  We examine the theoretical foundations of standard methods for computing
density perturbations in inflationary models. We find that: (1) the time-delay
formalism (introduced by Guth and Pi, 1982) is only valid when inflation is
well-described by the de Sitter solution and the equation-of-state is nearly
unchanging; and, (2) the horizon-crossing/Bessel approximation extends to
non-exponential inflation, but only if the equation-of-state is changing
slowly. Integration of the gauge-invariant perturbation equations mode-by-mode
is the only method reliable for general models. For models with rapidly varying
equation-of-state, the correction leads to significantly different predictions
for the microwave background anisotropy. An important corollary is that methods
proposed for "reconstruction" of the inflaton potential from anisotropy data
are unreliable for general models.
