  We explore perturbative double field theory about time-dependent
(cosmological) backgrounds to cubic order. To this order the theory is
consistent in a weakly constrained sense, so that for a toroidal geometry it
encodes both momentum and genuine winding modes. We give a self-contained
discussion of the consistency problems and their resolution, including the role
of cocycle factors and the $O(d,d,\mathbb{Z})$ duality. Finally, as a first
step toward the computation of cosmological correlation functions, we propose a
generalized scalar-vector-tensor decomposition and use it to construct gauge
invariant generalized Bardeen variables. Compared to standard cosmology there
are fewer tensor modes but more vector and scalar modes.
