  Primordial black holes could constitute part or all of dark matter but they
require large inhomogeneities to form in the early universe. These
inhomogeneities can strongly backreact on the large scale dynamics of the
universe. Stochastic inflation provides a way of studying this backreaction and
getting an estimation of the abundance of primordial black holes. Because
stochastic inflation focuses on large scale dynamics, it rests on the separate
universe approach. However, the validity of this approach has only been checked
in single field models, but not in multifield models in which we expect strong
boosts in the power spectrum, leading to the formation of primordial black
holes. We will check the validity of a separate universe approach in multifield
models by matching it with a complete cosmological perturbation theory approach
at large scales. In particular, we wish to compare these two paradigms and
their differences in the adiabatic and entropic directions of the phase space.
This will give us a range of validity and conditions one needs to verify in
order to apply the separate universe approach and stochastic inflation in
multifield models.
