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An interesting cross-check on whether the modeled sky actually matches the W MAP observations is whether theand ILC coefficients are similar to those derived from the analysis of the observed data or not. To assess this point, we studied the distributions of the weights obtained from the internal analysis of the raw simulated data, and compared them to those derived from the analysis of the real ones. These are shown in Figure 5.3 and 5.4 respectively for the model which does not include the WIM emission and the one that includes it.

It is easy to note that the  results are more spread than the ILC ones. Furthermore, in the latter case, the mean is generally not consistent with the values expected from the real data analysis: for a frequency larger than 40 GHz (Q band), the values are outside the distributions. This is not the case foralthough this result is mainly due to the fact that the distributions are very spread: the real weights are still consistent with the mean of the distributions or, at least, they lie on the tail. This discrepancy in the width of the distributions is probably the effect of the different statistic employed by the two methods as well as the fact that the ILC imposes a constraint on the CMB spectrum. On the other side, the inconsistency of the mean of the distributions with the weights derived from the real data analysis, suggests the idea that the foreground model used for simulations does not realistically reproduce the observations. Thus, our poor knowledge about foregrounds partially compromises the usage of simulations to predict real data analysis results.

We also examined the distribution of the derived coupling coefficients of all the components with respect to the mock data, and the cross-talk among them. As stated before, although the attention was focused on the free-free component, the distribution of the synchrotron and dust coefficients have been analysed as well, in order to have a complete picture of the simulation response, and to check that the results obtained for the free-free emission were not just a consequence of a bad reconstruction of the other components. For a non-standard analysis, we corrected the values for the bias due to the CMB subtraction. This value has been derived by fitting all the coefficients to the models assumed, as follows:

Figure 5.3: Simulations without WIM emission. Statistical distribution of the weights derived to compute the CMB map using(red) and the ILC code (blue). The first one is computed outside the K p2 mask and the latter has been computed using the partition of the sky proposed by the W MAP science team. The green (ICA) and yellow (ILC) lines show the values of the weights recovered by the real data analysis.

Imod,synchrotron = Async

ν 23GHz −3.1 +C0a(ν) ν 23GHz 2 , (5.5)

Imod,dust = AT hDust

ν 94GHz 1.7 +ACN M(CN M)+C0a(ν) ν 23GHz 2 (5.6) where Asynch, AT hDust and ACN M are the amplitudes of the synchrotron, the thermal and spinning dust emissions respectively, and C0is the offset. Equation 5.4 is the equivalent one for the free-free emission.

The coefficients of the synchrotron and dust emission derived from the standard analysis (without subtraction of the CMB) have a symmetric and not biased distribution. In the cases where the CMB is subtracted, instead, there is a bias which can be significant, if compared with the error on the mean. This is probably the consequence of a cross-talk between the two emissions, which is further enhanced by the bias introduced by the ICA and ILC maps. Such a deviation is not present in the distributions of the free-free coefficients, meaning that they are not involved in the cross-talk.

Finally, comparing the coupling coefficients obtained from simulations and those derived from the real data analysis (see Section 5.4), we detected a disagreement between them. When the CMB is pre-

Figure 5.4: Simulations with WIM emission. Statistical distribution of the weights derived to compute the CMB map using(red) and the ILC code (blue). The first one is computed outside the K p2 mask and the latter has been computed using the partition of the sky proposed by the W MAP science team. The green (ICA) and yellow (ILC) lines show the values of the weights recovered by the real data analysis.

subtracted, the coefficients are generally not consistent with the real ones, in both the cases where we include the WIM emission and we do not include it, and with both the andχ2 methods. This is a further indication of the fact that the adopted foreground model is not perfectly consistent with the real observations. The presence of the bias in the CMB tracers makes the discrepancy even larger.

A critical issue is to disentangle the different contributions encoded in the free-free coefficients following the prescription proposed by Dobler et al. (2009) for theand ILC corrected data. For both the sets of simulations, we fitted the coefficients with both the models, looking for the genuine free-free emission as well as for the correlated spinning dust component. In practice, in the case where the model A is adopted, we checked that there were no fake detections: the mean of the coefficients for the spinning dust component should be consistent with zero. On the other side, when the B model is used, fitting the coefficients under the hypothesis that they can be described as purely free-free, is interesting in order to see how the fits behave. In other words, in the first case we can test whether we can obtain a bump when there is none, and, in the case where there is a bump, whether we do not have evidence of this.

Figures 5.5, 5.6, 5.7 show the spectrum behaviour derived from the scaling coefficients for the three components when using the input models. It is clear that for all the cases of analysis, the spectral

behaviour for both the components is recovered according to the expectations. Specifically the bump in the free-free spectrum is visible, if the WIM emission is actually present in the data. This is true for either the case where the CMB contribution is subtracted from the data or not. Therefore, the subtraction of the CMB contribution does not seem to induce any artifact on the free-free spectrum, excluding the hypothesis that the bump is due to the contamination of the CMB map by residual foregrounds. Indeed, the simulations where the WIM contribution is not included in the foreground model, return spectra perfectly consistent with the theoretical expectations, also when the CMB is subtracted from the data. This is why we retained only the fit with the model used to simulate the data.

Looking at the plots of the frequency spectra, it is also easy to note that, when the CMB emission is not subtracted from the data, the uncertainty of the coefficients is much larger then in the other cases, due to the cross-talk between the CMB and the templates. Thus, taking into account the amplitude of the error bars, several scenarios are possible for the spectral behaviour of the free-free coupling coefficients.

Furthermore there are not dramatic differences between the results obtained withand theχ2 analysis. However, there are cases of analysis whereseems to be unstable: when the ILC map (created according to the W MAP prescriptions) is assumed as CMB tracer, the algorithm sometimes fails in the reconstruction of the components. The same happens when the ILC map is computed using data without contribution from the thermal dust emission.