  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.
