# Difference between horizon and flatness problems & how inflation solves flatness (w/out math)

Layman here so I'm hoping for an answer for my query that doesn't involve math.

I'm reading about inflation and how it solves the flatness and horizon problems. I get that the horizon problem deals with the uniformity of matter/energy in the universe.

Flatness deals with the curvature. But isn't the universe flat because of the uniformity of matter/energy? Or is it the uniformity plus also the density (i.e. how far apart everything is)? In this way, is it correct to say that the flatness problem encapsulates the horizon problem?

Finally, is there any way to explain why inflation would have given this particular density without using math?

• A uniform energy distribution does not impoly that the spacetime must be flat, if that's what you were asking. About your other question: it seems pretty impossible to explain the mechanism of inflation precisely (and quantitatively) without resorting to mathematics.
– Danu
Feb 22 '14 at 17:06
• Hawking explains the mechanism without math in A Brief History which I sort of get. The gist is that a super-heated early universe allowed for the unification of the strong, weak, and electromagnetic forces and theoretically if the universe cooled fast enough the symmetry between these forces wouldn't have broken (i.e. they wouldn't have become different forces) which would have an anti-gravitational effect, causing the universe to expand faster than it already was. But what I don't get is why inflation is needed to explain flatness. Why would the universe not be flat without inflation? Feb 23 '14 at 9:52

In cosmology, "flat" doesn't mean the opposite of "rough." It means the opposite of "curved in a global sense." The surface of a sphere is not flat, even if it is smooth, as the surface everywhere has positive curvature. A saddle has everywhere negative curvature. An uncurved plane is flat in the cosmological sense, even if it has some bumps and ripples on it.

As you can (and should) read on Wikipedia, very basic cosmology just assumes a few different constituents of the universe - normal matter, dark matter, radiation, dark energy - uniformly distributed everywhere. You can then ask

1. What is the overall curvature of the universe with this composition, according to general relativity?
2. How do the relative amounts of different components (including curvature) evolve over time, again according to general relativity.

It is easily shown that flatness comes from a fine balance between the components, and, what's more, any slight deviation from perfect flatness will grow over time. If the universe started off slightly positively curved, it would be enormously positively curved today, and the same holds for negative curvature.

The problem is that we observe the present-day universe to be flat to within the precision of our measurements. Inflation is a proposed mechanism by which the universe was flattened out very well early on, so that curvature will not have grown out of control by the present era.

Note that in some sense this only pushes the question further back. After all, given a fixed amount of smoothing caused by inflation, you can always choose pre-inflation initial conditions to be curved enough such that inflation doesn't fix it. Ideas like this have led some cosmologists to question how well inflation really solves things (PDF here). The ideas behind inflation have changed a lot over the last several decades, and they are still evolving.

• Thanks for your response. So the balance of the existence of the constituents of the universe (which you've listed) is what causes flatness (on the scale of the universe) - got it! Now does the uniform distribution of these constituents also play a role? Could the universe still be flat if its constituents were not uniformly distributed? Furthermore, is density of these constituents (i.e. how far apart they are from themselves/each other) a factor as well? Thanks a ton! Feb 23 '14 at 11:36
• Also if it's the balance of the existence of the constituents of the universe that causes flatness, how did inflation cause this balance? Feb 23 '14 at 11:56
• I'd argue that flatness is only particularly meaningful if there is some scale larger than which everything is uniform (as is the case in our universe). As for how inflation achieves this: You can treat the curvature term in the equations exactly the same as another substance (matter, radiation, dark energy, and curvature simply have different scalings with time), and going through the math you see the relative contribution of the curvature term compared to the others goes to 0.
– user10851
Feb 23 '14 at 13:35
• So then uniformity does lead to flatness i.e. flatness and uniformity are intrinsically linked. And it sounds like there's no way to explain how inflation causes flatness without delving into the equations. Feb 24 '14 at 9:32
• I'd like to point out that the fine balance you refer to is used to rule out an open universe and it does the job well. But the only constraint is imposes on the actual curvature is that $\Omega+\Omega_k=1$. Most measurements which calculate this balance are performed under the assumption that the universe is flat as opposed to open. The data itself does permit a value for $\Omega_k$ of about -0.1, which is a closed universe. However,the larger one makes the curvature, the more dark energy is needed to offset it and thus one also must change the matter density.
– Jim
Mar 25 '14 at 13:59