Constraining exotic high-$z$ reionization histories with Gaussian processes and the Cosmic Microwave Background

TL;DR

We address constraining exotic high-$z$ reionization histories using a model-independent reconstruction of the free-electron fraction $X_e(z)$ from Planck low-$\ell$ polarization. We implement a Gaussian-process–driven reionization scheme, reio_gpr_tanh, in CLASS with non-uniform redshift binning and tanh transitions between GP nodes up to $z_{ m max}=800$, and we decompose the optical depth as $\tau_{\rm reio}=\tau_{\rm lowz}+\tau_{\rm highz}$ using a transition redshift $z_c$. From the reconstructed histories we derive posteriors on $\tau_{\rm highz}$, providing a robust, model-independent probe of early energy injection, and illustrate the method with a MeV-scale decaying axion scenario that would yield large $\tau_{\rm highz}$ inconsistent with Planck data. The analysis is public, and the framework offers a general tool to test beyond-$\Lambda$CDM physics with current CMB polarization measurements. The derived $\tau_{\rm highz}$ posterior can be applied to any energy-injection model for which $X_e(z)$ is computable, enabling broad constraints on exotic early-Universe scenarios.

Abstract

The large-angle polarization anisotropies in the Cosmic Microwave Background (CMB) arise from Thomson scattering of CMB photons off free electrons in the post-recombination Universe. In the standard $Λ$ cold dark matter cosmological model, the free electron density increases at redshifts $z \lesssim 10$ as the first stars form, reionizing the intergalactic medium. We use \emph{Gaussian processes} to perform a model-independent reconstruction of the cosmic reionization history constrained by \textit{Planck} CMB data. Our approach recovers the standard reionization at $z \lesssim 10$ and places stringent limits on any additional high-$z$ reionization. From this reconstruction, we define a new derived parameter, the high-redshift contribution to the CMB optical depth, $τ_{\mathrm{highz}}$, whose posterior distribution provides robust constraints on exotic energy injection scenarios. We demonstrate this for decaying dark matter with particle masses in the range $\mathcal{O}(1\,\text{MeV})$. A companion paper applies this framework to multi-axion models. All data and code are publicly available at: \href{https://github.com/Cheng-Hanyu/CLASS_reio_gpr}{github.com/Cheng-Hanyu/CLASS\_reio\_gpr}.

Figures (8)

• Figure 1: Model independent reconstruction of the free electron fraction, $X_e(z)$, from Planck large angle E-mode polarization data, using the Gaussian process method. All other cosmological parameters are fixed to their best-fit values from the Planck 2018 TT,TE,EE+lowE analysis Planck:2018vyg. The shaded regions denote the 68% (blue) and 95% (light blue) confidence level (CL), and the solid line marks the median reconstruction.
• Figure 2: Normalized posterior distributions $P/P_{\mathrm{max}}$ for the low-redshift (left) and high-redshift (right) contributions to the optical depths, shown for different critical redshifts $z_{\rm c}$ separating the two regimes, as listed in Table \ref{['tab:tau_constraints']}. Results are based on Planck low-$\ell$ EE data alone.
• Figure 3: Ionization fraction in the $\Lambda$CDM scenario compared to $\Lambda$CDM with one additional axion component, assuming instantaneous energy injection with the "beyond on-the-spot" assumption and using the full energy deposition efficiency function provided by DarkHistoryLiu:2019bbm. In the model shown, the axion constitutes a fraction $\approx 10^{-7}$ of the DM (produced by freeze-in with the minimum reheating temperature consistent with BBN), before decaying at redshift $z_{\rm decay} < z_{\rm rec}$. The axion decay leads to efficient ionization of the IGM at high redshift, resulting in an unacceptably large CMB optical depth $\tau_{\mathrm{reio}} \approx 1.89$.
• Figure 4: High-redshift optical depth $\tau_{\mathrm{highz}}$ (color scale) as a function of axion mass and axion–photon coupling. Solid lines show 95% CL exclusion contours from different methods: yellow from our GP-based $\tau_{\mathrm{highz}}$ reconstruction ($z \in [30,800]$), red from low-redshift constraints ($z \in [0,30]$) consistent with Planck 2018 Planck:2018nkj, blue from Ref. Langhoff:2022bij ("Langhoff et al."), and white from the full $\chi^2$ analysis.
• Figure S1: The free electron fraction $X_e(z)$ (left) and marginalized constraints on the optical depth $\tau_{\mathrm{reio}}$ (right), obtained using lowE data alone. The amplitude of scalar perturbations, $A_{\mathrm{s}}\,e^{-2 \tau_{\mathrm{reio}}}$, as well as other cosmological parameters, are fixed to their best-fit values from the Planck TT,TE,EE+lowE datasets. A flat prior on $\tau_{\mathrm{reio}}$ is adopted, with a lower bound of $\tau_{\mathrm{reio}} > 0.043$ imposed to enforce observational constraints from the Gunn-Peterson effect.