Logarithm of the scale factor as a generalised coordinate in a lagrangian for dark matter and dark energy
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A lagrangian for the $k-$ essence field is set up with canonical kinetic terms and incorporating the scaling relation of [1]. There are two degrees of freedom, {\it viz.},$q(t)= ln\enskip a(t)$ ($a(t)$ 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 {\it negative} for time $t\to \infty$ and the sound velocity $c_{s}^{2}={\partial p\over\partialρ} << 1$. When dark energy dominates, the deceleration parameter $Q\to -1$ while in the matter dominated era $Q\sim {1\over 2}$. The equation of state parameter $w={p\over ρ}$ is shown to be consistent with $w={p\overρ}\sim -1$ for dark energy domination and during the matter dominated era we have $w\sim 0$. Bounds for the parameters of the theory are estimated from observational data.
Keywords: k-essence models, dark matter, dark energy
PACS No: 98.80.-k
16 pages, latex, paper shortened by 2 pages for journal publication
16 pages, latex, paper shortened by 2 pages for journal publication