Derivative Interactions during Inflation: A Systematic Approach

TL;DR

This paper develops a systematic method for calculating cosmological correlation functions in models with derivative interactions, comparing it with the in-in formalism, and applies it to compute the scalar trispectrum.

Contribution

It introduces a new prescription for derivative interactions in inflationary models, including an off-shell in-in formalism and a detailed comparison with existing methods.

Findings

The path integral over conjugate momenta recovers the classical action with additional divergences.

The off-shell in-in formalism simplifies calculations for derivative couplings.

Explicit calculation of the scalar trispectrum in a model with b1 b1'^{3} derivative interaction.

Abstract

We present a systematic prescription for calculating cosmological correlation functions for models with derivative interactions through the wavefunction of the universe and compare this result with the "in-in" formalism -- canonical approach. The key step in this procedure is to perform the path integral over conjugate momenta after which a straightforward generalisation of Feynman's Rules can be applied. We show that this integral recovers the classical action plus some additional divergent contributions which are necessary to cancel other divergences that arise due to loop diagrams involving time derivatives. As a side project, for the first time, we introduce the "off-shell" version of the in-in formalism that is sometimes more straightforward, especially for the models with derivative coupling. To examine our prescription, as a specific example, we work out the trispectra of the scalar fluctuation in the model with the $\lambda {\phi'}^3$ derivative coupling.