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

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4. Results of the data analysis

We analyse the simulated maps of secondary fluctuations due to the moving lens effect, for the three cosmological models described in Sect. 3, and we quantify their contributions. We also make attempts at detecting and extracting the secondary fluctuations from the entire signal (primary CMB, SZ kinetic effect and moving lenses).

4.1. Statistical analysis

We show the histogram of the secondary fluctuations for the moving lens effect (randomly generated) in the three cosmogonies (Fig. 2). In all cases, the amplitude of anisotropies ranges roughly between [FORMULA] and [FORMULA]. The rms value of the anisotropies varies a little with the cosmological model [FORMULA], [FORMULA] and finally [FORMULA]. Our results are in general agreement with those of Tuluie, Laguna & Anninos (1996). In all the cosmological models, the rms value of the anisotropies is about a factor 10 lower than the rms amplitude of the fluctuations due to the SZ kinetic effect associated with the same structures, which is about [FORMULA] ; and is about 30 times lower than the [FORMULA] of the primary fluctuations in a standard CDM model. The distribution of the temperature fluctuations induced by moving lenses exhibits a highly non Gaussian signature (Fig. 2). The fourth moment of the distribution, called the kurtosis, measures the peakedness or flatness of the distribution relative to the normal one. We find that the kurtosis for the standard CDM, OCDM and [FORMULA] CDM models are positive and respectively equal to about 51, 97 and 41. The distributions are thus peaked (leptokurtics).

[FIGURE] Fig. 2. Histograms showing the distributions of the secondary fluctuations in the simulated maps. The solid, dashed and dotted lines are for respectively the standard, Open and Lambda CDM model.

In the context of our statistical analysis of the secondary anisotropies, we also compute the fitted angular power spectra (Fig. 3) of the three main sources of anisotropies: primary CMB fluctuations (in the standard CDM model) and both the predicted power spectra of the fluctuations due to the moving lenses (thin lines) and the SZ kinetic effect (thick lines). In Fig. 3, the solid lines are for the standard CDM model, dashed and dotted lines are respectively for the open and non zero cosmological constant models.

[FIGURE] Fig. 3. Power spectra of the primary fluctuations obtained using the CMBFAST code compared to the fitted power spectra of the secondary fluctuations due to the Sunyaev-Zel'dovich kinetic effect (thick lines) and to the moving lens effect (thin lines). The power spectra for the standard CDM model (solid line), open CDM model (dashed line) and lambda CDM model (dotted line) are shown.

We fit the power spectra of the secondary anisotropies due to moving lenses with the general expression:

[EQUATION]

in which the fitting parameters for every cosmological model are given in Table 1.


[TABLE]

Table 1. Fitting parameters for the power spectrum of the fluctuations induced by moving lenses as a function of the cosmological model.


The SZ kinetic anisotropies are fitted with the following expression:

[EQUATION]

with the fitting parameters for the cosmological models gathered in Table 2.


[TABLE]

Table 2. Fitting parameters for the power spectrum of the fluctuations induced by the Sunyaev-Zel'dovich kinetic effect as a function of the cosmological model.


The power spectra of the SZ kinetic effect exhibit the characteristic [FORMULA] dependence on small angular scales for the point-like source dominated signal. All the power spectra have rather similar amplitudes, at large scales, in particular up to [FORMULA] where we notice an excess of power at small angular scales in the OCDM model. This is because low [FORMULA] models produce higher counts than [FORMULA] models (Barbosa et al. 1996).

The moving lens power spectra, for both CDM and [FORMULA] CDM models, exhibit a plateau at [FORMULA] with a decrease at larger angular scales. For the OCDM model, the dependence is roughly constant at all scales. We also note that the highest and lowest power are obtained, at small angular scales, for respectively the standard CDM and OCDM models. At large scales, the opposite is true.

In order to interpret this behaviour, we distinguish between what we refer to as the resolved and unresolved structures. The spatial extent of the resolved structures is much greater than the pixel size (or analogously the beam size). Whereas, the unresolved objects have extents close to, or smaller than, the pixel size. At the pixel size an unresolved structure generates a SZ kinetic anisotropy which is averaged to a non-zero value. Whereas the dipolar anisotropy induced by the moving lens effect is averaged to zero (except what remains from the side effects). A pixel size anisotropy thus does not contribute to the signal in the moving lens effect; while it contributes with its [FORMULA] amplitude in the SZ kinetic effect. As a result, the distribution of the moving lens anisotropies does not reflect the whole population of objects, but only the distribution of the resolved ones. In the OCDM model the structures are more numerous and form earlier than in a standard CDM model. Consequently, the distribution of unresolved objects in OCDM thus shows a large excess compared with the standard CDM and there are less resolved structures in the OCDM model than in the CDM. The excess of power in the the moving lens fluctuations spectrum (Fig. 3, solid line) reflects the dependence of the size distribution upon the cosmological model.

At a given large scale and for the SZ kinetic effect, there is more power on large scales in a standard CDM model compared with the OCDM. This is because the contribution to the power comes from low redshift resolved structures, which are less numerous in an OCDM model. Consequently, in the case of the fluctuations induced by the moving lens effect at large scale, the power in the OCDM model is greater than in the standard CDM. In addition, at a given large scale the power of the moving lens effect accounts for the cumulative contribution from the massive objects, with high amplitude, and from the less massive ones, with lower amplitudes.

A comparison between the CMB and the moving lens power spectra obviously shows that primary CMB fluctuations dominate at all scales larger than the cut-off scale, whatever the cosmological model (Fig. 3). Furthermore in the OCDM and [FORMULA] CDM models the cut-off is shifted towards smaller angular scales making the CMB the dominant contribution over a larger range of scales. The most favourable configuration to study and analyse the fluctuations is therefore the CDM model since it gives the largest cut-off scale compared to the other cosmological models and since it gives the highest prediction for the power of the moving lens effect. The level of spurious additional signal associated with the moving lens effect is negligible compared to both the primary and SZ kinetic fluctuations. Below the scale of the cut-off in the CMB power spectrum, the [FORMULA] dependence of the SZ fluctuations is dominant over the moving lens effect. Moreover, contrary to the thermal effect, the SZ kinetic, moving lens and primary fluctuations have black body spectra. This makes the spectral confusion between them a crucial problem. At small angular scales, the SZ kinetic effect represents the principal source of confusion.

Nevertheless, the contribution of the SZ kinetic effect is very dependent on the predicted number of structures that show a gas component. In other words, some objects like small groups of galaxies may not have a gas component, and therefore no SZ thermal or kinetic anisotropy is generated, but they still exhibit the anisotropy associated with their motion across the sky. We attempt to study a rather wide range of models. We therefore use two prescriptions to discriminate between "gaseous" objects and "non gaseous" ones. These prescriptions correspond to arbitrary limits on the masses of the structures. Namely: in the first model, we assume that all the dark matter halos with masses greater than [FORMULA] have a gas fraction of 20% and exhibit SZ thermal and kinetic anisotropies; while in the second model, it is only the structures with masses [FORMULA] which produce SZ anisotropies. We ran the simulations with both assumptions in the standard CDM model and computed the corresponding power spectra (Fig. 4). The power spectrum associated with the SZ kinetic effect shows, as expected, that the cut-off in masses induces a decrease in the power of the SZ kinetic effect on all scales, and in particular on very small scales with a cut-off at [FORMULA]. The power spectrum of the SZ kinetic anisotropies can be fitted with the following expression:

[EQUATION]

Despite this cut-off in mass and the decrease in power, the SZ kinetic effect remains much larger than the moving lens effect. Therefore at small angular scales, the SZ kinetic point like sources are still the major source of confusion. In order to get rid of this pollution in an effective way, one would need a very sharp but unrealistic cut-off in mass.

[FIGURE] Fig. 4. Randomly generated power spectra of the primary fluctuations in the standard CDM model (solid line) compared to the secondary fluctuations due to the moving lens effect (dashed-dotted line) and the Sunyaev-Zel'dovich kinetic effect with a cut-off at [FORMULA] (dashed line) and at [FORMULA] (dotted line).

4.2. Detection and extraction

We analyse the simulated maps in order to estimate the amplitudes of the anisotropies associated with each individual moving structure. In such an analysis both primary CMB and SZ kinetic fluctuations represent spurious signals with regards to the moving lens. Fig. 3 shows that these signals contribute at different scales and at different levels. The primary CMB contribution vanishes on scales lower than the cut-off whereas the SZ kinetic contribution shows up at all scales and its power increases as [FORMULA] on small scales. This indicates clearly that the most important problem with the analysis of the maps (extraction and detection of the moving lens anisotropy) is the confusion due to the point-like sources. This problem is made worse by spectral confusion. A compromise must be found between investigating scales smaller than the CMB cut-off, which maximises the pollution due to SZ kinetic effect, and exploring larger scales where the SZ contribution is low (but still 10 times larger than the moving lenses). The main problem here is that on these scales the primary fluctuations are 100 times larger than the moving lenses which makes their detection hopeless.

Nevertheless, the signal has two characteristics that make the attempts at detection worthy at small scales. The first advantage is that the anisotropy induced by a moving lens exhibits a particular spatial signature which is seen as the dipole-like patterns shown in Fig. 5. The second, and main advantage is that we know the position of the center of the structures thanks to the SZ thermal effect.

[FIGURE] Fig. 5. Plot showing the contours superimposed over a simulated map of the fluctuations induced by moving lenses in the standard CDM model (pixel size= 1.5'). The contour levels and grey scales shown in the plot are: [FORMULA]

In fact, the objects giving rise to a dipole-like anisotropy are either small groups or clusters of galaxies with hot ionised gas which also exhibit SZ thermal distortions. The latter, characterised by the so-called Comptonisation parameter y, have a very specific spectral signature. It is therefore rather easy to determine the position of the center of a structure assuming that it corresponds to the maximum value of the y parameter. In the context of the Planck multi-wavelength experiment for CMB observations, it was shown (Aghanim et al. 1997) that the location of massive clusters will be well known because of the presence of the SZ thermal effect.

We based our detection strategy for the moving lens effect on these two properties (spatial signature and known location). We also assumed that the SZ thermal effect was perfectly separated from the other contributions thanks to the spectral signature. The problem is therefore eased since it lies in the separation of moving lens, SZ kinetic and primary CMB anisotropies at known positions. Nevertheless the clusters and their gravitational potential wells are likely to be non-spherical, making the separation difficult. In the following, we will show that even in the simple spherical model we adopt the separation remains very difficult because of the spectral confusion of the moving lens, SZ kinetic and primary CMB fluctuations. Separation is even more difficult because of the numerous point-like SZ kinetic sources corresponding to weak clusters and small groups of galaxies for which we do not observe the SZ thermal effect.

4.2.1. Method

In order to clean the maps from the noise (SZ kinetic and CMB fluctuations), we filter them using a wavelet transform. Wavelet transforms have received significant attention recently due to their suitability for a number of important signal and image processing tasks. The principle behind the wavelet transform, as described by Grossmann & Morlet (1984), Daubechies (1988) and Mallat (1989) is to hierarchically decompose an input image into a series of successively lower resolution reference images and associated detail images. At each level, the reference image and detail image contain the information needed to reconstruct the reference image at the next higher resolution level. So, what makes the wavelet transform interesting in image processing is that, unlike Fourier transform, wavelets are quite localised in space. Simultaneously, like the Fourier transform, wavelets are also quite localised in frequency, or more precisely, on characteristic scales. Therefore, the multi-scale approach provides an elegant and powerful framework for our image analysis because the features of interest in an image (dipole pattern) are generally present at different characteristic scales. Furthermore, the wavelet transform performs contemporaneously a hierarchical analysis in both the space and frequency domains.

The maps are decomposed in terms of a wavelet basis that has the best impulse response and lowest shift variance among a set of wavelets that were tested for image compression (Villasenor et al. 1995). These two characteristics are important if we want to identify the locations and the amplitudes of the moving lenses. Since the moving lenses induce very small scale anisotropies compared to the CMB, we filter the largest scales in order to separate these two contributions. We note that this also allows us to separate the contributions due to the large scale SZ kinetic sources. In the following we describe our analysis method, first applied to an unrealistic study case and then to a realistic case.

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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