Randall Munroe and the Size of the Observable Universe

Posted on October 4th, 2008 — permalink

Randall Munroe of the fabulous webcomic xkcd has a great logarithmic height poster showing the size of everything from folks all the way up through the edge of the Solar System, on to the radius of the observable Universe. As a logarithmic plot, each gap of the same size vertically on the plot represents a doubling of the distance from the surface of the Earth; this is why he can show things of such vastly different scales as people and the whole Universe in the same plot.

But, wait, I thought you cosmologists kept saying that the Universe was infinite! How can this picture show the whole Universe then?

It doesn’t… it shows the observable Universe. Because the Universe is only 14 billion years old, and the speed of light is finite, we can only see things that are as far away as light has had time to reach us from. There is more Universe beyond that, but the light hasn’t reached us from it yet; the part of the Universe beyond our horizon is not the observable Universe.

But wait… if the Universe is only 14 billion years old, then, we should only be able to see things that are 14 billion light-years away… yet the xkcd pictures says the top of the Universe is 46 billion years away. What’s up with that?

Remember that the Universe has been expanding in the time since the light from distant galaxies (or the distant plasma of the early Universe) was emitted and started its journey on the way to us. During the time the light was making its way towards us, the Universe continued to expand… so the things light from which we are just now seeing for the first time have moved a lot farther away since that light was emitted. They’re much farther now than they were then.

OK… but I thought nothing could move faster than the speed of light?

That’s right.

Why then isn’t the observable Universe at most 28 billion light-years? If something emitted light and it took 14 billion years to reach us, and it was moving the other way as fast as it could, it would only be 28 billion light-years away right now. What’s with the 46?

This is the most subtle point. This is the reason why I cringe whenever I hear a popular cosmology talk refer to galaxies as flying away from us. For nearby galaxies (i.e. within a billion light years or so), it’s fine to treat them that way… but that treatment doesn’t work over universe scales. In fact, it’s far better to think of it that galaxies are not flying away from us, but that space itself is expanding. Light from a very distant galaxy was emitted when the galaxy was quite a bit closer than 14 billion light-years away. Then, as the light made its way towards us, the Universe expanded. The space between us and that distant galaxy got bigger. As a result, two things happened. First, the distant galaxy got further away. Second, the photon, always moving at the speed of light, had more and more space to cross making its overall journey. As such, even though the galaxy was initially at a distance of quite a bit less than 14 billion light-years, it still took nearly 14 billion light-years for the light to reach us.

The advantage of thinking about the expansion of the Universe as space itself expanding, rather than as a bunch of galaxies flying apart from each other, is that it’s closer to the mathematics of General Relativity when applied to the Universe as a whole. It also is conceptually cleaner. Things really can’t go faster than the speed of light… but velocity is a local concept, and only had meaning when measured pretty close to the thing which you’re measuring the velocity relative to. Very distant galaxies (more than a billion or so light years away) are far enough away that it’s not meaningful to talk about their velocity. (You can talk about the rate at which the distance is increasing, but even though that has the same dimensionality, it isn’t the same thing as a velocity.) The rate at which the distance between us and a far-away galaxy is increasing can be greater than the speed of light… but that does not violate Relativity, because of the meaningless of the concept of velocity when compared over such great distances.

If you calculate the proper distance right now to the edge of the observable Universe– that is, to the bit of matter whose light took 14 billion years to reach us– you get 46 billion years. Randall Munroe is a smart guy.

That’s most of what I wanted to say about it, but for the advanced readers, one more question:

Okay, wise guy, how can you talk about “distance” and Relativity at the same time? Relativity tells us that lengths contract along the direction of motion… in whose frame of reference are you measuring this distance? Huh?

Yes, that’s a good point. However, two things about it. First, the frame of reference of any given observer is only good over a relatively small range in space and time. This is for exactly the same reasons that there is no such thing as “velocity” for a very distant galaxy relative to us. (Anywhere where the frame of reference for an observer is valid– meaning the numbers you get using that frame of reference aren’t beyond your acceptable tolerance from the real numbers– it would be OK to talk about velocity.)

So, since I’ve just said that there’s no single frame of reference, why is it that I can talk about “the” distance at all, especially considering that I’ve already thrown out velocity? It turns out that for the Universe, there is a natural way to measure distance. Consider the following thought experiment. Lay out in the Universe a huge number of rulers, set up just right so that right now they are all end-to-end between here and the galaxy to which you want to measure distance, and the “time since the Big Bang” clock attached to each ruler has the identical reading. That is, shortly after the Big Bang you set out a bunch of rulers (all probably bunched together and running into each other), and you wait for the Universe to expand (moving the rulers apart from each other) so that they’re all end-to-end when they’re all the same age. Count up the readings on all of the rulers, and you get that thing that I call “proper distance”. Indeed, the calculation you do in General Relativity is pretty much exactly described by this thought experiment.

4 Responses to “Randall Munroe and the Size of the Observable Universe”

  1. Pseudonym Says:

    Lay out in the Universe a huge number of rulers, set up just right so that right now [...]

    I was taught that “right now” is also a concept that’s only defined relative to an observer. Can you expand on what you mean by that?

  2. rknop Says:

    Pseudonym — yes, that’s a good point, and related to the point that it’s hard to talk about “the” distance. I actually go on to explain what that means a sentence or two later — attach a clock to each ruler, and set that clock to 0 at the moment of the big bang (or, as may be safer, at the end of Inflation). Read the relative positions of the rulers to measure this distance with all of those clocks reading exactly the same thing that the clock closest to you reads. That’s what I really meant by “right now”.

  3. rknop Says:

    Another addendum — because of the tangled nature of distance, I generally don’t like to talk about “the distance” to very distant galaxies. Things less than a billion light-years away… sure. That’s fine. But when things (like the edge of the Universe) get really far away, I prefer to talk about “lookback time”, because that has a cleaner definition. (I.e. the difference in the age of the Universe between time of emission and time of detection.) In a Universe that’s monotonically expanding, there’s a 1:1 correlation between redshift and lookback time. (Well, for a given set of cosmological parameters… redshift doesn’t directly measure lookback time (nor does the observed brightness of a standard candle), but it is 1:1 in a given monotonically expanding Universe.)

  4. Brandon Says:

    “it still took nearly 14 billion light-years for the light to reach us.”