GRAVITATIONAL LENSES JOHN A. PEACOCK Gravitational lenses are in a sense a prediction of Albert Einstein's theory of general relativity. Given the idea that light rays will be bent by the gravitational pull of a massive body (so spectacularly confirmed during the solar eclipse of 1919), it may seem obvious that a sufficiently massive body will allow light to pass from a source to an observer by more than one path, thereby forming a lens of sorts. However, it took until 1936 before Einstein himself published such an idea, and until 1979 before the effect was first seen by astronomers-60 years after that first detection of gravitational light deflection. Since then the subject has flourished because it provides an unmatched probe of the dark matter that we now believe dominates the mass content of the universe. Because the subject was purely theoretical for so long, we describe the theory first and then some of the lenses that have actually been found. Reviews containing background references for this material can be found in the additional reading at the end of this entry. HOW LENSES WORK LIGHT DEFLECTION Light bends in response to a gravitational acceleration perpendicular to its path. The formula for this is *******************, where * is the angle of deflection, g is the acceleration, and dl is the length of path traversed. This is exactly twice the result obtained without using general relativity, and it provided a crucial test of that theory. The bending is not quite this simple in very strong gravitational fields, such as close to black holes, but this expression suffices for most purposes. LENSING GEOMETRY The operation of a single gravitational lens is illustrated in Fig. 1. For simplicity, source, lens, and observer are shown lying in the same plane, although this is not generally the case. The parameters D are angular-diameter distances that may be related to redshift in a given cosmological model. The lens is thin when the light bending takes place within a small enough distance that the deflection can be taken as sudden. Thick lenses correspond to the combined effects of deflections at several different distances, and are harder to treat. Equating distances at the left-hand side of Fig. 1 yields the fundamental lensing equation ***************************, where the angles **, **, and * are in general two-dimensional vectors on the sky. In principle, this equation can be solved to find the mapping between the object plane and the image plane [i.e.,**,(**)] and hence the positions of the images. It is clear that gravitational lenses are not very appealing from an optician's point of view. A perfect lens would have a deflection angle that increases linearly with distance from the optic axis, whereas the opposite is often true with gravitational deflections. In the French literature, the term "gravitational mirage" is used, and this is arguably more appropriate. AMPLIFICATIONS The only other thing needed in order to find the appearance of lensed images is the fact that gravitational lensing does not alter surface brightness (rate of reception of energy per unit area of sky). This is well known to be a quantity that is independent of distance in Euclidean space, but in fact is conserved generally. This comes about from the relativistic invariant **/**, which is proportional to the photon phase-space density and is conserved along light rays through Liouville's theorem. Hence, the amplification of image flux densities is given simply by a ratio of image areas: Larger images are brighter. In these circumstances, magnification might be a better word than amplification; however, one is often dealing with objects such as quasars where the image distortion cannot be observed; when the only observable is a change in the flux density of a point source, the term amplification is appropriate. IMAGE STRUCTURES The solution of the lensing equation for a symmetrical lens may be visualized graphically as the intersection of a straight line with the bend-angle curve. Images are displaced from their intrinsic positions along radial lines; although this produces some radial distortion, the principal effect is of a transverse stretching, leading to crescent-shaped images. For cases of close alignment, the principal outer pair of images has an angular separation (M/10**ù**M*)*******)*** arcsec. In cases of perfect alignment, the image becomes perfectly circular, forming an Einstein ring of this diameter. Although galaxies are not well described by point masses, in the case of circular symmetry, the size of the Einstein ring simply measures the projected mass interior to it. Thus, apart from uncertainties introduced by cosmology via the distance measures, lensing provides a clean and powerful method of estimating the masses of astronomical objects. ODD-NUMBER THEOREM One powerful theorem that governs the general appearance of the results of a lens event states that the number of separate images produced must be odd. The only exceptions to this occur when the lensing mass is singular (of infinite density at some point). This type of argument may, in principle, reveal the existence of a black hole or cosmic string. TIME DELAYS A potentially important aspect of gravitational lensing is that the lens alters the time taken for light to reach the observer, producing a time lag between any multiple images. There are two effects at work: a geometrical delay caused by second-order increase in the path-length and a second term due to the gravitational potential (light travels more slowly in a gravitational field, partly due to gravitational time- dilation effects and partly due to spatial curvature). Sadly, although this second term is rather more model-dependent than the geometrical term, it is usually of the same order of magnitude. Where this is not the case, it would be possible to use time delays to measure the Hubble constant H* quite accurately. This arises because the geometry of the lensing event can usually be found from the observed images. The differential time delay between various light paths then scales with the overall size of the system. Because cosmological distances inferred from redshifts scale as H**, an estimate of H* then follows. For well- constrained lens models, one can do quite well in this respect; see the section below on observations. Q - o CAUSTICS AND CATASTROPHES 0 Gravitational optics has an important connection with the branch of mathematics known as catastrophe theory. This provides a powerful means of analyzing general properties of lenses by concentrating on the mapping from the sky before lensing to that after lensing. The most important concept is that of critical lines or caustics: source positions where the amplification becomes infinite. New multiple images are created or destroyed in pairs as the source crosses a caustic. This allows important general properties to be proved: In particular, the cross section or probability of production of a high-amplification event behaves as *(> A)* A**. STATISTICAL LENSING Because many classes of astronomical object have a distribution of luminosities where faint objects far outnumber bright ones, the nonzero probability of high amplification alluded to previously opens the possibility that selection effects can cause lensed objects to dominate. This will tend to happen whenever the intrinsic luminosity function is as steep or steeper than the power-law tail of the lensing distribution: **L** in integral terms. It can be shown that the probability of strong lensing of a quasar at high redshift is roughly **, the density parameter of the universe. Because the contribution to ** by the cores of galaxies and rich clusters is small, such events are rare. However, they may be much more common if we consider microlensing by objects of much lower mass. This is because lines of sight through galaxy haloes may pass many potential lenses in the form of Stars; also, there is the exciting possibility that the dominant dark matter may consist of clumps. For quasars, where the intrinsic angular sizes are very small, effective lenses can be as small as 1% of a solar mass. Microlensing can thus be difficult to prove, but does leave a characteristic signature in variability as the stellar lens moves. OBSERVATIONS OF LENSES Observed lens events may conveniently be divided into two according to the type of object that is being lensed: ordinary galaxies or rare objects (active galaxies and quasars). Lensing in the latter is easier to recognize, not only because quasars can readily be detected to very high redshifts, but also because the typical separation of quasars on the sky is very large by comparison with the angular scale of lensing effects. A close pair of quasars then quickly raises the suspicion of lensing, whereas galaxy pairs are very common. MULTIPLE QUASARS In fact, the first detection of lensing was made when optical identification of a radio source yielded two stellar images 6 arcsec apart. Normally, one would be a quasar, the other a foreground galactic star. It is to the great credit of the discoverers that they took a spectrum of the second object even when the first had turned out to be "the" quasar. Similar objects have since been found at the rate of about one per year. The slow pace of discovery reflects not only the intrinsic rareness, but also the necessity for caution: True physical pairs of quasars are not completely unknown, and quite accurate spectroscopy is required to establish that the spectra are precisely identical in shape and redshift (give or take the effects of variability and differential time delays). Probably the best example of this genre is still the first one: The double quasar 0957+561. This only displays two images, but is important because it is a bright radio source. Both images are bright enough to allow the position angle of the nuclear jets to be measured via long-baseline interferometry; this information sets much tighter constraints on the lensing potential than is available just from image positions and luminosities. Long-term monitoring has also revealed correlated variability in the two images, establishing the differential time delay between the two paths to be about 420 days. For the best lens models, this corresponds to a Hubble constant of about 60 km s** Mpc**, but the true uncertainty in this will probably not reveal itself until this estimate can be repeated using other lenses. EINSTEIN RINGS Searches for lensed quasars in the radiowave band have also revealed more spectacular objects. Several thousand radio sources have been examined in the hope of finding multiple point images suggestive of lensed quasars, but in many cases the radio sources possess extended lobes that improve the chance of good alignment with a lensing galaxy. Two cases have been found to date of such lobes being imaged into a near-perfect Einstein ring. One of these is illustrated in Fig. 2. Here, the proof of lensing is not redshift coincidence because the radio emission has no spectral lines, but arises in showing that the ring can be deprojected to a simple source structure. CLUSTER ARCS There are also several cases known where high-redshift galaxies have been strongly lensed by clusters of galaxies into arcs tens of arcseconds in extent. Here, the confirmation of lensing consists of showing that all the arc has the same redshift. Such lenses that probe the cluster mass distribution are very useful because clusters are the largest relaxed self-gravitating systems that exist. Deducing their masses from galaxy velocities and x-ray data on intracluster gas is of great cosmological importance; lensing adds an invaluable extra constraint. QUASAR-GALAXY ASSOCIATIONS Even when no multiple imaging occurs, quasars seen through the outer parts of galaxy halos may be amplified (especially if microlensing is important). This can lead to an apparent association between quasars and low-redshift galaxies. At the time of writing, evidence is starting to emerge for the existence of such associations. Although it is hard to prove that one is seeing lensing (rather than, say, noncosmological redshifts), this type of observation has promise statistically, because weak lens events are so much more common than strong ones. THE FUTURE There are signs that the number of known lenses will increase more rapidly in the future. This will come not through searches for arcs in clusters, which will always be hard to recognize, but through automated surveys for quasars and radio sources using either radio maps or optical quasar searches. Present statistics suggest about one quasar in 1000 is lensed, so there must be thousands of lenses awaiting discovery. The problem is that only the bright ones are really useful for the detailed modeling that may yield the distance scale. Nevertheless, statistical studies may be just as important for understanding dark matter. In particular lensing may be the very best way of detecting cosmic strings: relics from the early phases of the Big Bang that would manifest themselves as a line of double galaxy images. Detection of even one string would be sufficient cosmological importance to justify any amount of effort in lens searches. Come what may, then, gravitational lensing seems likely to remain an important cosmological tool for the foreseeable future. Additional Reading Blandford, R.D.(1990). Gravitational lenses. Q.J. Royal Astron. Soc. 31 305. Blandford, R.D. and Kochanek, C.S.(1987). Dark Matter in the Universe, J. Bahcall, T. Piran, and S. Weinberg, eds. World Scientific, Singapore, p. 133. Peacock, J.A.(1983). Ouasars and Gravitational Lenses, J.-P. Swings, ed. Universite Liege, p. 86. Schneider, P., Ehlers, J., and Falco, E.E.(1991). Gravitational Lenses. Springer, Heidelberg. Turner, E.L.(1988). Gravitational lenses. Scientific American 259 (No.1) 54