Doppler effect, redshift and the Hubble constantEdexcel International A Level Physics: Revision notes

The Doppler effect is the change in the observed frequency and wavelength of a wave when there is relative motion between the source and the observer.

• Source moving towards the observer: wavefronts are closer together, so the observed wavelength is shorter and the frequency is higher.
• Source moving away: wavefronts are further apart, so the wavelength is longer and the frequency is lower.

The source moves between emitting one wavefront and the next, which is why the spacing changes. The wave speed does not change. The effect is heard in the siren of a passing ambulance.

Light from a source moving away from us is shifted to longer wavelengths. This is redshift. Light from a source approaching us is blueshifted.

For a source moving at speed $v$ much less than $c$, the redshift $z$ is

$z=\frac{\Delta\lambda}{\lambda}\approx\frac{\Delta f}{f}\approx\frac{v}{c}$

where $\Delta\lambda$ is the change in wavelength and $\lambda$ is the wavelength measured in the laboratory. $z$ has no units.

A hydrogen line has $\lambda=656$ nm in the laboratory and 662 nm in light from a galaxy.

$z=\dfrac{662-656}{656}=9.1\times10^{-3}$

$v=zc=9.1\times10^{-3}\times3.00\times10^{8}=2.7\times10^{6}$ m s⁻¹

The galaxy is moving away from us because the wavelength has increased.

Observations show that the light from distant galaxies is redshifted, and more distant galaxies have greater redshifts. The recession speed is proportional to distance:

$v=H_0d$

where $H_0$ is the Hubble constant. Distances are found using standard candles. In SI units $H_0$ is in s⁻¹; it is often quoted in km s⁻¹ Mpc⁻¹. The gradient of a graph of $v$ against $d$ is $H_0$.

This shows that the universe is expanding, so it must have started from a very small region, the Big Bang.

Worked example. For $v=2.7\times10^6$ m s⁻¹ and $H_0=2.3\times10^{-18}$ s⁻¹, $d=\dfrac{v}{H_0}=1.2\times10^{24}$ m.

If galaxies have always moved at a constant speed from the same point, the time since they were together is $t=\dfrac{d}{v}=\dfrac{1}{H_0}$.

For $H_0=2.3\times10^{-18}$ s⁻¹, $t=4.3\times10^{17}$ s, about $1.4\times10^{10}$ years.

A larger $H_0$ gives a younger universe. The value of $H_0$ is controversial, because it relies on distances found from standard candles, which are uncertain. The age must also be greater than the ages of the oldest stars, so an estimate that is too small is a problem.

Gravity from matter slows the expansion. The fate depends on the density of matter:

• Above a critical density the expansion stops and the universe collapses.
• Below it, the universe expands for ever.

Dark matter is matter that cannot be seen but has a gravitational effect. It adds to the mass, so it could slow the expansion more. Since the amount of dark matter is uncertain, the fate is uncertain. Slowing of the expansion also means that $1/H_0$ is only an estimate of the age.