d ℓ = a d χ

d t

. Thus the signed radial peculiar velocity is

v = a \frac{d χ}{d t} = \frac{a χ ˙}{t ˙} .

Substitute

a^{2} \overset{χ}{˙} = C

and the massive normalization from part (b):

v (a) = \frac{C / a}{1 + C ^{2} / ( a ^{2} c ^{2} )}, aγ v = C .

Equivalently, for velocity

v_{0}

at scale factor

a_{0}

v (a) = \frac{v _{0} a _{0}}{a ^{2} ( 1 - v _{0}^{2} / c ^{2} ) + a _{0}^{2} v _{0}^{2} / c ^{2}} .

v \propto a^{- 1}

; an initially relativistic speed stays below

c

and decreases as the universe expands.

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