Krylov complexity of thermal state in early universe

TL;DR

The paper investigates Krylov complexity and Krylov entropy of a thermal state across the early universe’s inflation, radiation, and matter-dominated eras by employing a purification-based open-system framework. It builds a two-mode purified wave function for the thermal state, derives closed- and open-system Krylov dynamics through Lanczos coefficients, and shows that complexity and entropy grow during inflation (a strongly dissipative regime) and saturate in RD/MD (weak dissipation) due to preheating and particle production. The analysis connects the growth rate of Lanczos coefficients to chaos indicators and demonstrates that open-system dynamics reproduce the closed-system results in the appropriate limit, providing a coherent quantum-information perspective on cosmological evolution. The results offer a novel lens on early-universe dynamics with potential implications for decoherence, multi-field inflation, and non-quadratic potentials within an open quantum systems context.

Abstract

Thermal interactions are ubiquitous in the cosmos, driving systems toward equilibrium. In this work, we investigate the evolution of thermal states across the early universe, encompassing the inflationary, radiation-dominated (RD), and matter-dominated (MD) eras, through the lens of Krylov complexity. Utilizing a purification scheme, we map the thermal state to a two-mode pure state, facilitating an open-system analysis of Krylov complexity in contrast to closed-system methodologies. Our numerical results demonstrate that Krylov complexity grows exponentially during inflation, indicating chaotic behavior, before saturating at nearly constant values in the RD and MD eras due to particle production via preheating. Furthermore, we analyze the Krylov entropy, which exhibits an evolutionary trend analogous to that of complexity. Crucially, our analysis reveals a dynamical transition in the universe's dissipative nature: with the universe acting as a strongly dissipative system during inflation and transitioning to a weakly dissipative regime in the subsequent eras. These findings provide a novel quantum information perspective on early universe dynamics.

Figures (7)

• Figure 1: The numerical solution for $T_{k}$ as a function of conformal time $\eta$ across three cosmological eras (inflation, RD, and MD) is presented. For computational simplicity, we adopt the parameter values: $\omega = 1$, $H_0 = 10^{-2.5}$, and $m = 10^{-6}$. The momentum scale $k$ assumes distinct characteristic values in each era, as illustrated in the respective panels.
• Figure 2: The numerical solution for the squeezed angle $\phi_k$ as a function of conformal time $\eta$ is presented across three cosmological eras: inflation, RD era, and MD era. For simplicity, we adopt the parameter values $\omega = 1$, $H_0 = 10^{-2.5}$, and $m = 10^{-6}$. The momentum scale $k$ takes distinct characteristic values in each era, as illustrated in the respective panels.
• Figure 3: The temporal evolution of Krylov complexity $C_K(\eta)$ is computed numerically for three cosmological phases: inflationary, RD, and MD. Parameter choices are $\omega = 1$, $H_0 = 10^{-2.5}$, and $m = 10^{-6}$, with era-specific momentum scales as shown in the corresponding panels.
• Figure 4: We present the numerical evolution of the K-entropy with respect to $\eta$ during inflation, RD, and MD. For simplicity, the parameters are set as follows: $\omega=1$; $H_0=10^{-2.5}$; and $m=10^{-6}$. The momentum takes on different typical values in each of these distinctive periods.
• Figure 5: Numerical results for the Lanczos coefficients $b_n(\eta)$ are shown for inflation, RD, and MD eras. We set $b_n=2\alpha=\lambda$ (at $n=2$), $H_0=10^{-2.5}$, and $m=10^{-6}$. The momentum scale $k$ is chosen appropriately for each era.