#### Cosmic scale factor

The cosmic scale factor is defined as the size of the universe at a given time relative to its size now.

\(z={R\over R_0} -1\)

- \(z\) is the factor by which the universe has expanded in size (dimensionless)
- \(R\) is the size of the universe at time \(t\)
- \(R_0\) is the size of the universe now

#### Hubble's law

Hubble's law is based on analysis of observational data from thousands of galaxies. It states that the recessional velocity of galaxies is approximately proportional to their distance from Earth.

\(v=H_0d\)

- \(v\) is the recessional velocity (kms
^{-1}) - \(H_0\) is the Hubble constant (=68 kms
^{-1}Mpc^{-1}) - \(d\) is the distance from Earth (Mpc)

We can use Hubble's law to calculate the age of the universe, by calculating the time over which the universe would have expanded to reach its current size.

NB: time = distance / speed

\(T\approx {d\over v}\)

\(T\approx {1\over H_0}\)

- \(T\) is the age of the universe
- \(H_0\) is the Hubble constant (=68 kms
^{-1}Mpc^{-1})

Converting all of the necessary units gives an estimate of the age of the universe as 14.4 billion years. This is based on the assumption that the universe has expanded at a constant rate.

However, observations of *type 1a supernovae* suggest that the unvierse is, in fact, accelerating in its expansion.

A type 1a supernova is an example of a *standard candle* with known luminosity. Using redshift to determine the distance of a type 1a supernova from Earth, we can calculate the expected apparent brightness.

The apparent brightness of distant type 1a supernovae is lower than calculated, which suggests that their distance from us is larger than expected. The universe has expanded at an increasing rate in the time since the light left the supernova!

The accelerating expansion of the universe is hypothesized as being a result of *dark energy*.