Simple examples of Thomson scattering with the charge being illuminated from a single direction. The two cases correspond to the incident light being either (a) linearly polarized or (b) unpolarized. For clarity, only the electric field of the radiation propagating along the coordinate axes is illustrated.  

Thomson scattering with an anisotropic incident radiation field. The incident (unpolarized) radiation is more intense along one axis than the other, which causes the charge to oscillate more strongly along one axis than the other. The radiation emitted by the accelerating charge along the third axis consequently has a finite polarization. Only a quadrupole moment in the intensity of the unpolarized radiation field can generate polarization in this way, since other moments do not define unique axes along which the motion of the charge is a maximum or a minimum.

Heuristically, Thomson scattering occurs when an incident electromagnetic wave causes a charged particle to oscillate back and forth, and the accelerated charge re-emits radiation over a wide range of angles. The intensity and polarization of the scattered light depends on both the direction and polarization of the incident radiation, as given by the cross section.

For a simple example of this process, consider the case in which a charge is illuminated from a single direction with linearly polarized light, as illustrated in Figure 1.2a. In this situation, the charge moves along a single axis, and the light emitted by the moving charge is simple dipole radiation. If, on the other hand, the incident light is unpolarized, as shown in Figure 1.2b., then the motion of the charge may be decomposed into oscillations along two orthogonal axes. With the charge confined to move in a single plane, radiation scattered into this plane must be polarized parallel to it, thus demonstrating that Thomson scattering can indeed generate polarization from initially unpolarized light. Of course, the electrons in the early universe were not illuminated from a single direction, but were instead bathed in a nearly isotropic sea of photons. If this incident radiation were perfectly isotropic, then there would be no favored direction, and the scattered light could not be polarized. However, since the primordial plasma was not completely homogeneous, the incident radiation on the charged particles was not, in general, perfectly isotropic. In particular, there could have been quadrupole moments in the intensity of the radiation incident upon these charges. Such a quadrupole moment, as shown in Figure 1.3, produces scattered light with a finite polarized component. At first, it might appear that these polarization-generating quadrupole moments were the result of variations in the temperature of the primordial plasma. In fact, such temperature gradients were unable to generate detectable amounts of cosmological polarization. Imagine that the two sources of the incident radiation illustrated in Figure 1.3 are from thermal sources at different temperatures. In this case, the induced motion of the charge along the third axis (not shown) scatters radiation between the two sources, bringing them into thermal equilibrium and washing out the quadrupole moment before much polarized radiation is generated. The expected polarized signals from the CMB were therefore produced by more dynamic aspects of the early universe. One of the more exotic sources of cosmological polarization was primordial gravity waves. Gravity waves passing through the primordial plasma caused the wavelengths of photons propagating in orthogonal directions to be alternately stretched and compressed, producing a quadrupole moment in the redshifts of the radiation (see Figure 1.4). These quadrupole moments in the apparent temperature of the plasma in turn generated a polarized signal with a distinct signature, that, in principle, allows the gravity wave content of the early universe to be quantified (which has relevance to the dynamics of inflation, etc.). Unfortunately, the polarized signal due to gravity waves is expected to be very small (less than 10% of the still undetected polarization signal), and therefore this particular component of the cosmological polarization may remain elusive for some time to come.

Generating polarization with gravity waves. The central figure illustrates a circular patch of homogeneous plasma in the absence of gravity waves. The right and left hand figures show how this patch is distorted by the passage of a (very intense) gravity wave. Just as the shape of the region is stretched along one axis and squashed along the other, the wavelengths of the photons propagating along these axes are distorted. This yields quadrupole moments in the apparent temperature of the radiation, as indicated by the shading. These quadrupole moments can in turn generate polarized signals as shown in Figure 1.3.

Velocity fields and quadrupole moments. (a) The velocity field for a small region of plasma. Material is converging to a point to the left under the influence of gravity, but is slowed down due to photon pressure. (b) The same region in the reference frame of one of the charged particles in the plasma. In this frame the fluid is moving in towards this particle from all directions, and the relative velocities have a distinct quadrupole moment. This results in a quadrupole moment in the redshifts of the radiation arriving at the charge, providing the quadrupole moment in the intensity of the radiation field necessary for the production of polarized radiation (see Figure 1.3)

The primary source of cosmological polarization is a far more mundane process involving velocity gradients in the primordial plasma. Consider a flow field with a gradient, such as that shown in Figure 1.5, which might occur when a plasma flows into a potential well. In the reference frame of one of the charged particles in this plasma, the fluid appears to be converging on this charge. Due to the gradient in the flow, the relative velocities are different along different directions. The redshifts of the radiation emitted from the surrounding plasma towards the particle consequently have a quadrupole moment. This anisotropy in the redshifts creates a quadrupole moment in the apparent temperature of the incident radiation, which, in turn, yields polarization. The polarization produced in this manner comprises most of the polarized component of the CMB.  

The Origin of Cosmological Polarization

The Structure of Cosmological Polarization

        Power Spectra and the Structure of CMB Anisotropies

        Initial Conditions and Inflation

        The Evolution of Perturbations and Acoustic Oscillations

        Acoustic Oscillations and Apparent Temperature Variations

        Acoustic Oscillations and CMB Anisotropies

Foregrounds