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

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2. Weak lensing effects on CMB maps

2.1. The basis of the physical mechanism

The effect of a gravitational lens is to induce a displacement of the light path, thus moving the apparent position of a sky patch on the last scattering surface by a given angle. The temperature of this patch is not affected itself, i.e. lenses do not created new structures, and a perfectly isotropic sky would remain so. The patch of the sky observed at the position [FORMULA] is thus actually coming from the position [FORMULA] on the "primordial sky", and the displacement, [FORMULA], is induced by the mass concentration on the line of sight. More precisely [FORMULA] is given by the transverse derivative of the projected potential [FORMULA] of the mass fluctuations,


where [FORMULA] is the angular distance, [FORMULA] is the distance of the lenses along the line of sight and [FORMULA] is the distance of the last scattering surface (see Kaiser 1992, Seljak 1996, Bernardeau, van Waerbeke & Mellier 1996 for more details on this equation). It is interesting to rewrite this equation in terms of the Fourier transform of the mass density fluctuation field. The Fourier transforms [FORMULA] are defined by,


where the linear growth factor [FORMULA] and the Fourier transforms are normalized to the present time. Then the potential reads,


which implies that the displacement can be written,




The function [FORMULA] gives the efficiency function of lenses for sources located on the last scattering surface. It will be investigated in more details in the last section.

Note that in the following I will amply use the small angle approximation. It implies in particular that a given patch of the sky can be decomposed in flat waves and also that, in moment calculations, the component of [FORMULA] along the line of sight can be neglected compared to the norm of [FORMULA].

2.2. The effects on CMB maps

Compared to detections on background galaxies, the investigation of lens effects on the last scattering surface is very attractive, because this surface is at a well defined redshift, and has a negligible width. The analysis of the lens effects requires however more sophisticated tools since the induced shear cannot be directly measured. The primordial temperature patches on the CMB sky are indeed known only statistically and have a large angular correlation length. In which way, then, can the lens effects be revealed? Actually lensed CMB maps can be seen as collections of temperature patches of different sizes and shapes, which or only a fraction of which are displaced or deformed. Although this is slightly arbitrary, two effects can be distinguished in the way sizes and shapes of patches are affected,

  • the shear effect that deforms, stretches out temperature patches in the shear direction,
  • the magnification effect that globally enlarges or shrinks those patches.

The local deformations of the temperature patches are however a priori difficult to disentangle from the actual primordial intrinsic temperature fluctuations 2. What will make then the effects detectable is the fact that close patches will be deformed in a similar way (when they are seen through a unique lens), and the excess of these close rare features cannot be accounted from a Gaussian field. It is thus possible to quantify their presence by statistical indicators. The power spectrum is of course not adapted to take into account the apparition of such non-Gaussian features. For that matter the high-order correlation functions, that are all identically zero for pure Gaussian fields, are extremely precious. Indeed these higher-order correlation functions contain informations about shapes, and their derivations can be pursued completely with Perturbation Theory techniques. In the following I focus my analysis on the first non vanishing correlation function, the four-point one.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998