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In the recent years, non-Gaussianity and statistical isotropy of the Cosmic Microwave Background
(CMB) was investigated with various statistical measures, first and foremost by means of the measurements
of the WMAP satellite. In this paper, we focus on the analyses that were accomplished
with a measure of local type, the so-called

The inflationary phase, first proposed in 1981, is an important part of what is called the Standard Model of Cosmology. Since inflation occurred already a few moments after the Big Bang, where the Universe was extremely hot, dense and thus opaque, it is not possible to observe this short-time period directly. The best way to achieve information about and to test theories for inflation is to look at the temperature anisotropies of the Cosmic Microwave Background (CMB). While the simplest single-field slow-roll inflationary scenario predicts these fluctuations to be nearly Gaussian [

In recent years, the observations of the CMB accomplished by the WMAP satellite offered researchers the possibility to analyse high-resolution full-sky data maps of this relic radiation. A multiplicity of analyses addressed the challenge of non-Gaussianity, thereby applying several different statistical measures as, for example (but far from complete), the angular bispectrum [

In this

This

Quite similar to wavelets, weighted scaling indices can be used to perform a

Scaling indices investigate the spatial distribution of a previously prepared d-dimensional data set. In CMB investigations, however, the fluctuations of the temperature maps are characterised by the values of the pixelised sky on a sphere. To be able to apply an analysis by means of scaling indices, one has to combine the temperature information with the two-dimensional spatial information of the map to create a three-dimensional point set, which includes all the information of the original map as spatial information only. This can be done by performing a preprocessing step, namely, a transformation of the pixelised spherical sky

WMAP 3-year data after application of the transformation into a three-dimensional point distribution. In the first and the third plots, the full set of points is presented, while the second and fourth show an

After this preprocessing step, the actual scaling index technique can be applied. In general, the SIM is a mapping that calculates for every point

A simulated CMB map, in which the central regions were masked out and filled with nearly white noise, whereby the spatial noise patterns are preserved (see Section

From (

In the publications concerning scaling index analysis with CMB to date, different WMAP data sets as well as different techniques to handle foreground contaminated regions were applied. To test for the amount of non-Gaussianity in the WMAP data, simulations based on Gaussian random fields were constructed, which is the most common way in CMB analysis. In addition, the so-called surrogate maps were generated. With the help of these surrogates, it is possible to test for the more specific hypothesis of uncorrelated phases. The surrogate method also offers the possibility to analyse the data in a scale-dependent but model-independent way. In the following, we will give an overview of the used sets of maps.

In the current fourth data release of the WMAP team (

The 7-year foreground-cleaned internal linear combination (ILC) map [

For comparison, we also included the map produced in [

Unlike to the two ILC maps from above, the single Q-, V-, and W-bands of the WMAP satellite as well as a coadded VW-map can shed light on the influence of the different wavelength depending foregrounds onto the CMB signal. Although we work with the maps that are reduced by means of the Foreground Template Model proposed in [

As for the ILC maps above, we decrease the resolution to

The two plots on the left-hand side illustrate the original 5-year WMAP data of the coadded VW-band (above) and the related colour-coded

Just cutting out the masked regions like above spoils the results of the scaling index method. Instead of a more or less uniform distribution, the

At first, we fill the masked regions with Gaussian noise, whose standard deviation for each pixel corresponds to the pixel noise made available by the WMAP team:

A simple approach to evaluate the amount of non-Gaussianity in the WMAP data is to compare the measured data with maps that fulfil the Gaussian hypothesis. For the band-wise analysis, it is important to create simulations for each respective band. The proceeding is hereby as follows. We take the best fit

A comparison with simulated CMB maps represents the most obvious and common approach to search for non-Gaussianities in the data set. However, it is also possible to create maps, the so-called surrogates, that are similar to the original map except for one (or more) previously selected feature(s) which is (are) randomised. By comparing the data with this set of maps, one focuses on the deviations caused by the randomisation of these feature(s). The whole proceeding is therefore

Consider a CMB map

However, the Gaussian shape of the histogram of the temperature distribution and the randomness of the set of Fourier phases in the sense that they are uniformly distributed in the interval

To ensure the randomness of the set of Fourier phases, we performed a rank-ordered remapping of the phases onto a set of uniformly distributed ones followed by an inverse Fourier transformation. These two preprocessing steps only have marginal influence to the maps (see Figure

The ILC map in its original form (a) and after remapping of the temperatures and phases (b). First-order (c) and respective second-order surrogate (d) for

At first, one generates a first-order surrogate map, in which any phase correlations for the scales, which are not of interest, are randomised. This is achieved by a random shuffle of the phases

In a second step, a chosen number of realisations of second-order surrogate maps are generated for the first order surrogate map, in which the remaining phases

In [

Besides this two-step procedure aiming at a dedicated scale-dependent search for non-Gaussianity, one can also test for non-Gaussianity using surrogate maps without specifying certain scales. In this case, there are no scales, which are not of interest, and the first step in the surrogate map making procedure becomes dispensable. The zeroth order surrogate map is to be considered here as first-order surrogate and the second-order surrogates are generated by shuffling all phases with

Finally, for calculating scaling indices to test for higher-order correlations, the surrogate maps were degraded to

Finding differences between observed and simulated CMB maps which fulfil the Gaussian hypothesis of the best fitting

Probability density

These effects can more exactly be quantified by calculating the mean and standard deviation for the distribution of scaling indices as calculated for different scaling ranges. For scales larger than

Besides the mean and standard deviation, we additionally considered a combination of these two test statistics, namely, a diagonal

Deviations of the (combined) moments of the

For the standard deviation, we find slightly different results. In a transition regime

Some readers might argue that the selection of certain moments (mean, standard deviation,

There is an ongoing discussion, whether a diagonal

If we select only those pixels for

The empirical probability densities

The probability densities

We also calculated the

Evidence for north-south asymmetry in the WMAP data was already detected using the angular power spectrum [

Thus, the colour of each pixel in the corresponding Figure

The

While the Q-band is heavily foreground-affected, first of all by synchrotron radiation as well as radiation from electron-ion scattering (free-free emission), the W-band is mainly distorted by dust emission. The V-band is affected by all three of these foregrounds, even though less than the other bands. Despite the different influences on the different bands, we obtain the same signatures of non-Gaussianity in all single bands as well as in the coadded VW-band. The correlations

An interesting anomaly in the CMB data is that there are small regions which show very high or very low values in some local structure analysis. One of the first of these local features, the well-known

For our investigations concerning spots in the WMAP data, we use the mask-filling method of Section

The pixel-wise deviations

The second approach smoothes the

The first approach clearly shows the Cold Spot and indicates some secondary spots in the southern as well as in the northern hemisphere. These get confirmed in the plot of the smoothing method, where we obtain a deviation of up to

If the considered spots really depend on some yet not completely understood, maybe secondary, physical effect, they should not be implemented in a testing for intrinsic non-Gaussianity. For this reason, we modify the 5-year KQ75-mask by additionally excluding all above-mentioned spots. A small peculiarity at the edge of the mask next to the Cold Spot as well as three very small blurs in the right half of the lower left Mollweide projection in Figure

We now apply this new mask to the

The

We compare the first- and second-order surrogate maps by calculating the

Deviations

First, various deviations representing features of non-Gaussianity and asymmetries can be found in the

Same as Figure

Second, we find for the scale-independent surrogate test (top rows in Figures

Third, for the scale-dependent analysis, we obtain for the largest scales (

Fourth, we also find for the smallest considered scales (

Fifth, we do not observe very significant anomalies for the two other bands (

Figure

The probability densities

The density distributions derived from the ILC7 and NILC5 map are clearly shifted against each other. The differences between these two maps can be attributed to, for example, the smoothing of the ILC7 map. However, the systematic differences between first- and second-order surrogates induced by the phase manipulations prevail in all cases, irrespective of the input map.

The results for the deviations

Deviations

Same as Figure

To test whether all these signatures are of intrinsic cosmic origin or more likely due to foregrounds or systematics induced by, for example, asymmetric beams or map making, we performed the same surrogate and scaling indices analysis for five additional maps described in [

In this

By comparing the 3-year and 5-year measurements of the WMAP satellite with simulated CMB maps, several clear non-Gaussianities as well as asymmetries were detected.

The spectrum of scaling indices of the data is systematically broader and shifted towards higher values than the one of the simulations, yielding highly significant deviations of the mean, the standard deviation, and a

All these results are consistent in different ways. Since the detected effects are the same for the 3-year as well as for the 5-year WMAP data, they can be concluded to be time-independent. In addition, the findings are nearly the same for the different bands that were analysed for the 5-year data, which leads to the conclusion that the foreground influence only plays a minor role. Furthermore, the usage of the mask-filling method, again applied to the 5-year data only, reduces the distorting influence of the mask. Since this leads to similar results as well, the detected deviations from Gaussianity and statistical isotropy must therefore be taken to be of cosmological origin so far.

In addition to these findings, several local features including the Cold Spot could be detected with the scaling indices, which turns out to be another advantage of this method. The fact that most of them are located in the southern hemisphere confirms the conclusions concerning the asymmetries from above. Nearly all detected spots are in agreement with former analyses (e.g., [

By accomplishing a comparison of the CMB data with surrogate maps, one focuses on the more specific assumption of random and uncorrelated phases, which is part of the Gaussian hypothesis. In addition, this method offers the possibility of a scale-dependent analysis. The scaling indices are the first measure which is used in combination with this surrogates approach. For an analysis of the 5- and 7-year observations of the WMAP satellite, the results are as follows.

Highly significant non-Gaussianities could be detected, again by performing an analysis of rotated hemispheres, for the very large scales and for the

For smaller scales (i.e., higher

The SIM is the only measure in CMB analysis that was used in combination with the surrogates technique so far. Further studies that combine the surrogates method with different measures, as, for example, Minkowski functionals, could support these investigations and produce even more reliable results. In addition, the upcoming data of the PLANCK satellite offers an independent measurement of the CMB and will allow investigations concerning higher

Many of the results in this paper have been derived using the HEALPix [