Annu. Rev. Astron. Astrophys. 1999. 37:
127-189
Copyright © 1999 by . All rights reserved |

**4.4. Critical Issues**

The critical issues already discussed for the weak lensing and mass reconstruction of clusters of galaxies are valid and even more critical for large-scale structures. This will not be discussed again here, but we should mention that systematic effects are a real concern which could be an ultimate limitation of the weak lensing capabilities, at least on ground-based telescopes.

Equations 27 and 28 show a
strong dependence of the variance and the skewness of the convergence on
the redshift distribution of the lensed sources. As far as weak lensing is
concerned, from the investigations of the effect of redshift
distribution of
sources on cluster mass reconstruction, it seems that only the averaged
redshift of the galaxies and the width of their distribution are needed
(Seitz & Schneider
1997), even with a
bad precision (say,
*z*_{s}
0.5). This
requirement is therefore not severe and can be obtained using
photometric redshifts. The work by
Connolly et al (1995)
demonstrates that galaxies brighter
than *I* = 22.5 can easily be calibrated using
spectroscopic redshifts and that photometric information in 4 different
filters constrain the redshift of these galaxies with an accuracy of
about
*z*_{s}
0.05. The future
surveys with 10-meter class telescopes will calibrate photometric redshift
up to *I* = 23.5 and even fainter, by using
Lyman-break galaxies
(Steidel et al 1998)
and near-infrared photometry.
This is deep enough for shallow wide field surveys so one can be
reasonably confident that the redshift distribution of the lensed
galaxies will not be a major problem. Conversely, this is another
argument in favor of shallow rather than deep surveys which would
reach limiting magnitudes beyond the capabilities of
spectrographs. Indeed, as shown by
Van Waerbeke et al
(1999) the signal to noise ratio of the variance and the skewness
does not strongly depend on the redshift of the sources, so it is useless
to reach very faint magnitudes.

Due to the intrinsic clustering of the galaxies, the redshift distribution can be broad enough to mix together the population of lensed galaxies and the galaxies associated with the lensing structures. In particular, an extended massive structure at high redshift can play simultaneously the role of a lens and a reservoir of lensed galaxies. The average redshift distribution of the sources can therefore be biased by the galaxies located within the massive structure, which can bias as well the estimated value of the convergence in a similar way. Indeed, the variance of the convergence is not affected by this clustering (Bernardeau 1998a). However, the skewness is much more affected, mainly because the overlapping acts exactly as a non-linear evolution of the projected density. Bernardeau (1998a) shows that most of these effects are negligible on scales beyond 10 arcminutes. It would also be interesting to investigate more deeply how the source clustering may contribute to spurious signals on small scales.

The apparent change of the two-point correlation function by magnification bias can also change the local redshift distribution of lensed sources. This effect, though mentioned by Bernardeau (1998a), has not been investigated in detail.

When ray bundles cross two lenses by accident, the cumulative convergence is given by the product of the magnification matrix, and not simply the sum of the two convergences. The resulting convergence contains additional coupling terms that must be estimated. Fortunately, in the weak lensing regime, this coupling appears to be negligible. It does not change the value of the variance, and the skewness is only weakly modified (Bernardeau et al 1997, Schneider et al 1997).

**4.4.4. Validity of the Born Approximation**

The effects of mass density fluctuations caused by large-scale structures on the deformation of the ray bundles are computed assuming that the Born approximation is valid, that is, the deformation can be computed along the unperturbed geodesic. In the case of linear perturbations, this assumption is valid at the lowest order. As discussed by Bernardeau et al (1997), the correction on the skewness should be at the percent level. However, this is less obvious once lens couplings are taken into account. The validity of the Born approximation has not been tested in detail, so far. This certainly deserves inspection using high-resolution numerical simulations. The simulations done by Jain et al (1998) or Wambsganss et al (1998) could provide valuable information on this issue.

**4.4.5. Intrinsic Correlated Polarization of
Galaxies**

It is possible that the intrinsic orientations of galaxies are not randomly distributed but have a coherent alignment correlated to the geometry of the large-scale structures in which they are embedded (Bingelli 1982). If so, the coherent alignment produced by weak lensing will be contaminated by the intrinsic alignment of the galaxies, and a mass reconstruction based on the shear pattern will be impossible. Such alignments could appear during the formation of large-scale structures or could result from dynamical evolution of galaxies within deep potential wells, such as superclusters or clusters of galaxies (see Coutts 1996, Garrido et al 1993 and references therein). However, the most recent numerical simulations do not show such correlations. Many attempts have been made to search for signatures of these intrinsic coherent patterns. So far, no convincing observations of nearby structures have demonstrated that there are large-scale coherent alignments. This possibility has to be investigated thoroughly, in particular in nearby large-scale structures where a coherent alignment from gravitational lensing effect is negligible. It would be valuable to have more quantitative estimates of possible trends for alignments from future very high-resolution numerical simulations of structure formation.