**B.7.1. The Effects of Halo Structure and the Power Spectrum**

Estimating the structure of clusters using gravitational lensing is
primarily a topic for Part 3, so we include only an abbreviated
discussion of lensing by clusters here. For a fixed cosmological
model, two parameters largely control the abundance of cluster
lenses. First, the abundance of clusters varies nearly exponentially
with the standard normalization
_{8}
1 of the power
spectrum on 8*h*^{-1} Mpc scales. Second, the cross sections of
the individual clusters depend strongly on the exponent of the
central density cusp of the cluster. There are recent studies of
these issues by Li & Ostriker
([2002],
[2003]),
Huterer & Ma
([2004]),
Kuhlen, Keeton & Madau
([2004]),
Oguri et al.
([2004]),
and Oguri & Keeton
([2004]).

We can understand the general effects of halo structure very easily from
our simple power law model in Eqn. B.9. In
Section B.3
we normalized the models to have the same Einstein radius, but we now
want to normalize them to all have the same total mass interior to
some much larger radius *R*_{0}. This is roughly what
happens when we keep the virial mass and break radius of the halo
constant but vary the central density exponent
*r*^{-n}. The deflection profile becomes

(B.121) |

where *b*_{0} << *R*_{0} sets the mass
interior to *R*_{0} and we recover our old
example if we let *b* = *b*_{0} =
*R*_{0}. The typical image separation is determined
by the tangential critical line at
_{t} =
*R*_{0}(*b*_{0} /
*R*_{0})^{2/(n-1)}, so
more centrally concentrated lenses (larger *n*) produce larger
image separations when *b*_{0} / *R*_{0}
<< 1. The radial caustic lies at
_{r}
= *f* (*n*)
_{t} where
*f* (*n*) is a not very interesting function of the index
*n*, so the cross section for multiple imaging
_{r}^{2}
*R*_{0}^{2}(*b*_{0} /
*R*_{0})^{4/(n-1)} - for an SIS profile
*b*^{4}
/ *R*_{0}^{2}, while the cross section for a
Moore profile (*n* = 3/2)
*b*^{8}
/ 16*R*_{0}^{6} is significantly smaller.
We cannot go to the limit of an NFW profile
(*n* = 1) because our power law model has a constant surface
density rather than
a logarithmically divergent surface density in the limit as
*n* 1,
but we can see that as the density profile becomes shallower the
multiple image cross section drops rapidly when the models have constant
mass inside a radius which is much larger than their Einstein
radius. As a result, the numbers of group or cluster lenses depends
strongly on the central exponent of the density distribution even when
the mass function of halos is fixed. Magnification bias
will weaken the dependence on the density slope because the models with
shallower slopes and smaller cross sections will generally have higher
average magnifications. The one caveat to
these calculations is that many groups or clusters will have central
galaxies, and the higher surface density of the galaxy can make the
central density profile effectively steeper than the CDM halo in isolation.