CMB Polarization and E/B Modes

CMB linear polarization requires two ingredients at the same event: Thomson scattering and a quadrupole in the radiation incident on the electron. The result is a spin-2 field on the sky, whose parity-even and parity-odd parts are called E and B. Scalar perturbations generate primordial E but no primordial B at linear order; tensors, vector modes, lensing, rotation, foregrounds, and instrumental leakage change that conclusion in physically distinct ways.

Required background. Probability and Inference for Gravity supplies Probability and Statistical Inference for random fields and cross-spectra.

Helpful background. The Einstein Equation supplies Einstein Equations and scalar/tensor metric sources; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects radiation moments; Relativistic Cosmology supplies recombination and perturbation transfer; and The Wave Equation and Characteristics supplies Waves, Green Functions, and Geometric Optics for polarization transport.

Thomson scattering selects a quadrupole

Choose a right-handed orthonormal basis $(\mathbf e_1,\mathbf e_2,\mathbf n)$, with the first two axes on the screen perpendicular to the outgoing direction $\mathbf n$. For incident direction $\mathbf n'$ and linear-polarization vectors $\boldsymbol\epsilon',\boldsymbol\epsilon$, the Thomson kernel is

$$\frac{d\sigma}{d\Omega} =\frac{3\sigma_T}{8\pi} |\boldsymbol\epsilon'\cdot\boldsymbol\epsilon|^2.$$

Let $(\beta,\varphi)$ describe $\mathbf n'$ about $\mathbf n$, with azimuth measured from $\mathbf e_1$ toward $\mathbf e_2$. Define $Q=J_{11}-J_{22}$ and $U=2\operatorname{Re}J_{12}$ from the electric-field coherency matrix $J_{ab}=\langle E_aE_b^*\rangle$, so that $Q\gt0$ means polarization along $\mathbf e_1$. For an optically thin electron screen with Thomson optical depth $\Delta\tau=\sigma_T\int n_e\,dl\ll1$, initially unpolarized incident intensity produces the scattered contributions

$$\begin{aligned} Q_{\rm sc}(\mathbf n)&=-\frac{3\Delta\tau}{16\pi} \int d\Omega'\,I(\mathbf n')\sin^2\!\beta\cos2\varphi,\\ U_{\rm sc}(\mathbf n)&=-\frac{3\Delta\tau}{16\pi} \int d\Omega'\,I(\mathbf n')\sin^2\!\beta\sin2\varphi. \end{aligned}$$

The dimensionless $\Delta\tau$ makes $I,Q_{\rm sc},U_{\rm sc}$ have the same intensity units. The signs follow by projecting the incident unpolarized matrix onto the screen: its contribution is proportional to $\delta_{ab}-(\mathbf e_a\cdot\mathbf n')(\mathbf e_b\cdot\mathbf n')$. For a beam incident along $\mathbf e_1$, the scattered electric field lies along $\mathbf e_2$, giving $Q_{\rm sc}\lt0$, as the formula predicts. Rotating the screen rotates both the Stokes components and their angular basis; a sign change of $Q,U$ alone is not a harmless change to a temperature– polarization cross-spectrum.

The angular kernels are $\ell=2$ harmonics. An isotropic monopole cancels between orthogonal polarization states; a dipole also integrates to zero. Only an incident quadrupole survives. Frequent scattering suppresses precisely that quadrupole, so polarization is generated most efficiently while the plasma becomes optically thin and again when reionization exposes a large-scale quadrupole.

The scattering geometry and Stokes interpretation are explained by Hu and White 1997, §2.1, pp. 2–3, eqs. (1)–(3) (Open PDF). In a cosmological calculation the infinitesimal source is integrated through the visibility function, rather than treating the entire last-scattering region as one thin screen.

In the brightness normalization used in the previous chapter, Thomson scattering couples the intensity quadrupole to

$$\Pi_\gamma=F_{\gamma2}+G_{\gamma0}+G_{\gamma2}.$$

The complete collision terms and their numerical factors are given in Ma and Bertschinger 1995, §5.5, printed pp. 21–22, eqs. (59)–(64) (Open PDF). In tight coupling the quadrupole is of order $kv_\gamma/\dot\kappa$, making the small polarization fraction a controlled mean-free-path effect rather than a separate primordial amplitude.

A spin-2 field on the sphere

The Stokes pair depends on the screen basis. Under a right-handed basis rotation by $\psi$,

$$(Q\pm iU)'=e^{\mp2i\psi}(Q\pm iU).$$

Expand these spin-weighted fields as

$$Q\pm iU=\sum_{\ell m}a_{\pm2,\ell m} \,{}_{\pm2}Y_{\ell m}(\mathbf n).$$

The basis-independent gradient-like and curl-like coefficients are defined by

$$a^E_{\ell m}=-\frac12 (a_{2,\ell m}+a_{-2,\ell m}), \qquad a^B_{\ell m}=\frac{i}{2} (a_{2,\ell m}-a_{-2,\ell m}).$$

They are nonlocal combinations of $Q$ and $U$: a single pixel does not possess an E or B label. Under parity,

$$a^E_{\ell m}\longrightarrow(-1)^\ell a^E_{\ell m}, \qquad a^B_{\ell m}\longrightarrow(-1)^{\ell+1}a^B_{\ell m}.$$

For $X,Y\in\{T,E,B\}$, statistical isotropy defines

$$C_\ell^{XY}=\frac{1}{2\ell+1} \sum_m\langle a^X_{\ell m}a^{Y*}_{\ell m}\rangle.$$

A parity-invariant ensemble has $C_\ell^{TB}=C_\ell^{EB}=0$; that is a physical assumption, not part of the definition of B. Hu and Dodelson introduce the polarization spectra and parity distinction in Hu and Dodelson 2002, §2.3, preprint printed pp. 5–8, especially eq. (5) (Open PDF).

Why a scalar mode has no linear primordial B

Orient one scalar Fourier mode along $\hat{\mathbf z}$. Its perturbations are axisymmetric about $\mathbf k$ and contain no second transverse direction or handedness. Thomson scattering therefore creates an axisymmetric spin-2 pattern aligned with or orthogonal to the projected wavevector. That pattern is pure E. Rotating the mode and linearly superposing scalar modes does not create B, because the E/B decomposition itself is linear.

This is the precise boundary of the familiar statement:

• scalar perturbations produce no primordial linear B modes;
• tensor and vector perturbations have transverse structure and can produce both E and B at last scattering;
• gravitational lensing remaps an existing E field and converts part of it to B even when the primordial perturbations were scalar;
• polarization rotation, including a calibration-angle error or physical birefringence, mixes E and B;
• Galactic foregrounds and temperature-to-polarization leakage can imitate B unless they are included in the measurement model.

The local quadrupole argument and scalar E-only result are developed in Hu and Dodelson 2002, §3.7, preprint printed pp. 21–22 (Open PDF). The Particle Data Group gives a current observational orientation in Scott and Smoot 2026, §29.7 and §§29.7.1–29.7.3, printed pp. 10–14 (official PDF). Those dated measurements do not alter the geometric definitions above.

Physical interpretation and limits

Temperature density extrema and fluid velocity are approximately a quarter period out of phase in the acoustic limit. Because the polarization quadrupole is fed by velocity gradients, EE peaks are correspondingly out of phase with the dominant TT acoustic extrema, while TE changes sign across correlated phases. Reionization generates an additional large-angle E feature and suppresses already-formed small-scale primary power by rescattering.

Circular polarization $V$ is absent from this Thomson, linear, parity-even discussion. Partial-sky E/B separation has ambiguous modes and leakage that do not occur in the full-sky definitions. Nonlinear scattering, patchy reionization, lensing, frequency-dependent foregrounds, and instrumental systematics must not be labeled primordial merely because they contribute to $C_\ell^{BB}$.

Exercises

Use parity to show that $TB$ and $EB$ vanish for a parity-invariant ensemble, while $TE$ need not vanish.

Solution

Statistical isotropy makes each two-point function diagonal in $\ell,m$. Under parity, a $T$ or E coefficient contributes $(-1)^\ell$, whereas a B coefficient contributes $(-1)^{\ell+1}$. Hence a $TB$ or $EB$ correlator changes by the factor $(-1)^{2\ell+1}=-1$. Parity invariance requires it to equal its transformed value, so it must be zero. A $TE$ correlator instead acquires $(-1)^{2\ell}=1$ and is allowed. Parity violation, cosmic or instrumental polarization rotation, or an inadequately modeled systematic can invalidate the vanishing-cross-spectrum check.

References

• Hu, W., and M. White, “A CMB Polarization Primer,” New Astronomy 2, 323–344 (1997), DOI, Open PDF.
• Hu, W., and S. Dodelson, “Cosmic Microwave Background Anisotropies,” Annual Review of Astronomy and Astrophysics 40, 171–216 (2002), DOI, Open PDF.
• Ma, C.-P., and E. Bertschinger, “Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges,” The Astrophysical Journal 455, 7–25 (1995), DOI, Open PDF.
• Scott, D., and G. F. Smoot, “Cosmic Microwave Background,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), Official PDF.