Radiation Question: Estimating Ωm & Time to t2×

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The discussion revolves around estimating the matter density parameter Ωm at a future time t2× when the universe has doubled in size. Given that the current values are Ωm,0 = 0.3 and ΩΛ,0 = 0.7, the relationship between Ωm and ΩΛ is crucial for understanding how they evolve over time. Participants are encouraged to use the flatness constraint and integrate the Friedmann equation to estimate the time until t2× and the Hubble distance at that time. There is a request for more detailed explanations on how to express the ratio of Ωm(t) to ΩΛ(t) in terms of their current values and the scale factor a(t). Overall, the thread seeks clarity on these cosmological calculations.
bolahab
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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×?
 
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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)?
 
Yeah it can be expressed like that
 
bolahab said:
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.
 
Thnx George that really helped, but can you please give more details ?
 
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