Chapter 11: Beyond the Standard Lore

When the evolution of the comoving Hubble radius is not accurately known the spectral energy density and the other observables can only be evaluated with approximate methods and the considerations presented in this chapter are a natural extension of the results obtained in the conventional lore. The Wentzel–Kramers–Brioullin (WKB) approach is particularly handy for the analysis of the graviton spectra when the early completion of the concordance paradigm does not coincide with the conventional inflationary models. A relevant area of applications also involves the decelerated stages whose rate are either faster or slower than radiation. The results of this chapter are useful whenever an accurate knowledge of the transition regimes between the different stages of the dynamics is lacking. After some specific examples both in the unpolarized and in the polarized case, we discuss a different class of approximation methods based on a modification of the time parametrization. While the notations followed hereunder are standard and coincide with the ones previously established, in the first part of the chapter we present a general discussion that can be applied, with the necessary modifications, also to the analysis of the scalar modes. This choice is practical also in view of the analysis presented in Chapter 12 where the scalar modes of the geometry will be shown to produce an effective anisotropic stress which may eventually affect the tensor modes and produce secondary spectra of relic gravitons. In the first three sections of the chapter we present a number of general results whereas the final two sections discuss some applications with particular attention to the comparison between the exact and the WKB results. In the first part a key step involves the analysis of the turning points that may be either regular or singular. In the two complementary situations the spectral energy density must be estimated is different terms.