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In the previous section, we discussed the physical dynamics of the primordial Universe that imprinted temperature anisotropies on the Cosmic Microwave Background as decoupling took place at the surface of last scattering, associated with acoustic oscillatory behavior

and the Sachs-Wolfe effect in areas of over- and underdensity in the early Universe. It turns out that the dynamics of these mechanisms leading to CMB temperature anisotropy also lead to a polarization signal to be manifest in measurements of the CMB. In particular, the primary mechanism resulting in the polarization of the CMB temperature signal is Thompson scattering of CMB photons off of free electrons at the surface of last scattering, resulting in linear polarization of CMB temperature radiation. In order to source the linear polarization of the CMB, we require that the incident radiation have quadrupolar anisotropy (Hu and White 1997). That quadrupolar anisotropy arises in incident radiation at the surface of last scattering is specifically due to the mechanisms described for the development of CMB temperature anisotropies from oscillatory motion into and out of over- and underdense in the previous section in which photons will either gain or lose energy depending on the region from which they are sourced. An additional source of quadrupolar anisotropy is associated with anisotropies created by compression or expansion within the primordial plasma due to gravitational waves (in which a passing gravitational wave would compress the plasma more in one direction than another), which would be expected in an inflationary cosmology regime. Thus, this leads to radiation of quadrupolar anisotropy incident on free electrons from these two sources at the surface of last scattering. The Thompson scattering of such a quadrupolar radiation field is given in Figure 1.2, in which we define hotter or more intense unpolarized radiation (colored in blue to represent more intense blue shifted regions) and cooler or less intense unpolarized radiation (colored in red to represent less intense red shifted regions) incident on a free electron. Thus, following scattering off of the free electron we are left with resultant polarized radiation that is linearly polarized along the line of sight according to the incident radiation intensities (Carroll and Ostlie 2007; Hu and White 1997; Newburgh 2010).

Viewed broadly, the characterization of CMB temperature and polarization anisotropies require knowledge of three quantities for any given observed point on the sky. These three quantities are: the observed temperature of the CMB blackbody spectrum T, the degree

Figure 1.2: The Thompson scattering of a CMB field. The CMB field is quadrupolar, and here which we define hotter or more intense unpolarized radiation (colored in blue to represent more intense blue shifted regions) and cooler or less intense unpolarized radiation (colored in red to represent less intense red shifted regions) incident on a free electron. Thus, following scattering off of the free electron we are left with resultant polarized radiation that is linearly polarized along the line of sight according to the incident radiation intensities (Hu and White 1997)

of polarization, or polarization intensity P, and the angle between the direction of polar- ization and a given coordinate system on the sky, denoted as α. P and α are traditionally parametrized in terms of the Stokes Parameters, Q and U, defined in Equations 1.30 and 1.31 as:

Q=P cos(2α) (1.30)

U =P sin(2α) (1.31)

It should be pointed out here that Stokes Q and U vectors are elongated from each other by 45◦, which is important for our later discussion of the design of the ACTPol polarimeter array package, which is designed with polarimeters aligned at 0◦ and 45◦ for coverage of Stokes Q and U. These definitions do not quite tell the full story, however, since we can also decompose polarized radiation with Stokes V parameter (though in this case, since the Stokes V has circular polarization and thus can not be sourced from a quadrupolar radiation field undergoing Thompson scattering), so we can define this decomposition of CMB radi- ation in terms of its intensity and polarization more formally, according to (Kamionkowski et al. 1997), in which for radiation propagating along the z-axis, we can decompose its electric fields into Ex and Ey and then yield the corresponding time-averaged Stokes Pa-

rameters P, Q, U, V corresponding to amplitudes ax and ay according to Equations 1.32,

1.33, 1.34, 1.35, 1.36, and 1.37 (Kamionkowski et al. 1997).

Ex =axcos[ω0t−θx(t)] (1.32)

Ey =aycos[ω0t−θy(t)] (1.33)

Q=⟨a2_x⟩ − ⟨a2_y⟩ (1.35)

U =⟨2axaycos(θx−θy)⟩ (1.36)

V =⟨2axaysin(θx−θy)⟩ (1.37)

Similar to our treatment of the temperature anisotropies in the previous section, given that we observe the CMB emanating from the surface of last scattering, we are observ- ing the CMB thus emanating from a spherical shell, so that we can then define the CMB polarization in terms of a decomposition into spherical harmonics for the relevant Stokes pa- rameters for CMB polarization, namely Stokes Q and U as we have discussed. As described in (Zaldarriaga and Seljak 1997), the all-sky expansion in spherical harmonics for CMB polarization must have a spin-weighted basis, so that the expansions in spherical harmonics go as in Equation 1.38 and 1.39. (Q+iU)(θ, ϕ) =∑ ℓm a2,ℓmY2,ℓm(θ, ϕ) (1.38) (Q−iU)(θ, ϕ) =∑ ℓm a₋2,ℓmY−2,ℓm(θ, ϕ) (1.39)

When an instrument like ACTPol makes a polarization-sensitive map of a given field of the CMB, we measure the Stokes Q and U polarization on the sky and then decompose them into two linear orthogonal bases, which have: (i) curl-free E-mode polarization, which arises from plasma accelerating into and out of overdense regions of the early Universe; and (ii) divergence-free B-modes, which, if directly detected, would provide evidence of gravitational waves propagating through space and a verification of inflationary cosmology. We can express the expansion for E-mode and B-mode signal according to Equation 1.40 and 1.41 as described by (Zaldarriaga and Seljak 1997).

E(θ, ϕ) =−1 2 ( δ2(Q+iU) +δ2(Q−iU))=∑ ℓm ( (ℓ+ 2)! (ℓ−2)! )1/2 aE,ℓmYℓm(θ, ϕ) (1.40) B(θ, ϕ) = i 2 ( δ2(Q+iU)−δ2(Q−iU))=∑ ℓm ( (ℓ+ 2)! (ℓ−2)! )1/2 aB,ℓmYℓm(θ, ϕ) (1.41)

With the mathematical expansion for E- and B-modes now in hand, we can jump to their associated power spectra similar to the previous section. As described by Kaminkowski in (Kamionkowski 2002), the two-point statistics of a combined CMB temperature and polarization map is described by six power spectra, given in Equation 6.3.

C_ℓKL, f orK, L=T, E, B (1.42)

However given constraints of parity invariance, C_ℓT B and C_ℓEB are trivial, leaving four relevant power spectra to describe CMB maps in temperature and polarization: C_ℓT T,C_ℓT E, C_ℓEE, andC_ℓBB (Kamionkowski 2002). The corresponding non-trivial power spectra defini- tions forC_ℓT T,C_ℓT E,C_ℓEE, andC_ℓBB, then go as defined in Equations 1.43, 1.44, 1.45, 1.46 as defined by (Zaldarriaga and Seljak 1997).

C_ℓT T = 1 (2ℓ+ 1) ∑ m ⟨a∗_T,ℓmaT,ℓm⟩ (1.43) C_ℓT E = 1 (2ℓ+ 1) ∑ m ⟨a∗_T,ℓmaE,ℓm⟩ (1.44) C_ℓT T = 1 (2ℓ+ 1) ∑ m ⟨a∗_E,ℓmaE,ℓm⟩ (1.45) C_ℓBB = 1 (2ℓ+ 1) ∑ ⟨a∗_B,ℓmaB,ℓm⟩ (1.46)

For the purposes of this manuscript, we shall simplify notation forC_ℓT T,C_ℓT E,C_ℓEE, and C_ℓBB, to TT, TE, EE, and BB, respectively. Figure 1.4 provides a visual representation of the sort of polarization geometry characteristic of E- and B-mode polarization. In particular, the top left inset shows a simulated CMB temperature map with both E- and B-modes represented in a linear combination across the field, with the corresponding decomposed E- and B-mode polarization maps shown for the right insets. The lower left inset shows curl-free E-modes in which E-mode polarization consists of tangential vectors around more intense, or warmer, regions of the CMB (designated in red), and radial vectors around less intense, or cooler, regions of the CMB (designated in blue). Furthermore, the divergence-free B-mode polarization is shown, which shows counter-clockwise vectors around more intense, warmer regions of the CMB, and clockwise vectors for less intense, cooler regions of the CMB. Whereas previous experimental cosmology platforms such as MBAC were able to better constrain characteristics of the primary temperature anisotropy, or TT, power spectrum of the CMB to good agreement withλCDM cosmology models (as did the early temperature results from the Planck mission), next generation, polarization-sensitive instruments like ACTPol would then be able to further probe the other three characteristic power spectra for temperature and polarization maps of the CMB, the temperature-polarization cross power spectrum, TE, and the so-called E-mode, and B-mode power spectra, EE and BB, respectively. Figure 1.3 represents the theoretical predictions of the TT, TE, EE, and BB power spectra, in which the significant drop in intensity for the polarization signals expected for both EE and BB power spectra set the principal technological challenge requiring CMB polarimeters operating with appropriate sensitivities to effectively measure these spectra.

Finally, as described in detail in (Hu and White 1997) and following the physical descrip- tion earlier in this section, we find that while scalar perturbations associated with acoustic oscillations produce solely E-mode polarization, tensor perturbations associated with gravi- tational waves, which are necessary for an inflationary period of the primordial Universe can produce both E-mode and B-mode polarization. Thus, the measurement of the tensor-to-

Figure 1.3: The theoretical predictions of the TT, TE, EE, and BB power spectra, in which the significant drop in intensity for the polarization signals expected for both EE and BB power spectra set the principal technological challenge requiring CMB polarimeters operating with appropriate sensitivities to effectively measure these spectra. (Hu and White 1997)

scalar ratio, denoted asr remains at the forefront of CMB polarization studies, and requires the measurement of the BB power spectrum, which, as yet, has been undetected, though measurements of the TT, TE, and EE power spectra have been made and results for the ACTPol instrument will be presented later in this work, along with a description of the implications of inflationary cosmology and some of the next-generation instruments that seek to confirm this model through the measurement of B-modes. If confirmed, inflationary cosmology would extend our understanding of ΛCDM cosmology and would help to explain the measured spatial curvature of the Universe as well as to explain how quantum mechani- cal density fluctuations in the primordial Universe could be stretched to large scales during inflation, preserving apparent thermal equilibrium between areas of the CMB that should not be in causal contact, which is often described as the so-called horizon problem (Car- roll and Ostlie 2007). It should also be noted that the CMB temperature and polarization anisotropies described in this section are not the only anisotropic features that experimental cosmology platforms observing the Cosmic Microwave Background measure, since what we are describing here are ideal CMB temperature and polarization signals detected without interaction with any structure throughout the expansion history of the Universe, so-called primary anisotropies. There are also secondary anisotropies, associated with small-angular scale measurements of the CMB, that describe effects on temperature and polarization sig- nal following interaction with structures in the Universe, including gravitational lensing of CMB temperature signal, and E-mode polarization that is induced to B-mode polarization from gravitational lensing from large scale structure. In the next section, we will briefly describe another such source of secondary anisotropy, describing the Sunyaev-Zel‘dovich effect resulting from the interaction of CMB radiation with ionized gas in galaxy clusters.