# 16.19 Gravitational Lensing

Einstein's general relativity is the theory that accurately describes the way light is affected by mass. In Einstein's theory, mass distorts space — light then bends to follow this distortion of space. More correctly, mass-energy affects the geometry of space-time. Or, as the eminent physicist John Wheeler, who coined the term black hole, said: "Mass tells space-time how to curve, and space-time tells mass how to move." General relativity received its first experimental confirmation in the observed deflection of starlight as it passed close to the edge of the Sun.

The equation that describes the deflection of light by mass is a fundamental result of general relativity. The square of the deflection angle in radians is given by:

Θ_{E}^{2} = 4GM / c^{2}D

In this equation, G is the gravitational constant, c is the speed of light, M is the mass of the deflector, and D is the distance to the deflector. A quasar at a typical redshift of z = 1 has a distance of about 2000 Mpc. Let’s calculate the deflection caused by a typical massive galaxy — roughly 10^{12} solar masses if we include the dark matter. Converting to metric units of meters and kilograms, we get Θ_{E}^{2} = (4 x 6.7 x 10^{-11} x 10^{12} x 2 x 10^{30}) / ((3 x 10^{8})^{2} x 2 x 10^{9} x 3 x 10^{16}) = 10^{-10} radians. So Θ_{E} = √ 10^{-10} = 10^{-5}. We then multiply by 206,265 to get the answer in arc seconds, Θ _{E} = 2.0 arc seconds.

A single galaxy can deflect distant quasar light by 2 arc seconds. It's a happy coincidence that this angle is a bit larger than the amount by which the atmosphere blurs incoming light; if it had been five or ten times smaller we might never have detected lensing! The first lensed quasar was detected in 1979 at the Multiple Morrow Telescope in Arizona, when the images of the double quasar 0957+561 were found to be 6 arc seconds apart. The multiple images are a kind of mirage — there is only one object but the light is bent on two different paths around the intervening galaxy so two images are seen. If the quasar is perfectly aligned behind the galaxy and the galaxy is symmetric (a good approximation for an elliptical galaxy), then the deflection will occur equally in every orientation and the deflected rays will trace back to form the image of a ring on the sky. Thus a point source of light — a quasar — is transformed into an Einstein ring. The angular scale Θ_{E} is known as the Einstein radius.

Lensing is rare. Only 1 in 300 quasars is gravitationally lensed because the odds of a galaxy or galaxy cluster lying directly in front of a distant quasar are low. However, it is not as rare as you might expect because a second consequence of lensing is the fact that the quasar light is amplified. A gravitational lens will magnify light just as an optical lens will. Lensing therefore makes a quasar brighter than it would have been in the absence of a foreground galaxy. This means that a lensed quasar can be seen to a greater distance than a non-lensed quasar. The greater volume out to that larger distance means a larger potential population to be lensed. Thus there is a selection effect in favor of detecting lensed quasars. Lensing is "nature's telescope." Since lensing can amplify the brightness of the background galaxies by factors of 10-20 or more, lensing can let astronomers galaxies that would be too faint or too distant to detect otherwise.

With spectra and images available for tens of thousands of quasars, over a hundred examples of quasar lensing have been cataloged. A perfect alignment is of course rare, so astronomers would not expect to see Einstein rings very often. Only a few have ever been observed. A more likely outcome is a slight misalignment, which leads to the ring splitting up into four distinct images. If the misalignment is more severe, the result is a single magnified arc and a de-magnified image on the other side of the lens. Formally, lensing always creates an odd number of images, but one is usually highly demagnified so the most common lensed systems are doubles, followed by quads. The same deflection equation applies to distant galaxies that are lensed by nearby galaxies. A cluster can produce numerous lensing arcs of background galaxies. Lensing is an excellent method for "weighing" a galaxy, since the deflection of light is caused by all the mass, including the invisible dark matter.

Considering deflection angles leads to an understanding of the phenomenon of gravitational microlensing. Imagine looking for dark matter in the halo of the Milky Way with the suspicion that it is made of sub-stellar objects, like white dwarfs or brown dwarfs. What is the lensing signature of a 0.01 solar mass object observed at a distance of 10 kpc against a backdrop of more distant stars? Scaling from our previous result, we note that Θ_{E}^{2 }∝ M/D, so Θ_{E} ∝ (M/D)^{1/2}. The mass is 10^{14} times smaller and the distance is 2 x 10^{5} times smaller, so the deflection angle is (10^{14}/2 x 10^{5})^{1/2} = 22,000 times smaller than the deflection angle for lensing by a galaxy. We get 2/22,000 = 10^{-4} arc seconds or 100 micro-arc seconds — hence the term micro lensing. Since the deflection angle is much smaller than the smallest angle that can be resolved by any telescope (including the Hubble Space Telescope), the Einstein ring is not visible and the image is not noticeably distorted.

If micro lensing produces such a small deflection angle, how can the effect be measured? The answer is that astronomers look for the amplification of the light source rather than image splitting. As a dark lens passes in front of a more distant star, the star will brighten considerably for the time it takes the lens to cross the area defined by the Einstein radius. Given the typical velocity of objects in the halo of the Milky Way, the background star will be brightened for about a week. Lower-mass dark objects have smaller Einstein radii, so the duration of the light amplification is shorter. Micro lensing has been an effective way to detect low mass extra solar planets or exo planets. Over a dozen have been detected, and in principle, planets smaller than the Moon can be detected.

What is the recipe for detecting micro lensing? Look for a one-time brightening and fading of a background star with a light curve that does not depend on wavelength (the deflection angle predicted by general relativity is the same for any wavelength of light). This distinctive signature is unlike any variation micro lensed at any particular time, so at least a million stars per night must be observed to have a chance of catching these rare events. Modern CCD detectors allow this kind of a wide-angle survey, and micro lensing events have been detected in the halo of our galaxy. However, the rate of micro lensing is too low for the dark halo of the Milky Way to be composed mostly of sub-stellar objects.