The luminosity function of Swift long gamma-ray bursts

The accumulation of {\it Swift} observed gamma-ray bursts (GRBs) gradually makes it possible to directly derive a GRB luminosity function (LF) from observational luminosity distribution, where however two complexities must be involved as (i) the evolving connection between GRB rate and cosmic star formation rate and (ii) observational selection effects due to telescope thresholds and redshift measurements. With a phenomenological investigation on these two complexities, we constrain and discriminate two popular competitive LF models (i.e., broke-power-law LF and single-power-law LF with an exponential cutoff at low luminosities). As a result, we find that the broken-power-law LF could be more favored by the observation, with a break luminosity $L_b=2.5\times10^{52}\rm erg s^{-1}$ and prior- and post-break indices $ν_1=1.72$ and $ν_2=1.98$. For an extra evolution effect expressed by a factor $(1+z)^δ$, if the matallicity of GRB progenitors is lower than $\sim0.1Z_{\odot}$ as expected by some collapsar models, then there may be no extra evolution effect other than the metallicity evolution (i.e., $δ$ approaches to be zero). Alternatively, if we remove the theoretical metallicity requirement, then a relationship between the degenerate parameters $δ$ and $Z_{\max}$ can be found, very roughly, $δ\sim2.4(Z_{\max}/Z_{\odot}-0.06)$. This indicates that an extra evolution could become necessary for relatively high metallicities.

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