Characterizing inflationary perturbations: The uniform approximation

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 percent. This meets the accuracy requirement for interpreting next-generation cosmic microwave background observations.

Figure 1

Sketch of the potential barrier for density perturbations. The vertical dashed lines delineate the three different regions where the solution for $u_k$ is investigated as explained in the text. $\overline{\eta }$ marks the point where the $k$-mode enters into the potential barrier, the turning point. Inflation lasts over a wide $\eta$-range, extending Region II as indicated by the arrows. In the far right of Region III radiation dominates.

Figure 2

Comparison of the exact power spectrum for power-law inflation with the uniform approximation and the slow-roll approximation in leading-order.

Figure 3

Comparison of the exact power spectrum for power-law inflation with the uniform approximation and the slow-roll approximation in next-to-leading-order. The uniform approximation and the exact solution are indistinguishable in this plot.