5. Breaking the mass invariance
In Sect. 4, we have shown that there exists an invariance transformation for the surface mass density map which leaves the observed local mean image ellipticities constant (4.5, 4.9 & 4.20). The occurrence of the invariance can be traced back to the fact that the inversion equation was derived from a first-order differential equation - see (4.1) - which has a free integration constant. We now discuss several ideas to determine this constant.
5.1. Using only shape information
In the case of as single source redshift the invariance transformation for the surface mass density can be formulated most generally by saying that the probability distribution of the observed image ellipticities is unchanged under the transformation (4.5); this was explicitly shown in Paper I. The invariance transformation as expressed in (4.9) and (4.20) is of a different kind, since it does not leave invariant, but only its first order moment . Therefore, one might be tempted to try to break the invariance transformation by using information of the image ellipticity distribution only.
5.1.1. A maximum-likelihood approach
We consider a fixed set of `observed' galaxies with ellipticities and positions . Then we reconstruct the surface mass density of a cluster as described in Sect. 4.2 assuming a value for the mean mass density in the field [see Eq. (4.9)]. From that we calculate the mass density and shear at the positions (). Next we calculate the probability to observe the image ellipticity at the galaxy position . Finally we maximize the likelihood using different values for in the mass reconstruction. We find that the mass degeneracy is broken, but unfortunately only weakly, and only in those regions of the cluster where the weak lensing regime is violated. From our simulation we find that the number density of galaxy images is orders of magnitudes too low to allow for the breaking of the mass sheet degenracy in practice. Furthermore, the likelihood depends much stronger on the assumed redshift distribution - which in practice is weakly constrained - than on the value of the mean mass density and is therefore a much better tool to investigate the redshift distribution of the sources than to determine the mean surface mass density of a particular cluster.
5.1.2. Second moments of image ellipticities
For non-critical clusters, the ratio
depends only on the local surface mass density . We derive (5.1) in Appendix 3. Measuring R could therefore provide a direct estimate for . However, we can not measure and accurately enough to determine the local mass density or the average mass density from a resonable number density of image ellipticities (see Appendix 3).
5.2. Magnification information through number density of images
Let us assume that the number density of the unlensed faint galaxies with flux S and redshift z is given through
where is the normalized redshift distribution and the distribution in flux. Note that the factorization of as given in (5.2) is a fairly special assumption for the galaxy distribution, which will not be valid in general. However, for simplicity we shall make use of (5.2) in this paper. This factorization may in fact be an approximate distribution over a limited range of flux S. Since the magnifications are not very large except perhaps in the very central parts of the cluster, the `dynamic range' over which (5.2) is applied is not large and might be a valid description. It should be noted that this factorization has already been used implicitly in (3.5). If (5.2) is not assumed, then the redshift distribution of sources locally - where the surface mass density is and the shear is - will differ from due to the magnification, so that the functions and will attain an additional dependence on and . This will not cause any additional conceptual problem.
Then, consider a position in the cluster with surface mass density and shear . The observed number density of galaxies with redshift z and flux larger than S is (see Broadhurst et al. 1995)
with the magnification defined in (2.5). The total number density of galaxies observed with flux larger than S is obtained through integration of (5.3) and yields
If , then we obtain from (5.4)
From Eq. (5.5) we conclude that the number density is not changed if and it is increased (decreased) for (). Next, averaging (5.5) over the data field with area U, we obtain
Hence, the ratio of the number of observed galaxies in the data field to the number which would be observed in the absence of a foreground lens gives . The number density of the unlensed galaxies is regarded as an universal function and has been measured in several colours down to very faint magnitudes (see for example Smail et al. 1995). Because of this, the local observables in the case of cluster lensing are the moments of the image ellipticities and . We note that the result that is a local observable is only true for the ansatz (5.2) with . If either the ansatz (5.2) does not hold, or if is not a power law, then the observable quantity is a different one and can in particular have a much more complicated dependence on the magnification. However, can still be compared - may be not analytically - with and provides information on the local magnification.
In the following we consider a fixed value of the flux threshold S and use for notational simplicity and instead of and .
5.2.1. Do galaxy positions break the mass invariance?
The probability to observe a galaxy within the area around is
Using Eqs. (5.5) and (5.6) we obtain this probability in terms of the local (redshift-averaged) magnification
The likelihood for observing N galaxies with positions () is
On the other hand, we can perform the mass reconstruction according to Eq. (4.9) and obtain as a function of the assumed mean mass density in the field. From that we can calculate with Eq. (5.8) the probability density and finally according to (5.9) the likelihood for the assumed mean mass density. However this does not work. To understand this, assume for a moment that all galaxies are at the same redshift; for simplicity we take this redshift to be `infinity' here. Then the reconstructed mass density is related to the true one through the invariance transformation (4.5) which gives and . For the magnification we obtain
and from that we conclude that is independent of , or, equivalently, independent of . This means that for all sources at the same redshift, the positions of the observed galaxies do not give any information on the mean mass density in the field.
If the sources are distributed in redshift then the mean mass assumed for the reconstruction changes the ratio
However, the dependence on is very weak as can be verified for non-critical clusters through calculating, for fixed and , the function using the invariance transformation (4.20). As a result, the positions of the observed galaxies provide no or not enough information to break the mass degeneracy.
Up to here all our attempts to break the mass degeneracy in practice have failed. This is because for all cases considered, the mass degeneracy is - in theory - broken because of the presence of a redshift distribution of the sources and is no longer broken if all sources are at the same redshift. We conclude that in order to derive one has to use a method which breaks the mass degeneracy not because of the presence of a redshift distribution of the sources, but independent of that.
5.2.2. Breaking the mass degeneracy through the total number of observed galaxies
We shall now consider the magnification effect on the distribution of galaxy images. We do not use the magnification effect locally as proposed by Broadhurst et al. (1995) and Broadhurst (1995) in order to derive the local mass density from it, but we use the `global' magnification effect, i.e., we compare the total number of observed galaxies with the expected (unlensed one), and relate that to the mean mass density inside the observed field.
In the above subsection we have not used the number density of galaxies observable in the absence of lensing. If we use this, then we can calculate from Eq.(5.6) the expected number of galaxies as a function of the mean mass density assumed in the mass reconstruction,
and compare that with the number N of galaxies actually observed. We compare the deviation in terms of the standard deviation and obtain
The value for which shows a minimum (in fact, at which ) gives the most probable value for the mean surface mass density of the cluster.
We note that in the case of a single redshift of the sources (see Sect. 4.1) the determination of the mean mass density is even simpler: the mass reconstruction is done according to (4.4) with an assumed value for , say . From the resulting mass and shear maps and , one can calculate .
Because of Eq. (5.6) we can relate that to the true magnification and from Eq. (5.10) we conclude that
Thus, we can immediately calculate from (5.14), and with the invariance transformation (4.5) the true mass density .
In Sect. 6.2, we apply this method to break the mass invariance to a numerically simulated cluster (), for the case that the sources are distributed in redshift. We demonstrate that the method yields good estimates for the mean surface mass density, i.e., it yields good estimates on the total mass inside the observed field. However, we point out that the method is not applicable in the case that , because then no magnification bias or antibias occurs, and the observed number density of galaxy images provides no information on the mean mass density in the field. We also remark that we do not use the magnification effect to determine the shape of the mass distribution, but only for the overall normalization of the mass.
We note that the method suggested by Bartelmann & Narayan (1995) still works in this case: they made use of the magnification of individual galaxy images, at fixed surface brightness which is known to be unchanged by gravitational light deflection. Their method will become extremely useful once sufficient HST data become available to obtain the mean size-surface brightness relation for these faint galaxies.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998