## 5. Breaking the mass invarianceIn 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 informationIn 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 ## 5.1.1. A maximum-likelihood approachWe 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 ellipticitiesFor non-critical clusters, the ratio depends only on the local surface mass density
. We derive (5.1) in Appendix 3. Measuring
## 5.2. Magnification information through number density of imagesLet us assume that the number density of
the unlensed faint galaxies with flux 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 Then, consider a position in the cluster
with surface mass density and shear
. The observed number density of galaxies with
redshift with the magnification defined in (2.5). The total number density
of galaxies observed with flux larger than 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 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
## 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 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 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 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 galaxiesWe 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 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 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 |