  Developing our understanding of how correlations evolve during inflation is
crucial if we are to extract information about the early Universe from our
late-time observables. To that end, we revisit the time evolution of scalar
field correlators on de Sitter spacetime in the Schrodinger picture. By direct
manipulation of the Schrodinger equation, we write down simple "equations of
motion" for the coefficients which determine the wavefunction. Rather than
specify a particular interaction Hamiltonian, we assume only very basic
properties (unitarity, de Sitter invariance and locality) to derive general
consequences for the wavefunction's evolution. In particular, we identify a
number of "constants of motion": properties of the initial state which are
conserved by any unitary dynamics. We further constrain the time evolution by
deriving constraints from the de Sitter isometries and show that these reduce
to the familiar conformal Ward identities at late times. Finally, we show how
the evolution of a state from the conformal boundary into the bulk can be
described via a number of "transfer functions" which are analytic outside the
horizon for any local interaction. These objects exhibit divergences for
particular values of the scalar mass, and we show how such divergences can be
removed by a renormalisation of the boundary wavefunction - this is equivalent
to performing a "Boundary Operator Expansion" which expresses the bulk
operators in terms of regulated boundary operators. Altogether, this improved
understanding of the wavefunction in the bulk of de Sitter complements recent
advances from a purely boundary perspective, and reveals new structure in
cosmological correlators.
