Transformations with the Mukhanov variable (Cosmology)

[ergospherical]

Homework Statement

See below

Relevant Equations

N/A

I've been going around in circles for a while. We have a parameter $z$, defined through$$z^2 = 2a^2 \epsilon$$where $a$ is the scale factor and $\epsilon$ is the slow-roll parameter. Considering the action$$S = \frac{1}{2} \int d\tau d^3 x \ z^2 \left[(\mathcal{R}')^2 - (\partial_i \mathcal{R})^2 \right]$$with $\mathcal{R}$ the comoving curvature perturbation. We want to write this in terms of a new 'Mukhanov' variable $v \equiv z\mathcal{R}$. This is where the confusion starts - specifically with $\mathcal{R}'$. You can find easily that$$\mathcal{R}' = \frac{v'}{z} - \frac{z' v}{z^2}$$We are looking to arrive at $$S = \frac{1}{2} \int d\tau d^3 x \ \left[(v')^2 - (\partial_i v)^2 + \frac{z''}{z}v^2 \right]$$It looks like I need another equation, to get from $z'$s to $z''$s. I've previously showed that $\tfrac{z''}{z} = \mathcal{H}^2(2-\epsilon + \tfrac{3}{2}\eta)$, just from the definition of $z$ and taking some care to keep only first order perturbations, but can't see whether this is useful.

 

[billtodd]

Is it taken from some book or paper this question of yours?

 

[billtodd]

What is $\partial_i R$?
Derivative w.r.t what?

 

[ergospherical]

$\partial_i = \partial/\partial x^i$