= Solution
Choose the null <affine parameter> so that the <photon> tangent is its <four-momentum> $P^\mu$. A <fundamental observer> has $U^\mu=(1,0,0,0)$ in cosmic-time coordinates, so the measured energy is $E=P_\mu U^\mu=c^2P^t$. The null normalization and conserved radial quantity in part (b) give
$$
P^t=\frac{|C|}{ac},\qquad E=\frac{c|C|}{a}.
$$
Thus <redshifting of peculiar momentum> gives $E\propto a^{-1}$. Since $E=h\nu$, with $h$ the <Planck constant>, the received-to-emitted <frequency> ratio is $\nu_R/\nu_E=a(t_E)/a(t_R)$. By the definition of <cosmological redshift>,
$$
\boxed{1+z=\frac{\nu_E}{\nu_R}=\frac{a(t_R)}{a(t_E)}.}
$$
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