Probing Helium Reionization Through the $^{3}\mathrm{He}^{+}$ Hyperfine Transition Line

We investigate the hyperfine transition of $^{3}\mathrm{He}^{+}$ as a promising probe of the IGM during the final stages of helium reionization. Utilising the most recent helium reionization simulation, we generate three-dimensional maps of the 3.5cm ($8.67$ GHz) differential brightness temperature and analyze its evolution. Our results show that the volume-averaged brightness temperature declines rapidly from $\sim 1 \mu$K at $z = 4$ to $\sim 2.5 \times 10^{-3} \mu$K by $z = 2.3$, tracing the HeII to HeIII transition driven by quasars. The power spectrum of the 3.5cm signal exhibits a scale-dependent evolution, peaking on small scales and declining as reionization progresses. We explore the cross-correlation of the 3.5cm transition line with the distribution of AGNs, which shows a transition from positive to negative correlation as ionized regions grow. We also examine the 3.5cm forest and demonstrate that absorption features persist down to $z \sim 2.90$, even when more than $85\%$ of HeII is ionized. Although current observational upper limits lie several orders of magnitude above theoretical predictions, future radio arrays such as $\mathrm{SKA-mid}$ offer promising prospects. Overall, this study highlights the $^{3}\mathrm{He}^{+}$ hyperfine transition as a sensitive tracer of the thermal and ionization history of the IGM during helium reionization.