The $k$-essence scalar field in the context of Supernova Ia Observations

TL;DR

This paper demonstrates that a specific $k$-essence scalar field model aligns with supernova luminosity distance data, deriving the field's time evolution and its relation to the model's kinetic function.

Contribution

It introduces a $k$-essence model consistent with supernova data and derives the field's time dependence and the form of the kinetic function $F(X)$.

Findings

The $k$-essence model fits supernova luminosity data.

The scalar field's time dependence is quadratic in time.

The $X$-dependence of $F(X)$ is determined from the model.

Abstract

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 $F(X)$.