# Redshift, expansion and cosmological distances

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

Course: [Astrophysics, cosmology and inference](https://studybrix.com/en/courses/astrophysics-and-cosmology/index.md)
Level: advanced
Prerequisites: Spectra, relativity and basic cosmological modelling.

## 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.

## Practice questions

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

Answer: 0.2

Explanation: z=(600−500)/500.

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

Answer: 2

Explanation: 1+z=3.

### 3. Is z measured in kilometres per second?

A. Only in visible light
B. Yes, always
C. Only at z=0
D. No; it is dimensionless

Answer: No; it is dimensionless

Explanation: 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.

Answer: 700 km/s

Explanation: v≈H0d=700 km/s.

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

Sample answer / 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.

Explanation: Credit approximation scope and expansion model.

### 6. What must accompany a quoted cosmological distance?

A. Its distance definition and modelling assumptions
B. Only the object's nickname
C. Only its font
D. A guarantee all distance types coincide

Answer: Its distance definition and modelling assumptions

Explanation: Different distance definitions serve different observables.

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

Answer: 1

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

Explanation: Credit data versus inference.

## Further reading

- https://science.nasa.gov/universe/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.