1 parsec (pc) = 3.086 × 10¹⁶ m; 1 Mpc = 3.086 × 10²² m. Hubble's law does not mean we are at the centre — in an expanding universe, every galaxy sees every other galaxy receding, regardless of position.
H₀ is measured with difficulty and some uncertainty. Current measurements cluster around 67–73 km/s/Mpc depending on the method (this "Hubble tension" is an active area of research as of 2026).
Cosmological Redshift
The recession of galaxies causes their light to be redshifted — wavelengths are stretched as space expands. This is not a Doppler effect in the classical sense, but a stretching of space itself.
For recession velocities much less than c (v << c):
z ≈ v / c → v = zc
Combining with Hubble's law:
d = v / H₀ = zc / H₀
Higher redshift means greater distance and greater recession velocity. Distant quasars can have z > 6, meaning their light has been redshifted by a factor of more than 7 since emission.
Age of the Universe from H₀
If the universe has been expanding at a constant rate since the Big Bang, then the time since the Big Bang is approximately:
t ≈ 1 / H₀
Converting H₀ = 70 km/s/Mpc to SI:
H₀ = 70 000 m/s / (3.086 × 10²² m) = 2.27 × 10⁻¹⁸ s⁻¹
t ≈ 1 / H₀ = 1 / (2.27 × 10⁻¹⁸) = 4.41 × 10¹⁷ s ≈ 14 billion years
The actual age of the universe (13.8 billion years from CMB observations) differs slightly from this estimate because the expansion rate has not been constant — it was decelerating early on and is now accelerating due to dark energy.
Evidence for an Expanding Universe
Two key pieces of evidence support the expanding universe model:
1. Galactic redshifts (Hubble's observations):
Spectra of galaxies show characteristic absorption lines shifted to longer wavelengths
More distant galaxies show larger redshifts
The linear relationship v = H₀d confirmed observationally
2. Cosmic Microwave Background (CMB):
Uniform thermal radiation filling all space at T ≈ 2.73 K
Relic radiation from when the universe was ~380,000 years old and first became transparent
The CMB shows tiny temperature fluctuations (ΔT/T ~ 10⁻⁵) that seeded structure formation
Running the expansion backwards in time implies all matter and energy originated from a single point ~13.8 billion years ago — the Big Bang.
Cosmological Distances: Unit Conversions
Unit
Value in metres
Value in light-years
1 light-year (ly)
9.461 × 10¹⁵ m
1 ly
1 parsec (pc)
3.086 × 10¹⁶ m
3.26 ly
1 kiloparsec (kpc)
3.086 × 10¹⁹ m
3260 ly
1 megaparsec (Mpc)
3.086 × 10²² m
3.26 × 10⁶ ly
The observable universe has a radius of about 46 billion light-years (14,000 Mpc) — much larger than 13.8 billion light-years because space itself has been expanding while the light travelled.
Example 1: Recession velocity from Hubble's law
A galaxy is 500 Mpc from Earth. Calculate its recession velocity. (H₀ = 70 km/s/Mpc)
1 v = H₀ × d = 70 × 500
v = 35 000 km/s = 3.5 × 10⁴ km/s ≈ 0.117c
Example 2: Distance from redshift
A galaxy's hydrogen Lyman-alpha line (emitted at 121.6 nm) is observed at 147.1 nm. Calculate (a) the redshift z, (b) the recession velocity, and (c) the distance in Mpc. (H₀ = 70 km/s/Mpc, c = 3.0 × 10⁵ km/s)
(a) z = 0.21 (b) v = 63 000 km/s ≈ 0.21c (c) d = 900 Mpc
Example 3: Age of the universe
Calculate the age of the universe from H₀ = 70 km/s/Mpc, giving your answer in years.
1 Convert H₀ to SI: H₀ = 70 × 10³ m/s / (3.086 × 10²² m) = 2.269 × 10⁻¹⁸ s⁻¹
2 t = 1/H₀ = 1 / (2.269 × 10⁻¹⁸) = 4.41 × 10¹⁷ s
3 Convert to years: t = 4.41 × 10¹⁷ / (3.156 × 10⁷) = 1.40 × 10¹⁰ years
t ≈ 14 billion years (consistent with independent measurements of ~13.8 Gyr)
Example 4: Observed wavelength from redshift
A quasar at redshift z = 2.5 emits the H-alpha line at 656 nm. At what wavelength is this line observed?
1 z = Δλ/λ₀ → Δλ = z × λ₀ = 2.5 × 656 = 1640 nm
2 λ_observed = λ₀ + Δλ = 656 + 1640 = 2296 nm
λ_observed = 2296 nm ≈ 2.3 μm (infrared) — the visible H-alpha line is redshifted far into the infrared
Q1. A galaxy at distance 200 Mpc from Earth has recession velocity (using H₀ = 70 km/s/Mpc):
Q2. A spectral line normally at 500 nm is observed at 510 nm from a distant galaxy. What is the redshift z?
Q3. Why does Hubble's law not mean that we are at the centre of the universe?
Q4. What is the approximate age of the universe estimated from H₀ = 70 km/s/Mpc?
Q5. Cosmological redshift is best described as:
Challenge 1. The Andromeda galaxy (M31) shows a blueshift: its Ca H line, normally at 396.8 nm, is observed at 396.5 nm. (a) Calculate the redshift z. (b) Find the velocity (is it towards or away?). (c) Does this violate Hubble's law? Explain.
(a) z = Δλ/λ₀ = (396.5 − 396.8)/396.8 = −0.3/396.8 = −7.56 × 10⁻⁴
(b) v = zc = −7.56 × 10⁻⁴ × 3 × 10⁵ = −227 km/s (negative = blueshifted = approaching)
Andromeda is moving towards us at ~227 km/s.
(c) This does NOT violate Hubble's law. Hubble's law describes the large-scale expansion of the universe. Nearby galaxies have significant peculiar velocities (motion through space relative to the Hubble flow) due to local gravitational interactions. Andromeda is close enough (only ~0.78 Mpc) that its gravitational attraction towards the Milky Way exceeds the Hubble recession. They are expected to merge in ~4.5 billion years.
Challenge 2. Quasar 3C 273 has redshift z = 0.158. (a) Calculate its recession velocity. (b) Calculate its distance in Mpc and in light-years. (c) Calculate the look-back time (time for light to travel from 3C 273 to us). (H₀ = 70 km/s/Mpc, c = 3 × 10⁵ km/s, 1 Mpc = 3.26 × 10⁶ ly)
(a) v = zc = 0.158 × 3 × 10⁵ = 4.74 × 10⁴ km/s = 47 400 km/s
(b) d = v/H₀ = 47 400/70 = 677 Mpc
In light-years: 677 × 3.26 × 10⁶ = 2.21 × 10⁹ ly ≈ 2.2 billion light-years
(c) Look-back time ≈ d/c (approximate for z << 1):
= 2.2 × 10⁹ light-years / (1 light-year per year) = 2.2 × 10⁹ years
We see 3C 273 as it was ~2.2 billion years ago, when the universe was roughly 11.6 billion years old.
Challenge 3. Explain the "Hubble tension": two methods of measuring H₀ give different values (~67 vs ~73 km/s/Mpc). Describe the two methods and why the discrepancy is significant for cosmology.
Method 1 — Early universe (CMB): The Planck satellite measured tiny temperature fluctuations in the cosmic microwave background. Using the standard ΛCDM cosmological model, these give H₀ ≈ 67.4 ± 0.5 km/s/Mpc.
Method 2 — Late universe (distance ladder): Using Cepheid variable stars to calibrate distances to nearby galaxies, then using Type Ia supernovae as standard candles for greater distances. This gives H₀ ≈ 73 ± 1 km/s/Mpc.
The discrepancy is ~5σ (5 standard deviations), well beyond experimental uncertainty — it is statistically significant.
Significance: If both measurements are correct, the standard cosmological model (ΛCDM with constant dark energy) is incomplete. Possible explanations include: early dark energy, extra neutrino species, modified gravity, or systematic errors in one or both methods. Resolving the Hubble tension is one of the biggest open problems in modern cosmology as of 2026.