## The Expanding Universe: Doppler Shift To Dark Energy

Introduction

A previous tutorial discussed how the Doppler Effect can be used to determine the velocity of an astronomical object towards or away from us. This follow-on tutorial covers its use in perhaps the greatest astronomical discovery of the twentieth century, namely that the entire universe is expanding. Key to this was the Doppler Effect as observed in distant galaxies.

The discovery of the expansion

At the start of the twentieth century, Henrietta Leavitt was studying a particular class of regular variable stars known as Cepheids. Leavitt discovered there was a direct relationship between the time the Cepheid took to vary in brightness and its absolute luminosity. This meant that if you measured the period of a Cepheid, you could determine its absolute brightness. Comparing this to its apparent brightness meant that it was possible to calculate the star’s distance.

In the late 1920s Edwin Hubble used the 100” telescope on Mount Wilson outside Los Angeles (figure 2), at the time the largest in the world, to identify and measure the periods of Cepheids in a number of galaxies. From this he was able to calculate how far away the galaxies were. Comparing this with Doppler redshifts previously obtained by Vesto Slipher and Milton Humason he found, broadly speaking, that the galaxies were all receding from us and the further away they were the faster they were moving. The equation Hubble derived is very simple:

Recessional velocity = H0 * distance.

Where H0 is the Hubble constant.

What is this telling us? Firstly, in whichever direction we look we will see that the majority of galaxies are receding from us. Secondly the speed of recession is proportional to how far away a galaxy is. Together this tells us that the universe is expanding at a constant rate defined by the Hubble constant and we appear to be at the centre of that expansion. We are not, read on!

We can run the expansion backwards in time. When we do, the universe gets smaller and smaller until about 13.8 billion years ago the universe has shrunk to a tiny point. This represents the beginning of the universe and its expansion from there is the famous Big Bang.

Running the expansion forwards, we find the galaxies becoming ever further and further apart.

Some sample redshifts

With large redshifts, radiation that is normally in the visible part of the spectrum will be shifted into the infrared or even further into the microwave region. Conversely, radiation that is normally unseen in the ultraviolet can be shifted into the visible part of the spectrum or even beyond into the infrared. This is one of the reasons that the James Webb Space Telescope is optimised for infrared observations. Very distant galaxies are brightest in this wavelength range because of the effects of redshift.

Let us recall the formula from the earlier tutorial on the Doppler Effect.

Velocity = redshift * speed of light (roughly 300,000 km/sec).

As can be seen from the above table, redshifts can exceed a value of one by a considerable amount.

Going back to the last equation and setting the redshift equal to 1 gives a recessional velocity equal to the speed of light and if the redshift is greater than this then the galaxy must be travelling faster than light. This seems to be a problem as nothing can travel faster than light. So, what is going on? Perhaps there is an issue with the calculation. There is an answer but we will need to discuss some other features of the universe first.

The inconstant constant

The value of the Hubble constant is usually given in units of km/sec/Mpc. In other words, by how much in km/sec does the recessional velocity increase for every million parsecs of distance. So, what is the value of the Hubble constant? That is a good question and one that has exercised the minds of scientists for the best part of a century. Over the years the perceived value has oscillated wildly and even now although it has been constrained much better than before the matter is still not settled.

There are two competing values in 2022. One derived from redshifts and direct distance measures gives a value of 72 km/sec/Mpc. The other derived from the cosmic microwave background is 67 km/sec/Mpc. Obviously, these are quite close. The problem is that they are both quite tightly constrained and although there may be errors in the stated values these are not large enough to cause the two results to overlap and potentially reconcile the difference. Clearly something is going on that we do not understand!

Parsecs When discussing distances in the universe astronomers often use the parsec rather than the light year. One parsec equals roughly 3.26 light years. One megaparsec (Mpc) is one million parsecs or around 3,260,000 light years

Although the Hubble Constant is often referred to as that, it is in fact not actually constant at all. The value we derive is the current value. Over the age of the universe the value of the ‘constant’ will change. The correct term for the ‘constant’ is actually the Hubble Parameter. H0 is used to describe the Hubble Parameter at the present time.

When red is blue

A common misconception is that all galaxies are receding from us. This is not the case, for example the Andromeda Galaxy M31 (Figure 3) is actually approaching us and has a blueshift of 0.001.

The reason for this is quite straightforward. We can think of individual galaxies as having two different types of motions. Firstly, there is the recession caused by the expansion of the universe. Secondly, galaxies can have their own intrinsic motions. If we consider the case of our own Milky Way galaxy and M31 the cause of this motion will become clear. In cosmological terms these two galaxies are comparatively close together, so close in fact that their mutual gravitational attraction is pulling them towards each other faster than the universe’s expansion is pulling them apart. It is this that results in the observed blue shift.

Now recall that the speed of expansion increases with distance. With galaxies that are close to us the expansion speed is quite small and is dwarfed by the gravitational motions of the galaxies. However, with increasing distance, the expansion speed outstrips any intrinsic motions galaxies may have and hence results in an overall redshift.

Space itself is expanding

One of the misconceptions sometimes encountered is the impression that the galaxies, in moving apart, are expanding into empty space. This is not the case. What is actually happening is rather that space is itself expanding and carrying the galaxies along with it. There is nothing for the universe to expand into as space is an integral part of the universe, born along with matter and energy in the Big Bang.

Let’s have an analogy to make this clearer. Consider two people, Alice and Bob. They are standing on a gigantic rubber sheet that is just pulled taut (Figure 4a). Notice they are two squares apart. There are two ways that they can move further apart.

Firstly, they can just walk away from each other as in Figure 4b and they are now four squares apart. The second way is for the rubber sheet they are standing on to be stretched tighter (Figure 4c). In this latter case Alice and Bob have been carried further apart as the space they are standing on (the rubber sheet) has expanded. They have not themselves moved, being still only two squares apart. With respect to the underlying sheet, they are standing in the same positions.

The first example where they got up and walked away is analogous to the intrinsic motions of the galaxies. The second with the stretching of the sheet is equivalent to the expansion of space itself carrying the galaxies along with it.

Suppose that Alice and Bob are actually walking towards each other as the sheet is being stretched. If they are walking faster than the stretching, then they will move closer to each other. This is comparable to what is happening with the Andromeda Galaxy and why it is approaching us. Conversely, if the stretching speed is greater than their walking pace, then they will continue to move apart as do the majority of galaxies.

Expanding or not?

Now you may think from what has gone before that if you had an enormous and impossible ruler stretching between two galaxies then it would always record the same distance as the ruler would expand along with space. This however is not the case. The forces that bind the molecules together in the ruler are far stronger than those due to the expansion of space.

The same is true for individual stars, planets, galaxies and clusters of galaxies. Here too the forces of molecular and gravitational attraction override the shearing effects of the universe’s expansion.

Redshifts in expanding space

We can now reconsider the redshifts we discussed earlier. The redshift due to the intrinsic motions of the galaxies is exactly that of the conventional Doppler Effect. The redshift due to the expansion of space is subtly different. Here the galaxies are not moving through space, and in this case the redshift is due to expanding space ‘stretching’ the wavelength. Imagine a wavy line painted on the rubber sheet, as the sheet is stretched, the separation of the wave peaks becomes further apart. This is more than just an academic distinction; it has an important effect that we will discuss later.

Expansion, expansion, expansion

So how does a universe expanding at a constant rate give rise to the observation that the more distant objects are receding faster? Also, as everything appears to be receding from us, does this mean we are truly at the centre of the universe?

Consider the four galaxies in Figure 5a. At some point in time, they are each say 100 million light years apart.

Therefore, we have the following distances

After a certain time, the expansion of the universe has doubled the separation of the individual galaxies to 200 million light years (Figure 5b).  As a result, the distances are now as follows

We can see that in the same time period A to B has increased by 100 million light years, A to C by 200 million light years and A to D by 300 million light years. In this way, the further away a galaxy is the more its distance has increased in a given time, in other words the faster its recession velocity.

Figure 5a and 5b also illustrate some other important points.

1. The expansion of the universe moves every galaxy away from every other galaxy.
2. Although we have measured everything with respect to galaxy A, we would get the same result if we took any of the other galaxies and measured the recession from them. In short, the expansion has no centre but everything is expanding away from everything else. We are not at the centre of the universe, nor is anywhere else.

Faster than light? – No

Einstein’s Special Theory of Relativity states that nothing can move faster than the speed of light. Armed with what we know about the expansion of the universe, we can now address the conundrum raised earlier about redshifts and the calculated faster than light velocity of receding galaxies.

Faster than light? – Yes

As we have seen the Hubble parameter tells us that the further away a galaxy is the faster it is receding. If we take this to extremes then at some distance a galaxy will appear to be travelling at the speed of light and if further away its velocity will then exceed the speed of light and its light will never reach us. Does this not break the rules of Special Relativity?

Here is another common misconception. It is often stated that nothing can move faster than the speed of light. This is true but perhaps carelessly worded (I am guilty of this in the section above). The speed limit of the velocity of light refers to objects moving through space. In the expanding universe the galaxies are not moving through space at all but are being carried along by space itself. There is no restriction on the rate that space can expand.

Cosmic acceleration

A reasonable question to ask is, will the expansion continue forever? After all, gravity is pulling on every star and galaxy in the universe. Surely this gravitational force will gradually overcome the expansion, slowing down the receding galaxies in their flight.

There seemed to be two options. Firstly, gravity would completely overcome the expansion and the galaxies would slow to a halt, reverse their direction and head back towards each other. The second option was that gravity would continually slow down the galaxies but never quite enough to bring them to a standstill. In this scenario, which was the favoured one, the universe continues expanding but at an ever-decreasing rate.

In the 1990s two teams set out to measure the expansion rate and resolve this question. To everyone’s surprise the expansion was found not to be slowing at all but actually increasing. This is a bit like throwing a ball in the air and rather than it falling back to earth it continues to fly away faster and faster.

The explanation put forward for this is the so-called Dark Energy. This is an unexplained force intrinsic to space itself. As space expands it creates yet more space thereby creating more dark energy and more expansion.

At the moment, while there are a number of theories put forward to explain Dark Energy, none of them seem satisfactory. It’s worth mentioning that Dark Energy is important and comprises around 71% of the universe. A Nobel Prize awaits the first scientists to successfully explain it!

Conclusion

The discovery that the galaxies were receding from each other led to the understanding that the universe itself was expanding. Running this backwards resulted in the realisation that 13.8 billion years ago the universe came into existence in what we now call the Big Bang. All of this was based on two simple phenomena, the variation in light of Cepheid variables and the redshift caused by the Doppler Effect.