The real shape of non-Gaussianities

TL;DR

The paper develops a real-space intuition for cosmological non-Gaussianity by linking bispectrum and trispectrum shapes to concrete density patterns, such as filaments, pancakes, and line-like features. It derives and discusses the general squeezed-bispectrum form, its angular decomposition, and the connection to local modulations and inflation, including the single-field consistency relation and multi-field possibilities. It also explores tetrahedral and diagonal-squeezed trispectra, flattened quadrilaterals from line-like objects (e.g., cosmic strings), and the role of statistical anisotropy and modulation reconstruction in probing primordial and late-time non-Gaussian signals. The work highlights practical implications for distinguishing physical origins of non-Gaussianity and for designing estimators in CMB and large-scale structure analyses, with attention to fundamental inequalities like $\tau_{\rm NL} \ge (6f_{\rm NL}/5)^2$ and the impact of cosmic variance on detectability.

Abstract

I review what bispectra and trispectra look like in real space, in terms of the sign of particular shaped triangles and tetrahedrons. Having an equilateral density bispectrum of positive sign corresponds to having concentrated overdensities surrounded by larger weaker underdensities. In 3D these are concentrated density filaments, as expected in large-scale structure. As the shape changes from equilateral to flattened the concentrated overdensities flatten into lines (3D planes). I then focus on squeezed bispectra, which can be thought of as correlations of changes in small-scale power with large-scale fields, and discuss the general non-perturbative form of the squeezed bispectrum and its angular dependence. A general trispectrum has tetrahedral form and I show examples of what this can look like in real space. Squeezed trispectra are of particular interest and come in two forms, corresponding to large-scale variance of small-scale power, and correlated modulations of an equilateral-form bispectrum. Flattened trispectra can be produced by line-like features in 2D, for example from cosmic strings, and randomly located features also give a non-Gaussian signal. There are relationships between the squeezed types of non-Gaussianity, and also a useful interpretation in terms of statistical anisotropy. I discuss the various possible physical origins of cosmological non-Gaussianities, both in terms of primordial perturbations and late-time dynamical and geometric effects.

Figures (12)

• Figure 1: Equilateral bispectrum: a field can be decomposed into plane-wave modes, and the three components with wavevectors that form an equilateral triangle may have different relative signs. The sign of the bispectrum tells you which combination of signs is more likely (on average gives a positive or negative product of the three modes). A positive reduced bispectrum corresponds to being likely to have waves combining to have strong overdensities surrounded by larger areas of milder underdensity. A negative equilateral bispectrum corresponds to being likely to have concentrated underdensities surrounded by areas of milder overdensity. Note that in 3D the figures extend into the page, and hence the positive bispectrum corresponds to concentrated overdense filaments surrounded by larger areas of milder underdensity.
• Figure 2: A snapshot of non-linear large-scale structure from the millennium simulations Springel:2005nw. Dynamical non-linear collapse of very dense filaments (surrounded by milder underdensities, voids) generates a large positive roughly equilateral density bispectrum.
• Figure 3: As an approximately equilateral bispectrum triangle flattens, the round areas of overdensity become flattened into pancakes. In 3D a positive flattened bispectrum with $k_1=k_2=k_3/2$ corresponds to being likely to have overdense pancakes with larger mildly underdense planes in between.
• Figure 4: Two short-scale modes combine giving interference patterns. These look like a large-scale modulation in the amplitude of the small-scale modes, with the modulation having wavevector ${{\mathbf{k}}}_1=-{{\mathbf{k}}}_2-{{\mathbf{k}}}_3$.
• Figure 5: Squeezed bispectrum: two small-scale modes with nearly-equal wavelength ($k_2\sim k_3$) interfere with each other, giving some regions with lots of small-scale power and others with destructive interference giving little small-scale power. The sign of the bispectrum tells you whether a region of high small-scale power is more likely to be associated with a large-scale overdensity or a large-scale underdensity. If the correlation is independent of the relative orientation (upper and lower figures have the same signal) the bispectrum is isotropic, but in general it is not.