Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 55 / 2 / i

The comoving observer's four-velocity has $u_0=-a$, so its measured photon energy is $E=-u_{\mu }{p}^{\mu }=a{p}^0$, and the comoving momentum is $q=aE=a^2{p}^0$. Let $P^i=p^i/p^0=d{x}^i/d\tau$. The null geodesic condition isDivide the temporal geodesic equation by $p^0=d\tau /d\lambda$, and use the supplied Christoffel symbols:Differentiating $q=a^2{p}^0$ cancels the homogeneous redshift terms. In the remaining first-order term, replace $P^i$ by the unperturbed unit propagation direction $n^i$:This is the synchronous photon momentum redshift. Background physical momentum decreases as $a^{-1}$, while the background comoving momentum stays constant.

The spatial geodesic equation gives $d{p}^i/d\tau =-2\mathcal{H}{p}^i$ in the unperturbed universe. Since the temporal equation gives the identical factor for $p^0$, their ratio $P^i$ is constant at zeroth order. More explicitly, at first order,Every displayed term contains a metric perturbation or its derivative. The orthonormal-frame unit direction differs from $P^i$ by a first-order metric correction, $n^i=P^i+h^i{}_j{P}^j/2+O(h^2)$, whose derivative is also first order. Hence $d{n}^i/d\tau =O(h)$, with no background change of direction. Direction corrections multiplied by an already first-order source contribute only at second order.