The Size of the Universe

Popular science articles and discussion forums on the internet are full of conflicting reports of the size of the universe, ranging from 13 billion light years to numbers with too many zeroes to write down here. Some of this confusion is because people are talking about slightly different things, while not being specific what they’re talking about. Some is because people genuinely don’t understand how big the universe is, and are just intuiting it from some statistic they have read, such as the age of the universe. So I put together this page to clear up some of the confusion.

First, let me say that everything that follows is derived from Big Bang cosmology. This is the currently accepted model of how the universe developed, and posits that that the universe began as a tremendous explosion from an infinitesimal point, which created space and ultimately everything that is in it. Space (and therefore the universe), have been expanding ever since. The discovery that space is expanding is credited to Edwin Hubble through his study of the red shifts of distant galaxies.

The Expanding Universe

Before we talk about the size of the universe, we need to understand this concept of space expansion thoroughly, because it’s crucial to the discussion. So here is an abbreviated explanation.

Hubble noted two things:

These two things in combination present the picture of an expanding universe. The red shift data indicates that objects that are far from us are moving away from us, no matter in which direction we look. And it shows that the farther these objects are, the faster they are moving away from us.

Now if you assume that the Earth is nothing special in the universe, that it has no special location, then it follows that no matter where you are in the universe, you would still observe the same thing – objects are moving away from you, and that the farther the objects, the faster they are moving away. If every object in the universe is moving away from every other object, it can only mean that the space between the objects is expanding.

It's easy to understand why farther objects are moving away from us faster than nearer objects, if you think of space as expanding at a certain rate. Suppose that the rate of expansion is 1 inch per foot of space per year (purely arbitrary numbers, don't take them seriously). Then every foot of space expands by 1 inch per year. If there is more space between two objects, obviously it will expand more, than if there is less space. If one of those objects is Earth, and you are standing on it, then the other object appears to move away from you as the space between you expands. The farther the object is, the more space there is between you to expand, and so the faster it appears to move away.

Before we go much further with this idea, let's clarify one more point. There are many cases where objects aren't moving away from each other. For example, the Sun isn't moving away from the Earth. While the moon is moving away from the Earth, it's not because the space between the Earth and the moon is expanding, it's because of the tides. If we look in our own neighborhood, yes there are stars that are moving away from us, but there are also stars moving towards us. Even in our galactic neighborhood, the Andromeda galaxy is moving towards us, not away, and is due to collide with the Milky Way in a few billion years. So what gives, is space expanding or not?

The short answer is that yes, space is expanding. However, at short ranges, gravity is a powerful force and can hold stuff together so it doesn't drift apart with the expansion of space. It can even pull things closer to each other, regardless of the expansion of space between them. This is exactly what happens within the Solar System, or within our galaxy. The distances are just too short, and gravity predominates. While the space occupied by the Milky Way is expanding, just like the rest of the universe, objects are not drifting apart because they are held together by gravity.

From this we can conclude that the expansion of the universe becomes apparent only when we look at much larger scales. If you look at galaxies that are tens, or hundreds of millions of light years away from us, or even billions of light years, then the expansion of space is very evident. At such extreme distances, gravity plays no role, and the expansion becomes observable in the red shifts of objects.

Hubble's Parameter

Hubble's observation, that distant objects are moving away from us, and that the more distant they are, the faster they are moving away, can be written mathematically as:

v ∝ D

v=H0 × D

The velocity with which objects are moving away from us is proportional to their distance D from us. You can change the proportionality to an equality by inserting a constant of proportionality, H0 . This is known as the Hubble Constant, which can be experimentally determined. There are several estimates of the Hubble Constant, ranging from about 50 - 100, but a commonly accepted value is about 73 km/s per Mpc. This means that a galaxy at a distance of 1 megaparsec (about 3.2 million light years) is moving away from us at a speed of 73 kilometers per second due to the expansion of the intervening space.

You can derive some interesting results from this about the rate of expansion. Taking the value of 73 km/s per Mpc, you can calculate H0 to be:

73,000 / 3.0857 ×1016 (convert both 73 kilometers and 1 megaparsec into meters) = 2.3658 × 10-18

So space is expanding at the rate of about 2.3 attometers per meter per second (an attometer is 10-18 meter). While this may seem like a tiny amount, it can add up given both the very large distances and very large times common in astronomical calculations.

Consider something familiar, the Earth and the Sun. The distance between them is about 147 million kilometers. How fast is this space between the Earth and Sun expanding? The answer is about 0.34 micrometers per second. That doesn't seem like a whole lot. But consider that the Earth has been around for 4.5 billion years. If we assume that the Earth and Sun have maintained a constant distance to each other, how much has the space between them expanded since the Earth was formed? Just multiply the rate by 4.5 billion years, and it comes to almost 50 million kilometers! So that space has expanded by about a third in this time. And yet, the Earth and Sun haven't drifted apart, because gravity holds them together.

Hubble Length and Volume

Related to the Hubble Constant are two more terms - the Hubble Length and the Hubble Volume.

Consider that Hubble's law says that the farther an object is from us, the faster it's moving away from us. This implies that there must be some distance at which the speed with which an object is moving away from us must be equal to the speed of light. This distance is called the Hubble Length. It can be calculated very easily as c/H0 where c is the speed of light. It comes to about 13.4 billion light years. In other words, objects that are 13.4 billion years away from us are moving away from us at the speed of light. Note that this does not violate relativity in any way, because the objects aren't really moving, the motion is entirely due to the expansion of space between us. There are no limits to this, and you may well have objects moving apart at speeds of several times the speed of light, if there is enough space between them. Again, this is not the movement of an object, in space, but rather the expansion of space itself.

The exact value of the Hubble Length is not known, since it is derived from the Hubble Constant, which itself hasn't been accurately measured. Depending upon the value of the Hubble Constant used, you may come across various values of the Hubble Length, ranging from 13 to 14 billion light years.

You may have noticed that the value approximates the age of the universe, which is currently believed to be about 13.7 billion years. Is this a coincidence or are the two related? The answer is that they are sort of related, but not an exact match.

The Hubble Constant is the rate of expansion of the universe, which is constant in space, but not in time. This means that it has some definite value now, and that this value applies everywhere in the universe, but it didn't always have this value and it might not have this value in the future. Current cosmological models say that the universe has not always expanded at a steady rate, that the rate of expansion has accelerated and slowed down in the past. In fact, even today, the rate of expansion is slowing down, and consequently, the value of the Hubble Constant is decreasing.

So you can see how a straightforward extrapolation from the current value of the Hubble Constant won't give you an exact age of the universe. This is why the numeric values of the Hubble Length and age of the universe won't quite match. This is expected.

The Hubble Volume is simply the volume of space around Earth within a radius of the Hubble Length. It comes to about the order of 1031 cubic lightyears.

The significance of the Hubble Length and the Hubble Volume is that objects that lie within the Hubble Volume (that are closer to us than the Hubble Length) are moving away from us at speeds slower than the speed of light. So a ray of light leaving these objects today will some day reach us. On the hand, objects outside the Hubble Volume are moving away from us faster than the speed of light. Light leaving these objects today will never reach us.

The important thing to note in the last couple of sentences is the word "today". Light leaving objects within the Hubble Volume today will eventually reach us, but light leaving objects outside the Hubble Volume today will never reach us. This is important because light that left objects outside the Hubble volume in the past, can still reach us.

This leads to the concept of the observable universe, described below.

Observable Universe

The observable universe is larger than the Hubble Volume. Objects outside the Hubble Volume, that are today receding from us at speeds faster than the speed of light, are still observable on Earth. Why?

The answer has to do with the expansion of space. Big Bang cosmology says that the universe started as an infinitesimal point, and expanded from there to reach its current dimensions. This implies that the universe was once much smaller, and therefore objects in it were much closer than they are today.

Consider a galaxy that was 8 billion light years away from us, 5 billion years ago. At the time, it was within our Hubble Volume (which has a radius of about 13.4 billion light years - it may have been a bit more or less due to the changes in the Hubble Constant over time, but somewhere in that ballpark). So at the time, it was receding from us at speeds slower than the speed of light.

Therefore, light emitted from that galaxy arrived on Earth 5 billion years later, because light travels faster than the galaxy was receding at the time. However, now 5 billion years have passed, and meanwhile the galaxy has continued to recede from us. According to Hubble's law, the more it recedes, the faster it recedes. In fact, now it is outside the Hubble Volume, and therefore receding from us faster than light. Light leaving it today will never reach us. But light that left it 5 billion years ago does reach us, and will continue to reach us until the point when it exited our Hubble Volume.

This is what makes the observable universe bigger than the Hubble Volume. It's a bit of a cheat. Yes, you can see stuff outside our Hubble Volume, but only as it looked in the distant past. How it looks now is forever outside our reach, because light leaving it today will never reach Earth.

Using some math, you can calculate the size of the observable universe. Figure out the distance to the farthest galaxy that could have emitted light which reaches us today, then figure out how far that galaxy is now, given the expansion of the universe in the period between when the light was emitted and when it was observed. Astrophysicists have done this, and calculated that the observable universe is about 91 billion light years in diameter, or 45.5 billion light years in radius. This is quite a bit larger than the Hubble radius of 13.4 billion light years. It goes to show how much a huge volume of space can expand over billions of years.

The Observable Universe is Growing

Consider the edge of the observable universe. It represents the farthest possible galaxies, which when they were within our Hubble Volume, could have emitted a light beam that arrived on Earth today. Let's pick one of these galaxies, and call it Galaxy A.

Now think of another galaxy that was 10 light years farther away from us at the time than Galaxy A. Let's call it Galaxy B. Note that when I say "at the time", I mean when Galaxy A emitted that light beam that reached us today.

Obviously, since Galaxy A represents the edge of the observable universe, light from Galaxy B (which was farther than Galaxy A) hasn't reached us. Yet. But in another 10 years it will - it's still on it way. So in 10 years, the observable universe will have grown by that amount, and its new edge will be represented by Galaxy B. In 20 years it will have grown even more, and in a billion years that edge will be much farther than it is today. The observable universe is constantly growing.

How far can it grow? Will it keep growing forever? The answer is no. Some objects will forever remain outside our observable universe, even if we wait forever. The reason is because there are objects much farther than Galaxy A or B in the examples above, that were already outside our Hubble Volume when they first started emitting light. Since they were already moving away from us faster than the speed of light, the light from such objects can never reach us.

It's possible to calculate the limits of growth of the observable universe. It comes to about 124 billion light years in diameter, or about 62 billion light years in radius. If we were to wait billions of years for the light from the most distant observable objects to reach us, such objects would be 62 billion light years away from us then. Note, of course, that we would also be seeing those objects in the past, as they appeared when the light left them, just before they passed outside the Hubble Volume at the time.

The Observable Universe is Theoretical

It's important to note that "observable universe" is a theoretical term, it doesn't actually mean we can see anything that far. What we can see depends on the sensitivity of our instruments, which are not very good. Perhaps in some scifi future, there will be planet sized space telescopes, and then we'll be able to see much farther.

Another thing to note is that farther objects get redshifted, so at extreme distances, objects may well redshift into invisibility. What this means is that:

So the limits of theoretical observability and the practical limits of the observable given current technologies are quite different.

Beyond the Observable Universe

As mentioned earlier, there are regions of space beyond our observable universe. These are the regions that were already outside our Hubble Volume when their stars appeared and started to emit light. Light from these regions will therefore never reach us.

The vast bulk of the universe is probably beyond the observable universe. Nobody knows exactly how large it is. Several astrophysicists have made calculations, but these are based on assumptions that have not been fully proven, so they are not exact. Alan Guth calculated (by extrapolating from the early moments of the big bang) that the universe must be at least 1023 times larger than the observable universe. Other people have made their own calculations and reached different results. Nobody really knows.

Putting it all Together

So what does it mean when you read in the news that astronomers have found the farthest object ever seen? The story says:

Astronomers have pushed the NASA/ESA Hubble Space Telescope to its limits by finding what is plausibly the most distant and ancient object in the Universe ever seen. Its light has travelled for 13.2 billion years to reach Hubble, which corresponds to a redshift around 10. The age of the Universe is 13.7 billion years.

The dim object, called UDFj-39546284, is likely to be a compact galaxy of blue stars that existed 480 million years after the Big Bang, only four percent of the Universe’s current age. It is tiny. Over one hundred such mini-galaxies would be needed to make up our own galaxy, the Milky Way.

The description carries several snippets of information:

What does all this mean? Exactly how far is this object? How far was it when the light that we are seeing now left it? Unfortunately, these paragraphs contain no answers to these questions. This galaxy is not 13.2 billion years from us now, if it still exists, it is probably near the edge of the observable universe, probably 30-40 billion light years away from us. Nor was this galaxy 13.2 billion light years away from us when the light left it. It was probably much closer. How close, we can't say without doing the calculations, and getting more observational information. But it could have been only a couple of billion years away from our location in space at the time.

How does this happen? To understand this, consider the diagram below in the light of what we've covered about the expansion of the universe.

The figure shows two objects: a light emitter (call it galaxy A) and an observer (a human on Earth) at different points in time. The numbers are for illustration purposes only, and the diagram is not to scale.

The top figure shows a very early stage in the universe. Suppose, for example, that the universe is only half a billion years old. Obviously the Earth or Solar System did not exist at the time, but we're pretending it did. It doesn't really matter, the point is that there was some spot in space that corresponds to the position of the Earth now, and that's the spot we're interested in.

So at this stage, there was a separation of 2 billion years between galaxy A and Earth, which is well within the Hubble length. Galaxy A was moving away from Earth, but not at relativistic speeds. We will consider what happened to a beam of light that left the galaxy at this time, in an earthwards direction. This is represented by the red squiggle.

The middle diagram shows some intermediate period, let's say 6 billion years after the light left galaxy A. You may wonder why the light is still in transit, since 6 billion years have passed, and galaxy A was only 2 billion light years from Earth to begin with, when the light started its journey. The answer is that it hasn't arrived at Earth because the space between the Earth and galaxy A has constantly been expanding during this period.

There is an interesting and very important thing to note here, which is why I made this middle diagram in the first place and didn't go straight to the 3rd. Note that at this stage (and in fact throughout the journey), the galaxy is farther from Earth than the front of the light wave (the end of the red squiggle nearer Earth). Light is traveling towards the Earth, and has therefore left galaxy A behind. This means that the front of the light wave (from where the light propagates as the wave moves forward) is closer to Earth than the distance between galaxy A and Earth. This means that there is more space between galaxy A and Earth, compared to the space between the light wave front and Earth. More space equals more space to expand, as we have seen earlier.

So what does this imply? It means that the light travel time and the actual distance between the galaxy and Earth can be completely different. To understand why this is so, look at that middle diagram again. There is a lot of space between the galaxy and Earth; all this space is expanding, therefore the galaxy must be moving away from Earth very fast. But the light beam has already covered much of this distance, and the front end of the light beam is quite close to Earth. There is much less space between it and Earth, and so while this space is also expanding, it's at a much reduced rate as compared to the space between the galaxy and Earth. There is less space to expand here, so it expands at a slower rate.

The 3rd figure represents the situation 13 billion years after the light beam left the galaxy. Now the beam has finally arrived on Earth, and people on Earth can see galaxy A. So what has in effect happened is that:

Light left the galaxy 13 billion years ago, when the galaxy was only 2 billion light years from Earth.

Light took 13 billion years to arrive at Earth.

Meanwhile, the galaxy itself has receded to a distance of 25 billion light years from Earth.

People on Earth are seeing the galaxy as it appeared 13 billion years ago, just 0.7 billion years after the big bang.

The reason for these disparate numbers is that space expanded during the 13 billion years that the light beam took to travel. The distance between the galaxy and Earth now, after the beam has arrived, is the comoving distance, that is distance measured along the Hubble flow, taking the expansion of the universe into account. In fact, the galaxy and Earth are now 25 billion years apart, and this is the comoving distance between them at this time.

The initial distance between the Earth and this galaxy was 2 billion years. That's how far they were when the light beam left the galaxy and headed towards Earth.

Meanwhile, the light travel distance is 13 billion light years, because that's how long the light actually took to travel the distance. Why is it neither 2 billion years or 25 billion years? Because the light is constantly traveling towards the Earth, therefore the amount of space between the front of the light beam and the Earth is constantly shrinking, therefore the rate of expansion of this space is constantly decreasing, therefore the "extra" amount that the light beam has to travel to compensate for the expansion of space is constantly decreasing.

So in the news story linked above, when the story says "farthest object discovered at 13.2 billion light years", what they really mean is that the light travel distance is 13.2 billion light years. This is not the original distance between the galaxy and Earth when the light beam left - that would probably have been only a couple billion light years. This is not the distance between the Earth and where the galaxy is now - that would probably be 35 billion light years or more. It's simply the distance that the light path had to travel, in order to reach from the point of origin to Earth, given constantly expanding space and constantly decreasing distance.

One effect of this is the "stretching out" of the light beam, which you can see in the picture. As space expands, the beam is stretched out, which is the reason for the redshift.

Another effect is something called the angular diameter distance. To understand this, think of some object in front of you, perhaps a computer screen. This object has a certain angular diameter, meaning if you were to draw light beams from two opposite edges of the monitor to your eye, there would be a certain angle between them, which represents the angular diameter. If the monitor were bigger, then the angular diameter would be bigger. If you keep the same size of monitor but move it 10 feet farther away from you, then its angular diameter becomes smaller, it looks smaller to you.

The same thing happens with astronomical objects. Their angular diameter increases with increasing size, decreases with increasing distance. So let's bring this to our example of galaxy A that we've been describing. What kind of angular diameter would we expect to see? Let's say we know the actual size of the galaxy. In terms of angular diameter, would it look like it was 2 billion light years away (when the light left the galaxy), 13 billion light years away (how long it took for the light to reach us), or 35 billion light years away (how far it actually is, now)?

Obviously, the only important event of these 3 is when the light actually left the galaxy, and so that's the angle that counts. This makes for surprising images when you look at these galaxies in telescopes. The light has traveled for 15 billion light years, it looks like it's traveled 15 billion light years, being red shifted and dim. And yet, the galaxies look big, as if they were only 2 billion light years away. Big and faint.

Angular diameter is therefore a very useful measure to determine how close the galaxy was when it first emitted the light that we see today.