Newtonian limit of cosmological perturbation

[Libra82]

Homework Statement

Problem in question is problem 5.6 in Dodelson's Modern Cosmology (https://www.amazon.com/dp/0122191412/?tag=pfamazon01-20)
Take the Newtonian limit of Einstein's equations. Combine the time-time equation (5.27) with the time-space equations of exercise 5 to obtain the algebraic (i.e. no time derivatives) equation for the potential given in Eq. (5.81). Show that this reduces to Poisson's equation (with the appropriate factors of a) when the wavelength is much smaller than the horizon ( k\eta >> 1).

Homework Equations

The equations mentioned in the problem are:
(5.27): $k^2 \Phi + 3\frac{\dot{a}}{a}\left(\dot{\Phi} - \psi \frac{\dot{a}}{a}\right) = 4\pi G a^2 (\rho_m \delta_m + 4\rho_r \Theta_{r,0})$
(5.81): $k^2 \Phi = 4\pi G a^2 \left(\rho_m \delta_m + 4\rho_r\Theta_{r,0} + \frac{3aH}{k}\left[i \rho_m v_m + 4\rho_r \Theta_{r,1} \right] \right)$
(5.84): $aH\psi - \dot{\Phi} = \frac{4\pi G a^2}{k}\left( i\rho_m v_m + 4\rho_r \Theta_{r,1} \right)$

We are working in a scalar perturbed metric so $g_{00} = -(1-2\psi)$, $g_{ij} = \delta_{ij} a^2(1+2\Phi)$.
$k$ is the wave vector of the perturbation.
Dot derivative indicate differentiation with respect to conformal time, $\dot{\Phi} = \frac{\partial \Phi}{\partial \eta}$.

The Attempt at a Solution

As I read the problem I am to find a combination of equations (5.27) and (5.84) to arrive at (5.81) and already here I run into problems. I've been trying various combinations for a couple of hours and I have yet to arrive at anything useful.

I can easily get the right hand side of equation (5.81) by multiplying (5.84) with $3aH$ and then adding this equation to (5.27) but the left hand side is always the problem. I seem to either have too few factors of $a$ or too many.

This is where I currently stand:
$k^2 \Phi + 3H\dot{\Phi} - 3H^2 \psi + 3a^2H^2\psi - 3aH\dot{\Phi} = \text{same as r.h.s. in eq. (5.81)}$.

To show the reduction to Poisson's equation I'd assume that $a = const.$ so $\dot{a} = 0$ thus arriving at Poisson's equation in Fourier space.

 

[Libra82]

Apparently I just need enough coffee to get it right.

I had forgotten that the Hubble rate which is defined as $H = a_{,0}/a$ also has to be converted to a conformal time derivative. By using $H = \dot{a}/a^2$ where dot-derivative means $\frac{\partial}{\partial \eta}$ I arrived at the equation I wanted. :)