  Particle production by slow-changing gravitational fields is usually
described using quantum field theory in curved spacetime. Calculations require
a definition of the vacuum state, which can be given using the adiabatic (WKB)
approximation. I investigate the best attainable precision of the resulting
approximate definition of the particle number. The standard WKB ansatz yields a
divergent asymptotic series in the adiabatic parameter. I derive a novel
formula for the optimal number of terms in that series and demonstrate that the
error of the optimally truncated WKB series is exponentially small. This
precision is still insufficient to describe particle production from vacuum,
which is typically also exponentially small. An adequately precise
approximation can be found by improving the WKB ansatz through perturbation
theory. I show quantitatively that the fundamentally unavoidable imprecision in
the definition of particle number in a time-dependent background is equal to
the particle production expected to occur during that epoch. The results are
illustrated by analytic and numerical examples.
