Measuring Distances in Cosmology

I’ve recently written an article for my native newbie astronomical society about distance measurements in cosmology. There is lots of confusion about what we imply by “distance” on the huge such scales and the article was written to assist clear up of those misunderstandings.

It is an expanded model of a shorter weblog submit I wrote on this subject in early 2021. As I assumed it could be of curiosity to lots of my readers, I’ve revealed it here.


I just lately learn an article from a popular astronomy website called Universe Today.

It said 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.


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



 Reading the article prompted me to write about what we actually mean by “distance” after we are dealing on the huge scales which happen in cosmology. This is an space round which there’s lots of confusion. For instance, the furthest galaxy from us GN-z11 is usually quoted as mendacity at a “distance of 32 billion light years”. This regularly causes puzzlement as a result of the Universe is usually believed to be solely 13.8 billion years previous, and nothing can journey quicker than mild. So, certainly nothing will be additional away than 13.8 billion mild years?  The state of affairs arises due to the way the primary distance measure  utilized by cosmologists, known as the correct distance, is outlined.

In reality, the correct distance is simply one of many many various methods to outline distance used in cosmology and in this text I’ll discuss these definitions and the way they differ.

The correct distance and the sunshine journey distance

When we take a look at a distant galaxy then the finite speed of sunshine implies that its mild could have taken thousands and thousands (and even billions) of years to have reached us.  if we might assemble a cosmological-sized ruler between the galaxy and the Earth then we’d measure a amount generally known as the correct distanceto the galaxy. The correct distance is probably the most extensively used measure in cosmology and when cosmologists used the time period “distance” with out additional qualification they normally imply the correct distance.


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


The correct distance isn’t after all straight measurable! It is estimated through the use of different methods and making assumptions in regards to the way the Universe is increasing. Because the Universe is increasing, the correct distance between two distant objects (which aren’t held collectively by gravity) will enhance over time. As an illustration:

  • Suppose {that a} photon of sunshine is obtained at the moment from a galaxy, which 300 million years in the past was at a correct distance of 297 million mild years from Earth.
  • The Earth has been shifting away from this photon on a regular basis it has been travelling in direction of us, so the photon must journey additional than 297 million mild years to achieve us.
  • In reality, the photon could have travelled 300 million mild years by the point it reaches Earth and could have taken 300 million years to take action.
  • When the photon reaches us at the moment, the galaxy is now at a correct distance of 303 million mild years from us .

In the instance above the space which the sunshine has truly travelled, generally known as the sunshine journey distance, is 300 million mild years. This is three million mild years decrease than the present correct distance.


Definition: the sunshine journey distance  between two objects at a given time is the space travelled by photons emitted by one object (in the instance above the distant galaxy) and detected on the opposite object (in this case the Earth)


When we’re coping with extraordinarily distant objects there are giant variations between the sunshine journey distance and the correct distance. If we take the instance of the furthest identified galaxy GN-z11, then its mild we see at the moment was emitted 13.4 billion mild years in the past, when the Universe was solely 3% of its present age. When this mild was emitted, the Earth, and the Solar System didn’t exist and the Milky Way was nonetheless in the Early phases of formation.

GN-z11 – Image credit score Wikimedia Commons

In the case of GN-z11.

  • The mild we see at the moment was emitted 13.4 billion years in the past when GN-z11 was at a correct distance of two.67 billion mild years from the Milky Way.
  • This mild has been travelling for 13.4 billion mild years to achieve us. So, the sunshine journey distance to GN-z11 is 13.4 billion mild years.

If we might assemble an imaginary ruler between the Milky Way and GN-z11 it could be 32 billion mild years lengthy. So, its present correct distance is  32 billion mild years

The Comoving distance

The comoving distance is one other distance measure which is typically utilized by cosmologists. It is a variation on the correct distance which components out the growth of the Universe, giving a distance that doesn’t change in time because of the growth of space.  The comoving distance between two objects will change if they’ve any further relative movement on high of that because of the growth of the Universe.

Another way to visualise the comoving distance between two distant objects is the space measured by an imaginary cosmological sized ruler, which is stretching at precisely the identical price the Universe is increasing.


Definition: the comoving distance between two distant objects at any given time in the previous or future is outlined to be the correct distance on the current time.


So, if we contemplate two objects that are shifting aside solely because of the growth of the Universe (and for no different purpose), the correct distance between them will increase over time however the comoving distance doesn’t change.

The comoving distance is mentioned in extra element in the notes on the finish of this text.

Other distance measures

If we measure the speed that galaxies are shifting away from us because of the growth of the Universe and plot it as a perform of their distance, then we get a graph like that proven beneath.

The distance models on the x- axis are in megaparsecs (Mpc) – a unit utilized by astronomers when measuring distances on intergalactic scales. One Mpc is the same as 3.26 million mild years.

There is a transparent relationship between the recessional velocity (v) and the correct distance (D) of a galaxy.

v = HoD

the place

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

So, to get an estimate of the correct distance you must divide the speed the galaxy is shifting away from us by the Hubble fixed.

This relationship is understood as Hubble’s Law after the American astronomer Edwin Hubble (1889-1953) who found it in 1929. However, as you’ll be able to see most galaxies don’t lie on the straight line; some lie above and a few beneath it. This is especially true for clusters of galaxies the place there’s a giant unfold of recessional velocities.

Definition: the redshift distance to an object is the space calculated assuming that it obeys Hubble’s legislation precisely.  
For a galaxy belonging to a bunch or cluster, the typical velocity of all of the galaxies in the group or cluster is generally used to calculate its redshift distance, slightly than the speed of the person galaxy. 
As an instance, the redshift distance to an object shifting away from us at 2800 km/s is:
2800 /70 = 40 Mpc

This is the same as about 130 million mild years.  The redshift distance has the benefit over different distance measures in that it’s straightforward to calculate. It offers an affordable estimate of the correct distance to close by objects.

To complicate issues, at very giant distances (i.e., billions of sunshine years) the speed towards distance graph is definitely a curve. This implies that the Hubble fixed isn’t a real fixed, it varies over time. In most fashions of the Universe the Hubble fixed was larger billions of years in the past than it’s at the moment. So, after we take a look at very distant objects (the place we’re trying again in time billions of years), we measure a bigger worth of the Hubble fixed.

The Hubble Sphere

If we glance once more on the method:

v = HoD

and assume for simplicity that the Hubble fixed doesn’t fluctuate over time, then for a big sufficient correct distance (D)  it will give a recessional velocity (v) larger than the speed of sunshine.

 For instance, if we contemplate GN-z11  at roughly 32 billion mild years (10 000 Mpc) from us, From Hubble’s legislation this galaxy will likely be receding from us at a velocity of
70  x 10 000 = 700 000 km/s.

This is greater than twice the speed of sunshine. This usually causes confusion as a result of there’s a standard false impression that Einstein’s concept of relativity implies that a galaxy can’t recede from us quicker than the speed of sunshine.  In reality, relativity says no such factor!

Special relativity says that inside a given reference body nothing can journey quicker than mild. However, GN z11 and ourselves are in two completely different reference frames and there’s no contradiction with particular relativity for GN z-11 to be shifting away from us quicker than the speed of sunshine. Because GN z-11 is receding from us so quick, any mild it’s emitting at the moment won’t ever be capable to attain us.

The Hubble sphere is an imaginary sphere centred on Earth the place a galaxy on the floor of the sphere will likely be receding from us at precisely the speed of sunshine (assuming a continuing worth of the Hubble fixed). If the Hubble fixed is 70 km/s per Mpc and the speed of sunshine is 300 000 km/s, the radius of the Hubble sphere is  300 000/70 = 4 285 Mpc which is the same as 14 billion mild years.

Finally, to finish the record, two different distance measures that are generally utilized by astronomers are the luminosity distance and the angular dimension distance.


Definition: the luminosity distance is how far-off a distant object of identified precise brightness could be to have the obvious brightness we observe.

Definition: the angular dimension distance is how far-off a distant object of identified precise dimension could be to have the obvious dimension we observe.


How far-off is the Cosmic Microwave background?

Going again to the subject at first of this text, the early Universe was too sizzling 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 mild is an instance, can not go via the plasma. When the Universe was about 400 000 years previous (an period which astronomers name the recombination time*) it had cooled to round 2700 levels C and all of the ions and electrons had mixed to make atoms. The Universe grew to become clear to radiation. Photons might go unhindered via the hydrogen and helium gases.


* Although extensively used, the time period recombination time is deceptive. It implies (wrongly) that atoms beforehand existed in the very early Universe, had been ionised later and, on the recombination time, the ions and electron had been recombined again into atoms. This just isn’t the case it was far too sizzling for atoms to have existed in the very early Universe.


The weak radiation we observe at the moment is a relic from the recombination time. The photons we detect had been final scattered 13.8 billion years in the past by the recent plasma. But, because of the growth of the Universe, the area they had been scattered from is a spherical shell of factors mendacity at a correct distance of 46 billion mild years from Earth.

In the diagram beneath a slice has been taken via this spherical shell. The areas contained in the sphere (shaded pale yellow) lie nearer than 46 billion mild years.  The CMB photons from these areas could have taken lower than 13.8 billion years to achieve us and could have arrived in the previous. The areas exterior the sphere (shaded pale blue) lie additional away than 46 billion mild years. The photons from these areas will take longer than 13.8 billion years to achieve us and can arrive in the long run.

Appendix A Further element on the Comoving distance

If we contemplate any two objects that are far sufficient away from one another in order that they aren’t sure collectively by gravity then the cosmic scale issue is the ratio of their correct distance at time t (D(t)) to their present correct distance (Do).

It is given the image a(t) and is outlined as:

 a(t) = D(t) / Do

The cosmic scale issue is.

  • equal to zero at t=0, the moment of the Big Bang
  • equal to one on the present age of the Universe

Clearly, because the Universe expands, and objects transfer additional aside the cosmic scale issue will increase.

So the comoving distance (DCM ) will be outlined as the correct distance divided by the dimensions issue.

DCM = D(t)/a(t)

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

For objects which solely get additional aside (i.e., their correct distance will increase) because of the growth of the Universe the comoving distance between them is not going to change over time.

As the Universe expands, the correct distance to the galaxy will increase however the comoving distance doesn’t change. This can be proven in the graph beneath.

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