Accelerating universe in terms of Hankel function index

In this paper, [Formula: see text] cosmology is proposed for the accelerating universe with asymptotic de Sitter expansion in terms of Hankel function index [Formula: see text]. To some extent, both the initial expansion during early inflation and the current accelerated expansion can be studied with a vacuum cosmic fluid, i.e. [Formula: see text] in the pure de Sitter phase. Observational data further support the notion of a quasi-vacuum fluid, rather than a pure vacuum, contributing to the quasi-de Sitter acceleration in both the early and late universe. By examining the asymptotic expansion of the Henkel function as an approximate solution of the Mukhanov–Sasaki equation, we seek a more detailed study of quasi-de Sitter solutions in cosmology containing vacuum-like fluid.