Ranku said:
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.