4. The reconstruction of the cluster mass distribution
4.1. Inversion relations for a single source redshift
The reconstruction of the surface mass density of a cluster from lensed images of background sources has been described in several papers (see introduction) for sources at the same redshift z. Let be the dimensionless surface mass density of the cluster, scaled by the appropriate critical surface mass density, and let be the corresponding complex shear. Both quantities are given as second partial derivatives of the deflection potential, and it was shown by Kaiser (1995) that the following relation between third partial derivatives of the deflection potential, or first partial derivatives of and , is valid:
Hence, the surface mass density can be obtained in terms of the shear by integrating this first-order differential equation. However, the shear is not an observable in general (Kochanek 1990; Paper I), but the quantity , if we confine our considerations to non-critical clusters. (For critical clusters only is observable.) Inserting into (4.1) yields (Kaiser 1995)
Hence, it is possible to derive the gradient of the quantity K in terms of the observable quantity g. Obviously, the integration of (4.2) allows an arbitrary integration constant, i.e., can only be determined up to an additive constant. Several methods exist to perform an integration of (4.2); see Schneider (1995), Kaiser et al. (1995), Bartelmann (1995), and Seitz & Schneider (1996). All of them would be equivalent if were a gradient vector field. However, since is determined observationally, it is not exact and thus in general not rotation-free. The integration derived in Seitz & Schneider (1996),
was explicitly constructed to account for this `rotational noise component' and has been demonstrated in Seitz & Schneider (1996) to work better than the other proposed methods. Here, is a vector field explicitly constructed in Seitz & Schneider (1996), and is the average of over the data field , i.e., the region where image ellipticities have been measured. Of course, is an undetermined constant, so that the surface mass density is determined up to the transformation
4.2. General inversion method
In the general case of a redshift distribution of the sources, we again make use of (4.1). The formal integration of (4.1) proceeds in the same way as that of (4.2), i.e.,
where is the average of over the data field . The vector field is defined in (4.1) and given by first partial derivatives of the shear. The shear in turn is related to the mean image ellipticity via (3.5),
Note that Eq. (4.7) is a local relation, valid at every point . The complicated dependence of on and suggests an iterative approach for the solution of the inversion problem: let be an `measured' estimate for and let and be an estimate for the shear field and the surface mass density. From that, an updated estimate for the shear field can be obtained, using (4.7):
Then, by differentiation, the vector field can be calculated from (4.1), by using the shear field . And finally, an updated estimate for the surface mass density field is obtained from (4.6),
This iteration process is started by chosing , . It is clear that the integration constant is still a free variable, i.e., with the method described here there remains a global invariance transformation of the resulting surface mass density field; in contrast to the case considered in Sect. 4.1, this transformation cannot be explicitly determined, due to the highly nonlinear relations occurring here. For critical clusters we need at most 10 steps to achieve a convergence of the iteration algorithm. For less massive clusters about 5 iteration steps are sufficient. We find that the iteration algorithm is more stable and converges faster than in the case of a single source redshift (see Paper II), mainly because there are no well-defined critical curves as function of , since their location depends on the source redshift.
4.3. The case for non-critical clusters
In the case of weak lensing (, ), the mass reconstruction depends only on the mean value of w, so that (Kaiser & Squires 1993). This ceases to be true if the cluster is not weak. Here we show how the generalization of (4.5) reads in the case of non-crtitcal clusters and a redshift distribution of sources.
If the cluster is non-critical for all redshifts of the sources, the inversion problem can be simplified because for all n - see (3.6) - and the depend only on ,
Then, (3.5) simplifies to
As a result, the iteration procedure described in the last subsection can be applied in a somewhat simpler way, by using (4.11) instead of (4.7); in addition, one can use the approximation (A2.4) derived in the Appendix 2 for which yields
This approximation is found to be sufficiently accurate to describe for (generic) non-critical clusters (see Fig. 8 in Appendix 2).
Combining (4.11) for , (4.12), and replacing the expectation value with the observed local average , we obtain
with the definition
Inserting this expression for into (4.1), one obtains after some manipulations
with the matrix
with the inverse
If we now define
(4.16) can be written as
Note the similarity between (4.19) and (4.2). Thus, the vector field can be constructed directly from the observable , as in the case of all sources being at the same redshift, and the same inversion equation (4.4) should be used, but with the current definitions of K and . Thus, by using the approximation (4.12), the inversion of a non-critical cluster is no more complicated than in the case of a single source redshift. In particular, (4.18) immediately shows that K can be determined only up to an additive constant, which implies the invariance transformation
in other words, can be determined only up to a multiplicative constant.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998