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

TL;DR

This work investigates the early universe under a non-trivial sound speed by marrying open quantum-system dynamics with Krylov (Arnoldi-Lanczos) complexity. It derives evolution equations for the open two-mode squeezed state parameters $r_k$ and $\phi_k$ and analyzes the Krylov complexity $\mathcal{C}_K$ and Krylov entropy $K_E$ across inflation, radiation domination, and matter domination. The results show $\mathcal{C}_K$ grows exponentially during inflation but does not saturate due to cosmic expansion, while $K_E$ reveals a clear distinction between standard inflation and models with $c_S\neq1$, notably around $\xi\approx0.02$. This open-system, information-theoretic lens treats the early universe as a maximally chaotic, non-equilibrium quantum system and suggests decoherence and extensions to multi-field or modified-gravity scenarios as fruitful directions.

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 $φ_k$ 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.

Figures (8)

• Figure 1: The plot of $c_S^2$ in different cosmological epochs with different $\xi$ parameters. The horizontal axis is $\log_{10}{(a)}$, and the vertical axis is $c_S^2$ . The black ($\xi=0$), blue ($\xi=0.01$), red ($\xi=0.02$), and green ($\xi=0.04$) lines represent various evolution of $c_S^2$. We also have set $k=1$ during inflation, $k=0.01$ during the RD epoch, and $k=0.005$ during the MD epoch.
• Figure 2: The plots $b_2$ against $\log_{10}a$ for $n=2$, showing numerical values across three cosmological epochs: inflation, RD, and MD. For simplicity, we adopt $H_0=1$ and set the $k=1$ during inflation, $k=0.01$ during RD, and $k=0.005$ during MD.
• Figure 3: The numerical results for the dissipation coefficient $u_2$ as a function of $\log_{10}a$ a during the RD and MD epochs. For simplicity, we adopt $H_0=1$ and set the $k=1$ during inflation, $k=0.01$ during RD, and $k=0.005$ during MD.
• Figure 4: The numerical of $r_{k}$ in terms of $\log_{10}a$ for three different periods (inflation, RD, and MD), where we set $H_{0}=1$, $k=1$ in inflation, $k=0.01$ at RD and $k=0.005$ at MD for simplicity.
• Figure 5: The numerical of $\phi_{k}$ in terms of $\log_{10}a$ for inflation period , where we set $H_{0}=1$, $k=1$ for simplicity.