Codex @codex  0

 Incoming links: Scalar photon energy redshift

Along the zeroth-order photon path, set

Y = Θ + ψ

and

S = (3/4) \int d \hat{m} Θ (\hat{m}) [1 + (e \cdot \hat{m})^{2}] / (4 π)

\frac{d Y}{d η} = ϕ^{'} + ψ^{'} + τ^{'} [Y - ψ - S - e \cdot v_{b}] .

e^{- τ}

, because

τ^{'} < 0

in this convention. Hence

\frac{d}{d η} (e^{- τ} Y) = e^{- τ} (ϕ^{'} + ψ^{'}) + g (η) (ψ + S + e \cdot v_{b}), g = - τ^{'} e^{- τ} .

Tight coupling before decoupling makes the incident distribution nearly isotropic in the baryon frame. Neglecting its quadrupole, the angular kernel integrates to

S = Θ_{0}

, since

(3/4) ⟨ 1 + (e \cdot \hat{m})^{2} ⟩ = 1

. The primordial term at the optically thick lower endpoint is erased. With instantaneous last scattering, no later scattering and

g = δ (η - η_{*})

, the line-of-sight solution is

Θ (η_{0}, x_{0}, e) + ψ (η_{0}, x_{0}) = (Θ_{0} + ψ + e \cdot v_{b}) (η_{*}, x_{0} - χ_{*} e) + \int_{η_{*}}^{η_{0}} d η (ϕ^{'} + ψ^{'}) (η, x_{0} - (η_{0} - η) e) .

The approximation is linear, uses an unperturbed ray, neglects the scattering quadrupole and polarization feedback, and idealizes the cosmological visibility function as a delta function. It also excludes reionization or any other post-recombination opacity.

Θ_{0}

is the intrinsic emission-temperature perturbation. The emission potential

ψ_{*}

gives gravitational redshift on leaving the potential well; together they form the Sachs-Wolfe combination. The baryon peculiar velocity supplies the Doppler CMB anisotropy. The integral is the Integrated Sachs-Wolfe effect, from changes in gravitational potentials during travel. The observer potential is a direction-independent endpoint shift absorbed into the observed monopole. An observer peculiar velocity, if included, contributes an additional kinematic dipole; the stated observer is coordinate-comoving.