Solutions of Friedmann Equations

[petmal]

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

 

[marcus]

I changed a theta to a psi. Is this what you intended?

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

for an open universe.

my apologies if this is wrong, or isn't what you meant.

It seems to me that you are asking if someone can express psi analytically as a function of t. Offhand I don't see how to do this.

 

[petmal]

I changed a theta to a psi. Is this what you intended?


my apologies if this is wrong, or isn't what you meant.

It seems to me that you are asking if someone can express psi analytically as a function of t. Offhand I don't see how to do this.

Simply, say, I want to make a plot of evolution of the scale factor R vs. t. Such plots are probably in every book which has something to do with cosmology.

To do this I need R as a function of t, unfortunatelly have no idea how to get it.

When k = 0, it's a simple differential equation solved on a piece of paper in a few moments, but what about k != 0 (open/closed universe)... :shy:

Just to be complete, I am adding the Friedmann Equation I am talking about:

[tex]\noindent\(\pmb{\left(\frac{dR[t]}{dt}\right)^2-\frac{8 \pi \rho _0 G}{3 R[t]}=-k c^2}\)[/tex]

where [tex]\rho _0[/tex] is the density as measured today...

 

[ziad1985]

Sometimes parametric equation like the one you have can't be transformed into simple explicit function...
This maybe just the case.
Still you can plot it , shouldn't be very hard.

 

[petmal]

Sometimes parametric equation like the one you have can't be transformed into simple explicit function...
This maybe just the case.

Thanks, you confirmed what I was thinking about.

Still you can plot it , shouldn't be very hard.

I guess this is what I don't know to do so I need R in terms of t...

 

[ziad1985]

on a second look I think you can express t as function of r
You have r as a function of psi, you can write psi=acosh((r+gamma)/Gamma)
and then replace psi in the second equation containing psi and t.
just came trough my mind , try it.

 

[petmal]

on a second look I think you can express t as function of r
You have r as a function of psi, you can write psi=acosh((r+gamma)/Gamma)
and then replace psi in the second equation containing psi and t.
just came trough my mind , try it.

I doubt the result could be solved explicitly for R...

Right now I've got three books in Astrophysics/Cosmology here, every of them provides the parametric solutions (above) and clearly explains how to get the age of the universe from it (expressing parameters in terms of [tex] \noindent\(\pmb{\Omega _0}\) [/tex] and so on)... But none of them explains how they got to graphing R(t) vs t...

 

[ziad1985]

... But none of them explains how they got to graphing R(t) vs t...

Graphing parametric equations is something I took in my calculus course in my first year in college...
It shouldn't be that hard, I suppose you can learn how to graph it, by looking into a calculus book...

try this link to draw: http://www.ies.co.jp/math/java/calc/sg_para/sg_para.html

 

[petmal]

Graphing parametric equations is something I took in my calculus course in my first year in college...
It shouldn't be that hard, I suppose you can learn how to graph it, by looking into a calculus book...

try this link to draw: http://www.ies.co.jp/math/java/calc/sg_para/sg_para.html

Thanks a lot, you opened my eyes... For some reason I didn't see the simple solution of plotting it as a parametric equation where x = t[[tex]\Theta[/tex]] and y = R[[tex]\Theta[/tex]]...

I was probably expecting something more complicated...