# Curvature, Omega, the Flatness problem, and the evolving shape of the universe

I'm a little confused by this:

http://en.wikipedia.org/wiki/Flatness_problem

Which seems to imply the universe is more curved now than it was soon after the Big Bang. Look at the graph on the right side. It's not drawn to scale but clearly implies that with continued expansion the mass-energy density to critical density ratio (Omega) will change which should mean a difference in curvature... no?

In other words the term |Ω − 1| is currently less than 0.01, and therefore must have been less than 10−62 at the Planck era.

So while it's still close to flat, it was closer to flat in the early period of the universe than it is now? And this is due to the mass-energy density decreasing quicker (due to expansion) than the curvature?

• And this is a problem because this implies the curvature had to be closer to flat (Omega had to be closer to 1) very early on, and this low level of curvature could not have contributed enough to the expansion of the universe to match the contribution of the mass-energy density? When they should have matched? Implying curvature was being governed by some other factor? (Assuming one didn't go the philosophical route of invoking the anthropic principle)

• OR am I reading the Wiki's language incorrectly and it merely says that curvature and mass-energy density should match but they don't as mass-energy density decreases faster than curvature which implies curvature in the past approaches the necessary value for the critical density? Which then implies that curvature is being affected by more than just mass-energy density?

So the inflationary model says there's an inflaton field which drove the sharp exponential inflation during the early period of the universe which quickly flattened out the universe (overriding the effect of mass-energy density?) and then the curvature began to deviate away again slowly as the universe continued to expand?

So now the curvature will increase slightly to match our current Omega which is slightly off the critical density? Should this not then mean that the universe is actually not flat but spherical (due to omega slightly greater than one) and soon to be hyperbolic because of the decreasing energy density? (due to expansion driven by dark energy)?

Does the expansion from dark energy affect flatness?

• – Qmechanic Apr 8 '12 at 11:58
• – John Rennie Apr 9 '12 at 14:37

## 1 Answer

"So the inflationary model says there's an inflaton field which drove the sharp exponential inflation during the early period of the universe which quickly flattened out the universe (overriding the effect of mass-energy density?) and then the curvature began to deviate away again slowly as the universe continued to expand?"

Popular articles tend to be make of mess of this, so I'll try to be precise.

Firstly, the curvature of the universe people talk about is that of a "co-moving section" - think of it as a "slice of the universe at a specific time point". (The "co-moving" bit is just to make precise sense of "at a specific time point".) This is different from the curvature of space-time which GR talks about. An expanding universe where co-moving sections are perfectly flat is still curved in this sense.

Anyway, imagine the co-moving section looks like a sphere. (This is a special case, but easiest to illustrate. Technically, this is the case where curvature is positive). Now, the curvature parameter simply measures the ratio of the distance to the observable horizon ("the observable universe") to the radius of the sphere.

Note that the distance to the horizon is directly proportional to t = Time since Big Bang.

Now when we throw in expansion, what happens is that in a universe dominated by matter and radiation, the radius of the sphere increases in proportion to a lower power of t, generally t^1/2 or t^2/3. So, it follows that the curvature will increase with time. Its just like as you can see more and more of the Earth, its curvature becomes more apparent.

During inflation, the sphere radius increases as exp(t), which means that the horizon becomes really small compared to the radius. Hence the curvature is "pushed towards zero". But yes, once matter and energy take over it gets pushed away from zero again.

Now this is where popular articles REALLY mess up, so let me belabor this a bit.

Inflation doesn't really predict that the universe should be flat.

All it says is: If the universe is approximately flat today - ie, the radius of curvature is of the same order of magnitude as the horizon - then the radius needed to be much, much bigger than the horizon very soon after the Big Bang (ie curvature was very close to zero)

But the mechanism for inflation is such that we generally expect that the radius is much, much bigger than the horizon even today - despite the fact that the curvature has been increasing due to the presence of matter and energy.

And yes, if dark energy is just a cosmological constant then the curvature will be driven to zero again in the long term.

Hope that helps.