High order correlation functions for self interacting scalar field in de Sitter space

TL;DR

This work derives exact tree-order expressions for the $3$- and $4$-point correlation functions of a self-interacting light scalar field in a de Sitter background for cubic and quartic potentials, and analyzes their super-horizon limits. Using a quantum-field-theoretic treatment in curved space with a Bunch-Davies vacuum, it expresses the higher-point functions in terms of mode functions and wave-number invariants, and shows that on superhorizon scales the results converge to those obtained from the classical stochastic approach, with a leading vertex $\nu_3 = -\lambda N_e/(3H^2)$ in the large-$N_e$ limit. The findings illuminate the link between quantum fluctuations during inflation and classical stochastic evolution, quantify subhorizon suppression of mode coupling after a few $e$-folds, and provide closed-form expressions that can inform analyses of primordial non-Gaussianity, including finite-volume effects discussed in related work. Overall, the paper strengthens the connection between quantum field theory in de Sitter space and stochastic inflationary descriptions, delivering concrete tools for interpreting higher-order correlators in early-universe cosmology.

Abstract

We present the expressions of the three- and four-point correlation functions of a self interacting light scalar field in a de Sitter spacetime at tree order respectively for a cubic and a quartic potential. Exact expressions are derived and their limiting behaviour on super-horizon scales are presented. Their essential features are shown to be similar to those obtained in a classical approach.

Figures (4)

• Figure 1: Dependence of the function $\zeta$ as a function of $k_4$ for fixed values of $k_1$, $k_2$ and $k_3$. The solid line corresponds to configuration $k_1$=0, $k_2$=$\frac{1}{2}$, $k_3$=$\frac{1}{2}$, $k_4$; the dashed line to $k_1$=$\frac{1}{3}$, $k_2$=$\frac{1}{3}$, $k_3$=$\frac{1}{3}$, $k_4$ and the dotted line to $k_1$=$\frac{1}{6}$, $k_2$=$\frac{1}{3}$, $k_3$=$\frac{1}{6}$, $k_4$.
• Figure 2: Behaviour of the function $Q_4$ as of function of time. The transition to the superhorizon behavior (dashed line) is shown. The function $Q_4$ is shown here for a "square" configuration ($k_1=k_2=k_3=k_4$) as a function of $k_t\eta=\sum k_i\eta$.
• Figure 3: Same as Fig. \ref{['Q4square']} for a "rectangular" configuration ($k_1=k_2=4,\, k_3=4 k_4$).
• Figure 4: Same as Fig. \ref{['Q4square']} for a "triangular" configuration ($k1=0, k_2=k_3=k_4$).