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 ). 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  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.  in Q0957+561), cluster galaxy velocity dispersions (e.g., Angonin-Willaime, Soucail, & Vanderriest  for Q0957+561, Hjorth et al.  for RXJ0911+0551) or X-ray temperatures/luminosities (e.g., Morgan et al.  for RXJ0911+0551 or Chartas et al.  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. ) 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.
|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. t is the longest of the measured delays and its 1 error; delays in parenthesis require further confirmation. The "Astrometry" column indicates the quality of the astrometric data for the system: + (good), + / - (some problems), - (serious problems). The "Model" column indicates the type of model needed to interpret the delays. "Simple" lenses can be modeled as a single primary lens galaxy in a perturbing tidal field. More complex models are needed if there is a satellite galaxy inside the Einstein ring ("satellite") of the primary lens galaxy, or if the primary lens belongs to a cluster. References: (1) Ofek & Maoz , Wyrzykowski et al. ; (2) Barkana , based on Schechter et al. ; (3) Burud et al. [2002b]; (4) Burud et al. , also Koopmans et al. ; (5) Burud et al. [2002a]; (6) Hjorth et al. ; (7) Kundic et al. , also Schild & Thomson  and Haarsma et al. ; (8) Fassnacht et al. ; (9) Biggs et al. , also Cohen et al. ; (10) Lovell et al. ; (11) Patnaik & Narasimha .