Cosmology · Complementary probes of cosmic history

CMB temperature, polarisation and lensing information

Separate a temperature pattern, a polarization basis and a foreground remapping using one supplied analytic patch.

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

Describe the different information in temperature and linear polarization.
Show how changing polarization axes changes components while preserving amplitude.
Explain how a supplied deflection remaps an existing temperature pattern.

Intensity and polarization describe different properties of light

CMB temperature anisotropy describes small differences in the microwave radiation received from different directions, using its frequency-dependent intensity to express a temperature pattern.

Linear polarization also describes an orientation. Its components Q and U depend on the chosen pair of axes on the sky; their values need not be determined by the temperature fluctuation at the same position.

Thomson scattering can produce linear polarization when the incident radiation has a quadrupolar directional anisotropy. The resulting pattern carries information about the radiation and scattering environment, alongside shared cosmological and observational assumptions.

Lensing supplies another physical response: foreground gravitational deflection changes which part of an existing radiation pattern is seen along an observed direction. It does not require that the temperature pattern was newly generated at the foreground lens.

Declare the pattern and its units

Use a small flat patch and a one-dimensional slice x = θ/θ_*, with θ_* = 0.01 radians and 0 ≤ x ≤ 1. Supply an arbitrary positive temperature reference T_*, rather than a measured CMB temperature.

Define t = (T−T_*)/T_* = 0.001 cos(πx). Represent Q and U in thermodynamic-temperature units and define q = Q/T_* = 0.0002 cos(2πx) and u_pol = U/T_* = 0.0002 sin(2πx).

The linear-polarization amplitude is √(q²+u_pol²) = 0.0002 at every x. The separate temperature and polarization amplitudes and phases are stipulated functions, not a recombination calculation or an observed sky.

E and B polarization patterns are nonlocal spatial combinations of a polarization field; they are not simply new names for the two local numbers Q and U. No E/B decomposition or angular power spectrum is calculated here.

A deflection samples a different part of the same pattern

Supply a smooth weak angular deflection α_θ = θ_*d(x), with d(x) = 0.02 sin(πx). The source coordinate is x_source = x+d(x).

The remapped temperature is t_lensed(x) = 0.001 cos[π(x+d(x))]. At x = 1/2, the original t is zero, but the shifted coordinate is 0.52 and the remapped t is about −0.000062791.

The coordinate-map derivative is 1+0.02π cos(πx), ranging from about 0.937168 to 1.062832 and remaining positive. The endpoints map to themselves, so the full slice stays within the supplied domain without a fold.

This construction applies a chosen deflection to temperature only. It is not a recovered lens potential from data, a self-consistent foreground simulation or a calculation of lensed polarization.

Supplied analytic temperature slice; t columns are multiplied by 1000 and are not observed fluctuations
x	d(x)	x_source	1000t	1000t_lensed
0	0	0	1	1
0.25	0.014142	0.264142	0.707107	0.675003
0.5	0.02	0.52	0	−0.062791
0.75	0.014142	0.764142	−0.707107	−0.737815
1	0	1	−1	−1

Different components can describe the same polarization

As a separate basis example, rotate the polarization axes clockwise by γ = π/8 using the stated convention. The components obey q′ = q cos(2γ)−u_pol sin(2γ) and u_pol′ = q sin(2γ)+u_pol cos(2γ).

At x = 0, the original components are (q,u_pol) = (0.0002,0). The rotated components are both 0.0002/√2 ≈ 0.000141421.

Their squared sum remains 0.0002². The same invariance holds at every x because the component transformation is a rotation through twice the basis angle.

Changing coordinate axes is different from gravitationally remapping a sky pattern. The component rotation here does not calculate a new physical polarization field or a full lensed sky.

This original analytic small flat patch supplies θ_* = 0.01 radians, an arbitrary positive T_*, x = θ/θ_* from 0 to 1, t = (T−T_*)/T_* = 0.001 cos(πx), and d = 0.02 sin(πx). Panel A has linear x from 0 to 1 and linear scaled fractional temperature 1000t from −1 to 1. Solid blue circles show the original cosine; dashed orange squares show t_lensed = 0.001 cos[π(x+d)]. At x = 0, 0.25, 0.5, 0.75, 1 the original scaled values are 1, approximately 0.707107, 0, −0.707107, −1, and remapped scaled values are 1, approximately 0.675003, −0.062791, −0.737815, −1. Endpoints intentionally coincide. Panel B has the same x domain and a separate magnified dimensionless difference 10⁶(t_lensed−t) from −70 to 0, identified by a dotted violet curve with diamonds. At x = 0.5, x_source = 0.52 and the difference is approximately −62.790520. This is a deterministic difference, not an uncertainty band. The map derivative 1+0.02π cos(πx) is at least 0.937168, and fixed endpoints plus monotonicity keep source coordinates within the supplied domain. Only scalar temperature is remapped. Panel C is a separate basis example at x = 0. Stokes components in thermodynamic-temperature units obey q = Q/T_*, u_pol = U/T_*; their amplitude is 0.0002. The component axes 10⁴q and 10⁴u_pol both run from −0.5 to 2.5 with exactly equal pixel scales. A solid blue vector with a circle ends at (2,0). A clockwise basis rotation γ = π/8 under the stated convention gives q′ = q cos(2γ)−u_pol sin(2γ), u_pol′ = q sin(2γ)+u_pol cos(2γ); a dashed orange vector with a square ends at (√2,√2). A dotted green quarter-circle of radius 2 passes through both endpoints and records the unchanged amplitude; it is a reference locus, not data or a physical polarization-angle arrow. All plotted quantities are dimensionless after their explicit scaling. Panel C changes coordinate components, not the physical polarization field, and does not calculate lensing of polarization or E/B decomposition. These supplied patterns do not calculate primordial generation, recombination, a spectrum, a recovered lens, measured CMB amplitudes or three independent datasets. Distinct lines, shapes, panel labels and exact formulas provide noncolor interpretation. Original Ark teaching model.

Three responses do not guarantee three independent datasets

Temperature, polarization and lensing can respond differently to physical ingredients, but their signals may be correlated. Calibration, foreground subtraction, shared sky coverage and cosmological assumptions also connect their interpretation.

A real inference requires a joint response and uncertainty model for the maps or statistics used. The small analytic slice supplies no measured covariance, fitted parameter, primordial amplitude or detection significance.

Optional tensor rotation and differentiation of a deflection potential can verify the mathematics of this patch. They do not replace the physical calculation of primordial generation, scattering and line-of-sight evolution.

The useful first distinction is what the light carries: intensity variation, polarization strength and orientation, and a pattern modified by intervening geometry. Their complementarity is assessed through those responses and dependencies.

Worked reasoning

Rotate the components at x = 0

Can Q and U change while the linear-polarization amplitude stays fixed?

At x = 0, q = 0.0002 and u_pol = 0.
For clockwise γ = π/8, both rotated components are 0.0002/√2.
Their amplitude is √[2(0.0002/√2)²] = 0.0002.

Yes: changing the stated basis changes the components while preserving the polarization amplitude.

Try the reasoning

At x = 1/2, does the remapped temperature equal the original zero value?
Show the answer and reasoning
No: t_lensed ≈ −0.000062791.
The supplied deflection samples the original pattern at x_source = 0.52.
Are E and B simply the two local component values Q and U?
Show the answer and reasoning
No.
E/B decomposition uses spatial information in the polarization field and is not performed by relabelling local components.
Does listing temperature, polarization and lensing establish three independent datasets?
Show the answer and reasoning
No.
Physical correlations and shared calibration, sky and model assumptions require an explicit joint treatment.

What to carry forward

Temperature and polarization encode different properties of received radiation.
A polarization-basis change preserves amplitude while changing components.
A deflection remaps an existing pattern, and complementary responses can still share information and systematics.

References

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