# How To Define Distance In An Expanding Universe

In Cosmology by Brian Koberlein6 Comments

Recently the Sloan Digital Sky Survey (SDSS) has completed the largest map of the universe thus far. The map focuses on the positions of quasars. These objects are powered by supermassive black holes in the centers of galaxies, and are so bright they can be seen from the farthest regions of the cosmos. Most quasars are so far away that we have to redefine what “distance” means. In an expanding universe, distance can be defined in a variety of ways.

For the stars we see in the night sky, their distance is just what you’d expect: the physical distance from the Sun to the star. The bright star Sirius, for example, is 2.6 parsecs away. A parsec is defined by the method used to measure stellar distances, known as parallax. As the Earth orbits the Sun, its view of the stars shifts very slightly. Nearby stars shift more than distant ones, and this is known as a parallax shift. The bigger the parallax, the closer the star. If a star were one parsec away, its parallax would be 1 arcsecond. There are 360 degrees in a circle. If you took a single degree and divided it into 3600 parts, each part would be an arcsecond.

The parallax of nearby stars is small because they are so very far away. While astronomers often use parsecs for distance, a more common measure is the time it takes light from the star to reach us. For Sirius, that is about 8.6 light years, meaning the starlight we observe left Sirius about 8.6 years ago. Of course that distance changes a bit over time. Sirius is moving relative to the Sun. Even if we could travel to Sirius at the speed of light, we would have to make accommodations for its changing location. But this change in distance is small compared to its current distance.

Because stellar parallax is so small, it can only be used for stars out to about 10,000 light years or so. Beyond that the parallax is simply too small to measure. For more distant objects such as galaxies we have to use other methods. One popular method is to use variable stars known as Cepheid variables. Cepheid variables have a particular relation between their overall brightness and how quickly they vary from bright to dim. By watching them vary over time we can calculate their distance. Observations of Cepheid variables in the Andromeda galaxy, for example, shows that it is about 766,000 parsecs away, or 2.5 million light years. Just as with stars, the distance of a galaxy changes over time. Over the course of a 2.5 million year journey to Andromeda, the galaxy would have moved by about 1,500 light years. That’s still a small fraction of its overall distance, but its not insignificant.

With more distant galaxies distance becomes much more complicated. If we measure the motion of various galaxies, usually through the redshift of their light, we find that the more distant the galaxy the greater its redshift. This is due to the overall expansion of the cosmos. Through dark energy, the overall distance between galaxies is increasing, and this cosmic expansion puts a serious wrench in the meaning of cosmic distance.

The most direct quantity we can measure for a distant galaxy is its redshift. Usually this is expressed as z, which is the fractional amount a particular wavelength has changes from its unshifted wavelength. The upper range of z we have observed is about 12, so lets consider a galaxy that has about half that amount, or z = 6. Just how far away is such a galaxy?

Redshift can be caused by two things: the motion of a galaxy through space (often called Doppler redshift) and the expansion of space itself (often called cosmological redshift). We can’t distinguish between them observationally, but we know from various observations that the motion of local galaxies that the Doppler shift tends to be rather small. So its safe to assume that for distant galaxies redshift is almost entirely due to cosmic expansion. To calculate distances, we then have to look at how the universe expands over time, and this relies on which particular cosmological model we use. Typically this is the concordance model, or ΛCDM model, which is your standard dark matter, dark energy dominated universe model. Assuming this model is accurate (and we have lots of reasons to think it is) then we can calculate galactic distances. But we have to be careful about how we define distances.

Suppose we use the parsec definition above. That is, based upon the light we currently see, how far away is a quasar with redshift z = 6? Another way to say this would be “How far away was the quasar when the light left it?” This turns out to be about 1.2 billion parsecs. It’s tempting to convert this to light years, and thus say it was about 3.9 billion light years away, but this is misleading. Because the cosmos was expanding as the light traveled to us, it actually took the light about 12.8 billion years to reach us. So its light travel time distance is actually 12.8 billion light years. This is the most common “distance” used, since it’s easy to compare with the age of the light. When we observe a quasar with a redshift of z = 6, we see the universe as it was 12.8 billion years ago.

Unfortunately this can also make things more confusing. Given that the travel time distance is 12.8 billion years, one might assume that the quasar is about 12.8 billion light years away, rather than 3.9 billion light years when the light left it. But the light we observe was traveling toward us as the universe expanded, while the quasar it left behind moved away from us with the expansion of space. This comoving distance is about 8.4 billion parsecs, which is equivalent to 27 billion light years.

Each of these distances is valid in its own way, even though they are all quite different. That’s why astronomers often stick to redshift.

1. Thanks for explaining this. Either I’ve just had a Krell brain boost, or this is the best explanation of co-moving distance that I’ve heard….!

2. We love that you take the effort to explain things as clearly as possible Brian. It’s not easy!!

3. I’m deeply perplexed by this discussion.
I wonder about the validity of assuming ‘C’ in Gravitationally bound, Dark Matter vacuum (Minkowski space), to be the same as ‘C’ in the Dark Energy driven expanding vacuum (De Sitter space) between Galaxies.
Lorentz and Einstein himself, declared that Special Relativity ONLY applies in non-accelerating reference frames, so it seems to me that light in that accelerating expansion of De Sitter space, should not be considered to be bound by the same constraints as it might be in non-expanding Minkowski space.

The ^CDM model rounds off far too much for my liking, trying to quantify non-fixed variables with constant values, only ONE of which is the “speed” of light, whilst another glaring problem is the variation in our best estimates of the expansion rate itself. These estimates vary by just over 9% and all three appear to be equally valid.
However the one factor which can be clearly defined as a difference in the methods used is the supposed time frame between events observed/measured. This time lag should have been analysed as evidence of the changed rate of expansion over time, but alas, appears to get swept under some kind of cosmic carpet and we persist with our insistence that the expansion can be offset with a fixed constant value.

There’s more, but this broadly sums up my confusion.

1. Author

The expanding universe model uses general relativity, not special relativity, which does apply to both accelerating and non-accelerating reference frames.

4. This article re-enforces a question that I have. While the speed of light is independent of the direction it is traveling, it is dependent on the density of the media that it is traveling through. From the big bang to some point, the density of the universe was high and with expansion dropping. What would the effect of the density be on measuring distances, determining the age of the universe, and what is the effect of gravity on the speed of light since it can bend light? Big multi-question and I expect a bigger set of answers–thank you!

5. I don’t think gravity affects the speed of light at all, since the lower speed in a denser medium is because the photons interact with the atoms on their way through.
The early universe part’s an interesting one. I guess it depends on the sensitivity of the telescope. I don’t know what altitude sunlight starts bending in our atmosphere to a measurable degree, but there would be a density point where we wouldn’t detect it anymore. So you can measure how much it bends at a given altitude to sea level.
I don’t imagine the ratio would be any differently in the early universe, I don’t think atoms reflect light any differently if they already emit light.