Asymptotic-de Sitter Modes for Early Universe Inflation

The recent data from observational Planck data about inflation indicate that our universe was nearly de Sitter space-time in the early times which results in an approximate scale-invariant spectrum. These observational data from scalar spectral index and curved space-time symmetry condition stimulate us to use quasi-de Sitter modes during inflation. To obtain the alternative modes, for first time, we consider the asymptotic expansion of the Hankel functions up to the higher order of 1⁄kτ . In this paper, we review the results due to selection of the generalized form of the asymptotic de Sitter inflationary modes, such as higher order corrections of power spectrum and particle creation, trans-Planckian corrections, equation of state, and non-minimal coupling inflation. These results are general and in flat and de Sitter limit confirm the conventional results. Finally, to reconstruction of the modes we fit the power spectrum of temperature anisotropies for model with ΛCDM model and Planck data.