Cosmology · Light, redshift and first distance clues

Comparing emitted and observed wavelengths

Redshift begins with an identified feature and a dimensionless ratio. Calculate that ratio, test its uncertainty and learn what it does and does not establish before interpreting a cosmic history.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

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After reading, you can

Calculate z and 1 + z from wavelengths in consistent units.
Distinguish a common fractional shift from a common additive displacement.
Keep the measured ratio, its uncertainty and the interpretation separate.

Specify the feature before calculating its shift

Suppose an identified spectral feature has reference wavelength λᵣ and observed wavelength λₒ. The subscripts mean reference and observed. The reference is the wavelength expected for the same transition in the stated rest-frame convention, normally established through laboratory measurements or an appropriate physical template.

Defining the reference does not mean we observed the actual source at the instant it emitted the photon. We measured radiation here and compared it with an identification supported by a model of the source or intervening matter. A mistaken identification can produce a precise but incorrect ratio.

Use the same length unit and wavelength convention in both entries. Comparing 600 nm with 0.5 micrometres is legitimate only after recognizing that 0.5 micrometres equals 500 nm. Mixing vacuum and air wavelength conventions can introduce a systematic offset in a small-shift measurement.

Divide the change by the reference wavelength

The redshift parameter is z = (λₒ − λᵣ)/λᵣ. It is the fractional wavelength change relative to the reference, not relative to the observed wavelength. Rearranging gives 1 + z = λₒ/λᵣ and λₒ = (1 + z)λᵣ.

Because numerator and denominator are both lengths, z has no unit. A value z = 0.20 means the wavelength is 20% longer than the reference. It does not mean a speed of 0.20 metres per second, a distance of 0.20 light-years or an age of 0.20 billion years.

For λᵣ = 500 nm and λₒ = 600 nm, the change is 100 nm and z = 100/500 = 0.20. Dividing by 600 instead gives about 0.167, which is a different fraction and is not the defined z. Writing the definition before inserting numbers prevents that common denominator mistake.

The sign describes the direction of the wavelength change

Positive z means a longer observed wavelength: a redshift. Negative z means a shorter observed wavelength: a blueshift. For a 500-nm reference feature observed at 450 nm, z = −50/500 = −0.10. The terminology applies outside visible light too; a shifted radio feature need not become visibly red.

Since local vacuum wavelength and frequency satisfy λν = c, the frequency ratio is νₒ/νᵣ = 1/(1 + z). A redshift reduces the observed frequency relative to the reference. It does not reduce it by exactly z as a fraction of the reference unless a suitable small-z approximation is made.

At z = 1, wavelengths are doubled and frequencies are halved. At z = 0.20, νₒ/νᵣ = 1/1.20 ≈ 0.8333. That is a frequency decrease of about 16.7%, despite the wavelength increasing by 20%. Reciprocal relationships need not produce equal percentage changes.

For finite positive observed and reference wavelengths, 1 + z must be positive, so z > −1. A proposed z = −1.2 would imply a negative wavelength ratio and cannot describe these measurements. This is a consistency check on the definition, not a statement that every allowed number is produced by any chosen physical model.

Synthetic wavelength comparisons, using a 500-nm reference
Observed wavelength	z	Frequency/reference frequency
450 nm	−0.10	About 1.111
500 nm	0	1
600 nm	0.20	About 0.8333
1,000 nm	1	0.5

Propagate a stated measurement uncertainty

Take a synthetic observation λₒ = 600 ± 1 nm and, for this calculation only, treat λᵣ = 500 nm as exact. Because z = λₒ/500 nm − 1 is linear in λₒ, the same uncertainty interval becomes z = 0.200 ± 0.002. Dividing the 1-nm uncertainty by 500 nm gives 0.002.

This calculation carries forward the supplied type of uncertainty. If the input ±1 nm denotes a one-standard-deviation estimate, the ±0.002 has that interpretation under the stated linear model. If it denotes a bounded interval, it remains an interval. The symbol ± alone is not a substitute for saying what the uncertainty means.

A real reference wavelength can have uncertainty, and the instrument calibration can affect several lines together. The simple calculation above does not account for those additional terms or their correlations. Averaging many lines does not automatically remove a common calibration error. Check the uncertainty model as well as the arithmetic.

A measured redshift is not yet a distance

A consistent fractional shift across identified features establishes a radiation comparison, with uncertainty and conventions attached. To interpret it, we still ask how source and observer moved, how their gravitational environments differed and what spacetime model connects the events.

In an ideal homogeneous expanding model for comoving emitter and observer, the cosmological relation is 1 + z = aₒ/aₑ, with aₒ and aₑ the scale factors at observation and emission. Setting aₒ = 1 gives aₑ = 1/(1 + z). That relation is a physical interpretation under stated conditions; the ratio definition itself does not choose those conditions.

Even when the cosmological interpretation is appropriate, a distance or lookback time requires the intervening expansion history. Two histories can share the same endpoint scale-factor ratio while producing different light-travel intervals. Report the ratio honestly before adding the model-dependent length or age.

Worked reasoning

Check two identified features

Reference features at 400 and 600 nm are observed at 500 and 750 nm. Calculate z from each and predict the observed position of a 520-nm feature from the same shifted pattern.

The first ratio is 500/400 = 1.25, giving z = 0.25.
The second is 750/600 = 1.25, giving the same z.
Multiply the third reference by 1.25: 520 × 1.25 = 650 nm.

Both identified features give z = 0.25; the third is predicted at 650 nm. The prediction does not by itself establish the physical cause.

Try the reasoning

A 200-nm reference feature is observed at 300 nm. Is z equal to one third or one half?
Show the answer and reasoning
One half, z = 0.5.
The change is 100 nm and the definition divides it by the 200-nm reference. Dividing by the observed value gives a different fraction.
At z = 2, what are the observed/reference wavelength and frequency ratios?
Show the answer and reasoning
3 and 1/3, respectively.
The wavelength ratio is 1 + z. The frequency ratio is its reciprocal for local vacuum wavelength-frequency measurements.
If z is known exactly, is the light's travel time determined without an expansion history or another physical interpretation?
Show the answer and reasoning
No.
A radiation ratio does not supply the full spacetime history or a unique time interval. Further physical assumptions and data enter the inference.

What to carry forward

z = (observed wavelength − reference wavelength)/reference wavelength.
1 + z is a dimensionless wavelength ratio; frequency uses its reciprocal.
Precision, identification and calibration are distinct parts of a redshift measurement.
A physical model is needed to turn a redshift into a cosmic scale factor, distance or lookback time.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.