MODERN COSMOLOGY

398 Gravitational lensing

profileI(r/R),whereris the distance of a source point from the centre of the
source. Adopting polar coordinates centred on the circular source, we obtain

μe(y)=

[

2 π

∫∞

0

I(r/R)rdr

]− 1 ∫∞

0

I(r/R)rdr

×

∫ 2 π

0

dφ

μp(y)y

√

y^2 +r^2 + 2 rycosφ

. (14.96)

For a uniform brightness profile the maximum ofμeis aty=0 (withμe( 0 )=
2 /Rifμpis the magnification of a point source, since thenμp(y)y →0for

y→0). Indeed, for a Schwarzschild lens withμp=(y^2 + 2 )/(y

√

y^2 + 4 )one
finds

μmaxe =

√

4 +R^2

R

. (14.97)

14.3.6 Two point-mass lens

A natural generalization of the Schwarzschild lens is to consider a lens with two
point masses. This case is also of relevance for the applications, since many binary
microlensing events have been observed. For several point massesMilocated at
transversal positionsξithe general formula equation (14.10) for the deflection
angle gives

α(ξ)=

∑N

i= 1

4 GMi

c^2

ξ−ξi

|ξ−ξ|^2

. (14.98)

LetM=

∑N

i Mibe the total mass andMi=ηiM. For the typical length scaleξ^0
we choose the Einstein radius equation (14.67) for the total mass. Then the lens
map becomes

y=x−

∑N

1 = 1

ηi

|x−xi|^2

(x−xi), (14.99)

wherexi=ξi/ξ 0. For a detailed discussion see [4].

14.4 Galactic microlensing

14.4.1 Introduction

There are cases in which the deflection angles are tiny, of the order of milli-
arcseconds or smaller, such that the multiple images are not observable. However,
lensing magnifies the affected source, and since the lens and the source are
moving relative to each other, this can be detected as a time-variable brightness.
This behaviour is referred to as gravitational microlensing, a powerful method
to search for dark matter in the halo of our own galaxy, if it consists of massive