Hamiltonian formalism for cosmological perturbations: the separate-universe approach

TL;DR

This paper reformulates the separate-universe approach within the Hamiltonian formalism to analyze large-scale cosmological perturbations, enabling a detailed phase-space understanding and quantum analysis, while establishing conditions for its validity.

Contribution

It introduces a Hamiltonian phase-space formulation of the separate-universe approach, extending its applicability to non-attractor regimes and quantum properties of perturbations.

Findings

The approach reproduces full perturbation dynamics for large-scale modes.

A lower wavelength bound for the approach's validity is derived.

Matching procedures between approaches require specific prescriptions.

Abstract

The separate-universe approach provides an effective description of cosmological perturbations at large scales, where the universe can be described by an ensemble of independent, locally homogeneous and isotropic patches. By reducing the phase space to homogeneous and isotropic degrees of freedom, it greatly simplifies the analysis of large-scale fluctuations. It is also a prerequisite for the stochastic-inflation formalism. In this work, we formulate the separate-universe approach in the Hamiltonian formalism, which allows us to analyse the full phase-space structure of the perturbations. Such a phase-space description is indeed required in dynamical regimes which do not benefit from a background attractor, as well as to investigate quantum properties of cosmological perturbations. We find that the separate-universe approach always succeeds in reproducing the same phase-space dynamics for homogeneous and isotropic degrees of freedom as the full cosmological perturbation theory, provided that the wavelength of the modes under consideration are larger than some lower bound that we derive. We also compare the separate-universe approach and cosmological perturbation theory at the level of the gauge-matching procedure, where the agreement is not always guaranteed and requires specific matching prescriptions that we present.