Realising Interactions Between Dark Matter and Dark Energy Using $k$-essence Cosmology

In this paper we exploit dynamics of a $k-$essence scalar field to realise interactions between dark components of universe resulting in a evolution consistent with observed features of late time phase of cosmic evolution. Stress energy tensor corresponding to a $k-$essence Lagrangian $L=V( \phi)F(X)$ (where $X=\frac{1}{2}g^{\mu\nu}\nabla_\mu\phi \nabla_\nu\phi$) is shown to be equivalent to an ideal fluid with two components having same equation of state. Stress energy tensor of one of the components may be generated from a constant potential $k-$essence Lagrangian of form $L_1=V_0F(X)$ ($V_0$ constant) and that of other from another Lagrangian of form $L_2=V_1(\phi)F(X)$ with $V=V_0 + V_1(\phi)$. We have shown that, the unified dynamics of dark matter and dark energy described by a single scalar field $\phi$ driven by a $k-$essence Lagrangian $L= V(\phi)F(X)$ may be viewed in terms of diffusive interactions between the two hypothetical fluid components `1' and `2' with stress energy tensors equivalent to that of Lagrangians $L_1$ and $L_2$ respectively. The energy transfer between the fluid components is determined by functions $V(\phi)$, $F(X)$ and their derivatives. Such a realisation is shown to be consistent with the Supernova Ia data with certain constraints on the temporal behaviour of $k-$essence potential $V(\phi)$. We have described a methodology to obtain such constraints.