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Implications of JLA data for $k-$essence model of dark energy with given equation of state

We investigated implications of recently released `Joint Light-curve Analysis' (JLA) supernova Ia (SNe Ia) data for dark energy models with time varying equation of state of dark energy, usually expressed as $w(z)$ in terms of variation with corresponding redshift $z$. From a comprehensive analysis of the JLA data, we obtain the observational constraints on the different functional forms of $w(z)$, corresponding to different varying dark energy models often considered in literature, \textit{viz.} CPL, JBP, BA and Logarithmic models. The constraints are expressed in terms of parameters ($w_a, w_b$) appearing in the chosen functional form for $w(z)$, corresponding to each of the above mentioned models. Realising dark energy with varying equation of state in terms of a homogeneous scalar field $ϕ$, with its dynamics driven by a $k-$essence Lagrangian $L=VF(X)$ with a constant potential $V$ and a dynamical term $F(X)$ with $X=(1/2)\nabla^μϕ\nabla_μϕ$ we reconstructed form of the function $F(X)$. This reconstruction has been performed for different varying dark energy models at best-fit values of parameters ($w_a, w_b$) obtained from analysis of JLA data. In the context of $k-$essence model, we also investigate the variation of adiabatic sound speed squared, $c_s^2(z)$, and obtained the domains in ($w_a, w_b$) parameter space corresponding to the physical bound $ c_s^2>0$ implying stability of density perturbations.