Equation of state and density parameter

[Ranku]

TL;DR Summary

Relation between equation of state and density parameter.

Does $w$ = -1 by itself imply that spacetime is flat, or is it only due to $\Omega$_Matter + $\Omega$_Λ = 1 ?

 

[PeterDonis]

spacetime is flat

You are talking about space, not spacetime; more precisely, "space" in standard comoving coordinates.

 

[kimbyd]

Summary: Relation between equation of state and density parameter.

Does $w$ = -1 by itself imply that spacetime is flat, or is it only due to $\Omega$_Matter + $\Omega$_Λ = 1 ?

It doesn't imply that space is flat. But a universe which has $w = -1$ will, over time, become more and more spatially flat (as PeterDonis notes, the space-time is still curved regardless).

This arises due to the fact that the effect of the curvature scales as $1/a^2$. In the presence of a cosmological constant which does not change with the expansion, the impact of the curvature will eventually dilute away to nothing. This is the source of the claim that inflation predicts a nearly-flat universe: inflation has an early expansion phase with $w \approx -1$ that drives the early universe to flatness.

 

[Ranku]

It doesn't imply that space is flat. But a universe which has $w = -1$ will, over time, become more and more spatially flat (as PeterDonis notes, the space-time is still curved regardless).

This arises due to the fact that the effect of the curvature scales as $1/a^2$. In the presence of a cosmological constant which does not change with the expansion, the impact of the curvature will eventually dilute away to nothing. This is the source of the claim that inflation predicts a nearly-flat universe: inflation has an early expansion phase with $w \approx -1$ that drives the early universe to flatness.

I am trying to understand the relation between spatial flatness as in $w$ = -1, and spacetime flatness as in $\Omega$_Matter + $\Omega$_Λ = 1.
I am reading in 'Oxford Companion to Cosmology' about the cosmological constant: "The inflationary cosmology predicted a flat universe, while observations indicated that the density of matter, including dark matter, fell well short of the critical density needed to achieve this. The cosmological constant was able to plug this gap." So, isn't a correlation being made between $w$ = -1 and $\Omega$_Matter + $\Omega$_Λ = 1?

 

[kimbyd]

I am trying to understand the relation between spatial flatness as in $w$ = -1, and spacetime flatness as in $\Omega$_Matter + $\Omega$_Λ = 1.
I am reading in 'Oxford Companion to Cosmology' about the cosmological constant: "The inflationary cosmology predicted a flat universe, while observations indicated that the density of matter, including dark matter, fell well short of the critical density needed to achieve this. The cosmological constant was able to plug this gap." So, isn't a correlation being made between $w$ = -1 and $\Omega$_Matter + $\Omega$_Λ = 1?

No, these are unrelated.

In any flat universe, $\Omega = 1$. That's just the way spatial flatness works. One way to think of it is to consider the first Friedmann equation (you can assume the cosmological constant is wrapped into $\rho$ here):

$$H^2 = {8\pi G \over 3} \rho - {k c^2 \over a^2}$$

Here we see that the rate of expansion is proportional to density plus a spatial curvature term. That constant $k$ is determined solely by the initial conditions of the expansion. If the rate of expansion is fast but there isn't a lot of matter, then $k$ is a large negative value. If the rate of expansion is slow but there is a lot of matter, then $k$ is a large positive value. This is basically the same idea as throwing a ball near the Earth. If you don't throw the ball very fast, it will fall back to Earth. If, however, you have a cannon which can launch the ball at high enough velocity, it will escape the Earth entirely and fly off through space. It's all about how much speed and how much mass you have around, and parameter $k$ represents that relationship.

Because GR is all about geometry, this parameter manifests as spatial curvature.

So, going back to the inflationary cosmology quote, what we have is a process in the early universe (inflation) which sets the initial conditions for the later expansion. It basically requires that one component of those initial conditions is near-perfect spatial flatness. So whatever the contents of the universe, they have to match with the expansion just because of the dynamics of inflation. The remaining argument basically amounts to, "Okay, we've measured $\Omega_m \approx 0.25$. What makes up the rest?"

The remaining stuff could, at this stage of the discussion, be literally anything, as long as it doesn't cluster with matter. To go further you have to include more lines of evidence, and from those the cosmological constant seems to be a reasonable explanation, with most other alternatives failing to explain the evidence. But it has nothing to do with why the universe is spatially flat in the first place: that's down to the behavior of the early universe.

 

[Ranku]

But it has nothing to do with why the universe is spatially flat in the first place: that's down to the behavior of the early universe.

So to be consistent with $k$ = 0, we need the cosmological constant, whose equation of state is $w$ = -1, and which also participates in $\Omega$_Matter + $\Omega$_Λ = 1. Isn't that like a chain of correlations?

 

[kimbyd]

So to be consistent with $k$ = 0, we need the cosmological constant, whose equation of state is $w$ = -1, and which also participates in $\Omega$_Matter + $\Omega$_Λ = 1. Isn't that like a chain of correlations?

To be consistent with $k=0$, we need something. The cosmological constant is the simplest possibility (it's just a constant), and it fits the data. But it's still technically possible for something other than the cosmological constant to be the culprit.