Measuring the Three Point Correlation Function of the Cosmic Microwave Background

Details Measuring the Three Point Correlation Function of the Cosmic Microwave Background Measuring the Three Point Correlation Function of the Cosmic Microwave Background

2005-08-15

arxiv.org/abs/astro-ph/0508316v1

Astrophys.J.635:743-749,2005

Astronomy and Astrophysics

Astrophysics

20 pages, 5 figures

Scientific paper

10.1086/497619

We present a new method to estimate three-point correlations in Cosmic Microwave Background maps. Our Fast Fourier Transform based implementation estimates three-point functions using all possible configurations (triangles) at a controlled resolution. The speed of the technique depends both on the resolution and the total number of pixels $N$. The resulting $N\log N$ scaling is substantially faster than naive methods with prohibitive $N^3$ scaling. As an initial application, we measure three-point correlation functions in the First Year Data Release of the Wilkinson Anisotropy Probe. We estimate 336 cross-correlations of any triplet of maps from the 8 differential assemblies, scanning altogether 2.6 million triangular configurations. Estimating covariances from Gaussian signal plus realistic noise simulations, we perform a null-hypothesis testing with regards to the Gaussianity of the Cosmic Microwave Background. Our main result is that at the three-point level WMAP is fully consistent with Gaussianity. To quantify the level of possible deviations, we introduce false discovery rate analysis, a novel statistical technique to analyze for three-point measurements. This confirms that the data are consistent with Gaussianity at better than 1-$\sigma$ level when jointly considering all configurations. We constrain a specific non-Gaussian model using the quadratic approximation of weak non-Gaussianities in terms of the $f_{NLT}$ parameter, for which we construct an estimator from the the three-point function. We find that using the skewness alone is more constraining than a heuristic suboptimal combination of all our results; our best estimate is $f_{NLT} = -110\pm 150$ assuming a $\Lambda$CDM concordance model.

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