A CMB Polarization Primer

TL;DR

The paper provides a pedagogical overview of CMB polarization, showing how Thomson scattering converts last-scattering quadrupoles into polarization and how scalar, vector, and tensor perturbations imprint distinct $E$- and $B$-mode patterns. It emphasizes the utility of polarization for isolating last-scattering physics, breaking degeneracies in cosmological parameters, and testing inflation versus defects and adiabatic versus isocurvature initial conditions. The authors discuss how the temperature–polarization cross-correlation, reionization signatures, and future polarization measurements (notably $E$-mode, $B$-mode) can constrain the early universe and the nature of primordial fluctuations, while addressing foregrounds and data-analysis challenges. Overall, polarization offers a complementary and powerful probe of cosmology with the potential to reveal the physics of the early universe and the processes that shaped structure formation.

Abstract

We present a pedagogical and phenomenological introduction to the study of cosmic microwave background (CMB) polarization to build intuition about the prospects and challenges facing its detection. Thomson scattering of temperature anisotropies on the last scattering surface generates a linear polarization pattern on the sky that can be simply read off from their quadrupole moments. These in turn correspond directly to the fundamental scalar (compressional), vector (vortical), and tensor (gravitational wave) modes of cosmological perturbations. We explain the origin and phenomenology of the geometric distinction between these patterns in terms of the so-called electric and magnetic parity modes, as well as their correlation with the temperature pattern. By its isolation of the last scattering surface and the various perturbation modes, the polarization provides unique information for the phenomenological reconstruction of the cosmological model. Finally we comment on the comparison of theory with experimental data and prospects for the future detection of CMB polarization.

Figures (15)

• Figure 1: Thomson scattering of radiation with a quadrupole anisotropy generates linear polarization. Blue colors (thick lines) represent hot and red colors (thin lines) cold radiation.
• Figure 2: The scalar quadrupole moment ($\ell=2,m=0$). Flows from hot (blue) regions into cold (red), ${\bf v} \parallel {\bf k}$, produce the azimuthally symmetric pattern $Y_2^0$ depicted here.
• Figure 3: The transformation of quadrupole anisotropies into linear polarization. (a) The orientation of the quadrupole moment with respect to the scattering direction $\hat{\bf n}$ determines the sense and magnitude of the polarization. It is aligned with the cold (red, long) lobe in the $\hat{\bf e}_\theta \otimes \hat{\bf e}_\phi$ tangent plane. (b) In spherical coordinates where $\hat{\bf n} \cdot \hat{\bf k} = \cos\theta$, the polarization points north-south ($Q$) with magnitude varying as $\sin^2\theta$ for scalar fluctuations.
• Figure 4: Polarization pattern for $\ell=2$, $m=0$, note the azimuthal symmetry. The scattering of a scalar $m=0$ quadrupole perturbation generates the electric $E$ (yellow, thick lines) pattern on the sphere. Its rotation by $45^\circ$ represents the orthogonal magnetic $B$ (purple, thin lines) pattern. Animation (available at http://www.sns.ias.edu/$\sim$whu/polar/scalaran.html): as the line of sight $\hat{\bf n}$ changes, the lobes of the quadrupole rotate in and out of the tangent plane. The polarization follows the orientation of the colder (red) lobe in the tangent plane.
• Figure 5: The vector quadrupole moment ($\ell=2$, $m=1$). Since ${\bf v} \perp {\bf k}$, the Doppler effect generates a quadrupole pattern with lobes $45^\circ$ from ${\bf v}$ and ${\bf k}$ that is spatially out of phase (interplane peaks) with ${\bf v}$.