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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.

Let $q=\mathbf{k}\cdot \hat{\mathbf{n}}$ and retain the negative-spatial-perturbation convention of the preceding part. Since $h=h_S$, the isotropic pieces of the supplied Fourier mode contraction cancel:For $h=A(\mathbf{k}){\tau }^2{k}^2$, the factor $1/2$ in the line-of-sight integral cancels the derivative's factor two. ThusAn antiderivative convenient even at $q=0$ isIt follows directly, or by two integrations by parts, thatThe source sign and both endpoint coefficients are fixed by the metric convention; no division by $q$ is needed.

To interpret the large-angle Cosmic microwave background anisotropy, restore the observer-centered propagation phase $e^{-iq{\tau }_0}$, which can equivalently be absorbed into the definition of each Fourier mode amplitude. Put $D={\tau }_0-{\tau }_{\mathrm{dec}}$. One mode contributesThe first term is a local angular monopole plus a dipole, so it is removed when discussing observed angular multipoles $\ell \ge 2$. For modes outside the horizon at decoupling, $k{\tau }_{\mathrm{dec}}\ll 1$, the higher-multipole emission signal is dominated byIts transfer amplitude has no leading positive power of $k$: the two explicit powers in $h$ have canceled in the integral. Projection onto the sky supplies Spherical Bessel functions $j_{\ell }(kD)$, with modes of order $kD\sim \ell$ contributing to each angular scale. This is the large-angle Sachs-Wolfe effect; the emission velocity-like correction is relatively of order $k{\tau }_{\mathrm{dec}}$. For wavelengths much larger than the entire observed region, $kD\ll 1$, removing the monopole and dipole leaves a signal starting at order $A(kD{)}^2$, rather than a finite observable anisotropy from a spatially uniform mode.

The stochastic scale dependence of $A(\mathbf{k})$ is not specified by the deterministic metric solution. If one additionally assumes a scale-invariant primordial spectrum, $k^3{P}_A(k)=\text{constant}$, thenfor $\ell \ge 1$, giving the familiar large-angle Sachs-Wolfe plateau, $\ell (\ell +1)C_{\ell }\simeq \text{constant}$, for the measured $\ell \ge 2$ multipoles. A tilted primordial spectrum changes that trend. The full numerical temperature transfer also includes intrinsic last-scattering and Doppler terms, omitted from the stipulated metric-only integral.

For two neighboring comoving points separated along the zeroth-order direction $\hat{\mathbf{n}}$ by coordinate distance $\delta \chi$, the synchronous gauge spatial metric gives the cosmological proper distanceDifferentiate with respect to cosmic time at fixed coordinate separation. To first order,A photon crosses this neighboring interval in $\delta t=\delta r$ when $c=1$, so its local relative recession velocity isAt each infinitesimal step, the local Doppler effect gives $d\log \nu =-\Delta v$. ThereforeThe first term integrates to the homogeneous cosmological redshift, $\bar{\nu }\propto a^{-1}$. Subtracting it isolates the metric contribution to the fractional frequency shift. A shifted blackbody radiation spectrum has its temperature shifted by the same fraction as its photon frequencies, givingThe dot denotes a partial time derivative at fixed comoving position, evaluated along the zeroth-order null geodesic. It is not the total derivative of $h_{ij}$ along the ray, so the integral cannot in general be replaced by a difference of $h_{ij}$ at the endpoints. Deflection of the ray or its direction multiplies an already first-order metric perturbation and is second order here. This is the metric part of the Sachs-Wolfe effect in the negative-spatial-perturbation convention; an intrinsic emission temperature fluctuation or an emitter velocity would add separate boundary terms. The time derivative is present in the original PDF, although missing from the converted TeX's displayed integral.

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.

The supplied matter-era solution has $h^{\prime }=h_S^{\prime }=A{k}^{1/2}\tau /2$, so the source reduces to $A{k}^{1/2}{\mu }^2\tau$. Put $D={\tau }_0-{\tau }_{\ast }$ and $W=e^{-ik\mu D}$. The integrating factor for the streaming equation gives the exact matter-era photon endpoint solutionFor $b=k\mu \ne 0$, the integral is $({\tau }_0-{\tau }_{\ast }W)/(ib)+(1-W)/b^2$. Substitution and collection of endpoint terms givesAt $\mu =0$ the original source is zero and the expression has the continuous limit ${\Delta }_0={\Delta }_{\ast }$. The integral form is numerically preferable near that limit, where the separated endpoint terms cancel.

The last two terms are respectively a local observer monopole and dipole. For measured higher angular multipoles, remove them. The remaining emission bracket contains initial brightness, an emission velocity-type term, and a gravitational Sachs-Wolfe effect term. Under ordinary adiabatic initial conditions for the growing synchronous matter mode, ${\Delta }_{\ast }\sim A{k}^{1/2}{\tau }_{\ast }^2$, while the velocity endpoint scales as $A{k}^{-1/2}{\tau }_{\ast }$. Relative to $A{k}^{-3/2}$ these are suppressed by $(k{\tau }_{\ast }{)}^2$ and $k{\tau }_{\ast }$ respectively. Thus for modes outside the horizon at last scattering,Its $k^{-3/2}$ amplitude is the scale-invariant large-angle contribution. In Newtonian language it is the ordinary endpoint Sachs-Wolfe effect, not a late integrated effect in a constant-potential matter universe. With completely arbitrary prescribed ${\Delta }_{\ast }$, no unconditional dominance theorem is possible: an independently large initial anisotropy could dominate. The exact solution above remains valid without that adiabatic assumption.

The emission term comes from the local photon temperature. A blackbody radiation density is proportional to $T^4$, so linearization gives $\delta T/T={\delta }_{\gamma }/4$. It is evaluated at last scattering, with the source position on the observed ray.

The velocity term is the Doppler CMB anisotropy: motion of the emitting plasma changes the measured photon frequency by the component of its peculiar velocity along the line of sight. Its sign here follows the paper's choice of the sky direction $\hat{\mathbf{n}}$.

The metric integral follows the null geodesic energy equation in synchronous gauge. Time dependence of the spatial metric perturbation changes the photon energy by $-h_{ij}^{\prime }{n}_i{n}_j/2$ along the ray. It includes gravitational redshift endpoint terms as well as the part due to evolving gravitational potentials. In particular, the entire synchronous metric integral is not just the integrated Sachs-Wolfe effect. Integrating by parts to express it in conformal Newtonian gauge separates the ordinary Sachs-Wolfe effect from the evolving-potential contribution.

Write the growing density contrast as ${\delta }_{m,\mathbf{k}}=A_{\mathbf{k}}{\tau }^2$, with $A_{\mathbf{k}}$ having the indicated $k^{n/2}$ spectral amplitude. In the dust-comoving synchronous gauge, $h^{\prime }=-2{\delta }_m^{\prime }$, and the supplied $h_s^{\prime }=h^{\prime }$ therefore implies $h_s^{\prime \prime }=-4A_{\mathbf{k}}$. On the same comoving density slicing, the cosmological Poisson equation isbecause $\mathcal{H}=2/\tau$. This is the comoving-density form of the relativistic scalar constraint, so it can be used on these large scales; replacing its density by the Newtonian-gauge density would not be justified there. Combining the two results givesThus the large-angle Sachs-Wolfe effect has source $\Phi /3$. The potential is constant for this matter-era growing mode, and its power spectrum obeys $P_{\Phi }(k)\propto k^{n-4}$.

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.

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.

Let $R=\Delta \tau >0$, $L={\ell }_1+{\ell }_2+{\ell }_3$ and $D={\prod }_{r=1}^3(2{\ell }_r+1)$. Use exactly the given constant primordial bispectrum normalization, without an additional local-template convention factor. Combining it with the large-angle Sachs-Wolfe effect cosmological transfer function gives the reduced CMB bispectrumwhere $I_{\ell }(x,R)={\int }_0^{\infty }{j}_{\ell }(kR)j_{\ell }(kx)dk$. Each $1/5$ transfer factor has been retained; their product is $1/125$.

Apply the supplied spherical Bessel product integral after rescaling $q=kR$:The two branches agree at $x=R$. In the radial integral put $r=x/R$. The factor $R^3$ from $x^{2}dx$ cancels the $R^{-3}$ from the three kernels. The remaining integral isprovided $L>0$. This evaluation uses the momentum integrals at fixed radial parameter in the projection, then the radial integral; it does not require an unjustified global exchange of all oscillatory integrals.

The factors $(2/\pi {)}^3$ and ${\pi }^3/8$ cancel exactly. Hence the Sachs-Wolfe projection of a constant bispectrum isThe all-monopole case $L=0$ has a logarithmically divergent outer radial integral and is not covered by the printed finite formula. Observable CMB analyses remove the monopole and dipole, normally using ${\ell }_i\ge 2$, so this issue is absent there. Angular triangle and parity selection are carried by the triple-spherical harmonic geometric factor multiplying the reduced bispectrum.

There is no dependence on the last-scattering distance $R$, and no extra physical scale appears when ${\Delta }_{\zeta }^2$ is held constant. At fixed triangle shape and in the range $1\ll {\ell }_i\ll 200$, common rescaling of all multipoles givesThis is angular scale invariance in the usual weighted sense; a constant primordial shape does not make the unweighted reduced CMB bispectrum independent of angular scale. The exact finite-multipole expression retains the $+1$ and $+3$ terms shown above.

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 $-\mathbf{e}$.

The conventional vertical variable is $D_{\ell }=\ell (\ell +1)C_{\ell }/(2\pi )$ in $\mu {\mathrm{K}}^2$, plotted against the dimensionless angular multipole $\ell$. The Sachs-Wolfe effect supplies a nearly flat large-angle plateau for $\ell \lesssim 30$. The total spectrum has its first acoustic peak near $\ell \simeq 220$ with $D_{\ell }$ of order $5\times 10^3\mu {\mathrm{K}}^2$, followed by further Cosmic microwave background acoustic peaks; the Doppler CMB anisotropy is phase shifted relative to the photon-density oscillation. At $\ell \gtrsim 1000$, Cosmic microwave background diffusion damping lets photons random-walk across perturbations during recombination and produces the rapidly falling damping tail.

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.

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.

For a scale-invariant potential spectrum with ${\mathcal{A}}_{\Phi }=k^3{P}_{\Phi }/(2{\pi }^2)$ constant, the matter-era Sachs-Wolfe effect gives $C_{\ell }=(4\pi {\mathcal{A}}_{\Phi }/9)\int j_{\ell }(x{)}^{2}dx/x$. The logarithmic spherical Bessel square integral then gives $\ell (\ell +1)C_{\ell }=2\pi {\mathcal{A}}_{\Phi }/9$. This plateau applies to large-angle temperature anisotropies, neglecting acoustic evolution and late evolving potentials; it is not a constant $C_{\ell }$ spectrum.