The GRB luminosity function: Predictions from the internal shock model and comparison with observations

We compute the expected luminosity function of gamma-ray bursts (GRBs) in the context of the internal shock model. We assume that GRB central engines generate relativistic outflows characterized by the respective distributions of injected kinetic power and contrast in Lorentz factor κ = Γ_max/Γ_min. We find that if the distribution of contrast extends down to values close to unity (i.e. if both highly variable and smooth outflows can exist), then the luminosity function has two branches. At high luminosity it follows the distribution of while at low luminosity it is close to a power law of slope -0.5. We then examine if existing data can constrain the luminosity function. Using the log N-log P curve, the E _p distribution of bright Burst and Transient Source Experiment (BATSE) bursts and the X-ray flash (XRF)/GRB ratio obtained by High Energy Transient Explorer 2 (HETE2), we show that single and broken power laws can provide equally good fits of these data. Present observations are therefore unable to favour one form or the other. However, when a broken power law is adopted they clearly indicate a low-luminosity slope ≃ -0.6 ± 0.2, compatible with the prediction of the internal shock model.