Cosmological complexity of the modified dispersion relation

TL;DR

This paper investigates how modified dispersion relations, arising from quantum gravity theories, influence the evolution of complexity in the early universe, revealing non-linear patterns, increased Lyapunov indices, and irregular oscillations during inflation.

Contribution

It introduces a numerical analysis of complexity evolution under modified dispersion relations, highlighting non-linear behavior and potential implications for quantum gravity frameworks.

Findings

Complexity exhibits non-linear growth after horizon exit.

Lyapunov index is larger compared to the standard case.

Complexity oscillates irregularly during inflation, with shorter scrambling times.

Abstract

Complexity will be more and more essential in high-energy physics. It is naturally extended into the very early universe. Considering the universe as a quantum chaotic system, the curvature perturbation of the scalar field is identified with the two-mode squeezed state. By solving the Schr$\ddot{o}$dinger equation, one can obtain the numerical solutions of the angle parameter and squeezing parameter. The solution of the squeezing parameter mainly determines the evolution of complexity. Our numeric indicates that the complexity of the modified dispersion relation will have a non-linear pattern after the horizon exits. Meanwhile, its corresponding Lyapunov index is also larger compared with the standard case. During the inflationary period, the complexity will irregularly oscillate and its scrambling time is also shorter compared with the standard case. Since the modified dispersion relation can be dubbed as the consequences of various frameworks of quantum gravity, it could be applicable to these frameworks. Finally, one can expect the framework of quantum gravity will lead to the fruitful evolution of complexity, which guides us in distinguishing various inflationary models.