Obviously, sound (like every other causal phenomena) may not travel faster than the speed of light. I know that materials with a high bulk modulus and low density will typically have faster speeds of sound, but is there a theoretical limit due to either a condition relating the density and bulk modulus, or some relativistic condition beneath the propagation of sound?
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There is no restriction other than $c_s<c$. Relativistic plasmas and fluids explore this regime. A weakly coupled quark gluon plasma has $c_s=c/\sqrt{3}$. Even higher speeds are reached in neutron stars, see Is the speed of sound almost as high as the speed of light in neutron stars? .
The speed of sound is related to the adiabatic compressibility $$ c_s^2 = \left(\frac{\partial P}{\partial \rho}\right)_s \, . $$ This quantity also enters the neutron structure via the TOV equation. It constrains, in particular, the maximum mass and the mass-radius relation. The recent observation of a 2-solar mass neutron star implies that $c_s$ becomes quite large, certainly bigger than $0.5c$.
Adendum: See here for a more quantitative analysis based on the existence of a 2 M(solar) neutron star, and equ.(10) of this paper for a theoretical limit in which we can shows that $c_s\to c$.
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Per the above link, the theoretical limit for the speed of sound in condensed matter is $$ v_{max} = \alpha c \sqrt{\frac{m_e}{2m_p}} \approx \frac{c}{8304} $$
where $\alpha$ is the fine structure constant, $c$ is the speed of light, $m_e$ is the mass of the electron and $m_p$ is the mass of the proton.
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