Complete Hamiltonian analysis of cosmological perturbations at all orders II: Non-canonical scalar field

TL;DR

This paper develops a Hamiltonian framework for analyzing cosmological perturbations in non-canonical scalar fields, introducing new phase-space variables, and deriving higher-order interaction Hamiltonians to facilitate advanced cosmological studies.

Contribution

It introduces a new phase-space variable for non-canonical scalar fields and extends Hamiltonian analysis to higher orders, including interaction Hamiltonians and a generalized speed of sound.

Findings

New phase-space variable simplifies Hamiltonian expression.

Consistent Hamiltonian and Euler-Lagrange equations inversion method.

Derived third and fourth order interaction Hamiltonians.

Abstract

In this work, we present a consistent Hamiltonian analysis of cosmological perturbations for generalized non-canonical scalar fields. In order to do so, we introduce a new phase-space variable that is uniquely defined for different non-canonical scalar fields. We also show that this is the simplest and efficient way of expressing the Hamiltonian. We extend the Hamiltonian approach of [arXiv:1512.02539] to non-canonical scalar field and obtain a new definition of speed of sound in phase-space. In order to invert generalized phase-space Hamilton's equations to Euler-Lagrange equations of motion, we prescribe a general inversion formulae and show that our approach for non-canonical scalar field is consistent. We also obtain the third and fourth order interaction Hamiltonian for generalized non-canonical scalar fields and briefly discuss the extension of our method to generalized Galilean scalar fields.