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

We present a new method to estimate three-point correlations in CosmicMicrowave Background maps. Our Fast Fourier Transform based implementationestimates three-point functions using all possible configurations (triangles)at a controlled resolution. The speed of the technique depends both on theresolution and the total number of pixels $N$. The resulting $N\log N$ scalingis substantially faster than naive methods with prohibitive $N^3$ scaling. Asan initial application, we measure three-point correlation functions in theFirst Year Data Release of the Wilkinson Anisotropy Probe. We estimate 336cross-correlations of any triplet of maps from the 8 differential assemblies,scanning altogether 2.6 million triangular configurations. Estimatingcovariances from Gaussian signal plus realistic noise simulations, we perform anull-hypothesis testing with regards to the Gaussianity of the Cosmic MicrowaveBackground. Our main result is that at the three-point level WMAP is fullyconsistent with Gaussianity. To quantify the level of possible deviations, weintroduce false discovery rate analysis, a novel statistical technique toanalyze for three-point measurements. This confirms that the data areconsistent with Gaussianity at better than 1-$\sigma$ level when jointlyconsidering all configurations. We constrain a specific non-Gaussian modelusing the quadratic approximation of weak non-Gaussianities in terms of the$f_{NLT}$ parameter, for which we construct an estimator from the thethree-point function. We find that using the skewness alone is moreconstraining than a heuristic suboptimal combination of all our results; ourbest estimate is $f_{NLT} = -110\pm 150$ assuming a $\Lambda$CDM concordancemodel.Comment: 20 pages, 5 figure