Polarization observations of the CMB provide an extremely important probe of fluctuations in the early universe.

(a) By invoking the electromagnetic features of Thomson scattering by free electrons, give a heuristic demonstration of why a net linear polarization signal is expected.

(b) Using the Monte Carlo formalism sketched in Sec. 28.6.1, calculate the polarization expected from a single fluctuational mode Φ̃e^iκ·χ of given amplitude.

(c) The description of polarization has many similarities with the formalism we have outlined to describe cosmic shear. Linear polarization is unchanged by rotation through π just like a shear deformation. In addition, the polarization pattern that should be seen from a single inhomogeneity mode should have an electric vector that alternates between parallel and perpendicular to the projection of the mode’s wave vector k on the sky, just like the elongation of the images of background galaxies in weak gravitational lensing. We do not expect to produce a signal in either case along a direction at 45◦ to the projection of k. These predicted polarization/shear patterns are commonly called “E-modes.” However, as we discuss in Sec. 28.7.1, primordial gravitational wave modes may also be present. Explain qualitatively how a single gravitational wave mode can produce a “B-mode” polarization pattern with electric vectors inclined at ±45◦ to the direction of the projection of k on the sky.

(d) When one sums over inhomogeneity modes, and over gravitational-wave modes, the resulting polarization E-modes and B-modes have distinctive patterns that differ from each other. In what ways do they differ? Read, explain, and elaborate the discussion of E-modes and B-modes near the end of Sec. 28.6.1.

(e) Outline how our perturbed metric, Eq. (28.44), would have to be modified to accommodate the presence of primordial gravitational waves.

Equation 28.44.

Transcribed Image Text:

ds² = (1+20)dt² + a²(1-24)8;jdx'dx'. (28.44)