Quantum standard clocks in the primordial trispectrum

Abstract

We calculate the primordial trispectrum of curvature perturbation in quasi-single field inflation, with general sound speeds for both the inflaton and the massive scalar. Special attention is paid to various soft limits of the trispectrum, where the shape function shows characteristic oscillatory pattern (known as the quantum primordial standard clock signal) as a function of the momentum ratio. Our calculation is greatly simplified by using the "mixed propagator" developed under a diagrammatic representation of the in-in formalism.

Schwinger–Keldysh mechanism in extended quasi-single field inflation

“…In this section following the method introduced in [75,76] for quasi-single-field inflation, but, regarding the QSMF inflationary regime we want to review briefly the S-K formalism and propagators. The diagrammatic approach based on in-in formalism is reviewed in [75,76] as well.…”

“…In this section following the method introduced in [75,76] for quasi-single-field inflation, but, regarding the QSMF inflationary regime we want to review briefly the S-K formalism and propagators. The diagrammatic approach based on in-in formalism is reviewed in [75,76] as well. Additionally, it can be showed that the diagrammatic method presented in [75,76] is very close to the usual Feynman rules in diagrammatic method, except that the space and the time are treated the same, and that here the model contains two types of propagators introduced by either + or −.…”

“…The diagrammatic approach based on in-in formalism is reviewed in [75,76] as well. Additionally, it can be showed that the diagrammatic method presented in [75,76] is very close to the usual Feynman rules in diagrammatic method, except that the space and the time are treated the same, and that here the model contains two types of propagators introduced by either + or −. Accordingly, as it was mentioned in [76] we know the internal vertex in the diagrams is associated to conserved three-momentum and also an integration of time, and to consider the effects of all interactions appearing in the Lagrangian one should sum over diagrams for all vertices with all possible types of them.…”

“…To determine the definitions of the tree level propagators we can consider the following twopoint functions, Γ pn , [75,76] , namely…”

“…where Θ(z) is the step function. To calculate the required three-point function of δφ with intermediate exchange of δσ one would regard the propagators (3.9) and the mode functions (3.3) or (3.4), and follow the diagrammatic rules as reviewed in [75,76]. Based on diagrammatic rules were presented in Chen et al [75], the three-momentum related to the self interacting cases could be summarized as bellow, Also from figure 1 it is realized that the external points are distinguished by little squares, while the shaded dots in each vertex means that it is related to two different types of propagators.…”