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Astron. Astrophys. 364, 1-16 (2000)

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2. The convergence and aperture mass fields

In weak lensing observations, background galaxy deformations can be used to reconstruct the local gravitational convergence field. We recall here how the local convergence is related to the line-of-sight cosmic density fluctuations. As a photon travels from a distant source towards the observer its trajectory is perturbed by density fluctuations close to the line-of-sight. This leads to an apparent displacement of the source and to a distortion of the image. In particular, the convergence [FORMULA] magnifies (or de-magnifies) the source as the cross section of the beam is decreased (or increased). One can show (Kaiser 1998) that the convergence along a given line-of-sight is,

[EQUATION]

when lens-lens couplings and departure from the Born approximation are neglected (e.g., Bernardeau et al. 1997). This equation states that the local convergence is obtained by an integral over the line-of-sight of the local density contrast. The integration variable is the radial distance, [FORMULA], (and [FORMULA] corresponds to the distance of the source) such that

[EQUATION]

while the angular distance [FORMULA] is defined by,

[EQUATION]

Then, the weight [FORMULA] used in (1) is given by:

[EQUATION]

where z corresponds to the radial distance [FORMULA]. Thus the convergence [FORMULA] can be expressed in a very simple fashion as a function of the density field. We can note from (1) that there is a minimum value [FORMULA] for the convergence of a source located at redshift [FORMULA], which corresponds to an "empty" beam between the source and the observer ([FORMULA] everywhere along the line-of-sight):

[EQUATION]

In practice, rather than the convergence [FORMULA] it can be more convenient (Schneider 1996) to consider the aperture mass [FORMULA]. It corresponds to a geometrical average of the local convergence with a window of vanishing average,

[EQUATION]

where [FORMULA] is the local convergence at the angular position [FORMULA] and the window function U is such that

[EQUATION]

In this case, [FORMULA] has the interesting property that it can be expressed as a function of the tangential component [FORMULA] of the shear (Kaiser et al. 1994; Schneider 1996) so that it is not in principle necessary to build local shear maps to get local aperture mass maps. More precisely we can write,

[EQUATION]

with

[EQUATION]

Nonetheless considering such a class of filters is interesting because convergence maps are always reconstructed to a mass sheet degeneracy only. Therefore, to some extent, any statistical quantities that can be measured in convergence mass maps correspond to smoothed quantities with compensated filters. In the following, for convenience rather than due to intrinsic limitation of the method, we consider filters that are defined on a compact support.

2.1. Choice of filter

It is convenient to write the filter function in terms of reduce variable, [FORMULA] where [FORMULA] is the filter scale,

[EQUATION]

so that the evolution with [FORMULA] of the properties of [FORMULA] only depends on the behavior of the density field seen on different scales (while the shape and the normalization of the angular filter [FORMULA] remains constant). In the following we shall use two different filters, which satisfy (10). One, which we note [FORMULA], has been explicitly proposed by Schneider (1996),

[EQUATION]

and [FORMULA] otherwise. The Fourier form of this filter corresponds to,

[EQUATION]

The other, which we note [FORMULA], corresponds to a simpler compensated filter that can be built from two concentric discs. It is built from the difference of the average convergence in a disc [FORMULA] and the average convergence in a disc unity:

[EQUATION]

where [FORMULA] is the characteristic function of a disc unity. In Fourier space it is simply related to the Fourier transform of a disc of radius unity,

[EQUATION]

and reads,

[EQUATION]

The ratio of the 2 radii has been chosen so that the 2 filters are close enough, as shown in Fig. 1. Note that the normalizations are somewhat arbitrarily. In the plot they have been chosen to give the same amplitude for the aperture mass fluctuations in case of a power law spectrum with index [FORMULA]. This is obtained by multiplying the expression (12) by 1.459. Fig. 1 shows actually that the two filters are very close to each other. In particular they have their maximum at the same k scale, and except for the large k oscillations they exhibit a similar behavior. In the following we will take the freedom to use either one or the other for convenience.

[FIGURE] Fig. 1. Shape of the filter functions we use in Fourier space. The solid line corresponds to the shape proposed by Schneider, Eq. (12), multiplied by 1.459 and the dashed line corresponds to the compensated filter we introduce in this paper, Eq. (15).

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© European Southern Observatory (ESO) 2000

Online publication: December 15, 2000
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