Mean Optical Depth

By definition, ionization separates electrons from their atoms, resulting in an increase in the number of free electrons. This boost in the number density of free electrons causes an increase in Thomson scattering of the CMB photons, producing a detectable signal. This signal is parametrized byτ, the effective optical depth to reionization:

τe=

Z

ne(z)σT(cdt/dz)dz, (1.4)

wheren_e is the number density of free electrons,σT =6.65×10−25cm2is the Thomson cross section andc, of course, is the speed of light. The integral is over the line of sight distance the photons travel. Only periods in the history of the Universe when there are significant numbers of free electrons contribute to this integral. Thus, the period from recombination to reionization has little effect onτe. This calculation is dominated by the free electrons produced in the ionization of hydrogen and the single ionization of helium; the double ionization of helium contributes only a few percent of the total value of τe (Reichardt 2016)

Because of the integral nature of this calculation,τeis insensitive to the precise details of reionization. It is, however, sensitive to shifts in the number density of free electrons. If we assume reionization is an instantaneous process, there is only one such seismic shift. This is the simplest approximation, and following Loeb and Furlanetto (2013), equation 1.4 can be solved analytically for a flat universe:

τe=4.44×10−3× {[ΩΛ+Ωm(1+zreion)

3_]1/2_{−1}. (1.5)}

In this calculation, it has been assumed that helium singly-ionizes at the same time as hydrogen. z_reion is the redshift of instantaneous reionization which corresponds roughly to the midpoint of an extended reionization period. It is often reported alongsideτe.

This Thomson scattering has two general effects on the CMB. First, it washes out small scale anisotropies in the CMB. After a moment’s reflection on the scattering pro- cess, this is somewhat intuitive. Photons traveling along a line of sight to the observer

may be scattered out of that path; conversely, photons may be scattered into that line of sight from other directions. Together, these effects work to suppress the CMB anisotropy power. Unfortunately, this effect is highly degenerate with the amplitude of the primordial power spectrum of scalar perturbations as shown in Fig. 1.7 (Reichardt 2016).

Second, Thomson scattering can also produce polarization in the CMB. This process is slightly more complicated. Consider light scattering off of a free electron as shown in Fig. 1.8. While the incident light may have any polarization, only certain polarizations will be scattered. Specifically, the polarization of the scattered light must be perpendicular to its direction of travel. Thus, if we examine Fig. 1.8, for light incident from the left, only the vertically polarized photons will be scattered into the observer’s line of sight, which points out of the page in the illustration. Similarly, for light incident from above, only the horizontally polarized light will be scattered into the line of sight. Since the electrons scatter light from all directions, the result is a mixture of horizontal and vertical polarizations. Now, if the incident light is not symmetric, specifically if the light is more intense coming from one direction than the other, the resulting scattered beam will be polarized. More precisely, if the incident field has a quadrupole anisotropy between the initial intensities along the horizontal and vertical axises, the resulting scattered light will have a net polarization (Loeb and Furlanetto 2013).

Since the CMB has such a quadrupole, Thomson scattering due to reionization leads to linear polarization. This signal peaks at large scales, leading to the reionization “bump" as shown in Fig. 1.7. This feature cannot be produced by any other parameter in stan- dard cosmological models, making it a clear signal of τe. However, the “bump” only tells about optical depth; it is insensitive to changes in the duration of reionization. Cur- rent measurements from the Planck Collaboration et al. (2016) placeτeat 0.058±0.012, corresponding toz_reion'8.8±0.9.

Figure 1.7: Figures from Reichardt (2016) showing the effect the Thomson optical depth has on observations of the CMB. (Left)The effects of varying optical depth, τ, and the

amplitude of scalar perturbations,A_s, on the CMB temperature power spectrum. Increas- ingτ results in a decrease in the power spectrum. Specifically, the power is reduced by

a factor ofe−2τ _{on scales`&20, smaller than the horizon at EoR. However, varyingA} s can produce a signal that is degenerate with this decrease due toτ. Here,As, the dashed red line, has been tuned to matchτ=0.08, the dark blue. (Right)The effects of varying

optical depth on the CMBE-mode polarization power spectrum. Increasingτ produces

a bump on large scales, `_.20, scales greater than the horizon size at the EoR. Unlike in the figure on the right, A_s, the red dashed line, cannot be tuned to mimic this bump at large scales. The dashed blue line and the solid dark blue line have the same optical depth,τ=0.08, but the duration of reionization is six times longer for the dashed versus

the solid line. As is shown here, the duration of reionization has virtually no effect on E-mode power as long as the total optical depth is not changed.

Figure 1.8: Figure from Hu and White (1997). This cartoon illustrates how CMB polar- ization is generated from Thomson scattering. The central electron scatters photons, here coming in from the left and above, into the line of sight to the observer, out of the page. Only polarizations that are perpendicular to both the line of sight and to a line connecting the incoming emission and the scattering electron are allowed. This means that for the light from the left, only the vertical polarization is allowed; similar, for the light from above, only the horizontal polarization is allowed. The resulting emission along the line of sight is a mix of those two states. However, if the incoming field is asymmetric, in par- ticular, if it has quadrupole anisotropy between the initial intensities along the horizontal and vertical axises, the resulting field will be polarized.