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General Relativity

The Gravitational Lenses of Galaxies

Like a star, a galaxy has a gravitational lens that affects the appearance of more distant objects. Far from a galaxy, the lens is identical to the lens of a point source, but within the galaxy, the lens's characteristics are set by the distribution of the galaxy's mass. For a nearly spherical galaxy, the behavior is not too complex. When such a galaxy is close, we see only one image of a more distant object, which we see by looking through the galaxy; if this object is on a direct line with the center of the galaxy, it will appear at the center of the galaxy. If the galaxy is moved back, a distance is eventually reached where we see an Einstein ring in addition to the image at the galaxy's center. This ring appears at a point from the center of the galaxy where the density of the galaxy begins falling rapidly. Pushing the galaxy farther out, the Einstein ring splits into two rings; the the inner ring shrinks to the center as the galaxy is pushed back, and the outer rings shrinks more slowly, so that it becomes larger than the source galaxy. As with a star, when the object behind the lens is moved off-axis, each Einstein ring splits and collapses into a pair of images that flank the center of the galaxy. This means that objects behind the gravitational lens of a galaxy cam appear as five images.

While the time delay between images created by a point source lens is set by the mass of the point source, the time delay between images created by a galaxy's lens depends on whether the light passes through the inner regions of the galaxy or through regions far from the galaxy's center. This itself depends on the distance of the galaxy from us. If the light passes far from the galaxy, for instance 20 Mpc from a galaxy similar to our own, the lens introduces a time delay that is proportional to the total mass of the galaxy. For a 1012 solar mass galaxy, this time delay is about a month. If the galaxy is closer to us, so that the light passes through the galaxy, the light is deflected from a straight line primarily by the mass that is close to the center of the galaxy; this smaller mass means that the characteristic time delay for the lens inside of the galaxy is shorter than for the lens outside of the galaxy. For instance, if the galaxy creating the lens is similar to our own Galaxy, with stars in the disk of the galaxy orbiting at a velocity of 250 km s-1, light passing within 3kpc of the center is delayed by about 2 days. These various time delays will be seen as different arrival times for the different images produced by the lens.

Galaxy Lens Minimum Distance

The large size of a galaxy means that we must be very far away to see its lens. The minimum distance is given by the same equation that sets the minimum distance we must be from a star to see it's lens. If we take the radius where the density of a galaxy fall rapidly to be 3 kpc, and give it a mass of 1012 solar masses, which are the values that characterize our own Milky Way Galaxy, we find that we see the galaxy's lens when the galaxy is more than 100 Mpc away, which is much farther than our neighboring galaxies—less than 1Mpc—but much farther than the edge of the universe—more than 4,000 Mpc.

The limit on how far a galaxy must be for its lens to be visible is proportional to R2/M. Most galaxies are much smaller than our own, but many of these would still have visible lenses.

Stellar Lenses Filling a Galaxy

Each star within our own galaxy is far enough away that we can always see its gravitational lens if it lies in front of a more distant source. These stellar lenses are tiny, so despite living in a galaxy of a trillion or more stars, we seldom see these lenses magnify a background source. But what of the stars in a distant galaxy? The distance on the sky between the stars on the sky decreases proportionally with the distance of a galaxy, but the size of each star's lens decreases in radius on the sky as the inverse square root of the distance. If the galaxy is far enough away, the area covered by the lens of each star in the galaxy covers more area than the galaxy. When this transition occurs, the individual stellar lenses no longer act as single lenses, but as part of the larger gravitational lens of the galaxy.

The area on the sky filled by the gravitational lens of a star is proportional to the star's mass. This is a nice property, because it allows us to calculate the total area spanned by a galaxy that is covered by gravitational lenses without knowing the masses of the individual stars. We do not need to know the fraction of stars with masses less than 0.1 solar masses versus the number with masses over 10 solar masses; if we know the mass of the galaxy contained in stars and if we know the distance to the galaxy, we know the area of the galaxy covered by stellar gravitational lenses.

As we move farther away from a galaxy, the fraction of the galaxy's area on the sky that is covered by stellar gravitational lenses must increase. The area on the sky of a star's Einstein ring decreases proportionally with distance, but the area covered by the galaxy containing the star decreases at the square of the distance. At some point the galaxy will be distant enough that the area covered by the stellar lenses in the galaxy equals the area covered by the galaxy. If a galaxy is composed only of stars, this limit occurs at the same point that we see the gravitational lens of the galaxy. If only part of the mass of a galaxy is in stars, brown dwarfs, and Jupiters, with the rest in diffuse gas or in some other diffuse form of matter or energy, such as neutrinos, then the distance where the lens of the galaxy becomes visible is closer than the distance where the stellar lenses overlap.

Lenses Filling the Universe

The stars in our own galaxy are not numerous enough to cover the sky significantly, but with all of the galaxies in our visible universe, the odds that a significant portion of the sky is covered by gravitational lenses are very high. This is particularly the case if all of the matter in our universe is condensed into point sources. Then the probability that an object with redshifts greater than 1 lies behind a gravitational lens is approximately the ratio of the density of the universe to the closure density. In other words, the fraction of the sky at a redshift of z = 1 that is covered by gravitational lenses is approximately Ω = ρ/ρc. A popular assumption within the astronomy community is that Ω = 1, although the observations are more consistent with a smaller values. Regardless of the precise value, the chances that an object with a redshift much smaller than 1 lies behind a gravitational lens are very small, but the chances that an object at high redshift lies behind a gravitation lens are very high, with the chances 100% for a universe at the critical density and with all of its mass condensed into stars, brown dwarfs, Jupiters, and galaxies that are sufficiently compact.

The probability of a high-redshift object lying behind a lens drops with the density of the universe, so if we live in a very low-density open universe, relatively few high-redshift objects will have their appearances modified by intervening gravitational lenses. The probability of this is also small if only a very small fraction of the universe's mass is in point masses and galaxies that are compact enough to display their lenses.

We should readily see the effects of gravitational lenses on the appearance of high-redshift objects, and the prevalence of these effects is a measure of the conditions in our universe at high redshift.

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