Mapping Dark Matter and Dark Energy

by

Anna Kathinka Dalland Evans

Submitted
in partial fulfilment of the requirements
for the degree of

Philosophiae Doctor

Institute of Theoretical Astrophysics

Faculty of Mathematics and Natural Sciences
University of Oslo

Oslo, Norway March, 2009

Preface

In science one tries to tell people, in such a way as to be understood by everyone, something
that no one never knew before. But in poetry, it’s the exact opposite.
Paul Dirac

In astronomy, we learn about the universe by studying the light that is emitted
from distant stars and galaxies. Light rays travel through the far reaches of
space until they arrive here on Earth and are recorded by our telescopes and
other observation equipment. The light we observe can give information about
objects located millions of light years away, but it also contains information about
what lies between us and these objects. Whether it be interstellar dust or strong
gravitational fields, the light rays are affected and carry the signatures back for
us to unscramble. As a twist of fate, it appears that most of what is out there is
actually invisible, or dark.
Chapter 1 gives a brief introduction to cosmology and the general physical frame-
work within which research for this thesis has been conducted, the so-called stan-
dard - or concordance - cosmological model. This is the simplest model which is
in general agreement with observations. However, the two major constituents of
the standard model remain enigmatic. Cosmologists call them dark matter and
dark energy, and in this thesis we are concerned with investigations into both of
these most mysterious parts of the Universe.
The focus of Chapter 2 is the dark matter. How do we know that it exists
and what do we actually know about it so far? In Chapter 3 I describe some
important techniques for mapping out the dark matter distribution at different
scales, based on a natural occurring phenomenon known as gravitational lensing.
Specifically, the shapes and possible ellipticities of galaxy cluster dark matter
halos are discussed, as observed in data from galaxy surveys (paper I) and from
large-scale simulations (paper II).
In Chapter 4 I give an introduction to the other – and perhaps even more myste-
rious – dark component of the Universe: dark energy. Many different models of
this largely unknown but major contributor to the energy budget of the Universe
have been suggested. As one way of classifying dark energy models, a set of
so-called statefinder parameters were introduced in 2004. In paper III we inves-
tigate the usefulness of these parameters by applying them to real and simulated
supernovae observations.

i
Acknowledgements

I would like to thank my supervisor, prof. Øystein Elgarøy, for the opportunity to have
all this fun doing research while actually getting paid for it, and for his bravery in getting
me started (and forcing me to finish!) on a topic which was partly new territory for him,
but very exciting to me.
My biggest, biggest thanks to the truly amazing dr. Sarah Bridle who has spent an
enormous amount of time teaching me how to do science in general and weak lensing
in particular. Your dedication and humour have been inspiring and your enthusiasm
contagious. And thank you for letting me bombard you with frantic e-mails, you’re a
star!
I spent the autumn of 2005 at the Astrophysics Group at University College London, and
would like to thank prof. Ofer Lahav for making this visit possible. Many thanks to all
the people in this group for making my stay not only productive but also extremely enjoy-
able: special thanks to drs. Rassat, Lintott, Dunkley and Bridle for cosmo-conversations,
dinners, concerts at the Barbican, haggis(!) and whisky tastings.
A ton of thanks to dr. Laurie Shaw, who not only provided excellent large-scale simu-
lations but also many interesting discussions as well as a huge effort in speed-finishing
paper II.
I would also like to thank all my office (in)mates at the Institute of Theoretical Astro-
physics (ITA) in rooms 502 and 201 during the long haul that working on this thesis has
sometimes been. Special thanks are due to dr. Morad Amarzguioui who was the very
best of room companions before he unfortunately finished his phd and was lost to the
world of industry.
Also to everyone eating lunch in the ITA kitchen at 12.30: thank you for many many
diverting conversations on important subjects such as which side of a crispbread is up,
whether glass is a liquid or a solid, and the assumed popularity of ice cream with goat
cheese bits in it.
From the bottom of my soul, thanks to all the beautiful, hard-working people in my
daughter’s kindergarten who allow me to relax and concentrate on a daily basis! And to
Nerina for providing me with chocolate in the crazy final stages of writing.
My most personal thanks to my husband Terje who takes care of me, persuades IDL to
work and makes life fun, and to our wonderful daughter Ronja, who reminds me every
day that funny-looking stones and deep puddles are more important things in life than
a phd.

iii
List of papers

Paper I: Evans, A. K. D., Bridle, S.,

Detection of dark matter halo ellipticity from lensing by galaxy clusters in SDSS,
Astrophysical Journal 695 (2009), 1446-1456

Paper II: Evans, A. K. D., Shaw, L., Bridle, S.,

Predictions of dark matter halo ellipticities measured by lensing,
To be submitted to Astrophyscial Journal (Ap.J.)

Paper III: Evans, A. K. D., Wehus, I. K., Grøn, Ø., Elgarøy, Ø.,
Geometrical constraints on dark energy,
Astronomy and Astrophysics, 430 (2005), 399-410

v
Contents

Preface i

Acknowledgements iii

List of papers v

I Introduction 1

1 Cosmological Framework 3
1.1 A very brief history of cosmology . . . . . . . . . . . . . . . . . . 3
1.2 The concordance model . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Invisible Universe I:
Dark Matter 9
2.1 Evidence for dark matter . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Different types of dark matter . . . . . . . . . . . . . . . . . . . . 11

3 Probing the unseen with gravitational lensing 13

3.1 Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Basic lensing theory . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Weak lensing by gravitational clusters (paper I) . . . . . . . . . . 22
3.3.1 The glow that illuminates: Ellipticity and shear . . . . . . 24
3.3.2 The glare that obscures: Systematic errors . . . . . . . . 28
3.4 Predictions of weak lensing from N -body simulations (paper II) . 29

4 Invisible Universe II:

Dark Energy 31
4.1 Evidence for dark energy . . . . . . . . . . . . . . . . . . . . . . . 31

vii
viii

4.2 Cosmological models with dark energy . . . . . . . . . . . . . . . 33

4.3 Investigating the statefinder parameters
(paper III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Bibliography 44

II Papers 45
 Part I

Introduction

1
Chapter 1

Cosmological Framework

1.1 A very brief history of cosmology

On a philosophical level, cosmology is a field in which researchers’ ultimate aims

are to answer the fundamental questions: Where do we come from? How did it
all begin? How will it all end?
In ancient times, people looked to philosophers, historians and story-tellers when
they wanted to know the workings of the world. Throughout the Middle Ages,
the appeal to the authority of Aristotle was the answer to questions of why
natural phenomena occurred. Why does a stone fall to the ground? Because
it is attracted to its natural element and wants to return to it. Why does a
feather fall more slowly than a stone? Because its natural element is the air.
Aristotelian views, such as the final cause, telos, of things (the final cause of a
pen is to produce good writing), lasted long into the Renaissance.
Towards the end of the Renaissance there was a period of great activity and
change in the natural sciences; Copernicus, Kepler, Newton and Galileo all lived
and worked in this time period, which came to be known as ‘the scientific revo-
lution’ (16th–17th centuries). It was a time which dramatically changed the way
we look at problem-solving.
One of the main outcomes of the Scientific Revolution was to supplant Aristo-
tle’s final cause with a mechanical philosophy; that all natural phenomena can
be explained by physical causes. Gradually, the Sun replaced the Earth as the
centre of the Universe. The idea that heavy bodies by their nature move down
while lighter bodies naturally move up, was replaced by the idea that all bodies
– both on Earth and in the heavens – move according to the same physical laws.
The concept that a continued force is necessary to keep a body on motion, was
abandoned by the inertial concept that motion, once started, continues indefi-
nitely unless acted on by another force. For a brief but comprehensive review,
see [1].
Yet the most remarkable development of this time was not a specific discovery
but a revolutionary new method of investigation. The new scientific method
focussed on mathematics as a language in which to formulate the laws of physics,
and empirical research as a reference frame from which to make hypotheses and

3
4

conduct experiments. Experimental outcomes replaced Aristotle’s teachings as

the new authority for truth.
Cosmology is the astrophysical study of the history, structure, and dynamics
of the Universe. Hence it was, for much longer than other branches of physics
and astronomy, a field more dominated by speculation and conjecture than by
observations and experiments. There were few possibilities for the accuracy of
tests and predictions starting to emerge in other disciplines in science laboratories
around the world. In very recent years, however, cosmology too has entered the
world of precision sciences.
In November 1989, NASA launched the satellite COBE (COsmic Background Ex-
plorer) to investigate the already detected [2] cosmic microwave radiation (CMB)
predicted by the hot Big Bang theory. COBE confirmed that the CMB spectrum
is that of a nearly perfect blackbody with a temperature of 2.725 ± 0.002K.
Subsequently, COBE data revealed tiny variations of temperature in different
directions on the sky, at a level of one part in 100 000. These variations offer im-
portant clues to early structure formation in the Universe. For these discoveries,
John C. Mather and George F. Smoot were awarded the Nobel Prize in Physics
in 2006. The Nobel committee wrote in a public statement [3] that “[. . . ] the
COBE-project can also be regarded as the starting point for cosmology as a preci-
sion science: For the first time cosmological calculations [. . . ] could be compared
with data from real measurements. This makes modern cosmology a true science
[. . . ] ”.
In the 20 years that have passed since the launch of COBE, cosmology has indeed
grown to a true precision science that now places strong boundary conditions on
cosmological models.
In this chapter I will give a very brief presentation of the cosmological framework
that is now considered the standard concordance cosmological model, and within
which the work of this thesis has been conducted.

1.2 The concordance model

concord —
From The Concise Oxford Dictionary
noun
1 agreement or harmony between people or things.

The Big Bang theory is today the sovereign cosmological model for the initial
conditions and subsequent evolution of the Universe. Ironically, its now famous
nickname originated by a man who spent a great deal of his life supporting
the once competing steady state model. Sir Fred Hoyle (1915-2001) unwittingly
coined the name in a radio broadcast in 1949, when he stated that current evi-
dence was in conflict with theories that required all matter to have been created
in ‘one Big Bang’ [4].
The Λ-Cold Dark Matter (ΛCDM) model is often referred to as the standard,
or concordance, model in Big Bang cosmology. It is the simplest model that is
in general agreement with observations. Based on overwhelming corroborating
evidence from many different types of observations, it is believed that this model
 5

describes the general characteristics and evolution of the Universe very well.
The Greek letter Λ stands for the cosmological constant, which is the dark en-
ergy term believed to be causing the expansion of the Universe to accelerate.
The idea that the Universe is accelerating became generally accepted after 1998,
when studies of large data sets of supernovae type Ia (SNIa) indicating this were
published by two independent groups [5, 6]. Cold dark matter denotes that most
of the matter in this model of the Universe is of a type that cannot be observed by
its electromagnetic radiation. In addition to being dark, this matter component
is non-relativistic (i.e. cold). In Chapters 2 and 4 I will describe the dark matter
and dark energy in a little more detail.
In the ΛCDM model, the Universe is spatially flat, meaning that it has an energy
density equal to the critical density. The critical density today is given by

3H02
ρcrit,0 = . (1.1)
8πG

The subscript ‘0’ indicates measurements of the quantity today, as opposed to at

earlier times (i.e higher redshifts) in the history of the Universe. H0 is the value
Hubble parameter today, recent measurements giving H0 ∼ 72 ± 3 km s−1 Mpc−1
[7]. The critical density today is equivalent to a mass density of 9.9×10−30 g/cm3 ,
which is equivalent to only ∼ 6 protons per cubic meter [7].
Based on observational evidence, there is currently general agreement in the cos-
mological community that our Universe closely resembles the ΛCDM model. The
COBE results [8] provided increased support for the Big Bang scenario in which
the universe has expanded from a primordial hot and dense initial condition at
some finite time in the past, and continues to expand to this day. The Big Bang
is the only theory that predicts the kind of cosmic microwave background ra-
diation measured by COBE. Anisotropies in the cosmic microwave background
temperature were later measured to greater precision with data from the Wilkin-
son Microwave Anisotropy Probe (WMAP) satellite [9] with the conclusion that
the Universe is flat with only a 2% margin of error. From this it follows that the
density of the Universe equals the critical density given by Eq. 1.1.
Several probes of the large-scale matter distribution such as the Sloan Digital
Sky Survey (SDSS) [10] show that the contribution of standard sources of en-
ergy density, whether luminous or dark, is only a fraction of the critical density.
Combinations of the CMB and large-scale structure show that an extra unknown
component, referred to as dark energy, is needed to explain the observations
[11, 12]. The dark energy is believed to cause the observed acceleration of the
expansion of the Universe.
For completeness, I mention here that the concordance model has 6 basic pa-
rameters [13]. The Hubble parameter H0 determines the rate of expansion of
the universe. Density parameters for baryons, dark matter and dark energy are
given as Ωb , Ωm and ΩΛ , which are ratios of these densities to the critical den-
sity: Ωm = ρm /ρcrit etc. Because the ΛCDM model assumes a flat Universe, the
density of dark energy is not a free parameter but is related to the other densities
by
ΩΛ = 1 − Ωb + Ωm . (1.2)
6

The optical depth τ to reionization determines the redshift, and hence the time,
when the Universe was reionized due to the formation of stars and galaxies that
radiated with enough energy to ionize neutral Hydrogen. Information about
density fluctuations is determined by the amplitude of the primordial fluctuations
from cosmic inflation As and the spectral index ns , which measures how the
fluctuations change with scale (ns = 1 corresponds to a scale-invariant spectrum).
Although the concordance model gives a consistent description of the dynamics
of the Universe, two large mysteries remain. The ordinary (baryonic) matter
that makes up our bodies, everything we see around us, the planets and stars,
seems to be a very small fraction of the total energy density in the Universe. The
concordance model seems to require the presence of about five times as much
dark matter (see Chapter 2) and a resounding fifteen times more dark energy
(see Chapter 4). Understanding these two major components of the Universe is
the primary goal of cosmology today.
Extensions of the simplest ΛCDM model allow the dark energy component to
vary with time. It is then sometimes referred to as quintessence, borrowed from
the ancient Greeks who used the term to describe a mysterious ‘fifth element’ -
in addition to air, earth, fire and water.

1.2.1 Theory

Einstein’s field equations form the basis for our interpretation of the Universe:

1 8πG
Rµν − gµν R = 4 Tµν (1.3)
2 c

The left-hand side of Eq. 1.3 expresses geometry by means of the Ricci tensor
Rµν and the Ricci scalar R. These are given by the form of the metric of the
space-time gµν . The right-hand side of Eq. 1.3 corresponds to the matter content
of the Universe, specified by the energy-momentum tensor Tµν .
To apply Einsteins general relativity to the dynamics of the Universe as a whole,
we need to make an assumption about how matter is distributed. The sim-
plest assumption is that the matter distribution in the Universe is homogeneous
(i.e. the same everywhere) and isotropic (i.e. the same in all directions) when
averaged over very large scales. This is called the Cosmological Principle. In a
homogeneous and isotropic Universe filled with one or more perfect fluids, we can
describe the metric through the Robertson-Walker line element which defines the
distance between two infinitesimally separated points:

ds2 = −dt2 + a2 (t) [ dω 2 + fK (ω)2 (dθ2 + sin2 θ dφ2 ) ] , (1.4)

where units have been chosen so that c = 1. (ω, θ, φ) are comoving spherical
polar coordinates, ω is the comoving radial distance, and fK (ω) depends on the
curvature K of the Universe. For a flat model, fK (ω) = ω. The scale factor
a(t) gives the rate of expansion of the Universe and is expressed in terms of the
redshift as
a
= (1 + z)−1 . (1.5)
a0
 7

That is, if we observe at a redshift of e.g. z = 1, the scale factor at that time was
only half of the value it has today.
From Einstein’s field equation for a homogeneous and isotropic Universe we can
derive the Friedmann equations for the time derivatives of the scale factor:

ä 4πG  
= − ρ + 3p (1.6)
a 3
 ȧ 2 8πG k
H2 = = ρ− 2 . (1.7)
a 3 a

The density ρ is the sum of all contributing parts (baryons, radiation and cos-
mological constant-term):

ρ = ρm + ρrad + ρΛ . (1.8)

To produce acceleration, we must have ä > 0. For Eq. 1.6 this means that
ρ
p<− . (1.9)
3
Since density is a positive quantity, this means that the dark energy component
must have negative pressure. In terms of the equation of state,
p
w= , (1.10)
ρ
we can write the general density development as

ρ ∝ a−3(1+w) , (1.11)

where the value of w will depend upon the component of the Universe. For
baryonic matter, the pressure is essentially zero giving w = 0 ⇒ ρm ∝ a−3 . For
radiation, w = 1/3 and ρrad ∝ a−4 . For a cosmological constant, w = −1 giving
ρΛ = constant. So for baryonic matter and radiation, the density decreases with
time but the density of the cosmological constant-term remains unchanged with
time.
Chapter 2

Invisible Universe I:
Dark Matter

The concordance model described in Chapter 1 gives a consistent description of

the dynamics of the Universe that agree well with most of the existing observa-
tions. However, the model seems to demand the presence of large amounts of
new and exotic types of energy and matter, and much of the effort of cosmolo-
gists today is directed towards understanding the so-called dark matter and dark
energy. In this chapter the focus is on dark matter, while in Chapter 4 I describe
the dark energy.
Although the idea of dark matter dates back to the 1930’s, its true nature still
eludes cosmologists. Dark matter exhibits the same gravitational attraction as
ordinary matter, but it does not interact electromagnetically at all. So it is
invisible, but can still be detected through its gravitational effects.

2.1 Evidence for dark matter

Most of our knowledge of the Universe comes from electromagnetic waves emitted
by astronomical objects. Dark matter, on the other hand, does not interact with
the electromagnetic spectrum. It is completely invisible, it does not emit, absorb
or reflect light on any frequency. In fact, it does not interact with ordinary
matter at all, and hence it is very difficult to detect. The evidence that dark
matter actually exists comes from its gravitational effect on other objects that
we can observe. Visible matter behaves in ways that cannot be explained in
standard gravitational theories without additional, invisible matter.
Individual galaxies have been found to contain dark matter. In visible light,
galaxies generally show a massive core, see Fig. 2.1. If the visible matter dom-
inated, one would therefore expect that the rotational velocity of stars in the
galaxy would decrease with radial distance from the core. Stars towards the
edge of the spiral galaxy in Fig 2.1 would be rotating slower than stars closer to
the centre. However, the velocity is found [14] to be very nearly constant as a
function of radial distance from the centre of the galaxy. This implies that the
galaxy that we see is embedded in a gigantic halo of dark matter, a halo which

9
10

Figure 2.1: The Andromeda galaxy in visible light. Image credits: John Lanoue.

extends far beyond the edges of the visible galaxy. The dark matter needed to
uphold the observed rotational curves of stars is generally found to be five to ten
times the mass of the visible galaxy.
Galaxy clusters are large conglomerations of 50-1000 galaxies, each galaxy con-
taining hundreds or thousands of million stars. When observed visually, clusters
appear to be collections of galaxies held together by mutual gravitational attrac-
tion. However their velocities are too large for the galaxies to remain gravita-
tionally bound by their mutual attractions, implying the presence of the invisible
mass component. More than 50 years ago, Fritz Zwicky was the first to find
evidence that dark matter exists. His findings in a study of the Coma cluster
[15] showed that the cluster would disperse if the entire attractive gravitational
force only came from the luminous, visible matter.
X-ray studies have revealed the presence of large amounts of intergalactic gas
in clusters. Since this gas is very hot, around 108 K, it emits X-rays. The total
mass of this gas is greater than that of the galaxies by roughly a factor of two.
However this is still not enough mass to keep the galaxies in the cluster.
An effect known as gravitational lensing can be used to predict how much material
there must be in a cluster to create its gravitational field. It turns out that
the total mass deduced from this measurement is much larger than the ordinary
matter of the stars and gas combined. In a typical cluster roughly 5% of the total
mass is in the form of visible galaxies, 10% is in the form of hot X-ray emitting
gas and the remainder is dark matter. It appears that the visible galaxy clusters
are embedded in proportionally huge dark matter halos. In Chapter 3 we look
at gravitational lensing in more detail, a very powerful tool with which to probe
the dark matter distribution in the Universe.
Because dark matter seems to be the dominating matter component in the Uni-
verse, it is desirable to find out how it is distributed in the Universe. In Fig. 2.2
results from the Millennium Simulation by the international Virgo consortium
are reproduced [16, 17]. This is the largest simulation ever of cosmic structure
growth. It uses more than 10 billion particles to trace the evolution of the matter
distribution in a cubic region of the Universe over 2 billion light-years on a side.
 11

Figure 2.2: The distribution of dark matter in the universe on different scales.
The figure shows a projected density field for a 15h−1 Mpc thick slice of the
redshift z = 0 output. The largest scales have a total extension of more than 9
billion light years on a side. The overlaid panels zoom in by factors of 4 in each
case, enlarging the regions indicated by the white squares. Image credits: The
Millennium Simulation Project, Max Planck Institute for Astrophysics.

On the largest scales shown in Fig. 2.2, the universe appears nearly homogeneous.
The series of enlargements overlaid show a complex web of dark matter up to the
‘smallest’ scales of order ∼ 100 Mpc (about 300 million light years). The largest
of these halos are rich clusters of galaxies, containing more than one thousand
galaxies which are still resolved as halo substructure in the simulation.

2.2 Different types of dark matter

Some of the dark matter in the Universe could be – and probably is – Jupiter
sized planets, brown dwarfs, or other such objects that we know exist but that
simply does not shine sufficiently for us to be able to observe them. These types
of objects have been nicknamed MACHOs (MAssive Compact Halo Objects).
However, the nucleosynthesis theory of light element production in the Big Bang
[18] places limits on how much baryonic matter (i.e. specifically protons and
neutrons and in general atoms of any sort) that can exist in the universe. There
is simply not enough baryons to make up for the missing matter content. A large
portion of the dark matter must be so-called non-baryonic.
Non-baryonic matter is any sort of matter that is not primarily composed of
12

baryons. This might include ordinary matter such as neutrinos or free electrons.
However, it may also include exotic species of supersymmetric particles. The
supersymmetric extension of the standard model of particle physics contains sev-
eral hypothetical candidates for dark matter particles, sometimes referred to as
WIMPS (Weakly Interacting Massive Particles). These particles must interact
only through the weak nuclear force and gravity, or at least with interaction
cross-sections no higher than the weak scale. They must also have a large mass
compared to standard particles. In standard theory, the particles that have little
interaction with normal matter, such as neutrinos, are all very light and would
therefore be moving at ultrarelativistic speeds. This hot dark matter can not ex-
plain observations of galaxy-sized clustering. So far, there has been no confirmed
detection of a WIMP.
Unlike normal, everyday matter, non-baryonic dark matter particles are assumed
to be collisionless, i.e. they are very hard to detect because they pass through
practically everything. Observing dark matter particles is extremely difficult due
to the fact that they are so weakly interacting. The collisionless nature of the
dark matter has recently been confirmed from a study of 16 galaxy clusters by
Høst et al. [19]. At the Large Hadron Collider at CERN, the intensive hunt for
supersymmetric particles will soon be on its way.
As dark matter particles have not yet been observed, the existence of dark matter
remains theoretical. Alternative gravity theories, such as Modified Newtonian
Dynamics (MOND), claim to explain the observed gravitational fields without
resorting to any mysterious dark matter. In these alternative theories, gravity
behaves differently on cosmological distance scales than on smaller distances. If
Einsteins theory of gravity is incomplete, then there might be no missing matter,
only ordinary everyday matter that gives rise to gravitational fields that look
differently than we expect them to.
However, the most direct evidence to date for the existence of dark matter poses
severe problems for alternative gravity theories. This evidence comes from the
Bullet Cluster [20], in actuality two colliding galaxy clusters. The stars, gas and
dark matter components in the clusters will behave differently during collision,
allowing for them to be studied separately. The stars in the cluster galaxies were
not greatly affected by the collision, and most of them passed right through,
gravitationally slowed but not otherwise altered. The hot gas of the two colliding
components, seen in X-rays, represents most of the mass of the baryonic matter
in the cluster pair. The gases interact electromagnetically, causing the gases
of both clusters to slow much more than the stars. The third component, the
dark matter, was detected indirectly by the gravitational lensing of background
objects. In theories without dark matter, the lensing effect would be expected
to be strongest around the X-ray gas (the major component of baryonic matter).
However, the lensing proved to be strongest elsewhere, supporting the idea that
most of the mass in the cluster pair is in the form of collisionless, non-baryonic
dark matter. At a statistical significance of 8σ, it was found that the spatial
offset of the center of the total mass from the center of the baryonic mass peaks
cannot be explained with an alteration of the gravitational force law.
Chapter 3

Probing the unseen with

gravitational lensing

There are two kinds of light – the glow that illuminates, and the glare that obscures.
James Thurber

Gravitational lensing is the bending of light rays from a distant source due to
the gravitational field of a massive object (e.g. a galaxy or a galaxy cluster)
that lies between us and the source. We can therefore never see the source
itself, as it would have looked without the lens effect. Instead, we see what we
call images of the source. These images can be shifted in position, multiplied,
magnified, de-magnified and also deformed in appearance due to the presence
of the gravitational field of the intervening material between us and the source.
Analogues of these cosmological effects can be observed in daily life. Looking
through a wine glass at a distant street light, looking in a fun-house mirror,
mirages of atmospheric lensing due to a heated ground all illustrate the same
concepts1 applying to gravitational lensing in the Universe. Fig. 3.1 illustrates
the concept of lensing in the laboratory, while Fig. 3.2 shows the distortions
caused by the galaxy cluster Abell2218.
Gravitational lensing is a unique tool in the exploration of the Universe. It
is a naturally occurring effect that can be used to detect mass independent of
its nature, i.e. whether it is shining or not. The degree to which an object is
lensed by intervening matter simply depends on the gravitational field of the
mass concentration. In other words, the light paths respond equally to ordinary
(luminous) matter as to the mysterious dark matter discussed in Chapter 2. This
is important because there seems to be much more dark than luminous matter.
Therefore, if we can measure the strength and distribution of the distortions
caused by lensing, we might be able to infer something about the actual mass
distribution in the Universe.
Other methods that probe the distribution of matter have to rely on luminous
markers of various kinds, e.g. X-ray radiation from the hot intracluster gas in a
galaxy cluster. The distribution of the gas traces the overall cluster gravitational
field, and therefore allows calculation of the total mass distribution in the clus-
1
For some fun examples, see
http://www.astro.ulg.ac.be/themes/extragal/gravlens/bibdat/engl/DE/didac.html

13
14

ter. However, this last step necessarily involves some sort of mass-to-luminosity
relation. Using gravitational lensing, there is no need to assume a relationship
between the visible and the invisible matter in order to trace the dominating
matter component. In addition, lensing can be used on scales were luminous
markers do not even exist.

Figure 3.1: Illustrating what we can see through a gravitational lens: The grav-
itational lens can be illustrated by the cut-off bottom of a wine glass. The glass
acts as a lens, analogous to a massive object in the Universe. We see through the
lens (the piece of glass) towards the galaxy (the black dot on the graph paper)
in the background. Left panel: What we would have observed without the lens
(not possible to reconstruct in astronomical situations!). Middle panel: the lens
is introduced, perfect alignment between lens and source. We see a ring (Ein-
stein ring) around the original position of the lens, as well as the source (dot)
in the middle. Right panel: Lens and source are displaced a little, producing
two arcs on either side of the source. In astronomical observations the source
in the middle would not be visible due to the lens obscuring it. Image Credits:
Talk on-line by Patricia Burchat: The search for dark energy and dark matter,
http://www.ted.com

Measurements of statistical properties of the lensing distortion can be related to

properties of the dark matter distribution at different times in the history of the
Universe, i.e. at different redshifts. The evolution of the dark matter (Chapter
2) distribution can yield information about dark energy (Chapter 4). It is often
concluded that gravitational lensing holds great promise for understanding the
nature of both dark matter and dark energy, see the report from the Dark Energy
Task Force [21] and the report by the ESA-ESO Working Group on Fundamental
Cosmology [22].
In this chapter I will first give a short introduction to some different types and
applications of lensing phenomena (Sec. 3.1), and the basic equations underlying
gravitational lensing theory (Sec 3.2). In Sec. 3.3 I focus more specifically on
lensing by galaxy clusters, which is the topic of paper I. In Sec. 3.4 I describe
how large-scale simulations of the projected matter distribution in the Universe
can be used to simulate the lensing signal, which is what we investigated in
paper II. For excellent reviews of lensing, see [23–25], and for reviews specifically
concerning weak lensing, see [26–32].
 15

Figure 3.2: Looking at the effects of dark matter: The famous strong lensing
caused by the galaxy cluster Abell2218, which is about 2 billion light years distant
in the constellation Draco. The golden ‘fuzzy’ galaxies are galaxies in the cluster
itself. The arcs and streaks are distorted shapes of galaxies that are actually much
further away. Image credits: NASA, A.Fruchter and the ERO Team (STScI).

3.1 Classifications

For a few hundred years, gravitational lensing was a theoretical possibility. Both
Newton’s and Einstein’s theory of gravity predict bending of light from nearby
stars by the massive Sun, see Fig. 3.3, but the bending angle is different in the
two theories. A Newtonian calculation, treating light as particles, predicts a
shift in position of 0.87 arcseconds for the stars closest to the Sun. The general
relativity calculation, taking into account the curvature of space, predicts a shift
twice as large. A solar eclipse in 1919 enabled the first observational confirmation
of gravitational lensing as well as a test of Newtonian versus Einsteinian gravity
theories. With the Moon blocking out the glaring sunlight, astronomers could
note the positions of stars close to the line of sight between the Earth and the
Sun, and compare this to the positions of the same stars at other times of the year
when they were visible at night. Observations taken during the eclipse in 1919
[33], though of poor quality by today’s standards, were convincing enough at the
time to confirm Einstein’s predictions and was a major contribution towards the
acceptance of general relativity.
The observations needed to confirm gravitational lensing by the Sun were diffi-
cult because the predicted shift in position was so small, less than two arcsec-
onds2 . Mass concentrations much larger than the Sun will produce gravitational
lensing effects that are more spectacular. Galaxy clusters have total masses of
∼ 1014 − 1015 times the mass of the Sun. Such a gigantic mass concentration
will drastically alter the curvature of space, gathering and distorting light from
incredibly distant light sources, focussing them into beautiful arcs in the sky like
those seen in Fig. 3.2. However, observing gravitational lensing on this larger
scale has not been possible until more recently.

2
The angular size of the Moon is about 1800 arcseconds.
16

Strong gravitational lensing is the name given to the production of multiple im-
ages and large arcs of very distant background galaxies, caused by very massive
lenses such as a galaxy cluster. The image of a background galaxy can be greatly
magnified or distorted depending on its position relative to the lens. The first
doubly-imaged object was found by Walsh et al. in 1979 [34], and since then hun-
dreds of multiple-imaged sources have been found [35], as well as many stunning
examples of strong distortions as shown in Fig. 3.2. The arcs get their squashed
shape because light rays are bent toward the lens, so extrapolating backward
shows that the source appears farther away, see Figs. 3.3 and 3.4. Therefore,
light coming from areas that pass closer to the lens, appear to us to be further
away from it, hence the arcs appear to bend around the cluster. Arcs or multiple
images from strong lensing can provide powerful constraints on the projected
mass in the lens that is contained within the arc, typically on scales confined to
the central regions of the cluster, see [36].
Weak gravitational lensing is the introduction of a small coherent shape distor-
tion in faint background galaxies due to lensing by foreground objects. The effect
of weak lensing is more subtle, there are no multiple images or dramatic distor-
tions as for strong lensing. The weak distortion is typically of the order of only
a few percent. In fact, variations in the intrinsic elliptical shapes of the galaxies
themselves is generally larger than the weak lensing signal. Because of the ex-
tremely low signal-to-noise for each individual galaxy, it is necessary to combine
data from thousands of lensed galaxies in order to detect the weak lensing signal.
Weak lensing can occur where the lens itself is an individual galaxy (so-called
galaxy-galaxy lensing) as well as where the lens is a galaxy cluster. The first
tentative detections of weak lensing by the tangential distorted images of cosmo-
logically distant, faint galaxies due to lensing by foreground galaxies was made
by Brainerd et al. [37] and confirmed by Fischer et al. [38]. Detection of cluster
weak lensing dates from 1990, and was reported by Tyson et al. [39] from the
detection of a coherent alignment of the ellipticities of faint blue galaxies behind
clusters Abell 1689 and CL 1409+52.
Weak lensing can be used to find constraints on the mass profiles of the lens.
Combining strong and weak lensing can significantly improve constraints on the
mass density profile [40]. Statistics of the shear field can be used to measure
the evolution of structures, measure cosmological parameters and explore models
beyond the standard ΛCDM model, see Chapter 4.
Microlensing is the study of peaks and troughs in charted brightness curves
of background stars or quasars, caused by individual stars or planetary sized
objects, e.g. brown dwarf stars. As the term implies, the multiple images induced
by microlensing have a separation of the order of microarcseconds, which is too
small for us to resolve with present-day technology. However, microlensing can
be detected where it causes uncorrelated brightness variations in multiple macro-
images of a source. Typically, light rays that are emitted from a source in different
directions at the same time and which travel different routes towards us due to
lensing, will arrive here at different times. We can measure this time delay using
multiply imaged sources with variable emission, by compiling and comparing light
curves as observed from the different images. In 1964, S. Refsdal [41] proposed
a geometrical method to calculate the Hubble parameter today, H0 , based on
this time delay and the deformation of wavefronts as light waves move past the
 17

Image
(apparent position)

Earth Source
(true position)
Sun

Figure 3.3: Gravitational deflection of a light ray from a distant source (a star).
The drawing illustrates the principle involved in gravitational lensing, and is not
drawn to scale. The light ray is bent towards the Sun (or another massive object),
causing the apparent position of the image of the star to be located further away
from the Sun. In reality, the angle of deflection is very small.

deflector towards the observer. If the mass distribution is known, an expression

for H0 can be derived solely in terms of observable quantities, see [24].
In some cases, a small lens, e.g. a planetary sized object, moves into the line
connecting a source to us. When this happens, the image of the source is mag-
nified in such a way that a distinct variability can be observed in the brightness
curve of the distant source. In recent years, microlensing has been used to find
MACHOs in our galaxies, as well as detecting extra-solar planets [42].
Cosmic shear is the weak lensing introduced by the deflections that light rays
encounter by passing all of the large-scale structure on its way from the last
scattering surface to us. This distortion is at ∼ 0.1 − 1% even more subtle than
cluster weak lensing or galaxy-galaxy lensing. In addition, the approximation
called the thin-lens approximation, see Fig. 3.4, does not always work in this
case because the large-scale structure can be elongated along the line of sight.
To first order cosmic shear can measure a combination of the matter density of
the Universe in units of the critical density, Ωm , and the rms mass fluctuation in
spheres of radius 8 Mpc also known as the normalisation of the power spectrum,
σ8 .
The first detections of cosmic shear were reported less than ten years ago, in
the spring of 2000, when four independent groups published their results [43–46].
The most recent results [47, 48] use millions of galaxies to measure the clumpiness
of dark matter to around 5 percent accuracy. Knowledge of the source redshifts
is currently the dominating limiting factor for cosmic shear studies. In future
surveys, the acquirement of large photometric redshift catalogues can remedy this
and allow future studies to divide the survey into several redshift bins. Sources
in redshift bins far away, i.e. at high redshift, will encounter (and be lensed
by) much more intervening structure than sources in bins closer to us. This
technique is called cosmic tomography [49] and makes it possible to map the
three dimensional distribution of mass. Since the third dimension involves both
distance and cosmic time, the results will be sensitive to the expansion history
of the Universe during that time. Many proposed experiments that aim to study
the properties of dark matter (Chapter 2) and dark energy (Chapter 4) have
18

focussed on weak lensing, such as the Dark Energy Survey (DES), Pan-STARRS
and the Large Synoptic Survey Telescope (LSST).

O M
β optical axis

θ ξ S
α
α̂

Observer plane Lens plane Source/Image plane

Ds

Dd Dds

Figure 3.4: Geometry of the lensing situation and illustration of the thin-lens
approximation. The source at S sends out a light ray (solid line) towards the
observer located at O. The ray passes a massive object at M and is deflected
by an angle α̂ as shown. The entire deflection is assumed to occur at the point
where the light ray is passing through the lens plane. Position angle of the source
with respect to the optical axis (along which the lens is located) is β

3.2 Basic lensing theory

Lensing can be described as a mapping of a source to an image of the source. In

some cases the lensing is strong enough to produce striking arcs (see Fig. 3.2) or,
in the case of perfect observer-lens-source alignment, rings.
Gravitational lensing changes the apparent solid angle of a source, but the surface
brightness is conserved. A magnification of source size will lead to an amplifi-
cation of the source signal. The total flux received from a gravitationally lensed
image is therefore changed by a factor equal to the ratio of the solid angles of
the image and the source:
image area
magnification = . (3.1)
source area

Fig. 3.4 illustrates the gravitational lensing situation. A distant source S located
at β, sends a light ray towards the observer at O. Passing through the lens plane,
the ray is deflected by an angle α̂ so that, as seen from O, the source appears to
be located at I. The position(s) of the images I are the solutions θ of the lens
equation:
 19

Dds
θ − β = α(θ) = α̂ . (3.2)
Ds

Dds , Dd and Ds are angular diameter distances between, respectively, deflector

(lens) and source, observer and deflector and observer and source. In general,
because we might have curved spacetimes, we have that: Dds 6= Ds − Dd . The
solutions θ of the lens equation yield the angular positions of the images of the
source.
To describe the shift in angle experienced by a light ray during lensing, it is
conventional to define the 2 × 2 symmetric transformation matrix

∂β
A= . (3.3)
∂θ
In general, the magnification is given by the inverse of the determinant of the
Jacobian matrix A. Locations at which detA = 0 have formally infinite mag-
nification. They are called critical curves in the lens plane. The corresponding
locations in the source plane are called caustics.
Distances between source, lens and observer are so large that most of the deflec-
tion of light rays takes place at a very small part of the total extent of the light
path. The lens can therefore be considered thin compared to the light path. This
is called the thin-lens – or thin screen – approximation. The mass distribution of
the lens can then be projected along the line of sight and replaced by a mass sheet
(the lens plane) orthogonal to the line-of-sight and with surface mass density Σ:
Z
Σ(ξ) = ρ(ξ, z) dz , (3.4)

where ξ is a 2D vector in the lens plane, indicating the distance from the light
ray to the centre of mass of the lens, as seen in Fig. 3.4.
Using the thin-lens approximation, we are assuming that all of the deflection
occurs in the lens plane. The deflection angle α̂ at position ξ is then the sum of
the deflections due to all the mass elements in the plane:

(ξ − ξ 0 ) Σ(ξ 0 ) 2 0
Z Z
4G
α̂(ξ) = 2 d ξ . (3.5)
c |ξ − ξ 0 |2

The so-called convergence, κ, is a rescaling of the surface mass density with the
critical density, so that
Σ(θ)
κ(θ) = . (3.6)
Σcrit

We have here defined the critical surface mass-density:

def c2 Ds
Σcr = (3.7)
4πG Dd Dds

A lens which has Σ > Σcr is referred to as being supercritical. Usually, multiple
imaging occurs only if the lens is supercritical.
20

For a circular symmetric lens Eq. (3.5) reduces to the scalar angle
4GM (ξ)
α̂(ξ) = , (3.8)
c2 ξ

with M (ξ) given by

Z ξ
M (ξ) = 2π Σ(ξ 0 )ξ 0 dξ 0 . (3.9)
0

For the special case of a symmetric lens, we consider a lens with constant surface-
mass density. The lens is then a disk with Σ = Σ0 , M = πξ 2 Σ0 . In practice, this
works as a normal converging lens, producing one image only. We then have,
from Eq. 3.2:

Dds 4GΣπξ 2 Dd Dds 4πGΣ Σ

α(θ) = 2
= 2
θ= θ = κθ . (3.10)
Dd c ξ Ds c Σcr
where we have also substituted ξ = θDd , which can be seen from Fig. 3.4.
This solution can be written
β
θ − κθ = β ⇒ θ = . (3.11)
1−κ

Due to rotational symmetry, a source which lies exactly on the optical axis,
i.e. β = 0, is imaged as a ring if the lens is supercritical. This is called the
Einstein ring, and its radius is θE , which can be calculated as follows. From
Eq. 3.2 – assuming a symmetric lens with arbitrary mass profile – and from
Eq. 3.8:

Dds Dds 4GM (θ)

α=θ−β = α̂ = . (3.12)
Ds Ds Dd c2 θ
The Einstein radius is the solution of this equation by solving for image position
θ and setting β = 0, i.e. assuming that the source lies on the optical axis, see
Fig. 3.4. We find that
s
4GM (< Dd θE )Dds
θE = . (3.13)
c2 Dd Ds

The Einstein radius provides a natural angular scale to describe the lensing ge-
ometry. For multiple imaging, the typical angular separation of images is of order
2θE . In addition, sources which are closer than about θE to the optical axis expe-
rience strong lensing in the sense that they are significantly magnified, whereas
sources which are located well outside the Einstein ring are magnified very little.
For β = 0 we have that α = θ ⇒ Σ = Σcr . So, the mean surface mass-density
inside the Einstein ring is the critical density Σcr , as defined in Eq. 3.7.
Convergence alone causes a magnification of the source, while the presence of the
so-called shear causes the image to be stretched. The shear is a two-component
quantity that can be written as a complex number:
γ = |γ|e2iφ . (3.14)
 21

The amplitude of the shear, |γ|, describes the degree of distortion and the phase
φ yields the direction of the distortion. The factor 2 in the phase is due to the
fact that the ellipse transforms back onto itself for a rotation of 180 degrees.
Introducing ψ as the scaled, projected Newtonian potential of the lens:

Z
Dds 2
ψ(θ) = Φ(Dd θ, z) dz , (3.15)
Dd Ds c2

where the Newtonian potential Φ has been integrated along the line of sight z
and scaled by the ratio of distances.
Writing
def ∂2Ψ
Ψ,ij = , (3.16)
∂θi ∂θj

the shear components can be expressed as linear combinations of Ψ,ij :

1
γ1 = (ψ,11 − ψ,22 ) γ2 = ψ,12 = ψ,21 . (3.17)
2

Using the lens equation, Eq. 3.2, we can write the Jacobian matrix A from Eq. 3.3
as
∂αi (θ)
Aij = δij − = δij − ψ,ij , (3.18)
∂θj

where δij is the Kronecker delta so that δij = 1 if i = j and δij = 0 if i 6= j.

The last part of Eq. 3.18 shows that the matrix of second derivatives of the
scaled potential ψ describes the deviation of the lens mapping from the identity
mapping. Using the definition of the shear components and

1 1
∇2θ ψ = 2κ ⇒ κ = ∇ · (∇ψ) = (ψ,11 + ψ,22 ) , (3.19)
2 2
we can further write the matrix as
 
1 − γ1 − κ −γ2
A= , (3.20)
−γ2 1 − κ + γ1

which is a common way of writing it. We see how the lens mapping depends on
both the convergence and the shear. In the absence of shear, a circular source will
be mapped onto a circle with modified radius, depending on κ. In other words,
the presence of convergence causes a magnification (the source is mapped onto
an image with the same shape but larger size), the shape-independent increase in
size of the galaxy image. Shear introduces anisotropy into the lens mapping and
causes an axis ratio different from unity: The circular source will be deformed
into an ellipse. The orientation of the resulting ellipse depends on the phase of
the shear.
For weak lensing, the diagonal of the Jacobian matrix in Eq. 3.20 will be ap-
proximately 1, implying weak distortions and small magnifications that cannot
be identified in individual sources but only in a statistical sense.
22

3.3 Weak lensing by gravitational clusters (paper I)

Weak gravitational lensing can be observed either from galaxy-galaxy lensing,

lensing by galaxy clusters or as cosmic shear. In this thesis the focus is on
lensing by clusters, but the techniques of measuring the shear of the distorted
ellipticities of lensed (background) galaxies are similar whatever the object acting
as lens.
Most galaxies in the Universe are gravitationally bound to a number of other
galaxies, forming a hierarchy of clustered structures, with the smallest such as-
sociations being termed groups, with masses around 1013 Msun . Galaxy clusters
are the largest gravitationally bound structures in the Universe and high-mass
clusters can be up to 1015 Msun . Clusters are often centered around a specific
galaxy that is brighter and larger than the rest, the so-called brightest cluster
galaxy (BCG). Typical cluster galaxies are red (i.e. old) ellipticals with little
star-formation activity. Clusters contain stars, gas and dark matter in the ap-
proximate mass ratio 1:10:100, see Chapter 2. The visible cluster, consisting of
galaxies and hot gas, is embedded in a huge, invisible dark matter halo.
Dark matter halo profiles are often parametrized by the Navarro-Frenk-White
(NFW) [50] profile, which can be written as:

ρs
ρ(r) = , (3.21)
( rrs )(1+ r 2
rs )

where

ρs = δc ρcrit,z (3.22)
200 c3
δc = c (3.23)
3 ln(1 + c) − 1+c
(3.24)

and
R200
rs = . (3.25)
c
ρcrit,z is the critical density of the Universe at the redshift of the cluster, and c
is called the concentration parameter and is a measure of the density of the halo
in the inner regions. According to simulations c ∼ 5 for clusters, but generally
depends on halo mass [51]. An NFW profile decreases as r−1 near the centre, as
r−2 at intermediate radii characterized by the scale radius rs , and as r−3 further
out. The transition region can be specified by the concentration parameter. R200
is the radius within which the density is 200 times that of the critical (or mean)
density at the redshift of the cluster.
Galaxy clusters in general cause weak lensing of galaxies that are further away
than the cluster. While strong lensing is a rare phenomenon, weak lensing is
much more common. It happens in principle everywhere we look, it is only a
question of how accurate we can measure.
The shapes of dark matter halos can give important insight into the nature of
dark matter. Studies find that cluster ellipticities are interesting because of
 23

their implications for differentiating between cosmological models [52, 53], their
dependence on cosmological parameters [54–56] and their effects on estimates of
cluster density profiles [57]. In general, cosmological N -body simulations predict
that gravitation causes matter on cluster scales to collapse into aspherical shapes
[58], in particular triaxial prolate ellipsoids [59, 60].

Carter & Metcalfe [61] showed in 1980 that the aspherical shape of a cluster
as measured by the positions of cluster galaxies, was connected to the velocity
anisotropy of the cluster member galaxies’ orbits. Later observations of both the
cluster galaxy distribution [62] and X-ray measurements of the intra-cluster gas
[53] indicates an elliptical shape, as seen projected on the sky.

The ellipticity of the dark matter distribution in galaxy sized halos using gravita-
tional lensing has been studied observationally using data from the Red-Sequence
Survey (RCS), the Sloan Digital Sky Survey (SDSS) and the Canada-France-
Hawaii Telescope Legacy Survey (CFHTLS) [63–65]. This measurement has
proven to be extremely difficult. However, as Mandelbaum et al. [64] also com-
mented, the stronger signal in clusters of galaxies means that there is more chance
of making a detection of ellipticity in this higher mass data set.

In paper I we investigate the shapes of galaxy cluster dark matter halos, as well
as the shape of the luminous matter as traced by the distribution of galaxies in
the cluster. We have detected, for the first time, an ellipticity in the dark matter
component of galaxy clusters.

We use a cluster catalogue compiled from the Sloan Digital Sky Survey (SDSS)
[66]. This is the largest existing cluster catalogue to date, consisting of almost 14
000 clusters. Because the shear induced by a cluster will affect the shear field of
its neighbours, we attempt to isolate the clusters by removing close neighbouring
clusters. For each cluster, we determine whether there are any other clusters
within a radius of 5h−1 Mpc, and remove them if they are less massive. By
retaining the most massive cluster we hope that the shear field will be dominated
by the remaining cluster. This process reduces the number of available clusters
to 4281 between 0.1 < z < 0.3.

Since the shear signal from a single cluster is very weak, we need to combine the
shear patterns produced by many clusters. We produce one image consisting of
a combination of the shear signal from the background galaxies of many clusters.
We divide this image up into a grid with square pixels the size of 0.4 arcmin.
The final result was found to be reasonably stable when changing the pixel size.

If this stacking was done without heed to the rotation angle of the clusters, it
would tend to erase any ellipticity that was there in the first place. We therefore
rotate and stack our clusters according to the method which is described by
Natarajan & Refregier [67] for galaxy-galaxy lensing. This is the first time that
this method has been applied to galaxy clusters.

Through the pattern imprinted on the shear maps by intervening massive cluster
halos, we detected a dark matter ellipticity of these halos by ruling out a circular
distribution at 99.6% confidence. The axis ratio f = b/a, where b is the semi-
minor axis of the ellipse and a is the semi-major axis, of the combined dark
matter halo was fDM = 0.48+0.14 2
−0.09 from a joint χ -analysis for an NFW model.
24

Similarly, the distribution of the luminous matter, as traced by the number den-
sity of individual cluster members, is also clearly elliptical, with a joint axis ratio
fLM = 0.60+0.004
−0.005 . Thus the ellipticity of the luminous matter is consistent with
the ellipticity of the galaxy number density distribution.
In order to interpret our results we need to take into account possible misalign-
ments during stacking, i.e. we do not actually know whether the light and dark
matter are aligned with each other. When we rotate clusters according to the
distribution of the cluster members before stacking, we assume that the orienta-
tion of the light is correlated with the orientation of the mass. A result of fDM
consistent with 1 could indicate either a circular mass distribution or a random
alignment between the light used for stacking and the dark matter. In the latter
case, the circular result would be caused by the stacking of many clusters with
different misalignments between the dark and the light matter.

3.3.1 The glow that illuminates: Ellipticity and shear

Weak lensing by clusters causes the intrinsic ellipticities of the background galax-
ies to be sheared, i.e. stretched and distorted. The shear caused by lensing can be
determined from measuring the (deformed) elliptical images of distant galaxies.
We do not have theoretical knowledge of the undistorted galaxy distribution, but
a good assumption is that unlensed galaxies are randomly oriented. This is a
reasonable assumption if there is no direction singled out in the Universe. In
this section I describe how the ellipticity of background galaxies is related to the
shear signal that we analysed in paper I.
Consider an isolated galaxy as seen on the sky, with surface brightness I(x, y)
centered at the origin. To quantify elongation in the horizontal, vertical or di-
agonal direction, we can use the quadrupole moments Qij of this galaxy, defined
as:
I(x, y)x2 dxdy
R
Qxx = R (3.26)
I(x, y) dxdy
R
I(x, y)xy dxdy
Qxy = R (3.27)
I(x, y) dxdy
I(x, y)y 2 dxdy
R
Qyy = R . (3.28)
I(x, y) dxdy

In practice, it is not always straightforward to say exactly where the image of

one galaxy stops and the image of a neighbouring galaxy begins. To reduce
contamination from neighbouring galaxies, a weight function W (x, y) is often
included in Eqs. 3.26 - 3.28. This is also the case in the oldest and most widely
used method for cosmic lensing analysis, often referred to as the KSB method
[68–70].
We can define the complex ellipticity e of the galaxy in terms of Eqs. 3.26 - 3.28:
Qxx − Qyy + 2iQxy
e= . (3.29)
Qxx + Qyy

Since e = e1 +ie2 , we see that the ellipticity components e1 and e2 can be written
 25

in terms of the quadrupole moments as

Qxx − Qyy
e1 = , (3.30)
Qxx + Qyy
2Qxy
e2 = . (3.31)
Qxx + Qyy

Figure 3.5: Ellipticity parameters e1 and e2 calculated for a series of ellipses using
the definition of e in Eq. 3.32. Ellipses located further from the centre of this
coordinate system are more elongated. Using the definition of Eq. 3.35 would
produce a similar plot.

Fig. 3.5 shows how a circular source is transformed as a function of the ellipticity
components e1 and e2 . A positive (negative) e1 will stretch an image along the
x (y) axis, while a positive (negative) e2 will stretch it along the line y = x
(y = −x). In terms of the axis ratio f = b/a, where a is the semi-major axis
of the ellipse and b is the semi-minor axis so that f < 1, the ellipticity can be
written:
(1 − f )
e= . (3.32)
(1 + f )

From this it follows that Eqs. 3.30 and 3.31 can be written in terms of a and b as
(a − b)
e1 = cos(2θ) (3.33)
(a + b)
(a − b)
e2 = sin(2θ) , (3.34)
(a + b)
26

where θ is the angle measured anticlockwise from the positive x axis to the major
axis of the ellipse.
Note that there is also a complementary definition of ellipticity often used, from
which completely similar deductions can be made:

(1 − f 2 )
= . (3.35)
(1 + f 2 )

This definition is sometimes referred to as the polarization. In paper I we use the

axis ratio f to measure the ellipticity. We note here that, for an ellipse aligned
along the abscissa (horizontal axis), 0 ≤ f ≤ 1, where f = 1 represents a circle
and a lower f -value gives a stronger ellipticity, f = 0 being a horizontal line.
f -values greater than 1 produces an ellipse oriented along the ordinate (vertical
axis).
In order to measure the distortion induced by lensing, we need to relate the
ellipticity of the image to the ellipticity of the source. Due to the fact that
surface brightness is invariant in gravitational light deflection, the transformation
between the quadrupole moment tensor (Eqs. 3.26 - 3.28) of the image versus
that of the source is
Qs = A Q AT , (3.36)
e.g. see [71, 72]. The matrix A is the Jacobi matrix (Eq. 3.3) of the lens map-
ping. Q is the matrix of quadrupole moments for the galaxy image and Qs is
the corresponding matrix for the source, i.e. representing the intrinsic, unlensed
ellipticity.
The transformation between the ellipticity of the source es and the ellipticity of
the image e is obtained from Eq. 3.36 and can be written, for |g| << 1, as

es + g
e= (3.37)
1 + g ∗ es

see [73]. The asterix ∗ denotes complex conjugation, and g is defined as the
reduced shear:
γ
g= . (3.38)
1−κ
We see that the condition |g| << 1 corresponds to the weak lensing regime, where
|γ| << 1 and κ << 1.
Unfortunately we do not know have knowledge of the intrinsic ellipticities in
the absence of lensing. The strategy is to locally average many galaxy images,
assuming that the intrinsic ellipticities on average are randomly oriented, in other
words that in the absence of lensing there is no preferred orientation for the
shapes of galaxies. When averaged over a large population of galaxies, this gives
hes1 i = hes2 i = 0. Taylor expanding Eq. 3.37 to first order in g q for the two
components ei shows that ei is roughly a noisy estimate of gi , as hesi 2 i is an
order of magnitude larger than the typical value of gi . Applying the symmetries
for a large population, we find that

hei ≈ g . (3.39)
 27

Using the ellipticity definition of Eq. 3.35 and a similar transformation to the
above, we get
2 + 2g + g 2 ∗
= (3.40)
1 + |g|2 + R(gs ∗)
where R denotes that the real part should be taken. Averaging over a population
for which hs1 i = hs2 i = 0, hs1 i = hs2 i and hs1 s2 i = 0, we arrive at the relation
g
hi ≈ , (3.41)
2(1 − σ2 )

where σ2 is the variance of the unlensed ellipticities . In terms of the two
components of the ellipticity, σ2i = h2i i. To sum up, then, the average ellipticity
of many galaxies can be used to estimate the shear at the position of that galaxy.

Figure 3.6: Fig.1 from Bridle & Abdalla [74]. Shading and contours show the
convergence κ constructed for an elliptical NFW mass distribution (e = 0.3,
M200 = 1.2 × 1012 h−1 Msun ) at a redshift of zl = 0.3, with sources at redshift
zs = 0.8. Shear sticks (lines) show the resulting shear map, where the length of
the stick represents the shear strength and the direction of the shear is given by
the angle of the stick with respect to the x-axis. For an elliptical lens aligned
along the x-axis, the mass density falls more steeply in the y than in the x
direction. The result of this is that the sticks on the x-axis are larger than sticks
on the y-axis for the same angular distance from the lens center.

When studying weak lensing around galaxy clusters the shear signal is referred
to as tangential shear since the distortion effect of a cluster is to introduce an
alignment of background galaxy images tangentially to the line from the cluster
centre to the galaxy.
From the mathematical definition of the shear in Eq. 3.14, we can define the
tangential (γT ) and cross (γ× ) components of the shear of the background galaxies
28

as

γT = γ cos(2φ) (3.42)
γ× = γ sin(2φ) , (3.43)

where φ is the angle between the galaxy major axis and a tangent through the
centre of the shear galaxy, with respect to the cluster centre. The cross compo-
nent γ× is not expected to occur in nature, but can be a useful test for systematic
errors.
The convergence and shear map for an elliptical NFW lens of ellipticity e = 0.3
aligned along the x-axis is shown in Fig. 3.6, a reproduction of fig.1 in Bridle
& Abdalla [74]. The convergence map (shading and contours) is the projected
mass density in units of the critical lens density. The shear field is shown by the
overlaid shear sticks (red lines), where the length of the stick gives the amplitude
of the shear and the direction of the stick gives the direction of the shear field
at that point. We see that the shear at 45 degrees around from the major axis
is approximately tangential to the centre of the lens. For a singular isothermal
ellipsoid (SIE) model, the shear is exactly tangential.

3.3.2 The glare that obscures: Systematic errors

Weak lensing measurements are subject to a number of potential sources of sys-

tematic error, both from the lensing shape measurements and from interpreta-
tions of the lensing signal. In this section I will briefly touch upon some of the
practical obstacles encountered in lensing analysis, and refer to [25, 72] for a more
detailed descriptions of these issues.
The relationship between ellipticity and shear outlined in Sec. 3.3.1 would be
adequate for calculating the shear in the absence of disturbances introduced
by the atmosphere, pixelisation from the detector CCD chip and noise. One
major observational problem that we have already mentioned in Sec. 3.3.1, is that
the intrinsic shape of galaxies is not round, and different galaxies have different
ellipticities. The observed ellipticity of the image of a galaxy is therefore a
combination of intrinsic ellipticity and induced shear that it is not straightforward
to separate. In addition, seeing by atmospheric turbulence as well as effects from
the camera optics will tend to blur the images, making them appear more circular.
The blurring effects of the atmosphere can be described by a convolution. One
tries to correct for the distorting effects by using the point spread function (PSF),
which gives information on how a star is broadened by the convolution. Since
the PSF represents the disturbance encountered by a point source (a star), it
gives information on the convolution kernel itself and can therefore be used to
deconvolve the extended source (galaxy). On large cameras, the PSF can vary
substantially across the image introducing further complications for PSF recon-
struction [75]. Corrections for anisotropic PSFs is one of the major sources of
systematic errors challenging weak lensing measurements.
The perhaps least well characterized source of systematic error for weak lensing
measurements [31] is called intrinsic alignments. There are reasons to believe that
unlensed galaxy ellipticities are not randomly oriented, as assumed in Sec 3.3.1.
 29

Galaxies at approximately the same redshift can align with other galaxies in re-
sponse to a gravitational field, thus adding to the shear signal, so-called intrinsic-
intrinsic alignments [76, 77]. This source of systematics can be greatly reduced
with photometric redshift information, separating galaxies in the z direction so
the tidal effects will be weak. On the other hand, the shapes of galaxies at differ-
ent redshifts may be correlated with their surrounding density field. This same
density field contributes to the lensing distortion of more distant galaxies [64, 78],
leading to a suppression of the measured shear signal. This second type of corre-
lation, often called gravitational-intrinsic alignments, affects galaxies at different
redshifts and is therefore more difficult to correct for. However, it is important
to overcome systematics introduced by both types of intrinsic alignments as it
can bias cosmic shear results [79].
As regards cluster lensing, which is the topic of papers I and II, possibly the
biggest potential systematic error is intrinsic alignment of cluster members point-
ing at the cluster centre [80–84]. A thorough assessment of the intrinsic alignment
effect went beyond the scope of the papers, but does represent a problem because
we did not make a significant attempt to remove cluster members from our anal-
ysis. To first order intrinsic alignments would make our observed shear maps less
elliptical since the contamination by cluster members is greatest along the cluster
major axis, and thus the strong gravitational shears expected along this axis will
be partially cancelled out by the cluster members since they tend to point at
the cluster centre. However, a full assessment of this effect would also have to
take into account the variation in intrinsic alignment with respect to the cluster
major axis, which appears to be more complicated [81] and possibly weakens the
degree of cancellation preferentially along the cluster major axis.

3.4 Predictions of weak lensing from N -body simula-

tions (paper II)

In paper II we apply the method for measuring cluster dark matter ellipticity
from paper I to output from a large-scale N -body simulation. In general, we find
excellent agreement with the observational results from paper I.
Some earlier numerical results provide a two-dimensional projected ellipticity
[85, 86], which is useful for direct comparison with observations using gravita-
tional lensing. However, the best-fit ellipse in previous work is conventionally
found by taking moments of particle positions about the center of mass. We
have simulated the expected ellipticity that would be observed using gravita-
tional lensing, instead of isolating particles cut off at some circular radius (i.e. the
virial radius or R200 ) in the simulation box. As lensing measures the projected
mass along the line of sight, ellipticities measured using gravitational lensing are
potentially increased by the inclusion of physically close neighbors and filaments.
However they may also be decreased by the contribution from coincidental over-
laps by physically separated objects projected along the line of sight. By properly
simulating gravitational lensing measurements, we can compare results from pa-
per II directly to results from paper I.
We started with the output of a large (N = 10243 particles) cosmological dark
30

matter simulation of a cube of size L = 320h−1 Mpc. The cosmology was chosen to
be consistent with that measured from the 3rd-year WMAP data combined with
large-scale structure observations, see [87], namely a spatially flat ΛCDM model.
Convergence maps were produced by projecting down a lightcone covering one
octant of the sky and summing up the contribution of all the clusters lying along
the line of sight of each map pixel. The projection was normalized by a factor
depending on the redshift distribution n(zs ) of background (lensed) galaxies, as
measured by the Sloan Digital Sky Survey (SDSS).
We find that surrounding large-scale structure causes a large scatter between
ellipticities as measured by lensing and ellipticities calculated directly from par-
ticles in the cluster simulation. When repeating the method of paper I of stacking
many clusters to obtain the shear signal for an average cluster, the large-scale
structure tends to cancel out. The resulting axis ratios agree well with those
calculated using the conventional method.
Chapter 4

Invisible Universe II:

Dark Energy

Until the end of the previous century, it was generally assumed that the Uni-
verse would expand slower and slower as gravity pulled on all existing matter.
Depending upon how much matter there was in the Universe, it would either ex-
pand forever, or reach a point where it would begin to contract again. Completely
contrary to expectations, in the late 1990’s results were published indicating the
exact opposite. The expansion of the Universe appears to be accelerating. This
revolutionary discovery prompted a new era in cosmology. In order for the Uni-
verse to be expanding at an ever increasing rate, there must be something that
is causing the acceleration. Currently, gravity is the only known force which can
act at cosmological distances, but gravity is an attractive force. What then, is
‘pushing’ the Universe to expand faster and faster? Cosmologists call it dark
energy and it turns out to be the major component of the Universe today.

4.1 Evidence for dark energy

Current estimates from NASA’s Wilkinson Microwave Anisotropy Probe

(WMAP) [13] show that the Universe consists of ∼ 4% baryons, 21% cold dark
matter and an astonishing 75% dark energy. WMAP measures the amplitude
of temperature fluctuations in the cosmic microwave background radiation as
a function of angular size on the sky, and from this information cosmological
parameters can be inferred.
Early indications of the acceleration of the Universe actually dates back to a result
from measuring the redshifts and magnitudes of galaxies by J.-E. Solheim as far
back as 1966 [88]. He used cluster galaxies to measure the magnitude-redshift
relation and found that the best-fit model describing the data included a non-zero
cosmological constant and a negative deceleration parameter. During the 1980’s,
more tantalizing hints were reported, but the evidence was highly uncertain.
Cosmic acceleration was not taken seriously by the cosmological community until
results of luminosity distance versus redshift measurements of the more reliable
supernova type Ia were published by two independent teams in 1998 [5, 6].

31
32

Supernovae of type Ia (SNIa) are the explosive deaths of white dwarf stars. These
explosions are relatively uniform in their luminosities, making them ideal stan-
dard candles for measuring luminosity distances. To date, supernovae remain the
most direct and well established method for constraining dark energy [21]. Cur-
rent surveys include the SuperNova Legacy Survey (SNLS) [89], which expects to
discover ∼ 700 high-redshift supernovae, and ESSENCE Supernova Survey [90],
which aims to measure the dark energy equation of state parameter to ±0.1 by
following ∼ 200 type Ia supernovae over a period of five years. Future surveys
such as The SuperNova Acceleration Probe (SNAP) will study more supernovae
at higher redshifts as well as bringing more precise information on sources of sys-
tematic uncertainties such as a possible redshift-dependent luminosity evolution
of the supernovae. SNAP [91] is now being proposed as part of the Joint Dark
Energy Mission (JDEM).
Galaxy cluster and weak lensing surveys are also sensitive to dark energy. Since
the number density of clusters as a function of mass and redshift is sensitive to the
underlying cosmological model, cluster counts can be used to constrain cosmo-
logical parameters. For example, the formation epoch of galaxy clusters depends
on the matter density parameter Ωm of the Universe [92]. One way of detecting
clusters is to measure peaks in the shear fields caused by weak lensing. Although
the shear peaks-method is subjected to projection effects because lensing mea-
sures mass along the line of sight, new studies show that the projected peak and
the 3-dimensional mass function scale with cosmology in an astonishingly close
way [93].
The acceleration of the Universe will try to counteract the formation of structure.
Therefore, in a universe with a strong dark energy term, the clusters we observe
today must have formed earlier, and hence are expected to be more relaxed
and contain less substructure, than if the Universe was matter-dominated. Like
cluster counting, gravitational weak lensing observations probe the dark energy
via both the expansion history of the Universe and the growth history of density
fluctuations. There is a multitude of weak lensing statistics that can be performed
(power spectra, cross-spectra, bispectra, etc.) which allows for the correction of
many potential systematic errors [21].
In order to obtain precise weak lensing results for cosmological purposes, it will
be necessary to measure the distortion of the shapes of galaxies to extremely
high accuracy for millions of galaxies [72]. Future surveys will be bringing in the
potential for a substantial increase in the accuracy of dark energy measurements
if the distortion can be measured better than an accuracy of 0.0003. 2008 saw
the launching of the GRavitational lEnsing Accuracy Testing 2008 (GREAT08)
challenge, aimed at solving this practical problem.
Baryon Acoustic Oscillations (BAO) is a series of peaks and troughs predicted in
the matter power spectrum, or equivalently a characteristic scale in the galaxy
distribution. In the very early Universe the environment was so hot and dense
that baryons and photons were coupled in a plasma. Fluctuations in this plasma
were due to the competing forces of gravity and radiation pressure. In early
2005 a peak in the galaxy correlation function at 100h−1 Mpc was reported by
the SDSS team [94], which was an excellent match to the predicted imprint
of acoustic oscillations on low-redshift clustering of matter. The scale of the
 33

acoustic oscillations provide a standard ruler with great potential to be used for
constraining dark energy from future galaxy surveys [21, 95] through its effect on
the angular distance versus redshift-relation.
The integrated Sachs-Wolfe (ISW) effect is caused by gravitational red- or
blueshifts affecting CMB photons propagating from the last scattering surface
and encountering gravitational potentials (i.e. large matter concentrations). With
dark energy dominating the Universe today, the gravitational potential will be
time dependent. In other words, the energy gained by photons falling into a
potential well does not cancel out the energy loss as the photons climb out of
the well. This large-scale effect can be measured by cross-correlating large-scale
structure observations with CMB anisotropies caused by the ISW-effect [96, 97].
By modifying Einstein’s field equations it might be possible to explain current
and future observations without introducing an explicit ‘dark energy’. As an
example, gravity may be modified on large scales due to the presence of a large
extra dimension. A proper treatment of this subject is beyond the scope of this
thesis.

4.2 Cosmological models with dark energy

One of the simplest ways to incorporate the acceleration of the Universe into
theoretical models, is to revive the cosmological constant Λ term, (see Chapter
1). Ironically, Λ was originally introduced by Einstein as a way of preventing his
model of the Universe from evolving. In its simplest form, general relativity pre-
dicted that the Universe must either expand or contract. As seemed natural at
the time, Einstein thought the universe was static, so he added the Λ term to his
equations to stop the expansion. Alexander Friedmann (1888-1925) realized that
this was an unstable fix, like balancing a pencil on its point. He proposed an ex-
panding universe model. When in 1929 Hubble’s study of nearby galaxies showed
evidence that the universe was indeed expanding, Einstein saw no further use of
the cosmological constant and rejected it. Years later, he famously commented
to George Gamow that he viewed the introduction of the cosmological constant
term as “the greatest blunder of my life” [98]. Modern field theory, however,
associates the cosmological constant with the energy density of the vacuum, and
on these grounds it has been revived as a driving force for the acceleration of the
Universe.
A cosmological constant is not the only possibility for producing an accelerating
expansion, and a large number of models have been proposed. These models are
mostly phenomenological in nature, i.e. they lack a basis in the current framework
of particle physics and cosmology. The hope is that some of these models for dark
energy can be connected to extensions of the Standard Model of particle physics,
e.g. supersymmetric models or string theory.
Models where the dark energy is a scalar field rolling slowly down a potential, the
so-called quintessence models, have become particularly popular. Quintessence
differs from the cosmological constant in that it can vary in space and time. It is
possible to find classes of scalar-field models which have the interesting property
that the scalar fields approach a common evolutionary track from a wide range of
34

initial conditions. In these so-called ‘tracker’ models the scalar field density (and
its equation of state) remains close to that of the dominant background matter
during most of the cosmological evolution. A scalar field with these properties
may arise naturally in models with large extra dimensions.
Although it is possible to compare models of dark energy to each other and to
observations on a model-by-model basis, it is often useful to classify models using
some sort of general parameters. The dark energy equation of state parameter
w is often used, where the equation of state is parametrized by p = wρ (see
Chapter 1). A cosmological constant has w = −1 while more general models of
dark energy will have a time dependent w.

4.3 Investigating the statefinder parameters

(paper III)

Several theoretical models describing an accelerated universe have been sug-

gested. They are often tested against the SNIa data using the relationship be-
tween luminosity distance and redshift, dL (z):
z
dz 0
Z
dL (z) = (1 + z) . (4.1)
0 H(z 0 )

In 2003 a method of classifying dark energy models using geometrical quantities

derived from the third derivative of the scale factor was proposed by Sahni and
coworkers [99, 100]. The so-called statefinder parameters r and s are defined as
...
a Ḧ Ḣ
r = = 3 +3 2 +1 (4.2)
aH 3 H H
r−1
s = , (4.3)
3(q − 12 )

where a = a(t) is the scale factor of the expansion of the Universe defined in
Chapter 1 and dots represent the derivatives with respect to time. We have that
H = ȧ/a, and q is the deceleration parameter

aä ä
q=− =− . (4.4)
ȧ2 aH 2

The introduction of the statefinders is motivated by the fact that they simply
reduce to (r, s) = (1, 0) for the ΛCDM model. This makes it easy to detect any
deviation from ΛCDM.
In paper III, we investigate the usefulness of the statefinder parameters as theoret-
ical classification of dark energy models. We also consider their use in obtaining
constraints on dark energy from present and future SNIa data sets.
In order to extract the statefinders in a model-independent way from the data,
one has to parametrize H(z) in an appropriate way. The ansatz of [100] was:
p
H(x) = H0 Ωm0 x3 + A0 + A1 x + A2 x2 , (4.5)
 35

where x = (1 + z). The luminosity distance-redshift relationship is then given by

Z 1+z
1+z dx
dL = √ . (4.6)
H0 1 Ωm0 x3 + A0 + A1 x + A2 x2

After fitting some or all (depending on the model) of the parameters Ωm0 , A0 ,
A1 and A2 using both existing and simulated supernovae data, we could then
reconstruct the statefinders by substituting Eq. 4.5 into the definitions for r
(Eq. 4.2) and s (Eq. 4.3) using the best-fit value of the parameters.
We reconstructed the statefinders from several dark energy models, including
ΛCDM with and without curvature, flat quiessence models (i.e. models where w
is a constant but can be different from −1), the modified polytropic Cardassian
model and the generalized Chaplygin Gas model. For comparison, we also cal-
culate the statefinders directly from Eq. 4.2 and Eq. 4.3 using the expression for
H(z) specified by each model.
We conclude that even with SNAP-quality data there may be difficulties in distin-
guishing between dark energy models solely based on the statefinder parameters.
Our findings also indicate that it is non-trivial to extract the statefinders from
the data in a model-independent way. While a useful theoretical and visual tool,
applying the statefinders to observations is not straightforward.
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Part II

Papers

45
Paper I

Evans, A. K. D.; Bridle, S.,

A detection of dark matter halo ellipticity using galaxy cluster lensing in SDSS,
Astrophysical Journal 695 (2009), 1446-1456
The Astrophysical Journal, 695:1446–1456, 2009 April 20 doi:10.1088/0004-637X/695/2/1446

C 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

A DETECTION OF DARK MATTER HALO ELLIPTICITY USING GALAXY CLUSTER LENSING IN THE SDSS
Anna Kathinka Dalland Evans1 and Sarah Bridle2
1
Institute of Theoretical Astrophysics, University of Oslo, Box 1029, 0315 Oslo, Norway
2 Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK
Received 2008 June 18; accepted 2009 January 22; published 2009 April 7

ABSTRACT
We measure the ellipticity of isolated clusters of galaxies in the Sloan Digital Sky Survey (SDSS) us-
ing gravitational lensing. We stack the clusters, rotating so that the major axes of the ellipses determined
by the positions of cluster member galaxies are aligned. We exclude the signal from the central 0.5 h−1
Mpc to avoid problems with stacking alignment and cluster member contamination. We fit an elliptical
Navarro–Frenk–White (NFW) profile and find a projected, two-dimensional axis ratio for the dark matter
of f = b/a = 0.48+0.14 −0.09 (1σ ), and rule out f = 1 at 99.6% confidence thus ruling out a spherical
halo. We find that the ellipticity of the cluster galaxy distribution is consistent with being equal to the dark
matter ellipticity. The results are similar if we change the isolation criterion by 50% in either direction.
Key words: cosmology: observations – dark matter – galaxies: clusters: general – galaxies: halos – large-scale
structure of universe
Online-only material: color figures

1. INTRODUCTION Plionis et al. 1991; Rhee et al. 1991; Strazzullo et al. 2005).
Most recently, Wang et al. (2008) studied groups of galaxies in
Cosmological simulations can be used to predict many the Sloan Digital Sky Survey (SDSS) and found an alignment
different statistics of the mass distribution in the universe. The between the brightest cluster galaxy (BCG) and the distribution
most commonly employed statistic is the two-point correlation of galaxies, which was strongest in the most massive groups and
function, or its Fourier counterpart, the power spectrum. Three- between red BCGs and red group member galaxies.
point statistics are much harder to predict and measure, and In this paper, we focus on measuring the dark matter ellip-
higher orders are rarely discussed. A more popular statistic is ticity directly using gravitational lensing. We also compare this
the number of peaks in the mass distribution, as given by the ellipticity to the ellipticity of the cluster member galaxy dis-
number of clusters of galaxies. The dark matter power spectrum tribution to see how reliable a tracer of ellipticity the light is,
and the number of clusters of galaxies are often cited as among and therefore whether it can be used by itself for cosmological
the best ways to constrain the properties of dark energy (e.g., studies.
Albrecht et al. 2006). Gravitational lensing has been used very successfully to
Uncertainties on cosmological parameters are decreased measure the mass and profile of clusters of galaxies by many
when two measurements have different parameter degeneracies authors. The work most relevant to our study is that of Sheldon et
and are often referred to as “complementary.” In this paper, we al. (2001); Sheldon et al. (2007b) and Sheldon et al. (2007a) who
consider a statistic which may offer complementary constraints stack the lensing signal from many clusters of galaxies to find
on cosmology: the shapes of peaks in the mass distribution, as an average signal. Natarajan & Refregier (2000) proposed that
probed by the ellipticity of galaxy cluster dark matter halos. In when stacking the shear signal from many halos the stack could
addition, this may place important constraints on modifications be made while retaining information about the major axis of
to the law of gravity since we may compare the results from the halo, as observed from the distribution of light. The stacked
both dark and light matter, as also tested by studying the dark shear map should then provide a constraint on the ellipticity of
and light matter distributions in the bullet cluster (Clowe et al. the dark matter halo, if indeed the mass and light were aligned.
2006). They considered lensing by galaxies, but we apply the same
Predictions of cluster ellipticities come mostly from numeri- technique to clusters of galaxies here.
cal simulations (West et al. 1989; de Theije et al. 1995; Jing & The ellipticity of the dark matter distribution has been studied
Suto 2002; Floor et al. 2003; Ho & White 2004; Flores et al. observationally for galaxy-sized halos using gravitational lens-
2007; Rahman et al. 2006). The ellipticity is expected to depend ing using data from the Red-Sequence Cluster Survey (Hoek-
on cosmological parameters (Evrard et al. 1993; Splinter et al. stra et al. 2004), the SDSS (Mandelbaum et al. 2006), and the
1997; Buote & Xu 1997; Suwa et al. 2003; Rahman et al. 2004) Canada–France–Hawaii Telescope Legacy Survey (CFHTLS;
and to evolve with redshift (Kasun & Evrard 2005; Allgood et Parker et al. 2007). This has proved to be extremely difficult
al. 2006), an evolution which itself might depend on cosmology and we present here, for the first time, results from stacking
(Hopkins et al. 2005; Ho et al. 2006). The distribution of subha- galaxy cluster halos. Cypriano et al. (2004) has previously made
los within a cluster halo is found to be an indicator of the overall measurements of the cluster dark matter ellipticity using grav-
halo ellipticity (Bode et al. 2007) but is slightly less elliptical. itational lensing measurements from individual clusters. They
The ellipticity of the brightest cluster galaxy and the ellipticity found a good agreement between the dark matter halo orienta-
of the distribution of cluster member galaxies are much easier tion and the orientation of the brightest cluster galaxy.
to observe than any other cluster ellipticity measure. This has When calculating angular diameter distances and the mean
been studied by a number of authors (West & Bothun 1990; density of the universe, we assume a flat cosmology with

1446
No. 2, 2009 A DETECTION OF DARK MATTER HALO ELLIPTICITY 1447
Ωm = 0.3. The Navarro–Frenk–White (NFW) halo profile has a We do not want to include in the shear catalogue galaxies
very weak dependence on the fluctuation amplitude. We assume which are already in the cluster member catalogue. Therefore
σ8 = 0.8 for this calculation. we remove from the shear catalogue all galaxies which are close,
The structure of this paper is as follows. In Section 2, as seen on the sky, to a cluster member. We cut on a physical (as
we describe our data set and operations we have performed opposed to angular) distance of 0.012 h−1 Mpc, as calculated at
on the data such as rotating and stacking and removal of the redshift of the cluster. This distance corresponds to 5 arcsec
neighboring clusters. In Section 3, we use two theoretical at redshift z = 0.2. In principle, this cut may also remove some
models—a Singular Isothermal Ellipsoid (SIE) and NFW (see real background galaxies; however, the shear measurements of
Navarro et al. 1997) model—of the mass and light distribution, these galaxies will in any case likely be adversely affected by
and look at correction factors from the redshift distribution and the light contamination by the cluster members. Note that we
cluster decontamination. Our results, and some interpretations do not eliminate all cluster members from the shear catalogue
of these, are presented in Section 4. Conclusions are summarized with this cut, only those clusters that fit the selection criteria set
and discussed in Section 5. by the maxBCG selection method (red galaxies, brighter than
0.4 L*; Koester et al. 2007a).
2. DATA
2.2. Postage Stamp Size and Close Neighbor Removal
In this section, we describe the catalogues used and the
In order to avoid contamination of the shear signal from
operations we carried out before comparing with models,
neighboring structures, we only include in our analysis galaxies
including cluster selection and stacking and rotating.
which are sufficiently close to a cluster, as seen on the sky. We
decide this distance by considering the predicted contribution
2.1. Catalogues
of neighboring clusters to the shear signal, and then use this to
We use the cluster catalogue of Koester et al. (2007a), which is decide the required separation of neighboring clusters.
at the present time the largest existing galaxy cluster catalogue, We include in our shear catalogue galaxies in a square postage
consisting of 13,823 galaxy clusters from the SDSS (York stamp of 10 × 10 h−1 Mpc centered on each cluster. This choice
et al. 2000). The cluster galaxies are red-sequence members was made using Figure 8 of Johnston et al. (2007), which shows
(occupying the so-called E/S0 ridgeline in color–magnitude model fits for the lensing signal split up into contributions
space), brighter than 0.4L∗ in the i band and between redshifts from several components. We want the lensing signal in our
0.1 < z < 0.3. They also lie within a circular aperture of radius postage stamp to be dominated by the central cluster, not by
R200 , which is the estimated radius within which the density of the contribution from neighboring mass concentrations such
galaxies with 24 < Mr < 16 is 200 times the mean density as nearby clusters and filaments. Our interest therefore lies in
of such galaxies. The number of cluster members inside this comparing their NFW profile (green line) to the contribution
aperture is 10  Ngals
r200
 188; the lower limit is a requirement from neighboring halos (blue line). The results in their Figure 8
r200 r200
for inclusion into the catalogue. We refer to Koester et al. (2007a) are shown for 12 richness (Ngals ) bins. The mean of the Ngals
for details of the catalogue and to Koester et al. (2007b) for a values of our cluster sample is ∼24, directing us to the panel
r200
description of the cluster selection algorithm. Ngals [18–25]. We match our postage stamp limit at the radius
In order to define an isolated sample, we removed clusters where the NFW contribution is roughly equal to that of the
found to be too close to each other, as seen on the sky. Details contribution from neighboring clusters. Note that we are also
on this close neighbor removal can be found in Section 2.2. removing close neighbors (see below), so the contribution from
Clusters that are too close to the survey edge are also removed, neighboring halos would actually be even lower for our case, so
with the requirement that the minimum distance from the cluster our cut may be conservative.
center to the survey edge is 7 h−1 Mpc. We are left with a total Clusters that are too close to each other in projected separation
of 4281 clusters to analyze for our purposes. We have also represent a challenge for our analysis. The mass distribution
organized the clusters into four redshift bins between z = 0.10 of a cluster will affect the shear field of its neighbor. This
and z = 0.30, each of width 0.05 in z. Since these cluster could have a significant effect on the measured ellipticity. While
redshifts are photometric, with uncertainties of ∼0.01 (Koester it is possible to isolate clusters in three dimensions within
et al 2007a), the spectroscopic bin width will be very slightly a simulation, we are plagued by overlaps on the sky since
larger than the photometric bin width. We do not take into gravitational lensing measures the projected mass. We therefore
account this small broadening in our analysis, because we expect endeavor to remove this complication by selecting relatively
it to have negligible effect on the ellipticity results. isolated clusters for this analysis. For proper comparison with
The shear galaxy catalogue is the same as used in Sheldon theory, a similar sample should be made from simulations.
et al. (2004), except that the area covered is larger, ∼6325 deg2 . However, in this paper, we are primarily concerned with the
There is approximately one galaxy per square arcminute in first significant detection of ellipticity in a large sample.
this catalogue. Galaxies in the shear catalogue have extinction- We do not want the shear from a neighbor to appear within
corrected r-band Petrosian magnitudes less than 22. Stars have our postage stamp, but this is unavoidable to some extent,
been removed from the catalogue by the Bayesian method because the shear at one point is affected by mass far away
discussed in Scranton et al. (2002). Unresolved galaxies and (shear is nonlocal). The effect on the measured ellipticity of
objects with photo-z errors greater than 0.4 have also been having neighbors is twofold. (1) It can make the distribution
removed. To correct the shapes of galaxies for effects of point- more circular, if the neighbor position is uncorrelated with the
spread function (PSF) dilution and anisotropy, the techniques cluster major axis. This will occur due to chance alignments
of Bernstein & Jarvis (2002) were used, with modifications close to the line of sight, particularly from clusters in different
specified in Hirata & Seljak (2003). We refer to Sheldon redshift bins. (2) It can make the distribution more elliptical, if
et al. (2004) for full details of the compilation of the shear the neighbor position is correlated with a major axis direction.
catalogue. Pairs of clusters which are physically close will probably be
1448 EVANS & BRIDLE Vol. 695

Shear galaxies Members Total

1 1 1

0.5 0.5 0.5

Mpc
0 0 0
/h

0 1 0 1 0 1
/h Mpc /h Mpc /h Mpc
Figure 1. Total number of galaxies per unit area for redshift bin 3 (0.20 < z < 0.25) for illustration. From left to right: shear galaxies, cluster members, and total
galaxies (shear galaxies plus cluster members). The left and right panels include galaxies which are uncorrelated with the cluster. We draw our contours relative to
the constant background level (which is the same number in both left and right panels, since the cluster member catalogue does not contribute). For the left and right
panels, contours are equally spaced from 1.2 to 2.6 times the background level, in steps of 0.2. The dashed contour shows 2 times the background level. For the central
panel the contours are 0.2, 0.4, 0.6, 0.8 times the same background level. The dashed circle shows location of mask, see text. For comparison, a 3 × 1014 h−1 M
cluster at a redshift of 0.2 has R200 ∼ 1.4 h−1 Mpc, where R200 is the radius within which the mean density of the cluster is 200 times the mean matter density at the
cluster redshift. We have zoomed in to show the central part of the postage stamp for clarity.
(A color version of this figure is available in the online journal.)

aligned along the major axis of the cluster (e.g., Plionis et al. to lie along an x-axis, which is the major axis as defined by the
1991). ellipticity of the cluster members (see Figure 1). This rotate-
We therefore remove clusters that are closer than 5 h−1 and-stack method is described by Natarajan & Refregier (2000)
Mpc (corresponding to half the width of the postage stamp) to for use in galaxy–galaxy lensing, and we have, for the first time,
neighboring clusters. When a cluster has one or more neighbors applied this technique for use on cluster lensing.
with an angular separation corresponding to less than 5 h−1 We calculate the direction of a cluster major axis from the
Mpc, calculated at the middle of the redshift bin of the cluster positions of the cluster members, as defined in the cluster
(see later), we discard it if any of the neighbors have a higher catalogue. We do not take into account the luminosity of each
r200
Ngals . The neighbor removal was done consecutively from low- cluster member. The cluster center (xc , yc ) was taken to be the
to-high redshift. This process reduced the number of available position of the BCG as defined by the maxBCG algorithm
clusters to 4281. In a later section, we analyze the use of different (Koester et al. 2007b). However, the position of the BCG is
minimum close neighbor distances. not necessarily coincident with the cluster’s actual center of
Although we remove a cluster’s less rich neighbors from our mass. For comparison, we therefore calculate the center of each
sample, the shear pattern of the remaining cluster will already cluster as given by the mean position of the cluster members.
have been affected by its neighbor(s). However, by retaining the The mean physical offset (for redshift bin 0.20 < z < 0.25)
richest of the neighboring clusters, we hope that the shear field between the two center definitions is ∼0.15 h−1 Mpc, and the
is dominated by this cluster. standard deviation 0.09 h−1 Mpc. Compared to our mask radius
of 0.5 h−1 Mpc, therefore, the shift in center position is relatively
2.3. Stacking and Rotating small. Any effect this mis-centering does have will increase the
ellipticity of the members, which we do not focus on here, and
On average, we have only around 1 shear galaxy per square
cause the misalignment angle to tend toward the direction from
arcminute and the uncertainty on the shear for a single galaxy
the BCG to the center of the cluster member distribution. This
is an order of magnitude larger than the shear we are trying to
would in itself be an alternative and potentially useful way to
measure. Therefore we need to use the shear signal from many
stack the clusters to obtain the results presented here. Therefore,
clusters in order to obtain a significant signal. We therefore stack
we do not consider this effect further.
the clusters on top of each other to improve the signal-to-noise
To find the ellipticity angle of rotation of the cluster, we
ratio. In other words, we use information from the postage stamp
use the quadrupole moments of the cluster members. The
field of shear galaxies for all clusters simultaneously.
quadrupole moments are given by
The stacking could be carried out in either physical or angular
space. For our method, the two approaches are exactly equivalent Qxx = (xi − xc )2 i (1)
if the redshift bins are small enough. We stack in angular space
Qxy = (xi − xc ) (yi − yc )i (2)
and use redshift bins of redshift width 0.05. This causes a radial
blurring, because two clusters of the same physical size will be Qyy = (yi − yc )2 i , (3)
stacked on top of each other in angular space to have different
angular sizes. The blurring of the shear and light maps will be of where the summation i is over the cluster members. We convert
at worst plus and minus 20% (for the lowest redshift bin). Since this into the ellipticity components e1 and e2 of the cluster
superposing ellipses of different scalings retains the original through the relations
ellipticity, this results in a slightly smoother cluster profile but Qxx − Qyy
will not affect our ellipticity results. We find that our results are e1 =  (4)
fairly similar even when comparing two very different profiles, Qxx + Qyy + 2 Qxx Qyy − Q2xy
see Section 3.
Straightforward stacking of elliptical clusters with random 2Qxy
orientations would erase any ellipticity and produce a circular e2 =  . (5)
average cluster. Before stacking, we therefore rotate each cluster Qxx + Qyy + 2 Qxx Qyy − Q2xy
No. 2, 2009 A DETECTION OF DARK MATTER HALO ELLIPTICITY 1449
The angle of the cluster, anticlockwise from the positive x axis, The object of this paper is to calculate the ellipticity of the dark
is then matter, as measured from gravitational lensing, and compare it
1 e  with that of the light-emitting galaxies. Any misalignment in
2
θ rot = atan . (6) the stacking and rotating will tend to make the dark matter
2 e1
appear less elliptical. However, this misalignment will have the
We rotate the positions of the cluster member galaxies same effect on the ellipticity of the light, as measured from the
and the positions of the shear galaxies using the following extra cluster members (left-hand panel) alone. Therefore, we
transformation: can compare like with like, and assess the relative ellipticity of
x rot = d cos(θ − θ rot ) (7) the dark and light matter, despite any misalignment.
We have chosen the contour levels so that a fair comparison
can be made between the left and central panels: the outermost
y rot = d sin(θ − θ rot ), (8) contour corresponds to the same cluster member density in each.
Therefore, we see that at large radii most of the cluster members
where (d, θ ) are the polar coordinates of the galaxy to be rotated, are not included in the cluster member catalogue. For all the axis
relative to the cluster center. For clusters with ellipticities close ratio measurements reported in this paper we exclude the central
to zero, θ rot in Equation (6) has little meaning. From the regions (see Section 3.3), this is shown by the dashed circle in
cluster members alone, 18% of our clusters have an ellipticity Figure 1. Outside this excluded region, we see that the contours
(e = e12 + e22 , using Equations (4) and (5)) of less than change only a little from the left to the right panel. This is
convenient because it means that it is not too important whether
0.1 and only 5% have ellipticities less than 0.05. Therefore,
we compare the dark matter ellipticity with the light ellipticity
the angle is reasonably well defined. Two per cent of the
derived from either the left- or the right-hand panel.
clusters have an ellipticity greater than 0.5. The cluster selection
We calculate the tangential, γ̂T , and cross, γ̂X , components
criteria by Koester et al. (2007a) and/or our isolation criteria
of the shear γ̂ for each shear galaxy
have therefore done a reasonable job of identifying isolated
clusters. In addition, the clusters seem relatively undisturbed,
γ̂T = γ̂ cos(2α) (9)
i.e., have low ellipticity. This also illustrates that it would not
be particularly useful to bin the clusters according to the cluster
member ellipticity because the range is relatively small (and our
γ̂X = γ̂ sin(2α), (10)
final signal to noise is quite low).
Figure 1 shows the number of galaxies per unit area, for z where α is the angle between the shear galaxy major axis and
bin 3, after stacking and rotating. The left panel shows galaxies a tangent through the center of the shear galaxy, with respect
from the shear galaxy catalogue, with cluster member catalogue to the cluster center. We calculate the shear estimate for each
galaxies removed, the middle panel shows cluster galaxies from galaxy from the polarizations () in the shear galaxy catalogue
the cluster member catalogue and the right panel shows the sum
of shear galaxies and cluster members. 1
The left-hand panel clearly contains a significant number of γ̂ = , (11)
2SSh
cluster members, despite the fact that the galaxies from the
cluster member catalogue are not included. As described in where the factor SSh ∼ 0.88 is the average responsivity of
Koester et al. (2007b), cluster members for this catalogue were the source galaxies to a shear, see Sheldon et al. (2001) and
identified using the maxBCG algorithm. This algorithm employs references therein, and  = (a 2 − b2 )/(a 2 + b2 ), where a is the
the red-sequence method, based on the observational fact that semimajor and b is the semiminor axis of the shear galaxy. We
cluster galaxies occupy a narrow region (a so-called ridgeline) have redone our main NFW dark matter ellipticity result using
in a color–magnitude diagram. This method is designed to a value SSh which is 20% higher, and find only a negligible
conservatively select red galaxies at the central area of a cluster. change. Note that the tangential and cross shears are invariant
As an illustration, we investigate the central region of the with respect to rotation of the cluster coordinates. We bin the
stacked cluster in the third redshift bin (0.2 < z < 0.25). shear galaxy catalogue into square pixels of size 0. 4 × 0. 4 on
In the central square arcminute, the number of galaxies in the sky, but our main results are insensitive to this exact value.
the member catalogue is ∼40% of the number of galaxies in We average the shear estimates in each pixel.
the (members removed) shear catalogue, after subtracting the Figure 2 shows the smoothed tangential shear. As expected,
constant background level, i.e., most of the cluster members are the tangential shear is largest in the cluster center. To interpret
not in the cluster member catalogue. this figure further it is helpful to consider the tangential and cross
The existence of these extra members allows a very con- shears pattern predicted from popular cluster models, discussed
venient check on our stacking and rotating: the rotation further in Section 3. Cluster ellipticity causes little, if any, cross
angles were calculated from the cluster member catalogue alone, shear, depending on the cluster profile. For an elliptical SIE,
whereas the left-hand panel does not include the galaxies used the cross shear is exactly zero. For an elliptical NFW with a
to decide the rotation angle. Therefore, the fact that we see el- major to minor axis ratio of 0.5 (our best fit result), a mass
lipticity in this panel means that the angle calculated from the of 1 × 1014 h−1 M and cluster redshift 0.15, the maximum
members catalogue is correlated with the angle of the extra clus- cross shear occurs when approaching the center of the lens.
ter members. If, for example, we had calculated rotation angles Just outside our mask radius of 0.5 h−1 Mpc, the maximum
from a very small number of galaxies from the cluster member cross shear is 0.0012 for a source redshift of 0.3. This value is
catalogue, there would be a large degree of randomness due to 10% of the maximum tangential shear outside the mask, and is
shot noise, and the alignment of measured and true ellipticity therefore small compared to our uncertainties. The main effect
would be random to a large extent, resulting in a circular pattern of cluster ellipticity is to produce an ellipticity in the tangential
in the left panel of Figure 1. shear map. Therefore, the tentative visual indication of some
1450 EVANS & BRIDLE Vol. 695
3 catalogue without weighting. Due to the cuts already made in
0.02
creating the shear galaxy catalogue, the difference in weight
2 between the noisiest and least noisy galaxies is only 30%;
therefore, the weighting would not make a large difference to
1 0.015 our analysis.
Mpc

T
0 3. MODELING

γ
0.01
/h

3.1. Mass and Light Distributions

0.005 The shear of galaxies depends on the mass and structure of the
cluster acting as a lens. To model the cluster mass distribution,
0
we use two alternative theoretical models: an SIE and an NFW
model. The NFW model is preferred from simulations, but we
0 1 2 3 also include results from the simpler SIE model to show an
/ h Mpc extreme and simple example of the dependence of our results
Figure 2. Tangential shear, γ̂T , as measured from all galaxies in the background on the cluster profile.
galaxy catalogue for a zoomed-in view of the postage stamp. To reduce the The SIE model corresponds physically to a distribution of
noise, we have smoothed with a Gaussian with standard deviation 1 h−1 Mpc. self-gravitating particles with a Maxwellian velocity distribution
The location of the mask is shown by the dashed circle.
with one-dimensional velocity dispersion σv . The convergence
(A color version of this figure is available in the online journal.)
(normalized mass density) κ = Σ/Σcrit of an SIE is given by

horizontal elongation in this figure is our first hint of dark matter σv2 Dds −1
ellipticity. κ = 2π r , (14)
c 2 Ds
To compare the data with models, we must incorporate
the errors on the shear estimates. The errors on the shear
where r is the generalized radius
measurements are given by

σγ2T = σi2 + σSN

2
, (12) r = (x 2 f + y 2 f −1 )1/2 (15)

where σi is the uncertainty in the shape measurement due to and f = b/a is the axis ratio of the ellipse (b < a), σv is
the finite number of photons falling in each detector element, the velocity dispersion, and c is the speed of light. x and y are
plus detector noise, and σSN is the “shape noise” due to intrinsic coordinates in the plane of the sky at the cluster redshift. The
variance in the unlensed galaxy shapes (assumed the same for distances Dds and Ds are the angular diameter distances between
all galaxies). the lens (deflector) and the source, and from the observer to the
We calculate the shape measurement uncertainty σi from the source, respectively. The SIE peaks sharply in the central parts,
uncertainties in the two components of the ellipticity e1 and e2 , but we mask out the central regions (see Section 3.3). For the
using SIE model, we have the simple relation that the normalized
1 σe1 + σe2 surface density equals the tangential shear γT = κ (Kassiola &
σi = . (13) Kovner 1993; Kormann et al. 1994).
2SSh 2
The NFW model is a more complicated but more realistic
The first factor converts from e to γ and the second part model based on numerical simulations. In order to implement
assumes that the uncertainty on the two ellipticity components is the NFW, we need the projected mass of the cluster. To calculate
essentially equal and uncorrelated, which is approximately true the projected mass, we use the equations given in Wright &
for some shear estimators (e.g., Figure 2 of Bridle et al. 2002). Brainerd (2000) and Bartelmann (1996) using our generalized
If these assumptions are true then it follows that the ellipticity radius of Equation (15) to make the cluster mass distribution
uncertainty on a component is independent of rotation. elliptical. We use M200 , the mass enclosed within the radius at
We estimate the shape noise σSN by calculating the rms which the density is 200 times the mean density of the universe,
dispersion in γ̂T values as a function of σi . We compare this for consistency with simulations. We derive the concentration
to σγT for various σSN values and find σSN = 0.24 to be the best parameter, c, where c ∝ M β according to Equation (12) of
fit. Seljak (2000), where we interpret the virial mass M as M200 . We
To find the error on the shear for each pixel, we take the mean use β = −0.15, as appropriate for an NFW model. The shear for
of the errors σγT and divide by the square root of the number an elliptical mass distribution is calculated using the equations in
of galaxies in the pixel. Galaxies will have a range of sizes and Keeton (2001) which are derived from those in Schramm (1990).
therefore of measurement errors. When we calculate the average A shear map using these equations is illustrated in Figure 1 of
shear in each pixel we do not weight the galaxies according Bridle & Abdalla (2007).
to their ellipticity errors. This is because this would tend to We calculate probability as a function of our free parameters
upweight the better measured galaxies. As better measured in each redshift bin, and marginalize over all but the axis ratio f.
galaxies may preferentially tend to be cluster members that have We then obtain a single result for f from combining all the
leaked into the shear galaxy catalogue, this might preferentially redshift bins by multiplying the probabilities from different
weight up cluster members. This would have to be taken into redshift bins together for each f value. This is the correct
account when removing the bias on the shear due to cluster calculation if we believe that the other parameters have different
member contamination (see Section 3.3), which would be values in each redshift bin, but that f is the same for all redshift
difficult. We therefore use all the galaxies in the shear galaxy bins.
No. 2, 2009 A DETECTION OF DARK MATTER HALO ELLIPTICITY 1451
3.1.1. Estimation of the Light Matter Axis Ratio (fLM ) using Galaxy 3.2. Redshift Distributions
Positions
As discussed in Section 2, we divide our cluster sample up
When stacking the clusters (Section 2.3), we calculated into four redshift bins. Due to the large photometric redshift
individual cluster ellipticities based on the cluster galaxies in uncertainties we decided not to use the redshift information
the members catalogue. We do not use these results as our in the shear galaxy catalogue. Therefore the “shear galaxies”
measure for the light matter ellipticity for the stacked cluster, may be in front of, behind, or part of the cluster. The shear
because they are relatively noisy due to the small number of for each lens–source pair depends on the redshift of both the
members (∼10 for the least rich clusters). Furthermore, we lens (cluster) and the source (shear galaxy). To calculate our
know that the members catalogue does not in fact contain all the theoretical model, we need a prediction for the distance ratio in
cluster members, and has some selection criteria that may affect Equation (14) at each possible redshift. This must be averaged,
the ellipticity (requirement on proximity to cluster center). We weighted by the number of galaxies at each redshift. In other
therefore use a χ 2 analysis to find the light matter ellipticity words, we need to calculate
of the clusters. We model the light map as coming from (1) ∞
D 
noncluster galaxies which have a constant density across the ds z (Dds /Ds ) n(zs ) dzs
postage stamp plus (2) a contribution from the cluster galaxies = L ∞ . (18)
Ds 0 n(zs ) dzs
which is assumed to have a galaxy density proportional to the
mass profiles of Equation (14) Note that the integration in the nominator starts at the lens
redshift, zL , so that galaxies between us and the lens do not
npred (r) = Kκ(r) + n0 , (16) contribute to the shear signal, as they are not influenced by the
presence of the cluster. We estimate n(zs ) from Figure 3 and
where n0 is the background level of galaxies per pixel and K Equation (8) in Sheldon et al. (2001), and as a result we use
is a constant. Note that we do not assume that the light is zc = 0.22. However, our results are quite insensitive to these
some constant multiple of the mass, only that the light map numbers because we focus only on the ellipticity of the dark
is proportional to an SIE or NFW profile (which may have matter halo and not on its mass.
different parameters than the dark matter distribution). We do We obtain Dds /Ds  = [0.51, 0.37, 0.26, 0.17] for the four
not tie the dark and light map parameters together, because redshift bins. The values are low for high redshift bins, because a
we wish to investigate whether the dark and light distributions large fraction of the galaxies are between us and the cluster, and
both have the same ellipticity. For practical purposes, we fix therefore do not contribute to the shearing effect. The calculation
the mass at M200 = 1014 h−1 M in this calculation, which of Dds /Ds is approximate, because we assume all clusters to be
corresponds approximately to clusters of the mean richness we located at the center of their redshift bin. However, this does not
used (Johnston et al. 2007, Table 6). The value used affects the affect the ellipticity of the theory prediction. The distance ratio
concentration parameter and therefore the mass profile of the is incorporated into the predictions using Equation (14).
cluster, which affects the weighting of the map. If we use a
value a factor of 10 higher, our light ellipticity results change by 3.3. Cluster Decontamination
less than 1σ and in any case our main results, on the dark matter Because spectroscopic redshifts are not available for all
ellipticity, are changed imperceptibly because the uncertainties shear catalogue galaxies, there will always be a degree of
on those are dominated by the much larger uncertainty on the contamination by cluster members in the shear signal (as seen in
dark matter quantities. Figure 1(a)). Since we assume that the cluster members have no
We calculate probabilities in the resulting three-dimensional systematic alignment (but see Section 5 for an assessment of the
space (fLM , K, and n0 ) by calculating a χ 2 between the predicted implications of this assumption), members that have leaked into
number (Equation (16)) and the observed number the shear galaxy catalogue will tend to dilute the shear signal.
We correct for this dilution in our analysis. The corrected shear
 (npred (ri ) − nobs (ri ))2 is given by
χ2 = (17) n(r)
i
σn2i γcor = γ̂ , (19)
n0
and calculating probabilities from this, Pr = e−χ /2 .
2
where n(r) is the total number of galaxies in the shear galaxy
Our assumption is that the errors are Poisson, therefore in the catalogue a two-dimensional position r from the center, n0 is the
limit of large numbers the error onthe number of galaxies, as number of galaxies not in the cluster (see Equation (16)) and γ̂
used in the χ 2 calculation, is σni = npred (ri ). We calculate our is the observed, uncorrected shear. We use the best fit n0 value
main results for each of the three stacked light maps shown in from the χ 2 fit to the light matter distribution. Inside the cluster,
Figure 1. we have n(r) > n0 , so the observed shear will be boosted by
correcting for the contamination.
3.1.2. Estimation of the Dark Matter Axis Ratio (fDM ) from the
We mask the central regions from our analysis for several
Stacked Shear Map reasons. (1) In the very central regions, cluster members will
obscure the shear galaxies. (2) The cluster center may be
To estimate the ellipticity of the dark matter distribution, we incorrect and thus the central parts may appear erroneously
calculate probability as a function of the cluster axis ratio fDM circular after the stacking. (3) There is actually an uncertainty
and cluster mass. We calculate the probability from χ 2 between associated with the correction factor that we have not taken into
the observed and predicted shears, using the uncertainty on the account. This error is due to Poisson fluctuations in the true
shear values from Equation (12). We marginalize over the cluster number of noncluster members per pixel, and will be more
mass with a flat prior to obtain the probability as a function of significant when the value of the correction factor is large,
the two-dimensional axis ratio fDM . which occurs in the central region where the observed number
1452 EVANS & BRIDLE Vol. 695

1 NFW Total 1 NFW

SIE Total SIE
0.9 NFW Shear galaxies 0.9

0.8 0.8

Normalised probability
Normalised probability

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0
0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0 0.5 1 1.5
f f
LM DM

Figure 3. Left panel: relative probability distribution of the axis ratio fLM = b/a for the galaxy number density map from the shear catalogue plus member catalogue,
thus including all cluster members. This includes clusters at all redshifts for an NFW model (dark, red line) and an SIE model (light, green line). The dashed line
shows the result from shear catalogue galaxies only (for an NFW model). Right panel: relative probability distribution for the dark matter axis ratio fDM . The dark, red
line shows result from the NFW model, and the light, green line shows result from the SIE model. The dotted vertical line shows fDM = 1 corresponding to a circular
distribution.
(A color version of this figure is available in the online journal.)

Table 1
Axis Ratio Results for the Light Maps

Population z SIE (fLM ) NFW (fLM )

Shear galaxies 0.10  z  0.15 0.548 + 0.013 − 0.013 0.620 + 0.015 − 0.015
Total 0.10  z  0.15 0.555 + 0.014 − 0.014 0.614 + 0.012 − 0.013
Members 0.10  z  0.15 0.302 + 0.006 − 0.006 0.544 + 0.006 − 0.007
Shear galaxies 0.15  z  0.20 0.672 + 0.011 − 0.011 0.649 + 0.012 − 0.012
Total 0.15  z  0.20 0.657 + 0.012 − 0.011 0.631 + 0.010 − 0.010
Members 0.15  z  0.20 0.261 + 0.005 − 0.005 0.475 + 0.006 − 0.006
Shear galaxies 0.20  z  0.25 0.581 + 0.012 − 0.012 0.583 + 0.011 − 0.010
Total 0.20  z  0.25 0.578 + 0.011 − 0.010 0.575 + 0.009 − 0.009
Members 0.20  z  0.25 0.254 + 0.004 − 0.004 0.456 + 0.005 − 0.005
Shear galaxies 0.25  z  0.30 0.637 + 0.010 − 0.009 0.619 + 0.008 − 0.008
Total 0.25  z  0.30 0.624 + 0.008 − 0.008 0.602 + 0.007 − 0.007
Members 0.25  z  0.30 0.235 + 0.003 − 0.003 0.434 + 0.004 − 0.004
Joint (total) 0.10  z  0.30 0.610 + 0.005 − 0.005 0.602 + 0.004 − 0.005

of galaxies in the shear galaxy catalogue peaks sharply. For all Figure 1, for the NFW model. This shows that the light matter
these reasons, we mask out the central region using a circular distribution is clearly elliptical with an axis ratio of fLM ∼ 0.6.
mask with rmask = 0.5 h−1 Mpc. The radius of the mask was set The errors on fLM are calculated by finding the 68% iso-
where the correction factor increases above 1.5 (as calculated probability limits from the probability distribution P (fLM ) after
for 0.20 < z  0.25), shown by the dashed circle in Figure 1. marginalizing over the other fit parameters. We find an error of
This corresponds to 3.3 arcmin at z = 0.225. ∼0.005.
Table 1 shows three light matter ellipticity results for each
4. RESULTS redshift bin and for each theoretical model (SIE and NFW): (1)
for the shear catalogue galaxies with cluster members removed;
We now present our results on the stacked cluster ellipticity,
(2) for the total cluster members; and (3) for the cluster members
focusing first on the NFW profile.
and (members from the cluster catalogue plus the extra members
4.1. Results for the Light Matter Axis Ratio (fLM ) included in the shear galaxy catalogue). The results from the
NFW and SIE are qualitatively similar. The last line in Table 1
The left panel of Figure 3 shows the one-dimensional relative shows the joint results combining all bins, for the total cluster
probability of the axis ratio fLM , marginalized over K and members.
n0 (see Equation (16)). The solid lines represent results from There is a clear detection of ellipticity based on the number
using the total cluster members, i.e., galaxies in both the shear density of shear catalogue galaxies alone. We see that the
galaxy catalogue and cluster member catalogue (corresponding distribution of galaxies in the members catalogue is more
to the right-hand panel of Figure 1). The blue, dashed line elliptical than that of the shear catalogue. This is not surprising
shows results from using only the (members-removed) shear because we stacked the galaxies according to the members
catalogue galaxies, corresponding to the left-hand panel of catalogue. Even if the galaxies in the members catalogue had
No. 2, 2009 A DETECTION OF DARK MATTER HALO ELLIPTICITY 1453
Table 2
Axis Ratio Results for the Dark Matter Distribution SIE
NFW
z SIE (fDM ) NFW (fDM )
0.10  z  0.15 0.754 + 0.230 − 0.186 0.614 + 0.400 − 0.176 1
0.15  z  0.20 0.522 + 0.159 − 0.115 0.269 + 0.158 − 0.054
0.20  z  0.25 0.958 + 0.251 − 0.199 0.599 + 0.379 − 0.208
0.25  z  0.30 0.853 + 0.306 − 0.215 0.614 + 0.444 − 0.266 0.8
Joint analysis 0.747 + 0.102 − 0.094 0.480 + 0.136 − 0.086

DM
0.6

f
been sampled from a circular distribution then the finite number
of galaxies would provide a rotation direction, so we would have
in effect stacked the random noise to produce an ellipticity for 0.4
the cluster members, while making the light map from the shear
catalogue more circular.
This extra induced ellipticity could be simulated, but in 0.2
fact there is no need, because the axis ratio from the shear
galaxy catalogue alone is very similar to that based on the total
catalogue containing both the shear catalogue and the cluster 0
member catalogue. Therefore, we do not consider the results 0.6 0.61 0.62 0.63 0.64
from only the cluster member catalogue any further. f
LM
The similarity of the ellipticities from the shear galaxy
Figure 4. 68% and 95% contours of the two-dimensional probability distribution
catalogue and the total catalogue is largely due to the fact that for the axis ratio of the dark matter (fDM ) vs. the axis ratio of the total galaxy
there are very few galaxies from the members catalogue outside number density distribution (fLM , representing the light matter). Light (green)
our mask radius. The important conclusion for our paper is that contours show the result from using an SIE profile as modeling the cluster,
any misalignment in the rotating and stacking will affect the light and dark (red) contours show result from using an NFW profile. The horizontal
dotted line shows fDM = 1, and the dotted line rising towards the right shows
and dark matter ellipticity the same. The ellipticity observed in fDM = fLM .
the light map makes us believe that the rotation angles are not (A color version of this figure is available in the online journal.)
completely random and therefore that we can hope to detect
some ellipticity in the dark matter, if indeed the dark matter
halo is elliptical. Figure 4 shows 68% and 95% contours of the two-
dimensional probability distribution for the axis ratio of the
4.2. Results for the Dark Matter Axis Ratio (fDM ) dark matter (fDM ) versus the axis ratio of the light matter (fLM ).
The dotted lines also shown on the plot are fDM = 1 (hori-
Ellipticity results for the dark matter analysis can be found zontal dotted line) and fDM = fLM (dotted line rising toward
in Table 2, for each individual redshift bin as well as for the the right). The two colors shown represent the different profiles
joint result combining all redshift bins. For the NFW model, used as cluster models: light (green) contours are for the SIE
the joint result of fDM = 0.48+0.14−0.09 excludes a circular mass model, and dark (red) contours are for the NFW. We see that
distribution (fDM = 1) by over 3σ (if the probability distribution fDM = 1, that is, a circular dark matter distribution, crosses just
is Gaussian). The lowest axis ratio is in the second redshift within the outermost (95%) contour for the SIE model and the
bin (0.15 < z < 0.20). We re-analyzed redshift bin 2 by NFW contours are both below the fDM = 1 line.
dividing it into sub-bins, but found the result unchanged on Figure 5 shows the one-dimensional probability distribution
removing the clusters with the lowest axis ratio. The joint result as a function of the ratio of axis ratios of the dark and light
for the remaining redshift bins was then fDM = 0.607+0.21 −0.14 . matter: fDM /fLM . Again, the light (green) line is for the SIE
This is a weaker detection than our final result including the model profile and the dark (red) line is for the NFW model. The
second redshift bin, but still a tentative detection of dark matter vertical dotted line shows where we have fDM = fLM , that is, the
ellipticity. axis ratio as deduced from the shear values equals the axis ratio
The right panel of Figure 3 shows the probability as a function as deduced from the distribution of the cluster galaxies. Both
of the axis ratio for the dark matter distribution. The dark (red) profiles are consistent with having the same ellipticity in the
line shows the result from using an NFW model, and the light light and in the dark matter, fDM /fLM = 1. We find fDM /fLM =
(green) line shows the result from an SIE model. The vertical
−0.25 for the NFW model and fDM /fLM = 1.26−0.17 for the
0.91+0.19 +0.16
dotted line is showing fDM = 1, which represents a circular SIE. The higher SIE result is driven by the larger fDM for the SIE,
mass distribution. which makes the dark matter distribution appear more round.
We can see from the figure that the probability is not totally
Gaussian, with a tail to larger fDM values. (Note that an axis ratio 4.3. Dependence on Close Neighbor Distance
greater than unity corresponds to an ellipse aligned along the y
axis.) We find that 99.6% of the probability is below fDM = 1; We made a decision to exclude all clusters with a more
therefore we consider this to be a reasonable detection of dark massive neighbor within an angular size corresponding to 5 h−1
matter ellipticity. Note that the best-fit NFW result is quite a Mpc at the cluster redshift. In this section, we test how dependent
bit more elliptical than the SIE result, but the NFW result has our results are on this decision. Table 3 shows that our main
a longer tail extending to higher fDM values than the SIE. For result changes little as the cut is made 50% smaller or larger. As
the SIE model, 98.2% of the probability is below fDM = 1. The expected, when a larger radius is used, fewer clusters survive
NFW is a more realistic profile, so we trust these results most. in the catalogue, therefore the uncertainties become larger.
1454 EVANS & BRIDLE Vol. 695

1 SIE
NFW
0.9

0.8
Normalised probability

0.7

0.6

0.5

0.4

0.3

0.2

0.1 Figure 6. Contours show output axis ratios for simulated misalignments between
the light and dark matter. Axes show the input axis ratio fin and standard
0
0 0.5 1 1.5 2 deviation σθ (in degrees) of the angle distribution of the simulated clusters.
fDM / fLM

Figure 5. One-dimensional probability distributions as a function of the ratio misalignment between the dark and light matter must be ∼50◦
fDM /fLM . Light (green) line is for the SIE result while dark (red) line is for the
NFW result.
for our dark matter fDM of ∼0.5 (see Table 2). We conclude that
(A color version of this figure is available in the online journal.)
the misalignment angle must be less than ∼50◦ . If there is any
misalignment between the light and the dark matter, the dark
Table 3 matter will be even more elliptical than the results shown in
Effect of Neighbor Removal on Measured Ellipticity. Results are Shown for the table and can be read off Figure 6 for a given misalignment
the NFW Profile Only. angle.
Cut/ h−1 Mpc No. of Clusters Light Matter fLM Dark Matter fDM
2.5 6934 0.587 + 0.004 − 0.004 0.459 + 0.155 − 0.077 5. CONCLUSIONS AND DISCUSSION
5 4281 0.602 + 0.004 − 0.005 0.480 + 0.136 − 0.086
We have used galaxy clusters from the catalogue of Koester
7.5 2542 0.625 + 0.006 − 0.006 0.614 + 0.205 − 0.139
et al. (2007a) and shear maps as used in Sheldon et al. (2004)
to investigate the light and dark matter ellipticities of galaxy
cluster halos. We rotate the selected clusters so that their major
Using a smaller radius does increase the ellipticity which is
axes are aligned, and stack them according to the method which
to be expected if the main effect is to include more physically
is described in Natarajan & Refregier (2000) for galaxy–galaxy
associated clusters which are more likely to lie along the major
lensing. This is the first time that this method has been applied
axis of the cluster, perhaps due to formation along an intervening
to cluster lensing.
filament. We find similar results for the light ellipticity. Our
Through the pattern imprinted on the shear maps by inter-
main results are unchanged by changing the cluster isolation
vening massive cluster halos, we have detected a dark mat-
criterion.
ter ellipticity of these halos at a 99.6% level, with an axis
4.4. Misalignment Simulations ratio of fDM = 0.48+0.14 2
−0.09 from a joint χ analysis for an
NFW model, using 4281 clusters between 0.10 < z  0.30.
In order to interpret our results, we need to take into We have corrected for dilution of the shear signal by cluster
account possible misalignments during stacking. When we members left in the shear galaxy catalogues but have masked
rotate clusters according to the distribution of the cluster out the central areas where this correction factor grows too
members before stacking, we assume that the orientation of large.
the light is correlated with the orientation of the mass. A result The light matter distribution of the clusters, as traced by
of fDM consistent with 1 could indicate either a circular mass the number density of individual cluster members, is also
distribution or a random alignment between the light used for clearly elliptical, with a joint axis ratio fLM = 0.60+0.004 −0.005
stacking and the dark matter. In the latter case, the circular result for the NFW model. Using the shear catalogue alone gives
would be caused by the stacking of many clusters with different very similar results to using the shear catalogue concatenated
misalignments between the dark and the light matter. with the cluster member catalogue, which means that we are
In order to quantify this effect, we simulate many clusters essentially comparing like with like when comparing dark and
all with the same input axis ratio fin . The misalignment angles light map ellipticities. This is because both the light matter
between the light and the dark matter for the simulated clusters (shear catalogue) and the dark matter have been stacked in the
are random, with a standard deviation σθ . For each value of fin same way.
and σθ , we rotate and stack the clusters in the same way as for We find that the ellipticity of the dark matter distribution
the SDSS data. Therefore, if σθ is large, the input ellipticity will is consistent with the ellipticity of the galaxy number density
be smeared out a great deal, and the output axis ratio fout will be distribution. Our result is limited by the uncertainty in the
close to unity. If σθ is small the output ellipticity will be more ellipticity of the dark matter distribution. The results for the
similar (or in the case of zero misalignment, equal) to fin . NFW and SIE agree within the errors, but any differences could
Figure 6 shows the output dark matter axis ratio as a function be attributed to a changing ellipticity with radius. We have not
of degree of misalignment σθ and the input axis ratio fin . In attempted to measure this effect since the uncertainties are too
order for the input ellipticity to be very elliptical (f → 0), the large.
No. 2, 2009 A DETECTION OF DARK MATTER HALO ELLIPTICITY 1455
Shear maps from neighboring clusters will influence each result. However, to make a proper comparison of our result with
other, making the pattern more elliptical or less, depending theory, it would be necessary to make a theoretical prediction
on the redshift of the neighbors. In order to reduce this effect that takes into account our observation method, especially the
as much as possible, we use only the cluster with the highest overlaps in the shear field due to close neighbors and the impact
r200
number of members (Ngals ) when two (or more) clusters are of selecting the most isolated clusters.
−1 Hoekstra et al. (2004) report on a first weak-lensing detection
closer together than 5 h Mpc, as seen on the sky. We find
that increasing or decreasing the minimum distance to a close of the flattening of galaxy dark matter halos, using data from
neighbor by 50% does not significantly affect our main results. the Red-Sequence Cluster Survey. They find an average galaxy
We have also simulated the effect that a possible misalignment dark matter halo ellipticity of  = 0.33+0.07
−0.09 . They also find a
between the light and dark matter could have, and concluded detection that dark matter halos are rounder than the light map.
that, according to our results, the light and dark matter must be In a later work, Mandelbaum et al. (2006) did not find a definite
misaligned by less than ∼50◦ . detection of this effect in the larger SDSS data set. However,
Our measurement is very insensitive to overall calibration Parker et al. (2007), measuring the ratio of tangential shear in
biases in shear measurements, since these are degenerate with regions close to the semiminor versus that close to the semimajor
cluster mass, rather than cluster ellipticity. It is unlikely that axes of the lens, find some evidence of a halo ellipticity of ∼0.3
biases in shear measurement would vary with an angle around using early data from the CFHTLS. Mandelbaum et al. (2006)
the cluster: residual PSF anisotropies would be oriented at commented that the stronger signal in clusters of galaxies means
random with respect to the cluster major axis and would cancel that there is more chance of making a detection of ellipticity
out on stacking; the cluster member light may leak into the in this higher mass data set. Therefore, we have confirmed
postage stamps used to measure shears of background galaxies, the detection of dark matter halo ellipticity, extending the
but we remove the central region where the number of confirmed measurement to cluster scales. We find no evidence for different
cluster members is significant. ellipticities for the light and dark matter distribution on cluster
Possibly the biggest potential systematic is intrinsic align- scales.
ment of cluster members pointing at the cluster center (Ciotti Mandelbaum et al. (2006) focus on measuring the ratio
& Dutta 1994; Kuhlen et al. 2007; Pereira et al. 2008; Knebe of (dark matter) halo ellipticity to galaxy (light) ellipticity
et al. 2008; Faltenbacher et al. 2008) which would be a prob- h /g , where  = (a 2 − b2 )/(a 2 + b2 ). They find h /g =
lem for us, because we have not made a significant attempt to 0.60 ± 0.38 (68% confidence) for elliptical galaxies, which is
remove cluster members from our analysis and our background most comparable with our result of fDM /fLM = 0.91+0.19 −0.25 (68%
catalogue is not particularly deep. A thorough assessment of confidence), where f = b/a. Repeating our calculation using
this effect is beyond the scope of the current work. To first or- the different ellipticity parameters, we find h /g = 1.37+0.35−0.26 .
der, this would make our observed shear maps less elliptical It is expected that our value is greater than unity, since we find
since the contamination by cluster members is greatest along that the dark matter is more elliptical (lower axis ratio) than the
the cluster major axis, and thus the strong gravitational shears light matter. However, it is not significantly larger. The result
expected along this axis will be partially canceled out by the of Mandelbaum et al. (2006) is the opposite side of unity, but
cluster members which have the opposite ellipticity since they the difference is not significant and we note that our result is for
tend to point at the cluster center. However, a full assessment of clusters and that of Mandelbaum et al. (2006) is for galaxies.
this effect would also have to take into account the variation in The anisotropy of the lensing signal around individual galax-
intrinsic alignment with respect to the cluster major axis, which ies, if definitely detected, has been said to pose a serious prob-
appears to be more complicated (Kuhlen et al. 2007) and possi- lem for alternative theories of gravity (Hoekstra et al. 2004). For
bly weakens the degree of cancellation preferentially along the galaxy clusters it is more complicated, as the dominant source
cluster major axis. of baryons is the intracluster gas. Lensing by clusters has been
In Hopkins et al. (2005), the authors use a large-scale, high- found to pose a problem for Modified Newtonian Dynamics
resolution N-body simulation to predict cluster ellipticities and (MOND), as there seems to be a need to include a dark matter
alignments in a Λ cold dark matter (ΛCDM) universe. They find component (Sanders 2003; Takahashi & Chiba 2007).
an ellipticity Because the gas—the dominant baryonic component of
clusters—is collisional, we can suppose that on cluster scales
 = 1 − f  = 0.33 + 0.05z (20) it will be less elliptical than the light. We may end up con-
cluding that either (1) the gas distribution is elliptical (which
for the redshift range 0 < z < 3. This redshift evolution is we would not expect) or (2) our result is inconsistent with
negligible for the redshift range considered here, and due to MOND. However, further study and simulations of this is clearly
the large uncertainties we do not try to detect any trends with needed.
redshift. For a redshift in the middle of the redshift range of
our cluster sample (z = 0.2), this formula yields an axis ratio
We thank Erin Sheldon and Benjamin Koester for providing
of f = 0.66 which is in good agreement with our result of
us with galaxy catalogues and for very helpful comments and
fDM = 0.48+0.14
−0.09 . suggestions. We also thank the anonymous referee for many
Ho et al. (2006) have used numerical simulations of cluster
helpful comments and suggestions. We also thank Timothy
formation with the aim of investigating the possibility of using
McKay, Alexandre Refregier, Jochen Weller, Håkon Dahle, An-
cluster ellipticities as a cosmological probe. They find that the
dreas Jaunsen, Margrethe Wold, Shirley Ho, Eduardo Cypriano,
mean ellipticity of high-mass clusters is approximated by
Laurie Shaw, Richard Cook, David Sutton, Andrey Kravtsov,

σ8 (z) Ωm  Øystein Elgarøy, Morad Amarzguioui, Terje Fredvik, and Stein
ē(z) = 0.245 1 − 0.256 + 0.00246 . (21) Vidar Hagfors Haugan for helpful discussions. A.K.D.E. ac-
0.9 0.3
knowledges support from the Research Council of Norway,
Using σ8 (z) = 0.8 and Ωm = 0.3, this gives an ellipticity of Project No. 162830. S.L.B. acknowledges support from the
e ∼ 0.2, or f ∼ 0.8. This result is more circular than our main Royal Society in the form of a University Research Fellowship.
1456 EVANS & BRIDLE Vol. 695
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60
Paper II

Shaw, L., Evans, A. K. D.; Bridle, S.,

Predictions of dark matter halo ellipticities measured by lensing,
To be submitted to Astrophyscial Journal (Ap.J.)
 Draft version April 29, 2009
Preprint typeset using LATEX style emulateapj v. 03/07/07

PREDICTIONS OF DARK MATTER HALO ELLIPTICITIES MEASURED BY LENSING

Anna Kathinka Dalland Evans1 , Laurie Shaw2 , Sarah Bridle3
Draft version April 29, 2009

ABSTRACT
We study the ellipticity of galaxy cluster dark matter halos using N-body simulations. Unlike previous
work we simulate the expected ellipticity that would be observed using gravitational lensing. Our analysis
is tuned to match the recent detection of cluster halo ellipticity by Evans & Bridle (2008), and we find
results which are very consistent with those observed. We investigate the ellipticity measured using
lensing, as compared to the more conventional measure of ellipticity that is applied to simulations. We
find good agreement, although the lensing ellipticity appears to show some dependence on the isolation
criteria applied.
Subject headings: cosmology: dark matter — cosmology: galaxy clusters — cosmology: large-scale
structure of universe — cosmology: simulations — galaxies: clusters: general galaxies:
halos —

1. INTRODUCTION a universe with a low matter density, clusters form earlier

Galaxy clusters are gravitationally bound aggregations and are hence expected to be more relaxed today. Sev-
of galaxies, hot gas and dark matter ranging in richness eral studies also look for patterns in the cluster-to-cluster
from a few tens to several thousands of galaxy members. alignment (Kasun & Evrard 2005; Hopkins et al. 2005)
Methods utilizing clusters to probe the large-scale struc- as a means to investigate the idea of hierarchical cluster-
ture in the universe have proliferated in recent years. For ing, wherein clusters would be expected to be aligned with
example, the number of clusters has a useful dependence their neighbours due to matter having converged in dense
on cosmological model (Peacock et al. 2006; Albrecht et al. regions along filamentary structures (Plionis & Basilakos
2006). Since clusters trace the highest peaks in the mat- 2002; West et al. 1995).
ter distribution in the universe, their number density as a A property of clusters that has perhaps been studied in
function of mass and redshift can be exploited to investi- less detail is their ellipticity. There have been some ob-
gate the growth rate of structure, and thus the dark energy servational studies of the ellipticities of galaxy clusters in
equation of state (see e.g. Holder et al. 2001; Levine et al. recent years (Wang et al. 2007; Cypriano et al. 2004; Evans
2002). & Bridle 2008), but most of the work has been theoretical,
Weak gravitational lensing can be used to infer the dis- using large-scale simulations of structure formation (e.g.
tribution of mass in clusters (e.g. Clowe et al. 2006, 2004; Jing 2002; Floor et al. 2003). In general, cosmological N -
Hoekstra et al. 2004; Marshall et al. 2002; Bridle et al. body simulations predict that gravitation causes matter
1998). The three-dimensional mass distribution in halos is on cluster scales to collapse into aspherical shapes (Kasun
complex due to differences in substructure and merger his- & Evrard 2005), in particular triaxial prolate ellipsoids
tories. To simplify comparisons with simulations, one of- (Dubinski & Carlberg 1991; Warren et al. 1992). Hopkins
ten just considers the radially-averaged mass profile. This et al. (2005) and Paz et al. (2006) also measure the two-
can be done on a per cluster basis (e.g. Pedersen & Dahle dimensional projected cluster ellipticity in their simula-
2007; Hoekstra 2007), however the mass estimates of in- tions, thus enabling a direct comparison with observations
dividual clusters are typically very noisy due to the small via gravitational lensing.
number of lensed galaxies. The precision of these mea- Studies find that ellipticities of clusters are interesting
surements can be greatly improved by stacking clusters to because of their implications for differentiating between
obtain a mean mass profile (Johnston et al. 2007; Mandel- cosmological models (Suwa et al. 2003; Flores et al. 2005),
baum et al. 2008; Dahle et al. 2003; Sheldon et al. 2001). their dependence on cosmological parameters (Ho et al.
The general conclusion is that cluster profiles largely agree 2006; Rahman et al. 2004; Splinter et al. 1997) and their
with the Navarro-Frenk-White (NFW; Navarro et al. 1997) effects on estimates of cluster density profiles (Meneghetti
profile predicted by N-body simulations. et al. 2007).
Internal properties of clusters themselves have also been Evans & Bridle (2008) reported a detection of dark mat-
used to place constraints on cosmology. Richstone et al. ter ellipticity using the combined weak lensing shear signal
(1992) explores how the presence of substructure in clus- from about 4000 clusters from the Sloan Digital Sky Sur-
ters, as a measure for the formation epoch of the cluster, vey (SDSS) using the cluster catalogue of Koester et al.
can be used to constrain the density parameter of the uni- (2007). In the current paper, our aim is to compare those
verse (see also Evrard et al. 1993; Mohr et al. 1995). In observational results to results from applying the same
weak lensing method to clusters extracted from a large-
1 Institute of Theoretical Astrophysics, University of Oslo, Box scale N -body simulation. In Sec. 2.1 we describe these
1029, 0315 Oslo, Norway. simulations. In the rest of Sec. 2, we briefly review the
2 Department of Physics, McGill University, 3600 rue University,
method of Evans & Bridle (2008), highlighting the modi-
Montreal, QC, Canada
3 Department of Physics and Astronomy, University College Lon- fications to the method needed in order to facilitate anal-
don, Gower Street, London, WC1E 6BT, UK. ysis of the simulated data set. Our results are presented
in Sec. 3, and a discussion of the results follows in Sec. 4.
2. ANALYSING SIMULATED DATA
Evans & Bridle (2008) measured the ellipticities of clus-
ter dark matter halos using the tangential shear signal
resulting from weak gravitational lensing of background
galaxies by foreground clusters. To increase the signal-to-
noise they combined the measurements from many clusters
of galaxies by stacking the shear maps according to the
cluster center, while retaining any ellipticity information
by rotating the clusters before stacking so that the major
axes of the light were aligned. The ellipticity of the dark
matter halo was then estimated by fitting an elliptical mass
density profile to the stacked shear map. The ellipticity of
the cluster member distribution was estimated by fitting
an elliptical profile to the stacked galaxy distribution. Fig. 1.— Histogram of the projected minor to major axis ratio fp
The main purpose of the present paper is to apply this for our sample of simulated halos. The sample was chosen using our
same technique on simulated clusters. Earlier work on default parameters, such that no halo had a larger neighbor closer
than 5h−1 Mpc, and with a threshold mass of 6 × 1013 h−1 M⊙ .
simulated cluster ellipticities (Hopkins et al. 2005; Flores Note that fp in this particular plot was calculated without applying
et al. 2005; Ho et al. 2006) has used the distribution of a central mask.
simulation particles in halos to measure their ellipticity. saved in 315 time slices. Dark matter halos were identified
However the only way to observe the halo ellipticity is using the Friends-of-Friends (FoF) algorithm with a co-
from gravitational lensing, which measures the projected moving linking length parameter b = 0.2. We set the min-
mass along the line of sight. Ellipticities measured using imum halo mass to 50 particles (MFoF < 1.09 × 1011 M⊙ );
gravitational lensing are therefore potentially increased by all particles in halos with MFoF less than this are dis-
the inclusion of physically close neighbors and filaments. carded. For the remainder, a 3D mesh (with cell side-
However they may also be decreased by the contribution length 4ǫ = 17.6h−1 kpc) is placed around each halo, onto
from coincidental overlaps by physically separated objects which we interpolate the halo particles. By summing up
projected along the line of sight. Until now, such effects the mass in each mesh cell along one direction, we pro-
have not been accounted for in the simulations. duce a 2D ‘postage-stamp’ projected mass-map for each
The convergence κ is defined in General Relativity by cluster. Note that each postage-stamp is produced having
rescaling the projected mass per unit area in the lens, and rotated the 3D mesh around the cluster so that one face
is given in terms of the projected surface mass density of lies tangential to the plane of the lightcone ‘sky’. Following
the lens, Σ, as: this step, we are left with a library containing the angu-
Σ(θ) lar positions, redshifts and mass planes for the one million
κ(θ) = , (1)
Σcrit lightcone halos above our mass threshold, encompassing
the redshift range 0 < z < 0.5.
where θ is an axis vector on the plane of the sky and Σcrit
Convergence maps are produced by projecting down the
is the critical surface density of the lens, given by
lightcone, summing up the contribution of all the clusters
c2 Ds along the line of sight of each map pixel. The value of each
Σcrit = . (2) map pixel is given by
4πG Dd Dds
X
In this expression, Ds , Dd and Dds are, respectively, the κ(θ, φ) = g(zi )Σi (zi ) , (3)
angular diameter distances between observer and source, i
observer and lens (deflector) and between lens and source.
where the the sum is over each cluster i that intersects the
2.1. Description of the simulations line of sight of the pixel (with angular co-ordinates θ, φ), zi
is the redshift of the cluster and Σi is the projected mass
To create our sample of convergence (or κ) maps, we density of the cluster at this position. The factor g(zi ) is
begin with the output of a large (N = 10243 parti- given by
cles) cosmological dark matter simulation. The cosmol- Z
ogy was chosen to be consistent with that measured from 4πG zs DA (zi )(DA (zs′ ) − DA (zi )) ′
g(zi ) = 2 n(zs′ ) dzs ,
the 3rd-year WMAP data combined with large-scale struc- c 0 DA (zs′ )
ture observations (Spergel et al. 2007), namely a spa- (4)
tially flat LCDM model with parameters: baryon density where n(zs ) is the redshift distribution of background
Ωb = 0.044; total matter density Ωm = 0.26; cosmolog- (lensed) galaxies (normalised such that the integral over
ical constant ΩΛ = 0.74; linear matter power spectrum 0 < z < 0.5 is unity), and DA (z) is the angular diameter
amplitude σ8 = 0.77; primordial scalar spectral index distance at redshift z. In this work we use the same n(zs )
ns = 0.95; and Hubble constant H0 = 72km s−1 Mpc−1 as in Evans & Bridle (2008), estimated from Fig. 3 and
(i.e. h = 0.72 = H0 /100km s−1Mpc−1 ). The simulated Eq. 8 in Sheldon et al. (2001), with zc = 0.22.
volume is a periodic cube of size L = 320h−1Mpc; the We produce a base sample of 28 10x10 deg convergence
particle mass mp = 2.2 × 109 h−1 M⊙ , and the cubic spline maps encompassing the full mass and redshift range of our
softening length ǫ = 3.2h−1 kpc. halo catalogue, with a pixel size of 0.25 arcminutes. Each
The matter distribution in a light cone covering one oc- map has an accompanying halo catalogue giving R200 (the
tant of the sky (≈ 5200 deg2 ) extending to z = 0.5 was radius in which the spherically-average density is 200ρcrit ),

2
 Fig. 2.— Illustrating a typical individual cluster (top row) as well as a stacked cluster (bottom row), in which the clusters have been rotated
to have their major axes along the x-axis. The left hand panels show the convergence maps, and the right hand panels show the tangential
shear. Contours for the convergence maps are at 10, 20 and 50 % of the maximum individual and stacked convergence values, while contours
for the shear maps are at 20 and 50% of the maximum individual and stacked shear values. The individual cluster (top panels) has fp = 0.6
and a rotation angle of ∼ 35 deg anticlockwise to the positive horizontal axis, as calculated using two-dimensional quadrupole moments of
particles isolated in three-dimensions around the cluster. The stacked cluster contains ∼ 200 clusters in the redshift range 0.15 < z < 0.20.
The dashed circle shows the location of the mask.
M200 and the redshift of every cluster in the map. We also simulation, from all clusters above the mass threshold of
include in this catalogue an estimate for the projected el- 6 × 1013 h−1 M⊙ that are accepted for the main analysis
lipticity and orientation of each cluster. This is calculated (see Section 2.2). We fit a Gaussian to the histogram, and
by measuring the 2D moment of inertia tensor from the in- in Fig. 1 show the best fit plot which is centred at f = 0.6
dividual mass-planes for each cluster, including the contri- and has a standard deviation of 0.2. Note that, in order
bution of all matter within 3Mpc (comoving) of the cluster to facilitate comparisons with other simulation studies, the
potential minimum. Solving for the eigenvectors gives the axis ratios in Fig. 1 are calculated without applying the
minor to major principle axis ratio for each halo. Before mask, whereas in the rest of this paper the mask has been
calculating the axis ratio, a central mask of radius 0.5h−1 applied to the simulated halos.
Mpc was applied to the halos, see Sec. 2.4. One advantage of our map-making procedure is that it
The orientiation of a cluster is defined as the angle be- is straightforward to reproduce maps that contain only
tween the principle axis and the line of constant decli- the contribution of halos that lie within a particular mass
nation at the angular position of the cluster in the map. or redshift range. In order to assess the impact of sur-
This information is used in the following section to stack rounding and intervening structure on the weak-lensing
clusters so that their principle axes are aligned before mea- estimates of cluster ellipticity, we also have produced a
suring ellipticity from their combined shear profiles. sub-sample of 48 maps (at the same resolution) , each con-
In Fig. 1 we show a histogram of the fp axis ratio cal- taining only a single cluster. These single cluster κ maps
culated from quadrupole moments of the particles in the are discussed further in Section 3.1.

3
 2.2. From projected mass maps to shear maps We extract a postage stamp of shears in a region 10 ×
Our aim is to study the shear maps made from our sim- 10h−1 Mpc around each cluster centre. This matches the
ulated convergence maps to quantify the accuracy of mea- size used in Evans & Bridle (2008) which was motivated by
surements of cluster ellipticity, both on a cluster-by-cluster reducing the contamination from neighboring large-scale
basis and via stacking.The first step is thus to convolve the structure.
convergence maps to make corresponding shear maps. The The halo ellipticity constraint from individual clusters
shear pattern can be expressed as a convolution in terms is usually very noisy and therefore more useful constraints
of the convergence κ. We use can be obtained by stacking many clusters on top of each
Z other. To retain the ellipticity information we must rotate
1 the clusters before stacking so that the features of interest
γ(θ) = D(θ − θ′ )κ(θ′ )d2 θ′ , (5)
π are aligned (as suggested by Natarajan & Refregier 2000).
θ2 − θ12 − 2iθ1 θ2 In this way, a single ellipticity for a typical cluster can be
D(θ) = 2 (6) obtained for the cluster sample. We rotate the clusters
|θ|4
according to the rotation angle calculated from particles
(e.g. Bridle et al. 1998) where θ1 and θ2 are components in the simulation, as described in Section 2.1. We then
of θ (axis vectors on the sky). calculate the mean of the tangential shear in each pixel in
We use all halos with a FoF mass above 6 × 1013 h−1 M⊙ , the stacked shear map.
which corresponds to the minimum number of galaxies Fig. 2 illustrates projected convergence (left panels) and
used in Evans & Bridle (2008) , which was 10 galaxies tangential shear (right panels) for a single cluster (upper
within R200 . We restrict the redshift range of our halo row) and a stack of ∼ 200 clusters (lower row). The upper
sample to 0.1 < z < 0.3, as in Evans & Bridle (2008). Due panels show that the single cluster is aligned at an angle
to the redshift-dependant galaxy number density distribu- with respect to the horizontal axis (∼ 35 deg anticlockwise
tion we have adopted, (n(zs ), see Sec. 2.1) higher redshift for this particular cluster), whereas in the lower panel all
clusters have very small shear signals. We further divide clusters in the stack have been rotated to align with the
this redshift range into four bins, each of width 0.05 in z. horizontal axis in order to retain any ellipticity information
We did not use clusters that were too close to the projected when stacking. When running our main analysis we apply
mass map edge, using the requirement that the distance a mask at the centre of the cluster, see Section 2.4, which
from a cluster centre to the map edge be greater than 12 completely removes this central region. The dashed circle
h−1 Mpc (at the redshift of the cluster). shows the location of the mask.
2.3. Isolating, stacking and rotating the shear maps 2.4. Modelling
Clusters that are close to each other on the sky require To model the cluster mass distribution, we use the
careful consideration. As the shear field of a cluster is Navarro-Frenk-White (NFW) model. We fix the NFW
contaminated by neighbouring clusters, a study interested concentration parameter to 5 for the analysis presented
in making a raw measurement of cluster ellipticities must here. The shear for an elliptical mass distribution is cal-
attempt to find clusters which are isolated on the sky. culated using the equations in Keeton (2001) which are
An alternative option would be to include all clusters; derived from those in Schramm (1990). A shear map using
The stacked shear map would then be a two-dimensional these equations is illustrated in Fig. 1 of Bridle & Abdalla
galaxy shear - cluster position two point correlation func- (2007).
tion. Extraction of the cluster ellipticities would then re- To estimate the ellipticity of the dark matter distribu-
quire careful modelling including the two-dimensional two tion we calculate probability as a function of (projected)
point function of cluster positions, taking into account the cluster minor-to-major axis ratio fl and cluster mass M200 .
tendency of cluster neighbors to sit preferentially along We calculate the probability of the predicted (or model)
the cluster major axis, for physically close clusters. We shear using the χ2 statistic and a Jeffrey’s prior on the
choose the first option here, but show results for different mass, as appropriate when the order of magnitude of the
neighbor distance criteria. mass is unknown. Because the simulated shear maps have
We follow the isolation criteria of Evans & Bridle (2008) no noise, we set the uncertainty in the χ2 -calculation to an
who required that there be no larger neighbouring clusters arbitrary value that still gives a well-defined peak in the
within a projected physical distance of 5 h−1 Mpc from any probability distribution. We marginalize over the cluster
one cluster. Procedurally, we work through all the clus- mass to obtain the probability as a function of the axis
ters from low to high redshift, and for each we calculate ratio fl .
the angular separation of the nearest cluster, considering We make stacked shear maps in four redshift bins be-
all clusters from all redshifts. We compare this to our tween 0.1 < z < 0.3 and fit each as a function of mass
physical cut-off distance at the redshift of the cluster in and axis ratio, marginalising over mass to obtain the one
question. If the separation between two clusters is below 5 dimensional probability as a function of fl . A single result
h−1 Mpc then we remove the least massive from our anal- for fl is then obtained by multiplying the probabilities for
ysis. Evans & Bridle (2008) defined cluster ‘size’ using all four redshift bins. In using this procedure we implicitly
the richness measured by the number of cluster members assume that fl is identical in all redshift bins but that the
within a circular aperture of radius R200 . In the current others parameters can vary.
paper, we use the same isolation distance but compare We use the angular diameter distance at the center of
size as measured by the halo mass M200 measured from each redshift bin to calculate for each cluster the size of
the simulations. This process resulted in a list of 1364 the postage stamp, the size of the mask and the minimum
clusters. We also investigate the effect on our results by closest distance to a neighbouring cluster.
decreasing or increasing this radius by 50%. When using shear catalgoues without spectroscopic red-

4
 Fig. 3.— The squares and diamonds in the two panels show the axis ratio measured using lensing (fl ) versus the conventional method for
measuring cluster ellipticities from simulations (fp ). There is one point per cluster. The shear maps used for the analysis in the left hand panel
are the 48 maps each containing only a single cluster i.e. no surrounding large-scale structure. The right panel contains all individual clusters
(after neighbour removal) from the 28 original projected mass maps including surrounding large-scale structure. Clusters that are marked with
squares in the left panel are also marked with squares in the right panel. The dotted line in each panel shows the line fl = fp . Left panel:
square symbols show a good match between fp and fl . Number of clusters in each fp bin: 4, 20, 13 and 11, giving 48 in total. Filled circles
joined by dashed line show the mean fl from the individual cluster results (i.e. √ filled circles are mean of the square symbols in each fp bin).
Error bars on these points show the standard deviation of fl values divided by nclust in each fp bin. Right panel: There is a large scatter
in the individual analysis, showing that the surrounding structure affects our results even though we try to isolate the clusters using the close
neighbour distances. We excluded from the right-hand panel the 520 clusters that resulted in fl > 2. Crosses joined by a solid line show fl
obtained from stacking the respective individual clusters in five bins of fp , versus the mean fp value in each bin.

shift information, there will always be some degree of con- tive radial weighting in a lensing analysis than in the
tamination of the shear catalogue by cluster members. quadrupole moments. By fitting an NFW profile we ef-
This can to a large extent be corrected for a posteriori fectively measure weighted quadrupole moments in which
using a correction factor which increases towards the cen- the importance of the outer regions is reduced. The filled
tre, but has a non-negligible uncertainty and therefore the circles joined by a dashed line are the mean of the square
central regions are best excluded from the lensing analy- shapes (fl ) in each fp bin, plotted versus the mean of fp in
sis. The central regions are also best avoided due to ob- each bin. The error bars show the error on this mean, cal-
scuration and contamination of the shear measurements culated by dividing the standard deviation of the points
by cluster members, and potential problems identifying by the square root of the number of clusters in the bin.
the center on which to stack the shear maps. The cen- We can see that on average the two ellipticity estimation
tral regions in Evans & Bridle (2008) were omitted using methods give consistent results within the errors.
a circular mask with rmask = 0.5h−1 Mpc. In the present We now produce lensing convergence maps for the whole
paper we have also applied the mask in order to mimic field – including fore- and background structures along the
observational results. line-of-sight – and convert them into shear maps. The
shear map around a cluster will now contain contamina-
3. RESULTS tion from surrounding structures. We now use all 28 fields
We first measure cluster ellipticities from shear maps of available, in which 1364 clusters survive our mass, red-
individual clusters (i.e. omitting all foreground and back- shift and isolation criteria. The diamonds in the right-
ground halos from the simulated maps). We compare the hand panel of Fig. 3 show the axis ratio estimated from
average of such ellipticities to the ellipticity measured from each postage stamp around each individual cluster. We
stacked shear maps. We calculate this as a function of the see that there is a huge scatter between the fp calculated
ellipticity calculated directly from the cluster particles in from the simulations and the fl calulated from the lensing
the simulation. We then investigate how the stacked ellip- analysis. This scatter is due to the disturbing influence of
ticity depends on the isolation criteria. the surrounding large-scale structure.
A large number (520) of clusters for the individual anal-
3.1. Results from individual clusters ysis shown in the right-hand panel of Fig. 3 resulted in fl
We start by analysing shear maps containing signal > 2. An axis ratio above unity means that the best-fit
purely from the cluster of interest. This is achieved by cluster is aligned along the y-axis, despite the fact that
making a convergence map using only the particles identi- we have rotated the shear map so the major axis of the
fied to be in the cluster. This is performed for a subset of cluster particles lies along the x-axis. This is to be ex-
the whole cluster catalogue, using just the first three of the pected when there is significant uncorrelated neighboring
28 fields, yielding 48 clusters. The squares in the left hand structure that survives our isolation criteria. For example,
panel of Fig. 3 show the lensing axis ratios (fl ) found from clusters that appear close in projection, but which are at
each individual cluster, without stacking. This is plotted different redshifts.
as a function of the axis ratio found from the quadrupole There are also a large number of diamonds at fl ∼ 0.2.
moments of the individual projected mass-maps of each Surrounding structure predominantly along the horizon-
cluster (fp ), as described in Section 2.1. We see that there tal axis will give a highly elliptical fl . Structure that is
is a good correlation between the two. physically associated with the cluster will tend to lie along
The scatter is expected because of the different effec- this axis, possibly due to formation along a filament. This

5
 The clusters appear slightly more circular for larger cut
radii which is to be expected if the main effect is to exclude
more physically associated clusters which are more likely
to lie along the major axis of the cluster. For a smaller
close neighbor distance value, we expect more clusters to
be accepted. For the crosses in Fig. 4, the number of clus-
ters accepted for each cut radius was 6934, 4281, 2542 in
increasing order of radius size. The corresponding number
of clusters used in the analysis of the simulated data is
2193, 1364, 815.
The dashed line in Fig. 4 shows the mean value of fp for
all clusters in the sample for each close neighbour distance,
as discussed in Section 2.1. This is roughly a constant with
close neighbour distance. We expect this line to be flatter
than that for the lensing stack because these axis ratios
Fig. 4.— Stacked results for different minimum close neighbour have been calculated from the particles identified to be
distances. Crosses with error bars: Axis ratio fDM combined from within the cluster and therefore are not contaminated by
all redshift bins, reproduced from table 3 of Evans & Bridle (2008), neighboring structures.
as a function of minimum distance allowed to the closest neighbour-
ing cluster. Number of clusters used in that analysis for each close
This result suggests that it may not always be reasonable
neighbor distance (i.e. 2.5, 5 and 7h−1 Mpc): 6934, 4281, 2542. Solid to use the ellipticities from the conventional method, using
line shows results fl from lensing, combining results from all redshift quadrupole moments of cluster particles, as a proxy for the
bins. Number of clusters used for each value of the closest neigh- ellipticity expected from lensing.
bour distance: 2193, 1364, 815. Dashed line shows the mean of all
fp from particles in the simulation. This is practically a constant The dotted line shows the results from Hopkins et al.
since the calculations of fp were independent of close neighbour dis- (2005) (for clusters at redshift z = 0.2) and is in very
tance. Dotted line shows results from Hopkins et al. (2005) in good good agreement with our results from a similar method.
agreement with our fp -line (dashed).
structure is not included in the fp -values. 4. CONCLUSIONS AND DISCUSSIONS
The clusters in the left-hand panel of Fig. 3 are a subset
of those in the right. In order to see the difference that Through the pattern imprinted on the shear maps by the
the large-scale structure makes for a particular sample, we intervening massive cluster halos, Evans & Bridle (2008)
have identified the clusters in the sub-sample (individual detected a dark matter ellipticity of these halos at a 99.6%
points in the left panel) with a square symbol in the right level, with an axis ratio of f = 0.48+0.14 2
−0.09 from a joint χ -
panel. analysis for an NFW model. We have repeated this anal-
In summary, Fig. 3 illustrates the fact that, although ysis for simulated clusters and found excellent agreement
the simulations are noise-free, we still need to stack the with the observational results.
clusters in order to obtain a reasonable result. If we We have illustrated the the need to stack cluster shear
stack enough clusters, the substructures in the surround- maps to average out the effect of neighboring structures,
ing structure will tend to cancel out. and found that on doing so, the resulting axis ratios agree
well with those calculated using the conventional method
3.2. Results from stacking clusters for a given isolation criterion. The trend with isolation
criterion appears slightly different for the conventional
The crosses connected by a solid line in the right panel method than the lensing method though, as might be ex-
of Fig. 3 shows the result of stacking the clusters five bins, pected due to contamination by uncorrelated structures
each of width 0.2 in fp . Resulting lensing axis ratios fl are when more lax isolation criteria are applied.
plotted against the mean fp -value in each bin. We stacked Hopkins et al. (2005) use N -body simulations out to
the clusters in four bins of redshift within each fp bin, and z = 3 to examine the evolution of ellipticity and align-
then combined them to obtain a single, one-dimensional ment of clusters with redshift in a ΛCDM cosmology. They
marginalised probability function for fl for each fp bin. have results for the full 3-dimensions of their simulations
We see that there is broad agreement between the stacked as well as projected on the sky in two dimensions. Their
lensing axis ratios and the conventional method, although results show that the mean dark matter ellipticity lies at
there is some noise. hǫi = 1 − f ∼ 0.3 − 0.5. Their results show an increasing
These figures suggest that the cluster ellipticity as mea- elliptcity with redshift, i.e. a decreasing axis ratio from
sured from particles in the simulations is a reasonable hf i ∼ 0.7 at z = 0 to hf i ∼ 0.5 at z = 0.5. The authors
proxy for the ellipticity as measured by lensing. also conclude that ellipticities increase with cluster mass
3.3. Lensing ellipticity as a function of isolation criterion and cluster radius, and that higher mass clusters are more
elliptical at all z. For our redshift range, the increase of
The solid line in Fig. 4 shows the ellipticity of the stacked ellipticities with redshift reported by Hopkins et al. (2005)
simulated shear map as a function of the closest neighbor is imperceptible. However, our simulated results fp agree
distance used in our isolation criterion. The equivalent extremely well with theirs.
results obtained using SDSS data are shown by the points
and error bars, which are taken from table 3 in Evans &
Bridle (2008). We see excellent agreement between the two We would like to thank Terje Fredvik, Stein Vidar Hag-
studies. Note that the error bars are strongly correlated fors Haugan and Øystein Elgarøy for helpful discussions.
since there is an overlap between the samples of clusters AKDE acknowledges support from the Research Council
used. of Norway, Project No. 162830. SLB acknowledges sup-

6
port from the Royal Society in the form of a University Research Fellowship.
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7
70
Paper III

Evans, A. K. D.; Wehus, I. K.; Grøn, Ø.; Elgarøy, Ø.,

Geometrical constraints on dark energy,
Astron. and Astrophys., 430 (2005), 399-410
A&A 430, 399–410 (2005)
DOI: 10.1051/0004-6361:20041590
Astronomy


&
c ESO 2005 Astrophysics

Geometrical constraints on dark energy

A. K. D. Evans1 , I. K. Wehus2 , Ø. Grøn3,2 , and Ø. Elgarøy1

1
Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029, Blindern, 0315 Oslo, Norway
e-mail: oelgaroy@astro.uio.no
2
Department of Physics, University of Oslo, PO Box 1048, Blindern, 0316 Oslo, Norway
3
Oslo College, Faculty of Engineering, Cort Adelers gt. 30, 0254 Oslo, Norway

Received 3 July 2004 / Accepted 23 September 2004

Abstract. We explore the recently introduced statefinder parameters. After reviewing their basic properties, we calculate the
statefinder parameters for a variety of cosmological models, and investigate their usefulness as a means of theoretical classi-
fication of dark energy models. We then go on to consider their use in obtaining constraints on dark energy from present and
future supernovae type Ia data sets. We find that it is non-trivial to extract the statefinders from the data in a model-independent
way, and one of our results indicates that parametrizing the dark energy density as a polynomial of second order in the redshift
is inadequate. Hence, while a useful theoretical and visual tool, applying the statefinders to observations is not straightforward.

Key words. cosmology: theory – cosmology: cosmological parameters

1. Introduction state w(z) ≡ px /ρx , where px and ρx are the pressure and the en-
ergy density, respectively, of the dark energy component in the
It is generally accepted that we live in an accelerating uni- model. One can then Taylor expand w(z) around z = 0. The cur-
verse. Early indications of this fact came from the magnitude- rent data allow only relatively weak constraints on the zeroth-
redshift relationship of galaxies (Solheim 1966), but the re- order term w0 to be derived. A problem with this approach is
ality of cosmic acceleration was not taken seriously until the that some attempts at explaining the accelerating Universe do
magnitude-redshift relationship was measured recently using not involve a dark component at all, but rather propose modi-
high-redshift supernovae type Ia (SNIa) as standard candles fications of the Friedmann equations (Deffayet 2001; Deffayet
(Riess et al. 1998; Perlmutter et al. 1999). The observations can et al. 2002; Dvali et al. 2000; Freese & Lewis 2002; Gondolo
be explained by invoking a contribution to the energy density & Freese 2003; Sahni & Shtanov 2003). Furthermore, it is pos-
with negative pressure, the simplest possibility being Lorentz sible for two different dark energy models to give the same
Invariant Vacuum Energy (LIVE), represented by a cosmolog- equation of state, as discussed by Padmanabhan (2002) and
ical constant. Independent evidence for a non-standard contri- Padmanabhan & Choudhury (2003).
bution to the energy budget of the universe comes from e.g. the Recently, an alternative way of classifying dark energy
combination of the power spectrum of the cosmic microwave models using geometrical quantities was proposed (Sahni et al.
background (CMB) temperature anisotropies and large-scale 2003, Alam et al. 2003). These so-called statefinder parame-
structure: the position of the first peak in the CMB power spec- ters are constructed from the Hubble parameter H(z) and its
trum is consistent with the universe having zero spatial curva- derivatives, and in order to extract these quantities in a model-
ture, which means that the energy density is equal to the critical independent way from the data, one has to parametrize H in
density. However, several probes of the large-scale matter dis- an appropriate way. This approach was investigated at length
tribution show that the contribution of standard sources of en- in Alam et al. (2003) using simulated data from a SNAP1 -type
ergy density, whether luminous or dark, is only a fraction of the experiment. In this paper, we present a further investigation of
critical density. Thus, an extra, unknown component is needed this formalism. We generalize the formalism to universe mod-
to explain the observations (Efstathiou et al. 2002; Tegmark els with spatial curvature in Sect. 2, and give expressions for
et al. 2004). the statefinder parameters in several specific dark energy mod-
Several models describing an accelerated universe have els. In the same section, we also take a detailed look at how
been suggested. Typically, they are tested against the SNIa data the statefinder parameters behave for quintessence models, and
on a model-by-model basis using the relationship between lu- show that some of the statements about these models in Alam
minosity distance and redshift, dL (z), defined by the model. et al. (2003) have to be modified. In Sect. 3 we discuss what can
Another popular approach is to parametrize classes of dark en-
ergy models by their prediction for the so-called equation of 1
see http://snap.lbl.gov

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20041590

400 A. K. D. Evans et al.: Geometrical constraints on dark energy

be learned from current SNIa data, considering both direct χ2 with k = 0 corresponding to a spatially flat universe. The dust
fitting of model parameters to data, and statefinder parameters. component is pressureless, so the equation of energy conserva-
In Sect. 4 we look at simulated data from an idealized SNIa sur- tion implies
vey, showing that reconstruction of the statefinder parameters
ρm = ρm0 a−3 . (8)
from data is likely to be non-trivial. Finally, Sect. 5 contains
our conclusions. This gives for the density of dark energy:
3k
2. Statefinder parameters: Definitions ρx = ρc − ρm −
8πGa2
and properties 3
= (H 2 + kx2 − Ωm0 H02 x3 ), (9)
The Friedmann-Robertson-Walker models of the universe have 8πG
earlier been characterized by the Hubble parameter and the where and Ωm0 and Ωx0 are the present densities of matter and
deceleration parameter, which depend on the first and second dark energy, respectively, in units of the present critical den-
derivatives of the scale factor, respectively: sity ρc0 = 3H02 /8πG. In the following, we will use the notation
Ωi ≡ 8πGρi (t)/3H 2 (t), Ωi0 ≡ Ωi (t = t0 ), where t0 is the present
ȧ 
H = (1) age of the Universe, and also Ω = i Ωi . From Friedmann’s
a acceleration equation
äa Ḣ
q = − = − 2 − 1, (2) ä 4πG 
ȧ 2 H =− (ρi + 3pi ), (10)
a 3 i
where dots denote differentiation with respect to time t. The
proposed SNAP satellite will provide accurate determinations where pi is the contribution to the pressure from component i,
of the luminosity distance and redshift of more than 2000 su- it follows that
pernovae of type Ia. These data will permit a very precise de-    
H2 Ω 3 1 2 k
termination of a(z). It will then be important to include also px = q− = (H ) x − x2 − H 2 · (11)
4πG 2 8πG 3 3
the third derivative of the scale factor in our characterization of
different universe models. Hence, if dark energy is described by an equation of state px =
Sahni and coworkers (Sahni et al. 2003; Alam et al. w(x)ρx , we have
2003) recently proposed a new pair of parameters (r, s) called 2 
3 (H ) x − H − 3 x
1 2 k 2
statefinders as a means of distinguishing between different w(x) = · (12)
types of dark energy. The statefinders were introduced to char- H 2 + kx2 − H02 Ωm0 x3
acterize flat universe models with cold matter (dust) and dark In the following subsections, we calculate statefinder parame-
energy. They were defined as ters for universe models with different types of dark energy.
...
a Ḧ Ḣ
r = = +3 2 +1 (3)
aH 3 H 3 H 2.1. Models with an equation of state p = w (z)ρ
r−1
s =
· (4) First we consider dark energy obeying an equation of state
3 q − 12 of the form px = wρx , where w may be time-dependent.
Introducing the cosmic redshift 1 + z = 1/a ≡ x, we have Quintessence models (Wetterich 1988; Peebles & Ratra 1988),
Ḣ = −H  H/a, where H  = dH/dx, the deceleration parameter where the dark energy is provided by a scalar field evolving in
is given by time, fall in this category. The formalism in Sahni et al. (2003)
and Alam et al. (2003) will be generalized to permit universe
H models with spatial curvature. Then Eq. (4) is generalized to
q(x) = x − 1. (5)
H
r−Ω
Calculating r, making use of a = −a2 , we obtain s= , (13)
3(q − Ω/2)
 2 
H H H  2 where Ω = Ωm + Ωx = 1 − Ωk , and Ωk = −k/(a2 H 2 ).
r(x) = 1 − 2 x + + x. (6)
H H2 H The deceleration parameter can be expressed as
The statefinder s(x), for flat universe models, is then found by 1 1
inserting the expressions (5) and (6) into Eq. (4). The general- q= [Ωm + (1 + 3w)Ωx ] = (Ω + 3wΩx ). (14)
2 2
ization to non-flat models will be given in the next subsection.
After differentiation of Eq. (2) and some simple algebra one
The Friedmann equation takes the form2
finds
8πG k q̇
H2 = (ρm + ρx ) − 2 , (7) r = 2q2 + q − , (15)
3 a H
where ρm is the density of cold matter and ρx is the density and further manipulations lead to
of the dark energy, and k = −1, 0, 1 is the curvature parameter  
9 3 ẇ
2
Throughout this paper we use units where the speed of light c = 1.
r = Ωm + 1 + w(1 + w) Ωx − Ωx · (16)
2 2H
 A. K. D. Evans et al.: Geometrical constraints on dark energy 401
2
Inserting Eq. (16) into Eq. (13) gives
1 ẇ
s =1+w− · (17)
3 wH 1.5

For a flat universe Ωm + Ωx = 1 and the expression for r sim-

plifies to 1
r

9 3 ẇ
r = 1 + w(1 + w)Ωx − Ωx . (18)
2 2H 0.5

Note that for the case of LIVE, w = −1 = constant, and one

finds r = Ω, s = 0 for all redshifts. For a model with curva-
ture and matter only one gets r = 2q = Ωm , s = 2/3. The 0

same result is obtained for a flat model with matter and dark –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1
energy with a constant equation of state w = −1/3, which is q

the equation of state of a frustrated network of non-Abelian Fig. 1. The q − r-plane for flat matter+quiessence models. The hori-
cosmic strings (Eichler 1996; Bucher & Spergel 1999). Thus, zontal curve has w = −1 (ΛCDM). Then w increases by 1/10 coun-
the statefinder parameters cannot distinguish between these two terclockwise until we reach w = 1 in the upper right. When Ωx0 = 0
models. However, neither of these two model universes are all models start at the point q = 0.5, r = 1 (Einstein-de Sitter model).
favoured by the current data (for one thing, they are both de- As Ωx0 increases every model moves towards the solid curve which
celerating), so this is probably an example of academic interest marks Ωx0 = 1. The crosses mark the present epoch.
only.
For a constant w, and Ωm0 + Ωx0 = 1, the q–r plane for Then,
different values of Ωx and w is shown in Fig. 1. Quintessence φ̇2 V̇
with w = constant is called quiessence. The relation between q r = Ω + 12πG + 8πG 3 , (24)
H2 H
and r for flat universe models with matter+quiessence is found
and furthermore,
by eliminating Ωx between Eq. (14), with Ω = 1, and Eq. (16).  
This gives Ω 3 px 4πG 1 2
q − = wΩx = 4πG 2 = 2 φ̇ − V · (25)
2 2 H H 2
1
r = 3(1 + w)q − (1 + 3w), (19) Hence the statefinder s is
2

2 φ̇2 + 23 HV̇
which is the equation of the dotted straight lines in Fig. 1. When s= · (26)
Ωx = 1, all models lie on the solid curve given by φ̇2 − 2V
For models with matter+quintessence+curvature, the
3 1
q = w+ (20) Friedmann and energy conservation equations give
2 2  
9 1 1 2
r = w(1 + w) + 1, (21) Ḣ = −3H + 2
ρm − V(φ) + ρk (27)
2 2M 2 2 3
or 1 2
φ̇ = 3H 2 M 2 − ρm − V(φ) − ρk (28)
2
r = 2q + q,
2
(22) ρ̇m = −3Hρm (29)
ρ̇k = −2Hρk , (30)
in accordance with Eq. (15) since q̇ = 0 for these models. This
curve is the lower bound for all models with a constant w. For and
−1 ≤ w ≤ 0, all matter+quiessence models will at any time 1
q = Ωm + 2Ωkin − Ωpot (31)
fall in the sector between this curve and the r = 1-line which 2
corresponds to ΛCDM. The results shown in Alam et al. (2003) MV 
r = Ωm + 10Ωkin + Ωpot + 3 6Ωkin · (32)
seem to indicate that all matter+quintessence models will fall ρc
within this same sector as the matter+quiessence models do. As customary when discussing quintessence, we have intro-
However, as we will show below, this is not strictly correct. duced the Planck mass M 2 = 1/8πG. Furthermore, we have
defined Ωkin = φ̇2 /2ρc , and Ωpot = V(φ)/ρc . For an exponen-
2.2. Scalar field models tial potential, V(φ) = A exp(−λφ/M), looking at values at the
present epoch, and eliminating Ωpot0 , using Ωm0 +Ωkin0 +Ωpot0 +
If the source of the dark energy is a scalar field φ, as in the Ωk0 = 1, one obtains
quintessence models (Wetterich 1988; Peebles & Ratra 1988),
3
the equation of state factor w is q0 = Ωm0 − (1 − Ωk0 ) + 3Ωkin0 (33)
2
φ̇2 − 2V(φ) r0 = (1 − Ωk0 ) + 9Ωkin0
w= · (23)
φ̇2 + 2V(φ) −3λ 6Ωkin0 (1 − Ωk0 − Ωm0 − Ωkin0 )· (34)
402 A. K. D. Evans et al.: Geometrical constraints on dark energy

12
1

10

0.8
8

6
0.6

4
r_0 r
0.4
2

0 0.2

–2
0

–4

–0.2
–6
–1 –0.5 0 0.5 1 1.5 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4

q_0 q
2

Fig. 3. Time-evolution of q and r for models with matter and

1.5
quintessence with an exponential potential. The crosses mark the
present epoch. The diamond represents the present ΛCDM model.
The curve on top has λ = 0.2 and then λ increases by 0.2 for
each curve going counter-clockwise until we reach λ = 2 to
1

the right. The corresponding values for Ωkin today are Ωkin0 =
r_0 0.5 0.002, 0.01, 0.02, 0.04, 0.06, 0.09, 0.12, 0.165, 0.22, 0.29. The dotted
curve shows the area all matter+quiessence models must lie within
0 at all times. We see that all models will eventually move towards this
curve.
–0.5

in the potential chosen to give Ωkin0 as stated in the caption

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6
of Fig. 3. This corresponds to the universe being matter domi-
q_0 nated at earlier times. When Ωpot0  Ωkin0 we have high accel-
eration today. Choosing Ωkin0 = 0 will again give us ΛCDM.
Fig. 2. Present values of q and r for matter+quintessence with an ex-
The three rightmost curves in the figure have λ2 > 2 and no
ponential potential. Top panel: from top to bottom the different curves
have λ = −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5. They all start at the point
eternal acceleration, although the λ = 1.6 universe accelerates
(q0 (Ωkin = .73) = 1.595, r0 (Ωkin = .73) = 7.57) (matter+Zeldovich today. It seems that in order to get a universe close to what we
gas (px = ρx )). As Ωkin decreases when we move to the left, they join observe, r and q for models with matter+quintessence with an
at the point (q0 (Ωkin = 0) = −0.595, r0 (Ωkin = 0) = 1) (ΛCDM, exponential potential will essentially lie within the same area
marked with a diamond). The dotted curve shows the area all mat- as matter+quiessence models. In Fig. 4 we have plotted the tra-
ter+quiessence models must lie within at all times. Bottom panel: jectories in the s0 –r0 -plane and the s0 –q0 -plane for the same
zoom-in of the figure above. Here the curve having λ2 = 2 is also models as in Fig. 2, to be compared with Figs. 5c and 5d in
plotted (thick line). Alam et al. (2003).
Choosing instead a power-law potential V(φ) = Aφ−α gives
V = − αφ V and

By choosing for instance Ωm0 = 0.27 and Ωk0 = 0 we can
plot the values of q0 and r0 for varying Ωkin0 ; see Fig. 2. As 1
we can see from Eqs. (33)–(34), when Ωkin0 = 0, q0 and q = Ωm + 2Ωkin − Ωpot (35)
2
r0 are independent of λ, and have the same values as in the M
ΛCDM model. This is obvious, since taking away the kinetic r = Ωm + 10Ωkin + Ωpot − 3α 6Ωkin Ωpot . (36)
φ
term will reduce quintessence to LIVE. However, when Ωkin0
is slightly greater then 0 we can make r0 as large or as small We see that for φ0 = M we get the same curves in the
as we like, by choosing |λ| sufficiently large. There is no rea- q0 −r0 -plane when varying α as we got when varying λ in the
son all quintessence models should lie inside the constant-w- exponential potential, see Fig. 2. We also see that varying φ0 for
curve. However, in order to get an accelerating universe today a given value of α is essentially the same as varying α. Figure 5
we must have λ2 < 2. But also for λ2 < 2 the present val- shows the q0 −r0 -plane for the case α = 2. Figure 6 shows an
ues of q0 and r0 can lie outside the constant-w-curve. In fact, example of time-evolving statefinders (φ0 = M, Ωkin0 = 0.05,
when we move on to the time-evolving statefinders, plotting Ωm0 = 0.27 Ωk0 = 0, h = 0.71). If one compares this plot with
q and r as functions of time for given initial conditions, we Fig. 1b in Alam et al. (2003), the two do not quite agree. Alam
obtain plots like Fig. 3. Here we have chosen as initial condi- et al. (2003) do not give detailed information about the initial
tions Ωm0 = 0.27 and Ωk0 = 0 as above, and h = 0.71. The conditions for the quintessence field. Our initial conditions cor-
last initial condition, for the quintessence field, we have cho- respond to a universe which was matter-dominated up to now,
sen to be φ0 = M/100 combined with the overall constant A when quintessence is taking over.
 A. K. D. Evans et al.: Geometrical constraints on dark energy 403
4

5
3

0
2

1 –5

s_0 0 r_0
–10

–1
–15

–2

–20
–3

–25
–4 –4 –2 0 2 4 6 8 10 –1 –0.5 0 0.5 1 1.5

r_0 q_0
2

Fig. 5. Present values of q and r for matter+quintessence with a power-

law potential with α = 2. From top to bottom the different curves have
1
φ0 = 8M, 4M, 2M, M, M2 , M4 , M8 .The diamond represents the ΛCDM
model. The dotted curve shows the area all matter+quiessence models
must lie within at all times.

s_0 0

–1

r
–2 –0.6 –0.4 –0.2 0 0.2 0.4
–2
q_0

Fig. 4. Present values of the statefinder parameters and the decel-

eration parameter for models with matter and quintessence with an –4

exponential potential. The diamond represents the ΛCDM model.

Top panel: from left to right the different curves have λ =
−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5. Bottom panel: from top to bottom –6
–1 –0.5 0 0.5 1

the different curves have λ = −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5. q

Fig. 6. Time-evolution of q and r for models with matter and

2.3. Dark energy fluid models quintessence with a power-law potential. The crosses mark the present
epoch, the diamond represents the present ΛCDM model. All models
We will now find expressions for r and s which are valid even if start out from the horizontal ΛCDM line and will eventually end up
the dark energy does not have an equation of state of the form as a de Sitter universe (q = −1, r = 1). The curve going deepest down
px = wρx . This is the case e.g. in the Chaplygin gas models has α = 5 and moving upwards we have α = 4, 3, 2, 1. The dotted
(Kamenshchik, Moschella & Pasquier 2001; Bilic et al. 2002). curve shows the area all matter+quiessence models must lie within at
The expression for the deceleration parameter can be written as all times. Obviously the same is not the case for matter+quintessence
  models.
1 px
q= 1+3 Ω, (37)
2 ρx
The Generalized Chaplygin Gas (GCG) has an equation of state
and using this in Eq. (15) we find
(Bento et al. 2002)
 
3 ṗx
r = 1− Ω (38) A
2 Hρx p=− , (42)
ρα
1 ṗx
s = − · (39)
3H px and integration of the energy conservation equation gives
For a universe with cold matter and dark energy one finds
 1
  ρ = A + Ba−3(1+α) 1+α , (43)
9 ρx + px ∂px
r = 1+ Ω (40)
2 ρm + ρx ∂ρx where B is a constant of integration. This can be rewritten as
 
ρx ∂px
 1
s = 1+ · (41) ρ = ρ0 As + (1 − As )x3(1+α) 1+α ,
px ∂ρx (44)
404 A. K. D. Evans et al.: Geometrical constraints on dark energy

where ρ0 = (A+B)1/(1+α) , and As = A/(A+B). For a flat universe 2.5. The luminosity distance to third order in z
with matter and a GCG, the Hubble parameter is given by
The luminosity distance is given by
H 2 (x)
 1
= Ωm0 x3 + (1 − Ωm0 ) As + (1 − As )x3(1+α) 1+α . (45) 1+z
H0 dL = √ Sk ( |Ωk0 |I), (55)
H0 |Ωk0 |
This leads to the following expressions for q(x) and r(x):
where Sk (x) = sin x for k = 1, Sk (x) = x for k = 0, Sk (x) =
3 Ωm0 x + (1 − Ωm0 )(1 − As ) v
3
x β −1
3
β sinh x for k = −1, and
q(x) = −1 (46)  z
2 Ωm0 x3 + (1 − Ωm0 ) v3/β dz
I = H0 · (56)
x 3 x2  0 H(z)
r(x) = 1 − 3 2 f (x) + f (x), (47)
h (x) 2 h2 (x) The statefinder parameters appear when one expands the lumi-
where β = 3(1 + α), h(x) = H(x)/H0 , and nosity distance to third order in the redshift z. This expansion
has been carried out by Chiba & Nakamura (1998) and Visser
v = As + (1 − As )xβ (48) (2003). The result is
β −1
3
f (x) = Ωm0 x2 + (1 − Ωm0 )(1 − As )v xβ−1 . (49) 
z  1 1
dL ≈ 1 + (1 − q0 )z − (1 + r0 − q0
In the r−s plane, the GCG models will lie on curves given by H0 2 6
(see Gorini et al. 2003) 

9 s(s + α) −3q20 − Ωk0 )z2 . (57)
r =1− · (50)
2 α
One can also find an expression for the present value of the
We note that a recent comparison of GCG models with SNIa
time derivative of the equation of state parameter w in terms of
data found evidence for α > 1 (Bertolami et al. 2004).
the statefinder r0 . A Taylor expansion to first order in z gives
 
2.4. Cardassian models 2 9 Ωm0 − r0
w(z) ≈ w0 − 1 + w0 (1 + w0 ) + z. (58)
3 2 Ωx0
As an alternative to adding a negative-pressure component to
the energy-momentum tensor of the Universe, one can take the
view that the present phase of accelerated expansion is caused 3. Lessons drawn from current SNIa data
by gravity being modified, e.g. by the presence of large ex- In this section we will consider the SNIa data presently avail-
tra dimensions. For a general discussion of extra-dimensional able, in particular whether one can use them to learn about
models and statefinder parameters, see Alam & Sahni (2002). the statefinder parameters. We will use the recent collection of
As an example, we will consider the Modified Polytropic SNIa data in Riess et al. (2004), their “gold” sample consisting
Cardassian ansatz (MPC) (Freese & Lewis 2002; Gondolo & of 157 supernovae at redshifts between ∼0.01 and ∼1.7.
Freese 2003), where the Hubble parameter is given by
  1/ψ 3.1. Model-independent constraints
H(x) = H0 Ωm0 x3 1 + u , (51)
The approximation to dL in Eq. (57) is independent of the
with cosmological model, the only assumption made is that the
−ψ Universe is described by the Friedmann-Robertson-Walker
Ωm0 − 1 metric. We see that, in addition to H0 , this third-order ex-
u = u(x) = , (52)
x3(1−n)ψ pansion of dL depends on q0 and the combination r0 − Ωk0 .
and where n and ψ are new parameters (ψ is usually called q Fitting these parameters to the data, we find the constraints
in the literature, but we use a different notation here to avoid shown in Fig. 7. The results are consistent with those of sim-
confusion with the deceleration parameter). For this model, the ilar analyses in Caldwell & Kamionkowski (2004) and Gong
deceleration parameter is given by (2004). In Fig. 8 we show the marginalized distributions for
  q0 and r0 − Ωk0 . We note that the supernova data firmly sup-
3  1 + nu  port an accelerating universe, q0 < 0 at more than 99% confi-
q(x) =  −1 (53)
2 1+u  dence. However, about the statefinder parameter r0 , little can
be learned without an external constraint on the curvature.
and the statefinder r by Imposing a flat universe, e.g. by inflationary prejudice or by
 invoking the CMB peak positions, there is still a wide range of
9 1 + nu  u(1 − n) − (1 + nu) allowed values for r0 . This is an indication of the limited abil-
r(x) = 1 − 1 +
4 1+u 1+u ity of the current SNIa data to place constraints on models of
 dark energy. There is only limited information on anything be-
(1 − n) u 
2
−2q · (54) yond the present value of the second derivative of the Hubble
(1 + u)(1 + nu)  parameter.
 A. K. D. Evans et al.: Geometrical constraints on dark energy 405
5 2

1.5
4

3
0.5
r0-Ωk0

w1
2 0

-0.5
1

-1

0
-1.5

-1 -2
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6
q0 w0

Fig. 7. Likelihood contours (68, 95 and 99%) resulting from a fit of Fig. 9. Likelihood contours (68, 95 and 99%) for the coefficients w0
the expansion of the luminosity distance to third order in z. and w1 in the linear approximation to the equation of state w(z) of
dark energy, resulting from a fit of the expansion of the luminosity
distance to third order in z.

2 subsection we will consider the following models:

1. The expansion of dL to second order in z, with h and q0 as

parameters.
2. The third-order expansion of dL , with h, q0 , and r0 − Ωk0 as
pdf

parameters.
1
3. Flat ΛCDM models, with Ωm0 and h as parameters to be
varied in the fit.
4. ΛCDM with curvature, so that Ωm0 , ΩΛ0 (the contribution
of the cosmological constant to the energy density in units
of the critical density, evaluated at the present epoch), and
0
−3 −1 1 3 5 h are varied in the fits.
q0 , r0−Ωk0 5. Flat quiessence models, that is, models with a constant
Fig. 8. Marginalized probability distributions for q0 (full line) and r0 − equation of state w for the dark energy component. The pa-
Ωk0 (dotted line). rameters to be varied in the fit are Ωm0 , w, and h.
6. The Modified Polytropic Cardassian (MPC) ansatz, with
Ωm0 , q, n, and h as parameters to be varied.
7. The Generalized Chaplygin Gas (GCG), with Ωm0 , As , α,
and h as parameters to be varied.
Under the assumption of a spatially flat universe, Ωk0 = 0, 8. The ansatz of Alam et al. (2003),
with Ωm0 = 0.3, one can use Eq. (58) to obtain constraints
on w0 and w1 in the expansion w(z) = w0 + w1 z of the equa- H = H0 Ωm0 x3 + A0 + A1 x + A2 x2 , (59)
tion of state of dark energy. The resulting likelihood contours
where we restrict ourselves to flat models, so that A0 =
are shown in Fig. 9. As can be seen in this figure, there is no
1 − Ωm0 − A1 − A2 . The parameters to be varied are Ωm0 , A1 ,
evidence for time evolution in the equation of state, the obser-
A2 , and h.
vations are consistent with w1 = 0. The present supernova data
show a slight preference for a dark energy component of the Note that these models have different numbers of free pa-
‘phantom’ type with w0 < −1 (Caldwell 2002). Note, however, rameters. To get an idea of which of these models is actu-
that the relatively tight contours obtained here are caused by ally preferred by the data, we therefore employ the Bayesian
the strong prior Ωm0 = 0.3. It should also be noted that the Information Criterion (BIC) (Schwarz 1978; Liddle 2004).
third-order expansion of dL is not a good approximation to the This is an approximation to the Bayes factor (Jeffreys 1961),
exact expression for high z and in some regions of the parame- which gives the posterior probability of one model relative to
ter space. another assuming that there is no objective reason to prefer one
of the models prior to fitting the data. It is given by
3.2. Direct test of models against data B = χ2min + Npar ln Ndata , (60)
The standard way of testing dark energy models against data where χ2min is the minimum value of the χ2 for the given model
is by maximum likelihood fitting of their parameters. In this against the data, Npar is the number of free parameters, and Ndata
406 A. K. D. Evans et al.: Geometrical constraints on dark energy

Table 1. Results of fitting the models considered in this subsection to (A0 = A2 = 0), and w = −1/3 (A0 = A1 = 0), and the luminos-
the SNIa data. ity distance-redshift relationship is given by

Model χ2min # parameters B 1 + z 1+z dx
dL = · (63)
2. order expansion 177.1 2 187.2 H0 1 Ωm0 x + A0 + A1 x + A2 x2
3

3. order expansion 162.3 3 177.5

Having fitted the parameters A0 , A1 , and A2 to e.g. supernova
Flat ΛCDM 163.8 2 173.9 data using Eq. (63), one can then find q and r by substituting
ΛCDM with curvature 161.2 3 176.4 Eq. (62) into Eqs. (5) and (6):
Flat + constant EoS 160.0 3 175.2  
1 A2 x2 + 2A1 x + 3A0
MPC 160.3 4 180.5 q(x) = 1− (64)
2 Ωm0 x3 + A2 x2 + A1 x + A0
GCG 161.4 4 181.6
Ωm0 x3 + A0
Alam et al. 160.5 4 180.7 r(x) = , (65)
Ωm0 x3 + A0 + A1 x + A2 x2
and furthermore the statefinder s is found to be
is the number of data points used in the fit. As a result of the 2 A 1 x + A 2 x2
approximations made in deriving it, B is given in terms of the s(x) = , (66)
3 3A0 + 2A1 x + A2 x2
minimum χ2 , even though it is related to the integrated likeli-
hood. The preferred model is the one which minimizes B. In and the equation of state is given by
Table 1 we have collected the results for the best-fitting mod- 1 A1 x + 2A2 x2
els. When comparing models using the BIC, the rule of thumb w(x) = −1 + · (67)
3 A 0 + A 1 x + A 2 x2
is that a difference of 2 in the BIC is positive evidence against
the model with the larger value, whereas if the difference is 6 The simulations of Alam et al. (2003) indicated that the
or more, the evidence against the model with the larger BIC statefinder parameters can be reconstructed well from simu-
is considered strong. The second-order expansion of dL is then lated data based on a range of dark energy models, so we will
clearly disfavoured, thus the current supernova data give in- for now proceed on the assumption that the parametrization in
formation, although limited, on r0 − Ωk0 . We see that there is Eq. (62) is adequate for the purposes of extracting q, r and s
no indication in the data that curvature should be added to the from SNIa data. We comment this issue in Sect. 4.
ΛCDM model. Also, the last three models in Table 1 seem to In Fig. 10 we show the deceleration parameter q and the
be disfavoured. One can conclude that there is no evidence in statefinder r extracted from the current SNIa data. The error
the current data that anything beyond flat ΛCDM is required. bars in the figure are 1σ limits. We have also plotted the model
This does not, of course, rule out any of the models, but tells predictions for the same quantities (based on best-fitting pa-
us that the current data are not good enough to reveal physics rameters with errors) for ΛCDM, quiessence, and the MPC.
beyond spatially flat ΛCDM. A similar conclusion was reached The figure shows that all models are consistent at the 1σ level
by Liddle (2004) using a more extensive collection of cosmo- with q and r extracted using Eq. (62). Thus, with the present
logical data sets and considering adding parameters to the flat quality of SNIa data, the statefinder parameters are, not sur-
ΛCDM model with scale-invariant adiabatic fluctuations. prisingly, no better at distinguishing between the models than
a direct comparison with the SNIa data. We next look at simu-
lated data to get an idea of how the situation will improve with
3.3. Statefinder parameters from current data future data sets.
If the luminosity distance dL is found as a function of red-
shift from observations of standard candles, one can obtain the 4. Future data sets
Hubble parameter formally from
We will now make an investigation of what an idealized SNIa
  −1 survey can teach us about statefinder parameters and dark en-
d dL
H(x) = · (61) ergy, following the procedure in Saini et al. (2004).
dx x
A SNAP-like satellite is expected to observe ∼2000 SN
However, since observations always contain noise, this relation up to z ∼ 1.7. Dividing the interval 0 < x ≤ 1.7 into
cannot be applied straightforwardly to the data. Alam et al. 50 bins, we therefore expect ∼40 observations of SN in each
(2003) suggested parametrizing the dark energy density as a bin. Empirically, SNIa are very good standard candles with a
second-order polynomial in x, ρx = ρc0 (A0 + A1 x + A2 x2 ), lead- small dispersion in apparent magnitude σmag = 0.15, and there
ing to a Hubble parameter of the form is no indication of redshift evolution. The apparent magnitude
is related to the luminosity distance through

H(x) = H0 Ωm0 x3 + A0 + A1 x + A2 x2 , (62) m(z) = M + 5 log DL (z), (68)

and fitting A0 , A1 , and A2 to data. This parametrization repro- where M = M0 + 5 log[H0−1 Mpc−1 ] + 25. The quantity
duces exactly the cases w = −1 (A1 = A2 = 0), w = −2/3 M0 is the absolute magnitude of type Ia supernovae, and
 A. K. D. Evans et al.: Geometrical constraints on dark energy 407
2 15

1
10
q (from data)

r (from data)
0

−1

0
−2

−3 −5
0 1 2 0 1 2
z z
2 15

1
10
q (ΛCDM)

r (ΛCDM)

−1

0
−2

−3 −5
0 1 2 0 1 2
z z
2 15

Fig. 11. Binned, simulated data set for a Cardassian model with ψ = 1,
1
10
n = −1 (upper curve), a flat ΛCDM universe with Ωm0 = 0.3 (middle
curve), and for a Generalized Chaplygin Gas with A s = 0.4, α = 0.7
q (quiessence)

r (quiessence)

0
(lower curve). The 1σ error bars are also shown.
5

−1

0 to the simulated dL , and hence our results give the ensemble

−2
average of the parameters we fit to the simulated data sets.
−3 −5
0 1 2 0 1 2
z z
2 15 4.1. A ΛCDM universe
1
10
We first simulate data based on a flat ΛCDM model with
Ωm0 = 0.3, h = 0.7, giving the data points shown in the mid-
0
q (MPC)

r (MPC)

dle curve in Fig. 11. To this data set we first fit the quiessence
5

−1
model, the MPC, the GCG, and the parametrization of H from
Eq. (62). Since all models reduce to ΛCDM for an appropriate
0
−2 choice of parameters, distinguishing between them based on
the χ2 per degree of freedom alone is hard. Based on the best-
−3 −5
0 1
z
2 0 1
z
2 fitting values and error bars on the parameters A0 , A1 , and A2
in Eq. (62) we can reconstruct the statefinder parameters from
Fig. 10. The deceleration parameter q and the statefinder r extracted Eqs. (64)–(66). In Fig. 12 we show the deceleration parame-
from current SNIa data using the Alam parametrization of H (top row), ter and statefinder parameters reconstructed from the simulated
for ΛCDM (second row), quiessence (third row), and the Modified
data. The statefinders can be reconstructed quite well in this
Polytropic Cardassian ansatz (bottom row)
case, e.g. we see clearly that r is equal to one, as it should for
flat ΛCDM. In Fig. 13 we show the statefinders for a selec-
tion of models, obtained by fitting their respective parameters
DL (z) = H0 dL (z) is the Hubble constant free luminosity dis-
to the data, and using the expressions for q and r for the respec-
tance. The combination of absolute magnitude and the Hubble
tive models derived in earlier sections, e.g. Eqs. (46) and (47)
constant, M, can be calibrated by low-redshift supernovae
for the Chaplygin gas. Since all models reduce to ΛCDM for
(Hamuy et al. 1993; Perlmutter et al. 1999). The dispersion in
the best-fitting parameters, their q and r values are also con-
the magnitude, σmag , is related to the uncertainty in the dis-
sistent with ΛCDM. Thus, if the dark energy really is LIVE, a
tance, σ, by
SNAP-type experiment should be able to demonstrate this.
σ ln 10
= σmag , (69)
dL (z) 5 4.2. A Chaplygin gas universe
and for σmag = 0.15, the relative error in the luminosity dis- We have also carried out the same reconstruction exercise us-
tance is ∼7%. If we assume that the dL we calculate for each ing simulated data based on the GCG with As = 0.4, α = 0.7,
z value is the mean of all√dL s in that particular bin, the errors re- see Fig. 11. Figure 14 shows q and r reconstructed using
duce from 7% to 0.07/ 40 ≈ 0.01 = 1%. We do not add noise the parametrization of H. The same quantities for the models
408 A. K. D. Evans et al.: Geometrical constraints on dark energy

Fig. 12. The statefinder parameters and the deceleration parameter for Fig. 14. The statefinder parameters and the deceleration parameter for
the best-fitting reconstruction of the simulated data based on ΛCDM, the best-fitting reconstruction of the simulated data based on the GCG,
using the parametrization of Alam et al. The 1σ error bars are also using the parametrization of Alam et al. The 1σ error bars are also
shown. shown.

Fig. 13. The statefinder parameters for a selection of models, evaluated Fig. 15. The statefinder parameters for a selection of models, evaluated
at the best-fitting values of their respective parameters to the simulated at the best-fitting values of their respective parameters to the simulated
ΛCDM dataset, with 1σ errors included. Chaplygin gas data set, with 1σ errors included.

considered, based on their best-fitting parameters to the simu-

lated data, are shown in Fig. 15. For the Cardassian model, the chose to impose a prior n > −1, producing the results shown in
best-fitting value for the parameter n, nbf , depends on the extent Fig. 15. The best-fitting values for ψ and n were, respectively,
of the region over which we allow n to vary. Extending this re- 0.06 and −0.94.
gion to larger negative values for n moves nbf in the same direc- Figure 16 shows the deceleration parameter extracted from
tion. However, the minimum χ2 value does not change signifi- the Alam et al. parametrization (full line), with 1σ error bars.
cantly. This is understandable, since H(x) for the MPC model Also plotted is the best fit q(z) from the quiessence (squares),
is insensitive to n for large, negative values of n. The quantities Cardassian (triangles) and Chaplygin (asterisk) models. We
r(x) and q(x) also depend only weakly on the allowed range for note that the q(z) from the Alam et al. parametrization has a
n, whereas their error bars are sensitive to this parameter. We somewhat deviating behaviour from the input model, especially
 A. K. D. Evans et al.: Geometrical constraints on dark energy 409

Fig. 16. Comparison of q(z) extracted using the parametrized H(z) Fig. 17. Comparison of r(z) extracted using the parametrized H(z)
with q(z) for the various best-fitting models. The input model is a with r(z) for the various best-fitting models. The input model is a
GCG model with As = 0.4, α = 0.7. Error bars are only shown on GCG model with As = 0.4, α = 0.7. Error bars are only shown on
the values extracted using the Alam et al. parametrization, but in the the values extracted using the Alam et al. parametrization, but in the
other cases they are roughly of the same size as the symbols. See text other cases they are roughly of the same size as the symbols. See text
for more details. for details.

at larger z. Also, no model can be excluded on the basis of their

predictions for q(z)
Figure 17 shows the same situation for the statefinder
parameter r(z). Note again that for large z, the recovered
statefinder from the Alam et al. parametrization does not corre-
spond well with the input model. As with the case for q(z), the
quiessence and Cardassian models follow each other closely.
These, however, do not agree with the input model for low val-
ues of z (similar to the case for q(z) they diverge for low z).
Comparing the statefinder r for the quiessence and Cardassian
models with that of the input GCG model, indicates that, not
surprisingly, neither of them is a good fit to the data.

4.3. A Cardassian universe

We repeated the analysis described in Sects. 4.1 and 4.2, this
time based on an underlying Cardassian model. The values of
the input parameters were chosen to be ψ = 1, n = −1. The lu-
minosity distance for this model is shown in Fig. 11. Figures 18 Fig. 18. Comparison of q(z) extracted using the parametrized H(z)
and 19 show, respectively, the deceleration parameter q(z) and with q(z) for the input Cardassian model.
the statefinder r(z) for the input Cardassian model (triangles)
compared to the reconstructed parameters (full line) using the
Alam et al. parametrization for H(z). For clarity, only the er- reconstructed statefinder. For the Cardassian universe, the dis-
ror bars for the reconstructed parameters are shown. As before, crepancy between input and reconstructed parameter is most
the error bars for the input model are roughly the size of the conspicuous for low z (z < 0.7). This further corroborates the
symbols, except in the case of z = 0−0.7 for r(z) where they conclusion in Sect. 4.2 that a better parametrization for H(z)
are somewhat larger (up to two symbol sizes in each direc- is needed. The best fit quiessence and Chaplygin gas models
tion). We see that the deceleration parameter is reconstructed are not shown in these figures. We only remark in passing that
quite well. However, the behaviour of the reconstructed r(z) with the quiessence model we managed to reproduce the input
does not seem to agree well with the input model, although the model quite well, while the Chaplygin gas model was a very
input model is more or less within the 1 σ errors bars of the poor fit to these simulated data.
410 A. K. D. Evans et al.: Geometrical constraints on dark energy

state w(x) from SNIa data using Eq. (67). They found that this
parametrization forces the behaviour of w(x) onto a specific set
of tracks, and may thus give spurious evidence for redshift evo-
lution of the equation of state. Since there are intrinsic correla-
tions between the statefinders, finding an unbiased reconstruc-
tion procedure, and demonstrating that it really is so, is likely
to be very hard.
Acknowledgements. We acknowledge support from the Research
Council of Norway (NFR) through funding of the project 159637/V30
“Shedding Light on Dark Energy”. The authors wish to thank Håvard
Alnes for interesting discussions and the anonymous referee for valu-
able comments and suggestions.

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