Population III.1 and III.2 gamma-Ray Bursts: Constraints on the event rate for future radio and X-ray surveys

We calculate the theoretical event rate of gamma-ray bursts (GRBs) from the collapse of massive first-generation (Population III; Pop III) stars. The Pop III GRBs could be super-energetic with the isotropic energy up to $E_{\rm iso} \gtrsim 10^{55-57}$ ergs, providing a unique probe of the high-redshift Universe. We consider both the so-called Pop III.1 stars (primordial) and Pop III.2 stars (primordial but affected by radiation from other stars). We employ a semi-analytical approach that considers inhomogeneous hydrogen reionization and chemical evolution of the intergalactic medium. We show that Pop III.2 GRBs occur more than 100 times more frequently than Pop III.1 GRBs, and thus should be suitable targets for future GRB missions. Interestingly, our optimistic model predicts an event rate that is already constrained by the current radio transient searches. We expect $\sim10-10^4$ radio afterglows above $\sim$ 0.3 mJy on the sky with $\sim 1$ year variability and mostly without GRBs (orphans), which are detectable by ALMA, EVLA, LOFAR, and SKA, while we expect to observe maximum of $N < 20$ GRBs per year integrated over at $z>6$ for Pop III.2 and $N < 0.08$ per year integrated over at $z>10$ for Pop III.1 with EXIST, and $N < 0.2$ for Pop III.2 GRBs per year integrated over at $z > 6$ with \textit{Swift}.