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 ${\widehat{\tau }}_{\mathrm{nl}}$ for the amplitude ${\tau }_{\mathrm{nl}}$ of the CMB trispectrum both for the null hypothesis (i.e., for Gaussian maps with ${\tau }_{\mathrm{nl}}=0$) and for maps with a nonvanishing trispectrum (${\tau }_{\mathrm{nl}}\ne 0$). 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, $⟨\Delta {\widehat{\tau }}_{\mathrm{nl}}^2⟩$, as a function of its underlying value, ${\tau }_{\mathrm{nl}}$. We find a strong dependence of this variance on ${\tau }_{\mathrm{nl}}$. 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 ${\widehat{\tau }}_{\mathrm{nl}}=0$ then at the 95% confidence level our calculations show that we can conclude ${\tau }_{\mathrm{nl}}\le 1005$; assuming a Gaussian PDF but with the correct ${\tau }_{\mathrm{nl}}$-dependent variance we would incorrectly conclude ${\tau }_{\mathrm{nl}}\le 4225$; further neglecting the ${\tau }_{\mathrm{nl}}$-dependence in the variance we would incorrectly conclude ${\tau }_{\mathrm{nl}}\le 361$.

Figure 1

The scaling of the variance of ${\widehat{\tau }}_{\mathrm{nl}}$ under the null hypothesis ${\tau }_{\mathrm{nl}}=0$. Each red point shows the result of a Monte Carlo simulation for 1000 realizations. The black line is a fit to those simulations given by Eq. (16).

Figure 2

The black-solid curve shows the PDF of ${\widehat{\tau }}_{\mathrm{nl}}$ under the null hypothesis, ${\tau }_{\mathrm{nl}}=0$ for $l_{\max }=50$ and calculated with ${10}^6$ realizations. The red-dashed curve shows a Gaussian PDF with the same variance. The upper panel shows the PDF on a linear scale, the lower panel on a logarithmic scale. When scaled by its variance, the PDF is identical for $l_{\max }=100$ showing that it takes on a universal form in the $l_{\max }\gg 1$ limit. We give a fitting formula for this PDF in Eq. (17).

Figure 3

The variances ${\sigma }_1^2$ and ${\sigma }_2^2$ as a fraction of the zeroth-order variance ${\sigma }_0^2$. The points are the results of the Monte Carlo simulations and the curves show the power-law fits given in Eqs. (22, 23).

Figure 4

The ratio $f_{\mathrm{nl}}{}^2/{\sigma }_{f_{\mathrm{nl}}}^2$ using the CMB bispectrum (black, from Ref. 29) and $\tau _{\mathrm{nl}}{}^2/{\sigma }_{{\tau }_{\mathrm{nl}}}^2$ using the CMB trispectrum [blue, Eq. (21)] as a function of ${\tau }_{\mathrm{nl}}=f_{\mathrm{nl}}{}^2$. These ratios can be interpreted as an estimate for the $S/N$ for a constraint to local-model non-Gaussianity in the CMB. The dashed curves show the scaling of the $S/N$ without taking into account the dependence of the variance on $f_{\mathrm{nl}}$ and ${\tau }_{\mathrm{nl}}$; the solid curves show the correct $S/N$ scaling. As in Ref. 14, from the dashed curves we would (incorrectly) conclude that the trispectrum estimator is more sensitive to a non-Gaussian signal for $\sqrt{{\tau }_{\mathrm{nl}}}\gtrsim 40$.

Figure 5

The scaling of the parameters for the fitting formula in Eq. (17) with $\sqrt{{\tau }_{\mathrm{nl}}}$ for $l_{\max }=600$ (dotted curves) $l_{\max }=1500$ (dashed curves) and $l_{\max }=3000$ (solid curves). For ${\tau }_{\mathrm{nl}}\ne 0$ the non-Gaussianity of the map imparts additional non-Gaussianity to the PDF. As ${\tau }_{\mathrm{nl}}$ increases the asymmetry of the PDF increases with the power-law index of the PDF for ${\widehat{\tau }}_{\mathrm{nl}}<{\tau }_{\mathrm{nl}}$ going from $n=3$ to $n=4$ in the case of $l_{\max }=1500$, and $n=4.24$ in the case of $l_{\max }=3000$.