Consistency test of the fine-structure constant from the whole ionization history

In cosmology, the fine-structure constant can affect the whole ionization history. However, the previous works confine themselves to the recombination epoch and give various strong constraints on the fine-structure constant. In this paper, we also take the reionization epoch into consideration and do a consistency test of the fine-structure constant from the whole ionization history. From the data combination of Planck 2018, BAO data, SNIa samples, SFR density from UV and IR measurements, and the $Q_\text{HII}$ constraints, we find the constraint on the fine-structure constant during the recombination epoch is $\alpha_{\text{rec}}/\alpha_{\text{EM}}=1.001494^{+0.002041}_{-0.002063}$ and its counterpart during the reionization epoch is $\alpha_{\text{rei}}/\alpha_{\text{EM}}=0.854034^{+0.031678}_{-0.027209}$ at 68$\%$ C.L.. They are not consistent with each other by $4.64\sigma$. A conservative explanation for such a discrepancy is that there are some issues in the data we used. We prefer a calibration of some important parameters involved in reconstructing the reionization history.