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

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Abstract
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 on both the resolution and the total number of pixels N. The resulting N log N scaling is substantially faster than naive methods with prohibitive N3 scaling. As an initial application, we measure three-point correlation functions in the first-year data release of the Wilkinson Microwave Anisotropy Probe (WMAP). We estimate 336 cross-correlations of any triplet of maps from the eight differential assemblies, scanning altogether 2.6 million triangular configurations. Using 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 three-point measurements. This confirms that the data are consistent with Gaussianity at better than the 1 σ level when jointly considering all configurations. We constrain a specific non-Gaussian model using a quadratic expansion of the temperature field in terms of the fNLT parameter, ΔT/T = (ΔT/T)L + fNLT[(ΔT/T) - (ΔT/T) ], for which we construct an estimator from 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 fNLT = -110 ± 150, assuming a ΛCDM concordance model.
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