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 Mentor P: 8,325 The problem is that the literature often uses $c_{\rm{s}}^2$ to mean two different things, sometimes simultaneously. Looking at things from a thermodynamic perspective, one can write $P=P(\rho,S)$, and then perturb to give $$\delta P=\frac{\partial P}{\partial\rho}\delta \rho +\tau \delta S$$ where $\frac{\partial P}{\partial\rho}$ is then identified as the adiabatic sound speed-- i.e. the speed with which perturbations travel through the background. Now, for a scalar field we can parametrise as $P=P(X,\phi)$. Then, the adiabatic sound speed can be written as $$c_{\rm{s}}^2=\frac{\partial P}{\partial\rho}=\frac{\partial_X P +\partial_\phi P}{\partial_X\rho+\partial_\phi\rho}$$. By writing things like this, it should be apparent that this is not the same as the first expression you quote. It turns out that, for a scalar field, the speed of propagation is not the adiabatic sound speed, but in fact a different speed (say, the "effective sound speed"), which is defined as $$\tilde{c_{\rm{s}}}^2=\frac{\partial_X P}{\partial_X\rho}$$. If you like, you can show this by calculating the Klein-Gordon equation for the perturbation of the field and looking at the term in front of the spatial derivative.