The Expanding Universe and Age Estimation Revision Notes | A Level AQA Physics

1. The Expanding Universe — What the Evidence Tells Us

Every galaxy we observe beyond our own Local Group is moving away from us. Not just a few — all of them. And the further away a galaxy is, the faster it recedes. This single observation reshaped our entire picture of the cosmos.

The evidence comes from redshift. When a light source moves away from an observer, its wavelengths are stretched toward the red end of the spectrum — the same Doppler effect that drops the pitch of a passing ambulance siren. Astronomers measure the shift in known spectral lines (such as the hydrogen Balmer series) and calculate the recession speed of the source.

The key insight is this: the redshift is not caused by galaxies flying through space from a central explosion. Space itself is stretching. Every region of the universe is expanding relative to every other region, so every galaxy appears to recede from every other galaxy simultaneously. There is no centre of expansion.

When Edwin Hubble plotted recession speed against distance for a sample of galaxies in 1929, he found a striking linear relationship. Galaxies twice as far away recede roughly twice as fast. This proportionality is the foundation of modern observational cosmology.

2. Hubble's Law and the Hubble Constant

The linear relationship Hubble observed is now formalised as Hubble's law: the recession speed $v$ of a galaxy is proportional to its distance $d$ from the observer. The constant of proportionality is the Hubble constant, $H_0$.

The currently accepted value of $H_0$ is approximately $65$–$73$ km s$^{-1}$ Mpc$^{-1}$, depending on the measurement method. For AQA exam calculations, you will typically be given a value to use — so do not memorise a specific number; focus on the method.

Notice the units: km s$^{-1}$ Mpc$^{-1}$ means "kilometres per second per megaparsec". When you cancel the distance units (km and Mpc are both distances), $H_0$ reduces to units of s$^{-1}$ — a pure inverse time. That is the key to estimating the age of the universe.

3. Estimating the Age of the Universe

Here is where the physics becomes genuinely elegant. If the universe is expanding at a constant rate, you can run the clock backwards: at some point in the past, all matter was in the same place. The time elapsed since that moment is the age of the universe.

Consider a galaxy at distance $d$ receding at speed $v$. If the expansion rate has always been constant, the time taken for that galaxy to reach its current distance is $t = d / v$. Substituting Hubble's law $v = H_0 d$ gives a result that is independent of which galaxy you choose.

The critical step in any exam calculation is converting $H_0$ from km s$^{-1}$ Mpc$^{-1}$ into SI units of s$^{-1}$ before taking the reciprocal. Getting this conversion right is where most marks are lost — so the worked example below takes it one step at a time.

Worked example: Estimating the Age of the Universe from $H_0$

The Hubble constant is measured as $H_0 = 65$ km s$^{-1}$ Mpc$^{-1}$. Estimate the age of the universe in years. (1 pc = 3.09 × 10$^{16}$ m; 1 year $\approx$ 3.15 × 10$^7$ s)

1. $65 \text{ km s}^{-1} = 65 \times 10^3 \text{ m s}^{-1} = 6.5 \times 10^4 \text{ m s}^{-1}$ — 1 km = 1000 m = 10³ m. This converts the numerator of $H_0$ into SI metres. [M1]
2. $1 \text{ Mpc} = 10^6 \text{ pc} = 10^6 \times 3.09 \times 10^{16} \text{ m} = 3.09 \times 10^{22} \text{ m}$ — 1 Mpc = 1 megaparsec = 10⁶ parsecs. Multiply by the given conversion for 1 pc. [M1]
3. $H_0 = \frac{6.5 \times 10^4 \text{ m s}^{-1}}{3.09 \times 10^{22} \text{ m}} = \frac{6.5 \times 10^4}{3.09 \times 10^{22}} \text{ s}^{-1}$\n$H_0 = 2.104\ldots \times 10^{-18} \text{ s}^{-1}$ — Dividing m s⁻¹ by m cancels the metre units, leaving s⁻¹. Keep full precision here to avoid rounding errors in the next step. [M1]
4. $t = \frac{1}{H_0} = \frac{1}{2.104 \times 10^{-18}} \text{ s} = 4.75 \times 10^{17} \text{ s}$ — Reciprocal of $H_0$ in s⁻¹ gives time in seconds. Check: 1 ÷ (2.104 × 10⁻¹⁸) = (1 ÷ 2.104) × 10¹⁸ = 0.4753 × 10¹⁸ = 4.75 × 10¹⁷ s. [A1]
5. $t = \frac{4.75 \times 10^{17}}{3.15 \times 10^7} \text{ years} = \frac{4.75}{3.15} \times 10^{17-7} \text{ years} = 1.508 \times 10^{10} \text{ years}$ — Divide by the number of seconds in a year. 4.75 ÷ 3.15 = 1.508, and 10¹⁷ ÷ 10⁷ = 10¹⁰. So the estimate is approximately 1.5 × 10¹⁰ years, or about 15 billion years. [A1]
6. $t \approx 1.5 \times 10^{10} \text{ years}$ — Round to 2 significant figures to match the precision of the given data. Note that this estimate is slightly above the accepted age of ~13.8 billion years; the discrepancy arises because the constant-$H_0$ assumption does not account for the varying expansion rate throughout cosmic history. [B1]

Answer: $t \approx 1.5 \times 10^{10}$ years (approximately 15 billion years — slightly above the accepted ~13.8 billion years due to the constant-$H_0$ assumption)

4. The Constant $H_0$ Assumption — What It Means and Why It Matters

The derivation $t \approx 1/H_0$ rests entirely on one assumption: that the expansion rate of the universe has always been the same as it is today. This is a significant simplification, and the AQA specification requires you to recognise it explicitly.

In reality, the expansion rate has changed over cosmic time. In the early universe, gravity was decelerating the expansion; more recently, evidence suggests a mysterious dark energy is accelerating it. A constant $H_0$ ignores both effects, so $1/H_0$ is an approximation — not an exact age.

Despite this, the Hubble time gives an order-of-magnitude estimate that is remarkably close to the accepted age of ~13.8 billion years. For AQA purposes, the key phrase to include in any written answer is: "assuming $H_0$ has remained constant throughout the history of the universe".

There is also an observational uncertainty: the value of $H_0$ itself is not precisely agreed upon. Different measurement techniques (using the cosmic microwave background versus using Type Ia supernovae as standard candles) give slightly different values, a discrepancy known as the Hubble tension. This means the estimated age carries an inherent uncertainty even before the constant-rate assumption is considered.

5. Unit Conversion Mastery — The Calculation Trap

The unit conversion from km s$^{-1}$ Mpc$^{-1}$ to s$^{-1}$ is the single most common source of lost marks in this topic. The logic is straightforward once you see it as a fraction: $H_0$ has a speed (km s$^{-1}$) on top and a distance (Mpc) on the bottom. Convert both to SI, and the metres cancel.

6. Exam Technique and Mark-Winning Strategies

AQA questions on this topic tend to fall into two categories: a multi-step calculation (convert $H_0$, find $t$, convert to years) and a written explanation of what the result means or what assumptions were made. Both reward precise language and clear working.

For calculation questions, always show the unit conversion as a separate, labelled step. Examiners award method marks for the conversion even if your final numerical answer contains an arithmetic slip. Writing "$1 \text{ Mpc} = 3.09 \times 10^{22} \text{ m}$" on its own line is worth a mark.

For written questions asking you to evaluate the age estimate, the mark-winning phrase is: "This is an estimate because it assumes $H_0$ has been constant throughout the history of the universe, which is not the case." Adding that the actual expansion rate has varied — decelerating early on and accelerating more recently — lifts a two-mark answer to three marks.

Taken together, Hubble's law and the age estimate $t \approx 1/H_0$ represent the core of AQA's cosmology requirement: the redshift of galaxies is interpreted as evidence for an expanding universe, and — assuming constant expansion — the reciprocal of the Hubble constant gives an order-of-magnitude estimate for the age of that universe.