:THE COSMOLOGICAL OTOC: Formulating New Cosmological Micro-Canonical Correlation Functions for Random Chaotic Fluctuations in Out-Of-Equilibrium Quantum Statistical Field Theory

Sayantan Choudhury

doi:10.3390/sym12091527

Abstract

The out-of-time-ordered correlation (OTOC) function is an important new probe in quantum field theory which is treated as a significant measure of random quantum correlations. In this paper, using for the first time the slogan “Cosmology meets Condensed Matter Physics”, we demonstrate a formalism to compute the Cosmological OTOC during the stochastic particle production during inflation and reheating following the canonical quantization technique. In this computation, two dynamical time scales are involved—out of them, at one time scale, the cosmological perturbation variable, and for the other, the canonically conjugate momentum, is defined, which is the strict requirement to define the time scale-separated quantum operators for OTOC and is perfectly consistent with the general definition of OTOC. Most importantly, using the present formalism, not only one can study the quantum correlation during stochastic inflation and reheating, but can also study quantum correlation for any random events in Cosmology. Next, using the late time exponential decay of cosmological OTOC with respect to the dynamical time scale of our universe which is associated with the canonically conjugate momentum operator in this formalism, we study the phenomenon of quantum chaos by computing the expression for the Lyapunov spectrum. Furthermore, using the well known Maldacena Shenker Stanford (MSS) bound on the Lyapunov exponent, λ≤2π/β, we propose a lower bound on the equilibrium temperature, T=1/β, at the very late time scale of the universe. On the other hand, with respect to the other time scale with which the perturbation variable is associated, we find decreasing, but not exponentially decaying, behaviour, which quantifies the random quantum correlation function out-of-equilibrium. We have also studied the classical limit of the OTOC and checked the consistency with the large time limiting behaviour of the correlation. Finally, we prove that the normalized version of OTOC is completely independent of the choice of the preferred definition of the cosmological perturbation variable.

Highlights

Perturbation to express the whole dynamics in terms of a gauge invariant description through a variable: Curvature perturbation : ζ(x, t) = −H(t) φ (t) δφ(x, t) +perturbation Higher order contribution
We have explicitly shown the behaviour of the two-point spectrum for α = 1/2 and α = 1, where it is clearly observed that the correlation decays very slowly with the increase in the magnitude of the mass parameter value and for very large value of |ν it will saturate to a non-zero large amplitude for a fixed time time scale
of-time ordered correlation (OTOC) computed from both the sides are exactly same and this is really good that after normalisation we do not need to think about the explicit origin or any preferred cosmological perturbation variable

Summary

Introduction

For the further analysis we restrict ourself only up to the first order contribution and will going to neglect all other higher order contributions as they are sufficiently small enough compared to the first order contribution as appearing in the above equation. In this context, geometrically the curvature perturbation ζ(x, t) measures the spatial curvature of constant hypersurface, which is represented by the following equation: Spatial curvature : R(3) 4 a2(t) ∂2 ζ (x, t) ∂2 δφ(x, t) [92].

Objectives

Results

Conclusion