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Along the zeroth-order photon path, set $Y=\Theta +\psi$ and $\mathcal{S}=(3/4)\int d\hat{\mathbf{m}}\Theta (\hat{\mathbf{m}})[1+(\mathbf{e}\cdot \hat{\mathbf{m}}{)}^2]/(4\pi )$. The scalar photon energy redshift and the Photon Boltzmann equation with Thomson scattering giveThe integrating factor is $e^{-\tau }$, because ${\tau }^{\prime }<0$ in this convention. HenceTight coupling before decoupling makes the incident distribution nearly isotropic in the baryon frame. Neglecting its quadrupole, the angular kernel integrates to $\mathcal{S}={\Theta }_0$, since $(3/4)⟨1+(\mathbf{e}\cdot \hat{\mathbf{m}}{)}^2⟩=1$. The primordial term at the optically thick lower endpoint is erased. With instantaneous last scattering, no later scattering and $g=\delta (\eta -{\eta }_{\ast })$, the line-of-sight solution isThe 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.

The photon monopole ${\Theta }_0$ is the intrinsic emission-temperature perturbation. The emission potential ${\psi }_{\ast }$ 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.

To first order, propagate along the unperturbed ray $\mathbf{x}(\eta )={\mathbf{x}}_0-({\eta }_0-\eta )\mathbf{e}$ and keep its orthonormal tetrad direction fixed. Corrections to direction or trajectory multiply an already first-order temperature source and are second order. Define the angular scattering sourceUsing the photon energy redshift from a tensor metric perturbation, the Photon Boltzmann equation with Thomson scattering along this ray becomesMultiply by the survival probability $e^{-\tau }$. The cosmological visibility function is $g(\eta )=-\dot{\tau }{e}^{-\tau }$, so integration givesEvery source here is evaluated on the same ray, and $\tau ({\eta }_0)=0$.

The instantaneous-visibility approximation makes $e^{-\tau }$ a step from zero to one at ${\eta }_{\ast }$. It suppresses the early boundary term and localizes the scattering source to $S_{\ast }$. To obtain only the requested tensor CMB line-of-sight source, additionally neglect this last-scattering angular source: in leading tight coupling, the tensor-induced incident quadrupole is small, and a tensor mode has no scalar monopole or dipole. Ignore later reionization and its scattering as well. ThenInstantaneous visibility alone would leave the term $S_{\ast }$; neglect of that term is an additional approximation. The time derivative on $h$ is essential and is present in the PDF, although it was lost in the extracted TeX.

Let $\Gamma =-\dot{\tau }=a{\bar{n}}_e{\sigma }_T$ and let $D={\partial }_{\eta }+\mathbf{e}\cdot \nabla$ differentiate along the unperturbed photon trajectory. Use $1+(\mathbf{e}\cdot \hat{\mathbf{m}}{)}^2=4/3+(2/3)P_2(\mathbf{e}\cdot \hat{\mathbf{m}})$ to decompose the Thomson scattering gain term:The Orthogonality of Legendre polynomials identifies $Q$ as a pure photon quadrupole. Dropping $Q$ is an approximation, justified when tight coupling makes the scattering quadrupole small; the supplied unpolarized equation already neglects polarization corrections. The Cosmic microwave background line-of-sight solution requested here omits that quadrupole source.

Set $Y=\Theta +\psi$. Adding $D\psi =\dot{\psi }+\mathbf{e}\cdot \nabla \psi$ to the Photon Boltzmann equation with Thomson scattering yieldsAlong $\mathbf{x}(\eta )={\mathbf{x}}_0-({\eta }_0-\eta )\mathbf{e}$, the integrating factor is $e^{-\tau }$ because $d{e}^{-\tau }/d\eta =\Gamma e^{-\tau }$. ConsequentlyAn early optically thick boundary makes $e^{-\tau }Y$ vanish for finite initial perturbations, and $e^{-\tau ({\eta }_0)}=1$. ThereforeHere $\dot{\varphi }$ and $\dot{\psi }$ are partial time derivatives, not total derivatives along the ray. We work at first order in scalar cosmological perturbations, evaluate sources along the background radial null geodesic in FLRW spacetime, and ignore observer peculiar velocity. Perturbed paths and lensing affect this first-order source solution only at higher perturbative order. A moving observer adds its local Doppler dipole.

The original PDF, unlike the transcribed TeX, also asks for the instantaneous-last-scattering limit and its interpretation. For $g(\eta )={\delta }^{(D)}(\eta -{\eta }_{\ast })$ with no later scattering, and a matter-era growing mode with constant $\varphi$ and $\psi$, the Integrated Sachs-Wolfe effect vanishes. HenceThe four terms are the intrinsic emission temperature, the gravitational potential at emission, the observer's potential, and the Doppler CMB anisotropy. The two potentials describe the gravitational redshift; the observer term is independent of direction and is absorbed into the measured temperature monopole. The intrinsic temperature and emission potential form the Sachs-Wolfe combination. On superhorizon scales for adiabatic initial conditions in matter domination, ${\Theta }_0=-2\psi /3$, giving the ordinary Sachs-Wolfe effect ${\Theta }_0+\psi =\psi /3$. The Doppler sign is positive with the paper's propagation direction $\mathbf{e}$; the viewing direction is $-\mathbf{e}$.

Let $\hat{\mathbf{n}}$ point from the observer toward the emission point, so the photon propagation direction is $\hat{\mathbf{p}}=-\hat{\mathbf{n}}$ andAlong this path $D=d/d\eta ={\partial }_{\eta }-\hat{\mathbf{n}}\cdot \nabla$. Substituting $\hat{\mathbf{p}}=-\hat{\mathbf{n}}$ into the Photon Boltzmann equation with Thomson scattering givesBecause ${\partial }_{\eta }{e}^{-\tau }=\Gamma e^{-\tau }$, multiplication by the integrating factor giveswhereIntegrating along the ray and using $e^{-\tau }\to 0$ at sufficiently early time yields the Cosmic microwave background line-of-sight solution