3. Local observables and their dependence on local lens parameters and
Throughout this section we assume that these lensing parameters can be considered as constant over a small solid angle around , and we will therefore suppress the argument in all equations.
where are the components of the tensor of second brightness moments, as defined in Paper I. Defining the intrinsic source ellipticity in complete analogy to (3.1), and using the locally linearized lens equation, yielding , one finds for the transformation of intrinsic to observed ellipticity:
The condition () in Eq. (3.2) is equivalent to the condition ().
Now, let be the probability that the source ellipticity is within around . Then, for fixed redshift z, the expectation value of the n -th moment is given through
The remarkable fact that the expectation values do not depend on the source ellipticity distribution, whereas does (Papers I & II), is the reason for choosing 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 , the expectation values of the moments 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 for galaxies with and is given by for galaxies with . For a single redshift of the sources, the expectation values reduce to given in (3.4).
Generally, the boundaries of the integrals in (3.5), and therefore and , depend on and ,
One sees that for some values of the parameters and there exist two intervals of w for which , seperated by that interval for which .
In the case of weak lensing we obtain from (3.5) that , and the shear is - modulo - 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 and defining the means
where is an appropriately chosen weight factor (see Paper II or Seitz & Schneider 1996). These mean values are statistically distributed around the expectation values , and we use them as an estimate for .
To summarize, the mean image ellipticity and their higher moments do no longer directly provide us with a local estimate for 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 on the local lens parameters and 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 , from we have to know the redshift distribution 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 for , but contribute to the noise which is inherent in the estimate of the local lens parameters derived from . 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 . 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 for , for and for .
In Fig. 1 we show this distribution for and the corresponding distribution for a lens with redshift .
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998