Separate universe in multifield inflation: a phase-space approach

In this article we extend a study of the validity conditions of the separate-universe approach of cosmological perturbations to models of inflation with multiple fields. The separate-universe approach consists in describing the universe as a collection of homogeneous and isotropic patches, giving us an effective description of perturbation theory at large scales through phase-space reduction. This approximation is a necessary step in stochastic inflation, an effective theory of coarse-grained, super-Hubble, scalar fields fluctuations. One needs a stochastic inflation description in the context of primordial black hole productions since it needs enhancements of the curvature power spectrum. It easily achievable in multifield inflation models but necessarily comes with strong diffusive effects. We study and compare cosmological perturbation theory and the separate-universe approach in said non-linear sigma models as a typical framework of multifield inflation and employing the Hamiltonian formalism to keep track of the complete phase space (or the reduced isotropic phase space in the separate-universe approach). We find that the separate-universe approach adequately describes the cosmological perturbation theory provided the wavelength of the modes considered is greater that several lower bounds that depend on the cosmological horizon and the inverse of the effective Hamiltonian masses of the fields; the latter being fixed by the coupling potential and the field-space geometry. We also compare gauge-invariant variables and several gauge fixing procedures in both approaches. For instance, we showed that the uniform-expansion gauge is nicely described by the separate-universe picture, hence qualifying its use in stochastic inflation as commonly done.