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We investigate the transformation of initial conditions for primordialcurvature perturbations under two types of transformations of the associatedaction: simultaneous redefinition of time and the field to be quantised, andthe addition of surface terms. The latter encompasses all canonicaltransformations, whilst the time- and field-redefinition is a distinct,non-canonical transformation since the initial and destination systems usedifferent times. Actions related to each other via such transformations yieldidentical equations of motion and preserve the commutator structure. Theyfurther preserve the time-evolution of expectation values of quantum operatorsunless the vacuum state also changes under the transformation. These propertiessuggest that it is of interest to investigate vacuum prescriptions that alsoremain unchanged under canonical transformations. We find that initialconditions derived via minimising the vacuum expectation value of theHamiltonian and those obtained using the Danielsson vacuum prescription are notinvariant under these transformations, whereas those obtained by minimising thelocal energy density are. We derive the range of physically distinct initialconditions obtainable by Hamiltonian diagonalisation, and illustrate theireffect on the scalar primordial power spectrum and the Cosmic MicrowaveBackground under the just enough inflation model. We also generalise theanalogy between the dynamics of a quantum scalar field on a curved,time-dependent spacetime and the gauge-invariant curvature perturbation. Weargue that the invariance of the vacuum prescription obtained by minimising therenormalised stress--energy tensor should make it the preferred procedure forsetting initial conditions for primordial perturbations. All other proceduresreviewed in this work yield ambiguous initial conditions, which is problematicboth in theory and practice.

Quantum initial conditions for inflation and canonical invariance