What is the time dependence of the Omegas?

[Amanheis]

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.

 

[Chronos]

I don't 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.

 

[Amanheis]

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]

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 don't 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.

 

[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.

 

[Amanheis]

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.

 

[protonchain]

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.

 

[Amanheis]

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.

 

[protonchain]

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.