The $k$-essence scalar field in the context of Supernova Ia Observations
  A $k$-essence scalar field model having (non canonical) Lagrangian of the
form $L=-V(\phi)F(X)$ where $X=1/2g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$
with constant $V(\phi)$ is shown to be consistent with luminosity
distance-redshift data observed for type Ia Supernova. For constant $V(\phi)$,
$F(X)$ satisfies a scaling relation which is used to set up a differential
equation involving the Hubble parameter $H$, the scale factor $a$ and the
$k$-essence field $\phi$. $H$ and $a$ are extracted from SNe Ia data and using
the differential equation the time dependence of the field $\phi$ is found to
be: $\phi(t) \sim \lambda_0 + \lambda_1 t + \lambda_2 t^2$. The constants
$\lambda_i$ have been determined. The time dependence is similar to that of the
quintessence scalar field (having canonical kinetic energy) responsible for
homogeneous inflation. Furthermore, the scaling relation and the obtained time
dependence of the field $\phi$ is used to determine the $X$-dependence of the
function $F(X)$.
