PhDthesis Valente

PhDthesis Valente
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
Put forward by
Ana Valente
born in
São João da Madeira, Portugal
Oral examination: December 19th, 2012
On the cross-correlation between
Weak Gravitational Lensing and the Sunyaev–Zel’dovich effect
by Ana Valente
under the supervision of Prof. Dr. Matthias Bartelmann
Referees:
Prof. Dr. Matthias Bartelmann, Institut für Theoretische Astrophysik, Universität Heidelberg
Prof. Dr. Luca Amendola, Institut für Theoretische Physik, Universität Heidelberg
Zusammenfassung
Die Verteilung und Entwicklung von kosmischen Strukturen - sowohl in ihren dunklen als auch baryonischen Komponenten - ist immer noch ein großer Unsicherheitsfaktor im gegenwärtigen Modell unseres
Universums und ein aktives Forschungsfeld. Um dieses Thema anzugehen, krosskorrelieren wir die Signale
des schwachen Gravitationslinsen- und des thermischen Sunyaev-Zel’dovich-Effektes von Galaxienhaufen
als eine Funktion der Rotverschiebung. Wir benutzen das Halomodell, dass die großräumige kosmische
Struktur beschreibt, um die Zweipunktkorrelationsfunktion zwischen der Dichte der dunklen Materie
und des Gasdruckes in Halos abzuschätzen. Nachdem wir das dreidimensionale Leistungsspektrum
berechnet und mit Hilfe der Gleichung von Limber auf den Himmel projiziert haben, schätzen wir ab,
wie sich die Krosskorrelation zwischen diesen beiden Datentypen mit zunehmender Rotverschiebung
aufbaut. Wir berechnen die Kovarianzmatrix für eine gegebene Krosskorrelationsfunktion und werten
zu erwartende Fehlerbalken für realistische Surveys aus. Außerdem untersuchen wir zum einen, wie die
Wahl von kosmologischen Parametern unsere Ergebnisse beeinflusst, und zum Anderen den Einfluss der
Eigenschaften von Galaxienhaufen auf das Krosskorrelationssignal. Wir finden heraus, dass - obwohl
das Krosskorrelationssignal nicht geeignet scheint, um kosmologische Parameter einzuschränken - es
hochgradig von den intrinsischen Eigenschaften der Galaxienhaufen abhängt und deshalb eine Möglichkeit
bietet, die Entwicklung der Gaskomponente von Halos zu beschreiben.
Abstract
The distribution and evolution of cosmic structures, in their dark and baryonic components, remains a
source of uncertainty in the current model of the Universe and is an active field of research. To address this
subject, we cross-correlate the weak gravitational lensing and thermal Sunyaev-Zel’dovich effects of galaxy
clusters as a function of redshift. We use the halo model of large-scale structure to estimate the two-point
correlation function between the dark matter density and the gas pressure in halos. After obtaining the
three-dimensional power spectrum and projecting it onto the sky by means of Limber’s approximation, we
estimate how the cross-correlation between these two types of data builds up as redshift increases. We
calculate the covariance matrix for a measured cross-correlation function and evaluate expected error bars
for realistic surveys. Further, we examine how the choice of cosmological parameters affects our results
and inspect the influence of cluster properties on the cross-correlation signal. We find that, although the
cross-correlation signal does not seem to be suitable for constraining cosmological parameters, it is highly
sensitive to the intrinsic properties of the clusters and thus provides a way to characterise the evolution of
the halo gas component.
Contents
Motivation
1
1
Theoretical background
3
1.1
Historical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Foundations of Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.1
History of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.2
The cosmological standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
The cosmic microwave background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.1
Temperature fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.3.2
Spectrum anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Structure formation and evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.4.1
Density fluctuations and structure growth . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.4.2
Non-linear evolution of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.3
1.4
2
3
The Halo Model revisited
23
2.1
Concepts of the original Halo Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2
Definition of the correlation function and power spectrum . . . . . . . . . . . . . . . . . . . .
24
2.2.1
Mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.2.2
Dark matter density profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.3
Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.3
Substructure and the extention of the Halo Model . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.4
Potential uses of the Halo Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Probes of the large-scale structure
35
3.1
Dark matter and weak lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.1.1
Principles of gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.1.2
Gravitational lensing phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.1.3
The weak lensing power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
Baryonic physics and the Sunyaev-Zel’dovich effect . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2.1
Inverse Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.2.2
The Sunyaev–Zel’dovich effects and applications . . . . . . . . . . . . . . . . . . . . .
47
3.2.3
The thermal SZ effect power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.2
i
CONTENTS
ii
4
5
6
7
The cross-correlation
55
4.1
The cross-power spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.1.1
3-dimensional spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.1.2
Angular spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
4.1.3
Substructure in the cross-correlation between dark matter and gas . . . . . . . . . . .
60
4.2
Correlation function of the power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.3
Covariance matrix of the correlation function and correlated errors . . . . . . . . . . . . . . .
61
4.3.1
Definitions: signal estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.3.2
Estimator of the cross-correlation power spectrum . . . . . . . . . . . . . . . . . . . .
63
4.3.3
Covariance of the cross-correlation power spectrum . . . . . . . . . . . . . . . . . . . .
64
4.3.4
Correlated errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Redshift dependence of the cross-correlation signal
69
5.1
Redshift dependence of the 3-dimensional spectra . . . . . . . . . . . . . . . . . . . . . . . . .
69
5.2
Redshift-binned signal and correlated errors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
Constraints on cosmological parameters
75
6.1
How the parameters affect the spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
6.2
Likelihood and Fisher analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
Heating history of baryons
81
7.1
81
Additional modelling of the cluster temperature . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions and further work
87
A Random fields and Limber’s Equation
91
List of Figures
95
List of Tables
97
Acknowledgments
99
References
101
Motivation
The lack of a definitive, thorough understanding of the matter distribution and the formation of structure
in the Universe continues to be a central motivation for most contemporary cosmological studies. Be it
dark or baryonic matter, the ever ongoing appearance of increasingly powerful surveys and cosmological
simulations promises more and better insight into this particular field, allowing the measurement and
the analysis of many effects described by previous theoretical and observational research. Two of such
phenomena are addressed in the work presented here.
Searching for a way to assess how the dark-matter distribution relates to the hot baryonic content of the
large scale structure of the universe, we choose the weak gravitational lensing (Bartelmann and Schneider
2001) and the thermal Sunyaev-Zel’dovich (SZ) (Sunyaev and Zel’dovich 1970, 1972) effects as privileged
probes of the dark matter and of the diffuse baryonic gas, respectively. Gravitational lensing, taken here as
describing the line-of-sight integrated effect of the gravitational potential of galaxy clusters on the images
of background galaxies, provides a detectable signal and thus a useful tool to describe structures we cannot
observe directly and which are believed to dominate the matter content of the Universe (Mellier 1999;
Refregier 2003; Munshi et al. 2008; Bacon et al. 2000; Laureijs et al. 2011). The thermal Sunyaev-Zel’dovich
effect has developed over decades into one of the preferred tools to trace the distribution of hot, ionised gas
which is trapped in the same potential wells (Birkinshaw 1999; Carlstrom et al. 2002; Planck Collaboration
et al. 2011b).
In this study, we exploit the simplicity and usefulness of the halo-model framework (Cooray and Sheth
2002) to obtain the power spectrum of the cross-correlation between the weak gravitational-lensing and
thermal SZ signals. Over the past couple of decades, this formalism for auto- and cross-correlation studies
has become an important supplementary tool to probe cosmology, either because of its predictive power
or because it paves a straightforward path towards relating theoretical expectations on dark-matter halos
to observations. We focus on the low-order statistics of this cross-correlation signal and examine the
covariance matrix and the redshift dependence of the correlation function based on the angular cross-power
spectrum. Furthermore, we investigate the possibility of constraining cosmological models with the results
obtained. Additionally, we constrain the halo mass integration to evaluate the effect of only considering hot
clusters at given redshift ranges.
Previous works have dealt with the computation and analysis of cross-correlations between a wide
range of signals. In particular, many of these studies have been devoted to the joint analysis of the weak
gravitational lensing and thermal SZ signals. The promising nature of complementing information from
the baryonic and dark-matter components of galaxy clusters prompted several works (Doré et al. 2001;
Seljak et al. 2001; Munshi et al. 2011). To carry out this particular cross-correlation within the halo model
1
Motivation
2
framework can be a desirable choice as presented in Cooray (2000) and Cooray and Sheth (2002). Most of
this analytic work has focused not only on a low-correlation level but also on higher-order statistics.
Our goal is to remain on the analysis of the two-point correlation but go beyond preceding studies
and, by making use of some of the same techniques, approach the subject of how we can constrain the
cosmological evolution of the matter-gas cross-correlation signal. We ultimately aim to achieve a clearer
picture of the thermal history of the baryonic component of the large-scale structure.
Chapter 1
Theoretical background
M
odern cosmology rests on decades of scientific research and discovery through which the image
one has of the Universe nowadays became the standard. We start this thesis by succinctly addressing
the most important cosmological concepts from history and literature which are relevant to understand
the present study. In Section 1.1, we provide an overview of two chronological sequences of events: the
scientific path that led to the building of the present-day cosmological paradigm and the sequence of events
that this model entails. On the other hand, the beginning of Section 1.2 follows the stages that make up the
evolutional history of the Universe. With these accounts, we aim to put our work into context and point out
which is the regime of evolution history we wish to investigate. The fundamental quantities and equations
of the standard cosmological model are presented. The follow-up Section 1.3 is devoted to characterising
the cosmic microwave background (CMB). The main concepts of structure formation and evolution are
accounted for in Section 1.4, at the end of this chapter.
All concepts presented in this chapter can be found in a more detailed account in general cosmology
textbooks (Kolb and Turner 1990; Dodelson 2003; Liddle 2003; Weinberg 2008) and reviews (Lyth 1993;
Trodden and Carroll 2004; Bartelmann 2010), to give a few examples.
1.1
Historical introduction
The tale of modern Cosmology is bound to always begin with the derivation of the Theories of Special
and General Relativity by Albert Einstein (Einstein 1905, 1916), the core component of our current view
of the Universe. His works unified the descriptions of space and time and the way these are affected
by cosmological distribution of energy, respectively. The field equations of general relativity were then
solved according to simplifying assumptions on the general characteristics of the cosmos. Firstly, using a
cosmological constant Λ that Einstein had included in his models to achieve a stationary Universe, and
famously rejected afterwards. de Sitter too included a non-zero value of Λ, a term which dominates his
empty model.
The solutions of Einstein equations regarded nowadays as the most accurate description of the Universe,
are the ones derived by Friedman (Friedman 1924). These include the cosmological constant and suggest
an expanding fabric of the Universe. In the late 1920s, the concept of expansion was subject of further
3
Chapter 1. Theoretical background
4
study. Lemaître too derived solutions of the field equations and additionally suggested the existence of a
relation between distance and redshift (Lemaître 1931). After observing a large number of galaxies, Hubble
confirmed the distance-redshift relation, thus confirming the existence of expansion. Lemaître interpreted
Λ as depicting the energy fluctuation of a fluid in vacuum and provided an equation of state to describe it.
As the first steps were being given towards the description of the Universe dynamics as a whole,
observations by Zwicky suggested that characteristics of clusters of galaxies were incompatible with the
amount of matter that could be observed. To explain the existence of such structures, he theorised that
a different kind of matter should be present in vast amounts although it could not be directly measured
(Zwicky 1937c). Other studies suggested the presence of this dark matter in galaxies (Oort 1940).
Works on the mechanisms of structure growth were conducted by Lifshitz, with a linear perturbation
approach (Lifshitz 1946). At approximately the same time, the first studies on structure formation at the
atomic level were published. Alpher et al. (1948) presented the theory of nucleosynthesis consistent with a
scenario where the Universe starts to evolve from a very hot, dense state which after a period of expansion
cools down enough to allow the combination of primordial nucleons into nuclei.
In 1948 we can find the first predictions of an uniform radiation that permeates all the volume of the
Universe, a relic from the first moments when the primordial fluid released photons. Known as the Cosmic
Microwave Background radiation as it peaks in that part of the electromagnetic spectrum, it was found by
chance by Penzias and Wilson (1965) and promptly indentified by Dicke et al. (1965).
Shortly after, independent groups predicted structures of low amplitude in the CMB (Peebles and Yu
1970; Sunyaev and Zel’dovich 1970). During the time between its detection and the detection of structures,
the CMB was intensively researched theoretically, with Peebles (1982) hypothesising that the low amplitude
of the CMB fluctuations could be explained by the majority of matter being a form of dark matter which
doesn’t interact with radiation.
Computational works shed light on other particularities of the Universe matter content: the use of
numerical simulations found that the existence of cold (non-relativistic) dark matter explains the structure
formation within the Friedman cosmological models (e.g. Davis et al. (1985)).
One of the most incisive moments of the last decades was the proposal of an inflationary phase of
expansion in the early time of the Universe. The theory of inflation (Guth 1981) was motivated by the need
to solve fundamental problems in the Friedman models and gained strength with the work of Mukhanov
and Chibisov (1981) which stated that the origin of structures could reside in the inflation of quantum
fluctuations.
What followed was the advent of an era devoted to observations of the Universe on all its scales, which
lasts still today. Supernovae type Ia were found to be standardisable candles and are used as probes of
the expansion and overall cosmological model since they provide a method for measuring cosmological
distances (Howell 2011). Most important to this phase were a series of measurements of the cosmic
microwave background radiation. First with the Cosmic Background Explorer (COBE), which measured
the radiation in the whole sky, as well as the anisotropy pattern for the first time (Smoot et al. 1992;
Boggess et al. 1992). The combination of these measurements with observations of galaxies was found
to support the vision that links the cosmological constant to a kind of Dark Energy that will eventually
drive the expansion of the Universe. Later, with the more precise observations by the Wilkinson Microwave
Anisotropy Probe (WMAP) (Komatsu et al. 2011), not only was the ΛCDM model supported; the density
fluctuations imprinted in the CMB are in agreement with what is predicted by inflationary theories. Further,
Chapter 1. Theoretical background
5
Figure 1.1: From top left to bottom right, some of the key scientists to the development of the present cosmological
picture: Albert Einstein, Alexander Friedman, Georges Lemaître, Edwin Hubble, Fritz Zwicky and George Gamow.
Chapter 1. Theoretical background
6
the detection of baryonic acoustic oscillations supports the models where dark matter is treated as a
non-relativistic fluid. The origins and evolution of the Universe, as modelled by the conclusions of the
studies mentioned in this historical introduction are summarised in the next section.
1.2
Foundations of Cosmology
Here we present the basic formalism of the cosmological standard, or concordance, model as well as its
most important corollaries: the description of time, space, dynamics and content of the Universe. The
Universe, as interpreted at present day, is built upon two fundamental concepts. The first of them is the
isotropy and homogeneity of the Universe on large sales: it should look about the same in all directions
and points. This is a direct consequence of the Cosmological Principle, according to which any observer
should, at any given time, look at the Universe and find the same general characteristics. The second belief
is that the dynamics of space and time are fully described by Einstein’s Theory of General Relativity (GR)
(Einstein 1916). Here we show how these two simple but powerful assumptions come together to form the
standard model of cosmology.
1.2.1
History of the Universe
During the described succession of discoveries and scientific advances towards a clearer picture of the
cosmos, the standard model of cosmology was built up to account for the succession of events that occurred
after the beginning of the Universe until the present day. Figure 1.2 depicts the basic steps believed to
have been taken through the formation and evolution of structures in the Universe, in chronological order,
which we here review.
It is consensually believed that the early state of the Universe can be characterised by a very dense state
of a primordial plasma. It was in a very hot state, dominated by quantum fluctuations – General Relativity
breaks when transported to such early stages of the cosmological history. It is speculated that the plasma
expanded and cooled allowing the separation of previously symmetric interactions: gravity, strong and
weak interactions and electromagnetism forces become independent. It is at this stage that inflationary
theories expect an epoch of exponential expansion to have occurred. The main outcome of this expansion is
the creation, via quantum fluctuations, of primordial curvature perturbations uniformly distributed over a
quasi-homogeneous background. The precise mechanism that could have given rise to such a period is still
an active area of research (see, e.g., Liddle and Lyth (2000) or Weinberg (2008)) but it is generally described
by a particle called the inflaton.The inflationary period is expected to have ended with the decay of the
inflaton into the whole a range of particle types.
After this point, the picture of the Universe starts to get more precise, as the studies of the last century
have substantially clarified the chronology of events from this stage on, as well as detailed the processes
taking place to a large extent.
The lowering temperatures at the photon era provide the conditions for nuclei formation, with protons
and neutrons building the first nuclei of hydrogen, helium and few other elements atoms. This period is
aptly known as nucleosynthesis. It is believed that at this point, radiation was the dominant energy source
of the Universe. Subsequently, the decrease of density allowed the combination of the nuclei and free
electrons – the so-called recombination period; during this time, the optical depth of the medium starts to
Chapter 1. Theoretical background
7
Figure 1.2: Highlights of the structure formation and evolution according to the standard cosmological model. From top
left to bottom right: nucleosynthesis, details of the temperature fluctuations on the cosmological microwave background
(NASA/WMAP), first stars (NASA/WMAP), galaxy (Hubble Space Telescope), galaxy cluster (Hubble Space Telescope)
and the web-like configuration of the large-scale structure (Millenium simulation).
Chapter 1. Theoretical background
8
diminish and the mean paths of free photons increase steadily until they can travel through space without
being subjected to scattering: this is the origin of the cosmic microwave background.
Left from the recombination epoch is a very homogeneous radiation visible over the whole sky. The
primordial density perturbations created during inflation give rise to temperature fluctuations in this
background. These are the seeds for structure formation. As structure slowly starts forming during this
dark period of time, a re-ionization of particles occurs. In the cold dark matter scenario – the most widely
accepted by now as it allows the existence of very small density perturbations as the ones observed in the
CMB – structure forms from the bottom up. This means that the first structures to form are the smallest.
As conditions are created for the gravitational collapse of element densities, the first stars and quasars are
born. The intense radiation created by such objects re-ionises the surrounding material. The formation of
structures produces progressively large objects. Stars agglomerate in galaxies; galaxies are attracted to each
other and build clusters; clusters of galaxies come into contact with others nearby, creating super-clusters,
in a web-like structure with empty areas permeating filaments that meet where clusters are.
The present state of the Universe seems to be characterised by an accelerated expansion phase driven by
the Λ-associated density (or dark energy) of negative pressure.
There is no clear picture of the ultimate fate of the Universe. This is under debate and depends on a
better assessment of the cosmological model and parameters.
In the next sections we will present the fundamental framework of the cosmological standard model
and focus on particular stages of the Universe evolution. We pay special attention to: firstly, the origin of
structures and the processes that characterise their growth; secondly, the observed background radiation
released at the time of recombination, the CMB.
1.2.2
The cosmological standard model
The isotropy and homogeneity assumptions can be introduced in Einstein’s equations of GR with the
following description of space and time, the Robertson-Walker metric:
h
i
ds2 = − c2 dt2 + a2 (t) dr2 + f k (r )2 dθ 2 + sin2 θ dφ2
(1.1)
where ds2 is the line element which for the description of light is 0. c is the speed of light and r, θ and φ are
co-moving spatial co-ordinates. a is the scale-factor which depends on proper time t (time as measured by
a co-moving observer) and describes the expansion or contraction of the three-dimensional space. Physical
(or proper) scales are related to co-moving ones by the scale factor as:
`(t) =
a(t)
`0 .
a0
(1.2)
a0 is commonly taken as the scale-factor at present day and given the value of 1. The parameter k is the
curvature constant and its value determines the space geometry: 0 characterises it as flat and positive or
negative values correspond to positively and negatively curved space, respectively. The radial function f
depends on the radial co-ordinate r in different ways, for given curvature values:

−1/2 sin ( k1/2 r )

k>0
 k
f k (r ) =
r
k=0 .


−
1/2
1/2
|k|
sinh(|k| r )
k<0
(1.3)
Chapter 1. Theoretical background
9
Figure 1.3: Proper and angular distances as functions of redshift. Replotted from Bartelmann (2010).
Further, the relative velocity v between two co-moving observers can be written as depending on the
scale factor:
v(t) =
d`
ȧ(t)
=
`(t) ≡ H (t) `(t) .
dt
a(t)
(1.4)
This is the Hubble law, first discovered during the study of the velocities of distant galaxies relative to
an observer (Hubble 1929): observed galaxies were receding from the observer; the relation between the
velocity of recession and the distance defines the rate of expansion of the Universe at a given time, the
Hubble parameter:
H (t) ≡
ȧ(t)
.
a(t)
(1.5)
This parameter is also commonly used in its reduced version h, in units of 100 km s−1 Mpc−1 . As the
Universe expands, the spectrum of distant objects is shifted from an initial wavelength λ to an observed
wavelength λ0 . This change can be quantified by the redshift value z, given simply by:
z≡
λ0 − λ
1
=
−1
λ
a
(1.6)
providing a useful relation between the redshift and the scale factor:
a = (1 + z ) −1 .
(1.7)
The radial function (1.3) is used to define the angular-diameter distance Dang , which connects the
physical and angular sizes of an object via the scale factor:
Da = a f k (r ( a)) .
(1.8)
The dependence of the angular-diameter distance on the redshift is shown in Figure 1.3: after a sharp
increase from redshifts 0 to 1.5, the distance slowly but steadily decreases.
Chapter 1. Theoretical background
10
Returning to the general description of the Universe dynamics, one can use the Einstein’s field equations,
here written in the customary Einstein notation:
Gµν = Rµν −
=
1
R gµν
2
8πG
Tµν + Λ gµν .
c4
(1.9)
(1.10)
Here G and Λ correspond to the gravitational and cosmological constants, respectively. (Λ, which was
initially introduced to obtain a static Universe description, is now supported by observations, as seen later
on.) gµν describes the metric and several tensors are also depicted: the Einstein tensor Gµν , the Ricci tensor
Rµν (with associated scalar R) and the energy-momentum tensor Tµν . The latter, combining the energy and
momemtum densities of all forms of matter, is given by:
p
p
Tµν = ρ + 2 Uµ Uν − 2 gµν
c
c
(1.11)
with ρ and p as density and pressure, both functions of time. This relation hold true for an ideal fluid. U is
the 4-velocity field, normalised such that Uµ U µ = −1. Obeying the isotropy/homogeinity assumptions, the
tensor becomes fully described by:
T00 = ρ
and
Tij = pgij .
(1.12)
Introducing the Robertson-Walker metric (1.1) to the field equations (1.10) we get two relations – the
Friedman equations:
2
8πG
ȧ
Λc2
kc2
=
ρ+
− 2
a
3
3
a
ä
4πG p
Λc2
=−
ρ+3 2 +
a
3
3
c
(1.13)
(1.14)
The combination of these two equations yields the following equality:
ρ̇ = −3
ȧ p
ρ+ 2
a
c
(1.15)
describing the first law of thermodynamics, giving the relation between density and pressure. An equation
of state of the form:
p = wρc2
with
−1 ≤ w ≤ 1
(1.16)
where the parameter w may be constant, can characterise the Universe during phases of domination of the
different components of the cosmological fluid. Going through with the integration of (1.15), the solutions
for ρ are of the general form:
ρ ( t ) = ρi
a(t)
ai
−3(1+ w )
(1.17)
where the index designates initial values of the density and scalar factor. Replacing w by 0, 1/3 and −1,
we get the evolution of the density field under the major influence of matter, radiation and the Λ field,
respectively.
Chapter 1. Theoretical background
11
The Friedman equation (1.13) can be further re-written in terms of densities:
Λc2
kc2
8πG
ρ
+
−
= 1.
3H 2
3H 2
a2 H 2
(1.18)
Defining a critical density by:
ρcritical ≡
3H 2
8πG
(1.19)
we introduce the matter density parameter:
Ω≡
ρ
ρcritical
(1.20)
For universes with Λ = 0, the density parameter defines the spatial geometry of the Universe:

k = +1

 >1
Ω=
=1
k=0


<1
k = −1
(1.21)
as closed, flat and open, respectively. Otherwise, defining:
ΩΛ =
Λc2
3H 2
and
Ωk = −
kc2
a2 H 2
(1.22)
(1.18) simply becomes:
Ω + ΩΛ + Ωk = 1
(1.23)
describing the evolution of the density. We note that Ω contains the contributions of matter and radiation
Ω = Ωm + Ωr and in turn, Ωm = ΩDM + Ωb . ’DM’ stands for dark matter, the dominant component of
structures which although not directly observed – as it is believed to not interact with electromagnetic
radiation – is inferred by the gravitationally induced behaviour of the directly observable baryonic matter,
labelled here with ’b’. In the standard model of cosmology, the dark matter component is assumed to be
cold, meaning that the particles were well non-relativistic when they de-coupled from thermal equilibrium.
This has been supported by many and diverse studies (more decisively by the cosmic microwave background
measurements) hence, the standard model is also known as Λ-Cold Dark Matter model, or ΛCDM. The
nature of dark matter has been heavily debated and is still poorly understood.
Combining Eq (1.14) with the above relations for the density parameters and the solutions for the
equation of state (1.17), we define the acceleration parameter q, describing the rate at which the expansion
of the Universe occurs:
q≡ −
äa
ä
1 + 3w
= − 2 =
Ω + ΩΛ .
2
ȧ2
aH
(1.24)
Evaluating the Friedman equation with the solutions for the different fluid components given by (1.17) in a
Big Bang scenario (meaning that the scale factor tends to zero some finite time in the past), we find that the
expansion of the Universe starts with a radiation dominated epoch:
"
#
4
3
2
1
1
1
H (t)2 = H02 Ωr0
+ Ωm0
+ Ωk0
+ ΩΛ0
a
a
a
(1.25)
Chapter 1. Theoretical background
12
parameter
estimate
error
Ωm
0.272
±0.014
Ωb
0.0456
±0.0016
ΩΛ
0.728
H0
70.4
+0.015
−0.016
+1.3
−1.4
σ8
0.809
±0.024
km s−1 Mpc−1
Table 1.1: Cosmological parameters of the ΛCDM model from the 7-year release of the Wilkinson Microwave Anisotropy
Probe. Measurements include baryonic acoustic oscillations and type-Ia supernovae data (Komatsu et al. 2011). The
parameter σ8 corresponds to the normalisation of the power spectrum and is a measure of the clustering level, as
discussed further in Section 1.4.
where the index ’0’ indicates the values at present time. As the expansion proceeds, the scale factor increases
and the other terms become gradually relevant. At a certain point, there is an equilibrium between the
influence of radiation and matter. The redshift at which this occurs, known as the equivalence redshift,
can be easily computed and is approximately 1 + zeq = Ωm0 /Ωr0 ' 2.4 × 104 Ω0 h−2 . The stage at which
the radiation is released from the matter, in a process described in the next section, happens after this
equilibrium period, at a redshift of about 1100.
Following this description, observations indicate that we are now in an Universe dominated by the
cosmological constant, so that ΩΛ is the driving source of expansion, associated to the energy type known
as Dark Energy.
The extensive cosmological observations of the last decades yielded estimates of all the parameters
involved in the description of the Universe. The Hubble parameter can be measured by a range of methods,
from galaxy distances, Cepheid variable stars, supernovae, the Sunyaev-Zel’dovich effect or gravitational
lensing. The total matter density as well as the radiation level can most accurately be estimated from the
cosmic microwave background. The cosmological constant associated energy has in supernovae type Ia its
most valuable proponent.
Table 1.1 shows the most recent values for these parameters, calculated from the data of the 7-year
release of the Wilkinson Microwave Anisotropy Probe (Komatsu et al. 2011).
1.3
The cosmic microwave background
After an initial phase where it is dense and hot, the Universe expands. This allows the temperature to
decrease, creating conditions for ions and free electrons to form a neutral fluid, releasing electrons, in a
period known as recombination. During this process, the medium becomes transparent, as the mean free
path of the photons increases dramatically, and by the end of recombination photons are able to travel
without being scattered. Once this stage is achieved, a highly isotropic radiation field is emitted – this is
known as the Cosmic Microwave Background radiation. There is a spherical radius separating the observer
from the period when this emission is released defining the so-called last scattering surface.
This relic radiation of the recombination period is predicted by the Big Bang theory. It was first
Chapter 1. Theoretical background
13
Figure 1.4: Contraints from the joint data from Supernovae Ia, galaxy clusters and the CMB in the matter-Λ density
plane point to flatness of the Universe. Reprinted from Amanullah et al. (2010).
hypothesised by Lemaître and estimated by Osterbrock et al. in the search for measurements of the
inter-stellar medium. By chance detected in the 1960s (Penzias and Wilson 1965) via a radio antenna, with
an emission corresponding to a temperature of ∼ 3K, the radiation was readily and rightfully interpreted
by Dicke et al. (1965) as the signature from the recombination period. It has since been extensively used
as one of the most important probes of cosmological properties. The intrinsic characteristics of the CMB
can reveal the structure of the Universe at the time the radiation was released. Moreover, the interaction
of these photons throughout time and space with evolved structures, sheds light upon the cosmological
history as well as its state at present day.
The relevance of the CMB in the current cosmological model is invaluable. Combining the output from
CMB surveys with data from galaxy clusters and Supernovae Ia it was possible to constrain the cosmological
parameters sufficiently to characterise the Universe as spatially flat (see Figure 1.4 by Amanullah et al.
(2010)).
In the next sections we will first characterise the basic properties of the CMB radiation, focusing on
the intensity and power spectra. Afterwards we shortly describe how the original radiation can be altered
during its journey from the last scattering surface, through processes that create distortions in its spectrum.
1.3.1
Temperature fluctuations
The cosmic background radiation globally dominates the emission in the Universe. This emission has been
extensively and precisely measured since the years of its discovery and displays a black-body intensity
Chapter 1. Theoretical background
14
Figure 1.5: CMB all-sky map from the WMAP 7- year data release showing the temperature fluctuations of the
microwave emission. Credit: NASA/WMAP Science Team.
spectrum, commonly given by the Planck distribution:
ν 3
1
dν
P(ν, T ) dν = 8πh
c
exphν/kB T −1
(1.26)
with a corresponding temperature of TCMB = 2.725 ± 0.002 K (Mather et al. 1999). The parameters that
define the shape of the black-body intensity spectrum are the frequency ν, light-speed c, Boltzmann constant
kB and Planck’s constant h.
Although the intensity of the CMB seems to be the same in every direction, a rigourous analysis to its
intensity discloses the presence of very small temperature fluctuations in a seemingly isotropic distribution.
The origin of these fluctuations can be divided in three main anisotropy types. The first and most intense
source of temperature fluctuations is the mirroring of the density perturbations of the primordial fluid at
the last scattering surface. The second source of anisotropy comprises all the mechanisms which affect
the propagation of the CMB photons once they are released. The third type is related to the noise of
observations. The overall anisotropy field has also been observed and Figure 1.5 displays one of these
measurements by the Wilkinson Microwave Anisotropy Probe (WMAP), where all sources of noise have
been removed.
The CMB temperature fluctuations are commonly statistically characterised by the correlation of the
fluctuations given different positions in the sky – the power spectrum. The deviation from the average
background temperature can be decomposed into spherical harmonics Y`m :
∆T
=
T
∞
`
∑ ∑
a`m Y`m (θ, φ)
(1.27)
`=0 m=−`
with
∆T
dΩ0 .
(1.28)
T
θ and φ are spherical angles and dΩ stands for the solid angle displacement. The coefficients defined by
a`m =
Z
Y`∗m
a`m are multi-polar moments. It is assumed that the fluctuations are a Gaussian random field, which allows
the full description of the temperature fluctuations by the angular power spectrum:
∆T
∆T 0
0
C (n̂, n̂ ) ≡
(n̂)
(n̂ ) = ∑ ∑ h a∗`m a`0 m0 i Y`∗m (n̂) Y`∗0 m0 (n̂0 )
T
T
``0 mm0
(1.29)
Chapter 1. Theoretical background
15
Multipole moment l
10
Temperature Fluctuations [µK2]
6000
100
500
1000
5000
4000
3000
2000
1000
0
90°
2°
0.5°
0.2°
Angular Size
Figure 1.6: The CMB power spectrum as observed by WMAP. Credit: NASA/WMAP Science Team.
where n̂ is a unit vector defined as the direction of the sky given by the spherical angles θ and φ. The
correlation is averaged over an ensemble of similar perturbation configurations. Assuming symmetry over
all angles (isotropy), the coefficients (1.28) are simply related to the angular power spectrum C` by:
h a∗`m a`0 m0 i = C` δ``0 δmm0
(1.30)
C` = h| a`m |2 i .
(1.31)
with
Introducing these assumptions in (1.29) we get the spectrum dependent on the spherical angles:
C (n̂, n̂0 ) =
∑
`
(2` + 1)
C` P` (cos ϑ )
4π
(1.32)
with cos ϑ = n̂.n̂0 . Through C` , theoretical models and the observations can be compared. The power
spectrum of the temperature fluctuations, also observed by WMAP, is shown in Figure 1.6. The CMB
radiation traces many cosmological properties while travelling through space and different scales of the
power spectrum shed light on different medium characteristics. This subject will be further discussed in
the next section.
1.3.2
Spectrum anisotropies
A closer look at the CMB power spectrum of Figure 1.6 allows the analysis of the primary anisotropy by
dividing the curve into sections corresponding to particular scale ranges as shown in Figure 1.7. This shape
is theoretically predicted by the cosmological standard model. Each range is characterised by the effect of
different physical processes which are imprinted in the power spectrum.
Starting at the large-scales regime we identify the region of the spectrum below angular multipoles of
` ∼ 100 as being under the Sachs-Wolfe effect: the phenomenon predicted by Sachs and Wolfe (1967) and
Chapter 1. Theoretical background
16
Figure 1.7: Scheme of the theoretically predicted CMB power spectrum; different parts of the spectrum are labelled
according to type. Re-plotted here from Scott and Smoot (2010).
related to the gravitationally induced redshift arising from the initial potential fluctuations when photons
de-coupled from matter. This region shows the influence of the initial conditions on the shape of the
spectrum as it portrays temperature fluctuations before they evolved significantly. The signal at very low
multipoles is the result of time variation of the gravitational potential and this rise is designated, as it is
integrated over the potential changing in time, by the rise of the integrated Sachs-Wolfe (ISW) effect.
At intermediate scales of 100 ≤ ` ≤ 1000, the observed peak in the spectrum is created by the acoustic
oscillations of the primordial fluid before recombination occurred. These oscillations arise from the
competing of increasing gravity and counteracting pressure forces in the plasma: photons coupled with
baryons felt the growth of gravitational instabilities linked to the dark matter component of the fluid.
As the recombination epoch set in, the oscillations became a pattern created by the paths of photons
out of the plasma and projected onto the sky we observe. The measurement of the imprint of baryons
(Baryon Acoustic Oscillations, or BAOs) can be combined with data from large-scale surveys to evaluate
the acceleration of the Universe – see Eisenstein (2005).
The small-scale limit of the CMB power spectrum, at ` > 1000, reflects the finite duration of the
recombination epoch. Although photons can start to freely travel through space, beyond the influence of
the rest of the primordial fluid, this does not happen instantaneously. Rather, the recombination happens
with the gradual increase of the free mean path of the photons, creating a damping tail, or Silk damping
region (Silk 1968).
The description presented until now amounts to the characterisation of the primary sources of anisotropy
in the CMB power spectrum. There are, however, other events that affect the power spectrum of fluctuations
while it travels through space between the last scattering surface and the observer. These created the
commonly named secondary anisotropies as they act as sources of contamination to the primary temperature
fluctuations.
The strongest source of interaction between the CMB photons and the environment they travel through
Chapter 1. Theoretical background
17
Figure 1.8: Secondary anisotropies plotted against the CMB power spectra. Adapted and reprinted from Cooray and
Sheth (2002).
Chapter 1. Theoretical background
18
is the thermal Sunyaev-Zel’dovich effect (Sunyaev and Zel’dovich 1970). This signal is created when the
photons exchange energy with free electrons in ionised, hot gas trapped in gravitational potential wells, in
a scattering event that ultimately traces the distribution of baryonic material in the Universe – the inverse
Compton scattering. As one of the phenomena evaluated in this thesis, it will later be discussed in more
detail in Chapter 3. Also part of the Sunyaev-Zel’dovich (SZ) formalism is the signal arising due to the bulk
motion of the hot gas relative to the rest frame of the propagating photons (Sunyaev and Zeldovich 1980).
This kinetic effect, although difficult to measure directly from the spectrum, modifies the power spectrum
via a Doppler shift and provides information on the dynamics of galaxy clusters and general motion of gas
in the cosmos. See top panel of Figure 1.8 for a comparison between the CMB power spectra and the SZ
effect signal.
The passage of the CMB photons through structures in the middle of their growth phase creates another
signature in the power spectrum (see bottom plot in Figure 1.8). This late-time integrated Sachs-Wolfe, or
Rees-Sciama (RS) effect (Rees and Sciama 1968) evaluates how the gravitational potentials change while the
photons are passing through evolving structures and is thus intrinsically related to this stage of structure
development.
It is not surprising that the path of the primordial radiation may be affected by the gravitational power
of the structures it encounters, creating a lensed signal of the original spectrum. As seen in Figure 1.8 the
lensing only significantly changes the spectrum at very small scales. This distortion can be used to probe
the cosmological model via its parameters. For details on the effect of gravitational lensing on the CMB
spectrum see, e.g. Seljak (1996).
1.4
Structure formation and evolution
The Universe described by the cosmological standard model started as an extremely dense and hot fluid
which at some point started to expand. It is believed that after a first inflationary phase where it expanded
exponentially, the Universe was governed sequentially by radiation, matter and dark-energy. As the
expansion progresses, the primordial plasma cooled down and different energy components de-couple
from the fluid. Structures are expected to have originated in the density fluctuations arising in during
inflation. Small energy and density perturbations were gravitationally unstable. Such perturbations
could be generated by quantum vacuum fluctuations (see, e.g., Liddle and Lyth (2000).) The growth
and evolutionary path of these fluctuations until today could then create the large-scale configuration we
observe nowadays: a web of filamentary structures which meet at galaxy clusters, populated by many
smaller-scale structures, surrounding large voids.
The way small density fluctuations grow into larger structures was first studied by Jeans (1902) while
trying to explain the mechanisms through which planets or stars are created in a nebula. According to
this framework, the gravitational instabilities in a homogeneous and isotropic self-gravitating fluid will
provoke in-fall of the surrounding material, thus locally increasing the density. Lifshitz (1946) introduced
this description into General Relativity, devising the first linear perturbation theory.
The linear perturbation theories are adequately used when describing the growth and evolution of
small fluctuations. With the rise of densities, this characterisation becomes increasingly inaccurate and
higher-order perturbations need to be taken into account as well. The complexity of the configuration will
Chapter 1. Theoretical background
19
eventually become too high and demand sophisticated techniques to describe the evolution of structures at
later times.
In this section, we present a brief overview of the fundamental equations describing the linear growth
of matter perturbations and write a few words on the non-linear analysis of structure evolution.
1.4.1
Density fluctuations and structure growth
Density perturbations are typically expressed by the density contrast field δ:
δ(x, t) =
ρ(x, t) − ρ̄(t)
=
ρ̄(t)
∑ δk (k, t)e−ik.x
(1.33)
with ρ being the density distribution and ρ̄ representing the mean background density. k is here the
wavenumber, depicting the scale in Fourier space. In the second equality, the excess distribution is described
as a Fourier series, where:
δk =
1
V
Z
d3 x δ(x, t) eik.x .
(1.34)
To describe the evolution of the density fluctuation field, we use the equations of motion describing
dark matter dynamics:
∂ρ
∇.(ρv) = 0
∂t
p
∂v
+ (v.∇)v = −∇ φ +
∂t
ρ
∇2 Φ = 4πGρ .
(1.35)
(1.36)
(1.37)
These are continuity, Euler and Poisson equations, respectively. v describes the velocity field which is
here assumed to have a negligible dispersion. Combining the three equations within a linear regime, we
conclude that, in the matter-dominated epoch, δ of sub-horizon perturbations evolves according to:
δ̈ + 2H δ̇ − 4π G ρ̄ δ = 0
(1.38)
From the two solutions of this relation, one increases with a and describes structure growth:
δ( a) = δ0 D+ ( a)
with
D+ =
G ( a)
G ( a0 )
(1.39)
where D+ is the growth factor and a0 and δ0 correspond to the values of the scale factor and the density
contrast at present day. The growth term is well fit by the formula:
1
Ωm
ΩΛ
4/7
G ( a) ≡ a Ωm Ωm − ΩΛ + 1 +
1+
.
2
70
(1.40)
It is assumed that the primordial density field is a Gaussian random field, as the consequence of the
central limit theorem applied to the superposition of quantum fluctuations in the initial field. As the mean
of the density contrast is zero by definition, the whole system is to be completely defined by the variance
of the field. Analysing the two-point correlation of δ and going to Fourier space, one obtains a simple
correspondence between the power spectrum and the variance of δk given in (1.34):
δ̂(k) δ̂∗ (k0 ) ≡ (2π )3 Pδ (k) δD (k − k0 ) .
(1.41)
Chapter 1. Theoretical background
20
similarly calculated as in (A.4). The evolution of the primordial power spectrum for cold dark matter is
well described by the following power law:
(
Pδ (k) =
A kn
k keq
A k n −4
k keq
(1.42)
where n is the spectral index of scalar perturbations. A is a constant amplitude and the wavenumber keq
is defined as the horizon scale at which the matter and radiation have equal densities. Moreover, we can
define the variance of the density contrast:
σR2
= 4π
Z
dk
W 2 ( k ) k2 P ( k )
(2π )3 R
(1.43)
where WR is a window function selecting the scales of interest. Traditionally, this scale is set at 8 h−1 Mpc.
This parameter, σ8 , quantified the amplitude of cosmic structures.
Until now we have focused on the linear stages of structure growth. However, the evolution of the
largest gravitationally bound structures observed in the Universe in equilibrium at present day – the clusters
of galaxies – cannot be fully described by linear approximations. The need for a better understanding of
how they form leads to the non-linear perturbation evolution techniques presented next.
1.4.2
Non-linear evolution of perturbations
As structures grow, the description of their evolution is no longer accurately evaluated by linear perturbation
theory. As the density contrast increases, structures contract. Power will thus be transported towards larger
wavenumbers.
To study the evolution of structures beyond the linear regime, N-body numerical simulations are the
most powerful tool at our disposal. However, a few analytical approaches can clarify the onset of non-linear
structure growth.
The spherical collapse assumes, as the denomination suggests, a spherically symmetric distribution of
homogeneous overdensity and is used to estimate the threshold after which the configuration will collapse
due to gravity towards its centre, independently of the surrounding influence of other overdensities. The
model can also be used to determine the linear density at which the structure becomes large enough to
collapse, although it doesn’t account for any processes that could establish virial equilibrium. Variations of
this method include removing the spherical symmetric assumption, as in the case of the elliptical collapse
model.
Another approach to non-linear density growth is the Zel’dovich approximation (Zel’dovich 1970).
In this method, particles are described by their co-moving co-ordinates and velocities instead of their
overdensities. The initial velocity of these particles is used to define the density and the laws that govern its
evolution. This approximation is reasonably accurate for as long as density perturbations are not high and
particle trajectories do not cross. The particles do not notice each other’s presence in this approximation.
Nevertheless, the method successfully mimics the anisotropy of the growing structures that would give rise
to the formation of the characteristic cosmic filaments. Moreover, it sheds light on how structures acquire
angular momentum.
Nowadays, numerical simulations are widely used to study structure formation and evolution, as
technical and methodical improvements allow increasingly large and complex set-ups. Simulations are not
Chapter 1. Theoretical background
21
bound to many of the assumptions made in the analytical models, which mostly simplify the case-studies
and may therefore affect the accuracy of the results and application range of the conclusions drawn. These
studies can include a virtually unlimited amount of models (describing adiabatic evolution, including or
not the effects of radiative phenomena as cooling or feedback, etc) and track how they affect the evolution
of structures.The equations of motion are numerically integrated and the particles trajectories are followed
over times, provided a set of initial conditions. For the definition of the initial conditions, the Zel’dovich
formalism mentioned above is commonly used.
22
Chapter 1. Theoretical background
Chapter 2
The Halo Model revisited
T
he characterisation of structure and its distribution in the Universe is an ambitious and heavily
populated field of work. From analytical approaches like perturbation theory to numerical methods
such as N-body simulations of large-scale structure, there are several options for defining cosmological
core features that can be used to predict and/or explain observational evidence. For the present study,
we choose to use an analytical technique which provides a fairly simple and yet comprehensive, flexible
framework to describe the signal of a given cosmological source: the halo model. This technique has
been extensively used as the kernel for a great number of studies on dark matter, galaxy and velocity
distributions, to name just a few.
Our description of the large scale structure and its overall signal is based on a specialised version of the
halo model. Below we list the main characteristics of this tool, starting in Section 2.1 with a brief account of
the context and ideas from which it arose. The formalism is reviewed in some detail in Section 2.2 and
followed by a few notes on the possibility of extending the model to account for substructure in halos
(Section 2.3). We finish this chapter by acknowledging the versatility of the halo model, which allows the
study of many properties beyond what was its original intent.
A most detailed account of the halo model specifics can be found, for instance, in the extensive review
by Cooray and Sheth (2002).
2.1
Concepts of the original Halo Model
The fundamental ideas of what is now known as the halo model were developed out of the need for a tool
to characterise the distribution of galaxies in the universe. Half-way through the last century, Neyman and
Scott (1952) theorised that galaxies clustered in halos throughout space and – finding discrete statistics
appropriate to describe galaxies – developed a formalism under which halos were to be distinguished by
basic properties: their size, inner structure and relative position to such other objects. It wasn’t, however,
until the last few decades that the collection of a series of results on these particular attributes yielded
enough information to allow successful applications of the halo model formalism.
It is most consensual at this point in the history of modern cosmology that the clustered structures we
observe in the universe today evolved from perturbations in a primordial, mostly uniform distribution of
23
Chapter 2. The Halo Model revisited
24
dark matter. Their distribution in space has been the subject of numerous phenomenological and numerical
studies, providing extensive and, on the whole, concordant methods of mimicking this property. A few
examples for reference: the mass function predicting halo abundances by Press and Schechter (1974)
or Sheth and Tormen (1999), N-body simulations of large-scale structure by Springel et al. (2005) and
observational catalogues of galaxies in Miller et al. (2005).
The inner structure of halos has also been described by several groups, being the most remarkable and
pervasive models the ones of Navarro et al. (1997), Moore et al. (1999) or Einasto and Haud (1989). All of
them parameterise the matter density in dark matter halos in radial distributions and most were mainly
motivated by the identical behaviour found in numerical simulations like the above-mentioned while others
relied on models developed from luminosity functions. In the former cases, although there is no obvious
basis as far as the theoretical framework is concerned, they are nevertheless widely used and observational
results of present-day surveys that seem to be in agreement with these fits support their continuous future
employment (Umetsu et al. 2011, 2012).
The reasoning behind the process through which halos evolve and relate to others has also been
researched intensively. This field bloomed in particular after the study published by White and Rees (1978)
on the clustering properties of galaxy halos. This very popular work proposes a hierarchical formation
of structures where galaxies should form inside and bear characteristics deeply connected to those of
their host dark matter halos. Models of the relation between the large-scale organisation of dark matter
halos we see today (through the direct or indirect methods that allow us to trace them) and correspondent
observables abound, mostly evolved from the framework described by Mo and White (1996).
As light was shed on the very properties of halos that allow to define the halo model, it could then be
used broadly despite a few changes of some original assumptions without changing their fundamental
purpose. The discrete approach was adapted to the description of a continuous density distribution within
which peaks correspond to halos. The bottom-line of the resulting halo model framework consists then of a
description of the universe where all mass is contained in individual halos, characterised by their spatial
abundances, a wide range of masses and with given density profiles. Moreover, it relies on a fully analytical
technique whose results can be directly compared to both observations and numerical simulations. It is so
a valuable alternative modelling tool.
Next, we present the method through which the correlation function of a dark matter halo distribution
can be calculated under the halo model formalism.
2.2
Definition of the correlation function and power spectrum
The halo model assumes as one of its most crucial features the fact that all matter is confined to halos,
characterised by their masses as well as the way these masses are arranged within them. It furthermore
relies on time and spatial distribution of halos along with how these characteristics relate to those of
neighbouring halos. The description we present next is a short summary of what can be found in initial
chapters of the review by Cooray and Sheth (2002).
We start by defining a general matter distribution as the sum of densities over all halos and their
Chapter 2. The Halo Model revisited
25
Figure 2.1: Illustration of the halo model ingredients. From left to right: spatial distribution of halos, inner distribution
of density and how the density peaks can be off-set from the initially over-dense region (biasing).
positions x:
ρ(x) =
∑ ρ ( x − x i , m i ) = ∑ m i ρn ( x − x i , m i )
i
(2.1)
i
where ρn is the normalised halo density profile which depends on its mass mi and the distance to the halo
R
centre. The normalisation is defined such that dx0 ρn (x − x0 ) = 1. One can additionally introduce delta
functions to achieve a useful mass-integral form:
ρ(x) =
∑
Z
dmdx0 m δ(m − mi ) ρn (x − x0 , m) δ(x0 − xi ).
(2.2)
i
This form allows us to define the number density of halos by averaging the sum over the product of the
delta functions:
*
n(m) =
∑ δ ( m − mi ) δ ( x
+
0
− xi )
.
(2.3)
i
The mean density of the distribution can then be simply given by a simple formulation:
ρ̄ = h ρ(x) i =
Z
dm n(m) m
Z
0
0
dx ρn (x − x , m) =
Z
dm n(m) m.
(2.4)
The key-function that the halo model stipulates is the two-point correlation function or its Fourier
conjugate – the power spectrum – of a given dark matter halo distribution. The two-point correlation
function is established as the simple sum of a Poisson term accounting for the contribution of individual
halos (commonly named as 1-halo term), and a second term giving the two-point correlation between
different halos (the 2-halo term):
ξ (x) = ξ 1h (x − x0 ) + ξ 2h (x − x0 ).
(2.5)
The two terms are determined as mass and space integrals over the number density of halos and their
density profiles:
Z
m 2
n(m, z) dy ρn (y, m) ρn (y + x − x0 , m)
ρ̄
Z
Z
Z
m n ( m1 )
m2 n ( m2 )
ξ 2h (x − x0 ) = dm1 1
dm2
dx1 ρn (x − x1 , m1 )
ρ̄
ρ̄
ξ 1h (x − x0 ) =
Z
Z
dm
dx2 ρn (x − x2 , m2 ) ξ hh (x1 − x2 , m1 , m2 )
(2.6)
(2.7)
Chapter 2. The Halo Model revisited
26
where, apart from the spatial number density of halos quantified by the mass function n(m) and the inner
structure of the halo, given by the density profile ρn , we have the two-point correlation between halos 1
and 2. As briefly discussed later on Subsection 2.2.3, in the large-scale regime this can be approximated by
the simple product of the bias b of each halo with the correlation function of linearly evolved primordial
fluctuations, ξ lin , yielding:
ξ hh (x1 − x2 , m1 , m2 ) ≈ b(m1 ) b(m2 ) ξ lin (x1 − x2 ).
(2.8)
The presence of many convolutions in these calculations suggests using Fourier transforms. Therefore,
the power spectrum is the preferable tool to describe and evaluate two-point correlations. After Fouriertransforming the correlation functions:
P(k, z) = P1h (k, z) + P2h (k, z)
(2.9)
which are dependent on wavenumber k and redshift z. For a given mass distribution, the model yields the
following expressions for the 1-halo and 2-halo terms of the 3-dimensional two-point correlation function
in Fourier space:
P1h (k, z) =
Z
dm
P2h (k, z) = Plin
Z
m
ρ̄
2
dm1
n(m, z) u2 (k, m)
m1
n(m1 , z) u(k, m1 ) b(m1 )
ρ̄
(2.10)
Z
dm2
m2
n(m2 , z) u(k, m2 ) b(m2 ).
ρ̄
(2.11)
These integrals over mass ranges depend on ρ̄ as the background density appearing in the normalisation
of the mass function n(m), the normalised Fourier transform of the density profile u(k, m) and the bias
parameter b. Plin refers to the linearly evolved power spectrum of primordial fluctuations.
In this section, we intended solely to present the basic framework of the halo model. The features of the
power spectrum of the dark matter halo distribution, computed with the above expressions, are discussed
with some detail in Subsection 4.1.1, within the next chapter dedicated to the 3-dimensional power spectra.
It is noteworthy the fact that although, it provides a fairly adequate method to describe the statistics
of dark matter distributions, the halo model does have limitations. Most of these arise from simplifying
assumptions which are mostly taken for practicality and clarity of thought. The accuracy suffers from these
approximations and there are a few concepts that can be improved: the parametrisation of halos according
to which there is spherical symmetry; how the dark matter profiles should depend on the concentration
distribution as well as on the halo mass; how halo profiles are not completely smooth and may contain
lower but independent peaks building an inner-halo substructure. The latter issue will be briefly addressed
in Section 2.3.
The subsections ahead give a glimpse of the most commonly chosen models for the distribution, profile
and bias of halos on halo model-based studies found in the literature, including the ones chosen for this
work.
2.2.1
Mass function
Press and Schechter (1974) derived a simple model, popularly used afterwards, which used the spherical
collapse model formalism to describe the mass distribution of halos in space and time. The co-moving
Chapter 2. The Halo Model revisited
27
Figure 2.2: Halo mass functions estimated from numerical simulations obtained by Jenkins et al. (2001), plotted against
the Press and Schechter (1974) (dashed line) and Sheth and Tormen (1999) (dotted line) results.
number density n(m) of halos of mass m at redshift z was then defined with the help of the co-moving
density of the background ρ̄, the critical density required for spherical collapse at the given redshift δsc (z)
and the variance of the primordial density fluctuation field σ2 (m):
dν
m2 n(m) dm
= ν f (ν) .
m
ν
ρ̄
(2.12)
2 ( z ) /σ2 ( m ), with both quantities being calculated by linearly extrapoThe parameter ν is defined as ν ≡ δsc
lating their initial values to present time. The shape of the function is given by:
r
ν
ν
ν f (ν) =
exp −
.
2π
2
(2.13)
A better fit to CDM models was later calculated by Sheth and Tormen (1999), taking into account the
shape of halos in an elliptical description of the halo collapse. According to this work, the mass function is
in better agreement with modern cosmological models when parameterised in the following way:
r
qν qν
ν f (ν) = A( p) (1 + (qν))− p
exp −
(2.14)
2π
2
√
where p ≈ 0.3, q ≈ 0.75 and A( p) = (1 + 2− p Γ(1/2 − p)/ π )−1 . This approach expanded the PressSchechter formalism – attainable with p = 0.5 and q = 1 – into a more flexible model.
With the advent of numerical simulations, the effort to find a mass function that could comprehensively
describe the widest possible range of halos in the mass-redshift plane, was significantly simplified. The
Chapter 2. The Halo Model revisited
28
shape of the mass functions can be compared in Figure 2.2, taken from the study Jenkins et al. (2001), where
halo mass profiles from a numerical simulations of large-scale structure are plotted and the average fit is
well parameterised into the above mass function, with the following re-arrangement of the formula:
f (σ, z) ≡
m dn(m, z)
.
ρ̄ d ln σ−1
(2.15)
Given the very good general agreement of the mass function by Sheth and Tormen (1999) with the ones
found in such studies, we chose to use it in the present work .
2.2.2
Dark matter density profile
One of the most important concepts of the halo model is the continuous distribution of matter whose peaks
correspond to halos where matter is confined. In this subsection, we briefly discuss the description of how
the matter is arranged within halos. We will here concentrate our description of the halo density profile on
the NFW model for it was the chosen one to be used throughout the work whose results are presented in
this thesis.
The already classic Navarro, Frenk and White (NFW) profile, first presented in Navarro et al. (1997), is a
fully analytical description and one of the most successful versions of the more general model:
ρ(r, m) =
ρs
(r/rs ) (1 + r/rs )γ
(2.16)
α
with α = 1 and γ = 2. This density profile is parameterised by a halo-constant characteristic density ρs at
the scale radius rs , which introduces the dependence of the profile on the concentration c of the halo:
rs ≡
rvir
c
(2.17)
where rvir is the halo virial radius and c is the concentration parameter. The characteristic density of a
given halo of mass m can be determined by the total mass, a radial integration up to the virial radius:
Z rvir
dr 4πr2 ρs
3
= 4π ρs rs ln(1 + c) −
mtotal ≡
(2.18)
0
c
1+c
.
(2.19)
As we need to describe the two-point correlation in terms of its Fourier conjugate, it is useful to define
the normalised Fourier transform of the density profile:
R 3
d x ρ(x, m) e ik.x
u(k, m) = R 3
.
d x ρ(x, m)
(2.20)
On the assumption that halos are spherically symmetric and considering that all mass is contained
within the virial radius, the Fourier transform becomes:
Z
Z rvir
sin(kr )
dr 4π r2 ρ(r, m)
kr
sin(ckrs )
= 4π rs3 ρs sin(krs ) [ Si( (1 + c) krs ) − Si(krs ) ] −
(1 + c) krs
+ cos(krs ) [ Ci( (1 + c) krs ) − Ci(krs ) ]
d3 x ρ(x, m) e ik.x =
0
(2.21)
(2.22)
Chapter 2. The Halo Model revisited
29
101
1010 M ¯
1015 M ¯
uDM(k,m)
100
10-1
10-2
10-3 -2
10
10-1
101
k [Mpc−1 h ]
100
102
Figure 2.3: Normalised NFW profile and its dependence on the halo mass at redshift zero.
where Si and Ci are sine and cosine integrals, respectively:
Z x
sin(t)
t
0
Z x
cos(t)
Ci( x ) =
dt
.
t
0
Si( x ) =
dt
(2.23)
(2.24)
Figure 2.3 shows the normalised Fourier transform of the NFW profile for halos of different masses at
redshift 0. The general behaviour of these curves follows a steeper drop of density for more massive halos
on small scales. This means that low mass halos dominate the contribution at small scales or, conversely,
that the most massive halos contribute to the total power at the largest scales.
Although this and sibling profiles are not theoretically motivated, they often very accurately describe
the distribution of matter around the centres of stable, gravitationally bound structures. Nevertheless, as
mentioned earlier in this chapter, not only simulations are in good agreement with these density profiles.
Observational data are too supporting this formalism (Umetsu et al. 2011, 2012). For alternative models
motivated by observational estimates of the density profile like the works of Einasto and Haud (1989) or
Sérsic (1963), the reader is referred, for instance, to the works in Merritt et al. (2006) or Coe (2010) where
comparisons in between all classic and recent models are extensively studied. Figure 2.4 shows a glimpse
of how the most commonly used models compare with each other and simulations.
2.2.3
Bias
With fairly good descriptions of how halos are distributed spatially and how matter is distributed within
halos, we only lack information on how the configuration of dark matter halos is biased relative to that of
the mass. It is believed that from the initial fluctuation field, only the configurations with enough density
were able to collapse over time. Following the work by Mo and White (1996), the halo bias relation δh can
Chapter 2. The Halo Model revisited
30
Figure 2.4: Dark matter density profiles of different models compared to simulation results. NFW and model by Moore
et al. (1999) are plotted against profiles from simulated halos (left panel) and the estimated profiles of the Milky Way,
M31 and the Virgo Cluster (right panel). Plot adapted from Hayashi et al. (2004).
be estimated through the expression:
δh (m, z1 | mV , V, z0 ) =
N (m, z1 | mV , V, z0 )
− 1.
n(m, z1 )V
(2.25)
This overdensity is defined in a conditional manner and dependent on two halo number densities which
trace the halo distribution at different redshifts z0 and z1 with z1 > z0 . N denotes a conditional mass
function of halos with a given mass m at a given collapse redshift z1 , which at redshift z0 are within a
volume V with mass mV . The above equality contains a link in between the distribution of overdensities
and that of the matter, since M/V = ρ̄(1 + δ).
What we need now is to introduce models to describe N, where the obvious choice would be any of the
presented earlier in Subsection 2.2.1. n depends on the way halos evolved in between z0 and the collapse
and so the quantity δsc (z1 ) needs to be taken into account in order to describe the mass function. The
description of ν in Eq. (2.12) can be replace by:
ν10 =
δsc (z1 ) − δ0 (δ, z0 )
σ2 ( m ) − σ2 ( mv )
(2.26)
where δ0 (δ, z0 ) corresponds to the initial overdensity evolved linearly so that at z0 it has an overdensity of
δ. Plugging the above parameter in model of Eq. (2.14) in Eq. (2.25) and taking first order approximations,
one can simply calculate the overdensity resorting to the expression:
qν − 1
2p/δsc (z1 )
δh ≈ δ 1 +
+
= b1 (m, z1 ) δ
δsc (z1 )
1 + (qν) p
(2.27)
stating a proportionality between the overdensity of halos and the overdensity of the mass, via the bias
parameter b which depends on the halo masses and their virialisation redshifts. This is the result we use in
Chapter 2. The Halo Model revisited
31
the present work. The halo-halo term in the correlation (2.7), between points in two halos of masses m1 and
m2 becomes simply:
ξ hh (r, m1 , m2 ) ≈ b(m1 ) b(m2 ) ξ lin (r )
(2.28)
with the variable r stating its spatial dependency. Hence, the power spectrum counterpart of this term is
calculated through the following approximation:
Phh (k, m1 , m2 ) ≈, b(m1 ) b(m2 ) Plin (k )
(2.29)
which can be recognised in Eq. (2.11).
2.3
Substructure and the extention of the Halo Model
The halo model has been extended to account for the existence of substructure in halos, as suggested by
observations (e.g. galaxy halos within galaxy clusters halos) and numerical simulations alike (see Springel
et al. (2008) or Giocoli et al. (2010a,b) and references therein). Unlike the original premise of the model,
halos in simulations of hierarchical clustering do not evolve into a purely smooth and spherically symmetric
density distribution of matter.
In the scenario where sub-structure is accounted for, each of the halo terms presented in Eqs. (2.10)
and (2.11) is split in two additional components. These describe the contribution of individual sub-halos
plus their mutual autocorrelation, in a very similar manner to what is done when describing the spatial
distribution of their host halos. Adapting the derivation results by Sheth and Jain (2003):
1h
1h
1h
1h
P1h (k) = Pss
(k) + Psc
(k) + P1c
(k) + P2c
(k)
2h
P (k)
2h
= Pss
(k)
+
2h
Psc
(k)
+
2h
P2c
( k ).
(2.30)
(2.31)
The sums in the above expressions are the result of two-point correlations between the different elements
of a halo, now divided in a smooth component and the clumps that populate it. These correlations are
illustrated for the one-halo term in Figure 2.5, and the subscripts in the above equations correspond to:
l ss: smooth-smooth correlation in between two points of smooth components
l sc: smooth-clump correlation in between a point in a smooth component and another in a clump
l 1c: correlation between two points within one halo clump
l 2c: correlation between points of different clumps.
In the case of the 2-halo term, an identical exercise can be done, leading to consider the same kind of
correlations, though now with points in two different host halos. This means that the two-halo component
of the power spectrum has three instead of four sub-components.
Sheth and Jain (1997) modelled large-scale structure considering the probable occurrence of sub-halos
and concluded that the inner structure of halos should have predominance in the signal that arises from
clustering, especially on the small-scale regime. On large-scales, however, it is expected that the correlation
in between particles from different halos contributes the most to the overall signal.
In the particular case of the work presented in this thesis, the characterisation of structure with hostand sub-halos was tested and decided against. As it will later be shown in Subsection 4.1.3, the inclusion of
Chapter 2. The Halo Model revisited
32
Figure 2.5: Illustration of correlation terms in the extended halo model. Depiction of area points to be correlated are,
for the one-halo term, from left to right: smooth-smooth, smooth-clump, 1-clump (points in the same sub-halo) and
2-clump (points in distinct sub-halos) components.
the substructure in the cross-correlation between dark matter and gas does not substantially add insight to
the signal source.
2.4
Potential uses of the Halo Model
The halo model provides a versatile method to analytically investigate the statistics of signals arising from
virtually all the components of the Universe. Here we list a few of its most important applications while
leaving an open door to others.
n Galaxies
The work by White and Rees (1978) introduces the notion that galaxies are always enclosed
by parent halos and that the properties of the latter are indelibly related to the way galaxies form and
evolve. Following the original purpose of the halo model, the study of the galaxy distribution is mostly
focused on the questions about the origins mechanisms of clustering. Observations of galaxies from the last
decades point to a galaxy-distribution correlation function described by a power law (see, e.g. Connolly
et al. (2002)). Although the slope of the power spectrum from dark matter halos can coincide with such a
curve on small scales, the nature of the differences found on larger scales is an interesting riddle, one that
the halo model might help to decipher. By using the mass functions to describe the distribution of galaxies
within halos, the halo model not only provides a method to calculate the respective auto-correlation power
spectrum. It additionally allows for cross-correlating the galaxy population with the parent halos via the
formalism presented previously in this chapter. With scale-bound characteristics of these spectra, one might
hypothesise how small can the mass of a galaxy be, what is the relation between the galaxy distribution
and its masses or the mean dark matter mass in regions inhabited by galaxies. Most importantly, clues can
be found about the mechanisms that fuel the formation and evolution of galaxies.
n Velocities
The study of large-scale velocities under the framework of the halo model is motivated
substantially by the ability to discern between the linear and non-linear contributions to the signal. The
total velocities (and, more importantly, their dispersions) are typically the sum of a virial component with
an intrinsic halo one, and they depend mainly on the halo masses and overdensities. The halo model
provides a natural way of explaining the shape of the non-linear distribution of velocities: a Gaussian core
Chapter 2. The Halo Model revisited
33
with exponential wings. Looking into the separation in between galaxies and their mean velocities, it is
possible to analyse the aspects of the gravitational interaction among them and the dependence of this
mechanism with scales. Moreover, as the masses and velocities are the main components of this method,
it is intuitive and easy to estimate the momentum and its cross-correlations with overdensities or the
velocity distribution. It is also possible to examine the redshift-space distortions in the non-linear regime of
clustering.
n Weak gravitational lensing
One of the two measurable effects to which the next chapter will be
dedicated is the weak gravitational lensing, tracing the large-scale structure through the slight light
deflection of background sources. Its main observable is the shear, the distortion of lensed galaxies. The
projection of the shear is directly related to the 2-dimensional convergence parameter, the angular power
spectrum of which is easily obtained via the projection of the expressions presented in Eqs. (2.10) and
(2.11). This use of the halo model can shed light upon how the different masses of halos may affect the
power spectrum and helps to evaluate the characteristics of the signal from surveys with given physical
properties. A more detailed description of this method is found in the next chapters. The formalism used
for determining the power spectrum through the halo model can also be used for higher-order statistical
studies. In the weak lensing field, these statistics allow the study of non-Gaussianities and provide an
alternative method to estimate cosmological parameters or to examine the non-linear evolution of the largescale structure. Furthermore, it is possible to cross-correlate galaxy and mass distributions by the two-point
correlation function of the cosmic shear and galaxies as well as through the foreground-background source
correlation. The choice of probe depends on which scale one is interested in studying.
n The Sunyaev-Zel’dovich Effect
The thermal form of the Sunyaev-Zel’dovich effect traces the distribu-
tion of hot, ionised gas within gravitational potential wells. An effective tracer of large-scale structure, it is
a valuable addition to the list of signals which can be evaluated through the halo model. The introduction
of baryons in this context does, however, involve cumbersome models and calculations which are not
always fully analytical. By defining a gas density model and its temperature profile for a given halo,
one can compute the gas pressure power spectrum and project it into the 2D form: the SZ effect power
spectrum. From this scheme, it is possible to analyse and infer the different contributions of different
masses at different scales. In the core of the study presented in this thesis is the cross-correlation signal in
between the gas pressure (evaluated through the SZ effect) and the dark matter distribution (measured
via weak lensing). Further, Tte temperature fluctuations created by the movement of clusters through the
CMB radiation are studied under the designation of kinetic SZ effect. With the definition of a line-of-sight
velocity distribution and matter density it is possible to predict the power spectrum of this phenomenon.
Although this is not an easily observable effect, the theoretical approach enables the study how cluster
velocities relate to density peaks or estimates of a possible cross-correlation with the thermal effect.
n Non-linear Integrated Sachs-Wolfe Effect
The temperature fluctuations due to the integrated Sachs-
Wolfe effect, briefly mentioned in Section 1.3.2, are related to the time derivative of the gravitational
potential while the light is crossing them. The power spectrum of the ISW can be estimated by using density
field and scale factor within the halo model formalism. Conveniently, the power spectrum of the density
field variation is directly related to that of the momentum-density field so that further cross-correlations
34
Chapter 2. The Halo Model revisited
with any of these quantities are quite straightforward. On small scales, the main contributor to this overall
effect is the non-linear component, the Rees-Sciama effect. This result is also directly related to the study of
the velocity dispersion through the halo model framework.
Chapter 3
Probes of the large-scale structure
F
rom all the cosmological observables at our disposal we initially picked those providing complementary information from large-scale structure: the weak gravitational lensing and the Sunyaev-
Zel’dovich (SZ) effect. Both measurements trace galaxy clusters through different components, dark matter
and hot gas, respectively. Through their shared property, we eventually correlated the two signals in the
expectation to gain insight on the way the dark component of halos relates to their baryonic distribution.
We further investigated the connection between the SZ effect with the x-ray emission from the same hot gas
in large-scale structures, attempting to make use of the different dependences of each phenomenon on the
cluster properties.
In this chapter, we review the observables used in this thesis to some detail as well as their description
via the halo model. Each section of this chapter starts with the presentation of the basic features of the
respective observable effect followed by the auto-correlation power spectrum within the formalism used
in this work. Section 3.1 deals with the weak gravitational lensing formalism and Section 3.2 shows the
properties and description of the SZ effect framework.
For more details on the theoretical aspects mentioned in this chapter we refer the reader to the following
reviews: for weak gravitational lensing, Bartelmann and Schneider (2001) or Refregier (2003) and on the SZ
effect, Birkinshaw (1999) or Carlstrom et al. (2002).
3.1
Dark matter and weak lensing
We start our chapter on cosmological phenomena by addressing the measurement that allows the characterisation of the dark matter distribution that forms the large-scale structure. The many historical and
theoretical aspects documented in this section can be found on several reviews on the subject of general
gravitational lensing (Narayan and Bartelmann 1996; Schneider 1996; Blandford and Narayan 1992) or weak
gravitational lensing (Bartelmann and Schneider 2001; Refregier 2003; Mellier 1999).
The deflection of light by the gravitational influence of cosmological objects while following the paths
from source to the observer is a corollary of combining Maxwell’s works and Einstein’s Theory of General
Relativity. The idea that the gravitational potential of an object can modify the trajectory of light was
not new before Einstein (1915) and Einstein (1936) quantified the deflection of light by the sun or wrote
35
Chapter 3. Probes of the large-scale structure
36
lens
plan
e
sour
ce
plan
e
Figure 3.1: Illustration of the gravitational lensing basic mechanism.
down the formalism of the lensing effect of a prototype star, respectively. Not much later, Zwicky (1937a,b)
hypothesised that the deflection of light from background sources by galaxies should be observable and
stated with certainty the possibility of observations of gravitational lensing by nebulae. Several detections
followed these and other publications over the years: gravitational lensing by stars (Dyson et al. 1920;
Alcock et al. 1993), then by increasingly large objects (Walsh et al. 1979) until the first observations of
lensing by galaxy clusters were successfully obtained (Soucail et al. 1987; Lynds and Petrosian 1986).
Proved to be a effective and powerful probe of the Universe on many scales, gravitational lensing has
been used extensively in diverse astrophysical studies. In this section, we introduce the fundamental
framework of gravitational lensing and its most important sub-genres, as well as applications. Most
importantly to our work, we review the basics of weak gravitational lensing and express this cosmological
phenomenon in halo model formalism terms.
3.1.1
Principles of gravitational lensing
The essential concept of gravitational lensing depends on the fact that light travelling throughout the
Universe is subjected to the influence of neighbouring objects and their gravitational fields. The deflection
of light coming from a given background source by a foreground object acting like a lens is described by
basic geometrical considerations. Figure 3.1 shows the basic scheme of the lensing set-up: an observer
and a single source with a deflecting object in between the two, along the line-of-sight, an optical axis
perpendicular to the planes containing the lens and the source. In this simple example where no other
objects are involved, we consider that the position of the source on the source plane is given by:
η=
Ds
ξ − Dds α̂(ξ )
Dd
(3.1)
with Ds , Dd and Dds representing the distances from observer to source plane, from the observer to the
lens (or deflector) plane and between the lens and source planes, respectively. α̂ represents the deflection
angle, and this value depends on the distance in the lens plane between the optical axis and the deflected
light path. To define α̂, one can resort to predictions of General Relativity according to which when gravity
Chapter 3. Probes of the large-scale structure
37
is weak and so the lensing system stays far outside the Schwarzschild radius RS - always the case in
gravitational lensing, even if lenses are galaxy clusters - the deflection angle created by a point mass M
simply becomes:
α̂ =
4 GM
.
ξ c2
(3.2)
As only small angular separations are being evaluated, the field equations of General Relativity can
be linearised and the overall lensing effect arising from several lenses becomes the simple sum of each
contribution. Considering a mass distribution, the resulting deflection angle is obtained through the
following integral over the entire lens plane:
α(ξ ) =
Z
d2 ξ 0
R2
4 G Σ(ξ 0 ) ξ − ξ 0
c2
| ξ − ξ 0 |2
(3.3)
where the quantity Σ(ξ 0 ) is the surface mass density, a projection of the mass. Replacing physical by
angular co-ordinates with η = Ds β and ξ = Dd θ, the source position from Eq. (3.1) is converted to:
α(θ) =
1
Dds
α̂( Dd θ) =
Ds
π
Z
R2
d2 θ 0 κ (θ0 )
θ − θ0
.
|θ − θ0 |2
(3.4)
Here we have defined the dimensionless surface density as:
κ (θ) =
Σ( Dd θ)
Σcr
with
Σcr =
c2
Ds
4πG Dds Dd
(3.5)
being the critical surface density. The value of Σcr quantifies the threshold surface density classifying lenses
as strong or weak, depending on whether κ 1 is valid everywhere (therefore creating weak lenses) or
κ 1 for some angles (originating strong deflectors). One can finally define the lens equation as:
β = θ − α(θ).
(3.6)
The origin of multiple images produced by gravitational lenses is enclosed in this equality: if the solution
of this equation is multiple for the same position in the source plane, multiple images are produced by the
lens.
The definition of the deflection angle α(θ) presented in Eq. (3.4) can be associated the corresponding
deflection potential ψ(θ). With α = ∇ψ:
ψ(θ) =
1
π
Z
R2
d2 θ 0 κ (θ0 ) ln |θ − θ0 | .
(3.7)
This potential is useful as it satisfies the Poisson equation:
∇2 ψ = 2 κ (θ) .
(3.8)
Gravitational lensing not only shifts the light path in the simple way presented above. The deflection of
light is differential and that creates an observed image which deviates from that of the source. The value of
the surface brightness should however remain conserved, as the number of photons is not perturbed by the
deflection and following the reasoning of Liouville’s Hamiltonian theorem. Nevertheless, the incoming
flux of observed light is changed due to its dependence on angular span. The flux of the source image can
Chapter 3. Probes of the large-scale structure
38
be defined as S = µ(θ)S0 , with S0 being the original source flux and µ the so-called magnification of the
image. The magnification is naturally dependent on the variation of the separation β with the angular scale:
µ = |detA(θ)|−1
with
A(θ) =
∂β
.
∂θ
(3.9)
A(θ) effectively characterises the distortions of the observed image of the source and these are described
by the following symmetric Jacobian matrix:
A=
=
∂2 ψ
δij −
∂θi ∂θ j
!
1 − κ − γ1
−γ2
(3.10)
!
−γ2
.
1 − κ + γ1
(3.11)
Here we introduce parameter γ, which depicts the shear of the image:
γ ≡ γ1 + iγ2 = |γ| e2i φ
(3.12)
we the shear components γ1 and γ2 described by:
γ1 =
1
(ψ,11 − ψ,22 )
2
and
γ2 = ψ,12 .
(3.13)
The magnification of the source image will then be given by:
µ=
1
(1 − κ )2 − | γ |2
(3.14)
while the inverse eigenvalues of A provide the information on its shape: if the source original image is
circular, the eigenvalues give the ratio between the axis of an ellipse, which will be the shape of the lensed
image. The lines defined by the values for which the determinant of matrix A is null represent critical
curves which, in the source plane, correspond to so-called caustics. The closer the light gets to the caustics,
the more deformed will be the shape of the image.
As a conclusion, one acknowledges two kinds of distortions of the source radiation by gravitational
lensing: the change in the size of the image and the change in the shape of the image relative to that of the
source. Next, we briefly discuss the usefullness and some of the applications of these effects.
3.1.2
Gravitational lensing phenomenology
The characteristics of the gravitational lensing phenomenon provide the a practical source of information
for several cosmological studies. Lensing is measured through a projected quantity, dependent only on the
luminosity and composition of the lens. This effect is highly dependent on the mass content of the source
and therefore particularly suitable for detecting and studying the distribution of dark matter in the Universe
and growth of collapsed structures. As the observed images get magnified through the deflection, that
makes them easier to access even if they belong to remote regions of the Universe which would otherwise
remain undetectable (see, e.g. Zheng et al. (2012)). Although it is a rather serendipitous effect as one cannot
control where to look through a "lens", it is nonetheless very useful when identified. Moreover, as the effect
depends intrinsically on redshift, distance and age of sources and lenses, these properties provide a way
Chapter 3. Probes of the large-scale structure
39
Figure 3.2: Hubble Telescope Image of Abell 370, example of strong gravitational lensing by a galaxy cluster. Image
credits: NASA, ESA, the Hubble SM4 ERO Team, and ST-ECF.
to study and constrain cosmological parameters, the Hubble constant or the general characteristics of the
source distribution in space and time.
This said, we turn to how the gravitational lensing study field can be sub-divided in three main kinds
of phenomena, each of which depends on the scale where the effect occurs. Here we summarise the most
important facts about each of these observation classes.
n Strong Lensing
As mentioned in the previous section, the occurrence of surface densities equal or
above its critical value create powerful lenses. The observational consequences of this kind of lenses are
the ones for which gravitational lensing is most well know - the arcs. Figure 3.2 shows an example of this
manifestation. The shapes of the observable distortions in the presence of strong gravitational lensing
vary depending on how foreground (lens) and background (source) objects are aligned with respect to
the observer, so depending on the distance to the caustic lines mentioned previously. In the extreme case
when observer, lens and and extended source are all aligned by the optical axis, the giant arcs created by
a sufficiently massive deflector might form a circle known as Einstein ring. For point-like sources (e.g.
quasars), the distortion creates multiple images instead.
Strong manifestations of gravitational lensing are sources of information for a variety of tasks in cosmological studies. The intrinsic dependence of the gravitational lensing phenomenon on the mass of the
deflector object is used to study and quantify its dark-matter distribution (see, e.g. Kochanek and Narayan
(1992)). This is done by modelling the relation between the deflection angle and the lens mass for a given
observation. Another possible use is the determination of the Hubble constant through the gravitational
Chapter 3. Probes of the large-scale structure
40
lensing of variable sources (see Refsdal (1964)): the different lensed images are produced with a time shift
that is proportional to the difference in light path lengths and this is proportional to the inverse of the H0 .
n Microlensing
On the low-mass end of the scale of gravitational lenses are the microlensing events.
These refer to the deflection of light by small objects as stars and are therefore transitional phenomena. The
analysis of the magnification of the image of a galactic object by an inner-galactic dark halo occasionally
positioned in between the source and the observer can provide information on the distribution and content
of mass within the Galaxy (Paczynski 1986). Moreover, in recent years, this technique has proved to be
valuable on the detection of extra-solar planets (Albrow et al. 1995; Wambsganß 1997; Sumi et al. 2011).
When a star that is being observed through a lens is part of a planetary system, it is possible that a planet
happens to position itself between the observer and source. Then, little but measurable changes occur in
light-curve, leading to the detection and further characterisation of the planet.
n Weak Lensing
When a case of strong gravitational occurs, not only evident distortions are taking
place. Smaller distortions of background sources are often observable too, as noticeable in Figure 3.2. These
subtle shape deviations classify the lensing manifestation as weak. Common weak gravitational lensing
studies deal with the set-up of a massive foreground object placed in front of an ensemble of background
sources, from which a series of subtle distortions appears. The inspection of these weakly deformed images
allows a statistical evaluation of the dark-matter distribution, which we will proceed with next.
Furthermore, weak gravitational lensing can be used together with strong lensing to more accurately
study the distribution of dark matter in massive deflectors. Cluster mass reconstruction with information
on these two scales is a very active field at present time (see, e.g. Merten et al. (2009) or Bradač et al. (2006)).
The cross-correlation work presented in this thesis deals with the dark matter distribution and its
relation to the signal from baryonic origin in large-scale structures such as galaxy clusters. The choice of
weak gravitational lensing presented itself as the most promising source of such information. Extensive
reviews with considerably more detail on this subject can be found in e.g. Bartelmann and Schneider (2001)
or Refregier (2003).
3.1.3
The weak lensing power spectrum
The weak gravitational lensing effect is commonly described by the convergence parameter κ. This
parameter is directly related to the measurable shear, reflected in the observable distortions of the galaxies,
as both quantities depend on the second derivatives of the deflection potential (see relations (3.13)):
1
κ (θ) =
2
∂2 ψ(θ)
∂2 ψ(θ)
+
2
∂θ1
∂θ22
!
(3.15)
which in Fourier space are simply related in the following way:
1 2
k1 + k22 ψ̂(k)
2
1 2
γˆ1 (k) = −
k1 − k22 ψ̂(k)
2
γˆ1 (k) = − k1 k2 ψ̂(k) .
κ̂ (k) = −
(3.16)
(3.17)
(3.18)
Chapter 3. Probes of the large-scale structure
41
10-4
W (χ)
10-5
10-6
10-7
101
102
104
103
χ [ Mpc h ]
−1
Figure 3.3: Geometrical weight function of the convergence parameter.
After the shear is observed, Eqs. (3.17) and (3.18) are solved for the potential. The two solutions are
combined by a minimal-variance estimation. Using relation (3.16), the convergence can then be expressed
in terms of a convolution of the shear:
1
π
κ (θ) =
Z
d2 θ 0 Re D ∗ (θ − θ0 )γ(θ0 )
(3.19)
where D is the complex convolution kernel given by:
D (θ) =
(θ22 − θ12 ) 2iθ1 θ2
θ4
(3.20)
and the shear is described as a complex variable as defined in 3.12.
To easily describe the convergence, we can alternatively return to Eq.(3.8) which stipulates a direct
relation between κ and the gravitational potential. The potential and the density contrast δ are related by
the Poisson equation (1.37). The convergence becomes:
κ=
3Ωm H0
2c2
Z
χ − χs χ
δ=
χs a
dχ
Z
dχ Wκ (χ) δ.
(3.21)
The distortion level depends of the Hubble constant H0 , the matter density Ωm and the speed of light c.
The integral along co-moving distances is performed over the considered matter distribution through a
dimensionless source field, the density contrast δ:
δ=
ρ − ρ̄
ρ̄
(3.22)
and weighted geometrically by the relation:
Wκ (χ) =
3Ωm H0 χ − χs χ
χs a
2c2
(3.23)
Chapter 3. Probes of the large-scale structure
42
104
103
∆2δδ
102
101
100
10-1
10-2 -2
10
10-1
100
k [ Mpc−1 h ]
101
102
Figure 3.4: Three-dimensional power spectrum of density contrast at redshift 0. Solid line shows the total power
spectrum as the result of adding the contributions from the 1-halo term (dashed line) and the 2-halo term (dotted line).
where χs is the co-moving distance to the source and a is the scale factor and behaves as shown in Figure
3.3. This geometrically weighs the integral along the line-of-sight by averaging over the source distances.
The weight function peaks at the scale of highest lensing efficiency.
The parameterisation of the convergence in the second step of Eq. (3.21) with a weight function and a
source field is done with the intention of applying the same method for different signals in a consistent
and practical way. This criterion will be used in the next section as well.
Applying the halo model formalism to the weak gravitational lensing phenomenon is rather straight
forward. Eqs. (2.10) and (2.11) can be directly used to calculate the power spectrum of the field that is in
the origin of the shear/convergence:
1h
2h
Pδδ (k, z) = Pδδ
(k, z) + Pδδ
(k, z).
(3.24)
The models chosen to build the dark-matter power spectrum within the halo framework were: the Sheth &
Tormen mass function for n(m), the NFW halo profile for normalised Fourier transform density u(k, m)
and the Mo & White model for the bias parameter b(m). The linearly evolved primordial power spectrum
is computed according to the calculations by Bardeen et al. (1986).
The input of these ingredients in the above-mentioned equations yields the 3-dimensional power
spectrum of the density contrast shown in Figure 3.4. The power is presented as a dimensionless quantity,
the power spectrum per logarithmic interval in wavenumber, ∆ = k3 P(k)/2π 2 . The curves for redshift
0 indicate that both halo terms show the same level of contribution to the total spectrum on relatively
large scales, k ∼ 10−1 , below which single halos clearly dominate the source of power revealing its strong
dependence on the density profiles chosen. On the other hand, for very large scales, the Poisson term
is slightly dominated by halo-halo correlation, mostly influenced by the bias parameter. This result for
redshift 0 is in good agreement with those of Cooray (2000) and Cooray and Sheth (2002), where an identical
method was used. The total curve amplitudes and the scale dependence of the 1- and 2-halo terms are
Chapter 3. Probes of the large-scale structure
43
10-3
`(` +1)C (`) / 2π
10-4
10-5
10-6
10-7
10-8
101
102
103
`
104
105
106
Figure 3.5: Convergence power spectrum, integrated from redshift 0 to 1. Scheme follows that from Figure 3.4.
reasonably similar. Amplitude disparities on the small scale level are related to the implementation of a
different halo mass-concentration relation in those works.
The 3-dimensional power spectrum is a valuable result when it comes to testing theory against
simulations. The projection of the power spectrum into an angular quantity allows the comparison
between theoretical predictions and observations. We apply Limber’s approximation (see Appendix) to
compute the convergence power spectrum from the density contrast 3-dimensional result. Using the flat-sky
approximation and setting the wavenumber as k = `/χ, where ` is the Fourier position in the sky, we get:
Cκκ (`) =
Z χs
dχ
0
χ2
Wκ2 (χ) Pδδ (`/χ, χ)
(3.25)
Figure 3.5 shows the convergence power spectrum integrated from redshift 0 to 1 (meaning that the
maximum value of the co-moving distance corresponds to putting the source at χs = χ(z = 0). The shape
of the spectrum is in good agreement with the many studies which can be found in the literature. As the
projection is mostly a geometrical weighting of the 3-dimensional counterpart, there are no intense changes
in the contribution level of each halo term from the dark-matter power spectrum. The decrease of power on
smaller scales reflects the smaller structures (∼ below 1 arcmin) do not contribute as significantly to the
overall lensing signal.
3.2
Baryonic physics and the Sunyaev-Zel’dovich effect
The journey of cosmic microwave background (CMB) photons through the Universe can be perturbed
in several ways. Gravitational potential fluctuations created by the presence of the web-like large scale
structure, composed by dark-matter halos connected by filaments, host most of the Universe’s baryonic
content. The high-temperature, ionised gas present in these dense regions is responsible for the scattering
Chapter 3. Probes of the large-scale structure
44
of the CMB photons through the inverse Compton process, causing a measurable distortion of the CMB’s
black-body spectrum. This effect, first predicted by Sunyaev and Zel’dovich (1970) and widely observed
ever since, forms a powerful probe for the presence of hot baryonic matter.
First observed in the early 1980s, the strongest of the secondary sources of CMB anisotropies has become
a reliable and fertile method to study the baryonic content of the Universe and other cosmological subjects
of interest. Besides tracing the hot gas in large-scale structures, the unique characteristics of the SZ effect
make it suitable to constrain cosmological models, estimate distances or the Hubble constant, all this while
not intrinsically depending on the redshift of the sources.
In this section we address the method we use to evaluate the gas component of halos and how it may
be useful to constrain cosmological parameters or the evolution of temperature in galaxy clusters. We first
introduce the basics of the scattering of photons by a non-relativistic population of electrons and briefly
characterise the sub-types of the SZ effect. The ending part of this chapter describes the thermal SZ effect
within the halo model framework. For further detail on the SZ effect, the reader is referred to reviews in
the literature such as Rephaeli (1995), Birkinshaw (1999) or Carlstrom et al. (2002).
3.2.1
Inverse Compton scattering
The early-20th-century works of Compton on the scattering of X-rays (Compton 1923a,b), following the
discovery of the photo-electric effect by Einstein a few years before, describe the simple but fundamental
way of how matter and radiation interact. His results showed how a distribution of hot, ionised gas
exchanges energy when in contact with photons.
The left panel of Figure 3.6 shows the process known as Compton scattering, an inelastic interaction
between a stationary electron and a low-energy photon. The electron receives energy from the photon,
whose energy is reduced. The conservation of energy-momentum allows the calculation of the energy shift
in the photons when scattered by an electron:
ε0 =
1+
ε
me c2
ε
1
( − cos φ12 )
(3.26)
with φ12 being the angle difference between the incoming and outgoing photon trajectories. ε and ε0 are the
initial and final energies of the photon, respectively.
Alternatively, one may consider the encounter of a low-energy photon with a high-energy electron.
What results from that interaction, under the same assumption of energy-momentum conservation, is
the increase of frequency of the photon resulting from an energy transfer by the electron, which loses
momentum (right panel of Figure 3.6). This process describes what happens to the CMB photons when
they come across distributions of high-energy electrons of the hot, ionised gas trapped in the potential
wells of the large-scale structure as filaments and galaxy clusters. Usual analyses include estimating the
scattering probability. It is also possible to extend this description to the scattering of a photon spectrum by
a given gas distribution which can in turn be used to estimate the intensity of the scattered radiation.
In this work, we present the results of the spectral changes in an electron distribution where no
relativistic arguments are considered. In most cases of inverse Compton scattering, to consider a nonrelativistic population of electrons is enough to accurately describe the processes involved in the origin of
the SZ effect. In most galaxy clusters, the energies do not rise much above the few keV. Energies of about
Chapter 3. Probes of the large-scale structure
45
Figure 3.6: On the left: illustration of the Compton scattering. A photon interacts with a low-energy electron at rest.
The electron gains the energy lost by the photon (ν0 < ν). On the right: the SZ effect basic mechanism, the inverse
Compton scattering. In this case, a low energy photon encounters a high-energy electron and the exchange increases
the frequency of the photon (ν0 > ν) while decreasing the energy of the electron.
an order of magnitude higher are necessary for relativistic corrections to become pertinent. For details on
this topic see, e.g., conclusion of Section 3.4 in the review by Birkinshaw (1999).
For our purposes, we consider that taking the non-relativistic approach (that the electron energy
h̄ν me c2 ) is sufficient and adequate. The short description of the photon spectral dependence lightly
follows that presented in, e.g., Weinberg (2008). Consider a gas distribution with a certain photon occupation
number N. The Kompaneets equation (Kompaneets 1956, 1957) states that, when Compton scattering
occurs, N varies at the following rate:
∂N
ne σT ∂
=
∂t
me c ν2 ∂ν
∂N
kB Te ν
+ h̄ ν N (1 + N )
∂ν
4
(3.27)
where ν is the photon frequency, σT is the Thompson cross-section, me c2 corresponds to the electron rest
mass, h̄ is the reduced Planck constant, ne is the electron gas density and Te its temperature. Considering
that N is a function of frequency ν and line-of-sight depth `, and the density and temperature are
additionally only dependent of `, the above relation can be re-written simply as:
ne (`) Te (`) kB σT ∂
∂N
4 ∂N
=
ν
∂`
∂ν
∂ν
me c2 ν2
Re-arranging this equality, we get an expression for the variation of N:
y ∂
∂N
∆N (ν) = 2
ν4
∂ν
ν ∂ν
(3.28)
(3.29)
which depends on the so-called Comptonization parameter here introduced:
y≡
σT kB
me c2
Z
d` ne (`) kB (`) Te (`)
For a black-body radiation with a temperature T̄, the photon occupation number is given by:
−1
N = exp−1 (h̄ν/kB T̄ )
and its variation can be re-written to yield the spectral shape of the electron population:
!
− x + ( x2 /4) cot( x/2)
∆N = y
sinh2 ( x/2)
(3.30)
(3.31)
(3.32)
Chapter 3. Probes of the large-scale structure
46
Figure 3.7: The initial CMB spectrum (dashed line) is shifted towards higher energies due to the thermal SZ effect. The
result is a decrease of intensity on lower frequencies (the Rayleigh-Jeans regime) and an increase of intensity at higher
frequencies (Wien regime). Here, an exaggerated case is shown with the considered cluster being approximately 1000
times more massive than a typical such object. Plot reprinted from Carlstrom et al. (2002).
with x ≡ h̄ν/kB T̄ as the dimensionless frequency/energy. This particular spectral shape makes it easy to
identify the SZ effect anisotropy and distinguish it from the underlying CMB radiation. In the limit of very
small energies (x 1), the Rayleigh-Jeans regime:
∆N →
−2 y
x
and
N →
1
x
(3.33)
meaning a decrement in the low-energy part of the spectrum. Alternatively, in the Wien regime (x 1),
the energy is increased. The thermal-SZ effect creates a lateral shift towards higher energies and there is a
transitional point at which the initial and final spectra show the same intensity (see Figure 3.7.) Another
practical property becomes evident from the above relations. The shape of the black-body spectrum is
preserved:
∆ T̄
∆N
= −2y .
=
N
T̄
(3.34)
and so, reliably easy to identify and select from the overall CMB radiation.
Although this is the phenomenon we are interested in for our study, the very robust SZ effect is not
confined to thermal events. Next, we give a brief overview of the different types of the Sunyaev-Zel’dovich
effects and how they are practical resources to probe a whole range of different cosmological fields.
Chapter 3. Probes of the large-scale structure
47
Figure 3.8: Simulated all-sky map of the thermal SZ effect for the Planck mission showing a cluster distribution as well
as tracing some filamentary structures. The colour-scale is proportional to arcsinh(106 × y). Image from Schäfer et al.
(2006).
3.2.2
The Sunyaev–Zel’dovich effects and applications
The Sunyaev-Zel’dovich effect – the scattering of CMB photons by the hot ionised gas of large-scale
structure – provides a useful and effective method to trace the distribution of baryonic matter in the
Universe. This property and the fact that the Comptonisation parameter is a projected measurement (and
therefore relatable to the weak gravitational lensing observable) makes it a desirable tool for our study.
Measurements of these events have been performed in the radio, microwave and X-ray wavelengths.
Many experiments have been conducted until present day (Wilkinson Microwave Anisotropy Telescope
(WMAP), Planck Mission, South Pole Telescope (SPT), Atacama Cosmology Telescope (ACT)), given the
robustness and maturity of the method, as well as the increasing technology advances in the detection field.
For an extensive review of the employed techniques, we once again refer the reader to the descriptions in
Birkinshaw (1999) or Carlstrom et al. (2002).
Here we assess the scope of phenomenology associated with the photon scattering by hot cluster gas
and the distinct applications of these sub-effects to the study of baryons.
n The thermal SZ effect
After describing the SZ effect formalism from the Kompaneets equation, we have all the tools to
characterise the thermal SZ effect: signal originated by thermal interaction of large reservoirs of ionised
gas and the crossing of CMB photons. Figure 3.8 shows a simulated all-sky map by Schäfer et al. (2006)
of the kind of signal distribution that should be observable with the Planck Surveyor (the actual data
results will be published in early 2013). The signal is dominated by the most massive clusters in the sky.
The paradigmatic characteristic of this effect is that the observable, the y-Compton parameter, shows no
direct dependence on the redshift, making it a particularly suitable tool to study the redshift evolution of
structures. The claim of redshift independence does however not take into account how the measurement
techniques in spatially unresolved observations – beam-smoothing effects – do indeed depend on the
redshift. Further, the thermal SZ effect is a useful source of information on the number density of structures.
Chapter 3. Probes of the large-scale structure
48
Although this can be achieved with X-ray emission, the near independence of the SZ effect on redshift
makes it less vulnerable to the diminution of signal because of the distance of clusters. Because the hot
electrons in the ionised gas produce X-ray radiation and this measurable signal depends proportional
on the square of the electron density, one can combine it with the intensity of the SZ-signal. This allows
the elimination of the electron density term and the consequent description of a distance relation which
provides a method to estimate the Hubble constant H0 . This parameter is found to depend mostly on the
CMB temperature shift and the X-ray determined cluster temperature. The investigation of the intra-cluster
medium properties is improved by the dependence of the thermal SZ effect on the electron temperature
and density of clusters. Moreover, the very amount of baryonic matter within the dark matter halos that
enclose galaxy clusters can be estimated via this effect.
n The kinetic SZ effect
The motion of the high-temperature and ionised gas in relation to the CMB
radiation also produces distortions in its spectrum and such features are different than those created by
the thermal SZ effect. The shift occurs in the same fashion as a Doppler-shift and it is evaluated through
the optical depth of the gas distribution, which depends on the Thomson cross-section σT and electron
density ne and has a spectral signature different than that accompanying the Compton parameter. The
overall temperature shift depends primarily on the optical depth and the peculiar velocity of the considered
object, while it does not depend on the frequency. The kinetic SZ effect was first established in literature
by Sunyaev and Zel’dovich (1972); Sunyaev and Zeldovich (1980) and causes very subtle and difficult to
detect changes in the spectrum. It was only this year that the attempts to trace the kinetic SZ effect started
to be successful (Hand et al. 2012). The estimation of the peculiar velocities of clusters can, for example,
be supplemented by observations of gravitational lensing, which provides estimates on the value of other
velocity components, given insight on the large-scale structure dynamics.
n Polarization
The SZ effect signal can additionally be show polarisation since the CMB irradiating from
clusters is partially polarised. Also first referred to in Sunyaev and Zeldovich (1980), this effect depends
strongly on cluster optical depths and may be useful to estimate its value. The polarisation of the SZ effect
is responsible for a very faint and difficult to detect signal and requires the super-positioning of many
clusters along the line-of-sight in order to expect a measurable signal.
3.2.3
The thermal SZ effect power spectrum
The thermal SZ effect is most commonly described by the Compton-y parameter as shown earlier in this
section. We here reproduce and re-arrange the terms in the following line-of-sight integral:
y = g( x )
Z
dχ a
kB σT
Te ne =
me c2
Z
dχ Wy (χ) ζ
(3.35)
The function g( x ) contains the dependence of the effect on the dimensionless frequency x = hν/kB T where
ν is the frequency at which temperature T is measured, h and k B are the Planck and Boltzmann constants,
respectively. In our study we consider g( x ) to be −2, corresponding to the limit of the Rayleigh-Jeans
regime where hν kB T. Although we make this assumption, our implementation is not bound to a
particular part of the CMB spectrum and can be used in a general way. Given the fact that the Compton
parameter is not more than the projection of the gas density times the temperature, one can define a
Chapter 3. Probes of the large-scale structure
49
dimensionless source field as the relative perturbation of the pressure:
ζ=
ne Te
n̄e T̄e
(3.36)
with the barred parameters corresponding to the mean electron density of the Universe n̄e and a mean
value for the temperature of an ionised gas T̄e . We assume kB T̄e = 1 keV and define the mean electron
density of the Universe:
n̄e =
f b ρcritical
.
µ mp
(3.37)
Here f b is the baryon fraction, ρcritical is the cosmological critical density, µ gives the effective mass of a
particle releasing one electron and mp is the proton rest-mass.
We define the remaining non-fluctuating terms as a weight function along the line-of-sight:
Wy (χ) = 2 a(χ)
kB σT
n̄e T̄e .
me c2
(3.38)
At this point, we have defined two quantities in a formally similar way. Both the convergence and the
Compton-y parameter describe essentially the projected values of the two complementary fields we wish
to study. In both cases, the defined source terms and weight functions have matching dimensions and
therefore play similar parts in the evaluation of both weak lensing and the SZ effect. In the next sections,
we show how they can be studied jointly and provide us with knowledge on the correlation level between
dark and baryonic matter.
The halo model framework can be used to estimate the power spectrum of the gas-pressure density
field ζ . Eqs. (2.10) and (2.11) are here reproduced with the few changes required to assess the baryon
distribution instead of the dark-matter component of a halo:
1h
Pζζ
(k, z) =
Z
dm
2h
Pζζ
(k, z) = Plin
m
ρ̄
2
n(m, z) u2gas (k, m) t(m, z)
m1
n(m1 , z) ugas (k, m1 ) t(m1 , z) b(m1 )
ρ̄
Z
m2
n(m2 , z) ugas (k, m2 ) t(m2 , z) b(m2 ).
dm2
ρ̄
Z
dm1
(3.39)
(3.40)
New in this description is the normalised Fourier transform of the baryon distribution ugas and a normalised
mass-temperature relation.
To model the distribution of baryonic matter in the halos, we use the β-profile to define the density
distribution of electrons ne . This simple model is described by:
ne (r, m) = ne0
1 + (r/rc )2
3β/2
(3.41)
where ne0 describing the normalisation of the density and rc being the core radius. Throughout this study,
we always use β = 1.
To calculate the normalised Fourier transform of the electron density, we use Eq. (2.20) from the previous
chapter. For this value of β, the Fourier transform of ne does not have an analytical solution, hence we
Chapter 3. Probes of the large-scale structure
50
101
1010 M ¯
1015 M ¯
ugas(k,m)
100
10-1
10-2
10-3 -2
10
10-1
101
100
k [Mpc−1 h ]
102
Figure 3.9: Normalised beta profile and its dependence on the halo mass at redshift zero. Value of β is 1.
compute it numerically. Setting R := r/rc :
Z
d3 x ne (x, m) e ik.x =
4 π ne0
k
Z Rvir
0
dR R sin(kRrc )
1 + R2
−3/2
.
(3.42)
On the other hand, the normalisation of the transform (which corresponds to counting the total number of
electrons within the halo) can be analytically determined:
Z
d3 x ne (x, m) = 4 π rc3
Z Rvir
0

dR R2
1 + R2
−3/2
(3.43)

q
R
= 4 π rc3 ln Rvir + R2vir + 1 − q vir 
R2vir + 1
(3.44)
To ensure that the density profile falls to zero at some point, as to avoid computational problems, we force
the distribution to have a smooth decrease after a sufficiently large radius. For that purpose, we use the
SPH smoothing kernel used in the GADGET-2 code by Springel (2005). The normalised gas density profile
is shown in Figure 3.9 for different mass values.
In order to calculate the temperature profile of the gas distribution, we use a simple power-law model
for the temperature. For each halo mass, the temperature starts at a minimum temperature plateau and
increases with mass and redshift:
t(m, z) =
tmin
+ (1 + z )
t0
m
m0
2/3
.
(3.45)
We use as normalisation temperature t0 = 1.16 × 107 K as it is a typical value to ensure ionisation and
as minimum temperature tmin = 1.46 × 107 K, corresponding to the temperature of a cluster of mass
m0 = 1014 M . The remaining ingredients are used as stated for the computation of the power spectrum of
the dark matter distribution.
Chapter 3. Probes of the large-scale structure
51
Figure 3.10: Mass-redshift distribution of detectable clusters with Planck. Contours depict the number density of
observable clusters and are logarithmically spaced by 0.2 decimal points; the lowest contour corresponds to a number
density of 10−10.5 M −1 . The dotted contours do not take into account beam convolution nor noise from background
fluctuations. In this work, we constrain our mass-redshift integration to the limits inferred from this result. Plot
reprinted from Bartelmann (2001).
Additionally, the mass integration is performed in accordance to the mass-redshift cut-offs corresponding
to observable-only halos, as described in Bartelmann (2001) and shown in Figure 3.10. This means that for
a given redshift, only the potentially measurable halos in a given mass range are taken into account. The
effect of a beam is the smoothing of the overall SZ signal, wiping out the low signal peaks. The smoothing
is strongest at the low mass and low redshift regime. The beam smoothing will decrease beyond the
redshift ∼ 1.25, where the angular-diametre distance peaks and all clusters massive enough to be seen at
that redshift can be seen throughout the Universe.
Figure 3.11 shows the resulting total power and the contributions from single halos and inter-halo
correlations, at redshift 0. The contribution from single halos is the clear dominating source of power
on scales smaller than k ∼ 0.2 while in the large-scale regime, most of the signal comes from halo-halo
correlations. The drop of power at the small end of the scales occurs as after a certain size, no matter how
massive the halo is, it will become too small to be resolved – as the result of the mass cut explained above –
and thus will not contribute to the signal.
With the computed 3-dimensional power spectrum of the baryon distribution, it is easy to compute its
projection, the Sunyaev-Zel’dovich power spectrum, again using Limber’s and the flat-sky approximations
as in Section 3.1.3:
Cyy (`) =
Z
dχ 2
W (χ) Pζζ (`/χ, χ)
χ2 y
(3.46)
The tendency observed at large-scales of the 3-dimensional power spectrum is not present after the
projection. As shown in Figure 3.12, where the power spectrum is integrated from 0 to χ(z = 1) the
single-halo contribution becomes the most important source of signal to the total curve on all scales. As
Chapter 3. Probes of the large-scale structure
52
104
103
∆2ζζ
102
101
100
10-1
10-2 -2
10
10-1
101
100
k [ Mpc−1 h ]
102
Figure 3.11: Three-dimensional power spectrum of gas pressure distribution at redshift 0. Solid line depicts the total
power, dashed and dotted lines correspond to the 1- and 2-halo contributions.
10-10
`(` +1)Cyy(`) / 2π
10-11
10-12
10-13
10-14
10-15
10-16
101
102
103
`
104
105
106
Figure 3.12: Sunyaev-Zel’dovich effect (angular) power spectrum, integrated from redshift 0 to 1. The line-style scheme
is the same as in Figure 3.11.
Chapter 3. Probes of the large-scale structure
53
one integrates over large volumes of the Universe, the halo-halo correlation becomes less relevant. The
reason why this effect doesn’t reflect in the weak gravitational lensing power spectrum is related to how
the the geometrical weight functions of Limber’s approximation shape the signal. A narrower peak of the
lensing weight function case infers a larger contribution of halo-halo correlations.
54
Chapter 3. Probes of the large-scale structure
Chapter 4
The cross-correlation
I
n the previous chapters we presented the background upon which we define the Universe and how
the signal from cosmological sources it can be analytically characterised through the halo model. We
furthermore revealed the phenomena we wish to correlate in the hope that the result will shed light
on the properties and history of the large-scale structure. Now that our method is presented and all
main ingredients are introduced, we are ready to estimate the cross-correlation signal between the weak
gravitational lensing and the thermal Sunyaev-Zel’dovich effect.
Section 4.1 recovers the most important aspects of the process of calculating the cross-correlation power
spectrum, recalls the assumptions on models and parameters which are taken to compute it and shows the
results for both the 3-dimensional power spectrum and its projection. With the latter quantity, we define
and determine the correlation function in Section 4.2, from which follows the detailed calculation of the
covariance matrix of the cross-correlation power spectrum in Section 4.3.
4.1
The cross-power spectra
The first goal of this work was to successfully specialise the halo model to compute the cross-correlation of
interest. Here we show the cross-correlation power spectra between the weak gravitational lensing signal
tracing the dark matter distribution in halos and their gas content originating the thermal SZ effect. In
the present section, we display the derived expressions and the outcome obtained for the 3-dimensional
and angular power spectra by specialising the halo model formalism from Chapter 2 via the results from
Chapter 3. Our results should, to some extent, match those of Cooray (2000) or Cooray and Sheth (2002).
Differences are expected to arise as we use a different parameterisation of the gas halo density and the
normalisation methods are not the same as implemented here. Also, the arbitrary way we calibrate the
temperature-mass relation might be a source of disagreement.
4.1.1
3-dimensional spectrum
The auto-correlation spectra of dark-matter and baryon fields are computed directly from Eqs. (2.10) and
(2.11) by assigning specific density distributions. uDM (k, m) and u gas (k, m) with mass-temperature relation
t(m, z) respectively set the normalised Fourier-transformed dark matter and electron gas density profiles
55
Chapter 4. The cross-correlation
56
with a normalised temperature function. The cross-correlation between the dark-matter density fluctuations
that create weak gravitational lensing, and the gas density that ultimately gives rise to the thermal SZ effect
can be theoretically evaluated by use of a specialised version of the same set of equations, as presented in
the following power spectra:
1h
Pδζ
(k, z)
=
2h
Pδζ
(k, z) =
Z
dm
Plin
2
m
ρ̄
2
n(m, z)uDM (k, m)u gas (k, m)t(m, z)
m1
n(m1 , z) uDM (k, m1 ) u gas (k, m2 ) t(m2 , z) b(m1 )
ρ̄
Z
m2
n(m2 , z) uDM (k, m2 ) ugas (k, m1 ) t(m1 , z) b(m2 )
dm2
ρ̄
Z
(4.1)
dm1
(4.2)
with all variables as defined in previous chapters. In our experiment with the halo formalism, we consider
a set of assumptions regarding the halo-model ingredients. Here we briefly summarise the models already
presented in Chapters 2 and 3. Throughout this work, we assume:
l n(m, z): for the mass function we use the Sheth-Tormen model (Sheth and Tormen 1999), which
takes into account elliptical collapse of structures and very well fits numerical simulation results;
l uDM : we choose the NFW (Navarro et al. 1997) profile to characterise the dark matter density
distribution. This model is empirically motivated and the computation of the normalised Fourier
transform, where the integration is done up to the virial radius, is fully analytical;
l ugas : we take the conventional isothermal β profile to describe baryonic densities. The Fourier
transform requires an integration up to the virial radius. The result is not fully analytical and a
smooth decrease to zero is added to the gas density, borrowing the properties of the smoothing kernel
of the GADGET-2 code (Springel 2005). We adopt a β = 1;
l t(m, z): the gas temperature is taken to depend on the halo mass as a simple power law of the kind
t(m, z) ∝ m2/3 (z + 1) with an arbitrary initial plateau of the order of 107 K to ensure gas ionisation;
l b(m, z): the biasing level is computed as in Mo and White (1996);
l Plin : the linear power spectrum is obtained by evolving the initial perturbations with the transfer
function of Bardeen et al. (1986);
l Cosmological parameters: we use the standard ΛCDM model with the seven-year WMAP results
(Komatsu et al. 2011), where parameters are estimated through the combination of CMB and BAO
data as well as external measurements of the Hubble parameter. Most importantly: matter density
Ωm = 0.227, dark energy density ΩΛ = 0.728, baryon density Ωb = 0.0456, Hubble constant H0 = 70.4
kms−1 Mpc−1 and fluctuation amplitude σ8 = 0.809.
l Mass integration: while performing the mass integration in the calculation of the 3-dimensional
power spectrum, we restrict the mass-redshift plane so that only halos that are observable via both
weak gravitational lensing and Sunyaev-Zel’dovich effect are taken into account. This follows the
work by Bartelmann (2001).
Chapter 4. The cross-correlation
57
104
103
∆2
102
101
100
10-1
10-2 -2
10
10-1
100
k [Mpc−1 h]
101
102
Figure 4.1: Three-dimensional power spectrum of the density contrast (in red) and gas pressure (in black) distributions
at redshift 0. Dashed lines correspond to 1-halo terms and dotted lines depict the signal from halo-halo correlations.
104
103
∆2δζ
102
101
100
10-1
10-2 -2
10
10-1
100
k [ Mpc−1 h ]
101
102
Figure 4.2: Three-dimensional power spectrum of the cross-correlation between dark matter density contrast and gas
pressure, at redshift 0. Solid line shows the total power as the sum of the 1-halo term (dashed curve) and 2-halo term
(dotted curve).
Chapter 4. The cross-correlation
58
Figures 4.1 and 4.2 show the three-dimensional auto- and cross-power spectra of dark matter and hot
gas at redshift zero, respectively, for the total signal and its one- and two-halo terms contributions.
We re-plot the auto-correlation curves to more closely compare the different features of the dark and
baryonic signals, which is relevant to better understand the cross-correlation behaviour. For all signals, the
single halo contribution is dominant for the most part of the scale range and the clustering term is mainly
relevant at the larger scales. The strong dependence on the 1-halo term as small angular scales is related
to how influential is the choice of density model whereas the dominance of the halo-halo correlations in
the large scales point to the importance of having an adequate bias model. Only at scales of the order of
k ∼ 10−1 do both halo terms show the same amount of contribution to the signal.
The cross-correlation spectra are more similarly shaped to that of the gas pressure than the dark matter
signal, which makes the former seem to bear the strongest features when compared to the latter.
Our results for the 3-dimensional power spectra of auto- and cross-correlation are in general good
agreement with those of Cooray (2000) and Cooray and Sheth (2002), where an identical method was used.
Minor differences are likely the result from the different choices of inputs other than those picked for the
present study, i.e. mainly the model for gas-density profile and the cosmological parameters. Overall, the
total curve amplitudes and the scale dependence of the 1- and 2-halo terms are reasonably similar.
The power spectrum obtained in this way provides a valuable tool for the comparison of a theoretical
result with that of numerical simulations. The next section will yield the projection of this spectrum, which
in turn gives a signal directly comparable to observations.
4.1.2
Angular spectrum
The 3-dimensional description of the signal is directly comparable to results from cosmological simulations.
In order to be able to directly test theoretical predictions of the signal with actual observational data it is
required that those predictions are projected onto the sky.
To compute the projected two-point correlation function, in Fourier space, we again use Limber’s
approximation (Limber 1954), according to which the statistics of a given projected quantity can be obtained
by integrating over the statistical description of its three-dimensional counterpart. We once more take the
flat-sky approximation and set the wavenumber as k = `/χ, with ` as the Fourier angular position.
Thus, given the power spectrum of the three-dimensional field calculated in the previous section
and the definitions of the two effects we aim to cross-correlate – see Eqs. (3.21) and (3.35) – it is rather
straightforward to project our results into the angular power spectrum based on the above-mentioned
assumption:
dχ 2
W (χ) Pδδ (`/χ, χ)
χ2 κ
Z
dχ
Cκy (`) =
Wκ (χ) Wy (χ) Pδζ (`/χ, χ) .
χ2
Cκκ (`) =
Z
(4.3)
(4.4)
These are line-on-sight integrals with all the parameters as previously defined. We choose to extend our
integration along co-moving distances up to a source at χs = χ(z = 1) because the lensing efficiency peaks
at a lower redshift, after which structures become too small to contribute to the overall signal.
Figures 4.3 and 4.4 show the auto- and cross-correlation angular power spectra of/between the weak
lensing convergence and the thermal SZ effect, respectively, with total signal (solid line) and halo terms.
`(` +1)C(`) / 2π
Chapter 4. The cross-correlation
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
10-16
59
101
102
103
`
104
105
106
Figure 4.3: Angular power spectra of weak lensing convergence and the SZ effect integrated up to redshift 1. Colour
and line-style schemes follow those of Figure 4.1.
10-5
10-6
10-7
`(` +1)C y(`) / 2π
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
101
102
103
`
104
105
106
Figure 4.4: Cross-correlation (weak gravitational lensing and the SZ effect) angular power spectrum, integrated from
redshift 0 to 1. Line-style is the same as in Figure 4.2.
Chapter 4. The cross-correlation
60
The auto-correlation spectra are here reproduced once again for the same reasons specified in the previous
section: it is easier to compare the amplitudes and shapes of both spectra and infer which signal and scale
has the most influence the final cross-correlation curves.
Regarding the cross-power, we observe a minimal contribution of the cross-correlation two-halo term
(dotted line) to the total power at low multipole orders, to a total signal otherwise strikingly dominated by
the contributions from individual halos (dashed line). After a peak in the cross-correlation at a multipole
order near ` ∼ 104 , the value decreases steeply towards smaller scales mainly due to the most massive, and
therefore rare, halos, as dictated by the mass function.
In the remaining sections we continue with the statistical evaluation of the angular power spectrum by
investigating the limits of correlation and detection for a given survey.
4.1.3
Substructure in the cross-correlation between dark matter and gas
We investigated the effect of introducing substructure into the halo description. However, early results did
not seem to add any useful or strikingly different information to that already obtained with the original halo
model. Accordingly, we continue our study without a detailed characterisation of the matter distribution
interior to the halo.
4.2
Correlation function of the power spectrum
We start our statistical analysis of the cross-correlation signal by leaving the Fourier formalism and returning
to the real space. The second-order measurements are computed by assuming that the signal is measured
by averaging the convergence and the Compton parameter over circular apertures of radius θ. Either signal
is assumed to be convolved with a top-hat beam. This means we are filtering the signal with a quite broad
window function, allowing us to probe the correlation level in a wide range of angles and, most importantly,
to potentially measure a substantial amount of signal.
In this effort, we follow Schneider et al. (2002) and references therein, using the formalism applied to
the computation of the shear dispersion measurements to define the correlation function with an integrand
which depends linearly on the angular cross-correlation power spectrum calculated in the previous section:
ξ (θ ) =
Z ∞
d` ` 4 J12 (`θ )
Cκy (`)
2
0
2π
(`θ )
(4.5)
where J1 is the Bessel function of the first kind and order 1. It arises from the convolution of the power
spectrum with a top-hat filter. Although choosing different convolution profiles would return different
functions, the results wouldn’t change significantly. This calculation yields Fig. 4.5 where the correlation
function of the total weak lensing/SZ cross-correlation power spectrum integrated from redshift 0 to 1 is
shown.
As the correlation function is just the conversion of the Fourier space angular power spectrum quantity
into the real space, the overall behaviour of the signal is not profoundly changed: the cross-correlation level
is higher at smaller scales and lower for large angles.
We follow the definition of the correlation function with the calculation of the covariance matrix of the
power spectrum which will provide the errors of our estimations.
Chapter 4. The cross-correlation
61
10-6
10-7
ξ
10-8
10-9
10-10
10-11
10-12
10-13
101
100
θ [arcmin]
102
Figure 4.5: Cross-correlation function between weak lensing and the SZ effect.
4.3
Covariance matrix of the correlation function and correlated errors
In this section, we will use the framework shown in Joachimi et al. (2008) to compute the covariance matrix
of our cross-correlation power spectrum. Although the reasoning behind this method is similar to that
of the shear estimation, we need to adapt the formalism to our particular case of having two different
observables: the shear and the Compton parameter. Following this method, we first define the signal
estimators and subsequently derive the power spectrum estimator. This is the one ingredient necessary to
the calculation of the covariance matrix which then becomes possible. In the end, some words are devoted
to how correlated errors are computed from the covariance matrix. We thoroughly present our calculations
in order to make all steps clear and free-of-doubt.
4.3.1
Definitions: signal estimators
n Weak Lensing
Take ε := γ + es as an estimator of the shear γ, with es being the intrinsic ellipticity of
an observed galaxy and therefore a source of noise to the lensing signal. We consider the estimator to be
unbiased since averaging over all angles yields:
h ε i = h γ i + h es i = h γ i
(4.6)
Chapter 4. The cross-correlation
62
The Fourier transform of the shear estimator is given by:
N
¯
ε̂ = ∑ ε i ei`.θ̄i
i =1
N
N
¯
¯
= ∑ γi ei`.θ̄i + ∑ ei ei`.θ̄i
i =1
=
Z
i =1
N
¯
N
¯
∑ δ(2) (θ̄ − θ̄i ) γ(θ̄ ) ei`.θ̄ + ∑ ei ei`.θ̄i
d2 θ
i =1
(4.7)
i =1
where ε i and θi are the ellipticity and position of the i-th galaxy, respectively. ` corresponds to the Fourier
conjugate of the position in the sky. Shear is measured within circular apertures but it can be described as
a continuous field which is the reason for the appearance of the delta function in the following definition:
N
n(θ̄ ) :=
∑ δ(2) (θ̄ − θ̄i ).
(4.8)
i =1
Separately carrying out the above integral:
Z
2
d θ n(θ̄ ) γ(θ̄ ) e
i `¯ .θ̄i
=
=
=
=
=
Z
2
d θ
Z
d2 ` 1
(2π )2
Z
d2 `2
¯
¯
¯
n̂(`¯ 1 ) e−i`1 .θ̄ γ̂(`¯ 2 ) e−i`2 .θ̄ ei`.θ̄
(2π )2
Z
d2 ` 1
(2π )2
Z
d2 `2
n̂(`¯ 1 ) γ̂(`¯ 2 )
(2π )2
Z
d2 ` 1
(2π )2
Z
d2 `2
n̂(`¯ 1 ) γ̂(`¯ 2 ) (2π )2 δ(`¯ − `¯ 1 − `¯ 2 )
(2π )2
Z
d2 ` 2
n̂(`¯ − `¯2 ) γ̂(`¯ 2 )
(2π )2
Z
d2 `0
n̂(`¯ − `¯0 ) γ̂(`¯ 0 ) .
(2π )2
Z
¯ ¯
¯
d2 θ ei(`−`1 −`2 ).θ̄
(4.9)
With this result, the shear estimator in Fourier space is finally given by:
¯ =
ε̂(`)
¯ =
with n̂(`)
R
¯
d2 `0
n̂(`¯ − `¯0 ) γ̂(`¯ 0 ) +
(2π )2
Z
N
¯
∑ ei ei`.θ̄i
(4.10)
i =1
¯
d2 θ n(θ̄ ) ei`.θ̄ = ∑iN=1 ei`.θ̄i . Its ensemble average is:
*
+
E
N
N D
¯
i
`
.θ
¯ = ∑ e i = ∑ ei`¯ .θi
n̂(`)
i =1
D
=N e
i =1
i `¯ .θi
E
= N
1
A
Z
¯
A
d2 θ Π(θ̄ ) ei`.θ̄
Here, the survey area is given by A and N is the number of galaxies at a position θi . Further:
(
1 in survey area
Π(θ̄ ) =
0 outside
R 2
so that A = A d θ Π(θ̄ ). The Fourier transform of the aperture is then given by
¯ =
Π̂(`)
Z
¯
R
d2 θ Π(θ̄ ) ei`.θ̄ =
Z
¯
A
d2 θ Π(θ̄ ) ei`.θ̄
(4.11)
(4.12)
(4.13)
Chapter 4. The cross-correlation
For
√
63
A 2π/`:
Z
A
¯
¯
d2 θ Π(θ̄ ) ei`.θ̄ −→ (2π )2 δ(`)
(4.14)
¯ = n̄ Π̂(`)
¯ is, in this regime, given by
This result yields that hn̂(`)i
¯ = (2π )2 n̄δ(`)
¯
hn̂(`)i
(4.15)
Up to this point, we have just repeated the calculations performed by Kaiser (1998).
The definition of the SZ effect is rather easier to assess than that of the
n Sunyaev-Zel’dovich effect
weak gravitational lensing. We define a SZ measurement as η := y + ny where y is the intrinsic Compton
parameter and ny is the noise contributed by unresolved clusters. Additionally, we assume that the noise
contribution is a non-vanishing one:
hη (θ̄ )i = hy(θ̄ )i + ny (θ̄ )
(4.16)
hη (θ̄ )i is a biased estimator.
4.3.2
Estimator of the cross-correlation power spectrum
Having calculated the shear and Compton parameter estimators, respectively Eqs. (4.10) and (4.16), and
the average of the aperture distribution given by Eq. (4.15), we are now ready to compute the estimator
cross-correlation power spectrum for weak lensing and the Sunyaev-Zel’dovich effect. The calculation
follows:
¯ η̂ (`¯ 0 )∗ =
ε̂(`)
=
*"Z
d2 `0
n̂(`¯ − `¯0 ) γ̂(`¯ 0 ) +
(2π )2
N
∑ ei e
i `¯ .θ̄i
#
h
∗
¯ + n̂∗ (`)
¯
ŷ (`)
y
i
+
i =1
d2 `0 ¯ ¯ 0 ¯ 0 ∗ ¯ n̂(` − ` ) γ̂(` ) ŷ (`)
(2π )2
Z
d2 `0 ¯ ¯ 0 D ¯ 0 ∗ ¯ E
+
n̂(` − ` ) γ̂(` ) n̂y (`) .
(2π )2
Z
(4.17)
At this point, we assumed that the intrinsic ellipticities from shear measurements
do not correlate with
D
E
s ∗
s
∗
any of the components of the SZ effect observations, i.e. ei ŷ = 0 = ei n̂y . However, even though they
are not individually accounted by SZ measurements, unresolved sources might produce a lensing signal,
meaning that a cross-correlation term in between weak lensing and SZ noise is needed to fully describe the
power spectrum. Using the result from (4.15) we continue the calculation:
Z
¯ η̂ (`¯ 0 )∗ = n̄ d2 `0 δ(`¯ − `¯0 ) γ̂(`¯ 0 ) ŷ∗ (`)
¯
ε̂(`)
Z
D
E
¯
+ n̄ d2 `0 δ(`¯ − `¯0 ) γ̂(`¯ 0 ) n̂∗y (`)
h
i
= n̄ (2π )2 δ(0) Pγy + Pγny
= An̄ Pγy + Pγny
(4.18)
Chapter 4. The cross-correlation
64
¯ =
with, (2π )2 δ(`)
R
¯
d2 θei`.θ̄ so that (2π )2 δ(0) = A. The above results allow the definition of an unbiased
estimator for the cross-correlation power spectra:
P̃γy :=
4.3.3
1
hε̂η̂ ∗ i − Pγny .
An̄
(4.19)
Covariance of the cross-correlation power spectrum
We have now all the necessary ingredients to compute the covariance of the power spectrum. We have
defined an unbiased power spectrum estimator P̃, i.e. h P̃i = P, so the covariance is given by:
Cov = h( P̃ − P)( P̃ − P)0 i − PP0 .
(4.20)
Applying this expression to our particular case using the result of Eq. (4.19) , we get:
*
0 +
1
1
∗
∗
0
Cov =
hε̂η̂ i − Pγny
hε̂η̂ i − Pγny
− Pγy Pγy
An̄
An̄
=
1 ∗ 0 0∗ 0
ε̂η̂ ε̂ η̂ + Pγny Pγn
y
( An̄)2
0 0∗ 1
0
0
+
ε̂
η̂
P
− Pγy Pγy
.
−
hε̂η̂ ∗ i Pγn
γn
y
y
An̄
(4.21)
According to Wick’s theorem, we can decompose the four-point correlation of the first term:
ε̂η̂ ∗ ε̂0 η̂ 0∗ = ε̂η̂ ∗ ε̂0 η̂ 0∗ c + hε̂η̂ ∗ i ε̂0 η̂ 0∗ + ε̂ε̂0 η̂ ∗ η̂ 0∗ + ε̂η̂ 0∗ ε̂0 η̂ ∗
(4.22)
where the subscript ‘c’ indicates the connected four-point correlator which we here assume to be negligible.
We are then left with terms concerning two-point correlators only. These are related to power spectra
Px = Px (`), with the index ’x’ refering to the correlated fields and given by the following expressions:
hε̂η̂ ∗ i = An̄ Pγy + Pγny
(4.23)
hε̂η̂ 0∗ i = (2π )2 δ(` − `0 ) n̄ Pγy + Pγny
(4.24)
hε̂ε̂0 i = (2π )2 δ(` + `0 ) n̄2 Pγγ + n̄σe2 (2π )2 δ(` + `0 )
hη̂ ∗ η̂ 0∗ i = (2π )2 δ(` + `0 ) ( Pyy + Pny ny ) .
(4.25)
(4.26)
with σe corresponding to the dispersion of intrinsic ellipticities. Introducing these equalities in expression
(4.21), we obtain:
1
Cov =
( An̄)2 ( Pγy + Pγny )( Pγy + Pγny )0
( An̄)2
+ (2π )2 δ(` + `0 ) n̄2 Pγγ
+ n̄σe2 (2π )2 δ(` + `0 ) (2π )2 δ(` + `0 ) ( Pyy + Pny ny )
2 + (2π )2 δ(` − `0 ) n̄( Pγy + Pγny )
0
0
0
0
+ Pγny Pγn
−
P
+
P
P
−
P
+
P
Pγny − Pγy Pγy
.
γy
γn
γy
γn
γny
y
y
y
(4.27)
Chapter 4. The cross-correlation
65
Carrying out the calculation a little further, we are left with a covariance which is the sum of two terms.
One term corresponds to the product of the auto-correlation spectra of weak lensing and SZ effect and the
other hosts the cross-correlation power spectrum:
Cov =
+
1
( An̄)2
(2π )2 δ(` + `0 ) n̄2 Pγγ + n̄σe2 (2π )2 δ(` + `0 ) ·
(2π )2 δ(` + `0 ) ( Pyy + Pny ny )
2
1 (2π )2 δ(` − `0 ) n̄( Pγy + Pγny ) .
2
( An̄)
(4.28)
Averaging over `-bands, the covariance becomes:
Z
A`
d2 `
Z
A`0
d2 `0 Cov =
d2 `0
(2π )4 δ(2) (` + `0 ) n̄2 Pγγ
A `0
4
2 (2)
0
+ (2π ) n̄ σe δ (` + ` ) · ( Pyy + Pny ny )
1
( An̄)2
+
Z
1
( An̄)2
d2 `
A`
Z
d2 `
A`
Z
Z
2
d2 `0 (2π )2 δ(` − `0 ) n̄( Pγy + Pγny ) .
A `0
(4.29)
For the covariance of the cross-power not to be zero, ` = `0 must be satisfied. Furthermore, the
transformation ` → −`0 should not change the result if the bands are defined by the modulus of the wave
vector. Given that (2π )2 δ(0) = A, the integration becomes:
Z
1
2
4 2
4
2
d
`
(
2π
)
n̄
δ
(
0
)
P
+
(
2π
)
n̄
δ
(
0
)
σ
γγ
e ( Pyy + Pny ny )
( An̄)2 A2`
Z
2
1
2
2
d
`
(
2π
)
δ
(
0
)
n̄
(
P
+
P
)
+ δ``0
γy
γn
y
( An̄)2 A2`
Z
σe2 (2π )2
2
d
`
P
+
Pyy + Pny ny
= δ``0
γγ
2
n̄
AA`
Z
2
2
(2π )
2
d
`
P
+
P
.
+ δ``0
γy
γn
y
AA` 2
Cov(`, `0 ) =δ``0
(4.30)
Introducing the band power, the covariance of the cross-correlation power spectrum is finally completely
described by:
(2π )2
Cov(`, ` ) = δ``0
AA`
0
σ2
P̄γγ + e
n̄
P̄yy + P̄ny ny + P̄γy + P̄γny
2 .
(4.31)
The barred values correspond to band power spectra and A` = 2π `∆`.
Finally, taking the covariance to real space using the definition of the correlation function of the
cross-correlation power spectrum:
ξ (θ ) =
Z ∞
`d`
0
2π
Pγy (`) J1 (`θ )
(4.32)
Chapter 4. The cross-correlation
66
×10
600
0.90
500
0.75
400
θ2 [arcmin]
17
1.05
0.60
300
0.45
200
0.30
100
0.15
100
200
300
400
θ1 [arcmin]
500
600
0.00
Figure 4.6: Covariance matrix of the cross-correlation power spectrum between weak lensing and the SZ effect. Contours
are linearly spaced and range from a minimum value of 10−18 up to 10−15 .
we get that the transformation of the covariance into real space becomes:
Z `d` Z ` 0 d` 0
∆ξ (θ ) ∆ξ (θ 0 ) =
Cov J1 (`θ ) J1 (`0 θ 0 )
2π
2π
Z
2 σ2 `d`
P̄yy + P̄ny ny + P̄γy + P̄γny
P̄γγ + e
·
=
n̄
2π A
J1 (`θ ) J1 (`θ 0 )
(4.33)
Given the dependence of the above quantity on the measurement specifications, we consider the
following parameters: a survey covering a field area A = 1000 square degrees, a galaxy background density
n̄ = 30 arcmin−1 and an ellipticity dispersion of σe = 0.3. The choice of area range was inspired by the
possibility of overlap of ranges reached by present and future weak lensing and SZ effect surveys such as the
ongoing South Pole Telescope measures (Williamson et al. 2011), the Planck mission (Planck Collaboration
et al. 2011a) or the upcoming Euclid space telescope (Laureijs et al. 2011).
In Figure 4.6 we present our estimate of the covariance matrix of the lensing-SZ cross-correlation power
spectrum. It shows a very broad distribution of uncertainties as proven by the lack of an obvious diagonal
feature, unlike what happens in the shear-only power covariance matrix. This is due to the fact that the
shot-noise term embodied by σe2 /n̄ dominates the source of error in the shear calculation. In our case, the
shear power spectrum is coupled with the SZ effect spectrum and is subsequently added to the cross-term,
diminishing the influence of shot-noise in the covariance matrix values.
4.3.4
Correlated errors
The correlated errors are obtained directly from the covariance matrix and are useful to evaluate the
error bars associated with each angular bin in the calculation of the correlation function. These will be
Chapter 4. The cross-correlation
67
10-6
10-7
ξ
10-8
10-9
10-10
10-11
10-12
10-13
100
101
θ [arcmin]
102
Figure 4.7: Perturbed correlation function of the fiducial model.
of particular use in the next chapter, when we focus on how the cross-correlation signal depends on the
redshift. Moreover, in order to do likelihood analysis with our data, we need to perturb the model we are
testing and for that task we also use the correlated errors.
The covariance matrix of a given correlation function contains the dispersion of the values of said
function. To compute the correlated errors, we calculate the eigenvalues of the covariance matrix and
perturb them with a Gaussian distribution of randomly generated numbers centred in zero, meaning that
the average of the error distribution should vanish, and rotate the errors back to the original frame.
Figure 4.7 shows how our fiducial model can be perturbed by a set of correlated errors which were
computed in this way.
68
Chapter 4. The cross-correlation
Chapter 5
Redshift dependence of the
cross-correlation signal
F
ollowing the computation of the cross-correlation power spectrum, we investigated the dependence
of the signal with redshift. By gaining insight on how dark and hot baryonic matter correlate in redshift
one can better understand the evolution of structures through the observable signal of the cross-correlation.
Through the redshift dependence of the signal, we aim to trace the thermal evolution of gas and thus,
structure growth.
In Section 5.1, we start by looking into the degree of dependence on redshift presented by the autocorrelation signal, both for a dark matter and a gas distribution. Afterwards we inspect the cross-correlation
3-dimensional signal for different redshift values. We then integrate the auto- and cross- power spectra
in redshift bins, in Section 5.2, to compare the contribution to the signal from different distance intervals.
Finally, we evaluate the correlation function of the cross-correlation power spectrum in redshift bins and
compute the correspondent correlated errors.
5.1
Redshift dependence of the 3-dimensional spectra
To create the power spectrum data, we compute the values of halo terms 1 and 2 for a range of redshifts and
store them in an information object with dimensions of [ spectra × wavenumber range × redshi f t range ].
Figure 5.1 shows the total 3-dimensional power spectrum (top panel) as well as the 1- and 2- halo
contributions (middle and bottom panels, respectively) of the dark matter (on the left) and the gas
distributions (on the right). The dark-matter density signal will overall decrease with increasing redshift.
The shapes of the total and halo terms remain fundamentally unchanged by the variation of z.
However, the power of the baryonic distribution behaves somewhat more eccentrically with the increase
of redshift: the halo-halo correlation is negligibly affected up to wavenumbers of about k ∼ 1 after which
the signal increases with redshift; the contribution of individual halos presents an inversion of this trend as
for wavenumbers below a few Mpc−1 h the signal is weaker for higher redshifts. At the small scales limit,
as redshift increases, the 1-halo term (and consequently, the total power) tends to become constant. As a
result of the combination of both halo terms, the shape of the overall power varies to some degree. This is
69
Chapter 5. Redshift dependence of the cross-correlation signal
70
due to the peculiar behaviour of the angular-diameter distance as it increases until redshift 1.25, reaching a
broad peak and then decreasing towards higher redshifts (see Section 1.2.2.)
Figure 5.2 shows the variation with redshift of the total and halo terms cross-correlation signal between
the dark matter and gas distributions. Here, the 1-halo term increases its contribution at larger redshifts
while the less influent two-halo term presents the opposite trend. The total signal is however clearly
dominated by the power from individual halos and therefore the two share the spectrum shape with the
exception of some minor deviations caused by the halo-halo correlation at very large scales. It seems like
the cross-correlation spectrum will borrow more characteristics from the baryon distribution signal while,
at the same time, presenting smoother curves. Since the gas has finite pressure, it develops a core which
smoothens it the dark matter cusp.
5.2
Redshift-binned signal and correlated errors
We further computed the angular power spectrum in sequential redshift bins to evaluate the contribution
to the signal from sources within different redshift intervals. The results are presented in Figure 5.3. In
all cases, the large scale regime is dominated by sources which are closer to the observer, a trend that is
more intense in the SZ spectrum with the closest bin being the largest contribution up to ` ∼ 200. This is a
natural projection effect as closer sources appear larger in the sky. For the convergence power spectrum,
past the large scales regime, the signal arises mostly from the intermediate redshifts, an effect intrinsic to
the geometrical sensitivity of gravitational lensing itself. The intermediate bin is prevalent in the SZ power
for a brief range of intermediate scales and the farthest bin dominates the source of power in the small
scale limit. This too is a particular feature of the SZ effect, as it arises mostly from the smaller and much
more abundant clusters. The behaviour of the cross-correlation power spectrum curves is very close to that
of the SZ effect, indicating that the dependence on scales, source characteristics and redshift ranges is more
closely related to the gas distribution than the dark matter abundance and sensitive to the imposed mass
cut-off.
We computed the correlation function via Eq (4.5) and investigated the influence of the redshift on the
observed signal signal. Figure 5.4 shows the correlation function of the measures in the different redshift
ranges within the interval of redshift 0 to 1. The correspondent correlated errors per angular bin, calculated
from the covariance matrix data as explained in 4.3.4, are also plotted. To compute the errors, we used the
same survey parameters and assumptions as presented in Sections 4.3.3. For very small angles, there is a
slight predominance of the signal originated from sources at the intermediate redshift range for very small
angles, while above approximately 4 arcminutes the closest sources give the highest contribution to the
correlation. The correlation signal steadily decreases with the increase of angle.
Chapter 5. Redshift dependence of the cross-correlation signal
103
102
100
10-1
10-1
102
100
k [ Mpc−1 h ]
101
10-2 -2
10
102
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
104
103
102
∆2ζζ
103
10-1
101
100
10-1
10-1
104
103
102
10-1
100
k [ Mpc h ]
−1
101
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
104
103
102
101
100
10-1
10-1
10-1
100
k [ Mpc h ]
−1
101
102
101
102
10-1
100
k [ Mpc−1 h ]
101
102
10-1
100
k [ Mpc−1 h ]
101
102
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
101
100
10-2 -2
10
100
k [ Mpc−1 h ]
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
10-2 -2
10
102
10-1
101
100
10-2 -2
10
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
101
100
104
∆2δδ
103
101
10-2 -2
10
∆2δδ
104
∆2ζζ
∆2δδ
102
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
∆2ζζ
104
71
10-2 -2
10
Figure 5.1: Variation of the 3-dimensional dark-matter density (left panel) and gas pressure (right panel) auto-correlation
power spectra with redshift. Top plots show the total power spectra for different redshift values; middle and bottom
plots show the variation of 1- and 2-halo term contributions, respectively.
Chapter 5. Redshift dependence of the cross-correlation signal
72
104
103
∆2δζ
102
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
101
100
10-1
10-2 -2
10
104
103
∆2δζ
102
10-1
100
k [ Mpc−1 h ]
101
102
10-1
100
k [ Mpc−1 h ]
101
102
10-1
100
k [ Mpc−1 h ]
101
102
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
101
100
10-1
10-2 -2
10
104
103
∆2δζ
102
z =0.0
z =0.2
z =0.4
z =0.6
z =0.8
z =1.0
101
100
10-1
10-2 -2
10
Figure 5.2: Variation of the 3-dimensional cross-correlation power spectrum with redshift. Top panel shows the total
signal and middle and bottom panels correspond to the 1- and 2- halo terms, respectively.
Chapter 5. Redshift dependence of the cross-correlation signal
10-3
`(` +1)C (`) / 2π
10-4
73
total
0.0 - 0.3
0.3 - 0.6
0.6 - 1.0
10-5
10-6
10-7
10-8
101
10-10
`(` +1)Cyy(`) / 2π
10-11
102
103
102
103
102
103
`
104
105
106
104
105
106
104
105
106
total
0.0 - 0.3
0.3 - 0.6
0.6 - 1.0
10-12
10-13
10-14
10-15
10-16
10-5
10-6
`(` +1)C y(`) / 2π
10-7
101
`
total
0.0 - 0.3
0.3 - 0.6
0.6 - 1.0
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
101
`
Figure 5.3: Redshift binned angular power spectrum. The top plot depicts the convergence power spectrum, the middle
plot corresponds to the Sunyaev-Zel’dovich power spectrum and the bottom plot depicts the cross-correlation. All
curves are the sum of both halo terms. Redshift ranges from 0 to a maximum of 1 and the bins are limited as shown in
the legends.
Chapter 5. Redshift dependence of the cross-correlation signal
74
0.0 - 0.3
0.3 - 0.6
0.6 - 1.0
10-6
10-7
ξ
10-8
10-9
10-10
10-11
10-12
10-13
100
101
θ [arcmin]
102
Figure 5.4: Redshift binned correlation function of the cross-correlation power spectrum with correlated errors. We
assume a survey area of 1000 square degrees, a galaxy background density of 30 n̄ = 30 arcmin−1 and an ellipticity
dispersion of σe = 0.3.
Chapter 6
Constraints on cosmological parameters
T
his chapter is devoted to assessing if the statistical analysis of the cross-correlation power spectrum
between the cosmic shear and the baryonic distribution thermal imprint in the CMB radiation may be
able to constrain the parameters of the cosmological model. First, we take a look at how the cross-spectrum
is affected by the change of these two parameters, by simply varying one of them at a time, for which the
results and plots are presented in Section 6.1. The correlation function and corresponding covariance matrix
computed in the previous sections allow an attempt to investigate constraints on the parameter plane we
naturally focus on given our raw signals: Ωm – σ8 . The analysis of the χ2 (or likelihood) distribution and
the Fisher-Matrix from the likelihood distribution are calculated in Section 6.2.
6.1
How the parameters affect the spectra
As a first step towards understanding how different cosmological parameters affects the cross-correlation
signal, we vary the matter density parameter Ωm while the fluctuation amplitude σ8 is kept fixed and
vice-versa. Figure 6.1 shows the 3-dimensional (top panel) and angular (bottom panel) power spectra of the
fiducial model – with Ωm = 0.272 and σ8 = 0.809 – and the spectra computed with the extreme values of
the cosmological parameters:
f id
f id
min
max
( Ωm
, Ωm
) =( Ωm (1 − 50%) , Ωm (1 + 50%))
f id
f id
( σ8min , σ8max ) =( σ8 (1 − 20%) , σ8 (1 + 20%))
All spectra is presented as a ratio with the fiducial model signal. Most curves show minor deviations from
the fiducial model when the cosmological parameters are varied. The deviations from the fiducial model
are more evident with the variation of σ8 , were an increased value of the fluctuation amplitude translates
into an increase in signal while the converse is also true. In the matter density case, we observe an increase
of power with smaller values of Ωm in the small-scale regime. The opposite behaviour is found on the
curves for the variation of the spectra with σ8 , with more obvious deviations from the fiducial values, for
most of the angular range. With the projection from the three-dimensional to the angular power spectrum,
the tendency found in the 3D spectra for the Ωm variation changes, as the weight function depends on the
matter density explicitly and proportionally.
75
Chapter 6. Constraints on cosmological parameters
76
102
102
fiducial
Ωmax
Ωmin
m
101
∆2δζ / ∆2δζ,fid (k)
∆2δζ / ∆2δζ,fid (k)
101
100
10-1
100
10-1
10-2 -2
10
101
10-1
100
k [ Mpc h ]
−1
101
10-2 -2
10
102
101
fiducial
Ωmax
C y / C fidy (`)
C y / C fidy (`)
Ωmin
m
100
101
10-1
100
k [ Mpc−1 h ]
101
102
fiducial
σ8max
σ8min
m
10-1
fiducial
σ8max
σ8min
m
102
103
`
104
105
106
100
10-1
101
102
103
`
104
105
106
Figure 6.1: Dependence of the 3-dimensional (top panel) and angular (bottom panel) cross-correlation power spectra on
cosmological parameters. All plots show the ratio between the spectra with alternative parameters and that of the
fiducial model. On the left, Ωm is varied while σ8 remains fixed to the fiducial value and one the right σ8 changes,
maintaining the fiducial value for Ωm .
Chapter 6. Constraints on cosmological parameters
77
The angular power spectrum for different σ8 values shows minor shifts in amplitude from the fiducial
model, maintaining the tendency of the 3D power spectrum for the same amplitude fluctuation levels.
Overall, these results reflect what is already known from the literature: the spectra of both signals (weak
lensing and SZ effect) increase with higher values of these cosmological parameters. A high matter density
naturally and a higher clustering level means more mass and more structures to contribute to the final
signal.
6.2
Likelihood and Fisher analysis
We consider that our data is given by two-dimensional cross-correlation functions whose underlying model
is specified by the cosmological parameters (and the background galaxy distribution.) We estimate the χ2
distribution, as defined in Schneider et al. (2002), which depends directly on the correlation function (4.5)
and on the inverse of the covariance matrix (4.33) from the cross-correlation power spectrum:
χ2 ( p) = ∑ ξ i ( p) − ξ ifid Covij−1 ξ j ( p) − ξ fid
j
(6.1)
ij
p illustrates the set of parameters we vary, ξ fid is the correlation function of our fiducial model perturbed
by correlated errors, mimicking the data, as shown in Section 4.3.4 and the indices label the correlation
function angular bins.
In the computation of the likelihood, we vary the values of matter density Ωm and fluctuation amplitude
σ8 while keeping all other cosmological parameters fixed. Nonetheless, we further assume a spatially-flat
Universe, meaning that our models obey Ωm + ΩΛ = 1.
Following the formalism presented in Fisher (1935) and adapted to our calculation, we can compute the
Fisher matrix of the χ2 distribution by estimating the expectation value of its second-order derivative with
the cosmological parameters, at the minimum value of χ2 :
Fij =
1 ∂2 χ2
2 ∂pi ∂p j
(6.2)
meaning that the Fisher matrix is the expectation value of the lowest-order non-vanishing term in the Taylor
expansion of the χ2 distribution in the model parameters:
χ2 = χmin + ( Fij δpi δp j ) |χmin
(6.3)
Our results are jointly shown in Figure 6.2, where (6.1) and (6.3) are depicted against one another.
Unlike what happens in the case of weak lensing, the signal doesn’t have a very defined dependence on
the Ω–σ8 plane. The contours of the χ2 distribution hint for a high level of non-Gaussian-like features. To
better understand the shape of the χ2 function, we show the contour plot evaluated at more levels than the
previous one. In Figure 6.3 we confirm that the distribution is indeed somewhat tilted. This behaviour
explains why the expansion of the χ2 up to its second derivative does not yield small Fisher-matrix contours,
as the curvature is not straightforward.
Moreover, we conclude that this cross-correlation is not suitable to estimate or constrain the cosmological
parameter space. The Sunyaev-Zel’dovich power spectrum is typically very sensitive to the value of σ8 .
This effect is however damped by the lack of angular resolution in the weak gravitational lensing signal.
Chapter 6. Constraints on cosmological parameters
78
2σ
0.86
0.84
3σ
σ8
1σ
1σ
2σ
0.82
0.80
0.78
3σ
0.76
0.15
0.20
0.25
Ωm
0.30
0.35
0.40
Figure 6.2: Fisher matrix ellipses plotted against χ2 contours. The confidence levels of the χ2 contours colour-match
those of the filled ellipses. those of the
χ2min +
2 +0.5
χmin
0.84
χ2min + 3
2 +2
χmin
0.86
4
0.82
.25
σ8
+5
+0
1
2
χχmi
2 n +4
5
min +
χ2min +
0.78
χ2min
in
0.80
χ2min + 2
2 +3
χmin
χm 2
0.76
0.15
0.20
0.25
Ωm
0.30
0.35
0.40
Figure 6.3: A more refined depiction of the χ2 distribution contours showing the eccentric behaviour of the function.
Chapter 6. Constraints on cosmological parameters
79
We conclude that the lack of cosmological sensitivity rather opens ways to study parameters characterising the evolution of the cluster population.
80
Chapter 6. Constraints on cosmological parameters
Chapter 7
Heating history of baryons
I
n this chapter, we an effort to further use the potential of this cross-correlation to give us information
on the structure evolution; particularly on how the gas temperature in cluster-sized halos evolved over
time, and the behaviour of gas. We further modelled our description of the temperatures in gas halos by
introducing a term that sorts hot halos. In Section 7.1 we present this heating term and investigate the effect
it has on the 3-dimensional and angular power spectra. We additionally bin the signal into two redshift
bins to infer how the signal is distributed along the line-of-sight integration.
7.1
Additional modelling of the cluster temperature
The halo mass-redshift space contributing to the cross-correlation signal is further constrained by considering collapsed structures only. For that purpose, an additional term is introduced to the temperature power
law, taking into account whether a halo of a given mass has already reached its virial temperature after
formation at a given redshift.
We assume that the heating of the gas is not instantaneous, but rather takes a certain multiple or fraction
of the free-fall time:
s
τff =
1
Gρ
(7.1)
with G being the gravitational constant. ρ can be taken as the scale density introduced as the density
within the scale radius, as previously defined in Subsection 2.2.2. Here, we reproduce and re-arrange Eq.
(2.18), which provides an expression for the scale density:
Z r200
dr 4πr2 ρs
3
= 4π ρs rs ln(1 + c) −
m200 =
(7.2)
0
≡ 4π ρs rs3 log C (c)
81
c
1+c
Chapter 7. Heating history of baryons
82
At the same time,
4π 3
r ρcr
3 200
4π
3H 2
=
200 c3 rs3
.
3
8πG
m200 = 200
(7.3)
(7.4)
Combining both of the above results, we can describe the scale density as:
ρs =
200
c3
3H 2
3 log C (c) 8πG
≡ δs
(7.5)
3H 2
.
8πG
It is now possible to the free-fall time in the following way:
s
τff =
r
=
1
=
G ρs
s
1
3H 2
8πG δs
(7.6)
8π H0 1 1
√
3 H H0 δs
and by introducing τHubble = 1/H0 , we can express the free-fall time in units of the present Hubble
time:
τff
r
=
τHubble
8π H0
.
3δs H
(7.7)
We plug this description of the free-fall time in Eqs. (4.1) and (4.2) by multiplying the temperature-mass
function by a heating factor:
∆t
theat (m, z, f ) = exp − f
τff
(7.8)
where ∆t is the difference between the halo redshift and its collapse redshift. The formation redshift is
estimated by following the prescription by Navarro et al. (1997). This term introduces a new variable, a
ratio f between the heating and free-fall time-scales, into the estimation of the signal, one that allows us to
get insight on the dependence of the signal on the heating history of the baryonic matter present in dark
matter halos.
Figure 7.1 shows the variation of the heating factor with mass and redshift, for different values of the
factor f as labelled. theat will weight the temperature increasingly with the halo mass, as well as with
redshift. For the fiducial model computed in this work, f is zero. Deviations from the fiducial value mean
an increasingly sharp cut in the less massive structures.
In Figure 7.2, we plot the total 3-dimensional cross-correlation power spectra for different values of
parameter f at fixed redshift zero. We find that the signal decreases very significantly with increasing f
values, meaning that constraining the integration to collapsed halos only will seriously affect the power
spectrum. The same behaviour is found on the curves of the angular power spectra (see Figure 7.3.) A
small but noticeable decrease of contribution by the 2-halo term to the total power on the large-scale regime
is also observed.
Chapter 7. Heating history of baryons
83
100
theat(m)
10-1
10-2
10-3
0.05
0.15
0.3
1016
10-4
1014
1013
m [M ¯]
1015
Figure 7.1: Heating factor variation with mass, redshift and factor f . Black lines correspond to redshift 0 and red lines
to redshift 1. The legend indicates the value of the parameter f .
104
103
fiducial model
f =0.05
f =0.15
f =0.3
∆2δζ(k)
102
101
100
10-1
10-2 -2
10
10-1
100
k [ Mpc−1 h ]
101
102
Figure 7.2: Variation of the total 3-dimensional cross-correlation power spectrum with f values. Power spectrum taken
at redshift 0.
Chapter 7. Heating history of baryons
84
10-5
10-6
`(` +1)C y(`) / 2π
10-7
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
101
102
103
`
104
105
106
Figure 7.3: Variation of the cross-correlation angular power spectrum with f values. Solid lines portray the total signal,
dashed and dotted lines show the 1- and 2-halo terms. Black lines depict the fiducial model with f = 0. Grey, red and
yellow lines show the angular spectra for f of increasing values: 0.05, 0.15 and 0.3, respectively.
10-6
10-7
total
z = 0.0 - 0.4
z = 0.4 - 1.0
`(` +1)C y / 2π
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
101
102
103
`
104
105
106
Figure 7.4: Cross-correlation signal within two redshift bins as labelled. Total lines are depicted as solid curves.
Greylines correspond to f = 0.05, red lines have f = 0.15 and yellow lines are the spectrum with f = 0.3.
Chapter 7. Heating history of baryons
85
We further analysed the dependence of the signal in redshift by integrating the 3-dimensional power
spectrum in two different redshift bins from redshift 0 to 1. We plot our findings in Figure 7.4. For every f
value, most of the signal is collected in the farthest redshift bin, with the closest bin contributing more in
the large-scale range. The latter contribution becomes less intense as the value of f increases.
After inspecting the above results, we conclude that the overall signal is highly sensitive to slight
variations of the factor f . This behaviour points to the fact that to undergo tests on the constraining power
of this model in the cosmological parameter space, one would need to only very slightly vary the value of
f around the fiducial model value and carefully sample the area around it.
86
Chapter 7. Heating history of baryons
Conclusions and further work
T
o wrap up this thesis, we review the main steps of our work as well as the most important interpretations of our results. We further speculate on the investigations which naturally follow the method
presented in this thesis.
Our present picture of the Universe can be characterised by the spatial distribution of halos, spread
in filamentary structures that cross each other in nodes composed of halo clusters. Outside of this
configuration there are only large volumes of mostly empty space. This image is supported both by
observations of galaxies and galaxy clusters as well as by cosmological numerical simulations which begin
with the Gaussian initial conditions of an early state in structure evolution, allowing a system of particles
to evolve, while governed by the laws of gravitation.
We set out to carry this study motivated by the still largely undefined characteristics of the behaviour
of structures throughout their formation and evolution periods. In order to investigate the nature of
the changes that occur during halo growth and collapse, a large number of observational, theoretical
and numerical techniques have been extensively used to model the evolution of structures. One of these
methods is the so-called halo model which provides an analytic framework to compute the two-point
correlation function of a given density field. This formalism, reviewed in Chapter 2, was based on an
early attempt to statistically illustrate the galaxy distribution, and originally developed to describe the
distribution of dark matter. In the model, the correlation function is the result of adding a Poisson term,
accounting for the contribution to the signal by individual halos, and a halo-halo term, representing the
correlations between halos. These terms are the product of a mass integration over the models describing
the distribution, structure and biasing of halos.
The analytic estimation of the correlation function makes the halo model an additional source of insight
to understand conclusions from numerical and observational data: the correlation function can be directly
compared to results from simulations, while it can also be projected into an angular signal which is directly
analysed against observations.
The halo model parameterises the computation of the two-point correlation function such that it is
rather straightforward to adapt the framework in order to describe a cross-correlation signal. In this thesis,
we took advantage of the good description of dark matter halos by the halo model, and cross-correlate
the dark matter signal with that arising from the halo gas component. The resulting two-point correlation
function – or its Fourier conjugate, the 3-dimensional power spectrum – relates the signals from a dark
matter density-contrast distribution with the gas pressure within each halo. Projecting this cross-correlated
signal into the angular power spectrum, while choosing adequate geometrical weight functions, yields the
cross-correlation power spectrum between two phenomena: the weak gravitational lensing (accounting for
87
Conclusions and further work
88
the dark component of halos) and the thermal Sunyaev-Zel’dovich effect (tracing the gas component.)
In Chapter 3, we introduced the two effects and computed the auto-correlation 3-dimensional power
spectra of each of them making use of the halo model formalism. We stated the models we choose to
parameterise the halo description with. In addition to the commonly assumed models that are extensively
used in such studies, we constrain our mass integration to only account for halos which are observed
through both weak lensing and the thermal SZ effect. We verified the resulting spectra to be in accordance
with the findings of previous studies where the halo model was used with the same purpose. Additionally,
we projected the power spectrum into its angular signal by applying Limber’s Equation (see Appendix A).
The general behaviour of the power spectra curves reveals that the term related to individual halos (1-halo
term) is the most significant source of signal overall, with the halo-halo correlations (2-halo term) being
important only in the regime of large scales. The differential dependence of the signal on large and small
scales discloses how the choice of the bias and density models strongly influences the power, respectively.
We specialised the halo model formalism to cross-correlate the dark matter and gas distributions. The
3-dimensional and angular cross-correlation signals, were presented in Chapter 4. The behaviour of the
resulting spectra shows a smoothed signal which borrows characteristics from both the dark matter and the
gas spectra. Overall, the halo-halo correlations become less decisive to the total signal and the observational
limit imposes a decrease on small scales as halos become too small to be resolved. Further statistical
analysis was performed by using the angular power spectrum of the cross-correlation. We computed the
correlation function by integrating the angular spectrum over a top-hat filter. This function was used in
the derivation of the covariance matrix: the expression for the matrix in real space relates the correlation
functions with the angular auto- and cross-correlation power spectra of weak gravitational lensing and the
SZ effect. The covariance matrix was then computed for a hypothetical survey with realistic parameters,
yielding a strikingly non-diagonal result. With the covariance matrix we computed correlated errors of the
correlation function which would be of use later on in our study.
The next step in our analysis was to investigate the signal evolution by inspecting the dependence of
the cross-correlation power spectrum on redshift. In Chapter 5, we first looked into the changes sustained
by the 3-dimensional auto- and cross-correlation power spectra. As expected, the overall signal increases
with redshift. The most remarkable features reside in the gas pressure spectra, which show a stark increase
of signal towards a stable amount of signal (imprint of the angular-diameter distance) in the small-scale
regime while maintaining the amplitudes in the very large scales. The cross-correlation signal exhibits a
behaviour very close that of the gas distribution, although with smoother curves (presents smooth peaks);
this is explained by the structural differences of the halo densities of gas and dark matter, as well as by
the fact that weak lensing lacks the angular resolution for resolving the gas distribution. We additionally
projected the 3-dimensional spectra integrating over different redshift bins. Again, we performed this
analysis in the auto- and cross-correlation cases. Again to no surprise, we found that the signal from
different redshift bins is bound to increased contributions from different scales. The large-scale regime
is dominated by the closer sources of power, as they occupy wider angles in the sky by the projection
effect. The peak of the cross-correlation power relies on sources within intermediate redshift scales and the
contribution at small-scales by the farthest sources is an inherited feature from the SZ effect. To conclude
the redshift analysis, we computed the correlation function obtained in Chapter 4 in the same redshift
bins and calculated the correlated errors associated to them. The signal mostly arises from intermediate
redshifts and invariably decreases with the increase of scales, as naturally does the amplitude of the errors.
Conclusions and further work
89
We further use the cross-correlation power spectrum in an attempt to constrain the cosmological
parameter space by doing likelihood and Fisher analyses in Chapter 6. The cosmological parameter space
is not well constrained by the cross-correlation signal between dark matter and gas pressure. This is due to
the limiting nature of the angular resolution in the weak lensing signal, which severely damps the strong
dependence of the SZ effect signal on the fluctuation amplitude given by σ8 . Although the cross-correlation
signal is not suitable to make assumptions on the underlying cosmological model, it hints at the possibility
of characterising the intrinsic properties of the cluster population.
The closing chapter of this thesis was devoted to a further constraint of the mass-redshift plane: we
introduced a term to include only hot structures. We delay the heating of the gas assuming that the
heating time-scale is a multiple of the free-fall time-scale. We analysed the signal for different values
of the parameter f , characterising the ratio between the heating and free-fall time-scales, and how that
signal is distributed in redshift and we found out that relatively small variations of the parameter induce
rather intense amplitude changes. Further work with this modelling method would include the tentative
evaluation of the constraining power of f against cosmological parameters. For this, a field of extremely
small deviations from the fiducial value f = 0 should be sampled.
The halo model is constructed in such a manner that it allows to test a wide range of signals arising
from the different components of the cosmological paradigm. Choosing different models to describe these
components is a way to gain insight into how they affect the signal and, most importantly, on how they
perform against observations. Many possible cross-correlations can be investigated with the halo model, as
shown in Cooray (2000); Cooray and Sheth (2002). A natural transition from this work could be to build the
cross-correlation between the SZ effect and the X-ray emission from galaxy clusters. Both signals depend on
the same halo components but do, however, depend on the gas pressure in different ways. This distinction
can be useful to further evaluate the thermal history of baryons within collapsed structures.
90
Conclusions and further work
Appendix A
Random fields and Limber’s Equation
We describe the formalism that allows the simple projection of the power spectrum into its angular
description, which we used extensively in this work. The original framework was devised by Limber (1954)
after which the projection relations are named. Presented here is the derivation of the equation following
Bartelmann and Schneider (2001).
The definition of a random field g( x ) comes with several properties. First
n Properties of a random field
of all, the average of the field is zero: h g( x ) i = 0, ∀ x. Secondly, the field is homogeneous, meaning that
g( x ) and a translation given by g( x + y) are not statistically distinguishable. The field is also isotropic, for
g( x ) and g(R x ) share the same statistical properties, with R standing for a rotation matrix of dimension n.
The two-point correlation function of the field is given by:
h g( x ) g∗ (y) i = Cgg (| x − y |)
(A.1)
where Cgg is a real number even if g ∈ C. The Fourier pair of the field is given by:
g( x ) =
Z
Rn
dn k
ĝ(k) eix.k
(2π )n
ĝ(k) =
and
Z
Rn
dn x g( x ) e−ix.k
(A.2)
so that the correlation function in Fourier space is given by:
h ĝ(k) ĝ∗ (k0 ) i =
Z
Rn
dn x eix.k
Z
0 0
Rn
dn x 0 e−ix .k h g( x ) g∗ ( x 0 ) i .
(A.3)
Using (A.1) and replacing x 0 = x + y, this relation becomes:
h ĝ(k) ĝ∗ (k0 ) i =
Z
Rn
dn x eix.k
Z
0
Rn
dn y e−i(x+y).k Cgg (| y |)
= (2π )n δD (k − k0 )
Z
Rn
dn y e−iy.k Cgg (| y |)
(A.4)
(A.5)
with δD standing for Dirac’s delta function. The power spectrum of an homogeneous and isotropic field g
is then defined as:
Pg (|k|) =
Z
Rn
dn y e−iy.k Cgg (| y |)
91
(A.6)
Appendix A. Random fields and Limber’s Equation
92
thus yielding a direct relation between the two-point correlation function and the power spectrum:
h ĝ(k) ĝ∗ (k0 ) i ≡ (2π )n δD (k − k0 ) Pg (|k|).
(A.7)
The objective of this equation is to establish a relation between the two-point
n Limber’s Equation
correlation function (or power spectrum) of a homogeneous, isotropic, 3-dimensional random field and
its projection in 2D. We define a 3D field characterised by a density contrast dependent on a radial and
angular co-ordinates r and θ as well as on a function of the r such that δ[ f k (r )θ, r ]. θ is a 2-dimensional
vector. We set the density contrast as having two different projections along the light-cone corresponding to
an observer at r = 0 and t = t0 :
gi ( θ ) =
Z
dr qi δ[ f k (r )θ, r ]
(A.8)
with i = 1,2 and r ranging from zero to a given scale rH standing for the horizon, the limit of the integration.
Then the correlation function if given by:
C12 (θ ) =h g1 (θ ) g2 (θ 0 ) i
=
Z
dr q1 (r )
Z
(A.9)
dr 0 q2 (r 0 ) hδ[ f k (r )θ, r ] δ[ f k (r 0 )θ 0 , r 0 ]i
(A.10)
In the large-scale regime, where k → 0, the density contrast power spectrum decreases proportionally with
k, so we can assume that there is a coherence scale Lcoh beyond which there are no fluctuations. For the
correlation not to be zero, we need to take into account only the scales for which |r − r 0 | . Lcoh stands,
within the horizon distance. We assume that weight functions qi do not vary substantially in over a scale
∆r ≤ Lcoh . With such assumptions, and setting f k (r 0 ) ≈ f k (r ) and q2 (r 0 ) = q2 (r ), the correlation function
becomes:
C12 (θ ) =
Z
dr q1 (r ) q2 (r )
Z
d(∆r ) Cgg (
q
f k2 θ 2 + (∆r )2 , r )
(A.11)
This is Limber’s Equation, relating the two-point correlation of the projected field to that of the 3dimensional field. Another useful form of this equation is achieved when going into Fourier space:
providing the relation between the two-point correlation and the projection of the 3-dimensional power
spectrum. Replacing the density contrasts of Eq. (A.10) with (A.3), we get:
C12 (θ ) =
Z
dr q1 (r )
Z
dr 0 q2 (r 0 )
Z
d3 k
(2π )3
Z
d3 k 0
(2π )3
(A.12)
h δ̂(k, r ) δ̂∗ (k0 , r 0 ) i exp(−i f k (r ) k⊥ . θ) exp(i f k (r 0 ) k0⊥ . θ) exp(−i k3 r ) exp(i k03 r 0 )
(A.13)
where k ⊥ is a 2-dimensional vector perpendicular to the line-of-sight. Using (A.4) we introduce Pδ into the
above relation:
C12 (θ ) =
Z
dr q1 (r ) q2 (r )
Z
d3 k
P (|k|, r ) exp(−i f k (r ) k⊥ . (θ − θ0 )) exp(−i k3 r )
(2π )3 δ
Z
dr 0 exp(i k03 r 0 )
(A.14)
Appendix A. Random fields and Limber’s Equation
93
This result implies that only 2πδD (k3 ) (solution of the last integral) will contribute to the projected
correlation function; performing the integration over k3 , the correlation becomes:
d2 k ⊥
P (|k |, r ) exp(−i f k k⊥ .θ)
(2π )2 δ ⊥
Z
Z
dk
= dr q1 (r ) q2 (r )
J [ f (r )θk ] .
(2π ) 0 k
C12 (θ ) =
Z
dr q1 (r ) q2 (r )
Z
(A.15)
(A.16)
Following both the definition of power spectrum (A.6) and (A.2), we can finally relate projected power
P12 (`) as:
P12 (`) =
Z
d2 θ C12 (θ )ei`.θ =
Z
dr q1 (r ) q2 (r )
Z
d2 k ⊥
P (k , r ) (2π )2 δD (` − f k (r )k⊥ )
(2π )2 δ ⊥
(A.17)
yielding the Fourier space version of Limber’s Equation, used in this thesis:
P12 (`) =
Z
dr
q1 (r ) q2 (r )
Pδ (`/ f k (r ), r ) .
f k (r )2
(A.18)
94
Appendix A. Random fields and Limber’s Equation
List of Figures
1.1
Illustration to the most important scientific achievements in the history of Cosmology. . . .
1.2
Illustration of the Universe evolution chronology. . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
Distances and redshift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.4
Contraints from the joint data of Supernovae Ia, galaxy clusters and CMB surveys. . . . . .
13
1.5
CMB all-sky map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.6
Observed CMB power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.7
Scheme of the CMB power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.8
CMB and secondary anisotropies power spectra. . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.1
Illustration of the halo model ingredients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2
Halo mass functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3
Normalised NFW profile and its dependence on the halo mass. . . . . . . . . . . . . . . . . .
29
2.4
Dark matter density profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.5
Illustration of correlation terms in the extended halo model. . . . . . . . . . . . . . . . . . . .
32
3.1
Illustration of the gravitational lensing basic mechanism. . . . . . . . . . . . . . . . . . . . . .
36
3.2
Example of strong gravitational lensing by a galaxy cluster. . . . . . . . . . . . . . . . . . . .
39
3.3
Geometrical weight function of the convergence parameter. . . . . . . . . . . . . . . . . . . .
41
3.4
Three-dimensional power spectrum of density contrast at redshift 0. . . . . . . . . . . . . . .
42
3.5
Convergence power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.6
Illustration of the inverse Compton scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.7
CMB intensity spectrum shift due to the thermal Sunyaev-Zel’dovich effect. . . . . . . . . .
46
3.8
Simulated all-sky map of the SZ effect from the Planck mission. . . . . . . . . . . . . . . . . .
47
3.9
Normalised beta profile and its dependence on the halo mass. . . . . . . . . . . . . . . . . . .
50
3.10 Mass-redshift distribution of detectable clusters with Planck. . . . . . . . . . . . . . . . . . .
51
3.11 Three-dimensional power spectrum of gas pressure distribution. . . . . . . . . . . . . . . . .
52
3.12 Sunyaev-Zel’dovich effect (angular) power spectrum. . . . . . . . . . . . . . . . . . . . . . . .
52
4.1
Three-dimensional power spectrum of the density contrast and gas pressure. . . . . . . . . .
57
4.2
Three-dimensional power spectrum of the cross-correlation between dark matter density
4.3
5
contrast and gas pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Angular power spectra of weak lensing convergence and the SZ effect. . . . . . . . . . . . . .
59
95
LIST OF FIGURES
96
4.4
Weak lensing-SZ effect cross-correlation angular power spectrum. . . . . . . . . . . . . . . .
59
4.5
Cross-correlation function between weak lensing and the SZ effect. . . . . . . . . . . . . . . .
61
4.6
Covariance matrix of the cross-correlation power spectrum between weak lensing and the SZ
effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4.7
Perturbed correlation function of the fiducial model. . . . . . . . . . . . . . . . . . . . . . . .
67
5.1
Variation of the 3-dimensional auto-correlation power spectra with redshift. . . . . . . . . .
71
5.2
Variation of the 3-dimensional cross-correlation power spectrum with redshift. . . . . . . . .
72
5.3
Redshift binned angular power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.4
Redshift binned correlation function of the cross-correlation power spectrum with correlated
errors. We assume a survey area of 1000 square degrees, a galaxy background density of 30
n̄ = 30 arcmin−1 and an ellipticity dispersion of σe = 0.3. . . . . . . . . . . . . . . . . . . . . .
74
6.1
Dependence of the cross-correlation power spectrum on Ωm and σ8 . . . . . . . . . . . . . .
76
6.2
Fisher matrix ellipses plotted against χ2 contours. . . . . . . . . . . . . . . . . . . . . . . . . .
78
6.3
χ2
distribution contours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
7.1
Heating factor variation with mass, redshift and factor f . . . . . . . . . . . . . . . . . . . . . .
83
7.2
Variation of the 3-dimensional cross-correlation angular power spectrum with f values. . . .
83
7.3
Variation of the cross-correlation angular power spectrum with f values. . . . . . . . . . . .
84
7.4
Redshift-binned cross-correlation angular power spectrum. . . . . . . . . . . . . . . . . . . .
84
List of Tables
1.1
Cosmological parameters of the ΛCDM model from the 7-year release of the Wilkinson
Microwave Anisotropy Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
12
98
LIST OF TABLES
Acknowledgments
The closing stretch of any thesis is a compendium of lists and, for me, none is more relevant than this one.
I acknowledge the financial support of the International Max Planck Research School for Astronomy and
Cosmic Physics at the University of Heidelberg and the Heidelberg Graduate School of Fundamental
Physics.
I thank Prof. Luca Amendola for kindly agreeing to be the second corrector of this thesis and Profs. Eva
Grebel and Hans-Christian Schultz-Coulon for their availability to be part of my defence committee.
I deeply thank Matthias Bartelmann for the opportunity to attempt a career in Physics but mostly for the
patience and support that made me feel so much more confident after each meeting. His kindness and
savoir faire kept me going through the many occasions when all I could think about was giving up.
I thank all the colleagues at the Institut für Theoretische Astrophysik that in one way or another contributed
to my work or general sanity. I especially wish to express my gratitude to Björn Malte Schäfer and Matteo
Maturi for being available to help me with my everyday work problems and to Carlo Giocoli for prodiving
me with his code and thus a good head start. As a source of distraction, silliness and friendship, I thank in
particular the company of Christian Angrick, Federica Capranico, Alexander Gelsin, Alessandra Grassi,
Gero Jürgens and Eleonora Sarli.
The people I am about to mention are dear friends who have stayed close to me, and to my heart, before
and/or during the last 4 years. Since I do not know if I will ever again be able to thank them on paper for
being in my life and making me a happier person, I will now take my chance. To Elisabete da Cunha I
thank for the two balconies, the smell and taste of homely food and fun dinner parties. I thank Mischa
Gerstenlauer, for the companionship and for reminding me of the reasons why bicycles are important and
useful. I might get one, one day. I thank Alexander Karim for always being such a fun, understanding
and devoted friend. I thank Benoît Knecht, my dear partner of relaxing drinks, cafe-studying and movie
day-dreaming. Julian Merten, for the traditions he helped me start, be it the Bar Centrale beer, the
Bundesliga Saturdays or the slaps in my face. And for being a sweet and constant friend, even from the
distance of his new home, all the way across an entire ocean and continent. The sweet and caring nature
of Milica Mićić made my days at work, and beyond, more pleasant and worthy. I thank Adi Zitrin, my
wonderful office mate who provides me with fun evenings, uplifting company and extraordinary cities to
visit.
A very special thank you to Catarina Cruzeiro and Carolina Rodrigues for always making me feel truly
at home when I stay at theirs. To Gonçalo Gregório who, with his wonderful crew in Porto, always
99
Acknowledgments
100
recharges my batteries with craziness and laughter. Filipe Gomes, my dear, dear friend, relentlessly and
unconditionally there for me. And it is because of Mafalda Dias that, for the last ten years, I have been
more of a dreamer and paradoxically a more pragmatic person. Together with Sandra Castiço, she always
supplied me with inspiration, optimism and the (by now incurable) need for wonder.
Para com o meu pai António, a minha mãe Alcina e minha irmã Lena, tenho uma extraordinária dívida
de gratidão pelo apoio, calor familiar constante e amor incondicionais.
Finally, in a last, warmest of thank you notes, I address the one person who made it possible for this thesis
to ever be real. My heart is immersed in all he has given me throughout these years. I would need a whole
other list to thank him properly. Dearest Emanuel, my lover and my love.
Ana Valente | Heidelberg | October 2012
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