# Light travel time since dark energy - matter equivalence

1. Dec 3, 2014

### 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 Ω0m = 0.315 , Ω0 = 1
Thus: Ω will equal Ω0 - Ω0m= 0.685

ρ0m (1 + zeq)3 = ρ

where: ρ0m = ρ0c * Ω0m
and: ρ = ρ0c * Ω

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?

2. Dec 3, 2014

### Staff: Mentor

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.

3. Dec 4, 2014

### 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

R3/2 ∝ 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.