The $\beta$-Dependence of Particle Spectra in Relativistic Turbulent Reconnection
We perform numerical simulations of particle acceleration in relativistic, self-driven turbulent magnetic reconnection using the MHD-PIC method. We systematically investigate the dependence of the non-thermal particle spectral exponent on the plasma $\beta$. We find that particle acceleration proceeds in two stages: an initial, efficient first-order Fermi phase where momentum gains are comparable in parallel and perpendicular directions, followed by a slower drift-dominated phase. The power-law slope of the non-thermal spectrum is established during the Fermi phase, as found in previous studies. Our results demonstrate a systematic steepening of the accelerated particle energy spectrum with increasing $\beta$. We derive empirical scaling relations: the spectral exponent $\alpha \propto \beta^{0.5}$ in the relativistic regime, compared to $\alpha \propto \beta^{0.3}$ in the non-relativistic case. This marked difference is rooted in relativistic physics: the increased inertial mass density ($\rho h$) in high-$\beta$ plasmas acts as an energy sink, reducing the Alfv\'en velocity and thereby altering the dynamics of magnetic energy release and its partition efficiency. The derived scaling provides a unified physical framework for interpreting the diversity of non-thermal radiation spectra observed in astrophysical sources, including black hole corona X-ray flares, gamma-ray bursts, and active galactic nucleus jets.