# Cosmic Scale Factor

By , Tuesday 19th May 2009 in Cosmology

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 the 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.

To compare 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 to the universe.

Cosmic Scale Factor

Consider 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 in 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 clearly see that they are separated by a larger distance. We can also see that the length intervals have also expanded with time.

There are two possible definitions of a 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 coordinate 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 are able to 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 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 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

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 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.