Non-gaussianity in the foreground-reduced CMB maps

Non-gaussianity in the foreground-reduced CMB maps

A. Bernui bernui@unifei.edu.br Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro – RJ, Brazil Instituto de Ciências Exatas, Universidade Federal de Itajubá, 37500-903 Itajubá – MG, Brazil    M.J. Rebouças reboucas@cbpf.br Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro – RJ, Brazil
July 15, 2019
Abstract

A detection or nondetection of primordial non-Gaussianity by using the cosmic microwave background Radiation (CMB) data is crucial not only to discriminate inflationary models but also to test alternative scenarios. Non-Gaussianity offers, therefore, a powerful probe of the physics of the primordial universe. The extraction of primordial non-Gaussianity is a difficult enterprise since several effects of non-primordial nature can produce non-Gaussianity. Given the far-reaching consequences of such a non-Gaussianity 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 non-Gaussianity that can be present in CMB data. Most of the Gaussianity analyses of CMB data have been performed by using part-sky frequency, where the masks are used to deal with the galactic diffuse foreground emission. However, full-sky 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 non-Gaussianity analyses. Here we use two recent large-angle non-Gaussianity indicators, based on skewness and kurtosis of large-angle patches of CMB maps, to examine the question of non-Gaussianity in the available full-sky five-year and seven-year Wilkinson Microwave Anisotropy Probe (WMAP) maps. We show that these full-sky foreground-reduced maps present a significant deviation from Gaussianity of different levels, which vary with the foreground-reducing procedures. We also make a Gaussianity analysis of the foreground-reduced five-year and seven-year WMAP maps with a KQ75 mask, and compare with the similar analysis performed with the corresponding full-sky foreground-reduced maps. This comparison shows a significant reduction in the levels of non-Gaussianity when the mask is employed, which provides indications on the suitability of the foreground-reduced maps as Gaussian reconstructions of the full-sky CMB.

pacs:
98.80.Es, 98.70.Vc, 98.80.-k

I Introduction

A key prediction of a number of simple single-field slow-roll inflationary models is that they cannot generate detectable non-Gaussianity of the cosmic microwave background (CMB) temperature fluctuations within the level of accuracy of the Wilkinson Microwave Anisotropy Probe (WMAP) Gauss_Single-field (). There are, however, several inflationary models that can generate non-Gaussianity at a level detectable by the WMAP. These non-Gaussian 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 Bunch-Davies vacuum state. Thus, although convincing detection of a fairly large primordial non-Gaussianity 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 (); WP-Komatsu-at-al09 (); Inflation-reviews ()). Moreover, a null detection of deviation from Gaussianity would rule out alternative models of the early universe (see, for example, Refs. NonGaus-Alternative-Models ()). Thus, a detection or nondetection of primordial non-Gaussianity 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 non-primordial effects that can also produce non-Gaussianity such as, e.g., unsubtracted foreground contamination, unconsidered point sources emission and systematic errors Chiang-et-al2003 (); Naselsky-et-al2005 (); Cabella-et-al2009 (). Thus, the extraction of a possible primordial non-Gaussianity is not a simple endeavor. In view of this, a great deal of effort has recently gone into verifying the existence of non-Gaussianity by employing several statistical estimators Some_non-Gauss-refs () (for related articles see, e.g., Refs. Non-Gauss-related ()). Different indicators can in principle provide information about multiple forms of non-Gaussianity 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 non-Gaussian 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_non-Gauss-refs ()). In a recent paper Bernui-Reboucas2009 () we proposed two new large-angle non-Gaussianity indicators, based on skewness and kurtosis of large-angle patches of CMB maps, which provide measures of the departure from Gaussianity on large angular scales. We used these indicators to search for the large-angle deviation from Gaussianity in the three and five-year 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 Monte-Carlo (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 full-sky 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. Bernui-Reboucas2009 () is modified if whole-sky foreground-reduced CMB maps are used. Our primary objective in this paper is to address this question by extending the analysis of Ref. Bernui-Reboucas2009 () in three different ways. First, we use the same statistical indicators to carry out a new analysis of Gaussianity of the available full-sky foreground-reduced five-year and seven-year CMB maps ILC-5yr-Hishaw (); HILC-Kim (); NILC-Delabrouille (); ILC-7yr-Gold (). 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 non-Gaussianity for each foreground reduction method. Third, we study quantitatively the consequences for the Gaussianity analysis of masking the foreground-reduced maps with the KQ75 mask. An interesting outcome is that this mask lowers significantly the level of deviation from Gaussianity even in the foreground-reduced maps, rendering therefore information about the suitability of the foreground-reduced maps as Gaussian reconstructions of the full-sky CMB.

Ii Non-Gaussianity Indicators

The chief idea behind our construction of the non-Gaussianity 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 non-Gaussianity 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. Bernui-Reboucas2009 ()):

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

  2. 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 non-Gaussianity in the direction of the center of the cap ;

  3. 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 foreground-reduced maps obtained from the five-year WMAP data.

Iii Non-Gaussianity

Figure 1: Skewness indicator maps calculated from the five-year foreground-reduced NILC full-sky (left panel) and KQ75 masked (right panel) maps.
Figure 2: Kurtosis indicator maps calculated from the five-year foreground-reduced NILC full-sky (left panel) and KQ75 masked (right panel) maps.

iii.1 Foregound-reduced maps

The WMAP team has released high angular resolution five-year 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 full-sky foreground-reduced 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 first-year ILC map is inappropriate for CMB scientific studies Bennett2003 (). However, in the five-year (also in the three-year and seven-year) 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 non-Gaussian tests on this version of the ILC map ILC-3yr-Hishaw (); ILC-5yr-Hishaw (). Notwithstanding the many merits of the five-year 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 HILC-Kim (), while in the second the foreground is reduced by using needlets as the basis of the cleaning process NILC-Delabrouille (). Thus, two new full-sky foreground-cleaned maps have been produced with the WMAP five-year data, namely the harmonic ILC (HILC) HILC-Kim () and the needlet ILC (NILC) (for more details see Refs. HILC-Kim (); NILC-Delabrouille ()).

In the next section, we use the full-sky foreground-reduced 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.111The 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.univ-paris7.fr/APC_CS/Recherche/Adamis/cmb_wmap-en.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 foreground-reduced maps, we have scanned the celestial sphere with spherical caps of aperture , centered at points homogeneously generated on the two-sphere by using the HEALPix code Gorski-et-al-2005 (). In other words, the point-centers 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 above-mentioned 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 foreground-reduced NILC full-sky and KQ75 maps. The panels of these figures clearly show regions with higher and lower values (’hot’ and ’cold’ spots) of and , which suggest large-angle multipole components of non-Gaussianity. 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 non-Gaussianity and maps obtained from the available full-sky foreground-reduced five-year 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 Monte-Carlo-generated statistically Gaussian CMB maps.222Each Monte-Carlo scrambled map is a stochastic realization of the WMAP best-fitting 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 foreground-reduced 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.

Figure 3: Differential power spectrum of skewness (left) and kurtosis (right) indicators calculated from the full-sky foreground-reduced ILC, HILC, and NILC maps obtained from the WMAP five-year data. The and confidence levels are indicated, respectively, by the dashed and dash-dotted lines.
    for     for
HILC        
ILC        
NILC        
Table 1: Results of the test to determine the goodness of fit for and multipole values calculated from the full-sky foreground-reduced HILC, ILC, and NILC maps as compared to the expected multipoles values from the Gaussian MC maps.

Figure 3 shows the differential power spectra calculated from full-sky five-year foreground-reduced 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 large-scale angular characteristics of the and maps. This figure shows a first indication of deviation from Gaussianity in five-year foreground-reduced 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 foreground-reduced map. This calculations are made in detail in the Appendix A. Thus, for example, we have for the full-sky 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 foreground-reduced maps. This indicates how unlikely are the occurrences of the values obtained from these foreground-reduced 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 foreground-reduced 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 full-sky 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.333We 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 non-Gaussian 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).

Figure 4: Differential power spectrum of skewness (left) and kurtosis (right) indicators calculated from the five-year foreground-reduced KQ75 masked ILC, HILC and NILC maps. The and confidence levels are indicated, respectively, by the dashed and dash-dotted lines.
    for [KQ75]     for [KQ75]
HILC        
ILC        
NILC        
Table 2: Results of the test to determine the goodness of fit for and multipole values calculated from the foreground-reduced HILC, ILC, and NILC maps with a KQ75 mask as compared to the expected multipoles values from the Gaussian MC masked maps.
    for [Kp0]     for [Kp0]
HILC        
ILC        
NILC        
Table 3: Results of the test to determine the goodness of fit for and multipole values calculated from the and maps obtained from the foreground-reduced HILC, ILC, and NILC maps with a Kp0 mask as compared to the expected multipoles values from the Gaussian MC masked maps.

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 foreground-reduced 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 foreground-reduced 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 full-sky 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 full-sky foreground-reduced 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 five-year full-sky foreground-reduced (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 full-sky foreground-reduced maps is modified if one uses the KQ75 mask, which was recommended by the WMAP team for tests of Gaussianity of the five-year band maps. Furthermore, the combination of the full-sky and mask analyses should provide information on the reliability of the foreground-reduced maps as appropriate reconstructions of the full-sky CMB.

Figure 4 shows the power spectra (left) and (right) calculated from five-year foreground-reduced KQ75 masked maps. This figure along with Fig. 3 show a significant reduction in the level of deviation from Gaussianity when the foreground-reduced 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.444Incidentally, 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 five-year 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 foreground-reduced maps as Gaussian reconstructions of the full-sky CMB, and makes apparent the relevant role of the mask KQ75 in reducing significantly the level of non-Gaussianity in these foreground-reduced maps.

The calculations of our non-Gaussianity 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 point-centers of the caps with values , and (see Ref. Bernui-Reboucas2009 () 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 full-sky 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 full-sky maps. We note, however, that full-sky foreground-reduced analysis does not change significantly if one uses smaller apertures as, for example, .

Iv Concluding remarks

The detection or nondetection of primordial non-Gaussianity 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 non-primordial effects that can also produce non-Gaussianity. This makes the extraction of a possible primordial non-Gaussianity a rather difficult endeavor. Since different indicators can in principle provide information about distinct forms of non-Gaussianity, it is important to test CMB data for non-Gaussianity 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 full-sky foreground-reduced map seems to be potentially more appropriate to test for Gaussianity the CMB data.555In reality, the full-sky 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 North-South asymmetry in CMB data, and signatures of a possible nontrivial cosmic topology, for example. The five-year version of the ILC map has been suggested as a full-sky map suitable for large angular scales analyses ILC-3yr-Hishaw (), even though the WMAP team has not performed a battery of non-Gaussianity tests on this map ILC-5yr-Hishaw ().

In this paper we have performed an analysis of Gaussianity of the available five-year full-sky foreground-reduced maps. To this end, we have used two new non-Gaussianity indicators based on skewness and kurtosis of large-angle patches of CMB maps, which provide a measure of departure from Gaussianity on large angular scales Bernui-Reboucas2009 (). We have shown that the full-sky five-year foreground-reduced maps (ILC, HILC and NILC) present a significant deviation from Gaussianity, which varies with the foreground-reducing procedures. We have established which of these full-sky foreground-reduced maps exhibit the highest and the lowest level of non-Gaussianity.

We have also masked the foreground-reduced maps with KQ75 and Kp0 masks and performed a quantitative analysis of deviation from Gaussianity of these maps. The comparison of the full-sky and masked analyses (see Fig. 3 and Fig. 4; and Tables 1, 2 and 3) shows a significant reduction in the levels of non-Gaussianity when the masks are employed, which in turn provides indications on the suitability of the foreground-reduced maps as Gaussian reconstructions of the full-sky 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 seven-year WMAP CMB data were released, including a new version of the full-sky foreground-reduced ILC map ILC-7yr-Gold (). We have considered this latest foreground-reduced ILC map, and performed a complete additional analysis of the Gaussianity of the five and seven-year versions of the ILC maps, whose details are given in Apenddix B.666Note that there is no available seven-year HILC and NILC maps. The main result of this appendix is that the full-sky seven-year foreground-reduced 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 seven-year WMAP CMB data.

Acknowledgements.
This work is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) – Brasil, under Grant No. 472436/2007-4. 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 Gorski-et-al-2005 ().

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 [full-sky]   HILC [full-sky]   ILC [full-sky]
           
           
           
           
           
           %
           
           
           
           
Table 4: The probability (percentage) of occurrence of the multipole values calculated from the data in the set {} of values computed from MC Gaussian CMB maps. The data from the five-year full-sky foreground-reduced NILC, HILC and ILC maps were used.
  NILC [KQ75]   HILC [KQ75]   ILC [KQ75]
           
           
           
           
           
           
           
           
           
           
Table 5: The probability (percentage) of occurrence of multipole values calculated from the data in the set {} of values computed from MC Gaussian CMB maps. The data from the five-year NILC, HILC and ILC KQ75 masked maps were used.
  NILC [full-sky]   HILC [full-sky]   ILC [full-sky]
           
           
           
           
           
           
           
           
           
           
Table 6: The probability (percentage) of occurrence of the multipole values calculated from the data in the set {} of values computed from MC Gaussian CMB maps. The data from the five-year full-sky foreground-reduced NILC, HILC and ILC maps were utilized.
  NILC [KQ75]   HILC [KQ75]   ILC [KQ75]
           
           
           
           
           
           
           
           
           
           
Table 7: The probability (percentage) of occurrence of the multipole value calculated from the data in the set {} of values computed from MC Gaussian CMB maps. The data from the five-year foreground-reduced NILC, HILC and ILC KQ75 masked maps were employed.

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 (full-sky 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 foreground-reduced CMB map. In Tables 4, 5, 6 and 7 we collect together the results of such calculations.

Figure 5: Differential power spectrum of skewness (left) and kurtosis (right) indicators calculated from the full-sky foreground-reduced ILC input map obtained from the WMAP five-year and seven-year data. The and confidence levels are indicated, respectively, by the dashed and dash-dotted lines.
    for [full-sky]     for [full-sky]
ILC5        
ILC7        
Table 8: Results of the test to determine the goodness of fit for and multipole values calculated from the foreground-reduced ILC5 and ILC7 full-sky maps as compared to the expected multipoles values from the MC Gaussian full-sky maps.

Thus, for example, from Table 4 we have for the full-sky 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 full-sky 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 foreground-reduced 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 full-sky foreground-reduced ILC map was released by the WMAP team ILC-7yr-Gold (). Since there is no available version of the NILC and HILC maps obtained from the seven-year WMAP data to be considered, here we present the results of a comparative analysis of deviation from Gaussianity performed by using the five and seven-year 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 full-sky five and seven-year foreground-reduced 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.

Figure 6: Differential power spectrum of skewness (left) and kurtosis (right) indicators calculated from the five-year and seven-year foreground-reduced ILC input maps with a KQ75 mask. The and confidence levels are indicated, respectively, by the dashed and dash-dotted lines.
    for [KQ75]     for [KQ75]
ILC5        
ILC7        
Table 9: Results of the test to determine the goodness of fit for and multipole values calculated from the foreground-reduced ILC5 and ILC7 maps with a KQ75 mask as compared to the expected multipoles values from the Gaussian MC masked maps.

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 small-scale differences are consistent with the pixel noise, but with a large-scale dipolar component, with the large-scale differences being consistent with a change in dipole of 6.7 KILC-7yr-Gold (). 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 five-year and seven year version of the foreground-reduced 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.

References

  • (1) V. Acquaviva, N. Bartolo, S. Matarrese, and A. Riotto, Nucl. Phys. B 667, 119 (2003); J. Maldacena, JHEP 0305 013 (2003); M. Liguori, F.K. Hansen, E. Komatsu, S. Matarrese, and A. Riotto, Phys. Rev. D73, 043505 (2006).
  • (2) N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto, Phys. Rept. 402, 103 (2004).
  • (3) E. Komatsu et al., arXiv:0902.4759v4 [astro-ph.CO].
  • (4) B.A. Bassett, S. Tsujikawa, and D. Wands, Rev. Mod. Phys. 78, 537 (2006); A. Linde, Lect. Notes Phys. 738, 1 (2008).
  • (5) K. Koyama, S. Mizuno, F. Vernizzi, and D. Wands, JCAP 11 (2007) 024; E.I. Buchbinder, J. Khoury and B.A. Ovrut, Phys. Rev. Lett. 100, 171302 (2008); J.-L. Lehners and P.J. Steinhardt, Phys. Rev. D77, (2008) 063533; Y.-F. Cai, W. Xue, R. Brandenberger, and X. Zhang, JCAP 05 (2009) 011; Y.-F. Cai, W. Xue, R. Brandenberger, and X. Zhang, JCAP 06 (2009) 037.
  • (6) L.-Y. Chiang, P.D. Naselsky, O.V. Verkhodanov, and M.J. Way, Astrophys. J. 590, L65 (2003).
  • (7) P.D. Naselsky, L.-Y. Chiang, I.D. Novikov, and O.V. Verkhodanov, Int. J. Mod. Phys. D 14, 1273 (2005).
  • (8) P. Cabella, D. Pietrobon, M. Veneziani, A. Balbi, R. Crittenden, G. de Gasperis, C. Quercellini, and N. Vittorio, arXiv:0910.4362.
  • (9) E. Komatsu et al., Astrophys. J. Suppl. 148, 119 (2003); D.N. Spergel et al., Astrophys. J. Suppl. 170, 377 (2007); E. Komatsu et al., Astrophys. J. Suppl. 180, 330 (2009); P. Vielva, E. Martínez-González, R.B. Barreiro, J.L. Sanz, and L. Cayón, Astrophys. J. 609, 22 (2004); M. Cruz, E. Martínez-González, P. Vielva, and L. Cayón, Mon. Not. R. Astron. Soc. 356, 29 (2005); M. Cruz, L. Cayón, E. Martínez-González, P. Vielva, and J. Jin, Astrophys. J. 655, 11 (2007); L. Cayón, J. Jin, and A. Treaster, Mon. Not. R. Astron. Soc. 362, 826 (2005); Lung-Y Chiang, P.D. Naselsky, Int. J. Mod. Phys. D 15, 1283 (2006); J.D. McEwen, M.P. Hobson, A.N. Lasenby, and D.J. Mortlock, Mon. Not. R. Astron. Soc. 371, L50 (2006); J.D. McEwen, M.P. Hobson, A.N. Lasenby, and D.J. Mortlock, Mon. Not. R. Astron. Soc. 388, 659 (2008); A. Bernui, C. Tsallis, and T. Villela, Europhys. Lett. 78, 19001 (2007); L.-Y. Chiang, P.D. Naselsky, and P. Coles, Astrophys. J. 664, 8 (2007); C.-G. Park, Mon. Not. R. Astron. Soc. 349, 313 (2004); H.K. Eriksen, D.I. Novikov, P.B. Lilje, A.J. Banday, and K.M. Górski, Astrophys. J. 612, 64 (2004); M. Cruz, M. Tucci, E. Martínez-González, and P. Vielva, Mon. Not. R. Astron. Soc. 369, 57 (2006); M. Cruz, N. Turok, P. Vielva, E. Martínez-González, and M. Hobson, Science 318, 1612 (2007); P. Mukherjee and Y. Wang, Astrophys. J. 613, 51 (2004); D. Pietrobon, P. Cabella, A. Balbi, G. de Gasperis, and N. Vittorio, Mon. Not. Roy. Astron. Soc. 396, 1682 (2009); D. Pietrobon, P. Cabella, A. Balbi, R. Crittenden, G. de Gasperis, and N. Vittorio, Mon. Not. Roy. Astron. Soc. 402, L34 (2010); Y. Ayaita, M. Weber, and C. Wetterich, Phys. Rev. D81, 023507 (2010); P. Vielva and J.L. Sanz, Mon. Not. Roy. Astron. Soc. 397, 837 (2009); B. Lew, JCAP 08 (2008) 017; A. Bernui and M.J. Rebouças, Int. J. Mod. Phys. A 24, 1664 (2009); M. Kawasaki, K. Nakayama, T. Sekiguchi, T. Suyama, and F. Takahashi, JCAP 11 (2008) 019 ; M. Kawasaki, K. Nakayama, and F. Takahashi, JCAP 01 (2009) 026; M. Kawasaki, K. Nakayama, T. Sekiguchi, T. Suyama, and F. Takahashi, JCAP 01 (2009) 042; M. Cruz, E. Martínez-González, and P. Vielva, arXiv:0901.1986 [astro-ph].
  • (10) E. Martínez-González, arXiv:0805.4157 [astro-ph]; Y. Wiaux, P. Vielva, E. Martínez-González, and P. Vandergheynst, Phys. Rev. Lett. 96, 151303 (2006); L.R. Abramo, A. Bernui, I.S. Ferreira, T. Villela, and C.A. Wuensche, Phys. Rev. D74, 063506 (2006); P. Vielva, Y. Wiaux, E. Martínez-González, and P. Vandergheynst, New Astron. Rev. 50, 880 (2006); P. Vielva, Y. Wiaux, E. Martínez-González, and P. Vandergheynst, Mon. Not. R. Astron. Soc., 381, 932 (2007); C.J. Copi, D. Huterer, D.J. Schwarz, and G.D. Starkman, Phys. Rev. D75, 023507 (2007); C.J. Copi , D. Huterer, D.J. Schwarz, and G.D. Starkman, Mon. Not. R. Astron. Soc. 399, 295 (2009); K. Land and J. Magueijo, Mon. Not. R. Astron. Soc. 378, 153 (2007); B. Lew, JCAP 09 (2008) 023; A. Bernui, Phys. Rev. D78, 063531 (2008); A. Bernui, Phys. Rev. D80, 123010 (2009); P.K. Samal, R. Saha, P. Jain, and J.P. Ralston, Mon. Not. R. Astron. Soc. 385, 1718 (2008); P.K. Samal, R. Saha, P. Jain, and J.P. Ralston, Mon. Not. R. Astron. Soc. 396, 511 (2009); A. Bernui, B. Mota, M.J. Rebouças, and R. Tavakol, Astron. & Astrophys. 464, 479 (2007); A. Bernui, B. Mota, M.J. Rebouças, and R. Tavakol, Int. J. Mod. Phys. D 16, 411 (2007); T. Kahniashvili, G. Lavrelashvili, and B. Ratra, Phys. Rev. D78, 063012 (2008); L.R. Abramo, A. Bernui, and T.S. Pereira, JCAP 12 (2009) 013; F.K. Hansen, A.J. Banday, K.M. Górski, H.K. Eriksen, and P.B. Lilje, Astrophys. J. 704, 1448 (2009).
  • (11) A. Bernui and M.J. Rebouças, Phys. Rev. D79, 063528 (2009).
  • (12) G. Hinshaw et al., Astrophys. J. Suppl. Ser. 180, 225 (2009).
  • (13) J. Kim, P. Naselsky, and P.R. Christensen, Phys. Rev. D77, 103002 (2008).
  • (14) J. Delabrouille et al., Astron. & Astrophys. 493, 835 (2009).
  • (15) B. Gold et al., arXiv:1001.4555 [astro-ph.GA].
  • (16) C.L. Bennett et al. Astrophys. J. Suppl. Ser. 148, 1 (2003).
  • (17) G. Hinshaw et al. Astrophys. J. Suppl. Ser. 170, 288 (2007).
  • (18) K.M. Górski, E. Hivon, A.J. Banday, B.D. Wandelt, F.K. Hansen, M. Reinecke, and M. Bartelman, Astrophys. J. 622, 759 (2005).
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