The field of gravitational lensing has evolved from a theoretical fantasy to a robust astrophysical tool in the past few decades. In this chapter, we introduce the basics of gravitational lensing. We explain what gravitational lensing is in Section 1.1 and briefly recount the history of lensing in Section 1.2. We continue with the basic theory for lensing in Section 1.3 and work through properties of simple lens mass distributions in Section 1.4.

Introduction

A perhaps familiar example of lensing is the bending of light by optics, such as the glasses that some people wear, or binoculars that some people use for viewing wildlife or events. These two examples of optical lenses are linear in the sense that one sees only a single (perhaps magnified) image of the object of interest. In contrast, the base of a wine glass is a non-linear lens so that through the glass one can see multiple images of the background object. Figure 1.1 is an illustration of this phenomenon. In the top-left panel, we see the original source of light that emanates from the candle. In the top-right panel, we see four images of the source as lensed by the wine glass: one in the lower left, two close pairs on the lower right and one behind the stem of the wine glass. By tilting the base of the wine glass, we change the properties of the optical lens and thus the light paths of the images we see. For example, in the bottom-right panel, there are only two multiple images of the background source. In the case where the stem of a symmetric wine glass is aligned perfectly along of the line of sight to the source, the source is lensed into a ring, as shown in the bottom-left panel.

In gravitational lensing, a massive object takes on the role of the lens, similar to the wine glass in optical lensing. According to Einstein's General Theory of Relativity, mass curves spacetime. Light, taking the shortest path in this curved spacetime, thus ‘bends’ around massive objects. Anything that has mass (e.g. planets, stars, galaxies, and clusters of galaxies) can thus act as a gravitational lens.

This chapter discusses computational and statistical methods for fitting models to strong lens data. It centres on parametric models of point-like lenses but includes extensions to composite models, free-form models, and extended sources. It describes how to use statistical tools including Monte Carlo Markov chains and nested sampling to explore the range of models that are consistent with data.

Introduction

Strong lensing is a versatile tool for astrophysics that can be used to study the physical properties and environments of lensing galaxies, to dissect the structure of source quasars and galaxies, to constrain cosmological parameters, and much more. Other chapters in this volume review the theory of strong lensing, the status of observations, and the variety of astrophysical applications that result. The goal of this chapter is to outline methods for fitting models to strong lens data. Since modelling is required for most applications of strong lensing, understanding the strengths and weaknesses of the analysis is key for drawing robust conclusions.

When discussing methodology, we need to distinguish between point-like and extended images. Point-like images (in a lensed quasar, for example) provide constraints on the potential and its derivatives at discrete positions, which can be described with a modest number of constraint equations or a straightforward χ 2 goodness of fit statistic. Established statistical methods can then be used to find the best fit and explore the range of allowed models. In this case the barrier to entry is low in the sense that fitting basic models does not require tremendous expertise, yet the potential for growth is high in the sense that advanced analyses can combine lensing with other astrophysical probes to draw conclusions that have broad reach. Extended images, by contrast, provide many more pixels of data but require many more free parameters (associated with the unknown shape of the source). Specialized methods must be used to simultaneously fit a mass model for the lens and a light model for the source. For pedagogical purposes, I focus on analysis methods that are applicable to point-like sources but include an overview of methods for modelling extended images (Section 7.5.4).

Active galactic nuclei (AGN) and quasi-stellar radio sources (quasars) are very luminous compact objects at cosmological distances. Right after their discovery in the 1960s, Sjur Refsdal realized that these properties made them ideal targets for determining the Hubble constant with a measurement of the time delay in a gravitationally lensed quasar system. The discovery of the first double quasar Q0957+561 in 1979 (Walsh, Carswell and Weymann 1979) paved the way for monitoring of multiple quasars. Chang and Refsdal (1979) immediately realized that individual stars in a lensing galaxy can act as microlenses and modify the magnification on time scales of years or months. Today, a few hundred gravitationally lensed quasars are known. Time delays have been determined with an accuracy of a few per cent in a few dozen systems. Averaged over an ensemble of lenses, the Hubble constant H 0 can be determined with an uncertainty of about 5%, the error budget being usually dominated by the mass model of the lensing galaxy. Uncorrelated fluctuations in the multiple images of a lensed quasar originate from microlensing and contain information on the lensing objects as well as on the quasar luminosity profile and size. Originally, quasar microlensing studies focused on the visual light; more recently, microlensing fluctuations in the broad emission lines have been analysed as well. Microlensing is a natural explanation for the flux-ratio anomaly in some of the quadruply imaged quasars: A smooth dark matter component produces an asymmetric magnification distribution between the two images in a close pair, with a relatively high probability of high demagnification of the saddle point (negative parity) image. Comparison of the observed flux ratios with microlensing simulations even allows us to quantify the most likely dark matter fraction in such systems.

This chapter summarizes the four lectures that the author presented at the XXIV Canary Islands Winter School of Astrophysics in Puerto de La Cruz, Tenerife, which took place over November 4–16, 2012. A very brief introduction to AGN/quasars is followed by a section on the basics of (micro)lensing and the relevant length and time scales. The two main sections then present results on time delay measurements in multiple quasar systems and subsequent determinations of the Hubble constant on the one hand and on various applications of quasar microlensing on the other.

Gravitational lensing offers a unique tool to study dark matter on a broad range of scales, from galaxies to clusters, to the large-scale matter distribution in the Universe. Density fluctuations on large scales have a small amplitude, hence their lensing effects are weak. In this chapter, after introducing the concepts of weak gravitational lensing – which are quite different from those used in strong lensing – I will concentrate on three main topics: the mass distribution of galaxy clusters as obtained from combining strong and weak lensing results, the lensing effects of the large-scale matter distribution in the Universe and lessons to be learned from them, and the weak lensing studies of the mass profiles in the outskirts of galaxies, together with the correlation of galaxies and the underlying dark matter distribution.

Introduction

Since gravitational light deflection is independent of the nature and state of matter, and in particular is equally sensitive to luminous and dark matter, it provides a unique tool for studying the total mass distribution of objects in the Universe. The study of the mass properties in the inner part of galaxies is covered by other chapters in this volume, as well as that of stellar- and planetary-mass objects. In my chapter, the mass distribution on larger scales is treated, namely that of galaxy clusters (Section 5.3), the statistical properties of the mass of galaxies, groups and clusters on large scales (Section 5.5), and that of the large-scale structure in the Universe in Section 5.4.

In preparing this chapter, I have assumed that the reader has read the contribution by Sherry Suyu (Chapter 1, this volume), where the basic theory of gravitational lensing is treated thoroughly, or is otherwise familiar with the concepts of gravitational lensing. Furthermore, I assume a certain familiarity with the cosmological standard model. It should also be mentioned here that I have written about these topics before, with extended lecture notes published as Schneider (2006). For more details on many issues discussed below, I refer the reader to this work. For the same reason, only a few papers from before 2005 are cited. I have made no efforts to present a balanced, or even complete presentation of the subject; therefore, I apologize to all those colleagues whose work has not been discussed or cited.

Magnification maps are an essential tool in microlensing studies. Their calculation is based on very simple principles and it is therefore quite straightforward to implement. This tutorial is intended to show how these calculations are done by using a basic rayshooting procedure. The tutorial assumes some basic knowledge of any programming language, but no previous knowledge of the specific language used here or experience with gravitational lensing computations is needed. The very basics of gravitational lensing are also implicitly assumed at some points. From the computational point of view, the tutorial covers topics ranging from the simplest ray-shooting program for generating images of an object through a simple lens system to the production of magnification maps for quasar microlensing. Source size effects and how to deal with them are also briefly discussed. We finish by also briefly discussing the main improvements that have been introduced into this technique to make calculations faster.

Introductory remarks

It is particularly difficult to put into writing what was intended as a couple of practical sessions on inverse ray-shooting techniques during the Winter School. Unfortunately, there is not much choice but to illustrate the ideas with some code. At this point, a choice has to be made whether to use pseudocode or to choose a given programming language. I have chosen the latter option here in the hope that the reader may actually use the code snippets presented here straight away and be able to produce some useful programs from it. During the lectures, the Python programming language was used for the tutorial. I chose it for a couple of reasons that are enumerated below. I have therefore also used Python here for presenting the techniques introduced in this tutorial. Nevertheless, the reader is not expected to know any Python in advance as I shall introduce all the required information on Python syntax in Section 8.2. Readers should therefore find no problem in following the explanations and/or in translating the code to their favourite programming language. Finally, these lectures were intended as practical sessions and, as such, the focus is on producing some useful code and on understanding the key steps in that process. Performance will be dealt with at some point in order introduce the necessary Python ingredients to make the programs fast enough to produce results in a reasonable time.

We are on the verge of an explosion in data volume owing to recently started or upcoming surveys of the skies. One of the benefits of these new programmes will be the vastly increased number of known strong gravitational lens systems. In this chapter I will discuss three main topics: lens discovery in these surveys; the use of lensing to determine the mass distribution in galaxies, and in particular substructure in massive galaxy haloes; and cosmological measurements with large lens samples.

Introductory remarks

The next few decades will present an especially exciting time for strong gravitational lensing. This is because a combination of new instrumentation and, in some cases, brand new telescopes have come online, or are at an advanced stage of planning. With the enhanced observing capability enabled by these new facilities, a number of large-scale astronomical surveys are planned. These surveys will provide unprecedented combinations of depth, area, angular resolution and, in some cases, will open up poorly explored wavelength regimes. As a result, they should lead to orders of magnitude increases in the number of known strong lens systems. Indeed, although dedicated observational surveys for lenses have proved productive in the past, it is likely that the vast majority of future lenses will be discovered by mining the data produced by the new large surveys. The resulting large samples of lenses will lead to two major advantages: (1) improved statistics for investigations of galaxy properties, evolution in these properties and cosmology etc., and (2) the discovery of rare lens systems that are especially interesting and useful. As a complement to the large surveys, the planned construction of significantly larger ground- and space-based telescopes will provide enhanced follow-up capabilities of the new discoveries. Furthermore, advances in modelling and analysis codes will allow researchers to exploit more of the information available in observations of lens systems.

As is obvious at this point, this chapter has a focus on the field of strong gravitational lensing, and how it can be affected by large recently started and upcoming astronomical surveys. Taking this approach necessarily ignores other aspects of lensing that also have an exciting future. The new surveys will have a strong impact on investigations that utilize weak lensing or microlensing.

This book gathers together the lectures and practical sessions imparted during the XXIVth Canary Islands Winter School of Astrophysics, held at Puerto de la Cruz, Tenerife (4–16 November 2012).

The basic phenomena of gravitational lenses, light deflection and time dilation by gravitational fields, are two essential predictions of Einstein's General Theory of Relativity. Both effects played a prominent role in the classical tests of General Relativity through famous experiments such as the deflection of light by the Sun measured by Eddington during the 1919 solar eclipse and the radar time delays first measured by Shapiro from the echoes of planets and space probes in the Solar System. Owing to rapid developments in technology, these once exotic and difficult-to-measure effects can nowadays be tested millions of times per second with a very popular device, the GPS (gravitational lensing in everyday life). The present and future importance of gravitational lenses is therefore no longer related exclusively to fundamental General Relativity but also (this is our motivation) to its use in probing the properties of astrophysical objects and of the Universe itself.

The optical bench is one of the most common pieces of laboratory apparatus in modern physics. A source emits photons, alpha particles, neutrons or some other kind of ‘bullets’ that interact with a test object (the target) and are subsequently detected by the observer. This set-up enables the researcher to change and move at will any of the components of the experiment to check hypotheses being tested. Astrophysical sources and targets (planets, stars, galaxies, etc.) are too big and distant for the astronomer to be able to manipulate them. Nevertheless, in certain rare cases a distant source (a star, galaxy, or quasar) appears to be almost aligned with an intervening target (a planet, star, galaxy, or galaxy cluster), thus allowing the observer to measure the deflection of the light rays caused by the gravitational field of the target. This is a gravitational lens system (or simply ‘gravitational lens’), an astronomical optical bench that can be used as a tool to study both the source and the deflecting target.