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Several types of extra-galactic high-energy transients have been discovered,which include high-luminosity and low-luminosity long-duration gamma-ray bursts(GRBs), short-duration GRBs, supernova shock breakouts (SBOs), and tidaldisruption events (TDEs) without or with an associated relativistic jet. Inthis paper, we apply a unified method to systematically study theredshift-dependent event rate densities and the global luminosity functions(ignoring redshift evolution) of these transients. We introduce some empiricalformulae for the redshift-dependent event rate densities for different types oftransients, and derive the local specific event rate density, which alsorepresents its global luminosity function. Long GRBs have a large enough sampleto reveal features in the global luminosity function, which is bestcharacterized as a triple power law. All the other transients are consistentwith having a single power law luminosity function. The total event ratedensity depends on the minimum luminosity, and we obtain the following valuesin units of ${\rm Gpc^{-3}~yr^{-1}}$: $0.8^{+0.1}_{-0.1}$ for high-luminositylong GRBs above $ 10^{50}~{\rm erg~s^{-1}}$, $164^{+98}_{-65}$ forlow-luminosity long GRBs above $5\times 10^{46}~{\rm erg~s^{-1}}$,$1.3^{+0.4}_{-0.3}$, $1.2^{+0.4}_{-0.3}$, and $3.3^{+1.0}_{-0.8}$ above $10^{50}~ {\rm erg~s^{-1}}$ for short GRBs with three different merger delaymodels (Gaussian, log-normal, and power law), $1.9^{+2.4}_{-1.2}\times 10^4$above $ 10^{44}~{\rm erg~s^{-1}}$ for SBOs, $ 4.8^{+3.2}_{-2.1}\times10^2 $ fornormal TDEs above $10^{44}~ {\rm erg~s^{-1}}$, and $0.03^{+0.04}_{-0.02}$ above$ 10^{48}~ {\rm erg~s^{-1}}$ for TDE jets as discovered by Swift. Intriguingly,the global luminosity functions of different kinds of transients, which coverover 12 orders of magnitude, are consistent with a single power law with anindex of -1.6.

Extra-galactic high-energy transients: event rate densities and luminosity functions