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In the negative-spatial-perturbation convention, a matter-era scalar solution $h=h_s+C=A{k}^{1/2}{\tau }^2/4$ gives the displayed exact photon brightness perturbation. Integration gives emission bracket ${\Delta }_{\ast }+iA{k}^{-1/2}\mu {\tau }_{\ast }-A{k}^{-3/2}$ times the propagation phase, plus observer monopole $A{k}^{-3/2}$ and dipole $-iA{k}^{-1/2}\mu {\tau }_0$. Under ordinary adiabatic growing-mode initial conditions and $k{\tau }_{\ast }\ll 1$, the emission gravitational term dominates higher angular multipoles. Arbitrary freely specified initial brightness need not obey this dominance.

To match the positive source printed in the question, choose synchronous gauge metricThe commonly used convention with ${\delta }_{ij}+h_{ij}$ gives the opposite source sign; switching conventions means replacing $h$ and $h_S$ by their negatives together. This distinction is necessary when comparing with the usual synchronous photon brightness equation.

Let $p^{\alpha }=d{x}^{\alpha }/d\lambda$ be the photon tangent for an affine parameter. The covariant geodesic equation isFor $\nu =0$, the homogeneous scale-factor contributions cancel by the null geodesic condition, leaving $d{\tilde{p}}_0/d\lambda =-a^2{h}_{ij}^{\prime }{p}^i{p}^j/2$. Since ${\tilde{p}}_0=-a^2{p}^0=-q$ and $d\tau /d\lambda =p^0$, use the background $p^i=p^0{n}^i$ in this first-order term to obtainUsing the affine form also avoids confusing a covariant $p_0$ with the contravariant factor $p^0$ required when changing the geodesic parameter to coordinate time.

In conformal time, the equation from part(a) becomes $f_1^{\prime }+n^i{\partial }_i{f}_1+q^{\prime }{f}_0^{\prime }=0$. A Fourier mode proportional to $e^{i\mathbf{k}\cdot \mathbf{x}}$ has streaming term $ik\mu f_1$. Multiply by $-4/(q{f}_0^{\prime })$; its prefactor is background-time independent. With $\Delta =-4f_1/(q{f}_0^{\prime })$,Contracting the scalar decomposition explicitly givessoThis is the photon brightness equation for a negative spatial perturbation. For a perturbation of blackbody-temperature form, $f_1=-q{f}_0^{\prime }\Theta$, so $\Delta =4\Theta$ is the photon brightness perturbation. A more general initial spectral distortion can retain $q$ dependence; the gravitational forcing is still independent of $q$ in this linear massless equation.

Write $T=\overline{T}(1+\Theta )$, with $\Theta =\Delta T/\overline{T}$ at linear order. Since $\overline{T}\propto a^{-1}$, the background Planck function depends only on comoving momentum. Expanding its temperature-shifted argument givesUsing $T$ rather than $\overline{T}$ in the perturbation denominator changes only higher-order terms.

The photon brightness perturbation is the energy-weighted angular perturbationIntegration by parts gives $\int q^4{f}_0^{\prime }dq=-4\int q^3{f}_{0}dq$; the boundary term vanishes at both endpoints for a Planck spectrum. Therefore $\Delta =4\Theta$. Multiplying the transport equation by $q^3$ and integrating yields the synchronous photon brightness equationThe PDF's definition is $4\Delta T/T$; the TeX's denominator $\hslash$ is a transcription error.

Expand $\Delta$ in Legendre moments in $\mu$. The monopole is the density/temperature perturbation, and the dipole is the bulk velocity. Before decoupling, rapid Thomson scattering isotropizes photons in the electron rest frame. In the tight-coupling approximation, the scattering rate is much larger than $k$ and $\mathcal{H}$, so the quadrupole and higher moments are suppressed by the mean-free-path parameter, with the quadrupole typically of order $k/\dot{\kappa }$ times the dipole. To leading order only monopole and dipole are needed. This suppression uses the collision term restored before decoupling; it does not follow from the collisionless equation by itself. Finite quadrupole corrections generate polarization and diffusion damping.

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.

Before decoupling, rapid Thomson scattering makes the photon distribution nearly isotropic in the local photon-fluid rest frame. A local blackbody temperature perturbation changes its energy density by ${\delta }_{\gamma }=4\delta T/T$. A first-order boost by the fluid velocity adds a temperature photon dipole $\mathbf{n}\cdot \mathbf{v}$; it does not generate a photon quadrupole until second order in that boost. Thus the photon brightness perturbation at instantaneous decoupling isHigher angular moments are suppressed by the photon mean-free-path to perturbation-length ratio in the tight-coupling approximation. The idealized equilibrium assumption neglects those finite-mean-free-path corrections, polarization effects and finite thickness of the last-scattering surface. It is not an assertion that collisionless photons remain isotropic after decoupling. The distinction between a fluid treatment and the subsequent hierarchy is discussed in the primary synchronous and Newtonian perturbation treatment.

Let $\mu =\hat{\mathbf{k}}\cdot \mathbf{n}$ and $W(\tau )=e^{ik\mu (\tau -{\tau }_0)}$. Multiplication of the synchronous photon brightness equation by its integrating factor givesIntegration from ${\tau }_{\ast }={\tau }_{\mathrm{dec}}$ to ${\tau }_0$ and division by four yieldFor the real-space observation point ${\mathbf{x}}_0$, the corresponding background ray is $\mathbf{x}(\tau )={\mathbf{x}}_0-\mathbf{n}({\tau }_0-\tau )$. Fourier inversion gives the synchronous Sachs-Wolfe line-of-sight formulaEvery source is evaluated at its appropriate point on the ray, rather than at the observation position. Here $\mathbf{n}$ is photon propagation direction, opposite to the outward observer-to-source sky direction; changing that convention changes the written sign of the Doppler projection consistently.

The first term is the intrinsic photon temperature at last scattering. On wavelengths larger than the sound horizon, it retains primordial adiabatic information; inside that horizon the photon-baryon fluid has acoustic energy density oscillations, producing the acoustic temperature peaks. The second term is the Doppler effect from the fluid's bulk motion at emission. A regular long-wavelength velocity is gradient-suppressed; the Doppler contribution is important near and below the acoustic horizon, with its acoustic phase displaced from that of the energy density oscillation.

The integral is the gravitational energy shift produced by the changing spatial metric tensor along the photon path. In synchronous gauge it includes endpoint gravitational redshifts as well as evolving-potential effects. It must not be identified solely with the Integrated Sachs-Wolfe effect: in Newtonian gauge the same total separates into the ordinary last-scattering Sachs-Wolfe effect, local observer terms, and an integral of time-varying potentials. The ordinary contribution is important on scales outside the sound horizon at decoupling. A late integrated contribution is strongest on large angular scales when dark energy or curvature changes the potentials; evolution around matter-radiation equality also contributes near the first acoustic scales.

The relevant comoving sound scale is $r_s({\tau }_{\ast })={\int }^{{\tau }_{\ast }}{c}_s(\tau )d\tau$, while the projection distance is $D_{\ast }\simeq {\tau }_0-{\tau }_{\ast }$. Modes project roughly to photon temperature multipole $\ell \sim k{D}_{\ast }$, and the acoustic scale is $\ell \sim D_{\ast }/r_s$, with successive peak spacing of order $\pi D_{\ast }/r_s$. At much shorter wavelengths Silk damping and the finite last-scattering thickness suppress energy density and velocity anisotropies; the instantaneous-decoupling approximation alone does not model that cutoff. These scale distinctions explain why the three terms cannot be assigned one common range of important wavelengths.

In the tight-coupling approximation, repeated Thomson scattering drives the photon distribution toward isotropy in the local baryon rest frame. At linear order a temperature perturbation gives a monopole and a small boost gives only a dipole. A photon propagating along $\mathbf{n}$ has rest-frame energy $p(1-\mathbf{n}\cdot \mathbf{v})$, soSince the rest-frame density contrast is ${\delta }_{\gamma }=4{\Theta }_{\mathrm{rest}}$, the initial photon brightness perturbation isThe quadrupole and higher photon temperature multipoles are small when the conformal collision rate $\kappa =a{n}_e{\sigma }_T$ greatly exceeds both $k$ and $\mathcal{H}$. For example the quadrupole is suppressed by a factor of order $k/\kappa$ relative to the dipole. Instantaneous decoupling idealizes this strongly coupled state as the initial data for collisionless propagation; finite mean free path and polarization corrections are omitted from that initial truncation.

Multiply the synchronous photon brightness equation by $e^{ik\mu \tau }$ and integrate from ${\tau }_{\ast }$ to ${\tau }_0$. The line-of-sight solution for free-streaming photons isThe phase is essential: it specifies the emission point and the points traversed by the ray. In real space define $\mathbf{x}(\tau )={\mathbf{x}}_0-\mathbf{n}({\tau }_0-\tau )$ and ${\mathbf{x}}_{\ast }=\mathbf{x}({\tau }_{\ast })$. Dividing by four gives the synchronous Sachs-Wolfe line-of-sight formula,The spatial arguments suppressed in a shorthand version must be understood in this retarded sense; evaluating the emission terms at ${\mathbf{x}}_0$ instead would discard free streaming. Here $\mathbf{n}$ is propagation direction. The observer-to-source direction on the sky is $-\mathbf{n}$, explaining the opposite Doppler sign often used with that convention. The formula describes the synchronous-frame observer; an additional observer peculiar velocity contributes a dipole.

The first term is the intrinsic temperature perturbation at emission, since ${\delta }_{\gamma }=4\Theta$. The second is the Doppler CMB anisotropy from the plasma bulk velocity. The third is gravitational frequency shifting by the time-dependent spatial metric along the ray, and can include scalar and tensor metric contributions. In scalar Newtonian gauge, rearranging that metric contribution supplies the emission gravitational potential as well as the Integrated Sachs-Wolfe effect, up to the observer terms. It should not be identified solely with the integrated Sachs-Wolfe term: a constant Newtonian potential need not give a vanishing synchronous metric integral.

On large angles, modes are outside or comparable to the sound horizon at emission. For the growing adiabatic mode during matter domination, the intrinsic and emission-potential contributions give the ordinary Sachs-Wolfe effect, ${\Theta }_0+\Psi =\Psi /3$ in Newtonian variables. Velocity is a gradient effect, of order $k{\tau }_{\ast }\Psi$ on superhorizon scales, so its Doppler contribution is suppressed there. A scale-invariant primordial curvature spectrum produces approximately the familiar large-angle plateau in $\ell (\ell +1)C_{\ell }$. Evolving potentials during radiation-to-matter transition or late acceleration can add integrated gravitational anisotropy; constant matter-era scalar potentials do not generate a genuine scalar integrated Sachs-Wolfe source.

On smaller angles, the photon-baryon fluid has undergone acoustic oscillations. Its density and velocity are approximately in cosine and sine phases, producing acoustic structure in the intrinsic and Doppler sources and their projection onto angular multipoles. Silk damping suppresses sufficiently short wavelengths before decoupling, while the finite width of last scattering also smooths the real small-angle signal. These corrections refine the instantaneous-decoupling model rather than follow from a collisionless equation alone. The separate intrinsic, velocity and metric pieces depend on gauge convention; the complete observed anisotropy, including the appropriate observer terms and removal of the unobservable mean, is the physical quantity.

Use the photon phase-space coordinates $(\mathbf{x},q,\mathbf{n})$, with $q=ap$ and $\mathbf{n}$ its propagation direction. Divide the Collisionless Boltzmann equation by $d\tau /d\lambda$ and apply the chain rule. The homogeneous blackbody distribution is time independent at fixed comoving momentum because its physical temperature scales as $a^{-1}$. ThereforeOn the unperturbed ray $d{x}^i/d\tau =n^i$; its correction is first order and multiplies the first-order spatial gradient of $f_1$. Similarly $q^{\prime }$ and $(n^i{)}^{\prime }$ are first order, so their products with derivatives of $f_1$ are second order. The zeroth-order distribution has no directional dependence. Keeping only linear terms and using $q^{\prime }=-q{h}_{ij}^{\prime }{n}^i{n}^j/2$ givesAfter Fourier transforming with $e^{i\mathbf{k}\cdot \mathbf{x}}$ and $\mu =\hat{\mathbf{k}}\cdot \mathbf{n}$, the Free-streaming photon Boltzmann equation in synchronous variables isDirection deflection is a real first-order geodesic effect, but its effect on the already perturbed distribution starts at second order here.

Let $I_0={\int }_0^{\infty }{q}^3{f}_0(q)dq$, so in the phase-space normalization being used $a^4{\bar{\rho }}_{\gamma }=4\pi I_0$. The photon brightness perturbation is $\Delta =I_0^{-1}\int q^3{f}_{1}dq$. This agrees with the stated density normalization, and $I_0$ is independent of conformal time. Integrating the distribution equation givesFor a Planck distribution, the endpoint term $[q^4{f}_0{]}_0^{\infty }$ vanishes. Integration by parts yields $\int q^4{f}_0^{\prime }dq=-4I_0$. Thus the synchronous photon brightness equation isFor a small directional blackbody temperature shift $\Theta =\Delta T/T$, expansion gives $f_1=-q{f}_0^{\prime }(q)\Theta$. The same integral then gives $\Delta =4\Theta$, also following from blackbody energy density proportional to $T^4$. This temperature interpretation presumes a perturbed blackbody shape; frequency-integrated brightness can still be defined for more general distributions.