Estimate cosmic age from the reciprocal Hubble constant
Use d/v = 1/H₀ as a simple estimate for the Universe’s age, with consistent time units.
Extended only (Supplement); Physics 0625, 2026–2028. Use supplied distances, periods and stated orbital models. Drawings are not to scale. Written practice is self-checked.
Before you begin
- Rearrange H₀ = v/d.
- Convert seconds into years.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the unit of d/v using SI distance and speed?
What you will learn
- Use d/v = 1/H₀ as a simple estimate for the Universe’s age, with consistent time units.
- Explain the denser-origin inference and the limits of a constant-rate backward extrapolation.
Derive a time scale from the ratio
Since H₀ = v/d, the reciprocal is d/v = 1/H₀. Metres divided by metres per second gives seconds. In a simple backward extrapolation that keeps the present recession rate constant, this is a time scale for separations to have been much smaller.
Using H₀ = 2.2 × 10⁻¹⁸ s⁻¹ gives 1/H₀ ≈ 4.55 × 10¹⁷ s. With the supplied conversion 1 year = 3.156 × 10⁷ s, the estimate is approximately 1.44 × 10¹⁰ years, or 14.4 billion years. Keep unrounded values until the final conversion.
State what the estimate supports
The syllabus uses this backward time scale as evidence for an origin in which matter was concentrated together, described in its simple model as a single point. The scientific inference is an earlier much denser state, not a claim that Earth identifies that point or that a present spectrum directly images it.
The actual expansion rate changes over cosmic history. Therefore 1/H₀ is an estimate, not an exact model-independent age or a proof of an ordinary explosion into pre-existing empty space. A larger positive H₀ gives a smaller reciprocal estimate. A detailed cosmic history needs more than this constant-rate calculation.
Pause and explain
Why is 1/H₀ an age estimate rather than an exact universal age?
Worked example
Use H₀ = 2.2 × 10⁻¹⁸ s⁻¹ and 3.156 × 10⁷ s per year to estimate age.
Show the worked solution
- t = 1/H₀ = 4.54545… × 10¹⁷ s.
- Divide by 3.156 × 10⁷ s/year.
- Round only after conversion.
Answer Approximately 1.44 × 10¹⁰ years, or 14.4 billion years.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find 1/H₀ in seconds for H₀ = 2.2 × 10⁻¹⁸ s⁻¹.
Give me a hint
Invert both the coefficient and the power of ten.
Compare my reasoning
- t = (1/2.2) × 10¹⁸ s.
- Calculate 0.454545… × 10¹⁸.
- Normalise and round.
Approximately 4.55 × 10¹⁷ s.
Look for these in your work
- Use the reciprocal.
- Give seconds.
Using a supplied alternative H₀ = 2.5 × 10⁻¹⁸ s⁻¹ and 3.156 × 10⁷ s/year, estimate age.
Give me a hint
Invert the supplied alternative, not the default reference.
Compare my reasoning
- t = 4.0 × 10¹⁷ s.
- Divide by 3.156 × 10⁷.
- Round to three significant figures.
Approximately 1.27 × 10¹⁰ years, or 12.7 billion years.
Look for these in your work
- Use the supplied value.
- Keep the estimate qualifier.
Compare the 2.2 and 2.5 × 10⁻¹⁸ s⁻¹ age estimates and evaluate “the larger H₀ means an older exact Universe”.
Give me a hint
The age estimate is inversely related to H₀ and has a model boundary.
Compare my reasoning
- The estimates are about 14.4 and 12.7 billion years.
- The larger H₀ gives the smaller time scale.
- Neither reciprocal alone is an exact full-history age.
The claim is false: larger H₀ gives a younger reciprocal estimate, and both are approximate.
Look for these in your work
- State the inverse trend.
- Retain the constant-rate limitation.
Common mistakes
- Invert H₀, not multiply by it.
- 1/H₀ is an estimate, not exact history.
- A denser origin does not privilege Earth as the centre.
What can you explain now?
Why is 1/H₀ an age estimate rather than an exact universal age?
Compare with the explanation
It is a simple backward extrapolation of a present rate
Cosmic expansion changes over time, so the reciprocal alone is not a complete history model.
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Make it stick
Explain the denser-origin inference and the limits of a constant-rate backward extrapolation.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.