7. Discussion and conclusions
Using Perturbation theory techniques, we have calculated the variance and the skewness of the local filtered convergence in an inhomogeneous cosmology in the FRW framework. The quantity can be calculated from the measurement of the magnification µ and/or the distortion .
The exploration of the dependence of the variance of on the parameters of interest have shown that it is, as expected, strongly dependent on , but also on . It is of course proportional to the magnitude of the power spectrum of the matter fluctuations. These dependences, although degenerate with each other, are of immediate cosmological interest. Note that these dependences are given for a given power spectrum normalization at redshift zero. As pointed out in other studies (Kaiser 1996, Jain & Seljak in preparation) if the power spectrum is normalized to the present number density of clusters the degeneracy with almost vanishes. More worrying is the fact that the variance is also strongly dependent on the redshift of the sources , making difficult to analyze quantitative results in a cosmological context if the sources are not well known.
It is possible to disentangle the dependence of the variance on the magnitude of the power spectrum and the cosmological parameters by the use of two different populations of sources. Thus, the ratio of the variances for these two populations could provide constraints on the cosmological parameters independently of the power spectrum (the dependence on its shape is only weak). Of course the problem mentioned previously for the knowledge of the population of sources is even more critical since both populations have to be perfectly known.
More interestingly, the skewness of the local convergence, expressed as the ratio of the third moment and the square of the second, can also provide a way to separate the dependence on the cosmological parameters. We indeed found that this quantity does not depend on the amplitude of the power spectrum, is weakly dependent on its shape and varies almost as the inverse of the density parameter, with a slight degeneracy with the cosmological constant. Moreover the skewness is a probe of the initial Gaussian nature of the density field. If physical mechanisms such as cosmic strings, topological defects, symmetry breaking, etc... generate non Gaussian features in the initial density field, they will modify the behavior of (see Gaztañaga & Mähönen 1996), inducing a strong angular variation of . The dependence of on the redshift of the sources is surprisingly large, stressing the absolute necessity of knowing the population of sources used to do such measurements. It is however interesting to note that the product of the variance and the skewness is almost independent of the redshift of sources, so in any case we have a robust estimator of the product . It might reveal to be an interesting quantity to consider.
An important concern was the contribution of multiple lenses configurations in reducing the significance of these statistical estimators. We have shown that the two main quadratic couplings between two lenses are ineffective, with contributions of a few percent, and do not strongly depend on the cosmology. Thus, whatever the cosmological model, the cosmological signal contained in the variance and the skewness could be slightly weakened, but is not washed out by these additional couplings.
We have not considered higher order moments, basically because we think that they are not going to improve the situation, i.e. to raise the degeneracy between the cosmological parameters we have with the second and third moments. A rough calculation (taking into account only the dominant contribution) shows interestingly that the to ratio is expected to be close to 2 for any cosmological model (, and the shape of the power spectrum). In principle it could be used as a test for the gravitational instability scenario with Gaussian initial conditions. We are however far from a reliable measurement of all these quantities.
A concern that has not been addressed in this paper is the validity domain of these results with respect to the nonlinear corrections due to the dynamical evolution of the cosmic fields. These corrections are expected to intervene at small angular scale where the physical scale of the lenses is expected to enter the nonlinear regime. The question is naturally, at which angular scale? The answer is not straightforward, and actually the situation is slightly different for the second moment and for the skewness. Indeed the second moment is always a measure of the projected power spectrum whether or not it is in the linear regime. This point was clearly made in a recent work (Jain and Seljak, in preparation) where the rms polarization is calculated including the nonlinear evolution of the power spectrum using prescriptions as the ones developed by Hamilton et al. (1991), Jain, Mo & White (1995) and Peacock & Dodds (1996). For there are no such transformations that would take into account the nonlinear effects. But one should have in mind that this parameter has been observed to be very robust against nonlinear corrections (this is not necessarily the case for the product of the variance and ). It has been amply checked in numerical simulations for the 3D filtered density field (see the remarkable results of Baugh, Gaztañaga & Efstathiou, 1995). The situation for the projected density field has not been investigated in as many details, and extended numerical work would be necessary for that respect. In particular it would be interesting to extend the work of Colombi et al. (1996), who propose a prescription to correct the high order moments of the local 3D filtered density PDF, to the projected density. Another arduous line of investigation would be to take into account in the analytic calculations the so called "loop corrections" as it has been pioneered by Scoccimarro & Frieman (1996) for the un-smoothed density field. Such calculations could quantify the errors introduced by the nonlinear effects at small angular scale.
In the last section we have briefly explored the observational requirements for completing a large observational program. We have in particular tried to estimate the cosmic noise that would affect a catalogue covering an area of 25 . This calculation is at most indicative and should not be considered too realistic. For instance, in this analysis we have made a fairly crude hypothesis for the noise. In particular we have assumed that the noise due to the intrinsic ellipticities of the galaxies is local. If this is true for the local distortion measurement, this is not true for the convergence if it has been obtained from the reconstruction scheme proposed by Kaiser since it involves the gradient of . In this analysis we have also not taken into account the fluctuations in the number density of sources, that could be significant at the angular scale we are interested in.
On the other hand we have assumed that the moments were calculated directly, i.e. with the average of the proper power of the different measured convergences. This may not be the most robust way to do it. Indeed, taking advantage of the fact that we expect the Probability Distribution Function of the local convergence to be close to a Gaussian distribution it is possible to estimate the variance and the skewness simply from the shape of the PDF around its maximum. More precisely one can use the Edgeworth expansion (see Juszkiewicz et al. 1995 and Bernardeau & Kofman 1995) to look for the best fitting values of the low order moments without having to actually compute them. With such a method the results should be less sensitive to the cosmic noise, because less sensitive to the rare events in the tails.
We leave for a coming paper the difficult task of defining what we think could be an optimal strategy for observing the shear induced by the large-scale structures and obtaining constraints of cosmological interest out of it. But we stress in any case that such a project demands a lensing survey at 25 square degrees scale with a very high image quality. Such a lensing survey will be used for the statistical work described in this paper, but also to probe the angular power spectrum at scales from the Mpc to few hundred Mpcs. This program is accessible with the UH8K camera or with the future 16K 16K (MEGACAM) camera at the Canada-France-Hawaii Telescope, provided optimal algorithms for measuring very weak shear are used. The pixel to pixel autocorrelation function (ACF) proposed by Van Waerbeke et al. (1996) which consists in analyzing the ellipticity of the ACF induced by the gravitational shear seems a promising approach because it does not require any more to define object centroids and to compute shape parameters for each individual faint galaxy from noisy images. The shape of the sample also impacts on the signal to noise: reconstruction method à la Kaiser is expected to introduce finite size errors along the edges of the catalogue, so it would be interesting to have a catalogue as compact as possible. On the other hand an elongated catalogue has less cosmic variance.
The drawback of this project is that it requires the knowledge of the redshift distribution of the sources, especially for faint objects (). A spectroscopic survey to get the redshift of these objects is unrealistic with present-day ground based telescopes; fortunately many other methods are possible, such as the photometric redshifts, the lensing inversion technique, and the depletion method (see Mellier 1996 for a review). None of them have proved their ability to get secure redshifts, but they are potentially very attractive because they are not time consuming.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998