Doppler effect, redshift and the expanding universeEdexcel A-Level Physics: Revision notes

When a wave source and an observer move relative to each other, the observed frequency differs from the frequency emitted. This is the Doppler effect.

If the source moves towards the observer, it moves forward between emitting successive wavefronts, so the wavefronts are bunched together. The observed wavelength is shorter and the observed frequency is higher. If the source moves away, the wavefronts are stretched, the wavelength is longer and the frequency is lower.

The wave speed is set by the medium and does not change, so $f = v/\lambda$ means a shorter wavelength gives a higher frequency. Only the relative motion matters. A driver travelling with a siren hears no shift, but a roadside listener does.

The effect applies to sound (an ambulance siren changing pitch as it passes) and to electromagnetic radiation (light from a moving star).

Light from a galaxy moving away from us is redshifted: every wavelength is stretched, so spectral lines move towards the red end of the spectrum. Astronomers identify known lines, such as hydrogen lines, and compare their observed wavelengths with laboratory values.

The redshift is defined by

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

where $\Delta\lambda$ is the observed wavelength minus the emitted (laboratory) wavelength, $v$ is the speed of the source along the line of sight and $c$ is the speed of light. The approximation $z \approx v/c$ holds only when $v \ll c$. A source moving towards us is blueshifted, with a negative $z$.

Worked example: a line of wavelength 656.3 nm is observed at 662.9 nm. $z = 6.6/656.3 = 0.0101$, so $v = zc = 0.0101 \times 3.00 \times 10^{8} = 3.0 \times 10^{6}$ m s⁻¹.

Almost every galaxy shows a redshift, so almost every galaxy is moving away from us. Edwin Hubble found that the recession speed is proportional to distance:

$v = H_0 d$

where $H_0$ is the Hubble constant, with a value of about $2.2 \times 10^{-18}$ s⁻¹ (roughly 70 km s⁻¹ per megaparsec). With $d$ in metres and $H_0$ in s⁻¹, $v$ is in m s⁻¹.

The more distant a galaxy, the faster it recedes. This is the pattern expected if space itself is expanding, with no special centre: observers in any galaxy would see the others moving away. It is evidence for the Big Bang, in which the universe began from a very hot, dense state about 14 billion years ago.

Worked example: a galaxy $4.5 \times 10^{24}$ m away recedes at $v = 2.2 \times 10^{-18} \times 4.5 \times 10^{24} = 9.9 \times 10^{6}$ m s⁻¹.

If a galaxy has moved away from us at a constant speed $v$ since the Big Bang, then $d = vt$. Combining this with $v = H_0 d$ gives $t = 1/H_0$.

$t \approx \frac{1}{H_0} = \frac{1}{2.2 \times 10^{-18}} = 4.5 \times 10^{17}\text{ s} \approx 1.4 \times 10^{10}\text{ years}$

This is only an estimate. It assumes the expansion rate has been constant, so no gravitational slowing and no speeding up, and it depends directly on the measured value of $H_0$. A larger $H_0$ gives a younger universe.

The Hubble constant is hard to pin down. Distances to galaxies are difficult to measure, and different methods give values of $H_0$ that disagree. Since the age is about $1/H_0$, the age is uncertain too.

The fate depends on the total mass of the universe. Gravity attracts galaxies and slows the expansion. If there is enough mass, the expansion could halt and reverse, ending in a collapse. If not, the universe expands for ever.

Dark matter is matter that gives out no detectable radiation but whose gravity is detected, for example in the way galaxies rotate. It means the visible mass is only part of the total, so the total mass, and hence the fate, is uncertain.

Until $H_0$ and the amount of dark matter are measured more precisely, the age and fate of the universe remain open to debate.