What is the theoretical limit to sound speed in a plasma?
In a monoatomic gas, it can be demonstrated that the equation of state in the relativistic regime is given by $\rho = \rho_0 + 3p/c^2$, which would indicate the maximum speed of sound for this gas is $c/\sqrt{3}$ instead of $c$. In a plasma, the mechanism for sound is based on the pressure of electrons and inertia of the ions, which is typically given in the non-relativistic regime by $\sqrt{\frac {\gamma_e T_e + \gamma_i T_i}{m_i}}$.
In the ultra-relativistic regime, I would suspect the total pressure and inertia to involve significant contributions from both ions and electrons, and the expression to be modified since the mechanism is based on fluctuating electric fields which may be enhanced at scales below the DeBye length when the charged electrons and ions are ultra-relativistic.
