###### By Don Lincoln**, **Fermilab

## The geometry we learn in school is what mathematicians call Euclidian, or what most would call the geometry on a plane. This is what cosmologists call flat space. However, we also know that general relativity can distort space, thus we can’t assume that our space (indeed our entire universe) is flat. And yet, when it comes to distorted spaces, there are three simple ones that are relevant to cosmology.

### Calculating Wavelength

The nonuniformities in the cosmic microwave background radiation (CMB) are a fossil remnant of sound waves in the early universe. Since we know the amount of dark and ordinary matter in the cosmos, with the makeup of elements in the early universe, plus their temperature right before the universe became transparent, physicists can calculate the wavelength of those sound waves.

As we know, when sound passes from a source to our ear, the wave compresses and rarifies air. The same thing happened in when the universe was young. The compressed regions were a little hotter and the rarified regions were a little cooler. Those hotter and cooler regions translate into the pattern of red and blue speckles in the CMB. And the distance between adjacent hot or adjacent cool spots was set by the sound prevalent at the time.

And yet, there wasn’t just one wavelength in the early universe. There were many. But some of the sounds were louder than others and some were more common. What we’re seeing now in the CMB is the sound generated in a sphere around the Earth at a distance where the sound is just getting to us now.

### Types of Curved Two Dimensional Spaces

It is easily possible to determine the distance at which that sphere is located. If we know the wavelength of the dominant sound and the distance that the sphere was from the Earth, we can set up a triangle. And, from that, we can figure out the most common angle between adjacent hot or cold spots. It turns out that if space acts like we learn about in introductory geometry class, where the sum of the angles of a triangle add to 180°, the most common angle subtended by adjacent spots is one degree.

Space is, of course, three dimensional, but thinking about curved three dimensional space is tricky, so in order to simplify it, we’ll talk about two dimensional space. This approach is imperfect, but is a helpful choice.

There are three main types of curved two dimensional space that seem to be relevant to cosmology. There is flat space, which is like the surface of a table. There is closed space, which is like the surface of a sphere. Then there is open space, which like the surface of a saddle, which goes up in the front and back, and downward on the sides.

This article comes directly from content in the video seriesThe Evidence for Modern Physics: How We Know What We Know. Watch it now, on Wondrium.

### Effect of Explosion

So, let’s imagine what happens if we cause something to explode on each kind of surface.

Let’s start with the flat space. One might have a shock wave that is circular in shape. Let’s assume that the circle never dissipates. So, the circle starts out at a point and then gets bigger and bigger and more and more diffuse.

So, what happens on a sphere? Suppose we have an identical explosion, located at the North Pole. The circle radiates out from the origin, getting bigger and bigger. Eventually, the circle will encircle the equator, where the circle is the biggest it gets. After that, the circle travels toward the South Pole, getting smaller and smaller, until eventually the circle coalesces into a point.

On a saddle, a different thing unfolds. This time we start the explosion at the place where we sit. The circle spreads out, but it gets bigger than it would in a flat space. That’s because even though the circle expands like in flat space, the space is warped, with parts going up and parts going down. That means that those up and down points are separated by larger distances than they would be in a flat space. In this space, the circle gets bigger than it does in a flat space and at a faster rate. Plus, the circle never coalesces.

### Mass, Energy, and the Geometry of the Universe

We’ve heard similar sorts of words in discussions pertaining to the expansion of the universe, talking about the amount of matter in the universe, with a specific amount being the critical mass. If the universe has the critical mass, it expands forever, slowing constantly. If it has too much, the universe will expand and then reverse, contracting with a big crunch. Too little mass, and the universe will expand and slow, but never stop. That’s important concept number one.

A second important concept is that the amount of matter and energy in space will distort both space and time, something we learn in general relativity.

Now, it is possible to prove that the geometry of the universe and the amount of mass and energy it contains are perfectly linked. If the universe contains the critical amount of mass, it is flat on huge distance scales. If the universe has too much mass, its shape is closed, like the surface of a sphere. If the universe has less mass than the critical mass, its shape is open, like the surface of a saddle. Of course, space is really three dimensional, but the big idea is still valid. If we can prove that space is flat, closed, or open, we will learn something about the amount of matter and energy in the universe.

So, that brings us back to the size of the spots of the CMB. In a flat space, the most common separation between adjacent hot or cold spots will be one degree. If space is closed or spherical, or if there is more mass than the critical mass, the separation between adjacent spots will be bigger than one degree. If space is open or saddle shaped, or if there is too little mass, then the separation between adjacent spots will be smaller than a degree.

### Common Questions about CMB and the Geometry of the Universe

**Q: What are the nonuniformities in the CMB a fossil remnant of?**

The nonuniformities in the cosmic microwave background radiation or CMB are a fossil remnant of sound waves in the early universe.

**Q: What happens if the universe has too much critical mass?**

If the universe has critical mass, it expands forever, slowing constantly. If it has too much, the universe will expand and then reverse, contracting with a big crunch.

**Q: What shape will the universe have if it has less mass than the critical mass?**

If the universe has less mass than the critical mass, its shape is open, like the surface of a saddle.