Can cosmological observations uniquely determine the nature of dark energy?

The observational effect of all minimally coupled scalar field models of dark energy can be determined by the behavior of the following two parameters: (1) equation of state parameter $w$, which relates dark energy pressure to its energy density, and (2) effective speed of sound ${c}_{e}^{2}$, which relates dark energy pressure fluctuation to its density fluctuation. In this paper we show that these two parameters do not uniquely determine the form of a scalar field dark energy Lagrangian even after taking into account the perturbation in the scalar field. We present this result by showing that two different forms of scalar field Lagrangian can lead to the same values for these paired parameters. It is well known that from the background evolution the Lagrangian of the scalar field dark energy cannot be uniquely determined. The two models of dark energy presented in this paper are indistinguishable from the evolution of background as well as from the evolution of perturbations from a Friedmann-Robertson-Walker metric.