 Solution

Express the Collisionless Boltzmann equation in variables $(\mathbf{x},q,\mathbf{n},\tau )$ and divide the derivative along the photon ray by $d\tau /d\lambda$. Since $f_0(q)$ has no explicit position, direction or conformal-time dependence,This is the chain rule in momentum magnitude and direction. At linear order, $P^i=n^i+O(h)$, while the last two terms are products of first-order perturbations and can be dropped. The synchronous photon momentum redshift from part i then gives, for spatial Fourier convention $e^{i\mathbf{k}\cdot \mathbf{x}}$,The factor $q$ is essential. It is missing in the PDF's displayed intermediate equation, but follows directly from part i and is needed to obtain its final brightness equation.

Let $I_0={\int }_0^{\infty }{q}^3{f}_0(q)dq$. Absorbing the common phase-space normalization into $f$, the homogeneous photon energy density is ${\rho }_{\gamma }=4\pi I_0/a^4$, so the photon brightness perturbation isA small directional temperature change $\Theta =\Delta T/T$ in a Planck distribution gives $f_1=-q{f}_0^{\prime }(q)\Theta$. Integration by parts yieldsbecause the Planck distribution behaves as $q^{-1}$ at low momentum and decays exponentially at high momentum. Thus $\Delta =4\Theta$. Equivalently, frequency-integrated blackbody radiation energy scales as $T^4$. Identifying brightness with a single temperature perturbation assumes this blackbody spectral form; the brightness integral itself is defined more generally.

Finally, multiply the corrected transport equation by $q^3$, integrate and divide by $I_0$. The spatial streaming factor is independent of $q$, while its gravitational source integrates to $(1/2)(-4)h_{ij}^{\prime }{n}^i{n}^j$. ThereforeThis derives the synchronous photon brightness equation with the normalization and sign fixed, rather than inferring it from the dimensionally inconsistent intermediate formula.