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Astron. Astrophys. 318, 687-699 (1997)

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3. Local observables and their dependence on local lens parameters [FORMULA] and [FORMULA]

Throughout this section we assume that these lensing parameters can be considered as constant over a small solid angle around [FORMULA], and we will therefore suppress the argument [FORMULA] in all equations.

In contrast to Papers I & II, we use here the ellipticity parameter


where [FORMULA] are the components of the tensor of second brightness moments, as defined in Paper I. Defining the intrinsic source ellipticity [FORMULA] in complete analogy to (3.1), and using the locally linearized lens equation, yielding [FORMULA], one finds for the transformation of intrinsic to observed ellipticity:




The condition [FORMULA] ([FORMULA]) in Eq. (3.2) is equivalent to the condition [FORMULA] ([FORMULA]).

Now, let [FORMULA] be the probability that the source ellipticity [FORMULA] is within [FORMULA] around [FORMULA]. Then, for fixed redshift z, the expectation value of the n -th moment [FORMULA] is given through


The remarkable fact that the expectation values [FORMULA] do not depend on the source ellipticity distribution, whereas [FORMULA] does (Papers I & II), is the reason for choosing [FORMULA] as the ellipticity parameter in this paper. We derive (3.4) in the Appendix 1.

If the galaxies are distributed in redshift according to the probability density [FORMULA], the expectation values of the moments [FORMULA] become:


In the third line we used the transformation of the source redshift distribution into their w -distribution, which consists of a delta `function' at [FORMULA] for galaxies with [FORMULA] and is given by [FORMULA] for galaxies with [FORMULA]. For a single redshift [FORMULA] of the sources, the expectation values reduce to [FORMULA] given in (3.4).

Generally, the boundaries of the integrals in (3.5), and therefore [FORMULA] and [FORMULA], depend on [FORMULA] and [FORMULA],


One sees that for some values of the parameters [FORMULA] and [FORMULA] there exist two intervals of w for which [FORMULA], seperated by that interval for which [FORMULA].

In the case of weak lensing we obtain from (3.5) that [FORMULA], and the shear is - modulo [FORMULA] - an observable.

An estimate of these expectation values for the moments of the image ellipticities can be obtained by considering (locally) an ensemble of sources with ellipticities [FORMULA] and defining the means


where [FORMULA] is an appropriately chosen weight factor (see Paper II or Seitz & Schneider 1996). These mean values [FORMULA] are statistically distributed around the expectation values [FORMULA], and we use them as an estimate for [FORMULA].

To summarize, the mean image ellipticity and their higher moments [FORMULA] do no longer directly provide us with a local estimate for [FORMULA] or its inverse as in the case where all sources are at the same redshift, which was assumed in previous papers. In fact, the dependence of [FORMULA] on the local lens parameters [FORMULA] and [FORMULA] can be quite complicated, depending on the redshift distribution of the sources and on the local lens parameters itself.

To obtain information on the local lens parameters [FORMULA], [FORMULA] from [FORMULA] we have to know the redshift distribution [FORMULA] of the galaxy population which enters Eq. (3.5). We point out that the average in (3.5) extends over all galaxies, i.e., also over those situated in front of the lens, which, however, do not provide any information on the local lens parameters, since [FORMULA] for [FORMULA], but contribute to the noise which is inherent in the estimate of the local lens parameters derived from [FORMULA]. We want to stress that galaxies situated in the foreground of the cluster do not affect the shear estimate systematically, as long as their fraction is known and properly included in (3.5) via [FORMULA]. In the rest of this paper we assume that this redshift distribution of galaxies is known or at least can be estimated, e.g., by the lensing effect itself (see Bartelmann & Narayan 1995, Kneib et al. 1995).

For illustration we assume that the redshift distribution is given by a function of the form


taken from Brainerd et al. (1995), with moments [FORMULA] for [FORMULA], [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA].

In Fig. 1 we show this distribution [FORMULA] for [FORMULA] and the corresponding distribution [FORMULA] for a lens with redshift [FORMULA].

[FIGURE] Fig. 1. The lower panel (solid line) shows the redshift distribution given in Eq. (3.8) for [FORMULA] and [FORMULA]. The dashed line shows the function [FORMULA] defined in Eq. (2.4) for a cluster redshift of [FORMULA]. The upper panel shows the corresponding distribution [FORMULA]. Note that [FORMULA] has a delta-`function' peak at [FORMULA], with an amplitude given by the probability that a source has a redshift smaller than that of the lens
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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998