Questions about Density Parameter and Critical Density

[pdm]

Hi,I'm reading about critical density and I'm a bit confused about it's derivation.Solving the Einstein equations using the cosmological principle we get the (second) Friedmann equation:

$$
\bigg( \frac{\dot{a}}{a} \bigg)^2

= H^2

= \frac{8\pi G}{3}\rho -\frac{Kc^2}{a^2}

+ \frac{\Lambda c^2}{3} \tag{1}
$$Now the evolution of the universe depends on two things:

• I: The matter/energy content
• II: The curvature

Question 1: I'm confused as to why we can vary both, the matter/energy content and the curvature. Don't the Einstein equation introduce a direct relationship between those two? Or asked different: How could a universe with fixed energy/matter content have different curvature?

Now we make the following "assumptions".

• A1: We are in a flat universe i.e. $K=0$
• A2: There is no dark energy i.e. $\Lambda=0$
• A3: $\Omega_/text{total} = 1$

Question 2: What is the reason for A1? I know we "know" it from observation but some details would be nice.
Question 3: The value of the critical density will depend heavily on A2. So why do we assume A2? Is it just for historical reasons?
Question 4: Why do we assume unity here? Has it to do with the initial condition problem?

Using A1 and A2, (1) turns into
$$
\bigg( \frac{\dot{a}}{a} \bigg)^2
= H^2(t)
= \frac{8\pi G}{3}\rho(t) \tag{2}
$$

We solve for $\rho$ and call it critical density:

$$
\rho_c(t) = \frac{3 H^2(t)}{8 \pi G} \tag{3}
$$

Furthermore we define the density parameter

$$
\Omega := \frac{\rho}{\rho_c}
$$

Furthermore we know that our universe consists of matter (baryonic and dark matter), radiation and dark energy, thus:

$$
\Omega_\text{total} = \Omega_{m,0} + \Omega_{\Lambda,0} + \Omega_{r,0}
$$

Now if

• $\Omega_\text{total} < 1$ we have negative curvature i.e. $K<-1$
• $\Omega_\text{total} = 1$ we have flat curvature i.e. $K=0$
• $\Omega_\text{total} > 1$ we have positive curvature i.e. $K<1$

Question 5: This is related to question 1. I'm very unsure about the relation dang of the curvature and the mass/energy content. I think this comes from the fact that cosmology is taught in a weird historical way where they start without dark energy and sudendly it's there but somehow we didn't adjust our equations. At least that's what it feels like. So the question is: Why can we do what we do in the above list? Didn't we already fix the curvature by using the critical density to being flat?

So if I see it correctly we basically do the following: We look at the universe without dark energy (for whatever reason), we fix the curvature (for whatever reason) and we figure out what energy/matter density that would lead to. We call that the critical density. (Probably because it marks the "cross-over point".)We then basically switch to a universe that contains dark energy but keep using everything we "derived" for a universe without dark energy.I hope my questions are clear. In general I'm just confused about the motivation behind all this.No idea if it's me or the topic but I find cosmology somehow "tedious" to learn, so feel free to suggest a good book about this topic.

 

[Orodruin]

Question 1: I'm confused as to why we can vary both, the matter/energy content and the curvature. Don't the Einstein equation introduce a direct relationship between those two? Or asked different: How could a universe with fixed energy/matter content have different curvature?

Don't confuse the Riemann curvature tensor with the spatial curvature of a Robertson-Walker spacetime.

Question 2: What is the reason for A1? I know we "know" it from observation but some details would be nice.

Why do you need another reason than knowing this from observation? Basically, the CMB would look very different if it were not the case (or almost the case).

Question 3: The value of the critical density will depend heavily on A2. So why do we assume A2? Is it just for historical reasons?

Assuming $\Lambda = 0$ clearly does not correspond to our current universe. I am not sure what reference you are using but there is no a priori need to exclude the cosmological constant from the definition of the critical density. You should start by defining what you mean by "critical density" and it is usually defined exactly like that: The energy density for which the spatial curvature is zero.

A3 is a consequence of the universe being flat.

You do not need to exclude the cosmological constant, you can perfectly well include it as part of the energy density (just remove the $\Lambda$ term and bake it into the density term, i.e., associate an energy density with the cosmological constant).

 

[pdm]

Don't confuse the Riemann curvature tensor with the spatial curvature of a Robertson-Walker spacetime.

My GR isn't very good (that course is only coming next semester for me) but IIRC didn't the Riemann curvature just basically boil down to K no? I know the Riemann curvature talks about the 4D space-time while the FRW just talks about 3D space. IIRC that comes from the fact that the cosmological principle doesn't apply for time. I fail to see the conclusion here though. Let me ask again in a very simple way.

Assume the energy/matter content of the universe is fixed. Does that imply its curvature or can it have any curvature? Could it be that it fixes the spatial curvature of the FRW-spacetime but not the "real" spacetime curvature?

Why do you need another reason than knowing this from observation? Basically, the CMB would look very different if it were not the case (or almost the case).

Of course I don't need any reasoning, I'm just confused so I basically attack anything that can be attacked.

Assuming clearly does not correspond to our current universe. I am not sure what reference you are using but there is no a priori need to exclude the cosmological constant from the definition of the critical density. You should start by defining what you mean by "critical density" and it is usually defined exactly like that: The energy density for which the spatial curvature is zero.

A3 is a consequence of the universe being flat.

You do not need to exclude the cosmological constant, you can perfectly well include it as part of the energy density (just remove the term and bake it into the density term, i.e., associate an energy density with the cosmological constant).

I didn't mean to say that the current universe has no dark energy.

Okay so this should be an rather easy concept no? No idea why I have such a hard time really getting it. So let's take a step back: What is the motivation to introduce the critical density?

 

[Orodruin]

IIRC didn't the Riemann curvature just basically boil down to K no?

No, not at all. They are very different concepts even.

Assume the energy/matter content of the universe is fixed. Does that imply its curvature or can it have any curvature? Could it be that it fixes the spatial curvature of the FRW-spacetime but not the "real" spacetime curvature?

No, rather the opposite. The energy content (or more specifically, the stress-energy tensor which contains energy and momentum densities as well as stresses) form the right-hand side of Einstein's equations and the left-hand side is the Einstein tensor, which depends on the Riemann curvature tensor. Spatial curvature is not an object of 4-dimensional spacetime but rather a property of a (rather arbitrary) choice of coordinates where the cosmological principle happens to apply.

What is the motivation to introduce the critical density?

It is just a reference point. It is the energy density for which the universe is flat. If you have more energy density the universe is closed and if you have less it is open.

Edit: Compare with the concept of escape velocity of a celestial body. If you have more velocity you escape to infinity with finite asymptotic velocity. If you have less, you fall back. What is the motivation to introduce the escape velocity? Nothing really, except that it is a convenient tool for telling you some qualitative things about freely falling objects.

 

[pdm]

No, not at all. They are very different concepts even.

No, rather the opposite. The energy content (or more specifically, the stress-energy tensor which contains energy and momentum densities as well as stresses) form the right-hand side of Einstein's equations and the left-hand side is the Einstein tensor, which depends on the Riemann curvature tensor. Spatial curvature is not an object of 4-dimensional spacetime but rather a property of a (rather arbitrary) choice of coordinates where the cosmological principle happens to apply.

Okay thanks, I'll hopefully understand this distinction better once I take the GR course.

It is just a reference point. It is the energy density for which the universe is flat. If you have more energy density the universe is closed and if you have less it is open.

Yeah, of course it's just a reference point. It's the energy density for which a universe without dark energy is flat.

Do we define the critical density like I do in (3) just for historical reasons?

And to sum it up and make sure I got it:

We have a universe. That universe consists of matter and energy. Depending on the density introduced by this matter and energy content we get one possibility for curvature of our universe: closed, flat, open.
Depending on the curvature and the kind of matter/energy content we get different scenarios for the evolution of the universe.
For convenience, we use the arbritary but historical motivated reference value (3).

 

[Orodruin]

Yeah, of course it's just a reference point. It's the energy density for which a universe without dark energy is flat.

As I said, the dark energy would typically also be included in the energy density. Then it is simply the energy density for which a universe is flat. Much simpler.

That universe consists of matter and energy.

Energy is not a thing, it is a property of things. In the simplest versions of cosmology, the universe is assumed to contain different ideal fluids. The relevant properties for those are described by the stress-energy tensor, which includes energy density.

Depending on the curvature and the kind of matter/energy content we get different scenarios for the evolution of the universe.

This is true only if you have no dark energy though. If you introduce dark energy, the border between being open/closed is no longer what determines the final evolution.

 

[pdm]

As I said, the dark energy would typically also be included in the energy density. Then it is simply the energy density for which a universe is flat. Much simpler.

AH! So people actually don't use (3)? Like if I'd do research now then I'd define my critical density in term of whatever model I use?

Thanks, I really got the impression we always use (3) and that just seemed a bit weird to me. Especially since some books explilictly set $\Lambda=0$.

Energy is not a thing, it is a property of things. In the simplest versions of cosmology, the universe is assumed to contain different ideal fluids. The relevant properties for those are described by the stress-energy tensor, which includes energy density.

I actually never thought too much about why my book keeps writing matter/energy. Thanks for pointing that out.

This is true only if you have no dark energy though. If you introduce dark energy, the border between being open/closed is no longer what determines the final evolution.

I remember that yeah, the whole discussion about evolution of the universe and dark energy was a bit weird and not very concrete. But that's something I didn't look into too much.

Thanks for discussing this a bit here - I think I'm more confident in the whole topic now. We'll see. Have a nice day/evening.