The 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 tau_nle for the amplitude tau_nl of the CMB trispectrum both for the null-hypothesis (i.e., for Gaussian maps with tau_nl = 0) and for maps with a non-vanishing trispectrum (|tau_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, <tau_nle^2>, as a function of its underlying value, tau_nl. We find a strong dependence of this variance on tau_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 lmax = 1500 (such as the Planck satellite) measures tau_nle = 0 then at the 95% confidence our calculations show that we can conclude tau_nl < 1005; assuming a Gaussian PDF but with the correct tau_nl-dependent variance we would incorrectly conclude tau_nl < 4225; further neglecting the tau_nl-dependence in the variance we would incorrectly conclude tau_nl < 361.