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Particle production by slow-changing gravitational fields is usuallydescribed using quantum field theory in curved spacetime. Calculations requirea definition of the vacuum state, which can be given using the adiabatic (WKB)approximation. I investigate the best attainable precision of the resultingapproximate definition of the particle number. The standard WKB ansatz yields adivergent asymptotic series in the adiabatic parameter. I derive a novelformula for the optimal number of terms in that series and demonstrate that theerror of the optimally truncated WKB series is exponentially small. Thisprecision is still insufficient to describe particle production from vacuum,which is typically also exponentially small. An adequately preciseapproximation can be found by improving the WKB ansatz through perturbationtheory. I show quantitatively that the fundamentally unavoidable imprecision inthe definition of particle number in a time-dependent background is equal tothe particle production expected to occur during that epoch. The results areillustrated by analytic and numerical examples.Comment: 14 pages, RevTeX, 5 figures; minor changes, a clarification in Sec.  II

Cosmological particle production and the precision of the WKB approximation