# Trispectrum during Inflation

By **
Ian
**

After a lot of work, “Large trispectrum in two-field slow-roll inflation” was released on the arXiv yesterday as arXiv:1203.6844. In this article Joe Elliston, Laila Alabidi, David Mulryne, Reza Tavakol and I look at the generation of higher order statistics during inflation in the early universe.

In the early universe the curvature perturbations, which later are seen as temperature fluctuations in the Cosmic Microwave Background (CMB), are initially thought to be Gaussian, but can become skewed during inflation depending on the physics of their evolution. In the last few years a lot of work has been done to both find evidence of this non-Gaussianity, and to construct physical models in which it is generated in the early universe.

In the past most of the focus has been on the 3-point function or bispectrum, and discussion of non-gaussianity has boiled down to finding bounds on the parameter $latex f*{\mathrm{NL}}$. In terms of the CMB the bispectrum in essence considers whether the temperature of three points on the sky is correlated. The WMAP satellite has not seen any definitive evidence of a non-zero value for $latex f*{\mathrm{NL}}$ but the Planck satellite should be able to detect a signal if it is moderately large. In this work we look beyond the bispectrum to the 4-point function or trispectrum. For the trispectrum the correlation we attempt to measure is between four different points on the sky.

In this work we have tried to find models which generate a large value for the trispectrum during inflation. We have found some new expressions for the parameters $latex f*{\mathrm{NL}}$, $latex \tau*{\mathrm{NL}}$ and $latex g_{\mathrm{NL}}$. The last two of these parametrise two different contributions to the trispectrum. The bottom line is that it is quite difficult to find conditions where the trispectrum can be large, at least under the assumptions we made of sum- and product-separable potentials using the $latex \delta N$ formalism.

In the course of searching for models which give large values to these parameters we plotted the coefficient functions which need to be large as heatmaps, following Byrnes et al (arXiv:0807.1101). In order to generate these heatmaps I relied on the combination of Python, Numpy and Matplotlib, which I have used before on Pyflation. The script I used to generate the heatmap figures in the paper is now available as a repository on Bitbucket.

PS Someone really needs to work on the Wikipedia Non-Gaussianity page!