MODERN COSMOLOGY

400 Gravitational lensing

with the satellite Infrared Space Observatory (ISO) also seem to exclude M-
dwarfs as significantly contributing to halo dark matter [29].
A scenario with white dwarfs as a major constituent of the galactic halo dark
matter has been explored [30]. However, it requires a ratherad hocinitial mass
function sharply peaked around 2–6M. Future Hubble Deep Field exposures
could either find the white dwarfs or put constraints on their fraction in the halo
[31]. A substantial component of neutron stars and black holes with masses higher
than∼ 1 M is also excluded, for otherwise they would lead to an overproduction
of heavy elements relative to the observed abundances.
A further possibility is that the hydrogen gas is in molecular form, clumped
into very cold clouds, as we proposed some years ago [32, 33]. Indeed, the
observation of such clouds is very difficult and, therefore, at present there are
no stringent limits on their contribution to the halo dark matter [34].

14.4.1.1 Microlensing probability

When a MACHO of massMis sufficiently close to the line of sight between
us and a more distant star, the light from the source star suffers a gravitational
deflection and we see two images of the source (figure 14.3). For most
applications we can consider the lens and the source as point-like and thus use
the Schwarzschild lens approximation previously discussed.REis then defined in
equation (14.67).
For a cosmological situation, where the lens is a galaxy or even a cluster
of galaxies and the source is a very distant quasar, one indeed sees two or more
images which are typically separated by an angle of some arcseconds. However,
in the situation being considered here, namely of a MACHO of typically∼ 0. 1 M
and a source star located in the LMC at about 50 kpc from us, the separation
angle turns out to be of the order of some milli-arcseconds. Thus, the images
cannot be seen separately. However, the measured brightness of the source star
varies with time. It increases until the MACHO reaches the shortest distance from
the line of sight between the observer on Earth and the source star. Afterwards,
the brightness decreases and eventually returns to its usual unlensed value. The
magnification of the original star brightness turns out to be typically of the order
of 30% or even more, corresponding to an increase of at least 0.3 magnitudes of
the source star (see figures 14.4 and 14.5). Such an increase is easily observable.
An important quantity is the optical depth τ due to gravitational
microlensing, which is the probability that a source is found within a circle of
radiusr≤REaround a MACHO. It is defined as follows

τ=

∫ 1

0

dx

4 πG

c^2

ρ(x)Ds^2 x( 1 −x) (14.100)

withρ(x)being the mass density along the line of sight at distances=xDsfrom
the observer.