# Distances in Cosmology

This post has now been superseded by a more detailed post Measuring Distances in Cosmology

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I recently read an article on the popular astronomy website site Universe Today.

It stated that

‘…the CMB [cosmic microwave background radiation]  is visible at a distance of 13.8 billion light years in all directions from Earth,

leading scientists to determine that this is the true age of the Universe. ‘ This statement isn’t quite correct. A light year is the distance light travels in a year (roughly 9.46 trillion km) and so is a unit of distance not of time.

Reading it prompted me to write this article about what we actually mean by ‘distance’ when we are dealing on the vast scales which occur in cosmology.  As I’ll explain later,  the proper distance to the radiation we see as the CMB is  actually 46 billion light years.

### The proper distance vs the light travel distance

It has been known since the late 1920’s that the Universe is expanding. By this we mean that distances between objects which are not bound together by another force, such as gravity, increase over time.

. Diagram showing the expansion of a small region of the Universe

When we look at a distant galaxy then the finite speed of light means that its light will have taken millions (or even billions) of years to have reached us.  if we could construct a cosmological-sized ruler between the galaxy and the Earth then we would measure a quantity known as the proper distance to the galaxy. This is the separation between the Earth and the galaxy at a given time. Because the Universe is expanding, the proper distance between two distant objects (which aren’t held together by gravity) will increase with time. For example, suppose that a photon of light was emitted from a galaxy which 300 million years ago was at a proper distance of 294 million light years from Earth, because the Earth is moving away from this photon all the time it is travelling towards us, the photon will have to travel 300 million light years to reach Earth. When it reaches us the proper distance of the galaxy is now 306 million light years. This is shown in the diagram below. In the example above, the distance travelled by the photons is called the  light travel distance and is equal to 300 million light years.

### The Comoving distance

If we consider any two objects which are far enough away from each other so that they are not bound together by gravity then the cosmic scale factor  is the ratio of their proper distance at time t  which will call DP(t)  to  their current proper distance DPo.

It is given the symbol a(t) is defined as:

a(t) = DP(t) / DPo

The cosmic scale factor is

• equal to zero at t=0, the instant of the Big Bang
• equal to one at the current age of the Universe

Clearly, as the Universe expands and objects move further apart the cosmic scale factor increases.

, The comoving distance (DCM ) is defined as the proper distance divided by the scale factor.

DCM = DP(t)/a(t)

For objects which only get further apart (i.e. their proper distance increases)  as a result of the expansion of the Universe the comoving distance between them does not change over time. The comoving distance changes if the proper distances change due to a factor other than the expansion of the Universe As the Universe expands, the proper distance to the galaxy increases but the comoving distance does not change. This is also shown in the graph below. ### Other distance measures

If we measure the velocity that galaxies are moving away from us due to the expansion of the Universe and plot it as a function of their distance, then we get a graph like that shown below.

. The Megaparsec (Mpc) is a unit used by astronomers when measuring distances at galactic scales. One Mpc is equal to 3.26 million light years

There is a clear relationship between the recessional velocity (v) and the proper distance (DP) of a galaxy

v = HoDP

where

• Ho is a constant known as the Hubble constant. If v is measured in km/s and DP in megaparsecs then Ho is approximately 70 km/s per Mpc. The Hubble constant measures how the recessional velocity of an object varies as a function of its distance.

This relationship is known as Hubble’s Law after the American astronomer Edwin Hubble (1889-1953) who discovered it in 1929.

The redshift distance (Dz ) to an object is a distance calculated assuming that it obeys Hubble’s law exactly. So, the redshift distance to an object moving away from us at 3000 km/s would be:

3000 /75 = 40 Mpc which is equal to about 130 million light years.

Two other distance measures which are sometimes used by astronomers are:

• The luminosity distance which is estimated from how far away a distant object of known actual brightness would need to be to have its apparent brightness.
• The angular size distance estimated from how far away a distant object of known actual diameter would need to be to have its observed apparent diameter.

### How far away is the Cosmic Microwave background

As readers of my earlier post will know, the early Universe was far too hot for atoms to exist. It contained a plasma of positively charged hydrogen and helium ions and negatively charged electrons. Electromagnetic radiation, of which light is an example, cannot pass through plasma. Photons are absorbed and re-emitted by  the electrically charged particles (a process known as Thompson scattering). When the Universe was about 400 000 years old (which astronomers call the recombination time*) it had expanded and cooled to around 2700 degrees C and all the ions and electrons had combined to make atoms. The Universe became transparent to radiation. Photons could pass unhindered through the hydrogen and helium gases.

The weak radiation we observe today is a relic from the recombination time. The photons we detect were last scattered 13.8 billion years ago by the hot plasma. But, due to the expansion of the Universe, the region they were scattered from is a spherical shell of points at a proper distance of 46  billion light years from Earth. The CMB photons which are reaching us now were Thompson scattered 13.8 billion years ago when the Universe was only 400 000 years old. The region of space where they were scattered is 46 billion light years away from us.

In the diagram below a slice has been taken through this spherical shell. The locations inside the sphere (shaded pale yellow) lie closer than 46 billion light years.  The CMB photons from these regions will have taken less than 13.8 billion years to reach us and will have arrived in the past. The locations outside the sphere (shaded green) lie further away than 46 billion light years. The photons from these regions will take longer than 13.8 billion years to reach us and will arrive in the future. In 3 billion years’ time when the Universe has expanded further. The CMB photons that will reach  us will have been emitted from a spherical shell which will lie at a proper distance of 57 billion light years from Earth and these photons will have been travelling for 16.8 billion years. * Although widely used, the term recombination time is misleading. It implies (wrongly) that atoms previously existed in the very early Universe, were later ionised and at the recombination time  were recombined back into atoms. This is not the case it was far too hot for atoms to have existed in the early universe.

## 3 thoughts on “Distances in Cosmology”

1. Hi Steve,
Great article – and your account & diagrams of expansion-related matters, e.g. co-moving distances, is far more understandable than most textbooks manage to achieve!
One point that occurs to me: the usual accounts of recombination (so-called!) make it appear a relatively abrupt process. But the difference in ionization energy between He (24.5 eV) and H (13.8 eV) mean that ~25% of the plasma disappeared at a much higher temperature than 2700C, & with the lower ionization-density, space would have been rather less opaque from then onwards.
Regards, David.

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1. Thank you for your comment. Yes you are right… there would have been a decrease in opacity when helium atoms formed, which would have been at a higher temperature than hydrogen atoms – given the higher first ionisation energy of helium. The formation of hydrogen atoms from protons wasn’t instantaneous but was essentially complete 380 000 years after the big bang.

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