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Astron. Astrophys. 324, 15-26 (1997)

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1. Introduction

A robust prediction of inflationary scenarios is that the temperature fluctuations of the Cosmic Microwave Background (CMB) are expected to obey Gaussian statistics. Actually this prediction has been challenged recently by several authors (Falk, Rangarajan & Srednicki 1993, Munshi, Souradeep & Starobinsky 1995) who calculated the skewness induced by nonlinear couplings in the primary 1 stage of the temperature fluctuation generation. The skewness induced at this level has been found, however, to be entirely negligible compared to the cosmic variance, and thus not accessible to detections. Therefore, the primary temperature maps are entirely defined, in a statistical sense, by the power spectrum of the temperature fluctuations or equivalently by the shape of the two-point correlation function. A number of other statistical indicators are thus set up by this a priori hypothesis. In particular Bond & Efstathiou (1987) have investigated expected properties of such temperature maps, as the number density of temperature peaks, their correlation functions... Moreover, the exploration of the CMB physics has been boosted recently after it has been realized that it would be possible to determine all the cosmological parameters with a remarkable precision from an accurate, and accessible, measurement of the temperature power spectrum (Jungman et al. 1996). In particular, the effects of the secondary sources of temperature fluctuations (Sunyaev-Zel'dovich effects, nonlinear Doppler effects, lenses..) and foregrounds (point sources, galactic dust) on the power spectrum have been investigated in more details (see Cobras/Samba report, 1996, for a general discussion on these problems). These calculations have shown that most, if not all, of these effects have a relatively small impact on it. In these calculations, however, the impact of secondary effects on the Gaussian nature of the temperature field has not been considered. Particularly interesting are the the higher order correlation functions that are identically zero for Gaussian fields, and are thus direct indicators of any, even small, non-Gaussian features.

In this paper the calculations will be focused on the effects of weak-lensing on CMB maps. They, indeed, constitute a particularly attractive mechanism because it comes from a coupling between the primary temperature fluctuation field and the mass concentration on the line of sight acting as deflectors. Their impact on the power spectrum has been investigated primarily by Blanchard & Schneider, (1987) who found the effect to be negligible. More recent works (Kashlinsky 1988, Cole & Efstathiou 1989, Sasaki 1989, Tomita & Watanabe 1989, Linder 1990, Cayón, Martínez-González & Sanz 1993a, b, Fukushige, Makino & Ebisuzaki 1994, Seljak 1996) eventually confirmed this conclusion, although the point was debated for a while. In particular Seljak (1996) made a detailed calculations of these effects for realistic models of CMB anisotropies and using a semi-analytic calculation based on a power spectrum approach that includes nonlinear corrections. In this text I will follow a rather similar approach to investigate the apparition of non-Gaussian features caused by weak lensing effects.

In Sect. 2, I present the basis of the physical mechanisms describing the CMB map deformations induced by weak lensing effects. In Sect. 3, I give the explicit expression of the first non-vanishing correlation function, the four-point, in different remarkable geometries. In Sect. 4, quantitative predictions are given for two different cosmological models. The dependence of the results on the cosmological parameters, and the practical interests that such a measurement could have, are discussed in the last section.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998