How Can the Universe Be Curved?

In the study guide notes and in my lecture class talks we learn of the “Omega” parameter, or the ratio of the observed density of the Universe to the density that would just exactly make the Universe geometrically “flat.”

But what does it mean to say that the Universe is “flat”? Astronomers, physicists and mathematicians use the term “flat” (as well as “curved”) in this context basically for lack of a better term. They are speaking not so much to the physical shape as to the mathematical meaning. We define “geometries” based on how the math works under those circumstances, not necessarily how things “look” visually. Mathematically, geometries define what things can do under those particular circumstances. For example, we can define the geometries of how things can move on a 2-dimensional surface like a uniform tabletop or wall or floor tile. Under such circumstances, the math works out in what we call “flat” or Euclidean geometry, such as you might have experienced in middle or high school. In this case, things (imagine ants crawling across a flat wall) can move forward and backward or side to side, but not up into the air or into the wall. There are only two axes of motion, which is why it is called 2-dimensional or 2D.

But we can also define the geometry for other types of 2-dimensional surfaces. Imagine those same ants crawling on the surface of a large ball or an outwardly curved surface called a “saddle shape.”

(This graphic I found a NASA website helps visualize the various possible “geometries” of the Universe. )

In all of these cases, the ants are restricted to only two directions of motion relative to the surface on which they are crawling. As far as they are concerned, that’s their Universe, and they are unaware of anything else. We, being outside in 3-dimensional space, can see things a bit differently.

(I know, you’re saying that this is incomplete and not realistic because the ants are really in 3-dimensional space like we are — they could crawl off the wall or onto the floor or whatever, so they aren’t really restricted to 2-dimensional motion. But you are missing the point. Work with me a bit. Try to imagine these as 2-dimensional ants who are just as restricted to motion in 2-dimensions are we are 3-dimensional humans who are restricted to motion along 3 axes of movement.)

We can imagine the entire Universe as being plotted on the surface of a flat plane, a spherical ball, or a saddle-shape. (In fact you could image many other shapes and multiple dimensions as well, but that is way beyond us at this point.)  The limitation here is that we are trying to imagine a full 3-dimensional Universe on a 2-dimensional surface. That is, here we have to confine things to the surface. There would be no such thing as above or below the plane, or inside or outside the ball or saddle-shape. Here we are 3-D beings trying to imagine the reality of a 2-D world. (That’s the situation we have. If we were 4-dimension or higher dimension beings, we could imagine something like this for the beings in a 3-D world.)

In a world of “flat geometry” or zero curvature, two parallel lines always stay parallel. In a world of “closed geometry” or a positive curvature, initially parallel lines would converge. In a world of “open geometry” or a negative curvature, parallel lines would diverge. Another way of looking at this is with a triangle. In normal, “Euclidean” geometry, the sum of all the angles inside a triangle is always 180 degrees. But in spherical geometry, such as the surface of the Earth, the sum of angles in a triangle is always less than 180 degrees. And in a Universe with a saddle-shaped geometry, the sum of the angles in a triangle is always less than 180 degrees.

Can we measure this? Why yes, yes we can! Just as standard Euclidean geometry — which you can work out and measure on a flat piece of paper — works in 2-dimensions in a high school geometry class, it works in 3 dimensions as well. If we scale this 2-D visualization up to 3 dimensions, it works the same. For example, if the Universe is “flat,” then the sum of the angles in any triangle always amount to exactly 180 degrees. Draw a triangle and check it out. And we can scale that up to a much large triangle. Astronomers can measure the angles between, say, 3 galaxies out in space. They can measure the angles and voila! they count up to 180 degrees. This says that the Universe we live in has a “flat” geometry.

OK, what does this all have to do with anything? Actually there are profound implications for the evolution and future of the Universe. It does not affect your everyday life, but it gives us important information as to how the Universe got to this point, and what the likely future is going to be.

The geometry depends on the densities of mass and energy in the Universe. (You can call it the mass/energy density.) It is based on conditions at the time of the Big Bang, nearly 14 billion years ago, These conditions could have been pretty much anything, but if they were even a tiny percentage different from what they were, the Universe we live in would have a very strong curvature, either open or closed.

But instead the Universe is flat. As mentioned above, the interior angles of a triangle always add up to 180 degrees. Thus we say that the Universe is “flat,” even though it is 3 dimensions. Weird, huh? But what is even more weird is that is could be cured outwardly or inwardly and in fact those choices are vastly more likely. We live in a Universe whose geometry seems almost impossibly unlikely.

But it may not be so strange as it seems. Cosmologists have an answer, which makes our “flat” Universe far more likely, perhaps even inevitable. It is called “Cosmic Inflation,” but perhaps we are getting ahead of ourselves now. That’s for Test 5!