Causality, Stability and Sound Speed in Scalar Field Models

The result from the SN1A projects suggest that the dark energy can be represented by a fluid with $w<-1$. However, it is commonly argued that a fluid with $|w|>1$ contradicts causality. Here, we will show that a fluid with $|w|>1$ does not contradict causality if $w$ is not constant. Scalar field are the most promising candidates for describing the dark energy and they do not have a constant equation of state parameter $w$. For typical scalar potentials there are regions where $w $ is larger or smaller than one and even regions where $d p/d\rho$ diverges. We study the evolution of scalar field perturbations and we show that the "sound speed" is always smaller than the speed of light independently of the value of $w=p/\rho$ or $d p/d\rho$. In general, it is neither the phase velocity nor the group velocity that gives the "sound speed". Our results can be applied to all fluids and allows for a fluid with $w<-1$ without contradicting causality as long as there is dispersion.