Faculty of Physical SciencesUniversity of Iceland

2012

Faculty of Physical SciencesUniversity of Iceland

2012

Gravitational Lensing Statistics

Kjartan Marteinsson

GRAVITATIONAL LENSING STATISTICS

Kjartan Marteinsson

60 ECTS thesis submitted in partial fulfillment of aMagister Scientiarum degree in Astrophysics

AdvisorsEinar H. Guðmundsson

Páll JakobssonGunnlaugur Björnsson

Faculty RepresentativeEinar H. Guðmundsson

M.Sc. committeeEinar H. Guðmundsson

Páll JakobssonGunnlaugur Björnsson

Faculty of Physical SciencesSchool of Engineering and Natural Sciences

University of IcelandReykjavik, June 2012

Gravitational Lensing Statistics60 ECTS thesis submitted in partial fulfillment of a M.Sc. degree in Astrophysics

Copyright c© 2012 Kjartan MarteinssonAll rights reserved

Faculty of Physical SciencesSchool of Engineering and Natural SciencesUniversity of IcelandHjarðarhagi 2-6107 Reykjavik, ReykjavikIceland

Telephone: 525 4000

Bibliographic information:Kjartan Marteinsson, 2012, Gravitational Lensing Statistics, M.Sc. thesis, Faculty ofPhysical Sciences, University of Iceland.

Printing: Háskólaprent, Fálkagata 2, 107 ReykjavíkReykjavik, Iceland, June 2012

Fyrir pabba

Abstract

Models of the formation and growth of the large-scale structure of the universe pre-dict that a large fraction of the cosmic matter resides in small dark matter halostructures of mass < 108M. Confirming the presence of such dwarf structureswould further strengthen current cosmological models. However, these structureshave low luminosity or are non luminous and cannot be detected by standard ob-servational methods. Thus an indirect method of detection is required. In thisthesis we have used gravitational lensing statistics to investigate whether the dwarfstructures could be discovered through their gravitational lensing effects and thus bedetectable by their influence on the light from background sources. Using models ofthe density distribution of dark matter halo structures as well as the time evolutionof their number density we compare the lensing effects of the dwarf structures tothat of a common type of well known, and much larger, lensing objects. We findthat dwarf structures very rarely act as strong lenses. Also, the lensing effects ofthe dwarfs on source spectra will most likely be undetectable by current detectors.

Úrdráttur

Samkvæmt fræðilegum líkönum um uppruna og þróun stórgerðar alheimsins er mikillhluti alls efnis bundinn í dvergvöxnum hulduefnishjúpum með massa < 108M. Efunnt væri að staðfesta tilvist slíkra dverghjúpa myndi það renna styrkari stoðumundir hina viðteknu heimsmynd nútímans. Ljósið sem dverghjúpar senda frá sérer hinsvegar lítið sem ekkert og þar af leiðandi er illmögulegt að finna þá meðhefðbundum mælingum. Því er nauðsynlegt að notast við óbeinar mæliaðferðir.Í þessarri ritgerð eru líkindi á þyngdarlinsuhrifum notuð til þess að kanna hvorthægt sé að greina tilvist dverghjúpa út frá þyngdarlinsuhrifum þeirra á fjarlægariljósuppsprettur. Með því að nota líkön fyrir massadreifingu hulduefnishjúpa og fjöldaþeirra sem fall af tíma getum við borið linsuhrif dverghjúpa saman við tilsvarandihrif stórra vetrarbrauta. Niðurstaðan er sú að dverghjúparnir sýna mjög sjaldansterk þyngdarlinsuhrif. Einnig er sennilegt að linsuhrif dverghjúpa á róf fjarlægariuppspretta séu ekki mælanleg með núverandi tækni.

v

Contents

List of Figures ix

Acknowledgments 1

1 Introduction 3

2 History of gravitational lensing 5

3 Basics of gravitational lensing 93.1 The thin lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Outside the lens plane . . . . . . . . . . . . . . . . . . . . . . 113.1.2 The lens plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Angular diameter distance . . . . . . . . . . . . . . . . . . . . . . . . 143.3 General Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . 183.3.2 Scalar potential . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.3 Time delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.4 Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Gravitational lensing statistics 234.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Number densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.1 The Singular Isothermal Sphere . . . . . . . . . . . . . . . . . 284.3.2 The Non-Singular Isothermal Sphere . . . . . . . . . . . . . . 294.3.3 The Singular Isothermal Ellipsoid . . . . . . . . . . . . . . . . 304.3.4 The Navarro-Frenk-White profile . . . . . . . . . . . . . . . . 31

5 Results 355.1 Comparison of lensing cross sections . . . . . . . . . . . . . . . . . . . 355.2 Lensing Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3 Conditional probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Conclusions 55

Bibliography 57

vii

List of Figures

3.1 Model of a gravitational lensing system. . . . . . . . . . . . . . . . . 10

3.2 Model of the angular diameter distance . . . . . . . . . . . . . . . . . 15

3.3 Angular diameter distance as a function of redshift. . . . . . . . . . . 16

3.4 Angular diameter distance as a function of redshift on a logscale. . . . 16

3.5 Angular diameter distance between source and lens as a function ofredshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Co-moving number density of dark matter halos as a function of redshift. 26

5.1 The cross sections for different density models as a function of lensredshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 The SIE cross section as a function of lens redshift for different ec-centricities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 The NIS cross section as a function of lens redshift for different coreradii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.4 The SIS cross section as a function of lens redshift for different sourceredshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.5 The NIS cross section as a function of lens redshift for different sourceredshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.6 The SIS cross section as a function of lens redshift for the limits ofthe velocity dispersion range . . . . . . . . . . . . . . . . . . . . . . . 41

5.7 Comparison of the two-image and four-image cross sections of the SIEmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

ix

LIST OF FIGURES

5.8 The NFW cross section as a function of lens redshift for the limits ofthe dark halo mass range of giants . . . . . . . . . . . . . . . . . . . . 42

5.9 The NFW cross section as a function of lens redshift for differentsource redshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.10 The SIS cross section as a function of lens redshift for a dwarf lens . . 43

5.11 The NFW cross section as a function of lens redshift for a dwarf lens 44

5.12 The NFW cross section as a function of lens redshift for differentsource redshifts for dwarfs . . . . . . . . . . . . . . . . . . . . . . . . 44

5.13 Line-of-sight probability for the SIS and NFW models as functions ofsource redshift for giant lenses . . . . . . . . . . . . . . . . . . . . . . 45

5.14 Line-of-sight probability for the SIS model as a function of sourceredshift for the limits of the velocity dispersion range . . . . . . . . . 46

5.15 Line-of-sight probability for the NIS model as a function of sourceredshift for different core radii . . . . . . . . . . . . . . . . . . . . . . 47

5.16 Line-of-sight probability for the NFW model as a function of sourceredshift for the dark halo mass range of giant lenses . . . . . . . . . . 48

5.17 Total probability for the SIS model as a function of source redshiftfor giant lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.18 Total probability for the NFW model as a function of source redshiftfor giant lenses with 1013M mass halos . . . . . . . . . . . . . . . . 49

5.19 The line-of-sight probability for the SIS and NFW models as a func-tion of source redshift for dwarf lenses . . . . . . . . . . . . . . . . . 50

5.20 The cumulative conditional probability distribution of the SIS modelfor time delays in the case of giant lenses . . . . . . . . . . . . . . . . 52

5.21 The cumulative conditional probability distribution of the NFWmodelfor time delays in the case of giant lenses . . . . . . . . . . . . . . . . 52

5.22 The cumulative conditional probability distribution of the SIS modelfor time delays in the case of dwarf lenses . . . . . . . . . . . . . . . . 53

x

LIST OF FIGURES

5.23 The cumulative conditional probability distribution of the NFWmodelfor time delays in the case of dwarf lenses . . . . . . . . . . . . . . . . 53

xi

Acknowledgments

I would like to start by thanking Einar, along with Gulli and Palli, for all theirguidance, help and support. I would also like to thank Nial Tanvir for his insightsand comments and Zarija Lukić for data he provided. I wish to extend my gratitudeto Annalisa, Bob, Steve along with Gulli Jr. and Paul for always being willing tolisten and lend a helping hand to a confused master student. Thanks to Árdís andJens for useful discussions at the early stages of this work. Finally I would like tothank my family and friends for their support and their efforts in keeping me (moreor less) sane.

1

1 Introduction

The most common type of matter in the universe is dark matter, named such becauseit does not seem to emit, or even interact, via the electromagnetic force. Dark mattermakes up most of the mass of galaxies, including our own Milky Way, and formsso-called dark matter halos. Numerical studies of large scale structure growth inthe universe have shown that such halos have a wide range of masses, from massiveones associated with galaxies and clusters to the halos of dwarf galaxies and smallerstructures. These same studies have shown that such small dark matter structuresare found at all redshifts and make up a sizeable fraction of the total cosmologicalpopulation of massive objects. They have however low luminosity or are even non-luminous and so would not be detectable by traditional methods. We must thereforeturn to an indirect detection technique to confirm their presence.

One of the predictions of Einstein’s theory of general relativity is that mass affectsthe space surrounding it, causing massive objects to curve their local spacetime. Thismeans that a beam of light travelling in a straight line near a massive object appears,to a distant observer, to curve around the object. This phenomenon can focuslight from background sources, making massive objects behave like cosmologicalmagnifying glasses. This is known as gravitational lensing and can be detected dueto the distinct effects the massive object, or lens, has on the light of the source.Using this phenomenon it might be possible to detect small dark matter structuresby looking for their effects on background sources.

To investigate whether gravitational lensing could be of use in detecting thesesmall structures we turn to gravitational lensing statistics. By using models ofthe number of lenses and their density profiles we can predict the probability of adetectable lensing event and the likely properties of such events. Similar methodshave been used successfully for a long time to predict the number of lenses, evenbefore the discovery of the first gravitational lens.

3

2 History of gravitational lensing

Before going into the science of gravitational lensing it is enlightening to first quicklyreview its history. For a more comprehensive overview of gravitational lensing his-tory we point the reader to the excellent books by Schneider, Ehlers and Falco [37],hereafter referred to as SEF, and Petters, Levine and Wambsgnass [30].

The beginnings of gravitational lensing lie with the German astronomer J. Soldner.Motivated by earlier work on theoretical objects similar to what we now know asblack holes, he looked at the deflection of light by celestial bodies using the gravita-tional theories of Newton [39]. Treating a light ray as a stream of particles he foundthat a light ray passing a spherically symmetric object of mass M with an impactparameter r will be deflected with an angle of deflection,

α =2GM

c2r, (2.1)

where G is the gravitational constant and c is the speed of light. As it turns out,this purely Newtonian value of the deflection angle is off by a factor of 2 [37].

In 1911, more than a century after Soldner’s work, Einstein used his principleof equivalence, and the assumption of an Euclidean metric, to investigate this phe-nomenon. While unaware of the previous work done on this subject, he re-derivedSoldner’s results. In a paper [6] detailing his calculations, Einstein also expressedhis wish that astronomers tried to test his predictions. Indeed at least two attemptswere made. The first was during an eclipse in Brazil in 1912, but it failed due tounfavourable weather conditions. Another was planned for an eclipse in the Crimeanin 1914 by a German expedition. However due to the outbreak of the first worldwar they were unable to perform their observations [30]. Einstein’s value for theexpected deflection of starlight by the sun therefore stood untested until he finishedhis new theory of gravitation. With this completed theory he correctly calculatedthe deflection angle as being twice that of the Newtonian value [7], i.e.

α =4GM

c2r. (2.2)

For the Sun this translates into a deflection of roughly 1.7" for a light-ray grazingits surface [24]. This value was later verified in a famous, and once controversial,measurement by a group lead by Eddington in 1919 [16]. This result was within 30%of Einsteins predicted value and newer results have reduced this uncertainty muchmore [30]. This experimental verification of the general theory of relativity was a

5

2 History of gravitational lensing

great boon for its acceptance. The earlier mishaps in the measurements thereforeturned out to be very fortuitous for Einstein and his credibility.

In the following decade more work was done on this new phenomenon. FirstEddington [5] and later Chwolson [4] considered a more general lensing system, thatof a foreground star lensing a more distant background star. They found that fora sufficiently well aligned lensing system, more than one image of the source wouldappear. In fact, Chwolson also posited that for a perfect alignment the images wouldform a ring-like structure. Such images are called Einstein rings. One might wonderwhy, if Chwolson was the first to consider these structures, they had been namedafter Einstein and not him. As it turns out Einstein, during his earlier work in 1912,had already derived some of the more important properties of this rudimentarylensing system, i.e. the existence of a double image, the lens equation and the factthat the images would be magnified [36]. But he did not publish these results until1936 [8], and then only due to the request of a Czech amateur scientist named RudiW. Mandl. Mandl had asked Einstein to consider, in greater detail, the effect oflensing by a star. In response Einstein redid his calculations from 1912, but he wasnot very impressed with the value of this work, stating in the paper that "there is nogreat chance of observing this phenomenon". More telling, he wrote a letter to theeditor of the journal, saying "Let me also thank you for your cooperation with thelittle publication, which Mister Mandl squeezed out of me. It is of little value, butit makes the poor guy happy". While it is true that history would prove Einsteinwrong, he was not unjustified in his statements. With the technology availableat the time the observation of this effect on distant stars would indeed have beenimpossible.

In the year following Einstein’s papers, Fritz Zwicky, in two landmark papers[45, 46] considered the gravitational lensing of galactic nebulae instead of stars. Inhis calculations he used the virial theorem to estimate the masses of the Virgo andComa clusters. He then used these estimates to find the probability of lensing. Hisconclusion was that "Present estimates of masses and diameters of cluster nebulaeare such that the observability of gravitational lens effects among the nebulae wouldseem insured" [46]. While his estimates of the deflection angles were too large,Zwicky’s prediction nevertheless came true. As well as correctly predicting thediscovery of galaxy lenses, he also stated other things in his paper that later provedto be accurate, i.e. that sources that would normally be too faint to detect wouldbe visible by gravitational lensing due to magnification and that this could be usedto constrain their masses. Also, Zwicky, like Einstein, was indirectly motivated toinvestigate this possibility because of Mandl. Mandl had contacted others besidesEinstein about his ideas, one of whom had forwarded them to Zwicky [37]. It mighttherefore be prudent to name Mandl among the founders of gravitational lensingtheory.

After Zwicky’s papers in 1937 the subject of gravitational lensing died out, withalmost no new ideas or papers being put forward for over two decades. It was notuntil the 1960s, that interest in the field began again. Several people publishedwork on lensing independently, among them the Norwegian Sjur Refsdal. He noted

6

that there will be a time delay between the multiple images formed and that forcertain sources, such as supernovae, this could be measured [32, 33]. Since this timedelay would be inversely proportional to the Hubble constant H0, he suggested thatgravitational lensing could be used to measure it. He went even further, statingthat gravitational lensing could be applied in general to test different cosmologicaltheories [34, 35].

More theoretical work on lensing was done in the decade following this revival,e.g. on the properties of specific lensing potentials. However a detection of an actuallensing system still eluded observers. That changed in 1979 with the discovery ofthe quasars 0957+561 A,B by Walsh, Carswell and Weymann [42]. They foundthat two quasars, with an angular separation of about 6 arcseconds, shared thesame redshift. This, along with the similarity of the spectra and the detection ofa foreground, or lensing, galaxy lead them to the conclusion that these were twoimages of the same quasar. This opened the floodgates on gravitational lensingdetections, with many more multiply imaged quasar systems being found in theyears that followed. Many different lensing systems have now been discovered, likegiant luminous arcs, which are highly deformed images of galaxies at high redshiftand Einstein rings, the ring like images formed when the source and the lens areperfectly aligned. Even microlensing, the star-star lensing systems that Einsteinhad dismissed, has been used to detect exoplanets and probe for candidates of darkmatter.

7

3 Basics of gravitational lensing

In this chapter we review some of the basics of gravitational lensing. The set-uphere mostly follows the excellent reviews by SEF [37], Narayan and Bartelmann [24]and Schneider, Kochanek and Wambsganss [38].

It is illustrative to start with the simplest lensing system, that of a point lens. Aschematic of this type of lensing system is shown in figure 3.1. A light ray from asource (e.g. a star or a galaxy) at point S passes a point mass M at the minimumdistance ξ and is deflected. This angle of deflection is, as stated earlier, given by

α =4GM

c2ξ. (3.1)

This perturbed light ray then travels to the observer at point O, who sees the lightas coming from an image I at an angle θ, instead of the actual source at an angleβ. Calling the distances to the source and the lens from the observer DS and DL

respectively, and the distance between the lens and the source DLS, and assumingsimple two dimensional Euclidean geometry, one can easily get the relation

θDS = βDS + αDLS. (3.2)

By introducing a reduced deflection angle α by

α =DLS

DS

α, (3.3)

one can then restate the above relation as

β = θ − α. (3.4)

This is generally known as the lens equation, and is the fundamental equation ofgravitational lensing theory.

Using relation (3.4), the position of the source can be found from the positionof the image(s) and the angle of deflection. Alternatively, specifying the deflectingmass, the position of images formed by the lensing of a source can be calculated.To do this for a point mass, we simply substitute the value of the deflection angle(3.1) into the lens equation (3.4). This gives

β = θ − DLS

DSDL

(4GM

c2θ

)= θ − θ2

E

θ, (3.5)

9

3 Basics of gravitational lensing

Figure 3.1: Model of a gravitational lensing system. A source at S emits a light-ray that gets deflected due to a point mass M , with the deflection happening adistance ξ from the point mass. Due to this the position of the source as seen bythe observer at O is changed. Figure modified from [24].

where we have used the fact that ξ = DLθ, as can be seen from figure 3.1. Here wehave also defined the angle

θE =

√DLS

DSDL

4GM

c2, (3.6)

which is called the Einstein radius, and has a fundamental connection to the pointlens system, as we will see later. Solving equation (3.5) for the image positions θ weget two solutions, θ+ and θ− given by

θ± =1

2

(β ±

√β2 + 4θ2

E

). (3.7)

From this we see that two images are formed, with an angular separation of

∆θ =√β2 + 4θ2

E. (3.8)

These images appear on opposite sides of the point mass M , with one image beinginside the Einstein radius and the other outside it. In the case of β = 0, i.e. when

10

3.1 The thin lens

the source is right behind the lens, the images will merge at an angular distance θE

from the lens forming a single ring, the so-called Einstein ring.In realistic lensing systems we do not have a simple point mass, or a flat Euclidean

metric. In general we deal with extended lenses and cosmological metrics. These canbe simplified by using the thin-lens approximation and angular diameter distances,which we will detail in the following sections. Using these, the lens equation retainsthe simplistic form of equation (3.4) even for more complex systems, i.e.

β = θ −α(θ), (3.9)

where the angles are now two dimensional vectors on the observers sky and thereduced deflection angle α is defined by equation (3.3).

3.1 The thin lens

In all real gravitational lensing systems the distances from the lens to the ob-server and from the lens to the source are much bigger than the size of the lens.We can therefore treat the deflection as happening only in a thin plane at thelens. This plane, which we call the lens plane, is perpendicular to the line ofsight. The rest of the universe is assumed to be described by the standard Fried-mann–Lemaître–Robertson–Walker (FLRW) cosmological model. We now treatboth of these regions in turn.

3.1.1 Outside the lens plane

The FLRW metric is given by

ds2 = −c2dt2 + a(t)2dχ2, (3.10)

where

dχ2 =

(dr2

1− kr2+ r2dθ2 + r2sin(θ)2dφ2

). (3.11)

Here k is the measure of curvature of the universe and a(t) is known as the scalefactor. There are three possibilities for the curvature, with the universe being flat(k = 0), closed (k > 0) or open (k < 0). The scale factor relates the proper distancebetween two objects at different times. It is given by the Friedmann equations:

H2(t) =

(a(t)

a(t)

)2

=8πG

3ρ(t)− kc2

a(t)2, (3.12)

H(t) +H2(t) =a(t)

a(t)= −4πG

3

(ρ(t) +

3p(t)

c2

). (3.13)

11

3 Basics of gravitational lensing

Here we have introduced several new variables. The first of these, the Hubbleparameter H(t), is simply the expansion rate of the universe. Its value at currenttime, H0, is known as the Hubble constant. We also have in equations (3.12) and(3.13) terms representing the pressure, p(t), and energy-mass density, ρ(t), of theuniverse. These two terms are made up of contributions of each of the componentsthat constitute our universe. Of most importance to our studies are the energy-matter densities of matter, both dark and normal, and dark energy. It is useful todefine these densities in terms of dimensionless values, called the density parameters.These are defined for each component as the ratio of the energy-mass density of thecomponent to the critical density, i.e. the total density of a spatially flat universe:

Ωx =ρx(t)

ρc(t), (3.14)

for component x. We take ΩΛ, ΩB and ΩDM to be the density parameters for darkenergy, baryonic matter and dark matter, respectively. The critical density is givenby

ρc(t) =3H(t)2

8πG. (3.15)

We can use these density parameters and equation (3.12) to find a simple expressionfor the Hubble parameter

H(z) = H0

√ΩΛ + k(1 + z)2 + ΩM(1 + z)3 ≡ H0E(z), (3.16)

where ΩM = ΩDM + ΩB and z is redshift. The definition of redshift is simply

1 + z =a(t0)

a(t). (3.17)

It is a dimensionless measure of distance (and consequently time) in cosmology. Thevalues we use in this work for the cosmological parameters come from the seven yearWMAP results [14]:

ΩB = 0.0456± 0.0016,

ΩDM = 0.227± 0.014,

ΩΛ = 0.728+0.015−0.016,

Ωtot = 1.0023+0.0056−0.0054,

H0 = 70.4+1.3−1.4 km s−1Mpc−1.

Since the total density is so close to the critical value, Ωtot = ΩΛ + ΩM = 1, we setk = 0 to simplify some of the calculations in the following sections.

12

3.1 The thin lens

3.1.2 The lens plane

Near the lens we assume that the geometry is described by the Minkowski metricperturbed by the gravitational potential Φ of the lens:

ds2 = −(

1 +2Φ

c2

)c2dt2 +

(1− 2Φ

c2

)(dx2 + dy2 + dz2). (3.18)

This approach is valid if |Φ| c2 and the velocity of the components of the lens issmall, i.e. v c. This can be described as the metric being weak and static, andholds for all lenses of interest in this thesis. For a metric of this type it can be shownthat it is possible to define an effective index of refraction

n = 1 +2|Φ|c2

, (3.19)

for the propagation of light. Using Fermat’s principle of least time and Snell’s lawwe can get the angle of deflection by

α = kin − kout =

∫∇⊥n dl =

2

c2

∫∇⊥Φ dl, (3.20)

where k is the wave vector and ∇⊥ is the projection of the gradient operator ontoa plane orthogonal to the wave vector k. The use of equation (3.20) is simplifiedsomewhat by the fact that since the deflection is very small we can safely integratewith respect to an unperturbed light ray with the same impact parameters.

We can use this to derive the deflection angle due to a single point mass M .Writing the potential as

Φ(ξ) = −GM|ξ|

, (3.21)

where ξ is the impact vector orthogonal to the unperturbed wave vector kin, asdefined in figure 3.1. Putting this in equation (3.20) we get

α =4GM

c2

ξ

|ξ|2. (3.22)

This can be simplified to Einsteins result as given by (3.1). For a more general lensdescribed as a thin mass sheet we then simply sum over all the mass elements of thesheet to get the deflection angle

α(ξ) =4G

c2

∫(ξ − ξ′)Σ(ξ′)

|ξ − ξ′|2d2ξ′. (3.23)

where Σ(ξ) is the surface mass density, found by projecting the density ρ(ξ, z) ofthe lens onto the lens plane:

Σ(ξ) =

∫ρ(ξ, z) dz. (3.24)

13

3 Basics of gravitational lensing

It is useful to look at the special case when Σ(ξ) is constant. We then get fromequation (3.23) that the reduced deflection angle is

α(θ) =4πGΣ

c2

DLSDL

DS

θ, (3.25)

where we have taken θ = DLξ. We can see that it is possible to define a surfacedensity so that the reduced deflection angle is α(θ) = θ. From the lens equation wesee that this means β = 0 for all image positions. The constant surface density forsuch a lens is known as the critical surface mass density

Σc =c2

4πG

DS

DLDLS

. (3.26)

We can then use this to define a dimensionless surface mass density

κ(ξ) ≡ Σ(ξ)

Σc

. (3.27)

This is usually called the convergence, for reasons that will become clear later.In our work, using equation (3.23) is generally cumbersome. We therefore intro-

duce a simpler method in the following sections.

3.2 Angular diameter distance

As we discussed earlier, the space outside the lens plane is describe by the FLRWmetric. In order to maintain the simplicity of the lens equation we use a specialdistance measure known as the angular diameter distance. We follow the review byHogg [13] in our derivation.

As can be seen in figure 3.2 this angular diameter distance is defined as the ratiobetween the objects transverse and angular sizes, DA = S

θ, which fits the require-

ment needed to derive equation (3.9).To calculate angular diameter distances we must first discuss co-moving dis-

tances. A co-moving distance is defined as the distance between two objects thatremains constant with respect to the expansion of the universe. Since the Hubble pa-rameter H(z) is a measure of this expansion we can define a co-moving line-of-sightdistance by

DC =

∫ z

0

cdz′

H(z′)=

c

H0

∫ z

0

dz′

E(z), (3.28)

where E(z) is defined in equation (3.16). For a flat universe the co-moving distancebetween two objects at the same redshift but separated on the sky by an angle θcan then be shown to be DCθ. The actual transverse size is then

S =DCθ

1 + z, (3.29)

14

3.2 Angular diameter distance

Figure 3.2: The angular diameter distance of an object DA is defined as the ratiobetween the objects transverse size S and its angular size θ.

so the angular diameter distance as a function of redshift is simply

DA =DC

1 + z. (3.30)

In the calculations that follow we use this formula to derive the distances from theobserver (us) to the lens and the source.

Figure 3.3 shows how the angular diameter distance varies with redshift. It isclear from the figure that this distance measure is very different from the well knownEuclidean distance. The reason is that it depends on the angular size as well as thephysical transverse size. Because of this two objects at different redshifts can havethe same angular diameter distance.

Equation (3.30) can only be used to calculate the angular diameter distances fromus to another object. The lens equation, however, also requires the distance betweenthe source and the lens, DLS. In general, for non-Euclidean distances DLS 6= DS−DL

and in a flat universe, it is given by

DLS =1

1 + zS

(DS(1 + zS)−DL(1 + zL)) , (3.31)

where zS is the redshift of the source and zL that of the lens. We have plotted thisdistance in figure 3.5. For a constant zL the general behaviour of DLS resemblesthat of a normal angular diameter distance, as in figure 3.4, except with a shiftedorigin. This seems obvious since we are simply looking at the same thing as we didearlier for an observer stationed at the lens. For a constant zS the distance decreasesmonotonically with lens redshift.

15

3 Basics of gravitational lensing

0

500

1000

1500

2000

0 2 4 6 8 10

DA [M

pc]

z

Figure 3.3: Angular diameter distance as a function of redshift.

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10

DA [M

pc]

z

Figure 3.4: Angular diameter distance as a function of redshift on a logarithmicscale.

16

3.2 Angular diameter distance

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10

DLS

[Mpc

]

z

zS = 10zL = 1

Figure 3.5: The angular diameter distance DLS between the source and lens as afunction of redshift. The red curve is for a fixed source at redshift 10. The greencurve is for a fixed lens at redshift 1. The scale is logarithmic.

17

3 Basics of gravitational lensing

3.3 General Lensing

3.3.1 Coordinate systems

Before going into the more complex aspects of gravitational lensing it is necessary toclarify the coordinate systems we use. The coordinates we have used so far are theangular positions of the source and images, β and θ, along with the impact vectorin the lens plane, ξ. Alternatively we can introduce a vector in the plane of thesource, η, which specifies the location of the source there. The angles and positionvectors are related in a simple way by

ξ = DLθ η = DSβ, (3.32)

where DS and DL are the angular diameter distances to the source and lens respec-tively. It is also useful to introduce dimensionless coordinates x and y by

x =ξ

ξ0

, (3.33)

y =ηDL

ξ0DS

, (3.34)

where ξ0 is a length scale dependant on the lensing system we are treating.

3.3.2 Scalar potential

At this juncture it is useful to introduce the deflection potential, ψ(ξ), the scaledprojection of the Newtonian potential, Φ, of the lens onto the lens plane

ψ(θ) ≡ DLS

DLDS

2

c2

∫Φ(ξ, z) dz. (3.35)

This has some interesting properties, such as

∇θψ = DL∇ξψ =DLS

DS

2

c2

∫∇⊥Φ dz = α, (3.36)

where we have used equations (3.20) and (3.3). The Laplacian of ψ is then

∇2θψ = D2

L∇2ξψ =

2

c2

DLDLS

DS

∫∇2ξΦ dz = 2

Σ(θ)

Σc

= 2κ(θ). (3.37)

Here we have used the definition of Σc given by (3.26) and Poisson’s equation for aNewtonian potential, i.e. ∇2Φ = 4πGρ.

18

3.3 General Lensing

3.3.3 Time delays

Using the deflection potential ψ from the last section we can restate the lensingequation (3.9) using relation (3.36):

∇θ

(1

2(θ − β)2 − ψ(θ)

)= 0. (3.38)

If we look at this in the context of Fermat’s principle the function inside the gradientis related to the travel time by [37]

t(θ,β) =(1 + zL)DLDS

cDLS

(1

2(θ − β)2 − ψ(θ) + constant

). (3.39)

Since this function is indicative of the extra time it takes a light-ray to travel alongthe deflected path it is most commonly referred to as the time-delay function. Thefirst term, 1

2(θ − β)2 is a measure of the geometric travel time, i.e. how long, in the

absence of a gravitational field, would it take the light-ray to travel along the de-flected path. The second, ψ(θ), is the due to the Shapiro effect, that light travellingthrough a gravitational field experiences a time delay due the effects of the field.

In equation (3.39) we see that there is also an indeterminate constant term, thepresence of which means that the time delay for a single image cannot be founddirectly. Instead we look at the difference in time delays between two images

∆t(θ1,θ2) = t(θ1,β)− t(θ2,β). (3.40)

In all later discussions of time delays in gravitational lensing systems we always havethis definition in mind. As a side note, since the only dimensional factor presentin this equation is the Hubble constant H0 (from the expression for the angulardiameter distances (3.28) and (3.30)) it can be, and has been, used to constrain itsvalue.

3.3.4 Magnification

An important property of gravitational lensing is the fact that while the deflectionof the lens can change the apparent solid angle of the source, the surface brightnessof the source is conserved. This means that an image of the source can be magnified,or de-magnified. This magnification is simply the ratio between the solid area of thesource and the image

µ =δθ2

δβ2 . (3.41)

This magnification factor can be either positive or negative, with the sign represent-ing the parity of the image. An image with negative magnification will therefore

19

3 Basics of gravitational lensing

appear mirror-inverted with respect to the source. We can also define a total mag-

nification of all N images as µtot =N∑i=0

|µi|.

Due to the fact that we have a mapping from the source plane to the lens plane,in the form of the lens equation, we can transform the solid angle of the source tothe lens plane with

δβ2 = |J | δθ2, (3.42)

where |J | is the Jacobian. This relation is applicable if the solid angles are smallenough, which is the case in gravitational lensing. The magnification is then theinverse of the Jacobian

µ =1

|J |. (3.43)

The Jacobian matrix is given by

J =∂β

∂θ= δij −

∂αi(θ)

∂θj= δij −

∂2ψ(θ)

∂θi∂θj= δij − ψij, (3.44)

where we have used equations (3.9) and (3.36). From equation (3.37) we see thatthe convergence can be written as

κ =1

2(ψ11 + ψ22) . (3.45)

We can also introduce a new quantity known as the shear

γ1 =1

2(ψ11 − ψ22) = γ cos(2φ) (3.46)

γ2 = ψ12 = ψ21 = γ sin(2φ). (3.47)

where

γ =√γ2

1 + γ22 (3.48)

φ = arctan

(γ2

γ1

). (3.49)

Using the convergence and the shear the Jacobian matrix can then be simplified as

J =

(1− κ− γ1 −γ2

−γ2 1− κ+ γ1

)= (1−κ)

(1 00 1

)−γ(

cos(2φ) sin(2φ)sin(2φ) − cos(2φ)

), (3.50)

and the magnification becomes

µ =1

(1− κ)2 − γ2. (3.51)

We see from this that the convergence causes an isotropic focusing of the light raysof the source, i.e. it does not change the shape of the source but simply increases its

20

3.3 General Lensing

size. The shear on the other hand causes anisotropy in the images created, with γdescribing the magnitude and φ the orientation.

The values θ that give |J | = 0 form a critical curve in the lens plane. An exampleof this is the Einstein radius. This can be mapped, using the lens equation, toanother curve in the source plane, known as a caustic. It can be shown that eachtime a source crosses a caustic, two new images are formed [37] . This means that inorder for multiple images of the same source to be produced the source must cross atleast one caustic, and that there is always an odd number of images formed (thoughthis rule is broken for the point lens).

21

4 Gravitational lensing statistics

We now turn to the statistics of gravitational lensing. The method we use to calcu-late the gravitational lensing probabilities is based on the work done by Turner etal. [41] and Fukugita et al. [9].

4.1 General theory

We start by taking an arbitrary line of sight that ends on a source. The probabilityof a single lensing event occurring on a part dR of this line for a light-ray from thesource is

dτ =dR

l=a(t)2dχ2

l, (4.1)

where l is the proper mean free path of the ray with respect to lensing. We alsohave a(t) and dχ which are the scale factor and space part of the FLRW metricrespectively (as described in §3.1.1). This is the familiar equation for optical depth,with the obstruction here being lenses. The mean free path will thus depend on thenumber density of lenses present, n(zL,M), and the cross section of lensing for eachof them, σ(zL, zS,M), as l = [n(zL,M)σ(zL, zS,M)]−1. These cross sections dependon the source, as well as the lens, position. This is due to the fact that the crosssectional area must be measured in the source plane, not in the lens plane. Usingthe definition of l and the FLRW metric (3.10) for light-rays, ds = 0, we can writethe optical depth as

dτ = n(zL,M)σ(zL, zS,M)cdt

dzL

dzL. (4.2)

In general this density will depend on the mass of the lenses in question and theirredshifts. Part of the redshift dependence is due to the fact that the universe isexpanding, i.e. that the average distances between lensing objects increases withredshift. It is therefore simpler to use the co-moving number density instead, definedas

n(zL,M) =n(zL,M)

(1 + zL)3. (4.3)

The redshift dependence of n is then only due to the evolution of the lens populationwith time.

23

4 Gravitational lensing statistics

Using the definition of redshift given by equation (3.17) and the Friedmann equa-tion (3.12) we get

cdt

dzL

=c

1 + zL

1

H(zL). (4.4)

The equation for the optical depth thus becomes

dτ =c

H(zL)n(zL,M)σ(zL, zS,M)(1 + zL)2dzL. (4.5)

By integrating this equation over the whole line of sight we get the probability forlensing of a given source at redshift zS by a population of lenses with a given massM (or a mass range ∆M):

τ(zS,M) =c

H0

∫ zL

0

n(zL,M)σ(zL, zS,M)(1 + zL)2

E(z)dzL. (4.6)

It is possible to modify this equation to consider the probability of lensing for allsources at redshift zS [37]. To do this we simply consider all the lenses in a shellat each zL and the cross sections due to these lenses in the source plane. Thencomparing the area covered with cross sections at the source plane to the total areaof the source plane, a shell at redshift zS, we get the total lensing probability

P (zS,M) =c

H0

1

D2S

∫ zL

0

n(zL,M)σ(zL, zS,M)D2L(1 + zL)2

E(z)dzL. (4.7)

This equation assumes that no two cross sections overlap, and is only applicable ifP 1. We can go further, and look at conditional probabilities τ(A |B), i.e. theprobability that an event B will have the property A [27]. To do this we simplyreduce the cross section in question to only those source areas that produce a lensingsystem with the property A. In the case of time delays e.g. this means removingsource positions that do not create multiple images with a time delay ∆t. Howeverfor time delays it is more interesting to consider cumulative conditional probabilities,i.e. the probability that a lensing system will have a time delay greater than ∆t.Such cumulative conditional probabilities are given by

τ(> ∆t | zS,M) =τ(> ∆t, zS,M)

τ(zS,M), (4.8)

where τ(> ∆t, zS,M) is the line-of-sight probability for a lensing event with a timedelay greater than ∆t. We discuss this in greater detail in §4.3.

4.2 Number densities

As discussed in the previous section the co-moving number density, as defined byequation (4.3), is one measure of the density of lenses in the universe. To get an

24

4.2 Number densities

accurate description of this co-moving number density we turn to models of structuregrowth in the universe. In simplistic terms, the origin of structure in the universe,i.e. galaxies and clusters, can be traced back to primordial density fluctuations in theearly universe [21, 44]. These density perturbations were magnified by gravitationalinstability, causing overdense regions, if they were massive enough, to form galaxies.Analytical models and approximations can then be used to derive the co-movingdensity using the results of numerical calculations of density perturbations. Lookingonly at the redshift evolution of dark matter halos the co-moving numerical densitycan be expressed as function of halo massM , redshift and cosmology by the equation[22]:

dn(M, z)

d logM=ρc(z)ΩM

Mf(σR)

d lnσ−1R

d logM. (4.9)

Here σR(M, z) is the mean square value of the fractional density perturbations overa sphere with co-moving volume V given by [15, 44]:

σ2R =

⟨(1

V

∫V

δmdV′)2⟩, (4.10)

where the fractional density perturbation (or density contrast) is [21]

δm(r, z) =δρm(r, z)

ρc(z)ΩM

. (4.11)

The density perturbation δρm(r, z) is the primordial density fluctuation extrapolatedto redshift z and ρc(z)ΩM is the average matter density.

The factor f(σR) in (4.9) is a scaled differential mass function of dark matter halos[15]. There exist various fits for f , the most common of which is the Press-Schechtermass function [31], given by

fPS(σR) =

√2

π

(δc

σR

)exp

(− δ2

c

2σ2R

), (4.12)

where δc is usually taken to be a linear overdensity of a spherical perturbation atthe time it starts to collapse. It is commonly taken to be δc = 1.686 [22]. Whilethis function gives good results around z = 0 it breaks down at higher redshifts [15].Therefore, in this work we use the so-called Warren mass function [43] given as

fWarren(σR) = 0.7234(σ−1.625R + 0.2538) exp

(−1.1982

σ2R

). (4.13)

For the actual values of the number density we used data provided to us by Lukić etal. [22]. Figure 4.1 shows this data, i.e. the co-moving density evolution for differenthalo mass ranges calculated using the Warren mass function. We can see that atearly times (high z) the density increases (with decreasing z), with halos that havehigher mass forming later than low mass ones. At late times (low z) the density ofthe low mass halos starts to decrease, due to them merging into higher mass halos.

25

4 Gravitational lensing statistics

0 5 10 15 20

Redshift

1e-15

1e-12

1e-09

1e-06

0.001

1

1000

n[(

Mp

c/h

)-3]

(107 - 10

8) M

O/h

(108 - 10

9) M

O/h

(109 - 10

10) M

O/h

(1010

- 1011

) MO

/h

(1011

- 1012

) MO

/h

(1012

- 1013

) MO

/h

(1013

- 1014

) MO

/h

(1014

- 1015

) MO

/h

.

.

.

.

.

.

.

.

Figure 4.1: Co-moving number density of dark matter halos as a function of redshiftfor different mass ranges. Figure from Lukić et al. [22].

4.3 Cross sections

We now turn to a discussion of lensing cross sections. A lensing cross section is acollection of areas dA(y) (where y is given by equation (3.34)) in the source planeinside of which the source will experience a lensing event, i.e.

σ =

∫dA(y). (4.14)

First we must define what we consider lensing. In our case we take this to meanstrong lensing, i.e. the creation of multiple images. The cross section will then bedefined by the outermost caustic of the lens. As we saw in §3.3.4 the location of thecaustic will depend on the convergence and shear of the lens, both of which dependon the deflection potential. Thus in order to calculate a cross section we must firstspecify a deflection potential, or more generally a density distribution, for the lenseswe are interested in. We have already treated the point-mass model, but that is nota good model for extended lenses, like the galaxies and clusters of interest to us.Instead we look at the simplest extended lens model, the singular isothermal sphere(SIS), and enhancements to it. We also use the more realistic Navarro-Frenk-White

26

4.3 Cross sections

(NFW) density distribution. It is also important to stress here that the cross sectionis defined in the plane of the source, and not in the plane of the deflector itself. Thisis due to the fact that it is the caustics, and not the critical curves, that define theareas where strong lensing occurs.

Before considering individual models we start by discussing a simplification thatcan be used to find the caustics of circularly symmetric profiles. In this case it canbe shown (see SEF [37]) that the deflection at dimensionless radius x (see equation(3.33)) is only affected by the mass inside that radius. The deflection will thenbehave as if the mass was all positioned at the center of mass. Hence by equations(3.36) and (3.37) the reduced angle of deflection will be

α(x) =2

x

∫ x

0

x′κ(x′)dx′ =m(x)

x, (4.15)

where m(x) is the dimensionless mass within a circle of radius x, and κ is thedimensionless surface density, or convergence, as defined in equation (3.27). Usingequations (3.43) and (3.44) we can then express the magnification as

µ =1(

1− m(x)x2

)(1 + m(x)

x2 − 2κ(x)) . (4.16)

From this we see that for circularly symmetric lenses two caustics are formed. Oneis known as the tangential caustic and is defined by the critical curve xt =

√m(x),

i.e.yt = xt −

m(xt)

xt

= 0. (4.17)

This means that all circularly symmetric lenses have a caustic that degenerates toa single point at the center of the source plane.

Of more interest to us is the other caustic, the radial caustic, defined by thecritical curve x2

r = m(xr)/(2κ(xr)−1). This means the equation defining the causticwill be

yr = 2xr (1− κ(xr)) . (4.18)

The cross section for a circularly symmetric lens is then simply the area of a circlein the source plane of radius ηr, i.e.

σ = π

(DS

DL

ξ0yr

)2

. (4.19)

Finally, as discussed earlier, it is possible to modify the lensing cross section bylooking at conditional probabilities. To do this we simply incorporate not only thecriteria of multiple lensing, but also new constrains arising from the conditionalprobability we are investigating. In this work we only consider one such criteria,that of the time delays. In this case we modify the cross sectional area by removingthose source positions that form images with time delays less than ∆t, i.e.

σ(M,> ∆t) =

∫dA−

∫dA(< ∆t). (4.20)

27

4 Gravitational lensing statistics

We do this because we expect that any images formed due to lensing by a small darkmatter structure will form so close to each other that they will be unresolvable bydetectors. However the time delays between the different images will still be presentand might therefore be measurable in the spectra of the single combined image.For circularly symmetric lensing models equation (4.20) can be simplified to [27]

σ(M,> ∆t) =

πξ20

(DS

DL

)2

y2r

(2y2r

∫ yrymin

y′dy′)

= πξ20

(DS

DL

)2

(y2r − y2

min) ymin < yr

0 ymin > yr

(4.21)where ymin is defined as the minimum distance a source must be from the origin soits images have a time delay greater than ∆t, that is ∆t(ymin) = ∆t. We considerthis conditional cross section for only two models, the SIS and NFW profiles.

4.3.1 The Singular Isothermal Sphere

One of the simplest approximation we can make for extended lenses is to assumetheir components behave as an ideal gas. Assuming circular symmetry, the densityfor this type of lens can be shown to be [24]

ρ(r) =σ2v

2πGr2, (4.22)

where r is the distance from the origin (the center of the lens) and σv is the velocitydispersion, the root-mean-square value of the random velocities around the meanvelocity of the components of the lens in question. From equation (3.24) we thenget the surface mass density as

Σ(ξ) =σ2v

2Gξ. (4.23)

Using equations (3.3) and (3.23) we also get the reduced deflection angle:

α(θ) = 4πσ2vDLS

c2DS

=ξ0

DL

. (4.24)

Then the dimensionless lens equation y = x − α(x) gives that the system has twoimages at x = |y| ± 1 if |y| < 1 and only one otherwise [27]. The radial caustic forthe SIS is therefore yr = 1 and from equation (4.19) we get that the cross section is

σSIS = 16π3(σvc

)4

D2LS. (4.25)

The deflection potential for this system can be found using equation (3.36). It is

ψ(x) =ξ2

0

D2L

|x|. (4.26)

28

4.3 Cross sections

Using equation (3.40) we can then get the time delay as a function of y for the regionwhere |y| < 1 [27]:

c∆t(y) = 32π2(σvc

)4 DLDLS(1 + zL)

DS

|y|. (4.27)

We can then use equation (4.21) to find the modified cross sections and conditionalprobabilities.

4.3.2 The Non-Singular Isothermal Sphere

One of the problems of the SIS-model is, as the name suggest, the singularity at theorigin. This non-physical behaviour can be fixed by assuming a constant densitycore with some radius ξc. We follow here a paper by Kormann et al. [18] in derivingthe properties of this modification. The density profile given in (4.22) is modifiedto

ρ(r)NIS =σ2v

2πG(r2 + ξ2c ), (4.28)

where NIS stands for Non-Singular Isothermal Sphere. The surface mass density isthen given by

Σ(ξ) =σ2v

2G

1√ξ2 + ξ2

c

. (4.29)

To find the locations of the caustics for this modified potential we use equation(4.16). First note that the convergence is

κ(x) =1

2√x2 + x2

c

, (4.30)

where we take ξ0 to be the same as in the SIS model and xc = ξc/ξ0. We can nowcalculate the reduced deflection angle from equation (4.15)

α(x) =

√x2 + x2

c − xc

x=m(x)

x, (4.31)

and hence the magnification is given by

µ =

[(1−

√x2 + x2

c − xc

x2

)(1 +

√x2 + x2

c − xc

x2− 1√

x2 + x2c

)]−1

. (4.32)

To get multiple images it has been shown that |xc| < 12[18]. If this is the case then

we get both radial and tangential critical curves [37], with the radial one defined by

x2r =

1

2

(2xc − x2

c − xc

√x2

c + 4xc

), (4.33)

29

4 Gravitational lensing statistics

so the radial caustic is

yr =

12

(4xc − x2

c − xc

√x2

c + 4xc

)−√

12

(2xc + x2

c − xc

√x2

c + 4xc

)√

12

(2xc − x2

c − xc

√x2

c + 4xc

) . (4.34)

Putting this into equation (4.19) gives the cross section for the NIS-model.

4.3.3 The Singular Isothermal Ellipsoid

One of the drawbacks of the SIS and NIS models is that they both have circularsymmetry. This can be fixed by considering a more general elliptical potential,of which the circular potential is a special case. Again we follow the derivationdone by Kormann et al. [18]. They replace the radial coordinate in equation (4.23)with the more general expression ξ = |ξ|

√cos2(ϕ) + (1− ε2) sin2(ϕ), where ε is the

eccentricity of the lens and the impact vector is in polar coordinates ξ = (|ξ|, ϕ),with ϕ = [0, 2π]. The surface density for this Singular Isothermal Ellipsoid (SIE) istherefore

Σ(ξ) =σ2v

2G|ξ|

4√

(1− ε2)√cos2(ϕ) + (1− ε2) sin2(ϕ)

. (4.35)

Using equation (3.27) we get that the convergence is

κ(x) =1

2|x|

4√

(1− ε2)√cos2(ϕ) + (1− ε2) sin2(ϕ)

, (4.36)

where we have again used the same distance scale ξ0 as we did in the SIS model. Aswe can see the formulas for Σ and κ reduce to those of the SIS model if ε = 0.

We can solve equation (3.37) to get the deflection potential [18]:

ψ(x) = |x|4√

1− ε2ε

(sin(ϕ) arcsin(ε sin(ϕ)) + cos(ϕ)arcsinh

(ε√

1− ε2cos(ϕ)

)),

(4.37)and then use the deflection to calculate the reduced deflection angle using equation(3.36):

α =4√

1− ε2ε

(arcsinh

(ε√

1− ε2cos(ϕ)

), arcsin(ε sin(ϕ))

). (4.38)

Using equations (3.46), (3.47) and (3.48) the shear is found to satisfy the equation

γ(x) = κ(x). (4.39)

30

4.3 Cross sections

This means according to equation (3.51), that the magnification of an image in thissystem is given by

µ =1

1− 2κ(x). (4.40)

We therefore have only one critical curve defined by κ(x) = 12and the caustic for

this system is thus given by ycaustic = (y1, y2) where

y1 =4√

1− ε2[

cos(ϕ)√cos2(ϕ) + (1− ε2) sin2(ϕ)

− 1

εarcsinh

(ε√

1− ε2cos(ϕ)

)],(4.41)

y2 =4√

1− ε2[

sin(ϕ)√cos2(ϕ) + (1− ε2) sin2(ϕ)

− 1

εarcsin (ε sin(ϕ))

].(4.42)

Along with this caustic there is also a "quasi"-caustic, or cut, that defines an areawhere multiple imaging exist without creating a critical curve. A source crossingthis cut will create a single new infinitely faint image at the origin, thus creating aneven number of images. The strange nature of this cut arises from the singularitypresent at the origin, if it is removed as we did in the NIS-model it changes to a truecaustic. The location of the cut in the source plane can be found by setting x = 0in the lens equation (3.9), so ycut = −α. Due to the symmetry of the caustic andthe fact that the lens equation is a conformal mapping [37] the cross section can beshown to be:

σcaustic = 4

(DS

DL

)2

ξ20

∫ π/2

0

y2

(dy1

dϕ

)dϕ. (4.43)

In the same way the cross section for the cut is:

σcut = 4

(DS

DL

)2

ξ20

∫ π/2

0

α2

(dα1

dϕ

)dϕ. (4.44)

In the future, following [18], we refer to the cross sections due to the cut and causticas the two- and four-image cross sections respectively. We then choose the largercross section of the two to calculate the probability.

4.3.4 The Navarro-Frenk-White profile

While the SIS model and its variations are in some ways a good first approximationfor extended lenses, we also want to look at more realistic density profiles for darkmatter halos. One of the most commonly used profile is that of Navarro, Frenk andWhite (NFW). In two papers [25, 26] Navarro et al. found, using numerical mod-elling, that dark matter halos seem to follow a universal density profile, independentof mass. They found that this density profile has the form

ρ(r) =ρs

(r/rs)(1 + r/rs)2, (4.45)

31

4 Gravitational lensing statistics

where ρs is a characteristic density and rs a scaled radius.Following Navarro et al. we introduce a dimensionless number, the concentration

parametercvir =

rvir

rs

. (4.46)

Here rvir is the virial radius, defined as the radius within which the mean density is∆vir times larger than the mean density of the universe ρu. Due to the fact that inthis work we assume a flat universe the universal density is just the critical densitydefined by equation (3.15), i.e. ρu = ρc. The virial over-density factor ∆vir can beapproximated in the cosmology we use by [2]

∆vir '18π2 + 82(ΩM − 1)− 39(ΩM − 1)2

ΩM

. (4.47)

The virial mass is [3]

Mvir =4π

3∆virρcr

3vir. (4.48)

This choice of the virial radius means that the characteristic density is given by [40]

ρs =∆virρcc

3vir

3

(∫ cvir

0

x

(1 + x)2dx

)−1

=∆virρc

3

(c3

vir

ln(1 + cvir)− cvir

1+cvir

), (4.49)

and the scaled radius by

rs =1

cvir

3

√3Mvir

4π∆virρc

. (4.50)

As for the concentration parameter, cvir, due to the significant scatter in the therelation between Mvir and cvir we approximate it by the median value [3]:

cmedianvir = cnorm

(1

1 + z

)(H0

1016

Mvir

M

)−0.13

, (4.51)

where cnorm = 8 as in [27]. Using equations (4.49) and (4.50) as well as (4.51) we canrewrite the density function in a form that only depends on the virial mass and theredshift of the halo. This allows us to use the NFW profile in gravitational lensingproblems. Since this is a circularly symmetric profile we can use equation (4.16) tosimplify our search for the caustics. We start by calculating the surface mass densityusing equation (3.24), and choosing the length scale ξ0 = rs [12]:

Σ(x) = 2ρsrsF (x), (4.52)

where

F (x) =

1

x2−1

(1− 1√

1−x2 arccosh(

1x

))x < 1

13

x = 11

x2−1

(1− 1√

x2−1arccos

(1x

))x > 1.

(4.53)

32

4.3 Cross sections

By equation (3.27) the convergence is

κ(x) =2ρsrsF (x)

Σc

. (4.54)

From equation (4.15) we find the reduced deflection angle [1, 12]:

α(x) =m(x)

x=

4ρsrs

Σc

g(x)

x, (4.55)

where

g(x) =

ln(x2

)+ 1√

1−x2 arccosh(

1x

)x < 1

1 + ln(

12

)x = 1

ln(x2

)+ 1√

x2−1arccos

(1x

)x > 1.

(4.56)

From equation (4.16) we see that the radial critical curve is given by the equation

F (xr)−g(xr)

x2r

=Σc

4ρsrs

, (4.57)

and the caustic is therefore

yr = 2xr

(1− 2ρsrs

Σc

F (xr)

). (4.58)

By inserting this in equation (4.19) we get the cross section

σ = 4π

(DS

DL

)2

r2sx

2r

(1− 2ρsrs

Σc

F (xr)

)2

. (4.59)

The time delay for this system has been discussed by Oguri et al. [27], who find thatit can be approximated by

c∆t(y) = 2r2sxt

DS

DLDLS

(1 + zL)y, (4.60)

where xt is the radius of the tangential critical curve, which we get from equation(4.16):

g(xt)

x2t

=Σc

4ρsrs

. (4.61)

33

5 Results

5.1 Comparison of lensing cross sections

As explained in the introduction, we are interested in investigating whether we canuse strong lensing to detect small dark matter structures. Motivated by the num-ber densities shown in figure 4.1 we take these structures to be in the lowest massbin shown, i.e. with masses around 108 M. This also fits the mass range of dwarfspheroidals (dSph), a well known type of dark matter dominated object. We there-fore assume that the properties of our small dark matter structures will be similarto those of dSph. As can be seen from the same figure 4.1, such objects shouldbe very common at all redshifts and have the potential to act as lenses. Whetherthese structures will have a noticeable effect on the light of background sources willdepend on their lensing properties, such as the size of their strong lensing crosssections and the time delays between images. Comparing such properties to thoseof a commonly observed type of lensing object will tell us whether we should expectto observe lensing effects on sources from such small dark matter structures. Wechoose large elliptical galaxies as a baseline since they are one of the most commonlyobserved lensing objects and have been studied extensively. For clarity we refer tolarge ellipticals as giants (or giant lenses) and to the small dark matter structuresas dwarfs (or dwarf lenses) for the reminder of this chapter.

We take the mass of dark matter halos of giants to be between 1012 M and1013 M, and assume the numerical density distribution shown in figure 4.1. Wetake the dark matter velocity dispersion to be v = 225+20

−12 km/s, which is the aver-age central velocity dispersion for giants as calculated by Fukugita and Turner [10].It has been found from observations that the central velocity dispersion is a goodapproximation of the dark matter velocity dispersion of galaxies [17]. We take thecore radius to be the radius at which the surface density falls to half the centralvalue. Lauer [19] has a list of measured core radii for giants, with the radii beingaround 50 to 1000 pc and we consider the same range here.

For the dwarf lenses we take the velocity dispersions to be 5 km/s. We derive thisvalue from Peñarrubia et al. [29] who looked at the properties of local group dSphand their dark matter halos. As we stated earlier the mass of the halos of someof these local dSph are of the same order of magnitude as the structures we areinterested in (the dSph spread in the paper being 108 to 1010 M) and so we assume

35

5 Results

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0 1 2 3 4 5 6 7 8 9 10

σ [k

pc2 ]

zL

SISNIS

NFW

Figure 5.1: The cross sections for three different models as functions of lens redshift.The source is at redshift 10 and the lens is a giant with a dark halo mass of 1013Mand a velocity dispersion of 225 km/s. For the NIS model the core radius is 500pc. The SIE cross section is not included because it is not visibly different fromthe SIS model for all but the largest eccentricities.

that the velocity dispersions of these two objects is similar. Since the lower limitof their inferred masses matches the mass of interest to us we take the lower limitof their central velocity dispersion spread, which is around 5 km/s. Peñarrubia etal. [29] also state the measured core radii of these dwarf galaxies, which are roughlythe same as in the elliptical case (around 50 to 1000 pc).

We begin by looking at the four different models discussed in chapter §4.3, con-sidering first giant lenses at different redshifts. In figure 5.1 the source is positionedat redshift 10. From the figure we see that the behaviour of the cross section withlens redshift is different between the three density profiles. Most notably the NFWmodel, unlike the SIS and NIS models, falls to zero at zL = 0. As for the SIE model,as can be seen from figure 5.2, it does not differ markedly from the SIS model exceptat very high eccentricities and thus we do not include it in figure 5.1. The shape ofthe SIS cross section is similar to that of the angular diameter distance between thesource and lens with lens redshift, as shown in figure 3.5. This is to be expected,since the only redshift dependence in equation (4.25) is from the D2

LS factor. Thesame is true of the SIE model, with both the two-image (4.44) and the four-image(4.43) cross section having the same redshift dependence as the SIS model.

Looking at the behaviour of the NIS cross section for a core radius of 500 pcin figures 5.1 and 5.3, we see that it is monotonically decreasing with redshift, with

36

5.1 Comparison of lensing cross sections

0.01

0.1

1

10

100

0 1 2 3 4 5 6 7 8 9 10

σ [k

pc2 ]

zL

ε = 0.0ε = 0.9

ε = 0.99

Figure 5.2: The SIE cross section as a function of lens redshift for different eccen-tricities. The source is at a redshift 10 and the lens is a giant with a velocitydispersion of 225 km/s.

the curve behaving similarly to the SIS cross section at a lower source redshift (seefigure 5.4). This is somewhat surprising since from the equation for the NIS radialcaustic (4.34) we see that the redshift dependence is much more complex than in theSIS and SIE cases, and thus it is not straightforward that it should follow the samegeneral shape. The low lens redshift cut-off is due to the fact that multiple lensing inthe NIS model only occurs if |xc| < 1/2. As xc depends on both the lens and sourceredshifts this bound is automatically broken above certain lens redshifts. This alsocauses the size of the NIS cross section to be smaller, the difference between the SISand NIS profile for a 500 pc core radius being about a factor 4.

Figure 5.3 shows the NIS cross section for four core radii. Since the bound|xc| < 1/2 must be fulfilled, increasing the core radius decreases the cut-off red-shift, and also the cross section. Thus for a core with a radius of 1 kpc the crosssection is negligible above zL ≈ 3 but for a core radius of a 100 pc the cross sectionis non-negligible up to zL ≈ 8. It should be noted that newer observations [20] havefound that giants do not in general have a well defined core of constant density.Instead the density profiles behave more like two power laws connected with a breakin the slope or just as a single uniform power law with no special core region. Thusthe NIS profile might not be an accurate model for giant lenses.

In figures 5.4 and 5.5 we see how the cross section of the isothermal models changeswhen the source redshift is varied. Both the redshift range and the slope of the σ-curve change. Looking specifically at the NIS model we see that the cut-off redshift

37

5 Results

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0 1 2 3 4 5 6 7 8 9 10

σ [k

pc2 ]

zL

rc = 0 pcrc = 100 pcrc = 500 pc

rc = 1000 pc

Figure 5.3: The NIS cross section as function of lens redshift for different core radii.The source is at redshift 10, and the lens is a giant with a velocity dispersion of225 km/s.

differs by only ∆z ≈ 1 for the sources at z = 5 and z = 10. This seems to indicatethat below a certain source redshift the |xc| < 1/2 requirement does not greatlyeffect the cut-off radius.

Figure 5.6 shows the effects of the upper and lower limits of the velocity disper-sion spread we are using. Since the scale factor ε0 is the same for all the isothermalmodels, we expect the changes for the NIS and SIE models to be similar to those ofthe SIS. From figure 5.6 we see that the difference in the cross sections at z = 0 isaround a factor of 2, and it decreases with increasing lens redshift. Comparing thelower curve to the ε = 0.99 curve in figure 5.2 we see that the differences betweenthe SIS and SIE models are comparable to those resulting from the spread in thevelocity dispersion. We thus consider it justified in the remainder of this thesis innot treating the SIE model separately, and that the presence of ellipticity in lensesshould not introduce a larger change in the lensing probabilities than e.g. the spreadof velocity dispersions. It should be noted that the SIE model has two differentcaustics, as shown in §4.3.3, and so this applies only to the larger two-image crosssection. If we look at figure 5.7 we see that the four-image cross section is alwaysmuch lower, and thus differs significantly from the normal SIS cross section. How-ever, we assume that the number of images formed should not affect the overalldetectability of a lens and therefore do not treat the four-image cross section sepa-rately.

Turning to the NFW cross section we see from figure 5.1 that it behaves quite

38

5.1 Comparison of lensing cross sections

0.001

0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10

σ [k

pc2 ]

zL

zS = 1zS = 5

zS = 10

Figure 5.4: The SIS cross section as a function of lens redshift for different sourceredshifts. The lens is a giant with a velocity dispersion of 225 km/s.

differently from the isothermal models. It does not have a monotonically decliningcurve with increasing lens redshift but instead has a peak similar to the one in theangular diameter distance plot in figure 3.3. This is due to the evolution of theradial caustic with redshift. Unlike the isothermal models, in which the caustic iseither constant or a monotonic curve, the caustic of the NFW model has a peakand vanishes at small and large lens redshifts. This translates to a peak in the crosssection. The value of the NFW cross section is less than the corresponding valuein the isothermal case by one to two orders of magnitude. However comparing thesize of these two cross sections directly is difficult due to the difference in the waythe size of the lens is determined. For the isothermal models this is done throughthe velocity dispersion of the galactic dark matter, rather than the total mass ofthe dark matter halo as in the NFW model. Therefore it is difficult to tell whetherthis difference in cross sections is only due to the difference in the models or to adifference in the size of the modelled galaxies themselves.

Figure 5.8 shows the cross section for the dark halo mass range of giant lenses.We see that lowering the mass by an order of magnitude reduces the cross section bytwo orders of magnitude. In figure 5.9 we see how the NFW cross section changeswith source redshift. We see that the peak for a source at zS = 4 is higher thanin the zS = 10 case, similar to what happens for the isothermal models, althoughthe peak in the latter case is wider. For a source at zS = 1 we see that the crosssection is very small and does not increase in size at low redshift, as happens forthe isothermal models. Also the change from zS = 4 to zS = 1 is larger than from

39

5 Results

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0 1 2 3 4 5 6

σ [k

pc2 ]

zL

zS = 1zS = 5

zS = 10

Figure 5.5: The NIS cross section as a function of lens redshift for different sourceredshifts. The lens is a giant with a velocity dispersion of 225 km/s and a coreradius of 500 pc.

zS = 10 to zS = 4. Finally we see that for all three source redshifts the cut-off ofthe cross section happens before the lens reaches the source, and is similar to whatwe observed for the NIS model in figure 5.5.

Figures 5.10 and 5.11 show the cross section values of dwarf lenses for the SISand the NFW models respectively. We see that the difference between these twomodels is much larger than it was for giant lenses. This large a difference, around 8orders of magnitude, can not be solely due to the differences between the spreads ofthe velocity dispersions and the masses, and so must arise, in part, from differencesin the accuracy of these two models. From equation (4.25) we see that the size ofthe SIS cross section depends on σ4

v while for the NFW the size of the cross sectionsis controlled by the dark halo mass in a complex way. In Peñarrubia et al. [29] thevelocity dispersion of the dSph they study ranges from about 5 km/s to 10 km/s,while the dark halo mass range is two orders of magnitude. Comparing this to thevelocity dispersion range of giants, which have a velocity dispersion range of around30 km/s for a similar mass range, and we see that the SIS models seems to get moreinaccurate as the mass (and the velocity dispersion difference) decreases. Thereforeit seems that for low mass objects the SIS model breaks down and a more completetreatment in the form of the NFW profile is needed. Finally comparing the NFWcross section for giant and dwarf lenses we see that the peak in the dwarf case ismuch narrower. In fact almost no lensing occurs for a source at zS = 10 if the lensis at a redshift greater zL = 2.

40

5.1 Comparison of lensing cross sections

0.001

0.01

0.1

1

10

100

0 2 4 6 8 10

σ [k

pc2 ]

zL

σv = 237 km/sσv = 225 km/sσv = 205 km/s

Figure 5.6: The SIS cross section as a function of lens redshift for the limits of thevelocity dispersion range used for giant lenses in this thesis. The green curve isthe mean value of the giant lens velocity dispersion range. We take the source tobe at redshift 10.

Figure 5.12 shows the change in the NFW cross section with redshift for dwarflenses. We see that the behaviour is completely different from what we saw for thegiant lenses (e.g. figure 5.9). The cross section peak gets both narrower and lowerwith decreasing source redshift. This indicates that lowering the lensing mass doesnot simply scale the system down but in fact alters the behaviour of the lens signif-icantly.

For the NIS model we find, for ξc from 100 pc to 500 pc, that |xc| > 1/2 for allzL, i.e. no strong lensing by dwarfs occurs according to the NIS model. Thereforeif the core radii of our dwarf lenses matches those of dSph we should not expect tosee any multiple imaging.

41

5 Results

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0 2 4 6 8 10

σ [k

pc2 ]

zL

ε = 0.99, 2-imageε = 0.99, 4-image

ε = 0.5, 4-imageε = 0.1, 4-image

Figure 5.7: Comparison of the two-image and four-image cross sections of the SIEmodel. The four-image cross section is plotted for several values of the eccentricity,but the two-image cross section is only plotted for an eccentricity of 0.99. The lensis a giant with a velocity dispersion of 225 km/s.

1e-10

1e-08

1e-06

0.0001

0.01

1

0 2 4 6 8 10

σ [k

pc2 ]

zL

M = 1013 MSunM = 1012 MSun

Figure 5.8: The NFW cross section as a function of lens redshift for the limits ofthe dark halo mass range we use for giant lenses. The source is at redshift 10.

42

5.1 Comparison of lensing cross sections

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

0 1 2 3 4 5 6 7 8 9 10

σ[kp

c2 ]

zL

zS = 1zS = 4

zS = 10

Figure 5.9: The NFW cross section as a function of lens redshift for different sourceredshifts. The lens is a giant with a dark matter halo mass of 1013M.

0

5e-06

1e-05

1.5e-05

2e-05

2.5e-05

3e-05

0 1 2 3 4 5 6 7 8 9 10

σ [k

pc2 ]

zL

zS = 10, σv = 5 km/s

Figure 5.10: The SIS cross section as a function of lens redshift. The source is atredshift 10, and the lens is a dwarf with a velocity dispersion of 5 km/s. Theσ-axis is on a linear rather than logarithmic scale.

43

5 Results

0

2e-14

4e-14

6e-14

8e-14

1e-13

1.2e-13

1.4e-13

1.6e-13

1.8e-13

0 1 2 3 4 5 6 7 8 9 10

σ [k

pc2 ]

zL

zS = 10, M = 108 MSun

Figure 5.11: The NFW cross section as a function of lens redshift. The source isat redshift 10, and the lens is a dwarf of mass 108M.The σ-axis is on a linearrather than a logarithmic scale.

0

2e-14

4e-14

6e-14

8e-14

1e-13

1.2e-13

1.4e-13

1.6e-13

1.8e-13

0 0.5 1 1.5 2 2.5 3 3.5 4

σ [k

pc2 ]

zL

zS = 10zS = 6zS = 4

Figure 5.12: The NFW cross section as a function of lens redshift for different sourceredshifts. The lens is a dwarf with a dark matter halo mass of 108M. The σ-axisis on a linear rather than a logarithmic scale. The curve for zS = 10 is the sameas in figure 5.11

44

5.2 Lensing Probability

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 1 2 3 4 5 6 7 8 9 10

τ

zS

SISNFW

Figure 5.13: Line-of-sight probability for the SIS and NFW models as functions ofsource redshift. The lenses are giants with velocity dispersions of 225 km/s and adark matter halo mass of 1013M.

5.2 Lensing Probability

We begin by looking at the line-of-sight probability given by equation (4.6), i.e. theprobability of a given source being lensed once by an intervening object. We as-sume that all lensing objects with dark matter halo masses between 1012 to 1013 Mare large elliptical galaxies, due to the fact that most galaxy lenses are early typegalaxies, with late type galaxy lenses being rare [38]. Figure 5.13 shows the lensingprobability for the SIS and NFW models. We see from this figure that the proba-bility that a source will be lensed by a SIS object decreases at high redshifts. Thusthe probability for lensing is highest if the source has a redshift between 2 and 5.As we discussed in §5.1 the value of the lensing cross section depends on the sourceredshift. As such, if the source redshift is high, the cross section will be lower at lowlens redshifts than if the source is closer to the observer. This can be seen in figure5.4. Combining this with the redshift distribution of the density of lensing objectsshown in figure 4.1 we see that the high redshift sources will have a lower probabilityof being lensed because the lensing cross sections are lower at the redshifts wherethe lensing population is high.

Figure 5.14 shows the effect of the lens velocity dispersion on the lensing prob-ability. The difference in probability between the high and low dispersion limits isabout a factor of 2 at the peak. This is similar to the difference found for the cross

45

5 Results

1e-05

0.0001

0.001

0.01

0 1 2 3 4 5 6 7 8 9 10

τ

zS

σv = 237 km/sσv = 225 km/sσv = 205 km/s

Figure 5.14: Line-of-sight probability for the SIS model as a function of sourceredshift for the limits of the velocity dispersion range. The green curve is theaverage value. The lenses are giants and have a dark matter halo mass of 1012 −1013M.

sections in figure 5.6. This follows from the line-of-sight probability (equation (4.6)),since the differential probability depends linearly on the cross section.

In figure 5.15 we have plotted the line-of-sight probability for the NIS model atdifferent core radii. Its shape is similar to the SIS model, with the probability beinglower due to the lower value of the cross section. The inclusion of a constant densitycore in the lens thus decreases the lensing probabilities. The peak in the probabilitycurve shifts with changing core size, with the peak appearing at lower redshifts forlarger core radii. This is due to the shift in the NIS cross section with core radius,shown in figure 5.3. The cross section curves decrease faster for higher core radiiand terminate at a lower redshift resulting in a shifting of the entire shape of theprobability curve to lower source redshifts. We also see that the decrease in proba-bility at higher source redshift becomes steeper than for the SIS with increasing coreradius. This comes from the bound |xc| < 1/2, which causes a cut-off in the crosssection at higher lens redshifts as we saw in figure 5.5. Decreasing and increasingthe core radius then causes the slope at high source redshift to decrease or increase,respectively.

The lensing probability for NFW giant lenses is plotted in figure 5.16. Due todifferent cross section behaviour the probability distributions are quite different forthe NFW and isothermal models. While the NFW probability distribution also hasa peak, it is at a much higher redshift. The peak of the NFW cross section curves

46

5.2 Lensing Probability

1e-05

0.0001

0.001

0.01

0 1 2 3 4 5 6 7 8 9 10

τ

zS

rc = 0 pcrc = 100 pcrc = 500 pc

rc = 1000 pc

Figure 5.15: Line-of-sight probability for the NIS model as a function of sourceredshift for different core radii. The lenses are giants with velocity dispersions of225 km/s and a dark matter halo masses of 1012 − 1013M.

means that the highest cross sections are usually around the same lens redshiftsfor most source redshifts, as shown in figure 5.9. So unlike the isothermal modelsthe lensing cross section is always highest at the redshifts where the number den-sity is high. However, as can be seen from the same figure, the cross section peakgets wider at high source redshifts, which causes the probability to decrease at thehighest redshifts. For both masses the NFW probability is much lower than forthe isothermal cases, thus a more realistic lensing potential significantly lowers thelensing probability.

In figures 5.17 and 5.18 we plot the total probability of lensing by SIS and NFWgiant lenses using equation (4.7). We see that the low line-of-sight probabilitiestranslate to low total probabilities. As we would expect from the differences in theline-of-sight probabilities, the evolution of the total probabilities with redshift is notthe same in the two models. While both curves are increasing with redshift, sincethe increase in the number of lenses at high redshift compensates for the loweredline-of-sight probability, the shapes differ noticeably. The slope of the curve for theSIS model changes, decreasing at higher redshifts. Meanwhile the slope of the NFWremains almost constant with redshift, indicating that the change in line-of-sightprobabilities follows precisely the increase in the number of lenses. Most notablythough, we see that the line-of-sight probabilities and total probabilities are of thesame order of magnitude. Thus for the purpose of comparing the relative probabilityvalues either distribution will do.

47

5 Results

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0 1 2 3 4 5 6 7 8 9 10

τ

zS

M = 1013 MSunM = 1012 MSun

Figure 5.16: Line-of-sight probability for the NFW model as a function of sourceredshift for the dark halo mass range of giant lenses.

Figure 5.19 show the SIS and NFW line-of-sight probabilities for dwarf lenseswith a mass of 108 M and a dark matter velocity dispersion of 5 km/s. While theshape of the SIS probability curve is similar to that of a giant lens, the same isnot true of the NFW curve. It has a monotonic rise and no peak is present. Asfigure 5.11 shows this is due to the fact that for low masses the NFW cross sectionat high source redshift has a very narrow peak. For this narrow redshift range thenumber density is essentially constant and so no peak is formed in the probabilitydistribution.

The probability values for the SIS dwarf lenses are about two orders of magnitudeslower than for the SIS giant lenses but around seven to nine orders of magnitudeslower in the NFW case. This large difference in model predictions follows the onefor the cross sections. From the NFW results we can state that the probability ofdwarf lenses producing a strong lensing effect is extremely rare, that is, for everybillion giant lenses we should on average see one dwarf lens. So even though thesesmaller objects are far more numerous than the larger ones, the decrease in thestrong lensing cross section means that they are overall much less likely to act aslenses.

48

5.2 Lensing Probability

0

0.001

0.002

0.003

0.004

0.005

0 1 2 3 4 5 6 7 8 9 10

P

zS

σv = 225 km/s

Figure 5.17: Total probability for the SIS model as a function of source redshift.The lenses are giants with velocity dispersions of 225 km/s and dark matter halomasses of 1012 − 1013M.

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0 1 2 3 4 5 6 7 8 9 10

P

zS

M = 1013 MSun

Figure 5.18: Total probability for the NFW model as a function of source redshift.The lenses are giants with dark matter halo masses of M = 1013M.

49

5 Results

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0 1 2 3 4 5 6 7 8 9 10

τ

zS

SISNFW

Figure 5.19: The line-of-sight probability for the SIS and NFW models as a functionof source redshifts. The lenses are dwarfs with masses of 108M and velocitydispersions of 5 km/s. The noise in the curves are due to inaccuracies in ournumerical calculations.

50

5.3 Conditional probabilities

5.3 Conditional probabilities

In figures 5.20 and 5.21, we see the cumulative conditional probabilities of time de-lays for giant lenses calculated using equations (4.6) and (4.8) (see the discussions in§3.3.3 and at the end of §4.1, §4.3, §4.3.1 and §4.3.4). We use the line-of-sight prob-ability rather than the total probability, since the difference between these two withrespect to the cumulative conditional probability is minimal. Since these are cumu-lative probabilities, a time delay of ∆t having a cumulative probability τ means thatthe chance that a lens will have a time delay greater than ∆t is τ (equation (4.8)).For the SIS giant lenses we see that the time delays are of the order of 100 days,while the NFW delays are an order of magnitude lower. From Paraficz et al. [28]we see that this distribution is close to observed time-delay distribution, with bothbeing of the order of days. We thus feel confident in stating that our calculationsshould, at least, reflect the scale of the expected time delays.

Looking at the results for dwarf lenses, shown in figures 5.22 and 5.23 we see thatthe time delay here is much shorter. For the SIS model the time delay is foundto be of the order of a second, and for the NFW model it is of the order of tensof nanoseconds. We would thus need a rapidly variable source in order to confirmlensing in the case of dwarf lenses.

One possible source to use to detect such small time delays are gamma ray bursts(GRBs). Their prompt emission can last from tens of microseconds to tens of min-utes [11] and have been observed to be variable down to a time scale of a fewmilliseconds [23]. While this is sufficient to detect effects from lensing if the SISmodel is accurate, it falls short of the nanosecond variability needed to confirm lens-ing if the NFW model is correct. Current detectors, on satellites like Fermi andSwift, only have a time resolution of ≈ 100 ms indicating that a time delay of ≈ 100ns (a typical value shown in figure 5.23) will remain undetected in the near future.

51

5 Results

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250

τ(>

∆t)

∆t [days]

zS = 5, σv = 225 km/s

Figure 5.20: The cumulative conditional probability distribution of the SIS model fortime delays. The lenses are giants with velocity dispersions of 225 km/s and thesource is at redshift 5.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

τ(>

∆t)

∆t [days]

zS = 5, M = 1013 MSun

Figure 5.21: The cumulative conditional probability distribution of the NFW modelfor time delays. The lenses are giants with dark matter halo masses of 1013Mand the source is at redshift 5.

52

5.3 Conditional probabilities

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5

τ(>

∆t)

∆t [s]

zS = 5, σv = 5 km/s

Figure 5.22: The cumulative conditional probability distribution of the SIS model fortime delays. The lenses are dwarfs with velocity dispersions of 5 km/s and thesource is at redshift 5.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250

τ(>

∆t)

∆t [ns]

zS = 5, M = 108 MSun

Figure 5.23: The cumulative conditional probability distribution of the NFW modelfor time delays. The lenses are dwarfs with masses of 108M and the source isat redshift 5.

53

6 Conclusions

In this thesis we have used a probabilistic approach to determine the feasibility ofusing strong gravitational lensing to detect small dark matter structures (or dwarflenses). These lenses, comparable in size to local dwarf spheroidals (dSph), areaccording to numerical models quite numerous, especially at early times. We calcu-lated the lensing cross sections of these dwarfs lenses, along with the lensing crosssections of a more common lensing object, a large elliptical galaxy (or giant lens),using three isothermal density profiles. These profiles are the singular isothermalsphere (SIS), the non-singular isothermal sphere (NIS) and the singular isothermalellipsoid (SIE). We also used the more realistic Navarro-Frenk-White (NFW) den-sity profile of dark matter halo masses. We found that the cross sections calculatedusing the SIS and SIE models were similar, while the NIS profile predicted almostno strong lensing for dwarf lenses like dSph. Thus if these dwarf lenses have an innercore of constant density comparable to those measured for dSph the NIS profile pre-dicts they will not produce strong lensing effects. Then using a Warren numericaldensity distribution of dark matter halos, we calculated the line-of-sight and totalprobabilities for both giant and dwarf lenses using the SIS and NFW profiles. Theprobability values for the dwarf lenses are very different for the two models, indicat-ing that the simpler SIS approximation breaks down at low masses. Comparing theprobabilities for lensing by giant and dwarf lenses we found that if the NFW modelis correct one can only expect around one lensing event by a dwarf lens for everybillion such events by giant lenses. We also looked at the distribution of time delaysfor dwarf lenses and found that they should be of the order of seconds in the SIScase, and of the order of tens of nanoseconds in the NFW case. This again shows avery large difference between the results of the two models. From this we concludethat if the NFW model is accurate, dwarfs are unlikely to be detected in the nearfuture by means of their gravitational lensing effects.

55

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