  In this paper, $F(\nu)$ cosmology is proposed for the accelerating universe
with asymptotic de Sitter expansion in terms of Hankel function index $\nu$. 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. $\Lambda$
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.
