Expansion of the Universe, Cosmic Scale Factor and Hubble's Law

Evidence suggests that the expansion of the universe is accelerating from observations Edwin Hubble made in 1925.

The accelerating expansion of the universe is the observation that the universe appears to be expanding at an increasing rate so that the velocity at which a distant galaxy is receding from the observer is continuously increasing with time.

American astronomer Edwin Hubble made the observations of the expansion of the universe in 1925 and was the first to prove that the universe is expanding. He proved that there is a direct relationship between the speeds of distant galaxies and their distances from Earth. This is now known as Hubble's Law.

Hubble was the first to prove that the universe is expanding. He did this by measuring the redshift of several distant galaxies together with their relative distances by measuring the apparent brightness of Cepheid variables in each galaxy. When he plotted the redshift against relative distance, he found that the redshift of distant galaxies increased as a linear function of their distance. This is now called Hubble's law. The only explanation for this observation is that the universe was expanding.

The linear relationship between recession velocity, v, and the distance, x, of a local galaxy, can be expressed as:

Equation 47 - Hubble's Law

Where H₀ is the Hubble constant.

A plot of the Hubble law shows that all distant galaxies seem to be moving away from us, which is not what we might initially expect since the gravitational attraction between the various bodies in the universe would act to pull galaxies together. Applying the cosmological principle, which is one of the guiding principles in modern cosmology, we may conclude that all distant galaxies are moving away from each other. This indicates that the Hubble law needs to have a cosmological origin - that is, one producing global behaviour.

Do all observers in the universe, independent of their position, measure the same phenomenon (the Hubble Law) or is our position special?

Although we observe detail in the local universe, on very large scales the universe seems to have a uniform appearance and we conclude that our location in the universe has no special significance. The universe changes its appearance with time but we believe that at any given time it looks the same - on average - everywhere.

Given the cosmological principle, we conclude that the universe is expanding, or more precisely, that space and spacetime are expanding.

The distances between objects are changing with time due to the expansion of the universe. When we observe objects at large distances, we are effectively looking back in time because of the light travel time. To compare distances and sizes from different epochs (times) is not straightforward since we must remove the effects of the expansion.

Cosmic Scale Factor

The cosmic Scale Factor is a function of time which represents the relative expansion of the universe. It relates the comoving distances for an expanding universe with the distances at a reference time arbitrarily taken to be present.

The distance between any two objects is changing over time due to the expansion of the universe. When we look at a distant object, we are effectively looking back through time, since the speed of light is finite and takes time to reach us. For example, when we look at the Sun, we see it as it was 8 minutes ago. When we look at the Andromeda galaxy, the light has taken about 2.2 million years to get to our galaxy, so we see the Andromeda galaxy as it was 2.2 million years ago. This relationship between time and distance is called Cosmic Scale Factor.

Comparing distances and sizes from different times is not easy since we must remove the effects of expansion. In this article, we are going to assume a flat geometry of the universe.

Consider the left-hand image in the graphic above. This represents an arbitrary point in time in the past (t₁). We can see a galaxy at point (0,0) and another one at point (3,2). These points are comoving coordinates. Simple trigonometry tells us that the distance between the two comoving coordinates is 3.6 units (a² = b² + c²).

Now consider the image to the right. This represents another point in time, let's say it's the present (t₀). The galaxies are still at the same coordinates, and the distance is still 3.6 units, but we can see that they are separated by a larger distance. We can also see that the longer intervals have also expanded with time.

There are two possible definitions of comoving coordinates, and both are used in cosmology. Unfortunately, the same symbol r is often used for both. Comoving distance coordinates are used as follows.

1. The comoving radial distance coordinates for calculating proper distances between objects at two different epochs (i.e. large time separation).
2. The comoving angular diameter distance coordinate for calculating proper distances between two objects at the same epoch.

The coordinates are exactly that - coordinates. They are not distances, but proper distance may be calculated from them. Think of comoving coordinates as labels attached to the galaxies for all time. Different galaxies have different comoving coordinates, but a particular galaxy keeps the same comoving coordinates forever. Using comoving coordinates, we can describe the position of any object independently of expansion.

We can define the proper distance x(t), corresponding to different times, in terms of the comoving radial distance coordinate r using the equation:

Equation 35 - Proper Distance

The notation R(t) indicates that the cosmic scale factor is a function of time and its value changes with time (epoch).

In an expanding universe, the scale factor R(t) corresponding to a past epoch is smaller than 1, and greater than 1 for future epochs. A scale factor of 1 represents the current epoch (now)

• R(t) <1 in past
• R(t) = 1 now
• R(t) > 1 in future

The behaviour of the cosmic scale factor R(t) with time tells how the universe itself evolves with time. Knowing this we can construct a relationship between redshift and the scale factor for another epoch.

Equation 36 - Redshift and Scale Factor

This shows that the redshift can be used to specify the size of the universe relative to the size today. Astronomers and cosmologists talk about redshifts of objects rather than distances or emission epochs.

Worked example of Cosmic Scale Factor

A distant galaxy has been analysed and it was found that it has a redshift of z = 2. What was the size of the universe at the time when the light left the galaxy relative to the size of the universe now?

Equation 37 - Scale factor worked

Thus the linear size of the universe was one-third of what it is now.

The Hubble Constant

The Hubble Constant (H₀) gives a value for the expansion rate of the universe. This expansion rate is equal to the rate of the change of the scale factor.

The rate of change in scale factor it is given the symbol

The dot above the R is the mathematical notation for "rate of change". Using the comoving radial distance coordinate, r, we can derive the expansion "velocity" v:

which is Hubble's law with

So the Hubble "constant" is actually time-dependent, and its value can be determined by the measurement of redshifts and distances to galaxies.