# Solutions of Friedmann Equations

• petmal
In summary, you are asking how to get the scale factor as a function of time from the Friedmann Equations when k != 0, lambda = 0. Mathematica 5.2 won't give you a solution, and wherever you search for the best solution, you find the parametric solution in terms of \Theta: \pmb{R[\Theta ]=\alpha (1-\text{Cos}[\Theta ]);}\\\pmb{t=\beta (\Theta -\text{Sin}[\Theta ]);}\f

#### 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, $$\Lambda$$ = 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 $$\Theta$$:

$$\noindent$$\pmb{R[\Theta ]=\alpha (1-\text{Cos}[\Theta ]);}\\$$ $$\pmb{t=\beta (\Theta -\text{Sin}[\Theta ]);}$$$$

for a closed universe and:

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

for an open universe.

Where $$\noindent$$\pmb{\alpha , \beta , \gamma , \delta }$$$$ 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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I changed a theta to a psi. Is this what you intended?
$$\noindent$$\pmb{R[\psi ]=\gamma (\text{Cosh}[\psi ]-1);}\\$$ $$\pmb{t=\delta (\text{Sinh}[\psi ]-\psi );}$$$$

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.

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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:

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

where $$\rho _0$$ is the density as measured today...

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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.

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...

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.

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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 $$\noindent$$\pmb{\Omega _0}$$$$ and so on)... But none of them explains how they got to graphing R(t) vs t...

... 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 [Broken]

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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 [Broken]

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[$$\Theta$$] and y = R[$$\Theta$$]...

I was probably expecting something more complicated...

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