Calculating RH & Rp in an Open Dust-Filled Universe

[sltungle]

1. Using the parametric solution of the Friedmann equations for a open, dust-filled universe

i) Calculate the radius of the Hubble sphere (R_H) for a 'dust filled' open universe.
ii) Compare this with the radius of the particle horizon (R_p) for the same universe and determine if there exists a time when R_H = R_p

Homework Equations

The Attempt at a Solution

For part (i) I've used the fact that,

to attempt to determine R_H.

Given that a is not in terms of t and the equations can't be rearranged to give a in terms of t I've opted to use the chain rule to determine a'(t)

This is the correct way to proceed with the question (according to my lecturer).

Here's where I get confused though. Because t is already a function of x, do I simply use a(x) as my a(t)? That being, is the 't' in a(t) already accounted for by the fact that a(t) is actually (a(t(x))), or do I have to take da/dt, 'multiply' both sides of my equation by dt (don't get angry at me, pure mathematicians), and then integrate (where dt is actually dt(x)) to find a(t)?

To answer part (ii) I intended to find the equation for the particle horizon and then plot both functions (R_H and R_p) on the same axes and check to see if there was ever an intercept, but I need to get R_H correct in order to do that, so I'm not going to jump ahead of myself just yet. However, that said, just from the looks of things I don't think all of my Ω₀ terms are going to drop out of the equation, so I'm not sure how I'm going to approach this if they stick around. I guess I might need to plot a 3D graph with x values, omega values, and respective R_H and R_p values.

I apologise if this is a bit convoluted. I intended to use latex, but I'm not sure how to get it working on here.

 

[sltungle]

Nobody? This question is really bugging me. I'd really appreciate the help.