Gravitational Lensing 73different sheets of the same wavefront. In principle, the time delays in Equation (4.3)
provides a tool for measuring퐻 0.
Surface Brightness and Microlensing. Since photons are neither emitted nor
absorbed in the process of gravitational light deflection, the surface brightness of
lensed sources remains unchanged. Changing the size of the cross-section of a light
bundle therefore only changes the flux observed from a source and magnifies it at
fixed surface-brightness level. For a large fraction of distant quasars the magnifica-
tion is estimated to be a factor of ten or more. This enables objects of fainter intrinsic
magnitudes to be seen. However, lensing effects are very model dependent, so to learn
the true magnification effect one needs very detailed information on the structure of
the lens.
If the mass of the lensing object is very small, one will merely observe a magnifi-
cation of the brightness of the lensed object. This is calledmicrolensing, and it has
been used to search for nonluminous objects in the halo of our Galaxy. One keeps
watch over several million stars in the Large Magellanic Cloud (LMC) and records
variations in brightness. Some stars are Cepheids, which have an intrinsic variability,
so they have to be discarded. A star which is small enough not to emit visible light and
which is moving in the halo is expected to cross the diameter of a star in the LMC in a
time span ranging from a few days to a couple of months. The total light amplification
for all images from a point-mass lens and point source is (Problem 3)
퐴=
1 +^12 푥^2
푥
√
1 +^14 푥^2
,푥=
휃S
휃E
. (4.12)
As the relative positions of the source, lens and observer change,휃Swill change.
Simple geometrical arguments give휃Sas a function of the relative velocities, and thus
the amplification as a function of time (see [4], pp. 106, 118–120). During the occul-
tation, the image of the star behind increases in intensity according to this function
and subsequently decreases along the time-symmetric curve. A further requirement
is that observations in different colors should give the same time curve. Several such
microlensing events have been found in the direction of the LMC, and several hun-
dred in the direction of the bulge of our Galaxy. The number of such occurrences is,
however, to small play any role as nonluminous dark matter later.
Cosmic Shear. The large-scale distribution of matter in the Universe is inhomoge-
neous in every direction, so one can expect that everything we observe is displaced
and distorted by weak lensing. Since the tidal gravitational field, and thus the deflec-
tion angles, depend neither on the nature of the matter nor on its physical state, light
deflection probes the total projected mass distribution. Lensing in infrared light offers
the additional advantage of sensing distant background galaxies, since their number
density is higher than in the optical. The idea of mapping the matter distribution using
thecosmic shearfield was already proposed (in 1937) byFritz Zwicky(1898–1974),
who also proposed looking for lensing by galaxies rather than by stars.