Mass and Light distributions in Clusters of Galaxies - Henry A ...

TELAVIVUNIVERSITY SCHOOLOFPHYSICS&ASTRONOMYaaia`-lzzhiqxaipe` RAYMONDANDBEVERLYSACKLER FACULTYOFEXACTSCIENCES xlw`qilxaaecpeniixy"r dinepexhq`edwiqitlxtqdzia miwiiecnmircnldhlewtd
MASS AND LIGHT DISTRIBUTIONS
IN
CLUSTERS OF GALAXIES
Thesis submitted for the degree doctor of philosophy
by
Elinor Medezinski
Submitted to the senate of Tel-Aviv University
June 24, 2011

This work was carried out under the supervision of
Dr. Tom Broadhurst & Prof. Yoel Rephaeli
The Raymond and Beverly Sackler School of Physics and Astronomy
Tel-Aviv University

To my parents

Contents
Abstract
V
Acknowledgments
IX
1 Introduction 1
1.1 Clusters of Galaxies . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Cluster Mass from the Virial Theorem . . . . . . . . . 3
1.1.2 The “Missing Mass Problem” - Dark Matter . . . . . . 5
1.1.3 Spatial Mass Distribution . . . . . . . . . . . . . . . . 6
1.1.3.1 NFW Profile . . . . . . . . . . . . . . . . . . 6
1.1.4 Luminosity Function of Galaxies in Clusters . . . . . . 8
1.1.5 X-ray Observations of Clusters . . . . . . . . . . . . . . 9
1.1.6 Sunyaev-Zel’dovich Measurements . . . . . . . . . . . . 10
1.2 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Historical Overview . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Principles of Gravitational Lensing . . . . . . . . . . . 13
1.2.2.1 The Deflection Angle . . . . . . . . . . . . . . 13
1.2.2.2 The Lens Equation . . . . . . . . . . . . . . . 15
1.2.2.3 Magnification and Distortion . . . . . . . . . 16
I

1.2.3 Weak Gravitational Lensing . . . . . . . . . . . . . . . 18
1.2.3.1 Measuring Galaxy Shapes and Sizes . . . . . 19
1.2.3.2 The KSB Method . . . . . . . . . . . . . . . . 20
1.2.4 Clusters of Galaxies as Gravitational Lenses . . . . . . 21
1.2.4.1 Distortion measurements in Clusters . . . . . 21
1.2.4.2 Systematics of the Methods . . . . . . . . . . 22
1.2.4.2.1 Projection Effects . . . . . . . . . . . 22
1.2.4.2.2 Mass-Sheet Degeneracy . . . . . . . 23
1.2.4.2.3 PSF Effects . . . . . . . . . . . . . . 23
1.2.4.2.4 Weak Lensing Dilution . . . . . . . . 24
1.3 The Structure of the Thesis . . . . . . . . . . . . . . . . . . . 25
2 Weak Lensing Dilution in A1689 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Subaru imaging reduction and sample selection . . . . . . . . 31
2.3 Distortion analysis of subaru images . . . . . . . . . . . . . . . 34
2.4 Distortion analysis of ACS/HST images . . . . . . . . . . . . . 40
2.5 Photometric redshifts . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Weak lensing dilution . . . . . . . . . . . . . . . . . . . . . . . 49
2.7 Cluster light and color profiles . . . . . . . . . . . . . . . . . . 51
2.8 Cluster luminosity functions . . . . . . . . . . . . . . . . . . . 53
2.9 M/L profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.10 Cluster mass profile . . . . . . . . . . . . . . . . . . . . . . . . 60
2.11 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . 65
Appendix 2.A Cluster galaxy fraction from the lensing dilution effect 67
Appendix 2.B Non-linear effect in the reduced shear estimate . . . 69
II

3 Mass and Light of A1703, A370 & RXJ1347-11 75
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 Subaru data reduction . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Sample selection from the color-color diagram . . . . . . . . . 81
3.3.1 Cluster Members . . . . . . . . . . . . . . . . . . . . . 81
3.3.2 Red Background Galaxies . . . . . . . . . . . . . . . . 82
3.3.3 Blue Background and Foreground Galaxies . . . . . . . 84
3.4 Evolutionary color tracks . . . . . . . . . . . . . . . . . . . . . 86
3.5 Depth estimation from SDF/COSMOS . . . . . . . . . . . . . 88
3.6 Weak lensing analysis . . . . . . . . . . . . . . . . . . . . . . . 93
3.7 Weak lensing dilution . . . . . . . . . . . . . . . . . . . . . . . 96
3.8 Cluster light profiles . . . . . . . . . . . . . . . . . . . . . . . 97
3.9 M/L profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.9.1 Tidal and Ram-Pressure Stripping . . . . . . . . . . . 103
3.10 Cluster luminosity functions . . . . . . . . . . . . . . . . . . . 106
3.11 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . 109
4 WL Determination of the Distance-Redshift Relation 113
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2 Subaru data reduction . . . . . . . . . . . . . . . . . . . . . . 116
4.3 Sample selection from the color-color diagram . . . . . . . . . 117
4.4 Weak lensing measurements . . . . . . . . . . . . . . . . . . . 122
4.4.1 Formalism: Relative Distortion Strength . . . . . . . . 123
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5.1 Weak lensing profiles . . . . . . . . . . . . . . . . . . . 125
4.5.2 COSMOS photometric redshifts . . . . . . . . . . . . . 127
4.5.3 Lensing strength dependence on magnitude . . . . . . . 132
4.5.4 Lensing strength vs. redshift . . . . . . . . . . . . . . . 134
4.6 Constraining cosmological parameters . . . . . . . . . . . . . . 136
4.7 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . 137
III

5 Discussion 139
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2 Further application of the methods in other work . . . . . . . 142
5.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.1 MCT/CLASH . . . . . . . . . . . . . . . . . . . . . . . 144
References 145
IV

Abstract
Clusters of galaxies are the most massive bound objects in the Universe.
Their deep potential wells deflect the light of background galaxies creating
lensed galactic images. Gravitational lensing can be used as a diagnostic tool
to determine the mass profiles of clusters. In the weak lensing regime, the
measurement of the cluster mass can be badly compromised by the presence
of unlensed foreground and cluster members which dilute the true lensing
signal. In this thesis, I study galaxy clusters mass and light properties via
weak lensing observations. In the first part, I show the importance of carefully
isolating background galaxies from foreground and cluster galaxies for a clean
weak lensing measurement, avoiding dilution of the signal. The dilution is
in turn used to determine the light properties of the clusters. The second
focus of the work shows the use of the detailed measurements to determine
the increment of the weak lensing signal with redshift, which is an important
cosmological probe.
Following an introduction to the field of galaxy clusters and gravitational
lensing in Chapter 1, I demonstrate my methods and results for four massive
galaxy clusters in Chapters 2 and 3. In Chapter 2 I study cluster A1689
(z = 0.183) with deep Subaru imaging in the V and i ′ bands, using the
color-magnitude diagram and the weak lensing signal to derive the luminous
properties of cluster galaxies. The tangential distortion of galaxies bluer
than the E/S0 sequence falls rapidly toward the cluster center relative to the
lensing signal of the background galaxies redder than the sequence. I use
this dilution effect to derive the cluster light profile and luminosity function
to large radius, with the advantage that no subtraction of far-field background
counts is required. The light profile has a constant declining slope
, d log(L)/d log(r) = −1.12 ± 0.06 to the limit of the data, r < 2 h −1 Mpc,
whereas the mass profile steepens continuously with radius, such that M/L
peaks at an intermediate radius, r ∼ 100 h −1 kpc. The cluster luminosity
function has a flat faint-end slope, α = −1.05 ± 0.05, independent of radius
and with no faint upturn to M i ′ < −12. I show that the bluest objects are
V

negligibly contaminated by the cluster (V − i ′ < 0.2). Combining both the
red and blue background populations I get an improved measurement of the
mass profile and derive constraints on the cluster concentration parameter,
clearly excluding low-concentration cold dark matter profiles.
As described in Chapter 3, the work was extended to three more intermediate
redshift clusters, A1703 (z = 0.258), A370 (z = 0.375) and RXJ1347-11
(z = 0.451) imaged with Subaru in at least three bands. With the multiband
data available here, I use the color-color diagram to demonstrate how
the lensing signal can be harnessed to separate cluster members from the
foreground and background populations in a more secure and robust way,
and identify several background galaxy populations of different nature for a
weak lensing measurement. The luminosity functions of these clusters, when
corrected for dilution, all have similar faint-end slopes, α ≃ −1.0, with no
marked faint-end upturn to the magnitude limit of M R ≃ −15.0, but a mild
radial gradient. Here too, for each of our clusters the M/L profile peaks
at intermediate radius, r ≃ 0.2r vir , at a level of 300 − 500(M/L R ) ⊙ , and
then falls steadily towards ∼ 100(M/L R ) ⊙ at the virial radius, similar to
the mean field level. This behavior is likely due to the intrinsic dearth of
late-type galaxies relative to early types nearer the center, whereas for the
E/S0-sequence only a mild radial decline in M/L is found for each cluster.
This can be explained by the continued conversion of disk galaxies into S0
galaxies, via tidal and gas stripping processes. I discuss this behavior in the
context of detailed simulations where predictions for tidal stripping may now
be tested accurately with observations.
The amplitude of weak lensing increases with source redshift, and is
proportional to the lensing distance ratio, D ds /D s , rising steeply behind a
lens and saturating at higher redshift. It provides, in principle, a modelindependent
way of measuring cosmological parameters. Using the multiband
Subaru dataset of the cluster A370, I explore this relation in Chapter
4. Applying the selection methods I developed I further separate and
select from the color-color diagram several background populations of differing
depths (0.6 < z < 3.8), including a prominent sample of distant dropout
galaxies, with little overlap in redshift. I accurately measure their lensing
amplitude, taking account of the radial variation of the weak lensing profile.
I define the depths of the lensed populations with reference to the COSMOS
and GOODS field surveys. The predicted distance-redshift relation is confirmed
for a wide range of cosmological models, albeit without the ability to
distinguish between models with only one cluster. Scaling this result to our
new approved HST /CLASH survey of ∼ 25 massive, relaxed clusters should
provide a useful cosmological constraint on the dark energy equation of state
VI

parameter, w, complementing existing techniques, with distance measurements
covering the untested redshift range, 1 < z < 5.
VII

Acknowledgments
I would like to thank my PhD advisor, Dr. Tom Broadhurst, for his guidance
and support. With great patience he taught me how to be an inquisitive
researcher, find my niche in the world of astronomy, work independently,
and above all gave me great inspiration with his unique insight on science.
I thank my advisor Prof. Yoel Rephaeli, who guided and mentored me,
helped with all the bumps along the way (and there were many...) and most
importantly encouraged me to strive for excellence, in the major topics as
well as the fine print.
A special gratitude goes to my collaborator Dr. Keiichi Umetsu, who was
every bit of an advisor to me. His skills, generosity, and endless patience in
hours of correspondence has made me a more careful and thorough scientist.
I thank Txitxo Benítez for his important part in our work and his codes, but
especially for opening his home and family to me and giving me a home away
from home during my visits to Spain. I thanks Dan Coe for his support and
his great code which I used. I also thank all my other collaborators - Holland
Ford, Emilio Falco, Masamune Oguri, Nobuo Arimoto, Xu Kong, Masafumi
Yagi and Andy Taylor.
The past 8 years I spent in the department have passed quickly, due to
the embracing surroundings I have spent them in. My office has become my
home, and my office mates my family. I owe the biggest gratitude to my
closest friend and “twin brother”, Assaf Horesh, who tolerated me through
thick and thin, spent hours in discussions about important scientific issues
and not-so-important “sessions”, and kept my sanity in check. I am especially
grateful to Benny Trakhtenbrot, for his close friendship and for always
making me laugh the hardest. I am also thankful for Omer Bromberg for his
friendship and support. I am grateful to Eran “papi” Ofek and “dod” Yiftah
Lipkin, who practically raised me with lots of patience and care. I thank all
the current and former students in the department with whom I have shared
my time – Orly Gnat, Smadar Naoz, Keren Sharon, Evgeny Gorbikov, David
Polishook, Ofer Yaron, Avi Shporer, Yifat Dzigan, Irina Dvorkin, Sharon
IX

Sadeh, Yinon Arieli, Doron Lemze, Adi Zitrin, Oded Spector, Or Graur, Ido
Finkelman and Yossi Shvartzvald.
I would like to thank all the faculty and research members at Tel Aviv
University – Rennan Barkana, Sara Beck, Noah Brosch, Ben-Zion Kozlovsky,
Elia Leibowitz, Amir Levinson, Dan Maoz, Tsevi Mazeh, Ehud Nakar, Hagai
Netzer, Amiel Sternberg, Elhanan Almoznino, Marcella Contini, Itzhak
Goldman and Shai Kaspi. A special thanks goes to the Wise Observatory
staff and observers – Assaf Bervald, John Dan, Haim Mendelson, Sami Ben-
Gigi, Ezra Mashal, Friedel Loinger, and Shai Kaspi.
I am eternally grateful to my mom, who raised, educated, loved and
believed in me. She is my rock, my inspiration, and my heart. To my Father
I am grateful for providing me with the thirst for knowledge, the love for
science, and the tools and freedom to pursue my passion. I thank Motti
Avrech, who has been a second father figure to me for most of my life.
I am grateful to N. Kaiser for making the IMCAT package publicly available.
Work at Tel-Aviv University was supported by Israel Science Foundation
grant 214/02. This research was based on data collected at Subaru
Telescope and obtained from the SMOKA, which is operated by the Astronomy
Data Center, National Astronomical Observatory of Japan.
X

Chapter 1
Introduction
Clusters of galaxies are the most massive virialized objects in the Universe,
and as such they gravitationally “pull” on the light traveling from galaxies
lying behind them. This is an example of an effect called “gravitational
lensing”. It has prompted a wide field of research and was established as a
robust method to explore galaxy clusters. The dependence on various angular
distances of the lens system means cosmology can be diagnosed via lensing.
In the past decade, with the improvement of observational instruments, and
the development of codes, many clusters have been studied in great detail
with lensing.
The focus of my PhD was the study of mass and light properties of several
clusters, using observations of both ground-based and space-based telescopes.
I developed techniques to carefully isolate background galaxies for
a clean weak lensing measurement, and to identify the cluster based on its
dilution of the weak lensing signal. The reconstructed mass and light profiles
of the clusters, which were resolved with unprecedented accuracy and detail,
were used to explore inner cluster properties, as well as to be compared
with cosmological simulations of structure formation and test current cosmological
models. The quality of the data and the extensive analysis allowed
for a unique determination of the increment of the weak lensing signal with
redshift.
1

Introduction
In this introduction, I will present an overview of galaxy clusters in the
next section, followed by a historical overview and the basic theory of gravitational
lensing. In Chapter 2 I present the work done to derive the mass and
light properties of A1689 using the weak lensing dilution effect. In Chapter 3
I present the further application of these methods to three more clusters:
A1703, A370, and RXJ1347-11, and in Chapter 4 results are presented for
the derived distance-redshift scaling from measurements of weak lensing of
galaxies in the background of A370. Finally, in chapter 5 I summarize the
work I have accomplished and present future surveys I am jointly carrying
out to extend the study done here to 25 more clusters.
1.1 Clusters of Galaxies
Galaxy clusters are the largest gravitationally bound structures formed in
the universe, containing hundreds to thousands of galaxies, and masses in the
range 10 14 − 10 15.5 M⊙. Growth of structure in a Cold Dark Matter (CDM)
model is hierarchical, with smaller scale perturbations collapsing first to form
galaxies, which then merge to form more massive objects. Clusters of galaxies
are the most recent, largest phase of hierarchical clustering, and arise from the
gravitational collapse of rare, high density perturbation peaks. Clusters are
still dynamically evolving, relaxing, and accreting matter and occasionally
small clusters mostly along large scale filaments and superclusters. They can
be detected out to large distances, serving as beacons probing the mass in the
large scale structure, and to high redshifts, providing important information
on the early universe.
Galaxy clusters can provide constraints on cosmological parameters and
models, complementary to other methods such as CMB, SNe-Ia and cosmic
shear measurements. As the most massive structures in the Universe,
they form the high-mass end of the mass function of gravitationally-bound
halos (Press & Schechter 1974), and its development as a function of redshift
is an effective test of the hierarchical structure formation scenario and
is highly sensitive to cosmological scenarios (see Voit 2005, for a review).
2

1.1 Clusters of Galaxies
The space density of clusters in the local universe has been used to measure
the amplitude of density perturbations on ∼ 10 Mpc scales. Studying the
mass distribution within clusters is another probe of the non-linear growth
of structures. Numerical N-body and hydrodynamical simulations make clear
predictions for the structure of clusters, for example that they should have
a universal density profile (Dubinski & Carlberg 1991; Navarro et al. 1996)
and that their observable properties should satisfy some predicted scaling relations.
Comparing these predictions with observations is an important test
of CDM models.
Clusters form through the collapse of cosmic matter over a region of several
megaparsecs. Cosmic baryons, which represent approximately 10–15%
of the mass content of the Universe, follow the dynamically dominant dark
matter during the collapse. They fall into the gravitational potential of the
cluster dark matter halo so formed, while the collapse and the subsequent
adiabatic compression and shocks heat the intra-cluster medium (ICM). Hot
gas permeating the cluster potential well is then assembled, reaching temperatures
of several 10 7 K, and as it becomes fully ionized emits via thermal
bremsstrahlung in the X-ray band. Typically, clusters of galaxies have total
masses which exceed 1 × 10 14 M ⊙ , with some ∼ 85% in DM, ∼ 12% in IC
gas, and the rest in galaxies.
The main methods used to determine the cluster mass are through the
cluster galaxy velocities, through the effect of gravitational lensing, through
X-ray observations of the hot diffuse gas component, and through the Sunyaev-
Zel’dovich (SZ) effect. In this section, I review our understanding of the basic
properties of clusters formation, their mass, distributions and luminosity
functions. I review the methods of X-ray and SZ observations of clusters.
The field of gravitational lensing, which is the effect I rely on in my work, is
reviewed more thoroughly in the next section.
1.1.1 Cluster Mass from the Virial Theorem
Zwicky (1933) was the first to apply the virial theorem to estimate cluster
masses. A cluster mass is estimated by assuming all cluster galaxies are
3

Introduction
bound in a self-gravitating structure.
Were they not bound, they would
disperse in a typical crossing time of ∼ 10 9 ! yr. Since they appear relaxed,
this assumption seems likely. A limit on the mass can therefore come from
the binding condition,
where E is the total energy, T is the kinetic energy,
and W is the gravitational energy,
E = T + W < 0 (1.1)
T = 1 ∑
m i vi 2 , (1.2)
2
W = − 1 2
i
∑
i≠j
Gm i m j
r ij
, (1.3)
m i and v i are the galaxy mass and velocity, and r ij is the separation between
galaxies i and j.
Differentiating the system’s moment of inertia I = ∑ i m iri
2
respect to time gives the equation of motion of galaxies,
twice with
1 d 2 I
2 dt = ∑ 2 i
m i r˙
2 i + ∑ i
m i ¨r i · r i = 2T + W. (1.4)
In a stationary system, the left hand side is set to zero, so
W = −2T, (1.5)
which is known as the Virial relation. The mass of an isolated, spherically
symmetric, bound system is therefore
M tot = R G〈v 2 〉
G
, (1.6)
where we define a mass-weighted velocity dispersion 〈v 2 〉 ≡ ∑ i m iv i /M tot ,
4

1.1 Clusters of Galaxies
and a gravitational radius
R G ≡ 2M 2 tot
[ ∑
i≠j
m i m j
r ij
] −1
≈ 2M 2 tot
[ ∑
i≠j
m i m j
r ⊥,ij
] −1
. (1.7)
The approximation gives R G in terms of the observable projected galaxy
separation r ⊥,ij in the plane of the sky (Limber & Mathews 1960). The
velocities can be evaluated from the radial velocity distribution 〈v 2 〉 = 3σ 2 r,
where σ r is the radial velocity dispersion. Substituting this in eq. 1.6 we have
M tot = 3R [
]
Gσr
2 2 [ ]
G = 7 × σ r RG
1014 M ⊙ . (1.8)
1000 km/sec Mpc
For a typical rich cluster, σ r ∼10 3 km/sec and R G ∼1 Mpc, so the mass is of
order M tot ∼ 10 15 M ⊙ .
1.1.2 The “Missing Mass Problem” - Dark Matter
Analyses of clusters measure surprisingly high masses, especially compared
to the typical total luminosity of a cluster, L tot ∼ 10 13 L ⊙ . Commonly, mass
is compared to luminosity by calculating the mass-to-light ratio, typically,
( M
L
)
tot
( )
M⊙
≃ 300 h . (1.9)
L ⊙
This exceeds the value of luminous galaxies which ranges from M/L B ≈
1 − 12 M ⊙ /L ⊙ by more than a factor of 10. This discrepancy means less
than 10% of the cluster mass is in stars in luminous galaxies. This is the
“missing mass” problem, first suggested by Zwicky (1933), who measured
the nearby Coma cluster mass from galaxy velocity dispersions.
Zwicky
(1937) postulated that there is a need for a non-standard mass component
to explain these measurements. Current measurements indicate that about
85% of the mass is made of this dark component. As of yet, the nature of
DM is unknown, but is fairly certain to be non-baryonic, non-radiating, cold
matter interacting practically through gravitation alone.
5
Quantifying the

Introduction
nature and distribution of DM are two of the most important open issues in
cosmology.
1.1.3 Spatial Mass Distribution
Most regular clusters show a centrally condensed number density distribution
of cluster galaxies, i.e., the galaxy density increases towards the center. If
the cluster is not significantly elliptical (see caveats, Sheth et al. 2001; Jing &
Suto 2002; Oguri et al. 2005; Corless & King 2007), then it is usually taken
to be spherically symmetric. Most models that describe the distribution
require at least five parameters, such at the position of the cluster center,
the central projected density, and two scale distances, the core radius r c , and
the truncation radius, e.g., r vir for DM halos. Assuming galaxies trace DM
(or vice versa), their distributions should generally be similar. One useful
DM density profile that was used in my work is the NFW model.
1.1.3.1 NFW Profile
N-body simulations of standard CDM predict a universal DM halo profile,
with a low-concentration profile for cluster-sized halos. The logarithmic slope
of the density profile decreases monotonically toward the center, and is much
shallower in the center than a pure isothermal profile inside a characteristic
radius of r 100 kpc/h, but lacking a constant density core. A supposedly
universal form for DM halos was deduced from N-body simulations (Navarro
et al. 1997, hereafterNFW),
ρ(r) =
δ c(M vir ) ¯ ρ(z)
(r/r s )(1 + r/r s ) 2 (1.10)
where r s is a scale radius, δ c is a (dimensionless) characteristic density excess,
¯ρ(z) = ρ crit Ω M (z) is the mean density at collapse, and ρ crit ≡ 3H(z) 2 /8πG is
the critical density on the Universe. The virial radius r vir is defined according
6

1.1 Clusters of Galaxies
to the spherical collapse model so that
M vir = 4π 3 ∆ vir ¯ρr 3 vir (1.11)
Where ∆ vir is the critical overdensity. It has been approximated by Kitayama
& Suto (1996) to be
∆ vir ≃ 18π 2 (1 + 0.4093w 0.9052
f ), (1.12)
and w f ≡ 1/Ω f − 1, and Ω f is the density parameter at z f . The condition
that the total mass inside r vir is M vir relates the characteristic density δ c and
the concentration parameter c = c vir ≡ r vir /r s via
δ c = ∆ vir
3
c 3
ln(1 + c) − c/(1 + c) . (1.13)
Both optical and X-ray observations seem to indicate that this profile is a
good representation of the underlying cluster mass profile, at least outside the
innermost region (Carlberg et al. 1997). Is it shallower than an isothermal
profile at smaller radii and steeper at large radii, changing gradually from
an approximate power law p = −1 falloff near the center to p = −3 beyond
r s . The transition of the density profile from shallow to steep can also be
expressed in terms of the concentration parameter, c. This free parameter
varies systematically with mass, where lower-mass halos tend to have higher
concentrations since they formed earlier in time, when the overall density of
the Universe was much higher. Typical concentration parameters for simulated
clusters are in the range c ∼4 − 10 (Navarro et al. 1997) with a rather
wide dispersion (Jing 2000).
The NFW profile was deduced also from higher resolution simulations
(Navarro et al. 2004), albeit somewhat steeper inner profiles are also claimed
(Moore et al. 1999), where some measure intrinsic variation in slopes that
is related to the assembly history of a cluster (Tasitsiomi et al. 2004). This
characteristic can be examined by measuring the mass distribution in clusters,
and can add to our understanding of the evolution processes in cluster
7

Introduction
assembly.
1.1.4 Luminosity Function of Galaxies in Clusters
The luminosity function (LF) of galaxies in a cluster gives the distribution
of luminosities of galaxies. LFs are often defined in terms of magnitudes,
m ∝ −2.5 log 10 (L). There are several LF forms commonly used. Schechter
(1976) introduced an analytic approximation for the differential LF
φ(L)dL = N ∗ (L/L ∗ ) −α e −L/L∗ d(L/L ∗ ) (1.14)
where L ∗ is a characteristic luminosity, and α is the faint-end slope. The parameter
N ∗ is a useful measure of the richness of the cluster. The parameter
L ∗ usually denotes the break in the LF, hence it represents a characteristic
luminosity of cluster galaxies. It is found that L ∗ is similar for many clusters,
corresponding to an absolute magnitude of MV
∗ ≈ −21.9 + 5 log 10 h 50 . The
advantages of the Schechter function are that it is analytical and continuous.
An expression for the mass function similar to the Schechter form was predicted
by a simple analytical model for galaxy formation (Press & Schechter
1974).
While the Schechter function serves as a good fit to the majority of clusters,
there are a number of clusters that show a departure from it. These
departures include variations in the values of M ∗ and α, corresponding to
the cluster morphology (Oemler 1974; Dressler 1978). They probably stem
from the conditions of the cluster at its formation epoch. Variations in the
steepness of the bright end may reflect evolutionary changes, such as tidal
interactions or merging of massive galaxies (Richstone 1975; Hausman & Ostriker
1978). For example, steep bright-end often fits to clusters that have
bright cD galaxies, indicating many massive galaxies were merged to form
the cD (Dressler 1978).
8

1.1 Clusters of Galaxies
1.1.5 X-ray Observations of Clusters
X-ray observations probe the gas component of the cluster. It has been
established that the majority of the baryonic matter is in the form of hot
intracluster (IC) gas, with only about 3% residing within stars in galaxies.
Probing IC gas can reveal unique insight into the physical processes that
govern galaxy formation, and the gas properties can be used to determine
the total cluster mass, in clusters that are in hydrostatic equilibrium.
Spectral X-ray observations yield the gas temperature, in the range of
3−15 keV, while its density is determined from the spatial profile of the (thermal
Bremsstrahlung) emission. Spectroscopic measurements of the emission
lines (mostly of the strong K α line complex of Fe 24+ ) reveal that IC gas
is metal enriched with typically 1/3 solar abundances. X-ray imaging has
progressed significantly in the past decade by detailed measurements with
the Chandra and XMM-Newton telescopes that yield high-resolution spatial
mapping of the distribution of intracluster gas. From measurement of the
temperature and density profiles the gas pressure and entropy distributions
can be derived. When combining density and temperature profiles the total
mass of the cluster can be deduced from a solution of the hydrostatic
equilibrium equation. Detailed X-ray images can also yield information on
whether the cluster is relaxed or has substructure due to ongoing dynamical
processes, such as merging of galaxies or groups.
Many clusters seem to be in hydrostatic equilibrium. Assuming spherical
symmetry, the total gravitating mass within the virial radius R is then,
M(< R) = − k (
BT R d ln ρg
Gµm p d ln R + d ln T )
, (1.15)
d ln R
where ρ g is the gas density, and µm p is the mean mass per plasma particle.
Thus, the total cluster mass can be inferred from X-ray measurements of the
gas density and temperature profiles. IC gas is commonly modeled with a
9

Introduction
β density profile, (Cavaliere & Fusco-Femiano 1976), where
ρ g (r) = ρ g,0
[1 +
( r
r c
) 2
] −3β/2
, (1.16)
and β ≡ µm p σr/k 2 B T is the ratio of the kinetic (gauged by the velocity dispersion
σ r , mostly of the DM) energy and thermal gas energy, and r c is
the core radius. This profile describes isothermal gas in hydrostatic equilibrium
within the potential well associated with a King (1962) DM density
profile ρ(r) ∝ [1 + (r/r c ) 2 ] −3/2 . Beta models describe the observed X-ray
surface-brightness reasonably well at radii r c − 3r c with a 〈β〉 ≈ 2/3, but underestimate
the density in central cooling cores of clusters (Jones & Forman
1984). There is also a possible trend toward lower β in poor clusters. These
discrepancies arise partly since the IC gas is not strictly isothermal. Acquiring
a full description of the temperature distribution is difficult, especially
for high-redshift clusters.
There are inherent limitations in constructing X-ray density profiles and
total mass estimates; these stem from the assumptions of hydrostatic equilibrium
and spherical symmetry, and gas isothermality. Nonetheless, the
methodology of determining the gas properties from X-ray observations, and
the total mass profiles from strong and weak lensing measurements is very
useful when applied to relaxed, low-redshift clusters for which spatially resolved
Chandra and XMM-Newton images of the temperature and density
distributions indicate relatively low level of (projected) structure.
1.1.6 Sunyaev-Zel’dovich Measurements
The SZ effect is a unique observational probe of galaxy clusters, detectable
at radio and microwave wavelengths. It is a small spectral distortion in the
cosmic microwave background (CMB) spectrum caused by Compton scattering
of CMB photons off hot ICM electrons (Sunyaev & Zeldovich 1972). As
the number of photons is conserved in the scattering, the SZ effect causes
a decrease in the intensity of CMB at low frequencies ( 218 GHz) and an
10

1.2 Gravitational Lensing
increase at higher frequencies. The shape of the distorted spectrum of the
thermal component of the effect depends on the Compton y-parameter,
∫
y =
n e
k B T e
m e c 2 σ T dl, (1.17)
which is essentially a line of sight integral of the gas pressure.
The important features of the SZ effect are that its signal is independent
of redshift, unlike optical or X-ray surface brightnesses, and that it directly
probes the gas pressure, or thermal energy density, along the line of sight.
These properties make the SZ effect a unique and powerful observational tool
for detecting distant clusters.
SZ and X-ray observations are complementary to each other, specifically
in the outskirts of clusters where the X-ray surface-brightness is difficult to
observe, but the SZ signal remains substantial. The SZ distortion is linear
in the gas density, whereas X-ray emission is proportional to n 2 e. This means
X-ray measures the gas in the center of the cluster with better resolution, but
that the SZ effect is more suitable for tracing the gas across a much larger
region of the cluster.
1.2 Gravitational Lensing
1.2.1 Historical Overview
The deflection of light rays by gravity was predicted by Einstein’s General
Relativity (GR). Einstein’s calculation of the magnitude of deflection by the
gravitatioal field of the sun was confirmed in 1919 when angular shifts of
stars close to the Sun’s edge were measured during a total solar eclipse.
This served as a confirmation of GR, and light deflection and other related
phenomena are called “Gravitational Lensing”.
Observing multiple images of sources being lensed is another result of
the theory. Although Einstein had thought that multiple images due to
11

Introduction
stellar-scale lensing are unlikely to be resolved, Zwicky (1937) argued that
multiple images due to lensing by galaxies will lie at large enough angles to
be observed and would serve as another measure of galaxy masses. Such
an independent method was needed in order to test the emerging evidence
that galaxies weigh about two orders of magnitude more than their stellar
content. The first example of multiple images due to gravitational lensing was
discovered when a double-imaged quasar was measured by Walsh, Carswell, &
Weymann (1979). The two images of QSO 0957+561 were shown to be of the
same quasar being lensed by a foreground galaxy. This discovery transformed
gravitational lensing from a theoretical idea to a valid experimental field with
cosmological applications.
More examples of lensing soon followed. In the massive Abell 370 cluster,
a peculiar “ring-like” shape was discovered by Soucail et al. (1987a) and
independently by Lynds & Petrosian (1986), thought at first to be the result
of galaxy/galaxy interactions in the cluster itself. It was only later recognized
to be a strong lensing “giant arc”, occurring when the source is lying behind
the lens close to a caustic (Soucail et al. 1987b; Paczynski 1987).
Apart from these spectacular examples of strong lensing which are quite
rare and are only seen in the center, clusters overall coherently distort (and
amplify) fainter background galaxies on a weaker scale, referred to as “weak
lensing” (Fort et al. 1988). Using these observations to reconstruct in a
model-independent way the two-dimensional mass map of the cluster was
suggested by Kaiser & Squires (1993) and has been subsequently measured
for many clusters (Bonnet et al. 1993; Smail & Dickinson 1995; Broadhurst
et al. 1995). In the past decade major improvements have been made in
the field of weak lensing, due to the development of algorithms that robustly
measures accurate galaxy shapes by overcoming inherent systematics (Kaiser
et al. 1995), and on the other level the recent construction of large telescopes
with wide-field cameras that enable imaging of clusters to their virial radius
with increasingly higher resolution. This set the ground for numerous cluster
mass reconstruction studies both in the strong (Kneib et al. 1996; Hammer
et al. 1997; Sand et al. 2002; Gavazzi et al. 2003; Sand et al. 2004; Broadhurst
et al. 2005a; Sharon et al. 2005) and weak lensing regimes (Kneib et al. 2003;
12

1.2 Gravitational Lensing
Broadhurst et al. 2005b; Limousin et al. 2007; Jee et al. 2009).
The cosmic shear technique is based on measuring the distortions to
galaxies induced by the Large Scale Structure (LSS). Formalisms that crosscorrelate
the foreground structures along the line of sight with the distribution
of background images were developed (Wittman et al. 2001; Jain
& Taylor 2003; Bacon et al. 2003; Taylor et al. 2007). Cosmic shear was
recently statistically detected in large random patches of the sky including
the COMBO-17 fields (Brown et al. 2003; Kitching et al. 2007) and the
COSMOS field (Massey et al. 2007). These cosmic shear surveys provide a
direct measurement of the LSS, and can be compared to theories of structure
formation, thereby opening wide prospects for studying cosmology. To usefully
derive cosmological parameters from general cosmic shear work, deep
all-sky surveys have been proposed (e.g., LSST 1 , DES 2 , JDEM 3 , EUCLID 4 )
and will commence in the coming decade.
1.2.2 Principles of Gravitational Lensing
In Figure 1.1 a typical case of gravitational lensing is portrayed. If a mass
at redshift z d deflects light from a source at z s , and its size is much smaller
than the angular diameter distances D d (observer to lens) and D ds (lens to
source), the deflection can be describes by two straight light rays as in the
figure. The angle between the two lines is called the “deflection angle” ˆ⃗α,
which depends on the mass.
1.2.2.1 The Deflection Angle
In the simple case of deflection by a point mass M, where the light ray travels
far enough from the horizon of the lens, i.e., it’s impact parameter is much
1 http://www.lsst.org/lsst
2 https://www.darkenergysurvey.org/
3 http://jdem.gsfc.nasa.gov
4 http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=42266
13

Introduction
Figure 1.1 – Sketch of a typical gravitational lens system.
larger than the Schwarzschild radius of the lens, ξ ≫ R S = 2GM/c 2 , GR
gives the deflection angle,
ˆα = 4GM
c 2 ξ , (1.18)
which is twice the classical Newtonian gravity result (e.g., Futamase 1995).
Considering a more general case of three-dimensional mass distribution,
with volume density ρ(⃗r), the total deflection angle is then
ˆ⃗α( ξ) ⃗ = 4G ∑
dm(ξ
′
c 2 1 , ξ 2, ′ r 3) ⃗ ′ ξ − ξ ⃗′
| ξ ⃗ − ξ ⃗′ | 2
= 4G ∫ ∫
d 2 ξ ′ dr
c
3ρ(ξ ′ 1, ′ ξ 2, ′ r ′ ξ
3) ⃗ − ξ ⃗′
2
| ξ ⃗ − ξ ⃗′ | . (1.19)
2
14

1.2 Gravitational Lensing
If we instead define a two-dimensional surface mass density
∫
Σ( ξ) ⃗ ≡ dr 3ρ(ξ ′ 1, ′ ξ 2, ′ r 3) ′ (1.20)
which is the mass density projected on the image plane, then the deflection
angle is simply
ˆ⃗α( ⃗ ξ) = 4G
c 2
∫
d 2 ξ ′ Σ( ⃗ ξ ′ ) ⃗ ξ − ⃗ ξ ′
| ⃗ ξ − ⃗ ξ ′ | 2 . (1.21)
1.2.2.2 The Lens Equation
The basic lens equation relates the true angular position of the source, ⃗ β, to
its lensed image position on the sky, ⃗ θ
⃗β = ⃗ θ − D ds
D s
⃗ˆα(Ds ⃗ θ) ≡ ⃗ θ − ⃗α( ⃗ θ) , (1.22)
(see Figure 1.1).
expressed as
Here ⃗α( ⃗ θ) is the scaled deflection angle, which can be
⃗α( ⃗ θ) = ⃗ ∇ θ ψ = 1 π
∫
κ( ⃗ θ ′ ) ⃗ θ − ⃗ θ ′
| ⃗ θ − ⃗ θ ′ | 2 d2 θ ′ , (1.23)
where ψ is the deflection potential, and κ( ⃗ θ), called the convergence, is the
dimensionless surface mass density, defined as
κ( θ) ⃗ ≡ Σ(D ⃗ dθ)
with Σ cr = c2
Σ cr 4πG
D s
D ds D d
. (1.24)
Σ cr is a characteristic value of surface mass density that marks the distinction
between “strong” and “weak” lenses.
It can be readily shown that κ( ⃗ θ) satisfies Poisson’s Equation,
⃗∇ 2 θψ = 2κ( ⃗ θ) , (1.25)
since ψ( ⃗ θ) is equivalent to a two-dimensional gravitational potential, which
15

Introduction
in turn can be written as,
⃗ψ( ⃗ θ) = 1 π
∫
κ( ⃗ θ ′ ) ln | ⃗ θ − ⃗ θ ′ |d 2 θ ′ . (1.26)
1.2.2.3 Magnification and Distortion
As presented above, ⃗ θ describes the image angular position of a source at ⃗ β.
The image shape will differ from the source shape because light bundles are
deflected differentially. Giant arcs in clusters are an extreme example of such
a distortion.
The local properties of the lens mapping from the source plane to the lens
plane are expressed by the Jacobian matrix,
A( θ) ⃗ ≡ ∂⃗ (
) (
)
β
∂θ ⃗ = δ ij −
∂2 ψ 1 − κ − γ 1 −γ 2
=
∂θ i ∂θ j −γ 2 1 − κ + γ 1
, (1.27)
where the components of the shear tensor, γ = γ 1 + γ 2 = |γ|e 2iφ , are defined
as the linear combinations of ψ ij ,
γ 1 ( ⃗ θ) = 1 2 (ψ 11 − ψ 22 ) ≡ γ( ⃗ θ) cos[2φ( ⃗ θ)] , (1.28)
γ 2 ( ⃗ θ) = ψ 12 = ψ 21 ≡ γ( ⃗ θ) sin[2φ( ⃗ θ)] .
and as in eq. 1.25, κ( ⃗ θ) satisfies
κ( ⃗ θ) = 1 2 (ψ 11 + ψ 22 ) = 1 2 trψ ij (1.29)
The shear describes the anisotropy of the lens mapping, turning a circle
into an ellipse, where |γ| is the magnitude of the shear and φ is its orientation.
The convergence κ results in isotropic magnification of a source. Figure 1.2
describes such mapping, in the presence of both κ and γ, turning a circle
into an ellipse with major and minor axes
a = (1 − κ + γ) −1 , b = (1 − κ − γ) −1 (1.30)
16

1.2 Gravitational Lensing
Figure 1.2 – Illustration of the effects of convergence and shear on a circular source.
Convergence magnifies the image isotropically, and shear deforms it to an ellipse.
The magnification is then,
µ = 1
det A = 1
(1 − κ) 2 − |γ| 2 . (1.31)
Points in the lens plane where the determinant equals zero, i.e. locations
of infinite magnification, form closed curves, which are called critical curves.
Their equivalent curves in the source plane are called caustics. Images along
these curves will be magnified and distorted substantially.
Strong lensing occurs when the gravitational potential is extremely large,
called a supercritical lens, where Σ ≥ Σ cr , and the shear and convergence are
strong enough [see][](Hattori et al. 1999), often leading to multiple images of
a suitably positioned source. The effect causes great amplification and shear,
often visible in centers of massive clusters as large distorted arcs lying near
the critical curves. These have been observed with high quality and great
detail with the Advanced Camera for Surveys (ACS) aboard the Hubble Space
Telescope (HST ). Some examples of giant arcs can be seen in the clusters
Abell 370 and Abell 2218 (see Figure 1.3).
17

Introduction
Figure 1.3 – Examples of gravitationally lensed arcs in galaxy clusters. Left: Galaxy
cluster Abell 370. Right: Giant arcs in Abell 2218.
1.2.3 Weak Gravitational Lensing
Beyond this region, where κ ≪ 1 and |γ| ≪ 1, a weaker effect is induced,
therefore called weak-lensing. The distortion and magnification here are so
small that the effect is statistical in nature – it can only be measured by
averaging over thousands of galaxies. As galaxies are elliptical by nature,
one cannot deduce the potential field from a single image (except for giant
arcs as shown above, where the distortion is extreme). However, assuming
galaxy orientations are randomly distributed, a coherent distortion can be
measured from a large sample of galaxies, if the net ellipticity is higher than
the Poisson noise of the intrinsic galaxy ellipticity distribution.
lensing the reduced shear g is defined as,
In weak
g( ⃗ θ) ≡ γ(⃗ θ)
1 − κ( ⃗ θ) . (1.32)
The reduced shear is the actual quantity accessible through measurements of
image ellipticities. I will demonstrate below that in the weak lensing limit,
γ ≈ g ≈ 〈e〉 . Using this relation, measuring image ellipticities can be used to
2
deduce the lensing quantities.
18

1.2 Gravitational Lensing
1.2.3.1 Measuring Galaxy Shapes and Sizes
With the surface brightness I( ⃗ θ) measured for a galaxy at ⃗ θ, we define the
tensor of second brightness moments,
Q ij =
∫
d 2 θW [I( θ)](θ ⃗ α − ¯θ i )(θ j − ¯θ j )
∫
d2 θW [I( θ)] ⃗ , i, j ∈ 1, 2, (1.33)
where the center ¯⃗ θ of the shape is defined as
¯⃗θ =
∫
d 2 θW [I( ⃗ θ)] ⃗ θ
∫
d2 θW [I( ⃗ θ)]
(1.34)
and W [I( θ)] ⃗ is a properly chosen weight function, e.g., in our case a Gaussian
window function.
The size of the object can be found from the determinant of the tensor
Q,
ω = (Q 11 Q 22 − Q 2 12) 1/2 . (1.35)
We define the shape of the image by the complex ellipticity
e ≡ Q 11 − Q 22 + 2iQ 12
Q 11 + Q 22
. (1.36)
For the case of an elliptical galaxy, this simplifies to
e = 1 − r2
1 + r 2 e2iϑ (1.37)
where r ≤ 1 is the axis ratio of the elliptical isophote, and the phase is twice
the position angle of the major axis, ϑ.
Thus, the ellipticity is the same
if the galaxy image is rotated by π, since a rotation of an ellipse leaves it
unchanged.
Defining the complex ellipticity of the intrinsic source in analogy to
eq. 1.36 with in terms of the tensor Q (s)
ij of the source, leads to
e (s) =
e − 2g + g2 e ∗
1 + |g| 2 − 2R[ge ∗ ]
(1.38)
19

Introduction
(Schneider & Seitz 1995), where the asterisk denotes complex conjugation,
and g is the reduced shear as defined in eq. 1.32. This equation demonstrates
that the reduced shear is the only lensing-related quantity accessible through
image ellipticity measurements. Based on the assumption that the sources are
randomly oriented, the expectation value of the intrinsic (unlensed) source
ellipticity e (s) is assumed to vanish, 〈e (s) 〉 = 0. Replacing this expectation
value by the average over a local ensemble of image ellipticities, Schneider &
Seitz (1995) showed that this is equivalent to
∑
i
w i
e i − δ g
1 − R[δ g e ∗ i ] = 0 . (1.39)
Here, δ g ≡ 2g/(1 + |g| 2 ) is the complex distortion (Schneider & Seitz 1995),
which is a function of g that is invariant under g → 1/g ∗ , and w i is a statistical
weight for the ith object.
e (s) = 0, we have
Thus, for an intrinsically circular source with
e =
2g
1 + |g| 2 . (1.40)
In the weak limit, where κ ≪ 1 and |γ| ≪ 1, eq. 1.38 reduces to e (s)
e − 2g ≈ e − 2γ.
Assuming the random orientations of the background
sources, we average observed ellipticities over a sufficient number of images
to obtain
〈e〉 ≈ 2g ≈ 2γ. (1.41)
≈
1.2.3.2 The KSB Method
For practical applications of weak lensing shape measurements, we must
take into account various observational effects such as noise in the shape
measurement due to readout and/or sky background, and smearing of the
lensing signal due to isotropic/anisotropic point-spread function (PSF) effects.
Thus, one cannot simply use eq. 1.41 to measure the gravitational
shear field. Kaiser, Squires, & Broadhurst (1995, hereafter KSB) used explicitly
the Gaussian weight function in calculations of noisy shape moments
to calibrate the effect of seeing, and derived in the limit of linear anisotropies
20

1.2 Gravitational Lensing
a transformation that relates the unlensed (intrinsic) and lensed (observed)
ellipticities, expressed as
e α = e (s)
α + P g αβ g β + P q αβ q β (1.42)
where q β is a kernel which measures the PSF anisotropy, P q αβ
is the “smear
polarizability tensor”, and P g αβ
is the “shear polarizability tensor”, both of
which can be estimated. In practice, q( θ) ⃗ can be estimated using image
ellipticities of foreground stars, for which both e (s) and g vanish, so,
q ∗ α = (P ∗ q ) −1
αβ eβ ∗ (1.43)
As before, assuming 〈e (s) 〉 vanishes, we have a linear relation between the
averaged image ellipticity and the reduced shear,
g α = 〈(P g ) −1
αβ (e − P q q) β 〉. (1.44)
A careful calibration of P g is crucial for accurate measurements of the weak
lensing signal.
1.2.4 Clusters of Galaxies as Gravitational Lenses
Here I summarize basic quantities and techniques used for measuring cluster
weak lensing profiles.
1.2.4.1 Distortion measurements in Clusters
The shape distortion of an object due to lensing is described by the complex
reduced shear, g = g 1 + ig 2 ≡ γ/(1 − κ), as described above (eq. 1.32). To
measure the cluster induced distortion, we form a cluster-oriented coordinate
system, and measure the tangential distortion and the 45 ◦ -rotated component
(Tyson & Fischer 1995),
g + ≡ g T = −(g 1 cos 2ϕ + g 2 sin 2ϕ), g × = −(g 1 cos 2ϕ + g 2 sin 2ϕ) (1.45)
21

Introduction
where ϕ is the position angle of an object relative to the cluster center, and
the uncertainty in the g + and g × measurement is
σ + = σ × = σ g / √ 2 ≡ σ (1.46)
in terms of the RMS error σ g on the complex distortion measurement, g. The
+-component, g + , is a measure of the tangential coherence of the shape distortions
of background sources due to weak lensing. On the other hand, the
×-component, g × , corresponds to divergence-free, curl-type distortion patterns.
To improve the statistical significance of the distortion measurement,
we calculate the weighted average of the g + and its weighted error as
∑
i
〈g + (θ n )〉 ≡ 〈g T (θ n )〉 =
u g,i g
∑ T,i
i u , (1.47)
g,i
√ ∑
i
σ + (θ n ) ≡ σ T (θ n ) =
u2 g,i σ2 i
( ∑ i u g,i) 2 , (1.48)
where the index i runs over all of the galaxies located within the n-th annulus
with a median radius of θ n , and u g,i is the inverse variance weight for i-th
object, u g,i = 1/(σ 2 g,i + α 2 ), where α 2 is a softening constant variance. In our
work, we choose α = 0.4, which is a typical value of the mean RMS ¯σ g over
the background sample.
The tangential reduced shear g + (θ) as a function of radius is a useful,
direct observable in cluster weak lensing, being free from the mass sheet
degeneracy (see 1.2.4.2.2): it does not require a non-local mass reconstruction,
and one can easily assess its error propagation. Furthermore, the g ×
component can be used as a useful null check for systematic effects.
1.2.4.2 Systematics of the Methods
1.2.4.2.1 Projection Effects The main systematic afflicting mass reconstruction
using gravitational lensing is that lensing measures projected
mass. Hence, interpreting lensing measurements into three-dimensional masses
necessitates the assumption of spherical symmetry of the system. Projection
22

1.2 Gravitational Lensing
effects can range from slight biases arising from triaxiality of clusters, to large
uncertainties in the mass estimates due to line-of-sight mergers of clusters of
comparable sizes.
1.2.4.2.2 Mass-Sheet Degeneracy Adding a constant mass sheet to
κ does not affect the image ellipticities, since it causes only an isotropic
expansion of the images. This transformations is equivalent to scaling the
Jacobian matrix A by some scalar multiple λA, resulting in
A ′ = λA = λ
(
)
1 − κ − γ 1 −γ 2
−γ 2 1 − κ + γ 1
which is equivalent to the following transformations of κ and γ,
, (1.49)
1 − κ ′ = λ(1 − κ), γ ′ = λγ. (1.50)
This transformation evidently leaves the reduced shear g = γ/(1 − κ) invariant
under this scaling. This invariance ambiguity is referred to as the
mass-sheet degeneracy. It was originally discovered by Falco et al. (1985)
and further highlighted by Schneider & Seitz (1995). The ambiguity can
be broken by measuring the magnification effect, because µ is not invariant
under the transformation, as
µ( ⃗ θ) → λ 2 µ( ⃗ θ). (1.51)
Measuring the magnification bias has been suggested by Broadhurst et al.
(1995) using galaxy number-counts. Another approach is the use of wide-field
imaging with field-of-view larger than the clusters, in which case the lensing
effect vanishes at the boundaries of the field, thus breaking the degeneracy.
1.2.4.2.3 PSF Effects The main difficulty of weak lensing measurements
is due to point-spread function (PSF) effects, arising from the telescope/CCD
response and/or atmospheric seeing. The atmospheric seeing,
which is a term describing the result of atmospheric turbulence, causes smear-
23

Introduction
ing of the image, thus diluting or even erasing the real lensing signal. Since
galaxies have angular sizes below 1 ′′ , the required seeing for observations
should also be below 1 ′′ . Such a limit is not commonly met in many observational
sites. The anisotropy in the PSF induced by the telescope optics
results in systematic image ellipticities. The PSF has an effect of 8 − 10%,
whereas the weak lensing signal is comparable or less (only about 1 − 3% for
cosmic shear). It is therefore crucial to model and remove the systematic effects
of the PSF with high accuracy. Given that PSF varies significantly with
different telescope/CCD configurations and atmospheric situation, this is not
a simple task. A number of lensing methods are now able to model this quite
well, e.g., the Kaiser, Squires, & Broadhurst (1995) method described above
(1.2.3.2), generally using stars along the field to sample the PSF, with robust
algorithms for modeling and deconvolving the effect (Heymans et al. 2006;
Massey et al. 2007a, for a recent review and comparison of all methods).
1.2.4.2.4 Weak Lensing Dilution It is crucial in cluster weak lensing
analysis to carefully select background galaxies in order to avoid contamination
by the lensing cluster and/or foreground galaxies. The inclusion of such
unlensed galaxies reduces the lensing signal, particularly at smaller distances
from the cluster center, where the cluster is relatively dense. When excluding
only a narrow band containing the obvious E/S0 sequence, following common
practice, the lensing signal of the remainder is found to fall rapidly
towards the cluster center, in contrast to the uncontaminated population of
background galaxies lying redward of the cluster sequence (Broadhurst et al.
2005b).
Other calibration issues in the cluster mass measurement arise from the
redshift distribution of the background population which has to be taken into
account. In the work presented here I elaborate on and attempt to correct
for both these effects.
24

1.3 The Structure of the Thesis
1.3 The Structure of the Thesis
The thesis consists of two main scientific efforts: A detailed study of several
clusters of galaxies with weak gravitational lensing, and an initial attempt to
make cosmological deductions from the study. On one level, I developed new
procedures to maximize and improve the way weak lensing is measured in
clusters, focusing on a careful selection of galaxies for weak lensing analysis,
while benefiting from the “dilution” to learn about the cluster light properties.
This work is presented in Chapter 2 for the first cluster examined, Abell
1689. This work was extended to three more high-mass clusters, as described
in Chapter 3. On the other level, I examined the distance-redshift relation
from the weak lensing of selected samples of background galaxies behind one
of the explored clusters, A370, and led the way to a new cosmological probe
based on this detectable behavior, as discussed in Chapter 4.
25

Chapter 2
Using Weak Lensing Dilution
to Improve Measurements of
the Luminous and Dark Matter
in A1689
A version of this chapter has been published as Medezinski, E. et al. 2007,
ApJ, 663, 717.
2.1 Introduction
The influence of “Dark matter” is strikingly evident in the centers of massive
galaxy clusters, where large velocity dispersions are measured and giant
arcs are often visible. Cluster masses may be estimated by several
means, leading to exceptionally high central mass-to-light ratios, M/L R ∼
100 − 300h(M/L R ) ⊙ (Carlberg et al. 2001), far exceeding both the mass of
stars comprising the light of the cluster galaxies and the mass of plasma
derived from X-ray emission and the SZ effect. Reasonable consistency is
claimed between dynamical, hydrodynamical and lensing-based estimates of
27

Weak Lensing Dilution in A1689
cluster masses, supporting the conventional understanding of gravity. However,
the high mass-to-light ratio requires that an unconventional nonbaryonic
dark material of unclear origin dominates the mass of clusters.
In detail, discrepancies are often reported between masses derived from
strong lensing and X-ray measurements, with the claimed X-ray masses often
lower in the centers of clusters. High resolution X-ray emission and temperature
maps reveal that the majority of local clusters undergo repeated merging
with sub-clumps, and obvious shock fronts are seen in some cases (Markevitch
et al. 2002; Reiprich et al. 2004). The double cluster 1E 0657–56 (a.k.a.,
the “bullet” cluster, z = 0.296) is the most extreme example studied, where
the associated gas forms a flattened luminous shock-heated structure lying
between the two large distinct clusters, clearly indicating these two bodies
collided recently at a high relative velocity (Markevitch et al. 2004), with
the gas remaining in between while the cluster galaxies have passed through
each other relatively collisionlessly. Very interestingly, the weak lensing signal
follows the double structure of the clusters, rather than the gas in between,
cleanly demonstrating that the bulk of the matter is collisionless and
dark (Clowe et al. 2004). This system favors the standard cold dark matter
(CDM) scenario, places a restrictive limit on the interaction cross-section of
any fermionic dark matter, and disfavors a class of alternative gravity theories
in which only baryons are present (Clowe et al. 2004). Smaller but significant
discrepancies of the same sort are also claimed for other interacting clusters
(Natarajan et al. 2002; Jee et al. 2005).
Many clusters show no apparent signs of significant ongoing interaction;
these have centrally symmetric X-ray emission and little obvious substructure
(Allen 1998), and for some of these the lensing and X-ray (or dynamical)
derived masses are claimed to agree, as would be expected for relaxed systems
(e.g., Arabadjis et al. 2002; Rines et al. 2003; Diaferio et al. 2005).
More generally, since the dyanmics of dark matter and most cluster galaxies
is essentially collisionless, we would expect them to have similar radial profiles.
Biasing inherent in hierarchical growth may significantly modify this
similarity (Kauffmann et al. 1997), and dynamical friction is expected to
concentrate the relatively more massive galaxies in the core. These together
28

2.1 Introduction
with tidal effects may help to account for the unique properties of cD galaxies.
Hence comparisons of the mass profile with the light profile of the cluster
galaxies are expected to provide an additional insight into the formation of
clusters and the nature of dark matter.
Recent improvements in the quality of data useful for weak and strong
lensing studies now allow the construction of much more definitive mass profiles
that are sufficiently precise to test the predictions of popular models,
relatively free of major assumptions. The inner mass profiles of several clusters
have been constrained in some detail via lensing, using multiply lensed
background galaxies (Kneib et al. 1996; Hammer et al. 1997; Sand et al. 2002;
Gavazzi et al. 2003; Sand et al. 2004; Broadhurst et al. 2005a; Sharon et al.
2005). The statistical effects of weak lensing have been used to extend the
mass profile to larger radii. The mass profiles derived from these observations
have been claimed to show NFW-like behaviour (Navarro, Frenk, & White
1997), with a continuous flattening towards the center, but with higher density
concentrations than expected. This is particularly evident for A1689, for
which we have constructed the highest quality lensing based mass profile to
date, combining over 100 multiply-lensed images and weak lensing effects of
distortion and magnification from Subaru (Broadhurst et al. 2005a,b).
A lesson learned from this earlier work is the importance of carefully
selecting a background population to avoid contamination by the lensing
cluster. It is not enough to simply exclude a narrow band containing the
obvious E/S0 sequence, following common practice, because the lensing signal
of the remainder is found to fall rapidly towards the cluster center, in
contrast to the uncontaminated population of background galaxies lying redward
of the cluster sequence (Broadhurst et al. 2005b). In the well studied
case of A1689, there has been a long standing discrepancy between the strong
and weak lensing effects, with the weak lensing signal underpredicting the observed
Einstein radius by a factor of ∼ 2.5 (Clowe & Schneider 2001; Bardeau
et al. 2005), based only on a minimal rejection of obvious cluster members
using one or two-band photometry.
In our recent weak lensing analysis of Subaru images of A1689 the above
29

Weak Lensing Dilution in A1689
behavior was found when we rejected only the cluster sequence in the same
way as others. This resulted in a relatively shallow trend of the weak lensing
signal with radius and consequently an underprediction of the Einstein
radius accounting for the earlier discrepancy (Broadhurst et al. 2005b) . If,
however, one selects only objects redder than the cluster sequence for the
lensing analysis, then the weak tangential distortion continues to rise all the
way to the Einstein radius in very good agreement with the strong lensing
strength. This red population is naturally expected to comprise only
background galaxies, made redder by relatively large k-corrections and with
negligible contamination by cluster members, since the bulk of the reddest
cluster members are the early-type galaxies defined by the cluster sequence.
However, for galaxies with colors bluer than the cluster sequence, cluster
members will be present along with background galaxies since the cluster
population extends to bluer colors of the later-type members, overlapping in
color with the blue background. The effect of the cluster members is simply
to reduce the strength of the weak lensing signal when averaged over a
statistical sample, in proportion to the fraction of cluster members whose
orientations are randomly distributed, therefore diluting the lensing signal
relative to the reference background level derived from the red background
population.
We can turn this dilution effect to our advantage and use it to derive
properties of the cluster population, in particular the radial light profile, for
comparison with the dark matter profile. Deriving a light profile this way
has advantages over the usual approach to defining cluster membership. The
inherent fluctuations in the number counts of the background population are
a significant source of uncertainty in the usual approach of subtracting the
far-field level when defining the cluster population (e.g., Paolillo et al. 2001;
Andreon et al. 2005; Pracy et al. 2005). This uncertainty is often cited as
a potential explanation for the substantial variation reported between luminosity
functions derived for different clusters, particularly in the outskirts
of clusters, where not only is the density of galaxies lower, but their colors
are bluer, thus harder to distinguish from the background using photometry
alone.
30

2.2 Subaru imaging reduction and sample selection
In § 2.2 we describe the observations and photometry of the Subaru images
of A1689. In § 2.3 we describe the distortion analysis applied to the Subaru
data. The distortion analysis of ACS images of A1689 is described in § 2.4.
In § 2.5 we describe the photometric redshift analysis of the Subaru and ACS
images with reference to the Capak et al. (2004) sample of deep multi-color
Subaru images. Our weak lensing dilution analysis of the Subaru images
is described in § 2.6. In § 2.7 we go on to derive the cluster luminosity
profile and color, and in § 2.8 the cluster luminosity function is deduced at
several radial positions. In § 2.9 we determine the M/L profile, and in § 2.10
we do a consistency check for the mass derived in this paper with previous
estimations. Our conclusions are summarized in § 2.11.
The concordance ΛCDM cosmology is adopted (Ω M = 0.3, Ω Λ = 0.7 but
h is left in units of H 0 /100 km s −1 Mpc −1 , for easier comparison with earlier
work).
2.2 Subaru imaging reduction and sample selection
We have retrieved Suprime-Cam imaging of A1689 in V (1920s) and SDSS i ′
(2640s) from the Subaru archive, SMOKA. 1 Reduction software developed by
Yagi et al. (2002) is used for flat-fielding, instrumental distortion correction,
differential refraction, PSF matching, sky subtraction, and stacking. The
resulting FWHM is 0 ′′ .82 in V and 0 ′′ .88 in i ′ with 0 ′′ .202 pix −1 , covering a
field of 30 ′ × 25 ′ .
Photometry is based on a combined V + i ′ image using the program SExtractor
(Bertin & Arnouts 1996). The limiting magnitudes are V = 26.5 and
i ′ = 25.9 for a 3σ detection within a 2 ′′ aperture. We define three galaxy
samples according to color and magnitude – “red”, “green”, and “blue”
(see table 2.1 for summary), and for all our samples we define a limiting
magnitude of i ′ < 26.5, to avoid incompleteness, as shown in Figure 2.1.
1 http://smoka.nao.ac.jp.
31

Weak Lensing Dilution in A1689
Figure 2.1 – Color vs. magnitude diagram for A1689 cluster galaxies. The E/S0 sequence
is apparent at (V − i ′ ) ∼ 0.8 where there is an overdensity of bright galaxies. The red
points represent the background sample of galaxies redder than the E/S0 sequence. The
blue points represent a background sample of the bluest objects in the field. The green
points cover a range of color chosen to include the cluster sequence and bluer cluster
members, but in addition background galaxies are also present whose colors fall in this
range.
The red galaxy sample consists of galaxies 0.2 mag redder than the E/S0
sequence of the cluster, which is accurately defined by the linear relation
(V − i ′ ) E/S0 = −0.03525i ′ + 1.505, and up to 2.5 mag redder than this line
to include the majority of the background red population. Very red dropout
galaxies may be detected beyond this point. Indeed, one spectroscopically
confirmed example at z = 4.82 has been detected behind this cluster (Frye
et al. 2002), and such cases are excluded by this upper limit, so that we do
not need to make an uncertain correction for the level of their weak lensing
signal which will be significantly larger than for the bulk of the background
red galaxy population. As we will show in § 2.5, most of these red background
galaxies are at a much lower mean redshift of 〈z〉 ∼ 0.85. The red
32

2.2 Subaru imaging reduction and sample selection
Table 2.1 – Sample Selection
Sample mag limit color limit N n [h 2 Mpc −2 ] < z > < D >
red 18 < i ′ < 26.5 0.2

Weak Lensing Dilution in A1689
bluer colors.
The green galaxy sample is simply selected to lie between the red and
blue samples defined above, (V − i ′ +0.1
) E/S0 (Fig. 2.1), with generous limits
−0.3
set to include the vast majority of cluster galaxies, since - as we have established
- both the red and blue samples are negligibly contaminated by cluster
members, and hence the vast majority of cluster members must lie within
this intermediate range of color. A narrow gap on each side of these samples
is left out of our analysis to ensure that the definition of the background does
not encroach on the cluster population. Note that unlike the green sample
containing the cluster population, the background populations do not need to
be complete in any sense but should simply be well defined and contain only
background. Increasing the green sample to cover these narrow gaps does not
lead to any particularly significant change in our conclusions, but only increases
somewhat the level of noise by including relatively more background
galaxies. Within the green sample there are of course background galaxies,
and the purpose of this paper is to make use of the relative proportion of
these cluster and background populations via weak lensing to establish the
properties of the cluster galaxy population, by using the the dilution of the
weak lensing signal of the background galaxies due to the cluster members.
2.3 Distortion analysis of subaru images
We use the IMCAT package developed by N. Kaiser 2 to perform object detection,
photometry and shape measurements, following the formalism outlined
in (Kaiser, Squires, & Broadhurst 1995, hereafter KSB). We have modified
the method somewhat following the procedures described in Erben et al.
(2001, see Section 5).
To obtain an estimate of the reduced shear, g α = γ α /(1 − κ), we measure
the image ellipticity e α from the weighted quadrupole moments of the surface
brightness of individual galaxies. Firstly the PSF anisotropy needs to be
2 http://www.ifa.hawaii/kaiser/IMCAT
34

2.3 Distortion analysis of subaru images
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6
color−(red sequence)
Figure 2.2 – To establish the boundaries of the color distribution free of cluster members
we calculate the mean tangential distortion averaged over the full radial extent of the
cluster, done separately for the blue and red samples. On the right the red curve shows
that g T drops rapidly when the bluer limit of the entire red sample is decreased below
a color while lies +0.2 mag redward of the cluster sequence. This sharp decline marks
the point at which the red sample encroaches on the E/S0 sequence of the cluster. On
the blue side the red limit of the blue sample is chosen to lie −0.45 mag blueward of the
cluster sequence, marking the point at which significant contamination by the cluster acts
to dilute the weak lensing signal from later-type bluer cluster members.
corrected using the star images as references:
e ′ α = e α − P αβ
smq ∗ β (2.1)
where P sm is the smear polarizability tensor being close to diagonal, and
qα ∗ = (P ∗sm ) −1
αβ eβ ∗ is the stellar anisotropy kernel. We select bright, unsaturated
foreground stars identified in a branch of the half-light radius (r h ) vs.
magnitude (i ′ ) diagram (20 < i ′ < 22.5, 〈r h 〉 median
= 2.38 pixels) to calculate
q ∗ α. In order to obtain a smooth map of q ∗ α which is used in equation (2.1),
we divided the 9K × 7.4K image into 5 × 4 chunks each with 1.8K × 1.85K
pixels, and then fitted the q ∗ in each chunk independently with second-order
bi-polynomials, q α ∗ ( ⃗ θ), in conjunction with iterative σ-clipping rejection on
each component of the residual e ∗ α − P αβ
∗smq ∗ β (⃗ θ). The final stellar sample consists
of 540 stars, or the mean surface number density of n ∗ = 0.72 arcmin −2 .
From the rest of the object catalog, we select objects with 2.4 r h 15
pixels as an i ′ -selected weak lensing galaxy sample, which contains 61, 115
35

Weak Lensing Dilution in A1689
galaxies or ¯n g ≃ 81 arcmin −2 . It is worth noting that the mean stellar ellipticity
before correction is (ē 1 ∗ , ē 2 ∗ ) ≃ (−0.013, −0.018) over the data field,
while the residual e ∗ α after correction is reduced to ē ∗res
1 = (0.47±1.32)×10 −4 ,
ē ∗res
2 = (0.54 ± 0.94) × 10 −4 . The mean offset from the null expectation is
|ē ∗res | = (0.71 ± 1.12) × 10 −4 . On the other hand, the rms value of stellar
ellipticities, σ e∗ ≡ 〈|e ∗ | 2 〉, is reduced from 2.64% to 0.38% when applying the
anisotropic PSF correction.
Second, we need to correct the isotropic smearing effect on image ellipticities
caused by seeing and the window function used for the shape measurements.
The pre-seeing reduced shear g α can be estimated from
g α = (P −1
g ) αβ e ′ β (2.2)
with the pre-seeing shear polarizability tensor P g αβ
. We follow the procedure
described in Erben et al. (2001) to measure P g (see also § 3.4 of Hetterscheidt
et al. 2007). We adopt the scalar correction scheme, namely,
P g αβ = 1 2 tr[P g ]δ αβ ≡ P g s δ αβ (2.3)
(Hudson et al. 1998; Hoekstra et al. 1998; Erben et al. 2001; Hamana et al.
2003) The P s g measured for individual objects are still noisy especially for
small and faint objects. We thus adopt a smoothing scheme in object parameter
space proposed by Van Waerbeke et al. (2000, see also Erben et al.
2001; Hamana et al. 2003). We first identify thirty neighbors for each object
in r g -i ′ parameter space. We then calculate over the local ensemble the median
value 〈P s g〉 of P s g and the variance σ 2 g of g = g 1 +ig 2 using equation (2.2).
The dispersion σ g is used as an rms error of the shear estimate for individual
galaxies. The mean variance ¯σ 2 g over the sample is obtained as ≃ 0.171, or
√¯σ
2
g ≈ 0.41.
In the previous study by Broadhurst et al. (2005b), those objects that
yield a negative value of the raw Pg
s estimate were removed from the final
galaxy catalog to avoid noisy shear estimates. On the other hand, in the
present study, we use all of the galaxies in our weak lensing sample including
36

2.3 Distortion analysis of subaru images
galaxies with Pg s < 0. After smoothing s in the object parameter space,
all of the objects yield positive values of 〈 Pg〉 s , with the minimum of ≈ 0.04.
The median value of 〈 Pg〉 s over the weak lensing galaxy sample, including
galaxies with Pg s < 0, is calculated as ≈ 0.32. For a reference, the subsample
of galaxies with Pg s > 0 gives the median average of ≈ 0.33, mostly
weighted by galaxies with r g = 2 − 2.5 pixels. Finally, we use the following
estimator for the reduced shear:
g α = e ′ α/ 〈 P s g〉
. (2.4)
The quadratic shape distortion of an object is described by the complex
reduced-shear, g = g 1 + ig 2 . The tangential component g T is used to obtain
the azimuthally averaged distortion due to lensing, and computed from the
distortion coefficients g 1 , g 2 :
g T = −(g 1 cos 2θ + g 2 sin 2θ), (2.5)
where θ is the position angle of an object with respect to the cluster center,
and the uncertainty in the g T measurement is σ ≡ σ g / √ 2 in terms of the
rms error σ g for the complex shear measurement. The cluster center is well
determined from symmetry of the strong lensing pattern (Broadhurst et al.
2005a). The estimation of g T only has significance when evaluated statistically
over large number of galaxies, since galaxies themselves are not round
objects but have a wide spread in intrinsic shapes and orientations. In radial
bins we calculate the weighted average of the g T s and the weighted error:
〈g T (r n )〉 =
∑
gT /σ 2
∑ 1/σ
2
(2.6)
σ T (r n ) =
(∑ ) −1/2
1/σ
2 . (2.7)
It has been shown that such weights depend on the size of the objects
but mostly on their magnitudes (see e.g. Hoekstra et al. (2006)). Therefore,
as apparent magnitude increases with redshift the redshift distribution of
sources will be modified to some extent by this weighting scheme. We have
37

Weak Lensing Dilution in A1689
0.5
0.5
0.2
0.2
Distortion g +
0.1
Distortion g +
0.1
0.02
0.02
0.01
0.01
2 5 10 20
θ [arcmin]
2 5 10 20
θ [arcmin]
Figure 2.3 – Tangential shear profiles,
g T (r) of the red and blue background populations.
The tangential distortion profile
of both decline smoothly from the center,
remaining positive to the limit of the data.
The red galaxies are fitted well with a simple
power-law, d log g T /d log r = −1.17 ±
0.1. The blue sample is more noisy but
also well represented by the same relation,
only offset in amplitude by 23 ± 17% and
is related to the greater depth of the blue
population relative to the red, § 2.5
Figure 2.4 – Tangential shear vs. radius.
The green points represent the intermediate
color sample, containing both cluster
and background galaxies. The black points
show the level of tangential distortion of
the combined red+blue sample of the background.
The green points fall close to
the background level defined at large radii,
indicating the green sample is dominated
by background galaxies, and falls short towards
the cluster center where cluster members
increasingly dilute the lensing signal.
investigated this using the catalog of Capak et al. (2004) as here photometric
redshifts are estimated (see § 2.5 for a fuller description of the photometric
properties of this sample). First, we generated from the Capak catalog
blue/red background galaxy samples with the same color-magnitude criteria
as the present study. We then derived an i’-magnitude vs. photo-z relation
for each galaxy sub-sample. We subdivided the data into magnitude (i’) bins
and derived a magnitude (i’) vs. photo-z relation using median averaging.
We then assume this magnitude-redshift relation holds in our A1689 data
and obtain for each galaxy in A1689 an estimate of redshift via the magnitude
– photo-z relation. It is then straightforward to have an effective redshift
distribution taking into account weak lensing statistical weights, w. We can
see a qualitative feature that although low-z background galaxies are more
strongly weighted than higher-z ones, the effect on our observed redshift distribution
is negligible because our redshift selection window does not sample
38

2.3 Distortion analysis of subaru images
these larger angle, lower redshift objects, but rather the more distant faint
population whose small angular sizes are heavily influenced by the seeing.
In Figure 2.3 we compare the radial profile of g T of the red and blue
samples defined above. These have a very similar form implying that the
blue sample, like the red sample, is dominated by background galaxies with
negligible dilution by cluster members even at small radius where the cluster
overdensity is large. Very interestingly a clear offset is visible between these
profiles over the full range of radius, with the amplitude of the blue sample
lying systematically above that of the red sample, as shown in Figure 2.3.
This is readily explained as a depth related effect, as we show below in § 2.5
where we evaluate the redshift distributions of these two populations. We
find the blue sample to be deeper than the red, and since lensing scales with
depth, so should the lensing profiles be offset from each other by the same
scale.
The tangential distortion of the green population behaves quite differently
(Fig. 2.4) falling well below the background level near the cluster center.
The green sample has a maximum signal at intermediate radius, 3 ′ − 5 ′ ,
and then declines quickly inside this radius as the unlensed cluster galaxies
dominate over the background in the center. Notice that the green sample
does not fall to zero at the outskirts but rises up to almost meet the level
of the background sample, indicating that the majority of the green sample
at large radius comprises background rather than cluster members. We go
on to use the ratio of the distortion of the green sample compared with the
background level to determine the proportion of cluster members in § 2.6,
but to do so we first evaluate the expected depths of our samples in § 2.5,
in order to make a precise comparison of the lensing signals between them,
since the lensing signal scales geometrically with increasing source distance
and must be accounted for in any comparisons.
The results of this paper depend on the ratio of the background distortion
to the cluster contaminated distortion, (g (B)
T
/g(G) T
), so that the 5-10% level
calibration correction factors estimated from simulations done by the STEP
project (Heymans et al. 2006) for the various weak distortion methods are
39

Weak Lensing Dilution in A1689
not of major concern for the bulk of our work.
2.4 Distortion analysis of ACS/HST images
In the center of the cluster inside a radius of approximately 1 ′ the Subaru
data become limited in depth by the extended bright halos of the many luminous
central galaxies. This region is far better resolved and more deeply
imaged with HST/ACS in 20 orbits of imaging shared between the g ′ r ′ i ′ z ′
passbands. Many multiple images are known here, defining accurately the
shape of both the tangential and radial critical curves (Broadhurst et al.
2005a). Here we analyze the statistical distortion of the shapes of the many
galaxies recorded in these images, to extend our analysis of the properties
of the cluster galaxies into the center, allowing an accurately defined central
cluster luminosity function to faint luminosities. In addition, it will be interesting
to see how consistent the distortion profile derived here independently
matches the mass profile obtained previously from the strongly lensed multiple
images. We stick to very similar definitions of the three-color selected
population as with Subaru, but extend their depths by an additional 1.5 magnitudes
(see Figure 2.5), since the ACS data are so much deeper. The ACS
images are limited to m=28.5 (5σ) in each of the passbands. The reduction
of the ACS image and the photometry for the faint sources is described in
detail in Broadhurst et al. (2005a), including the subtraction of the bright
central galaxies in the cluster which is essential for obtaining accurate photometry
and shape measurements of central lensing images including radial
arcs and demagnified central images. For the distortion analysis we prefer
the im2shape method developed by Bridle et al. (2002) for dealing in particular
with relatively elongated images produced by lensing in the strongly
lensed region. This is an improvement over the standard KSB method which
we used in the weak lensing regime appropriate for everything except the
central region r < 2 ′ , and used for the Subaru analysis described above.
Using this method, galaxies are fit to a sum of two sheared Gaussians
convolved with a PSF. Each Gaussian has two free parameters, amplitude
40

2.4 Distortion analysis of ACS/HST images
3.5
3
2.5
2
g’−i’
1.5
1
0.5
0
−0.5
−1
16 18 20 22 24 26 28
i’ AB
Figure 2.5 – Color vs. magnitude diagram for the central region covered by ACS photometry
with magnitudes transformed to AB system to match the V,I photometry from
Subaru. Notice the prominent E/S0 sequence and the greater depth of these data compared
with Subaru shown in Figure 1. A one-to-one comparison of magnitudes for objects
in both datasets is shown in Figure 2.15.
and width. The centroid and the shear are also allowed to vary, but these
are restricted to be the same for both Gaussians. Meanwhile, the PSF for
each galaxy is determined based on models described in Jee et al. (2005).
These PSF models for ACS’s Wide Field Camera (WFC) were derived from
observations of the globular cluster 47 Tuc (PROP 9656, P.I. De Marchi).
As the distortion measurements are performed in the detection image, each
galaxy is assigned an ”average” PSF based on the different filters and chip
positions in which it was observed.
We plot the resulting values of g T (r) (Fig. 2.6) for the blue, green and
red galaxies defined in the same color ranges as the Subaru data, but to
fainter magnitudes. A very well defined saw-tooth pattern is visible, showing
that images are maximally radially aligned at about 17 ′′ and then maximally
tangentially aligned at about 47 ′′ . This is a very clear signature of strong
41

Weak Lensing Dilution in A1689
lensing, where the maximum corresponds to the location of the tangential
critical curve (Einstein radius), and the minimum to the radial critical curve,
where images are maximally stretched in the radial direction generating a
ring of long images pointing to the center of mass, as found in Broadhurst
et al. (2005a). The location of these critical radii agrees very well with those
derived from the model to the strong lensing data for this cluster, fitted by
Broadhurst et al. (2005a).
Another clearly defined radius can also be identified from the point where
the images distortion goes through zero, g T = 0, at a radius in between these
two critical radii at about r ≃ 27 ′′ . It is important to note that in this region,
between the two critical curves, the parity is p = −1 (odd parity), and here
g = 1 e ∗ =
e
|e| 2 (2.8)
(see Kaiser 1995). Here, instead of measuring g we are measuring −1 < e < 1.
(This is different than in the weak lensing region, outside the tangential
critical curve, where p = +1 and g = e, and therefore no distinction needed
to be made). Since
e = 1 ∝ 1 − κ, (2.9)
g∗ zero distortion corresponds to κ = 1 curve, which lies in between the tangential
and critical curves. Note, for several reasons we cannot expect that the
data will reach the theoretically extreme value of g T = 1 at the tangential
critical radius, meaning that the images are infinitely stretched tangentially,
and also g T = −1 at the radial critical radius where they are stretched infinitely
in the radial direction. By definition, weak lensing measurements will
underestimate the strong distortions near critical curves. In addition, convolution
by the redshift distribution of the background sources will smooth
these features out. Nonetheless, we can define their locations in radius rather
precisely and these positions must be reproduced in any satisfactory model,
at r ≃ 17 ′′ and r ∼ 48 ′′ . In addition the radius at which g T = 0 is also well
defined at about r ≃ 28 ′′ and corresponds to a surface density where κ = 1,
supplying another important constraint on model mass profiles.
42

2.4 Distortion analysis of ACS/HST images
0.8
0.6
0.4
Distortion g +
0.2
0
−0.2
−0.4
−0.6
0.1 0.4 0.9 2 5 10 20
θ [arcmin]
Figure 2.6 – Distortion profile for the central r < 2 ′ area covered by ACS (filled circles)
together with the weaker distortions measured by Subaru at larger radius (open symbols).
The black points represent the combined blue and red sample of background galaxies, and
the green points include the cluster members. A remarkable saw-tooth pattern is visible
for the background galaxy distortions in the strong lensing region where the tangential and
radial critical radii are clearly visible corresponding to a maximum and a minimum in the
value of g T , respectively. In between the distortion passes through zero, where the degree
of tangential and radial distortion is equal, leaving images unchanged in shape at a radius
where κ = 1. The distortion of the green sample is consistent with zero in the inner region
where the cluster members dominate the sample but at larger radius the green and black
points merge for r > 3 ′ , indicating that there the green sample comprises predominately
background galaxies.
Outside the tangential critical curve we find that the tangential shear
of the background sample (black circles in Fig. 2.6) drops to g T,B ∼ 0.2 at
r = 2 ′ , in good agreement with the Subaru analysis at this radius, giving
us confidence in the consistency of our work. For the color range defined
above, which includes the cluster sequence and all the bluer members of the
cluster galaxy population, g T,G ∼ 0 over the full range of radius of the ACS
data (green circles in Fig. 2.6), indicating - as expected - that the galaxy
population in the this color range is dominated by cluster members with
negligible background contamination.
43

2.5 Photometric redshifts
Weak Lensing Dilution in A1689
We need to estimate the respective depths of our color-magnitude selected
samples when estimating the cluster mass profile, because the lensing signal
increases with source distance, and therefore must differ between the samples.
The effect of this difference in distance on the weak lensing signal is simply
linear as we can see from the relation between the dimensionless surface mass
density,
where
and the tangential distortion:
κ(r) = Σ(r)/Σ crit , (2.10)
Σ crit =
c2
4πGd l
D s
D ds
(2.11)
〈g T (r)〉 = (¯κ(r) − κ(r))/(1 − κ(r)) (2.12)
so that in the weak limit where κ is small,
〈g T (r)〉 ∝ D ds
D s
(¯Σ(r) − Σ(r)) (2.13)
and hence for an individual cluster, with a fixed redshift and a given mass
profile, the observed level of the weak distortion simply scales with the lensing
distance ratio. Further details are presented in the appendix. The mean ratio
of D ds /D s , which is weighted by the redshift distribution of the background
population corresponding to our magnitude and color cuts, is calculated using
the expression
〈D〉 ≡ 〈 D ds
D s
〉 =
∫ D ds
D s
(z)N(z)dz
∫
N(z)dz
. (2.14)
Since we cannot derive complete samples of reliable photometric redshifts
from our limited 2-color V, i ′ images of A1689, we instead make use of other
deep field photometry covering a wider range of passbands, sufficient for photometric
redshift estimation of the faint field redshift distribution appropriate
for samples with the same color and magnitude limits as our red, green and
44

2.5 Photometric redshifts
Figure 2.7 – Color–magnitude diagram for Capak galaxy catalog for the same passbands
as the photometry of A1689 shown in Fig 2.1. This photometry is of a field region and
serves as our reference for evaluating the expected depth of the background samples defined
in § 2.2.
blue populations.
The photometry of Capak et al. (2004) is very well suited for our purposes,
consisting of relatively deep multi-color photometry over a wide field taken
with Subaru, producing reliable photometric redshifts for the majority of
field galaxies to faint limiting magnitudes. The Capak et al. (2004) galaxy
catalog contains almost 50,000 galaxies over 0.2 sq. deg. with UBV RIz ′
photometry. We have estimated photometric redshifts for this catalog using
the Bayesian based method of Benítez (2000), with a prior based on the
redshift and spectral type distributions of the HDF-N, with a spectral library
containing the templates of Benítez et al. (2004) with an additional two blue
starburst galaxies as described in Coe et al. (2006). A full redshift probability
45

Weak Lensing Dilution in A1689
0.85
0.8
0.75
1.4
1.3
1.2
1.1
0.696
0.694
0.692
0.69
0.78
0.77
0.76
0.75
1.95
1.9
1.85
24.5 25 25.5 26 26.5 27
Limiting magnitude
0.849
0.846
0.843
0.84
24.5 25 25.5 26 26.5 27
Limiting magnitude
Figure 2.8 – Mean redshift as a function of
the apparent i’-magnitude limit of the red,
green & blue backgrounds, calculated using
the photometric redshifts of the Capak
et al. (2004) sample. The average redshift
differs significantly between the three samples,
being lowest for the red background
〈z〉 ∼ 0.85 and highest for the blue 〈z〉 ∼ 2.
Figure 2.9 – Similar to the previous figure,
but here the weighted mean lensing depth
D ds /D s is calculated as a function of the
apparent i’-magnitude limit. The expected
depth of the samples differs significantly between
the samples and in general the distance
ratio grows only slowly with increasing
apparent magnitude for each sample.
distribution is produced for each galaxy of the form:
p(z|C) ∝ ∑ T
p(z, T |m 0 )p(C|z, T ), (2.15)
where p(C|z, T ) is the redshift likelihood obtained by comparing the observed
colors C with the redshifted library of templates T . The factor p(z, T |m 0 ) is
a prior which represents the redshift/spectral mix distribution as a function
of the observed I−band magnitude. We use a prior which describes the redshift/spectral
type mix in the HDF-N, which has been shown to significantly
reduce the number of “catastrophic” errors (∆z > 1) in the photometric
redshift catalog (see Benítez et al. 2004, and references therein). For each
galaxy we look at its redshift probability distribution p(z) and identify up
to 3 redshift local maxima. Each of these maxima corresponds to a redshift
z i , spectral type t i , and a discretized probability p i (z i , t i ) ≤ 1. Using
this information we generate a mock observation of all the z i , t i combinations
in the Subaru filters, and then build a redshift histogram by selecting
galaxies using the same color cuts and adding up their probabilities in each
redshift bin. The color-magnitude diagram for the Capak catalog galaxies is
46

2.5 Photometric redshifts
shown in Figure 2.7, where the equivalent color-magnitude selected samples
are displayed. The resulting mean redshift of the background galaxies in
each of our three color-selected samples is calculated as a function of limiting
magnitude of the sample (Fig. 2.8), by using the redshift distribution
from the Capak catalog. The redshift distribution is also used to evaluate
the weighted mean depths 〈D〉 (shown in Fig. 2.9 as a function of sample
limiting magnitude), for comparing the weak lensing amplitudes between the
green and the red+blue samples. This is done by dividing up each sample
into 81 independent bins of 2 ′ , calculating the weighted mean redshift and
depth in each, and taking the mean value and variance over the bins. The
mean redshift of the red sample is only 〈z red 〉 ∼ 0.871 ± 0.045, whereas the
blue sample is calculated to have, 〈z blue 〉 ∼ 2.012 ± 0.124. The green sample
lies in between with, 〈z green 〉 ∼ 1.429 ± 0.093. The weighted relative depths
of these samples using equation (2.14), for samples selected to our magnitude
limit of i < 26.5, are 〈D red 〉 = 0.693 ± 0.012, 〈D green 〉 = 0.728 ± 0.012,
and 〈D blue 〉 = 0.830 ± 0.011, and the corresponding redshifts z D equivalent
to these mean depths are, z D,red = 0.68, z D,green = 0.79, z D,blue = 1.53, respectively.
Hence the ratio of the mean depth of the blue sample to the red
sample is 〈D blue 〉/〈D red 〉 = 1.20, accounting well for the observed offset seen
in Figure 2.3.
We also make use of the Capak “green” sample to investigate the level of
“cosmic variance” in D ds /D s , and although there is variation in the redshift
distribution the variance of the mean redshift is remarkably tight, and as
quoted above we find a very small variance associated with the mean lensing
depth, σ(〈D ds /D s 〉) = 0.015. This stability is also a feature noticed in pencil
beam redshift surveys in general, that the mean depth is stable to spikes in
the redshift distribution, e.g., Broadhurst et al. (1988).
The form of the distance ratio D can be expressed in terms of the redshifts
of the source and lens for a given set of cosmological parameters. In the main
case of interest, that of a flat model with a nonzero cosmological constant,
47

Weak Lensing Dilution in A1689
the relation is given by
D ds
= 1 − ζ(z l)
(2.16)
D s ζ(z s )
∫ x
dz
ζ(x) =
. (2.17)
[Ω Λ + Ω M (1 + z) 3 ]
1/2
0
General expressions for the dependence of this distance ratio on arbitrary
combinations of Ω M and Ω Λ are lengthy and can be found in Fukugita et al.
(1990). For a low redshift cluster like A1689 (z=0.183), the form of this
function is rather flat for sources at z > 1, see Broadhurst et al. (2005a).
Therefore the main uncertainty in determining the cosmological parameters
from a comparison of g T between samples of different redshifts, is small compared
with clustering noise along a given line of sight behind the cluster,
as examined in detail by (Broadhurst et al. 2005a).
Thus, we do not seriously
examine this effect here but rather simply adopt the recent (three
year) WMAP cosmological parameters (Spergel et al. 2007) when making
the above depth correction. With sufficient number of clusters and similar or
better photometric redshift information (from multiple filter observations)
one can hope to examine the trend of redshift vs. lensing distance in the
future.
Note that lensing magnification, µ, will modify g T slightly by increasing
the depth to a fixed magnitude. But the magnification is small, µ < 0. m 2
over most of the cluster, r > 3 ′ . In any case, the dependence of the mean
redshift on depth is a slow function of redshift, so that it is safe for our main
purposes to ignore the effect of magnification on the depth of our samples.
Furthermore, since we are only interested in the proportion of the cluster
relative to the background for our purposes, we are not affected by the modification
of the background number counts caused by lensing, which has been
shown to significantly deplete the surface density of background red galaxies
in A1689, and found to be consistent with the predicted level of magnification
based on the distortion measurements (Broadhurst et al. 2005b).
48

2.6 Weak lensing dilution
0.06
0.04
0.02
Distortion g +
0
−0.02
−0.04
−0.06
2 5 10 20
θ [arcmin]
Figure 2.10 – Tangential distortion of the bright cluster sequence (i ′ < 21.5) galaxies
plotted against radius from the cluster center. By choosing the bright part of the sequence
we minimize the background contamination, and can therefore check that the tangential
distortion of the cluster members is negligible, which indeed is very clear from this figure.
2.6 Weak lensing dilution
We can now estimate the number density of cluster galaxies by taking the
ratio of the weak lensing signal between the green sample and the background
sample, with the background including both red and blue galaxies selected
from the Subaru catalog as explained in § 2.2, and accounting for differences
in the relative depths of these samples, as explained in § 2.5.
Cluster members are unlensed and hence assumed to have random orientations,
so they are expected to contribute no net tangential lensing signal.
This assumption can be examined for the brighter (i ′ < 21.5) cluster sequence
galaxies whose tight sequence protrudes beyond the faint field background
(Fig. 2.1) with negligible background contamination, so that we are secure
in selecting this subset to test the assumption that the cluster galaxies are
randomly oriented. Indeed, the net tangential signal of this population is
consistent with zero (Fig. 2.10), g (G)
T
= 0.0043 ± 0.0091. For a given radial
bin (r n ) containing objects in the green sample, whose width in color has
been chosen to encompass the full range of cluster galaxies (§ 2.2), the mean
value of g (G)
T
(eq. 2.6) is an average over background and cluster members.
Thus, its mean value 〈g (G) 〉 will be lower than the true background level de-
T
49

Weak Lensing Dilution in A1689
1.4
1.2
1
Fraction 1−g +
(G)/g +
(B)
0.8
0.6
0.4
0.2
0
−0.2
0.9 2 5 10 20
θ [arcmin]
Figure 2.11 – Fraction of cluster membership vs. radius. Cluster membership is proportional
to the dilution of the distortion signal of the green sample, relative to the expected
distortion of the background galaxies set by the red and blue samples. As expected, close
to the cluster center the fraction of cluster members in the green sample becomes maximal,
whereas at larger radius r > 3 ′ , the fraction of cluster members is small, indicating that
most of the green sample comprises background galaxies.
noted by 〈g (B) 〉 (Fig. 2.4) in proportion to the fraction of unlensed galaxies
T
in the bin that lie in the cluster (rather than in the background), since the
cluster members on average will add no net tangential signal. Therefore,
f cl (r n ) ≡
N cl
= 1 − 〈g T (r n ) (G) 〉
N Green 〈g T (r n ) (B) 〉
〈D (B) 〉
〈D (G) 〉
(2.18)
is the cluster membership fraction of the green sample (see full derivation in
the appendix).
Thus, we can use this effect to quantify statistically the number of cluster
galaxies by comparing g (G)
T
with the true background level derived from the
pure background red and blue samples g (B) , at a fixed radius. This is shown
T
in Figure 2.11, where we have also taken into account the effect of the relative
depths of the differing samples. We find that the fraction of cluster members
drops smoothly from ∼ 100% within r < 2 ′ to only ∼ 20% at the limit of the
data, r ∼ 15 ′ .
50

2.7 Cluster light and color profiles
2.7 Cluster light and color profiles
To determine the luminosity profile of the cluster galaxies, we need to go
further, because in general the brightness distribution of the cluster members
is different than that of the background galaxies; specifically, it is skewed
to brighter magnitudes, certainly for the bulk of the cluster sequence. To
account generally for any difference in the brightness distributions we can
subtract a “g T -weighted” luminosity contribution of each galaxy, which when
averaged over the distribution will have zero contribution from the unlensed
cluster members. We first calculate the “g-weighted” correction in arbitrary
flux units. We estimate the total flux of the cluster in the n th radial bin,
F cl (r n ) = ∑ i
F (G)
i − 〈D(B) 〉/〈D (G) 〉
〈g T (r n ) (B) 〉
∑
i
F (G)
i
· g (G)
T,i
(2.19)
where the sum is over all galaxies in the radial bin.
The flux is then translated back to apparent magnitude, and from that
the luminosity is derived. First we calculate the absolute magnitude,
M i ′ = i ′ − 5 log d L − K(z) + 5, (2.20)
where the K-correction is evaluated for each radial bin according to its V −i ′
color, which - after the correction is made for each of the bands - is now the
cluster color. The luminosity is then
L i ′ = 10 0.4(M i ′ ⊙ −M i ′) L i ′ ⊙, (2.21)
where M i ′ ⊙ = 4.54 is the absolute i ′ magnitude of the Sun (AB system).
The result yields the luminosity profile of the cluster as shown in Figure
2.12 (red squares). Here we can see that the cluster luminosity profile
is well approximated by a simple power-law with a projected slope of
d log(L i ′)/d log(r) ∼ −1.12 ± 0.06, to the limit of the data. We also show in
Figure 2.13 the unweighted luminosity profile with no correction for the field,
demonstrating that the g-weighted correction is negligible at small radius as
51

Weak Lensing Dilution in A1689
expected, since the cluster dominates numerically over the background, but
becomes increasingly more important at larger radius where the background
dominates. Note that we derive a more accurate inner luminosity profile using
the ACS photometry for the central region (Fig. 2.12, red circles), and
here there is only a negligible correction for the background due to the high
central density of galaxies in this cluster.
In a careful study of clusters and groups identified in the SDSS survey,
Hansen et al. (2005) find a similar slope for the most massive clusters, in
terms of the composite surface density profile of d log(n)/d log(r) ≃ −1.05 ±
0.04, over the radius range r < 2 Mpc, with slightly shallower slopes occurring
in the less overdense clusters and groups. This may be compared directly
with our slope derived above, assuming a constant M ∗ /L, for the ratio of
galaxy mass to galaxy luminosity. More directly we derive a density profile
using the f cl n (G) which also gives a slope of ≃ −0.9 ± 0.09.
In the same manner, we construct a “g-weighted” color profile of the
cluster, V − i ′ , which we have corrected as described above. We obtain a
color profile that shows a weak tendency towards bluer colors with increasing
radius, as expected, indicating a tendency towards later-type galaxies at
large radius (Fig. 2.14). Also shown is the unweighted color profile (green
points), which again is steeper due to the uncorrected field component which
dominates numerically over the cluster at larger radius and is generally bluer
in color than the cluster. This change in color with radius corresponds to
a significant radial gradient in spectral type, from predominantly early-type
with (V −i ′ ) AB ≃ 0.84, to mid sequence type, Sb, with (V −i ′ ) AB ≃ 0.73, and
indicates that for this cluster very blue starburst and Scd galaxies are not the
dominant population at the limiting radius of our sample (r ∼ 2 h −1 Mpc),
where otherwise the color would tend to (V − i ′ ) AB ≃ 0.5, using standard
template sets (Benítez et al. 2004). We go on to make use of this color-radius
relation in § 2.9, when examining the radial profile of the ratio of total cluster
mass to the stellar mass in galaxies. We do this by correcting the luminosity
profile for the tendency towards more luminous early-type stars that are
responsible for the bluer galaxy colors at large radius and which otherwise
bias the interpretation of the M/L ratio, as described below.
52

2.8 Cluster luminosity functions
10 15 θ [arcmin]
Luminosity density [L iο
h 2 Mpc −2 ]
10 14
10 13
10 12
Luminosity density [L i’ο
h 2 Mpc −2 ]
10 11
0.1 0.4 0.9 2 5 10 20
10 12 1 2 5 10 20
θ [arcmin]
Figure 2.12 – The “g-Weighted” luminosity
density vs. cluster radius. Each
galaxy’s flux F i (green sample) is weighted
by its tangential distortion g T,i with respect
to the background distortion signal
(red points). The filled circles represent the
ACS data, and the empty squares are for
the Subaru data. The blue points are derived
from integrating over the luminosity
functions of the same radial bins (see § 2.8),
and serve as a consistency check, showing
good agreement between these differing calculations.
The dashed line is the best fitting
linear relation for the blue points.
Figure 2.13 – The “g-Weighted” luminosity
density vs. cluster radius (red
squares), compared to the unweighted luminosity
density (green circles), showing
as expected the increasing size of the correction
with increasing radius, where the
sample becomes increasingly dominated by
background galaxies.
2.8 Cluster luminosity functions
The data allow the luminosity function to be usefully constructed in several
independent radial and magnitude bins, and hence we can examine the
form of the luminosity function of cluster members as a function of projected
distance from the cluster center. For this we combine the ACS and the Subaru
photometry. The ACS has the advantage of extending two magnitudes
fainter in the i ′ -band than the Subaru photometry for r < 2 ′ . As can be
seen in Figure 2.15, the ACS i ′ magnitudes agree with Subaru i ′ magnitudes
for galaxies found and matched in both catalogs. The background correction
to evaluate f cl , must be made in each magnitude bin independently, since
the relative proportion of background galaxies increases with apparent magnitude,
so that the lower luminosity bins of the green luminosity function
53

Weak Lensing Dilution in A1689
0.86
0.84
0.82
Cluster V−i [mag]
0.8
0.78
0.76
0.74
0.72
0.7
0.68
0.66
0.9 2 5 10 20
θ [arcmin]
Figure 2.14 – Galaxy color profile after weighting the color of each object by its individual
distortion, g i , accounting for any difference between the color distribution of the cluster
and background populations comprising the green galaxy population. The color of the
cluster members becomes slowly bluer with increasing radius moving from E/S0 colors in
the center to mid-type galaxy colors at the limit of the data, r ∼ 2 h −1 Mpc. The green
points represent the uncorrected V − i ′ profile of the green sample.
are expected to contain a greater fraction of background galaxies and hence
should have a relatively higher value of g (G)
T
. This trend is apparent in Figure
2.16 (left panels), where we plot the recovered mean tangential distortion
(here the average is over a magnitude bin) for each of the four radial bins,
as a function of absolute magnitude. A clear trend is found at all radii towards
higher levels of g T at fainter luminosities. Note that the mean level
of the background distortion (black solid line) drops with increasing radius
so that the proportion g (G)
T
(M)/g(B) T
is generally an increasing function of
radius and a decreasing function of luminosity. To correct for this we simply
apply equation (2.18) to each magnitude bin:
Φ cl (M k ) = Φ(M k ) · [1 − 〈g (G)
T
(M k)〉/〈g (B)
T
(r)〉] (2.22)
(Note that the background signal is averaged over the whole range of magnitudes
at that radius.)
We then construct the luminosity function for four independent radial
bins, as shown in Figure 2.16 (middle panel) and fit a Schechter (1976) function
to each (dashed lines). It can be seen that there is no obvious tendency
54

2.8 Cluster luminosity functions
28
d(i’ ACS
)/d(i’ Subaru
)=1.0039±0.0002
26
24
i’ acs
22
20
18
16
16 18 20 22 24 26 28
i’ subaru
Figure 2.15 – Comparison between ACS and Subaru photometry for objects in common
in the central region covered by both datasets, r < 2 ′ . There is very good agreement
between magnitudes of the two datasets which have independent zero-points and of course
independent photometry.
for the shape of the luminosity function to change with radius. The faint-end
slope of a Schechter function fit is α = −1.05 ± 0.07 in the i ′ -band. This
constancy with radius has been argued with somewhat less significance in
other well studied massive clusters (e.g., Pracy et al. 2005), based on similar
deep 2-color imaging, where the limiting radius is more restricted. We also
construct a composite luminosity function for the whole cluster (Fig. 2.17),
for r < 10 ′ , which shows clearly the effect of our “g-weighted” background
correction, without which the faint-end slope would be considerably steeper,
α ∼ 1.4.
Our approach is of course essentially free of uncertainties in the subtraction
of background galaxies by its nature. While qualitative similarity
between the results of the various studies is clear, agreement in detail is not
necessarily expected, given the likely dispersion in the strength of this effect
between clusters. Also, the question of background contamination is always
an issue in the standard approach due to the inherent fluctuations in the surface
density of background galaxies, and the need to establish the background
counts at a sufficiently large radius from the cluster to avoid self-subtraction
of the cluster at the boundaries of the data, a subject explored in depth, e.g.
Adami et al. (2000); Paolillo et al. (2001); Andreon et al. (2005); Hansen
55

Weak Lensing Dilution in A1689
et al. (2005); Popesso et al. (2005).
We also integrate our luminosity functions as a consistency check of the
luminosity density profiles derived earlier. This is done by calculating Φ cl (M)
in the same radial bins as our luminosity profile above, and summing over
the magnitude bins:
L cl (r n ) = ∑ k
Φ cl (M k ) · ∆M k · 10 0.4(M i ′ ⊙ −M k)
(2.23)
The results shown above in Figure 2.12 (blue points) agree very well with
those of L cl described in the previous section.
Note that in constructing these luminosity density profiles we have implicitly
assumed that the luminosity function is integrated over fully. Fortunately
our data is complete to a sufficient depth (i ′ < 26.5) so that the contribution
of the integrated luminosity density from undetected objects is very small, as
evaluated when we examine the luminosity functions. The difference between
integrating up to a limiting magnitude of M i ′ < −14 and extrapolating up
to M i ′ < −10 is only about 0.1%.
The lack of any obvious upturn in the cluster luminosity function to very
faint luminosities, M i ′ < −12, in the cluster core, is in agreement with several
other studies based on deep photometry of large cluster samples, e.g., the
composite cluster luminosity function derived by Gaidos (1997); Garilli et al.
(1999); Paolillo et al. (2001); Hansen et al. (2005); Pracy et al. (2005), where
wide field imaging is employed for several Abell clusters and overdensities
identified in the SDSS survey, and where careful attention can be paid to the
background counts and their uncertainty. The study of Pracy et al. (2005)
is the most similar to our own, containing three rich and fairly distant Abell
clusters, and here the LF’s show no obvious upturn to M < −13 with a
generally flat Schechter function slope in the range -1.1 to -1.25.
For the well studied Coma cluster, a steeper slope has been claimed,
α ≃ −1.4, by Bernstein et al. (1995), though subsequent faint spectroscopy
by Adami et al. (2000), has revealed the presence of a background cluster at
z ≃ 0.5, which when corrected for leads to a flat faint-end slope. In contrast
56

2.8 Cluster luminosity functions
an upturn is claimed for a composite sample of 25 SDSS selected clusters by
Popesso et al. (2005), though an individual examination shows considerable
variation, with only a minority of ∼ 6 displaying a distinct upturn which
varies in amplitude, so that one may wonder about the role of anomalous
background count fluctuations in these cases.
57

Weak Lensing Dilution in A1689
g +
−1
0’−1’, ACS
0.2
−1.05
1’−2’, ACS
0.1
2’−4’, Subaru
−1.1 α
0
10 3 4’−8’, Subaru −1.15
α=−1.046 M *
=−21.46
−1
0.3
−1.05
g +
0.2
0.1
−1.1 α
0
10 2
−1.15
α=−1.059 M *
=−20.77
−1
0.08
g + 0.04
−1.05
−1.1 α
−1.15
0
10 1
α=−1.094 M *
=−21.21
0.06
−1
0.04
−1.05
g +
−1.1
0.02
α
−1.15
0
10 0
α=−1.108 M *
=−21.41
−24 −22 −20 −18 −16 −14 −12 −24 −22 −20 −18 −16 −14 −12 −23 −22 −21 −20
M i’
M i’
M *i’
Φ(M) [h 2 Mpc −2 mag −1 ]
Figure 2.16 – Center: Luminosity functions are shown for four independent radial bins,
indicating little trend with radius. The left-hand panel is the degree of tangential distortion
g T as a function of magnitude used in the derivation of the corresponding luminosity
function. Notice that in general g T increases with decreasing luminosity because the level
of background increases at fainter magnitudes. The right-hand panel shows the 1σ, 2σ, and
3σ contours for the Schechter function parameters M ∗ and α, for each of the corresponding
luminosity functions.
10 3 uncorrected
corrected
−10 0.151
10 2
−10 0.152
Φ(M) [h 2 Mpc −2 mag −1 ]
10 1
10 0
10 −1
uncorrected, α=−1.421
M i’
−10 0.153
−1
−1.02
−1.04
α
−1.06
−1.08
10 −2
−24 −22 −20 −18 −16 −14
M i’
corrected, α=−1.048
−24 −22
M i’*
−20 −1.1
Figure 2.17 – Composite luminosity function of the cluster (black squares), for r < 10 ′ ,
with the fit to a Schechter function displayed in the lower right panel. The green circles
show the luminosity distribution without the correction made for the background dilution.
At bright magnitudes, there is little difference between the uncorrected and the corrected
points, however, at the faint end the uncorrected distribution rises, which if uncorrected
would overestimate the faint end slope α ∼ 1.4 (top right panel), showing clearly the
magnitude of the background correction, which when accounted for by our method results
a flat faint end slope, α ∼ 1.

2.9 M/L profiles
1400
180
180
1200
160
160
M/L i
[ hM o
/L iο
]
1000
800
600
400
M(

Weak Lensing Dilution in A1689
Note that the peak value is rather large, equivalent to M/L B ∼ 400h(M/L) ⊙
in the restframe B-band often used as a reference, but the mass of A1689
is at the extreme end of the cluster population, M ∼ 2 × 10 15 hM ⊙ (Broadhurst
et al. 2005b), and given the general tendency of M/L to increase with
increasing mass, from galaxies through groups to clusters, we may not be
surprised to find the peak to somewhat exceed the typical range for clusters,
150 − 300hM/L B ⊙
(Carlberg et al. 2001). The general profile of M/L is
similar in form to that derived for CL 0152-1357 by Jee et al. (2005) based
on a careful weak lensing analysis of recent deep ACS images.
We have also constructed a profile of the total mass to stellar mass M/M ∗
ratio (red curves in Fig. 2.18 and Fig.2.19). This is arguably a more physically
useful indicator of the relationship between dark and luminous matter
compared to the ratio of M/L, because the starlight can be strongly influenced
by the presence of relatively small numbers of luminous hot stars. To
calculate M/M ∗ we make use of the color profile derived in § 2.7, and an empirical
relationship between color and the ratio M ∗ /L for stellar populations
established for local galaxies in the the SDSS survey by Bell et al. (2003). The
slope of the projected stellar mass profile, d log(M ∗ )/d log(r) = −1.15 ± 0.13,
derived this way is slightly steeper than the luminosity profile, as expected.
The observed relation we derive this way is somewhat flatter than for M/L
and the mean contribution by mass for stars is about 1.25% for this cluster
and similar to a mean value of ∼ 2% derived from a carefully selected sample
of local clusters by Biviano & Salucci (2006).
2.10 Cluster mass profile
We use the combined distortion information obtained from the ACS and Subaru
imaging, as described above (Fig. 2.6) and compare with models for the
mass distribution. We have improved on our earlier distortion measurements
made with Subaru, with the addition of the background blue galaxy population
defined here, so that the significance of the distortion measurements
is somewhat greater than our earlier work which was based only on the red
60

2.10 Cluster mass profile
sample (Broadhurst et al. 2005b). In addition, we have extended the distortion
measurements to the central region using the HST /ACS information, as
described in § 2.4, where we have clearly identified a maximum and a minimum
value of g T , which accurately correspond to the tangential and radial
critical curves (Fig. 2.20), independently derived from the the many giant
tangential and radial arcs observed for this cluster (see Broadhurst et al.
2005a).
Here we test the universal parameterization of CDM-based mass profiles
advocated by Navarro et al. (1997). This model profile is weighted over the
differing results from sets of halos identified in N-body simulations. A cluster
profile is summed over all the mass contained within the main halo, including
the galactic halos. Hence, we compare the integrated mass profile we deduced
directly with the NFW predictions without having to invent a prescription
to remove the cluster galaxies.
NFW have shown that massive CDM hales are predicted to be less concentrated
with increasing halo mass, a trend identified with collapse redshift,
which is generally higher for smaller halos following from the steep evolution
of the cosmological density of matter. The most massive bound structures
form later in hierarchical models and therefore clusters are anticipated to
have a relatively low concentration, quantified by the ratio C vir = r virial /r s .
In the context of this model, the predicted form of CDM dominated are predicted
to follow a density profile lacking a core, but with a much shallower
central profile (r ≤ 100 h −1 kpc) than a purely isothermal body.
The fit to an NFW profile is made keeping r s , and ρ s , the characteristic
radius and the corresponding density, as free parameters.
These can be
adjusted to normalize the model to the observed maximum in the distortion
profile at the tangential critical radius of ≃ 45 ′′ . The combination of these
parameters then fixes the degree of concentration, and the corresponding
lensing distortion profile can then be calculated.
Integrating the mass along a column, z, where r 2 = (ξ r r s ) 2 +z 2 gives:
∫ ξ ∫ ∞
M(ξ) = ρ s rs(ξ)
3 d 2 1 dz
ξ
. (2.24)
o −∞ (r/r s )(1 + r/r s ) 2 r s
61

Weak Lensing Dilution in A1689
Using this mass, a bend-angle of α = 4GM(

2.10 Cluster mass profile
tangential critical radius of 47 ′′ to match the mean critical radius derived
from the data (Broadhurst et al. 2005a).
To normalize the models we choose to reproduce the observed Einstein
radius of 47 ′′ and compare the predicted location of the radial critical curve
and the κ = 1 curve. Figure 2.20 shows that both these radii decrease slowly
as the concentration parameter is increased. We have marked the observed
values of these radii as determined by two independent observational means.
We can use the statistical distortion measurements as described in § 2.4,
and the same values derived from the multiple image analysis presented in
Broadhurst et al. (2005a). These differing estimates are closely consistent
with each other, and by comparison with the model curves bracket intermediate
values of concentration in the range 5 < C vir < 15 (Fig. 2.20), a range
consistent with the results from a detailed fit to the inner profile measured
in Broadhurst et al. (2005a). This is found to be very similar to the independently
derived central profile of A1689 from Diego et al. (2005); Zekser
et al. (2006); Halkola et al. (2006)
Outside the tangential critical radius, for r > 1.5, the distortion measurements
are small enough not to suffer from any significant underestimation
due to the finite source sizes and we may compare the observed distortion
profile, g T (r), out to the limit of our data, r < 15 ′′ . We find this distortion
profile is reasonably well fitted by an NFW profile particularly at large radius,
4.0 ′′ < r < 15 ′′ , but with a relatively large concentration, C vir = 27.2 +3.5
−5.7, as
shown in Figure 2.21. Note that we have used here a linear radial binning
when measuring the concentration parameter and therefore the result here
is more weighted by large radius signal than for the analysis of Broadhurst
et al. (2005b) where we used logarithmic binning, yielding a smaller value
of C vir . This difference in the derived value of C vir is not because of any
revision in our estimates of the distortion, in fact both analyses yield very
consistent distortion profiles at large radius, but rather that the form of the
NFW profile is not consistent with our data over the full radial range - the
best fitting NFW model is either too shallow at large radius or too steep at
small radius depending where one prefers to fit the data.
63

Weak Lensing Dilution in A1689
We also plot lower concentration profiles, including C vir = 14, which was
found previously by Broadhurst et al. (2005a), to fit best the overall lensing
derived mass profile from combining the mass profile derived from the
multiply lensed images in the central region, r < 2 ′ , with the mass distribution
derived from weak lensing distortion and magnification measurements
from the red background galaxy sample. This model fit, as pointed out by
Broadhurst et al. (2005b), is not as pronounced as the observed surface mass
profile, being too shallow at larger radius and too steep at small radius (see
figures 1&3 of Broadhurst et al. 2005b). Here we see more clearly that this
fit with C vir = 14 increasingly overpredicts the observed distortion profile
with radius. We also plot C vir = 8.2 which best fits the central strongly
lensed region (Broadhurst et al. 2005a) r < 2 ′ , derived from 106 multiply
lensed images. Again this fit overpredicts the g T profile in the weak lensing
regime, as pointed out in Broadhurst et al. (2005b). We clearly exclude
the low concentration profile generally predicted by CDM based models of
structure formation. A value of C vir ∼ 5 is generally anticipated for massive
clusters, although the scatter in concentration at a given mass is considerable
(e.g., Bullock et al. 2001). Figure 2.21 shows clearly how this profile is much
too shallow to generate the relatively steeply declining observed distortion
profile. The triaxiality of realistic halos means that projection effects will
bias somewhat the derived distortion profile, as examined carefully by Oguri
et al. (2005) and Hennawi et al. (2005), showing that the level of such bias
effect is expected to enhance the derived concentration by approximately
∼ 20% on average. Whilst A1689 is clearly an anomalous cluster in terms
of the size of the Einstein radius, the cluster is very round in terms of the
projected X-ray emission, with only minimal substructure observed in the
optical near the center. Hence we are left with a clearly unresolved problem,
that the observed concentration would seem to far exceed any reasonable
estimate. Other independent work on the combined profile from strong and
weak lensing measurements for the clusters Cl0024+17 (Kneib et al. 2003)
and MS2137-23 (Gavazzi et al. 2003) also point to surprisingly high concentrations,
and it is therefore important to extend this type of detailed work
to other clusters to test the generality of the profile derived here.
64

2.11 Discussion and conclusions
For reference we also plot the distortion profile for a singular isothermal
body in Figure 2.21, which is simply expressed as
g T =
1
2 θ
θ E
− 1 , (2.26)
and normalized to the observed Einstein radius, θ E = 47 ′′ . This model also
overpredicts the data at large radius, indicating the outer mass profile is
steeper than 1/θ in projection.
2.11 Discussion and conclusions
We have explored a new approach to deriving the luminous properties of
cluster galaxies by utilizing lensing distortion measurements, based on the
dilution of the lensing distortion signal by unlensed cluster members which
we assume are randomly oriented. We have tested this assumption for a
restricted sample of bright cluster galaxies which project beyond the faint
galaxy background population so the level of background contamination is
negligible, confirming that the cluster galaxies are randomly oriented with a
negligible net tangential distortion for the purposes of our work.
This dilution approach is applied to A1689 to derive the radial light profile
of the cluster, a color profile and radial luminosity functions. The light
profile is found to be smoothly declining and fitted with a power-law slope
d log(L)/d log(r) = −1.12 ± 0.06. We also see a mild color gradient corresponding
to a change in the cluster population from early- to mid-type
galaxies in moving from the center out to the limit of our data at 2 h −1 Mpc.
Unlike the light profile the gradient of mass profile is continuously steepening,
such that the ratio of M/L peaks at intermediate radius. We find that
the cluster luminosity function has a flat faint-end slope of α = −1.05±0.07,
nearly independent of radius and with no faint upturn to M i ′ < −12.
A major advantage of our approach is that we do not need to define farfield
counts for subtracting a background, as in the usual method where there
is a limitation imposed by the clustering of the background population that
65

Weak Lensing Dilution in A1689
limits the radius to which a reliable subtraction of the background can be
made.
We have also established that the bluest galaxies in the field of A1689
lie predominantly in the background, as their radial distortion profile follows
closely the red galaxies, but with an offset indicating the blue population lies
at a greater mean distance than the red background galaxies, and consistent
with the estimated mean redshifts of these two populations. With a larger
sample of clusters, this purely geometric effect can potentially be put to use to
provide a simple model-independent measure of the cosmological curvature.
The mass profile of A1689 was reexamined using our combined background
sample of red and blue galaxies. The distortion profile derived from
this sample is consistent with our earlier work, but somewhat more statistically
significant, so we have examined the mass profile more carefully out
to a larger radius. We have found that the distortion profile is steeper than
predicted for CDM halos appropriate for cluster sized masses C vir ∼ 5. This
discrepancy is particularly clear at large radius r > 2 ′ , where an acceptable
fit is found to an NFW profile but with a concentration C vir = 27 +3.5
−5.7. This
finding is consistent with our earlier work which showed that although an
overall best fit profile of C vir ≃ 14 to the joint strong and weak lensing based
data presented in Broadhurst et al. (2005b), the curvature of the data is
more pronounced than an NFW profile being shallower in the inner region
and steeper at larger radius, so that the derived value of the concentration
increases with radius depending on the radial limits being examined.
This result is surprising and may require a significant departure from the
standard CDM model, either in terms of the mass content, or the epoch at
which the bulk of the cluster was assembled. For example, one possibility to
achieve earlier formation of clusters is to allow deviation from Gaussianity
of the primordial density fluctuation field, as has been considered recently
by, e.g., Sadeh et al. (2006).
A1689 is amongst the most massive known
clusters, and projection effects may play a role in boosting somewhat the
lensing signal along the line of sight. We therefore aim to test the generality
of this result with a careful study of a statistical sample of clusters.
66

2.A Cluster galaxy fraction from the lensing dilution effect
Upcoming spatially resolved SZ measurements will add a significant new
ability to determine cluster mass profiles over a large range of radius, and allow
for improved consistency checks between the various independent means
of estimating masses. The combination of X-ray, lensing and SZ measurements
will soon lead to far greater accuracy in understanding the nature of
cluster mass profiles.
We plan an improvement to the weak lensing work with deeper multi-color
imaging from Subaru for measuring reliable photometric redshifts for a sizable
fraction of the background population. This added dimension of depth will
enhance the weak lensing signal and reduce the systematic problems of cluster
and foreground contamination of the lensing signal. We also aim to extend
this work to well studied clusters at lower redshift with archived Subaru
imaging and detailed X-ray and upcoming SZ observations as in principle, the
lensing signal should be equally strong for lower redshift clusters, given the
maximal ratio of lens to source distances, D ds /D s ≃ 1, for faint background
sources.
Appendix
2.A Cluster galaxy fraction from the lensing
dilution effect
Let us derive eq. (2.18). For simplicity, here we assume the weak lensing limit
so that the reduced shear is approximated by the gravitational shear, g ≈ γ.
It is useful to factorize the lensing signal with the geometry-dependent factor
such that (Seitz & Schneider 1997):
g T (r) = w(z)g T,∞ (r) (2.27)
where g T,∞ (r) denotes the tangential shear calculated for hypothetical sources
at an infinite redshift, and w(z) is the lensing strength of a source at z
67

Weak Lensing Dilution in A1689
relative to a source at z → ∞, w(z) = D(z)/D(z → ∞); D(z) ≡ D ds /D s
as introduced in §5. The relative lensing strength vanishes for cluster and
foreground galaxies, that is, w(z) = 0 for z ≤ z l .
As the tangential shear is obtained by averaging over an annular region,
it can be formally written in the following form:
〈g T (r)〉 = g T,∞ (r)
∫
d 2 x dz dn/dz w(z)
∫
d2 x dz dn/dz
(2.28)
= g T,∞ (r)
∫ ∞
z l
dz dN/dz w(z)
N tot
= g T,∞ (r) N bg
N tot
〈w〉 z>zl
where dn/dz is the surface number density distribution of galaxies per
unit redshift interval per steradian, dN/dz = ∫ d 2 x dn/dz is the mean redshift
distribution of galaxies in the annulus, N tot = ∫ ∞
dz dN/dz is the total
0
number of galaxies in the annulus, N bg = ∫ ∞
z l
dz dN/dz is the number of background
galaxies in the annulus, 〈w〉 z>zl = ∫ ∞
z l
dz dN/dz w(z)/ ∫ ∞
z l
dz dN/dz is
the mean lensing strength without including the dilution effect; here we have
assumed that the lensing properties are constant over the annulus where we
take the ensemble averaging. Note that the factor N bg /N tot accounts for the
dilution effect on the lensing signal strength due to contamination by foreground
and cluster-member galaxies. In general, there is a contribution from
foreground galaxies to the total number of galaxies N tot . However, for the
case of A1689 at a low redshift of z l = 0.183, this contribution is negligible.
That is, N tot ≈ N cl + N bg with N cl being the number of cluster galaxies in
the annulus. For a background galaxy sample, N tot = N bg .
Since we are to compare galaxy samples with different redshift distributions,
we need to account for different values of the mean lensing strength,
〈w〉 z>zl . As explained above, our green sample (denoted with G) comprises
both cluster and background galaxies. Hence, according to eq. (2.28), the
68

2.B Non-linear effect in the reduced shear estimate
expectation value for the mean tangential shear estimate is
〈g (G)
T
(r)〉 = g T,∞(r)
N (G)
bg
N cl + N (G)
bg
〈w (G) 〉 z>zl . (2.29)
As for our background sample (denoted with B), including the red and blue
samples, this is
〈g (B)
T (r)〉 = g T,∞(r)〈w (B) 〉 z>zl . (2.30)
By taking the ratio of the two tangential shear estimates, we obtain the
following expression:
〈g (G)
T
(r)〉/〈g(B)
T (r)〉 = N (G)
bg
N cl + N (G)
bg
〈w (G) 〉 z>zl
〈w (B) 〉 z>zl
. (2.31)
Alternatively, we have the expression for the cluster galaxy fraction as
f cl (r) ≡
N cl
N cl + N (G)
bg
= 1 − 〈g(G) T
(r)〉
〈g (B)
T
〈w (B) 〉 z>zl
(r)〉 = 1 − 〈g(G) T
(r)〉 〈D (B) 〉 z>zl
〈w (G) 〉 z>zl 〈g (B) (r)〉 .
〈D (G) 〉 z>zl
T
(2.32)
This is the desired formula for the cluster galaxy fraction from the weak
lensing dilution effect. In order to take into account different populations of
background galaxies in the two samples, one needs to estimate the correction
factor, 〈D (B) 〉 z>zl /〈D (G) 〉 z>zl .
2.B Non-linear effect in the reduced shear estimate
In Appendix A, we assume that the observable reduced shear is linearly proportional
to the lensing strength factor, w(z). However, the reduced sear, defined
as g = γ/(1−κ), is non-linear in κ, so that the averaging operator with
respect to the redshift generally acts non-linearly on the redshift-dependent
components in g.
69

Weak Lensing Dilution in A1689
To see this effect, we expand the reduced shear with respect to the convergence
κ as
g = γ(1 − κ) −1 = wγ ∞ (1 − wκ ∞ ) −1 = wγ ∞
∞
∑
k=0
(wκ ∞ ) k (2.33)
where κ ∞ and γ ∞ are the lensing convergence and the gravitational shear, respectively,
calculated for a hypothetical source at an infinite redshift. Hence,
the reduced shear averaged over the source redshift distribution is expressed
as
〈g〉 = γ ∞
∞
∑
k=0
〈w k+1 〉κ k ∞. (2.34)
In the weak lensing limit where κ ∞ , |γ| ∞ ≪ 1, then 〈g〉 ≈ 〈w〉γ ∞ . Thus,
the mean reduced shear is simply proportional to the mean lensing strength,
〈w〉. The next higher-order approximation for eq. (2.34) is given by
〈g〉 ≈ γ ∞
(
〈w〉 + 〈w 2 〉κ ∞
)
≈
〈w〉γ ∞
1 − κ ∞ 〈w 2 〉/〈w〉 . (2.35)
Seitz & Schneider (1997) found that eq. (2.35) yields an excellent approximation
in the mildly non-linear regime of κ ∞ 0.6. Defining f w ≡ 〈w 2 〉/〈w〉 2 ,
we have the following expression for the mean reduced shear valid in the
mildly non-linear regime:
〈g〉 ≈
〈γ〉
1 − f w 〈κ〉
(2.36)
with 〈κ〉 = 〈w〉κ ∞ and 〈γ〉 = 〈w〉γ ∞ (Seitz & Schneider 1997). For lensing
clusters located at low redshifts of z l 0.2, 〈w 2 〉 ≃ 〈w〉 2 or f w ≈ 1, so that
〈g〉 ≈ 〈γ〉/(1 − 〈κ〉).
The ratio of tangential shear estimates using two different populations B
and G of background galaxies, in the mildly non-linear regime, is given as
〈g (G)
T 〉
〈g (B)
T 〉 ≈ 〈w(G) 〉
〈w (B) 〉
≈ 〈w(G) 〉
〈w (B) 〉
1 − f w
(B) 〈w (B) 〉κ ∞
1 − f (G)
(2.37)
w 〈w (G) 〉κ ∞
{ (
1 − f
(B)
w 〈w (B) 〉 − f w (G) 〈w (G) 〉 ) κ ∞ + O(〈κ〉 2 ) } .
70

2.B Non-linear effect in the reduced shear estimate
(
The lowest-order correction term is proportional to
f (B)
w
)
〈w (B) 〉 − f w (G) 〈w (G) 〉 κ ∞ ,
which is much smaller than unity for the galaxy samples of our concern in the
mildly non-linear regime. In conclusion, it is therefore a fair approximation
to use eq. (2.18) for measuring the cluster galaxy fraction via the dilution
effect.
71

Weak Lensing Dilution in A1689
30
28
Radial critical curve
κ=1 curve
26
24
θ [arcsec]
22
20
18
16
14
12
0 5 10 15 20 25 30
C Vir
Figure 2.20 – The curves shows how the radius of the radial critical curve, shown in
blue, varies with the concentration parameter of an NFW profile, where the model is normalized
to generate the observed Einstein radius of 47 ′′ for all values of the concentration
parameter, C vir . The radial critical curve shrinks as the mass profile becomes steeper.
This is also the case for the radius at which the distortion g T = 0, corresponding to the
radius where the surface density of matter is equal to the critical surface density (κ = 1),
here the degree of tangential and radial distortion is always equal, independent of the
form of the mass profile. We have marked the observed values of the radial critical curves
and the radius where the distortion is seen to be zero as measured independently in two
ways using the statistical distortion measurements (dotted lines) as described in § 2.4, and
the same values derived from the multiple image analysis presented in Broadhurst et al.
(2005a) (dashed lines). These differing measurements are consistent with each other and
by comparison with the model curves bracket intermediate values of concentration in the
range 5 < C vir < 15 for the inner strongly lensed region.

2.B Non-linear effect in the reduced shear estimate
10 0 θ [arcmin]
c vir
=27
c vir
=14
c vir
=8
c vir
=5
Distortion g +
10 −1
10 −2
0 2 4 6 8 10 12 14 16
Figure 2.21 – NFW models are compared with the measured values of g T for the red+blue
background sample described in § 2.2. The models are normalized to match the observed
Einstein radius of 47 ′′ . A relatively high concentration is preferred with the best fit
corresponding to C vir = 27 −5.7 +3.5 (black curve), for the bulk of data, r > 1.5′ , excluding the
strong region where the measured distortions required a significant correction for the finite
source sizes. The anticipated low concentration of C ∼ 5 (pink curve) is obviously excluded
by the data. For reference we also overplot the purely isothermal profile normalized to the
observed Einstein radius (red dotted curve). It also overpredicts the data.

Chapter 3
Detailed Cluster Mass and
Light Profiles of A1703, A370
& RXJ1347-11 from Deep
Subaru Imaging
A version of this chapter has been published as Medezinski, E., Broadhurst,
T., Umetsu, K., Oguri, M., Rephaeli, Y., & Benítez, N. 2010, MNRAS, 405,
257.
3.1 Introduction
Galaxy Clusters are the largest virialised objects whose properties reflect the
initial spectrum of density perturbations and the evolutionary history of dark
matter (DM) and baryons. Wide ranging observations provide a wealth of
information for examining the relation between dark and visible matter and
their radial profiles. Unlike galaxies, where substantial cooling is predicted to
have concentrated baryons in the centers of DM halos, clusters are observed
to cool relatively inefficiently via thermal emission so that the gas remains
close to the virial temperature, making cluster mass profiles easy to interpret,
75

Mass and Light of A1703, A370 & RXJ1347-11
as corrections for gas cooling are not significant (Blumenthal et al. 1986;
Broadhurst & Barkana 2008).
Lensing work is now able to achieve measurements of projected cluster
mass profiles with unprecedented detail, sufficient to usefully address the distinctive
prediction of a relatively shallow mass profiles for CDM dominated
halos (Navarro, Frenk, & White 1997, hereafter NFW; Duffy et al. 2008).
Combined weak and strong lensing measurements have shown that the continuously
steepening form of the NFW profile is a reasonable description for
the mass profiles of three careful, independent studies (Kneib et al. 2003;
Gavazzi 2005; Broadhurst et al. 2005b), although with surprisingly high values
derived for the profile concentration parameter in each case (Broadhurst
et al. 2005b). More recently, the concordance ΛCDM cosmology has been
examined critically with a somewhat larger sample of clusters, indicating
that the concentrations derived are significantly higher than predicted over
a wide range of cluster mass, after accounting for statistical corrections for
lensing-induced biases. This is seen both in terms of the size of the Einstein
radius (Broadhurst & Barkana 2008; Oguri & Blandford 2009; Zitrin
et al. 2009b) and the weak lensing (WL) mass profiles of several well known
clusters (Broadhurst et al. 2008; Sadeh & Rephaeli 2008; Lapi & Cavaliere
2009).
Current detailed cluster mass and light profiles can now be used to examine
the ratio of M/L all the way out to the virial radius. Comparisons of
the general DM distribution with the galaxy distribution are potentially of
great interest as the distribution of galaxies is thought to be influenced by
significant tidal forces which may substantially modify massive galaxy halos
reducing their masses (Ghigna et al. 1998, 2000; Colín et al. 1999; Springel
et al. 2001; De Lucia et al. 2004; Gao et al. 2004; Nagai & Kravtsov 2005;
Limousin et al. 2009). Mass loss calculated over Hubble time in ΛCDM predict
losses of ∼ 30% for orbits extending to the virial radius, with typically
a loss of ∼ 70% near cluster centers, so that the ratio of integrated galaxy
mass to the general DM distribution falls steadily towards the cluster center
(Nagai & Kravtsov 2005).
76

3.1 Introduction
Stars are not expected to be stripped significantly in these simulations
because of their location deep in the central potential of a galaxy. Hence,
for clusters the galaxy light should simply trace mass, assuming both are
collisionless, so that a fairly constant M/L is expected (Gao et al. 2004;
Nagai & Kravtsov 2005). The presence of high-mass galaxies in the inner
core, resulting from both higher merger rates and the effect of dynamical
friction, explains the reduced value of the integrated M/L ratio within a
radius of ≤ 200 h −1 kpc for the highest mass clusters (Nagai & Kravtsov
2005; El-Zant et al. 2001).
Reliable M/L profiles of galaxy clusters have begun to emerge recently
based on dynamical (Rines et al. 2000, 2004) and lensing based methods
(Medezinski et al. 2007). For the local cluster A576, a careful dynamicallybased
M/L profile has been obtained with caustic work by (Rines et al.
2000), who note a puzzling declining behavior, with a peak around M/L ∼
600M/L R at r ∼ 100 h −1 kpc, declining to a limiting 100M/L B , at r ∼
1 h −1 Mpc and beyond. Other dynamical estimates from stacking together
samples of clusters also show this overall behavior, where the M/L ratio
may peak around 10% of the virial radius (Katgert et al. 2004) and declines
thereafter (Rines et al. 2000, 2004).
Our earlier lensing based determination of the M/L profile of A1689
(Medezinski et al. 2007) shows behavior similar to the above dynamical work,
with a peak of M/L B ∼ 400(M/L) ⊙ at r ∼ 100 h −1 kpc, and a steady decline
to ∼ 100M/L B at the virial radius, r ∼ 2 h −1 Mpc (Medezinski et al.
2007). In the case of A576 (Rines et al. 2000) and in our own work on A1689
(Medezinski et al. 2007), careful corrections are made for the trend of M/L of
the stellar population with cluster radius, using the measured colors of cluster
member galaxies which favor predominately redder, earlier-type galaxies in
the central region. In both these differing analyses the color trend was found
to be weak and not nearly sufficient to account for the measured variation
of the M/L profile, leaving a puzzling result which is generally at odds with
the known result that M/L is an increasing function of scale. Clearly, it is
important to extend the investigation of the M/L profile to more clusters
to examine the generality of our earlier work and to examine possible trends
77

Mass and Light of A1703, A370 & RXJ1347-11
with other observables in a more statistically robust way.
Our previous work demonstrated the importance of carefully selecting
a background population to avoid contamination by the lensing cluster. It
is not sufficient to simply exclude a narrow band containing the obvious
E/S0 sequence, following common practice, because the lensing signal of the
remainder bluer objects is found to fall rapidly towards the cluster center
relative to the background of red lensed galaxies indicating that unlensed
cluster members are present blueward of the sequence diluting the true lensing
signal (Broadhurst et al. 2005b). Due to this effect, many WL studies
have underestimated the masses and concentrations of cluster mass profiles
(Lu et al. 2009), a point now widely acknowledged (Limousin et al. 2007;
Oguri et al. 2009; Okabe et al. 2009). This effect resolved the long standing
discrepancy between the strong and WL effects, with the WL signal
underpredicting the observed Einstein radius by a factor of ∼ 2.5 (Clowe
& Schneider 2001; Bardeau et al. 2005), based only on a minimal rejection
of obvious cluster members using one or two-band photometry. Red background
galaxies behind clusters have since been explored carefully with deep
wide-field spectroscopy by Rines & Geller (2008), who emphasize that the red
population in their cluster data comprises exclusively background objects.
This lensing “dilution” effect may be turned to our advantage as a means
of correcting the cluster population for background contamination, extending
the work of Medezinski et al. (2007). Here we analyze very high quality Subaru
images of A1703, A370, & RXJ1347-11 for which we have accurate WL
measurements. For these clusters, a wider range of colors is available than
for A1689, which we make full use here to improve the separation of cluster
members from foreground and background populations using the color-color
(CC) diagram and application of our dilution method. Using the dilution
method has the advantage that it does not require the actual subtraction
of background galaxies which introduces the inherent uncertainty caused by
the clustering of the background signal and the attendant requirement for
the accurate photometry of control fields.
In § 3.2 we present the cluster observations, the data reduction, and in
78

3.2 Subaru data reduction
§ 3.3 we explain how we select cluster galaxies, background samples, and
foreground samples. In § 3.4 we examine evolutionary tracks to reinforce the
sample selection criteria, and compare with blank-field surveys to determine
redshift distributions in § 3.5. In § 3.6 we describe the WL analysis, and
in § 3.7 we describe how the WL measurements are used to determine the
cluster fraction using our WL dilution method. In § 3.8 we derive the light
profile of each cluster, and compare with the mass profile to obtain the M/L
profile in § 3.9. In § 3.10 we investigate the shape and variation of the cluster
luminosity functions (LF) and its radial dependence. Finally, we summarize
our findings in § 3.11.
The ΛCDM cosmological model is adopted, with Ω M = 0.3, and Ω Λ = 0.7,
but with h left in units of H 0 /100 km s −1 Mpc −1 , for easier comparison with
earlier work.
3.2 Subaru data reduction
We analyze deep images of three intermediate-redshift clusters, A1703 (z =
0.258), A370 (z = 0.375) and RXJ1347-11 (z = 0.451), observed with the
wide-field camera Suprime-Cam (Miyazaki et al. 2002) in several optical
bands, at the prime focus of the 8.3m Subaru telescope. The cluster A1703
was observed on 2007 June 15 and the WL signal analyzed by Broadhurst
et al. (2008) and Oguri et al. (2009). The observations of the other two clusters
are available in the Subaru archive, SMOKA 1 . Subaru reduction software
(SDFRED; Yagi et al. 2002; Ouchi et al. 2004) is used for flat-fielding, instrumental
distortion correction, differential refraction, sky subtraction, and
stacking. Photometric catalogs are created using SExtractor (Bertin &
Arnouts 1996) in dual-image mode, for which we use our deepest band as
the detection image. Since our work relies heavily on the colors of galaxies,
we prefer using isophotal magnitudes. Astrometric correction is done with
the scamp tool (Bertin 2006) using reference objects in the NOMAD catalog
1 http://smoka.nao.ac.jp.
79

Mass and Light of A1703, A370 & RXJ1347-11
Table 3.1 – The Cluster Sample: Redshift and Filter Information
Cluster z r vir
1
Filters Total exposure m lim 2 Seeing Detection
(h −1 Mpc) time (sec) (AB mag) (arcsec) Band
A1689 0.183 1.9 V J 1920 26.5 0.82 i ′
i ′ 2640 25.9 0.88
A1703 0.258 1.7 g ′ 1200 27.4 0.89 r ′
r ′ 2100 26.9 0.78
i ′ 1200 26.1 0.8
A370 0.375 2.1 B J 7200 27.4 0.72 R C
R C 8340 26.9 0.6
z ′ 14221 26.1 0.7
RXJ1347-11 0.451 1.6 V J 1800 26.5 0.7 R C
R C 2880 26.7 0.76
z ′ 4860 25.5 0.6
1 based on Broadhurst et al. (2008)
2 limiting magnitude for a 3σ detection within a 2 ′′ aperture
(Zacharias et al. 2004) and the SDSS DR6 (Adelman-McCarthy et al. 2008)
where available. The clusters observation details are listed in Table 3.1.
Photometric zeropoints were calculated from associated standard star
observations and also from independent well calibrated photometry, where
available. In the case of A1703 and A370, no standard star observations were
available, so zeropoints were derived independently in similar passbands by
comparing to magnitudes of stars observed in SDSS fields (Oguri et al. 2009).
Conversions from SDSS g, r bands to Subaru B, R C bands were done using
the relations given in Lupton (2005) for the SDSS. For RXJ1347-11, standard
stars observed during the observation nights were used to derive photometric
zeropoints in the B, R C and z ′ bands. For the z’-band only three Standard
stars were unsaturated in the Subaru image. To better constrain the z’-band,
and as a consistency check for the B J & R C -bands, unsaturated stars were
compared with the HST /ACS images of RXJ1347-11 in the F 475W, F 814W
& F 850LP filters, finding good consistency to within ±0.05 mag.
80

3.3 Sample selection from the color-color diagram
3.3 Sample selection from the color-color diagram
For each cluster we use Subaru observations in three broad-bands, which differ
in their choice of passbands between the clusters, though all observations
are deep and taken in conditions of good seeing, representing some of the
highest quality imaging of any target by Subaru in terms of depth, resolution,
and color coverage. Using three bands in terms of two-color space for
object selection will help separate different populations and improve subsequent
WL and light measurements of the cluster, as we now demonstrate.
3.3.1 Cluster Members
For all three clusters we construct a CC diagram and first identify in this
space where the cluster lies. We do this by calculating the mean distance of all
objects from the cluster center in a given CC cell, as shown in Fig. 3.1. This
turns out to be a very clear way of finding the cluster as the mean radius is
markedly lower in a well defined region of CC space for each cluster (region of
bluer (darker) colors, the approximate boundary of which is marked in black).
This small region clearly corresponds to an overdensity of galaxies in CC
space comprising the red sequence of each cluster and a blue trail of later type
cluster members (Fig. 3.2, dashed white curve). Of course, some background
galaxies must also be expected in this region of CC space, and the proportion
of these will be established in § 3.7 by looking at their WL signal – which
should be consistent with zero in the case of no background contamination.
In general, a positive signal will be expected at some level, in proportion
to the fraction of lensed background galaxies occupying that same region of
CC space as the cluster. Indeed, it is clear that the level of the tangential
distortion seen in Fig. 3.6 is consistent with zero almost to the outskirts
of each cluster radial span, and only rises to meet the background level at
large radius where the proportion of cluster members is small compared with
the background counts. In Fig. 3.1 (black curve), in Fig. 3.2 (dashed white
81

Mass and Light of A1703, A370 & RXJ1347-11
curve) and in Fig. 3.3 (left panels, green points) we mark the boundaries
of this region, selected relative to the red sequence in this CC space, and
conservatively to safely encompass the region occupied by the cluster.
The above sample, which includes all cluster member galaxies, we term
the “green” sample, as distinct from the well separated populations of red and
bluer galaxies discussed below (Fig. 3.3, left panels). In defining this green
sample we apply a magnitude limit of m ≃ 26.5 in the reddest band available
for each cluster (i ′ -band for A1703 and z ′ -band for A370 and RXJ1347-11),
to which the data are complete for all three clusters. Beyond this limit incompleteness
creeps into the bluer bands, complicating color measurements,
in particular of red galaxies. We also set a bright magnitude limit, according
to the magnitude of the brightest cluster member in the same equivalent
band.
3.3.2 Red Background Galaxies
The mean radius used above is useful for locating the cluster in CC space
but does not help in the definition of background or foreground galaxy populations,
which are of course relatively uniform over the field of view, apart
from the relatively small magnification bias which tends to deplete the number
of red background galaxies towards the center of mass, as described in
Broadhurst et al. (2008) and is visible here in Fig. 3.3 (central panels, black
squares) as a decrease in number counts towards the center.
To define the foreground and background populations we look instead
at the combination of the WL signal and the distribution of galaxies in the
CC plane. For each cluster, we see a dense cloud of red objects (Fig. 3.2,
lower right overdensity). This is particularly obvious for A370 where we have
the widest wavelength coverage, B J , R C & z ′ . The cloud is relatively red in
R C − z ′ but blue in B J − R C , so very well separated from the cluster member
region defined above. This red cloud is known from wide-field survey work
(Capak et al. 2007) and established to comprise early- to mid-type galaxies
spanning a broad redshift range, 0.5 < z < 2, as discussed below (§ 3.5).
82

3.3 Sample selection from the color-color diagram
2
A1703
10
2
A1703
x 10 −6
7
1.8
1.6
1.4
9
8
1.8
1.6
1.4
6
5
g − r
1.2
1
0.8
7
Radius
g − r
1.2
1
0.8
4
3
Number/pix 2
0.6
6
0.6
2
0.4
5
0.4
1
0.2
0.2
−0.4 −0.2 0 0.2 0.4 0.6 0.8
r − i
−0.4 −0.2 0 0.2 0.4 0.6 0.8
r − i
0
A370
13
A370
x 10 −6
7
2
12
2
6
11
5
B − R
1.5
1
10
9
Radius
B − R
1.5
1
4
3
Number/pix 2
8
2
0.5
7
0.5
6
1
0
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
R − Z
0
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
R − Z
0
1.4
RXJ1347−11
13
1.4
RXJ1347−11
x 10 −6
7
1.2
12
1.2
6
1
11
1
5
V − R
0.8
0.6
0.4
10
9
8
Radius
V − R
0.8
0.6
0.4
4
3
Number/pix 2
0.2
7
0.2
2
0
6
0
1
−0.2
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
R − Z
5
−0.2
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
R − Z
0
Figure 3.1 – Averaged radius from cluster
center displayed in CC space for A1703,
A370 & RXJ1347-11 (Top to bottom). The
bluer (darker) colors imply average lower
radius, hence correspond to the location of
the cluster in CC space. The black box
marks the boundaries of the green sample
we select, which conservatively includes all
cluster members.
Figure 3.2 – Number density in CC space
for A1703, A370 & RXJ1347-11 (Top to
bottom). The four distinct density peaks
(seen especially in the A370 plot) are shown
to be different galaxy populations - the reddest
peak in the upper right corner of the
plots (dashed white line) depicts the overdensity
of cluster galaxies, whose colors are
lying on the red sequence; the middle peak
with colors bluer than the cluster shows the
overdensity of foreground galaxies; the two
peaks in the bottom part (bluest in B − R)
can be demonstrated to comprise of blue
and red (left and right, respectively) background
galaxies.

Mass and Light of A1703, A370 & RXJ1347-11
For this red sample we define a conservative diagonal boundary relative to
the red sequence as for the green sample (see Fig. 3.3, left panels, red points),
to safely avoid contamination by cluster members and also foreground galaxies
as discussed below (see § 3.3.3). We can check if there is any significant
contamination of this red sample by unlensed galaxies by measuring the WL
amplitude of these red galaxies as a function of distance from this boundary.
As shown in Fig. 3.6 (red triangles), no evidence of dilution of the lensing
signal is visible. We also define a blue color limit to separate this red population
from what appears to be a distinct density maximum of very blue
objects – the so called “blue cloud” (see Fig. 3.2, in the extreme blue corner
of CC space (see below § 3.3.3). We further limit the red sample to m < 26
AB mag in the reddest band for each cluster to avoid incompleteness and
so that we can rely on good detections in all bands when defining our color
samples.
The boundaries of the red sample as defined above are marked on Fig. 3.3
(red points), and can be seen to lie well away from the “green” cluster sample
previously defined (green points). For this red sample a clearly rising WL
signal is seen all the way to the smallest radius accessible (r 1 ′ ) for each
cluster (Fig. 3.6, red triangles), with no sign of a central turnover which
would indicate the presence of unlensed cluster members, as discussed fully
in § 3.6.
3.3.3 Blue Background and Foreground Galaxies
Above we have been able to define the region of CC space occupied by cluster
members by virtue of their clustered distribution. We have also defined with
confidence a sample of red background galaxies lying well separated from
cluster members and for which there is a clear WL signal. For bluer galaxies
the situation is more complex as these may comprise blue cluster members,
foreground objects and background blue galaxies. This is of particular concern
where only one color is available, and which we have shown in our earlier
work can lead to a dilution of the WL signal of blue selected objects relative
84

3.3 Sample selection from the color-color diagram
to the red background galaxies by unlensed foreground galaxies and cluster
members (Broadhurst et al. 2005b; Medezinski et al. 2007).
We can see in the CC plane of each cluster that there are two regions
lying blueward of the cluster within which most blue objects lie. For A370,
where we have the deepest dataset, these density peaks are located around
R C − z ′ ∼ 0 and B J − R C ∼ 0.8, 1.5. The redder of these two blue peaks lies
in the center of the CC plane, relatively close to the blue-end of the cluster
sequence (see Fig. 3.3, left panels, magenta points). The spatial distribution
of objects in this CC region shows no significant clustering about the cluster
center (see Fig. 3.3, middle panels, magenta circles). In addition, the WL
signal of these objects (see Fig. 3.3, right panels, magenta circles) shows
very little signal, indicating this is a predominately unlensed population,
and together with the lack of clustering means this blue peak is dominated
by galaxies lying in the foreground of the cluster. The lensing signal here
is not strictly zero but marginally positive, implying a relatively low level
of 10 − 20% lensed background galaxies. This region of CC space is clearly
of no use for our purposes as it does not contain cluster members and is
mostly unlensed. It is convenient, however, that the foreground unlensed
population is so well defined in CC space as a clear overdensity and therefore
can be readily excluded from our analysis. The redshift distribution of this
population using deep field surveys (discussed in § 3.5) supports our finding
that the bulk of this population lies at low redshift.
The bluer of the two peaks, in the extreme blue corner of CC space seen
in Fig. 3.2 (lower left peak) is also unclustered on the sky, indicating no
significant cluster members lie in this region (Fig. 3.3, middle panels, black
squares). This blue overdensity is the well known “blue cloud” identified in
deep field images. For each cluster, a clear WL signal is found for objects
lying in this blue cloud (Fig. 3.6, blue circles), rising toward the center of
each cluster with a radial trend very similar to that of the red population
((Fig. 3.6, red triangles), indicating minimal foreground or cluster dilution of
the WL signal. Objects lying in this region are expected to fall in the redshift
range 1 < z < 2.5 (discussed below in § 3.5). Hence, we can safely conclude
that these objects lie in the background with negligible cluster or foreground
85

Mass and Light of A1703, A370 & RXJ1347-11
contamination, which would otherwise drag down the central WL signal. The
boundaries we have chosen for this blue sample and plotted in Fig. 3.3 (left
panels, blue points), are extended to include objects lying outside the main
blue cloud but well away from the foreground and cluster populations defined
above, in order to maximize the size of the blue sample but steering well clear
of contamination by unlensed low redshift objects. We further limit the blue
sample magnitude in the range 22 < m AB < 26 to avoid contamination by
very bright foreground galaxies, and to avoid incompleteness at the faint end.
3.4 Evolutionary color tracks
In order to examine in more detail the sample selection technique, we overlay
evolutionary and empirical tracks on the CC plane, for different classes
of galaxies. We first calculate non-evolving magnitudes in Subaru filters
B J , R C , z ′ as a function of redshift using different galaxy templates constructed
empirically for elliptical, Scd and starburst types from the Kinney-
Calzetti library (Calzetti et al. 1994; Kinney et al. 1996). The resulting color
tracks are overplotted in Fig. 3.4 (top) on the B J − R C vs. R C − z ′ plane
of A370. These templates show how the foreground population of galaxies
in the center of the CC plane quickly jumps above a redshift of ∼ 1
to the lower blue density peak. For the E/S0 it appears that even the noevolution
track is a good fit to a distribution of red galaxies which fall on
a well-defined loop visible in the CC plane with a maximum red color of
B J − R C = 2.5, R C − z ′ = 1.5 at a redshift of z ∼ 0.8. The remainder of
the loop is not well constrained (and we therefore leave it out), likely where
evolution kicks in making the SED of early-type objects bluer.
To generate evolving tracks representing the evolution of galaxy SED’s,
the Galev 2 (Kotulla et al. 2009) code was used to represent galaxy types
with different star-formation histories – a short 3 Gyr starburst (Z ⊙ ), E-
type (exponentially declining SFR with Z ⊙ ), S0 (gas-related SFR with Z ⊙ ),
Sa (gas-related SFR with 2.5Z ⊙ ), Sb (gas-related SFR with 0.4Z ⊙ ) and Sd
2 http://www.galev.org/
86

3.4 Evolutionary color tracks
0.25
n [arcmin −2 ]
Distortion g +
0.2
0.15
0.1
0.05
0
10 1
10 0
0 2 4 6 8 10 12 14 16 18 20
θ [arcmin]
(a) A1703
−0.05 0.2
−0.2
1 2 5 10 20
θ [arcmin]
g x
0
0.8
0.7
0.6
n [arcmin −2 ]
Distortion g +
0.5
0.4
0.3
0.2
0.1
10 1 θ [arcmin]
0
0.2
g x
0
0 2 4 6 8 10 12 14 16 18 20
(b) A370
−0.2
1 2 5 10 20
θ [arcmin]
0.5
n [arcmin −2 ]
Distortion g +
0.4
0.3
0.2
0.1
0
10 1
10 0
0 2 4 6 8 10 12 14 16 18
θ [arcmin]
(c) RXJ1347
−0.2
1 2 5 10 20
θ [arcmin]
−0.1 0.2
g x
0
Figure 3.3 – Sample selection and analysis. Left: CC diagrams, for A1703 (a), A370
(b) and RXJ1347-11 (c) (top to bottom), displaying the green sample, comprising mostly
cluster member galaxies, and the red and blue samples, comprising of background galaxies.
The galaxies that we identify as predominantly foreground lie in between the cluster and
background galaxies are marked in magenta. Center: Surface number density of galaxies
vs. radius. Background (red+blue galaxies) density (black squares) is fairly constant
with radius, except for a slight decrease in the central region (depletion due to lensing
magnification effect). The green sample (green pentagrams) rises steeply toward the center
as expected for cluster members. The magenta sample (magenta diamonds) has a flat
density profile, meaning these are not part of the cluster. Right: Tangential distortion vs.
radius. The background (black squares) rises continuously toward the center, representing
real WL signal. The green sample distortion (green pentagrams) is essentially in agreement
with zero, as expected from unlensed cluster galaxies, and the magenta sample distortion
(magenta diamonds) is significantly lower than the background WL signal, indicating
object selected in this CC range are mostly unlensed foreground galaxies. However, the
magenta sample also contains to some extent background galaxies, therefore the WL signal
is slightly higher than zero.

Mass and Light of A1703, A370 & RXJ1347-11
(constant SFR with 0.2Z ⊙ ) (Fig. 3.4, bottom). It is evident that at low redshifts
the modeled galaxies coincide with the high central peak we recognized
above with our WL technique to be foreground, and as they are redshifted
the colors of E/S0 galaxies agree at z ∼ 0.4 with the cluster colors. At higher
redshifts the early types objects become bluer due to the effect of evolution,
and also the late-type galaxies jump quickly across the valley and fall close
to the observed trail of red and blue background galaxies at 1 < z < 2. At
even higher redshifts (z > 2.5), they fade out from the B J -band and become
redder in B J − R C , agreeing with observed colors of dropout galaxies. Accounting
for evolution, the tracks follow more carefully the trend around the
lower-left blue corner (B J − R C ∼ 0.1, R C − z ′ ∼ 0), seen clearly from the
galaxy color distribution.
In general, similar behavior is seen for both methods, giving compelling
evidence that in this CC plane the foreground and cluster galaxies are rather
well-separated from the higher-redshift background galaxies, and support the
picture we presented above for the safe selection of background galaxies for a
WL study. The slight differences in color (offset seen from the Galev models
to bluer B J − R C colors for example) is interesting and may arise from the
inclusion of evolution, the presence of dust or minor model uncertainties. The
shape of the early-type galaxy loop is certainly worth exploring in greater
detail and may help to constrain the general evolutionary history of earlytype
galaxies.
3.5 Depth estimation from SDF/COSMOS
The lensing signal is dependent on the source distance, scaling linearly with
D ds /D s , the lensing distance ratio. We thus need to estimate and correct
for the respective depths of the different samples. To this end we rely on
accurate photometric redshifts derived for two deep, multi-band field surveys,
the Subaru Deep Field (SDF; Kashikawa et al. 2004), and COSMOS
(Capak et al. 2007). For the SDF, photometric redshifts are calculated with
BPZ (Benítez 2000), using a new template library, generated using a set of
88

3.5 Depth estimation from SDF/COSMOS
Figure 3.4 – Top: Empirical color tracks overlaid on the galaxy distribution of A370 in
B J − R C vs. R C − z ′ plane, for a range of galaxy templates: elliptical, Scd, SB1 and SB3
SED’s. Bottom: Synthetic color tracks including evolution, calculated with the Galev code
for a single 3 Gyr burst, elliptical, S0, Sa, Sb and Sd type models.

Mass and Light of A1703, A370 & RXJ1347-11
0.35
SDF
0.3
0.25
0.2
P(z)
0.15
0.1
0.05
0
0 0.5 1 1.5 2 2.5 3 3.5 4
z
0.35
COSMOS
0.3
0.25
0.2
N(z)
0.15
0.1
0.05
0
0 0.5 1 1.5 2 2.5 3 3.5 4
z
Figure 3.5 – Redshift distributions of CC-selected samples. Two deep field surveys are
used to estimate mean redshifts and depth: SDF (top), and COSMOS (bottom). The
green (green thin solid line), red (red thick solid line), blue (blue thick dashed line)
and foreground (magenta dotted-dashed line) samples are selected according to A370
CC/magnitude limits (see Fig. 3.3).

3.5 Depth estimation from SDF/COSMOS
Table 3.2 – CC-selected Sample Properties
Cluster Sample magnitude limits N ¯n ¯z s < D ds /D s >
arcmin −2 COSMOS SDF COSMOS SDF
A1703 green 16.3 < i ′ < 26.5 5678 6.2 0.54 - 0.45 -
red 21 < i ′ < 26 4742 5.2 1.00 - 0.67 -
blue 22 < i ′ < 26 4467 4.9 1.59 - 0.74 -
A370 green 16.5 < z ′ < 26.5 5369 5.3 0.46 0.69 0.25 0.29
red 21 < z ′ < 26 13372 13.7 1.14 1.21 0.57 0.57
blue 22 < z ′ < 26 8481 8.7 1.74 1.65 0.64 0.56
RXJ1347 green 17.5 < z ′ < 26.5 2512 3.5 0.67 0.79 0.28 0.31
red 21 < z ′ < 26 2943 4.1 1.09 1.26 0.49 0.52
blue 22 < z ′ < 26 1691 2.3 1.45 1.67 0.44 0.46
PEGASE templates (Fioc & Rocca-Volmerange 1997) which approximates
the library described in Benítez et al. (2004) and Coe et al. (2006). This new
library has an accuracy of δz/(1+z) = 0.037 with the FIREWORKS catalog
(Wuyts et al. 2009) and a low catastrophic error rate. Using those photo-z,
and the spectral classification provided by BPZ, we generate magnitudes in
the required Subaru filters to match the cluster observations. We select samples
according to CC/magnitude limits described above (Fig. 3.3) for A370
and RXJ1347-11 (the SDF filters are not matched with the filters used for
A1703). The resulting redshift distributions of the green (thin solid line),
red (thick solid line), blue (dashed line) and foreground (dash-dotted line)
samples selected to match A370 samples using SDF are displayed in Fig. 3.5
(top).
For COSMOS, photometric redshifts have been derived by Ilbert et al.
(2009) using 30 bands in the UV to mid-IR. Since the COSMOS photometry
does not cover the Subaru R C band, we need to estimate R C -band magnitudes
for it. We use the HyperZ (Bolzonella et al. 2000) template fitting code to
obtain the best-fitting spectral template for each galaxy, from which the R C
magnitude is derived with the transmission curve of the Subaru R C -band
filter (see Umetsu et al. 2010). We then select samples by applying the same
CC/magnitude limits as for A1703, A370 and RXJ1347-11. The resulting
91

Mass and Light of A1703, A370 & RXJ1347-11
redshift distributions of the green (thin solid line), red (thick solid line), blue
(dashed line) and foreground (dash-dotted line), samples selected to match
A370 samples using COSMOS are displayed in Fig. 3.5 (bottom). The results
for each sample using SDF or COSMOS are summarized in table 3.2.
Although SDF is deeper, COSMOS redshifts are based on more bands
over a wider spectral range. However, as stated in Ilbert et al. (2009), COS-
MOS photo-z’s are reliable only to a magnitude of i ′ < 25, which is below
our magnitude cutoff for the sample selection. We therefore make sure to
limit our redshift estimation for the green sample to z < 3, to avoid mistakenly
including high-z dropouts that lie above the cluster in CC due to
slight photometric offsets or unreliable photo-z’s. Note we are only interested
in the mean D ds /D s of background objects, and therefore estimate
the depth of background galaxies present in the green sample in the range
z cluster < z < 3. Overall, the redshift distributions of both field surveys look
quite similar, and there is good agreement between values derived with either
SDF or COSMOS, to ∼ 10% level in the depth, which can be due to the
differences mentioned above. These depth values will be used later to correct
the lensing signal (see § 3.7).
Importantly, this redshift analysis independently supports our assessment
in both sections 3.3 & 3.4 that the various regions of the CC space correspond
to differing populations. In particular the foreground population isolated in
the center of the CC diagram (shown in Fig. 3.3, magenta points, whose
redshift distribution is shown in Fig. 3.5, magenta dotted-dashed curve) corresponds
to predominantly low redshift galaxies ¯z ≃ 0.4, most of which are in
the foreground of the clusters examined here, or at such low redshift behind
the cluster that the lensing signal is small by virtue of the small separation,
D ds , between the lens and the source. In contrast, the red and blue populations
are much more distant lying well behind our clusters, supporting
our conclusion that the continuously rising WL signal of these populations is
not significantly contaminated by cluster members. This also indicates that
the predicted level of unlensed foreground galaxies is negligible for the red
sample, as expected, and with only a possible ∼ 10% dilution in the blue.
92

3.6 Weak lensing analysis
3.6 Weak lensing analysis
We use the IMCAT package developed by N. Kaiser 3 to perform object detection
and shape measurements, following the formalism outlined in Kaiser,
Squires, & Broadhurst (1995, hereafter KSB). Our analysis pipeline is described
in Umetsu et al. (2010). We have tested our shape measurement and
object selection pipeline using STEP (Heymans et al. 2006) data of mock
ground-based observations (see Umetsu et al. 2010, § 3.2). Full details of
the methods are presented in Umetsu & Broadhurst (2008), Umetsu et al.
(2009) and Umetsu et al. (2009). In short, we find that we can recover the
weak lensing signal with good precision, typically, m ∼ 5% of the shear
calibration bias, and c ∼ 10 −3 of the residual shear offset which is about
one-order of magnitude smaller than the weak-lensing signal in the cluster
outskirts (|g| ∼ 10 −2 ). We emphasize that this level of calibration bias is subdominant
compared to the statistical uncertainty (∼ 15%) in the mass due
to the intrinsic scatter in galaxy shapes. Rather, the most critical source of
systematic uncertainty in cluster weak lensing is dilution of the distortion signal
due to the inclusion of unlensed foreground and cluster member galaxies,
which can lead to an underestimation of the true signal for R < 400 h −1 kpc
by a factor of 2−5, as demonstrated in Broadhurst et al. (2005b), Medezinski
et al. (2007) and here.
The shape distortion of an object is described by the complex reducedshear,
g = g 1 + ig 2 , where the reduced-shear is defined as:
g α ≡ γ α /(1 − κ). (3.1)
The tangential component of the reduced-shear, g T , is used to obtain the
azimuthally averaged distortion due to lensing, and computed from the distortion
coefficients g 1 , g 2 :
g T = −(g 1 cos 2θ + g 2 sin 2θ), (3.2)
3 http://www.ifa.hawaii.edu/ kaiser/imcat
93

Mass and Light of A1703, A370 & RXJ1347-11
1.5
0.25
Distortion g +
0.2
0.15
0.1
0.05
0
−0.05 0.2
Fraction 1−g +
(G)/g +
(B)
1
0.5
0
g x
0
−0.2
1 2 5 10 20
θ [arcmin]
−0.5
1 2 5 10 20
θ [arcmin]
0.8
1.5
0.7
0.6
Distortion g +
0.5
0.4
0.3
0.2
0.1
0
0.2
Fraction 1−g +
(G)/g +
(B)
1
0.5
0
g x
0
−0.2
1 2 5 10 20
θ [arcmin]
−0.5
1 2 5 10 20
θ [arcmin]
1.5
0.5
Distortion g +
0.4
0.3
0.2
0.1
0
−0.1 0.2
Fraction 1−g +
(G)/g +
(B)
1
0.5
0
g x
0
−0.2
1 2 5 10 20
θ [arcmin]
−0.5
1 2 5 10 20
θ [arcmin]
Figure 3.6 – Tangential reduced shear vs.
radius for A1703, A370 and RXJ1347-11
(top to bottom). The lower panels show the
45 ◦ -rotated (g X ) component of the reduced
shear. The green pentagrams represent the
green sample, containing mostly cluster and
some background galaxies whose colors are
similar to the cluster. The black squares
show the level of tangential distortion of the
combined red+blue (triangles and circles,
respectively) samples of the background.
The green pentagrams slightly rise to the
background level at large radii, indicating
the green sample still contains some level of
background galaxies at the outskirts of the
cluster, and is consistent with zero towards
the cluster center where cluster members
dominate the green sample and dilute the
lensing signal.
Figure 3.7 – Fraction of cluster membership
vs. radius for A1703, A370 and
RXJ1347-11 (top to bottom). Cluster
membership is proportional to the dilution
of the distortion signal of the green sample,
relative to the expected distortion of the
background galaxies set by the combined
red and blue samples. The fraction departs
from unity only at large radius.

3.6 Weak lensing analysis
where θ is the position angle of an object with respect to the cluster center,
and the uncertainty in the g T measurement is σ T = σ g / √ 2 ≡ σ in terms of
the RMS error σ g for the complex reduced-shear measurement. To improve
the statistical significance of the distortion measurement, we calculate the
weighted average of g T and its weighted error, as
∑
i
〈g T (θ n )〉 =
u g,i g
∑ +,i
, (3.3)
i u g,i
σ T (θ n ) =
√ ∑
i u2 g,i σ2 i
( ∑ i u g,i) 2 , (3.4)
where the index i runs over all of the objects located within the nth annulus
with a median radius of θ n , and u g,i is the inverse variance weight for ith
object, u g,i = 1/(σg,i 2 + α 2 ), softened with α. We choose α = 0.4, which
is a typical value of the mean RMS ¯σ g over the background sample. The
case with α = 0 corresponds to an inverse-variance weighting. On the other
hand, the limit α ≫ σ g,i yields a uniform weighting. We have confirmed that
our results are insensitive to the choice of α (i.e., inverse-variance or uniform
weighting) with the adopted smoothing parameters.
In Fig. 3.6 we plot the radial profile of g T of the green (pentagrams),
red (triangles) and blue (circles) samples defined above. The black squares
represent the red+blue combined sample, showing the best estimate of the
lensing signal. The red and blue samples profiles rise continuously toward
the center for each of the clusters, and are in good agreement with each
other, demonstrating that both are dominated by background galaxies and
are not contaminated by the cluster at all radii. The g T profile of the green
sample lies close to zero, especially at small radii, but shows a small positive
signal at large radius, close in amplitude to the background signal measured
above, meaning at beyond ∼ 5 ′ there are few cluster members in comparison
to “green” background galaxies. Indeed, the virial radius derived from our
best fitting NFW model lies at about 10 ′ − 15 ′ and so physically it is reasonable
that the cluster population is small here. The measured zero level
of tangential distortion interior to this radius for the green sample reinforces
95

Mass and Light of A1703, A370 & RXJ1347-11
our CC selection.
3.7 Weak lensing dilution
We can now estimate the fraction of cluster galaxies by taking the ratio of
the WL signal between the green sample and the background sample, with
the background including both red and blue galaxies as explained in § 3.3.
For a given radial bin (r n ) containing objects in the green sample (§ 3.2),
the mean value of g (G)
T
(eq. 3.3) is an average of the signal over cluster members
and some background galaxies. Thus, its mean value 〈g (G)
T
〉 will be lower
than the true background level denoted by 〈g (B) 〉 (Fig. 3.6) in proportion to
T
the fraction of unlensed galaxies in the bin that lie in the cluster (rather
than in the background), since the cluster members on average will add no
net tangential signal. Therefore,
f cl (r n ) ≡
N cl
= 1 − 〈g T (r n ) (G) 〉
N Green 〈g T (r n ) (B) 〉
〈D (B) 〉
〈D (G) 〉
(3.5)
is the cluster membership fraction of the green sample, where 〈D (B) 〉 and
〈D (G) 〉 are the mean lensing depths, D ds /D s , of the background and green
galaxy samples, respectively, derived in § 3.5 (see full derivation in the appendix
of Medezinski et al. 2007).
In Fig. 3.7 we show this ratio for each of our new clusters. In our previous
analysis of A1689, where only color-magnitude selection was done, many
background galaxies were present in the green sample, so that only in the first
radial bins the fraction of cluster galaxies was high, and gradually declined
with radius (due to increasing fraction of background galaxies) at larger
radii. Here, since we select a tight region around the cluster in CC and are
able to better resolve it, our results show that the fraction is close to unity
throughout the radial range of the cluster. Only in the outer bins, usually
at r > 5 ′ − 10 ′ does the fraction decline, and we see a smaller percentage of
cluster members. Large errorbars are introduced at outer radii since there
96

3.8 Cluster light profiles
600
A1689: −1.5±0.08
Luminosity density [L ο
h 2 Mpc −2 ]
10 12
A1703: −0.95±0.1
A370: −1.4±0.06
RXJ1347:−0.9±0.1
M/L [ hM o
/L ο
]
500
400
300
200
100
10 11
10 13 r/r vir
0.1 0.2 0.5 1 2
0
0.1 0.2 0.5 1 2
r/r vir
Figure 3.8 – “g-Weighted” luminosity
density vs. r/r vir for A1689 (magenta
circles), A1703 (cyan asterisks), A370
(red squares) and RXJ1347-11 (green diamonds).
The flux F i (green sample) of each
galaxy is weighted by its tangential distortion
g T,i with respect to the background
distortion signal. The dashed lines are the
best fitting lines whose slopes are given in
the legend. For clarity, the luminosity profiles
have been shifted by a factor: A1703
shifted up by a factor of 2, A370 by a factor
of 3, and RXJ1347-11 by a factor of 1.2.
Figure 3.9 – Mass-to-light ratio vs.
r/r vir for A1689 (magenta circles), A1703
(cyan asterisks), A370 (red squares) and
RXJ1347-11 (green diamonds). The profiles
are similar to each other, peaking
around 0.2r v ir, and declining continuously
to the virial radius, which seems to be a
general property of all well-studied clusters.
are few cluster galaxies present, making it harder to use this effect in order
to determine cluster membership.
3.8 Cluster light profiles
We now turn to translate the deduced profile of membership fraction to a
cluster luminosity profile.
To account for the background contribution in
the green sample and deduce only the cluster luminosity, we first measure
the flux of all the galaxies in the green sample, ∑ F (G)
i , and then subtract
i
the flux of the i-th galaxy weighted by it’s tangential distortion, g i , relative
to the background sample distortion, 〈g T (r n ) (B) 〉. When averaged over the
entire green sample this will have zero contribution from the unlensed cluster
members, but will remove the contribution of background galaxies present
97

Mass and Light of A1703, A370 & RXJ1347-11
in the green sample. The total flux of the cluster in the n th radial bin is
therefore,
F cl (r n ) = ∑ i
F (G)
i − 〈D(B) 〉/〈D (G) 〉
〈g T (r n ) (B) 〉
∑
i
F (G)
i g (G)
T,i
(3.6)
and the flux is then translated to luminosity. First we calculate the absolute
magnitude,
M = m − 5 log d L − K(z) + 5, (3.7)
where the K-correction is evaluated for each radial bin according to its color.
The luminosity is simply,
L = 10 0.4(M ⊙−M) L ⊙ , (3.8)
where M ⊙ is the absolute magnitude of the Sun (AB system) in the relevant
band.
The results for all four clusters, including A1689, are shown in Fig. 3.8.
As explained in § 3.3, the green sample comprises mostly of cluster galaxies,
therefore the correction is not big for r < 10 ′ , and only present at large radii,
increasing the error. The cluster luminosity profiles mostly have a steady
linear decline with radius, and are well described by a power-law fit, but with
different slopes of d log(L)/d log(r) ∼ −0.95 ± 0.1, −1.4 ± 0.06, −0.9 ± 0.1 for
A1703, A370 and RXJ1347-11, respectively.
3.9 M/L profiles
We may now derive a radial profile of the differential mass-to-light ratio
(often referred to as δM(r)/δL(r)) by dividing the best-fit NFW projected
mass density profile by the newly derived luminosity density profile obtained
in § 3.8. The four M/L profiles are shown together in Fig. 3.9 to the limit of
our data, ∼ 3 h Mpc. The mass profiles for these clusters and their NFW fits
were derived in our earlier work, Broadhurst et al. (2005b) for A1689 and in
98

3.9 M/L profiles
Broadhurst et al. (2008) for A370, A1703 and RXJ1347 based on the sample
selection scheme described in this paper.
For A1689, we found previously that the M/L ratio peaks at intermediate
radius around r ∼ 100 h −1 kpc and then falls off to larger radius.
Here we find a similar behavior for the other clusters, especially evident for
A370, where M/L peaks at about r ∼ 350 h −1 kpc with values of M/L R ∼
300 − 400 h(M/L) ⊙ , and declines to values of M/L R ∼ 100 h(M/L) ⊙ at the
outskirts. A decline in M/L is also seen towards the center of the cluster,
reaching M/L R ∼ 200 h(M/L) ⊙ below r 200 h −1 kpc. The M/L
profile of A1703 also shows a similar trend, where it peaks at a radius of
r ∼ 200 − 300 h −1 kpc with values of M/L r ′ ∼ 300 h(M/L) ⊙ and declines
gradually to values of M/L r ′ ∼ 100 h(M/L) ⊙ at large radii. The profile
of RXJ1347-11 declines from values of M/L R ∼ 400 − 500 h(M/L) ⊙ at
r ∼ 200 − 300 h −1 kpc to values of M/L R ∼ 100 h(M/L) ⊙ at large radius.
The decrease in M/L values toward the center of the cluster is only really
evident for two clusters, A1689 and A370. For these clusters the luminosity
profiles are relatively steep, and it this is continuation of a steep power-law
towards the center that causes M/L to decline, as the mass profiles are not
power-law but have a continuously declining gradient towards the cluster centers
(NFW-like), so that the proportion of M/L declines markedly interior
to the characteristic radius for these two clusters.
The decrease in M/L toward large radius seen for all four clusters is
possibly explained by the increasing proportion of later type galaxies at larger
radii, a manifestation of the morphology-density relation (Dressler 1980).
Similar results have been shown by Rines et al. (2000) and Katgert et al.
(2004); in Rines et al. (2000) where the differential M/L profile is similar
to ours, with a decreasing M/L with radius, and their outermost value of
M/L R ∼ 100 h(M/L) ⊙ is in good agreement with the values we get for all our
four clusters. Katgert et al. (2004) show a decrease at small radius, and by
removing the brightest cluster members this decrease is shown to disappear.
The declining behavior is not simply accounted for by the change of stellar
population with distance from the cluster center. The galaxy colors show a
99

Mass and Light of A1703, A370 & RXJ1347-11
10 4 r [Mpc/h]
late−types
early−types
all types
M/L R
[ hM o
/L r’ο
]
10 3
10 2
0.2 0.5 1 2 3
10 4 r [Mpc/h]
late−types
early−types
all types
M/L R
[ hM o
/L Rο
]
10 3
10 2
0.2 0.5 1 2 3
10 4 r [Mpc/h]
late−types
early−types
all types
M/L R
[ hM o
/L Rο
]
10 3
10 2
0.2 0.5 1 2 3
Figure 3.10 – Mass-to-light ratio vs. radius for A1703, A370 and RXJ1347 (top to
bottom plots) divided into two different cluster sub-populations. The black squares mark
the entire green sample (same M/L profiles shown above in Fig. 3.9); the red circles mark
a subsample from a tighter selection of cluster galaxies around the red sequence (white
solid line marked in Fig. 3.2), mostly early-type elliptical galaxies. The blue hexagrams
correspond to later-type cluster members (the rest of the green population inside the white
dashed line, but outside the solid white line marked in Fig. 3.2). The slope of the declining
outer profile is significantly steeper for the blue profiles, whereas for the red profiles the
M/L is much flatter.

3.9 M/L profiles
10 5 r [Mpc/h]
bright early−types
bright late−types
faint early−types
faint late−types
M/L R
[ hM o
/L r’ο
]
10 4
10 3
10 2
0.2 0.5 1 2 3
10 5 r [Mpc/h]
bright early−types
bright late−types
faint early−types
faint late−types
M/L R
[ hM o
/L Rο
]
10 4
10 3
10 2
0.2 0.5 1 2 3
10 5 r [Mpc/h]
bright early−types
bright late−types
faint early−types
faint late−types
M/L R
[ hM o
/L Rο
]
10 4
10 3
10 2
0.2 0.5 1 2 3
Figure 3.11 – Mass-to-light ratio vs. radius for A1703, A370 and RXJ1347 (top to
bottom plots) for early (red circles) and late type (blue hexagrams) clusters members, as
in Fig. 3.10, but divided into high- (solid) and low-luminosity (empty) subsamples. Clearly
the behavior of the early and later types is not a strong function of luminosity.

Mass and Light of A1703, A370 & RXJ1347-11
10 3 r [Mpc]
a early,bright
=−1.2±0.2
a early,faint
=−0.79±0.3
n [Mpc −2 ]
10 2
10 1
a late,bright
=−0.42±0.2
a late,faint
=−0.42±0.1
10 3 r [Mpc]
n [Mpc −2 ]
10 2
10 1
0 0.5 1 1.5 2 2.5 3
a early,bright
=−1.3±0.2
n [Mpc −2 ]
10 3
10 2
r [Mpc]
a early,faint
=−1±0.2
10 1
n [Mpc −2 ]
10 3
10 2
r [Mpc]
a late,bright
=−0.65±0.3
a late,faint
=−0.64±0.2
10 1
0 0.5 1 1.5 2 2.5 3 3.5 4
a early,bright
=−1.2±0.2
a early,faint
=−0.89±0.2
n [Mpc −2 ]
10 2
10 1
r [Mpc]
a late,bright
=−0.31±0.4
a late,faint
=−0.52±0.3
n [Mpc −2 ]
10 2
10 1
r [Mpc]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Figure 3.12 – Number density profiles for A1703, A370 and RXJ1347-11 (top to bottom
plots) for early (upper panels, red circles) and late (lower panels, blue hexagrams) type
cluster members further divided into high (solid) and low-luminosity (empty) subsamples
(as in Fig. 3.11). The redder profiles are more centrally concentrated than the bluer
profiles, suggesting the proportion of early- to late-type members is a clear function of
radius but not as much of luminosity, favoring early-types at smaller radius.

3.9 M/L profiles
very moderate decline with radius. In the case of A370, the radial color
gradient is d(B − z ′ )/d log(r) = −0.13 ± 0.04, equivalent to a total change
of δ(B − z ′ ) = 0.12 over the virial radius. This corresponds to a change
in the mean stellar mass-to light ratio, M ∗ /L R , of only ∼ 17%. Therefore,
such a small correction alone cannot explain the declining trend of M/L with
radius.
We now further examine the behavior of M/L by separating cluster member
galaxies by color and by luminosity. There is a clear distinction between
the M/L profiles based on color selection as shown in Fig. 3.10. Tightly
selecting just the sequence galaxies (inside the white solid line marked in
Fig. 3.2) the M/L profiles are relatively flatter for all the clusters (Fig. 3.10,
red circles), showing mild decline with radius, whereas the remaining nonsequence
objects (the rest of the green population between the white dashed
line and the solid while line marked in Fig. 3.2) have a steep dependence
(Fig. 3.10, blue hexagrams) which is clearly responsible for the majority of
the trend of M/L found above for the cluster population as a whole (Fig. 3.10,
black squares). If we now further divide these subsamples by luminosity into
two broad bins as shown in Fig. 3.11, we see that this difference between the
sequence and non-sequence populations is not strongly dependent on luminosity.
If we also look at the number-density profile of each of these four
subsamples (Fig. 3.12), we see that the early-type galaxies (red circles) are
more concentrated than the later types (blue hexagrams), independently of
luminosity.
It is evident from the above comparisons that later-type galaxies are relatively
rare in the cluster center, an effect which most likely has its origins in
tidal and/or stripping of infalling later-type galaxies and their transformation
into redder and rounder galaxies predominating in cluster centers. We
now explore tidal effects in this context and comment on other related work.
3.9.1 Tidal and Ram-Pressure Stripping
In CDM-dominated cosmological models, the growth of structure is hierarchical,
with clusters evolving through many merger events during which
103

Mass and Light of A1703, A370 & RXJ1347-11
galaxies are dynamically affected to a varying degree based on their formation
times and spatial locations in their parent systems. This process can
be described by the excursion set formalism (Lacey & Cole 1993) with more
detailed insight gained from hydrodynamical simulations. It has been clearly
demonstrated that during these merger events DM halos suffer pronounced
tidal stripping (e.g., Moore et al. 1996). Detailed N-body simulations indicate
that up to 70% of the mass may be stripped from halos orbiting close to
the center of massive clusters when integrated over the Hubble time, and with
∼ 20% loss even near the virial radius (Nagai & Kravtsov 2005). Statistical
evidence for halo truncation is claimed also from lensing based measurements
(Natarajan et al. 2002, 2009).
As can be expected, simulations show that despite the substantial effect of
tidal forces in removing the outer DM from galaxy halos, the stellar content
of early-type galaxies is only weakly affected by tidal interactions due to the
centralized location of most stars within the DM halos (Nagai & Kravtsov
2005; Reed et al. 2005). Hence, the ratio of total mass to galaxy luminosity
is predicted to be approximately independent of radius in this context. In
detail, the above simulations show a mild radial decline of the ratio of the
total mass density over the number density of DM halos, M/n, as shown in
Fig. 3.13 (dashed curve). The relatively gentle declining trend in this ratio is
the result of tidal forces (Nagai & Kravtsov 2005; Reed et al. 2005), which act
over time to reduce the number of sub halos in the inner region as compared
to the cluster outskirts.
This M/n ratio is a very useful prediction of the simulations which may
be readily compared to our data. For each cluster we indeed find a gentle
decline in the ratio of the total mass to the number of early type galaxies, the
mean slope of which is in very good agreement with the prediction as shown
in Fig. 3.13. The declining trend in this ratio is seen in the region 0.1−1 r vir ,
in good agreement with the prediction by Nagai & Kravtsov (2005).
As we showed in § 3.9 the ratio of the total mass density to bluer cluster
members light density has a very steep decline with radius – much greater
than the mild trend observed above for early-type red sequence galaxies. The
104

3.9 M/L profiles
0.6
0.4
0.2
log 10
(M/n)
0
−0.2
−0.4
−0.6
−0.8
0.1 0.2 0.5 1 2
r/r vir
Figure 3.13 – Radial profile of the ratio of the cluster mass density over the galaxy number
density for A1703 (cyan stars), A370 (red squares) and RXJ1347-11 (green diamonds).
Dashed line shows this ratio as calculated from the simulations of Nagai & Kravtsov
(2005). The observed trend is similar to the theoretical prediction for each of the clusters
and can be attributed theoretically to tidal effects by the cluster acting on the galaxy DM
halos.
tendency towards an increasing proportion of early-type galaxies near cluster
centers (also evident in Fig. 3.12) is of course well known and is understood
empirically to result from the disruption of the gaseous disks of infalling
galaxies. Radio continuum observations reveal that “wide-angle tails” of
gaseous material appearing to trail behind disk galaxies (e.g., Marvel et al.
1999). Recently spectacular large trails of H α emission have been detected
around disk galaxies in the Coma and Virgo clusters (Bravo-Alfaro et al.
2000; Vollmer et al. 2004; Kenney et al. 2008). The most impressive object
is a large spiral galaxy (NGC 4438) with obvious disrupted arms and a long
trail (∼ 120 kpc) of H α emission revealing the orbit of this galaxy has been
bent around the massive central galaxy, M86, with a large relative velocity
of ∼ 500 km/s (Kenney et al. 2008).
Ram-pressure stripping has long been calculated to be important (Gunn
& Gott 1972) for disk galaxies (in clusters), removing preferentially the relatively
low-density gas within a characteristic stripping radius, the scale of
which can be estimated from basic considerations, and which was verified
105

Mass and Light of A1703, A370 & RXJ1347-11
by careful simulations (Abadi et al. 1999; Kronberger et al. 2008). The loss
of gas has a transformative effect on both the morphology and the colors
of such objects, a process which is claimed as a natural explanation for the
cluster populations of S0 galaxies, (Homeier et al. 2005; Postman et al. 2005;
Poggianti et al. 2009; Huertas-Company et al. 2009) and which is also related
to the evolution of cluster gas metallicity (Schindler et al. 2005; Arieli et al.
2008).
3.10 Cluster luminosity functions
The data allow the LF to be usefully constructed in several independent
radial and magnitude bins, and hence we can examine the form of the LF of
cluster members as a function of projected distance from the cluster center.
By measuring the degree of dilution separately in each magnitude bin we can
construct LFs without resorting to uncertain background subtraction.
Here, the dilution correction is evaluated as a function of magnitude and
radius. Left panels of Fig. 3.14 show the tangential distortion of the green
sample versus absolute magnitude bin, and the reference background distortion
level is marked in the black solid line; see (Medezinski et al. 2007) for
more details. To correct for this we apply equation (3.5) to each magnitude
bin:
Φ cl (M k ) = Φ(M k )[1 − 〈g (G)
T
(M k)〉/〈g (B)
T
(r)〉] (3.9)
(Note that the background signal is averaged over the whole range of magnitudes
at that radius). The correction for background present in the green
sample is in fact not large for the brighter bins, but is more essential at the
fainter bins, with magnitude M R −18. This is similar to the observation
we made in § 3.7 for the radial profile of the fraction correction, only there
the correction was a function of radius, and here it is a function of magnitude.
We construct LFs in three broad radial annuli, except for RXJ1347-11
for which this is feasible – due to its small angular size – only in two bins.
The results are shown in Fig. 3.14 (center panels). Each LF is well fitted
106

3.10 Cluster luminosity functions
by a Schechter (1976) function (dashed lines), and the confidence contours
for the Schechter function parameters M ∗ and α are shown on the righthand
panels of Fig. 3.14. The Schechter fit parameters are also specified in
table 3.3. Clearly there is no evidence for a distinct upturn, to the limits of
the photometry of any of our clusters. The slope of the Schechter function
is well defined in each case, lying in a tight range, from α = −0.95 ± 0.05
to α = −1.24 ± 0.2. Also, there is no significant radial dependence in the
shape of the LFs with distance from the cluster center. There may however
be some evidence that in the outer radial bin the LFs are somewhat steeper
than in the center of the cluster for A1703, as can be seen in Fig. 3.14, where
the slope changes from α ∼ −0.95 to α ∼ −1.1 to α ∼ −1.25 at larger radius.
The knee of the LFs, namely the bright-end break, is very distinct, and is
around M ∗r ′ ∼ −21.5 for A1703, M ∗R ∼ −22.5 for A370, and M ∗R ∼ −22
for RXJ1347-11.
For A1689 we also saw no upturn at the faint magnitude limit, with
α = −1.05 ± 0.07 (Medezinski et al. 2007). There has been some significant
disagreement regarding the shapes and faint-end slopes of LFs in clusters.
Popesso et al. (2006) analyzing 69 RASS-SDSS clusters found a steep upturn
at the faint-end by fitting a composite of two Schechter functions, with the
faint-end fit slope being α ∼ −2. Rines & Geller (2008) made a comprehensive
study of two local galaxy clusters, using spectroscopic data to identify
securely the cluster galaxies, for a more reliable measurement of cluster LFs.
They found no distinct upturn for either of the clusters, with α = −1.13 for
A2199 and α = −1.28 for the Virgo cluster. When measuring the overall
slope of the composite cluster LF, i.e., up to the virial radius, we find slopes
of α ∼ −1.1 ± 0.1 for all three clusters, in good agreement with the results of
Rines & Geller (2008). We note that general cluster-to-cluster variation in
the total LF fits is not necessarily unexpected, but probably depends on the
specific characteristics of individual clusters. Our clusters all show relatively
flat LFs, with only a weak radial dependence.
107

Mass and Light of A1703, A370 & RXJ1347-11
0−0.3 Mpc/h
0.2
0.3−0.8 Mpc/h
0.8−2.5 Mpc/h
0.1
0
0.1
Φ(M) [h 2 Mpc −2 mag −1 ]
−0.93
α=−0.95±0.05 M *
=−21.4±0.6
−0.94
−0.95α
10 2
−0.96
−1.06
α=−1.08±0.08 M *
=−21.7±0.8
−1.07
g + 0.05
10 1
−1.08
−1.09α
−1.1
0
0.02
α=−1.24±0.2 M
10 0
*
=−21.8±3
−1.2
10 3 M R
g +
−24 −22 −20 −18 −16 −14
g +
−24 −23 −22 −21
0.01
0
−0.01
−24 −22 −20 −18 −16 −14
M r’
10 −1
M *r’
−1.25α
−1.3
g
g
0.6
10 3 −1.03
0−0.3 Mpc/h
α=−1±0.04 M *
=−22.4±0.6
0.3−1 Mpc/h
−1.035
0.4
+
1−2.2 Mpc/h
−1.04 α
0.2
10 2
−1.045
0
−1.11
0.15
α=−1.12±0.05 M *
=−22.1±0.7
g + 0.1
10 1
−1.12
α
0.05
−1.13
0
−1.14
0.06
α=−1.18±0.09 M *
=−22.8±1
10 0
−1.16
0.04 +
−1.18α
0.02
−1.2
0
10 −1
−1.22
−25 −20 −15
−24 −22 −20 −18 −16 −14 −23.5 −23 −22.5 −22
M R
M R
M *R
Φ(M) [h 2 Mpc −2 mag −1 ]
10 3
g
−23 −21
0.25
0−0.8 Mpc/h
α=−1±0.06 M *
=−21.6±0.5
0.8−2 Mpc/h
0.2
0.15
+
10 2
0.1
0.05
10 1
0
0.06
α=−1.12±0.1 M *
=−22±2
10 0
0.04
0.02
10 −1
0
−0.02
10 −2
−25 −20 −15
M R
−24 −22 −20
M R
−18 −16 −14
−22
M *R
g +
Φ(M) [h 2 Mpc −2 mag −1 ]
−0.98
−0.985
−0.99
−0.995α
−1
−1.005
−1.01
−1.08
−1.1
−1.12α
−1.14
−1.16
Figure 3.14 – Luminosity functions of A1703 (top), A370 (middle) and RXJ1347-11
(bottom). Center part: LFs and their Schechter fits are shown for several independent
radial bins, indicating little trend of the shape of the LF with radius. Left part: tangential
distortion g T of the green sample as a function of magnitude used in the derivation of
the corresponding LF. Right part: 1σ, 2σ, and 3σ contours for the Schechter function fits
parameters M ∗ and α, for each of the corresponding LFs of each radial bin.

3.11 Discussion and conclusions
Table 3.3 – Parameters of Schechter fits to the Luminosity function
Cluster radial range α M ∗
(h −1 Mpc)
A1703 0 − 0.3 −0.95 ± 0.05 −21.4 ± 0.6
0.3 − 0.8 −1.08 ± 0.08 −21.7 ± 0.8
0.8 − 2.5 −1.24 ± 0.2 −21.8 ± 3
A370 0 − 0.3 −1 ± 0.04 −22.4 ± 0.6
0.3 − 1 −1.12 ± 0.05 −22.1 ± 0.7
1 − 2.2 −1.18 ± 0.09 −22.8 ± 1
RXJ1347-11 0 − 0.8 −1 ± 0.06 −21.6 ± 0.5
0.8 − 2 −1.12 ± 0.1 −22 ± 2
3.11 Discussion and conclusions
In this paper we have compared in detail the dark and luminous properties of
three high mass galaxy clusters, A1703, A370 and RXJ1347-11, and examined
the relation between mass and light all the way out to the virial radius.
We have further applied the WL dilution approach previously developed for
A1689 (Medezinski et al. 2007) as an alternative means of obtaining the
LF and light profiles of galaxy clusters. This is based on the dilution of the
lensing distortion signal by unlensed cluster members, and has the advantage
of being independent of density fluctuations in the background.
We have examined clusters A1703, A370 and RXJ1347-11 for which data
of exceptional quality has been taken with Subaru/Suprime-Cam in multiple
passbands. We have shown how a careful separation of the cluster,
foreground, and background galaxies can be achieved using weak lensing information
and galaxy colors. We also make use of the clustering of cluster
members, finding the cluster members are confined to a distinct region of
CC space with relatively small mean radial distances. We have constructed
a reliable background sample for WL measurements, by selecting red background
galaxies whose WL signal rises all the way to the center of the cluster,
and adding to those the faint cloud of very blue galaxies, whose WL signal
we have shown is in good agreement with the red galaxies. We have avoided
109

Mass and Light of A1703, A370 & RXJ1347-11
dilution of the WL signal by identifying and excluding the foreground population
in the CC diagram for which no clustering or WL signal is detected.
Together, the red and blue samples WL measurements can be used to reconstruct
accurate mass maps of all four clusters (see Broadhurst et al. 2008).
By applying this sample selection and WL dilution correction, the LFs
are derived for each of the clusters, in several radial bins, without the need
of statistical background subtraction. The LFs extend to very faint magnitude
limits, M R ∼ −14, and each shows a very flat faint-end slopes,
−0.95 ≤ α ≤ −1.24, in agreement with our previous study for A1689. There
are various claims in the literature regarding the shape of the faint end of
the galaxy LF (Popesso et al. 2006; Rines & Geller 2008; Adami et al. 2008;
Crawford et al. 2009), with relatively flat slopes measured and found consistent
with a simple Schechter function, and other work claiming a distinctive
upturn at faint magnitudes. Differences may conceivably represent intrinsic
differences between clusters, but a worry here is the uncertain subtraction of
the steep background counts of faint galaxies when evaluating the background
using control fields. Uncertainties may result from inherent clustering of the
background galaxies and also variations in photometric incompleteness, or
differential extinction between the cluster and control fields.
Redshift measurements may remove concerns about the background for
complete surveys, but in practice spectroscopy can only reach interestingly
faint limiting luminosities for fairly local clusters, requiring multiplexing capability
over a wide-fields of view on large telescopes. Rines & Geller (2008)
recently provided the tightest constraints from spectroscopy out to a sizable
fraction of the virial radius, finding no significant upturn for the clusters
studied. Confidence is gaining in the use of photometric redshifts, from careful
deep multi-color imaging, which in the work of Adami et al. (2008) a
relatively steep LF is claimed for Coma, but flatter slopes are measured for
a sample of intermediate-redshift clusters (Crawford et al. 2009). Clearly, it
is essential that there is no significant confusion of background galaxies with
cluster members or foreground, as is achieved with our approach.
The radial light profiles of these clusters are well characterized by a power-
110

3.11 Discussion and conclusions
law slope of approximately d log(L)/d log(r) ≃ −1 over all radii. These are
compared with the mass profiles from WL to derive the radial profile of
the M/L ratio, a quantity which has hitherto not been constrained in much
detail, due to the difficulty of measuring mass profiles with sufficient precision
and light profiles free of systematic effects. Interestingly, we find that all
our clusters show a very similar radial trend of M/L, peaking in the range
M/L R ∼ 500 h(M/L) ⊙ at intermediate radius ≃ 0.15R vir , and declining to
approximately a limiting level of M/L R ∼ 100 h(M/L) ⊙ toward the virial
radius of each cluster.
The decline of M/L towards large radius is very interesting and seems
to be a universal feature of clusters, or at least it is clearly found in the
clusters we have been able to study here in sufficient detail and for A1689
(Medezinski et al. 2007). Additionally, in the careful dynamical-based study
of Rines et al. (2000) a similar declining trend was found for A576. We
have shown that such behavior is not simply accounted for by the change
in the stellar content of galaxies with distance from the cluster center. The
radial trend of galaxy colors found is very mild and when corrected for using
standard stellar population synthesis falls well short of explaining the marked
trend of M/L with radius.
The variation of M/L is however seen to be almost entirely stemming
from the much steeper density profile of blue cluster members compared to
the red sequence galaxies. The higher relative abundance of blue galaxies
in the outer region of rich clusters is well known from local morphological
studies and deeper work with high resolution (Rines et al. 2000; Katgert
et al. 2004; Postman et al. 2005) and is part of the general morphologydensity
relation (Dressler 1980). Such behavior is commonly explained by
the continued conversion of disk galaxies into S0 galaxies, via tidal and gas
stripping processes (Abadi et al. 1999), and thought to be consistent with
the observed evolution of the numbers of S0 galaxies relative to other cluster
populations measured in the accessible redshift range z < 1 (Dressler 1980;
Ellis et al. 1997; Postman et al. 2005; Poggianti et al. 2009; Simard et al.
2009). Disk galaxies are expected to be relatively more affected than earlytype
objects as their stellar content is relatively less bound gravitationally
111

Mass and Light of A1703, A370 & RXJ1347-11
and the gas is susceptible to ram pressure from the ICM.
In contrast, for early-type galaxies the M/L profile we obtained for each
cluster is fairly flat, with only a shallow radial decline for r > 200 h −1 kpc.
Interestingly CDM based N-body simulations aimed at evaluating the tidal
influence of the cluster potential do claim to predict a shallow radial decline in
the ratio of cluster mass to the number of sub-halos (Nagai & Kravtsov 2005;
Reed et al. 2005; Natarajan et al. 2009), matching well our observed trend.
The simulations indicate that although up to 70% of a galaxy halo mass can
be stripped over a Hubble time for those objects with orbits bringing them
close to the cluster center, the stellar content is not likely to be significantly
affected, at least for early-type galaxies, as the stars are more tightly bound.
Hence, a relatively flat relation is expected reflecting the relatively collisionless
behavior of both galaxies and DM. Therefore, even though tidal effects
are significant, the luminous component that we measure when examining
the M/L profile of early-type galaxies is expected to be only slightly affected,
in good agreement with the mild decline we observe for each cluster.
Further work on the tidal evolution of clusters is obviously needed. This
is particularly important in view of the interesting tension that has now
emerged between the relatively concentrated mass profiles measured for many
high mass clusters, compared to the much shallower mass profiles predicted
by standard ΛCDM (Duffy et al. 2008; Broadhurst et al. 2008; Oguri et al.
2009). Steeper mass profiles lead to stronger tidal effects and compounded
with the greater ages implied by the relatively high concentrations should lead
to more significant tidal effects. Measuring directly the tidal truncation of
luminous cluster member galaxies should be feasible via lensing in favorable
cases where multiple images are locally affected by individual halos. The
statistical influence on the general lensing deflection field has been claimed
to be detected for two clusters independently by Natarajan et al. (2009)
for Cl0024+16 and by Halkola et al. (2007) for A1689. These first results
encourage further deeper work with an emphasis on the degree of correlation
of the lensing signal with cluster member galaxies, a formidable challenge
but one well worthy of pursuit.
112

Chapter 4
A Weak Lensing Determination
of the Cosmological
Distance-Redshift Relation
Behind Abell 370
A version of this chapter has been submitted for publication in MNRAS as
Medezinski et al. (2010)
4.1 Introduction
Constraining cosmological parameters has been the focus of major surveys
in the last decade, via precision cosmic microwave background (CMB) temperature
correlations (Spergel et al. 2007; Brown et al. 2009) and SN-Ia light
curves (Riess et al. 1998; Perlmutter et al. 1999).
A standard, ΛCDM, cosmological model has been defined by this work,
albeit at the price of accepting an accelerating expansion driven by a cosmological
constant, and non-baryonic DM of an unknown nature dominating
the mass density of the Universe. Measurements of the angular diameter
113

WL Determination of the Distance-Redshift Relation
distance of the CMB refer to z ∼ 1100, and luminosity distances are derived
from SN-Ia in the range z < 1. In principle, lensing can provide a
complementary distance measurements in the range, z > 1, from the purely
geometric deflection of light, which increases with source distance behind a
lens.
For lensing clusters, the bend-angle of light scales linearly with angular
diameter distance ratio, D ds /D s , the separation between the lens and the
source, divided by distance to the source. This distance ratio has a characteristic
geometric dependence on redshift, rising steeply behind the lens
and then saturating at large source redshift (e.g., Fig. 1 of Broadhurst et al.
1995). This effect has been detected behind massive lensing clusters, where
the separation in angle between multiple images of higher redshift sources
is noticeably larger than for lower redshift sources. For example, for the
well studied cluster SDSSJ1004+4112 (z = 0.68), five images of a QSO at
z = 1.734 are within an Einstein radius of about θ E ∼ 7 ′′ (Inada et al. 2003;
Oguri et al. 2004), whereas a more distant multiply lensed galaxy behind
this cluster at z = 3.332 is at a much larger Einstein radius of θ E ∼ 16 ′′
(Sharon et al. 2005). Other sets of multiple images show this increasing angular
scaling with source redshift, following well the expected general form
of the redshift-distance relation in careful strong lensing analyses of deep
Hubble data (Soucail et al. 2004; Broadhurst et al. 2005a; Zitrin et al. 2009b;
Zitrin & Broadhurst 2009; Zitrin et al. 2009a).
However, these studies are not able to distinguish between the relatively
subtle changes between cosmologies in the range of interest, due to the inherent
insensitivity of the distance ratio D ds /D s to the cosmological parameters.
Moreover, the bend-angle is particularly sensitive to the gradient of the mass
profile, requiring many sets of multiple images in the strong regime to simultaneously
solve for both the cosmological model and the mass distribution.
Instead in practice, strong lens modeling usually adopts the standard cosmological
relation in order to better derive the mass distribution, with multiply
lensed sources forced to lie on the lensing distance-redshift relation. This
helps eliminate the otherwise considerable degeneracy in constraining the
slope of the lensing mass profile of a galaxy cluster (Broadhurst et al. 2005a;
114

4.1 Introduction
Zitrin et al. 2009b; Zitrin & Broadhurst 2009; Zitrin et al. 2009a). The hope
of constraining the cosmography from strong lensing data is remote with
current tools (Gilmore & Natarajan 2009).
Weak lensing (WL), by contrast, offers a model independent way of constraining
the cosmological parameters via the distance-redshift relation (Taylor
et al. 2004). Image distortions and magnification depend on gradients of
the deflection field and in the WL limit, these are just proportional to D ds /D s .
The mass profile enters only in the stronger regime as a second order correction
(Medezinski et al. 2007). Here we are concerned with the amplitude of
cluster WL dependence on source redshift, for which no mass reconstruction
is required when evaluating the distance-redshift relation. However, although
this observed effect is independent of the mass distribution in the weak limit,
the sensitivity to cosmological parameters is still inherently very small.
The analogous effect in the field has been explored in terms of the cosmic
shear, to measure the general mass distribution. Optimal formalisms
have been developed which cross-correlate the foreground distribution of
galaxies along a given line of sight with the distribution of background images
(Wittman et al. 2001; Jain & Taylor 2003; Bacon et al. 2003; Taylor
et al. 2004, 2007) with clear detections of large scale structure, including the
COMBO-17 fields (Brown et al. 2003; Kitching et al. 2007) and the COSMOS
field (Massey et al. 2007b; Schrabback et al. 2010). To usefully derive cosmological
parameters from general cosmic shear work, deep all-sky surveys
have been proposed (e.g., LSST 1 , DES 2 , JDEM 3 , EUCLID 4 ).
Here we make use of detailed color-color (CC) information for an intermediate
redshift lensing cluster, Abell 370 (hereafter A370, z = 0.375), and
define several samples of galaxies of differing background depths with which
to explore the dependence of WL distortion on source redshift. With only
three bands we cannot reliably define photometric redshifts for a sizable proportion
of objects, but instead, by reference to the redshift disributions of
1 http://www.lsst.org/lsst
2 https://www.darkenergysurvey.org/
3 http://jdem.gsfc.nasa.gov
4 http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=42266
115

WL Determination of the Distance-Redshift Relation
the well studied deep field surveys, we may reliably define several galaxy
populations of differing mean depths in the CC-space covered by the filters
used for our cluster.
Three colour selection has also been applied in a similar context to simulations
aimed at forecasting the capabilities of WL tomograghy (Jain et al.
2007; Medezinski et al. 2010). These simulations convincingly demonstrate
that greater efficieny is likely by using limited 3-band imaging for WL tomorgraphy,
rather than by investing greater imaging time in additional bands
to improve photometric redshift precision. For our purposes too we show here
that a judicious choice of non-orthogonal boundries in CC-space allows the
definition of several distinct redshift samples of differing mean depth, with
relatively little overlap in redshift. We rely on well studied deep field surveys
to estimate the redshift distribution of these different background populations,
using the wide-field COSMOS 30-band photometric redshift survey
(Ilbert et al. 2009) and also the deeper GOODS-MUSIC survey which has
wide multi-wavelength coverage in 14 bands (from the U band to the Spitzer
8 µm band) (Grazian et al. 2006).
In § 4.2 we present the cluster observations and data reduction and in
§ 4.3 we explain the selection of background galaxy samples and in § 4.4 we
describe the WL analysis and outline the formalism. In § 4.5 we derive the
WL amplitude and mean redshift information of the background samples,
presenting our results regarding the lensing distance-redshift relation. We
discuss the requirements for constraining the cosmological model with this
method in 4.6. We summarize and conclude in § 4.7.
4.2 Subaru data reduction
We analyze deep images of the intermediate-redshift cluster, A370, observed
with the wide-field camera Suprime-Cam (Miyazaki et al. 2002) in three
optical bands, at the prime focus of the 8.3m Subaru telescope. The cluster
data are publicly available from the Subaru archive, SMOKA 5 . Subaru
5 http://smoka.nao.ac.jp
116

4.3 Sample selection from the color-color diagram
Table 4.1 – The A370 Subaru Data
Filters used 1 Exposure Time Seeing
(sec)
(arcsec)
B J 7200 0.7
R C 8340 0.6
z ′ 14221 0.7
1 Detection band marked in bold.
reduction software (SDFRED) developed by Yagi et al. (2002) is used for
flat-fielding, instrumental distortion correction, differential refraction, sky
subtraction, and stacking. Photometric catalogs are created using SExtractor
(Bertin & Arnouts 1996). Since our work relies much on the colors of
galaxies, we prefer using isophotal magnitudes. We use the Colorpro (Coe
et al. 2006) program to detect in the R C -band and measure colors through
matched isophotes in the other two bands. Astrometric correction is done
with Scamp (Bertin 2006) using reference objects in the NOMAD catalog
(Zacharias et al. 2004) and the SDSS-DR6 (Adelman-McCarthy et al. 2008)
where available. The observational details are listed in Table 4.1.
4.3 Sample selection from the color-color diagram
We use Subaru observations in three broad optical passbands and all observations
are of good seeing, representing some of the highest quality imaging
by Subaru in terms of depth, resolution, and color coverage. We first describe
how we separate the background galaxies from foreground and cluster galaxies,
combining WL measurements and the distribution of objects in the CC
plane and their clustering relative to the center of the cluster. We then examine
the redshift distribution of objects selected to lie in the background using
the CC plane with reference to the COSMOS field where deep photometric
redshifts are established to faint limits using 30 independent passbands
117

WL Determination of the Distance-Redshift Relation
covering a very wide range of wavelength.
Note that with only three bands, only a small fraction of the objects have
well defined photometric redshifts, in the sense of having a single, narrow
peak in their probability p(z), and therefore their use to separate among
different redshift populations is very limited. Using the BRz ′ CC space
in this way with reference to the now well established redshift surveys is
arguably more reliable in separating galaxy populations of differing depths,
in agreement with the simulations of Jain et al. (2007).
In our previous analysis of this cluster, (Medezinski et al. 2010, hereafter
M10), we demonstrated how the cluster and foreground galaxies can be reliably
identified and separated from background galaxies in the CC diagram,
using B J , R C , z ′ bands (Fig. 4.1). We found in the field of A370 the prominent
overdensity of galaxies centered on B J − R C ∼ 2 and R C − z ′ ∼ 0.8 denotes
the red-sequence of cluster galaxies (see Fig. 4.1, left-hand panel, where this
overdensity is enclosed by white dashed line). This was also shown by the
relatively small mean distance from cluster center of galaxies in that region
in CC space (Fig. 1 in M10). The WL measurements for this population of
objects is very close to zero, with a measured tangential distortion profile,
g T (r), consistent with zero all the way out to the virial radius, as expected
for cluster galaxies which are unlensed. The main central overdensity around
B J − R C ∼ 1 and R C − z ′ ∼ 0.3 consists of many foreground galaxies (see
Fig. 4.1, left-hand panel, with an overdensity enclosed by solid white line).
These galaxies show a very low level g T compared to the reference background,
marked in gray on Fig. 4.1 (right panel), and their surface density
profile shows only modest central clustering (see M10) indicating that most of
these objects lie in the foreground of the cluster. We use a bright subsample
of these foreground galaxies to demonstrate the zero-level signal at a redshift
point below the cluster (see below § 4.5), marked in gray on Fig. 4.1 (bottom
panel). When selecting background galaxies we stay well away from these
regions to minimize contamination by these unlensed galaxies, as described
below.
To identify background populations in M10, we took into account both
118

4.3 Sample selection from the color-color diagram
the WL signal and the density distribution of galaxies in the CC plane. In
CC space a relatively red population can be rather well defined, dominated
by an obvious overdensity (around B J − R C ∼ 0.5 and R C − z ′ ∼ 0.8), and
the bluest population is confined to a separate cloud (around B J − R C ∼ 0.3
and R C − z ′ ∼ 0.2) of faint galaxies with a clear lensing signal. Looking at
the WL profiles of the red and blue samples, we see very similar behavior,
with a continuous rising signal toward the cluster center. Both cases show
good agreement with each other (M10). Combined together, they form our
reference background sample, to which all the other samples we derive
below will be normalized.
The validity of our selection was also demonstrated in M10 by comparison
with the spectral evolution of galaxies calculated with the stellar synthesis
code Galev 6 (Kotulla et al. 2009). Here we overlay the CC diagram with
color-tracks of galaxy models: E-type (exponentially declining SFR with
Z ⊙ ), S0 (gas-related SFR with Z ⊙ ), Sa (gas-related SFR with 2.5Z ⊙ ), and
Sd (constant SFR with 0.2Z ⊙ ). These evolutionary tracks originate in the
low-redshift cloud we established as foreground in CC space and evolve to
higher redshift passing through the cluster redshift and then to bluer colors
as shown in Fig. 4.5 (right), and finally end at the top-left corner of our CC
space, dropping out of the B J -band at a redshift of z ∼ 3.5
In this paper we add additional samples of background galaxies. Two
samples will consist of galaxies at redshifts not far beyond that of the cluster,
which we term “orange” and “green”. These are selected to lie on the
upper-right of the CC space, corresponding reasonably to color-tracks of E/S0
galaxies which are predicted to show a bend in the CC plane, becoming bluer
(B J − R C ∼ 2 − 2.5 and R C − z ′ ∼ 1 − 1.5) toward higher redshift, matching
well our observed distribution. A third group comprises the red-cloud
described above, centered on B J − R C ∼ 0.5 and R C − z ′ ∼ 0.8, we call the
“red” sample. This sample consists of an overdensity of background galaxies
from the known “red” branch of the bimodality of field galaxies colors
(Capak et al. 2007). A Forth sample consists of the prominent blue peak
6 http://www.galev.org/
119

WL Determination of the Distance-Redshift Relation
Table 4.2 – CC-selected Sample Properties
Sample magnitude limits N ¯n Γ χ 2 /dof < z s >
arcmin −2 (g T -amplitude ratio) (PL)
foreground 18

4.3 Sample selection from the color-color diagram
x 10 −6
7
3
6
2.5
5
B − R
2
1.5
4
3
Number/pix 2
1
2
0.5
1
0
0 0.5 1 1.5
R − Z
0
3.5
3.5
3
E−type
3
2.5
2.5
B − R
2
1.5
2
1.5
redshift
1
S0
0.5
Sa
Sd
0
−0.5 0 0.5 1 1.5 2
R − Z
1
0.5
0
Figure 4.1 – Top: Number density in B J − R C vs. R C − z ′ CC space. The four distinct
density peaks are shown to be different galaxy populations - the reddest peak in the upper
right corner of the plots (dashed white line) depicts the overdensity of cluster galaxies,
whose colors lay on the red sequence; the middle peak lying blueward of the cluster
comprises mainly foreground galaxies (solid white line); the two peaks in the bottom part
(bluest in B J −R C ) can be demonstrated to comprise of blue and red background galaxies.
Together, the red+blue galaxies will serve as our “reference” background sample for WL
purposes. Bottom: B J − R C vs. R C − z ′ CC diagram, showing the distribution of galaxies
in A370. Marked are the selected foreground sample (gray) and background samples:
orange (orange), green (green), red (red), blue (blue) and dropout (magenta) background
galaxies, selected to include galaxies lying away from the cluster and foreground regions.
Overlaid are synthetic color tracks including evolution, calculated with the Galev code for
an elliptical, S0, Sa and Sd type models.

WL Determination of the Distance-Redshift Relation
4.4 Weak lensing measurements
To make the WL catalogs, we use the IMCAT package developed by N.
Kaiser 7 to perform object detection and shape measurements, following the
formalism outlined in Kaiser, Squires, & Broadhurst (1995, hereafter KSB).
Our analysis pipeline is described in Umetsu et al. (2010). We have tested
our shape measurement and object selection pipeline using STEP (Heymans
et al. 2006) data of mock ground-based observations (see Umetsu et al. 2010,
§ 3.2). Full details of the methods are presented in Umetsu & Broadhurst
(2008), Umetsu et al. (2009) and Umetsu et al. (2010).
The shape distortion of an object is described by the complex reducedshear,
g = g 1 + ig 2 , where the reduced-shear is defined as:
g α ≡ γ α /(1 − κ). (4.1)
The tangential component g T is used to obtain the azimuthally averaged
distortion due to lensing, and computed from the distortion coefficients g 1 , g 2 :
g T = −(g 1 cos 2θ + g 2 sin 2θ), (4.2)
where θ is the position angle of an object with respect to the cluster center,
and the uncertainty in the g T measurement is σ T = σ g / √ 2 ≡ σ in terms of the
RMS error σ g for the complex shear measurement. To improve the statistical
significance of the distortion measurement, we calculate the weighted average
of g T and its weighted error, as
∑
i
〈g T (θ n )〉 =
u g,i g
∑ T,i
, (4.3)
i u g,i
σ T (θ n ) =
√ ∑
i u2 g,i σ2 i
( ∑ i u g,i) 2 , (4.4)
where the index i runs over all of the objects located within the n-th annulus
with a median radius of θ n , and u g,i is the inverse variance weight for i-th
7 http://www.ifa.hawaii.edu/~kaiser/imcat
122

4.4 Weak lensing measurements
object, u g,i = 1/(σg,i 2 + α 2 ), where α 2 is the softening constant variance.
We choose α = 0.4, which is a typical value of the mean RMS ¯σ g over
the background sample. We accurately combine the photometry with weaklensing
measurements of as many galaxies as possible, discarding objects
below the seeing limit (given in table 4.1) plus two standard deviation of
that value in the detection band, to remove stars and avoid unreliable shape
measurements.
4.4.1 Formalism: Relative Distortion Strength
For a given source redshift z s and a fixed lens redshift z l , the observable
(complex) reduced gravitational shear g(z s ) in the subcritical regime is expressed
in terms of the gravitational shear γ and the lens convergence κ as
(e.g., Seitz & Schneider 1997; Medezinski et al. 2007)
g(z s ) = γ(z s )(1 − κ[z s ]) −1 = γ ∞
∞
∑
k=0
w k+1 (z s )κ k ∞ (4.5)
where κ ∞ and γ ∞ are the lensing convergence and the gravitational shear,
respectively, calculated for a hypothetical source at z s → ∞, and w(z s )
is the lensing strength of a source at z s relative to a source at z s → ∞,
w(z s ) = D(z s )/D(z s → ∞); D(z s ) ≡ D ds /D s . Hence, the reduced shear
averaged over the source redshift distribution is expressed as
〈g〉 = γ ∞
∞
∑
k=0
〈w k+1 〉κ k ∞, (4.6)
where 〈w k 〉 is defined such that
〈w k 〉 ≡
∫
dzs N(z s )w k (z s )
∫
dz N(zs )
(4.7)
with the redshift distribution N(z s ). In the WL limit where |κ ∞ |, |γ| ∞ ≪ 1,
then
〈g〉 ≈ 〈w〉γ ∞ = 〈γ〉. (4.8)
123

WL Determination of the Distance-Redshift Relation
Thus, the mean reduced shear is simply proportional to the mean lensing
strength, 〈w〉 ∝ 〈D〉. The next order approximation is
〈g〉 ≈ 〈γ〉 (1 + f w 〈κ〉) ≈
〈γ〉
1 − f w 〈κ〉 , (4.9)
where f w ≡ 〈w 2 〉/〈w〉 2 is a redshift-moment ratio of the order of unity (Seitz
& Schneider 1997).
Since the tangential distortion signal, g T , is a function of cluster radius,
we first decompose the tangential distortion profile of our background sample
(B), defined above as our reference (see § 4.3), into the following form:
g T,B (θ) = a B × θ −b B
, (4.10)
where the radial shape of g T (θ) is assumed to be a single power-law with a
power index b B , and a B represents the distortion amplitude of our reference
background. In practice, we fit the outer profile of g T (θ), excluding the nonlinear
regime (θ 1), to the power-law model, constraining simultaneously
the distortion amplitude a B and the outer slope, b B . Next, for each of our
other defined samples, we fit a power law with the same slope b i = b B , but
allow the amplitude a i to vary:
g T,i (θ) = a i × θ −b B
. (4.11)
Therefore, if we calculate the lensing signal of i-th sample relative to the
reference background (B),
Γ i ≡ g T,i (θ)/g T,B (θ) = a i /a B . (4.12)
From equation (4.8), we obtain the following expression in the WL approximation
(|〈κ〉|, |〈γ〉| ≪ 1):
Γ i = a i /a B ≈ 〈w〉 i /〈w〉 B = 〈D〉 i /〈D〉 B , (4.13)
where 〈 〉 i (i = 1, 2, ..., B) represents averaging over the redshift distribu-
124

4.5 Results
tion N i (z s ) of i-th galaxy sample. The relative distortion strength Γ i can
be regarded as a function of the discrete background sample i with the redshift
distribution N i (z s ), which is observationally available and calibrated by
deep, multi-band blank surveys such as the COSMOS survey. For a given
cosmological model, one can readily construct its theoretical prediction Γ i
(i = 1, 2, ..., B) using a set of redshift distribution functions N i (z s ). The
function Γ i can be formally labeled by its mean redshift
∫
〈z s 〉 i ≡
∫
dz s N i (z s ) z s /
dz s N i (z s ) (4.14)
which is independent of the cosmological model.
To the next order of approximation, the distortion amplitude ratio is
written as (Appendix B of Medezinski et al. 2007)
Γ i = 〈D〉 i
〈D〉 B
{
1 + (fw,i 〈w〉 i − f w,B 〈w〉 B ) κ ∞ (θ) + O(κ 2 ∞) } (4.15)
with f w,i ≡ 〈w 2 〉 i /〈w〉 2 i and f w,B ≡ 〈w 2 〉 B /〈w〉 2 B . The next order correction
term is proportional to (f w,i 〈w〉 i − f w,B 〈w〉 B )κ ∞ (θ), which is much smaller
than unity for the galaxy samples of our interest in the mildly nonlinear
regime (θ 1).
We thus simply adopt equation (4.13) obtained in the
WL approximation. This can be further justified by the fact that the slope
parameter b B is constrained by a least χ 2 fit to the outer distortion profile.
The mean weighted cluster radius 〈θ〉 ≡ ∑ i u g,iθ i / ∑ i u g,i used for a fit is
〈θ〉 ∼ 10 − 11 arcmin for our clusters, where the weak lensing approximation
is valid.
4.5 Results
4.5.1 Weak lensing profiles
In the background of A370 a reference background sample has been defined
(explained above in § 4.3). The WL tangential distortion, g T , vs. distance
125

WL Determination of the Distance-Redshift Relation
1
0.9
0.8
0.7
χ 2 /dof = 19/8
0.6
g T
0.5
0.4
0.3
0.2
0.1
0
1 2 5 10
θ [arcmin]
Figure 4.2 – Tangential distortion g T vs. distance from cluster center for A370 reference
background samples. Overlaid is the power-law fit (dashed black line) with 1−σ confidence
levels (shaded region) and the PL-fit χ 2 is indicated.
from cluster center, θ, is plotted in Fig. 4.2 (black crosses). The reference
background sample is fitted by a power-law according to Eq. 4.10, but only
in the WL regime, i.e., outside θ 1 ′ . The fit is estimated using two ways –
we fit the entire sample dataset, weighting each galaxy g T,i by u g,i (see § 4.4),
and we also fit the binned g T profile, 〈g T (θ n )〉, weighting by the bin error,
1/σ 2 T (θ n ). We find good consistency between the two fitting schemes. The
goodness-of-fit χ 2 values of the binned fit are displayed next to each powerlaw
fit, and also in Table 4.2. As can be seen, a simple power-law serves as
a reasonable fit in all cases.
For each of our defined samples (foreground, orange, green, red, blue
and dropouts) we again plot g T vs. radius in Figure 4.4 (top: foreground,
orange, green, red, blue bright samples, bottom: green, red, blue dropout
faint samples). We fit each sample with the same power-law index, b B , given
by its relevant reference background (the sample fit is shown as dashed black
line with 1 − σ confidence bounds, and the reference background fit is also
shown as a dotted line). The WL amplitude of each sample relative to the
reference background is given in each panel as Γ i ≡ a i /a B (as defined by
Eq. 4.12) and detailed in Table 4.2. We see that indeed, for the foreground
126

4.5 Results
n [arcmin −2 ]
10 1
10 0
θ [arcmin]
0 2 4 6 8 10 12 14 16 18 20
Figure 4.3 – Galaxy surface number density vs. radius for A370 foreground sample (gray
triangles), orange (orange diamonds), green (green pentagrams), red (red squares), blue
(blue circles) and dropout (magenta hexagrams) background samples.
samples, the g T (θ) profile agrees with zero throughout, and gives a relative
WL amplitude of zero.
As another consistency check, we plot the galaxy surface number density
vs. radius for A370 samples in Figure. 4.3. As can be seen, no clustering is
observed toward the center for any of the samples, which demonstrate that
there is no contamination by cluster members in the samples comprising
only background members. The foreground sample (gray triangles) shows
a modest increase in number density (factor of 2 increase from θ = 20 to
θ = 2) compared to the cluster, despite the exclusion of the cluster earlytype
galaxies by colour. Bluer later-type cluster members are to be expected
here given the redshift window sampled by reference to Figure 4.6, which
shows that the tail of the distribution reaches just beyond the redshift of
A370 § 4.5.2).
4.5.2 COSMOS photometric redshifts
To estimate the respective depths of the different samples defined above from
our Subaru photometry, we make use of the accurate photometric redshifts
127

WL Determination of the Distance-Redshift Relation
g T
g T
g T
g T
g T
1
0.5
0
1
0.5
0
1
0.5
0
1
0.5
0
1
0.5
0
θ [arcmin]
θ [arcmin]
θ [arcmin]
θ [arcmin]
1 2 5 10
θ [arcmin]
χ 2 Γ=−0.04±0.1
/dof = 13/6
χ 2 Γ=0.41±0.2
/dof = 1/6
χ 2 Γ=0.85±0.2
/dof = 11/6
χ 2 Γ=0.87±0.1
/dof = 17/8
χ 2 Γ=0.95±0.2
/dof = 6/8
g T
g T
g T
g T
1
0.5
0
1
0.5
0
1
0.5
0
1
0.5
0
θ [arcmin]
θ [arcmin]
θ [arcmin]
1 2 5 10
θ [arcmin]
χ 2 Γ=0.88±0.2
/dof = 6/7
χ 2 Γ=1±0.1
/dof = 27/8
χ 2 Γ=1.1±0.2
/dof = 23/8
χ 2 Γ=1.3±0.2
/dof = 13/8
Figure 4.4 – g T vs. cluster radius for A370 bright samples (top: foreground, orange,
green, red & blue) and faint (bottom: green, red, blue & dropouts), where the fixed
power-law fit is overlaid (dashed black line) with 1 − σ confidence levels (shaded region).
Also plotted is the power-law fit of the equivalent “reference” background sample (dotted
curve). In each case the resulting normalized g T amplitude ratio, Γ, is denoted next to
the profile.

4.5 Results
3.5
3
2.5
4
3.5
3
B − R
2
1.5
2.5
2
redshift z
1
0.5
0
−0.5 0 0.5 1 1.5 2
R − z
1.5
1
0.5
Figure 4.5 – Left: B J − R C vs. R C − z ′ CC diagram, showing the distribution of galaxies
in COSMOS field. Applying the same CC-cuts as our selection for A370, we define galaxy
samples: foreground (gray), orange (orange) ,green (green), red (red), blue (blue), and
dropout (magenta) galaxies, in order to estimate the mean redshifts and mean depths
of these samples using COSMOS photo-z’s. Right: Average COSMOS redshift in CC
bins. Overlaid are the boundaries of the foreground (thin dashed gray line), orange(thick
dotted-dashed orange line), green (thin solid green line), red (thick dashed red line), blue
(thin dotted-dashed blue line) and dropout (thick solid magenta line) samples selected in
the COSMOS field according to A370 CC-cuts. Evidently, different regions of CC space
correspond to different redshift populations of galaxies. Most notably, The top left corner
denotes dropout galaxies of z 3.5, corresponding well to our selection of highly distorted
dropout galaxies.

WL Determination of the Distance-Redshift Relation
derived for the well studied multi-band field survey, COSMOS (Capak et al.
2007). For COSMOS, photometric redshifts have been derived by Ilbert et al.
(2009) using 30 bands in the UV to mid-IR. Since the COSMOS photometry
does not cover the Subaru R C band, we estimate R C -band magnitudes for
it. For this we use the HyperZ (Bolzonella et al. 2000) template fitting code
to obtain the best-fitting spectral template for each galaxy, from which the
R C magnitude is derived with the transmission curve of the Subaru R C -band
filter (see Umetsu et al. 2010).
We then select samples by applying the same CC/magnitude limits as
we did above for A370. This is shown in Fig. 4.5 (left) for the COSMOS
catalog and plotted in terms of the same CC plane as A370. The color
distribution of COSMOS field galaxies seen in this B, R, z ′ CC-space is very
similar to that of A370, displaying the same morphology, including red, blue
and dropout populations, but without the density peak associated with the
massive cluster A370. We also show how redshift varies in this CC-plane by
calculating the mean photo-z redshift from COSMOS in fine bins over the CC
plane (Fig. 4.5, right), with the samples boundaries displayed as well. This
demonstrates that the main overdensity in the CC plane near B J − R C ∼ 1
and R C − z ′ ∼ 0.3 has a mean redshift of around z 0.5, which agrees with
our estimation for A370 where we found very little WL signal implying these
object lie predominantly in the foreground of the cluster. We also see from
this figure that the region where we picked “red” galaxies corresponds to
z ∼ 1 − 1.5, and the “blue” galaxies occupy a region of mean redshift around
z ∼ 2. Most notably, the top left corner of “dropout” galaxies corresponds
to high-z with z 3.5. We further plot the redshift distribution of all the
samples in Fig. 4.6. We calculate the median redshift of each sample and
summarized in Table 4.2.
However, if we look at the distribution of COSMOS redshifts of the
dropout sample (Fig. 4.7, red), we find it is somewhat double-peaked, with
most galaxies lying around z ∼ 3.5, but a significant fraction identified as
having z ∼ 0.4. Since we are certain most of the galaxies in this region are
in fact high redshift dropout galaxies, justified by the apparent high WL distortions
measured for these galaxies (see Fig. 4.4) and by reference to deep
130

4.5 Results
0.4
0.35
0.3
0.25
foreground
orange
green−b
green−f
red−b
red−f
blue−b
blue−f
drops
N(z)
0.2
0.15
0.1
0.05
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
z
Figure 4.6 – Redshift distribution of all A370 samples: foreground (dashed gray), orange
(dotted-dashed orange), green (green) bright (solid) and faint (dashed), red (red) bright
(solid) and faint (dashed), blue (blue) bright (solid) and faint (dashed), and dropout
(dotted-dashed magenta) using COSMOS photo-z’s.
Figure 4.7 – Redshift distribution of the dropout-selected sample using COSMOS photoz’s
(light red) and using GOODS-MUSIC photo-z’s (dark blue). The low-z peak seen more
notably from the COSMOS photo-z’s is most likely due to misclassified galaxy redshifts,
supported by the smaller numbers found when using the GOODS-MUSIC photo-z catalog.

WL Determination of the Distance-Redshift Relation
spectroscopic work in this rising plume of “dropout” galaxies (Steidel et al.
1999), we may conclude the low redshifts assigned to some of these galaxies
may be misclassified as low redshift early type/dusty galaxies in the photo-z
catalog of the COSMOS field. This is not surprising, since at faint magnitudes
the COSMOS photo-z have a relative high catastrophic failure rate
(Ilbert et al. 2009).
We further examine this issue using the somewhat deeper GOODS-MUSIC
catalog (Grazian et al. 2006; Santini et al. 2009), which has 15 bands, including
high quality ACS photometry (GOODS-S) and deep IRAC imaging
which is very helpful in reducing the outlier rate a high-z. Using the z ′ -band
selected GOODS-MUSIC photo-z, we derive a spectral classification for each
galaxy using a library of ∼ 200 PEGASE (Fioc & Rocca-Volmerange 1997)
templates very similar to that described in Grazian et al. (2006), as included
in the EAZY software (Brammer et al. 2009), and then calculate Subaru
magnitudes for all the galaxies using the BPZ (Benítez 2000) code.
By making the same CC selection, we plot the redshift distribution of the
same dropout sample (Fig. 4.7, blue), but here no low-z peak is observed,
and all galaxies in this region are estimated to have high redshifts, z 2.5.
This comparison between COSMOS and GOODS-MUSIC redshifts allows us
to securely set a conservative lower B − R limit to avoid inclusion of real
low-z objects. We can thus safely assume all objects identified as low-z in
the COSMOS catalog are largely mistakenly classified, a point also made in
relation to this by Schrabback et al. (2010). Estimating the median redshift
of the dropout sample gives z ≃ 3.8, a value very close to the mean redshift
of all galaxies lying above z > 1.5 galaxies. This further demonstrates the
low-z peak is a low-significance contamination. A further examination of the
photometric redshift estimation for such objects in the COSMOS field seems
worthwhile in view of these results.
4.5.3 Lensing strength dependence on magnitude
As another check, we plot Γ, the g T -amplitude ratio, vs. z ′ -band magnitude
for A370 selected samples, to see if there is any trend of the lensing amplitude
132

4.5 Results
Γ (g T
amplitude ratio)
2
1
0
−1
2
1
0
2
0
2
03
2
1
0
20 21 22 23 24 25 26
z’ [mag]
Figure 4.8 – Γ, the g T amplitude ratio, vs. magnitude (z ′ band) for A370 orange,
green, red, blue and dropout galaxy samples (top to bottom). Overlaid is the lensing
depth, D ds /D s , vs. magnitude calculated from COSMOS photo-z’s for each of the samples
(dashed black line) with 1 − σ confidence levels (shaded region).
with magnitude. We also plot the mean D ds /D s as a function of magnitude
(all values normalized to the reference background values) calculated in independent
magnitude bins for each sample using the COSMOS photo-z catalog.
This serves as a further consistency check. Interestingly, for the blue galaxies
(and the dropout galaxies to some extent), the mean signal seems to drop
slightly with fainter magnitudes. This trend is somewhat counter-intuitive,
since we expect that fainter galaxies will be at higher redshifts, and therefore
have on average a higher signal.
The diminished WL signal could possibly hint at a problem with estimating
the WL signal from faint blue galaxies, which are in general quite
irregular in morphology, and empirical simulations with higher space based
resolution can be made to examine this better. The declining trend of the
predicted D ds /D s in the case of blue galaxies, based on the COSMOS photo-z
estimation, shown in Fig. 4.8 (dashed curves), could possibly point to a missclassification
of blue galaxies with photo-z methods, a well known problem
133

WL Determination of the Distance-Redshift Relation
for blue galaxies, or even simply a limiting magnitude beyond which the
photo-z method fails in this catalog. We set our magnitude limits conservatively
to minimize these possible problems, with 19 < z ′ < 23 for the orange
sample, 20 < z ′ < 24.5 for the green sample, 23 < z ′ < 25 for the blue sample,
22 < z ′ < 26 for the red sample, and 24.5 < z ′ < 26.5 for the dropout
sample.
4.5.4 Lensing strength vs. redshift
We may now finally combine the WL distortion information with the redshift
information for all the samples of background galaxies defined above. In order
to compare the resulting trend with the cosmological trend, we plot Γ, the
g T -amplitude ratio, against median redshift, 〈z s 〉, in Fig. 4.9. Each point in
the figure represents an independent sample, where the first point represents
the foreground sample in front of the cluster, and the other points represent
background galaxy samples. For each sample, we also calculate the median
lensing distance ratio, 〈D ds /D s 〉, for each cosmological model – ΛCDM
(empty circles), Einstein-de Sitter (crosses), and an empty universe (empty
squares), using the COSMOS photo-z measurements of galaxies within the
same CC-magnitude boundaries defined for each background sample and thus
obtain the predicted depth. We interpolate between these discrete predicted
values to provide the theoretical relation, D ds /D s (z) for each cosmological
model. We can thus compare how the WL strength Γ agrees with predicted
D ds /D s . A clear trend of increasing WL amplitude is seen with redshift. A
null result of no distance-redshift increase is easily excluded with low significance
– χ 2 /N = 91/8. It is evident that the WL amplitude agrees well with
the theoretical relations for D ds /D s as a function of redshift, rising steeply
behind the lens and saturating at high redshifts, albeit currently consistent
with a wide range of cosmologies.
134

4.5 Results
1.6
1.4
1.2
Γ (g T
amplitude ratio)
1
0.8
0.6
0.4
0.2
0
ΛCDM; χ 2 /N=3.4/8
EdS; χ 2 /N=3.3/8
Ω M
=0,Ω Λ
=0; χ 2 /N=3.5/8
−0.2
0 0.5 1 1.5 2 2.5 3 3.5 4
Figure 4.9 – g T amplitude ratio, Γ, vs. redshift for A370 bright (stars) and faint (circles)
samples. Also plotted is the lensing depth, D ds /D s , vs. redshift for different cosmologies
- ΛCDM (circles+solid line), Einstein-de Sitter (crosses+dashed line), and an empty Universe
(squares+dashed-dotted line) estimated using the COSMOS photometric redshift
catalog.

WL Determination of the Distance-Redshift Relation
4.6 Constraining cosmological parameters
The clear detection here of the distance-redshift relation from our WL analysis
of A370, prompts the question of how many such clusters would be
required in order to provide a useful cosmological constraint. This issue has
been explored more generally in the context of planned field and cluster surveys
by Jain & Taylor (2003); Taylor et al. (2007); Kitching et al. (2007).
Taylor et al. (2007) present a detailed analysis of the sensitivity of cosmological
cluster surveys to the ratio of shear values measured in independent
redshift bins, finding that a large fraction of the potential integrated signal on
the sky is contributed by abundant small, cluster mass range (M ≈ 10 14 M ⊙ ),
but that a large contribution also comes from the largest clusters, like those
studied here.
Here we use an order-of-magnitude calculation to estimate how well our
newly approved MCT/CLASH 8
survey (P.I. M. Postman) can do in this
context, for which we aim to complete very high quality WL data for approximately
25 massive clusters, similar in quality to the B J R C z ′ imaging of
A370.
As defined above in Eq.(4.12), Γ, is a shear ratio statistic between any
two independent redshift bins summed over N cl clusters, given by:
Γ ij ≈ γ i
γ j
= r[χ(z j)]r[χ(z i ) − χ(z l )]
r[χ(z i )]r[χ(z j ) − χ(z l )] , (4.16)
where r = f K (χ) is the comoving angular diameter distance in the given
curvature K, χ(z) is the comoving distance and z l is the redshift of the
lens, and Γ scales with the dark energy equation of state parameter, w, as
Γ ≈ |w| −0.02 (Taylor et al. 2007). The fractional error on w is given by
∆w
w
= 2 ( ) −1 d ln Γ σ
√ e
, (4.17)
γ t d ln w Nb
where γ T is the typical mean tangential shear of each cluster, and σ e = 0.3 is
the measured intrinsic scatter in galaxy ellipticity per mode (KSB), and N b
8 http://www.stsci.edu/~postman/CLASH/
136

4.7 Discussion and conclusions
is the total number of galaxies summed up behind all the clusters co-added
for this purpose.
Assuming γ ≈ 0.05 and N b ≈ 1.25 × 10 6 , summed over the available
background for 25 clusters, (taking A370 as our guide to the number of
background galaxies detected per cluster) we find the expected precision on w
from our sample is ∆w ≈ 0.6. While this seems a relatively large uncertainty,
the first application of the method by Kitching et al. (2007) suggests that the
error distribution is non-Gaussian, with a rather sharp cut-off at high values
of w placing a relatively tight upper limit. Other geometric probes currently
do not succeed much better than this individually, e.g., from Baryon acoustic
oscillations and SN-Ia, ∆w ≈ 0.3. Furthermore, the shear-ratio test has a
different degeneracy with respect to the cosmological parameters to other
probes, making even a crude measurement worthwhile.
4.7 Discussion and conclusions
Using deep observational data, the dependence of the amplitude of WL with
source distance has been measured for the massive galaxy cluster A370, using
independent samples of background galaxies of differing depths with a visible
increasing trend. For A370 we have Subaru imaging of relatively high quality
in B, R and z ′ bands, allowing us to further subdivide these samples into
independent bright and faint populations. Still, small number statistics and
possible dilution, especially in the case of the dropout sample, may lead to
large uncertainties and underestimated values. Our photometry comprises
only three optical bands and so we do not rely on photo-z estimates, but
instead determine the depth of these background populations with reference
to the very well studied COSMOS and GOODS-MUSIC fields, by applying
CC and magnitude cuts equal to that of our background populations.
The trend of increasing WL-amplitude with redshift uncovered here follows
the expected form of the lensing distance-redshift relation but with
uncertainties presently too large to distinguish between cosmologies. The
137

WL Determination of the Distance-Redshift Relation
encouraging results found here merit further application of this approach to
a larger sample of clusters.
The recently approved MCT/CLASH program will observe 25 clusters
with HST ACS/WFC3, most of which have deep multi-color Subaru imaging,
so that we estimate a WL based precision on w of ∆w ≈ 0.6, but
with a different degeneracy relative to other probes, complementing existing
methods. Combining this WL estimate with the distance-redshift relation
from strong lensing will provide an enhanced geometric-based cosmological
constraint.
To extract the cosmological parameters from such accurate data will require
further refinement of the method. We must take account of the greater
mean depth that lensing magnification generates whose effect on cluster lensing
has been explored in some detail previously (Broadhurst et al. 1995), and
also a second order correction for the surface density described in § 4.4.1.
These effects, although small in terms of the lensing amplitude, are comparable
with the relatively small differences of interest between competing
cosmologies and thus must be explored in any serious study of cosmology
with this method.
138

Chapter 5
Discussion
5.1 Summary
In this thesis, I presented a weak lensing study of four massive galaxy clusters.
I combined observations from the ground-based telescope Subaru, and also
from space-based HST /ACS observations for some of the cluster centers.
I developed and employed new techniques to determine cluster mass and
light properties based on the weak lensing effect measured from background
galaxies, and also from the weak lensing dilution caused by including cluster
galaxies, in a new and innovative way. The deep and wide multi-band data
were used to determine accurate cluster mass and light profiles. With these
detailed cluster profiles, I was able to make useful constraints on cosmological
predictions of structure formation models, and better examine the physics.
cluster environment.
In Chapter 2, I showed how we utilized the color-magnitude diagram and
the weak lensing signal to derive the luminous properties of cluster galaxies.
We used the E/S0 sequence of a cluster to obtain a reliable weak lensing signal
from background galaxies lying redward of the sequence, uncontaminated by
cluster members. For bluer colors, both background and cluster members are
present, reducing the distortion signal by the proportion of unlensed cluster
members. The new method we have devised relies on this dilution of the
139

Discussion
lensing distortion signal by unlensed cluster members was employed in turn
to measure the cluster light. It was applied to deep Subaru and HST /ACS
imaging of A1689, a massive galaxy cluster at a relatively low redshift (z =
0.183), which is a known strong lens. We derived radial light profile of the
cluster, a color profile and radial luminosity functions more reliably than
the usual approach of simply subtracting background counts. the data, r <
2 h −1 Mpc, with a constant slope, d log(L)/d log(r) = −1.12±0.06, unlike the
lensing mass profile which steepens continuously with radius. We derived the
radial profile of the M/L ratio, which has not been previously constrained
in such detail, due to the difficulty of measuring mass and light profiles with
such accuracy and free of systematic effects. We found that M/L peaks
at an intermediate radius, ∼ 100 h −1 kpc, with a somewhat high value of
M/L B ∼ 400h(M/L) ⊙ , and then falls at larger radius. The cluster luminosity
function was found to have a flat slope, α = −1.05 ± 0.05, independent of
radius and with no faint upturn to M i ′ < −12 as claimed in previous work.
We established that the very bluest objects are negligibly contaminated by
the cluster (V − i ′ < 0.2), because their distortion profile rises towards the
center following the red background, but offset higher by ≃ 20%, consistent
with the greater estimated depth of the faint blue galaxies, 〈z〉 ∼ 2 compared
to 〈z〉 ∼ 0.85 for the red background. Finally, we improved upon our earlier
mass profile by combining both the red and blue background populations,
clearly excluding low-concentration CDM profiles.
In Chapter 3 I further employed my methods to explore three massive
clusters where excellent multi-band data are available. We discovered that
the color-color space can be used to better separate cluster members from the
foreground and background populations behind clusters A1703 (z = 0.258),
A370 (z = 0.375) and RXJ1347-11 (z = 0.451) imaged with Subaru. The
clustering of cluster members reveals that they are confined to a distinct
region of CC space. A reliable background sample for weak lensing measurements
was obtained by selecting red background galaxies from the CC
diagram whose weak lensing signal is undiluted, and adding to those faint
blue galaxies, whose weak lensing signal is in good agreement with the red
galaxies. We avoided dilution of the weak lensing signal by identifying and
140

5.1 Summary
excluding the foreground population in the CC diagram for which no clustering
or weak lensing signal is detected. The combined red and blue samples
were used to reconstruct accurate mass maps of all four clusters in a subsequent
paper (Broadhurst et al. 2008). The luminosity functions of these
clusters, when corrected for dilution, showed similar faint-end slopes to what
we found for A1689, α ≃ −1.0, with still no marked faint-end upturn to our
limit of M R ≃ −15.0, and only a mild radial gradient of the slope. claims in
the literature for a distinctive upturn at faint magnitudes was thus clearly
excluded, and may represent uncertain subtraction of the steep background
counts of faint galaxies when evaluating the background using control fields
Interestingly, here we also find that all our clusters show a very similar radial
trend of M/L, peaking at an intermediate radius, ≃ 0.2r vir , to a value of
M/L R ∼ 500 h(M/L) ⊙ and declining outward to a typical field value. The
radial trend of galaxy colors is very mild and falls short of explaining the observed
radial trend of M/L. We concluded this behavior is likely due to the
relative paucity of central late-type galaxies, whereas for the E/S0-sequence
only a mild radial decline in M/L is found for each cluster. This may stem
from disk galaxies having their stellar content more susceptible to stripping
processes than early-type galaxies, whose stars are more tightly bound.
In Chapter 4 I explored the scaling of the weak lensing signal with source
redshift. Using three-color Subaru imaging of A370 we selected several independent
samples of galaxies of differing depths (0.6 < z < 3.8) in the
background of the cluster. Their weak lensing amplitude vs. redshift was
measured and compared to the lensing distance-redshift relation. We defined
the depth of lensed populations with reference to the COSMOS and GOODS
fields, providing a consistency check of photo-z estimates over a wide range
of redshifts and magnitudes. The predicted distance-redshift relation is followed
well for a wide range of cosmologies, but currently without the ability
to make clear distinction using results from only one cluster due to large uncertainties.
We estimated that with a dataset as deep as that of A370, scaled
to our new survey of ∼ 25 massive clusters, we should be able to provide a
useful cosmological constraint on w, of about ∆w ≈ 0.6, complementing existing
techniques, with distance measurements covering an untested redshift
141

Discussion
range, 1 < z < 5.
5.2 Further application of the methods in other
work
By applying my methods to carefully separate and select background galaxies
for a weak lensing study, we measured distortion effects in all four massive
clusters studied in my work, A1689, A1703, A370 & RXJ1347, and derived
the most accurate published mass profiles to date (Broadhurst et al. 2008).
In producing these profiles, we combined the strong lensing information obtained
from sets of multiple images found in high-quality HST/ACS imaging,
with the weak lensing distortion to background galaxies in deep, wide-field,
multi-band Subaru/Suprime-Cam imaging. The mass profiles all agree with
the predicted form for CDM-dominated halos, with continuously steepening
profiles, consistent with the expectation that DM is cold. In detail, however,
these profiles are somewhat more concentrated than predicted, 8 < c vir < 15
(< c vir >= 10.39 ± 0.91). If this behavior proves to be typical it will require
modification of the underlying model.
In (Broadhurst et al. 2008) we also examined the lensing magnification
effect, seen as a depletion in the surface number density profiles of red background
galaxies near the cluster center. The measured depletion compared
to the expected depletion for the best-fitting NFW density profile (derived
from the corresponding distortion profiles) were found to be consistent, which
strengthens our conclusions and more importantly establishes the utility of
the background red galaxies for measuring magnification. By combining
magnification and distortion we improve upon the accuracy of lensing mass
determinations.
Further implementation of the color-color selection method was used to
study the mass structure of the galaxy cluster Cl0024+17 in Umetsu et al.
(2009) and in Zitrin et al. (2009b). Based on deep Subaru BR c z ′ imaging
and our recent comprehensive strong lensing analysis of HST/ACS/NIC3
142

5.3 Future work
observations we obtained strong+weak lensing distortion and magnification
profiles. Unlike previous lensing studies of this cluster, the weak and strong
lensing were found to be in excellent agreement where the data overlap.
This together with the anomalously low X-ray emission and complex internal
dynamics strongly indicate that this cluster system is in a post collision state,
which is consistent with our mass profile for a major merger occurring along
the line of sight, viewed approximately 2 − 3 Gyr after impact when the
gravitational potential has had time to relax in the center, but before the
outer cluster region is virialized.
5.3 Future work
I plan to continue and apply our robust approach to other X-ray (e.g., Ebeling
et al. 2001, MACS) and SZ (e.g., Staniszewski et al. 2009, SPT; Geisbüsch
et al. 2005, Planck) selected clusters. Such samples are arguably less prone
to selection-biases than optically-selected, and therefore may provide a more
secure determination of the concentration-mass relation for comparison with
theoretical ΛCDM predictions.
I have so far obtained and reduced data from the Subaru/Suprime-Cam
archive, consisting of seven MACS cluster in multiple bands - B J , V J , R C , I C , z ′ ,
and at least a dozen more MACS clusters are available in the archive. Highresolution
imaging of the central parts of the clusters will be required from
space in order to determine the strong lensing properties of the cores.
An integral part of this study will be to also determine the light properties
of these clusters using the dilution method I have developed. I intend to
further explore the declining M/L characteristic by examining the connection
between morphology and the color-color plane. To that end higher resolution
imaging and spectroscopy of close-by (z < 0.1) clusters will aid in identifying
cluster galaxy morphologies.
143

Discussion
5.3.1 MCT/CLASH
Our recently approved multi-cycle treasury (MCT) HST program CLASH
will accurately measure the distribution of mass profiles from 10 kpc to beyond
2 Mpc for an unbiased and representative sample of 25 relaxed galaxy
clusters observed with 525 orbits to 1) definitively derive the representative
equilibrium mass profile shape, and 2) robustly measure the cluster DM mass
concentrations and their dispersion as a function of cluster mass and their
evolution with redshift.
We have selected our clusters to be relaxed on the basis of their symmetric
and smooth X-ray emission (Allen et al. 2004) with no lensing information
used a-priori. This selection ensures that this sample of clusters is
not preferentially aligned along the line of sight, in contrast with a purely
lensing-selected sample, where the surface mass density is, on average, biased
upward along the line of sight by intrinsic triaxiality (Oguri et al. 2005; Hennawi
et al. 2007). We will match the characteristics of our observed cluster
sample with those from simulations of relaxed DM halos (e.g., Neto et al.
2007). Our study will, for the first time, place statistically significant limits
on the distribution of the parameters of the DM mass profile.
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156

כפי שמתואר בפרק השלישי,‏ העבודה הורחבה לעוד שלושה צבירים,‏
, A1703 z=0.258
RXJ1347−11 z=0.451 ו-‏ , A370 z=0.375
שצולמו עם סובארו בלפחות שלושה פילטרים.‏
עם הנתונים מרובי הפילטרים הזמינים כאן,‏ השתמשתי בתרשים צבע-צבע כדי להדגים כיצד ניתן
לרתום את העידוש על מנת להפריד את הגלקסיות שבצביר מהאוכלוסיות שבחזית וברקע באופן
בטוח יותר,‏ ולזהות כמה אוכלוסיות רקע בעלות אופי שונה למדידת העידוש החלש.‏ לפונקציות
הבהירות של כל הצבירים הללו,‏ לאחר תיקון עבור דילול,‏ יש שיפוע דומה המתאר את התפלגות
≃−1.0
הגלקסיות החיוורות,‏ α, ללא השתנות מעלה בקצה החיוור עד גבול בהירות הנתונים,‏
, אבל עם שינוי רדיאלי קל.‏ גם כאן,‏ לכל אחד מהצבירים שלנו פרופיל יחס המסה
M R
≃−15.0
לבהירות M/L מגיע לשיא ברדיוס אמצעי,‏
r≃0.2r vir
, ברמה של
, 300−500 M / L R
o
M / L~100 M / L R
o
ואז נופל
ברדיוס הויריאלי,‏ ערך השדה הממוצע.‏ התנהגות זו
בהתמדה לעבר ערך של
מחסור אינטרינזי קרוב למרכז של גלקסיות מסוג מאוחר יחסית לסוג
נובעת ככל הנראה בשל הקדום יותר,‏ ואילו עבור רצף ה-‏ E/S0 רק ירידה קלה עם הרדיוס M/L נצפית עבור כל צביר.‏ זו
יכולה להיות מוסברת על ידי המרה מתמשכת של גלקסיות דיסק לגלקסיות S0, דרך תהליכים של
הסרת הגז וכוחות גיאות.‏ אנחנו דנים בהתנהגות זו בהקשר של סימולציות מפורטות בהן תחזיות
לקריעה באמצעות כוחות גאות ניתנות כעת לבדיקה באופן מדויק עם תצפיות.‏
המשרעת של העידוש החלש עולה עם היסט לאדום של המקור,‏ ופרופורציונאלית ליחס מרחקי
העידוש,‏
D ds
/ D s
, העולה בתלילות מאחורי העדשה ומגיע לרוויה בהיסט לאדום גדול.‏ יחס זה
מספק,‏ באופן עקרוני,‏ דרך עצמאית למדידת פרמטרים קוסמולוגיים.‏ באמצעות התצפיות מסובארו
של A370, אני חוקרת קשר זה בפרק הרביעי.‏ על ידי יישום שיטות הבחירה שפיתחתי אנחנו
מפרידים ובוחרים מתוך דיאגרמת צבע-צבע עוד אוכלוסיות רקע במרחקים שונים )
,( 0.6z3.8
כולל מדגם בולט של גלקסיות ‏"נשירה”‏ רחוקות,‏ עם חפיפה מועטה בין הקבוצות של ההיסט
לאדום.‏ אנו מודדים במדויק את משרעת העידוש שלהם,‏ תוך שאנו לוקחים בחשבון את השינוי
הרדיאלי של פרופיל העידוש החלש.‏ אנחנו מגדירים את העומקים של האוכלוסיות המעודשות
ביחס לסקרי השדה COSMOS ו-‏GOODS‏.‏ יחס המרחק-היסט לאדום החזוי מאושש עבור מגוון רחב
של מודלים קוסמולוגים,‏ אם כי ללא יכולת להבחין בין מודלים באמצעות צביר אחד בלבד.‏ התאמת
תוצאה זו לסקר CLASH החדש והמאושר שנערוך באמצעות טלסקופ החלל ‏"האבל"‏ עבור כ-‏‎25‎
צבירים רגועים,‏ צפויה לספק חסם קוסמולוגי שימושי על פרמטר משוואת המצב של האנרגיה
האפלה,‏ w, שישלים את הטכניקות הקיימות,‏ עם מדידות מרחק שמכסות טווח ההיסט לאדום
שטרם נבדק,‏
. 1z5

תקציר
צבירי גלקסיות הם המבנים הקשורים כבידתית המאסיביים ביותר ביקום.‏ בורות הפוטנציאל
העמוקים שלהם מסיטים את האור של גלקסיות ברקע ויוצרים תמונות מעודשות של הגלקסיות.‏
עידוש כבידתי יכול לשמש ככלי אבחון כדי לקבוע את פרופילי המסה של צבירים.‏ בתחום העידוש
החלש,‏ המדידה של מסת הצביר יכולה להיפגע מאוד מנוכחות של גלקסיות בצביר ובחזיתו שאינן
מעודשות,‏ הגורמות לדילול האות האמיתי מהעידוש.‏ בתיזה זו,‏ אני חוקרת את מאפייני המסה והאור
של צבירי גלקסיות באמצעות תצפיות של עידוש כבידתי חלש . בחלק הראשון,‏ אני מראה את
החשיבות של בידוד גלקסיות הרקע מגלקסיות בחזית ובצביר למען מדידה נקייה של העידוש
החלש,‏ על מנת למנוע את דילול האות.‏ הדילול עצמו משמש כדי לקבוע את תכונות האור של
הצבירים.‏ המוקד השני של העבודה מראה את השימוש במדידות המפורטות כדי לקבוע את
התגברות אות העידוש החלש עם ההיסט לאדום,‏ שיכול לשמש כבחן קוסמולוגי חשוב.‏
לאחר מבוא לתחום צבירי הגלקסיות ועידוש כבידתי בפרק הראשון,‏ אני מדגימה את השיטות
והתוצאות שקיבלתי עבור ארבעה צבירי גלקסיות מאסיביים בפרקים 2 ו-‏‎3‎‏.‏ בפרק השני אני חוקרת
את הצביר
A1689 z=0.183
מתצפיות אופטיות עמוקות מהטלסקופ סובארו בפילטרים
, V ,i '
באמצעות תרשים הבהירות-צבע ואות העידוש החלש בכדי למצוא את תכונות האור של צביר
הגלקסיות.‏ העיוות המשיקי של גלקסיות הכחולות יותר מאשר קו רצף גלקסיות הצביר (E/S0) נופל
במהירות לכיוון מרכז הצביר ביחס לאות העידוש של גלקסיות הרקע האדומות יותר מהרצף.‏ אנו
משתמשים באפקט הדילול להפיק את פרופיל האור של הצביר ואת פונקציית התפלגות בהירות
גלקסיות הצביר עד לרדיוס גדול,‏ כשהיתרון של שיטה זו הוא כי לא נדרש חיסור של מניות הרקע
משדה רחוק.‏ לפרופיל האור יש נטיית דעיכה קבועה,‏
של הנתונים,‏
באמצע,‏
dlog L/dlogr=−1.12±0.06
rh −1 Mpc
, ואילו פרופיל המסה נהיה תלול עם הרדיוס,‏ כך ש
M/L
r~100h −1 kpc
הגלקסיות החיוורות,‏
בהירות
עד לגבול
מגיע לשיא
. לפונקציית הבהירות של הצביר יש שיפוע שטוח המתאר את התפלגות
=−1.05±0.05 ,α בלתי
M i '
−12
גלקסיות הצביר )
תלוי ברדיוס וללא התגברות בתחום החיוור עד
. אנחנו מראים כי עצמים כחולים מאוד מזוהמים במידה אפסית על ידי
V −i '0.2
). משילוב של שתי האוכלוסיות – גלקסיות הרקע האדומות
והכחולות – אנו מקבלים מדידה משופרת של פרופיל המסה כמו גם אילוצים על פרמטר ריכוזיות
הצביר,‏ כשאנו שוללים בבירור פרופילי חומר אפל פחות מרוכזים.‏

עבודה זו נעשתה בהנחייתם של
ד"ר טום ברודהרסט
ופרופ'‏ יואל רפאלי
בית הספר לפיזיקה ואסטרונומיה
אוניברסיטת תל­אביב

התפלגות המסה והאור
בצבירי גלקסיות
חיבור לשם קבלת התואר ‏"דוקטור לפילוסופיה"‏
מאת
אלינור מידזינסקי
הוגש לסנאט של אוניברסיטת תל­אביב
תשרי תשע"א