Cosmological perturbation theory with matter time

TL;DR

This paper develops a Hamiltonian framework for cosmological perturbation theory using matter time, incorporating scalar fields and dust, and derives gauge-invariant variables with a time-dependent oscillator structure.

Contribution

It introduces a novel Hamiltonian formulation of cosmological perturbations employing dust as a matter-time gauge, enabling gauge-invariant analysis with a time-dependent oscillator analogy.

Findings

Constructed a momentum space Hamiltonian for linear perturbations.

Demonstrated the first class nature of the constraint system.

Expressed the Hamiltonian in gauge-invariant canonical variables as a time-dependent oscillator.

Abstract

We study cosmological perturbation theory with scalar field and pressureless dust in the Hamiltonian formulation, with the dust field chosen as a matter-time gauge. The corresponding canonical action describes the dynamics of the scalar field and metric degrees of freedom with a non-vanishing physical Hamiltonian and spatial diffeomorphism constraint. We construct a momentum space Hamiltonian that describes linear perturbations, and show that the constraints to this order form a first class system. We then write the Hamiltonian as a function of certain gauge invariant canonical variables and show that it takes the form of an oscillator with time dependent mass and frequency coupled to an ultralocal field. We compare our analysis with other Hamiltonian approaches to cosmological perturbation theory that do not use dust time.