Nongaussianity in the foregroundreduced CMB maps
Abstract
A detection or nondetection of primordial nonGaussianity by using the cosmic microwave background Radiation (CMB) data is crucial not only to discriminate inflationary models but also to test alternative scenarios. NonGaussianity offers, therefore, a powerful probe of the physics of the primordial universe. The extraction of primordial nonGaussianity is a difficult enterprise since several effects of nonprimordial nature can produce nonGaussianity. Given the farreaching consequences of such a nonGaussianity for our understanding of the physics of the early universe, it is important to employ a range of different statistical tools to quantify and/or constrain its amount in order to have information that may be helpful for identifying its causes. Moreover, different indicators can in principle provide information about distinct forms of nonGaussianity that can be present in CMB data. Most of the Gaussianity analyses of CMB data have been performed by using partsky frequency, where the masks are used to deal with the galactic diffuse foreground emission. However, fullsky map seems to be potentially more appropriate to test for Gaussianity of the CMB data. On the other hand, masks can induce bias in some nonGaussianity analyses. Here we use two recent largeangle nonGaussianity indicators, based on skewness and kurtosis of largeangle patches of CMB maps, to examine the question of nonGaussianity in the available fullsky fiveyear and sevenyear Wilkinson Microwave Anisotropy Probe (WMAP) maps. We show that these fullsky foregroundreduced maps present a significant deviation from Gaussianity of different levels, which vary with the foregroundreducing procedures. We also make a Gaussianity analysis of the foregroundreduced fiveyear and sevenyear WMAP maps with a KQ75 mask, and compare with the similar analysis performed with the corresponding fullsky foregroundreduced maps. This comparison shows a significant reduction in the levels of nonGaussianity when the mask is employed, which provides indications on the suitability of the foregroundreduced maps as Gaussian reconstructions of the fullsky CMB.
pacs:
98.80.Es, 98.70.Vc, 98.80.kI Introduction
A key prediction of a number of simple singlefield slowroll inflationary models is that they cannot generate detectable nonGaussianity of the cosmic microwave background (CMB) temperature fluctuations within the level of accuracy of the Wilkinson Microwave Anisotropy Probe (WMAP) Gauss_Singlefield (). There are, however, several inflationary models that can generate nonGaussianity at a level detectable by the WMAP. These nonGaussian scenarios comprise models based upon a wide range of mechanisms, including special features of the inflation potential and violation of one of the following four conditions: single field, slow roll, canonical kinetic energy, and initial BunchDavies vacuum state. Thus, although convincing detection of a fairly large primordial nonGaussianity in the CMB data would not rule out all inflationary models, it would exclude the entire class of stationary models that satisfy simultaneously these four conditions (see, e.g., Refs. Bartolo2004 (); WPKomatsuatal09 (); Inflationreviews ()). Moreover, a null detection of deviation from Gaussianity would rule out alternative models of the early universe (see, for example, Refs. NonGausAlternativeModels ()). Thus, a detection or nondetection of primordial nonGaussianity in the CMB data is crucial not only to discriminate (or even exclude classes of) inflationary models but also to test alternative scenarios, offering therefore a window into the physics of the primordial universe.
However, there are various nonprimordial effects that can also produce nonGaussianity such as, e.g., unsubtracted foreground contamination, unconsidered point sources emission and systematic errors Chiangetal2003 (); Naselskyetal2005 (); Cabellaetal2009 (). Thus, the extraction of a possible primordial nonGaussianity is not a simple endeavor. In view of this, a great deal of effort has recently gone into verifying the existence of nonGaussianity by employing several statistical estimators Some_nonGaussrefs () (for related articles see, e.g., Refs. NonGaussrelated ()). Different indicators can in principle provide information about multiple forms of nonGaussianity that may be present in WMAP data. It is therefore important to test CMB data for deviations from Gaussianity by using a range of different statistical tools to quantify or constrain the amount of any nonGaussian signals in the data, and extract information on their possible origins.
A number of recent analyses of CMB data performed with different statistical tools have provided indications of either consistency or deviation from Gaussianity in the CMB temperature fluctuations (see, e.g., Ref. Some_nonGaussrefs ()). In a recent paper BernuiReboucas2009 () we proposed two new largeangle nonGaussianity indicators, based on skewness and kurtosis of largeangle patches of CMB maps, which provide measures of the departure from Gaussianity on large angular scales. We used these indicators to search for the largeangle deviation from Gaussianity in the three and fiveyear single frequency maps with a KQ75 mask, and found that while the deviation for the Q, V, and W masked maps are within the expected values of MonteCarlo (MC) statistically Gaussian CMB maps, there is a strong indication of deviation from Gaussianity ( off the MC) in the K and Ka masked maps.
Most of the Gaussianity analyses with WMAP data have been carried out by using CMB temperature fluctuation maps (raw and clean) in the frequency bands Q, V and W or some combination of these maps. In these analyses, in order to deal with the diffuse galactic foreground emission, masks such as, for example, KQ75 and Kp0 have been used.
However, sky cuts themselves can potentially induce bias in Gaussianity analyses, and on the other hand fullsky maps seem more appropriate to test for Gaussianity in the CMB data. Thus, a pertinent question that arises is how the analysis of Gaussianity made in Ref. BernuiReboucas2009 () is modified if wholesky foregroundreduced CMB maps are used. Our primary objective in this paper is to address this question by extending the analysis of Ref. BernuiReboucas2009 () in three different ways. First, we use the same statistical indicators to carry out a new analysis of Gaussianity of the available fullsky foregroundreduced fiveyear and sevenyear CMB maps ILC5yrHishaw (); HILCKim (); NILCDelabrouille (); ILC7yrGold (). Second, since in these maps the foreground is reduced through different procedures each of the resulting maps should be tested for Gaussianity. Thus, we make a quantitative analysis of the effects of distinct cleaning processes in the deviation from Gaussianity, quantifying the level of nonGaussianity for each foreground reduction method. Third, we study quantitatively the consequences for the Gaussianity analysis of masking the foregroundreduced maps with the KQ75 mask. An interesting outcome is that this mask lowers significantly the level of deviation from Gaussianity even in the foregroundreduced maps, rendering therefore information about the suitability of the foregroundreduced maps as Gaussian reconstructions of the fullsky CMB.
Ii NonGaussianity Indicators
The chief idea behind our construction of the nonGaussianity indicators is that a simple way of accessing the deviation from Gaussianity distribution of the CMB temperature fluctuations is by calculating the skewness , and the kurtosis from the fluctuations data, where and are the third and fourth central moments of the distribution, and is its variance. Clearly calculating and from the whole sky temperature fluctuations data would simply yield two dimensionless numbers, which are rough measures of deviation from Gaussianity of the temperature fluctuation distribution.
However, one can go further and obtain a great number of values associated to directional information of deviation from Gaussianity if instead one takes a discrete set of points homogeneously distributed on the celestial sphere as the center of spherical caps of a given aperture and calculate and from the CMB temperature fluctuations of each spherical cap. The values and can then be taken as measures of the nonGaussianity in the direction of the center of the spherical cap . Such calculations for the individual caps thus provide quantitative information ( values) about possible violation of Gaussianity in the CMB data.
This procedure is a constructive way of defining two discrete functions and (defined on from the temperature fluctuations data, and can be formalized through the following steps (for more details, see Ref. BernuiReboucas2009 ()):

Take a discrete set of points homogeneously distributed on the CMB celestial sphere as the centers of spherical caps of a given aperture ;

Calculate for each spherical cap the skewness () and kurtosis () given, respectively, by
(1) and (2) where is the number of pixels in the cap, is the temperature at the pixel, is the CMB mean temperature in the cap, and is the standard deviation. Clearly, the values and obtained in this way for each cap can be viewed as a measure of nonGaussianity in the direction of the center of the cap ;

Patching together the and values for each spherical cap, one obtains our indicators, i.e., discrete functions and defined over the celestial sphere, which can be used to measure the deviation from Gaussianity as a function of the angular coordinates . The Mollweid projection of skewness and kurtosis functions and are nothing but skewness and kurtosis maps, hereafter we shall refer to them as map and map, respectively.
Now, since and are functions defined on they can be expanded into their spherical harmonics in order to have their power spectra and . Thus, for example, for the skewness indicator one has
(3) 
and can calculate the corresponding angular power spectrum
(4) 
which can be used to quantify the angular scale of the deviation from Gaussianity, and also to calculate the statistical significance of such deviation. Obviously, similar expressions hold for the kurtosis .
In the next section we shall use the statistical indicators and to test for Gaussianity the available foregroundreduced maps obtained from the fiveyear WMAP data.
Iii NonGaussianity
iii.1 Foregoundreduced maps
The WMAP team has released high angular resolution fiveyear maps of the CMB temperature fluctuations in the five frequency bands K ( GHz), Ka ( GHz), Q ( GHz), V ( GHz), and W ( GHz). They have also produced a fullsky foregroundreduced Internal Linear Combination (ILC) map which is formed from a weighted linear combination of these five frequency band maps in which the weights are chosen in order to minimize the galactic foreground contribution.
It is well known that the firstyear ILC map is inappropriate for CMB scientific studies Bennett2003 (). However, in the fiveyear (also in the threeyear and sevenyear) version of this map a bias correction has been implemented as part of the foreground cleaning process, and the WMAP team suggested that this map is suitable for use in large angular scales (low ) analyses although they admittedly have not performed nonGaussian tests on this version of the ILC map ILC3yrHishaw (); ILC5yrHishaw (). Notwithstanding the many merits of the fiveyear ILC procedure, some cleaning features of this ILC approach have been considered, and two variants have been proposed recently. In the first approach the frequency dependent weights were determined in harmonic space HILCKim (), while in the second the foreground is reduced by using needlets as the basis of the cleaning process NILCDelabrouille (). Thus, two new fullsky foregroundcleaned maps have been produced with the WMAP fiveyear data, namely the harmonic ILC (HILC) HILCKim () and the needlet ILC (NILC) (for more details see Refs. HILCKim (); NILCDelabrouille ()).
In the next section, we use the fullsky foregroundreduced ILC, HILC and NILC maps with the same smoothed resolution (which is the resolution of the ILC map) as the input maps from which we calculate the and maps, and then we compute the associated power spectra in order to carry out a statistical analysis to quantify the levels of deviation from Gaussianity.^{1}^{1}1The ILC, HILC and NILC maps are available for download from: http://lambda.gsfc.nasa.gov/product/map/dr3/ilc_map_get.cfm, http://www.nbi.dk/jkim/hilc/ and http://www.apc.univparis7.fr/APC_CS/Recherche/Adamis/cmb_wmapen.php.
iii.2 Analysis and results
In order to minimize the statistical noise, in the calculations of skewness and kurtosis maps (map and map) from the foregroundreduced maps, we have scanned the celestial sphere with spherical caps of aperture , centered at points homogeneously generated on the twosphere by using the HEALPix code Gorskietal2005 (). In other words, the pointcenters of the spherical caps are the center of the pixels of a homogeneous pixelization of the generated by HEALPix with . We emphasize, however, that this pixelization is only a practical way of choosing the centers of the caps homogeneously distributed on . It is not related to the pixelization of the abovementioned ILC, HILC and NILC input maps that we have utilized to calculate both the and maps from which we compute the associated power spectra.
Figures 1 and 2 show examples of and maps obtained from the foregroundreduced NILC fullsky and KQ75 maps. The panels of these figures clearly show regions with higher and lower values (’hot’ and ’cold’ spots) of and , which suggest largeangle multipole components of nonGaussianity. We have also calculated similar maps (with and without the KQ75 mask) from the ILC and HILC maps. However, since these maps provide only qualitative information, to avoid repetition we only depict the maps of Figs. 1 and 2 merely for illustrative purpose.
In order to obtain quantitative information about the large angular scale (low ) distributions for the nonGaussianity and maps obtained from the available fullsky foregroundreduced fiveyear maps, we have calculated the (low ) power spectra and for these maps. The statistical significance of these power spectra is estimated by comparing with the corresponding multipole values of the averaged power spectra and calculated from maps obtained by averaging over MonteCarlogenerated statistically Gaussian CMB maps.^{2}^{2}2Each MonteCarlo scrambled map is a stochastic realization of the WMAP bestfitting angular power spectrum of the CDM model, obtained by randomizing the temperature components within the cosmic variance limits. Throughout the paper the mean quantities are denoted by overline.
Before proceeding to a statistical analysis, let us describe with some detail our calculations. For the sake of brevity, we focus on the skewness indicator , but a completely similar procedure was used for the kurtosis indicator . We generated MC Gaussian (scrambled) CMB maps, which are then used to generate skewness maps, from which we calculate power spectra: ( is an enumeration index, and ). In this way, for each fixed multipole component we have multipole values from which we calculate the mean value . From this MC process we have at the end ten mean multipole values , each of which are then used for a comparison with the corresponding multipole values (obtained from the input map) in order to evaluate the statistical significance of the multipole components . To make this comparison easier, instead of using the angular power spectra and themselves, we employed the differential power spectra and , which measure the deviation of the skewness and kurtosis multipole values (calculated from the foregroundreduced maps) from the mean multipoles and (calculated from the Gaussian maps). Thus, for example, to study the statistical significance of the quadrupole component of the skewness from HILC map (say) we calculate the deviation , where the mean quadrupole value is calculated from the quadrupole values of the MC Gaussian maps.
for  for  

HILC  
ILC  
NILC 
Figure 3 shows the differential power spectra calculated from fullsky fiveyear foregroundreduced maps, i.e., it displays the absolute value of the deviations from the mean angular power spectrum of the skewness (left panel) and kurtosis (right panel) indicators for , which is a range of multipole values needed to investigate the largescale angular characteristics of the and maps. This figure shows a first indication of deviation from Gaussianity in fiveyear foregroundreduced ILC, HILC and NILC maps in that the deviations and for these maps are not within of the mean MC value.
To obtain additional quantitative information regarding the deviation from Gaussianity, we can also calculate the percentage of the deviations calculated from MC Gaussian maps, which are smaller than obtained from each foregroundreduced map. This calculations are made in detail in the Appendix A. Thus, for example, we have for the fullsky NILC, HILC and ILC maps, respectively, that , , and of the multipole values obtained from the MC maps are closer to the mean than the value calculated from the data, i.e. from each of the foregroundreduced maps. This indicates how unlikely are the occurrences of the values obtained from these foregroundreduced maps for the multipole in the set of values of from MC simulated maps. In other words, the probability of occurrence of the values (in the set of MC values) for the NILC, HILC and ILC maps is only , and , respectively. Similarly, the probability of occurrence of , for example, is for all these foregroundreduced maps, while for are respectively (NILC), (HILC) and (ILC). In Tables 4 and 6 of the Appendix A we collect together the probability of occurrence of each of the values and () calculated from and maps obtained from the fullsky NILC, HILC and ILC maps. In Tables 5, and 7 we present these probabilities calculated from the same input maps but now with KQ75 mask.^{3}^{3}3We emphasize that, throughout this paper, in the implementation of the mask we do not take for the temperature fluctuation of the pixels inside the masked region. This would clearly induce nonGaussian contribution. In our scan of the CMB sky when the spherical cap move into the masked area the pixels of the cap inside the masked do not contribute to the values of the indicators in the center of the cap. In these cases, the values and for a cap are calculated with small number of pixels. The comparison of Table 4 with Table 5, and of Table 6 with Table 7, makes apparent the role of the KQ75 mask in reducing the level of deviation from Gaussianity (see the Appendix A for more details).
for [KQ75]  for [KQ75]  

HILC  
ILC  
NILC 
for [Kp0]  for [Kp0]  

HILC  
ILC  
NILC 
Although the set of ’local’ (fixed ) estimates collected together in the tables of Appendix A gives an indication of deviation from Gaussianity as measured by each multipole component to have an overall assessment of low power spectra and calculated from each CMB foregroundreduced map, we have performed a test to find out the goodness of fit for and multipole values as compared to the expected multipole values from the MC Gaussian maps. In this way, we can obtain one number for each foregroundreduced map that collectively (’globally’) quantifies the deviation from Gaussianity. For the power spectra and we found that the values given in Table 1 for the ratio (dof stands for degrees of freedom) for the power spectra calculated from HILC, ILC and NILC fullsky input maps. Clearly a good fit occurs when . Moreover, greater are the values, the smaller the probabilities, that is the probability that the multipole values and and the expected MC multipole values agree. Thus, regarding the skewness indicator Table 1 shows that the HILC presents the greatest level of deviation from Gaussianity (), as captured by the indicator , while the NILC map has the lowest level.
Regarding the deviation from Gaussianity as detected by the kurtosis indicator , Table 1 shows again that the HILC presents the largest deviation followed by the ILC and NILC. To the extent that is considerably greater than one, all these fullsky foregroundreduced maps also present a significant deviation from Gaussianity as captured here by the kurtosis indicator.
The above results of our statistical analysis given in Fig. 3 and gathered together in Table 1 (and also supported by Tables 4 and 5 of Appendix A) show a significant deviation from Gaussianity in fiveyear fullsky foregroundreduced (ILC, NILC and HILC) maps as detected by both the skewness and the kurtosis indicators and . A pertinent question that arises here is how this analysis of Gaussianity for the fullsky foregroundreduced maps is modified if one uses the KQ75 mask, which was recommended by the WMAP team for tests of Gaussianity of the fiveyear band maps. Furthermore, the combination of the fullsky and mask analyses should provide information on the reliability of the foregroundreduced maps as appropriate reconstructions of the fullsky CMB.
Figure 4 shows the power spectra (left) and (right) calculated from fiveyear foregroundreduced KQ75 masked maps. This figure along with Fig. 3 show a significant reduction in the level of deviation from Gaussianity when the foregroundreduced ILC, HILC, and NILC maps are masked. To quantify this reduction we have recalculated for these input maps with the KQ75 mask, and have collected the results in Table 2. The comparison of Table 1 and Table 2 shows quantitatively the reduction of the level of Gaussianity for the case of CMB masked maps.^{4}^{4}4Incidentally, this reduction is also revealed through (and agrees with) the comparison of Table 4 with Table 5, and of Table 6 with Table 7.
In the above analyses we have followed the fiveyear WMAP recommendation for tests of Gaussianity and thus used the mask the KQ75, which is slightly more conservative than the Kp0 (theKQ75 sky cut is while the Kp0 cut is ). A pertinent question at this point is how the above results are modified if the less conservative Kp0 mask is used. We have examined this issue by calculating the power spectra and and the from and maps obtained from the ILC, NILC, and HILC input maps with the Kp0 mask. The result of this analysis is given in Table 3.
A comparison between Tables 2 and 3 shows that in general the value of increases for both indicators when the less conservative mask Kp0 is used. We note that the changes in values are greater for the HILC, though.
The comparison between Fig. 3 and Fig. 4, and Tables 1 and Table 2 along with the tables of the Appendix A clearly provides quantitative information on the suitability of the foregroundreduced maps as Gaussian reconstructions of the fullsky CMB, and makes apparent the relevant role of the mask KQ75 in reducing significantly the level of nonGaussianity in these foregroundreduced maps.
The calculations of our nonGaussianity indicators require the specification of some quantities whose choice could in principle affect the outcome of our calculations. To test the robustness of our scheme, hence of our results, we studied the effects of changing in the parameters employed in the calculation of our indicators. We found that the and angular power spectra do not change appreciably as we change the resolution of CMB temperature maps used and the number of pointcenters of the caps with values , and (see Ref. BernuiReboucas2009 () for more details on the robustness of this method).
Concerning the robustness of the above analyses with the KQ75 mask some additional words of clarification are in order here. First, we note that the calculations of the maps and maps by scanning the CMB masked maps sometimes include caps whose center is within or close to the KQ75 masked region. In these cases, the calculations of the and indicators are made with a smaller number of pixels, which clearly introduce additional statistical noise as compared to the fullsky map cases. In order to minimize this effect we have scanned the CMB masked sky with spherical caps of aperture , and for the sake of uniformity we have used caps with the same aperture for the fullsky maps. We note, however, that fullsky foregroundreduced analysis does not change significantly if one uses smaller apertures as, for example, .
Iv Concluding remarks
The detection or nondetection of primordial nonGaussianity in the CMB data is essential to discriminate or even exclude classes of inflationary models. It can also be used to test alternative scenarios of the primordial universe. There are, however, several nonprimordial effects that can also produce nonGaussianity. This makes the extraction of a possible primordial nonGaussianity a rather difficult endeavor. Since different indicators can in principle provide information about distinct forms of nonGaussianity, it is important to test CMB data for nonGaussianity by using different estimators to quantify and/or constrain its amount in order to extract information about their possible sources.
Most of the Gaussianity analyses of CMB data have been performed with frequency band maps. In these studies, to deal with the galactic diffuse foreground emission, masks have been employed. However, a fullsky foregroundreduced map seems to be potentially more appropriate to test for Gaussianity the CMB data.^{5}^{5}5In reality, the fullsky map seems to be the most suitable for a number of other issues, including the test of statistical isotropy, the search for evidence of a NorthSouth asymmetry in CMB data, and signatures of a possible nontrivial cosmic topology, for example. The fiveyear version of the ILC map has been suggested as a fullsky map suitable for large angular scales analyses ILC3yrHishaw (), even though the WMAP team has not performed a battery of nonGaussianity tests on this map ILC5yrHishaw ().
In this paper we have performed an analysis of Gaussianity of the available fiveyear fullsky foregroundreduced maps. To this end, we have used two new nonGaussianity indicators based on skewness and kurtosis of largeangle patches of CMB maps, which provide a measure of departure from Gaussianity on large angular scales BernuiReboucas2009 (). We have shown that the fullsky fiveyear foregroundreduced maps (ILC, HILC and NILC) present a significant deviation from Gaussianity, which varies with the foregroundreducing procedures. We have established which of these fullsky foregroundreduced maps exhibit the highest and the lowest level of nonGaussianity.
We have also masked the foregroundreduced maps with KQ75 and Kp0 masks and performed a quantitative analysis of deviation from Gaussianity of these maps. The comparison of the fullsky and masked analyses (see Fig. 3 and Fig. 4; and Tables 1, 2 and 3) shows a significant reduction in the levels of nonGaussianity when the masks are employed, which in turn provides indications on the suitability of the foregroundreduced maps as Gaussian reconstructions of the fullsky CMB.
Finally, when we were in the process of rewriting a revised version of this paper, by taking into account the referee’s recommendations, the sevenyear WMAP CMB data were released, including a new version of the fullsky foregroundreduced ILC map ILC7yrGold (). We have considered this latest foregroundreduced ILC map, and performed a complete additional analysis of the Gaussianity of the five and sevenyear versions of the ILC maps, whose details are given in Apenddix B.^{6}^{6}6Note that there is no available sevenyear HILC and NILC maps. The main result of this appendix is that the fullsky sevenyear foregroundreduced ILC map also present a significant deviation from Gaussianity, which again is reduced substantially when the KQ75 mask is employed. In this way, our results are robust with respect to sevenyear WMAP CMB data.
Acknowledgements.
This work is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) – Brasil, under Grant No. 472436/20074. M.J.R. and A.B. thank CNPq for the grants under which this work was carried out. We are also grateful to A.F.F. Teixeira for reading the manuscript and indicating the omissions and misprints. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Some of the results in this paper were derived using the HEALPix package Gorskietal2005 ().Appendix A
Clearly from MC maps one can calculate for each one thousand values of both and () and the corresponding mean values and . For the sake of brevity in what follows we focus on the skewness indicator , but completely similar calculations were used to have the probabilities for kurtosis indicator .
NILC [fullsky]  HILC [fullsky]  ILC [fullsky]  

%  
NILC [KQ75]  HILC [KQ75]  ILC [KQ75]  

NILC [fullsky]  HILC [fullsky]  ILC [fullsky]  

NILC [KQ75]  HILC [KQ75]  ILC [KQ75]  

With the MC values and the mean one can calculate the percentages of values of the deviations calculated from MC Gaussian maps which are smaller than with obtained from the data (fullsky and masked maps). For each multipole this number indicates how unlikely are the occurrences of the values obtained from the data (input maps) for that multipole in the set of values obtained from MC Gaussian maps. In this way one can calculate the probability of occurrence of a given multipole value (obtained from the data) in the set of MC values (obtained from the MC maps) for each foregroundreduced CMB map. In Tables 4, 5, 6 and 7 we collect together the results of such calculations.
for [fullsky]  for [fullsky]  

ILC5  
ILC7 
Thus, for example, from Table 4 we have for the fullsky NILC, HILC, and ILC maps, respectively, the probability of occurrence of the values (in the set of MC values) is , whereas from Table 6 the probability for is, respectively, , and for the fullsky NILC, HILC, and ILC input maps.
The comparison of Table 4 with Table 5, and of Table 6 with Table 7 shows that the role of the KQ75 mask is to cut down significantly the level of deviation from Gaussianity for all multipoles and obtained from the foregroundreduced input maps. This is clear because the probabilities of occurrences for these multipoles values in the set of MC multipole values increase substantially when the mask is employed.
Although the estimates of probabilities collected in these tables give a clear quantitative indication of deviation from Gaussianity an overall assessment of the power spectra and can be obtained through test of the goodness of fit for and from the data as compared to the expected multipoles values obtained from the Gaussian MC maps. This point is discussed in Section III.2.
Appendix B
While we were in the final phase of writing a modified version of this paper a new version of the fullsky foregroundreduced ILC map was released by the WMAP team ILC7yrGold (). Since there is no available version of the NILC and HILC maps obtained from the sevenyear WMAP data to be considered, here we present the results of a comparative analysis of deviation from Gaussianity performed by using the five and sevenyear versions of the ILC as input maps. As the calculations are similar to those of Section III.2 we refer the readers to that section for more details.
Figure 5 shows the differential power spectra calculated from the fullsky five and sevenyear foregroundreduced ILC input maps (ILC5 and ILC7, for short). Apart from some local deviation of the deviations and this figure shows a deviation from Gaussianity, which is quantified in Table 8.
for [KQ75]  for [KQ75]  

ILC5  
ILC7 
It is interesting to note that the deviation from Gaussianity as measured by our indicators is greater for the ILC7 than for the ILC5 input map. Concerning this point some words of clarification are in order here. First, we note that the details of the algorithm used to compute the ILC7 maps are the same as those of the ILC5 map. However, to take into account the most recent updates to the calibration and beams, the frequency weights for each of the 12 regions (in which the sky is subdivided in the ILC method) are slightly different in the calculation of the ILC7 map. Second, the difference between the ILC7 and ILC5 maps is a map whose smallscale differences are consistent with the pixel noise, but with a largescale dipolar component, with the largescale differences being consistent with a change in dipole of 6.7 KILC7yrGold (). Thus, the resultant ILC7 map is not indistinguishable from the ILC5 map, and the differences between them have been captured by our indicators.
Figure 6 shows the differential power spectra calculated from a fiveyear and seven year version of the foregroundreduced ILC maps with a KQ75 mask. This figure along with Fig. 5 show a significant reduction in the level of deviation from Gaussianity when both ILC5 and ILC7 are masked. To quantify this reduction we have calculated for these input maps with the KQ75 mask, and have collected the results in Table 9. The comparison of Table 8 and Table 9 shows quantitatively the reduction of the level of Gaussianity for the case of CMB masked maps.
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