Abstract: Gamma-ray bursts (GRBs) are among the most luminous electromagnetic events in the universe. These are divided into two subtypes: long GRBs (LGRBs), originating from the collapse of massive stars with a typical duration exceeding two seconds, and short GRBs, associated with the merger of binary compact objects. LGRBs at high redshifts serve as key indicators of star formation during the early stages of cosmic reionization, driven by the first Population III (Pop III) stars. The rate of GRBs from isolated Pop III stars remains poorly understood, hinging on uncertainties in their initial mass function (IMF), rotation rates, stellar evolution, and mass loss. A subset of massive Pop III stars (M_ZAMS ≥ 20 M⊙) is expected to undergo core-collapse, potentially launching relativistic jets that power GRBs. This study examines a grid of stellar evolution models for Pop III stars with masses ranging from 20 to 100 M⊙, initially rotating at surface angular velocities between 60% and 90% of the critical rotation limit (0.6 ≤ Ω₀/Ω_crit ≤ 0.9). By using realistic accretion and jet propagation models, we derive initial black hole masses, spins, and jet breakout times. GRB production efficiency is calculated across a phase space of progenitor initial mass, rotation, and wind efficiency, yielding values of η_GRB ~ 10⁻⁵ to 3 × 10⁻⁴ M⊙⁻¹ for a top-heavy IMF. This value corresponds to an observable all-sky equivalent rate of approximately 2–40 events yr⁻¹, as estimated by Swift, with 75% of GRBs located at z ≤ 8. If the actual observed rate is significantly lower, it suggests that the wind efficiency must be greater than 50%, implying substantial mass and angular momentum loss that may render isolated Pop III stars incapable of producing detectable GRBs with current instruments.
