2. The expanding universe

We can see the past of the universe because speed of light is finite. Let us try to understand this.

In the olden times, people used to mail handwritten letters through post offices. Consider that Atik writes and posts a letter to Bithi and it takes one month for the letter to reach Bithi. When Bithi opens the letter, she only learns how Atik was a month ago. Bithi is literally learning about the past of Atik by looking at something sent by Atik.

Now replace Atik with the star Alpha Centauri (A) and Bithi with a scientist (B) on earth. A produces light by juggling its electric charges and radiates part of the light in all directions. A is 4 light-years away from B and, hence, it takes 4 years for a light ray to reach B from A. When B analyzes the light using a spectrometer, she actually sees A as it was 4 years ago.

We can say, the farther we look in space the further we see in time.

Speed of light is $3\times 10^8$ m/s. That means if something is at a distance of 1 m, we see it as it was 3 ns (nano-second) ago:

If light travels $3\times 10^8$ m in 1 s, then it travels 1 m in 3 ns. So looking at something 1 m away amounts to seeing something 3 ns ago. Similarly,

$$ 3 \text{ m away } \rightarrow 10 \text{ ns ago}. $$

Now consider the star Betelgeuse (يد الجوزاء) in the Orion constellation which is 548 ly (light-years) away. How far in the past we are looking at when we see this star? Compare this with a historical event. 548 years ago in the year 1474, some parts of Spain was still in Muslim control and queen Isabella and king Ferdinand were still fighting a civil war against their relatives.

How about the Andromeda Galaxy seen here on the upper-left of the Very Large Telescope (VLT) in Chile? Andromeda is about 2.5 Mly (mega or million ly) away from us. When we see Andromeda we are looking at an object as it was 2.5 million years ago, when homo habilis was just beginning to use tools.

How far back in time you can see depends on how far you look.

The diagram above shows the timeline of the universe from the big bang (far left) to the present (far right). From the present times, if we look back toward the left at an object that is 12 billion ly away, we will be observing the universe as it was 12 billion years ago, that is when it was only around 2 billion years old.

By analyzing the light coming from a galaxy we can measure its distance and speed using an instrument called a spectrometer. For now we will not discuss the details of how this is exactly done.

Edwin Hubble measured the distances and speeds of a lot of far-away galaxies in the 1920s and came to the surprising conclusion that

the farther a galaxy is from us the faster it is receding away from us, that is the receding speed of a galaxy is directly proportional to its distance from us, or

$$ \text{speed } \propto \text{ distance}. $$

This could only mean one thing. The universe was created by an explosion and everything is expanding in all directions ever since the explosion. Before uncovering the exact nature of this explosion, let us compare it poetically with an ordinary explosion.

After the explosion of an ordinary bomb, everything stored within the bomb are scattered in all directions as fragments or shrapnel. If we take a snapshot of the scene 30 seconds after the explosion, we will see that some fragments are closer to the position of the original bomb and some fragments are farther away. All fragments have taken the same 30 seconds to travel their individual distances. So the fragment that is the farthest from the initial position of the bomb must have had the highest speed. And the fragment that is closest to the initial position must have had the lowest speed.

You see that the fragment that is farther from the initial position of the bomb is receding faster from that position. This is exactly the case with the universe. Replace the shrapnel with the galaxies. Each galaxy is like a fragment of a bomb. The galaxy that is farther from the initial point of the explosion (called the big bang) has been able to come this far only because it had a higher speed compared to the galaxies that are closer.

But we have previously said that all galaxies seem to be receding from us. Does it mean we are at the initial point of the explosion from which everything is fleeing? No. We are just like another fragment of the explosion. Each fragment sees all other fragments receding from it. This will be clear in a bit. But first let us use the analogy of explosion to measure the age of the universe.

Speed is nothing but distance traveled per unit time. A bus is going at 60 km/h means the bus is traveling 60 km in 1 hour. We write

$$ \text{speed } = \frac{\text{distance}}{\text{time}}. $$

If you know that a bus was running at a constant speed of 60 km/h and it is now 360 km away from its starting point, then you can immediately calculate how long ago the bus began its journey, that is how long ago it was at its starting point. The bus started $360/60=6$ hours ago. We can write

$$ \text{time } = \frac{\text{distance}}{\text{speed}}. $$

Keeping this definition of speed in mind, let us go back to our analogy of an explosion. Each fragment in the explosion traveled a different distance within a given time (30 seconds in our example above) because each had a different speed. But if we divide each fragments distance by its speed, we get the same time for all the fragments (do not forget that the snapshot is taken after this same time).

The same is true for the universe. If we divide the distance of any far-away galaxy by its receding speed, we get the time it took for this galaxy to arrive at its current location (the analogy is wrong but useful as will be clear later below) from the initial point of the big bang, that is we get the age of the universe.

$$ \text{age of the universe } = \frac{\text{distance of a galaxy}}{\text{receding speed of the galaxy}}. $$

This is how we have been able to calculate the age of the universe to be 13.8 billion years (Gy).

In practice, it is not enough to calculate the speed and distance of just one galaxy because of measurement errors. Scientists measure the speeds and distances of a lot of galaxies and calculate the average ratios. Note that the distance-speed ratio should be the same for all galaxies, in principle.

So far we have used the analogy of explosion to describe the expansion of the universe without any second thought. But the explosion we call big bang is nothing like an ordinary explosion. It is not an explosion of matter, but an inflation and expansion of space itself. We came to understand all these after Einstein revealed his special and general theories of relativity which we will briefly summarize here before having a go at visualizing an expanding universe.

German scientist Albert Einstein published three papers in 1905 each more revolutionary than the other. One of the papers began formalizing the special theory of relativity which basically entails two things: (1) space and time are equivalent, (2) matter and energy are equivalent:

$$ \text{space } \equiv \text{ time}, $$

$$ \text{energy } = \text{mass } \times (\text{speed of light})^2. $$

Before Einstein, we believed space, time, matter and energy to be different things. Special theory of relativity related time with space and matter with energy. After 1905, Einstein worked ten more years to relate space-time with matter-energy and come up with a single theory that explains it all. This new theory, published in 1915, is called the general theory of relativity.

The general theory basically says: “matter-energy determines how space-time is curved and space-time determines how matter-energy moves.”

The period between 1905 and 1920 in the life of Einstein is nicely captured in the television film Einstein and Eddington directed by Philip Martin. Eddington was an astronomer who proved Einstein’s general theory of relativity by observing the bending of starlight by the sun during a solar eclipse in 1919. The title of this chapter in our lecture note is taken from the title of a famous book written by Eddington, The Expanding Universe. Now let us try to understand this expansion in light of theories of relativity.

This diagram shows how matter (a form of energy) affects space and time. The massive object at the center (white ball) is curving space and slowing down clocks. Space gets more and more curved and time becomes slower and slower as we go closer to the object. Curvature of time is equivalent to slowing down or speeding up of clocks here.

In this new theory, gravity is related to curvature of space:

$$ \text{gravity } \equiv \text{ curvature}. $$

If you are outside an object,

$$ \text{curvature } \equiv \text{ mass} $$

and if you are inside an object

$$ \text{curvature } \equiv \text{ density}. $$

We are located inside the universe. So the curvature of space (and equivalently of time) in the universe as a whole on average must be related to the average density of the universe. It turns out that the

1. Universe is globally flat, with a lot of local curvatures (blac kholes).
2. Universe is globally dark, with a lot of local sources of light (stars).
3. Universe is globally cold ($-270^\circ$ C), with a lot of local sources of heat (Sun, $6000^\circ$ C).

But Einstein created a thought-experiment of a globally curved universe that is expanding into time like a balloon. We can explain the explosive expansion of the universe more truly by using this untrue model of the universe, as shown below.

Universe has three dimensions of space and one dimension of time, it is four-dimensional or 4D. But we cannot visualize four space-like dimensions because we live in only 3 dimensions. 3D creatures cannot visualize 4D space. However, the 4 dimensions of spacetime can be understood qualitatively by using a 3D analogy or allegory.

In this analogy of the 4D universe, the 3D space is imagined to be the 2D surface of a spherical balloon and the 1D time is imagined to be the radius (distance from the center to the surface) of the balloon. In total we have 3 dimensions, 2 for space and 1 for time, as shown below.

Assume that the radius of this sphere is the current age of the universe, 14 Gy, and the 2D surface represents the 3D space in the universe at present time. Then several properties of space and time become clear which are actually true.

Surface of a sphere is finite but does not have any edge, beginning or end. If you start from any point on the surface, you will come back to that point. The universe could be like this, but in 3D. The space in the universe does not have an edge or center. All points are alike. The universe is homogeneous and isotropic.

On the other hand, the radius of a sphere do have a beginning at the center and also an edge at the surface. Real time in the universe also had a beginning 14 Gy ago, and an edge at present.

Of course, there is a big difference between the space on the surface of this balloon and real space. Space in the real universe is not curved on average. The universe is globally flat with a lot of local curvatures only. Let us forget this dissimilarity for now.

Let us try to understand the expanding universe using this analogy. Expansion of the universe is equivalent to the inflation of this balloon. As the balloon is inflated, the radius increases. Note that the space (surface) does not expand into other space outside the balloon. There is no space outside the balloon, all space is imagined to be on the surface. In this analogy, the balloon expands into time.

In this picture, the gridlines on the surface are used to measure 2D space and each red dot represent a galaxy. Assume that the radius of the smaller balloon (a) is 10 Gy. Now inflate the balloon using a pump and assume that the radius of the bigger balloon (b) is 14 Gy. The space between the dots (galaxies) has expanded from (a) to (b) and, hence, the distance between all the galaxies on the surface has increased. None of the galaxies are at the center, space does not have a center, but an observer in each galaxy (dot) sees all other galaxies going away from it.

The balloon is only a metaphor of the the expanding universe, and the metaphor works well, especially because our observable universe is very similar to this as shown below in 2D.

From our position in the universe whenever we look into outer space we see the past. Our line of sight is like the radius of a spherical balloon and the surface of this sphere is the edge or limit of our observable universe, beyond which we cannot see. We can see equal distance in all directions, so the distance to the edge of the observable universe is indeed equal in all directions just like the radius of a sphere.

We are located inside the Virgo supercluster of galaxies, and from that point the horizontal diameter of the obsevable universe is shown above. The diameter has gridlines in light years and parsecs. Let us forget about the parsec unit for now. The separation between any two gridlines above the diameter corresponds to 1 billion or giga light year (Gly). There are 93 such units, and, indeed the diameter of our observable universe is around 93 Gly.