  We show that the amount of particle production in an arbitrary cosmological
background can be determined using only the late-time positive-frequency modes.
We don't refer to modes at early times, so there is no need for a Bogolubov
transformation. We also show that particle production can be extracted from the
Feynman propagator in an auxiliary spacetime. This provides a first-quantized
formalism for computing particle production which, unlike conventional
Bogolubov transformations, may be amenable to a string-theoretic
generalization.
