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I am really puzzled. I have several questions about how Omega_M and Omega_Lambda evolve with time. Ultimately I want to reconstruct figure 1 one Sean Carroll's Website http://nedwww.ipac.caltech.edu/level5/March01/Carroll/Carroll1.html.

First of all, I will assume a flat universe throughout this post. As I understand, this means the universe was and will be always flat, although I couldn't proof this. If someone has a quick proof for that, I'd appreciate it.

That also means, that for all times, Omega_M + Omega_Lambda = 1. (By Omega_M0 etc. I mean the density parameter or whatever now, anything else means it is dependent on a.)

Now my problems:

1. Omega_M = rho_M/rho_c

2. rho_M = rho_M0/a^3

3. rho_c ~ a^2 because of H^2 in the denominator of rho_c (fixed a sign there...)

4. Hence Omega_M = Omega_M0/a^5

5. Therefore Omega_Lamda = 1 - Omega_M0/a^5 ??

But since rho_Lambda = const, how is Omega_Lambda defined such that eq. 5 holds for all times? It can't be a simple a^n dependence, especially not rho_Lambda/rho_c (except for n=5, but why would that be).

Also, if Omega_M = const/a^5, what happens if a is small enough early in the universe such that Omega_M > 1? I feel really stupid.

First of all, I will assume a flat universe throughout this post. As I understand, this means the universe was and will be always flat, although I couldn't proof this. If someone has a quick proof for that, I'd appreciate it.

That also means, that for all times, Omega_M + Omega_Lambda = 1. (By Omega_M0 etc. I mean the density parameter or whatever now, anything else means it is dependent on a.)

Now my problems:

1. Omega_M = rho_M/rho_c

2. rho_M = rho_M0/a^3

3. rho_c ~ a^2 because of H^2 in the denominator of rho_c (fixed a sign there...)

4. Hence Omega_M = Omega_M0/a^5

5. Therefore Omega_Lamda = 1 - Omega_M0/a^5 ??

But since rho_Lambda = const, how is Omega_Lambda defined such that eq. 5 holds for all times? It can't be a simple a^n dependence, especially not rho_Lambda/rho_c (except for n=5, but why would that be).

Also, if Omega_M = const/a^5, what happens if a is small enough early in the universe such that Omega_M > 1? I feel really stupid.

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