3.9.3.2 - Hubble's Law

Hubble's law and redshift

A galaxy's redshift is found by comparing identified spectral lines with their laboratory wavelengths. For nearby galaxies in the large-scale expansion, a small redshift gives recession speed $v\approx zc$. Here $v$ is positive away from us, unlike the approach-positive convention in the Doppler lesson.

The Hubble constant $H_0$ is the present recession speed per unit distance. With $v$ in $\mathrm{km\,s^{-1}}$ and $d$ in megaparsecs, its unit is $\mathrm{km\,s^{-1}\,Mpc^{-1}}$. A graph of recession speed against distance has gradient $H_0$. It is not a velocity–time graph: its gradient is not acceleration, and the area is not a distance travelled.

[DIAGRAM: asset_name: A.3.2 - Hubble's Law - Diagram 1; asset_slug: A.3.2 - Hubble's Law - Diagram 1; caption: Recession speed against distance with a classroom Hubble constant of 65 km s⁻¹ Mpc⁻¹; description: Illustrative galaxy points and best-fit line v=65d. Both axes use linear scales; horizontal axis is distance, not time.]

In the example, $H_0=65\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ means $65\,\mathrm{km\,s^{-1}}$ more recession speed for each extra Mpc. Use the value supplied in a question. A line through $(20\,\mathrm{Mpc},1300\,\mathrm{km\,s^{-1}})$ and $(80\,\mathrm{Mpc},5200\,\mathrm{km\,s^{-1}})$ has gradient $(5200-1300)/(80-20)=65$ in those units.

Individual galaxies also have peculiar velocities caused by local gravitational interactions. These produce scatter; a nearby galaxy can even approach us. A gravitationally bound pair is not a reliable sample of the smooth Hubble flow. This does not overturn the large-scale expansion pattern.

Estimating the age of the universe

An age estimate comes from extrapolating the observed expansion backwards at constant recession speed. If the present separation is $d$ and the speed is $v$, the time to remove that separation is $t\approx d/v$. Substituting Hubble's law gives

This is the Hubble time, an approximate age scale. AQA describes the approximation as assuming $H$ is constant: use a fixed supplied value and ignore changes in the expansion history. The actual distance-over-speed extrapolation assumes constant recession speeds; it is not an exact cosmological solution in which $H(t)$ stays fixed. Real expansion has changed, so $1/H_0$ is not an exact age.

Convert the units before taking the reciprocal

For $H_0=65\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $1\,\mathrm{Mpc}=3.08\times10^{22}\,\mathrm m$,

The length units cancel, leaving inverse seconds for $H_0$ and seconds for its reciprocal. Taking $1/65$ and calling it seconds loses the Mpc/km conversion. A larger $H_0$ gives a smaller age estimate. Because $t\propto H_0^{-1}$, a small percentage uncertainty in $H_0$ produces approximately the same percentage uncertainty in the age scale, but does not account for the model's simplifying assumption.