Eliminating the optical depth nuisance from the CMB with 21 cm cosmology

Amongst standard model parameters that are constrained by cosmic microwave background (CMB) observations, the optical depth $\tau$ stands out as a nuisance parameter. While $\tau$ provides some crude limits on reionization, it also degrades constraints on other cosmological parameters. Here we explore how 21 cm cosmology—as a direct probe of reionization—can be used to independently predict $\tau$ in an effort to improve CMB parameter constraints. We develop two complementary schemes for doing so. The first uses 21 cm power spectrum observations in conjunction with semianalytic simulations to predict $\tau$. The other uses global 21 cm measurements to directly constrain low redshift (post-reheating) contributions to $\tau$ in a relatively model-independent way. Forecasting the performance of the upcoming hydrogen epoch of reionization array, we find that significant reductions in the errors on $\tau$ can be achieved. These results are particularly effective at breaking the CMB degeneracy between $\tau$ and the amplitude of the primordial fluctuation spectrum $A_s$, with errors on $\ln ({10}^{10}{A}_s)$ reduced by up to a factor of 4. Stage 4 CMB constraints on the neutrino mass sum are also improved, with errors potentially reduced to 12 meV regardless of whether CMB experiments can precisely measure the reionization bump in polarization power spectra. Observations of the 21 cm line are therefore capable of improving not only our understanding of reionization astrophysics, but also of cosmology in general.

Figure 1

Fractional error in $\tau$ induced by uncertainties in helium reionization as a function of the redshift of helium reionization $z_{\mathrm{ion},\mathrm{He}}$ and the uncertainty in this redshift $\delta z_{\mathrm{ion},\mathrm{He}}$. For reasonable values of these parameters, the errors arising from uncertainties in helium reionization are subdominant to those arising from cosmological parameter uncertainty. It is thus permissible to neglect uncertainties in helium reionization.

Figure 2

Cumulative contribution to the optical depth $\tau$ from low to high redshift, for several different models of reionization. The crucial astrophysical quantity for a precise determination of $\tau$ is the density-weighted ionized fraction. This depends on the correlation between the ionization field and the density field. The different reionization models shown here reflect different models for this correlation, which must be known for a precise prediction of $\tau$, given the spread seen here.

Figure 3

Top row: Simulation of the nonlinear density field over the past light cone that is observed by a 21 cm experiment. Second row: Corresponding ionization fraction, assuming $(T_{\mathrm{vir}},R_{\mathrm{mfp}},\zeta )=(6\times {10}^4\mathrm{K},35\mathrm{Mpc},30)$ to match the optical depth of $\text{Planck TT},\mathrm{TE},\mathrm{EE}+\mathrm{lowP}+\text{lensing}+\mathrm{ext}$. Third row: Corresponding ionized fraction history ${\overline{x}}_{\mathrm{HII}}$ (red solid curve) and the density-weighted ionization history $\overline{x_{\mathrm{HII}}(1+{\delta }_b)}$ (black solid curve). The averaged ionized fraction is also seen to be a poor approximation for the density-weighted ionized fraction, which is the crucial quantity for determining $\tau$. Bottom row: Corresponding 21 cm power spectra (black) at various redshifts, plotted as ${\mathrm{\Delta }}_{21}^2(k)\equiv k^3{P}_{21}(k)/2{\pi }^2$. Blue and orange curves show power spectra for different values of $T_{\mathrm{vir}}$. Note that this figure is intended for illustrative purposes only, and that the scales on the top two rows do not correspond exactly to the redshift axis on the third row. In our proposed analysis, one measures the bottom row through observations, constraining underlying model parameters that are then fed into simulations to produce the top two rows. The density-weighted ionization fraction (third row) is then extracted and inserted into Eq. (10) to determine $\tau$.

Figure 4

Forecasted 68% and 95% confidence regions (black ellipses) in the $T_{\mathrm{vir}}-\zeta$ parameter space for HERA observations, along with 21cmfast-predicted optical depth $\tau$ (filled color contours). The rough alignment of the degeneracy directions suggest that uncertainties in astrophysical parameters arising from 21 cm power spectrum measurements are unlikely to seriously compromise one’s ability to make highly precise predictions of $\tau$.

Figure 5

Likelihood contours for $\mathrm{\Lambda }\mathrm{CDM}$ cosmological parameters as defined in the publicly released $\text{Planck TT},\mathrm{TE},\mathrm{EE}+\mathrm{lowP}+\text{lensing}+\mathrm{ext}$ data set. Black ellipses show 68% and 95% confidence regions for every parameter against $\tau$. Red lines indicate values of $\tau$ as predicted in 21cmfast and approximated by Eq. (17), holding all other parameters fixed. The blue dashed line in the $\tau \text{−}\ln ({10}^{10}{A}_s)$ plot indicates constant $A_s{e}^{-2\tau }$, illustrating the strong degeneracy inherent in CMB observations that we expect to be broken by 21 cm observations.

Figure 6

Likelihood contours on the $\tau \text{−}\ln ({10}^{10}{A}_s)$ plane, with bold lines signifying 95% confidence regions and light lines signifying 68% confidence regions. Blue contours denote the constraints using $\text{Planck TT},\mathrm{TE},\mathrm{EE}+\mathrm{lowP}+\text{lensing}+\mathrm{ext}$ data only, while the red contours show the effect of adding 21 cm power spectrum and—crucially—self-consistency between the CMB-measured and 21 cm-predicted $\tau$. The 21 cm observations break the CMB degeneracy between $A_s$ and $\tau$, enabling much better constraints on both parameters. The grey band shows a width of optical depths representative of the spread of models shown in Fig. 2, and is indicative of a scenario where the ionization history is known, but the density-ionization correlation is unknown. Even in the midst of such modeling uncertainty, one sees an improvement in $A_s$ errors, although we stress that such a scenario is rather pessimistic since early 21 cm measurements will place constraints on the correlation.

Figure 7

Likelihood contours on the ${\mathrm{\Omega }}_m\text{−}{\sigma }_8$ plane, with bold lines signifying 95% confidence regions and light lines signifying 68% confidence regions. Blue contours denote the constraints using $\text{Planck TT}+\mathrm{lowP}$ data only, while red incorporates 21 cm power spectrum and self-consistent $\tau$ information. Dashed contours denote constraints from combining Planck’s SZ cluster counts, BAO, and BBN (as published in Ref. [3]). In green are constraints using a CMB lensing-calibrated prior on the cluster mass bias factor (CMBlens). In orange and purple are constraints based on calibrations using gravitational shear mass measurements from the Canadian cluster comparison project (CCCP) and Weighing the Giants (WtG) program, respectively.

Figure 8

Likelihood contours on the $\tau \text{−}\sum m_{\nu }$ plane, with bold lines signifying 95% confidence regions and light lines signifying 68% confidence regions. Blue contours denote the constraints using $\mathrm{S}4(\ell >50)+\text{Planck polarization}+\mathrm{DESI}$ data only, while the red contours show the effect of adding 21 cm power spectrum and self-consistency between the CMB-measured and 21 cm-predicted $\tau$. Early 21 cm observations will confirm models of reionization, allowing high sensitivity measurements to predict $\tau$. This will break the CMB degeneracy between $\tau$ and $\sum m_{\nu }$ and enable improved constraints on the neutrino mass.

Figure 9

Fiducial global signal history (black solid lines) chosen to match the fiducial model tied to the $\text{Planck TT},\mathrm{TE},\mathrm{EE}+\mathrm{lowP}+\text{lensing}+\mathrm{ext}$ data set. Dashed lines show perturbations about the fiducial history driven by excitations of the deviation eigenmodes (first eigenmode on top plot; second eigenmode on bottom plot) of our power spectrum-informed principal component basis.

Figure 10

Fractional error in a global signal measurement of $\tau (z=8.5)$ as a function of the number of foreground parameters $N_p$ and number of signal parameters $N_d$ that are necessary for an adequate fit to the data. The assumed reionization scenario is chosen to match parameters from $\text{Planck TT}+\mathrm{lowP}$. A discretized contour of the fractional error on Planck measurement of $\tau$ is given by the thick black line for reference. As long as the number of parameters in a global signal remains small, global signal experiments can provide direct, model-independent constraints on relatively low-redshift portions of $\tau$.

Figure 11

Similar to Fig. 10, but for the $\text{Planck TT},\mathrm{TE},\mathrm{EE}+\mathrm{lowP}+\text{lensing}+\mathrm{ext}$ data set.