Supernova data may be unable to distinguish between quintessence and k-essence

We consider the efficacy of using luminosity distance measurements of deep redshift supernovae to discriminate between two forms of dark energy, quintessence (a scalar field with canonical kinetic terms rolling down a potential) and $k$-essence (a scalar field whose cosmic evolution is driven entirely by non-linear kinetic terms). The primary phenomenological distinction between the two types of models that can be quantified by supernova searches (at least in principle) is that the equation of state $w\equiv p/\rho$ of quintessence is falling today while that of $k$-essence is rising. By simulating $10^5$ possible datasets that SNAP could obtain, we show that even if the mass density $\Omega_m$ is known exactly, an ambiguity remains that may not allow a definitive distinction to be made between the two types of theories.