We can calculate an estimate for the age of the universe based on the velocity at which galaxies are receding from our observations.

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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

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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.

In the early stages of the universe, photons had enough energy to ionise newly formed atoms.

At what temperature did photons in the early universe stop ionising atoms?

The cosmic microwave background is now a temperature of 2.73 K.

The graph shows the spectrum of cosmic microwave background radiation.

The temperature is:

3000 K was the original temperature before the expansion of CMB radiation. The peak wavelength can be used to determine the temperature today:

\(T = {0.0029\over λ_\text{peak}}\)

Using 1 mm, we obtain an answer of 2.9 K so must choose the closest.

The image shows two galaxies photographed with the same telescope from the Earth.

If the distance to A is 9 Mpc, the distance to B is:

For small angles (i.e. those subtended by galaxies at great distance from Earth!), the angular width of a body is proportional to its width. B has an angular width approximately one third of A.

The distance away is inversely proportional to angular width, so B is three times further from Earth than A.

The image shows three spectra. The top one is from a stationary source. A and B are the spectra from two galaxies. Wavelength increases from left to right.

The ratio \(\text{speed}_A \over \text{speed}_B\) is approximately:

Speed is proportional to \(Δλ\). The shift in wavelength for B is approximately twice that of A. Therefore, B is travelling at twice the speed of A.

This Algodoo simulation represents particles moving after an explosion.

If the speed of the pink ball is 20 ms^{-1}, the speed of the yellow ball is approximately:

The pink ball has traveled 5x further so must have 5x the speed.

This Algodoo simulation represents particles moving after an explosion.

The graph shows the velocity of a selection of balls plotted against their distance from the centre.