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Astron. Astrophys. 342, 15-33 (1999)

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5. Discussion

In this paper, we have focussed our investigations on scales larger than [FORMULA] thus allowing ourself the use of a simplified dynamics. This is a complementary approach to the ongoing investigations by Jain et al. (1998) who analyzed the high order moments and the power spectrum of the calculated from ray tracing in high resolution simulations. Although they only analyze the statistical properties of the convergence without noise, their approach completes our own towards the smaller angular scales. They have shown in particular that the skewness of the convergence is significantly higher at small scales (0.1 arcmin) than the theoretical expectations of perturbation theory due to highly non-linear structures, (this was already mentioned by Gaztañaga & Bernardeau 1998 and whether this behavior can still be described by means of perturbation theory with the help of loop correction terms is still an open question). Unfortunately, even at these small scales, they are not able to analyze the cosmic variance because of the small number of realizations. The use of high resolution simulations probably prevents a detailed analysis of this quantity.

The preceding sections provide quantitative estimates of the capability of mass reconstructions from weak lensing measurements at large scale to probe the large scale structures, as well as the cosmological parameters. We have shown how the projected mass distribution can be reconstructed accurately from the observed shape of the galaxies. Two complementary analysis have been examined, the power spectrum and the non-Gaussian features through the high order moments.

For the power spectrum estimation, the best sampling strategy (i.e. the question of sparse or compact surveys) is not discussed in this paper (see Kaiser 1998 for a discussion), but our results show that in order to probe the smallest scales of mass fluctuations a deep (but narrow) survey is required that diminishes the cosmic variance caused by the shot noise. Once this is done, the survey can be extended in a more shallower manner to probe the power spectrum at scale where the cosmic variance caused by shot noise becomes unimportant. At this stage, the question of a sparse or compact survey is a matter of choice, depending on scientific interests.

Concerning the moments measurement, a summary is given in Table 2 (for a top-hat filter) and 3 (for a compensated filter) which shows the smallest accessible error on the measurement of the variance and the skewness of the convergence for different observational contexts. This demonstrates that weak lensing measurements can reach a few percent precision on [FORMULA] for a reasonable survey size. We pointed out that the results are not much deteriorated for a number density of galaxies of 30 gal/arcmin2 compared to 50 gal/arcmin2, whereas the realization of the survey in the latter case requires a factor 3 more observing time. It suggests that large shallow surveys would be more adequate since it efficiently reduces the cosmic variance. Such a strategy would be much more comfortable with respect to some systematics like the redshift distribution of the sources (which can be determined easily if the sources are closer), the source clustering effects that are limited if the source distribution is narrow (Bernardeau 1998), and possibly the morphological evolution of distant galaxies, in particular if most distant galaxies are composed of many merging substructures.


Table 2. Relative error on the measured variance and skewness of the convergence in different observational contexts (in tables [FORMULA]gal/arcmin2 and [FORMULA]gal/arcmin2) for a top-hat filter. The numbers correspond to the smallest error within the range of scales considered in this work.


Table 3. Same as Table 2 for a compensated filter

Note that there is still room for potential improvement of the signal to noise ratio we are obtaining. Indeed, since we are limited by construction to simulations of scales larger than [FORMULA] we do not know whether the cosmic variance can be reduced by observing the moments at smaller scales, as it is suggested by the results obtained with a top-hat filter (for which the signal to noise curves never bend down at small scales). The optimal size for the measurement of the variance and the skewness might in fact correspond to the arcmin scale. However, there are several issues that are generally believed to be irrelevant for weak lensing at large scales, but are probably potential difficulties at small scales:

  • Our work implicitly assumes a constant power spectrum below the pixel size. A mass reconstruction from real data should include small scale features such as cluster lensing, and the propagation of the noise from this peaks of signal is not known. Moreover that ability of the [FORMULA] method to match the noise properties correctly at small scales should be reinvestigated.

  • Born approximation and lens-lens coupling terms are stronger at small scales (see SvWJK). A quantification of this effect on simulations would be necessary to decide at which scale the measurement of moments is optimal.

  • Source clustering (investigated at large scales by means of perturbation theory by Bernardeau 1998) could also have a significant impact at small scale.

The possible systematics caused by all these effects have to be investigated in order to estimate the precision of weak lensing surveys.

Moreover the moments are not necessary the best means to distinguish between different cosmological models (not to mention the fact that the results are anyway sensitive to the choice of the filter). For example, it seems that by comparing open and flat universes on Fig. 1 topological tools should be as well a strong discriminant of cosmological models. The statistical instruments are extremely diverse. Among the possible tools the results of multi-scale filtering, as provided by wavelet transforms, is likely to be a good candidate.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998