MODERN COSMOLOGY

416 Gravitational lensing

ellipticity allowsγ, to be determined and, making use of equation (14.130) one
can get the surface mass densitykof the lens. Recently, several groups have
reported the detection of cosmic shear, which clearly demonstrates the technical
feasibility of using weak lensing surweys to measure dark matter clustering and
the potential for cosmological measurements, in particular with the upcoming
wide-field CCD cameras [67, 68].

14.6.2 Comparison with results from x-ray observations

Beside the lensing technique, there are two other methods for determining mass
distributions of clusters:

(1) the observed velocity dispersion, combined with the Jeans–equation from

stellar dynamics gives the total mass distribution, if it is assumed that light

traces mass; and

(2) x-ray observations of the intracluster gas, combined with the condition of

hydrostatic equilibrium and spherical symmetry also lead to the total mass

distribution as well as to the baryonic distribution.

If the hydrostatic equilibrium equation for the hot gas

dPg

dr

=−ρg

GMt(r)

r^2

(14.135)

is combined with the ideal equation of statePg=(kBTg/μmH)ρgand assuming
spherical symmetry, one easily finds for the total mass profile

Mt(r)=−

kBTg

GμmH

(

dlnρg

dlnr

+

dlnTg

dlnr

)

r. (14.136)

The right-hand side can be determined from the intensity distribution and some
spectral information. (At present, the latter is not yet good enough, because of
relatively poor resolution which, however, will change with the XMM survey.)
Weak lensing, together with an analysis of x-ray observations, offers a
unique possibility for probing the relative distributions of the gas and the dark
matter, and for studying the dynamical relationship between the two. As an
example consider the cluster of galaxies A2163 (z=0.201) which is one of the
two most massive clusters known so far.
ROSAT measurements reach out to 2. 3 h−^1 Mpc(∼15 core radii)(hbeing
the Hubble constant in units of 100). The total mass is 2.6 times greater than
that of COMA, but the gas mass fraction,∼ 0. 1 h−^3 /^2 is typical for rich clusters.
The data together suggest that there was a recent merger of two large clusters.
The optical observations of the distorted images of background galaxies were
made with the CFHT telescope. The resulting lensing and x-ray mass profiles
are compared in figure 14.12. The data-sets only overlap out to a radius of
200
′′
 500 h−^1 kpc to which the lensing studies were limited. It is evident