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In a local orthonormal tetrad, anisotropic stress is the trace-free spatial part of the stress-energy tensor, conventionally denoted ${\pi }^{ij}$. Some linear cosmological perturbation theory conventions instead use ${\Pi }^{ij}=-{\pi }^{ij}$ and write $T^{ij}=P{\delta }^{ij}-{\Pi }^{ij}$. The sign must be specified before using a photon Euler equation. The photon quadrupole determines the linear photon anisotropic stress.

Thomson scattering generates linear Cosmic microwave background polarization when the incident radiation has a photon quadrupole. The cross-section prefers electric fields perpendicular to the scattering direction. Contributions from an isotropic field cancel, and a dipole alone does not supply the required spin-two angular pattern. Schematically the outgoing Stokes combination is proportional towhich selects the incident quadrupolar angular dependence. Tight coupling nearly isotropizes the radiation, making the quadrupole small; it develops when the photon mean free path becomes finite near decoupling. At leading tight-coupling order its size scales as $(k/\dot{\kappa })v_{\gamma }$.

The polarization is therefore weaker than the temperature anisotropy: a characteristic amplitude is of order a few percent to roughly ten percent of the temperature signal, or several microkelvin compared with temperature fluctuations of several tens of microkelvin on acoustic scales. This is an order-of-magnitude comparison, not a scale-independent ratio of every power-spectrum multipole.

The photon monopole follows the acoustic displacement, approximately $\cos (k{r}_s)$; the velocity sourcing the quadrupole follows $\sin (k{r}_s)$. Hence polarization maxima occur near displacement zeros, between the main temperature peaks. In the leading coherent-oscillator approximation the shift is half the peak spacing, $\Delta \ell /2$, although projection and gravitational driving modify exact locations. Scalar perturbations generate E-mode polarization at linear order, not B-mode polarization. Tensor perturbations and lensing can give B modes, and reionization generates an additional large-angle polarization feature.

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.

The tight-coupling approximation requires the positive scattering rate $\kappa =-\dot{\tau }$ to exceed both $k$ and the conformal Hubble parameter. The photon-baryon velocity slip and photon quadrupole are then small, and the leading fluid motion has $v_b=v_{\gamma }=v$. One must combine the equations before setting the slip to zero: a small slip times a large collision rate can exert a finite force.

Add the photon Euler equation to $R$ times the baryon Euler equation with Thomson drag. The collisions cancel and, neglecting ${\Theta }_2$ at this order, the result isSince $\dot{R}=\mathcal{H}R$, the common velocity obeysThe inertia of the baryons reduces the photon-baryon sound speed to $c_s^2=1/[3(1+R)]$.

Differentiate the photon continuity equation and substitute this velocity equation:Thus the photon-baryon acoustic oscillator isThe terms on the right drive the oscillation gravitationally; the first-derivative term comes from changing baryon inertia. Diffusion damping enters only beyond this leading tight-coupling approximation.

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 $-\mathbf{e}$.

Write $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{p}}$ and use the Legendre polynomial recurrence relation. The definition in the question is inverted byFor Newtonian-gauge potentials the scanned geodesic equation is the standard identityIts monopole and dipole are $S_0={\Phi }^{\prime }$ and $S_1=k\Psi /3$. Projecting the photon Boltzmann hierarchy onto $P_0$ and $P_1$ therefore givesThe first is the photon continuity equation; the second is the photon Euler equation, with the photon quadrupole providing the anisotropic-stress term. Indeed, with photon density contrast ${\delta }_{\gamma }=4{\Theta }_0$ and velocity divergence ${\theta }_{\gamma }=3k{\Theta }_1$,

Project the photon Boltzmann equation onto Legendre polynomials. The angular average uses $⟨\Theta ⟩={\Theta }_0$ and $⟨\mu ⟩=0$, so the collision term has no monopole andFor the dipole, the recurrence relation makes free streaming couple ${\Theta }_1$ to ${\Theta }_0$ and ${\Theta }_2$. The gravitational term $-ik\mu \Psi$ contributes $k\Psi /3$, while projection of $i\mu v_b$ gives the baryon-velocity source. ThusNeglecting the photon quadrupole gives

When gravity and expansion can be neglected and $R$ is constant, the photon monopole satisfiesThe first damping contribution comes from photon-baryon velocity slip; the second comes from the photon quadrupole. This is a local form of Silk damping.

For the expansion $\Theta (\mu )={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }(\mu )$ without a $(2\ell +1)$ factor, the leading polarization relation is $E_2=-\sqrt{6}{\Theta }_2/4$. The photon Boltzmann hierarchy then gives ${\Theta }_2=-8k{\Theta }_1/(9\dot{\tau })$. This small photon quadrupole supplies the shear-viscosity part of Silk damping.

For a scalar photon quadrupole emitted at distance ${\chi }_{\ast }$, the meridian-basis source is $Q\pm iU\propto {\Theta }_2{e}^{-ik{\chi }_{\ast }\mu }(1-{\mu }^2)$. Integrating its scalar-potential equation and removing the low-multipole kernel of polarization potentials yields $P_E\propto {\sum }_{\ell \ge 2}(-i{)}^{\ell }(2\ell +1){\Theta }_2{j}_{\ell }(k{\chi }_{\ast })P_{\ell }(\mu )/(k{\chi }_{\ast }{)}^2$, with no scalar $P_B$. The apparent $k^{-2}$ divergence lies in discarded low multipoles; $j_{\ell }(x)=O(x^{\ell })$ makes the physical small-$x$ limit regular.