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Astronomy & space · lesson 2 of 4

Redshift, expansion and cosmological distances

Define z=(λobserved−λemitted)/λemitted. Redshift is dimensionless and can arise through different physical mechanisms.

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Prerequisites: Spectra, relativity and basic cosmological modelling.

Redshift

Define z=(λobserved−λemitted)/λemitted. Redshift is dimensionless and can arise through different physical mechanisms.

Expansion

In a homogeneous expanding model, 1+z=a(now)/a(emission). Large cosmological redshifts are not generally converted to speed by the low-speed Doppler formula.

Local law

At small redshift and in an appropriate distance regime, v≈H0d provides a useful approximation. Stated H0 values in exercises are model inputs, not claims to resolve measurement tensions.

Distance choices

Luminosity distance, angular-diameter distance and comoving distance are different definitions. Specify which a calculation uses.

Worked examples

Example 1

A line emitted at 500 nm is observed at 600 nm. Find z.

z=(600−500)/500.

0.2

Example 2

In an expanding model, a(now)/a(emission)=3. Find z.

1+z=3.

2

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 A line emitted at 500 nm is observed at 600 nm. Find z.

Show answer and explanation

0.2

z=(600−500)/500.

2 In an expanding model, a(now)/a(emission)=3. Find z.

Show answer and explanation

2

1+z=3.

3 Is z measured in kilometres per second?

Show answer and explanation

No; it is dimensionless

It is a ratio of wavelengths.

4 Use H0=70 km/s/Mpc as a stated model input and d=10 Mpc in the local linear regime. Find approximate recession speed.km/s

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700 km/s

v≈H0d=700 km/s.

5 Why should v=cz not be applied as an exact rule to arbitrarily large cosmological redshift?

Sample answer and marking guidance

It is a small-redshift approximation and mixes distance and expansion interpretation outside its regime. A specified cosmological model is needed for larger redshifts.

Credit approximation scope and expansion model.

6 What must accompany a quoted cosmological distance?

Show answer and explanation

Its distance definition and modelling assumptions

Different distance definitions serve different observables.

7 If scale factor at emission is 0.5 of its current value, find z.

Show answer and explanation

1

The ratio a(now)/a(emission) is 2, so z=1.

8 Distinguish a measured spectral shift from a model-derived expansion history.

Sample answer and marking guidance

The shift is obtained from wavelength comparison; relating it to cosmic history requires an expansion model and other measurements.

Credit data versus inference.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.