Schwinger-Keldysh Diagrammatics for Primordial Perturbations

TL;DR

The paper develops a self-contained Schwinger-Keldysh diagrammatic framework for calculating in-in correlators of primordial perturbations in inflation, showing that diagram rules can be derived from a classical Lagrangian even with derivative couplings. It introduces the mixed propagator technique in quasi-single-field inflation, dramatically reducing the complexity of computing the bispectrum and trispectrum. The resulting framework yields both known results for light spectator masses and new predictions for heavier fields, including observable clock signals that encode the primordial expansion history. The approach offers substantial computational speedups, provides guidance for gauge and gravity contexts, and outlines limitations for loop-level calculations and general FRW scenarios.

Abstract

We present a systematic introduction to the diagrammatic method for practical calculations in inflationary cosmology, based on Schwinger-Keldysh path integral formalism. We show in particular that the diagrammatic rules can be derived directly from a classical Lagrangian even in the presence of derivative couplings. Furthermore, we use quasi-single-field inflation as an example to show how this formalism, combined with the trick of mixed propagator, can significantly simplify the calculation of some in-in correlation functions. The resulting bispectrum includes the lighter scalar case ($m<3H/2$) that has been previously studied, and the heavier scalar case ($m>3H/2$) that has not been explicitly computed for this model. The latter provides a concrete example of quantum primordial standard clocks, in which the clock signals can be observably large.

Figures (4)

• Figure 1: Integral contour in (\ref{['3pt']}). The left panel is helpful for a theoretical understanding of IR divergence, while the right panel is suitable for numerical calculation.
• Figure 2: We plot the dimensionless shape functions. Near the equilateral limit $k_1=k_2=k_3$, from top to down, the top 3 layers are $\nu=i$ (blue), $\nu=5i$ (green), $\nu=3i$ (red), respectively. Here the shape functions are normalized to be 1 at the equilateral limit. In the equilateral limit, before normalization, the three-point function is not monotonic as a function of $i\nu$. Thus the layers with different values of $i\mu$ intersects with each other due to this normalization. We also plotted the factorizable ansatz of the equilateral shape Creminelli:2005hu (orange). This ansatz is a good approximation of, but not identical to, the equilateral shape originating from models with a small sound speed. The equilateral shape (from small sound speed models) would look more similar to the other layers of the plot.
• Figure 3: The squeezed limits for $\nu=i,2i,3i,4i$. The vertical axis is the dimensionless shape function $S$ magnified by a factor of $k_\mathrm{short}/k_\mathrm{long}$ for visual effect. The oscillatory components, namely the clock signals Chen:2015lza, take the same form as those in Fig.4 of Ref. Chen:2015lza but their relative amplitudes to the non-oscillatory components are larger. This is because the couplings used in these two examples are different. In Chen:2015lza, a simple example of coupling is studied; while here we have used the full leading coupling of the quasi-single-field model in Chen:2009weChen:2009zp.
• Figure 4: Upper-left panel:$s_\pm(\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu)$. Numerically, one finds that at $\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu \gg 1$, $s_+(\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu) \sim e^{-\pi\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu}$, and $s_-(\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu) \sim e^{-2\pi\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu}$ up to polynomial factors (the power of the polynomial is chosen to better fit the curve overall for the range of $\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu$ plotted). As noticed in Chen:2009zp, the apparent divergence at $\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu \rightarrow 0$ is an indication of change of shape, instead of anything physical blowing up. Upper-right panel:$s_{\{1,2\}}(\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu)$ for $0<\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu<1$. Lower-left panel: Large $\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu$ behavior for $s_1(\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu)$. Lower-right panel: Large $\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu$ behavior for $s_2(\mkern 2mu \widetilde{\mkern -2mu \nu \mkern -2mu}\mkern 2mu)$.