Purely kinetic k-essence description of $c_s^2(w)$ barotropic fluid
  models
  Purely kinetic k-essence models have been shown in the literature to be a
field theory equivalent of barotropic fluid models of dark energy or dark
matter-dark energy unification. In the modeling framework where the speed of
sound squared of a barotropic fluid is modeled as a function of its Equation of
State parameter, a systematic procedure of obtaining the Lagrangian density of
an equivalent purely kinetic k-essence model is presented. As this modeling
approach starts from the speed of sound, purely kinetic k-essence models can be
constructed for which the speed of sound is in agreement with the observational
constraints. Depending on the chosen functional form for the barotropic fluid
speed of sound squared, analytically tractable examples of solutions for the
purely kinetic k-essence Lagrangian density in parametric and closed form are
obtained.
