Light travel time since dark energy - matter equivalence

[leonmate]

So I'm looking to find the distance light has traveled since matter - dark energy equivalence. Assume dark energy dominance from equivalence.

Space-time has flat geometry and Ω_0m = 0.315 , Ω₀ = 1
Thus: Ω_0Λ will equal Ω₀ - Ω_0m= 0.685

ρ_0m (1 + z_eq)³ = ρ_0Λ

where: ρ_0m = ρ_0c * Ω_0m
and: ρ_0Λ = ρ_0c * Ω_0Λ

I used this to find z = 0.296

Next, I need to use this somehow to find the light travel time. My guess is that I should use a solution of the Friedmann equation and use it to find a time? I haven't been able to figure out how to do this so far. I think I may just need to find the right Friedmann equation for dark energy dominance and work it out from there. But, it's missing from my notes and I don't know how to get there.

Any pointers here guys?

 

[mfb]

Next, I need to use this somehow to find the light travel time. My guess is that I should use a solution of the Friedmann equation and use it to find a time?

That would help. You know the matter and dark energy density as function of the scale factor, this allows to calculate its derivatives (using the values observed today).
I guess assuming a linear expansion wouldn't be so completely wrong, however.

 

[leonmate]

Ok,

Not entirely sure this is the best way to get a result but here's how I did it:

I showed in a previous question on this assignment that if you take the friedmann equation for a matter dominated universe you get:

(dR/dT)² * R^-2 = 8*π*G*ρ / 3 * R₀³/R³

If you solve this you eventually get to

R^3/2 ∝ t

Using this relationship I found the age of the Universe at z = 0.296 (9.29 billion years) and assumed that after this moment dark energy dominates.

In order to find my light travel time do I need to take into account that the universe is expanding? Use co-moving distance maybe? We're taking a flat universe btw.