(x, q, n, τ)

and divide the derivative along the photon ray by

d τ / d λ

. Since

f_{0} (q)

has no explicit position, direction or conformal-time dependence,

0 = \frac{\partial f _{1}}{\partial τ} + P^{i} \frac{\partial f _{1}}{\partial x ^{i}} + \frac{d q}{d τ} \frac{d f _{0}}{d q} + \frac{d q}{d τ} \frac{\partial f _{1}}{\partial q} + \frac{d n ^{i}}{d τ} \frac{\partial f _{1}}{\partial n ^{i}} .

(1)

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 k \cdot x}

f_{1}^{'} + ik μ f_{1} = \frac{1}{2} q \frac{d f _{0}}{d q} h_{ij}^{'} n^{i} n^{j}, μ = k \cdot n .

(2)

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) d q

. Absorbing the common phase-space normalization into

f

, the homogeneous photon energy density is

ρ_{γ} = 4 π I_{0} / a^{4}

, so the photon brightness perturbation is

Δ = \frac{\int _{0}^{\infty} q ^{3} f _{1} d q}{I _{0}} .

(3)

A small directional temperature change

Θ = Δ T / T

in a Planck distribution gives

f_{1} = - q f_{0}^{'} (q) Θ

. Integration by parts yields

\int_{0}^{\infty} q^{4} f_{0}^{'} (q) d q = [q^{4} f_{0} (q)]_{0}^{\infty} - 4 I_{0} = - 4 I_{0},

(4)

because the Planck distribution behaves as

q^{- 1}

at low momentum and decays exponentially at high momentum. Thus

Δ = 4Θ

. 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}^{'} n^{i} n^{j}

. Therefore

Δ^{'} + ik μ Δ = - 2 h_{ij}^{'} n^{i} n^{j} .

(5)

This derives the synchronous photon brightness equation with the normalization and sign fixed, rather than inferring it from the dimensionally inconsistent intermediate formula.