Janek asks

I would be interested in a further post detailing roughly how we’ve calculated the size of the observable universe.

Okay. Actually I’m more than happy to answer this because the answer is relatively short and it’ll let me bunk off science posts for a week.

So once upon a time there was an astronomer called Edwin Hubble. Hubble’s career is notable for a number of firsts – such as the identification and use of the standard candles known as Cepheid variables found within the Andromeda and Triangulum galaxies to prove that these were indeed galaxies and not merely nebulae as had previously been assumed, thus incidentally proving that the universe extended beyond the bounds of the Milky Way – but the one that ended up getting his name attached to it was the discovery of something called the Hubble constant which basically defines our universe. Astronomers spend a lot of their time nailing down increasingly accurate values for the Hubble constant; part of the motivation for the whole CMB thing is to get better measurements of H_o.

So the Hubble constant is kind of a big deal, and what it is and how it works is best explained by telling the story of how Hubble found it in the first place. After having such success with the Cepheid variable approach to Andromeda and Triangulum (if you need Cepheid variables explained I go over them in one of my first posts here, and the Wikipedia article is here) Hubble decided to see just how many other galaxies he could find with Cepheid variables in them in order to build up a picture of what our universe really looked like. The Cepheids let him measure how far away these particular galaxies were, but while he was observing them he noticed something odd: the light from some of the more distant galaxies displayed a significant degree of redshift. It was already known that galactic redshift was related to their recession velocity thanks to work done by Vesto Slipher a few years earlier, and so Hubble was able to make a graph of galactic distance versus recession velocity that looked much like the following:

It’s a straight line relationship (albeit one with a fair amount of scatter), which is convincing evidence that the recession velocity of a particular galaxy is directly proportional to its distance away from . In other words,

where v is the recession velocity of the galaxy, d is its distance from us, and Ho is the Hubble constant. The expression above forms Hubble’s law, which states that the further away from us something is the faster it will be moving – and since that movement is caused by the expansion of space, Hubble’s law is effectively a relationship which governs that phenomenon. This means that what exactly the precise value of the Hubble constant is is going to have a profound impact on our views of how the universe is expanding, which is why scientists are so interested in obtaining increasingly accurate measurements of it. It also incidentally allows us to get a rough idea of galactic distance by measuring the redshift, converting it into a recession velocity, and plugging it into Hubble’s law along with the current value of the Hubble constant (67.80 ± 0.77 km s^-1 Mpc^-1 as of the most recent measurements obtained by the Planck probe).

So in theory when we look at things we have a fairly good way of figuring out how far away they are from us. All we have to do is find the thing that’s furthest away, plug its redshift into Hubble’s law, and from that calculate the approximate radius of our observable universe, right? Unfortunately — as with most things in astronomy – the reality is nowhere near that simple. Whenever you read an article reporting the discovery of the most distant astronomical object yet discovered, that article will likely quote two things: the object’s redshift, which is a thing that can be directly measured via observation, and its age, which can be reliably calculated from its redshift. What it won’t quote, unless it’s been written by an idiot, is its distance from us. This is due to several factors which complicate Hubble’s law immensely, most notably:

• That the simplified version of the law I described above – which incidentally is the one we teach to reasonably intelligent people up to and including undergraduate level¹    – assumes that the redshifted light we see from galaxies moving away from us is caused by a simple Doppler effect; that is, that the galaxies are moving away from us through space, like a boat sailing on the ocean. What’s actually happening is that the space itself is expanding and carrying that boat away from us on a tidal wave, which requires a far more complex treatment of the redshift.
• That the rate of galactic expansion has been constant and unchanging throughout its entire history. This is not true, as we now think that the universe is accelerating in its expansion. You know that value of Hubble’s constant we calculated just this year using the Planck probe? That’s the value of Hubble’s constant now. We have little to no idea what the value of Hubble’s constant was ten billion years ago, and the various cosmological models that attempt to establish this differ wildly in their predictions of how it’s changed over time.

These two factors together not only make the use of galactic redshift to calculate distance a rather iffy proposition at the best of times, but that second one also makes it effectively impossible to gauge the distance of the really old stuff we find. And if we can’t put an accurate number on those distances, this means it’s impossible to directly measure the volume of our observable universe.

If that’s the case, though, then why do astronomers throw around a precise number for the radius of the observable universe (currently about 46 billion light years)? It’s because they’re not using the term observable in the same way that you (or I) might. To them, “observable” does not mean “Can we see it right now?” Instead, the observable universe is simply the radius inside which light emitted from objects at some point over the entire history of the universe could theoretically reach us here on Earth. It doesn’t matter if the light actually does get here or not, or whether somebody manages to detect it, or whether we can make sense of what we eventually see; if the light has the potential ability to reach us, then the part of the universe that would hypothetically emit it falls within the bounds of the observable universe.

This simplifies the calculation of the radius of the observable universe immensely; instead of trying to establish actual physical boundaries you simply dial things back all the way to the oldest thing that could possibly exist and be observed and try to derive a distance measurement to that. In the case of our universe this happens to be the cosmic microwave background radiation, which is a collection of extremely redshifted particles released at the moment of photon decoupling². That 46 billion light-year number is derived from our observations of the surface of last scattering, a collection of points in the sky where photons from the CMB are only just now reaching us here on Earth. The great advantage of using the CMB to calculate this radius is that it isn’t just one object; since it exists over the entire sky it provides an awful lot of data which allows astronomers to average out those huge uncertainties in using redshift to gauge distance I described above. This means that it’s a reasonably solid number, although there’s no reason a more detailed observation of the CMB wouldn’t cause it to change in future.

So that’s how we arrived at a definition for our observable universe. There’s stuff inside it that we haven’t seen — and likely never will – because our instruments aren’t good enough, but that doesn’t change the fact that we could see it with sufficiently advanced equipment. It’s possible that the boundaries of the observable universe could move outwards if we figure out a way to see stuff without relying on electromagnetic radiation (observations of neutrino emissions and gravity waves could allow us to look back beyond the moment of photon decoupling, for example) but I wouldn’t hold your breath; that stuff is very definitely future science.

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