Krylov Complexity in early universe

TL;DR

This work investigates Krylov complexity across the entire early-universe evolution, treating the cosmos as an open quantum system and contrasting closed- and open-system Lanczos formalisms from inflation through radiation and matter domination. It develops an open-system two-mode squeezed-state framework built with the second kind of Meixner polynomials, deriving coupled evolution equations for the squeezing parameters $r_k$ and $\phi_k$ and explicit expressions for the Lanczos/data-dynamics coefficients. Across three representative inflationary potentials, the authors find that Krylov complexity and Krylov entropy exhibit similar qualitative trends, growing during inflation and saturating in RD/MD, while open-system dissipation ($\mu_2$) strongly suppresses operator growth compared to the closed-system case. Inflation is identified as strongly dissipative, whereas RD and MD behave as weakly dissipative regimes, with decoherence-like tendencies becoming pronounced in the open-framework results. The study provides a quantum-information perspective on early-universe dynamics and offers a versatile, model-robust platform for probing inflationary physics and potential links to holographic complexity conjectures in an open-system setting.

Abstract

The Lanczos algorithm offers a framework for constructing wave functions in closed and open quantum systems from their Hamiltonians. Since the early universe is inherently an open system, we employ this algorithm to investigate Krylov complexity across various cosmological phases: inflation, radiation domination (RD), and matter domination (MD). Our results highlight a clear distinction in Krylov complexity between the closed- and open-system methodologies. To accurately capture the influence of potentials during RD and MD, we examine a set of inflationary potentials, including the Higgs potential, $R^2$ inflation, and chaotic inflation, while incorporating violations of slow-roll conditions. This study is conducted in conformal time through the preheating stage. Numerically, we find that the evolution of Krylov complexity and Krylov entropy shows remarkable similarity across different potentials during RD and MD. Furthermore, we rigorously construct an open two-mode squeezed state using the second kind of Meixner polynomial. Based on this construction, we derive for the first time the evolution equations for the squeezing parameter $r_k$ and phase $φ_k$ in terms of the scale factor. Our analysis indicates that dissipative effects lead to rapid decoherence-like behavior. In addition, we observe that the inflationary universe behaves as a strongly dissipative system, whereas during the RD and MD epochs the universe exhibits weak dissipative characteristics. This work opens new perspectives for studying the universe from a quantum-informational viewpoint.

Figures (10)

• Figure 1: The numeric of $V_{,\phi\phi}$ at first 50 iterations of $V_{\rm R^2,\phi\phi}$ and $V_{\rm Higgs,\phi\phi}$ in $\log_{10}a=0.2$ and $\log_{10}a=2.8$, in which we set $M_{p}=1$, $m=10^{-6}$; $M^{2}=10^{-12}$ at $V_{\rm R^2}$, and $\lambda A^{2}=10^{-4}$ at $V_{\rm Higss}$.
• Figure 2: The numeric of $V_{,\phi\phi}$ in terms of $\log_{10}a$ for potential $V_{\rm R^2}$, and $V_{\rm Higgs}$ in RD and MD, in which we replace its exact value with the iteration of $V_{\rm R^2}$, and $V_{\rm Higgs}$. We set $M_{p}=1$, $m=10^{-6}$; $M^{2}=10^{-12}$ for $V_{\rm R^2}$, and $\lambda A^{2}=10^{-4}$ for $V_{\rm Higgs}$.
• Figure 3: The numerical solution 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, $k=0.01$ for RD, $k=0.005$ for MD. The effect of potential is always ignored in slow-roll inflation. But in RD and MD periods, the impact of potential is taken into account.
• Figure 4: The numerical of Krylov complexity 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 Krylov entropy 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.