and Cosmology

2.5 The Galactic Microlensing Effect: The Quest for Compact Dark Matter
close to the shaded Solar disk. Its agreement with the
value predicted by Einstein made him world-famous
over night, because this was the first real and challenging
test of General Relativity. Although the precision
of the measured value back then was only ∼ 30%, it
was sufficient to confirm Einstein’s theory. By now the
law (2.71) has been measured in the Solar System with
a precision of about 0.1%, and Einstein’s prediction has
been confirmed.
Not long after the discovery of gravitational light
deflection at the Sun, the following scenario was considered.
If the deflection of the light were sufficiently
strong, light from a very distant source could be visible
at two positions in the sky: one light ray could
pass a mass concentration, located between us and the
source, “to the right”, and the second one “to the left”, as
sketched in Fig. 2.21. The astrophysical consequence of
this gravitational light deflection is also called the gravitational
lens effect. We will discuss various aspects of
the lens effect in the course of this book, and we will
review its astrophysical applications.
The Sun is not able to cause multiple images of distant
sources. The maximum deflection angle ˆα ⊙ is much
smaller than the angular radius of the Sun, so that two
beams of light that pass the Sun to the left and to the
right cannot converge by light deflection at the position
of the Earth. Given its radius, the Sun is too close to produce
multiple images, since its angular radius is (far)
larger than the deflection angle ˆα ⊙ . However, the light
deflection by more distant stars (or other massive celestial
bodies) can produce multiple images of sources
located behind them.
Lens Geometry. The geometry of a gravitational lens
system is depicted in Fig. 2.22. We consider light rays
from a source at distance D s from us that pass a mass
concentration (called a lens or deflector) at a separation
ξ. The deflector is at a distance D d from us. In Fig.
2.22 η denotes the true, two-dimensional position of the
source in the source plane, and β is the true angular position
of the source, that is the angular position at which
it would be observed in the absence of light deflection,
β = η D s
. (2.72)
65
The position of the light ray in the lens plane is denoted
by ξ,andθ is the corresponding angular position,
θ = ξ D d
. (2.73)
Hence, θ is the observed position of the source on the
sphere relative to the position of the “center of the lens”
which we have chosen as the origin of the coordinate
system, ξ = 0. D ds is the distance of the source plane
from the lens plane. As long as the relevant distances are
much smaller than the “radius of the Universe” c/H 0 ,
which is certainly the case within our Galaxy and in
the Local Group, we have D ds = D s − D d .However,
this relation is no longer valid for cosmologically distant
sources and lenses; we will come back to this in
Sect. 4.3.3.
Fig. 2.21. Sketch of a gravitational lens system. If a sufficiently
massive mass concentration is located between us and a distant
source, it may happen that we observe this source at two
different positions on the sphere
Lens Equation. From Fig. 2.22 we can deduce the condition
that a light ray from the source will reach us from
the direction θ (or ξ),
η = D s
D d
ξ − D ds ˆα(ξ), (2.74)