AcademyParticles and Cosmology
Academy
Expanding Universe
Level 1 - Physics topic page in Particles and Cosmology.
Principle
The universe expands: distant galaxies show redshift, and on large scales recessional speed is proportional to distance.
Notation
\(v\)
recessional speed
m s^{-1}, km s^{-1}
\(H_0\)
Hubble constant
s^{-1}, km s^{-1} Mpc^{-1}
\(d\)
proper distance
m, Mpc
\(z\)
redshift
1
\(a\)
scale factor
1
\(\lambda\)
wavelength
\(\mathrm{m}\)
Method
Derivation 1: Hubble law
For nearby galaxies in the smooth large-scale universe, recessional speed is proportional to distance.
Hubble law
\[v=H_0d\]
Expansion age scale
\[t_H\sim\frac1{H_0}\]
Derivation 2: Redshift from stretched wavelengths
Cosmic expansion stretches photon wavelengths by the same factor as the scale factor.
Redshift definition
\[z=\frac{\lambda_{\mathrm{obs}}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}}\]
Scale factor relation
\[1+z=\frac{a_0}{a_{\mathrm{emit}}}\]
Derivation 3: Redshift and speed for small redshift
For small \(z\), the redshift can be interpreted approximately as a Doppler speed.
Low-redshift limit
\[v\approx cz\]
Combine with Hubble law
\[d\approx\frac{cz}{H_0}\]
Rules
Hubble law
\[v=H_0d\]
Redshift
\[1+z=\frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{emit}}}\]
Scale factor
\[1+z=\frac{a_0}{a_{\mathrm{emit}}}\]
Examples
Question
Using
\[H_0=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\]
find \(v\) for \[d=20\,\mathrm{Mpc}\]
Answer
\[v=H_0d=70(20)=1400\,\mathrm{km\,s^{-1}}\]
Checks
- Hubble law applies to large-scale expansion, not bound systems like the Solar System.
- Larger redshift means the light was emitted when the scale factor was smaller.
- The Hubble time \(1/H_0\) is an age scale, not an exact age by itself.
- For large \(z\), use cosmological redshift rather than simple Doppler reasoning.