Questions: Cosmological Redshift and the Hubble Law

Question 1

A galaxy is observed with a recession velocity that appears to exceed the speed of light based on the Hubble law. What is the correct interpretation?

• A. The measurement must be wrong — nothing can recede faster than light
• B. The galaxy is actually moving through space faster than light, which is possible for massive objects
• C. The space between us and the galaxy is expanding, and metric expansion is not limited by the speed of light
• D. The Hubble law has broken down and a different formula must be used instead

Answer: (C)

Cosmological redshift is caused by the expansion of spacetime itself, not by galaxies moving through pre-existing space. The speed-of-light limit applies to objects moving through space; the expansion of the metric has no such restriction. Galaxies at redshift z > 1 have recession velocities exceeding c in the naive Hubble formula, but this is perfectly consistent with general relativity. The key insight is distinguishing motion through space (kinematic, limited by c) from the stretching of space itself (metric expansion, not so limited).

Question 2

A galaxy is observed at redshift z = 2. What does this tell us about the scale of the universe when that light was emitted?

• A. The universe was twice as large as it is today
• B. The universe was one-third its current size when the light was emitted
• C. The galaxy was moving at twice the speed of light when it emitted the light
• D. The observed wavelength is twice the emitted wavelength

Answer: (B)

z = (λ_observed − λ_emitted)/λ_emitted = 2 means λ_observed = 3 × λ_emitted. This tells us the universe has expanded by a factor of 1 + z = 3 since that light was emitted — the universe was one-third its current size. Option A (twice as large) is wrong; option D only captures part of the definition. The redshift parameter directly encodes the ratio of the scale factor at observation to the scale factor at emission: a_observed/a_emitted = 1 + z.

Question 3

Cosmological redshift and Doppler redshift are essentially the same phenomenon — both involve the stretching of light wavelengths as source and observer separate.

• T. True
• F. False

Answer: False

They are fundamentally different despite producing similar spectral shifts. Doppler redshift is a kinematic effect: the source moves through space, and the wavefronts are physically stretched by that motion. Cosmological redshift arises from metric expansion: the source and observer are not moving through space; rather, space itself is expanding between them, stretching the photon wavelengths during transit. The distinction has real observational consequences — at large redshifts, the Doppler interpretation gives velocities exceeding c and breaks down, while the metric expansion interpretation remains physically consistent.

Question 4

The Hubble constant H₀ encodes the expansion rate of the universe, and inverting it (1/H₀) gives a rough estimate of the age of the universe.

• T. True
• F. False

Answer: True

If the expansion rate has been roughly constant, the time elapsed since all galaxies were at the same location is approximately 1/H₀. With H₀ ≈ 70 km/s/Mpc, this gives 1/H₀ ≈ 14 billion years — close to the more precise value of 13.8 billion years from detailed cosmological modeling. The estimate is only approximate because the expansion rate has not been constant: it was decelerating early (matter-dominated era) and has been accelerating recently (dark energy-dominated). A full treatment requires integrating over the entire expansion history.

Question 5

Explain why cosmological redshift differs fundamentally from the Doppler effect, and why the distinction matters for interpreting observations of very distant galaxies.

Model answer: Doppler redshift occurs when a source moves through space — the motion compresses or stretches wavefronts relative to a stationary observer. Cosmological redshift occurs when space itself expands: a photon's wavelength is stretched in proportion to the expansion of the universe during the photon's travel time. The distinction matters because the speed-of-light limit applies to objects moving through space, not to the expansion of space. At large redshifts (z > 1), naive application of the Doppler formula implies velocities exceeding c, which would be physically impossible for a kinematic effect but is perfectly valid for metric expansion.

The practical consequence is that the simple Hubble law v = H₀d is only valid for nearby galaxies where z << 1 and the two interpretations converge. At cosmological distances, the full general-relativistic treatment is required, relating redshift to the integral of the scale factor history. Understanding this distinction is also essential for interpreting the cosmic microwave background (z ≈ 1100), where photons have been traveling since the universe was 1/1101 its current size.