Gravitational Lensing Statistics
Kjartan Marteinsson
Faculty
Faculty of
of Physical
Physical Sciences
Sciences
University
University of
of Iceland
Iceland
2012
2012
GRAVITATIONAL LENSING STATISTICS
Kjartan Marteinsson
60 ECTS thesis submitted in partial fulfillment of a
Magister Scientiarum degree in Astrophysics
Advisors
Einar H. Guðmundsson
Páll Jakobsson
Gunnlaugur Björnsson
Faculty Representative
Einar H. Guðmundsson
M.Sc. committee
Einar H. Guðmundsson
Páll Jakobsson
Gunnlaugur Björnsson
Faculty of Physical Sciences
School of Engineering and Natural Sciences
University of Iceland
Reykjavik, June 2012
Gravitational Lensing Statistics
60 ECTS thesis submitted in partial fulfillment of a M.Sc. degree in Astrophysics
c 2012 Kjartan Marteinsson
Copyright All rights reserved
Faculty of Physical Sciences
School of Engineering and Natural Sciences
University of Iceland
Hjarðarhagi 2-6
107 Reykjavik, Reykjavik
Iceland
Telephone: 525 4000
Bibliographic information:
Kjartan Marteinsson, 2012, Gravitational Lensing Statistics, M.Sc. thesis, Faculty of
Physical Sciences, University of Iceland.
Printing: Háskólaprent, Fálkagata 2, 107 Reykjavík
Reykjavik, Iceland, June 2012
Fyrir pabba
Abstract
Models of the formation and growth of the large-scale structure of the universe predict that a large fraction of the cosmic matter resides in small dark matter halo
structures of mass < 108 M . Confirming the presence of such dwarf structures
would further strengthen current cosmological models. However, these structures
have low luminosity or are non luminous and cannot be detected by standard observational methods. Thus an indirect method of detection is required. In this
thesis we have used gravitational lensing statistics to investigate whether the dwarf
structures could be discovered through their gravitational lensing effects and thus be
detectable by their influence on the light from background sources. Using models of
the density distribution of dark matter halo structures as well as the time evolution
of their number density we compare the lensing effects of the dwarf structures to
that of a common type of well known, and much larger, lensing objects. We find
that dwarf structures very rarely act as strong lenses. Also, the lensing effects of
the dwarfs on source spectra will most likely be undetectable by current detectors.
Úrdráttur
Samkvæmt fræðilegum líkönum um uppruna og þróun stórgerðar alheimsins er mikill
hluti alls efnis bundinn í dvergvöxnum hulduefnishjúpum með massa < 108 M . Ef
unnt væri að staðfesta tilvist slíkra dverghjúpa myndi það renna styrkari stoðum
undir hina viðteknu heimsmynd nútímans. Ljósið sem dverghjúpar senda frá sér
er hinsvegar lítið sem ekkert og þar af leiðandi er illmögulegt að finna þá með
hefðbundum mælingum. Því er nauðsynlegt að notast við óbeinar mæliaðferðir.
Í þessarri ritgerð eru líkindi á þyngdarlinsuhrifum notuð til þess að kanna hvort
hægt sé að greina tilvist dverghjúpa út frá þyngdarlinsuhrifum þeirra á fjarlægari
ljósuppsprettur. Með því að nota líkön fyrir massadreifingu hulduefnishjúpa og fjölda
þeirra sem fall af tíma getum við borið linsuhrif dverghjúpa saman við tilsvarandi
hrif stórra vetrarbrauta. Niðurstaðan er sú að dverghjúparnir sýna mjög sjaldan
sterk þyngdarlinsuhrif. Einnig er sennilegt að linsuhrif dverghjúpa á róf fjarlægari
uppspretta 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
3.1 The thin lens . . . . . . . . .
3.1.1 Outside the lens plane
3.1.2 The lens plane . . . . .
3.2 Angular diameter distance . .
3.3 General Lensing . . . . . . . .
3.3.1 Coordinate systems . .
3.3.2 Scalar potential . . . .
3.3.3 Time delays . . . . . .
3.3.4 Magnification . . . . .
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23
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5 Results
5.1 Comparison of lensing cross sections . . . . . . . . . . . . . . . . . . .
5.2 Lensing Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Conditional probabilities . . . . . . . . . . . . . . . . . . . . . . . . .
35
35
45
51
6 Conclusions
55
Bibliography
57
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4 Gravitational lensing statistics
4.1 General theory . . . . . . . . . . . . . . . .
4.2 Number densities . . . . . . . . . . . . . . .
4.3 Cross sections . . . . . . . . . . . . . . . . .
4.3.1 The Singular Isothermal Sphere . . .
4.3.2 The Non-Singular Isothermal Sphere
4.3.3 The Singular Isothermal Ellipsoid . .
4.3.4 The Navarro-Frenk-White profile . .
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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 of
redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 lens
redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2
The SIE cross section as a function of lens redshift for different eccentricities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3
The NIS cross section as a function of lens redshift for different core
radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.4
The SIS cross section as a function of lens redshift for different source
redshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.5
The NIS cross section as a function of lens redshift for different source
redshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.6
The SIS cross section as a function of lens redshift for the limits of
the velocity dispersion range . . . . . . . . . . . . . . . . . . . . . . . 41
5.7
Comparison of the two-image and four-image cross sections of the SIE
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ix
LIST OF FIGURES
5.8
The NFW cross section as a function of lens redshift for the limits of
the dark halo mass range of giants . . . . . . . . . . . . . . . . . . . . 42
5.9
The NFW cross section as a function of lens redshift for different
source 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 different
source redshifts for dwarfs . . . . . . . . . . . . . . . . . . . . . . . . 44
5.13 Line-of-sight probability for the SIS and NFW models as functions of
source redshift for giant lenses . . . . . . . . . . . . . . . . . . . . . . 45
5.14 Line-of-sight probability for the SIS model as a function of source
redshift for the limits of the velocity dispersion range . . . . . . . . . 46
5.15 Line-of-sight probability for the NIS model as a function of source
redshift for different core radii . . . . . . . . . . . . . . . . . . . . . . 47
5.16 Line-of-sight probability for the NFW model as a function of source
redshift for the dark halo mass range of giant lenses . . . . . . . . . . 48
5.17 Total probability for the SIS model as a function of source redshift
for giant lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.18 Total probability for the NFW model as a function of source redshift
for giant lenses with 1013 M mass halos . . . . . . . . . . . . . . . . 49
5.19 The line-of-sight probability for the SIS and NFW models as a function of source redshift for dwarf lenses . . . . . . . . . . . . . . . . . 50
5.20 The cumulative conditional probability distribution of the SIS model
for time delays in the case of giant lenses . . . . . . . . . . . . . . . . 52
5.21 The cumulative conditional probability distribution of the NFW model
for time delays in the case of giant lenses . . . . . . . . . . . . . . . . 52
5.22 The cumulative conditional probability distribution of the SIS model
for time delays in the case of dwarf lenses . . . . . . . . . . . . . . . . 53
x
LIST OF FIGURES
5.23 The cumulative conditional probability distribution of the NFW model
for 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 their
guidance, help and support. I would also like to thank Nial Tanvir for his insights
and comments and Zarija Lukić for data he provided. I wish to extend my gratitude
to Annalisa, Bob, Steve along with Gulli Jr. and Paul for always being willing to
listen and lend a helping hand to a confused master student. Thanks to Árdís and
Jens for useful discussions at the early stages of this work. Finally I would like to
thank my family and friends for their support and their efforts in keeping me (more
or less) sane.
1
1 Introduction
The most common type of matter in the universe is dark matter, named such because
it does not seem to emit, or even interact, via the electromagnetic force. Dark matter
makes up most of the mass of galaxies, including our own Milky Way, and forms
so-called dark matter halos. Numerical studies of large scale structure growth in
the universe have shown that such halos have a wide range of masses, from massive
ones associated with galaxies and clusters to the halos of dwarf galaxies and smaller
structures. These same studies have shown that such small dark matter structures
are found at all redshifts and make up a sizeable fraction of the total cosmological
population of massive objects. They have however low luminosity or are even nonluminous and so would not be detectable by traditional methods. We must therefore
turn to an indirect detection technique to confirm their presence.
One of the predictions of Einstein’s theory of general relativity is that mass affects
the space surrounding it, causing massive objects to curve their local spacetime. This
means 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 focus
light from background sources, making massive objects behave like cosmological
magnifying glasses. This is known as gravitational lensing and can be detected due
to 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 structures
by looking for their effects on background sources.
To investigate whether gravitational lensing could be of use in detecting these
small structures we turn to gravitational lensing statistics. By using models of
the number of lenses and their density profiles we can predict the probability of a
detectable lensing event and the likely properties of such events. Similar methods
have been used successfully for a long time to predict the number of lenses, even
before 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 quickly
review its history. For a more comprehensive overview of gravitational lensing history 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 as
black holes, he looked at the deflection of light by celestial bodies using the gravitational theories of Newton [39]. Treating a light ray as a stream of particles he found
that a light ray passing a spherically symmetric object of mass M with an impact
parameter r will be deflected with an angle of deflection,
α̂ =
2GM
,
c2 r
(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 principle
of equivalence, and the assumption of an Euclidean metric, to investigate this phenomenon. While unaware of the previous work done on this subject, he re-derived
Soldner’s results. In a paper [6] detailing his calculations, Einstein also expressed
his wish that astronomers tried to test his predictions. Indeed at least two attempts
were made. The first was during an eclipse in Brazil in 1912, but it failed due to
unfavourable weather conditions. Another was planned for an eclipse in the Crimean
in 1914 by a German expedition. However due to the outbreak of the first world
war they were unable to perform their observations [30]. Einstein’s value for the
expected deflection of starlight by the sun therefore stood untested until he finished
his new theory of gravitation. With this completed theory he correctly calculated
the deflection angle as being twice that of the Newtonian value [7], i.e.
α̂ =
4GM
.
c2 r
(2.2)
For the Sun this translates into a deflection of roughly 1.7" for a light-ray grazing
its 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 much
more [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 therefore
turned out to be very fortuitous for Einstein and his credibility.
In the following decade more work was done on this new phenomenon. First
Eddington [5] and later Chwolson [4] considered a more general lensing system, that
of a foreground star lensing a more distant background star. They found that for
a sufficiently well aligned lensing system, more than one image of the source would
appear. In fact, Chwolson also posited that for a perfect alignment the images would
form a ring-like structure. Such images are called Einstein rings. One might wonder
why, if Chwolson was the first to consider these structures, they had been named
after 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 rudimentary
lensing system, i.e. the existence of a double image, the lens equation and the fact
that the images would be magnified [36]. But he did not publish these results until
1936 [8], and then only due to the request of a Czech amateur scientist named Rudi
W. Mandl. Mandl had asked Einstein to consider, in greater detail, the effect of
lensing by a star. In response Einstein redid his calculations from 1912, but he was
not very impressed with the value of this work, stating in the paper that "there is no
great chance of observing this phenomenon". More telling, he wrote a letter to the
editor of the journal, saying "Let me also thank you for your cooperation with the
little publication, which Mister Mandl squeezed out of me. It is of little value, but
it makes the poor guy happy". While it is true that history would prove Einstein
wrong, he was not unjustified in his statements. With the technology available
at the time the observation of this effect on distant stars would indeed have been
impossible.
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. In
his calculations he used the virial theorem to estimate the masses of the Virgo and
Coma clusters. He then used these estimates to find the probability of lensing. His
conclusion was that "Present estimates of masses and diameters of cluster nebulae
are such that the observability of gravitational lens effects among the nebulae would
seem insured" [46]. While his estimates of the deflection angles were too large,
Zwicky’s prediction nevertheless came true. As well as correctly predicting the
discovery of galaxy lenses, he also stated other things in his paper that later proved
to be accurate, i.e. that sources that would normally be too faint to detect would
be visible by gravitational lensing due to magnification and that this could be used
to constrain their masses. Also, Zwicky, like Einstein, was indirectly motivated to
investigate this possibility because of Mandl. Mandl had contacted others besides
Einstein about his ideas, one of whom had forwarded them to Zwicky [37]. It might
therefore be prudent to name Mandl among the founders of gravitational lensing
theory.
After Zwicky’s papers in 1937 the subject of gravitational lensing died out, with
almost no new ideas or papers being put forward for over two decades. It was not
until the 1960s, that interest in the field began again. Several people published
work 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 for
certain sources, such as supernovae, this could be measured [32, 33]. Since this time
delay would be inversely proportional to the Hubble constant H0 , he suggested that
gravitational lensing could be used to measure it. He went even further, stating
that gravitational lensing could be applied in general to test different cosmological
theories [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 actual
lensing system still eluded observers. That changed in 1979 with the discovery of
the quasars 0957+561 A,B by Walsh, Carswell and Weymann [42]. They found
that two quasars, with an angular separation of about 6 arcseconds, shared the
same redshift. This, along with the similarity of the spectra and the detection of
a foreground, or lensing, galaxy lead them to the conclusion that these were two
images of the same quasar. This opened the floodgates on gravitational lensing
detections, with many more multiply imaged quasar systems being found in the
years that followed. Many different lensing systems have now been discovered, like
giant luminous arcs, which are highly deformed images of galaxies at high redshift
and Einstein rings, the ring like images formed when the source and the lens are
perfectly aligned. Even microlensing, the star-star lensing systems that Einstein
had dismissed, has been used to detect exoplanets and probe for candidates of dark
matter.
7
3 Basics of gravitational lensing
In this chapter we review some of the basics of gravitational lensing. The set-up
here 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. A
schematic of this type of lensing system is shown in figure 3.1. A light ray from a
source (e.g. a star or a galaxy) at point S passes a point mass M at the minimum
distance ξ and is deflected. This angle of deflection is, as stated earlier, given by
α̂ =
4GM
.
c2 ξ
(3.1)
This perturbed light ray then travels to the observer at point O, who sees the light
as coming from an image I at an angle θ, instead of the actual source at an angle
β. 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 assuming
simple two dimensional Euclidean geometry, one can easily get the relation
θDS = βDS + α̂DLS .
(3.2)
By introducing a reduced deflection angle α by
α=
DLS
α̂,
DS
(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 of
gravitational lensing theory.
Using relation (3.4), the position of the source can be found from the position
of the image(s) and the angle of deflection. Alternatively, specifying the deflecting
mass, 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
DLS
4GM
θE2
β =θ−
=
θ
−
,
(3.5)
DS DL
c2 θ
θ
9
3 Basics of gravitational lensing
Figure 3.1: Model of a gravitational lensing system. A source at S emits a lightray that gets deflected due to a point mass M , with the deflection happening a
distance ξ from the point mass. Due to this the position of the source as seen by
the observer at O is changed. Figure modified from [24].
where we have used the fact that ξ = DL θ, as can be seen from figure 3.1. Here we
have also defined the angle
r
DLS 4GM
,
(3.6)
θE =
DS DL c2
which is called the Einstein radius, and has a fundamental connection to the point
lens system, as we will see later. Solving equation (3.5) for the image positions θ we
get two solutions, θ+ and θ− given by
q
1
2
2
β ± β + 4θE .
(3.7)
θ± =
2
From this we see that two images are formed, with an angular separation of
q
∆θ = β 2 + 4θE2 .
(3.8)
These images appear on opposite sides of the point mass M , with one image being
inside 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 can
be simplified by using the thin-lens approximation and angular diameter distances,
which we will detail in the following sections. Using these, the lens equation retains
the 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 the
reduced 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 observer 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 the
lens. This plane, which we call the lens plane, is perpendicular to the line of
sight. The rest of the universe is assumed to be described by the standard Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological model. We now treat
both of these regions in turn.
3.1.1 Outside the lens plane
The FLRW metric is given by
where
2
dχ =
ds2 = −c2 dt2 + a(t)2 dχ2 ,
(3.10)
dr2
2
2
2
2
2
+ r dθ + r sin(θ) dφ .
1 − kr2
(3.11)
Here k is the measure of curvature of the universe and a(t) is known as the scale
factor. 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 distance
between two objects at different times. It is given by the Friedmann equations:
2
8πG
kc2
ȧ(t)
2
H (t) =
=
ρ(t) −
,
(3.12)
a(t)
3
a(t)2
ä(t)
4πG
3p(t)
2
Ḣ(t) + H (t) =
.
(3.13)
=−
ρ(t) + 2
a(t)
3
c
11
3 Basics of gravitational lensing
Here we have introduced several new variables. The first of these, the Hubble
parameter H(t), is simply the expansion rate of the universe. Its value at current
time, 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 the
universe. These two terms are made up of contributions of each of the components
that constitute our universe. Of most importance to our studies are the energymatter densities of matter, both dark and normal, and dark energy. It is useful to
define 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 the
component 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 dark
energy, baryonic matter and dark matter, respectively. The critical density is given
by
3H(t)2
ρc (t) =
.
(3.15)
8πG
We can use these density parameters and equation (3.12) to find a simple expression
for the Hubble parameter
p
(3.16)
H(z) = H0 ΩΛ + k(1 + z)2 + ΩM (1 + z)3 ≡ H0 E(z),
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. The
values we use in this work for the cosmological parameters come from the seven year
WMAP results [14]:
ΩB
ΩDM
ΩΛ
Ωtot
H0
= 0.0456 ± 0.0016,
= 0.227 ± 0.014,
=
0.728+0.015
−0.016 ,
=
1.0023+0.0056
−0.0054 ,
=
70.4+1.3
−1.4
km s−1 Mpc−1 .
Since the total density is so close to the critical value, Ωtot = ΩΛ + ΩM = 1, we set
k = 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 metric
perturbed by the gravitational potential Φ of the lens:
2Φ
2Φ 2 2
2
(3.18)
ds = − 1 + 2 c dt + 1 − 2 (dx2 + dy 2 + dz 2 ).
c
c
This approach is valid if |Φ| c2 and the velocity of the components of the lens is
small, i.e. v c. This can be described as the metric being weak and static, and
holds for all lenses of interest in this thesis. For a metric of this type it can be shown
that 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 law
we can get the angle of deflection by
Z
Z
2
(3.20)
α̂ = kin − kout = ∇⊥ n dl = 2 ∇⊥ Φ dl,
c
where k is the wave vector and ∇⊥ is the projection of the gradient operator onto
a plane orthogonal to the wave vector k. The use of equation (3.20) is simplified
somewhat by the fact that since the deflection is very small we can safely integrate
with 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
GM
,
(3.21)
Φ(ξ) = −
|ξ|
where ξ is the impact vector orthogonal to the unperturbed wave vector kin , as
defined in figure 3.1. Putting this in equation (3.20) we get
α̂ =
4GM ξ
.
c2 |ξ|2
(3.22)
This can be simplified to Einsteins result as given by (3.1). For a more general lens
described as a thin mass sheet we then simply sum over all the mass elements of the
sheet to get the deflection angle
Z
4G (ξ − ξ 0 )Σ(ξ 0 ) 2 0
α̂(ξ) = 2
dξ.
(3.23)
c
|ξ − ξ 0 |2
where Σ(ξ) is the surface mass density, found by projecting the density ρ(ξ, z) of
the lens onto the lens plane:
Z
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 from
equation (3.23) that the reduced deflection angle is
α(θ) =
4πGΣ DLS DL
θ,
c2
DS
(3.25)
where we have taken θ = DL ξ. We can see that it is possible to define a surface
density so that the reduced deflection angle is α(θ) = θ. From the lens equation we
see that this means β = 0 for all image positions. The constant surface density for
such a lens is known as the critical surface mass density
Σc =
DS
c2
.
4πG DL DLS
(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 introduce 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 FLRW
metric. In order to maintain the simplicity of the lens equation we use a special
distance measure known as the angular diameter distance. We follow the review by
Hogg [13] in our derivation.
As can be seen in figure 3.2 this angular diameter distance is defined as the ratio
between the objects transverse and angular sizes, DA = Sθ , which fits the requirement needed to derive equation (3.9).
To calculate angular diameter distances we must first discuss co-moving distances. A co-moving distance is defined as the distance between two objects that
remains constant with respect to the expansion of the universe. Since the Hubble parameter H(z) is a measure of this expansion we can define a co-moving line-of-sight
distance by
Z z
Z z
c
dz 0
cdz 0
DC =
=
,
(3.28)
0
H0 0 E(z)
0 H(z )
where E(z) is defined in equation (3.16). For a flat universe the co-moving distance
between two objects at the same redshift but separated on the sky by an angle θ
can then be shown to be DC θ. The actual transverse size is then
S=
14
DC θ
,
1+z
(3.29)
3.2 Angular diameter distance
Figure 3.2: The angular diameter distance of an object DA is defined as the ratio
between 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 the
observer (us) to the lens and the source.
Figure 3.3 shows how the angular diameter distance varies with redshift. It is
clear from the figure that this distance measure is very different from the well known
Euclidean distance. The reason is that it depends on the angular size as well as the
physical transverse size. Because of this two objects at different redshifts can have
the same angular diameter distance.
Equation (3.30) can only be used to calculate the angular diameter distances from
us to another object. The lens equation, however, also requires the distance between
the 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
(DS (1 + zS ) − DL (1 + zL )) ,
1 + zS
(3.31)
where zS is the redshift of the source and zL that of the lens. We have plotted this
distance in figure 3.5. For a constant zL the general behaviour of DLS resembles
that of a normal angular diameter distance, as in figure 3.4, except with a shifted
origin. This seems obvious since we are simply looking at the same thing as we did
earlier for an observer stationed at the lens. For a constant zS the distance decreases
monotonically with lens redshift.
15
3 Basics of gravitational lensing
2000
DA [Mpc]
1500
1000
500
0
0
2
4
6
8
10
z
Figure 3.3: Angular diameter distance as a function of redshift.
DA [Mpc]
1000
100
10
1
0
1
2
3
4
5
z
6
7
8
9
10
Figure 3.4: Angular diameter distance as a function of redshift on a logarithmic
scale.
16
3.2 Angular diameter distance
zS = 10
zL = 1
1000
DLS [Mpc]
100
10
1
0
1
2
3
4
5
z
6
7
8
9
10
Figure 3.5: The angular diameter distance DLS between the source and lens as a
function of redshift. The red curve is for a fixed source at redshift 10. The green
curve 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 to
clarify the coordinate systems we use. The coordinates we have used so far are the
angular positions of the source and images, β and θ, along with the impact vector
in the lens plane, ξ. Alternatively we can introduce a vector in the plane of the
source, η, which specifies the location of the source there. The angles and position
vectors are related in a simple way by
ξ = DL θ η = DS β,
(3.32)
where DS and DL are the angular diameter distances to the source and lens respectively. It is also useful to introduce dimensionless coordinates x and y by
ξ
,
ξ0
ηDL
,
y=
ξ0 DS
x=
(3.33)
(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 scaled
projection of the Newtonian potential, Φ, of the lens onto the lens plane
Z
DLS 2
ψ(θ) ≡
Φ(ξ, z) dz.
(3.35)
D L D S c2
This has some interesting properties, such as
DLS 2
∇θ ψ = DL ∇ξ ψ =
DS c2
Z
∇⊥ Φ dz = α,
(3.36)
where we have used equations (3.20) and (3.3). The Laplacian of ψ is then
Z
2 DL DLS
Σ(θ)
2
2
2
∇θ ψ = DL ∇ξ ψ = 2
∇2ξ Φ dz = 2
= 2κ(θ).
(3.37)
c DS
Σc
Here we have used the definition of Σc given by (3.26) and Poisson’s equation for a
Newtonian potential, i.e. ∇2 Φ = 4πGρ.
18
3.3 General Lensing
3.3.3 Time delays
Using the deflection potential ψ from the last section we can restate the lensing
equation (3.9) using relation (3.36):
1
2
(θ − β) − ψ(θ) = 0.
(3.38)
∇θ
2
If we look at this in the context of Fermat’s principle the function inside the gradient
is related to the travel time by [37]
(1 + zL )DL DS 1
2
t(θ, β) =
(θ − β) − ψ(θ) + constant .
(3.39)
cDLS
2
Since this function is indicative of the extra time it takes a light-ray to travel along
the deflected path it is most commonly referred to as the time-delay function. The
first term, 21 (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 deflected path. The second, ψ(θ), is the due to the Shapiro effect, that light travelling
through 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, the
presence of which means that the time delay for a single image cannot be found
directly. Instead we look at the difference in time delays between two images
∆t(θ 1 , θ 2 ) = t(θ 1 , β) − t(θ 2 , β).
(3.40)
In all later discussions of time delays in gravitational lensing systems we always have
this definition in mind. As a side note, since the only dimensional factor present
in this equation is the Hubble constant H0 (from the expression for the angular
diameter distances (3.28) and (3.30)) it can be, and has been, used to constrain its
value.
3.3.4 Magnification
An important property of gravitational lensing is the fact that while the deflection
of the lens can change the apparent solid angle of the source, the surface brightness
of 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 the
source and the image
δθ 2
.
(3.41)
µ=
δβ 2
This magnification factor can be either positive or negative, with the sign representing 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 magN
P
nification of all N images as µtot =
|µi |.
i=0
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 to
the lens plane with
δβ 2 = |J| δθ 2 ,
(3.42)
where |J| is the Jacobian. This relation is applicable if the solid angles are small
enough, which is the case in gravitational lensing. The magnification is then the
inverse of the Jacobian
1
.
(3.43)
µ=
|J|
The Jacobian matrix is given by
∂β
∂αi (θ)
∂ 2 ψ(θ)
J=
= δij −
= δij −
= δij − ψij ,
∂θ
∂θj
∂θi ∂θj
(3.44)
where we have used equations (3.9) and (3.36). From equation (3.37) we see that
the convergence can be written as
κ=
1
(ψ11 + ψ22 ) .
2
(3.45)
We can also introduce a new quantity known as the shear
1
(ψ11 − ψ22 ) = γ cos(2φ)
2
γ2 = ψ12 = ψ21 = γ sin(2φ).
γ1 =
(3.46)
(3.47)
where
q
γ12 + γ22
γ2
.
φ = arctan
γ1
γ=
(3.48)
(3.49)
Using the convergence and the shear the Jacobian matrix can then be simplified as
1 − κ − γ1
−γ2
1 0
cos(2φ) sin(2φ)
J=
= (1−κ)
−γ
, (3.50)
−γ2
1 − κ + γ1
0 1
sin(2φ) − cos(2φ)
and the magnification becomes
µ=
1
.
(1 − κ)2 − γ 2
(3.51)
We see from this that the convergence causes an isotropic focusing of the light rays
of 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 γ
describing the magnitude and φ the orientation.
The values θ that give |J| = 0 form a critical curve in the lens plane. An example
of this is the Einstein radius. This can be mapped, using the lens equation, to
another curve in the source plane, known as a caustic. It can be shown that each
time a source crosses a caustic, two new images are formed [37] . This means that in
order for multiple images of the same source to be produced the source must cross at
least one caustic, and that there is always an odd number of images formed (though
this 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 calculate the gravitational lensing probabilities is based on the work done by Turner et
al. [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 probability
of a single lensing event occurring on a part dR of this line for a light-ray from the
source is
a(t)2 dχ2
dR
=
,
(4.1)
dτ =
l
l
where l is the proper mean free path of the ray with respect to lensing. We also
have a(t) and dχ which are the scale factor and space part of the FLRW metric
respectively (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 the
number density of lenses present, ñ(zL , M ), and the cross section of lensing for each
of them, σ(zL , zS , M ), as l = [ñ(zL , M )σ(zL , zS , M )]−1 . These cross sections depend
on the source, as well as the lens, position. This is due to the fact that the cross
sectional area must be measured in the source plane, not in the lens plane. Using
the definition of l and the FLRW metric (3.10) for light-rays, ds = 0, we can write
the optical depth as
dτ = ñ(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 their
redshifts. Part of the redshift dependence is due to the fact that the universe is
expanding, i.e. that the average distances between lensing objects increases with
redshift. It is therefore simpler to use the co-moving number density instead, defined
as
ñ(zL , M )
n(zL , M ) =
.
(4.3)
(1 + zL )3
The redshift dependence of n is then only due to the evolution of the lens population
with time.
23
4 Gravitational lensing statistics
Using the definition of redshift given by equation (3.17) and the Friedmann equation (3.12) we get
cdt
c
1
=
.
(4.4)
dzL
1 + zL H(zL )
The equation for the optical depth thus becomes
c
dτ =
n(zL , M )σ(zL , zS , M )(1 + zL )2 dzL .
(4.5)
H(zL )
By integrating this equation over the whole line of sight we get the probability for
lensing of a given source at redshift zS by a population of lenses with a given mass
M (or a mass range ∆M ):
Z zL
n(zL , M )σ(zL , zS , M )(1 + zL )2
c
dzL .
(4.6)
τ (zS , M ) =
H0 0
E(z)
It is possible to modify this equation to consider the probability of lensing for all
sources at redshift zS [37]. To do this we simply consider all the lenses in a shell
at each zL and the cross sections due to these lenses in the source plane. Then
comparing the area covered with cross sections at the source plane to the total area
of the source plane, a shell at redshift zS , we get the total lensing probability
Z zL
n(zL , M )σ(zL , zS , M )DL2 (1 + zL )2
c 1
dzL .
(4.7)
P (zS , M ) =
H0 DS2 0
E(z)
This equation assumes that no two cross sections overlap, and is only applicable if
P 1. We can go further, and look at conditional probabilities τ (A | B), i.e. the
probability that an event B will have the property A [27]. To do this we simply
reduce the cross section in question to only those source areas that produce a lensing
system with the property A. In the case of time delays e.g. this means removing
source positions that do not create multiple images with a time delay ∆t. However
for 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 time
delay 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 by
equation (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 structure
growth 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 the
early universe [21, 44]. These density perturbations were magnified by gravitational
instability, causing overdense regions, if they were massive enough, to form galaxies.
Analytical models and approximations can then be used to derive the co-moving
density using the results of numerical calculations of density perturbations. Looking
only at the redshift evolution of dark matter halos the co-moving numerical density
can be expressed as function of halo mass M , redshift and cosmology by the equation
[22]:
ρc (z)ΩM
d ln σR−1
dn(M, z)
=
f (σR )
.
(4.9)
d log M
M
d log M
Here σR (M, z) is the mean square value of the fractional density perturbations over
a sphere with co-moving volume V given by [15, 44]:
* Z
2 +
1
σR2 =
,
(4.10)
δm dV 0
V V
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 extrapolated
to 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-Schechter
mass function [31], given by
r 2 δc
δc2
exp − 2 ,
(4.12)
fPS (σR ) =
π σR
2σR
where δc is usually taken to be a linear overdensity of a spherical perturbation at
the time it starts to collapse. It is commonly taken to be δc = 1.686 [22]. While
this 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
1.1982
−1.625
fWarren (σR ) = 0.7234(σR
+ 0.2538) exp − 2
.
(4.13)
σR
For the actual values of the number density we used data provided to us by Lukić et
al. [22]. Figure 4.1 shows this data, i.e. the co-moving density evolution for different
halo mass ranges calculated using the Warren mass function. We can see that at
early times (high z) the density increases (with decreasing z), with halos that have
higher mass forming later than low mass ones. At late times (low z) the density of
the low mass halos starts to decrease, due to them merging into higher mass halos.
25
4 Gravitational lensing statistics
1000
7
8
8
9
(10 - 10 ) MO. /h
(10 - 10 ) MO./h
9
10
10
11
0.001
(10 - 10 ) MO. /h
1e-06
(10 - 10 ) MO. /h
-3
n[(Mpc/h) ]
1
1e-09
11
12
(10 - 10 ) MO. /h
12
13
(10 - 10 ) MO. /h
1e-12
13
14
15
14
(10 - 10 ) MO. /h
(10 - 10 ) MO. /h
1e-15
0
5
10
15
20
Redshift
Figure 4.1: Co-moving number density of dark matter halos as a function of redshift
for 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 a
collection of areas dA(y) (where y is given by equation (3.34)) in the source plane
inside of which the source will experience a lensing event, i.e.
Z
σ = dA(y).
(4.14)
First we must define what we consider lensing. In our case we take this to mean
strong lensing, i.e. the creation of multiple images. The cross section will then be
defined by the outermost caustic of the lens. As we saw in §3.3.4 the location of the
caustic will depend on the convergence and shear of the lens, both of which depend
on the deflection potential. Thus in order to calculate a cross section we must first
specify a deflection potential, or more generally a density distribution, for the lenses
we are interested in. We have already treated the point-mass model, but that is not
a 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 section
is defined in the plane of the source, and not in the plane of the deflector itself. This
is due to the fact that it is the caustics, and not the critical curves, that define the
areas where strong lensing occurs.
Before considering individual models we start by discussing a simplification that
can be used to find the caustics of circularly symmetric profiles. In this case it can
be 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 then
behave 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
Z
m(x)
2 x 0
,
(4.15)
x κ(x0 )dx0 =
α(x) =
x 0
x
where m(x) is the dimensionless mass within a circle of radius x, and κ is the
dimensionless surface density, or convergence, as defined in equation (3.27). Using
equations (3.43) and (3.44) we can then express the magnification as
µ= 1
1+
1 − m(x)
x2
m(x)
x2
.
− 2κ(x)
(4.16)
From this we see that for circularly symmetric lenses two caustics are formed.
p One
is known as the tangential caustic and is defined by the critical curve xt = m(x),
i.e.
m(xt )
= 0.
(4.17)
y t = xt −
xt
This means that all circularly symmetric lenses have a caustic that degenerates to
a single point at the center of the source plane.
Of more interest to us is the other caustic, the radial caustic, defined by the
critical curve x2r = m(xr )/(2κ(xr ) − 1). This means the equation defining the caustic
will be
yr = 2xr (1 − κ(xr )) .
(4.18)
The cross section for a circularly symmetric lens is then simply the area of a circle
in the source plane of radius ηr , i.e.
2
DS
σ=π
ξ0 yr .
(4.19)
DL
Finally, as discussed earlier, it is possible to modify the lensing cross section by
looking at conditional probabilities. To do this we simply incorporate not only the
criteria of multiple lensing, but also new constrains arising from the conditional
probability 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 removing
those source positions that form images with time delays less than ∆t, i.e.
Z
Z
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 dark
matter structure will form so close to each other that they will be unresolvable by
detectors. However the time delays between the different images will still be present
and 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]

2
 2 DS 2 2 2 R yr 0 0
DS
2
(yr2 − ymin
) ymin < yr
πξ0 DL yr y2 ymin y dy = πξ02 D
L
r
σ(M, > ∆t) =
0
y
>y
min
r
(4.21)
where ymin is defined as the minimum distance a source must be from the origin so
its images have a time delay greater than ∆t, that is ∆t(ymin ) = ∆t. We consider
this 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 assume
their components behave as an ideal gas. Assuming circular symmetry, the density
for this type of lens can be shown to be [24]
ρ(r) =
σv2
,
2πGr2
(4.22)
where r is the distance from the origin (the center of the lens) and σv is the velocity
dispersion, the root-mean-square value of the random velocities around the mean
velocity of the components of the lens in question. From equation (3.24) we then
get the surface mass density as
Σ(ξ) =
σv2
.
2Gξ
(4.23)
Using equations (3.3) and (3.23) we also get the reduced deflection angle:
α(θ) = 4π
σv2 DLS
ξ0
=
.
2
c DS
DL
(4.24)
Then the dimensionless lens equation y = x − α(x) gives that the system has two
images at x = |y| ± 1 if |y| < 1 and only one otherwise [27]. The radial caustic for
the SIS is therefore yr = 1 and from equation (4.19) we get that the cross section is
4
3 σv
2
.
(4.25)
σSIS = 16π
DLS
c
The deflection potential for this system can be found using equation (3.36). It is
ψ(x) =
28
ξ02
|x|.
DL2
(4.26)
4.3 Cross sections
Using equation (3.40) we can then get the time delay as a function of y for the region
where |y| < 1 [27]:
σ 4 D D (1 + z )
v
L LS
L
c∆t(y) = 32π
|y|.
c
DS
2
(4.27)
We can then use equation (4.21) to find the modified cross sections and conditional
probabilities.
4.3.2 The Non-Singular Isothermal Sphere
One of the problems of the SIS-model is, as the name suggest, the singularity at the
origin. This non-physical behaviour can be fixed by assuming a constant density
core with some radius ξc . We follow here a paper by Kormann et al. [18] in deriving
the properties of this modification. The density profile given in (4.22) is modified
to
σv2
,
(4.28)
ρ(r)NIS =
2πG(r2 + ξc2 )
where NIS stands for Non-Singular Isothermal Sphere. The surface mass density is
then given by
σ2
1
Σ(ξ) = v p
.
(4.29)
2
2G ξ + ξc2
To find the locations of the caustics for this modified potential we use equation
(4.16). First note that the convergence is
1
κ(x) = p
,
2 x2 + x2c
(4.30)
where we take ξ0 to be the same as in the SIS model and xc = ξc /ξ0 . We can now
calculate the reduced deflection angle from equation (4.15)
p
x2 + x2c − xc
m(x)
=
,
(4.31)
α(x) =
x
x
and hence the magnification is given by
"
!
!#−1
p
p
x2 + x2c − xc
x2 + x2c − xc
1
µ=
1−
1+
−p
.
x2
x2
x2 + x2c
(4.32)
To get multiple images it has been shown that |xc | < 21 [18]. If this is the case then
we get both radial and tangential critical curves [37], with the radial one defined by
x2r
p
1
2
2
=
2xc − xc − xc xc + 4xc ,
2
(4.33)
29
4 Gravitational lensing statistics
so the radial caustic is
1
2
yr =
4xc −
x2c
r p
p
2
− xc xc + 4xc − 12 2xc + x2c − xc x2c + 4xc
r .
p
1
2
2
2xc − xc − xc xc + 4xc
2
(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 circular
symmetry. This can be fixed by considering a more general elliptical potential,
of which the circular potential is a special case. Again we follow the derivation
done by Kormann et al. [18]. They replace
p the radial coordinate2 in equation (4.23)
with the more general expression ξ = |ξ| cos2 (ϕ) + (1 − 2 ) sin (ϕ), where 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) is
therefore
p
4
(1 − 2 )
σv2
p
.
(4.35)
Σ(ξ) =
2G|ξ| cos2 (ϕ) + (1 − 2 ) sin2 (ϕ)
Using equation (3.27) we get that the convergence is
p
4
(1 − 2 )
1
p
κ(x) =
,
2|x| cos2 (ϕ) + (1 − 2 ) sin2 (ϕ)
(4.36)
where we have again used the same distance scale ξ0 as we did in the SIS model. As
we can see the formulas for Σ and κ reduce to those of the SIS model if = 0.
We can solve equation (3.37) to get the deflection potential [18]:
√
4
1 − 2
ψ(x) = |x|
sin(ϕ) arcsin( sin(ϕ)) + cos(ϕ)arcsinh √
cos(ϕ)
,
1 − 2
(4.37)
and then use the deflection to calculate the reduced deflection angle using equation
(3.36):
√
4
1 − 2
α=
cos(ϕ) , arcsin( sin(ϕ)) .
(4.38)
arcsinh √
1 − 2
Using equations (3.46), (3.47) and (3.48) the shear is found to satisfy the equation
γ(x) = κ(x).
30
(4.39)
4.3 Cross sections
This means according to equation (3.51), that the magnification of an image in this
system is given by
1
.
(4.40)
µ=
1 − 2κ(x)
We therefore have only one critical curve defined by κ(x) = 12 and the caustic for
this system is thus given by ycaustic = (y1 , y2 ) where
"
#
√
1
cos(ϕ)
4
y1 = 1 − 2 p
− arcsinh √
cos(ϕ) ,(4.41)
1 − 2
cos2 (ϕ) + (1 − 2 ) sin2 (ϕ) "
#
√
sin(ϕ)
1
4
y2 = 1 − 2 p
− arcsin ( sin(ϕ)) .(4.42)
cos2 (ϕ) + (1 − 2 ) sin2 (ϕ) Along with this caustic there is also a "quasi"-caustic, or cut, that defines an area
where multiple imaging exist without creating a critical curve. A source crossing
this cut will create a single new infinitely faint image at the origin, thus creating an
even number of images. The strange nature of this cut arises from the singularity
present at the origin, if it is removed as we did in the NIS-model it changes to a true
caustic. The location of the cut in the source plane can be found by setting x = 0
in the lens equation (3.9), so ycut = −α. Due to the symmetry of the caustic and
the fact that the lens equation is a conformal mapping [37] the cross section can be
shown to be:
2 Z π/2 dy1
DS
2
ξ0
y2
dϕ.
(4.43)
σcaustic = 4
DL
dϕ
0
In the same way the cross section for the cut is:
2 Z π/2 DS
dα1
2
σcut = 4
ξ0
α2
dϕ.
(4.44)
DL
dϕ
0
In the future, following [18], we refer to the cross sections due to the cut and caustic
as the two- and four-image cross sections respectively. We then choose the larger
cross 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 approximation
for extended lenses, we also want to look at more realistic density profiles for dark
matter halos. One of the most commonly used profile is that of Navarro, Frenk and
White (NFW). In two papers [25, 26] Navarro et al. found, using numerical modelling, that dark matter halos seem to follow a universal density profile, independent
of mass. They found that this density profile has the form
ρs
ρ(r) =
,
(4.45)
(r/rs )(1 + r/rs )2
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
parameter
rvir
.
(4.46)
cvir =
rs
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 in
this work we assume a flat universe the universal density is just the critical density
defined by equation (3.15), i.e. ρu = ρc . The virial over-density factor ∆vir can be
approximated in the cosmology we use by [2]
∆vir '
18π 2 + 82(ΩM − 1) − 39(ΩM − 1)2
.
ΩM
(4.47)
The virial mass is [3]
4π
3
∆vir ρc rvir
.
(4.48)
3
This choice of the virial radius means that the characteristic density is given by [40]
Mvir =
∆vir ρc c3vir
ρs =
3
Z
cvir
0
x
dx
(1 + x)2
−1
∆vir ρc
=
3
c3vir
ln(1 + cvir ) −
!
cvir
1+cvir
,
(4.49)
and the scaled radius by
rs =
1
cvir
s
3
3Mvir
.
4π∆vir ρc
(4.50)
As for the concentration parameter, cvir , due to the significant scatter in the the
relation between Mvir and cvir we approximate it by the median value [3]:
−0.13
H0 Mvir
1
median
,
(4.51)
cvir
= cnorm
1+z
1016 M
where cnorm = 8 as in [27]. Using equations (4.49) and (4.50) as well as (4.51) we can
rewrite the density function in a form that only depends on the virial mass and the
redshift of the halo. This allows us to use the NFW profile in gravitational lensing
problems. Since this is a circularly symmetric profile we can use equation (4.16) to
simplify our search for the caustics. We start by calculating the surface mass density
using equation (3.24), and choosing the length scale ξ0 = rs [12]:
Σ(x) = 2ρs rs F (x),
where
F (x) =
32

1


 x2 −1 1 −
√ 1
arccosh
1−x2
(4.52)
1
x
1
3


 1
1−
2
x −1
x<1
x=1
√ 1
x2 −1
arccos
1
x
x > 1.
(4.53)
4.3 Cross sections
By equation (3.27) the convergence is
κ(x) =
2ρs rs F (x)
.
Σc
(4.54)
From equation (4.15) we find the reduced deflection angle [1, 12]:
α(x) =
where
m(x)
4ρs rs g(x)
=
,
x
Σc x

x
1
√ 1
ln
+
x<1
arccosh

2

2
x
1−x
1
g(x) = 1 + ln 2
x=1


x
1
1
ln 2 + √x2 −1 arccos x
x > 1.
(4.55)
(4.56)
From equation (4.16) we see that the radial critical curve is given by the equation
F (xr ) −
g(xr )
Σc
,
=
x2r
4ρs rs
(4.57)
and the caustic is therefore
2ρs rs
F (xr ) .
yr = 2xr 1 −
Σc
(4.58)
By inserting this in equation (4.19) we get the cross section
σ = 4π
DS
DL
2
rs2 x2r
2
2ρs rs
F (xr ) .
1−
Σc
(4.59)
The time delay for this system has been discussed by Oguri et al. [27], who find that
it can be approximated by
c∆t(y) = 2rs2 xt
DS
(1 + zL )y,
DL DLS
(4.60)
where xt is the radius of the tangential critical curve, which we get from equation
(4.16):
g(xt )
Σc
=
.
(4.61)
2
xt
4ρs rs
33
5 Results
5.1 Comparison of lensing cross sections
As explained in the introduction, we are interested in investigating whether we can
use strong lensing to detect small dark matter structures. Motivated by the number densities shown in figure 4.1 we take these structures to be in the lowest mass
bin shown, i.e. with masses around 108 M . This also fits the mass range of dwarf
spheroidals (dSph), a well known type of dark matter dominated object. We therefore assume that the properties of our small dark matter structures will be similar
to those of dSph. As can be seen from the same figure 4.1, such objects should
be very common at all redshifts and have the potential to act as lenses. Whether
these structures will have a noticeable effect on the light of background sources will
depend on their lensing properties, such as the size of their strong lensing cross
sections and the time delays between images. Comparing such properties to those
of a commonly observed type of lensing object will tell us whether we should expect
to observe lensing effects on sources from such small dark matter structures. We
choose large elliptical galaxies as a baseline since they are one of the most commonly
observed lensing objects and have been studied extensively. For clarity we refer to
large ellipticals as giants (or giant lenses) and to the small dark matter structures
as 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 and
1013 M , and assume the numerical density distribution shown in figure 4.1. We
take the dark matter velocity dispersion to be v = 225+20
−12 km/s, which is the average 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 good
approximation of the dark matter velocity dispersion of galaxies [17]. We take the
core radius to be the radius at which the surface density falls to half the central
value. Lauer [19] has a list of measured core radii for giants, with the radii being
around 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 this
value from Peñarrubia et al. [29] who looked at the properties of local group dSph
and their dark matter halos. As we stated earlier the mass of the halos of some
of these local dSph are of the same order of magnitude as the structures we are
interested in (the dSph spread in the paper being 108 to 1010 M ) and so we assume
35
5 Results
SIS
NIS
NFW
100
10
1
σ [kpc2]
0.1
0.01
0.001
0.0001
1e-05
1e-06
0
1
2
3
4
5
zL
6
7
8
9
10
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 1013 M
and a velocity dispersion of 225 km/s. For the NIS model the core radius is 500
pc. The SIE cross section is not included because it is not visibly different from
the SIS model for all but the largest eccentricities.
that the velocity dispersions of these two objects is similar. Since the lower limit
of their inferred masses matches the mass of interest to us we take the lower limit
of their central velocity dispersion spread, which is around 5 km/s. Peñarrubia et
al. [29] also state the measured core radii of these dwarf galaxies, which are roughly
the 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, considering first giant lenses at different redshifts. In figure 5.1 the source is positioned
at redshift 10. From the figure we see that the behaviour of the cross section with
lens redshift is different between the three density profiles. Most notably the NFW
model, 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 except
at very high eccentricities and thus we do not include it in figure 5.1. The shape of
the SIS cross section is similar to that of the angular diameter distance between the
source and lens with lens redshift, as shown in figure 3.5. This is to be expected,
2
since the only redshift dependence in equation (4.25) is from the DLS
factor. The
same 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 pc
in figures 5.1 and 5.3, we see that it is monotonically decreasing with redshift, with
36
5.1 Comparison of lensing cross sections
ε = 0.0
ε = 0.9
ε = 0.99
100
σ [kpc2]
10
1
0.1
0.01
0
1
2
3
4
5
zL
6
7
8
9
10
Figure 5.2: The SIE cross section as a function of lens redshift for different eccentricities. The source is at a redshift 10 and the lens is a giant with a velocity
dispersion of 225 km/s.
the curve behaving similarly to the SIS cross section at a lower source redshift (see
figure 5.4). This is somewhat surprising since from the equation for the NIS radial
caustic (4.34) we see that the redshift dependence is much more complex than in the
SIS and SIE cases, and thus it is not straightforward that it should follow the same
general shape. The low lens redshift cut-off is due to the fact that multiple lensing in
the NIS model only occurs if |xc | < 1/2. As xc depends on both the lens and source
redshifts this bound is automatically broken above certain lens redshifts. This also
causes the size of the NIS cross section to be smaller, the difference between the SIS
and 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 redshift, and also the cross section. Thus for a core with a radius of 1 kpc the cross
section is negligible above zL ≈ 3 but for a core radius of a 100 pc the cross section
is non-negligible up to zL ≈ 8. It should be noted that newer observations [20] have
found 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 break
in the slope or just as a single uniform power law with no special core region. Thus
the 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 changes
when the source redshift is varied. Both the redshift range and the slope of the σcurve change. Looking specifically at the NIS model we see that the cut-off redshift
37
5 Results
rc = 0 pc
rc = 100 pc
rc = 500 pc
rc = 1000 pc
100
10
σ [kpc2]
1
0.1
0.01
0.001
0.0001
1e-05
0
1
2
3
4
5
zL
6
7
8
9
10
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 of
225 km/s.
differs by only ∆z ≈ 1 for the sources at z = 5 and z = 10. This seems to indicate
that below a certain source redshift the |xc | < 1/2 requirement does not greatly
effect the cut-off radius.
Figure 5.6 shows the effects of the upper and lower limits of the velocity dispersion spread we are using. Since the scale factor 0 is the same for all the isothermal
models, we expect the changes for the NIS and SIE models to be similar to those of
the SIS. From figure 5.6 we see that the difference in the cross sections at z = 0 is
around a factor of 2, and it decreases with increasing lens redshift. Comparing the
lower curve to the = 0.99 curve in figure 5.2 we see that the differences between
the SIS and SIE models are comparable to those resulting from the spread in the
velocity dispersion. We thus consider it justified in the remainder of this thesis in
not treating the SIE model separately, and that the presence of ellipticity in lenses
should not introduce a larger change in the lensing probabilities than e.g. the spread
of velocity dispersions. It should be noted that the SIE model has two different
caustics, as shown in §4.3.3, and so this applies only to the larger two-image cross
section. If we look at figure 5.7 we see that the four-image cross section is always
much lower, and thus differs significantly from the normal SIS cross section. However, we assume that the number of images formed should not affect the overall
detectability of a lens and therefore do not treat the four-image cross section separately.
Turning to the NFW cross section we see from figure 5.1 that it behaves quite
38
5.1 Comparison of lensing cross sections
1000
zS = 1
zS = 5
zS = 10
100
σ [kpc2]
10
1
0.1
0.01
0.001
0
1
2
3
4
5
zL
6
7
8
9
10
Figure 5.4: The SIS cross section as a function of lens redshift for different source
redshifts. The lens is a giant with a velocity dispersion of 225 km/s.
differently from the isothermal models. It does not have a monotonically declining
curve with increasing lens redshift but instead has a peak similar to the one in the
angular diameter distance plot in figure 3.3. This is due to the evolution of the
radial caustic with redshift. Unlike the isothermal models, in which the caustic is
either constant or a monotonic curve, the caustic of the NFW model has a peak
and vanishes at small and large lens redshifts. This translates to a peak in the cross
section. The value of the NFW cross section is less than the corresponding value
in the isothermal case by one to two orders of magnitude. However comparing the
size of these two cross sections directly is difficult due to the difference in the way
the size of the lens is determined. For the isothermal models this is done through
the velocity dispersion of the galactic dark matter, rather than the total mass of
the dark matter halo as in the NFW model. Therefore it is difficult to tell whether
this difference in cross sections is only due to the difference in the models or to a
difference 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 by
two orders of magnitude. In figure 5.9 we see how the NFW cross section changes
with source redshift. We see that the peak for a source at zS = 4 is higher than
in the zS = 10 case, similar to what happens for the isothermal models, although
the peak in the latter case is wider. For a source at zS = 1 we see that the cross
section is very small and does not increase in size at low redshift, as happens for
the isothermal models. Also the change from zS = 4 to zS = 1 is larger than from
39
5 Results
zS = 1
zS = 5
zS = 10
100
10
1
σ [kpc2]
0.1
0.01
0.001
0.0001
1e-05
1e-06
0
1
2
3
zL
4
5
6
Figure 5.5: The NIS cross section as a function of lens redshift for different source
redshifts. The lens is a giant with a velocity dispersion of 225 km/s and a core
radius of 500 pc.
zS = 10 to zS = 4. Finally we see that for all three source redshifts the cut-off of
the cross section happens before the lens reaches the source, and is similar to what
we 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 SIS
and the NFW models respectively. We see that the difference between these two
models is much larger than it was for giant lenses. This large a difference, around 8
orders of magnitude, can not be solely due to the differences between the spreads of
the velocity dispersions and the masses, and so must arise, in part, from differences
in the accuracy of these two models. From equation (4.25) we see that the size of
the SIS cross section depends on σv4 while for the NFW the size of the cross sections
is controlled by the dark halo mass in a complex way. In Peñarrubia et al. [29] the
velocity 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 the
velocity dispersion range of giants, which have a velocity dispersion range of around
30 km/s for a similar mass range, and we see that the SIS models seems to get more
inaccurate as the mass (and the velocity dispersion difference) decreases. Therefore
it seems that for low mass objects the SIS model breaks down and a more complete
treatment in the form of the NFW profile is needed. Finally comparing the NFW
cross section for giant and dwarf lenses we see that the peak in the dwarf case is
much narrower. In fact almost no lensing occurs for a source at zS = 10 if the lens
is at a redshift greater zL = 2.
40
5.1 Comparison of lensing cross sections
σv = 237 km/s
σv = 225 km/s
σv = 205 km/s
100
σ [kpc2]
10
1
0.1
0.01
0.001
0
2
4
6
8
10
zL
Figure 5.6: The SIS cross section as a function of lens redshift for the limits of the
velocity dispersion range used for giant lenses in this thesis. The green curve is
the mean value of the giant lens velocity dispersion range. We take the source to
be at redshift 10.
Figure 5.12 shows the change in the NFW cross section with redshift for dwarf
lenses. We see that the behaviour is completely different from what we saw for the
giant lenses (e.g. figure 5.9). The cross section peak gets both narrower and lower
with decreasing source redshift. This indicates that lowering the lensing mass does
not simply scale the system down but in fact alters the behaviour of the lens significantly.
For the NIS model we find, for ξc from 100 pc to 500 pc, that |xc | > 1/2 for all
zL , i.e. no strong lensing by dwarfs occurs according to the NIS model. Therefore
if the core radii of our dwarf lenses matches those of dSph we should not expect to
see any multiple imaging.
41
5 Results
ε = 0.99, 2-image
ε = 0.99, 4-image
ε = 0.5, 4-image
ε = 0.1, 4-image
100
10
1
σ [kpc2]
0.1
0.01
0.001
0.0001
1e-05
1e-06
0
2
4
6
8
10
zL
Figure 5.7: Comparison of the two-image and four-image cross sections of the SIE
model. 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 lens
is a giant with a velocity dispersion of 225 km/s.
M = 1013 MSun
M = 1012 MSun
1
σ [kpc2]
0.01
0.0001
1e-06
1e-08
1e-10
0
2
4
6
8
10
zL
Figure 5.8: The NFW cross section as a function of lens redshift for the limits of
the dark halo mass range we use for giant lenses. The source is at redshift 10.
42
5.1 Comparison of lensing cross sections
10
zS = 1
zS = 4
zS = 10
1
0.1
σ[kpc2]
0.01
0.001
0.0001
1e-05
1e-06
1e-07
1e-08
0
1
2
3
4
5
zL
6
7
8
9
10
Figure 5.9: The NFW cross section as a function of lens redshift for different source
redshifts. The lens is a giant with a dark matter halo mass of 1013 M .
3e-05
zS = 10, σv = 5 km/s
2.5e-05
σ [kpc2]
2e-05
1.5e-05
1e-05
5e-06
0
0
1
2
3
4
5
zL
6
7
8
9
10
Figure 5.10: The SIS cross section as a function of lens redshift. The source is at
redshift 10, and the lens is a dwarf with a velocity dispersion of 5 km/s. The
σ-axis is on a linear rather than logarithmic scale.
43
5 Results
1.8e-13
zS = 10, M = 108 MSun
1.6e-13
1.4e-13
σ [kpc2]
1.2e-13
1e-13
8e-14
6e-14
4e-14
2e-14
0
0
1
2
3
4
5
zL
6
7
8
9
10
Figure 5.11: The NFW cross section as a function of lens redshift. The source is
at redshift 10, and the lens is a dwarf of mass 108 M .The σ-axis is on a linear
rather than a logarithmic scale.
1.8e-13
zS = 10
zS = 6
zS = 4
1.6e-13
1.4e-13
σ [kpc2]
1.2e-13
1e-13
8e-14
6e-14
4e-14
2e-14
0
0
0.5
1
1.5
2
zL
2.5
3
3.5
4
Figure 5.12: The NFW cross section as a function of lens redshift for different source
redshifts. The lens is a dwarf with a dark matter halo mass of 108 M . The σ-axis
is on a linear rather than a logarithmic scale. The curve for zS = 10 is the same
as in figure 5.11
44
5.2 Lensing Probability
SIS
NFW
0.01
0.001
τ
0.0001
1e-05
1e-06
1e-07
1e-08
0
1
2
3
4
5
zS
6
7
8
9
10
Figure 5.13: Line-of-sight probability for the SIS and NFW models as functions of
source redshift. The lenses are giants with velocity dispersions of 225 km/s and a
dark matter halo mass of 1013 M .
5.2 Lensing Probability
We begin by looking at the line-of-sight probability given by equation (4.6), i.e. the
probability of a given source being lensed once by an intervening object. We assume that all lensing objects with dark matter halo masses between 1012 to 1013 M
are large elliptical galaxies, due to the fact that most galaxy lenses are early type
galaxies, with late type galaxy lenses being rare [38]. Figure 5.13 shows the lensing
probability for the SIS and NFW models. We see from this figure that the probability that a source will be lensed by a SIS object decreases at high redshifts. Thus
the 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 source
redshift. As such, if the source redshift is high, the cross section will be lower at low
lens redshifts than if the source is closer to the observer. This can be seen in figure
5.4. Combining this with the redshift distribution of the density of lensing objects
shown in figure 4.1 we see that the high redshift sources will have a lower probability
of being lensed because the lensing cross sections are lower at the redshifts where
the lensing population is high.
Figure 5.14 shows the effect of the lens velocity dispersion on the lensing probability. The difference in probability between the high and low dispersion limits is
about a factor of 2 at the peak. This is similar to the difference found for the cross
45
5 Results
0.01
σv = 237 km/s
σv = 225 km/s
σv = 205 km/s
τ
0.001
0.0001
1e-05
0
1
2
3
4
5
zS
6
7
8
9
10
Figure 5.14: Line-of-sight probability for the SIS model as a function of source
redshift for the limits of the velocity dispersion range. The green curve is the
average value. The lenses are giants and have a dark matter halo mass of 1012 −
1013 M .
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 at
different core radii. Its shape is similar to the SIS model, with the probability being
lower due to the lower value of the cross section. The inclusion of a constant density
core in the lens thus decreases the lensing probabilities. The peak in the probability
curve shifts with changing core size, with the peak appearing at lower redshifts for
larger 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 radii
and terminate at a lower redshift resulting in a shifting of the entire shape of the
probability curve to lower source redshifts. We also see that the decrease in probability at higher source redshift becomes steeper than for the SIS with increasing core
radius. This comes from the bound |xc | < 1/2, which causes a cut-off in the cross
section at higher lens redshifts as we saw in figure 5.5. Decreasing and increasing
the 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 to
different cross section behaviour the probability distributions are quite different for
the NFW and isothermal models. While the NFW probability distribution also has
a peak, it is at a much higher redshift. The peak of the NFW cross section curves
46
5.2 Lensing Probability
0.01
rc = 0 pc
rc = 100 pc
rc = 500 pc
rc = 1000 pc
τ
0.001
0.0001
1e-05
0
1
2
3
4
5
zS
6
7
8
9
10
Figure 5.15: Line-of-sight probability for the NIS model as a function of source
redshift for different core radii. The lenses are giants with velocity dispersions of
225 km/s and a dark matter halo masses of 1012 − 1013 M .
means that the highest cross sections are usually around the same lens redshifts
for most source redshifts, as shown in figure 5.9. So unlike the isothermal models
the lensing cross section is always highest at the redshifts where the number density is high. However, as can be seen from the same figure, the cross section peak
gets wider at high source redshifts, which causes the probability to decrease at the
highest redshifts. For both masses the NFW probability is much lower than for
the isothermal cases, thus a more realistic lensing potential significantly lowers the
lensing probability.
In figures 5.17 and 5.18 we plot the total probability of lensing by SIS and NFW
giant lenses using equation (4.7). We see that the low line-of-sight probabilities
translate to low total probabilities. As we would expect from the differences in the
line-of-sight probabilities, the evolution of the total probabilities with redshift is not
the same in the two models. While both curves are increasing with redshift, since
the increase in the number of lenses at high redshift compensates for the lowered
line-of-sight probability, the shapes differ noticeably. The slope of the curve for the
SIS model changes, decreasing at higher redshifts. Meanwhile the slope of the NFW
remains almost constant with redshift, indicating that the change in line-of-sight
probabilities follows precisely the increase in the number of lenses. Most notably
though, we see that the line-of-sight probabilities and total probabilities are of the
same order of magnitude. Thus for the purpose of comparing the relative probability
values either distribution will do.
47
5 Results
M = 1013
MSun
M = 1012 MSun
0.0001
1e-05
1e-06
τ
1e-07
1e-08
1e-09
1e-10
1e-11
0
1
2
3
4
5
zS
6
7
8
9
10
Figure 5.16: Line-of-sight probability for the NFW model as a function of source
redshift for the dark halo mass range of giant lenses.
Figure 5.19 show the SIS and NFW line-of-sight probabilities for dwarf lenses
with a mass of 108 M and a dark matter velocity dispersion of 5 km/s. While the
shape of the SIS probability curve is similar to that of a giant lens, the same is
not true of the NFW curve. It has a monotonic rise and no peak is present. As
figure 5.11 shows this is due to the fact that for low masses the NFW cross section
at high source redshift has a very narrow peak. For this narrow redshift range the
number density is essentially constant and so no peak is formed in the probability
distribution.
The probability values for the SIS dwarf lenses are about two orders of magnitudes
lower than for the SIS giant lenses but around seven to nine orders of magnitudes
lower in the NFW case. This large difference in model predictions follows the one
for the cross sections. From the NFW results we can state that the probability of
dwarf lenses producing a strong lensing effect is extremely rare, that is, for every
billion giant lenses we should on average see one dwarf lens. So even though these
smaller objects are far more numerous than the larger ones, the decrease in the
strong lensing cross section means that they are overall much less likely to act as
lenses.
48
5.2 Lensing Probability
0.005
σv = 225 km/s
0.004
P
0.003
0.002
0.001
0
0
1
2
3
4
5
zS
6
7
8
9
10
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 halo
masses of 1012 − 1013 M .
0.00012
M = 1013 MSun
0.0001
P
8e-05
6e-05
4e-05
2e-05
0
0
1
2
3
4
5
zS
6
7
8
9
10
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 = 1013 M .
49
5 Results
SIS
NFW
0.0001
1e-06
1e-08
τ
1e-10
1e-12
1e-14
1e-16
1e-18
1e-20
0
1
2
3
4
5
zS
6
7
8
9
10
Figure 5.19: The line-of-sight probability for the SIS and NFW models as a function
of source redshifts. The lenses are dwarfs with masses of 108 M and velocity
dispersions of 5 km/s. The noise in the curves are due to inaccuracies in our
numerical 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 delays 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 probability rather than the total probability, since the difference between these two with
respect to the cumulative conditional probability is minimal. Since these are cumulative probabilities, a time delay of ∆t having a cumulative probability τ means that
the 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 both
being of the order of days. We thus feel confident in stating that our calculations
should, 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 that
the time delay here is much shorter. For the SIS model the time delay is found
to be of the order of a second, and for the NFW model it is of the order of tens
of nanoseconds. We would thus need a rapidly variable source in order to confirm
lensing 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 minutes [11] and have been observed to be variable down to a time scale of a few
milliseconds [23]. While this is sufficient to detect effects from lensing if the SIS
model is accurate, it falls short of the nanosecond variability needed to confirm lensing if the NFW model is correct. Current detectors, on satellites like Fermi and
Swift, only have a time resolution of ≈ 100 ms indicating that a time delay of ≈ 100
ns (a typical value shown in figure 5.23) will remain undetected in the near future.
51
5 Results
1
zS = 5, σv = 225 km/s
0.8
τ(>∆t)
0.6
0.4
0.2
0
0
50
100
150
200
250
∆t [days]
Figure 5.20: The cumulative conditional probability distribution of the SIS model for
time delays. The lenses are giants with velocity dispersions of 225 km/s and the
source is at redshift 5.
1
zS = 5, M = 1013 MSun
0.9
0.8
0.7
τ(>∆t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
∆t [days]
20
25
30
Figure 5.21: The cumulative conditional probability distribution of the NFW model
for time delays. The lenses are giants with dark matter halo masses of 1013 M
and the source is at redshift 5.
52
5.3 Conditional probabilities
1
zS = 5, σv = 5 km/s
0.9
0.8
0.7
τ(>∆t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
∆t [s]
Figure 5.22: The cumulative conditional probability distribution of the SIS model for
time delays. The lenses are dwarfs with velocity dispersions of 5 km/s and the
source is at redshift 5.
1
zS = 5, M = 108 MSun
0.9
0.8
0.7
τ(>∆t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
∆t [ns]
Figure 5.23: The cumulative conditional probability distribution of the NFW model
for time delays. The lenses are dwarfs with masses of 108 M and the source is
at redshift 5.
53
6 Conclusions
In this thesis we have used a probabilistic approach to determine the feasibility of
using strong gravitational lensing to detect small dark matter structures (or dwarf
lenses). These lenses, comparable in size to local dwarf spheroidals (dSph), are
according to numerical models quite numerous, especially at early times. We calculated the lensing cross sections of these dwarfs lenses, along with the lensing cross
sections of a more common lensing object, a large elliptical galaxy (or giant lens),
using three isothermal density profiles. These profiles are the singular isothermal
sphere (SIS), the non-singular isothermal sphere (NIS) and the singular isothermal
ellipsoid (SIE). We also used the more realistic Navarro-Frenk-White (NFW) density profile of dark matter halo masses. We found that the cross sections calculated
using the SIS and SIE models were similar, while the NIS profile predicted almost
no strong lensing for dwarf lenses like dSph. Thus if these dwarf lenses have an inner
core of constant density comparable to those measured for dSph the NIS profile predicts they will not produce strong lensing effects. Then using a Warren numerical
density distribution of dark matter halos, we calculated the line-of-sight and total
probabilities for both giant and dwarf lenses using the SIS and NFW profiles. The
probability values for the dwarf lenses are very different for the two models, indicating that the simpler SIS approximation breaks down at low masses. Comparing the
probabilities for lensing by giant and dwarf lenses we found that if the NFW model
is correct one can only expect around one lensing event by a dwarf lens for every
billion such events by giant lenses. We also looked at the distribution of time delays
for dwarf lenses and found that they should be of the order of seconds in the SIS
case, and of the order of tens of nanoseconds in the NFW case. This again shows a
very large difference between the results of the two models. From this we conclude
that if the NFW model is accurate, dwarfs are unlikely to be detected in the near
future by means of their gravitational lensing effects.
55
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