## Friday, November 4, 2011

### The size of the observable universe

Previously, I analogized the expansion of the universe to a stretching of a rubber band.  Light that travels across the universe is like an ant crawling on that rubber band.  Whether the ant ever reaches the other side of the rubber band depends on how fast the rubber band is stretching.

Here, I will introduce something called the scale factor.  The scale factor is similar in concept to the total length of the rubber band.  The tricky part is that there is no "total length" of the universe, because the universe (as far as we know) is infinitely long. But even if the rubber band were infinitely long, we can still imagine it stretching.

Even without referring to the "ends" of the rubber band, we can still see that the ants are getting further apart.  Similarly, even though the universe has no "ends", we can still observe galaxies getting farther apart.  There are some complications (like if the ants are crawling around while the rubber band is stretching), but let's assume that clever physicists have found a way to correct for this.  The scale factor is proportional to the distance between these two ants/galaxies.

I say "proportional" because it doesn't really matter what the distance is exactly; we just care about how that distance changes over time.  Is the scale factor increasing as a constant rate?  Or is it slowing down?  Or is it accelerating?  The answer ultimately depends on the theory of General Relativity, which is outside the scope of this post.  There are, at least, some results of General Relativity which make intuitive sense (and others that do not make intuitive sense).

If the universe had no energy, we'd expect the scale factor to increase (or decrease) at a constant rate.  "Inertia" is the intuitive explanation.  As I explained in my previous post, a constant stretching rate implies that light from any galaxy will eventually reach us.

But since the universe has energy in it, that mass pulls itself together with gravity.  So we'd expect the scale factor to slow down.  But the rate at which the scale factor slows down depends on what kind of energy it is.  If that energy comes from matter, the scale factor slows down at a certain rate.  If that energy comes from massless particles like light, the scale factor slows down even at a greater rate.*  The difference between matter and light is that they dilute at different rates as the universe stretches.

Now, if the energy comes from something that doesn't dilute at all, then the scale factor actually accelerates over time.  I realize this is quite counterintuitive, but I'm postponing the explanation indefinitely until I understand it myself!  In any case, that's what dark energy is.  Dark energy is a transparent form of energy which doesn't get more dilute as the universe grows.

In summary,
Universe without energy: Scale factor increases at constant rate
Universe dominated by radiation: Scale factor slows down over time
Universe dominated by matter: Scale factor slows down (but not by quite as much)
Universe dominated by dark energy: Scale factor accelerates

Of course, the universe has a mix of the different kinds of energies.  The early universe was dominated by radiation.  But the radiation diluted away as the universe expanded, so the later universe was dominated by matter.  But now the matter has diluted away, leaving a universe dominated by dark energy.  So at first the scale factor slows down, but it eventually speeds up again.

You want more details, with graphs?  There was this great paper called "

Click to enlarge.  This is based on the ΛCDM (Lambda Cold Dark Matter) model, assuming that the energy of the universe is currently 70% dark energy and 30% matter.

Some definitions to understand the graphs:

The comoving distance is simply the distance divided by the scale factor.

The dotted lines represent the paths of objects, provided they are not "crawling around", but just following the expansion of space.

The particle horizon is the answer to the question: how far can we see?  Some 380,000 years after the big bang, the universe became transparent. (Update: the particle horizon is not defined from the point when the universe becomes transparent, but from the "beginning".) From that point on, light traveled from far away objects, and eventually it reached us.  Simultaneously, those objects were getting further from us.  The particle horizon refers to the current distance of those objects whose light from billions of years ago is just reaching us now.  Currently, the particle horizon is 46 billion light years away.  Therefore, the size of the observable universe is 93 billion light years across!

The hubble sphere is the distance at which the rate of expansion is equal to the speed of light.  Beyond the hubble sphere, objects are receding from us faster than the speed of light.  It's allowed to be faster than light, because it's an expansion of space, not motion through space.  The relativistic speed limit only applies to the relative motion of objects that are near each other.

The light cone is the region of the universe's past that we can presently see.  Remember that the further away we look, the further into the past we look.  And the further back into the past, the more dense the universe used to be.  This is not the same as the particle horizon, which refers to the current position of galaxies whose past we can see, not their past positions.

The event horizon is the largest region that will ever be in the light cone.  If the scale factor were increasing at a constant rate (or decelerating), there would be no event horizon.  That is, light from every part of the universe from any time would eventually reach us.  But since the scale factor is accelerating, there is an event horizon.  There are some parts of the universe that we will never see!

I hope this illuminates some of the many confusing aspects of cosmology.  As the paper's title suggests, there are a variety of misconceptions about the expanding universe.  One of those misconceptions is that the expansion of the universe isn't allowed to exceed the speed of light.  This mistake is even made by physicists when speaking to the public!  If you want to read more, I recommend a more popular version of this article, which appeared in Scientific American.

*These are based on solutions to the Friedmann equations in a flat universe.