\$\lambda_{\text{observed}}\$ = wavelength measured on Earth
\$\lambda_0\$ = wavelength emitted at the source (laboratory value)
\$z\$ = redshift (dimensionless)
From Redshift to \cdot elocity
For velocities that are small compared with the speed of light, the Doppler approximation gives:
\$v \approx c\,z\$
where \$c = 3.00 \times 10^8\ \text{m s}^{-1}\$ is the speed of light.
Hubble’s Law
Edwin Hubble discovered that the recession velocity of a galaxy is proportional to its distance from Earth:
\$v = H_0 d\$
where:
\$v\$ = recession velocity (m s⁻¹)
\$d\$ = distance to the galaxy (Mpc or light‑years)
\$H_0\$ = Hubble constant (≈ 70 km s⁻¹ Mpc⁻¹ in modern measurements)
Evidence for an Expanding Universe
Spectra of distant galaxies show systematic redshift.
Redshift increases with distance, exactly as predicted by Hubble’s Law.
The linear relationship \$v = H_0 d\$ implies that space itself is stretching.
Extrapolating backwards in time leads to a hot, dense origin – the Big Bang.
Summary Table
Observation
Measured Quantity
Interpretation
Shifted spectral lines
Redshift \$z > 0\$
Galaxy receding from us
Redshift vs. distance plot
Linear trend \$v = H_0 d\$
Space expanding uniformly
Extrapolation to \$t = 0\$
All points converge
Universe began in a hot, dense state (Big Bang)
Suggested Diagram
Suggested diagram: Comparison of a laboratory spectrum (rest wavelengths) with the spectrum of a distant galaxy showing the same lines shifted to longer wavelengths (redshift).
Key Points to Remember
Redshift is a direct observational signature of recession.
Hubble’s Law quantifies the expansion and provides a method to estimate distances.
The expanding Universe model underpins the Big Bang Theory, explaining the observed cosmic microwave background and the abundance of light elements.