Reconstructing the epoch of reionisation with Planck PR4

The epoch of reionisation is a key phase in cosmic history, characterised by the ionisation of the intergalactic medium by the first luminous sources. In this work, we constrain the reionisation history of the Universe using data from the cosmic microwave background, more specifically the latest Planck Public Release 4 (PR4) dataset. We investigate a wide range of reionisation models, from simple parametric descriptions to more flexible non-parametric approaches, systematically evaluating their impact on the inferred constraints. Special attention is given to implicit priors introduced by each model and their influence on the derived reionisation optical depth, $τ$. To achieve this, we employ both Bayesian and frequentist methods to derive robust constraints. We obtain consistent estimates of $τ$ across models, highlighting the robustness of the constraints on the integrated optical depth derived from the Planck PR4 data. Averaging across models, the posterior means and best-fit values, respectively, yield $τ= 0.0576 \pm 0.0060$ and $τ= 0.0581$, highlighting the presence of small volume effects. Based on our analysis, we estimate that an additional uncertainty, associated with the modelling of reionisation, contributes an error of approximately $σ_τ\!\sim\!0.0006$. Beyond the integrated optical depth, our analysis reveals that the inferred ionisation fraction as a function of redshift is highly model-dependent. While current CMB data do not favour significant early ionisation, they are consistent with a modest contribution from ionised gas at very early times ($z>15$). Although indicative upper bounds can be placed on such contributions, these limits remain strongly dependent on the assumed model.