The Size of the Universe: A Hard Question

I get a certain question every so often, and it’s one of the most difficult questions any cosmologist faces. Today, I try to tackle it. It goes something like this:

If the Universe is 13.7 billion years old, and nothing can go faster than the speed of light, how is it that we see things that are 46.5 billion light years away?

First off — and I want to clarify this — everything in this question is legit.

1.) The Universe is 13.7 billion years old. There are small errors there — no one would be surprised if it was 13.5 billion or 14.0 billion years old — but it’s definitely not 12 billion years or younger and definitely not 16 billion years or older.

2.) Nothing can move faster than the speed of light. If you’ve got a mass, whether you’re a galaxy, a spaceship, a bullet or a neutrino, you’re going to go slower than the speed of light. And if you don’t have a mass, you’re going to move exactly at the speed of light. No exceptions.

3.) The farthest things in the Universe — the things that emitted their light 13.7 billion years ago — are 46.5 billion light years away from us now.

So how did this happen? Two things, one simple and one not-so-simple. The simple thing is that the Universe has been expanding this entire time. Imagine you’ve got an ant on a deflated balloon, and the ant moves at a rate of 1 cm/second. When the balloon is totally deflated, the ant is only 2 cm away from the top of the balloon, her destination. But as she starts walking towards the top, something inflates the balloon. As she walks towards the top, she notices that the balloon around her is expanding.

How does this expansion work? Well, this is the not-so-simple part. Expansion isn’t a velocity. It’s a velocity-per-unit-distance. Let’s say that it’s 0.4 cm/second per centimeter. This means that if the ant is 1 cm away from something, it expands away from her at 0.4 cm/second. The top of the balloon, initially, since it’s 2 cm away, expands away at 0.8 cm/second. And something that’s 15 cm away would be expanding away at 6 cm/second.

So if I run through the math of this ant walking at 1 cm/second to a point 2 cm away on this expanding balloon, it doesn’t take 2 seconds to get there. In fact — doing the math correctly — it takes just a shade over 3 seconds for the ant to reach her destination. Moreover, the balloon has continued to expand, so when she looks back at her starting point, do you know how far away it is? Over 6 cm away! When she looks back at her starting point, not only is it more than three times as far away as it was when she started, but the entire balloon is bigger than it was before.

And that’s what our Universe is doing: expanding while the light is traveling towards us from distant sources. There is, of course, one more caveat in our Universe. The expansion rate is mind-bogglingly slow, 72 kilometers per second per Megaparsec. In the ant’s terms, that’s 2.3 x 10^-18 cm / second / cm. It’s just that our Universe is so big that as you get far enough away — just under 13 billion light years — the expansion rate eventually becomes greater than the speed of light.

But this is okay. It’s only that space (i.e., the balloon) is expanding; there’s no matter that’s moving. So, in principle, space can expand as quickly as it wants, even faster than the speed of light, because there’s nothing moving. And that’s why, even though the Universe is only 13.7 billion years old, we can see things that are 46.5 billion light years away.

Any questions?