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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.

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.

The comoving observer measures $E=a{p}^0$, so $q=aE=a^2{p}^0$. Write $\mathcal{H}=a^{\prime }/a$. The null geodesic condition impliesThe time component of the geodesic equation, using $d\tau /d\lambda =p^0$, is thereforeDifferentiate $q=a^2{p}^0$. The homogeneous expansion terms cancel:At zeroth order $p^i/p^0=n^i$, with ${\delta }_{ij}{n}^i{n}^j=1$. The difference between coordinate velocity and physical unit direction is already first order and multiplies $h^{\prime }$, so it can be discarded in this expression. We obtain the synchronous photon momentum redshiftIn the unperturbed universe $q$ is constant and physical photon energy redshifts as $a^{-1}$; the displayed term is the additional anisotropic first-order redshift.

To check the direction, let $v^i=p^i/p^0$. Combining the spatial and temporal geodesic equations cancels their homogeneous Hubble terms and givesEvery remaining connection or explicit metric-derivative term is first order. A local orthonormal spatial frame defines the physical unit vector by $n^i=v^i+\frac{1}{2}{h}^i{}_j{v}^j+O(h^2)$. Its derivative adds only first-order terms, soConsequently direction deflection multiplied by an already first-order temperature anisotropy is second order. The first-order brightness calculation may use a straight unperturbed ray without neglecting the first-order gravitational frequency change.