Gravitational Lenses I:           Galaxies as Lenses

Gravitational Deflection of Light

Einstein's Theory of General Relativity predicts that light in the gravitational field of massive bodies will be deflected. Measurements of this deflection of light near the edge of the sun during a solar eclipse in 1919 provided the breakthrough to support Einstein's Theory of Gravitation (see Fig.1). The deflection of a light ray increases with the mass of the body, and decreases with distance of the light from the body. Using the gravitational deflection of light, now called the gravitational lens effect, this dependence of the deflection angle allows the measurement of mass or the general distribution of mass in the universe to be studied. Of special importance is that the deflection of light is independent of the nature of the matter. This means that with the use of the gravitational lens effect, both known, luminous matter, as well as dark matter can be measured. Therefore, in recent years this effect has proven to be an important tool in cosmology.
 
 

Fig.1: A representation of the deflection of light at the edge of the sun: In the gravitational field of the sun, light rays deviate from a straight path, whereby the position of a star in the background appears displaced. At the edge of the sun, the deflection of light is about 1.75 arc seconds. If the lens were much more massive, and thus the deflection angle much larger, then it would be possible that additional light rays from a distant source would reach the observer.

Sober, but Nevertheless Seeing Multiple Images ...

If the deflector is compact and massive enough, it would be possible for light from one source to reach the observer via more than one path. In this case, the observer would see the same source at several positions in the sky. Should a galaxy act as a lens, the expected separation of the images would be about an arc second, comparable with the resolution capability of modern, ground-based telescopes. The discovery of the first gravitational lens system was not made until 1979, when the first double quasar was found - a galaxy lying between us and the distant quasar yields two separate light paths from the quasar two us, hence it is seen in two different directions. In the meantime, around 50 such multiple image systems have been discovered, two of which are shown here.
 
 

Fig.2: Due to the influence of an elliptical galaxy lens, the quasar 1422+231 with a redshift z=3.62 is split into four images. The picture shows a map of this lens system in radio wave frequencies. As can be clearly seen, three of these images are much brighter than the fourth. The varying degrees of brightness result from the magnifying effect of the lens; the brightest image is about 20 times brighter than the quasar seen without a lens.

Fig.3: The lens system 2237+0305 was discovered during a spectroscopic study of a spiral galaxy. Besides the spectrum of the galaxy are strong emission lines which originate from a quasar with a redshift z=1.7. Later it was shown that the "galactic core" is divided into four quasar images. The spectroscopy of each image shows without a doubt that all originate from the same source. The four images almost trace a circle around the core of the spiral galaxy, which characterizes the Einstein radius of this lens. The mass of the lens within this Einstein radius can be determined within an accuracy of 3%.

(Einstein) Rings in the Sky

If source, lens and observer are almost exactly on a straight line and the distribution of matter of the lens is almost symmetrical, then the light from the source appears as a ring-shaped image to the observer. Such a ring - called an Einstein ring - was first observed in 1986 in a radio source. Presently around a dozen such rings are known, two of which are shown here. The radius of such a ring is called the Einstein radius. It depends only on the distance of the lens and of the source as well as on - and this is crucial - the mass of the lens within the ring. It is important that from an observed ring one can immediately find out the mass of the lensing galaxy inside the ring, with an unsurpassed accuracy in (extragalactic) astronomy.

Similarly, the mass estimation can be done in multiple image systems; particularly for four image systems, the Einstein radius of the lens can be obtained very precisely. For a few lens systems the mass inside the Einstein radius can be determined to within a few percent!

Both the investigation of individual lens systems, and the statistics of multiply imaged quasars, clearly indicates that spiral galaxies and elliptical galaxies have a halo of dark matter, which is very difficult to verify with other methods.
 
 

Fig.4: The Einstein Ring 1938+666. The left panel is an infrared image of this lens system, showing a ring-shaped image of the host galaxy of this quasar. In a radio image (right panel), the radio emission from the quasar is mapped into a partial ring only - the size of the radio source is smaller than the infrared source, and so no full ring is formed

Fig.5: The contours show the radio map of MG1654 with redshift z=1.7. Superposed in colors is an optical image of the system. The radio lobe is seen to be deformed into a ring-shaped imaged by a spiral galaxy with redshift z=0.25, centered on the ring.

Cosmic magnifying 'glasses'

It can be seen in Fig.2 that the individual images of the multiple image system are unequally bright. This comes from the fact that the light bundle from the source is not only deflected on the whole, but rather experiences differential deflection. The shape and size of the images thus change compared to the unlensed source. From a larger image it appears that more light arrives at us from the source; it is said that the aforementioned image is magnified. One sees for example in the quasar 1422+231 that the flux ratio of the images may be very large.
 
 
 

Big Lenses, Small Lenses

The distribution of matter  in galaxies is not "smooth" but instead grainy: galaxies contain stars, molecular clouds, star clusters, spiral arms etc. The scale of the mass of these smaller structures is small enough that it barely influences the angle of deflection of the light bundles as a whole. However, if the sources of these light bundles are small enough, the graininess of the distribution of matter  becomes noticable by the differential light deflection, and the magnification of individual images will be influenced. The area from which the optical light from quasars originates is so small (probably smaller than a light day), that even stars in lens galaxies can disturb the brightness observed in images of quasars. Since the relative position of source, lens and observer changes in time through their peculiar motion, the brightness of the images also changes. This effect can be proven for multiple quasars through observation of the temporal variation of the images' brightnesses. If the variation would be due to a fluctuation in the brightness of the source, all images would show the same light curve, while uncorrelated fluctuations in brightness would be due to this so-called micro-lensing effect.
 
 

Fig.6: Models of the effects of micro lensing predict the type of changes in brightness. The typical result of  such simulations is shown in the form of magnification maps, through which synthetic light curves can be determined. They, in turn, can be compared statistically with observed light curves. The characteristics of such magnification maps strongly depend on the density of the microlenses, i.e. of the stars, located at the point of transit of the light bundle through the galaxy.

Fig.7: Light curves of four images of quasars in the lens system 2237+0305 show different variations. From models of the lens it is known that differences in the arrival time of light between images is less than a day, so that intrinsic changes in the source's brightness must appear practically simultaneously in the four images. This is obviously not the case. Thus the different variations in brightness are interpreted to be a time-varying magnification, caused by the motion of stars within the lens galaxy. From the timescale of the variation, an upper limit can be placed on the size of the region from which the optical light is emitted.

How big is the Universe?

If a source that is imaged several times experiences intrinsic fluctuations in brightness, these variations can be seen in all images of the source - but not simultaneously. The time needed for the light to travel from the source to the observer is different for different images, since the length of the paths of the light and gravitational time delay in the lens potential differ. This difference in the light travel time depends on the configuration of the images relative to the lens galaxy, in particular on the angle of each image from the center of the lens. A simple consideration shows that the difference is directly proportional to the size of the universe. If the geometry of the lens system is understood well enough, then this size can be directly calculated from the measurement of the time delay of the light. It is normally expressed in terms of the Hubble constant Ho, the expansion rate of the universe. This method of determining Ho is completely independent of all other methods, which are much more indirect and are based on a so-called distance ladder. In contrast, measuring Ho with lenses is purely of a geometric nature. Such a measurement has already been carried out for five lens systems. The resulting values for Ho are at the lower end of the regime obtained by other methods.
 

Fig.8: The optical light curves of the double quasar 0957+561 are shown. Here, the light curve of the B-component was delayed by 417 days. As can be seen, these two light curves fit together very well, while the ones without this delay differ greatly. This means the fluctuations in brightness of both images are compatible with the intrinsic variations of the source and with a light time delay of about 417 days. From this, the Hubble constant is estimated to be around 65 km/s/Mpc.

Gravitational Lenses III: The Weak Lensing Effect