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Astron. Astrophys. 334, 409-419 (1998)

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Realistic case

When this method is applied to filter a map containing all contributions (CMB + SZ kinetic + moving lenses), we are no longer able to identify or locate the moving lens fluctuations, as shown in Fig. 6 lower right panel. Here, the CMB which dominates at large scales is cleaned, whereas the SZ kinetic effect, which is mainly a point-like dominated signal, at least one order of magnitude larger than the power of the moving lenses, is not cleaned and remains in the filtered signal. We have filtered at several angular scales without any positive result. On large scales the extended dipole patterns are polluted by the CMB, as mentioned above, and on small scales the SZ kinetic fluctuations are of the same scale as the moving lens anisotropies. We also tried the convolution of the total map (CMB + SZ kinetic + moving lenses) with the dipole pattern function but we were still unable to recover the moving lens fluctuations. In fact, the combination of two SZ kinetic sources, one coming forward and the other going backward, mimics a dipole-like pattern. In order to distinguish between an intrinsic dipole due to a lens and a coincidence, one needs to know a priori the direction of the motion which is of course not possible. During our analysis, we investigated two cases for the cut off in mass as describe in Sect 4.1. For the simulations with cut-off mass [FORMULA] the resulting background due to point-like SZ fluctuations is lower than the cut-off at [FORMULA] case; but we were still unable to recover the moving lens fluctuations.

In our attempt at taking advantage of the spatial signature of the moving lens fluctuations, we have located the coefficients in the wavelet decomposition that are principally associated with the moving lenses and selected them from all the wavelet coefficients. Our study case procedure is the following. We make the wavelet transform for the moving lens fluctuations and, separately, we also make the transform for the remaining signals (CMB + SZ kinetic). We locate the wavelet coefficients for the moving lenses whose absolute values are higher than the absolute values of CMB+SZ kinetic coefficients. Then, we select, in the transform of the total fluctuation map (CMB + SZ + lenses), the coefficients corresponding to the previously located ones. Finally we perform the inverse transform on the map (CMB + SZ + lenses) according to the selected coefficients. When we compare the average peak to peak amplitudes of the recovered (Fig. 7: dashed line and Fig. 8: dotted-dashed line) and input (Fig. 7 and Fig. 8, solid line) lens fluctuations, we find a very good correlation between the amplitudes of the original and the reconstructed moving lens fluctuations. The correlation factor is of the order of 0.7 for CMB + SZ + lenses with a cut-off mass at [FORMULA] and higher than 0.9 with the cut-off at [FORMULA]. This difference between the correlation factors is an effect of the cut-off in masses. In fact, for the [FORMULA] cut-off, the filtered maps are cleaner than for the [FORMULA] cut-off. Therefore, in the latter case some of the lenses have very little or no signature in the wavelet decomposition, hence they are not recovered and the correlation factor decreases.

[FIGURE] Fig. 7. Average peak to peak amplitude of the secondary anisotropies due to the moving lenses (cut-off mass [FORMULA]) for lenses with a decreasing y parameter. The solid line represents the amplitudes in the original simulated lenses map. The dashed line represents the extracted amplitudes after sorting the wavelet coefficients and filtering all contributions (CMB+SZ+lenses). The correlation factor is equal to 0.9.

The results of our study case confirm that the moving lens fluctuations have a significant spatial signature in the total signal although their amplitudes are very low compared with the CMB and SZ fluctuations. However, it is worth noting that such a "good" result is obtained only because we use sorted coefficients from two separated maps, one containing the lens signal and the other containing the polluting signals. In a real case, there is no way to separate the contributions because of the spectral confusion and therefore there is no a priori knowledge of the "right" coefficients in the wavelet decomposition. In our analysis, we tried several sorting criteria for the coefficients but we could not find a robust and trustworthy criterion to reproducibly discriminate between the wavelet coefficients belonging to the moving lens fluctuations and the coefficients belonging to the noise (SZ kinetic and CMB fluctuations). During the analysis, we could not overcome the physical limitation corresponding to the presence of sources of SZ kinetic anisotropies at the same scale and with amplitudes at least 10 times higher than the signal (moving lens fluctuations).

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998