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.

on 14/11/2013 at 08:47 |hitchens67Reblogged this on hitchens67 Atheism WOW!! Campaign.

on 14/11/2013 at 20:00 |Phillip Helbig“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.”OK, you can use this as a catch-all to excuse all errors above, but some things are seriously misleading.

First, you need to specify what you mean by Omega. Does it include the cosmological constant or not?

If you assume the cosmological constant is 0, then what you say is true: Omega=1 divides spatially open from spatially closed and expanding forever from collapsing in the future. If the cosmological constant is not 0, however, then “geometry” and “destiny” don’t have a simple relationship.

If your Omega includes the cosmological constant, which seems to be the case, then what you say about geometry is still true (Omega=1 divides spatially open and infinite from spatially closed and finite (for experts: assuming a trivial topology)), but what you say about the future of the universe is not true. Suppose, for example, that Omega_matter and the cosmological constant have the values of the “standard model”, 0.27 and 0.73 or whatever, but that the sum is just slightly more than 1. In this case, the universe is spatially closed and finite, but will expand forever. It is also possible that the universe is infinite with negative curvature yet collapses in the future.

Rolf Stabell and Sjur Refsdal wrote the classic paper on this topic. This is the paper in classical cosmology which everyone should read. If you are interested in cosmology and want to read just one refereed-journal paper, this should be it. I have read it several times myself. It contains more stuff in some of the footnotes than entire papers by other people.

The Stabell and Refsdal paper uses the old-fashioned notation with sigma and

q, so might be a bit difficult to follow for those used to other choices of parameters (though their notation does have its advantages). For this reason, I discuss the most basic concepts from Stabell and Refsdal in Sect. 2. of this paper.Please get rid of the relativistic Doppler formula. Simple justification: It does not contain any cosmological parameters. However, a little thought shows that the velocity as a function of

zcannot be the same whatever the cosmological parameters, though this formula implies that it is. Also, it implies thatvcannot exceedc, but it can, and does, for some distances and some redshifts. This is even the case for your example: The Hubble’s law you mention above holds exactly, but neither the distance nor the velocity is “directly observable”. In particular, it holds for the comoving radial distance and its temporal derivative. Plugging in the numbers, you see that this galaxy is receding at 655,423.2 km/s. This is more than twice the speed of light.Essential reading: my paper on distance calculation, which includes a discussion of various distances. This piece of correspondence in

The Observatorysheds some light on what is meant by Hubble’s law. In particular, the reference to Harrison clears up the confusion about the relativistic Doppler formula. (The Bunn and Hogg paper describes a situation in which the relativistic Doppler formula is actually valid in cosmology, but this is one of only 2 examples I know of where this is the case. Moreover, both cases are more pedagogical exercises than anything practical. All other references to the relativistic Doppler formula in cosmology (including at least one by Hogg) are wrong.)on 14/11/2013 at 20:01 |Phillip HelbigThere are several links in my post above. Unfortunately, they are not very visible due to the colour scheme.

on 14/11/2013 at 20:16Phillip HelbigThe links in your posts are somewhat more visible because they are underlined with dashes, but the ones in the comments are almost invisible!

on 14/11/2013 at 20:05 |RhEvansThank you for this expert comment Phillip. It may raise more questions than it answers, but I appreciate your contribution!

on 14/11/2013 at 20:03 |Phillip HelbigI think it would be fair to mention Ned Wright’s cosmology calculator and provide a link to it. 🙂

on 14/11/2013 at 20:16 |RhEvansYes I was going to do that in a future blog on different definitions of distance 😛

on 06/02/2014 at 06:34 |Measuring the distances to stars | thecuriousastronomer[…] my blog on the supernova in Messier 82, I quoted the distance to M82 in Mega parsecs (Mpc), and in my blog on the most distant galaxy yet discovered, I quoted Hubble’s constant in km/s per […]

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