Probability distribution for non-Gaussianity estimators

One of the principle efforts in cosmic microwave background (CMB) research is measurement of the parameter $f_{\mathrm{nl}}$ that quantifies the departure from Gaussianity in a large class of nonminimal inflationary (and other) models. Estimators for $f_{\mathrm{nl}}$ are composed of a sum of products of the temperatures in three different pixels in the CMB map. Since the number $\sim N_{\mathrm{pix}}^2$ of terms in this sum exceeds the number $N_{\mathrm{pix}}$ of measurements, these $\sim N_{\mathrm{pix}}^2$ terms cannot be statistically independent. Therefore, the central-limit theorem does not necessarily apply, and the probability distribution function (PDF) for the $f_{\mathrm{nl}}$ estimator does not necessarily approach a Gaussian distribution for $N_{\mathrm{pix}}\gg 1$. Although the variance of the estimators is known, the significance of a measurement of $f_{\mathrm{nl}}$ depends on knowledge of the full shape of its PDF. Here we use Monte Carlo realizations of CMB maps to determine the PDF for two minimum-variance estimators: the standard estimator, constructed under the null hypothesis ($f_{\mathrm{nl}}=0$), and an improved estimator with a smaller variance for $f_{\mathrm{nl}}\ne 0$. While the PDF for the null-hypothesis estimator is very nearly Gaussian when the true value of $f_{\mathrm{nl}}$ is zero, the PDF becomes significantly non-Gaussian when $f_{\mathrm{nl}}\ne 0$. In this case we find that the PDF for the null-hypothesis estimator $\widehat{f_{\mathrm{nl}}}$ is skewed, with a long non-Gaussian tail at $\widehat{f_{\mathrm{nl}}}>|f_{\mathrm{nl}}|$ and less probability at $\widehat{f_{\mathrm{nl}}}<|f_{\mathrm{nl}}|$ than in the Gaussian case. We provide an analytic fit to these PDFs. On the other hand, we find that the PDF for the improved estimator is nearly Gaussian for observationally allowed values of $f_{\mathrm{nl}}$. We discuss briefly the implications for trispectrum (and other higher-order correlation) estimators.

Figure 1

Numerical evaluations of $P(\widehat{f_{\mathrm{nl}}};f_{\mathrm{nl}}=0,l_{\max })$. The left (right) two panels show the PDF for $l_{\max }=5$ and $l_{\max }=25$ for ${10}^6$ realizations for a scale-invariant power spectrum. In all panels the PDF has been normalized to have a unit variance, and the corresponding Gaussian PDF (with the same variance) is shown as the red dashed curve. As $l_{\max }$ gets larger, the PDF tends toward a Gaussian. This is not guaranteed by the central-limit theorem since the majority of the terms that appear in the estimator are not statistically independent.

Figure 2

The PDF $P(\widehat{f_{\mathrm{nl}}})$ when $\widehat{f_{\mathrm{nl}}}=1500$ using the estimator in Eq. (8) with $l_{\max }=25$. The upper (lower) panel shows the PDF on a linear (log) scale. We can see that the PDF is significantly non-Gaussian with an exponential drop-off to the left of mean and a power-law to the right. We provide a fitting formula for $P(\widehat{f_{\mathrm{nl}}};f_{\mathrm{nl}},l_{\max })$ in the text.

Figure 3

The dependence of $⟨(\Delta E_1{)}^2⟩$ on $l_{\max }$. The points correspond to the results of our Monte Carlo simulations for 1000 realizations at different values of $l_{\max }$. The solid curve shows the analytic calculation of the variance presented in Appendix  which is well fit by the function $⟨(\Delta E_1{)}^2⟩=[14.0(l_{\max }{)}^{0.433}]/[{ln}^{5.1}(l_{\max })]\approx 4.5{ln}^{-3}(l_{\max })$.

Figure 4

The PDF of $E_1$ calculated using ${10}^6$ realizations. The thin solid curves correspond to $P(E_1)$ with $l_{\max }=25$ (green, middle curve), $l_{\max }=50$ (purple, right curve), and $l_{\max }=100$ (black, left curve). Since the functional form of the PDF for each choice of $l_{\max }$ is nearly identical, we conclude that $P(E_1)$ approaches an asymptotic functional form in the large-$l_{\max }$ limit. The thick red dashed curve corresponds to a fit to $P(E_1)$, accurate to $\sim 10\%$ out to 3 times the root variance, using the fitting formula in Eq. (17) with parameter values $x_p=-0.22$, $\sigma =0.80$, and $c=0.91$.

Figure 5

The skewness, $⟨(\Delta \widehat{f_{\mathrm{nl}}}{)}^3⟩$, as a fraction of the variance of $P(\widehat{f_{\mathrm{nl}}};f_{\mathrm{nl}},l_{\max })$ as a function of $l_{\max }$ for $f_{\mathrm{nl}}=100$. We provide an analytic fitting formula in Eqs. (22, 23) as a function of $f_{\mathrm{nl}}$ and $l_{\max }$.

Figure 6

The PDF $P({\mathcal{E}}_0)$ (left panels) and $P({\mathcal{E}}_1)$ (right panels) for $l_{\max }=25$ determined with ${10}^6$ non-Gaussian realizations. The top panels show the PDF on a linear scale; the bottom panels show the PDF on a log scale. We have confirmed that the shape of the PDF is unchanged for larger values of $l_{\max }$. The PDF of ${\mathcal{E}}_0$ (left panels) is well approximated by a Gaussian. However, the PDF of the first-order correction ${\mathcal{E}}_1$ (right panels) has significant non-Gaussian wings. This implies that the full PDF of $\widehat{f_{\mathrm{nl}}^n}$ is also non-Gaussian, even if the true value of $f_{\mathrm{nl}}$ matches that assumed in the construction of the CSZ estimator. Quantitatively, however, the level of non-Gaussianty will be small for Planck, as the variance of ${\mathcal{E}}_1$ is $⟨(\Delta {\mathcal{E}}_1{)}^2⟩\approx 9{ln}^2(l_{\max })/(l_{\max }^3)$.