Shape of primordial non-Gaussianity and the CMB bispectrum

We present a set of formalisms for comparing, evolving, and constraining primordial non-Gaussian models through the CMB bispectrum. We describe improved methods for efficient computation of the full CMB bispectrum for any general (nonseparable) primordial bispectrum, incorporating a flat sky approximation and a new cubic interpolation. We review all the primordial non-Gaussian models in the present literature and calculate the CMB bispectrum up to $l<2000$ for each different model. This allows us to determine the observational independence of these models by calculating the cross correlation of their CMB bispectra. We are able to identify several distinct classes of primordial shapes---including equilateral, local, warm, flat, and feature (non-scale invariant)---which should be distinguishable given a significant detection of CMB non-Gaussianity. We demonstrate that a simple shape correlator provides a fast and reliable method for determining whether or not CMB shapes are well correlated. We use an eigenmode decomposition of the primordial shape to characterize and understand model independence. Finally, we advocate a standardized normalization method for ${f}_{\mathrm{NL}}$ based on the shape autocorrelator, so that observational limits and errors $\ensuremath{\Delta}{f}_{\mathrm{NL}}$ can be consistently compared for different models.