r(t) = ω̃ a(t)

v(t) = \frac{dr}{dt} = ω̃ \frac{da}{dt}

v ≡ H r

H = \frac{1}{a} \frac{da}{dt}

\frac{1}{H} = \frac{a}{da/dt} = \frac{size of universe}{speed of expansion}

(m − M) = 5 log_{10} d − 5

v ∼ c z