D C F N

D C F N
DISSERTATION
SUBMITTED TO THE
COMBINED FACULTIES OF THE NATURAL SCIENCES AND MATHEMATICS
OF THE RUPERTO-CAROLA-UNIVERSITY OF HEIDELBERG, GERMANY
FOR THE DEGREE OF
DOCTOR OF NATURAL SCIENCES
DISSERTATION
PUT FORWARD BY
SURHUD SHRIKANT MORE
BORN IN: ALIBAG, INDIA
ORAL EXAMINATION: JULY 27th , 2009
GALAXY-DARK MATTER CONNECTION
INSIGHTS FROM SATELLITE KINEMATICS
REFEREES:
PROF. DR. HANS-WALTER RIX
PROF. DR. FRANK C. VAN DEN BOSCH
asto mA sd^gmy tmso mA >yoEtgmy।
From ignorance, lead me to the truth, from darkness lead me to light
mAJF aAяF
к
{ slocnA sхArAm sAv\t
EhlA smEpt
Dedicated to the fond memory of my late grandmother
Kai. Sulochana Sakharam Sawant
Abstract
A thorough knowledge of the connection between the mass of dark matter haloes and the
properties of their central galaxies is crucial to understand the physics of galaxy formation.
The kinematics of satellite galaxies is an excellent technique to measure the dark matter halo
masses. However, the kinematics can be measured with high signal-to-noise only by stacking
the signal around central galaxies with similar properties, which results in various systematic
biases and complicates the interpretation of the signal. This thesis presents an analytical framework that accounts for systematic biases and selection effects and aids in the interpretation of
the kinematics of satellite galaxies. A new method is established to obtain the average scaling
relations between halo mass and central galaxy properties, and the scatter in these relations simultaneously. After a thorough testing of this method using a realistic mock galaxy catalogue,
it is applied to the Sloan Digital Sky Survey to extract the halo mass-luminosity and halo massstellar mass relationship of central galaxies and their scatter. Comparisons with other probes
of these scaling relations, such as galaxy−galaxy lensing, show good agreement which implies
that these scaling relations are well established and supported by various astrophysical probes.
Physical insights about these scaling relations, in particular their scatter, gained by the analysis of a semi-analytical model for galaxy formation are also presented. Finally, the inferred
scaling relations crucially depend on the transparency of the Universe. By performing a test
of the “Etherington relation” between the distances measured by standard rulers and by standard candles, a quantitative measure of the cosmic transparency, which is relatively free from
astrophysical assumptions, is obtained.
Zusammenfassung
Um die physikalischen Mechanismen der Galaxienentwicklung zu verstehen, müssen die
Zusammenhänge zwischen der Masse eines Dunkle-Materie-Halos und den Eigenschaften seiner
Zentralgalaxie bekannt sein. Die Bestimmung der Kinematik von Satellitengalaxien ist eine
etablierte Methode zur Messung von Halomassen. Allerdings sind kinematische Messungen
mit hoher Signifikanz nur dadurch möglich, dass die Signale über viele Zentralgalaxien mit
ähnlichen Eigenschaften gemittelt werden. Dies hat systematische Fehler zur Folge und erschwert die Interpretation kinematischer Beobachtungen. In dieser Dissertation wird ein analytisches Modell vorgestellt, mit dessen Hilfe systematische Verzerrungen und Auswahleffekte
korrigiert werden können, was die Auswertung kinematischer Daten von Satellitengalaxien erleichtert. Hieraus folgt ein Verfahren zur Bestimmung der Skalierungsvorschrift zwischen Halomasse und Eigenschaften der Zentralgalaxie, die gleichzeitig die Streuung über diese Relation
vorhersagt. Diese Gültigkeit dieses Verfahrens wird anhand eines künstlichen Galaxienkatalogs mit bekannten, aber realistischen Eigenschaften verifiziert. Aus der Anwendung auf die
Beobachtungen der Sloan Digital Sky Survey werden die Skalierungsvorschriften zwischen
Halomasse einerseits und Leuchtkraft sowie stellarer Masse andererseits für Zentralgalaxien
mitsamt dazugehöriger Streuung bestimmt. In Vergleichen zeigt sich eine gute übereinstimmung
mit anderen Messverfahren für diese Vorschriften, wie z.B. der Beobachtung von Gravitationslinseneffekten der Zentralgalaxien auf andere Galaxien. Dies bestätigt die Zuverlässigkeit der
hier vorgestellten Skalierungen. Darüber hinaus ergeben sich aus der Analyse eines semiempirischen Modells für Galaxienentstehung neue Erkenntnisse über die physikalischen Ursachen der vorgestellten Skalierungen und insbesonderere ihrer Streuung. Schlielich hängen
die aus unserem Modell erhaltenen Skalierungen entscheidend von der Transparenz des Universums ab. Vermittels einer überprüfung des “Etherington’schen Gesetzes” für den Zusammenhang zwischen Entfernungen, die aus Standardlängen beziehungsweise Standardkerzen bestimmt werden, wird die kosmische Transparenz in einer Weise bestimmt, die relativ unabhängig
von astrophysikalischen Annahmen ist.
Contents
Table of Contents
i
List of Figures
v
List of Tables
vii
1
2
3
Introduction
1
1.1
Galaxy Formation in a Dark Universe . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Satellite Kinematics: The Analytical Formalism
7
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Weighting Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.4
Selection Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.5
More Realistic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.A Comparison with van den Bosch et al. . . . . . . . . . . . . . . . . . . . . . .
23
Satellite Kinematics: Tests on a Mock Catalogue
25
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.2
Mock Catalogue Construction . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.3
Selection Criteria to identify Centrals and Satellites . . . . . . . . . . . . . . .
29
3.4
Satellite Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.4.1
Unbinned Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.4.2
Binned Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.4.3
Analytical Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.5
Number Density Distribution of Satellites . . . . . . . . . . . . . . . . . . . .
39
3.6
Mass−Luminosity Relationship . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.6.1
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.6.2
Monte-Carlo Markov Chain . . . . . . . . . . . . . . . . . . . . . . .
43
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.7
i
ii
4
CONTENTS
3.A Sampling of Central Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
The Halo Mass−Luminosity Relationship
49
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2
Central and Satellite Samples from the SDSS . . . . . . . . . . . . . . . . . .
51
4.3
Velocity Dispersion−Luminosity Relation . . . . . . . . . . . . . . . . . . . .
52
4.4
Number Density Distribution of Satellites in SDSS . . . . . . . . . . . . . . .
57
4.5
Results from the MCMC Analysis . . . . . . . . . . . . . . . . . . . . . . . .
60
4.5.1
The Halo Mass−Luminosity Relation . . . . . . . . . . . . . . . . . .
60
4.5.2
The Colour Dependence of the Halo Mass−Luminosity Relation . . . .
68
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.6
5
The Halo Mass−Stellar Mass Relationship
73
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.2
Central and Satellite Samples from the SDSS . . . . . . . . . . . . . . . . . .
74
5.3
Velocity Dispersion−Stellar Mass Relation . . . . . . . . . . . . . . . . . . .
75
5.4
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.5
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
5.5.1
The Halo Mass−Stellar Mass Relationship . . . . . . . . . . . . . . .
83
5.5.2
Comparison of MSR with Other Studies . . . . . . . . . . . . . . . . .
89
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
5.6
6
7
8
On the Stochasticity of Galaxy Formation
93
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.2
Semi-analytical Models of Galaxy Formation . . . . . . . . . . . . . . . . . .
94
6.3
Numerical Simulation and Halo Merger Trees . . . . . . . . . . . . . . . . . .
97
6.4
Halo Formation Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
6.5
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Cosmic Transparency
107
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2
Data, Procedure, and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.3
Future Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Summary
8.1
119
Future possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.1.1
Properties of Satellite Galaxies . . . . . . . . . . . . . . . . . . . . . . 121
8.1.2
Shapes of Dark Matter Haloes . . . . . . . . . . . . . . . . . . . . . . 122
8.1.3
Redshift Evolution of the Halo Occupation Distributions . . . . . . . . 122
CONTENTS
iii
Acknowledgments
125
Bibliography
127
List of Figures
2.1
Velocity dispersion of satellites in the satellite-weighted and the host-weighted
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
Comparison of velocity dispersions for models with different scatter . . . . . .
18
2.3
Illustration of the MLR of central galaxies. . . . . . . . . . . . . . . . . . . . .
21
2.4
Comparison with the analytical estimates of van den Bosch et al. (2004) . . . .
24
3.1
Schematic diagram of a selection criterion . . . . . . . . . . . . . . . . . . . .
30
3.2
Scatter plot of ∆V (Lc ) for MOCKV . . . . . . . . . . . . . . . . . . . . . . .
33
3.3
Satellite-weighted P (∆V ) distributions of satellites in MOCKV . . . . . . . .
34
3.4
Velocity dispersion of satellites in MOCKV . . . . . . . . . . . . . . . . . . .
35
3.5
Projected number density distributions of satellites in MOCKV . . . . . . . . .
38
3.6
Results of the MCMC analysis of the velocity dispersions obtained from MOCKV 41
3.7
Sampling of central galaxies by different selection criteria . . . . . . . . . . .
47
4.1
Colour−Luminosity plot of galaxies in the SDSS . . . . . . . . . . . . . . . .
52
4.2
Scatter plot of ∆V (Lc ) using satellites of all central galaxies . . . . . . . . . .
54
4.3
Scatter plots of ∆V (Lc ) for satellites of red and blue centrals . . . . . . . . . .
54
4.4
Satellite velocity dispersions around centrals stacked by luminosity . . . . . . .
55
4.5
Dependence of the velocity dispersion−luminosity relation on the colour of
centrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.6
Projected number density distributions of satellites in Sample LA . . . . . . . .
57
4.7
Projected number density distributions of satellites in Sample LR . . . . . . . .
58
4.8
Projected number density distributions of satellites in Sample LB . . . . . . . .
59
4.9
Results of the MCMC analysis of the velocity dispersions from Sample LA . .
61
4.10 Posterior distribution of the parameter σlog L obtained from the MCMC analysis
64
4.11 Results of the MCMC analysis of the velocity dispersions from Sample LR . .
65
4.12 Results of the MCMC analysis of the velocity dispersions from Sample LB . .
66
4.13 Colour dependence of the MLR of central galaxies . . . . . . . . . . . . . . .
68
4.14 Comparison of the MLR of central galaxies by different methods . . . . . . . .
70
Scatter plot of ∆V (M∗c ) using satellites of all central galaxies . . . . . . . . .
76
5.1
v
vi
LIST OF FIGURES
5.2
Scatter plots of ∆V (M∗c ) for satellites of red and blue centrals . . . . . . . . .
76
5.3
Satellite velocity dispersions around centrals stacked by stellar mass . . . . . .
77
5.4
Dependence of velocity dispersion−stellar mass relation on the colour of centrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5.5
Projected number density distributions of satellites in Sample SA . . . . . . . .
80
5.6
Projected number density distributions of satellites in Sample SR . . . . . . . .
81
5.7
Projected number density distributions of satellites in Sample SB . . . . . . . .
82
5.8
Results of the MCMC analysis of the velocity dispersions from Sample SA . .
85
5.9
Results of the MCMC analysis of the velocity dispersions from Sample SR . .
86
5.10 Results of the MCMC analysis of the velocity dispersions from Sample SB . .
87
5.11 Colour dependence of the MSR of central galaxies . . . . . . . . . . . . . . .
88
5.12 Comparison of the MSR of central galaxies obtained by different methods . . .
90
6.1
Schematic diagram of a merger tree . . . . . . . . . . . . . . . . . . . . . . .
99
6.2
Scatter plot of the stellar mass of central galaxies against their halo masses . . . 100
6.3
Dependence of formation redshifts on the halo mass . . . . . . . . . . . . . . . 101
6.4
Scatter plot of the stellar mass of central galaxies versus the formation redshifts
of their haloes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.5
Scatter plot of residuals in the formation redshift−halo mass relation versus the
residuals in the stellar mass−halo mass relation . . . . . . . . . . . . . . . . . 104
6.6
Dependence of stellar mass−halo mass relation on formation redshifts . . . . . 105
7.1
Distance-modulus–redshift relation . . . . . . . . . . . . . . . . . . . . . . . . 112
7.2
Posterior distribution of ∆τ between z = 0.35 and z = 0.20 . . . . . . . . . . 114
List of Tables
2.1
Different models for the HOD of centrals . . . . . . . . . . . . . . . . . . . .
17
3.1
Selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2
Selection criteria parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.3
MOCKV: Parameters recovered from the MCMC . . . . . . . . . . . . . . . .
42
4.1
Selection criteria parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.2
Sample LA: Velocity dispersion measurements . . . . . . . . . . . . . . . . .
53
4.3
Sample LR: Velocity dispersion measurements . . . . . . . . . . . . . . . . .
56
4.4
Sample LB: Velocity dispersion measurements . . . . . . . . . . . . . . . . .
56
4.5
MLR of central galaxies: Parameters recovered from the MCMC . . . . . . . .
62
4.6
Sample LA: MLR of central galaxies . . . . . . . . . . . . . . . . . . . . . . .
62
4.7
Sample LR: MLR of red central galaxies . . . . . . . . . . . . . . . . . . . . .
67
4.8
Sample LB: MLR of blue central galaxies . . . . . . . . . . . . . . . . . . . .
67
5.1
Selection criteria parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.2
Sample SA: Velocity dispersion measurements . . . . . . . . . . . . . . . . .
78
5.3
Sample SR: Velocity dispersion measurements . . . . . . . . . . . . . . . . . .
79
5.4
Sample SB: Velocity dispersion measurements . . . . . . . . . . . . . . . . . .
79
5.5
MSR of central galaxies: Parameters recovered from the MCMC . . . . . . . .
84
5.6
Sample SA: MSR of central galaxies . . . . . . . . . . . . . . . . . . . . . . .
84
5.7
Sample SR: MSR of red central galaxies . . . . . . . . . . . . . . . . . . . . .
88
5.8
Sample SB: MSR of blue central galaxies . . . . . . . . . . . . . . . . . . . .
89
vii
Chapter 1
Introduction
1.1
Galaxy Formation in a Dark Universe
The vast ocean of space is full of starry islands called galaxies, such as our own Milky Way.
Galaxies act as lighthouses in this vast ocean, serving as an interface with which we can explore
and understand our Universe. The majesty and the variety of galaxies has often boggled the
human mind. It is more curious however that, in our current understanding, galaxies form a
very small portion of the energy content of the Universe. Most of the energy content of the
Universe today is “dark” − the two dominant components “dark energy” and “dark matter”
account for nearly 95 percent of the energy density of the Universe. The rest is ordinary matter
primarily present in the form of gas in the intergalactic medium and around galaxies (referred
to as baryons). How do galaxies come into existence in this dark Universe and how do they
evolve? What is the relation of galaxies to the dark components in the Universe? What shapes
the properties of different galaxies? How are different properties of galaxies correlated with
each other and what is the physics that drives these correlations? These questions, among
others, currently drive the research field of galaxy formation and evolution.
The origin and the nature of both the dark components of the Universe is still a mystery.
Although the presence of dark matter can be motivated theoretically from (currently untested)
ideas in particle physics that are based upon supersymmetry (Preskill et al. 1983; Ellis et al.
1984), the presence of dark energy and its ubiquitous nature has little theoretical motivation
(see e.g., Dolgov 2004). The evidence for the presence of both the components is purely astrophysical. Dark energy manifests itself through the recently-discovered accelerated expansion
of the Universe (Riess et al. 1998; Perlmutter et al. 1999; Kowalski et al. 2008) while dark
matter makes its presence felt only through its gravitational effects (Zwicky 1933; Rubin et al.
1982). All attempts to detect the elusive dark matter through a wide range of non-gravitational
experiments have largely been inconclusive (see e.g., Benoit et al. 2002; Akerib et al. 2003,
2004; Sanglard et al. 2005). These negative results have led some to favour the radical approach
of modifying Newton’s theory of gravity in the weak field limit (Milgrom 1983a,b,c) and its
relativistic version (Bekenstein 2004). Whether this approach can explain all the observed as1
2
1. INTRODUCTION
trophysical phenomena which suggest the presence of dark matter − the jury is still out (see
e.g., McGaugh & de Blok 1998; Sanders & McGaugh 2002; Sanders 2003; Clowe et al. 2004;
Klypin & Prada 2009).
In the simplest picture, that conforms to a wide range of astrophysical observations, dark
energy is assumed to be an all-pervading, non-clumpy form of energy and is generally attributed
to the vacuum. Dark matter, on the other hand, is supposed to be dynamically cold and collisionless, but clumpy due to the effects of gravity. These two dark components form the backbone
of the ΛCDM theory (Λ stands for dark energy, CDM for cold dark matter). According to this
theory, the early Universe started off as a dense hot soup of elementary particles and underwent
a rapid inflationary phase where the tiny fluctuations of a (hypothesized scalar) quantum field
were stretched to cosmologically large scales. These fluctuations were imprinted onto the initial
density field of the particles. Dark matter, the most abundant gravitationally unstable component in the Universe, was then responsible for the formation of structure in the Universe. The
tiny initial fluctuations in the density field grew over time by the action of gravity and formed
bound structures (haloes). Baryons were trapped within the gravitational potential of these dark
matter haloes and they underwent a series of complex physical processes to form the galaxies
that we observe today.
The great thing about the ΛCDM theory is its ability to make testable predictions. Given
the power spectrum of the initial density fluctuations and the energy density parameters of the
various components of the Universe, the statistical properties of the dark matter distribution can
be accurately predicted (see e.g., Eisenstein & Hu 1999). As the dark matter distribution is not
directly observable, establishing the link between the observable galaxies to their dark matter
haloes is central to test this prediction (see e.g., Tegmark et al. 2004). The theory also predicts
that the properties of dark matter haloes, in particular the mass, should shape the properties of
galaxies that form within them. Precise measurements of the scaling relations between different
galaxy properties with the mass of a dark matter halo can also provide key insights into the
galaxy-dark matter connection predicted by this theory. This is precisely the aim of this thesis.
We wish to investigate the connection between different galaxy properties and their dark matter
haloes.
There are various approaches to study the galaxy-dark matter connection. One approach is
to study such a connection via direct numerical simulations (see e.g., Katz et al. 1992; Evrard
et al. 1994; Frenk et al. 1996, 1999; Katz et al. 1996; Navarro et al. 1997; Pearce et al. 1999;
Kravtsov 1999). This involves following the evolution of the density field and its fluctuations,
and the various astrophysical processes that transform the baryons into luminous galaxies. In
this approach, the gravitational and hydrodynamical equations need to be solved in full generality. The resultant population of galaxies can then be compared with the observed population of
galaxies in the Universe. One drawback of this approach is the tremendous computational expense of simulating a cosmologically meaningful volume with sufficient resolution and within
a reasonable amount of time.
1.1. GALAXY FORMATION IN A DARK UNIVERSE
3
The second approach, called semi-analytical modelling, improves upon the former by separating the evolution of the dark matter component and the baryonic component (see e.g., White
& Rees 1978; White & Frenk 1991). The evolution of the dark matter component is followed
numerically (or by using Monte-Carlo techniques) while the evolution of the baryonic component in the distribution of dark matter is followed by using simple analytical recipes. This
approach of modelling the formation of galaxies can be used to compute the properties of a large
population of galaxies and establish their link to the underlying dark matter distribution (e.g.,
White & Frenk 1991; Kauffmann & White 1993; Cole et al. 1994; Kauffmann 1996; Kauffmann et al. 1997; Baugh et al. 1998; Somerville & Primack 1999; Cole et al. 2000; Benson
et al. 2002; Springel et al. 2005; Croton et al. 2006; De Lucia & Blaizot 2007). The advantage
of this method is its flexibility. It is relatively easy to test the effects of the various assumptions
and parameters involved in the modelling on the final properties of the modelled galaxies (see
e.g., Cole et al. 2000). The first approach is complementary to the semi-analytical approach,
because the simple analytical recipes often have to be calibrated against high resolution hydrodynamical simulations which focus on a small volume. One drawback of the semi-analytical
approach is that any back reaction of the baryons on the dark matter haloes are either neglected
or are included a posteriori.
The third approach to investigate the link between galaxies and their dark matter haloes is
statistical in its nature. In this approach the connection between galaxies and dark matter haloes
is specified by a halo occupation model (for an excellent review, see Cooray & Sheth 2002).
The model uses a few parameters to specify the distribution of various properties of galaxies as
a function of the mass of the halo in which they reside. Given the properties of the dark matter
haloes, such as their abundance, their clustering strength and their density profiles (usually
obtained from dark matter only numerical simulations), these models can be easily used to make
analytical predictions for various observational properties of galaxies. The observed properties
of the real-world galaxies can then be used to constrain the parameters of the halo occupation
model and thus establish the link between galaxies and their dark matter haloes (Bullock et al.
2002; Berlind & Weinberg 2002; Berlind et al. 2003; Wang et al. 2004; Abazajian et al. 2005;
Zheng et al. 2005; van den Bosch et al. 2007; Zheng et al. 2007; Cacciato et al. 2009). In this
thesis, we will focus on this third approach and infer the halo occupation distribution of galaxies
by probing the dark matter haloes of galaxies.
Various observational probes can be used to measure the masses of dark matter haloes and
subsequently connect it to the properties of the galaxies. These include various methods that
focus on measuring the kinematics of a tracer population in the halo around individual galaxies.
For example, kinematic measurements of the gas around spiral galaxies observed in the optical
and the radio wavelengths have been traditionally used as a strong evidence for the presence of
dark matter haloes (e.g., Shostak 1973; Roberts & Whitehurst 1975; Bosma 1978; Rubin et al.
1978, 1982; Sofue & Rubin 2001). Unfortunately, these probes do not trace the entire extent
of the dark matter halo and hence can at best be used to measure the masses of only the inner
4
1. INTRODUCTION
regions of the haloes. The X-ray emission from hot gas in clusters can also be used to measure
the dark matter halo masses in these systems under simplifying assumptions of hydrostatic
equilibrium and spherical symmetry (e.g., Mushotzky et al. 1978). Strong gravitational lensing,
manifested by the presence of multiple images or highly magnified arcs of background objects,
is yet another important probe of the halo masses in individual systems (see e.g., Schneider
et al. 1992). Being a purely gravitational effect, lensing has the advantage of being able to
probe mass without the need for simplifying assumptions about the relaxedness of the system
under consideration. However, degeneracies in the modelling of lens systems and projection
effects can cause some trouble in the interpretation of the lensing observations.
With the advent of large scale galaxy redshift surveys in the last decade, such as the the
Sloan Digital Sky Survey (SDSS; York et al. 2000) and the Two degree Field Galaxy Redshift
Survey (2dFGRS; Colless et al. 2001), new methods have been developed to investigate the
galaxy-dark matter connection. These methods do not focus on individual systems but rather
examine the statistical properties of the galaxy distribution to infer the halo mass properties
on average. The galaxy redshift surveys can be used to reliably determine the abundance of
galaxies as a function of their properties such as luminosity or stellar mass (e.g., Norberg et al.
2002b; Bell et al. 2003; Blanton et al. 2003b; Panter et al. 2004). Another statistical property is
the clustering of the galaxy distribution, given by the two-point correlation function measured
as a function of galaxy properties (e.g., Zehavi et al. 2002; Norberg et al. 2002a; Madgwick
et al. 2003; Zehavi et al. 2004, 2005; Wang et al. 2007). Since the abundance and clustering
of dark matter haloes is a function of the mass of the halo, the abundance and clustering of
galaxies can also be used to constrain the halo occupation distribution of galaxies (e.g., Jing &
Suto 1998; Peacock & Smith 2000; Bullock et al. 2002; Berlind & Weinberg 2002; Wang et al.
2004; Abazajian et al. 2005; van den Bosch et al. 2007; Zheng et al. 2007; Cacciato et al. 2009).
The dark matter haloes around galaxies can cause weak tangential distortions in the shapes
of background galaxies. This effect known as galaxy-galaxy lensing is yet another statistical
way to probe the dark matter haloes around galaxies (e.g., Tyson 1987; Brainerd et al. 1996;
dell’Antonio & Tyson 1996; Hudson et al. 1998; Hirata et al. 2004; Mandelbaum et al. 2006b).
Measurements of the galaxy−galaxy lensing signal can be used to constrain the properties of
dark matter haloes around galaxies (e.g., Schneider & Rix 1997; Wilson et al. 2001; Guzik &
Seljak 2002; Parker et al. 2007; Cacciato et al. 2009). Since this effect is very weak, a stacking
procedure has to be adopted in which the signal around galaxies of similar properties is added
to improve the signal-to-noise ratio. Such a measurement then probes the average dark matter
halo of galaxies as a function of the property used to stack the galaxies.
In this thesis, we scrutinize another powerful method that involves measuring the kinematics
of satellite galaxies that orbit the dark matter haloes of central galaxies to measure the masses
of these haloes. Satellite galaxies trace the dark matter halo in its entirety and hence are useful
to probe the dark matter haloes around galaxies. This method is of historical significance as
its application to the Coma cluster of galaxies had led to the discovery of dark matter (Zwicky
1.2. THESIS OVERVIEW
5
1933). Precise measurement of the kinematics of satellite galaxies is only possible in systems,
such as clusters, that have a large number of satellites (e.g., Carlberg et al. 1996, 1997). The
number of satellites in low mass systems is too small to provide a reliable measure of the
kinematics. However, stacking methods were soon pioneered that enabled the measurement of
the kinematics of satellites of central galaxies in low mass haloes stacked by their luminosities
(Erickson et al. 1987; Zaritsky et al. 1993; Zaritsky & White 1994; Zaritsky et al. 1997). These
studies involved a modest number of satellite galaxies (. 100), but were nevertheless succesful
in establishing the presence of extended dark matter haloes around spiral galaxies.
The sample of satellite galaxies used to measure the kinematics received an order of magnitude boost in number after data from large scale redshift surveys became available (McKay
et al. 2002). This has led to a number of interesting studies that have measured the scaling
relations between galaxy properties and their dark matter haloes (Brainerd & Specian 2003;
Prada et al. 2003; van den Bosch et al. 2004; Becker et al. 2007; Conroy et al. 2007; Norberg
et al. 2008). Qualitatively all studies agree on the fact that the velocity dispersion of satellites
correlates positively with the property (luminosity/stellar mass) used to stack central galaxies
which in turn implies that the mass of dark matter haloes increases as a function of the stacking
property. However, there are quantitative disagreements about the exact scaling relations that
are inferred by these studies. The contrast between the results of different studies was recently
highlighted in Norberg et al. (2008). The differences between various studies were attributed to
the different criteria used to select the samples of centrals and satellites by the previous studies.
The research work presented in this thesis aims to understand how selection effects and
systematic biases affect the kinematics of satellite galaxies and establish a new method based
on satellite kinematics that can be used to measure the scaling relations between dark matter
haloes and central galaxies in an unbiased manner. We also discuss the implications of these
results on the physics of galaxy formation.
1.2
Thesis Overview
Chapter 2 introduces the reader to the theoretical framework that can be used to analyse the
kinematics of satellites around central galaxies stacked according to their properties. In this
chapter, we present a degeneracy problem that has been hitherto ignored in the analysis of
satellite kinematics. We show that the kinematics of satellite galaxies cannot be used to infer
a unique relation between halo masses and the property used to stack central galaxies if the
scatter in this relation is unknown. In this chapter, we also present a novel method that has the
potential to break this degneracy and measure both the average scaling relation and its scatter
from the kinematics of satellite galaxies.
In Chapter 3, we test the feasibility of the application of the method presented in Chapter 2
to realistic galaxy surveys. For this purpose, we first create an artificial universe by populating
galaxies in a dark matter only simulation. By mimicking the flux limited nature of galaxy
6
1. INTRODUCTION
observations in redshift surveys, we create a mock galaxy catalogue from this artificial universe.
This mock catalogue is then used to test how various selection effects affect the measurement
of the kinematics of satellite galaxies. We also show that the new method we proposed can
reliably recover both the average scaling relation between halo masses and luminosity and its
scatter originally present in the mock catalogue.
The next two chapters deal with the application of these methods to actual data from the
SDSS. Chapter 4 focusses on the halo mass−luminosity relationship of central galaxies and
infer this relation from the kinematics of satellite galaxies. The focus of Chapter 5 is the halo
mass-stellar mass relationship of central galaxies. We show that the average scaling relations
derived by our method are in excellent agreement with several other probes of these relations.
We find that both the relations demonstrate an appreciable scatter and present quantitative measurements of the same. The scatter in these relations is a result of the stochasticity in galaxy
formation.
In Chapter 6, we attempt to gain physical insights on the origin of the stochasticity in galaxy
formation that we constrained using satellite kinematics. For this purpose, we use a semianalytical model of galaxy formation. We analyse the scatter in the merger histories of haloes
of similar masses and explore its effects on the properties of the galaxies that form at its center.
We quantify the merger histories of haloes by considering various definitions for the formation
times of haloes and show that haloes that form earlier on average host central galaxies that have
a larger stellar mass.
The properties of galaxies that we observe such as their luminosity are often based upon
the assumption that the Universe is transparent. In Chapter 7, we obtain a quantitative measure of the transparency of the local Universe in the optical bands. We perform a test of the
“Etherington relation” by checking for the consistency of the luminosity distances obtained by
supernova Ia experiments and the angular diameter distances obtained by experiments that detect the baryon accoustic feature in the power spectrum of galaxies. Note that, such measures
of the transparency of the universe are important given the fact that the only evidence of the
presence of dark energy is the dimming of distant supernovae in the Universe and this could be
mimicked by the presence of opacity in the Universe.
Finally, in Chapter 8, a summary of the results obtained in this thesis is presented with a
short discussion on the possibilities of future work in this field.
Chapter 2
Satellite Kinematics: The Analytical
Formalism
The contents of this chapter are based upon the article More et al. (2009b) published in the
Monthly Notices of the Royal Astronomical Society. The reference is
More, S., van den Bosch, F. C., & Cacciato, M. 2009, MNRAS, 392, 917.
The introduction from the article published above has been slightly modified to avoid repetition
of material from the introductory chapter in the thesis.
2.1
Introduction
According to the current paradigm, the mass of a dark matter halo is believed to strongly influence the process of galaxy formation and thus shape the properties of the galaxies that form and
reside at their centres (hereafter referred to as central galaxies). Hence, a reliable determination
of scaling relations between halo mass and properties of their central galaxies can provide important constraints on the physics of galaxy formation. Determination of such scaling relations
require precise measurements of the halo mass. Numerous methods are available to probe the
masses of dark matter haloes (see Chapter 1). In this thesis, we scrutinize in detail the method
which uses the kinematics of satellite galaxies that orbit within the halo of their central galaxies,
to measure the masses of dark matter haloes.
The kinematics of satellite galaxies in individual cluster-sized haloes can be reliably measured as they host a large number of satellite galaxies which properly sample the line-of-sight
(hereafter los) velocity distribution of their haloes. The extension of this analysis to group-scale
and galaxy-scale haloes necessitates the use of stacking methods (Erickson et al. 1987; Zaritsky
et al. 1993; Zaritsky & White 1994; Zaritsky et al. 1997). Under the assumption that galaxies with similar properties (e.g. luminosities) reside in haloes of similar mass, these methods
combine the velocity information of satellite galaxies that revolve around such central galaxies. The kinematics of such a stacked system is then used to infer the average halo mass of the
8
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
stacked central galaxies. More recent studies (McKay et al. 2002; Prada et al. 2003; Brainerd &
Specian 2003; van den Bosch et al. 2004; Conroy et al. 2007; Becker et al. 2007) apply similar
stacking procedures to central galaxies selected from the large homogeneous galaxy redshift
surveys such as the Sloan Digital Sky Survey (SDSS; York et al. 2000) and the Two degree
Field Galaxy Redshift Survey (2dFGRS; Colless et al. 2001). All these studies find that the los
velocity dispersion of satellite galaxies, σsat , increases with the luminosity of the host (central)
galaxy, Lc . This is in agreement with the expectation that more massive haloes host more luminous centrals. In a recent study, Norberg et al. (2008) have shown that there exist quantitative
discrepancies between these previous studies and these discrepancies arise mainly due to the
differences in the criteria used to select central hosts and their satellites. This underscores the
necessity for a careful treatment of selection effects in order to extract reliable mass estimates
from satellite kinematics.
Except for van den Bosch et al. (2004), all previous studies have been extremely conservative in their selection of hosts and satellites. Consequently, despite the fact that the redshift
surveys used contain well in excess of 100,000 galaxies, the final samples only contained about
2000 − 3000 satellite galaxies. This severely limits the statistical accuracy of the velocity dispersion measurements as well as the dynamic range in luminosity of the central galaxies for
which halo masses can be inferred. The main motivation for using strict selection criteria is to
select only ‘isolated’ systems, with satellites that can be treated as tracer particles (i.e., their
mass does not cause significant perturbations in the gravitational potential of their host galaxy).
Let P (M |Lc ) denote the conditional probability distribution that a central galaxy of luminosity
Lc resides in a halo of mass M . If the scatter in P (M |Lc ) is sufficiently small, preferentially
selecting ‘isolated’ systems should yield an unbiased estimate of hM i(Lc ), which is the first
moment of P (M |Lc ). However, very little is known about the actual amount of scatter in
P (M |Lc ) and different semi-analytical models for galaxy formation make significantly different predictions (see discussion in Norberg et al. 2008). If appreciable, the scatter will severely
complicate the interpretation of satellite kinematics, and may even cause a systematic bias (van
den Bosch et al. 2004; More et al. 2009c). Furthermore, even if the scatter is small, in practice,
satellites of central galaxies stacked in finite bins of luminosity are used to measure the kinematics. If the satellite sample is small, one has to resort to relatively large bins in order to have
sufficient signal-to-noise. Therefore, even if the distribution P (M |Lc ) is relatively narrow, this
still implies mixing the kinematics of haloes spanning a relatively large range in halo masses.
In this chapter, we demonstrate that whenever the scatter in P (M |Lc ) is non-negligible,
the σsat (Lc ) inferred from the data has to be interpreted with great care. In particular, we
demonstrate that there is a degeneracy between the first and second moments of P (M |Lc ), in
that two distributions with different hM i(Lc ) and different scatter can give rise to the same
σsat (Lc ). Therefore, a unique hM i(Lc ) cannot be inferred from satellite kinematics without a
prior knowledge of the second moment of P (M |Lc ). However, not all hope is lost. In fact, we
demonstrate that by using two different methods to measure σsat (Lc ), one can actually break
2.2. WEIGHTING SCHEMES
9
this degeneracy and thus constrain both the mean and the scatter of P (M |Lc ). In this chapter
we introduce the methodology, and present the analytical framework required to interpret the
data, taking account of the selection criteria used to identify the central host galaxies and their
satellites. In Chapter 4, we apply this method to the SDSS to infer both the mean and the scatter
of P (M |Lc ), which we show to be in good agreement with the results obtained from clustering
and galaxy−galaxy lensing analyses. In addition, in Chapter 4 we demonstrate that (i) the scatter in P (M |Lc ) can not be neglected, especially not at the bright end, and (ii) the strict isolation
criteria generally used to select centrals and satellites result in a systematic underestimate of the
actual hM i(Lc ).
This chapter is organized as follows. In Section 2.2, we present two different schemes to
measure the velocity dispersion, the satellite-weighting scheme and the host-weighting scheme.
In Section 2.3, we present a toy model which serves as a basis for understanding the dependence
of velocity dispersion estimates on the different parameters of interest. In Sections 2.4 and 2.5
we refine our toy model by including selection effects and by using a realistic halo occupation
distribution (HOD) model for the central galaxies. We use these more realistic models to investigate how changes in the halo occupation statistics of central galaxies affect the velocity
dispersion of satellite galaxies, and we demonstrate how the combination of satellite-weighting
and host-weighting can be used to infer both the mean and the scatter of the mass−luminosity
relation. We summarize our findings in Section 2.6. Throughout this chapter, M denotes the
halo mass in units of h−1 M .
2.2
Weighting Schemes
In order to estimate dynamical halo masses from satellite kinematics one generally proceeds as
follows. Using a sample of satellite galaxies, one determines the distribution P (∆V ), where
∆V is the difference in the line-of-sight velocity of a satellite galaxy and its corresponding
central host galaxy. The scatter in the distribution P (∆V ) (hereafter the velocity dispersion),
is then considered to be an estimator of the depth of the potential well in which the satellites
orbit, and hence of the halo mass associated with the central. In order to measure the velocity
dispersion as a function of central galaxy luminosity, σsat (Lc ), with sufficient signal-to-noise,
one has to combine the los velocity information of satellites which belong to centrals of the
same luminosity, Lc . This procedure is influenced by two effects, namely mass-mixing and
satellite-weighting, which we now discuss in turn.
Mass-mixing refers to combining the kinematics of satellites within haloes of different
masses. The mass−luminosity relation (hereafter MLR) of central galaxies can have an appreciable scatter, i.e., the conditional probability distribution P (M |Lc ) is not guaranteed to be
narrow. In this case, the satellites used to measure σsat (Lc ) reside in halo masses drawn from
this distribution, and σsat (Lc ) has to be interpreted as an average over P (M |Lc ).
In most studies to date, the technique used to measure σsat (Lc ) implies satellite weighting.
10
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
This can be elucidated as follows. Let us assume that one stacks Nc central galaxies, and that
P c
the j th central has Nj satellites. The total number of satellites Nsat is given by N
j=1 Nj . Let
∆Vij denote the los velocity difference between the ith satellite and its central galaxy j. The
average velocity dispersion of the stacked system, σsw , is such that
PNc PNj
2
σsw
=
2
i=1 (∆Vij )
j=1
PNc
j=1 Nj
=
Nc
1 X
Nj σj2 .
Nsat
(2.1)
j=1
Here σj is the velocity dispersion in the halo of the j th central galaxy. The velocity dispersion
measured in this way is clearly a satellite-weighted average of the velocity dispersion σj around
each central galaxy 1 . Although not necessarily directly using Eq. (2.1), most previous studies
have adopted this satellite-weighting scheme (McKay et al. 2002; Brainerd & Specian 2003;
Prada et al. 2003; Norberg et al. 2008).
In principle, the satellite-weighting can be undone by introducing a weight wij = 1/Nj for
each satellite−central pair in the los velocity distribution (van den Bosch et al. 2004; Conroy
et al. 2007). The resulting host-weighted average velocity dispersion, σhw , is such that
PNc PNj
2
σhw
2
i=1 wij (∆Vij )
PNc
j=1 wij Nj
j=1
=
Nc
1 X
σj2 ,
=
Nc
(2.2)
j=1
and it gives each halo an equal weight.
Consider a sample of central and satellite galaxies with luminosities L > Lmin . The velocity
dispersions in the satellite-weighting and host-weighting schemes can be analytically expressed
(see also van den Bosch et al. 2004) as follows:
2
σsw
(Lc )
=
R∞
2
σhw
(Lc )
0
=
2 i dM
P (M |Lc ) hNsat iM hσsat
M
R∞
,
P
(M
|L
)
hN
i
dM
c
sat M
0
R∞
0
2 i dM
P (M |Lc ) hσsat
M
R∞
.
0 P (M |Lc ) dM
(2.3)
(2.4)
Here hNsat iM denotes the average number of satellites with L > Lmin in a halo of mass M , and
2 i
hσsat
M is the square of the los velocity dispersion of satellites averaged over the entire halo.
Consider a MLR of central galaxies that has no scatter, i.e. P (M |Lc ) = δ(M − M0 ),
where M0 is the halo mass for a galaxy with luminosity Lc . In this case both schemes give an
2 = σ 2 = hσ 2 i
equal measure of the velocity dispersion, i.e., σsw
sat M0 . Most studies to date
hw
have assumed the scatter in P (M |Lc ) to be negligible, and simply inferred an average MLR,
2 (L ) = hσ 2 i
M0 (Lc ) using σsw
c
sat M0 (McKay et al. 2002; Brainerd & Specian 2003; Prada et al.
2003; Norberg et al. 2008). However, as shown in van den Bosch et al. (2004), and as evident
from the above equations (2.3) and (2.4), whenever the scatter in P (M |Lc ) is non-negligible,
1
Note that the velocity dispersion should always be averaged in quadrature as is evident from Eq. (2.1).
2.3. TOY MODEL
11
2 (L ) and σ 2 (L ) can differ significantly2 (see also Chapter 3).
σsw
c
c
hw
In this chapter, we show that ignoring the scatter in the MLR of central galaxies can result
in appreciable errors in the inferred mean relation between mass and luminosity. We show,
2 (L ) and σ 2 (L ).
though, that these problems can be avoided by simultaneously modeling σsw
c
c
hw
In particular, we demonstrate that the ratio of these two quantities can be used to determine the
actual scatter in the MLR of central galaxies.
2.3
Toy Model
2 (L ) and σ 2 (L ) can be analytically
In the previous section, we have shown that both σsw
c
c
hw
expressed in terms of the probability function, P (M |Lc ), the satellite occupation, hNsat iM ,
2 i . In
and the kinematics of the satellite galaxies within a halo of mass M specified by hσsat
M
fact, the inversion of equations (2.3) and (2.4) presents an opportunity to constrain P (M |Lc )
2 and σ 2 . In this section we use a simple toy model to demonstrate that
using the observable σsw
hw
2 and σ 2 can be used to constrain the first two moments (i.e., the mean
the combination of σsw
hw
and the scatter) of P (M |Lc ).
For convenience, let us assume that P (M |Lc ) is a lognormal distribution
2 
 
1
ln(M/M0 )   dM
P (M |Lc ) dM = q
exp −  q
.
M
2
2
2πσln
2
σ
M
ln M
(2.5)
Here M0 is a characteristic mass scale which obeys
ln M0 =
∞
Z
P (M |Lc ) ln M dM = hln M i ,
(2.6)
0
2
and σln
M reflects the scatter in halo mass at a fixed central luminosity and is given by
2
σln
M =
Z
∞
P (M |Lc )(ln M − ln M0 )2 dM.
(2.7)
0
2 i
In addition, let us assume that both hσsat
M and hNsat iM are simple power laws,
hNsat iM = Ñ
2
hσsat
iM
= S̃
2
M
1012
α
M
1012
β
,
(2.8)
.
(2.9)
with α and β two constants, Ñ the average number of satellites in a halo of mass 1012 h−1 M ,
and S̃ the corresponding los velocity dispersion.
2
2
Note that σsw
6= σhw
is a sufficient but not a necessary condition to indicate the presence of scatter in P (M |Lc );
2
2
after all, if hNsat iM does not depend on mass then σsw
= σhw
independent of the amount of scatter.
2
12
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
Substituting Eqs. (2.5), (2.8) and (2.9) in Eqs. (2.3) and (2.4) yields
2
σsw
(Lc )
= S̃
2
2
σhw
(Lc )
M0
1012
= S̃
2
β
σ2 β 2
exp ln M
2
M0
1012
β
α
1+2
,
β
σ2 β 2
exp ln M
2
(2.10)
.
(2.11)
The velocity dispersions σsw (Lc ) and σhw (Lc ) depend on both M0 and σln M , elucidating
the degeneracy between the mean mass M0 (Lc ) and the scatter σln M (Lc ) of the distribution
P (M |Lc ). In particular, if only σsw (Lc ) or σhw (Lc ) is measured, one cannot deduce M0 (Lc )
without having an independent knowledge of the scatter σln M (Lc ). However, the latter can be
inferred from the ratio of the satellite-weighted to the host-weighted velocity dispersion. In
particular, in the case of our toy model,
2
σln
M
1
=
ln
αβ
2
σsw
2
σhw
(2.12)
Thus, by measuring both σsw (Lc ) and σhw (Lc ) one can determine both M0 (Lc ) and its scatter
σln M (Lc ), provided that the constants α and β are known. Since virialized dark matter haloes
all have the same average density within their virial radii, β = 2/3 (e.g. Klypin et al. 1999; van
den Bosch et al. 2004). Previous studies have obtained constraints on α that cover the range
< α < 1.1 (e.g. Yang et al. 2005a; van den Bosch et al. 2007; Tinker et al. 2007; Yang et al.
0.7 ∼
∼
−1
2
2007). Since σln
M ∝ α , this uncertainty directly translates into an uncertainty of the inferred
scatter. Therefore, in Chapter 4, we do not use the constraints on α available in the literature to
infer the mean and scatter of the MLR from real data. Instead, we treat α as a free parameter
and use the average number of observed satellites as a function of the luminosity of central as an
additional constraint. Note that the relation between σln M and the ratio of σsw to σhw specified
by Eq. (2.12) is model-dependent, i.e. we have assumed particular functional forms for the halo
occupation statistics of centrals and satellites to arrive at Eq. (2.12). Furthermore, we have not
accounted for any selection effects. In what follows, we present a careful treatment of selection
effects and more realistic halo occupation models.
2.4
Selection Effects
The toy model presented in the previous section illustrates that measurements of the satelliteweighted and host-weighted kinematics of satellite galaxies can be used to infer the mean and
scatter of the MLR of central galaxies, P (M |Lc ). However, in practice one first needs a method
to select central galaxies and satellites from a galaxy redshift survey. In general, central galaxies
are selected to be the brightest galaxy in some cylindrical volume in redshift space, and satellite
galaxies are defined as those galaxies that are fainter than the central by a certain amount and
located within a cylindrical volume centered on the central. In this section we show how these
2.4. SELECTION EFFECTS
13
2 and σ 2 , and how this can be accounted for in the analysis.
selection criteria impact on σsw
hw
No selection criterion is perfect, and some galaxies will be selected as centrals, while in
reality they are satellites (hereafter ‘false centrals’). In addition, some galaxies will be selected
as satellites of a certain central, while in reality they do not reside in the same halo as the
central (hereafter ‘interlopers’). The selection criteria have to be tuned in order to minimize the
impact of these false centrals and interlopers. Here we make the assumption that interlopers can
be corrected for, and that the impact of false centrals is negligible. Using mock galaxy redshift
surveys, van den Bosch et al. (2004) have shown that one can devise adaptive, iterative selection
criteria that justify these assumptions (see also Chapter 3). Here we focus on the impact of
these iterative selection criteria on the satellite kinematics in the absence of interlopers and
false centrals. Our analytical treatment for selection effects follows the one presented in van
den Bosch et al. (2004) except for the averaging of velocity dispersions in quadrature and the
inclusion of an extra selection effect. We state and quantify these differences in Section 2.A.
For completeness, we outline our treatment below.
In general, satellite galaxies are selected to lie within a cylindrical volume centered on
its central galaxy, and specified by Rp < Rs and |∆V | < (∆V )s . Here Rp is the physical
separation from the central galaxy projected on the sky and ∆V is the los velocity difference
between a satellite and its central. Usually, (∆V )s is chosen sufficiently large, so that it does not
exclude true satellites from being selected. However, in the adaptive, iterative selection criteria
of van den Bosch et al. (2004), which we will use in the subsequent chapters, the aperture radius
is tuned so that Rs ' 0.375 rvir , where rvir is the virial radius of the dark matter halo hosting
2 i
the central−satellite pair. This means that hNsat iM and hσsat
M in Eqs. (2.3) and (2.4) need to
2 i
be replaced by hNsat iap,M and hσsat
ap,M , respectively. Here hNsat iap,M is the average number
2 i
of satellites in a halo of mass M that lie within the aperture, and hσsat
ap,M is the square of the
los velocity dispersion of satellite galaxies averaged over the aperture.
The number of satellites present within the aperture, hNsat iap,M , is related to the number of
satellites given by the halo occupation statistics, hNsat iM , via
(
hNsat iap,M =
with
fcut
4π
=
hNsat iM
Z
Rs
fcut hNsat iM
if Rs < rvir
hNsat iM
if Rs ≥ rvir
R dR
0
Z
rvir
R
nsat (r|M ) √
r dr
.
r 2 − R2
(2.13)
(2.14)
Here nsat (r|M ) is the number density distribution of satellites within a halo of mass M , which
is normalized so that
hNsat iM = 4π
Z
rvir
nsat (r|M ) r2 dr .
(2.15)
0
Under the assumption that the satellites are in virial equilibrium within the dark matter
halo, and that the velocity dispersion of satellite galaxies within a given halo is isotropic, the
14
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
los velocity dispersion of satellites within the cylindrical aperture of radius Rs is given by
Z Rs
4π
dR R
=
hNsat iap,M 0
Z rvir
r dr
2
nsat (r|M ) σsat
(r|M ) √
.
r 2 − R2
R
2
hσsat
iap,M
(2.16)
Here σsat (r|M ) is the local, one-dimensional velocity dispersion which is related to the potential Ψ of the dark matter halo via the Jeans equation
2
σsat
(r|M )
1
=
nsat (r|M )
Z
∞
nsat (r0 |M )
r
∂Ψ 0
(r |M ) dr0 .
∂r0
(2.17)
The radial derivative of the potential Ψ represents the radial force and is given by
∂Ψ
4πG
(r|M ) = 2
∂r
r
Z
r
ρ(r0 |M ) r02 dr0 ,
(2.18)
0
with ρ(r|M ) the density distribution of a dark matter halo of mass M . The assumptions of virial
equilibrium and orbital isotropy are supported by results from numerical simulations which
show that dark matter subhaloes (and hence satellite galaxies) are in a steady state equilibrium
within the halo and that their orbits are nearly isotropic at least in the central regions (Diemand
et al. 2004). Furthermore, van den Bosch et al. (2004) have demonstrated that anisotropy has a
negligible impact on the average velocity dispersion within the selection aperture.
Finally, there is one other effect of the selection criteria to be accounted for which has not
been considered in van den Bosch et al. (2004) or in Eq. (2.4). When selecting central−satellite
pairs, only those centrals are selected with at least one satellite inside the search aperture. This
has an impact on the host-weighted velocity dispersions that needs to be accounted for. The
probability that a halo of mass M , which on average hosts hNsat iap,M satellites within the
aperture Rs , has Nsat ≥ 1 satellites within the aperture, is given by
P (Nsat ≥ 1) = 1 − P (Nsat = 0)
= 1 − exp [−hNsat iap,M ]
≡ P(hNsat iap,M ).
(2.19)
Here, for the second equality, we have assumed Poisson statistics for the satellite occupation
numbers. Note that, in the satellite-weighting scheme, haloes that have zero satellites, by definition, get zero weight. Therefore only the host-weighted velocity dispersions need to be corrected for this effect.
Thus, in light of the selection effects, Eqs. (2.3) and (2.4) become
2
σsw
(Lc )
=
R∞
0
2 i
P (M |Lc ) hNsat iap,M hσsat
ap,M dM
R∞
,
0 P (M |Lc ) hNsat iap,M dM
(2.20)
2.5. MORE REALISTIC MODELS
and
2
σhw
(Lc )
=
R∞
0
15
2 i
P (M |Lc ) P(hNsat iap,M ) hσsat
ap,M dM
R∞
.
0 P (M |Lc ) P(hNsat iap,M ) dM
(2.21)
Note that P(hNsat iap,M ) ' hNsat iap,M when hNsat iap,M → 0. This implies that |σsw −σhw | →
0 for faint centrals (i.e. when Lc becomes comparable to Lmin , the minimum luminosity adopted
to select the satellites). Therefore, the ability to detect the difference between σsw and σhw
depends on how bright Lc is compared to Lmin . In principle, this can be overcome by decreasing
Lmin (detecting faint satellite galaxies), such that hNsat iap,M 0 and P(hNsat iap,M ) → 1.
However, faint satellite galaxies can only be detected out to a very small distance due to the flux
limit of a survey. The number of galaxies in a volume-limited sample with low Lmin is small.
This in turn makes the detection of the difference between σsw and σhw difficult due to small
number statistics. Therefore, there is a trade-off involved in the choice of Lmin , which limits the
significance with which one can detect the difference between σsw and σhw . Since this selection
effect was not taken into account in Section 2.3, Eq. (2.12), which relates the ratio σsw /σhw to
the scatter in halo masses, σln M , does not reveal this dependence on Lmin .
2.5
More Realistic Models
Using the methodology described above, we now illustrate how satellite kinematics can be
used to constrain the mean and the scatter of the MLR of central galaxies, P (M |Lc ). We
improve upon the toy model described in Section 2.3 by considering a realistic model for the
halo occupation statistics and take the impact of selection criteria into account.
2 (L ) and σ 2 (L )
As is evident from the discussion in the previous section, calculating σsw
c
c
hw
requires the following input:
• the density distributions of dark matter haloes, ρ(r|M )
• the number density distribution of satellites, nsat (r|M )
• the halo occupation statistics of centrals, P (M |Lc ).
We assume that dark matter haloes follow the NFW (Navarro et al. 1997) density distribution
M
ρ(r|M ) =
4πrs3 µ(c)
r
rs
−1 r −2
.
1+
rs
(2.22)
Here, rs is a characteristic scale radius, c = rvir /rs is the halo’s concentration parameter, and
µ(x) ≡ ln(1 + x) −
x
.
1+x
Throughout we use the relation between c and M given by Macciò et al. (2007).
(2.23)
16
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
Figure 2.1: The satellite-weighted (σsw ) and host-weighted (σhw ) velocity dispersions of satellite galaxies for model G1. Note that σsw (Lc ) > σhw (Lc ) at the bright end, indicating that the
MLR of central galaxies, P (M |Lc ), has a non-negligible amount of scatter.
2.5. MORE REALISTIC MODELS
17
Table 2.1: Different models for the HOD of centrals
Model σlog L
γ1
γ2
L0
M1
G1
0.14 3.27 0.25 9.94 11.07
G2
0.25 3.27 0.25 9.94 11.07
G3
0.14 1.80 0.40 9.80 11.46
Three different models describing the MLR of centrals used to predict σsw (Lc ) and σhw (Lc ).
We assume that the number density distribution of satellite galaxies is given by the generalised NFW profile,
nsat (r|M ) ∝
r
Rrs
−γ r γ−3
1+
,
Rrs
(2.24)
where γ represents the slope of the number density distribution of satellites as r → 0 and R is a
free parameter. In this chapter, we assume γ = 1 and R = 1, i.e. the number density distribution
of satellite galaxies is spatially unbiased with respect to the distribution of dark matter particles.
Note that this is a fairly simplistic assumption. We address the issue of potential spatial antibias
of satellite galaxies in Chapter 4.
Substituting ρ(r|M ) and nsat (r|M ) in Eqs. (2.18) and (2.17) gives
2
σsat
(r|M )
cV 2
= 2 vir
R µ(c)
r
Rrs
γ Z
r 3−γ ∞
µ(x)dx
,
1+
γ+2
Rrs
(1 + x/R)3−γ
r/rs (x/R)
where Vvir = (GM/rvir )1/2 is the circular velocity at rvir .
The final ingredient is a realistic model for the halo occupation statistics of centrals and
satellites. To that extent, we use the conditional luminosity function (CLF) presented in Cacciato et al. (2009). The CLF, denoted by Φ(L|M )dL, specifies the average number of galaxies
with luminosities in the range L ± dL/2 that reside in a halo of mass M , and is explicitly
written as the sum of the contributions due to central and satellite galaxies, i.e. Φ(L|M ) =
Φc (L|M ) + Φs (L|M ). From this CLF, the probability distribution P (M |Lc ) follows from
Bayes’ theorem according to
Φc (Lc |M ) n(M )
,
Φc (Lc |M ) n(M ) dM
P (M |Lc ) = R ∞
0
(2.25)
with n(M ) the halo mass function, while the average number of satellites with L ≥ Lmin in a
halo of mass M is given by
hNsat iM =
Z
∞
Lmin
Φs (L|M ) dL .
(2.26)
18
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
Figure 2.2: Comparison of three models with different HODs for the central galaxies. In all
panels the solid line corresponds to model G1, the dotted line to model G2 and the dashed
line to model G3 (see Table 1 for the parameters). Panels (a), (b) and (c) show hlog Lc i(M ),
hlog M i(Lc ) and σlog M (Lc ), respectively. Panels (d) and (e) show the predicted satelliteweighted and host-weighted velocity dispersions as function of luminosity, and panel (f) shows
the logarithm of the ratio between σsw and σhw . See text for a detailed discussion.
2.5. MORE REALISTIC MODELS
19
The parametric forms for Φc (L|M ) and Φs (L|M ) are motivated by the results of Yang et al.
(2008, hereafter YMB08), who determined the CLF from the SDSS group catalogue of Yang
et al. (2007). In particular, Φc (L|M ) is assumed to follow a log-normal distribution
 "
Φc (L|M )dL = √
log e
exp −
2π σlog L
log(L/L∗c )
#2 
√
2σlog L
 dL ,
L
(2.27)
with σlog L a free parameter that we take to be independent of halo mass, and
L∗c (M ) = L0
(M/M1 )γ1
[1 + (M/M1 )]γ1 −γ2
(2.28)
which has four additional free parameters: two slopes, γ1 and γ2 , a characteristic halo mass,
M1 , and a normalization, L0 . Note that, L∗c ∝ M γ1 for M M1 and L∗c ∝ M γ2 for M M1 .
Cacciato et al. (2009) constrained the free parameters, σlog L , γ1 , γ2 , M1 and L0 , by fitting the
SDSS luminosity function of Blanton et al. (2003b) and the galaxy−galaxy correlation lengths
as a function of luminosity from Wang et al. (2007). The resulting best-fit parameters are listed
in the first row of Table 1, and constitute our fiducial model G1. We also consider two alternative
models for Φc (L|M ), called G2 and G3, the parameters of which are also listed in Table 1. For
Φs (L|M ) we adopt the model of Cacciato et al. (2009) throughout, without any modifications:
i.e. models G1, G2, and G3 only differ in P (M |Lc ) and have the same nsat (r|M ).
Having specified all necessary ingredients, we now compute the satellite weighted and hostweighted satellite kinematics for our fiducial model G1 using Eqs. (2.20) and (2.21). The results
are shown as solid and dotted lines in Fig. 2.1, where we have adopted a minimum satellite
luminosity of Lmin = 109 h−2 L . At the faint-end, the velocity dispersions σsw and σhw
are equal, this simply reflects the fact that hNsat iM → 0 if Lc → Lmin . At the bright end,
though, the non-zero scatter in P (M |Lc ) causes the difference between σsw and σhw to increase
systematically with increasing Lc . This is a generic trend for any realistic halo occupation
model (see also van den Bosch et al. 2004). It is important to note here that the difference
between σsw and σhw depends upon the central galaxy luminosity and how bright this luminosity
is compared to Lmin (see discussion at the end of Section 2.4). In Chapter 3, we show that
the difference between the velocity dispersions in the two schemes is detectable from current
datasets. Previous studies (McKay et al. 2002; Brainerd & Specian 2003; Prada et al. 2003;
Conroy et al. 2007; Norberg et al. 2008) did not have sufficient number statistics to detect the
difference between the two schemes given the measurement errorbars.
The upper panels of Fig. 2.2 show the mean and scatter of the MLR of central galaxies in
models G1 (solid lines), G2 (dotted lines) and G3 (dashed lines). Panel (a) plots hlog Lc i(M ) =
log(L∗c ), which reveals the double power-law behavior of Eq. (2.28), panel (b) shows the inverse
relation,
hlog M i(Lc ) =
Z
0
∞
P (M |Lc ) log M dM .
(2.29)
20
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
and panel (c) shows the scatter in the MLR, σlog M (Lc ), deduced by using
2
σlog
M
=
Z
∞
P (M |Lc ) [log M − hlog M i(Lc )]2 dM .
(2.30)
0
Note that σlog M (Lc ) increases with increasing Lc , even though the scatter σlog L is constant
with halo mass. This simply owes to the fact that the slope of hlog Lc i(M ) becomes shallower
with increasing Lc , as illustrated in Fig. 2.3.
The comparison between models G1 and G2 illustrates the effect of changing the scatter
σlog L in Φc (L|M ). Both models have exactly the same hlog Lc i(M ) (the solid line overlaps
the dotted line in panel a). However, because the scatter σlog L in G2 is larger than in G1 (see
Table 1), the hlog M i(Lc ) of G2 is significantly lower than that of G1 at the bright end (∼
0.5 dex at the bright end). This is due to the shape of the halo mass function. Increasing the
scatter adds both low mass and high mass haloes to the distribution P (M |Lc ) (cf. Eqs. [2.25]
and [2.29]), and the overall change in the average halo mass depends on the slope of the halo
mass function. Brighter galaxies live on average in more massive haloes where the halo mass
function is steeper. In particular, when the halo mass range sampled by P (M |Lc ) lies in the
exponential tail of the halo mass function, an increase in the scatter adds many more low mass
haloes than massive haloes, causing a shift in the average halo mass towards lower values. On
the other hand, fainter galaxies live in less massive haloes, where the slope of the halo mass
function is much shallower. Consequently, a change in the scatter does not cause an appreciable
change in the average mass. Finally, as expected, the scatter in the MLR, σlog M (Lc ), in G2 is
higher than for G1 at all luminosities (see panel c).
Panels (d) and (e) of Fig. 2.2 show the analytical predictions for σsw (Lc ) and σhw (Lc ),
respectively. Note that models G1 and G2 predict satellite kinematics that are significantly
different (which can be distinguished given the typical measurement errors in Chapter 4), even
though both have exactly the same hlog Lc i(M ). In particular, model G2 predicts larger σsw
and σhw at the faint end, but lower σsw and σhw at the bright end. The trend at the faint is
due to the fact that the scatter σlog M (Lc ) is higher in G2 than in G1. Quantitatively, this is
evident from Eqs. (2.10) and (2.11), which demonstrate that both the satellite weighted and host
weighted satellite kinematics increase with increasing scatter. At the bright end, however, the
drastic decrease in hlog M i(Lc ) for G2 with respect to G1 overwhelms this boost and causes
σsw and σhw to be lower in G2.
Now consider model G3. This model has the same amount of scatter as model G1, but
we have tuned its parameters (γ1 , γ2 , M1 , L0 ) that describe hlog Lc i(M ) such that its σsw (Lc )
closely matches that of model G2 (the dotted and dashed curves in panel (d) are almost overlapping). As is evident from panels (a)−(c), though, the MLR of G3 is very different from that
of G2. Note that the higher values of hlog M i(Lc ) for G3 are compensated by its lower values
of σlog M (Lc ), such that the satellite-weighted kinematics are virtually identical. This clearly
illustrates the degeneracy between the mean and the scatter of the MLR: One can decrease the
2.5. MORE REALISTIC MODELS
21
Figure 2.3: Illustration of the MLR of central galaxies. The solid black line indicates the mean
of the Lc -M relation, while the gray scale region reflects the scatter. In this particular case the
scatter in P (Lc |M ) (indicated by vertical arrows) is taken to be constant with halo mass. Note,
though, that the scatter in P (M |Lc ) (indicated by horizontal arrows) increases with increasing
Lc ; this simply is due to the fact that the slope of the mean Lc -M relation becomes shallower
with increasing halo mass.
22
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
mean of the MLR and yet achieve the same σsw by increasing the scatter of the MLR. It also
shows that σsw alone does not yield sufficient information to uniquely constrain the MLR.
Note, though, that although σsw is the same for models G2 and G3, their host-weighted
satellite kinematics, σhw (Lc ), are different at the bright end. In fact, the ratios σsw /σhw for
models G2 and G3 are clearly different. The logarithm of this ratio, shown in panel (f), follows
the same trend as σlog M (Lc ), i.e. it is higher for model G2 than for G3. This is in agreement
with our toy model, according to which the ratio σsw /σhw increases with the scatter σlog M (Lc )
(cf. Eq. [2.12]). This illustrates once again that the combination of σsw and σhw allows one to
constrain both the mean and the scatter of the MLR simultaneously (see also Chapter 4).
2.6
Summary
The kinematics of satellite galaxies is a powerful probe of the masses of the dark matter haloes
surrounding central galaxies. With the advent of large, homogeneous redshift surveys, it has
become possible to probe the mass−luminosity relation (MLR) of central galaxies spanning a
significant range in luminosities. Unfortunately, since most centrals only host a few satellite
galaxies with luminosities above the flux limit of the redshift survey, one generally needs to
stack a large number of central galaxies within a given luminosity bin and combine the velocity information of their satellites. Because of the finite bin-width, and because the MLR has
intrinsic scatter, this stacking results in combining the kinematics of satellite galaxies in haloes
of different masses, which complicates the interpretation of the data. Unfortunately, most previous studies have ignored this issue, and made the oversimplified assumption that the scatter is
negligible.
Using realistic models for the halo occupation statistics, and taking account of selection
effects, we have demonstrated a degeneracy between the mean and the scatter of the MLR: one
can change the mean relation between halo mass, M , and central galaxy luminosity, Lc , and
simultaneously change the scatter around that mean relation, such that the observed satellite
kinematics, hσsat i(Lc ), are unaffected.
We have also presented a new technique to break this degeneracy, based on measuring the
satellite kinematics using two different weighting schemes: host-weighting (each central galaxy
gets the same weight) and satellite weighting (each central galaxy gets a weight proportional
to its number of satellites). In general, for central galaxies close to the magnitude limit of the
survey, the average number of satellites per host is close to zero, and the satellite-weighted
velocity dispersion, σsw , is equal to the host-weighted velocity dispersion, σhw . This is because
only those centrals with at least one satellite are used to measure the satellite kinematics. For
brighter centrals, however, σsw > σhw and the actual ratio of these two values is larger for
MLRs with more scatter (see Eq. [2.12] and panels c and f of Fig. 2.2). Hence, the combination
of σsw (Lc ) and σhw (Lc ) contains sufficient information to constrain both the mean and the
scatter of the MLR of central galaxies. In Chapter 3, we apply this method to a mock catalogue,
2.A. COMPARISON WITH VAN DEN BOSCH ET AL.
23
and show that the difference between σsw and σhw can be detected with sufficient significance
to constrain both the mean and the scatter of the MLR of central galaxies. In Chapter 3, we also
address the issues of measurement errors, sampling effects and interlopers. In Chapter 4, we
apply this method to data from the SDSS and show that the MLR and its scatter inferred from
the data are in excellent agreement with other, independent constraints.
In a recent study, Becker et al. (2007) analyzed the kinematics of MaxBCG clusters (Koester
et al. 2007) and inferred the mean and the scatter of the mass−richness relation (here richness is
a measure for the number of galaxies that reside in the cluster). Becker et al. (2007) combined
the kinematics of satellite galaxies in finite bins of cluster richness and measured the second and
fourth moments of the host-weighted velocity distribution. They used these two moments simultaneously to determine the mean and the scatter of the mass−richness relation. This method
is complementary to that presented here, and it will be interesting to compare both methods and
investigate their relative strengths and weaknesses. We intend to address this in a future study.
Finally we emphasize that the scatter in the conditional probability function P (M |Lc ) is
expected to increase with increasing Lc . This is due to the fact that the slope of hLc i(M ),
which is the mean of P (Lc |M ), becomes shallower with increasing halo mass. Hence, when
stacking haloes according to the luminosity of the central galaxy, one cannot ignore the scatter
in M , even when the scatter in P (Lc |M ) is small. This has important implications for any
technique that relies on stacking, such as satellite kinematics and galaxy−galaxy lensing (see
e.g. Tasitsiomi et al. 2004; Cacciato et al. 2009)
Appendix
2.A
Comparison with van den Bosch et al.
The analytical treatment of the selection effects presented in Section 2.4 closely follows van den
Bosch et al. (2004, hereafter vdB04) except for two subtle differences. First of all, vdB04 incorrectly averaged the velocity dispersion directly rather than in quadrature as done here (Eqs. 2.16,
2.20 and 2.21). Secondly, vdB04 failed to account for the factor P(hNsat iap,M ) (hereafter P
for brevity) that corrects for the centrals that do not host any satellite and hence do not contribute
to the host-weighted velocity dispersion of satellites (Eq. 2.21).
Fig. 2.4 quantifies the error in the estimates of vdB04 due to these differences. The dotted
line shows the relative error on σhw caused due to direct averaging instead of averaging in
quadrature. Direct averaging leads to an underestimate of the velocity dispersion which is
negligible at the faint end, but grows to ∼ 5% at the bright end. The satellite-weighted velocity
dispersion is also underestimated by a similar amount. In their paper, vdB04 only compared
the analytical estimate of the satellite-weighted velocity dispersion to real data. Fortunately, the
small error in the estimate of the satellite-weighted velocity dispersion due to direct averaging
does not change any of their conclusions.
24
2. SATELLITE KINEMATICS: THE ANALYTICAL FORMALISM
Figure 2.4: The relative error in the estimates of the host-weighted velocity dispersion by
vdB04. The dotted line shows the error caused due to direct averaging of the velocity dispersion.
The dashed and the dot-dashed lines show the error when P is ignored for Lmin = 109 h−2 L
and Lmin = 3 × 107 h−2 L respectively.
The dashed line in Fig. 2.4 shows that σhw is underestimated by ∼ 10% if the factor P in
Eq. (2.21) is ignored. The factor P depends on the minimum luminosity, Lmin , adopted to select
the satellites: lower values of Lmin result in larger number of satellites, which imply P → 1. For
the dashed line, Lmin = 109 h−2 L . However, vdB04 adopted Lmin = 3 × 107 h−2 L in their
analysis. In this case (shown with the dot-dashed line), the relative error decreases to < 5%.
Note that the factor P affects only the host-weighted velocity dispersion (see Section 2.4). Since
vdB04 only compared their estimates of the satellite-weighted velocity dispersion to data, their
results are not influenced by the fact that they failed to account for P in their equations for
host-weighting.
Chapter 3
Satellite Kinematics: Tests on a Mock
Catalogue
The contents of this chapter are based upon the article More et al. (2009c) published in the
Monthly Notices of the Royal Astronomical Society. The reference is
More, S., van den Bosch, F. C., Cacciato, M., et al. 2009b, MNRAS, 392, 801.
The original article also contains the results of the analysis with SDSS data. To maintain a coherent flow in this thesis these results are presented in the next chapter. In addition, unnecessary
repetition of certain equations is avoided by referring to the previous chapter.
3.1
Introduction
According to the standard picture of galaxy formation, galaxies form in dark matter haloes. The
complex astrophysics of galaxy formation and evolution is primarily believed to be governed
by the mass of the dark matter halo in which it occurs. Quantifying scaling relations between
central galaxy properties and their dark matter halo masses is, hence, an important stepping
stone towards understanding galaxy formation. The kinematics of satellite galaxies is a powerful
probe of the halo masses of central galaxies and can be used to determine the scaling relations
between central galaxy properties and their halo masses.
In the past, several studies have used the kinematics of satellite galaxies to determine the
halo mass−luminosity relation (MLR) of central galaxies (McKay et al. 2002; Brainerd & Specian 2003; Prada et al. 2003; van den Bosch et al. 2004; Conroy et al. 2005, 2007) and to study
the density profiles of dark matter haloes (Prada et al. 2003; Klypin & Prada 2009). Although
the results obtained by these studies appear consistent with each other, Norberg et al. (2008,
hereafter N08) have demonstrated a quantitative disagreement in the kinematice obtained by
these studies and showed that this disagreement is largely due to subtle differences in the selection criteria used to identify central and satellite galaxies. Therefore, it is crucial to understand
how selection effects bias the MLR of central galaxies inferred from satellite kinematics and
26
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
to test the methods used to quantify the kinematics of satellites in order to identify potential
systematic biases that can affect the measurements.
The present chapter is aimed at understanding the various selection biases that affect the
analysis of the kinematics of satellite galaxies and the subsequent determination of the halo
mass−luminosity relationship (MLR). We construct a realistic mock galaxy catalogue for this
purpose. In this chapter, we create an analysis pipeline, which takes a galaxy redshift catalogue
as input, performs the analysis of the kinematics of satellite galaxies and outputs both the mean
and the scatter of the MLR of central galaxies based upon the the novel method presented in
Chapter 2. This analysis pipeline is rigorously tested using the mock galaxy catalogue. In
particular, we show that our central−satellite selection criteria and the method to measure the
kinematics reliably recover the true kinematics present in the mock catalogue. We also show
that the mean and the scatter of the MLR inferred from the kinematics match the corresponding
true relations in the mock catalogue. In the subsequent chapters, this analysis pipeline will be
used on data from the Sloan Digital Sky Survey (SDSS).
This chapter is organized as follows. In Section 3.2 we describe the construction of the
mock catalogue that is used to test our method of analysis of the kinematics of satellites. In
Section 3.3 we briefly outline the iterative selection criteria used to select centrals and satellites.
In Section 3.4 we describe and test the method used to measure the kinematics of satellites as
a function of the central luminosity. The inference of the MLR from the kinematics of satellite
galaxies requires the knowledge of the number density distribution of satellites within a halo.
In Section 3.5 we show that this distribution can also be inferred from the selected satellites. In
Section 3.6 we describe our model to interpret the measured velocity dispersions and show that
this model is able to recover the true mean and scatter of the MLR of central galaxies from the
mock catalogue. We summarize our results in Section 3.7.
3.2
Mock Catalogue Construction
It is important to carefully identify central and satellite galaxies from a redshift survey in order to
study the kinematics of satellite galaxies. Furthermore, it is also important to reliably quantify
the kinematics of the selected satellites as a function of central luminosity which in turn can
yield the MLR of central galaxies. We monitor the performance of our method of analysis for
each of these tasks using a realistic mock galaxy catalogue (MGC) which serves as a control
dataset. The halo occupation of galaxies in the MGC is known a priori, thereby allowing an
accurate assessment of the level of contamination of the selected sample of centrals and satellites
due to false identifications and also a comparison between the kinematics recovered from the
selected satellites and the actual kinematics present in the MGC.
The two essential steps to construct a MGC are to obtain a distribution of dark matter haloes
and to use a recipe to populate the dark matter haloes with galaxies. For the former purpose, we
use a numerical simulation of dark matter particles in a cosmological setup. For the latter, we
3.2. MOCK CATALOGUE CONSTRUCTION
27
use the conditional luminosity function (CLF) which describes the average number of galaxies
with luminosities in the range L ± dL/2 that reside in a halo of mass M .
A distribution of dark matter haloes is obtained from a N −body simulation for a ΛCDM
cosmology with the following parameters, matter density Ωm = 0.238, energy density in the
cosmological constant ΩΛ = 0.762, the linearly extrapolated root mean square variance of the
density fluctuations on scales of 8 h−1 Mpc σ8 = 0.75, the spectral index of the initial density
fluctuations ns = 0.95 and the Hubble parameter h = H0 /100 km s−1 Mpc−1 = 0.73. The
simulation consists of N = 5123 particles within a cube of side Lbox = 300 h−1 Mpc with
periodic boundary conditions. The particle mass is 1.33 x 1010 h−1 M . Dark matter haloes are
identified using the friends−of−friends algorithm (Davis et al. 1985) with a linking length of
0.2 times the mean inter−particle separation. Haloes obtained with this linking length have a
mean overdensity of 180 (Porciani et al. 2002). We consider only those haloes which have at
least 20 particles or more.
To populate the dark matter haloes with galaxies, we need to know the number and the
luminosities of galaxies to be assigned to each halo. Furthermore, we also need to assign phase
space coordinates to each of these galaxies. We use the CLF described in Cacciato et al. (2009)
for the first purpose. The CLF is a priori split into a contribution from centrals and satellites,
i.e. Φ(L|M ) = Φc (L|M ) + Φs (L|M ). Here, Φc (L|M )dL denotes the conditional probability
that a halo of mass M harbours a central galaxy of luminosity between L and L + dL, and
Φs (L|M )dL denotes the average number of satellites of luminosity between L and L + dL.
The parameters that describe the CLF are constrained using the luminosity function (Blanton
et al. 2005) and the luminosity dependence of the correlation length of galaxies (Wang et al.
2007) in SDSS.
Let us consider a halo of mass M . The luminosity of the central galaxy within this halo is
sampled from the distribution Φc (L|M ). The average number of satellites that have a luminosity
greater than Lmin = 109 h−2 L and reside within haloes of mass M is given by
hNsat i(M ) =
Z
∞
Φs (L|M )dL .
(3.1)
Lmin
We assume Poisson statistics for the occupation number of satellites (Kravtsov et al. 2004; Yang
et al. 2005a, 2008) and assign Nsat galaxies to the halo where Nsat is drawn from
P (Nsat |M ) = exp(−µ)
µNsat
,
Nsat !
(3.2)
with µ = hNsat i(M ). The luminosities of these satellite galaxies are drawn from the distribution Φs (L|M ).
Phase space coordinates are assigned to the galaxies in the following manner. The central
galaxy is assumed to reside at rest at the centre of the halo. Therefore, it has the same phase
space coordinates as the parent dark matter halo. As in Chapter 2, we assume that the halo
28
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
is spherical and that the dark matter density distribution, ρ(r|M ), follows the universal NFW
profile (Navarro et al. 1997) given by Eq. (2.22). Similarly the number density distribution of
satellite galaxies, nsat (r|M ), is assumed to follow the profile given by Eq. (2.24) which has
two additional parameters γ and R which allow the satellite galaxies to be spatially biased
with respect to dark matter particles. For populating the MGC, we adopt γ = R = 1 which
implies that the satellites trace the dark matter density distribution in an unbiased manner. The
distribution, nsat (r|M ), is normalized such that
hNsat i(M ) = 4π
Z
rvir
nsat (r|M ) r2 dr .
(3.3)
0
The radial coordinates of the satellite galaxies with respect to the center of the halo are sampled
from the distribution nsat (r|M ). The satellite distribution around centrals is assumed to be
spherically symmetric and random angular coordinates are assigned to the satellite galaxies. At
the assigned position for every satellite galaxy, velocities along each of the three axes are drawn
from a Gaussian,
f (vj ) = √
1
2 πσsat (r|M )
"
exp −
vj2
2 (r|M )
2σsat
#
,
(3.4)
where vj denotes the relative velocity of the satellite with respect to the central along axis j
2 (r|M ) denotes the radial velocity dispersion at a distance r from the centre of the
and σsat
halo. Here isotropy of orbits is assumed, i.e. the velocity dispersion along the j th axis, σj2 ,
2 (r|M ). The radial velocity dispersion is related to ρ(r|M ) and n (r|M ) via the
equals σsat
sat
Jeans equation. We use Eq. (2.25) from Chapter 2 to determine the radial velocity dispersion,
2 (r|M ), within the halo. The radial velocity dispersion is used in the distribution given by
σsat
Eq. (3.4) to assign velocities to satellites. The entire procedure of assigning central and satellite
galaxies is repeated for all the dark matter haloes within the simulation.
Our aim is to construct a mock redshift survey that mimics the SDSS. Therefore, 2x2x2
identical galaxy−populated simulation boxes (which have periodic boundary conditions) are
stacked together. A (RA, DEC) coordinate frame is defined with respect to a virtual observer
at one of the corners of the stack. The apparent magnitude of each galaxy is computed according
to its luminosity and distance from the observer. The line-of-sight (los) velocity of the galaxy
is calculated by adding its peculiar velocity to the velocity of the cosmological flow. A random
velocity drawn from a Gaussian distribution with a dispersion of 35 km s−1 is further added
along the los to account for the spectroscopic redshift errors present in the SDSS. The redshift as
seen by the virtual observer is then computed using the total velocity. We only consider galaxies
with an observed redshift z < 0.15 and an apparent magnitude brighter than 17.77. This flux
limited catalogue is denoted henceforth by MOCKF and has 289,500 galaxies above an absolute
luminosity of 109 h−2 L . MOCKF is used in Appendix 3.A to investigate potential selection
biases associated with the selection of central galaxies. In addition to MOCKF, we construct a
volume limited sample, MOCKV, of galaxies that lie in the redshift range 0.02 ≤ z ≤ 0.072 and
3.3. SELECTION CRITERIA TO IDENTIFY CENTRALS AND SATELLITES
29
have luminosities greater than 109.5 h−2 L . It consists of 69,512 galaxies. In what follows, we
use the volume limited sample MOCKV to validate our method for quantifying the kinematics
(Section 3.4), to validate the method to infer the number density distribution of satellites (Section 3.5) and finally to confirm that the mean and scatter of the MLR can be reliably recovered
using the kinematics of satellites (Section 3.6).
Note that we have made the simplifying assumption that the satellites are unbiased tracers
of the dark matter for the construction of our mock catalog. The effects of a bias in the satellite
number density distribution were investigated in detail by van den Bosch et al. (2004). In
particular, they have shown that if the satellites are spatially antibiased with respect to the dark
matter, then the velocity dispersion of satellites is systematically higher than the dark matter
velocity dispersion. In Chapter 4, when we use data from SDSS to constrain the MLR, we do
take into account the fact that satellite galaxies may be spatially antibiased with respect to the
dark matter. We have also assumed that the angular distribution of satellites is uniform which
is not realistic. Note that this is not a concern as the random stacking of haloes to infer the
kinematics of satellite galaxies will wash away any non-uniformities.
3.3
Selection Criteria to identify Centrals and Satellites
Large-scale galaxy redshift surveys such as the SDSS allow the selection of a statistically significant sample of satellites. Since the observed galaxies cannot be a priori classified as centrals
and satellites, it is important to use selection criteria that can correctly identify central galaxies
and the satellites which orbit around them. In this section, we describe the selection criteria that
we use to identify the central and satellite galaxies.
A galaxy is identified as a central if it is at least fh times brighter than every other galaxy
within a cylindrical volume specified by R < Rh and |∆V | < (∆V )h (see Fig. 3.1). Here,
R is the physical separation from the candidate central galaxy projected on the sky and ∆V is
the los velocity difference. Around each of the identified centrals, satellites are those galaxies
that are at least fs times fainter than their central galaxy and lie within a cylindrical volume
specified by R < Rs and |∆V | < (∆V )s . The identification of the central galaxies depends on
the parameters Rh , (∆V )h and fh , while the selection of satellites depends on the parameters
Rs , (∆V )s and fs . The values of these parameters also determine the level of contamination
of the sample due to falsely identified centrals and falsely identified satellites (hereafter interlopers). The false identification of centrals can be minimized by choosing large values of
Rh , (∆V )h and fh so that the selected central is the dominant galaxy in a large volume. On the
other hand, minimizing the interlopers requires small values of Rs and (∆V )s . A large value of
fs further guarantees that the selected satellites are small and do not dominate the kinematics
of the halo (i.e. can safely be considered as test particles). Although stricter restrictions yield
cleaner samples, they also reduce the sample size significantly. This makes the velocity dispersion measurements noisy. Thus, there is a tradeoff between the contamination level and the
30
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
Rh
fh
Rs
fs
(∆V )s
(∆V )h
Figure 3.1: Schematic diagram of a selection criterion. Two coaxial cylinders are defined around
each galaxy (represented by a solid dot). The axis is along the los while the face of each cylinder
is parallel to the plane of the sky.
sample size.
Most authors have chosen fixed values for the selection criteria parameters, independent of
the luminosity of the galaxy under consideration (McKay et al. 2002; Prada et al. 2003; Brainerd & Specian 2003; Norberg et al. 2008). Since brighter centrals on average reside in more
extended haloes, van den Bosch et al. (2004) advocated an aperture which scales with the virial
radius of the halo around the galaxy. They used iterative criteria which scale the cylindrical
aperture based upon the estimate of the velocity dispersion around the central after every iteration. In this thesis, we also use these iterative criteria to select centrals and satellites. In
Appendix 3.A, we compare the performance of our iterative criteria with the restrictive selec-
SC
ITER
N08
Rh
Mpc/h
2.0
0.8σ200
1.0
Table 3.1: Selection criteria
(∆V )h
fh
Rs
km/s
Mpc/h
4000
1.0
0.5
1000σ200 1.0 0.15σ200
2400
2.0
0.4
(∆V )s
km/s
4000
4000
1200
fs
1.0
1.0
8.0
The parameters used to specify the inner and the outer cylinders around a galaxy for the selection criteria used in this chapter (ITER) and the selection criteria used in N08. The first row
for ITER denotes the parameters used in the first iteration, while the second row denotes the
parameters used in subsequent iterations. The velocity dispersion, σsat in units of 200 km s−1
is denoted by σ200 and is used to scale the cylinders in every iteration.
3.4. SATELLITE KINEMATICS
31
tion criteria used by N08 in there analysis. The parameter set {Rh , (∆V )h , fh , Rs , (∆V )s , fs }
that defines the inner and outer cylinders for the iterative criteria (ITER) is listed in Table 3.1.
The first row lists the parameters for the first iteration while the next row lists the scaling of
these parameters in the subsequent iterations. In short, we proceed as follows:
1. Use fixed values of the aperture size to select centrals and satellites in the first iteration.
2. Fit the velocity dispersion of the selected satellites as a function of the central galaxy
luminosity, σsat (Lc ), with a simple functional form (see Section 3.4.1).
3. Select new centrals and satellites by scaling the inner and the outer cylinders based on the
estimate of the velocity dispersion.
4. Repeat 2 and 3 until σsat (Lc ) has converged to the required accuracy.
For step 3, we adopt the aperture scalings used in van den Bosch et al. (2004). These aperture
scalings were optimised to yield a large number of centrals and satellites, but at the same time
reduce the interloper contamination. The values chosen for Rh and Rs approximately correspond to 2 and 0.375 times the virial radius, rvir .
3.4
Satellite Kinematics
In this section, we describe how to measure and model the velocity dispersion−luminosity
relation, σsat (Lc ), using the satellites identified by the selection criteria. The relation σsat (Lc )
can be measured either by binning the satellites by central galaxy luminosity or by using an
unbinned estimator. We use the unbinned estimate after every iteration of the selection criteria
to scale the selection aperture. However, to quantify the kinematics of the final sample of
satellites, we use the binned estimator, for reasons which we describe further in the text. In
the following subsections, we describe the unbinned and the binned estimators for σsat (Lc ) and
finally an analytical model for the same.
3.4.1
Unbinned Estimates
We use a maximum likelihood method to estimate the relation σsat (Lc ) from the velocity information of the selected satellites after every iteration of the selection criteria. Let σ200 denote
σsat (Lc ) in units of 200 km s−1 and L10 denote the luminosity of the central galaxy in units of
1010 h−2 L . Following van den Bosch et al. (2004), we parametrize σ200 as,
σ200 (log L10 ) = a + b (log L10 ) + c (log L10 )2 .
(3.5)
Let fint denote the interloper fraction and assume that this fraction is independent of the luminosity of the central galaxy and ∆V . The probability for a selected satellite to have a los
32
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
Table 3.2: Selection criteria parameters
Sample
a
b
c
MOCKV 2.06 0.45 0.25
MOCKF 2.05 0.50 0.23
The parameters used in Eq. (3.5) to define σ200 as a function of the luminosity of a galaxy in
the final iteration for samples MOCKV and MOCKF.
velocity difference of ∆V km s−1 with respect to the central is then given by
1 − fint
fint
(∆V )2
+
,
P (∆V ) =
exp −
2
2(∆V )s
ω̄
2σeff
(3.6)
2 + σ 2 ]1/2 is the effective velocity dispersion in the presence of the redshift
where, σeff = [σsat
err
errors and the factor
ω̄ =
√
(∆V )s
,
2πσeff erf √
2σeff
(3.7)
is such that the P (∆V ) is properly normalized to unity. In our attempt to mimic SDSS, we
have added a Gaussian error of 35 km s−1 to the velocity of each galaxy in the mock catalog.
Therefore, the error on the velocity difference, ∆V , of the central and satellite galaxies is σerr =
√
2 × 35 km s−1 which adds in quadrature to σsat to yield σeff .
We use Powell’s direction set method to determine the parameters (a, b, c, fint ) that maxiP
mize the likelihood L = i ln[P (∆V )]i , where the summation is over all the selected satellites.
This yields a continuous estimate of σsat (Lc ) without the need to bin the los velocity information of satellites according to the luminosity of the central galaxy. The parameter set (a, b, c)
fitted in the last but one iteration determines the size of the apertures used to select the final
sample of satellites. The values of these parameters for the samples investigated in this chapter
are listed in Table 3.2.
3.4.2
Binned Estimates
We use a binned estimator to quantify the kinematics of the final sample of satellites. The binned
estimator allows us to relax the simplistic assumption of fint being independent of Lc . More
importantly, the binned estimator allows us, in a straightforward manner, to measure σsat (Lc )
using two different weighting schemes − satellite-weighting and host-weighting. Most studies
in the literature have used one of these two weighting schemes to infer the mean of the MLR.
However, as demonstrated in Chapter 2, the mean of the MLR inferred from the velocity dispersion in any one of these two schemes is degenerate with the scatter in the MLR. This degeneracy
can be broken by modelling the velocity dispersions in both schemes simultaneously. In what
follows, we briefly explain these two weighting schemes in turn and then verify that the velocity
dispersions in both schemes can be accurately recovered from the MGC.
3.4. SATELLITE KINEMATICS
33
Figure 3.2: Scatter plot of the velocity difference, ∆V , between the satellites and their centrals
as a function of the central galaxy luminosity. The satellites were obtained by applying the
iterative selection criteria to MOCKV.
To measure the velocity dispersion of satellites in the satellite-weighting scheme, we obtain
the distribution of velocities of the satellites, P (∆V ), with respect to their centrals for several
bins of central galaxy luminosity. Each bin has a width ∆ log[Lc ] = 0.15. In this scheme, the
centrals that have a larger number of satellites clearly contribute more to the P (∆V ) distribution than those which have a smaller number of satellites. Therefore, the resulting scatter in
P (∆V ) is a satellite-weighted average of the velocity dispersions around the stacked centrals
(see Chapter 2 for a detailed discussion). The dispersion obtained using this scheme is denoted
henceforth by σsw .
One has to undo the satellite-weighting described above in order to measure the hostweighted velocity dispersion. This can be accomplished by introducing a weight w = N −1
for each central−satellite pair while constructing the P (∆V ) distribution (van den Bosch et al.
2004; Becker et al. 2007; Conroy et al. 2007). Here, N denotes the number of satellites selected
around the central under consideration. Therefore, in this scheme each central receives a total
weight of unity irrespective of the number of satellites it hosts. The scatter in this weighted
P (∆V ) distribution is the host-weighted velocity dispersion and is denoted henceforth by σhw .
The procedure to obtain the scatter in the P (∆V ) distributions is the same for both the
satellite-weighted and the host-weighted case. This procedure must account for the interlopers
and the redshift errors present in MOCKV. In what follows, we illustrate this procedure only for
the satellite-weighted case.
34
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
Figure 3.3: The satellite-weighted P (∆V ) distributions of satellites around centrals selected in
several luminosity bins from MOCKV. The average log(Lc / h−2 L ) for each bin is indicate
at the upper right corner of every panel. The (brown) dot-dashed line at the bottom of each
distribution shows the contamination of the P (∆V ) distributions due to the interlopers. The
(blue) dashed lines indicate the double-Gaussian fits.
3.4. SATELLITE KINEMATICS
35
Figure 3.4: Upper panels show the satellite-weighted and the host-weighted velocity dispersions
recovered from MOCKV. The (red) circles show values recovered from a single Gaussian fit
while the (black) triangles show those from the double Gaussian fit. The solid line shows the
variance of the true satellites and the dot-dashed line shows the analytical prediction using the
halo occupation statistics of centrals from the MGC. The bottom panels show the percentage
deviation of the single and double Gaussian fits from the variance of the true satellites.
Fig. 3.2 shows the scatter plot of velocity difference ∆V of the selected satellites and the
centrals as a function of the luminosity of the centrals. The satellite-weighted P (∆V ) distributions of the satellites selected from MOCKV for several central luminosity bins are shown
in Fig. 3.3. Dot-dashed lines show the contamination of the P (∆V ) distributions due to interlopers and are barely visible at the bottom of each distribution. This confirms the claims in
van den Bosch et al. (2004) that the iterative criteria yield a small fraction of interlopers with a
weak dependence on Lc and that the interlopers can be modelled as a constant contribution to
the velocity distribution independent of ∆V .
A simple way to estimate the scatter of a P (∆V ) distribution is to fit a Gaussian plus a
36
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
constant model given by
−(∆V )2
.
P (∆V ) = a0 + a1 exp
2
2σeff
(3.8)
Here, a0 denotes the constant (with respect to ∆V ) interloper background, a1 is the normalization of the Gaussian and σeff is the effective dispersion in the presence of the redshift errors.
The velocity dispersion obtained using a single Gaussian plus constant model fit can be
systematically affected if the P (∆V ) distribution is intrinsically non-Gaussian. Diaferio &
Geller (1996) demonstrated that the velocity distribution can be non-Gaussian partly due to
mass mixing (which is the result of stacking haloes of different mass) and partly due to the
unrelaxed state of a halo. The second moment of such a non-Gaussian distribution can be
estimated with a double Gaussian plus a constant model (Becker et al. 2007) given by
P (∆V ) = a0 + a1 exp
−(∆V )2
−(∆V )2
+
a
exp
.
2
2σ12
2σ22
(3.9)
The scatter, σeff , in this case is such that
2
=
σeff
a1 σ13 + a2 σ23
2
2
= σsw
+ σerr
.
a1 σ1 + a2 σ2
(3.10)
The dashed lines in Fig. 3.3 show the double-Gaussian fits to the P (∆V ) distributions, respectively.
Fig. 3.4 shows the velocity dispersions obtained from the satellite-weighted P (∆V ) distributions in the upper left panel and those obtained from the host-weighted P (∆V ) distributions
in the upper right panel. The (red) circles and the (black) triangles indicate the single and the
double Gaussian fits respectively. Since the true satellites of centrals selected from MOCKV
are known, they can be used to judge the goodness of the fits. The satellite-weighted and the
host-weighted velocity dispersions of the true satellites (among the satellites selected using the
iterative criteria) are obtained using
PNc PNj
2
σtrue
=
2
i=1 wij (∆V )ij
PNc PNj
j=1
i=1 wij
j=1
2
− σerr
.
(3.11)
Here, Nc denotes the number of true centrals, Nj denotes the number of true satellites of the
j th central and (∆V )ij denotes the los velocity difference of the j th central with respect to
its ith satellite. The weight wij = 1 for the satellite-weighted case and wij = Nj−1 for the
host-weighted case. The true velocity dispersions thus obtained are shown as solid curves in
Fig. 3.4.
The bottom panels of Fig. 3.4 show the percentage deviation of both the single and the
double Gaussian fits from the velocity dispersions of the true satellites. The single Gaussian
fit (the dotted line) underestimates the dispersions systematically by about 5 − 10%. The dou-
3.4. SATELLITE KINEMATICS
37
ble Gaussian fit (the solid line) on the other hand gives an unbiased estimate of both velocity
dispersions. Therefore, in what follows, we use the double Gaussian fit for measuring both the
satellite-weighted and the host-weighted velocity dispersions (cf. Becker et al. 2007).
3.4.3
Analytical Estimates
We now compare the velocity dispersions obtained from the satellite-weighted and the hostweighted schemes to their analytical expectation values. As detailed in Chapter 2, the satelliteweighted and the host-weighted velocity dispersions depend on the distribution of halo masses
of central galaxies specified by P (M |Lc ). The analytical expressions describing the velocity
dispersion in these two weighting schemes are
2
σsw
(Lc ) =
2
(Lc )
σhw
=
R∞
0
R∞
0
2 i
P (M |Lc ) hNsat iap,M hσsat
ap,M dM
R∞
,
0 P (M |Lc ) hNsat iap,M dM
(3.12)
2 i
P (M |Lc ) P(hNsat iap,M ) hσsat
ap,M dM
R∞
.
0 P (M |Lc ) P(hNsat iap,M ) dM
(3.13)
Here, the average number of satellites and the average velocity dispersion of satellites, within
2 i
the aperture Rs in a halo of mass M , are denoted by hNsat iap,M and hσsat
ap,M , respectively.
We use results from Chapter 2 to describe these quantities. In particular, we use Eqs. (2.13) and
2 i
(2.14) to describe hNsat iap,M , and Eq. 2.16 to describe hσsat
ap,M .
Note that, when measuring the host-weighted velocity dispersions only satellites of those
centrals that have at least one satellite within the search aperture are used. The fraction of such
centrals is denoted by P(hNsat iap,M ) and is given by the probability that a halo of mass M ,
which on average hosts hNsat iap,M satellites within the aperture Rs , has Nsat ≥ 1 within the
aperture. We assume that the satellite occupation numbers (cf. Eq. 3.2) follow Poisson statistics,
which is supported by numerical simulations (Kravtsov et al. 2004) and by results from group
catalogs based on SDSS (Yang et al. 2005a, 2008) and use the expression for P(hNsat iap,M )
given by Eq. 2.19 from Chapter 2. The factor P(hNsat iap,M ) is not considered in the analytical
estimate in the satellite-weighting scheme as haloes with zero satellites, by definition, contribute
zero weight.
From the analytical description presented in Chapter 2, it is clear that the analytical estimates for the velocity dispersions require the knowledge of
• the density distribution of dark matter haloes, ρ(r|M )
• the number density distribution of satellites, nsat (r|M )
• the halo occupation statistics of centrals, P (M |Lc ), and the halo occupation number of
satellites, hNsat i(M ).
We assume that the density distribution of dark matter haloes is given by Eq. (2.22) and that
the number density distribution of satellites is given by Eq. (2.24) with γ = R = 1. The halo
38
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
Figure 3.5: The projected number density distributions of the satellites selected from MOCKV
as a function of the projected radius in the brightest central luminosity bins. The average luminosity of the bin is indicated at the top right corner in each panel. The errorbars assume Poisson
statistics for the number of satellites in each bin. The (black) solid lines indicate the analytical
predictions and assume that the satellite number density distribution follows the dark matter
distribution in an unbiased manner, i.e. R = 1 and γ = 1 in eq. (2.24). For comparison, the
(red) dotted lines show the analytical predictions that assume R = 2 and γ = 0.
3.5. NUMBER DENSITY DISTRIBUTION OF SATELLITES
39
occupation statistics of centrals, P (M |Lc ) is given by
P (M |Lc ) = R
Φc (Lc |M )n(M )
Φc (Lc |M )n(M )dM
(3.14)
where Φc (Lc |M ) is the conditional luminosity function of central galaxies and n(M ) is the
halo mass function. The number of satellite galaxies in a halo of mass M is given by Eq. (3.1).
We adopt the Φc (Lc |M ) and Φs (L|M ) that were used in Section 3.2 to populate the MGC.
With this input, we compute the analytical estimates for the velocity dispersions of satellites
as a function of luminosity using Eqs. (3.12) and (3.13). The results thus obtained are shown
as dot-dashed curves in the corresponding panels of Fig. 3.4. Overall the agreement with the
velocity dispersions obtained from the satellites in the MGC is very good, except at intermediate
luminosities where the analytical estimates are ∼ 5 percent higher than σtrue . This indicates that
the central galaxies selected from the MGC do not properly sample the full P (M |Lc ). This can
be due to two reasons: (i) a systematic problem with the criteria used to select central galaxies,
or (ii) cosmic variance due to the finite volume probed by MOCKV. As we demonstrate in
Appendix 3.A our iterative criteria accurately sample the true P (M |Lc ), except for the fact
that it misses the haloes of those centrals which have zero satellites. However, this sampling
effect is accounted for in our analytical model via Eq. (2.19). In fact detailed tests show that the
discrepancies between σtrue and our analytical estimates are entirely due to cosmic variance in
the MGC.
In Appendix 3.A, we also show that the strict selection criteria, that have been abundantly
used in the literature, lead to a sample of central galaxies that is biased to reside in relatively
low mass haloes. Consequently, the resulting MLR of central galaxies is similarly biased, and
has to be interpreted with great care.
3.5
Number Density Distribution of Satellites
As described above, the number density distribution of satellite galaxies, nsat (r|M ), is a necessary input to analytically compute the velocity dispersions. The projected number density
distribution of satellites, Σ(R|Lc ), around centrals of a given luminosity, directly reflects the
functional form of nsat (r|M ). The distribution Σ(R|Lc ) can be directly measured by combining the satellites around centrals of a given luminosity, Lc , chosen by the selection criteria. However, it is necessary to first assess the impact of the interloper contamination on the
measurement of Σ(R|Lc ), for which we again make use of the satellite sample selected from
MOCKV.
Fig. 3.5 shows, for the five brightest luminosity bins, the azimuthally averaged projected
number density distributions of the satellites selected from MOCKV. The errorbars reflect the
Poisson noise on the number of satellites in each radial bin. The abrupt cutoff at large R is an
artefact due to the parameter Rs in the selection criteria which describe the maximum projected
40
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
radius within which satellites get selected. Note that, since Rs depends upon the luminosity of
central galaxies under consideration, this cutoff shifts to larger R with increasing central galaxy
luminosity.
The projected number density distribution of satellites around centrals stacked according to
luminosity, Σ(R|Lc ), can be analytically expressed as,
Σ(R|Lc ) = R
P (M |Lc ) Σ(R|M ) dM
.
P (M |Lc ) P(hNsat i(M )ap,M ) dM
R
(3.15)
Here, Σ(R|M ) is the projection of nsat (r|M ) along the line-of-sight and is given by
Σ(R|M ) =
Z
rvir
R
nsat (r|M ) 2r dr
√
,
r 2 − R2
(3.16)
Using nsat (r|M ) given by Eq. (2.24) with R = γ = 1 and the true P (M |Lc ) present in the
MGC, we analytically compute the expected number density distribution of satellites around
centrals of a given luminosity. The solid lines in Fig. 3.5 show the results of this analytical
expectation. The small differences between the measured and the analytically obtained distributions are due to the interlopers in the sample. However, the differences become negligible in
the brighter luminosity bins. For comparison, the (red) dotted lines show the expected Σ(R|Lc )
for R = 2 and γ = 0. This shows that the parameters R and γ, that characterize the number
density distribution of satellites, can be inferred from the projected number density distributions
of the selected satellites.
3.6
Mass−Luminosity Relationship
In the previous sections, using a mock catalog, we have demonstrated that the satellite-weighted
velocity dispersions, the host-weighted velocity dispersions and the projected number density
distributions of satellites around centrals of a given luminosity can be reliably measured starting
from a volume limited redshift catalogue of galaxies. Next, we attempt to infer the MLR of
central galaxies from the velocity dispersions measured from MOCKV. The aim is to invert
Eqs. (3.12) and (3.13) which describe the dependence of the velocity dispersions on the MLR
of central galaxies. In addition to the velocity dispersions, we also measure the average number
of satellites per central of a given luminosity, hNsat i(Lc ), and use this as a constraint. The
dependence of hNsat i(Lc ) on the MLR of central galaxies is given by
R∞
hNsat i(Lc ) = R ∞0
0
P (M |Lc )hNsat iap,M dM
.
P (M |Lc )P(hNsat iap,M )dM
(3.17)
In this section, we first describe the model we use to infer the mean and the scatter of the MLR
from the observables σsw , σhw and hNsat i. Next, we use this model to infer the mean and the
scatter of the MLR in the MGC and compare it to the true relations present in the MGC.
3.6. MASS−LUMINOSITY RELATIONSHIP
41
Figure 3.6: The results of the MCMC analysis of the velocity dispersions obtained from
MOCKV. Crosses with errorbars in the upper panels denote the data used to constrain the
MCMC; the satellite-weighted velocity dispersions in panel (a), the host-weighted velocity dispersions in panel (b) and the average number of satellites per central in panel (c). The relations
recovered from the MCMC analysis are shown in the bottom panels; hlog Lc i(M ) in panel (d),
hlog M i(Lc ) in panel (e) and σlog M (Lc ) in panel (f). In each panel, the blue and purple colours
denote the 68% and the 95% confidence levels. The solid lines in the lower panels denote the
true relations present in MOCKV.
42
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
Table 3.3: MOCKV: Parameters recovered from the MCMC
Parameter Input 16%
50%
84%
log(L0 )
9.93
9.64 10.01 10.32
log(M1 ) 11.04 10.48 11.28 11.69
γ2
0.25
0.18
0.26
0.32
σlog L
0.14
0.13
0.15
0.17
The input parameters that describe P (Lc |M ) are compared to the 16th , 50th and the 84th percentiles of the corresponding distributions of parameters obtained from the MCMC.
3.6.1
The Model
As mentioned earlier, the analytical computation of σsw , σhw and hNsat i requires the knowledge
of the density distribution of dark matter haloes, the number density distribution of satellites and
the halo occupation statistics of centrals and satellites. We assume that the density distribution
of dark matter haloes follows the NFW profile given by Eq. (2.22). For the number density
distribution of satellites within a halo of mass M , nsat (r|M ), we use Eq. (2.24) with R = γ =
1. As shown in Section 3.5, the projected number density distributions of satellites selected
from MOCKV is consistent with this analytical expression. Next, we describe our model for
the halo occupation statistics of the centrals, specified by P (M |Lc ), and the satellites, specified
by hNsat i(M ).
The distribution P (M |Lc ) is related to the complementary distribution, P (Lc |M ), by Bayes’
theorem
P (M |Lc ) = R
n(M )P (Lc |M )
,
n(M )P (Lc |M )dM
(3.18)
where n(M ) is the halo mass function. We follow Cacciato et al. (2009) and parametrize the
distribution P (Lc |M )1 as a lognormal in Lc ,
"
#
log(e)
(log[Lc /L̃c ])2 dLc
P (Lc |M )dLc = √
exp −
.
2
Lc
2σlog
2πσlog L
L
(3.19)
Here, log L̃c (M ) denotes the mean of the lognormal distribution and σlog L is the scatter in this
distribution. We use four parameters to specify the relation L̃c (M ): a low mass end slope γ1 , a
high mass end slope γ2 , a characteristic mass scale M1 , and a normalisation L0 , such that
L̃c = L0
(M/M1 )γ1
.
[1 + (M/M1 )]γ1 −γ2
(3.20)
We assume the scatter σlog L to be independent of mass. We do not explore the faint end slope
(γ1 ) in our analysis as the velocity dispersions at the faint end are very uncertain due to low
1
Note that the distribution P (Lc |M ) is equivalent to the CLF for central galaxies. We model P (Lc |M ) using a
few parameters and use Eq. (3.18) to infer P (M |Lc ).
3.6. MASS−LUMINOSITY RELATIONSHIP
43
number statistics. Instead, we keep it fixed at 3.273, which is the value obtained from the analysis of the abundance and clustering of galaxies (see Cacciato et al. 2009). This parametrization
is motivated by results of Yang et al. (2008) who measure the conditional luminosity function
from the SDSS group catalogue described in Yang et al. (2007).
We model the satellite occupation number, hNsat i(M ), as a power law distribution, given
by
hNsat i(M ) = Ns
M
12
10 h−1 M
α
,
(3.21)
which adds two more parameters (Ns , α). Thus, in total, our model has six free parameters
(σlog L , L0 , M1 , γ2 , Ns , α). Given these parameters and the radial number density distribution
of satellites (specified by R and γ), the velocity dispersions σsw (Lc ) and σhw (Lc ) as well as the
number of satellites per central, hNsat i(Lc ) in an aperture of a given size can be computed using
Eqs. (3.12), (3.13) and (3.17) and compared to the measured values. Crosses with errorbars in
panels (a), (b) and (c) of Fig. 3.6 show σsw , σhw and hNsat i as a function of the luminosity of
the central obtained from MOCKV, respectively. We use these measurements to constrain the
six free parameters of our model.
3.6.2
Monte-Carlo Markov Chain
To determine the posterior probability distributions of the 6 free parameters in our model, we
use the Monte-Carlo Markov Chain (hereafter MCMC) technique. The MCMC is a chain of
models, each with 6 parameters. At any point in the chain, a trial model is generated with the
6 free parameters drawn from 6 independent Gaussian distributions which are centred on the
current values of the corresponding parameters. The chi-squared statistic, χ2try , for this trial
model, is calculated using
χ2try = χ2sw + χ2hw + χ2ns ,
(3.22)
with
χ2sw
χ2hw
=
=
10 X
σsw (Lc [i]) − σ̂sw (Lc [i]) 2
i=1
10 X
i=1
χ2ns
=
∆σ̂sw (Lc [i])
σhw (Lc [i]) − σ̂hw (Lc [i])
∆σ̂hw (Lc [i])
,
(3.23)
,
(3.24)
2
"
#2
10
X
hNsat i(Lc [i]) − N̂sat (Lc [i])
i=1
∆N̂sat (Lc [i])
.
(3.25)
(3.26)
44
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
Here, X̂ denotes the observational constraint X and ∆X̂ its corresponding error. The trial step
is accepted with a probability given by
(
Paccept =
1.0,
ifχ2try ≤ χ2cur
(3.27)
exp[−(χ2try − χ2cur )/2], ifχ2try > χ2cur
where χ2cur denotes the χ2 for the current model in the chain.
We initialize the chain from a random position in the parameter space and discard the first
104
models allowing the chain to sample from a more probable part of the distribution. This is
called the burn-in period for the chain. We proceed and construct a chain of models consisting
of 10 million models. We thin this chain by a factor of 104 to remove the correlations between
neighbouring models. This leaves us with a chain of 1000 independent models that sample
the posterior distribution. We use this chain of models to estimate the confidence levels on the
parameters and relations of interest.
In Table 3.3, we compare the 16th , 50th and 84th percentiles of the distributions of parameters, which characterize P (Lc |M ), obtained from the MCMC with the corresponding true
values of these parameters present in MOCKV. The true parameter values have been recovered
within the 68% confidence intervals. The 68 and 95% confidence levels in panels (a), (b) and
(c) of Fig. 3.6 show that the models from the MCMC accurately fit the velocity dispersions,
σsw and σhw , as well as the average number of satellites per central, hNsat i as a function of
central galaxy luminosity. The confidence levels for the average luminosity of the centrals as
a function of the halo mass, L̃c (M ), are shown in panel (d). The confidence levels on the
mean, hlog M i(Lc ), and the scatter, σlog M (Lc ), of the distribution P (M |Lc ), i.e. the MLR of
central galaxies are shown in panels (e) and (f), respectively. They have been calculated using
Eq. (3.18) and
hlog M i(Lc ) =
Z
∞
log M P (M |Lc ) dM ,
(3.28)
0
σlog M (Lc ) =
Z
∞
(log M − hlog M i) P (M |Lc ) dM
2
1/2
.
(3.29)
0
The solid lines in the lower panels show the corresponding true relations present in MOCKV.
Clearly, our method is able to accurately recover the true MLR.
This completes our tests with the MGC. Employing a variety of tests on a realistic MGC,
we have established a proof-of-concept that, starting from a redshift survey of galaxies, one can
reliably select central and satellite galaxies, quantify the kinematics of the selected satellites
around central galaxies and use this information to infer an unbiased estimate of the mean and
the scatter of the MLR of central galaxies.
3.7. SUMMARY
3.7
45
Summary
The kinematics of satellite galaxies can be used to statistically relate the mean halo masses of
central galaxies to their extensive properties. In this chapter, using a realistic mock catalogue,
we showed that it is indeed possible to recover the average and the scatter of the scaling relation
between halo mass and a central galaxy property (such as the luminosity) using the kinematics
of satellites. We thoroughly tested the analysis method at every step. We first tested the performance of the iterative selection criteria, advocated by van den Bosch et al. (2004), to identify
central and satellite galaxies and our method to measure the kinematics of the selected satellites.
We showed that the kinematics recovered from the selected satellites are a fair representation of
the true kinematics of satellite galaxies present in the mock catalogue. We presented an analytical model that properly accounts for the selection biases and showed that the predictions of this
analytical model are in good agreement with the measured kinematics of the selected satellites.
In Chapter 2, we have shown that the velocity dispersion of satellites can be measured using
two different weighting schemes: satellite-weighting and host-weighting. We have demonstrated a degeneracy between the mean and the scatter of the MLR obtained from either the
satellite-weighted or the host-weighted velocity dispersion alone. However, we have also shown
that this degeneracy can be broken by using the velocity dispersions in the two schemes simultaneously. In this chapter, we first tested our method using a mock galaxy catalogue. We
fitted the measured satellite-weighted and host-weighted velocity dispersions simultaneously
using a parametric model for the halo occupation statistics of central and satellite galaxies, and
demonstrated that we can reliably obtain confidence levels on the true mean and scatter of the
mass−luminosity relation of central galaxies.
In the next two chapters of this thesis, we apply the method developed in this chapter to
data from the Sloan Digital Sky Survey. In Chapter 4, we use satellite kinematics to infer the
scaling relation between halo mass and luminosity of central galaxies and quantify the scatter
in this scaling relation. In Chapter 5, we use satellite kinematics to learn more about the scaling
relation between halo mass and the stellar mass of central galaxies along with the scatter in this
relation.
We would like to conclude this chapter by emphasizing that, satellite kinematics need not
be restricted to the study of isolated haloes of galaxies as is routinely done in the literature.
By using a relaxed criteria to identify central−satellite systems and properly accounting for the
selection biases, we have shown that satellite kinematics can be effectively used to probe the
halo masses in a wide range of environments.
46
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
Appendix
3.A
Sampling of Central Galaxies
The ultimate goal of satellite kinematics is to probe the halo mass−luminosity relationship
(MLR) of central galaxies. In principle, an unbiased estimate for the MLR requires that the
central galaxies identified by the selection criteria are an unbiased (sub-)sample with respect to
their corresponding dark matter haloes. In this section, we investigate, using the MGC, how
our iterative criteria perform in this respect and compare them with the strict criteria used in the
literature.
For reasons that will become clear later, we use the flux-limited sample MOCKF for this
test. The solid lines in Fig. 3.7 show the distributions of halo masses, P (M |Lc ), for all central
galaxies in MOCKF divided in 5 luminosity bins. The average logarithm of the luminosities
of central galaxies in each bin is indicated at the top right corner. The dotted lines show the
distributions, P (M |Lc ), for all central galaxies that have at least one satellite in the selection
aperture defined by our iterative selection criteria. Finally, the histograms show the distributions, P (M |Lc ), of the centrals selected by our iterative criteria. Clearly, the centrals selected
by our iterative criteria sample the distribution of halo masses from the dotted lines (and not the
solid lines). However, as discussed in Section 3.4.3, this bias is taken into account while modelling the kinematics (see Eq. [2.19]) and therefore allows us to make an unbiased estimate. As
shown in Section 3.6, we indeed recover an unbiased MLR from the kinematics of the selected
satellites measured around the centrals selected by our iterative criteria.
For comparison, we now repeat this exercise using the strict criteria employed in previous
studies. In particular, we adopt the criteria N08 (see Table 3.1) used in Norberg et al. (2008).
These criteria identify a galaxy as a central if it is at least fh = 2 times brighter than any other
galaxy in a fixed (irrespective of the luminosity of the galaxy) aperture cylinder (see Table 3.1)
around itself. Satellites are identified as those galaxies that are at least fs = 8 times fainter
than the centrals and reside in a smaller aperture cylinder defined around the centrals. The
values of fh and fs in the N08 criteria are conservative, as the principle goal of their study
was to select isolated central galaxies. Applying the N08 criteria to MOCKV selects only 126
satellites around 96 central galaxies. Therefore, to do a meaningful comparison, we apply the
N08 criteria to MOCKF for which it selects 657 satellites around 395 centrals. For comparison,
our iterative criteria yields 39, 951 satellites around 21, 206 centrals.
Solid lines in the lower panels of Fig. 3.7 are the same as in the upper panels and show
the distributions of halo masses, P (M |Lc ), for all central galaxies in MOCKF divided in 5
central luminosity bins. The dotted lines show the P (M |Lc ) for those centrals that have at
least one satellite around them which is fs (= 8) times fainter than themselves. There is a
negligibly small difference in the dotted lines in the two rows due to different values of fs .
Finally, the histograms show the P (M |Lc ) distributions of the sample of centrals selected by
the N08 criteria. Clearly, these do not sample the distributions shown by the dotted lines and the
3.A. SAMPLING OF CENTRAL GALAXIES
47
Figure 3.7: Comparison of the sampling of central galaxies using the iterative selection criteria
(ITER) used in this chapter and the criteria used in N08. The histograms in the upper (bottom)
panel show the distributions of halo masses of central galaxies selected according to ITER
(N08). The average log(Lc / h−2 L ) for central galaxies in each bin is indicated at the top
right corner of each panel. The solid lines show the true distributions of halo masses for all
the central galaxies and the dotted lines show the distribution of halo masses of those central
galaxies that have at least one satellite more than fs times fainter than themselves in the inner
cylinder defined by the selection criteria.
48
3. SATELLITE KINEMATICS: TESTS ON A MOCK CATALOGUE
distributions are clearly biased towards the low mass end, especially, in the bright luminosity
bins. This owes to the fact that Norberg et al. (2008) adopt fh = 2, which preferentially
selects centrals that do not have satellite galaxies of comparable brightness. This biases the
distributions towards the low mass end. Note, though, that this is not a critique regarding
their selection criteria; after all, as Norberg et al. (2008) clearly described in their paper, their
principal goal is to study the kinematics around isolated galaxies. However, it does mean that it
is not meaningful to compare their MLR, which is only applicable to isolated galaxies, to that
obtained here, which is representative of the entire central galaxy population.
Chapter 4
The Halo Mass−Luminosity
Relationship
The contents of this chapter are partially based upon the article More et al. (2009c) published
in the Monthly Notices of the Royal Astronomical Society. The reference is
More, S., van den Bosch, F. C., Cacciato, M., et al. 2009, MNRAS, 392, 801
The original article does not contain the results of the colour dependence of the Halo mass−Luminosity
relationship of central galaxies. This analysis is part of a manuscript in preparation. The
manuscript will be submitted as:
Satellite Kinematics III: Colour and Stellar Mass Dependence
More, S., van den Bosch, F. C., Cacciato, M., et al. 2009
4.1
Introduction
According to the standard framework of galaxy formation, dark matter haloes form gravitational
potential wells in which baryons collapse, dissipate their energy and form stars and galaxies
(White & Rees 1978; Blumenthal et al. 1984). The complex process of galaxy formation and
evolution is believed to be governed by the mass of the dark matter halo in which it occurs. To
understand the halo mass dependence of this process, it is important to statistically relate the
observable properties (e.g. luminosity) of galaxies to the masses of their dark matter haloes.
The kinematics of satellite galaxies, stacked according to the property of their centrals (e.g.
luminosity), can be used to determine the scaling relation between central galaxy properties
and halo mass. However, the stacking procedure complicates the interpretation of the kinematic
signal in terms of the halo mass.
The scaling relation between halo mass and the luminosity of the central galaxies (halo
mass−luminosity relation MLR) can, more generally, be specified in terms of the conditional
probability P (M |Lc ), which describes the probability for a central galaxy with luminosity Lc
to reside in a halo of mass M . For a completely deterministic relation between halo mass and
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
50
central luminosity, P (M |Lc ) = δ(M −M0 ), where δ denotes the Dirac−delta function and M0
is a characteristic halo mass corresponding to centrals of luminosity Lc . The velocity dispersion, σ(Lc ), measured by stacking centrals with luminosity Lc , then translates into a mass M0
according to the scaling relation σ 3 ∝ M . However, galaxy formation is a stochastic process
and the distribution P (M |Lc ) is expected to have non-zero scatter. If this scatter is appreciable then the stacking procedure results in combining the kinematics of haloes spanning a wide
range in halo mass. This complicates the interpretation of the velocity dispersion. We addressed
this issue in Chapter 2 (More et al. 2009c); where we investigated a method to measure both the
mean and the scatter of the MLR of central galaxies1 using satellite kinematics. We outlined two
different weighting schemes to measure the velocity dispersion of satellites, satellite-weighting
and host-weighting, and showed that the mean and the scatter of the MLR can be inferred by
modelling the velocity dispersion measurements in these two schemes simultaneously.
In Chapter 3, we carried out a series of tests on a realistic mock galaxy catalogue to validate
our method to infer the MLR of central galaxies from a redshift survey using satellite kinematics. In particular, we showed that our central−satellite selection criterion and the method to
measure the kinematics reliably recovers the true kinematics present in the mock catalogue. We
also showed that the mean and the scatter of the MLR inferred from the kinematics match the
corresponding true relations in the mock catalogue.
In this chapter, we apply this method to the spectroscopic galaxy catalogue from SDSS
(Data Release 4) in order to determine both the mean and the scatter of the MLR of central
galaxies. In addition, we also analyse the kinematics of the satellites of central galaxies separated on the basis of their colour. This allows us to investigate the impact of the scatter in
the colour of central galaxies at a particular luminosity on the scatter in the MLR of central
galaxies.
This chapter is organized as follows. In Section 4.2, we describe the SDSS data and apply
the iterative criteria described in Section 3.3 to identify central and satellite galaxies for our
analysis of the kinematics. We form three different samples to analyse the kinematics of satellites around (i) all central galaxies, (ii) red central galaxies only and (iii) blue central galaxies
only. The measurements of the velocity dispersion as a function of the luminosity for central
galaxies in each of these samples are presented in Section 4.3. The number density distribution of satellites is an essential ingredient to infer the MLR from the kinematic measurements
of the satellites around their centrals. In Section 4.4, we present the projected number density
distribution of satellites around centrals from the SDSS data. The results obtained from the
MCMC analysis of the kinematics of satellites are presented and compared with results from
independent studies in Section 4.5. A summary of all the results is presented in Section 4.6.
For the analysis presented in this chapter, we assume the cosmological parameters from
the 3 year data release of WMAP (Spergel et al. 2007), Ωm = 0.238, ΩΛ = 0.762, h =
1
The term MLR refers to the distribution P (M |Lc ). The mean and the scatter of the MLR refer to the average
of this distribution and its scatter respectively.
4.2. CENTRAL AND SATELLITE SAMPLES FROM THE SDSS
51
H0 /100 km s−1 Mpc−1 = 0.734, the spectral index of initial density fluctuations ns = 0.951
and normalization σ8 = 0.744.
4.2
Central and Satellite Samples from the SDSS
The SDSS (York et al. 2000) is a joint five−passband (u, g, r, i and z) imaging and medium
resolution (R ∼ 1800) spectroscopic survey. The observations are carried out using a dedicated
2.5-m telescope at the Apache Point Observatory in New Mexico. The SDSS is designed to
cover one quarter of the entire sky and obtain images of around 100 million objects and obtain
the spectra of around 1 million objects. Data are made available to the scientific community
through periodic data releases.
In this study, we use the New York University Value Added Galaxy Catalogue (Blanton
et al. 2005, hereafter NYU−VAGC), which is based upon SDSS Data Release 4 (AdelmanMcCarthy et al. 2006) but includes a set of significant improvements over the original pipelines.
The magnitudes and the colours of the galaxies are based upon the standard SDSS Petrosian
technique and have been k−corrected and evolution corrected to z = 0.1 using the method
described in Blanton et al. (2003a,b). The notations 0.1 (g − r) and 0.1 Mr − 5 log h are used to
denote the resulting (g − r) colour and the absolute magnitude of the galaxies. The magnitude
limit of the spectroscopic sample is 17.77 in the 0.1 r band.
From this catalogue, we select all galaxies in the main galaxy sample with redshifts in the
range 0.02 ≤ z ≤ 0.072 and with a redshift completeness limit C > 0.8. We construct a volume
limited sample of galaxies with r-band luminosities above Lmin = 109.5 h−2 L . This sample
consists of 57, 593 galaxies and is henceforth denoted as SDSSV.
In Fig. 4.1, we show the scatter plot of the 0.1 (g − r) colour and the luminosity of galaxies
in this sample. The galaxy distribution follows a bimodal distribution in the colour−luminosity
scatter plot (Baldry et al. 2004; Blanton et al. 2005; Li et al. 2006). The solid line shows the
separation criteria between red and blue galaxies obtained in Yang et al. (2009) by fitting a
binormal distribution to the 0.1 (g − r) colour as a function of luminosity. The separation criteria
is given by
0.1
(g − r) = 1.022 − 0.0651 x − 0.00311 x2
(4.1)
where x =0.1 Mr − 5 log h + 23.0. The sample SDSSV consists of 30, 383 red galaxies and
27, 210 blue galaxies.
We apply the iterative criteria (ITER) outlined in Section 3.3 of Chapter 3 to select centrals
and their satellites from SDSSV for the analysis of the MLR of central galaxies without a
split in colour. For every iteration, the selection criteria is adjusted based upon the velocity
dispersion estimate in the previous iteration. The sample of centrals and satellites identified in
this manner is denoted by LA. For the analysis of the MLR around red (blue) central galaxies,
a separate sample of central and satellite galaxies is formed by applying a selection criteria
which is iteratively tuned using the estimate of velocity dispersions around the red (blue) central
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
52
Figure 4.1: Scatter plot of the 0.1 (g − r) colour and the absolute luminosities of galaxies in the
SDSS. To avoid overcrowding, a random subset of 10,000 galaxies was used to make this plot.
The red line shows the separation criterion from Yang et al. (2009) that was used to classify the
galaxies into red and blue.
galaxies only. The sample that contains the red (blue) central galaxies and their satellites is
denoted by LR (LB).
The parameters (a, b, c) in Eq. (3.5) that define the aperture used in the final iteration of
the central−satellite selection for samples LA, LR and LB are listed in Table 4.1. Sample LA
consists of 3, 863 central galaxies that host at least one satellite galaxy. The total number of
satellite galaxies in Sample LA is 6, 101. The number of red central galaxies that host at least
one satellite in Sample LR (LB) is 2, 503 (1, 221). The number of satellites in Sample LR (LB)
is 4, 599 (1, 449).
4.3
Velocity Dispersion−Luminosity Relation
The scatter plot of the velocity difference, ∆V , between the satellites and the corresponding
centrals as a function of the central luminosity in Sample LA is shown in Fig. 4.2. The scatter
plots obtained from Samples LR and LB are shown in the left and the right panels of Fig. 4.3
respectively. The scatter in the velocities of satellites with respect to their centrals clearly increases with central galaxy luminosity for all the three samples. The scatter plot of Sample LB
is markedly different from that of Sample LR, not only in terms of the number of satellites but
also the amount of scatter in the velocity differences, ∆V at fixed luminosity.
To quantify the velocity dispersion−luminosity relation, we obtain the P (∆V ) distribu-
4.3. VELOCITY DISPERSION−LUMINOSITY RELATION
53
Table 4.1: Selection criteria parameters
Parameters
Samples
LA
LR
LB
a
2.20
2.25
2.12
b
0.38
0.37
0.44
c
0.33
0.31
0.32
The parameters a, b and c that define the criteria used
to select central and satellite galaxies for the three samples used in this Chapter.
Table 4.2: Sample LA: Velocity dispersion measurements
log(Lc )
σsw
∆σsw
σhw
∆σhw
−2
−1
−1
−1
h L km s
km s
km s
km s−1
9.61
108
20
107
20
9.73
148
26
146
25
9.88
159
19
155
18
10.03
162
14
159
12
10.17
214
43
203
20
10.31
254
11
220
11
10.45
272
15
247
13
10.59
412
35
287
39
10.72
470
28
378
73
10.87
650
54
574
254
The velocity dispersion measurements in the satellite-weighted and host-weighted schemes together with the associated errors for sample LA.
tions in both the satellite-weighting and the host-weighting schemes by combining the velocity
differences, ∆V , of satellites within luminosity bins of uniform width ∆ log[Lc ] = 0.15 for
Samples LA and LR, and 0.13 for Sample LB. The satellite-weighted and the host-weighted
velocity dispersions are estimated from these distributions by fitting a model that consists of
two Gaussians and a constant as described in Section 3.4.2. Fig. 4.4 shows these dispersions
as a function of central luminosity for Sample LA. The values of σsw , σhw and their associated errors are listed in Table 4.2. Both the satellite-weighted and the host-weighted velocity
dispersions increase with the luminosity of the central galaxy. Note that the satellite-weighted
velocity dispersions are systematically larger than the host-weighted velocity dispersions. As
is evident from Eqs. (2.20) and (2.21), this is a sufficient condition to indicate the presence of
scatter in the MLR of central galaxies (see Chapter 2 for a detailed discussion).
The left (right) panel of Fig. 4.5 shows the comparison between the satellite-weighted
(host-weighted) velocity dispersions around red and blue central galaxies separately. Both the
satellite-weighted and the host-weighted velocity dispersions of satellites around red centrals
54
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
Figure 4.2: Scatter plot of the velocity difference, ∆V , between the satellites and their central
galaxies in sample LA as a function of the central galaxy luminosity.
Figure 4.3: Scatter plot of the velocity difference, ∆V , between the satellites and their central
galaxies in sample LR sample LB as a function of the central galaxy luminosity are shown in
the left and the right panels respectively.
4.3. VELOCITY DISPERSION−LUMINOSITY RELATION
55
Figure 4.4: The satellite-weighted (red triangles) and the host-weighted (blue squares) velocity
dispersions as a function of the central galaxy luminosity obtained from satellites in the sample
LA.
Figure 4.5: Comparison between the satellite-weighted (host-weighted) velocity dispersions as
a function of the luminosity of the central galaxy obtained from Sample LR and Sample LB is
shown in the left (right) panel.
56
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
Table 4.3: Sample LR: Velocity dispersion measurements
log(Lc )
σsw
∆σsw
σhw
∆σhw
h−2 L km s−1 km s−1 km s−1 km s−1
9.61
99
27
95
28
9.74
157
19
155
19
9.88
166
19
166
21
10.03
177
16
169
12
10.17
230
19
209
15
10.31
272
11
238
17
10.45
292
19
251
16
10.59
436
29
315
39
10.72
471
29
406
83
10.87
683
64
603
253
The velocity dispersion measurements in the satellite-weighted and host-weighted schemes together with the associated errors for sample LR.
Table 4.4: Sample LB: Velocity dispersion measurements
log(Lc )
σsw
∆σsw
σhw
∆σhw
h−2 L km s−1 km s−1 km s−1 km s−1
9.60
67
27
67
27
9.72
113
26
113
25
9.86
106
19
107
19
9.99
134
14
132
15
10.12
159
13
154
12
10.25
180
23
159
16
10.38
206
17
189
19
10.52
220
32
201
36
10.65
302
113
272
137
10.78
260
166
561
310
The velocity dispersion measurements in the satellite-weighted and host-weighted schemes together with the associated errors for sample LB.
4.4. NUMBER DENSITY DISTRIBUTION OF SATELLITES IN SDSS
57
Figure 4.6: The projected number density distributions of satellites around centrals in Sample
LA for the five bright luminosity bins (average log(Lc / h−2 L ) in right corner). The (black)
solid curves indicate the expected distributions if the number density distribution of satellites
follows the dark matter density, i.e. R = γ = 1 in Eq. (2.24). The (red) dotted curves in turn
indicate the expected distributions for a model in which the satellite galaxies are a factor 2 less
concentrated than dark matter and have a central core in the number density distribution i.e.
R = 2, and γ = 0.
are larger than those around blue centrals. This figure captures the difference between Sample
LR and LB observed in the scatter plots of Fig. 4.3. The velocity dispersion measurements for
Sample LR and for Sample LB are listed in Tables 4.3 and 4.4, respectively.
4.4
Number Density Distribution of Satellites in SDSS
The model to infer the MLR of central galaxies from the kinematics requires the radial number
density distribution of satellites, nsat (r), as an input. In Chapter 3, to infer the MLR of central
galaxies from the mock catalog, we used a model of nsat (r) that follows the density distribution
of the dark matter in an unbiased manner, i.e., γ = R = 1 in Eq. (2.24). However with SDSS, it
is not clear what functional form of nsat (r) should be used. In fact, various studies have shown
that the satellite galaxies are spatially antibiased with respect to the dark matter (Yang et al.
2005b; Chen 2007, 2008). Rather than including γ and R as free parameters in our model, we
seek to constrain these parameters using the observable Σ(R|Lc ). Fig. 4.6 shows the projected
58
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
Figure 4.7: The projected number density distributions of satellites around red centrals from
Sample LR for the five bright luminosity bins (average log(Lc / h−2 L ) in right corner).
4.4. NUMBER DENSITY DISTRIBUTION OF SATELLITES IN SDSS
59
Figure 4.8: The projected number density distributions of satellites around blue centrals from
Sample LB for the five bright luminosity bins (average log(Lc / h−2 L ) in right corner).
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
60
number density distributions of the satellites in Sample LA for the five brightest luminosity
bins. As can be seen from the analytical expressions that describe Σ(R|Lc ) (see Eq. 3.15),
predicting Σ(R|Lc ) requires the knowledge of P (M |Lc ), which is the principle goal of our
study. Furthermore, it also requires the knowledge of hNsat i(M ). Both these quantities are
unknown.
To proceed, we use the P (M |Lc ) and hNsat i(M ) from the CLF model of Cacciato et al.
(2009) which was also used to populate the mock catalogue in Chapter 3. We explore two
different models for nsat (r), one with R = γ = 1, where the number density distribution of
satellites follows the dark matter density distribution, and the other with R = 2 and γ = 0,
where the number density distribution of satellites is spatially antibiased with respect to the
dark matter distribution. The former model is shown as a (black) solid line while the latter with
a (red) dotted line in Fig. 4.6. Clearly, the latter model is favoured by the data. Therefore, we
use R = 2 and γ = 0 to specify nsat (r) for the analysis of the velocity dispersions to infer the
MLR of central galaxies from Sample LA.
The projected number density distributions of satellites in Sample LB and LR are shown
in Figs. 4.7 and 4.8, respectively. These distributions also show a similar behaviour with a
flattening of the projected number density distributions at the center. We use the same values of
R = 2 and γ = 0 for the analysis of both Sample LR and Sample LB.
4.5
Results from the MCMC Analysis
Next, we use the parametric model described in Section 3.6.1 and constrain it using the measured velocity dispersions, σsw and σhw , and the number of satellites per central, hNsat i as a
function of the luminosity of centrals for the Samples LA, LR and LB. As shown in Chapter 3,
this allows us to determine both the mean and the scatter of the MLR of central galaxies in these
three samples. We use a Monte−Carlo Markov Chain to recover the relations hlog M i(Lc ),
σlog M (Lc ) and hlog Lc i(M ).
4.5.1
The Halo Mass−Luminosity Relation
First, we present the results obtained from the MCMC analysis of the velocity dispersions measured from Sample LA. The 16th , 50th and the 84th percentiles of the distributions of the parameter values that describe the distribution P (Lc |M ) for the central galaxies in this sample are
listed in Table 4.5. Fig. 4.9 shows the results of the MCMC analysis. The upper row of panels
shows the data that was used to constrain the parameters. In the bottom row, panel (d) shows
the confidence levels on the relation L̃c (M ) while panels (e) and (f) show the confidence levels
on the inferred mean and scatter of the distribution P (M |Lc ) (i.e the MLR), respectively. The
values of hlog M i(Lc ) and σlog M (Lc ) together with their 1-σ errors are listed in Table 4.6.
Clearly, the average masses of dark matter haloes increase with central galaxy luminosity,
as expected. Interestingly, the scatter in halo masses also increases systematically with the lu-
4.5. RESULTS FROM THE MCMC ANALYSIS
61
Figure 4.9: Results of the MCMC analysis of the velocity dispersions obtained from Sample LA. Crosses with errorbars in the upper panels show the data points used to constrain the
MCMC; the satellite-weighted velocity dispersions in panel (a), the host-weighted velocity dispersions in panel (b) and the mean number of satellites per central as a function of luminosity
in panel (c), all measured by using the satellites in Sample LA. The blue and purple bands represent the 68% and 95% confidence regions respectively. The bottom panels show the relations
inferred from the MCMC; the average log(Lc ) is in panel (d), and the mean and the scatter in
the MLR of central galaxies in panels (e) and (f) respectively. The median relations obtained
from the MCMC are shown using dashed lines. The relations obtained from the best-fit CLF
model of Cacciato et al. (2009), are shown using solid lines. The squares in panels (e) and (f)
indicate the values obtained from the semianalytical model of Croton et al. (2006).
62
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
Table 4.5: MLR of central galaxies:
Sample Parameter
LA
log(L0 )
log(M1 )
γ2
σlog L
Ns
αs
LR
log(L0 )
log(M1 )
γ2
σlog L
Ns
αs
LB
log(L0 )
log(M1 )
γ2
σlog L
Ns
αs
Parameters recovered from the MCMC
16%
50%
84%
9.69 10.05 10.33
10.85 11.74 12.01
0.19
0.28
0.35
0.12
0.16
0.19
0.09
0.14
0.21
1.17
1.32
1.46
9.58
9.78 10.05
10.75 11.35 11.83
0.26
0.31
0.37
0.17
0.20
0.21
0.07
0.11
0.17
1.27
1.41
1.54
9.63
9.91 10.27
10.87 11.34 11.71
0.10
0.26
0.47
0.18
0.27
0.35
0.29
0.47
0.68
0.56
0.83
1.07
The 16th , 50th and 84th percentiles of the distributions of our model parameters for the three
samples analysed in this chapter.
Table 4.6: Sample LA: MLR of central galaxies
log(Lc ) hlog Mi ∆hlog Mi
σlog M
∆σlog M
−2
−1
−1
−1
h L h M
h M
h M h−1 M
9.61
12.06
0.35
0.12
0.06
9.73
12.16
0.32
0.13
0.07
9.88
12.28
0.29
0.15
0.08
10.03
12.44
0.26
0.18
0.10
10.17
12.60
0.23
0.22
0.10
10.31
12.80
0.21
0.26
0.11
10.45
13.01
0.19
0.30
0.10
10.59
13.24
0.19
0.34
0.09
10.72
13.47
0.21
0.36
0.08
10.87
13.74
0.23
0.38
0.07
The mean and scatter of the halo masses as a function of the central galaxy luminosity inferred
from the MCMC analysis. The errors on each of the inferred quantities correspond to the 68%
confidence levels.
4.5. RESULTS FROM THE MCMC ANALYSIS
63
minosity of the central galaxy. At the bright end, this scatter is roughly half a dex. Therefore,
stacking central galaxies by luminosity amounts to stacking haloes that cover a wide range in
masses. This justifies the need to account for this scatter in the analysis of the satellite kinematics. Neglecting this scatter leads to an overestimate of the halo mass at a given central luminosity (see Chapter 2). Most previous studies dealing with satellite kinematics have neglected this
scatter which has resulted in a biased estimate of the halo mass-luminosity relationship. As we
have shown in the Appendix 3.A of Chapter 3, their use of strict selection criteria to identify the
centrals and satellites have further biased their estimate of the MLR of central galaxies.
In a recent study, Cacciato et al. (2009) have constrained the CLF using the abundance and
clustering of galaxies in SDSS. They have shown that this CLF is also able to reproduce the
galaxy−galaxy lensing signal and is further consistent with the MLR obtained from a SDSS
group catalogue (Yang et al. 2008). As a consistency check, we compare the results of their
study with the results obtained here from satellite kinematics. In panels (e) and (f) of Fig. 4.9,
dashed lines show the mean and scatter of the distribution P (M |Lc ) (MLR), obtained from
satellite kinematics, while the solid lines show the relations obtained from the best−fit CLF
model of Cacciato et al. (2009). The agreement with the results obtained here using the kinematics of satellite galaxies is not perfect. However, given the errorbars it is certainly consistent
with 68% confidence. Amongst others this consistency provides further support that the halo
mass assignment in the SDSS group catalogue of Yang et al. (2007) is reliable (Wang et al.
2008).
Since the mean and scatter of the MLR reflect the physics, and in particular the stochasticity, of galaxy formation, it is interesting to compare the results obtained here to predictions
from semi-analytical models (SAM) of galaxy formation. To that extent we use the SAM of
Croton et al. (2006), which has been shown to match the observed properties of the local galaxy
population with reasonable accuracy2 . Using a volume limited sample of galaxies selected from
the SAM with the same luminosity and redshift cuts as SDSSV, we measure the mean and the
scatter of the distributions of halo masses for central galaxies in several bins of r-band luminosity. The results are shown in panels (e) and (f) of Fig. 4.9 as open squares. The agreement
with our constraints from the satellite kinematics is remarkably good. It is both interesting and
encouraging that a semi-analytical model, which uses simple, physically motivated recipes to
model the complicated baryonic physics associated with galaxy formation, is able to reproduce
not only the mean of the MLR of central galaxies but also the correct amount of stochasticity in
this relation.
In our model, the stochasticity of galaxy formation is best described by the parameter σlog L ,
which indicates the amount of scatter in the luminosity of central galaxies given the mass of a
halo, i.e. the scatter in the distribution P (Lc |M ). The histogram in Fig. 4.10 shows the posterior
probability of σlog L , obtained from our MCMC, which yields σlog L = 0.16 ± 0.04 (68%
2
Note that Croton et al. (2006) adopted a slightly different cosmology than the one used in our data analysis
which can have a small impact on the MLR.
64
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
Figure 4.10: Posterior distribution of the parameter σlog L as obtained from the MCMC analysis
of the satellite velocity dispersions. The 1-σ constraints on the parameter σlog L obtained from
other independent methods are shown as shaded regions. Region GC indicates the SDSS group
catalogue result by Yang et al. (2008), region CLF indicates the result obtained by Cooray
(2006) with an independent CLF analysis and region SAM shows our measurement of σlog L
from the semi-analytical model of Croton et al. (2006).
4.5. RESULTS FROM THE MCMC ANALYSIS
65
Figure 4.11: Results of the MCMC analysis of the velocity dispersions obtained from Sample LR. Crosses with errorbars in the upper panels show the data points used to constrain the
MCMC; the satellite-weighted velocity dispersions in panel (a), the host-weighted velocity dispersions in panel (b) and the mean number of satellites per central as a function of luminosity in
panel (c), all measured by using the satellites from Sample LR. The blue and purple bands represent the 68% and 95% confidence regions respectively. The bottom panels show the relations
inferred from the MCMC; the average log(Lc ) is in panel (d), and the mean and the scatter in
the MLR of central galaxies in panels (e) and (f), respectively.
confidence levels). Note that we have made the assumption that σlog L is independent of halo
mass. The same assumption was made by Cooray (2006), who obtained that σlog L = 0.17+0.02
−0.01
using the luminosity function and clustering properties of SDSS galaxies (see also Cacciato
et al. 2009). Using a large SDSS galaxy group catalogue, Yang et al. (2008) obtained direct
estimates of the scatter in P (Lc |M ), and found that σlog L = 0.13 ± 0.03 with no obvious
dependence on halo mass. Finally, we also determined σlog L in the SAM of Croton et al.
(2006): using several bins in halo mass covering the range 1010 h−1 M ≤ M ≤ 1016 h−1 M ,
we find that σlog L = 0.17 ± 0.02, once again with virtually no dependence on halo mass. All
these results are summarized in Fig. 4.10. Not only do they support our assumption that σlog L
is independent of halo mass, they also are in remarkable quantitative agreement with each other.
66
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
Figure 4.12: Results of the MCMC analysis of the velocity dispersions obtained from Sample LB. Crosses with errorbars in the upper panels show the data points used to constrain the
MCMC; the satellite-weighted velocity dispersions in panel (a), the host-weighted velocity dispersions in panel (b) and the mean number of satellites per central as a function of luminosity in
panel (c), all measured by using the satellites from Sample LB. The blue and purple bands represent the 68% and 95% confidence regions respectively. The bottom panels show the relations
inferred from the MCMC; the average log(Lc ) is in panel (d), and the mean and the scatter in
the MLR of central galaxies in panels (e) and (f), respectively.
4.5. RESULTS FROM THE MCMC ANALYSIS
67
Table 4.7: Sample LR: MLR of red central galaxies
log(Lc ) hlog Mi ∆hlog Mi
σlog M
∆σlog M
−2
−1
−1
−1
h L h M
h M
h M h−1 M
9.61
11.81
0.36
0.20
0.06
9.74
11.95
0.33
0.22
0.07
9.88
12.11
0.31
0.26
0.07
10.03
12.30
0.28
0.30
0.08
10.17
12.49
0.25
0.33
0.08
10.31
12.70
0.23
0.37
0.09
10.45
12.92
0.20
0.40
0.08
10.59
13.15
0.18
0.42
0.08
10.72
13.38
0.16
0.43
0.08
10.87
13.64
0.15
0.44
0.07
The mean and scatter of the halo masses as a function of the central galaxy luminosity inferred
from the MCMC analysis of red central galaxies. The errors on each of the inferred quantities
correspond to the 68% confidence levels.
Table 4.8: Sample LB: MLR of blue central galaxies
log(Lc ) hlog Mi ∆hlog Mi
σlog M
∆σlog M
−2
−1
−1
−1
h L h M
h M
h M h−1 M
9.60
11.66
0.25
0.25
0.10
9.72
11.76
0.24
0.27
0.10
9.86
11.88
0.23
0.31
0.11
9.99
12.02
0.22
0.34
0.12
10.12
12.15
0.22
0.38
0.12
10.25
12.28
0.23
0.41
0.12
10.38
12.42
0.25
0.43
0.12
10.52
12.57
0.27
0.46
0.12
10.65
12.70
0.30
0.48
0.12
10.78
12.84
0.33
0.49
0.12
The mean and scatter of the halo masses as a function of the central galaxy luminosity for blue
central galaxies inferred from the MCMC analysis. The errors on each of the inferred quantities
correspond to the 68% confidence levels.
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
68
Figure 4.13: Comparison of the average halo masses and the scatter in halo masses of central
galaxies split by colour as a function of their luminosity. The results for red central galaxies are
shown with (red) squares and those for blue central galaxies are shown with (blue) triangles.
4.5.2
The Colour Dependence of the Halo Mass−Luminosity Relation
Next, we present the results of the MCMC analysis of the velocity dispersions obtained from
Samples LR and LB respectively. The 16th , 50th and 84th percentiles of the distributions of the
parameter values that describe the distribution P (Lc |M ) for the red central galaxies and for blue
central galaxies are listed in Table 4.5. In Fig. 4.11, we show the results of the MCMC analysis
of velocity dispersions around red central galaxies while in Fig. 4.12 we show the corresponding
results for blue central galaxies. The values of hlog M i(Lc ) and σlog M (Lc ) together with their
1-σ errors for samples LR and LB are listed in Table 4.7 and Table 4.8, respectively.
The mean luminosity of red central galaxies scales with halo mass as Lc ∝ M 0.31±0.06
while that of blue central galaxies scales as Lc ∝ M 0.3±0.2 at the bright end (see Panel (d) in
Figs. 4.11 and 4.12). The scaling between the luminosity and halo mass is poorly constrained
for the blue galaxies at the bright end as the data does not contain many bright blue centrals. The
value of the scatter in the distribution P (Lc |M ) for red central galaxies is σlog L = 0.20+0.02
−0.02
and that for blue central galaxies is σlog L = 0.27+0.09
−0.07 . The value of the scatter for blue central
galaxies is rather poorly constrained due to the poor quality of the velocity dispersion data
(smaller satellite sample).
The average halo mass of central galaxies increases with the luminosity of the central galaxy
irrespective of their colour (Panel e in Figs. 4.11 and 4.12). However, the MLR of blue central
galaxies differs from the MLR of red central galaxies. The blue central galaxies exhibit a shallower scaling relation than their red counterparts (see Fig 4.13). This shows that the luminosity
of a central galaxy alone is not a good proxy for its halo mass. Furthermore, the scatter in
halo masses at fixed luminosity is non-negligible, and increases as a function of the luminosity
4.6. SUMMARY
69
for both the red and the blue central galaxies. Thus, even after central galaxies are stacked on
the basis of their luminosities and colours, the MLR shows a considerable amount of scatter.
This further emphasizes the need to model this scatter when interpreting results of studies that
involve stacking.
Next, we compare the results obtained using satellite kinematics with results from the group
catalogue of Yang et al. (2007). These authors have assigned group halo masses based upon
either the total stellar mass or the total luminosity content of each group. We use this group
catalog and investigate the MLR of central galaxies, with and without the split into red and blue
by colour. The solid lines in Fig. 4.14 correspond to the MLR of central galaxies in their group
catalogue where the halo masses have been assigned using the total stellar mass content of the
group. The dashed lines, in turn, show the MLR of central galaxies in the group catalogue
where the halo masses have been assigned according to the luminosity content of the group.
The MLR for central galaxies obtained in this chapter are shown with shaded areas (68 percent
confidence region).
The MLR of all central galaxies (top left panel in Fig. 4.14) inferred from satellite kinematics is in good agreement with the results from the group catalog. There exists a slight tension
at the bright end between the MLR obtained from the group catalogue and that from satellite
kinematics for red central galaxies and this problem is worse for the blue central galaxies. Since
the central galaxy colour information was not used in the construction of the galaxy group catalogue, we believe this to be an artefact in the group catalogue. The group catalogue can reliably
reproduce the average properties of the entire galaxy sample, however it may not be reliable to
deduce the average properties of a subset of galaxies.
Galaxy−galaxy lensing is yet another technique to probe the halo masses and hence the
MLR of central galaxies. Mandelbaum et al. (2006) presented the weak lensing signal around
galaxies stacked by luminosity. The galaxies were split into early (red) and late (blue) types
based upon their morphology. The results obtained from their analysis of the weak lensing signal are shown in Fig. 4.14 as squares with errorbars. These results are in excellent agreement
with our results from satellite kinematics. The potential disagreement for the brightest luminosity bin of red centrals from their sample is most likely a result of the different criteria used to
separate red and blue central galaxies.
4.6
Summary
The kinematics of satellite galaxies have been widely used to statistically relate the mean halo
masses of central galaxies to their luminosities (Zaritsky et al. 1993; Zaritsky & White 1994;
Zaritsky et al. 1997; McKay et al. 2002; Brainerd & Specian 2003; Prada et al. 2003; Norberg
et al. 2008). These studies use strict criteria to identify central and satellite galaxies that reside
preferentially in isolated environments. van den Bosch et al. (2004) have advocated the use
of a relaxed but adaptive selection criterion to identify centrals and their satellites, not only in
70
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
Figure 4.14: Comparison of the MLR of central galaxies by different methods. The results obtained from the weak lensing analysis are shown as squares with errorbars. The results obtained
from the group catalogue are shown with solid and dashed lines. The shaded areas represent the
results obtained in this chapter using the kinematics of satellites.
4.6. SUMMARY
71
isolated environments but also in massive groups and clusters. This has the potential to allow
the study of the kinematics of satellites over a wide range of central galaxy luminosity. In
this chapter, we applied a relaxed but adaptive selection criterion to a volume limited sample
from SDSS to identify centrals and their satellites, not only in isolated environments but also
in massive groups and clusters which allowed us to study the kinematics of satellites over a
wide range of central galaxy luminosity. We inferred both the mean and the scatter of the
mass−luminosity relationship of central galaxies from the kinematics of satellite galaxies both
with and without a split in the colour of central galaxies.
The analysis of the kinematics of satellites around centrals stacked without a split in their
colour shows that the mean of the mass−luminosity relation increases as a function of the
central host luminosity indicating that, as expected, brighter centrals reside in more massive
haloes. This result is in quantitative agreement with a recent study by Cacciato et al. (2009),
who use the abundance and the clustering properties of galaxies in SDSS to constrain the CLF,
and with the SAM of Croton et al. (2006). The satellite kinematics obtained in our study are
consistent with a model in which P (Lc |M ) has a constant scatter, σlog L , independent of the
halo mass M . We obtain σlog L = 0.16 ± 0.04 in excellent agreement with other independent
measurements suggesting that the amount of stochasticity in galaxy formation is similar in
haloes of all masses. This is also suggested by the SDSS group catalogue of Yang et al. (2008)
and by the SAM of Croton et al. (2006). It is important to note that a constant scatter in
the distribution P (Lc |M ) leads to a scatter in the distribution P (M |Lc ) that systematically
increases with luminosity (see Chapter 2).
We also analysed the kinematics of satellite galaxies around red and blue central galaxies
separately to investigate the colour dependence of the MLR. We found that the MLR of central
galaxies is different for central galaxies of different colours. Red central galaxies on average
occupy more massive haloes than blue central galaxies. This shows that the scatter in the colour
of central galaxies is, to a certain extent, responsible for the scatter in halo masses at fixed
central galaxy luminosity. However, we also found that both red and blue central galaxies of
a given luminosity reside in haloes with a large scatter in their masses (∼ 0.4 dex) especially
at the bright end. Hence it is imperative to account for the scatter in any analysis that involves
stacking.
We compared the average MLR inferred from satellite kinematics with those inferred from
the SDSS group catalogue of Yang et al. (2008) for central galaxies. The average MLR of
central galaxies without a split in colour is in excellent agreement with the group catalogue
results. The average MLRs of central galaxies split by colour are in slight tension with those
present in the group catalogue which we believe to be an artefact of the group catalogue. The
group catalogue may not be reliable for deducing the average properties of subsets of galaxies.
We also compared the average MLRs of red and blue central galaxies obtained by us with the
results obtained using galaxy−galaxy lensing by Mandelbaum et al. (2006b) and showed that
these results are in excellent quantitative agreement with each other.
72
4. THE HALO MASS−LUMINOSITY RELATIONSHIP
Chapter 5
The Halo Mass−Stellar Mass
Relationship
The contents of this chapter are based upon an article which is in preparation. The article will
be submitted to the Monthly Notices of the Royal Astronomical Society as:
Satellite Kinematics III: Colour and Stellar Mass Dependence
More, S., van den Bosch, F. C., Cacciato, M., et al. 2009.
5.1
Introduction
Establishing scaling relations between the properties of central galaxies and their dark matter
halo properties is central to understanding the process of galaxy formation. In this thesis, we
have presented a systematic method that can be used to probe the halo masses of galaxies
that reside at the centre of dark matter haloes using the kinematics of their central galaxies
(Chapters 2 and 3). In Chapter 5, we have established the scaling relation between halo mass
and luminosity of central galaxies and the scatter in this scaling relation. In this chapter, we infer
the halo mass−stellar mass relationship of central galaxies (hereafter MSR) with and without a
split in their colour.
This chapter is organized as follows. In Section 5.2, we describe the samples of central and
satellite galaxies used in this chapter. In Section 5.3, we present the measurement of the kinematics of the satellites as a function of the stellar mass of the central galaxy. In Section 5.4, we
present the model we use to infer the MSR of central galaxies from the kinematics of satellites.
We present our results in Section 5.5 and also compare them with results from other independent
studies. Finally, we summarize our findings in Section 5.6.
For the analysis presented in this chapter, we assume the cosmological parameters from
the 3 year data release of WMAP (Spergel et al. 2007), Ωm = 0.238, ΩΛ = 0.762, h =
H0 /100 km s−1 Mpc−1 = 0.734, the spectral index of initial density fluctuations ns = 0.951
and normalization σ8 = 0.744. We consider the dark matter halo mass to be M180 , i.e. the mass
73
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
74
enclosed within a spherical overdensity δρ/ρ̄ = 180, where ρ̄ denotes the mean matter density
of the universe.
5.2
Central and Satellite Samples from the SDSS
As in Chapter 4, we again use data from the NYU−VAGC (Blanton et al. 2005) which is
based upon Data Release 4 (Adelman-McCarthy et al. 2006) of the SDSS. We start from sample SDSSV, the volume limited sample of galaxies described in Section 4.2. The galaxies in
this sample were assigned a colour (red or blue), according to the separation criteria given by
Eq. (4.1) which was based upon the bimodal distribution of galaxies in the 0.1 (g − r) colour at a
given luminosity. The stellar masses of these galaxies (denoted as M∗ ) are computed using the
relation between the stellar mass-to-light ratio and the 0.0 (g − r) color provided by Bell et al.
(2003),
log
M∗
= −0.306 + 1.097 [0.0 (g − r)] − 0.10
h−2 M
−0.4 (0.0 Mr − 5 log h − 4.64).
(5.1)
Here, 0.0 (g−r) and 0.0 Mr −5 log h denote the (g−r) colour and the r-band absolute magnitude
of galaxies k−corrected and evolution corrected to z = 0.0; 4.64 is the r-band magnitude of
the Sun in the AB system; and the −0.10 term is a result of adopting the Kroupa (2001) initial
mass function (see Borch et al. 2006).
For the analysis of the MSR, we use a slightly modified version of the iterative criteria
(ITER) outlined in Section 3.3 of Chapter 3 to identify centrals and their satellites from SDSSV.
Firstly, the modified criteria requires the central galaxy to be the largest in terms of the stellar
mass (instead of the brightest in terms of luminosity) in its neighbourhood (specified by (∆V )h
and Rh ). Secondly, the unbinned estimate of the velocity dispersion as a function of the stellar
mass of the central galaxy is used to refine the search cylinders of the iterative criteria. We
use the method described in Section 3.4.1 to measure the unbinned estimate of the velocity
dispersion. The velocity dispersion (in units of 200 km s−1 ), σ200 , is parametrized as,
σ200 (log M∗10 ) = a + b (log M∗10 ) + c (log M∗10 )2 .
(5.2)
Here M∗10 denotes the stellar mass of the galaxy in units of 1010 h−2 M . During every iteration,
the parameters (a, b, c) are fit by using the velocity difference information between the satellites
and their centrals (see Section 3.4.1 for details). The parameters that define the search cylinder
are scaled for the subsequent iteration based upon the estimate of σ200 (M∗10 ) (see criteria ITER
in Table 3.1).
Similar to Chapter 4, we form three different samples of centrals and their satellites for the
analysis of the MSR. The sample used to analyse the MSR around all central galaxies (without
5.3. VELOCITY DISPERSION−STELLAR MASS RELATION
75
Table 5.1: Selection criteria parameters
Parameters
Samples
SA
SR
SB
a
2.06
2.12
1.97
b
0.22
0.18
0.45
c
0.20
0.21
-0.07
The parameters a, b and c that are used to define the criteria used to select central and satellite
galaxies for the samples used in this chapter (see Table 3.1 for details of the selection criteria).
a split by colour) is denoted by SA. The samples used to analyse the MSR around red and
blue central galaxies are denoted by SR and SB, respectively. Note that the difference between
the three samples is the estimate of the velocity dispersion used to scale the search cylinders.
The search criteria used for sample SR (SB) are tuned based upon the estimate of the velocity
dispersion around the red (blue) central galaxies only. The parameters (a, b, c) in Eq. (5.2) that
define the aperture used in the final iteration of the central−satellite selection for samples SA,
SR and SB are listed in Table 5.1.
The number of central galaxies with at least one satellite is 3, 778 for Sample SA, 2, 877
for Sample SR and 805 for Sample SB. The number of satellite galaxies in Sample SA is 6, 104
while that in Sample SR and SB are 5, 061 and 912, respectively.
5.3
Velocity Dispersion−Stellar Mass Relation
The scatter plot of the velocity difference, ∆V , between the satellites and the corresponding
centrals as a function of the central galaxy stellar mass in Sample SA is shown in Fig. 5.1, while
those in Samples SR and SB are shown in the left and the right panels of Fig. 5.2, respectively.
For all the three samples, the scatter in the velocities of satellites with respect to their centrals
increases with central galaxy stellar mass. Its relatively clear from these figures that the scatter
of the velocities of satellites in Sample SB is smaller than that of satellites in Samples SR and
SA.
The P (∆V ) distributions for all the three samples, in both the satellite-weighting and the
host-weighting schemes are obtained in 10 bins of uniform width. For Sample SA the binwidth is ∆ log[M∗c ] = 0.18 while for Samples SR and SB we choose the bin widths to
be 0.16 and 0.15, respectively. All the P (∆V ) distributions are fitted using a model that
consists of two Gaussians and a constant, for reasons discussed in Section 3.4.2. The resultant velocity dispersions and their errors for Sample SA are listed in Table 5.2 and those
for Samples SR and SB in Tables 5.3 and 5.4, respectively. The satellite-weighted and the
host-weighted velocity dispersions as a function of stellar mass for Sample SA are shown in
76
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
Figure 5.1: Scatter plot of the velocity difference, ∆V , between the satellites and their central
galaxies in sample SA as a function of the central galaxy stellar mass.
Figure 5.2: Scatter plot of the velocity difference, ∆V , between the satellites and their central
galaxies in sample SR sample SB as a function of the central galaxy stellar mass are shown in
the left and the right panels respectively.
5.4. THE MODEL
77
Figure 5.3: The satellite-weighted (open hexagons) and the host-weighted (filled hexagons)
velocity dispersions as a function of the central galaxy stellar mass obtained from satellites in
the sample SA.
Fig. 5.3. The velocity dispersion−stellar mass relation shows a similar behaviour to the velocity
dispersion−luminosity relation. The satellite-weighted velocity dispersioins are systematically
larger than the host-weighted velocity dispersions at any given stellar mass, which signals the
presence of a non-negligible scatter in the MSR as well. The comparison between the velocity
dispersions around red and blue centrals is shown in Fig. 5.4. The velocity dispersions around
red centrals in both the schemes are systematically larger than the velocity dispersions around
blue centrals. As in Chapter 3, we also measure the average number of satellites as a function
of stellar mass and use it to constrain the MSR.
5.4
The Model
We use a very similar model to the one used in Chapters 3 and 4 to determine the MSR from the
measured velocity dispersions. The analytical expressions that describe the velocity dispersions
and the the average number of satellites as a function of stellar mass are the same as Eqs. (3.12),
(3.13) and (3.17) with P (M |Lc ) replaced by P (M |M∗c ), i.e.,
2
σsw
(M∗c )
=
R∞
R∞
0
2 i
P (M |M∗c ) hNsat iap,M hσsat
ap,M dM
R∞
,
0 P (M |M∗c ) hNsat iap,M dM
2 i
P (M |M∗c ) P(hNsat iap,M ) hσsat
ap,M dM
R∞
,
0 P (M |M∗c ) P(hNsat iap,M ) dM
R∞
P (M |M∗c )hNsat iap,M dM
hNsat i(M∗c ) = R ∞0
.
0 P (M |M∗c )P(hNsat iap,M )dM
2
σhw
(M∗c ) =
0
(5.3)
(5.4)
(5.5)
78
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
Figure 5.4: Comparison between the satellite-weighted (host-weighted) velocity dispersions as
a function of the stellar mass of the central galaxy obtained from Sample SR and Sample SB is
shown in the left (right) panel.
Table 5.2: Sample SA: Velocity dispersion measurements
log(M∗c )
σsw
∆σsw
σhw
∆σhw
h−2 L km s−1 km s−1 km s−1 km s−1
9.62
83
28
83
28
9.81
121
16
121
16
9.99
97
27
98
26
10.18
134
14
134
14
10.37
181
39
173
24
10.55
184
10
176
9
10.74
248
17
225
13
10.92
294
12
258
18
11.10
425
30
312
34
11.28
530
34
376
90
Velocity dispersion measurements in the satellite-weighted and host-weighted schemes together
with the associated errors for sample SA.
5.4. THE MODEL
79
Table 5.3: Sample SR: Velocity dispersion measurements
log(M∗c )
σsw
∆σsw
σhw
∆σhw
h−2 L km s−1 km s−1 km s−1 km s−1
9.91
162
64
162
64
10.05
122
24
123
25
10.21
171
38
157
38
10.36
172
14
168
15
10.53
193
20
182
15
10.68
216
23
209
17
10.84
289
15
255
22
10.99
311
18
260
14
11.14
429
30
334
47
11.31
567
38
434
129
Velocity dispersion measurements in the satellite-weighted and host-weighted schemes together
with the associated errors for sample SR.
Table 5.4: Sample SB: Velocity dispersion measurements
log(M∗c )
σsw
∆σsw
σhw
∆σhw
h−2 L km s−1 km s−1 km s−1 km s−1
9.57
129
92
129
92
9.73
74
41
74
41
9.88
109
23
109
23
10.03
114
42
102
40
10.18
114
14
114
14
10.33
120
14
121
15
10.48
154
18
148
14
10.62
172
15
166
16
10.76
179
39
155
38
10.91
192
24
174
35
Velocity dispersion measurements in the satellite-weighted and host-weighted schemes together
with the associated errors for sample SB.
80
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
Figure 5.5: Projected number density distributions of satellites around centrals in Sample SA
for the five massive stellar mass bins (average log(M∗c / h−1 M ) in right corner).
2 i
The notations hNsat iap,M and hσsat
ap,M describe the average number of satellites in the
aperture used to select the satellites and their aperture−averaged velocity dispersion in a halo of
mass M used to select the satellites. The factor P adjusts the distribution P (M |M∗c ) to account
for those centrals that do not host any satellite. We use the expressions derived in Section (2.4)
2 i
to calculate these quantities. Note that the calculation of hNsat iap,M and hσsat
ap,M requires
us to specify the number density distribution of satellites in the halo. We use the generalised
NFW profile given by Eq. (2.24) for this purpose, keeping in mind that the parameters, R and γ
have to be fixed using the projected number density distribution, Σ(R), of satellites around their
centrals. Figs. (5.5)−(5.7) show the distributions Σ(R) of satellites in the samples SA, SR and
SB, respectively. These distributions resemble the projected number density distribution shown
in Fig. 4.6 in that they show a flattening of the distribution in the central parts. Based upon this
behaviour, we use the values of R = 2 and γ = 0 in the further analysis.
We use the Bayes’ theorem to relate the distribution P (M |M∗c ) to P (M∗c |M ),
P (M |M∗c ) = R
n(M )P (M∗c |M )
,
n(M )P (M∗c |M )dM
(5.6)
where n(M ) is the halo mass function. The distribution P (M∗c |M ) is modelled as a lognormal
5.4. THE MODEL
81
Figure 5.6: Projected number density distributions of satellites around red centrals from Sample
SR (average log(M∗c / h−1 M ) in right corner).
82
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
Figure 5.7: Projected number density distributions of satellites around blue centrals from Sample SB (average log(M∗c / h−1 M ) in right corner).
5.5. RESULTS
83
distribution in M∗c ,
P (M∗c |M )dM∗c
#
"
log(e)
(log[M∗c /M˜∗c ])2 dM∗c
.
=√
exp −
2
M∗c
2σlog
2πσlog M∗
M∗
(5.7)
The quantities log M˜∗c (M ) and σlog M∗ are the mean and the scatter of the lognormal distribution at a particular halo mass M . The scatter σlog M∗ is assumed to be independent of the halo
mass M and the relation log M˜∗c (M ) is specified using four parameters: a low mass end slope
γ1 , a high mass end slope γ2 , a characteristic mass scale M1 , and a normalisation M∗0 ;
M˜∗c = M∗0
(M/M1 )γ1
.
[1 + (M/M1 )]γ1 −γ2
(5.8)
Thus, the distribution P (M |M∗c ) is completely specified by the five parameters (σlog M∗ , M∗0 ,
M1 , γ1 , γ2 ) and the halo mass function. The last ingredient of our model is the satellite occupation number, hNsat i(M ). We use a simple power law distribution given by,
hNsat i(M ) = Ns
M
12
10 h−1 M
α
.
(5.9)
to specify the occupation number of satellites. Thus in total our model has seven free parameters. We use a MCMC to constrain the parameters and infer the MSR. Since the velocity
dispersions at the low stellar mass are not well measured, we do not expect to properly constrain the low mass end slope, γ1 . Therefore, we impose a flat prior on γ1 and allow it to vary
in the interval [2.0, 4.0].
5.5
5.5.1
Results
The Halo Mass−Stellar Mass Relationship
We analyse the kinematics of satellite galaxies around their centrals in Sample SA to infer the
MSR for all central galaxies. Samples SR and SB are analysed to infer the MSR of central
galaxies split by colour into red and blue, respectively. The results from the MCMC analysis
are shown in Figs. 5.8, 5.9 and 5.10. The 16, 50 and 84 percentiles of the posterior distributions
of the parameters obtained in the MCMC analysis are listed in Table 5.5.
+0.08
The stellar mass of all central galaxies scales as M 0.38−0.07 at the high mass end. This is
in excellent agreement with recent results from Moster et al. (2009) who fit the stellar mass
function of SDSS galaxies using a halo occupation model and obtain that the stellar mass of
central galaxies scales as M 0.370±0.014 (see table 3 in Moster et al. 2009)1 . In the case of red
central galaxies, the stellar mass scales as M∗ ∝ M 0.39±0.08 and in the case of blue central
+0.3
galaxies the stellar mass scales as M∗ ∝ M 0.7−0.2 . The halo mass−stellar mass scaling for
1
Note that the parameter γc in the analysis of Moster et al. 2009 is related to the parameter γ2 in our analysis
such that γc = γ2 − 1.
84
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
Table 5.5: MSR of central galaxies: Parameters recovered from the MCMC
Sample Parameter 16 percent 50 percent 84 percent
SA
log(M∗0 )
9.57
9.97
10.39
log(M1 )
10.25
10.83
11.68
γ2
0.31
0.38
0.46
σlog P
0.18
0.21
0.23
Ns
0.11
0.15
0.20
αs
1.19
1.28
1.37
SR
log(M∗0 )
9.63
9.99
10.36
log(M1 )
10.28
10.82
11.55
γ2
0.31
0.39
0.47
σlog P
0.21
0.23
0.25
Ns
0.06
0.09
0.13
αs
1.34
1.47
1.62
SB
log(M∗0 )
9.35
9.98
10.60
log(M1 )
10.73
11.29
11.78
γ2
0.33
0.68
0.89
σlog P
0.13
0.27
0.41
Ns
0.16
0.22
0.28
αs
0.69
1.10
1.48
The 16, 50 and 84 percentile values of the posterior distribution for the parameters of our model
obtained from the MCMC analysis of the velocity dispersion data from Sample SA, Sample SR
and Sample SB.
Table 5.6: Sample SA: MSR of central galaxies
log(Lc ) hlog Mi ∆hlog Mi
σlog M
∆σlog M
−2
−1
−1
−1
h L h M
h M
h M h−1 M
9.62
11.18
0.44
0.17
0.05
9.81
11.35
0.41
0.20
0.07
9.99
11.54
0.37
0.23
0.08
10.18
11.77
0.32
0.27
0.09
10.37
12.03
0.27
0.32
0.09
10.55
12.30
0.23
0.36
0.09
10.74
12.59
0.20
0.39
0.09
10.92
12.90
0.17
0.41
0.09
11.10
13.22
0.15
0.42
0.08
11.28
13.53
0.14
0.43
0.07
The mean and scatter of the halo masses as a function of the central galaxy stellar mass for all
central galaxies inferred from the MCMC analysis. The errors on each of the inferred quantities
correspond to the 68% confidence levels.
5.5. RESULTS
85
Figure 5.8: The results of the MCMC analysis of the velocity dispersions obtained from Sample SA. Crosses with errorbars in the upper panels show the data points used to constrain the
MCMC; the satellite-weighted velocity dispersions in panel (a), the host-weighted velocity dispersions in panel (b) and the mean number of satellites per central as a function of stellar mass
in panel (c), all measured by using the satellites in Sample SA. The blue and purple bands represent the 68% and 95% confidence regions, respectively. The bottom panels show the relations
inferred from the MCMC; the average log(M∗c ) is in panel (d), and the mean and the scatter in
the MSR of central galaxies in panels (e) and (f), respectively.
86
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
Figure 5.9: The results of the MCMC analysis of the velocity dispersions obtained from Sample SR. Crosses with errorbars in the upper panels show the data points used to constrain the
MCMC; the satellite-weighted velocity dispersions in panel (a), the host-weighted velocity dispersions in panel (b) and the mean number of satellites per central as a function of stellar mass
in panel (c), all measured by using the satellites in Sample SR. The blue and purple bands represent the 68% and 95% confidence regions, respectively. The bottom panels show the relations
inferred from the MCMC; the average log(M∗c ) is in panel (d), and the mean and the scatter in
the MSR of central galaxies in panels (e) and (f), respectively.
5.5. RESULTS
87
Figure 5.10: The results of the MCMC analysis of the velocity dispersions obtained from Sample SB. Crosses with errorbars in the upper panels show the data points used to constrain the
MCMC; the satellite-weighted velocity dispersions in panel (a), the host-weighted velocity dispersions in panel (b) and the mean number of satellites per central as a function of stellar mass
in panel (c), all measured by using the satellites in Sample SB. The blue and purple bands represent the 68% and 95% confidence regions, respectively. The bottom panels show the relations
inferred from the MCMC; the average log(M∗c ) is in panel (d), and the mean and the scatter in
the MSR of central galaxies in panels (e) and (f), respectively.
88
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
Figure 5.11: Comparison of the average halo masses and the scatter in halo masses of central
galaxies split by colour as a function of their stellar mass. The results for red central galaxies
are shown with (red) squares and those for blue central galaxies are shown with (blue) triangles.
Table 5.7: Sample SR: MSR of red central galaxies
log(Lc ) hlog Mi ∆hlog Mi
σlog M
∆σlog M
−2
−1
−1
−1
h L h M
h M
h M h−1 M
9.91
11.41
0.40
0.23
0.06
10.05
11.54
0.38
0.26
0.07
10.21
11.72
0.35
0.29
0.07
10.36
11.92
0.32
0.33
0.08
10.53
12.14
0.28
0.36
0.08
10.68
12.36
0.25
0.40
0.08
10.84
12.61
0.22
0.43
0.08
10.99
12.86
0.19
0.44
0.08
11.14
13.12
0.16
0.46
0.07
11.31
13.39
0.14
0.46
0.07
The mean and scatter of the halo masses as a function of the central galaxy stellar mass inferred
from the MCMC analysis of red central galaxies. The errors on each of the inferred quantities
correspond to the 68% confidence levels.
5.5. RESULTS
89
Table 5.8: Sample SB: MSR of blue central galaxies
log(Lc ) hlog Mi ∆hlog Mi
σlog M
∆σlog M
−2
−1
−1
−1
h L h M
h M
h M h−1 M
9.57
11.51
0.30
0.17
0.09
9.73
11.61
0.29
0.18
0.09
9.88
11.71
0.27
0.19
0.10
10.03
11.82
0.27
0.20
0.11
10.18
11.94
0.26
0.22
0.12
10.33
12.07
0.25
0.24
0.13
10.48
12.19
0.24
0.26
0.14
10.62
12.32
0.25
0.27
0.15
10.76
12.45
0.25
0.29
0.16
10.91
12.60
0.26
0.31
0.17
The mean and scatter of the halo masses as a function of the central galaxy stellar mass for blue
central galaxies inferred from the MCMC analysis. The errors on each of the inferred quantities
correspond to the 68% confidence levels.
blue galaxies at the massive end is poorly constrained as the sample does not consist of massive
blue central galaxies. The value of the scatter in stellar masses of all central galaxies at fixed
+0.02
halo mass is 0.21+0.03
−0.02 . The value of the scatter is 0.23−0.02 for red central galaxies while it is
0.27+0.14
−0.13 for blue central galaxies.
The average halo mass of all central galaxies and that of red and blue central galaxies increases as a function of the stellar mass and so does the scatter in halo masses. In Fig. 5.11,
we overplot the mean and the scatter of the MSR from Samples SA, SR and SB together. The
difference in the scaling relation between the average halo mass and the stellar mass for red and
blue galaxies is much less pronounced than that seen in the MLR. This demonstrates that when
stacked by stellar mass central galaxies on average occupy haloes of similar mass independent
of their colour and in this respect stellar mass is a better proxy for halo mass. We find that the
scatter in halo masses also increases as a function of the stellar mass, similar to the behaviour
seen as a function of the luminosity, and reaches about ∼ 0.4 dex at the massive end.
Finally, we would like to point out here that the velocity dispersion of satellite galaxies
around red central galaxies is always systematically higher than that around blue central galaxies
of the same stellar mass. However, the average MSR inferred from the analysis is roughly
similar in both cases. This shows that it is not straightforward to use the velocity dispersion as
a proxy for halo mass and that inferring the average halo masses demands a careful modelling
similar to the one presented in this thesis.
5.5.2
Comparison of MSR with Other Studies
In this section, we compare the MSR obtained using satellite kinematics with those obtained
with other independent methods. We first investigate the MSR of central galaxies, with and
90
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
Figure 5.12: Comparison of the MSR of central galaxies by different methods. The shaded
areas represent the results obtained in this chapter using the kinematics of satellites. The results
obtained by analysing the group catalogue are shown with solid and dashed lines. The results
obtained from the weak lensing analysis are shown as squares with errorbars. The results obtained by the analysis of satellite kinematics by Conroy et al. (2007) are shown using circles
with errorbars.
5.5. RESULTS
91
without the split into red and blue colours in the SDSS group catalogue of Yang et al. (2007).
The solid lines in Fig. 5.12 correspond to the MSR of central galaxies in their group catalogue
where the halo masses have been assigned using the total stellar mass content of the group.
The dashed lines, in turn, show the MSR of central galaxies in the group catalogue where the
halo masses have been assigned according to the luminosity content of the group. The MSR
for central galaxies obtained in this chapter are shown with shaded areas (68 percent confidence
region). The MSR of all central galaxies (top left panel in Fig. 5.12) inferred from satellite
kinematics is overall in good agreement with the results from the group catalog. The MSR for
red centrals obtained in our study are in slight tension at the massive end while those for blue
centrals are in a fair agreement. As mentioned in the previous chapter, one has to be careful
while interpreting results from the group catalogue for a subset of galaxies because it may not
be reliable to deduce the average properties of a subset of galaxies (such as central galaxies split
by their colour).
The MSR of central galaxies from the galaxy−galaxy lensing by Mandelbaum et al. (2006b)
are shown in Fig. 5.12 as squares with errorbars. These results are also in fair agreement with
the results from satellite kinematics. The potential disagreement for the most massive stellar
mass bin from their sample is most likely a result of the different criteria used to separate red
and blue central galaxies. Unlike our sample, Mandelbaum et al. split the galaxies into early
(red) and late (blue) types based upon their morphology.
Using data from the SDSS and the DEEP2 survey, Conroy et al. (2007) used the kinematics
of satellite galaxies to determine the evolution of the stellar mass-to-light ratio of central galaxies from z ∼ 1 to z ∼ 0. They measured and modelled the radial dependence of the velocity
dispersion to infer the average mass of the halo as a function of the stellar mass of the central
galaxy. We compare the MSR at z ∼ 0 obtained by Conroy et al. (2007) with our results. Note
that the halo mass definition used by Conroy et al. (2007) corresponds to M200 , i.e., the mass
of the halo within a radius which encloses an average density which is 200 times the critical
density of the Universe. We have converted their definition of the halo mass to our definition of
the halo mass following the procedure outlined in Hu & Kravtsov (2003). The stellar masses of
the galaxies were calculated using a stellar mass-to-light ratio based on a Chabrier initial mass
function. We have also converted these stellar masses to the ones based on the initial mass function obtained by Kroupa (2001) for consistency using results from Bell et al. (2003). The halo
mass−stellar mass relationship for all central galaxies obtained from the analysis of Conroy
et al. (2007) is shown in Fig. 5.12 using circles with errorbars. The average halo masses obtained by their analysis are in agreement at the 2-σ level with the ones obtained in this chapter.
However, their halo mass measurements are always systematically larger than those obtained by
us. This could be a result of the fact that the scatter in the halo mass−stellar mass relationship is
assumed to be negligible in their analysis. This tends to overestimate the halo masses of central
galaxies (see Chapter 2).
5. THE HALO MASS−STELLAR MASS RELATIONSHIP
92
5.6
Summary
In this chapter, we used the kinematics of satellites galaxies to investigate the scaling relation
between the halo mass and the stellar mass of a central galaxy (the halo mass−stellar mass
relationship or MSR). We also investigated the dependence of the MSR on the colour of central galaxies. The MSR shows similar trends as seen in the halo mass−luminosity relationship
(MLR, see Chapter 4). Both the mean and the scatter of the MSR of central galaxies increase
with the stellar mass of the central galaxy. However, when split by colour, the difference between the mean MSR of the red and the blue central galaxies is less pronounced than that seen
in the MLR. This implies that when stacked by stellar mass, the red and the blue central galaxies on average occupy similar mass haloes. We also found that the MSR of both red and blue
central galaxies, individually, have an appreciable scatter at the massive end.
We compared the average MSR of central galaxies obtained by our analysis of the kinematics of satellite galaxies with other independent studies. The average MSR we obtain is in good
quantitative agreement with results from galaxy−galaxy lensing (Mandelbaum et al. 2006b) and
with results from the SDSS group catalogue of Yang et al. (2007). This shows that the average
scaling relations that relate the stellar mass of galaxies to their dark matter halo masses are well
established and are supported by several different astrophysical probes.
The scatter in the MLR and the MSR of central galaxies that we have inferred from satellite
kinematics reflect the stochasticity of galaxy formation. In the next chapter, we investigate the
physical processes that are responsible for this scatter. With the help of a semi-analytical model
of galaxy formation, we investigate the scatter in the merger histories of dark matter haloes and
its effect on the properties of the central galaxies that form at their centres.
Chapter 6
On the Stochasticity of Galaxy
Formation
The contents of this chapter are based upon an article which is in preparation. The article will
be submitted to the Monthly Notices of the Royal Astronomical Society as:
Stochasticity of Galaxy Formation: Insights from Galaxy Formation Models
More, S., More, A., van den Bosch, F. C., et al. 2009
6.1
Introduction
Dark matter is the most abundant gravitationally unstable component in the Universe and is
therefore responsible for the formation of structure. The tiny initial fluctuations in the dark
matter density field grow over time by the action of gravity and form bound structures (haloes).
The baryons within these haloes undergo cooling and gradually transform into stars and form
galaxies (White & Rees 1978). The ratio of the mass in baryons to the mass in dark matter is
universal and each halo collapses with its “fair share” of baryons. If the efficiency with which
the baryons transform into stars is independent of the halo mass, it is a natural expectation that
extensive properties of galaxies such as the luminosity and the stellar mass correlate positively
with halo mass. However, each halo is unique, each has its own merger history and its own star
formation history and thus, haloes with similar mass need not harbour galaxies with the same
properties. It is certainly interesting to quantify and study this difference as it directly reflects
the stochasticity in the physics of galaxy formation.
In this thesis, we have used the kinematics of satellite galaxies to constrain the scaling
relations between halo mass and central galaxy properties. In Chapter 4, we inferred the mean
and scatter of the relationship between halo mass and the luminosity of central galaxies (MLR)
which occupy the centres of dark matter haloes. In Chapter 5, we inferred the mean and scatter
of the relationship between halo mass and the stellar mass of central galaxies (MSR). The results
show that the average halo mass increases with both the luminosity and the stellar mass of
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a central galaxy. In addition, the scatter in halo masses for galaxies of a particular luminosity
(stellar mass) also increases as a function of the galaxy luminosity (stellar mass). Understanding
the origin of this scatter can help us learn more about the physics of galaxy formation.
There exists a large scatter in the assembly histories of different dark matter haloes (see e.g.,
Lacey & Cole 1993; Kauffmann & White 1993; Lemson & Kauffmann 1999; van den Bosch
2002; Gao et al. 2005; Wechsler et al. 2006; Li et al. 2007, 2008). This scatter in the assembly
history is also reflected in the clustering properties of dark matter haloes such that haloes that
assemble earlier are more strongly clustered than those which assemble later (Gao et al. 2005;
Wechsler et al. 2006; Harker et al. 2006; Jing et al. 2007; Gao & White 2007). The scatter in
the assembly history of haloes possibly also reflects in a scatter in the star formation histories
of the galaxies, which may finally result in a scatter in their properties. Therefore, the scatter in
the assembly history of haloes is a plausible origin of the stochasticity of galaxy formation. A
semi-analytical model of galaxy formation (SAM) is an excellent tool to investigate the origin of
the scatter in the MLR/MSR of galaxies. A SAM considers the hierarchical merger histories of
dark matter haloes and implements simple physical recipes to model the astrophysical processes
that affect the baryons in these haloes (see e.g., White & Frenk 1991; Kauffmann & White
1993). The most recent versions of these models include processes such as radiative cooling,
star formation, energetic feedback from supernovae and active galactic nuclei, and galactic
outflows (see Section 6.2).
In this chapter, we only focus on the MSR of central galaxies and investigate the origin of
the scatter in the MSR using the semi-analytical model of De Lucia & Blaizot (2007, hereafter
DL07 ). This chapter is organized as follows. In Section 6.2, we explain the ideology behind the
use of semi-analytic methods to model galaxy formation and briefly explain the processes that
are commonly modelled in SAMs. In Section 6.3, we describe the numerical simulation used
by DL07 and describe the procedure used to construct the merger histories of haloes from this
simulation. The merger histories of haloes can be characterized by the formation times of the
haloes. In Section 6.4, we present various definitions for the formation times of haloes based
upon their merger histories. In Section 6.5, we investigate the effect of the scatter in the merger
histories of similar mass haloes on the stellar mass of central galaxies that form in these haloes.
We conclude in Section 6.6 with a discussion and a short summary of the results.
6.2
Semi-analytical Models of Galaxy Formation
The last decade has seen a remarkable progress on the observational front which has helped us
further our understanding of galaxy formation. Detailed observations of the local galaxy population (redshift z ∼ 0.0) via large scale galaxy redshift surveys coupled with observations of
galaxies at high redshifts (redshift 1 ≤ z ≤ 5) in various bands have resulted in a wealth of
observational results: e.g. the luminosity functions of local galaxies (Blanton et al. 2003b; Norberg et al. 2002b) and galaxies at high redshifts (Drory et al. 2003; Gabasch et al. 2004; Drory
6.2. SEMI-ANALYTICAL MODELS OF GALAXY FORMATION
95
et al. 2005; Faber et al. 2007), the spatial clustering of galaxies (Norberg et al. 2002a; Tegmark
et al. 2004; Zehavi et al. 2005), the stellar mass−metallicity relation of galaxies (Tremonti et al.
2004; Gallazzi et al. 2005), the bimodal distribution of galaxies in the colour−magnitude plane
(Baldry et al. 2004), the correlation between the mass of the central black hole and the bulge
in a galaxy (Häring & Rix 2004), the Tully−Fisher relation (Giovanelli et al. 1997) and the
the star formation rate history of the Universe (Madau et al. 1996). To understand and connect
these observations in the cosmological framework and to sketch a picture of the formation and
evolution of galaxies through cosmic time is a holy grail for the modern astrophysicist.
The initial conditions that describe a ΛCDM Universe are fairly simple and can be specified
by a handful of parameters (Dunkley et al. 2009): the energy density parameters of the different
components of the Universe (dark matter, dark energy and baryons), the parameters that specify
the power spectrum of small fluctuations in the initial density field (the normalization and the
spectral index), and the present rate of the expansion of the Universe (specified by the Hubble
parameter). The fluctuations in the dark matter component evolve purely due to gravity and can
be followed in a straightforward manner by using a numerical simulation (Davis et al. 1985).
However, following the evolution of baryons and resolving the formation of individual galaxies
in a cosmological volume is computationally very expensive. SAMs use a hybrid approach, in
which the evolution of the dark matter skeleton is followed by a numerical simulation and the
evolution of baryons is followed by using simple analytical recipes, to model the formation of
galaxies in a cosmological context (see e.g., White & Frenk 1991; Kauffmann & White 1993;
Cole et al. 1994; Kauffmann 1996; Kauffmann et al. 1997; Baugh et al. 1998; Somerville &
Primack 1999; Cole et al. 2000; Benson et al. 2002; Springel et al. 2005; Croton et al. 2006; De
Lucia & Blaizot 2007).
The internal structure of dark matter haloes, specified by the density profile and the angular momentum, is an important ingredient to model the galaxies that form within these haloes.
High resolution simulations have shown that the density profile of dark matter haloes is universal and can be described by Eq. 2.22 (Navarro et al. 1997) which has only one free parameter for a given mass, the concentration c. The concentration depends very weakly on the
mass (Bullock et al. 2001; Macciò et al. 2007), however this relation has a considerable scatter
which is correlated with the formation histories of haloes (Navarro et al. 1997; Wechsler et al.
2002; Zhao et al. 2003). Large scale tidal torques impart angular momentum to the dark matter
haloes. The angular momentum of a halo is quantified by the dimensionless spin parameter,
λ = J|E|0.5 /GM 2.5 , where J, M and E are the angular momentum, mass and the energy of
the halo. The spin parameter follows a lognormal distribution with a mean and scatter which is
relatively independent of the mass of the halo (Barnes & Efstathiou 1987; Cole & Lacey 1996;
Lemson & Kauffmann 1999; Bullock et al. 2001; Macciò et al. 2007).
The amount of baryons in each dark matter halo is initially a fixed fraction of the halo mass
and is present in a diffuse form. The elemental composition of this gas in the halo is set by
the big bang nucleosynthesis. As haloes merge, the infalling gas is shock−heated. Numerical
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simulations show that this hot gas settles down to a isothermal density profile which has a
core at roughly one third the scale radius of the halo and with a temperature which is close
to the virial temperature (Navarro et al. 1995; Eke et al. 1998; Frenk et al. 1999). The rate
of cooling of gas depends upon the cooling function which is a function of the temperature,
density and the metallicity of the gas (Sutherland & Dopita 1993). By the virtue of its angular
momentum, the cold gas settles down into a disc which then fragments due to gravitational
instabilities and leads to the formation of stars (Kennicutt 1998). The initial mass function
specifies the distribution of masses of the newly formed stars (Salpeter 1955; Chabrier 2003;
Kroupa 2001). Stars are reservoirs of energy that constantly heat up their surroundings. They
also lose a fraction of their mass to the surroundings by stellar winds. In addition, young massive
stars undergo supernovae explosions and heat up the surrounding cold gas. The supernova ejecta
also enrich the metallicity of the surrounding gas. These baryonic processes which take place
in haloes are modelled using simple analytical prescriptions in a SAM.
Simple recipes are also required to determine the fate of the baryons in two haloes that
merge. Mergers are commonly classified as major or minor based upon the ratio of masses of
haloes that merge together. In case of a minor merger, the central galaxies and satellites in the
smaller halo become satellites of the bigger halo. Their individual (sub)haloes can be resolved
in a numerical simulation if they are above the detection limit of substructure specified in the
substructure finding routines. Once the subhalo masses fall below this limit, these galaxies are
merged with the central galaxy on a dynamical friction timescale (Chandrasekhar 1943). Any
cold gas in the satellite galaxies is added to the disc of the central galaxy. Minor mergers of
galaxies which have some cold gas are accompanied by minor bursts of star formation. Major
mergers between galaxies are more violent and can result in an extreme episode of star formation if any of the galaxies have a reservoir of cold gas. Such mergers destroy the discs present
in any of the haloes and the stars so formed are distributed in a spheroid.
The final ingredient we discuss are the supermassive blackholes which reside at the centres
of galaxies. The process of gas accretion on to supermassive blackholes results in the release
of a significant amount of energy. A high accretion rate, which can be a result of perturbations
such as bar instabilities induced during mergers, can cause the blackhole to be in a quasar phase.
In this phase, the blackhole emits a tremendous amount of energy which is transferred to the
surrounding gas via massive jets. A more quiescent accretion of the hot gas by the black hole
can cause low energy radio activity which pumps energy into the surroundings (Tabor & Binney
1993; McNamara et al. 2000, 2005). Both feedbacks have been shown to be effective in shaping
the bright end of the luminosity function of galaxies and preventing star formation in clusters
by suppressing cooling flows (Croton et al. 2006).
This concludes our very brief description of the various processes that are commonly modelled in a SAM. We refer the reader to Cole et al. (2000) for an excellent overview on this
subject and to the papers by Springel et al. (2005), Croton et al. (2006) and De Lucia & Blaizot
(2007) for the details of the specific model that we use in our analysis. With this description,
6.3. NUMERICAL SIMULATION AND HALO MERGER TREES
97
we hope to have conveyed the diversity of processes that shape the properties of galaxies and
that results in the stochasticity of galaxy formation in haloes of similar masses.
6.3
Numerical Simulation and Halo Merger Trees
The semi-analytical recipes that model the processes described in the previous section are implemented in halo merger trees. DL07 use the halo merger trees derived from the Millennium
simulation which was carried out by the Virgo consortium (Springel et al. 2005). The Millennium simulation tracks the positions and velocities of dark matter particles in a comoving cube
with length equal to 500 h−1 Mpc on each side. The simulation is carried out assuming a flat
ΛCDM cosmology with the following cosmological parameters: density parameter for the cosmological constant ΩΛ = 0.75, matter density parameter Ωm = 0.25, baryon density parameter
Ωb = 0.045, h = H0 /100 km s−1 Mpc−1 , the linearly extrapolated root mean squared variance of density fluctuations on a scale of 8 h−1 Mpc σ8 = 0.9 and the spectral index of the
initial density fluctuations ns = 1. The number of particles used to carry out the simulation is
21603 and the particle mass is 8.6 × 108 h−1 M .
The particle positions and velocities from the Millennium simulation are stored at 64 different epochs. For each snapshot, a friends-of-friends (FOF) algorithm with a linking length
of 0.2 times the mean inter-particle separation is used to construct a catalogue of haloes. The
SUBFIND algorithm (Springel et al. 2001) is run on each of these FOF haloes to identify the
substructure in each halo. The smooth background halo is also identified as a substructure by
this algorithm. This catalogue which groups the simulation particles as subhaloes is the input
for the construction of the merger trees. Hereafter, we denote the smooth background haloes
identified by SUBFIND as central haloes and the rest of the substructure as satellite haloes. The
number of particles in each of these haloes is used to calculate the mass of the halo. Whenever
required, we explicitly use the notation “FOF haloes” to denote the haloes identified originally
by the FOF algorithm.
For every halo at a given snapshot, one must identify its progenitors at an earlier snapshot
to construct a merger tree. A halo H1 at redshift z1 is considered to be a progenitor of a halo H0
at a latter epoch with redshift z0 , if a certain fraction of the most bound particles of the halo H1
are part of the halo H0 . In this manner, any given halo at redshift 0 branches into its progenitors
at the previous redshift and these progenitors subsequently branch into their own progenitors at
higher redshifts. By following the growth of mass in dark matter haloes, merger trees capture
the assembly history of these haloes.
The merger trees of haloes from the Millennium simulation can be accessed online at the
URL http://www.g-vo.org/Millennium from a database table using the Sequential
Query Language (SQL). Another database table contains the output of the SAM of DL07. The
query execution times on the database are limited to 30 (500) seconds for unregistered (registered) users. The website also offers similar tables for a milli-version of the entire simulation
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which is 8 × 8 × 8 smaller than the original simulation but has the same mass resolution as the
larger simulation. The SQL queries that are run on the milli-version require a shorter execution
time compared to that on the original simulation because of the smaller volume. Therefore, we
carry out our analysis only using the milli-version. Eventually, we would like to use the entire
simulation to capture the full diversity of the merging histories of haloes.
6.4
Halo Formation Timescales
The merger trees of haloes with the same masses can be quite different in appearance. Our aim
is to quantify if the scatter in the properties of the central galaxies such as their luminosities
or stellar masses at fixed halo mass correlate with the scatter in the merger histories of their
haloes. One way to characterise the merger history of a halo is to consider its formation time.
The formation time of a halo is generally defined as the time when the halo has assembled a
fixed fraction of its final mass.
The growth of assembled mass in any merger tree can be studied in two different ways.
Starting from a halo at present time one can consider its most massive progenitor in the previous snapshot (see the red arrows in Fig. 6.1. This most massive progenitor has its own most
massive progenitor and so on. The linked list of the most massive progenitors of the most massive progenitors which lead to the formation of the final halo is called the main branch of the
merger tree. Alternatively, one can consider the linked list which connects the most massive
progenitor at every snapshot with the most massive progenitor at an earlier snapshot (see the
blue arrows in Fig. 6.1). Such a linked list may not always be a continous merging branch. This
list represents the growth of the “maximum progenitor”. Li et al. (2008) have presented eight
different definitions of the formation times for any given halo based either upon the main branch
or the linked list of the maximum progenitors and shown how these formation times vary with
the mass of the halo.
We use the following 4 different definitions of the formation redshift of the halo for the purpose of our analysis. We have chosen to use these definitions for the relative ease of extracting
these formation redshifts from the database with the help of SQL queries.
1. z1/2,mp : This is the highest redshift at which the “maximum progenitor” has accumulated
a mass which is greater than or equal to one half of the final halo mass.
2. z1/2,t1 : This is the highest redshift at which the progenitors with at least 2% of the final
mass have collectively assembled a mass which is greater than or equal to one half of
the final halo mass. As noted by Li et al. (2008), such a definition of the formation time
of the halo has also been used by Navarro et al. (1997) to study the dependence of halo
concentrations on their formation times.
3. z1/2,t2 : This is the highest redshift at which half of the final halo mass has assembled
into progenitors above a fixed minimum mass, Mc . We use Mc = 1011.5 h−1 M because
6.4. HALO FORMATION TIMESCALES
99
7
8
10
9
z3
Redshift
4
6
5
z2
2
3
z1
1
z0
Figure 6.1: Schematic diagram of a merger tree. The haloes at different epochs are shown as
circles with radii roughly proportional to the cuberoot of the mass. The x-axis in this figure
is arbitrary. Each merger event is represented by a black solid line. The main branch of the
merger tree is shown with red arrows while the most massive progenitors at each redshift are
linked with blue arrows.
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Figure 6.2: Scatter plot of the stellar mass of central galaxies against their halo masses at redshift
z = 0 predicted by the SAM of DL07.
star formation is the most efficient at this mass scale and this time marks the beginning of
the epoch when star formation prevails in the assembly history of a halo.
4. zcore : This is the highest redshift at which the maximum progenitor has assembled a mass
equal to Mc . This formation redshift thus marks the time when the maximum progenitor
becomes capable of hosting a bright central galaxy.
We use these definitions for the formation redshift of each halo to quantify its merger history
and investigate how it shapes the stellar mass of the central galaxy that forms in it.
6.5
Results
In Fig. 6.2, we first show a scatter plot of the stellar masses of central galaxies with respect to
the masses of their corresponding dark matter haloes predicted by the SAM of DL07. Note that
we are using the galaxies from the milli-Millennium database only. The use of a small volume
results in the plot being sparsely populated at the massive end. The stellar mass of the central
galaxy increases with the halo mass with roughly a constant scatter of ∼ 0.16 ± 0.06 dex at
fixed halo mass. This is in fair agreement with results for the MSR inferred from the kinematics
of satellite galaxies presented in Chapter 5.
The different formation redshifts of the halo defined in the previous section show a varied
dependence on the halo mass. In Fig. 6.3, we show the scatter plot of the logarithm of the scale
factor at the formation redshift (multiplied by -1) against the final mass of the halo for all haloes
6.5. RESULTS
101
Figure 6.3: Dependence of various formation redshifts of dark matter haloes on their final mass.
The solid curves show the median relation between the formation redshifts and the final mass
of the halo while the dashed curves show the 20 and 80 percentiles.
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in the milli-Millennium database. The median of the distribution of formation redshifts as a
function of the halo mass is shown with a solid line in each panel. The dashed lines show the 20
and 80 percentiles of the distribution of the formation redshifts obtained by using bins of equal
width in the logarithm of the halo mass log Mh . The different formation times of the haloes are
quite different from each other and each of these capture different aspects of the formation of
haloes.
The formation redshifts z1/2,mp and z1/2,t1 show a decreasing trend with the mass of a halo,
i.e. according to these definitions the most massive haloes on average form later than the low
mass haloes. The formation redshift, z1/2,t2 on the other hand shows a relatively flat behaviour
while the formation time zcore is highly correlated with the mass of the halo with a scatter which
is slightly less than that seen in the other definitions of the formation redshift.
In Fig. 6.4, we show the scatter plot of the logarithm of the scale factor at the formation
redshifts against the stellar mass of the central galaxy at the final epoch. The formation redshifts
z1/2,mp , z1/2,t1 and z1/2,t2 show a very little correlation with the stellar mass of the central
galaxy. However the formation redshift zcore is strongly correlated with the final stellar mass of
the central galaxy. This shows that the earlier the maximum progenitor reaches the mass Mc ,
the more stars the central galaxy has at redshift zero.
We now investigate a possible correlation between the scatter in the formation redshifts of
the haloes and the scatter in the stellar masses of the central galaxies that form in these haloes.
Fig. 6.5 shows the scatter plot of the residuals around the mean relation between the stellar mass
and halo mass against the residuals around the mean relation between formation redshifts and
halo mass. The correlation between these residuals is quantified by the correlation coefficient
which is denoted at the bottom right corner of each panel in the figure. This figure shows that
at fixed halo mass the stellar mass of central galaxies is correlated with the formation time of
the halo. The formation redshift zcore shows the highest correlation with the formation time
of the halo. This correlation of the residuals can also be seen from Fig. 6.6, where we show
the stellar mass halo mass relation. In this figure, the points in each panel are colour coded
according to the residuals in the corresponding formation redshift halo mass plot. The red
points which correspond to haloes that form earlier lie preferentially above the mean relation
between halo mass and stellar mass while blue points which correspond to haloes that form later
lie below the mean relation. This shows that at fixed halo mass, haloes that form early typically
host central galaxies that have more stellar mass. However, as expected, the correlation is not
perfect. This implies that the scatter in the formation times of haloes is not the only reason for
the stochasticity of galaxy formation. In future, we plan to investigate this issue with the help of
semi-analytical modelling to find out the main reasons for the presence of scatter in the galaxy
property-halo mass scaling relations.
6.5. RESULTS
103
Figure 6.4: Scatter plot of the stellar mass of central galaxies versus the various formation
redshifts of their haloes.
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Figure 6.5: Scatter plot of residuals in the formation redshift−halo mass relation versus the
residuals in the stellar mass−halo mass relation. The correlation coefficients between these
residuals are shown at the bottom right corner in each panel.
6.5. RESULTS
105
Figure 6.6: Scatter plot of the stellar mass−halo mass relation colour coded according to the
residuals in the formation redshift−halo mass relation. The points in red correspond to haloes
with residuals ∆[log(1 + zf )] less than -0.05, green for haloes with residuals in the range
[−0.05, 0.05] and blue for haloes with residuals greater than 0.05. The red (blue) points lie
preferentially above (below) the mean relation.
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6. ON THE STOCHASTICITY OF GALAXY FORMATION
Summary
Extensive properties of galaxies such as the luminosity and the stellar mass correlate with the
mass of the halo in which they reside. However, there is a certain amount of scatter in these scaling relations which reflects the stochasticity in the galaxy formation and evolution processes.
We investigated the origin of the scatter in the halo mass−stellar mass relationship of galaxies
that reside at the centre of dark matter haloes using the SAM of galaxy formation from DL07.
The stellar mass of galaxies can grow via two processes: (i) by the accretion of the stellar
content of satellite galaxies that merge with it and, (ii) by the formation of stars from its own
reservoir of cold gas and that brought in by its satellites. Mergers of galaxies are often accompanied with bursts of star formation. Therefore, merger histories of haloes play a central role
in building up the stellar mass of central galaxies. We explored the effect of the scatter in the
merger histories of haloes of similar masses on the properties of the central galaxies that form
in these haloes.
The merger history of a halo can be characterized by its formation time. We used several
different definitions of the halo formation time and investigated their behaviour with the halo
mass. The halo formation times show different behaviours as a function of mass depending upon
the definition used. However, at fixed halo mass, there is a large scatter in the formation times in
all the definitions. We correlated the residuals around the mean relation of halo formation time
and halo mass with the residuals around the halo mass−stellar mass relationship. We found a
positive correlation between these residuals which suggests that, on average, haloes that form
earlier harbour central galaxies that have more stellar mass. The correlation we find between
these residuals is certainly dependent on the particular SAM that we have used. In future, we
would like to investigate the aspect by considering results from several different SAMs and by
comparing them to each other.
We conclude by pointing out that a correlation between two variables does not necessarily
imply a causal relationship between the two. To give a definite answer to what causes the
scatter in the properties of central galaxies at fixed halo mass, it would be interesting to turn
off by hand the various stochastic components in a SAM (e.g., use the same spin parameter for
each halo) and explicitly quantify the effect each component has on the halo mass−stellar mass
relationship of central galaxies. We aim to perform such an analysis and present its results in a
forthcoming paper.
Chapter 7
Cosmic Transparency
This research work originated from a Galaxy Coffee talk given by David Hogg when he was
visiting MPIA last summer. The contents of this chapter are entirely based upon the article
More et al. (2009a) which has been accepted for publication in the Astrophysical Journal. The
reference is
More, S., Bovy, J., & Hogg, D. 2009, ApJ, in press, arXiv:0810.5553
7.1
Introduction
The transparency of the Universe is extremely good. A typical astronomical camera has a
shutter whose thickness is measured in microns; that shutter is far more opaque than the entire
line of sight to the majority of extragalactic sources, even at extremely high redshifts, despite—
in many cases—considerable column depths of dark matter, plasma, gas, and dust. There are,
however, very few quantitative measures of the transparency with contemporary astronomical
data.
There are several sources for photon attenuation that are clustered with matter. For example, as stars eject heavy elements, they also eject photon-absorbing ash (called “dust”). The gas
and plasma in and around galaxies absorbs, scatters, and re-emits at longer wavelengths some
fraction of incident radiation. More speculatively, if the dark matter is an axion or axion-like
particle, it will in general have photon interactions, which can in principle produce effective absorption of photons in regions of high dark matter density and high magnetic fields. The sources
of attenuation—such as these—that are clustered with matter will be correlated with galaxies
and large-scale structure, and can be found with “angular difference” measurements that compare the apparent properties of sources whose lines of sight have different impact parameters
with the correlated structure.
The Sloan Digital Sky Survey has permitted very sensitive angular difference measurements, which find that the attenuation correlated with large-scale structure is very small and
consistent with being caused by dust, presumably the dust emitted by the stars in the galaxies
that populate the structure. Measurements in the literature constrain this in visible bandpasses
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at the part in 103 level (Ménard et al. 2008; Bovy et al. 2008). To be specific, these studies constrain differences in opacity along different lines of sight caused by absorbers correlated with
galaxies.
It is possible, however, that there might be unclustered or “monopole” sources of attenuation, that affect all lines of sight equally, for example if the non-matter contributors to the
cosmological energy-momentum tensor (the “dark energy” in modern parlance) have interactions with photons, or if there are small violations of Lorentz invariance on cosmological scales.
These sources of attenuation are much harder to detect with differential experiments, but they
can be detected by “radial difference” experiments that compare cosmological sources of radiation of known physical properties at different redshifts or radial distances.
A number of different mechanisms have been proposed during the last decade to explain
the observed dimming of type-Ia supernovae (SNeIa; Riess et al. 1998; Perlmutter et al. 1999)
without cosmic acceleration by employing exactly such unclustered sources of attenuation. The
mixing of photons with axions in extragalactic magnetic fields could lead to photons oscillating
into axions with a non-negligable probability over cosmological distances, thus reducing the
flux of SNeIa at large distances (Csáki et al. 2002). Alternatively, “gray” intergalactic dust
could be so gray as to evade detection through its reddening, while still being cosmologically
important because of its overall opacity (Aguirre 1999). In order to account for the observed
SNeIa dimming, these models predict violations of transparency at the order-unity level out to
redshifts of unity (Mörtsell et al. 2002).
Furthermore, even if there are no exotic absorbers in the Universe, it is difficult (and usually
model-dependent) to infer the total mean opacity from any absorbers that have been found
by angular difference experiments. Radial difference and angular difference experiments are
complementary, even when the absorbers are mundane; although radial difference experiments
are usually less precise, they provide irreplaceable information for measuring total opacity.
Radial difference experiments are sometimes known as “Tolman tests” because they are
variants of the test proposed by Tolman (1930) of the expansion of the Universe: a test of
the (1 + z)−4 (where z is redshift) dimming of bolometric intensity (energy per unit time per
unit area per unit solid angle; also called “bolometric surface brightness”) with redshift. The
intensity is closely related to the phase-space density of photons, which is conserved (in a
transparent medium) along the light path; that conservation plus Lorentz invariance implies the
(1 + z)−4 relation: one factor of (1 + z)−1 comes from the decrease in energy of each photon
due to the redshift, another factor comes from the decrease in photons per unit time, and two
more factors arise from the solid-angle effects of relativistic aberration. The Tolman test does
not really test for the expansion of the Universe—the result does not depend on cosmological
model, or even assumptions of isotropy or homogeneity—but rather for the combination of
conservation of photon phase-space density and Lorentz invariance.
In addition to models that violate transparency, there are models that violate Lorentz invariance. Generically these models produce an energy-dependent speed of light and birefringence,
7.1. INTRODUCTION
109
breaking the perfect non-dispersiveness of the vacuum (Amelino-Camelia et al. 1998; Gambini
& Pullin 1999). These effects generally become larger with increasing energy, and observations
of high-energy sources such as active galactic nuclei (Biller et al. 1999; Aharonian et al. 2008;
Albert et al. 2008) and gamma-ray bursts (Schaefer 1999; Ellis et al. 2006) have shown that the
linear dispersion relation for photons is preserved to good accuracy at these energies. Therefore,
while these models do fail the Tolman test because of their non-trivial dispersion relations, the
effect will be unmeasurably small for low-energy (visible-band) photons.
By far the most precise radial difference test to date has been performed in the radio with
the cosmic background radiation. In contemporary cosmological models, the CBR comes from
redshift ∼ 1100 and is a near-perfect blackbody. The COBE DMR experiment established
that the spectrum and amplitude of this radiation is consistent with the blackbody expectation
at the < 10−2 level at 95-percent confidence (Mather et al. 1994). A source of attenuating
material, unless in perfect thermal equilibrium with the CBR, would tend to change either the
spectrum or the amplitude, so this result provides a very strong constraint on the transparency
at cm wavelengths (Mirizzi et al. 2005). Another test of transparency at cm wavelengths is the
increase in the CMB temperature TCMB according to the relation TCMB ∝ (1 + z). Srianand
et al. (2000) find consistency with a transparent Universe by measuring the CMB temperature
at z = 2.3. Of course, many sources of attenuation are expected to be wavelength-dependent,
so these beautiful results may not strongly constrain the opacity in the visible.
At visible wavelengths, there have been much less precise radial difference tests performed
with galaxies, whose properties would deviate from naive predictions under extreme attenuation. After correcting for the evolution of stellar populations in galaxies, these studies find
consistency with transparency at the 0.5 mag level at 95-percent confidence (Pahre et al. 1996;
Lubin & Sandage 2001), which correspond to optical depth limits < 0.5 out to redshift z ∼ 1.
Unfortunately, the precision of these tests is not limited by the precision of the measurements,
but rather by the precision with which the evolution of galaxy stellar populations is known; the
results will not be improved substantially with additional or more precise observations.
Another test of transparency at visible wavelengths involves the measurement of the Cosmic
Infrared Background (hereafter CIB). The absorption of visible photons by a diffuse component
of intergalactic dust and its re-emission in the infrared contributes to the CIB. The amount of
dust required to explain the systematic dimming of SNeIa would produce most of the observed
CIB (Aguirre & Haiman 2000). However, discrete sources (e.g., dusty star-forming galaxies)
also emit in the infrared and account for almost all of the CIB, strongly constraining the role
of dust in the dimming of SNeIa. Any constraint on the transparency from the CIB requires a
careful subtraction of the discrete sources (Hauser & Dwek 2001).
The Tolman test can be re-written as a relationship among cosmological distance measures.
There are several empirical definitions of distance in cosmology (e.g., Hogg 1999); the most important for contemporary observables are the luminosity distance DL and the angular diameter
distance DA . The luminosity distance DL to an object is defined to be the distance that relates
110
7. COSMIC TRANSPARENCY
bolometric energy per unit time per unit area S (flux) received at a telescope to the energy per
unit time L (luminosity or power) of the source, or
S=
L
4π DL2
.
(7.1)
The angular diameter distance DA is the distance that relates the observed (small) angular size
Θ measured by a telescope to the proper size R of an object, or
Θ=
R
DA
.
(7.2)
Because the ratio of flux to the solid angle is essentially the intensity, the (1 + z)−4 redshiftdependence of the intensity is reflected in these distance measures by
DL = (1 + z)2 DA
.
(7.3)
Both distance measures are strong functions of world model, but this relationship between
them—known sometimes as the “Etherington relation” (after Etherington 1933, who showed
that the result is valid in arbitrary spacetimes)—depends only on conservation of phase-space
density of photons (transparency) and Lorentz invariance. Fortunately, for some fortuitous types
of objects, these distances can be measured nearly independently.
A test of this type for transparency has been proposed and carried out previously (Bassett &
Kunz 2004a,b), with luminosity distances from SNeIa and angular diameter distances estimated
from FRIIb radio galaxies, compact radio sources, and x-ray clusters (Uzan et al. 2004; Jackson
2008). The results were imprecise because there are many astrophysical uncertainties in the
proper diameter estimates of these exceedingly complex astrophysical sources.
In the contemporary adiabatic cosmological standard model, there is a feature in the darkmatter auto-correlation function (or the power spectrum) corresponding to the communication
of density perturbations by acoustic modes during the period in which radiation dominates
(Peebles & Yu 1970; Eisenstein et al. 2005). This feature has a low amplitude in presentday structure (that is, the distribution of galaxies), but because it evolves little in comoving
coordinates, it provides a “standard ruler” for direct measurement of the expansion history. A
measurement of the baryon acoustic feature (BAF) in a population of galaxies at a particular
redshift provides a combined measure of the angular diameter distance to that redshift (from the
transverse size of the feature) and the Hubble Constant or expansion rate at that redshift (from
the line-of-sight size of the feature). As we discuss below, as signal-to-noise improves, the
BAF can be used to measure the angular diameter independently of the local Hubble rate. Most
importantly, because the BAF arises from extremely simple physics in the early Universe when
the growth of structure is linear and electromagnetic interactions dominate, the BAF measures
the angular diameter distance with far fewer assumptions than any method based on complex
astrophysical sources in the highly non-linear regime.
7.2. DATA, PROCEDURE, AND RESULTS
111
At the same time, SNeIa have been found to be standard — or really “standardizable” —
candles, which can be used to make an independent direct measurement of the expansion history
(Baade 1938; Tammann 1979; Colgate 1979; Riess et al. 1998; Perlmutter et al. 1999). Up to an
over-all scale and some uncertainties about the intrinsic spectra and variability among SNeIa, a
collection of SNeIa measure the luminosity distance.
Given overall scale uncertainties, the most robust test of global cosmic transparency that can
be constructed from these two distance indicators is a measurement of the ratio of the distances
to two redshifts z1 and z2 . That is, transparency requires
DL (z2 )
[1 + z2 ]2 DA (z2 )
=
DL (z1 )
[1 + z1 ]2 DA (z1 )
.
(7.4)
This expression cancels out overall scale issues and is independent of world model. We perform
a very conservative variant of this test below, where we measure the left-hand side with SNeIa
and the right-hand side with the BAF, marginalizing over a broad range of world models.
The tests presented here are not precise, simply because at the present day BAF measurements are in their infancy, and we make use of no cosmological data other than the BAF and
SNeIa. As we discuss below, when these measurements are made at higher redshifts and with
higher precisions, our limits on transparency and Lorentz invariance will improve by orders of
magnitude. Eventually they may be limited not by the data quality but by the cosmic variance
limit on the BAF measurement itself (Seo & Eisenstein 2007).
7.2
Data, Procedure, and Results
In surveys to date, where the BAF is measured at low signal-to-noise, the optimal extraction of
the signal best constrains not the angular diameter distance directly, but rather a hybrid distance
DV
2
c z [1 + z]2 DA
DV =
H(z)
1/3
,
(7.5)
where DA is the angular diameter distance and H(z) is the Hubble expansion rate (velocity per
unit distance) at redshift z (Eisenstein et al. 2005).
Using data from the Sloan Digital Sky Survey and the Two Degree Field Galaxy Redshift
Survey, the power spectrum and BAF have now been measured in samples of massive, red
galaxies at two different redshifts: z = 0.20 and z = 0.35. The measured BAF at each redshift
z translates to a distance measure DV (z). Accounting for covariances in the measurements at
the two redshifts (which are not based on entirely independent data sets), the ratio of distances
is DV (0.35)/DV (0.20) = 1.812 ± 0.060 (68-percent confidence; Percival et al. 2007). .
We formed two samples of SNeIa data from a recent compilation (Davis et al. 2007). “Sample A” consists of all 7 SNeIa in the redshift range 0.15 < z < 0.25 and “Sample B” consists
of all 22 SNeIa in the redshift range 0.3 < z < 0.4. We estimate the distance-modulus DM at
112
7. COSMIC TRANSPARENCY
Figure 7.1: The distance-modulus–redshift relation. Filled black squares with uncertainty bars
show the SNeIa data (from Davis et al. 2007) used in Samples A (left panel) and B (right panel).
Open red squares show the distance moduli DM (0.20) = 40.14 ± 0.06 and DM (0.35) =
41.48 ± 0.07 (68-percent confidence) inferred from the fits to the data.
z = 0.20 and z = 0.35 by fitting a straight line to Samples A and B separately (Fig. 7.1), and
obtain a distance-modulus difference
∆DM obs = DM obs (0.35) − DM obs (0.20) = [1.34 ± 0.09] mag
,
(7.6)
where we are indicating that this is an observed value, and might differ from the true value if
there is opacity.
The distance modulus derived from the SNeIa is systematically affected by the presence of
any intervening absorber. Let τ (z) denote the opacity between an observer at z = 0 and a source
at redshift z due to such extinction effects. The flux received from this source is reduced by the
factor e−τ (z) . The inferred (“observed”) luminosity distance differs from the “true” luminosity
distance:
DL 2obs (z) = DL 2true (z) eτ (z)
.
(7.7)
The ratio of the luminosity distances at two different redshifts z1 and z2 depends upon the factor
e[τ (0.35)−τ (0.20)]/2 . The inferred (“observed”) distance modulus differs from the “true” distance
modulus:
DM obs (z) = DM true (z) + [2.5 log e] τ (z) .
(7.8)
Taking differences of distance moduli at the two redshifts:
∆DM obs = ∆DM true + [2.5 log e] ∆τ
,
(7.9)
where ∆τ ≡ [τ (z2 ) − τ (z1 )]. If the distance indicator from the BAF is unaffected by the
7.2. DATA, PROCEDURE, AND RESULTS
113
absorption as we expect, then
ln(10)
DV (z2 )
z2 [1 + z1 ]2 H(z1 )
∆τ =
∆DM obs − 7.5 log
+ 2.5 log
2.5
DV (z1 )
z1 [1 + z2 ]2 H(z2 )
.
(7.10)
The above equation can be used to determine ∆τ from z = 0.35 to z = 0.20 in light of the
ratio of the distances DV obtained from the BAF observations (hereafter B) and the difference in
distance moduli obtained from the SNeIa observations (hereafter S) at these redshifts. However,
the last term in the above equation makes the result cosmology-dependent. Therefore, we follow
a Bayesian approach and assign posterior probabilities to 100 uniformly spaced values of ∆τ ∈
[0,0.5] by marginalising over 100×100 ΛCDM cosmologies uniformly spaced in the (ΩΛ , ΩM )
plane with ΩΛ ∈ [0,1] and ΩM ∈ [0,1]. Thus,
P (∆τ |S, B) =
Z
Z
ΩΛ
P (ΩΛ , ΩM |B) P (∆τ, ΩΛ , ΩM |S) dΩM dΩΛ
,
(7.11)
ΩM
where P (ΩΛ , ΩM |B) and P (∆τ, ΩΛ , ΩM |S) are the posterior probabilities of the set of model
parameters given B and S respectively. We assume that the uncertainties on B and S are Gaussian and calculate the likelihood of B and S for different sets of parameters in the (∆τ, ΩΛ , ΩM )
space. Assuming flat priors on ΩΛ and ΩM in the ranges 0 < Ω < 1, and flat prior on ∆τ in
the range 0 < ∆τ < 0.5, the posterior probabilities P (ΩΛ , ΩM |B) and P (∆τ, ΩΛ , ΩM |S) are
calculated from the likelihoods of the two datasets. Equation (7.11) yields the posterior for ∆τ ,
marginalized over all world models. Fig. 7.2 shows the posterior P (∆τ |S, B) for the difference in optical depths between redshifts 0.35 and 0.20 obtained from the procedure outlined
above. The posterior peaks at 0 and yields ∆τ < 0.13 at 95-percent confidence. The result
demonstrates the transparency of the Universe between these two redshifts, although not at high
precision.
The abundance and absorption properties of absorbers can be constrained using the difference in optical depths measured above. Let n(z) denote the comoving number density of
absorbers, each with a proper cross-section σ(z) at redshift z. The difference in optical depths
between redshifts z1 and z2 is then given by
∆τ =
Z
z2
n(z) σ(z) DH
z1
(1 + z)2
dz
E(z)
,
(7.12)
where DH is c/H0 and
E(z) ≡
H(z) p
= ΩM (1 + z)3 + Ωk (1 + z)2 + ΩΛ
H0
.
(7.13)
In detail, the output of this integral depends on world model. For the concordance model,
Hubble Constant H0 = 100 h km s−1 Mpc−1 , and assuming n(z) and σ(z) to be independent
of redshift, ∆τ measured between redshifts 0.35 and 0.20 constrains n σ < 2 × 10−4 h Mpc−1
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7. COSMIC TRANSPARENCY
Figure 7.2: Posterior distribution of ∆τ between z = 0.35 and z = 0.20 obtained from the
Bayesian analysis described in Section 7.2. The 68, 95 and 99-percent confidence upper limits
are indicated by the corresponding dashed lines.
at 95-percent confidence.
A naive calculation of ∆τ using Equation (7.10) for the concordance ΛCDM model (ΩM =
0.258, ΩΛ = 0.742) obtained from the analysis of the 5-year WMAP data (Dunkley et al.
2009), yields ∆τ = −0.30 ± 0.26 at 95-percent confidence. This shows that there is a slight
tension between the results of current measurements of the BAF and of the SNeIa under the
currently accepted world model. More generally, a similar tension, i.e. a brightening of the SNe,
between measurements of the cosmological parameters by using standard rulers and standard
candles has been reported before (Bassett & Kunz 2004a,b; Percival et al. 2007; Lazkoz et al.
2008). SNe brightening is not impossible in models that involve axion-photon mixing (Bassett
& Kunz 2004b) or chameleon-photon mixing (Burrage 2008) if the corresponding particles
are abundantly produced during SNeIa explosions. However, a negative value of ∆τ could
also indicate the presence of a systematic bias in the distance measurements based upon the
SNeIa brightness or the BAF, e.g., overcorrection for extinction in the host galaxy of the SNeIa
brightnesses or magnification bias in the SNeIa selection (Williams & Song 2004). Note that
the prior, ∆τ > 0, improves the magnitude of the uncertainty on ∆τ (from 0.26 to 0.13). The
95-percent confidence interval shrinks with the prior because we sample only from the rapidly
falling tail of the posterior.
7.3
Future Constraints
In the future, the constraints from both the SNeIa and the BAF observations will improve in
accuracy and will cover a wider redshift range. The Baryon Oscillation Spectroscopic Survey
7.3. FUTURE CONSTRAINTS
115
(BOSS) is currently underway and plans to measure the BAF in luminous red galaxies at redshifts z = 0.35 and z = 0.6. The key improvements would be the larger redshift range and the
power to resolve the BAF both in the line-of-sight direction (constrains H) and the transverse
direction (constrains DA ). This would remove the weak world-model dependence in our present
analysis. The angular diameter distances to these redshifts would be measured to an accuracy
of ∼ 1 percent (http://www.sdss3.org/). In parallel, the Supernovae Legacy Survey (SNLS),
when complete, expects ∼ 700 SNeIa in the redshift range 0 < z < 1.7 (Astier et al. 2006).
The uncertainty on the estimate of the distance moduli to redshifts z = 0.35 and z = 0.6 will
be roughly four times better with the increased numbers. Using the test of the duality relation
described above, ∆τ between z = 0.35 and z = 0.6 would be constrained to better than 0.07
(95-percent confidence), independent of the adopted cosmological model. The constraint on
n σ would become n σ < 5.4 × 10−5 h Mpc−1 .
BOSS will also use the Ly-α forest in the spectra of bright QSOs to measure the BAF at
redshift z ∼ 2.5 with an accuracy of ∼ 1.5 percent. No current or planned SNeIa surveys expect
to detect SNeIa at such a high redshift. However the highest redshift (∼ 1.7) measurements of
DL from the SNLS could potentially be used in conjunction with the DA measurement to get
a constraint on the transparency of the Universe by marginalizing over different world models.
Interestingly, there have been recent efforts to calibrate gamma-ray bursts (hereafter GRBs) as
standard candles and to extend the Hubble diagram to higher redshifts (Lazkoz et al. 2008).
The SNeIa at low redshift and the GRBs at high redshifts can provide a measurement of the
difference between the DM between redshifts 0.35 and 2.5. We optimistically assume that the
difference in the DM to these redshifts can be measured with an accuracy of ∼ 0.1 similar to
the one obtained from the analysis of SNeIa at z = 0.2 and z = 0.35 in Section 7.2. These
measurements shall then constrain ∆τ between redshifts 2.5 and 0.35 to an accuracy of 0.2 with
95 percent confidence. This translates into an accuracy on n σ of ∼ 1.1 × 10−5 h Mpc−1 .
In the optimistic future, the uncertainty on DL (z) could, in the absence of damaging systematics, diminish arbitrarily as the number of observed SNeIa grows. However, the precision
of any BAF measurement is limited by sample variance (the number of independent wavelengths of a given fluctuation that can fit in the finite survey volume is limited), even when the
uncertainty caused by incomplete sampling of the density field (shot noise) is negligible (Seo &
Eisenstein 2007). The sample variance error goes down with the square root of the volume of the
survey. To calculate a representative limit, we consider an optimistic all-sky survey covering the
redshift range 2.45 < z < 2.55. Such a survey can be used to determine DA (z = 2.5) to a fractional accuracy of ∼ 0.004 (95-percent confidence). This will ultimately constrain the optical
depth to redshift z = 2.5 to τ < 0.008 and hypothetical absorbers to n σ < 4 × 10−7 h Mpc−1 .
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7.4
7. COSMIC TRANSPARENCY
Discussion
We have advocated and analyzed the expected future performance of a simple Tolman test or
test of the Etherington relation, that is, that the luminosity distance is larger than the angular
diameter distance by two powers (1 + z), using type Ia supernovae to measure the luminosity
distance and the baryon acoustic feature to measure the angular diameter distance. We have
shown that this test will eventually provide very precise measurements of the conservation of
photon phase-space density.
We performed the test with the limited data available at the present day. We used only the
ratio of distances at redshifts of z = 0.20 and 0.35 to remove uncertainties about the overall scale. We find consistency with a Lorentz-invariant, transparent Universe. Our results are
consistent with all other measures of transparency to date. This is in part because they are not
extremely precise. Our Tolman test also assumes that the measurements of the SNeIa and of
the BAF are not affected by systematic biases with magnitudes that are a significant fraction of
the magnitudes of the uncertainties. Our test is limited by the precision of the BAF measurement and the redshift range over which it has been measured. As we have shown, experiments
planned and underway will increase the redshift range and improve the overall precision by an
order of magnitude.
The most precise transparency measurements at visible wavelengths today are statistical
angular difference measurements, which can only constrain attenuation correlated with specific
types of absorbing structures in the Universe (e.g., MgII absorbers, Ménard et al. 2008; clusters
of galaxies, Bovy et al. 2008). The simple Tolman test performed here limits the full, unclustered, line-of-sight attenuation between two redshifts.
The technique used in this paper provides a test of transparency that is not very sensitive to
astrophysical assumptions, both because the BAF has a straightforward origin during an epoch
in which growth of structure is linear and the dominant physics is well understood, and because
there is no significant “evolution” with cosmic time for which we must account. This is in
contrast to other methods for measuring angular diameter distances and brightnesses, where
there are no precisely “standard” rulers, and evolution is dramatic with redshift. On the other
hand, the ultimate precision of any test of this type may come from the finite comoving volume
in the observable Universe. Cosmic variance will dominate the BAF error budget eventually.
The SNeIa samples have been corrected as best as possible for line-of-sight extinction by
fitting an empirical correlation of extinction with a change in color. However, there are a few
problems with this approach. First, this approach cannot correct for “gray” dust (Aguirre 1999).
Second, this approach can also not correct for a monopole component; it only corrects for
components that show variations around the mean level. Third, these corrections will be wrong
or fooled if there are intrinsic relationships between color and luminosity for SNeIa. Fourth, the
empirical corrections found by these projects tend to be odd in the context of what is expected
from the reddening and attenuation by dust in the Milky Way (Jöeveer 1982; Conley et al.
2007; Ellis et al. 2008; Nobili & Goobar 2008). The Tolman test is sensitive to any kind of
7.4. DISCUSSION
117
absorber and makes no assumptions about the wavelength-dependence or fluctuations of the
opacity. Given that the SNeIa results have been corrected for a color–brightness relation, the
test presented here looks at the mean opacity towards SNeIa of the fiducial color to which the
compiled SNeIa have been corrected.
The best-fit value of ∆τ obtained from our analysis is negative, i.e. SNeIa are brighter than
expected from the angular diameter distance measurements using the BAF. A conversion of
dark sector particles into photons could provide a physical explanation for this result. However,
a systematic bias in either the SNeIa or BAF experiments cannot be ruled out and the Tolman
test is a useful tool to identify such biases.
At present, because the differences among competitive world models are not large over the
redshift range 0.20 < z < 0.35, our test is not yet sensitive enough to rule out extreme axion
or “gray” dust models that reconcile SNeIa results with an Einstein-de Sitter Universe by using
effective opacity to adjust the inferred redshift–luminosity-distance relation. However, these
models will all be severely constrained within the next few years (see also Corasaniti 2006).
118
7. COSMIC TRANSPARENCY
Chapter 8
Summary
It is now a well established fact that most of the matter in the Universe is dynamically cold,
collisionless and dark. It forms an ever-changing cosmic web of gravitationally bound structures
called haloes. Galaxies form and evolve in these dark matter haloes and are shaped by various
astrophysical processes which depend upon the properties of the dark matter halo they reside
in. A precise knowledge of the connection between galaxies and their dark matter haloes is
therefore crucial to improve our understanding of the physics of galaxy formation. In this
thesis, we have established scaling relations between the mass of dark matter haloes and the
properties of galaxies that reside at their centre.
Satellite galaxies are excellent tracers of the dark matter haloes around their central galaxies.
The kinematics of these satellite galaxies reflect the depth of the potential well in which they
orbit. Therefore, they can be used to measure the mass of the halo of central galaxies. A large
number of satellite galaxies is required to precisely measure the kinematics in individual haloes,
a condition which is easily met in cluster-sized haloes, but is rarely satisfied in haloes of low
mass. Under the assumption that central galaxies with similar properties reside in similar mass
haloes, one can stack central galaxies with similar properties, such as luminosity or stellar mass,
and combine the velocity information of their satellites that will allow a precise measurement
of the kinematics of satellite galaxies. However, galaxy formation is a stochastic process and
one expects a scatter in the relation between central galaxy properties and halo masses. This
implies that the stacking procedure results in combining the kinematics of satellite galaxies in a
wide range of halo masses. This complicates the interpretation of the kinematics signal.
Most studies that use the kinematics of satellite galaxies to probe the halo masses have made
the simplified assumption that the scatter in the halo masses of stacked centrals is negligible. In
this thesis, we have demonstrated that the inference of the halo masses from the kinematics of
satellite galaxies faces a degeneracy problem. The average relation between the halo mass of
central galaxies and their properties cannot be inferred without the knowledge of the scatter in
this relation. We have presented a novel method that can break this degeneracy. The method
involves the measurement of the kinematics of satellite galaxies using two different weighting
schemes: the satellite-weighting scheme and the host-weighting scheme. The ratio of the mea119
120
8. SUMMARY
surements in these two different schemes is sensitive to the scatter in the masses of the stacked
haloes. Therefore, a simultaneous modelling of the kinematics obtained with these two weighting schemes can be used to measure both the average and the scatter in the scaling relation
between halo mass and the property of the central galaxy used for stacking.
The interpretation of the kinematics of stacked centrals is also complicated as a result of various selection effects which can bias the final determination of the halo masses. To understand
these biases carefully, a realistic mock catalogue of galaxies was constructed and analysed. We
compared the strict isolation criteria that have been abundantly used in the literature to select
central and satellite galaxies with our adaptive iterative criteria. The comparison shows that the
strict isolation criteria result in a preferential selection of low mass haloes. Another selection
bias that has been previously neglected is due to the fact that the kinematics of satellite galaxies
is always averaged over those haloes that host at least one satellite. This bias preferentially
misses low mass haloes. This implies that the kinematic studies carried out previously have
selected a sample which is not representative for the entire population of galaxies. The scaling
relations derived so far are, at best only valid for the sample of the isolated galaxies. We have
improved this situation and presented an analytical model which accounts for all these selection
effects. Tests using the mock catalogue have shown that the method proposed and used by us in
this thesis can be used to identify a representative population of central galaxies from a redshift
survey, quantify the kinematics of their satellites and use this kinematic information to reliably
infer the scaling relation between halo mass and central galaxy properties.
This method was then applied to data from the SDSS galaxy catalogue. The velocity
dispersion-luminosity relation for central galaxies was measured from the data and used to infer
the mean and the scatter of the halo mass−luminosity relation of central galaxies (MLR). The
results show that brighter central galaxies on average reside in more massive haloes and that the
scatter in halo masses is an increasing function of central galaxy luminosity. The investigation
of the colour dependence of the MLR showed that at fixed luminosity, red galaxies on average
occupy haloes that are more massive than their blue counterparts. A similar method was also
applied to infer the halo mass−stellar mass relationship of central galaxies (MSR) and its colour
dependence. Central galaxies that have more stellar mass on average reside in more massive
haloes and the scatter in halo masses increases as a function of the stellar mass too. The analysis
of satellite kinematics around centrals separated by colour shows that at fixed stellar mass, the
red and the blue centrals on average occupy similar mass haloes.
The existence of scatter in halo masses at fixed central galaxy property implies that haloes of
equal masses harbour central galaxies with a scatter in their properties. To investigate the origin
of this stochasticity in galaxy formation, a semi-analytical model was analysed. Haloes of similar masses show a large scatter in their formation times. Our analysis shows that the residuals
around the average formation time−halo mass relation positively correlate with the residuals
around the halo mass−stellar mass relationship predicted by the semi-analytical model. This
implies that haloes that form early, on average, host central galaxies which contain more stel-
8.1. FUTURE POSSIBILITIES
121
lar mass. Thus, the scatter in the merger histories of haloes is a plausible explanation for the
stochasticity of galaxy formation. The dependence of this statement on the particulars of the
semi-analytical model are under investigation and a subject of future work.
The transparency of the Universe is also crucial to understand the galaxy-dark matter connection as the observed properties of galaxies can be biased if the Universe is significantly
opaque. Transparency can be affected by intergalactic dust or interactions between photons
and the dark sector. Such effects cause a deviation from the Etherington relation which relates
the distances measured using standard candles to the distances measured using standard rulers
at a particular redshift. A test of this relation was carried out by using the currently available
observations of these distance measures to obtain a quantitative measure of the transparency of
the Universe. With the limited amount of data available, we find consistency with a transparent
Universe between redshifts 0.2 and 0.35. We analyzed the expected future performance of this
test and showed that as better distance measurements covering a wider range in redshift become
available, the test can provide a very precise measurement of the transparency of the Universe.
Such precise measurements ultimately can also be used to limit the cross-section of interactions
of the photons with the dark sector.
8.1
8.1.1
Future possibilities
Properties of Satellite Galaxies
In this thesis, we have used satellite galaxies to probe the connection between the properties of
their central galaxies and dark matter haloes. However, we did not investigate various properties of satellite galaxies themselves. In the hierarchical structure formation scenario, satellite
galaxies are interesting in their own right. The satellite galaxies may be affected due to various
physical process, such as dynamical friction, effects due to tidal fields, ram pressure stripping,
harrassment and strangulation (see e.g., Chandrasekhar 1943; Gunn & Gott 1972; Farouki &
Shapiro 1980, 1981; Larson et al. 1980; Byrd & Valtonen 1990). It would be interesting to
investigate the effects that these processes have on the properties of satellite galaxies.
The satellite galaxies selected in our samples can be used to study the abundance and radial
distribution of satellites in haloes. In Chapter 3, we have shown that our selection criteria is able
to recover the projected number density distribution of satellite galaxies around bright centrals
quite accurately. An analysis of how these number density distributions depend on the colour
of the satellite galaxies and the luminosities/stellar masses of the centrals can be carried out
with the help of our sample. Very preliminary analyses indicate that at fixed luminosity of
centrals, the number density distribution of blue satellites may be less concentrated than that of
the red satellite galaxies. However, the effects of fiber collisions and stacking biases need to be
carefully checked before interpreting these distributions. Our sample of satellites can also be
used to analyse the dependence of the blue satellite fraction on the colour and luminosity/stellar
mass of the centrals.
122
8. SUMMARY
In this thesis, we have always focussed on the aperture averaged velocity dispersions. However, it would also be interesting to study the radial decline of the velocity dispersions and
its dependence on the satellite-colour. Other interesting studies such as the measurements of
the higher order moments of the velocity distribution and their colour dependence can also be
carried out to investigate the orbital properties of satellite galaxies.
8.1.2
Shapes of Dark Matter Haloes
The existence of triaxial dark matter haloes is a firm prediction of the ΛCDM theory. These
triaxial haloes should appear flattened when seen in projection. If observed, the flattening of
haloes at large radii has a potential to discriminate between alternative theories of gravity such
as MOND because the potentials at large radii predicted by such theories are isotropic far away
from the stellar content of galaxies. Statistical measurements of dark matter halo shapes often
rely on the statistical alignment of the dark matter halo axis and the ellipticity of the central
galaxies that reside in them. There have been recent claims of the detection of ellipticities of
dark matter haloes inferred from the modelling of the azimuthal dependence of the weak lensing
signal around galaxies that are stacked by aligning their major axes, (see e.g., Hirata et al. 2004;
Parker et al. 2007). However, there exist several sources of systematics which can contaminate
this signal and lead to a spurious detection (Mandelbaum et al. 2006a). This makes it important
to have alternative confirmations of these results. It would be interesting to study the azimuthal
dependence of the velocity dispersions around central galaxies stacked by their properties and
aligned along their major axis and its interpretation.
8.1.3
Redshift Evolution of the Halo Occupation Distributions
The halo occupation distribution (HOD) is the end result of the complex baryonic physics involved in the formation of galaxies. The halo mass-to-light ratio and the halo mass-to-stellar
mass ratio obtained from the HOD quantify the efficiency of dark matter haloes to turn the
baryons into stars (see e.g., Yang et al. 2003). The halo occupation distribution of galaxies is
expected to evolve with redshift. The time evolution of the HOD can reflect interesting changes
in the physics that takes place in dark matter haloes of different masses. Therefore, it is important to obtain observational constraints on such evolution. Such constraints have the ability to
uncover new aspects in the physics of galaxy formation.
The satellite kinematics analysis presented in this thesis can be applied to high redshift
datasets such as that provided by the DEEP2 redshift survey. Conroy et al. (2007) have already
used the kinematics of satellite galaxies to derive the mass-to-light ratio (and the mass-to-stellar
mass ratio) of central galaxies in SDSS and DEEP2. They use these results to constrain the
evolution of the mass-to-light ratios. In light of the fact that their criteria were tuned to only
select isolated central galaxies and their method to analyse the kinematics of satellites may
have been systematically biased due to simplistic assumptions (see Section 5.5.2), it would be
8.1. FUTURE POSSIBILITIES
123
certainly interesting to revisit some of their conclusions on the redshift evolution of the massto-light ratio. We are currently carrying out such an analysis. Preliminary investigation seems
to indicate that at high redshift the small number statistics will be a big problem. It may be
possibly to overcome this problem by an appropriate tuning of the parameters of the selection
criteria and/or by using flux limited samples.
I would like to conclude by pointing out the quotation from the Brihadaranyaka Upanishad
that was used to open this thesis. It is perfectly possible that dark components in our current
theory are figments of our imagination and a result of our ignorance about nature. The spirit of
curiosity and a constant questioning of our beliefs should always be kept alive to take us from
ignorance to the Truth, and from darkness towards Light.
124
8. SUMMARY
Acknowledgements
I am greatly indebted to my supervisor Frank van den Bosch for being extremely supportive
and helpful right through the first day I arrived in Heidelberg. He has been a constant source
of energy, ideas and an inspiration for hard work. I am thankful to Hans-Walter Rix for his
support, encouragement and guidance during the phase of my PhD. I am extremely grateful
to him, to Thomas Henning and the entire staff at MPIA for a wonderful research atmosphere
through which I have learned a lot just by interacting with people around me. I am also thankful
to all my collaborators for their help in the research work presented in this thesis. Many thanks
to David Hogg for excellent discussions during the work on transparency.
It is a pleasure being part of the theory group at the MPIA. It was great to have Ramin
Skibba, Andrea Macciò, Kris Blindert and Xi Kang available to answer any questions I had. I
enjoyed a lot having particle physics discussions with Suchita Kulkarni. My heartfelt thanks to
Anna Pasquali for her generous affection and support. I am also thankful to Svitlana Zhukovska
for a careful reading of the introductory and the concluding chapters of this manuscript and to
Sebastian Jester for help in translating the abstract into German. I hardly had any astronomy
courses before I came to Heidelberg. I thoroughly enjoyed the lecture courses that I took during
the time of my PhD and I am thankful to all the lecturers for their efforts. In particular, it was
a great experience to learn from Matthias Bartelmann, Hans-Walter and Eric Bell. I must also
thank Christian Fendt for his wonderful skills in managing the IMPRS and being always ready
for help. I am also thankful to the administration department in the institute, especially Frau
Apfel, Frau Seifert and Frau Schleich for always being helpful and making administrative tasks
easy.
I had a great partner in Marcello from whom I have learnt a lot both about work and life.
I shall always remember and cherish our “back to the basics” discussions on the whiteboard. I
appreciate his friendliness, care and support during the not-so-good phases of my time here. I
could not have had better officemates than Cassie and Marcello and virtual officemates in Ros
and Christian, who maintained constant giggles and laughter in the office and cheered me up. I
am also thankful to all the IMPRS guys for a wonderful time during IMPRS seminars, outings,
barbeques and Christmas parties.
I am thankful to Osman Oezalp for providing me an accomodation in Kohlhof during the
first two years of my PhD. I thoroughly enjoyed my stay in the woods. Many thanks also to my
neighbour Frau Solweig for her support and grandmotherly love. A great thanks goes to Kris
125
126
Acknowledgements
Blindert for allowing me to use her guest room. I learnt a great deal of culinary skills from her.
It was wonderful being friends with Pepper, who is a thorough entertainment package. He has
always surprised me with his smartness.
Yaroslav has been a great companion and teacher for me. I have greatly benefitted from his
zeal and enthusiasm for science. I hope we finish writing at least one chapter of our book on
fundamental problems. I am sorry that even after his best efforts I could not master the skills
of table tennis that he tried so hard to teach me. His constant support and motivation has been
precious. I will always cherish our friendship.
I really cannot imagine myself here without all the love, hard work and efforts of my family.
Aai and Baba have been my strongest pillars of support. Maushi, Aaji, Maai, Bhaiya, Chamu
and my relatives in Dehugaon and Mushet, you all are an inseparable part of me. Mrinalini and
Madhu, your love and support has been invaluable. A hearty thanks to my in-laws for their love
and belief in me. I am also grateful to Gau, Anu, Aru, Anji, Mani, Jui, Deepu, Sweety, Jyoti
and Bharu for the short but sweet time we spent together.
Life can really spring surprises in unexpectedly short amounts of time. Had it not been
for the IMPRS Christmas party of 2007, I would never have met my sweetheart, Anupreeta. I
feel really lucky to have a loving, caring and supportive ardhangini in her. You make my life
meaningful, Nupur!
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