CMB Polarization and E/B Modes

The cosmic microwave background is not only anisotropic in temperature but also weakly linearly polarized, at roughly the ten-percent level of the temperature anisotropies. That polarization is manufactured, photon by photon, at the surface of last scattering and again during reionization, whenever a free electron Thomson-scatters radiation whose intensity has a quadrupole pattern on the sky. A monopole or a dipole scatters to nothing; only the $\ell=2$ moment survives, so the polarization is a direct fingerprint of the flow and shear of the photon-baryon fluid at the moment the universe turned transparent — information partly independent of, and partly complementary to, what temperature alone can tell us. Polarization is a spin-2 field: the Stokes parameters $Q$ and $U$ rotate into one another under a change of axes, so neither is a good observable alone. The fix is to decompose the field into two rotationally invariant scalars of definite parity — the gradient-like, parity-even E-mode and the curl-like, parity-odd B-mode. This split is the polarization analogue of the electric and magnetic parts of a vector field, and it carries the physics: to linear order, scalar density perturbations make E-modes only, while tensor perturbations — primordial gravitational waves from inflation — make both E and B, and gravitational lensing later shears some E into B. This article builds the microphysics of polarization from Thomson scattering, constructs the E/B decomposition and its parity rules, derives the power spectra, and shows how the reionization bump, the EE/TE peaks, lensing B-modes, and the still-hunted primordial B-mode each serve as a cosmological probe.

Key takeaways

CMB polarization arises when free electrons Thomson-scatter radiation carrying a local quadrupole ( $\ell=2$ ) anisotropy; velocity gradients at last scattering imprint linear polarization at $\sim 10\%$ of the temperature anisotropies.
The polarization is a spin-2 field described by Stokes $Q,U$ ; on the sphere it splits into a curl-free E-mode (parity even) and a divergence-free B-mode (parity odd), the two rotationally invariant observables.
Scalars source only E-modes and TE at linear order; primordial B-modes require tensor perturbations (gravitational waves), so their amplitude measures the tensor-to-scalar ratio $r$ and the inflationary energy scale.
The low- $\ell$ EE reionization bump ( $\ell \lesssim 10$ ) measures the optical depth $\tau \approx 0.054$ , breaking the $A_s\,e^{-2\tau}$ degeneracy temperature data cannot resolve.
Gravitational lensing shears E-modes into a secondary B-mode peaking at $\ell \sim 1000$ — a confirmed detection and a foreground floor beneath the primordial signal.
EE and TE are measured precisely; the primordial B-mode remains undetected, with $r < 0.032$ (BICEP/Keck + Planck).
Key idea: Thomson scattering of a quadrupole anisotropy generates linear polarization; decomposing the spin-2 polarization field into parity-even E-modes and parity-odd B-modes cleanly separates scalar density perturbations (E only) from tensor gravitational waves (E and B), while lensing mixes the two.

Where this lives today

CMB polarization has moved from curiosity to precision industry. Planck's low- $\ell$ polarization anchors $\tau=0.054$ , and its high- $\ell$ EE and TE spectra match $\Lambda$ CDM with no free parameters beyond those fixed by temperature — one of the model's most stringent consistency tests. Ground-based experiments dominate small scales: SPT-3G, ACT, POLARBEAR and the Simons Array have mapped the lensing B-mode and now routinely delens their maps to clear the way for a primordial signal. The primordial hunt is led by BICEP/Keck at the South Pole, whose degree-scale sensitivity combined with Planck's frequency coverage (to remove Galactic dust) sets $r<0.032$ , already excluding the simplest large-field ( $\phi^2$ ) models. The next generation is built for B-modes: the Simons Observatory and CMB-S4 target $r\sim 10^{-3}$ , while LiteBIRD is designed to measure the reionization B-bump from space. Meanwhile the parity-forbidden spectra are a frontier of their own: analyses of Planck and ground data report hints of a nonzero $EB$ correlation consistent with an isotropic "cosmic birefringence" rotation $\beta\approx 0.3^\circ$ , which — if confirmed — would signal a parity-violating coupling of a new light field to electromagnetism.

The physical picture

Put yourself at a free electron in the last-scattering plasma, bathed in radiation from every direction. Thomson scattering re-radiates with the classical dipole pattern: an incoming wave polarized along some direction drives the electron that way and re-emits most strongly perpendicular to it. Isotropic incident intensity, or a pure dipole (hotter on one side, colder on the exactly opposite side), averages away in the scattering integral and leaves the light unpolarized. Net polarization appears only when the radiation is hotter along one axis and colder along the perpendicular axis: a quadrupole. The excess along the hot axis, re-radiated perpendicular to it, leaves a residual polarization aligned with the cold axis.

Figure

Thomson scattering off a free electron turns a temperature quadrupole into linear polarization: radiation hot along one axis (red) and cold along the perpendicular axis (blue) scatters to a net polarization aligned with the cold direction, at roughly 10 percent of ΔT/T

The quadrupole the electron needs is supplied by the velocity field of the photon-baryon fluid: the electron sees photons Doppler-shifted differently from different directions, and that shear is a quadrupole. This is why polarization traces the fluid velocity rather than its density — and the velocity is $90^\circ$ out of phase with the density, so the TE correlation and the offset of the EE peaks from TT follow directly. The E/B split is cleanest by analogy to electromagnetism: E-modes are like the field of a static charge (radial or tangential, curl-free), B-modes like the field around a current (a handed swirl, divergence-free). A scalar makes only the radial or tangential pattern — pure E; the swirl of a B-mode needs the handedness of a gravitational wave or the shear of lensing. The second figure shows the power spectra an experiment measures.

Figure

The polarization power spectra versus multipole ℓ: the EE spectrum (blue) shows the reionization bump at ℓ ≲ 10 fixing τ ≈ 0.054 and acoustic peaks phase-shifted from TT; lensing shears E into a B-mode (green) peaking near ℓ ≈ 1000; a primordial tensor B-mode (amber) would appear at ℓ ≈ 80 with amplitude ∝ r, bounded r ≲ 0.032

The core equations

A polarized beam is specified by the Stokes parameters; its degree of linear polarization is the length of the $(Q,U)$ vector normalized by the intensity — the observable an instrument reports before any mode decomposition.

P = \frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}} = \frac{\sqrt{Q^2+U^2}}{I} \qquad (1)

where: $P$ — degree of linear polarization ( $0\le P\le 1$ ); $I$ — total-intensity Stokes parameter; $Q,\,U$ — the frame-dependent linear-polarization Stokes parameters. For the CMB $P\sim 10\%$ of the $\sim 10^{-5}$ temperature anisotropy, so the signal is at the $\mu\mathrm{K}$ level. The microphysics is Thomson scattering, whose differential cross section depends on the angle between incident and scattered polarization vectors. That angular structure is what filters out everything but the quadrupole of the incident field.

\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} = \frac{3\sigma_T}{8\pi}\,|\hat{\boldsymbol\epsilon}\cdot\hat{\boldsymbol\epsilon}'|^2 \qquad (2)

where: $\sigma_T = 6.65\times10^{-29}\,\mathrm{m^2}$ — Thomson cross section; $\hat{\boldsymbol\epsilon},\hat{\boldsymbol\epsilon}'$ — polarization unit vectors of incident and scattered radiation. Averaged over the sky, the $|\hat{\boldsymbol\epsilon}\cdot\hat{\boldsymbol\epsilon}'|^2$ dipole weighting projects onto the $\ell=2$ harmonic alone. Because $Q$ and $U$ are not scalars, the natural object is the spin-2 combination $Q\pm iU$ , expanded in spin-weighted spherical harmonics of spin $\pm 2$ — the polarization counterpart of $\Delta T = \sum a_{\ell m}Y_{\ell m}$ .

(Q \pm iU)(\hat n) = \sum_{\ell m} {}_{\pm 2}a_{\ell m}\; {}_{\pm 2}Y_{\ell m}(\hat n) \qquad (3)

where: ${}_{\pm2}Y_{\ell m}$ — spin- $\pm2$ spherical harmonics; ${}_{\pm2}a_{\ell m}$ — spin-2 multipole coefficients; $\hat n$ — line-of-sight direction. The two signs correspond to the two helicities. From the spin-2 coefficients one forms two parity eigenstates: the even combination is the E-mode, the odd combination (with an extra $i$ ) the B-mode. These are the physical, frame-independent fields.

a_{\ell m}^{E} = -\tfrac12\big({}_{2}a_{\ell m} + {}_{-2}a_{\ell m}\big),\quad a_{\ell m}^{B} = -\tfrac{1}{2i}\big({}_{2}a_{\ell m} - {}_{-2}a_{\ell m}\big) \qquad (4)

where: $a_{\ell m}^{E}$ — parity-even E-mode; $a_{\ell m}^{B}$ — parity-odd B-mode. Under parity $a_{\ell m}^{E}\to(-1)^{\ell}a_{\ell m}^{E}$ (like temperature) while $a_{\ell m}^{B}\to(-1)^{\ell+1}a_{\ell m}^{B}$ — the extra sign is the whole distinction. A Gaussian isotropic sky is fully described by the angular power spectra, one per pair of fields — the numbers plotted above and confronted with data.

C_\ell^{XY} = \frac{1}{2\ell+1}\sum_{m} \langle a_{\ell m}^{X\,*}\, a_{\ell m}^{Y}\rangle,\quad X,Y \in \{T,E,B\} \qquad (5)

where: $C_\ell^{TT},C_\ell^{EE},C_\ell^{BB}$ — auto-spectra; $C_\ell^{TE}$ — the temperature–E cross, the only nonzero cross in the standard model. EE and TE oscillate at the acoustic frequency but are velocity-sourced, so their peaks sit offset from TT. Parity forbids any correlation between a parity-even and a parity-odd field, killing two of the six possible spectra outright.

C_\ell^{TB} = C_\ell^{EB} = 0 \quad(\text{parity conservation}) \qquad (6)

where: $C_\ell^{TB},C_\ell^{EB}$ — cross-spectra that vanish unless parity is violated (e.g. cosmic birefringence). A measured nonzero $EB$ is a clean null test for new physics. The prize observable is set by one number: the ratio of primordial tensor to scalar power at a pivot scale, which fixes the primordial B-mode amplitude.

r \equiv \frac{A_t}{A_s} = \frac{\mathcal{P}_t(k_0)}{\mathcal{P}_s(k_0)} \qquad (7)

where: $A_t,A_s$ — tensor and scalar amplitudes at pivot $k_0=0.05\,\mathrm{Mpc^{-1}}$ ; $r$ — the tensor-to-scalar ratio. Since $A_t$ is set by the inflationary Hubble rate, $r$ reads off the energy scale (Worked example 2). The large-angle EE and TE signal is dominated by a second scattering epoch — reionization — whose strength is the Thomson optical depth accumulated since reionization.

\tau = \int_0^{z_{\rm re}} n_e(z)\,\sigma_T\, \frac{c\,\mathrm{d}t}{\mathrm{d}z}\,\mathrm{d}z \approx 0.054 \qquad (8)

where: $\tau$ — Thomson optical depth to reionization (Planck 2018: $0.054\pm0.007$ ); $n_e$ — free-electron number density; $z_{\rm re}\approx 7.7$ . A fraction $\approx\tau$ of photons rescatter, so polarization scales as $\tau$ , hence $\tau^2$ in power. Finally, gravitational lensing by intervening matter shears pure-E patterns into ones with a B-component; the lensing B-spectrum is a convolution of the deflection-potential power with the E-mode power.

C_\ell^{\tilde B\tilde B} = \int \frac{\mathrm{d}^2\boldsymbol\ell'}{(2\pi)^2}\,W(\boldsymbol\ell,\boldsymbol\ell')\;C_{|\boldsymbol\ell-\boldsymbol\ell'|}^{\phi\phi}\,C_{\ell'}^{EE} \qquad (9)

where: $C_\ell^{\phi\phi}$ — lensing deflection-potential power; $W$ — geometric mode-coupling kernel; $C_{\ell'}^{EE}$ — unlensed E-spectrum sheared into B. Mixing scales, the lensing B-spectrum is broad, peaking near $\ell\sim 1000$ .

Derivation APolarization from the local temperature quadrupole

The goal is to prove the central claim of the subject: Thomson-scattered light is linearly polarized only if the radiation incident on the electron carries a quadrupole anisotropy, and to trace that quadrupole to the velocity gradient of the photon-baryon fluid. We integrate the scattered intensity against the correct angular weight and use harmonic orthogonality to show only the $\ell=2$ moment survives.

Step 1. Write the scattered intensity as the incident field against the Thomson weight, $I_{\rm sc}(\hat n)\propto\int\mathrm{d}\Omega'\,(1+\cos^2\Theta)\,I_{\rm in}(\hat n')$ , with $\Theta$ the scattering angle. Why: the $\cos^2\Theta$ piece is the dipole re-radiation pattern, the only part that gives a direction-dependent, polarized response.

Step 2. Split the kernel, $1+\cos^2\Theta = \tfrac{4}{3} + (\text{quadrupole terms})$ : the constant $\tfrac{4}{3}$ scatters isotropically and produces no polarization, leaving a pure $\ell=2$ remainder. Why: it separates the part that averages away from the part that can survive as net $Q,U$ .

Step 3. Expand the incident field, $I_{\rm in}(\hat n')=\sum_{\ell m}a_{\ell m}Y_{\ell m}$ ; the $Q$ Stokes parameter picks up the weight $\sin^2\theta'\cos 2\phi'$ , which is exactly the real quadrupole harmonic $Y_2^{\pm2}$ . Why: harmonics turn the scattering integral into an overlap evaluable term by term.

Step 4. Orthogonality gives $\int\mathrm{d}\Omega'\,Y_2^{\pm2}Y_{\ell m}=0$ unless $\ell=2$ , so monopole and dipole vanish and $Q-iU\propto a_{2m}^{T,\,\rm local}$ . Why: net polarization is impossible without a local temperature quadrupole at the electron.

Step 5. Identify the source at last scattering, $a_{2m}^{\rm local}\sim\partial_i v_j$ (velocity shear); in tight coupling the quadrupole is suppressed as $\sim k/\dot\tau$ and only grows as coupling loosens. Why: this ties polarization to the fluid velocity, explaining its $\sim 10\%$ smallness and its $90^\circ$ phase offset from the density.

Go deeper · Derivation B · 1 minE-modes from scalars, both E and B from tensors

Derivation BE-modes from scalars, both E and B from tensors

The aim is the discriminant that makes polarization so valuable: a scalar (density) perturbation produces polarization with no B-component, whereas a tensor perturbation (a gravitational wave) produces both E and B. We work in the flat-sky Fourier picture, write the polarization a single plane wave induces in terms of the angle to the wavevector, and apply the E/B projectors.

Step 1. A scalar mode with $\mathbf k$ along $\hat z$ is azimuthally symmetric about $\mathbf k$ , so its polarization is locked parallel or perpendicular to the projected wavevector: $Q(\boldsymbol\ell)\propto\cos 2\varphi_\ell$ , $U(\boldsymbol\ell)\propto\sin 2\varphi_\ell$ , with $\varphi_\ell$ the angle of $\boldsymbol\ell$ from $\mathbf k$ . Why: a scalar has no handedness or transverse direction other than $\mathbf k$ .

Step 2. The flat-sky transform is $\tilde E=Q\cos 2\varphi_\ell+U\sin 2\varphi_\ell$ , $\tilde B=-Q\sin 2\varphi_\ell+U\cos 2\varphi_\ell$ — the small-angle limit of the spin-2 decomposition. Why: E collects polarization aligned with or across $\boldsymbol\ell$ ; B collects the $45^\circ$ , handed part.

Step 3. Substitute the scalar forms into B: $\tilde B\propto-\cos 2\varphi_\ell\sin 2\varphi_\ell+\sin 2\varphi_\ell\cos 2\varphi_\ell=0$ identically. Why: the scalar $\varphi_\ell$ -dependence matches the E projector exactly, so density perturbations give E and TE but zero primordial B.

Step 4. A tensor mode carries intrinsic $\pm2$ helicity, so its shear rotates as $e^{\pm 2i\gamma}$ and is not locked to $\mathbf k$ ; the induced $Q,U$ are misaligned with $\boldsymbol\ell$ and the B projector no longer cancels. Why: the wave's spin-2 handedness is the reflection-odd ingredient a scalar lacks.

Step 5. Hence $C_\ell^{BB,\,\rm scalar}=0$ while $C_\ell^{BB,\,\rm tensor}\propto r$ , peaking at $\ell\sim 80$ (recombination) and $\ell\lesssim 10$ (reionization). Why: a degree-scale primordial B-mode is a near-unambiguous signature of inflationary gravitational waves. *The scalar E-only rule is a linear-theory theorem, broken only by second-order effects such as lensing.*

Go deeper · Derivation C · 1 minThe reionization bump and the optical depth

Derivation CThe reionization bump and the optical depth

This derivation explains why the reionization of the intergalactic medium around $z\sim 8$ produces a distinctive large-angle feature in the EE and TE spectra whose amplitude scales as $\tau^2$ , and therefore measures the optical depth directly. We track the small fraction of photons that rescatter off newly liberated electrons and follow the induced polarization to the power spectrum.

Step 1. The rescattering fraction is small, $P_{\rm scatter}=1-e^{-\tau}\approx\tau\ll 1$ . Why: only a first-order-in- $\tau$ population is reprocessed, so the induced polarization amplitude is linear in $\tau$ .

Step 2. Each rescattered photon is polarized in proportion to the quadrupole $Q_2$ the reionization-era electron sees, $Q,U\propto\tau\,Q_2$ , where $Q_2$ is the free-streamed anisotropy on the horizon scale at $z_{\rm re}$ . Why: the same quadrupole-sourcing physics as at recombination, now with the large-scale anisotropy.

Step 3. That horizon subtends tens of degrees today, so the signal sits at very low multipole, $\ell_{\rm re}\sim\pi/\theta_{\rm horizon}(z_{\rm re})\sim 2\text{–}10$ — the "reionization bump". Why: late rescattering across a large horizon imprints only large-angle structure, cleanly separated from the $\ell\gtrsim 100$ acoustic peaks.

Step 4. Squaring gives power, $C_\ell^{EE}\big|_{\ell\lesssim 10}\propto|Q|^2\propto\tau^2$ . Why: the bump amplitude fixes $\tau$ with almost no other cosmological dependence — the cleanest handle on reionization we have.

Step 5. In TT the amplitude scales as $C_\ell^{TT}\propto A_s\,e^{-2\tau}$ , degenerate in $A_s$ and $\tau$ ; the independent $\tau^2$ bump lifts it, fixing $A_s$ and hence $\sigma_8$ . Planck low- $\ell$ EE gives $\tau=0.054\pm0.007$ . Why: without polarization the primordial normalization, and thus structure growth, stays badly uncertain.

Limiting cases

The polarization signal partitions cleanly by physical source and by angular scale — which is exactly what makes it so diagnostically powerful.

Case	Result	Meaning
Scalar perturbations	$C_\ell^{BB,\,\mathrm{scalar}} = 0$	To linear order, density perturbations produce E-modes and the TE correlation but zero primordial B-modes — the defining discriminant.
Tensor perturbations	$C_\ell^{BB} \propto r\,A_s$	Gravitational waves source B-modes peaking at $\ell \sim 80$ (recombination) and $\ell \lesssim 10$ (reionization); the amplitude is set by the tensor-to-scalar ratio $r$ .
Lensing conversion	$C_\ell^{BB,\,\mathrm{lens}} \neq 0$	Weak lensing shears E-mode photons into a nearly white B-spectrum peaking at $\ell \sim 1000$ ; a confirmed detection and a foreground floor beneath the primordial signal.
Reionization limit	$C_\ell^{EE}\big\vert _{\ell \lesssim 10} \propto \tau^2$	Rescattering off reionization electrons at $z\sim 8$ makes a large-scale bump whose amplitude measures the optical depth $\tau$ .

Worked examples

1 · The reionization optical depth from a fully worked estimate (fully worked)

The IGM reionizes sharply at $z_{\rm re}=7.7$ and is fully ionized thereafter, with present-day free-electron density $n_{e,0}\approx 2.0\times10^{-7}\,\mathrm{cm^{-3}}$ (H fully ionized plus singly-ionized He). With a matter-dominated expansion, $\Omega_m=0.315$ and $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , estimate $\tau$ and compare with Planck.

Step 1. With $n_e(z)=n_{e,0}(1+z)^3$ and, in matter domination, $\mathrm{d}t = -\mathrm{d}z/[H_0\sqrt{\Omega_m}\,(1+z)^{5/2}]$ , the optical depth becomes $\tau=\dfrac{\sigma_T\,c\,n_{e,0}}{H_0\sqrt{\Omega_m}}\displaystyle\int_0^{z_{\rm re}}(1+z)^{1/2}\,\mathrm{d}z$ . Why: the $(1+z)^3$ density and the $(1+z)^{-5/2}$ expansion factor combine to a gentle $(1+z)^{1/2}$ integrand.

Step 2. Evaluate the integral: $\int_0^{7.7}(1+z)^{1/2}\mathrm{d}z=\tfrac{2}{3}\big[(1+z_{\rm re})^{3/2}-1\big]=\tfrac{2}{3}(8.7^{3/2}-1)=\tfrac{2}{3}(25.66-1)=\mathbf{16.44}$ . Why: the steep $(1+z)^{3/2}=25.7$ growth of density with lookback makes even a small ionized fraction matter.

Step 3. Assemble the prefactor with $c/H_0 = 1.38\times10^{28}\,\mathrm{cm}$ and $\sqrt{\Omega_m}=0.561$ : $\dfrac{\sigma_T\,c\,n_{e,0}}{H_0\sqrt{\Omega_m}} = \dfrac{(6.65\times10^{-25})(1.38\times10^{28})(2.0\times10^{-7})}{0.561}=\mathbf{3.26\times10^{-3}}$ (with $\sigma_T$ in $\mathrm{cm^2}$ ). Why: the constants and the density fix the absolute normalization of the scattering probability.

Step 4. Multiply: $\tau = 3.26\times10^{-3}\times 16.44 = \mathbf{0.054}$ , landing on the Planck 2018 value $\tau = 0.054\pm0.007$ . A sharp reionization at z ≈ 7.7 gives τ ≈ 0.054, matching the value read off the low-ℓ EE bump. The precise agreement is partly fortuitous — a realistic ionization history shifts it by tens of percent — but a few-percent $\tau$ is exactly what a $z_{\rm re}\sim 8$ reionization delivers.

2 · The inflationary energy scale from the B-mode bound

The current bound is $r<0.032$ . With $A_s = 2.1\times10^{-9}$ and reduced Planck mass $M_{\rm Pl}=2.435\times10^{18}\,\mathrm{GeV}$ , what upper limit does this place on the inflationary energy scale $V^{1/4}$ ?

Step 1. Use the single-field relation $V^{1/4}=\big(\tfrac{3\pi^2}{2}A_s\,r\big)^{1/4}M_{\rm Pl}$ , from the tensor power $\mathcal{P}_t = \tfrac{2}{\pi^2}(H/M_{\rm Pl})^2$ and $V\simeq 3H^2M_{\rm Pl}^2$ in slow roll. Why: the tensor amplitude is set by the inflationary Hubble rate, so a B-mode is a direct scale reading of the potential.

Step 2. Evaluate the bracket at the bound: $\tfrac{3\pi^2}{2}=14.80$ ; times $A_s=2.1\times10^{-9}$ gives $3.11\times10^{-8}$ ; times $r=0.032$ gives $9.95\times10^{-10}$ , whose fourth root is $5.62\times10^{-3}$ . Why: the one-quarter power turns a tiny dimensionless combination into an order-unity fraction of the Planck mass.

Step 3. Multiply by $M_{\rm Pl}$ : $V^{1/4} = 5.62\times10^{-3}\times 2.435\times10^{18}\,\mathrm{GeV} = \mathbf{1.4\times10^{16}\ GeV}$ , i.e. $V^{1/4}\approx 1.0\times10^{16}\,(r/0.01)^{1/4}\,\mathrm{GeV}$ , near the grand-unification scale. The bound r < 0.032 caps the inflationary energy scale at V^{1/4} ≲ 1.4×10¹⁶ GeV, with Hubble rate H_inf ≲ 4×10¹³ GeV — read directly off a polarization map.

Physics of the portrait

The E/B decomposition looks inevitable once seen — yet it took thirty years from the first prediction of CMB polarization to the language that now organizes the field, a story best told about the concept itself.

Origins · 1 mine and b modes and how this was discovered

e and b modes

the parity split of CMB polarization, 1997 · microwave-background cosmology

The idea that the microwave background should be polarized is old: Martin Rees pointed out in 1968 that an anisotropic radiation field at last scattering would leave the light linearly polarized — the quadrupole mechanism at the heart of this article. For decades that stayed a footnote, since the predicted signal was a fraction of a $\mu\mathrm{K}$ . The conceptual leap came in 1997, when two groups independently saw that the frame-dependent $Q$ and $U$ were the wrong variables and that the field should be split by parity into a gradient part and a curl part. Marc Kamionkowski, Arthur Kosowsky and Albert Stebbins called them "gradient" and "curl"; Uroš Seljak and Matias Zaldarriaga called them E and B — and the name stuck. Both saw the payoff at once: scalars make only E, so a curl-type B-mode would be a fingerprint of inflationary tensor perturbations. Zaldarriaga and Seljak (1998) soon showed lensing converts some E into B; detections followed — DASI's E-modes and TE in 2002, the lensing B-mode in 2013 — and the primordial B-mode became a central quarry of cosmology. What began as Rees's footnote is now the sharpest knife we have for cutting inflation apart from its alternatives.

History

1968

— Rees predicts the CMB should be linearly polarized by an anisotropic radiation field at last scattering

1997

— Kamionkowski, Kosowsky and Stebbins, and independently Zaldarriaga and Seljak, define the E and B decomposition

2002

— DASI reports the first detection of CMB E-mode polarization and the TE correlation

2013

— SPTpol and POLARBEAR detect the lensing-induced B-mode

2021

— BICEP/Keck combined with Planck tighten the primordial bound to r < 0.032

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., Ch. 10 · Ryden, Introduction to Cosmology, 2nd ed., Ch. 8 · Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 6 · Hu and White, "A CMB Polarization Primer", New Astronomy 2 (1997) 323. Key values used throughout: $\tau=0.054\pm0.007$ , $z_{\rm re}\approx 7.7$ , $A_s=2.1\times10^{-9}$ (Planck 2018); $r<0.032$ (BICEP/Keck); $\sigma_T=6.65\times10^{-29}\,\mathrm{m^2}$ . The central analytic results are: (i) monopole and dipole vanish in the Thomson integral, leaving only $\ell=2$ ; (ii) the flat-sky projector gives $\tilde B=0$ for a scalar; (iii) the EE reionization bump scales as $\tau^2$ ; and the worked numbers $\tau\approx 0.054$ and $V^{1/4}\approx 1.0\times10^{16}(r/0.01)^{1/4}$ GeV.

Prerequisites. The CMB Anisotropy Power Spectrum · Acoustic Oscillations in the Photon-Baryon Fluid · Density Perturbations and the Density Contrast · The Primordial Power Spectrum · The ΛCDM Concordance Model. Leads to. Baryon Acoustic Oscillations · The Transfer Function and the Matter Power Spectrum · Dark Energy and the Cosmological Constant.

Ask the tutor: Ask me to walk through why a quadrupole — and nothing lower — is what Thomson scattering needs to polarize the CMB, or to show how a single measured B-mode would fix the energy scale of inflation.

Why is the CMB only about ten percent polarized, not fully polarized?

Because the quadrupole that drives polarization is small, being suppressed while the plasma is tightly coupled. In tight coupling, frequent scattering keeps the photon distribution nearly isotropic, so the local quadrupole is only of order $k/\dot\tau$ — the ratio of wavelength to mean free path — which is small. Polarization is generated only in the brief window as coupling loosens at recombination, so it inherits that suppression: roughly $10\%$ of the temperature anisotropy, about a $\mu\mathrm{K}$ — which is why it was detected four decades after the temperature signal.

What, in plain terms, is the difference between an E-mode and a B-mode?

Both describe how the little polarization "sticks" are arranged on the sky. An E-mode pattern is symmetric under reflection: around a hot or cold spot the sticks point straight at it (radial) or straight around it (tangential), like the spokes of a wheel or its rim. A B-mode pattern is handed — the sticks curl at $45^\circ$ , forming pinwheels that look different in a mirror. The formal definitions are global spin-2 transforms, but the mirror test captures it: E is what gradients build (parity even), B needs a curl (parity odd). Scalars have no handedness, so they make only E.

Why does the reionization signal appear only at very low multipoles?

Because the electrons that rescatter during reionization sit at redshift $z\sim 8$ , where the causal horizon subtends tens of degrees on today's sky. Polarization can only be imprinted on scales up to that horizon, i.e. multipoles of just a few ( $\ell\lesssim 10$ ). The recombination polarization, by contrast, comes from a horizon at $z\approx 1100$ subtending about a degree, landing at $\ell\gtrsim 100$ . The two epochs thus occupy cleanly separated parts of the spectrum, making the low- $\ell$ "reionization bump" an isolated, clean measurement of $\tau$ .

If a B-mode is detected, how do we know it is primordial rather than dust or lensing?

By where it sits and how it behaves across frequency. Lensing B-modes peak near $\ell\sim 1000$ with a known shape set by the measured E and lensing spectra, so they can be modeled and subtracted (delensing). Galactic dust and synchrotron are separated by their spectra — dust brightens toward high frequency, synchrotron toward low, while the CMB is a blackbody — so multi-frequency observations fit and remove them. Only after delensing and foreground cleaning can a residual degree-scale ( $\ell\sim 80$ ) B-mode be attributed to primordial gravitational waves — which is why the BICEP/Keck analysis leans so heavily on Planck's frequency coverage.

What does a measured value of r actually tell us about inflation?

It measures the amplitude of the primordial gravitational-wave background, which in single-field inflation is fixed by the Hubble rate — hence the energy density — during inflation. Through $V^{1/4}\approx 1.0\times10^{16}(r/0.01)^{1/4}$ GeV, a detection reads off the inflationary energy scale directly, typically near grand unification. It also constrains the potential's shape: by the Lyth bound, an observable $r$ implies the inflaton rolled over a super-Planckian distance, favoring large-field models — the simplest of which $r<0.032$ already rules out, while $r\sim 10^{-3}$ would decisively select among the survivors.

Is the TB or EB cross-spectrum really always exactly zero?

In the standard model, yes — parity conservation forbids any correlation between the parity-even fields (T, E) and the parity-odd field (B), so $C_\ell^{TB}=C_\ell^{EB}=0$ by symmetry, independent of the perturbations, which makes these spectra powerful null tests. A measured nonzero $EB$ would signal parity violation, the leading candidate being "cosmic birefringence" — a rotation of the polarization plane in transit, as from a light axion-like field coupled to electromagnetism. Recent Planck reanalyses report a hint of such a rotation at about $0.3^\circ$ , which, if confirmed, would be physics beyond the standard model.