 + (r/rs)3/2) . (2.26)
An modified version of the isothermal profile, fitted to observations, is the Burkert profile given by [33]
ρBurkertDM (r) = ρsr
3
(r + rs)(r2+ r2s)
A different parametrization is for example the Einasto profile which reads [34]: ρEinastoDM (r) = ρ−2exp −2 α r r−2 α − 1 . (2.28)
In this expression ρ−2and r−2 are the density and radius for which the logarithmic slope equals
−2 i.e. at which ρ(r) ∝ r−2.
To compare these different profiles by eye and visualize the classification of cuspy versus cored models, they are shown in Figure 2.4, having all constants set to unity. The horizontal axis represents the radius R and the vertical axis the density ρ.
Figure 2.4: Five dark matter density profiles: NFW, isothermal, Moore, Burkert and Einasto. Units are arbitrary
In this chapter the Standard Model of Cosmology is introduced and evidence for Dark Matter is discussed together with several candidates. In the next chapter will be discussed how DM can be detected and specifically what the current results are of the astrophysical objects of interest in this research: dwarf spheroidal galaxies.
Indirect dark matter detection
3.1
Detection methods
The WIMP scenario reasons that dark matter can interact with SM particles through collisions such as χχ ↔ f ¯f , W+W−, ZZ. Here χ represents the DM particle and since it needs to be electrically neutral it is equal to its antiparticle. On the right-hand side are SM particles, a fermion f with its antifermion ¯f , the W-boson and the Z-boson.
This DM-SM particle interaction allows for three different options to study DM: indirect detec-tion, direct detection and collider searches. These three methods are illustrated in Figure 3.1.
Figure 3.1: Three detection methods for dark matter: indirect detection, direct detection and collider searches
On the left-hand side in Figure 3.1 there are two DM particles shown and on the right-hand side two SM particles, the nature of the interaction between the two is unknown. For now will be assumed that interactions are possible without being able to explain how and why.
As Figure 3.1 is a Feynman diagram one can choose how time flows. First, let time flow from left to right. One starts at the left-hand side with two DM particles these annihilate or decay into SM particles. The study of the final stable products is called indirect detection. Second, let time flow in the upward, vertical direction. In this case the aim is to analyze scatterings of DM particles off of SM particles in a detector. This type of search is referred to as direct detection. Lastly, let time flow from right to left. Starting with high energy SM particles, the objective is to search for the DM production process by letting the SM particles collide. This type of dark matter detection is called collider searches.
This thesis focuses on indirect detection for which there are several messengers and wavelengths such as gamma-rays, charged cosmic rays, neutrinos and X-rays [10]. More specifically, this thesis focuses on indirect detection from gamma-rays and therefore this will be discussed more thoroughly in the next section.
3.2
Dark matter from gamma-rays
Among the DM annihilation messengers, gamma-rays are interesting since they travel in straight lines and are not absorbed in the local Universe [10]. The detected gamma-rays per square centimeter per second is the detected flux φs. The expected signal from dark matter annihilation
for a solid angle ∆Ω is given by:
φs(∆Ω) = 1 4π hσνi 2m2DM Z Emax Emin dNγ dEγ dEγ Z ∆Ω Z l.o.s. ρ2DM(r)dldΩ0 , (3.1)
where hσνi is the thermally-averaged annihilation cross section, mDM the particle mass and
dNγ/dEγ the number of photons per energy interval produced in the annihilation process and
ρDM(r) is the dark matter mass density distribution. The first part of the expression i.e. 1 4π hσνi 2m2 DM REmax Emin dNγ
dEγdEγ contains the spectral information and is dependent on particle physics
properties. Consequently this term is dubbed the particle physics factor. The second term, R
∆Ω
R
l.o.s.ρ 2
DM(r)dldΩ0, contains the angular information and is referred to as the astrophysical
factor or the J-factor.
The gamma-ray instruments can be classified in to two groups: the space-based telescopes (aboard satellites) and ground-based telescopes. The former (e.g. Fermi-LAT) are pair-conversion telescopes, the latter (e.g. H.E.S.S., MAGIC and VERITAS) are Cherenkov telescopes. There are several DM search targets such as the Galactic center, the Sun, galaxy clusters, Milky Way
halo and dwarf spheroidal galaxies. As this research aims to analyze dwarf spheroidal galaxies, these objects will be discussed in the next section.
3.3
Dwarf Spheroidal Galaxies
Dwarf spheroidal galaxies (dSphs) are highly DM dominated as suggested by kinematic data and have low astrophysical gamma ray backgrounds [35]. Moreover they are proximate objects with known locations [36]. Therefore dSphs are promising targets for indirect DM gamma-rays searches.
There has not been found any significant excess of gamma rays from dSphs [37]. These non-detections, together with an assumption for the DM profile, lead to constraints on the cross section hσνi which are on the order of the value for a thermal relic: 3 · 10−26cm3s−1 of
mass . 100 GeV [38], see Section 2.4.
The Fermi-LAT collaboration used six years of data and have recently published their upper limits on the cross section from a combined analysis of 15 dSphs [36]. In Figure 3.2 these results are shown in black for the b¯b and τ+τ− channels as a function of mass. For comparison other results are included in the figure as well. In solid gray the constraints found by the Fermi-LAT collaboration from an analysis of the Milky Way halo are shown. The results from H.E.S.S. obtained from 112 hours of observations of the Galactic Center are presented in dashed red. In solid orange the constraints derived from 157.9 hours of observations of the dwarf galaxy Segue 1 with MAGIC are plotted. The closed contours and the marker with error bars represent the best-fit cross section and mass from different analyses of the Galactic Center excess. The dashed gray curve shows the thermal relic cross section.
Figure 3.2: Constraints on the DM annihilation cross section as function of DM mass. The left panel shows the results for the b¯b channel, right for the τ+τ− channel. Taken from [36]
From Figure 3.2 can be seen that for the b¯b channel the best-fit contours are in a parameter space somewhat above the most recent limits derived by the Fermi-LAT collaboration. However, as argued in [36], due to uncertainties in the structure of the Galactic dark matter distribution these best-fit contours can significantly enlarge.
In this chapter both the detection method as the astrophysical objects of interest are discussed. In the next chapter will be examined how baryons can alter the picture.
Impact of baryons
4.1
Problems with the ΛCDM model
Whereas the ΛCDM is successful on cosmological scales, explaining the CMB and galaxy clus-tering, it has problems on galactic scales. These problems are the missing satellite problem, the too-big-to-fail problem and the cusp-core discrepancy.
Starting with the missing satellite problem. It was found that there is a discrepancy between the number of observed satellites and the number of predicted dark matter halos, the latter being larger than the former [39]. This mismatch is referred to as the missing satellite problem. As the Sloan Digital Sky Survey (SDSS) discovered new, very faint dwarf galaxies and nearly doubled the amount of known satellites, the problem was somewhat alleviated [40]. A possible solution is that the lowest DM halos scarcely have star formation, due to early reionization of the intergalactic medium [41].
The too-big-to-fail problem is connected to the missing satellite problem. In the high mass range, where halos are too massive to have suppressed star formation due to reionization pro-cesses i.e. halos that are too big to fail, there is a discrepancy between the predicted and observed kinematics of galaxies [42]. The expected velocities of stars in the satellites are higher than observed.
Finally there is the cusp-core discrepancy. As already discussed in Section 2.5, dark matter density profiles that are inferred from collisionless N-body simulations are cuspy, meaning that they are steep towards the halo center [29]. However, observational evidence suggests that the inner slope of the DM distribution is flat or cored [3].
4.2
A baryonic solution
Different studies have shown that baryons can affect the dark matter density profile [43], since it can either contract or expand a halo. The former is the result of gas cooling to the center of a halo which leads to a strengthened cuspy profile and therefore increases the mismatch between the theoretical models and observations. On the other hand, expanded halos can alleviate this mismatch. There are two mechanisms through which baryons can expand halos: outflow driven by stellar feedback and dynamical friction. The former is effective at expanding low mass halos and the latter is effective at expanding massive halos [44]. Since the focus of this thesis is on dwarf galaxies, stellar feedback is the mechanism of interest here.
Stars are formed in the center of the halo as a result of gas cooling. These stars cause repeated energetic outflows in the form of supernovae feedback. As has been shown in e.g. [43],[45] this feedback flattens the central dark matter density profile creating a core.
Therefore it can be concluded that a dark matter density profile that takes into account baryons is a well motivated choice. Such a profile has been proposed by Di Cintio et al. [46] (hereafter DC14) and will be discussed in Section 4.3.
4.3
Density profile with baryons
Di Cintio et al. [46] proposed a density profile that depends on the stellar-to-halo mass ratio M?/Mhalo, i.e. the star formation efficiency. They argue that this ratio controls the extent of
the baryonic impact.
Di Cintio et al. analyzed smoothed-particle hydrodynamics (SPH) simulated galaxies taken from the MaGICC project [47], the initial conditions are taken from MUGS [48]. In these sim-ulated galaxies stellar feedback by supernovae, stellar winds and energy from young, massive stars were implemented. The used sample comprised ten galaxies with five different initial con-ditions, covering the mass range 9 · 109M − 7 · 1011M .
The dark matter profiles of these SPH simulated galaxies were analyzed using a generic form of the Hernquist-Zhao profile
ρDM(r) =
ρs
(r/rs)γ· [1 + (r/rs)α](β−γ)/α
It was found that dark matter halos having a M?/Mhalo ratio below 0.01 per cent maintain the
cuspy NFW profile. The amount of stellar feedback is too small to alter the profile. As the ratio increases, the stellar feedback becomes strong enough to expand the halo and this results in the profile becoming progressively flatter. The most cored profiles are found at M?/Mhalo ≈ 0.4 per
cent. For halos having a larger ratio return again to the cuspy NFW profile. The explanation for this is that in the high mass range more mass collapses at the center which opposes the expansion process.
In Figure 4.1 is shown how the transition slope α (black), the outer slope β (red) and the inner slope γ (green) vary as function of the stellar-to-halo mass ratio. The symbols represent the different initial conditions and their sizes indicate the mass of the halo, a larger symbol implies a larger halo mass.
Figure 4.1: Transition slope α (black), outer slope β (red) and inner slope γ (green) of the DC14 density profile. Taken from [46]
The correlations between α, γ and M?/Mhalo were fitted with a four parameter, double power
law function. Whereas for β a parabola was used. The resulting expressions are given by
α = 2.94 − log10[(10X+2.33)−1.08+ (10X+2.33)2.29] ,
β = 4.23 + 1.34X + 0.26X2 ,
γ = −0.06 + log10[(10X+2.56)−0.68+ (10X+2.56)] ,
with X = log10(M?/Mhalo).
Therefore the DC14 profile is a specific form of Equation 4.1, where the slopes are given by 4.2. The scale density ρs can be obtained by normalizing the mass integral i.e. by stating that
ρs= Mhalo/ 4π Z Rvir 0 r2 (rr s) γ[1 + (r rs) α](β−γ)/αdr ! . (4.3)
The new quantity appearing in Equation 4.3 is the viral radius Rvir which is defined as the
radius at which the mass of a sphere containing ∆ times the critical density ρcrit at z = 0 is
equal to the mass of the halo, i.e.
Mhalo=
4 3πR
3
vir∆ρcrit. (4.4)
The remaining parameter in Equation 4.1 is the scale radius rs which should be left as a free
parameter.
4.4
DC14 for dwarf galaxies
The too-big-to-fail problem and the cusp-core problem can be cast as a single problem: the former being an effect of the latter i.e. the mismatch between the predicted and observed kine-matics is due to the existence of a cored density profile. Therefore in [41] the kinematical data of 40 Local Group (LG) galaxies have been used, both isolated and satellite, to examine whether the too-big-to-fail problem might be alleviated using the mass dependent DC14 profile. This was done by finding the halo mass that provided the best fit for each galaxy for both a cuspy NFW profile as the cored DC14 profile. The approach taken by the authors was to compare the effect of the two different density profiles on the LG members as a whole, this differs from the method that will be used in this thesis: a Jeans analysis of the velocity dispersions. The advantage of the method chosen by [41] is that the mass - anisotropy degeneracy is avoided. Jeans analysis will be discussed in Section 5.1.
Having halo masses the abundance matching technique can be applied. This method constraints the relation between the stellar mass and the halo mass of galaxies: the heaviest galaxies are matched with the heaviest halos and one continues doing this with the less massive galaxies and halos up to the point that all observed galaxies are assigned to a dark matter halo.
In [41] abundance matching is performed for the found halo masses. With this result they argue that the DC14 profile rather than the NFW profile alleviates the too-big-to-fail problem since
the dwarf galaxies are assigned to more massive halos having a cored density distribution.
In this chapter is discussed how baryons can have an impact on DM. This combined with all the previous chapters results in a well-motivated decision for examining the J-factors of dSphs for a density profile that takes baryons into account.
Analysis
5.1
Jeans Analysis
Closely following [49] for performing a Jeans analysis we assumed that dSphs are steady-state collisionless systems that can be described by their stellar phase-space density f (x, v, t) = d3xd3v. To the latter we applied the collisionless Boltzmann equation and thus obtained: ∂f∂t + v · ∇f − ∇Φ∂f∂v where Φ is a smooth gravitational potential. Then, assuming spherical symmetry and negligible rotational support, the second-order Jeans equation was found by integrating moments of the former expression:
d dr(ν ¯v 2 r) + ν r 2 ¯v2 r − ( ¯vθ2+ ¯vφ2) = −νdΦ dr , (5.1)
where ν is the stellar number density and ¯v2
xthe velocity dispersion in the x ∈ {r, θ, φ} direction.
This can be rewritten by introducing an expression for the velocity anisotropy βani ≡ 1 − ¯ v2 θ ¯ v2 φ . Implementing this and taking the derivative of the gravitational potential yields:
1 ν(r) d drν(r) ¯v 2 r(r) + 2 βani(r) ¯v2r(r) r = − GM (r) r2 . (5.2)
The enclosed mass at radius r, M (r), of the dSphs has contributions from both the stars and the dark matter. The former can be neglected since the objects are highly DM dominated. Therefore we have
M (r) = 4π Z r
0
ρDM(s)s2ds , (5.3)
here ρDM is the dark matter mass density profile.
The generic solution of the Jeans equation relates M (r) to ν(r) ¯v2
r(r) and is given by:
ν(r) ¯v2 r(r) = 1 f (r) Z ∞ r f (s)ν(s)GM (s) s2 ds , (5.4) where f (r) = fr1exp Z r r1 2 tβani(t)dt . (5.5)
The last expression contains a new variable r1 which after integration gives a normalization
factor that cancels out in the generic solution of the Jeans equation.
However the proper motions of the tracer stars cannot be resolved, but the quantities projected along the line-of-sight can. Therefore the Jeans solution, Equation 5.4, needs to be projected. In general, for a spherically symmetric system, the quantity f (r) can be projected into F (R) by applying an Abel transformation (and de-projected by an inverse Abel transformation):
F (R) = 2 Z ∞ R f (r)rdr √ r2− R2 , f (r) = Z ∞ r dF dR −dR π√R2− r2 . (5.6)
By applying this to the Jeans solution, we obtained the following expression:
σ2p(R) = 2 Σ(R) Z ∞ R 1 − βani R2 r2 ν(r) ¯v2 r(r)r √ r2− R2 dr , (5.7)
with R the projected radius, σ2
p the projected stellar velocity dispersion and Σ(R) the projected
light profile of the surface brightness. The latter is given by
Σ(R) = 2 Z ∞ R ν(r)rdr √ r2− R2 . (5.8)
The quantities σp(R) and Σ(R) can be measured directly. We used parametric models for the
dark matter density profile ρDM(r), the light profile ν(r) and the anisotropy profile βani(r) in
order to fit the velocity dispersion σp(R) to the data.
5.1.1 Used profiles
For the dark matter density profile we used the DC14 profile as introduced in Section 4.3. We used a Planck cosmology and therefore set ∆ = 104.2 [50] in Equation 4.4. Moreover we set the velocity anisotropy βani to zero. The impact of this choice will be discussed in Section
7.3.1. For the light profile we used the Hernquist-Zhao profile, given in Equation 2.23. The Hernquist-Zhao profile is analytical for the density profile ν(r). Since the observed quantity is the projected light profile Σ(R), the profile ν(r) has to be projected. We did this numerically
by using the Abel transform given in Equation 5.6.
From Equations 5.4, 5.5 and 5.7 can be seen that three successive integrations are required. However it is shown in [51] that for several velocity anisotropy profiles the expression can be rewritten as a single integration, which enormously speeds up the calculation. For the case of βani= 0 the expression becomes
σp2(R) = 2G Σ(R) Z ∞ R K r R ν(r)M (r)dr r , (5.9) with K(u) = r 1 − 1 u2 . (5.10)
5.2
MCMC
In order to explore the parameter space efficiently a Markov chain Monte Carlo (MCMC) tech-nique is used. Therefore this method will be discussed in this section.
A sequence of random elements X1, X2, . . . is a Markov chain provided that Xn+1 only depends
on Xn [52]. The state space of the chain is the set in which the Xi take values. Changes
xi → xi+1 are called transitions and the corresponding transition probabilities are given by the
transition matrix Wxy. Since the sum of probabilities of going from an state Xi to any other
state is equal to one, the columns of Wxy are normalized:
X
y
Wxy = 1 . (5.11)
Therefore the Markov process conserves the total probability.
For an MCMC simulation a suitable Markov chain is constructed such that its equilibrium distribution is the desired target distribution. When such a chain is constructed, one can start from an arbitrary point and iterate the chain as many times as wanted. Eventually the generated draws would appear as if they were drawn from the target distribution.
One way to construct a appropriate Markov chain is by using the Metropolis-Hastings algorithm which works as follows [53]:
1. Start with xi, choose randomly one out of a fixed number N changes ∆x, such that the
new state is x0 = xi+ ∆x
2. Calculate the ratio of probabilities P = px0/px i
3. Draw a random number R ∈ [0, 1]
4. If R < P accept the step, add the new values to the chain. Else reject the step, re-add the old values to the chain
For this analysis one has to choose prior ranges and a likelihood function. Starting with the former. In order to not favor the values at the high end, we chose the ranges in log space: log10(Mhalo/M ) ∈ [7, 12] and log10(rs/kpc) ∈ [−1, 2]. The range for the scale radius was based
on [54]. As the MCMC returns samples from the models posterior probability distribution function (PDF), the credibility intervals (CIs) were computed by drawing random samples from the distribution.
We used the likelihood function for a binned analysis from [55] which is given by
Lbin= Nbins Y i=1 (2π)−1/2 ∆σi(Ri) exp " −1 2 σobs(Ri) − σp(Ri) ∆σi(Ri) 2# , (5.12) where ∆2σi = ∆2σobs(Ri) + 1 2[σp(Ri+ ∆Ri) − σp(Ri− ∆Ri)] 2 . (5.13) In this expression ∆σp is the model value for the velocity dispersion calculated with Equation
5.7, ∆σi is the error on the observed velocity dispersion, Ri is the mean radius of the i -th bin
and ∆Ri the standard deviation of the radii distribution in this bin.
5.3
J-factor
As discussed in Section 3.2 the J-factor is the dark matter density profile squared integrated along the line-of-sight l and over a solid angle Ω:
J (Ω) = Z ∆Ω Z l.o.s. ρ2DM(r)dldΩ0 . (5.14)
In order to integrate ρ(r) along the line-of-sight it has to be slightly rewritten. Following [38], let z denote the shortest distance between the line-of-sight and the center of the dwarf galaxy and define the impact parameter b = Dsin(θ) where D is the distance between the observer and the center of the dwarf. This situation is displayed in Figure 5.1.
Figure 5.1: Sketch of the situation for calculating the J-factor. D is the distance between the observer and the dwarf galaxy, l is the line-of-sight and z the shortest distance between the
dwarf and the line-of-sight
For small angles the impact parameter can be approximated by b = Dθ. Then dl = dz and r = √b2+ z2 such that ρ(r) = ρ(√b2+ z2). The integration limits become z = −∞ and
z = +∞.
In the Fermi-LAT analysis [36] the J-factors are calculated using a solid angle of ∆Ω ∼ 2.4 · 10−4 sr i.e. an angular radius of 0.5◦. In this analysis we set the angle to 0.5 degrees in order to be able to compare the results to those of the Fermi-LAT collaboration.
5.4
Data
In this research we used the classical dwarf galaxies Fornax, Carina, Draco, Leo I, Leo II, Sculptor and Sextans. We took the stellar masses for all dSphs from [41]. These values combined with their Galactic longitude and latitude, l and b respectively, and their distance all taken from [36] are shown in Table 5.1.
Name galaxy l b Distance Mstar
(deg) (deg) (kpc) (106M ) Fornax 237.1 -65.7 147 24.5 Carina 260.1 -22.2 105 0.513 Draco 86.4 34.7 76 0.912 Leo I 226.0 49.1 254 4.90 Leo II 220.2 67.2 233 1.17 Sculptor 287.5 -83.2 86 3.89 Sextans 243.5 42.3 86 0.851
Table 5.1: Properties of the used dSphs
5.4.1 Stellar kinematic data
We used the stellar kinematic data as presented in [55] for all but one dwarf in the form of pro-jected positions and line-of-sight velocities for individual stars. For Leo II we used a different
set, as theirs is in preparation and has not been made public yet.
For each star the probability for its membership of the dSph Pi has to be taken into account.
For clarity the direct references will now be given as well as a brief explanation of how the membership had to be determined.
The data for Leo I did we obtain from [56]. A star is defined to be a member if the heliocentric velocity is in the range +250 to +320 km s−1.
The data for Carina, Fornax, Sculptor and Sextans originates from [57]. For all individual stars the probability Pi was provided. This value is estimated using an expectation-maximization
algorithm. Following [55] all stars having a probability lower than 0.95 were discarded.
The data set for Draco we used is presented in [58]. The members are the stars having a line-of-sight velocity vlos∼ −290km s−1 and a relatively low log10g and [F e/H].
We used the stellar kinematic data for Leo II from [59]. The membership is defined by having a velocity ∼ 76.0 km s−1.
All data sets originate from a similar period, i.e. ∼ 2008.
For each dwarf we binned the data into√N bins where N is defined as the sum of the member-ship probabilities i.e. N =P
i=1Pi. To estimate the velocity dispersion for each bin we used
the maximum-likelihood algorithm, this will be discussed in Section 5.4.2.
The number of stars we used for each dwarf galaxy is given in Table 5.2. Name galaxy Number of stars used Fornax 2273 Carina 746 Draco 453 Leo I 328 Leo II 172 Sculptor 1349 Sextans 395
Table 5.2: Number of used stars per dwarf galaxy
5.4.2 Velocity dispersion profiles
For all the dwarfs we had to calculate the velocity dispersion from their measured radial ve-locities. This we did for each bin by using the maximum-likelihood procedure described in [60] which will be explained in the next paragraph.
For each bin consisting of N member stars, we let vi be the observed radial velocity for the i -th
star. Assuming that these values follow a Gaussian distribution centered on the mean velocity hui, the probability function is given by
p = N Y i=1 1 q 2π(σi2+ σ2 p) exp " −1 2 (vi− hui)2 (σ2 i + σp2) # , (5.15)
where σiis the internal measure uncertainty and σp the intrinsic radial velocity dispersion which
is the quantity of interest. The values for hui and σp we numerically obtained by maximizing
the logarithm of Equation 5.15. We could determine the confidence intervals from the 2x2 covariance matrix A given by
A = a c c b
!
. (5.16)
The elements a and b are the variances of hui and σp respectively. The numerical expressions for
the diagonal elements can be calculated by using the inverse covariance matrix A−1 evaluated at the estimates of hui and σp denoted by hˆui and ˆσp
A−1= ∂2log(p) ∂hui2 ∂2log(p) ∂σp∂hui ∂2log(p) ∂hui∂σp ∂2log(p) ∂σ2 p (hˆui,ˆσp) . (5.17)
Having the variances a and b, the 68% confidence intervals are given by hˆui ±√a and ˆσp±
√ b.
5.4.3 Surface brightness data
The surface brightness data sets we used for all the dSphs are taken from [61]. Here the stellar surface density profiles are listed as the number of stars counted in elliptical annuli. Each ellipse has its semi-major and semi-minor axis, a and b respectively, equal to the estimated axis of the dSph as a whole. Since this research uses spherical symmetry for the Jeans analysis, the ellipti-cal annuli had to be converted into circular annuli. We did this by setting the radius R ≡√ab [55].
Results
The MCMC simulation returns posterior probability distributions for the two free parameters. These we plotted in a corner plot which also shows the covariance between the parameters. In Figure 6.1 the corner plot for each dwarf consists of three frames. The upper left frame shows the posterior probability distribution for the halo mass Mhalo in units solar mass. The lower
right frame presents this distribution for the scale radius rs in kpc. The third frame, the lower
left frame, shows the two dimensional posterior distribution. For both parameters the values are in Log10 and for both the solid blue line represents the median value.
(a) Fornax (b) Carina
(c) Draco (d) Leo I
(g) Sextans
Figure 6.1: One and two dimensional posterior distributions for the halo mass Mhaloand the scale radius rs. The solid blue line represents the median value.
In Figure 6.2 the best fit result and the 95% confidence intervals (CIs) to the velocity dispersion profiles σp for each dwarf are presented. The horizontal axis represents the radius in kpc, the
vertical axis the velocity dispersion in km/s. The CIs were chosen such that one can compare them by eye to the fit results of [55] which are shown in their Figure 1.
(c) Draco (d) Leo I
(e) Leo II (f) Sculptor
(g) Sextans
Figure 6.2: Velocity dispersion profiles for seven classical dwarfs. The black dots represent the data, the blue solid line the median and the dashed the 95% CIs
The J-factors we obtained by applying the DC14 prescription are shown in green in Figure 6.3. In order to compare them to those of the Fermi-LAT collaboration published in [36], these are plotted in blue. The J-factors can be expressed in particle physics units GeV2/cm5 (see e.g. [36], [38]) or in astrophysical units M2 /kpc5 (see e.g. [2], [54]). For this plot both units are
used, the particle physics units on the left side and the astrophysical units on the right side.
Figure 6.3: J-factors for seven dwarf galaxies. In green are the J-factors obtained from this research, in blue are those of the Fermi-LAT collaboration
The errors on the J-factors are derived by propagating the uncertainties on the free parameters Mhalo and rs, the uncertainties on the light profile are not taken into account. The effect of this
In Figure 6.4 the derived J-factors by using the DC14 profile are presented as a function of the log stellar-to-halo mass ratio Mstar/Mhalo. For each dwarf the corresponding J-factor from the
Fermi-LAT collaboration is plotted at the same mass ratio. Similar to Figure 6.3 both particle physics as astrophysics units are used.
Figure 6.4: Jfactors msmh
The dwarf galaxies on the left side are respectively Leo I, Leo II, Sculptor, Carina and Draco. The two dwarfs on the right side are respectively Sextans and Fornax.
Discussion and conclusion
7.1
Derived halo masses
The derived halo masses can be read off of Figure 6.1 and are on the order of 109M . In the
literature values are found, when assuming an NFW profile, on the order of 107 and 108M [62]
and [63]. Hence, the DC14 masses are larger.
As explained in Section 4.4 the DC14 profile was applied to several dwarf galaxies in the Local Group in [41]. In Figure 1 of this reference the fits to the circular velocities V (r1/2), r1/2 being the deprojected half-light radii, are presented. The circular velocity is related to the velocity dispersion σ used in this work through V (r1/2) =
√
3σ2. Moreover, the obtained halo masses
are listed in their Table 2.
Table 7.1 compares the masses for the DC14 profile we obtained in this research to those presented by Di Cintio et al. in [41].
Mhalo(108M )
Name of galaxy This research Di Cintio
Fornax 25 368 Carina 21 20 Draco 37 88 Leo I 577 289 Leo II 91 70 Sculptor 294 152 Sextans 2 7
Table 7.1: Halo masses for seven dwarf galaxies derived in this research and By Di Cintio [41]
As the authors used a different approach, their results serve as an independent test. From Table 7.1 can be seen that the obtained values for the masses are similar, except for Fornax where
there is a difference of factor 10. To see how this affects the density profile one has to turn to Figure 4.1 where the evolution of the inner slope γ of the DC14 profile is depicted. The curve for the inner slope follows a parabolic shape and due to symmetry the value for the inner slope is in both cases γ ∼ 0.6. Therefore can be concluded that in both studies similar behavior for the inner region of Fornax is found. This agrees with observations of Fornax for which a non cuspy density profile has been measured [64].
7.2
Derived J-factors
As can be seen from Figure 6.3 the derived J-factors are for most dwarfs similar to those of the Fermi-LAT collaboration. Except for Fornax and Sextans where the J-factor is lower when the DC14 density profile is used.
This might be explained by Figure 6.4 where the J-factors are shown as a function of their stellar-to-halo mass ratio. From this figure can be seen that for most of the dwarfs the ratio is in the range Log10 Mstar/Mhalo ∈ [−3.5, −4.0], which is where the inner slope γ of the DC14
profile returns to the cuspy value of γ = 1. The same value the inner slope of the NFW profile has. However for Fornax and Sextans the mass ratio is respectively −2.0 and −2.4, indicating a more cored profile.
In [41] it is argued that environmental effects might have occurred in Sextans which results in an incorrect halo mass and therefore the factor might not be correct. Other papers that derive J-factors for several dwarf galaxies (e.g. [38], [55], [54]) however do not mention anomaly behavior.
Most of the errors we obtained on the J-factors are on the same order as those of the Fermi-LAT collaboration except for Fornax and Carina. In the case of the former the error is very small. An explanation for this is that the simulation got stuck in a local maximum. We tried to solve this by repeating the MCMC analysis with different prior ranges, however these attempts were unsuccessful. For Carina the corner plot in Figure 6.1 showed a clear bimodal distribution which causes a somewhat larger error.
As we identified a potential problem with the J-factor of Fornax, further investigation is currently in progress to identify the origin of this discrepancy.
7.3
Impact of parametrizations
7.3.1 Impact of velocity anisotropy profile
The velocity anisotropy profile βaniis a key ingredient of the Jeans analysis. It is degenerate with
the mass profile and cannot be measured directly. In [2] the impact of different parametrizations is studied by comparing the results of a constant velocity anisotropy profile, the Osipkov-Merrit profile [65] and the Baes and van Hese profile [66]. It is concluded that the Baes and van Hese profile alleviates some of the biases in the Jeans analysis as it has the most free parameters. Therefore the authors recommend this profile.
As pointed out in [67] this method of examining the impact of the velocity anisotropy profile relies on marginalization over a parameter space and integration measure which are arbitrary choices. Consequently it is not transparent what the impact is of selecting a given parametric form, priors and integration measures. Therefore the authors argue a different method of assess-ing the impact. They suggest that the Jeans equation can be rewritten such that the dependence of the anisotropy profile in the dark matter mass profile becomes explicit. By doing this they show that for minimizing the J-factor, the results do not alter significantly by introducing a radial dependence in the anisotropy profile (e.g. Osipkov-Merrit profile and Baes and van Hese profile) compared to a constant profile as long as the projected stellar velocity dispersion σp(R)
is mildly varying in R.
The effect of setting the velocity anisotropy profile to zero is that the corresponding J-factor is the minimum value. By setting the velocity anisotropy to a non-zero constant, the J-value increases [67]. In this research we set the velocity anisotropy profile to zero and therefore the obtained J-factors are the minimum values.
7.3.2 Impact of the light profile
Another important ingredient of the Jeans analysis is the light profile which appears both in its projected form Σ(R) and deprojected form ν(r) in the projected solution of the Jeans equation, see Equation 5.7. In [2] the impact of the light profile has been examined by fitting simulated data with five different profiles: Plummer [68], exponential [69], S´ersic [70], King [71] and Hernquist-Zhao which is given in Equation 4.1. It is concluded that the Hernquist-Zhao parametrization provides the best possible fits to the data and reduces biases in the calculated J-factors, since this profile is the most flexible i.e. has the most degrees of freedom. Therefore can be concluded that the choice for using the Hernquist-Zhao profile in this research is well-motivated.
As mentioned in Chapter 6 the errors on the light profile have not been propagated to the J-factor. The effect of this is examined in [2] by using three sets of mock data that differ in sample size. It is found that for any sample size the obtained J-factor is only weakly affected by including the uncertainties on the light profile when using the Hernquist-Zhao profile.
7.4
Effect of binned analysis
In this research we binned the data as is explained in Section 5.4.1. The effect of binned analysis versus unbinned analysis is studied in [2]. The authors conclude that the results obtained by both methods are very similar. The unbinned analysis reduces the statistical uncertainties on the J-factors, namely on the ultrafaint dwarf galaxies. In this research we only used classical dSphs and therefore the derived values for the J-factors will not change significantly when using unbinned data.
7.5
Assessing the DC14 fit
In order to be able to asses the DC14 fit we calculated the reduced χ2 values for each dwarf. These values allows one the describe the goodness of fit by comparing observed values to values of the model in question. For comparing how well the DC14 profile fits the data relative to other profiles, we repeated the procedure of fitting the data but now using a NFW profile. The reduced χ2 values for both density profiles are presented in Table 7.2.
Name of galaxy DC14 NFW Fornax 0.9 1.2 Carina 2.4 2.4 Draco 1.0 1.2 Leo I 0.6 0.6 Leo II 1.3 1.3 Sculptor 1.2 1.2 Sextans 1.3 1.3
Table 7.2: Reduced χ2 for fits with DC14 and NFW profile
As can be seen from this table the reduced χ2 for each dwarf are very similar, indicating an
equally goodness of fit for the two profiles. Therefore this does not advocate the DC14 profile over the NFW profile.
7.6
Future work
At present there has not been detected an excess of gamma-rays from dwarf spheroidal galaxies resulting in upper limits on the cross section, see Section 3.3. Now having the J-factors for a profile that takes into account baryons would allow one to put constraints on the dark matter annihilation cross section that encapsulates this. This is a suggestion for future work.
Another suggestion for further research would be to examine the effect of introducing a non-zero constant velocity profile and see how the J-factors change. Moreover, as this parametrization is used in other papers e.g. [38] and [54], the results can be compared more straightforwardly to the values they find.
For improving the derived results it would be interesting to see what the J-factors are when an unbinned analysis is performed and the uncertainties of the light profile are propagated to the J-factor.
7.7
Conclusion
In this research the J-factors for seven classical dwarf galaxies have been calculated by fitting their velocity dispersion with the DC14 density profile, the velocity anisotropy being set to zero and a power law for the light profile. The J-factors we obtained are in general somewhat lower than those derived by the Fermi-LAT collaboration and much lower for the two dSphs with the largest ratio between stellar and halo mass. Putting constraints on the dark matter annihilation cross section with these J-factors is for future work.
Acknowledgments
I would like to express my gratitude to my supervisors Gianfranco, Christoph and Jennifer for their guidance, useful conversations and patience. Moreover I would like to thank Shin’ichiro Ando for being my second examiner. For their useful conversations and suggestions I would like to thank Djoeke and Arianna di Cintio. Besides I thank Patrick Decowski for helping me throughout my master with sorting out all UvA rules and regulations. Finally, I am thankful to my friends, my mother and my sister for their unfailing support.
[1] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ash-down, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al. Planck 2013 results. XVI. Cosmological parameters. Astronomy and Astrophysics, 571:A16, 2014. [2] V. Bonnivard, C. Combet, D. Maurin, and M. G. Walker. Spherical Jeans analysis for dark matter indirect detection in dwarf spheroidal galaxies - impact of physical parameters and triaxiality. Monthly Notices of the Royal Astronomical Society, 446:3002–3021, 2015. [3] S.-H. Oh, W. J. G. de Blok, E. Brinks, F. Walter, and R. C. Kennicutt, Jr. Dark and
Luminous Matter in THINGS Dwarf Galaxies. The Astronomical Journal, 141, 2011. [4] L. Bergstr¨om and A. Goobar. Cosmology and Particle Astrophysics. Praxis Publishing,
2004.
[5] J. C. Kapteyn. First attempt at a theory of the arrangement and motion of the sidereal system. Astrophysical Journal, 55, 1922.
[6] J. H. Oort. The force exerted by the stellar system in the direction perpendicular to the galactic plane and some related problems. Bulletin of the Astronomical Institutes of the Netherlands, 6, 1932.
[7] F. Zwicky. Die rotverschiebung von extragalaktischen nebeln. Helvetica Physica Acta, 6:110–127, 1933.
[8] K. G. Begeman, A. H. Broeils, and R. H. Sanders. Extended rotation curves of spiral galaxies - Dark haloes and modified dynamics. Monthly Notices of the Royal Astronomical Society, 249:523–537, 1991.
[9] Y. Mellier. Probing the Universe with Weak Lensing. Annual Review of Astronomy and Astrophysics, 37:127–189, 1999.
[10] G. Bertone. Particle Dark Matter : Observations, Models and Searches. Cambridge Uni-versity Press, 2010.
