Beyond Optical Depth: Future Determination of Ionization History from the Cosmic Microwave Background

In standard ΛCDM, the CMB Stokes parameters ${\boldsymbol{m}}\,=(I,Q,U)$ are a realization of a Gaussian random process. The spherical harmonic transforms of these maps ${{\boldsymbol{a}}}_{{\ell }m}=({a}_{{\ell }m}^{{\rm{T}}},{a}_{{\ell }m}^{{\rm{E}}},{a}_{{\ell }m}^{{\rm{B}}})$ are therefore also Gaussian distributed. Neglecting B-modes, the ${{\boldsymbol{a}}}_{{\ell }m}$ are distributed as a complex Gaussian ${{\boldsymbol{a}}}_{{\ell }m}\,\sim { \mathcal N }\left({\bf{0}},{{\mathsf{\bf{C}}}}_{{\ell }}\right)$ with mean 0 and covariance

$$\begin{eqnarray}{{\mathsf{\bf{C}}}}_{{\ell }}=\left(\begin{array}{cc}{C}_{{\ell }}^{\mathrm{TT}} & {C}_{{\ell }}^{\mathrm{TE}}\\ {C}_{{\ell }}^{\mathrm{TE}} & {C}_{{\ell }}^{\mathrm{EE}}\end{array}\right).\end{eqnarray} \tag{ 1 }$$

As demonstrated in Hamimeche & Lewis (2008), the sample covariance matrix of measured power spectra ${\hat{{\mathsf{\bf{C}}}}}_{{\ell }}$ drawn from a theory covariance matrix ${{\mathsf{\bf{C}}}}_{{\ell }}$ is given by a Wishart distribution,

$$\begin{eqnarray}&&(2{\ell }+1){\hat{{\mathsf{\bf{C}}}}}_{{\ell }}\equiv \sum _{m}{{\boldsymbol{a}}}_{{\ell }m}^{\dagger }{{\boldsymbol{a}}}_{{\ell }m}^{}\sim {W}_{n}(2{\ell }+1,{{\mathsf{\bf{C}}}}_{{\ell }})\end{eqnarray} \tag{ 2 }$$

where n is the number of dimensions in ${{\boldsymbol{a}}}_{{\ell }m}$. A Wishart distribution is a multivariate gamma distribution. A gamma distribution is a two-parameter probability distribution of which the χ² distribution is a special case. When considered as a likelihood ${ \mathcal L }({{\mathsf{\bf{C}}}}_{{\ell }})\equiv P({\hat{{\mathsf{\bf{C}}}}}_{{\ell }}| {{\mathsf{\bf{C}}}}_{{\ell }})$, this is often normalized such that ${\chi }_{\mathrm{eff},{\ell }}^{2}\equiv -2\mathrm{ln}{ \mathcal L }({{\mathsf{\bf{C}}}}_{{\ell }})=0$ when ${{\mathsf{\bf{C}}}}_{{\ell }}={\hat{{\mathsf{\bf{C}}}}}_{{\ell }}$, i.e.,

$$\begin{eqnarray}&&-2\mathrm{ln}{ \mathcal L }({{\mathsf{\bf{C}}}}_{{\ell }})=(2{\ell }+1)\left[{Tr}[{\hat{{\mathsf{\bf{C}}}}}_{{\ell }}{{\mathsf{\bf{C}}}}_{{\ell }}^{-1}]-\mathrm{ln}| {\hat{{\mathsf{\bf{C}}}}}_{{\ell }}{{\mathsf{\bf{C}}}}^{-1}| -n\right].\end{eqnarray} \tag{ 3 }$$

In the single-dimensional case, this reduces to the more familiar χ² distribution,

$$\begin{eqnarray}&&-2\mathrm{ln}{ \mathcal L }({C}_{{\ell }})=(2{\ell }+1)\left[\displaystyle \frac{{\hat{C}}_{{\ell }}}{{C}_{{\ell }}}-\mathrm{ln}\displaystyle \frac{{\hat{C}}_{{\ell }}}{{C}_{{\ell }}}-1\right],\end{eqnarray} \tag{ 4 }$$

in agreement with Equation (8) of Hamimeche & Lewis (2008) when normalized such that $\mathrm{ln}{ \mathcal L }=0$ when ${C}_{{\ell }}={\hat{C}}_{{\ell }}$.

We also use the distribution of ${\hat{C}}_{{\ell }}^{\mathrm{TE}}$, i.e., the mean of the product of correlated Gaussian random variables. This was derived in Mangilli et al. (2015) and independently in Nadarajah & Pogány (2016) and Gaunt (2018), and is given by a variance-gamma distribution (also called a generalized Laplace distribution or a Bessel function distribution) with functional form

$$\begin{eqnarray}&&P({\hat{C}}_{{\ell }}^{\mathrm{TE}}| {\boldsymbol{\theta }})=\displaystyle \frac{{N}^{(N+1)/2}| \hat{c}{| }^{(N-1)/2}{e}^{N\rho \hat{c}/\xi }{K}_{{\ell }}\left(\tfrac{N| \hat{c}| }{\xi }\right)}{{2}^{(N-1)/2}\sqrt{\pi }{\rm{\Gamma }}(N/2)\sqrt{\xi }{\left({\sigma }_{{\ell }}^{\mathrm{TT}}{\sigma }_{{\ell }}^{\mathrm{EE}}\right)}^{N/2}},\end{eqnarray} \tag{ 5 }$$

where ${\boldsymbol{\theta }}=\{{C}_{{\ell }}^{\mathrm{TT}},{C}_{{\ell }}^{\mathrm{TE}},{C}_{{\ell }}^{\mathrm{EE}}\}$, $\hat{c}={\hat{C}}_{{\ell }}^{\mathrm{TE}}$, $\rho ={C}_{{\ell }}^{\mathrm{TE}}/({\sigma }_{{\ell }}^{\mathrm{EE}}{\sigma }_{{\ell }}^{\mathrm{TT}})$ is the correlation coefficient between the two noisy vectors, ${\sigma }_{{\ell }}^{\mathrm{XX}}=\sqrt{{C}_{{\ell }}^{\mathrm{XX}}+{N}_{{\ell }}^{\mathrm{XX}}}$ is the total uncertainty on the power spectrum ${C}_{{\ell }}^{\mathrm{XX}}$, ${N}_{{\ell }}^{\mathrm{XX}}$ is the noise power spectrum, N = 2ℓ + 1 is the number of modes per multipole, $\xi =(1-{\rho }^{2}){\sigma }_{{\ell }}^{\mathrm{TT}}{\sigma }_{{\ell }}^{\mathrm{EE}}$ is a useful auxiliary variable, Γ is the gamma function, and K_ν is the modified Bessel function of the second kind of order ν.

To better understand the variance-gamma distribution, we show how it reduces to the χ² distribution when taking a cross spectrum of identical vectors, i.e., ρ → 1. This distribution P(x) is proportional to $| x{| }^{(N-1)/2}{e}^{N\rho x/\xi }{K}_{{\ell }}\left(\tfrac{N| x| }{\xi }\right){\xi }^{-1/2}$. The modified Bessel function of the second kind decays exponentially, and its zeroth order expansion is given by ${K}_{\nu }(x)\approx \sqrt{\tfrac{\pi }{2x}}{e}^{-x}$ (Abramowitz & Stegun 1964). In the limit of large x, the functional form of the variance-gamma distribution goes to

$$\begin{eqnarray}&&P(x)\propto | x{| }^{(N-1)/2}\exp \left(\displaystyle \frac{N\rho x}{\xi }\right)\exp \left(-\displaystyle \frac{N| x| }{\xi }\right)\sqrt{\displaystyle \frac{\pi \xi }{2N| x| }}{\xi }^{-1/2}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\propto \,| x{| }^{N/2-1}\exp \left(\displaystyle \frac{\rho x-| x| )}{1-{\rho }^{2}}\right).\end{eqnarray} \tag{ 7 }$$

For perfectly correlated variables, the correlation ρ = 1 and the data are positive definite with x ≥ 0, giving $P(x)\,\propto {x}^{N/2-1}{e}^{-x/2}$, the χ² distribution with N degrees of freedom.

This parameterization of the variance-gamma distribution has mean and variance per multipole

$$\begin{eqnarray}&&\left\langle {\hat{C}}_{{\ell }}^{\mathrm{TE}}\right\rangle ={C}_{{\ell }}^{\mathrm{TE}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\mathrm{var}\left({\hat{C}}_{{\ell }}^{\mathrm{TE}}\right)=\displaystyle \frac{1}{2{\ell }+1}\left[{\left({C}_{{\ell }}^{\mathrm{TE}}\right)}^{2}+{C}_{{\ell }}^{\mathrm{TT}}{C}_{{\ell }}^{\mathrm{EE}}\right],\end{eqnarray} \tag{ 9 }$$

in agreement with the mean and variance of the off-diagonal component of the Wishart distribution and the Gaussian distribution of ${a}_{{\ell }m}^{{\rm{T}}}$ and ${a}_{{\ell }m}^{{\rm{E}}}$. We have also validated the functional form using 10⁴ realizations of ${{\boldsymbol{a}}}_{{\ell }m}$ vectors, and find that the distribution of ${\hat{C}}_{{\ell }}^{\mathrm{TE}}$ agrees with Equation (5).

To demonstrate the relative constraining power of the Wishart, χ², and variance-gamma likelihoods, we start with the theoretical power spectra as a function of the reionization optical depth τ in the case of instantaneous reionization, ${C}_{{\ell }}^{\mathrm{TT}/\mathrm{TE}/\mathrm{EE}}=f(\tau ,{A}_{s})$, with ${A}_{s}{e}^{-2\tau }$ fixed. Additionally, we include a white noise component that is uncorrelated between I, Q, and U and whose amplitude ${w}_{p}^{-1/2}$ varies between 0–230 μK arcmin. Using this formalism allows us to make predictions for the best-case constraining power on τ for future experiments, assuming instantaneous reionization.

We characterize the likelihood of τ by evaluating ${ \mathcal L }(\tau | \{{\hat{C}}_{{\ell }}^{\mathrm{TT}/\mathrm{TE}/\mathrm{EE}}\})$ for many realizations of the CMB sky. We create 50,000 realizations of ${{\boldsymbol{a}}}_{{\ell }m}$ with 2 ≤ ℓ ≤ 100 to test this formalism using the HEALPix¹ routine synalm. In Figure 2, we show the averaged likelihood of these different spectra in the case of a full-sky cosmic variance-limited measurement, and obtain ${\sigma }_{\tau }^{\mathrm{TT}+\mathrm{TE}+\mathrm{EE}}=0.0017$, ${\sigma }_{\tau }^{\mathrm{EE}}=0.0021$, and ${\sigma }_{\tau }^{\mathrm{TE}}\,=0.0072$. The TE-only constraint is comparable to the uncertainty from Planck, ${\sigma }_{\tau }^{{Planck}}=0.007$, which only includes E-mode data. The distribution for ${\hat{C}}_{{\ell }}^{\mathrm{TE}}$ in Figure 2 is visibly skewed. This is a manifestation of the underlying skewed distributions that the ${\hat{C}}_{{\ell }}^{\mathrm{TT}/\mathrm{TE}/\mathrm{EE}}$ are themselves drawn from.

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Since the uncertainty on ${\hat{C}}_{{\ell }}^{\mathrm{TE}}$ in Equation (9) is a function of $({C}_{{\ell }}^{\mathrm{EE},\mathrm{th}}+{N}_{{\ell }}^{\mathrm{EE}})({C}_{{\ell }}^{\mathrm{TT},\mathrm{th}}+{N}_{{\ell }}^{\mathrm{TT}})$, it is reasonable to ask whether there is a combination of uncertainties that makes ${\sigma }_{\tau }^{\mathrm{TE}}$ competitive with ${\sigma }_{\tau }^{\mathrm{EE}}$. Figure 3 demonstrates that at a given polarized white noise level ${w}_{p}^{-1/2}$, the constraining power on τ from the ${\hat{C}}_{{\ell }}^{\mathrm{TE}}$ alone is a factor of ∼3.5 weaker than the ${\hat{C}}_{{\ell }}^{\mathrm{EE}}$ constraint. This means that using ${\hat{C}}_{{\ell }}^{\mathrm{TT}}+{\hat{C}}_{{\ell }}^{\mathrm{TE}}+{\hat{C}}_{{\ell }}^{\mathrm{EE}}$ results in an approximately 20% increase in precision compared to using ${\hat{C}}_{{\ell }}^{\mathrm{EE}}$ data alone.

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The white noise temperature component is functionally negligible for this analysis. We can see this by looking at the components of Equation (9) contributing to the white noise in ${\hat{C}}_{{\ell }}^{\mathrm{TE}}$, $({C}_{{\ell }}^{\mathrm{TT},\mathrm{th}}+{N}_{{\ell }}^{\mathrm{TT}})({C}_{{\ell }}^{\mathrm{EE},\mathrm{th}}+{N}_{{\ell }}^{\mathrm{EE}})$. The theory-noise cross-terms are comparable when ${N}_{{\ell }}^{\mathrm{TT}}{C}_{{\ell }}^{\mathrm{EE},\mathrm{th}}\approx {N}_{{\ell }}^{\mathrm{EE}}{C}_{{\ell }}^{\mathrm{TT},\mathrm{th}}$. Since ${C}_{{\ell }}^{\mathrm{TT},\mathrm{th}}/{C}_{{\ell }}^{\mathrm{EE},\mathrm{th}}\simeq {10}^{4}$ for ℓ ≲ 100, the polarization sensitivity ${w}_{p}^{-1/2}$ would have to be ${ \mathcal O }({10}^{-2})$ times that of temperature for the temperature spectrum's white noise component to noticeably contribute to the ${\hat{C}}_{{\ell }}^{\mathrm{TE}}$ variation.

We explore the constraining power of low-multipole CMB polarization data using a specific parameterization of the reionization history. We parameterize the global reionization history x_e(z) using the the ratio of free electrons to hydrogen nuclei as a function of time, ${x}_{e}\equiv {n}_{e}/{n}_{{\rm{H}}}$,² and write the contribution to the reionization optical depth between two redshifts z₁ and z₂ as

$$\begin{eqnarray}&&\tau ({z}_{1},{z}_{2})\equiv {\int }_{t({z}_{1})}^{t({z}_{2})}c{\sigma }_{{\rm{T}}}{x}_{e}\left[z(t)\right]{n}_{{\rm{H}}}\left[z(t)\right]{dt}.\end{eqnarray} \tag{ 10 }$$

We parameterize the reionization history using a similar model to that used in Equation (A3) of Heinrich & Hu (2018),

$$\begin{eqnarray}\begin{array}{rcl}{x}_{e}(z) & = & \displaystyle \frac{1+{f}_{\mathrm{He}}-{x}_{e}^{\min }}{2}\left\{1+\tanh \left[\displaystyle \frac{{y}_{\mathrm{re}}-y}{\delta y}\right]\right\}\\ & & +\,\displaystyle \frac{{x}_{e}^{\min }-{x}_{e}^{\mathrm{rec}}}{2}\left\{1+\tanh \left[\displaystyle \frac{{y}_{{\rm{t}}}-y}{\delta y}\right]\right\}+{x}_{e}^{\mathrm{rec}}\end{array}\end{eqnarray} \tag{ 11 }$$

where $y{(z)\equiv (1+z)}^{3/2}$ and $\delta y=\tfrac{3}{2}{(1+z)}^{1/2}\delta {z}_{\mathrm{re}}$. The ionization fraction from recombination alone is ${x}_{e}^{\mathrm{rec}}$, the second transition step is given at the redshift z_t, the amplitude of reionization from the second transition is x_e^min, and the fraction of electrons from singly ionized helium is given by ${f}_{\mathrm{He}}\,\equiv {n}_{\mathrm{He}}/{n}_{{\rm{H}}}$. We use this form because it parameterizes a small but nonzero early ionization fraction. An upper limit on x_e(15 ≤ z ≤ 30) was first inferred by Millea & Bouchet (2018) and further constrained by Planck Collaboration VI (2018). Figure 45 of Planck Collaboration VI (2018) shows that above z ≳ 10, Planck measurements do not rule out ${x}_{e}^{\min }\approx 10 \%$. Motivated by this result, we choose a fiducial value of ${x}_{e}^{\min }=0.05$ to demonstrate the potential effects this ionization fraction can have on CMB measurements. We also choose z_re = 6.75 so that the total optical depth τ = 0.06 is consistent with the Planck Collaboration VI (2018) values. We highlight the parameters of this model in Figure 4, with ${x}_{e}^{\min }$ set to 0.2 for visibility purposes.

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There are several measurements of the neutral hydrogen fraction, which we summarize in the inset of Figure 4. These direct measurements are in general agreement with the reionization constraints from CMB measurements, but do not yet provide strong evidence for or against reionization models more complex than the instantaneous tanh-like model. More concretely, these direct measurements do not rule out a free electron fraction ${x}_{e}^{\min }=0.05$ at redshifts 7 ≲ z ≲ 8.³

We show the dependence of ${C}_{{\ell }}^{\mathrm{EE}}$ and ${C}_{{\ell }}^{\mathrm{TE}}$ on these reionization histories in Figure 5. We choose the ranges of the parameters such that they induce roughly equivalent changes in the amplitude of the output power spectrum. The equivalent white noise powers are labeled on the right-hand side of Figure 5. We also vary δz_re to show that although this parameter does affect the power spectra, unphysically large widths δz_re ≳ 5 are needed to affect the power spectra as much as z_re and ${x}_{e}^{\min }$. Variations in δz_re induce a high-redshift tail, similar to the empirical extended phenomenological models described in Lapi et al. (2017), Roy et al. (2018), and Kulkarni et al. (2019). The rightmost column of Figure 5 shows these models are not strongly constrained by large-scale CMB polarization data. We fix the width of these transitions to δz_re = 0.5 because it is weakly constrained by E-mode power spectra for a reionization history that is complete by z = 0.

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In the parameterization of Equation (11), it is natural to compare to Equation (10) and constrain the parameters ${\tau }_{\mathrm{lo}}\equiv \tau (0,{z}_{\mathrm{split}})$ and ${\tau }_{\mathrm{hi}}\equiv \tau ({z}_{\mathrm{split}},{z}_{\mathrm{dark}})$. We choose z_dark = 100 as a redshift sufficiently far removed from both recombination and reionization effects. We define ${z}_{\mathrm{split}}\equiv {z}_{\mathrm{re}}+1$. This parameterization essentially allows a one-dimensional mapping such that ${\tau }_{\mathrm{lo}}=f({z}_{\mathrm{re}})$ and ${\tau }_{\mathrm{hi}}=g({x}_{e}^{\min })$. In the case of standard tanh-like reionization, ${\tau }_{\mathrm{lo}}\to \tau$ and ${\tau }_{\mathrm{hi}}\to 0$, or equivalently ${x}_{e}^{\min }\to {x}_{e}^{\mathrm{rec}}$.

The primary effect of adding a second component to x_e(z) is an increase in the total reionization optical depth τ, and therefore the rough amplitudes of the polarized power spectra, specifically ${C}_{{\ell }}^{\mathrm{EE}}\propto {\tau }^{2}$ and ${C}_{{\ell }}^{\mathrm{TE}}\propto \tau$ at the lowest multipoles ℓ ≲ 10. The second and more distinguishing effect is that both of these power spectra change shape due to the different angular sizes of local quadrupoles at the primary and secondary reionization redshifts. This provides an opportunity to go beyond τ in probing the nature of reionization. We demonstrate the effects of varying ${x}_{e}^{\min }$ and z_re on the polarized power spectra (see Figure 5) using the Boltzmann code CLASS (Blas et al. 2011). For every reionization history, we compute τ and vary A_s such that ${A}_{s}{e}^{-2\tau }$ is held constant.

Using the CLASS code, we set reio_parameterization equal to reio_many_tanh with δz_re = 0.5, fixing z_t = 30, f_He = 1. 324 using the fiducial helium mass fraction Y_p = 0.25, and ${x}_{e}^{\mathrm{rec}}=2\times {10}^{-4}$. We vary z_re and ${x}_{e}^{\min }$ to write the cosmological power spectrum as a function of two parameters, ${C}_{{\ell }}^{\mathrm{TT}/\mathrm{TE}/\mathrm{EE}}=f({z}_{\mathrm{re}},{x}_{e}^{\min })$.

In Figure 6, we plot ${\chi }_{\mathrm{eff},{\ell }}^{2}$ using Equation (3). By varying z_re and ${x}_{e}^{\min }$ separately, we can observe a few noteworthy features. First, although there is more variation in the power spectra at the very largest scales, the constraining power peaks at ℓ ≃ 10, corresponding to fluctuations on scales of tens of degrees. Second, the two different reionization histories have notably different ${\chi }_{\mathrm{eff},{\ell }}^{2}$, demonstrating that the partial degeneracy between these two modifications to reionization history can be broken with high signal-to-noise measurements across this range of angular scales. The very largest scales ℓ < 10 are much more constraining for z_re than ${x}_{e}^{\min }$, whereas the 10 ≤ ℓ < 20 range is very sensitive to both parameters.

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We quantify this multipole dependence by performing Fisher forecasts on subsets of cosmic variance-limited data in Figure 7. As expected, there is relatively little constraining power in the 30 ≲ ℓ ≲ 100 multipole range, but the majority of constraining power comes from the 10 ≲ ℓ ≲ 20 range, in agreement with the shape of the ${\chi }_{\mathrm{eff},{\ell }}^{2}$ curves in Figure 6. While there is significant constraining power in the 2 ≲ ℓ ≲ 10 and 20 ≲ ℓ ≲ 30 ranges, the 10–20 range is by far the most important for quantitative assessment for this reionization scenario.

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We show the uncertainty on the optical depth parameters τ_lo, τ_hi, ${\tau }_{\mathrm{tot}}\equiv {\tau }_{\mathrm{lo}}+{\tau }_{\mathrm{hi}}$, and τ from tanh-like reionization as a function of white noise level in Figure 8. This demonstrates that the optical depth from high-redshift reionization can be meaningfully constrained with relatively high white noise levels, and that the uncertainty on any additional optical depth from high-redshift sources can be improved by an order of magnitude above current measurements with white noise levels as high as 10 μK arcmin.

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The constraining power of low-ℓ polarization data can most clearly be seen in the Fisher contours in Figure 9. At current noise levels, the constraints on the reionization redshift are relatively weak, and the presence of high-redshift reionization cannot be distinguished from instantaneous reionization. As expected, there is a negative degeneracy between x_e^min and z_re that is most pronounced at high noise levels where only the lowest multipoles contribute to the variation of the power spectra, while the degeneracy becomes less severe as the noise level decreases.

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Figure 9 demonstrates the possible advances in our understanding of reionization from the CMB. The ultimate sensitivity to $({z}_{\mathrm{re}},{x}_{e}^{\min })=(6.75,0.05)$—equivalent to τ_tot = 0.06—from the CMB is shown in blue, using a Fisher forecast with zero instrumental noise. This noiseless measurement represents the fundamental limits for constraining these reionization parameters with large-scale CMB polarization measurements. We plot Fisher contours for noise levels ${w}_{p}^{-1/2}\,=\{10,60,100\}\,\mu {\rm{K}}\,\mathrm{arcmin}$. The smallest number corresponds to the projected Cosmology Large Angular Scale Surveyor (CLASS) white noise level over 70% of the sky in Essinger-Hileman et al. (2014) using all four of its observing bands. This value is a benchmark for detecting primordial gravitational waves with a tensor-to-scalar ratio r ∼ 0.01, a goal for the current generation of ground-based CMB measurements. The ${w}_{p}^{-1/2}=60\,\mu {\rm{K}}\,\mathrm{arcmin}$ white noise level corresponds to the sensitivity of the CLASS Q-band (40 μK arcmin) cleaned using the WMAP K-band (280 μK arcmin) as a synchrotron template. The 100 μK arcmin value corresponds to the geometric mean of the 100 GHz and 143 GHz white noise levels reported in Table 4 of Planck Collaboration I (2018).

We transform the contours in Figure 9 to the integrated quantities τ_lo and τ_hi, both approximately single-variable functions of z_re and x_e^min, respectively, in Figure 10. We also plot lines of constant τ = τ_lo + τ_hi to show the total integrated contribution of this two-parameter reionization model.

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We summarize the results of this section in Table 1. We highlight data rows that are particularly constraining. This includes the full resolution cosmic variance measurement, the noiseless measurement with 10 ≤ ℓ < 20, and the ${w}_{p}^{-1/2}=10\,\mu {\rm{K}}\,\mathrm{arcmin}$ measurements. We highlight these to emphasize the relative importance of future data with these properties to constrain reionization histories.

Table 1.  Fisher Forecasts for z_re and ${x}_{e}^{\min }$ when Varying Multipole Range Analyzed and White Noise Levels, Using the Fiducial Model $({z}_{\mathrm{re}},{x}_{e}^{\min })=(6.75,0.05)$

${w}_{p}^{-1/2}$	Multipole Range	${\sigma }_{{z}_{\mathrm{re}}^{}}$	${\sigma }_{{x}_{e}^{\min }}$	${\sigma }_{{\tau }_{\mathrm{lo}}}$	${\sigma }_{{\tau }_{\mathrm{hi}}}$
0	${\bf{2}}\leqslant {\ell }\lt {\bf{100}}$	0.3	0.005	0.003	0.001
0	2 ≤ ℓ < 10	0.8	0.020	0.007	0.005
0	${\bf{10}}\leqslant {\ell }\lt {\bf{20}}$	1.0	0.005	0.009	0.001
0	20 ≤ ℓ < 30	5.9	0.024	0.054	0.006
0	30 ≤ ℓ < 100	10.8	0.152	0.991	0.038
10	${\bf{2}}\leqslant {\ell }\lt {\bf{100}}$	0.4	0.005	0.003	0.001
60	2 ≤ ℓ < 100	0.7	0.018	0.006	0.004
100	2 ≤ ℓ < 100	1.0	0.031	0.009	0.008

Note. We have highlighted the three forecasts that are most constraining in bold text.

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With our fiducial parameters, the ultimate sensitivity to this model with large-scale CMB anisotropy measurements is ${\sigma }_{{z}_{\mathrm{re}}}=0.3$ and ${\sigma }_{{x}_{e}^{\min }}=0.005$. Remarkably, this constraint does not weaken appreciably either when examining only the multipole range 10 ≤ ℓ < 20, or when the data are contaminated by white noise at the 10 μK arcmin level.