  The large-scale power deficit in the cosmic microwave background fluctuations
might be relevant with the physics of pre-inflation, a bounce, or a
superinflationary phase preceding slow-roll inflation, which can provide a
singular-free realization of inflation. We investigate the primordial
perturbations from such pre-inflationary evolutions, which generally may
consist of multiple phases with different background dynamics, and give a
universal formula for the power spectrum of primordial perturbations in terms
of the recursive Bogoliubov coefficients. We also apply our formula to
corresponding cases and show how the intensity of large-scale power suppression
is affected by the pre-inflationary physics.
