# Time dependence of the Omegas

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.

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Chronos
Gold Member
I dont think time dependency is the right approach if you allow for the possibility time is an emergent propoerty of the universe. The formula you cite does not resolve this issue.

I am sorry, I don't understand. What formula do you mean? Which one of them is wrong? And what do you think is the right approach? I mean, call it a-dependence instead of time-dependence, but the question remains: How got Carroll to his little figure of the a-dependence of d/da Omega_Lambda?

marcus
Gold Member
Dearly Missed
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.

I dont see the reason for your step 3.
Step 3. seems wrong. You seem to be acting as if you thought that H is proportional to a.

But H is much larger in the past
while the scalefactor a is smaller in the past.
So there can be no simple proportionality between H and a.

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Chalnoth
Yes, step 3 is wrong. It's not quite that easy. You have to take into account the full contents of the universe to compute rho_c in terms of said contents.

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Well my reasoning was that H is defined as "a dot over a". And I somehow assumed that "a dot" is independent of "a", just like a general coordinate "q dot" is independent of "q" in classical mechanics.

But thanks, I'll take a closer look on the critical density.

Second thing you should note is that Omega_M + Omega_L isn't = 1 at all times. You have to consider Omega_R (radiation) and Omega_K (curvature). The sum of all these values is = 1.

Well I obviously neglected radiation (as it is very common) and stated at the beginning that I am assuming a flat universe.

Also, I am now assuming that rho_c = rho_M0/a^3 + rho_Lambda.

With this, rho_M/rho_c + rho_Lambda/rho_c is always 1 and neither Omega_M nor Omega_Lambda leave the intervall [0,1]. This also yields the same graph as the one by Sean Carroll, which is why I think I am on the right track.

Note that this presumes that k=0 at any given moment. I am still not exactly sure why k is constant for k=0 and not constant if k>/<0.

Ah I apologize I didn't see the assumption of a flat universe. I know radiation isn't a big deal right now, and is almost negligible but at different times of the Universe's lifespan, radiation had been a factor, and sometimes even dominant (when you delve into the past). So that's why I thought I'd put that out there. I'm sure they're assuming that radiation has always been 0 or negligible.