Reionization effective likelihood from Planck 2018 data

We release relike (reionization effective likelihood), a fast and accurate effective likelihood code based on the latest Planck 2018 data that allows one to constrain any model for reionization between $6<z<30$ using five constraints from the CMB reionization principal components (PC). We tested the code on two example models which showed excellent agreement with sampling the exact Planck likelihoods using either a simple Gaussian PC likelihood or its full kernel density estimate. This code enables a fast and consistent means for combining Planck constraints with other reionization data sets, such as kinetic Sunyaev-Zeldovich effects, line-intensity mapping, luminosity function, star formation history, quasar spectra, etc., where the redshift dependence of the ionization history is important. Since the PC technique tests any reionization history in the given range, we also derive model-independent constraints for the total Thomson optical depth ${\tau }_{\mathrm{PC}}={0.0619}_{-0.0068}^{+0.0056}$ and its $15\le z\le 30$ high redshift component ${\tau }_{\mathrm{PC}}(15,30)<0.020$ (95% C.L.). The upper limits on the high-redshift optical depth is a factor of $\sim 3$ larger than those reported in the Planck 2018 cosmological parameter paper using the FlexKnot method and we validate our results with a direct analysis of a two-step model which permits this small high-$z$ component.

Figure 1

Top: rank-ordered CMB reionization principal components $S_a$ for $a=1\dots 5$ for $z_{\max }=30$. We assume a fiducial model with fully ionized hydrogen and singly ionized helium by $z<6$, and doubly ionized helium at $z=3.5$. Bottom: the cumulative optical depth from $z$ to $z_{\max }$ of a unit amplitude PC.

Figure 2

Constraints from Planck data on the amplitudes of the five reionization principal components that describe all physical ionization models up to $z_{\max }=30$. We show the 68% and 95% $m_a-m_b$ constraints as well as marginal constraints on individual $m_a$ for Planck 2018 in blue, using SRoll2 for the low-$\ell EE$ likelihood, and compare that to the Planck 2015 constraints in red. The corresponding marginal distributions for individual $m_a$ are also shown. When deriving model independent optical depth constraints we impose an additional physicality prior that excludes the gray region in $m_1$ and regions beyond the box limits in the others. The black solid line indicates the trajectory of tanh models with varying $\tau$. The red and black crosses indicate the best-fit tanh models from Planck 2015 and 2018 respectively. Note that the best-fit tanh model has moved to a lower $\tau$ in Planck 2018, and in particular, is now within the 68% C.L. contours of the constraints.

Figure 3

Changes in the best constrained $m_1-m_2$ PC plane between Planck 2015 (red) and Planck 2018 (blue, SRoll2; green, SimAll). The default SRoll2 likelihood moderately improves the best constrained direction and raises the optical depth for Planck 2018.

Figure 4

Total optical depth ${\tau }_{\tanh }$ posterior for the tanh model in Sec. 4a for the exact (Planck 2018) likelihood (shaded), KDE (blue solid) and Gaussian (black dashed) effective PC likelihoods. The overall agreement is excellent and the KDE results even capture the small skewness of the exact result.

Figure 5

Cumulative optical depth in the tanh model as in Fig. 4 (68% and 95% C.L. contours). Overall agreement is good in regions where the contribution to ${\tau }_{\tanh }$ is high but the model form forbids high-$z$ contributions. Oscillations around zero at high-z in the KDE and Gaussian PC representations are due to the effective low pass filter from the truncation at 5 PCs which does not produce any observable difference in the CMB.

Figure 6

Two-step ionization history $x_e(z)$ for the model of Sec. 4b. In addition to a tanh transition at low-redshift as in Sec. 4a, there is an additional ionization plateau at high redshifts with a fixed transition at $z_{\mathrm{hi}}=28$ and $\mathrm{\Delta }{z}_{\mathrm{hi}}=1$. Here we show the best-fit model in the Planck 2018 data $({\tau }_{\mathrm{lo}},{\tau }_{\mathrm{hi}})=(0.053,0.006)$, as well as a model on the 95% C.L. contour of Fig. 7 which maximizes ${\tau }_{\mathrm{hi}}$, $({\tau }_{\mathrm{lo}},{\tau }_{\mathrm{hi}})=(0.043,0.026)$.

Figure 7

Two-step parameter constraints (marginalized posteriors and 68% and 95% C.L. contours). The exact, KDE PC and Gaussian PC likelihoods again agree very well and a small contribution from high-$z$ through ${\tau }_{\mathrm{hi}}$ is still allowed but not significantly favored. Models of Fig. 6 are marked and correspond to the best fit parameters and parameters which maximize ${\tau }_{\mathrm{hi}}$ on the 2D 95% C.L. contour.

Figure 8

Cumulative optical depth in the two-step model (as in Fig. 5). Agreement between the exact, KDE PC and Gaussian PC likelihoods is again excellent and show that a small amount of high-$z$ ionization is allowed. The form of the model requires that the cumulative optical depth steadily declines above the first step at $z_{\mathrm{re}}\sim 6–7$ especially for $z\gtrsim 15$.

Figure 9

Total optical depth posterior for the model-independent PC analysis (shaded) compared with the exact tanh model (blue solid line) and the two-step model (black dashed line) from Figs. 4 and 7. The PC analysis captures the ability to raise the optical depth through the high-$z$ contributions in models whose form allow it like in the two-step case.

Figure 10

Total optical depth ${\tau }_{\mathrm{PC}}$ posterior (shaded) without (top) and with (bottom) the additional physicality prior of Eq. (11). Imposing the physicality prior eliminates negative ionization at high $z$ and shifts the distribution to a higher total $\tau$. This unphysical contribution comes from the $m_1$ PC, as can be traced through the approximate ${\tau }_{12}\approx {\tau }_{\mathrm{PC}}$ built from a linear combination of the first two components defined in Eq. (19) (blue lines).

Figure 11

Priors on the $m_1-m_2$ parameter space: the default physicality prior (inset square, solid lines) and an alternate prior which is flat in its marginal ${\tau }_{12}$ distribution (rotated rectangle, dashed lines) given boundaries which are aligned with lines of constant ${\tau }_{12}$ (light gray lines). Note that although the physicality prior allows for more parameter space at higher ${\tau }_{12}$, that is also the region excluded by the data constraints (ellipses). The only difference in the prior across the allowed region is that the physicality prior removes cases where $m_1$ is so small that it requires negative ionization at high-$z$.

Figure 12

Cumulative optical depth ${\tau }_{\mathrm{PC}}(z,z_{\max })$ from the model-independent PC approach with and without the physicality prior. The prior helps exclude models with negative contributions at high-$z$ but conservatively allows some negative values at $z\sim z_{\max }=30$ where the oscillations in the PCs no longer cancel due to the cumulative integration. Upper limits are nearly unaffected.

Figure 13

Changes in the cumulative optical depth ${\tau }_{\mathrm{PC}}(z,z_{\max })$ (as in Fig. 12) between Planck 2015 and 2018 (top panel) and between 2018 SRoll2 (default) and SimAll. Planck 2015 favored contributions at high $z$ which are disallowed in 2018 whereas the changes due to SRoll2 mainly tighten the lower limits at low $z$.

Figure 14

Robustness of the cumulative optical depth ${\tau }_{\mathrm{PC}}(z,z_{\max })$ to changes from the default $z_{\max }=30$ to $z_{\max }=50$. Upper limits at $z\lesssim 30$ are nearly unaffected whereas extending $z_{\max }$ allows the high-$z$ component to come from even higher-$z$.