Cosmology · Recombination, last scattering and the CMB

Polarisation and information beyond temperature

Compare two specified four-beam Thomson-scattering examples with identical outgoing intensity but different linear polarisation.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Explain how scattering geometry selects an outgoing electric-field orientation.
Calculate intensity and Stokes components for a supplied four-beam model.
Distinguish basis-dependent local Q/U from spatial E/B polarisation patterns.

Light can carry orientation information

An intensity measurement records one aspect of radiation. Linear polarisation adds information about the electric-field orientation in a plane perpendicular to propagation, allowing two radiation fields with the same intensity to differ.

Thomson scattering by free electrons can generate linear polarisation from suitable incident angular anisotropy. In the continuous-sky CMB treatment, the local incident quadrupole is the relevant source of this effect.

The geometric calculation below samples only four ideal incoming beams and one outgoing direction. It demonstrates cancellation and imbalance without claiming a full-sky quadrupole normalization or a real CMB polarisation amplitude.

A propagation arrow and an electric-field orientation are different objects. Keeping them distinct is essential for understanding why a bright incoming beam can strengthen an outgoing polarisation perpendicular to that beam’s scattering plane.

Observe along z and compare two transverse components

Observe single optically thin Thomson scattering toward +z from an electron at rest. Four unpolarised incoming pencil beams propagate along ±x and ±y, so each scatters through 90 degrees into the chosen outgoing direction.

At this angle, the outgoing surviving polarisation is perpendicular to the scattering plane. An x-directed incident beam has an x–z plane and contributes to the y-polarised outgoing intensity. A y-directed input similarly contributes to the x-polarised output.

Absorb the common positive cross-section and unpolarised half-intensity factor into A. For supplied beam weights, I_x/A = w_(+y) + w_(−y), while I_y/A = w_(+x) + w_(−x).

Define total I = I_x + I_y and Stokes Q = I_x − I_y in this chosen x,y basis. The symmetric cases here have U = 0 and circular Stokes V = 0. Their linear polarisation fraction is p = √(Q² + U²)/I.

The same total intensity can hide an angular imbalance

First give every incoming beam weight 2. The two outgoing components are I_x = I_y = 4A, so I = 8A, Q = 0 and p = 0 in the selected outgoing direction.

Next choose weights 3, 3, 1, 1 along +x, −x, +y, −y respectively. Now I_x = 2A and I_y = 6A. The total remains 8A, but Q = −4A and p = 1/2.

The negative Q sign means the y component is larger in the declared convention. The brighter x-directed incoming beams feed that y component because of the perpendicular scattering geometry.

Equal weights in four sampled directions are not a perfectly isotropic full sky. They cancel the two selected outgoing linear components in this construction. The chosen 50 percent polarisation is a visible toy value, not an observed CMB fraction.

Four ideal unpolarised incoming beams, one +z outgoing direction, common positive response A; weights are ordered +x, −x, +y, −y
Case	Incoming weights	I_x/A	I_y/A	I/A	Q/A	U/A	p
Equal four directions	2, 2, 2, 2	4	4	8	0	0	0
Contrasted directions	3, 3, 1, 1	2	6	8	−4	0	1/2

The components change while the physical fraction remains

Q and U depend on the transverse axes used to describe the same radiation. Under a passive counter-clockwise basis rotation by α, use Q′ = Q cos(2α) + U sin(2α) and U′ = −Q sin(2α) + U cos(2α).

For the contrasted case, a 45-degree rotation has α = π/4 and therefore Q′ = 0 and U′ = 4A. The linear fraction remains √(Q′² + U′²)/I = 1/2.

A zero Q in the rotated basis does not mean that the light has become unpolarised; U now carries the same linear-polarisation magnitude. The factor of two in the rotation is tied to the orientation symmetry of linear polarisation.

The basis rotation does not change the outgoing propagation direction or cause a new scattering event. It changes only the coordinate description of the already specified transverse field statistics.

Original four-pencil-beam Thomson geometry, not a full CMB sky. A resting electron in a single optically thin scattering model is viewed along +z, out of the diagram toward the viewer. Incoming arrows denote propagation along +x, −x, +y and −y, not electric orientation. The beams are unpolarised and all scatter through 90 degrees. Projection makes the x-directed beams feed outgoing y electric intensity, while y-directed beams feed outgoing x electric intensity. A > 0 absorbs the common cross-section, incident intensity unit and unpolarised half-intensity factor. Equal input weights 2, 2, 2, 2 give I_x = I_y = 4A. Contrasted weights 3, 3, 1, 1 give I_x = 2A and I_y = 6A. Four bars compare the components on one intensity/A scale from 0 to 6; solid blue and dashed orange outlines supplement colour. Both cases have I = 8A. With Q = I_x − I_y and U = V = 0, the equal case has Q = 0 and p = 0; the contrasted case has Q = −4A and p = 1/2. A passive counter-clockwise 45-degree basis rotation gives Q′ = 0 and U′ = +4A, preserving p = 1/2. It causes no propagation change or new scattering. Four equal directions are not full-sky isotropy; local Q/U are not E/B spatial patterns. No measured CMB fraction or primordial gravitational-wave detection is inferred. Original Ark teaching model.

E and B require information across the sky

CMB polarisation maps contain a spatial pattern of Q and U relative to a stated basis convention. E and B describe particular spatial pattern information derived from that field, rather than simply renaming Q and U at one point.

Polarisation can therefore test aspects of the radiation and perturbation history that intensity alone does not determine. Quantitative predictions require the full angular source, propagation and measurement models.

A local direction or one toy scattering fraction cannot identify a primordial gravitational-wave signal. Foregrounds, lensing, instrumental response and the wider spatial pattern enter actual interpretation.

The four-beam calculation establishes a narrower concrete result: equal outgoing intensity can coexist with different linear polarisation, and the scattering geometry explains which electric-field component is enhanced.

Worked reasoning

Follow the brighter incident beams to the outgoing component

Why does the contrasted example have a larger y-polarised output when its brighter incident beams run along ±x?

An x-directed incoming beam scattered toward z defines the x–z scattering plane.
The surviving 90-degree outgoing component is perpendicular to that plane, along y.
The two x-directed weights 3 + 3 give I_y = 6A, while the two y-directed weights 1 + 1 give I_x = 2A.

Scattering selects the perpendicular electric-field orientation, producing Q = −4A and p = 1/2 in the declared basis.

Try the reasoning

Is the equal four-beam input a perfectly isotropic full sky?
Show the answer and reasoning
No.
It is a limited angular sampling that cancels the two outgoing linear components in the selected direction.
May Q and U at one point be relabelled E and B?
Show the answer and reasoning
No.
E/B information requires the spatial polarisation pattern and a stated convention.
After the stated 45-degree basis rotation, does Q′ = 0 imply no linear polarisation?
Show the answer and reasoning
No; U′ = 4A and p stays 1/2.
The same physical linear polarisation is represented by different basis components.

What to carry forward

Scattering geometry can turn an angular intensity imbalance into linear polarisation.
Two fields with identical outgoing intensity can carry different polarisation.
Local Q/U components depend on basis, while E/B concern the spatial pattern.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.