An MAD explanation for the correlation between bulk Lorentz factor and minimum variability time-scale

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Abstract
We offer an explanation for the anticorrelation between the minimum variability time-scale (MTS) in the prompt emission light curve of gamma-ray bursts (GRBs) and the estimated bulk Lorentz factor of these GRBs, in the context of a magnetically arrested disc (MAD) model. In particular, we show that previously derived limits on the maximum available energy per baryon in a Blandford–Znajek jet leads to a relationship between the characteristic MAD time-scale in GRBs and the maximum bulk Lorentz factor: tMAD ∝ Γ−6, somewhat steeper than (although within the error bars of) the fitted relationship found in the GRB data. Similarly, the MAD model also naturally accounts for the observed anticorrelation between MTS and gamma-ray luminosity L in the GRB data, and we estimate the accretion rates of the GRB disc (given these luminosities) in the context of this model. Both these correlations (MTS−Γ and MTS−L) are also observed in the Active Galactic Nuclei (AGN) data, and we discuss the implications of our results in the context of both GRB and blazar systems.
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