I've been reading about inflation and i encountered that one can always define the sound's speed as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]c_s^2 \equiv \frac{\partial_X P}{\partial_X \rho}[/tex]

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

[tex]P+\rho=2 X[/tex] and

[tex]\rho-P= 2 V[/tex]

taking variations of these last to equations i obtain

[tex]\delta P = - \delta \rho + 2 \delta X[/tex] (1) and

[tex]\delta P = \delta \rho - 2 \partial_\phi V \delta \phi[/tex] (2)

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

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# The speed of sound of the inflaton field

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