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

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6. Conclusion

We have investigated in detail how weak lensing observations could be used to measure the projected power spectrum and to discriminate among different cosmological models. In order to analyze the cosmic variance of the results (which requires a large number of realizations), we used a two dimensional, second order Lagrangian dynamics to generate the convergence fields. In each of the observational situation and cosmological model, 60 different maps has been generated. Cosmological models include: flat model ([FORMULA]), open model ([FORMULA]), Baugh & Gaztañaga initial power spectrum, standard CDM spectrum. Observational contexts include: high noise ([FORMULA]gal/arcmin2) and low noise ([FORMULA]gal/arcmin2) level, "small" survey size ([FORMULA] degrees) and large survey size ([FORMULA] degrees), two different mean redshift of the sources ([FORMULA] and [FORMULA]), and the use of top-hat and compensated filters. From these convergence maps, a shear map is calculated, on which the noise is introduced according the selected observational context, the mass map is finally reconstructed (using a [FORMULA] method) and analyzed.

Our results are:

  • At scales larger than [FORMULA] the [FORMULA] reconstruction method is a very stable process which does not produce any boundary effects, spurious signal, and which leaves the noise properties unchanged (the noise does not propagates and remains close to the theoretical prediction using the linearized lens equation). This permits to work with the convergence (which is the physical field of interest) instead of the shear, with no loss of information.

  • The shot noise contribution can be removed simply by measuring the observed ellipticities of the galaxies, and this leads to unbiased estimates of the power spectrum and the moments of the convergence (at least up to the kurtosis).

  • The precision obtained on the normalization can be as low as 2% with survey of [FORMULA], and 5% for [FORMULA] (see Tables 1 and 2) for the most favorable cases (i.e. low [FORMULA]).

  • The larger shallower surveys are more promising to recover the cosmological quantities. In addition to that, the redshift distribution of the brighter galaxies is known better and their light distribution is more regular, making ellipticity measurement from their weighted second moment easier and more relevant.

  • The compensated filter yields by far smaller cosmic variance than the top-hat filter for the variance of the convergence, while it gives unsatisfactory results for the skewness. However the cosmic variance decreases more rapidly with an increasing survey size for a compensated filter.

The MEGACAM project offers the possibility to perform these observations. Indeed the large field of the CCD device (1 square degree) and the high image quality at CFHT (extended to the edges of the field with the future field corrector) provides the ideal instrument to perform such a scientific program.

In the context of the rapid changes occurring in observational cosmology it is worth stressing that weak lensing surveys offer precious complementary views of our Universe and unique tools to probe directly the dark matter and to compare with the light distribution at any scales. The perspective of determining the projected power spectrum independently of biases is indeed attractive. Moreover the possibility of determining [FORMULA] in a way which is independent of the power spectrum, and independent of all the methods that have been suggested so far, is also extremely precious. We remind that this determination relies only on dynamical effects assuming that the large-scale structures originate from Gaussian initial conditions through gravitational instabilities. In Fig. 13 examples of constraints in the ([FORMULA]) plane with weak lensing as describes in this work (shaded areas) are presented 10 together with the location of the major constraints that are expected to be brought either with CDM experiments, (constant curvature density, solid lines), or from Type-Ia supernovae measurements (the other straight lines, describing constant [FORMULA] values).

[FIGURE] Fig. 13. Constraints that can be brought by weak lensing survey in an [FORMULA] plane. The grey areas are the location of the 1 and 2-[FORMULA] bands (respectively darker and lighter bands) allowed by a measured skewness that would be obtained with either [FORMULA] (left bands ) or [FORMULA] (right bands ). The solid straight lines corresponds to a zero curvature universe, and the dot-dashed lines to a fixed acceleration parameter, [FORMULA]. The panels correspond to survey of either [FORMULA] (top ) or [FORMULA] degrees (bottom ).

Future CMB experiments can determine the cosmological parameters with a remarkable accuracy, but only when some prior is put on the shape of the initial power spectrum. The curvature will probably be determined with a good accuracy in the near future from the position of the first Doppler peak on the [FORMULA] curves. But it appears that it can be very difficult to disentangle [FORMULA] from [FORMULA] (see for instance Zaldariagga, Spergel & Seljak 1997) from the mere temperature (and polarization) fluctuations. This might be only possible from a very detailed analyses and the power spectra which require not only a good understanding of the possible systematics that may affect the measurements, but also specific hypothesis on the regularity of the primordial power spectrum. The requirements to measure [FORMULA] from weak lensing survey are less strict provided the instrumental systematics can be controlled correctly.

Moreover weak lensing surveys are able to constraint [FORMULA] if it is possible to select efficiently several populations of sources (see Villumsen 1996, BvWM for clues of such possibilities). We thus think that weak lensing survey will be a major mean for probing the global cosmological parameters. It is complementary to the CMB experiments and indispensable for breaking the parameter degeneracy.

The feasibility of the weak lensing by large scale structure program and The full scientific exploitation of the weak lensing effect is reduced to the capability to control instrumental systematics, where there are still open crucial points. The most dedicated problem comes probably from the need of a PSF correction able to avoid artificial large scale coherent alignments of galaxies. Related to this problem is the pixelisation effect. Galaxies shapes are indeed determined from a finite number of pixels with the possible introduction of errors and biased estimation on the shapes.

These effects can all be investigated independently. Taking several images of the same portion of the sky with a shifted position for the camera, or with different cameras would definitely test the robustness of the observed distortion maps. Moreover we have numerous statistical tests at our disposal that can be done on the maps: comparison of the 2-point correlation function of the distortion with the one of the shear field or with the one of the [FORMULA] field, or even by the investigation of quantities that are a priori sensitive to the systematics in a different manner, such as the correlation of the orientations of the distortion field. That all these quantities have related properties is somehow due to the fact that in the thin lens approximation the [FORMULA] field should be curl-free (see for example Luppino & Kaiser 1997 for a used of these tests). This is true only when lens-lens couplings are neglected but it should be possible to investigate those effects in numerical simulations.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998