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Astron. Astrophys. 335, 1-11 (1998)

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5. Comparison with numerical simulations

In this section we compare our predictions with the results obtained by Seitz & Schneider (1996) from numerical simulations. Simulations start by defining a lens mass distribution and a random sample of source galaxies. Each galaxy is traced to the lens plane and the reduced shear g is then calculated from the observed ellipticities using Eq. (5). Finally the shear map is inverted into the lens mass distribution [FORMULA] using various methods. For [FORMULA] Seitz & Schneider take a square [FORMULA]. Source galaxies have random orientations and their ellipticities follow truncated Gaussian distributions. Simulations have been performed with three different variances for [FORMULA]: [FORMULA], [FORMULA] and [FORMULA].

The reconstruction method used by Seitz & Schneider is similar to the one described in Sect. 2, with the following differences:

  • i. Their weight function is not invariant upon translations because it is a Gaussian of argument [FORMULA] with the variance depending on [FORMULA].
  • ii. An outer smoothing is added to the final lens mass distribution. The first point is a device introduced in order to have better resolution in the stronger parts of the lens. The second point is used in order to have a smooth lens distribution from a discrete map of [FORMULA].

Our result (47) for the power spectrum can be easily generalized in order to take into account the outer smoothing:

[EQUATION]

Here [FORMULA] is the variance associated with the outer smoothing. Note however that the expression given above does not take into account the variable-scale smoothing used by Seitz & Schneider. Even if this result has been derived with some approximations (weak lensing limit, large area A of [FORMULA], fixed inner smoothing [FORMULA]), a comparison with the simulations shows that Eq. (48) can reproduce the main features of the simulated power spectrum.

Fig. 2 shows the results of simulations together with the power spectrum predicted by Eq. (48). Thus Eq. (48) underestimates the error. This difference can be attributed to the following factors:

  1. The weight function W considered is not precisely of the form of Eq. (3), because of the change of normalization near [FORMULA]. This last factor should increase the variance of [FORMULA] near the boundary of [FORMULA] (the variance of [FORMULA], with direct influence on [FORMULA], should double near a side of [FORMULA] and quadruple near a corner; cf. top-right frame of Fig. 10 in Seitz & Schneider 1996).
  2. The weight function W is not of the form of Eq. (3) also because of differential smoothing.
  3. The constant term [FORMULA] has not been estimated properly (see first paragraph of Sect. 4.3). In principle, this should be traced to a counterpart in [FORMULA], but for finite sets [FORMULA] there is an additional term in [FORMULA].
  4. The set [FORMULA] is not the whole plane.
  5. The lens is not weak (see Eq. (25) and the extra contribution in Eq. (D4)).
  6. Poisson noise is associated with the finite number of source galaxies.
  7. The population of source galaxies is characterized by sizable c.
[FIGURE] Fig. 2. Comparison of predicted power spectrum (dashed lines) with measured power spectrum (solid lines) for the simulations made by Seitz & Schneider (1996). All frames refer to a source population characterized by [FORMULA]. Top frame: [FORMULA], [FORMULA], [FORMULA]. Middle frame: [FORMULA], [FORMULA], [FORMULA]. Bottom frame: [FORMULA], [FORMULA], [FORMULA].

In spite of these limitations, the general behavior of [FORMULA] is reasonably well reproduced by Eq. (48). In particular, the maximum of the simulated points corresponds exactly to the maximum of our theoretical curve.

5.1. Curl-free kernels

In order to check the result of Eq. (35), we have considered different kernels used by various authors and we have compared our predictions with other aspects of the numerical simulations performed by Seitz & Schneider (1996).

The first kernel considered is the noise-filtered "SS-inversion" (Seitz & Schneider 1996) described above in Eqs. (18-20). This method has been especially designed to reduce the statistical errors and performs very well in simulations. In fact, as stated in Eq. (21), this kernel is curl-free.

Another kernel considered is the "S-inversion" (see Schneider 1995). Simulations show that errors of the S-inversion are nearly the same as for the SS-inversion. The S-inversion operates by averaging over radial paths made of two segments. Inside the segments the kernel is easily shown to be curl-free.

The last kernel is the so called "B-inversion" (Bartelmann 1995; see also Squires & Kaiser 1996). From the results of the simulations it is clear that the B-inversion leads to larger errors on the map distribution. This behavior is once again explained by Eq. (35), since the B-inversion kernel is not curl-free. Notice that in this case it is difficult to estimate analytically the exact error on [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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