Hubble Law

Local Law

For sufficiently nearby galaxies,

$$v \simeq H_0 d.$$

This is the first-order expansion law around the present epoch. $H_0$ is the current value of the Hubble parameter.

From Scale Factor

For proper distance $D=a(t)\chi$ at fixed comoving coordinate $\chi$,

$$\dot D = \dot a\chi = \frac{\dot a}{a}D = HD.$$

Thus Hubble’s law is the local form of metric expansion.