  We investigate the transformation of initial conditions for primordial
curvature perturbations under two types of transformations of the associated
action: simultaneous redefinition of time and the field to be quantised, and
the addition of surface terms. The latter encompasses all canonical
transformations, whilst the time- and field-redefinition is a distinct,
non-canonical transformation since the initial and destination systems use
different times. Actions related to each other via such transformations yield
identical equations of motion and preserve the commutator structure. They
further preserve the time-evolution of expectation values of quantum operators
unless the vacuum state also changes under the transformation. These properties
suggest that it is of interest to investigate vacuum prescriptions that also
remain unchanged under canonical transformations. We find that initial
conditions derived via minimising the vacuum expectation value of the
Hamiltonian and those obtained using the Danielsson vacuum prescription are not
invariant under these transformations, whereas those obtained by minimising the
local energy density are. We derive the range of physically distinct initial
conditions obtainable by Hamiltonian diagonalisation, and illustrate their
effect on the scalar primordial power spectrum and the Cosmic Microwave
Background under the just enough inflation model. We also generalise the
analogy between the dynamics of a quantum scalar field on a curved,
time-dependent spacetime and the gauge-invariant curvature perturbation. We
argue that the invariance of the vacuum prescription obtained by minimising the
renormalised stress--energy tensor should make it the preferred procedure for
setting initial conditions for primordial perturbations. All other procedures
reviewed in this work yield ambiguous initial conditions, which is problematic
both in theory and practice.
