The GRB luminosity function in the internal shock model confronted to
  observations
  We compute the expected luminosity function of 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 Edot and contrast in Lorentz factor Kappa = Gamma_max/Gamma_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) the luminosity
function has two branches. At high luminosity it follows the distribution of
Edot 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 Ep distribution of bright BATSE bursts and the XRF/GRB ratio
obtained by HETE2 we show that single and broken power-laws can provide equally
good fits of these data. Present observations are therefore unable to favor one
form of 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.
