research30 min studyLesson 875 in Physics

CMB polarization and E–B mode decomposition

Distinguish local Stokes measurements from spatial E/B patterns, including rotation conventions and foreground or lensing contamination.

Before you begin

On this page

The question for today

What you’ll understand

Compute E and B for a declared flat-sky Fourier convention.
Distinguish coordinate rotation from physical polarization rotation.
Explain why observed B power is not uniquely primordial tensor power.

Linear polarization adds directional information to the microwave sky, but a local polarization stick is not itself an E mode or a B mode. Those labels describe how polarization varies across the sky. They are useful because different perturbations and propagation effects generate different spatial patterns. We will declare a simple flat-sky Fourier convention, calculate its components by hand, and separate a coordinate change from physical mixing. This prevents a common inference error: interpreting any nonzero observed B pattern as an unambiguous detection of primordial gravitational waves.

Thomson scattering needs a quadrupole

Thomson scattering generates net linear polarization when the incoming radiation has a quadrupolar angular anisotropy. An isotropic radiation bath has no preferred outgoing polarization after its contributions are combined. Near last scattering, a finite photon mean free path allows the required quadrupole to develop. Reionization supplies another scattering epoch and affects large angular scales. Polarization therefore depends on the visibility history and radiation perturbations, not merely on the existence of free electrons somewhere along the line of sight.

Stokes Q and U describe linear polarization relative to chosen perpendicular axes on the sky. Rotating those axes by an angle psi changes the pair with twice that angle, because a polarization orientation is unchanged by a half turn. Their numerical values depend on basis choice. A whole-sky treatment uses spin-weighted spherical harmonics; the following small-patch Fourier equations retain the essential spin-two geometry while making all signs explicit.

\begin{pmatrix}E(\boldsymbol\ell)\\B(\boldsymbol\ell)\end{pmatrix}=\begin{pmatrix}\cos2\varphi_\ell&\sin2\varphi_\ell\\-\sin2\varphi_\ell&\cos2\varphi_\ell\end{pmatrix}\begin{pmatrix}Q(\boldsymbol\ell)\\U(\boldsymbol\ell)\end{pmatrix}

Phi ell is the Fourier wavevector’s angle in the same fixed sky basis as Q and U. Fourier amplitudes may be complex; the examples choose real ones. E and B are pattern components, not the two local electric-field components of a photon.↔ Scroll wide equations horizontally.

Scalar sources and propagation have different roles

At first order, scalar density perturbations generate E polarization but no B polarization in a parity-respecting standard treatment. Tensor perturbations can generate both. This source distinction motivates searching for primordial B power. However, gravitational lensing remaps an E pattern and converts some of its power into B. Galactic polarized dust and synchrotron emission also contribute, so the measured B spectrum must be separated into modeled components with frequency and spatial information.

A physical rotation of polarization relative to a fixed sky coordinate system mixes E and B. An instrument’s polarization-angle calibration error can mimic such a rotation. This differs from rotating the coordinate axes and the wavevector description together, which leaves the E/B decomposition invariant. On a masked finite sky, boundary information is incomplete and naive estimators also leak E into B. Pure-mode estimators, simulations, and window treatment address that analysis effect rather than changing the underlying spin-two definitions.

Q'=Q\cos2\psi+U\sin2\psi,\quad U'=-Q\sin2\psi+U\cos2\psi,\quad\varphi_\ell'=\varphi_\ell-\psi;\qquad B_{\mathrm{rot}}=E\sin2\alpha+B\cos2\alpha

Primed quantities describe a passive basis rotation by psi and yield invariant E and B. The final expression instead describes an active physical rotation by alpha in the fixed basis, with the stated sign convention.↔ Scroll wide equations horizontally.

A B-spectrum claim requires a component likelihood

A polarization analysis must propagate beam response, gain and angle calibration, filtering, noise, and sky masks into its prediction. Multifrequency observations help distinguish microwave and Galactic spectra, but foreground spectral variation prevents treating one universal subtraction coefficient as exact. Lensing templates can reduce lensing B variance; their uncertainty and correlations with the original maps must also enter the likelihood. A clean-looking map alone does not establish the origin of a weak residual.

The absence of a primordial B detection would constrain tensor models under the adopted foreground and cosmological assumptions, rather than prove that every possible inflationary history is false. Conversely, a B detection requires consistency in angular shape, frequency behavior, cross-spectra, and null tests before a primordial interpretation is secure. We make no numerical claim about the latest tensor limit here. The exercises test geometric definitions and an illustrative rotation, leaving observational status to a specifically dated analysis.

Work it through

A pure E Fourier mode and an active rotation

In the displayed convention a real Fourier mode has wavevector angle 45 degrees and amplitudes Q = 0, U = 2 microkelvin. Find E and B. Then actively rotate its polarization by 45 degrees in the fixed sky basis while keeping its wavevector unchanged. Find the new E/B amplitudes.

The doubled wavevector angle is 90 degrees, so its cosine is zero and sine is one. Therefore $E=0(0)+2(1)=2\,\mathrm{\mu K}$ and $B=-0(1)+2(0)=0$ .
An active rotation by alpha sends $E_{\mathrm{rot}}=E\cos2\alpha-B\sin2\alpha$ and $B_{\mathrm{rot}}=E\sin2\alpha+B\cos2\alpha$ . Here twice alpha is 90 degrees.
Thus $E_{\mathrm{rot}}=0$ and $B_{\mathrm{rot}}=2\,\mathrm{\mu K}$ . Total squared amplitude stays four square microkelvin. This deliberately large geometric rotation is not a realistic calibration-error estimate or a tensor inference.

The result

The initial mode is pure E with amplitude 2 microkelvin; the stated active rotation makes it pure B with the same amplitude.

The living notebook

Explore the model

Interactive lab

Interactive lab · vectors

A double-angle component analogy

Use the vector angle as the doubled polarization angle. Reset, set Angle = 0° and Magnitude = 2, and predict projections (2, 0). For a fresh comparison Reset, set Angle = 90° and Magnitude = 2, and predict projections (0, 2). In this analogy that vector turn corresponds to a 45° physical polarization turn.

Vectorx / y projectionsTrace marker

x component: 1.393
y component: 0.975
Magnitude: 1.7

Angle35°Magnitude1.7

t = 0 s

Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forOne vector illustrates component rotation only. It has no sky map, spin-weighted harmonics, Fourier wavevector, lensing, or foreground separation and cannot perform an E/B analysis.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Several physical and instrumental mechanisms generate or mimic B power. The scalar-source statement is restricted to first-order generation, and does not remove lensing conversion.

Your turn

Try it yourself

A real Fourier mode has wavevector angle zero, Q = 3 and U = 4 in common units. Find E and B. Rotate only the coordinate basis by 45 degrees, obtaining the primed Stokes components and wavevector angle. Verify the same E and B and their total squared amplitude.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

The passive rotation changes both the Stokes pair and the wavevector angle; twice the primed wavevector angle is minus 90 degrees.

Compare with the solution

Initially E = 3 and B = 4, so squared amplitude is 25. The basis rotation gives $Q'=4$ , $U'=-3$ , and $\varphi_\ell'=-45^\circ$ . Therefore $E'=4(0)+(-3)(-1)=3$ and $B'=-4(-1)+(-3)(0)=4$ . The unchanged result distinguishes a passive coordinate change from the active physical rotation in the worked example.

Keep the idea

Take it with you

Polarization generation needs an incident radiation quadrupole.
E/B are spatial spin-two patterns with declared basis conventions.
Lensing, foregrounds, masks, and angle calibration complicate primordial B inference.

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