Inflationary power spectrum from the Lanczos algorithm

Abstract The generalized Lanczos algorithm can provide a universal method for constructing a wave function under the the Hamiltonian group structure. 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 $$ r_k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> and the squeezing rotation angle $$ \phi _k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> . 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 a Meixner polynomial of the second kind (equivalent to the generalized Lanczos algorithm) and the symmetry of the Hamiltonian. Therefore, our research may offer a new insight into the calculation of correlation functions through a group-theoretic perspective.