Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 64 / 2 / i

Use the photon phase-space coordinates $(\mathbf{x},q,\mathbf{n})$, with $q=ap$ and $\mathbf{n}$ its propagation direction. Divide the Collisionless Boltzmann equation by $d\tau /d\lambda$ and apply the chain rule. The homogeneous blackbody distribution is time independent at fixed comoving momentum because its physical temperature scales as $a^{-1}$. ThereforeOn the unperturbed ray $d{x}^i/d\tau =n^i$; its correction is first order and multiplies the first-order spatial gradient of $f_1$. Similarly $q^{\prime }$ and $(n^i{)}^{\prime }$ are first order, so their products with derivatives of $f_1$ are second order. The zeroth-order distribution has no directional dependence. Keeping only linear terms and using $q^{\prime }=-q{h}_{ij}^{\prime }{n}^i{n}^j/2$ givesAfter Fourier transforming with $e^{i\mathbf{k}\cdot \mathbf{x}}$ and $\mu =\hat{\mathbf{k}}\cdot \mathbf{n}$, the Free-streaming photon Boltzmann equation in synchronous variables isDirection deflection is a real first-order geodesic effect, but its effect on the already perturbed distribution starts at second order here.

Let $I_0={\int }_0^{\infty }{q}^3{f}_0(q)dq$, so in the phase-space normalization being used $a^4{\bar{\rho }}_{\gamma }=4\pi I_0$. The photon brightness perturbation is $\Delta =I_0^{-1}\int q^3{f}_{1}dq$. This agrees with the stated density normalization, and $I_0$ is independent of conformal time. Integrating the distribution equation givesFor a Planck distribution, the endpoint term $[q^4{f}_0{]}_0^{\infty }$ vanishes. Integration by parts yields $\int q^4{f}_0^{\prime }dq=-4I_0$. Thus the synchronous photon brightness equation isFor a small directional blackbody temperature shift $\Theta =\Delta T/T$, expansion gives $f_1=-q{f}_0^{\prime }(q)\Theta$. The same integral then gives $\Delta =4\Theta$, also following from blackbody energy density proportional to $T^4$. This temperature interpretation presumes a perturbed blackbody shape; frequency-integrated brightness can still be defined for more general distributions.