Returning to my Physics for Non-Physicists series, I will temporarily switch to a new topic: cosmology. I’d like to talk about dark matter next, and a little knowledge about modern cosmology is necessary to understand some things about dark matter. As the title suggests, this discussion is meant to be qualitative and pretty brief. I may return to the subject in the future to discuss it in further detail, but that wouldn’t be for a while.

## Introduction: The Expansion of the Universe

Many early physicists traditionally saw the universe as a static, fixed thing. However, by some time during the 20th century, astrophysicists were able to measure objects many millions of light years away from Earth. While the Milky Way galaxy contains huge numbers of stars gravitationally bound to one another, and the galaxy is even bound to other galaxies in a galactic cluster, it turned out that on large enough scales everything appeared to be moving away from us, as if in the distant past an enormous explosion sent all these things away from us.

That is not the whole story, though. It also seemed like on large enough scales, everything is moving away from everything else. In fact, the rate at which the distance between two objects is increasing is proportional to the distance between the objects. The proportionality constant is called Hubble’s constant. So, it almost seems like in the distant past this motion was set in motion by a huge explosion that existed everywhere all at once (the Big Bang).

It turns out that this is what we would expect if instead of what we might consider an explosion (a violent chemical reaction), the recession of everything from everything else were actually due to an expansion of space. So what does this mean? Consider an empty toy universe, where you can step outside the universe and place two test particles at some known distance. Also let these two test particles be noninteracting and have no relative velocity. If space expands or contracts uniformly, the distance between the particles will expand and contract as well, even with no forces acting on the particles. This may seem at first glance to violate Newton’s First Law (the Law of Inertia). It doesn’t. Many of the laws of physics describe local phenomena. It only matters that on a small enough (even infinitesimally small) scale the particles look stationary.

This expansion of space was something that was predicted by Einstein’s general theory of relativity (GR), first released in 1916. GR describes space and time as dynamical, rather than static, objects. The equations governing the dynamics of space and time are intrinsically related to the distribution of energy in the universe. In GR, the geometry of space and time give rise to gravity, which exerts forces on energy, which affects the geometry of space and time.

## The Shape of the Universe

Then how do we go from GR (a local theory whose dynamics depends only on knowledge of what’s going on in some finite region of space) to talking about the history of the whole universe?

We first invoke the principles homogeneity and isotropy. Homogeneity means that on large enough scales (hundreds of millions of light years), everywhere in the universe looks the same. Obviously, we exist so homogeneity does not hold on all scales, but on truly cosmological scales we assume this to be true. Isotropy means that on large enough scales every direction looks the same as well. Thus, the universe has no preferred position and no preferred direction.

With these assumptions, we can easily deduce that on large enough scales, the small structures in the universe from the uneven distribution of mass/energy smooth out, leaving us with a universe that is smooth, homogeneous and isotropic. Everything else is just minor details. It turns out that there are then only 3 possibilities for the shape of the universe. Now consider a two-dimensional universe. In two dimensions, the 3 possibilities are a sheet (a flat/Euclidean universe), the surface of a sphere (positive curvature), or a shape similar to a potato chip (negative curvature). The positive curvature universe has a finite size, while a flat or negative curvature universe is literally infinite in size. Our universe must be shaped like the 3-dimensional analog of one of these.

It’s actually in principle possible to directly differentiate the 3 types of universes. The “shape” is characterized mathematically by what’s known as the metric tensor. The metric tensor defines the length of an infinitesimal line segment given some set of coordinates. The different geometries basically just have different (and non-equivalent) definitions of a straight line. In the 2D case, the line on a sheet is what we wold term a line. The “line” on the surface of a sphere is a great circle. If you could draw a perfect triangle and perfectly measure its angles, you would see that in a flat universe, the sum of the three angles is 180 degrees, in a positive curvature universe, the sum is greater than 180, and in a negative curvature universe, the sum is less than 180. Universes with non-zero curvature have some characteristic length scale governing the curvature. On scales much less than this scale, everything looks flat. As far as we can tell, the universe is flat, but this really just means that the characteristic length scale of the universe is very large.

## The Observable Universe

Remember that two of the three global geometries of the universe are infinite in size. You may also know that we can only look so far away (a few tens of billions of light years). It seems to us like the universe really is finite. How do we reconcile these facts without just declaring the universe to have positive curvature?

Our best estimates of the age of the universe tell us that the Big Bang occurred 13.8 billion years ago. Light in a vacuum travels at speed of light (obviously), so in those 13.8 billion years, any given photon can only have traveled so far. Looking far away is also looking backward in time, so we can’t see farther away from us than the maximum distance light could have traveled throughout the whole age of the universe. It turns out that this distance is actually larger than 13.8 billion light years. The light traveled that far, but the objects have been pulled away from us since the light was emitted. We also can’t really see light from the Big Bang as at some point the universe was dense and hot enough to be opaque to light.

Basically, as long as the universe – or at least the universe as we know it – has a finite age, we will have a finite horizon beyond which we can never see.

That’s enough for today. I’ll continue later with discussions of the overall history of the universe and the thermal history of the universe.