Improved WKB analysis of cosmological perturbations

Improved Wentzel-Kramers-Brillouin (WKB)-type approximations are presented in order to study cosmological perturbations beyond the lowest order. Our methods are based on functions which approximate the true perturbation modes over the complete range of the independent (Langer) variable, from subhorizon to superhorizon scales, and include the region near the turning point. We employ both a perturbative Green’s function technique and an adiabatic (or semiclassical) expansion (for a linear turning point) in order to compute higher order corrections. Improved general expressions for the WKB scalar and tensor power spectra are derived for both techniques. We test our methods on the benchmark of power-law inflation, which allows comparison with exact expressions for the perturbations, and find that the next-to-leading order adiabatic expansion yields the amplitude of the power spectra with excellent accuracy, whereas the next-to-leading order with the perturbative Green’s function method does not improve the leading order result significantly. However, in more general cases, either or both methods may be useful.

Figure 1

The frequency ${\omega }^2$ for $\beta =-2$. The turning point is at $x_{*}=\ln (3/2)\simeq 0.4$

Figure 2

The function $\xi$ for $\beta =-2$.

Figure 3

The “perturbation” $\sigma$ for $\beta =-2$.

Figure 4

The total error on the power spectrum to leading order and to next-to-leading order for the perturbative correction (superposed upper lines) and to next-to-leading order for the adiabatic correction (lower line).

Figure 5

The functions ${\varphi }_{(1)}(x)$ (solid line) and ${\gamma }_{(1)}(x)$ (dotted line) for $\beta =-2$.