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Hello everybody,

could somebody tell me how to get the scale factor as a function of time [R(t)] from the Friedmann Equations for a simple dust, pressureless universe when k != 0, [tex]\Lambda[/tex] = 0.

Mathematica 5.2 doesn't want to give me any solution and wherever I searched the best thing I got was the parametric solution in terms of [tex]\Theta[/tex]:

[tex]\noindent\(\pmb{R[\Theta ]=\alpha (1-\text{Cos}[\Theta ]);}\\[/tex]

[tex]\pmb{t=\beta (\Theta -\text{Sin}[\Theta ]);}\)[/tex]

for a closed universe and:

[tex]\noindent\(\pmb{R[\psi ]=\gamma (\text{Cosh}[\psi ]-1);}\\[/tex]

[tex]\pmb{t=\delta (\text{Sinh}[\psi ]-\psi );}\)[/tex]

for an open universe.

Where [tex]\noindent\(\pmb{\alpha , \beta , \gamma , \delta }\)[/tex] are some constants...

Everywhere I also found plots of R vs. t for various parameters but without showing the solution for R(t).

I guess I just need to somehow invert expressions for time...

Thanks for help.

Petr

could somebody tell me how to get the scale factor as a function of time [R(t)] from the Friedmann Equations for a simple dust, pressureless universe when k != 0, [tex]\Lambda[/tex] = 0.

Mathematica 5.2 doesn't want to give me any solution and wherever I searched the best thing I got was the parametric solution in terms of [tex]\Theta[/tex]:

[tex]\noindent\(\pmb{R[\Theta ]=\alpha (1-\text{Cos}[\Theta ]);}\\[/tex]

[tex]\pmb{t=\beta (\Theta -\text{Sin}[\Theta ]);}\)[/tex]

for a closed universe and:

[tex]\noindent\(\pmb{R[\psi ]=\gamma (\text{Cosh}[\psi ]-1);}\\[/tex]

[tex]\pmb{t=\delta (\text{Sinh}[\psi ]-\psi );}\)[/tex]

for an open universe.

Where [tex]\noindent\(\pmb{\alpha , \beta , \gamma , \delta }\)[/tex] are some constants...

Everywhere I also found plots of R vs. t for various parameters but without showing the solution for R(t).

I guess I just need to somehow invert expressions for time...

Thanks for help.

Petr

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