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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 (<kappa>) and the external shear/quadrupole (gammaext) 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, Psic(vector{x}) = B|vector{x} - vector{x}c|, where B is the Einstein radius of the cluster and vector{x}c = RC(cos thetac, sin thetac) 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 / Rc, dropping constant and linear terms that have no observable consequences, to find that

Equation 10 (10)

The first term has the form (1/2)kappac R2, which is the potential of a uniform sheet whose surface density kappac = B / 2Rc is that of the cluster at the lens center. The second term has the form (1/2) gammac R2cos 2(theta - thetac), which is the (external) tidal shear gammac = B / 2Rc that would be produced by putting all the cluster mass inside a ring of radius Rc 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 (gammac approx 0.1). (5)

The problems with nearby objects arise when the convergence kappac 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 Deltat0 and then add a sheet of constant surface density kappac, then the time delay is changed to (1 - kappac) Deltat0 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 <kappa> = kappac and eta = 1. The parameters of the lens, in particular the mass scale b, are also rescaled by factors of 1 - kappac, 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 kappac and the shear gammac that is controlled by the density distribution of the cluster (for our isothermal model kappac = gammac). 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 kappac > 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 / Rc)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 kappa contributed by material along the line of sight (Barkana 1996). This introduces small uncertainties in the H0 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 kappac leq gammac, as most mass distributions are more centrally concentrated than isothermal. Back.

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