Introduction to Cosmology

72 Tests of General Relativity

Equation (4.4) we have

훼g(푀b,휃I)=

4 퐺푀b

푐^2 휃I퐷LS

. (4.7)

Equation (4.5) is the fundamental lensing equation giving the geometrical relation
between the bend angle훼land the source and image positions:

훼l(휃S,휃I)=

퐷S

퐷LS

(휃I−휃S). (4.8)

There will be an image at an angle휃I∗that simultaneously solves both equations:

훼g(푀b,휃∗I)=훼l(휃S,휃I∗). (4.9)

For the case of a symmetric (or point-mass) lens,휃I∗will be the two solutions to the

quadratic

2 휃퐼∗=휃S+

√

휃S^2 +

16 퐺푀b퐷LS

푐^2 퐷L퐷S

. (4.10)

This reduces to the radius of the Einstein ring when휃S=0. The angle corresponding

to the radius of the Einstein ring we denote휃E.

Equation (4.10) describes a pair of hyperbolas so there will always be two images

for a point-mass lens. When the source displacement is zero (휃S=0) the images will

be at the positive and negative roots of Equation (4.6)—the Einstein ring. When휃Sis

large the positive root will be approximately equal to휃S,whilethenegativerootwill

be close to zero (on the line of sight of the lens). This implies that every point-mass

lens should have images of every source, no matter what the separation in the sky.

Clearly this is not the case. The reason is that the assumption of a point mass and

hyperbolic훼gcannot be maintained for small휃I.

A more realistic assumption for the mass distribution of a galaxy would be that

the density is spherically symmetric, with density as a function of distance from the

galactic core,푅, given by

휌(푅)=휌core

(

1 +

푅^2

푅^2 core

)− 1

, (4.11)

The density is approximately constant (equal to휌core) for small radii (푅≪푅core)and

falls off as푅−^2 for large radii. This roughly matches observed mass-density distribu-

tions (including dark matter) as inferred from galaxy rotational-velocity observations.

The mass will grow like푅^3 for푅≪푅coreand like푅for푅≫푅core.

For a nonsymmetric mass distribution, the function훼gcan become quite compli-

cated (see, e.g., [4]). Clearly, the problem quickly becomes complex. An example is

shown in Figure 4.1, where each light ray from a lensed object propagates as a spher-

ical wavefront. Bending around the lens then brings these wavefronts into positions

of interference and self-interaction, causing the observer to see multiple images. The

size and shape of the images are therefore changed. From Figure 4.1 one understands

how the time delay of pairs of images arises: this is just the time elapsed between