Fast and precise way to calculate the posterior for the local non-Gaussianity parameter $f_{\mathrm{nl}}$ from cosmic microwave background observations

We present an approximate calculation of the full Bayesian posterior probability distribution for the local non-Gaussianity parameter $f_{\mathrm{nl}}$ from observations of cosmic microwave background anisotropies within the framework of information field theory. The approximation that we introduce allows us to dispense with numerically expensive sampling techniques. We use a novel posterior validation method (DIP test) in cosmology to test the precision of our method. It transfers inaccuracies of the calculated posterior into deviations from a uniform distribution for a specially constructed test quantity. For this procedure we study toy cases that use one- and two-dimensional flat skies, as well as the full spherical sky. We find that we are able to calculate the posterior precisely under a flat-sky approximation, albeit not in the spherical case. We argue that this is most likely due to an insufficient precision of the used numerical implementation of the spherical harmonic transform, which might affect other non-Gaussianity estimators as well. Furthermore, we present how a nonlinear reconstruction of the primordial gravitational potential on the full spherical sky can be obtained in principle. Using the flat-sky approximation, we find deviations for the posterior of $f_{\mathrm{nl}}$ from a Gaussian shape that become more significant for larger values of the underlying true $f_{\mathrm{nl}}$. We also perform a comparison to the well-known estimator of Komatsu et al. [Astrophys. J. 634, 14 (2005)] and finally derive the posterior for the local non-Gaussianity parameter $g_{\mathrm{nl}}$ as an example of how to extend the introduced formalism to higher orders of non-Gaussianity.

Figure 1

Normalized posterior distributions for $f$ in a one-dimensional test case with data generated from $f_{\mathrm{gen}}=3$ [panels (a) and (c)] and $f_{\mathrm{gen}}=3000$ [panels (b) and (d)]. The (a), (b) upper [(c), (d) lower] panels show the numerically calculated posterior including a parabola (Gaussian) fit. For $f_{\mathrm{gen}}=3000$, the PDF is negatively skewed and thus significantly non-Gaussian. The fitting curves in the upper panels arise from translating the Gaussian fit of the lower panels into a quadratic function.

Figure 2

DIP distribution of calculated $x$ values for one- (a) and two-dimensional (b) test cases. The histograms show the un-normalized distribution of 500 $x$ values within eight bins. The standard deviation interval ($1\sigma$) around the expectation value as calculated from Poissonian statistics is also shown.

Figure 3

DIP distribution of calculated $x$ values for the one-dimensional test case with Gaussian convolution. The histogram shows the un-normalized distribution of 500 $x$ values within eight bins. The standard deviation interval ($1\sigma$) around the expectation value as calculated from Poissonian statistics is also shown.

Figure 4

DIP distribution of calculated $x$ values for the test case on the sphere. The histogram shows the un-normalized distribution of 500 $x$ values within eight bins. The standard deviation interval ($1\sigma$) around the expectation value as calculated from Poissonian statistics is also shown. The green curve shows the analytical shape of a Gaussian posterior distribution whose standard deviation deviates by 35% from the one of the true Gaussian posterior.

Figure 5

Reconstruction of the primordial gravitational potential $\phi$ or the auxiliary Gaussian field $\varphi$ by using the maximum of the Hamiltonian and by applying a Wiener Filter, respectively. The upper four panels [(a)–(d)] are showing the generation of the mock data $d$, whereas the last two panels [(e), (f)] are showing the reconstructions of the original fields. Note the different color codes.

Figure 6

DIP distributions of calculated $x$ values for the KSW estimator [(a) upper left, (b) upper right] and approximated posterior according to Eq. (20) [(c) lower] and according to the intervals $I_1=[-6000,6000]\ni f_{\mathrm{nl}}$ (left, right), $I_2=[-2000,2000]\ni f_{\mathrm{nl}}$ (middle). The histograms show the un-normalized distributions of 500 $x$ values within eight bins. The standard deviation interval ($1\sigma$) around the expectation value as calculated from Poissonian statistics is also shown. The (green) fit in the (a) upper-left [(b) upper-right] panel is a theoretical DIP distribution calculated with a Gaussian posterior, whose standard deviation was underestimated by 35% (20%).