**B.6.4. Optical Depth**

The optical depth associated with a cross section is the fraction of the
sky in which you can place a source and see the effect. This simply
requires adding up the contributions from all the lens galaxies between
the observer and the redshift of the source. For the SIS lens we simply
need to know the comoving density of lenses per unit dark matter
velocity dispersion *dn* /
*d*
(which may be a function of redshift)

(B.106) |

where *dV* / *dz*_{l} is the comoving volume element per
unit redshift (e.g. Turner, Ostriker & Gott
[1984]).
For a flat cosmology, which
we adopt from here on, the comoving volume element is simply
*dV* = 4
*D*_{d}^{2} *dD*_{d} where
*D*_{d} is the comoving distance to the lens
redshift (Eqn. B.2).
As with most lens calculations, this means that the
expression simplifies if expressed in terms of the comoving angular
diameter distances,

(B.107) |

(Gott, Park & Lee [1989], Fukugita, Futamase & Kasai [1990]). If the comoving density of the lenses does not depend on redshift, the integrals separate to give

(B.108) |

(Fukugita & Turner [1991]). If we now assume that the galaxies can be described by the combination of Schechter luminosity functions and kinematic relations described in Section B.6.2, then we can do the remaining integral to find that

(B.109) |

where [*x*] is
a Gamma function, *r*_{H} = *c* / *H*_{0}
is the Hubble radius and the optical depth scale is

(B.110) |

Thus, lens statistics are essentially
a volume test of the cosmology (the *D*_{s}^{3}),
predicated on knowing the comoving density of the lenses
(*n*_{*}) and their average mass
(_{*}).
The result does not depend on the Hubble constant - all determinations of
*n*_{*}
scale with the Hubble constant such that *n*_{*}
*D*_{s}^{3} is independent of *H*_{0}.

Two other distributions, those in image separation and in lens redshift at
fixed image separation, are easily
calculated for the SIS model and useful if numerical for any other
lens. The SIS image separation is
=
8(_{v} /
*c*)^{2} *D*_{ds} / *D*_{s}, so

(B.111) |

where =
(
/
_{*})^{FJ/2} and

(B.112) |

is the maximum separation produced by an *L*_{*}
galaxy. The mean image separation,

(B.113) |

depends only on the properties of the lens galaxy and not on cosmology. If the cosmological model is not flat, a very weak dependence on cosmology is introduced (Kochanek [1993c]). For a known separation , the probability distribution for the lens redshift becomes

(B.114) |

(we present the result only for Schechter function
= - 1 and Faber-Jackson
_{FJ}
= 4). The location of the exponential cut off
introduced by the luminosity function has a strong cosmological dependence,
so the presence or absence of lens galaxies at higher redshifts dominates
the cosmological limits. The structure of this function is quite different
from the total optical depth, which in a flat cosmology is a slowly varying
function with a mean lens distance equal to one-half the distance to the
source. The mean redshift changes with cosmology because of the changes
in the distance-redshift relations, but the effect is not as dramatic as
the redshift distributions for lenses of known separation.

We end this section by discussing the Keeton
([2002])
"heresy". Keeton
([2002])
pointed out that if you used a luminosity function derived at intermediate
redshift rather than locally, then the cosmological sensitivity of the
optical depth effectively vanishes when the median redshift of the lenses
matches the median redshift of the galaxies used to derive the
luminosity function. The following simple thought experiment shows that
this is true at one level. Suppose there was only one kind of galaxy
and we make a redshift survey and count all the galaxies in a thin
shell at redshift *z*, finding *N* galaxies between *z*
and *z* + *z*.
The implied comoving density of the galaxies, *n* = *N* /
( *z dV* /
*dz*), depends on the cosmological model with the same volume
factor appearing in the
optical depth calculation (Eqn. B.106). To the extent that
the redshift ranges and weightings of the galaxy survey and a lens survey
are similar, the cosmological sensitivity of the optical depth vanishes
because the volume factor cancels and the optical depth depends only on
the number of observed galaxies *N*. This does not occur when we use a
local luminosity function because changes in cosmology have no effect on
the local volume element. The problem with the Keeton
([2002])
argument is that it basically says that
if we could use galaxy number counts to determine the cosmological model
then we would not need lensing to do so because the two are redundant.
To continue our thought experiment, we also have local estimates
*n*_{local}
for the density of galaxies, and as we vary the cosmology we would find
that *n* and *n*_{local} agree only for a limited
range of cosmological
models and this would restore the cosmological sensitivity. The problem
is that the comparison of
near and distant measurements of the numbers of galaxies is tricky because
it depends on correctly matching the galaxies in the presence of galaxy
evolution and selection effects - in essence, you cannot use this
argument to eliminate the cosmological sensitivity of lens surveys unless
you think you understand galaxy evolution so well that you can use galaxy
number counts to determine the cosmological model, a program of research
that has basically been abandoned.