A few weeks ago it was announced that a team had discovered what seems to be the most distant galaxy yet discovered. You can read the BBC story about it here, or if you like you can read the Nature science paper here to get as much detail as you could wish for. The galaxy, which has the catchy name *z8_GND_5296*, was discovered using the Hubble Space Telescope, with its distance being determined using the Keck 10m telescope on the summit of Mauna Kea.

In fact, what astronomers measure is not the distance of a distant galaxy, but its redshift, which astronomers denote with the letter . Redshift is the movement of the spectral lines of a galaxy to longer wavelengths due to the expansion of the Universe, the expansion discovered by Edwin Hubble in 1929. The redshift of this newly discovered galaxy has been found by Keck to be , beating the previous record of . But how do astronomers translate this into a distance?

## The cosmological definition of redshift

It turns out that measuring distances in astronomy is one of the most difficult things to do for several reasons. Not only are there very few direct ways to measure the distance to an object, after all we can hardly lay down a measuring tape between us and the stars and galaxies! But, to make it even worse, there also are various definitions of distance! In a future blog I will talk about the most direct ways we have to measure distance, but how we translate from these measurements to a distance also depend on the *geometry* of the Universe, which Einstein showed in his General Theory of Relativity is determined by the effects of gravity.

The geometry of the Universe is determined by its average density , and how this relates to something called the *“critical density”* , which is the dividing line between whether the Universe will carry on expanding forever, or stop expanding and start to collapse. If average density the Universe will stop expanding and collapse. If the Universe will carry on expanding forever, and if the average density the Universe is on the dividing line between the two, and is said to have a *flat* geometry. Without going into the details here, most cosmologists believe that we live in a Universe where , that is a flat Universe.

The preferred method for measuring large distances “directly” is to use something called a Type Ia Supernova, I will blog about this method again in a future blog. But, we can only see Type Ia supernovae out to distances corresponding to a redshift of about . The galaxy in this story is *much* further away than this, . So, to calculate its distance we have to use a model for the expansion of the Universe, and something called *Hubble’s law*.

The measured redshift of a galaxy (or any object) is just given by

where is the observed wavelength and would be the wavelength of a spectral line (usually for a galaxy it is a line called the Lyman-alpha line) in the laboratory.

As long as the redshift is much less than 1, we can then write that

where is the recession velocity of the galaxy and is the speed of light. In the case of not being less than 1, we need to modify this equation to the relativistic version, so we write

In our case, , so we need to use this relativistic formula, and when we do we get that the recession velocity of the galaxy is , the speed of light.

Re-arranging equation 1 we can write . In principle, the distance and redshift are just related via the Hubble law

,

where is the recession velocity of the galaxy, is the Hubble constant, and is the distance of the galaxy.

Things get a lot more complicated, however, when we take into account General Relativity, and its effects on the curvature of space, and even the definition of distance in an expanding Universe. I will return to this in a future blog, but here I will just quote the answer one gets if one inputs a redshift of into a “distance calculator” where we specify the value of Hubble’s constant to be and we have a flat Universe () with a value of (the relative density of the Universe in the form of matter) and (the relative density of the Universe in the form of dark energy).

Putting these values in gives a *co-moving radial distance* to the galaxy of . (I will define what *“co-moving radial distance”* is in a future blog, but it is the distance quoted in this story, and is the measurement of distance which is closest to what we think of as “distance”).

The redshift also gives a time when the galaxy was formed, with being the present. We find that it was formed some 13.1 billion years ago, when the Universe was only about 700,000 years old.

## A galaxy 30 billion light years away??

Going back to the *“co-moving radial distance”*, I said it is about 30 billion light years. A light year is, of course, the distance light travels in one year. So how can a galaxy be 30 billion light years away, implying the light has taken 30 billion years to reach us, if the Universe is only 13.7 billion years old?? This sounds like a contradiction. The solution to this apparent contradiction is that the Universe has expanded since the light left the galaxy. This is what causes the redshift. In fact, the size of the Universe now compared to the size of the Universe when the light left the galaxy is simply given by

where is known as the *scale factor* of the Universe, or its relative size. For we have , so the Universe is 8.51 times *bigger* now than when light left the galaxy (this is what causes the redshift, it is the expansion of space, not that the galaxy is moving through space with a speed of 97% of the speed of light). It is the fact that the Universe is over 8 times bigger now than when the light left the galaxy which allows its distance measured in light years to be more than a distance of 13.7 billion light years that one would naively think was the maximum possible! So, there is no contradiction when one thinks about things correctly.