Research Collection
Doctoral Thesis
Dark Matter Distribution in Dwarf Spheroidals
Author(s): Steger, Pascal S.P.
Publication Date: 2015
Permanent Link: https://doi.org/10.3929/ethz-a-010476521
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ETH Library DARK MATTER Distribution in Dwarf Spheroidals
PhD Thesis Pascal Stephan Philipp Steger March 2015 DISS. ETH NO. 22628
Dark Matter Distribution in Dwarf Spheroidals
A thesis submitted to attain the degree of
DOCTOR OF SCIENCES of ETH ZURICH
(Dr. sc. ETH Zurich)
presented by
Pascal Stephan Philipp Steger
MSc Physics ETH
born on 04. 08. 1986
citizen of Emmen / Ettiswil LU, Switzerland
accepted on the recommendation of
Prof. Dr. Simon Lilly Prof. Dr. Justin I. Read Prof. Dr. Jorge Pe˜narrubia
2015 Acknowledgements
I warmly thank Justin Read for the pleasant environment. You guided me around numer- ous obstacles, and showed me what science really is all about. You helped me with inputs for simplifications and generalisations, whenever they were sorely needed. And some cute integral transformations. You helped me personally by showing me what counts in life, and how adventurous ones dreams might be. I want to thank Simon Lilly for the hassle- free administrative takeover. Thank you for your patience and for enabling me to show my work at all the conferences, too. Frank Schweitzer, thank you for your succinct and energetic clarifications on the scientific method. It has been proven to be an invaluable guide to perform research.
Thank you, Hamish Silverwood, for the effort to take over the disc geometry part of my mass-modelling code. Silvia Sivertsson is thanked for asking questions that led to the extinction of some bugs. And back to Hamish again: Thank you for your insights into New Zealand’s culture. It’s valued. I thank Matthew Walker for all the exquisite data and mock galaxies he provided for free. Thank you as well for the warm welcomes at all the conferences we met.
Alexander Hobbs broadened my horizon on numerical astrophysics. Thank you for all the important lessons on CBM and strong frames, too. That is knowledge that will not be forgotten. Ouch. The precursor non-equilibrium chemistry simulation was performed by Aaron Boley, who gave useful technical hints as well. Silvia Garbari provided a prototype algorithm for the correction of the prospective halo centres. Might be simple, but makes all the difference when searching for cusps.
A warm thank-you goes to Michael Mayer, who showed me all about teaching. Your policy of freedom for the exercise composition allowed me to express myself, but I never failed to know the boundaries, either. I had several discussions with Jorge Penarrubia on general topics of thermodynamics, dynamics, and physics itself. Thank you for the overview this gave me. Thank you as well for your down-to-earth hints for code opti- misation. Thank you, Vincent Henault, for your introduction to MultiNest with high number of parameters, and the interest for applying the method to globular clusters – it broadened my horizon. Mark Gieles contributed to the spherical Jeans modelling by critically assessing the method. Thank you for your friendly welcome at Surrey.
iii Thank you, Endre and Niculin. Your enthusiasm for big data and our project on modelling correlations has opened doors. Some special kudos go to Niculin for sharing with me the knowledge about Dvorak-for-Programmers, and the boost his GPG usage has indirectly given my work-flow. Thank you all from the Akademischer Mittelbau des Physikdepartements, who showed me the broader physics science world, PSI, CERN, ABB, IBM, and some of the world aside of physics with canoeing and soccer. You’re great sports!
Thank you, Annina. It has been a most gratifying PhD time thanks to your loving affection, all the happiness, and motivation. You are my muse!
Z¨urich, March 2015 Pascal S.P. Steger
iv Contents
Acknowledgements iv
Contents ix
Abstract xvii
Kurzfassung xix
1 Introduction1
1.1 Dark Matter...... 2
1.1.1 Evidence for Dark Matter...... 2
1.1.2 What is Dark Matter?...... 6
1.1.3 Direct and Indirect Detection Experiments...... 12
1.1.4 Cold Dark Matter vs Warm Dark Matter...... 14
1.1.5 Cusps and Cores...... 16
1.1.6 The Need for Small Scales...... 17
1.1.7 Predictions from Simulations...... 19
1.1.8 Dwarf Galaxies...... 21
1.2 Mass Modelling...... 23
v CONTENTS
1.2.1 Jeans Modelling...... 23
1.2.2 Other Mass Modelling Approaches...... 24
1.3 Aim of This Thesis...... 25
2 Mass Modelling Spherical Systems 27
2.1 Introduction...... 27
2.2 Method...... 30
2.2.1 Derivation of the Key Equations...... 30
2.2.2 The Mass Distribution...... 32
2.2.3 The Tracer Density Profile...... 33
2.2.4 The Velocity Anisotropy...... 34
2.2.5 Comparison with Data...... 35
2.2.6 Priors...... 36
2.2.7 Parameter Space Sampling...... 36
2.3 Mock Data...... 37
2.4 Results...... 39
2.4.1 Single Tracer Population...... 39
2.4.2 Two Tracer Populations...... 42
2.4.3 Triaxial Mock Data...... 43
2.5 Conclusions...... 45
2.6 Appendix...... 46
2.6.1 Convergence of the MultiNest Model Ensemble...... 46
2.6.2 Influence of Binning Choices...... 46
3 Mass Modelling of Fornax 49
3.1 Introduction...... 50
3.2 Method...... 51
vi CONTENTS
3.2.1 The GravImage Code...... 51
3.2.2 Priors in Use...... 53
3.3 Data...... 54
3.3.1 Photometry...... 55
3.3.2 Splitting Populations...... 55
3.4 Results...... 56
3.4.1 Single Component Models without Anisotropy Priors...... 56
3.4.2 Single Component Models with Central Isotropy Prior...... 58
3.4.3 Two Populations split by Mg without Anisotropy Priors...... 60
3.5 Conclusions...... 60
4 Mass Modelling of other Dwarf Spheroidals 61
4.1 Introduction...... 61
4.2 Baryonic Density Profiles...... 63
4.3 Data...... 63
4.4 Results...... 63
4.5 Conclusions...... 65
4.6 Further Work...... 66
4.7 Appendix...... 67
4.7.1 Priors in Use...... 67
5 Simulations of Dwarf Galaxies 69
5.1 Simulation Methods...... 69
5.1.1 Non-Equilibrium Chemistry...... 70
5.1.2 Star Formation and Feedback...... 74
5.1.3 Halo Finding...... 75
5.1.4 Bound Structures...... 76
vii CONTENTS
5.1.5 Radial Profile...... 77
5.1.6 Relaxation Radius...... 77
5.2 Simulation Suite...... 78
5.2.1 Initial Conditions...... 79
5.2.2 Temporal Coverage...... 85
5.3 Results...... 87
5.3.1 Halo Finding...... 89
5.3.2 Dark Matter Density Profile...... 95
5.3.3 Stars...... 99
5.3.4 Forming Globular Clusters and Dwarfs at High Redshift...... 103
5.4 Conclusions...... 103
6 Conclusions and Future Prospects 105
6.1 Conclusions...... 105
6.1.1 Addressing the Questions posed at the Start of the Thesis...... 105
6.1.2 Future Prospects...... 106
Appendix 109
A.1 Plummer Profile...... 109
A.2 NFW Profile...... 110
A.3 King Profile...... 110
A.4 Sersic Profiles...... 110
A.5 Prigniel-Simien Profile...... 111
A.6 Double-power law Profile...... 111
A.7 Einasto Profile...... 111
A.1 Local Dark Matter Density...... 115
A.2 Geometry...... 116
viii CONTENTS
A.3 Representation...... 117
ix CONTENTS
x List of Figures
1.1 NGC 3198 Rotation Curve...... 4
1.2 Milky Way Rotation Curve...... 5
1.3 WDM Simulations...... 15
1.4 Cosmic Web...... 17
1.5 Power Spectrum Fit...... 18
1.6 Cooling Functions...... 19
1.7 Fornax Dwarf Spheroidal...... 21
2.1 Fitting of Analytic Anisotropy...... 35
2.2 GravImage Data Fit...... 39
2.3 Cusped profiles...... 40
2.4 Cored Profiles...... 41
2.5 Cored Profiles...... 43
2.6 Cored Triaxial Model Profiles with Central Isotropy...... 44
2.7 Convergence with Split Populations...... 46
2.8 3 Binning Choices...... 47
xi LIST OF FIGURES
3.1 Fornax Metallicity Split...... 56
3.2 Fornax Profiles 1pop without Isotropy Prior...... 57
3.3 Prior Combinations on Fornax...... 59
4.1 Sculptor 1pop with Monotonicity Prior...... 64
4.2 Draco with Monotonicity Prior...... 64
4.3 Sextans with Monotonicity Prior...... 65
4.4 Carina with Monotonicity Prior...... 65
4.5 Draco Surface Gravity...... 66
5.1 Cooling Rate...... 71
5.2 Chemical Abundances over Recombination...... 74
5.3 Power Spectrum...... 81
5.4 Initial Conditions...... 84
5.5 Refinement Mask...... 86
5.6 Dark matter distribution at z = 10...... 87
5.7 Zoomed dark matter distribution at z = 12...... 88
5.8 Visualisation of pSIM Run...... 89
5.9 SOD vs AHF Positions...... 90
5.10 Density Projections for Halo pSIM 1...... 91
5.11 Density Projections for Halo Hydro 5...... 92
5.12 Particle Positions Close-Up...... 93
5.13 Halo pSIM 1: ρ(r), d ln ρ(r)/d ln(r)...... 95
5.14 Halo pSIMDMonly 1, pSIM 1: ρ, d ln ρ/d ln r ...... 97
5.15 Halo Hydro 1: ρDM ...... 98
5.16 Density Profiles in pSIM 1 and pSIM 5...... 100
5.17 Projection of Stars in pSIM 1...... 101
xii LIST OF FIGURES
5.18 Stellar Density Profile in Hydro 1...... 102
5.19 Stellar Mass vs Half-Light Radius...... 104
A.1 Einasto profiles...... 112
xiii LIST OF FIGURES
xiv List of Tables
1.1 Cosmological Parameters for Simulation...... 6
2.1 Gaia Mock Data...... 38
3.1 GravImage Priors on Fornax...... 54
4.1 Priors for Other Dwarf Spheroidals...... 67
5.1 NEC Reaction Rates Network...... 73
5.2 Simulation Parameters...... 79
5.3 pSIM Major Halo Properties...... 94
5.4 Hydro Major Halo Properties...... 94
xv LIST OF TABLES
xvi Abstract
Our Universe is filled with a mysterious component called dark matter that appears to act gravitationally only. It remains a key challenge of modern physics to understand its nature. One approach to probing the nature of dark matter is to predict its distribution for an assumed dark matter model and compare it to the observed Universe. This allows us to constrain or even rule out models.
Most information about dark matter is available on the smallest scales, in the number and internal dark matter distribution of tiny dwarf galaxies. This has led to the birth of a new field – Near-Field Cosmology – that uses these nearby galaxies to constrain galaxy formation and cosmology. Two early successes of Near-Field Cosmology uncovered tensions with the standard ΛCDM cosmological model that could point to new physics. Simulations modelling only the dark matter fluid predicted many more satellite galaxies for the local group than have been found to date: the missing satellites problem. These same simulations predicted internal dark matter density profiles that were cuspy, in stark contrast to observed constant density cores: the cusp-core problem. However, both prob- lems could be solved by baryonic physics – star formation and feedback during galaxy formation.
The desire to evade the complication of difficult-to-model baryonic physics has driven astronomers to measure the dark matter distribution in ever smaller dwarfs. There it is hoped that so few stars form, with so few supernovae explosions that there is no longer enough energy to transform an initial dark matter cusp to a core. However, there is a problem. These tiny galaxies lie close to the Milky Way and are devoid of gas: the dwarf spheroidals. Having only stars, when viewed in projection they suffer a strong mass anisotropy degeneracy. This has led to conflicting results in the literature with claims of cusps, cores and everything in between.
xvii ABSTRACT
In the first part of this thesis, I develop a non-parametric Jeans framework GravImage to model the dark matter profiles of spherical systems based on kinematic data alone. I determine the data quality required to distinguish cusps from cores using a suite of mock galaxies. I then apply this method to the Fornax dwarf spheroidal – the most luminous Milky Way dwarf – to show that it has a dark matter core, consistent with earlier timing arguments. Application of the method to four dwarf spheroidals with fewer stars – Sextans, Sculptor, Carina and Draco – is ongoing, with the first results showing a core for Sextans, and cusps for Sculptor, Carina, and Draco. If these cusps hold up to further tests, they will be strong evidence in favour of the prevailing cold dark matter paradigm.
In the second part of this thesis, I attempt to calculate the distribution of dark matter in the smallest galaxies, accounting for baryonic effects. To achieve this, I run a suite of cosmological simulations of the formation of dwarf galaxies at high redshift. Focusing on high redshifts allows me to reach unprecedented resolution, resolving individual super- novae. Such high resolution requires non-equilibrium chemistry to model cooling at low temperatures. In a first phase, I study a lower resolution pilot simulation ( 1000 M ) ∼ per dark matter particle). I show that when including baryons, the central dark matter density becomes slightly steeper at radii where the simulation is well resolved. A small core ( 10 pc) forms in the innermost regions, but the numerical resolution is too poor ∼ for this to be trustworthy. At redshift z = 10, massive star clusters form alongside lower stellar density dwarf galaxies. The relationship between half light radius and stellar mass for these shows a striking resemblance to the dichotomy in half light radius versus stel- lar mass observed between globular clusters and dwarfs in the Local Group today. The massive star clusters are devoid of dark matter, suggesting that globular clusters are not good sites to search for dark matter annihilation or decay signals. In a second phase, I consider much higher resolution simulations ( 100 M per dark matter particle). This ∼ work is on-going and so far only the runs without non-equilibrium chemistry have com- pleted. Nonetheless the results are already interesting. I find that, similarly to the lower resolution pilot simulation, the dark matter density in the hydrodynamic run is steeper than in the pure dark matter simulation. This suggests that dwarf galaxies that survive to the present day as untouched fossils from reionisation may be expected to retain a pristine dark matter cusp – the smoking gun for non-relativistic cold dark matter. This is particularly exciting given the results from the first half of my thesis that favour such cusps in nearby dwarfs that contain only old stellar populations.
xviii Kurzfassung
Unser Universum ist gef¨ulltmit einer mysteri¨osenMaterie, der Dunklen Materie. Sie scheint nur gravitationell zu wirken. Es ist eine der dringendsten Herausforderungen unserer Zeit, deren Natur zu verstehen. Eine Herangehensweise ist es, die Verteilung der Dunklen Materie f¨urein bestimmtes Modell vorherzusagen und sie mit dem beobachtbaren Universum zu vergleichen. Dies erlaubt, die Modellparameter einzugrenzen, oder das Modell komplett zu falsifizieren.
Der Grossteil der uns zug¨anglichen Informationen befindet sich in den kleinsten kosmol- ogischen Skalen, der Anzahl von Zwerggalaxien und der Verteilung der Dunklen Materie in ihnen. Das f¨uhrtezur Entstehung eines neuen Feldes, der Nahfeld-Kosmologie. Die Nahfeld-Kosmologie nutzt diese Galaxien, um Modelle f¨urdie Galaxienentstehung und Kosmologie einzugrenzen. Zwei fr¨uheErfolge der Nahfeld-Kosmologie deckten Unstim- migkeiten des aktuellen ΛCDM Modells auf, welche vielleicht neue Physik zur Kl¨arung ben¨otigen. W¨ahrendSimulationen der Dunklen Materie viele Satellitengalaxien in der Lokalen Gruppe vorhersagen, finden sich nur wenige – das missing satellite Problem. Eine andere Diskrepanz besteht darin, dass Zwerggalaxien einen konstanten Anstieg der Dichte bis zum Zentrum verzeichnen. Dies steht im direkten Gegensatz zu Beobachtungen, wo konstante Dichten im Zentrum gemessen werden – das cusp-core Problem. Beide Prob- leme k¨onnten gel¨ostwerden wenn baryonische Effekte beachtet werden – Sterngeburt und Feedback in den ersten Phasen der Galaxienentstehung.
Der Wunsch, die komplizierten baryonischen Prozesse zu modellieren hat dazu gef¨uhrt, dass die Verteilung Dunkler Materie auf immer kleineren Zwerggalaxien gemessen wird. Man erhofft sich, dass dort so wenige Sterne entstehen, und so wenige Supernovae ex- plodieren, dass nicht mehr gen¨ugendEnergie f¨ureine cusp-core Transformation zur Verf¨ugung steht. Jedoch gibt es ein Problem: Diese sph¨arischen Zwerggalaxien liegen nahe der Milchstrasse und besitzen nur sehr wenig Gas. Die Analyse der auf die Himmelskugel pro- jezierten Sternbewegungen f¨uhrenzur Masse-Anisotropie Entartung, und widerspr¨uchliche
xix KURZFASSUNG
Resultate zum zentralen Dichteabfall von cusps, cores, und vielen Werten dazwischen wur- den publiziert.
Im ersten Teil dieser Doktorarbeit entwickle ich GravImage, eine nicht-parametrische Methode zur Massebestimmung in sph¨arischen Systemen, beruhend auf den Jeans-Gleichungen und rein kinematischen Gr¨ossen.Ich bestimme mithilfe von simulierten Zwerggalaxien die ben¨otigtenAnforderungen an die Datenqualit¨at,um cusps von cores zu unterscheiden. Ich wende die Methode anschliessend auf die Fornax Zwerggalaxie an – die leuchtkr¨aftigsteZw- erggalaxie der lokalen Gruppe – und zeige, dass sie einen core aus Dunkler Materie besitzt, in Ubereinstimmung¨ mit fr¨uherenArgumenten basierend auf Orbits von galaxie-internen Clustern. Eine Anwendung der Methode auf vier weitere Sph¨arische Zwerggalaxien zeigt einen core f¨urSextans; f¨urSculptor, Carina und Draco hingegen einen cusp.
Im zweiten Teil der Arbeit versuche ich die Verteilung der Dunklen Materie in den kleinsten Galaxien unter Ber¨ucksichtigung der baryonischen Effekte von Beginn weg zu berechnen. Dazu verwende ich eine Gruppe von kosmologischen Simulationen, welche die Bildung von Zwerggalaxien bei hohen Rotverschiebungen berechnen. Der Fokus auf hohe Rotverschiebungen erlaubt mir, bisher unerreichte Aufl¨osungenanzustreben. Bei Aufl¨osungen,in denen einzelne Supernova-Ereignisse sichtbar werden, ben¨otigtman ko- rrekte Modelle f¨urtiefe Temperaturen und damit eine Nicht-Gleichgewichts-Chemie der wichtigsten Elemente, welche Energie abgeben k¨onnen.
In einer ersten Phase untersuche ich eine Simulation geringer Aufl¨osung, 1000 M ∼ Dunkle Materieteilchen, und zeige Folgendes: Wenn die baryonischen Effekte ber¨ucksichtigt werden, wird die Dichte Dunkler Materie ein wenig steiler an gut aufgel¨ostenSkalen. Ein kleiner core von 10 pc Gr¨osse bildet sich in der innersten Region, innerhalb unserer ∼ Aufl¨osungsgrenzen. Bei Rotverschiebungen um z = 10 entstehen schwere Sternhaufen gleichzeitig wie Zwerggalaxien mit geringerer Sternendichte. Das Verh¨altnis zwischen Hal- blichtradius und Masse in Sternen zeigt eine verbl¨uffende Ahnlichkeit¨ zu den beiden Grup- pen der heute in der Lokalen Gruppe beobachteten Sternhaufen und Zwerggalaxien. Die simulierten Sternhaufen enthalten sehr wenig Dunkle Materie, weshalb sie keine guten Ob- servationsobjekte abgeben zur indirekten Entdeckung Dunkler Materie ¨uber Annihilation oder Zerfall.
In einer zweiten Phase untersuche ich eine Simulation viel h¨ohererAufl¨osung,mit 100 M pro Dunklem Materieteilchen. Diese laufenden Untersuchungen haben ergeben, ∼ dass Simulationen, welche die hydrodynamischen Gr¨ossenberechnen, aber ohne Nicht- gleichgewichts-Chemie einen steileren Abfall von Dunkler Materie bilden, als das eine Simulation, welche nur Dunkle Materie berechnet. Das l¨asstvermuten, dass Zwerggalax- ien, die von der Reionisation bis heute ohne Interaktionen mit anderen Galaxien ¨uberlebt haben noch einen urspr¨unglichen cusp besitzen – was ein schlagender Beweis daf¨urw¨are, dass Dunkle Materie nicht-relativistisch ist. Das ist besonders aufregend, da ich im ersten Teil dieser Arbeit genau solche cusps in nahegelegenen Zwerggalaxien ausfindig gemacht habe, welche nur alte Sternpopulationen enthalten.
xx 1 Introduction
“It cannot be seen, cannot be felt, Cannot be heard, cannot be smelt, It lies behind stars and under hills, And empty holes it fills” – J.R.R. Tolkien
In this chapter, I begin with a motivation for the existence of Dark Matter. Based on observations, I will list proofs of its existence from the largest structures and the dynamics of the Universe itself down to the smallest dwarf galaxies. The latest observational con- straints on dark matter suggest that dark matter is a new particle of nature, for which I present the leading candidates and strategies for detecting or creating them in the labora- tory. I show that the highest probability of an indirect detection lies in dwarf spheroidal galaxies, which forms part of the motivation for my thesis.
Further introductory remarks on structure formation and the evolution of galaxies set the scene for understanding the origin and diverse formation histories of dwarf galaxies. Predictions from ΛCDM simulations and observations of dwarf galaxies are pitched against each other, and the discrepancy between central dark matter slopes in both cases – the cusp-core problem – is introduced.
Even if no indirect detection is possible while observing dwarf spheroidals, the central dark matter density profile encodes information about the nature of dark matter. Warm or self-interacting dark matter models predict shallower central slopes than the cold dark matter typically assumed. This forms another part of the motivation for this work.
A closer look at existing mass modelling schemes with their advantages and drawbacks helps to understand the differences in profiles and especially the central slopes derived for the same systems, and sometimes even the same data. Going beyond some of these
1 CHAPTER 1. Introduction limitations is the main aim of this work.
The thesis is organised as follows. In chapter2, I develop a new non-parametric Jeans modeller for spherical systems, GravImage . Chapter3 presents an application to obser- vations of Fornax, and chapter4 to observations of four other dwarf spheroidal galaxies. Chapter5 turns to simulations of dwarf galaxies, with analysis of low resolution non- equilibrium chemistry simulations, and high resolution hydrodynamical ones with a focus on predicting the central density profile of the very smallest dwarf galaxies. I finish in chapter6 with conclusions and outlook to further work.
1.1 Dark Matter
1.1.1 Evidence for Dark Matter
Galaxy Clusters
The first evidence for dark matter came from Fritz Zwicky’s work in 1933 (Zwicky 1933). He was interested in weighing the Coma cluster of galaxies, finding a large discrepancy in the mass implied if the galaxies obeyed the virial theorem and the sum of all the luminous matter. This led him to introduce a hitherto unknown mass component dubbed dark matter. Modern studies continue to confirm Zwicky’s findings both in Coma (Kent & Gunn 1982; Merritt 1987; Lokas & Mamon 2003a) and all other galaxy clusters studied to date (Vikhlinin et al. 2006; Limousin et al. 2008; Newman 2013). At the time, Zwicky assumed that the dark matter would comprise faint stars or difficult-to-detect hot gas. However, no emission or absorption of light of any frequency by dark matter has been detected so far. I will discuss this further in 1.1.2 §
Rotation Curves
Although first identified in the 1930s, it wasn’t until missing mass was discovered in disc galaxies via rotation curve measurements that the search for dark matter was recognised as a key problem of modern physics (Volders 1959; Rubin 1983). If gas is moving on circular orbits in a spherical galaxy, the balance between centripetal and gravitational forces gives:
GM(< r) v(r)2 = (1.1) r for the circular speed v(r) of a star at radius r from the centre of the galaxy in a potential generated from a mass M(< r) inside that radius. G is Newton’s gravitational constant.
2 1.1. Dark Matter
Outside of the visible part of the galaxy the mass should asymptote to a constant value M(r ) const and v(r) r 1/2 should follow a Keplerian decline. Instead, most → ∞ → ∝ − galaxies show a rotation curve with v const out to the outermost stars or gas, see Figure ' 1.1. Rearranging equation 1.1 for the mass gives:
v2r M(r) = r (1.2) G ∝ Thus, a flat rotation curve implies linearly increasing mass. More gravitating mass at high radii where less luminous mass is seen makes it clear that dark matter is unrelated to the visible mass. The Milky Way rotation curve, Figure 1.2, which is consistent with Oort’s constants at the Sun’s position, is a prominent example. Oort’s constants:
1 V0 dv 1 1 A = = (14.82 0.84) km s− kpc− (1.3) 2 R0 − dr ± R0 !
1 V0 dv 1 1 B = + = ( 12.37 0.64) km s− kpc− (1.4) −2 R0 dr − ± R0 !
with distance R0 from the Sun to the galactic centre and rotational velocity V0 at the position of the Sun, and values by Feast & Whitelock(1997), describe the shearing and rotational motion of the Milky Way disc and are directly related to observable quantities of distances, galactic longitudes, and radial and tangential velocities of objects within the Milky Way. Determining A and B from these single objects and noting that the measurements approximately fulfill A = V /2R = B and dv/dr = 0, one finds that 0 0 − the rotational motion of the stars in the vicinity of the Sun all share the same rotational velocity, and cannot be described by Keplerian rotation.
Gravitational Lensing
Further evidence for dark matter comes from gravitational lensing, allowing us to infer the enclosed mass of a galaxy cluster in the case of strong deflections of background sources without information on the dynamical state of the system. This is of importance, as the assumption of dynamic equilibrium for the observed stellar systems might turn out to be wrong.
Chwolson(1924) proposed a that photons passing a massive object would be deflected. The approach was later formalized by Einstein(1936) , but at the time, Einstein could not conceive a way to observe a lensed star. Zwicky(1937) then calculated the probability to observe a lensing nebula – a lensing galaxy, by modern naming conventions. If the light of the background galaxy passes closely enough by the foreground galaxy, the image of the background source will be a ring around the front galaxy. The probability of such an alignment was considered good enough to be observable. Another effect of strong lensing
3 CHAPTER 1. Introduction
Fig. 1.1 —: Rotation curve of NGC 3198 (van Albada et al. 1985). Dark matter in the halo is needed to explain the high rotation speed at large radii. is that one light path may be longer than the other, and thus changes in the background source are observed later in the deflected image, than in the direct one. Refsdal(1964) calculated the time difference between direct and deflected optical path of a background supernova event, and found it to depend on the mass enclosed within the light path. The general interest in lensed systems was raised when Walsh et al.(1979) reported the first observation of a lensed system, Q0957+561. The lensing mass is consistently found to be higher than expected from the baryonic components: Limousin et al.(2008) e.g. find a minimum mass of 2.8 1014 M for the Coma cluster, which is at least 3 times higher · than the baryonic component – this fact was dubbed the baryonic catastrophe when first discovered, and lends further evidence for dark matter.
Large Scale Structure
Going beyond the galactic scale, the influence of dark matter is visible in the large scale structure of voids, sheets, filaments and clusters. This structure would not have formed by the current epoch without dark matter. This is because during the era of matter-radiation- equilibrium Silk damping from free photons washed out the small-scale fluctuations in baryons (Silk 1968). Dark matter is generally not affected by Silk damping on small scales, and thus the dark matter structures continued to grow during that epoch, dragging in the baryons in a later stage of the evolution of the Universe, and leaving them clustered
4 1.1. Dark Matter
Fig. 1.2 —: Rotation curve of the Milky Way (data compiled, Figure by Honma & Sofue (1997)). Dark matter in the halo is needed to explain the high rotation speed at large radii. as observed today.
Cosmic Microwave Background Radiation
The cosmic microwave background probes dark matter on the largest scales – the size of the Universe – and constrains parameters in the Friedman equation of the expansion of the Universe:
a˙ 2 8πG k Λ H2 = = ρ + , (1.5) a 3 − a2 3 3H2 ρ = , (1.6) crit 8πG Ω ρ/ρ , (1.7) ≡ crit where H = hH0 is Hubble’s parameter, defined as H =a/a ˙ where a = a(t) denotes 11 3 1 2 the expansion factor of the Universe; G = 6.67300 10 m kg− s is the gravitational · − − constant; ρ = ρ(t) gives the mean energy density of the Universe; and k 1, 0, 1 ∈ {− } denotes the curvature (open, flat, closed) of the Universe depending on whether ρ < ρ , ρ = ρ , ρ > ρ ; the cosmological constant Λ = 0 encodes the effect of dark energy. crit crit crit 6 The mean energy density is observed to lie very close to the critical value,ρ ¯ ρ , ' crit
5 CHAPTER 1. Introduction and thus we live in a flat Universe with Ω 1. The cosmic microwave background ' has temperature fluctuations of order one part in 105 (Penzias & Wilson 1965; Partridge 1995). These encode information about the growth of structure in the Universe when it was just 400, 000 years old. Fits to these fluctuations constrain parameters in the ∼ Friedmann equations. Of key importance is that the fluctuations have an amplitude that is difficult to understand without dark matter. If the Universe were filled only with normal or baryonic dark matter, then this matter fluid would couple strongly to the photon fluid in the early Universe. This leads to a damping of power of the photon free streaming scale – Silk Damping (Silk 1968) – and imprints acoustic oscillations. Such baryonic acoustic oscillations (BAO) have now been observed in the matter power spectrum (Eisenstein et al. 2005; Cole et al. 2005), but the oscillations are too small to be explained by a Universe devoid of dark matter (e.g. Dodelson 2011). Both the cosmic microwave background and the baryonic acoustic oscillations point to a significant dark matter component that does not interact with photons.
The currently favoured cosmological model is referred to as ΛCDM, and explains the accelerated expansion of the Universe (Riess et al. 1998; Perlmutter et al. 1999), the cosmic microwave background, baryonic acoustic oscillations (Slosar et al. 2013), gravitational lensing (Walsh et al. 1979; Clowe et al. 2006), and primordial element abundances assuming non-relativistic dark matter and a cosmological constant Λ (e.g. Peacock 1999; Spergel et al. 2007). Primordial nucleosynthesis requires that the baryonic density is a factor 6 smaller than the overall matter density (Bludman 1998, e.g.). Current values of the cosmological parameters are summarised in Table 1.1,(Planck Collaboration et al. 2014).
Parameter Symbol Value angular size of the sound horizon at recombination θ (1.04112 0.00029) 10 2 ∗ ± · − physical densities of baryons Ω h2 0.02230 0.00014 b ± physical densities of cold dark matter Ω h2 0.1188 0.0010 c ± scalar spectral index ns 0.9667 0.0040 ± 1 1 Hubble constant H (67.74 0.46) km s Mpc− 0 ± − matter density parameter Ω h2 0.14170 0.00097. m ± Tab. 1.1 —: Cosmological Parameters. Quoted values are TT, TE, EE+lowP+lensing+ext values from Planck, with 68% confidence limits.
1.1.2 What is Dark Matter?
Faint Stars or Gas
When first proposed, dark matter was believed to consist of gas or undetected, faint stars. In the Hubble Deep Field, only 3 red dwarfs were found, whereas 30 red dwarfs would ≤ be necessary in the angular patch spanned by the Hubble Deep Field if they were to make up the dark matter in the Milky Way (Flynn et al. 1996). Hot gas in giant elliptical galaxies emits X-rays via bremsstrahlung radiation from hydrogen plasma at 107 K gas, ∼
6 1.1. Dark Matter which is detectable from space-based telescopes. Integrating the flux, one gets constraints on the total gas mass of these objects, and the baryonic fraction. Burstein & Blumenthal (2002) find M M for ellipticals. But even if the luminous masses of these galaxies gas ≤ stars are doubled by assuming the maximum allowed gas content, they cannot make up all the mass required for a flat rotation curve at their edges.
On smaller scales, gas is detectable through its 21cm emission lines (if the gas consists of HI) or CO emission lines (for molecular gas). These components, too, give too little mass to make the predominant dark matter baryonic: Combes(1991) e.g. finds only 2 .5 1010 M 11 · for the total H2 gas mass in the Milky Way instead of the necessary 10 M in dark ∼ matter. Pfenniger et al.(1994) propose that the cold gas could reside in small dense clouds. If so, they would emit γ-rays when being hit by cosmic rays, and thus be detectable – and yet, they have not been found.
Another constraint can be made on cold gas: if it resides in the form of neutral hydro- gen, it absorbs light from distant quasars as it passes through. If there is some residual luminosity left, we can constrain the column density of cold gas in between the quasar and the Earth, and find Ω = ρ /ρ 0.03, an order of magnitude too low coldgas coldgas crit ≤ (Coc et al. 2002). The only way to evade detection by absorption lines would come from fractally-distributed cold gas, which could yield the required amount (Pfenniger 1994). An important problem with that setup, though, is that the cold gas clumps would be detected as MACHOs by microlensing (see next). We are forced to exclude gas as the sole component of dark matter.
MACHOs
Griest & Hu(1992) propose a model where dark matter comprises MAssive Compact Halo Objects (MACHOs). All astrophysical objects that do not emit detectable light and are not inside a planetary system count as MACHOs: white dwarfs, brown dwarfs, black holes, and disassociated planets. The tool of choice for detecting faint stars and MACHOs is microlensing (Paczynski 1986): The idea is to search for short increases in brightness of background stars originating from gravitational lensing by small, massive objects passing in the foreground. Milsztajn(2002) constrains the masses of MACHOs to lie below 1 M , giving a maximum contribution of MMACHO/M 0.4, with typical values ∗ ≤ of 0.4 M and MMACHO/M = 0.2, which is an order of magnitude below the required ∗ value. Observations by the OGLE, EROS, and MACHO program (Wyrzykowski et al. 2011; Tisserand et al. 2007; Graff & Freese 1996; Aubourg et al. 1993; Najita et al. 2000; Alcock et al. 1993) suggest that dark matter cannot be formed from MACHOs alone. So while these dark baryonic components may make up for a fraction of undetected mass, it is necessary to introduce something else for dark matter (Bi & Davidsen 1997; Choudhury et al. 2001; Richter et al. 2006).
7 CHAPTER 1. Introduction
MOND and TeVeS
Theories of modified gravity argue that the laws for gravitational forces have been mea- sured and tested on small scales in the solar system (Bertotti et al. 2003) and binary pulsars (Antoniadis et al. 2013). They could differ from that on larger scales, though. Modified Newtonian Dynamics (MOND, Milgrom 1983, Bekenstein & Milgrom 1984) ex- plains the effect of dark matter in low acceleration regions by a modification to Newton’s law:
a F~ = mµ ~a (1.8) N a 0 10 2 with F~N : Newtonian force, m : gravitating mass, ~a : acceleration, a0 = 10− ms− a new fundamental constant, and an interpolating function:
a 1 µ = (1.9) a 2 0 1 + (a0/a) constructed in such a manner that the rotationp curve of a galaxy follows:
2 vc = a0GM(< r) (1.10) in the case a << a0, where M(< r) denotesp the enclosed mass.
Tensor Vector Scalar gravity (TeVeS) puts that theory in a relativistic framework (Bekenstein 2004), which is needed for cosmological and lensing studies. The usual metric of general relativity gµν is modified according to:
2φ g˜ = e− g 2U U sinh(2φ) (1.11) µν µν − µ ν µ with φ a new scalar field and Uµ a new vector field, UµU = 1. The total modified Lagrangian is:
c4 c4 c4F = g˜R˜ + g˜[KA Aµν 2λ(U U µ + 1)] + 0 gF˜ (`2σ) (1.12) L 16πG − 32πG µν − µ 8πG`2 p p p with a Lagrange multiplier λ, A = U U , F and K constant, σ = (˜g µν ∇µ ν − ∇ν µ 0 µν − U µU ν) φ φ, scale length `, F some free function for interpolation. Bekenstein(2004) ∇µ ∇ν show that in the Newtonian limitg ˜ g , there holds φ, K, F 0. µν → µν 0 →
Swaters et al.(2010) measured a0 for low surface brightness galaxies, and found a 10 2 scatter around a0 = 10− ms− . Aguirre et al.(2001) find that temperature profiles of
8 1.1. Dark Matter galaxy clusters do not agree with MOND, which can be solved by including dark matter, in which case the idea of modified gravity to account is led ad absurdum. In addition, Ibata et al.(2011) show that for some globular cluster in the deep MOND regime, classical Newtonian gravity gives a much better fit for the kinematics. Zhao et al.(2006) test TeVeS by calculating the mass of lensed systems, and find that for a handful of lenses, the deduced stellar mass would be up to two orders of magnitude wrong under the assumption that µ is a constant and the same as in galactic rotation curves. Blanchet & Novak(2011) use the best fitting a0 value to calculate the expected precession of perihelion of Saturn in the Solar system stemming from external gravitational fields, and find that ephemerides of Saturn contradict the expected precession.
The bullet cluster (Clowe et al. 2006) shows a mismatch of the location of gravitating mass determined by (weak) lensing of background galaxies and emission of a considerable fraction of gas in an ongoing merger. It thus indicates that the effect of dark matter is generated by freely streaming particle masses rather than by any modification of gravity. This is corroborated by other colliding cluster systems like MACS J0025.4-1222 (Bradaˇc et al. 2008). The analysis is complicated by a complex geometry, though, and for special combinations of potentials and mass distributions, different positions on mass and light can be found (Angus et al. 2010). Additionally, as mentioned in the CMB section above, the absence of baryonic acoustic peaks in the mass power spectrum at the largest scales of the Universe excludes TeVeS as a source for all dark matter effects over a large part of the evolution (Dodelson 2011). In this work, I will thus concentrate on Einstein‘s theory of gravity which is Newtonian in the weak-field limit.
Particle Dark Matter
The key evidences for dark matter to be made up of particles are the missing acoustic peaks in the mass spectrum at large scales, and the locus of gas in interacting galaxy clusters as detailed in the last section. What particles could then make up dark matter? A particle candidate needs to fulfill the astrophysical constraints discussed ΩDM = 0.26, must have lower bounds on the lifetime as indicated in Ibarra et al. 2013, and ideally solve other problems in particle physics. The main candidates are WIMPs, sterile neutrinos, and axions:
1. WIMPs: Weakly Interacting Massive Particles (WIMP) are proposed to be the most likely candidate genre of particles to make up for dark matter. WIMPs only interact via gravity and the weak force or other forces on the scales of the weak interaction force by design, and thus will stop interactions in the first phases of expansion of the young Universe. If the annihilation cross-section is 1pb, the WIMP density ∼ is about the same as required by the cosmological model, a fact known as WIMP miracle (Peacock 1999; Griest & Kamionkowski 2000). The corresponding WIMP mass would be 100 GeV/c2 – and the WIMPs would have sub-relativistic speed ∼ at decoupling, and thus appear as Cold Dark Matter.
9 CHAPTER 1. Introduction
The electroweak symmetry is broken at that exact energy, and extra dimensions or supersymmetry predict there to be particles generated of that mass, which are massive, weakly interacting, and stable (Jungman et al. 1996). In the early stages of the Universe, dark matter annihilation processes from WIMPs could have violated CP-invariance and thus given rise to the lepton asymmetry observed today (Kumar & Stengel 2013). These are examples of models for dark matter that, in addition to give the right dark matter density, also solve problems of the standard model, and would thus make good candidates for dark matter. Jungman et al.(1996) summarise three main approaches for WIMP detection: cre- ating them in laboratories like the Large Hadron Collider in Cern, detecting their interactions with baryons in sensitive earth-bound detectors, and looking for in- direct signals in outer space. If WIMPs fulfill the Majorana criterion they will self-annihilate and produce lepton-antilepton or quark-antiquark pairs, which will annihilate and produce a detectable range of γ-rays. The different channels will pro- duce a power-law distribution in energies of the γ-rays, which sets the WIMP signal off from γ-rays of sterile neutrinos, see next section. The self-annihilation requires 2 two particles to interact, and thus the signal will be sensitive to ρDM – which in turn will be highest in the gravitational centre of galaxies, centre of the Sun, or even the Earth.
2. Sterile Neutrinos: If one wants to solve several problems of the standard model of elementary particles – especially, the neutrino oscillations, baryon asymmetry, and a missing candidate for dark matter – without introducing a new energy scale and many new particles above the electroweak energy range of 102 GeV to 1016 GeV, one could add a single new particle, the sterile neutrino. The resulting theory is named νMSM, the Neutrino Minimal Standard Model. The sterile neutrino is postulated to be right-handed, of considerable mass around keV, and to have only gravitational ≥ interactions with other particles owing to its lack of charge. It would well fit the role of a dark matter particle, with relativistic speeds at decoupling. See Asaka et al. 2006, Boyarsky et al. 2009 and references therein for further properties. The sterile neutrino is unstable and is expected to decay and produce keV-photons in a sharp line. This line would be visible at the centres of dense dark matter regions ( ρ) all over the Universe, at the same frequency. ∝ 3. Axions: The charge conjugation & parity symmetry (CP symmetry) states that physics should be unchanged if we swap a particle with its antiparticle and flip its parity. This symmetry is broken for the weak force, and although it could easily be broken for the strong force, too, we do not see the resulting strong electric dipole moment for the neutron. In order to solve this conundrum, Peccei & Quinn(1977) introduced a new particle, the axion. It would have low cross-sections for weak and strong forces, no electric charge, and low masses. If its decay constant is at 1012 GeV, it would constitute all dark matter in the Universe (Preskill et al. 1983). As this parameter is not fixed, one cannot constrain whether axions are relativistic or not at decoupling. Its supersymmetric particle, if existing, would constitute the lightest supersymmetric particle, and make a good candidate for dark matter. Duffy & van Bibber(2009) describe the Primakoff effect, an axion-photon mixing,
10 1.1. Dark Matter
which could be observed by applying strong magnetic fields in a microwave cavity. 2 If an axion of mass ma passes through the resonance hν = mac , it interacts with the magnetic field photons and decays, producing photons with a specific frequency. No axions were detected so far, but a large frequency range remains to be explored (Asztalos et al. 2010).
4. Further Particles: To finish the tour d’horizon of dark matter, I give examples of more exotic particle extensions to the standard model which have claimed dark matter properties.
Axinos are postulated elementary particles which are stable or almost stable for • a Hubble time. They are assumed to have very low interaction cross-sections (extremely weak interacting massive particles, EWIMPs). See Choi et al. 2013 for a recent review. Wino, Zino, Higgsino: A wino can be used to explain an excess of positrons • from cosmic rays. These are thought to originate in dark matter annihilations in local over-densities via WW , hh, ZZ, tt¯, b¯b, qq¯ channels (Hektor et al. 2014). Kozaczuk & Profumo(2013) explore the regions of parameter space in NMSSM for a light neutralino dark matter particle with large spin-independent nucleon scattering cross section, assuming a Higgs mass of 126 GeV consistent with LHC findings (Aad et al. 2012). Future measurements with CDMS (Cryogenic Dark Matter Search) or variants thereof could probe these regions. Fourth Generation Fermions. These are constructed to be fermions with no • interaction with the three fermion families of the Standard Model. A fourth neutrino fulfilling the Majorana criterion could build a sub-dominant part of all dark matter. Its mass would lie in the range 41 59 GeV (Bao et al. 2013). − Asymmetric Dark Matter: Dark matter is about five times more abundant • than baryonic matter. Baryonic matter was left over in the early universe due to a matter-antimatter asymmetry, and the close relationship between the two densities points to a common process in the generation, namely another asym- metry between particles and antiparticles. Dark matter mass and asymmetry parameter are proposed to fall in the same sector as the baryonic ones. Volkas (2013) for an overview of generation, general properties, and bounds. Self-Interacting Dark Matter. Dark matter with a finite cross section for inter- • actions with itself is known as self-interacting dark matter, with many candi- dates discussed (Carlson et al. 1992; de Laix et al. 1995; Firmani et al. 2000; Pospelov et al. 2008; Peter 2012). It has a finite free-streaming length, and thus any structure below that length is smoothed out (Spergel & Steinhardt 2000). With high enough cross sections, any cusp would be smoothed out before being formed. Hogan & Dalcanton(2000) find that these cores are unstable and decay via relaxation, and consequently SIDM cannot explain the observed cores. Ad- ditionally, two-body interactions grow in strength with the cross-section, and evaporation would destroy dark matter halos faster than they form.
11 CHAPTER 1. Introduction
1.1.3 Direct and Indirect Detection Experiments
So how to detect dark matter particles? If we manage to do this, we have done the first step to rule out some of the proposed extensions to the standard model. Three strategies exist to find dark matter: (i) create the dark matter particles in the lab, and look for missing momentum and energy in collisions (ii) detect photons from interactions between dark matter and a carefully shielded detector (direct detection) (iii) detect decay or annihilation products (indirect detection).
One of the big challenges for many direct detection experiments is the lack of on- off-source measurements: Dark matter is continually flowing through the detectors. If searching for dark matter signals with astronomical methods – indirect detection –, this is no problem anymore: One can train a telescope at a prospective source, or away from it, and thus deduce the background noise. Indirect detection methods assume a finite cross- section for self-interaction of dark matter particles with one another. Gunn et al.(1978) e.g. calculate the expected flux of γ-rays from a cosmological background annihilation of undetected leptons of mass 10 GeV at an order of magnitude of 6 10 11 photons/(cm2s ≥ · − sr).
Because indirect dark matter detection is mostly based on the assumption that dark matter annihilation or decay produces additional photons, the best places to look for these photons is at places of high dark matter density, and low baryonic mass density, giving a high signal-to-noise or signal-to-background ratio. By this requirement, galaxy clusters make poor detection sites, because two sources of gamma rays exist: both the hot coronal gas and accretion discs around black holes in the centre add to the overall flux. Prokhorov & Churazov(2013) find that at 4.3 σ, there is an enhancement of gamma-rays in direction to the cores of galaxy clusters, with only weak evidence for a spatial extent. They find that the dominant part stems from AGNs, thus the centre of galaxy clusters does not lend itself for the search of gamma-rays coming from dark matter annihilation.
For similar reasons, less distant loci with abundant predicted dark matter density like the Milky Way centre or the Sun, are difficult targets: Any background created by baryonic processes must be well understood before it can be subtracted from the measured signal to determine the signal coming from dark matter interactions. Lake(1990) proposes Milky Way dwarfs as a promising site to search for dark matter signals. The absence of astrophysical background sources for gamma rays produced via dark matter annihilation and/or decay makes them the systems of choice for mass modelling in this thesis.
Least contamination is expected from superdense dark matter clumps (SDMC), as de- fined by Berezinsky et al. 2013 to be dark matter objects which are gravitationally bound, in virial equilibrium, and formed in the radiation-dominated epoch. Any dark matter object forming around such an SDMC is dubbed an ultra-compact mini-halo (UCMH). See Berezinsky et al. 2013 for formation scenarios of these mini-halos.
Currently, many indirect detection experiments for dark matter are under way:
12 1.1. Dark Matter
A space-bound detection gamma-ray experiment is the Large Area Telescope aboard • the Fermi satellite. Forecasts predict a 2.7σ excess for detection of decaying neutral pions. See The Fermi-LAT Collaboration et al. 2013 for more detailed information on background rejection. Bergstr¨omet al.(2013) find better constraints on WIMPs with masses below 300 GeV. Astrophysical constraints on dark matter from Fermi-LAT is summarised in the • thesis of Oscar Ramirez1 , highlighting the influence of unresolved γ-ray sources in the Virgo cluster. Furthermore, a claimed detection of 2-5 GeV at the centre of the Milky Way is fit with different background models and an NFW dark matter profile (see later), concluding that it is hard to tell whether dark matter of 30 GeV or 103 unresolved γ-ray sources generate the signal. Especially the signal of self-annihilation of dark matter is searched for in underground • experiments, which decrease the background of cosmic rays. See e.g. Aartsen et al. 2013 for recent results on the IceCube experiment, and Fermani 2013 for results of the ANTARES experiment in the northern hemisphere). Esmaili & Dario Serpico(2013) find hints for Dark Matter to be a heavy particle • with a few PeV rest-energy and a lifetime of 1027 s, based on the observation of two PeV cascades above the atmospheric background. On the other hand, AMS-02 and GAPS are used to detect cosmic-rays, concentrating • on antideuterons with small kinetic energies stemming from dark matter annihila- tion. Fornengo et al.(2013). An overview of the respective sensitivities are given in Vittino et al. 2014. If no signal is detected, one formulates bounds instead. One recent finding by Moulin • et al. 2013 summarises the bounds on decaying dark matter found by gamma-ray emissions of the extragalactic background measured by FERMI, and the signal com- ing from the Fornax galaxy cluster, measured by HESS. Most dark matter particles would need to have a half-lifetime of at least 1026 s to be consistent with the mea- surements. Inoue et al.(2013) calculate the expected secondary gamma-ray measurements for • a cascade after decays of ultra-high-energy cosmic rays. A statistically significant detection is expected for the Cerenkov Telescope Array (CTA) even in the high- energy range of 1 TeV. Bulbul et al.(2014a); Boyarsky et al.(2014) report the detection of a weak emission • at (3.55 0.03) keV in stacked observations of galaxy clusters and propose it is due ± to decaying of a sterile neutrino, though several emission lines are in the same region (Jeltema & Profumo 2014; Bulbul et al. 2014b). Boyarsky et al.(2014) observe that for a preferred X-ray emission line of 3.5 keV, the dark matter particle masses are already quite high, and the corresponding free streaming length of a few kpc in the case of sterile neutrino dark matter is below the Plummer length at z = 10. This motivates to first consider cold dark matter models for any structure formation simulation. 1http://ir.canterbury.ac.nz/bitstream/10092/9857/1/thesis.pdf
13 CHAPTER 1. Introduction
And finally, a measurement of radio emissions from the Ursa Major II dwarf spheroidal • galaxy with the Green Bank Telescope made by Natarajan et al. 2013 excludes an- + nihilation of 10 GeV WIMPs into e e− at a 2σ level.
1.1.4 Cold Dark Matter vs Warm Dark Matter
Cold Dark Matter (CDM) is defined to have non-relativistic speeds at decoupling, and thus to contribute to the matter content of the universe rather than the radiation content. As soon as an overdensity of matter – and as they are proportional to each other to a good degree, an overdensity of dark matter – crosses the horizon, it decouples from the expansion around it and collapses. WIMPs and possibly axions are examples of cold dark matter.
Warm Dark Matter (WDM) on the other hand is postulated to have a finite veloc- ity at decoupling, and thus a finite free-streaming scale (Bode et al. 2001; Avila-Reese et al. 2001). The main proposed WDM candidates are sterile neutrinos and axions. The free-streaming smears out any small-scale structures like dwarf galaxy cusps (Hogan & Dalcanton 2000). Figure 1.3 shows the effects of increasingly warmer dark matter on the structure of the cosmic web.
When including also a model for gas cooling and star formation in a Warm Dark Matter cosmology, Herpich et al.(2014) show that warm dark matter
affects low-mass halos more than high-mass ones, leaving them less gas rich; • reduces star formation overall; • produces stellar density profiles with smaller concentration. •
These effects are strongest for the lightest particle mass considered, 1 keV, and do not alter the disc galaxy formation within the observational constraints of mWDM > 2 keV. See Destri et al. 2013 for primordial power spectra of WDM as function of different masses.
However, numerical issues with Warm Dark Matter simulations still remain. These are known to produce spurious, fake halos (Wang & White 2007). Hahn et al.(2013) propose a tetrahedral decomposition of the density field to circumvent that problem. Lovell et al. 2014 on the other hand identify spurious halos by comparing the position of protohalos in the initial conditions for runs with varying resolutions, and match the number of non- spurious subhalos produced in their simulations to the number of observed galaxies. Thus they restrict the warm dark matter to have a mass of around 1.5 keV.
Hogan & Dalcanton(2000) and more recently Lovell et al.(2014) show that sterile neutrinos or axions as warm dark matter particles would create a cutoff in the power
14 1.1. Dark Matter
Fig. 1.3 —: . Projected dark matter density for a suite of CDM (left column), WDM (350 eV, middle column), and (135 eV, right column) simulations at z = 3, 2, 1 top to bottom. Image taken from (Bode et al. 2001). CDM forms a plethora of small-scale cusps, which are smoothed out in WDM. spectrum at about the masses of dwarf galaxies, based on the effects of free streaming. The sizes of the so created cores depend on the free-streaming length of the particles, see Robinson & Tsai 2014. In general they are found to lie below the sizes of observationally determined dwarf satellites cores in the local group (Macci`oet al. 2012; Shao et al. 2013; Macci`oet al. 2013).
Mixtures of cold and warm dark matter are possible as well, and sterile neutrinos and axions of specific masses can act as such (Boyarsky et al. 2009). Mixed dark matter proposed before the advent of ΛCDM consisted of 80% cold dark matter and 20% hot dark matter, and was rejected because it could not explain the clustering of galaxies at large scales. Hot dark matter neutrinos exist alright, but can not make up the dominant fraction of the dark matter: The free-streaming at the earliest epoch of structure formation would have halted galaxy formation too long for today’s observed high-z galaxy clusters to have formed in time.
15 CHAPTER 1. Introduction
Latest Astrophysical Constraints
Anderhalden et al.(2013) use Milky Way satellites as probes to the smallest cosmological scales to constrain the warm dark matter particle mass to be above MWDM > 2 keV. That constraint is found by simultaneously fitting the abundance of satellites, their radial distribution, and their density profiles. The strongest constraints on WDM to date come from the Lyman-alpha forest. Neutral hydrogen on the line of sight towards a bright background quasar absorbs its light of the frequency of the Lyman-α line and transitions the electron from n = 1 to n = 2. As each gas cloud sits at a given redshift, the absorption line will be shifted by a characteristic amount, and the final collection of absorption lines – the Lyman-α forest – encodes information on the gas distribution at large scales towards the quasar. Seljak et al. 2006; Viel et al. 2008 use the Lyman-alpha forest to constrain the free-streaming length of WDM, fitting the flux power spectrum out to redshifts of z = 5.5. The latest constraints suggest m 3.3 keV (Viel et al. 2008). WDM ≥
1.1.5 Cusps and Cores
After recombination, collapse led to large structures decoupling from the general expansion of the Universe, and collapsing on their own. In the early stages, collapse progressed scale- invariant, along the shortest axis first, and created a whole range of sheets of enhanced density, which intersect with each other to form filaments. The filaments in turn meet at nodes and create halos. The halos collapse further and generate cuspy profiles (Dubinski & Carlberg 1991).
In the overdense regions, dark matter creates a deep enough potential well to capture gas and allow it to cool radiatively and form galaxies, visible today in the cosmic web structure (Fabian 1992; Bond et al. 1996; Hahn et al. 2007), see Figure 1.4. The collapse leaves the linear regime early on, and subsequently, numerical simulations are needed to model the evolution of overdensities.
We are left with a complex halo network with a mass spectrum of:
α 2 2α N(M)dM M − exp( (M/M ) )dM (1.13) ∝ − c where the number of halos N as function of mass M is given in terms of the slope of the power spectrum α > 0 right after recombination (Gott & Turner 1977).
Baryons are trapped in these dark matter potentials, and potentially form stars and galaxies.The distribution of galaxies follows the Press-Schechter function, Press & Schechter 1974; Bond et al. 1991, Gan et al. 2010 and references therein, and we are left with many more small galaxies than massive ones.
The first population of stars was made of primordial gas, mostly hydrogen and helium,
16 1.1. Dark Matter
Fig. 1.4 —: Cosmic Web from the Millennium Simulation with sheets, filaments, and clusters. Image taken from Springel et al. 2005. very little deuterium, and a trace lithium. They are called pop III for historical reasons. After a proposedly short life, these massive stars exploded as Supernovae, creating elements of higher mass numbers, and injecting energy via shockwaves and heat in the surrounding medium. The enhancement of the abundance of higher-order elements is called metal enrichment, and provides the basis for the formation of popII stars, which by definition have only few heavy elements. Further star formation will use the reprocessed gas, and produce pop I with more heavy elements.
A sequence of star formation and supernova explosions followed, driving out the gas in several strong shocks. Gas cooling, star formation and subsequent gas blow-out can lead to an irreversible heating of the dark matter transforming cusps into constant density cores (Navarro et al. 1996a; Read & Gilmore 2005; Pontzen & Governato 2012).
1.1.6 The Need for Small Scales
Figure 1.5 from Tegmark & Zaldarriaga 2002 shows that ΛCDM gives a good fit to the power spectrum from the largest scales to the Lyman-α forest at around k = 1 Mpc/h, on our scales of interest. Baryonic effects are important at the scales of Mpc and below (Tegmark et al. 2002), and the simulations need to take them into account. As they act on the smallest scales in isolated galaxies, we can concentrate on a small patch of the Universe, and stop at high redshift (Read et al. 2006; Boley et al. 2009; Revaz & Jablonka 2012; Sawala et al. 2010). This approach misses part of the large scale environment, and
17 CHAPTER 1. Introduction
Fig. 1.5 —: Power spectrum of ΛCDM fit to the SDSS data (Tegmark & Zaldarriaga 2002). interactions between massive systems are neglected, but gains on the resolution.
To resolve dwarf galaxies and globular clusters in simulations, one needs high resolution at their centre. For statistical purposes, one aims for 105 106 particles inside a virial − radius. The lowest scales needed to see single stars forming basically encompass the Jeans length of a protostellar gas core at temperatures of 300 K. Looking at such low mass scales, one should sample the initial mass function as well, and have a model for collisional kinematics in the centre of clusters. This would require a prohibitive amount of computing resources. A compromise is to start with lower resolution, at 150 M . With that resolution, each supernova in nature corresponds to a supernova in the simulation. By doing this one can overcome many problems inherent in the subgrid physics.
Cooling from metal lines and radiative processes in hydrogen and helium gas stagnates 4 at 10 K. To cool further down to 300K in molecular clouds, H2 cooling is required. As the abundance of H2 species depends on a number of non-equilibrium processes (Glover & Abel 2008), one has to include them. To compare with today’s dwarfs the effects of heating from reionisation need to be taken into account, which will be done in later work.
Richardson et al.(2013) investigate formation and evolution of a large population of small dark matter mini-halos around larger galaxies in a cosmological context. Outflows
18 1.1. Dark Matter
4 Fig. 1.6 —: Examples of cooling functions for collisions with H2, below 10 K as needed for formation of protostellar gas clumps (Glover & Abel 2008). from galaxies were found to lead to collapse of these objects, ultimately giving rise to dense homogeneous star-forming gas.
1.1.7 Predictions from Simulations
Cosmological simulations aim to reproduce the observed matter distribution of the current epoch and earlier on by following the evolution of structure buildup and star formation from the primordial power spectrum. The simulations usually begin at redshift z 50 ∼ when the Universe is still in the linear regime. The initial conditions are matched to the observed temperature fluctuations in the cosmic microwave background. In the simplest form, they are set up assuming gaussianity. Higher order corrections incorporate f = 0, NL 6 where f is a measure for the non-Gaussianity of Bardeen’s potential, Φ = ϕ + f (ϕ2 NL NL − ϕ2 ), where ϕ denotes an auxiliary Gaussian field. h i These initial conditions are power spectra for the modes of displacement in density and velocity, modeled on a grid of N 3 particles in a box. The box is assumed to repeat itself in each cartesian coordinate, effectively forming a hyper-torus. In general, the simulation is started and begins to calculate iteratively
19 CHAPTER 1. Introduction
gravitational forces between particles • hydrodynamical quantities, either for hydrodynamic particles or on a mesh • the resulting changes for the next step •
Bagla(2005) gives a review of how structure formation can be modeled by numerical means. Collisional codes follow each of the particles directly in systems where close en- counters play a significant role and need to be resolved. Collisionless simulations follow a smaller number of particles that are actually expected to be in the system, e.g. for a continuum distribution by sampling only a small number of particles. Dark matter as a pressure-less fluid is treated by this paradigm. One method for solving Poisson’s equation is to estimate the density of particles at the positions of a grid and calculate the forces on the particles from this background mesh. See Dehnen & Read 2011 for a review.
Smoothed particle hydrodynamics (SPH) represents the fluids by particles, and cal- culates the hydrodynamical forces from a discrete form of the Euler equations. Grid codes in contrast follow the gas dynamics in small patches and updates the hydrodynamic variables in the neighboring cells according to the flux through boundaries and volume changes. Meshes can take a diversity of geometries, depending on the physical system under consideration. Adaptive mesh refinement (AMR) uses an advanced technology to follow the fluid more accurately: The mesh is refined in the vicinity of high overdensity, allowing for correct capture of accretion, shock waves, and other small features, while keeping the mesh at low resolution where density is small, which increases efficiency con- siderably. The high dynamic range reached by this method gives the reason that AMR is used for this work.
Boundary conditions are needed for any cosmological simulation. They are given by assuming a power spectrum motivated by the CMB, and the fact that the Universe or a patch thereof is represented as a box of size a3, where a is typically of order Mpc or above. It is repeated in each direction, giving the topology of a hyper-torus, meaning that whenever a particle is moving out of the box at one end, it will come in on the other side again. Forces are calculated over this boundary in the same manner, s.t. any distance along an axis x between two particles at x1, x2 is determined via:
∆x = min( x x , x + L x ) (1.14) | 1 − 2| 1 − 2 where Lbox denotes the boxsize.
There is thus a minimal size of the box needed to follow the evolution of a representative fraction of the Universe. With decreasing redshift, ever lower modes of the initial power spectrum enter the expanding box. Using the Sheth-Tormen mass function, we expect several 108 M halos inside a box of 1 Mpc at z = 10 (Boley et al. 2009).
Independently of the method employed, pure dark matter ΛCDM simulations find a steep central cusp in well-resolved halos (Heitmann et al. 2008). Dubinski & Carlberg
20 1.1. Dark Matter
Fig. 1.7 —: Fornax dwarf spheroidal. It resides at a distance of 140 kpc, with a mass distribution according to a low-concentration King model with core radius of 460pc. Its mass is 1.6 107 M . Credits: NASA/STScI. ·
(1991) proposed a split power-law profile with central slope of 1, and outer slope of 4 − − (a Hernquist profile, see appendix 6.1.2), Navarro et al.(1996a) found that profile to with a lower outer slope of 3 (Moore et al. 1999), independent of the Ω ,Ω , and largely − m Λ independent of the halo size. Even if baryons are added, without extended subgrid physics, or too small feedback, adiabatic contraction results from energy loss through cooling of baryons. Increased resolution and upturn of feedback was found necessary to circumvent that artificially high concentration.
1.1.8 Dwarf Galaxies
The effects of dark matter are best seen in small scale systems like dwarf galaxies. They differ from other gravitationally bound systems in that they consist mostly of dark matter; more luminous halos show comparable densities of baryonic and dark matter in the inner regions. Dwarf galaxies are perfect to study dark matter for three reasons: they represent the low-mass end of galaxy formation, enable direct detection of dark matter clustering properties through mass models, and serve as possible detection sites of Majorana dark matter self-annihilation signals. Future experiments as e.g. GAMMA-400 (Moiseev et al. 2013) are planned to resolve features with angular resolutions down to 0.01 deg at E ≈ 100 GeV.
For observational purposes, any galaxy with absolute magnitude M < 16.0(M < B − V
21 CHAPTER 1. Introduction
17) and spatial extension larger than a typical globular cluster (a few kpc) is called a − dwarf (Tammann 1994). A more theoretical definition is a bound stellar system, where the kinematics of the visible part reveals the need for a dark matter halo, whose mass is constrained to be M < 1011 M . A varying group of objects with these properties are found, including early-type dwarf spheroidals (Figure 1.7), elliptical star-forming dwarf irregulars (dIs), ultra-faint dwarfs (uFd) with very low surface brightness, actively star- forming BCDs, and ultra-compact dwarfs (UCDs) (Tolstoy et al. 2009).
In galaxies with a mass as low as dwarf spheroidals the luminosity is expected to be very low as well. They are so faint in fact that they are only detected in the Local Group (Belokurov et al. 2007; Whiting et al. 1997), showing some 10 classical dwarfs in the Milky Way halo. Mateo(1998) gives an overview of the local group dwarf galaxies. Star formation histories of the dwarfs is covered e.g. in Skillman 2005; Dolphin et al. 2005, kinematics in Walker et al. 2009a, and their orbits in Lux et al. 2010. Recent wide area surveys discovered more and more dwarfs, (Belokurov et al. 2007), (Belokurov et al. 2010), totalling some 25 around the Milky Way in a 20% coverage of the sky (Koposov & Belokurov 2008). Following predictions of ΛCDM, there should be thousands of them, though (Diemand et al. 2007). This discrepancy is generally known as the missing satellites problem, see Moore et al. 1999; Klypin et al. 1999.
Moreover, observations give hints to the existence of a constant dark matter density core in the central parts of larger dwarf galaxies (Oh et al. 2011). First, observations had to be carefully checked, but nowadays there is mounting evidence for smaller dwarfs as well, see Kleyna et al. 2003; Goerdt et al. 2006; Battaglia et al. 2008; Walker & Pe˜narrubia 2011; Amorisco & Evans 2012; Kuzio de Naray & Kaufmann 2011, and other small systems like ultra compact high velocity clouds like recently discovered by the ALFALFA 21cm survey. Faerman et al.(2013) propose underlying dark matter models with flat cores and masses up to 107 M .
This is in contradiction to theory predicting cuspy profiles for dark matter only simu- lations (Moore 1994; Flores & Primack 1994; Moore et al. 1999). One proposed solution to this cusp-core problem is that baryonic physics is non-negligible, as is seen clearly in recent simulations (Mashchenko et al. 2008a; Governato et al. 2010), correcting the early assumption that baryonic physics has only small influence (Navarro et al. 1996a; Gnedin & Zhao 2002). Feedback from star formation and supernovae (or even long-duration gamma-ray bursts, as shown in Kelly et al. 2013) can remove significant amounts of the dominant baryonic gas from the central regions, having the dark matter realign to the overall broadened potential. Repeated star formation bursts can lead to a shallow profile (Read & Gilmore 2005), see also Pontzen & Governato 2011. Another effect might be dynamical stir-up during infall of gas clumps, measurable via phase space density of the central region (El-Zant et al. 2001).
I concentrate on dwarf spheroidals as the smallest galaxies in the Universe, with dynam- ical masses 106 M 108 M determined from the observed kinematics of the stars, while − star masses range from 1000 M to 107 M . These galaxies typically have no detected ∼
22 1.2. Mass Modelling gas, and provide a group of very promising indirect dark matter search sites (Charbonnier et al. 2011).
1.2 Mass Modelling
In this thesis, I am interested in measuring the distribution of dark matter inside tiny dwarf galaxies. The total mass distribution can be calculated from the kinematic motions of stars and/or gas within the galaxy – a procedure called mass modelling. Subtracting the baryonic part will give the dark matter distribution directly. Here I focus on purely spher- ical stellar systems. In section 1.2.1, I review the “Jeans” method that I use throughout the thesis. I briefly mention other approaches in section 1.2.2.
1.2.1 Jeans Modelling
To deduce the mass of a system of mass particles, one needs to find the gravitational potential Φ first. The Boltzmann equation relates the 6-dimensional distribution func- tion f(r, θ, φ, vr, vθ, vφ) in spherical coordinates to that potential Φ. We are interested in spherical systems where Φ = Φ(r), and assuming a collision-less system in steady state, we get that:
df ∂f ∂f ∂Φ ∂f 0 = = + ~v (1.15) dt ∂t · ∂~x − ∂~x · ∂~v as developed by Jeans 1915 for a general distribution function f, and after multiplication by vr and integration over velocities we get for the second moments (Binney & Tremaine 2008):
d(νσ2) 2β dΦ r + νσ2 + ν = 0 (1.16) dr r r dr where ν(r) = fd~v is the stellar tracer density, σ2 = v2 is the velocity dispersion along r h r i the radial direction, and β(r) = 1 σ2(r)/σ2(r) is the anisotropy parameter (Binney & R − t r Tremaine 2008).
Similar expressions hold for higher order moments (Merrifield & Kent 1990; An & Evans 2011). After solving this equation (system) for Φ(r), one can calculate the density profile ρ(r) from Poisson’s equation:
2Φ(r) = 4πGρ(r) (1.17) ∇ where G denotes the gravitational constant.
23 CHAPTER 1. Introduction
1.2.2 Other Mass Modelling Approaches
Schwarzschild 1979, 1993 propose to start mass modelling by assuming a density distribu- tion in N bins, calculating its gravitational potential, and then create a library of many orbits within this potential. Linear programming will then produce the best fitting linear superposition of occupation numbers in stars for each orbit to reproduce the observed system. If no solutions are found in the end, one must conclude that there is no dynamical equilibrium for the chosen density distribution and exclude it. Schwarzschild models have been applied to spherical, axisymmetric, and triaxial geometries, and to both continuous and discrete measurements (Rix et al. 1997; Gebhardt et al. 2005; van de Ven et al. 2006; Chanam´eet al. 2008). The main advantage is that it does not rely on any symmetry of the observed system, and that one can directly extract the distribution function (H¨afner et al. 2000). Drawbacks of the method as summarised in Chanam´e 2010 are that
large orbit libraries mean expensive calculations, • initial conditions for orbits need to be sampled carefully, • the best solution for superposition might change if more orbits are included. •
A mass modelling tool closely related to Schwarzschild modelling is the Made2Measure (M2M) method. The main idea behind the made2measure n-body method as introduced by Syer & Tremaine 1996 is to adapt an n-body model for the system under study iter- atively until the best-fitting system has been found. Instead of integrating orbits inde- pendently first, and then superimposing them as in the Schwarzschild modelling, M2M calculates orbits and their weights concurrently. de Lorenzi et al.(2007) incorporate ob- servational errors for the first time, and use this to elliptical galaxies (de Lorenzi et al. 2008).
If one does not want to have any restriction on the distribution function from, say, taking only moments of it to compare to data, or line-of-sight velocities, distribution- function based methods are available to help (Binney & Tremaine 2008). They are more challenging than moment-based methods because they require the solution of an integral equation instead of a differential equation, and most often only applicable in case of special geometries like spherical symmetry or axisymmetric systems, where an additional integral of motion – the angular momentum – is known.
24 1.3. Aim of This Thesis
1.3 Aim of This Thesis
I split my efforts into two main investigation lines:
Can we constrain the central dark matter density slope in spherical sys- tems non-parametrically? Chapter2 presents a non-parametric mass-modelling tool dubbed GravImage . The application to a first observed dwarf galaxy, Fornax, is shown in chapter3. A suite of four other observed dwarf galaxies – Draco, Sculptor, Sextans, and Carina – is presented in chapter4.
Can we predict the dark matter distribution on small scales? Two regions are of interest: dwarf spheroidals and globular clusters. Chapter5 describes the ingredients of non-equilibrium chemistry, its validation over the epoch of reionisation, and feedback mechanisms in play. A suite of simulations is presented. A further section shows the differences between a classical hydrodynamical code and a non-equilibrium chemistry patch on a small dwarf galaxy at high redshift. The influence of baryons on the dark matter distribution within simulated dwarfs is investigated.
25 CHAPTER 1. Introduction
26 2 Mass Modelling Spherical Systems
We present a new non-parametric method based on Jeans modelling to determine the mass distribution in spherical systems with single or multiple tracer populations. A high dimensional parameter space encoding tracer density, line of sight velocity dispersion and total mass density is sampled with MultiNest.
With minimal assumptions on the functional form of these profiles, we show that we can successfully recover the radial density profile of spherical mock dwarf galaxies. With two populations, we begin to disentangle the degeneracy between dark matter density and tracer velocity anisotropy, determining the dark matter cusp slope at the half light radii at nearly 2-σ confidence. This improves to nearly 99% confidence if we introduce a physically motivated prior on the central velocity anisotropy. We also test our method on triaxial mock data for which our method is expected to become biased. We find that with 10, 000 tracers, this bias remains smaller than our other uncertainties unless we are ∼ staring down the barrel of the triaxial figure. This leads to the perhaps counter-intuitive result that GravImage can be reliably applied to triaxial systems so long as they do not appear circularly symmetric on the sky.
Keywords: galaxies: dwarf – galaxies: fundamental parameters – galaxies: kinematics and dynamics – cosmology: dark matter
The work of this chapter will be submitted as a paper to MNRAS.
2.1 Introduction
Cosmological ΛCDM simulations predict a hierarchical self-similar assembly of dark matter halos (e.g. White & Rees 1978; Navarro et al. 1996b). Modelling only the dark matter
27 CHAPTER 2. Mass Modelling Spherical Systems
fluid in the absence of baryons (stars and gas), Dubinski & Carlberg(1991) found that the density profile of resulting halos are best described by a split power law that diverges 1 as r− at the centre. Navarro et al.(1996b) demonstrated that this profile is universal, giving a good match to halos of all mass from those hosting dwarf galaxies to those hosting galaxy clusters; they suggested a fitting function:
1 2 r − r − ρ(r) = ρ 1 + (2.1) 0 r r 0 0 that has become known as the NFW profile.
While ΛCDM has performed remarkably well on large scales (e.g. Tegmark & Zaldar- riaga 2002), the above prediction has long been at tension with observational data from dwarf and low surface brightness (LSB) galaxies. Flores & Primack(1994) and Moore (1994) were the first to show that fits to dwarf galaxy rotation curves favour a dark mat- ter density profile with a central constant density core: d ln ρ/d ln r α = 0, rather r=0 ≡ than a cusp with α 1. Since then, similar results have been reported for a wide range ∼ − of gas rich dwarf and low surface brightness (LSB) galaxies (e.g. de Blok et al. 2001; McGaugh et al. 2001; de Blok et al. 2008; Hague & Wilkinson 2014); a result that is robust to known observational uncertainties and model systematics (e.g. Kuzio de Naray & Kaufmann 2011). This now long-standing discrepancy between theory and observation has become known as the cusp-core problem (for a review see e.g. de Blok 2010).
One proposed solution to the cusp-core problem is to invoke stellar feedback, not mod- elled in the pure dark matter simulations discussed above. Navarro et al.(1996) suggest that impulsive winds driven by supernovae could cause the dark matter halo to expand erasing the central cusp. However, Gnedin & Zhao(2002) demonstrated that, once the angular momentum barrier to gas collapse is taken into account, the maximum effect of a blow out is small. Read & Gilmore(2005) showed that this problem can be overcome if star formation proceeds in multiple bursts, gradually transforming a cusp to a core over several cycles of star formation. This mechanism appears to be what is at play in recent high resolution simulations of dwarf galaxies that model both the dark matter fluid and the baryons (Mashchenko et al. 2008b; Governato et al. 2010; Teyssier et al. 2013). For an elegant analytic treatment of the effect see Pontzen & Governato 2012; and for a review see Pontzen & Governato 2014.) Teyssier et al.(2013) point to two key observational predictions of such a scenario: (i) star formation should be bursty with a duty cycle of ∼ a dynamical time; and (ii) the stars should be collisionlessly heated along with the dark matter, producing vertically hot stellar discs even in isolated galaxies. Such predictions appear to be supported by the latest data (Leaman et al. 2012; Kauffmann 2014).
If baryons really do transform cusps to cores then this is bad news for constraining dark matter models. Exotic models that produce cores (e.g. Zavala et al. 2013) would become indistinguishable from vanilla cold dark matter. This motivates pushing to ever smaller scales where the baryonic effects should diminish. Pe˜narrubiaet al.(2012) have recently suggested that supernova feedback will no longer provide enough energy for cusp-
28 2.1. Introduction
< 6 core transformations below some critical stellar mass Mc 10 M , though the precise ∼ value of Mc remains under investigation (e.g. Madau et al. 2014). The dwarf spheroidal galaxies that orbit the Milky Way and Andromeda straddle this critical stellar mass, mak- ing them prime targets for measuring their dark matter density profiles. They also have the added advantage that most of their gravitating mass is dark with typically negligible contributions from stars and/or gas. The most massive, Fornax, has some 107 M in ∼ stars (Coleman & de Jong 2008), lying above Mc; while the smallest, Segue 1 – if it is indeed a galaxy; (Niederste-Ostholt et al. 2009; Martinez et al. 2011) – has just 1000 M ∼ (Belokurov et al. 2007), lying well below Mc.
Because of their proximity to their host galaxies, dwarf spheroidals differ from the dwarf and LSB galaxies discussed above in that they are almost completely devoid of gas (e.g. Gatto et al. 2013). This presents a challenge because stellar orbits, unlike gas, can cross. If only radial velocities are available, this leads to a strong degeneracy between the dark matter density profile and the orbit distribution of the stars, typically parameterised by the velocity anisotropy (e.g. Wilkinson et al. 2002; Binney & Tremaine 2008):
2 2 σθ + σφ β 1 2 (2.2) ≡ − 2σr where σr,θ,φ are the velocity dispersions in spherical polar coordinate directions r, θ and φ, respectively. An anisotropy β = 0 corresponds to isotropic velocity dispersions, while β = 1 is pure radial and β = pure tangential. −∞ Battaglia et al.(2008) suggested that the above degeneracy could be broken if stars were split into multiple tracer populations by either metallicity or abundance. They performed such an analysis on the Sculptor dSph, weakly favouring a dark matter core. This analysis was further refined and applied to four dwarf spheroidals by Walker & Pe˜narrubia(2011) (hereafter WP11). They also favour cores but with higher statistical significance, supporting earlier timing arguments for UMi and Fornax (Kleyna et al. 2003; Goerdt et al. 2006; Read et al. 2006; Cole et al. 2012). However, recently the literature has become divided on the issue of cusps or cores in the dwarf spheroidals. An analysis of four dwarfs using a non-parametric Schwarzschild method (Schwarzschild 1979) favoured a wide range of density profiles (Jardel & Gebhardt 2013); simple distribution function models (Amorisco & Evans 2012) or virial parameter models (Evans et al. 2011) support the findings of WP11; while single component Schwarzschild (Breddels & Helmi 2013) and higher order Jeans (Richardson & Fairbairn 2013) analyses conclude that the data are simply not sufficient to say one way or the other (see also the discussion in Breddels & Helmi 2014). Finally, Strigari et al.(2014) have recently pointed out that some of the key assumptions in the WP11 analysis may be violated if the density profile of one or more of the tracers is cuspy, leading to central cores being artificially favoured over cusps.
In light of the above discrepancies, in this paper we present a new non-parametric Jeans modelling tool: GravImage . Our goal is to assume only equilibrium and spher- ical symmetry, allowing the model full freedom otherwise. We refer to this approach as
29 CHAPTER 2. Mass Modelling Spherical Systems non-parametric, though really what we mean is that we have a model with far more pa- rameters than the available data constraints. Since this means that we are necessarily under-constrained, we are then forced to build model ensembles and explore parameter degeneracies. For this, we use the efficient MultiNest technique (Feroz et al. 2009). We support multiple tracers that can be simultaneously modelled in a single underlying po- tential (each with its own velocity anisotropy). Similar approaches have been attempted in the literature before. Diakogiannis et al. 2014 build on early work by Ibata et al. 2011 to present a non-parametric spherical Jeans solver for modelling globular clusters; Jardel & Gebhardt(2013) present a non-parametric Schwarzschild method; and Chakrabarty & Saha(2001) present a non-parametric method for modelling the black hole at the centre of our Galaxy. However, our new method is more general than these previous works. Diakogiannis et al.(2014) – since they are modelling globular star clusters – assume that the mass distribution is known, fitting only for the mass to light ratio and β(r); Jardel & Gebhardt(2013) present a method very different from that discussed here (and there- fore complementary); however, they do not allow for multiple populations or test their methodology on mock data. Finally, Chakrabarty & Saha(2001) present an elegant non- parametric distribution function method – but it relies on assuming an isotropic velocity distribution function, β = 0. This may be reasonable for stars orbiting a super-massive black hole, but it is likely a poor assumption when modelling dwarf spheroidals. Using spherical mock data, we set out to determine what type and quality of data is sufficient to constrain the logarithmic cusp slope within dwarf spheroidals. We also use triaxial mock data to test what happens when our method is pushed beyond its regime of validity. We will present applications of our method to real data for dwarf spheroidals in forthcoming papers.
Finally, we note that while our clear science goal here is to measure the dark matter distribution in dwarf spheroidals, GravImage can be applied to any spheroidal stellar system, including globular clusters, giant elliptical galaxies, star orbits around supermas- sive black holes, and even galaxies in galaxy clusters (e.g. Ibata et al. 2011; Chakrabarty & Saha 2001; Lokas et al. 2006; Napolitano et al. 2014). We will explore such applications in future work.
This paper is organised as follows. In 2.2, we introduce the GravImage method. In § 2.4, we test our method on mock data. Finally, in 2.5 we present our conclusions. § §
2.2 Method
2.2.1 Derivation of the Key Equations
To a very good approximation, stars in galaxies move as a collisionless fluid that obeys the collisionless Boltzmann equation (e.g. Binney & Tremaine 2008):
30 2.2. Method
df ∂f = + f ~v f Φ = 0, (2.3) dt ∂t ∇~x · − ∇~v · ∇~x where f(~x,~v ) is the distribution function of stars in phase space; ~x and ~v are the position and velocity of the stars; and Φ is the gravitational potential.
In spherical coordinates (r, θ, φ), the collisionless Boltzmann equation becomes:
∂f ∂f ∂f ∂f ∂f ∂f ∂f +r ˙ + θ˙ + φ˙ +v ˙r +v ˙θ +v ˙φ = 0 (2.4) ∂t ∂r ∂θ ∂φ ∂vr ∂vθ ∂vφ wherer ˙ = vr, θ˙ = vθ and φ˙ = vφ/r sin θ are the velocities in spherical coordinates.
In principle, given the positions and velocities of many stars, we could directly solve equation 2.4 for the force field Φ. In practice, this is impractical as it involves derivatives ∇~x of f that is six dimensional and thus very poorly sampled. One way around this problem is to use instead velocity moments of equation 2.4 – the Jeans equations (e.g. Binney & Tremaine 2008). Assuming a non-rotating steady state (∂f/∂t = 0, v = v = v = h ri h θi h φi 0); spherical symmetry; and defining the tangential velocity dispersion σ2 σ2 = σ2, the φ ≡ θ t second order moment is given by:
1 ∂ 2β(r)σ2 ∂Φ GM(< r) νσ2 + r = = , (2.5) ν ∂r r r − ∂r − r2 where M(< r) is the total cumulative mass; ν is the density of a set of tracer star particles moving in the potential Φ(r); G = 6.67398 10 11m3/kg s2 is Newton’s gravitational · − constant; and β is the velocity anisotropy as in equation 2.2.
Integrating both sides of equation 2.5 gives the radial velocity dispersion as function of radius r, and correspondingly the fourth order moment:
1 r β(s) GM(˜r)ν(˜r) r˜ β(s) σ2(r) = exp 2 ds ∞ exp 2 ds d˜r (2.6) r ν(r) − s · r˜2 s Zrmin ! Zr Zrmin ! and:
3 r β (s) v4 (r) = exp 2 0 ds h r i ν(r) − s · Zrmin ! GM(˜r)νσ2 r˜ β (s) ∞ r exp 2 0 ds d˜r (2.7) r˜2 s Zr Zrmin !
31 CHAPTER 2. Mass Modelling Spherical Systems where β (r) = 1 3 v2v2 / v4 is similar to the velocity anisotropy β(r), but at fourth 0 − h r t i h r i order, and gives the relative strength of the kurtosis of the radial and tangential velocity distributions. Note that rmin leads to an integration constant that cancels and so the choice of rmin is arbitrary. Typically, only projected velocities are observable. Projecting equation 2.6 along the line of sight, we obtain:
2 R2 ν(r)σ2(r)r σ2 (R) = ∞ 1 β r dr, (2.8) LOS 2 2 2 Σ(R) R − r ! √r R Z − where Σ(R) denotes the surface mass density at projected radius R.
From equation 2.6, it is clear that the velocity anisotropy β(r) trivially degenerates with the cumulative mass distribution M(< r) that we would like to measure. This is problematic because with only line of sight velocities β(r) is poorly constrained, unless many thousands of velocities are available (e.g. Wilkinson et al. 2002). This has motivated considering the higher order moment equations as these may break the M(< r) β(r) − degeneracy. We will explore this in a later paper, but note that there is cause already for pessimism. For every higher order moment equation, there are new anisotropy pa- rameters, in this case β0. The hierarchy of Jeans equations are not closed (e.g. Binney & Tremaine 2008). Since each new equation adds yet another new unknown, unless there are strong theoretical reasons for a relationship between β, β0 and similar, the higher order moments are unlikely to yield improved constraints (e.g. Richardson & Fairbairn 2013). More promising are the virial parameters discussed by Richardson & Fairbairn(2014) that involve fourth order moments but are independent of β0. We will consider these in a separate publication.
2.2.2 The Mass Distribution
In the following, we present a non-parametric method for the solution of equation 2.8 for the total gravitating mass density ρ(r), given a single or multiple tracer density profiles νi(r) with corresponding line-of-sight velocity dispersions σLOS,i(R). We write the overall density profile ρ(r) as:
M ρ(r) = ρDM(r) + ∗ L (r) (2.9) L · ∗ where ρDM(r) is the dark matter contribution; L (r) is the visible light profile; and (M /L) ∗ ∗ is a constant mass-to-light ratio. The assumption that this is constant with radius could be relaxed if required by the data. For our mock data, the tracer particles have negligible mass and so M /L 0. The enclosed mass M(< r) then follows from the density via: ∗ ≈
r 2 M(< r) = ρ(r0)r0 dr0, (2.10) Z0
32 2.2. Method
Note that in principle our method can be generalised to investigate alternative gravity models if the acceleration GM(r)/r2 is replaced by ∂Φ/∂r in equation 2.5. −
The dark matter density ρDM(r) is represented in terms of the logarithmic density slope:
d ln ρ(r) n(r ) = (2.11) j − d ln r r=rj for radial bins j where 1 j N , as: ≤ ≤ bin
ln r ρ(r) = ρ exp n(s)ds , 1/2 · − " Zln r1/2 # with the density at half-light radius ρ(r1/2) = ρ1/2, and n(r) interpolated in between bin radii r < r < r . We prescribe two times three buffer bins n(r ) for j min,j max,j j ∈ N + 1,N + 2,N + 3 outside of the range where data is given to enable sensible { bin bin bin } extrapolations towards small and high radii, and two additional slopes n0 < 3, n > 3 for ∞ the asymptotic density slopes towards r = 0 and r = , which are reached at half the ∞ smallest radius and r = 5rmax. ∞ If the n(r) are allowed to change freely between a minimum value of 0 and a maximum value nmax in each bin, we find a tendency towards significant wiggling from bin to bin that is likely unphysical. To counteract this behaviour and enable faster convergence to sensible profiles, we introduce a prior on the maximal change in n(r) that is allowed, by sampling – instead of n(r) directly – its derivative dn(r)/d ln(r). To enforce faster convergence and prevent over-fitting we favour smooth n(r) profiles by sampling dn(r)/d ln r from a normal distribution with width w > 0, rather than using a uniform prior in the range [ w, w]. − The full range in n(r) can still be sampled if the data demand it.
2.2.3 The Tracer Density Profile
The tracer density is represented similarly to the dark matter profile, but we take ad- vantage of our additional knowledge of its distribution. We know the surface density of our tracer stars a priori, and can derive a three-dimensional density profile for the stellar tracers by de-projecting under the assumption of spherical symmetry:
1 dΣ(R) dR ν(r) = ∞ (2.12) 2 2 −π r dR √R r Z − However, once we allow for errors on Σ this deprojection is not unique. For this reason, we simply use it as a starting point for priors on the nν(r) parameters that describe ν(r) similarly to the dark matter profile (see above). Over radii where we have data,
33 CHAPTER 2. Mass Modelling Spherical Systems
we sample a small range around each nν(ri) assuming flat priors. For the extension bins where we have no data, the range for nν(r) is enlarged by a factor five, as we have no prior information.
To speed up convergence, we use an initial Σ-burn-in phase, ignoring the projected velocity dispersion at first. The 3D light distribution for each tracer νi(r) is projected to give the surface density:
∞ νi(r)r Σi(R) = 2 dr (2.13) 2 2 0 √r R Z − which is then compared with the observational data (see 2.2.5). After 1000 models have 2 been found to achieve a χΣ < 1, we terminate the Σ-burn-in phase, and calculate velocity dispersions, as discussed next.
2.2.4 The Velocity Anisotropy
The velocity anisotropy β(r) is allowed to vary broadly in the interval [ , 1] by sampling −∞ a modified, symmetric β∗ (e.g. Read et al. 2006):
2 2 σr σt β β∗ = − = [ 1, 1] (2.14) σ2 + σ2 2 β ∈ − r t − with the following function:
a0 a β∗(r) = − ∞ + a (2.15) 1 + κ exp(α ln(r/rs)) ∞ a0 a κ = − ∞ 1 (2.16) β∗(rs) a − − ∞ where a0, a [ 1, 1] are the inner and outer asymptotic values for β∗(r 0, ), rs is ∞ ∈ − → ∞ the scale radius of a0 a transition with a β∗ at that point of β∗(rs), α the speed of → ∞ the transition.
This approach allows us to sample many qualitatively different models – isotropic, radially biased, tangentially biased, and any transitions between those – with very few parameters. In Figure 2.1, we compare our functional form in equation 2.15 with a typical Osipkov-Merritt profile taken from our mock. With our 5 parameters, we obtain an excellent recovery of the input profile, while being able also to capture constant, and both rising and falling models. The fact that an asymptotic value is reached at high radii allows us to recover the correct β(s)/sds. In principle, if the data demand it even more general functional forms for β(r) could be easily implemented in GravImage. R
34 2.2. Method
Fig. 2.1 —: Fitting of the analytic β profile using the form in equation 2.15. The upper plot shows an analytic Osipkov-Merritt anisotropy profile as dashed, black curve, and our fitted profile. Shown below is the difference between analytic and fitted profile. We see that the analytic profile is fit well over the range of input data, and extrapolated agreeably well to higher radii.
2.2.5 Comparison with Data
Given the above functional forms, σLOS,i(r) is calculated from ρ(r), νi(r), and βi(r) ac- cording to equation 2.8. This is done numerically, involving three integrations, which are performed with polynomial extrapolations of the integrands up to infinity, such that miss- ing contributions from r > rmax do not lead to an artificial falloff of σLOS. We ensured that our integration is sufficiently accurate by explicitly checking our routines against an analytically known Hernquist profile. The last step involves comparison of the projected surface density Σi(r) – calculated from the 3D tracer density νi(r) – as well as σLOS,i(r), to the respective 2D data profiles for the tracer populations. We use a likelihood based on the overall goodness of fit:
N 2 2 2 χ = χΣ,i + χσ,i, (2.17) Xi=1 Nbin 2 2 Σdata,i(rj) Σmodel,i(rj) χΣ,i = − , (2.18) εΣ(rj) ! Xj=1
2 with error εΣ(rj) on the data Σdata,i(rj). Analogous expressions hold for χσ,i. In the 2 absence of a measured βi(r), we set χβ,i = 0.
35 CHAPTER 2. Mass Modelling Spherical Systems
2.2.6 Priors
In addition to our regularisation priors on the dark matter density profile ( 2.2.2), we § investigate two theoretically motivated priors – on the velocity anisotropy β(r), and a monotonicity prior for n(r):
1. β∗(0) = 0, and thus β(0) = 0, the asymptotic central velocity anisotropy is set to 0, and thus isotropy is assumed for the inner parts of our the models. This is motivated by a recent attractor solution for spheroidal systems that is maximally stable to potential fluctuations (Pontzen et al. 2015). It furthermore applies to our mock data, and allows us to eliminate part of the mass-anisotropy problem.
2. Monotonicity prior on n(r): Given that the broad range allowed in n(r) takes a long time to sample, we investigate the use of a monotonicity prior dn(r)/d ln r 0. This ≥ is a strong prior, as it favours rising n(r) to high values. All our mock data fulfills that criterium, and many theoretically motivated parametrised profiles like NFW and Einasto do so as well, see 6.1.2. §
The two priors are used in two combinations – i) monotonicity in n(r), no β∗(0) = 0 prior and ii) β∗(0) = 0, no monotonicity in n(r). Either one of them introduces constraints into our system; constraints which act over more than just a fixed radial range.
We use weak flat priors on all other parameters:
1) log ρ is sampled uniformly from 1 dex around the estimated value based on 10 1/2 ± half-mass and half-mass radius. This provides a reasonably large error margin for the initial mass estimator;
2)0 n(r) 5 for the profile parameters of ρ; ≤ ≤ 3)0 n (r) 5 for the profile parameters of all ν profiles. ≤ ν ≤
2.2.7 Parameter Space Sampling
Given the freedom in our functional forms for the matter density; tracer densities; and the velocity anisotropy profiles of these, we are left with a large number of free parameters:
N = N + N (N + N ) (2.19) dim bin pop · bin beta with the number of bins N = N + 2 (3 + 1) with 3 extension bins towards small bin bin,data · and high radii each, and associated asymptotic values. This is significantly higher than
36 2.3. Mock Data the number of data bins with observational constraints. As discussed in 2.1, we refer to § this as non-parametric mass modelling.
To efficiently sample the above high dimensional parameter space, we use the Multi- Nest code1 (Feroz & Hobson 2008; Feroz et al. 2009, 2013). MultiNest is a Bayesian nested sampling algorithm to generate posterior samples from non-trivial distributions in high dimensions. It samples the n-dimensional hypercube κ = [0, 1]Ndim , which needs to be translated into physical prior distributions for each of the parameter profiles. We set the number of live points to be 10 times the number of free parameters, such that we get a denser sampling of the highly dimensional parameter space while still keeping the computational cost reasonably low. The optimal sampling efficiency for a high number of parameters N > 30 as proposed by the MultiNest authors is 5 percent. We increase this fraction by a factor of 4, to get a higher output frequency after the initial burn-in phase.
2.3 Mock Data
We apply our method to a set of mock data, consisting of multiple tracers moving in spherical or triaxial potentials with radial or tangential velocity anisotropy. All mock data are available on the Gaia Challenge wiki site2. The first set of mocks have cusped or cored dark matter density profiles with radial anisotropy – the Walker mocks (WP11), with tracer density:
(γ β )/α γ α ∗− ∗ ∗ r − ∗ r ∗ ν (r) = ν0 1 + (2.20) ∗ r r ∗ " ∗ # inside dark matter halos of the form:
γ α (γDM βDM)/αDM r − DM r DM − ρ = ρ 1 + (2.21) DM 0 r r DM " DM # with scale radii r , rDM; central slopes of γ , γDM; transition parameters β , βDM; and outer ∗ ∗ ∗ slopes α , αDM. ∗ The anisotropy follows the functional form of Osipkov(1979) and Merritt(1985):
2 2 σθ r β(r) = 1 2 = 2 2 . (2.22) − σr r + ra with scale radius r , turning over from nearly isotropic at r 0 to radially biased at a → r = ra. ∗ 1http://ccpforge.cse.rl.ac.uk/gf/project/multinest/ 2http://astrowiki.ph.surrey.ac.uk/dokuwiki/
37 CHAPTER 2. Mass Modelling Spherical Systems
Tab. 2.1 —: Top: Parameters of the 1-population Gaia challenge mock data-set by M. Walker. The central stellar density slope for both models is γ , the stellar turnover slopes ∗ are β ; r is the stellar characteristic radius; and the anisotropy scale radii are named ∗ ∗ ra,1,2 = 0.1. Middle: Parameters of the 2-population Gaia mock data-set under study. Bottom: Parameters of the 1-population triaxial mock data.
ID geometry γDM γ r ra,1 ra,2 ∗ ∗ 1pop core sphere 0 0.1 250 pc 0.1 0.1 1pop cusp sphere 1 0.1 250 pc 0.1 0.1
ID geometry γDM r1/2,DM 2pop core sphere 0 1000 pc 2pop cusp sphere 1 1000 pc
ID geometry γDM Triax01 triaxial, 45o 0 Triax02 triaxial, along x 0
Of these distributions, finite samplings are taken, giving our first set of mock data – Table 2.1 – and then converted to mock observational data including observational parameters like spectral indices, systemic velocities, proper motions, and binary motions. The full suite of mock observational data is much larger than that used here. Our particular subset is given in the table. In addition to the above Walker mocks, we add also a mock dwarf with cusped triaxial density viewed along two different projection angles: down the barrel; and along a line tilted by 45o, the intermediate axis.
38 2.4. Results
2.4 Results
2.4.1 Single Tracer Population
We first apply our method to dark matter halos hosting a single population of tracer stars. We consider both cored and cusped models. As default, we assume that each tracer star is a member of the dwarf galaxy, and thus the full number of tracer stars – on the order of 104 – contributes to the tracer density profile and velocity dispersion. We consider poorer sampling in the appendix 2.6.2. The parameters of our mock data are listed in Table 2.1. § Our results for the recovered mass distributions and a comparison of the model tracer surface density Σ(R) and projected velocity dispersion σLOS(R) with are shown in Figure 2.2. Notice that we successfully recover the input data Σ(R), σLOS(R) within our quoted uncertainties. The three extension bins to the left and right of our data range show generally a broader distribution of models than the points where data is given, which is expected as no data are available there to restrict these values. Furthermore, the model band spans of order the range seen in the errors of the data, and does not follow every up- and downturn in the profile, a clear sign that overfitting has not occured. The goodness of fit as measured by χ2 shows a first concentration of high χ2 > 10 values, which corresponds 2 2 to the burn-in phase. The models with log10 χ > 1 (the first minimum in χ after the peak) are excluded from further analysis.
103 4.0 25 2 3.5 ]
10 s] 2 / 20 3.0 pc 101 / 2.5 [km 15 100 2.0 1 [M , 10 1.5 10 1 − frequency 1
LOS 1.0 Σ
2 σ 5 10− 0.5 0 0.0 101 102 103 101 102 103 2 1 0 1 2 3 4 5 6 − − 2 R [pc] R [pc] log10 χ
Fig. 2.2 —: Example data fit for a 1pop cored model run with monotonicity prior on n(r), and without β∗ priors. Left: Surface density profile Σ(r) of tracer stars. The dashed blue line gives the analytic profile from projecting equation 2.20. The vertical blue line gives the 2D half-light radius. Models are shown in red (median), and three shades of gray encompassing 68%, 95%, 99% of all models in each bin. Center: Projected velocity dispersion profile σLOS(r). The blue band gives data as read out from the mock data, with errors on the velocity dispersion determined via a bootstrapping algorithm. Right: Normalized distribution of the goodness of fit χ2 given for all 96k accepted models, before cutting the burn-in phase.
39 CHAPTER 2. Mass Modelling Spherical Systems
1pop cusp; monotonic n(r); no β∗ prior; 44k accepted models:
5 1.0 101 4 ]
3 0.5 100 3 pc / ) ∗ r 1
1 ( 2 0.0 10− β n [M 10 2 1 − 0.5 ρ − 3 0 10− 1.0 1 − 101 102 103 − 101 102 103 101 102 103 R [pc] R [pc] R [pc]
1pop cusp; no monotonicity prior; β∗(0) = 0 prior; 46k accepted models:
4.0 1.0 101 3.5 ] 3.0 3 0.5 100 2.5 pc / )
2.0 ∗ r 1
1 ( 0.0 10− β
n 1.5 [M 10 2 1.0 − 0.5 ρ 0.5 − 3 10− 0.0 1.0 0.5 − 101 102 103 − 101 102 103 101 102 103 R [pc] R [pc] R [pc]
2pop cusp; monotonic n(r); no β∗ prior; 39k accepted models:
4.0 1.0 1.0 101 3.5
] 0 3.0 3 10 0.5 0.5 1 2.5 pc 10− / 2 ) 2.0 ∗ ∗ r 1 2 10− ( 0.0 0.0 β β
3 n 1.5
[M 10− 4 1.0 10− 0.5 0.5 ρ 0.5 − − 10 5 − 0.0 6 10− 1.0 1.0 0.5 − − 102 103 104 − 102 103 104 102 103 104 102 103 104 R [pc] R [pc] R [pc] R [pc]
2pop cusp; no monotonicity prior; β∗(0) = 0 prior; 190k accepted models:
4.0 1.0 1.0 101 3.5
] 0 3.0 3 10 0.5 0.5 2.5 pc 1 / 10− )
2.0 ∗ ∗ r 1 2
2 ( 0.0 0.0 10− β β
n 1.5 [M 3 10− 1.0 0.5 0.5 ρ 4 0.5 10− − − 5 0.0 10− 1.0 1.0 0.5 − − 102 103 104 − 102 103 104 102 103 104 102 103 104 R [pc] R [pc] R [pc] R [pc] Fig. 2.3 —: 1 pop and 2 pop cusped models: Left: Reconstructed Density profiles. Center: Logarithmic density slope n(r). Right: Velocity anisotropy profile β∗. The input model profile is marked by the blue dashed line; the red line and gray contours show the median, 68, 95, and 99 percent confidence intervals for our chains, respectively; the vertical green line marks the 3D half-light radius of the stellar tracer populaiton (blue: projected 2D half-light radius); and the fine gray lines show a random sub-set of 30 individual models. Not all models have run for an equal number of iterations and therefore they are not all equally converged.
40 2.4. Results
1pop core; monotonic n(r); no β∗ prior; 96k accepted models:
4.0 1.0 100 3.5 ] 3.0 3 1 0.5 10− 2.5 pc / )
2.0 ∗ r 1
2
10 ( 0.0 − β
n 1.5 [M 10 3 1.0 − 0.5 ρ 0.5 4 − 10− 0.0 1.0 0.5 − 101 102 103 − 101 102 103 101 102 103 R [pc] R [pc] R [pc]
1pop core; no monotonicity prior; β∗(0) = 0 prior, a slabs; 63k accepted models: ∞
4.0 1.0 3.5
] 0 3.0 3 10 0.5 2.5 pc / )
2.0 ∗ r 1 1
10 ( 0.0
− β
n 1.5 [M 1.0 10 2 0.5 ρ − 0.5 − 0.0 3 1.0 10− 0.5 − 101 102 103 − 101 102 103 101 102 103 R [pc] R [pc] R [pc]
2pop core; monotonic n(r); no β∗ prior; 101k accepted models:
4.0 1.0 1.0 1 3.5 10− ] 3.0 3 0.5 0.5 2.5 pc 10 2
/ − )
2.0 ∗ ∗ r 1 2
( 0.0 0.0 β β
3 n 1.5
[M 10 − 1.0 0.5 0.5 ρ 0.5 10 4 − − − 0.0 1.0 1.0 0.5 − − 102 103 104 − 102 103 104 102 103 104 102 103 104 R [pc] R [pc] R [pc] R [pc]
2pop core; no monotonicity prior; β∗(0) = 0 prior; 47k accepted models:
4.0 1.0 1.0 1 3.5 10− ] 3.0 3 0.5 0.5 2.5 pc / 10 2 )
− 2.0 ∗ ∗ r 1 2
( 0.0 0.0 β β
n 1.5 [M 3 1.0 10− 0.5 0.5 ρ 0.5 − − 0.0 4 10− 1.0 1.0 0.5 − − 102 103 104 − 102 103 104 102 103 104 102 103 104 R [pc] R [pc] R [pc] R [pc]
Fig. 2.4 —: As Figure 2.3, but for cored profiles.
41 CHAPTER 2. Mass Modelling Spherical Systems
Figure 2.3 highlights the model results for a cusped profile, with two sets of prior choices: a) monotonicity in n(r), and no β∗ prior, and b) no monotonicity in n(r), but with β∗(0) = 0 prior. All models are still running to test convergence; some like the 1pop cusp no monotonicity prior, β∗(0) = 0 have not yet fully converged.
The 1pop cusped model is recovered well within our quoted 95% confidence intervals, see the first row in with monotonicity prior on n(r) and no β∗ prior. The models are particularly well-constrained around the half light radius of the stars (vertical lines), as has been noted previously in the literature (Walker et al. 2009; Wolf et al. 2010). This owes to the fact that the stars obey the Virial theorem (Agnello & Evans 2012).
In the second row of Figure 2.3, we explore the effect of switching on the prior β∗(0) = 0. The effect of this prior is to hone in more tightly on the correct solution (since the mock data also satisfy the β∗(0) = 0 prior), and was found to be needed when running without the monotonicity prior on n(r). The broad parameter bands around the mean value indicate that this model has not yet fully converged. This model is still running to further test convergence which can be rather slow in some cases given the high dimensional parameter space we explore.
The cored model with the monotonicity prior on n(r) (top row in Figure 2.4) is signifi- cantly biased with respect to the input model (blue dashed line). This highlights a problem that MultiNest can become stuck for long times around a local minimum. Given enough iterations, it will properly explore the full parameter space, but convergence can be very slow. It also demonstrates the mass-anisotropy degeneracy, since we have found a per- fectly good solution that is offset from the input mock. To combat this problem, we explore switching off the monotonicity prior which allows MultiNest to more easily ex- plore the parameter space for n(r). At the same time, we enforce a proper exploration of the mass-anisotropy degeneracy by running 8 different chains with the asymptotic values of β∗ fixed in narrow bins: a [n 0.25, (n + 1) 0.25], n [0, 7]. We then combine the ∞ ∈ · · ∈ models from each a bin, ensuring that each bin contributes an equal number of models. ∞ 2 We remove all models with log10 chi > 1 from the final set. The results are shown in the second row of Figure 2.4. As can be seen, after 40k iterations MultiNest has not yet converged. However, the results are encouraging with a much broader sampling of the parameter space. The mass-anisotropy degeneracy is now visible in the error envelopes.
The good news, as we shall show shortly, is that biased fitting problems are solved by population splitting. This works because each split population might favour a biased fit, but since both have to live within the same underlying gravitational potential their biases pull in different random directions.
2.4.2 Two Tracer Populations
In this section, mock dwarfs with a two populations of tracer particles are analysed. This is done in the following manner: Each particle in the mock data-set is flagged as
42 2.4. Results
4.0 1.0 1 3.5 10− ] 3.0 3 0.5 2.5 pc 2 / 10− )
2.0 ∗ r 1
( 0.0 β
n 1.5
[M 3 10− 1.0 0.5 ρ 0.5 − 4 10− 0.0 1.0 0.5 − 102 103 104 − 102 103 104 102 103 104 R [pc] R [pc] R [pc]
Fig. 2.5 —: 2 pop cored model with monotonicity prior on n(r) and β∗(0) = 0 and β (r) 0 priors. ∗ ≥ a member of population 1, 2, or a foreground population. We assume here that we can perfectly identify which star belongs to which population, and use the flagged value to split the tracers into two populations. The results are shown in the third and fourth rows of Figures 2.3 and 2.4 where we consider a monotonic n(r) with no β∗ prior (3rd row) and no monotonicity prior with a β∗(0) = 0 prior. Notice that we are not as badly biased as in the single population case with the monotonicity prior on n(r). This occurs for three reasons. Firstly, in splitting the populations we have lowered the sampling in each leading to larger statistical uncertainties. Secondly, we now properly sample the mass-anisotropy degeneracy. We have successfully traded systematic bias for random error. This leads to larger but more faithful errors. Thirdly, the split population runs without monotonicity prior (4th row) are almost certainly not converged yet.
Finally, note that with the monotonicity prior and using split populations (Figures 2.3 and 2.4; third row), we find that we can reliably distinguish cores and cusps at nearly 95% confidence. This highlights the combined effects of population splitting and the monotonicity prior. In Figure 2.5, I explore the effect of combining monotonicity with population splitting and a stronger prior on β (0) = 0 and β 0 for the 2pop cored ∗ ∗ ≥ model. This raises the ability to distinguish cusps and cores to 99% confidence giving truly powerful constraints on the mass profile.
2.4.3 Triaxial Mock Data
In this section, we test the dependency of GravImage on the assumption of spherical sym- metry by applying it on a mildly triaxial mock dwarf galaxy. The models were generated with the Made2Measure algorithm of Dehnen 2009 and are tailored to follow a similar profile to the profiles specified above for the dwarf galaxies. They show a spherically averaged density profile of approximately:
43 CHAPTER 2. Mass Modelling Spherical Systems
ρ ρ(r) = S (2.23) α(β γ) γ 1/α − r 1 + r rS rS with radius r, scale radius rS = 1.5 kpc, α = 1, β = 4. For the cusped profiles we 7 3 have an inner logarithmic slope of γ = 1, scale density ρS = 5.522 10 M / kpc , and 9 · 8 3 Mtot = 1.171 10 M , while for the cored one we have γ = 0.23, ρS = 1.177 10 M / pc , · 9 · Mtot = 1.802 10 M . The axis ratios are b/a = 0.8 and c/a = 0.6. The stars have · negligible mass and follow the same functional form in the density profile as dark matter, with α = 0.34, β = 5.92, γ = 0.23, rS = 0.81 kpc.
The velocity anisotropy of the stellar part is calculated via:
η η r β0 + r β s,β ∞ β(r) = η η , (2.24) r + rs,β with rs,β = 0.81 kpc, β0 = 0, β = 0.5 and η = 0.5, going from isotropic to radially ∞ anisotropic with increasing radius.
1pop triax along intermediate axis; monotonic n(r); β (0) = 0 and β (r) 0 priors: ∗ ∗ ≥
4.0 1.0 1 3.5 10− ] 3.0 3 0.5 2 2.5 pc 10− / )
2.0 ∗ r 1
( 0.0 3 β 10− n 1.5 [M 1.0 4 0.5 ρ 10− 0.5 − 0.0 5 10− 1.0 0.5 − 102 103 104 − 102 103 104 102 103 104 R [pc] R [pc] R [pc] 1pop triax down the barrel, monotonic n(r); β (0) = 0 and β (r) 0 priors: ∗ ∗ ≥
4.0 1.0 3.5 1 ] 10− 3.0 3 0.5 2.5 pc 2 / 10− )
2.0 ∗ r 1
( 0.0 β
n 1.5
[M 3 10− 1.0 0.5 ρ 0.5 10 4 − − 0.0 1.0 0.5 − 102 103 104 − 102 103 104 102 103 104 R [pc] R [pc] R [pc]
Fig. 2.6 —: Density profile of a cored triaxial mock dwarf, for which the line of sight is inclined with 45 degrees with respect to all axes (top) or along the major axis (bottom). The vertical line indicates the 3D half-light radius.
44 2.5. Conclusions
The retrieved ρ profile (Figure 2.6 top panels) for a line of sight inclined by 45 degrees with respect to all symmetry axes is actually a remarkably good match to the input model model: we recover the correct answer within our 68% confidence interval. We do, however, get a small bias in β∗. By contrast, when staring down the barrel of the triaxial galaxy, we get the central density slope wrong (Figure 2.6 central and lower panels for the two prior sets). This leads to the result that GravImage – for the data quality explored in this paper – works just fine on mildly triaxial systems so long as they do not look spherical on the sky. This result might seem counterintuitive at first; but makes sense, since the projection along the major axis of the triaxial figure aligns all of the most energetic orbits along the line of sight, leading to an overestimate in the enclosed mass when we assume spherical symmetry. We note that in this extreme case, GravImage produces rather narrow confidence intervals that are indicative of an attempt to fit the wrong model to the data. We may be hopeful, then, of detecting this scenario occurring when applying GravImage to real data.
2.5 Conclusions
We have presented a new non-parametric Jeans method – GravImage – and tested it on diverse mock data. Our key results are as follows:
With a single tracer population consisting of 10, 000 stars, we correctly recover the • ∼ mass distribution within our 95% confidence interval if we do not use a monotonicity prior on the density exponent n(r). Adding this prior can cause MultiNest to become stuck in a local minimum leading to highly biased results. We found that this problem is significantly alleviated if the monotonicity prior is excluded. We began to explore an alternative solution where the mass-anisotropy degeneracy is manually explored; however, the models are not yet converged enough to be conclusive.
Using two tracers of 5, 000 stars each, each with different half light radii, we • ∼ start to break the mass-anisotropy degeneracy. This gives an unbiased measure of both the mass distribution and the velocity anisotropy profile even when using a monotonicity prior. Indeed, the monotonicity prior for the split population case is key for reducing model uncertainties.
Introducing a physically motivated prior on the central velocity anisotropy β (0) = • ∗ 0 improves both the single and multi-component tracer results. While this prior does not break the mass-anisotropy degeneracy, it substantially reduces the range of models that fit the data. Our strongest constraints come from split populations, with monotonicity prior on n(r) and β (0) = 0, β (r) 0 priors. With these we are ∗ ∗ ≥ able to distinguish cusps and cores at 99% confidence, giving truly strong constraints on the dark matter profile.
45 CHAPTER 2. Mass Modelling Spherical Systems
5 3 10 10 1 4 2 101 10 103 10 ] 102 ] 101 ] 0 ] 0 3 3 3 10 3 10 101 0 10 10 1
pc 0 pc pc 1 pc 10 10 10 1 10− − / 1 / − / / 10− 2 2 2 10− 2 10− 10−3 3 10− 10− 10− 3 [M 4 [M 4 [M 3 [M 10− 10− 10− 10− 10 5 5 4 −6 10− 4 10− ρ 10− ρ 6 ρ 10− ρ 10 7 10− 5 −8 7 5 10− 10− 10− 10 9 8 − 6 10− 10− 10− 102 103 104 102 103 104 102 103 104 102 103 104 R [pc] R [pc] R [pc] R [pc]
1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5
1 0.0 1 0.0 1 0.0 1 0.0 β β β β
0.5 0.5 0.5 0.5 − − − −
1.0 1.0 1.0 1.0 − − − − 102 103 104 102 103 104 102 103 104 102 103 104 R [pc] R [pc] R [pc] R [pc]
Fig. 2.7 —: Testing the convergence of MultiNest for GravImage models. The panels show the density profile (top) and velocity anisotropy (bottom) of the 2pop core model with n(r) monotonicity, and without β∗ priors after (3k, 30k, 300k, 868k) iterations (left to right). In both cases, there is no significant change of the position of the 2σ area swept by the models after 300k iterations.
2.6 Appendix
2.6.1 Convergence of the MultiNest Model Ensemble
In this appendix, we check the convergence of MultiNest. First, we consider the range of density profiles encompassed by our 68/95% confidence intervals after 3k, 30k, 300k, and the full 868k iterations. This is shown in Figure 2.7, upper panels. Notice that the error envelope converges already after 300k iterations.
Secondly, we consider the range of the four velocity anisotropy profile parameters. These also converge, as shown in the lower panels of Figure 2.7 that focus on β1∗ (the parameters for β2∗ behave similarly).
2.6.2 Influence of Binning Choices
In this appendix, we explore the effect of our choice of binning. In developing GravImage, we explored three different binning strategies for the tracer stars: constant bin size in linear R space, constant bin size in logarithmic R space, and constant number of tracer stars
46 2.6. Appendix
105 4 1 10 10 101 103 ] ] ] 2
3 3 3 10 100 1 100 10 pc pc pc 100 / 1 / / 1 10− 10− 1 2 10− 10− 2 10 3 [M 10− [M [M −4 2 10− 10− 5 10 3 10− ρ − ρ ρ 10 6 3 −7 4 10− 10− 10− 10 8 −9 10− 101 102 103 101 102 103 101 102 103 R [pc] R [pc] R [pc]
Fig. 2.8 —: Exploring the effect of the number of bins: 8 bins (left); 12 bins (centre, default); and 18 bins (right) for Gaia 1pop cusped model with monotonicity prior on n(r). In each case, we show results for the same number of iterations (208k) for all three, without cuts in χ2. As more bins mean a much higher dimensionality for the parameter space to be sampled, the 18 bin case is not as converged as the models with smaller bin choices. Otherwise, the results agree well between the three cases.
per bin. We show results only for the latter in this paper. This is because the first choice of linearly spaced bins is not practical for many interpolation steps during the integration routines, as these rely on power-law extensions between bin centres. Binning stars in logarithmically spaced bins works, but gives too much weight to higher radii, where our sampling is poor, and thus only a handful of measurements can gain an over-proportional influence on χ2.
We show the influence of the number of bins by running the same model with 8, 12, and 18 bins in Figure 2.8. Each of the three models is plotted with all sampled models (i.e. without any cut in χ2 and thus using all the burn-in models), after 210k iterations. If the bin number decreases, so does the parameter space to sample. With the same number of iterations, we find that the 18 bins model converges much more slowly, as each of the 18 dn(r)/d ln r parameters has to be adjusted. Apart from speed of convergence, however, we find that the number of bins does not significantly affect our results which is encouraging.
47 CHAPTER 2. Mass Modelling Spherical Systems
48 3 Mass Modelling of Fornax
We use a new non-parametric mass modelling tool – GravImage – to measure the dark matter density profile of the Fornax dwarf spheroidal galaxy. GravImage assumes only dynamical equilibrium and spherical symmetry, and nothing on the underlying density profile; in a separate paper, we have extensively tested it on a wide range of mock data, including triaxial mocks (Steger & Read 2015).
We develop a new algorithm for splitting Fornax’s stars into two populations using only their measured Magnesium abundance. We then show that we obtain a consistent mass distribution for Fornax whether modelling a single tracer population; split tracers; and/or including a physically motivated priors on the velocity anisotropy β∗(0) = 0 and β (r) 0. In all cases, we find that a dark matter core is favoured, however the statistical ∗ ≥ significance of this grows when using our β∗ priors. For single population models, the a core of slope < 0.1 is favoured at 95% confidence; for split populations this grows to < 0.5; − − including β (0) = 0 and β (r) 0 it is < 0.1 at 99% confidence. We find also evidence ∗ ∗ ≥ − at 95% confidence for tangential anisotropy at Fornax’s edge, as reported previously in the literature – if using no β∗ prior. The tangential anisotropy cautions against the use of a β (r) 0 prior and could owe to the onset of tides; unmodelled rotation and/or remaining ∗ ≥ foreground contamination. We will explore this further in future work.
Our results show that both population splitting and the choice of priors are critical for breaking the mass-anisotropy degeneracy and obtaining robust constraints on the dark matter distribution. This likely explains the broad range of results in the literature re- ported to date. Independently of the number of populations or priors on the anisotropy, we rule out a central slope of n(r) = d ln ρ /d ln r > 0.5 at 99% confidence. We conclude − DM that the Fornax dwarf spheroidal galaxy almost certainly has a central dark matter core.
Keywords: galaxies: dwarf – galaxies: fundamental parameters – galaxies: kinematics and dynamics – cosmology: dark matter
49 CHAPTER 3. Mass Modelling of Fornax
The work of this chapter will be submitted as a paper to MNRAS.
3.1 Introduction
The cusp-core problem is a now long-standing tension between the standard Λ Cold Dark Mater (ΛCDM) cosmological model and observations. Structure formation simulations that model only the dark matter fluid predict that dark matter halos should have cuspy central density profiles (ρ r 1; Dubinski & Carlberg 1991) in stark contrast with obser- ∝ − vations that favour constant density cores (Flores & Primack 1994; Moore 1994; Oh et al. 2011). This could point to exciting new physics like Self-Interacting Dark Matter (SIDM; Rocha et al. 2013). However, it has been argued that repeated cycles of star formation can transform dark matter cusps to cores (Navarro et al. 1996; Read & Gilmore 2005; Mashchenko et al. 2008a; Pontzen & Governato 2011, 2014), bringing ΛCDM into better agreement with observations. While uncertainties in the physics of star formation remain, if cusp-core transformations do take place, Teyssier et al.(2013) have recently argued that there should be two unavoidable observational consequences: (i) star formation should be bursty with a duty cycle of order the local dynamical time; and (ii) the stars, being a collisionless fluid, should be heated along with the dark matter. Both predictions appear to be supported by recent data (Leaman et al. 2012; Kauffmann 2014).
If all cusps are transformed to cores by star formation, then all of the information about the nature of dark matter would be erased within dwarf galaxies – bad news for probing the nature of dark matter with dwarf galaxies. However, Pe˜narrubia et al.(2012) have recently argued that the very smallest galaxies by stellar mass may well retain their cusps, even in the face of bursty star formation. This is because galaxies at or below the mass of the Fornax dwarf spheroidal would simply not have had enough supernovae to produce the energy required to heat their central dark matter. Similarly, O˜norbe et al. (2015) have argued that galaxies with truncated star formation may also retain a pristine cusp. Both arguments motivate attempts to measure the dark matter distribution in ever smaller galaxies; the smallest being nearby gas-poor dwarf spheroidals.
While the motivation for measuring the dark matter distribution in dwarf spheroidals is clear, there is a key complication. Unlike their more massive and gas rich counterparts, dwarf spheroidals do not host a cold rotating gas disc; their stars cannot be assumed to move on circular orbits. Viewed in projection on the sky, the dwarf spheroidals suffer a strong degeneracy between their mass distribution and their stellar orbit distribution: the mass-anisotropy degeneracy (Wilkinson et al. 2002). Recent claims that this could be broken using multiple tracer populations (split by metallicity and/or abundance) with different half light radii (Battaglia et al. 2008; Walker & Pe˜narrubia 2011) have been questioned (Breddels et al. 2012; Strigari et al. 2014), leading to substantial confusion in the literature. Using different data and different techniques, dark matter cores, cusps and everything in between have now been claimed for the dwarfs (e.g. Battaglia et al. 2008; Walker & Pe˜narrubia 2011; Amorisco & Evans 2012; Agnello & Evans 2012; Jardel &
50 3.2. Method
Gebhardt 2013; Strigari et al. 2014; Richardson & Fairbairn 2014).
In this paper, we use a new non-parametric Jeans modeller – GravImage (Steger & Read 2015) to revisit the question of whether the Fornax dwarf spheroidal (the most luminous of the Milky Way dwarf spheroidals; e.g. McConnachie 2012) has a dark matter cusp or core. Early indirect timing arguments have favoured a core for Fornax – otherwise, its globular clusters would rapidly sink to its centre creating a bright nucleus that is not observed (Goerdt et al. 2006; Cole et al. 2012). Results from Jeans modelling of the stars in Fornax, split into a metal poor and metal rich population, support this analysis (Walker & Pe˜narrubia 2011; Amorisco & Evans 2012). However, both of these studies rely on prior assumptions that may not be valid for Fornax (Strigari et al. 2014). Notably, Schwarzschild models of the dwarf still permit cuspy solutions within quoted uncertainties (Breddels et al. 2012; Jardel & Gebhardt 2013). Our new method assumes only spherical symmetry and equilibrium and has been well tested on a wide array of mock data (Steger & Read 2015). By switching on multiple tracers and priors on the velocity anisotropy one at a time, we aim to shed light on the range of results presented in the literature to date. In doing so, we will obtain a robust measure of the dark matter distribution in Fornax.
This paper is organised as follows. In section 3.2, we briefly review the GravImage method. In section 3.3, we discuss the data used for Fornax in this paper and we present our algorithm for population splitting on Magnesium abundance. In section 3.4, we present our main findings. Finally, in section 3.5 we present our conclusions.
3.2 Method
3.2.1 The GravImage Code
GravImage is described and tested in detail in Steger & Read(2015); here we review the salient features. GravImage solves the spherically symmetric steady state Jeans equation:
d(νσ2) 2β(r) GM(< r) r + νσ2 + ν = 0 (3.1) dr r r r2 with spherical density distribution of tracer stars ν(r); radial velocity dispersion σr; grav- itational constant G; mass enclosed in radius r of M(< r); and velocity anisotropy:
2 σt (r) β(r) = 1 2 (3.2) − σr (r) Projecting along the line of sight, this leads to a projected velocity dispersion of:
51 CHAPTER 3. Mass Modelling of Fornax
2 2 2 ∞ R ν(r)σr (r)r σLOS(R) = 1 β dr (3.3) 2 2 2 Σ(R) R − r ! √r R Z − with the radial velocity dispersion:
1 r β(s) σ2(r) = exp 2 ds r ν(r) − s · Zrmin ! GM(˜r)ν(˜r) r˜ β(s) ∞ exp 2 ds d˜r (3.4) r˜2 s Zr Zrmin ! and stellar surface density profile:
ν(r)r Σ(R) = 2 ∞ dr (3.5) 2 2 0 √r R Z − that can both be compared with data.
GravImage solves equations 3.3 and 3.5 non-parametrically. As discussed in Steger & Read 2015 this really means that we use parameterisations with many more parameters than model constraints such that our system of equations is deliberately under-constrained. We use MultiNest (Feroz et al. 2009) to explore this high dimensional parameter space. The dark matter density is given in bins by setting the density slope:
d ln ρ(r) n(r ) = (3.6) j − d ln r r=rj for N radial bins j: 1 j N , and interpolated linearly in between bin radii bin ≤ ≤ bin rmin,j < r < rmax,j. The density follows from this as:
ln r ρ (r) = ρ exp n(s)ds , dm 1/2 · − " Zln r1/2 # with the dark matter density at half-light radius ρ(r1/2) = ρ1/2, We prescribe three buffer bins n(r ) for j N +1,N +2,N +3 to either side, outside the range where data j ∈ { bin bin bin } is given, to enable sensible extrapolations towards low and high radii, and two additional slopes n0 < 3, n > 3 for the asymptotic density slopes towards r = 0 and r = , which ∞ ∞ are reached at half the smallest radius and r = 5rmax. The total mass distribution is ∞ then a sum of the dark matter density and the baryonic mass density, which in this case is on the order of the stellar mass density.
The stars are treated in the following manner: In addition to the dark matter density ρDM, a baryonic contribution ρbary is accounted for in our mass model:
52 3.2. Method
ρtot = ρDM + ρbary (3.7) where ρ = ΥL(r); L(r) is the Abel-deprojected light profile (see 3.3); and Υ is the bary § mass to light ratio which is allowed to vary in the range [0.8, 3]. As the kinematic data we have is rather sparse – not each star in the dSph has its line-of-sight velocity measured – and misses other contributions from gas, we add a photometrically defined baryonic density. The errors on this density are small compared to the errors on the velocity dispersion, so we only use the de-projected 3D photometric density profile for the subtraction, without sampling over its errors, as is done for the kinematic tracer population profiles. The tracer density is determined via:
1 dΣ(R) dR ν(r) = ∞ (3.8) 2 2 −π r dR √R r Z − from the measured surface density profile Σ(R), and is internally represented via a derived quantity:
ln(ν(r)) ln(ν(r + ∆r)) n (r) = − (3.9) ν ln(r + ∆r) ln(r) − approximating the first derivative of the tracer density of a population (and analogously for an optional second population). The nν parameters are based on the measured surface density profile Σ(R) and use these values as starting points, with each nν,model(ri) sampled inside a range n (r ) ε , where ε = 0.1 max(n (r)). A symmetrised velocity anisotropy ν i ± ν ν defined via:
2 2 σr σt β β∗ = − = [ 1, 1] (3.10) σ2 + σ2 2 β ∈ − r t − is modelled by the following function:
a0 a β∗(r) = − ∞ + a (3.11) 1 + κ exp(α ln(r/rs)) ∞ a0 a κ = − ∞ 1 (3.12) β∗(rs) a − − ∞ with the inner and outer asymptotic values for β∗(r 0, ) given by a0, a [ 1, 1]; → ∞ ∞ ∈ − rs the scale radius of the a0 a transition with a β∗ at that point of β∗(rs), and the → ∞ speed of the transition measured by α.
3.2.2 Priors in Use
We use the priors outlined in (Steger & Read 2015), with numerical values in Table 3.1:
53 CHAPTER 3. Mass Modelling of Fornax
Quantity type characteristics Nbin fixed 12 nβ fixed 4 dn(r)/d ln r Gaussian mean 0, width σ = 1.5 n(r 0) flat [0, 2.999] → ln ρ(r ) flat centre 1, width 2.5 dex 1/2 − M/L flat [0.8, 3]
Tab. 3.1 —: Priors for the GravImage method used on Fornax.
1. β∗(0) = 0: Pontzen et al. 2015 claim that a central isotropy of β∗(0) = 0 is favourable for analyses assuming spherical symmetry. It corresponds to an attractor of a maxi- mally stable system, motivated by the observation that tightly bound particles reach a spherically ergodic limit for the mean angular momentum j const, faster for h i ∼ particles inside a region where local dynamical times are short compared to dynam- ical time of the host system. Thus ergodicity for tightly bound tracers is to be expected. This translates to β(0) = 0. All their numerical experiments reach this limiting case, and this gives our motivation to include a β∗(0) = 0 prior.
2. β (r) 0, and thus β(r) 0 is motivated by the Osipkov-Merritt anisotropy pro- ∗ ≥ ≥ files, which describe a velocity anisotropy rising with radius. The prior in use will exclude any tangentially anisotropic models. Both of these priors are motivated by the observation that stellar systems undergoing gravitational collapse in n-body simulations end up with isotropic cores and radial anisotropy in the outskirts (van Albada 1982). After combining the prior with the β∗(0) = 0 prior in earlier runs, we detected only weak improvements on the asymptotic anisotropy parameter a ∞ for the strongly radial biased mocks.
3. Monotonic n(r): n(r) = d ln ρ(r)/d ln r is calculated by integrating its derivative − dn(r)/d ln r – which is given at discrete bin centres, and drawn from a Gaussian prior of width σ – with the additional constraints that dn/d ln r 0. The density profile is ≥ thus represented by its (positive) second derivatives. This enforces a monotonically increasing n(r). Looking only at a subset of all possible parametrizations of n(r) translates into faster convergence, and helps further to circumvent strong oscillations from numerical effects.
3.3 Data
The kinematic data used for our study is based on Walker et al. 2009b (see Walker et al. 2009c for a detailed description of the acquisition). It contains 2, 500 measurements of ∼ the kinematics of Fornax member stars. Additional measurements for red giant stars from Mateo et al. 1991 and Walker et al. 2006 are included as well.
54 3.3. Data
3.3.1 Photometry
For the stellar density profile of Fornax, we use photometric data from de Boer et al. (2012). It encompasses data for 270,000 individual stars in B, V and I bands. We are only interested in the position of each star, and thus do not use the information on the different photometric bands further. We have to make the simplifying assumption of a constant stellar mass for each of the 270,000 stars, which in general is not true, but approximately so when they are binned into 12 bins, giving us 22500 stars per bin – as is the case in our work. We use these same photometric data to define the centre of Fornax.
3.3.2 Splitting Populations
Battaglia et al.(2008) were the first to suggest that splitting the stars in a dwarf galaxy into metal rich and metal poor subpopulations could act to break the mass-anisotropy degeneracy. If the two populations have different spatial distributions then each will measure well an enclosed mass at a different radius. Walker & Pe˜narrubia(2011) used this idea to argue that the Fornax dwarf has a dark matter core. However, the method used by Walker & Pe˜narrubia(2011) is not suitable for our non-parametric GravImage method. They used the Magnesium index; positions; and line-of-sight velocities together, assuming that Fornax could be decomposed into two populations, each well-fit by a Plummer sphere. Our method makes no a-priori assumption about the form of the tracer density profile and so we must population split using only non-kinematic data, i.e. the metallicity or abundance of stars.
We split the tracer stars into two populations based on their Magnesium index, using a separate Monte Carlo Markov Chain before the main MultiNest run. The iron mea- surements in Walker & Pe˜narrubia 2011 are not reliable enough and a sizeable fraction of stars have no metallicity measurements, thus we do not use that information. The overall Magnesium index distribution is assumed to be well represented by a sum of two Gaus- sians with means µ1,2 and widths σ1,2. The parameter space for µ1,2, σ1,2, and fraction p of stars in population 1 is sampled to yield the most likely representation. After storing these parameters, each tracer star is assigned a population based on the likelihood of its Mg index belonging to said population. An example splitting result applied on Fornax Mg measurements is shown in Figure 3.1.
We restrict the method to Mg indices only, as the iron measurements in Walker & Pe˜narrubia 2011 are not reliable enough, and a sizeable fraction has no metallicity mea- surements. Walker & Pe˜narrubia(2011) assume a Plummer-like profile for both popu- lations, which must be dropped in our non-parametric profile. We additionally let drop the velocity information, and concentrate on non-kinematic properties of the stars only. Foreground stars are accounted for by weighting the probabilities of membership in each population with the overall dSph probability of membership.
55 CHAPTER 3. Mass Modelling of Fornax
4 3.0 180 10− 160 2.5 140 2.0 120 ) 100 5 1.5 R 10− ( pdf pdf 80 ν 1.0 60 40 0.5 20 6 0.0 0 10− 0.20.0 0.2 0.4 0.6 0.8 1.0 1.2 20 40 60 80 100 120 140 160 103 − Mg ∆R R
Fig. 3.1 —: Left: Reconstruction of two populations from Mg. The retrieved best-fitting Gaussians are shown as white lines in front of the overall distribution of Mg indices, and the reconstructed Mg distribution, after drawing the stars based on their Mg index, is depicted as red line. This is close to the green line, which shows the smoothed distribution of Mg measurements if the Mg witdths errors are Gaussian. Center: Distribution of differences in 2D half-light radii for repeated samplings of stellar tracer particles based on their Mg index. Right: Tracer density profiles for populations 1 and 2 (red, blue) after assigning stars randomly according to their Mg index, for a model that gives ∆R = 100pc. Vertical lines indicate the respective half-light radii. The profiles are clearly Non-Plummer.
We first assume two populations, and from the pdf of the Mg indices, we calculate the tracer density profiles for both populations. Each star contributes to both populations, with a fraction given by its probability of belonging to the chemical population i. If we then see no significant difference between the half-light radii distribution of all models with two populations, we conclude that population splitting for that particular galaxy will not give us a significant information gain in GravImage . If we on the other hand get a distinct peak in the pdf of the half-light radii we know that there are two – or possibly more – distinct populations we can use for further analysis. Optimally, we assign each star randomly in proportion to fi to population i for each step in the MultiNest procedure. For Fornax, we see that the half-light radii nicely cluster around 80 and 100 pc, a clear sign that indeed two or more chemically distinct populations reside at different positions in the gravitational potential of Fornax.
3.4 Results
3.4.1 Single Component Models without Anisotropy Priors
Figure 3.2 shows the results for a single population GravImage run without any β∗ 2 2 2 priors. We plot only profiles that satisfy χ < χmax, where χmax is the first minimum in the histogram of all χ2, or the first bin in the histogram that has no models assigned (in 2 this figure, log10χmax = 0.5). This removes models from the MultiNest initial burn-in
56 3.4. Results
4.0 1.0 1 3.5 10− ] 3.0 3 0.5 2.5 pc 2 / 10− )
2.0 ∗ r 1
( 0.0 β
n 1.5 [M 3 10− 1.0 0.5 ρ 0.5 − 10 4 0.0 − 1.0 0.5 − 102 103 104 − 102 103 104 102 103 104 R [pc] R [pc] R [pc]
101 12
] 25 10 s] 2 /
pc 20 8 / 0 10 [km 15 6 1 [M , 10 4 frequency 1 LOS
Σ 1 10− σ 5 2 0 0 102 103 104 102 103 104 2 1 0 1 2 3 4 5 6 − − 2 R [pc] R [pc] log10 χ
Fig. 3.2 —: Top row: Reconstructed dark matter density ρ, dark matter density slope n(r), and velocity anisotropy β∗ of Fornax (red shows median, shaded areas show the 68, 95, and 99 percentiles) for 3150 tracer particles, after 3.5 million iterations, with monotonic n(r), and without any priors on β∗. The vertical lines give the projected half-light radius (blue for 2D quantities), and the half-light radius for the median model (green for 3D quantities). Bottom row: Surface density Σ1; velocity dispersion σLOS with data as a blue band covering errors determined with bootstrapping velocity errors; χ2 of all 3.5 million models.
phase. For each plot, we stack the corresponding profiles, and evaluate the median, and the 68, 95, and 99 percent envelope for each bin independently. This gives us a good representation of the full distribution of densities (or other quantities) at a given radius. Beware, however: a given median profile might not correspond to any sampled profile (though it is approached in the case of convergence). We show the median profile in red, and subsequently lighter shaded areas indicate 68%, 95%, 99% of all models at a given radius. Vertical lines indicate the half-light radius – green for the derived 3D half-light radius – estimated from the tracer density fall-off. In the lower row of Figure 3.2, we show the refitted profiles of surface density Σ1 and velocity dispersion σ1, and the distribution of χ2. The data are well fit by the models. Over-fitting has not occurred, as can be seen by the range of the models, which is about the same size as the data errors. The three extension bins to smaller and higher radii are needed for continuation of the profiles to 0 and and were allowed to take 5 times bigger sampling space than the data errors were ∞ imposing for the radii in between. Consequently, the range of profiles outside the data range is much broader than within.
57 CHAPTER 3. Mass Modelling of Fornax
Firstly, notice that the dark matter density ρdm(r) shows a pinch-point at a radius slightly below the half-light radius r1/2. This agrees well with earlier works that noted that, owing to the Virial theorem, the strongest constraints are at r (Walker et al. ∼ 1/2 2009; Wolf et al. 2010; Agnello & Evans 2012).
The corresponding n(r) = d ln ρ /d ln r (central panel) indicates that the data − dm prefers a central logarithmic slope of 0.1, with a maximum value of 0.3 at 99% confi- dence. Towards larger radii, the slope increases and at 500 pc – where the pinch-point ∼ in ρ lies – we reach a slope of n(500 pc) 1. A further increase outwards becomes less dm ' certain as the data points thin out and the confidence intervals broaden correspondingly. Interestingly, we find a clear deviation from a power-law n(r), illustrating the value of our non-parametric approach.
The velocity anisotropy in this model ensemble was left completely free. Remarkably, an isotropic distribution is preferred in the central parts. Deviations from that start at the same r = 500 pc, and tend to prefer tangential models. Interestingly, similar tangential bias at large radii have been reported by Lokas 2009. Such tangential bias – that is favoured at 95% confidence – could be indicative of tides acting at Fornax’s edge (Read et al. 2006); unmodelled rotation; or residual foreground contamination. We will explore this further in future work.
3.4.2 Single Component Models with Central Isotropy Prior
A second run with β (0) = 0 and β (r) 0 priors is shown in Figure 3.3, second row. The ∗ ∗ ≥ first constraint on β was shown to yield better estimates for mock dwarf galaxies with ∗ Osipkov-Merritt type anisotropies, and β (r) further reduces the parameter space for all ∗ ≥ models by 1 dimension, and thus injects information helping to sort the mass-anisotropy degeneracy. The pinch point in ρdm has shifted to slightly higher radii; while n(r) remains qualitatively the same, with tighter constraints on the central slope. The anisotropy profile still prefers isotropy in the centre, out to the half-light radius, where it starts to broaden to sample the full freedom of the extension bins. The fact that β1∗ is consistently at the lowest allowed value indicates that the assumption of β 0 might not be a valid one for ∗ ≥ Fornax.
58 3.4. Results
Single population; monotonic n(r); no β∗ priors:
4.0 1.0 1 3.5 10− ] 3.0 3 0.5 2.5 pc 2 / 10− )
2.0 ∗ r 1
( 0.0 β
n 1.5 [M 3 10− 1.0 0.5 ρ 0.5 − 10 4 0.0 − 1.0 0.5 − 102 103 104 − 102 103 104 102 103 104 R [pc] R [pc] R [pc] Single population; monotonic n(r); β (0) = 0 and β (r) 0 priors: ∗ ∗ ≥
4.0 1.0 1 10− 3.5
] 3.0 3 0.5 2.5 pc / )
2.0 ∗ r 1
2
10 ( 0.0 − β
n 1.5 [M 1.0 0.5 ρ 0.5 − 3 10− 0.0 1.0 0.5 − 102 103 − 102 103 102 103 R [pc] R [pc] R [pc] Split populations; monotonic n(r); no β∗ priors:
4.0 1.0 1 10− 3.5
] 3.0 3 0.5 2.5 pc / )
2.0 ∗ r 1
2 ( 0.0 10− β
n 1.5 [M 1.0 0.5 ρ 0.5 − 10 3 0.0 − 1.0 0.5 − 102 103 − 102 103 102 103 R [pc] R [pc] R [pc] Fig. 3.3 —: Different models for Fornax exploring split/single populations and the use of a β (0) = 0 and β (r) 0 priors. ∗ ∗ ≥
59 CHAPTER 3. Mass Modelling of Fornax
3.4.3 Two Populations split by Mg without Anisotropy Priors
In this section, we explore modelling Fornax with two populations split on Mg as described in 3.3. The results are shown in Figure 3.3, third row. The main difference to the corresponding 1pop run is the fact that there are fewer stellar tracers in any given bin, and thus the errors are broader for Σi and σi. Additionally, almost twice as many calculations are necessary, and the dimensionality of the parameter space is almost twice as large as in the 1pop case. All these contribute to a broader range in ρ , n(r), and β profiles dm ∗ after the same runtime. In return, we gain more information on the mass-anisotropy degeneracy – any change in n(ri) to a wrong value is penalised twice in σLOS,1 and σLOS,2. It is encouraging that the n(r) profile is consistent with the previous runs.
3.5 Conclusions
We developed a new algorithm for splitting Fornax’s stars into two populations using only their measured Magnesium abundance. This is required for our non-parametric GravIm- age method as previous methods in the literature relied on assumed functional forms for the tracer density profile.
We obtain a consistent mass distribution for Fornax whether modelling a single tracer population; split tracers; and/or including a physically motivated prior on the central velocity anisotropy β (0) = 0 and β (r) 0. In all cases, we find that a dark matter core ∗ ∗ ≥ is strongly favoured, mostly so for the runs with β∗ priors. For single population models, the a cored profile with slope < 0.1 is favoured at 95% confidence; for split populations − this grows to < 0.5 at 95%; including β (0) = 0 and β (r) 0 it is < 0.1 at 99% − ∗ ∗ ≥ − confidence. We find also evidence at 95% confidence for tangential anisotropy at Fornax’s edge, as reported previously in the literature – if using no β∗(0) = 0 prior. The tangential anisotropy could cautions againts the use of a β (r) 0 prior and could owe to the onset of ∗ ≥ tides; unmodelled rotation and/or remaining foreground contamination. We will explore this further in future work.
We conclude that the Fornax dwarf spheroidal galaxy almost certainly has a central dark matter core.
60 4 Mass Modelling of other Dwarf Spheroidals
I show first applications of the new mass modelling tool GravImage to the four dwarf galaxies Sculptor, Sextans, Carina, and Draco. All dwarfs are consistent with cusped profiles, with a slight preference to smaller slopes for the light dwarfs Carina and Sextans, n(r 0) 0.5. No dwarf is as shallow as Fornax, the most luminous MW dwarf galaxy. → → However, I caution that many of the models are likely not yet converged, while results for split population models for these dwarfs are still running. If these early results hold up to further convergence tests and are consistent with the split population models (as is the case for Fornax), then this will constitute exciting evidence in favour of the prevailing ΛCDM paradigm.
Keywords: galaxies: dwarf – galaxies: fundamental parameters – galaxies: kinematics and dynamics – cosmology: dark matter
4.1 Introduction
In the previous two chapters, we have set the framework for mass modelling spherical systems with GravImage . In this chapter, we are applying the method to other dwarf galaxies: Sculptor; Carina; Sextans; and Draco. The data for the first three is taken from the literature; the data for Draco includes many new radial velocity measurements to be published shortly by Prof. Matthew Walker (see 4.3 for details). I briefly review previous measurements of the dark matter distribution in these dwarfs, next.
For Sculptor, Battaglia et al. 2008 introduce population splitting, and find a core with low significance, and confirm that later, together with a radial anisotropy profile
61 CHAPTER 4. Mass Modelling of other Dwarf Spheroidals
(Battaglia et al. 2012). Amorisco & Evans 2012 use population splitting and a distribution function approach, to find a core at high significance. Breddels et al. 2012 use orbit based Schwarzschild modelling of Sculptor, and find that the scale radius for an assumed NFW profile lies in a range more than 1.5 dex wide, and cannot confine the central slope very well either – the central slope they find has non-vanishing probabilities for values in the range α [ 1, 1]. Richardson & Fairbairn 2014 with virial parameters on one population ∈ − find a cusp for Sculptor. Strigari et al. 2014 on the other hand use population splitting and a distribution function approach with separable f(E,J) = g(J)h(E), finding a cusp.
Draco – if isotropic – was measured to have a core (Wolf & Bullock 2012) on one hand, but on the other hand to not be matched by an extended harmonic core (Kleyna et al. 2002), based on slightly tangentially biased models. A 6-parameter density profile for Draco by Charbonnier et al. 2011 finds values for the central density slope which varies between 0 and 1.5, with most values around γ = 1.
The central density profile in Sextans was matched to the Aquarius simulations (Stri- gari et al. 2010), and also found to be compatible with a dark matter cusp. This is in contradiction with Kleyna et al.(2004) who find a very small central velocity dispersion and deduce a kinematic core in Sextans. Battaglia et al.(2011) find, based on extended LOS velocities, that Sextans is compatible both with a large core, and a low concentration cusp.
Carina is a relatively young satellite galaxy of the Milky Way with a complex star formation history (Hurley-Keller et al. 1998). We can thus hope to find multiple pop- ulations. It has been shown that Carina is tidally disrupted by the Milky Way (Kuhn et al. 1996), which is confirmed by Lokas 2009, finding a slightly tangential bias on the velocity anisotropy for Carina. Mu˜nozet al. 2006 detect rotation, which could explain the tangential bias measured, but this finding is challenged by Lokas 2009. Wolf & Bullock (2012) find that for isotropic models, Carina prefers a core. Strigari et al.(2010) on the other hand match Carina to a subhalo in Aquarius simulations, finding that cuspy models can give a good fit to its kinematic data.
It is clear from the above brief review of the literature that for all of these dwarf spheroidals there have been claims of dark matter cores, cusps, and everything in between. The situation is made especially confusing by the use of different data, different priors, and different techniques by all of the above authors. We hope to shed some light on this by applying a single well tested non-parametric techniquge – GravImage – to kinematic and photometric data for these dwarfs.
62 4.2. Baryonic Density Profiles
4.2 Baryonic Density Profiles
Our Jeans-based method finds constraints on the overall mass density profile:
ρtot = ρDM + ρbary (4.1) where ρDM is the dark matter density profile, and ρbary is the baryonic density profile. For the baryonic density profile ρbary, we assume mass follows light and take the tracer densities with a variable mass-to-light ratio Υ: ρbary = ΥL(r), where L(r) is the deprojected light profile. Υ is added as a parameter to MultiNest and sampled in the range [0.8, 3], assuming that the stars make up most of the baryonic matter.
4.3 Data
We use data from Walker et al. 2007, who took several observations with the Magellan/Clay telescope and the Michigan/MIKE Fiber System (MMFS), yielding high-resolution spectra of around 6000 individual red giant stars. The spectral region was centred around the Mg-triplet, covering 5140-5180 A, with a resolution of R 20000. Additional stellar ∼ measurements were added from Mateo et al. 1993 and Mu˜nozet al. 2006 for Carina. Additional stars from (Westfall et al. 2006) are included in the Walker data release as well, and Hargreaves et al. 1994; Kleyna et al. 2004 contributed additional data-sets for Sextans.
A second data-set from Walker(2015) is used with 1565 stars for the dwarf spheroidal galaxy Draco. Non-kinematic data for Draco includes metallicity, effective temperature, and surface gravity, which we will later use for population splitting. A final note about velocity measurement errors is needed to understand the errors on the quoted velocity dispersions: Velocity errors were determined from multiple measurements of the same stars, and found to be around 2 km/s. The velocity errors are assumed to be Gaussian with that width, and sampled repeatedly to determine our errors on the velocity dispersion.
4.4 Results
The following results are shown at an early stage in convergence. Results for the most massive dwarf galaxy of the sample, Sculptor, are shown in figure 4.1. GravImage is still very broadly sampling the parameter space, and I caution that it has likely not yet converged. For this reason, the results presented here should be seen as work in progress rather than the final definitive answer.
After Sculptor, we turn to the second most luminous dwarf galaxy in our sample, Draco.
63 CHAPTER 4. Mass Modelling of other Dwarf Spheroidals
104 4.0 1.0 103 3.5 2 ] 10 3.0 3 101 0.5 0 2.5 pc 10 / 1 )
2.0 ∗ r 10 1 −
( 0.0 2 β
10− n 1.5
[M 3 10− 4 1.0 10− 0.5 ρ 5 0.5 − 10− 6 10− 0.0 7 1.0 10− 0.5 − 101 102 103 − 101 102 103 101 102 103 R [pc] R [pc] R [pc]
Fig. 4.1 —: Reconstructed density, density slope, and velocity anisotropy of Sculptor with 1pop, monotonic n(r), and β (0) = 0, β (r) 0 priors. The red line shows the median ∗ ∗ ≥ of all models in each bin, shaded areas show the 68, 95, and 99 percentiles for 1541 tracer particles, after 2840 accepted models. The vertical lines give the half-light radius for the median model (green for 3D quantities).
Application of our method to Draco yields the profiles shown in figures 4.2. Draco seems to prefer a slightly radially anisotropic profile for β∗ in the center, with exclusion of any strongly radial profiles β 0.6 with 99% certainty. ∗ ≥
4.0 2 1.0 10 3.5 101 ] 3.0 3 100 0.5 1 2.5 pc 10− / 2 )
2.0 ∗ r 10− 1
( 0.0 10 3 β − n 1.5
[M 4 10− 5 1.0 10− 0.5 ρ 6 0.5 − 10− 10 7 0.0 − 1.0 0.5 − 102 103 104 − 102 103 104 102 103 104 R [pc] R [pc] R [pc]
Fig. 4.2 —: Models for 1565 tracer particles of Draco after 800 accepted models. Mono- tonic n(r); β (0) = 0 and β (r) 0 priors in use. ∗ ∗ ≥
Next, we apply the method to Sextans, figure 4.3. Without β∗ priors, our method prefers a very tangential anisotropy model for 1 population, and consistently so: β = 1 ∗ − up to the half-light radius. It has not reached a high number of iterations, though. With 139k iterations as compared to the several million iterations for the other dwarfs it is probable that MultiNest is restricting the models to too small a parameter space.
The final dwarf galaxy studied is Carina, figure 4.4. Without β∗ priors, we have trouble converging on a special β∗ profile, and sample a very broad range in n(r), even after 3.5 million iterations.
64 4.5. Conclusions
4.0 103 1.0 102 3.5 ] 1 3.0 3 10 0.5 0 2.5 pc 10 / 1 )
2.0 ∗ r 10− 1
( 0.0 10 2 β − n 1.5
[M 3 10− 1.0 10 4 0.5 ρ − 0.5 5 − 10− 6 0.0 10− 1.0 0.5 − 102 103 − 102 103 102 103 R [pc] R [pc] R [pc]
Fig. 4.3 —: Same as figure 4.2, for 947 tracer particles of Sextans, after 580 accepted models. Monotonic n(r); β (0) = 0, β (r) 0 priors in use. ∗ ∗ ≥
104 5 1.0 103 4 ] 102 3 1 0.5 10 3 pc 0 / 10 ) ∗ r 1
1 10 ( 2 0.0 − β
2 n
[M 10− 3 1 10− 0.5
ρ 4 10− − 5 0 10− 10 6 1.0 − 1 − 102 103 104 − 102 103 104 102 103 104 R [pc] R [pc] R [pc]
Fig. 4.4 —: Same as figure 4.2, for 1982 tracer particles of Carina, after 3860 accepted models. Monotonic n(r); β (0) = 0 and β (r) 0 priors in use. ∗ ∗ ≥
4.5 Conclusions
The new non-parametric method samples the profiles of the overall density bin-wise, and was shown to reconstruct the density of diverse mock data. The runs for the four dwarf galaxies under study have not converged with the same speed as seen for the mock galaxies, probably owing to the smaller number of tracer stars.
All dwarfs are cuspier than Fornax; only Sextans shows evidence for a central core, but significantly smaller than that in Fornax. If these results hold up to further scrutiny, this could be strong evidence in favour of the prevailing ΛCDM paradigm.
65 CHAPTER 4. Mass Modelling of other Dwarf Spheroidals
4.6 Further Work
I will continue running the models to test for convergence. After that, I will run models with population splitting; with and without the monotonicity prior on n(r); and with and without β∗ priors to check for consistency, similarly to the work already completed for Fornax.
For Draco, we miss Magnesium triplet measurements. Rather, we have measurements of temperature, mean metallicity, and surface gravity. Surface gravity is found to be bimodal, see figure 4.5, and lends itself well for population splitting. Our method is prepared to use the surface gravity to split the populations for Draco, with the same setup of a separate MCMC to split it before the main MultiNest run as in chapter3: The best-fitting split of the log g indices in two Gaussians is used to assign each star randomly in proportion to the fraction of the respective Gaussian curves to population 1 or 2. This is done in a one-time initial step. Repeated runs will not show the same assignments, though, so we have to check that our results are independent of the exact assignment of single stars by verifying that the results stay the same over several runs.
Fig. 4.5 —: Histogram of surface gravity measurements log g for all stars in our Draco data-set.
Armed with the above, we will be able to determine whether these dwarfs really do show varying central dark matter density profiles and with what statistical significance.
66 4.7. Appendix
Quantity type characteristics Nbin fixed 12 dn(r)/d ln r Gaussian mean 0, width σ = 0.5 n(r) bound min(n(r)) = 0, max(n(r)) = 5 ln ρ(r ) flat centre 1, width 1.5 dex 1/2 − Υ flat [0.8, 3]
Tab. 4.1 —: Priors for the GravImage method used on Sculptor, Draco, Sextans, and Carina.
4.7 Appendix
4.7.1 Priors in Use
We use the priors outlined in Steger & Read 2015, with numerical values in Table 4.1:
1. β∗(r) is given in the form:
2 2 σr σt β β∗ = − = [ 1, 1] (4.2) σ2 + σ2 2 β ∈ − r t − with the following function:
a0 a β∗(r) = − ∞ + a (4.3) 1 + κ exp(α ln(r/rs)) ∞ a0 a κ = − ∞ 1 (4.4) β∗(rs) a − − ∞ 2. n(r) = d ln ρ(r)/d ln r is calculated by integrating its derivative dn(r)/d ln r – − which is given at discrete bin centres, and drawn from a Gaussian prior of width σ – with the additional constraint that n(r) 0. The density profile: ≥
ln r ρ(r) = ρ exp n(s)ds , 1/2 · − " Zln r1/2 # is thus represented by its second derivatives. This enforces a monotonically decreas- ing ρ(r), and further circumvents strong oscillations from numerical effects.
67 CHAPTER 4. Mass Modelling of other Dwarf Spheroidals
68 5 Simulations of Dwarf Galaxies
After the previous chapter on the mass modelling of dwarf spheroidal galaxies, we are < 8 interested in modelling the smallest galaxies with Mvir 10 M ab initio. What processes contribute to what extent to the formation of the diverse∼ cusps/cores we found for real dwarf galaxies? The exact simulation setup is a direct consequence of the system under study, and the processes and parameters that need to be modelled. The methodology of simulations in general and of rezoom simulations in special is given in section 5.1. I set the stage in particular by stating the aim of the simulations, and motivating the need for non-equilibrium chemistry in section 5.1.1. I analyse the dark matter only run in section 5.2. After identifying halos as prospective dwarf galaxies in section 5.3 I analyse the dark matter density profile for both the dark matter and hydrodynamics run.
5.1 Simulation Methods
Cosmological simulations in general aim to reproduce the observed dark matter, star, and gas distribution of the current epoch and earlier on statistically by following the evolution of structure buildup and star formation from the primordial power spectrum. The ini- tial conditions are constrained by the temperature fluctuations in the cosmic microwave background. In their simplest form, they are set up assuming gaussianity. Higher or- der corrections incorporate f = 0, where f is a measure for the non-Gaussianity of NL 6 NL Bardeen’s potential, Φ = ϕ + f (ϕ2 ϕ2 ), where ϕ denotes an auxiliary Gaussian NL − h i field. These initial conditions are power spectra for the modes of displacement in density and velocity, modeled on a grid of N 3 particles in a box. The box is usually assumed to co-move with the expanding Universe, and to repeat itself in each cartesian coordinate, effectively forming a hypertorus (Bagla 2005).
69 CHAPTER 5. Simulations of Dwarf Galaxies
The simulation is then started and begins to calculate iteratively
the gravitational forces between particles • all hydrodynamical quantities, either for hydrodynamic particles or on a mesh • the resulting changes in velocity, density, temperature, and any other quantity of • interest for the next step
Collisional codes follow each of the particles directly, while collisionless simulations follow a smaller number of particles than are actually expected to be in the system. Collisional codes are thus mainly used to follow systems in which close encounters play a significant role and need to be resolved, i.e. when the relaxation time is of order of the simulation timestep. Collisionless codes on the other hand are used in collisionless simulations where one tries to model a continuum distribution by sampling only a small number of particles. Dark matter as a pressureless fluid is treated this way. One approach to solving Poisson’s equation (the method used in this thesis) is to estimate the density at the positions of a grid and calculating the forces on a particle from the background mesh. This method, Particle-Mesh (PM), has to assign particle masses to a grid, which can be accomplished via assignment of all mass to the nearest grid point (NGP), division of the mass between cells depending on the position of the particle (cloud in cell, CIC), or with triangular shaped clouds (TSC) assuring continuous changes in mass and first derivatives as the particle moves through the grid.
In astrophysics, typically one of two methods is used to deal with hydrodynamic quan- tities: Smoothed Particle Hydrodynamics (SPH) (see e.g Monaghan 1992; Springel 2005) approaches the fluids by particles representing volume elements, and calculates the hy- drodynamical forces from a discrete form of the Euler equations. Grid codes in contrast follow the gas dynamics in small patches and update the hydrodynamic variables in the neighboring cells according to the flux through boundaries and volume changes. Meshes can take a diversity of geometries, depending on the physical system under considera- tion. Adaptive mesh refinement (AMR) (Kravtsov et al. 1997; Teyssier 2010) as a special grid-based code class uses an advanced technology to follow the fluid more accurately: Typically, the mesh is refined in the vicinity of high overdensity, though other refinement schemes are possible. This allows for the correct capture of accretion, shock waves, and other small features, while keeping the mesh at low resolution where the density is small, which increases efficiency considerably. The high dynamic range reached by this method gives the reason that AMR with density-based refinement is used for this work.
5.1.1 Non-Equilibrium Chemistry
Primordial density fluctuations with associated gas collapse under the influence of gravity. Gas in the region where the gravitational attraction from the overdensity is higher than
70 5.1. Simulation Methods
Fig. 5.1 —: Cooling rate for collisions of several chemical species of our reaction network 4 4 3 with H2, going below 10 K at an assumed density of n = 10− cm− . Figure taken from Glover & Abel 2008.
from other overdensities falls in and gets compressed and thus heated. If the heat can be dissipated through dissociation, ionisation, or excitation and radiation, the gas collapses further, on the free-fall timescale of:
3π t = (5.1) 16Gρ r 0 The galaxies we are interested in have halos with virial temperature T 104 K, roughly vir ∼ the ionisation temperature of hydrogen. Any gas cloud below 108 M thus requires some cooling process to efficiently dissipate energy for further collapse and star formation. One of the most powerful cooling mechanisms in that temperature range – exciting metals and radiating the energy through metal lines – is not sufficient at high redshifts, because the mean metallicity of primordial gas is too low. Most energy can be released from primordial gas by exciting H2,(Peebles & Dicke 1968). H2 cooling is proportional to the density of H2, which is very rare right after recombination and formed so slowly that it is not in collisional ionisation equilibrium.
We are thus forced to model the chemical evolution in gas if we want to study globular
71 CHAPTER 5. Simulations of Dwarf Galaxies clusters and dwarf galaxies at high redshift. It is also necessary to take into account non- Local Thermodynamic Equilibrium (LTE) effects. Local thermodynamic equilibrium is valid if there are no macroscopic flows of temperature, mass, radiation, and/or chemical abundance. Several of these conditions are not fulfilled in our case: cooling changes the temperature; gas is ionised by ultraviolet radiation; and the surrounding gas is enriched with metals by each supernova. Non-equilibrium reactions start to be important when the H2 cooling time-scale becomes similar to or larger than dynamical time-scales, or time- scales of the expansion of the Universe, 1/H (Abel et al. 1997). That is the case in regions of gas of temperature around and below 104 K and at high redshift. Once considering non-LTE effects, we must explicitly model the chemical reaction network between all key elements/molecules that can contribute to the cooling, or that produce or destroy key coolants. The full network studied here is taken from Glover & Abel 2008 and is described in Table 5.1. The cooling rate for several important chemical species is plotted in figure 5.1 from Glover & Abel 2008, and we see that the cooling rate below 104 K does not vanish for several species in our network from Table 5.1.
Solving the interaction between all these species amounts to the solution of a large set of coupled partial differential equations. I solve these in Ramses as follows. We compute a cooling function for each reaction, which in turn is based on the dissociation and ionisation rates for each species, and abundances calculated iteratively with the temperature of the gas as free parameter. As soon as the cooling rates have converged, we use them to calculate the cooling affected on each cell.
72 5.1. Simulation Methods 2 3 1,7 1,9 1,10 1,11 1,8 1,3 1,15 1,15 3 4 5 71 . 2 8 . − 2 1 ) / 2 1 T / 1 hν 1) 23 ( ) 1,8 1) − 14 − /T − − 10 x x 10 × = ( × = ( 21050 85 6353 α . 5 . . − 0 α − : 2 T exp( 10 eV − 10 7 416 eV); . . − 6700 K: 1 598eV); 10 . 17 ) 5 10 × (54 ν (13 × 5 T < . hν/ hν > hν/ T = ln( = , for 13 = x ) )) + 1 − x ; 2 x , for ) 10 /T )); 17 17 )); ) 1,13 /T 10000 K: 3 56200)) − − × 11 . 4 3 152 T/ 10 10 . π/α . ( π/α 3 2 eV 587eV) 1,16 /x 2 10 T < × × . 940000 0512032)))) T (7500 − 2 . + 4 ] for the non-equilibrium chemistry network under − / 4 − log 2 . (1 3 2 − 48 ) 1,3 (24 )) 1,2 1,6 . 9 x × / ) 1,14 ) 2 2 1 1) 1 2472 exp( ) 1,2 / / . ), for − /T exp( T 1 1 2 5 5 /T − − /T / 6657 hν/ − . T T 1 11 3 exp( − 5 0 /T . x hν 73693+ − ( = T . − (1 hν ( 18 (1 / 16 15100 x (1 + 0 10 18 / ) − 32200 (4060 (1 + (1 + / − ) (148 − − ); ) / / × x − − 14000 10 (1 + /α ) ) 10 05 /α ) )(1 + 0 10 2 / . eV − . ) ) 3 × α 35 56200) 9 × . /T /T exp( )) 1 α ( × − ) 1,12 /T 2 0 /T exp( exp( 1 4 ( 7 1 )) 1,2 . 4 6431+ . . . /T x ) 1,3 4 . . − 1 T/ 7 T . 0 − exp( . 1 6 ( 14 15 ) − /T 8 0 486 6 66 − T 463 . /T − − . . − 16 10 T 4 7898 16200) 1,8 0 T . tan 4 /T − − 0 4 ) 1,3 − 10 10 4 10 470000 ( − − − − 4 tan 1: 3 T/ 6 eV: 6 157890 285335 631515 x 10 10 (1 + . 43000 8750 × × − − × 7 eV: 1 /T T 2 − (1 + . − − − 05 10 . − > 8 − − . × × 29 2 3 16 0 . . 3 33+ . 2 22 17 − exp( . 0 17 0 . . 5 × 3 102000 T − ) . exp( 81 2 T 8750 . exp( x exp( exp( exp( − / 5 2 + 6 012 exp( exp( exp(4 . 1 . − /T / 2 2 2 1 exp(4 2 2 2 4 66 1 / / / eV T . 928 / / 17 74 eV) − 1 1 1 4 . . − (186603 . ( T 2 K: 5 2 K: 3 1 3 T 2 0 2 exp( . . x − < hν < ( / T T T T (1 x / − − 10 (0 exp( (300 1 < hν < T / x T /s] and cross sections (22-28) in [cm 10 13 8 11 11 12 10 18 18 T T 7+ − . 3 T 11 10 18 9 6 8 18 20 − − − − − − − − − /T 6700 K: 5 7291 7291 10000 K: 1 10000 K: 4 10000 K: 10 − − − − − 10 − − − hν/ 6 12 10 10 10 10 10 10 10 10 10 5 eV 42 eV − − . . 10 10 10 10 10 × 10 10 10 = × × × × × × × × × 10 10 1654771 T > T > T < T > T > T < x × × × × × × × × − 4 36 4 3 93 68 7 7 067 38 4 89 68 38 3 42 3 58 × × ...... for 3 1 1 1 2 8 2 5 5 6 4 for 4 7 5 1 for for for 16 for for 15 e e hν e e e e hν hν e e e e e − e + e hν + hν + HI for + + HII 6 + + 2HI for + 2 2 + 2 2 2 + 2 HeIII + + 2 + HI 5 2HI 5 H HeII + HeII + H H HI + HI + HII 10 H H 2HI + HeI + HeIII + 2 H − 2 H 2HI see reference 3HI 1 HII + HeII + 2 HI + 2HI 1 HI + 2 → HI + H 2HI + → → H HII + 2 H → → → → → → → → → → → → → → → e → → → → → e e hν 2 → → − → → → e hν e e e hν hν e H e hν hν e e − hν + HI + + HII + + HI + HII + + + + + H + + HII + + H + + 2 + 2 − + 2 + 2 − 2 2 2 2 2 − − − 2 2 H 7. HI + HII 6. HeIII + 8. H 9. HI + 5. HeII + 4. HII + 1. HI + 2. HeI + 3. HeII + 10. HI11. + H H 12. H 13. H 14. H 15. H 16. H 18. H 22. 17. H 19. H 23. HeI + 20. H 21. H 24. HeII + 25. H 27. H 26. H 28. H Tab. 5.1 —: Reactionstudy. rates References (1-20) on in the( Abel [cm et right: al. 1.1997 ); 6.11. ( Oh ( Black ( Dalgarno &1978 ); & Haiman 7. Lepp 2002 );( Haardt1987 ); ( Karpas & 2. 12. et Madau al. ( Hirasawa ( Cen ) 1996 1979 );1969 );1992 ); 8. 13. 3. ( Galli ( Dove & & ( Shapiro Palla Mandy &1998 );1986 ); Kang 9. 14.1987 ); ( de ( Lepp 4. Jong &1972 ); Shull ( Tegmark 10.1983 ); et ( Nakashima 15. al. et ( Osterbrock 1997 ); al. 1974 );1987 ); 5. 16.
73 CHAPTER 5. Simulations of Dwarf Galaxies
To test the workings of the chemical network, I simulate a simple cosmological box of size 100 Mpc/h from redshift z = 104 over the recombination era at z 103 down to ∼ redshift z = 102 and follow the abundances, see figure 5.2. As expected, the abundance of charged particles drops rapidly, after recombination at z = 1500, and then hydrogen ions form.
Fig. 5.2 —: Abundances of the nine chemical species in the cosmological simulation NEC- CHECK over the epoch of recombination. Plotted are the number densities divided by the sum of number densities of H and H+. The evolution progresses from the right hand of the plot to the left hand: High abundances of ionised helium, hydrogen and electrons 4 3 at z = 10 , which recombine before z = 10 and start to build H2.
5.1.2 Star Formation and Feedback
The Boley et al. 2009 physics was designed to be relevant at 0.5 pc resolution in the gas, ∼ which is the resolution I employ here. This leads to stars forming with a mass of 200 M .
However, Boley et al. 2009 have a much lower dark matter particle resolution ( 1000 M ) ∼ than the stars. I will start by analysing their simulations before running a suite of higher resolution simulations with a dark matter particle mass of 200 M similar to the stellar ∼ mass.
The same physics as in Boley et al. 2009 was used for the high resolution hydrody- namical simulation. This includes ram pressure, star formation, SN feedback, and AGN.
74 5.1. Simulation Methods
Ramses has been used in a augmented version to follow non-equilibrium physics as well, + namely the gas of e, HI, HII, HeI, HeII, HeIII, H−,H2, and H2 (Abel et al. 1997).
Star formation has to be modelled as subgrid physics, still, as our resolution limit of on order 200 M is still above the mass of single stars. We employ a star formation efficiency 5 3 of = 0.1 and a threshold density of 10 mH / cm . This is high in comparison to what lower resolution simulations use, and motivated by the fact that stars are only allowed to form in the highest resolution elements. See Boley et al. 2009 for the exact popIII star formation recipe.
The feedback by supernovae is split into thermal energy, kinetic energy, and ionisation of the surrounding gas. A time delay between debris particle and dumping of energy is computed based on the mass, and thus the distance covered between explosion and onset of energy output on the whole cell. The surrounding cells of a supernova II are identified to be used for the feedback. Thermal energy of 1051 erg and the kinematic energy of the star is dumped as the next step, and metallicities are updated consistently. A UV background according to Boley et al. 2009 is assumed. No further influence of radiative transfer is considered.
The simulation starts with background metallicity of Zave = 0.0 at z = 1000, and follows the evolution of the 9 element abundances self-consistently. Re-ionisation models normally take into account the fact that the strong UV radiation of the first generations of stars quickly re-ionised the surrounding gas while exploding as a supernova at some fixed redshift z, and thus enriched the interstellar medium with a metallicity floor at that redshift. We sample the metallicity build-up from first principles, and thus set no such metallicity floor.
5.1.3 Halo Finding
A simulation snapshot by itself shows only properties of individual dark matter and star particles as well as gas particles, or – in the case of an AMR simulation – a mesh for the hydrodynamic constituents. We are interested in bound substructures, which have to be found first. To do so, a halo finder needs to be invoked. There are many options available:
FOF The friend-of-friends algorithm as e.g. described in Press & Davis 1982 starts with a particle and searches for its nearest neighbor, out to a predefined radius. After doing this iteratively, all particles connected by such a chain are considered to lie within the same halo. This procedure works well for isolated halos only; two nearby halos connected by a filament could be detected as one structure only. Recent findings (More et al. 2011) show that the overdensity δ = (ρ ρ¯)/ρ¯ inside the virial radius − is typically of order 80 instead of the generally expected 180, though, and systems of halos connected by filements can be determined as one halo instead of two. This givus us reason to look for more advanced halo finders.
75 CHAPTER 5. Simulations of Dwarf Galaxies
HOP Hop from the Ramses toolkit starts off by computing the density around each particle with an adaptive kernel, then hops to the neighbor particle at highest density, assigning all particles ending at the same point to the same structure. Breaking up large structures into local density maxima is overcome via merging of two groups if the density of the boundary layer between them lies above a given threshold. Still two close structures end up in the same halo. SOD The spherical overdensity algorithm as introduced by Lacey & Cole 1994 grows a sphere around particles at density maxima and stops if the mean density falls below a threshold. This procedure is repeated with the remaining particles until no structure with a minimally required number of particles is found anymore. It does not handle correctly the cases of mergers, yielding a halo position in the middle of both merging halos. AHF The AHFstep algorithm of the simulation code Amiga (Knollmann & Knebe 2009; Knebe & Doumler 2010; Knebe et al. 2011, 2013) showed superior capabilities with mock halos and subhalos, and can be configured to detect substructures and star clusters, too. It is the tool of choice for our problem. In its early stages it was designed for Sph simulations using the Gadget file format, which imposes an addi- tional conversion of any simulation snapshots by Ramses to an SPH analog before halo finding.
An additional iterative procedure is needed after the first halo centres have been iden- tified, namely to exclude all kinematically unbound particles: if the kinetic and internal energy for dark matter or star particles – or converted SPH gas particles – exceed the potential energy, the particle is removed, the potential recalculated and further unbound particles excluded. The prospective halo centres can be defined in several ways: by the position of the centre of mass, the centre of the potential or the density maximum. Vi- sual best agreement with particle density projection is found for the position of maximal density. This approach will be used throughout the chapter.
5.1.4 Bound Structures
We identify not only the positions of halos, but want to know their size and mass, too. The virial radius rvir is determined to be the distance from the densest point of a halo to a point at which the mean enclosed density ρ(r) falls below:
ρ(r) ∆(z)ρ (5.2) ≤ crit,0 ∆(z) = 18π2 + 82x 39x2 (5.3) − x = Ω(z) 1 (5.4) − 3 Ω(z) = Ω0(1 + z )/E(z) (5.5) 3 2 E(z) = Ω0(1 + z ) + ΩR(1 + z) + ΩΛ (5.6) p
76 5.1. Simulation Methods
according to Bryan & Norman 1998. All particles at r > rvir are prone to influences from the encompassing large scale environment, especially tides. After the tidal radius rt, these influences are bigger than the gravitational forces from the bound structure, and thus particles start to strip off the host halo. The above-mentioned iterative removal of unbound particles helps to restrict further analysis to the inner parts of the halo, and keep it unaffected by spurious interlopers and unbound particles at the edges. We then define a further quantity, the virial mass, to be the mass within the virial radius.
5.1.5 Radial Profile
Not only the overall size and mass of the halo are of interest. Most information is available within the halo. The radial density profile ρ(r) is determined from logarithmically spaced shells – to keep the density of shells high where we have most data – around the previously detected centres. The shells lie in between ri and ri+1, where:
i r /N r r = r 10 ∗ min − min (5.7) i max ·
with rmin, rmax the minimal and maximal radius considered, and N the number of bins in between the two. Vera-Ciro et al.(2012) find that Einasto profiles give a better fit to dark matter density than NFW in Aquarius simulations, which take into account a semi-analytic baryon model on top of dark matter. A tour d’horizon for other profiles can be found in the appendix 6.1.2. We will concentrate on the central parts and determine the asymptotic slope towards r 0, and the extent of any interesting features like cores. →
5.1.6 Relaxation Radius
All quantities determined from particles inside a characteristic radius rrelax are expected to be widely independent of the properties of the surrounding halo, as any memory was erased from numerous encounters. The relaxation radius is calculated according to definitions and methods described in Read 2009. Consider N particles enclosed in a sphere of radius r around the centre of a system with a mass of M(< r), bmax and bmin are maximal and minimal impact parameter, which are set to virial radius and gravitational force softening length in a first step. The relaxation radius can be found by setting the relaxation time trelax – which depends on the orbit time torbit equal to the simulation time tsim:
77 CHAPTER 5. Simulations of Dwarf Galaxies
N t = n t t (5.8) relax cross cross ∼ 16π ln Λ orb
bmax torb = 2πbmax (5.9) sGM(< r) b ln Λ = ln max (5.10) bmin 3/2 Nbmax trelax(r) = (5.11) 8 GM(< r) ln(bmax/bmin) tsim = trelaxp (rrelax) (5.12)
The last equation is solved for rrelax by a basic root finding algorithm. The relaxation radius can be defined in several ways for simulated systems, with best matches to observa- tions when bmax = r, allowing the Coulomb logarithm ln Λ to change slowly with radius. Any material in spherical shells above the current radius r will not have a net gravita- tional effect on matter inside the shell, and may be thus neglected. This definition of rrelax will constitute our definition of relaxation radius, and finally set the limiting radius below which our simulations results should not be trusted.
5.2 Simulation Suite
Five simulations were run for this project: 1) A precursor non-equilibrium chemistry run by A. Boley, and 2) a purely dark matter run with the same initial conditions, 3) a cos- mological box at low resolution over the epoch of recombination with non-equilibrium chemistry, 4) a dark matter only simulation for the rezoomed initial conditions (section 5.2.1), and 5) a full hydrodynamics run for the same initial conditions. The main simu- lation 6) with non-equilibrium chemistry at full resolution was started, but was hindered by insurmountable memory restrictions. 12 compute nodes with each 20 CPUs and an assigned RAM of 256GB per node totalling a whopping 3TB of RAM were needed for the hydrodynamic run at full resolution, and we would have needed at least a factor (9 + 5)/6 more memory to hold the additional 9 variables of chemistry abundances per cell. The total number of fat nodes on the supercomputer share could not accomodate that number. Lowering the accuracy of the force calculations to use double precision instead of quad precision – which would be needed for any cell refinement level above 19 – did not help to alleviate the problem. See Table 5.2 for size, resolution, physics and feedback parameters, and redshift reached for each simulation.
78 5.2. Simulation Suite
ID status size resolution physics zend pSIMDMonly finished 0.701 Mpc/h 1800 M DM only 10
pSIM finished 0.701 Mpc/h 1800 M hydro,NEC 10
NECCHECK finished 100 Mpc/h 3.2 1010 M hydro,NEC 10 · DMonly finished 1 Mpc/h 133 M DM only 10
Hydro finished 1 Mpc/h 133 M Hydro 12
NEC aborted 1 Mpc/h 133 M NEC –
NEC+Strom planned 1 Mpc/h 133 M NEC+Stromgren spheres –
NEC+Delayed planned 50 Mpc/h 133 M NEC+delayed cooling –
Tab. 5.2 —: Properties of the simulations under study. ε = 5 pc/h is the equivalent force softening length on the highest refinement level of dark matter.
5.2.1 Initial Conditions
Assumptions
Boundary conditions are needed for any cosmological simulation. They are generally given by assuming a power spectrum motivated by the cosmic microwave background, and the 3 fact that the Universe or a patch thereof is represented as a box of size Lbox, where Lbox is typically of order Mpc or above. It is repeated in each direction, giving the topology of a hypertorus, meaning that whenever a particle is moving out of the box at one end, it will come in on the other side again. Forces are calculated over this boundary in the same manner, s.t. any distance along an axis x between two particles at x1, x2 is determined via:
∆x = min( x x , x + L x ) (5.13) | 1 − 2| 1 box − 2 There is a minimal size of the box needed to follow the evolution of a representative fraction of the Universe. With decreasing redshift, smaller and smaller modes of the initial power spectrum enter the expanding box.
Following assumptions are made for the simulations under study:
The content of the Universe is distributed as predicted by ΛCDM, • initial conditions follow the Eisenstein & Hu(1998) power spectrum, • the Universe is representable by repeated boxes, thus it shows large scale homogene- • ity, with no scales of importance larger than the box size, all hydrodynamical interaction happening in gas and stars ensembles with masses • below 150 M is covered well by sub-grid modules for star-formation and feedback
as described in Boley et al. 2009,
79 CHAPTER 5. Simulations of Dwarf Galaxies
gas rich dwarf galaxies forming at high redshift can be compared with dwarf galaxies • in the Local Group at the present epoch. Ram pressure and tidal stripping of gas as it falls into the halo of another galaxy is thus missed. We argue that most stars in dwarf galaxies are very old and thus formed early, before an infall happens. Gas left over by star formation in isolated galaxies is assumed to have negligible mass and will thus not alter the overall mass profile by much if stripped. Future work should model the infall of established dwarf galaxies into their hosts, though.
Initial conditions were generated by using a strongly modified version of Quick – a parallelised initial condition generator by Doug Potter and Markus Wetzstein– which pre- pares refined initial conditions starting from a high-resolution initial condition map and a reference dark matter only run of reduced resolution. Particles ending up in the final region of interest are tracked back to their initial positions and are replaced consistently with a cloud of higher resolution particles. This puts computational power into the pre- viously determined interesting regions. If the whole box were sampled with the highest resolution of (10243), the computation would be around 7 times more expensive.
Cosmological Parameters
We assume a cosmological model in this work based on ΛCDM and the WMAP 7-year cosmological parameters (Peacock 1999; Weinberg 2008; Komatsu et al. 2011). The high resolution hydrodynamical simulation was run with a ratio of dark energy density to critical density ΩΛ = 0.728, ΩM = 0.272 for matter density, Ωb = 0.045 for baryon density, and 1 1 1 1 Hubble parameter H = 70.2 km s− Mpc− = 100h km s− Mpc− , h = 0.719.
We use a higher value for the primordial fluctuation normalisation parameter at 8 kpc of σ8 = 1.0 as compared to the canonical value of σ8 = 0.807, in order to sample an over-dense region with our small box, similar to what was done in Boley et al. 2009. For the dark matter only simulations, the baryonic contribution to the energy budget of the Universe was put into ΩM – assuming that baryons would behave like dark matter, while keeping the other parameters fixed, such that ΩΛ = 0.742, ΩM = 0.258, and Ωb = 0.
Power Spectrum
The CDM power spectrum is computed using the analytic fit by Eisenstein & Hu 1998. For the 5123 sampling of the 1 Mpc/h box, this gives figure 5.3.
For the precursor run, initial conditions were generated with Cosmics (Bertschinger 1995), for a box of 1 Mpc/h box-length, at z = 1000, with a low resolution boundary layer of 1283 equivalent particles and an inner high resolution region of 2563 particles. Moreover, 5123 was prepared for the dark matter only run at highest resolution. Higher resolution maps (up to 20483) were prepared for follow-up simulations. For the full resolution runs on
80 5.2. Simulation Suite
Fig. 5.3 —: Power spectrum on small scales. The green curve is a fit to an adiabatic cold dark matter model from Peacock 1999, equation 15.82(1). The Bardeen Bond Kaiser Szalay (BBKS) curve uses an approximation to the transfer function to calculate the current-day power spectrum, and starts oscillating when the baryon content gets too large. We show the power spectrum of a the 5123 sampling of our box, where the comoving wave number k is plotted up to the Nyquist frequency of 256. We are offset from the green curve at low k, or the large scales, owing to the finite size of our box, but otherwise capture the power on small scales well. the other hand, the initial conditions are generated using grafic++ by D. Potter, a parallel version of Edmund Bertschinger’s grafic (Bertschinger 2001). We run a low-resolution dark matter only precursor simulation to centre the initial conditions on the centre of mass of the most massive central halo, and generate masks for refinements.
Simulation Resolution & Box Size
We describe here the required resolutions for the simulation suite. We expect to form a 108 M halo at redshift z = 10, before re-ionisation. For simulations to lower redshifts, ≤ we need the simulation box to be approximately the size of the scales that start to collapse at today, 50 Mpc. This is roughly the scales on which the Universe looks homogeneous.
Our aim is to resolve star formation from single molecular clouds. They reach a typical
81 CHAPTER 5. Simulations of Dwarf Galaxies temperature of T = 300K, where they hit the polytopes, and sub-clumps form if they surpass the density ρJ over the Jeans scale-length λJ , which we want to sample with four ∆x resolution elements, corresponding to the Jeans mass MJ :
3 3 a ρJ MJ = ρJ λJ = , (5.14) 3/2 1/2 G ρJ
15kBT λJ atf f = a/ Gρ = ; (5.15) ≈ s4πGµρ p a2 ρ = 104/cm3, (5.16) J G (4∆x)2 ≈ · (5.17) where a 1.5km/s is the sound speed in the gas of temperature T 300K. This gives an ≈ ≈ estimated ∆x = λJ /4 0.3pc, and a resolution mass of mdm,res = 6 M for dark matter ≈ 3 by requiring mdm,res = ρJ ∆x , and m? = 2 M for stars by ρ? ρJ /3. ≈ This should give us a minimal number of particles of 104 for globular clusters, which are the smallest systems we are interested in. Read et al.(2006) show that one should expect some dwarf galaxy precursors of mass 108 M at that time in a box with comoving side length of 1 Mpc/h. Metal line cooling allows to form stars in these halos, and the overall mass keeps the gas inside, against the pressure from stellar winds and supernovae. With a value for the over-density at scales of 8 Mpc, σ8 = 1.0, slightly higher than the concordance value of σ8 = 0.807 I ensure that I map a region of higher density in the Universe (Boley et al. 2009), and mimic a region that will most likely end up inside the halo of a 1012 M galaxy at z = 0.
The resolution goal for this simulation suite is set by imposing the condition that each supernova in the real Universe gets sampled by one supernova in the simulation. We look at supernova of type II, into which all stars with a mass exceeding 8 M will turn eventually. The fraction of stars above 8 M is calculated assuming a universal power law
IMF (Kroupa 2002) with Salpeter slope 7/3 in the range 0.01 M < M < 20 M , we get − that a fraction:
20 M 20 M 7/3 8 M N(M)dM 8 M M − dM fN = = = 0.002 (5.18) 20 M 20 M 7/3 R0.01 M N(M)dM R0.01 M M − dM will explode as supernova,R injecting an averageR energy of 1051erg in the surrounding medium. They are rare, but very massive. The fraction of stellar mass in SN is then:
20 M 20 M 7/3 8 M M N(M)dM 8 M M M − dM fM = · = · = 0.072 (5.19) 20 M 20 M 7/3 R 0.01 M M N(M) R 0.01 M M M − · · R R
82 5.2. Simulation Suite
The average supernova progenitor mass is:
20 M 8 M MN(M)dM M = = 12 M (5.20) h i 20 M R 8 M N(M)dM
R The typical stellar mass needed to yield one supernova gives our resolution for star particles as:
M = 12 M /0.072 = 167 M (5.21) ∗
The dark matter mass resolution should be comparable to this value, such that analysis of the dark matter content on small scales is not lacking from poor statistics. This directly prescribes the scales for the highest resolution initial conditions, Mres,DM, given that a star 3 forms out of a cell with mass Mcell on a grid of dimension Ngrid,max:
M N = ρ (z)L3 (5.22) cell · grid,max crit box 3H2 = L3 (5.23) 8πG box 3L3 = box H2[Ω (1 + z)3 + Ω ] (5.24) 8πG 0 M Λ
Lbox is the box size, ρcrit is the critical density of the concordance cosmology (ΩM , ΩΛ, Ωk = 10 0). We chose to take Lbox = 1 Mpc/h, and a power of 2 for Ngrid,max = 2 , which gives a slightly smaller value for Mcell = Mres,DM = 133 M .
The force resolution (Plummer’s equivalent length) in the highest resolution grid is 4 pc/h at z = 10, which sets a restriction on the determination of densities in the inner regions for the smallest halos. It is indicated as a vertical line in the graphs of radial profiles.
Using quick, the initial conditions are generated in the following way: A first set of initial over-density and velocity fields is sampled from a Gaussian. With this, a dark matter only simulation is performed. The most massive halo is identified, all belonging particles tagged and traced back to the initial condition files. Filling in holes, and accounting for the overall motion of the halo during its generation leads to a mask, in which spatial resolution is doubled. The process is repeated to get a second refined region. The initial conditions are refined within these masks, honouring the lower modes across the boundaries. A final dark matter run ensures that less than 1 percent of the mass in the final halo at r < rvir comes from lower resolution particles.
83 CHAPTER 5. Simulations of Dwarf Galaxies
Fig. 5.4 —: Initial conditions in highest resolution, used for DMonly, Hydro NEC simu- lations. Shown is a colour-coded overdensity for the full simulation box, with red showing high overdensity, in the central slab of 1024 along the z-axis. The central overdensity will collapse to form the most massive galaxy at z = 12.
Re-zoom Technique
The rezoom technique is applied two times to put resolution into the regions of interest, while keeping the tidal forces and inflows from larger structures consistent with the cosmic web. The first step is to find the halo(s) of interest. I concentrate on the three most massive ones, which do not include any substructure of more than half the virial mass inside five times their virial radius. Thus, no upcoming merger of two halos will enlarge the refinement mask, possibly including holes in it. As a next step, I centre up the white noise file on the given halo position, such that the zoomed region lies in the centre of the simulation. Figure 5.4 shows a cut through the recentered overdensities at the highest resolution. The centering eliminates the need for workarounds for interactions across the borders of the simulation box. With a repeated run of grafic I then generate centred initial conditions up and down to the desired resolution, and determine the refined resolution part. These initial conditions are stitched together such that the highest resolution of 10243 cells lies in the central region of 250 kpc/h, encompassed by an intermediate resolution (5123 cells) box of 500 kpc/h, and finally filling the rest of the 1000 kpc/h box with the low resolution initial conditions of 2563 cells. This way, each initial condition files are needed at 2563 cells each.
84 5.2. Simulation Suite
The tree of initial conditions together with the initial mask is only a first guess at the required initial conditions. The initial mask can have holes in it, which will end up very close or inside the collapsed structure in the end, adding low resolution particles to the final halo. Another effect is that as the halo collapses, it may also move in physical coordinates and attract particles from the surrounding lower resolution regions flowing by. Both effects need to be counteracted. A first dark matter only simulation with the setup determines which low resolution particles end up in the virial radius. These particles are then tracked back to the initial conditions, and replaced with higher resolution particles, enlarging the mask accordingly. The mask can still be lumpy at that stage, and needs an iterative smoothing step to ensure cohesion. We resimulate the refined mask shown in Figure 5.5 with dark matter only to correct for possible asymmetries, and to verify a low contamination with low resolution dark matter particles.
5.2.2 Temporal Coverage
The precursor simulation pSIM and pSIMDMonly were performed with 270 outputs spaced non-uniformly across the redshift range z = 1000..10. The simulations DMonly, Hydro , and NEC run from z = 1000 down to z = 10, with outputs on 90 equally spaced intervals in expansion number aexp = 1/(1 + z).
It is likely that at least some of the progenitors of today’s dwarf spheroidal galaxies are formed before re-ionisation (Ricotti et al. 2004; Ricotti & Gnedin 2005; Kravtsov et al. 2004). Boley et al.(2009) show that in a typical volume of 1 Mpc /h at z 10 there ∼ should be a few dwarf spheroidal galaxies present. I stop the simulation at that specific high redshift, where – for dwarfs like Draco with truncated star formation – most of the stellar populations should have formed. The influence of the re-ionisation, which completes at a later step, z 8.5 on the baryonic component should be considered in future work. ∼
85 CHAPTER 5. Simulations of Dwarf Galaxies
Fig. 5.5 —: Final refinement mask used for DMonly, Hydro , NEC in projections along x, y, z axes. No holes are left, and the main elongation follows the dark matter overdensity that emerges at z = 10.
86 5.3. Results
5.3 Results
The projected dark matter distribution for the pSIM simulation at z = 10 can be seen in figure 5.6. Clearly visible are the central halo(s) under study, and a filament to the right as part of the cosmic web. Note the sparse dark matter sampling – individual dark matter particles are visible up to 100 kpc to the centre. Note further that no artificial overdensity in the form of a cross is visible, giving us proof that the scales of order 1 Mpc have not yet started to collapse at z = 10.
Fig. 5.6 —: Dark Matter distribution at the final redshift z = 10 for simulation pSIM . The plotted region spans 1 Mpc. We show a slab of width 100 kpc around the median z coordinate, projected along z.
We verify that the rezoom technique worked in Figure 5.7: Higher resolution dark matter particles in the center show up as slightly distorted rectangle of smaller dots. All particles inside the central halo were replaced with high-resolution particles.
87 CHAPTER 5. Simulations of Dwarf Galaxies
Fig. 5.7 —: Re-zoom of the central region, dark matter distribution at z = 12 for the refined simulation DMonly. Shown is the projection along the z-axis of a cube of sidelength 250 kpc, a quarter of the region shown in Figure 5.6.
88 5.3. Results
Fig. 5.8 —: Visualisation of the simulation box in the pSIM run, with dark matter density in the left panel, and gas density in the right box. The visible box spans 1 Mpc/h. The red box shows the refined region in the gas. We restrict the adaptive mesh refinement to regions inside the refinement mask, see later.
5.3.1 Halo Finding
Figure 5.8 gives a visualisation of dark matter and gas distribution. The low resolution dark matter particles are visible as small density enhancements. Gas is smoother, with refinements of the Amr grid constrained to the high resolution box in the centre. For comparative reasons both resolutions were used for dark matter only simulations, and convergence was investigated by comparing the halos found in both simulations. A mini- mum particle number of 300 for the halo finder was enough to remove spurious halos from the low-resolution run that do not feature in the higher resolution simulation.
It is important to center the halos correctly, as a little offset implicates an artificial over-density at the radius of the offset, thus turning a cusp into a core numerically. Figure 5.9 shows positions and virial radii detected with two of the above mentioned methods, spherical over-density (Sod, pink circles) and Ahf (red crosses), plotted on the dark matter density map. Virial radii detected by this method can extend out to the nearest neighbouring structure. Ahf detects many more structures, which do correlate with dark matter over-density on large scales. On smaller scales, it detects over-densities from gas and stars as well, finding substructures like star clusters.
89 CHAPTER 5. Simulations of Dwarf Galaxies
Fig. 5.9 —: Positions of the halos detected by Ahf (red crosses) and SOD (pink circles) on top of dark matter density projection along z for the pSIM simulation. The radii of the circles correspond to the values of rvir as detected by Sod. We see that Ahf detects smaller overdensities and substructure much more reliably than Sod. The visible box spans 30 kpc/h.
The method mentioned so far detect a local density maximum as the centre of a halo. This does not coincide with the classical definition of system centre as motivated from an observational point of view, where e.g. star clusters are exempt. Several of them show up in the central kiloparsec of the most massive halo, causing a density maximum centre to fail. For this reason, I find the centre instead using a shrinking sphere algorithm: It starts from the centre of mass of all dark matter particles in a sphere of radius rvir, then rescales the radius to r = f r , 0 < f < 1, and considers only particles inside the i+1 · i smaller sphere to find the centre of mass in the next step, until the positions converge to (∆xi i+1 ∆xi 1 i)/∆xi i 1 < ε. Similar restrictions on the convergence of centre → − − → → − of mass velocity were not considered. The two parameters f and ε are not restricted by a physical argument and thus have been chosen such that the most massive halos converged before more than 90% of the particles were excluded. Erroneous shifts towards 1 substructures in the outer parts of the halo are corrected by starting with r0 = e− rvir.
90 5.3. Results
Figure 5.10 and 5.11 give an overview of halos pSIM 1 and pSIM 5 as the most massive and least massive halo considered in pSIM . Projections along the three axes x, y, z of dark matter density, star density and gas density are shown. The most massive halo resides near the centre of the simulated box and exhibits a complex structure of sub-halos and star clusters. The least massive halo has no near neighbours and is nearly spherical. On the small scales considered by this simulation, only few substructures are detected. Satellites are not observed in excessive numbers. They do not interfere with analysis, since their distance is on the order of the virial radius of the larger structure.
Fig. 5.10 —: Projections of dark matter density (top row), star density (middle row) and gas density (bottom row) along x (left column), y (middle column), and z (right column) for halo pSIM 1 for the precursor run pSIM . Colours are inverted to enhance contrast. Blue dots give the centres of stellar overdensities. We see a lumpy gas distribution which has not settled yet.
91 CHAPTER 5. Simulations of Dwarf Galaxies
Fig. 5.11 —: Same as Figure 5.10, but for the least massive halo considered, pSIM 5. The stars have had time to concentrate centrally, and only show one stellar halo instead of the full distribution of subhalos in pSIM 1. The gas is still broken into several components.
92 5.3. Results
Fig. 5.12 —: Close up view of raw particle positions (dark matter in blue, stars in green) in pSIM 1, with centres as proposed by Ahf before (red circle) and after (black circle) the shrinking sphere algorithm on dark matter particles only has completed. The position after the shrinking sphere algorithm corresponds much better with the stellar centre. If taken as centre, the density distribution around the red point would give an artificial core of at least 500 pc. It is important to use the refined centre.
Baryons though, with a mean mass fraction of 15% are clustered strongly and boost the local density considerably. Centring techniques searching for most bound particle, highest density or centre of mass from shrinking sphere get locked up in one of these positions. Centring on dark matter only is less sensitive to small scale noise, as it neglects contributions of clustering baryons. This is the conservative way to define the centre of a dwarf, as e.g. star clusters in Fornax represent peaks above a continuously rising background density and are generally not considered to sit directly at the centre. See figure 5.12 for the offset found for the most massive halo.
We are interested in the inner density profile of dwarf spheroidals, therefore only halos with virial masses above 107 M /h and no detected host halo are considered further.
Halos with prospective centres outside the highest resolution region of the AMR box are excluded as well. The main properties for pSIM and pSIMDMonly halos are listed in Table 5.3. The corresponding quantities for the Hydro run are listed in Table 5.4. The halo identification routine found only the central halo again in the pSIMDMonly halo, and then continued to classify massive halos outside the central refinement region. The Hydro run was performed with newly created initial conditions, and thus shows other halo positions. Interestingly, we find pretty consistent gas fractions of around 0.16, also when looking at the Hydro run. The stellar mass fraction is consistent at low values of 1 percent for pSIM , and only shows values > 0 for the most massive halo in Hydro.
93 CHAPTER 5. Simulations of Dwarf Galaxies
That is a consequence of the missing NEC cooling, keeping the gas too hot to collapse in the shallower gravitational wells of smaller halos. Our analysis of any quantity concerning effects of stars will thus be restricted to pSIM , which correctly models cooling, albeit at low resolution.
Sim. ID Mvir[M /h] Rvir[pc/h] x y z fDM fg fs
pSIM 1 4.070 107 8.52 103 0.509 0.506 0.494 0.85 0.13 0.02 · · pSIM 2 1.522 107 6.14 103 0.521 0.504 0.480 0.80 0.19 0.01 · · pSIM 3 1.410 107 5.98 103 0.398 0.632 0.641 0.83 0.16 0.01 · · pSIM 4 1.336 107 5.88 103 0.388 0.616 0.637 0.84 0.15 0.01 · · pSIM 5 1.108 107 5.52 103 0.433 0.395 0.441 0.91 0.08 0.01 · · pSIMDMonly 1 5.579 107 9.46 103 0.505 0.505 0.498 1.0 0.0 0.0 · ·
Tab. 5.3 —: Properties of the selected halos for pSIM and pSIMDMonly simulations. Mvir and Rvir are the virial mass and radius of the halos, fi gives the fraction of mass in component i.
Sim. ID Mvir[M /h] Rvir[pc/h] x y z fDM fg fs
Hydro 1 3.641 107 8.25 103 0.499 0.498 0.496 0.83 0.10 0.07 · · Hydro 2 1.304 107 5.86 103 0.478 0.521 0.535 0.83 0.17 0 · · Hydro 3 1.242 107 5.76 103 0.508 0.493 0.468 0.82 0.18 0 · · Hydro 4 7.600 106 4.89 103 0.511 0.493 0.450 0.83 0.17 0 · ·
Tab. 5.4 —: Properties of the selected halos for the Hydro simulation.
94 5.3. Results
Fig. 5.13 —: Dark matter density profiles for the most massive halo in both pSIMDMonly (green) and pSIM (blue). The vertical lines indicate the force softening lengths in the colours of the respective run; the red line represents the relaxation radius for the pSIM run. Errors are Poissonian. We see that the run with hydrodynamics steepens the overall slope of the dark matter profile with respect to the dark matter only run, and no core outside the relaxation radius shows up. Inside the relaxation radius, we see a possible cusp-core transformation. However, this is on a scale where we cannot trust our numerical results.
5.3.2 Dark Matter Density Profile
Dark matter density profiles are plotted in figure 5.13 for the most massive halo in each simulation at z = 10. The vertical lines indicate the force softening length as given by the highest refinement levels for dark matter in both runs. The relaxation radius is indicated as well. All values below the resolution limit are dominated by numerical effects and do not show information about physical processes. The dark matter only halo follows an NFW profile, with a turn to α = 2 at 500pc/h. The force softening length in the hydrodynamic − run is larger owing to the lower central dark matter density. Additionally, its relaxation radius at 28pc/h requires that the innermost 30 particles are excluded from analysis. This implies that the apparent flattening of the profile to a core cannot be interpreted as being purely physical. On scales that we do resolve well, the dark matter density rises more steeply in the run with baryons than in the pure dark matter simulation.
An explanation for the change of the density slope in the dark matter halos is given by
95 CHAPTER 5. Simulations of Dwarf Galaxies adiabatic contraction as described in Blumenthal et al. 1986: Baryons lose energy through cooling and sink to much lower radii. If angular momentum is conserved, we have the constraint:
2 2 j1 = GM1(r1) = GM2(r2) = j2 (5.25)
for a mass element M1 at r1 falling down to r2. This process perturbs the underlying dark matter potential such that dark matter particles are pulled inwards as well. Figure 5.13 shows this effect to be significant, especially for higher radii. To actually check whether this contraction is adiabatic, one would need to use equation 5.25 for the dark matter only simulation profile as shown in figure 5.14 plus the baryonic distribution from the hydro run, to compare the resulting contracted dark matter profile with the actual dark matter profile in the hydrodynamic simulation. This will be explored in future work.
Several function forms have been suggested to provide good fits to the spherically averaged dark matter density profile in pure dark matter simulations. The two most popular are the Einasto profile given by:
α ρ(r) r− (5.26) ∝ ln ρ(r) α ln r (5.27) ∝ − and the Navarro, Frenk and White (1996) profile, given by:
1 ρ(r) 2 , (5.28) ∝ r(1 + r/rs) 1 ρ(r rs) r− , (5.29) ∝ 3 ρ(r r ) r− , (5.30) s ∝ In figure 5.14 (bottom panel), I show the logarithmic slope of the dark matter density profilee for the most massive halos in pSIM. If we write the dark matter density as ρ r α(r), then we have that: ∝ −
d ln ρ = α(r) (5.31) d ln r − and we see that within the numerical noise both an Einasto or NFW profile provide a reasonable match to the simulation data.
The higher resolution run for the hydrodynamic simulation gives the dark matter den- sity distribution of Figure 5.15. The core of size 10 pc is bigger than the 4 pc resolution
96 5.3. Results limit, but not by much. Again, this closely resembles a dark matter only simulation, if not steeper in the 10 pc region, because cooling allowed adiabatic contraction of gas, and insufficient star formation caused missing feedback in the centre, keeping the gas inside.
Fig. 5.14 —: Dark matter density of the two most massive halos in pSIM (upper panel) and corresponding change of slope (lower panel). The vertical blue line indicates force resolution length, the other two show the relaxation radii of the halos. Both profiles are close to an NFW profile above the relaxation radius.
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Fig. 5.15 —: Dark matter density of the most massive halo in the Hydro simulation. For each particle inside the halo, the dark matter density is estimated and plotted at the locus (r, ρDM). We see a core of size 10 pc. The increased density at radii beyond 3 kpc is due to substructure. ρ(r) r 2 outside the core. ∝ −
98 5.3. Results
5.3.3 Stars
In this section, I show the results for the distribution of stars in the pSIM simulation. Figure 5.16 shows both dark matter and stellar density profiles for pSIM 1 and pSIM 5. The most massive halo shows an enhancement of the density – owing to several star clusters – out to a typical radius of 30pc/h, over which region both the stars and dark matter have a constant density core. Interestingly, the dynamical friction time for these star clusters in the core should be very short if we consider Chandrasekhar(1943) friction:
v dv 2 2 v | | M = 4πG M ln Λ dv0 M(v0) (5.32) c dt − c v 3 | | Z0 1 2 6 2.64 10 1 ri vc 10 M t · (5.33) fric ∼ ln Λ 2 kpc 250 km/s M ! which results in t 0.04 Gyr for a typical star cluster. However, it is known that in fric ∼ constant density cores dynamical friction ceases, which might explain the dominance of these star clusters at small projected radii (Read et al. 2006). This remains to be fully explored.
Figure 5.17 gives a projection of all stars in pSIM 1, plotted on the overall density distribution. Metallicity is encoded in colour (white for low metallicity, black for high metallicity). One can distinguish different populations of stars, with very few stars of medium metallicity. High metallicity stars form groups of 30 star particles. ∼
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Fig. 5.16 —: Density profile of dark matter and star particles in halos pSIM 1 and pSIM 5. In the most massive halo, stars cluster in the centre, and contribute roughly equally to the overall density as dark matter does. For the less massive halo, stars follow the distribution of the dominant and more centrally concentrated dark matter.
100 5.3. Results
Fig. 5.17 —: Projection of stars (circles, empty and filled) in pSIM 1, centred on the overall centre of mass. The red background gives the overall density in patches. The stars are coloured according to metallicity, white showing primordial metallicity, and black giving 0.1Z . There are different clusters of stars with distinct metallicity compared to the background.
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Fig. 5.18 —: Stellar density profile of the most massive halo of the Hydro simulation. For each particle inside the halo, the stellar density is estimated and plotted at the locus (r, ρ ). ∗
102 5.4. Conclusions
5.3.4 Forming Globular Clusters and Dwarfs at High Redshift
Figure 5.19 plots the stellar mass versus the half light radius of bound stellar structures in the pSIM simulation at redshift z = 10. This is a high redshift version of the similar obser- vational plot for Local Group globular clusters and dwarf galaxies compiled by Belokurov et al. 2007, their Figure 8. Fascinatingly, I find that massive, concentrated star clusters with little dark matter (blue points) form alongside more extended and dark-matter rich dwarf galaxies (red triangles), similar to what is observed today. The precise physical reason for these two distinct but coeval stellar populations remains to be explored. This is a key goal of my future research plans.
5.4 Conclusions
With the precursor NEC simulation we confirm that dark matter only halos show a NFW density profile with a cusp in the center. The inclusion of hydrodynamics only, without non-equilibrium chemistry, triggers contraction that enhances the central density, and thus dark matter cusp-core transformations do not occur in the low mass ( 107 M ) dwarfs at ∼ redshift z = 10 that I simulate here. This suggests that if such tiny dwarfs survive to the present day as fossils of reionisation, then they can be expected to retain their central dark matter density cusp. The perfect candidates for such fossils are the recently discovered ultra-faint dwarfs around the Milky Way and M31 (Simon & Geha 2007; Belokurov et al. 2007; Chiboucas et al. 2013). These will then be prime locations to search for dark matter annihilation or decay signals (e.g. Geringer-Sameth et al. 2015).
I find that massive star clusters form alongside low stellar density dwarf galaxies at z = 10 (Figure 5.19). The star clusters are completely devoid of dark matter. If these are the progenitors of the globular clusters in the Milky Way today, then this suggests that globular clusters will not be good sites to hunt for dark matter annihilation or decay signals.
The physics of massive star clusters – the possible progenitors of today’s globular clusters – remains to be elucidated in future work.
103 CHAPTER 5. Simulations of Dwarf Galaxies
Fig. 5.19 —: Stellar mass as function of half-light radius for halos of pSIM . This plot shows a striking qualitative resemblance to observed globular clusters and dwarfs in the Local Group today (see top panel from Belokurov et al. 2007). Spheres show subhalos, triangles show halos without host halo. Red colored datapoints are dark matter dominated systems.
104 6 Conclusions and Future Prospects
6.1 Conclusions
In the first part of this thesis, I developed a non-parametric Jeans framework GravImage to model the dark matter profiles of spherical systems based on kinematic data alone. I tested it successfully on mock data consisting of a) a single tracer population and b) split populations, under all combinations of population number, central isotropy prior, and monotonicity prior on n(r).
GravImage was then applied to Fornax observations, encorporating a new population splitting method. Further application to our four smaller dwarf spheroidals shows an interesting preliminary result: these all appear to be substantially cuspier than Fornax.
In the second part of this thesis, I attempted to calculate the distribution of dark matter in the smallest galaxies, accounting for baryonic effects. To achieve this, I incorporated a non-equilibrium chemistry cooling module in a modern version of Ramses. Analysis of a precursor run was finished, and two high-resolution simulations performed.
6.1.1 Addressing the Questions posed at the Start of the Thesis
I am now in a position to return to the questions outlined in 1.3 at the start of this thesis. §
Can we constrain the central dark matter density slope in spherical systems non-parametrically? I have shown that the answer to this is “yes”, and at high signifi- cance (99% confidence). But only if populations are split on metallicity and/or abundance;
105 CHAPTER 6. Conclusions and Future Prospects the power law exponent of the dark matter density is assumed to monotonically rise; and if we assume priors on the velocity anisotropy β (0) = 0 and/or β (r) 0. With fewer ∗ ∗ ≥ assumptions, the constraints weaken.
Using the above, I showed that the Fornax dwarf – the brightest Milky Way dwarf spheroidal – almost certainly has a core. Four other dwarfs that are fainter and show truncated star formation favour a more cuspy dark matter profile. If this latter result holds up to further tests, it could be striking evidence in favour of the prevailing ΛCDM cosmological model.
Can we predict the dark matter distribution on small scales? I have used very high resolution simulations of dwarfs at high redshift to resolve individual supernovae. The results suggest that such early forming dwarfs do not significantly alter their dark matter profiles when baryonic effects (star formation and feedback) are taken into account. At least not on the mass scales (107 M ) simulated here. If anything, the dark matter profiles are slightly steeper when baryons are included than in the pure dark matter runs. This result suggests that galaxies that are true ”fossils” of reionisation may be expected to retain a pristine dark matter cusp. This is particularly exciting given the results from the first half of my thesis that may support the existence of such cusps in nearby dwarfs that have only old stellar populations.
6.1.2 Future Prospects
There are many avenues for future research following on from the work presented in this thesis. Here I highlight a few key areas.
Application of GravImage to the Milky Way Dwarfs A first next-step would be to complete the work started in chapter 4 modelling four Milky Way dwarfs. Current § models should be run further to ensure convergence; split populations models remain to be explored, as do the effects of switching off the β∗ priors and/or the monotonicity prior on n(r). Such tests will make the preliminary results presented here substantially more robust.
Application of GravImage beyond the Dwarfs GravImage is a non-parametric spherical Jeans modeller. It assumes sphericality and dynamic equilibrium – conditions that not only apply to dwarf spheroidal galaxies. It can be interestingly applied to a wide range of astrophysical systems from old globular clusters (Binney 1982; Ibata et al. 2011), stellar clusters around black holes (Chakrabarty & Saha 2001) to evolved galaxies like giant ellipticals (Merritt 1999; Napolitano et al. 2014). Even galaxy clusters that have most of their baryonic mass in hot spherical gas clouds, possibly aligned with the overall
106 6.1. Conclusions gravitational potential, would fall in that regime (e.g. Lokas & Mamon 2003b; Arnaud et al. 2010). I will explore such applications in future papers. A final novel application of GravImage is to model the vertical mass distribution in the Milky Way disc. This is work in progress in collaboration with Prof. Gianfranco Bertrone and Hamish Silverwood at the University of Amsterdam. It requires some substantial changes to the code to model the vertical rather than spherical Jeans equation. But much of the work is already completed and papers are in preparation, see appendix A.7.
High Resolution NEC Simulations of Dwarfs at High Redshift Completing the work begun in Chapter 5 is another clear research avenue. The results from the pilot § simulation are already interesting. It would be exciting to run similar simulations again with higher time frequency in the outputs to study the physics of massive star cluster formation at high redshift. This would shed light on the formation mechanisms of metal poor old globular clusters. Furthermore, I would like to run and complete the NEC high resolution dwarf simulation. This would determine whether or not the small dark matter cores observed in the pilot simulation are physical; and further elucidate the physics of globular cluster formation at high redshift.
107 CHAPTER 6. Conclusions and Future Prospects
108 Appendix A: Profiles
A.1 Plummer Profile
Dwarf Spheroidals show a surface density profile which was traditionally fit by the Plum- mer profile (Plummer 1911):
L I(R) = 2 2 2 (A-1) πR1/2(1 + R /R1/2) with R the projected radius on the sky and L the total luminosity of the dwarf spheroidal. By multiplying with a mass-to-light ratio M/L, we get the mass surface density:
M Σ(R) = M/L I(R) = 2 2 2 (A-2) · πR1/2(1 + R /R1/2)
The 3D density distribution is obtained via deprojection under the assumption of spher- icality, giving:
1 dΣ dR ν(r) = ∞ (A-3) −π dR √R2 r2 Zr 3M − = 3 2 3 (A-4) 4πr1/2(1 + r /r1/2)
109 APPENDIX.
A.2 NFW Profile
Navarro et al.(1995) proposed following spherical density profile for X-ray clusters:
1 ρ(r) 2 (A-5) ∝ r(1 + r/rS) where r is radius from cluster centre, and rS is a scale radius. As seen in the seminal paper of Navarro et al. 1996b, this is also the form found for halos in ΛCDM simulations without baryonic feedback. Hanyu & Habe(2001) argument that the NFW profile follows automatically from a given fractional mass distribution.
A.3 King Profile
More complicated models exist which encompass the complexity of the profile by several parameters. That enables a better fit for many profiles, but complicates the physical rea- soning for the generation of said profiles. Furthermore, the danger of overfitting increases with increasing number of parameters.
A profile to fit globular clusters is the King profile (King 1962):
2 1 1 Σ(R) (A-6) 2 2 ∝ 1 + (R/Rc) − 1 + (Ri/Rc) ! p p where Rc,Ri are two scale radii.
A.4 Sersic Profiles
Surface density profiles for massive elliptical galaxies and disc galaxies alike were found to follow:
ln Σ(R) = ln Σ kR1/n (A-7) 0 − with central surface density Σ0, Sersic index n and scale factor k. Elliptical galaxies generally have a good fit to n = 4 profiles, also named a de Vaucouleurs profile, whereas exponential profiles with n = 1 often fit spiral galaxies.
110 A.5. Prigniel-Simien Profile
A.5 Prigniel-Simien Profile
ρ 2 ρPS(r) = p − (A-8) r exp n(2 p) (r/r )(1/n) 1 r 2 2 − − − − where n describes the curvature of the profile,n andp a fit parameter,o given by p = 1.0 − 0.6097/n + 0.05463/n2 for 0.6 n 10 (Lima Neto et al. 1999). ≤ ≤
A.6 Double-power law Profile
Di Cintio et al.(2012) propose a more general profile:
ρ ρ (r) = S (A-9) α,β,γ γ α (β γ)/α (r/rS) [1 + (r/rS) ] − with rS the scale radius, ρS the scale density, α regulating the transition between inner logarithmic slope β and outer logarithmic slope γ. − − The NFW profile is recovered from that general profile by setting (α, β, γ) = (1, 3, 1), Moore et al.(1999) use (1 .5, 3, 1.5).
A.7 Einasto Profile
ρ 2 ρ (r) = − (A-10) E (1/n) exp r 1 r 2 − − with r 2 the radius at logarithmic density slope 2, n the shape parameter. This has − − the same functional form as the Sersic profile above. Again, NFW can be recovered, by setting ρ 2 = ρS,NF W , ρ 2 = ρS/4. Merritt et al.(2005) and Merritt et al.(2006) argue − − that the Einasto profile is a better fit to simulated galaxies than the NFW profile.
111 APPENDIX.
Fig. A.1 —: Einasto profiles for ρ(r) exp( αr). Source: Wikipedia 2015. ∝ −
112 Appendix B: Fitting
If we are interested in finding the best empirical profile, we need to compare fits of all of these models via a goodness of fit measure:
Nbin 2 1 2 ∆ = (log10 ρsim,k log10 ρfit,k) (A-11) Nbin − Xk=1 which is summing up the differences between densities ρsim,k and ρfit,k over all Nbin bins in logarithmic space. I use an adapted form of this error measure for calculation of the χ2 value in the mass-modelling routines.
The problem with fixed form functions of these profiles lies in the fact that real galaxies might have undergone a series of interactions. The strongest interactions are shown to happen where the 3D densities of matter in the interacting systems are of the same order. Depending on the trajectories of the interacting structures, that might lead to changes at several radii, with or without relaxation. Observed at one specific time, we might not see the final relaxed state after a single simple interaction, which would be fit well by one of the relaxed system’s profiles. To circumvent that restriction, I look at non-parametric profiles instead in chapter2.
113 APPENDIX.
114 Appendix C: Mass-Modelling of Disc Systems
The Jeans equations are 1-dimensional equations between projected quatities of the dis- tribution function. Another geometry than a spherical one can be constructed, and if the integration routines for the line-of-sight velocity dispersion is adapted, GravImage can be used to model them, too. I show an extension of GravImage to 1-dimensinal cylindrical coordinates. This allows me to model the mass distribution of the Milky Way perpendicular to its disc. I start with the derivation of the Jeans equations for the disc case. Next, I show the choice of representation for each profile of interest.
A.1 Local Dark Matter Density
Garbari et al.(2012) use a model with a minimal set of assumptions (MA-method) to determine the local dark matter density from the velocity dispersion of a set of 15 tracer populations perpendicular to the Milky Way disc. Their value of 0.033 M / pc3 with no assumption of isothermality of the tracer stars is several times higher than the canonical value of 0.003 M / pc3 (Garbari et al. 2011). Read et al.(2008) use a suite of hydrodynamic simulations to show that an enhancement of dark matter is expected around the central plane of the Milky Way after merging satellites are drawn to the over-density of the thin disc. A thick disc of stars forms during the same period. See Bensby 2013 for a review of the thick disc (in visible matter) that directly relates to the formation history of the galaxy in question.
115 APPENDIX.
A.2 Geometry
As in (Silverwood et al. 2015), I concentrate on the simple geometry of a razor-thin baryonic disc in a massive spherical dark matter halo. Any stars in the disc will have negligible mass compared to the overall potential and thus act as tracer particles. The vertical motion of the stars obeys the collision-less Boltzmann equation, much the same as in the spherical case (Binney & Tremaine 2008):
df ∂f = + ~ f ~v ~ f ~ Φ = 0 (A-12) dt ∂t ∇~x · − ∇~v · ∇~x or in cylindrical polar coordinates (R, φ, z):
1 ∂(Rν σ ) 1 d(νσ2) dΦ i Rz + z = = K (A-13) Rν ∂R ν dz − dz z where it is assumed that the system is in dynamic equilibrium (∂f/∂t = 0). ν(z) is the 2 3D tracer density profile as function of height z above the plane, σz (z) is the vertical velocity dispersion, and Kz defined as above represents the vertical acceleration. The first term in this sum is called tilt, and constitutes the only place where R and z are mingled. Integrating over z and solving for the vertical velocity dispersion one gets the expression:
z 2 1 C σ (z) = ν(z0)[K (z0) (z0)]dz0 + (A-14) z ν(z) z − T ν(z) Zzmin where is the above-mentioned tilt term, and C = ν(z )σ2(z)(z ) represents the T min z min integration constant, with the physical meaning of a vertical velocity dispersion at the minimal integration height normalised by the tracer density at that point. This corre- sponds to equation 10 in (Smith et al. 2012). The velocity dispersion is connected to the overall matter density ρ by the Poisson equation:
∂2Φ 1 ∂V 2(R) ~ 2Φ = + c = 4πGρ (A-15) ∇ ∂z2 R ∂R with rotation speed V (R) and rotation curve term as the second term in the summation. c R Setting this latter to 0 – as V const at the radius of the Sun – and integrating gives c ≈ the overall surface density:
K Σ (z) = | z| (A-16) z 2πG as a function of height z above (and below) the disc.
116 A.3. Representation
A.3 Representation
A first approach used the same representation of dark matter density as in the spherical case (chapter2), namely by parametrising its second derivative and integrating twice to get the profile. Instead of representing ρ(r) and ν(r) profiles with the second derivatives as in the spherical case, we can then represent the acceleration Kz(z) in the same way:
z κ(z) = k(z) (A-17) 0 Z z Kz(z) = κ(z) (A-18) Z0 The testing of these representations on mock data from simulations and inclusion of models for the different baryonic components of the Milky Way disc is part of a current collabo- ration.
117 APPENDIX.
118 Bibliography
Aad G., Abajyan T., Abbott B., Abdallah J., Abdel Khalek S., Abdelalim A. A., Abdinov O., Aben R., Abi B., Abolins M., et al. 2012, Physics Letters B, 716, 1
Aartsen M. G., Abbasi R., Abdou Y., Ackermann M., Adams J., Aguilar J. A., Ahlers M., Altmann D., Andeen K., Auffenberg J., 2013, Phys. Rev. D, 87, 062002
Abel T., Anninos P., Zhang Y., Norman M. L., 1997, Nature, 2, 181
Agnello A., Evans N. W., 2012, ArXiv e-prints
Aguirre A., Schaye J., Quataert E., 2001, ApJ, 561, 550
Alcock C., Akerlof C. W., Allsman R. A., Axelrod T. S., Bennett D. P., Chan S., Cook K. H., Freeman K. C., Griest K., Marshall S. L., Park H.-S., Perlmutter S., Peterson B. A., Pratt M. R., Quinn P. J., Rodgers A. W., Stubbs C. W., Sutherland W., 1993, Nature, 365, 621
Amorisco N. C., Evans N. W., 2012, MNRAS, 419, 184
An J. H., Evans N. W., 2011, MNRAS, 413, 1744
Anderhalden D., Schneider A., Macci`oA. V., Diemand J., Bertone G., 2013, JCAP, 3, 14
Angus G. W., Famaey B., Diaferio A., 2010, MNRAS, 402, 395
Antoniadis J., Freire P. C. C., Wex N., Tauris T. M., Lynch R. S., van Kerkwijk M. H., Kramer M., Bassa C., Dhillon V. S., Driebe T., Hessels J. W. T., et al. 2013, Science, 340, 448
Arnaud M., Pratt G. W., Piffaretti R., B¨ohringerH., Croston J. H., Pointecouteau E., 2010, A&A, 517, A92
Asaka T., Shaposhnikov M., Kusenko A., 2006, Physics Letters B, 638, 401
119 APPENDIX.
Asztalos S. J., Carosi G., Hagmann C., Kinion D., van Bibber K., Hotz M., Rosenberg L. J., Rybka G., Hoskins J., Hwang J., Sikivie P., Tanner D. B., Bradley R., Clarke J., ADMX Collaboration 2010, Physical Review Letters, 104, 041301
Aubourg E., Bareyre P., Br´ehinS., Gros M., Lachi`eze-Rey M., Laurent B., Lesquoy E., Magneville C., Milsztajn A., Moscoso L., Queinnec F., et al. 1993, Nature, 365, 623
Avila-Reese V., Col´ınP., Valenzuela O., D’Onghia E., Firmani C., 2001, ApJ, 559, 516
Bagla J. S., 2005, Current Science, 88, 1088
Bao S.-S., Li H.-L., Si Z.-G., Yang Y.-B., 2013, Communications in Theoretical Physics, 59, 472
Battaglia G., Helmi A., Tolstoy E., Irwin M., Hill V., Jablonka P., 2008, ApJ, 681, L13
Battaglia G., Irwin M., Tolstoy E., de Boer T., Mateo M., 2012, ApJ, 761, L31
Battaglia G., Tolstoy E., Helmi A., Irwin M., Parisi P., Hill V., Jablonka P., 2011, MNRAS, 411, 1013
Bekenstein J., Milgrom M., 1984, ApJ, 286, 7
Bekenstein J. D., 2004, Phys. Rev. D, 70, 083509
Belokurov V., Walker M. G., Evans N. W., Gilmore G., Irwin M. J., Just D., Koposov S., Mateo M., Olszewski E., Watkins L., Wyrzykowski L., 2010, ApJ, 712, L103
Belokurov V., Zucker D. B., Evans N. W., Kleyna J. T., Koposov S., Hodgkin S. T., Irwin M. J., Gilmore G., Wilkinson M. I., Fellhauer M., Bramich D. M., 2007, ApJ, 654, 897
Bensby T., 2013, ArXiv e-prints
Berezinsky V. S., Dokuchaev V. I., Eroshenko Y. N., 2013, ArXiv e-prints
Bergstr¨omL., Bringmann T., Cholis I., Hooper D., Weniger C., 2013, Physical Review Letters, 111, 171101
Bertotti B., Iess L., Tortora P., 2003, Nature, 425, 374
Bertschinger E., 1995, ArXiv Astrophysics e-prints
Bertschinger E., 2001, ApJS, 137, 1
Bi H., Davidsen A. F., 1997, ApJ, 479, 523
Binney J., 1982, ARA&A, 20, 399
Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press
Black J. H., 1978, ApJ, 222, 125
Blanchet L., Novak J., 2011, ArXiv e-prints
120 BIBLIOGRAPHY
Bludman S. A., 1998, ApJ, 508, 535
Blumenthal G. R., Faber S. M., Flores R., Primack J. R., 1986, ApJ, 301, 27
Bode P., Ostriker J. P., Turok N., 2001, ApJ, 556, 93
Boley A. C., Lake G., Read J., Teyssier R., 2009, ApJ, 706, L192
Bond J. R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379, 440
Bond J. R., Kofman L., Pogosyan D., 1996, Nature, 380, 603
Boyarsky A., Ruchayskiy O., Iakubovskyi D., Franse J., 2014, Physical Review Letters, 113, 251301
Boyarsky A., Ruchayskiy O., Shaposhnikov M., 2009, Annual Review of Nuclear and Particle Science, 59, 191
BradaˇcM., Allen S. W., Treu T., Ebeling H., Massey R., Morris R. G., von der Linden A., Applegate D., 2008, ApJ, 687, 959
Breddels M. A., Helmi A., 2013, A&A, 558, L3
Breddels M. A., Helmi A., 2014, ArXiv e-prints
Breddels M. A., Helmi A., van den Bosch R. C. E., van de Ven G., Battaglia G., 2012, ArXiv e-prints
Bryan G. L., Norman M. L., 1998, ApJ, 495, 80
Bulbul E., Markevitch M., Foster A., Smith R. K., Loewenstein M., Randall S. W., 2014a, ApJ, 789, 13
Bulbul E., Markevitch M., Foster A. R., Smith R. K., Loewenstein M., Randall S. W., 2014b, ArXiv e-prints
Burstein D., Blumenthal G., 2002, ApJ, 574, L17
Carlson E. D., Machacek M. E., Hall L. J., 1992, ApJ, 398, 43
Cen R., 1992, ApJS, 78, 341
Chakrabarty D., Saha P., 2001, AJ, 122, 232
Chanam´eJ., 2010, Highlights of Astronomy, 15, 190
Chanam´eJ., Kleyna J., van der Marel R., 2008, ApJ, 682, 841
Chandrasekhar S., 1943, ApJ, 97, 255
Charbonnier A., Combet C., Daniel M., Funk S., Hinton J. A., Maurin D., Power C., Read J. I., Sarkar S., Walker M. G., Wilkinson M. I., 2011, MNRAS, 418, 1526
Chiboucas K., Jacobs B. A., Tully R. B., Karachentsev I. D., 2013, AJ, 146, 126
121 APPENDIX.
Choi K.-Y., Kim J. E., Roszkowski L., 2013, Journal of Korean Physical Society, 63, 1685
Choudhury T. R., Srianand R., Padmanabhan T., 2001, ApJ, 559, 29
Chwolson O., 1924, Astronomische Nachrichten, 221, 329
Clowe D., BradaˇcM., Gonzalez A. H., Markevitch M., Randall S. W., Jones C., Zaritsky D., 2006, ApJ, 648, L109
Coc A., Vangioni-Flam E., Cass´eM., Rabiet M., 2002, Phys. Rev. D, 65, 043510
Cole D. R., Dehnen W., Read J. I., Wilkinson M. I., 2012, MNRAS, 426, 601
Cole S., Percival W. J., Peacock J. A., Norberg P., Baugh C. M., Frenk C. S., Baldry I., Bland-Hawthorn J., Bridges T., Cannon R., Colless M., Collins C., et al. 2005, MNRAS, 362, 505
Coleman M. G., de Jong J. T. A., 2008, ApJ, 685, 933
Combes F., 1991, ARA&A, 29, 195
Dalgarno A., Lepp S., 1987, in Vardya M. S., Tarafdar S. P., eds, Astrochemistry Vol. 120 of IAU Symposium, Chemistry in the early universe. pp 109–118 de Blok W. J. G., 2010, Advances in Astronomy, 2010 de Blok W. J. G., McGaugh S. S., Rubin V. C., 2001, AJ, 122, 2396 de Blok W. J. G., Walter F., Brinks E., Trachternach C., Oh S.-H., Kennicutt Jr. R. C., 2008, AJ, 136, 2648 de Boer T. J. L., Tolstoy E., Hill V., Saha A., Olszewski E. W., Mateo M., Starkenburg E., Battaglia G., Walker M. G., 2012, ArXiv e-prints de Jong T., 1972, A&A, 20, 263 de Laix A. A., Scherrer R. J., Schaefer R. K., 1995, ApJ, 452, 495 de Lorenzi F., Debattista V. P., Gerhard O., Sambhus N., 2007, MNRAS, 376, 71 de Lorenzi F., Gerhard O., Saglia R. P., Sambhus N., Debattista V. P., Pannella M., M´endezR. H., 2008, MNRAS, 385, 1729
Dehnen W., 2009, MNRAS, 395, 1079
Dehnen W., Read J. I., 2011, European Physical Journal Plus, 126, 55
Destri C., de Vega H. J., Sanchez N. G., 2013, Nature, 22, 39
Di Cintio A., Knebe A., Libeskind N. I., Brook C., Yepes G., Gottloeber S., Hoffman Y., 2012, ArXiv e-prints
Diakogiannis F. I., Lewis G. F., Ibata R. A., 2014, MNRAS, 443, 598
122 BIBLIOGRAPHY
Diemand J., Kuhlen M., Madau P., 2007, ApJ, 657, 262
Dodelson S., 2011, International Journal of Modern Physics D, 20, 2749
Dolphin A. E., Weisz D. R., Skillman E. D., Holtzman J. A., 2005, ArXiv Astrophysics e-prints
Dove J. E., Mandy M. E., 1986, ApJ, 311, L93
Dubinski J., Carlberg R. G., 1991, ApJ, 378, 496
Duffy L. D., van Bibber K., 2009, New Journal of Physics, 11, 105008
Einstein A., 1936, Science, 84, 506
Eisenstein D. J., Hu W., 1998, ApJ, 496, 605
Eisenstein D. J., Zehavi I., Hogg D. W., Scoccimarro R., Blanton M. R., Nichol R. C., Scranton R., Seo H.-J., Tegmark M., Zheng Z., Anderson S. F., et al. 2005, ApJ, 633, 560
El-Zant A., Shlosman I., Hoffman Y., 2001, ApJ, 560, 636
Esmaili A., Dario Serpico P., 2013, JCAP, 11, 54
Evans N. W., An J., Deason A. J., 2011, ApJ, 730, L26
Fabian A. C., ed. 1992, Clusters and superclusters of galaxies Vol. 366 of NATO Advanced Science Institutes (ASI) Series C
Faerman Y., Sternberg A., McKee C. F., 2013, ApJ, 777, 119
Feast M., Whitelock P., 1997, MNRAS, 291, 683
Fermani P., 2013, ArXiv e-prints
Feroz F., Hobson M. P., 2008, MNRAS, 384, 449
Feroz F., Hobson M. P., Bridges M., 2009, MNRAS, 398, 1601
Feroz F., Hobson M. P., Cameron E., Pettitt A. N., 2013, ArXiv e-prints
Firmani C., D’Onghia E., Avila-Reese V., Chincarini G., Hern´andezX., 2000, MNRAS, 315, L29
Flores R. A., Primack J. R., 1994, ApJ, 427, L1
Flynn C., Gould A., Bahcall J. N., 1996, ApJ, 466, L55
Fornengo N., Maccione L., Vittino A., 2013, JCAP, 9, 31
Galli D., Palla F., 1998, A&A, 335, 403
Gan J., Kang X., van den Bosch F. C., Hou J., 2010, MNRAS, 408, 2201
123 APPENDIX.
Garbari S., Liu C., Read J. I., Lake G., 2012, MNRAS, 425, 1445
Garbari S., Read J. I., Lake G., 2011, MNRAS, 416, 2318
Gatto A., Fraternali F., Read J. I., Marinacci F., Lux H., Walch S., 2013, MNRAS, 433, 2749
Gebhardt K., Rich R. M., Ho L. C., 2005, ApJ, 634, 1093
Geringer-Sameth A., Koushiappas S. M., Walker M., 2015, ApJ, 801, 74
Glover S. C. O., Abel T., 2008, MNRAS, 388, 1627
Gnedin O. Y., Zhao H., 2002, MNRAS, 333, 299
Goerdt T., Moore B., Read J. I., Stadel J., Zemp M., 2006, MNRAS, 368, 1073
Gott III J. R., Turner E. L., 1977, ApJ, 216, 357
Governato F., Brook C., Mayer L., Brooks A., Rhee G., Wadsley J., Jonsson P., Willman B., Stinson G., Quinn T., Madau P., 2010, Nature, 463, 203
Graff D. S., Freese K., 1996, ApJ, 456, L49
Griest K., Hu W., 1992, ApJ, 397, 362
Griest K., Kamionkowski M., 2000, Phys. Rep., 333, 167
Gunn J. E., Lee B. W., Lerche I., Schramm D. N., Steigman G., 1978, ApJ, 223, 1015
Haardt F., Madau P., 1996, ApJ, 461, 20
H¨afnerR., Evans N. W., Dehnen W., Binney J., 2000, MNRAS, 314, 433
Hague P. R., Wilkinson M. I., 2014, ArXiv e-prints
Hahn O., Abel T., Kaehler R., 2013, MNRAS, 434, 1171
Hahn O., Porciani C., Carollo C. M., Dekel A., 2007, MNRAS, 375, 489
Hanyu C., Habe A., 2001, ApJ, 554, 1268
Hargreaves J. C., Gilmore G., Irwin M. J., Carter D., 1994, MNRAS, 269, 957
Heitmann K., Luki´cZ., Fasel P., Habib S., Warren M. S., White M., Ahrens J., Ankeny L., Armstrong R., O’Shea B., Ricker P. M., Springel V., Stadel J., Trac H., 2008, Computational Science and Discovery, 1, 015003
Hektor A., Raidal M., Strumia A., Tempel E., 2014, Physics Letters B, 728, 58
Herpich J., Stinson G. S., Macci`oA. V., Brook C., Wadsley J., Couchman H. M. P., Quinn T., 2014, MNRAS, 437, 293
Hirasawa T., 1969, Progress of Theoretical Physics, 42, 523
124 BIBLIOGRAPHY
Hogan C. J., Dalcanton J. J., 2000, Phys. Rev. D, 62, 063511
Honma M., Sofue Y., 1997, PASJ, 49, 453
Hurley-Keller D., Mateo M., Nemec J., 1998, AJ, 115, 1840
Ibarra A., Tran D., Weniger C., 2013, International Journal of Modern Physics A, 28, 30040
Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., Dalessandro E., 2011, ApJ, 738, 186
Inoue S., Granot J., O’Brien P. T., Asano K., Bouvier A., Carosi A., Connaughton V., Garczarczyk M., Gilmore R., Hinton J., Inoue Y., Ioka K., Kakuwa J., Markoff S., Murase K., Osborne J. P., Otte A. N., Starling R., Tajima H., 2013, Astroparticle Physics, 43, 252
Jardel J. R., Gebhardt K., 2013, ApJ, 775, L30
Jeans J. H., 1915, MNRAS, 76, 70
Jeltema T. E., Profumo S., 2014, ArXiv e-prints
Jungman G., Kamionkowski M., Griest K., 1996, Phys. Rep., 267, 195
Karpas Z., Anicich V., Huntress W. T., 1979, J. Chem. Phys., 70, 2877
Kauffmann G., 2014, MNRAS, 441, 2717
Kelly P. L., Filippenko A. V., Fox O. D., Zheng W., Clubb K. I., 2013, ApJ, 775, L5
Kent S. M., Gunn J. E., 1982, AJ, 87, 945
King I., 1962, AJ, 67, 471
Kleyna J., Wilkinson M. I., Evans N. W., Gilmore G., Frayn C., 2002, MNRAS, 330, 792
Kleyna J. T., Wilkinson M. I., Evans N. W., Gilmore G., 2004, MNRAS, 354, L66
Kleyna J. T., Wilkinson M. I., Gilmore G., Evans N. W., 2003, ApJ, 588, L21
Klypin A., Kravtsov A. V., Valenzuela O., Prada F., 1999, ApJ, 522, 82
Knebe A., Doumler T., , 2010, AMIGA: Adaptive Mesh Investigations of Galaxy Assembly, Astrophysics Source Code Library
Knebe A., Knollmann S. R., Muldrew S. I., Pearce F. R., Aragon-Calvo M. A., Ascasibar Y., Behroozi P. S., Ceverino D., Colombi S., Diemand J., Dolag K., Falck B. L., Fasel P., Gardner J., et al. 2011, MNRAS, 415, 2293
Knebe A., Libeskind N. I., Pearce F., Behroozi P., Casado J., Dolag K., Dominguez- Tenreiro R., Elahi P., Lux H., Muldrew S. I., Onions J., 2013, MNRAS, 428, 2039
Knollmann S. R., Knebe A., 2009, ApJS, 182, 608
125 APPENDIX.
Komatsu E., Smith K. M., Dunkley J., Bennett C. L., Gold B., Hinshaw G., Jarosik N., Larson D., et al. 2011, ApJS, 192, 18
Koposov S., Belokurov V., 2008, Observational Constraints on the “Missing Satellite” Problem from SDSS. -, pp 195–+
Kozaczuk J., Profumo S., 2013, ArXiv e-prints
Kravtsov A. V., Gnedin O. Y., Klypin A. A., 2004, ApJ, 609, 482
Kravtsov A. V., Klypin A. A., Khokhlov A. M., 1997, ApJS, 111, 73
Kroupa P., 2002, Science, 295, 82
Kuhn J. R., Smith H. A., Hawley S. L., 1996, ApJ, 469, L93
Kumar J., Stengel P., 2013, ArXiv e-prints
Kuzio de Naray R., Kaufmann T., 2011, MNRAS, 414, 3617
Lacey C., Cole S., 1994, MNRAS, 271, 676
Lake G., 1990, MNRAS, 244, 701
Leaman R., Venn K. A., Brooks A. M., Battaglia G., Cole A. A., Ibata R. A., Irwin M. J., McConnachie A. W., Mendel J. T., Tolstoy E., 2012, ApJ, 750, 33
Lepp S., Shull J. M., 1983, ApJ, 270, 578
Lima Neto G. B., Gerbal D., M´arquezI., Capelato H. V., 1999, in Merritt D. R., Valluri M., Sellwood J. A., eds, Galaxy Dynamics - A Rutgers Symposium Vol. 182 of Astronomical Society of the Pacific Conference Series, S´ersicProfile and Specific Entropy of Elliptical Galaxies. p. 211
Limousin M., Richard J., Kneib J.-P., Brink H., Pell´oR., Jullo E., Tu H., Sommer-Larsen J., Egami E., Michalowski M. J., Cabanac R., Stark D. P., 2008, A&A, 489, 23
Lokas E. L., 2009, MNRAS, 394, L102
Lokas E. L., Mamon G. A., 2003a, MNRAS, 343, 401
Lokas E. L., Mamon G. A., 2003b, MNRAS, 343, 401
Lokas E. L., Wojtak R., Gottl¨ober S., Mamon G. A., Prada F., 2006, MNRAS, 367, 1463
Lovell M. R., Frenk C. S., Eke V. R., Jenkins A., Gao L., Theuns T., 2014, MNRAS, 439, 300
Lux H., Read J. I., Lake G., 2010, MNRAS, 406, 2312
Macci`oA. V., Paduroiu S., Anderhalden D., Schneider A., Moore B., 2012, MNRAS, 424, 1105
126 BIBLIOGRAPHY
Macci`oA. V., Ruchayskiy O., Boyarsky A., Mu˜noz-CuartasJ. C., 2013, MNRAS, 428, 882
Madau P., Shen S., Governato F., 2014, ApJ, 789, L17
Martinez G. D., Minor Q. E., Bullock J., Kaplinghat M., Simon J. D., Geha M., 2011, ApJ, 738, 55
Mashchenko S., Wadsley J., Couchman H. M. P., 2008a, Science, 319, 174
Mashchenko S., Wadsley J., Couchman H. M. P., 2008b, Science, 319, 174
Mateo M., Olszewski E., Welch D. L., Fischer P., Kunkel W., 1991, AJ, 102, 914
Mateo M., Olszewski E. W., Pryor C., Welch D. L., Fischer P., 1993, AJ, 105, 510
Mateo M. L., 1998, ARA&A, 36, 435
McConnachie A. W., 2012, AJ, 144, 4
McGaugh S. S., Rubin V. C., de Blok W. J. G., 2001, AJ, 122, 2381
Merrifield M. R., Kent S. M., 1990, AJ, 99, 1548
Merritt D., 1985, MNRAS, 214, 25P
Merritt D., 1987, ApJ, 313, 121
Merritt D., 1999, PASP, 111, 129
Merritt D., Graham A. W., Moore B., Diemand J., Terzi´cB., 2006, AJ, 132, 2685
Merritt D., Navarro J. F., Ludlow A., Jenkins A., 2005, ApJ, 624, L85
Milgrom M., 1983, ApJ, 270, 365
Milsztajn A., 2002, Space Sci. Rev., 100, 103
Moiseev A. A., Galper A. M., Adriani O., Aptekar R. L., Arkhangelskaja I. V., Arkhangel- skiy A. I., 2013, ArXiv e-prints
Monaghan J. J., 1992, ARA&A, 30, 543
Moore B., 1994, Nature, 370, 629
Moore B., Ghigna S., Governato F., Lake G., Quinn T., Stadel J., Tozzi P., 1999, ApJ, 524, L19
Moore B., Quinn T., Governato F., Stadel J., Lake G., 1999, MNRAS, 310, 1147
More S., Kravtsov A. V., Dalal N., Gottl¨ober S., 2011, ApJS, 195, 4
Moulin E., Cirelli M., Panci P., Serpico P., Viana A., 2013, ArXiv e-prints
127 APPENDIX.
Mu˜nozR. R., Majewski S. R., Zaggia S., Kunkel W. E., Frinchaboy P. M., Nidever D. L., Crnojevic D., Patterson R. J., Crane J. D., Johnston K. V., Sohn S. T., Bernstein R., Shectman S., 2006, ApJ, 649, 201
Najita J. R., Tiede G. P., Carr J. S., 2000, ApJ, 541, 977
Nakashima K., Takagi H., Nakamura H., 1987, J. Chem. Phys., 86, 726
Napolitano N. R., Pota V., Romanowsky A. J., Forbes D. A., Brodie J. P., Foster C., 2014, MNRAS, 439, 659
Natarajan A., Peterson J. B., Voytek T. C., Spekkens K., Mason B., Aguirre J., Willman B., 2013, Phys. Rev. D, 88, 083535
Navarro J. F., Eke V. R., Frenk C. S., 1996, MNRAS, 283, L72
Navarro J. F., Frenk C. S., White S. D. M., 1995, MNRAS, 275, 720
Navarro J. F., Frenk C. S., White S. D. M., 1996a, ApJ, 462, 563
Navarro J. F., Frenk C. S., White S. D. M., 1996b, ApJ, 462, 563
Newman A. B., 2013, PhD thesis, California Institute of Technology
Niederste-Ostholt M., Belokurov V., Evans N. W., Gilmore G., Wyse R. F. G., Norris J. E., 2009, MNRAS, 398, 1771
O˜norbe J., Boylan-Kolchin M., Bullock J. S., Hopkins P. F., KerˇesD., Faucher-Gigu`ere C.-A., Quataert E., Murray N., 2015, ArXiv e-prints
Oh S.-H., Brook C., Governato F., Brinks E., Mayer L., de Blok W. J. G., Brooks A., Walter F., 2011, AJ, 142, 24
Oh S. P., Haiman Z., 2002, ApJ, 569, 558
Osipkov L. P., 1979, Pisma v Astronomicheskii Zhurnal, 5, 77
Osterbrock D. E., 1974, Astrophysics of gaseous nebulae. Osterbrock, D.
Paczynski B., 1986, ApJ, 304, 1
Partridge R. B., 1995, 3K: The Cosmic Microwave Background Radiation
Pe˜narrubiaJ., Pontzen A., Walker M. G., Koposov S. E., 2012, ApJ, 759, L42
Peacock J. A., 1999, Cosmological Physics. Cambridge University Press
Peccei R. D., Quinn H. R., 1977, Physical Review Letters, 38, 1440
Peebles P. J. E., Dicke R. H., 1968, ApJ, 154, 891
Penzias A. A., Wilson R. W., 1965, ApJ, 142, 419
128 BIBLIOGRAPHY
Perlmutter S., Aldering G., Goldhaber G., Knop R. A., Nugent P., Castro P. G., Deustua S., Fabbro S., Goobar A., Groom D. E., Hook I. M., et al. 1999, ApJ, 517, 565
Peter A. H. G., 2012, ArXiv e-prints
Pfenniger D., 1994, in Durret F., Mazure A., Tran Thanh Van J., eds, Clusters of Galaxies Fractal cold gas as dark matter in galaxies and clusters. p. 35
Pfenniger D., Combes F., Martinet L., 1994, A&A, 285, 79
Planck Collaboration Ade P. A. R., Aghanim N., Armitage-Caplan C., Arnaud M., Ash- down M., Atrio-Barandela F., Aumont J., Baccigalupi C., Banday A. J., et al. 2014, A&A, 571, A16
Plummer H. C., 1911, MNRAS, 71, 460
Pontzen A., Governato F., 2011, ArXiv e-prints
Pontzen A., Governato F., 2012, MNRAS, 421, 3464
Pontzen A., Governato F., 2014, Nature, 506, 171
Pontzen A., Read J., Teyssier R., Governato F., Gualandris A., Roth N., Devriendt J., 2015, ArXiv e-prints
Pospelov M., Ritz A., Voloshin M., 2008, Physics Letters B, 662, 53
Preskill J., Wise M. B., Wilczek F., 1983, Physics Letters B, 120, 127
Press W. H., Davis M., 1982, ApJ, 259, 449
Press W. H., Schechter P., 1974, ApJ, 187, 425
Prokhorov D. A., Churazov E. M., 2013, ArXiv e-prints
Read J. I., 2009, Astrophysical Dynamics. University Leicester, UK
Read J. I., Gilmore G., 2005, MNRAS, 356, 107
Read J. I., Lake G., Agertz O., Debattista V. P., 2008, MNRAS, 389, 1041
Read J. I., Wilkinson M. I., Evans N. W., Gilmore G., Kleyna J. T., 2006, MNRAS, 366, 429
Refsdal S., 1964, MNRAS, 128, 307
Revaz Y., Jablonka P., 2012, A&A, 538, A82
Richardson M. L. A., Scannapieco E., Gray W. J., 2013, ApJ, 778, 80
Richardson T., Fairbairn M., 2013, MNRAS, 432, 3361
Richardson T., Fairbairn M., 2014, MNRAS, 441, 1584
129 APPENDIX.
Richter P., Fang T., Bryan G. L., 2006, A&A, 451, 767
Ricotti M., Gnedin N. Y., 2005, ApJ, 629, 259
Ricotti M., Haehnelt M. G., Pettini M., Rees M. J., 2004, MNRAS, 352, L21
Riess A. G., Filippenko A. V., Challis P., Clocchiatti A., Diercks A., Garnavich P. M., Gilliland R. L., Hogan C. J., Jha S., Kirshner R. P., et al. 1998, AJ, 116, 1009
Rix H.-W., de Zeeuw P. T., Cretton N., van der Marel R. P., Carollo C. M., 1997, ApJ, 488, 702
Robinson D. J., Tsai Y., 2014, Phys. Rev. D, 90, 045030
Rocha M., Peter A. H. G., Bullock J. S., Kaplinghat M., Garrison-Kimmel S., O˜norbe J., Moustakas L. A., 2013, MNRAS, 430, 81
Rubin V. C., 1983, Science, 220, 1339
Sawala T., Scannapieco C., Maio U., White S., 2010, MNRAS, 402, 1599
Schwarzschild M., 1979, ApJ, 232, 236
Schwarzschild M., 1993, ApJ, 409, 563
Seljak U., Makarov A., McDonald P., Trac H., 2006, Physical Review Letters, 97, 191303
Shao S., Gao L., Theuns T., Frenk C. S., 2013, MNRAS, 430, 2346
Shapiro P. R., Kang H., 1987, ApJ, 318, 32
Silk J., 1968, ApJ, 151, 459
Silverwood H., Read J. I., Steger P., 2015, MNRAS, 000, 0
Simon J. D., Geha M., 2007, ApJ, 670, 313
Skillman E. D., 2005, Nature, 49, 453
Slosar A., IrˇsiˇcV., Kirkby D., Bailey S., Busca N. G., Delubac T., Rich J., Aubourg E.,´ Bautista J. E., Bhardwaj V., et al. 2013, jcap, 4, 26
Smith M. C., Whiteoak S. H., Evans N. W., 2012, ApJ, 746, 181
Spergel D. N., Bean R., Dor´eO., Nolta M. R., Bennett C. L., Dunkley J., Hinshaw G., Jarosik N., et al. 2007, ApJS, 170, 377
Spergel D. N., Steinhardt P. J., 2000, Physical Review Letters, 84, 3760
Springel V., 2005, MNRAS, 364, 1105
Springel V., White S. D. M., Jenkins A., Frenk C. S., Yoshida N., Gao L., Navarro J., Thacker R., Croton D., Helly J., Peacock J. A., Cole S., Thomas P., Couchman H., Evrard A., Colberg J., Pearce F., 2005, Nature, 435, 629
130 BIBLIOGRAPHY
Steger P. S. P., Read J. I., 2015, MNRAS, 000, 0
Strigari L. E., Frenk C. S., White S. D. M., 2010, MNRAS, 408, 2364
Strigari L. E., Frenk C. S., White S. D. M., 2014, ArXiv e-prints
Swaters R. A., Sanders R. H., McGaugh S. S., 2010, ApJ, 718, 380
Syer D., Tremaine S., 1996, MNRAS, 282, 223
Tammann G. A., 1994, in G. Meylan & P. Prugniel ed., European Southern Observa- tory Conference and Workshop Proceedings Vol. 49 of European Southern Observatory Conference and Workshop Proceedings, Dwarf Galaxies in the Past. p. 3
Tegmark M., Dodelson S., Eisenstein D. J., Narayanan V., Scoccimarro R., Scranton R., Strauss M. A., Connolly A., Frieman J. A., et al. 2002, ApJ, 571, 191
Tegmark M., Silk J., Rees M. J., Blanchard A., Abel T., Palla F., 1997, ApJ, 474, 1
Tegmark M., Zaldarriaga M., 2002, Phys. Rev. D, 66, 103508
Teyssier R., , 2010, RAMSES: A new N-body and hydrodynamical code, Astrophysics Source Code Library
Teyssier R., Pontzen A., Dubois Y., Read J. I., 2013, MNRAS, 429, 3068
The Fermi-LAT Collaboration : Ackermann M., Ajello M., Albert A., Allafort A., 2013, ArXiv e-prints
Tisserand P., Le Guillou L., Afonso C., Albert J. N., Andersen J., Ansari R., Aubourg E.,´ Bareyre P., Beaulieu J. P., Charlot X., Coutures C., Ferlet R., Fouqu´eP., Glicenstein J. F., Goldman B., Gould A., 2007, A&A, 469, 387
Tolstoy E., Hill V., Tosi M., 2009, ARA&A, 47, 371 van Albada T. S., 1982, MNRAS, 201, 939 van Albada T. S., Bahcall J. N., Begeman K., Sancisi R., 1985, ApJ, 295, 305 van de Ven G., van den Bosch R. C. E., Verolme E. K., de Zeeuw P. T., 2006, A&A, 445, 513
Vera-Ciro C. A., Helmi A., Starkenburg E., Breddels M. A., 2012, ArXiv e-prints
Viel M., Becker G. D., Bolton J. S., Haehnelt M. G., Rauch M., Sargent W. L. W., 2008, Physical Review Letters, 100, 041304
Vikhlinin A., Kravtsov A., Forman W., Jones C., Markevitch M., Murray S. S., Van Speybroeck L., 2006, ApJ, 640, 691
Vittino A., Fornengo N., Maccione L., 2014, Nuclear Instruments and Methods in Physics Research A, 742, 145
131 APPENDIX.
Volders L. M. J. S., 1959, Bull. Astron. Inst. Netherlands, 14, 323
Volkas R., 2013, in SnowCDARK 2013 Review of Asymmetric Dark Matter Models. p. 5
Walker M., 2015, MNRAS, 000, 0
Walker M. G., Mateo M., Olszewski E. W., 2009a, AJ, 137, 3100
Walker M. G., Mateo M., Olszewski E. W., 2009b, AJ, 137, 3100
Walker M. G., Mateo M., Olszewski E. W., 2009c, AJ, 137, 3100
Walker M. G., Mateo M., Olszewski E. W., Bernstein R., Wang X., Woodroofe M., 2006, AJ, 131, 2114
Walker M. G., Mateo M., Olszewski E. W., Gnedin O. Y., Wang X., Sen B., Woodroofe M., 2007, ApJ, 667, L53
Walker M. G., Mateo M., Olszewski E. W., Pe˜narrubiaJ., Wyn Evans N., Gilmore G., 2009, ApJ, 704, 1274
Walker M. G., Pe˜narrubiaJ., 2011, ApJ, 742, 20
Walsh D., Carswell R. F., Weymann R. J., 1979, Nature, 279, 381
Wang J., White S. D. M., 2007, MNRAS, 380, 93
Weinberg S., 2008, Cosmology. Oxford University Press
Westfall K. B., Majewski S. R., Ostheimer J. C., Frinchaboy P. M., Kunkel W. E., Pat- terson R. J., Link R., 2006, AJ, 131, 375
White S. D. M., Rees M. J., 1978, MNRAS, 183, 341
Whiting A. B., Irwin M. J., Hau G. K. T., 1997, AJ, 114, 996
Wilkinson M. I., Kleyna J., Evans N. W., Gilmore G., 2002, MNRAS, 330, 778
Wolf J., Bullock J. S., 2012, ArXiv e-prints
Wolf J., Martinez G. D., Bullock J. S., Kaplinghat M., Geha M., Mu˜nozR. R., Simon J. D., Avedo F. F., 2010, MNRAS, 406, 1220
Wyrzykowski L., Skowron J., Kozlowski S., Udalski A., Szyma´nskiM. K., Kubiak M., Pietrzy´nskiG., Soszy´nskiI., Szewczyk O., Ulaczyk K., Poleski R., Tisserand P., 2011, MNRAS, 416, 2949
Zavala J., Vogelsberger M., Walker M. G., 2013, MNRAS, 431, L20
Zhao H., Bacon D. J., Taylor A. N., Horne K., 2006, MNRAS, 368, 171
Zwicky F., 1933, Helvetica Physica Acta, 6, 110
Zwicky F., 1937, Physical Review, 51, 679
132 Curriculum Vitae Pascal S.P. Steger
Personal Data
Address: Ackersteinstr. 93, 8049 Zürich, Switzerland Phone: +41 77 43 43 0 53 email: [email protected] homepage: http://steger.aero Place and Date of Birth: Zürich, Switzerland | 04 August 1986 Work Experience
Current Substitute Lecturer at Department of Physics, ETH, Zürich 2014 Extragalactic Astronomy Group Physics I
Spring Term 2014 Lead Teaching Assistant at Department of Physics, ETH, Zürich Star and Planet Formation Group Physics II, creation of exercises with solutions, and final exam
Fall Term 2013 Teaching Assistant at Department of Physics, ETH, Zürich Star and Planet Formation Group Physik-Praktikum
Spring Term 2012, 2013 Teaching Assistant Astroweek, ETH, Zürich with Dr. C. Monstein Gurnigel and Diavolezza, Radio-Astronomy: Mapping of Neutral Hydrogen using a remotely-controlled Radio Telescope
2010 IT Responsible, ETH, Zürich 2008 Dark Matter Group
2009 Scientific Collaborator, ETH, Zürich 2008 Extragalactic Astrophysics Group Data Analysis of Cosmological Simulations
2007 Typesetter of Lecture Scripts, ETH, Zürich 2006 Prof. Dr. J. Blatter Electrodynamics, Thermodynamics
2008-2009 Extended Semesterarbeit at IfA ETH, Zürich Development of analysis program package in C++ for Smoothed Par- ticle Hydrodynamics simulations. Education
March 2015 PhD sc. ETH in Astronomy, ETH Zürich, Switzerland | Astrophysics, Thesis: “Dark Matter in Small Structures” Advisors: Prof. Dr. S. Lilly, Prof. Dr. J. I. Read Dynamical Jeans modelling of spherical and axisymmetric systems, high-resolution cosmological simulations with non-equilibrium chemistry
July 2011 Master of Science, ETH Zürich, Switzerland | Major: Physics. Thesis: “Dark Matter in Dwarf Spheroidals” | Advisor: Prof. Dr. J. Read Cum Laude Grade: 5.78/6.0 | Detailed List of Exams
July 2009 Bachelor of Science ETH Zürich, Switzerland Physics, Correlations between Dark Matter, Gas and Stars of Halos Advisor: Prof. Dr. M. Carollo Grade: 4.84/6.0 | Detailed List of Exams
July 2005 Literargymnasium Rämibühl, Switzerland. | Matura. Final Grade: 5.89/6.0 July 2002 Bezirksschule Aarau, Switzerland | Final Grade: 5.9/6.0 Research Interest My main areas of research are in astrophysical probes of dark matter, gravita- tional dynamics, and computational astrophysics. I study the smallest galaxies in the Universe, where we can measure how dark matter clusters. Literature • Steger, Read, in prep.: GravImage: Non-Parametric Mass Modelling Tool for Spherical Systems • Steger, Read, in prep.: GravImage in action: Non-Parametric Mass Modelling of observed Dwarf Spheroidals • Silverwood, Steger, Read, in prep: GravImage: Non-Parametric Mass Mod- elling Tool for Disk Systems • Steger, Read, in prep.: ramsesNEC: Non-Equilibrium Chemistry in Comsmolog- ical Simulations of Dwarf Galaxies Extra-Curricular Projects
Theme/Lecture Project Big Data Correlations between Weather and Child Mortality Mining Massive Datasets MOOC, Statement of Attendance with Distinction Hackathon Zurich Predicting Timeseries for River Flows in Central Asia Agent-Based Modelling Generations in Bounded Confidence Model Autonomous Mobile Robotics Obstacle Avoidance with Inverted Fish-Eye Camera Artificial Intelligence Function Classification with a 2-layer Neural Nework Network Security Practicals: CVEs, SSH, spoofing, john IT Skills
Advanced Knowledge: Python, C, C++, Emacs, LATEX MPI, HPC, Linux, gentoo, Intermediate Knowledge: Big Data algorithms, Hadoop, CDAP, Spark, Scala, Java, Mathematica, Matlab, Octave, Fortran, Pandas, html, php, mysql, Apache, Assembler, ubuntu Basic Knowledge: Word, Excel, PowerPoint Languages
German: Mother tongue English: Fluent French: Fluent Russian: Basic Spanish, Italian: Basic Interests and Activities Amateur Radio, HB9FDO Piloting, PPL CH.FCL.54781 Technology, Open-Source, Programming Paradoxes in Decision Making, Robotics Sports (Ballroom Dancing, Tennis, Jogging) Our Universe is filled with a mysterious component called dark matter that appears to act gravitationally only. It remains a key challenge of mod- ern physics to understand its nature. One approach to probing the nature of dark matter is to predict its distribution for an assumed dark matter model and compare it to the observed Universe. This allows us to constrain or even rule out models. In the first part of this thesis, I develop a non-parametric Jeans frame- work GravImage to model the dark matter profiles of spherical systems based on kinematic data alone. I determine the data quality required to distinguish cusps from cores using a suite of mock galaxies. I then apply this method to the Fornax dwarf spheroidal – the most luminous Milky Way dwarf – to show that it has a dark matter core, consistent with earlier tim- ing arguments. Application of the method to four dwarf spheroidals with fewer stars – Sextans, Sculptor, Carina and Draco – is ongoing, with the first results showing a core for Sextans, and cusps for Sculptor, Carina, and Draco. If these cusps hold up to further tests, they will be strong evidence in favour of the prevailing cold dark matter paradigm. In the second part of this thesis, I attempt to calculate the distribu- tion of dark matter in the smallest galaxies, accounting for baryonic effects. To achieve this, I run a suite of cosmological simulations of the formation of dwarf galaxies at high redshift. Focusing on high redshifts allows me to reach unprecedented resolution, resolving individual supernovae. Such high resolution requires non-equilibrium chemistry to model cooling at low temperatures. In a first phase, I study a lower resolution pilot simulation ( 1000 M ) per dark matter particle). I show that when including baryons, the∼ central⊙ dark matter density becomes slightly steeper at radii where the simulation is well resolved. A small core ( 10 pc) forms in the innermost ∼ regions. At redshift z = 10, massive star clusters form alongside lower stellar density dwarf galaxies. The relationship between half light radius and stellar mass for these shows a striking resemblance to the dichotomy in half light radius versus stellar mass observed between globular clusters and dwarfs in the Local Group today. The massive star clusters are devoid of dark matter. In a second phase, I consider much higher resolution simulations ( 100 M per dark matter particle). This work is on-going and so far only∼ the runs⊙ without non-equilibrium chemistry have completed. Nonetheless the results are already interesting. I find that, similarly to the lower resolution pilot simulation, the dark matter density in the hydrodynamic run is steeper than in the pure dark matter simulation. This suggests that dwarf galaxies that survive to the present day as untouched fossils from reionisation may be expected to retain a pristine dark matter cusp – the smoking gun for non- relativistic cold dark matter. This is particularly exciting given the results from the first half of my thesis that favour such cusps in nearby dwarfs that contain only old stellar populations.