Analytical reconstruction of equivalent purely kinetic k-essence description for $c_s^2(w)$ barotropic fluid models

A novel approach to the class of cosmic barotropic fluids in which the speed of sound squared is defined as a function of the Equation of State parameter, so called $c_s^2(w)$ models, is examined. For this class of models, a new analytical reconstruction method is introduced for finding their equivalent purely kinetic k-essence formulation. The method is explicitly demonstrated for several $c_s^2(w)$ models. A distinguished feature of the studied models is a peak in $c_s^2$ during cosmic expansion. A possible role that such model features could play in understanding of so far unexplained aspects of cosmic evolution is discussed. The application of the obtained explicit or closed form solutions in understanding dark sector unification models is outlined.