Inflationary power spectrum from the Lanczos algorithm

TL;DR

This paper introduces a novel method using the Lanczos algorithm and Bogoliubov transformation to compute the inflationary power spectrum from an open two-mode squeezed state, aligning with the Bunch-Davies vacuum.

Contribution

It develops a new approach to calculate the inflationary power spectrum by explicitly incorporating Bogoliubov coefficients and leveraging group-theoretic methods with the Lanczos algorithm.

Findings

Power spectrum matches the Bunch-Davies vacuum numerically.

Explicit retention of Bogoliubov coefficients enhances calculation accuracy.

Utilizes Meixner polynomial and Hamiltonian symmetry for state derivation.

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 \( \phi_k \). 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.