Eddington-LeMaitre model (expanding universe,cosmological constant)

[shooride]

I'm trying to solve Friedmann equations for ##k=1 ## & ##p=0## (matter).
These are equations:
$$3(\dot{R}^2 +1)/R^2 -λ =8πGε$$
$$(2R\ddot{R}+\dot{R}^2 +1)/R^2 -λ=0$$
If we use ##(\dot{R}^2 +1) /R^2## and put it in the second equation:

##\ddot{R}+ω^2/6=0## & ##ω^2=8πGε-2λ##

If ##ω^2=0##, then we have Einstein's static model.
Furthermore, for ##ω^2>>0##, the solution is like ##R=A\exp(ωt)## or ##R=A\sin(ωt)+B\cos(ωt)##with suitable constants. But I know Eddington-LeMaitre model (expanding universe) and the solution above, it's not that and it doesn't have coasting period or others property of that model. I don't know that what is the my wrong?!
Can anyone help me?
Thanks for your consideration on my problem.

 

[AndyW]

Hello,

Thank you for sharing your equations and your attempts at solving them. It looks like you are on the right track, but there are a few things that need to be clarified.

Firstly, the Friedmann equations are a set of equations that describe the evolution of the universe in terms of its expansion rate and energy density. The first equation you provided is the Friedmann equation for a flat universe (##k=0##), while the second equation is the acceleration equation. The Friedmann equations are valid for all values of ##k##, including ##k=1##.

Next, let's take a closer look at your second equation. You have correctly substituted ##(\dot{R}^2 +1)/R^2## for ##\ddot{R}##, but you seem to have made a mistake in your substitution for ##ω^2##. The correct substitution is ##ω^2=8πGε/3-λ/R^2##. This is because the energy density term in the Friedmann equation is divided by 3, and the curvature term is divided by ##R^2##. This means that the value of ##ω^2## will depend on the value of ##R##, and will not be constant.

Finally, it is important to note that the solutions you have provided are for a matter-dominated universe (##p=0##). This means that the energy density term in the Friedmann equation will only depend on the matter density, and not on the curvature term (##λ##). This is why the value of ##ω^2## does not change for different values of ##λ## in your solutions. However, if you were to consider a universe with other forms of energy, such as radiation or dark energy, then the value of ##ω^2## would change and you would see different solutions.

I hope this helps to clarify your understanding of the equations and their solutions. Please let me know if you have any further questions. Good luck with your research!