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Astronomy & space · Advanced university

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.

Study notes

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

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

z=(600−500)/500.

0.2

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

1+z=3.

2

Check your understanding

  1. Is z measured in kilometres per second?

    1. Only in visible light
    2. Yes, always
    3. Only at z=0
    4. No; it is dimensionless

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

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

  4. What must accompany a quoted cosmological distance?

    1. Its distance definition and modelling assumptions
    2. Only the object's nickname
    3. Only its font
    4. A guarantee all distance types coincide

Answers and marking guidance

Answers and marking notes
  1. No; it is dimensionless

    It is a ratio of wavelengths.

  2. 700 km/s

    v≈H0d=700 km/s.

  3. 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.

  4. Its distance definition and modelling assumptions

    Different distance definitions serve different observables.

Further reading and review references

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