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*Astron. Astrophys. 324, 15-26 (1997)*
## 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.
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998
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