Radiation Question: Estimating Ωm & Time to t2×

[bolahab]

Hey guys, can someone please help with this question.

We now live in a time when the energy balance of the universe is dominated
by Λ. A long time from now, at t2× the universe will have doubled
in size, i.e.: a(t2×) = 2. At present, radiation is negligible: Ωr,0 ~ 0,
and will remain so, while Ωm,0 = 0.3 and ΩΛ,0 = 0.7 now. Note that
since ΩΛ,0 + m,0 = 1 now, their sum will remain the same.
a) Estimate the value of Ωm at t2×. Consider how the ratio Ωm/ΩΛ
changes with time, and use the flatness constraint mentioned above.
b) Estimate the time from now until t2×.
c) What will the Hubble distance be at t2×?

 

[George Jones]

Can you express

$$\frac{\Omega_m \left( t \right)}{\Omega_\Lambda \left( t \right)}$$

in terms of

$$\frac{\Omega_{m,0}}{\Omega_{\Lambda,0}}$$

and $a \left( t \right)$?

 

[bolahab]

Yeah it can be expressed like that

 

[George Jones]

Yeah it can be expressed like that

This with $a \left( t \right) = 2$ and $\Omega_\Lambda \left( t \right) + \Omega_m \left( t \right) = 1$ give two equations with two unknowns.

For part b), integrate the Friedmann equation.

 

[bolahab]

Thnx George that really helped, but can you please give more details ?