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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 $E=P_{\mu }{U}^{\mu }=c^2{P}^t$. The null normalization and conserved radial quantity in part (b) giveThus 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,

A nearby fundamental observer measures radial proper distance $d\ell =ad\chi$ and proper time $dt$. Thus the signed radial peculiar velocity isSubstitute $a^2\dot{\chi }=C$ and the massive normalization from part (b):Equivalently, for velocity $v_0$ at scale factor $a_0$,This is the exact law of redshifting of peculiar momentum. In the nonrelativistic limit, $v\propto a^{-1}$; an initially relativistic speed stays below $c$ and decreases as the universe expands.