The CMB Bispectrum

TL;DR

The paper develops and validates a general late-time CMB bispectrum estimator based on separable modal expansions to reconstruct and constrain a broad class of non-Gaussian models from WMAP7 data. By expanding the full bispectrum in an orthonormal modal basis, it achieves efficient reconstruction and model-agnostic tests, yielding new constraints across constant, local, equilateral, flat, and warm-inflation shapes, and introducing a total integrated bispectrum measure $\bar{F}_{\rm NL}$. The analysis finds no significant deviation from Gaussianity in the WMAP domain, including for non-scaling feature models, with one anomalous mode that warrants further study with higher-resolution data. The approach provides a flexible, scalable framework for Planck and future surveys, enabling broad, model-independent searches for primordial non-Gaussianity and potential extensions to the trispectrum.

Abstract

We use a separable mode expansion estimator with WMAP data to estimate the bispectrum for all the primary families of non-Gaussian models. We review the late-time mode expansion estimator methodology which can be applied to any non-separable primordial and CMB bispectrum model, and we demonstrate how the method can be used to reconstruct the CMB bispectrum from an observational map. We extend the previous validation of the general estimator using local map simulations. We apply the estimator to the coadded WMAP 5-year data, reconstructing the WMAP bispectrum using $l<500$ multipoles and $n=31$ orthonormal 3D eigenmodes. We constrain all popular nearly scale-invariant models, ensuring that the theoretical bispectrum is well-described by a convergent mode expansion. Constraints from the local model $ \fnl=54.4\pm 29.4$ and the equilateral model $\fnl=143.5\pm 151.2$ ($\Fnl = 25.1\pm 26.4$) are consistent with previously published results. (Here, we use a nonlinearity parameter $\Fnl$ normalised to the local case, to allow more direct comparison between different models.) Notable new constraints from our method include those for the constant model $\Fnl = 35.1 \pm 27.4 $, the flattened model $\Fnl = 35.4\pm 29.2$, and warm inflation $\Fnl = 10.3\pm 27.2$. We investigate feature models surveying a wide parameter range in both the scale and phase, and we find no significant evidence of non-Gaussianity in the models surveyed. We propose a measure $\barFnl$ for the total integrated bispectrum and find that the measured value is consistent with the null hypothesis that CMB anisotropies obey Gaussian statistics. We argue that this general bispectrum survey with the WMAP data represents the best evidence for Gaussianity to date and we discuss future prospects, notably from the Planck satellite.

Figures (18)

• Figure 1: The reduced CMB bispectrum for the constant model $b_{l_1l_2l_3}^{\rm const}$ arising from the convolution of the primordial shape function $S(k_1,k_2,k_3) =1$ with transfer functions (normalised relative to the large-angle constant solution $b_{l_1l_2l_3}^{const(la)}$ given in (\ref{['eq:constbispect']})). On the left, the 3D bispectrum is plotted over the allowed tetrahedral region of multipole triples (see fig. \ref{['fig:tetrapyd']}) using several density contours (light blue positive and magenta negative) out to $l_i\le 2000$. On the right, a transverse triangular slices through the bispectrum is shown for $l_1+l_2+l_3= 4000$ (Planck resolution). Note the coherent pattern of acoustic peaks with a dominant primary peak in a broad diagonal region around $l_1=l_2=l_3=220$. This constant model bispectrum plotted is the analogue of the angular power spectrum $C_l$'s for a purely scale-invariant model.
• Figure 2: Observational domain (\ref{['eq:tetrapydl']}) for the CMB bispectrum $b_{l_1l_2l_3}$. Allowed multipole values $(l_1,\,l_2,\,l_3)$ lie inside the shaded 'tetrapyd' region, satisfying both the triangle condition and $l <L$$\,\equiv\,$$l_\textrm{max}$.
• Figure 3: Recovered mode coefficients $\bar{\beta}^{{\cal R}}_n$ (\ref{['eq:cmbestmodes']}) from the WMAP7 coadded V and W maps. Error bars (1$\sigma$) are also shown for each mode as estimated from 144000 Gaussian map simulations in WMAP-realistic context.
• Figure 4: Recovered 3D bispectrum from WMAP7 data showing the result using the reconstructed mode coefficients $\bar{\beta}^{{\cal R}}_n$ shown in fig. \ref{['fig:reconalphaWMAP5']} with the partial sum (\ref{['eq:cmborthmodes']}). Several isodensity surfaces are shown for the bispectrum out to $l_i\le 500$ (light blue positive and magenta negative).
• Figure 5: Recovered 3D bispectrum from WMAP7 data showing slices through the data at $\tilde{l}\equiv l_1+l_2+l_3= \hbox{const.}$. Slices shown are $\tilde{l} = 250, 500, 750, 1000$, using the same colour scale as fig. \ref{['fig:3dreconWMAP5']}.