A&A 395, 409415 (2002)
DOI: 10.1051/00046361:20021264
H. K. Eriksen^{1}  A. J. Banday^{2}  K. M. Górski^{3,4}
1  Institute of Theoretical Astrophysics, University of Oslo, PO Box
1029 Blindern, 0315, Norway
2 
MaxPlanckInstitut für Astrophysik, Garching bei München, Germany
3 
European Southern Observatory, Garching bei München, Germany
4 
Warsaw University Observatory, Aleje Ujazdowskie 4, 00478 Warszawa, Poland
Recieved 19 June 2002 / Accepted 28 August 2002
Abstract
We calculate the two, three and (for the first time)
fourpoint correlation functions of the COBEDMR 4year sky
maps, and search for evidence of nonGaussianity by comparing the
data to Monte Carlosimulations of the functions. The analysis is performed for
the 53 and 90 GHz channels, and five linear combinations thereof.
For each map, we simulate an ensemble of 10 000 Gaussian realizations
based on an a priori bestfit scaleinvariant cosmological power spectrum,
the DMR beam pattern and instrumentspecific noise properties.
Each observed COBEDMR map is compared to the ensemble using a
simple
statistic, itself calibrated by simulations.
In addition, under the assumption of Gaussian fluctuations, we find explicit
expressions for the expected values of the fourpoint functions in
terms of combinations of products of the twopoint functions,
then compare the observed fourpoint statistics to those predicted by
the observed twopoint function, using a redefined
statistic.
Both tests accept the hypothesis that the DMR maps are consistent
with Gaussian initial perturbations.
Key words: cosmic microwave background  cosmology: observations  methods: statistical
The study of CMB temperature anisotropies and their statistical properties has become an important theme in modern cosmology. In its most conventional interpretation, the distribution of anisotropies reflects the properties of the universe approximately 300 000 years after the Big Bang, at the surface of last scattering. Thus, by measuring statistical quantities such as the angular power spectrum or the angular twopoint correlation function, we can infer the values of many interesting cosmological parameters.
For both theoretical and practical purposes, it is convenient to
expand the temperature anisotropy field into a sum of (complex)
spherical harmonics:
(1) 
However, testing for nonGaussianity is anything but trivial, and several qualitatively different tests are required in order to perform a complete analysis. At present the arsenal of available tests which have been applied to the COBEDMR data consists of at least the following: bi and trispectrum based analysis (Ferreira et al. 1998; Magueijo 2000; Sandvik & Magueijo 2001; Komatsu et al. 2002; Kunz et al. 2001), 3point correlation function based tests (Kogut et al. 1996), methods utilizing wavelets (Cayón et al. 2001; Barreiro et al. 2000) and Minkowski functionals (Schmalzing & Górski 1998; Novikov et al. 2000). Indeed, there has been a small resurgence in interest in the possibility of nonGaussian signals in the COBEDMR maps as a consequence of the bispectrum work of Ferreira et al. (1998) and Magueijo (2000). These papers find nonGaussian contributions using harmonic analyses at the 98% confidence limit, and although Banday et al. (2000) explain these tentative detections by appealing to the presence of a specific residual systematic artifact in the data, additional investigation is warranted.
In this paper we adopt Npoint correlation functions as probes for nonGaussianity. For a Gaussian field all odd Npoint functions (such as the threepoint function) have vanishing expectation values, while all even Npoint functions can be reduced to expressions involving the twopoint function. Thus, if the observed threepoint function is significantly nonzero when compared to a Gaussian ensemble, its native distribution is probably nonGaussian. Further, if the fourpoint function does not reduce into twopoint functions, the same conclusion can be made.
The first part of this paper builds on ideas demonstrated in Kogut et al. (1996) and Hinshaw et al. (1995). We study the 4year COBEDMR sky maps, computing the two and threepoint functions, as has been performed previously, then proceeding to extend the analysis for the first time to the determination of several fourpoint functions. The definitions of these new functions are given in Sect. 2.
In Sect. 3 we compute the various correlation functions for the four DMR channels and five linear combinations thereof. Next, we compute the same functions for 10 000 Monte Carlo simulated Gaussian maps, which are used as the basis of the various statistical tests of nonGaussianity. Initially, we apply the test as defined by Kogut et al. (1996), by comparing the observed data value to the distribution generated from the application of the statistic for each map in the simulated ensemble.
Subsequently, in Sect. 5, we provide expressions for the expected value of several fourpoint functions in terms of the twopoint function, then explicitly compare the observed fourpoint functions to those predicted by the observed twopoint function. We define a suitable statistic in order to quantitatively measure the degree of deviation, once again calibrated by Monte Carlosimulations.
An Npoint correlation function is defined as the average product of
N temperatures with a fixed relative orientation on the sky:
(2) 
Although the Npoint functions are easily defined and relatively simple to implement computationally, their evaluation is generally CPUintensive, which is especially problematic since detailed assessment of results requires large Monte Carlo simulation data sets. The full computation of an Npoint correlation function scales as , and is therefore virtually impossible to compute for highresolution maps for any order N greater than two. For this reason we choose to compute only a subset of the possibilities in the Ndimensional configuration space, designed to reduce the complexity of the problem. As an example consider the pseudocollapsed threepoint function for which we require two points to coincide, effectively reducing the geometry to that of the twopoint function. Such subsets typically scale somewhere between and . Thus, with some effort put into the implementation these functions can be computed even for rather highresolution maps.
Previous work has considered two special threepoint functions, namely the collapsed and the equilateral functions (Kogut et al. 1996; Hinshaw et al. 1995). As mentioned above, the collapsed function is defined by requiring two of the three point to coincide, while the equilateral function requires the three points to span an equilateral triangle on the sphere.
In this paper, we shall also consider several simple fourpoint configurations. These functions are, in order of complexity:
Several of the functions defined above are socalled collapsed functions, i.e. one pixel is multiplied one or more times with itself. Unfortunately, for noisy maps this renders the function completely noise dominated. To remedy this problem we substitute the collapsed functions by socalled pseudocollapsed versions, as introduced by Hinshaw et al. (1995). For the COBEDMR experiment the beam size is approximately , while the pixel size is  necessarily for adequate sampling  1.8 (for the HEALPix pixelization used here). Therefore the CMB signal component between two neighboring pixels is highly coherent, whereas the noise contributions are independent. Thus, instead of multiplying a given pixel by itself several times, we multiply the pixel by one or more of its immediate neighbors, then sum over all such possible products, effectively multiplying by an average over the nearest neighbors. Hence, we more generally define a pseudocollapsed function as an average product of pixels where at least one pixel is multiplied in the pseudocollapsed sense, i.e. by an average over its neighbors. The golden rule for our analysis is that no pixel is ever multiplied with itself. This definition is then not completely equivalent to that introduced by Hinshaw et al. (1995) They defined the pseudocollapsed function as the average product of 1) a center pixel, 2) one of its neighbors and 3) a far point, where the far point was not allowed to be the center pixel. However, it was allowed to be the neighboring pixel. Although not a major problem for the threepoint function, we have determined that the inclusion of such a product renders the first bin of the fourpoint functions completely noise dominated.
Figure 1: Three and fourpoint correlation functions of the coadded 4year DMR map. Solid line shows the most likely value for each bin, dark shading shows the 68% confidence region and light shading the 95% confidence region, as computed by Monte Carlosimulations. Dots represent functions for the uncorrected coadded map, and boxes shows the functions for the map for which highlatitude Galactic emission has been removed. Note the different angular units on the horizontal axis, reflecting the fact that the various functions are defined on different angular intervals.  
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We also introduce one further small change compared to Hinshaw et al. (1995) in that we exclude the zeroth angular bin (for which all N pixels coincide) as any cosmological information here is heavily suppressed due to the low signaltonoise ratio. Indeed, the inclusion of the zeroth bin only acts to increase the variance of the statistic, and is therefore better omitted.
The COBEDMR experiment resulted in six independent maps, two for each of the three frequencies at 31.5, 53 and 90 GHz. In this work we only include the maps from the 53 and 90 GHz channels, as they are superior in terms of the signaltonoise ratio. The maps are analyzed in the HEALPix^{} pixelization scheme (Hivon et al. 1998), with a resolution parameter of , corresponding to 12 288 pixels on the sky. At each frequency we compute the "sum'' (A+B)/2 and "difference'' (AB)/2 combinations, which yield, respectively, maps with enhanced signaltonoise or noise content alone. In addition, we also generate a coadded map from the four basic channels using weights to achieve the optimal signaltonoise ratio.
The two, three and fourpoint correlation functions for these nine map combinations are then computed. All pixels corresponding to the extended Galactic cut (Banday et al. 1997 but recomputed explicitly for the HEALPix scheme) are rejected from the analysis, leaving a total of 7880 accepted pixels. The bestfitting monopole, dipole and quadrupole are subtracted from each map before the Npoint functions are evaluated. The observed correlation functions are also computed after correction for the diffuse foreground emission at high Galactic latitude, using information from the (appropriately scaled) DIRBE map (Górski et al. 1996).
For our Monte Carlo ensemble, we simulate 10 000 individual realizations of the CMB sky, based on an a priori bestfit cosmological power spectrum. In particular, we consider scaleinvariant Gaussian temperature fluctuations ( with n=1) with (Górski et al. 1996). The powerspectrum is filtered through the DMR beam and pixel window functions. To each simulated CMB sky, we add four noise realizations based on the rms noise levels and observation patterns of the observed 53 and 90 GHz sky maps. These are then combined to generate the corresponding sum, difference and coadded sky maps. These are then processed in an identical fashion to the DMR data.
We note that we have also assumed that there are no significant pixelpixel noise correlations, although some will indeed be present as a consequence of the differential nature of the radiometers which couple observations separated by on the sky. Lineweaver et al. (1994) have investigated this effect in detail, and find a small excess signal is present in the 2point correlation function at for maps containing noise signal alone. However, we do not expect our results to be compromised by this assumption.
Figure 1 shows the results from these calculations for the coadded maps. The observed functions lie comfortably within the confidence region defined by the Monte Carlosimulations and there are no striking deviations visible by simple inspection.
In order to quantitatively measure the agreement between the DMR maps
and the simulated ensemble, we utilize the same
methodology
described by Kogut et al. (1996):

In Kogut et al. (1996) the results for the pseudocollapsed and the equilateral threepoint functions are given for the 53 GHz (A+B)/2 map; they find the fractions to be respectively 0.66 and 0.31, while we find 0.65 and 0.29. Considering the minor changes in the definitions of the correlation functions and the different pixelizations used, the agreement is most satisfactory.
Overall, the numbers indicate that the COBEDMR maps agree very well with the simulations. The optimal coadded map, for which the signaltonoise ratio is the highest, returns results comfortably in the accepted range, as does the combined analysis of all Npoint functions. We conclude that the DMR maps are compatible with the Gaussian hypothesis as measured by this test.
Figure 2: Comparison of the observed and the "reduced'' fourpoint functions for the coadded DMR sky map. Boxes indicate the function predicted by the twopoint functions, while shaded areas represent the 68% and 95% confidence intervals computed from Monte Carlosimulations. The observed fourpoint functions are shown with a solid line.  
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(5) 
(6) 
Thus, if the CMB temperature anisotropy field is in fact Gaussian distributed
with zero mean, then all even Npoint functions can be reduced to combinations
of products of the twopoint function. In particular, the fourpoint function reduces
to:
(12) 
This procedure has one major advantage compared to the one described in Sect. 3: the power spectrum only mildly affects the result. That is, the two most important contributions to the analysis come from the map itself, in the form of a twopoint and a fourpoint function. The assumed power spectrum is only used for estimating the acceptable deviations, not the overall shape. Therefore, this procedure provides a more direct test for Gaussianity than the previous one.

The results are shown for the coadded map in
Fig. 2. The observed function lies well
within the confidence regions about the predicted function
for all four cases.
For a more quantitative measure of the perceived agreement,
we define a
statistic, incorporating the new degree of freedom provided
by the predicted fourpoint function by simply replacing the average
correlation function with that new function:
(13) 
(14) 
Table 2 summarizes the results, which again support the hypothesis that the DMR sky maps are consistent with a scaleinvariant cosmological model with Gaussian initial fluctuations.
By performing Monte Carlosimulations we have studied the statistical properties of the COBEDMR 53 GHz and 90 GHz channels. The basic ingredients for this analysis were various Npoint correlation functions, and, in particular, four different fourpoint functions which have been presented for the first time. We have additionally taken advantage of a result from statistical theory, relating all even Npoint functions to reductions in terms of the twopoint function. This allowed us to define a test for Gaussianity in which the assumed power spectrum only plays a secondary role. This test could therefore prove better suited for situations in which we do not have access to the optimal power spectrum.
Comparison of the DMR Npoint correlation functions with the MonteCarlo ensemble indicates no evidence for possible nonGaussian behavior, in agreement with the earlier analysis of Kogut et al. (1996). Furthermore, the agreement between the observed DMR functions and the simulated ensembles also supports the validity of our model assumptions, namely that of a scaleinvariant power law model for the anisotropies, and uncorrelated noise.
On the other hand, the excellent agreement between the simulated and the observed correlation functions poses an intriguing problem: tests of Gaussianity based on a harmonic analysis of the DMR data  the bispectrum work of Ferreira et al. (1998) and trispectrum results of Kunz et al. (2001)  show compelling evidence for nonGaussian features (although these have subsequently been associated with systematic artifacts in the DMR data by Banday et al. 2000), while tests based on realspace highorder statistics such as those presented here do not. The resolution of such apparently contradictory results is most likely rather mundane: the source of the nonGaussian signal was found to be strongly located at the multipole order l=16. Since the correlation functions are by definition (weighted) averages over the full multipole range, the reduced sensitivity to this type of nonGaussian structure is certainly not unexpected.
Acknowledgements
We acknowledge use of the HEALPix software and analysis package for deriving the results in this paper. H.K.E. acknowledges useful discussions with Per B. Lilje.