Investigating the Dark Universe
through Gravitational Lensing
Teresa Riehm
Department of Astronomy
Stockholm University
Cover image:
Abell 1689, one of the most massive galaxy clusters known, acts as a
gravitational lens, distorting the images of galaxies that lie behind it.
Image credit: NASA, ESA, E. Jullo (Jet Propulsion Laboratory),
P. Natarajan (Yale University), and J.-P. Kneib (Laboratoire d’Astrophysique
de Marseille, CNRS, France)
c Teresa Riehm, Stockholm 2011
ISBN 978-91-7447-281-3
Universitetsservice, US-AB, Stockholm 2011
Department of Astronomy, Stockholm University
Doctoral Dissertation 2011
Department of Astronomy
Stockholm University
SE-106 91 Stockholm
Abstract
A variety of precision observations suggest that the present universe is dominated by some unknown components, the so-called dark matter and dark energy. The distribution and properties of these components are the focus of
modern cosmology and we are only beginning to understand them.
Gravitational lensing, the bending of light in the gravitational field of a
massive object, is one of the predictions of the general theory of relativity. It
has become an ever more important tool for investigating the dark universe,
especially with recent and coming advances in observational data.
This thesis studies gravitational lensing effects on scales ranging over ten
orders of magnitude to probe very different aspects of the dark universe. Implementing a matter distribution following the predictions of recent simulations, we show that microlensing by a large population of massive compact
halo objects (MACHOs) is unlikely to be the source of the observed longterm variability in quasars. We study the feasibility of detecting the so far
elusive galactic dark matter substructures, the so-called “missing satellites”,
via millilensing in galaxies close to the line of sight to distant light sources. Finally, we utilise massive galaxy clusters, some of the largest structures known
in the universe, as gravitational telescopes in order to detect distant supernovae, thereby gaining insight into the expansion history of the universe. We
also show, how such observations can be used to put constraints on the dark
matter component of these galaxy clusters.
To m y fam ily
List of Papers
This thesis is based on the following publications:
I
High-redshift microlensing and the spatial distribution of
dark matter in the form of MACHOs
Zackrisson E., & Riehm T., 2007, A&A, 475, 453
II
Strong lensing by subhalos in the dwarf galaxy mass range.
I. Image separations
Zackrisson E., Riehm T., Möller O., Wiik K., & Nurmi P., 2008,
ApJ, 684, 804
III
Strong lensing by subhalos in the dwarf galaxy mass range.
II. Detection probabilities
Riehm T., Zackrisson E., Mörtsell E., & Wiik K., 2009, ApJ, 700,
1552
IV
Near-IR search for lensed supernovae behind galaxy clusters.
I. Observations and transient detection efficiency
Stanishev V., Goobar A., Paech K., Amanullah R., Dahlén T.,
Jönsson J., Kneib J. P., Lidman C., Limousin M., Mörtsell E.,
Nobili S., Richard J., Riehm T., & von Strauss M., 2009, A&A,
507, 61
V
Near-IR search for lensed supernovae behind galaxy clusters.
II. First detection and future prospects
Goobar A., Paech K., Stanishev V., Amanullah R., Dahlén T.,
Jönsson J., Kneib J. P., Lidman C., Limousin M., Mörtsell E.,
Nobili S., Richard J., Riehm T., & von Strauss M., 2009, A&A,
507, 71
VI
Near-IR search for lensed supernovae behind galaxy clusters.
III. Implications for cluster modeling and cosmology
Riehm T., Goobar A., Mörtsell E., Amanullah R., Dahlén T.,
Jönsson J., Limousin M., Paech K., & Richard J., 2011,
submitted to A&A
The articles are referred to in the text by their Roman numerals. Reprints were made
with permission from the publishers.
Contents
1
Introduction
2
The Dark Universe
2.1 Dark matter and structure formation . . . . . . . . . . . . . . . . . . . .
2.1.1 Cold dark matter subhalos . . . . . . . . . . . . . . . . . . . . .
2.2 Dark energy and the expansion of the Universe . . . . . . . . . . . .
2.2.1 The Hubble constant . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The cosmic acceleration . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
8
9
10
3
Gravitational lensing
3.1 Strong and weak lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Arrival time and images . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Einstein radius and lens equation . . . . . . . . . . . . . . . . . . . . . .
3.4 Image magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Lensing on different scales . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
13
13
15
16
17
4
MACHOs vs WIMPs
4.1 Microlensing in the local group . . . . . . . . . . . . . . . . . . . . . . .
4.2 Extragalactic microlensing . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
19
21
5
Missing satellites
5.1 Flux ratio anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Astrometric effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Small-scale structure in macroimages . . . . . . . . . . . . . . . . . . .
5.4 Time delay effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
26
30
32
34
6
Gravitational telescopes
6.1 Cluster models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Multiply imaged SNe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
42
44
7
Summary and Outlook
47
8
Svensk sammanfattning
49
Publications not included in this thesis
1
51
10
CONTENTS
Acknowledgements
53
Bibliography
55
1
1
Introduction
Over the last decades, astronomers have realised that the universe had for a
long time been hiding a large fraction of its energy density. In the quest to
uncover the properties of these dark components, gravitational lensing has
proven to be a valuable tool. In this thesis, we show how gravitational lensing effects from the smallest to the largest scales can be used to probe very
different aspects of what we believe to be the standard cosmology.
This thesis is organized as follows. Chapter 2 gives a short introduction into
the standard cosmological model. In Chapter 3, the basic concepts of gravitational lensing are described. In the following chapters we discuss gravitational
lensing effects on different scales. Chapter 4 deals with constraining some of
the smallest dark matter structures predicted, so-called MACHOs, using microlensing. In Chapter 5, several approaches, commonly titled as millilensing,
in the search for galactic dark matter substructure, the “missing satellites” are
described. Finally, Chapter 6 deals with lensing by some of the largest structures formed in the universe, galaxy clusters, and their use as gravitational
telescopes for studying the history of the universe. Chapter 7 gives a summary
of the thesis and the papers. Finally, a short summary in Swedish is given in
Chapter 8.
My contribution to the papers included in this work
For Paper I, I implemented the different mass distribution scenarios and assisted in the writing of the text.
In Paper II, I wrote the code for computing the surface mass density profiles and expected image separations for different subhalo profiles and contributed to the scientific discussion.
Most of the work in Paper III was done by me. I prepared all the figures
and text. The results were interpreted together with the other authors.
For Papers IV and V, I computed the magnification and time delay maps
used for the preparation of the observations and the analysis of the data and
contributed to the scientific discussion.
In Paper VI, I carried out the statistical analysis and prepared most of the
figures and text.
3
2
The Dark Universe
One of the key questions in modern cosmology is what the universe is made
of. In the past few decades, there has been increasing evidence that suggests
that only a small fraction of the energy density in the universe is in the form
of baryonic matter which we can directly observe.
The standard model of Big Bang cosmology is the so-called Lamda-Cold
Dark Matter (ΛCDM) model. Following this model, recent observations estimate that the present energy content of the universe is made up of (Komatsu
et al. 2011):
• 4% baryonic matter
• 23% cold dark matter (CDM)
• 73% dark energy, namely a cosmological constant (Λ)
This implies that more than 95% of the energy density in the universe is
in the form of some dark components which have, so far, not been directly
detected in the laboratory. In this chapter, we outline some of the observational
and theoretical evidence for the properties of these components and discuss
some of their possible problems.
2.1
Dark matter and structure formation
The first evidence for the existence of dark matter was collected already in the
1930s. Zwicky (1933) was the first to notice that the velocities of galaxies in
the Coma cluster were unexpectedly high. Zwicky concluded, that the cluster
must contain a large amount of “dark matter” in order to keep the galaxies in
orbit. Since then, dark matter has been used for explaining the kinematics of
spiral galaxies and clusters (e.g. Smith 1936; Oort 1940; Rubin & Ford 1970;
Rubin et al. 1980), the gravitational lensing potentials of galaxy clusters (e.g.
Broadhurst et al. 1995; Clowe et al. 2006; Dahle 2006; Zitrin et al. 2011a)
and the large-scale structure in the universe (e.g. Davis et al. 1985; Efstathiou
et al. 1990; Springel et al. 2005, 2006).
CDM is thought to be made up of non-baryonic particles, which interact
predominantly through gravity and have moved with non-relativistic velocities
since the earliest epochs of structure formation. While this scenario has been
very successful in explaining the formation of large-scale structures (galaxies,
galaxy groups and galaxy clusters), its predictions on subgalactic scales have
not yet been observationally confirmed in any convincing way. On the contrary, there are at least two features of current CDM simulations that appear
4
The Dark Universe
to be in conflict with empirical data: the existence of high-density cusps in the
centres of dark matter halos (e.g. Gentile et al. 2004; Diemand et al. 2005a; de
Souza & Ishida 2010), and a rich spectrum of substructures within each dark
matter halo (see Kravtsov 2010, for a recent review). There is, however, no
consensus on how serious these problems really are for the CDM paradigm.
2.1.1 Cold dark matter subhalos
Massive CDM halos are assembled hierarchically from smaller halos. As these
subunits fall into the potential well of larger halos, they suffer tidal stripping
of material which ends up in the smooth dark matter component of the halo
that swallowed them. Since this is a process that may take several billion years
to complete, many of these smaller halos temporarily survive in the form of
substructures (also known as subhalos or subclumps) within the larger halo.
According to current simulations, around 10% of the virial mass of a Milky
Way-sized CDM halo should be in the form of subhalos at the current epoch
(Springel et al. 2008; Zemp et al. 2009, compare Figure 2.1). Naively, one
might expect dwarf galaxies to form in these low-mass halos prior to merging,
which would result in large numbers of satellite galaxies within the CDM
halo of each large galaxy. A long-standing problem with this picture is that
the number of subhalos predicted by simulations greatly exceeds the number
of dwarf galaxies seen in the the vicinity of large galaxies like the Milky Way
and Andromeda (Klypin et al. 1999; Moore et al. 1999). This has become
known as the “missing satellite problem”. A similar lack of dwarf galaxies
compared to the number of dark halos predicted is also evident within groupsized dark matter halos (Tully et al. 2002). While most of the efforts in this
field have been focused on the discrepancy between the number of subhalos
and observed satellite galaxies, this problem persists in the field population
as well. In a sense, the missing satellites is just one aspect of a more general
problem – the mismatch between the low-mass end of the dark matter mass
function and the luminosity function of dwarf galaxies (Verde et al. 2002).
A number of potential solutions to the missing satellites problem have been
suggested in the literature. These can be sorted into three different categories,
depending on how they propose to answer the question “Do subhalos exist in
the numbers predicted by CDM simulations?”:
• No. In the first category, we find modifications of the properties of dark
matter that reduce the numbers of low-mass halos and subhalos, including
warm dark matter (Bode et al. 2001), self-interacting dark matter (Spergel
& Steinhardt 2000), fuzzy dark matter (Hu et al. 2000) and dark matter in
the form of superWIMPs (Cembranos et al. 2005), but also models of inflation that produce the required cut-off in the primordial fluctuation spectrum
(Kamionkowski & Liddle 2000).
• Yes. Here, we find processes for inhibiting star formation in low-mass halos (Bullock et al. 2000; Somerville 2002; Benson et al. 2002; Kravtsov
2.1 Dark matter and structure formation
5
Figure 2.1: Dark matter density map visualising the large amount of substructure
predicted by simulations around any galactic halo. The main halo, which is resolved
with more than 1 billion particles, and the over 105 numerically resolved subhalos it
contains are visible as bright peaks. Figure adopted from Zemp et al. (2009).
et al. 2004; Moore et al. 2006; Nickerson et al. 2011) and observational
biases that would put the resulting “dark galaxies” (Trentham et al. 2001;
Verde et al. 2002) outside the reach of current surveys (Simon & Geha
2007; Tollerud et al. 2008; Walsh et al. 2009; Koposov et al. 2009; Macciò et al. 2010). While these mechanisms may be able to solve the missing
satellites problem as it was originally defined, solutions of this type imply
that a vast population of low-mass CDM subhalos (hosting very faint stellar
populations or none at all) should still be awaiting discovery.
• Yes, but not in our neighbourhood. The final possibility is that the large
halo-to-halo scatter in subhalo mass fraction may have left the Milky Way
and Andromeda sitting inside CDM halos with unusually few subhalos
compared to the cosmic average (Ishiyama et al. 2008, 2009). This would
imply that large numbers of CDM subhalos (either bright or dark) should
be awaiting discovery in the vicinity of more distant galaxies.
Gravitational lensing may play an important role in the quest to determine
which of these different solutions is the correct one. If subhalos do exist, lensing can in principle be used to detect even those that are too faint to be observed through other means. If subhalos do not exist, the absence of lensing
effects associated with subhalos should be able to tell us so. Chapter 5 ex-
6
The Dark Universe
plains how this can be achieved and points out some potential pitfalls along
the way.
As computing power has increased over the last decade, N-body simulations
have been able to make predictions for important properties of dark matter
subhalos at higher and higher resolution. Still, essentially all high resolution
simulations are CDM only, i.e. the effect of a possible baryonic component
has not been taken into account. Although the possible consequences of this
have been discussed in the literature (Barkana & Loeb 1999; Gnedin et al.
2004; Macciò et al. 2006; Gustafsson et al. 2006; Kampakoglou 2006; Read
et al. 2006; Weinberg et al. 2008; Romano-Díaz et al. 2010), it is, however,
still not clear how strong this effect is likely to be. This should be kept in mind
when discussing the properties of dark matter substructures.
Mass function
N-body simulations indicate that the subhalos within a galaxy-sized CDM
halo follow a mass function of the type:
dN
−α
∝ Msub
,
dMsub
(2.1)
with α ≈ 1.9 (Gao et al. 2004; Springel et al. 2008), albeit with non-negligible
halo-to-halo scatter at the high mass end (Msub & 5 × 108 M⊙ ) (Ishiyama et al.
2009; Springel et al. 2008). Current simulations of entire galaxy-sized dark
matter halos can resolve subhalos with masses down to Msub ∼ 105 M⊙ , but
the mass function may extend all the way down to the cut-off in the density fluctuation power spectrum, which is set by the detailed properties of
the CDM particles. For many types of WIMPs (e.g. neutralinos) this cut-off
lies at ∼ 10−6 M⊙ (Green et al. 2005; Loeb & Zaldarriaga 2005; Diemand
et al. 2005a; Profumo et al. 2006; Berezinsky et al. 2008; Bringmann 2009),
but other CDM candidates may alter this truncation mass considerably. As
an example, axions may allow the existence of halos with masses as low as
10−12 M⊙ (Hogan & Rees 1988), whereas very few halos with masses below
104 –107 M⊙ are expected in the case of MeV mass dark matter (Hooper et al.
2007). The overall mass contained in resolved subhalos (i.e Msub & 105 M⊙ )
within a galaxy-sized CDM halo amounts to a subhalo mass fraction around
fsub ≈ 0.1, and extrapolating the mass function given by eq. (2.1) towards
lower masses does not boost this by much (Springel et al. 2008).
Spatial distribution
Since subhalos are more easily disrupted in the central regions of their parent
halo, the subhalo population tends to be less centrally concentrated than the
smooth CDM component. The spatial distribution of subhalos within r200 , the
radius at which the density of the halo drops below 200 times the critical
2.1 Dark matter and structure formation
7
density of the Universe, can be described as (Madau et al. 2008):
N(< x) = N(< r200 )
12x3
,
1 + 11x2
(2.2)
where x = r/r200 and N denotes the number of subhalos within a specific radius. It should be noted that this result is based on CDM-only simulations,
and that the presence of baryons within the subhalos may make them more
resistant to tidal disruption, thereby boosting their number densities in the inner regions of their parent halos (Weinberg et al. 2008). Some simulations of
cluster-mass halos have moreover indicated that the spatial distribution of subhalos may be a function of subhalo mass, in the sense that high-mass subhalos
would tend to avoid the central regions more than low-mass ones (Ghigna
et al. 2000; De Lucia et al. 2004), but this has not been confirmed by the latest
simulations of galaxy-sized halos (Springel et al. 2008; Ludlow et al. 2009).
While the term subhalo is typically used to denote clumps located within
the virial radius (or, alternatively, r200 ) of a large CDM halo, there is also a
large number of low-mass clumps located just outside this limit (Diemand
et al. 2007; Ludlow et al. 2009). Some of these have previously been bona
fide subhalos, and others are bound to venture inside the virial radius in the
near future. Such objects can through projection appear close to lines of sight
passing through the centres of large galaxies, and may therefore be important
in certain lensing situations.
Density profile
The internal structure of subhalos is still a matter of much debate. As lowmass halos are accreted by more massive ones and become subhalos, substantial mass loss occurs, preferentially from their outer regions. The shape of the
outer part of the subhalo density profile may therefore be seriously affected by
stripping, whereas the inner regions are left more or less intact. In many lensing studies, CDM subhalos are considered to be singular isothermal spheres
(SIS) or even point masses. This is mainly for simplicity – the lensing properties of such objects are well-known, but neither observations, theory nor
simulations favour models of this types for the subhalos predicted by CDM
(e.g. Ma 2003; Gilmore et al. 2007; Springel et al. 2008).
An SIS has a density profile given by:
ρSIS =
σv2
,
2π Gr2
(2.3)
where σv is the line of sight velocity dispersion and G is Newton’s gravitational constant. This model has proved to be successful for the massive
galaxies responsible for strong lensing on arcsecond scales (Rusin et al. 2003;
Koopmans et al. 2006). The SIS density profile has a steep inner slope (ρ ∝ rβ
with β = −2), which in the case of massive galaxies is believed to be due
8
The Dark Universe
to the luminous baryons residing in their inner regions. This baryonic component contributes substantially to the overall mass density in the centre, and
its formation over cosmological time scales may also have caused the CDM
halo itself to contract, thereby steepening the inner slope of its density profile
(Gnedin et al. 2004; Macciò et al. 2006; Gustafsson et al. 2006; Kampakoglou
2006).
Low-mass CDM halos which never formed many stars are unlikely to have
density profiles this steep. Instead, they should resemble the halo density profiles derived from CDM-only simulations. The NFW density profile (Navarro
et al. 1996), with an inner slope of β = −1, has for a number of years served
as the standard density profile for CDM halos, and is given by:
ρi
,
(2.4)
ρNFW (r) =
(r/rS )(1 + r/rS )2
where rS is the characteristic scale radius of the halo and ρi is related to the
density of the Universe at the time of collapse. Modifications of this formula
are required once halos become subhalos and are tidally stripped. Attempts to
quantify the effects of this mass loss on their density profile have been made
by Hayashi et al. (2003) and Kazantzidis et al. (2004)
Controversy remains, however, over whether the NFW profile gives the best
representation of CDM halos (and, consequently, of subhalos prior to stripping). Based on recent high-resolution simulations, some have argued for a
slightly steeper inner slope (β ≈ −1.2; Diemand et al. 2008) with significant halo-to-halo variations, whereas other have favoured a far shallower inner
slope (Navarro et al. 2004; Springel et al. 2008; Stadel et al. 2009; Navarro
et al. 2010). Inner density profiles as steeps as that of the SIS model (2.3)
are, however, unanimously ruled out. While the internal structure of subhalos
may be relatively unimportant in certain lensing situations, it can be crucial in
others (Moustakas et al. 2009). For lensing tests that are sensitive to the exact
slope of the inner density profile of subhalos, the subhalo-to-subhalo scatter
in this quantity may also be very important (compare chapter 5, Papers II and
III).
2.2
Dark energy and the expansion of the Universe
In the standard model of cosmology, dark energy is in the form of a cosmological constant Λ, a vacuum energy which is constant in space and time. The
cosmological constant was first proposed by Albert Einstein as a modification of his field equations in order to achieve a stationary universe (Einstein
1917). However, Friedman (1922), realised that this was an unstable solution
and proposed an expanding universe model, now called the Big Bang theory.
When Hubble (1929) showed that the universe was in fact expanding, Einstein regretted modifying his elegant theory and abandoned the cosmological
constant.
2.2 Dark energy and the expansion of the Universe
9
However, the cosmological constant gained renewed interest in the late
1990s when two teams of astronomers combined observations of nearby and
distant Type Ia supernovae (SNe Ia) in order to study the expansion rate of the
universe with time. These measurements led to the unexpected discovery of
the accelerating universe (Riess et al. 1998; Perlmutter et al. 1999).
The expansion of the universe can be described by the Friedmann equation
2
8π G
kc2 Λc2
ȧ
=
ρ− 2 +
H =
a
3
a
3
2
(2.5)
where H is the Hubble parameter, i.e. the rate of expansion of the universe, a
is the scale factor where the overdot denotes a time derivative, ρ is the mass
density of the universe and k is the normalised spatial curvature of the universe
equal to -1, 0 or +1.
The value of the Hubble parameter changes over time, either at an accelerating or decelerating rate, depending on the sign of the parameter
Ḣ
(2.6)
q = − 1+ 2
H
which had been named the deceleration parameter, illustrating the expectation
of a deceleration of the universe prior to the discoveries in the late 1990s.
2.2.1 The Hubble constant
The present value of the Hubble parameter, H0 , is called the Hubble constant.
Traditionally, the value of the Hubble constant is measured by comparing the
redshift of a source (like a galaxy or a SN) with the distance to the source. The
redshift parameter z describes how much the light emitted from a source has
been stretched due to the expansion of the universe during a time t when the
light was emitted and the current time t0 at which the source is observed
1+z =
λ0 a(t0 )
=
.
λ
a(t)
(2.7)
As the redshift of a source can be measured with high precision, the main
uncertainties in determining the value of H0 are caused by uncertainties in the
physical assumptions used to determine the distances to the sources.
For a source of intrinsic luminosity L, its luminosity distance dL is defined
by the measured energy flux F from the source
r
L
.
(2.8)
dL (z) =
4π F
For sources with known intrinsic luminosity L, so called standard candles, dL
can be determined by measuring the observed flux F . Known standard candles
10
The Dark Universe
include Cepheid variable stars with a well-defined relationship between the luminosity and pulsation period (e.g. Russell 1927; Sandage & Tammann 1968;
Udalski et al. 1999), and Type Ia SNe with a standardizable peak magnitude
(Phillips 1993; Tripp 1998; Tripp & Branch 1999). The technique of using a
succession of standard candles in galaxies at increasing redshift to determine
distances is known as the “cosmic distance ladder”.
Hubble’s initial value for the present epoch expansion rate, H0 , was
∼ 500 km s−1 Mpc−1 using data from Cepheid variable stars in nearby
galaxies (Hubble 1929). The most recent estimate using Hubble Space
Telescope observations of Cepheids and Type Ia SNe gives a value of
H0 = 73.8 ± 2.4 km s−1 Mpc−1 (Riess et al. 2011). This value agrees
well with those of alternative measurements of the Hubble constant from
gravitational lensing (e.g. Suyu et al. 2010, compare chapter 6 and Paper VI),
recent WMAP results from anisotropies of the cosmic microwave background
(Komatsu et al. 2011) and the Sunyaev-Zeldovich effect combined with
X-ray observations of galaxy clusters (Allen et al. 2008).
2.2.2 The cosmic acceleration
Combining high-redshift observations of Type Ia SNe with the cosmic distance ladder technique, enabled the discovery of the acceleration of the cosmic expansion in the late 1990s. Riess et al. (1998) and Perlmutter et al. (1999)
found that distant SNe were dimmer than what would have been expected in a
decelerating universe, indicating that the expansion of the universe has in fact
been speeding up for the past ∼ 5 Gyrs. As SNe Ia are intrinsically bright objects, they can be observed out to high redshift and are therefore an excellent
tool for probing the expansion history of the universe. Subsequent SN observations, extending to redshifts z ∼ 1.5, have reinforced the original results
(compare Figure 2.2, see Amanullah et al. 2010, for a recent compilation). For
higher redshifts, even Type Ia SNe are not bright enough to be detected with
current observational facilities. Therefore, we have suggested a technique of
utilising the magnification power of massive galaxy clusters in order to study
the expansion history of the universe to a redshift of z ∼ 3 (compare chapter 6,
Papers IV and V).
Other observational probes have contributed new evidence for the cosmic
acceleration and the existence of an additional energy component making up a
large fraction of the energy density of the universe. These include the growth
of large scale structure through galaxy cluster counts (Haiman et al. 2001) and
weak lensing shear (Hoekstra et al. 2006; Massey et al. 2007), studying the
anisotropies of the cosmic microwave background (Komatsu et al. 2011) and
measurements of the baryon acoustic oscillations through galaxy clustering
(Eisenstein et al. 2005; Percival et al. 2010).
Although, all observations today are compatible with Einstein’s cosmological constant, Λ, being responsible for the accelerated expansion of the uni-
2.2 Dark energy and the expansion of the Universe
11
Figure 2.2: The Hubble diagram of Type Ia SNe from the recent Union2 compilation
(Amanullah et al. 2010). The linear expansion in the local universe, described by the
Hubble constant H0 , can be traced out to a redshift z < 0.1. Distance is measure by
the distance modulus µ = m − M, the difference between the apparent magnitude m
and the absolute magnitude M of the SNe. The distance relative to an empty universe
model (ΩM = ΩΛ = 0) is shown in the lower panel for binned data points. It fits
well with the prediction from a ΛCDM model (ΩM = 0.3, ΩΛ = 0.7) shown by the
overplotted curve. Figure adopted from (Goobar & Leibundgut 2011).
verse, there are some unexplained curiosities. Associating Λ with a vacuum
zero-point energy, its corresponding density does not match any scale predicted from standard particle physics theory but might be too low by up to
120 orders of magnitude. Furthermore, there is no compelling argument why
the present fraction of the energy density of the universe in matter and dark energy are so close in spite of mass density being diluted ∝ a3 while the vacuum
energy density remains constant. This is known as the coincidence problem.
Alternative theories to the cosmic expansion being driven by a cosmological
constant Λ, can be tested by precision measurements of the expansion history
of the universe.
13
3
Gravitational lensing
Einstein’s general theory of relativity predicts that the path of light is bent
in a gravitational field, i.e. a ray of light passing close to a massive body is
deflected towards that body. In this chapter, we will describe some of the basic
features of gravitational lensing used in this thesis.
3.1
Strong and weak lensing
Gravitational lensing effects can be divided into two regimes, strong and weak
lensing, depending on the alignment of the lens and source (see Fig. 3.1).
Strong lensing occurs when the line of sight from the observer to source is very
close to the lens, a situation that gives rise to high magnifications, multiple
images, arcs and rings in the lens plane. Weak lensing occurs when the lens is
located further away from the line of sight, resulting in small magnifications
and mild image distortions. Generally speaking, weak lensing is extremely
common in the cosmos (at some level, every single light source is affected)
but inconspicuous, and can only be detected statistically by studying a large
number of lensed light sources. Strong lensing effects, on the other hand, are
rare but dramatic, and can readily be spotted in individual sources. The lensing
effects studied in this thesis can all be attributed to the strong lensing regime.
3.2
Arrival time and images
Fermat’s principle states that the path taken by a ray of light between two
points must be stationary. Thus images will appear where the arrival time surface has either a minimum, maximum or a point of inflection (a saddle point).
This condition can be written as
~∇t(~θ ) = 0
(3.1)
with the arrival time t(~θ ). The arrival time t(~θ ) is proportional to the light
travel time of a ray with fixed angle ~β , the true position of the source on the
sky, as a function of ~θ , the apparent position of the source measured by an
observer due to lensing (compare figure 3.2).
The arrival time consists of two components, a ‘geometrical’ and a ‘gravitational’ part:
(3.2)
t(~θ ) = tgeom (~θ ) + tgrav (~θ )
14
Gravitational lensing
Figure 3.1: Weak and strong lensing. a) Weak lensing occurs when the lens (here illustrated by a gray elliptical galaxy surrounded by a dark matter halo) lies relatively
far from the line of sight between the observer (eye) and the background light source
(star). In this case, only a single image is produced, subject to mild magnification and
distortion. The signatures of this are only detectable in a statistical sense, by studying
the weak lensing effects on large numbers of background light sources. b) Strong lensing can occur when the dense central region of the lens is well-aligned with the line of
sight. The light from the background light source may then reach the observer along
different paths, corresponding to separate images in the sky. This case is also associated with high magnifications and strong image distortions. The angular deflection in
this figure, as in all subsequent ones, has been greatly exaggerated for clarity.
The geometrical part is given by the difference between the unlensed and
the lensed path (dotted and dashed line in figure 3.2, respectively),
DL DS ~ ~ 2
1
(θ − β ) ,
tgeom (~θ ) = (1 + zL )
2
cDLS
(3.3)
where zL is the lens redshift and DS , DL and DLS denote the angular diameter
distances from observer to source, observer to lens and from lens to source,
respectively. The angular diameter distance to an object is defined by the ratio
of its actual size l to the angle it spans on the sky δ θ ,
D≡
l
.
δθ
(3.4)
For a flat universe in the standard cosmological model with a matter density,
ΩM , and a cosmological constant, Λ, the angular diameter distance between
objects at redshift zA and zB is given by
D zA zB
c
=
(1 + zB )H0
Z zB
zA
dz
p
ΩM (1 + z)3 + ΩΛ
.
(3.5)
The gravitational part is also known as the Shapiro time delay. It arises from
relativistic effects in a gravitational field and depends on the surface density
Σ(~θ ) of the lens at the apparent position ~θ ,
8π G
1
tgrav (~θ ) = − (1 + zL ) 3 ∇−2 Σ(~θ ).
2
c
(3.6)
3.3 Einstein radius and lens equation
15
Figure 3.2: Light deflection of a background source by a lens. Without the presence of
the lens, the light from the background source would arrive at the observer following
the dotted line and appear at the sky position ~β . Due to the lens, the light is deflected
(dashed line) and is observed at the apparent position ~θ . DS , DL and DLS denote the
angular diameter distances from observer to source, observer to lens and from lens to
source, respectively.
3.3
Einstein radius and lens equation
In the special case of a point-mass lens aligned with a point source, i.e. Σ(~θ ) =
M δ (~θ ) and ~β = 0, the arrival time t(~θ ) has a minimum at θ = θE where
θE2 =
4GM DLS
,
c2 DL DS
(3.7)
is known as the Einstein radius. This lensing situation results in a ring image
centered on the lens with angular radius θE , called an Einstein ring.
Although this is not a very common situation, the Einstein radius is still a
very useful concept since the angular image separation in a multiple-image
systems typically is of the order of 2θE . This can be derived from the twodimensional analog of Gauss’s flux law: for any circular mass distribution
Σ(θ ), ~∇t(~θ ) will only depend on the enclosed mass. This implies that any
circular distribution of mass M will produce multiple images from a colinear
source provided it fits within θE which is known as the ‘compactness’ criterion. Equation 3.7 states that the area within θE depends linearly on the lens
mass M . Thus, for each lens situation there is a critical surface mass density
Σcrit =
c2 DL DS
4π G DLS
(3.8)
which needs to be exceeded to satisfy the compactness criterion and produce
an Einstein ring or multiple images.
We can use the critical surface mass density Σcrit , as defined above, to introduce the dimensionless scaled surface mass density κ (~θ ) which is also known
16
Gravitational lensing
as the convergence
Σ(~θ )
κ (~θ ) =
.
Σcrit
(3.9)
Defining the time scale
T0 = (1 + zL )
DL DS
,
cDLS
(3.10)
we can also express the arrival time t(~θ ) dimensionless as the scaled arrival
time τ (~θ ) with
t(~θ )
.
(3.11)
τ (~θ ) =
T0
Equation 3.2 can then be rewritten as
1
τ (~θ ) = (~θ − ~β )2 − 2∇−2 κ (~θ ).
2
(3.12)
The second term, 2∇−2 κ (~θ ), depends on the mass distribution of the lens and
is called the lens potential ψ (~θ ).
Images will appear for ~∇τ (~θ ) = 0, which can then be expressed as
~β = ~θ − ~∇ψ (~θ ),
(3.13)
the so-called lens equation.
3.4
Image magnification
The gravitational potential of a foreground lens can not only change the apparent position ~θ and arrival time t(~θ ) of a source but also affect its observed
flux. Since surface brightness is preserved by lensing (Liouville’s theorem),
but gravitational lensing changes the apparent solid angle of a source, the total flux received from a gravitationally lensed image of a source is therefore
changed. The ratio of the flux of an image to the flux of the unlensed source
is called the magnification µ and is proportional to the ratio between the solid
angles of the image and the source in absence of a lens.
The source-plane displacement needed to produce a given small image displacement, i.e. the inverse of the magnification, is given by (combining equations 3.12 and 3.13)
~∇~β = ~∇~∇τ (~θ ) = M−1
(3.14)
where M is a 2D tensor which depends on ~θ but not ~β . This equation implies
that the magnification µ = det M is equal to the inverse of the curvature of the
arrival time surface. Examining M closer, one sees that it can be devided into
a trace (1 − κ ) and a traceless part γ . The scalar magnification µ can then be
computed using
(3.15)
µ = det M = [(1 − κ )2 − γ 2 ]−1 .
3.5 Lensing on different scales
17
Here κ is the convergence and identical to the scaled surface mass density
already defined in equation 3.9. It can also be interpreted as an isotropic magnification. γ is called the shear which changes the shape of an image but not
its size.
3.5
Lensing on different scales
Strong lensing has traditionally been divided into subcategories, depending on
the typical angular separation of the multiple images produced: macrolensing
(& 0.1 arcseconds), millilensing (∼ 10−3 arcseconds), microlensing, (∼ 10−6
arcseconds), nanolensing (∼ 10−9 arcseconds) and so on. When large galaxies (M ∼ 1012 M⊙ ) or galaxy clusters (M ∼ 1014 –1015 M⊙ ) are responsible
for the lensing, the image separation typically falls in the macrolensing range,
whereas individual solar-mass stars give image separations in the microlensing regime. Since all objects with resolved multiple images due to gravitational lensing have image separations of & 0.1 arcseconds, the term strong
lensing is often used synonymously with macrolensing.
19
4
MACHOs vs WIMPs
In general, dark matter refers to any matter that is undetectable by absorbed,
emitted or scattered electromagnetic radiation. Due to the observed large-scale
distribution of galaxies in universe, dark matter must be cold, i.e. move at nonrelativistic speed. The most likely candidates for cold dark matter (CDM) at
present are elementary particles called weakly interacting massive particles
(WIMPs). However, CDM might to some part consist of compact objects,
so-called MACHOs (massive compact halo objects). Although the MACHO
acronym was originally invented with baryonic objects in mind several nonbaryonic dark matter candidates can also manifest themselves in this way –
e.g. axion aggregates (Membrado 1998), mirror matter objects (Mohapatra
1999), primordial black holes (Green 2000), preon stars (Hansson & Sandin
2005), scalar dark matter miniclusters (Zurek et al. 2007) and ultracompact
minihalos (Ricotti & Gould 2009). Hence, MACHOs could in principle make
up a substantial fraction of the dark matter of the universe. As there is still
a large fraction of the baryonic matter density unaccounted for (Fukugita &
Peebles 2004; Bregman 2007), the so-called “missing baryons”, there might
even be a substantial population of baryonic MACHOs like faint stars and
stellar remnants awaiting detection.
Gravitational lensing provides a way to test if and what fraction of the dark
matter density in the universe consists of MACHOs.
4.1
Microlensing in the local group
Our galaxy, the Milky Way, is expected to reside inside a CDM halo which
comprises ∼ 95% of the total mass (Battaglia et al. 2005). If at least some fraction of the CDM is in the form of MACHOs, gravitational lens signals should
be observable in background sources (stars) whenever a MACHO passes close
to the line of sight. As most MACHOs are expected to have masses on (sub)solar scales, the resulting image splitting would be on the order of microarcseconds and thus below the detection threshold of any available telescope.
However, this lensing event would cause a well measurable magnification of
the background source (Paczynski 1986) on the time scale it takes the source
to cross the Einstein radius of the lensing object,
s
s
r
M
DL
DLS 200kms−1
DL θE
≈ 0.214yr
.
(4.1)
t=
v
M⊙ 10kpc DS
v
20
MACHOs vs WIMPs
Thus, the duration of such a microlensing event depends on the mass of the
lens, as well as the relative velocity v between source and lens and the lensing
geometry via the angular diameter distances DL , DS and DLS . The expected
time scales are on the order of hours up to a few hundred days for MACHOs
in the mass range 10−6 – 102 M⊙ .
Due to the degeneracy of the time scale with the different parameters in
equation 4.1, it is not possible to determine the MACHO mass from one single microlensing event. Instead, a number of microlensing detections combined with the modelling of the spatial and velocity distribution of the lensing
objects will be needed in order to put constrains on MACHO masses and densities.
The probability for observing a microlensing event by MACHOs can be
expressed in terms of its optical depth, τ , i.e. the probability for the line of
sight to a background star to pass within the Einstein radius, θE , of a MACHO.
The optical depth is given by
τ=
Z Ds
4π Gρ (DL ) DL DLS
0
c2
DS
dDL
(4.2)
where ρ (DL ) is the matter density in MACHOs at distance DL from the observer. Unlike the microlensing time scale, t , the microlensing optical depth,
τ , depends only on the mass density ρ and not the specific mass M of the
MACHOs. The optical depth for MACHO microlensing is estimated to be of
order 10−6 – 10−7 . Thus, the continuous monitoring of millions of stars is
required for such microlensing events to be detected.
Furthermore, as a large fraction of stars are intrinsically variable, such luminosity variations will have to be ruled out if flux variations are observed
in a background star in order to confirm the detection of an a microlensing
event. Fortunately, these scenarios can be distinguished due to the fact that the
light curves of lensed stars are, contrary to those of variable stars, expected to
be both symmetric in time and show achromatic magnifications (in the small
source regime) as gravitational lensing does not depend on wavelength.
Several groups have put in an extensive effort in the search for MACHOs
through their lensing effect by studying millions of stars in the galactic center,
the Large and Small Magellanic Clouds (LMC and SMC) and Andromeda
(M31). Here, we summarise some of the constraints gained in these studies.
The MACHO project analysed data taken from 11.9 million stars over 5.7
years in the LMC and found 13 – 17 events with time scales, t , ranging from
34 to 230 days (Alcock et al. 2000). This is in contrast to the ∼ 2 – 4 events
expected from lensing by known stellar populations in the Milky Way and the
LMC. The microlensing optical depth from events with 2 < t < 400 days was
+0.4
× 10−7 . Although this rules out a model in
then estimated to be τ = 1.2−0.3
which all of the dark matter halo consists of (uniformly distributed) MACHOs
at 95% confidence level, a maximum-likelihood analysis gives a MACHO
halo fraction of 20% for a typical halo model with a 95% confidence interval
4.2 Extragalactic microlensing
21
of 8% – 50%. However, supernovae (SNe) in galaxies behind the LMC are
expected to be an important source of background and some of the microlensing events have been challenged to be SNe or variable stars. Depending on the
halo model, the most likely MACHO mass ranges between 0.15 and 0.9 M⊙ .
The EROS-2 project monitored 33 million stars over 6.7 years in the LMC
and SMC and found only one microlensing event, whereas ∼ 39 events would
have been expected if the Milky Way dark matter halo was entirely populated
by MACHOs of mass M ∼ 0.4M⊙ (Tisserand et al. 2007). Combined with the
results of EROS-1 (Lasserre et al. 2000), this implies a microlensing optical
depth τ < 0.36 × 10−7 at 95% confidence level, corresponding to less than 8%
of the halo mass being in the form of MACHOs.
The OGLE project observing the LMC and central bulge of the Milky Way
detected only two possible microlensing events, translating into an optical
depth of τ = 0.43 ± 0.33 × 10−7 , although these events seem to be consistent with the self-lensing scenario, i.e. microlensing by normal stars inside the
target galaxy (Wyrzykowski et al. 2009). However, if both events were due to
MACHOs, this results in upper limits on their abundance in the galactic halo
of 19% for objects of mass M ∼ 0.4M⊙ and 10% for objects in the mass range
0.01 – 0.2 M⊙ .
The effect of self-lensing of stars in the LMC and M31 has been discussed
in several works arriving at very different conclusions regarding its contribution to the observed microlensing rates (Aubourg et al. 1999; Gyuk et al. 2000;
Jetzer et al. 2002; Ingrosso et al. 2006; de Jong et al. 2006; Calchi Novati et al.
2009).
Thus, although microlensing events by MACHOs may have been detected
within the local group, they are insufficient to explain the total amount of
dark matter expected in the Milky Way halo within the probed mass range
(∼ 10−7 – 10 M⊙ ) . However, local microlensing observations are only able to
detect the densest, most compact MACHOs at a given mass (e.g. Zurek et al.
2007), whereas more diffuse objects will give rise to detectable microlensing
effects only at higher redshifts.
4.2
Extragalactic microlensing
Press & Gunn (1973) suggested a technique for constraining a cosmologically
significant density of compact objects by searching for their gravitational lensing effects on distant sources. After the discovery of the first multiply imaged
quasar (QSOs), it was proposed that compact objects like stars and MACHOs
in the lens galaxy should act as microlenses on the individual images, giving
rise to uncorrelated flux variations (Chang & Refsdal 1979; Gott 1981).
Contrary to microlensing by compact objects within the local group, the
optical depth, τ , for microlensing of strongly lensed QSOs is of order unity.
Thus, essentially all of these systems should be affected by microlensing at
22
MACHOs vs WIMPs
any time with a non-negligible probability of several microlenses simultaneously contributing to an observed flux variation. Depending on the size of the
source, the caustic crossings of the microlenses can be resolved individually
with relatively high maxima in the lightcurves (small sources) or are smoothed
out (large sources). To evaluate the effects of this, numerical methods were developed (see Wambsganss 2006, for a review on the different techniques).
Studying the properties of such variations, constraints on the abundance of
compact objects in the halo of the lensing galaxy can be obtained, assuming
a model for the QSO size. A number of multiple quasar systems have been
monitored, searching for brightness fluctuations in the different components
which can not be mapped to the other multiple images, taking time delays into
account. Such fluctuations can not be due to intrinsic luminosity variations.
Because QSOs are unresolved in optical imaging, appearing as point
sources, gravitational microlensing offers at the same time an interesting
possibility of producing information about the luminous structure.
Early lightcurves of the first gravitational lens discovered, double QSO
Q0957+561, showed an almost linear change in brightness between the two
images of ∆mAB ≈ 0.25 mag, which can be interpreted as potential evidence
for microlensing (Schild 1996). However, since then, differential flux variations of only ∆mAB < 0.05 mag have been observed in this system (e.g. Schild
1996; Schmidt & Wambsganss 1998). Combining extensive numerical simulations with monitoring data of the double QSO, it has been possible to put
constraints on the fraction of dark matter in MACHOs with masses ∼ 10−7 –
1 M⊙ , as well as the source size of the QSO (Wambsganss et al. 2000; Refsdal
et al. 2000; Colley et al. 2003; Colley & Schild 2003).
The Einstein Cross, QSO Q2237+0305, consisting of four images arranged
symmetrically around the lens galaxy has been intensely monitored by a number of groups in the hope of observing high-magnification events (e.g. Corrigan et al. 1991; Alcalde et al. 2002; Eigenbrod et al. 2008b). This system
is particularly well suited for microlensing studies since it shows very short
time delays between the images, due to its symmetric configuration, as well
as a high expected optical depth for microlensing from stars, due to the positions of the images being located closly to the lens (e.g. Wambsganss et al.
1990, compare Figure 4.1). Flux variations have been observed for all of the
images with strenghts up to ∆m ∼ 1 mag, allowing to resolve QSO structures
on spatial scales several orders of magnitudes below the resolution of exisiting telescopes (Woźniak et al. 2000a,b; Anguita et al. 2008b; Eigenbrod et al.
2008a).
Several other multiply imaged QSOs have been monitored for microlensing
effects in order to put constraints on QSO source structure and the fraction of
galaxy halo mass in MACHOs (e.g. Koopmans & de Bruyn 2000; Jackson
et al. 2000; Anguita et al. 2008a). Recently, Mediavilla et al. (2009) combined
microlensing measurements of a sample of 29 QSO image pairs and derived a
4.2 Extragalactic microlensing
23
Figure 4.1: The Einstein Cross, QSO Q2237+0305, which has been strongly lensed
into four images by a galaxy acting as a gravitational lens (positioned in the center).
Credit: NASA, ESA, and STScI.
fraction of halo dark matter mass in MACHOs with masses 0.1 – 10 M⊙ of .
0.1 assuming reasonable QSO source sizes.
Thus, similar to the results obtained in the local group, microlensing studies
of multiply imaged QSOs support the hypothesis of only a low content in
MACHOs.
In addition to microlensing searches in multiply imaged sources, it has been
suggested to look for flux variability in individual high-redshift sources such
as QSOs, SNe and gamma-ray bursts (GRBs). Although this might prove to be
difficult in the case of QSOs as they exhibit a varying degree of intrinsic variability, microlensing should be detectable through its effect on SN light curves
(e.g. Kolatt & Bartelmann 1998; Metcalf & Silk 2007) and GRB afterglows
(e.g Wyithe & Turner 2002a; Baltz & Hui 2005).
In the case of multiply imaged light sources, it is customary to adopt the thin
lens approximation, as most of the microlensing can be assumed to take place
in the dark halo responsible for the macrolensing effect. For high-redshift
sources which are not multiply-imaged, the situation becomes more complicated, as compact objects at vastly different distances along the line of sight
may contribute to the microlensing. Most studies of the latter situation (Rauch
1991; Schneider 1993; Loeb & Perna 1998; Minty et al. 2002; Zackrisson
& Bergvall 2003; Zakharov et al. 2004; Baltz & Hui 2005) adopt the Press
& Gunn (1973) approximation, in which the microlenses are assumed to be
randomly and uniformly distributed with constant comoving density along
the line-of-sight. Due to the strong clustering of matter (e.g. Springel et al.
2006), this approximation may not be entirely suitable and may give rise to
misleading results (Wyithe & Turner 2002b). In Paper I, we relaxed this approximation in favour of a model where the compact objects instead follow the
spatial distribution predicted for CDM, thereby clustering into cosmologically
24
MACHOs vs WIMPs
distributed halos and subhalos, as would be expected for non-baryonic MACHOs. We demonstrate that this gives rise to substantial sightline-to-sightline
scatter in microlensing optical depths, and quantify this scatter as a function
of source redshift.
It has been claimed that the optical variability of QSOs on long time scales
(years to decades) can be caused by microlensing by MACHOs in the subsolar mass range (e.g Hawkins 1996). In principle, this microlensing variability
scenario has several attractive features. It provides a natural explanation for
the statistical symmetry (Hawkins 2002), achromaticity (Hawkins 2003) and
lack of cosmological time dilation (Hawkins 2001) in the optical light curves
of QSOs. Zackrisson et al. (2003) argued that the long-term optical variability
of QSOs could not primarily be caused by microlensing because the number of predicted high-amplitude magnifications was much lower than what is
observed in the light curves.
By applying the model developed in Paper I to the long-term optical variability of QSOs which are not multiply-imaged, we show that relaxing the
Press & Gunn approximation only has a modest effect on the predicted distribution of light curve amplitudes. Hence, the problems associated with microlensing as the dominant mechanism for the long-term variability are in no
way diminished, but instead slightly augmented, once the large-scale clustering of MACHOs predicted in the ΛCDM cosmology is taken into account.
In summary, studying the effects of microlensing on both local sources as
well as sources at cosmological redshifts, suggests that at most a small fraction of the dark matter density can consist of MACHOs in a mass range of
∼ 10−7 − 10 M⊙ . Therefore, the hypothesis of MACHOs as the dominant
matter component in the universe has most probably to be abandoned in favour
of e.g. the WIMPs.
25
5
Missing satellites
The ΛCDM standard model seems to predict a large amount of galactic substructure in the mass range of dwarf-galaxy masses (∼ 106 – 1010 M⊙ ), the socalled missing satellites, which has not (yet) been observationally confirmed
(compare chapter 2.1.1). Lensing by such objects has been estimated to give
rise to millilensing, although the exact image separation depends on the internal structure of these objects. In this chapter, we will discuss the different
possible approaches for the search of these presumably dark objects using
gravitational lensing.
In principle, any distant light source may be affected by subhalos along the
line of sight. The situation is schematically illustrated in Fig. 5.1a. A random
line of sight towards a light source outside the local volume passes within the
virial radius of numerous galaxy-sized CDM halos (Paper I), and may hence
intersect subhalos anywhere along the line of sight. The probability of intersecting a subhalo is, however, rather small in this situation, and sightlines of
this type may also pass through low-mass field halos, i.e. the progenitors of
subhalos (Chen et al. 2003; Metcalf 2005a,b; Miranda & Macciò 2007). It will
therefore be difficult to distinguish between the effects produced by these two
types of lenses. While the low-mass end of the field halo population may be
very interesting in its own right, lensing by such objects is more often considered an unwanted “background” when attempting to address the missing
satellite problem as it is currently defined.
Instead, the main targets for attempts to constrain the CDM subhalo population using lensing have so far been sources that are already known to be
macrolensed (see Fig. 5.1b), which in practice means observing either multiply imaged QSOs or galaxies lensed into arcs or Einstein rings. By doing
so, one preselects a sightline where one knows that there is a massive dark
matter halo (with subhalos) located along the line of sight. Whether several
distinct, point-like images or elongated arcs that approach the form of a ring
are seen, mainly depends on the source size: point-like sources (QSOs in the
optical, but potentially also SNe, GRBs and their afterglows) give distinct
images whereas extended sources (galaxies) give rise to arcs and rings (see
Fig. 5.2). The strong magnification produced by the macrolens (large foreground galaxy) acts to boost the probability for lensing by the subhalo, and
typically augments the observable consequences of such secondary lensing.
Transient light sources, like SNe or GRBs can in principle also be used for
26
Missing satellites
Figure 5.1: Singly-imaged and macrolensed sources. a) The sightline towards a distant light source passes through many halos with subhalos, but too far from the halo
centres for macrolensing to occur. A subhalo in one of these halos happens to intersect the line of sight, thereby potentially producing millilensing effects in a singlyimaged light source. b) One of the halos happens to lie almost exactly on the line of
sight, thereby splitting the background light source into separate macroimages. One
of the subhalos in the main lens crosses the sightline towards one of the macroimages,
thereby producing millilensing effects in this macroimage.
this endeavour, but no macrolensed sources of this type have so far been detected.
Lensing by subhalos can give rise to a number of observable effects, which
we describe in the following sections: flux ratio anomalies, astrometric effects,
small-scale structure in macroimages and time-delay effects. In what follows,
we will focus on subhalos in the mass range from globular clusters to dwarf
galaxies (∼ 105 –1010 M⊙ ), since current predictions indicate that subhalos at
lower masses may be very difficult to detect through lensing effects.
5.1
Flux ratio anomalies
It was noticed quite early that simple, smooth models of galaxy lenses usually
fit the image positions of macrolensed systems well, while the magnifications
of the macroimages are more difficult to explain (Kochanek 1991). To see how
this works, some more simple lens theory is required.
Specific relations are expected to apply for the magnifications of macroimages close to each other and a critical line. Formally, critical lines are the
curves in the lens plane where the magnification tends to infinity. If critical curves are mapped into the source plane, a set of caustic curves is obtained. These separate regions in the source plane that give rise to different
numbers of images (see Fig. 5.3). The smooth portions of a caustic curve are
called folds, while the points where two folds meet are referred to as cusps.
For a background source which is close to either a fold (Fig. 5.3a) or a cusp
(Fig. 5.3b) in the caustic of a smooth lens, two respectively three close images
5.1 Flux ratio anomalies
27
Figure 5.2: Small and large sources. a) A galaxy surrounded by a dark matter halo
produces multiple images of a small background light source (e.g an optical QSO). b)
For a larger background source (e.g. a galaxy or a radio-loud QSO), the macroimages
may blend into arcs or even a complete Einstein ring.
will be produced near the critical line in the lens plane. If the source is placed
in the center of the caustic, the macroimages will form a cross configuration
(Fig. 5.3c).
All macroimages can furthermore be described as having either positive
parity (meaning that the image has the same orientation as the source) or negative parity (the image is mirror flipped relative to the source). When taking
the image parity into account and assigning negative magnifications to negative parity images, the sum of the magnifications of the close images should
approach zero (Mao 1992; Schneider & Weiss 1992; Zakharov 1995). The following relations should then apply for the flux ratio R of a fold configuration:
Rfold =
| µA | − |µB |
→ 0,
| µA | + |µB |
(5.1)
when the separation between the close images is asymptotically small. Here,
µ represents the magnification of a specific image. For the cusp configuration,
the corresponding relation is:
Rcusp =
|µA | − |µB | + |µC |
→ 0.
|µA | + |µB | + |µC |
(5.2)
However, most observed lensing systems violate these relations. This has
been interpreted as evidence of small-scale structure in the lens on approximately the scale of the image separations between the close images. Magnifications of individual macroimages due to millilensing by subhalos would
indeed cause the values for Rfold and Rcusp to differ from zero fairly independently of the form of the rest of the lens (Mao & Schneider 1998; Metcalf &
Madau 2001; Chiba 2002; Dalal & Kochanek 2002; Metcalf & Zhao 2002;
Keeton et al. 2003; Kochanek & Dalal 2004).
A notable problem with this picture is that recent high-resolution ΛCDM
simulations seem to be unable to reproduce the observed flux ratio anomalies,
28
Missing satellites
Figure 5.3: Different configurations of a four-image lens: a) Fold, b) Cusp and
c) Cross. The upper row shows the caustics and position of the source (star) in the
source plane. The solid line indicates the inner caustic and the dashed line the outer
caustic. A source positioned inside the inner caustic produces five images. A source
positioned between the inner and outer caustic produces three images, whereas a
source positioned outside the outer caustic will not be multiply imaged. In the case
of multiple images, one of the images is usually highly de-magnified, so that only
four- and two-image lens systems are observed, respectively. The lower row shows
the corresponding critical lines and resulting observable images in the lens plane. Note
that the image labeling differs for the different cases. The inner caustic maps on the
outer critical line and vice versa. A close pair (A, B) and a close triplet (A, B, C) are
produced in the fold (a) and cusp (b) configurations, respectively.
.
since the surface mass density in substructure is lower than that required close
to the Einstein radius where the multiple images are produced (Zentner &
Bullock 2003; Amara et al. 2006; Macciò et al. 2006; Macciò & Miranda
2006; Xu et al. 2009).
Several alternative causes for the observed flux anomalies have been discussed, such as propagation effects like absorption, scattering or scintillation
in the interstellar medium of the lens galaxy (Mittal et al. 2007) and microlensing by stars in the lensing galaxy (Schechter & Wambsganss 2002).
Since some sources, like QSOs, can exhibit intrinsic flux variations on different timescales depending on wavelength, flux ratios may also be difficult to
interpret if the time delay between the macroimages is not well known.
The relevance of propagation effects can be tested by supplementary observations of flux ratios at different wavelengths, since flux losses due to such
mechanisms should vary as a function of wavelength. Microlensing by stars
can be checked for using long-term monitoring, as this type of lensing is transient and expected to introduce extrinsic variability on the order of months.
Millilensing by halo substructure can on the other hand be treated as stationary (Metcalf & Madau 2001). Extended sources (e.g. QSOs at mid-infrared
and radio wavelengths) should also be far less affected by microlensing than
5.1 Flux ratio anomalies
29
small, point-like sources (QSOs in the optical and at X-ray wavelengths). Even
though it is often assumed that radio observations of QSOs are essentially
microlensing-free, some caution should be applied, since substantial shortterm microlensing variability is possible in the special case of a relativistic
radio jet oriented close to the line of sight. This phenomenon has been detected in at least one multiply-imaged system (Koopmans & de Bruyn 2000).
Mid-infrared imaging of lenses is attractive because the flux is free from differences in extinction among the macroimages, in addition to being free from
microlensing by stars due to the extended source size. Such observations can
therefore be used to test some of the alternative causes for flux ratio anomalies. Recent studies have used this technique to examine several macrolensed
QSOs with known flux ratio anomalies in the optical (Chiba et al. 2005; Sugai
et al. 2007; Minezaki et al. 2009; MacLeod et al. 2009). The mid-infrared flux
ratios of about half of these systems can be fitted with smooth lensing models, which means that only the remaining half of these anomalies are due to
millilensing by substructures.
There are also other observational features that argue for substructures as
the cause of (at least some) flux ratio anomalies. Negative parity images (so
called saddle images – e.g. the middle image of a close triplet, like image
B in Fig. 5.3b) are often fainter than predicted by smooth lens models. This
is expected from millilensing, as the magnification perturbations induced by
substructure lensing have been shown to depend on image parity (Schechter
& Wambsganss 2002; Kochanek & Dalal 2004; Chen 2009). In contrast, such
anomalies cannot be attributed to propagation effects since those should statistically affect all types of images similarly, regardless of their parity. Whether
the lensing is due to luminous or dark substructures is, however, a different
matter.
Luminous substructures have been identified in many of the lens systems
with known flux ratio anomalies. Including such substructures in the lens
model tends to greatly improve the fit to observations. One example of such
a lens system is the radio-loud quadruple QSO B2045+265 (Fassnacht et al.
1999) which exhibits one of the most extreme anomalous flux rations known.
Recent deep imaging of this system has revealed the presence of a small satellite galaxy which is believed to cause the flux ratio anomaly (McKean et al.
2007). Nearly half of the lenses detected in the Cosmic Lens All-Sky Survey
(CLASS) show luminous satellite galaxies within a few kpc of the primary
lensing galaxy (Xu et al. 2009).
Recently, there have been studies combining the results from simulations
and semi-analytical models of galaxy formation to investigate if luminous
dwarf galaxies might be able to explain the frequency of flux ratio anomalies
observed (Shin & Evans 2008; Bryan et al. 2008). They find that the fraction of luminous satellites in group-sized halos is roughly consistent with the
observational data within a factor of two, while the results for galaxy-sized
halos seem too low to explain the frequency of luminous satellites within the
30
Missing satellites
Figure 5.4: Astrometric perturbations. a) One of the multiple sightlines towards a
distant light source passes through a dark subhalo. b) The images of the macro-lensed
source are observed at the positions of the bright source symbols. Modelling of the
lens system with a smooth lens potential only, predicts the position of the upper image
at the dark source symbol. The subhalo close to the sightline of the image causes a
deflection on the order of a few tens of milliarcseconds.
observed systems. The lensing effect of these luminous dwarf galaxies is also
somewhat unclear since most satellites found in the inner regions of larger
galaxies are expected to be ‘orphan’ galaxies stripped of their dark matter halos. To investigate this further, higher-resolution simulations involving a realistic treatment of the gas processes are required. Possible explanations for the
discrepancy between the expected and observed fraction of luminous satellites
include dwarf galaxies elsewhere along the line of sight mistakenly identified
as the lens perturber (Metcalf 2005b). Luminous substructures may moreover
be more efficient in producing flux ratio anomalies, since they are likely to be
denser than dark substructures due to baryon cooling and condensation (Shin
& Evans 2008).
Projection effects are potentially also important for flux ratio anomalies
caused by completely dark substructures as one expects a large amount of
line of sight structure. Although those structures are less effective than substructures within the lens galaxy in inducing magnification perturbations, the
overall effect of line of sight clumps may be significant (Chen et al. 2003;
Mao et al. 2004; Metcalf 2005b; Miranda & Macciò 2007).
5.2
Astrometric effects
In macrolensed systems, the presence of halo substructure may perturb the
angular deflection caused by the lens galaxy and thereby the position of
macroimages at observable levels, so called astrometric perturbations (see
Fig. 5.4).
This method for detecting subhalos has the advantage of being relatively
unaffected by propagation effects (absorption, scattering or scintillation by
5.2 Astrometric effects
31
the interstellar medium) and stellar microlensing that may contaminate the
flux ratio measurements. Furthermore, since the astrometric perturbation is
a steeper function of subhalo mass than flux ratio perturbations, it is mostly
sensitive to intermediate and high mass substructures and therefore probes a
distinct part of the subhalo mass function (Chen et al. 2007; Moustakas et al.
2009).
However, the overall size and probability of such a perturbation by a subhalo are expected to be rather small. Metcalf & Madau (2001) used lensing
simulations of random realisations of substructure in regions near images
and found that it would take substructures with masses & 108 M⊙ that are
very closely aligned with the images to change the image positions by a few
tens of milliarcseconds. Such an alignment would be rare in the CDM model.
Therefore, they suggest to deploy lensed jets of QSOs observed at radio wavelengths, as such sources would cover more area on the lens plane. This would
increase the probability of having a large subhalo nearby, but still allow for
pronounced distortions due to the thinness of the jet. Metcalf & Zhao (2002)
investigated this technique further and used it to show that the lens system
B1152+199 is likely to contain a substructure of mass ∼ 105 − 107 M⊙ .
Further observational evidence for astrometric perturbations from small
scale structure was found in the detailed image structures for B2016+112
(Koopmans et al. 2002; More et al. 2008) and B0123+437 (Biggs et al. 2004).
In the latter system, a substructure of at least ∼ 106 M⊙ would be needed in
order to reproduce the observed image positions.
The CDM scenario predicts that there are far more low-mass subhalos than
high-mass ones (see equation 2.1) and their summed effect could in principle
add up to a substantial perturbation. Conversely, since perturbers positioned
on opposite sides around the macrolens generate equal but opposite perturbations, the net effect of a large number of substructures may cancel out, ensuring that rare, massive substructures dominate the position perturbation of the
images. Chen et al. (2007) have investigated this by modelling the effects of
a wide range of subhalo masses and found that all residual distributions had
very large peak perturbations (& 10 milliarcseconds). Since the simulation
models predict extremely few or no substructures in the inner region of the
lens, the perturbers must be located further away. Therefore, it was also inferred that position perturbations of different images in any lens configuration
may be strongly correlated. Although these results suggested that rare, massive clumps may cause larger perturbations than the more abundant smaller
clumps, even in the models where no such massive substructures were present,
the astrometric perturbations of the images were still considerable.
Since astrometric perturbations are expected to manifest themselves at
(sub-)milliarcsecond levels, high spatial resolution observations are required
which so far are mainly achieved by Very Long Baseline Interferometry
(VLBI) observations of radio-bright sources.
32
Missing satellites
However, recent studies have shown that perturbation effects of substructure
should also be detectable on larger scales (∼ 0.1 arcseconds) and at shorter
wavelengths in extended Einstein rings and arcs produced by galaxy-galaxy
lensing. Peirani et al. (2008) used the perturbative method and lens distributions from toy models as well as cosmological simulations to predict the
possible signatures of substructures. They show that when a substructure is
positioned near the critical line, not only astrometric but also morphological effects, i.e. breaking of the image, will occur which are approximately 10
times larger and should be easier to detect. Other studies have suggested to
use non-parametric source and lens potential reconstructions to probe small
perturbations in the lens potential of highly magnified Einstein rings and arcs
(e.g. Blandford et al. 2001; Koopmans 2005). Vegetti & Koopmans (2009a)
have used an adaptive-grid method and shown that for substructures located on
or close to the Einstein ring, perturbations with masses & 107 M⊙ respectively
109 M⊙ can be reconstructed. This technique may then be used to constrain the
substructure mass fraction and their mass-function slope, once a larger sample
of high-resolution lenses becomes available (Vegetti & Koopmans 2009b).
With the upcoming generation of high spatial resolution telescopes – e.g.
the Large Synoptic Survey Telescope (LSST) and the Atacama Large Millimeter Array (ALMA) – a significant increase in the number a high-resolution
lenses is expected which will allow the use of astrometric perturbations to
study the dark satellite population.
5.3
Small-scale structure in macroimages
When dark objects in the dwarf-galaxy mass range intersect the line of sight
towards distant QSOs, image-splitting or distortion on characteristic scales of
milliarcseconds) may occur (Kassiola et al. 1991; Wambsganss & Paczynski
1992). At the current time, QSOs can only be probed on such small scales
using VLBI techniques at radio wavelengths, but future telescopes and instruments may allow similar angular resolution at both optical and X-ray wavelengths (Paper II).
Using VLBI, Wilkinson et al. (2001) reported no detections of millilensing
among 300 compact-radio sources and was able to impose an upper limit of
Ω < 0.01 on the cosmological density of point-mass objects (i.e., very compact objects, like black holes) in the 106 – 108 M⊙ range. However, this does
not convert into any strong limits on the subhalo population, since CDM halos and subhalos are not nearly as dense as black holes. Correcting for this
would decrease the expected image separations for a millilens of a given mass
and yield a probability for lensing that is much lower than assumed in their
analysis. The sources used were moreover not macrolensed, a situation which
would have made it difficult to make the distinction between subhalos and
5.3 Small-scale structure in macroimages
33
Figure 5.5: A foreground galaxy with a dark matter halo produces multiple macroimages of a background light source. A subhalo located in the dark halo intercepts one of
these macroimages, which may give rise to a) additional image small-scale splitting
of the affected macroimage if the source is sufficiently small, which is not seen in the
other macroimages, or b) a mild distortion in the affected macroimage, if the source
is large, which is not seen in any of the other macroimages.
low-mass field halos as the main culprits even if any signs of millilensing had
been detected (see Fig. 5.1a).
The effects that a subhalo can have on the internal structure of one of the
macroimages in a multiply-imaged QSO (Fig. 5.1b) are schematically illustrated in Fig. 5.5. For a small, point-like source (e.g. a QSO observed at optical wavelengths), the macroimage may split into several distinct images with
small angular separations (Fig. 5.5a). A larger source (e.g. a QSO at radio
wavelengths) may instead exhibit small-scale image distortions (Fig. 5.5b).
Even though QSOs may display complicated intrinsic structure when imaged
with high spatial resolution, such effects can at least in principle be separated
from the features imprinted by millilensing, since intrinsic structure will be reproduced in all macroimages, whereas millilensing effects are unique to each
macroimage. The distinction between these small-scale changes in the morphologies of macroimages, and the astrometric effects discussed in section 5,
becomes somewhat arbitrary in some cases, since image distortion may both
shift the centroid of the image and alter its overall appearance (e.g. through
the introduction of new, small-scale images). The distortion of macrolensed
jets is for instance often referred to as an astrometric effect.
Yonehara et al. (2003) have argued that a significant fraction of all
macrolensed optical QSOs may exhibit secondary image-splitting on
milliarcsecond scales due to CDM subhalos. Contrary, in Paper III, we
have estimated the expected optical depth for millilensing by CDM subhalos
and conclude that the prospects for observing such effects in point-sources,
like QSOs in the optical, is very low. However, we also show that the
probability for subhalo lensing is promising in the case of extended sources.
Inoue & Chiba (2005b,a) have explored a similar scenario in the case of the
extended images expected for macrolensed QSOs at longer wavelengths, and
34
Missing satellites
concluded that the small-scale macroimage distortions produced by CDM
subhalos may be detectable with upcoming radio facilities such as ALMA or
the VLBI Space Observatory Programme 2 (VSOP-2).
The merits of probing CDM subhalos through the small-scale structure of
macroimages is that, contrary to the case for flux ratio anomalies, there is little risk of confusion due to microlensing by stars or propagation effects in
the interstellar medium. Globular clusters may be able to produce similar effects (Baryshev & Ezova 1997), and so may luminous dwarf galaxies (i.e.
the subset of CDM subhalos that happen to have experienced substantial starformation), but subhalos are expected to outnumber both of these populations,
at least in most mass intervals. Instead, the main problem with this approach
seems to be that CDM subhalos may not be sufficiently dense to produce
strong lensing effects on scales that can be resolved by current technology.
Most studies of these effects have assumed that CDM subhalos can be treated
as SIS lenses, resulting in an gross overprediction of the image separations
compared to more realistic subhalo models (Paper II). The angular resolution by which macrolensed QSOs can be probed is on the other hand likely to
increase substantially in the coming years, in principle reaching ≈ 0.04 milliarcseconds with ALMA and the global VLBI array operating at millimeter
wavelengths.
5.4
Time delay effects
The images of a macrolensed light source (see Fig. 5.2a) are subject to different time delays, which become detectable when the source exhibits intrinsic
temporal variability over observable time scales. These time delays stem from
a combination of differences in the relativistic time delays (clocks running
slower in deep gravitational fields, also known as Shapiro time delays) and
the differences in photon path lengths (due to geometric deflection) among
the macroimages (compare section 3.2). Since QSOs are both non-transient
and known to vary significantly in brightness on time scales of hours and upwards, they are very convenient targets for observing campaigns aiming to
measure such time delays. At the current time, around 20 macrolensed QSOs
have measured time delays (with delay times of ∆t ∼ 0.1–400 days; see Oguri
(2007) for a compilation). Time delays have been used to constrain the Hubble
constant and the density profile of the macrolens (i.e. the overall gravitational
potential of the lens galaxy and its associated dark halo), but can also potentially be used to probe the CDM subhalos of the lens galaxy.
As shown by Keeton & Moustakas (2009), the presence of subhalos within
the macrolens will perturb the time delays predicted by smooth lens models,
and may also violate the predicted arrival-time ordering of the images. Such
violations would signal the presence of subhalos in a way that, unlike the
case for optical flux ratio anomalies, cannot be mimicked by dust extinction
5.4 Time delay effects
35
Figure 5.6: A galaxy with a dark matter halo produces distinct macroimages of a
background light source. If this source displays intrinsic variability, observable time
delays between the different macroimages may occur. If one of the macroimages experiences further small-scale image splitting due to a subhalo along the line of sight,
a light echo may be observable in the affected macroimage. This may serve as a signature of millilensing in cases where the small-scale images blend into one due to
insufficient angular resolution of the observations.
or microlensing by stars. The time delay perturbations due to subhalos are
typically on the order of a fraction of a day. By pushing the uncertainties in
the observed time delays to this level, strong constraints on CDM subhalo
populations may potentially be derived. One case of a time ordering reversal
which may possibly be attributed to subhalos has already been identified in the
macrolensed QSO RX J1131 - 1231 (Morgan et al. 2006; Keeton & Moustakas
2009).
If the subhalos themselves give rise to small-scale image splittings, short
time lags between the light pulses of the separate small-scale images would
be introduced. This imprints echo-like signatures in the overall light curve of
astronomical objects with short-term variability (such as gamma-ray bursts
and X-ray QSOs), even if the small-scale images cannot be spatially resolved.
These echos correspond to light signals transported through small-scale images with longer time delays than the leading image, and the flux ratios of
these peaks are given by the different magnifications of these images. The
light curves of gamma-ray bursts have been used to search for such light echos
in the interval ∼ 1–60 s, resulting in upper limits (Ω < 0.1) on compact dark
objects in the 105 –109 M⊙ range (Nemiroff et al. 2001) and even a few candidate detections of repeating flares due to millilensing (Ougolnikov 2003).
However, just like in the case of the search for spatial millilensing effects by
36
Missing satellites
Wilkinson et al. (2001), current investigations of this kind have little bearing
on CDM subhalos, since the probability for subhalo millilensing is too low
when the target objects are not macrolensed. Yonehara et al. (2003) instead
suggested monitoring of macrolensed QSOs, predicting that CDM subhalos
may produce light echos separated by ∼ 1000 s, which could potentially be
detected in X-rays, where rapid intrinsic flares have been observed. This lensing situation is schematically illustrated in Fig. 5.6.
Open questions and future prospects
Lensing can in principle be used to probe the CDM subhalo population, but
has so far not resulted in any strong constraints. Most studies have focused
on flux ratio anomalies, but a number of studies now suggest that subhalos by
themselves are unable to explain this phenomenon (Mao et al. 2004; Amara
et al. 2006; Miranda & Macciò 2007; Xu et al. 2009). If correct, this would
limit the usefulness of this diagnostic, since some other mechanism must also
be affecting the flux ratios. Luckily, results from other techniques, such as
astrometric perturbations, small-scale image distortions and time delay perturbations may be just around the corner.
Observationally, the future for the study of strong gravitational lensing is
looking bright. As of today, around 200 macrolensed systems have been detected with galaxies acting as the main lens. Planned observational facilities
such as the Square Kilometer Array (SKA) and the LOw Frequency ARray
for radio astronomy (LOFAR) at radio wavelengths and the LSST in the optical have the power to boost this number by orders of magnitude in the coming
decade (Koopmans et al. 2009). The spatial resolution by which these systems
can be studied is also likely to become significantly better, approaching ∼ 10
milliarcseconds in the optical and ∼ 0.1 milliarcseconds at radio wavelenghts.
On the modelling side, there are still a number of issues that need to be
properly addressed before strong constraints on the existence and properties
of CDM subhalos can be extracted from such data.
Input needed from subhalo simulations
The largest N-body simulations of galaxy-sized halos are now able to resolve
CDM subhalos with masses down to ∼ 105 M⊙ , but there are still a number
of aspects of the subhalo population that remain poorly quantified and could
have a significant impact on its lensing signatures:
• What is the halo-to-halo scatter in the subhalo mass function and how does
this evolve with redshift?
• What are the density profiles of subhalos? How does this evolve with subhalo mass and subhalo position within the parent halo? How large is the
difference from subhalo to subhalo?
5.4 Time delay effects
37
• What is the spatial distribution of subhalos as a function of subhalo mass
within the parent halo? What is the corresponding distribution outside the
virial radius?
• How do baryons affect the properties of subhalos? Can baryons promote
the survival of subhalos within the inner regions of their host halos?
The lensing effects discussed in this chapter are sensitive to the density
profiles and mass function of subhalos, albeit to varying degree (Moustakas
et al. 2009). Attempts to quantify the effects of different density profiles of
lensing signature have been made (Macciò & Miranda 2006; Chen et al. 2007;
Shin & Evans 2008; Keeton & Moustakas 2009, Paper II), but the models
used are still far from realistic, and many of those active in this field still cling
to SIS profiles for simplicity.
The role of other small-scale structure
CDM subhalos are not the only objects along the line of sight capable of producing millilensing effects. Many large galaxies are known to be surrounded
by 102 –103 globular clusters with masses in the 105 –106 M⊙ mass range.
While typically less numerous than CDM subhalos in the same mass range,
they are concentrated within a smaller volume (the stellar halo) and have more
centrally concentrated density profiles, thereby potentially making them more
efficient lenses. We also expect a fair share of luminous dwarf galaxies within
the dark halos of large galaxies. These dwarfs may represent the subset of
CDM subhalos inside which baryons were able to collapse and form stars, but
if so, this means that they may have density profiles significantly more centrally concentrated than their dark siblings. While the role of globular clusters and luminous satellite galaxies has been studied in the case of flux ratio anomalies (Chiba 2002; Shin & Evans 2008), their effects on many of the
other lensing situations discussed in previous sections have not been addressed
yet. Low-mass halos along the line of sight may also affect these lensing signatures, and sometimes appreciably so (Metcalf 2005b; Miranda & Macciò
2007).
Aside from these objects, there may of course be other surprises hiding in
the dark halos of galaxies. Intermediate mass black holes (M ∼ 102 –104 M⊙ ),
formed either in the very early Universe or as the remnants of population III
stars may inhabit the halo region (van der Marel 2004; Zhao & Silk 2005;
Kawaguchi et al. 2008) and could give rise to millilensing effects (Inoue &
Chiba 2003). If accretion onto such objects is efficient, the predicted X-ray
properties of such black holes already place very strong constraints on their
contribution to the dark matter (Ω . 0.005; Mapelli et al. 2006), but the more
generalised dynamical (Carr & Sakellariadou 1999) and lensing (Nemiroff
et al. 2001) constraints on other types of dark objects in the star cluster mass
range (∼ 102 –105 M⊙ ) are otherwise rather weak (Ω . 0.1). Lensing observations originally aimed to constrain the CDM subhalo population may therefore
38
Missing satellites
also lead to the detections of completely new types of halo substructure. As
telescopes attain better sensitivity and higher angular resolution in the next
decade, we can surely look forward to an exciting new era in the study of the
dark matter halos.
39
6
Gravitational telescopes
Galaxy clusters are the largest known gravitationally bound objects which
have formed in the universe through the process of hierarchical structure formation. They typically contain hundreds of galaxies and have total masses of
1014 – 1015 M⊙ . While only ∼ 5% of this mass is in the form of galaxies and
∼ 10% in hot X-ray emitting gas, as much as ∼ 85% is thought to be in the
form of dark matter. In this chapter, we will outline how gravitational lensing
can be used for studying these massive objects as well as high redshift light
sources located behind them.
Zwicky (1937) proposed the use of galaxies as gravitational lenses. However, it was not until the late 1970s that the first gravitationally lensed object
was identified (Walsh et al. 1979). Since then galaxy clusters have successfully been employed as so-called gravitational telescopes.
Massive galaxy clusters acting as powerful gravitational telescopes offer
unique opportunities to observe extremely distant objects. These include
galaxies out to a redshift of z ∼ 7 (e.g. Ellis et al. 2001; Kneib et al. 2004;
Bradley et al. 2008; Zheng et al. 2009; Richard et al. 2011), as well as distant
SNe (e.g. Kovner & Paczynski 1988; Kolatt & Bartelmann 1998; Gal-Yam
et al. 2002; Gunnarsson & Goobar 2003, Papers IV and V), too faint to
be detected otherwise. They can also be employed for detailed studies of
galaxies at more modest redshifts of z ∼ 3 – 5 (Bunker et al. 2000; Frye
et al. 2007, 2008). Gravitational lensing of distant objects with measured
redshifts can be used to put constraints on cosmological parameters (Gilmore
& Natarajan 2009; Jullo et al. 2010). Lensing magnifications of up to a
factor ∼ 100 have been observed for multiply lensed images, and typical
magnification factors of 5 – 10 are very common within the central one
arc-minute radius of massive cluster lenses.
However, nothing comes without a cost. Conservation of flux implies that
the survey area behind a massive cluster in the source plane is shrunk due to
large lensing magnifications. Therefore, a gravitational lens does not always
enhance the number of discoveries (Gunnarsson & Goobar 2003; Maizy et al.
2010). However, it does increase the limiting redshift of a magnitude-limited
survey. Thus massive galaxy clusters act as “high-redshift filters” and can be a
valuable tool for observing sources at redshifts beyond what would otherwise
be possible.
In Papers IV and V, we exploit this effect to search for SNe at redshifts beyond those explored by “traditional” SN searches (see Amanullah et al. 2010,
40
Gravitational telescopes
Figure 6.1: Predictions for the CC SN rate to high redshift derived from various estimates of the SFR (Mannucci et al. 2007; Dahlen et al. 2004; Kobayashi & Nomoto
2009). Figure adopted from Paper V.
for a recent review). SNe can be separated into two main physical classes.
Core collapse (CC) SNe include all type II SNe (which exhibit H lines in their
spectra) and Type Ib/c SNe (which lack H but show weak Si and S lines). Type
Ia SNe, which can be used as so-called standard candles to trace the expansion
history of the universe (compare chapter 2.2), lack H lines in their spectra but
show strong Si and S lines (see Filippenko (1997) and Leibundgut (2008) for
reviews on SN classification and their general properties).
So far, only approximately 20 well-measured SNe are known beyond a redshift of z ∼ 1. Type Ia SNe have been confirmed to redshift z ∼ 1.7 (Riess et al.
2001, 2007; Poznanski et al. 2007). Until recently, the most distant CC SNe
were confirmed to redshift z ∼ 0.8 (Soderberg et al. 2006; Poznanski et al.
2007; Botticella et al. 2008). Searching for the luminous subclass of SNe IIn,
which are very bright in the ultraviolet and thereby detectable in the optical
for redshifts of z ∼ 2 and higher, Cooke et al. (2009) discovered the highest
core collapse SN to date at z ∼ 2.4.
Finding SNe at high redshift is very interesting for several reasons. Due to
the short evolutionary lifetimes of their progenitors, the rate of CC SNe, rVCC ,
gives a direct measurement of the ongoing star-formation rate (SFR), for an assumed initial mass function. The estimates of the SFR as a function of redshift
from various data sets show a large span (e.g. Chary & Elbaz 2001; Giavalisco
et al. 2004; Hopkins & Beacom 2006; Mannucci et al. 2007; Bouwens et al.
41
Figure 6.2: Extrapolation of available SN Ia rate predictions to higher redshifts (Mannucci et al. 2007; Kobayashi & Nomoto 2009; Neill et al. 2006; Scannapieco & Bildsten 2005; Dahlen et al. 2008, corresponding to τ = 3.4 Gyr (D08) and τ = 1.0 Gyr
(TAU1)). Figure adopted from Paper V.
2009). Correspondingly, the predictions for the high redshift CC SN rate differ
substantially depending on the underlying assumptions (compare Figure 6.1).
Thus, measuring the CC SN rate, rVCC , out to high redshifts, will give valuable
information on the SFR in the early universe.
Conversely, if a star-formation history can be assumed, measuring the SN
Ia rate, rVIa , as a function of redshift and galaxy type can put constraints on the
progenitor scenarios for Type Ia SNe (e.g. Dahlén & Fransson 1999; Gal-Yam
& Maoz 2004; Strolger et al. 2004; Mannucci et al. 2005; Neill et al. 2006;
Sullivan et al. 2006; Botticella et al. 2008). While SNe Ia have been used
extensively for deriving cosmological parameters, it is unsatisfactory that the
progenitor scenario preceding the SN explosion is still mostly unknown. Existing models predict that different scenarios, such as the single degenerate
and the double degenerate models, should have different delay-time distributions, Φ(t), describing the time between the formation of the progenitor star
and the explosion of the SN, which should be reflected in the evolution of the
Type Ia SN rate, rVIa (compare Figure 6.2).
Furthermore, as has been discussed in chapter 2.2, Type Ia SNe can be
used to trace the expansion history of the universe. Using observations of high
redshift Type Ia SNe to extend the Hubble diagram to redshifts beyond z ∼ 1.5,
42
Gravitational telescopes
will help significantly to constrain the cosmological model and also to check
for systematic effects in the SNe.
In order to make predictions on the number of expected high redshift SN
discoveries behind galaxy clusters as well as for correcting any observations
for their lensing effect, accurate lens mass models are needed.
6.1
Cluster models
The study of the mass distribution within galaxy cluster proves to be interesting by itself for a number of reasons. Tracing their baryonic and dark matter components yields insights into structure formation (Umetsu et al. 2009;
Kawaharada et al. 2010) and possibly even the nature of dark matter particles
(Randall et al. 2008; Feng et al. 2009). Cluster number counts with measured
masses can be used to put constraints on cosmological parameters (Vikhlinin
et al. 2009; Mantz et al. 2010; Rozo et al. 2010; Oguri & Takada 2011).
Gravitational lensing analyses of the matter distribution within galaxy clusters have resulted in many exciting findings, like the so-called "Bullet Cluster", 1E 0657-56, providing strong evidence for the existence of dark matter
(Markevitch et al. 2004; Clowe et al. 2006; Randall et al. 2008).
Cluster mass profiles can be probed through several independent
techniques. The total gravitational mass of a galaxy cluster can be determined
by measurements of the Sunyaev-Zel’dovich effect (Sunyaev & Zeldovich
1972) towards galaxy clusters combined with information on the intra-cluster
medium temperature estimated from X-ray spectral analyses (e.g. Grego
et al. 2001; Hincks et al. 2010).
Using information from strong and weak gravitational lensing effects is
one of the most promising strategies for modeling the mass distribution in
galaxy clusters. So far, it has only been possible to produce reliable mass
maps for a few well-studied clusters enabling a comparison with predictions
(e.g. Kneib et al. 1993; Gavazzi et al. 2003; Broadhurst et al. 2008; Richard
et al. 2010; Okabe et al. 2010; Umetsu et al. 2010). The galaxy cluster A1689
is one of the most effective gravitational lenses known as well as one of the
best studied galaxy clusters to date with 114 strongly lensed images of 34
background galaxies (Broadhurst et al. 2005; Limousin et al. 2007, compare
Figure 6.3). Over the past decades, as the quality of observational data
improved, various strong lensing modeling methods have been developed.
These can usually be classified as “parametric” or “non-parameteric”.
Parametric lensing models assume certain model prescriptions characterising
the mass distribution (e.g. Limousin et al. 2007; Richard et al. 2010).
Non-parametric models, grid-based or interpolative, may instead adopt
arbitrary forms, however not necessarily physically realistic ones (e.g. Saha
et al. 2006; Coe et al. 2010). In this thesis, we have used parametric mass
6.1 Cluster models
43
Figure 6.3: Galaxy cluster Abell 1689 overlayed with a map of its dark matter distribution (shown in blue) as derived from gravitational lensing. Credit: NASA, ESA,
E. Jullo (Jet Propulsion Laboratory), P. Natarajan (Yale University), and J.-P. Kneib
(Laboratoire d’Astrophysique de Marseille, CNRS, France).
models developed using the LENSTOOL software package (Kneib 1993;
Jullo et al. 2007, http://www.oamp.fr/cosmology/lenstool/).
Strong lensing analyses of galaxy clusters, are usually constrained by the
positions of the multiple images. In Paper VI, we have investigated the power
of additional constraints from a Type Ia SN observed in a background galaxy.
Due to the standard candle nature of SNe Ia, this would give a direct measurement of the lensing magnification of the galaxy cluster at the position of
the SN. For galaxy clusters with no or only very few known multiply imaged
systems, this technique is especially interesting, breaking the so-called mass
sheet degeneracy (Kolatt & Bartelmann 1998, Paper VI). We have shown that
even for a well studied cluster as A1689, the precision of the parameters describing the overall dark matter component of the cluster might improve by
almost a factor 2.
44
Gravitational telescopes
Once, the mass distribution of a galaxy cluster has been modeled with good
precision, this enables several interesting predictions, e.g. in the case of SNe
exploding in multiply imaged background galaxy.
6.2
Multiply imaged SNe
So far, there are no known multiply imaged SNe. This situation might however
soon change, with several surveys targeting massive clusters both at optical
and near-IR wavelengths (e.g. Zitrin et al. 2011b, Papers IV and V). Strongly
lensed SNe are interesting for several reasons.
If a SN is strongly lensed, a measurement of the time delay between the
transient in the multiple images could potentially constrain the Hubble parameter (Refsdal 1964), and thus dark matter and dark energy parameters (Goobar
et al. 2002; Mörtsell & Sunesson 2006; Suyu et al. 2010). This can be seen
when analysing the expression for the arrival time (compare equation 3.2) for
its dependence on cosmological parameters. Assuming the lens potential is
well understood (which usually is the cause of the largest uncertainties) and
the lens redshift zL can be measured, the time delay between two images will
depend on the distance ratio
D≡
DL DS
,
DLS
(6.1)
where DL , DS and DLS denote the angular diameter distances from observer
to source, observer to lens and from lens to source, respectively. As this ratio
scales inversely with the Hubble constant, D ∝ H0−1 , a measurement of the
time delay can give an estimate on the Hubble constant. There is also a dependence in D on other cosmological parameters, such as ΩM and ΩΛ , although
this dependence is much weaker than that on the Hubble constant (e.g., Bolton
& Burles 2003; Coe & Moustakas 2009; Suyu et al. 2010).
Though not as precise as other cosmological tests to study dark energy,
the time delay technique has the major advantage of measuring cosmological parameters at redshifts where few other probes are currently available.
Given the transient nature of SNe, the time delay between multiple images
could potentially be measured to very high precision. The main limitation
of this technique is the strong degeneracy between the lens mass model and
the Hubble parameter, H0 (e.g., Wambsganss & Paczynski 1994; Witt et al.
2000; Kochanek 2002; Zhao & Qin 2003). However, observing cluster lenses
with potentials well constrained by a large number of already known multiple
images or, in the case of a strongly lensed SNe Ia, direct magnification information, this degeneracy can be diminished (Oguri & Kawano 2003, Paper
VI).
In the case of a SN exploding in a galaxy which is known to be multiply
imaged by a foreground galaxy cluster, one might even be able to predict when
6.2 Multiply imaged SNe
45
and where the explosion should show up in the corresponding images, thus
allowing for scheduling observations accordingly. Early observations of SNe
can give important clues on the physics involved. In Type Ia SNe, the shock
interaction with the companion star may be detected for the first time (Kasen
2010), as well as the UV shock from the stellar surface in both pair-instability
SNe (Kasen et al. 2011) and CC SNe (e.g. Suzuki & Shigeyama 2010).
47
7
Summary and Outlook
Thanks to recent improvements in observational data, gravitational lensing
has proven to be a valuable tool for investigating the dark components of the
universe. In this thesis, we have been studying gravitational lensing effects
from the smallest (microarcseconds) to the largest (several arcminutes) scales.
Paper I deals with lensing by so called MACHOs, some of the smallest
dark matter structures which have been predicted to have formed in the universe. We demonstrate that relaxing the Press-Gunn approximation, in which
such lenses are assumed to be randomly and uniformly distributed with constant comoving density, gives rise to substantial sightline-to-sightline scatter
in microlensing optical depths. By applying the model to the microlensing
contribution to the long-term optical variability of QSOs which are not multiply imaged, we also show that relaxing the Press-Gunn approximation, however, only has a modest effect on the prediction of light curve amplitudes.
Thus, microlensing by MACHOs is unlikely to be the dominant mechanism
for the observed long-term variability in QSOs.
In Papers II and III, we study the detectability of dark subhalos, the so
called missing satellites, via their gravitational lensing effect on distant light
sources. We estimate the image separations for the subhalo density profiles
favoured by recent N-body simulations and compare these to the angular resolution of both existing and upcoming observational facilities. We find that
the image separations produced are very sensitive to the exact subhalo density
profile assumed, but in all cases considerably smaller than previous estimates
based on the premise that subhalos can be approximated by singular isothermal spheres. We also estimate the overall probabilities for lensing by substructures in a host halo closely aligned to the line of sight to a light source.
Under the assumption of a point source, the optical depth for strong gravitational lensing by subhalos typically turns out to be very small (τ < 0.01),
contrary to previous claims. However, if one assumes the source to be spatially extended, as is the case for a QSO observed at radio wavelengths, there
is a reasonable probability for witnessing substructure lensing effects even at
rather large projected distances from the host galaxy, provided that the angular
resolution is sufficient. Detections of dark subhalos through image-splitting
effects will therefore be far more challenging than previously believed, albeit
not necessarily impossible. To investigate the feasibility of recognising smallscale image distortions from dark subhalos in a future survey, we are currently
conducting ray-tracing simulations of lensed radio jets observed with the Eu-
48
Summary and Outlook
ropean VLBI Network (EVN) or ALMA combined with the global VLBI array.
Finally, we investigate the lensing properties of some of the largest
structures known in the universe, massive galaxy clusters. Due to their large
masses, these clusters act as so called gravitational telescopes, magnifying
the flux of distant background sources by several magnitudes.
In Papers IV and V, we exploit this effect to search for lensed distant supernovae that may otherwise be too faint to be detected. The feasibility of finding
lensed supernovae in our survey was investigated using synthetic lightcurves
of supernovae and several models of the volumetric Type Ia and core-collapse
supernova rates as a function of redshift. We also estimate the number of supernova discoveries expected from the inferred star-formation rate in the observed galaxies. A first transient was discovered behind galaxy cluster A1689,
consistent with being a Type IIP supernova at redshift z = 0.59. The lensing
model predicts 1.4 mag of magnification at the location of the transient, without which this object would not have been detected in the near-IR groundbased search described in the papers. Since the publication of these papers,
other SN candidates have been found in a survey with the newer HAWK-I
camera at VLT, most interestingly a probable Type IIn SN at redshift z = 1.703
(Amanullah et al. in preparation).
In Paper VI, we analyze the possible constraints from information gained
by the detection of lensed SNe behind galaxy clusters. We show that the detection of a Type Ia SN exploding in one of the background galaxies can be
a valuable tool for improving the cluster models and gaining information on
the overall distribution of dark matter in galaxy clusters. Also, in the case
of multiply imaged SNe, well constrained cluster models can be used for an
independent estimate on the Hubble constant.
49
8
Svensk sammanfattning
Tack vare allt bättre observationella instrument, har gravitationella linseffekter
blivit mer och mer användbart för att undersöka universums mörka komponenter. I denna avhandling har linseffekter från de minsta till de största skalorna
studerats (från mikrobågsekunder till flera bågminuter) .
Artikel I behandlar linseffekten av så kallade MACHOs, några av de
minsta strukturer av mörk materia som förslagits finnas i universum. I
artikeln visas att det uppkommer en väsentlig spridning i det optiska djupet
för mikrolinsning av avlägsna ljuskällor när man antar att linserna följer den
materiefördelningen som förutsägs av simuleringarna. Vi visar också att den
observerade långsamma variationen i kvasarernas ljusstyrka, till skillnad mot
tidigare påståenden, troligtvis inte kan härröra från mikrolinsning på grund
av MACHOs.
I Artikel II och III undersöks möjligheten att upptäcka mörka subhalor
runt galaxer genom deras linseffekter på bakomliggande ljuskällor. Vi visar
att avståndet mellan de multipla bilderna är väldigt känsligt för subhalornas
exakta densitetsprofil. Vi visar också att sannolikheten för linsning av punktkällor är väldigt låg. För utsträckta källor bedöms sökandet efter dessa mörka
substrukturer genom linsning som mer utmanande än man tidigare trott, dock
inte omöjlig. Hur dessa linseffekter kan visa sig, till exempel i radioobservationer av linsade jetstrålar från aktiva galaxer, analyseras för närvarande
genom simuleringar.
Slutligen undersöker vi linseffekterna av några av de största kända strukturer i universum, tunga galaxhopar. På grund av deras stora massor, kan dessa
hopar användas som så kallade gravitationella teleskop då de förstärker ljuset
från bakomliggande objekt med flera magnituder.
I Artikel IV och V används denna effekt för att leta efter avlägsna linsade supernovor vilka annars skulle vara för ljussvaga för att upptäckas. En
första transient har hittats bakom hopen A1689 som överensstämmer med en
typ IIP supernova vid ett rödskift av z = 0.59. Linsmodellen förutsäger en
förstärkning med 1.4 magnituder vid transientens position, utan vilken objektet inte hade kunnat observeras i studien. Ytterligare transienter har hittats
sedan publiceringen av artiklarna, bland annat en sannolik typ IIn supernova
med rödförskjutning z = 1.703 (Amanullah et al. under förberedelse).
Artikel VI visar hur informationen från linsade Typ Ia supernovor bakom
galaxhopar kan användas för att förbättra modelleringen av massfördelningen
i dessa hopar. Om flera bilder av en supernova, linsad av en välmodellerad
50
Svensk sammanfattning
galaxhop, skulle observeras kan tidsfördröjningen mellan bilderna användas
för att bestämma Hubblekonstanten.
51
Publications not included in this thesis
• Detecting CDM substructure via gravitational millilensing
Riehm T., Zackrisson E., Mörtsell E., & Wiik K., 2008, idm conf, 54
• An analytical model of surface mass densities of cold dark matter
haloes - with an application to MACHO microlensing optical depths
Holopainen J., Zackrisson E., Knebe A., Nurmi P., Heinämäki P., Flynn C.,
Gill S., & Riehm T., 2008, MNRAS, 383, 720
• On the Probability for Sub-Halo Detection through Quasar Image
Splitting
Riehm T., Zackrisson E., Wiik K., & Möller O., 2008, IAUS, 244, 376
• The Detectability of Dark Galaxies Through Image-Splitting Effects
Zackrisson E., Riehm T., Lietzen H., Möller O., Wiik K., & Nurmi P., 2008,
IAUS, 244, 397
• Prospects for CDM sub-halo detection using high angular resolution
observations
Riehm T., Zackrisson E., Möller O., Mörtsell E., & Wiik K., 2008, JPhCS,
131, 2045
• Dark Matter Millilensing and VSOP-2
Wiik K., Zackrisson E., & Riehm T., 2009, ASPC, 402, 326
• Gravitational Lensing as a Probe of Cold Dark Matter Subhalos
Zackrisson E., & Riehm T., 2010, AdAst2010, 9
• A highly magnified supernova behind massive cluster Abell 1689
Amanullah, R., Goobar, A., Clément, B., Cuby, J.-G., Dahle, H.,
Dahlén, T., Hjorth, J., Fabbro, S., Jönsson, J., Kneib, J.-P., Lidman, C.,
Limousin, M., Mörtsell, E., Nordin, J., Paech, K., Richard, J., Riehm, T.,
Stanishev, V., & Watson, D., submitted
53
Acknowledgements
Of course, there are many people without which my PhD time would not have
been nearly the good experience it really was. Here, I would like to take the
opportunity to give some thanks.
First of all, I would like to thank the members of my thesis group for their
support during all those years. Ariel Goobar for taking me on as his student and being the perfect supervisor he is, always positive, giving me the
possibility to find my own way but also small pushes in the right direction
when needed. Erik Zackrisson for getting me into the astronomy business
and sticking by me, for all your guidance and for always looking out for me.
Edvard Mörtsell for your help in seeing things in a different way and getting
me excited about our results. Göran Östlin for being an enthusiastic mentor
and prefect.
I would also like to take the opportunity to thank all my other scientific
collaborators both within the snova group and abroad. Working with you has
always been very pleasant. I am also thankful to all the people at the astronomy department for providing a friendly and productive work environment.
We had our share of cookies and cakes over the years. Thanks as well to all
the nice cops people for welcoming me to their group.
A special thank you to Sergio Gelato for solving all my computational
problems at incredible speed. Your help will be greatly missed. Also big
thanks to Sandra Åberg and Lena Olofsson for taking care of all my paper
work and financial matters. I know that I have not made it easy for you.
Thanks to Magnus Näslund for your tireless efforts to get the fun of astronomy out to the public and letting me be a small part of it.
Michael, you have been the perfect room mate, always to count on when
needed. Thanks for making all that work with teaching, PR and the webpage
easy. Indeed, we should be proud.
I also had the pleasure to share many nice words, silly lunch conversations,
fun activities and good parties with the rest of the PhD group. Thank you
Anders, Angela, Gautam, Jaime, Javier, Jens, Laia, Magnus A., Martina,
Matthias, Sofia and all the others.
I am greatly thankful to my family. Thank you Matilda, for keeping me
from working off-hours many times, closing the lid of my laptop with a very
determining “mamma, inte!” (mummy, don’t!). Thank you Mats, for keeping
Matilda from keeping me from work all of the time. Thank you also for all of
your love and support over the past nine years.
54
Acknowledgements
Thanks to my parents, Mama und Papa, for always letting me make my
own decisions even if it means having to take a plane to see their grandchild.
Stockholm, May 2011
55
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