The spectrum of primordial fluctuations from inflation can be obtained using
a mathematically controlled, and systematically extendable, uniform
approximation. Closed-form expressions for power spectra and spectral indices
may be found without making explicit slow-roll assumptions. Here we provide
details of our previous calculations, extend the results beyond leading order
in the approximation, and derive general error bounds for power spectra and
spectral indices. Already at next-to-leading order, the errors in calculating
the power spectrum are less than a per cent. This meets the accuracy
requirement for interpreting next-generation CMB observations.Comment: 17 pages, 3 figure