 Solution

Let $p^0=d\tau /d\lambda$. A comoving observer measures photon energy $p=a{p}^0$, so $q=ap=a^2{p}^0$. The null condition is $({\delta }_{ij}+h_{ij})p^i{p}^j=(p^0{)}^2$. Using the stated Christoffel symbols in the time geodesic equation givesDivide by $p^0$ to differentiate with respect to conformal time. The background redshift term cancels when differentiating $a^2{p}^0$:In the perturbation term use the background relation $p^i/p^0={\hat{n}}^i$, givingWithout metric tensor perturbations, both $q$ and the propagation direction are constant along an unperturbed ray. Thus $q^{\prime }$ and $d{\hat{n}}^i/d\tau$ are first order. Multiplying them by derivatives of $f_1$, which is also first order, gives second-order terms. The term $q^{\prime }d{f}_0/dq$ is first order and must be retained. At linear order the Collisionless Boltzmann equation therefore readsTaking the Fourier transform in the spatial coordinates provesA first-order correction to the propagation velocity multiplying a spatial derivative of $f_1$ is likewise second order.