A quantum information method for early universe with non-trivial sound speed

Abstract: 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, our universe can be described as an open system. Under the non-trivial sound speed, we employ the method of open quantum systems combined with Arnoldi iterations to study the Krylov complexity throughout the early universe, including the inflationary, radiation-dominated, and matter-dominated epochs. A key ingredient in our analysis is the open two-mode squeezed state formalism and the generalized Lanczos algorithm. To numerically compute the Krylov complexity, we are the first time to derive the evolution equations for the parameters $r_k$ and $\phi_k$ within an open two-mode squeezed state. Our 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, we also investigate the Krylov entropy for completeness. 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, we find that the case of non-trivial sound speed behaves as a maximally chaotic system. However, our 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.