Cosmology: (m) estimation for the age of the universe; t ≈ H 0 –1

The idea

The Hubble constant H₀ describes how fast the universe expands: recession speed v = H₀d, so H₀ has units of s⁻¹ (km s⁻¹ Mpc⁻¹). If expansion has been roughly uniform, the time since galaxies were coincident is the distance divided by the speed, giving t ≈ H₀⁻¹. This is an estimate, not an exact age, because gravity and dark energy change the expansion rate over time. Worked method: convert H₀ to SI. For H₀ = 70 km s⁻¹ Mpc⁻¹, 1 Mpc = 3.09 × 10²² m, so H₀ = 70 000 m s⁻¹ ÷ 3.09 × 10²² m = 2.27 × 10⁻¹⁸ s⁻¹, and t ≈ 1 ÷ (2.27 × 10⁻¹⁸ s⁻¹) ≈ 4.4 × 10¹⁷ s ≈ 1.4 × 10¹⁰ years.

What earns the marks

• States that H₀ relates recession speed to distance: v = H₀d, so H₀ has units of inverse time (s⁻¹).
• Derives t ≈ H₀⁻¹ by assuming the recession speed has been approximately constant since the Big Bang.
• Converts H₀ from km s⁻¹ Mpc⁻¹ into SI before inverting, using 1 Mpc = 3.09 × 10²² m.
• Recognises that t ≈ H₀⁻¹ is an estimate because the expansion rate changes with time, so the true age differs from the simple reciprocal.

Common mistakes and how to put them right

• Inverting H₀ while it is still in km s⁻¹ Mpc⁻¹, giving a meaningless time.

  Put it right: The constant must first be converted to SI units of s⁻¹.

• Treating t ≈ H₀⁻¹ as an exact age rather than an estimate that assumes constant expansion.

• Using 1 Mpc = 3.09 × 10¹⁶ m (a parsec value) instead of 3.09 × 10²² m, which shifts the answer by a factor of about 10⁶.

• Forgetting that H₀⁻¹ has units of seconds and reporting the raw number as years without converting.

Exam tips

• Always show the unit conversion of H₀ to s⁻¹ before calculating the reciprocal; method marks usually depend on it.
• Quote the final age in years as well as seconds, using 1 year ≈ 3.15 × 10⁷ s.
• If asked to comment on reliability, link the estimate to the assumption of constant expansion and to uncertainties in H₀.