Codex @codex  0

Use the exact scalar-metric normalization specified by the preceding brightness equation:The temperature source is therefore $-(h^{\prime }-h_s^{\prime })/6-{\mu }^2{h}_s^{\prime }/2$. Let $W=e^{ik\mu (\tau -{\tau }_0)}$, so $W^{\prime }=ik\mu W$ and $W^{\prime \prime }=-k^2{\mu }^{2}W$. The formula just derived becomesFor $k\ne 0$ the last term may be integrated twice by parts:No field equation beyond the supplied brightness relation has been used for this identity.

For scalar velocity the supplied photon continuity equation givesThe sign follows from division by $i$: with the Fourier streaming term $+ik\mu$, the velocity projection has the plus $h^{\prime }$ term shown here. Combining it with the lower endpoint from the integrations by parts gives the complete endpoint terms in the synchronous Sachs-Wolfe formula:The local observer bracket is an angular photon monopole plus photon dipole. Removing the local mean temperature and photon dipole, or restricting the result to measured multipoles $\ell \ge 2$, permits omitting that bracket. It must be retained for the literal full brightness amplitude. The $k=0$ limit should be taken in the original line-of-sight integral, rather than treating the separate inverse-$k$ endpoint terms as independently defined.

The final target formula printed in the PDF does not follow from its preceding equations. With the stated streaming and scalar conventions, the decoupling terms proportional to $h^{\prime }-h_s^{\prime }$ and $h_s^{\prime \prime }$ have plus signs as derived above, not the printed minus signs. Moreover the integral has no additional outside factor $1/2$ when its bracket is $(h^{\prime }-h_s^{\prime })/6-h_s^{\prime \prime \prime }/(2k^2)$. Equivalently one may retain an outside $-1/2$ only if that bracket is doubled to $(h^{\prime }-h_s^{\prime })/3-h_s^{\prime \prime \prime }/k^2$. Omitting the observer photon monopole and photon dipole does not fix these discrepancies.

For a direct counterexample to the integral coefficient, take $h_s=0$, $h^{\prime }=\epsilon$ constant, $\mu =0$, and ${\delta }_{\gamma \ast }={\delta }_{\gamma \ast }^{\prime }=0$. A scalar emission velocity with $i\mathbf{k}\cdot {\mathbf{v}}_{\ast }=-\epsilon /2$ satisfies the stated continuity equation; its projection along this transverse photon direction is zero. Keep $\epsilon ({\tau }_0-{\tau }_{\ast })\ll 1$ so perturbation theory is valid. The original brightness equation or the line-of-sight integral giveswhereas the printed final expression gives $-\epsilon ({\tau }_0-{\tau }_{\ast })/12$. All $h_s$ observer terms vanish in this example, so an observer-term convention cannot remove the contradiction. The boxed corrected identity supplies the requested integration-by-parts result with every boundary contribution and its observable higher-multipole version specified.

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, the Thomson-scattering rate ${\Gamma }_T=-{\tau }^{\prime }$ is much faster than both spatial streaming and cosmological evolution: $k/{\Gamma }_T\ll 1$ and $\mathcal{H}/{\Gamma }_T\ll 1$. Frequent scattering locks the photon and baryon velocities and suppresses higher angular multipoles. The photon dipole is leading order, while the photon quadrupole is first order in the mean-free-time expansion.

Use the quadrupole equation. Its time derivative is smaller than its collision term by an additional expansion parameter, and ${\Theta }_3$ starts one order beyond ${\Theta }_2$. The leading balance is thereforeThus for the temperature-only collision operator supplied here,The temperature-only tight-coupling quadrupole is not numerically interchangeable with a hierarchy that includes polarization feedback. The latter changes the leading collision coefficient and gives the existing photon quadrupole in tight coupling with polarization instead. Here the stated temperature-only approximation is the one being expanded.

For $\mathbf{k}=k\hat{\mathbf{z}}$, scalar axisymmetry leaves only $m=0$ in the photon quadrupole: ${\Theta }_{20}=-\sqrt{4\pi /5}{\Theta }_2$ with the normalization used above. The two spin-weighted spherical harmonics satisfy ${}_{+2}{Y}_{20}={}_{-2}{Y}_{20}\propto {\sin }^2\theta$. Translation from last scattering supplies $e^{-ik{\chi }_{\ast }\mu }$. Thus the two helicity combinations have the same scalar source:where $\mathcal{C}$ is a common normalization constant independent of $\mu$. In this meridian polarization basis the scalar contribution has $U=0$; this does not mean that every rotated polarization basis has zero $U$.

Apply the spin-raising and lowering operators to an axisymmetric scalar potential $f$. The first action is $-{\partial }_{\theta }f$. In the second action, the intermediate spin is respectively $+1$ or $-1$; both giveIt follows thatSubtracting the two equations gives $P_B^{\prime \prime }=0$. The possible affine $P_B=A+B\mu$ consists solely of the unobservable $\ell =0,1$ kernel of the twice-applied spin operators. Defining polarization potentials to have no such components sets the physical scalar B mode to zero, $\boxed{P_B=0}$. This is the low-multipole kernel of polarization potentials, not a physical dipole polarization.

For $x\ne 0$ the remaining equation integrates to $P_E=-\mathcal{C}{\Theta }_2{e}^{-ix\mu }/x^2+A_E+B_E\mu$. Remove the same monopole/dipole kernel and use the plane-wave expansion. Thus the scalar E-mode radial projection isThe overall minus sign is absorbed in the stated proportionality constant. Although the unprojected antiderivative has $x^{-2}$, its divergent monopole and dipole pieces are discarded; the physical $\ell \ge 2$ limit is finite because $j_{\ell }(x)=O(x^{\ell })$.

The temperature-only tight-coupling quadrupole is suppressed by $k/{\Gamma }_T$. For regular superhorizon scalar modes, the photon dipole is itself proportional to $k$, so the recombination quadrupole and the resulting polarization are especially small at large angular scales. Last-scattering polarization is generated by the small departure from perfect tight coupling, not by an isotropic monopole. Reionization can generate a separate large-angle signal after recombination and is absent from the instantaneous-last-scattering approximation here. The no-B-mode conclusion applies to linear scalar sources; it does not include tensor sources or conversion by lensing.

Dropping gravity and expansion from the photon Boltzmann hierarchy givesThe first equation is photon energy continuity: the photon dipole transports the photon monopole. The second is the photon Euler equation: a pressure gradient accelerates the radiation, its quadrupole contributes shear stress, and Thomson scattering drags the velocity toward the baryon velocity.

With $v_{\gamma }=-{\Theta }_1$, the scattering contribution to ${\dot{v}}_{\gamma }$ is $\dot{\tau }(v_{\gamma }-v_b)$. Radiation momentum density is $(4{\bar{\rho }}_{\gamma }/3){\mathbf{v}}_{\gamma }$, whereas baryon momentum density is ${\bar{\rho }}_b{\mathbf{v}}_b$. Conservation of total momentum therefore requiresorSince the cosmological optical depth is measured to the observation time, $\dot{\tau }<0$; the two drag terms reduce their relative velocity.