Physics • Year 12 • Module 8 • Lesson 3
Hubble's Law and the Expanding Universe
Lock in the core vocabulary, key equations, and the meaning of redshift and scale factor before tackling harder application questions.
1. Term–definition match
The definitions below are shuffled. In the right-hand column write the matching term from this list: Hubble constant, recession velocity, redshift, blueshift, megaparsec, scale factor, lookback time, Hubble time, cosmological redshift, Hubble tension. 10 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The rate of expansion of the universe, approximately 70 km/s per megaparsec; its inverse gives an estimate of the age of the universe. | |
| 1.2 | The speed at which a galaxy moves away from an observer, as measured from the Doppler shift of its spectral lines. | |
| 1.3 | An increase in the observed wavelength of light from a source that is moving away from the observer. | |
| 1.4 | A decrease in the observed wavelength of light from a source that is approaching the observer. | |
| 1.5 | A unit of distance equal to approximately 3.086 × 1019 km or 3.26 million light-years, used for cosmic distances. | |
| 1.6 | A dimensionless number describing the relative size of the universe at a given time; by convention it equals 1 today. | |
| 1.7 | The time light from a distant object has been travelling before reaching us; the time we are “looking back” into the past. | |
| 1.8 | An estimate of the age of the universe equal to 1/H0, approximately 14 billion years. | |
| 1.9 | The redshift of a galaxy caused by the expansion of space itself stretching the wavelength of light in transit, rather than by the galaxy's peculiar motion. | |
| 1.10 | The discrepancy between the value of H0 measured from the early universe (CMB) and the value measured from nearby supernovae and Cepheid variable stars. |
2. True or false — with correction
Circle T or F for each statement. If the statement is false, write the corrected version on the line below it. 12 marks (1 T/F + 1 correction each)
2.1 Hubble's law states that galaxies that are farther away are moving faster, with recession velocity proportional to distance. T / F
2.2 A redshifted galaxy is moving toward the observer, so its observed wavelength is shorter than the rest wavelength. T / F
2.3 The formula for redshift is z = (λobs − λrest) / λrest, and z is always positive for receding galaxies. T / F
2.4 If the scale factor at emission was a = 0.5, the universe was twice its current size when that light was emitted. T / F
2.5 Using Hubble's law with H0 = 70 km/s/Mpc, a galaxy with recession velocity 7000 km/s is at a distance of 100 Mpc. T / F
2.6 Because all galaxies are receding from us, the Milky Way must be located at the centre of the universe. T / F
3. Fill-in-the-blank paragraph
Use the word bank to complete the passage. Each word is used once. 8 marks (1 per blank)
Word bank:
distance · expanding · Hubble · inversely · megaparsecs · proportional · recession · scale factor
Edwin Hubble's 1929 discovery showed that the ___________ velocity of a galaxy is directly ___________ to its ___________. This means the universe is ___________ uniformly in all directions, like dots on an inflating balloon. The Hubble constant H0 has units of km/s per ___________, where one megaparsec is approximately 3.26 million light-years. The ___________ time, equal to 1/H0, is ___________ proportional to H0 and gives an approximate upper limit for the age of the universe. The ___________ a relates redshift to the relative size of the universe: at redshift z = 1, the universe was half its present size.
4. Function recall
Answer each question in 1–2 sentences using precise terms from the lesson. 8 marks (2 each)
4.1 State Hubble's law in words and as an equation, defining every symbol.
4.2 Write the equation that relates redshift z to observed wavelength λobs and rest wavelength λrest. Explain what a value of z = 0.10 means physically.
4.3 Write the equation connecting redshift z to the scale factor athen at the time the light was emitted. Evaluate this equation when z = 4.
4.4 Why does Hubble's law imply the universe has a finite age, and how is the Hubble time related to this estimate?
5. Choose the correct equation
For each scenario, write the letter of the equation you would use first and briefly explain why. 6 marks (1 equation + 1 reason each)
Equation bank:
A. v = H0 d B. z = (λobs − λrest) / λrest C. v = cz D. 1 + z = 1 / athen
5.1 You are given the distance to a galaxy in Mpc and need to find its recession velocity.
5.2 You observe a spectral line at 540 nm; the rest wavelength is 486 nm. You need to find the redshift.
5.3 You have already calculated z = 0.08 for a galaxy and need to find its recession velocity (assume z < 0.1).
Q1 — Term–definition match
1.1 Hubble constant • 1.2 recession velocity • 1.3 redshift • 1.4 blueshift • 1.5 megaparsec • 1.6 scale factor • 1.7 lookback time • 1.8 Hubble time • 1.9 cosmological redshift • 1.10 Hubble tension.
Q2 — True / false with correction
2.1 True.
2.2 False. A redshifted galaxy is moving away from the observer; its observed wavelength is longer than the rest wavelength. A galaxy moving toward the observer would be blueshifted (shorter observed wavelength).
2.3 True. For a receding source λobs > λrest, so the numerator is positive and z > 0.
2.4 False. At a = 0.5 the universe was half its current size, not twice. A scale factor less than 1 means the universe was smaller in the past; a = 1 today by convention.
2.5 True. d = v / H0 = 7000 / 70 = 100 Mpc. Correct.
2.6 False. Hubble's law does not imply we are at the centre. In a uniformly expanding universe, every observer in every galaxy sees all other galaxies receding; there is no special centre to the expansion.
Q3 — Cloze paragraph
In order: recession / proportional / distance / expanding / megaparsecs / Hubble / inversely / scale factor.
Q4.1 — Hubble's law
Hubble's law: the recession velocity v of a galaxy is directly proportional to its distance d. Equation: v = H0d, where v = recession velocity (km/s), H0 = Hubble constant (~70 km/s/Mpc), d = distance (Mpc).
Q4.2 — Redshift equation and meaning
z = (λobs − λrest) / λrest. A value of z = 0.10 means the observed wavelength is 10% longer than the rest wavelength; the galaxy is receding at approximately 10% of the speed of light (v ≈ 0.10 × 3 × 105 = 30 000 km/s).
Q4.3 — Scale factor equation
1 + z = anow / athen = 1 / athen. At z = 4: athen = 1 / (1 + 4) = 1/5 = 0.20. The universe was one-fifth its current size when that light was emitted.
Q4.4 — Finite age and Hubble time
If galaxies are receding at speeds proportional to distance, running time backwards implies all matter converges at a single point at a finite time in the past — the Big Bang. The Hubble time tH = 1/H0 ≈ 1/(70 km/s/Mpc) ≈ 14 Gyr gives an approximate upper limit for this age; the actual age (~13.8 Gyr) is slightly less because gravity has slowed the expansion rate over time.
Q5 — Equation selector
5.1 A (v = H0d) — directly gives recession velocity from distance in Mpc.
5.2 B (z = (λobs − λrest) / λrest) — the definition of redshift; use spectral data to calculate z first.
5.3 C (v = cz) — for small z (< 0.1) this gives recession velocity from a known redshift.