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Consider a logarithmic band of wavenumbers around $k\sim R^{-1}$, where $R$ is a comoving scale. The dimensionless cosmological power spectrum of the density contrast is ${\Delta }_{\delta }^2(k)=k^3{P}_{\delta }(k)/(2{\pi }^2)$. For $P_{\delta }=Ak$ this scales as $k^4$, so the rms density amplitude in such a band scales as $R^{-2}$.

The cosmological Poisson equation gives $-k^2{\Phi }_{\mathbf{k}}=4\pi G{a}^2{\bar{\rho }}_m{\delta }_{\mathbf{k}}$, with $\Phi$ a dimensionful peculiar gravitational potential. Its spectrum and its potential fluctuations per logarithmic wavenumber are thereforeindependent of $k$. Equivalently, ${\Phi }_R\sim G{\bar{\rho }}_m(aR{)}^2{\delta }_R$ is independent of $R$ because ${\delta }_R\propto R^{-2}$. This is the scale-local meaning of the Harrison-Zeldovich spectrum. An uncut spectrum has a divergent total potential variance; equal amplitude per logarithmic band does not imply a finite unsmoothed variance over all scales. For wavelengths beyond the Hubble radius, the statement is formulated using the corresponding conserved primordial curvature mode and its adiabatic matter-era potential, rather than treating the subhorizon Poisson approximation as universally valid.

On angular scales larger than roughly a degree, the Sachs-Wolfe effect relates the last-scattering potential to the observed Cosmic microwave background anisotropy: $\Delta T/T\simeq \Phi /(3c^2)$ for adiabatic modes in matter domination. Thus the observed large-angle amplitude constrains $A$. Scale-invariant potential bands produce a Sachs-Wolfe plateau, $\ell (\ell +1)C_{\ell }$ approximately constant, rather than constant $C_{\ell }$. Its amplitude and any tilt test the normalization and spectral shape. Cosmic variance, Integrated Sachs-Wolfe effect and assumptions about adiabatic initial conditions limit this inference; the density spectrum's late-time transfer must also be included when comparing it with galaxy clustering.

There is a sign error in the printed synchronous line-of-sight integrand. Direct contraction of the stated decomposition gives $h_{ij}^{\prime }{n}^i{n}^j=(h^{\prime }-h_S^{\prime })/3+(\hat{\mathbf{k}}\cdot \mathbf{n}{)}^2{h}_S^{\prime }$, with a plus sign in its last term. In the positive spatial-perturbation convention of part a, the redshift term is half this contraction. The printed minus sign cannot yield the requested Newtonian expression with the supplied transformations.

The consistent identity can be derived explicitly. Work mode by mode with $q=\mathbf{k}\cdot \mathbf{n}$, $E(\tau )=e^{iq(\tau -{\tau }_0)}$ along the unperturbed ray; the shift of the phase origin is immaterial. Put $A=(h-h_S)/6$. Part a gives $A=\Psi -\mathcal{H}T$, while $\Phi =-\mathcal{H}T-T^{\prime }$. Therefore $A^{\prime }={\Phi }^{\prime }+{\Psi }^{\prime }+T^{\prime \prime }$. The corrected synchronous integral issince $[E(T^{\prime }-iqT){]}^{\prime }=E(T^{\prime \prime }+q^{2}T)$. This is the photon redshift identity with positive spatial perturbations.

The supplied fluid shifts are ${\delta }_{\gamma }^N={\delta }_{\gamma }+4\mathcal{H}T$ and ${\mathbf{v}}_{\gamma }^N={\mathbf{v}}_{\gamma }+i\mathbf{k}T$. At emission, the synchronous intrinsic and Doppler terms are consequently $E_d[{\delta }_{\gamma }^N/4+{\mathbf{v}}_{\gamma }^N\cdot \mathbf{n}-\mathcal{H}T-iqT{]}_d$. Adding the lower integral endpoint cancels the velocity shift and leaves $-\mathcal{H}T-T^{\prime }=\Phi$. The upper endpoint is the local observer monopole/dipole $[T^{\prime }-iqT{]}_0$, removed by temperature calibration and the observer-rest-frame convention. With zero anisotropic stress, $\Phi =\Psi$, the observable higher-multipole result isAll fields are evaluated along the ray; the integral contains the time derivative at fixed spatial position. Retaining the printed negative angular term would leave the additional $-2q^2\int ETd\tau$, generally a gauge-dependent, nonzero term. It is therefore not repaired merely by discarding observer monopole and dipole.

The first term is the intrinsic photon temperature perturbation, important in acoustic compressions and rarefactions. The velocity term is the Doppler shift and is prominent near acoustic, roughly degree and subdegree, scales; it is suppressed on very large scales. The potential at emission is the ordinary Sachs-Wolfe effect; combined with the intrinsic term it gives the familiar large-angle adiabatic matter-era result $\Phi /3$. The final term is the Integrated Sachs-Wolfe effect, caused by changing potentials. It vanishes for constant matter-era potentials, receives an early contribution near radiation-matter transition, and has an important late large-angle contribution when curvature or dark energy causes potentials to evolve. The sign of the Doppler term follows the photon-propagation direction convention used in the ray; reversing that direction reverses it consistently.

Use Newtonian gauge with scalar metric perturbationsLet $\hat{n}$ point from the observer toward the emitter, $\Theta =\delta T/T$, and evaluate the unperturbed ray at $\mathbf{x}(\tau )=({\tau }_0-\tau )\hat{n}$. In the sudden-decoupling approximation the Sachs-Wolfe effect and its linear companions giveHere ${\Theta }_0={\delta }_{\gamma }/4$ is the intrinsic photon monopole; ${\Psi }_{\ast }$ is the emission gravitational redshift; the velocity term is the Doppler CMB anisotropy of the baryon-photon plasma; and the integral is the Integrated Sachs-Wolfe effect from evolving potentials during travel. The last two local terms are an observer monopole and dipole, removed when discussing the higher measured multipoles. Changing the direction convention reverses both velocity signs. A finite cosmological visibility function replaces the sharp emission surface by a weighted source integral.

For scale-invariant primordial potentials, ${\mathcal{P}}_{\Psi }=k^3{P}_{\Psi }/(2{\pi }^2)$ is constant before transfer effects. In matter domination, the adiabatic superhorizon relation is ${\Theta }_0=-2\Psi /3$, so the observable Sachs-Wolfe combination is ${\Theta }_0+\Psi =\Psi /3$. Its amplitude per logarithmic scale is therefore approximately independent of angle. In a flat geometry $\ell \sim k{D}_{\ast }$ and $\theta \sim {\ell }^{-1}$, and projection gives the Sachs-Wolfe plateauBoth the intrinsic and gravitational emission terms have this leading large-angle scaling; they must be combined, not treated as unrelated random sources.

The Euler equation gives $v_b\sim k{\tau }_{\ast }\Psi$ for the growing large-scale matter mode. Thus the Doppler amplitude grows roughly as ${\theta }^{-1}$ relative to the large-scale potential amplitude, while its logarithmic power grows as ${\theta }^{-2}$, until acoustic dynamics and damping invalidate that simple estimate. On subhorizon pressureless scales the Poisson equation gives ${\delta }_m\sim (k\tau {)}^2\Psi$: the corresponding density amplitude has the often-used ${\theta }^{-2}$ scaling. This is a matter-density estimate, not a universal small-angle photon-temperature law; the actual intrinsic photon source undergoes the oscillations in part(b). On superhorizon scales it instead has the constant-potential behavior just described.

The integrated source vanishes for constant linear potentials in pure matter domination. Potential decay during radiation domination or late accelerated expansion makes it nonzero. Its broad, slowly evolving sources favor large angular scales and give a potential-sized contribution there; crossings of many structures cancel progressively on small scales. Scale invariance alone fixes no universal power of angle for the integrated term: its time evolution and transfer functions are also required. Consequently the large-angle plateau, velocity-gradient enhancement and subhorizon density-gradient estimate describe distinct physical regimes, rather than one extrapolated power law across the whole spectrum.

Three distinct low-redshift nonlinear contributions are:

Gravitational lensing supplies another nonlinear modification by remapping pre-existing temperature and polarization patterns. It preserves the local blackbody spectrum; it does not generate temperature differences by lensing a perfectly isotropic background.

The emission-density term is the intrinsic temperature fluctuation: blackbody radiation has $\delta {\rho }_{\gamma }/{\rho }_{\gamma }=4\Delta T/T$, hence the factor $1/4$. Adiabatic compression and rarefaction of the photon-baryon fluid create coherent density extrema at last scattering. Their projection produces acoustic peaks, with angular scale set by the sound horizon divided by the distance to last scattering. In a familiar nearly flat cosmology the first prominent temperature peak is around degree scales, $\ell$ of order a few hundred; precise positions depend on the background.

The velocity term is a Doppler effect from the emitting plasma's line-of-sight motion. The acoustic velocity is approximately a quarter-cycle out of phase with the density oscillation. Its projected contribution is strongest on acoustic and smaller angular scales and interleaves with density extrema, with projection broadening its features. Both sources are suppressed on sufficiently small scales by Silk damping and the finite thickness of last scattering.

The metric tensor integral describes photon-energy shifts during propagation through the changing perturbed geometry. In synchronous coordinates its split from the intrinsic endpoint term is gauge dependent. Rewriting the observable in Newtonian variables produces the usual emission Sachs-Wolfe effect and the Integrated Sachs-Wolfe effect from evolving potentials, with the appropriate endpoint terms. The ordinary large-scale Sachs-Wolfe combination gives a low-multipole plateau for nearly scale-invariant initial curvature power. Late-time evolving potentials chiefly affect large angles, often $\ell \lesssim 20$; evolution near radiation-matter equality also drives an early integrated contribution near the first acoustic peak. Static matter-era potentials do not produce that Newtonian integrated contribution.

All emission quantities are evaluated at the last-scattering endpoint and the metric tensor source along the unperturbed photon ray, not at a single fixed spatial point throughout the integral. Observer monopole/dipole conventions do not change this physical interpretation.

The sketch uses arbitrary units and representative scales. It displays source auto-power shapes, not measured data or an assertion that the total power is their simple sum. The terms are correlated; their cross-power contributions and the gauge-consistent endpoint combination are essential in an actual angular-spectrum calculation.

Let $q_{\ast }=k{\tau }_{\mathrm{dec}}/\sqrt{3}$ and $E_D=\exp (-k^2/k_D^2)$ at last scattering. For random oscillator amplitudes, define spectra $P_A,P_B$ and cross-spectrum $P_{AB}$ using the same Fourier normalization as the density spectrum. Averaging the solution's squared modulus givesConsequently the intrinsic-temperature source has variance spectrum $P_{{\delta }_{\gamma }}/16$. A common acoustic phase is necessary for pronounced peaks in the variance. Equal independent sine/cosine amplitudes would erase them. A regular growing adiabatic mode begins with negligible velocity outside the sound horizon and selects approximately the cosine phase, $B\simeq 0$, preserving a ${\cos }^2{q}_{\ast }$ sequence. This illustrates coherent initial phases are needed for acoustic variance peaks.

From continuity, ${\theta }_{\gamma }=3{\delta }_{\gamma }^{\prime }/(4k^2)$. Ignoring the small damping correction in this velocity relation, the coherent cosine solution givesThe Doppler CMB anisotropy source therefore has directional variance $3{\mu }^2{P}_A{E}_D^2{\sin }^2{q}_{\ast }/16$, where $\mu =\hat{\mathbf{n}}\cdot \hat{\mathbf{k}}$. Its velocity maxima lie between density maxima: compression and bulk motion are a quarter acoustic period apart.

Angular projection must also be included. Let ${\chi }_{\ast }={\tau }_0-{\tau }_{\mathrm{dec}}$, assume a thin last-scattering surface, and choose the emission phase $e^{ik{\chi }_{\ast }\mu }$. The coherent intrinsic-plus-Doppler radial response is proportional toThe derivative follows from $i\mu e^{ix\mu }={\partial }_x{e}^{ix\mu }$ and the velocity sign above. A reversed line-of-sight convention reverses the corresponding Doppler sign consistently. The CMB angular power spectrum is $C_{\ell }=(2/\pi )\int k^{2}dk{P}_A(k)|{\Theta }_{\ell }(k){|}^2$ in this simplified model. Squaring the combined response keeps its correlated density–velocity cross term; adding two independent angular powers would discard that correlation. The derivative Bessel projection also spreads the Doppler contribution differently from the density contribution.

The sound horizon here is $r_s\simeq {\tau }_{\mathrm{dec}}/\sqrt{3}$. Intrinsic acoustic extrema occur near $k{r}_s=n\pi$, mapping roughly to ${\ell }_n\simeq n\pi {\chi }_{\ast }/r_s$, with interleaved velocity extrema and projection broadening. Cosmic microwave background diffusion damping suppresses both contributions exponentially at $k\gtrsim k_D$, or multipoles $\ell \gtrsim k_D{\chi }_{\ast }$. Modes outside the sound horizon have no completed acoustic oscillation.

The metric integral represents gravitational redshift, including contributions that in another gauge are separated into endpoint Sachs-Wolfe effects and Integrated Sachs-Wolfe effects. It is not correct to call the entire synchronous metric integral only a late-time integrated effect. Large-scale metric/curvature fluctuations remain important where photon pressure cannot redistribute matter causally and produce the broad large-angle signal rather than just the pressure-oscillation sequence. Potential evolution during radiation-to-matter transition also drives and shifts acoustic amplitudes near horizon entry; late accelerating-era evolution adds a large-angle integrated signal. These effects are excluded by the oscillator's instruction to ignore metric perturbations. Even if the Newtonian-gauge potential is constant in matter domination, synchronous $h_{ij}^{\prime }$ need not vanish, and the complete observable requires the consistent density, velocity and metric combination.

There is first an angular-scale typo in the PDF: smaller angles correspond to larger multipoles, approximately $\theta \sim \pi /\ell$. Small scales relative to the first acoustic peak therefore have $\ell \gtrsim 200$, not the printed inequality in the other direction.

The photon-baryon acoustic oscillator produces rapidly varying density and velocity sources around and below the sound horizon. Treating the emission density as a simple long-wavelength gravitational source misses these acoustic peaks and their phases. Silk damping suppresses short-scale perturbations through diffusion before decoupling. The cosmological visibility function has finite width, so different emission times and acoustic phases are averaged rather than sampled on one sharp surface. As tight coupling breaks down, the photon quadrupole and higher moments are generated; Thomson scattering of the quadrupole produces polarization. These require the full photon Boltzmann hierarchy, rather than exactly vanishing emission moments above the dipole. They are the angular-scale limits of instantaneous photon decoupling.

The distinction is important: acoustic evolution and diffusion can already be included in correctly calculated emission ${\delta }_{\gamma }$ and $\mathbf{v}$. They do not themselves invalidate collisionless propagation after decoupling. The two-moment, single-time initial condition and omission of scattering after that surface are the approximations used in the requested line-of-sight formula.

For the stated assumptions, multiply the synchronous photon brightness equation by $e^{ik\mu \tau }$ and integrate:Here ${\tau }_d$ is the decoupling time. Inverse Fourier transformation shifts each source to the unperturbed rayInsert the initial brightness monopole and Doppler dipole, and divide by four:This is the synchronous Sachs-Wolfe line-of-sight formula. The first term is intrinsic emission temperature, the second is the Doppler effect, and the last is the accumulated gravitational redshift in this gauge. The derivative $h_{ij}^{\prime }$ is the partial conformal-time derivative evaluated on the ray, not its total derivative along the ray. Its integral contains endpoint gravitational effects as well as evolving-potential effects when rewritten in Newtonian gauge; it should not be identified solely with the Integrated Sachs-Wolfe effect.

The direction $\mathbf{n}$ is photon propagation toward the observer. The observer-to-source sky direction is its negative, so changing that convention also changes the displayed Doppler sign and the ray parametrization. Correct emitter positions and this direction convention are needed to interpret the 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.

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.

For the growing adiabatic mode in an epoch of constant $w$, the superhorizon potential is constant. The Friedmann equation gives $4\pi G{a}^2(\bar{\rho }+\bar{P})=\frac{3}{2}(1+w){\mathcal{H}}^2$, so the given relation impliesThus ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. Photon continuity, neglecting its gradient term on superhorizon scales, gives $({\Theta }_0-\varphi {)}^{\prime }=0$. The radiation-era initial condition fixes ${\Theta }_0-\varphi =\mathcal{R}$. Through the matter-radiation transition it follows that ${\Theta }_{0\ast }=\mathcal{R}+{\varphi }_{\mathrm{mat}}=2\mathcal{R}/5$. Since $\psi =\varphi$, the matched large-scale emission perturbation isKeeping the radiation value of ${\Theta }_0$ unchanged across the transition would give the wrong coefficient. This is the Sachs-Wolfe radiation-to-matter matching for the paper's sign convention for comoving curvature perturbation.

Neglect the Integrated Sachs-Wolfe effect, the superhorizon-suppressed Doppler term, observer monopole, and any observer kinematic dipole. Put ${\mathbf{x}}_0=0$ by statistical homogeneity, and defineThe plane-wave expansion on ${\mathbf{x}}_{\ast }=-{\chi }_{\ast }\mathbf{e}$ gives angular multipolesStatistical isotropy and orthonormal spherical harmonics then imply $⟨a_{\ell m}{a}_{{\ell }^{\prime }{m}^{\prime }}^{\ast }⟩={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}{C}_{\ell }$. Radial integration supplies $(4\pi {)}^2(2{\pi }^2)/(2\pi {)}^3=4\pi$, giving the large-angle angular power spectrumThis assumes adiabatic growing modes, matter domination at emission, negligible anisotropic stress, and wavenumbers that are outside the Hubble radius then. It treats recombination as instantaneous, omits late potential evolution and rescattering, and omits lensing at this linear order. The formal primordial contribution is defined for $\ell >0$; the observed dipole has a large additional kinematic contribution, so the clean large-angle primordial comparison is usually $\ell \ge 2$. A scale-invariant spectrum also yields the Sachs-Wolfe plateau $\ell (\ell +1)C_{\ell }=2\pi {\mathcal{P}}_{\mathcal{R}}/25$ for $\ell \ge 1$.

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 $-\mathbf{e}$.

Along the unperturbed photon path, combine the temperature and gravitational-redshift terms and use ${\mathcal{T}}^{\prime }=-\Gamma$:The integrating factor is $e^{-\mathcal{T}}$, and $g=\Gamma e^{-\mathcal{T}}$. Neglecting the exponentially hidden initial boundary term gives the Cosmic microwave background line-of-sight solutionFor instantaneous recombination, $g(\tau )=\delta (\tau -{\tau }_{\mathrm{dec}})$, and the observer potential contributes only an unobservable monopole. ThereforeThe first two terms form the ordinary Sachs-Wolfe effect, the velocity term is the Doppler CMB anisotropy at last scattering, and the integral is the Integrated Sachs-Wolfe effect produced by evolving potentials.

The line-of-sight solution for free-streaming photons follows the unperturbed ray $\mathbf{x}(\eta )={\mathbf{x}}_0-\hat{\mathbf{p}}({\eta }_f-\eta )$. If $\Phi =\Psi$ is spatially homogeneous, its gradient vanishes and $d\log ϵ/d\eta ={\Phi }^{\prime }$. Integration along the ray givesThe needed initial condition is the temperature pattern on the initial hypersurface; for a superhorizon adiabatic mode during matter domination one may use ${\Theta }_0({\eta }_i)=-2\Psi ({\eta }_i)/3$.

This calculation contains free streaming of the initial pattern and the gravitational temperature shift caused by the evolving potential, the homogeneous limit of the Integrated Sachs-Wolfe effect. A perfectly homogeneous potential changes only the unobservable sky monopole and therefore contributes no CMB angular power spectrum for $\ell \ge 1$. Direction-dependent Sachs-Wolfe effect, Doppler, acoustic, polarization, lensing, and rescattering contributions require spatially varying perturbations or collision terms and are outside this special calculation.

Write ${\mathbf{x}}_j={\mathbf{x}}_0+({\eta }_0-{\eta }_j)\hat{\mathbf{n}}$. The delta functions in $g$ evaluate the local last-scattering sources, while the piecewise $e^{-\tau }$ weights the integrated gravitational source. ThusThe first two lines are weighted Sachs-Wolfe combination and Doppler sources at the two last-scattering surfaces; the final lines are the corresponding Integrated Sachs-Wolfe effect.

To predict these quantities from primordial perturbations, one solves the coupled linear Einstein field equations and Boltzmann equation hierarchy for photons, baryons, cold dark matter, and neutrinos, together with the ionization history that determines $\Gamma$ and hence $g$. The solutions provide ${\Theta }_0$, ${\mathbf{v}}_e$, $\Phi$, and $\Psi$ at both surfaces and along the line of sight.

Let $q=\mathbf{k}\cdot \mathbf{n}$, $E=e^{iq(\tau -{\tau }_0)}$ and $T=h_S^{\prime }/(2k^2)$. The synchronous-to-Newtonian transformation with positive spatial perturbations gives $(h^{\prime }-h_S^{\prime })/6={\Phi }^{\prime }+{\Psi }^{\prime }+T^{\prime \prime }$. Since $[E(T^{\prime }-iqT){]}^{\prime }=E(T^{\prime \prime }+q^{2}T)$, the displayed identity follows by integration. The emission endpoint combines intrinsic and velocity shifts into the ordinary Sachs-Wolfe effect; the local observer endpoint is a monopole plus a dipole. Without anisotropic stress the remaining integral is the Integrated Sachs-Wolfe effect, $2\int E{\Phi }^{\prime }d\tau$. The angular term in the synchronous contraction is positive for this metric convention.

With instantaneous photon decoupling and tight-coupling initial brightness, integrate the synchronous photon brightness equation along the unperturbed ray $\mathbf{x}(\tau )={\mathbf{x}}_0-\mathbf{n}({\tau }_0-\tau )$. The direction $\mathbf{n}$ here is photon propagation, opposite to the observer-to-source sky direction. Emission terms are evaluated at the emitter, not the observer. The synchronous metric integral combines endpoint gravitational redshifts and evolving-potential effects when expressed in Newtonian gauge; it is not solely the Integrated Sachs-Wolfe effect.

// Target: cosmology.bigb