Gravitational lensing statistics

Candidatus Scientiarum thesis by

A. O. Jaunsen

Institute of Theoretical Astrophysics

University of Oslo Norway

May 1994

Preface

During the course of the work for this thesis I have had many valuable experiences, aside from the pure scientifical point of view. Various problems and tasks during the work, both in the practical course of reducing the data and in the past half year of theoretical work and the writing of the thesis, would have been solved in more efficient ways had I started the work today. I suppose this is among the many experiences a master thesis is meant to give a student. However, I have most of all appreciated the interesting and broad nature of the thesis task. For this I would like to thank my supervisor, Rolf Stabell, whose support and suggestions has been much appreciated throughout the complete process of the thesis. Furthermore I would like to thank Jean Surdej for his comments, suggestions and support during the course of this work and with the transcript. I am also thankful for valuable answers from Chris Kochanek to my questions on his recent papers on lens statistics.

Among the most exciting and memorable experiences were the observational runs, performed at Nordic Optical Telescope, La Palma, Spain. Marek Jablonski has been a great help and support in these observations and much fun on non-scientific occasions. Anlaug Amanda Kaas gave a lot of support and help throughout these observations, beside the social occasions with Marek and Yamuna. I would also like to thank Stein Vidar Haugan, who has helped me on numerous occasions throughout my studies in Oslo, for his suggestions/corrections on the thesis transcript and many enlightening discussions.

Finally I would like to dedicate this thesis to my wife, Yamuna, for all her support and help during this past time. I also wish to thank my parents for supporting my early interests in astronomy as a child and teenager.

Andreas ii

0Front cover depicts the title page at z=1.0 lensed by a spherically symmetric lens model at z=0.3 with velocity dispersion 1350 km/s and core radius 0.1 Mpc (h=0.5).

Contents

Preface i

List of Figures iv

List of Tables vii

1 Introduction 3 1.1 Historic notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Previous related work . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 About the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Gravitational lensing - theory and modeling 7 2.1 Gravitational lens optics . . . . . . . . . . . . . . . . . . . . . 7 2.2 The Lens Model . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Lens mass-distributions . . . . . . . . . . . . . . . . . . 12 2.2.2 Lens population . . . . . . . . . . . . . . . . . . . . . . 14 2.2.3 Optical depth for lensing . . . . . . . . . . . . . . . . . 15

2.3 The QSO Luminosity Function . . . . . . . . . . . . . . . . . 18 2.4 The Magnification Bias . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Magnification probability distribution . . . . . . . . . . 22 2.4.2 Differential magnification bias . . . . . . . . . . . . . . 24

3 Observations and the QSO sample 27 3.1 Selection and observation of the sample . . . . . . . . . . . . . 27 3.2 Image processing and data reduction . . . . . . . . . . . . . . 30

3.2.1 Pre-processing of direct images . . . . . . . . . . . . . 31 3.2.2 Spectroscopic observations and reductions . . . . . . . 34

3.3 Selecting the GL candidates . . . . . . . . . . . . . . . . . . . 35 3.3.1 False positives . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.2 False negatives . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Angular Selection Function . . . . . . . . . . . . . . . . . . . . 37 3.4.1 Small separations . . . . . . . . . . . . . . . . . . . . . 38 3.4.2 Large separations . . . . . . . . . . . . . . . . . . . . . 39 3.4.3 The ASF . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Uncovering small separation candidates . . . . . . . . . . . . . 41 3.5.1 The PSF . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5.2 Multiple PSF fitting . . . . . . . . . . . . . . . . . . . 42 3.5.3 Non-linear deconvolution . . . . . . . . . . . . . . . . . 43

4 Statistical results and probabilities 47 4.1 The SIS lens probability distribution . . . . . . . . . . . . . . 47 4.2 Results from the NOT sample . . . . . . . . . . . . . . . . . . 49 4.3 Cosmological results . . . . . . . . . . . . . . . . . . . . . . . 56 4.4 Survey completeness . . . . . . . . . . . . . . . . . . . . . . . 60

5 Conclusions 65 5.1 Deficiency of the model . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Cosmological applications . . . . . . . . . . . . . . . . . . . . 67 5.3 Further work using advanced models . . . . . . . . . . . . . . 68

Bibliography 70

List of Figures

2.1 Geometry of a GL-system, with the impact parameter ξ, reduced deflection angle α(θ), true deflection angle α̂(ξ) and the observer-lens, observer-source and lens-source angular distances A_{l}, A_{s} and A_{ls}, respectively. . . . . . . . . . . . . . . . 8

2.2 Ray tracing diagram for the point mass lens model . . . . . . 10 2.3 The magnification of lensed images alters the image surface

area, while the surface brightness (local flux density) is preserved. In this diagram we illustrate a typical lensing scenario, where the lens mass (50% shaded) is not aligned exactly with the line-of-sight (dotted). At left the individual unlensed light rays are shown (i.e. the observed image (light rays) if there were no lens present), while at right light rays deflected by the lens mass are refocused towards the observer. Notice that the image brightness with and without the lens mass are the same, while the lensed image is “magnified” (larger surface). . 11

2.4 Ray tracing diagram for the SIS lens model . . . . . . . . . . . 13 2.5 The hybrid luminosity function for different redshift values

(z = [1, 2, 3, 4, 5]), displaying the “sliding” of the redshift dependent break. Observational measurements of the LF for 1.9 < z < 2.2 from several surveys gathered in Hartwick and Schade (1990) is indicated by + . . . . . . . . . . . . . . . . . 20

2.6 Cumulative number-magnitude plot of our LF (valid for z < 5) adapted from Wallington and Narayan (1993) (solid) and the best-fit to the number-count (valid for z < 2.2) from Hartwick and Schade (1990) (dotted). The data points for the numbercount from Hartwick and Schade (1990) and Hawkins and Véron (1993) are indicated by + and ⋄, respectively. . . . . . 21

2.7 The overdensity of detectable QSOs due to the magnification by the SIS lens (left) and the differential magnification bias (right), i.e. the overdensity folded with the differential magnification probability. . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Magnitude-redshift distribution of the 166 NOT sample, each

object indicated by ∗. The small dots represent the QSOs from the Véron-Cetty and Véron (1991) and the Hewitt and Burbidge (1993) catalog. The solid lines indicate the absolute magnitudes −26,−27 & −30 and are computed, as described in Appendix ??, for an Einstein-de Sitter universe model (Ω_{M} = 1, Ω_{Λ} = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Apparent magnitude histogram for the 166 QSOs sample … 30

3.3 Redshift histogram for the 166 QSOs sample . . . . . . . . . . 30

3.4 Non-linearity in the NOT-CCD . . . . . . . . . . . . . . . . . 32

3.5 Illustration of slit-spectroscopy. Lower panel shows 2D-spectral image, while upper panel shows a single combined row of strongest component (QSO) along dispersion-axis. Numbers along the ordinate and abcissa axis are irrelevant. . . . . . . . 35

3.6 Two of the investigated QSOs, showing at left a typical field surrounding the QSO with a close companion. At right a field with no companions within r_{c} = 10′′. The dashed circle in the right panel marks the survey area πr_{c}^{2}. The fields are 100×100 arcseconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 The FWHM distribution of the observations of 166 QSOs … 38

3.8 The selection function for NOT observations at 0.7” and 1.0”(solid lines left and right, respectively) FWHM/seeing at left. Also included for comparison, are the ground-based observations with inspection by EYE (dotted) and PSF-subtraction (dashed) for 1.0” seeing and the HST snapshot survey (dash-dot). At right we reproduce the complete selection function showing the reduced mangitude difference at separations larger than 5′′ (solid) and without this modification (dotted). . . . . . . . 41

3.9 PSF-subtraction and multiple PSF fitting results on Q1208+1011. The diagram displays five combined resampled (5 times) images of the object(upper left) and PSF (upper right), best double PSF fit (lower left) and residual after model subtraction (lower right). Data are courtesy Jens Hjorth and Frank G.Jensen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.10 Result from 600 RL-deconvolution iterations (right) of a 5 times resampled combined image of Q1208+1011 (left)(data coutsey J.Hjorth & F.G.Jensen). . . . . . . . . . . . . . . . . . 46

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4.1 Distribution of separations, due to macro-lensing by a SIS-lens of a distant source (dotted line). The dashed line illustrates the distribution of image separations including the selection function for FWHM=1.0”, which will be discussed and deduced in Section 3.4. . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Differential lensing probability of a QSO at z = 2.0 at the north galactic pole. Lens probabilities for a companion within 10” of a QSO with m_{V} = 16, 17, 18, 19 is shown (solid lines) with a peak probability close to the QSO magnitude. The star, galaxy, associated QSO and unassociated QSO probabilities are illustrated by the dotted,dashed,dash-dot and dash-dot-dot-dot lines respectively. . . . . . . . . . . . . . . . . . . . . . 52

4.3 Spectroscopic observations of a few promising candidates, per-

formed at NOT. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4 Likelihood contour plot for various flat cosmologies (Ω_{M} =

1 − Ω_{Λ}) and velocity dispersions (σ_{∗}), assuming lensing is significant for all lens types (E/S0/S). . . . . . . . . . . . . . . . 61

4.5 Likelihood contour plot for various flat cosmologies (Ω_{M} = 1 − Ω_{Λ}) and velocity dispersions (σ_{∗}), assuming lensing is significant only for late type galaxies (E/S0). . . . . . . . . . . . 62

4.6 Completeness of the NOT survey as a function of image separation for a QSO with m_{B} = 17.0, 18.0, 19.0 and σ_{∗} = 245kms^{−1}. The upper most line corresponds to m_{B} = 17.0 and the lowest to m_{B} = 19.0. Left panel illustrate details at small separations, while the right panel shows the full survey range. The completeness with and without the reduced magnitude difference at large separations (see Section 3.4.2) is illustrated by the solid and dotted lines, respectively. Dashed lines indicate completeness using the ESO Key program ASF. . . . . . . . . 63

4.7 Completeness of the NOT survey as a function of characteristic velocity dispersion σ_{∗} for a QSO with m_{B} = 18.0. The completeness with and without the reduced magnitude difference at large separations (see Section 3.4.2) is illustrated by the solid and dotted lines respectively. . . . . . . . . . . . . . 64

viii  LIST OF FIGURES

List of Tables

2.1 The relative abundance, central velocity dispersions and computed F values of late-type E/S0 galaxies and early-type S spiral galaxies. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Constants describing the ASF for various seeing (FWHM) values. 40

4.1 List of GL-candidates in the NOT-sample, with individual probabilities of the companion being a lensed image, star/galaxy or an ass./unass. QSO. The two objects marked by an asterix are not part of the NOT sample, but two of the identified GLsystems from the SCYM-sample and are only included here for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Estimated σ_{∗} and F parameters from the Poisson likelihood’s of the observations. The first row is the results assuming that lensing is significant from E, S0 and S galaxy types, while the second row only for E and S0 (late-type) galaxies. The maximum likelihood value is indicated by F^{max} and σ_{∗}^{max}, respectively. The results are shown for the Einstein-de Sitter universe model (Ω_{M} = 1 and Ω_{Λ} = 0) at 90% confidence levels. 60

2  LIST OF TABLES

Chapter 1

Introduction

Cosmology has grown, over the present century, to become a substantial part of modern astrophysics. In many ways cosmology is the extension of the ancient astronomical science, when philosophy and science stood equal and the quest for understanding the heavens was the ultimate task. Today a great many of the heavenly mysteries have been resolved, but still the modern man is incapable of understanding the realm of the universe. In this thesis we shall investigate one of the many wonders of the universe, which was predicted in the early nineteen hundreds and was observationally confirmed in 1979. This phenomenon, known as gravitational lensing (GL), has become an important tool in probing the structure, size and age of the universe. In this work we investigate and illustrate one such method of determining the properties of the universe and its contents.

1.1 Historic notes

Einstein’s “General theory of relativity”, which he presented in 1916, marked a turning point in science. His famous prediction that light rays from a star close to the limb of the sun would be bent by the gravitational field, impressed the world when this was confirmed within 20-30% accuracy a few years later. Chwolson (1924) described how a star, which obscures a more distant star, might produce a “ring” around the optical axis (line-of-sight). He also pointed out the possible presence of “fiktiver Doppelstern” (false double stars). Later Einstein (1936) re-discovered the ring and double image possibilities, but pointed out that “Of course, there is no hope of observing this phenomenon directly”. Zwicky (1937b,a) discussed the possibility and probabilities of galaxies (nebulae) acting as gravitational lenses on other galaxies further away. With this new idea, it became “practically a certainty” to discover such phenomena and he went on to express the minimal probability of having a perfect alignment between the observer, lens and source taking place by p = ω_{m}/4π. The ω_{m} representing the total solid angle all background galaxies brighter than m span and p the conditional probability of alignment between a background nebulae and a given closer nebulae.

For decades little progress was made in the field of gravitational lensing, until renewed interest eventually was taken in the early sixties. Several fundamental papers appeared during a few years by Klimov (1963); Liebes (1964); Zel’dovich (1964); Refsdal (1964b,a) and Refsdal (1966). Furthermore Refsdal (1965, 1970) and Press and Gunn (1973) discussed matters on using lensing statistics to probe the universe.

There was, however, considerable doubt attached to the feasibility of this extravagant phenomenon. It was not until 1979 that (Walsh et al., 1979) serendipitously discovered the double quasar “0957+561”, giving the first observational evidence of the predicted multiple imaging of distant sources.

1.2 Previous related work

Several authors have published results of systematic search programs for gravitational lensing effects and presented statistical applications to cosmology. Probably the first systematic gravitational lensing (GL)-search program, initiated in 1986, was the ESO key program (Surdej et al., 1988b). This survey succeeded in finding several GL-systems, the first being UM-673 (Surdej et al., 1987, 1988a) and later H 1413+117 (Magain et al., 1988) and Q 1208+1011 (Magain et al., 1992). Lately the ESO-survey was combined with other similar projects (Crampton et al., 1992; Yee et al., 1993; Maoz et al., 1993) and the results from this work was recently presented in Surdej et al. (1993a). In this and other works (Maoz and Rix, 1993; Kochanek, 1993a,b) the same simple analytical galaxy model is applied to estimate characteristic parameters of the galaxy (the one-dimensional velocity dispersion) and universe model (the matter density, ρ_{M}, and the cosmological parameter, Λ). All four of these surveys used different instruments, telescopes and reduction methods. This and the fact that all survey images have limited detection capabilities of GL-systems, has led to the application of an angular selection function (ASF) describing these limitations for each of the surveys. Among the important tasks in the present work has been the combination of our sample with the objects investigated by these previous optical surveys and show how we might statistically deduce fundamental parameters of the universe from a large sample of quasi stellar objects (QSOs).

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1.3 About the thesis

The present work was initiated by Rolf Stabell, having received 4 observational runs at the Nordic Optical Telescope (NOT) for the investigation of about 200 quasi stellar objects (QSOs). These objects were observed between November 1990 and June 1992.

The aim of the thesis has not been to discriminate between cosmological models or to reach conclusive results, but merely to show how a full systematic and statistical reduction of GL-surveys can be carried out. The basic information of each QSO, the number of detected GL’s and the use of a simple analytical model of the lensing galaxy are key elements in this process, which we will present in the following pages of the thesis. One of the most intriguing and fundamental scientifical questions cosmologists seek to answer today is whether the universe is closed, and hence will recollapse upon itself sometime in the future; if it is flat and will ultimately stop expanding; or if it is open and therefore will continue its present expansion for all time, becoming eventually so diluted no galaxies will be visible. Ultimately gravitational lenses might answer this and other questions, or at least considerably constrain the possible universe model in which we exist.

In Chapter 2 we present the theoretical and fundamental aspects that govern the gravitational lens phenomenon. We offer a brief introduction to the geometrical and optical properties of a typical gravitational lens system. The most important element of gravitational lensing (GL)-systems are the lensing objects mass, form, distance and population characteristics drastically rule the frequencies and incidences of gravitational lensing (GL)-effects. A simple analytical lens/galaxy model is applied and the implications this model has on the magnification bias and probability distributions are discussed. Furthermore we present a model of the population and distribution of the QSOs themselves and discuss the effects this has on the frequency of lensing in the universe.

Chapter 3 discusses the observational and practical aspects of the work, including a general discussion of the detectable constraints imposed by observations. It is important to evaluate the sample of QSOs we choose to investigate because of observational selection effects and this is also discussed. Furthermore a significant proportion of the thesis work has been to reduced the observed data and later also to obtain new data, both direct and spectrographic. The data were reduced by methods described in Section 3.2. An alternative method of investigating QSOs at sub-arcsecond separations is also briefly presented in Section 3.5, but it is not applied in the statistical framework because of the ambiguous results this method presently gives. The observational constraints on separation and magnitude difference are investigated by means of synthetic data, and described by a selection function. We concentrate our present work within the observational window defined by this selection function.

Finally applying the theoretical model from Chapter 2 on the observational sample presented in Chapter 3, we present our results in Chapter 4. First we include all the elements from Chapter 2 into an expression for the relative conditional lensing probability. In fact all probability expressions are independent of the number of QSOs, because the probabilities are conditional, i.e given a certain QSO with magnitude, m and redshift, z. Secondly we evaluate the sample of GL-candidates found and compute relative probabilities for these candidates. Finally we illustrate how our sample in combination with previous GL-surveys, might constrain certain cosmological parameters. Even though we confine ourselves to a flat-universe model in the present work, larger samples should however also be able to determine the most probable curvature of the universe model in the future. Due to the restricted detectability (i.e. we are not able to observe all GL-systems) imposed by the observational limitations, we also discuss the completeness of the survey.

In Chapter 5 we summarize our results and discuss the credibility on the basis of the lens model and other assumptions made throughout the thesis. Suggestions as to how lens statistics could, in the future, be improved to yield more accurate estimates using e.g. more realistic lens models, such as the elliptical potential model, are also presented.

Gravitational lensing - theory and modeling

In the following sections we present the fundamental concepts of gravitational lensing statistics, laying the basis for the deduction of statistical results presented in chapter 4. A short introduction to the general aspects of lens optics is given in Section 2.1. Further on, in Section 2.3, we discuss the problems and uncertainties of the quasar luminosity function, and compare available data to an evolving model. A complete deduction of a simple analytic lens model is presented in Section 2.2, followed by discussion of the important magnification bias in Section 2.4. The lens model and its applicability is a frequently discussed subject in the gravitational lensing (GL)-community. Even though we shall make use of a somewhat simple lens model in our numerical applications, the increasing efforts to adopt a more realistic elliptical potential in gravitational lensing (GL)-statistics are also discussed.

2.1 Gravitational lens optics

The deformation of the space-time curvature by large masses can have significant effects on the observable objects in the universe. Distant sources are multiply imaged (macro-lensed) when massive objects close to the line-of-sight “bend” the light paths between the source and observer. Cosmologically distant objects, that is, objects at distances comparable to the scale of the universe, are candidates for observable macro-lensing. The most common candidates with such properties are quasars (QSOs). They are intrinsically bright and extremely distant objects, possibly fueled by a central supermassive black hole. Their extreme intrinsic brightness allows us to observe them, despite their huge distance. Precisely because of the large distance, the possibility of finding massive objects close to or along the line-of-sight of a QSO is largely enhanced.

If a massive galaxy is situated close to the line-of-sight of the QSO, the light-rays from the QSO are bent by the gravitational field, as by an optical lens. Depending on the impact parameter, ξ (see Figure 2.1), and the positions of the source and observer, one might observe not one, but two or more images of the QSO.

Figure 2.1: Geometry of a GL-system, with the impact parameter ξ, reduced deflection angle α(θ), true deflection angle α̂(ξ) and the observerlens, observer-source and lens-source angular distances A_{l}, A_{s} and A_{ls}, respectively.

Assuming the simplest case of a point mass lens situated close to the line-of-sight between the observer and the source, we easily derive the lens equation which connects the source- and image-positions. This relation can easily be verified from Figure 2.1 and is expressed as

θ = θ_{s} + α(θ)  (2.1)

where α(θ) is the reduced deflection angle. The reduced deflection angle is the observed deflection angle, while the true deflection angle α̂(ξ) is the “real” angle a light ray is deflected (in the lens plane). These two angles are related by

α(θ) = \frac{A_{ls}α̂(ξ)}{A_{s}}  (2.2)

where ξ = A_{l}θ. See appendix ?? for the definition of the angular size distances A_{l}, A_{s} and A_{ls}. Under the weak gravitational field approximation, we can express the deflection angle due to a point mass as

α̂(ξ) = \frac{2R_{sc}}{ξ} = \frac{4GM}{c^{2}A_{l}θ}  (2.3)

where R_{sc} is the Schwarzschild radius. Using Equations (2.1), (2.2) and (2.3), Chwolson (1924) and Einstein (1936) showed that a source on the optical axis (which implies θ_{s} = 0) would form a ring of angular radius

θ_{E} = α(θ_{E}) = \frac{4GMA_{ls}}{c^{2}A_{l}A_{s}} \frac{1}{θ_{E}}

⇓
θ_{E} = (\frac{4GM}{c^{2}A} )^{1/2}  (2.4)

where the effective lens distance A ≡ \frac{A_{l}A_{s}}{A_{ls}}. Equation 2.4 is, however, only valid for compact objects/point mass lenses. The Einstein-radius^{1} is nevertheless a useful parameter, which is often used as a measure for geometrical quantities.

In addition to splitting a distant source into multiple images, lenses also magnify the source, i.e. focusing more light towards the observer than would be the case in a no-lensing scenario. Gravitational lensing events are divided into two classes; macro-lensing and micro-lensing. Micro-lenses are typically produced by small masses (∼ M_{⊙}) and the image splitting is therefore small and geometrically unresolvable. The observable effect of magnification from micro-lensing is therefore simply amplification of light, while the magnification term is used for macro-lensing^{2}, because the original image of the source is distorted when lensed, changing the integrated brightness of the image. The surface brightness in an image is preserved in the lens plane, because lensing is purely a geometrical effect^{3}. The image surface area in the lens plane is however altered, hence a magnification effect. Specifically, the lens deflects light-rays (in the lens plane) from the source towards an observer^{4} in any of the focus points (see Figure 2.2). Such an observer will detect one or several enhanced images of the source, due to the effective focusing of light-rays. An image is magnified if more light-rays are focused towards the observer in the presence of a lens than without one. We have illustrated the effect of magnification in the lens plane in Figure 2.3.

1the angular radius of the Einstein ring

2producing geometrically resolvable multiple images of a distant source, as opposed to the unresolved images of micro-lensing

3the geometrical nature of lensing also imply the effect is independent of wavelength

4who observes multiple images

Figure 2.2: Ray tracing diagram for the point mass lens model

The magnification (demagnification) of a macro-lensed image, therefore, corresponds to an increase (or decrease) in the observed flux, giving rise to the magnification bias discussed in Section 2.4.

For a detailed introduction to gravitational lensing (GL)-optics, see Blandford and Narayan (1992); Refsdal and Surdej (1992, 1994) and Schneider et al. (1992).

2.2 The Lens Model

In this section we discuss the population of lenses (galaxies) and the individual galaxy’s probability of producing macro-lensing^{5} events. We are restricted to investigate lensing by isolated galaxies due to the present lens model’s simple and symmetrical properties. This problem and its implications will be discussed below and in Chapter 5. The uncertainties of Figure 2.3: The magnification of lensed images alters the image surface area, while the surface brightness (local flux density) is preserved. In this diagram we illustrate a typical lensing scenario, where the lens mass (50% shaded) is not aligned exactly with the line-of-sight (dotted). At left the individual unlensed light rays are shown (i.e. the observed image (light rays) if there were no lens present), while at right light rays deflected by the lens mass are refocused towards the observer. Notice that the image brightness with and without the lens mass are the same, while the lensed image is “magnified” (larger surface).

5as discussed in Section 2.1, this term is used for resolvable multiple-image lensing events, except in the special case of single image lensed objects

the results we arrive at in Chapter 4, due to the observational nature of the various constants and model-fits applied in the following section, have been discussed in Kochanek (1991, 1993b). The statistical lensing properties of a gravitational lens are, needless to say, crucial aspects of the lensingphenomena. We therefore discuss the properties that such a lens model is built on, specifically the lens mass-density distribution and the population of the lenses. Even though we only apply the simple analytical singular isothermal sphere (SIS)-model (defined in Section 2.2.1) to our statistical computations, progress and increasing efforts are still being made in this field (cf. the modeling of realistic elliptical galaxies), a short introduction to the modeling of realistic elliptical galaxies is offered in Section 5.3. The use of singular isothermal sphere (SIS)-models are nevertheless justified by most authors (despite its unrealistic simplicity), because the results are consistent with present available lens-statistics (Kochanek, 1991; Maoz and Rix, 1993; Surdej et al., 1993a; Kochanek, 1993b).

2.2.1 Lens mass-distributions

The determination of mass- or mass-density distributions of lensing galaxies, is naturally a vital step in shaping the lens model. Realistic distributions based on observational data are clearly favorable as opposed to a simplified model such as the isothermal sphere, applied in this work. Nevertheless, the isothermal sphere has been shown to be a good approximation to the velocityprofiles observed in most galaxies and moreover giving results consistent with present gravitational lensing (GL)-statistics. The isothermal sphere has a characteristic surface mass density distribution

Σ(ξ) = σ^{2}/2Gξ  (2.5)

which is characterized by a singular core and a one dimensional velocity dispersion, σ. The one-dimensional velocity dispersion, σ, indicates the average line-of-sight dispersion of the central matter in a galaxy. The velocitydispersion is, as we shall see later, a vital parameter in lensing models and its application in lens statistics has been discussed in Wu (1993) and Kochanek (1993b). The former paper discuss the different definitions of the velocity dispersion applied by various authors, but points out that all of these are equal to the average line-of-sight velocity dispersion for a singular isothermal sphere (SIS) model. In general, the average line-of-sight velocity dispersion corresponds to the observed/measured values of galaxies (listed in Table 2.1). Some models apply a defined core-radius, within which the surface mass density is constant. In our singular isothermal sphere (SIS) model, however, we apply (the more “special” case) of a singular core, implying that the mass density increases towards infinity as r_{c} → 0. However unreal this property might seem, it is the only definition of the model that gives an analytically tractable statistical lens model. The singular isothermal sphere (SIS) model mass distribution (Eq. (2.5)) sustains a constant amount of kinetic energy at all radii, i.e. giving a flat rotation curve. This has some very characteristic implications, which shall become apparent below. By generalising Eq. (2.3), it is possible to write

α̂ = ±\frac{4GM(≤ ξ)}{ξc^{2}}  (2.6)

which denotes the deflection angle of light rays due to the projected mass M(≤ ξ), within the impact parameter ξ (cf. Figure 2.1) Turner et al., 1984, hereafter TOG. We express the projected mass by

M(≤ ξ) = 2π ∫_{0}^{ξ} Σ(ξ)ξdξ  (2.7)

where Σ(ξ) is the surface mass density distribution of the lens given by Equation (2.5). Combining Equations (2.6) and (2.7) we get the deflection angle in the lens plane

α̂ = ±4π \frac{σ^{2}}{c^{2}}  (2.8)

which gives a constant deflection angle, characteristic of the singular isothermal sphere (SIS)-lens. The effects of a constant deflection angle (independent of ξ) are illustrated by a ray tracing diagram in Figure 2.4, which can be compared the the deflection from a point mass in Figure 2.2.

Figure 2.4: Ray tracing diagram for the SIS lens model

Transforming the true deflection angle α̂ (lens plane) to the reduced deflection angle α (in the observer plane), by inserting Equation (2.8) into Equation (2.2), introduces an additional factor A_{ls}/A_{s}

α = α̂\frac{A_{ls}}{A_{s}} = ±4π \frac{σ^{2}}{c^{2}} \frac{A_{ls}}{A_{s}}  (2.9)

which is the deflection angle measured in the observers plane and the expression we shall apply in the following text.

In most previous related work, it has been common to model the effects of dark matter halo masses in addition to the dynamics of central velocity dispersions. The one-dimensional velocity dispersion, σ, was weighted by a \sqrt{3/2} factor, to maintain dynamical consistency with the addition of dark matter halos (Turner et al., 1984) (Fukugita and Turner, 1991, hereafter F&T). This has, however, been claimed to be a poor method for modeling the dark halos of galaxies (Kochanek, 1993b, 1994). See the next Section 2.2.2 on how this is treated in the present work and Section 2.2.3 for our altered estimate of the lens effectiveness parameter F found in F&T.

2.2.2 Lens population

In the following work we only treat lensing by isolated galaxies, thus the additional shear of surrounding (dark) matter (e.g. in clusters) is neglected. To include such effects one would have to add a constant sheet of matter [TOG] or apply an elliptical lens, where the magnification bias then would have to be computed numerically^{6}. The population of potential lenses, situated between the source (the quasar) and the observer, mostly consists of earlytype E and S0 galaxies and late-type spiral galaxies. It has, however, been claimed that E and S0 galaxies are the most efficient lenses, while S (spiral) galaxies only contribute about 14% of the lensing events due to their small velocity dispersions [F&T]. The lens-effectiveness (i.e. the F parameter from F&T) of E and S0 galaxies may, however, be overestimated if the \sqrt{3/2} factor for dark matter halos does not exist (see Section 2.2.1). We re-estimate the F&T lens effectiveness parameter F in Section 2.2.3 without the dark matter (DM) dispersion factor, to illustrate the possible importance of spiral galaxies and for comparison of the F=0.047 value deduced in F&T. We return to the question of dark matter dispersion correction factor in Chapters 4 and 5.

Having estimated the individual probability for a galaxy to produce a lensing event, we now estimate the population distribution of these galaxies by the following Schechter galaxy-luminosity function (Schechter, 1976)

ϕ_{g}dL = n_{∗} (\frac{L}{L_{∗}} )^{α} exp(−\frac{L}{L_{∗}})dL/L_{∗}  (2.10)

where n_{∗} = (1.56 ± 0.4) × 10^{−2}h^{3}Mpc^{−3} and α = −1.1 ± 0.1, following Efstathiou et al. (1988). We have assumed h = H_{0}/100kms^{−1}. Separating the two types of lenses (early E, S0 and late S galaxies), we apply the fractional abundances n_{E} = 1.94 × 10^{−3}h^{3}Mpc^{−3}, n_{S0} = 3.06 × 10^{−3}h^{3}Mpc^{−3} and n_{S} = 1.1 × 10^{−2}h^{3}Mpc^{−3} for S galaxies (see also Table 2.1)(Postman and Geller, 1984). The characteristic galaxy luminosity, L_{∗}, is commonly used to normalize the galaxy luminosity, L. We also normalize the velocity dispersion, population, etc., by the corresponding values of a L_{∗}-galaxy. The ratio between these two observables is, however, parameterized/substituted by the one-dimensional central velocity dispersion, σ of that galaxy.

6see section 5.3 on how this is done using two-dimensional elliptical lens models

L = L_{∗} (\frac{σ}{σ_{∗}} )^{γ} ⇒ σ = σ_{∗} (\frac{L}{L_{∗}} )^{1/γ}  (2.11)

where the exponent is taken to be γ = 3.7 for E and S0 (late-type) galaxies and γ = 2.6 for S (spiral) galaxies (Tully and Fisher, 1977; Faber and Jackson, 1976). The measured central velocity dispersions for late-type E, S0 and early-type S galaxies and the relative abundances are summarized in Table 2.1. Assuming there are significant dark halos, we need to include the effects of its dispersion in our model. This is commonly achieved by multiplying the velocity dispersion by the TOG \sqrt{3/2}-correction factor. The contribution of dark matter can instead, however, simply be estimated by setting the dark matter dispersion equal to the central velocity dispersion (i.e. we just use the Tully-Fisher/Faber-Jackson relation, expressed by Equation (2.11)) (Kochanek, 1994). We return to this matter in the next section and estimate values for the lens effectiveness parameter F in Section 4.3.

Furthermore we assume that the number of lenses is constant in a comoving (i.e. non-evolving) volume, hence the density n_{L} of lenses is given by

n_{L} = n_{0}(1 + z_{l})^{3}  (2.12)

where the (1 + z_{l})^{3} factor arises from the fact that the co-moving volume decreases with increasing z, due to the expansion of the universe. The present density of isolated galaxies, n_{0}, has been found from observations (values quoted in next section).

2.2.3 Optical depth for lensing

The optical depth for lensing can be defined as the probability that a source (a quasar) lies within the (projected) Einstein radius of a lens (E, S0 or S galaxies). We follow the deduction of TOG for an isothermal sphere, using the expressions for angular distance A and metric (proper) distance D, as defined in Appendix ??. Note that we derive the lensing probability (optical depth) for an Einstein-de Sitter (Ω_{M} = 1, Ω_{Λ} = 0) model, but the result is valid for all flat (k=0) universe models (Gott et al., 1989). The differential probability distribution, dτ, for lensing by an object at a distance between X and X + dl (proper distance) in a flat (k=0) universe is simply expressed by the product of the probability for a single isolated galaxy to produce a lensing event, σ_{τ}, (lensing cross section) and the number of lenses, n_{L}, times the proper distance step dl

dτ = n_{L}σ_{τ}dl = n_{L}σ_{τ} \frac{cdt}{dx}dx  (2.13)

where x = 1 + z_{l} and σ_{τ} = π(b_{cr}A_{l})^{2} is the cross section for strong lensing^{7}. The critical radius b_{cr} is the angular radius corresponding to the effective Einstein radius produced by a singular isothermal sphere (SIS)-lens (see Eq. (2.9))

b_{cr} = 4π (\frac{σ}{c} )^{2} \frac{A_{ls}}{A_{s}} = 4π (\frac{σ}{c} )^{2} \frac{1/(1 + z_{l})(D_{s} − D_{l})}{1/(1 + z_{l})D_{s}} = 4π (\frac{σ}{c} )^{2} \frac{D_{ls}}{D_{s}} (2.14)

where the lens-source conversion from angular size distance to proper distance D_{ls} = [(1 + z_{s})A_{s} − (1 + z_{l})A_{l}]/(1 + z_{s}) is found in TOG (Eq.(A.3)). Note also that the critical radius (in the lens plane) is the product of the angular radius, b_{cr}, and the angular distance, A_{l}.

Using Equations (2.12), (2.13) and

\frac{dt}{dx} = H_{0}^{−1}x^{−5/2},

which is taken from TOG, we are able to deduce a differential expression for the optical depth, dτ

dτ = n_{0}x^{3}π(b_{cr}A_{l})^{2} \frac{c}{H_{0}} x^{−5/2}dx

and inserting for b_{cr}, yields

dτ = 16π^{3}n_{0} (\frac{σ}{c} )^{4} x^{1/2}R_{0}A_{l}^{2} \frac{D_{ls}^{2}}{D_{s}^{2}} dx

which, after insertion of the angular and proper distances for Ω_{M} = 1 and Ω_{Λ} = 0 (given in Appendix ??), gives

dτ = 4 × 16π^{3}n_{0}R_{0}^{3} (\frac{σ}{c} )^{4} \frac{(x^{1/2} − 1)^{2}}{x^{7/2}} [\frac{y^{1/2} − x^{1/2}}{y^{1/2} − 1} ]^{2} dx  (2.15)

7strong lensing is defined to be satisfied when the QSO is multiply imaged (into two images), which occurs when the source is projected within the critical radius, b_{cr} (the Einstein radius)

where R_{0} = \frac{c}{H_{0}} is the Hubble radius and y = 1 + z_{s}. We also find an expression for n_{0}σ^{4} from Equation (2.10) and (2.11)

n_{0}σ^{4} = σ^{4} ∫_{0}^{∞} ϕ_{g}dL = n_{∗}σ_{∗}^{4} ∫_{0}^{∞} (\frac{L}{L_{∗}} )^{α+4/γ} exp(−\frac{L}{L_{∗}})dL/L_{∗}

which can be simplified by using the gamma function Γ(a) = ∫_{0}^{∞} X^{−a−1} exp (−1/X)dX and substituting X = \frac{L}{L_{∗}}

n_{0}σ^{4} = n_{∗}σ_{∗}^{4}Γ(1 + α + \frac{4}{γ}).  (2.16)

Integrating Equation (2.15) over all redshifts from 0 to z_{s}, we have the following expression for the optical depth τ

τ = \frac{4}{15}16π^{3}n_{0}R_{0}^{3} (\frac{σ}{c} )^{4} \frac{(y^{1/2} − 1)^{3}}{y^{3/2}}

and inserting Equation (2.16)

τ = \frac{1}{30}16π^{3}n_{∗}R_{0}^{3} (\frac{σ_{∗}}{c} )^{4} Γ(1 + α + 4/γ)\frac{8(y^{1/2} − 1)^{3}}{y^{3/2}}

which after using the expression for the source proper distance, finally becomes

τ = \frac{1}{30}τ_{∗}Γ(1 + α + 4/γ)D_{s}^{3}  (2.17)

where τ_{∗} = 16π^{3}n_{∗}R_{0}^{3}(σ_{∗}/c)^{4}. The lens effectiveness parameter F, introduced by TOG, is now expressed as

F = \frac{16π^{3}}{cH_{0}^{3}} ⟨n_{0}σ^{4}⟩ = \frac{16π^{3}}{cH_{0}^{3}} Γ[1 + α + 4/γ]n_{∗}σ_{∗}^{4}  (2.18)

Note that, σ_{∗} or F, is the only free parameter in our statistical model. We shall see, in Section 4.2, that the statistics are quite sensitive to this parameter. Using Eq. (2.18), we compute a new value for the lens effectiveness parameter, F, without the \sqrt{3/2} correction factor, using the observational values of F&T. In their work they applied the DM dispersion correction factor, which yielded F = 0.047. Our estimated F parameter is therefore obviously lower than the F&T value. As we shall see in Section 4, the statistically deduced F parameter is significantly higher than the present value deduced directly from galaxy populations.

It is worth noting that if the \sqrt{3/2} DM correction factor is excluded, the spiral galaxies contribution to the total F -parameter (cf. Table 2.1) should Table 2.1: The relative abundance, central velocity dispersions and computed F values of late-type E/S0 galaxies and early-type S spiral galaxies.

type γ  abund. n_{∗}[h^{3}Mpc^{−3}] σ_{∗} [kms^{−1}] F
E  3.7_{−1}^{+1} 12 ± 2% 1.87 × 10^{−3} 225_{−20}^{+12}  0.0080 ± 0.0035
S0  3.7_{−1}^{+1} 19 ± 4% 2.96 × 10^{−3} 206_{−20}^{+12}  0.0089 ± 0.0045
S  2.6_{−1}^{+1} 69 ± 4% 1.08 × 10^{−2} 143_{−13}^{+8}  0.0066 ± 0.0023
total  100%  1.56 × 10^{−2}  0.0236 ± 0.0103

be accounted for. The new deduced value of the lens effectiveness parameter excluding the DM factor is F = 0.024 ± 0.01.

Equation (2.17) gives the probability for cosmologically distributed singular isothermal sphere (SIS)-galaxies to produce doubly imaged gravitational lensing events of distant QSOs. In Section 4.1 we apply the optical depth, given by Equation (2.17), to express the differential lensing probability.

2.3 The QSO Luminosity Function

When we model/estimate a population of an astronomical object-type (e.g. quasars, galaxies), it is most common to represent this population through what is known as a luminosity function (LF). The luminosity function (LF) gives the number count of an object-type as a function of the intrinsic/absolute magnitude. In Section 2.4 it will become apparent that we need knowledge of the quasar population or more precisely the quasar number count distribution as a function of apparent magnitude. The population of quasars is difficult to estimate from empirical data, because of various observational limitations such as magnitude limits and resolution^{8}. These observational limits impose uncertainties in the determination of the luminosity function (LF) for faint objects such as quasars and galaxies. It is thus a difficult task to estimate the QSO luminosity function (LF), when at the same time biasing must be prevented from distorting the true data. The problem of determining a consistent luminosity function (LF) has recently been illustrated by Hawkins and Véron (1993, hereafter H&V), where they suggest that the break^{9} in the QSO luminosity function (LF) could possibly be an artifact of limiting magnitudes in previous surveys such as Boyle et al. (1988)(hereafter BSP) and Hartwick and Schade (1990)(hereafter H&S). Lens statistics is, however, very sensitive to the break, because it gives rise to a strong magnification bias, and suggests there should be such a break in the luminosity function (LF) (Kochanek, 1993a). See Section 2.4 for a more thorough discussion on the magnification bias.

8see Section 3.1 for a discussion on observational limitations

9this is the much discussed break in the LF of quasars and can be seen both in Figures 2.5 and 2.6

The existing number counts of quasars from the different surveys (mentioned above) are incomplete in the redshift range 0.3 < z < 2.2, leaving out a faint sub-population of distant QSOs. Due to this incompleteness in redshift of the QSO surveys, the estimates of the faint end number counts are poor. We correct for this incompleteness by applying a hybrid model which expands the redshift interval to 0.3 < z < 5. Following Wallington and Narayan (1993) we use their hybrid QSO luminosity function (LF), Φ(z, M_{B}), which is taken from the BSP-data (0.3 < z < 2.2, model B)

Φ(z, M_{B}) = \frac{Φ^{∗}}{10^{0.4[M_{B}−M_{B}(z)](α+1)} + 10^{0.4[M_{B}−M_{B}(z)](β+1)}} Mpc^{−3}mag^{−1} (2.19)

for z ≤ 3, while the extended part for the redshift interval 3 < z < 5 is^{10}

Φ(z, M_{B}) = \frac{Φ^{∗} × 10^{−\{A+B[0.4(β+1)]\}}}{10^{0.4[M_{B}−M_{B}(z)](α+1)} + 10^{0.4[M_{B}−M_{B}(z)](β+1)}} Mpc^{−3}mag^{−1} (2.20)

where Φ^{∗} = 7.8 × 10^{−10}, α = −3.79 and β = −1.44. The BSP model is calculated assuming an Einstein-de Sitter universe model with H_{0} = 50 kms^{−1}Mpc^{−1}. Investigation of QSO surveys, such as BSP, shows that the QSO luminosity function evolves (with redshift). This mainly affects the assumed LF break, which tends to evolve towards brighter absolute magnitudes. The redshift dependent absolute magnitude of the break is given by

M_{B}(z) = M_{B}^{⋆} − 2.5k_{L} log(1 + z)  (2.21)

where k_{L} = 3.15 and M_{B}^{⋆} = −22.42. Equation (2.21) is valid for z < 3 apart from the flat interval^{11}, while for z > 3 the break continues to “slide” down along the high-luminosity end of the luminosity function (LF) (see Figure 2.5) by the following relation

10we have thus implicitly assumed that the observable universe does not exceed z = 5

11for 2.5 < z < 3 the LF is flat and does not evolve, i.e. the break is the same as for z = 2.5 (Wallington and Narayan, 1993, see)

M_{B}(z) = M_{B}^{⋆} − 2.5k_{L} log(1 + z) + B  (2.22)

where B = A/(0.4(α − β)) and A = (z − 3) log(3.2). The hybrid model is shown in Figure 2.5 for five different redshift values. Note the different locations of the break given by Equations (2.21) and (2.22).

Figure 2.5: The hybrid luminosity function for different redshift values (z = [1, 2, 3, 4, 5]), displaying the “sliding” of the redshift dependent break. Observational measurements of the LF for 1.9 < z < 2.2 from several surveys gathered in Hartwick and Schade (1990) is indicated by +

We now sum the different QSOs from the LF which correspond to a magnitude brighter than a certain apparent magnitude limit m_{B}. This cumulative number count of the luminosity function (LF) model is used to estimate how many QSOs we are able to observe at an apparent magnitude limit, m_{B} (see Section 2.4.2). The cumulative number count is computed by integrating the luminosity function (LF) over all redshifts and for all magnitudes up to the magnitude limit, m_{B}. The following expression gives the cumulative number counts taking into account the absolute blue magnitude and redshift for QSOs brighter than an apparent blue magnitude, m_{B}

N_{q}(< m_{B}) = ∫_{0.3}^{5.0} ∫_{−∞}^{M_{B}^{lim}} Φ(z, M_{B})dM_{B}dz  (2.23)

where M_{B}^{lim} is the absolute magnitude limit which corresponds to the survey magnitude limit m_{B} for each redshift. For these computations we have used the Einstein-de Sitter model (Ω_{M} = 1 and Ω_{Λ}) and H_{0} = 50kmsMpc. The conversion between apparent magnitudes and absolute magnitudes are given in Section ??. The resulting number counts of quasars for the hybrid luminosity function (LF) model is shown in Figure 2.6 along with the Hartwick and Schade (1990) (dotted line) best fit, which was used in Surdej et al. (1993a); Kochanek (1993a). Note that the discrepancy between the Hartwick and Schade (1990) fit and our applied LF is partly due to the fact that the latter is complete to z ≤ 5.0. Furthermore there is also a possibility that the overdensity in the observed data (’+’ and ’⋄’) at bright magnitudes is due to a bias effect. We do not claim this to be the case, but since the magnification is expected to be more significant at bright magnitudes (cf. the magnification bias in the next section) this might explain the discrepancy and the nature of highly luminous quasar (HLQ)s.

Figure 2.6: Cumulative number-magnitude plot of our LF (valid for z < 5) adapted from Wallington and Narayan (1993) (solid) and the best-fit to the number-count (valid for z < 2.2) from Hartwick and Schade (1990) (dotted). The data points for the number-count from Hartwick and Schade (1990) and Hawkins and Véron (1993) are indicated by + and ⋄, respectively.

2.4 The Magnification Bias

The gravitational lensing of distant quasars not only may produce multiple images, but also significantly magnify^{12} an image, which in effect increase the flux due to the larger area of the image. In fact all sources at larger distances than the lens in the vicinity of the line-of-sight to the quasar are magnified by a certain amount. In a survey such as ours, where we investigated quasars with magnitudes brighter than some given magnitude limit, m_{B},^{13} there is a biasing effect. The bias effect arises from the fact that QSOs with originally dimmer magnitudes than the survey limit are magnified to magnitudes above the limit. This implies that there should be an over-density of background objects^{14} in the vicinity of a gravitationally lensed quasar. Estimates of such over-densities could give an independent measure of the magnification bias, thus giving an alternative indirect method of estimating the QSO luminosity function. We will shortly discuss the possibility of estimating such overdensities by the use of star, galaxy and QSO counts in Section 4.2 (Schneider et al., 1992; Van Drom et al., 1993; Claeskens and Surdej, 1994, see also).

2.4.1 Magnification probability distribution

The singular isothermal sphere (SIS)-lens can only produce either a single magnified image^{15} or a triple image, which only consists of two observable images because the third image is strongly demagnified by the singular isothermal sphere (SIS) lens galaxy and is therefore “invisible” (Schneider et al., 1992, p.243-254). This image configuration is usually refered to as the “double image” and is the only configuration other than the magnified single image a singular isothermal sphere (SIS) lens is able to produce. The magnification of images can be found from the image positions, which are given by the lens equation (2.1) using Eqs. (2.2) and (2.9)

θ_{1,2} = θ_{s} + α = θ_{s} ± b_{cr}  (2.24)

where 1 and 2 refer to the two images produced of a source within the critical radius of a singular isothermal sphere (SIS) lens. The reduced deflection angle Eq. (2.9), which correspond to the critical angular radius, give the image positions at each side of the true source position, θ_{s}

12see Section 2.1 for a short explanation of the term magnification of sources

13the flux(magnitude) limit is discussed in Section 3.1

14the background objects are more distant than the lens, e.g. associated/unassociated QSOs or faint blue galaxies

15the only effect from the lens is the magnification of background objects close to the line-of-sight to the lens

θ_{1,2} = θ_{s} ± 4π \frac{σ^{2}}{c^{2}} \frac{D_{ls}}{D_{s}}  (2.25)

where the critical radius b_{cr} corresponds to the Einstein ring of a singular isothermal sphere (SIS)-lens and the magnification of each image is found by means of the relation

M_{1,2} = |\frac{θ_{1,2}}{θ_{s}} \frac{dθ_{1,2}}{dθ_{s}} | = |\frac{(θ_{s} ± b_{cr})}{θ_{s}} \frac{d(θ_{s} ± b_{cr})}{dθ_{s}} |  (2.26)

which for a two-dimensional lens-model (e.g. an elliptical lens model) corresponds to the inverse determinant of the Jacobian matrix (i.e the inverse determinant of the transformation θ_{s}(θ_{i})) (Blandford and Narayan, 1992). In the two-image case, however, Equation (2.26) leads us to the magnification M_{1} = 1 + b_{cr}/θ_{s} for the first image and M_{2} = b_{cr}/θ_{s} − 1 for the second. The total magnification, M is simply the sum of M_{1} and M_{2} and can also be expressed in terms of the flux ratio f

M = 2\frac{b_{cr}}{θ_{s}} = 2\frac{f + 1}{f − 1}  (2.27)

where the latter relation is found by using the obvious relation f = M_{1}/M_{2} for the total magnification M.

In the case of a lensing event^{16}, we must also estimate the magnification probability, which decreases as the source is situated closer to the observergalaxy line-of-sight. The slope of this probability distribution as a function of magnification, is determined by the caustics and cusps produced by the lens. The magnification probability therefore intrinsically depends on the lens model. The exact integral magnification probability distribution for the singular isothermal sphere (SIS) model is therefore, from the equations above, given by

P_{M}(> M) = 4/M^{2}  (2.28)

[Eq. (A.18) TOG]. The expression is normalized in relation to the definition of M, being greater then 2. As we will apply differential probabilities, we need the differential magnification probability distribution, which is found by differentiating the integral probability function (Eq. (2.28)).

16by lensing event we mean “strong” lensing, which occurs when the source is projected within the critical radius (the Einstein radius)

\frac{dP_{M}}{dM} = |\frac{d}{dM} (\frac{4}{M^{2}} ) | = \frac{8}{M^{3}}  (2.29)

This magnification distribution is an integral part of the magnification bias. The integral magnification probability can simply be thought of as the probability of obtaining a magnification larger than a certain value M. The differential probability for a SIS galaxy to produce a certain magnification in the range M, M + dM, is then represented by Equation (2.29).

2.4.2 Differential magnification bias

We will estimate the magnification bias directly from the QSO LF presented in Section 2.3. The differential magnification bias, simply being the overrepresentation of QSOs in a magnitude limited sample (e.g m_{B} < 18.5) due to the magnification of originally dimmer QSOs to magnitudes brighter than the magnitude limit m_{B}, is given by

B(m, M) = \frac{dP_{M}}{dM} \frac{N_{q}(m + 2.5 log M)}{N_{q}(m)}  (2.30)

where dP_{M}/dM is the magnification probability distribution which is expressed by Equation (2.29) for our particular singular isothermal sphere (SIS) lens model. The overdensity of lensed QSOs for a certain magnification M is represented in Equation (2.30) by the ratio between the number of QSOs at a magnitude including the magnification M, to that without a magnification (i.e. just the magnitude). The overdensity of lensed QSOs (the ratio between number counts including M and without) is illustrated for different magnifications in Figure 2.7 along with the corresponding differential magnification bias. The magnification bias will enter in the computations of the lensing probability in Section 4.1.

Figure 2.7: The overdensity of detectable QSOs due to the magnification by the SIS lens (left) and the differential magnification bias (right), i.e. the overdensity folded with the differential magnification probability.

26  Gravitational lensing - theory and modeling

Observations and the QSO sample

The goals of gravitational lensing (GL)-survey’s are to perform statistical analysis of the lensing phenomena, derive specific parameters that determine the frequency of lensing and ultimately fundamental parameters of the universe (see appendix ??). The chances of finding gravitational lensing (GL)-systems depend on the sample of quasars we choose to investigate. A sample of QSOs, however, is subject to favorable and/or un-favorable selection effects. We will therefore discuss these selection effects and their impact on our sample in Section 3.1. In Section 3.2 we discuss the procedures of image pre-processing and reduction of spectra. Section 3.3 discuss the procedures of selecting gravitational lensing (GL) candidates and the problem of false positives and false negatives. Finally, in Section 3.4, we investigate our abilities to detect gravitational lensing (GL)-systems and propose a selection function, accurately defining the area (in the magnitude difference - separation plane) in which our particular observational results are expected to be sufficiently correct.

3.1 Selection and observation of the sample

We begin with the determination of the sample that we shall investigate. A sample of QSOs (with our intentions) should be evaluated thoroughly, because it is often chosen among previously discovered QSOs. Among the lens surveys conducted in the previous years, most have been selected from already existing QSO catalogs. Furthermore it has been suggested that many QSO catalogs are biased against the inclusion of elongated or diffuse QSOs, because of the selection routines used in search for QSOs. According to Kochanek (1991), some GL candidates and real GL-systems are lost in surveys searching for GL’s as a direct consequence of the inherent selection effects in QSO catalogs. Elongated QSOs are interesting candidates to gravitational lensing, because closely separated GL-systems would appear elongated under (sub-)critical seeing conditions^{1}. Depending on the seeing conditions, which dictate the resolution and detection of small separations at ground-based observatories, the selection effect might play an important role in GL-surveys. Yet there is the possibility that the small separation candidates would in any case be excluded by a realistic selection function. We will therefore conduct tests through simulations in Section 3.4 to estimate the effects of limited resolution on the detectability of candidates.

Furthermore, a color criterion used in some QSO surveys when distinguishing between a star and a QSO candidate might also affect the selection of QSO candidates. Gravitationally lensed QSOs are expected to have small color variations between the individual images, because lensing is not wavelength dependent. A GL-system, however, where the QSO is obscured by the lensing galaxy is often subjected to a reddening effect (from the red color of the galaxy) thereby affecting the assumed color alikeness. This would obviously endanger a gravitationally lensed QSO from being recognized as a QSO (by its colors) in many QSO surveys, thereby affecting the completeness of the QSO catalog. This effect was pointed out by Kochanek (1991) where as much as 30% of the gravitationally lensed QSOs are believed to be excluded from QSO catalogs. (Surdej et al., 1993a) included this in their computations of the lensing frequencies from GL surveys, thus multiplying the optical depth by a correction factor S_{cat} = 0.7. This factor is also included in the present computations, as most of the QSOs investigated here are chosen under similar conditions.

Our QSO sample (listed in appendix ?? and illustrated in Figure 3.1) was selected from 3 different QSO lists compiled by, C. Hazard, D. Reimers and J. Surdej. The latter consisting of QSOs chosen from the Véron-Cetty and Véron (1991)-catalog, while the two others are independent QSO surveys. Reimer’s group (the Hamburg Sternwarte (HS) group) identifies QSO candidates first from objective prism plates and are reobserved spectroscopically. Unfortunately some of the QSOs in our sample were difficult to identify on the images taken, due to crowded or in some cases empty CCD-fields. Inaccurate telescope pointing, poor QSO position accuracy or wrong coordinates were the causes for these mishappenings, which thereby reduced the total sample by 10 QSOs. Finding charts for the QSOs would have reduced this amount and should therefore be available to future GL surveys. The recent availability of the digitized Palomar Sky Atlas makes it possible to produce deep (m_{V} < 21) small field finding charts of the area surrounding the QSO, allowing easy identification of the QSO.

1in fact bearing in mind the highly variable seeing and poor resolution of sky-plates, from which the QSOs often are chosen, the incompleteness accusations of QSO catalogs seem sensible

Figure 3.1: Magnitude-redshift distribution of the 166 NOT sample, each object indicated by ∗. The small dots represent the QSOs from the Véron-Cetty and Véron (1991) and the Hewitt and Burbidge (1993) catalog. The solid lines indicate the absolute magnitudes −26,−27 & −30 and are computed, as described in Appendix ??, for an Einstein-de Sitter universe model (Ω_{M} = 1, Ω_{Λ} = 0).

Most of the larger optical GL-surveys optimize the effectiveness of finding gravitational lensing (GL)’s in their surveys by selecting intrinsically bright QSOs (e.g. M_{V} ≤ −26). This is achieved by choosing high-redshift QSOs with bright apparent magnitudes (m_{V} ≤ 18.5). Such QSOs are often called highly luminous quasars (HLQs), because their intrinsic luminosities are unusually high. The strong correlation between lensing probabilities and bright absolute magnitudes is discussed in Surdej et al. (1993a). The effects of selecting a bright sample (m_{V} ≤ 18.5) with a limitation on the magnitude (setting a flux limit) will be discussed in Section 2.4. Most of the QSOs from our sample (marked by ∗) tend to lie in the lower part of Figure 3.1 (beneath the line indicating absolute magnitude M_{V} = −27), hence at the high intrinsic luminosities. The irregular pattern of the absolute magnitude lines is due to the correction for emission lines appearing in the V-filter for various redshifts. The emission line correction, K-correction and computation of absolute magnitudes is discussed in Véron-Cetty and Véron (1991).

The selection of a flux limited sample of HLQs (see Page 29) is consistent with the previous joint sample of Surdej et al. (1993a); Crampton et al. (1992); Yee et al. (1993) and Maoz et al. (1993) (hereafter the SCYM-sample). Among the 469 highly luminous quasar (HLQ)’s from the SCYM-sample, only 23 were duplicated in the 166 highly luminous quasar (HLQ) NOT sample, leaving 143 unduplicated. The apparent magnitude and redshift distribution of the NOT sample is shown in Figure 3.2 and 3.3 respectively. The median apparent magnitude is m_{V} ∼ 17.5, the median redshift is z ∼ 2.0 and the median absolute magnitude is M_{V} ∼ −27.7.

Figure 3.2: Apparent magnitude histogram for the 166 QSOs sample Figure 3.3: Redshift histogram for the 166 QSOs sample

3.2 Image processing and data reduction

The extensive use of Charge Coupled Device (CCD) detectors in modern astronomy has created a whole new field of research. Not only in the technical field of creating larger and more sensitive detectors, but also in the digital image-processing field. The Hubble Space Telescope (HST)’s spherically abberated images, especially boosted the need for powerful and effective post-processing software. In Section 3.2.1, however, we describe the basic procedures of image pre-processing which was performed on the direct images^{2} of the QSOs which we have investigated. We have used IRAF^{3} with SAO-image to pre-process the images and performed photometry using the IRAF/DAOPHOT II-package^{4}. IDL^{5} was also extensively used to process and investigate data using self-written procedures, especially for the resampling, shifting and combination technique described in Section 3.5. Modest reductions of spectra were also performed in IDL, and a short description of our reduction steps is presented in Section 3.2.2. Among the various software made, was also a database containing all relevant information of each object exposure for the organization and handling of the relatively large amount of data.

3.2.1 Pre-processing of direct images

Image pre-processing is the necessary task for correcting direct images for various artifacts arising from the CCD detector. This task is especially important when photometry is to be carried out on the faint QSOs (m_{V} ≈ 18). We shall briefly outline the most basic pre-processing tasks, used in the reductions of the direct images obtained using the Stockholm TEK512 CCD at Nordic Optical Telescope (NOT). Furthermore we also outline the stepwise process in performing these tasks using IRAF. Specifications and documentation of the S-CCD may be obtained in (Kjeldsen, 1990) or through the NOT Gopher menu using e.g. Mosaic

> Mosaic gopher://zoo.ll.iac.es/

which is an information menu^{6} for the various sites of Instituto de Astrofisica Canarias (IAC) at the Canary Islands, Spain (Tenerife and La Palma).

For an illustrative introduction and description of the basic reduction steps, see Kjeldsen (1992).

2by direct images we mean two dimensional digital image arrays

3Image Reduction and Analysis Facility (IRAF), NOAO, Tucson, Arizona, USA email:iraf@noao.edu

4DAOPHOT II is an aperture and PSF photometric package by Stetson (1992), interfaced to IRAF

5Interactive Data Language(IDL), Research Systems, Inc. (c)

6choosing the Nordic Optical Telescope (NOT) item from this menu will bring you to the Nordic Optical Telescope (NOT) menu, from which you can get documentation and specifications of the various instruments at Nordic Optical Telescope (NOT)

Preparing the data for processing

All images are stored in the common and convenient image format Flexible Image Transport System (FITS) (and were read into IRAF using the task rfits), which also consists of an image header. The image header consists of valuable information about the image such as exposure time, filter, date, time and object position (alternatively zenith distance). This header is of great importance, especially if there is a substantial number of images to be reduced. The ccdred procedure under IRAF takes advantage of the relevant information about the individual exposure and automatically performs the pre-processing tasks on all of the images. Unfortunately, the outdated software controlling the Stockholm CCD-camera at Nordic Optical Telescope (NOT), does not store this relevant information in the image-headers. We therefore had to insert this information into each image, using the task hedit. Basic relevant information inserted was filter type, exposure time and optionally the zenith distance(for determining the airmass). Each frame is also given an image type code zero, flat or object for bias, flatfield and object frames, respectively. The inclusion of these few items of information, significantly simplifies the otherwise tedious tasks of pre-processing. These tasks, described below, are performed in the order of which they appear and performed separately for each of the four observational runs of which the NOT GL-search program consisted of. The observational runs were performed in November 1990, July 1991, January 1992 and June 1992. In addition, two re-observation programs initated during the thesis, were also granted observations at the NOT. These two observational programs consisted of a run in December 1993 and a second in May 1994. These will also partially provide data for the investigation of small separation lens-system candidates.

Non-linearity

Figure 3.4: Non-linearity in the NOT-CCD Most CCDs should be fairly linear in their response to the amount of light they receive, but there are some that do have a noticeable amount of nonlinearity. This can be corrected by mapping the nonlinearity effect as a function of pixel value.

In the case of the Stockholm (NOT) CCD camera, this non-linearity effect was indeed present in the observations performed earlier than April 1993. To correct for non-linearity we have applied a program written by Jens Hjorth, which uses statistical data from Hans Kjeldsens (Figure 3.4) work on determining the non-linearity in the NOT-CCD (a Tektronix 512 chip) [(Kjeldsen, 1990)]. However, in the case of the reobservations, performed in December 1993, no correction for non-linearities were necessary. Due to the change of an amplifier unit in the S-CCD, tests show the non-linearity has been reduced to below 1.0% [(Nilsson, 1993)].

Bias

The CCD chip adds a (more or less) uniform level of counts (analog-digital units (ADU)) to the image before the image is read from the chip. This is to avoid potential negative counts (e.g. arising from bad pixels) when the chip is read out. All images are therefore subtracted by a single average or median scalar value of an overscan region on the chip (an unexposed region of the chip, called the overscan strip). Due to the non-uniform nature of the added bias level, there is also a need to readout a complete image which is not exposed (exposure time zero), to map the bias level’s spatial variation. This bias frame is also subtracted from the exposed image. Both of these tasks are carried out by the ccdproc procedure after the overscan region has been defined (epar ccdproc and enter [1:520,514:520] in the bias section) and the bias (zero) frames are combined using zerocombine. Note the bias/zero frames are also subtracted by the bias level scalar, therefrom the name “zero” frame. All bias frames from an observational run are thereby combined and subtracted from the remaining flatfield and object frames.

We do not correct for dark currents, as this is neglectable in the S-CCD [(Kjeldsen, 1990)], especially for our purposes.

Flat-Fielding

Due to the fact that each pixel in the CCD has a varying sensitivity to light, the photon-responsitivity varies over the CCD-chip’s field. This response pattern changes over time and with wavelength(filter), therefore the best way to correct for this effect is by exposing several frames while the CCD is evenly illuminated, thereby mapping the sensitivity. This is usually done by exposing an empty region of the sky in the twilight at dawn. The flat-field frames of the same night and filter were then combined, using flatcombine.

The object images are then divided by the combined flat-field frames of the corresponding night and filter to correct for the irregular sensitivity of the CCD chip.

Running the ccdproc task again divides e.g. R-filter object frames by the corresponding R-filter combined flat-field frame. The resulting object images can then be stored in FITS-format again, using the wfits command.

3.2.2 Spectroscopic observations and reductions

All spectroscopic data were obtained using the low dispersion spectrograph (LDS) at NOT. We used both the red [6000, 10000 Å] and blue [3900, 4900 Å] channels. Specifications of the Aarhus low dispersion spectrograph (LDS) can be obtained either in Thomsen (1990); Jensen (1992), or by retrieving documentation using Mosaic at the same address given in Section 3.2.1.

The image processing routines are somewhat different for spectroscopic data as for the direct images. The bias subtraction is done in a similar manner as described in Section 3.2.1. In the case of CCD non-linearities, corrections for this would have to be performed at this stage (as for the direct images), but fortunately it is unnecessary to perform such corrections for the LDS-CCD data. Next we correct for the CCD-detector’s irregular sensitivity over the chip and is, in similar ways as described in Section 3.2.1,corrected by dividing the image by an evenly luminated image. Using IDL, we have created a quick-look software to allow simple and easy reductions of the few spectra obtained at NOT. The two-dimensional^{7} images are first reduced into 2 one-dimensional arrays, one for each object’s dispersion axis. Next we must remove the night sky lines, which are (strong) emission (and absorption) lines from various elements/gasses in the earth’s atmosphere. These lines are quite easily removed by subtracting regions of the spectra which are only exposed to the scattered radiation from the atmosphere (e.g use dispersion axis between two objects in the spectra). The “empty” regions are the regions along the spatial axis of the spectra in which no objects lie, i.e. the dispersion axis is only exposed to the scattered radiation of the atmosphere. The final task is to correct for the spectrograph/CCD’s wavelength dependent sensitivity. For this, we make calibration images of standard stars, for which the continuum is known. The resulting spectra are also wavelength calibrated using either the night-sky lines, calibration lamps or spectra of standard stars. This task includes the correction for differential sampling of the spectrograph, i.e. there is a certain non-linearity in the wavelength dispersion due to imperfect grisms.

7the spectrographic-slit image consists of a one-dimensional spatial axis along the slit and a dispersion axis

Figure 3.5: Illustration of slit-spectroscopy. Lower panel shows 2D-spectral image, while upper panel shows a single combined row of strongest component (QSO) along dispersion-axis. Numbers along the ordinate and abcissa axis are irrelevant.

3.3 Selecting the GL candidates

A survey’s completeness is affected by inescapable limitations of the observations in the form of detectability (dynamical range) and image quality (resolution and seeing). It is therefore essential to estimate these effects through the definition of a selection function for the particular survey, if the statistical results are to be realistic. We will define such a selection function in Section 3.4. There is also the problem of false identification of gravitational lensing (GL) candidates, which can both exclude real gravitational lensing (GL)-systems and include confusing objects. Several criteria have been used when selecting gravitational lensing (GL)-candidates from a sample in the various GL-surveys conducted over the years. Small color variations, stellar appearance and a reasonable separation are typical selection criteria. It has, however, been shown that discriminating too much on behalf of the color, might exclude lenses which are contaminated by the (red) color of the lensing galaxy [(Surdej et al., 1993a; Kochanek, 1991)].

3.3.1 False positives

There are two effects that contribute to the identification of false positives. Firstly, there is the problem of confusing objects, such as stars, galaxies, and unrelated QSOs. This effect is discussed in Section 4.2, where it is shown that stars are the main source of confusing objects. We note here, however, that stars primarily contribute to the list of candidates, with companions having large separations and magnitude differences. We can eliminate many of these false positives by introducing a “filter”, which exclude such candidates and hence significantly reduces our list of false positives without significantly loosing real lensed companions. This is discussed further in Section 3.4. The second source of false positives is the difficult identification of small separation candidates, because there is no way to distinctly identify an elongated close-seperation system. The PSF-subtraction method, which has been used to some extent in several surveys (such as Surdej et al. (1993a); Crampton et al. (1992)), is one method of uncovering small separation systems. We discuss methods of determining the Point Spread Function (PSF) and describe the Point Spread Function (PSF) subtraction method in more detail in Section 3.5, but briefly mention here that the Point Spread Function (PSF) subtraction method applied by the ESO Key program made use of a numerical Point Spread Function (PSF) retrieved from one or several selected stars in the image. If a significant residual is leftover after subtraction, the QSO is considered interesting. Supplement techniques such as comparison of contour plots of the QSO and stellar objects in the field are also applied [(Yee et al., 1993)]. Promising as these techniques might seem, there are significant uncertainties attached to them. Kochanek (1993a) warns that the residuals of the Point Spread Function (PSF)-subtraction method mostly produce ambigous candidates(false positives) and that the selection functions of Surdej et al. (1993a) for ground based observations are too optimistic for small separations. The determination of a selection function for the NOT sample will be investigated in the following Section 3.4, and compared to that of the SCYM-sample.

Today the smallest accepted optical-lens-system is Q1208+1011, discovered independently by Magain et al. (1992) and Bahcall et al. (1992b) and has a separation of 0.45 arcseconds. It is expected from lensing probabilities, that about 50% of the lensed QSOs in the universe are to be found at sub-arcsecond separations (θ ≤ 1.0′′, see Figure 4.1) [TOG,FT]. This is also why it is important to increase the resolution of observations, if one wants to increase the number of identified gravitational lensing (GL)-systems.

3.3.2 False negatives

The case of false negatives is due to the limitations on dynamical range, which sets an upper limit to the detectable magnitude difference (corresponding to a minimum magnification M_{obs} > 2). Additionally we choose a radius cutoff, r_{c}, to accurately define the area in which we search for candidates. The choice of this survey radius cutoff must prevent the sample to become incomplete.

Lensed QSOs such as Q 0957+561, could be excluded from surveys using a cutoff radius which is too low, while increasing the cutoff radius value will mostly add unwanted stars, galaxies and QSOs (i.e. confusing objects). It is thus important to allow most gravitational lensing (GL)-systems to be included in the survey, while keeping confusing objects out. Our choice for the cutoff radius r_{c} = 10′′ is somewhat high, but matches that of Yee et al. (1993). To avoid many ambiguous candidates (false positives), we compromise be lowering the permissible magnitude difference at separations larger than 5.0” (see Section 3.4.2). We illustrate in Figure 3.6 two of the 166 QSOs observed in the NOT sample, the object at left being a GL-candidate (chosen by the above described procedures).

Figure 3.6: Two of the investigated QSOs, showing at left a typical field surrounding the QSO with a close companion. At right a field with no companions within r_{c} = 10′′. The dashed circle in the right panel marks the survey area πr_{c}^{2}. The fields are 100 × 100 arcseconds.

3.4 Angular Selection Function

Several important issues must be taken into consideration when we define the angular selection function (ASF). Firstly we must estimate the limitations on resolution and dynamical range (detectable magnitude differences), affecting the ability to resolve close components and large magnitude differences, respectively. The selection function will then simply describe our observational window in the separation-magnitude difference plane. As discussed in the previous section, selecting a small cutoff radius will possibly exclude some cases of large separated gravitational lensing (GL)-systems, while choosing a large cutoff radius will narrow (worsen) the confusion limit of stars and galaxies. Matters related to the confusion limit will be discussed and illustrated in Section 4.2. We deduce the selection function for our particular survey in the following subsections.

3.4.1 Small separations

Due to the atmospheric turbulence, images obtained from ground based observatories have reduced image quality. The image quality or resolution is measured by the seeing or equivalently the full width at half maximum (FWHM) of a stellar profile. The image quality of observations from space instruments such as HST (see Page 30), is therefore only governed by the quality of the instrument^{8}. Ground-based observations are, however, governed by the local atmospheric conditions at the time of observation. Figure 3.7 illustrates the distribution of the full width at half maximum (FWHM) values for each of the 166 objects carried out at the Nordic Optical Telescope (NOT), La Palma, Spain.

Figure 3.7: The FWHM distribution of the observations of 166 QSOs

The seeing is distributed in a range between 0.5 − 2.0 arc-seconds, indicating the highly variable image quality typical of ground-based observatories. This implies that each image has a different detectability range in the separation-magnitude difference plane. These differences obviously affect the detection of small separations (θ < 1.0′′). It is therefore necessary to weigh each observation/image by its image quality (seeing/full width at half maximum (FWHM) value). This is done by estimating the detectable (maximum) magnitude difference for a given image separation, which we evaluate by performing a simulation-test. Using a typical Point Spread Function (PSF), for different seeing values, we produce simulated images of various closely separated pair-scenarios, with an additional random flux ratio. The results of these test’s are reproduced in Figure 3.8, which also shows the selection function for the other surveys added in Chapter 4. We discuss these tests in more detail in Section 3.4.3.

8this is illustrated by comparing the pre- and post- COSTAR/WFPC-II images from HST

3.4.2 Large separations

In the identification of gravitational lensing (GL)-candidates we chose all QSOs with a companion weaker than the QSO within the area πr_{c}^{2}. If more than one close companion exists, we chose the brighter one, because it has the highest probability of being a gravitationally lensed image of the QSO. Further, we only identify companions within the cutoff radius of r_{c} = 10 arc-seconds. This limit was initially chosen to secure the inclusion of most GL-systems. But we must also be aware of the fact that using such a large cutoff radius, also include many star/galaxy companions, hence false positives. In optimizing/balancing the radius cutoff value between the dilemma of false positives (large r_{c} include star/galaxy companions) and false negatives (small r_{c} exclude large separation gravitational lensing (GL)-systems), we alter the large separation section of the selection function. By decreasing the magnitude difference for large separations from Δm = 5 to Δm = 2, we effectively lower the amount of false positives (star and galaxy companions) and at the same time avoid loosing real gravitational lensing (GL)-systems. We have hence introduced an angular selection function (ASF) were the detectable magnitude difference is reduced at large separations (θ ≥ 5.0′′). Even though we shall see in Section 4.3 that GL candidate pairs with very large separations, in any case are left out of the computations, we include them for the sake of completeness.

3.4.3 The ASF

To determine the angular selection function (ASF) of the NOT sample for a wide range of seeing values we performed detection tests on synthetic data, produced from numerical Point Spread Function (PSF)’s obtained from one or several stars in the respective image of a certain quality (we used seeing/FWHM: 0.6′′, 0.75′′, 1.0′′ and 1.2′′). The synthetic images were produced for two images with random separation and magnitude difference. Manual inspection by eye was performed for a series of synthetic images in each FWHM catagory. For these seeing values, a set of constants (C_{1}, C_{2}, A and B) were determined, describing the particular angular selection function (ASF) for that particular FWHM/seeing value. Cubic interpolation was applied to determine these constants for the intermediate seeing values, thereby estimating the angular selection function (ASF) for any given seeing value within reasonable range of the actual measured values. The ASF is valid for seeing values from these tests in the range 0.5′′ ≤ F W HM ≤ 1.5′′. We also found from these tests that the detection probabilities are altered for different position angles (PA) of the image pair. Specifically, alignment along the rows or columns of the detector in some cases decrease the probability of separating the pair. The complete angular selection function (ASF), which we apply in our work, is now represented by the function described in Eq. (3.1) and shown in Figure 3.8.

Δm(θ = 2b) = \{ \begin{matrix}no solution & for 0′′ < θ < C_{1} \\ B + A × θ & for C_{1} < θ < C_{2} \\ 5.0 mag & for C_{2} < θ < 5.0′′ \\ 2.0 mag & for 5.0′′ < θ < rc \end{matrix}  (3.1)

where C_{1} and C_{2} are determined for each full width at half maximum (FWHM) value and constitute the starting value and “break” point of the angular selection function (ASF), respectively.

Table 3.1: Constants describing the ASF for various seeing (FWHM) values.

\begin{matrix}FWHM & A & B & C_{1} & C_{2} \\ 0.5 & 22.318 & -4.305 & 0.192 & 0.416 \\ 0.6 & 17.700 & -3.520 & 0.198 & 0.482 \\ 0.7 & 14.193 & -3.185 & 0.224 & 0.576 \\ 0.8 & 12.845 & -3.536 & 0.276 & 0.664 \\ 0.9 & 13.390 & -4.508 & 0.336 & 0.710 \\ 1.0 & 14.176 & -5.712 & 0.402 & 0.756 \\ 1.1 & 13.809 & -6.854 & 0.496 & 0.858 \\ 1.2 & 12.599 & -8.000 & 0.636 & 1.032 \\ 1.3 & 11.526 & -9.385 & 0.814 & 1.248 \end{matrix}

Figure 3.8: The selection function for NOT observations at 0.7” and 1.0”(solid lines left and right, respectively) FWHM/seeing at left. Also included for comparison, are the ground-based observations with inspection by EYE (dotted) and PSF-subtraction (dashed) for 1.0” seeing and the HST snapshot survey (dash-dot). At right we reproduce the complete selection function showing the reduced mangitude difference at separations larger than 5′′ (solid) and without this modification (dotted).

3.5 Uncovering small separation candidates

The two factors restricting the dectability of lensed systems are, as we have discussed in Section 3.3, the dynamic range (setting a maximum detectable magnitude difference) and the resolution. The former, being a physical limitation of the instrument, is difficult to improve. It is, however, possible to enhance the resolution somewhat. Atmospheric turbulence cause luminous sources to be blurred, i.e lowering the effective resolution. The resolution of direct images obtained at ground based observatories are therefore not only restricted by the sampling of the CCD detector, but also by the local atmospheric conditions at the time of observation.

Though we have not included gravitational lensing (GL)-candidates which have separations and magnitude differences that lie outside the selection function (see Section 3.4), we here briefly present observational and image post-processing techniques which enhances the probability of finding lensed systems. It is stressed that though steps have been taken in the inspection of small separation candidates (see objects with a “2” in the Note column in Table ??), these candidates are not treated in the present work.

Contour plots are used by Yee et al. (1993), while multiple PSF fitting has been applied by Surdej et al. (1993a). We shortly present the latter of these methods. The presented resampling, shifting and combination (RSC) method, with subsequent deconvolution, takes advantage of the effectively higher sampling^{9} that multiple imaging delivers. The additional deconvolution of such images, restore the image by unfolding (deconvolving) the Point Spread Function (PSF) from the image using e.g. maximum likelihood methods.

3.5.1 The PSF

The Point Spread Function (PSF) describes how a point source (delta function) is disturbed by the atmospheric turbulence. The Point Spread Function (PSF) is really a photon probability distribution, which hence gives the probability of registrating a photon event in a certain spatial posistion (XY pixel position). The atmospheric disturbance can actually be thought of as “atmospheric lensing”, because the turbulence really changes the local refractive index and effectively bends the path of light rays. For a thorough discussion of Point Spread Function (PSF)’s, see Teuber (1989).

Because astronomical images usually contain several perfect point sources (stars), it is possible to estimate the Point Spread Function (PSF) quite accurately. Usually the Point Spread Function (PSF) is estimated by an analytical gaussian (or similar) profile to remove shot (Poisson) noise. The analytical profile is a good estimate at high values, but does not satisfactorily fit the stellar profile at lower values (e.g. below exp(−1)∗peakvalue). Packages such as DAOPHOT I/II, estimate semi-analytical Point Spread Function (PSF)’s consisting of an analytically fitted profile, which is substituted by a lookuptable at low values. The lookup-table is really just a residual table of the difference between the stellar-object and the analytical profile.

Alternatively Point Spread Function (PSF)’s can be resampled to a finer grid and e.g. simultaneously cubic-interpolated. This approach avoids the need to approximate the Point Spread Function (PSF)-profile by analytical functions and instead just use a numerical (“true”) Point Spread Function (PSF) which is “smoothed” by the cubic-interpolation on a finer grid (removing some of the Poisson noise).

3.5.2 Multiple PSF fitting

We have also investigated an additional multiple Point Spread Function (PSF) fitting scheme^{10} to the resampled images, which fit’s 3n parameters to the observed object profile. Here n represents the number of PSFs to fit and the code chooses the best step and dimension to change (highest derivative). For each n there are three parameters to fit, the X and Y position and the intensity. The code is general for n image configurations, but we have only investigated double image fitting (n = 2). Below is an illustration of the results from applying the multiple Point Spread Function (PSF) fitting (using double image configuration) on the gravitational lensing (GL)-system Q1208+1011. The 5 × 200s images were obtained at the Nordic Optical Telescope (NOT) by J. Hjorth and F.G. Jensen, and kindly made available for these “real” examples [(Hjorth and Jensen, 1993)]. The results yield a separation of θ = 0.44′′ and a magnitude difference dm = 1.2 (A:B=2.1). This is below their (using non-linear deconvolution) and HST estimates of θ = 0.48′′ and A:B=3.8.

9a series of n exposures will roughly give a maximum sampling boost of n times

10implemented by Stein Vidar Haugan, UiO, Norway (e-mail: steinhh@astro.uio.no)

3.5.3 Non-linear deconvolution

Following the successful results of Hjorth and Jensen (1993) in enhancing the resolution of a combined multiple exposure of Q1208+1011, we have extended the method to search for sub-arcsecond separated gravitational lensing (GL)- systems. We have applied for observational time at the Nordic Optical Telescope (NOT) to both follow up observations of interesting GL-candidates and search for small separation candidates using the outlined method. In Section 4.2, it will become apparent that, even though it is difficult to uncover small separation candidates, it is still worth the effort. Briefly, because the probability of a close companion (sub-arcsecond separation) being a gravitationally lensed image of the QSO is substantially higher than it being a confusing object (e.g. a star or galaxy).

In our particular case, we have exposed five individual frames of an object with a typical integration time of 200s. To ensure a different sampling of the object(a sub-pixel shift), we shifted the telescope position by an arcsecond between each exposure. Following the pre-processing, the images were resampled with a cubic interpolation and subsequently re-shifted to overlap. The shift was computed using fast-fourier-transform cross-correlation routines implemented in IDL. The advantages of using cross-correlations instead of plain (source) registration, is both speed/simplicity and the fact that many object frames only consist of the QSO source with a single PSF-star, which may give significant errors using image registration. The accuracy of cross-correlation should be at least that of plain registration. After combining the images, an accelerated iterative Richardson-Lucy deconvolution scheme, implemented in IDL by Hans-Martin Adorf, was applied to restore the atmospherically blurred combined image [(Adorf et al., 1992)]. The Point Spread Function (PSF) used in the deconvolution was simply a resampled, shifted and combined image of a stellar object in the field. Another approach would be to fit a semi-analytical profile to a stellar object in the frame(s), and subsequently resample it correspondingly. In Jaunsen (1994) we perform simulations and tests to determine the best possible approach of deconvolving resampling, shifting and combination (RSC)-images, and characterize optimal Point Spread Function (PSF) determinations (e.g. resampled raw stellar Point Spread Function (PSF)’s and semi-analytical DAOPHOT II Point Spread Function (PSF)’s).

Furthermore we discuss recent alternative deconvolution techniques, such as co-addition, and the expected numerical seeing improvements this image enhancement-method can supply. Our simulations and tests show the technique can improve the full width at half maximum (FWHM) of images by up to four-six times, in agreement with the results of Hjorth and Jensen (1993).

The method is, however, not flawless – if PSF-variations are signicant in the field, this and similar methods (such as the PSF-subtraction) may give ambiguous results. In addition it is dependent on a series of images (preferably five or more) from the field in the same filter, and is therefore only applicable in the investigation of already suspected (elongated) QSOs. Several QSOs have been found to be elongated or to result in a significant residual. These gravitational lensing (GL)-candidates are not treated in our present work, as they all lie outside the present selection function. Further observations are needed and are underway to obtain better quality images, i.e. better seeing/full width at half maximum (FWHM).

Figure 3.9: PSF-subtraction and multiple PSF fitting results on Q1208+1011. The diagram displays five combined resampled (5 times) images of the object(upper left) and PSF (upper right), best double PSF fit (lower left) and residual after model subtraction (lower right). Data are courtesy Jens Hjorth and Frank G.Jensen.

Figure 3.10: Result from 600 RL-deconvolution iterations (right) of a 5 times resampled combined image of Q1208+1011 (left)(data coutsey J.Hjorth & F.G.Jensen).

Statistical results and probabilities

In this chapter we will outline a astatistical framework on the basis of the theoretical aspects and lens model described in Chapter 2 and present the results of appling this framework to the observed and reduced sample of QSOs described in Chapter 3. Specifically, in Section 4.1 we present the lensing probability distributions from the singular isothermal sphere (SIS) model. In Section 4.2 we present the gravitational lensing (GL)-candidates found in the NOT sample, following the specifications described in Section 3.3. Estimated relative lensing probabilities of each gravitational lensing (GL) candidate will be computed, along with respective probabilities for the companion being a star, galaxy, unassociated or associated QSO. Furthermore we combine our sample with the complete SCYM-sample, to perform statistical analysis of the complete sample in Section 4.3. We finally discuss the completeness of our sample and compare it with the SCYM-sample in Section 4.4.

4.1 The SIS lens probability distribution

In this chapter we compute individual lensing probabilities, survey completeness and absolute lensing probability. These probabilities are computed assuming our particular model outlined in Chapter 2. We here compile this model into a set of differential probabilities. Following Kochanek (1993a), we express the differential probability of lensing for a given separation and magnification by the product of the differential magnification bias B(m_{i}, M), Equation (2.30) in Section 2.4.2 and the differential optical depth (the effective distribution of separations produced by singular isothermal sphere (SIS)-galaxies) as a function of image separation, θ. The latter differential distribution of image separations has been derived by Kochanek (1993c) (Eq. (3.5) in that paper), and is expressed by

p_{c}(b_{cr}) = 30\frac{b̂^{2}}{b_{∗}} \frac{Γ(1 + α − 2γ^{−1}, b̂^{γ/2}) − 2b̂Γ(1 + α − 4γ^{−1}, b̂^{γ/2}) + b̂^{2}Γ(1 + α − 6γ^{−1}, b̂^{γ/2})}{Γ(1 + α + 4γ^{−1})}

(4.1)

where b̂ = b_{cr}/b_{∗}. We expand the expression to obtain the observable parameter θ instead of the critical radius, b_{cr}, by writing out τ and b̂ (Kochanek, 1993a, used in)

\frac{dτ}{dθ} = \frac{τ_{∗}}{2b_{∗}} D_{s}^{3} (\frac{θ}{2b_{∗}} )^{2} [Γ(1 + α − 2γ^{−1}, (\frac{θ}{2b_{∗}} )^{γ/2})

− \frac{θ}{b_{∗}} Γ(1 + α − 4γ^{−1}, (\frac{θ}{2b_{∗}} )^{γ/2}) + (\frac{θ}{2b_{∗}} )^{2} Γ(1 + α − 6γ^{−1}, (\frac{θ}{2b_{∗}} )^{γ/2})] (4.2)

where θ = 2b_{cr} is the image separation and α and γ are given in Section 2.2.2. The critical radius of a characteristic L_{∗} galaxy and a source at infinity is b_{∗} = 4π(σ_{∗}/c)^{2}, i.e. using the characteristic velocity dispersion, σ_{∗}, and the ratio of the two proper distances, which for a source at infinity is D_{ls}/D_{s} = 1. The incomplete gamma-function Γ[x, y], is a gamma-function where the infinity limit has been substituted by the limit y. Note that τ is included in the separation distribution, Equation (4.2). We have illustrated the separation distribution of a SIS-lensed QSO (σ_{∗} = 245kms^{−1}) in Figure 4.1 using Equation (4.1) for various image separations.

The differential lensing probability is now finally expressed by

\frac{dP}{dθdM} = S_{cat} \frac{dτ}{dθ} \frac{dP_{M}}{dM} \frac{N_{q}(m + 2.5 log M)}{N_{q}(m)}  (4.3)

where the factors are given by Equation (4.2), (2.29) and (2.23). This relation expresses the individual conditional probability of finding that a close companion of a QSO is a lensed image of the QSO with separation θ and magnification, M.

It is, however, in some cases more appropriate to express the probability in terms of the magnitude difference, Δm, instead of the magnification, M. The differential lensing probability for a given angular separation and magnitude difference is found by expressing the magnification M in Equation (4.3) in terms of the magnitude difference Δm = 2.5 log f. Using this expression for Δm, Equation (2.27) and f = (M + 2)/(M − 2) we find Figure 4.1: Distribution of separations, due to macro-lensing by a SIS-lens of a distant source (dotted line). The dashed line illustrates the distribution of image separations including the selection function for FWHM=1.0”, which will be discussed and deduced in Section 3.4.

d(Δm) = 2.5d(log f) = 2.5/ ln(10)d(ln f) = 2.5/ ln(10)df/f
= 2.5/ ln(10)(M − 2)/(M + 2)d((M − 2)/(M + 2))
= 2.5/ ln(10)(1 − f)/(M + 2)dM

which substituted into (4.3) gives

\frac{dP}{dθdΔm} = 1.6 ln(10)S_{cat} \frac{dτ}{dθ} \frac{f(f − 1)}{(f + 1)^{3}} \frac{N_{q}(m + 2.5 log M)}{N_{q}(m)}  (4.4)

where the magnitude of the images can be expressed by the total magnitude m and the flux ratio f. The brighter image has magnitude m+2.5 log(1+1/f) and the fainter m + 2.5 log(1 + f), for f ≥ 1.

4.2 Results from the NOT sample

The probability of finding a lensed quasar depends on the cutoff radius used in the investigation, number of confusing objects and on the limitations imposed by the selection function (dynamic range and resolution/seeing). Among the previously discussed factors contributing to the number of false positives (see Section 3.3.1), is the problem of confusing objects such as galactic stars, etc. We wish to obtain relative conditional probabilities of a QSOs companion being either a star, galaxy, QSO or lensed image. Following Kochanek (1993a) we apply approximated counts of stars, galaxies, associated QSOs and unassociated QSOs in addition to the lensing probabilities presented in Section 4.1, to estimate these relative probabilities. We begin with the galactic star distribution model estimated by Bahcall and Soneira (1980) in the V band

\frac{dN_{∗}}{dm_{V}} = \frac{C_{1}10^{β(m_{V} −m^{∗})}}{(1 + 10^{α(m_{V} −m^{∗})})^{δ}} \frac{1}{(sin b − μ cos b cos l)^{3−5γ}}
+ \frac{C_{2}10^{η(m_{V} −m^{†})}}{(1 + 10^{κ(m_{V} −m^{†})})^{λ}} \frac{1}{(1 − cos b cos l)^{σ}}  (4.5)

where C_{1} = 200, C_{2} = 400, α = −0.2, β = 0.01, η = 0.065, κ = −0.26, δ = 2, λ = 1.5, m^{∗} = 15, m^{†} = 17.5, μ = 0.03(m_{V} < 12), 0.0075(m_{V} − 12) + 0.03(12 < m_{V} < 20), 0.09(m_{V} > 20), γ = 0.36(m_{V} < 12), 0.04(12 − m_{V}) + 0.36(12 < m_{V} < 20), 0.04(m_{V} > 20), and σ = 1.45 − 0.20 cos b cos l. The coordinates l and b are the galactic longitude and latitude coordinates, respectively. Note that this stellar distribution does not include absorption by interstellar dust/gas-clouds and is only valid for |b| ≥ 20^{∘}. This is satisfied by our and other’s QSO samples due to the selective effect observations implied on QSOs^{1}.

Furthermore we use a simple faint galaxy population model from Tyson (1988) combined with an empirical V − R = 1 color magnitude conversion to estimate the number of confusing galaxies present at a certain magnitude (Kochanek, 1993a)

\frac{dN_{g}}{dm_{V}} = 10^{0.39m_{V} −5.19}  (4.6)

which is valid for galaxies with z < 1.

The QSO number count model introduced in Section 2.3 has units number of QSOs per square degrees and should in addition to Equation (4.5) and (4.6) (which are also per. square degree), be converted to differential distributions in magnitude as well as in separation by using

\frac{dN′}{dθdm} = \frac{2πθ}{3600^{2}} \frac{dN}{dm}  (4.7)

where θ is in units of arc-second.

1this can be verified by inspecting the spatial distribution of the sample and how very few QSOs are discovered close to the galactic plane, shown in Figure ?? in appendix ??

The unassociated QSOs, which just happen to lie close to the line-of-sight, is simply estimated from the QSO number-magnitude counts in Eq. (2.23), whereas the clustered QSOs are estimated assuming the clustering of QSOs are similar to that of galaxies. Using the correlation function for clustering of galaxies (Phinney and Blandford, 1986), Kochanek (1993a) has applied this as an estimate of QSO clustering giving the following differential probability distribution for clustering of QSOs

\frac{dN_{c}}{dθdm_{B}} ≃ 10^{−7}N_{q}(< m_{B}) (\frac{θ}{3′′} )^{0.2} × (\frac{1}{x^{2}} )^{1.8} \frac{x^{1.9}}{(x^{1/2} − 1)^{0.9}}  (4.8)

where x = 1 + z.

Employing these population probability estimates, we are able to predict the relative probabilities of a QSO companion being either a gravitationally lensed image, a galactic star, a galaxy, an unassociated QSO or an associated (clustered) QSO. In Figure 4.2 we illustrate the differential probability of a QSO companion (within the cutoff radius r_{c} ≤ 10′′) being either of the types mentioned above, for a source with redshift z_{s} = 2.0 and situated at the galactic north-pole (i.e. a weakly stellar populated region of the sky).

Using the differential lensing probability for a separation and magnitude difference, Eq. (4.4), we are able to compute the probability of the companion being a lensed image of the QSO. This integrated probability (over the survey area with cutoff radius, r_{c} = 10′′) is plotted for four QSO magnitudes (m_{V} = 16, 17, 18, 19), illustrating the fading lensing probability for increasingly fainter companions. Assuming an observed QSO (with the above characteristics) has a companion within the cutoff radius and magnitude m_{V}^{c} = 18.0, then the relative probability of the companion being a lensed image, star, galaxy or QSO (un/ass.) can be read from the figure along a vertical line drawn for m_{V}^{c}. The probability for lensing is clearly concentrated within a few magnitudes of the QSO, and rapidly decreases for fainter QSOs. At fainter magnitudes (i.e. at larger Δm) the increasing surface density of stars, galaxies, associated and unassociated QSOs make it less probable of encountering a lensed image. At a certain magnitude, the probability of lensing becomes comparable/equal or less than that of other (confusing) objects and this is what we refer to as the confusion limit for stars, galaxies or QSOs.

Using the same lensing probability, Eq. (4.4), we are also able to compute the individual lensing probabilities for each candidate. Applying the previously presented expressions of star, galaxy and QSO surface density estimates, we can predict the relative object-type probabilities for each companion. Our complete list of objectively^{2} selected gravitational lensing (GL)- Figure 4.2: Differential lensing probability of a QSO at z = 2.0 at the north galactic pole. Lens probabilities for a companion within 10” of a QSO with m_{V} = 16, 17, 18, 19 is shown (solid lines) with a peak probability close to the QSO magnitude. The star, galaxy, associated QSO and unassociated QSO probabilities are illustrated by the dotted,dashed,dash-dot and dash-dot-dot-dot lines respectively.

2the candidates are chosen simply on the basis of finding companions within the cutoff

candidates is shown in Table 4.1 with the respective probability of being a lensed image, a star, a galaxy or an associated/unassociated QSO.

In the following, we discuss these candidates, some of which are not yet confirmed as false positives, and present additional data which might enable us to reject or approve the candidate. Some objects have been observed by other surveys, in which case we present their conclusions, while a few candidates have been observed spectroscopically at Nordic Optical Telescope (NOT) using the low dispersion spectrograph (LDS). Since no photometriccalibration exposures were taken for the Nordic Optical Telescope (NOT) sample, we are not able to supply color estimates. We have, however, compared the magnitude difference between the QSO and companion for the Table 4.1: List of GL-candidates in the NOT-sample, with individual probabilities of the companion being a lensed image, star/galaxy or an ass./unass. QSO. The two objects marked by an asterix are not part of the NOT sample, but two of the identified GL-systems from the SCYM-sample and are only included here for comparison.

radius/survey area (see Section 3.3)

Object:  z_{s}  m  θ  Δ m P_{lens} P_{star/gal} P_{QSO} comment
* Q1208+1011 3.800 17.500 0.460 1.500 0.967 0.033  0.000 GL
PG1715+535 1.930 16.500 3.700 0.116 0.816 0.184  0.000 star
TEX2127+176 2.010 18.000 2.100 1.300 0.508 0.488  0.004 red color
* UM 673  2.720 17.000 2.200 2.100 0.475 0.524  0.000 GL
Q1442+295  2.670 16.200 3.400 2.000 0.396 0.604  0.000 red color
TEX2116+203 1.680 17.000 3.700 2.750 0.039 0.960  0.001 red color
Q1222.7+22.6 2.290 18.000 4.300 0.800 0.034 0.963  0.003
4C 56.28 1.595 17.300 3.200 2.700 0.033 0.966 0.001 red color
H0905+1507 3.160 19.000 3.100 2.700 0.020 0.969  0.011
B3 1621+392 1.970 17.500 3.800 2.300 0.019 0.980  0.001 star
Q0308+1902 2.840 18.600 4.400 3.100 0.004 0.985  0.011 galaxy
H0817+1142 2.000 18.000 3.200 4.000 0.002 0.997  0.001 red color
HS0855+2549 1.800 18.000 3.300 4.300 0.002 0.997  0.001
HS1824+6507 0.300 16.500 5.200 1.500 0.001 0.998  0.001 red color
HS0850+2852 2.090 17.500 3.700 4.500 0.001 0.999  0.000
Q1722+34 2.200 18.500 6.100 1.900 0.001 0.991 0.009
HS0727+6342 2.380 18.500 4.700 3.800 0.000 0.997  0.003
Q1720+3470 2.400 18.400 7.700 1.300 0.000 0.986  0.014 red color
PC1158+4635 4.733 22.420 8.900 1.500 0.000 0.943  0.057

\underline{Notes}: Col. 1: HLQ identification Col. 2: Redshift z of the QSO Col. 3: V-magnitude of the QSO as estimated from our data or tabulated in other QSO catalogues Col. 4: Angular separation θ between the QSO and its companion (if more than 1, chosen as described in Section 3.3) Col. 5: Magnitude difference in V-band between the QSO and companion Col. 6: Relative lensing probability computed as described in the text, using the differential probability Eq. (4.4) Col. 7: Relative probability of the companion being a star or a galaxy Col. 8: Relative probability of the companion being an associated or unassociated QSO Col. 9: Short comment on the morphology identification of the companion

various filters and use this as an indication of color resemblance.

• PG 1715+535’s companion is confirmed by Yee et al. (1993); Crampton et al. (1992) to be a star, having also been investigated by these surveys spectroscopically. Its red color (Δm_{V} = 0.12, Δm_{R} = 0.12 and Δm_{I} = −0.05) confirms their conclusion. Its small magnitude difference (i.e. high magnification) is the main contribution of the high lensing probability.

• TEX 2127+176 has a significant color variation in magnitude difference (Δm_{V} = 1.3, Δm_{R} = 0.5 and Δm_{I} = 0.3) and is therefore not a strong candidate. Spectroscopic observations were unfortunately not properly achieved. Because of the small angle separating the QSO (m_{V} ∼ 18) and companion (m_{V} ∼ 19.5), we suspect the companion was not satisfactorily aligned on the slit. We were able to identify the rather strong CIV emission line in the spectra of the QSO, while nothing was detectable where the component should have been. Further spectroscopic observations are planned in May 1994 and will hopefully confirm or reject the lensing hypothesis.

• Q 1442+295 shows a significant color variation in magnitude difference (Δm_{V} = 2.0, Δm_{R} = 1.6 and Δm_{I} = 1.1), indicating the companion is quite redder than the QSO. Spectroscopic data show no sign of a distinct alikeness in emission line or continuum. We conclude that this object is not a strong GL candidate, most probably a star. Recent observations of this pair, carried out at the NTT, La Silla, Chile by J. Surdej also show little resemblance between the QSO and companion.

• TEX 2116+203 shows significant color variation in magnitude difference (Δm_{V} = 2.75, Δm_{R} = 2.2 and Δm_{I} = 1.5). Spectroscopic observations also indicate there is no sign of the companion being a lensed image of the QSO.

• Q 1222.7+22.6 has not been imaged in more than R and V filters. The magnitude differences in these colors are, however, quite similar (Δm_{V} ≈ 0.8 and Δm_{R} ≈ 0.65) and within what we expect to be a plausible uncertainty. The companion remains, therefore, an interesting candidate of lensing.

• 4C 56.28 has been observed by other surveys, but has not yet been investigated spectroscopically. Our photometric data indicate the companion is somewhat redder than the QSO (in agreement with previous surveys), but not more than can be expected from a reddening effect due to a lensing galaxy (Δm_{V} ≈ 2.7 and Δm_{R} ≈ 2.65). We therefore treat this pair as interesting and plan to perform spectroscopic observations in the middle of 1994.

• H 0905+1507 was observed spectroscopically by J.Surdej at the NTT and seems to be quite red, though no photometric data is available.

• B3 1621+392 and its companion shows fairly similar colors (Δm_{V} ≈ 2.3, Δm_{R} ≈ 2.25 and Δm_{I} ≈ 2.4), but the companion has been reported by Bahcall et al. (1992a) to be a G-type star.

• Q 0308+19 and its companion shows fairly similar colors (Δm_{V} ≈ 3.1, Δm_{R} ≈ 3.0 and Δm_{I} ≈ 3.0), but the companion has been reported by Crampton et al. (1992) to be a galaxy.

• H 0817+1142 show significant color variation (Δm_{V} ≈ 4.0, Δm_{R} ≈ 3.0 and Δm_{I} ≈ 2.1) and is therefore considered unlikely to be a GL-system.

• HS 0855+25 lacks photometric data and demands further investigation if its companion’s nature is to be revealed.

• HS 1824+65 shows fairly similar colors (Δm_{V} ≈ 1.5, Δm_{R} ≈ 1.2 and Δm_{I} ≈ 1.2), and has been observed spectroscopically at the NOT. These observations show little resemblance between the QSO and companion and we conclude that the companion is most likely a foreground star.

• HS 0850+28 shows fairly similar colors (Δm_{V} ≈ 4.5, Δm_{R} ≈ 4.5 and Δm_{I} ≈ 4.5), but has not yet been observed spectroscopically.

• Q 1722+34 lacks photometric data and demands further investigation if its companion’s nature is to be revealed.

• HS 0727+63 shows fairly similar colors (Δm_{R} ≈ 3.8 and Δm_{I} ≈ 3.5), but has not yet been observed spectroscopically.

• Q 1720+34 shows different colors (Δm_{V} ≈ 1.3, Δm_{R} ≈ 0.7 and Δm_{I} ≈ 0.4), indicating that the companion is quite redder than the QSO.

• PC1158+46 lacks photometric data and demands further investigation if its companion’s nature is to be revealed. Its companion is, however, highly unlikely to be a lensed image with the large separation and the fact that no other extended objects are present in the rather deep exposure.

Figure 4.3: Spectroscopic observations of a few promising candidates, performed at NOT.

Finally we briefly discuss the possibility of applying the sample data to determine if there are over-densities of background objects in the vicinity of QSOs and specifically HLQs. Kochanek (1993a) has performed Kolmogorov-Smirnov tests to estimate the background density of stars, galaxies and QSOs. These estimates could be compared with the observed density and should, if the sample is significantly large and the assumed population densities are correct, give indications towards an over-density in the close region surrounding the QSO. It should, however, be noted that it is possible that the assumed QSO luminosity function (from Section 2.3) is significantly altered due to the effect of magnification itself.

4.3 Cosmological results

Having presented the gravitational lens candidates of the NOT sample and presented the individual probability of a QSO companion either being a lensed image, star, galaxy or QSO, we here present the expected number of GL candidates using SIS-galaxy statistics and compare this to the observed results. The Nordic Optical Telescope (NOT) sample is, however, too small to be employed in quantitative lens statistics. We therefore combine the sample with the full optical SCYM-sample presented by Surdej et al. (1993a). Using likelihood functions, we estimate maximum likelihood velocity dispersions, σ_{∗}, or lens effectiveness parameters, F, from the combined sample of 469+143 HLQs for various flat cosmologies. In these computations we employ complete integral probabilities. This is the probability for a given QSO (with apparent magnitude, m_{i}, and redshift, z_{i}) to be lensed by an intervening isolated galaxy. Our previous computed probabilities (in Section 4.2) were relative, i.e. the probability of a companion detected within the survey area being a lensed image (with a particular observed separation and magnitude difference) is expressed by the relative probability. The probability of finding a lensed image for any QSO with a redshift z_{i} and B magnitude, m_{i}, can thus be expressed as the product of the lens optical depth for that redshift and the integral magnification bias

p_{i}(m_{i}, z_{i}) = S_{cat}τ(z_{i})Bias(m_{i}, 2, ∞)
= S_{cat} \frac{τ_{∗}}{30}Γ[1 + α + 4γ^{−1}]D_{s}^{3}(z_{i})Bias(m_{i}, 2, ∞)
= S_{cat} \frac{τ_{∗}}{30}Γ[1 + α + 4γ^{−1}]D_{s}^{3}(z_{i}) ∫_{2}^{∞} B(m_{i}, M)dM (4.9)

where B(m_{i}, M) is given by Eq. (2.30), τ_{∗} by Eq. (2.17), and the infinite range of the bias integral is numerically taken to be M_{2} = 10000, which corresponds to a magnitude interval of 10.

We wish now to express the probability of finding a lensed image of a QSO under the constraints imposed by the observational conditions. The observational conditions are characterized by the seeing or the full width at half maximum (FWHM) of point sources in the images. The detectable magnitude difference is dependent on the image separation, and closely related to the seeing. We introduced the angular selection function (ASF) in Section 3.4, which describes the relations of detectable magnitude differences, separation and seeing for our particular survey. Following Surdej et al. (1993a); Kochanek (1993b), we can express the probability, p_{i}^{SF}, of finding a gravitationally lensed image in the vicinity (within r_{c}) of a QSO within the observational window (defined by the selection function) by

p_{i}^{SF} (m_{i}, z_{i}) = ∫_{0}^{r_{c}} ∫_{M_{obs}}^{M_{2}} \frac{dP}{dMdθ}dMdθ
= p_{i} ∫_{0}^{r_{c}/2} p_{c}(b_{cr})\frac{Bias(m_{i}, M_{obs}(2b_{cr}), M_{2})}{Bias(m_{i}, 2, M_{2})}db_{cr} (4.10)

where p_{c}(b_{cr}) = \frac{dτ}{τdb_{cr}} = \frac{2dτ}{τdθ} (refer Eq. (4.2)), because of the relation between the critical radius b_{cr} and the image separation θ^{3}. Note that Bias(m_{i}, M_{obs}(2b_{cr}), M_{2}) is normalized so that ∫_{min(M_{obs})}^{M_{2}} dP_{M}/dMdM = 1. The minimum detectable magnification (i.e. maximum magnitude difference) is given by the expression for the magnification M_{obs}(2b_{cr}) = 2\frac{f(2b_{cr})+1}{f(2b_{cr})−1}, where f(2b_{cr}) = f(θ) is the flux ratio corresponding to the detectable magnitude difference Δm(θ) (given by Eq. (3.1)). Note also that in absence of any selection function the minimum detectable magnification would be 2.5 log 2 (TOG, Surdej et al. (1993a)).

3we have converted the lensing probability from units of θ to units of the critical radius, b_{cr}

The probability for finding a lensed pair with a certain redshift and magnitude within the selection function, is thus the probability for lensing weighted by the selection function Eq. (3.1) for each critical radius b_{cr}. The configuration probability (the relative probability of a separation) p_{c}(b_{cr}) multiplied by the ratio of the bias including selection functions to that with no selection function, is thus integrated over the survey area (r ≤ r_{c}) and multiplied by the general probability of lensing Eq. (4.9), leading to Equation (4.10). The sum of the probabilities of finding a gravitational lensing (GL)-system within the detectable ranges of the selection function for each observed object, ∑_{i} p_{i}^{SF}, will then give us the expected number of lensed QSOs in the

sample.

⟨p_{i}^{SF} ⟩ = \frac{N_{L}}{N_{Q}} = \frac{F S_{cat}}{30N_{Q}} ∑_{i}^{N_{Q}} D_{s}^{3}(z_{i}) ∫_{0}^{r_{c}} p_{c}(b_{cr})Bias(m_{i}, M_{obs}(2b_{cr}), M_{2})db_{cr}

(4.11)

where N_{Q} is the total number of investigated quasars and N_{L} is the number of lensed quasars.

The combined NOT+SCYM sample consists of 612 QSOs, which are applied for the computations below. Among these QSOs, four have been identified as being gravitationally lensed images of the source. We therefore have N_{L} = 4, which consists of the two double imaged lensed systems “Q 1208+1011” and “UM 673”, the quadrupole “H1413+117” and the triple image “PG 1115+080”, which were found in the SCYM-sample.

We do not included the well known “Q 0957+561”, because this system is not only lensed by a galaxy, but also by the surrounding cluster which contributes to the large separation of 6.1 arc-second. The four lensed systems are included because, even though our lens model is not able to reproduce the image configuration of two of them, the frequency of lensing is still preserved. If we wanted to consider solely the two image cases, the population of lenses responsible for the n > 2 image configurations would have to be removed (Surdej, 1994). See also Sections 5.1 and 5.3, for a brief discussion on the possibility of applying image configuration probabilities in the computations, in which case an elliptical model would have to be used for the n > 2 image lens-systems.

Kochanek (1993b) estimated σ_{∗} for E and S0 galaxy populations, while TOG,F&T and Surdej et al. (1993a) estimated the lens effectiveness F parameter for E,S0 and S galaxy populations. Since we do not know the importance of S spiral galaxies we estimate σ_{∗} values, which are independent of the lens population. For these σ_{∗} values we calculate appropriate F values for late type E/S0 galaxies and E/S0 and S galaxies. The σ_{∗} values are estimate from a likelihood function described below.

The distribution of a small quantity of registered events (in this case the number of lensed quasars) is usually approximated by a Poisson distribution. By applying our results (N_{L} = 4) and the individual lensing probability of a QSO (Eq. (4.11)), we investigate the implications these results impose on the various flat cosmological models (i.e. various combinations of Ω_{M} = 1 − Ω_{Λ}). We, hence, compute likelihoods for various values of the velocity dispersion, σ_{∗}, and flat cosmological models. The value of Ω_{M} affects the proper distance, which is part of the optical depth (Eq. (2.17)), while σ_{∗} govern the F parameter and the image separation distribution, p_{c}(b_{cr}). The Poisson distribution is expressed by

P [X = N_{L}] = \frac{e^{−⟨N_{L}⟩}⟨N_{L}⟩^{N_{L}}}{N_{L}!}  (4.12)

and converted into logarithmic probabilities, as in Kochanek (1993b), to give a more convenient expression

lnP_{tot} = N_{L}ln⟨N_{L}⟩ − ⟨N_{L}⟩ − lnN_{L}!  (4.13)

where N_{L} = 0, 1, 2 … is the observed number of lensed quasars and ⟨N_{L}⟩ = N_{Q}⟨p_{i}^{SF} ⟩ is the average number of lensed quasars.

For these estimates, we computed the integral of Equation (4.11) for each object and for 400 combinations of various Ω_{M} and σ_{∗} values (a 20×20 array). We present the likelihood results from these computations as contours of the 68%, 90%, 95% and 99% confidence levels in Figures 4.4 and 4.5. These two dimensional confidence levels are simply the corresponding percentage levels of the maximum likelihood. The computation of confidence intervals for the σ_{∗} or F parameter (a single parameter), can be determined from the χ^{2} distribution with one degree of freedom. The 90% confidence intervals in Table 4.2 are therefore determined from the 26% peak values of the two dimensional likelihood contours in Figures 4.4 and 4.5 (see §5 in Kochanek (1993b)).

From these results we see there is a significant discrepancy between the low F = 0.0236 parameter deduced in Section 2.2.3 and the higher F = 0.188 parameter (maximum likelihood) deduced from the lens statistical results.

Table 4.2: Estimated σ_{∗} and F parameters from the Poisson likelihood’s of the observations. The first row is the results assuming that lensing is significant from E, S0 and S galaxy types, while the second row only for E and S0 (late-type) galaxies. The maximum likelihood value is indicated by F^{max} and σ_{∗}^{max}, respectively. The results are shown for the Einstein-de Sitter universe model (Ω_{M} = 1 and Ω_{Λ} = 0) at 90% confidence levels.

galaxy types F^{low} F^{max} F^{high} σ_{∗}^{low} σ_{∗}^{max} σ_{∗}^{up}
E/S0/S  0.068 0.188 0.469 232 288 344
E/S  0.075 0.177 0.362 304 392 492

Constraining the cosmological models from these results may presently give too uncertain conclusions, but has been included below to illustrate how this might be done, while also estimating the σ_{∗} value for both E/S0 and E/S0/S galaxy populations. Note that our results are all computed without the DM velocity dispersion correction factor and that this is one reason for the discrepancy between our results on the F parameter and that of e.g. F&T’s F=0.047 computed value.

In Figures 4.4 and 4.5 contour plots of various confidence levels of the likelihoods computed using Eq. (4.13) is shown. The maximum likelihood from these computations gave Ω_{M} = 1.3 and σ_{∗} = 300 kms^{−1} using E/S0/S galaxy types, but the results are inaccurate because there is no significant maximum along the contours of the likelihood in the Ω_{M} dimension. It is thus difficult to discriminate between various flat cosmologies, while the possible values of σ_{∗} are more restricted. The values of σ_{∗} at various confidence levels can be read along the vertical line indicating a specific flat universe model, e.g. an Einstein-de Sitter model (Ω_{M} = 1 and Ω_{Λ} = 0).

4.4 Survey completeness

Our observations are confined by the selection function (defined by Eq. (3.1)), thereby reducing the fraction of observed pairs. The completeness of singular isothermal sphere (SIS)-models are only dependent on the magnitude, imageseparations and velocity dispersions. The magnitude and image-separation (or critical radius) dependence entering through the magnification bias, while the velocity dispersion affects the critical radius b_{cr}, because it is proportional to σ^{2}. The survey completeness is expressed by the ratio between the fraction of observationally detectable lensed pairs and the total population of Figure 4.4: Likelihood contour plot for various flat cosmologies (Ω_{M} = 1−Ω_{Λ}) and velocity dispersions (σ_{∗}), assuming lensing is significant for all lens types (E/S0/S).

lensed pairs. These quantities are estimated by integrating the differential probability function (4.3) over magnification with and without the selection function constraint on the magnification, respectively.

c(θ) = \frac{p_{obs}(θ)}{p_{0}(θ)} = \frac{∫_{M_{obs}(θ)}^{∞} \frac{dP}{dθdM} dM}{∫_{2}^{∞} \frac{dP}{dθdM} dM}  (4.14)

Note that the redshift dependency of the optical depth cancels out in the ratio, which implies that completeness for singular isothermal sphere (SIS)- models are independent on redshift. The completeness of our survey for a given QSO with magnitude m_{B} = 17.0, 18.0, 19.0 is illustrated as a function of image separation (given by Eq. (4.14)) in Figure 4.6.

Evaluating the completeness for different values of the characteristic velocity dispersions σ_{∗}, we integrate the probability over the survey area

c(σ_{∗}) = \frac{p_{obs}(σ_{∗})}{p_{0}(σ_{∗})} = \frac{∫_{0}^{r_{c}} dθ ∫_{M_{obs}(θ)}^{∞} \frac{dP}{dθdM} dM}{∫_{0}^{∞} dθ ∫_{2}^{∞} \frac{dP}{dθdM} dM}  (4.15)

Figure 4.5: Likelihood contour plot for various flat cosmologies (Ω_{M} = 1−Ω_{Λ}) and velocity dispersions (σ_{∗}), assuming lensing is significant only for late type galaxies (E/S0).

where the integration limit is set to r_{c} instead of infinity for the fraction of GL’s found within the definition range of the selection function, due to the fact that the selection function is not defined for separations larger than r_{c}. The total amount of existing GL’s, p_{0}(σ_{∗}), is naturally integrated over all separations, i.e. from zero to infinity. The completeness in σ_{∗} decreases at higher velocity dispersions, because the b_{∗} separation scale is proportional to σ_{∗}^{2} and hence the completeness is decreased by the survey cutoff radius at higher velocity dispersions (cf. Figure 4.7).

As we readily see from Figure 4.6 and 4.7 the difference in completeness between our altered and unaltered selection function^{4} has little significance. The completeness difference is only due to the loss of a few large-separation GL-systems, while we at the same time exclude a significant amount of false positives. If we, however, compare the completeness of our survey to that Figure 4.6: Completeness of the NOT survey as a function of image separation for a QSO with m_{B} = 17.0, 18.0, 19.0 and σ_{∗} = 245kms^{−1}. The upper most line corresponds to m_{B} = 17.0 and the lowest to m_{B} = 19.0. Left panel illustrate details at small separations, while the right panel shows the full survey range. The completeness with and without the reduced magnitude difference at large separations (see Section 3.4.2) is illustrated by the solid and dotted lines, respectively. Dashed lines indicate completeness using the ESO Key program ASF.

4with and without the lowered detectable magnitude difference for separations larger than 5”

of the combined sample, we find a more significant difference at small separations. This is due to our more pessimistic selection function at smaller separations.

Figure 4.7: Completeness of the NOT survey as a function of characteristic velocity dispersion σ_{∗} for a QSO with m_{B} = 18.0. The completeness with and without the reduced magnitude difference at large separations (see Section 3.4.2) is illustrated by the solid and dotted lines respectively.

Conclusions

In this final chapter we discuss the applications of our work to the field of cosmology and how our assumptions and simplicity of the models we use might affect the statistical outcome. Several weak points in the modeling and alternate ways of dealing with these, are presented. In the spirit of improving the credibility or reality of the models, we also present a short description of what might be achieved in future work, using the more complicated elliptical potentials and mass distributions.

5.1 Deficiency of the model

As we have pointed out throughout the thesis, several assumptions and simplifications have been employed in our work. We shall here summarize these and discuss the effects they might have on the statistical results.

• The analytical SIS model which we have employed in our work is an over-simplified model of the effects lens-galaxies produce. But because of the singular isothermal sphere (SIS)-models tractability and its consistency with previous lens statistical work by Kochanek (1993a,b); Maoz and Rix (1993) and Surdej et al. (1993a), we are justified in using this model for the moderate goals and purpose of the present thesis. The singular isothermal sphere (SIS)-model, which we have outlined in chapter 2, is the only feasible model which is analytically tractable in a lens statistical sense. Its most obvious lack is its inability to produce quadrupole-mirages of the source, which demands an elliptical lens (see section 5.3).

• Magnification bias plays a significant role in lens statistics, because it can alter the potential number of sources within the projected area of strong lensing and thereby the observable frequency of lensing in the universe. Since we are not presently able to accurately determine the QSO luminosity function^{1}, the magnification bias (as we estimate it) also suffers from these uncertainties (Narayan and Wallington, 1993). The magnification bias could also be determined in other (independent of the luminosity function (LF)) ways, as discussed in Sections 4.2, 2.4 and Claeskens and Surdej (1994).

• In the course of developing the lens-model, we have assumed that the population of lenses do not evolve (i.e. n_{L} = n_{0}(1 + z_{l})^{3}). Mao (1991) introduced a lens-redshift cutoff model, which might explain the observed overdensity of large separation lens-systems. Similar effects can, however, also be mimicked by introducing a core radius for the lensing galaxy.

• As pointed out throughout the thesis we have investigated lensing by isolated galaxies, thus leaving out the lensing effects of galaxy clusters. Multiple imaged lensed quasars are believed to be produced primarily by an isolated galaxy. In many cases, however, surrounding mass (dark matter and other galaxies in a cluster) contribute by e.g. altering the original magnification probability (from the isolated galaxy). Surdej et al. (1993a) discriminates between these two forms of lensing by excluding candidates with separations larger than 3.0 arcseconds. Many other surveys indirectly apply similar constraints by using a small (e.g. 3.0 − 5.0′′) cutoff radius. Our rather large cutoff radius does, however, only include comparably bright large separation pairs (from Table 4.1 we see only 4 candidates have been included with separations larger than 5.0 arcseconds). The difficulty of discriminating between these two events obviously constitute a problem. In the future, however, this problem should be possible to overcome using elliptical lens models and advanced configuration probabilities (see Kochanek (1993b)) which take into consideration the individual details of each lensed system. In the present work the configuration probability of a lens system is not possible to deduce since, having employed the SIS-lens model, we are unable to produce n > 2 image configurations.

• Implicitly we have also assumed that isolated galaxies are homogeneously distributed and because of the inability to discriminate between lensing by isolated galaxies and galaxy clusters (discussed in previous paragraphs), we have also made similar assumptions about the clusters.

1due to the various selection effects and biases of QSO surveys, see section 2.3

If biasing is substantial in QSO catalogs (discussed in Section 3.1), this might play an important role in the presently modest samples investigated by GL-surveys.

• Because our computations, and therefore our results, are sensitive to the selection function at small separations, future work should improve the determination methods of the selection function. The results are sensitive in this area, due to the high percentage of GL-systems, expected to have such image separations.

5.2 Cosmological applications

Several alternative ways of estimating and probing the depths and structure of the universe have been presented over the years. The seemingly complex structure and observable effects of the universe, might seem unsolvable by the use of simple models and gross assumptions. Nevertheless, the overly simplified models do give reasonable results in a statistical sense, due to the manifold populations of sources and lenses in the universe. The presented methods and data do not, however, give substantial evidence from which one can discriminate clearly between cosmological models, which is why we have confined ourselves to the flat universe model. But we believe that the present total sample of investigated QSOs combined with future observations and improvements will enable us to significantly constrain the possible cosmological universes in which we exist.

With the combined sample of previously investigated QSOs and our Nordic Optical Telescope (NOT)-sample, we will carefully begin to constrain the confidence levels of σ_{∗}. Using Monte-Carlo simulations and likelihood estimates, Kochanek (1993b) showed how this might be done in a careful way, while Wallington and Narayan (1993) presented an important step in modifying the statistical methods using a more realistic model for the lenses.

The computations in Section 4.3 for various flat cosmologies, considering both F and σ_{∗} for E/S0/S and E/S0 galaxies, illustrates a basic application of lens statistics to discriminate cosmological models. With the present sample, we can not expect significant signals for the most probable model parameters, because our sample is still too small and hampered by Poisson noise. Furthermore the application of a configuration likelihood function would have been a better likelihood function to apply than the present Poisson likelihood. But due to the lack of proper configuration probabilities (because we lack information about the individual GL-systems) we are unable to correctly apply this likelihood function fully. In agreement with Surdej et al. (1993a), we therefore conclude that the investigated samples are presently too small to yield significantly certain results from GL-statistics. Because we apply maximum likelihoods for the F parameter and exclude \sqrt{3/2}, our maximum likelihood value of F = 0.188 cannot be compared to the median value deduced in Surdej et al. (1993a). The confidence intervals, however, overlap significantly well.

If, however, our high estimates on the velocity dispersion, σ_{∗}, are assumed to be marginally correct, they would indicate that lensing is produced by far larger velocity dispersions than the directly observed values given in Table 2.1. This would imply that the observed σ_{∗} is not a good measure in lens statistics, but should e.g. be corrected by a DM correction factor.

Notes

Many valuable experiences have been made in the course of this work, and certain tasks and computations has therefore been necessary to perform repeatedly. An important task is therefore to implement optimal procedures for quick computations on workstations. In the present thesis we have made all implementations in the high-level language IDL, which proved eventually to be too slow for these computations. Future investigations should therefore be made using languages such as FORTRAN, C, or similar, which are also easier to transport between different platforms. The availability of code optimizers for such languages can additionally increase the performance.

5.3 Further work using advanced models

In the most recent years, there has been development towards a more realistic model of the lensing galaxies (Kochanek and Blandford, 1987; Wallington and Narayan, 1993; Kassiola and Kovner, 1993). Using two-dimensional elliptical potentials or mass-distributions, all lensed multiple-image configurations are possible to create. These models are however non-analytic (except Kassiola and Kovner (1993)) and demand extensive numerical calculations. Due to the non-analytical nature of these lens models, the steps in calculating statistical properties for various ellipticities, core radii and velocity dispersions are more intricate. The cross sections and magnifications (and magnification probabilities) are determined numerically from e.g. raytracing-schemes. Kochanek and Blandford (1987) discussed two-dimensional elliptical-potential models, while Wallington and Narayan (1993) applied a similar model to investigate several aspects of lensing and to estimate the core radius from lens statistics. There are however many difficulties attached to the use of more complicated elliptical galaxy models. This is due to the fact that all quantities must be computed numerically and therefore bring noise into the results. The inclusion of selection functions also become complicated and no real parametrical fit’s have yet been published from the use of such elliptical models. It will however, hopefully give more realistic and accurate details of the intricate effects from gravitational lensing, when these problems have been overcome sometime in the near foreseeable future. The use of elliptical lens models will also enable the computation of configuration probabilities, which can be successfully included in likelihood estimates and apply information of the individual lensed systems into the computations (see Section 4.3 and Kochanek (1993b)).

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