A Value of quintessence while studying black hole thermodynamics

[djymndl07]

It is often seen in research papers that values of quintessence state parameter $\omega$ is taken between -1 and -1/3. However $\omega=-1$ corresponds to the cosmological constant and -1/3 corresponds to dust phase. Why these two cases are excluded in studying BH extended phase space thermodynamics?

 

[ergospherical]

$\omega = -1$ corresponds to a cosmological constant, with de Sitter (exponential) expansion. $-1 < \omega < -1/3$ corresponds to quintessence-like stress energy -- these models have null singularities (i.e. genuine past singularities), whereas de Sitter models have regular null surfaces that can be smoothly extended (i.e. they can exist eternally into the past). $w = -1/3$ corresponds to inertial expansion. As you can see, different $\omega$ leads to different causal structures.