Modified scaling in $k$-essence model in interacting dark energy - dark matter scenario

It has been shown by $\mathit{\text{Scherrer}}\mathit{\text{and}}\mathit{\text{Putter}}\mathit{\text{et.al}}$ that, when dynamics of dark energy is driven by a homogeneous $k-$essence scalar field $\phi$, with a Lagrangian of the form $L=V_{0}F(X)$ with a constant potential $V_0$ and $X=\frac{1}{2}{\mathrm{\nabla }}^{\mu }\phi {\mathrm{\nabla }}_{\mu }\phi =\frac{1}{2}{\dot{\phi }}^2$, one obtains a scaling relation $X(dF\mathrm{/}dX{)}^2=C{a}^{-6}$ , where $C$ is a constant and $a$ is the FRW scale factor of the universe. The separate energy conservation in the dark energy sector and the constancy of $k-$essence potential are instrumental in obtaining such a scaling. In this paper, we have shown that even when considering time-dependent interactions between dark energy and dark matter, the constancy of $k-$essence potential may lead to a modified form of scaling. We have obtained such a scaling relation for a particular class of parametrisation of the source term occurring in the continuity equation of dark energy and dark matter in the interacting scenario. We used inputs from the JLA analysis of luminosity distance and redshift data from Supernova Ia observations, to obtain the modified form of the scaling.