A Quantum Information Method for Early Universe with Non-Trivial Sound Speed

Many quantum gravitational frameworks, such as DBI inflation, k-essence, and effective field theories obtained by integrating out heavy modes, can lead to a non-trivial sound speed. Meanwhile, universe can be described as an open system. Under the non-trivial sound speed, the method of open quantum systems combined with Arnoldi iterations is employed to study the Krylov complexity throughout the early universe, including the inflationary, radiation-dominated, and matter-dominated epochs. A key ingredient in the analysis is the open two-mode squeezed state formalism and the generalized Lanczos algorithm. To numerically compute the Krylov complexity, the evolution equations for the parameters (Formula presented.) and (Formula presented.) are derived for the first time within an open two-mode squeezed state. The results indicate that the Krylov complexity exhibits a similar trend in both the standard case and the case with non-trivial sound speed. To distinguish between these two scenarios, the Krylov entropy for completeness is also investigated. The evolution of the Krylov entropy shows a clear difference between the standard case and the non-trivial sound speed case. Furthermore, based on the behavior of the Lanczos coefficients, the case of non-trivial sound speed behaves as a maximally chaotic system. However, the numerical results suggest that the Krylov complexity does not saturate to a constant value due to the huge expansion of spacetime background. This study offers a new perspective for exploring the early universe through the quantum information.