Investigating the Dark Universethrough Gravitational Lensing

Teresa Riehm

Department of AstronomyStockholm University

Cover image:Abell 1689, one of the most massive galaxy clusters known, acts as agravitational lens, distorting the images of galaxies thatlie behind it.Image credit: NASA, ESA, E. Jullo (Jet Propulsion Laboratory),P. Natarajan (Yale University), and J.-P. Kneib (Laboratoire d’Astrophysiquede Marseille, CNRS, France)

c© Teresa Riehm, Stockholm 2011ISBN 978-91-7447-281-3Universitetsservice, US-AB, Stockholm 2011Department of Astronomy, Stockholm University

Doctoral Dissertation 2011Department of AstronomyStockholm UniversitySE-106 91 Stockholm

AbstractA variety of precision observations suggest that the present universe is domi-nated by some unknown components, the so-called dark matterand dark en-ergy. The distribution and properties of these components are the focus ofmodern cosmology and we are only beginning to understand them.

Gravitational lensing, the bending of light in the gravitational field of amassive object, is one of the predictions of the general theory of relativity. Ithas become an ever more important tool for investigating thedark universe,especially with recent and coming advances in observational data.

This thesis studies gravitational lensing effects on scalesranging over tenorders of magnitude to probe very different aspects of the dark universe. Im-plementing a matter distribution following the predictions of recent simula-tions, we show that microlensing by a large population of massive compacthalo objects (MACHOs) is unlikely to be the source of the observed long-term variability in quasars. We study the feasibility of detecting the so farelusive galactic dark matter substructures, the so-called“missing satellites”,via millilensing in galaxies close to the line of sight to distant light sources. Fi-nally, we utilise massive galaxy clusters, some of the largest structures knownin the universe, as gravitational telescopes in order to detect distant super-novae, thereby gaining insight into the expansion history of the universe. Wealso show, how such observations can be used to put constraints on the darkmatter component of these galaxy clusters.

To my fam ily

List of Papers

This thesis is based on the following publications:

I High-redshift microlensing and the spatial distribution ofdark matter in the form of MACHOsZackrisson E., & Riehm T., 2007,A&A, 475, 453

II Strong lensing by subhalos in the dwarf galaxy mass range.I. Image separationsZackrisson 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 probabilitiesRiehm 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 efficiencyStanishev 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 prospectsGoobar 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 cosmologyRiehm T., Goobar A., Mörtsell E., Amanullah R., Dahlén T.,

Jönsson J., Limousin M., Paech K., & Richard J., 2011,submitted toA&A

The articles are referred to in the text by their Roman numerals. Reprints were madewith permission from the publishers.

Contents

1 Introduction 1

2 The Dark Universe 32.1 Dark matter and structure formation . . . . . . . . . . . . . . . . .. . . 3

2.1.1 Cold dark matter subhalos . . . . . . . . . . . . . . . . . . . . .42.2 Dark energy and the expansion of the Universe . . . . . . . . . .. . 8

2.2.1 The Hubble constant . . . . . . . . . . . . . . . . . . . . . . . . .92.2.2 The cosmic acceleration . . . . . . . . . . . . . . . . . . . . . . .10

3 Gravitational lensing 133.1 Strong and weak lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 133.2 Arrival time and images . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 133.3 Einstein radius and lens equation . . . . . . . . . . . . . . . . . . . .. . 153.4 Image magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163.5 Lensing on different scales . . . . . . . . . . . . . . . . . . . . . . . . .. . 17

4 MACHOs vs WIMPs 194.1 Microlensing in the local group . . . . . . . . . . . . . . . . . . . . .. . 194.2 Extragalactic microlensing . . . . . . . . . . . . . . . . . . . . . . . .. . . 21

5 Missing satellites 255.1 Flux ratio anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 265.2 Astrometric effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 305.3 Small-scale structure in macroimages . . . . . . . . . . . . . . .. . . . 325.4 Time delay effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34

6 Gravitational telescopes 396.1 Cluster models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 426.2 Multiply imaged SNe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

7 Summary and Outlook 47

8 Svensk sammanfattning 49

Publications not included in this thesis 51

10 CONTENTS

Acknowledgements 53

Bibliography 55

1

1 Introduction

Over the last decades, astronomers have realised that the universe had for along time been hiding a large fraction of its energy density.In the quest touncover the properties of these dark components, gravitational lensing hasproven to be a valuable tool. In this thesis, we show how gravitational lens-ing effects from the smallest to the largest scales can be used to probe verydifferent aspects of what we believe to be the standard cosmology.

This thesis is organized as follows. Chapter 2 gives a short introduction intothe standard cosmological model. In Chapter 3, the basic concepts of gravita-tional lensing are described. In the following chapters we discuss gravitationallensing effects on different scales. Chapter 4 deals with constraining some ofthe smallest dark matter structures predicted, so-called MACHOs, using mi-crolensing. In Chapter 5, several approaches, commonly titled as millilensing,in the search for galactic dark matter substructure, the “missing satellites” aredescribed. Finally, Chapter 6 deals with lensing by some of the largest struc-tures formed in the universe, galaxy clusters, and their useas gravitationaltelescopes for studying the history of the universe. Chapter 7 gives a summaryof the thesis and the papers. Finally, a short summary in Swedish is given inChapter 8.

My contribution to the papers included in this workFor Paper I, I implemented the different mass distribution scenarios and as-sisted in the writing of the text.

In Paper II , I wrote the code for computing the surface mass density pro-files and expected image separations for different subhalo profiles and con-tributed to the scientific discussion.

Most of the work inPaper III was done by me. I prepared all the figuresand text. The results were interpreted together with the other authors.

For Papers IV andV, I computed the magnification and time delay mapsused for the preparation of the observations and the analysis of the data andcontributed to the scientific discussion.

In Paper VI, I carried out the statistical analysis and prepared most ofthefigures and text.

3

2 The Dark Universe

One of the key questions in modern cosmology is what the universe is madeof. In the past few decades, there has been increasing evidence that suggeststhat only a small fraction of the energy density in the universe is in the formof baryonic matter which we can directly observe.

The standard model of Big Bang cosmology is the so-called Lamda-ColdDark Matter (ΛCDM) model. Following this model, recent observations esti-mate that the present energy content of the universe is made up of (Komatsuet 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 isin the form of some dark components which have, so far, not been directlydetected in the laboratory. In this chapter, we outline someof the observationaland theoretical evidence for the properties of these components and discusssome of their possible problems.

2.1 Dark matter and structure formationThe first evidence for the existence of dark matter was collected already in the1930s. Zwicky (1933) was the first to notice that the velocities of galaxies inthe Coma cluster were unexpectedly high. Zwicky concluded, that the clustermust contain a large amount of “dark matter” in order to keep the galaxies inorbit. Since then, dark matter has been used for explaining the kinematics ofspiral galaxies and clusters (e.g. Smith 1936; Oort 1940; Rubin & Ford 1970;Rubin et al. 1980), the gravitational lensing potentials ofgalaxy 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 etal. 1985; Efstathiouet al. 1990; Springel et al. 2005, 2006).

CDM is thought to be made up of non-baryonic particles, whichinteractpredominantly through gravity and have moved with non-relativistic velocitiessince the earliest epochs of structure formation. While this scenario has beenvery successful in explaining the formation of large-scalestructures (galaxies,galaxy groups and galaxy clusters), its predictions on subgalactic scales havenot yet been observationally confirmed in any convincing way. On the con-trary, 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 thecentres of dark matter halos (e.g. Gentile et al. 2004; Diemand et al. 2005a; deSouza & Ishida 2010), and a rich spectrum of substructures within each darkmatter halo (see Kravtsov 2010, for a recent review). There is, however, noconsensus 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 thesesubunits fall into the potential well of larger halos, they suffer tidal strippingof material which ends up in the smooth dark matter componentof the halothat swallowed them. Since this is a process that may take several billion yearsto complete, many of these smaller halos temporarily survive in the form ofsubstructures (also known as subhalos or subclumps) withinthe larger halo.According to current simulations, around 10% of the virial mass of a MilkyWay-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, onemight expect dwarf galaxies to form in these low-mass halos prior to merging,which would result in large numbers of satellite galaxies within the CDMhalo of each large galaxy. A long-standing problem with thispicture is thatthe number of subhalos predicted by simulations greatly exceeds the numberof dwarf galaxies seen in the the vicinity of large galaxies like the Milky Wayand Andromeda (Klypin et al. 1999; Moore et al. 1999). This hasbecomeknown as the “missing satellite problem”. A similar lack of dwarf galaxiescompared to the number of dark halos predicted is also evident within group-sized dark matter halos (Tully et al. 2002). While most of theefforts in thisfield have been focused on the discrepancy between the numberof subhalosand observed satellite galaxies, this problem persists in the field populationas well. In a sense, the missing satellites is just one aspectof a more generalproblem – the mismatch between the low-mass end of the dark matter massfunction and the luminosity function of dwarf galaxies (Verde et al. 2002).

A number of potential solutions to the missing satellites problem have beensuggested in the literature. These can be sorted into three different categories,depending on how they propose to answer the question “Do subhalos exist inthe 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, includingwarm dark matter (Bode et al. 2001), self-interacting dark matter (Spergel& Steinhardt 2000), fuzzy dark matter (Hu et al. 2000) and dark matter inthe form of superWIMPs (Cembranos et al. 2005), but also models of infla-tion 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 ha-los (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 substructurepredicted by simulations around any galactic halo. The mainhalo, which is resolvedwith more than 1 billion particles, and the over 105 numerically resolved subhalos itcontains are visible as bright peaks. Figure adopted from Zemp et al. (2009).

et al. 2004; Moore et al. 2006; Nickerson et al. 2011) and observationalbiases that would put the resulting “dark galaxies” (Trentham et al. 2001;Verde et al. 2002) outside the reach of current surveys (Simon & Geha2007; Tollerud et al. 2008; Walsh et al. 2009; Koposov et al. 2009; Mac-ciò et al. 2010). While these mechanisms may be able to solve the missingsatellites problem as it was originally defined, solutions of this type implythat a vast population of low-mass CDM subhalos (hosting very faint stellarpopulations or none at all) should still be awaiting discovery.

• Yes, but not in our neighbourhood.The final possibility is that the largehalo-to-halo scatter in subhalo mass fraction may have leftthe Milky Wayand Andromeda sitting inside CDM halos with unusually few subhaloscompared to the cosmic average (Ishiyama et al. 2008, 2009).This wouldimply that large numbers of CDM subhalos (either bright or dark) shouldbe 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, lens-ing can in principle be used to detect even those that are too faint to be ob-served through other means. If subhalos do not exist, the absence of lensingeffects 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 potentialpitfalls alongthe way.

As computing power has increased over the last decade, N-body simulationshave been able to make predictions for important propertiesof dark mattersubhalos at higher and higher resolution. Still, essentially all high resolutionsimulations are CDM only, i.e. the effect of a possible baryonic componenthas not been taken into account. Although the possible consequences of thishave been discussed in the literature (Barkana & Loeb 1999; Gnedin et al.2004; Macciò et al. 2006; Gustafsson et al. 2006; Kampakoglou 2006; Readet 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 mindwhen discussing the properties of dark matter substructures.

Mass function

N-body simulations indicate that the subhalos within a galaxy-sized CDMhalo follow a mass function of the type:

dNdMsub

∝ M−αsub, (2.1)

with α ≈ 1.9 (Gao et al. 2004; Springel et al. 2008), albeit with non-negligiblehalo-to-halo scatter at the high mass end (Msub& 5×108M⊙) (Ishiyama et al.2009; Springel et al. 2008). Current simulations of entire galaxy-sized darkmatter halos can resolve subhalos with masses down toMsub∼ 105 M⊙, butthe mass function may extend all the way down to the cut-off inthe den-sity fluctuation power spectrum, which is set by the detailedproperties ofthe CDM particles. For many types of WIMPs (e.g. neutralinos) this cut-offlies at∼ 10−6M⊙ (Green et al. 2005; Loeb & Zaldarriaga 2005; Diemandet al. 2005a; Profumo et al. 2006; Berezinsky et al. 2008; Bringmann 2009),but other CDM candidates may alter this truncation mass considerably. Asan example, axions may allow the existence of halos with masses as low as10−12 M⊙ (Hogan & Rees 1988), whereas very few halos with masses below104–107 M⊙ are expected in the case of MeV mass dark matter (Hooper et al.2007). The overall mass contained in resolved subhalos (i.eMsub& 105 M⊙)within a galaxy-sized CDM halo amounts to a subhalo mass fraction aroundfsub ≈ 0.1, and extrapolating the mass function given by eq. (2.1) towardslower 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 parenthalo, the subhalo population tends to be less centrally concentrated than thesmooth CDM component. The spatial distribution of subhalos within r200, theradius 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)

wherex = r/r200 andN denotes the number of subhalos within a specific ra-dius. It should be noted that this result is based on CDM-onlysimulations,and that the presence of baryons within the subhalos may makethem moreresistant to tidal disruption, thereby boosting their number densities in the in-ner regions of their parent halos (Weinberg et al. 2008). Some simulations ofcluster-mass halos have moreover indicated that the spatial distribution of sub-halos may be a function of subhalo mass, in the sense that high-mass subhaloswould tend to avoid the central regions more than low-mass ones (Ghignaet al. 2000; De Lucia et al. 2004), but this has not been confirmed by the latestsimulations of galaxy-sized halos (Springel et al. 2008; Ludlow et al. 2009).

While the term subhalo is typically used to denote clumps located withinthe virial radius (or, alternatively,r200) of a large CDM halo, there is also alarge number of low-mass clumps located just outside this limit (Diemandet al. 2007; Ludlow et al. 2009). Some of these have previouslybeen bonafide subhalos, and others are bound to venture inside the virial radius in thenear future. Such objects can through projection appear close to lines of sightpassing through the centres of large galaxies, and may therefore be importantin certain lensing situations.

Density profile

The internal structure of subhalos is still a matter of much debate. As low-mass halos are accreted by more massive ones and become subhalos, substan-tial mass loss occurs, preferentially from their outer regions. The shape of theouter part of the subhalo density profile may therefore be seriously affected bystripping, whereas the inner regions are left more or less intact. In many lens-ing studies, CDM subhalos are considered to be singular isothermal spheres(SIS) or even point masses. This is mainly for simplicity – thelensing prop-erties of such objects are well-known, but neither observations, theory norsimulations 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=σ2

v

2πGr2 , (2.3)

whereσv is the line of sight velocity dispersion andG is Newton’s grav-itational constant. This model has proved to be successful for the massivegalaxies responsible for strong lensing on arcsecond scales (Rusin et al. 2003;Koopmans et al. 2006). The SIS density profile has a steep innerslope (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. Thisbaryonic com-ponent contributes substantially to the overall mass density in the centre, andits formation over cosmological time scales may also have caused the CDMhalo itself to contract, thereby steepening the inner slopeof its density profile(Gnedin et al. 2004; Macciò et al. 2006; Gustafsson et al. 2006; Kampakoglou2006).

Low-mass CDM halos which never formed many stars are unlikelyto havedensity profiles this steep. Instead, they should resemble the halo density pro-files derived from CDM-only simulations. The NFW density profile (Navarroet al. 1996), with an inner slope ofβ =−1, has for a number of years servedas the standard density profile for CDM halos, and is given by:

ρNFW(r) =ρi

(r/rS)(1+ r/rS)2 , (2.4)

whererS is the characteristic scale radius of the halo andρi is related to thedensity of the Universe at the time of collapse. Modifications of this formulaare required once halos become subhalos and are tidally stripped. Attempts toquantify the effects of this mass loss on their density profile have been madeby Hayashi et al. (2003) and Kazantzidis et al. (2004)

Controversy remains, however, over whether the NFW profile gives the bestrepresentation of CDM halos (and, consequently, of subhalos prior to strip-ping). Based on recent high-resolution simulations, some have argued for aslightly steeper inner slope (β ≈ −1.2; Diemand et al. 2008) with signifi-cant halo-to-halo variations, whereas other have favoureda far shallower innerslope (Navarro et al. 2004; Springel et al. 2008; Stadel et al. 2009; Navarroet 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 subhalosmay be relatively unimportant in certain lensing situations, it can be crucial inothers (Moustakas et al. 2009). For lensing tests that are sensitive to the exactslope of the inner density profile of subhalos, the subhalo-to-subhalo scatterin this quantity may also be very important (compare chapter5, Papers II andIII ).

2.2 Dark energy and the expansion of the UniverseIn the standard model of cosmology, dark energy is in the formof a cosmo-logical constantΛ, a vacuum energy which is constant in space and time. Thecosmological constant was first proposed by Albert Einstein as a modifica-tion of his field equations in order to achieve a stationary universe (Einstein1917). However, Friedman (1922), realised that this was an unstable solutionand proposed an expanding universe model, now called the BigBang theory.When Hubble (1929) showed that the universe was in fact expanding, Ein-stein regretted modifying his elegant theory and abandonedthe cosmologicalconstant.

2.2 Dark energy and the expansion of the Universe 9

However, the cosmological constant gained renewed interest in the late1990s when two teams of astronomers combined observations of nearby anddistant Type Ia supernovae (SNe Ia) in order to study the expansion rate of theuniverse with time. These measurements led to the unexpecteddiscovery ofthe accelerating universe (Riess et al. 1998; Perlmutter etal. 1999).

The expansion of the universe can be described by the Friedmann equation

H2 =

(

aa

)2

=8πG

3ρ −

kc2

a2 +Λc2

3(2.5)

whereH is the Hubble parameter, i.e. the rate of expansion of the universe,ais the scale factor where the overdot denotes a time derivative, ρ is the massdensity of the universe andk is the normalised spatial curvature of the universeequal to -1, 0 or +1.

The value of the Hubble parameter changes over time, either atan acceler-ating or decelerating rate, depending on the sign of the parameter

q =−

(

1+HH2

)

(2.6)

which had been named the deceleration parameter, illustrating the expectationof a deceleration of the universe prior to the discoveries inthe 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 measuredby comparing theredshift of a source (like a galaxy or a SN) with the distance to the source. Theredshift parameterz describes how much the light emitted from a source hasbeen stretched due to the expansion of the universe during a time t when thelight was emitted and the current timet0 at which the source is observed

1+ z =λ0

λ=

a(t0)a(t)

. (2.7)

As the redshift of a source can be measured with high precision, the mainuncertainties in determining the value ofH0 are caused by uncertainties in thephysical assumptions used to determine the distances to thesources.

For a source of intrinsic luminosityL, its luminosity distancedL is definedby the measured energy fluxF from the source

dL(z) =

√

L4πF

. (2.8)

For sources with known intrinsic luminosityL, so called standard candles,dL

can be determined by measuring the observed fluxF. Known standard candles

10 The Dark Universe

include Cepheid variable stars with a well-defined relationship between the lu-minosity 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 asuccession of standard candles in galaxies at increasing redshift to determinedistances is known as the “cosmic distance ladder”.

Hubble’s initial value for the present epoch expansion rate, H0, was∼ 500 km s−1Mpc−1 using data from Cepheid variable stars in nearbygalaxies (Hubble 1929). The most recent estimate using Hubble SpaceTelescope observations of Cepheids and Type Ia SNe gives a value ofH0 = 73.8 ± 2.4 km s−1 Mpc−1 (Riess et al. 2011). This value agreeswell with those of alternative measurements of the Hubble constant fromgravitational lensing (e.g. Suyu et al. 2010, compare chapter 6 andPaper VI),recent WMAP results from anisotropies of the cosmic microwave background(Komatsu et al. 2011) and the Sunyaev-Zeldovich effect combined withX-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 dis-tance ladder technique, enabled the discovery of the acceleration of the cos-mic 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 adecelerating universe, indicating that the expansion of the universe has in factbeen speeding up for the past∼ 5 Gyrs. As SNe Ia are intrinsically bright ob-jects, they can be observed out to high redshift and are therefore an excellenttool for probing the expansion history of the universe. Subsequent SN ob-servations, extending to redshiftsz ∼ 1.5, have reinforced the original results(compare Figure 2.2, see Amanullah et al. 2010, for a recent compilation). Forhigher redshifts, even Type Ia SNe are not bright enough to bedetected withcurrent observational facilities. Therefore, we have suggested a technique ofutilising the magnification power of massive galaxy clusters in order to studythe expansion history of the universe to a redshift ofz ∼ 3 (compare chapter 6,Papers IV andV).

Other observational probes have contributed new evidence for the cosmicacceleration and the existence of an additional energy component making up alarge fraction of the energy density of the universe. These include the growthof large scale structure through galaxy cluster counts (Haiman et al. 2001) andweak lensing shear (Hoekstra et al. 2006; Massey et al. 2007), studying theanisotropies of the cosmic microwave background (Komatsu et al. 2011) andmeasurements 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 cosmolog-ical 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 theHubble constantH0, can be traced out to a redshiftz < 0.1. Distance is measure bythe distance modulusµ = m−M, the difference between the apparent magnitudemand the absolute magnitudeM of the SNe. The distance relative to an empty universemodel (ΩM = ΩΛ = 0) is shown in the lower panel for binned data points. It fitswell with the prediction from aΛCDM model (ΩM = 0.3, ΩΛ = 0.7) shown by theoverplotted curve. Figure adopted from (Goobar & Leibundgut 2011).

verse, there are some unexplained curiosities. Associating Λ with a vacuumzero-point energy, its corresponding density does not match any scale pre-dicted from standard particle physics theory but might be too low by up to120 orders of magnitude. Furthermore, there is no compelling argument whythe present fraction of the energy density of the universe inmatter and dark en-ergy are so close in spite of mass density being diluted∝ a3 while the vacuumenergy density remains constant. This is known as the coincidence problem.

Alternative theories to the cosmic expansion being driven by a cosmologicalconstantΛ, can be tested by precision measurements of the expansion historyof the universe.

13

3 Gravitational lensing

Einstein’s general theory of relativity predicts that the path of light is bentin a gravitational field, i.e. a ray of light passing close to amassive body isdeflected towards that body. In this chapter, we will describe some of the basicfeatures of gravitational lensing used in this thesis.

3.1 Strong and weak lensingGravitational lensing effects can be divided into two regimes, strong and weaklensing, 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 veryclose to the lens, a situation that gives rise to high magnifications, multipleimages, arcs and rings in the lens plane. Weak lensing occurswhen the lens islocated further away from the line of sight, resulting in small magnificationsand mild image distortions. Generally speaking, weak lensing is extremelycommon in the cosmos (at some level, every single light source is affected)but inconspicuous, and can only be detected statistically by studying a largenumber of lensed light sources. Strong lensing effects, on the other hand, arerare but dramatic, and can readily be spotted in individual sources. The lensingeffects studied in this thesis can all be attributed to the strong lensing regime.

3.2 Arrival time and imagesFermat’s principle states that the path taken by a ray of light between twopoints must be stationary. Thus images will appear where the arrival time sur-face 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 timet(~θ ). The arrival timet(~θ ) is proportional to the lighttravel time of a ray with fixed angle~β , the true position of the source on thesky, as a function of~θ , the apparent position of the source measured by anobserver due to lensing (compare figure 3.2).

The arrival time consists of two components, a ‘geometrical’and a ‘gravi-tational’ part:

t(~θ ) = tgeom(~θ )+ tgrav(~θ ) (3.2)

14 Gravitational lensing

Figure 3.1: Weak and strong lensing.a) Weak lensing occurs when the lens (here il-lustrated by a gray elliptical galaxy surrounded by a dark matter halo) lies relativelyfar 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 anddistortion. The signatures of this are only detectable in a statistical sense, by studyingthe weak lensing effects on large numbers of background light sources.b) Strong lens-ing can occur when the dense central region of the lens is well-aligned with the line ofsight. The light from the background light source may then reach the observer alongdifferent paths, corresponding to separate images in the sky. This case is also associ-ated with high magnifications and strong image distortions.The angular deflection inthis figure, as in all subsequent ones, has been greatly exaggerated for clarity.

The geometrical part is given by the difference between the unlensed andthe lensed path (dotted and dashed line in figure 3.2, respectively),

tgeom(~θ ) =12(1+ zL)

DLDS

cDLS(~θ −~β)2, (3.3)

wherezL is the lens redshift andDS, DL andDLS denote the angular diameterdistances from observer to source, observer to lens and fromlens to source,respectively. The angular diameter distance to an object is defined by the ratioof its actual sizel to the angle it spans on the skyδθ ,

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 betweenobjects at redshiftzA andzB is given by

DzAzB =c

(1+ zB)H0

∫ zB

zA

dz√

ΩM(1+ z)3+ΩΛ. (3.5)

The gravitational part is also known as the Shapiro time delay. It arises fromrelativistic effects in a gravitational field and depends onthe surface densityΣ(~θ ) of the lens at the apparent position~θ ,

tgrav(~θ ) =−12(1+ zL)

8πGc3 ∇−2Σ(~θ ). (3.6)

3.3 Einstein radius and lens equation 15

Figure 3.2:Light deflection of a background source by a lens. Without thepresence ofthe lens, the light from the background source would arrive at the observer followingthe 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 andDLS denote theangular diameter distances from observer to source, observer to lens and from lens tosource, respectively.

3.3 Einstein radius and lens equationIn the special case of a point-mass lens aligned with a point source, i.e.Σ(~θ ) =Mδ (~θ ) and~β = 0, the arrival timet(~θ ) has a minimum atθ = θE where

θ2E =

4GMc2

DLS

DLDS, (3.7)

is known as the Einstein radius. This lensing situation results in a ring imagecentered on the lens with angular radiusθE , called an Einstein ring.

Although this is not a very common situation, the Einstein radius is still avery useful concept since the angular image separation in a multiple-imagesystems typically is of the order of2θE . This can be derived from the two-dimensional analog of Gauss’s flux law: for any circular massdistributionΣ(θ), ~∇t(~θ ) will only depend on the enclosed mass. This implies that anycircular distribution of massM will produce multiple images from a colinearsource provided it fits withinθE which is known as the ‘compactness’ crite-rion. Equation 3.7 states that the area withinθE depends linearly on the lensmassM. Thus, for each lens situation there is a critical surface mass density

Σcrit =c2

4πGDLDS

DLS(3.8)

which needs to be exceeded to satisfy the compactness criterion and producean Einstein ring or multiple images.

We can use the critical surface mass densityΣcrit , as defined above, to intro-duce the dimensionless scaled surface mass densityκ(~θ ) which is also known

16 Gravitational lensing

as the convergence

κ(~θ) =Σ(~θ)Σcrit

. (3.9)

Defining the time scale

T0 = (1+ zL)DLDS

cDLS, (3.10)

we can also express the arrival timet(~θ ) dimensionless as the scaled arrivaltime τ(~θ ) with

τ(~θ ) =t(~θ )T0

. (3.11)

Equation 3.2 can then be rewritten as

τ(~θ ) =12(~θ −~β )2−2∇−2κ(~θ ). (3.12)

The second term,2∇−2κ(~θ ), depends on the mass distribution of the lens andis called the lens potentialψ(~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 magnificationThe gravitational potential of a foreground lens can not onlychange the ap-parent position~θ and arrival timet(~θ ) of a source but also affect its observedflux. Since surface brightness is preserved by lensing (Liouville’s theorem),but gravitational lensing changes the apparent solid angleof a source, the to-tal flux received from a gravitationally lensed image of a source is thereforechanged. The ratio of the flux of an image to the flux of the unlensed sourceis called the magnificationµ and is proportional to the ratio between the solidangles of the image and the source in absence of a lens.

The source-plane displacement needed to produce a given small image dis-placement, i.e. the inverse of the magnification, is given by(combining equa-tions 3.12 and 3.13)

~∇~β =~∇~∇τ(~θ ) = M−1 (3.14)

whereM is a 2D tensor which depends on~θ but not~β . This equation impliesthat the magnificationµ = detM is equal to the inverse of the curvature of thearrival time surface. ExaminingM closer, one sees that it can be devided intoa trace(1−κ) and a traceless partγ. The scalar magnificationµ can then becomputed using

µ = detM = [(1−κ)2− γ2]−1. (3.15)

3.5 Lensing on different scales 17

Here κ is the convergence and identical to the scaled surface mass densityalready defined in equation 3.9. It can also be interpreted asan isotropic mag-nification.γ is called the shear which changes the shape of an image but notits size.

3.5 Lensing on different scalesStrong lensing has traditionally been divided into subcategories, depending onthe 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 galax-ies (M ∼ 1012 M⊙) or galaxy clusters (M ∼ 1014–1015 M⊙) are responsiblefor the lensing, the image separation typically falls in themacrolensing range,whereas individual solar-mass stars give image separations in the microlens-ing regime. Since all objects with resolved multiple imagesdue to gravita-tional lensing have image separations of& 0.1 arcseconds, the term stronglensing 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 theobserved large-scaledistribution of galaxies in universe, dark matter must be cold, i.e. move at non-relativistic speed. The most likely candidates for cold darkmatter (CDM) atpresent 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). Althoughthe MACHOacronym was originally invented with baryonic objects in mind several non-baryonic dark matter candidates can also manifest themselves in this way –e.g. axion aggregates (Membrado 1998), mirror matter objects (Mohapatra1999), primordial black holes (Green 2000), preon stars (Hansson & Sandin2005), scalar dark matter miniclusters (Zurek et al. 2007) and ultracompactminihalos (Ricotti & Gould 2009). Hence, MACHOs could in principle makeup a substantial fraction of the dark matter of the universe.As there is stilla large fraction of the baryonic matter density unaccountedfor (Fukugita &Peebles 2004; Bregman 2007), the so-called “missing baryons”, there mighteven be a substantial population of baryonic MACHOs like faint stars andstellar remnants awaiting detection.

Gravitational lensing provides a way to test if and what fraction of the darkmatter density in the universe consists of MACHOs.

4.1 Microlensing in the local groupOur galaxy, the Milky Way, is expected to reside inside a CDM halo whichcomprises∼ 95%of the total mass (Battaglia et al. 2005). If at least some frac-tion of the CDM is in the form of MACHOs, gravitational lens signals shouldbe observable in background sources (stars) whenever a MACHO passes closeto the line of sight. As most MACHOs are expected to have masses on (sub-)solar scales, the resulting image splitting would be on theorder of microarc-seconds and thus below the detection threshold of any available telescope.However, this lensing event would cause a well measurable magnification ofthe background source (Paczynski 1986) on the time scale it takes the sourceto cross the Einstein radius of the lensing object,

t =DLθE

v≈ 0.214yr

√

MM⊙

√

DL

10kpc

√

DLS

DS

(

200kms−1

v

)

. (4.1)

20 MACHOs vs WIMPs

Thus, the duration of such a microlensing event depends on themass of thelens, as well as the relative velocityv between source and lens and the lensinggeometry via the angular diameter distancesDL, DS andDLS. The expectedtime scales are on the order of hours up to a few hundred days for MACHOsin the mass range 10−6 – 102 M⊙.

Due to the degeneracy of the time scale with the different parameters inequation 4.1, it is not possible to determine the MACHO mass from one sin-gle microlensing event. Instead, a number of microlensing detections com-bined with the modelling of the spatial and velocity distribution of the lensingobjects will be needed in order to put constrains on MACHO masses and den-sities.

The probability for observing a microlensing event by MACHOscan beexpressed in terms of its optical depth,τ , i.e. the probability for the line ofsight to a background star to pass within the Einstein radius,θE , of a MACHO.The optical depth is given by

τ =

∫ Ds

0

4πGρ(DL)

c2

DLDLS

DSdDL (4.2)

whereρ(DL) is the matter density in MACHOs at distanceDL from the ob-server. Unlike the microlensing time scale,t, the microlensing optical depth,τ , depends only on the mass densityρ and not the specific massM of theMACHOs. The optical depth for MACHO microlensing is estimated to be oforder 10−6 – 10−7. Thus, the continuous monitoring of millions of stars isrequired for such microlensing events to be detected.

Furthermore, as a large fraction of stars are intrinsicallyvariable, such lu-minosity variations will have to be ruled out if flux variations are observedin a background star in order to confirm the detection of an a microlensingevent. Fortunately, these scenarios can be distinguished due to the fact that thelight curves of lensed stars are, contrary to those of variable stars, expected tobe both symmetric in time and show achromatic magnifications(in the smallsource regime) as gravitational lensing does not depend on wavelength.

Several groups have put in an extensive effort in the search for MACHOsthrough 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 starsover 5.7years in the LMC and found 13 – 17 events with time scales,t, ranging from34 to 230 days (Alcock et al. 2000). This is in contrast to the∼ 2 – 4 eventsexpected from lensing by known stellar populations in the Milky Way and theLMC. The microlensing optical depth from events with 2 <t < 400 days wasthen estimated to beτ = 1.2+0.4

−0.3 × 10−7. Although this rules out a model inwhich all of the dark matter halo consists of (uniformly distributed) MACHOsat 95% confidence level, a maximum-likelihood analysis gives a MACHOhalo 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 theLMC areexpected to be an important source of background and some of the microlens-ing events have been challenged to be SNe or variable stars. Depending on thehalo model, the most likely MACHO mass ranges between 0.15 and 0.9M⊙.

The EROS-2 project monitored 33 million stars over 6.7 years inthe LMCand SMC and found only one microlensing event, whereas∼ 39 events wouldhave been expected if the Milky Way dark matter halo was entirely populatedby MACHOs of massM ∼ 0.4M⊙ (Tisserand et al. 2007). Combined with theresults of EROS-1 (Lasserre et al. 2000), this implies a microlensing opticaldepthτ < 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 MilkyWaydetected only two possible microlensing events, translating into an opticaldepth ofτ = 0.43± 0.33× 10−7, although these events seem to be consis-tent with the self-lensing scenario, i.e. microlensing by normal stars inside thetarget galaxy (Wyrzykowski et al. 2009). However, if both events were due toMACHOs, this results in upper limits on their abundance in the galactic haloof 19% for objects of massM ∼ 0.4M⊙ and 10% for objects in the mass range0.01 – 0.2M⊙.

The effect of self-lensing of stars in the LMC and M31 has been discussedin several works arriving at very different conclusions regarding its contribu-tion 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 detectedwithin the local group, they are insufficient to explain the total amount ofdark matter expected in the Milky Way halo within the probed mass range(∼ 10−7 – 10M⊙) . However, local microlensing observations are only able todetect the densest, most compact MACHOs at a given mass (e.g.Zurek et al.2007), whereas more diffuse objects will give rise to detectable microlensingeffects only at higher redshifts.

4.2 Extragalactic microlensingPress & Gunn (1973) suggested a technique for constraining acosmologicallysignificant density of compact objects by searching for their gravitational lens-ing effects on distant sources. After the discovery of the first multiply imagedquasar (QSOs), it was proposed that compact objects like stars and MACHOsin the lens galaxy should act as microlenses on the individual images, givingrise to uncorrelated flux variations (Chang & Refsdal 1979; Gott 1981).

Contrary to microlensing by compact objects within the local group, theoptical 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 simultane-ously contributing to an observed flux variation. Dependingon the size of thesource, the caustic crossings of the microlenses can be resolved individuallywith relatively high maxima in the lightcurves (small sources) or are smoothedout (large sources). To evaluate the effects of this, numerical methods were de-veloped (see Wambsganss 2006, for a review on the different techniques).

Studying the properties of such variations, constraints onthe abundance ofcompact objects in the halo of the lensing galaxy can be obtained, assuminga model for the QSO size. A number of multiple quasar systems have beenmonitored, searching for brightness fluctuations in the different componentswhich can not be mapped to the other multiple images, taking time delays intoaccount. Such fluctuations can not be due to intrinsic luminosity variations.

Because QSOs are unresolved in optical imaging, appearing as pointsources, gravitational microlensing offers at the same time an interestingpossibility of producing information about the luminous structure.

Early lightcurves of the first gravitational lens discovered, double QSOQ0957+561, showed an almost linear change in brightness between the twoimages of∆mAB ≈ 0.25 mag, which can be interpreted as potential evidencefor microlensing (Schild 1996). However, since then, differential flux varia-tions of only∆mAB < 0.05mag have been observed in this system (e.g. Schild1996; Schmidt & Wambsganss 1998). Combining extensive numerical simu-lations with monitoring data of the double QSO, it has been possible to putconstraints 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;Refsdalet al. 2000; Colley et al. 2003; Colley & Schild 2003).

The Einstein Cross, QSO Q2237+0305, consisting of four imagesarrangedsymmetrically around the lens galaxy has been intensely monitored by a num-ber of groups in the hope of observing high-magnification events (e.g. Cor-rigan et al. 1991; Alcalde et al. 2002; Eigenbrod et al. 2008b). This systemis particularly well suited for microlensing studies sinceit shows very shorttime delays between the images, due to its symmetric configuration, as wellas a high expected optical depth for microlensing from stars, due to the po-sitions 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 theimages with strenghts up to∆m ∼ 1 mag, allowing to resolve QSO structureson spatial scales several orders of magnitudes below the resolution of exisit-ing telescopes (Wozniak et al. 2000a,b; Anguita et al. 2008b; Eigenbrod et al.2008a).

Several other multiply imaged QSOs have been monitored for microlensingeffects in order to put constraints on QSO source structure and the fraction ofgalaxy halo mass in MACHOs (e.g. Koopmans & de Bruyn 2000; Jacksonet al. 2000; Anguita et al. 2008a). Recently, Mediavilla et al. (2009) combinedmicrolensing 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 stronglylensedinto 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 studiesof multiply imaged QSOs support the hypothesis of only a low content inMACHOs.

In addition to microlensing searches in multiply imaged sources, it has beensuggested to look for flux variability in individual high-redshift sources suchas QSOs, SNe and gamma-ray bursts (GRBs). Although this might prove to bedifficult in the case of QSOs as they exhibit a varying degree of intrinsic vari-ability, 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 customaryto adopt the thinlens approximation, as most of the microlensing can be assumed to take placein the dark halo responsible for the macrolensing effect. For high-redshiftsources which are not multiply-imaged, the situation becomes more compli-cated, as compact objects at vastly different distances along the line of sightmay contribute to the microlensing. Most studies of the latter situation (Rauch1991; 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 berandomly and uniformly distributed with constant comovingdensity alongthe 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 tomisleading results (Wyithe & Turner 2002b). InPaper I, we relaxed this ap-proximation in favour of a model where the compact objects instead follow thespatial distribution predicted for CDM, thereby clustering into cosmologically

24 MACHOs vs WIMPs

distributed halos and subhalos, as would be expected for non-baryonic MA-CHOs. We demonstrate that this gives rise to substantial sightline-to-sightlinescatter in microlensing optical depths, and quantify this scatter as a functionof 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 inthe subso-lar mass range (e.g Hawkins 1996). In principle, this microlensing variabilityscenario has several attractive features. It provides a natural explanation forthe statistical symmetry (Hawkins 2002), achromaticity (Hawkins 2003) andlack of cosmological time dilation (Hawkins 2001) in the optical light curvesof QSOs. Zackrisson et al. (2003) argued that the long-term optical variabilityof QSOs could not primarily be caused by microlensing because the num-ber of predicted high-amplitude magnifications was much lower than what isobserved in the light curves.

By applying the model developed inPaper I to the long-term optical vari-ability of QSOs which are not multiply-imaged, we show that relaxing thePress & Gunn approximation only has a modest effect on the predicted dis-tribution of light curve amplitudes. Hence, the problems associated with mi-crolensing as the dominant mechanism for the long-term variability are in noway diminished, but instead slightly augmented, once the large-scale cluster-ing of MACHOs predicted in theΛCDM cosmology is taken into account.

In summary, studying the effects of microlensing on both local sources aswell as sources at cosmological redshifts, suggests that atmost a small frac-tion of the dark matter density can consist of MACHOs in a massrange of∼ 10−7 − 10 M⊙. Therefore, the hypothesis of MACHOs as the dominantmatter component in the universe has most probably to be abandoned in favourof e.g. the WIMPs.

25

5 Missing satellites

TheΛCDM standard model seems to predict a large amount of galactic sub-structure in the mass range of dwarf-galaxy masses (∼ 106 – 1010 M⊙), the so-called missing satellites, which has not (yet) been observationally confirmed(compare chapter 2.1.1). Lensing by such objects has been estimated to giverise to millilensing, although the exact image separation depends on the in-ternal structure of these objects. In this chapter, we will discuss the differentpossible approaches for the search of these presumably darkobjects usinggravitational lensing.

In principle, any distant light source may be affected by subhalos along theline of sight. The situation is schematically illustrated inFig. 5.1a. A randomline of sight towards a light source outside the local volumepasses within thevirial radius of numerous galaxy-sized CDM halos (Paper I), and may henceintersect subhalos anywhere along the line of sight. The probability of inter-secting a subhalo is, however, rather small in this situation, and sightlines ofthis type may also pass through low-mass field halos, i.e. theprogenitors ofsubhalos (Chen et al. 2003; Metcalf 2005a,b; Miranda & Macciò 2007). It willtherefore be difficult to distinguish between the effects produced by these twotypes of lenses. While the low-mass end of the field halo population may bevery interesting in its own right, lensing by such objects ismore often con-sidered an unwanted “background” when attempting to address the missingsatellite problem as it is currently defined.

Instead, the main targets for attempts to constrain the CDM subhalo pop-ulation using lensing have so far been sources that are already known to bemacrolensed (see Fig. 5.1b), which in practice means observing either mul-tiply imaged QSOs or galaxies lensed into arcs or Einstein rings. By doingso, one preselects a sightline where one knows that there is amassive darkmatter halo (with subhalos) located along the line of sight.Whether severaldistinct, point-like images or elongated arcs that approach the form of a ringare seen, mainly depends on the source size: point-like sources (QSOs in theoptical, but potentially also SNe, GRBs and their afterglows) give distinctimages whereas extended sources (galaxies) give rise to arcs and rings (seeFig. 5.2). The strong magnification produced by the macrolens(large fore-ground galaxy) acts to boost the probability for lensing by the subhalo, andtypically 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 dis-tant light source passes through many halos with subhalos, but too far from the halocentres for macrolensing to occur. A subhalo in one of these halos happens to inter-sect the line of sight, thereby potentially producing millilensing effects in a singly-imaged light source.b) One of the halos happens to lie almost exactly on the line ofsight, thereby splitting the background light source into separate macroimages. Oneof 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 haveso far been de-tected.

Lensing by subhalos can give rise to a number of observable effects, whichwe 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 dwarfgalaxies (∼ 105–1010 M⊙), since current predictions indicate that subhalos atlower masses may be very difficult to detect through lensing effects.

5.1 Flux ratio anomaliesIt was noticed quite early that simple, smooth models of galaxy lenses usuallyfit the image positions of macrolensed systems well, while the magnificationsof the macroimages are more difficult to explain (Kochanek 1991). To see howthis works, some more simple lens theory is required.

Specific relations are expected to apply for the magnifications of macroim-ages close to each other and a critical line. Formally, critical lines are thecurves in the lens plane where the magnification tends to infinity. If criti-cal curves are mapped into the source plane, a set of caustic curves is ob-tained. These separate regions in the source plane that give rise to differentnumbers of images (see Fig. 5.3). The smooth portions of a caustic curve arecalled 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 respectivelythree close images

5.1 Flux ratio anomalies 27

Figure 5.2: Small and large sources.a) A galaxy surrounded by a dark matter haloproduces 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-loudQSO), the macroimagesmay blend into arcs or even a complete Einstein ring.

will be produced near the critical line in the lens plane. If the source is placedin the center of the caustic, the macroimages will form a cross configuration(Fig. 5.3c).

All macroimages can furthermore be described as having either positiveparity (meaning that the image has the same orientation as the source) or neg-ative parity (the image is mirror flipped relative to the source). When takingthe image parity into account and assigning negative magnifications to nega-tive parity images, the sum of the magnifications of the closeimages shouldapproach zero (Mao 1992; Schneider & Weiss 1992; Zakharov 1995). The fol-lowing relations should then apply for the flux ratioR of a fold configuration:

Rfold =|µA |− |µB|

|µA |+ |µB|→ 0, (5.1)

when the separation between the close images is asymptotically small. Here,µ represents the magnification of a specific image. For the cuspconfiguration,the corresponding relation is:

Rcusp=|µA |− |µB|+ |µC|

|µA |+ |µB|+ |µC|→ 0. (5.2)

However, most observed lensing systems violate these relations. This hasbeen interpreted as evidence of small-scale structure in the lens on approx-imately the scale of the image separations between the closeimages. Mag-nifications of individual macroimages due to millilensing by subhalos wouldindeed cause the values forRfold andRcusp to differ from zero fairly indepen-dently 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 ΛCDMsimulations 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 andc) Cross. The upper row shows the caustics and position of the source (star) in thesource plane. The solid line indicates the inner caustic andthe dashed line the outercaustic. A source positioned inside the inner caustic produces five images. A sourcepositioned between the inner and outer caustic produces three images, whereas asource positioned outside the outer caustic will not be multiply imaged. In the caseof multiple images, one of the images is usually highly de-magnified, so that onlyfour- and two-image lens systems are observed, respectively. The lower row showsthe corresponding critical lines and resulting observableimages in the lens plane. Notethat the image labeling differs for the different cases. Theinner caustic maps on theouter critical line and vice versa. A close pair (A, B) and a close triplet (A, B, C) areproduced in the fold (a) and cusp (b) configurations, respectively.

.

since the surface mass density in substructure is lower thanthat required closeto the Einstein radius where the multiple images are produced(Zentner &Bullock 2003; Amara et al. 2006; Macciò et al. 2006; Macciò & Miranda2006; Xu et al. 2009).

Several alternative causes for the observed flux anomalies have been dis-cussed, such as propagation effects like absorption, scattering or scintillationin the interstellar medium of the lens galaxy (Mittal et al. 2007) and mi-crolensing by stars in the lensing galaxy (Schechter & Wambsganss 2002).Since some sources, like QSOs, can exhibit intrinsic flux variations on differ-ent timescales depending on wavelength, flux ratios may alsobe difficult tointerpret if the time delay between the macroimages is not well known.

The relevance of propagation effects can be tested by supplementary obser-vations of flux ratios at different wavelengths, since flux losses due to suchmechanisms should vary as a function of wavelength. Microlensing by starscan be checked for using long-term monitoring, as this type of lensing is tran-sient and expected to introduce extrinsic variability on the order of months.Millilensing by halo substructure can on the other hand be treated as station-ary (Metcalf & Madau 2001). Extended sources (e.g. QSOs at mid-infraredand 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). Eventhough it is often assumed that radio observations of QSOs are essentiallymicrolensing-free, some caution should be applied, since substantial short-term microlensing variability is possible in the special case of a relativisticradio jet oriented close to the line of sight. This phenomenonhas been de-tected in at least one multiply-imaged system (Koopmans & deBruyn 2000).

Mid-infrared imaging of lenses is attractive because the flux is free from dif-ferences in extinction among the macroimages, in addition to being free frommicrolensing by stars due to the extended source size. Such observations cantherefore be used to test some of the alternative causes for flux ratio anoma-lies. Recent studies have used this technique to examine several macrolensedQSOs with known flux ratio anomalies in the optical (Chiba et al. 2005; Sugaiet al. 2007; Minezaki et al. 2009; MacLeod et al. 2009). The mid-infrared fluxratios of about half of these systems can be fitted with smoothlensing mod-els, which means that only the remaining half of these anomalies are due tomillilensing by substructures.

There are also other observational features that argue for substructures asthe cause of (at least some) flux ratio anomalies. Negative parity images (socalled saddle images – e.g. the middle image of a close triplet, like imageB in Fig. 5.3b) are often fainter than predicted by smooth lens models. Thisis expected from millilensing, as the magnification perturbations induced bysubstructure lensing have been shown to depend on image parity (Schechter& Wambsganss 2002; Kochanek & Dalal 2004; Chen 2009). In contrast, suchanomalies cannot be attributed to propagation effects since those should statis-tically affect all types of images similarly, regardless oftheir parity. Whetherthe lensing is due to luminous or dark substructures is, however, a differentmatter.

Luminous substructures have been identified in many of the lens systemswith known flux ratio anomalies. Including such substructures in the lensmodel tends to greatly improve the fit to observations. One example of sucha 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 presenceof a small satel-lite 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 primarylensing galaxy (Xu et al. 2009).

Recently, there have been studies combining the results from simulationsand semi-analytical models of galaxy formation to investigate if luminousdwarf galaxies might be able to explain the frequency of flux ratio anomaliesobserved (Shin & Evans 2008; Bryan et al. 2008). They find that the frac-tion of luminous satellites in group-sized halos is roughlyconsistent with theobservational data within a factor of two, while the resultsfor galaxy-sizedhalos 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 adistant light source passes through a dark subhalo.b) The images of the macro-lensedsource are observed at the positions of the bright source symbols. Modelling of thelens system with a smooth lens potential only, predicts the position of the upper imageat the dark source symbol. The subhalo close to the sightlineof the image causes adeflection on the order of a few tens of milliarcseconds.

observed systems. The lensing effect of these luminous dwarfgalaxies is alsosomewhat unclear since most satellites found in the inner regions of largergalaxies are expected to be ‘orphan’ galaxies stripped of their dark matter ha-los. To investigate this further, higher-resolution simulations involving a real-istic treatment of the gas processes are required. Possibleexplanations for thediscrepancy between the expected and observed fraction of luminous satellitesinclude dwarf galaxies elsewhere along the line of sight mistakenly identifiedas the lens perturber (Metcalf 2005b). Luminous substructures may moreoverbe more efficient in producing flux ratio anomalies, since they are likely to bedenser than dark substructures due to baryon cooling and condensation (Shin& Evans 2008).

Projection effects are potentially also important for flux ratio anomaliescaused by completely dark substructures as one expects a large amount ofline of sight structure. Although those structures are lesseffective than sub-structures within the lens galaxy in inducing magnificationperturbations, theoverall 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 effectsIn macrolensed systems, the presence of halo substructure may perturb theangular deflection caused by the lens galaxy and thereby the position ofmacroimages at observable levels, so called astrometric perturbations (seeFig. 5.4).

This method for detecting subhalos has the advantage of beingrelativelyunaffected by propagation effects (absorption, scattering or scintillation by

5.2 Astrometric effects 31

the interstellar medium) and stellar microlensing that maycontaminate theflux ratio measurements. Furthermore, since the astrometric perturbation isa steeper function of subhalo mass than flux ratio perturbations, it is mostlysensitive to intermediate and high mass substructures and therefore probes adistinct 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 sub-halo are expected to be rather small. Metcalf & Madau (2001) used lensingsimulations of random realisations of substructure in regions near imagesand found that it would take substructures with masses& 108 M⊙ that arevery closely aligned with the images to change the image positions by a fewtens of milliarcseconds. Such an alignment would be rare in the CDM model.Therefore, they suggest to deploy lensed jets of QSOs observed at radio wave-lengths, as such sources would cover more area on the lens plane. This wouldincrease the probability of having a large subhalo nearby, but still allow forpronounced distortions due to the thinness of the jet. Metcalf & Zhao (2002)investigated this technique further and used it to show thatthe lens systemB1152+199 is likely to contain a substructure of mass∼ 105−107M⊙.

Further observational evidence for astrometric perturbations from smallscale structure was found in the detailed image structures for B2016+112(Koopmans et al. 2002; More et al. 2008) and B0123+437 (Biggset al. 2004).In the latter system, a substructure of at least∼ 106M⊙ would be needed inorder to reproduce the observed image positions.

The CDM scenario predicts that there are far more low-mass subhalos thanhigh-mass ones (see equation 2.1) and their summed effect could in principleadd up to a substantial perturbation. Conversely, since perturbers positionedon opposite sides around the macrolens generate equal but opposite perturba-tions, the net effect of a large number of substructures may cancel out, ensur-ing that rare, massive substructures dominate the positionperturbation of theimages. Chen et al. (2007) have investigated this by modelling the effects ofa wide range of subhalo masses and found that all residual distributions hadvery large peak perturbations (& 10 milliarcseconds). Since the simulationmodels predict extremely few or no substructures in the inner region of thelens, the perturbers must be located further away. Therefore, it was also in-ferred that position perturbations of different images in any lens configurationmay be strongly correlated. Although these results suggested that rare, mas-sive clumps may cause larger perturbations than the more abundant smallerclumps, 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 requiredwhich 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 substructureshould also be detectable on larger scales (∼ 0.1 arcseconds) and at shorterwavelengths in extended Einstein rings and arcs produced by galaxy-galaxylensing. Peirani et al. (2008) used the perturbative methodand lens distri-butions from toy models as well as cosmological simulationsto predict thepossible signatures of substructures. They show that when a substructure ispositioned near the critical line, not only astrometric butalso morphologi-cal effects, i.e. breaking of the image, will occur which areapproximately 10times larger and should be easier to detect. Other studies have suggested touse non-parametric source and lens potential reconstructions to probe smallperturbations 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 onor close to the Einstein ring, perturbations with masses& 107M⊙ respectively109M⊙ can be reconstructed. This technique may then be used to constrain thesubstructure mass fraction and their mass-function slope,once a larger sampleof 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 LargeMillime-ter Array (ALMA) – a significant increase in the number a high-resolutionlenses is expected which will allow the use of astrometric perturbations tostudy the dark satellite population.

5.3 Small-scale structure in macroimagesWhen dark objects in the dwarf-galaxy mass range intersect the line of sighttowards distant QSOs, image-splitting or distortion on characteristic scales ofmilliarcseconds) may occur (Kassiola et al. 1991; Wambsganss & Paczynski1992). At the current time, QSOs can only be probed on such small scalesusing VLBI techniques at radio wavelengths, but future telescopes and instru-ments may allow similar angular resolution at both optical and X-ray wave-lengths (Paper II ).

Using VLBI, Wilkinson et al. (2001) reported no detections ofmillilensingamong 300 compact-radio sources and was able to impose an upper limit ofΩ < 0.01 on the cosmological density of point-mass objects (i.e., very com-pact objects, like black holes) in the106 – 108 M⊙ range. However, this doesnot convert into any strong limits on the subhalo population, since CDM ha-los and subhalos are not nearly as dense as black holes. Correcting for thiswould decrease the expected image separations for a millilens of a given massand yield a probability for lensing that is much lower than assumed in theiranalysis. The sources used were moreover not macrolensed, a situation whichwould 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 macroim-ages of a background light source. A subhalo located in the dark halo intercepts one ofthese macroimages, which may give rise toa) additional image small-scale splittingof the affected macroimage if the source is sufficiently small, which is not seen in theother macroimages, orb) a mild distortion in the affected macroimage, if the sourceis 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 hadbeen detected (see Fig. 5.1a).

The effects that a subhalo can have on the internal structure of one of themacroimages in a multiply-imaged QSO (Fig. 5.1b) are schematically illus-trated in Fig. 5.5. For a small, point-like source (e.g. a QSOobserved at opti-cal wavelengths), the macroimage may split into several distinct images withsmall angular separations (Fig. 5.5a). A larger source (e.g. a QSO at radiowavelengths) may instead exhibit small-scale image distortions (Fig. 5.5b).Even though QSOs may display complicatedintrinsic structure when imagedwith high spatial resolution, such effects can at least in principle be separatedfrom the features imprinted by millilensing, since intrinsic structure will be re-produced in all macroimages, whereas millilensing effectsare unique to eachmacroimage. The distinction between these small-scale changes in the mor-phologies of macroimages, and the astrometric effects discussed in section 5,becomes somewhat arbitrary in some cases, since image distortion may bothshift the centroid of the imageand alter its overall appearance (e.g. throughthe introduction of new, small-scale images). The distortion of macrolensedjets is for instance often referred to as an astrometric effect.

Yonehara et al. (2003) have argued that a significant fraction of allmacrolensed optical QSOs may exhibit secondary image-splitting onmilliarcsecond scales due to CDM subhalos. Contrary, inPaper III , wehave estimated the expected optical depth for millilensingby CDM subhalosand conclude that the prospects for observing such effects in point-sources,like QSOs in the optical, is very low. However, we also show that theprobability for subhalo lensing is promising in the case of extended sources.Inoue & Chiba (2005b,a) have explored a similar scenario in the case of theextended images expected for macrolensed QSOs at longer wavelengths, and

34 Missing satellites

concluded that the small-scale macroimage distortions produced by CDMsubhalos may be detectable with upcoming radio facilities such as ALMA orthe VLBI Space Observatory Programme 2 (VSOP-2).

The merits of probing CDM subhalos through the small-scale structure ofmacroimages is that, contrary to the case for flux ratio anomalies, there is lit-tle risk of confusion due to microlensing by stars or propagation effects inthe interstellar medium. Globular clusters may be able to produce similar ef-fects (Baryshev & Ezova 1997), and so may luminous dwarf galaxies (i.e.the subset of CDM subhalos that happen to have experienced substantial star-formation), but subhalos are expected to outnumber both of these populations,at least in most mass intervals. Instead, the main problem with this approachseems to be that CDM subhalos may not be sufficiently dense to producestrong lensing effects on scales that can be resolved by current technology.Most studies of these effects have assumed that CDM subhaloscan be treatedas SIS lenses, resulting in an gross overprediction of the image separationscompared to more realistic subhalo models (Paper II ). The angular resolu-tion by which macrolensed QSOs can be probed is on the other hand likely toincrease substantially in the coming years, in principle reaching≈ 0.04 mil-liarcseconds with ALMA and the global VLBI array operating at millimeterwavelengths.

5.4 Time delay effectsThe images of a macrolensed light source (see Fig. 5.2a) are subject to differ-ent time delays, which become detectable when the source exhibits intrinsictemporal variability over observable time scales. These time delays stem froma combination of differences in the relativistic time delays (clocks runningslower in deep gravitational fields, also known as Shapiro time delays) andthe differences in photon path lengths (due to geometric deflection) amongthe macroimages (compare section 3.2). Since QSOs are both non-transientand known to vary significantly in brightness on time scales of hours and up-wards, they are very convenient targets for observing campaigns aiming tomeasure such time delays. At the current time, around 20 macrolensed QSOshave 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 Hubbleconstant and the density profile of the macrolens (i.e. the overall gravitationalpotential of the lens galaxy and its associated dark halo), but can also poten-tially be used to probe the CDM subhalos of the lens galaxy.

As shown by Keeton & Moustakas (2009), the presence of subhalos withinthe macrolens will perturb the time delays predicted by smooth lens models,and may also violate the predicted arrival-time ordering ofthe images. Suchviolations would signal the presence of subhalos in a way that, unlike thecase 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 abackground light source. If this source displays intrinsicvariability, observable timedelays between the different macroimages may occur. If one of the macroimages ex-periences 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 sig-nature of millilensing in cases where the small-scale images blend into one due toinsufficient angular resolution of the observations.

or microlensing by stars. The time delay perturbations due tosubhalos aretypically on the order of a fraction of a day. By pushing the uncertainties inthe observed time delays to this level, strong constraints on CDM subhalopopulations may potentially be derived. One case of a time ordering reversalwhich may possibly be attributed to subhalos has already been identified in themacrolensed QSO RX J1131 - 1231 (Morgan et al. 2006; Keeton & Moustakas2009).

If the subhalos themselves give rise to small-scale image splittings, shorttime lags between the light pulses of the separate small-scale images wouldbe introduced. This imprints echo-like signatures in the overall light curve ofastronomical objects with short-term variability (such asgamma-ray burstsand X-ray QSOs), even if the small-scale images cannot be spatially resolved.These echos correspond to light signals transported throughsmall-scale im-ages with longer time delays than the leading image, and the flux ratios ofthese peaks are given by the different magnifications of these images. Thelight curves of gamma-ray bursts have been used to search forsuch light echosin the interval∼ 1–60 s, resulting in upper limits (Ω < 0.1) on compact darkobjects in the105–109 M⊙ range (Nemiroff et al. 2001) and even a few can-didate 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 bearingon CDM subhalos, since the probability for subhalo millilensing is too lowwhen the target objects are not macrolensed. Yonehara et al.(2003) insteadsuggested monitoring of macrolensed QSOs, predicting thatCDM subhalosmay produce light echos separated by∼ 1000s, which could potentially bedetected in X-rays, where rapid intrinsic flares have been observed. This lens-ing situation is schematically illustrated in Fig. 5.6.

Open questions and future prospectsLensing can in principle be used to probe the CDM subhalo population, buthas so far not resulted in any strong constraints. Most studies have focusedon flux ratio anomalies, but a number of studies now suggest that subhalos bythemselves are unable to explain this phenomenon (Mao et al.2004; Amaraet al. 2006; Miranda & Macciò 2007; Xu et al. 2009). If correct, this wouldlimit the usefulness of this diagnostic, since some other mechanism must alsobe affecting the flux ratios. Luckily, results from other techniques, such asastrometric perturbations, small-scale image distortions and time delay per-turbations may be just around the corner.

Observationally, the future for the study of strong gravitational lensing islooking bright. As of today, around 200 macrolensed systemshave been de-tected with galaxies acting as the main lens. Planned observational facilitiessuch as the Square Kilometer Array (SKA) and the LOw FrequencyARrayfor radio astronomy (LOFAR) at radio wavelengths and the LSST in the opti-cal have the power to boost this number by orders of magnitudein the comingdecade (Koopmans et al. 2009). The spatial resolution by which these systemscan be studied is also likely to become significantly better,approaching∼ 10milliarcseconds in the optical and∼ 0.1 milliarcseconds at radio wavelenghts.

On the modelling side, there are still a number of issues thatneed to beproperly addressed before strong constraints on the existence and propertiesof CDM subhalos can be extracted from such data.

Input needed from subhalo simulations

The largest N-body simulations of galaxy-sized halos are nowable to resolveCDM subhalos with masses down to∼ 105 M⊙, but there are still a numberof aspects of the subhalo population that remain poorly quantified and couldhave 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 sub-

halo mass and subhalo position within the parent halo? How large is thedifference from subhalo to subhalo?

5.4 Time delay effects 37

• What is the spatial distribution of subhalos as a function ofsubhalo masswithin the parent halo? What is the corresponding distribution outside thevirial radius?

• How do baryons affect the properties of subhalos? Can baryons promotethe 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 (Moustakaset al. 2009). Attempts to quantify the effects of different density profiles oflensing signature have been made (Macciò & Miranda 2006; Chen et al. 2007;Shin & Evans 2008; Keeton & Moustakas 2009,Paper II) , but the modelsused are still far from realistic, and many of those active inthis field still clingto 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 pro-ducing millilensing effects. Many large galaxies are knownto be surroundedby 102–103 globular clusters with masses in the105–106 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 morecentrally concentrated density profiles, thereby potentially making them moreefficient lenses. We also expect a fair share of luminous dwarf galaxies withinthe dark halos of large galaxies. These dwarfs may represent the subset ofCDM subhalos inside which baryons were able to collapse and form stars, butif so, this means that they may have density profiles significantly more cen-trally concentrated than their dark siblings. While the role of globular clus-ters and luminous satellite galaxies has been studied in thecase of flux ra-tio anomalies (Chiba 2002; Shin & Evans 2008), their effects on many of theother lensing situations discussed in previous sections have not been addressedyet. Low-mass halos along the line of sight may also affect these lensing sig-natures, and sometimes appreciably so (Metcalf 2005b; Miranda & Macciò2007).

Aside from these objects, there may of course be other surprises hiding inthe dark halos of galaxies. Intermediate mass black holes (M ∼ 102–104 M⊙),formed either in the very early Universe or as the remnants ofpopulation IIIstars 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, thepredicted X-rayproperties of such black holes already place very strong constraints on theircontribution to the dark matter (Ω . 0.005; Mapelli et al. 2006), but the moregeneralised dynamical (Carr & Sakellariadou 1999) and lensing (Nemiroffet al. 2001) constraints on other types of dark objects in thestar cluster massrange (∼ 102–105 M⊙) are otherwise rather weak (Ω . 0.1). Lensing observa-tions 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. Astelescopes attain better sensitivity and higher angular resolution in the nextdecade, we can surely look forward to an exciting new era in the study of thedark matter halos.

39

6 Gravitational telescopes

Galaxy clusters are the largest known gravitationally bound objects whichhave formed in the universe through the process of hierarchical structure for-mation. They typically contain hundreds of galaxies and havetotal masses of1014 – 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 theform of dark matter. In this chapter, we will outline how gravitational lensingcan be used for studying these massive objects as well as highredshift lightsources located behind them.

Zwicky (1937) proposed the use of galaxies as gravitational lenses. How-ever, it was not until the late 1970s that the first gravitationally lensed objectwas identified (Walsh et al. 1979). Since then galaxy clusters have success-fully been employed as so-called gravitational telescopes.

Massive galaxy clusters acting as powerful gravitational telescopes offerunique opportunities to observe extremely distant objects. These includegalaxies out to a redshift ofz ∼ 7 (e.g. Ellis et al. 2001; Kneib et al. 2004;Bradley et al. 2008; Zheng et al. 2009; Richard et al. 2011), aswell as distantSNe (e.g. Kovner & Paczynski 1988; Kolatt & Bartelmann 1998;Gal-Yamet al. 2002; Gunnarsson & Goobar 2003,Papers IV and V), too faint tobe detected otherwise. They can also be employed for detailedstudies ofgalaxies at more modest redshifts ofz ∼ 3 – 5 (Bunker et al. 2000; Fryeet al. 2007, 2008). Gravitational lensing of distant objects with measuredredshifts can be used to put constraints on cosmological parameters (Gilmore& Natarajan 2009; Jullo et al. 2010). Lensing magnifications of up to afactor ∼ 100 have been observed for multiply lensed images, and typicalmagnification factors of 5 – 10 are very common within the central onearc-minute radius of massive cluster lenses.

However, nothing comes without a cost. Conservation of flux implies thatthe survey area behind a massive cluster in the source plane is shrunk due tolarge lensing magnifications. Therefore, a gravitational lens does not alwaysenhance the number of discoveries (Gunnarsson & Goobar 2003; Maizy et al.2010). However, it does increase the limiting redshift of a magnitude-limitedsurvey. Thus massive galaxy clusters act as “high-redshift filters” and can be avaluable tool for observing sources at redshifts beyond what would otherwisebe possible.

In Papers IV andV, we exploit this effect to search for SNe at redshifts be-yond 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 es-timates of the SFR (Mannucci et al. 2007; Dahlen et al. 2004; Kobayashi & Nomoto2009). Figure adopted fromPaper V.

for a recent review). SNe can be separated into two main physical classes.Core collapse (CC) SNe include all type II SNe (which exhibitH lines in theirspectra) and Type Ib/c SNe (which lack H but show weak Si and S lines). TypeIa SNe, which can be used as so-called standard candles to trace the expansionhistory of the universe (compare chapter 2.2), lack H lines in their spectra butshow strong Si and S lines (see Filippenko (1997) and Leibundgut (2008) forreviews on SN classification and their general properties).

So far, only approximately 20 well-measured SNe are known beyond a red-shift of z∼ 1. Type Ia SNe have been confirmed to redshiftz∼ 1.7 (Riess et al.2001, 2007; Poznanski et al. 2007). Until recently, the mostdistant CC SNewere confirmed to redshiftz ∼ 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 opticalfor redshifts ofz ∼ 2 and higher, Cooke et al. (2009) discovered the highestcore collapse SN to date atz ∼ 2.4.

Finding SNe at high redshift is very interesting for severalreasons. Due tothe short evolutionary lifetimes of their progenitors, therate of CC SNe,rCC

V ,gives a direct measurement of the ongoing star-formation rate (SFR), for an as-sumed initial mass function. The estimates of the SFR as a function of redshiftfrom various data sets show a large span (e.g. Chary & Elbaz 2001; Giavaliscoet al. 2004; Hopkins & Beacom 2006; Mannucci et al. 2007; Bouwens et al.

41

Figure 6.2:Extrapolation of available SN Ia rate predictions to higherredshifts (Man-nucci et al. 2007; Kobayashi & Nomoto 2009; Neill et al. 2006;Scannapieco & Bild-sten 2005; Dahlen et al. 2008, corresponding toτ = 3.4 Gyr (D08) andτ = 1.0 Gyr(TAU1)). Figure adopted fromPaper V.

2009). Correspondingly, the predictions for the high redshift CC SN rate differsubstantially depending on the underlying assumptions (compare Figure 6.1).Thus, measuring the CC SN rate,rCC

V , out to high redshifts, will give valuableinformation on the SFR in the early universe.

Conversely, if a star-formation history can be assumed, measuring the SNIa rate,rIa

V , as a function of redshift and galaxy type can put constraints on theprogenitor scenarios for Type Ia SNe (e.g. Dahlén & Fransson1999; Gal-Yam& Maoz 2004; Strolger et al. 2004; Mannucci et al. 2005; Neillet al. 2006;Sullivan et al. 2006; Botticella et al. 2008). While SNe Ia have been usedextensively for deriving cosmological parameters, it is unsatisfactory that theprogenitor scenario preceding the SN explosion is still mostly unknown. Ex-isting models predict that different scenarios, such as thesingle degenerateand the double degenerate models, should have different delay-time distribu-tions,Φ(t), describing the time between the formation of the progenitor starand the explosion of the SN, which should be reflected in the evolution of theType Ia SN rate,rIa

V (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 highredshift Type Ia SNe to extend the Hubble diagram to redshifts beyondz∼ 1.5,

42 Gravitational telescopes

will help significantly to constrain the cosmological modeland also to checkfor systematic effects in the SNe.

In order to make predictions on the number of expected high redshift SNdiscoveries behind galaxy clusters as well as for correcting any observationsfor their lensing effect, accurate lens mass models are needed.

6.1 Cluster modelsThe study of the mass distribution within galaxy cluster proves to be interest-ing by itself for a number of reasons. Tracing their baryonicand dark mat-ter 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 measuredmasses can be used to put constraints on cosmological parameters (Vikhlininet al. 2009; Mantz et al. 2010; Rozo et al. 2010; Oguri & Takada2011).Gravitational lensing analyses of the matter distributionwithin galaxy clus-ters have resulted in many exciting findings, like the so-called "Bullet Clus-ter", 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 independenttechniques. The total gravitational mass of a galaxy clustercan be determinedby measurements of the Sunyaev-Zel’dovich effect (Sunyaev &Zeldovich1972) towards galaxy clusters combined with information onthe intra-clustermedium temperature estimated from X-ray spectral analyses(e.g. Gregoet al. 2001; Hincks et al. 2010).

Using information from strong and weak gravitational lensing effects isone of the most promising strategies for modeling the mass distribution ingalaxy clusters. So far, it has only been possible to producereliable massmaps 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; Richardet al. 2010; Okabe et al. 2010; Umetsu et al. 2010). The galaxy cluster A1689is one of the most effective gravitational lenses known as well as one of thebest studied galaxy clusters to date with 114 strongly lensed images of 34background galaxies (Broadhurst et al. 2005; Limousin et al.2007, compareFigure 6.3). Over the past decades, as the quality of observational dataimproved, 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 characterisingthe mass distribution (e.g. Limousin et al. 2007; Richard et al. 2010).Non-parametric models, grid-based or interpolative, may instead adoptarbitrary forms, however not necessarily physically realistic ones (e.g. Sahaet 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 dis-tribution (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 thepositions of the multiple images. InPaper VI, we have investigated the powerof 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 mea-surement of the lensing magnification of the galaxy cluster at the position ofthe SN. For galaxy clusters with no or only very few known multiply imagedsystems, this technique is especially interesting, breaking the so-called masssheet degeneracy (Kolatt & Bartelmann 1998,Paper VI). We have shown thateven for a well studied cluster as A1689, the precision of theparameters de-scribing the overall dark matter component of the cluster might improve byalmost a factor 2.

44 Gravitational telescopes

Once, the mass distribution of a galaxy cluster has been modeled with goodprecision, this enables several interesting predictions,e.g. in the case of SNeexploding in multiply imaged background galaxy.

6.2 Multiply imaged SNeSo far, there are no known multiply imaged SNe. This situationmight howeversoon change, with several surveys targeting massive clusters both at opticaland near-IR wavelengths (e.g. Zitrin et al. 2011b,Papers IV andV). Stronglylensed SNe are interesting for several reasons.

If a SN is strongly lensed, a measurement of the time delay between thetransient in the multiple images could potentially constrain the Hubble param-eter (Refsdal 1964), and thus dark matter and dark energy parameters (Goobaret al. 2002; Mörtsell & Sunesson 2006; Suyu et al. 2010). This can be seenwhen analysing the expression for the arrival time (compareequation 3.2) forits dependence on cosmological parameters. Assuming the lens potential iswell understood (which usually is the cause of the largest uncertainties) andthe lens redshiftzL can be measured, the time delay between two images willdepend on the distance ratio

D ≡DLDS

DLS, (6.1)

whereDL, DS andDLS denote the angular diameter distances from observerto source, observer to lens and from lens to source, respectively. As this ratioscales inversely with the Hubble constant,D ∝ H−1

0 , a measurement of thetime delay can give an estimate on the Hubble constant. There is also a depen-dence inD on other cosmological parameters, such asΩM andΩΛ, althoughthis 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 cosmolog-ical parameters at redshifts where few other probes are currently available.Given the transient nature of SNe, the time delay between multiple imagescould potentially be measured to very high precision. The main limitationof this technique is the strong degeneracy between the lens mass model andthe Hubble parameter,H0 (e.g., Wambsganss & Paczynski 1994; Witt et al.2000; Kochanek 2002; Zhao & Qin 2003). However, observing cluster lenseswith potentials well constrained by a large number of already known multipleimages or, in the case of a strongly lensed SNe Ia, direct magnification in-formation, this degeneracy can be diminished (Oguri & Kawano 2003,PaperVI) .

In the case of a SN exploding in a galaxy which is known to be multiplyimaged by a foreground galaxy cluster, one might even be ableto predict when

6.2 Multiply imaged SNe 45

and where the explosion should show up in the corresponding images, thusallowing for scheduling observations accordingly. Early observations of SNecan give important clues on the physics involved. In Type Ia SNe, the shockinteraction with the companion star may be detected for the first time (Kasen2010), as well as the UV shock from the stellar surface in bothpair-instabilitySNe (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 lensinghas proven to be a valuable tool for investigating the dark components of theuniverse. In this thesis, we have been studying gravitational lensing effectsfrom the smallest (microarcseconds) to the largest (several arcminutes) scales.

Paper I deals with lensing by so called MACHOs, some of the smallestdark matter structures which have been predicted to have formed in the uni-verse. We demonstrate that relaxing the Press-Gunn approximation, in whichsuch lenses are assumed to be randomly and uniformly distributed with con-stant comoving density, gives rise to substantial sightline-to-sightline scatterin microlensing optical depths. By applying the model to themicrolensingcontribution to the long-term optical variability of QSOs which are not multi-ply imaged, we also show that relaxing the Press-Gunn approximation, how-ever, only has a modest effect on the prediction of light curve amplitudes.Thus, microlensing by MACHOs is unlikely to be the dominant mechanismfor the observed long-term variability in QSOs.

In Papers II and III , we study the detectability of dark subhalos, the socalled missing satellites, via their gravitational lensing effect on distant lightsources. We estimate the image separations for the subhalo density profilesfavoured by recent N-body simulations and compare these to the angular res-olution of both existing and upcoming observational facilities. We find thatthe image separations produced are very sensitive to the exact subhalo densityprofile assumed, but in all cases considerably smaller than previous estimatesbased on the premise that subhalos can be approximated by singular isother-mal spheres. We also estimate the overall probabilities forlensing by sub-structures in a host halo closely aligned to the line of sightto a light source.Under the assumption of a point source, the optical depth forstrong gravi-tational 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 spa-tially extended, as is the case for a QSO observed at radio wavelengths, thereis a reasonable probability for witnessing substructure lensing effects even atrather large projected distances from the host galaxy, provided that the angularresolution is sufficient. Detections of dark subhalos through image-splittingeffects will therefore be far more challenging than previously believed, albeitnot necessarily impossible. To investigate the feasibility of recognising small-scale image distortions from dark subhalos in a future survey, we are currentlyconducting 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 ar-ray.

Finally, we investigate the lensing properties of some of the largeststructures known in the universe, massive galaxy clusters.Due to their largemasses, these clusters act as so called gravitational telescopes, magnifyingthe flux of distant background sources by several magnitudes.

In Papers IV andV, we exploit this effect to search for lensed distant super-novae that may otherwise be too faint to be detected. The feasibility of findinglensed supernovae in our survey was investigated using synthetic lightcurvesof supernovae and several models of the volumetric Type Ia and core-collapsesupernova rates as a function of redshift. We also estimate the number of su-pernova discoveries expected from the inferred star-formation rate in the ob-served galaxies. A first transient was discovered behind galaxy cluster A1689,consistent with being a Type IIP supernova at redshiftz = 0.59. The lensingmodel predicts 1.4 mag of magnification at the location of thetransient, with-out which this object would not have been detected in the near-IR ground-based search described in the papers. Since the publicationof these papers,other SN candidates have been found in a survey with the newerHAWK-Icamera at VLT, most interestingly a probable Type IIn SN at redshiftz= 1.703(Amanullah et al. in preparation).

In Paper VI, we analyze the possible constraints from information gainedby the detection of lensed SNe behind galaxy clusters. We show that the de-tection of a Type Ia SN exploding in one of the background galaxies can bea valuable tool for improving the cluster models and gaininginformation onthe overall distribution of dark matter in galaxy clusters.Also, in the caseof multiply imaged SNe, well constrained cluster models canbe used for anindependent estimate on the Hubble constant.

49

8 Svensk sammanfattning

Tack vare allt bättre observationella instrument, har gravitationella linseffekterblivit mer och mer användbart för att undersöka universums mörka komponen-ter. I denna avhandling har linseffekter från de minsta tillde största skalornastuderats (från mikrobågsekunder till flera bågminuter) .

Artikel I behandlar linseffekten av så kallade MACHOs, några av deminsta strukturer av mörk materia som förslagits finnas i universum. Iartikeln visas att det uppkommer en väsentlig spridning i det optiska djupetför mikrolinsning av avlägsna ljuskällor när man antar att linserna följer denmateriefördelningen som förutsägs av simuleringarna. Vi visar också att denobserverade långsamma variationen i kvasarernas ljusstyrka, till skillnad mottidigare påståenden, troligtvis inte kan härröra från mikrolinsning på grundav MACHOs.

I Artikel II och III undersöks möjligheten att upptäcka mörka subhalorrunt galaxer genom deras linseffekter på bakomliggande ljuskällor. Vi visaratt avståndet mellan de multipla bilderna är väldigt känsligt för subhalornasexakta densitetsprofil. Vi visar också att sannolikheten för linsning av punk-tkällor är väldigt låg. För utsträckta källor bedöms sökandet efter dessa mörkasubstrukturer genom linsning som mer utmanande än man tidigare trott, dockinte omöjlig. Hur dessa linseffekter kan visa sig, till exempel i radioobser-vationer av linsade jetstrålar från aktiva galaxer, analyseras för närvarandegenom simuleringar.

Slutligen undersöker vi linseffekterna av några av de största kända struk-turer i universum, tunga galaxhopar. På grund av deras storamassor, kan dessahopar användas som så kallade gravitationella teleskop då de förstärker ljusetfrån bakomliggande objekt med flera magnituder.

I Artikel IV och V används denna effekt för att leta efter avlägsna lin-sade supernovor vilka annars skulle vara för ljussvaga för att upptäckas. Enförsta transient har hittats bakom hopen A1689 som överensstämmer med entyp IIP supernova vid ett rödskift avz = 0.59. Linsmodellen förutsäger enförstärkning med 1.4 magnituder vid transientens position, utan vilken ob-jektet inte hade kunnat observeras i studien. Ytterligare transienter har hittatssedan publiceringen av artiklarna, bland annat en sannoliktyp IIn supernovamed rödförskjutningz = 1.703 (Amanullah et al. under förberedelse).

Artikel VI visar hur informationen från linsade Typ Ia supernovor bakomgalaxhopar kan användas för att förbättra modelleringen avmassfördelningeni dessa hopar. Om flera bilder av en supernova, linsad av en välmodellerad

50 Svensk sammanfattning

galaxhop, skulle observeras kan tidsfördröjningen mellanbilderna användasför att bestämma Hubblekonstanten.

51

Publications not included in this thesis

• Detecting CDM substructure via gravitational millilensingRiehm T., Zackrisson E., Mörtsell E., & Wiik K., 2008,idm conf, 54

• An analytical model of surface mass densities of cold dark matterhaloes - with an application to MACHO microlensing optical depthsHolopainen 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 ImageSplittingRiehm T., Zackrisson E., Wiik K., & Möller O., 2008,IAUS, 244, 376

• The Detectability of Dark Galaxies Through Image-Splitting EffectsZackrisson E., Riehm T., Lietzen H., Möller O., Wiik K., & Nurmi P., 2008,IAUS, 244, 397

• Prospects for CDM sub-halo detection using high angular resolutionobservationsRiehm T., Zackrisson E., Möller O., Mörtsell E., & Wiik K., 2008,JPhCS,131, 2045

• Dark Matter Millilensing and VSOP-2Wiik K., Zackrisson E., & Riehm T., 2009,ASPC, 402, 326

• Gravitational Lensing as a Probe of Cold Dark Matter SubhalosZackrisson E., & Riehm T., 2010,AdAst2010, 9

• A highly magnified supernova behind massive cluster Abell 1689Amanullah, 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 havebeen nearly the good experience it really was. Here, I would like to take theopportunity to give some thanks.

First of all, I would like to thank the members of my thesis group for theirsupport during all those years.Ariel Goobar for taking me on as his stu-dent and being the perfect supervisor he is, always positive, giving me thepossibility to find my own way but also small pushes in the right directionwhen needed.Erik Zackrisson for getting me into the astronomy businessand 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 gettingme excited about our results.Göran Östlin for being an enthusiastic mentorand prefect.

I would also like to take the opportunity to thank all my otherscientificcollaborators both within thesnovagroup and abroad. Working with you hasalways been very pleasant. I am also thankful to all the people at theastron-omy department for providing a friendly and productive work environment.We had our share of cookies and cakes over the years. Thanks as well to allthe nicecopspeople for welcoming me to their group.

A special thank you toSergio Gelato for solving all my computationalproblems at incredible speed. Your help will be greatly missed. Also bigthanks toSandra Åberg andLena Olofssonfor taking care of all my paperwork and financial matters. I know that I have not made it easy for you.

Thanks toMagnus Näslund for your tireless efforts to get the fun of as-tronomy 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 whenneeded. Thanks for making all that work with teaching, PR and the webpageeasy. 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 youAnders, Angela, Gautam, Jaime, Javier, Jens, Laia, Magnus A., Martina,Matthias, Sofiaand all the others.

I am greatly thankful to my family. Thank youMatilda , for keeping mefrom working off-hours many times, closing the lid of my laptop with a verydetermining “mamma, inte!” (mummy, don’t!). Thank youMats, for keepingMatilda from keeping me from work all of the time. Thank you also for all ofyour love and support over the past nine years.

54 Acknowledgements

Thanks to my parents,Mama und Papa, for always letting me make myown decisions even if it means having to take a plane to see their grandchild.

Stockholm, May 2011

55

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