B.5.2. Time Delay Lenses in Groups or Clusters
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
(c) and the
external shear/quadrupole
(
c)
we discussed in Section B.4. In general the
effects of the external shear
c
are minimal because they
either have little effect on the delays (two-image lenses) or
are tightly constrained by either the astrometry or delay ratios
(four-image lenses or systems with lensed host galaxies see
Section B.10).
The problems arise from either the degeneracies associated with the
surface density
c or the need
for a complete, complicated cluster model.
The problem with
c is the
infamous mass-sheet degeneracy
(Part 1, Falco, Gorenstein & Shapiro
[1985]).
If we have a model predicting a time delay
t0
and add a sheet of constant surface density
c, then the
time delay is changed to
(1 -
c)
t0
without changing the image positions,
flux ratios, or time delay ratios. Its effects can be understood from
Section B.5.1 as a contribution to the
annular surface density with
<
> =
c and
= 1. Its only
observable effect other than that
on the time delays is a reduction in the mass of the lens galaxy that
could be detected given an independent estimate of the lens galaxy's mass
such as a velocity dispersion
(e.g. Section B.4.9 see Romanowsky &
Kochanek
[1998]
for an attempt to to this for Q0957+561). It can also be done given
an independent estimate of the properties of the group or cluster
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).
The accuracy of these methods is uncertain at present because each
suffers from its own systematic uncertainties, and they probably
cannot supply the 10% or higher precision measurements of
c
needed to strongly constrain models.
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 an
isothermal model
c =
c).
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 (see Eqn. B.8).
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. For
most time delay systems this correction is probably
10%.
If the cluster or any member galaxies are sufficiently close, then we
cannot ignore the higher-order perturbations in the expansion of
Eqn. (B.26). This is certainly true for Q0957+561 (as discussed in
Section B.4.6)
where the lens galaxy is the brightest cluster galaxy and located very
close to the center of the cluster. It is easy to gauge when they
become important by simply comparing the deflections produced by any
higher order moments of the cluster beyond the quadrupole with the
uncertainties being used for the image positions. For a cluster
of critical radius bc at distance
c from a lens of
Einstein radius b, these perturbations are of order
bc(b /
c)2 ~
b
c(b /
c). Because the
astrometric precision of the measurements is so high, these higher
order terms can be relatively easy to detect. For example, models of
PG1115+080 (e.g. Impey et al.
[1998])
find that using
a group potential near the optical centroid of the nearby galaxies
produces a better fit than simply using an external shear. In this
case the higher order terms are fairly small and affect the results
little, but results become very misleading if they are important
but ignored.
System | Nim | ![]() |
Astrometry | Model | Ref. |
HE1104-1805 | 2 | 161 ± 7 | + | "simple" | 1 |
PG1115+080 | 4 | 25 ± 2 | + | "simple" | 2 |
SBS1520+530 | 2 | 130 ± 3 | + | "simple" | 3 |
B1600+434 | 2 | 51 ± 2 | + / - | "simple" | 4 |
HE2149-2745 | 2 | 103 ± 12 | + | "simple" | 5 |
RXJ0911+0551 | 4 | 146 ± 4 | + | cluster/satellite | 6 |
Q0957+561 | 2 | 417 ± 3 | + | cluster | 7 |
B1608+656 | 4 | 77 ± 2 | + / - | satellite | 8 |
B0218+357 | 2 | 10.5 ± 0.2 | - | "simple" | 9 |
PKS1830-211 | 2 | 26 ± 4 | - | "simple" | 10 |
B1422+231 | 4 | (8 ± 3) | + | "simple" | 11 |
Nim is the number of images.
|