Studying Dark Matter Haloes with Weak Lensing

Studying Dark Matter Haloes with Weak Lensing
Studying Dark Matter
Haloes with Weak Lensing
Studying Dark Matter
Haloes with Weak Lensing
PROEFSCHRIFT
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van de Rector Magnificus prof. mr. P. F. van der Heijden,
volgens besluit van het College voor Promoties
te verdedigen op woensdag 20 juni 2012
klokke 11.15 uur
door
Malin Barbro Margareta Velander
geboren te Lund, Zweden
in 1983
Promotiecommissie
Promotor:
Prof. dr. K. H. Kuijken
Overige leden:
Prof. dr. M. Franx
Prof. dr. A. Taylor (Edinburgh University, UK)
Dr. H. Hoekstra
Dr. J. Brinchmann
For my husband
and our little one
Cover: Chinese Kite Cluster by Gabrian J. van Houdt
Table of Contents
1 Introduction
1.1 Cosmology . . . . . . . . . . . . . . . . . . .
1.1.1 The concordance model of cosmology .
1.1.2 Alternative models . . . . . . . . . . .
1.1.3 Cosmology probes . . . . . . . . . . .
1.2 Gravitational lensing overview . . . . . . . . .
1.2.1 Fundamentals of lensing . . . . . . . .
1.2.2 Microlensing . . . . . . . . . . . . . .
1.2.3 Strong lensing . . . . . . . . . . . . .
1.3 Weak lensing . . . . . . . . . . . . . . . . . .
1.3.1 Convergence, shear and flexion . . . .
1.3.2 Cosmic shear . . . . . . . . . . . . . .
1.3.3 Galaxy-galaxy lensing . . . . . . . . .
1.3.4 Bullets and train wrecks . . . . . . . .
1.3.5 Shape measurement methods . . . . .
1.4 This Thesis . . . . . . . . . . . . . . . . . . .
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1
1
1
6
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23
25
29
32
36
2 A new shape measurement method and its application to
galaxies with colour gradients in weak lensing surveys
38
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 Shear and flexion . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.2 Shapelets . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.3 The MV pipeline . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Monochromatic tests . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.1 GREAT08 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.2 FLASHES . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4 Non-monochromatic tests . . . . . . . . . . . . . . . . . . . . . . 55
2.4.1 Analytical prediction . . . . . . . . . . . . . . . . . . . . . 55
i
TABLE OF CONTENTS
2.5
2.4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
59
62
3 Probing galaxy dark matter haloes in COSMOS with weak
lensing flexion
64
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Shear and flexion . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Shapelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 The MV pipeline . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 Testing the pipeline . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.1 GREAT08 . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.2 FLASHES . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.3 Galaxy-galaxy simulations and bright object removal . . . 73
3.5 COSMOS analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5.1 The COSMOS data set . . . . . . . . . . . . . . . . . . . 75
3.5.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5.4 Removing bright objects . . . . . . . . . . . . . . . . . . . 78
3.5.5 The effect of substructure . . . . . . . . . . . . . . . . . . 78
3.5.6 Profile determination . . . . . . . . . . . . . . . . . . . . . 81
3.6 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . 83
3.A FLASHES results . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.B COSMOS data analysis . . . . . . . . . . . . . . . . . . . . . . . 85
3.B1 Catalogue creation . . . . . . . . . . . . . . . . . . . . . . 85
3.B2 PSF interpolation . . . . . . . . . . . . . . . . . . . . . . 87
3.B3 CTI correction . . . . . . . . . . . . . . . . . . . . . . . . 88
3.B4 Signal computation . . . . . . . . . . . . . . . . . . . . . . 89
3.C High redshift results . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.D Comparison with KSB . . . . . . . . . . . . . . . . . . . . . . . . 90
4 The relation between galaxy dark matter haloes
in the CFHTLS from weak lensing
4.1 Introduction . . . . . . . . . . . . . . . . . . . . .
4.2 Data . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Lens sample . . . . . . . . . . . . . . . . .
4.2.2 Source catalogue . . . . . . . . . . . . . .
4.3 Method . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Weak galaxy-galaxy lensing . . . . . . . .
4.3.2 The halo model . . . . . . . . . . . . . . .
4.4 Systematics tests . . . . . . . . . . . . . . . . . .
4.4.1 Verification of the shear catalogue . . . .
4.4.2 Seeing test . . . . . . . . . . . . . . . . .
4.5 Luminosity trend . . . . . . . . . . . . . . . . . .
4.5.1 Photometric redshift error corrections . .
4.5.2 Luminosity scaling relations . . . . . . . .
4.5.3 Satellite fraction . . . . . . . . . . . . . .
4.6 Stellar mass trend . . . . . . . . . . . . . . . . .
4.6.1 Stellar mass scaling relations . . . . . . .
4.7 Discussion and conclusions . . . . . . . . . . . . .
ii
and baryons
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93
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120
TABLE OF CONTENTS
4.A Detailed luminosity bins . . . . . . . . . . . . . . . . . . . . . . . 121
4.B Detailed stellar mass bins . . . . . . . . . . . . . . . . . . . . . . 124
5 Constraining cluster profiles with weak lensing shear and flexion
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Cluster lensing formalism . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Shear and flexion . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Contribution from the BCG . . . . . . . . . . . . . . . . .
5.2.3 Contribution from centred cluster dark matter haloes . .
5.2.4 Contribution from a cluster population with miscentred
BCGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.5 Other contributions . . . . . . . . . . . . . . . . . . . . .
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Mass dependence . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Concentration dependence . . . . . . . . . . . . . . . . . .
5.3.3 Offset width dependence . . . . . . . . . . . . . . . . . . .
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
128
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132
132
134
135
135
137
139
141
Bibliography
143
Nederlandse samenvatting
157
Svensk sammanfattning
162
Publications
166
Curriculum Vitæ
168
Afterword
170
iii
Introduction
1
The vast unknown that is our Universe has always fascinated mankind.
Though science is progressing fast and efficiently, still a large number of riddles remain unsolved. Every decade brings with it new discoveries, but more
often than not a breakthrough gives rise to yet more questions. Amongst the
main advancements of the past quarter-century is the uncovering of dark matter
and dark energy as the principal ingredients of our standard model of cosmology. With this model our understanding of the mechanisms behind the origin
and the evolution of our Universe has progressed immensely, but to advance
further we have do answer this: what is dark matter and dark energy? To
help bring clarity to the nature of these phenomena we study the distribution
of matter within galaxies, within galaxy clusters, and throughout our Universe.
A relatively recent technique, developed in the last couple of decades, has the
ability to map matter regardless of whether it is visible or dark and without it
having to be confined to large overdensities such as galaxy clusters. This technique is known as weak gravitational lensing and it is a highly powerful probe
of cosmology.
With this Thesis I aim to increase our knowledge of the distribution of matter in galaxies and galaxy clusters both by further developing the theoretical
framework for weak lensing, and by using large optical surveys to observe the
weak lensing signal directly. I therefore start with a brief introduction to cosmology and to gravitational lensing, with an emphasis on weak lensing and the
current status of lensing distortion software.
1.1
1.1.1
Cosmology
The concordance model of cosmology
Cosmologists study the Universe as a whole and are striving to understand how
it was formed and how it has arrived at the point where we are now. How
did the initially nearly smooth and homogeneous matter distribution evolve to
form the stars, galaxies and galaxy clusters that surround us today? To describe this process we use a template which we know to be a fairly accurate
description of reality. The one currently favoured by cosmologists is known as
ΛCDM, where Λ represents dark energy and CDM stands for cold dark matter.
This model attempts to simultaneously explain the growth of matter structure
1
1. INTRODUCTION
observed throughout the Universe, the temperature structure observed in the
cosmic microwave background (CMB), and the accelerated expansion of the
Universe indicated by e.g. supernova studies. In the process ΛCDM quantifies
the size of the mass-energy density constituents. Surprisingly, the known components of the Standard Model of Particle Physics, such as electrons, protons
and neutrons, compose only a minor part — about 5% — while the majority of
the matter constituents appears to be something new: dark matter. Even more
surprising is that the majority of the energy density appears to be composed of
the mysterious dark energy which makes up some 70% of the total. Dark matter
is necessary for structure formation as it adds gravity which holds large structures such as galaxies or clusters of galaxies together. Though we have not yet
determined exactly what dark matter is, there are some indications of what it
could possibly be. Traditionally there are three categories of dark matter: cold,
warm and hot. These labels refer to how fast the particles were able to move at
the very beginning of the Universe. Cold dark matter became non-relativistic
early on, while hot dark matter stayed relativistic until shortly before the epoch
known as recombination during which atoms formed. Since we know the temperature of the Universe at that time, this also sets limits on the masses of such
particles, with hot dark matter being much lighter than cold dark matter. The
most commonly known candidate for a hot dark matter particle is the neutrino.
Neutrinos are very light and conform to the constraint that dark matter has to
be only weakly interacting, making them hard to detect. A model dominated
by hot dark matter is inconsistent with hierarchical galaxy formation though,
so this alternative has effectively been ruled out via observations. Warm dark
matter is then more feasible, and behaves similarly to cold dark matter on large
scales though there may be differences on small scales. The most commonly
considered candidate warm dark matter particle is the sterile neutrino which is
more massive than its hot dark matter counterpart. However, since these sterile
neutrinos are not well motivated in particle physics, the current standard model
of cosmology prefers cold dark matter. There is now a plethora of candidates
for what cold dark matter could be since there is no real upper limit to the mass
allowed. Thus these candidates range from the hypothetical weakly interacting
particles (WIMPs), which may be massive neutrinos or so-called axions, to massive compact halo objects (MACHOs) which could refer to dwarf planets or black
holes. Observations have, however, ruled out MACHOs as the sole explanation
for dark matter (see e.g. Section 1.2.2) and so it is generally concluded that dark
matter must be a new type of cold particle, yet to be discovered. The nature of
dark energy requires some further introduction and is therefore discussed later
in this Section.
ΛCDM has gained great support due to its ability to successfully reproduce
a universe much like ours. Of the triumphs of the model, the results from
the Cosmic Background Explorer (COBE; Mather, 1982; Gulkis et al., 1990;
Mather et al., 1990) and its successor the Wilkinson Microwave Anisotropy
Probe (WMAP; Bennett et al., 2003; Spergel et al., 2003; Jarosik et al., 2011)
stand out. The two space missions have together accumulated 15 years’ worth of
CMB data, producing exceedingly accurate measurements of the echoes of the
Big Bang via the CMB angular power spectrum, shown in Figure 1.1. The bestfit model, assuming ΛCDM, is also shown in the Figure, clearly demonstrating
that ΛCDM describes current cosmological observations well. This is just one
example of ways to constrain cosmology though. Another very powerful probe is
2
1.1. COSMOLOGY
Figure 1.1 The CMB angular power spectrum from the 7-year WMAP data
release. The red curve represents the best-fit ΛCDM model and the grey band
shows the cosmic variance expected for that model. The first, second and third
acoustic peaks are highly constrained. Figure originally published in Larson et al.
(2011).
weak lensing which will be introduced later, and yet more probes are discussed
in Section 1.1.3.
Selected best-fit parameters from the WMAP 7-year data release are quoted
in Table 1.1 and this is also the cosmology assumed throughout this Thesis unless
explicitly stated otherwise. These are the aspects of ΛCDM that are relevant
to weak gravitational lensing, which is the focus of this work. Conversely, weak
gravitational lensing can be used to constrain most of these parameters. To
elaborate on the meaning of the parameters in Table 1.1, further background is
needed.
Our Universe is expanding which means that objects that are not gravitationally bound together will move away from each other. Therefore we see
galaxies and galaxy clusters receding from us in all directions and the further
away from us an object is, the faster it moves away from us. The Hubble constant
H0 relates this recessional speed to the distance from us via Hubble’s Law:
v = H0 D
(1.1)
The exact value of the Hubble constant is important for interpreting all cosmological results since it affects distances and thus volumes and densities. In most
applications, the dimensionless version of the Hubble parameter, h, is used, and
it is defined as
H0 ≡ 100 h km s−1 Mpc−1
(1.2)
so for the value of H0 given in Table 1.1 we have h ≃ 0.70. Furthermore, objects
that are moving away from us will have an electromagnetic spectrum which is
shifted towards the redder end due to a stretching of light waves (known as the
Doppler effect), and thus we can determine the distance to a distant object via
3
1. INTRODUCTION
Table 1.1 Cosmological parameters from the WMAP 7-year data release
(Jarosik et al., 2011). These parameters are the result of combining WMAP data
with priors from baryonic acoustic oscillations based on the Sloan Digital Sky
Survey Data Release 7 (Percival et al., 2010) and from the present-day Hubble
constant value determined using 240 Cepheid variables and as many supernovae
type Ia (Riess et al., 2009).
Parameter
H0
Ωb
Ωdm
ΩΛ
σ8
w
WMAP7+BAO+H0 value
−1
Mpc−1
70.4+1.3
−1.4 km s
0.0456 ± 0.0016
0.227 ± 0.014
0.728+0.015
−0.016
0.809 ± 0.024
−0.980 ± 0.053
its redshift z:
1+z =
Comment
The Hubble constant
Baryon density
Dark matter density
Dark energy density
Fluctuation amplitude at 8h−1 Mpc
Equation of state
λobs
λemit
(1.3)
where λobs and λemit are the observed and emitted wavelengths respectively.
Cosmologists often use redshift as a measure of distance to objects, and also —
somewhat confusingly — as a measure of time.
Figure 1.2 Influence of the two main energy density parameters on the overall
behaviour of the Universe. Here, Ωmatter and Ωvacuum are identical to the Ωm
and ΩΛ parameters mentioned in the text. Figure originally published in Peacock
(1999).
4
1.1. COSMOLOGY
We can define a critical density ρcrit at time t for which the local Universe
is flat, i.e. where the angles of a triangle add up to 180◦ , as
ρcrit (t) =
3H 2 (t)
8πG
(1.4)
Note here that the H parameter is time-dependent; this is the Hubble parameter
which defines the relative expansion rate at that point in time, and H0 ≡ H(t0 ).
G is the gravitational constant. Using this we can then derive a density parameter ΩX which represents the ratio of the actual density to the critical density:
ΩX =
8πGρ
ρX
=
ρcrit
3H 2 (t)
(1.5)
where the subscript X can represent any mass-energy density constituent. Ωb ,
Ωdm and ΩΛ are then the density of baryons, dark matter and dark energy compared to the critical density, and these constants evolve with time and therefore
with redshift. Adding the two first parameters together we get the total matter
density Ωm = Ωb + Ωdm . The influence of these parameters on the global behaviour of the Universe is illustrated in Figure 1.2. The solid straight line for
which Ωtot = Ωm + ΩΛ = 1 represents a flat universe. For Ωtot > 1 the model is
spatially closed which means that it has a finite volume and positive curvature
everywhere, i.e. the angles of a triangle add up to more than 180◦ like on the
surface of a sphere. Conversely, for Ωtot < 1 the Universe is spatially open,
has infinite volume and negative curvature everywhere. This type of universe is
usually illustrated by a saddle-like shape. Figure 1.2 also shows that a negative
ΩΛ causes the Universe to eventually recollapse while it will continue to expand
forever for a positive parameter (in most cases). There is also the possibility of a
‘loitering’ model with a maximum redshift and infinite age, and for high values
of ΩΛ there is no Big Bang. The current parameter estimates thus support a flat
universe which will be expanding forever. We can write the Hubble parameter
in terms of density parameters and redshifts:
(1.6)
H 2 (a) = H02 ΩΛ + Ωm a−3 + Ωr a−4 − (Ωtot − 1)a−2
where a = 1/(1 + z) is a scale factor, Ωr is the density of radiation energy and
Ωtot is equal to 1 for a spatially flat Universe. All density parameters are defined
at the present time, t0 .
The next parameter in Table 1.1, σ8 , is a crucial parameter for cosmology.
Formally it is the fluctuation amplitude within a sphere of radius 8 h−1 Mpc and
functions as a normalisation for the linear matter power spectrum. The value of
this parameter influences the growth of structure in the Universe. If it is too low,
the fluctuations in the early Universe were too small to feasibly allow for the formation of the stars and galaxies we see today. Finally, the parameter w ≡ p/ρc2 ,
where p is pressure, defines the equation of state of a postulated contribution to
the overall energy density from an unknown quantity. For w ≃ −1 this quantity
causes an accelerated expansion and the Universe is thus expanding at an everincreasing rate due to some unknown energy contribution, commonly referred
to as dark energy. The presence of this dark energy has been corroborated via
several observational indicators of an accelerated expansion (see Section 1.1.3).
Just as for dark matter, we have yet to confirm the exact nature of dark energy,
though candidates may be categorised as either a constant homogeneous energy
5
1. INTRODUCTION
density or scalar fields which may vary through space-time. The scalar fields
alternative is discussed further in the next Section, but the concordance model
ΛCDM assumes a constant energy density which is represented by the cosmological constant Λ. The use of this particular constant is a nod to Einstein and
his attempt to balance his field equations to obtain a static universe. Though
Einstein’s exact solution has since been proven unstable, the recycling of his
constant to represent an accelerating universe signifies the similarities between
his constant and the behaviour of modern dark energy.
1.1.2
Alternative models
Although ΛCDM is generally accepted as the most successful model for describing our Universe given current observations, there are alternative descriptions.
One of the main criticisms of ΛCDM is the need for unknown ‘dark’ quantities
and this has been the motivation for the development of alternatives. ΛCDM
assumes in general that the laws of physics hold true throughout the Universe.
A family of alternative models reason that this assumption may be false and
that laws of gravity require modifying at large distances. Amongst the most
well-known are Modified Newtonian Dynamics (MOND; Milgrom, 1983) and
Tensor-Vector-Scalar gravity (TeVeS; Bekenstein, 2004).
MOND was initially introduced as a way to model the flat rotation curves of
galaxies without the need for dark matter. The puzzle of galaxy rotation curves
was first noted by Rubin & Ford (1970). Studying the Andromeda galaxy,
they found that the velocities of stars in the disk stayed constant rather than
decreased with distance as would be expected from classical mechanics. The
stars in the outer regions of the disk were thus moving much faster around the
centre of the galaxy than should be possible. Freeman (1970) noted a similar
behaviour in their sample of disk galaxies, tentatively suggesting that there is
undetected matter beyond the optical extent of NGC 300. Rubin et al. (1980)
then used a larger galaxy sample to conclude, inspired by a remarkable prediction by Zwicky (1933), that there must be a significant amount of unseen
mass beyond the limit of the optical observations. The work of Vera Rubin
and colleagues on galaxy rotation curves thus constitutes the first real evidence
for dark matter — or an indication that classical mechanics is an inaccurate
description on galaxy scales. As the name implies, MOND modifies Newtonian
dynamics by allowing for some critical acceleration a0 below which the classical
Newtonian force-acceleration relation, F ∝ a, breaks down. The acceleration
close to massive structures thus obeys general relativity, but at large enough
distances force is related to acceleration via F ∝ a2 /a0 . This theory has been
highly successful in modelling the rotation curves of galaxies, particularly for
galaxies with low surface brightness which represent an extreme where ΛCDM
is currently not as powerful. However, on a galaxy cluster scale MOND still
requires more mass than what is observed in baryonic form, with massive neutrinos being suggested as a possibility (Sanders, 2007). Furthermore, because
MOND is non-relativistic it cannot reproduce gravitational lensing, or indeed
cosmology as a whole and is unable to model the CMB power spectrum. TeVeS
was then developed as a relativistic generalisation of MOND and successfully
models phenomena that MOND does not. It can reproduce the first and second acoustic peaks in the observed CMB power spectrum shown in Figure 1.1,
though this necessitates the inclusion of massive neutrinos (McGaugh, 2004;
6
1.1. COSMOLOGY
Skordis et al., 2006). For the tertiary and higher peaks, the amplitude is too
low even with added neutrinos because a baryon-only model necessarily predicts
that the peak amplitudes should be monotonically decreasing. As for MOND,
TeVeS is also relatively successful in modelling spherically symmetric clusters,
although again massive neutrinos are required for an essential dark halo and
the neutrino mass necessary is unrealistically large (Takahashi & Chiba, 2007).
Merging clusters, a few cases of which are discussed in Section 1.3.4, also pose
a problem.
MOND and TeVeS have primarily been developed as an alternative to dark
matter rather than attempting to replace the full ΛCDM description which includes considerations of the accelerated expansion of the Universe. Though
there has been some work on the acceleration implied in TeVeS (Zhao, 2007;
Hao & Akhoury, 2009), further exploration is needed in this area. Other models have been suggested as an alternative to the cosmological constant Λ as
mechanisms for accelerating the expansion of the Universe. Amongst the first
alternative explanations to be suggested is quintessence — a fifth fundamental
force which is repulsive. Evolved from string theory, another intriguing model
is that of the brane cosmology. Brane here is short for membrane, and in
this cosmology space-time as we know it is confined to a brane embedded in a
higher-dimensional space known as the bulk. The fundamental forces of nature
are localised to the brane while gravitational force is not, which means that our
brane can interact gravitationally with the bulk and with other branes. In one
version of brane theory, the Big Bang is the result of a collision between two
parallel branes (Khoury et al., 2001). The flavour that has gained most support
are the Randall-Sundrum models which assume a five-dimensional space in total, i.e. only one extra dimension for the bulk (Randall & Sundrum, 1999a,b,c).
The fifth dimension is finite and there are two branes in the model, although in
one version one of the two branes is placed infinitely far away, effectively leaving
a sole brane in the model. The energies of the two branes cause a severe warping
of spacetime along the fifth dimension. An effective cosmological constant is the
automatic result of this model (Cline et al., 1999).
Developing alternative methods to describe our Universe is ultimately beneficial to science because they do further our insight into physical mechanisms,
though a completely satisfactory version has yet to emerge. ΛCDM is currently
the model that is most successful at recovering what we see in observations on a
large range of scales and for many different types of structure. It has to be kept
in mind, however, that it is just a model and that it, too, has applications which
are not completely understood. Emphasis should also be put on the fact that
dark matter and dark energy are just descriptors for gravitational and accelerating fields which help us visualise these fields. Whether the forces involved are
due to actual dark particles or due to as yet unknown physics, the effect is the
same. And there is a lot of work to be done still before we can claim to fully
understand the Universe we live in. The probes of cosmology described below
are therefore vital for furthering that understanding.
1.1.3
Cosmology probes
There are several ways to test and constrain our cosmology models and often
each such probe is more sensitive to some parameters than others. Combining
several datasets will therefore result in much tighter constraints on cosmology
7
1. INTRODUCTION
2.0
No Big Bang
1.5
1.0
SNe
0.5
CM
B
Fl
at
BAO
0.0
0.0
0.5
1.0
Figure 1.3 Combining several independent datasets to constrain cosmology
(c.f. Figure 1.2). The datasets shown in this figure are the results from the cosmic microwave background (CMB; Dunkley et al., 2009), baryon acoustic oscillations (BAO; Eisenstein et al., 2005) and supernovae (SNe; Kowalski et al., 2008).
Though each dataset is degenerate in some sense, combining them all gives tight
constraints on Ωm and ΩΛ (contours at intersection). Figure originally published
in Kowalski et al. (2008).
than each on its own.
The CMB power spectrum is sensitive in shape, peak location and relative
peak heights to the underlying cosmology (see Figure 1.1, and e.g. Hu & White,
1996; Peacock, 1999). The location of the first acoustic peak is related to the
8
1.1. COSMOLOGY
curvature of the Universe; we now know that the Universe is essentially flat.
Furthermore, the location of each peak relative to the previous one is an indicator of the nature of the primordial density perturbations. The peak locations
measured by WMAP provide strong support for dark energy. Regarding amplitudes, the amplitude of the first peak compared to the second one (or odd
peaks versus even ones) holds information on the baryon density. The more
baryons present, the more relatively suppressed the second peak is. Finally, by
determining the height of the third peak we determine the ratio of dark matter
density to radiation density, and since we know the radiation density from other
measurements it gives us the dark matter density in the Universe. However, different parameters may affect the power spectrum in a similar way, which means
that we cannot tell whether the shift in one direction is due to the variation of
one parameter or another. This is what is known as a parameter degeneracy. As
an example, the Ωm and H0 parameters are degenerate which is why the spread
in allowed values for Ωm is so large (see Figure 1.3). To break such a degeneracy,
independent measurements of H0 are needed and these measurements may be
provided by e.g. studies of supernovae (SNe).
Historically, SNe provided one of the first indications of an accelerated expansion (Riess et al., 1998; Perlmutter et al., 1999). SNe is a collective name
for all types of stars exploding during, or at the end of, their life cycle. There
are several mechanisms that can cause such an explosion, but for cosmological
applications one mechanism is of particular interest: that which leads to a Type
Ia SN. This species inevitably results in a characteristic light curve, i.e. how
the luminosity resulting from the explosion decays with time is identical for
all SNe of a given brightness. By precisely measuring such a light curve and
comparing it to the observed brightness, the distance to the SN can be accurately inferred. The redshift of the SN host galaxy is then used to constrain
the relationship between distance and redshift which in turn constrains Ωm and
ΩΛ , breaking the degeneracy in the CMB power spectrum as described above
(again, see Figure 1.3).
Another probe which allows the breaking of the above degeneracy is the
study of large-scale structure (LSS). The way galaxies are distributed throughout the observable Universe is a measure of how matter is distributed and
how it clusters, something which is sensitive to Ωm . Galaxies have therefore
been mapped in redshift space through spectroscopic surveys such as the TwoDegree-Field Galaxy Redshift Survey (2dFGRS; see e.g. Cole et al., 2005, for
results from the final data set) and the Sloan Digital Sky Survey (SDSS; see
e.g. Tegmark et al., 2004). However, because we do not know exactly how the
locations of galaxies correspond to the location of the underlying dark matter,
interpreting the results in terms of Ωm is difficult. It requires a description of
how well galaxies trace the total mass distribution, and this description is quantified via the galaxy bias. The choice of bias constitutes an uncertainty in LSS
measurements and needs to be further investigated. In general though, we see a
pattern of clustered matter and filaments connecting the clusters, and between
the filaments we see voids where there is no matter. This pattern is commonly
known as the Cosmic Web and the voids are a signature of sound waves created by cosmological matter perturbations in the early Universe, identified as
baryon acoustic oscillations (BAO). The imprint of BAO on the matter power
spectrum provides a characteristic length scale, and measuring it constrains the
distance-redshift relation giving a measure of Ωm (e.g. Eisenstein et al., 2005,
9
1. INTRODUCTION
as in Figure 1.3).
Figure 1.4 Millennium simulation slices at progressively lower redshifts as
printed on each panel. The scale of all slices is the same. Images originally
published in conjunction with Springel et al. (2005).
A different approach to studying cosmology is to create a new cosmos using
the physical laws and properties we are aware of so far. This can be done using so-called N-body simulations which take (dark) matter particles, place them
according to some initial conditions and let them interact over the life-span of
the Universe. Comparing these simulations at different points in redshift to
real observations at the same redshifts tells us how well we have understood
the underlying physical processes. Currently, the most widely known and used
N-body simulation is that of the Millennium Simulation (Springel et al., 2005).
Shown in Figure 1.4 are four slices from this dark matter only simulation at
different redshifts which illustrate the growth of structure from a nearly homogeneous matter distribution at z = 18.3 to a galaxy cluster at z = 0.0. One
application of N-body simulations is, for instance, that the density profile of a
simulated cluster may be modelled and then compared to an equivalent observed
cluster studied using gravitational lensing, a technique with the power to map
the full mass distribution, to see how well the profiles agree. Studying clusters
at different redshifts allows us to investigate the evolution of structure as well.
A significant limitation of most N-body simulations is, however, that they use
only dark matter particles and disregard the influence of baryons. The reason
for this is partly that baryons are expected to follow the general distribution
of dark matter and partly that the processes involved are less well understood.
The comparison with lensing observations, which are sensitive to all mass, may
therefore be somewhat restricted but may also inform us of how the inclusion of
baryons affects the dark matter only Universe. This is far from the sole appli10
1.2. GRAVITATIONAL LENSING OVERVIEW
cation of gravitational lensing in the context of cosmology, and so the technique
will be more extensively discussed in the next Section.
1.2
Gravitational lensing overview
Gravitational lensing is the collective name given to a set of methods, all of
which have a common goal: to probe gravitational fields irrespective of whether
their source is visible or not. In some cases gravitational lensing is the only
way to detect what cannot be seen directly through telescopes. As such, the
methods are beneficial to the study of the dark components of our Universe
discussed in Section 1.1, as well as to the search for extrasolar planets otherwise
drowned in the flux of their surroundings.
In essence, gravitational lensing methods exploit the bending of light rays
caused by gravitational potentials. As the light travelling from background
sources gets lensed by foreground structures, the source appears displaced, magnified and distorted. Since this is a purely geometrical effect and since it depends
only on the total amount of matter in the intervening structure, no assumptions
on the physical state of the lens need be made. This makes gravitational lensing
exceedingly powerful.
1.2.1
Fundamentals of lensing
Before elaborating on the different applications of gravitational lensing, the fundamental ideas have to be understood. Here I give a brief introduction to the
different concepts involved; for a more in-depth review I refer the reader to
Bartelmann & Schneider (2001). The general geometry of gravitational lensing
is illustrated in Figure 1.5, and this simple image turns out to represent reality
well. The light from a background source is deflected by a foreground structure
which acts as a lens. A customary simplification of this theory is that of the thin
lens approximation: the light ray is instantaneously deflected at the lens plane.
Though this is not strictly correct, it is a valid assumption if the spread of the
lensing mass along the line-of-sight is much smaller than the angular diameter
distances involved, something which is true in most lensing systems (though for
the cosmic mass distribution a more general description is necessary; see Section 1.3.2). As is clear from Figure 1.5, the source image appears displaced with
respect to its true position as a result of gravitational lensing. Unfortunately
it is difficult to take advantage of this effect observationally since the intrinsic
position is not known. However, deflection angle α̂ is related to the impact
parameter ξ via
4GM
(1.7)
α̂ = 2
c ξ
where G is the gravitational constant, M is the mass of the lens and c is the
speed of light. Thus the amount of deflection is determined not only by the
lens mass but also by the impact parameter and this results in a distortion.
In extreme geometrical setups where the source is perfectly aligned to lie right
behind the lens, the image will be circular. This is known as an Einstein Ring,
the radius of which is known as the Einstein radius θE which is directly related
11
1. INTRODUCTION
η
Source plane
Dds
^
α
Ds
ξ
Lens plane
β
Dd
θ
Observer
Figure 1.5 Schematic of a typical gravitational lens system. A light ray travelling from a source at position η is deflected by a lens in the lens plane. If there
was no lens the source would have been observed at angle β. In the presence of a
lens, however, the deflection angle is α̂ with the impact parameter ξ which results
in the source being observed at angle θ instead. Ds , Dd (Dl in the text) and Dds
(Dls ) are the angular diameter distances to the source, to the lens and between
the lens and the source respectively. Figure originally published in Bartelmann &
Schneider (2001).
to the mass of the lens; for a point mass it is given by
θE =
12
Dls 4GM
D l D s c2
(1.8)
1.2. GRAVITATIONAL LENSING OVERVIEW
We can now define the lensing equation:
β = θ − α(θ)
(1.9)
This basic equation relates the observed position angle θ = ξ/Dl to the true
position β and the reduced deflection angle α = α̂Dls /Ds . Furthermore, the
thin lens approximation allows us to assume that the lensing mass lies on a 2D
lens plane and we can therefore define the 2D surface mass density Σ(ξ) of the
lens
Z
Σ(ξ) = ρ(ξ, z) dz
(1.10)
where ρ is the 3D mass density and z is the third dimension. Now, the lensing
equation (Equation 1.9) can have more than one solution resulting in multiple
images on the sky; if this happens the lens is said to be strong. This condition
may be quantified using a dimension-less surface mass density, or convergence,
κ:
Σ(ξ)
(1.11)
κ(θ) =
Σcrit
where Σcrit is the critical surface mass density which is defined as
Σcrit =
c2 D s
4πG Dl Dls
(1.12)
If the surface mass density is greater than this critical limit, i.e. if κ ≥ 1, then
multiple images are produced and we enter the strong lensing regime. The
convergence may also be integrated over to define the lensing potential ψ of the
system:
Z
1
ψ(θ) =
κ(θ ′ ) ln |θ − θ ′ | d2 θ′
(1.13)
π R2
which can be related to the reduced deflection angle via α = ∇ψ; the deflection
angle is thus the gradient of the deflection potential. It also satisfies Poission’s
equation ∇2 ψ(θ) = 2κ(θ).
Having introduced the basic concepts in gravitational lensing theory, I now
move on to observational applications. Though the main emphasis of this Thesis is weak lensing it is instructive to briefly touch upon the related topics of
microlensing and strong lensing as well.
1.2.2
Microlensing
In the beginning of last century, Einstein’s theory of General Relativity (GR)
was still new and required observational evidence for credibility. Gravitational
microlensing provided such early evidence when Eddington set out on an expedition to confirm Einstein’s prediction that a star passing close to the Sun
would appear displaced due to its gravitational field. The exact displacement
predicted by GR was 1.75 arcsec for a star at the solar limb, whereas Newtonian gravity predicted a mere 0.87 arcsec (i.e. half that of GR). To discriminate
between the two theories, Eddington took advantage of the full solar eclipse
on May 29, 1919. The displacement found by him and his collaborators was
1.61 ± 0.30 arcsec which clearly favours General Relativity (Dyson et al., 1920)
and shows the power of the gravitational lensing technique.
13
1. INTRODUCTION
Figure 1.6 Example lightcurve caused by a microlensing event. The top two
datasets are the same event observed in different filters, with the best-fit microlensing model shown as well. The bottom graph shows the residuals of the model fit.
Figure originally published in Wyrzykowski et al. (2011).
Microlensing is a transient effect, caused by a foreground object often located
in our own Milky Way galaxy passing in front of a bright background source.
As the alignment between the source, the lens and the observer changes, the
apparent brightness of the source is boosted and then diminished, causing the
characteristic shape of the light curve shown in Figure 1.6. Since the event is
transient, a potential source has to be monitored for some time to observe an
event, but unfortunately such an event cannot be predicted. The probability
of microlensing being observed is thus very low and therefore large dedicated
surveys regularly scanning millions of stars are crucial for detection. Generally
these surveys are trained towards areas with a high density of background stars,
such as the centre of the Galaxy or another nearby galaxy like the Large Magellanic Cloud (LMC) or Andromeda. The microlensing optical depth is a measure
of the probability of a source undergoing a microlensing event at a given time;
the optical depth towards the centre of our Galaxy is τ = 2.43 × 10−6 (Alcock
et al., 2000b) while the equivalent measure towards the LMC is τ = 3.6 × 10−8
(Tisserand et al., 2007). However, choosing a suitable backdrop is more dependent on which type of population is to be observed. If we are interested in
objects in the halo of our Galaxy for instance, the galactic bulge is unavailable
to us and an external galaxy is necessary.
Currently there are two major applications of microlensing: the search for
MACHOs and other dark transient objects, and the search for extrasolar planets. A MACHO passing in front of a star would produce a light curve such as
the one shown in Figure 1.6 but the lensing object itself would not be seen.
14
1.2. GRAVITATIONAL LENSING OVERVIEW
Therefore other potential causes of a change in brightness, such as the intrinsic
variability of the source star, have to be ruled out before a successful detection can be claimed. Two collaborations that have been working to identify
candidates in the Milky Way halo are the MACHO and EROS collaborations.
MACHO’s results contradict the hypothesis that our halo consists of MACHOs,
effectively ruling out the theory that dark matter is composed of such massive
objects (Alcock et al., 2000a). EROS provided agreement with these findings;
they found that the maximum fraction of the halo mass that could consist of
MACHOs is 8% (Tisserand et al., 2007). They also ruled out MACHOs in the
mass range 0.6 × 10−7 M⊙ < M < 15 M⊙ as the primary occupants of the halo.
MOA-2011-BLG-293
OGLE
MOA
CTIO I
Wise
Weizmann
Peak
15
15.0
16
I (mag)
I (mag)
15.5
17
16.0
18
19
20
0.050
0.025
0.000
-0.025
-0.050
5744
Residuals
Residuals
16.5
5745
5746
5747
5748
HJD-2,450,000
5749
5750
0.050
0.025
0.000
-0.025
-0.050
5747.3
5747.4
5747.5
HJD-2,450,000
5747.6
5747.7
Figure 1.7 Example lightcurve caused by the star-and-planet system MOA2011-BLG-293. The top two panels show the full light curve (left) and an enlarged
view of the peak (right), with the best-fit microlensing model also shown in each
case. The bottom panels show the residuals of each model fit. Figure originally
published in Yee et al. (2012).
If the lens consists of more than one object, such as a binary star or a star
and a planet, the light curve displays several peaks as illustrated in Figure 1.7.
This is a direct way of finding extrasolar planets and determining their properties. From the observed light curve alone, the mass distribution in the lens may
be deduced and thus the mass of the planet(s) can be determined. To date,
15 extrasolar planetary systems, with planets ranging in mass from 0.01 MJ
to 3.7 MJ and separations of 0.66 AU to 5.1 AU, have been discovered using
this technique (Yee et al., 2012; Bennett et al., 2012). This number is relatively low compared to the rival radial velocity detection method, but the list is
rapidly growing as surveys collect more data. Such surveys include the Optical
Gravitational Lensing Experiment (OGLE: Udalski et al., 1992; Udalski, 2003)
and Microlensing Observations in Astrophysics (MOA: Bond et al., 2001; Sumi
et al., 2003). An interesting discovery to come out of MOA is that of a population of planetary-mass objects that are seemingly not gravitationally bound to
host stars (Sumi et al., 2011). Such a population could be explained via various
scattering scenarios, but the number of candidate planets found indicates that
the size of the population is almost twice that of main-sequence stars. This is
larger than would be expected from scattering. However, the planets are only
defined as isolated because no corresponding star was detected during the mi15
1. INTRODUCTION
crolensing event. Sumi et al. (2011) offered the explanation that the planets
may simply be bound in a very large orbit which gives a lower bound on their
separation of 7 − 45 AU. None the less, this discovery could have an impact on
planet formation theories if the planetary objects are indeed orphaned (Bowler
et al., 2011). The power of microlensing thus lies with its ability to detect dark
compact objects in our own Galaxy and those nearby, thereby challenging theories of both dark matter and thus cosmology, and of planet and star formation.
For applications of the related theories of strong and weak lensing, however, we
have to move to a much grander scale.
1.2.3
Strong lensing
Figure 1.8 Example of strong lensing: the massive galaxy cluster Abell 2218
imaged by the Hubble Space Telescope in 1999. Image credit: NASA/ESA,
A. Fruchter and the ERO Team (STScI, ST-ECF).
Distant clusters of galaxies display remarkable arc-like images such as those
manifested in the stunning Abell 2218 (Figure 1.8). These source images are
clear examples of strong gravitational lensing and typically appear in massive
structures such as galaxy clusters or close to individual galaxies. As mentioned
in the introductory Section 1.2.1, a condition for strong lensing is that the surface density is greater than the critical limit Σcrit , i.e. that κ ≥ 1. Alternatively,
for a source which is much smaller than the angular scale on which lens properties change, the mapping between source and lens plane can be linearised using
a Jacobian matrix A(θ):
f (θ) = f s [β0 + A(θ0 ) · (θ − θ0 )]
(1.14)
where f is the observed surface brightness distribution in the lens plane, f s
is the corresponding brightness distribution in the source plane, θ0 is a point
within the image corresponding to the point β0 within the source and
Aij (θ) ≡
∂βi
= δij − ∂i ∂j ψ(θ)
∂θj
(1.15)
where we use the shorthand ∂i ≡ ∂/∂θi . The magnification µ is the ratio of
the observed flux from the image to that from the unlensed source, and this is
16
1.2. GRAVITATIONAL LENSING OVERVIEW
Figure 1.9 Strong lensing reconstruction of (part of) Abell 2218, with the cluster centre located at the origin. The crosses mark the two galaxies responsible
for the majority of the lensing effect, and the arcs being modelled are also shown.
The circle labelled S represents the size and location of the source. Dashed lines
represent critical lines in the lens plane, while the curves close to the source represent the corresponding caustics. Figure originally published in Saraniti et al.
(1996).
simply given by
1
(1.16)
det A
A more rigorous definition of strong lensing is then a system for which det A = 0,
and in which multiple images are produced. A strong lens will have a locus in
the image plane for which this condition holds true, and this locus is known as a
critical curve. This curve can be visualised as a smooth loop. When the critical
curve is mapped back to the source plane it is instead known as a caustic which
will, contrary to its corresponding critical curve, generally display cusps. Along
a critical curve, the magnification formally diverges and sources near these are
highly magnified and distorted, resulting in the long arcs visible in Figure 1.8.
The number of images associated with a particular source also depends on its
vicinity to a critical curve, providing additional constraints. These effects are
illustrated in Figure 1.9 which shows the reconstruction of a sub-cluster within
Abell 2218, as modelled by Saraniti et al. (1996).
The main advantages of studying galaxy cluster lenses were recognised very
early on by the bold visionary Zwicky who anticipated that we would be able to
use clusters to trace the unseen mass (Zwicky, 1937), also predicted by himself
(Zwicky, 1933). He further envisioned that given good enough imaging we
µ=
17
1. INTRODUCTION
could study the distant sources behind clusters. Both these predictions have
proved accurate, even though strong lensing was not observed until much later
(e.g. the double quasar Q0957+561 and giant arcs; Walsh et al., 1979; Lynds &
Petrosian, 1986; Soucail et al., 1987). Observers can use the arcs and multiple
images in clusters or around single galaxies to model the critical curves and
thus constrain the mass distribution within the lens. Lensing therefore offers
a unique way to probe substructure. Clusters consist of several galaxies that
are interacting now, or has done at some point in the past. Through these
interactions and via their traversing through the cluster core, the extended
dark matter haloes surrounding member galaxies are expected to be tidally
stripped. This view is corroborated by evidence that cluster galaxies undergo
strong morphological evolution including quenched star formation (e.g. Jones
et al., 2000; Kodama & Bower, 2001; Treu et al., 2003). By accurately modelling
the distribution of mass in the inner regions of clusters using individual cluster
members, direct evidence of such stripping can be gathered, providing support
for the theory of hierarchical merging as the main process in cluster assembly.
The accuracy of such analyses is further improved by including weak lensing
signals (see Section 1.3) since strongly lensed arcs are rare (e.g. Natarajan et al.,
2007, 2009). Strong lensing has also been used to tentatively detect substructure
in galaxy-size lenses consistent with predictions from ΛCDM (Vegetti et al.,
2010).
As already mentioned, another use for these massive lenses is to employ them
as Nature’s own telescopes. Due to the great magnification effects involved we
are privy to objects that would otherwise be too far away or too faint for us
to see. These background objects do most likely not suffer from any prominent
selection bias other than that related to the distances involved in the geometrical setup, although intrinsically brighter sources will produce brighter arcs for a
given geometry. Though rare, magnifications of up to 4 magnitudes have been
measured (Seitz et al., 1998) and increases in brightness of more than 1.5 magnitudes are relatively common (e.g. Richard et al., 2011). The magnification
is wavelength independent, so the background sources can be fully studied for
morphology and physical properties that would otherwise not be resolved. This
yields insight into a very high redshift regime which we could not study in such
detail directly. The cosmic telescope as a tool to detect high-redshift galaxies
has since its first use heralded the discoveries of the most distant galaxies of
their time (e.g. Franx et al., 1997; Ellis et al., 2001; Hu et al., 2002; Kneib et al.,
2004). Some studies have claimed detections of candidate galaxies at redshifts
as high as z = 10.2 using this technique (Stark et al., 2007), clearly on par
with the highest-redshift galaxy candidate ever discovered (z = 10.3; Bouwens
et al., 2011). Detecting and analysing such early galaxies is essential for our
understanding of the era when the first stars and galaxies were assembled and
objects such as quasars formed. It also provides vital clues to the process that
led to the cosmic reionization, a crucial phase during the evolution of the early
Universe.
Finally, strong lensing clusters have the power to constrain cosmology directly since the effect is dependent on angular diameter distances. These distances in turn are defined by the geometry of the Universe and in particular on
the parameters Ωm and w. For clusters with several arc sets due to sources at
known but different redshifts, the Einstein radii may be compared. The ratio of
the radii then holds information on the fundamental geometry of the Universe
18
1.3. WEAK LENSING
(e.g. Link & Pierce, 1998; Golse et al., 2002; Soucail et al., 2004; Jullo et al.,
2010). Encoded in arc properties is also the value of the Hubble parameter H0 .
It can be constrained independently of cosmology by measuring the time delay
between arcs originating from the same time-varying source (e.g. Blandford &
Narayan, 1986; Saha et al., 2006; Oguri, 2007; Paraficz & Hjorth, 2010; Riehm
et al., 2011). The number of giant arcs observed is also tied to the background
cosmology, and in particular to the σ8 parameter. Bartelmann et al. (1998)
showed that the observed arc statistics differs from that predicted by ΛCDM,
and this discrepancy has yet to be fully resolved. It may be explained by observational effects such as a poorly understood source population or substructure
(Horesh et al., 2005), or physical effects due to baryons like cooling and star
formation (Meneghetti et al., 2010). Furthermore, the observed giant-arc statistics may be an interesting indicator of primordial non-Gaussianity (D’Aloisio
& Natarajan, 2011). Whatever the origin of the excess, it is clear that strong
lensing has a lot to offer when it comes to confirming our understanding of
cosmology. The applications of this effect are naturally focussed on large structures and although we have given only a brief overview here, clusters are very
powerful probes of the geometry of the Universe (see Kneib & Natarajan, 2011,
for a recent review). To take full advantage of these cosmological behemoths,
however, we have to break away from the restrictions of strong lensing. Arcs are
rare and contingent on serendipitous alignments and high-density regions. Combining the strengths of this technique with weak lensing, which is ubiquitous,
will allow us to study clusters in ever more detail.
1.3
Weak lensing
Weak gravitational lensing is a relatively new study, with the first detection
recorded by Tyson et al. (1990). Given sufficient depth and area, and good
enough image quality, this statistical alignment of galaxies can be observed
anywhere on the sky. The power of this technique to explore the unseen matter
in clusters, in galaxies and even in the Cosmic Web is hence unrivalled. We will
therefore review the fundamentals of this method in a bit more detail than its
sister practices above, though for a thorough treatment we refer the reader to
Bartelmann & Schneider (2001) and Schneider (2005).
1.3.1
Convergence, shear and flexion
As described in Section 1.2.1, a distinguishing limit between strong and weak
lensing is the critical surface density Σcrit and the related convergence κ. Starting from Equation 1.15 we can write the mapping between source and lens plane
as
βi ≃ Aij θj
(1.17)
This holds true for small source galaxies where the convergence is constant
across the source image. Rewriting Equation 1.15 we can also get an alternative
description of the distortion matrix A:
1 − κ − γ1
−γ2
(1.18)
A=
−γ2
1 − κ + γ1
19
1. INTRODUCTION
convergence and
shear
S
A
β2
D
−1
ϕ
θ2
β1
ǫs
θ1
b
convergence only
O
ǫ
Figure 1.10 Effect of shear and convergence. On the left is the original circular
source, while the lensed image is on the right. Convergence only results in an
enlargement of the source image while the shear causes a stretch entailing a difference in axis ratio. The orientation of the resulting ellipse depends on the relative
amplitudes of γ1 and γ2 as illustrated in Figure 1.11. This figure was originally
published in Schneider (2005).
Figure 1.11 Orientation of the ellipse resulting from the relative amplitudes
of γ1 (on the x-axis) and γ2 (on the y-axis) applied to a circular source. Figure
originally published in Schneider (2005).
20
1.3. WEAK LENSING
where γ1 and γ2 are the two components of the shear induced by the lensing
potential: γ = γ1 + iγ2 . These shear components are related to the lensing
potential ψ via
γ1
=
γ2
=
1
(ψ11 − ψ22 )
2
ψ12
(1.19)
(1.20)
where e.g. ψ11 = ∂12 ψ is the second derivative of the lensing potential. Defining
the complex gradient operator
∂ = ∂1 + i∂2
(1.21)
using the same shorthand as before we can also relate the shear to convergence
in a way which compactly shows that shear is the second-order gradient of the
lensing potential:
γ = ∂∂ψ
(1.22)
The effect of shear on a source image is to stretch it in one direction as illustrated
in Figure 1.10 with the direction dependent on the relative amplitudes of the
γ1 and γ2 distortions. As is clear from Figure 1.11, the transformation γ → −γ
results in a 90◦ rotation and pure γ2 is at 45◦ to pure γ1 . Generally we also
assign a property known as spin to weak lensing distortions. A distortion type
with spin s is invariant under a rotation of φ = 360◦/s = 2π/s radians. Since an
ellipse rotated by 180◦ looks the same, shear is a spin-2 quantity. The lensing
displacement field α is a spin-1 quantity which is also the gradient of the spin-0
lensing potential:
α = α1 + iα2 = ∂ψ
(1.23)
We can now interpret ∂ as a spin-raising operator; applying it once to the lensing
potential results in a spin-1 quantity, while applying it twice results in spin-2.
Similarly the complex conjugate ∂ ∗ is a spin-lowering operator. For instance,
the convergence is related to the lensing displacement field and lensing potential
via
1
1
(1.24)
κ = ∂ ∗ α = ∂ ∗ ∂ψ
2
2
and is thus a spin-0 quantity.
Equation 1.17 is an approximation that is sufficiently accurate when shear is
constant across a source image. If this is is not the case, however, the equation
has to be extended to higher orders to encapsulate the variations in shear:
1
βi ≃ Aij θj + Dijk θj θk
2
(1.25)
Dijk = ∂k Aij
(1.26)
where
is a third-order distortion tensor. The lensed surface brightness of a source may
now be written
1
(1.27)
f (θ) ≃ 1 + (A − I)ij θj + Dijk θj θk ∂i f s (θ)
2
where I is the identity matrix. The tensor D captures the distortions responsible
for the arc-like shape of weakly lensed images, reminiscent of the giant arcs in
21
1. INTRODUCTION
Figure 1.12 Illustration of convergence, shear and flexion distortions as applied
to a circular source with a Gaussian density profile. The spin values, as described
in the main text, increase from 0 for convergence to 3 for G flexion. Figure
originally published in Bacon et al. (2006).
strong lensing (though the giant arcs are usually the result of several distorted
images merging, unlike in weak lensing). This tensor can be succinctly expressed
in terms of two new quantities, known as flexion:
Dijk = Fijk + Gijk
(1.28)
where F is the first flexion, or F flexion, and G is the second flexion, or G
flexion. The F flexion was first discovered and investigated a decade ago by
Goldberg & Natarajan (2002) with a tentative detection in Goldberg & Bacon
(2005). Bacon et al. (2006) then developed the notation further and included
the G flexion as well. The flexions are the third-order derivatives of the lensing
potential and the gradients of convergence and shear:
F
=
G
=
1 ∗
∂∂ ∂ψ = ∂ ∗ γ = ∂κ
2
1
∂∂∂ψ = ∂γ
2
(1.29)
(1.30)
which, following the above discussion, means that F flexion is a spin-1 quantity
while G flexion has spin-3. Convergence, shear and flexion are all illustrated in
Figure 1.12.
It should be noted that, unlike convergence and shear, flexion is not dimensionless but has units of inverse length. Furthermore, throughout this Thesis we
make the implicit assumption that we are working in the weak lensing regime,
i.e. κ ≪ 1. If this condition is broken, our observations would be biased since
22
1.3. WEAK LENSING
what we truly observe are the quantities g, G1 and G3 (Schneider & Seitz, 1995;
Schneider & Er, 2008):
g
G1
G3
=
γ
1−κ
F + gF
1−κ
G + gF
= ∂g =
1−κ
= ∂∗g =
(1.31)
∗
(1.32)
(1.33)
where g is the reduced shear and G1 and G3 are the reduced flexions. This is a
consequence of the mass-sheet degeneracy, a well-known potential source of bias
in gravitational lensing. The degeneracy arises from the fact that the addition of
a sheet of constant surface density in front of the lens will not alter the shear or
flexion measurements (Falco et al., 1985). Breaking this degeneracy is possible
with magnification measurements in principle, because the magnification reacts
differently to a mass sheet. It has also been pointed out that there is some
cross-talk between shear and flexion which has to be considered for an unbiased
measurement (Viola et al., 2012). Both these effects are significantly reduced
in impact in the weak lensing limit. Therefore I do not touch upon it further
in this Thesis which is mainly concerned with the lensing signal induced by
galaxy-sized halos, but use the approximation that the observed quantities are
equivalent to the non-reduced quantities.
1.3.2
Cosmic shear
As light travels through space to reach us it is continuously deflected by the
filaments and nodes of the Cosmic Web. Source galaxies are thus sheared and
weakly aligned even when there are no large structures in the way. The statistics of these distortions and alignments therefore reflect the statistics of the
underlying matter distribution. Though the distortion is minute at less than
∼ 1%, this effect was detected at the turn of the millennium (Bacon et al., 2000;
Kaiser et al., 2000; Van Waerbeke et al., 2000; Wittman et al., 2000). It is since
being measured with ever more refined accuracy using imaging data of ever
increasing area, depth and quality, and used to constrain cosmological parameters (e.g. Hoekstra et al., 2002; Brown et al., 2003; Jarvis et al., 2003; Massey
et al., 2005; Van Waerbeke et al., 2005; Hoekstra et al., 2006; Semboloni et al.,
2006; Benjamin et al., 2007; Schrabback et al., 2007; Fu et al., 2008; Schrabback
et al., 2010; Huff et al., 2011). The correlation between shears across the sky
as a function of angular scale can be used to derive the lensing power spectrum which is related to the three-dimensional matter power spectrum (e.g.
Kaiser, 1992; Bartelmann & Schneider, 2001; Schneider, 2005; Hoekstra & Jain,
2008). Technically, cosmic shear cannot make use of the thin lens approximation used to derive the lensing results quoted so far in this Introduction because
the deflection does not take place in a single lens plane. It turns out, however,
that under the assumption that the deflection angle is small the end result is
a redshift-dependent convergence κ which behaves just like in ordinary lensing
(see e.g. Schneider, 2005). We can therefore use ordinary shear measurements
to constrain the matter power spectrum, and thus in particular the cosmological
parameters Ωm and σ8 .
23
1. INTRODUCTION
Since the measurements do not rely on baryonic tracers there are no assumptions on e.g. galaxy bias necessary and this gives cosmic shear great value. Furthermore, the constraints resulting from cosmic shear intersect the constraints
from the CMB in a way that reduces degeneracies adding to the benefits of
such analyses. The task of observing this effect is a fairly substantial challenge
however, owing to the fact that the distortions are so small. It is impossible
to detect a signal on a single galaxy image since the intrinsic ellipticity of the
source galaxy is in general much larger than the induced distortion. Assuming
that galaxies have random intrinsic ellipticities and that they are randomly oriented on the sky we can discern the lensing signal in a statistical way though. If
we average over enough sources we can reduce this shape noise and essentially
reason that the mean intrinsic shape is circular. Under ideal conditions, any
ellipticity observed must then be produced by lensing.
Figure 1.13 Point-spread function (PSF) pattern of a typical field in the
Canada-France-Hawaii Telescope Legacy Survey (CFHTLS). Each tick represents
the observed magnitude and orientation of a stellar ellipticity. The artificially
induced ellipticity is most prominent in the corners in this case. Figure originally
published in Fu et al. (2008).
Unfortunately, such ideal conditions are also difficult to attain. A telescope
will in general produce a complicated pattern which correlates galaxy ellipticities
in a way that imitates a lensing signal. This pattern, illustrated in Figure 1.13, is
24
1.3. WEAK LENSING
usually referred to as the point-spread function (PSF), although other distinct
processes can be involved as well (such as wind-shake of the telescope). For
ground-based surveys the PSF is worsened by turbulence in the atmosphere
and these seeing conditions tend to blur the galaxy images and dilute their
ellipticities. Generally the PSF is corrected for by taking advantage of the fact
that stars should be circular. Any ellipticity observed for stars is therefore due
to the PSF and this information may be used to model the distortions. Spacebased telescopes face other trials, however, such as the gradual degradation of
CCD chips due to the constant bombardment of cosmic radiation (resulting in
charge-transfer inefficiency or CTI; see e.g. Rhodes et al., 2007, 2010). CTI
is the result of so-called charge traps in the silicon surface of a CCD which
reveal themselves as artificial trails behind objects on an astronomical image.
Again, this can cause a false shear detection if left unaccounted for. Recently
though, promising ways to correct for this effect have been suggested either at
an image reconstruction level (Massey et al., 2010) or parametrically (as in e.g.
Schrabback et al., 2010).
Additional difficulties include the fact that detectors collect photons in square
bins (or pixels) which places a fundamental lower limit on the size galaxy that
can be reliably analysed, and the fact that there is some intrinsic alignments of
galaxies due to them being affected by tidal fields during formation (e.g. Splinter et al., 1997; Faltenbacher et al., 2002; Lee & Pen, 2008) or lower-redshift
tidal fields affecting all higher-redshift sources (Hirata & Seljak, 2004; Heymans
et al., 2006b). Another limiting factor is the accuracy of the software used to
extract the shear signal from a given image. Great effort has been put into
developing reliable software and at the moment there are many alternatives
available. To take full advantage of future surveys, however, the accuracy has
to be improved even more. An overview of the current shape measurement
software state-of-the-art is given in Section 1.3.5, but first I will introduce a
different weak lensing application which is more robust against issues such as
PSF and CTI: galaxy-galaxy lensing.
1.3.3
Galaxy-galaxy lensing
The source images due to a lens galaxy will be aligned in a circular pattern
around the lens, and the distortions of the sources decrease in strength the
further from the lens they are. By measuring the average lensing distortion
in circular bins of successively increasing size centred on the lens, a function
will emerge that encodes the density profile of the lens, i.e. it tells us how the
mass is distributed within the lens. Since the distortions are weak in general,
we again have to average over many lenses and sources in order to decrease the
shape noise. This way we can study the density profiles of a galaxy population
in a statistical fashion, a technique known as galaxy-galaxy lensing. Galaxygalaxy lensing may also be applied to clusters to complement strong lensing
(where available) and to map the matter distribution in these more complicated
systems.
The shear components γ1 and γ2 and the equivalent flexion components are
defined with respect to a Cartesian coordinate system. For galaxy-galaxy lensing
studies it is more convenient to define components relative to the lens that the
sources are centred on. It is therefore common practice to adopt tangential and
25
1. INTRODUCTION
α = 0◦
ǫt = 0.3
ǫ× = 0.0
α = 45◦
ǫt = 0.0
ǫ× = 0.3
b
φ
O
α = 90◦
ǫt = −0.3
ǫ× = 0.0
Figure 1.14 Illustration of the tangential and cross components of shear for a
circular lens located at the origin O. The background source is located at angle φ
relative to the horizontal. For a tangentially aligned source, the tangential shear
ǫt (γt in the text) is positive and the cross term ǫ× (γ× ) is zero; if ǫt is negative
instead then the source is radially aligned. A positive or negative cross term with
no tangential component signifies a source image angled at 45◦ relative to the lens.
Figure originally published in Schneider (2005).
cross components, γt and γ× :
γt
γ×
= −ℜ[γe−2iφ ] = − cos(2φ)γ1 − sin(2φ)γ2
= −ℑ[γe−2iφ ] = sin(2φ)γ1 − cos(2φ)γ2
(1.34)
(1.35)
where φ is the angle between lens and source image as illustrated in Figure 1.14.
Similarly we can define the corresponding components for the flexions:
Ft
F×
Gt
G×
= − cos(φ)F1 − sin(φ)F2
(1.36)
= sin(3φ)G1 − cos(3φ)G2
(1.39)
= sin(φ)F1 − cos(φ)F2
= − cos(3φ)G1 − sin(3φ)G2
(1.37)
(1.38)
where the effect of the spin property is clear in the multiplication factor of the
angle φ. Averaging as described above, a lensing mass would produce a purely
positive tangential signal while a so-called void (underdensity) would cause a
purely negative signal. The cross terms can never be induced by lensing (for
an isolated circular lens) and measurement of such a signal therefore provides a
good null test for systematic errors (or systematics for short).
Since sources are averaged in circles relative to lens positions, galaxy-galaxy
lensing is less sensitive to systematics such as PSF or CTI which tend to induce
26
1.3. WEAK LENSING
correlation between shapes across an entire observed field. There are however
other systematics that have to be taken into account. The main concerns include
intrinsic alignments of satellite galaxies (see Section 1.3.2) diluting the signal,
and neighbouring galaxies at the same redshift having a similar effect simply
because they are not lensed. By using redshift information for both lenses and
sources these effects can be minimised however, illustrating the importance of
accurate redshift measurements for weak lensing analyses. Since weak lensing surveys are generally very large and there are millions of galaxies involved,
spectroscopic redshifts are unfeasible. Fortunately, state-of-the-art photometric
redshift software is able to produce reliable redshift estimates (see e.g. Hildebrandt et al., 2012, for recent CFHTLS results). Additionally, as suggested by
Rowe (2008) and further investigated in Velander et al. (2011) (Chapter 3 in
this Thesis), the light from lens galaxies may be bright enough to affect the
source shapes measured, particularly in the case of F flexion. This effect can
be avoided by not using sources too close to other bright light sources such as
the lens. The amplitude of the flexion signal falls off very quickly with distance,
however, and we therefore have to go close to the lens in order to detect it. An
alternative approach is to model the lens light and remove it from the image
before measuring the source shapes (see Section 3.4.3, page 73). Though this
seems to work well, we have to be careful not to introduce new artifacts.
Observing the density profiles of galaxies tells us about the total mass of
the lenses which provides constraints on various relations between halo mass
and the properties of the observed galaxy (see Chapter 4). It is also of interest
because N-body simulations predict specific profiles. Confirming or disproving
these profiles will provide clues to the underlying physics used when creating
the simulation. There are currently two main density profiles being used in
weak lensing to determine mass: the singular isothermal sphere (SIS) and the
Navarro-Frenk-White profile (NFW; Navarro, Frenk, & White, 1996). The SIS
is a fairly simplistic powerlaw for which the density is inversely proportional to
the square of the physical radius r, ρ(r) ∝ r−2 . For such a density profile the
shear and flexion profiles are given by
|γt (θ)|
=
|Ft (θ)|
=
|Gt (θ)|
=
θE
2θ
θE
2θ2
3θE
2θ2
(1.40)
(1.41)
(1.42)
where θ = ξ/Dl is the angular distance from the lens and θE is the Einstein
radius:
σ 2 D
ls
v
θE = 4π
(1.43)
c
Ds
with σv the velocity dispersion of the lens. The velocity dispersion of a halo is
directly related to its mass via
σv2 =
1 2
GM (r)
= vrot
2r
2
(1.44)
where M (r) is the mass interior to r and vrot is the rotational velocity. This
profile thus reproduces the flat rotation curves discussed in Section 1.1.2 since
27
1. INTRODUCTION
Figure 1.15 Comparison of the singular isothermal sphere (SIS) and NavarroFrenk-White (NFW) density profiles for shear, F flexion and G flexion (top to
bottom). The dashed (solid) line represents the SIS (NFW) profile in each case,
and the surface mass density is proportional to the shear.
the rotational velocity is constant, but it is clearly unphysical close to the lens
where the density profile approaches infinity.
The NFW profile is motivated by the properties of pure dark matter haloes in
28
1.3. WEAK LENSING
N-body simulations and has a softer inner profile but approaches the behaviour
of the SIS on larger scales. The density scales as
ρ(x) =
ρcrit (z)∆c
x(1 + x)2
(1.45)
where x = r/rs is the radius in units of a scaling radius rs and ∆c is a dimensionless scaling density. The shear profile for the NFW has been analytically
derived by Wright & Brainerd (2000) and the corresponding flexion expressions
are given in Bacon et al. (2006). A defining parameter of the NFW halo is also
its concentration c which is related to its virial radius r200 within which the
total mass is M200 (see e.g. Duffy et al., 2008, for a recent relation). The virial
radius defines the point where the density of the halo is 200 times the critical
density ρcrit . M200 is frequently used as a measure of the halo mass in weak
lensing analyses, and a relation between the NFW halo mass and the SIS Einstein radius is given by e.g. Bacon et al. (2006). A comparison of the SIS and
NFW profiles for each of shear, F flexion and G flexion is shown in Figure 1.15
for a halo of mass M = 1 × 1012 h−1 M⊙ at redshift zl = 0.35. The lensing
distortions shown are imprinted on sources at redshift zs = 0.8. It is clear that
the inner regions are important for distinguishing density profiles and this is an
interesting application of flexion since it has the potential to better distinguish
between the two profiles. In Chapter 3 we use space-based data to observe
galaxy-galaxy flexion and use the measurements to constrain the density profile
of an average galaxy.
These profiles are useful for studying isolated galaxies or for characterising
the mass distribution on small scales, close to the lens. However, we know
that galaxies in general cluster along dark matter filaments and in Cosmic Web
nodes. If we do not specifically select galaxies that are isolated, we will see
an excess signal on larger scales due to neighbouring galaxies and their haloes
adding their signature to the profile. To extract an accurate mass estimate
this fact has to be accounted for, and the established approach is to use a halo
model (e.g. Cooray & Sheth, 2002; Guzik & Seljak, 2002; Mandelbaum et al.,
2005; van Uitert et al., 2011). The model becomes more complicated when the
lensing signal contribution from satellite galaxies is also included, and when
striving to accurately account for the normal baryonic matter in galaxies as
well, rather than just the dark matter. In Chapter 4 we briefly review the
halo model introduced in van Uitert et al. (2011) and apply it to data from
the full Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) with the
aim of learning about the connection between dark matter haloes and their
corresponding host galaxies. For genuine galaxy clusters however, the approach
is slightly different. We discuss this in more detail in Chapter 5, but first I give
a qualitative example of the impressive cluster results that can be achieved with
weak lensing.
1.3.4
Bullets and train wrecks
The power of combining a weak lensing analysis with other types of mass observations is well illustrated by recent studies of merging galaxy clusters. The
merging process causes the matter distribution in such clusters to be highly
disturbed, allowing us to study the behaviour of their mass components under
such unusual circumstances. This Section will mainly be concerned with two
29
1. INTRODUCTION
such clusters which are in stark contrast to each other: the Bullet Cluster (1E
0657-558) at z = 0.296 and the Cosmic Train Wreck (Abell 520) at z = 0.201.
Figure 1.16 Bullet Cluster weak lensing results. The top panel shows the galaxies imaged with the Hubble Space Telescope and the bottom panel shows the
plasma imaged with Chandra. Overlaid on both panels are the green contours
from the weak lensing analysis, the peaks of which clearly coincide with the two
galaxy concentrations rather than with the plasma concentration which contains
more baryonic mass. Figures originally published in Clowe et al. (2006).
The Bullet Cluster was made famous by Clowe et al. (2006) when they
presented a study where a weak lensing mass reconstruction is compared to the
locations of baryons in the cluster. The baryonic components considered in that
paper are the galaxies themselves in each of the two merged clusters, and the
30
1.3. WEAK LENSING
plasma displaced from the clusters during the collision. Their results are shown
in Figure 1.16 and they are in excellent agreement with what ΛCDM predicts.
As the two main clusters pass through each other the galaxies act essentially as
collisionless particles and emerge on either side relatively intact. The plasma
from each cluster on the other hand is caught in the middle as evidenced by the
‘bullet’ or shock wave visible in the X-ray imaging (lower panel in Figure 1.16).
In the absence of dark matter we would expect the majority of the mass to be
contained in the plasma. As is clear from the weak lensing analysis, shown as
green contours in Figure 1.16, the mass peaks coincide perfectly with the two
galaxy concentrations and there is no evidence of a peak near the bullet. This
is consistent with the prediction that dark matter is collisionless and should
thus follow the galaxies as the two clusters pass through each other. Since the
observations are difficult to explain with models that just modify the gravity
strength of ordinary matter, this particular study has become a standard piece
of evidence for the existence of dark matter.
Figure 1.17 Cosmic Train Wreck weak lensing results. The weak lensing mass
contours are overlaid a composite image showing both the optical CFHT observations and the Chandra X-ray observations as a diffuse red cloud. Significant weak
lensing mass peaks are numbered 1-6 and number 3 coincides with a high plasma
concentration in this case, contrary to what is seen in the Bullet Cluster. Figure
originally published in Jee et al. (2012).
As a counter-example, the Cosmic Train Wreck which is a merging cluster
thought to be at a stage similar to that of the Bullet Cluster has been studied
using the same approach (Mahdavi et al., 2007; Jee et al., 2012). The resulting composite image (Figure 1.17) shows six distinct weak lensing mass peaks.
Most of these peaks coincide with concentrations of galaxies, with the glaring
exception of peak number 3. There are a few faint cluster galaxies in the vicinity of this peak but, more strikingly, that area seems dominated by the high
plasma density. This is very puzzling because it seems to indicate that there is
31
1. INTRODUCTION
dark matter, but no associated luminous matter, left at the centre of the cluster
suggesting that the dark matter particles have collided. Comparing the dark
matter collisional cross-section estimated from this analysis with the maximum
allowed value derived from the Bullet Cluster, it becomes clear that the two
analyses are incompatible at a 6σ level. There have been a few suggested explanations for this result, including a Cosmic Web filament centred on peak 3
and pointing straight towards or away from us, but there have to be a fair few
fortuitous circumstances for any of these explanations to hold. If all other explanations are ruled out and we are left with collisional dark matter as the only
possibility, then the Cosmic Train Wreck constitutes a strong counter-argument
to the conclusions drawn from the Bullet Cluster.
These two examples showcase the ability of weak lensing to reveal basic
properties of dark matter, thanks to its sensitivity to all matter structure in the
foreground. It also shows the scientific value in analysing not only well-behaved
large structures with settled features but also atypical cases with heavily distorted mass distributions. It is clear that there is a great deal to be learnt via
analyses like these.
1.3.5
Shape measurement methods
The challenge of correctly determining the lensing distortions is a significant one.
As already discussed, the distortions are small and imprinted on galaxies which
are not intrinsically circular. Furthermore, our view of the resulting source
images is somewhat muddled by the imaging systems we use to observe them
and, in the case of ground-based telescopes, by the atmosphere. Considerable
effort has gone into producing a method that can reliably measure shear and
flexion, though most of the effort has so far been focused on recovering the shear
only. As a result there is a wide variety of shape measurement methods in use
today, and I will here give an overview of the most common types.
Moments-based methods
A widespread approach to determining the shape of a galaxy image is to measure the moments of its surface brightness distribution. The first (or monopole)
moments x̄ and ȳ correspond to the centroid while the second (or quadrupole)
moments Qij encode the ellipticity and higher order (octupole and 16-pole) moments Qijk and Qijkl are related to the flexions. By combining these moments,
estimators for shear and flexion may be derived in the weak lensing limit (see
e.g. Okura et al., 2008):
γ
32
≃
F
≃
G
≃
1
hχi
2
ζ
(9/4) − 3(trQ)2 /ξ
4
hδi
3
(1.46)
(1.47)
(1.48)
1.3. WEAK LENSING
where χ, ξ, ζ and δ are moments combinations:
χ
ξ
ζ
δ
Q11 − Q22 + 2iQ12
Q11 + Q22
≡ Q1111 + 2Q1122 + Q2222
Q111 + Q122 + i(Q112 + Q222 )
≡
ξ
Q111 − 3Q122 + i(Q112 − Q222 )
≡
ξ
≡
(1.49)
(1.50)
(1.51)
(1.52)
In most applications a weight function is applied to the moments in order to limit
the effect of noise; these are then known as weighted moments and form the basis
of the currently most common family of shape measurement methods — the
Kaiser-Squires-Broadhurst method (KSB; initially suggested and subsequently
developed by Kaiser, Squires, & Broadhurst, 1995; Luppino & Kaiser, 1997;
Hoekstra, Franx, Kuijken, & Squires, 1998). A fundamental limitation of KSB is
the simplifying assumptions it makes regarding the PSF which are not applicable
to more realistic functions (Kaiser, 2000). Even so, the method has been highly
successful in practice and the recovered shear still compares well with newer
methods. KSB now comes in many flavours, most of them measuring shear only
(e.g. Bacon et al., 2000; Erben et al., 2001; Heymans et al., 2005; Schrabback
et al., 2007). The extension of KSB to higher orders and thus flexion is known as
higher-order lensing image characteristics (HOLICs; Okura et al., 2007, 2008).
Additionally, some methods keep to the general philosophy of KSB but vary the
weight function (e.g. DEIMOS; Melchior et al., 2011).
Model-fitting methods
The PSF limitation in KSB has inspired the development of alternative methods,
and many of them are based on characterising the galaxy brightness distribution
through model-fitting. The variation between techniques here is greater than
for the moments-based methods simply because there are many different ways
to model a galaxy. The general idea is the same though: create a circular or
elliptical model galaxy and compare it to an observed source image to determine
how much it has been sheared by. The PSF can be accounted for either by
convolving the model galaxy with the observed pattern or deconvolving the
observed image, though the latter is often discarded due to the difficulty of
performing such an operation and its detrimental effect on noise properties.
One way to compare the model galaxy to the source image is to decompose
them both into a series of so-called Shapelets (Bernstein & Jarvis, 2002; Refregier, 2003; Refregier & Bacon, 2003). The advantage of doing so is to gain
analytical expressions for PSF convolution, shear and flexion, making the calculations exact and fast. Shapelets also allow for a more realistic description of
the PSF than the one utilised by KSB. This method, however, fundamentally
assumes that galaxies are well described by a Gaussian brightness distribution
and this limits the reach of the components. Therefore the wings of galaxies are
often not well constrained unless very high orders are used, something which is
usually difficult because of a lack of pixels. Never the less this approach has
proven efficient and accurate and there are several implementations available
(Kuijken, 2006; Massey et al., 2007b; Nakajima & Bernstein, 2007; Velander
33
1. INTRODUCTION
et al., 2011). This Thesis recounts in part the development of the Velander
et al. (2011) version and the use of it to detect flexion in space-based data (see
Chapters 2 and 3). A related approach is that of Sérsiclets (Ngan et al., 2009;
Andrae et al., 2011) which uses a more realistic basis set derived from the Sérsic
description of galaxy brightness profiles (Sérsic, 1968). The galaxy can also be
modelled as a sum of elliptical Gaussians (Kuijken, 1999; Bridle et al., 2002;
Voigt & Bridle, 2010).
In general the above shape measurement pipelines use a least-squares fitting
technique, or equivalent, to determine the relevant parameters of the brightness
distribution. A different procedure is to use Bayesian statistics, taking the full
posterior probability in ellipticity into account. This is exemplified by lensfit
(Miller et al., 2007; Kitching et al., 2008), a shear measurement software suite
which has shown great promise in recent years, both in simulations and on real
data. The galaxy images are modelled individually using the sum of two Sérsic
profiles to represent the bulge and disk and a full likelihood surface is produced.
This likelihood is then used to estimate the shear of the galaxy. However,
Bayesian model fitting requires a prior (i.e. a best guess) and is therefore sensitive to the exact choice of such a prior. In principle it can be found iteratively
using the data at hand, but it is still not clear exactly how strongly a wrong
choice would affect the outcome. These worries are relevant for most modelfitting techniques though, since the introduction of additional information is
often required due to models being under-constrained by the data. Another
fundamental concern is that which also applies to all model-based methods: the
model used to imitate a galaxy may not accurately represent the morphology of
the true galaxy. To assess the impact of the choices and approximations made
in any shear measurement method (be it moments- or model-based), it is vital
to use simulated data where we know what the distortion should be.
Simulations to test shape measurement software
Throughout the past decade while the above methods were being developed,
lensing simulations designed to test them also evolved. Of particular significance
are the shape measurement challenges posed to the weak lensing community as
a whole. These blind challenges provide images of simulated sheared galaxies,
with the amount of shear unknown to the participants of the challenge. The
participants then analyse the images without any preconceptions, thus avoiding
confirmation bias, and submit an estimate of the shear to the trial organisers.
Because the challenge is blind, and because all entrants analyse the same images
under the same conditions, the participating methods can be compared and
contrasted with each other. So far these trials are limited to shear measurements
only, with no flexion applied to the simulated galaxies.
The first such large-scale challenge was the Shear Testing Program (STEP;
Heymans et al., 2006a) closely followed by its successor STEP2 (Massey et al.,
2007a). The bias of shear measurements was parameterised using a multiplicative bias factor m and an additive factor c:
hγim i − γit = mi γit + ci
(1.53)
where γ t is the true (input) shear, γ m is the measured shear and i = 1, 2 represents the component. A negative m thus indicates that the distortion is generally
34
1.3. WEAK LENSING
Figure 1.18 Results from STEP (left panel) and STEP2 (right panel) comparing m and c biases for participating methods. Note the difference in scale; the
entire figure for STEP2 is within the grey band signifying a calibration bias of less
than 7 per cent in the STEP1 figure. The closer to zero a method is, the greater
its ability to recover the input shear. Figures originally published in Heymans
et al. (2006a) and Massey et al. (2007a); for details on the different methods the
reader is referred there.
underestimated, something most entries in STEP suffer from. A systematic offset c may be caused by e.g. insufficient PSF correction but is in general small
for implementations in use today. The first STEP installment strived to provide as realistic simulations as possible, while the successor introduced some
simplifications to ensure biases were not due to e.g. shape noise. Already in
STEP, however, the most successful methods achieved percent-level accuracy
(see Figure 1.18) which is sufficient for current weak lensing surveys but needs
to reach sub-percentage accuracy in preparation for near future surveys.
The majority of shape measurement methods taking part in the STEP challenges were based on KSB, but by the time the next generation of challenges
emerged this picture had changed. These new sets of blind simulations, Gravitational Lensing Accuracy Testing (GREAT08 and GREAT10; Bridle et al., 2009,
2010; Kitching et al., 2010, 2012) had a somewhat different philosophy to STEP.
They were aimed not only at the weak lensing community but endeavoured to
entice other communities as well, such as computer scientists. Therefore the
simulations were stripped down to the core problem of estimating shear and
PSF from images, rather than them being as realistic as possible. Furthermore,
there were several branches which, although still kept blind for the participants,
allowed for a clear picture of which galaxy properties most affect the accuracy
of the measurements. The performance of each method was quantified via a
quality factor Q which is essentially a combination of the m and c parameters
of STEP. The higher the quality factor, the better the method performs. For
future surveys, a Q of about 1000 would be ideal, and current methods achieve
in general Q ∼ 20–100. The results from the GREAT08 challenge are shown in
Figure 1.19; for the GREAT10 results the reader is referred to Kitching et al.
(2012) as the analysis is too extensive to display succinctly here.
There are a vast number of lensing simulations available more or less publicly. STEP and GREAT have the benefit of being able to compare several
methods under the same conditions and thus providing a good measure of how
well a method can recover lensing distortions in general. They are however
35
1000
1000
100
100
Q
Q
1. INTRODUCTION
10
10
b+d b+d offcenter
Galaxy type
Fid rotated
1000
1000
100
100
Q
Q
b or d
10
Fid
PSF type
Fid ex2
1.4
Rgp/Rp
1.6
HB
AL
TK
CH
PG
MV
KK
HHS3
SB
HHS2
HHS1
MJ
USQM
10
10
20
SNR
40
1.22
Figure 1.19 Results from GREAT08 showing the Q-values for each branch and
each participating method. The higher the Q-value, the better the method performs. Figure originally published in Bridle et al. (2010); for details on the different
methods the reader is referred there.
fairly idealised and not attempting to mirror a particular survey. To acquire
an accurate impression of how well a method works for a specific survey, simulations imitating the exact observing conditions are necessary. Therefore there
are often detailed lensing simulations created for each weak lensing survey. We
will, however, always be limited by how well we understand all the effects that
influence our distortion measurements, and how closely our simulations mimic
reality.
1.4
This Thesis
This Thesis is concerned with studying dark matter haloes using weak lensing
through a variety of different applications, observational as well as theoretical.
The overall aim is to ascertain how galaxies populate their dark matter haloes,
and how the haloes affect the formation and evolution of their host galaxies. To
this end we create a new shape measurement pipeline based on the Shapelets
formalism with the ability to determine both shear and flexion simultaneously.
We also develop software to accurately model the lensing signal on large scales,
taking into account contributions from neighbouring and satellite galaxies and
galaxy haloes. This extensive software package is then applied to real data,
both ground-based and from the Hubble Space Telescope.
In Chapter 2 we describe in detail the shape measurement software known
as the MV pipeline and show that it is robust using both GREAT08 simulations,
and simulations created specifically for the purpose of testing flexion recovery.
We then apply it to galaxies which are not monochromatic as is standard in
36
1.4. THIS THESIS
simulations, but which have a colour gradient. This is something that could
potentially be an issue for broad-band imaging, though our findings indicate
that the effect is small. Chapter 3 provides the first detection of galaxy-galaxy
flexion using data from the Hubble Space Telescope. We use this detection in
combination with shear measurements to constrain the density profiles of galaxysized haloes, and show that the inclusion of flexion is advantageous to accurately
determining the halo mass. In Chapter 4 we describe our halo modelling software
and apply it to galaxies in the full CFHTLS survey. We study the halo mass
as a function of host galaxy properties such as luminosity and stellar mass.
Our constraints on the relations between light and dark matter are tighter than
earlier analyses thanks to the large area and depth of the CFHTLS survey.
Finally in Chapter 5 we investigate the mass distribution in clusters of galaxies
and how it translates into shear and flexion profiles when averaged over several
clusters. In clusters we are in general unable to accurately determine the true
gravitational center, but are compelled to use visible tracers such as the brightest
cluster galaxy to estimate it. As a result the observed density profile is offset
from the true profile. We provide predictions for what we expect to observe
when several randomly offset profiles are averaged, in shear and flexion space.
Furthermore, we show that the use of flexions is particularly valuable in this
application.
37
2
A new shape measurement
method and its application to
galaxies with colour gradients in
weak lensing surveys
Sections to be published in Semboloni E.,
Velander M., Hoekstra H., Kuijken K., et al, in
preparation
As one of the most powerful probes of cosmology, weak gravitational
lensing is now the main motivation behind some of the largest nearfuture optical surveys ever undertaken. The statistical nature of the
method requires analysis of a large number of sources, and the minute
distortions involved demand high-quality data and precise shape measurements. Weak lensing software therefore has to be both fast and accurate, and such a software suite is introduced and tested in this Chapter.
This MV pipeline is shown to be very competitive, with the added benefit of being able to measure higher-order lensing distortions, or flexion.
The tests described in this Chapter involve both monochromatic and
non-monochromatic simulations, where the latter have been included
to assess the amount of bias induced by a wavelength-dependent PSF.
Since most galaxies display colour gradients, with a core that has a
different colour from the outskirts, a wavelength-dependent PSF will
affect different parts of the galaxy image differently. Thus some additional shape bias may be introduced if the PSF is not precisely corrected
for. Creating simulations based on real galaxies observed in two different filters, we find that the additional bias induced by this effect is not
greater than the bias inherent in the shape measurement software itself.
We conclude from our tests that given enough training data we will
most likely be able to characterise the colour gradient bias sufficiently
accurately to correct for it in future Euclid-like surveys.
38
2.1. INTRODUCTION
2.1
Introduction
With weak gravitational lensing rapidly gaining traction as a powerful probe of
cosmology, new surveys are being designed with lensing as a main science goal.
Since weak lensing relies on the statistical properties of a galaxy population,
large surveys are necessary to minimise systematics such as the intrinsic shape
noise. The great number of precise measurements required for future weak lensing analyses increases the necessity for shape measurement software to be both
fast and accurate. Currently there is a fair amount of software available, most of
which is centred either around the determination of shapes from combinations
of weighted second-order brightness moments, such as the method introduced
in Kaiser, Squires, & Broadhurst (1995) (KSB hereafter), or around model fitting techniques such as lensfit (Miller et al., 2007; Kitching et al., 2008) or
Shapelets (Refregier, 2003; Refregier & Bacon, 2003).
The design of a survey also has to take systematics other than those due
to biases in shape measurements into account, and primary amongst them is
the shape distortion induced by the telescope and, in the case of ground-based
surveys, by the atmosphere. This shape distortion is known as the point-spread
function (PSF) and can cause coherent distortion across a survey field, biasing
the lensing signal. KSB methods have the inherent limitation of too simplistic a
description of the PSF and not all realistic PSFs can be accurately accounted for
using this description (see e.g. Hoekstra et al., 1998). lensfit has a more flexible
PSF model and has been proven to be accurate when applied to simulations such
as the Gravitational Lensing Accuracy Testing 2008 set (GREAT08; Bridle et al.,
2009, 2010), but due to the Bayesian approach it is unfeasibly slow for large nearfuture surveys. We have therefore chosen to base the new shape measurement
software introduced and tested in this Chapter, the MV pipeline, on Shapelets
which are both flexible and fast thanks to their analytical nature. Because
of their definition as a set of Gauss-Hermite polynomials, any distortion or
convolution may be done analytically. This makes it straight-forward to extend
the shape analysis to higher-order lensing distortions, known as flexions, without
loss of time or accuracy. Flexion, which quantifies variations in shear across
a source image, was first discussed in Goldberg & Natarajan (2002) and the
notation was then further developed in Goldberg & Bacon (2005) and Bacon
et al. (2006). Adding flexion to shear results in a weak arc-like shape which
is a better description of the true lensing-induced distortion than the shear
stretch alone. Flexion is sensitive to small-scale fluctuations so added detail to
mass reconstructions is gained by including it. This makes flexion a powerful
complement to shear, particularly for detecting substructure within dark matter
haloes (Okura et al., 2008; Bacon et al., 2010; Er et al., 2010; Leonard et al.,
2011), or for determining their profiles and shapes (Hawken & Bridle, 2009; Er
& Schneider, 2011; Er et al., 2011).
In this Chapter we convey the details of the MV pipeline and the tests performed on it using the GREAT08 simulations and simulations created specifically for the purpose of testing the MV pipeline in preparation for the analysis
of space-based data. Both sets of simulations are monochromatic in nature,
but recently the question of the impact of a wavelength-dependent PSF on
shape measurement accuracy was raised (Voigt et al., 2011). Since the PSF is
a function of wavelength, and since galaxies in general are expected to display
39
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
different colours in their cores and disks, the PSF will look different at different
points on a galaxy image. Thus two galaxies with dissimilar intrinsic shapes
and colour gradients may become indistinguishable after being convolved with
a wavelength-dependent PSF. Perfectly correcting for the PSF in such a case
is impossible without further information on the colour gradient of the galaxy.
This could present a challenge for surveys where observations are done using a
broad-band filter such as the planned space-based Euclid1 survey, scheduled for
launch in 2019. The ultimate impact of this effect on weak lensing analyses will
depend on the true intrinsic colours of the galaxy distribution and on the total
shape measurement bias induced by the wavelength-dependent PSF. For the
first part several studies into colour gradients of galaxies have been carried out
in the context of galaxy evolution, most of them at low redshifts (e.g. GonzalezPerez et al., 2011). To assess the impact of the second part, the bias induced
by a wavelength-dependent PSF, representative simulations have to be created.
In this Chapter we use real galaxies from the All-Wavelength Extended Groth
Strip International Survey (AEGIS; Davis et al., 2007) together with photometric redshifts from the third Canada-France-Hawaii Telescope Legacy Survey
Deep field (CFHTLS-Deep3) to create realistic broad-band simulations. AEGIS
is here assumed to provide a representative galaxy sample which has been observed through two filters with the Advanced Camera for Surveys (ACS) onboard the Hubble Space Telescope (HST). These two filters can be combined
to approximate the broad-band filter proposed for Euclid and therefore these
data form the ideal starting point for Euclid-like simulations. The MV pipeline
is then tested on these simulations to determine the level of bias induced by
colour gradients in galaxies, and to identify the galaxy properties that have the
greatest impact on this bias.
This Chapter is organised as follows: in Section 2.2 we introduce the theoretical background of shear and flexion, and of Shapelets, with the MV pipeline being described in detail in Section 2.2.3. Monochromatic tests of the MV pipeline
are carried out in Section 2.3 and the software is applied to non-monochromatic
simulations in Section 2.4. We conclude in Section 2.5.
2.2
2.2.1
Theoretical background
Shear and flexion
If the lensing convergence and shear are not constant across a given source image,
then we need to quantify how they vary. This can be done by measuring higherorder lensing distortions known as flexion. The formalism was first explored
by Goldberg & Bacon (2005) and then further investigated by Bacon et al.
(2006) (hereafter B06). In the weak lensing regime where convergence is small
the lensed surface brightness of a source galaxy, f (x), and the unlensed surface
brightness, f0 (x), are related through
∂
1
f0 (x).
(2.1)
f (x) ≃ 1 + (A − I)ij xj + Dijk xj xk
2
∂xi
where I is the identity matrix, xi denotes lensed coordinates, and A is a distortion matrix which may be expressed in terms of convergence κ and shear
1 http://www.euclid-ec.org
40
2.2. THEORETICAL BACKGROUND
γ:
A=
1 − κ − γ1
−γ2
−γ2
1 − κ + γ1
.
(2.2)
Dijk ≡ ∂Aij /∂xk describes how the lensing field varies across a source image.
Assuming that there are no such fluctuations, an assumption which may be
valid if e.g. the source image is very small, is equivalent to setting Dijk = 0. We
can now re-express this matrix as a sum of two quantities: Dijk = Fijk + Gijk .
These two quantities are referred to as first flexion, or F flexion, and second
flexion, or G flexion, respectively and similarly to shear have two components
each. To make the relation between convergence, shear and flexion clear we can
express all quantities in terms of derivatives of the lensing potential ψ (see e.g.
Hawken & Bridle, 2009):
κ
=
γ1
=
γ2
=
F1
=
F2
=
G1
=
G2
=
1
(ψxx + ψyy )
2
1
(ψxx − ψyy )
2
ψxy
1
(ψxxx + ψyyx )
2
1
(ψxxy + ψyyy )
2
1
(ψxxx − 3ψxyy )
2
1
(3ψxxy − ψyyy )
2
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
The full matrices Fijk and Gijk in terms of the four flexion components are
written explicitly in B06. Visually, if the shear is a stretch in one direction then
F flexion is a subtle skewness of the brightness profile reminiscent of a centroid
shift and the G flexion has three-fold rotational symmetry. When all the above
distortions are applied to a circular object, a weak arc is created.
2.2.2
Shapelets
The shape measurement pipeline presented in this Chapter is based on the
Shapelet formalism which makes possible the linear decomposition of a galaxy
image with surface brightness f (x) into a set of complete and orthogonal basis
functions Bab called Shapelets:
f (x) =
∞ X
∞
X
sab Bab (x; β)
(2.10)
a=0 b=0
where sab are the Shapelets coefficients. The formalism was first introduced by
Refregier (2003) and its application to weak lensing shape estimates was further
studied in Refregier & Bacon (2003). The basis functions employed consist of
Gauss-Hermite polynomials:
2
Bab (x; β) = kab β −1 e
− |x|
2β 2
Ha (x/β)Hb (y/β).
(2.11)
41
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
The Hermite polynomial of order n, Hn , depends on the coordinate on the image
plane and on the Shapelets scale radius β, and the basis functions are normalised
by a constant kab . What makes Shapelets powerful is not only their completeness but also their invariance under Fourier transforms which enables us to do
convolutions analytically. This makes Shapelets a very fast method for determining distortions which is essential to weak lensing, particularly for dedicated
surveys where a large number of objects have to be analysed. They are also
analogous to the eigenstates of the two-dimensional quantum harmonic oscillator, and thus any linear transformation such as translation, rotation, dilation
and shear and flexion can be expressed as a combination of ladder operators:
âi
≡
â†i
≡
1
√ (x̂i + ip̂i )
2
1
√ (x̂i − ip̂i )
2
(2.12)
(2.13)
where i = 1, 2 (for the x- and y-directions), x̂ ≡ x and p̂ ≡ ∂/∂x. The property
raised or lowered by these operators is known as spin; a quantity which is
invariant under rotation by an angle φ = 2π/s is said to be a spin-s quantity.
Thus shear (or ellipticity) is a spin-2 quantity, while F flexion is spin-1 and G
flexion is spin-3. The shear operators may be written in terms of raising and
lowering operators as
1 †2
2
2
â1 − â†2
−
â
+
â
(2.14)
Ŝ1 =
1
2
2
2
(2.15)
Ŝ2 = â†1 â†2 − â1 â2
or, in terms of the x̂ and p̂ operators
Ŝ1
=
Ŝ2
=
1
− (x̂1 p̂1 − x̂2 p̂2 )
2
1
− (x̂1 p̂2 + x̂2 p̂1 )
2
(2.16)
(2.17)
Using the same notation we can write simple analytical expressions for the
flexion operators:
F̂1
=
F̂2
=
Ĝ1
=
Ĝ2
=
1
8
1
−
8
1
−
8
1
−
8
−
3x̂21 p̂1 + 2x̂1 x̂2 p̂2 + x̂22 p̂1
x̂21 p̂2 + 2x̂1 x̂2 p̂1 + 3x̂22 p̂2
x̂21 p̂1 − 2x̂1 x̂2 p̂2 − x̂22 p̂1
x̂21 p̂2 + 2x̂1 x̂2 p̂1 − x̂22 p̂2
(2.18)
(2.19)
(2.20)
(2.21)
Applying these operators to circular Shapelets we thus get an image which is
‘flexed’.
Using Shapelets, the point-spread function (PSF) can be convolved with
a galaxy image in a similarly analytical fashion. The PSF is described by a
distortion matrix P:
X β β β
β
βobj βpsf
obj psf
(2.22)
pa3 b3
Cb1con
Ca1con
Pa1 a2 b1 b2 (βobj , βcon ) =
a2 a3
b2 b3
a3 ,b3
42
2.2. THEORETICAL BACKGROUND
Here pab are the Shapelets coefficients of the PSF and βpsf , βobj and βcon are
the scale radii of the PSF, the object and the resulting PSF convolved object
β1 β2 β3
is a convolution tensor which depends on the different scale
respectively. Cnml
radii. The full expression is given in Refregier (2003). The PSF convolution is
then done by multiplying the above matrix and the Shapelets expansion of the
object being convolved.
2.2.3
The MV pipeline
Figure 2.1 Polar Shapelets basis functions up to a maximum Shapelets order
of nmax = 10. For m ≥ 0, the real components of the basis functions are shown
while for m < 0 the imaginary components are shown. The solid purple (thick)
lines mark the coefficients used by the MV pipeline to estimate the shear and
flexions for an analysis with nmax = 10. The dashed purple (thick) lines mark the
coefficients not used by the KK06 implementation for the same nmax .
The software we introduce and test in this Chapter, the MV pipeline, is based
on an earlier Shapelets implementation described in Kuijken (2006) (hereafter
KK06), which we will refer to as the KK pipeline. The KK pipeline is a robust
piece of shear estimation software which has been thoroughly tested on simulation suits such as the two Shear TEsting Programmes (STEP1 and STEP2)
(Heymans et al., 2006a; Massey et al., 2007a). The MV pipeline keeps the core
43
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
Shapelet decompositions of KK, but extends the analysis package to enable
flexion measurement. The basic procedure for doing this is to
i) create a circular galaxy model
ii) apply shear and flexion to the model, and convolve with the measured
PSF
iii) decompose the true PSF convolved source image into Shapelets
iv) fit the model to the observed image.
In theory a galaxy image could be perfectly described by an infinite series of
Shapelets, but in practice we have to truncate this expansion. We choose to
truncate at order nmax = a + b (see Equation 2.10). In general we also keep the
choice of nmax constant for all galaxies in an analysis, rather than allowing for it
to vary according to some criteria such as size or brightness. This ensures that
we do not introduce artificial S/N-dependent biases. The trade-off is some noise
at higher-order coefficients for smaller or fainter sources but these coefficients
will remain unbiased.
Steps i) and ii) above can be summarised as follows to first order in ellipticity
s and flexions f and g:


Nc
X
X cn C n
(2.23)
ti T̂ i + si Ŝ i + fi F̂ i + gi Ĝi 
P · 1 +
i=1,2
even
where P is the PSF matrix and T̂ i , Ŝ i , F̂ i and Ĝi are the translation, shear, F
flexion and G flexion operators respectively, as specified in Equations 2.16–2.21.
ti , si , fi and gi are the corresponding coefficients which are determined through
step iv) above. The translation operators are included in the fit to allow for
some shifting to ensure that the fit is not spoilt by an inaccurate centroid. The
last term is the circular model in step i) which in this case is expressed as a sum
of circular Shapelets C n with coefficients cn . n is even (see the m = 0 Shapelets
in Figure 2.1) and the expansion is truncated at Nc = nmax − 2 to safeguard
against PSF structure at higher orders affecting the highest order Shapelets
used.
Once steps i), ii) and iii) have been carried out, the model galaxy and the
cartesian Shapelets representation of the true source image are both converted
into polar coordinates in preparation for the fit, as described in Refregier (2003).
For cartesian Shapelets of order n = a + b, the corresponding polar Shapelets
will have order n with angular order m ≤ n and n + m even. This conversion is
done in order to avoid truncation effects due to the mixing of orders. F flexion,
shear and G flexion operators acting on a polar Shapelet of order (n, m) generate
terms at order (n ± 1, m ± 1), (n ± 2, m ± 2) and (n ± 3, m ± 3) respectively
(see e.g. Massey et al., 2007b, Figure 2, for an illustration of the mixing of
Shapelet coefficients). We therefore truncate the polar Shapelets expansion in
the diamond shape shown in Figure 2.1, i.e. we only include terms up to order
(Nc , 0), (Nc − 1, ±1), (Nc − 2, ±2) and (Nc − 3, ±3) in the fit. This minimises
the impact of order mixing. As illustrated in Figure 2.1, the choice of which
Shapelets to include in the fit differs slightly between the MV and the KK
pipelines. The extra Shapelets included in the MV pipeline are necessitated by
the fact that the spin-3 information (i.e. G flexion) is encoded in the m ± 3
components.
44
2.3. MONOCHROMATIC TESTS
Finally in step iv), the model object is fit to the observed source using leastsquares. This gives us an estimate for all the relevant quantities simultaneously:
the ellipticity (s1 , s2 ), the F flexion (f1 , f2 ), and the G flexion (g1 , g2 ). This
technique is fast and adding the four flexion parameters does not significantly
increase the computation time compared to fitting for ellipticity alone. The
errors on the estimates originate from the errors on the Shapelet coefficients
derived from the photon noise. The χ2 is differentiated at the best-fit in order
to obtain covariances between the fit parameters. For further discussion on
errors see KK06.
2.3
Monochromatic tests
As part of the development of the KK pipeline, several aspects relevant to the
MV pipeline were thoroughly tested. We will therefore not delve further into
tests for details which are common between the two, such as the optimal choice
of scale radius β and the effect of noise. The distortion measurement routines
differ, however, and so we will in this Section thoroughly assess the shear and
flexion recovery performance of the MV pipeline. To this end we will use a series
of simulations which will be limited to one colour in this Section with the added
complication of colour gradients across galaxies and PSFs in the next.
2.3.1
GREAT08
The Gravitational Lensing Accuracy Testing 2008 (GREAT08) challenge (Bridle et al., 2009, 2010) was a competition continuing a tradition of challenges
designed to test the accuracy of current state-of-the-art shear measurement
software available to the weak lensing community (e.g. STEP1 and STEP2;
Heymans et al., 2006a; Massey et al., 2007a). Both the MV pipeline and the
KK pipeline were entered in the GREAT08 competition, allowing us to not only
test the performance of the MV pipeline under different observing conditions,
but also to compare and contrast its shear estimation capabilities to those of its
predecessor.
Simulations
The GREAT08 challenge provided simulations designed for testing the fundamentals of shape measurement. Since part of the philosophy of the project
was to entice the participation of other communities, such as computer programmers, the simulations were kept fairly simplistic and focused on the core
problem of taking a noisy distorted galaxy image and measuring how much it
has been sheared by. To avoid any deblending issues, the galaxies were created
in individual postage stamps which were then placed on a grid to create an image of 4000 × 4000 pixels and 10 000 galaxies. Each galaxy postage stamp was
created by i) simulating an elliptical and sheared galaxy; ii) convolving it with
a PSF; iii) binning the light to create a pixellised image; iv) applying a noise
model. During the course of the challenge there were four sets of simulations
released to participants; two sets with known shears (low and real noise) and
two blind sets (low and real noise). The main challenge consisted of the blind
real noise set which consisted of 2 700 composite images as described above.
45
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
Table 2.1 Different branches of the GREAT08 RealNoise Blind simulations.
Four parameters were varied between branches according to this table, with further
explanations in the text.
S/N
Rg /Rp
PSF type
Galaxy type
Fiducial
Variation 1
Variation 2
20
1.40
Fid
b+d
10
1.22
Fid rotated
b or d
40
1.60
Fid e × 2
b+d offcentre
Observing conditions were varied between images, one at a time, with the 9
different branches shown in Table 2.1. The fiducial branch had galaxies with
S/N = 20 and a ratio between the radius of the PSF convolved galaxy and that
of the PSF of Rg /Rp = 1.40. These numbers were varied to create four additional branches. The PSF used for all images was a truncated Moffat profile
which was mildly elliptical in the horizontal direction for the fiducial branch.
The PSF was rotated 45◦ or its ellipticity was doubled to create two additional
branches. The final variable to be altered was the galaxy type. In the fiducial
case galaxies were represented by the sum of two Sérsic profiles (Sérsic, 1968)
corresponding to the bulge and disk components. For one branch, the galaxies
consisted of only one Sérsic profile corresponding to either a bulge or a disk,
and in another the centroids of the bulge and disk did not coincide. For more
details on the different branches see the GREAT08 results paper (Bridle et al.,
2010).
Both the applied shear and the PSF was kept constant across each image,
although they were varied between images. The true shear values were concealed
from the participants, and so was the information pertaining to which image
was part of which of the 9 branches, but the PSF was provided as a star image.
Participants were thus told which of the three PSFs had been applied to which
image. The true applied shear values were perturbations around 5 root values,
root
both positive and negative, with |γ1,2
| ≤ 0.037. The GREAT08 team utilised
the paired rotation technique introduced in STEP2 whereby each simulated
galaxy has a twin galaxy which has been rotated by 90◦ before shearing. This
method minimises shape noise since the ellipticity estimates of each pair should
cancel in the absence of applied shear and PSF. The large number of simulated
galaxy images in combination with this shape noise minimisation technique
allowed for high precision assessment of current shape measurement methods in
preparation for future surveys.
Results
For the submitted results, we used the MV pipeline and a maximum Shapelets
order of nmax = 8. To average over all the galaxies in an image we used a
technique known as convex hull peeling (CHP). CHP works essentially like a 2D
median and is a way of eliminating outliers from a sample in a 2D parameter
space (e.g. in the (γ1 , γ2 ) plane). This is done by removing a so-called convex
hull, i.e. the minimal convex set of data points containing all other points.
By peeling away a number of convex hulls and averaging over the remaining
points, a mean unaffected by extreme results may be produced (see Figure 2.2).
46
2.3. MONOCHROMATIC TESTS
Figure 2.2 Illustration of the convex hull peeling procedure applied to the shears
measured on all galaxies in a single GREAT08 image. Each asterisk represents a
shear measurement, and the lines connecting data points show the points in the
convex hulls being removed before averaging. The final average, after removing
50% of the data points this way, is marked by a red star.
The choice of how many points are removed before averaging may be varied
according to their distribution. For GREAT08 we chose to remove 50% of the
measurements before averaging.
The GREAT08 team compare the different submissions using a quality factor, or Q-value, in an attempt to consolidate the m (multiplicative bias) and
c (additive bias) parameters of STEP into a single quantity. In this case, the
Q-value is defined as
kQ σ 2
Q=
(2.24)
m − γt i
2
h(hγij
ij j∈k ) iikl
2
2
where σ 2 = σstat
+ σsyst
is a combination of the statistical spread in the simulations and the expected systematic errors. The superscripts m and t denote
measured and true values respectively and γij is the shear component i for
simulation image j. The differences between the measured and true shears are
averaged over root shear sets k and simulation branches l. The whole expression is normalised by kQ such that a method with a purely statistical spread in
47
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
Figure 2.3 Submitted MV γ1 results for GREAT08. Each circle represents the
average γ1 in a single image containing 10 000 galaxies, and the crosses with
error bars show the resulting average for each root shear value. The fitted black
line would coincide with the zero-line if there were no biases in the measurements.
Also shown as a thinner pink line are the results submitted using the KK pipeline.
Each panel represents a different simulation branch as specified in the bottom left
corner of each panel (see also Table 2.1). The MV method does well in all cases
apart from low signal-to-noise (bottom left panel) and small galaxies (bottom right
panel).
the measured shears will have a Q-value of kQ which is the level desirable for
future surveys. In the case of GREAT08, kQ = 1000 and σ 2 = 10−7 , giving a
Q-value nominator of 10−4 . Established shape measurements at the time of the
challenge, such as those based on the KSB method (Kaiser, Squires, & Broadhurst, 1995), generally achieve 10 . Q . 100. For future surveys with greater
requirements on accuracy, we would ideally use methods with Q → 1000.
Both the MV and KK pipelines performed well for current surveys, reaching
an overall Q ∼ 25 in the RealNoise Blind simulations. Due to the definition of
Q however, a method is severely penalised if it presents issues in even one of
the nine branches. This ensures that a method with a stable high Q across all
branches wins the challenge. It is never the less instructive to look at the different branches separately to assess the impact of different observing conditions
on the performance of a particular method. In Figures 2.3 and 2.4 we show the
residual shear versus true shear for each branch for the submitted MV γ1 and γ2
results respectively. From this it is clear that the MV pipeline does very well in
7 of the 9 branches (100 . QMV . 500). For a perfect measurement, the black
solid line which has been fitted to the data would coincide with the zero-line
and any deviation is parameterised via the STEP m and c parameters, defined
as follows:
(2.25)
hγim i − γit = mi γit + ci
48
2.3. MONOCHROMATIC TESTS
Figure 2.4 Submitted MV γ2 results for GREAT08. Each circle represents the
average γ2 in a single image containing 10 000 galaxies, and the crosses with error
bars show the resulting average for each root shear value. The fitted line would
coincide with the zero-line if there were no biases in the measurements. Also shown
as a thinner pink line are the results submitted using the KK pipeline. Each panel
represents a different simulation branch as specified in the bottom left corner of
each panel (see also Table 2.1). The MV method does well in all cases apart from
low signal-to-noise (bottom left panel) and small galaxies (bottom right panel).
where i = 1, 2 represents the shear component. A negative multiplicative bias
mi thus indicates that the distortion is generally underestimated. A systematic
offset ci may be caused by e.g. insufficient PSF correction, and the Q-values and
m and c biases for each of the 9 branches for the MV pipeline may be found
in Table 2.2. Also shown for comparison in each panel of Figures 2.3 and 2.4
are the results of the KK pipeline, and the differences in accuracy between the
two is small in most cases. A minor distinction is that while the MV pipeline
seems to underestimate the shear in nearly all panels, the KK pipeline predominantly overestimates it. For the faint and barely resolved sources, however,
both pipelines show similar trends.
Although the MV pipeline performs very well in most cases, the results
for faint galaxies (bottom left panels in Figures 2.3 and 2.4) and, to a lesser
extent, barely resolved galaxies (bottom right panels) cause the overall Q-value
to not reach values adequate for future surveys. The KK pipeline also severely
underestimates the shear, and even more so than the MV pipeline in the case of
faint galaxies. The strong bias at S/N = 10 is however not consistent with the
results found in the STEP challenges (for the KK pipeline; see Heymans et al.,
2006a; Massey et al., 2007a) or in our own simulations (for the MV pipeline;
see Section 2.3.2). One of the reasons for this discrepancy may be due to the
definition of S/N. In GREAT08, the flux of a simulated object is set such that
49
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
Table 2.2 MV results for the different branches of the RealNoise Blind simulations. The Q, m and c parameters are defined in the text (Equations 2.24 and
2.25); the larger the Q-value and the smaller the m and c, the better the recovery
of the input shear.
Fiducial
b+d offset
b or d
PSF e × 2
PSF rot
S/N = 40
S/N = 10
Rg /Rp = 1.6
Rg /Rp = 1.22
Q
m1 (×10−2 )
m2 (×10−2 )
c1 (×10−4 )
c2 (×10−4 )
241
152
88.9
140
173
471
4.81
395
23.9
−3.05 ± 0.43
−3.64 ± 0.44
−4.42 ± 0.48
−3.37 ± 0.44
−3.09 ± 0.45
2.06 ± 0.32
−18.72 ± 0.70
−1.60 ± 0.46
−8.76 ± 0.62
−2.18 ± 0.56
−3.76 ± 0.56
−4.90 ± 0.67
−3.59 ± 0.56
−3.88 ± 0.60
2.20 ± 0.39
−20.79 ± 1.03
−2.39 ± 0.59
−8.41 ± 0.82
0.24 ± 1.02
0.83 ± 1.05
−2.96 ± 1.15
−3.36 ± 1.03
2.05 ± 1.10
−1.14 ± 0.76
−17.42 ± 1.74
3.09 ± 1.11
−8.05 ± 1.46
1.29 ± 0.99
−0.37 ± 1.13
2.69 ± 1.21
2.08 ± 1.03
−1.28 ± 1.08
−0.65 ± 0.70
5.75 ± 1.81
−0.68 ± 1.03
0.41 ± 1.57
the number quoted as the signal-to-noise ratio is equal to the total flux divided
by the uncertainty in the flux obtained if the true shape, but not normalization,
of the object is known. We find that this does not correspond to the S/N we
detect as observers, defined as the total observed flux divided by the uncertainty
in the flux measurement (as determined by e.g. SExtractor). With this
definition we find that GREAT08 simulations with S/NGREAT08 = 10, 20, 40
actually correspond to an observed S/Nobs = 6, 12, 23 respectively, and our
results are then more in agreement with previous tests. In real applications
we do generally exclude galaxies with S/Nobs < 10 precisely because we know
that the bias increases steeply below this level. It should be noted, however,
that most galaxies in a weak lensing survey are small and faint, so a shape
measurement which is unbiased down to low S/N is vital for future surveys, and
it is clear that more work is required in this area. In general though the MV
pipeline did exceptionally well under “good” observing conditions, e.g. for the
high S/N branch or for well resolved galaxies. Our own simulations described
in the next section will further test the dependence of the MV performance on
different observing conditions.
2.3.2
FLASHES
In GREAT08, no flexion field has been applied so we are not able to test that
aspect of the MV pipeline using those simulations. Because the addition of
flexion measurements is the main development since the KK pipeline and all
the tests performed on it, it is essential that flexion recovery is tested as well.
With no public flexion simulations available, we create our own Flexion and
Shear Simulations (FLASHES) using software closely related to the MonteCarlo selection software used to create the GREAT08 simulations. FLASHES
are created with the intent of testing the MV pipeline in preparation for an
analysis of the space-based Cosmic Evolution Survey (COSMOS; Scoville et al.,
2007), and so several observing conditions are optimised for that survey.
50
2.3. MONOCHROMATIC TESTS
Table 2.3 The different branches of FLASHES. Four parameters are varied
between the branches according to this table.
Fiducial
Shape branch
Profile branch 1
Profile branch 2
S/N branch 1
S/N branch 2
S/N branch 3
PSF branch
Intrinsic shape
Galaxy profile
S/N
PSF
Round
Elliptical
Round
Round
Round
Round
Round
Round
Gaussian
Gaussian
Exponential
de Vaucouleur
Gaussian
Gaussian
Gaussian
Gaussian
100
100
100
100
8
20
40
100
Round
Round
Round
Round
Round
Round
Round
Elliptical
Simulations
The simulation creation technique is not the only similarity between GREAT08
and FLASHES. We create images containing 10 000 galaxies on a grid, with a
pair-wise match of intrinsic ellipticities in the case of elliptical galaxies. And
just as in GREAT08, each galaxy is created by i) simulating a lensing distorted
(elliptical) galaxy; ii) convolving it with a PSF; iii) binning the light to create
a pixellised image; iv) applying a noise model. However, since this is not a
challenge but an investigation into the behaviour of our pipeline, we choose
different observational conditions to GREAT08 and generally the S/N is kept
high. An overview of the 8 branches of FLASHES is shown in Table 2.3. All
galaxies are approximated as single-component Sérsic intensity profiles (rather
than the bulge-plus-disk description of GREAT08), with the fiducial profile
being a circular Gaussian, i.e. an intensity profile with Sérsic index n = 0.5. For
one branch this index is set to n = 1 instead, corresponding to an exponential
profile, and in another the index is n = 4, creating a de Vaucouleur profile.
For the branch with intrinsic ellipticities we pick random ellipticities from the
distribution in COSMOS and to minimise shape noise we use the paired rotation
technique as described in the previous section. We do not, however, include any
intrinsic flexion in these simulations. While we allow the lensing distortion to
vary between images, it is kept constant for all galaxies across a single image.
The strength of the shear and flexion fields are picked randomly but we ensure
that the value never exceeds |γ1,2 | ≤ 0.05, |F1,2 | ≤ 0.008 pixel−1 and |G1,2 | ≤
0.02 pixel−1 .
Once a lensing distorted galaxy model has been created we convolve it with
a PSF which is described by a Moffat profile with an index m = 9, making
it nearly Gaussian. This PSF is circular in general, except for one branch
where it is elliptical in the horizontal direction with e1,PSF = 0.02. As these
simulations are intended to mimic COSMOS data, the size of the PSF is fairly
small with a full width at half maximum (FWHM) of 2.1 pixels, resulting in
a PSF convolved galaxy size of 5.8 pixels. Finally we use the definition of
S/Nobs from the previous section to define the four S/N branches. Most of the
tests are carried out under near-perfect noise conditions to highlight any noiseindependent biases, but the lower S/N branches have the function of showing the
impact of noise on shape measurement accuracy. The lowest S/N = 8 branch
shows the bias below the S/N = 10 cut we generally apply when using real data.
51
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
Results
Figure 2.5 The multiplicative bias m on the first component for each of shear, F
flexion and G flexion. The purple stars, pink circles and green triangles represent
shear, F flexion and G flexion respectively. The symbols and solid lines show the
weighted averages while the dashed lines show the CHP average. This is from
running the MV pipeline on FLASHES with nmax = 10.
The version of the MV pipeline tested on FLASHES is the same as that tested
in GREAT08, with one minor difference; since the galaxies are in general better
resolved we use a maximum Shapelets order of nmax = 10. The galaxy shapes
in each image are averaged using two separate methods: CHP as in GREAT08,
and a weighted average with weights inversely proportional to the measurement
errors. To quantify the performance we use the m and c parameters from STEP
only rather than calculating a Q-value, or equivalent for flexion. The results for
m1 and c1 are shown in Figures 2.5 and 2.6 respectively, and the fitted biases
are detailed in Table 2.4. The biases and their trends are all very similar for
the second component and so we choose not to show them here. It is clear from
this that though the shear can be recovered with an accuracy of a few per cent
in general, the flexions are likely to be underestimated. This is particularly true
for higher Sérsic indices and noisier data. The dependence on galaxy brightness
profile is most likely an effect of the fact that Shapelets consists of Gauss52
Table 2.4 First component multiplicative and additive biases in the MV pipeline
based on FLASHES. The top, middle and bottom tables show the shear, F flexion
and G flexion results respectively. Both results using a weighted average (superscript avg) and convex hull peeling (superscript CHP) are displayed. For details
on the branches, see Table 2.3.
Shear
Fiducial
Shape branch
Profile branch 1
Profile branch 2
S/N branch 1
S/N branch 2
S/N branch 3
PSF branch
mavg
1,γ
(×10−2 )
cavg
1,γ
(×10−4 )
mCHP
1,γ
(×10−2 )
cCHP
1,γ
(×10−4 )
0.29 ± 0.10
30.31 ± 0.10
−0.16 ± 0.06
2.92 ± 0.30
−15.73 ± 0.15
−3.05 ± 0.09
−0.51 ± 0.09
0.25 ± 0.07
1.75 ± 0.11
−2.52 ± 5.11
−6.85 ± 0.12
−3.39 ± 1.70
1.02 ± 1.86
2.15 ± 0.59
4.87 ± 0.32
3.36 ± 0.11
0.33 ± 0.10
8.68 ± 0.13
−0.10 ± 0.06
2.95 ± 0.32
−10.86 ± 0.10
−1.44 ± 0.10
−0.15 ± 0.09
0.31 ± 0.07
1.80 ± 0.11
−0.96 ± 5.11
−6.75 ± 0.12
−2.68 ± 1.70
6.09 ± 1.86
2.81 ± 0.59
5.01 ± 0.32
3.38 ± 0.11
mavg
1,F
(×10−2 )
Fiducial
Shape branch
Profile branch 1
Profile branch 2
S/N branch 1
S/N branch 2
S/N branch 3
PSF branch
−20.70 ± 0.05
−34.15 ± 0.03
−41.00 ± 0.01
−57.41 ± 0.03
−81.87 ± 0.02
−57.83 ± 0.02
−37.69 ± 0.04
−20.13 ± 0.03
mavg
1,G
(×10−2 )
Fiducial
Shape branch
Profile branch 1
Profile branch 2
S/N branch 1
S/N branch 2
S/N branch 3
PSF branch
−20.73 ± 0.04
−20.40 ± 0.04
−39.17 ± 0.03
−64.68 ± 0.15
−36.52 ± 0.08
−23.22 ± 0.03
−21.62 ± 0.04
−19.21 ± 0.03
F flexion
cavg
mCHP
1,F
1,F
(×10−4 )
(×10−2 )
4.83 ± 0.00
1.12 ± 0.00
−2.24 ± 0.00
−2.99 ± 0.00
1.41 ± 0.03
2.84 ± 0.01
3.16 ± 0.00
4.52 ± 0.00
−20.84 ± 0.05
−33.16 ± 0.03
−41.06 ± 0.01
−57.00 ± 0.04
−77.27 ± 0.03
−54.16 ± 0.02
−35.51 ± 0.04
−20.03 ± 0.03
G flexion
cavg
mCHP
1,G
1,G
−4
(×10 )
(×10−2 )
−7.48 ± 0.00
−2.18 ± 0.00
0.09 ± 0.00
3.78 ± 0.01
−7.77 ± 0.24
−6.90 ± 0.03
−7.88 ± 0.01
−8.37 ± 0.00
−20.81 ± 0.04
−20.47 ± 0.04
−39.16 ± 0.03
−64.03 ± 0.17
−28.07 ± 0.11
−21.80 ± 0.03
−21.51 ± 0.04
−19.22 ± 0.04
cCHP
1,F
(×10−4 )
4.85 ± 0.00
1.18 ± 0.00
−2.24 ± 0.00
−2.95 ± 0.00
2.14 ± 0.03
3.22 ± 0.01
3.29 ± 0.00
4.51 ± 0.00
cCHP
1,G
(×10−4 )
−7.41 ± 0.00
−2.20 ± 0.00
0.06 ± 0.00
4.29 ± 0.01
−8.71 ± 0.24
−7.43 ± 0.03
−7.88 ± 0.01
−8.47 ± 0.00
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
Figure 2.6 The additive bias c on the first component for each of shear, F
flexion and G flexion. The purple stars, pink circles and green triangles represent
shear, F flexion and G flexion respectively. The symbols and solid lines show the
weighted averages while the dashed lines show the CHP average. This is from
running the MV pipeline on FLASHES with nmax = 10.
Hermite basis functions so it will be optimised for profiles similar to Gaussian
profiles. As the Sérsic index increases, the central peak becomes sharper and
any skewness (e.g. F flexion) may get more affected by the diluting effect of the
PSF, effectively drowning out the F flexion signal. Additionally the wings of the
profile reach further which means that very high order Shapelets are required to
model them. However, a maximum Shapelets order of nmax = 10 necessitates
the fitting of 66 free parameters during the galaxy image decomposition stage,
and a typical galaxy in these simulations only covers an area of ∼ 80 pixels.
Including much higher orders than already done will therefore entail fitting
noise, and so we keep our maximum Shapelets order at 10. The consequence
is that some information in the outer wings is not modelled, and this in turn
leads to an underestimation of the flexions for higher Sérsic indices. The G
flexion also displays a non-negligible systematic offset c for galaxies with fiducial
Gaussian profiles, which may be a sign that this particular combination of galaxy
brightness and PSF profiles causes a spurious G flexion signal.
54
2.4. NON-MONOCHROMATIC TESTS
The F flexion is more sensitive to noise than shear or G flexion, as is evident
from the lower left panel of Figure 2.5. While shear and G flexion show more
bias for the lowest S/N value of 8, a dataset which would be deemed too noisy
in an analysis of real data, than for other values, the F flexion shows a trend of
greater underestimation even for reasonably high S/N galaxies. A S/N cut is
therefore essential, but for F flexion a more sophisticated treatment is necessary
to calibrate the measurements. FLASHES have been designed to test the performance of the MV pipeline under COSMOS-like observing conditions. We use
our findings to correct for any effects due to noise biases in our analysis of the
COSMOS survey (see Chapter 3). It should be noted, however, that as of yet too
little is known about potential biases under different observing conditions, so
calibrating shape measurements in any other survey based on FLASHES alone
is not recommended.
2.4
Non-monochromatic tests
As made clear, the PSF of a telescope will, if left uncorrected for, bias galaxy
shape measurements. For a broad-band filter, such as the one included in the
design of the future space-based mission Euclid, additional complications arise
from the fact that the PSF usually depends on wavelength. Hence, since the
colour generally varies across a galaxy, which is likely to have a redder central
bulge and a bluer disk, the PSF will as well. We therefore have to determine how
galaxy colour gradients affect our ability to recover the true lensing distortions,
represented by shear in this Section.
2.4.1
Analytical prediction
To assess the possibility to correct for the effect of a colour-dependent PSF, we
describe the observed intensity, I obs (θ), of a galaxy image observed in a filter
of finite bandwidth as an integral over wavelength:
Z
I obs (θ) =
dλ I obs (θ, λ)
(2.26)
Z
=
dλ I 0 (θ, λ) ⊗ T (θ, λ)
(2.27)
where we have made explicit that the observed intensity is the pre-seeing intensity I 0 (θ, λ) viewed through an imaging system with a PSF T (θ, λ). For a
broad filter, the observed centroid is
Z
Z
1
dλ dθ θi I obs (θ, λ)
(2.28)
θ̄i ≡
Ftot
Z
Z
Z
1
dλ dθ dϕ θi I 0 (ϕ, λ)T (θ − ϕ, λ)
(2.29)
=
Ftot
where Ftot is the total flux. By employing a change in variable, x = θ − ϕ, we
can derive the following expression:
Z
Z
Z
1
θ̄i =
dλ dθ dx I 0 (θ, λ)ϕi T (x, λ) + I 0 (θ, λ)xi T (x, λ) (2.30)
Ftot
Z
1
=
dλ [θi (λ)F (λ) + F (λ)pi (λ)T (λ)]
(2.31)
Ftot
55
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
where, for a given wavelength λ, F (λ) is the total flux, θiR(λ) is the centroid,
pi (λ) are the first-order moments
of the PSF and T (λ) = dθ T (θ, λ). For a
R
symmetric PSF the term dx I 0 (θ, λ)xi T (x, λ) will vanish; for more complicated PSFs (such as imaging affected by coma) the term may be evaluated if
the PSF moments are known.
Assuming that the centroids for all wavelengths coincide for each galaxy, we
can estimate the pre-seeing centroid using Equation 2.31. We therefore continue
our analysis using centred moments. The second-order unweighted moments of
the pre-seeing and post-seeing intensities, Q0ij and Qobs
ij , and of the PSF, Pij ,
are defined as
Z
Z
1
Q0ij =
dλ dθ θi θj I 0 (θ, λ)
(2.32)
Ftot
Z
Z
1
dλ
dθ θi θj I obs (θ, λ)
(2.33)
Qobs
=
ij
Ftot
Z
Z
Z
1
=
dλ dθ θi θj dϕ I 0 (ϕ, λ)T (θ − ϕ, λ)
(2.34)
Ftot
Z
1
dθ θi θj T (θ, λ)
(2.35)
Pij (λ) =
T (λ)
Using the same substitution as above we can rewrite Equation 2.34 as
Z
Z
Z
1
0
obs
dλ dϕ I (ϕ, λ) dx T (x, λ)(xi xj + ϕi ϕj + xi ϕj + ϕi xj )
Qij =
Ftot
(2.36)
The two latter terms can be eliminated since the centroid of the pre-seeing
galaxy is assumed to be independent of wavelength. We then have
Z
Z
Z
1
obs
Qij
=
dλ
dϕ I 0 (ϕ, λ) dx T (x, λ)xi xj
Ftot
Z
Z
0
(2.37)
+ dϕ I (ϕ, λ)ϕi ϕj dx T (x, λ)
which may be rewritten as before:
Z
1
obs
Qij =
dλ F (λ)Pij (λ)T (λ) + Q0ij (λ)F (λ)T (λ)
Ftot
i.e.
(2.38)
Z
1
dλ T (λ)F (λ)Pij (λ)
(2.39)
=
−
Ftot
Equation 2.39 shows that to measure second-order unweighted moments, and
thus shear, we only need to know F (λ) and Pij (λ). Assuming accurate knowledge of F (λ), we conclude that for a perfect shape measurement method colour
gradient will not cause systematic errors to dominate the error budget. However, using unweighted moments to estimate shear is not possible due to noise.
Methods in use today use either weighted moments or a fitting procedure such
as Shapelets, and both techniques are equivalent to a weighting scheme.
Now, assuming that we know the effective PSF, a valid assumption according
to Cypriano et al. (2010), we may use the wavelength-integrated image I obs (θ)
to derive an estimate of the pre-seeing image I est (θ):
Z
obs
est
I (θ) = I (θ) ⊗ dλ F (λ)T (θ, λ)
(2.40)
Q0ij
56
Qobs
ij
2.4. NON-MONOCHROMATIC TESTS
where as above, F (λ) is the flux of the galaxy at a given wavelength and I est (θ)
is not necessarily equal to I 0 (θ, λ). In the general case there will therefore be
a bias since we will not have enough information to reconstruct I 0 (θ, λ), even
with perfect knowledge of T (θ, λ) and I obs (θ). To quantify this discrepancy we
create simulations to mimic galaxies with colour gradients as observed through
a broad-band filter.
2.4.2
Simulations
Figure 2.7 Simulated image representative of the simulations created for each
galaxy in our sample. There are 8 identical galaxy images rotated in equal steps
of 22.5◦ before a shear has been applied. In the top right corner is a star image
representing the PSF.
The simulations we create to assess the impact of colour gradients on shape
measurement consist of nearly 20 000 real galaxies taken from the All-Wavelength
Extended Groth Strip International Survey (AEGIS; Davis et al., 2007) imaged
with the Advanced Camera for Surveys (ACS) onboard the Hubble Space Telescope (HST). Our specific aim here is to evaluate the significance of this bias for
Euclid, and the planned diffraction limit of this future telescope is twice the size
57
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
of the HST limit. We therefore use only AEGIS galaxies that are well resolved
and can thus disregard the small effect of the HST PSF on the images.
Each galaxy has been observed in both the F606W (V-band) and the F814W
(I-band) filters, and in general both the size and the flux of a galaxy will be
greater in the redder band due in part to morphology and in part to telescope
optics. We also have access to photometric redshifts for 11260 objects via the
overlap with CFHTLS-Deep3 which gives us the ability to evaluate the bias as a
function of redshift. This is in fact an important aspect of our tests because as
they evolve, galaxies change morphologies and therefore their colour gradients.
Our simulations then consist of a series of images with each one corresponding
to a single AEGIS galaxy, as exemplified in Figure 2.7. In order to minimise
noise in our simulations we choose not to use the galaxy image directly, but
we decompose the observed AEGIS galaxy in each band into Shapelets. This
Shapelets representation is then duplicated and rotated to create eight identical galaxy models with different orientations. The differing orientations of
the galaxy images allows us to perform a ‘ring test’ which reduces the noise
generated by the intrinsic galaxy morphology. After rotating, the same shear
is applied to each galaxy realisation before we convolve them with a PSF and
add them to the simulated image using a pixel-scale of 0.05 arcsec. To ensure
that their brightness distributions do not overlap we place the galaxies at set
positions on the image. A representation of the PSF acting on a point source
(a ‘star’) with the same flux as the galaxy is also inserted in each image of eight
galaxy realisations. This allows for the shape measurement software to be run
as it normally would be on real survey images.
The shear we apply to our simulated galaxies is relatively large compared
to the other simulation sets described in this Chapter, but still well within the
weak lensing regime at γ1 = 0.05, γ2 = 0.00. A subtlety of our approach is that
since we use real objects as a basis for our simulations, the original galaxies have
already been sheared by foreground structure which results in a slightly different
response compared to the true intrinsic galaxy. This effect is small however and
will not significantly impact our ability to quantify the bias induced by colour
gradients.
To simulate a broad-band Euclid-like PSF, we approximate a diffractionlimited Airy disk using a Gaussian profile with a frequency dependent FWHM.
606W
The FWHM is chosen to be FWHMF
= 0.17 arcsec for the bluer filter and
PSF
F 814W
FWHMPSF = 0.21 arcsec for the redder one, though we note that the Eculid
PSF has extended wings and may therefore effectively be slightly larger. We
convolve the galaxy image in the red filter with the red PSF, and similarly for
blue. Combining the two as described below results in a total PSF which is
the weighted mean of the blue and red PSFs, and which thus has a different
response depending on wavelength.
We now have eight sheared and PSF-convolved realisations for each galaxy
combined into a single image for each of two narrow filters. To simulate a broad
filter similar to the one proposed for Euclid, we stack the two narrow-band
images by adding them:
I obs (θ) = I obs,F 606W (θ) + I obs,F 814W (θ)
= I
0,F 606W
(θ) ⊗ T
F 606W
(θ) + I
0,F 814W
(2.41)
(θ) ⊗ T
F 814W
(θ)(2.42)
The wavelength-dependent PSF is thus approximated as the sum of two Gaus58
2.4. NON-MONOCHROMATIC TESTS
sian profiles of different width:
T (θ, λ) ≃
1 F 606W F 606W
F
T
(θ) + F F 814W T F 814W (θ)
Ftot
(2.43)
Since we do not have access to a perfect shape measurement method and
thus expect a bias even without colour gradients, we have to quantify the bias
associated with the method itself. To this end we create two control images
for each set of galaxy realisations. The control images consist of galaxies that
have no colour gradient, but that are subjected to the same PSF as the Euclidlike simulations above. Comparing our results on the broad-band simulations
to these control images will convey the additional bias induced through the
assumption of monochromaticity. We thus first convolve the F606W galaxy
image with a Gaussian PSF of the same width as the F814W simulation, and
vice versa:
I obs,1 (θ) = I 0,F 606W (θ) ⊗ T F 814W (θ)
I obs,2 (θ) = I 0,F 814W (θ) ⊗ T F 606W (θ)
(2.44)
(2.45)
To account for the normalisations of the PSFs, we ensure that the control images
are created using the appropriate proportions:
I ctrl,F 606W (θ) =
I ctrl,F 814W (θ) =
F F 814W obs,1
I
(θ)
F F 606W
F F 606W
I obs,F 814W (θ) + F 814W I obs,2 (θ)
F
I obs,F 606W (θ) +
(2.46)
(2.47)
I ctrl,F 606W is thus a galaxy with no colour gradient, but with the intensity
distribution observed in F606W and convolved with our approximate Euclid
PSF, and similarly for I ctrl,F 814W . As mentioned above, the important feature of
these two control images is that they both have the same PSF as our broad-band
simulation. This is crucial because we have to compare images with identical
PSFs in order to avoid introducing another source of bias discrepancy between
filters.
We note here that our approach does entail a simplification of the problem
since we base our simulations on galaxies observed in two filters which are
themselves fairly broad. One of our assumptions is therefore that the colour of a
galaxy in one filter is that of the central wavelength, and that the spectral energy
distribution (SED) can be approximated through an interpolation between the
two filters. Ideally we would use data from several narrower filters, but we do
find via analytical tests that this assumption does not impact our knowledge of
the bias significantly.
2.4.3
Results
We use the MV pipeline to estimate the shear in both the control images and
in the stacked simulated broad-band image. Since we do not have access to
a perfect shape measurement method, we want to minimise the bias inherent
in the method itself. Our simulations are created with Shapelets, and therefore a Shapelets shear measurement pipeline should be the optimal technique
for analysing these images. However, because the PSF consists of two stacked
59
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
Figure 2.8 Bias measured in the non-monochromatic simulations as a function
of different galaxy parameters. Filled black points represent the average multiplicative bias hmi = 0.5(m606 + m814 ) determined in the narrow-band filters,
while open green points represent the bias measured in the Euclid-like stack, mbb .
FWHMbb is the size measured in the stack, and the magnitude and redshift data
are obtained via CFHTLS-Deep3.
Gaussian profiles of different widths the resulting simulation is no longer as ideal
and thus there will still be a bias present. The control images are therefore vital in determining which bias is due to limitations of the shape measurement
method, and which is due to galaxy colour gradients and PSF wavelength dependence.
We determine the multiplicative bias by obtaining the mean shear in each
bin, dividing it by the input shear and subtracting 1 such that a perfectly
recovered shear would result in maeg = 0:
maeg,bin =
hγ1,bin i
−1
γ1,in
(2.48)
where we are only considering the first shear component as the second component has been set to zero at input. The bias we measure in these simulations, as
shown in Figure 2.8, is in general positive since the galaxies are very high S/N
and the MV pipeline is optimised for images with lower S/N. In the top two
panels of Figure 2.8 we display the average bias determined in each filter F606W
and F814W, and that measured in the broad-band simulations, as a function
of galaxy size and CFHTLS i′ -band magnitude. We see that the bias measured
initially increases as galaxies become bigger and brighter, and the trend then
plateaus. In the lower panel the bias is shown as a function of photometric
redshift and colour.
The biases measured in each band only carries information about the partic60
2.4. NON-MONOCHROMATIC TESTS
Figure 2.9 Difference between the multiplicative bias measured in the simulated
stack, mbb , and that measured in the narrow-band filters on average, hmi =
0.5(m606 + m814 ), as a function of different galaxy parameters. FWHMbb is the
size measured in the stack, and the magnitude and redshift data are obtained via
CFHTLS-Deep3.
Figure 2.10 Error on the multiplicative bias difference between the simulated
stack and the narrow-band images, as a function of galaxy properties.
61
2. THE MV PIPELINE AND GALAXY COLOUR GRADIENTS
ular shape measurement method used (the MV pipeline in this case). It is encouraging that the accuracy of the shear measurements is at percent level, but to
assess the impact of a wavelength-dependent PSF we have to contrast the accuracy in the broad-band filter with that in each individual narrow-band filter. In
Figure 2.9 we therefore show the difference in bias ∆m = mbb −0.5(m606 +m814 )
as a function of the same parameters as before. Though there is a positive signal, indicating a higher bias in the broad-band filter than in the narrower ones,
it is consistently sub-percentage in size so it is much smaller than the bias induced by the shape measurement software. There are some trends in the colour
gradient bias, particularly as a function of redshift and average colour. These
trends will need to be carefully modelled in order to account for this bias in
future surveys. It is still complicated to interpret the results, however, since
some of the difference in bias could still be explained by the results in individual bands. If for instance the galaxy is smaller when observed in F606W than
when observed in the broad-band filter then the bias in the broad-band filter
will be greater simply due to the size-dependence of the bias (see Figure 2.8).
The results we have presented here are an indication of what may be expected
in terms of the bias induced by a wavelength-dependent PSF. However, the
simulations have been created using two filters only and though these filters are
narrower than the one proposed for Euclid, the wavelength resolution may still
be too low to properly represent the colour gradient in the observed galaxy.
We have also used real galaxies which have been sheared before being observed
in AEGIS and this causes a small uncertainty in our bias. With more data
observed in several narrow bands we will be able to constrain the bias further,
but the results displayed here show that the shape measurement bias induced
by assuming monochromaticity despite the use of a broad-band and the loss
of colour information that entails, is lower than the bias inherent in the shape
measurement method. It is also important to note that due to the noisiness of
the shape measurements and due to the limited galaxy sample available, the
errors on the additional bias (as shown in Figure 2.10) are a good indication
of the true errors. Thus we are able to determine the level of bias, given the
constraints described above, accurately. By studying the bias in more detail we
will therefore most likely be able to correct for this small effect in surveys such
as Euclid.
2.5
Conclusion
We have in this Chapter described and tested a new weak lensing shape measurement software suite with the capability of measuring higher order distortions
known as flexion, as well as shear: the MV pipeline. Based on the Shapelets
formalism, it is a new incarnation and an extension of the software described
in Kuijken (2006) with which it was contrasted in the context of the GREAT08
challenge. The GREAT08 simulations provided an ideal testbed for testing the
shear recovery accuracy under different observing conditions. The MV pipeline
did very well in this challenge in nearly all regimes with very competitive quality
factor values of Q ∼ 100 and above. The exception was very faint and barely
resolved galaxies where the S/N was just too low and this resulted in an overall
Q-value of Q ∼ 25.
To test the MV pipeline for the accuracy of the flexion measurements we
62
2.5. CONCLUSION
created our own simulations which we named FLASHES. These simulations were
generated using software very similar to the one used to produce the GREAT08
simulations, but with the important difference of flexion distortions being added
to the lensing potential. FLASHES mimic the survey conditions of the spacebased COSMOS survey, and were kept generally low-noise to assess any biases
induced by other factors. We confirmed that the input shear could be recovered
with high accuracy, with a multiplicative bias of a few percent in most cases.
The flexions displayed a greater bias in general, and a greater sensitivity to
the intrinsic brightness profile of the sources. Additionally, the F flexion in
particular showed a trend with S/N which may need to be calibrated in lowerquality data.
While the GREAT08 and FLASHES simulation sets were monochromatic,
care is needed in future surveys where the PSF may be wavelength-dependent.
If a galaxy with an intrinsic colour gradient, such as a redder core and a bluer
disk, is observed through a broad filter with an imaging system which results
in such a PSF, then there may be additional shear measurement bias induced.
With perfect knowledge of the PSF and the intrinsic colour gradient, this may
be corrected for but such perfect knowledge is not feasible for surveys such as
Euclid. We therefore created simulations based on real galaxies observed in
two bands as part of the HST AEGIS survey. Comparing the bias measured
in each narrow-band with that measured in a simulated broad-band, we found
that the additional bias induced by the galaxy colour gradient was at most at
the percentage level, with some variation with redshift, magnitude, size and
overall colour. This additional bias may be partly explained by inherent biases
in the MV pipeline but the results indicate that it will be possible to accurately
determine the magnitude of this effect and thus correct for it. To get a more
precise bias estimate we will in the near future create simulations with more
realistic intrinsic colour gradients and observe them through several yet narrower
bands.
The MV pipeline has been shown here to be both accurate and versatile.
We will in the next Chapter apply it to the real COSMOS survey and measure
a flexion signal around galaxies for the first time.
Acknowledgements
This study makes use of data from AEGIS, a multiwavelength sky survey conducted with
the Chandra, GALEX, Hubble, Keck, CFHT, MMT, Subaru, Palomar, Spitzer, VLA, and
other telescopes and supported in part by the NSF, NASA, and the STFC.
The authors would like to thank Gary Bernstein for valuable input and discussions. MV
acknowledges support from the European DUEL Research-Training Network (MRTN-CT2006-036133) and from the Netherlands Organization for Scientific Research (NWO).
63
3
Probing galaxy dark matter
haloes in COSMOS with weak
lensing flexion
Velander M., Kuijken K., Schrabback T., 2011,
MNRAS, 412, 2665
Current theories of structure formation predict specific density profiles
of galaxy dark matter haloes, and with weak gravitational lensing we
can probe these profiles on several scales. On small scales, higherorder shape distortions known as flexion add significant detail to the
weak lensing measurements. We present here the first detection of a
galaxy-galaxy flexion signal in space-based data, obtained using a new
Shapelets pipeline introduced here. We combine this higher-order lensing signal with shear to constrain the average density profile of the
galaxy lenses in the Hubble Space Telescope COSMOS survey. We also
show that light from nearby bright objects can significantly affect flexion measurements. After correcting for the influence of lens light, we
show that the inclusion of flexion provides tighter constraints on density
profiles than does shear alone. Finally we find an average density profile
consistent with an isothermal sphere.
64
3.1. INTRODUCTION
3.1
Introduction
Weak gravitational lensing is a powerful technique for studying the distribution
of matter in the universe due to its ability to model the matter distribution
in foreground structures, independent of the nature of the matter present. As
the light from background sources is bent around foreground lenses, the galaxy
images get distorted by the tidal gravitational field. The first-order distortion
is known as shear and is essentially an elongation of the image causing the
source galaxy to appear stretched in one direction. This type of distortion measurement has been used in a wide variety of cosmological studies ranging from
modeling the large-scale structure using cosmic shear (see e.g. Van Waerbeke
& Mellier, 2003; Hoekstra & Jain, 2008; Munshi et al., 2008, for reviews) to
determining galaxy halo shapes using galaxy-galaxy lensing (Hoekstra et al.,
2004; Mandelbaum et al., 2006a; Parker et al., 2007).
First described by Goldberg & Natarajan (2002), the second-order distortion is a relatively new addition which has since been named flexion (Goldberg
& Bacon, 2005; Bacon et al., 2006). There are two types of flexion relevant
to weak lensing studies: the first flexion induces a skewness of the brightness
profile whilst the second flexion is a three-pronged distortion. In combination
with shear, these distortions cause the well-known banana shape of lensed source
images. As flexion is effectively the gradient of shear, it is sensitive on small
scales. This makes it an important complement to shear which is sensitive on
relatively large scales only. By virtue of this, and of the orthogonality of the
three measurements, flexion is highly beneficial to investigations of the inner
profiles of dark matter haloes, where baryons become important, and to the
detection of substructure in cluster haloes. Indeed, it was recently shown (Er
et al., 2010) that mass reconstructions profit from the use of flexions in combination with shear, and flexion has already been used to constrain the halo mass
distribution and to detect substructure in clusters of galaxies (Leonard et al.,
2011; Okura et al., 2008). To provide more information on substructure and
mass profiles, there are currently new statistical flexion tools being developed
(eg. Leonard et al., 2009; Leonard & King, 2010; Bacon et al., 2010). Another
application, as discussed in Hawken & Bridle (2009), is to use both flexions in
combination with shear to significantly tighten the constraints on galaxy halo
ellipticities compared to using shear alone.
The shape measurement technique known as Shapelets (Refregier, 2003; Refregier & Bacon, 2003) works by decomposing a galaxy image into a series of
2D Hermite polynomials. These provide a simple framework for describing the
main galaxy image distortion operators, such as shear and flexion, and the convolution with the point-spread function (PSF). Due to the flexible treatment
of the PSF, the Shapelets formalism has an advantage over the currently most
widely used shape measurement method, KSB (from Kaiser, Squires, & Broadhurst, 1995), since KSB uses an idealised model for the PSF whilst Shapelets
is more versatile. The KSB equivalent for flexion is known as HOLICs (Okura
et al., 2007).
Since the field of weak lensing is relatively new, lensing measurements are
continuously being improved in accuracy and applicability. Being a statistical
technique, however, the accuracy of the weak lensing results depends heavily
on the amount of data available. Galaxy-galaxy flexion has been tentatively
65
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
observed (Goldberg & Bacon, 2005) using the ground-based Deep Lens Survey (DLS), but to further investigate galaxy-size haloes more and better data is
needed. With large surveys such as the Canada-France-Hawaii Telescope Legacy
Survey (CFHTLS) and the Red Sequence Cluster Surveys (RCS, RCS2) available, and new surveys like the 1500 square degree Kilo-Degree Survey (KiDS)
imminent, the future looks bright. However, a space-based data set provides better resolution and such a data set is already accessible to us: the HST COSMOS
survey. Using this data we will in this Chapter improve on the galaxy-galaxy
flexion measurements of Goldberg & Bacon (2005).
This Chapter is organised as follows: in Section 3.2 we review the formalism
for shear and flexion, whilst we review the Shapelets method in Section 3.3 with a
description of our implementation (dubbed the MV pipeline) in Section 3.3.1. In
Section 3.4 we test the MV pipeline on simulations and in Section 3.5 the pipeline
is applied to data from the COSMOS survey. We conclude in Section 3.6.
Throughout this Chapter we assume the following cosmology (WMAP7; Komatsu et al., 2010):
(ΩM , ΩΛ , h, σ8 , w) = (0.27, 0.73, 0.70, 0.81, −1)
3.2
Shear and flexion
We begin by briefly reviewing the weak lensing formalism. Flexion is a secondorder lensing effect first introduced by Goldberg & Bacon (2005) and further
developed by Bacon et al. (2006) (hereafter B06). It arises from the fact that
convergence and shear are not constant across a source image, and can be used
to describe how these fields fluctuate. In the weak lensing regime, the lensed
surface brightness of a source galaxy, f (x), is related to the unlensed surface
brightness, f0 (x), via
f (x) ≃
1
∂
1 + (A − I)ij xj + Dijk xj xk
f0 (x).
2
∂xi
(3.1)
Here I is the identity matrix, xi denotes lensed coordinates, and
A=
1 − κ − γ1
−γ2
−γ2
1 − κ + γ1
(3.2)
with κ = 21 (ψxx + ψyy ) a second derivative of the lensing potential ψ, where
subscripts denote partial differentiation. γ1 = 12 (ψxx − ψyy ) and γ2 = ψxy are
the two components of the complex shear γ = γ1 + iγ2 . The matrix
Dijk =
∂Aij
∂xk
(3.3)
describes how convergence and shear vary across a source image. We can reexpress Dijk as the sum of two flexions: Dijk = Fijk + Gijk . The two flexions,
the first flexion F (known as F flexion or one-flexion) and the second flexion G
(known as G flexion or three-flexion), are the derivatives of the convergence and
shear fields. There are four flexion components, each of which may be written in
66
3.3. SHAPELETS
terms of the third derivatives of the lensing potential (Hawken & Bridle, 2009):
F1
=
F2
=
G1
=
G2
=
1
(ψxxx + ψyyx )
2
1
(ψxxy + ψyyy )
2
1
(ψxxx − 3ψxyy )
2
1
(3ψxxy − ψyyy )
2
(3.4)
(3.5)
(3.6)
(3.7)
where F = F1 + iF2 and G = G1 + iG2 are the complex F and G flexions respectively. The full matrices Fijk and Gijk in terms of the four flexion components
are written explicitly in B06.
3.3
Shapelets
The Shapelets basis function set was introduced by Refregier (2003) and is more
fully described there. In summary, the surface brightness of an object f (x) can
be expressed as a sum of orthogonal 2D functions
f (x) =
∞ X
∞
X
sab Bab (x; β)
(3.8)
a=0 b=0
where sab are the Shapelets coefficients and the Shapelets basis functions Bab (x; β)
are defined as
2
Bab (x; β) = kab β −1 e
− |x|
2β 2
Ha (x/β)Hb (y/β).
(3.9)
Here kab is a normalization constant, β is the Shapelets scale radius, (x, y) are
coordinates on the image plane and Hn (x) is a Hermite polynomial of order
n. The Shapelets basis functions are easily recognised as the energy eigenstates
of the 2D Quantum Harmonic Oscillator (QHO). The formalism developed for
the QHO can also be applied to Shapelets, providing analytical expressions for
transformations such as shear and flexion. In theory, an object can be perfectly
described through a decomposition into Shapelets up to order n → ∞ but in
practice the expansion has to be truncated. We truncate at combined order
nmax = a + b to avoid introducing a preferred direction.
Convolution with the point-spread function (PSF) can also be done analytically in the Shapelets formalism by simply multiplying the Shapelets expansion
by a PSF matrix P:
Pa1 a2 b1 b2 (βobj , βcon ) =
X
β
β
obj
Ca1con
a2 a3
βpsf
β
β
obj
Cb1con
b2 b3
βpsf
pa3 b3
(3.10)
a3 ,b3
where pab are the Shapelets coefficients of the PSF and βpsf , βobj and βcon are
the scale radii of the PSF, the object and the resulting PSF convolved object
β1 β2 β3
is a convolution tensor which depends on the different
respectively. Cnml
scale radii and the full expression is given in Refregier (2003).
67
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Figure 3.1 Polar Shapelets basis functions up to a maximum Shapelets order
of nmax = 10. For m ≥ 0, the real components of the basis functions are shown
whilst for m < 0 the imaginary components are shown. The solid purple (thick)
lines mark the coefficients used by the MV pipeline to estimate the shear and
flexions for an analysis with nmax = 10. The dashed purple (thick) lines mark the
coefficients not used by the KK06 implementation for the same nmax .
3.3.1
The MV pipeline
We introduce here an implementation of the Shapelets method which builds
on a previous implementation described in Kuijken (2006) (hereafter KK06).
This approach creates a Shapelets representation of the brightness profile of a
PSF-convolved galaxy image. It also creates a model circular source and applies
shear and flexion to it before convolving it with the point-spread function (PSF).
Finally it fits the galaxy image to this modeled source in order to find the amount
by which it has been sheared and flexed.
To first order in ellipticity s and flexions f and g, the model object can be
written as


Nc
X
X cn C n
(3.11)
ti T̂ i + si Ŝ i + fi F̂ i + gi Ĝi 
P · 1 +
i=1,2
even
where P is the PSF matrix, T̂ i , Ŝ i , F̂ i and Ĝi are the translation, shear, F
68
3.4. TESTING THE PIPELINE
flexion and G flexion operators respectively and ti , si , fi and gi are the corresponding coefficients. The translation terms here ensure that fits spoiled by
undue centroid shifts are caught. The operators are acting on a circular source
which can be expressed as a series of circular Shapelets C n with coefficients
cn where n is even and the series is truncated at Nc = nmax − 2. The reason
for truncating at Nc rather than nmax is to safeguard against PSF structure at
higher orders affecting the highest order Shapelets used. To avoid introducing
signal-to-noise (S/N) dependent biases, the nmax is kept constant for all galaxies rather than being allowed to vary according to size or brightness. For faint
sources, this means the higher-order coefficients will be noisy but unbiased.
Once we have a cartesian Shapelets representation of both the sheared, flexed
and PSF convolved circular model and of the PSF convolved object we want
to fit, we convert them both into polar Shapelets as described in KK06. Polar
Shapelets are simply cartesian Shapelets of order n = a + b expressed in polar
coordinates, resulting in polar Shapelets of order n with angular order m ≤ n
and n + m even. The construction of these is discussed in Refregier (2003) and
further investigated in Massey & Refregier (2005) and Massey et al. (2007b). In
our implementation, the purpose of converting the model and object Shapelets
expansions into polar Shapelets is to avoid truncation effects. F flexion, shear
and G flexion operators acting on a polar Shapelet of order (n, m) generate
terms at order (n ± 1, m ± 1), (n ± 2, m ± 2) and (n ± 3, m ± 3) respectively.
By truncating the polar Shapelets expansion in the diamond shape shown in
Figure 3.1, i.e. only including terms up to order (Nc , 0), (Nc −1, ±1), (Nc −2, ±2)
and (Nc − 3, ±3) in the fit, we minimise truncation effects from the mixing of
orders.
The model is fit to each source using least-squares, resulting in a simultaneous estimate for the ellipticity (s1 , s2 ), the F flexion (f1 , f2 ), and the G flexion
(g1 , g2 ). As explained in KK06, the errors on the Shapelet coefficients are derived from the photon noise and propagated through the χ2 function for this
fit. By differentiating the χ2 at the best-fit, we obtain the covariances between
the fit parameters, resulting in proper error estimates.
In essence, the main development since KK06 is the addition of flexion to
the model and the inclusion of higher order polar Shapelets (m = ±3) in the fit.
3.4
Testing the pipeline
Several aspects of the pipeline, such as the choice of scale radius β, the method
of PSF correction and the effect of noise on ellipticity estimates, have been
thoroughly tested in KK06 as part of the development of the KK06 pipeline. In
this section we will therefore focus on testing the recovery of shear and flexion.
3.4.1
GREAT08
As participants in the GRavitational lEnsing Accuracy Testing 2008 (GREAT08)
challenge (Bridle et al., 2009, 2010), we were able to contrast the shear measurement capability of the KK06 pipeline with that of the MV pipeline under
different observing conditions. The challenge provided a large number of simulated sheared and pixelated galaxy images with added noise. The performance
of the different shape measurement pipelines taking part was quoted in terms
69
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Table 3.1 The different branches of FLASHES. Four parameters are varied
between the branches according to this table.
Fiducial
Shape branch
Profile branch 1
Profile branch 2
S/N branch 1
S/N branch 2
S/N branch 3
PSF branch
Intrinsic shape
Galaxy profile
S/N
PSF
Round
Elliptical
Round
Round
Round
Round
Round
Round
Gaussian
Gaussian
Exponential
de Vaucouleur
Gaussian
Gaussian
Gaussian
Gaussian
100
100
100
100
8
20
40
100
Round
Round
Round
Round
Round
Round
Round
Elliptical
of a quality factor, or Q-value, defined as
Q=
m
h(hγij
kQ σ 2
t i
2
− γij
j∈k ) iikl
(3.12)
2
2
where σ 2 = σstat
+ σsyst
is a combination of the statistical spread in the simulations and the expected systematic errors. The superscripts m and t denote
measured and true (input) values respectively and γij is the shear component i
for simulation image j. The differences between the measured and true shears
are averaged over different input shear sets k and simulation branches l. The
whole expression is normalised by kQ so that a method with a purely statistical
spread in the measured shears will have a Q-value of kQ which is the level desirable for future surveys. In the case of GREAT08, kQ = 1000 and σ 2 = 10−7 ,
giving a Q-value nominator of 10−4 . With this definition current methods, like
those that took part in the earlier Shear TEsting Programme (STEP) (Heymans
et al., 2006a; Massey et al., 2007a), generally achieve 10 . Q . 100. This is
sufficient for current weak lensing surveys. For a more in-depth discussion on
the Q-value and its relation to the STEP parameters m (multiplicative bias)
and c (additive bias), we refer to Kitching et al. (2008).
The overall Q-value was similar for the KK06 and the MV pipelines, both
in the LowNoise Blind competition (Q ∼ 20) and in the RealNoise Blind (Q ∼
25). When broken down into the separate observing condition branches some
differences became apparent. In general the MV pipeline did exceptionally well
under “good” observing conditions, e.g. for the high S/N branch or for well
resolved galaxies. Our own simulations described in the next section will further
test the dependence of the MV performance on different observing conditions.
3.4.2
FLASHES
As there is no flexion simulation set publicly available to date, we create our
own FLexion And SHEar Simulations (FLASHES). FLASHES are very similar
to the GREAT08 simulations in several respects. First, each galaxy is generated
on a grid, ensuring that there is no overlap of objects, thus avoiding deblending
issues. Second, each simulation image consists of 10000 such objects. Third,
each galaxy is generated through the following sequence: (i) simulate a sheared
and/or flexed (elliptical) galaxy model (depending on simulation branch); (ii)
70
3.4. TESTING THE PIPELINE
convolve with the PSF; (iii) apply the noise model. Four parameters are varied
between the different FLASHES branches; the intrinsic galaxy shape, the light
profile of the galaxies, the S/N of the galaxies and the shape of the PSF. These
parameters are detailed below and summarised in Table 3.1.
Simulation details
All parameters except for the intrinsic ellipticities are kept constant in each
simulation image, and all images are created using Monte-Carlo selection. This
is very similar to the process described in KK06 and in Bridle et al. (2010), but
with the photon trajectories being influenced by flexion as well as by shear if
required.
1/n
The galaxies are modeled with Sérsic intensity profiles Igal ∝ e−kr
(Sérsic,
1968) with varying indices n. A Sérsic index of n = 0.5 is a Gaussian profile
whilst n = 1 and n = 4 are exponential and de Vaucouleur profiles respectively.
Half of the FLASHES branches have intrinsically round galaxies whilst the other
half consists of galaxies with intrinsic ellipticities picked randomly from the
ellipticity distribution of objects in the COSMOS survey. There is no intrinsic
flexion included. The PSFs applied to the simulations are nearly Gaussian with
a Moffat profile IPSF = (1 + r2 /a2 )−m of index m = 9. In half of the branches,
the PSF is round whilst in the other half it is elliptical in the horizontal direction
with e1,PSF = 0.02. To mimic the properties of the COSMOS survey, we use a
PSF FWHM of 2.1 pixels and a PSF convolved galaxy size of 5.8 pixels which
is the typical size of the galaxies we use in our COSMOS analysis. Finally
there are four S/N branches, with S/N being defined as Flux/(Flux error). It
is expected that shape measurements will be less accurate at low S/N. For this
reason the MV pipeline applies a S/N cut at 10 in general. The low S/N branch
of 8 is designed to test how biased measurements are below this cut. The high
S/N branch of 100 tests biases under near-perfect noise conditions.
The strength of the different distortions is picked randomly but with the
following maximum values: |γ1,2 | ≤ 0.05, |F1,2 | ≤ 0.008 pixel−1 and |G1,2 | ≤
0.02 pixel−1 . The value of each distortion component is kept constant across
each image, but differs between the 30 images in each set, and between different
sets.
Simulation results
To estimate the average distortion on each image we use two different techniques:
a weighted average with weights inversely proportional to the measurement
errors, and Convex Hull Peeling (CHP). CHP is an efficient way of eliminating
outliers and is essentially a 2D median. A convex hull, in the context of a point
cloud in e.g. the γ1 , γ2 plane, is the minimal convex set of points containing that
point cloud. Thus if all the points in this convex set were connected, a polygon
containing the entire point cloud would be produced. By peeling away convex
hulls, outliers are removed from the point cloud and the remaining points may
be averaged over to produce a mean unaffected by extreme results. This is the
averaging technique we used in GREAT08 where we peeled away 50% of the
measurements before averaging.
We employ the parameters m and c as used in STEP (Heymans et al., 2006a;
71
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Figure 3.2 The multiplicative bias on the first component for each of shear, F
flexion and G flexion. The purple stars represent shear, pink circles represent F
flexion and green triangles represent G flexion. The symbols and solid lines show
the weighted averages whilst the dashed lines show the CHP average. This is from
running the MV pipeline on FLASHES, with nmax = 10. For the results for m2 ,
c1 and c2 please refer to Appendix 3.A.
Massey et al., 2007a) to quantify the performance of the software:
hγimeasured i − γiinput = mi γiinput + ci
(3.13)
and similarly for the flexions, where i = 1, 2 represents the shear component.
A negative multiplicative bias mi thus indicates that the distortion is generally
underestimated. A systematic offset ci may be caused by e.g. insufficient PSF
correction.
In Figure 3.2 we show the multiplicative bias of the first component for
each of shear, F flexion and G flexion as a function of the different simulation
branches (please refer to Appendix 3.A for the remaining bias components). For
these results we use a Shapelets order of nmax = 10. We use SExtractor (Bertin
& Arnouts, 1996) to detect the objects in each simulation, which we then split
into clean star and galaxy catalogues by matching to the input catalogue. We
keep all properties apart from the one under investigation fixed at a fiducial
72
3.4. TESTING THE PIPELINE
value to allow for a fair comparison. The fiducial simulations in Figure 3.2 have
intrinsically round, high S/N galaxies with Gaussian light profiles and a circular
PSF.
From the above figure it is clear that both flexions are likely to be underestimated, especially for higher Sérsic indices. The bias is also strongly S/N
dependent, particularly for the F flexion. Thus a S/N cut is essential to improve the performance of the MV pipeline, but a bias correction should also be
implemented. Investigating the dependence of m on S/N further, we are able
to fit the following power-law to our FLASHES results:
m1,2 = −a(S/N)−b
(3.14)
where a and b are constants as follows: for shear (aγ , bγ ) = (6.48, 1.78); for
F flexion (aF , bF ) = (2.30, 0.48); for G flexion (aG , bG ) = (0.36, 0.13). We will
apply this bias correction to our shape measurements in COSMOS, but since
FLASHES have been tailored for this particular data the biases should be explored further before being applied to other surveys.
3.4.3
Galaxy-galaxy simulations and bright object removal
At the core of weak galaxy-galaxy lensing is the averaging of the signal in rings
centered on lenses consisting of single galaxies rather than a galaxy cluster.
This type of analysis is robust as numerous systematics, induced by e.g. the
PSF, cancel out. Different systematics may however be introduced, such as the
light from the central, often bright, lens causing biases in the shape measurements as discussed in Rowe (2008). To study this possible effect, we created
simple simulations with sources placed in evenly spaced rings around a central
lens. Apart from source numbers and positions, the simulations were created
in the same way as FLASHES. The S/N of the images was set to 200 to ensure
minimum bias, and for the same reason the source galaxies had Gaussian light
profiles. The size and profile parameters of the lens were varied between images.
The results for a lens with an exponential profile are shown in Figure 3.3
(black stars), where we have used nmax = 10. We recover a near-perfect average
signal in each source circle far from the lens. However, close to the lens the shear
and G flexion are slightly affected, but, more strikingly, the F flexion is severely
overestimated. The conclusion we draw from this is that bright objects can add
significantly to the F flexion signal, due to light ‘leaking’ into the Shapelets
fitting radius. This causes the pipeline to detect a source light profile that is
skewed towards the lens, and interpreting it as extra F flexion.
Our solution is to remove any bright objects sufficiently close to the source
being fit using a technique we introduce here as Bright Object Removal (BOR).
Before decomposing a galaxy image into Shapelets, we identify any bright objects that could conceivably intrude using selection criteria based on distance
between the two objects, Shapelets fitting radius of the source, and size and
brightness of the intruding object. We then create Sérsic models of the intruding objects using GALFIT (Peng et al., 2002) and subtract these models from
the Shapelets stamp before doing the fitting. It works well in these simulations,
provided one is careful with the parameters given to GALFIT as input. The sky
background value given to GALFIT is particularly important as a small error
in this estimate results in postage stamp artifacts when the stamps are subtracted from the original image. In Figure 3.3 we also show the results if BOR
73
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Figure 3.3 The shear (top panel), F flexion (middle panel) and G flexion
(bottom panel) results from galaxy-galaxy lensing simulations, with and without
Bright Object Removal (BOR). The black stars represent the tangential signal
without BOR and the green triangles represent the same measurement corrected
using BOR. The dashed pink line is the input signal and the purple circles are
the cross-signal, which is expected to vanish, for the uncorrected measurements.
Here, the FWHM of the lens is 14 pixels. Note the slight underestimation of the
shear, the slight overestimation of the G flexion and the massive overestimation
of the F flexion in the innermost bins when BOR is not applied.
is switched on whilst the rest of the analysis is kept identical to the previous run
(green triangles). There is still some excess F flexion signal around 44 pixels,
indicating that there may be some residual light remaining, but this excess is
smaller than for the uncorrected measurements. This provides a confirmation
that the measured signal reproduces the input signal well if BOR is applied,
and no new artifacts are introduced. We note, however, that the leaking light
does not affect the cross component of the measurements, with the consequence
that this effect cannot be detected through the usual systematic checks.
3.5
COSMOS analysis
Goldberg & Bacon (2005) made a first detection of galaxy-galaxy flexion using the ground-based DLS, proving that flexion can indeed be detected, but
ultimately they were hampered by the small size of their sample, the lack of
74
3.5. COSMOS ANALYSIS
redshifts and the extra blurring caused by the atmosphere. Therefore we choose
the space-based Cosmic Evolution Survey (COSMOS, Scoville et al. (2007)) as
the first real dataset for the MV pipeline. Thanks to the depth of this survey
we will have access to more than a thousand times as many lens-source pairs
as Goldberg & Bacon (2005) did. More than half of these have photometric
redshifts meaning that the division of the sample into lenses and sources will
be more accurate. The intention is to provide independent confirmation that
galaxy-galaxy flexion has high enough S/N to be detected, and that the software presented in this Chapter is able to do it. We will also look closer to the
lens than previous analyses and attempt to combine shear and flexion to give
constraints on galaxy dark matter halo profiles.
3.5.1
The COSMOS data set
COSMOS is to date the largest contiguous field imaged by the Hubble Space
Telescope (HST) with a total area of 1.64 deg2 . The 579 tiles were observed in
F814W (I-band) by the Advanced Camera for Surveys (ACS) between October
2003 and November 2005. Each tile consisted of 4 dithered exposures of 507 seconds each (2028 seconds in total) with about 95% of the survey area benefiting
from the full 4 exposures.
We use the images reduced by Schrabback et al. (2010) (hereafter S10) and
also their catalogues for stars and galaxies, detected using SExtractor. There
are a total of 446 934 galaxies with i814 < 26.7 in the mosaic catalogue, almost
half of which have COSMOS-30 photometric redshifts from Ilbert et al. (2009).
These redshifts are magnitude limited and cover the entire COSMOS field to a
depth of i+ < 25.
3.5.2
Data analysis
Galaxy-galaxy lensing is less affected by the problems plaguing cosmic shear
analyses, since most systematic shape distortions induced by instruments cancel
out when azimuthally averaged. Still, we have to be careful not to introduce new
systematic effects or biases, so correcting for the PSF and the charge-transfer
inefficiency (CTI) (e.g. Rhodes et al., 2007; Massey et al., 2010) is important.
We use all galaxies with redshifts of z < 0.6 as lenses. At higher redshifts
the light from the lensing galaxies becomes difficult to account for due to the
small angular separation on the sky, as explained further in Appendix 3.C.
Furthermore, imposing a lens redshift cut will ensure that the vast majority of
sources are truly background objects.
Our source catalogue is comprised of all objects with a shape measurement.
We clean this catalogue using a series of conditions on size and measured shape,
detailed in Appendix 3.B1, the most important of which is to remove objects
with S/N < 10. Roughly two-thirds of the remaining sources have individual
COSMOS-30 photometric redshifts assigned to them. For the remaining third
(redshift bin 6 in S10) we use the estimated redshift distribution employed by
S10 to assign mean angular diameter distance ratios (Ds /Dls ) to each lenssource pair. We are finally left with 216 873 sources, corresponding to a source
density of ∼ 37 arcmin−2 . For the pairs we use, the median lens redshift is
zlens = 0.27 and the median source redshift is zsource = 0.98.
75
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Despite the excellent space-based resolution, we need to correct the galaxy
shapes for the instrumental PSF. The ACS PSF is known to fluctuate both
spatially and temporally (e.g. Rhodes et al., 2007; Schrabback et al., 2007), a
variation mostly driven by changes in telescope focus caused for example by
the breathing of the telescope. We can map the PSF using stars, but, in highgalactic latitude ACS fields typically only ∼ 10 − 20 stars are present. This
number is too low for the standard approach of a polynomial interpolation.
Instead, we closely follow the analysis of S10, who conducted a principal component analysis (PCA) of the ACS PSF variation as measured in dense stellar
fields. Details for the Shapelets implementation of PCA may be found in Appendix 3.B2.
A challenge with using CCD detectors in space is that they are not protected
by the atmosphere. Exposed, they continuously get bombarded by radiation,
causing deterioration of the chip surface. The imperfections created in this
way act as charge traps which causes inefficiency in the moving of electrons to
read-out. This effect is known as CTI (e.g. Rhodes et al., 2007; Massey et al.,
2010). As the electrons get trapped and then released at a later point, charge
trails following objects are created in the read-out direction, effectively causing
a spurious shear signal in that direction. Our correction for CTI again closely
follows S10, who derive parametric corrections for the change in polarization
for both galaxies and stars. For more details on this correction, please refer to
Appendix 3.B3.
Once corrected, the galaxy-galaxy shear and flexion signals are weighted
according to the geometric lensing efficiency of each lens-source pair. In the
case of flexion there is an extra scale dependence of the signal. For the NavarroFrenk-White (NFW) profile (Navarro, Frenk, & White, 1996), the strength of
the shear signal scales as
Dl Dls
γNFW ∝
(3.15)
Ds
where Dl , Ds and Dls are the angular diameter distances to the lens, to the
source, and between lens and source respectively (Wright & Brainerd, 2000).
The flexion signals scale as
FNFW , GNFW ∝
Dl2 Dls
Ds
(3.16)
(B06). We therefore weight the signals accordingly, scale them to a reference
lens and source redshift and compute the weighted average in 25 logarithmic
distance bins (see Appendix 3.B4 for details). We use a reference lens redshift
of zl,ref = 0.27 since that is close to the effective median redshift of our lenses,
and a reference source redshift zs,ref = 0.98. To estimate the errors on each
bin and the covariances between them, we use 5000 bootstrap resamples of our
source catalogue.
3.5.3
Results
The results from our galaxy-galaxy lensing analysis of the full COSMOS lens
and source sample is shown in Figure 3.4. In the left panel we plot the shear
results as a function of physical distance from the lens. These results agree
very well with those from S10 (see Appendix 3.D), providing an independent
76
Figure 3.4 The galaxy-galaxy lensing results for the COSMOS data, using a
maximum Shapelets order of nmax = 10. Black solid points represent the tangential signal and green triangles represent the cross term. The pink circles represent
the tangential signal if we apply the multiplicative bias correction implied by
FLASHES. Note that the SIS and NFW profiles have been fitted to the shear
data and then translated into predictions for F and G curves.
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
consistency check. To this we fit a Singular Isothermal Sphere (SIS) profile and
a tentative NFW profile. Due to the dependence on mass and redshift of the
mean concentration parameter (e.g. Duffy et al., 2008), the NFW profile is only
an indication when the spread in lens masses and redshifts is as great as it is
in the above sample. Splitting the sample up into redshift and/or mass bins
would increase the confidence in the fit, but decrease the S/N of the signals
significantly.
The middle and right panels show the F and G flexion results respectively,
for the same lenses and sources. The profiles plotted here are identical to those
plotted in the shear panel but translated into predictions for F and G, as opposed
to fitted to the flexion data directly. The F flexion has a tendency to be overestimated compared to the predicted profile from the shear, and we investigate this
discrepancy further in the following sections. We also note that we measure a
G flexion that is very noisy and consistent with zero. This is most likely caused
by lack of information in higher m-order Shapelets for fainter sources, and we
choose to use only shear and F flexion in the continued analysis.
Also shown in pink circles in Figure 3.4 is the signal if we apply the multiplicative S/N-dependent bias correction implied by FLASHES. With this correction, the F flexion signal becomes slightly higher. This bias correction is
only based on one specific set of simulations and is thus rather preliminary;
this is also indicated in the increased size of the error bars. Correcting for the
morphology-dependent bias requires accurate source morphology determination.
Using the photometric galaxy type estimates from Ilbert et al. (2009) as an indicator of morphology we find that < 5% of our source sample consists of likely de
Vaucouleur objects. This type estimate is not accurate enough to implement a
morphology bias correction, but simply removing the de Vaucouleur candidates
we identified makes little difference to our results. It is clear, however, that
an accurate bias calibration of the flexion amplitude, taking into account both
source S/N and brightness profiles, requires further investigation.
3.5.4
Removing bright objects
We now explore the tendency of the F flexion points to lie above the predicted
profiles. As shown in Section 3.4.3, the shape measured may be affected by
bright objects nearby. We implement BOR in our COSMOS analysis to see the
effect on real data. For very well resolved objects, prominent spiral arms and
other complications cause GALFIT to reject the single Sérsic profile fit. Removing
these objects, and the residual light from the wings of the profile (Figure 3.3),
requires a more sophisticated model. For now we are only interested in a rough
indication of the impact this light leakage has on a galaxy-galaxy signal so we
will not correct for the few large objects in this Chapter. However, as shown
in Figure 3.5, the correction to the innermost F flexion bin is non-zero even
without accounting for the very large objects. The shear is largely unaffected,
but for flexion analyses in future deeper and larger surveys it will be important
to correct for this effect.
3.5.5
The effect of substructure
Since flexion is more sensitive to the underlying mass distribution on small scales
than shear is, we expect it to respond differently to the presence of substructure
78
3.5. COSMOS ANALYSIS
Figure 3.5 Comparison between the galaxy-galaxy shear and flexion signals
with and without Bright Object Removal, showing the non-zero correction to the
innermost F flexion bin (corresponding to roughly 40 px in Figure 3.3). Black solid
points represent the difference between the signals before and after correction, with
the F flexion in units of kpc−1 , whilst green circles represent the cross term.
in galaxy haloes. To test whether this has any impact on our analysis we
take a galaxy-size SIS halo (see B06, for shear and flexion expressions) and
populate it with subhaloes, allowing 20% of the mass to be in substructure.
The total mass of the halo is 1012 h−1 M⊙ and the galaxy is placed at z =
0.35 with Dl /Dls = 0.5. We spread the substructure mass over 100 subhaloes,
randomly distributed according to an SIS density profile. Finally we average
the azimuthally averaged signal over 100 such galaxies. Now, subhaloes are
generally stripped. To approximate this we use a Truncated SIS (TSIS) profile
for the subhaloes (see Hoekstra et al., 2004, for constraints on parameters). The
TSIS convergence is given by
!
θE
θ
κ(θ) =
1− p
(3.17)
2θ
θ2 + θS2
where θE is the Einstein radius and θS is a truncation scale where the profile
steepens. On small scales (θ ≪ θS ) the TSIS behaves like an SIS but at large
scales (θ ≫ θS ) the profile decreases as θ4 . The TSIS shear is given in Schneider
& Rix (1997) and the flexions are
θ3
θE
− 1 eiφ
(3.18)
F (θ) = 2
2θ
(θ2 + θS2 )3/2
and
G(θ) =
θE
2θ3
3θ4 + 12θ2 θS2 + 8θS4
3θ + 8θS −
ei3φ
(θ2 + θS2 )3/2
(3.19)
79
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Figure 3.6 Simulated shear and flexion signals azimuthally averaged in galaxy
haloes with and without TSIS subhaloes. Grey stars, circles and triangles represent the binned shear, F flexion and G flexion respectively. Purple, pink and green
lines represent the shear, F flexion and G flexion signal if the halo is a smooth SIS
(dashed). The solid lines are an SIS profile as fitted to the shear data points in
a simulated galaxy containing TSIS subhaloes and translated into predictions for
the flexions.
where φ is the position angle of the background source. Using the parameters
above and a truncation scale θS = 2 arcsec for the subhaloes we get the results
shown in Figure 3.6. The shear profile fit is pulled down slightly compared to
a smooth halo but the flexions are not similarly affected. Due to the substructure the flexions are more scattered, but the overall trend is for the points to
follow the smooth profile, or even slightly above in the F flexion case. Thus the
flexions seem overestimated compared to the shear fit. We stress however that
the fraction of substructure used in this test (20%) is high to exaggerate the
effect. The test does show that substructure may affect the flexions differently
to the shear, but its influence is likely less than the excess currently observed in
COSMOS.
80
3.5. COSMOS ANALYSIS
Figure 3.7 The correlation matrix between the shear and flexion bins, using
5000 bootstrap resamples. Please note the scale; to display the minute variations between off-diagonal elements we have artificially set the diagonal elements
(dark green) only to 0.1, whilst all other elements are unscaled and normalised to
diagonal elements of 1.0 as is customary.
3.5.6
Profile determination
One of the most interesting potential uses of flexion is as an aid to shear in
determining the inner density profiles of dark matter haloes. The two signals
are sensitive to the underlying density profile on different scales, so combining
the two will give us tighter constraints than either on their own. To combine
the shear and flexion signals we have to take any correlation between them
into account. B06 assumed that the shear and flexion measurements would be
uncorrelated. Here we confirm this assumption through the correlation matrix
between the shear and flexion bins, using 5000 bootstraps, shown in Figure 3.7.
This implies that it is trivial to combine the shear and flexion information to
find the profile of an average lens. We use the F flexion in conjunction with
the shear to fit density profiles to the measured signal. For this purpose we
try two different families of profiles: the power-law and the NFW. Our general
power-law is defined as
γ = −Ad−n
(3.20)
with d the distance from the lens, and the amplitude A and the index n free
parameters. An index of n = 1 would be equivalent to an SIS. The above
81
Figure 3.8 Joint profile constraints using shear and F flexion. The top (bottom)
panel shows the results for the power-law (NFW) fit. Purple (thin solid) lines
represent shear and green (dashed) represent F flexion. The contours show the
67.8%, 95.4% and 99.7% confidence limits respectively in terms of constant ∆χ2
(2.30, 6.17 and 11.8 respectively). The white (thick) contour marks the joint
confidence limits. The grey-scale is logarithmic in χ2 .
3.6. DISCUSSION AND CONCLUSIONS
expression is easily differentiated to give the F flexion
F = (n − 2)Ad−n−1 .
(3.21)
The expressions for the NFW profiles are somewhat more complicated but
they are given in full in Wright & Brainerd (2000) and B06 for shear and flexion
respectively. Here we leave the virial radius M200 and the concentration c as
free and independent parameters. We fit the power-law and NFW profiles to
the inner 100 kpc only as this is the region where F flexion becomes important
and the shear profile is not affected by halo-halo contamination.
The top panel in Figure 3.8 shows that both the shear and the F flexion are
consistent with an SIS (n = 1), although together they prefer a slightly lower
power-law index of n = 0.73+0.40
−0.43 . The bottom panel shows that it is difficult to
constrain the NFW concentration if it is left completely unrestricted. This analysis with two free and independent parameters is not completely representative,
however, since simulations indicate a fixed mean mass-concentration relationship (Duffy et al., 2008). It is also important to note that the average profile we
constrain here is a composite of lenses in a large redshift range. Detection at
the high end of the redshift distribution tend to be biased towards intrinsically
brighter objects than at the low end. We also combine measurements from lenses
of different sizes and morphologies. Nonetheless, combining shear and F flexion
does provide tighter constraints than shear alone on the density profiles, and this
is an important proof of concept. The resulting mass estimate for the average
11 −1
lens in COSMOS from the combined NFW fit is M200 = 2.12+3.60
h M⊙
−1.09 × 10
+7.04
with a concentration of c = 4.82−3.16 .
3.6
Discussion and conclusions
We have shown a significant detection of galaxy-galaxy F flexion for the first
time with Shapelets using the space-based COSMOS data set. We used this
flexion signal in conjunction with the shear to constrain the average density
profile of the galaxy haloes in our lens sample. We found a power-law profile
consistent with an SIS. Furthermore, we showed that the inclusion of F flexion
provides tighter constraints on both power-law and NFW profiles, an important
proof of concept.
The galaxy-galaxy F flexion signal measured in COSMOS is slightly higher
than expected from the shear signal, especially if we apply the multiplicative
bias correction. There is however no indication from the cross term that there
are systematics present. The discrepancy could be partly due to insufficient
nearby object light removal, but this is unlikely to explain the full offset. Substructure in galaxy haloes may cause excess F flexion compared to what the
shear measures. However, a large fraction of the galaxy halo mass has to be in
substructure in order for the effect to become significant. We note that Goldberg
& Bacon (2005) also find shear and F flexion signals that are inconsistent with
each other; the velocity resulting from an SIS profile fit to their F flexion signal
is nearly twice that found using shear. This is qualitatively consistent with our
findings, which leads us to believe that there is something more fundamental
affecting the signal. In the near future we would like to further investigate the
dependence of these discrepancies on lens properties.
83
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
We measure a galaxy-galaxy G flexion signal that is consistent with the
predicted profile, but due to the large measurement errors it is also consistent
with zero. This measurement is a lot noisier than the other two, an effect most
likely caused by the fact that there is less information available in the higher
m-order Shapelets for fainter sources. To measure a G flexion signal we thus
require many well-resolved sources, an extravagance not yet awarded us. Future
large space-based surveys such as EUCLID will enable us to investigate G flexion
further, but for now F flexion is a promising tool in its own right.
The software introduced in this Chapter, the MV pipeline, is able to detect
these higher order lensing distortions. We have shown that in practice, the
Shapelets F flexion measure is affected by light from nearby bright objects and
detailed a way to correct for this effect. This BOR does require further sophistication to account for large, well resolved galaxies, galaxies which are not well
described by the single Sérsic light profile employed here. From the FLASHES
simulations it is clear that there is more work required in order to improve the
accuracy of the F flexion measurements for future surveys. Noise related biases
are particularly significant for this type of shape measure, and we have modeled
these biases in COSMOS.
In the future we hope to measure flexion on a larger survey, enabling us
to reduce the noise so that we can investigate the trend with e.g. redshift and
lens mass. A larger number of sources would also enable us to further tighten
the profile constraints in the inner regions of dark matter haloes where baryons
become important. It is not yet clear how well we can measure flexion on
ground-based data, but surveys like KiDS, CFHTLS and RCS2 should provide
an excellent test-bed.
Acknowledgements
We would like to thank our colleagues Henk Hoekstra, Edo van Uitert and Elisabetta Semboloni at Leiden Observatory, and Peter Schneider at Bonn University, for useful discussions.
Gary Bernstein drew our attention to the possibility of using Convex Hull Peeling for our
averages, and for this we would like to thank him. MV is supported by the European DUEL
Research-Training Network (MRTN-CT-2006-036133). TS acknowledges support from the
Netherlands Organization for Scientific Research (NWO).
APPENDIX 3.A:
FLASHES results
The figures shown in this Appendix complement Figure 3.2 in the main Chapter 3 and provide additional detail on the results from running the MV pipeline
on FLASHES, with nmax = 10. The parameters m and c are defined through
hγimeasured i − γiinput = mi γiinput + ci
(3.22)
and similarly for the flexions, where i = 1, 2 is the component. We use two
different techniques to estimate the average distortion on each image: a weighted
average and Convex Hull Peeling (CHP).
In Figure 3.9 we show the multiplicative bias of the second component for
each of shear, F flexion and G flexion as a function of the different simulation
branches. For these results we use a Shapelets order of nmax = 10. This
bias behaves as the multiplicative bias of the first component (Figure 3.2), as
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3.B. COSMOS DATA ANALYSIS
Figure 3.9 The multiplicative bias on the second component for each of shear,
F flexion and G flexion. The purple stars represent shear, pink circles represent F
flexion and green triangles represent G flexion. The symbols and solid lines show
the weighted averages whilst the dashed lines show the CHP average.
expected. The biases of all distortion measurements, and in particular F flexion,
are severely dependent on S/N and brightness profile.
The additive bias c is minimal for shear and F flexion (see Figures 3.10 and
3.11) indicating that the PSF is either well corrected for or not significantly
affecting these two measurements. For the G Flexion the offset is larger.
APPENDIX 3.B:
3.B1
COSMOS data analysis
Catalogue creation
To maximise the number of lens-source pairs we use all objects with assigned
photometric redshifts as sources, but imposing a redshift cut of z < 0.6 for
lenses. Additionally we use sources without individual redshifts (S10 redshift
bin 6), assigning mean angular diameter distance ratios (Ds /Dls ) to these lenssource pairs according to the estimated redshift distribution employed by S10.
We then weight all pairs with their individual lensing efficiency, similar to the
85
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Figure 3.10 The additive bias on the first component for each of shear, F flexion
and G flexion. As before, the purple stars represent shear, pink circles represent F
flexion and green triangles represent G flexion. The symbols and solid lines show
the weighted averages whilst the dashed lines show the CHP average.
weighting scheme in e.g. Mandelbaum et al. (2006b) (see Appendix 3.B4). This
downweights pairs that are close in redshift and naturally removes pairs where
the “source” is in front of the “lens”. To the source catalogues we apply the
following cuts:
• S/N > 10. This cut is important as the F flexion measurement in particular gets heavily biased towards low S/N (see Section 3.4.2).
• If the centroid cannot be determined accurately the Shapelets decomposition will be inferior. Therefore objects where the code is forced to move
the centroid compared to the one estimated by SExtractor by more than
half a pixel are excluded.
• The summed power in constant m of the polar Shapelets provides an
indicator of the Shapelet fit being affected by a neighbouring object. If the
fractional power is particularly high at high orders the object is excluded
(see KK06, for more details).
86
3.B. COSMOS DATA ANALYSIS
Figure 3.11 The additive bias on the second component for each of shear, F
flexion and G flexion. As before, the purple stars represent shear, pink circles
represent F flexion and green triangles represent G flexion. The symbols and solid
lines show the weighted averages whilst the dashed lines show the CHP average.
• If the FWHM or scale radius of the object is too small compared to the
scale radius of the PSF the object is excluded.
• If γ 2 > 1.4, F 2 > 3.0 arcsec−1 or G 2 > 6.6 arcsec−1 then the object is
excluded. These numbers are based on the measured distributions and the
cuts are applied to remove outliers with very noisy shape measurements.
• Finally, we remove faint objects with an assigned photometric redshift of
z < 0.6 that have a prominent secondary peak at z2nd > 0.6, as discussed
in S10.
3.B2
PSF interpolation
The ACS PSF fluctuates both spatially and temporally (e.g. Rhodes et al.,
2007; Schrabback et al., 2007), a variation mostly driven by changes in telescope
focus caused for example by the breathing of the telescope. We can map the
PSF using stars, but in high-galactic latitude ACS fields typically only ∼ 10 −
87
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Figure 3.12 CTI-induced residuals on the stellar Shapelet coefficients s20 (left)
and s03 (right) in an example star field. The black stars show the mean of the
coefficients as a function of stellar flux after subtraction of a spatial third-order
polynomial model derived from bright stars to separate PSF and CTI effects.
Each coefficient has been scaled to a reference number of ytrans = 2048 parallel
readout transfers. The purple curves show the parametric CTI model, jointly
determined from 700 stellar field exposures. The horizontal dashed line indicates
an offset corresponding to the mean CTI model for the bright stars used for the
polynomial interpolation. The green triangles indicate the corrected coefficients
after subtraction of the CTI model.
20 stars are present. This number is too low for the standard approach of a
polynomial interpolation. Instead, we closely follow the analysis of S10, who
conducted a principal component analysis (PCA) of the ACS PSF variation as
measured in dense stellar fields. They found that ∼ 97% of the PSF variation
can be described with a single parameter (the first principal component). This
parameter is related to the HST focus position, and we therefore dub it ‘focus’1
Here we make use of the S10 measurement of the HST focus in all COSMOS
exposures and the investigated stellar field exposures. We also obtain Shapelets
versions of the focus-dependent S10 PSF models, by decomposing the dense stellar field stars into Shapelets and interpolating between them with polynomials
which are varied both spatially and with different powers of the focus principal
component coefficient. From these models and from the COSMOS focus estimates we then compute a Shapelets PSF model for each COSMOS exposure,
which we then combine to obtain a model for the stacked PSF at all galaxy
positions.
3.B3
CTI correction
Our correction for CTI again closely follows S10, who derive parametric corrections for the change in polarization for both galaxies and stars. The correction
for stars is important in order to measure the actual PSF, independent of the
1 The capturing of small additional variations beyond focus was relevant for the cosmic
shear analysis of S10. Here we can safely ignore these minor additional effects. Galaxy-galaxy
lensing is much less sensitive to PSF anisotropy residuals as they cancel out to first order.
88
3.B. COSMOS DATA ANALYSIS
non-linear CTI effects. In the stellar field analysis we therefore correct the PSF
cartesian Shapelet coefficients for CTI before generating the PCA PSF model.
In order to estimate the influence of CTI on the different Shapelet coefficients,
we follow S10 and spatially fit each coefficient within one exposure with polynomials. Due to the limited depth of the charge traps, CTI is non-linear, and has
a larger relative impact on faint sources than on bright ones. The CTI effect can
thus be estimated from the flux-dependent residuals, after the polynomial model
has been used to subtract both the flux-independent PSF and the flux-averaged
CTI signal.
Figure 3.12 shows these residuals as a function of stellar flux for the stellar
Shapelets coefficients s20 and s03 in one example stellar field. Here the residuals
were scaled to the same number of readout transfers (2048). The CTI effect on
the coefficients is clearly visible (black stars), but with our power law model
(curve) it can be well corrected for (green triangles). The model is fit simultaneously from all 700 stellar fields as a function of stellar flux, sky background,
time and number of readout transfers (see S10). CTI affects object shapes in
the readout direction, which also after drizzling roughly matches the y-direction.
Thus CTI residuals are expected to be roughly symmetric about the y-axis and
hence vanish for coefficients sab with odd a. In the drizzled images the readout
direction is up for the upper and down for the lower chip and the CTI trails
occur in the opposite directions. This leads to a sign switch for coefficients with
basis functions that are not symmetric about the x-axis (odd b), and we have
taken this into account for s03 in Figure 3.12. We have detected (and modeled)
a significant signature of CTI on the following stellar Shapelets coefficients: s00 ,
s02 , s03 , s04 , s05 , s20 , s21 , s22 , s40 , and s60 .
The correction of galaxy shapes for CTI again closely follows S10. Here we
fit power-law corrections to the shear and (now in addition) flexion estimates
as a function of galaxy flux, flux radius, sky background, time, and number of
readout-transfers. Note that Massey et al. (2010) introduced a more advanced
CTI correction scheme operating directly on the pixel level. This is expected to
yield higher precision, enabling for example the correction of the s01 component,
which cannot be estimated with our method due to its degeneracy with a simple
shift in object position. However, we are confident that our correction scheme
is sufficiently accurate for the analysis presented here, in particular as potential
residuals cancel to first order for the azimuthally averaged galaxy-galaxy lensing
signal.
3.B4
Signal computation
For the Navarro-Frenk-White (NFW) profile (Navarro, Frenk, & White, 1996),
the strength of the shear signal scales as
γNFW ∝
Dl Dls
Ds
(3.23)
where Dl , Ds and Dls are the angular diameter distances to the lens, to the
source, and between lens and source respectively (Wright & Brainerd, 2000).
The flexion signals scale as
FNFW , GNFW ∝
Dl2 Dls
Ds
(3.24)
89
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
(B06). We therefore weight the signals accordingly, scale them to a reference
lens and source redshift and compute the weighted average in 25 logarithmic
distance bins as follows:
P
Eγt,i wγt,i
hγt i = P
(3.25)
wγt,i
and similar for the flexions, with the shear estimator and weight
Eγt,i = γt,i
ηi
ηref
−1
where
η=
wγt,i
1
= 2
σγ,i
ηi
ηref
2
(3.26)
Dl Dls
Ds
(3.27)
2
2
is the geometric lensing efficiency and σγ,i
= σγ,intr,i
+ σγ21 ,i + σγ22 ,i is the error
on the shape measurement with σγ,intr the intrinsic shear noise. By contrast we
use the following F flexion estimator and weight:
EFt,i = Ft,i
Dl,i ηi
Dl,ref ηref
−1
wFt,i =
1
2
σF
,i
Dl,i ηi
Dl,ref ηref
2
(3.28)
and similarly for the G flexion.
APPENDIX 3.C:
High redshift results
As specified in the main Chapter 3, the lens catalogue we use has a redshift cut of
z < 0.6. This is to avoid having to go too close to the lens on the sky in order to
see a flexion signal. Within an angular radius of 2 arcsec we have low confidence
in the results; we are simply too close to the lensing galaxies and it becomes
difficult to account for effects induced by the lens light. BOR corrects for light
leakage at larger radii, but the correction is most likely incomplete very close
to the lens due to deviations from a smooth Sérsic profile. For objects beyond
our lens sample, the median redshift is close to 1.0. At this redshift the angular
distance limit of 2 arcsec on the sky corresponds to a physical distance of about
17 kpc. The F flexion falls off to low values already at about 20 kpc for a typical
galaxy, so we are left with a very low signal within a narrow ring around the
lens. Imposing the redshift cut of z < 0.6 on lenses gives us a median lens
redshift of z = 0.27 at which the inner limit corresponds to 9 kpc, leaving a
wider distance interval in which we can investigate the F flexion signal.
In Figure 3.13 we show the galaxy-galaxy signal for the high redshift sample,
i.e. for lenses with z > 0.6. The bins that are within 2 arcsec of the average
lens in this sample, and which are most likely contaminated by lens light, are
marked with dotted lines. The F flexion signal outside of this limit does agree
well with the profile predicted by the shear, but falls off quickly.
APPENDIX 3.D:
Comparison with KSB
We compare our galaxy-galaxy shear signal to the one we get using the shears
from S10, using all the cuts normally applied in each analysis so that only
90
Figure 3.13 The galaxy-galaxy lensing results from running the MV pipeline on
the COSMOS data, with nmax = 10. Black solid points represent the tangential
signal and green triangles represent the cross term. Open circles with dotted error
bars are bins that are too close to the lens on the sky. Please note that the SIS
and NFW profiles have been fitted to the shear data and then translated into
predictions for F and G curves.
3. GALAXY DM HALOES IN COSMOS WITH FLEXION
Figure 3.14 A comparison between the shears used in this Chapter and the
ones used in S10. Black points (green triangles) show the difference between the
tangential (cross) shear values in this Chapter and those produced by a KSB
pipeline for S10.
common objects are used. The bias correction described in their paper is incorporated in their shears whilst our measurements have no correction applied.
However, due to our S/N cut (see Appendix 3.B1) their correction is always less
than 4.2%.
As shown in Figure 3.14 the difference between the results from the two
pipelines, KSB and Shapelets, is negligible. This provides an independent confirmation that the MV pipeline produces shears of as high a quality as the
state-of-the-art weak lensing analysis presented in S10.
92
4
The relation between galaxy dark
matter haloes and baryons in the
CFHTLS from weak lensing
Velander M., van Uitert E., Hoekstra H. and the
CFHTLenS Collaboration, in prep.
Current theories of structure formation predict that galaxies are immersed in extensive dark matter haloes. To learn more about the
baryon-dark matter connection it is therefore imperative to probe large
scales as well as small. Weak galaxy-galaxy lensing has the power to
do this since it not only is sensitive on a large range of scales, but also
is independent of the type of matter studied. We present a study of
large-scale galaxy dark matter halo properties as a function of the characteristics of the baryonic host galaxies using data from one of the largest
completed weak lensing surveys to date, the CFHTLS. Dividing our lens
sample into red and blue subsamples, we find that for red galaxies, the
+0.10
1.28−0.08
halo mass scales with luminosity as M200 ∝ Lr
1.36+0.10
mass as M200 ∝ M∗ −0.06 , while
+0.06
0.54
and M200 ∝ M∗ −0.08 . We also
and with stellar
0.50+0.18
−0.12
for blue galaxies M200 ∝ Lr
find indications that blue galaxies
reside in less clustered environments than red galaxies do.
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4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
4.1
Introduction
In order to fully understand the mechanisms behind galaxy formation, the connection between galaxies and the extensive dark matter haloes in which they
are enveloped must be studied in exhaustive detail. In pursuit of this precision,
reliable mass estimates of both the baryonic and the dark matter content of
galaxies are required. The visible component may be evaluated using e.g. the
galaxy luminosity or the stellar mass, which can be derived using stellar synthesis models (Kauffmann et al., 2003; Gallazzi et al., 2005; Bell & de Jong, 2001;
Salim et al., 2007). The dark matter, on the other hand, cannot be observed
directly but must be examined through the influence it has on its surroundings.
At the largest scales reached by haloes, optical tracers such as satellite galaxies are scarce. Furthermore, estimates of halo mass from e.g. satellite galaxy
kinematics not only require spectroscopic measurements of a large number of
objects, which is unfeasible both in terms of time and from a financial perspective, but they also require the application of the virial theorem and all the
associated assumptions. To study any and all galaxies it is therefore desirable
to use probes independent of these tracers, and independent of the physical
state of the halo, but with the power to explore a large range of scales. These
requirements are all satisfied by weak gravitational lensing.
Weak gravitational lensing is fundamentally a consequence of general relativity. As light from distant objects travels through the Universe it is deflected
by intervening matter. This deflection causes the distant objects, or sources,
to appear distorted. In the weak regime the distortion is minute, and only by
correlating the shapes of a large number of sources can information about the
foreground gravitational field be extracted. There are a handful of different
ways of correlating these shapes, each resulting in data about a separate category of matter accumulation. By correlating the shape of sources with those of
other sources, finely detailed large-scale structure can be discerned in the foreground. The precise properties of this pattern are intimately connected with
the composition of our Universe and thus allows for constraints on cosmological
parameters via cosmic shear (see e.g. Van Waerbeke & Mellier (2003); Hoekstra
& Jain (2008); Munshi et al. (2008), for reviews, and e.g. Schrabback et al.
(2010) for recent results). Alternatively, source shapes may be correlated with
the positions of foreground objects, or lenses, through a technique known as
galaxy-galaxy lensing. The strength of the lensing signal as a function of the
distance from the lens holds information on the depth and shape of the potential well causing the distortion. Thus density profiles of the dark matter haloes
surrounding galaxies and, equivalently, galaxy clusters may be directly investigated. Simulations predict that dark matter haloes are well approximated by
Navarro-Frenk-White profiles (Navarro, Frenk, & White, 1996) and confirming
this would provide evidence for the concordance model of cosmology.
Generally, however, galaxies and their haloes are not isolated but reside in
clustered environments. The ramification is that the interpretation of the observed galaxy-galaxy lensing signal becomes more complicated since the signal
from nearby haloes influence the result. Over the past decade a new approach
has gained traction: the weak lensing halo model (e.g. Cooray & Sheth, 2002;
Guzik & Seljak, 2002; Mandelbaum et al., 2005; van Uitert et al., 2011; Leauthaud et al., 2011). Within the halo model framework, all haloes are represented
94
4.2. DATA
as distinct entities, each with a galaxy at the center. Enclosed in each main halo
are satellite galaxies surrounded by subhaloes. In this work we seek to employ
the halo model to gain a more accurate picture of galaxy-size dark matter haloes,
allowing for a more precise analysis of the link between galaxies and the dark
matter haloes they reside in. For this purpose we use data from the CanadaFrance-Hawaii Telescope Legacy Survey (CFHTLS). The CFHTLS consists of
just over 170 deg2 of images in five filters (u∗ g ′ r′ i′ z ′ ) to the impressive depth of
i′ = 24.5. The CFHTLS weak lensing collaboration (CFHTLenS) has extracted
8.7 × 106 galaxy shears, and provided all objects with reliable photometric redshifts, making this survey one of the most powerful completed weak lensing
surveys to date. This work thus improves on the preliminary galaxy-galaxy
lensing analysis carried out using a small subset of the CFHTLS data and a
single-halo model fit to the inner regions only (Parker et al., 2007).
This Chapter is organised as follows: we introduce the data in Section 4.2.
In Section 4.3 we review our halo model and the formalism behind it, and
in Section 4.4 we test our shear catalogue for systematic effects. We investigate the lensing signal as a function of luminosity in Section 4.5 and as a
function of stellar mass in Section 4.6, and we conclude in Section 4.7. The
following cosmology is assumed throughout (WMAP7; Komatsu et al., 2010):
(ΩM , ΩΛ , h, σ8 , w) = (0.27, 0.73, 0.70, 0.81, −1)
4.2
Data
In this Chapter we present a weak lensing analysis of the entire Wide Synoptic Survey of the Canada-France-Hawaii Telescope Legacy Survey (CFHTLSWide). The impressive expanse and depth of this survey makes it ideal for
weak lensing analyses, as we will show. The CFHTLS is a joint 5-year project
between Canada and France which commenced in 2003 and which is now completed. The data are imaged using the Megaprime wide field imager mounted at
the prime focus of the Canada-France-Hawaii Telescope (CFHT) and equipped
with the MegaCam camera. MegaCam comprises an array of 9 × 4 CCDs and
has a field of view of 1 deg2 . The wide synoptic survey covers an effective area
of about 155 deg2 in five bands: u∗ , g ′ , r′ , i′ and z ′ . This area is composed
of four independent fields, W1–4, each with an area of 25-72 deg2 and with a
full multi-colour depth of i′ = 24.7 (7σ detected source in the CFHTLenS1 catalogue). The images have been independently reduced within the CFHTLenS
Collaboration, and for details on this data reduction process, please refer to
Erben et al (in prep.).
4.2.1
Lens sample
The depth of the CFHTLS enables us to investigate lenses with a large range
of lens properties and redshifts, which in turn grants us the opportunity to
thoroughly study the evolution of galaxy-sized dark matter haloes. An initial
study was performed by van Uitert et al. (2011) (hereafter VU11) using the same
halo model as the one used here, described in Section 4.3.2. That study exploited
a 300 deg2 overlap between two major lensing surveys. The foreground sample
consisted of galaxies from the seventh data release of the Sloan Digital Sky
1 http://cfhtlens.org
95
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.1 Magnitude (left panel) and photometric redshift (right panel) distributions of all objects in the CFHTLenS catalogue, with our lens (source) redshift
selection marked with purple dashed (green dotted) lines. Also shown is the cutoffpoint in magnitude for all objects used in our analysis (purple dash-dotted).
Survey (SDSS DR7; Abazajian et al., 2009), and objects from the intermediatedepth Second Red Sequence Cluster Survey (RCS2; Gilbank et al., 2011) were
used as background sources, improving greatly on previous analyses based on
the shallow SDSS alone. However, while their lenses had accurate spectroscopic
redshift estimates, their sources did not have enough photometric data available
at the time to provide redshift estimates for the sources. Thus the CFHTLS
has, aside from the increased depth, a further advantage over the VU11 analysis
owing to the high-precision photometric redshifts available for all objects used
in our analysis.
Throughout this Chapter, we place an upper cut in apparent magnitude of
i′ ≤ 24.5 and select our lenses in redshift such that 0.2 ≤ zlens ≤ 0.4, unless
explicitly stated otherwise. These selections are illustrated in Figure 4.1. For
the full CFHTLS-Wide we achieve a lens count of Nlens = 1.53 × 106 , nearly a
hundred times the size of the lens sample used in VU11. We then further split
our lens sample in luminosity or stellar mass bins as described in Sections 4.5
and 4.6 to investigate the halo mass trends as a function of lens properties.
4.2.2
Source catalogue
The shear estimates for the sources used in this Chapter have been obtained
using lensfit as detailed in Miller et al (in prep.), and thoroughly tested by the
CFHTLenS Collaboration. During the testing process, the lensfit shears were
compared to those extracted using other shape measurement methods (such
as those introduced in Hoekstra et al., 1998; Kuijken, 2006; Schrabback et al.,
96
4.3. METHOD
2007; Velander et al., 2011) to successfully eliminate software-specific issues.
All sources also have multi-band photometric redshift estimates as described in
Hildebrandt et al. (2012).
To ensure photometric accuracy, we use only sources with redshifts of zlens ≤
zsource ≤ 1.3, and we impose the same cut in magnitude as we do for the lenses:
i′ ≤ 24.5. Furthermore, we ensure that our sources have been detected in at
least six exposures. Our source count for the full CFHTLS-Wide (excluding
masked areas) is then Nsource = 3.9 × 106 , corresponding to a source density of
9.3 arcmin−2 which is a factor of 1.5 greater than that of the RCS2.
4.3
Method
To analyse the dark matter haloes in the CFHTLS we use a method known as
weak galaxy-galaxy lensing, and compare the measured signal with a halo model.
In this section we will introduce the basic formalism and give an overview of
our halo model.
4.3.1
Weak galaxy-galaxy lensing
Weak gravitational lensing is the measure of weak distortions induced by foreground structure on background source galaxies. By correlating the shapes of
background galaxies, statistical properties of the matter in the foreground can
be inferred.
The first-order lensing distortion, shear, is a stretch in one direction which
is applied to the intrinsic shape of a source galaxy. By averaging over enough
randomly oriented sources we can assume the mean intrinsic galaxy to be circular, and thus any distortion measured is due to lensing. In this analysis we use
galaxy-galaxy lensing where source galaxy distortions are averaged in concentric
rings centered on lens galaxies. We measure the tangential shear as a function of
radial distance from the lens this way, and also the cross shear which is a 45deg
rotated signal. The cross shear can never be induced by a lens which means
that it may be used as a systematics check. The amplitude of the tangential
shear is directly related to the differential surface density ∆Σ(r) via
∆Σ(r) = Σcrit hγt (r)i
(4.1)
where Σcrit is the critical surface density
Σcrit =
c2 D s
4πG Dl Dls
(4.2)
with Ds , Dl and Dls the angular diameter distance to the source, to the lens
and between the lens and source respectively. By using differential surface
densities rather than tangential shears, the geometric factor is neutralised and
the amplitude of the signals can be directly compared between different samples.
The only caveat is that the properties of lenses depend on the lens redshift so
this difference still has to be taken into account.
The circular average makes this type of analysis robust against small-scale
systematics introduced by e.g. the telescope. None the less, there will be largescale systematics present, mainly due to areas being masked. By masking areas,
97
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
or by considering lenses close to the edge of the area covered, the circular average
is in fact not perfectly circular. This effect, most noticeable on large scales,
is important to correct for, particularly in the case of precision galaxy-galaxy
lensing such as the analysis in this Chapter. Our correction is done by measuring
the signal around random lens positions. Were there no systematics present the
measured signal would be zero. If it is not, it can easily be corrected for by
subtracting it from the observed lensing signal.
As for any concerns regarding the fidelity of the photometric redshifts, the
weights we employ use the geometric lensing efficiency Ds /(Dl Dls ) to downweight close pairs as described in e.g. Velander et al. (2011) (Chapter 3 of
this Thesis), effectively minimising any influence of redshift inaccuracies on
the measured signal. Additionally we calculate a correction factor based on
the redshift error distribution for each mass estimate to remove any remaining
redshift systematics. This calculation is described further in Section 4.5.1.
4.3.2
The halo model
Figure 4.2 Illustration of the halo model used in this Chapter. Here we have
used a halo mass of M200 = 1012 h−1 M⊙ , a stellar mass of M∗ = 3 × 1011 M⊙
and a satellite fraction of α = 0.2. The lens redshift is zlens = 0.5. Purple lines
represent quantities tied to galaxies which are centrally located in their haloes
while green lines correspond to satellite quantities. The purple dash-dotted line is
the baryonic component, the green dash-dotted line is the stripped satellite halo,
dashed lines are the 1-halo components induced by the main dark matter halo and
dotted lines are the 2-halo components originating from nearby haloes.
To accurately model the weak lensing signal observed around galaxy-size
haloes, we have to account for the fact that galaxies generally reside in clustered
environments. In this work we do this by employing the halo model software
first introduced in VU11. For full details on the exact implementation, please
98
4.3. METHOD
see VU11; here we give a qualitative overview.
Our halo model builds on work presented in Guzik & Seljak (2002) and Mandelbaum et al. (2005), where the full lensing signal is modelled by accounting
for the central galaxies and their satellites separately. Here we assume that a
fraction (1 − α) of our galaxy sample reside at the centre of a dark matter halo,
and the remaining objects are satellite galaxies surrounded by subhaloes which
in turn reside inside a larger halo. In this context α is the satellite fraction of a
given sample.
The lensing signal induced by central galaxies consists of two components:
the signal arising from the main halo (the 1-halo term ∆Σ1h ) and the contribution from neighbouring haloes (the 2-halo term ∆Σ2h ). The two components
simply add to give the lensing signal due to central galaxies:
2h
∆Σcent = ∆Σ1h
cent + ∆Σcent
(4.3)
In our model we assume that all main dark matter haloes are well represented by
a Navarro-Frenk-White density profile (NFW; Navarro, Frenk, & White, 1996)
with a mass-concentration relationship as given by Duffy et al. (2008).
We assume that satellite galaxies reside in subhaloes which have been tidally
stripped of dark matter in the outer regions. Adopting a truncated NFW profile which has been stripped of about 50% of its dark matter, we acquire a
satellite term which supplies signal on small scales. Thus satellite galaxies add
three further components to the total lensing signal: the contribution from the
stripped subhalo (∆Σstrip ), the satellite 1-halo term which is off-centred since
the satellite galaxy is not at the centre of the main halo, and the 2-halo term
from nearby haloes. Just as for the central galaxies, the three terms add to give
the satellite lensing signal:
1h
2h
∆Σsat = ∆Σstrip
sat + ∆Σsat + ∆Σsat
(4.4)
There is an additional contribution to the lensing signal, not yet considered
in the above equations. This is the signal induced by the lens baryons (∆Σbar ).
This last term is a refinement to the halo model presented in VU11, necessary
since weak lensing measures the total mass of a system and not just the dark
matter mass. The baryonic component is modelled as a point source with a mass
equal to the mean stellar mass of the lenses in the sample (as in e.g. Leauthaud
et al., 2011):
hM∗ i
(4.5)
∆Σbar =
πr2
where r is the physical distance from the lens. This term could technically be
decomposed into a central and a satellite component. In this work we do not
leave the baryon term as a free variable and so we do not need to distinguish
between the two. Thus we treat this term as a single entity.
Finally, to obtain the total lensing signal of a galaxy sample of which a
fraction α are satellites we combine the baryon, central and satellite galaxy
signals, applying the appropriate proportions:
∆Σ = ∆Σbar + (1 − α)∆Σcent + α∆Σsat
(4.6)
All components of our halo model are illustrated in Figure 4.2. In this
example the halo mass is M200 = 1 × 1012 h−1 M⊙ , the stellar mass is M∗ =
99
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
3 × 1011 M⊙ , the satellite fraction is α = 0.2, the lens redshift is zlens = 0.5 and
Dls /Ds = 0.5. On small scales the baryonic component is prominent, while on
large scales the 2-halo components dominate.
4.4
4.4.1
Systematics tests
Verification of the shear catalogue
Figure 4.3 Comparison of three data sets: the KSB catalogues from ∼ 22 deg2
CFHTLS (pink circles), the results from RCS2 (green stars) and our results (purple
dots). The lines show the best fit singular isothermal sphere for each dataset (with
green and pink nearly identical), and the grey triangles show the cross-shear from
our results which should be zero in the absence of systematic errors.
In this study we use lenses and sources from the 155 deg2 CFHTLS, with
high-quality photometric data and redshifts available for all objects. To validate
the quality of our shear catalogue we compare with the results from two previous
analyses of a very similar nature. The first is a preliminary weak galaxy-galaxy
lensing analysis of the CFHTLS-Wide conducted by Parker et al. (2007). At
that time, the survey was not yet finished, so they only had access to an area of
∼ 22 deg2 in i′ -band corresponding to about 15% of our area. Since they only
had data from one band their analysis also lacked redshift estimates for lenses
and sources, but they separated lenses from sources using magnitude cuts. They
then obtained shear estimates for their sources using a version of the technique
introduced by Kaiser, Squires, & Broadhurst (1995) as outlined in Hoekstra
et al. (1998). For their average lens they derived a best-fit velocity dispersion of
σv = 132 ± 10 km s−1 using a singular isothermal sphere profile (SIS) to model
the lensing signal, though there is some disagreement between this number and
our findings (see discussion below). The second analysis is based on the shear
100
4.4. SYSTEMATICS TESTS
Table 4.1
Sample
Details of the seeing bins.
Nfields
hr∗ i [arcsec]
θE [arcsec]
σθE
27
23
33
38
28
36
0.50
0.57
0.62
0.67
0.72
0.80
0.053
0.044
0.050
0.047
0.040
0.049
0.005
0.006
0.005
0.005
0.006
0.005
P1
P2
P3
P4
P5
P6
catalogue from VU11 (see Section 4.2). The data used in that study is from the
RCS2 which is slightly shallower than the CFHTLS and for which no redshifts
were available for the sources at the time of this analysis.
To compare and contrast our lensing signal with the one obtained by Parker
et al. (2007) we apply the same i′ -band magnitude cuts as they did, viz. 19.0 <
i′ < 22.0 for lenses and 22.5 < i′ < 24.5 for sources. A slight difference between
their analysis and ours is that Parker et al. (2007) boosts their signal by an
approximate factor to correct for contamination by sources that are in front of,
or physically associated with, the lens while we use our redshift information to
minimise this contamination. The resulting galaxy-galaxy signal, scaled with
the angular diameter distance ratio β = Dls /Ds = 0.67, is shown as purple
dots in Figure 4.3. The best-fit SIS profile corresponds to a velocity dispersion
of σv = 83.3 ± 1.6 km s−1 , which is somewhat lower than the one quoted
in Parker et al. (2007). However, we re-analysed the actual shear catalogues
used for the Parker et al. (2007) analysis and the results are shown as light
circles in Figure 4.3. For that signal, which is corrected for contamination
using the Parker et al. (2007) boost factor, we find a velocity dispersion of
σv = 79.4 ± 3.3 km s−1 using the redshifts stated in Parker et al. (2007). This is
in good agreement with our analysis of the full CFHTLS-Wide. The discrepancy
with the velocity dispersion quoted in Parker et al. (2007) remains unexplained,
but we have shown that the shear estimates are consistent between the two
CFHTLS catalogues.
Furthermore, also shown as green stars in Figure 4.3 is the signal obtained by
VU11 using RCS2 and the same magnitude selection as Parker et al. (2007). The
shears have been corrected for contamination by physically associated sources,
as described in VU11, and scaled with the β = 0.48 appropriate for the RCS2.
For this signal we find a velocity dispersion of σv = 79.7 ± 2.3 km s−1 which
is again in good agreement with our results. Based on this, and based on our
re-analysis of the original Parker et al. (2007) shear catalogues, we conclude
that our shear estimates are valid and reliable for further galaxy-galaxy lensing
studies.
4.4.2
Seeing test
In general a round PSF causes circularisation of source images which in turn
causes a multiplicative bias of the shapes measured. The amount of bias depends
on the size of the PSF. Assuming that the shapes of very well resolved galaxies
101
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.4 The weak galaxy-galaxy signal measured in each of 6 seeing bins,
according to Table 4.1.
can be accurately recovered we can model the effect of the PSF as
"
2 #
r∗
obs
true
γ
=γ
1+M
r0
(4.7)
where γ obs is the observed shear, γ true is the true shear, r∗ is the PSF size and
r0 is the intrinsic (Gaussian) size of the galaxy. The particular dependence on
PSF size is the result of a full moments analysis. M is a function close to zero
representing the multiplicative bias and may be separated into two components:
M = M′ + P
(4.8)
where M′ is the true limitation of the shape measurement method and P is the
bias contribution induced by the PSF. This last term depends on the statistical
and systematic errors in the size estimates. Thus if the shape measurement
software has no intrinsic bias and if the PSF is perfectly known, then the dominant source of shear bias is the accuracy of the size estimates of the faint, small
galaxies. For the Bayesian shape measurement method used in this Chapter,
lensfit, this term will be negligible.
This multiplicative bias may be related to the parameterisation used in the
Shear Testing Programme (STEP Heymans et al., 2006a; Massey et al., 2007a),
mSTEP , via
* +
2
r∗
(4.9)
mSTEP = M
r0
The smallest bias achieved by the pipelines taking part in STEP was mSTEP ∼
0.01 which, for the adopted size distribution and simulated ground-based data
used in STEP, corresponds to M ∼ 0.005.
102
4.5. LUMINOSITY TREND
Since the bias depends on the size of the PSF, data with a spread in seeing
should enable us to determine the bias M directly from the data, thus allowing
us to deduce the true performance of the shape measurement pipeline. The
CFHTLS has such a spread, with the best seeing being 0.44 arcsec and the
worst being 0.94 arcsec.
Galaxy-galaxy lensing provides us with a neat way of determining the bias.
Assuming that the systematic offset due to PSF anisotropy is negligible (a fair
assumption given our correction for spurious signal around random lenses; see
Section 4.3.1) the observed shear is related to the true shear via
γ obs = (1 + mSTEP )γ true
(4.10)
For a singular isothermal sphere (SIS), the amplitude of the shear signal as a
function of distance θ from the lens is
γ(θ) =
θE
2θ
(4.11)
where θE is the Einstein radius. Thus there is a simple relationship between
the observed Einstein radius and the true one:
* +!
2
r∗
true
obs
θE
(4.12)
θE
= 1+M
r0
By measuring the Einstein radius of the average lens as a function of seeing we
can therefore determine both the true Einstein radius and the performance of
the shape measurement pipeline.
We split the data according to Table 4.1, measure the galaxy-galaxy lensing
signal in each seeing bin and fit an SIS to the innermost 140 h−1 kpc. By fitting
only small scales we avoid the influence of neighbouring haloes. The results
are shown in Figure 4.4 and quoted in Table 4.1. We then fit the relation
described by Equation 4.12 to the resulting Einstein radii and find a value of
M = −0.048 ± 0.071 which implies a STEP bias of mSTEP = −0.094 ± 0.14.
This is consistent with no bias given the error bars, but with a greater range in
seeing we would be able to constrain this bias even further. According to this
analysis, the true Einstein radius of the average lens galaxy in our sample is
true
θE
= 0.052” ± 0.007”.
4.5
Luminosity trend
The luminosity of a galaxy is an easily obtainable indicator of its baryonic
content. To investigate the relation between dark matter halo mass and galaxy
mass we therefore split the lenses into 8 bins according to MegaCam r′ -band
magnitudes as detailed in Table 4.2 and illustrated in Figure 4.5. The choice
of bin limits follow the lens selection in VU11, a previous analysis carried out
using the shallower RCS2 and an earlier version of the halo model we use here.
This choice will allow us to compare our results to the results obtained by
VU11 because the RCS2 data have been obtained using the same filters and
telescope. Since the behaviour of early-type galaxies is expected to differ from
that of late-type galaxies, using only one luminosity estimate to characterise a
lens sample results in an average relation which may be difficult to interpret.
103
Table 4.2 Details of the luminosity bins. (1) Absolute magnitude range; (2) Number of lenses; (3) Mean redshift; (4) Fraction of lenses that are
blue; (5) Mean luminosity for red lenses [1010 L⊙ ]; (6) Mean stellar mass for red lenses [1010 M⊙ ]; (7) Redshift-corrected best-fit halo mass for red
lenses [1011 h−1 M⊙ ]; (8) Best-fit satellite fraction for red lenses; (9) Mean luminosity for blue lenses [1010 L⊙ ]; (10) Mean stellar mass for blue
lenses [1010 M⊙ ]; (11) Redshift-corrected best-fit halo mass for blue lenses [1011 h−1 M⊙ ]; (12) Best-fit satellite fraction for blue lenses
Sample
L1
L2
L3
L4
L5
L6
L7
L8
Mr (1)
nlens (2)
hzi(3)
fblue (4)
(5)
hLred
r i
hM∗red i(6)
Mhred (7)
αred(8)
hLblue
i(9)
r
hM∗blue i(10)
Mhblue (11)
αblue(12)
[-21.0,-20.0]
[-21.5,-21.0]
[-22.0,-21.5]
[-22.5,-22.0]
[-23.0,-22.5]
[-23.5,-23.0]
[-24.0,-23.5]
[-24.5,-24.0]
89215
31889
22492
13105
5840
1769
389
87
0.30
0.30
0.30
0.30
0.30
0.29
0.29
0.27
0.61
0.40
0.29
0.21
0.15
0.12
0.16
0.24
0.91
1.76
2.77
4.32
6.75
10.5
16.5
25.8
4.72
8.97
14.1
22.1
34.6
53.8
84.0
132
2.36+0.68
−0.53
4.39+1.40
−1.06
4.55+1.45
−1.10
10.4+3.30
−2.50
17.5+5.58
−4.23
+13.7
39.4−10.2
+46.1
132−34.2
167+137
−75.2
0.30+0.02
−0.02
0.20+0.03
−0.02
0.24+0.03
−0.03
0.20+0.03
−0.03
0.22+0.04
−0.04
0.17+0.06
−0.06
0.01+0.11
−0.11
0.38+0.26
−0.26
0.83
1.71
2.71
4.28
6.71
10.8
17.3
29.6
2.48
5.12
8.16
12.9
20.2
32.0
49.7
84.2
0.87+0.57
−0.35
1.78+1.18
−0.71
3.39+2.50
−1.44
0.87+1.53
−0.82
0.53+2.60
−0.52
0.99+6.50
−0.98
0.01+6.04
−0.00
+31.8
7.48−7.47
0.00+0.01
−0.00
0.00+0.03
−0.00
0.00+0.02
−0.00
0.13+0.06
−0.06
0.04+0.10
−0.10
0.27+0.22
−0.22
0.00+0.18
−0.00
0.00+0.25
−0.00
4.5. LUMINOSITY TREND
Figure 4.5 r ′ -band absolute magnitude distribution in the CFHTLS for lenses
with redshifts 0.2 ≤ zlens ≤ 0.4 (black solid histogram). The distribution of red
(blue) lenses is shown in dotted purple (dot-dashed green). Our lens bins are
marked with vertical lines.
Since we have access to multi-colour data, we are able to further divide our
lenses in each bin into a red and a blue sample, approximately corresponding to
early-type and late-type galaxies. In practice we do this using their photometric
types TBPZ . TBPZ is a number in the range of [1.0, 6.0] representing the best-fit
spectral energy distribution (SED) and we define our red and blue samples as
galaxies with TBPZ < 1.5 and TBPZ > 2.0 respectively where the latter captures
most spiral galaxies. A colour-colour comparison confirms that these samples
are well defined. We proceed to measure the galaxy-galaxy lensing signal for
each sample, and fitting it with our halo model, leaving the halo mass M200 and
the satellite fraction α as free parameters. The results are shown in Figure 4.6
for all luminosity bins and for each red and blue lens sample, with details of
the fitted halo model parameters quoted in Table 4.2. Qualitatively comparing
these results to the ones presented in VU11 we see that the amplitudes of the
signals agree well.
An overview of the broad trends in Figure 4.6 is given in Figure 4.7 for
red galaxies and Figure 4.8 for blue. As expected, the amplitude of the signal
increases with luminosity for both red and blue samples indicating an increased
105
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.6 The weak galaxy-galaxy signal around lenses which have been split
into luminosity bins according to Table 4.2, modelled using the halo model described in Section 4.3.2. The purple (green) dots represent the measured differential surface density of the red (blue) lenses, and the solid line is the best-fit
halo model. Triangles represent negative points that are included unaltered in
the model fitting procedure, but that have here been moved up to positive values
as a reference. The dotted error bars are the unaltered error bars belonging to
the negative points. The squares represent distance bins containing no objects.
For a detailed decomposition into the halo model components, please refer to
Appendix 4.A.
halo mass. In general, for identical luminosity selections blue galaxies have less
massive haloes than red do. For the red sample, lower luminosity bins display
a slight bump at scales of ∼ 0.5 h−1 Mpc. This is due to the satellite 1-halo
term becoming significant and indicates that a large fraction of the galaxies
in those bins are in fact satellite galaxies inside a larger halo. Thus brighter
galaxies are more likely to be centrally located in a halo. The blue galaxies
also display a bump for the lower luminosity bins, but this feature is at larger
scales than the satellite 1-halo term. The signal breakdown shown in Figure 4.20
(Appendix 4.A) reveals that this bump is due to the central two-halo term, i.e. it
106
4.5. LUMINOSITY TREND
is the contribution from nearby haloes.
To make a quantitative comparison with VU11, however, there are several
differences between the analyses that have to be taken into account. Firstly,
including the baryonic component in our halo model results in a lower halo
mass estimate than not doing so (see Section 4.5.2 and Figure 4.12 for a more
extensive discussion on this topic). This is intuitive since gravitational lensing
measures the total mass of a system and we are allowing some of that mass to
be baryonic, leaving less mass for the dark matter halo. Secondly, the red and
blue selection in our analysis does not necessarily correspond to the early- and
late-type classification in VU11, making a completely fair comparison difficult.
Thirdly, we have a strong enough signal to be able to limit our lens sample
to a small redshift range, which minimises any contamination of the relations
due to redshift evolution. Finally, we do not have spectroscopic redshifts for
our lenses which means that our lenses may have been assigned an inaccurate
redshift. This will cause a luminosity-dependent bias in the halo mass estimate,
as discussed in e.g. Hoekstra et al. (2005). We will examine this effect further
in the next Section.
Figure 4.7
4.5.1
Best-fit halo models for red lenses for all luminosity bins.
Photometric redshift error corrections
Before interpreting the luminosity results we have to take into account the redshift bias effect previously mentioned. The accuracy of our photometric redshifts
is high, but never the less the errors on the redshift estimates have to be taken
into account. If the true redshift differs from the estimated one, this will affect
all derived quantities. An underestimated redshift, for example, would cause the
estimated absolute magnitude to be fainter than the true absolute magnitude
and the lens would be placed in the wrong luminosity bin. As can be seen in Fig107
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.8
Best-fit halo models for blue lenses for all luminosity bins.
Figure 4.9 Bias as a function of luminosity induced through inaccuracies in
the photometric redshift estimates. The purple solid (green dashed) line with
dots (triangles) shows the bias for the red (blue) lens sample. The error bars are
obtained using ten lens catalogue realisations.
ure 4.5 there are more faint objects than bright, which means that more objects
will scatter from fainter bins into brighter bins than the other way around. This
will lower the lensing signal in each bin and bias the observed halo mass low, and
the amount of bias will be luminosity dependent. To estimate the impact of this
bias we create a simulated version of the CFHTLS-Wide as follows. We fit an
initial powerlaw mass-luminosity relation of the form M200 = M0,L (L/Lpivot)βL
to the estimated halo masses as described in VU11, with Lpivot = 1011 Lr′ ,⊙ .
108
4.5. LUMINOSITY TREND
This relation we then use to assign halo masses to our lenses. Constructing
NFW haloes from these halo masses at the photometric redshift of the lenses,
we create mock source catalogues with the observed source redshift distribution but with simulated shear estimates with strengths corresponding to those
which would be induced by our lens haloes. Finally we measure the mock signal
within 200 h−1 kpc of the lenses in each luminosity bin and take this to be the
‘true’ signal. We only use the small scales for our mass estimate to avoid complications due to insufficient treatment of clustering, and we force our satellite
fraction to zero to obtain a pure NFW fit. Scattering the lenses, assuming a
Gaussian error distribution of width ∆z = (0.004mi′ − 0.04)(1 + z) (see Hildebrandt et al. (2012) for a plot of the photometric redshift errors as a function of
magnitude), we then measure the signal in each of 10 realisations and compare
the resulting estimated halo masses to the ‘true’ halo masses. The average of
these realisations provides the observed halo mass given the bias, with errors
equal to the standard deviation. Since the starting point is a perfect signal, the
number of realisations given the area is adequate to retrieve the bias.
The results from this test are shown in Figure 4.9. The quality of our
photometric redshifts is high which means that the correction factor is small
overall, reaching only ∼ 15% for a luminosity of Lr′ ∼ 2.5 × 1011 L⊙ . Here the
contamination is largest due to the shape of the luminosity function causing a
larger fraction of low luminosity objects to scatter into the higher-luminosity bin.
For our faintest red luminosity bin the correction is ∼ 10%, in this case caused
by larger errors in the photometric redshift estimates. The correction factor is
less than unity for lower-luminosity bins due to the turn-over of the distribution
of red lenses at Mr′ ∼ −21.2 (see Figure 4.5). The small correction factor
for blue lenses is due to their flatter mass-luminosity relation (see Figure 4.10).
Because of the relative insensitivity of halo mass to changes in luminosity, minor
errors in luminosity measurements due to photometric redshift inaccuracies will
not strongly affect the halo mass estimate. The process described in this section
could in principle be iterated over, starting from the fitting of a mass-luminosity
relation, until convergence is reached. Since Hoekstra et al. (2005) find that
different choices for that relation yield similar curves, we choose not to iterate
further.
4.5.2
Luminosity scaling relations
The estimated halo masses for all luminosity bins, corrected for bias due to
errors in the photometric redshifts, are shown as a function of luminosity in the
top panel of Figure 4.10. Red lenses display a steeper relationship between halo
mass and luminosity than blue lenses do, and the higher luminosity bins contain
too few blue lenses to adequately constrain the mass. As done in VU11, we fit
a powerlaw of the form M200 = M0,L (L/Lpivot)βL to our lensing signal, with
Lpivot = 1011 Lr′ ,⊙ . Rather than fitting to the final mass estimates we fit this
relation directly to the lensing signals themselves. We do this because the error
bars are asymmetric in the former case, making a fit more complicated. The
difference in results between the two fitting techniques is small however.
12
For our red lenses we find M0,L = 3.53+0.29
h−1 M⊙ and βL =
−0.29 × 10
+0.10
1.28−0.08 , while for our blue lenses the corresponding numbers are M0,L =
11
h−1 M⊙ and βL = 0.50+0.18
3.45+0.98
−0.12 . The constraints for these fits
−1.47 × 10
109
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.10 Satellite fraction α and bias-corrected halo mass M200 as a function
of r ′ -band luminosity. Purple (green) dots represent the results for red (blue)
lens galaxies, and the dash-dotted lines show the powerlaw scaling relations as
described in the text.
are shown in Figure 4.11. Here we again see that the red lenses are better
constrained than the blue. This is partly because we have more red lenses,
and partly because red lenses in general are more massive making the lensing
signal stronger. Our powerlaws are shallower than the ones found by VU11, but
there are some differences between the analyses, making a direct comparison
difficult. The way we select our red and blue samples differs significantly from
the VU11 selection of early- and late-type samples (which is based on estimated
Sérsic profiles rather than colours). Furthermore, in our halo model we account
for the baryonic mass of each lens, something that was not done in VU11.
Removing the baryonic component from our model, we find that the masses
for some bins are overestimated by as much as 60%. It may appear counterintuitive that including a baryonic component with a mass which is of order 10%
of the total mass should result in such a significantly lowered halo mass estimate.
The explanation lies in the halo model fitting, and specifically in the way the
satellite fraction is allowed to vary. Adding a baryonic component on small
scales will result in a lowered central halo mass. The central halo profile reaches
further than the baryonic component however, and thus power on intermediate
110
4.5. LUMINOSITY TREND
Figure 4.11 Constraints on the powerlaw fits shown in Figure 4.10. In purple
(green) we show the constraints on the fit for red (blue) lenses, with lines representing the 67.8%, 95.4% and 99.7% confidence limits and stars representing the
best-fit value.
scales is also diminished. To compensate for this loss of power, the halo model
will increase the satellite 1-halo term by increasing the satellite fraction, which
also increases the stripped satellite halo term, lowering the central 1-halo term
further until an equilibrium is reached. These mechanisms are illustrated for
red galaxies in luminosity bin L4 in Figure 4.12, where we have allowed halo
mass, satellite fraction and stellar mass fraction to vary simultaneously for both
panels. This Figure also makes clear the degeneracies introduced to the halo
model if the stellar mass is left as a free parameter.
Higher-luminosity bins are more severely affected by this effect than the
lower-luminosity end due to the lack of a prominent satellite 1-halo feature.
The net effect is a steeper slope, which is exactly what VU11 are displaying.
The general overestimation of halo mass in VU11 also means that the mass
of a L = 1011 Lr′ ,⊙ galaxy is overestimated, partly explaining the discrepancy
between the M0,L estimates.
VU11 also convert their best-fit halo masses to mean halo masses, accounting
for the fact that the halo mass function is a declining function, causing us to
preferentially pick lower-mass haloes. The lensing best-fit halo mass therefore
111
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.12 Dependence of halo model fitting parameters halo mass M200 and
satellite fraction α on stellar mass, with fSM the fraction of true mean stellar
mass used in the halo model and contours showing the 67.8%, 95.4% and 99.7%
confidence intervals. The left panel shows that including a baryonic component
in the model (i.e. setting fSM = 1) will result in a significantly lower best-fit halo
mass than not doing so (fSM = 0), and the right panel shows that the reason for
this is an increased satellite fraction.
does not correspond to the mean halo mass in a given bin. The correction
factors they apply range between a few percent for the lower-luminosity bins to
∼ 30% at the highest luminosities.
Another important factor to take into account is the fact that we limit our
lens samples to redshifts of 0.2 < zlens < 0.4 keeping our mean lens redshift
fairly stable at hzlens i ∼ 0.3. This is not done in VU11 and as a result, the
median redshift of our lower-luminosity bins is higher than for the same bins
in VU11, with the opposite being true for the higher-luminosity bins. Recent
numerical simulations indicate that the relation between stellar mass and halo
mass will evolve with redshift (e.g. Conroy & Wechsler, 2009; Moster et al.,
2010). Lower-mass host galaxies (M∗ < 1011 M⊙ ) increase in stellar mass faster
than their halo mass increases, i.e. for higher redshifts the halo mass is lower for
the same stellar mass. The opposite trend holds for higher-mass host galaxies
(M∗ > 1011 M⊙ ). As a result, the relation between halo mass and stellar mass
(or an indicator thereof, such as luminosity) steepens with increasing redshift.
This means that for the lower-luminosity bins, where our redshifts are higher, we
may measure a steeper slope than VU11 and vice-versa for higher-luminosity
bins. There are other factors which could affect the measured slope, such as
the scatter between luminosity bins due to errors in the estimated luminosities.
VU11 find that this bias is only relevant for the two highest luminosity bins,
and that the correction factor is small compared to the error on the halo mass.
We therefore choose not to model this effect in this Chapter.
112
4.6. STELLAR MASS TREND
4.5.3
Satellite fraction
The lower panel of Figure 4.10 shows the satellite fraction α as a function of
luminosity for both the red and the blue sample. At lower luminosities the
satellite fraction is ∼ 40% for red lenses and as luminosity increases the satellite
fraction decreases. This indicates that a large number of faint red lenses are
satellites inside a larger dark matter halo, consistent with previous findings (e.g.
Mandelbaum et al., 2006b; van Uitert et al., 2011). In the highest luminosity
bins the satellite fraction is difficult to constrain due to the shape of the halo
model satellite terms (green lines in Figure 4.2) becoming indistinguishable from
the central 1-halo term (purple dashed), as discussed in Appendix 4.A. For blue
lenses, the satellite fraction remains low across all luminosities indicating that
almost none of our blue galaxies are satellites, again consistent with previous
findings. This may be a sign that blue galaxies in our analysis are in general
more isolated than red ones, a theory corroborated by the low signal on large
scales for blue galaxies (see Figure 4.20 in Appendix 4.A). Here we have made
no distinction between field galaxies and galaxies residing in a more clustered
environment.
4.6
Stellar mass trend
A more accurate indicator of the baryonic content of a galaxy than its luminosity
is its stellar mass, since luminosity is sensitive to recent star formation. We
therefore study the relation between stellar mass and the dark matter content
in this Section, dividing the lenses into 9 stellar mass bins as illustrated in
Figure 4.13 with details in Table 4.3. As in Section 4.5 we further split each
stellar mass bin into a red and a blue sample using their photometric types to
approximate early- and late-type galaxies.
We measure the galaxy-galaxy lensing signal for each sample as before, and
fit using our halo model with the halo mass M200 and the satellite fraction α
as free parameters. Similarly to the previous section, the results are shown in
Figure 4.14 for all stellar mass bins and for each red and blue lens sample, with
details of the fitted halo model parameters quoted in Table 4.3. In the case of
blue lenses, the two highest stellar mass bins are not well-constrained, due to
a lack of lenses, and we therefore remove them from our analysis. The same
issues with a direct comparison between this analysis and the one presented
in VU11 remain: (1) their halo masses are likely overestimated due to inadequate modelling of the baryonic component, (2) our red and blue samples do
not necessarily correspond to their early- and late-type galaxies, (3) we limit
our analysis to a narrow redshift range of 0.2 < zlens < 0.4 and (4) we have
photometric redshifts for all our objects while VU11 had access to spectroscopic
redshifts but for their lenses only. The latter effect we again account for in a
similar fashion to the procedure described in Section 4.5.1.
An overview of the trends in Figure 4.14 is given in Figure 4.15 for red lenses
and Figure 4.16 for blue. The mean mass in each bin increases with increasing
stellar mass as expected, resulting in an increased signal amplitude. Similar to
what we saw in the luminosity samples in the previous Section, the lower-mass
bins display a bump at scales of ∼ 0.5 h−1 Mpc. Here the lowest bins contain
less massive galaxies than the lowest luminosity bins and the bump is more
113
Table 4.3 Details of the stellar mass bins. (1) Stellar mass range [M⊙ ]; (2) Number of lenses; (3) Mean redshift; (4) Fraction of lenses that
are blue; (5) Mean luminosity for red lenses [1010 L⊙ ]; (6) Mean stellar mass for red lenses [1010 M⊙ ]; (7) Best-fit mean halo mass for red lenses
[1011 h−1 M⊙ ]; (8) Best-fit satellite fraction for red lenses; (9) Mean luminosity for blue lenses [1010 L⊙ ]; (10) Mean stellar mass for blue lenses
[1010 M⊙ ]; (11) Best-fit mean halo mass for blue lenses [1011 h−1 M⊙ ]; (12) Best-fit satellite fraction for blue lenses
Sample
log10 M∗ (1)
nlens (2)
hzi(3)
fblue (4)
(5)
hLred
r i
hM∗red i(6)
Mhred (7)
αred(8)
hLblue
i(9)
r
hM∗blue i(10)
Mhblue (11)
αblue(12)
S1
S2
S3
S4
S5
S6
S7
S8
S9
[9.00,9.50]
[9.50,10.00]
[10.00,10.50]
[10.50,11.00]
[11.00,11.25]
[11.25,11.50]
[11.50,11.75]
[11.75,12.00]
[12.00,12.50]
399730
240732
146657
91556
26942
13287
4481
890
147
0.29
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.29
0.88
0.85
0.73
0.39
0.15
0.07
0.04
0.04
0.20
0.04
0.11
0.37
1.14
2.60
4.51
7.82
13.5
24.5
0.20
0.56
1.95
6.05
13.3
23.2
40.2
68.9
127
0.01+0.01
−0.00
0.07+0.04
−0.03
0.43+0.28
−0.17
3.02+0.96
−0.73
5.75+1.83
−1.39
11.6+3.71
−2.81
25.7+8.17
−6.19
+42.2
121−31.3
+40.1
115−51.8
0.53+0.02
−0.03
0.78+0.02
−0.02
0.58+0.02
−0.02
0.26+0.01
−0.02
0.21+0.02
−0.02
0.20+0.03
−0.03
0.25+0.03
−0.04
0.26+0.07
−0.07
0.59+0.19
−0.19
0.06
0.19
0.57
1.66
4.20
7.31
13.6
25.3
51.9
0.18
0.56
1.73
5.15
12.8
22.1
40.0
72.3
144
0.35+0.10
−0.14
0.50+0.14
−0.20
1.16+0.33
−0.26
1.63+0.47
−0.65
1.16+1.32
−0.83
3.78+4.88
−2.53
0.01+2.31
−0.00
+30.5
10.2−10.2
0.01+9.86
−0.00
0.00+0.01
−0.00
0.02+0.01
−0.01
0.00+0.01
−0.00
0.00+0.01
−0.00
0.11+0.05
−0.05
0.10+0.09
−0.09
0.00+0.22
−0.00
0.15+0.36
−0.30
0.00+0.35
−0.00
4.6. STELLAR MASS TREND
Figure 4.13 Stellar mass distribution in the CFHTLS for lenses with redshifts
0.2 ≤ zlens ≤ 0.4 (black solid histogram). The distribution of red (blue) lenses
is shown in dotted purple (dot-dashed green). Our lens bins are marked with
vertical lines.
pronounced, indicating that most of the galaxies in these low-mass samples are
satellite galaxies. The contribution from nearby haloes is again clearly visible in
the lower-mass blue samples. The two highest-mass bins contain too few lenses
to constrain the signal and have therefore been removed.
4.6.1
Stellar mass scaling relations
The best-fit halo masses and satellite fractions for each stellar mass bin are
shown in Figure 4.17. We have corrected the halo masses for the bias induced
by errors in our photometric redshift estimates using the mean luminosity in
each bin as before. It is clear that the relation between dark matter halo and
stellar mass is different for red and blue lenses as expected. To quantify the
difference, we fit a powerlaw to the lensing signals in each bin simultaneously,
similarly to our treatment of the luminosity bins in the previous Section. The
form of the powerlaw is M200 = M0,M (M∗ /Mpivot )βM with Mpivot = 2×1011 M⊙
as in VU11. We note here that for the lowest red stellar mass bins, though the
halo model fits the data very well (see Figure 4.14), the sample consists of nearly
115
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.14 The weak galaxy-galaxy signal around lenses which have been split
into stellar mass bins according to Table 4.3, modelled using the halo model described in Section 4.3.2. The purple (green) dots represent the measured differential surface density of the red (blue) lenses, and the solid line is the best-fit
halo model. Triangles represent negative points that are included unaltered in
the model fitting procedure, but that have here been moved up to positive values
as a reference. The dotted error bars are the unaltered error bars belonging to
the negative points. The squares represent distance bins containing no objects.
For a detailed decomposition into the halo model components, please refer to
Appendix 4.B.
100% satellite galaxies. It is therefore not a central halo mass associated with
these lenses that is constrained by the halo model and so we exclude the two
116
4.6. STELLAR MASS TREND
Figure 4.15
Best-fit halo models for red lenses for all stellar mass bins.
Figure 4.16
Best-fit halo models for blue lenses for all stellar mass bins.
lowest stellar mass bins from our analysis.
+0.10
The resulting best-fit values for red lenses are M0,M = 1.07−0.06
×1012 h−1 M⊙
+0.10
+0.70
11 −1
and βM = 1.36−0.06 , and for blue lenses M0,M = 3.52−0.70 × 10 h M⊙ and
βM = 0.54+0.06
−0.08 . We show the constraints and best-fit values in Figure 4.18.
The red lenses are clearly better constrained than the blue due to the higher117
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.17 Satellite fraction α and halo mass M200 as a function of stellar
mass. Purple (green) dots represent the results for red (blue) lens galaxies.
quality signal generated by these generally more massive and more abundant
galaxies. Similarly to our luminosity relation, this powerlaw is shallower than
the one found by VU11 but as discussed in the previous section the two analyses differ in ways that make a direct comparison difficult. Primarily the object
selection differs, both in redshift and in defining red and blue lenses, and our
halo masses are in general lower since we account for the baryonic mass in the
lens while VU11 takes only dark matter into account. This also partly explains
the differences in M0,M . Furthermore, the stellar mass estimates we use here
are based on the luminosity-stellar mass relations derived in Bell et al. (2003).
It has since emerged that these relations tend to significantly overestimate the
stellar mass (Zibetti et al., 2009). The accuracy of our stellar masses is thus
somewhat limited, and new estimates will be derived in the near future using
the Zibetti et al. (2009) relations instead.
An effect VU11 does account for, however, is the scatter between mass bins
due to inaccuracies in the stellar mass estimate. Due to the shape of the mass
distribution (see Figure 4.13), objects will preferentially scatter from lowermass to higher-mass bins, biasing our halo masses low. The effect is greatest
at the highest mass end since the distribution tapers off there and as a result
fractionally more low-mass objects will scatter into the higher-mass bin. For
118
4.6. STELLAR MASS TREND
Figure 4.18 Constraints on the powerlaw fits shown in Figure 4.17. In purple
(green) we show the constraints on the fit for red (blue) lenses, with lines representing the 67.8%, 95.4% and 99.7% confidence limits and stars representing the
best-fit value.
their late-type galaxies, roughly corresponding to our blue sample, VU11 applies
a correction of up to 20%, while for their early-type galaxies the correction is
∼ 10% at the low-mass end and reaches ∼ 40% for higher stellar masses. Just
as for the luminosity results, they also convert their best-fit halo mass to mean
halo mass with corrections of up to ∼ 30% for the highest stellar masses. As a
result, their halo mass relation is steepened compared to the uncorrected case,
but this is not enough to explain the differences with our results.
The satellite fraction α as a function of stellar mass is shown in the lower
panel of Figure 4.17 for both red and blue lenses. In the lower-mass red bins,
nearly all lenses are satellites while for higher masses, nearly all are located
centrally in their halo as expected. As discussed in the previous section, this
fraction is difficult to constrain for high masses due to the shape of the satellite
terms. The overall low satellite fraction for blue galaxies, suggesting together
with low large-scale signal that most blue galaxies are isolated, is consistent
with the luminosity results.
119
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
4.7
Discussion and conclusions
In this Chapter we have used high-quality weak lensing data from the CFHTLS
to place constraints on the relation between dark matter halo mass and the
baryonic content of the lenses, quantified through luminosity and stellar mass
estimates. We verified the fidelity of our shear catalogue by comparing our lensing signal with that of two independent shear catalogues, and by using a method
based on seeing variations in the data. The impressive source number density
in this survey has made it possible to achieve tighter constraints than have so
far been attained using previous lensing surveys such as the SDSS or the RCS2.
Splitting our lens samples into red and blue subsamples, we approximated the
trends for early- and late-type galaxies separately. We also extended our study
to lower stellar masses than have been studied before using a halo model such
as the one described in this Chapter. We note, however, that the stellar mass
estimates used in this analysis are somewhat outdated, which may affect the
trends found. In the near future this will be remedied.
As luminosity and stellar mass increases, the halo mass increases as well. For
red lenses, the halo mass increases with greater baryonic content at a higher
rate than for blue galaxies, independent of whether the measure of baryonic
content is luminosity or stellar mass. The two measures thus produce consistent
results. For each we fit powerlaw relations to quantify the rate of increase in
halo mass. We find that for red galaxies, the halo mass scales with luminosity
1.28+0.10
−0.08
as M200 ∝ Lr′
1.36+0.10
−0.06
and with stellar mass as M200 ∝ M∗
, while for
0.50+0.18
0.54+0.06
blue galaxies M200 ∝ Lr′ −0.12 and M200 ∝ M∗ −0.08 . For a fiducial red
galaxy with a luminosity of L0 = 1011 Lr′ ,⊙ we find a halo mass of M200 =
12 −1
3.53+0.29
h M⊙ . This number is lower than the number found by VU11,
−0.29 ×10
but the two analyses differ significantly in object selection. Furthermore, in this
chapter we have included a component of our halo model which was neglected
by VU11: the baryonic component. Since the lensing signal is a response to
the total mass of a system, it is essential to account for baryons in order to
not overestimate the mass contained in the dark matter halo. However, we also
showed that great care has to be taken when including a baryonic component
since doing so has a greater impact on the fitted halo mass than one might
naı̈vely expect due to the complicated interplay between stellar mass, satellite
fraction and halo mass.
For our blue galaxy selection, the satellite fraction is low across all luminosities and stellar masses considered here. The signal at large scales for these
samples is also generally low, indicating that these galaxies are relatively isolated and reside in less clustered environments than the red galaxies do and that
we may be overestimating the bias for these samples. At low luminosity/stellar
mass, a considerable fraction of red galaxies are satellites within a larger dark
matter halo. This fraction decreases steadily with increasing luminosity or stellar mass.
The tight constraints on the relation between baryonic content indicators
and dark matter halo mass achieved in this work will help improve our understanding of the mechanisms behind galaxy formation. If the halo mass threshold
for galaxy formation is accurately known for all galaxies then cosmological simulations can be further improved and phenomena such as the missing satellite
problem may be better studied. Furthermore, by studying red and blue lenses
120
4.A. DETAILED LUMINOSITY BINS
separately we have determined that the bias description which works well for
red galaxies is not optimal for blue galaxies. The environments the two samples
reside in are thus radically different and the difference will have to be taken into
account in the future.
With currently ongoing (e.g. KiDS) and planned (e.g. Euclid) surveys, weak
lensing analyses will become yet more powerful than the one presented in this
Chapter. In preparation for the future there are therefore several sources of uncertainty that should be investigated. As mentioned above, the bias description
may not be optimal for blue lenses and with future data this bias can likely be
constrained directly using galaxy-galaxy observations. Recent simulations have
also indicated that there is a redshift evolution of the halo mass relations, and
this evolution can be studied with weak lensing. Other possible improvements to
the halo model used here include studies of the distribution of satellites within
a galaxy dark matter halo, and investigations of the stripping of satellite haloes.
The analysis presented in this Chapter is already a great improvement on recent
analyses, and with future surveys we will be able to use galaxy-galaxy lensing
to study the connection between baryons and dark matter in exquisite detail.
Acknowledgements
This Chapter is based on observations obtained with MegaPrime/MegaCam, a joint project
of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science
de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the
University of Hawaii. This work is based in part on data products produced at TERAPIX
and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope
Legacy Survey, a collaborative project of NRC and CNRS.
MV acknowledges support from the European DUEL Research-Training Network (MRTNCT-2006-036133) and from the Netherlands Organization for Scientific Research (NWO).
APPENDIX 4.A:
Detailed luminosity bins
In this Appendix we show the decomposition of the best-fit halo model for
red (Figure 4.19) and blue (Figure 4.20) lenses, split in luminosity according
to Table 4.2. Showing the full decomposition is highly informative because it
highlights some of the major trends and clarifies which effects dominate in each
case.
The baryonic component based on the mean stellar mass in each bin (purple dot-dashed line) becomes more dominant for higher luminosities, but the
luminous size of the lenses also increases, making measurement of background
source shapes in the innermost distance bins difficult. Thus it is not possible
to reliably constrain the baryonic component with our data. Never the less,
the effect of including the baryons in our model is an overall lowering of the
dark matter halo profile (purple dashed) compared to without baryons. For the
red lenses we see that a considerable fraction of the sample at lower luminosities necessarily consists of satellite galaxies, since there is a clear bump in the
signal at intermediate scales which has to be accounted for. This satellite fraction continuously drops as luminosity increases, and simultaneously becomes
more difficult to constrain since the combination of the stripped satellite profile
(green dash-dotted) and satellite 1-halo terms (green dashed) becomes almost
indistinguishable from a single NFW profile for high halo masses. This effect
was discussed in more detail in VU11, Appendix C.
121
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
Figure 4.19 The weak galaxy-galaxy signal around red lenses which have been
split into luminosity bins according to Table 4.2, and modelled using the halo
model described in Section 4.3.2. The black dots are the measured differential
surface density, and the black line is the best-fit halo model with the separate
components displayed using the same convention as in Figure 4.2. Grey triangles
represent negative points that are included unaltered in the model fitting procedure, but that have here been moved up to positive values as a reference. The
dotted error bars are the unaltered error bars belonging to the negative points.
The grey squares represent distance bins containing no objects.
For the blue lenses, the signal becomes very noisy for the two highestluminosity bins due to a lack of lenses. These two bins are therefore discarded
from the full analysis in Section 4.5. In general, blue galaxies produce a noisier signal than red galaxies for the same luminosity cuts. This is because blue
lenses are in general less massive, and there are fewer of them which results
in a weaker signal. We also notice that nearly all blue lenses are galaxies located at the centre of their halo, rather than being satellites. This is consistent
with previous findings. It is possible that satellite galaxies in general are redder because they have been stripped of their gas and thus have had their star
formation quenched. It could also mean that most blue galaxies in our analysis
122
4.B. DETAILED STELLAR MASS BINS
Figure 4.20 The weak galaxy-galaxy signal around blue lenses which have been
split into luminosity bins according to Table 4.2, and modelled using the halo
model described in Section 4.3.2. The black dots are the measured differential
surface density, and the black line is the best-fit halo model with the separate
components displayed using the same convention as in Figure 4.2. Grey triangles
represent negative points that are included unaltered in the model fitting procedure, but that have here been moved up to positive values as a reference. The
dotted error bars are the unaltered error bars belonging to the negative points.
The grey squares represent distance bins containing no objects.
are isolated; we have made no distinction between field galaxies and galaxies in
a more clustered environment. If blue galaxies are more isolated than red ones
then the contribution from nearby haloes (dotted lines) would also be less. It is
clear from Figure 4.20 that the large scales are not optimally fit by our model,
and isolation may be one of the reasons since we assume the same mass-bias
relation for blue galaxies as for red. With current data it is not possible to
constrain the bias as a free parameter, but with future wider surveys this could
be done.
123
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
APPENDIX 4.B:
Detailed stellar mass bins
Figure 4.21 The weak galaxy-galaxy signal around red lenses which have been
split into stellar mass bins according to Table 4.3, and modelled using the halo
model described in Section 4.3.2. The black dots are the measured differential
surface density, and the black line is the best-fit halo model with the separate
components displayed using the same convention as in Figure 4.2. Grey triangles
represent negative points that are included unaltered in the model fitting procedure, but that have here been moved up to positive values as a reference. The
dotted error bars are the unaltered error bars belonging to the negative points.
The grey squares represent distance bins containing no objects.
The decomposition of the best-fit halo model for red and blue lenses, divided
124
4.B. DETAILED STELLAR MASS BINS
Figure 4.22 The weak galaxy-galaxy signal around blue lenses which have been
split into stellar mass bins according to Table 4.3, and modelled using the halo
model described in Section 4.3.2. The black dots are the measured differential
surface density, and the black line is the best-fit halo model with the separate
components displayed using the same convention as in Figure 4.2. Grey triangles
represent negative points that are included unaltered in the model fitting procedure, but that have here been moved up to positive values as a reference. The
dotted error bars are the unaltered error bars belonging to the negative points.
The grey squares represent distance bins containing no objects.
using stellar mass as detailed in Table 4.3, is shown in Figures 4.21 and 4.22
respectively.
By construction the baryonic component amplitude (purple dash-dotted line)
125
4. RELATION BETWEEN GALAXY DM HALOES AND BARYONS IN CFHTLS
increases with increasing bin number, and so does the dark matter halo mass
(dashed lines). Note that with our stellar mass selections we push to smaller and
fainter objects, so the objects in the three lowest-mass bins are on average less
massive and less luminous than the galaxies in the faintest luminosity bin. In
these bins, nearly all red galaxies are satellites, while for higher stellar mass bins
the satellite fraction diminishes, a behaviour which is consistent with the trends
we saw for luminosity (Appendix 4.A). For the higher stellar mass bins, as for
the higher luminosity bins, the sum of the satellite stripped and 1-halo terms
result in a profile which resembles a single NFW profile, making the satellite
fraction more difficult to determine. For the blue lenses we run into the same
issues for the highest mass bin as for the highest luminosity bins; the number
of lenses is too small to constrain the halo model and so the bin has to be
discarded. Furthermore, the satellite fraction is low across all blue lens bins
indicating that these lenses are most likely isolated, which is consistent with the
low large-scale signal and with our findings for luminosity.
126
5
Constraining cluster profiles with
weak lensing shear and flexion
Clusters of galaxies are important probes of the background cosmology
since their numbers as a function of mass and redshift are sensitive to
cosmological parameters. Most of the mass in a cluster of galaxies is
in the form of dark matter, and in order to learn about cosmology this
mass has to be accurately estimated given visible observables only. Weak
lensing has the power to constrain mass profiles of dark matter haloes,
on scales ranging from a few kiloparsecs to several megaparsecs. To take
full advantage, the cluster lensing profile has to be accurately modelled
even if the centre of the cluster is not accurately known. In this Chapter
we model the misaligned profile for both the first-order lensing distortion
(shear) and for higher-order distortions known as flexions. We also show
that the slope of the differential surface density can be recovered via a
simple combination of flexions. These flexions are sensitive to smallscale density variations, making them important complements to the
shear. Through a series of tests we find that this is particularly true
when it comes to constraining the misaligned cluster profile.
127
5. CLUSTER PROFILES
5.1
Introduction
Clusters of galaxies are amongst the most massive gravitationally bound structures in the Universe. By studying how the masses and abundances of these
striking galaxy communities evolve, crucial constraints on current theories of
cosmology, and of structure formation, may be yielded. However, N-body simulations consistently show galaxy clusters to be immersed in expansive dark
matter haloes. Thus one of the main challenges of using clusters of galaxies as
a probe for cosmology becomes to accurately predict the total mass of a system
given baryonic observables only.
There are currently a number of methods in use for estimating cluster masses.
The gas trapped in the potential well of a cluster, the intracluster medium
(ICM), may be exploited for this purpose via e.g. the Sunyaev-Zel’dovic (SZ)
effect (Sunyaev & Zeldovich, 1972; Motl et al., 2005; Nagai, 2006) or X-ray
studies (e.g. Reiprich & Böhringer, 2002; Nagai et al., 2007; Mantz et al., 2010).
These techniques do however suffer inaccuracies due to e.g. intervening radio
point sources (Vale & White, 2006) or assumptions such as hydrostatic equilibrium (e.g. Evrard, 1990; Nagai et al., 2007). To estimate the cluster mass from
the dynamics of cluster members is theoretically straight-forward (Biviano &
Girardi, 2003; Rines et al., 2003; Katgert et al., 2004; Rines & Diaferio, 2006)
but it also requires several assumptions on e.g. the dynamical equilibrium of
the cluster. Finally, there is gravitational lensing. Strong gravitational lensing
is powerful when investigating the mass profiles of individual massive clusters
(e.g. Shu et al., 2008; Zitrin et al., 2011), but it requires high mass densities in
the inner regions. The clusters in which strong lensing is observed are therefore not necessarily representative of all clusters, or even of clusters of the same
mass. Weak gravitational lensing, on the other hand, obtains cluster masses
while making no assumptions on the physical state of the cluster, and is observable in all clusters with sufficient background source density (recently Holhjem
et al., 2009; Abate et al., 2009; Sheldon et al., 2009; Okabe et al., 2010; Hoekstra
et al., 2011; Lerchster et al., 2011; Jee et al., 2012). Weak lensing provides a
direct estimate of the total mass of a cluster, irrespective of whether that mass
is baryonic or non-baryonic. This is in essence what makes weak lensing one of
the most promising cluster mass determination methods in use today.
One of the main challenges with weak lensing is that it is subject to degeneracy since it is sensitive not only to the cluster in question but also to any other
intervening structure. However, for the purpose of determining mass-observable
relationships, clusters may be grouped according to their observable properties
and then averaged. This effectively eliminates the degeneracy since foreground
structures no longer add coherent distortions to the weak lensing signal. However, the centre of a cluster is generally difficult to determine. Often it is done
by identifying the brightest cluster galaxy (BCG), but a misidentification of said
BCG would cause the assumed cluster centre to be offset from the true centre
of the dark matter halo. As a result the lensing signal would peak at some distance from the assumed centre. Stacking several clusters where the BCG may
be offset from the true centre by a random amount would therefore affect the
cluster mass estimate.
Fortunately, the offset distribution can be statistically modelled and taken
into account when interpreting the weak lensing signal, as described in Johnston
128
5.2. CLUSTER LENSING FORMALISM
et al. (2007) (from here on referred to as J07). The effect of miscentred BCGs
is subtle in the case of first-order weak lensing (shear), but if not taken into
account it will cause an overestimation of the mass of the average halo in the
cluster sample. The analysis could therefore benefit from the use of higherorder weak lensing distortions known as flexion (Goldberg & Bacon, 2005; Bacon
et al., 2006). Flexion is a measure of the gradient of the lensing convergence,
making it much more sensitive to small-scale density fluctuations than shear
is. As it is an independent method for obtaining the lensing signal it may be
used in conjunction with shear to improve the mass estimate of an ensemble
of clusters. Previously, these higher-order distortions have been used to study
individual clusters (e.g. Okura et al., 2008; Leonard et al., 2011; Cain et al.,
2011) and galaxy-sized haloes (Velander et al. (2011), Chapter 3 of this Thesis).
This Chapter, however, is concerned with modelling the flexion signal for a
cluster ensemble where a significant fraction of BCGs are misaligned with their
associated dark matter haloes, following the general recipe outlined in J07. The
resulting flexion signal is then compared to that of shear, and studied as a
function of cluster mass, cluster redshift and offset distribution.
In Section 5.2 we describe our cluster model by first introducing shear and
flexion in Section 5.2.1 and then giving an overview of the contributions to
our model in Sections 5.2.2 through 5.2.5. In Section 5.3 we investigate the
dependence of our model on halo mass, halo concentration and offset distribution
properties. We conclude in Section 5.4. Throughout this Chapter we assume
the following cosmology (WMAP7; Komatsu et al., 2010): (ΩM , ΩΛ , h, σ8 , w) =
(0.27, 0.73, 0.70, 0.81, −1)
5.2
Cluster lensing formalism
To deduce the mass and density profile of an ensemble of clusters from weak
lensing is qualitatively similar to doing the same for a selection of galaxies (see
Chapters 3 and 4). By averaging the weak lensing signal over enough sources
we can assume that the average background source is circular, and any residual
coherent distortion must be due to lensing. If we then average the signal in
azimuthal bins centred on the lens we can acquire an accurate picture of the
density profile of the total mass in the lens. Worth noting here is that the circular
average makes this type of analysis robust against most systematic errors that
can affect weak lensing measurements, such as insufficiently corrected residual
point-spread function (PSF) anisotropy.
However, interpreting the signal we measure around clusters differs slightly
from the interpretation of the signal around galaxies, mainly because the position that the measured lensing signal should be centred on is generally not
as accurately determined. In clusters, the brightest cluster galaxy (BCG) is
typically assumed to be located at the very centre of the cluster dark matter
halo. This assumption does not hold in all cases. The cluster finding algorithm
may for instance have misidentified the BCG, or the true BCG may be offset
from the halo centre, causing the peak of the lensing signal to be shifted from
the assumed centre.
In the model we present here we will closely follow the procedure outlined in
J07, but extend it from the first-order lensing signal (shear) to the higher-order
signals known as F and G flexion.
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5. CLUSTER PROFILES
5.2.1
Shear and flexion
Before moving onto the interpretation of the lensing signal, we will give a brief
overview of shear and flexion. In the weak lensing regime, the lensed surface
brightness of a source galaxy, f (x), is related to the unlensed surface brightness,
f0 (x), via
1
∂
f (x) ≃ 1 + (A − I)ij xj + Dijk xj xk
f0 (x).
(5.1)
2
∂xi
Here I is the identity matrix, xi denotes lensed coordinates, and Aij is a combination of the lensing convergence κ and the shear γ. The matrix
Dijk =
∂Aij
∂xk
(5.2)
describes how convergence and shear vary across a source image and is the
sum of the two flexions: Dijk = Fijk + Gijk . These two quantities are thus the
derivatives of the convergence and shear fields. The full expressions for Aij ,
Fijk and Gijk may be found in e.g. Bacon et al. (2006).
The shear is a stretch in one direction which is applied to the intrinsic shape
of a source galaxy. This type of lensing distortion has been used in weak lensing
analyses for nearly two decades. Flexion on the other hand was first studied by
Goldberg & Bacon (2005) and then further developed by Bacon et al. (2006).
As observations are increasing in accuracy, flexion is now gaining in popularity
as a tool for cluster studies (e.g. Okura et al., 2008; Er et al., 2010; Bacon et al.,
2010; Leonard et al., 2011; Cain et al., 2011). The first flexion, F flexion, is a
skewness of the brightness profile which is reminiscent of a centroid shift, while
the second flexion, G flexion, is a triangular distortion. The signal-to-noise
(S/N) of flexion is proportional to the size of the source, and Goldberg & Bacon
(2005) estimate that the S/N of the shear and flexion signals will be comparable
at a distance θ ≃ 4rsource from the lens with the flexion S/N being higher within
this limit. Furthermore, for relaxed sources there is no intrinsic flexion signal
expected, making the shape noise small. Thus the theoretical flexion S/N will
exceed the shear S/N on small scales. However, measurement biases are still
fairly large for flexion while biases for shear are at sub-percentage level.
As noted in Chapter 4, assuming circular symmetry the amplitude of the
azimuthally averaged tangential shear, γt , is directly related to the differential
surface density ∆Σ(R) via
∆Σ(R) = Σ(< R) − Σ(R) = Σcrit γt (R)
(5.3)
where Σcrit is the critical surface density
Σcrit =
c2 D s
4πG Dl Dls
(5.4)
with Ds , Dl and Dls the angular diameter distance to the source, to the lens and
between the lens and the source respectively. The differential surface density is
the difference between Σ(< R), the average surface density inside a disk of radius R, and Σ(R), the azimuthal average of the surface density in a thin annulus
of radius R. This is a convenient measure of the amplitude of the shear signal
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5.2. CLUSTER LENSING FORMALISM
since it is independent of the specific configuration between lens and source.
Equivalent quantities for the flexion signals are F♦ (R) ≡ (Σcrit /Dl )Ft (R) and
G♦ (R) ≡ (Σcrit /Dl )Gt (R), where the dependence on lens redshift is more apparent due to the Dl factor. Σ and ∆Σ both have units of mass per area, while
the equivalent flexion quantities gain an extra inverse length unit (here Ft and
Gt have units of radians−1 ). For the rest of this Chapter, all shear and flexion
quantities will refer to the tangential signal, and so we will drop the subscript
t.
The two flexion quantities are related to the surface density Σ as follows
(Lasky & Fluke, 2009):
F♦ (R) =
G♦ (R) =
dΣ(R)
dR
dΣ(R) 4Σ(R) 4M (R)
−
+
dR
R
πR3
−
(5.5)
(5.6)
where M (R) is the projected mass distribution, i.e. the area integral of the
surface density Σ(R), and the negative sign for the F flexion signal depends on
the definition of tangential flexion.
The two flexion quantities may also be directly related to the differential
surface density ∆Σ:
F♦ (R) =
G♦ (R) =
2∆Σ(R)
+ ∆Σ′ (R)
R
2∆Σ(R)
− ∆Σ′ (R)
R
(5.7)
(5.8)
where ∆Σ′ (R) is the derivative of ∆Σ with respect to R. It is clear from this
that we can recover ∆Σ either through shear (Equation 5.3) or through a linear
combination of flexions:
∆Σ(R) =
R
(F♦ (R) + G♦ (R))
4
(5.9)
Similarly, we can obtain the slope of ∆Σ by combining the flexions in a different
way:
1
∆Σ′ (R) = (F♦ (R) − G♦ (R))
(5.10)
2
These equations highlight yet again that the flexions are an independent measure of the surface density, and a direct measure of its slope, and therefore an
important complement to shear.
5.2.2
Contribution from the BCG
Similarly to our treatment of the baryonic lens galaxies in our galaxy-galaxy
halo model (see Section 4.3.2, page 98), we model the contribution from the
BCG as a point source. The surface density for a point source is simply a delta
function which is a reasonable approximation since the individual galaxy is small
compared to the cluster halo:
Σ(R) = δD (R)
(5.11)
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5. CLUSTER PROFILES
The shears and flexions for the baryonic contribution are then given by (Lasky
& Fluke, 2009)
∆Σb (R) =
F♦b (R)
=
G♦b (R) =
M0
πR2
0
4M0
πR3
(5.12)
(5.13)
(5.14)
where M0 could be fixed to e.g. the estimated stellar mass of the BCG or left as
a free parameter when fitting the model. Note that the F flexion is zero in this
case since it is the gradient of the surface density, so there is no contribution
from the BCG to the F flexion cluster lensing signal.
5.2.3
Contribution from centred cluster dark matter haloes
To model the cluster dark matter halo, we will assume that the halo density
distribution is well described by an NFW profile (Navarro, Frenk, & White,
1996) in terms of the three-dimensional radius r:
ρ(r) =
δc ρc (z)
(r/rs )(1 + r/rs )2
(5.15)
where δc is an amplitude determined by the NFW concentration parameter,
cNFW , ρc (z) is the critical density for closure at the cluster redshift and rs is a
scale radius. This density profile is ∝ r−1 on small scales (r ≪ rs ) and ∝ r−3
on large scales (r ≫ rs ). In our description of the cluster model we will define
all quantities in terms of a virial radius r200 within which the mean density is
200 times the critical density. The enclosed mass is then
4 3
M200 = 200ρc(z) πr200
3
and the NFW concentration parameter at this radius is
−0.084
M200
r200
−0.47
= 5.71
(1 + z)
c200 =
rs
2 × 1012 h−1 M⊙
(5.16)
(5.17)
where the second relationship was determined by Duffy et al. (2008) using Nbody simulations based on a WMAP5 cosmology. In this Chapter we generally
use the more recent WMAP7 cosmology, but the difference is negligible. For
a discussion on other choices of virial radii, please refer to J07. Analytical
expressions exist for the NFW halo description, both for surface density Σ(R)
and shear γ(R) (see e.g. Wright & Brainerd, 2000), and for the two flexions
F (R) and G(R) (see Bacon et al., 2006; Lasky & Fluke, 2009). The contribution
from the well-centred population in our cluster sample is then simply Σc (R) =
ΣNFW (R).
5.2.4
Contribution from a cluster population with miscentred BCGs
It is likely that for a fraction of clusters in a given sample the assumed centre will
be offset from the true centre of the dark matter halo. In this Chapter, we will
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5.2. CLUSTER LENSING FORMALISM
Figure 5.1 The effect of miscentred BCGs. The lighter green lines represent a
centred NFW profile while the darker purple lines represent the corresponding average profiles for a miscentred population with offsets drawn from a Gaussian distribution with width σs = 0.42 h−1 Mpc. In order, the solid, dashed, dash-dotted
and dotted lines represent Σ, ∆Σ, F♦ and G♦ , where the two former signals have
units of h M⊙ pc−2 and the two latter signals have units of Mpc−1 h M⊙ pc−2 .
assume that the position of the centre of every cluster dark matter halo coincides
with the associated BCG position so that any offset between the assumed and
true halo centres stems from a misidentification of the BCG. There are other
possible reasons for a misalignment, such as the BCG truly being offset from
the centre of its halo. In what follows here, this assumption may affect the
miscentrering distribution used but the principles remain the same.
For our model we will largely follow the procedure outlined in J07. The
azimuthally averaged surface density Σ(R) of a miscentred halo is given by
Σ(R|Rs ) =
1
2π
Z
0
2π
Σ
p
R2 + Rs2 + 2RRs cos(θ) dθ
(5.18)
(Yang et al., 2006), where Rs is the 2D offset in the lens plane. This expression
applies to a single misaligned halo, but for an ensemble the offsets will follow
some distribution. Using N-body simulations, J07 find for their maxBCG sample that the distribution of miscentred BCGs is well described by a Gaussian
distribution:
"
2 #
1 Rs
Rs
(5.19)
P (Rs ) = 2 exp −
σs
2 σs
with width σs = 0.42 h−1 Mpc. For our model we will keep the shape of the
distribution the same as above but allow σs to vary. The resulting mean surface
133
5. CLUSTER PROFILES
density profile for the miscentred population is then given by
Z
Σs (R) = P (Rs )ΣNFW (R|Rs )dRs
(5.20)
where we have explicitly noted our assumption that all dark matter haloes are
NFW profiles.
From this surface density we can then numerically obtain the shear and flexion profiles using Equations 5.3, 5.5 and 5.6 (or, equivalently, Equations 5.7 and
5.8). In Figure 5.1 we show the four signals as a function of distance R from
the assumed cluster centre, for a cluster mass of 1013 h−1 Mpc, cluster redshift
zl = 0.22 and offset distribution dispersion (Equation 5.19) σs = 0.42 h−1 Mpc.
As noted by J07, the effect of miscentrering on the Σ(R) essentially flattens
the profile on small scales (purple solid), making it a mass-sheet which causes
very little shear. This is the reason why the miscentred shear profile (purple
dashed) is strongly suppressed at small scales. The shear then peaks at a radius R ∼ 2.5σs and, as we will show in Section 5.3.3, this approximate scale
relationship holds for all σs . The miscentred flexion signals are also strongly
suppressed at small scales. The F flexion signal (purple dash-dotted) peaks at
slightly lower cluster-centric distances than shear, while the peak of the G flexion signal (purple dotted) roughly coincides with the shear peak. The G flexion
is in fact the most strongly affected signal and would thus be an important aid
in determining the width of the offset distribution. Note that, given the appropriate units of Dl , the amplitudes of the four signals are similar at intermediate
scales. For reference we have also included the corresponding centred signals
(green) where we see the by now well-known behaviour of the flexion signals;
F♦ falls off quickly with distance while G♦ has a greater range.
5.2.5
Other contributions
There are other contributions to the lensing signal to be taken into account.
Neighbouring haloes contribute significant signal on large scales (& 5 h−1 Mpc)
similar to the 2-halo term in our galaxy-galaxy halo model (Section 4.3.2,
page 98), but since we are focusing this Chapter on the effect of BCG misalignment which is only significant on intermediate scales, we will not take this
term into account here. Similarly, we will not model the non-linear term due
to the basic weak lensing assumption of κ ≪ 1 no longer being valid, as noted
in J07 since this term is only relevant on small scales (. 0.1 h−1 Mpc), and in
all but the most massive cases contributes significantly less to the shear signal
than the baryonic component does. For flexion it could prove important though,
since this is the region in which this lensing measurement is most sensitive, so
in full cluster models it should be taken into account.
Finally we summarise our cluster model. Assuming that the BCG has been
correctly identified in a fraction fc of the clusters, the average profile will be
given by
∆Σ(R) = ∆Σb (R) + fc ∆Σc (R) + (1 − fc )∆Σs (R)
(5.21)
and similarly for the flexions, where ∆Σb (R) is the baryonic component, ∆Σc (R)
is the contribution from accurately centred haloes and ∆Σs (R) is the contribution from haloes that are not centred.
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5.3. RESULTS
Table 5.1
Fiducial values of the cluster model parameters.
BCG stellar mass M∗
Halo mass M200
Halo concentration c200
Cluster redshift z
Offset distribution width σs
Offset fraction fc
5.3
5.00 × 1011 h−1 M⊙
1.00 × 1014 h−1 M⊙
As in Equation 5.17
0.34
0.42 h−1 Mpc
0.50
Results
We now turn to studying the change in the lensing signal profiles as we vary
the properties of our cluster model. In the following sections we will vary the
halo mass M200 , the halo concentration c200 and the offset distribution width
σs to assess and contrast the impact of BCG misalignment on the different
types of lensing distortions, i.e. Σ, ∆Σ, ∆Σ′ , F♦ and G♦ . The first signal, the
surface density, is not directly observable through lensing but is included in this
study for reference since it is a more intuitive physical quantity than the other
four. The parameters that are not explicitly varied in each section will be kept
constant at the values listed in Table 5.1. The offset distribution width σs is set
to the value found by J07 to approximately fit most of their maxBCG cluster
ensembles. They also found that an offset fraction of fc ∼ 0.60 fit their samples
well, but we will set our fc to 0.5 to allow for a direct comparison of the centred
and miscentred profiles in the following sections.
In general, as can be seen in Figure 5.1, miscentred haloes will cause a
flattening of the Σ profile on small scales, and suppress the other three signals
(for clarity we do not include ∆Σ′ in that figure). On intermediate scales,
however, there is a slight increase, or ‘bump’, in the signal amplitude for Σ, F♦
and G♦ , something which is not noticeable or detectable in the ∆Σ profile. This
bump, as will be seen in the following sections, causes a fairly sharp feature in the
flexion profiles at the position of the offcentre profile peak, potentially allowing
for an improved estimate of the distribution width σs through its relationship
with the peak location.
5.3.1
Mass dependence
We first investigate how the different signals corresponding to our cluster model
change with halo mass. Figures 5.2 to 5.4 show the five lensing signals Σ, ∆Σ,
∆Σ′ , F♦ and G♦ as a function of distance R from the assumed cluster centre,
for five different halo masses M200 . All other parameters are kept constant at
the fiducial values given in Table 5.1. The solid line in each panel represents the
total signal for a cluster ensemble where a fraction fc = 0.5 of the clusters have
misaligned BCGs, while the dotted, dash-dotted and dashed lines represent the
components of the signal as specified in the captions.
As expected, the amplitude increases with increasing halo mass. We also
clearly see that the bump in the offcentre profile at ∼ 1 h−1 Mpc becomes
more pronounced for lower masses. This trend, which is visible in the Σ signal
(Figure 5.2), is not discernible in the ∆Σ (Figure 5.3) or G♦ (Figure 5.4) signals.
It is, however, clearly prominent in the F♦ signal (Figure 5.4) and, particularly,
135
5. CLUSTER PROFILES
Figure 5.2 Dependence of the cluster model surface density Σ on halo mass.
The dash-dotted and dashed lines represent the cluster model components (centred
clusters and miscentred clusters respectively) while the solid line is the sum of the
components. Here, the fraction of miscentred clusters is fc = 0.5. The signal
amplitude increases with mass, colour coded as per the legend. Note that in the
case of Σ there is no contribution from the BCG other than at R = 0.
Figure 5.3 Dependence of the cluster model differential surface density ∆Σ
(left panel) and its derivative ∆Σ′ (right panel) on halo mass. Note the different
units as specified in the legend, and that ∆Σ′ has been negated since it is mostly
negative. The dotted, dash-dotted and dashed lines represent the cluster model
components (BCG, centred clusters and miscentred clusters respectively) while the
solid line is the sum of the components. Here, the fraction of miscentred clusters
is fc = 0.5. The signal amplitude increases with mass, colour coded as per the
legend, with thick purple (thin green) representing positive (negative) quantities.
Only the inner dashed lines in the right panel are negative.
in the combination of the two flexion signals ∆Σ′ (Figure 5.3) where for the
latter it turns into a dip due to the switch between positive and negative slope
of ∆Σ for the offcentred distribution, further illustrating the sensitivity of these
types of distortion to the underlying density variations. The feature in the
136
5.3. RESULTS
Figure 5.4 Dependence of the cluster model flexions F♦ (left panel) and G♦
(right panel) on halo mass. The dotted, dash-dotted and dashed lines represent
the cluster model components (BCG, centred clusters and miscentred clusters
respectively) while the solid line is the sum of the components. Note that there
is no contribution from the BCG to F♦ . Here, the fraction of miscentred clusters
is fc = 0.5. The signal amplitude increases with mass, colour coded as per the
legend, with thick purple (thin green) representing positive (negative) quantities.
Only the highest mass bin has significant negative signal in the inner regions.
signal is always located at the same cluster-centric distance for all masses since
it only depends on the width of the misalignment distribution. This is also the
explanation for the bump becoming less prominent with increasing mass in all
signals but ∆Σ′ ; we force the distribution width to σs = 0.42 h−1 Mpc, meaning
that the Σs profile is roughly flat out to that distance. For the highest mass
profile, where M200 = 1015 h−1 M⊙ , the scale radius is rs = 0.49 h−1 Mpc & σs
and this allows the flat Σs distribution to transition into the large-scale powerlaw
in a smooth fashion. For the lowest mass bin, however, M200 = 1011 h−1 M⊙
and the scale radius is rs = 0.01 h−1 Mpc ≪ σs . This causes the apparent bump
at R ∼ 2.5σs , which in turn causes the turnover in the derived signals ∆Σs , F♦s
s
and G♦s , and the switch from positive to negative in ∆Σ′ . J07 do claim that the
offset width is roughly independent of cluster richness, so they give this fitting
parameter a strong prior centred on our fiducial value. For their lowest richness
bins the feature should thus be clearly discernible in a flexion measurement, and
particularly in the combination F♦ − G♦ ∝ ∆Σ′ , given high-quality data.
5.3.2
Concentration dependence
We now turn to studying the change in the cluster profiles as the NFW concentration parameter cNFW is varied. The resulting profiles are shown in Figures 5.5
to 5.7. As concentration increases the NFW scaling radius rs , or pivot point,
is forced to decrease in order to keep the physical scale of the halo, r200 , the
same (see Equation 5.17). Thus the amplitude of Σ increases on small scales
but decreases on large scales, seen most clearly in the accurately centred halo
profiles. The miscentred profiles have similar amplitudes independent of concentration within R ∼ 2.5σs , but on larger scales they follow the same trend as
the centred profiles. Thus the differences between different c200 become amplified on large scales, making the feature at ∼ 1 Mpc more prominent for higher
concentrations. This is also true for the flexions, but as before the trend is
reversed for ∆Σ′ ; the feature becomes less prominent as the transition between
137
5. CLUSTER PROFILES
Figure 5.5 Dependence of the cluster model surface density Σ on halo concentration. The dash-dotted and dashed lines represent the cluster model components
(centred clusters and miscentred clusters respectively) while the solid line is the
sum of the components. Here, the fraction of miscentred clusters is fc = 0.5. The
signal amplitude increases with concentration in the inner regions, colour coded
as per the legend. Note that in the case of Σ there is no contribution from the
BCG other than at R = 0.
Figure 5.6 Dependence of the cluster model differential surface density ∆Σ
(left panel) and its derivative ∆Σ′ (right panel) on halo concentration. Note
the different units as specified in the legend, and that ∆Σ′ has been negated
since it is mostly negative. The dotted, dash-dotted and dashed lines represent
the cluster model components (BCG, centred clusters and miscentred clusters
respectively) while the solid line is the sum of the components. Here, the fraction of
miscentred clusters is fc = 0.5. The signal amplitude increases with concentration,
colour coded as per the legend, with thick purple (thin green) representing positive
(negative) quantities. Only the inner dashed lines in the right panel are negative.
positive and negative ∆Σ slopes becomes sharper. Assuming the Duffy et al.
(2008) relationship between concentration, mass and redshift (Equation 5.17),
the c200 for a halo mass equal to our fiducial mass would range between ∼ 4 at
138
5.3. RESULTS
Figure 5.7 Dependence of the cluster model flexions F♦ (left panel) and G♦
(right panel) on halo concentration. The dotted, dash-dotted and dashed lines
represent the cluster model components (BCG, centred clusters and miscentred
clusters respectively) while the solid line is the sum of the components. Note that
there is no contribution from the BCG to F♦ . Here, the fraction of miscentred
clusters is fc = 0.5. The signal amplitude increases with concentration, colour
coded as per the legend.
low redshifts and ∼ 3 at z = 0.8.
5.3.3
Offset width dependence
Figure 5.8 Dependence of the cluster model surface density Σ on the width of
the miscentrering distribution. The dash-dotted and dashed lines represent the
cluster model components (centred clusters and miscentred clusters respectively)
while the solid line is the sum of the components. Here, the fraction of miscentred
clusters is fc = 0.5. The signal amplitude decreases with distribution width in the
inner regions, colour coded as per the legend. Note that in the case of Σ there is
no contribution from the BCG other than at R = 0.
139
5. CLUSTER PROFILES
Figure 5.9 Dependence of the cluster model differential surface density ∆Σ (left
panel) and its derivative ∆Σ′ (right panel) on the width of the miscentrering distribution. Note the different units as specified in the legend, and that ∆Σ′ has
been negated since it is mostly negative. The dotted, dash-dotted and dashed lines
represent the cluster model components (BCG, centred clusters and miscentred
clusters respectively) while the solid line is the sum of the components. Here, the
fraction of miscentred clusters is fc = 0.5. The signal amplitude changes with
distribution width, colour coded as per the legend, with purple (green) representing positive (negative) quantities. Only the inner regions in the right panel are
negative.
Figure 5.10 Dependence of the cluster model flexions F♦ (left panel) and G♦
(right panel) on the width of the miscentrering distribution. The dotted, dashdotted and dashed lines represent the cluster model components (BCG, centred
clusters and miscentred clusters respectively) while the solid line is the sum of the
components. Note that there is no contribution from the BCG to F♦ . Here, the
fraction of miscentred clusters is fc = 0.5. The signal amplitude decreases with
distribution width in the inner regions, colour coded as per the legend.
Finally we study the lensing signals due to misaligned BCGs as the offset
distribution width σs is varied, and assess the impact on the composite signal,
as shown in Figures 5.8 to 5.10. As the offset width increases, the Σs profile is
flattened to larger scales, causing the features in the other four signals to shift,
closely following the relation Rpeak ∼ 2.5σs pointed out by J07. What we also
notice is that for the greatest offset distribution widths, the offset signal never
quite reaches the amplitude of the centred signal, even on the largest scales. Additionally, the bump in the total Σ and flexion signals becomes more pronounced
for increased σs due to the same mechanisms described in the previous Section.
Here we can clearly see the sensitivity of the flexions to the relative sizes of σs
140
5.4. CONCLUSIONS
and the NFW scale radius rs . For the largest offsets, where σs ≫ rs , prominent
features are visible in both the flexion signals and ∆Σ′ , and as mentioned in the
previous section, these features could be visible in reasonable cluster samples.
5.4
Conclusions
In this Chapter we have closely followed the initial work of Johnston et al. (2007)
and derived model cluster profiles for the two flexions, F and G, for an ensemble
of clusters where a significant fraction have a misidentified dark matter halo
centre. We have compared these profiles to the corresponding surface density Σ
and shear γ (or ∆Σ) profiles and found that both flexions are promising tools
for determining properties of misalignment distributions, particularly if they are
combined to produce ∆Σ′ , and thus important in removing systematic errors
when determining the overall halo mass. In theory, the signal-to-noise for flexion
is higher than for shear on small scales, though this is currently limited by the
accuracy of shape measurement software.
We studied the lensing profiles as a function of halo mass, halo concentration and offset distribution width and found that for feasible cluster samples
the flexion signals in different combinations will display clear indicators of the
true offset distribution widths, something shear is less sensitive to. We also verified that the peak of the offset distribution will be located at a cluster-centric
distance of Rpeak ∼ 2.5σs , as claimed by J07.
Our cluster lensing profiles are a composite of the contribution from the
BCG, from the accurately centred haloes and from the misaligned haloes. In the
near future we plan to include further components in our flexion signals, such as
the contribution on large scales from neighbouring haloes, and the contribution
on small scales due to non-linearity in the shear. This latter component may
turn out to be more important for the flexions than it is for shear, since this
is at scales where flexion has the most power. We will also investigate the
use of combinations of the two flexion signals to produce new measures of the
underlying density distribution. In conclusion, modelling the cluster density
profile using all three orthogonal lensing observables in conjunction will better
constrain the offset distribution and improve halo mass estimates, which in turn
will yield a greater understanding of scaling relations important to cosmology.
Acknowledgements
MV acknowledges support from the European DUEL Research-Training Network (MRTNCT-2006-036133) and from the Netherlands Organization for Scientific Research (NWO).
141
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156
Nederlandse samenvatting
Donkere materie en kosmologie
Het lijkt misschien verwonderlijk, maar de sterren en sterrenstelsels die we zien
als we omhoog kijken zijn slechts een klein deel van waar ons universum werkelijk
uit bestaat. Zelfs de meest geavanceerde telescopen op aarde of in de ruimte kunnen de echte massa die zich rondom alle bekende lichtbronnen bevindt niet direct
waarnemen. Deze onzichtbare materie beı̈nvloedt echter door zijn zwaartekracht
alles wat we zien, van onze eigen melkweg tot de rand van het universum, en
dat is hoe we van het bestaan ervan af weten. Omdat het geen licht uitzendt
of absorbeert kunnen we het niet ‘zien’ in de traditionele zin, en het heeft dan
ook de naam donkere materie gekregen.
Kosmologie is de studie van de oorsprong en de ontwikkeling van het heelal. Aangezien donkere materie veel invloed heeft op dit proces vanaf het begin
13,7 miljard jaar geleden, zijn wij als kosmologen vooral geı̈nteresseerd in hoe
het zich gedraagt in verschillende omgevingen. Simulaties die gebruik maken
van alle ingewikkelde natuurkundige wetten die we tot nu toe hebben verzameld laten ons zien dat de donkere materie zich verdeelt in een netwerk van
filamenten dat bekend staat als het Kosmische Web (zie Figuur 7.1). Wanneer
de draden elkaar kruisen ontstaan grote concentraties, of halo’s, van donkere
materie waarin de sterren en sterrenstelsels zich verzamelen. Door observaties
kunnen we kaarten met de verdeling van donkere materie maken, en door het
vergelijken van deze kaarten met de voorspellingen van simulaties kunnen we
bepalen of onze natuurkundige wetten juist zijn of niet.
Er zijn veel manieren om donkere materie te bestuderen, en verschillende
technologieën worden gebruikt voor verschillende ruimte-omgevingen, maar een
techniek die in de meeste gevallen kan worden toegepast is zwakke zwaartekrachtlenswerking. Zwaartekrachtlenzen gebruiken, zoals de naam al aangeeft,
de zwaartekracht om het licht van sterrenstelsels in de verre achtergrond te
buigen in een proces dat vergelijkbaar is met een gewone lens in, bijvoorbeeld,
brillen. Een zwaartekrachtlens kan elke verzameling van massa zijn, zoals een
cluster van sterrenstelsels, een enkel sterrenstelsel of zelfs maar een dun donker
filament in het Kosmische Web. Wanneer de materie zeer geconcentreerd is,
veroorzaakt die een ernstige verstoring van het beeld van de erachter gelegen
sterrenstelsels. Dit resulteert in de prachtige bogen die men kan zien in het
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NEDERLANDSE SAMENVATTING
Figuur 7.1 De voorspelling van de Millenniumsimulatie voor het heelal zoals
het er vandaag uitziet, gemaakt door Volker Springel en zijn medewerkers. Wat
we zien zijn de verbonden filamenten van het Kosmische Web, met een ophoping
van donkere materie in het midden. De centrale concentratie komt overeen met
een zeer rijk cluster van sterrenstelsels.
Figuur 7.2 Voorbeeld van een zwaartekrachtlens: de rijke cluster van sterrenstelsels Abell 2218 afgebeeld door de ruimtetelescoop Hubble Space Telescope in
1999. Afbeelding: NASA/ESA, A. Fruchter en het ERO Team (STScI, ST-ECF).
cluster Abell 2218 (Figuur 7.2). Omdat een grotere massaconcentratie meer
vervorming veroorzaakt, kunnen we door gebruik te maken van deze bogen
afleiden waar de donkere materie zich bevindt. Wanneer de massaconcentratie
laag is, hetgeen bijna overal het geval is, kunnen we zwakke vervormingen door
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zwaartekrachtlenswerking, genoemd shear, gebruiken om statistisch te bepalen
waar de donkere materie is. Dit is het regime waar de zwakke lenswerkingtechniek zich mee bezig houdt.
Met dit Proefschrift heb ik ernaar gestreefd om onze kennis over de verdeling
van materie in sterrenstelsels en clusters van sterrenstelsels te vergroten, door
zowel het ontwikkelen van de theoretische kant van de zwakke zwaartekrachtlenswerking, als door het gebruik van grote optische waarneemprogramma’s om
de zwakke zwaartekrachtlenswerkingeffect in de werkelijkheid te observeren. Ik
zet mijn werk uiteen in de vier afzonderlijke maar verwante Hoofdstukken die
hieronder worden samengevat.
Dit Proefschrift
Ik begin dit Proefschrift met een kort overzicht van de huidige status van kosmologie in Hoofdstuk 1, en beschrijf de verschillende manieren om meer te leren
over het heelal als geheel. Dit Hoofdstuk bevat ook een gedetailleerdere kennismaking met de zwakke zwaartekrachtlenswerking en de software die beschikbaar
is om de zwakke vervormingen van de achtergrond sterrenstelsels te meten.
Daarna volgen de Hoofdstukken waarin ik mijn onderzoek van de afgelopen vier
jaar in detail beschrijf, waarvan sommige al zijn gepubliceerd in het tijdschrift
Monthly Notices of the Royal Astronomical Society (MNRAS). Ik vat hier deze
Hoofdstukken samen.
Hoofdstuk 2: Een nieuwe vormmeting-techniek, en de toepassing ervan op sterrenstelsels met kleurgradiënten in de context van zwakke
zwaartekrachtlenswerking waarneemprogramma’s
Zwakke zwaartekrachtlenswerking, als een van de meest krachtige methoden
van kosmologie, is de belangrijkste drijvende kracht achter de grootste optische waarneemprogramma’s die gepland zijn in de nabije toekomst. De statistische aard van de methode vereist de analyse van een groot aantal achtergrond
sterrenstelsels, en de minimale vervormingen vereisen waarnemingen van hoge
kwaliteit en uiterst precieze vervormingsmetingen. Software voor de zwakke
zwaartekrachtlenswerking techniek moet dus zowel snel als accuraat zijn, en
zo’n software pakket wordt geı̈ntroduceerd en getest in dit Hoofdstuk. Deze
MV pipeline heeft bewezen zeer competitief te zijn, met extra voordeel de mogelijkheid om lensvervormingen van hogere orde dan shear, oftewel flexion, te
meten. De tests beschreven in dit Hoofdstuk beslaan zowel monochromatische
als niet-monochromatische simulaties, waarbij de laatste zijn opgenomen om
het effect van een golflengte-afhankelijke puntspreidingsfunctie (PSF) veroorzaakt door de telescoop te bepalen. Aangezien de meeste sterrenstelsels kleurgradiënten hebben, met in de kern een andere kleur dan aan de buitenkant, heeft
een golflengte-afhankelijke PSF en variërende invloed op verschillende delen van
het waargenomen beeld van het sterrenstelsel. Hierdoor kan een bias worden
geı̈ntroduceerd wanneer niet precies voor de PSF kan worden gecorrigeerd. Door
het creëren van simulaties op basis van werkelijke sterrenstelsels, waargenomen
in twee verschillende filters, zien we dat de extra bias die wordt veroorzaakt door
dit effect niet erger is dan de bias in de meetsoftware. Wij stellen uit onze tests
vast dat als er voldoende trainingsgegevens beschikbaar zijn, we waarschijnlijk
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het kleurgradiënteffect voldoende nauwkeurig kunnen karakteriseren om hiermee
toekomstige onderzoeken, zoals de Euclid ruimtemissie, wiens lancering gepland
staat voor 2019, te corrigeren.
Hoofdstuk 3: Studie van de galactische donkere materie halo’s in
het COSMOS onderzoek door middel van zwakke zwaartekrachtlenswerking flexion
De huidige theorieën van structuurvorming voorspellen dat sterrenstelsels ingebed zijn in uitgestrekte donkere materie halo’s. Van deze halo’s wordt verwacht
dat zij een specifiek dichtheid profiel hebben, en met zwakke zwaartekrachtlenswerking kunnen we deze profielen meten op verschillende schalen. Op kleine
schaal voegen de hogere-orde vervormingen, bekend als flexion, belangrijke informatie toe aan de zwakke zwaartekrachtlenswerking metingen. We presenteren
in dit Hoofdstuk de eerste detectie van het flexionsignaal rond sterrenstelsels in
data die in de ruimte is waargenomen. Het signaal is gemeten met de MV
pipeline, geı̈ntroduceerd en getest in Hoofdstuk 2. Wij combineren dit flexionsignaal met shear om het gemiddelde dichtheid profiel van sterrenstelsels in het
Hubble Space Telescope COSMOS onderzoek nauwkeurig te bepalen. We tonen
ook aan dat het licht uitgezonden door nabijgelegen lichtbronnen een aanzienlijk
storend effect op flexion metingen kan hebben. Na correctie voor de invloed van
het licht van de lens sterrenstelsels, tonen we aan dat het opnemen van flexion
sterkere beperkingen geeft op de dichtheid profielen dan het gebruik van louter
shear.
Hoofdstuk 4: De relatie tussen baryonen en de donkere materie halo’s
van sterrenstelsels in de CFHTLS bepaald met zwakke zwaartekrachtlenswerking
Omdat donkere materie halo’s zo uitgebreid zijn is het belangrijk om ze zowel
op grote als kleine schaal te onderzoeken om zo meer te leren over de invloed die
donkere materie heeft op normale ‘baryonische’ materie, en vice versa. Zwakke
zwaartekrachtlenswerking biedt de mogelijkheid om dit te doen omdat de technologie niet alleen bruikbaar is op verschillende schalen, maar ook omdat ze
onafhankelijk is van de aard van de materie die onderzocht wordt. In dit Hoofdstuk presenteren we een studie van de algemene eigenschappen van donkere
halo’s van sterrenstelsels als functie van de eigenschappen van het sterrenstelsel
dat ermee geassocieerd wordt, op basis van gegevens uit een van de grootste
voltooide zwakke zwaartekrachtlenswerking waarneemprogramma’s tot nu toe:
CFHTLS. We verdelen de lens sterrenstelsels in een rode en een blauwe set en
bevestigen dat er een duidelijk verband bestaat tussen de donkere halo-massa
en de helderheid, en tussen halo-massa en stellaire massa. Deze relaties zijn
verschillend voor blauwe en rode sterrenstelsels, en we vinden ook aanwijzingen dat blauwe sterrenstelsels zich in minder geclusterde omgevingen dan rode
sterrenstelsels bevinden.
Hoofdstuk 5: Het beperken van cluster profielen door middel van
zwakke zwaartekrachtlenswerking shear en flexion
Clusters van sterrenstelsels zijn belangrijk voor ons begrip van de kosmologie,
omdat hun aantal als functie van massa en afstand van ons afhankelijk is van de
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NEDERLANDSE SAMENVATTING
kosmologische parameters. De meerderheid van de massa van een cluster van
sterrenstelsels is in de vorm van donkere materie. Om meer te leren over kosmologie willen we deze massa zorgvuldig bepalen, maar alleen zichtbare eigenschappen zijn beschikbaar. Zwakke zwaartekrachtlenswerking heeft, zoals reeds
vermeld, de mogelijkheid om de dichtheid profielen van donkere halo’s in kaart
te brengen op zowel grote als kleine schalen. Om hiervan ten volle te profiteren,
moeten de profielen van de donkere halo’s zorgvuldig zijn gemodelleerd, zelfs
als het centrum van het cluster niet nauwkeurig bekend is. In dit Hoofdstuk
modelleren we de profielen die niet correct gecentreerd zijn voor zowel shear als
flexion. We tonen ook dat de helling van de curve van de differentiële oppervlaktedichtheid kan worden verkregen door een eenvoudige combinatie van flexions.
Deze flexions zijn gevoelig voor kleine locale variaties in de dichtheid, waardoor
ze een belangrijke aanvulling op shear zijn. Door middel van een reeks tests
zien wij dat dit met name relevant is als het gaat om het bepalen van cluster
profielen die niet correct gecentreerd zijn.
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Svensk sammanfattning
Mörk materia och kosmologi
Det kan verka förbluffande, men de stjärnor och galaxer vi ser när vi blickar
uppåt är bara en liten del av det som verkligen finns ute i vårt universum.
Inte ens de mest avancerade teleskopen på jorden eller i rymden kan urskilja
den materia som omsluter alla kända ljuskällor eftersom den inte avger eller
absorberar något ljus. Därför kan vi inte ‘se’ den i ordets traditionella mening,
och den har därav fått namnet mörk materia. Denna osynliga materia påverkar
däremot genom sin tyngdkraft allt vi ser, från vår egen galax till universums
yttersta kant, och det är så vi känner till dess existens.
Kosmologi är läran om universums ursprung och utveckling. Eftersom mörk
materia har haft så stor inverkan på denna process från dess allra första början
för 13,7 miljarder år sedan, så är vi som kosmologer särskilt intresserade av hur
den beter sig i olika omgivningar. Simuleringar som utnyttjar alla de intrikata
fysikaliska lagar som vi hittills upptäckt visar oss att den mörka materien är
fördelad i ett nätverk av filament som på engelska kallas för the Cosmic Web (se
Figur 8.1). Där filamenten korsas skapas stora moln, eller haloer, av mörk materia där stjärnor och galaxer ackumuleras. Genom observationer kan vi skapa
kartor över fördelningen av mörk materia och genom att jämföra sådana kartor
med förutsägelserna från simuleringarna så kan vi avgöra om våra fysikaliska
lagar är korrekta.
Det finns många sätt att undersöka mörk materia på och olika tekniker
används för olika rymdmiljöer, men en teknik som kan tillämpas i de flesta fall
är svag gravitationslinsning. Gravitationslinser använder, som namnet antyder,
gravitation till att böja ljus från avlägsna bakgrundsgalaxer i en process som kan
liknas vid en vanlig lins i till exempel ett par glasögon. En gravitationslins kan
bestå av vilken ansamling av massa som helst, såsom en galaxhop, en enstaka
galax eller bara ett tunt mörkt filament. När materien är mycket koncentrerad
orsakar den en kraftig förvrängning av bilden av bakgrundsgalaxen, vilket resulterar i de vackra bågar som syns i galaxhopar som Abell 2218 (Figur 8.2). Eftersom en större masskoncentration innebär mer förvrängning så kan vi använda
dessa bågar till att härleda var den mörka materien finns. När masskoncentrationen är låg, vilket förekommer i huvudsak överallt annars i universum, så
kan vi använda svagare linsningsdistorsioner, kallade shear, till att statistiskt
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SVENSK SAMMANFATTNING
Figur 8.1 Millenniumsimuleringens vision av vårt universum som det ser ut
idag, skapad av Volker Springel och hans medarbetare. Det vi ser är filamenten
tillhörande det kosmiska nätverk som på engelska benämns the Cosmic Web, med
en ansamling av mörk materia i mitten. Ansamlingen motsvarar en mycket rik
galaxhop.
Figur 8.2 Exempel på gravitationslinsning: den rika galaxhopen Abell 2218
avbildad med hjälp av rymdteleskopet Hubble Space Telescope 1999. Bildkälla:
NASA/ESA, A. Fruchter och ERO-teamet (STScI, ST-ECF).
fastställa var den mörka materien finns; det är detta som den svaga linsningstekniken uträttar.
Med den här Avhandlingen har jag som målsättning att öka vår kunskap
om fördelningen av materia i galaxer och galaxhopar, både genom att utveckla
163
SVENSK SAMMANFATTNING
den teoretiska sidan av svag gravitationslinsning och genom att använda stora
optiska datainsamlingar till att observera den svaga linseffekten i verkligheten.
Jag skildrar mitt arbete i de fyra separata men sammanlänkade Kapitlen som
finns sammanfattade härnedan.
Denna Avhandling
Jag börjar Avhandlingen med att i Kapitel 1 ge en kort översikt av kosmologins
aktuella status och av de olika sätt som finns tillgängliga för att studera vårt
universum i sin helhet. Där ger jag även en mer ingående introduktion till svag
gravitationslinsning och till de programvaror som finns till hands för att mäta de
svaga linsningsdistorsionerna av bakgrundsgalaxer. Därefter följer Kapitel där
jag mer detaljerat beskriver min forskning under de senaste fyra åren, varav en
del redan har publicerats i Monthly Notices of the Royal Astronomical Society
(MNRAS). Jag sammanfattar här dessa Kapitel.
Kapitel 2: En ny distorsionsmätningsmetod och dess tillämpning på
galaxer med färggradienter i gravitationslinsningsundersökningar
Som en av kosmologins mest kraftfulla metoder så är svag gravitationslinsning
nu den främsta drivkraften bakom några av de mest omfattande aktuella optiska
datainsamlingarna som någonsin gjorts. Teknikens statistiska karaktär kräver
analys av ett stort antal bakgrundsgalaxer, och de minimala distorsionerna det
handlar om fordrar data av hög kvalitet och ytterst exakta distorsionsmätningar.
Mjukvara för svag linsningsteknik måste därför vara både snabb och korrekt,
och en sådan programsvit introduceras och testas i detta Kapitel. Denna MV
pipeline har visat sig vara mycket konkurrenskraftig, med den extra egenskapen
att kunna mäta linsförvrängningar av högre ordning än shear, även kallade flexion. De tester som beskrivs i detta Kapitel inbegriper både monokromatiska
och polykromatiska simuleringar, där de senare har inkluderats för att bedöma
effekten av en våglängdsberoende punktspridningsfunktion (PSF) förorsakad av
teleskopet. Eftersom de flesta galaxer uppvisar färggradienter, med en kärna
som har en annan färg än utkanten, kommer en våglängdsberoende PSF att
påverka skilda delar av galaxavbildningen olika. Således kan viss felaktighet introduceras om PSF:en inte korrigeras för exakt resultat. Genom att skapa simuleringar baserade på verkliga galaxer som observerats i två olika filter, finner vi
att extra felaktigheter orsakade av denna effekt inte är värre än de felaktigheter
som finns i själva mätningsprogramvaran. Vi drar slutsatsen från våra tester
att om tillräcklig mängd träningsdata ges så kommer vi sannolikt att kunna
karaktärisera färggradienteffekten tillräckligt noggrant för att kunna korrigera
för den i framtida undersökningar såsom det rymdbaserade Euclid-uppdraget
med planerad uppskjutning år 2019.
Kapitel 3: Studie av galaktiska mörkmateriehaloer i COSMOS-undersökningen med svag gravitationslinsningsflexion
Nuvarande teorier om strukturbildning förutspår att galaxer är inneslutna i
omfattande haloer bestående av mörk materia. Haloerna förväntas ha specifika
densitetsprofiler och med svag gravitationslinsning kan vi undersöka dessa profiler på flera skalor. På små skalor kompletterar distorsioner av högre ordning,
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SVENSK SAMMANFATTNING
som kallas flexion, distorsionsmätningarna med högre detaljnoggrannhet. Vi
presenterar i detta Kapitel den första detekteringen av en flexionsignal kring
galaxer i rymdbaserade data. Signalen har erhållits med hjälp av the MV
pipeline som introducerats och testats i Kapitel 2. Vi kombinerar denna flexionsignal med shear för att definiera den genomsnittliga densitetsprofilen hos
galaxerna i Hubble Space Telescopes COSMOS-undersökning. Vi visar också att
ljuset från närliggande ljuskällor kan ha en betydande effekt på flexionsmätningar.
Efter korrigering för påverkan av galaxlinsernas ljus visar vi att inkluderingen
av flexion ger strängare restriktioner på densitetsprofiler än vad enbart shear
gör.
Kapitel 4: Relationen mellan galaxers mörkmateriehaloer och baryoner i CFHTLS via svag gravitationslinsning
Eftersom mörkmateriehaloer är så omfattande är det viktigt att undersöka både
stora och små skalor för att förstå mer om det inflytande mörk materia har på
normala atomer, och vice versa. Svag linsning har förmågan att göra detta eftersom tekniken inte bara är känslig på flera avståndsskalor, utan också oberoende
av vilken typ av materia som studeras. I detta Kapitel presenterar vi en studie
av de generella egenskaperna hos galaxers mörka haloer som en funktion av
egenskaperna hos deras värdgalaxer med hjälp av data från en av de största
färdigställda svaga linsningsundersökningarna hittills: CFHTLS. Vi delar in
linsgalaxerna i en röd och en blå kategori och bekräftar att det finns ett klart
samband mellan den mörka halomassan och luminositet, och mellan halomassan
och stjärnmassan. Dessa relationer är olika för blå och röda galaxer, och vi finner
också indikationer på att blå galaxer vistas i mindre klustrade omgivningar än
vad röda galaxer gör.
Kapitel 5: Undersökning av galaxhopsprofiler via svag linsningsshear
och -flexion
Galaxhopar är viktiga för vår förståelse av bakgrundskosmologin eftersom deras
antal som en funktion av massa och avstånd från oss beror på kosmologiska
parametrar. Majoriteten av massan i en galaxhop är i form av mörk materia,
och för att vi ska lära oss mer om kosmologi måste denna massa noggrant
beräknas trots att bara synliga storheter finns att tillgå. Svag linsning har, som
redan nämnts, förmågan att kartlägga mörka haloers densitetsprofiler på både
små och stora skalor. För att kunna dra full nytta av detta måste galaxhopens
profil omsorgsfullt modelleras även om dess centrum inte är exakt fastställt. I
detta Kapitel har vi tagit fram ocentrerade profiler för både shear och flexion.
Vi visar också att lutningen på kurvan för den differentiella ytdensiteten kan
erhållas via en enkel kombination av flexioner. Dessa flexioner är känsliga för
små variationer i densitet, vilket gör dem till viktiga komplement till shear.
Genom en serie tester finner vi att detta är särskilt relevant när det gäller att
kartlägga den ocentrerade galaxhopsprofilen.
165
Publications
1. CFHTLenS: improving the quality of photometric redshifts with precision
photometry
Hildebrandt, Erben, Kuijken, van Waerbeke, Heymans, Coupon, Benjamin, Bonnett, Fu, Hoekstra, Kitching, Mellier, Miller, Velander, Hudson, Rowe, Schrabback, Semboloni, & Benı́tez 2012
MNRAS, 421, 2355-2367
This publication is publicly available at http://arxiv.org/abs/1111.4434
2. Galaxy-galaxy lensing constraints on the relation between baryons and dark
matter in galaxies in the Red Sequence Cluster Survey 2
van Uitert, Hoekstra, Velander, Gilbank, Gladders, & Yee 2011
A&A, 534, A14+
This publication is publicly available at http://arxiv.org/abs/1107.4093
3. Probing galaxy dark matter haloes in COSMOS with weak lensing flexion
Velander, Kuijken, & Schrabback 2011
MNRAS, 412, 2665-2677 (Chapter 3, page 64)
This publication is publicly available at http://arxiv.org/abs/1011.3041
4. Gravitational Lensing Accuracy Testing 2010 (GREAT10) Challenge Handbook
Kitching, Balan, Bernstein, Bethge, Bridle, Courbin, Gentile, Heavens,
Hirsch, Hosseini, Kiessling, Amara, Kirk, Kuijken, Mandelbaum, Moghaddam, Nurbaeva, Paulin-Henriksson, Rassat, Rhodes, Schölkopf, ShaweTaylor, Gill, Shmakova, Taylor, Velander, van Waerbeke, Witherick, Wittman,
Harmeling, Heymans, Massey, Rowe, Schrabback, & Voigt 2010
preprint (astro-ph/1009.0779)
This publication is publicly available at http://arxiv.org/abs/1009.0779
166
PUBLICATIONS
5. Results of the GREAT08 Challenge: an image analysis competition for
cosmological lensing
Bridle, Balan, Bethge, Gentile, Harmeling, Heymans, Hirsch, Hosseini,
Jarvis, Kirk, Kitching, Kuijken, Lewis, Paulin-Henriksson, Schölkopf, Velander, Voigt, Witherick, Amara, Bernstein, Courbin, Gill, Heavens, Mandelbaum, Massey, Moghaddam, Rassat, Réfrégier, Rhodes, Schrabback,
Shawe-Taylor, Shmakova, van Waerbeke, & Wittman 2010
MNRAS, 405, 2044-2061
This publication is publicly available at http://arxiv.org/abs/0908.0945
6. Evidence of the accelerated expansion of the Universe from weak lensing
tomography with COSMOS
Schrabback, Hartlap, Joachimi, Kilbinger, Simon, Benabed, Bradač, Eifler, Erben, Fassnacht, High, Hilbert, Hildebrandt, Hoekstra, Kuijken,
Marshall, Mellier, Morganson, Schneider, Semboloni, van Waerbeke, &
Velander 2010
A&A, 516, A63+
This publication is publicly available at http://arxiv.org/abs/0911.0053
167
Curriculum Vitæ
Born in 1983 in Lund of southern Sweden, I grew up in a small village
about a Swedish mile (10 km) from the city, surrounded by sweeping fields of
green, yellow and blue. It was when I started seventh grade, in 1996 at the
age of thirteen, that I first came in contact with science. I had always been
curious and always asked the big questions about how the world worked, and
now suddenly I was being taught the answers. My teacher at that time was
a warm and inspiring person who encouraged me to use my own approach to
internalise the material, such as creating a website about our Solar System.
I thoroughly enjoyed making the computer show what I wanted it to display
using the secret language of HTML, and the subject of my work — astronomy
— caught my attention as well.
Once I had finished ninth grade in 1999 I got accepted to the International
Baccalaureate (IB) in Malmö, the Big City. From then on my life changed
dramatically as I was now amongst others who asked as many questions as I did.
All subjects were taught in English, which is why I do not have an established
science vocabulary in Swedish. After the introductory year I was allowed to
deepen my knowledge in a few subjects of my choice, and apart from languages
and maths I also chose physics and economics. Both subjects held the sublime
promise of explaining the workings of the world, but in the end physics appealed
to me more. Therefore I chose What mass does the Hydra 1 cluster of galaxies
(Abell 1060) contain? as the subject of my Extended Essay for which I carried
out the research with the help of Lund University. This project constituted my
first encounter with the mysterious dark matter, a taunting enigma which has
fascinated me ever since.
I had decided early on during the IB that I wanted to attend university in
the UK, partly because of the language barrier I would otherwise face in my
own country, and partly because of the adventure it entailed. As I scrolled
down the UCAS list of subjects aiming for physics, a subject at the top of
the list once again caught my attention: astronomy & astrophysics. Going
through the directory of institutions offering such a specialised programme, I
knew straight away that Edinburgh University had to be my highest priority
choice, though since I had never been my main reason for this decision was
having seen Braveheart. To my great joy I was accepted and in 2002 my parents
and I took the ferry across the North Sea to the wonderful city of Edinburgh
which still holds a special place in my heart. As an interesting aside, the second
person I spoke to in this city was my beloved future husband though I did not
168
CURRICULUM VITÆ
yet know it.
In my fifth year at Edinburgh University, it was time to chose my master’s research project. My fascination with dark matter had only grown during
my time there, and amongst the choice of about thirty project titles only one
seemed appropriate to me: Probing dark matter and baryonic sub-structure with
gravitational lensing with Andy Taylor. I did not rest until I had secured that
project, and I thoroughly enjoyed the research the rest of that year and received
my Master of Physics degree in 2007. The project gave me a taste of what weak
lensing could do, and as a result I limited my PhD applications to institutes
where I knew I would be able to research further using this technique. The
only non-UK institute I applied to was Leiden, at the recommendation of Andy,
as there was an opportunity to become an Early Stage Researcher within the
European Marie Curie network the Dark Universe with Extragalactic Lensing
(DUEL) with Konrad Kuijken. When I went for a visit I instantly liked the
atmosphere at the department and the fact that I would be able to design large
parts of my PhD research myself. Once again I therefore took the ferry across
the North Sea, this time together with my then fiancé and our cat Trouble, and
with the Netherlands as destination.
Five years on I know I made the right decision as the Sterrewacht at Leiden
University has been a truly fantastic place to conduct my research. I have
been in charge of the direction to take and have at the same time received great
support from my supervisors and brilliant colleagues. The department also gave
me the chance to act as a teaching assistant on the master’s course Astrophysical
Accretion where I created solutions to problem sheets and corrected the students’
work. The plethora of international conferences and meetings I have been able,
and encouraged, to attend have also given me a broad knowledge of topical
cosmology and a large number of friends both within the weak lensing clique
and in the greater astrophysical community. My travels have also given me the
opportunity to present my work in settings as awe-inspiring as OZ Lens 2008
or the winter AAS meeting in 2012. On a more personal level, I have through
these conferences also had the honour to meet and talk to many of my heroes.
Most notable of these is Vera Rubin who fifty years ago helped change the world
of cosmology forever even though she, as a woman, had to face and overcome
obstacles every step of the way. Although I do not face the same obstacles, as
women in science now enjoy much the same opportunities as men, I find her
story highly inspirational.
My husband and I have now taken the ferry back across the North Sea and
I am currently a post-doctoral Beecroft Fellow at Oxford University, working
with Lance Miller. Already I have tutored Optics at Oriel College as I am
a fellow there. Though I did act as a teaching assistant in Leiden, here my
responsibilities towards the students are greater and include being in charge of
tutorials and revision sessions, and setting and correcting collections papers. I
have also quickly gotten involved in the one thing I was unable to do in Leiden
due to my own lack of Dutch skills: outreach activities. Connecting with the
public, and interacting with school children who are as inquisitive as I was, truly
inspires me and provides me with the motivational sustenance that makes doing
research so fulfilling. I do hope I will have the opportunity to do both for many
years to come.
169
Afterword
The Sterrewacht at Leiden University has been a truly outstanding environment in which to conduct the research for this Thesis. The international focus
of the Sterrewacht is, I believe, one of the main reasons why this department
attracts so many great scientists and why it is so successful. And not only is the
academic staff both encouraging and highly skilled, but so is the IT support.
Were there any computer related issues, something which did not happen often,
they were soon sorted thanks to the considerable efforts of Eric, David, Aart
and Tycho. For this I am eternally grateful. Other support staff also helped
make things run smoothly, and amongst them I would particularly like to thank
Jan, Jeanne and Evelijn.
The lensing group consisted of an incredible collection of people. Soon after I
arrived, several inspiring individuals were recruited to the group, creating a very
stimulating environment for a green PhD-student like myself. Amongst them
was Tim who became both a friend and a mentor to me. Tim, I am not sure if
you are aware of how much you actually taught me. You sped up my learning
significantly and equipped me with all the tools I needed to finish this research.
You also made certain that I was not completely absorbed with work through
games’ nights and the occasional all-night Lord of the Rings (though not as often
as you would have liked). I cannot even begin to tell you how much I appreciate
your support and friendship. In fact, I consider everyone in the lensing group a
friend above all else. Edo, your path to a PhD degree ran parallel to mine for
more than four years, with you just pipping me to the post. Though I am not
entirely happy about this, I do know that you fully deserve it. Working with you
was a true pleasure; your intelligence always shone through and you were never
afraid to argue your (often correct) point. You also became a close friend of
mine, and I thoroughly enjoy the gossip sessions we both still indulge in. Once
Stefania and Elisabetta joined us the group felt complete, although the noise
level in my corridor certainly increased. Stefania, you are great fun and I wish I
had gotten the chance to know you better. And Elisabetta, oh Elisabetta. With
your arrival the Sterrewacht improved a hundredfold, thanks to your special take
on life. You always make me see things in a different (and often better) light,
both in science and otherwise. You are a fun and loud person and completely
my opposite, which is probably why I find your company so refreshing. Finally,
I would like to thank the lensing group as a whole and any sporadic add-ons
as well, including Freeke, Berenice, Merijn, Remco and Hendrik, for being such
a sociable bunch and for making coffee breaks and evenings that much more
170
AFTERWORD
enjoyable.
Actually Hendrik, you were part of another side to my life in Leiden as well:
AA-Awesome. This band truly lived up to its name, arguably not in talent but
definitely in terms of band members. You were all awesome Rik, Hendrik, Rob
and Craig and I had an awesome time while it lasted. But Craig, you were
probably the awesomest of them all. Your harsh sarcasm coupled with your
strange politeness made me feel right back in Britain, and I often indulged in
taking the mickey out of you. I think you did intensify my own sarcasm though,
which means that I now frequently accidentally offend my own family. This is
to your credit.
I would also like to take this opportunity to thank the two extended groups
I have been a part of: the DUEL network, and the CFHTLenS collaboration.
Through frequent DUEL meetings I was continuously spurred on to do my
research, and I made many social connections with other young researchers in
the field. CFHTLenS has also been very good to me and provided me with a
great network that now spans the globe. Catherine, you in particular have been
very encouraging. Thank you for believing in me; it has without doubt helped
build my confidence as a scientist. You have also always been there to lend
advice on a more personal level, something I greatly appreciate.
Of course, my family has also always been there for me and have always
managed to muster the proper amount of awe at each one of my achievements.
Thank you, Mum and Dad, for never pressuring me unnecessarily but nudging
me in the right direction when appropriate. Mum, you always read through
all my essays for school and university to make sure that the language was
correct and fluent. As a thank you for that, I will not make you read my Thesis.
Dad, thank you for coaxing me into doing things I would never have dared do
otherwise, from joining that class at högstadiet which allowed me to be taught
by Arne who got me interested in science in the first place, to contacting Lund
University for guidance on my Extended Essay. These and many other choices
have had a long-lasting positive effect on my life and career. Kattis, and lately
also Ivar, thank you for always treating me without respect whenever I come
home to Sweden and that way making me feel like I never left. Jonas, you and
Ann have also supported me through this simply by making home still seem like
home. I do sometimes wonder who the real big sister is though, since you and
Kattis have turned into such sensible adults in my absence. Richelle, though
you are not officially my sister, I do consider you one and I am grateful that you
have remained my close friend even though I have been somewhat unavailable
over the past year.
My wonderful family-in-law has also helped see me through. In particular,
thank you Ellen for your kind-heartedness, thank you Gabrian for adding a
sense of art to my Thesis, and thank you Josha for keeping my motivation high
by being so fascinated by what I do that you decided to try it out for yourself.
Special thanks also to all of my cousins who have always been great friends,
and in particular to Tove with whom I have always been close. Furthermore,
I am grateful to Johanna for taking the time to read and correct my Swedish
summary. For this reason I also thank Fredrik who has been a friend ever since
I recognised him as a fellow Swede in Edinburgh because of his Loka bottle.
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AFTERWORD
Last, but most certainly not least, I would like to thank my own budding
family. Marcus, without your tireless efforts to keep our home tidy and me fed,
without your continuous and unfailing support, and without your endless love I
would never have been able to do this. And little one, still dwelling within me,
without your encouraging kicks to keep me motivated and focused towards the
end, this would be work in progress even now.
172
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