Sarli Waizmann dissertation

Sarli Waizmann dissertation
An algorithm for the reconstruction
of the projected gravitational potential
of galaxy clusters from galaxy kinematics
Eleonora Sarli-Waizmann
Dissertation
submitted to the
Combined Faculties of Natural Sciences and Mathematics
of the Ruperto-Carola-University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Eleonora Sarli-Waizmann
born in: Bergamo
Oral examination: December, 18th
An algorithm for the reconstruction of the projected gravitational
potential of galaxy clusters from galaxy kinematics
Referees:
Prof. Dr. Matthias Bartelmann
Prof. Dr. Luca Amendola
A mia nonna,
che mi ha insegnato a contare
Keine Experimente!
K. Adenauer
Summary: In this work we develop a method to incorporate the information from
galaxy kinematics into the reconstruction of the two-dimensional, projected gravitational potential of galaxy clusters. We start by deprojecting the observed line-of-sight
velocity dispersions of cluster galaxies with an application of Bayes’ theorem, the
Richardson-Lucy method, requiring the assumption of a shape for the cluster. Assuming spherical symmetry, after the deprojection we obtain an effective galaxy pressure,
i.e. the density-weighted radial velocity dispersions of the cluster galaxies, which is
then related to the three-dimensional gravitational potential by using the tested assumption of a polytropic relation between the effective galaxy pressure and the density. The
two-dimensional gravitational potential can finally be found by straightforward projection along the line of sight. We test the method with a numerically simulated triaxial galaxy cluster and the galaxies identified therein and perform the reconstruction
for three different lines of sight, initially assuming sphericity. Expanding the gravitational potential in the cluster’s geometrical ellipticities yields second-order corrections
to the spherical reconstruction. By comparing our results with the projected gravitational potential directly obtained from the simulation, we show that the deviation between the projected potential obtained with our reconstruction method and the potential
directly extracted from the simulation is . 10 % within approximately the virial radius
(1.5 h−1 Mpc) from the cluster centre in the case of a spherical cluster and remains moderate (below 10 % − 25 %) within the same radius in the case of an ellipsoidal cluster.
Zusammenfassung: In der vorliegenden Arbeit entwickeln wir eine Methode, um
die Information aus der Kinematik der Galaxien in die Rekonstruktion des zweidimensionalen, projizierten Gravitationspotentials von Galaxienhaufen einzubeziehen. Wir
beginnen mit der Deprojektion der beobachteten Geschwindigkeitsdispersionen der Haufengalaxien längs der Sichtlinie mithilfe einer Anwendung des Bayes’schen Theorems,
der Richardson-Lucy-Methode, die eine Annahme über die Form des Galaxienhaufens
voraussetzt. Unter der Annahme sphärischer Symmetrie ergibt die Deprojektion einen
effektiven Druck der Haufengalaxien, d.h. deren Dichte-gewichtete radiale Geschwindigkeitsdispersion, die dann durch eine angenommene und überprüfte polytrope Relation
zwischen dem effektiven Galaxiendruck und der Dichte mit dem dreidimensionalen
Gravitationspotential in Beziehung gesetzt wird. Das zweidimensionale Gravitationspotential ergibt sich schließlich durch Projektion längs der Sichtlinie. Wir überprüfen
die Methode anhand eines numerisch simulierten triaxialen Galaxienhaufens und darin
identifizierter Galaxien und führen die Rekonstruktion entlang dreier verschiedener Sichtlinien durch. Indem wir das Gravitationspotential in den geometrischen Elliptizitäten
entwickeln, leiten wir Korrekturen zweiter Ordnung an die sphärische Rekonstruktion
ab. Ein Vergleich unserer Ergebnisse mit dem aus der Simulation bekannten Gravitationspotential zeigt, dass die Abweichung zwischen dem projizierten Potential aus
unserer Rekonstruktionsmethode und dem bekannten Potential innerhalb des Virialradius (1.5 h−1 M pc) um das Haufenzentrum bei etwa 10% liegt und innerhalb desselben Radius auch bei einem rotationsellipsoidalen Galaxienhaufen vertretbar bleibt
(10% − 25%).
Contents
Abstract
i
Table of contents
i
List of figures
iv
List of tables
vii
1
Introduction and motivation
2
Foundations of cosmology and cosmic structure formation
2.1 An expanding Universe . . . . . . . . . . . . . . . . . .
2.2 The homogeneous Universe . . . . . . . . . . . . . . . .
2.2.1 The cosmological principle . . . . . . . . . . . .
2.2.2 Robertson-Walker geometry . . . . . . . . . . .
2.2.3 Redshift . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Dynamical properties of the metric . . . . . . . .
2.2.5 Matter models . . . . . . . . . . . . . . . . . .
2.2.6 Cosmological parameters . . . . . . . . . . . . .
2.2.7 Cosmological distances . . . . . . . . . . . . . .
2.3 Introducing inhomogeneities . . . . . . . . . . . . . . .
2.3.1 The growth of perturbations . . . . . . . . . . .
2.3.2 The power spectrum . . . . . . . . . . . . . . .
3
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Galaxy Clusters
3.1 Galaxy clusters: a brief overview . . . . . . . . . . . . . . . . . . . . .
3.2 Formation and abundance . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Spherical collapse model . . . . . . . . . . . . . . . . . . . . .
3.2.2 The density profile of galaxy clusters . . . . . . . . . . . . . .
3.2.3 The mass functions and the quest for a definition of the mass of
galaxy clusters . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Obervational properties of galaxy clusters . . . . . . . . . . . . . . . .
3.3.1 Gravitational lensing in clusters . . . . . . . . . . . . . . . . .
3.3.2 X-ray emission . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Thermal SZ effect . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Galactic dynamics . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
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Richardson-Lucy deprojection method
6.1 Inverse problems in astronomy . . . . . . . . . . . . . . . . . . . . . .
6.2 Implementation of the Richardson-Lucy algorithm . . . . . . . . . . .
73
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Spherical reconstruction of the lensing potential
7.1 Recovering the projected gravitational potential from the projected velocity dispersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Basic relations . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 The data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Testing the algorithm . . . . . . . . . . . . . . . . . . . . . . .
7.3 Application to MACS J1206.2 − 0847 . . . . . . . . . . . . . . . . . .
79
5
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The dynamical structure of galaxy clusters
4.1 The Vlasov equation . . . . . . . . . . . . . . . .
4.2 The Jeans equations . . . . . . . . . . . . . . . . .
4.2.1 The Jeans equation in spherical coordinates
4.3 Why gravitational lensing alone is not enough . . .
4.4 Why dynamics alone is not enough . . . . . . . . .
Introduction to the reconstruction method
5.1 The key idea . . . . . . . . . . . . . . . .
5.2 General calculation for the isotropic case .
5.3 Implementation of the method . . . . . .
5.3.1 Application to observational data
5.4 Application to the thermal SZ effect . . .
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Ellipsoidal reconstruction of the lensing potential
8.1 Introduction and motivation . . . . . . . . . . .
8.2 Description of the method . . . . . . . . . . .
8.3 Numerical tests . . . . . . . . . . . . . . . . .
8.3.1 The data . . . . . . . . . . . . . . . . .
8.3.2 Testing the algorithm . . . . . . . . . .
9
Summary and conclusions
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80
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107
Acknowledgements
111
Bibliography
113
iv
List of Figures
1.1
2.1
2.2
2.3
2.4
2.5
2.6
The projected gravitational potential is constrained by different observables at different scales. . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity-distance relation among extragalactic nebulæ [82]. . . . . . . .
Handwritten plot by Lemaı̂tre illustrating the time evolution of the radius of the universe with the cosmological constant (labelled a), for a
space with positive curvature. . . . . . . . . . . . . . . . . . . . . . . .
Visual comparison between the full-sky maps provided by COBE, WMAP
and Planck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Best-fit confidence regions in the Ω M − ΩΛ plane. . . . . . . . . . . . .
Joint confidence intervals for (Ω M , ΩΛ ) from SNe Ia. . . . . . . . . . .
The projected distribution of galaxies shown in this cone diagram is
evidently anisotropic. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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24
Key idea: Different phenomena can be described by a formally similar
algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Azimuthally averaged and normalised surface brightness profile of a
simulated galaxy cluster. . . . . . . . . . . . . . . . . . . . . . . . . .
Reconstructed and normalised projected potential of a simulated galaxy
cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Projected gravitational potential reconstructed using the method described
in [87] compared to the potential recovered from weak gravitational
lensing as a function of the projected radius s. . . . . . . . . . . . . . .
69
6.1
Sketch of the problem described in Richardson 1972. . . . . . . . . . .
76
7.1
7.2
Cluster geometry (see [26]). . . . . . . . . . . . . . . . . . . . . . . .
Relation (7.8) between the effective pressure and the density shown for
the Hernquist (Eq. (3.8)) and NFW (Eq. (3.9)) density profiles and
adopting the anisotropy parameter proposed by [79]. . . . . . . . . . .
(a) Surface-mass density of the simulated cluster g1 in the x-y plane.
(b) Two-dimensional gravitational potential obtained from the surfacedensity map solving Poisson’s equation via fast-Fourier transform. Both
images show regions with 10 h−1 Mpc side length. . . . . . . . . . . . .
80
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5.1
5.4
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18
Concentration parameter as a function of redshift. .
Abell 1689. . . . . . . . . . . . . . . . . . . . . .
Lensing geometry in the thin screen approximation.
Thermal Sunyaev-Zel’dovich effect . . . . . . . . .
5.3
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3.1
3.2
3.3
3.4
5.2
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82
v
List of Figures
7.4
Number density profile of the simulated cluster galaxies vs. the radius.
It is obtained by galaxy number counts in spherical shells. . . . . . . .
7.5 Radial profile of the anisotropy parameter β(r) defined in Eq. (7.3), obtained from the simulated cluster data. . . . . . . . . . . . . . . . . . .
7.6 Effective galaxy pressure vs. the density. The relation is represented
by a power law, supporting our assumption of an effective polytropic
relation. The mean polytropic index, as derived by linear regression, is
γ = 0.915 ± 0.022. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Results of the four different steps comprising our algorithm. . . . . . .
7.8 Reconstructed gravitational potentials in (a) three and (b) two dimensions are plotted as functions of radius and compared with the true potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9 The relative deviation between the reconstructed and true two-dimensional
gravitational potentials where the line of sight is taken along the z-axis
is shown here as a function of distance from the cluster center. The deviation remains moderate (below 10 %) within a radius of approximately
1.5 h−1 Mpc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10 Radial profiles of the gravitational potential of MACS J1206. . . . . . .
8.1
8.2
8.3
8.4
vi
83
85
86
87
89
90
91
Mass density profile of the simulated cluster galaxies vs. the radius. . . 100
Results of the four different steps comprising our algorithm. . . . . . . 101
Reconstructed gravitational potentials in two dimensions are plotted as
functions of radius and compared with the true potentials. . . . . . . . . 103
The relative deviation between the reconstructed and true two-dimensional
gravitational potentials where the line of sight is taken along the x-, yand z-axis is shown here as a function of distance from the cluster center. The deviation remains moderate (below 10 %−25 %) within a radius
of approximately the virial radius of the cluster 1.5 h−1 Mpc for all the
three cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
List of Tables
2.1
8.1
Constraints on Ωm0 (matter density parameter), ΩΛ0 (dark energy density parameter), Ωb0 (baryonic matter density parameter), H0 (Hubble
constant), nS (spectral index of the primordial power spectrum), σ8
(power spectrum normalisation), zeq (equality redshift) and the age of
the Universe estimated by [137]. . . . . . . . . . . . . . . . . . . . . .
31
Values of cluster parameters. . . . . . . . . . . . . . . . . . . . . . . . 104
vii
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1
Introduction and motivation
Over the course of millennia, any model of the cosmos, seen as an entity comprising
Heavens, Earth, natural phenomena, time and humanity, has mostly been rooted in religion. If one restricts oneself to an analysis of western history, it is already possible
to identify two main approaches to the Ancient Greek depiction of the Universe: the
one appealing to a divine order1 , such as in Homer and Hesiod, and the one anchoring
the discourse on the nature, formation and composition of the world to the logical reading of natural reality, such as in Anaxagoras, Empedocles, Democritus and Aristotle.
These two distinct views proceed along intertwining paths across history. Attempts to
disentangle the rational description of the Universe from its religious and metaphysical
aspects have met with intrinsic difficulties: the broad scope of the topic encompasses
cosmogony, questions on the origins and purpose of life and human consciousness, the
debate on the ultimate fate of the Universe and of mankind and, finally, knowledge arising from the collection of empirical data via careful observation of the firmament and
scrutiny of the skies by laymen and mathematicians, astronomers and philosophers.
Even with the birth of the scientific method and the development of modern science,
Cosmology was for a long time relegated to the rank of a largely speculative branch of
knowledge and it was not until the past century that it evolved into a rigorous scientific
discipline identifying the Universe as a whole as its main domain of study.
Despite its fast pace of development, though, many of the original questions about
the Universe remain unanswered and many more new questions arose.
One of the most profound problems that troubles the astronomers since the last century regards the actual composition of the Universe. The latest results from the Planck
mission [136] report 68.3% of dark energy, 26.8% of dark matter and 4.9% of ordinary,
baryonic matter for the energy-matter budget of the cosmos. The idea of an invisible
substance permeating the Universe is not new and the first account of a “dark matter”
can be traced back to a paper by Fritz Zwicky in 1933. In 1970, Vera Rubin conducted
what can be considered one of the first deductions of dark matter from observations.
Starting from an analysis of the spectra of spiral galaxies and the study of their rotation
1 The
same word κoσµoς in Greek stands for order.
1
Introduction
curves, she noticed a discrepancy between the theoretical expectation about those curves
and the actual observations. A way to resolve this conflict would have been questioning
the validity of Newton’s theory at large scales, another possible solution lay in admitting
that another kind of non-visible matter was present in the galaxies themselves and that
the actual mass of galaxies exceeded in good part the one predicted by the sole analysis
of their luminous content. In 2000, Yannick Mellier and his team analysed images from
the Canada-France-Hawaii Telescope’s high-resolution wide-field imaging camera and
published a cartography of the dark-matter distribution in a two-square-degree section
of sky containing around 200, 000 galaxies. This first publication led the way to a mapping of the dark-matter distribution in the local Universe. Today we know that the dark
matter in galaxies is at least ten times more than the visible matter. The ratio in galaxy
clusters is even more extreme: the dark matter content exceeds by thirty times that of
the luminous matter.
The success of the cosmological standard model implies that massive, gravitationally bound cosmic objects such as galaxy clusters should be dominated by dark matter
and characterised by its properties. Numerical simulations routinely find that the darkmatter distribution in galaxy clusters is expected to exhibit universal properties, for
example its radial density profile and its degree of substructure. For our understanding
of the nature of dark matter, it is important to test whether the mass distribution in real
clusters confirms the expectations from simulations.
Galaxy clusters are much less affected by baryonic physics than galaxies since the
baryonic cooling time exceeds their lifetime except in their innermost cores. Thus, they
represent an important probe for the nature of dark matter and play a key role in testing
our current understanding of cosmic structure formation.
A growing number of increasingly wide surveys in a broad range of wavebands
provide or will soon provide precise information on large galaxy-cluster samples. For
instance, one of the goals of the Cluster Lensing And Supernova survey with Hubble
(CLASH, [138]) is the mapping of dark matter in clusters based on strong and weak
gravitational lensing. The Dark Energy Survey (DES, [4]), started in September 2013,
will combine several probes of dark energy, and ESA’s Euclid mission [89] will mainly
focus on weak gravitational lensing measurements and galaxy clustering, but will also
provide data on galaxy clusters and the integrated Sachs-Wolfe effect. The Kilo Degree
Survey (KiDS) aims at mapping the matter distribution in the Universe by means of
weak gravitational lensing and photometric redshift measurements [46].
Strong and weak gravitational lensing are widely used as an effective tool for reconstructing the projected mass or gravitational-potential distributions in galaxy clusters
[18, 33, 37, 43, 105, 106, 109]. Lensing effects are due to light deflection alone and thus
(largely) insensitive to equilibrium and stability assumptions. They can be completely
characterised by the scaled and projected Newtonian gravitational potential of the lensing matter distributions and thus most directly constrain the projected, two-dimensional
2
Introduction
Figure 1.1: A quick look at the physics shows that all of these quantities constrain
the projected gravitational potential in a different way: strong lensing constrains the
determinant of the Hessian matrix of the lensing potential, ψ, i.e. the matrix of the
second-order partial derivatives of ψ; weak lensing constrains combinations of the second derivatives of ψ; X-rays and the thermal Sunyaev-Zel’dovich effect provide the
relation reported in the figure; galaxy kinematics constrain the gradient of the gravitational potential; gravitational flexion constrains the third-order derivatives of ψ. The
relation between galaxy kinematics and lensing is the main topic of this work and will
be investigated further in Chapters 7 and 8.
gravitational potential. Non-parametric, adaptive methods have been developed and are
now routinely being applied to recover cluster potentials. However, clusters provide a
multitude of other observables through X-ray emission, the thermal Sunyaev-Zel’dovich
effect and galaxy kinematics which can be used to investigate their internal structure.
X-ray emission and the thermal Sunyaev-Zel’dovich effect constrain respectively
the temperature, the density and the pressure of the intracluster gas, galaxy kinematics
constrain the gradient of the gravitational potential and gravitational lensing constrains
the curvature of the potential (i.e. the gravitational tidal field). The essential implication of these considerations is that each of these quantities provides information on
the projected gravitational potential at different scales, therefore allowing us to cover a
broad range of radii (from few kpc to over 1 Mpc) and thus to construct a much more
comprehensive picture of galaxy clusters, as sketched in Fig. 1.1.
Some of these observables have already been combined in joint methods to constrain the projected gravitational potential (strong and weak gravitational lensing, X-ray
emission and thermal Sunyaev-Zel’dovich effect, for instance) but what is missing is
3
Introduction
a consistent method to jointly constrain the projected gravitational potential of galaxy
clusters with as little prejudice as possible by making use of all these observables at
once. This is certainly an enticing prospect and represents the overarching theme of this
thesis. In particular, we will aim at devising an algorithm for the reconstruction of the
projected gravitational potential starting from some of the available observables listed
in Fig. 1.1.
As mentioned above, combining all observables has the substantial advantage that
all available information is bundled in single models, and that a range of linear scales
covering approximately two orders of magnitude can faithfully be bridged. We will start
by introducing the derivation of the projected, gravitational potential of a cluster from
its surface brightness using X-ray data as an input and, in a second moment, we will
sketch the steps needed to derive it from the observed relative changes in the intensity
of the CMB photons through the thermal SZ effect using SZ data. Subsequently, we will
focus on the relation between galaxy kinematics and gravitational lensing and present in
detail the development and implementation of an algorithm for the reconstruction of the
projected, gravitational potential of a galaxy cluster from the line-of-sight projections
of the velocity dispersions of its members with the goal of inserting this work into a
broader, joint method accounting for information from all available cluster observables
in the future. We will also argue that, despite the physics describing these three separate phenomena is profoundly different, it is possible to exploit their formal analogy to
develop one single reconstruction method.
A roadmap of the content of the following chapters can be sketched in this way.
The theoretical and physical framework for research on galaxy clusters is described in
Chapter 2, which is meant to offer a broad, panoramic view on cosmology. Chapter 3
discusses the main issues in introducing a basic model of structure formation and defining a density profile and a mass function for galaxy clusters, quantities that are central
in any attempt to describe them in a comprehensive way. The chapter ends with a brief
account of the observational properties of galaxy clusters that will be used in the course
of this work. The dynamical structure of clusters of galaxies is reviewed in Chapter 4.
The key idea which our reconstruction algorithm is based on is introduced in Section 5.1
and the method for the reconstruction of the projected gravitational potential of galaxy
clusters from the X-ray emission of thermal gas or its interaction with CMB photons is
outlined in the second part of Chapter 5. Sect. 5.3 presents the results for X-ray data
and section 5.4 delineates the steps necessary to implement the algorithm in the case
of the thermal SZ effect. Chapter 6 briefly presents the Richardson-Lucy deprojection
method, which will be used to retrieve three-dimensional quantities from projected ones
in order to insert them in the reconstruction algorithm. In Chapters 7 and 8, the problem of reconstructing the projected gravitational potential starting from the projected
velocity dispersions of the cluster galaxies along the line of sight is discussed. Chapter
7 presents the formulation of the reconstruction algorithm for a spherical galaxy cluster,
4
Introduction
while the assumption of spherical symmetry is relaxed in Chapter 8 in favour of triaxiality. The main results of this thesis are summarised in the Conclusions.
Part of the content and of the ideas presented in this thesis has appeared already, or
will soon appear, in the following publications:
• D. Stock, S. Meyer, E. Sarli, M. Bartelmann, I. Balestra, C. Grillo, A. Koekemoer,
A. Mercurio, M. Nonino, P. Rosati. The projected gravitational potential of the
galaxy cluster MACS J1206 derived from galaxy kinematics., submitted to A&A,
2015
• C. Tchernin, C. L. Majer, S. Meyer, E. Sarli, D. Eckert, M. Bartelmann. Reconstructing the projected gravitational potential of Abell 1689 from X-ray measurements., A&A, 574, id.A 122, 7 pp., 2015
• E. Sarli, S. Meyer, M. Meneghetti, S. Konrad, C. L. Majer, M. Bartelmann. Reconstructing the projected gravitational potential of galaxy clusters from galaxy
kinematics., A&A, 570, id.A 9, 9 pp., 2014
• C. L. Majer, S. Meyer, S. Konrad, E. Sarli, M. Bartelmann. Reconstruction the
mass distribution of galaxy clusters from the inversion of the thermal SunyaevZel’dovich effect., in prep.
• S. Konrad, C. L. Majer, S. Meyer, E. Sarli, M. Bartelmann. Joint reconstruction
of galaxy clusters from gravitational lensing and thermal gas. I. Outline of a
non-parametric method., A&A, 570, id.A 118, 7 pp., 2013
5
He who does not know what the world is, does
not know where he is. And he who does not
know for what purpose the world exists, does
not know who he is, nor what the world is.
Marcus Aurelius
2
Foundations of cosmology and cosmic
structure formation
A debate on the origins and structure of the Universe is present in every civilisation
and has been declined in various ways throughout all human history. For millennia,
though, religious beliefs and scientific understanding of the world have been blended
together in a discipline that combined observations with faith. Modern cosmology, in
its interpretation as the branch of astrophysics devoted to the study of the Universe as a
whole, is a very young science and took its actual form only over the past century. The
purpose of this chapter is to give an idea of the central concepts and discoveries at the
base of modern Cosmology. Section 2.1 provides a very broad historical overview on
the milestones that shaped the evolution of our current understanding of the Universe.
Section 2.2 deals with the theoretical framework (Sect. 2.2.1 to 2.2.5), the definitions
of distances (Sect. 2.2.7) and the parameters (Sect. 2.2.6) describing a homogeneous
Universe and presents the latest constraints obtained on them. Sect. 2.3 is meant to
introduce the basic theory illustrating the formation of cosmic structures: we start from
motivating the necessity of introducing inhomogeneities in our picture of the evolution
of the Universe and we proceed with an outline of the main tools at our disposal to
describe the growth of perturbations in Sect. 2.3.1. We discuss their statistical properties
in Sect. 2.3.2.
2.1 An expanding Universe
The birth of modern Cosmology is traditionally associated with the formulation of Einstein’s general theory of relativity in 1915 [59].
It is arguable whether pinpointing a given time in History in order to identify a
turning point in a specific discipline would not result in a pure intellectual exercise,
given the intrinsic non-linearity that characterises the development of physical sciences
in general. In the particular case of Cosmology, though, it is interesting to notice how
such an arbitrary and debateable choice rests on the grounds of the conceptual pitfalls
7
Chapter 2. Foundations of cosmology and cosmic structure formation
hidden in the principle of inertia, one of the cornerstones of Newtonian dynamics.
Newton’s theory postulated the existence of preferred motions in space, defined as
inertial reference frames. The simultaneous presence of both preferred reference frames
and a relativity principle in the theory led to an insoluble contradiction and codified
the distinction between inertial and non-inertial motions. In the latter case, the laws
of dynamics need the introduction of absolute space as a privileged reference frame
with respect to which definitions like the one of uniform and rectilinear motion could
hold. Newton labeled the absolute space sensorium dei [144], God’s sensory organ,
with respect to which bodies could be declared truly moving or truly at rest.
Since the publication of Isaac Newton’s work, Philosophiæ Naturalis Principia
Mathematica [122], in 1687, the concepts of absolute space, time and motion (including
rotations and accelerations) became pillars of the broadly accepted dynamical description of the world until the end of the 19th century and were hardly challenged, with
the notable exceptions of Gottfried Wilhelms Leibniz [41], George Berkeley [20] and
Carl Neumann [120]. In the General Scholium appended to the Principia in 1713, Newton described the famous rotating bucket argument, designed to support his position in
favour of absolute motion. The experiment is prepared by hanging a bucket filled with
water to a very long cord attached to the ceiling and turning the cord around until it is
strongly twisted. When the water is at rest with the bucket, the cord is left unwinding
and the bucket starts rotating in the opposite direction because of the torsion of the cord.
Newton observed four distinct stages of this process. In the very first one, the bucket is
standing still, the cord is tightly twisted and the water surface is flat. At this initial stage,
there is no deformation and no relative motion between the bucket and the water. In the
second moment, the cord is permitted to unwind and the bucket starts rotating. Since
the motion of the bucket is not yet communicated to the water, there is no deformation
of the surface of the water but there is a relative motion between the bucket and the
water. In the third phase, the water starts rotating and slightly ascending the walls of the
bucket. In the fourth and last phase, the increased rotation will impress a concave shape
to the surface of the water, deformation that will remain even when the water and the
bucket will have the same angular velocity and therefore no relative motion.
From this experiment Newton concluded the impossibility to identify a correlation
between the deformation of the water surface and the relative motion because one can
observe no deformation both when there is and when there is not a relative motion between water and bucket. This observation led him to postulate the existence of absolute
space as a reference frame for the centrifugal forces.
For more than a century, inertia was seen as an absolute property of nature. This
principle remained, as well, in the attempt of a mechanical interpretation of the electromagnetic phenomena and in the aether’s theory.
Mach’s principle. In the decades following the publication of the Principia, several
criticisms to Newton’s argument were expressed. All of them, though, proved
8
2.1. An expanding Universe
inconclusive. The very first cogent objection to the dominating idea of absolute
space was articulated by Ernst Mach. In a treatise published in 1872, History
and Root of the Principle of the Conservation of Energy [100], he manifested his
conviction that the law of inertia had been wrongly interpreted:
If we think of the Earth at rest and the other celestial bodies revolving around it,
there is no flattening of the Earth ... at least according to our usual conception of
the law of inertia. Now one can solve the difficulty in two ways; either all motion
is absolute, or our law of inertia is wrongly expressed ... I [prefer] the second.
In 1893 [99] he directly addressed the bucket argument and asserted that:
Newton’s experiment with the rotating vessel of water simply informs us, that the
relative rotation of the water with respect to the sides of the vessel produces no
noticeable centrifugal forces, but that such forces are produced by its relative
rotation with respect to the mass of the earth and the other celestial bodies. No
one is competent to say how the experiment would turn out if the sides of the
vessel increased in thickness and mass till they were ultimately several leagues
thick. The one experiment only lies before us, and our business is, to bring it into
accord with other facts known to us, and not with the arbitrary fictions of our
imagination.
In other words, Mach identified the origin of centrifugal forces as gravitational
and referred them to the action of the mass of celestial objects on other masses.
The notion that the distribution and motion of the matter in the Universe determines the inertial reference frames was later designated by Albert Einstein as
Mach’s principle [62, 128].
General Theory of Relativity. With the formulation of the special theory of relativity
[58], it became clearer that the fundamental question triggered by Newton’s interpretation of the concept of inertia (“with respect to what reference frames are
the laws of dynamics valid?”) was wrongly posed. Newton’s theory of dynamics could hold without absolute space as its axiomatic basis and offered a method
to construct an infinite class of equivalent inertial frames. Despite offering profound modifications to the “theory of space and time”, special relativity was not
immune to the same “epistemological defect” [60]: in a similar fashion to their
predecessor, absolute space, inertial frames singled out a preferred state of motion
and were not an “observable fact of experience”. In Einstein’s words, “it conflicts
with one’s scientific understanding to conceive of a thing which acts but cannot
be acted upon”.
This realisation accompanied Einstein in his work for more than a decade, during
which two major moments can be identified. The first achievement is the formulation of the principle of equivalence as described in a letter to Sommerfeld written
9
Chapter 2. Foundations of cosmology and cosmic structure formation
on 5th November 1908. A vital assumption is the equivalence of gravitational
(schwere) and inertial (träge) masses (letter to Wien, 10 July 1912 [63]). The second breakthrough is the final presentation of his new theory of gravity in a series
of lectures to the Prussian Academy of Sciences, the first of which was given on
the 25th of November 1915 [59] and introduced the field equations describing the
dynamics of the gravitational field. He published them in 1916 in the Annalen
der Physik in an article titled “Die Grundlage der allgemeinen Relativitätstheorie”1 [60]. There he noted that a definite answer to the lingering problem of inertia
could be offered by a theory that would focus on observable facts alone, relative
motions. Instead of limiting the validity of physical laws to the class of inertial
frames, of which it would be impossible to say whether they exist independently
of the rest of the Universe, the relativity principle should be extended and the new
theory should embrace the postulate of general covariance:
the laws of nature must be fully independent of the
choice of any coordinate system or reference frame.
This assumption implies that the physical laws must be covariant for all possible
transformations and substitutes the inertial frames with a class of freely-falling,
non-rotating reference frames, that are locally inertial.
The concept of local inertial frames is a direct consequence of the second pillar
on which Einstein’s theory of gravity was built, the principle of equivalence:
an accelerated, non-rotating laboratory in the absence of gravitational effects
and a non-accelerated laboratory endowed with a gravitational field are
equivalent.
The combination of general covariance and principle of equivalence is the backbone of this new worldview. On one hand, general covariance marks the exquisitely
geometric character of general relativity, whose geometric arena is differential geometry. On the other hand, the equivalence principle prescribes that a locally constant and linear (i.e. non-rotating) acceleration field is locally indistinguishable
from the gravitational field.
The role of the masses and of energy is now to curve the space-time, i.e. a pseudoRiemannian manifold whose metric tensor is treated as a dynamical field, in their
surroundings and the law of gravitation turns into the structure condition of a
curved and extremely flexible environment.
1 The
10
Foundation of the General Theory of Relativity
2.1. An expanding Universe
Einstein-de Sitter. The years between 1915 and 1930 saw the publication of some of
the most relevant and influential theoretical works in the history of relativistic
Cosmology. The formulation of the field equations paved the way to the application of general relativity to construct cosmological models. Driven by the persuasion that a Universe obeying Mach’s principle was epistemologically preferable,
Einstein [61] ensured that his field equations would not have an acceptable solution in the absence of matter by including the term −λgµν in the left-hand side,
where the coefficient λ was a “temporarily unknown universal constant”2 , today
known as cosmological constant and denoted with the Greek capital letter Λ. The
effectively repulsive cosmological term was supposed to counterbalance the effects of gravity and enforce a static and closed cylindrical Universe, in which
Einstein believed Mach’s principle would hold [21], without destroying the general covariance. The importance of Mach’s principle in the development of general relativity and of Einstein’s cosmological model is still matter of debate. 3
The essential point is that Mach managed to turn the attention from the motion
of a body with respect to another body to the study of the Universe as a whole,
effectively initiating the sequence of considerations and analyses that led to the
birth of modern Cosmology.
Less than seven weeks after the paper by Einstein was published, the Dutch astronomer Willem de Sitter [48, 49], with whom Einstein had frequently corresponded in the previous months, presented an exact solution of the generalised
field equations4 that proposed a non-Machian cosmological model based on a
high degree of space-time symmetry. De Sitter’s spherical solution offered the
possibility of a Universe in which the presence of matter was not required and
that stood in intense anthitesis with Einstein’s world view.
Friedmann-Lemaı̂tre-Robertson-Walker. In 1922 the Russian Alexander Friedmann5
sent to Paul Ehrenfest a manuscript accompanied by a letter in which he asked him
to publish his paper about the curvature of space. An excerpt of this letter6 reads:
I’m sending you a brief note regarding the question about the possible shape of the
universe more general than the cylindrical world of Einstein, and the spherical
[61] introduced λ first in the Poisson equation, i.e. into Newtonian theory, and subsequently
in the field equations.
3 Einstein noted later that Mach’s principle played a purely heuristic role in the development of general
relativity, for instance. For a very interesting account of the debates that accompanied the birth of modern
cosmology, the reader is referred to [21].
4 Therefore including the cosmological term.
5 The presence of multiple spellings of his surname is in part due to the confusion made from the
editors of Die Zeitschrift für Physik who typed Friedmann in 1922 and Friedman in 1924. The modern
transliteration into English of his last name is Fridman but we will adopt the original one of 1922 as it is
still the most commonly used.
6 https://www.lorentz.leidenuniv.nl/history/Friedmann_archive/
2 Einstein
11
Chapter 2. Foundations of cosmology and cosmic structure formation
world of De Sitter; aside from these two cases there appears also a world, the
space of which possesses a curvature radius varying with time; it seemed to me
that a question of this sort may interest you or De Sitter. In the near future I will
send you a German translation of this note, if you find the question considered in
it interesting, then please be so kind to have it placed in some journal.
The paper was published in the German journal Die Zeitschrift für Physik [72] and
showed that Einstein’s emphasis on the necessity of a static Universe was misplaced. If the spatial curvature of the Universe was taken to be time-dependent,
the field equations allowed for non-static cosmologies with positive or negative
curvature. Friedmann’s analysis included Einstein and de Sitter’s world models
as special cases in the assumption of a time-independent curvature and continued to a generalisation that led to explicitly evolving cosmological models. As
in Einstein and de Sitter’s solutions, the Universe is self-contained and unfolds
into nothing. A remarkable point of this consideration is set, in Friedmann’s first
paper, by the definition of the growth time of the curvature radius from 0 to a
certain value R0 as die Zeit seit die Erschaffung der Welt, the time since the creation of the world. This definition contains in nuce the idea of the Big Bang. The
fundamental contributions of Friedmann are not limited to the construction of a
general model that could incorporate the apparently conflicting cosmological applications elaborated by Einstein and de Sitter. In the 1922 paper, he formulated
the two basic assumptions which, together with the general theory of relativity,
represent the foundations of the cosmological standard model: the principles of
isotropy and homogeneity of the Universe. In a second paper [73], published
in 1924, he provided the solution for a Universe with negative spatial curvature
definitely proving that a static solution would not be stable.
The notion that the Universe originates from nothing was later formalised by
George Lemaı̂tre who, in the English translation of his article from 1927, speculated about a “primeval atom” [90, 91, 98], an initial point from which the Universe expanded. The relevance of Lemaı̂tre’s work, acknowledged only after the
publication in 1931 of his article in the Monthly Notices of the Royal Astronomical Society upon Eddington’s recommendation, is not restricted to the idea
of the Big Bang. In his 1927 paper, Un Univers homogène de masse constante
et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extragalactiques, he managed to mark the causal connection (as attested by its title) between the cosmic expansion deriving from the non-static solutions of Einstein’s
field equations and the recent observations of the recession velocities of extragalactic nebulæ. Lemaı̂tre derived differential equations for the curvature and
the density identical to those found by Friedmann by following a rather different
approach. For the first time in a generally relativistic cosmological model, he introduced a concept borrowed from thermodynamics, the conservation of energy,
12
determined. Distances can then be calculated from the velocities corrected for solar motion, and absolute magnitudes can be derived from the
apparent magnitudes. The results are given in table 2 and may be
compared with the distribution of absolute magnitudes among the nebulae
2.1. An
expanding
N.Universe
G. C. 404
criteria.
in table 1, whose distances are derived from other
0.
S0OKM
o~~~~~~~~~~~~~~~~
0
DISTANCE
0
IDPARSEC
FIGURE 1
S
2 ,10 PARSECS
Velocity-Distance Relation among Extra-Galactic Nebulae.
Figure
Velocity-distance
among
extragalactic
nebulæagainst
[82].
for solar
are plotted
motion,
Radial2.1:
velocities,
correctedrelation
distances estimated from involved stars and mean luminosities of
nebulae in a cluster. The black discs and full line represent the
and included both the radiation pressure and the matter density in the energysolution for solar motion using the nebulae individually; the circles
momentum tensor while assuming that the matter density contribution would be
and broken line represent the solution combining the nebulae into
negligible. These equations were the starting point from which he could derive
groups; the cross represents the mean velocity corresponding to
the
of proportionality
between
the recession
andbetheestidistances
whose
distancesvelocities
could not
therelation
of 22 nebulae
mean distance
that
is now
known as Hubble’s law and, by accounting for observations made by
mated
individually.
Strömberg with the Mount Wilson telescope, give an explicit expression to the
proportionality constant, today known as the Hubble constant.
can be excluded, since the observed velocity is so small that the peculiar
most
of homogeneous
andthe
isotropic
models
within the
frameThe
with
effect.
in comparison
distance
must
begeneral
motion The
object
large form
work
of
general
relativity
was
formalised
by
Robertson
[145]
and
Walker
[176]
in
is not necessarily an exception, however, since a distance can be assigned
landmark papers during the second half of the 1930s and the metric of space-time
for which
the peculiar motion and the absolute magnitude are both within
derived by these four scientists is today called Friedmann-Lemaı̂tre-Robertsondetermined. The two mean magnitudes, - 15.3
the range
previously
Walker
metric.
and - 15.5, the ranges, 4.9 and 5.0 mag., and the frequency distributions
for these two entirely independent sets of data; and
areHubble’s
similar
closelylaw.
Lemaı̂tre’s discovery, published in French in an obscure Belgian jourto the
can be attributed
mean
innot
difference
even the
magnitudes
nal,slight
was overlooked.
It was
until 1929
that the interpretation
of cosmological
unforced
This by
Cluster.
the Virgo
nebulaeofincosmic
entirely
bright,
selected,redshift
very as
a consequence
expansion
was formalised
Edwin Hubble
in aandvery
relation
of extragalactic
the velocity-distance
thea sample
validity
agreement
[82].supports
Hubble used
of 24
nebulæ for which
distances
velocities were available at Mount Wilson Observatory and found a “roughly linear relation between velocities and distances among nebulæ” that will later take
the form:
13
Chapter 2. Foundations of cosmology and cosmic structure formation
v = H0 d,
(2.1)
and is now known as Hubble’s law. Hubble’s law quickly became the strongest
observational evidence supporting the hypothesis of an expanding Universe. Distances and velocities were not measured directly. Hubble estimated the galaxy
distances from luminosities of “Cepheid variables, novæ and blue stars involved in
emission nebulosity” and the velocities from the redshift measurements by Vesto
Slipher [160] and Milton Humason, Hubble’s assistant at Mount Wilson. Due to
the large errors in the calibration of the distances and the fact that at that time it
was not known that different subcategories of Cepheids exist and have different
properties, the value obtained by Hubble for his eponymous constant was around
500 km/s/Mpc, notably higher than the one accepted today.7 The most recent
Planck results point in fact toward a value of 67.4 ± 1.4 km/s/Mpc [136].
Figure 2.1 shows the original plot of the relation between ”radial velocities and
distances among extragalactic nebulæ” that is now called Hubble diagram.
Dark Matter. Since 1844 and the discovery of Sirius B by Friedrich Bessel [22], astronomers used an observational strategy that relies on the gravitational effects
that ”hidden” matter exerts on visible matter in order to detect new astronomical
objects. The same strategy was applied to the case of the anomalous motions of
the planet Uranus and led to Leverrier’s discovery of Neptune in 1846 [74]. The
generalisation of this method came in the course of the 1930’s with the works of
Oort, Zwicky and Babcock. In 1932, Jan Oort [123] identified the motion perpendicular to the galactic plane 8 of the nearby stars as an indicator of the presence of
dark matter in the neighbourhood of the Sun and observed that ”it might be a quite
considerable quantity”, comparable with the quantity of the luminous matter. A
year later, Fritz Zwicky extended this method to the study of clusters of galaxies
[186]. The significancy of his contribution was due to the idea of applying for
the very first time the virial theorem to the Coma cluster. In this way he obtained
evidence of unseen mass, which he referred to as dunkle Materie, that totally
dominated the cluster and accounted for the missing mass necessary for the cluster to be held together by gravity. Similar evidence of a very high mass-to-light
ratio was found in 1936 by Sinclair Smith [161] for the Virgo cluster. One of the
earliest indications of dark matter in galaxies is contained in Horace Babcock’s
PhD dissertation [9]. In his work he studied the rotation curve of the Andromeda
7 The
estimate by Lemaı̂tre was 575 km/s/Mpc. Observing on the dimensions of the errors, he noted:
Tout ce que l’imprécision des observations permet de faire est de supposer v proportionell à r et d’essayer
d’éviter une erreur systématique dans la détermination du rapport v/r [90].
8 i.e. the vertical motion
14
2.1. An expanding Universe
galaxy and his measurements revealed a radial growth of the mass-luminosity ratio which he did not attribute to dark matter but either to absortion “in the outer
portions of the spirals” or to further unspecified vnew dynamical considerations,
which will permit of a smaller relative mass in the outer parts”.
Dark matter soared to the rank of a scientifically interesting problem only in the
1970’s [57, 124], from the observational side, when its presence was systematically noticed both in spiral and dwarf galaxies9 and in galaxy clusters and, from
the theoretical side, when its existence was incorporated in models of structure
formation. Today, despite the absence of a general consensus10 , dark matter is
considered by most to be the responsible factor for the discrepancy between the
predicted and observed rotation curve of spiral galaxies and has become one of
the fastest growing fields of research in astrophysics.
The necessity to explain rotation curves of galaxies and mass-to-light ratios of
galaxy clusters are only two of the main arguments that led to postulate the existence of dark matter. Other two crucial reasons in support of its introduction
are to be found in the Big Bang Nucleosynthesis and in the Cosmic Microwave
Background theories.
Big Bang Nucleosynthesis. In the 1940’s, physicists and astronomers who supported
the Big Bang model had at their disposal a coherent mathematical and physical
framework based on the FLRW metric, the cosmological principle and the idea
of an expanding Universe. Many questions, though, remained unanswered. One
of these pressing problems regarded the origin of the light element abundances.
George Gamow had pointed out that several nuclear species must have originated
“as a consequence of a continuous building-up process arrested by a rapid expansion and cooling of the primordial matter” [5, 75]. In order to find a quantitative agreement between the predicted and observed primordial composition of
the Universe, it was necessary to assume that the high temperatures of the early
Universe would trigger processes of nuclear fusion able to produce the required
proportions of 3 He, 4 He, 7 Li and 2 H. A first publication pointing in this direction was the letter sent on April 1, 1948 to the Physical Review by Ralph Alpher,
Hans Bethe and George Gamow. In their paper, later known as αβγ paper from the
surnames of its authors, they successfully managed to provide a correct account
of the relative abundances of the hydrogen and helium isotopes. Even though
they wrongly claimed the formation of heavier atoms via the successive capture
of neutrons by the newly formed nuclei, this letter marked the beginning of a new
theory: the Big Bang Nucleosynthesis (BBN), also called primordial nucleosyn9 Later
studies of the velocity dispersions in elliptical galaxies led to the same conclusions.s
exist models suggesting alternative explanations, such as the MoND (Modified Newtonian
Dynamics).
10 There
15
Chapter 2. Foundations of cosmology and cosmic structure formation
thesis. A further step towards an explanation of the heavy element abundances
was taken by Margaret Burbidge, Geoffrey Burbidge, William Fowler, and Fred
Hoyle who, in 1957, formulated the hypothesis that only hydrogen, helium and
deuterium are of cosmological origin, while heavier atoms are created in stars
[36]. Their report on the “Synthesis of the elements in stars” is today credited as
the birth of the stellar nucleosynthesis model.
The statement that the light elements form in the early Universe is a decisive
implication of the Big Bang theory and the observational evidence found in its
support represent one of the theory’s most striking triumphs. The primordial nucleosynthesis and the use of scaling arguments also allow an estimation of the
cosmic baryon density from the proton and neutron densities. The quantity obtained in this way, already in the 1970’s, appeared to be too low and in contrast
with the estimate deriving from calculations of the expansion rate of the Universe.
Also in this case a solution to the missing matter problem was prompted by the
introduction of dark matter into the theory.
Cosmic Microwave Background. Another fundamental prediction of Big Bang Cosmology is the existence of the Cosmic Microwave Background (CMB) radiation,
the relic thermal radiation of photons that started freely propagating when the
matter became neutral after the recombination of electrons and protons. The recombination occurred when the age of the Universe was 3 × 105 years11 and its
temperature had lowered to 3 × 103 K: photons were no longer prisoners of Thomson scattering and could decouple from matter. This event had three relevant consequences:
1. The thermodynamical equilibrium reached by photons and electrons before
decoupling imprinted the photons of the last scattering surface with a blackbody spectrum at a temperature of 2.72548 ± 0.00057 K [69].
2. The black-body spectrum was preserved by the cosmic expansion, whose
only effect is a systematic increase in the wavelengths.
3. The CMB appears, on large scales, isotropic and homogeneous to all fundamental observers, further endorsing the Cosmological Principle.
The first prediction of a cosmic radiation background was connected with the
requirement of the BBN theory to have a hot beginning. In 1948 Alpher and
Herman forecasted a temperature of around 5 K for the relic radiation [6]. Publications with different methods and values for the temperature of the CMB were
issued and discussed for more than a decade before a direct detection was made.
Finally, in 1964, Robert Dicke, who was setting up with his group at Princeton
11 At
16
a redshift z = 1100.
2.1. An expanding Universe
University a radiometer to try to measure it, received a phone call from Arno Penzias and Robert Woodrow Wilson, two physicists working at the Bell Telephone
Laboratories in a nearby town. The two Bell employees were working on a new
type of antenna and had found a higher noise temperature than expected. This
excess temperature of approximately 3.5 ± 1.0 K was described as “isotropic, unpolarized, and free from seasonal variations” [132]. The interpretation of it as
the signature of a hot Big Bang came from the Princeton group, whom they had
scooped, Robert Dicke, P. J. E. Peebles, P. G. Roll and D. T. Wilkinson [54].
The discovery of radiation from the “primeval fireball” offered new insights into
the origin of galaxies. Firstly, it set a time when bound structures like galaxies
could start to form and, secondly, it could be regarded as the sought-after mean to
imprint onto the power spectrum of initial density fluctuations some characteristic
features, such as lengths and masses, that could serve as observational probes for a
confirmation of the theory. The prediction of structures in the CMB was independently formulated in 1970 by Peebles and Yu [130] and Sunyaev and Zel’dovich
[167, 168]. In their works, the two groups derived what is today called baryon
acoustic oscillations in the galaxy power spectrum and suggested that their relative amplitude should be of the order of 10−4 . Subsequent attempts to measure
these fluctuations failed and in 1982 Peebles [126] proposed a model assuming
that “the mass of the Universe is dominated by a gas of particles that interact
weakly, if at all, with baryonic matter and radiation. These dark matter particles
have negligible primeval velocities, hence the eventual name, cold dark matter,
and the model name, CDM” [129]. The CDM model rested on the assumption
of general relativity as the correct theory of gravity at cosmological scales and
was based on a FLRW metric and the cosmological principle. It described the
geometry of the observable Universe on the basis of the ratio between the actual
matter density and the critical density for which the Universe is spatially flat, a
quantity that can be derived from Friedmann’s equations by assuming that the
cosmological constant is zero.
Inflation. A solid motivation for supporting a model based on the concept of a critical
density was provided by the theory of cosmic inflation. Friedmann cosmology
in the 1970’s was far from being a complete theory. In particular, two major
problems had been discovered which needed a careful explanation: the flatness
problem and the horizon problem.
The flatness problem is a question concerning the fine-tuning of the density of
matter and energy in the universe and was originally identified by Robert Dicke
in 1969. A key prediction of Friedmann’s world model, deriving from the combination of general relativity and the cosmological principle, is that three types
of universe can be possible, as sketched in Figure 2.2. If there is sufficient mat17
Chapter 2. Foundations of cosmology and cosmic structure formation
Figure 2.2: Handwritten plot by Lemaı̂tre illustrating the time evolution of the radius
of the universe with the cosmological constant (labelled a), for a space with positive
curvature. All the models start with a singularity in (x = 0, t = 0). For a sufficiently
large cosmological constant, the universe becomes open. The most recent cosmological
data are compatible with a Lemaı̂tre’s solution with positive curvature and accelerated
expansion (top curve) [98].
ter density for gravity to triumph over expansion, the universe will be spatially
closed; if the matter density is not high enough, it will be spatially open and if
gravity balances expansion, the Universe will be spatially flat or, in other words,
Euclidean. Dicke noted that even the tiniest departure from the critical density
in the early Universe would have been enormously magnified during the expansion, leading either to a closed or an open universe. In either case, no structure
formation in the way observed today would have been possible.
The horizon problem is an issue regarding the finiteness of the speed of light
on one hand and the homogeneity of the CMB spectrum on the other. It was
pointed out by Charles Misner that patches of the sky that are not in causal contact
with each other should exhibit different physical properties and yet the cosmic
background radiation appears isotropic and homogeneous everywhere in the sky.
These two problems, among others (such as the absence of magnetic mono-poles),
could be solved in 1981 by Alan Guth [78], who postulated a very early phase of
inflationary expansion in the evolution of the Universe (at some time between
10−36 s and 10−33 s after the Big Bang) which reconciled theory with observations. In the same year Mukhanov and Chibisov [113] noticed that inflated quantum fluctuations could lead to the formation of cosmic structures. In the following
18
2.1. An expanding Universe
years, numerous variants of the inflation theory were proposed and some of the
detailed predictions on the initial conditions of the Universe provided by inflationary theories could be experimentally verified only in recent years. The existence
of fluctuations in the CMB has been proven by missions like COBE, WMAP and
Planck. The detection of quantum fluctuations in the gravitational field, known as
gravitational waves, was announced by the BICEP2 collaboration in March 2014
but their extraordinary claims, if not completely dismissed, were proven to be
premature by the Planck collaboration [135].
ΛCDM. Right after its debut in 1982, the physical implications of the Cold Dark Matter
model were explored by many groups and several candidates were soon proposed
[1, 29, 32, 139]. The idea that cold dark matter would not interact with radiation,
while baryons were still coupled to the CMB, would provide an explanation of
two apparently contrasting observations: the clumpy distribution of matter in the
Universe on one side and the simultaneous presence of tiny anisotropies in the
CMB on the other. In 1984 Peebles [127] suggested a further modification to the
CDM model: by introducing a cosmological constant it would be possible to go to
lower mass density while permitting spatial flatness and meeting the requirements
of the inflationary scenario. The same strategy was suggested by Kofman and
Starobinskii [86] a year later and the modified framework they proposed took the
name of Lambda-CDM (ΛCDM).
In 1987 numerical simulations carried out by Bond and Efstathiou [31] and by
White, Davis, Efstathiou and Frenk [180] contributed to the promotion of ΛCDM
as a promising model for structure formation analyses.
In 1992 reports of the findings of the COsmic Background Explorer (COBE)
mission launched a few years earlier were announced. COBE had provided evidence that the CMB has a nearly perfect black-body spectrum and that very faint
anisotropies could be detected in it. These findings scored a crucial point in favour
of the Big Bang theory by fulfilling its predictions. In the course of the years, two
major follow-up missions were launched: the Wilkinson Microwave Anisotropy
Probe (WMAP) and Planck satellites.
For nearly ten years (February 2003 - December 2012) WMAP data releases,
containing a full-sky map of the CMB, helped refining the measurements of the
temperature fluctuations in it and provided further constraints on cosmological
parameters and other model values, such as the age of the Universe and the Hubble
constant. In the abstract of the 9-Year release paper [19], the authors conclude:
“With no significant anomalies and an adequate goodness-of-fit, the inflationary
flat ΛCDM model and its precise and accurate parameters rooted in WMAP data
stands as the standard model of cosmology.”
The last mission devoted to studying the CMB anisotropies in order of time is
19
Chapter 2. Foundations of cosmology and cosmic structure formation
Figure 2.3: Visual comparison between the full-sky maps provided by COBE, WMAP
and Planck. The three satellites provided solid observational evidence in support of the
idea of cosmic microwave background by identifying the predicted small temperature
fluctuations arising from early weak fluctuations in the matter distribution.
the European Planck cosmology probe. The three-year results were released in
February 2015 and the Planck all-sky map is to date the most detailed image of the
early Universe. For an overview of the Planck estimates of the main cosmological
parameters, see Table 2.1.
The published results provide both striking support to the notion of inflation and
show remarkable agreement with the ΛCDM model. The confirmation that the
Universe is spatially flat, together with the recent developments on inflation and
the discovery of the accelerated expansion of the Universe complete the cosmological standard model.
Accelerated expansion. A method to gain information on the geometry of the universe
was inaugurated by Hubble and is based on the observation of the standard candles, a class of objects of which either the absolute magnitude, or luminosity,
is known or the relation between their luminosity change and redshift is wellunderstood. The same method has been used in recent years on Type-Ia Supernovæ. The most stunning result of this application was published nearly simulta20
2.1. An expanding Universe
3
No Big Bang
ΩΛ
2
99%
95%
90%
68%
1
expands forever
lly
recollapses eventua
0
0
ed
os t
cl fla
en
-1
op
Flat
Λ=0
Universe
1
2
3
ΩΜ
Figure 2.4: Best-fit confidence regions in the Ω M − ΩΛ plane. The 68%, 90%, 95%
and 99% statistical confidence regions are shown. Note that the spatial curvature of
the universe -open, flat, or closed - is not determinative of the future of the universe’s
expansion, indicated by the near-horizontal solid line. In cosmologies above this nearhorizontal line the universe will expand forever, while below this line the expansion
of the universe will eventually come to a halt and recollapse. This line is not quite
horizontal, because at very high mass density there is a region where the mass density
can bring the expansion to a halt before the scale of the universe is big enough that
the mass density is dilute with respect to the cosmological constant energy density.
The upper-left shaded region, labeled ”no Big Bang”, represents ”bouncing universe”
cosmologies with no Big Bang in the past. The lower right shaded region corresponds
to a universe that is younger than the oldest heavy elements for any value of H0 >
50kms−1 Mpc−1 . [133].
21
Chapter 2. Foundations of cosmology and cosmic structure formation
neously by two competing groups in 1998 and 1999, the Supernova Cosmology
Project led by Saul Perlmutter [133] and the High-Z Supernova Search Team led
by Brian Schmidt and Adam Riess [143]. The main goal of their work was a systematical study of high-redshift supernovæ and, in the course of the data analysis,
the two teams found observational evidence that, contrary to any expectation, the
Universe is undergoing a phase of accelerated expansion. In Perlmutter et al.
[133], while discussing the meaning of Figure 2.4, the authors note that ”the data
are strongly inconsistent with the Λ = 0, flat universe model (indicated with a circle) that has been the theoretically favored cosmology. If the simplest inflationary
theories are correct and the universe is spatially flat, then the supernova data imply
that there is a significant, positive cosmological constant. Thus the universe may
be flat or there may be little or no cosmological constant 12 , but the data are not
consistent with both possibilities simultaneously. This is the most unambiguous
result of the current data set”.
In Figure 2.5 we show the results of the other collaboration.
The history of the cosmological constant is dramatic. From the day of its first
introduction by Einstein in 1917, it has been postulated, forgotten, removed from
the theory and resurrected several times whenever a new inexplicable problem in
cosmology arose. The publication of the results of Perlmutter, Riess and Schmidt
opened a completely new perspective on the evolution of the Universe and several
models were proposed to explain their unexpected findings. A possible solution
would be modified gravity models, which would not require the presence of a cosmological constant, and another one would exactly be the reintroduction into the
theory of the idea of a vacuum energy dominating the expansion, which could be
either dependent on time or constant. In the latter and simpler case, this vacuum
energy could be identified with the cosmological constant originally postulated by
Einstein. The acceleration of today’s cosmic expansion is generally considered as
a manifestation of an unknown, uniformly- distributed form of energy, labeled
dark energy, to which a negative pressure is associated and that is viewed as the
liable factor for this kind of repulsive gravity.
Currently, a flat FLRW Universe with a non-zero cosmological constant including
cold dark matter represents the broadly-accepted standard model of cosmology.
12 and
22
a more sophisticated inflationary scenario or even something completely different, we could add.
2.2. The homogeneous Universe
99 .
7%
N
o
Bi
g
Ba
ng
3
.4%
2
95
0
q 0=
1
95 68.
.4 3%
%
a
eler
ng
Acc
rati
e
l
e
c
De
Cl
Recollapses
99
os
Op
en
Ω
to
t
0.5
1.0
ΩM
^
ΩΛ=0
ed
MLCS
-1
0.0
5
0.
q 0=
Expands to Infinity
.7
0
.5
ting
%
ΩΛ
99
.7
%
-0
q 0=
1.5
=1
2.0
2.5
Figure 2.5: Joint confidence intervals for (Ω M , ΩΛ ) from SNe Ia. Regions representing
specific cosmological scenarios are illustrated. Contours are closed by their intersection
with the line Ω M = 0 [143].
2.2 The homogeneous Universe
2.2.1
The cosmological principle
As mentioned in Section 2.1, three main assumptions are used in Friedmann cosmology
in order to describe an expanding Universe. The first one is that general relativity is the
correct theory of gravity, the second and the third ones are that the Universe is spatially
homogeneous and isotropic. These last two hypotheses go under the common name of
cosmological principle.
The cosmological principle combines local isotropy with a Copernican principle:
23
Chapter 2. Foundations of cosmology and cosmic structure formation
Figure 2.6: The projected distribution of galaxies shown in this cone diagram is evidently anisotropic. Image taken from http://www2.aao.gov.au/ TDFgg/
• Local isotropy states that, on large scales, a freely-falling (i.e. fundamental) observer experiences the observables of the Universe (CMB, Hubble flow, galaxy
distribution) as statistically isotropic, i.e. the same in all directions.
• The Copernican principle posits that man is in no way a favoured observer and
that Earth is neither in a central nor in a privileged position in the Universe.
A Universe that appears isotropic to all its fundamental observers is necessarily homogeneous. An interesting effect descending naturally from the homogeneity principle
is the synchronisation of cosmological time, t, with the proper time of observers moving
with the mean motion of the Universe.
The cosmological principle was originally formulated by Friedmann in 1922 as a
foundation of his solution to Einstein’s field equations and has been retained as a working hypothesis until now. Its two main assumptions, though, are quite vigorous ones and
need to be justified in some way. The idea of a completely isotropic and homogeneous
universe, in fact, contrasts with the indisputable existence of structures. The question
that arises from these considerations regards the level of homogeneity and isotropy required for the cosmological principle to hold and is still an open issue [42].
On one hand, while offering a glimpse of the Universe in its early days, data collected by the COBE, WMAP and Planck collaborations in the past twenty years reveal
the extraordinary degree of isotropy of the cosmic microwave background, which corroborates the symmetry hypothesis.
24
2.2. The homogeneous Universe
On the other hand, the distribution of the nearby galaxies up to a redshift of z = 0.3
that emerged from the Two Degree Field Galaxy Redshift Survey (2dF survey, [44])
is unmistakably anisotropic. A reconciliation of these two apparently contradictory
remarks can be achieved by recalling that the only requirement of the cosmological
principle is that the Universe exhibits a homogeneous character when its properties are
averaged on sufficiently large scales. Despite its manifest inhomogeneities (voids, filaments and walls can be seen everywhere in Figure 2.6), a rather uniform pattern in the
large-scale structure (LSS) of the Universe can be identified.
2.2.2
Robertson-Walker geometry
The line element
The structure of spacetime in general relativity is described via the line element:
ds2 = gµν dxµ dxν ,
(2.2)
and the dynamics of the metric tensor gµν is governed by Einstein’s field equations
which couple matter and energy to curvature and, in their generalised version, read:
8πG
T µν − Λgµν ,
(2.3)
c2
where Gµν is the Einstein tensor, G is the gravitational constant, c is the speed of
light, T µν is the energy-momentum tensor and Λ is the cosmological constant.
The adoption of the cosmological principle allows a simplification of the form of the
metric through the definition of a global coordinate, the cosmological time, according
to which spacetime can be described as a unique foliation of 3-spaces at constant time.
Isotropy allows to find spatial coordinates such that g0i = 0. As we mentioned in Section
2.2.1, global isotropy implies uniform density. The existence of a global time t and
the synchronisation of clocks leads to g00 = −c2 and to the definition of fundamental
observers for whom dxi = 0 and whose proper time coincides with the coordinate time.
This ansatz reduces the line element to:
Gµν =
ds2 = −c2 dt2 + gi j dxi dx j .
(2.4)
Denoting the line element of the so-defined spatial hypersurfaces with dl and rescaling it with a time-dependent function a(t) permits to write the line element of the spacetime as:
ds2 = −c2 dt2 + a(t)2 dl2 .
(2.5)
The function a(t) is termed scale factor and measures the expansion rate of the
universe. By convention, it is normalised such that a(t0 ) = a0 = 1, where t0 is the present
time.
25
Chapter 2. Foundations of cosmology and cosmic structure formation
A further consequence of the cosmological principle is that the possible 3-space
geometries allowed are restricted to cases with constant curvature K. As seen in Section
2.1, the possibilities are:
• K > 0 : 3-sphere with positive curvature everywhere, spatially closed, finite volume, bounded;
• K < 0 : 3-saddle with negative curvature everywhere, spatially open, infinite volume, unbounded;
• K = 0 : Flat Euclidean 3-space, spatially open, infinite volume, unbounded.
The isotropy assertion imposes spherical symmetry and the line element that describes spatial shells of constant curvature, k, can be cast into the form:
dr2
+ r2 dΩ2 ,
1 − kr2
(2.6)
dΩ2 = dθ2 + sin2 θdφ2 .
(2.7)
dl2 =
where dΩ2 is the solid-angle element:
In spherical coordinates, therefore, the line element becomes:
#
dr2
2
2
ds = −c dt + a(t)
+ r dΩ .
1 − kr2
"
2
2
2
2
(2.8)
This is known as the Friedmann-Lemaı̂tre-Robertson-Walker metric.
For the sake of convenience, a new radial coordinate is usually introduced:
dw2 =
dr2
.
1 − kr2
(2.9)
In this way we can recast the FLRM metric as:
h
i
ds2 = −c2 dt2 + a(t)2 dw2 + fk2 (w)dΩ2 ,
(2.10)
where fk (w) parametrises the possible 3-space geometries in the following way:

√ 

√1 sin
kw ,
if k > 0,




 k
fk (w) = 
(2.11)
w,
if k = 0,

√



1

 √|k| sinh |k|w , if k < 0.
26
2.2. The homogeneous Universe
2.2.3
Redshift
In this framework, the scale factor, a(t), is the only degree of freedom left in the system and the dynamics of the metric is reduced to the dynamics of the scale factor. In
particular, a central concept in cosmology can be derived from the study of the light
propagation in an expanding universe. If one considers a radial light ray, its trajectory
will follow a null geodesic: dΩ = 0 and ds = 0. If we assume that the worldlines of
the photons we observe today were emitted at certain times te , from Equation (2.10) we
have:
Z t0
cdt0
.
(2.12)
w=
0
te a(t )
The coordinate distance w between emitter and observer must be constant and its
first derivative must vanish, which allows us to write:
Z t0 +∆t0
cdt0
,
(2.13)
w=
0
te +∆te a(t )
and
λ0 − λe
c∆t0 νe λ0 a(t0 )
= 1+
=
=
=
= 1 + z,
c∆te ν0 λe a(te )
λe
(2.14)
assuming that the cycle time of a light wave corresponds to the inverse of its frequency. This expression can be interpreted by noticing that wavelengths are also stretched
by expansion: if the Universe shrinks, they are blue-shifted; if it expands, they are redshifted. This is the reason why the factor z is called cosmological redshift. Since, by
convention, a(t0 ) is set to unity, we can retrieve the relation between the scale factor and
the redshift:
1
1+z = .
a
2.2.4
(2.15)
Dynamical properties of the metric
The assumption of the cosmological principle and of the FLRW metric (Eq. (2.10))
represents the basis of the exact solution of Einstein’s Field Equations that goes under
the name of Friedmann’s equations.
The next step is to take an ideal fluid as a source of the gravitational field. By
definition, an ideal fluid is isotropic in its rest frame [178], has neither viscosity nor
heat conduction [156] and is completely characterised by its energy density ρc2 and by
the pressure p. If we choose the density and the pressure to be functions only of time,
we can write the energy-momentum tensor in Equation 2.3 to be:
27
Chapter 2. Foundations of cosmology and cosmic structure formation
T µν = (ρc2 + p)ũµ ũν + pgµν ,
(2.16)
where ũµ is the 4-velocity of an element of the fluid.
These preliminary remarks allow us to finally write the Friedmann equations that
govern the dynamics of the scale factor:
8πG
kc2 Λ
=
ρ− 2 + ,
a
3
a ! 3
ä 4πG
3p Λ
ρ+ 2 + .
=
a
3
3
c
ȧ 2
(2.17)
(2.18)
These two equations can be combined to give the adiabatic equation:
i
dh 3
da3 (t)
a (t)ρ(t)c2 + p(t)
= 0,
dt
dt
(2.19)
which is the cosmological version of the first law of thermodynamics.13
2.2.5
Matter models
A fundamental hypothesis in the ΛCDM Cosmology is that the Universe, seen as a
fluid, is in thermodynamical equilibrium and its equation of state reads p = wρc2 with
constant14 w. If we remember that a particle can be labeled as non-relativistic if v c
and relativistic in the opposite condition, we can call a gas (at a given temperature
T ) non-relativistic if it satisfies the condition kB T mc2 and relativistic if, viceversa,
kB T mc2 , where m is the average mass of its particles. Following this definition, we
can broadly identify two different types of matter constituing the cosmic fluid: nonrelativistic matter, also called ordinary or baryonic matter and a relativistic component,
called radiation. A third constituent is the dark energy, associated to the cosmological
constant Λ. Accordingly, the equation of state parameter assumes the values:



0,
matter,




w=
1/3, radiation,




−1, cosmological constant.
13 For
(2.20)
a thourough derivation, please see http://www.ita.uni-heidelberg.de/research/
bartelmann/Lectures/cosmology/cosmology.pdf.
14 w is not constant in dark energy models, for instance.
28
2.2. The homogeneous Universe
2.2.6
Cosmological parameters
In cosmology it is common to encounter a number of parameters and functions and we
will define them briefly in this paragraph.
Hubble function: From Equation (2.2.4), it is possible to define the Hubble function
as the logarithmic change in the expansion rate:
ȧ = H(t).
(2.21)
a
Hubble constant: At present time the Hubble function becomes:
H(a(t0 )) = H(1) = H0 ,
(2.22)
and has the unit of an inverse time: H0 ≈ 3.2×10−18 h s−1 , where h is called Hubble
parameter, parametrises our lack of knowledge on the value of H0 and is given
by H0 /(100 km/s/Mpc).
Hubble time: The inverse of the Hubble constant is:
1
tH =
,
H0
(2.23)
and would represent the age of the Universe if its expansion rate was constant
throughout its evolution.
Critical density: Assuming k = 0 = Λ, the critical density for the Universe to be spatially flat is:
3H 2
,
(2.24)
ρcr =
8πG
and its value today is
ρcr,0 =
3H02
8πG
= 1.86 × 10−29 h2 g cm−3 .
(2.25)
Density parameter: The dimensionless ratio between the actual matter density and the
critical density provides a convenient way to specify the geometry of the Universe:
Ω(t) =
ρ(t)
.
ρcr (t)
(2.26)
The individual contributions of the different components of the Universe can also
be illustrated via density parameters, such as the contribution due to the presence
of luminous matter, Ω∗ , or the contribution due to the matter estimated to be contained in galaxies, ΩG . The most useful and recurring definitions are the matter
density parameter, the radiation density parameter and the cosmological constant
density parameter.
29
Chapter 2. Foundations of cosmology and cosmic structure formation
Matter density parameter: It is the contribution of both the baryonic and dark matter
to the critical density of the Universe:
ρm (t)
,
ρcr (t)
Ωm (t) =
(2.27)
and it is often split into the fraction of baryonic matter, Ωb , and the one due to
dark matter, ΩCDM .
Radiation density parameter: The evolution of the Universe depends also on electromagnetic energy, relativistic particles and neutrinos. In particular, the radiation
density is dominated by the energy density of the CMB:
Ωr (t) =
ρr (t)
.
ρcr (t)
(2.28)
Cosmological constant density parameter: Even though the nature of dark energy is
still an enigma, observations are able to determine its input to the cosmic density:
ΩΛ (t) =
Λ
.
3H 2
(2.29)
Curvature density parameter: A similar definition can be given for the curvature as
well:
kc2
Ωk (t) = − 2 2 .
(2.30)
a H
The introduction of these parameters permits to reformulate Friedmann’s equation as:
h
i
H 2 (a) = H02 E 2 (a) = H02 Ωr,0 a−4 + Ωm,0 a−3 + ΩΛ,0 + ΩK,0 a−2 ,
(2.31)
which tells us how the Universe changes as it expands or shrinks in relation to today and
at present time (a = 1) provides us with the identity:
Ωr,0 + Ωm,0 + ΩΛ,0 + ΩK,0 = 1.
(2.32)
In Table 2.1 [137] we report the values of these parameters found by the Planck
collaboration. The density parameter attributed to the cosmological constant nowadays
is around 70%, which leads us to view dark energy as the dominant influence on the
current phase of expansion of the Universe. The previous two phases in the history from
the Big Bang to today are considered to be first radiation dominated and subsequently
matter dominated. The relative importance of the various density parameters over time
can be easily seen by inspecting the square brackets of Equation 2.31: it is immediately
obvious that they scale with different powers of the scale factor, a.
30
2.2. The homogeneous Universe
Table 2.1: Constraints on Ωm0 (matter density parameter), ΩΛ0 (dark energy density
parameter), Ωb0 (baryonic matter density parameter), H0 (Hubble constant), nS (spectral index of the primordial power spectrum), σ8 (power spectrum normalisation), zeq
(equality redshift) and the age of the Universe estimated by [137]. The column labeled “Planck” gives results for the Planck temperature power spectrum data alone. The
column labeled “Planck + lensing” combines the Planck temperature data with Planck
lensing. The column labeled “Planck + WMAP pol.” includes WMAP polarization at
low multipoles. The results presented in [137] are the best fit parameters (i.e. the parameters that maximise the overall likelihood for each data combination) as well as the
68% confidence limits for the constrained parameters.
Parameter
Planck
Planck + lensing
Ωm0
0.314 ± 0.020
0.307 ± 0.019
ΩΛ0
0.686 ± 0.020
0.693 ± 0.019
Ωb0 h2
0.02207 ± 0.00033 0.02217 ± 0.00033
H0
67.4 ± 1.4
67.9 ± 1.5
nS
0.9616 ± 0.0094
0.9635 ± 0.0094
σ8
0.834 ± 0.027
0.823 ± 0.018
ze q
3386 ± 69
3362 ± 69
Age[Gyr]
13.813 ± 0.058
13.796 ± 0.058
2.2.7
Planck + WMAP pol.
0.315+0.016
−0.018
0.685+0.016
−0.018
0.02205 ± 0.00028
67.3 ± 1.2
0.9603 ± 0.0073
0.829 ± 0.012
3391 ± 60
13.817 ± 0.048
Cosmological distances
In cosmology and general relativity, the concept of distance is no longer unambiguous
because of the relative character of simultaneity and the finiteness of the speed of light.
Since the measurement of a distance is taken along the past light cone and the Universe
expands, it is not surprising that, in a cosmological framework, distances change in time.
A number of definitions are available and, even though they converge to each other
in the limit of a Minkowski spacetime, they considerably differ from one another in an
expanding spacetime.
Proper distance: It is possible to define the proper distance as the distance that light
travels between a source at redshift z2 and an observer at redshift z1 < z2 :
Z a(z2 )
Z t2
Z a(z2 )
c
da
cda
=
.
(2.33)
D prop (z1 , z2 ) =
cdt =
ȧ
H0 a(z1 ) aE(a)
t1
a(z1 )
Comoving distance: This is the distance scale that expands with the Universe and is
defined as the spatial distance between the intersections of the worldline of two
comoving sources at redshifts z1 < z2 with the spatial hypersurface t = t0 [152].
31
Chapter 2. Foundations of cosmology and cosmic structure formation
In GR light propagates along null geodesics and from the FRLW metric 2.10 we
can write 15 cdt = −adw and remembering that dt = da
ȧ :
Dcom (z1 , z2 ) = w(z1 , z2 ) = c
Z
a(z1 )
a(z2 )
Dang =
da
= w(z2 ) − w(z1 ).
ȧa
a(z2 )
fK [w(z1 , z2 )],
a(z1 )
(2.34)
(2.35)
Luminosity distance: The distance between a source and an observer can be also defined as:
L 2
,
(2.36)
DL =
4πF
where L is the intrinsic luminosity of the source and F is the flux measured from
an observer and both these quantities are bolometric, therefore integrated over
all frequencies. In the Euclidean limit, the definitions of luminosity and angular diameter distance would be equivalent. In a more general context, they are
connected via the Etherington relation [64]:
DL = (1 + z)2 Dang ,
(2.37)
which implies that, in an expanding Universe, the luminosity distance is larger
than the angular diameter one.
2.3 The formation of structures in the Universe: introducing inhomogeneities
2.3.1
The growth of perturbations
As mentioned in Sect. 2.2.1, the cosmological principle is valid when averaging on very
large scales. Nonetheless, it is also necessary to account for inhomogeneities such as the
galaxies (around 10 kpc h−1 ), galaxy clusters and superclusters (around 1 to 10 Mpc h−1 )
, filaments and voids (around 50 Mpc h−1 ) and all the other structures one can observe
when looking at the Universe on smaller scales. Even though the primordial distribution and the mechanism that triggered the formation of such inhomogeneities is not yet
completely clarified, many studies point toward a period of accelerated expansion that
goes under the name of inflation as a good candidate to explain the known features of
such perturbations.
we are in the position w = 0. The choice of the minus sign is due to the fact that light
emitted from a source at redshift z2 > z1 will have dt > 0 and dw < 0.
15 Assuming
32
2.3. Introducing inhomogeneities
Given the small scale of these inhomogeneities with respect to the Hubble radius,
i.e. the typical scale of the Universe, and typical velocities being much smaller than the
speed of light, Newtonian dynamics is in general considered a reasonable approximation
to use and provides us with a solid theory for describing them in the context of a nonrelativistic cosmic fluid.
The very first tool we can borrow from fluid mechanics is the continuity equation:
∂ρ(t, ~x) ~
+ ∇ · (ρ~v(t, ~x)) = 0,
∂t
(2.38)
which codifies the conservation of mass and the dependence of density and velocity on
time and position in an inhomogeneous Universe.
The conservation of momentum is prescribed by the Euler equation:
~
∂~v
~ v = − ∇p − ∇φ,
~
+ (~v · ∇)~
∂t
ρ
(2.39)
where the terms on the RHS contain the pressure gradient and the gravitational forces
and the Newtonian potential φ is related to the density field via the Poisson equation:
∇2 φ = 4πGρ.
(2.40)
The density perturbations are described by the dimensionless density contrast δ:
δ(t, ~x) =
ρ(t, ~x) − ρ̄(t)
,
ρ̄(t)
(2.41)
which represents the difference between the density at a given coordinate and the mean
density normalised by the mean density.
The evolution of the perturbations can be studied in two different regimes: the linear
and the non-linear regime.
The use of Eqs. (2.38) to (2.40) and linear perturbation theory permit to derive the
time evolution equation for the density contrast:
δ̈(t, ~x) + 2H δ̇(t, ~x) −
c2s 2
∇ δ(t, ~x) − 4πGδ(t, ~x) = 0,
a2
(2.42)
where the over-dots identify partial time derivatives and a transformation from physical
to comoving coordinates ~r = a~x is performed. Decomposing δ into plane waves:
Z
Z
d3 k
~
−i~k·~x
~
~
δ(t, ~x) =
δ̂(t, k)e
, δ̂(t, k) = d3 xδ(t, ~x)eik·~x ,
(2.43)
3
(2π)
with δ̂ the Fourier transform of δ, yields its equivalent form in Fourier space:
33
Chapter 2. Foundations of cosmology and cosmic structure formation
!
c2s 2
~
~
~
δ̈(t, k) + 2H δ̇(t, k) + δ(t, k) 2 k − 4πGρ0 = 0,
a
(2.44)
with ρ0 the background value of the density field. The sound speed c s is the proportionality factor relating pressure and density fluctuations in the equation of state:
δp = c2s δρ = c2s ρ0 δ. The definition of the Jeans length, λ J , naturally follows from
Eqs.(2.44) and (2.43):
r
2π
π
= cs
.
(2.45)
λJ =
kJ
Gρ0
This scale is the threshold where attraction and pressure balance each other: perturbations smaller than the Jeans length (therefore with k > k J ) oscillate, those larger either
decay or grow.
If we restrict ourselves to the case of perturbations much larger than the Jeans length,
we can observe how the growth rate of perturbations varies with the species dominating the energy density of the Universe. It can be shown that δ ∝ a2 in the radiationdominated era and afterwards δ ∝ a in the matter-dominated one.
The linear evolution of the density contrast obeys:
δ(a) = δ0 D+ (a),
(2.46)
and the linear growth factor D+ (a), which describes the evolution of the density contrast
with respect to the scale factor, in ΛCDM cosmologies is well approximated by the
fitting formula [39]:
"
!
!#−1
1
1
5a
4/7
.
D+ (a) = Ωm Ωm − ΩΛ + 1 + Ωm 1 + ΩΛ
2
2
70
(2.47)
For a δ & 1, the assumption of small fluctuations of the background density and velocity fields is no longer valid and the perturbations become non-linear. An analytic
treatment of these equations for a mildly non-linear regime requires different approximations such as the spherical collapse model, for instance, that will be mentioned in
Sect. 3.2.1. For treating perturbations at later times it is common to use numerical
simulations such as the Millenium simulation.16
2.3.2
The power spectrum
As we already discussed in the previous section, a central requirement of the current
standard cosmological theory is the ability to reconcile the high degree of uniformity
16 For
details about the Millenium simulation see http://www.mpa-garching.mpg.de/millennium/ and
refences therein.
34
2.3. Introducing inhomogeneities
seen at large scales and the incredible isotropy observed in the CMB that are prescribed by the cosmological principle with the structures that are unmistakably present
at smaller scales.
The density contrast δ is a random field and because of the cosmological principle it
can be assumed to be homogeneous and isotropic, which implies that its statistical properties need to be invariant under rotations and translations. The idea that the statistical
properties of a field describing inhomogeneities should be homogeneous sounds less
paradoxical when one assumes the existence of an ensemble of Universes and sees our
Universe as a statistical realisation of this random field. An issue that can be brought up
against this argument is that we find ourselves in the condition to observe only one realisation of this ensemble. This is usually overcome by resorting to the ergodic hypothesis
which states the equivalence between ensemble averages, i.e. taken on several different
realisations of the distribution, and sample averages, i.e. averages taken on uncorrelated
samples within the same realisation.
Adler (1981) [3] proved that a random field is ergodic if it can be described by Gaussian statistics and if its power spectrum is continuous. In linear perturbation theory, each
Fourier mode evolves independently. Thus, the density field in real space is a combination of independent random variables δ(k) drawn from the same distribution. According
to the central limit theorem, this distribution has to be Gaussian and a Gaussian field is
completely specified by two statistical properties: the mean and the variance.
The mean of the density contrast vanishes by definition:
hρi
ρ − ρ0
i=
− 1 = 0.
(2.48)
ρ0
ρ0
The second important property to study is the variance of δ, which measures on
average the departure of the density field from its mean: the more inhomogeneous the
density field is, the larger the variance. In Fourier space (see Eq.(2.43)) the variance is
expressed by:
hδ̂(~k)δˆ∗ (k~0 )i = (2π)3 P(k)δD (~k − k~0 ),
(2.49)
hδi = h
which contains the definition of power spectrum, P(k), and where δD is the Dirac’s
delta distribution which ensures homogeneity by prescribing that different modes are
uncorrelated in Fourier space. The isotropy argument of the cosmological principle
garantees that P(k) is independent of the direction of the wave vector. 17
In real space the variance is related to the autocorrelation function or 2-point correlation function, ξ(y):
ξ(y) = hδ(~x)δ(~x +~y)i,
(2.50)
where the average h·i extends over all positions ~x and orientations of ~y and the correlation function depends only on the magnitude y. The variance σ2 of the density contrast
17 From this relation we can observe that the dimensions of the power spectrum are [L]3
k−3 .
or, equivalently,
In general, one plots the dimensionless function of k: ∆2 (k) = k3 P(k)/(2π)3 .
35
Chapter 2. Foundations of cosmology and cosmic structure formation
is then the correlation function at ~y = 0:
σ2 = hδ2 (~x)i,
(2.51)
which makes sense if we observe that the mean is null.
Thanks to Eq. (2.49), it is possible to find an expression relating the variance to the
power spectrum:
Z 2
k dk
2
σR = 4π
P(k)WR (k),
(2.52)
(2π)3
which shows how the density field can be completely described by its first two moments (i.e. mean and variance). WR (k) is a window function restricting the variance
to density fluctuations of spatial scale R. We note that the 2-point correlation function
specifies a Gaussian random field completely. If the field was non-Gaussian, the use of
higher-order moments (skewness, kurtosis, up to N-point correlation functions) would
be necessary.
36
We would, indeed, be fortunate in science if the inaccuracy of observation
were never more than a small fraction of
the quantity observed.
V. M. Slipher, 1917
3
Galaxy Clusters
Galaxy clusters constitute the largest gravitational potential wells in the Universe. Their
extraordinary richness in observable properties makes them a valuable tool both for
astrophysics and for cosmology. On the one hand, they are the perfect laboratory in
which models of galaxy formation and evolution can be tested and the intergalactic
medium be studied. On the other hand, they are inevitably imprinted by the underlying
cosmology and can be exploited as magnifying lenses to gain information on objects
that would otherwise not be in reach of telescopes.
In this chapter, we review the main aspects of galaxy clusters that will be central
concepts for the scope of this work. Section 3.1 presents a brief overview of clusters
in general and of their relevance to observational astronomy and cosmology. Section
3.2 begins with the introduction of the spherical collapse model of structure formation,
proceeds in reviewing several possible models describing the density profile of galaxy
clusters and ends with a section dedicated to the mass function. The observational
properties of clusters that will be used in the upcoming chapters are examined in Section
3.3.
3.1 Galaxy clusters: a brief overview
Clusters of galaxies are the largest and most massive, gravitationally bound structures
in the Universe. A tendency of galaxies (then called “nebulæ”) to cluster had already
been noticed in the eighteenth century by Charles Messier and William Herschel. From
that moment on several catalogues of collections of galaxies started to be compiled and
their relevance as objects of astronomical study increased after the discovery that our
Universe is not limited to the Milky Way and the extragalactic origin of galaxies was
confirmed (See Section 2.1 and references therein for a more extensive treatment of the
topic).
As described in Chapter 2, the energy content in the Universe is mostly dark. Galaxies are embedded into dark matter haloes and the main constituent of a galaxy cluster
is dark matter, followed by a hot, X-ray luminous intracluster medium (ICM), which
37
Chapter 3. Galaxy Clusters
accounts for most of the luminous baryonic matter, and by the optical galaxies. More
precisely, the mass content of a galaxy cluster of 1014 − 1015 solar masses can be identified to be composed of 85 − 90% of dark matter and 10 − 15% of baryons [104], with
10% of the baryonic component to be ascribed to unbound, intracluster galaxies [184].
The depth of their potential wells makes clusters an ideal approximation of isolated
systems in which the physics of the intracluster medium (effects of major mergers, energy feedback from supernovæ and active galactic nuclei, outbursts from supermassive
black holes...) can be investigated and models of galaxy formation can be tested.
Furthermore, clusters are an excellent environment to test cosmological models.
Their dynamical timescales are very large (of the order of Gyrs) and this implies that,
on one side, we can study their evolution even at low redshifts because cluster formation
is an ongoing process and, on the other, that we can use them to constrain cosmological
parameters (such as the matter density parameter, Ωm , (Eq. 2.27) and the equation of
state for dark energy, w (Eq. (2.20)).
Another great service paid to cosmology is that, via gravitational lensing, clusters
can be used as natural cosmic telescopes to explore regions of space that would otherwise be unaccessible. This also explains why astronomers are so interested in the mass
of galaxy clusters, a precise determination of which would be beneficial both in the
study of objects at high redshift and in probing the validity of competing cosmological
scenarios. As we will see in Section 3.2.3, though, a rigorous and exact definition of the
concepts of cluster mass and mass function is as elusive as it is of fundamental importance. We will therefore argue for the necessity of adopting the concept of the cluster
lensing potential as the cardinal quantity to use instead of the mass in our reconstruction
method.
3.2 Formation and abundance
One of the primary problems in cosmology concerns the formation of dark matter haloes
in the ΛCDM picture of a hierarchical build-up of structures from smaller to larger
scales via a sequence of mergers and accretion. Given the complexity of the equations
describing this scenario, several numerical techniques have been developed in order
to account for gravitational interactions, collisions among gas particles and pressure
forces. In addition, it is possible to construct analytical models that illustrate the basic
processes that lead to halo formation. One of these analytical models, and probably
the most widely used, is the spherical collapse model, which becomes useful when the
linear theory described in Section 2.3.1 breaks down. As a rule-of-thumb, one assumes
that this occurs when the density contrast is no longer small compared to unity. In the
case of galaxy clusters, δ ≈ 103 , we can see therefore why a non-linear approach is
required to treat them.
38
3.2. Formation and abundance
3.2.1
Spherical collapse model
A dark matter halo can be assumed to be a self-gravitating, quasi-equilibrium system
of dark matter particles which formed by gravitational collapse [181]. Considering an
Einstein-de Sitter (spatially flat and matter dominated) universe with critical density,
ρcr , the spherical collapse (or top-hat) model studies the evolution of a homogeneous
overdense sphere of density ρcr + δρ in an expanding background. This density perturbation evolves like a closed universe with a matter density parameter Ωm = 1 + δ and
grows until it reaches a maximum radius r = rmax (turn-around) at a time t = tmax and
then collapses in a finite time. This implies that its scale factor reaches a maximum
as well. The evolution equation for the scale factor of such a perturbation is the first
of Friedmann equations (Eq. (2.2.4)) with a positive curvature term. The Newtonian
equation of motion for the radius of the perturbation is:
r̈ = −
GM
,
r2
(3.1)
3
where M = 4πρr
for a homogeneous sphere, and the integral of motion which corre3
sponds to the conservation of energy reads:
1 2 GM
ṙ −
= E,
(3.2)
2
r
where E is the total energy per unit mass of the system. A criterion for collapse can be
found in the above relation for E < 0. A parametric solution for Eq. (3.1) is given in
[125] in the special case of an Einstein-de Sitter universe1 and the relation between the
radius of the sphere and the time is:
r(θ) = A(1 − cos θ),
t(θ) = B(θ − sin θ),
(3.3)
(3.4)
where θ denotes the development angle and is a scaled form of the conformal time. The
relation between the constants A and B is given by A3 = GMB2 . From the first functional
form in Eq. (3.3) it is possible to determine the maximal size that the density fluctuation
will reach by setting:
dr
= A sin θ = 0,
(3.5)
dθ
for which three solutions are available: θ = 0, π, 2π. θ = 0 corresponds to t = 0, θ = π
corresponds to the maximum expansion radius at turn-around time, tmax = ta and θ = 2π
indicates the time in which the sphere is fully collapsed to a singularity. This happens
1 For
a derivation, please see http://www.ita.uni-heidelberg.de/research/bartelmann/
Lectures/cosmology/cosmology.pdf.
39
Chapter 3. Galaxy Clusters
because the model only accounts for gravitational interactions. A more realistic treatment would consider the scattering of the particles and, by including it, it is possible to
define a virialisation condition imposing that the spherical blob stops its collapse before
reaching an infinite density and enters virial equilibrium. A hypothesis to explain the
stability of the observed haloes is that they virialize, i.e. form a spherical system which
satisfies the scalar virial theorem:
1
Ek = − E pot ,
2
(3.6)
where Ek , E pot are the kinetic and potential energies respectively. The virial radius is
reached when the system has collapsed to approximately half its maximum radius and
the density contrast2 is δ ≈ 200.
3.2.2
The density profile of galaxy clusters
One of the central quantities in any comprehensive description of a galaxy cluster is
its density profile. Unfortunately, providing a realistic treatment of all the DM and
baryonic physics playing a role in clusters is not a challenge easy to face. If we restrict
ourselves to a DM-only scenario where gravity is the sole actor in a CDM Universe, we
can see from numerical simulations [116, 164] that the density profile of the DM halo,
ρDM , exhibits a central density cusp and scales with r−1 in the inner part and with r−3
at larger radii. In more recent years, N-body simulations have specified even more the
shape of this functional behaviour [76, 118].
Once we add baryons to the picture, things complicate further: cooling cores and
a higher concentration for the dark matter are phenomena that have to be investigated
[163], among many other effects.
Although it may appear to be a daunting task, being able to determine the density
profile of galaxy clusters and to decompose it in the profile describing the baryonic and
the dark matter components of the cluster is of immense importance in shaping a better
understanding of the nature, structure and formation mechanisms of clusters of galaxies
and on the nature of the dark matter.
In a later section, we will discuss the wealth in observational properties of galaxy
clusters. An accurate model for their density profile is in general deemed necessary for
studying the mass distributions at a wide range of radii [121].
In chapters 5 and 7, we will present a method that avoids putting too strongly constraining assumptions on the density profile. We will see, though, in Chapter 4 that
this choice, as any other legitimately different one, has a price in terms of the other
assumptions that one has to impose on the model.
2 To
40
be more precise, ∆v ≈ 178 where ∆v − 1 ≈ δ.
3.2. Formation and abundance
The Hernquist profile
In 1990 Lars Hernquist [80] introduced the profile that later became known under his
name. Starting from a spherical gravitational potential φ:
φ(r) = −
GM
,
r+a
(3.7)
in which he assumed the presence of a scale length, a, Hernquist derived a simple analytic form for the density profile, ρ:
ρ(r) =
Ma 1
,
2π r (a + r)3
(3.8)
which mimics the r−4 behaviour empirically determined for the observed luminosity
distribution of elliptical galaxies and bulges and described by the de Vaucoulers’ law
[50].
By inserting it in the Poisson equation and fixing the integration constant such that
the gravitational force vanishes at the centre (i.e., ∂r φ = 0 for r = 0), it is possible to
retrieve the form (Eq. (3.7)) of the potential.
The Navarro, Frenk and White profile
In a series of papers published between 1995 and 1997 [115, 116, 117], Julio Navarro,
Carlos Frenk and Simon White reported analyses of numerical simulations from which
it emerged that a functional form that fits well the density profile of spherically averaged
dark matter haloes is:
δc
r
ρ(r)
=
with x = ,
(3.9)
2
ρcr
rs
x(1 + x)
where ρcr is the critical density for closure, r s is the scale radius and δc is a dimensionless
density contrast defined as:
δc =
200
c3
.
3 ln(1 + c) − c/(1 + c)
(3.10)
This profile differs from Hernquist’s profile in its asymptotic behaviour for r > r s :
we can in fact notice that it tends to r−3 instead of r−4 .
The scale radius is defined as rs = r200 /c, with r200 taken as the fiducial virial radius,
the radius of a sphere whose mean density is 200 times the critical density, ρcr , and c
is a factor called concentration, the only parameter on which the model depends for a
given mass.
In the 1997 paper, Navarro, Frenk and White proposed that the density profile (that
today carries their name and is often referred to as NFW profile) could be universal.
41
Chapter 3. Galaxy Clusters
ΛCDM
9
simulation
NFW
Bullock et al.
ENS
polynomial fit
8
c200,av
7
6
5
4
3
2
0
0.5
1
1.5
<z>
2
2.5
3
Figure 3.1: Concentration parameter as a function of redshift [55]. Galaxy clusters with
the same mass show a higher concentration at higher redshifts.
All simulated and observed clusters of galaxies, in fact, exhibited the same shape of the
profile, despite different halo masses, redshift, initial fluctuation spectrum and cosmological parameters that characterised them.
The Einasto profile
More recent simulations suggest the presence of a flattening central density profile that
is better described by the Einasto profile [56]:
ρE (r/rs ) = ρs exp{−(2/αE )[(r/rs )αE − 1]},
(3.11)
where αE is a structural parameter describing the degree of curvature of the profile, ρ s
is the density at the radius rs that defines a volume containing half the total mass. At
r = rs the double-logarithmic slope:
α=
d ln ρ
= −2(r/rs )αE
d ln r
(3.12)
of the density profile has a value α = −2, which coincides with the value of the NFW
profile at r s .
42
3.2. Formation and abundance
The Einasto profile with αE = 1/5 [107] was found to be a good fit for the density
profile of numerically simulated dark matter haloes. While the NFW profile exhibits a
central density cusp, the Einasto profile does not. If dark matter particles decay or annihilate, the annihilation rates depend on the squared particle density [15]. A profile with
a finite-density core, as the Einasto profile is, would therefore lead to sensibly lower
predictions of the annihilation rates than the NFW and to different levels of substructuring.
3.2.3
The mass functions and the quest for a definition of the mass
of galaxy clusters
In the current paradigm of structure formation, clusters of galaxies, which can be viewed
as rare peaks in the density field, are thought to be the last objects to collapse under
the pull of gravity. While the struggle to identify a density profile that fits them in
an accurate way is relevant in order to understand their internal structure, the attempt
to predict the abundance of haloes is important in order to have information on their
statistical distribution and to characterise the objects we observe based on their mass.
The crucial quantity to do so is the mass function defined as:
dN = n(M, z)dM,
(3.13)
where dN is the number density of haloes per unit comoving volume with mass within
M and M + dM as a function of redshift. Clusters of galaxies form the massive end of
the halo mass function. An analytic expression for it was given by William H. Press
and Paul Schechter in a paper they published in 1974. Their model is known as Press &
Schechter formalism [140].
Their derivation relies on the assumption that the mass in all volumes in which the
threshold of the critical density contrast δc is exceeded belongs to collapsed haloes and
that the density fluctuation field, filtered by a top-hat window function, has a Gaussian
probability distribution. The critical density contrast, δc , corresponding to virialised
structures, is predicted by the spherical collapse model to have an approximate value of
1.69. The halo mass function, therefore, takes the shape:
r
!
δc
2 ρ0 dν
ν2
exp −
with ν =
,
(3.14)
n(M, z)dM =
π M dM
2
σR (M)D+ (z)
where ρ0 is the average matter density of the Universe today, D+ (z) is the growth factor
from Eq. (2.47) and σR (M) is the standard deviation of the mass fluctuations computed
on the scale R. The exponential cutoff above a mass M∗ defined by σR (M∗ ) = δc allows
one to constrain the cosmological parameters, in particular σ8 and Ωm .
The Press & Schechter formalism, though, is not devoid of complications. Chiefly, it
ignores the cloud-in-cloud problem: by assuming that only originally overdense regions
43
Chapter 3. Galaxy Clusters
can be included in collapsed objects, it neglects the finite probability that overdense
regions, enclosed within overdense regions, can enter the haloes as well. This problem
was well known to the scientific community, since it represented a reason against the
peak formalism proposed by Bardeen, Bond, Kaiser and Szalay [13], and it was solved
by Press and Schechter by artificially introducing a factor of 2 in their expression to
account for the missing matter. An alternative and more rigorous derivation of the mass
function, which offered a way out of this ad hoc introduction, was devised by Bond,
Cole, Efstathiou and Kaiser in 1991 [30]. They suggested the excursion set formalism,
also known as extended Press & Schechter (EPS) formalism, making use of the statistics
of random walks to infer the halo mass function.
Over the years, many shortcomings of the theory have been observed and modifications have been formulated. In particular, the EPS formalism is built upon the spherical
collapse model, whilst nowadays the collapse is believed to be ellipsoidal. Another
problem resides in the fact that it uses results from linear theory in order to predict
structures that form in a non-linear density field.
The EPS theory was further extended with fitting functions by Ravi Sheth and
Giuseppe Tormen to account for an ellipsoidal collapse in 1999 [158] and later by Jenkins et al. in 2001 [83]. Comparisons with numerical simulations, notably the Millenium
Simulation [164], showed a remarkable agreement with the predictions of the modified
versions of the EPS model.
The main point that we intend to make with this brief review is that the concept of
mass is not uniquely defined despite it is crucial for cosmological studies. In particular,
a galaxy cluster does not have a well-defined boundary, nor a well-defined centre and
sphericity is a poor assumption for its geometry. In the following section we will introduce a quantity that one can resort to in order to bypass these problems, the projected
potential.
3.3 Obervational properties of galaxy clusters
As we mentioned at the beginning of this chapter, galaxy clusters are very rich in observational properties. In this section we will limit ourselves to a brief overview of
the observational features relevant for this work: gravitational lensing, X-ray emission,
thermal Sunyaev-Zel’dovich effect and dynamics of member galaxies.
3.3.1
Gravitational lensing in clusters
When light rays propagate through a gravitational field, they are deflected. Despite
this effect had already been speculated about by Isaac Newton, John Mitchell (1784),
Pierre Simon de Laplace (1797) and Johann von Soldner (1804) [88, 162], to name
some, it was not until the formulation of Einstein’s general theory of relativity that a
44
3.3. Obervational properties of galaxy clusters
Figure 3.2: Abell 1689. Galaxy clusters are the largest gravitationally bound objects in
the Universe. Courtesy of NASA.
quantitative description of this phenomenon became available to the astronomical and
physical community. The term used to describe the deflection of light by massive bodies
and the phenomena stemming from it is gravitational lensing. Several reviews provide
an extensive treatment of the topic and its applications [16, 17, 114, 153].
Galaxy clusters, with their huge masses, are perfect to identify the observational
signatures of the strong and weak regimes of gravitational lensing: giant arcs in the case
of strong lensing and arclets in the case of weak lensing. An example is shown in Fig.
3.2.
Gravitational lensing is a versatile instrument and serves both as a probe to study the
physics of galaxy clusters and as a magnifying glass to explore objects that are further
away from the reach of telescopes [111, 185]. It is sensitive both to luminous and dark
matter. In fact, the gravitational field through which light propagates is related to the
mass density of the lens via Poisson’s equation, which implies that, on the one hand,
lensing cannot discriminate between luminous and dark matter and that, on the other,
light deflection is not affected by the nature of the matter nor by its equilibrium state
[152].
A well-known combination of differential geometry and electrodynamics requires
45
Chapter 3. Galaxy Clusters
Figure 3.3: Lensing geometry in the thin screen approximation [114]. The thickness
of the lens is negligible in comparison with the extent of the light path and the mass
distribution of the lens can be substituted by a thin mass sheet orthogonal to the line of
sight, the lens plane. The measurement of distances relies on the underlying cosmology,
as described in Chapter 2, and allows to pass from physical to angular scales.
light propagation to occur along the null geodesics of the space-time metric. However,
a typical lensing system is a localised matter distribution and can be fairly described
by resorting to the thin lens approximation sketched in Fig. 3.3. The thin lens approximation can be invoked when the thickness of the lens is negligible in comparison to
the extent of the path undertaken by light to reach the observer. This requirement is
frequently met: a galaxy cluster has a typical size of a few Mpc, while the distances
between the source, the lens and the observer are usually of the order of fractions of the
Hubble length cH0−1 = 3h−1 × 103 Mpc.
In the cartoon (Fig. 3.3), Ds is the distance between the observer and the source, Dd
represents the distance between the observer and the lens and Dds indicates the distance
between the lens and the source, α is the deflection angle, θ is the observed angular
position and β is the unlensed angular position of the source. The unlensed position of
the source and its images are related via the lens equation, or ray-tracing equation:
~β = ~θ − α
~ (~θ).
46
(3.15)
3.3. Obervational properties of galaxy clusters
The lens equation is non-linear in the general case, which implies that multiple images at positions ~θi can correspond to the same source position ~β.
The main quantity we are interested in here is the lensing (or deflection) potential,
ψ, defined as a scaled projection3 along the line of sight of the Newtonian gravitational
potential of the lens, φ:
Z
2 Dds
φ(Dd θ, z)dz.
(3.16)
ψ(θ) = 2
c Dd Ds
~ is related to the lensing potential ψ by:
The deflection angle α
~ θ ψ.
~ =∇
α
(3.17)
The Laplacian of the lensing potential describes the convergence κ(~θ), i.e. the surface
mass density of the lens plane scaled to its critical value:
1~2
κ(~θ) = − ∇
ψ.
2 θ
(3.18)
These two relations make the lensing potential a more straightforward and less errorprone quantity to address than the mass of a galaxy cluster, which, as we have seen,
relies on a more cumbersome definition.
As mentioned in the Introduction, in fact, even though traditionally the mass of
galaxy clusters is constrained making use of gravitational lensing, X-ray emission, the
thermal SZ effect and galaxy kinematics, none of these observables measures cluster
mass. On the other hand, all of them constrain in a way or another the projected gravitational potential. In particular, gravitational lensing constrains the curvature of the
gravitational potential (i.e. the gravitational tidal field). X-ray emission and the thermal
Sunyaev-Zel’dovich effect constrain respectively the temperature and the pressure of the
intracluster gas, which can be related to the lensing potential, as we will see in Chap.
5. Galaxy kinematics constrain the gradient of the gravitational potential. Another
negative aspect of the definition of cluster mass is that it is a global quantity requiring
integration over second derivatives of the gravitational potential [8]. This operation is
difficult to unequivocally perform on objects with irregular shapes and lacking precisely
identifiable boundaries, as in the case of galaxy clusters.
If a source is much smaller than the scale on which the properties of the lens change,
the lens mapping ~θ 7→ ~β is always locally linear and the distortion of images is encoded
in the Jacobian matrix:
!
∂(~β)
1 − κ − γ1
−γ2
~
=
(3.19)
A(θ) =
−γ2
1 − κ + γ1
∂(~θ)
3 We
will later reconstruct a projection of the gravitational potential along the line of sight, a quantity
that is proportional to the lensing potential defined in Eq. (3.16).
47
Chapter 3. Galaxy Clusters
which is written in terms of the lensing potential ψ, convergence κ and shear γ = γ1 +
iγ2 . The shear is the quantity that describes the shape distortion due to the gravitational
tidal field and is related to the lensing potential by:
1
(3.20)
γ(~θ) = ψ,11 − ψ,22 + iψ,12 .
2
The inverse of the Jacobian matrix, A(~θ), is called magnification matrix M(~θ) =
and its determinant is called magnification, µ:
A−1 (~θ)
µ = det M =
1
1
=
.
det A (1 − κ)2 − |γ|2
(3.21)
A galaxy cluster acts therefore as a lens and distorts the shape, size and position of
the background sources. Since lenses magnify background sources, clusters can be used
as natural telescopes. Surface brightness has been shown to be conserved [125]. In this
way, this magnification effect makes the background sources appear brighter than they
would otherwise be and provides a larger angular resolution for the sources, allowing in
many cases a detection that would not be possible without the presence of the cluster.
3.3.2
X-ray emission
Observations of clusters of galaxies show that they are some of the brightest X-ray
emitters (with luminousities around 1043 − 1045 erg/s) in the Universe, second only to
quasars. The main emission mechanism of diffuse X-rays has been identified to be
the thermal bremsstrahlung by a hot (T ∼ 108 K), low-density (10−2 − 10−3 atoms/cm3 )
plasma. The X-ray spectra of clusters also show line emission from ionised iron (Fe26+ )
and other metals, which have been ejected from the galaxies in the intracluster gas [148].
If we consider a spherical cluster as a first approximation, and we assume that the
ionised gas is in hydrostatic equilibrium, the mass profile of the cluster within the radius
r can be described by [66]:
!
kB T d ln ρ d ln T
+
r,
(3.22)
M(< r) = −
Gm d ln r d ln r
where ρ(r) and T (r) are the density and temperature profiles of the ionised gas and m is
the average mass of the gas constituents. The density profile ρ(r) can be observationally
determined by studying the surface brightness, S X :
Z ∞
r0
jX (r0 ),
(3.23)
S X (r) = 2
dr0 √
02
2
r
r −r
√
where jX (r) ∝ n2e T is the luminosity density or emissivity and ne is the electron number density [87]. In Chapter 5 we will show how to obtain information on the projected
gravitational potential of the cluster from the surface brightness map.
48
3.3. Obervational properties of galaxy clusters
Figure 3.4: Thermal Sunyaev-Zel’dovich effect: some low-energy CMB photons receive an energy boost during the collision with the electrons of the hot (T ∼ 108 K)
intracluster medium. Image of cluster courtesy of J.C. Waizmann.
3.3.3
Thermal SZ effect
The thermal Sunyaev-Zel’dovich (or tSZ) effect [166] originates from the interaction
between some of the CMB photons and high-energy electrons via inverse Compton
scattering: by colliding with the electrons of the hot gas in clusters, the CMB photons
scatter off and become more energetic. As a result, the initially perfect black-body
spectrum of the cosmic microwave background radiation is distorted by the clusters
of galaxies in its foreground. The observational signature of this effect is a transfer
of photons from the low-energy (Rayleigh-Jeans) portion to the high-energy (Wien)
section of the spectrum. This observable distortion from the original Planck spectrum
is a valuable tool to probe both the properties of the cluster and the gas clustering as
well as the cosmology. In particular, it is possible to use it to determine the value of H0 :
since the Compton-y parameter is inversely proportional to the squared angular diameter
distance, D2ang , and D2ang is inversely proportional to H02 , it follows that the Compton-y
parameter is directly proportional to H02 .
The thermal SZ effect depends on the integrated product of the number density of
the electrons and the temperature and is encoded in the thermal Compton-y parameter,
an integral of the electron pressure along the line-of-sight [141]:
Z
kB
σT dzT (~s, z)ne (~s, z),
(3.24)
y(~s) =
me c2
in which the thermal energy (kB T ) of the electrons is divided by the rest mass energy
(me c2 ), multiplied by the Thomson cross section, σT , and the electron number density,
49
Chapter 3. Galaxy Clusters
ne . The Compton-y parameter permits to quantify the intensity change, ISZ , of the CMB
radiation seen through a galaxy cluster with respect to the black-body spectrum Bω (T )
of the CMB:
∆IS Z
= g(x)y(~s),
(3.25)
Bω (T )
where x denotes the energy of the photon in units of the mean thermal energy, i.e. the
dimensionless frequency:
h̄ω
x=
,
(3.26)
kB T
and the spectral function g(x) is described by:
" x
#
x4 e x
x(e + 1)
g(x) = x
−4 .
(3.27)
(e − 1)2 e x − 1
It is negative for x < 3.83 and positive for larger values. This implies that the tSZ effect
causes a positive signal for frequencies above 217 GHz, frequency corresponding to
x = 3.83, and at the same time a negative one below. This signal represents a unique
spectral signature for SZ studies. It is necessary to notice that the distortion in the CMB
spectrum is tiny: the relative change in temperature of a CMB photon is of order 10−4
along a line of sight that crosses the centre of a rich cluster.
3.3.4
Galactic dynamics
Clusters are bound systems of galaxies and gas and are mainly dominated by their dark
matter content. This statement alone explains why studying their dynamical and statistical properties, such as the velocity dispersions of their member galaxies, and their
relation to the mass distribution is a powerful diagnostics to add relevant information
to the picture that we have of them. Since a major part of this work deals with galactic
dynamics, we chose to dedicate a separate chapter (Chapter 4) to it.
50
Nam tempus, spatium, locum et motum
ut omnibus notissima non definio.
Isaac Newton
4
The dynamical structure of galaxy
clusters
Given the breadth of the topic, writing a complete and comprehensive review of galactic
dynamics would certainly be a complex task and exceed the scopes of this thesis. Several excellent reviews of galactic dynamics are available and were used for the extension
of this chapter, in particular [25, 146]. The following chapter outlines the main theoretical foundations that will represent the starting point for the formulation and implementation of the method for the reconstruction of the lensing potential of galaxy clusters
that we will detail in Chapters 7 and 8. Section 4.1 discusses the Vlasov (or collisionless Boltzmann) equation. In section 4.2, moments of the Vlasov equation are taken to
obtain the Jeans equations. A specification in spherical coordinates is provided. Sections 4.3 and 4.4 introduce the concepts of mass-sheet degeneracy and mass-anisotropy
degeneracy.
4.1 The Vlasov equation
A galaxy can be pictured as a self-gravitating system of stars and dark matter particles
in a six-dimensional phase space and described via a particle distribution function or
phase-space density f (x, v, t).
We plan to use the study of the motion of the cluster member galaxies to infer the
shape of the gravitational potential within which they move. Throughout this thesis we
will assume that the cluster potential is solely sourced by the dark-matter distribution
and that the baryonic effects can be neglected to leading order. Since dark matter constitutes roughly 90% of the energy density of a galaxy cluster, this assumption seems
physically meaningful, at least as a first-order approximation. Hence, we will treat
galaxies as test particles in an external gravitational potential, a situation that is a priori
different from the case of a self-gravitating system typical for stellar kinematics studies. However, methods from stellar kinematics can be adapted to galaxy kinematics in
clusters because the gravitational potential φ contained in the equations is generally not
51
Chapter 4. The dynamical structure of galaxy clusters
restricted to the self-gravity of the system and can therefore be treated as an external
quantity that may be sourced by a different matter distribution than the test particles.
In contrast to hydrodynamics or plasma physics, which are dominated by short-ranged
interactions, gravitational forces are long ranged and cannot be shielded. Therefore, no
nearest-neighbour approximation is possible and the motion of a point-like test particle is affected by gravitational interactions with other particles in the full test volume.
Given a test particle with mass m0 moving in a volume filled by roughly equally distributed particles of mass m and particle number density n, the combined force of a
surrounding spherical shell with radius r and thickness dr is given by
dF = 4πr2
Gmm0
ndr.
r2
(4.1)
Assuming a constant particle number density, this relation can be integrated and yields:
Z r
Gmm0
F=
4πr02 02 ndr0 = 4πGmm0 r,
(4.2)
r
0
which diverges linearly for r → ∞. This consideration shows that long-range forces
have ample influence on the motion of the test particle and that the density distribution
of particles in the full test volume has to be taken into account. Long-ranged interactions
play therefore an important role and are strongly dominating over short-ranged ones.
This argument is supported by considering the typical relaxation time scales due
to the long-range interactions in galaxy clusters. For a generic astrophysical system,
the relaxation time describes the timescale for a test particle to substantially change the
direction of its motion due to gravitational interaction. For a stellar motion with initial
velocity v in a disk-like configuration of radius R that is composed of N particles, this
timescale can be estimated to be [14]:
trelax =
R N
.
v 8 ln N
(4.3)
This equation just constitutes an order-of-magnitude estimation and the fact that galaxy
clusters are obviously not disk-like objects will be ignored for the sake of this example.
For a typical cluster, masses are given by 1014 solar masses which can be split into a
system of approximately N = 1000 objects of galaxy mass. We assume a virial radius
of 1 Mpc and estimated galaxy velocities of 1000 km/s. Inserting these numbers into
Eq. (4.1) yields a relaxation time of about trelax ≈ 5 · 1011 years, which exceeds the age
of the Universe and therefore of the cluster lifetime by almost one order of magnitude.
Thus, one can safely assume that the test galaxy moves without being influenced by
small-scale encounters with local dark-matter distributions or by collisions with other
particles. Hence, the galaxy distribution in the cluster can be well approximated as a
collisionless system moving under the influence of an external gravitational potential.
52
4.1. The Vlasov equation
At a time t, the number dN of particles located in the volume element d3 x centred in
x and velocity in the d3 v centred in v is:
dN = f (x, v, t)d3 xd3 v,
(4.4)
and the number density of the system is:
Z
n(x) = d3 v f (x, v, t).
(4.5)
A statistical description of the system can be achieved by assuming that the distribution function f (x, v, t) satisfies Vlasov’s equation, also known as the collisionless
Boltzmann equation:
!
6
∂ f X ∂ f ∂φ ∂ f
+
vi
−
= 0,
(4.6)
∂t i=1 ∂xi ∂xi ∂vi
where φ is the smooth potential under whose influence the particles in the system are
moving. In vector notation,
!
∂f
∂f
+ v · ∇ f − ∇φ
= 0.
(4.7)
∂t
∂v
This is the fundamental equation of stellar dynamics, a special case of Liouville’s theorem, and states that the flow of stellar phase points through phase space is incompressible [25]. In other words, the phase-space density is conserved along flow-lines. If one
makes use of the Lagrangian derivative:
6
∂f
df ∂f X
≡
+
w˙α
,
dt
∂t α=1 ∂wα
(4.8)
~ = (w1 , ..., w6 ) are the phase-space coordinates, then the Vlasov equation
where (~x,~v) = w
can be expressed as:
df
= 0.
(4.9)
dt
When analysing spherical or axially symmetric cluster configurations, it is convenient to transform the Vlasov equation to coordinates that reflect the symmetry properties of the system. If we assume, for instance, the galaxy cluster to obey spherical
symmetry, choosing spherical polar coordinates allows to study the galaxy motion in a
centrally symmetric potential φ(r).
In spherical polar coordinates (r, θ, ϕ), Vlasov’s equation reads:
∂
∂
∂
∂
∂
∂
∂
f + ṙ f + θ̇ f + ϕ̇ + v˙r
f + v˙θ
f + v˙ϕ
f = 0,
∂t
∂r
∂θ
∂ϕ
∂vr
∂vθ
∂vϕ
(4.10)
53
Chapter 4. The dynamical structure of galaxy clusters
where the time derivatives of the coordinates can be written in terms of the velocity
components:
ṙ = vr
vθ
θ̇ =
r
vϕ
ϕ̇ =
.
r sin θ
(4.11)
(4.12)
(4.13)
and the components of the acceleration can be derived from the Euler-Lagrange equations:
v2θ − v2ϕ
∂
φ,
r
∂r
v2ϕ cot θ − vr vθ
∂
v˙θ =
φ,
−
r
∂vθ
−vϕ vr − vϕ vθ cot θ
1 ∂
−
φ.
v˙ϕ =
r
r sin θ ∂ϕ
v˙r =
−
(4.14)
(4.15)
(4.16)
4.2 The Jeans equations
Vlasov’s equation (Eq. (4.6)) is a partial differential equation governing the time evolution of a phase-space density, which in general is a function in six dimensions, all
parametrised by time. A complete solution is therefore not straightforward and is often impossible to achieve. In general, two strategies are available: either one simplifies
the problem by taking appropriate assumptions on the system or takes moments of the
equation by projecting out dimensions. This second strategy leads to a set of equations
that we will extensively use in the course of this work: the Jeans equations.
The zeroth moment of Eq. (4.6) is simply obtained by integrating Eq. (4.6) over all
possible velocities:
Z
Z
Z
∂f 3
∂φ
∂f 3
∂f 3
d v + vi d v −
d v = 0.
(4.17)
∂t
∂xi
∂xi
∂vi
For each velocity component, the mean velocity is defined by:
Z
1
d3 v f vi .
(4.18)
v̄i =
n(x)
Rearranging its first two terms and taking Eq. (4.5), Eq. (4.18) and the divergence
theorem into account, it is possible to rewrite Eq. (4.17) as:
∂n ∂(nv̄i )
+
= 0,
∂t
∂xi
54
(4.19)
4.2. The Jeans equations
a continuity equation for the spatial stellar density.
In order to get the next three equations, one can multiply Eq. (4.17) by v j and
integrate over all velocities:
Z
Z
Z
∂φ
∂f 3
∂f
∂
3
f v j d v + vi v j d v −
v j d3 v = 0.
(4.20)
∂t
∂xi
∂xi
∂vi
Following the same strategy as before, the result is:
∂(nv̄ j ) ∂(nvi v j )
∂φ
+
+n
= 0,
∂t
∂xi
∂x j
(4.21)
where
Z
1
vi v j =
vi v j f d3 v,
(4.22)
n
which can be written as the sum of the contribution of the streaming motion, v̄i v̄,
and the velocity dispersion tensor accounting for the distribution of stellar velocities
with respect to the mean at each point, σ2i j :
σ2i j = (vi − vi )(v j − v j ) = vi v j − vi v j .
(4.23)
The velocity dispersion tensor described by Eq. (4.23) is manifestly symmetric
(σi j = σ ji ) and, at any given point x, defines a velocity ellipsoid: an ellipsoid whose
principal axes are determined by the orthogonal eigenvectors of the dispersion tensor
and whose lengths of the semi-axes are defined by the square roots of its eigenvalues.
By subtracting v j times the continuity equation (Eq. (4.19)) from Eq. (4.21) and
making use of Eq. (4.23), one obtains a set of three equations known as Jeans equation
for a collisionless fluid:
∂v j
∂v j
∂φ ∂(nσi j )
+ nvi
= −n
−
,
n
∂t
∂xi
∂x j
∂xi
2
(4.24)
which is the equivalent of Euler’s equation for fluid dynamics, a comparison which
suggests to interpret the last term on the right-hand side as an anisotropic pressure. It is
nevertheless crucial to notice how in this case an equation of state is missing that relates
the pressure to the density.
In the case of fluid dynamics, there are five unknowns (three streaming motions, the
density and the pressure) and there are the continuity equation, three components of
the Euler equation and one equation of state to close the system. In the case of stellar
dynamics, there are ten unknowns (three streaming motions, the density and six independent components of the dispersion tensor) and three components of Jeans equation
and the continuity equation, leaving us with an underdetermined set of equations. We
will further analyse this point in Chapters 7 and 8.
55
Chapter 4. The dynamical structure of galaxy clusters
Another necessary remark to make is that, although any real distribution function
obeys the Jeans equation, not every solution of the Jeans equation is a physical distribution function. In order to be as such, it should be everywhere non-negative. We will see
in Chapter 6 that this requirement allows us to successfully apply the Lucy-Richardson
deprojection algorithm to the problem of identifying the lensing potential of spherical
and triaxial galaxy clusters.
4.2.1
The Jeans equation in spherical coordinates
We start from Vlasov’s equation in spherical coordinates, (r, θ, ϕ), Eq. (4.10). If we
substitute Eqs. (4.11) to (4.16) and integrate over the velocities to obtain the moment
equations, we get the Jeans equation for each component of the gradient of the potential.
The radial Jeans equation reads1 :
!
v̄ϕ
v̄θ
∂ϕ v̄r
n∂t v̄r + n v̄r ∂r v̄r + ∂θ v̄r +
r
r sin θ
1
1
(4.25)
∂ϕ (nσ2rϕ )
+∂r (nσ2rr ) + ∂θ (nσ2rθ ) +
r
r sin θ
n
+ [2σ2rr − (σ2θθ + σ2ϕϕ + v̄2θ + v̄2ϕ ) + σ2rθ cot θ] = −n∂r φ,
r
where ∂ xi = ∂x∂ i . Several assumptions can be taken to simplify this equation. The first
of which is to demand a stationary stellar-dynamical equilibrium, which sets the time
derivative to zero and excludes any explicit time dependence from the galaxy number
density and the mean radial velocity. Imposing spherical symmetry on the gravitational
potential makes it physically meaningful to assume that the mean velocities in the tangential and polar directions vanish. Additionally, the velocity components can be considered to be statistically independent of each other. The tangential and polar velocity
dispersions can be assumed to be equal since no preferred angular direction exists in a
centrally symmetric configuration. One can therefore write:
v̄θ = v̄ϕ = 0,
σ2rθ = σ2rϕ = σ2θϕ = 0,
σ2θθ = σ2ϕϕ .
(4.26)
(4.27)
(4.28)
The Jeans equation is at this point a linear, first-order inhomogeneous differential
equation for nσ2r :
nσ2
∂r (nσ2r ) + 2β r = −n∂r φ,
(4.29)
r
1 There exists two additional equations constraining the tangential and polar mean velocity components
respectively. Since they are of similar complexity and will be trivial for our further investigations once
we assume Eqs. (4.26), we decided to omit reporting them here.
56
4.3. Why gravitational lensing alone is not enough
where we have introduced the anisotropy parameter, β, measuring the departure from
isotropy of the velocity dispersion tensor:
β = 1−
σ2θθ
σ2rr
.
(4.30)
The anisotropy parameter takes values in the range −∞ < β < 1, where the extremes
correspond to the cases of purely circular and purely radial orbits respectively. If β = 0,
the velocity space is completely isotropic. A glance at Eq. (4.30) makes it evident
how difficult it proves to determine the anisotropy parameter, β(r), from observations.
In fact, observations do not provide information on the radial or tangential velocity
dispersions, σ2rr or σ2θθ , but only on the ones projected along the line-of-sight, σ2los ,
which are observed, among other quantities, in the Doppler distortion of the spectra: the
width of a line that experiences Doppler broadening is directly related to the velocity
dispersion averaged along the line of sight.
Furthermore, the observed spectral line is the result of the superposition of spectral
lines of many galaxies along the line of sight and hence it incorporates information
about the galaxy distribution as well. More precisely, brighter regions along the line of
sight contribute more strongly than fainter ones. Assuming that the distribution of light
along the line of sight is related to the galaxy density by some constant factor, we do not
infer the projected velocity dispersion alone, but rather its density-weighted average.
This is a most welcome property since, after appropriate deprojection, the results can
straightforwardly be inserted into the Jeans equation.
We will make use of this last consideration and of Eq. (4.29) in Chap. 7 to formulate
an algorithm able to recover the lensing potential of a galaxy cluster from the projection
along the line of sight of the velocity dispersions of its member galaxies.
A general formulation of the Jeans equation in spherical coordinates and its solution:

 Z r0
Z r

β(r00 ) 00 
 0

2
0
0
dr 
(4.31)
nσr = −
n(r )∂r φ(r ) exp 
 dr .
2
00
r
∞
r
can be found in [10].
4.3 Why gravitational lensing alone is not enough
As we discussed in Chapter 3, gravitational lensing is sensitive to the mass along the
line of sight. Ideally, one would have one, well-defined lens but, especially in clusters
of galaxies, this is often not the case. Additional shear and convergence from nearby
objects and mass along the line of sight can be present. [177] studied, for example, the
case of secondary matter planes producing strong lensing effects due to chance alignments along the line of sight and found, among other things, that these gravitational contaminations cause small but systematic effects leading to overestimation of the masses
57
Chapter 4. The dynamical structure of galaxy clusters
of individual clusters by gravitational lensing techniques as compared with the results
obtained with the velocity dispersions of the cluster members and/or the X-ray temperature.
The convergence of the lens is often degenerate with the morphology of the sources
and this phenomenon, known as mass-sheet degeneracy [67, 154], makes it strictly impossible to disentangle the convergence of the lens from the other contributions when
relying only on lensing with a single lens. Assuming that all the mass is concentrated
in the lens will therefore result in a bias either in the estimate of the mass of the galaxy
cluster or in the estimates of the cosmological parameters.
Over the years, several techniques have been devised to limit the effects of this
degeneracy. [35] proposed to lift the degeneracy by resorting to the magnification effect:
the local number counts of background sources is directly related to the magnification
of the lens, which allows an estimate of the convergence [71, 169].2 Another method
proposed by [34] relies on the use of distortion and redshift information of background
sources. Depending on the goal one has, a multi-wavelength/multi-probe approach can
be effective in increasing the information one can extract from the data, while limiting
the problem of dealing with the mass-sheet degeneracy. The same argument can be
made for dynamics.
4.4 Why dynamics alone is not enough
The exploitation of dynamics is of profound relevance because it provides information
on the mass of the clusters at large radii and gives additional constraints on quantities
that can be studied through several other techniques, as we will see in the next chapters.
Moreover, data come with different uncertainties and the dynamical and lensing masses
are affected by projection in different ways. However, it has to be said that dynamics rely
on an equilibrium assumption, while lensing does not, and that any attempt to determine
the mass of a galaxy or a cluster is not a trivial operation.
Depending on what kind of information is required, the density profile, the massto-light ratio or the velocity anisotropy need to be modeled, sometimes assumptions on
all these quantities have to be taken at once. The anisotropy profile is inevitably and
directly correlated with the three-dimensional shape of the galaxy distribution via the
tensor virial theorem and this implies that assumptions about the shape or viewing angle
of the galaxy distribution are necessary, or that, alternatively, a deprojection algorithm
has to be used. Other methods [38] require the phase-space distribution function to
determine the anisotropy profile.
A degeneracy that all the dynamical studies need to take into consideration is the
mass-anisotropy degeneracy: different assumptions regarding the anisotropy profile β(r)
From Eq. (3.3.1) one can see that to first order the magnification is related to 1 + 2κ, where κ is the
convergence of the lens.
2
58
4.4. Why dynamics alone is not enough
(Eq. (4.30)) lead to significantly different predictions for the mass distribution M(r)
of the cluster. Several methods have been presented to break or partially break this
degeneracy. Examples are a joint analysis of the observed profiles of velocity dispersion
and constraints from the fourth velocity moments (kurtosis) [93] or the use of flexible Bspline functions for the representation of the radial velocity dispersion in the spherically
symmetric Jeans equation assuming a given functional form of the mass density [52, 53].
In dynamics, there exists a vast arsenal of techniques for the determination of galaxy
and cluster masses, spanning from the Schwarzschild superposition of orbits [45] in the
case of stellar kinematics and the Jeans analysis [25] to the caustic method [51], the use
of analytical distribution functions, N-body simulations [47] and maximum-likelihood
fits of the distribution of observed tracers in the projected phase space [101]. All these
techniques have strengths and weaknesses and are successfully applied to a broad range
of data. The choice of a specific method is customarily tied to the kind of information
one wants to extract from the data and the kind of assumptions one is ready to make.
Our approach is based on the use of the Jeans equation and, in the spherical case,
does not require assumptions on the shape of the density profile. As we have seen in
the previous sections, the components of the Jeans equation do not constitute a closed
set. In the case of spherically symmetric clusters, presented in Chapter 7, an additional
constraint relating the density-weighted velocity dispersions, which we will call an effective galaxy pressure, to the density itself via a polytropic equation of state will be
taken. By adopting this polytropic relation, we will be able to invert the radial component of the Jeans equation relating the effective galaxy pressure to the dark matter
potential gradient.
In the case of a triaxial ellipsoid, a suitable form of a polytropic equation of state for
the effective pressure is so far not available. It has though been feasible to approach the
problem with perturbative techniques. Our results will be detailed in Chapter 8.
59
“Begin at the beginning,” the King said,
gravely, “and go on till you come to the
end; then stop.”
Lewis Carroll
5
Introduction to the reconstruction method
and application to the potential
reconstruction from thermal gas emission
Galaxy clusters are characterised at different scales by a wealth of observational properties. As we discussed in the Introduction and in Chapter 3, these properties can turn
into powerful diagnostics to obtain further insight on the physical nature of clusters.
Over the years, several methods combining two or more of them have become available
but none of these methods exploits the full spectrum of available cluster observables.
Starting from the consideration that it is possible to relate all of these observables to the
lensing potential, the essential goal of our approach is to construct a broad, joint method
for its reconstruction that combines the information encoded in all of them in order to
build a more comprehensive picture of galaxy clusters.
Section 5.1 introduces the key idea upon which we based the implementation of the
reconstruction method and discusses the overarching theme of this thesis.
The second part of this chapter is dedicated to the application of the method to the
reconstruction of the lensing potential from the observed surface brightness profile of
X-ray emission in clusters.
Approximately 15% of the mass of a cluster of galaxies is composed of a hot, diffuse
plasma that fills the cluster’s potential well, as we mentioned in Chapter 3. A loadbearing assumption of the standard model of galaxy clusters posits that the galaxies and
the gas in a cluster are in an approximate hydrostatic equilibrium with the gravitational
potential of the cluster and that they approximately trace each other in the cluster [11,
65, 70, 148].
Section 5.2 presents the general calculation for the density in the case of a galaxy
cluster in hydrostatic equilibrium and whose member galaxies form a polytropic gas
and have isotropic velocities. Section 5.3 summarises a method to reconstruct the lensing potential of galaxy clusters from the observed X-ray surface brightness profile of
galaxy clusters, described in detail in [87] and [170]. We start with the key points for
its implementation and present the results of its application to a simulated cluster. In
61
Chapter 5. Introduction to the reconstruction method
Figure 5.1: Key idea: Different phenomena can be described by a formally similar
algorithm.
the following, we proceed with the results of its application to observational data in
the case of Abell 1689. Section 5.4 reports on a method for recovering the projected
gravitational potential from the observed relative changes in the intensity of the CMB
photons through the thermal SZ effect, that will be published in Majer et al. 2015 (in
prep.).
5.1 The key idea
As mentioned in the Introduction, this work is part of a more general effort aiming at the
formulation of a unique, joint method for the reconstruction of the lensing potential of
galaxy clusters that incorporates information from all the available cluster observables.
The key idea on which we base our method is sketched in the cartoon shown in Fig.
5.1. The input to the algorithm comes from a cluster observable (in turn, the observed Xray surface brightness profile of galaxy clusters, the relative changes in the intensity of
the CMB photons through the thermal SZ effect and the projected velocity dispersions
of the cluster galaxies along the line of sight). This cluster observable is a projected
quantity and we deproject it by means of the Richardson-Lucy deprojection method that
we will discuss in Chapter 6. This method requires a shape to be assumed. For the
purposes of the deprojection, it turns out to be sufficient to assume a spherical galaxy
cluster, as it will be shown in Chapters 7 and 8. We have then a deprojected quantity
Q, which we want to relate to the three-dimensional gravitational potential φ of the
cluster. This is the step in which the physics describing the different phenomena ( X-ray
emission due to thermal bremsstrahlung, thermal SZ effect and galaxy kinematics) plays
a major role. At this point, we have a relation connecting the deprojected observable
to the gravitational potential. We can then invert it to obtain an expression for the
gravitational potential and subsequently project it along the line of sight to recover the
62
5.1. The key idea
quantity of interest, the projected potential.
A model-free, maximum-likelihood reconstruction method combining strong- and
weak-lensing information was developed by [18, 37, 109] and is called SaW-Lens, an
acronym standing for Strong And Weak Lensing.
The input to SaWLens is a catalogue containing the positions of the sources in an
arbitrary coordinate frame and the two corresponding components of the ellipticity. Information regarding the strong-lensing constraints can be provided by giving as an input
a catalogue containing the positions of multiple images and of estimates of critical-point
coordinates.
A two-component χ2 -function is then defined as:
χ2 (ψ) = χ2W (ψ) + χ2S (ψ),
(5.1)
where the individual terms depend on the lensing potential ψ and account for the information provided by the strong- (χ2S (ψ)) and weak-lensing (χ2W (ψ)) constraints, such
as multiple images, critical curves, ellipticity and, as an additional option, flexion measurements. Since the individual χ2 -functions are statistically independent, they can be
combined. The overall χ2 -function is then minimised with respect to ψ in every pixel:
∂χ2 (ψ)
! = 0, l ∈ [0, Npixels ],
∂ψl
(5.2)
where l indicates the grid position and Npixels the number of pixels.
The result of this minimisation is the lensing potential that reproduces the multiple
input constraints best. For a more detailed presentation of SaWLens, we refer the reader
to [108].
This maximum-likelihood approach can be extended to incorporate in the reconstruction of the lensing potential the information contained in all the cluster observables
at once, covering in this way a wide range of scales:
χ2 (ψ) = χ2W (ψ) + χ2S (ψ) + χ2X (ψ) + χ2SZ (ψ) + χ2kin (ψ) + ...
(5.3)
where χ2X (ψ) is the term accounting for the information provided by the X-ray emission,
χ2SZ (ψ) includes the reconstruction from the relative changes in the intensity of the CMB
photons through the thermal SZ effect and χ2kin (ψ) includes the reconstruction from the
line-of-sight projected velocity dispersions of the cluster galaxies.
For clarity, in the next sections we will briefly review the method for deriving the
lensing potential from the X-ray surface brightness profile of galaxy clusters and anticipate the method for the potential reconstruction from the relative changes in the
intensity of the CMB photons through the thermal SZ effect. Chapter 7 and Chapter 8
are instead dedicated to the detailed description of the algorithm for the reconstruction
of the lensing potential from kinematics, which is the main topic of this thesis.
63
Chapter 5. Introduction to the reconstruction method
5.2 General calculation for the isotropic case
In or near hydrostatic equilibrium and independently of the cluster shape, the hydrostatic
equation reads:
~
∇p
~
= −∇φ,
(5.4)
ρgas
where p and ρgas are the gas pressure and density, respectively. The gravitational potential φ of the cluster is dominantly sourced by the dark matter distribution.
We assume a polytropic stratification for the gas:
!
ρgas γ
p = p0
,
(5.5)
ρ0
where γ is the polytropic index and the quantities with a subscript 0 refer to an arbitrary
fiducial radius r0 that could, for example, be set to the virial radius.
Inserting this relation in Eq. (5.4) yields:
! #
"
ρgas γ
~
~ p0
= −ρgas ∇φ,
(5.6)
∇
ρ0
Since we know that:

! 
!
!
 ρgas γ−1 
ρgas γ−1 ρgas
~ 
~
 = (γ − 1)
∇
∇
,
 ρ
ρ0
ρ0
0
(5.7)
if we integrate both sides and rearrange the terms, we obtain:
"
ρgas = ρ0
1
# (γ−1)
ρ0 (γ − 1)
(φcutoff − φ)
,
p0 γ
(5.8)
where we introduced a cutoff radius rcutoff > r0 and fixed the gravitational potential such
that φcutoff = φ(rcutoff ).
5.3 Implementation of the method and application to
simulated X-ray data
As we introduced in Chapter 3, the main emission mechanism of diffuse X-rays has
been identified to be the thermal bremsstrahlung by a hot (T ∼ 108 K), low-density
(10−2 − 10−3 atoms/cm3 ) plasma. With the term bremsstrahlung, we designate the electromagnetic radiation produced by the acceleration of a charge (e.g., in a plasma, an
electron) in the Coulomb field of another charge (e.g. an ionised nucleus). A classical
64
5.3. Implementation of the method
treatment of this problem shows that an unbound electron coming from infinity and scattering off an ion with charge Ze, where Z is the atomic number and e is the elementary
charge, follows a hyperbolic orbit, where the distance of closest approach is referred
to as impact parameter, b. A Fourier transform of this orbit allows us to retrieve the
spectrum (i.e., the distribution of the energy over frequency) for a single electron. At
this point we can account for the fact that we do not deal with a single process but with
a superposition of such interactions.
Introducing the number density of ions and electrons ni and ne and integrating over
all impact parameters, we obtain the mean bremsstrahlung spectrum assuming that all
the electrons have the same velocity v∞ at infinite distance. Of course, this is not a
realistic description of the phenomenon. We therefore want to generalise these results
to a population of electrons with a given velocity distribution. The case in which we are
interested is a thermal population, namely an electron population with a locally uniform
temperature T . Then, the velocity distribution of the particles is Maxwellian. Such a
process occurring in a plasma in thermal equilibrium is called thermal bremsstrahlung.
In [14] we can find an expression for the spectra described above and for the frequency-dependent emissivity due to non-relativistic, thermal bremsstrahlung that one can
obtain after integrating the mean bremsstrahlung spectrum over a thermal electron population:
r
!
h̄ω
2m
16π2 Z 2 e6 ni ne
exp −
,
(5.9)
ḡ(ω)
jX (ω) = √
πκB T
kB T
3 3 m2e c3
where ω is the frequency, h̄ is the reduced Planck constant, kB is Boltzmann’s constant
and me is the mass of the electron. This spectrum is flat up to a cutoff at kh̄ω
and
BT
falls off exponentially at higher frequencies. The velocity-averaged Gaunt factor ḡ(ω)
encapsulates the quantum-mechanical correction to the classical formulae and can be
approximated by unity in many astrophysical situations.
If we substitute the electron and ion number densities with the gas density ρ and
integrate over all frequencies, we arrive at an expression for the frequency-integrated
emissivity due to bremsstrahlung:
r
2m 2 p
16π2 Z 2 e6
jX = √
ρ kB T = Cρ2 T 1/2 ,
(5.10)
π
3 3 m̄m2e c3 h̄
where C is the bremsstrahlung constant and m̄ is the mean gas-particle mass.
In [87] we develop a method for recovering the projected gravitational potential from
the observed X-ray surface brightness profile of galaxy clusters. We start by assuming
hydrostatic equilibrium. In this case, the density ρ and the temperature T of the gas are
fully characterised by the Newtonian potential φ. We can follow the logic exposed in
section 5.2 and, according to Eq. (5.8), we find for the density an expression:
1
ρgas = ρ0 ϕ (γ−1) ,
(5.11)
65
Chapter 5. Introduction to the reconstruction method
1
Sx(s)/Sx(smin)
0.1
0.01
0.001
0.0001
Surface brightness
0.2
0.4
0.6
0.8
1
1.2
s/Mpc/h
Figure 5.2: Azimuthally averaged and normalised surface brightness profile of a simulated galaxy cluster with a mass of 5 × 1014 h−1 M and a redshift of 0.2 as a function of
the projected radius s [87].
where we have introduced the dimensionless potential:
ϕ=
ρ0 (γ − 1)
(φcutoff − φ),
p0 γ
(5.12)
and where the quantity:
p0
= c2s,0
(5.13)
ρ0
is the squared sound speed at the cutoff radius.
The temperature of an ideal gas in thermal equilibrium with the potential ϕ is:
γ
T=
m̄ P m̄ P0
=
ϕ = T 0 ϕ,
kB ρ kB ρ0
(5.14)
where T 0 is the temperature at the fiducial radius.
If we combine Eq. (5.14) with Eq. (5.10), we can find the following relation:
jX = Cρ20 T 01/2 ϕη , η :=
66
3+γ
.
2(γ − 1)
(5.15)
5.3. Implementation of the method
2
(s)- (smax)
1.5
1
0.5
0
lensing potential (finite)
lensing potential (expected)
0
0.2
0.4
0.6
0.8
1
1.2
s/Mpc/h
Figure 5.3: Reconstructed and normalised projected potential of the simulated galaxy
cluster whose surface brightness profile is shown in Fig.5.2 as a function of the projected radius s. The potential was reconstructed assuming α = 0.4 (Eq. (6.10)) and
L = 0.3 h−1 Mpc in the Richardson-Lucy deprojection [87].
Assuming a polytropic relation between the temperature profiles and the integrated
emissivity profiles yields adiabatic indices that lie in the range 1.1 - γ - 1.2 [68].
If we look again at the idea sketched in Fig. 5.1, we can see that we obtained
an expression of the three-dimensional Newtonian potential in terms of a deprojected
quantity, the frequency-integrated emissivity due to bremsstrahlung, and that we are
now able to obtain an estimate for the projected gravitational potential.
Our algorithm to constrain the projected potential from the projected X-ray surface
brightness is composed of the following steps:
1. We deproject the projected X-ray surface brightness map, S X , by resorting to
the Richardson-Lucy method described in Chapt. 6 and find an estimate for the
frequency-integrated emissivity, j̃X ;
2. We use Eq. (5.15) to derive an estimate for the three-dimensional Newtonian
67
Chapter 5. Introduction to the reconstruction method
potential1 :


ϕ̃ = 
1/η
j̃X 
 ;
Cρ20 T 01/2
(5.16)
3. We project ϕ̃ along the line of sight and obtain an estimate ψ̃ for the two-dimensional
potential, which is proportional to the lensing potential and can therefore be combined with estimates of ψ derived from lensing and other reconstruction methods.
In order to test our algorithm, we simulate galaxy clusters in a spatially flat, standard
ΛCDM Universe with Ωm = 0.3, Ωb = 0.04 and ΩΛ = 0.7. We assume that the density
profile of the dark matter in the cluster potential well has the NFW form [117], described
by Eq. (3.9) in Sect. 3.2.2. The gas-mass fraction is set to equal the universal baryon
mass fraction fb = Ωb /Ωm and the gas is assumed to consist of 75% of hydrogen and
25% of helium, both completely ionised, with an effective adiabatic index γ = 1.2. The
virial radius of the cluster is approximated by r200 .
We use Eqs. (5.12) and (5.14) to compute the gas density and temperature profiles.
We set the cutoff radius for the gravitational potential to 100 r200 , which makes the
temperature profile drop to zero at a large radius.
For a detailed description of the properties of the simulated CCD image and of
further tests of our algorithm, please refer to [87]. The normalised surface brightness
profile for one realisation of a galaxy cluster with a mass 5 × 1014 h−1 M and a redshift
of 0.2, which represents the input to our algorithm2 , is shown in Fig.5.2.
The output of our algorithm, namely the reconstructed and normalised projected
potential, is shown in Fig. 5.3 and plotted against the expected result for reference.
The expected projected gravitational potential is taken to be the solution of Poisson’s
equation assuming an NFW density profile with the addition of Poissonian noise [87].
5.3.1
Application to observational data
Our method has been applied to X-ray observations of the cluster Abell 1689 in [170].
The cluster is located at a redshift of 0.183 in the Virgo constellation and is a wellknown strong-lensing cluster. Kinematics and X-ray studies ([94, 110, 112, 131, 157])
revealed a discrepancy between the hydrostatic mass and the mass estimated from gravitational lensing and offered several possible explanations, mostly agreeing on indicating
a merger aligned with the line of sight ([7] and references therein).
To recover the cluster potential from X-ray measurements, we first deproject the
X-ray data, then convert the X-ray three-dimensional profile into the three-dimensional
is probably worthwhile observing that the exponent η in Eq. (5.15) is a large number (its value lies
in the range 10 - η - 20) and that therefore the exponent 1/η in Eq.(5.16) is a small number, which helps
smoothing ϕ̃ and thus damping errors.
2 The concentration parameter (Eq. (3.10)) for the NFW profile has been set to c = 5.
1 It
68
5.3. Implementation of the method
reconstructed for γ=1.19 +/- 0.04
recovered from lensing
Lensing Potential
0.01
0.001
102
103
s[kpc]
Figure 5.4: Projected gravitational potential reconstructed using the method described
in [87] compared to the potential recovered from weak gravitational lensing as a function of the projected radius s. The projected potential reconstructed from the X-ray
emission has been obtained by assuming a polytropic index γ = 1.19 ± 0.04. The choice
of this value is based on a fit of the polytropic relation between the emissivity and the
temperature profile [170]. The potential recovered by lensing is shown in blue; the mean
projected potential obtained from X-ray observations with γ = 1.19 ± 0.04 is shown in
red, the uncertainties of the latter are shown in gray. The mean and errors have been
obtained using a Monte Carlo method to randomize the algorithm described in Konrad
et al 2013 [87]. The reconstructed lensing potential has been normalized to the lensing
data within a circular shell region delimited by the radii in the range [92; 1734] kpc.
gravitational potential using equilibrium assumptions, and project it again to compare
it with the two-dimensional gravitational potential obtained by gravitational lensing,
following the algorithm described in Sect. 5.1 and in this section. The reconstructed,
two-dimensional potential, assuming a polytropic index γ = 1.19 ± 0.04, is shown in Fig.
5.4. This profile is compared with the lensing potential obtained from weak lensing.
At radii larger than 500 kpc our reconstructed potential profile agrees well with the
reconstructed lensing potential within the error bars, even if the potential reconstructed
from the X-ray emission appears to be slightly less curved. The discrepancy at small
radii and the slight curvature change at larger radii may be caused by the lack of resolution in the weak-lensing measurement and by the other assumptions made (sphericity,
69
Chapter 5. Introduction to the reconstruction method
polytropic stratification, hydrostatic equilibrium, ideal-gas equation of state) and is expected to vanish if more potential observables are taken into account. For instance, a
combined strong- and weak-lensing reconstruction is expected to yield a better match
in the region close to the cluster centre because the strong lensing is sensitive to this
region and has the resolution to resolve it.
For a complete discussion of the application of the reconstruction method to Abell
1689 and an analysis of the possible reasons for such a discrepancy at small radii (R 500kpc), please refer to [170].
5.4 Application to the thermal SZ effect
The above-described reconstruction method can also be applied to recover the projected
gravitational potential from the observed relative changes in the intensity of the CMB
photons through the thermal SZ effect and is currently under study (Majer et al. 2015,
in prep.). As mentioned in Sect. 3.3.3, the thermal SZ effect is a small distortion of
the black-body CMB spectrum due to inverse Compton scattering of the CMB photons
with the more energetic ICM electrons.
By assuming hydrostatic equilibrium and a polytropic stratification of the intracluster gas, it is possible to find a relation between the Compton-y parameter and the
Newtonian gravitational potential. Given these assumptions, we can use Eq. (5.12) for
the binding potential of the cluster and derive Eq. (5.14) for the temperature of a gas in
thermal equilibrium from the ideal gas equation.
Combining Eqs. (5.11), (5.14) and (3.24) allows to rewrite the Compton-y parameter
in terms of the gravitational potential as:
Z
kB
σT T 0 ρ0 dzϕζ (~s, z),
(5.17)
y(~s) =
2
me c
with the exponent ζ = γ(γ − 1)−1 .
From Eqs. (3.25) and (5.17) we can infer the relation
P(~r) = P0 ϕζ (~s, z),
(5.18)
between an effective pressure P(~r) = P0 T (~r)ne (~r) and the dimensionless gravitational
potential ϕ, where the amplitude:
kB
σT T 0 ρ0
(5.19)
me c2
was introduced, where g(x) is the spectral function described in Eq. (3.27) and x is the
dimensionless frequency in Eq. (3.26). We can at this point notice a formal analogy
with Eq. (5.15).
We can therefore use Eq. (5.18) to implement a method analogous to the one presented in Sect. 5.3:
P0 = g(x)
70
5.4. Application to the thermal SZ effect
1. We deproject the measured, relative specific intensity change (Eq. (3.25)),
∆I¯S Z s(Bω (T ))−1 , by resorting to the Richardson-Lucy method described in Chapt.
6 and find an estimate for the three-dimensional effective pressure P;
2. We use Eq. (5.18) to derive an estimate for the three-dimensional Newtonian
potential:
!1/ζ̃
P(~r)
;
(5.20)
ϕ̃ =
P0
3. We project ϕ̃ along the line of sight and obtain an estimate ψ̃ for the two-dimensional
potential, which is proportional to the lensing potential and can therefore be combined with other estimates of ψ derived from lensing.
71
When my information changes, I alter
my conclusions. What do you do, sir?
John Maynard Keynes
6
Richardson-Lucy deprojection method
In Chapter 5 we introduced the working principles of our reconstruction method. The
first step of the algorithm sketched in Fig. 5.1 requires a deprojection: from a quantity projected along the line of sight we want to retrieve a three-dimensional quantity to
relate to the Newtonian, gravitational potential of the cluster, which we will then integrate along the line of sight to obtain an estimate for the two-dimensional gravitational
potential of the lens.
In this chapter we present the implementation of the deprojection method that we use
in the following chapters for the reconstruction of the two-dimensional gravitational potential starting from the projections of the velocity dispersions of cluster galaxies along
the line of sight. Its contents partially reproduce the section on deprojection contained
in the paper Reconstructing the projected gravitational potential of galaxy clusters from
galaxy kinematics [Sarli et al. 2014] [149]. Similar implementations have been used
for the reconstruction of the projected gravitational potential from the observed X-ray
surface brightness profile of galaxy clusters and from the observed changes in the intensity of the CMB photons using SZ data and are detailed in Konrad et al. 2013 [87] and
Majer et al. 2015 (in prep.).
6.1 Inverse problems in astronomy
In physics, and therefore in astronomy and astrophysics, methods to solve data-interpretation problems can be generally categorised in two ways: direct and inverse. The direct
method approaches the problems by identifying and following a causal sequence: starting from the formulation of hypotheses, their possible implications are analysed and
compared to simulations or available observed data. Although this forward procedure
is the most commonly and successfully used one and has the most predictive power,
problems arise in many branches of astronomy and astrophysics that require an inverse
formulation. For an interesting overview of forward and inverse problems in astronomy,
please see Lucy 1994 [96]. In this paper, Lucy specifies two criteria that designate the
inverse character of a problem. The first one is the lack of possibility of in situ measure73
Chapter 6. Richardson-Lucy deprojection method
ments of the quantity of interest, which is often the case in astronomy, and the second
one is the absence of a physical model on which the predictions can be based on due to
a poor theoretical understanding of the phenomenon. The first criterion is straightforwardly met in the study of galaxy clusters and, as seen in Chapter 4, the latter criterion
is met in the case of the anisotropy profile of cluster galaxies.
A large number of inverse problems in astronomy can be cast in the form of a Fredholm or a Volterra integral equation [2]. The main difference between the two is that a
Fredholm integral equation has constants as integration limits, while the Volterra integral equations have the independent variable of the equation as integration boundary. In
1-D notation, the Volterra integral equation of the first kind takes the generic form of:
Z x
φ(x) =
ψ(ξ)P(x|ξ)dξ,
(6.1)
0
and the Volterra integral equation of the second kind reads:
Z x
φ(x) = ψ(x) +
ψ(ξ)P(x|ξ)dξ,
(6.2)
0
where ψ is the function of interest, φ is the function available to observation and P is the
projection kernel. Lucy points out that for almost all of the inverse problems that can be
reduced to this standard form, the functions φ, ψ and P are non-negative. As mentioned
in Sect. 4.2, this is also the case for any solution of the Jeans equation that is at the same
time a physical distribution function. The topic we want to address belongs to this class
of problems.
We will see in Chapter 7 that a central role in the reconstruction of the projected
gravitational potential of galaxy clusters from the line-of-sight projected velocity dispersions of the cluster members is played by the inversion of a Volterra integral equation
of the second kind relating the density-weighted, three-dimensional velocity dispersions
of the cluster galaxies to the three-dimensional gravitational potential of the cluster. Observations, though, only yield the line-of-sight projection of the galaxy velocity dispersions and in order to be able to formulate our problem in terms of an equation of the
form of Eq. (6.2), we first need to perform a deprojection.
6.2 Implementation of the Richardson-Lucy algorithm
Different deprojection techniques by which it is possible to retrieve the three-dimensional
velocity dispersions are available. An option briefly discussed in [102, 147] is the deprojection by a Fourier transform. If proper integration variables are chosen, projection
can be considered as a convolution problem (see Eq. (6.4)). A Fourier transform of
this convolution leads to a product in Fourier space which can be inverted easily. Transforming back to real space yields the deprojected function1 . Another possibility is the
1 See
74
[84] for a more detailed discussion.
6.2. Implementation of the Richardson-Lucy algorithm
Abel inversion used by [28, 102, 183], which involves a differentiation of the data. This
can be very problematic in presence of noisy input data. We adopted as a third possible
method the Richardson-Lucy deconvolution [24, 95, 142]. A central requirement of this
method, due to its analogy to conditional probabilities, is a normalisation of the data.
A strength of this method is the presence of a regularisation that helps controlling the
noise.
A first version of this iterative deconvolution approximation method is presented in
Richardson 1972 [142]. There a probability technique to the restoration of noisy degraded images, another case of inverse problem, is applied. The main idea of the paper
is the use of Bayes’ theorem to retrieve the original image W starting from a degraded
image H and a point-spread function S (PSF) treating them as discrete probability distribution functions. Bayes’ theorem is formulated2 as the conditional probability of an
event W given an event H:
P(W)P(H|W)
,
(6.3)
P(H)
where P(W) is the prior on the original image, P(H) is the evidence and P(H|W) is the
likelihood which encapsulates the lack of knowledge about the PSF and its possible effects on the original image. Fig. 6.1 contains a schematic representation of the problem.
Richardson concludes his paper by remarking that no proof of convergence has been
devised but by observing that the process converged in all the cases he studied.
In [95, 96, 97] the same technique is further studied and developed by Lucy, hence
the name Richardson-Lucy algorithm.
The ingredients required for the implementation of this method in the case of the
reconstruction of the projected gravitational potential from galaxy kinematics are a lineof-sight projection g(s), assumed to be related to a three-dimensional quantity f (r) by
Z
Z
p
2
2
s + z = dr f (r)K(s|r)
(6.4)
g(s) = dz f
P(W|H) =
with a projection kernel K(s|r). As we will see in the next chapters, for g(s) we
will have the projected velocity dispersions along the line of sight and for f (r) the
three-dimensional density-weighted radial velocity dispersions. In the example reported
above (see Eq. (6.3)), the projection g(s) would be identified by H, f (r) by W and the
kernel K(s|r) by the PSF.
In spherical symmetry and with the anisotropy parameter β(r), the projection kernel
for the velocity dispersion is
!
2
r
s2
2
2
K(s|r) = √
Θ(r − s ) 1 − β(r) 2 ,
(6.5)
π r 2 − s2
r
2 We
decided to use the nomenclature chosen by Richardson, even if it can be misleading. When
presenting Bayes’ theorem, H is often used to denote the hypothesis or the model and D is taken to
identify the data. In our case the data are denoted with H and the true image with W.
75
Chapter 6. Richardson-Lucy deprojection method
Figure 6.1: Sketch of the problem described in Richardson 1972 [142]. The paper reports about the application of probability methods to the restoration of degraded images.
H is the degraded image, W is the original image, S is the point-spread function and the
symbol ∗ denotes the operation of convolution.
where Θ is the Heavyside step function. For isotropic velocity dispersions or an isotropic
gas pressure, the final factor in parentheses in Eq. (6.5) is unity. The kernel in Eq. (6.5)
can be easily derived from Fig. 7.1 as shown by [26].
Provided that the integrals of g(s), f (r) and K(s|r) are normalised to unity3 , the
following iterative scheme ensues from Bayes’ theorem (see [87, 95]):
Z
g(s)
K(s|r) ,
(6.6)
f˜i+1 (r) = f˜i (r) ds
g̃i (s)
where
g̃i (s) =
Z
dr K(s|r) f˜i (r) .
(6.7)
Thus, starting from a suitably guessed, normalised function f˜0 (r), the scheme given
by Eqs. (6.6) and (6.7) allows the recovery of the three-dimensional function f (r) from
its two-dimensional projection g(s), assuming the spherical symmetry incorporated into
the projection kernel K(s|r). Experience shows that even for guess functions f˜0 (r) that
are wildly different from the true f (r), the method converges surprisingly quickly within
a few iteration steps.
The deprojection algorithm can be interpreted as the result of a maximum-likelihood
problem obtained by variation,
#
"
˜i ]
˜i ] Z
δH[
f
δH[
f
(6.8)
f˜i+1 (r) = f˜i (r) + f˜i (r)
− dr f˜i (r)
,
δ f˜i (r)
δ f˜i (r)
3 The
76
kernel K(s|r) is normalised integrating over s.
6.2. Implementation of the Richardson-Lucy algorithm
where the functional derivatives of the functional H[ f˜]
Z
˜
H[ f ] = ds g(s) ln g̃(s)
(6.9)
occur, where g̃(s) is a functional of f˜ described by Eq. (6.7).
Both Richardson and Lucy observe that convergence is reached within few iterations. In order to apply this approach to realistic observational data containing small
scale fluctuations due to background or instrumental noise, though, a regularisation term
should be introduced as discussed in [97].
Following [97, 134, 174, 175], we introduce, therefore, a penalty functional S [ f˜]
decreasing as f˜ increases in complexity and weighted with a non-negative regularisation
parameter α:
H[ f˜] → Q[ f˜] = H[ f˜] + αS [ f˜] ,
(6.10)
where α adjusts the relevance that we attribute to a simple model (large S ) with respect
to the goodness of our fit of the data (large H).
The penalty functional S [ f˜] can have the entropic form4 :
Z
f˜(r)
˜
.
(6.11)
S [ f ] = − dr f˜(r) ln
χ(r)
Here, χ(r) is a smooth prior function, or default solution (i.e. the solution in absence of
data or in the limit α → ∞), chosen to suppress small-scale fluctuations. As a suitable
prior, we may take the smoothed version of the deprojection result from the previous
iteration step. This choice is known as floating default (see [81, 97]) and is defined by
Z
χ(r) = dr0 P(r|r0 ) f˜(r0 ) ,
(6.12)
with a normalised, usually sharply peaked convolution kernel P(r|r0 ) symmetric in r −r0 .
We use a properly normalised Gaussian with a smoothing scale L,
!
(r − r0 )2
0
P(r|r ) ∝ exp −
.
(6.13)
L2
It is crucial to remember that the adoption of a penalty functional and of a default solution introduces a bias in one’s estimate, since its role is effectively to parametrise one’s
ignorance.
Replacing the variation of H[ f˜] in Eq. (6.8) by the variation of Q[ f˜] from Eq. (6.10),
we obtain
(Z
g(s)
K(s|r)
(6.14)
f˜i+1 (r) = f˜i (r)
ds
g̃i (s)
"
! Z
#)
˜ 0
f˜i (r)
0 fi (r )
0
˜
+ α 1 + S [ fi ] + ln
− dr
P(r|r ) ,
χi (r)
χi (r0 )
4 For
a panorama of suitable penalty functions, we refer the reader to Titterington 1985 [172].
77
Chapter 6. Richardson-Lucy deprojection method
which we use henceforth. Since we work with discretised data sets, the integrals in
Eq. (6.14) need to be approximated by sums.
78
Any coincidence is always worth noticing. You can throw it away later if it is
only a coincidence.
Miss J. Marple
7
Spherical reconstruction of the lensing
potential
In this chapter, we extend the reconstruction method described in Chapter 5 towards
including information from galaxy kinematics. We assume that the motion of cluster
member galaxies is solely determined by the dark-matter (DM) gravitational potential.
The observables here are the line-of-sight velocity dispersions of the cluster galaxies as
a function of cluster-centric radius. The relation between three-dimensional galaxy velocity dispersions and the dark-matter gravitational potential is governed in equilibrium
by the Jeans equation (Eq. (7.5)). It resembles the equation of hydrostatic equilibrium
(Eq. (5.4)), but contains an additional term that takes a possible anisotropy in the velocity distribution into account. We describe the galaxy velocity dispersions here as
an effective, possibly anisotropic pressure related to the matter density by an effective
polytropic relation. Under this assumption, which we find well satisfied in simulations,
the effective galaxy pressure is related to the gravitational potential by a Volterra integral equation of the second kind (Eq. (6.2)), which can be solved iteratively. We can
then proceed as in Sect. 5.3 and [87]: We apply the Richardson-Lucy deprojection to
the observable line-of-sight velocity dispersions to obtain a three-dimensional effective
galaxy-pressure profile. This is then converted to the three-dimensional potential, from
which the two-dimensional potential is found by projection.
This chapter is organised as follows: We review in Sect. 7.1 the basic relations
underlying our analysis, in particular our treatment of the Jeans equation. The implementation of the Richardson-Lucy deprojection method in the case of galaxy kinematics
was presented in Sect. 6.2. Numerical tests of our method based on kinematic data of
a numerically simulated cluster and applications to observational data are described in
Sect. 7.2.
The contents of this chapter almost entirely reproduce the paper Reconstructing the
projected gravitational potential of galaxy clusters from galaxy kinematics [Sarli et al
2014] [149]. Parts of Sect. 7.3 will be published in Stock et al. 2015 [165].
79
Chapter 7. Spherical reconstruction of the lensing potential
Figure 7.1: Cluster geometry (see [26]).
7.1 Recovering the projected gravitational potential
from the projected velocity dispersions
7.1.1
Basic relations
To incorporate measurements of the kinematics of cluster galaxies into reconstructions
of the two-dimensional gravitational potential, we first require a relation between galaxy
velocity dispersions and the three-dimensional gravitational potential. The velocity dispersions are generally defined (see [26, 155]) as the mean squared deviations of the
velocities of the cluster members from the mean velocity hvi i of the population in each
radial bin i:
(7.1)
σ2i = hv2i i − hvi i2 .
Measured velocity dispersions are density-weighted projections of the three-dimensional
velocity dispersions along lines of sight through the cluster. Throughout this chapter,
we orient the coordinate system such that the z-axis coincides with the line of sight. A
projected velocity dispersion profile is constructed by averaging over concentric cylindrical shells drilled around the line of sight as a symmetry axis. This profile represents
the input of our method.
The final output of our algorithm is the projected Newtonian potential of the lens,
defined by
Z
~
ψ(θ) = φ(Dd~θ, z) dz ,
(7.2)
where φ is the three-dimensional gravitational potential of the cluster at distance Dd
from the observer. The projected potential is given as a function of the two-dimensional
position coordinate ~θ on the sky.
80
7.1. Recovering the projected gravitational potential from the projected
velocity dispersions
p/p0
100
Hernquist
NFW
10
1
0.1
1
ρ/ρ0
Figure 7.2: Relation (7.8) between the effective pressure and the density shown for the
Hernquist (Eq. (3.8)) and NFW (Eq. (3.9)) density profiles and adopting the anisotropy
parameter proposed by [79]. The very nearly straight lines (note the logarithmic axes)
demonstrate that the assumption of a polytropic relation is justified.
The geometry of the problem is sketched in Fig. 7.1 (see also [26]). For simplicity
during the construction of our method, we adopt a spherically-symmetric cluster model.
All equations derived in the following are thus formulated in spherical coordinates.
The possible anisotropy of the velocity distribution is described by the conventional
anisotropy parameter β (Eq. (4.30)) introduced in Sect. 4.2.1 and in [26],
β = 1−
σ2θ
σ2r
.
(7.3)
Our algorithm consists of three main steps:
• We deproject the observable, i.e. the velocity dispersions along the line of sight,
into a deprojected quantity that, for reasons to be clarified later, is called the effective galaxy pressure P.
81
Chapter 7. Spherical reconstruction of the lensing potential
(a)
(b)
Figure 7.3: (a) Surface-mass density of the simulated cluster g1 in the x-y plane. (b)
Two-dimensional gravitational potential obtained from the surface-density map solving Poisson’s equation via fast-Fourier transform. Both images show regions with
10 h−1 Mpc side length.
• Since this effective pressure P is related to the DM gravitational potential φ, we
can solve the Jeans equation using symmetry assumptions and a formal analogy
to a polytropic gas stratification. We thus obtain a relation between the DM gravitational potential φ and the effective galaxy pressure. This equation is a Volterra
integral equation of the second kind (Eq. (6.2)) for φ, which can be solved iteratively.
• Having obtained the gravitational potential, we simply project it to find the twodimensional potential ψ.
We described our deprojection method in Sect. 6.2. The relevant three-dimen-sional
quantities are the galaxy density ρ, the dark-matter density ρDM , the mean radial velocity
dispersion σ2r and the dark-matter gravitational potential φ. They are related via the
Poisson and Jeans equations. In spherical symmetry, the Poisson and Jeans equations
are
!
1 ∂ 2 ∂φ
r
= 4πGρDM (r) ,
(7.4)
∂r
r2 ∂r
and
σ2
∂φ
1 ∂(ρσ2r )
+ 2β r = − ,
(7.5)
ρ ∂r
r
∂r
respectively. We note here that the second term on the left-hand side of Eq. (7.5) is
the only formal difference to the hydrostatic equation for the hot intracluster gas (Eq.
(5.4)). This difference is important because it complicates the solution of the Jeans
equation considerably.
For solving the Jeans equation, we formally identify the density-weighted radial
velocity dispersion ρσ2r with an effective galaxy pressure P and assume that it is related
82
7.1. Recovering the projected gravitational potential from the projected
velocity dispersions
100000
number density
10000
n(r)
1000
100
10
1
0.01
0.1
1
10
r[Mpc/h]
Figure 7.4: Number density profile of the simulated cluster galaxies vs. the radius. It is
obtained by galaxy number counts in spherical shells.
to the density by a polytropic relation. This assumption can be justified by the following
calculation.
A given radial density profile ρDM (r) implies the mass profile
Z r
M(r) = 4π
x2 ρDM (x)dx
(7.6)
0
and, by Poisson’s equation, the gravitational-potential gradient
∂φ(r) GM(r)
=
.
∂r
r2
With this expression, the Jeans equation has the solution
!
Z r
2β
P(r) = P0 exp −
dx
r0 x
!
Z r
Z y
2β(x)
GM(y)ρ(y)
−
dy
exp
dx ,
x
y2
r0
r0
(7.7)
(7.8)
83
Chapter 7. Spherical reconstruction of the lensing potential
where the boundary conditions are set by the pressure P0 at the fiducial radius r0 . The
fiducial pressure P0 can be related to a fiducial density ρ0 via the virial theorem,
Z r0
P0
GM(y)
2
dy .
(7.9)
= hv i = −φ(r0 ) ⇒ P0 = ρ0
ρ0
y2
∞
This way, the effective pressure profile of the cluster galaxies can be expressed in
terms of the density and anisotropy profiles. At this point, we devise a simple way to
test our assumption in which we restrict ourselves to a dissipationless case study. We
evaluate Eq. (7.8) for two different mass density profiles, the Hernquist [80] and NFW
[119] profiles (Eqs. (3.8) and (3.9)) described in Sect. 3.2.2, adopting the anisotropy
profile derived by [79] from detailed numerical studies. As Fig. 7.2 shows, the effective
pressure-density relation is nearly polytropic, i.e. an approximate power law. Thus, our
assumption of an effectively polytropic galaxy stratification,
!γ
ρ
P = P0
,
(7.10)
ρ0
seems appropriate. In the numerical tests (see Sect. 7.2), we do not make use of the [79]
profile. We instead use the anisotropy profile directly obtained from the data.
With this result, we return to the Jeans equation (7.5), where we express the density
by the effective pressure by means of Eq. (7.10). For brevity, we further abbreviate
=
γ−1
,
γ
p=
P
,
P0
ϕ=
ρ0 φ,
P0
(7.11)
to cast the Jeans equation (Eq. (7.5)) into the form
dp 2β dϕ
+
p =− .
dr
r
dr
(7.12)
This linear, inhomogeneous, first-order differential equation with non-constant coefficients can be solved straightforwardly, e.g. by variation of constants. The solution
!
Z r
β
dx
p = −ϕ(r) + exp −2
r0 x
!
Z r
Z y
β
β
+2
dy ϕ(y) exp 2
dx
(7.13)
y
r0
r x
is the unique relation between the effective pressure and the gravitational potential we
were aiming at.
Equation (7.13) is a Volterra integral equation of the second kind [2] that can be
solved iteratively. In this way, we find a relation between the effective pressure p and the
gravitational potential ϕ. Projection along the line of sight leads to the two-dimensional
potential ψ.
84
7.2. Numerical tests
1
β(r)
0.5
0
-0.5
data points
mean β
fit Mamon-Lokas
-1
0
0.5
1
1.5
2
2.5
r[Mpc/h]
Figure 7.5: Radial profile of the anisotropy parameter β(r) defined in Eq. (7.3), obtained
from the simulated cluster data. The two-parameter model by [103] is fitted to the data
points. We find β∞ = 0.66 ± 0.36 and rβ = 0.82 ± 1.11.
7.2 Numerical tests
We now proceed to demonstrate that it is possible to recover the projected gravitational
potential ψ of a galaxy cluster from the measured velocity dispersions projected along
the line of sight. The algorithm described in Sect. 7.1.1 assumes spherical symmetry
and a polytropic relation between the effective galaxy pressure and the density.
If we feed the projected velocity dispersions into the Richardson-Lucy deprojection
described in Sect. 6.2, we obtain the effective pressure P related to the gravitational
potential φ by Eq. (7.13). Once this is achieved, the gravitational potential just needs
to be projected along the line of sight. We perform the reconstruction for three distinct
lines of sight, chosen to be parallel to the x-, y- and z-axes respectively.
85
Chapter 7. Spherical reconstruction of the lensing potential
1e+11
pr
1e+10
1e+09
1e+08
data points
linear regression
1e+07
1
10
100
1000
10000
100000
ρ
Figure 7.6: Effective galaxy pressure vs. the density. The relation is represented by a
power law, supporting our assumption of an effective polytropic relation. The mean
polytropic index, as derived by linear regression, is γ = 0.915 ± 0.022.
7.2.1
The data
For testing our algorithm with simulated data, we require a velocity-dispersion profile
projected along the line of sight and a two-dimensional gravitational potential obtained
independently. We obtain such data from one of the clusters (denoted g1) in the hydrodynamically simulated sample described in [150] and used previously in [106]. The g1
cluster has a virial mass M200 = 1.14 × 1015 h−1 M and is located at a redshift z = 0.297.
We start from a catalogue listing the Cartesian coordinates and the three Cartesian velocity components of simulation particles tracking the motion of cluster galaxies. All
information necessary for the kinematic analysis described in Sect. 7.2.2 can be extracted from this catalogue.
In addition, we convert the surface-mass density of the cluster into the two-dimensional potential, solving Poisson’s equation by means of a fast-Fourier transform. The
surface-density map and the two-dimensional gravitational potential are shown in Fig.
7.3. Evidently, the potential is considerably smoother than the mass density.
86
7.2. Numerical tests
10
10
proj. pressure
pressure
1
pr
pz
1
0.1
0.1
0.01
0.01
0.01
0.1
1
0.001
0.01
10
0.1
s[Mpc/h]
1
10
r[Mpc/h]
(a)
(b)
10
10
potential γ = const
lensing potential γ = const
1
φ(r)/φ(rmin)
ψ(s)/ψ(smin)
1
0.1
0.1
0.01
0.01
0.01
0.1
1
10
0.001
0.01
0.1
r[Mpc/h]
(c)
1
10
s[Mpc/h]
(d)
Figure 7.7: Results of the four different steps comprising our algorithm. (a) The input
to our pipeline is the normalised, line-of-sight projected velocity-dispersion profile as
a function of the projected radius and weighted by the galaxy number density. (b) The
Richardson-Lucy deprojection algorithm yields an effective galaxy-pressure profile. (c)
Solving the Volterra integral equation (Eq. (7.13)), we obtain the three-dimensional,
Newtonian potential. (d) The last step consists in projecting the gravitational potential
of panel (c) to find the two-dimensional potential. smin is set to 0.1 h−1 Mpc, smax is
chosen to be 2.0 h−1 Mpc.
87
Chapter 7. Spherical reconstruction of the lensing potential
To enforce axial symmetry, we azimuthally average the projected gravitational potential shown in Fig. 7.3 (b) around the centre, chosen to be the point with the deepest
(most negative) potential. To make it comparable to the three-dimensional gravitational
potential that we reconstruct from the kinematic data, it can be passed into the RL deprojection (with β = 0) yielding the gravitational potential which can then be appropriately
shifted and normalised.
7.2.2
Testing the algorithm
The assumption of spherical symmetry suggests spherical polar coordinates as a natural
choice for describing the system. After transforming the velocity components to this
coordinate frame, the number-density profile of the cluster galaxies is obtained within
radial bins ri centred on the cluster centre chosen to contain equal numbers of galaxies.
The number counts are then converted to a number-density profile by weighting with
the inverse volume of each radial shell,
n(r) ∝
counts
.
3 − r3
ri+1
i
(7.14)
Figure 7.4 reveals that the number-density profile for this particular cluster essentially follows a power law. In a first-order approximation that is sufficient for our purposes, galaxy biasing and variations of the galaxy mass function are neglected, allowing
us to adopt the number density as a direct tracer of the mass density.
Given the number-density profile and the radial bins, a mean galaxy velocity can
be obtained in each radial shell. The variance of the velocities about this mean yields
the galaxy velocity-dispersion profiles σ2r , σ2φ and σ2θ in the radial, azimuthal and polar
directions. These quantities enable us to derive the profile of the anisotropy parameter
according to the definition in Eq. (7.3). The result is shown in Fig. 7.5, where we fit the
two-parameter model
r
(7.15)
β(r) = β∞
r + rβ
proposed by [103], generalised by [171] and used in [101]. This fitted profile is the best
option available in the literature but the substantial scatter (also noted by [101, 182])
convinced us to use the profile-averaged mean value of β. We have also considered
a linear interpolation of the data points, but this leads to unconvincing results for the
gravitational potential, because the scatter in the interpolated anisotropy profile strongly
affects the results of the Volterra equation.
The observed velocity dispersions are quantities projected along the line of sight and
are thus implicitly weighted by the number density of galaxies along the line of sight.
Therefore, we weigh the radial velocity dispersions σ2r with the number density and
obtain in this way the effective galaxy pressure. Projecting this quantity along the line
88
7.2. Numerical tests
φ(r)
1
0.1
0.01
0.01
Simulation x-axis
Reconstruction x-axis
Simulation y-axis
Reconstruction y-axis
Simulation z-axis
Reconstruction z-axis
0.1
1
10
1
10
r[Mpc/h]
(a)
ψ(s)
1
0.1
0.01
0.01
Simulation x-axis
Reconstruction x-axis
Simulation y-axis
Reconstruction y-axis
Simulation z-axis
Reconstruction z-axis
0.1
s[Mpc/h]
(b)
Figure 7.8: Reconstructed gravitational potentials in (a) three and (b) two dimensions
are plotted as functions of radius and compared with the true potentials. In each panel,
the blue, light blue and sky blue points show the true potential determined from the
convergence map, while the red, orange and yellow points show the result of our reconstruction method. For better visibility, the additional point sets are multiplied by
factors of 0.5 for the y-axis case and 0.3 for the z-axis case. The corresponding relative deviation as a function of radius assuming the z-axis as a line of sight is shown in
Fig. 7.9.
89
Chapter 7. Spherical reconstruction of the lensing potential
0.8
relative deviation z-axis
relative deviation x-axis
relative deviation y-axis
10% level
0.6
0.4
0.2
reconstr
(s) - ψ
0.5
|ψ
sim
(s)|/|ψ
sim
(s)|
0.7
0.3
0.1
0
0
0.5
1
1.5
2
2.5
s[Mpc/h]
Figure 7.9: The relative deviation between the reconstructed and true two-dimensional
gravitational potentials where the line of sight is taken along the z-axis is shown here as
a function of distance from the cluster center. The deviation remains moderate (below
10 %) within a radius of approximately 1.5 h−1 Mpc.
of sight,
pz =
rmax
Z
dr pr (r)K(s|r) ,
(7.16)
s
with the projection kernel K(s|r) defined in Eq. (6.5), we finally complete the input for
our method which would be a proper observable provided by observations.
Figure 7.6 is a double-logarithmic plot of the effective galaxy pressure vs. the density, confirming that our polytropic assumption is reasonable. The mean polytropic
index γ, introduced in Sect. 7.1.1, is estimated from it by linear regression.
Figure 7.7 illustrates our complete algorithm. In the top left-hand panel, we show
the normalised, line-of-sight projected velocity dispersions as a function of the projected radius and weighted with the galaxy number density. In the top right-hand panel,
the normalised effective pressure profile obtained from the Richardson-Lucy deprojection algorithm is displayed. We then invert Eq. (7.13) to obtain the three-dimensional,
Newtonian gravitational potential shown in the bottom left-hand panel. In the bottom
90
Projected grav. potential Ψ(s)
7.2. Numerical tests
Lensing
X-Ray
Galaxy Kinematics
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
Projected radius s [Mpc]
2.5
3.0
Figure 7.10: Radial profiles of the gravitational potential of MACS J1206, reconstructed
from combined strong and weak gravitational lensing (blue), X-ray emission (yellow)
and from galaxy kinematics (red) with the parameters α = 0.3, L = 0.6, γ = 1.1 and β as
in Eq. (7.18). [165]
right-hand panel, we finally show the derived two-dimensional gravitational potential.
Figure 7.7 provides an overview of the first test we performed on our algorithm and
shows all steps needed to reconstruct the two-dimensional potential from the line-ofsight projected velocity dispersions. A comparison between our reconstructed profiles
and the true three- and two-dimensional gravitational potentials is now in order. We
define as “true” the potentials extracted directly from the numerically simulated cluster.
Figure 7.8 (a) shows the comparison between the reconstructed and the true gravitational potentials, while Fig. 7.8 (b) compares the reconstructed and the true projected
potentials. The reconstruction was performed three times with the line of sight axis
chosen to be in turn the x-, the y- and the z-axis. For easier comparison of both potentials, their zero points and normalisations are adjusted1 . To be consistent with [87], we
1 Note
that due to the normalisation constraint of the Richardson-Lucy deprojection, we obtain a func-
91
Chapter 7. Spherical reconstruction of the lensing potential
decided to normalise both functions to unity and fix φ(rmax ) = 0. In Fig. 7.9, the relative
deviation
ψreconstr (s) − ψsim (s)
,
(7.17)
∆ψ(s) =
ψsim (s)
between the reconstructed and the true two-dimensional potentials is shown as a function of projected radius. The deviation strongly increases at large radii, although it
remains below 10 % within 1.5 h−1 Mpc, which is similar to the virial radius of the cluster.
7.3 Application to MACS J1206.2 − 0847
The numerical tests presented above were run on line-of-sight projected data extracted
from a simulated galaxy cluster. Starting from spatial Cartesian coordinates and velocity components of a sample of member galaxies, the effective galaxy pressure was
obtained by weighting the squared radial velocity dispersion with the number density.
Afterwards, the required observable was constructed by projection along the line of
sight. The effective polytropic index γ and the profile β(r) of the anisotropy parameter could be easily obtained from the simulated sample. Unfortunately, neither of the
last two quantities is directly accessible to observations. Therefore a method needs to be
proposed that allows modelling the effective polytropic index and the anisotropy profile.
An application of the algorithm described in this chapter to the galaxy cluster MACS
J1206.2 − 0847 was performed by Stock et al. 2015 [165].
As we have seen in the previous sections, the reconstruction method requires the
setting of three parameters: the regularisation amplitude α, the smoothing scale L and
the polytropic index γ. In addition, the radial profile of the anisotropy parameter β(r)
has to be modelled.
Fig. 7.10 shows the reconstructed projected gravitational potential from galaxy kinematics (in red) with the parameters α = 0.3, L = 0.6, γ = 1.1. The value chosen for the
polytropic index γ is motivated by the results of [149] that γ would usually be around
unity, discussed in Sect. 7.2. The regularisation and reconstruction parameters are chosen such that the agreement with the lensing reconstruction is best. The β(r) is chosen
to be in agreement with the one presented in [27]:



0.1, if r < 0.2.




β(r) = 
(7.18)
0.4, if 0.2 < r < 0.5.




0.6, if 0.5 < r < 3.0.
tion proportional to φ. The potential extracted from the simulation has to be normalised in the same way.
Since both potentials are obtained from different datasets providing different spatial boundaries, they need
to be appropriately shifted (gauged) by adding an appropriate constant.
92
7.3. Application to MACS J1206.2 − 0847
In Sect. 4.4, we introduced the concept of mass-anisotropy degeneracy between the
anisotropy parameter β and the gravitational potential that ensues when only kinematic
data are used for the potential reconstruction. This degeneracy was lifted by fixing the
β-profile in order to obtain a non-parametric estimate for the gravitational potential.
To avoid an arbitrary, unmotivated guess for β(r), it was chosen such that the reconstructed potential agrees best with a reconstruction based on gravitational-lensing data.
Alternative methods for breaking the anisotropy-mass degeneracy and for recovering
the gravitational potential using galaxy kinematics have been discussed in Chapter 4.
Major efforts have been undertaken in recent years to either constrain the anisotropy
profile from observational and simulated data [92, 101] or to identify a general relation
describing it [103, 79]. The main objective of our work is to reconstruct the projected
gravitational potential of a galaxy cluster. However, we can envisage an inverse application of our algorithm. By adopting a technique similar to the one proposed in [92], it is
possible to constrain β(r) using simultaneously information from gravitational lensing
and galaxy kinematics. The existence of multiple constraints on the two-dimensional
potential ψ also allows developing an iterative method for constraining β(r) and γ.
93
Mathematics is the art of giving the
same name to different things.
Henri Poincaré
8
Ellipsoidal reconstruction of the lensing
potential
In Chapter 7 we presented a method for reconstructing the projected gravitational potential of a galaxy cluster starting from a catalogue of the velocity dispersions of its
members along the line-of-sight. Its main limitation lies in the assumption of sphericity
of the cluster. In this chapter we present the extension of this reconstruction method to
the case of a triaxial galaxy cluster.
8.1 Introduction and motivation
The main difficulty in extending our method to the case of a triaxial ellipsoid rests in
the form assumed by the Jeans equation (Eq.(4.24)). Whilst in the spherical approximation presented in Chapter 7, the Jeans equation can be reduced to one radial component
(Eq.(4.29)), in the ellipsoidal approximation, one has to work with a set of three components of the Jeans equation.
We have explored several possible approaches on how to further simplify the treatment of this problem. Following Chandrasekhar [40], we described a heterogeneous
ellipsoid via an ellipsoidal coordinate, λ, and formulated an expression for the gravitational potential as a function of λ, with the goal of reducing our problem to one component of the Jeans equation. The major difficulty that arose is that one should assume
a triaxial system in order to be able to correctly apply the definition of the ellipsoidal
coordinate λ and this would further complicate the set of equations rather than simplify
it. Another idea that we have investigated involves the use of methods from differential
geometry: we transformed the line element of the Euclidean space to prolate spheroidal
coordinates and formulated the Vlasov equation from the metric calculated in this way.
By taking velocity moments of this equation we obtained the Jeans equation in the new
set of coordinates (ρ, λ, ϕ). This procedure, though, yielded a set of two equations that
appeared to be more complicated than the classic ones in cylindrical coordinates. Probably a numerical solution could be set up for the Jeans equation for a spheroid, i.e.
95
Chapter 8. Ellipsoidal reconstruction of the lensing potential
a rotational ellipsoid that forms when an ellipse is rotated along its major axis. This
procedure would make sense if one wanted to investigate stellar kinematics in a galaxy
containing millions of stars. In our case, though, it would very likely not be worth the
effort since we are dealing with clusters of galaxies, which contain hundreds to thousand members in the case of rich clusters. In this case, we could calculate the velocity
dispersions in at most few tens of radial bins and the resulting resolution would be too
poor to solve a partial differential equation in an acceptable way.
8.2 Description of the method
A third and more successful approach is a perturbative expansion making use of the
general result for the gravitational potential of a body whose equidensity surfaces are
similar, coaxial ellipsoids, as presented in Binney and Tremaine [25] and Chandrasekhar
[40].
On the isodensity surfaces, the variable m2 is defined by:
m
2
:= a21
3
X
x2
i
,
a2
i=1 i
(8.1)
where ~x = (x1 , x2 , x3 ) are Cartesian coordinates and (a1 , a2 , a3 ) are the semi-axes of the
ellipsoid. Without loss of generality, we set a3 ≤ a2 ≤ a1 . We start considering the
homoeoid theorem extended to the case of heterogeneous ellipsoids:
Extended homoeoid theorem. A thin shell of uniform density (homoeoid), whose inner
and outer skins are the surfaces m and m + δm, generates an external potential that is
constant on the ellipsoidal surfaces:
3
X
xi2
m2 := a21
i=1
a2i + u
,
(8.2)
where u labels the surfaces. By breaking the gravitational potential of any body in which
the density is a function of m2 only, ρ = ρ(m2 ), into a series of thin triaxial shells, it is
thus possible to find [25]:
Z
a2 a3 ∞
ψ(∞) − ψ(m2 )
,
(8.3)
φ(~x) = −πG
du q
a1 0
(u + a21 )(u + a22 )(u + a23 )
where ψ(m2 ) is the integrated density:
ψ(m ) :=
Z
2
0
96
m2
ρ(x2 )dx2 .
(8.4)
8.2. Description of the method
We change the variable that labels the isodensity shells:
τ := u/a21 ,
and define the two ellipticities e1 and e2 :
v
v
t
t
2
a2
a2
e1 := 1 − 2 , e2 := 1 − 32 ,
a1
a1
(8.5)
(8.6)
which are well defined for a3 ≤ a2 ≤ a1 . We can then rewrite the potential φ in the
following way:
Z ∞
q
ψ(∞) − ψ(m2 )
2
2
φ(~x) = −πG (1 − e1 )(1 − e2 )
dτ q
.
(8.7)
0
(1 + τ)(1 + τ − e21 )(1 + τ − e22 )
The aim of our treatment is to find an approximate solution of the Jeans equation
by expanding the gravitational potential φ (Eq. (8.7)) to second order in the ellipticities
e1 , e2 . For readability, we define three auxiliary quantities T 1 , T 2 and A:
Z ∞
ψ(∞)
,
(8.8)
T 1 :=
dτ q
2
2
0
(1 + τ)(1 + τ − e1 )(1 + τ − e2 )
Z ∞
ψ(m2 )
T 2 (~x) :=
dτ q
,
(8.9)
)
0
2
(1 + τ)(1 + τ − e1 )(1 + τ − e2
q
A := (1 − e21 )(1 − e22 ),
(8.10)
and we rewrite the gravitational potential as:
φ(~x) = −πGA(T 1 − T 2 ).
Since our goal is to obtain an expression of the form:
∂φ ∂φ φ ≈ φ|e1,2 =0 + e1
+ e2
∂e1 e1,2 =0
∂e2 e1,2 =0
2 φ e21 ∂2 φ e22 ∂2 φ ∂
+
+
+ e1 e2
,
2 ∂e21 e1,2 =0 2 ∂e22 e1,2 =0
∂e1 ∂e2 e1,2 =0
many terms need to be calculated to proceed further.
We start with φ|e1,2 =0 :
#
"
Z ∞
ψ(m2 )
φ|e1,2 =0 = −πG 2ψ(∞) −
dτ
.
(1 − τ)3/2
0
(8.11)
(8.12)
(8.13)
97
Chapter 8. Ellipsoidal reconstruction of the lensing potential
After some algebra, one can see that the first-order derivatives of A, T 1 and T 2 (and
therefore of φ) with respect to the ellipticities vanish in the limit for e1,2 = 0. The same
happens for the mixed derivative, i.e. the last term in the Taylor expansion.
We now want to compute the second derivatives of A, T 1 and T 2 with respect to the
ellipticities in the limit for e1,2 = 0. In this limit, the second derivative of A with respect
to e1 is:
∂2 A q


 −e 1 − e2  1
2
∂ 
  q
=
= −1,
 
∂e

1
2
=0
e
=0
1,2
1,2
1 − e1
2
∂e e
1
(8.14)
In the case of T 1 we have:
2
∂2 T 1 = ψ(∞),
2
∂e1 e1,2 =0 3
(8.15)
∂2 T 1 2
= ψ(∞).
2
∂e2 e1,2 =0 3
(8.16)
and
In the case of T 2 we have:
Z ∞
Z ∞
ρ(m2 )x22
∂2 T 2 ψ(m2 )
=
,
dτ
+
2
dτ
(1 + τ)7/2
(1 + τ)5/2
∂e2 e =0
0
0
1
(8.17)
1,2
and
Z ∞
Z ∞
ρ(m2 )x32
∂2 T 2 ψ(m2 )
=
dτ
+
2
dτ
.
(1 + τ)7/2
(1 + τ)5/2
∂e22 e1,2 =0
0
0
(8.18)
Therefore, the second derivatives of φ with respect to the ellipticities in the limit for
e1,2 = 0 are:
Z ∞
ψ(∞) − ψ(m2 )
∂2 φ =
πG
dτ
(1 + τ)3/2
∂e21 e1,2 =0
0


Z ∞
Z ∞
2)
 2
x22 ρ(m2 ) 
ψ(m
 ,
−2
− πG  ψ(∞) −
dτ
dτ
7/2 
5/2
3
(1
+
τ)
(1
+
τ)
0
0
and, in a similar way,
98
(8.19)
8.3. Numerical tests
Z ∞
ψ(∞) − ψ(m2 )
∂2 φ =
πG
dτ
(1 + τ)3/2
∂e22 e1,2 =0
0


Z ∞
Z ∞
2)
 2
x32 ρ(m2 ) 
ψ(m
 ,
− πG  ψ(∞) −
dτ
−2
dτ
3
(1 + τ)7/2 
(1 + τ)5/2
0
0
(8.20)
A combination of these results with Eq. (8.13) in Eq. (8.12) gives the expansion for
the gravitational potential φ at the second order in the ellipticities e1 , e2 :


 2

 e1 e22  Z ∞ (2 − τ)ψ(m2 )

e21 e22 
φ(~x) = 1 − −  φ|e1,2 =0 + πG  + 
dτ
3 3
2 2 0
3(1 + τ)5/2
{z
}
|
{z
} |
Z ∞
Z ∞
2
2
2
2
ρ(m )x3
ρ(m )x2
2
+ πGe21
dτ
+
πGe
dτ
,
2
7/2
7/2
0 {z (1 + τ) } |
0 {z (1 + τ) }
|
(8.21)
where the first block incorporates the reconstruction in spherical symmetry formulated
in Chapter 7, φ|e1,2 =0 , and the second, third and fourth blocks represent the expansion
terms.
8.3 Numerical tests
As detailed in the previous chapter, we aim at a reconstruction algorithm that takes
a catalogue of velocity dispersions as an input and gives the projected gravitational
potential along the line-of-sight as an output. Our main objective in this chapter is to
extend the method we elaborated there for a spherical galaxy cluster to a triaxial object.
Our algorithm consists of the following steps:
• Starting from a catalogue of velocity dispersions, we first reconstruct the gravitational potential φ of the object of interest while assuming that it has spherical
symmetry with the reconstruction technique presented in Chapter 7.
• At this point, we are ready to use the expansion of the gravitational potential φ
described by Eq. (8.21).
• The last step of our algorithm consists in projecting the gravitational potential φ
obtained in this way along the line-of-sight to get the output we were aiming for,
the two-dimensional gravitational potential ψ of the cluster.
99
Chapter 8. Ellipsoidal reconstruction of the lensing potential
Normalised density profile z-axis
Normalised density ρ(r)
1e-24
Density
1e-25
1e-26
1e-27
1e-28
0.01
0.1
1
Radial position r[Mpc/h]
10
Figure 8.1: Mass density profile of the simulated cluster galaxies vs. the radius. The
normalisation accounts for M200 of the cluster at the given redshift. It is obtained following the procedure described by Eqs. (8.22) to (8.26). The values for r200 and the
redshift for the g1 cluster are specified in Table 8.1.
8.3.1
The data
In order to test our algorithm with simulated data, we need a velocity-dispersion profile
projected along the line of sight and a two-dimensional gravitational potential obtained
independently. We use the same data sample we adopted for testing the method in
spherical approximation: we take the g1 cluster from [150] and [106]. All information
regarding the data is presented in Sect. 7.2.1.
We perform the reconstruction for three distinct lines of sight, chosen to be parallel
to the x-, y- and z-axes respectively.
8.3.2
Testing the algorithm
As mentioned at the beginning of this section, there is a number of objects that we must
define before being able to implement our algorithm, the very first of which being the
density of the galaxies. For our cluster of interest we are provided with the value of
100
8.3. Numerical tests
10
10
proj. pressure
pressure
1
pr
pz
1
0.1
0.1
0.01
0.01
0.01
0.1
1
0.001
0.01
10
0.1
s[Mpc/h]
(a)
10
1
10
r[Mpc/h]
(b)
10
potential γ = const
lensing potential
φ(r)/φ(rmin)
ψ(s)/ψ(smin)
1
1
0.1
0.01
0.01
0.1
1
r[Mpc/h]
(c)
10
0.01
0.1
1
10
s[Mpc/h]
(d)
Figure 8.2: Results of the four different steps comprising our algorithm. (a) The input
to our pipeline is the normalised, line-of-sight projected velocity-dispersion profile as
a function of the projected radius and weighted by the galaxy number density. (b) The
Richardson-Lucy deprojection algorithm yields an effective galaxy-pressure profile. (c)
By solving the Volterra integral equation (Eq. (7.13)), we obtain the three-dimensional,
Newtonian potential for a spherical galaxy cluster. We use the spherical gravitational
potential of panel (c) in Eq. (8.21), from which we obtain the three-dimensional, Newtonian potential for a triaxial ellipsoid. (d) The last step consists in projecting the gravitational potential described by Eq. (8.21) to find the two-dimensional gravitational
potential assuming an ellipsoidal geometry for the galaxy cluster.
101
Chapter 8. Ellipsoidal reconstruction of the lensing potential
r200 , i.e. the radius where the mean matter density is 200 times the critical background
density of the Universe, ρb . We are therefore in the condition of computing the mass of
the cluster contained within this radius, M200 :
M200 = 4π
r200
Z
drr ρ(r) = 4πµ
Z
2
0
r200
drr2 ρgal (r),
(8.22)
0
where µ is the proportionality factor that accounts for the dark-matter fraction in the
cluster matter density budget and ρgal is the galaxy mass density.
Since our potential is normalised, we can assume that the galaxy density is proportional to the number density described by Eq. (7.14) taken from the spherical reconstruction and analysed in Sect. 7.2.2.
We know that by definition:
M200 = 200
Z
r200
dr4πr2 ρb (r) = 200
0
4π 2
r ρb ,
3 200
(8.23)
where ρb is the mean density of the background and is spatially constant in a FLRW
Universe:
3H02
3
ρb (z) = ρ(0)
(z)(1
+
z)
=
(1 + z)3 ,
(8.24)
b
8πG
with ρ(0)
(z) the background density today and z the redshift of the cluster given in Table
b
8.1.
Equating Eqs. (8.22) and (8.23), one can solve for the proportionality factor, µ, as:
2
200 4π
3 r200 ρb
,
µ = R r200
4π 0 drr2 ρgal (r)
(8.25)
which allows us to compute the total dark-matter density, ρ:
ρ(r) = µρgal (r).
(8.26)
Fig. 8.1 shows the normalised density profile that we later insert into Eq. (8.21).
In order to find a suitable expression for the ellipticities e1 and e2 , we define the
inertia tensor I component-wise:
I xx = Σnα=0 (xα )2 · wα ,
I xy = Σnα=0 xα yα · wα ,
...
102
(8.27)
8.3. Numerical tests
ψ(s)
1
0.1
0.01
0.01
Simulation x-axis
Reconstruction x-axis
Simulation y-axis
Reconstruction y-axis
Simulation z-axis
Reconstruction z-axis
0.1
1
10
s[Mpc/h]
Figure 8.3: Reconstructed gravitational potentials in two dimensions are plotted as functions of radius and compared with the true potentials. The red points show the true
potential, labeled “simulation”, determined from the convergence map, while the blue
points show the result of our reconstruction method. For better visibility, the additional
point sets are multiplied by factors of 0.5 for the y-axis case and 0.3 for the z-axis case.
The corresponding relative deviation as a function of radius assuming in turn the x-, yand z-axis as a line of sight is shown in Fig. 8.4.
where α denotes the particle and n is the number of particles and the weight function,
wα , is chosen to be1 :






 1 rα2 
,
(8.28)
wα = exp 
− cutoff 

2

 2(
)
4
where rα2 = xα2 + y2α + z2α . The cluster parameters and their values are summarised in Tab.
8.1.
We then calculate the eigenvalues a1 , a2 , a3 of the inertia tensor and we use them to
define the ellipticities e1 , e2 for the matter distribution according to the Eqs. (8.6). We
1 We
need for wα a function that goes like a density distribution. We chose a Gaussian of the form of
Eq. (8.28) and obtained a very good agreement with the axis-ratios listed in [106].
103
Chapter 8. Ellipsoidal reconstruction of the lensing potential
Parameter
Value
cutoff
1.5 Mpc/h
e1
0.73382
e2
0.934301
redshift
0.297
r200
1.5 Mpc/h
Table 8.1: Values of cluster parameters.
can see in Tab. 8.1 that the values obtained for them are rather large. We can though
notice that, even in case of very elliptic matter distributions, thanks to Poisson’s equation
and to the form of the isopotential surfaces given by the extended homoeoid theorem
(Eq. (8.2)), gravitational potentials are considerably less elliptic than their sourcing
density distributions.
At this stage we are ready to apply our algorithm to the sample of simulated data
described in Sect. 8.3.1.
Fig. 8.2 illustrates the four different steps of our algorithm for the z-axis in the
assumption of a triaxial galaxy cluster. The first three steps coincide with the procedure
we adopted in Chapter 7 for a spherical cluster. The top left panel shows the input of
our algorithm: the normalised, line-of-sight projected velocity dispersions as a function
of the projected radius and weighted with the galaxy number density, specified by Eq.
(7.16). In the top right panel, the normalised effective pressure profile obtained from the
Richardson-Lucy deprojection method described in Chapter 6 is displayed. The next
stage of the procedure involves inverting Eq. (7.13) to obtain the three-dimensional,
Newtonian gravitational potential shown in the bottom left panel. The reconstruction of
the gravitational potential we obtain in this way is then inserted into the first block of Eq.
(8.21) as φ|e1,2 =0 . At this point we can use all the objects obtained so far to implement the
expansion of the gravitational potential to second order in the ellipticities, as detailed in
Eq. (8.21). The bottom left panel displays its projection along the line-of-sight and the
final goal of our work, the projected potential ψ in the assumption of a triaxial galaxy
cluster.
Figure 8.2 gives an overview of the logic behind our algorithm and shows the single
steps that are to be taken to reconstruct the two-dimensional potential from the line-ofsight projected velocity dispersions. A comparison between our reconstructed profiles
and the true two-dimensional gravitational potentials is now in order. We call “true” the
potentials extracted directly from the numerically simulated cluster. Fig. 8.3 compares
the reconstructed (blue points) and the true (red points) projected potentials. The reconstruction was performed three times with the line-of-sight axis chosen to be in turn the
x-, the y- and the z-axis. For easier comparison of both potentials, their zero points and
normalisations are adjusted as in Chapter 7. To be consistent with [87] and [149], we
104
8.3. Numerical tests
relative deviation z-axis
relative deviation x-axis
relative deviation y-axis
10% level
0.2
0.15
|ψ
reconstr
(s) - ψ
sim
(s)|/|ψ
sim
(s)|
0.25
0.1
0.05
0
0
0.2
0.4
0.6
0.8
s[Mpc/h]
1
1.2
1.4
Figure 8.4: The relative deviation between the reconstructed and true two-dimensional
gravitational potentials where the line of sight is taken along the x-, y- and z-axis is
shown here as a function of distance from the cluster center. The deviation remains
moderate (below 10 % − 25 %) within a radius of approximately the virial radius of the
cluster 1.5 h−1 Mpc for all the three cases.
decided to normalise both functions to unity and to set φ(rmax ) = 0.
In Fig. 8.4, the relative deviation, Eq. (7.17), between the reconstructed and the true
two-dimensional potentials is shown as a function of the projected radius. The deviation
increases at large radii, although it remains below 25 % within the virial radius of the
cluster 1.5 h−1 Mpc for all the three axes and mostly below 10 %. It is possible to observe
a fluctuation in the reconstruction around 0.7 h−1 Mpc when taking the y-axis as the line
of sight.
105
Science never solves a problem without
creating ten more.
G.B. Shaw
9
Summary and conclusions
The central question addressed in this thesis is the problem of how to reconstruct the
projected gravitational potential of a galaxy cluster from the observed line-of-sight projected velocity dispersions of the cluster galaxies. We started by assuming spherical
symmetry in Chapter 7 and relaxed this assumption in Chapter 8, where we presented a
reconstruction algorithm valid for mildly elliptical, triaxial ellipsoids.
Chapter 5 begins by introducing the main argument on which we based our reconstruction method: clusters of galaxies exhibit a wealth of observational properties on a
wide range of scales and these properties can be used to obtain further insights into the
nature, structure and formation mechanisms of clusters as well as the nature of the dark
matter.
It was shown that a number of these cluster observables (X-ray emission due to
thermal bremsstrahlung, thermal SZ effect and galaxy kinematics) can be used to constrain the two-dimensional gravitational potential of the cluster, a quantity proportional
to the lensing potential. The essential goal of our approach is to construct individual
algorithms for each of these observables and to combine them at a later stage in a joint
method for the reconstruction of the lensing potential of galaxy clusters that incorporates information from all the available cluster observables.
A model-free, maximum-likelihood reconstruction method, called SaWLens, combining strong- and weak-lensing information is already available and was developed in
[18, 37, 109]. It is based on the minimisation of a χ2 -function composed of two contributions, one for strong and one for weak lensing. Since the individual χ2 -functions are
statistically independent, they can be combined and the method extended to incorporate
terms for all the cluster observables.
In Sect. 5.3 and in [87, 170], we have shown how the reconstruction of the projected
gravitational potential can be achieved with one of the two cluster observables based on
the hot intracluster gas, i.e. X-ray emission. An extension of our method to the thermal
Sunyaev-Zel’dovich effect can be formulated in an analogous way and the fundamental
steps to implement it are outlined in Sect. 5.4.
This thesis primarily focused on the formulation of an algorithm for the reconstruction of the projected gravitational potential starting from a catalogue containing the
107
Chapter 9. Summary and conclusions
observed line-of-sight projected velocity dispersions of the cluster members. In Chapter 7 we proposed a solution for spherical clusters that closely follows our interpretation
of the observables provided by the intracluster plasma described in Sect. 5.3, but necessarily differs from it in the crucial aspect that the velocity dispersion of the cluster
galaxies, unlike the gas pressure, can be anisotropic.
The algorithm we proposed rests on the following crucial assumptions. First, we
assumed that the effective galaxy pressure, by which we mean the product of the galaxy
number density and the local, squared radial velocity dispersion, can be related to the
density itself by a polytropic relation, i.e. a power law. We tested this assumption with
different density profiles and functional forms of the anisotropy parameter and found it
to be satisfied. Second, by adopting this polytropic relation, we solved the radial component of the Jeans equation, relating the effective galaxy pressure to the dark-matter
potential gradient. This solution can be analytically given in the form of a Volterra
integral equation of the second kind for the three-dimensional gravitational potential,
which can be inverted by iteration. Thus, we established a relation between the threedimensional gravitational potential and the effective galaxy pressure.
The effective galaxy pressure itself can be obtained from the observable, densityweighted line-of-sight projected galaxy velocity dispersions by means of the Richardson-Lucy deprojection introduced in Chapt. 6. The two-dimensional gravitational potential can finally be found by straightforward projection along the line of sight.
We have tested this algorithm on a simulated galaxy cluster in which galaxies have
been identified. This cluster is part of a sample of numerical hydrodynamical simulations described in [150] and used in [106]. The anisotropy profile β(r), as well as
the mean effective polytropic index γ, were obtained directly from the given member
galaxy catalogue. Although these quantities can hardly be directly measured from observations, and it has up to now been impossible to give general prescriptions of their
behaviour, it may well be justified to calibrate them on numerical simulations without
introducing an unacceptable bias.
Fig. 7.5 shows that the anisotropy profile may be poorly constrained. The fit performed by the model suggested by [103] covers a wide range of possible behaviour.
Reconstructing the two-dimensional potential based on the nominal mean value of the
anisotropy profile yields a result that closely follows the numerical expectation out to
≈ 1.5 h−1 Mpc, as demonstrated by the relative deviation of the reconstructed projected
gravitational potential from the true one (see Fig. 7.9). The deviation remains below
10 % in this radial range. Figure 7.6 confirms the approximate treatment of our system
as a polytropic stratification.
Up to this point, we have always worked on simulated data, so we have not applied
error propagation methods to our algorithm yet.
In Sect. 7.3 we showed the results of the application of the algorithm assuming
spherical symmetry to the galaxy cluster MACS J1206.2 − 0847 performed by Stock et
108
al. 2015 [165] on data provided by the CLASH-VLT collaboration (see Fig. 7.10). In
this publication, the propagation of errors through the Richardson-Lucy deconvolution
and through our reconstruction method is investigated numerically. Within the error
bars, the reconstructed potential profile is indistinguishable from the potential profile
obtained from the combination of weak- and strong-gravitational lensing and from the
X-ray analysis.
The main limitation of the algorithm exposed in Chapter 7 lies in the assumption
of spherical symmetry. Observations and numerical simulations, instead, suggest that a
triaxial ellipsoid would represent a more realistic hypothesis on the geometry of clusters
of galaxies [23, 12, 77] and of the velocity field of cluster members [85, 151, 159, 173,
179].
In Chapter 8, we relaxed the assumption of sphericity and extended it to the case of
an ellipsoidal body.
In contrast to stellar kinematics studies in galactic environments, the sample size
of objects is considerably smaller in galaxy clusters since, in the case of spectroscopic
studies of member galaxies, only a few hundred gravitationally-bound galaxies can be
identified. Furthermore, in the case of a triaxial ellipsoid, we cannot yet formulate a suitable form of a polytropic equation of state for the effective pressure. We have therefore
decided to approach the problem with perturbative techniques rather than pursuing a
rigorous modelling in ellipsoidal geometry. Our results seem to indicate that a thorough
ellipsoidal analysis might not be necessary.
Expanding the gravitational potential in the cluster’s geometrical ellipticities yields
second-order corrections to the spherical reconstruction. Due to Poisson’s equation,
gravitational potentials are considerably less elliptic than their sourcing density distributions and thus this approach seems applicable even in case of rather elliptic mass
distributions.
In particular, the effects of ellipticity on the cluster potential have been investigated
in the bachelor thesis of Dmitri Suharev and they were found to be minimal: the deviation from sphericity of a potential reconstructed from velocity dispersions in an axiallysymmetric (rather than radially-symmetric) cluster is below 10 − 15% for a radius larger
than 0.5 h−1 Mpc for an ellipticity e ∼ 0.2 − 0.3.
We tested the extended algorithm by comparing our results with the projected gravitational potential directly extracted from the surface mass density of the cluster by
inverting the Poisson equation in Fourier space. The data used for this comparison are
shown in Fig. 7.3. Fig. 8.3 displays the results of our comparison for the three x-, y-,
z-axes taken in turn as lines of sight and Fig. 8.4 shows the corresponding relative deviation as a function of radius. From them we can see that the deviation between the reconstructed and the simulated projected potential remains moderate (below 10 % − 25 %)
within a radius of approximately the virial radius 1.5 h−1 Mpc for the x-, y- and z-axis.
From the analysis presented here we can conclude that we can reconstruct the two109
Chapter 9. Summary and conclusions
dimensional, gravitational potential of a galaxy cluster starting from a catalogue of the
projections along the line of sight of the velocity dispersions of its member galaxies and
from the assumption of a spherical body, as discussed in Chapter 7. We can relax the
assumption of sphericity later on and extend our reconstruction to the case of an ellipsoidal galaxy cluster by expanding the gravitational potential in the ellipticities for the
purposes of a comparison with the projected potential derived from other observables,
as detailed in Chapter 8.
As mentioned in Sect. 5.1, the final goal of the work exposed here and in the publications of several members of our group [87, 149, 170] is the combination of the
algorithms presented in Chapters 5, 7 and 8 to allow a single, joint reconstruction of the
two-dimensional potential of galaxy clusters using lensing, X-ray, thermal SZ effect and
kinematic data.
A quantitative statement about the relative performance of the individual and joint
methods we propose is intrinsically bound to the quality of the data sets that will be used
to do the analysis and to the assumptions entering in the model. However, we can envisage at least three possible applications of the combination of potentials derived from
them. First, especially in clusters at low redshift, the gravitational potential obtained
from galaxy dynamics will extend to larger scales than the potential from gravitational
lensing. In order to recover the radial profile of the matter distribution and to measure
scale radii accurately, long lever arms are needed, and galaxy dynamics adds information from intermediate to large radii. Second, the dynamics of mass-less compared to
the dynamics of massive particles tests the theory of gravity. Therefore, combinations
of potentials reconstructed from lensing and from galaxy dynamics may be particularly
important for this purpose if the individual potentials can be reconstructed accurately
enough. Third, among the observables which cluster potentials can be recovered from,
strong and weak lensing neither need mechanical nor hydrostatic nor thermal equilibrium, galaxy dynamics needs mechanical equilibrium, and the X-ray emission and the
thermal SZ effect additionally need hydrodynamical and thermal equilibrium. Comparing the results will directly allow testing these equilibrium conditions.
110
Acknowledgements
You’ll need coffee shops and sunsets
and roadtrips. Airplanes and passports
and new songs and old songs, but people more than anything else. You will
need other people and you will need to
be that other person to someone else - a
living, breathing, screaming invitation to
believe better things.
J. Tworkowski
When I began working on my Ph.D. project in July 2011, I knew it would have been
a long journey. I knew that in the following three years I would have many exciting
moments and I guessed I would also live many, many frustrating days. This is somehow
what one expects from a Ph.D. programme: to learn a lot of new things and to learn
that it takes a good deal of patience and determination to obtain some results. What I
did not fully take into consideration was life. It did not take me three years to complete
my Ph.D. It will take me four years and a half. In this amount of time my life changed
forever several times.
Even though only my name appears on its cover, this thesis owes its existence to the
collaboration, help, support and inspiration of many people.
First and foremost, I would like to express my gratitude to my supervisor, Matthias
Bartelmann. Four years are a long time and many things have happened. Matthias’
door was always open. I could not have imagined having a better supervisor, from the
scientific or the personal point of view. In particular, I have to thank him for supporting
me in my decision to change the direction of my thesis when I realised I did not want
to continue working on the topic we had originally agreed on. His continuous guidance
was precious and he was a constant source of encouragement and inspiration. He supported my visits to the University of Bologna and my research interests. His scientific
curiosity, patience and enthusiasm have been an incredible example and taught me a
great deal about what it means to be a scientist. On a more personal note, six years ago
he introduced me to the man that later became my husband and, when I decided to take
a break of fifteen months after the birth of my son, he supported me and helped me with
additional funding to complete my Ph.D. thesis once I came back to work, things I am
very grateful for.
I wish to thank Luca Amendola for kindly accepting to be the second referee of my
thesis and for having me as a tutor in his course on general relativity a couple of years
ago.
I would like to thank as well Karlheinz Meier and Volker Springel for being examiners in my Ph.D. defence and Andrea Macciò for being a member of my IMPRS thesis
committee, together with Matthias and Luca.
Part of the work presented in this thesis has been carried out at the Observatory of
111
Acknowledgements
Bologna. I warmly thank Lauro Moscardini for the hospitality and for giving me the
possibility to present my work during the Tuesday seminar. A big thank you goes to
Massimo Meneghetti for the numerically simulated data that I used to test my method
and for the useful and interesting conversations about science and astronomy.
I would like to thank Anna Zacheus, Gesine Heinzelmann and Elizabeth Miller who
deserve credit for providing much needed assistance with administrative tasks and reminding me of impending deadlines.
This thesis has been written under very special circumstances. I had started it while
I was pregnant and had planned to conclude it before the birth of my baby but life is
unpredictable and he arrived two months before the due date. The weeks following his
birth were full of pain, fears and uncertainty. They were also full of love and wonder and
friends that gave us the space to adjust to this unexpected, scary situation while always
being next to us. It is a long list but they all deserve to be mentioned: Sven and Tini,
little Matthias and Barbara, Christian and Evi, Alex and Romy, Agnese, Ana, Emanuel,
Valentina, Mary, Maria, Charles, big Matthias, Anna, Emanuela, Fede and Ale, Daniela
and Giovanni. A special acknowledgement is dedicated to Arianna, who was so kind to
welcome my mother in her home and to feed my husband in the darkest hours.
After six months I started to feel better and I decided to go on writing my thesis. I
wrote it bit by bit, using the spare time that my lovely and patient baby was giving me,
and when I started to work again, I had one month to finish it. If I managed to do so, it is
also and especially thanks to the tireless support and help from two great colleagues and
friends, Sven and Christian. They helped me to check for every inconsistency, read all
the chapters and commented them with suggestions, questions and remarks that notably
improved the quality of my thesis. In particular, I want to thank Sven for going through
a non-negligible number of derivatives for a non-negligible amount of times in order to
make sure that my calculations were correct and for strongly advising me against showing Christian any page without the bibliographical references in the correct numerical
order. I have always appreciated Christian’s attention to the detail. In this last month
it helped me immensely. He always found the time to discuss the scientific content of
my thesis, an originally rather confused line of argument, a punctuation dilemma or an
editing rule.
I will always have fond memories of my years in Heidelberg. Not only I had the
chance to learn many things about physics and to take some important decisions about
my future but I also met many friends here. I want to thank the ITA boys and the
ARI girls for making me feel home immediately, for the uncountable coffee breaks, the
glorious movie nights and for helping so much with the organisation of my wedding!
A big thank you goes to my office mates, Felix, Sven, Agnese and Iva, for the relaxed
and pleasant working atmosphere, for sharing funny jokes and profound conversations
and for tolerating Sven and me discussing of spherical cows and sci-fi movies way too
often!
112
Acknowledgements
I would especially like to thank my dear friend and office mate, Agnese, who has
been sharing all the highs and lows of the Ph.D. with me, an incredible amount of coffee
and recently a good number of blank pages to write. These four years would have been
way harder without her company, friendship and support.
My years at ITA have been accompanied by many enjoyable breaks. Emanuela,
Anna, Agnese and Arianna made them very special moments.
Over the years, I met many good friends at ITA and ARI. Some of them are still here,
some now live in other countries. I wish to thank Federica for always being there and
for providing philosophical analyses of the Ph.D. life and support in written form when
she cannot do it in person, Alessandra because she knows that sometimes all you need
is a picture with a fluffy and furry animal, Ana for the uncountable chats and Massimo
for offering me soothing words (and alcohol) when I most needed them.
Eventually I would like to thank some non-cosmologists. I thank James and Felicity
for welcoming me in their home in Wales, for trying to elucidate the mystery of English punctuation rules and for initiating me to some embarassing yet legendary BBC
productions.
A big thank you to the generalised Teodolinda group: Valentina, because she is
always there and she shares my unhealthy passion for American TV shows; Chiara,
because she made a crazy trip to Germany in order to be at my wedding; Sara, Martina
and Teresa, because despite the distance they are still supporting me; Salvo, Matteo,
Andrea, ing. Pensini and Massimo, because they are Teodolinde, no matter what.
A dear thought goes to my former flatmate, Maria, for all the time spent together,
the support and the encouragement she always gives me.
Deep thanks also go to Mary, who is there for me since the very first day of high
school, and to Daniela, who was always my friend.
Ringrazio i miei genitori, Grazia e Renato, mia sorella Giulia e mia nonna Gina che
mi hanno sostenuto in questi anni di studio e di lontananza da casa.
My last words are for my two boys. Jean-Claude, I would like to say many things
that don’t belong here. The past years have been wonderful, full of joy, laughter and
adventures. You have been close to me in the hardest and saddest moments of my life
and never lost your sense of humour, even when mine was long gone. We have lived in
different countries and in the same tiny village. We got married, we crossed Cuba on
a not so solid-looking Chinese car, spent breakfasts talking about manifolds, literature
and art and now have the funniest, loveliest baby. Waking up every morning with you
two makes me immensely happy.
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