Cosmological diagrammatic rules

TL;DR

The paper addresses efficient perturbative evaluation of in-in correlators in cosmology using a concise diagrammatic rule set based on the Schwinger-Keldysh formalism. It outlines a bookkeeping scheme with a dotted line at $\eta_0$ that classifies vertices as above or below the line, pairing propagators into Wightman $W(x,y)$ and Feynman $G(x,y)$ types, and assigning $V=-ig$ or $V^{\dagger}=ig$ to vertices with final results obtained as $2\mathrm{Re}$ of the diagram sums, while conserving momentum and integrating over internal times and momenta in momentum space. The method is demonstrated on a simple $\phi^3$ theory in a RobertsonWalker background, using de Sitter propagators $W_k(\eta,\eta')$, $U_k(\eta)$, and $G_k(\eta,\eta')$, and is extended to graviton–scalar interactions via the graviton propagator in transverse traceless gauge. Applications to higher-point functions such as the trispectrum and tri-spectrum reproduce known results (e.g., in Giddings:2010nc, Adshead:2009cb, Seery:2008ax) and illustrate calculational efficiency relative to traditional closed-time-path approaches.

Abstract

A simple set of diagrammatic rules is formulated for perturbative evaluation of ``in-in" correlators, as is needed in cosmology and other nonequilibrium problems. These rules are both intuitive, and efficient for calculational purposes.

Figures (3)

• Figure 1: The bubble diagrams corresponding to expressions (\ref{['Aamp']}), (\ref{['Bamp']}).
• Figure 2: Diagrams contributing to the trispectrum, through the second line of (\ref{['trispec']}).
• Figure 3: Diagrams contributing to the trispectrum, through the third line of (\ref{['trispec']}).