Probability distribution for non-Gaussianity estimators constructed from the CMB trispectrum

Considerable recent attention has focussed on the prospects to use the cosmic microwave background (CMB) trispectrum to probe the physics of the early universe. Here we evaluate the probability distribution function (PDF) for the standard estimator τ^_nl for the amplitude τ_nl of the CMB trispectrum both for the null hypothesis (i.e., for Gaussian maps with τ_nl=0) and for maps with a nonvanishing trispectrum (τ_nl≠0). We find these PDFs to be highly non-Gaussian in both cases. We also evaluate the variance with which the trispectrum amplitude can be measured, ⟨Δτ^_nl²⟩, as a function of its underlying value, τ_nl. We find a strong dependence of this variance on τ_nl. We also find that the variance does not, given the highly non-Gaussian nature of the PDF, effectively characterize the distribution. Detailed knowledge of these PDFs will therefore be imperative in order to properly interpret the implications of any given trispectrum measurement. For example, if a CMB experiment with a maximum multipole of l_max⁡=1500 (such as the Planck satellite) measures τ^_nl=0 then at the 95% confidence level our calculations show that we can conclude τ_nl≤1005; assuming a Gaussian PDF but with the correct τ_nl-dependent variance we would incorrectly conclude τ_nl≤4225; further neglecting the τ_nl-dependence in the variance we would incorrectly conclude τ_nl≤361.