Hubble's Law and Redshift Revision Notes | A Level AQA Physics

1. The Universe Is Running Away From Us

Picture a galaxy millions of light-years away. You point a telescope at it and analyse the light. Every spectral line you expect to see is there — but each one has shifted towards the red end of the spectrum. The galaxy is not just sitting still; it is moving away from you. Now look at another galaxy, twice as far away. Its lines are shifted twice as much. This pattern, repeated across thousands of galaxies, is the foundation of modern cosmology.

This is the story of Hubble's Law: a deceptively simple equation that tells us the universe itself is expanding. Understanding it requires two interlocking ideas — redshift and the proportional relationship between velocity and distance — and both are examinable at A-level with real calculation demands.

2. Redshift — The Doppler Effect on a Cosmic Scale

You already know the Doppler effect from sound: an ambulance siren rises in pitch as it approaches and drops as it moves away. Light behaves the same way. When a source of light moves away from you, the waves are stretched — each successive wavefront is emitted from a position slightly further away, so the wavelength you receive is longer than the wavelength emitted.

Longer wavelength in the visible spectrum means a shift towards red — hence redshift. Astronomers detect this by comparing the observed wavelengths of known spectral lines (for example, the hydrogen Balmer series) against their laboratory values. The fractional shift in wavelength is directly related to the recessional velocity of the source.

The key observational fact is this: virtually every distant galaxy shows redshift, not blueshift. A handful of very nearby galaxies (such as Andromeda) show blueshift because local gravitational attraction dominates, but on the large scale the universe is overwhelmingly redshifted. This is not a coincidence — it is evidence that space itself is expanding, carrying galaxies apart like dots on the surface of an inflating balloon.

With redshift established as the observational tool, the next question is whether there is a pattern in how much different galaxies are redshifted. This is where Edwin Hubble's 1929 analysis of galaxy data produced one of the most important results in the history of science.

3. Hubble's Law — $v = H_0 d$

Hubble plotted the recessional velocities of galaxies (derived from their redshifts) against their distances (estimated from standard candles such as Cepheid variable stars). The result was a straight line through the origin: velocity is directly proportional to distance. Double the distance, double the recession speed. This relationship is Hubble's Law.

Notice that $H_0$ has units of s$^{-1}$ when $v$ is in m s$^{-1}$ and $d$ is in metres. You will also see $H_0$ quoted in km s$^{-1}$ Mpc$^{-1}$ in astronomical contexts, but AQA calculations use SI units, so always convert distances to metres before substituting.

The gradient of a velocity–distance graph for galaxies gives $H_0$ directly. A steeper gradient means a larger Hubble constant and a faster-expanding universe. This is also why $H_0$ is linked to the age of the universe: the reciprocal $1/H_0$ gives an estimate of the time since the Big Bang, assuming a constant rate of expansion.

4. Calculating with Hubble's Law

Exam questions on Hubble's Law almost always involve a calculation — either finding recessional velocity from a given distance, finding distance from a given velocity, or extracting $H_0$ from a graph. The equation $v = H_0 d$ is straightforward to rearrange, but unit handling is where marks are lost.

Worked example: Calculating Recessional Velocity Using Hubble's Law

A galaxy is at a distance of $9.5 \times 10^{25}$ m from Earth. Using $H_0 = 2.2 \times 10^{-18}$ s$^{-1}$, calculate the recessional velocity of the galaxy. [3 marks]

1. $v = H_0 d$ — [M1] Correct formula stated. This is the only formula needed for this type of question.
2. $H_0 = 2.2 \times 10^{-18}$ s$^{-1}$, $\quad d = 9.5 \times 10^{25}$ m — Both values are already in SI units, so no conversion is needed before substituting.
3. $v = 2.2 \times 10^{-18} \times 9.5 \times 10^{25}$ — [M1] Correct substitution with units carried through.
4. $2.2 \times 9.5 = 20.9$ — Multiply the coefficients first: $2.2 \times 9.5 = 20.9$.
5. $10^{-18} \times 10^{25} = 10^{(-18+25)} = 10^{7}$ — Add the exponents when multiplying powers of ten.
6. $v = 20.9 \times 10^{7} = 2.09 \times 10^{8}$ m s$^{-1}$ — [A1] Final answer in standard form with correct units. This is approximately 70% of the speed of light — physically plausible for a very distant galaxy.

Answer: $v \approx 2.1 \times 10^{8}$ m s$^{-1}$

Now consider the reverse: you are given a recessional velocity and asked to find the distance. Rearranging $v = H_0 d$ gives $d = v / H_0$. The arithmetic is the same; only the algebra changes.

Worked example: Finding Distance from Recessional Velocity

A galaxy has a recessional velocity of $1.4 \times 10^{7}$ m s$^{-1}$. Using $H_0 = 2.2 \times 10^{-18}$ s$^{-1}$, calculate the distance to the galaxy. [3 marks]

1. $d = \frac{v}{H_0}$ — [M1] Correct rearrangement. Divide both sides of $v = H_0 d$ by $H_0$.
2. $d = \frac{1.4 \times 10^{7}}{2.2 \times 10^{-18}}$ — [M1] Values substituted with units. Both are in SI units already.
3. $\frac{1.4}{2.2} = 0.6\overline{36} \approx 0.636$ — Carry enough decimal places to avoid rounding errors in the final answer.
4. $\frac{10^{7}}{10^{-18}} = 10^{7-(-18)} = 10^{25}$ — Subtracting a negative exponent is the same as adding: $7 - (-18) = 25$.
5. $d = 0.636 \times 10^{25} = 6.36 \times 10^{24}$ m — [A1] Correct answer in standard form with units of metres.

Answer: $d \approx 6.4 \times 10^{24}$ m

5. Reading a Hubble Graph and Determining $H_0$

A common exam question presents a graph of recessional velocity (y-axis, in m s$^{-1}$) against distance (x-axis, in metres) for a set of galaxies. Because $v = H_0 d$ is a straight line through the origin, the gradient of that line equals $H_0$.

To extract $H_0$ from such a graph, draw the best-fit straight line through the origin, then choose two widely separated points on the line (not data points) and calculate $\text{gradient} = \Delta v / \Delta d$. The result has units of (m s$^{-1}$) / m = s$^{-1}$, which is the correct SI unit for $H_0$.

• The line of best fit must pass through the origin — Hubble's Law predicts zero velocity at zero distance.
• Use the full range of the line when calculating the gradient, not just two adjacent data points.
• Quote $H_0$ in s$^{-1}$ for AQA calculations; convert to km s$^{-1}$ Mpc$^{-1}$ only if the question explicitly asks.
• A scatter of points around the line is expected — peculiar velocities (local gravitational effects) cause deviations.

6. Redshift as Evidence for an Expanding Universe

Hubble's Law is not merely a mathematical curiosity — it carries profound physical meaning. If every galaxy is receding from us, and the further away it is the faster it recedes, then the universe is expanding uniformly in all directions. Running this expansion backwards in time leads to the conclusion that all matter was once concentrated at a single point: the Big Bang.

Redshift is the observational evidence that underpins this conclusion. The systematic redshift of distant galaxies, combined with the linear velocity–distance relationship, provides strong support for the Big Bang model. No static or contracting universe could produce this pattern.

An important subtlety: the redshift of distant galaxies is not purely a Doppler effect in the classical sense. At cosmological scales, it is the expansion of space itself that stretches the wavelength of light as it travels. The Doppler formula $z \approx v/c$ is a valid approximation for recession speeds much less than $c$, and this is the level of treatment required at A-level.

7. Exam Technique and Common Mistakes

Hubble's Law questions are reliable mark-scorers if you handle units carefully and structure your working clearly. The following guidance consolidates the most frequent sources of lost marks in AQA mark schemes.

For extended-response questions asking you to evaluate the evidence for the Big Bang, structure your answer as: observation (redshift detected) → quantitative pattern (Hubble's Law, $v \propto d$) → physical interpretation (uniform expansion) → extrapolation (convergence to a single point in the past). Each link in that chain corresponds to a marking point.

Hubble's Law, $v = H_0 d$, is the quantitative expression of the expanding universe: recessional velocity is directly proportional to distance, with the Hubble constant as the constant of proportionality — and the redshift of galaxy spectra is the observational evidence that makes this law testable.