Measuring Distances in Cosmology

I have recently written an article for my local amateur astronomical society about distance measurements in cosmology. There is a lot of confusion about what we mean by “distance” on the vast such scales and the article was written to help clear up of these misunderstandings.

It is an expanded version of a shorter blog post I wrote on this topic in early 2021. As I thought it might be of interest to many of my readers, I have published it here.

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I recently read an article from a popular astronomy website called 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 and so is a unit of distance not time. So, the age of anything in the Universe cannot be measured in light years and, as I’ll explain later, the radiation we detect as the CMB actually lies at a distance of 46 billion light years.

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Definition: a light year is a unit of distance. It is the distance which light travels in a year and is equal to  approximately 9 461 000 000 000 km

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 Reading the article prompted me to write about what we actually mean by “distance” when we are dealing on the vast scales which occur in cosmology. This is an area around which there is a lot of confusion. For example, the furthest galaxy from us GN-z11 is often quoted as lying at a “distance of 32 billion light years”. This frequently causes puzzlement because the Universe is generally believed to be only 13.8 billion years old, and nothing can travel faster than light. So, surely nothing can be further away than 13.8 billion light years?  The situation arises because of the way the main distance measure  used by cosmologists, called the proper distance, is defined.

In fact, the proper distance is only one of the many different ways to define distance used in cosmology and in this article I’ll talk about these definitions and how they differ.

The proper distance and the light travel distance

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 distanceto the galaxy. The proper distance is the most widely used measure in cosmology and when cosmologists used the term “distance” without further qualification they usually mean the proper distance.

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Definition: the proper distance is the separation between two a distant objects at a given time which would be measured by an imaginary cosmological sized ruler.

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The proper distance isn’t of course directly measurable! It is estimated by using other techniques and making assumptions about the way the Universe is expanding. Because the Universe is expanding, the proper distance between two distant objects (which aren’t held together by gravity) will increase over time. As an illustration:

  • Suppose that a photon of light is received today from a galaxy, which 300 million years ago was at a proper distance of 297 million light years from Earth.
  • The Earth has been moving away from this photon all the time it has been travelling towards us, so the photon will have to travel further than 297 million light years to reach us.
  • In fact, the photon will have travelled 300 million light years by the time it reaches Earth and will have taken 300 million years to do so.
  • When the photon reaches us today, the galaxy is now at a proper distance of 303 million light years from us .

In the example above the distance which the light has actually travelled, known as the light travel distance, is 300 million light years. This is three million light years lower than the current proper distance.

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Definition: the light travel distance  between two objects at a given time is the distance travelled by photons emitted by one object (in the example above the distant galaxy) and detected on the other object (in this case the Earth)

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When we are dealing with extremely distant objects there are large differences between the light travel distance and the proper distance. If we take the example of the furthest known galaxy GN-z11, then its light we see today was emitted 13.4 billion light years ago, when the Universe was only 3% of its current age. When this light was emitted, the Earth, and the Solar System did not exist and the Milky Way was still in the Early stages of formation.

GN-z11 – Image credit Wikimedia Commons

In the case of GN-z11.

  • The light we see today was emitted 13.4 billion years ago when GN-z11 was at a proper distance of 2.67 billion light years from the Milky Way.
  • This light has been travelling for 13.4 billion light years to reach us. So, the light travel distance to GN-z11 is 13.4 billion light years.

If we could construct an imaginary ruler between the Milky Way and GN-z11 it would be 32 billion light years long. So, its current proper distance is  32 billion light years

The Comoving distance

The comoving distance is another distance measure which is sometimes used by cosmologists. It is a variation on the proper distance which factors out the expansion of the Universe, giving a distance that does not change in time due to the expansion of space.  The comoving distance between two objects will change if they have any additional relative motion on top of that due to the expansion of the Universe.

Another way to visualise the comoving distance between two distant objects is the distance measured by an imaginary cosmological sized ruler, which is stretching at exactly the same rate the Universe is expanding.

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Definition: the comoving distance between two distant objects at any given time in the past or future is defined to be the proper distance at the present time.

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So, if we consider two objects which are moving apart solely due to the expansion of the Universe (and for no other reason), the proper distance between them increases over time but the comoving distance does not change.

The comoving distance is discussed in more detail in the notes at the end of this article.

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 distance units on the x- axis are in megaparsecs (Mpc) – a unit used by astronomers when measuring distances on intergalactic 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 (D) of a galaxy.

v = HoD

where

·        Ho, the slope of the graph, is the Hubble constant. measures how the recessional velocity of an object varies as a function of its distance.
If v is measured in km/s and D in megaparsecs then Ho is approximately 70 km/s per Mpc.

So, to get an estimate of the proper distance you need to divide the velocity the galaxy is moving away from us by the Hubble constant.

This relationship is known as Hubble’s Law after the American astronomer Edwin Hubble (1889-1953) who discovered it in 1929. However, as you can see most galaxies do not lie on the straight line; some lie above and some below it. This is particularly true for clusters of galaxies where there is a large spread of recessional velocities.


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Definition: the redshift distance to an object is the distance calculated assuming that it obeys Hubble’s law exactly.  
For a galaxy belonging to a group or cluster, the average velocity of all the galaxies in the group or cluster is normally used to calculate its redshift distance, rather than the velocity of the individual galaxy. 
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As an example, the redshift distance to an object moving away from us at 2800 km/s is:
2800 /70 = 40 Mpc

This is equal to about 130 million light years.  The redshift distance has the advantage over other distance measures in that it is easy to calculate. It gives a reasonable estimate of the proper distance to nearby objects.

To complicate matters, at very large distances (i.e., billions of light years) the velocity against distance graph is actually a curve. This means that the Hubble constant isn’t a true constant, it varies over time. In most models of the Universe the Hubble constant was greater billions of years ago than it is today. So, when we look at very distant objects (where we are looking back in time billions of years), we measure a larger value of the Hubble constant.

The Hubble Sphere

If we look again at the formula:

v = HoD

and assume for simplicity that the Hubble constant doesn’t vary over time, then for a large enough proper distance (D)  this will give a recessional velocity (v) greater than the speed of light.

 For example, if we consider GN-z11  at approximately 32 billion light years (10 000 Mpc) from us, From Hubble’s law this galaxy will be receding from us at a velocity of
70  x 10 000 = 700 000 km/s.

This is more than twice the speed of light. This often causes confusion because there is a popular misconception that Einstein’s theory of relativity means that a galaxy can’t recede from us faster than the speed of light.  In fact, relativity says no such thing!

Special relativity says that within a given reference frame nothing can travel faster than light. However, GN z11 and ourselves are in two different reference frames and there is no contradiction with special relativity for GN z-11 to be moving away from us faster than the speed of light. Because GN z-11 is receding from us so fast, any light it is emitting today will never be able to reach us.

The Hubble sphere is an imaginary sphere centred on Earth where a galaxy on the surface of the sphere will be receding from us at exactly the speed of light (assuming a constant value of the Hubble constant). If the Hubble constant is 70 km/s per Mpc and the speed of light is 300 000 km/s, the radius of the Hubble sphere is  300 000/70 = 4 285 Mpc which is equal to 14 billion light years.

Finally, to complete the list, two other distance measures which are sometimes used by astronomers are the luminosity distance and the angular size distance.

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Definition: the luminosity distance is how far away a distant object of known actual brightness would be to have the apparent brightness we observe.

Definition: the angular size distance is how far away a distant object of known actual size would be to have the apparent size we observe.

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How far away is the Cosmic Microwave background?

Going back to the topic at the start of this article, the early Universe was too hot for atoms to have existed. It consisted of a plasma of positively charged hydrogen and helium ions and negatively charged electrons. Electromagnetic radiation, of which light is an example, cannot pass through the plasma. When the Universe was about 400 000 years old (an era which astronomers call the recombination time*) it had 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.

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* Although widely used, the term recombination time is misleading. It implies (wrongly) that atoms previously existed in the very early Universe, were ionised later and, at the recombination time, the ions and electron were recombined back into atoms. This is not the case it was far too hot for atoms to have existed in the very early Universe.

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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 lying at a proper distance of 46 billion light years from Earth.

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 pale blue) 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.

Appendix A Further detail on 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 (D(t)) to their current proper distance (Do).

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

 a(t) = D(t) / Do

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.

So the comoving distance (DCM ) can be defined as the proper distance divided by the scale factor.

DCM = D(t)/a(t)

(In many astronomy textbooks the comoving distance is denoted by the Greek letter χ (lowercase chi))

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 will not change over time.

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.

11 thoughts on “Measuring Distances in Cosmology”

  1. This is so very good, thank you! Whenever I’ve heard that a galaxy is for example, 5 billion years “away” and that the light we see was emitted 5 billion years ago, I’ve always wondered, “okay but where is that galaxy now? It’s been moving away for 5 billion years.” That seems rarely touched on in popular articles so I really like this post for touching on it and showing that the present distance can be calculated with some reliability. Also, it was especially cool to see calculations for how far the galaxy was in the past when the light on our eyes was emitted. I think this is the first time I’ve had a proper (!) appreciation of that side of the question.

    Fun and mind-stretching stuff!

    Like

  2. I came here trying to find an answer to the question of the definition of year in regards to the age of the universe. Is the usage here the same as an earth year? 13 billion earth years? If so, isn’t that arbitrary since earth years are simply a result of the orbit around the sun which is further constrained in definition by the Distance relative to the sun. An earth year, Saturn year, all measured by orbital speed but distance of the object seems indispensable in defining “year”.
    Or is the age of the universe version of year a light year? So again, distance. If it is a light year, what is the relative “location” ,which I know is a horrible word in this context- but what is the relative guage for determining what a light year would be? Light travels at a certain speed, regardless of if the observer is on earth or Saturn. Yet each of these are limited by themselves essentially it seems.
    So, what is the proper meaning of year in the age of universe. If the answer is that the definition isn’t really about time, then what is meant? I’m barely even an amateur on any of this so go easy if some of or all of what I said is incorrect. All I’m looking for is an answer to the “year” question applied to the age of the universe.

    Liked by 1 person

    1. Hi Josh,

      the question you ask is very interesting and perfectly valid. What we mean when we talk about a “year” is often glossed over.

      In astronomy, when we talk about years, we normally mean Julian years. A Julian year is a measurement of time and is defined as exactly 365.25 days. Where a “day” is defined as having exactly 86400 SI seconds.

      Because the Earth’s rotation speed varies a little and is slowing down gradually. an SI second is defined in terms of a transitions of a caesium atom rather than being a fraction of the mean solar day (the natural day due the apparent motion of the Sun across the sky).

      I hope this helps…

      Kind regards,

      Steve

      Like

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