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The parity-odd part of linear sky polarization is represented by the pseudoscalar potential $P_B$. At linear order an axisymmetric scalar cosmological perturbation gives no physical B mode. Tensor or vector sources and lensing conversion of E-mode polarization are outside that scalar-source statement.

Thomson scattering generates linear Cosmic microwave background polarization when the incident radiation has a photon quadrupole. The cross-section prefers electric fields perpendicular to the scattering direction. Contributions from an isotropic field cancel, and a dipole alone does not supply the required spin-two angular pattern. Schematically the outgoing Stokes combination is proportional towhich selects the incident quadrupolar angular dependence. Tight coupling nearly isotropizes the radiation, making the quadrupole small; it develops when the photon mean free path becomes finite near decoupling. At leading tight-coupling order its size scales as $(k/\dot{\kappa })v_{\gamma }$.

The polarization is therefore weaker than the temperature anisotropy: a characteristic amplitude is of order a few percent to roughly ten percent of the temperature signal, or several microkelvin compared with temperature fluctuations of several tens of microkelvin on acoustic scales. This is an order-of-magnitude comparison, not a scale-independent ratio of every power-spectrum multipole.

The photon monopole follows the acoustic displacement, approximately $\cos (k{r}_s)$; the velocity sourcing the quadrupole follows $\sin (k{r}_s)$. Hence polarization maxima occur near displacement zeros, between the main temperature peaks. In the leading coherent-oscillator approximation the shift is half the peak spacing, $\Delta \ell /2$, although projection and gravitational driving modify exact locations. Scalar perturbations generate E-mode polarization at linear order, not B-mode polarization. Tensor perturbations and lensing can give B modes, and reionization generates an additional large-angle polarization feature.

The cosmic microwave background is relic thermal radiation from the hot early universe. Its temperature anisotropy is expanded in spherical harmonics, with angular power spectrum $C_{\ell }^{TT}$. Thomson scattering of a local radiation quadrupole generates linear polarization around last scattering and again during reionization. Polarization maps are decomposed into E-mode polarization and B-mode polarization, giving spectra such as $EE$, $TE$ and $BB$ in addition to temperature information.

Cosmic microwave background acoustic peak positions compare the sound horizon with the angular-diameter distance to last scattering, constraining geometry and expansion history. Relative peak heights constrain baryon loading and matter-radiation equality; the damping tail constrains diffusion and other early-universe physics. The scalar spectral index and primordial scalar amplitude constrain inflationary perturbations. Coherent acoustic structure and temperature-polarization correlations support predominantly adiabatic initial perturbations. Large-angle E-mode polarization measures the reionization optical depth, helping break its degeneracy with the scalar amplitude inferred from small-scale temperature anisotropy.

Under a specified cosmological model, these data constrain ordinary and cold dark matter densities, the Hubble constant, spatial curvature and cosmic age. Complementary distance information helps break the geometrical degeneracy: temperature peaks alone do not uniquely establish a cosmological constant. The Planck 2013 parameter analysis used Planck temperature data and, in important combinations, WMAP low-multipole polarization; it should not be described as a full Planck polarization release.^[ref]

Linear scalar cosmological perturbations generate E-mode polarization but no primordial B-mode polarization in the simplest treatment. tensor cosmological perturbations can generate B-mode polarization, making them a possible probe of inflationary gravitational waves. gravitational lensing converts some E-mode power into B-mode polarization, however, and Galactic dust and synchrotron radiation also contribute polarized foregrounds. Multifrequency observations, foreground modelling, beam calibration and control of temperature-to-polarization leakage are therefore necessary. Polarization does not by itself establish a primordial tensor signal. temperature and polarization jointly test the expansion history and primordial perturbations, while complementary data and foreground control determine how uniquely parameters can be inferred.