The speed of sound of the inflaton field

[chronnox]

I've been reading about inflation and i encountered that one can always define the sound's speed as

c_s^2 \equiv \frac{\partial_X P}{\partial_X \rho}

where X \equiv \frac{1}{2} g^{ab} \partial_a \phi \partial_b \phi. In the case of a canonical scalar field P=X-V and \rho=X+V, so c_s^2=1. That is what is obtained by definition. But i can always consider P and \rho as a function of P=(X,\phi) and \rho=(X,\phi) so

P+\rho=2 X and

\rho-P= 2 V

taking variations of these last to equations i obtain

\delta P = - \delta \rho + 2 \delta X (1) and

\delta P = \delta \rho - 2 \partial_\phi V \delta \phi (2)


Recalling that in general P=(\rho,S) then \delta P = c_s^2 \delta \rho + \tau \delta S. Thus if i read the coefficient of \delta \rho of eq. (1) one obtains that c_s^2 = -1 and \tau \delta S = 2 \delta X, but if i read the coefficient of eq. (2) one obtains c_s^2 = 1 and \tau \delta S = - 2 \partial_\phi V \delta \phi, according to the definition the correct reading would be the one done by (2) but is there another explanation of why reading the coefficient c_s^2 from (2) is the correct way, or is there a motivation for the first definition for c_s^2?

 

[cristo]

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.