LECTURES ON GRAVITATIONAL LENSING RAMESH NARAYAN1 AND MATTHIAS BARTELMANN2 1HARVARD-SMITHSONIAN CENTER FOR ASTROPHYSICS,60GARDEN STREET,CAMBRIDGE, MA 02138, USA 2MAX-PLANCK-INSTITUT FURÈ ASTROPHYSIK,P.O.BOX 1523, D±85740 GARCHING,GERMANY in: Formation of Structure in the Universe Proceedings of the 1995 Jerusalem Winter School; edited by A. Dekel and J.P. Ostriker; Cambridge University Press ABSTRACT These lectures give an introduction to Gravitational Lensing. We discuss lensing by point masses, lensing by galaxies, and lensing by clusters and larger-scale structures in the Universe. The relevant theory is developed and applications to astrophysical problems are discussed. CONTENTS phenomena resulting therefrom, are now referred to as Gravita- tional Lensing. 1. Introduction 1 2. Lensing by Point Masses in the Universe 3 2.1.BasicsofGravitationalLensing......... 3 2.2.MicrolensingintheGalaxy........... 7 2.3.ExtragalacticMicrolenses............ 10 3. Lensing by Galaxies 11 3.1. Lensing by a Singular Isothermal Sphere . 11 3.2.EffectiveLensingPotential........... 12 3.3. Gravitational Lensing via Fermat's Principle . 13 3.4.CircularlySymmetricLensModels....... 15 3.5. Non-Circularly-Symmetric Lens Models . 15 FIG. 1.ÐAngular de¯ection of a ray of light passing close to the limb 3.6.StudiesofGalaxyLensing........... 17 of the Sun. Since the light ray is bent toward the Sun, the apparent po- 3.7. Astrophysical Results from Galaxy Lensing . 19 sitions of stars move away from the Sun. 4. Lensing by Galaxy Clusters and Large-Scale Struc- Eddington (1920) noted that under certain conditions there ture 21 may be multiple light paths connecting a source and an observer. 4.1. Strong Lensing by Clusters Ð Giant Arcs . 22 This implies that gravitational lensing can give rise to multiple 4.2.WeakLensingbyClustersÐArclets...... 25 images of a single source. Chwolson (1924) considered the cre- ation of ®ctitious double stars by gravitational lensing of stars by 4.3. Weak Lensing by Large-Scale Structure . 28 stars, but did not comment on whether the phenomenon could ac- tually be observed. Einstein (1936) discussed the same problem 1. INTRODUCTION and concluded that there is little chance of observing lensing phe- nomena caused by stellar-mass lenses. His reason was that the One of the consequences of Einstein's General Theory of Rel- angular image splitting caused by a stellar-mass lens is too small ativity is that light rays are de¯ected by gravity. Although this to be resolved by an optical telescope. discovery was made only in this century, the possibility that there Zwicky (1937a) elevated gravitationallensing from a curiosity could be such a de¯ection had been suspected much earlier, by to a ®eld with great potential when he pointed out that galaxies Newton and Laplace among others. Soldner (1804) calculated can split images of background sources by a large enough angle the magnitude of the de¯ection due to the Sun, assuming that to be observed. At that time, galaxies were commonly believed 9 light consists of material particles and using Newtonian grav- to have masses of 10 M . However, Zwicky had applied the ity. Later, Einstein (1911) employed the equivalence principle to virial theorem to the Virgo and Coma clusters of galaxies and had 11 calculate the de¯ection angle and re-derived Soldner's formula. derived galaxy masses of 4 10 M . Zwicky argued that the Later yet, Einstein (1915) applied the full ®eld equations of Gen- de¯ection of light by galaxies would not only furnish an addi- eral Relativity and discovered that the de¯ection angle is actu- tional test of General Relativity, but would also magnify distant ally twice his previous result, the factor of two arising because galaxies which would otherwise remain undetected, and would of the curvature of the metric. According to this formula, a light allow accurate determination of galaxy masses. Zwicky (1937b) ray which tangentially grazes the surface of the Sun is de¯ected even calculated the probability of lensing by galaxies and con- 00 by 1: 7. Einstein's ®nal result was con®rmed in 1919 when the cluded that it is on the order of one per cent for a source at rea- apparent angular shift of stars close to the limb of the Sun (see sonably large redshift. Fig. 1) was measured during a total solar eclipse (Dyson, Edding- Virtually all of Zwicky's predictions have come true. Lens- ton, & Davidson 1920). The quantitative agreement between the ing by galaxies is a major sub-discipline of gravitational lens- measured shift and Einstein's prediction was immediately per- ing today. The most accurate mass determinations of the cen- ceived as compelling evidence in support of the theory of Gen- tral regions of galaxies are due to gravitational lensing, and the eral Relativity. The de¯ection of light by massive bodies, and the cosmic telescope effect of gravitational lenses has enabled us to 1 study faint and distant galaxies which happen to be strongly mag- (Soucail et al. 1987a,b; Lynds & Petrosian 1986). PaczyÂnski ni®ed by galaxy clusters. The statistics of gravitational lensing (1987) proposed that the arcs are the images of background events, whose order of magnitude Zwicky correctly estimated, galaxies which are strongly distorted and elongated by the grav- offers one of the promising ways of inferring cosmological pa- itational lens effect of the foreground cluster. This explanation rameters. was con®rmed when the ®rst arc redshifts were measured and In a stimulating paper, Refsdal (1964) described how the Hub- found to be signi®cantly greater than that of the clusters. ble constant H0 could in principle be measured through gravita- Apart from the spectacular giant luminous arcs, which re- tional lensing of a variable source. Since the light travel times for quire special alignment between the cluster and the background the various images are unequal, intrinsic variations of the source source, clusters also coherently distort the images of other faint would be observed at different times in the images. The time de- background galaxies (Tyson 1988). These distortions are mostly lay between images is proportional to the difference in the ab- weak, and the corresponding images are referred to as arclets solute lengths of the light paths, which in turn is proportional (Fort et al. 1988; Tyson, Valdes, & Wenk 1990). Observa- 1 tions of arclets can be used to reconstruct parameter-free, two- to H0 . Thus, if the time delay is measured and if an accurate model of a lensed source is developed, the Hubble constant could dimensional mass maps of the lensing cluster (Kaiser & Squires be measured. 1993). This technique has attracted a great deal of interest, All of these ideas on gravitational lensing remained mere spec- and two-dimensional maps have been obtained of several galaxy ulation until real examples of gravitational lensing were ®nally clusters (Bonnet et al. 1993; Bonnet, Mellier, & Fort 1994; discovered. The stage for this was set by the discovery of quasars Fahlman et al. 1994; Broadhurst 1995; Smail et al. 1995; Tyson (Schmidt 1963) which revealed a class of sources that is ideal & Fischer 1995; Squires et al. 1996; Seitz et al. 1996). for studying the effects of gravitational lensing. Quasars are dis- As this brief summary indicates, gravitational lensing mani- tant, and so the probability that they are lensed by intervening fests itself through a very broad and interesting range of phenom- galaxies is suf®ciently large. Yet, they are bright enough to be ena. At the same time, lensing has developed into a powerful tool detected even at cosmological distances. Moreover, their optical to study a host of important questions in astrophysics. The appli- emission region is very compact, much smaller than the typical cations of gravitational lensing may be broadly classi®ed under scales of galaxy lenses. The resulting magni®cations can there- three categories: fore be very large, and multiple image components are well sep- ± The magni®cation effect enables us to observe objects arated and easily detected. which are too distant or intrinsically too faint to be ob- Walsh, Carswell, & Weymann (1979) discovered the ®rst ex- served without lensing. Lenses therefore act as ªcosmic ample of gravitational lensing, the quasar QSO 0957+561A,B. telescopesº and allow us to infer source properties far be- This source consists of two images, A and B, separated by 6 00 . low the resolution limit or sensitivity limit of current obser- Evidence that 0957+561A,B does indeed correspond to twin vations. However, since we do not have the ability to point lensed images of a single QSO is provided by (i) the similarity this telescope at any particular object of interest but have to of the spectra of the two images, (ii) the fact that the ¯ux ra- work with whatever nature gives us, the results have been tio between the images is similar in the optical and radio wave- only modestly interesting. bands, (iii) the presence of a foreground galaxy between the im- ages, and (iv) VLBI observations which show detailed corre- ± Gravitational lensing depends solely on the projected, two- spondence between various knots of emission in the two radio dimensional mass distribution of the lens, and is indepen- images. Over a dozen convincing examples of multiple-imaged dent of the luminosity or composition of the lens. Lensing quasars are known today (Keeton & Kochanek 1996) and the list therefore offers an ideal way to detect and study dark matter, continues to grow. and to explore the growth and structure of mass condensa- PaczyÂnski (1986b) revived the idea of lensing of stars by stars tions in the universe. when he showed that at any given time one in a million stars in the Large Magellanic Cloud (LMC) might be measurably mag- ± Many properties of individual lens systems or samples of ni®ed by the gravitational lens effect of an intervening star in lensed objects depend on the age, the scale, and the overall the halo of our Galaxy.