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The speed of sound of the inflaton field

  1. Jan 12, 2010 #1
    I've been reading about inflation and i encountered that one can always define the sound's speed as

    [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]?
  2. jcsd
  3. Jan 12, 2010 #2


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    The problem is that the literature often uses [itex]c_{\rm{s}}^2[/itex] to mean two different things, sometimes simultaneously. Looking at things from a thermodynamic perspective, one can write [itex]P=P(\rho,S)[/itex], and then perturb to give
    [tex]\delta P=\frac{\partial P}{\partial\rho}\delta \rho +\tau \delta S[/tex]
    where [itex]\frac{\partial P}{\partial\rho}[/itex] 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 [itex]P=P(X,\phi)[/itex]. Then, the adiabatic sound speed can be written as
    [tex]c_{\rm{s}}^2=\frac{\partial P}{\partial\rho}=\frac{\partial_X P +\partial_\phi P}{\partial_X\rho+\partial_\phi\rho}[/tex]. 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
    [tex]\tilde{c_{\rm{s}}}^2=\frac{\partial_X P}{\partial_X\rho}[/tex]. 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.
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