Most galaxies are not isolated, and many early-type lens galaxies are
members of groups or clusters, so we need to consider the effects of
the local environment on the time delays. Weak perturbations are
easily understood since they will simply be additional contributions
to the surface density
(<>) and the
external shear/quadrupole
(_{ext}) we discussed in
Section 3. In this
section we focus on the consequences of large perturbations.

As a first approximation we can assume that a nearby cluster (or galaxy)
can be modeled by an SIS potential,
_{c}() =
*B*| -
_{c}|,
where *B* is the Einstein radius of the cluster and
_{c} =
*R*_{C}(cos
_{c}, sin
_{c}) is the
position of the cluster relative to the primary lens. We can
understand its effects by expanding the potential as a
series in *R* / *R*_{c}, dropping constant and linear
terms that have no observable consequences, to find that

(10) |

The first term has the form
(1/2)_{c}
*R*^{2}, which is the potential of a uniform sheet whose
surface density
_{c} = *B*
/ 2*R*_{c} is that of the cluster at the lens
center. The second term has the form (1/2)
_{c}
*R*^{2}cos
2( -
_{c}),
which is the (external) tidal shear
_{c} =
*B* / 2*R*_{c} that would be produced by
putting all the cluster mass inside a ring of radius
*R*_{c} at the cluster center.
All realistic lens models
need to incorporate a tidal shear term due to objects near the lens or
along the line of sight
(Keeton, Kochanek, &
Seljak 1997),
but as we discussed in
Section 3 the shear does not lead to
significant ambiguities
in the time delay estimates. Usually the local shear cannot be associated
with a particular object unless it is quite strong
(_{c}
0.1).
^{(5)}

The problems with nearby objects arise when the convergence
_{c}
becomes large because of a global degeneracy known as the *mass-sheet
degeneracy*
(Falco, Gorenstein,
& Shapiro 1985).
If we have a model predicting a time delay
*t*_{0}
and then add a sheet of constant surface density
_{c}, then the
time delay is changed to
(1 - _{c})
*t*_{0}
without changing the image positions,
flux ratios, or time delay ratios. Its effects can be understood from
Section 3 as a contribution to the annular
surface density with
<> =
_{c} and
= 1. The
parameters of the lens, in particular
the mass scale *b*, are also rescaled by factors of
1 - _{c}, so the
degeneracy can be broken if there is an independent
mass estimate for either the cluster or the galaxy.
^{(6)}
When the convergence is due to an object like a cluster, there is a strong
correlation between the convergence
_{c} and the shear
_{c}
that is controlled by the density distribution of the cluster (for our
isothermal model
_{c} =
_{c}).
In most circumstances, neglecting the extra surface density coming
from nearby objects (galaxies, groups, clusters) leads to
an overestimate of the Hubble constant because these objects all have
_{c} > 0.

If the cluster is sufficiently close, then we cannot ignore the
higher-order perturbations in the expansion of Equation (1.10).
They are quantitatively important when they produce deflections at
the Einstein ring radius *b* of the primary lens,
*B*(*b* / *R*_{c})^{2}, that are larger
than the astrometric uncertainties. Because these uncertainties are
small, the higher-order terms quickly become important. If they are
important but ignored in the models, the results can be very misleading.

^{5} There is a small random component of
contributed by material
along the line of sight
(Barkana 1996).
This introduces small uncertainties
in the *H*_{0} estimates for individual lenses (an rms
convergence of 0.01 - 0.05,
depending on the source redshift), but is an unimportant source of
uncertainty in estimates from ensembles of lenses because it is a random
variable that averages to zero.
Back.

^{6} For the cluster this can be done using
weak lensing (e.g.,
Fischer et al. 1997
in Q0957+561), cluster galaxy velocity dispersions (e.g.,
Angonin-Willaime,
Soucail, & Vanderriest 1994
for Q0957+561,
Hjorth et al. 2002
for RXJ0911+0551) or X-ray temperatures/luminosities (e.g.,
Morgan et al. 2001
for RXJ0911+0551 or
Chartas et al. 2002
for Q0957+561). For the lens galaxy this can be done
with stellar dynamics
(Romanowsky &
Kochanek 1999
for Q0957+561 and PG1115+080,
Treu & Koopmans
2002b
for PG1115+080).
The accuracy of these methods is uncertain at present because each
suffers from its own systematic uncertainties. When the lens is in
the outskirts of a cluster, as in RXJ0911+0551, it is probably
reasonable to assume that
_{c}
_{c},
as most mass distributions are more centrally concentrated than isothermal.
Back.