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Search for non-Gaussianity in pixel, harmonic and wavelet space: compared and combined

We present a comparison between three approaches to test non-Gaussianity of cosmic microwave background data. The Minkowski functionals, the empirical process method and the skewness of wavelet coefficients are applied to maps generated from non-standard inflationary models and to Gaussian maps with point sources included. We discuss the different power of the pixel, harmonic and wavelet space methods on these simulated almost full-sky data (with Planck like noise). We also suggest a new procedure consisting of a combination of statistics in pixel, harmonic and wavelet space.


I. INTRODUCTION
The fluctuations of the cosmic microwave background (CMB) are expected to be close to Gaussian distributed. In view of the increasing quantity of CMB experiments, it is now possible to check this assumption on data with growing resolution and sky coverage.
Most models for the early universe predict some small deviations from Gaussianity; non-standard models of inflation [1, 2, 3,4,5,6,7], cosmic strings (See Ref. [8] for a review) and point sources. Detecting these small deviations would be of great importance for the understanding of the physics of the early universe. Also systematic effects like a non-symmetric beam and noise could give rise to non-Gaussian features. For this reason a non-Gaussianity check could reveal whether the impact of the instrumental effects on the data of the experiment is well understood.
The methods to search for non-Gaussianity in the literature mainly concentrate on implementing the test in three different spaces: (1) In pixel space: the Minkowski functionals [9,10] (which were used to set limits on the non-Gaussianity in the WMAP data [11]), temperature correlation functions [12], the peak to peak correlation function [13], skewness and kurtosis of the temperature field [14] and curvature properties [15,16], to mention a few. (2) In harmonic space: analysis of the bispectrum and its normalized version [17,18,19,20] and the bispectrum in the flat sky approximation [21]. The explicit form of the trispectrum for CMB data was derived in [22,23]. Phase mapping [24]. Applications to COBE, Maxima and Boomerang data have also drawn enormous attention and raised wide debate [25,26,27,28]. The empirical process method [29,30,31]. Finally, (3) wavelet space: [32,33,34,35]. Traditionally, these tests are performed separately in each space.
In this article, we will take methods in pixel-(the Minkowski functionals), harmonic-(the empirical process) and wavelet-space (skewness), and we will make a comparison for two different models of non-Gaussianity. We will also combine the methods in order to improve the total power. It should be noted that all the procedures we consider are non-parametric, that is they do not assume any a priori knowledge about the nature of non-Gaussianity.
We will use these methods on 100 maps generated from a non-standard inflationary model [6] and on 100 maps where we have included point sources. We assess the performance of the methods in the different spaces for the different types of non-Gaussianity. We also propose a combined test which turns out to be more robust.
In section II, we review the method of Minkowski functionals, in section III we describe our implementation of the method and in section IV we define our proposed statistic. In section V, we review the empirical process method while section VI is devoted to the wavelets. In section VII the methods are compared and applied to non-standard inflationary models, in section VIII to maps with point sources. Finally in section IX we summarize and comment on our results.

II. MINKOWSKI FUNCTIONALS
To analyze a spherical map in terms of Minkowski functionals, we consider the excursion sets, that is, the map subsets which exceed a given threshold value. The threshold is labelled ν, and it is treated as an independent variable, on which these functionals depend.
More precisely, considering the normalized random field of temperature fluctuations, u = ∆T /σ(∆T ); we can define the 'hot region' Q as the ensemble of pixels u i higher than the ν level : (1) The three functionals of interest then are, up to constant factors [36]: 1) Area: M 0 (ν) is the total area of all hot regions.
2) Boundary length: M 1 (ν) is proportional to the total length of the boundary between cold and hot regions 3) Euler characteristic or genus: M 2 (ν), a purely topological quantity, counts the number of isolated hot regions minus the number of isolated cold regions, i.e. the number of connected components in Q minus the number of 'holes'.
The rationale behind these statistics can be explained from mathematical results in Hadwiger (1959); in particular, these results can be interpreted by stating that all the morphological information of a convex body is contained in the Minkowski functionals (Winitzki and Kosowsky, 1997)); here, by morphological we mean the properties which are invariant under translations and rotations and which are additive [37,38,39]. The three statistics, normalized by the area density, can then be expressed as where ∂Q is the contour of the region Q; dA and dl are the differential elements of Q and ∂Q, respectively; κ is the geodetic curvature of dl .
The expected values for a given thresholds depends on a single parameter τ given for a Gaussian field by [39]: with : where semicolon indicates the covariant derivative on the sphere.
In the case of CMB (8) reduces to [38]: where C ℓ is the angular power spectrum.
An immediate consequence of the above formulae is that, although the expected value of the first Minkowski functional is invariant with respect to the dependence structure of ∆T , for the second and third Minkowski functional this is not the case and calibration for a given angular power spectrum C ℓ is needed. Moreover, even for the first Minkowski functional, knowledge of the angular power spectrum is required for a Monte Carlo evaluation of its variance. This can be viewed as a drawback, and because of this some effort has been undertaken to provide at least some crude upper bound for the functionals' variance (Winitzki and Kosowsky (1997)).

III. THE IMPLEMENTATION OF MINKOWSKI FUNCTIONALS
In order to estimate the three functionals we simulate a map of the CMB [40] with a known power spectrum, and then we cut the maximum number of independent tangent planes of dimension ≃ 12 • × 12 • ; in this way it is easy to calculate the values of the three Minkowski functionals in the flat-sky limit, taking into account the possibility of gaps (galactic cut, polar calottes). Due to projection effects, finite pixel size and the dimension of the tangent planes, we find a deviation of the simulated values with respect to the analytical spherical predictions (eqs.5). Note however that the shape of the curves is unaffected, which is not the case when non-Gaussianities are present (see fig. 2). In fig. 1 we show a comparison between analytical expectation values and the values computed on the tangent planes.

IV. TEST OF NON-GAUSSIANITY IN PIXEL SPACE
In order to test non-Gaussianity we use a test defined by: Our procedure is as follows; • Given an observed map, we estimate the power spectrum.
• Using the estimated power spectrum, we generate 200 Gaussian realizations [40] • For each map we calculate the Minkowski functionals, using the tangent planes as

V. THE EMPIRICAL PROCESS METHOD
The details of the empirical process approach to detect non-Gaussianity in the CMB were given in [29,30,31]. In short, the method consists of a family of tests which focus on the total distribution of a ℓm and check for dependencies between k ℓ-rows. The first step is to transform the spherical harmonic coefficients into variables u ℓm which have an approximate uniform distribution between 0 and 1, given that the a ℓm were initially Gaussian distributed. This is done using the Smirnov transformation, defined as where Φ n is the cumulative distribution function of a χ 2 with n degrees of freedom and C ℓ are the power spectrum coefficients estimated from the data. The error introduced by using estimatedĈ ℓ instead of the real underlying C ℓ is dealt with using a bias-subtraction, as described in [29]. Then the joint empirical distribution function for row ℓ is formed, where ∆ ℓ,i determines the spacing between the rows for which the dependencies are tested and ∆ mi denotes the difference in m for row i. The parameters α i run over the interval [0, 1]. The empirical process is expressed using the centered and rescaled F ℓ...ℓ+∆ ℓ,k−1 given as The intuition behind this procedure is as follows: if the a ℓm s are Gaussian, G converges to a well-defined limiting process, whose distribution can be readily tabulated. On the other hand, for non-Gaussian a ℓm s { F (α 1 , ..., α k ) − k i=1 α i } and thereby G will take 'high' values over some parts of α-space. Thus, the analysis of some appropriate functional of G can be used to detect non-Gaussianity. To combine the information over all multipoles into one statistic, we define where L is the highest multipole where the data is signal dominated.
The method can then be summarized as follows: the distribution of sup| K L | is found using Monte-Carlo simulations of Gaussian distributed a ℓm . Then, for a given observed set of a ℓm , the value k max = sup| K L | is found and compared to the distribution obtained from Monte-Carlo. The consistency of the data with a Gaussian distribution can then be estimated to any suitable σ-level. In [30] this simple approach was extended in three different ways. First of all, the fact that the above explained estimator is not rotationally invariant is exploited, using the k max value averaged over many rotations. Each rotation can be viewed as a resampling of the a ℓm . Secondly, we introduced three variations of the test, taking into account, not only the modulus of the a ℓm but also the phases. Finally, experimental effects like noise and galactic cut was accounted for using Monte Carlo calibration of the k max distribution with these effects included. It should be noted that the rotated maps are clearly dependent and the resulting statistic may thus depend on the shape of the angular power spectrum.

VI. TEST OF NON-GAUSSIANITY IN WAVELET SPACE
A third space where one could look for non-Gaussianity is the wavelet space. The use of wavelets for non-Gaussianity tests of the CMB has been investigated by several authors [32,33,34,35] and turns out to be a very powerful tool. We will here just briefly describe the wavelet method, and refer to the above references for more details.
An isotropic wavelet can be defined as with the properties where x = | x|, R represents a scale and b a translation. The Fourier transform of the wavelet is represented by ψ(ω). We will focus on the mexican hat wavelets given by: From the wavelet transform of a function f ( x) one can obtain the wavelet coefficients : and if f ( x) is Gaussian, w( b, R) will be Gaussian as well. We will here use the CMB temperature fluctuation field as the f ( x) function. We will implement the non-Gaussianity test in wavelet space as we did for the Minkowski functionals: • we generate a set of Gaussian CMB maps for calibration and a set of non-Gaussian maps for testing • we cut tangent planes (12 • × 12 • ) • we calculate for each plane the coefficients w( b, R) • we evaluate the skewness of the wavelet coefficients for each sky.
• finally, using the skewness of the wavelets from the Gaussian maps, we define the one and two σ detection levels as described above for the other methods.

VII. COMPARISON AND COMBINED TEST
This section aims at comparing the different methods described above. Applying the methods on the same maps , we will first compare the number of detections. We will in this paper use the non-standard inflationary model described in [6,7] which has the non-linear coupling parameter f N L as a measure of the strength of non-Gaussianity. We generated 100 'observed' skies with Planck-like noise (LFI 100 GHz), beam 20', pixel-size ≃ 6' (Nside 512 in Healpix language), using a pure Sachs-Wolfe spectrum with f N L values of 300 and 100.
In figures 2, 4 and 5 one can see the behavior of the Minkowski functionals in the presence of a non-zero f N L . In tables I and II we show the rejection rates. As the first Minkowski functional was giving the best results in this tests, we will focus only on M 0 for this kind of non-Gaussianity. In table III we list the results of the wavelet test on the same maps (for the wavelets we used the parameter R = 22.5 ′ ). For individual results of the empirical process test, we refer to [30].      For the combined test we suggest to use an indicator consisting of I 0 from the Minkowski functionals and k max from the empirical process. We chose to normalize the I 0 and k max so that they both have mean zero and variance one, using Monte-Carlo simulations of Gaussian maps. In this way, the two values can be averaged: whereĨ Here means mean value taken over 100 Gaussian simulations. The weights w 1 and w 2 were chosen proportionally to the power of each procedure. Using the rejection rates for f N L = 100 in table II, we arrived at w 1 ≈ 0.6 and w 2 ≈ 0.4. Of course, the threshold value for x needs to be evaluated anew. In figure (6) we plot the distribution of x for the Gaussian and f N L = 100 non-Gaussian maps. By a similar motivation, it is natural to combine also the wavelet method into a single procedure; it is easy to see (tables II and III) that the detection rate at 2σ is about two times higher for the wavelets, warranting the wavelet coefficient a very high weight in the combined analysis. Inspecting table IV, we detect a moderate improvement in the detection rate when combining Minkowski functionals and empirical process. The result of the empirical process + Minkowski functionals + wavelets combined test is shown in table V. A significant improvement of the number of detections at both confidence levels is evident; note that the combined procedure is to some extent model dependent, as the weights we used were tabulated from specific non-Gaussian models.

VIII. POINT SOURCES
We also simulated maps with point sources (with noise and beam as given above) to compare the power of the methods on a different kind of non-Gaussianity. We generated a toy model of point sources with a distribution building on formula (1) in [41] and formulae (1) and (2) in [42]. In table VI we show the results for the Minkowski functionals. We see immediately that for this kind of non-Gaussianity, the first Minkowski functional is not sensitive, whereas the other two functionals show a good rejection rate. This suggests that we might be able to discriminate between these two types of non-Gaussianity. The first  [43]. However as seen in figure (5) for non-standard inflation the curve has a particular shape.
One could imagine combining M 1 and M 2 in order to strengthen the power of the test, similarly to what we have done above. However, it turns out that the maps detected by M 2 are contained within the maps detected by M 1 , so that there is no additional information in combining the two estimators.
For the empirical process method, there was no detection for any of the tests, univariate, bivariate or trivariate. The numbers k max obtained for the maps with point sources were consistent with those for Gaussian maps. Also for the wavelet test, the number of detections was very small. Note that wavelets can be useful for detection of bright point sources [44], but in our source model these were excluded. This might suggest that the empirical process methods and the skewness of the wavelets can be used to probe primordial non-Gaussianity without confusion from point sources, making the combined test presented above more robust.

IX. COMMENTS AND CONCLUSIONS
We have compared three methods for detecting non-Gaussianity in observations of the cosmic microwave background. The methods were applied to non-Gaussian maps with two different kinds of non-Gaussianity, primordial non-Gaussianity with a varying f N L and point sources. It is important to note that the non-Gaussian maps used in this article were generated taking into account only the Sachs-Wolfe effect. In future work we will study maps where the full radiative transfer equations have been applied. For the time being, we stress that our results are broadly consistent with the power of the procedures adopted for WMAP data analysis. More precisely, a detailed comparison is unfeasible, as we are assuming a simplified non-Gaussian model (no radiative transfer) and Planck LFI like noise and beam. Moreover we are restricting the analysis to the first 500 multipoles . Broadly speaking, however, Monte-Carlo simulations suggest that a value of f N L about 100 represents the lower limit that can be detected at a 2σ level, by using a combined procedure: this seems consistent with the value f N L = 139 reported in ( [11]). As an estimator of non-Gaussianity for the Minkowski functionals we have introduced a statistic which gathers the information from different thresholds (eq.10). The estimator for the empirical process method is k max , the maximum value of the function K(α, r) obtained from a given map (eq.11) using the trivariate test. For the test in wavelet space, we use the skewness of wavelet coefficients. carrying complementary statistical information, should not be viewed neither as orthogonal nor as independent, so that some care is needed when merging them into a single statistic.
In any case, the fact that different methods detect different kinds of non-Gaussianity can be viewed as an advantage, in the sense that, for instance, primordial non-Gaussianity can be detected without confusion from point sources.