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James McKeets
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Homework Statement
Show mathematically that a model with:
Ω_M0 = 3
Ω_Λ0 = 0.01
Ω_R0 = 0
Ω_T0 = 3.01
is a model that re-collapses in the future. Be certain to indicate at what value of the scale factor 'a' the expansion reverses and becomes contraction.
Homework Equations
It's hinted pretty strongly that we should probably be using:
( (d(a)/dt ) / a ) = H^2 {Ω_M0 a^{-3} + Ω_Λ0 - (Ω_T0 - 1) a^{-2}}
And that we should be solving a cubic somewhere along the way
The Attempt at a Solution
So my first plan (I've spent many many hours on this) was to move the 'a' over to the RHS, and then expand out the (d(a)/dt ) using the Friedman equations. This allowed me to reduce the data and find a cubic eq in 'a' such that:
3a^3+0.01a-3=0
However, when I plot this, I get an exponentially increasing line. I was expecting kinda an arch, which would tell me that scale factor has increased, then decreased back to 0.
Any ideas?
If you need more information about the problem, I'd be happy to help, including further information about what I've tried so far.
Many thanks,
James
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