Free Pearson Edexcel International A Level Physics lesson

Redshift, expansion and the universe

Connect spectral observations with a model whose assumptions matter.

What you’ll learn

  • Define the spectral reference and use the small-speed approximation only in its range.
  • Fit a population with independent distances and distinguish an expansion timescale from an exact age.
  • Combine independent evidence and keep age and fate conclusions conditional on the expansion model.

Use spectral shifts as motion evidence

Relative motion between source and observer changes observed wave frequency. For a source approaching through a medium, successive wavefronts arrive closer together; recession spreads them farther apart. Light also shows a Doppler shift, but electromagnetic propagation does not require a material medium and high-speed calculations require relativity.

Define redshift z = (λobserved − λemitted)/λemitted. A receding source has positive z and a longer observed wavelength. For small speeds compared with c, z ≈ v/c. With frequency, the positive recession fractional decrease is (femitted − fobserved)/femitted; this approximately equals z for small shifts, but the finite-shift expressions are not exactly identical.

Identify spectral lines from their pattern and known rest wavelengths, rather than calling every red-looking object redshifted. A cool source can emit a redder continuum without recession. Cosmological redshift describes stretching as the universe expands; at low redshift the same approximate velocity language is useful, but applying v = cz at large z as a local speed can mislead.

z = (λobs − λemit)/λemit for |v| ≪ c: z ≈ v/c

Symbols and units

  • λobs and λemit are the observed and emitted reference wavelengths, in matching units such as m or nm; z is their dimensionless fractional shift.
  • v is radial recession speed in m s⁻¹, positive for recession in this convention; c = 3.00 × 10⁸ m s⁻¹ is the speed of light.
  • The symbol ≈ marks the low-speed approximation z ≈ v/c, valid when |v| is much smaller than c; it is not an exact large-redshift velocity law.

How the equation explains the idea

  1. Subtract the known emitted wavelength from the observed wavelength: Δλ = 505 − 500 = +5 nm. The positive change is a redshift of this identified spectral line.
  2. Divide by the emitted reference, not the observed value: z = 5/500 = 0.0100. Matching wavelength units cancel, so converting both to metres would give the same ratio.
  3. Rearrange the small-shift relation to v ≈ cz = (3.00 × 10⁸)(0.0100) = 3.00 × 10⁶ m s⁻¹. This is 0.01c, consistent with the stated low-speed approximation.
  4. The reference-line shift supplies the evidence; a red thermal continuum alone does not. For large cosmological redshift, use an appropriate expansion model rather than treating cz as an exact local speed.

Worked example

A shifted spectral line

  1. A known line at 500 nm is observed at 505 nm.
  2. z = (505 − 500)/500 = 0.0100.
  3. The low-redshift recession estimate is v ≈ 0.0100 × 3.00 × 10⁸ = 3.00 × 10⁶ m s⁻¹.

Use the emitted reference wavelength in the denominator.

Keep this idea. Define the spectral reference and use the small-speed approximation only in its range.

Infer an expansion rate from distances and redshifts

For suitably distant galaxies at low redshift, recession speed is approximately proportional to distance: v = H₀d. Plot speed vertically against independently estimated distance; its gradient estimates H₀. Nearby galaxies can have peculiar velocities from local gravitational interactions, so one nearby galaxy is not a reliable expansion-rate measurement.

H₀ is often quoted in km s⁻¹ Mpc⁻¹. This is an inverse-time quantity: convert kilometres to metres and one megaparsec to 3.086 × 10²² m before taking an inverse in seconds. A larger H₀ means faster present expansion for the same distance and a shorter inverse-Hubble timescale.

The inverse H₀ gives an order-of-magnitude age estimate if present separation speed is extrapolated back as constant. Real expansion has changed with the universe’s contents, so this is not an exact age formula. Distance calibration, selection effects and peculiar motion can affect the fitted gradient. Values in these exercises are supplied illustrative inputs, not a claim that one currently measured H₀ is uncontested.

v = H₀d 1 Mpc = 3.086 × 10²² m Hubble timescale ≈ 1/H₀

Symbols and units

  • v is model recession speed, in km s⁻¹ when using d in Mpc and H₀ in km s⁻¹ Mpc⁻¹; d is independently estimated galaxy distance.
  • H₀ is the present-epoch Hubble constant. Its SI unit is s⁻¹ because speed divided by distance is an inverse time.
  • 1 Mpc = 3.086 × 10²² m and 1 km = 1000 m. The example uses 3.156 × 10⁷ s per year.
  • The inverse-Hubble timescale 1/H₀ is in s after SI conversion. It becomes an age estimate only with a specified expansion-history assumption.

How the equation explains the idea

  1. For low-redshift galaxies following the expansion trend, v = H₀d gives a speed–distance gradient H₀. Local peculiar velocities and distance errors must be considered when fitting that trend.
  2. Convert the illustrative value before inversion: H₀ = 70 × 1000/(3.086 × 10²²) = 2.268 × 10⁻¹⁸ s⁻¹. The metre units cancel.
  3. Invert the rate: 1/H₀ = 4.409 × 10¹⁷ s. Divide by 3.156 × 10⁷ s per year to obtain 1.397 × 10¹⁰ years, approximately 14.0 billion years.
  4. The constant-speed extrapolation gives d/v = d/(H₀d) = 1/H₀. A changing expansion history breaks that direct age identification, so this is a reference timescale rather than an exact measured age.
The velocity–distance gradient is the Hubble constant

Conditions. Illustrative low-redshift models v = H₀d with H₀ = 60 or 80 km s⁻¹ Mpc⁻¹. These are computed comparison lines, not measured galaxy data or claims about the current best H₀. Peculiar velocities are omitted. 1 Mpc = 3.086 × 10²² m; each axis uses its displayed SI scale factor.

Distance / (10²⁴ m); Recession speed / (10⁶ m s⁻¹).

At 100 Mpc the two models predict 6000 and 8000 km s⁻¹. A steeper gradient means a larger present expansion rate and a shorter 1/H₀ timescale. Converting that timescale into a cosmic age requires an expansion-history model; nearby peculiar motion can obscure the trend.

Open the interactive lesson to view the plotted curves and their coordinate tables.

Worked example

Convert an illustrative Hubble constant

  1. H₀ = 70 km s⁻¹ Mpc⁻¹ = 70000/(3.086 × 10²²) s⁻¹.
  2. H₀ = 2.268 × 10⁻¹⁸ s⁻¹.
  3. 1/H₀ = 4.409 × 10¹⁷ s ≈ 14.0 billion years using 3.156 × 10⁷ s per year.

A converted inverse rate is a timescale whose age interpretation needs a history model.

Keep this idea. Fit a population with independent distances and distinguish an expansion timescale from an exact age.

Evaluate age, dark matter and possible futures

Large-scale redshift trends support an expanding universe. The cosmic microwave background and the abundance of light elements provide additional evidence for a hot dense early phase; they address different observations rather than merely repeating the same redshift measurement. Extrapolating expansion requires a model for how its rate changed, not an assumption that today’s rate has always applied.

Orbital speeds in galaxies, motions within clusters and gravitational lensing can imply more gravitating matter than luminous material accounts for. Dark matter names this missing gravitating component; it is not simply another name for a black hole or dark energy. Its distribution affects gravitational slowing and structure formation, while uncertainty in distance scales and expansion measurements also affects age estimates.

In a simplified matter-only model, density above a critical value can lead to recollapse, while lower density permits continuing expansion. As a labelled extension, ρcritical = 3H₀²/(8πG) gives the density associated with spatial flatness in the standard homogeneous framework. The real fate is not determined by matter density alone: a component associated with accelerated expansion, called dark energy, changes the dynamics. Treat future scenarios as conditional on measured contents and model assumptions.

Matter-only comparison is conditional; extension: ρcritical = 3H₀²/(8πG)

Symbols and units

  • ρcritical is a reference cosmic mass-equivalent density in kg m⁻³, not a directly inferred matter density from this equation alone.
  • H₀ is the present expansion-rate parameter in s⁻¹; G = 6.67 × 10⁻¹¹ m³ kg⁻¹ s⁻² is the gravitational constant.
  • The extension ρcritical = 3H₀²/(8πG) is the density associated with spatial flatness in the stated homogeneous cosmological framework.
  • Dark matter and dark energy identify different inferred cosmic components; their contents and evolution are additional model inputs, not interchangeable symbols in the density formula.

How the equation explains the idea

  1. Treat the displayed relation as a labelled model extension, not a deduction of cosmic fate. Insert the example’s H₀ only after expressing it in s⁻¹.
  2. Square the rate: H₀² = (2.3 × 10⁻¹⁸)² = 5.29 × 10⁻³⁶ s⁻². Then ρcritical = (3 × 5.29 × 10⁻³⁶)/(8π × 6.67 × 10⁻¹¹) = 9.47 × 10⁻²⁷ kg m⁻³.
  3. Check the units: s⁻² divided by m³ kg⁻¹ s⁻² is kg m⁻³. The calculation supplies a comparison scale; it does not measure the actual matter content.
  4. A density-only recollapse argument assumes a particular matter-only model. Dark energy and the history of the components can change the dynamics, so the reference density alone fixes neither the age nor the eventual fate.

Worked example

A critical-density scale

  1. For illustrative H₀ = 2.3 × 10⁻¹⁸ s⁻¹ and G = 6.67 × 10⁻¹¹ SI, use ρcritical = 3H₀²/(8πG).
  2. ρcritical ≈ 9.47 × 10⁻²⁷ kg m⁻³.
  3. This is a model reference density, not sufficient on its own to predict the universe’s fate.

Name the model before attaching a future scenario to a density comparison.

Keep this idea. Combine independent evidence and keep age and fate conclusions conditional on the expansion model.

Open the interactive lesson for free practice and reviewed three-question samples when available. New full question sets and timed unit checks are in preparation. Existing complete Physics revision packs retain paid access; existing free teaching PDFs remain free.

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