Logarithm of the scale factor as a generalised coordinate in a Lagrangian for dark matter and dark energy

A Lagrangian for the k-essence field is set up with canonical kinetic terms and incorporating the scaling relation of [R.J. Scherrer, Phys. Rev. Lett. 93 (2004) 011301]. There are two degrees of freedom, viz., q(t)=lna(t) (a(t) is the scale factor) and the scalar field ϕ, and an interaction term involving ϕ and q(t). The Euler-Lagrange equations are solved for q and ϕ. Using these solutions quantities of cosmological interest are determined. The energy density ρ has a constant component which we identify as dark energy and a component behaving as a−3 which we call dark matter. The pressure p is negative for time t→∞ and the sound velocity cs2=∂p∂ρ≪1. When dark energy dominates, the deceleration parameter Q→−1 while in the matter dominated era Q∼1/2. The equation of state parameter w=p/ρ is shown to be consistent with w=p/ρ∼−1 for dark energy domination and during the matter dominated era we have w∼0. Bounds for the parameters of the theory are estimated from observational data.