## Monday, October 24, 2011

### The ant and the universe

Because light in our universe has had a finite amount of time to travel, we can only see a finite part of our universe.  We call this the observable universe.  But did you know that, even though the age of the universe is 13.7 billion years, we can actually see further than 13.7 billion light-years?

According to Hubble's Law, distant galaxies are receding from us at a speed proportional to their distance.  In fact, beyond 13.8 billion light years away, objects are receding from us faster than light.  Did you know that the furthest objects we see are actually receding faster than light?

Believe it or not, these facts are easy to understand with only elementary knowledge of cosmology.

There is a classic puzzle involving an ant on a rubber band.  The ant tries to run from one end of the rubber band to the other, but every second we stretch the rubber band longer.  Will the ant ever reach the other side, or will it just get further and further from its destination as the rubber band stretches?

An ant comes across a rubber band... (Ant image from here)

The ant on the rubber band is a lot like the path of light through the expanding universe.  The ant crawls at a constant speed, but only relative to the rubber band.  As the rubber band stretches, the ant may become further from its destination.  Similarly, light travels at a constant speed.  But as space itself expands, light from far away galaxies may actually become further from us over time.

(Note that the far away galaxies aren't breaking any speed limit even if they recede faster than light.  They are receding because the space between us is expanding.  The relativistic speed limit only applies to the relative velocities of two objects at nearly the same location.  Analogously, we can say that even if we stretch the rubber band really fast, this doesn't mean the ant is breaking its own crawling-speed limit.  It just means the rubber band is stretching.)

So, will the ant make it or not?  It depends how quickly we stretch the rubber band.  If we were feeling devious, we could stretch the rubber band at exactly the right rate so that the ant is running in place, never getting any closer to its destination.  But let's consider a stretching rate which is a little more like our universe.  Suppose that the ant crawls one inch per second, and each second we stretch the total rubber band's length by one foot.

At first, the ant will gain an inch, and lose a foot.  But after the ant covers some distance, I waste a lot of effort stretching the part of the rubber band that is already behind the ant.  So eventually, the ant may catch up!  Indeed, we can show that it will catch up, if we consider the total fraction of rubber band that the ant covers each second.  In the first second, the ant covers 1/12 of the rubber band.  In the second second, it covers 1/12 * 1/2 of the rubber band (since now the rubber band has been stretched to twice its previous length.)  The next second, the ant covers 1/12 * 1/3.  And so forth.  We get a sequence like this:

The series in parentheses is known as the harmonic series.  The harmonic series can arbitrarily big, just by adding more terms (but it takes an exponentially large number of terms).  If we were to graph the ant's trajectory, it would look like this:

Note: I calculated this assuming continuous stretching, not by discrete stretching.  This is more true to cosmology.

So that's how the ant traverses the rubber band even though the rubber band was stretching faster than the ant could crawl.  Similarly, light can reach us from far away galaxies even if they're receding faster than light.

How far did the ant travel?  The ant is traveling an inch per second, and traveled for over 160,000 seconds.  Therefore, it traveled over 160,000 inches.  But there's another distance we can ask about: How far away is the point on the rubber band from which the ant started?  Because the opposite end of the rubber band moves at a foot per second, it must be over 160,000 feet away.

With light we can ask the same questions.  According to NASA, the furthest galaxy we know of is 13.2 billion light years away, by which they really mean that the light has traveled 13.2 billion light years to get to us.  But how far away is that galaxy now?  It is probably much further than 13.7 billion light years away, because it is receding from us very quickly.

But there are additional complications when we apply this to the universe.  As it turns out, the universe does not expand at a uniform rate, like our rubber band.  The two additional complications are matter and dark energy.  This will be explained in the next part.