Inflationary power spectrum from the Lanczos algorithm

TL;DR

The paper develops an open quantum-system description of inflationary curvature perturbations by applying a generalized Lanczos algorithm to a Lindblad dynamics, yielding an open two-mode squeezed state (OTMSS) built from second-kind Meixner polynomials. The BD vacuum is connected to the OTMSS via a Bogoliubov transformation with coefficients $\alpha_k=\cosh r_k$ and $\beta_k=-e^{-i\phi_k}\sinh r_k$, and the resulting power spectrum for the curvature perturbation is found to be observationally indistinguishable from the BD prediction (ratio $\gamma_k \approx 1$ after horizon exit). The framework explicitly separates open-system dissipation (parameterized by $\mu_2$) from closed-system squeezing, offering a group-theoretic, model-independent route to compute correlation functions and potential signatures in higher-order statistics, with extensions to multifield or modified-gravity contexts. Overall, the work provides a universal, symmetry-based description of quantum fluctuations in curved spacetime that recovers standard results while highlighting possible decoherence effects in non-Gaussian observables.

Abstract

The generalized Lanczos algorithm can provide a universal method for constructing the wave function under the group structure of Hamiltonian. Based on this fact, we obtain an open two-mode squeezed state as the quantum origin for the curvature perturbation. In light of this wave function in the open system, we successfully develop a new method to calculate its corresponding power spectrum by using the Bogoliubov transformation. Unlike traditional approaches, we explicitly retain the Bogoliubov coefficients in terms of the squeezing amplitude \( r_k \) and the squeezing rotation angle \( φ_k \). As a result, the power spectrum of the open two-mode squeezed state will match that of the Bunch-Davies vacuum numerically. Furthermore, the derivation of the open two-mode squeezed state relies on the second kind Meixner polynomial (equivalent to the generalized Lanczos algorithm) and the symmetry of the Hamiltonian. Therefore, our research may offer a new insight into the calculation of the correlation functions through a group-theoretic perspective.

Figures (5)

• Figure 1: The numeric of squeezed parameter $r_k$ in terms of $k$ in the horizon crossing. The dotted line is the reference point of $k_*=0.05\ \hbox{Mpc}^{-1}$.
• Figure 2: The numeric of squeezed parameter $\phi_k$ in terms of $k$ in the horizon crossing. The dotted line is the reference point of $k_*=0.05\ \hbox{Mpc}^{-1}$.
• Figure 3: The numeric of squeezed parameter $|\beta_k|^2$ in terms of $k$ in the horizon crossing. The dotted line is the reference point of $k_*=0.05\ \hbox{Mpc}^{-1}$.
• Figure 4: The numeric of squeezed parameter $\gamma_z(k)$ in terms of $k$ in the horizon crossing. The dotted line is the reference point of $k_*=0.05\ \hbox{Mpc}^{-1}$.
• Figure 5: The numeric of power spectrum $\Delta_{\mathcal{R}z}^2(k)$ in terms of $k$ in the horizon crossing. The dotted line is the reference point of $k_*=0.05\ \hbox{Mpc}^{-1}$.