Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 312 / 4 / ii / Solution

The Collisionless Boltzmann equation is $df/d\eta =0$. At first order, directional deflections act only on the isotropic background and hence give no angular contribution. Differentiating $f=\bar{f}-\Theta d\bar{f}/d\ln ϵ$ along the unperturbed trajectory givesUsing the tensor redshift of the comoving energy,The method of characteristics follows the backward ray $\mathbf{x}({\eta }^{\prime })=\mathbf{x}-(\eta -{\eta }^{\prime })\mathbf{e}$. With zero initial perturbation, the line-of-sight solution isTo this order the frame and coordinate direction components may be identified inside a quantity already proportional to $h$.