The burst intensity distribution is the convolution of the intrinsic luminosity function and the burst rate, both as functions of the distance to the burst source. The simplest (i.e., "minimal") model assumes that there has been no cosmological evolution in the burst rate per comoving volume or in the luminosity function. Further, the luminosity function is assumed to be a delta function for a certain intrinsic intensity measure (e.g., the total radiated energy or the peak photon luminosity), that is, all bursts are "standard candles." The standard candle has been the same at all cosmological epochs by the no-evolution assumption. The aesthetic beauty of the cosmological scenario is that the bend in the intensity distribution can be wholly explained by the curvature of space and time dilation at cosmological distances. A consequence of the minimal model is that there is a one-to-one mapping between the burst intensity and the distance, with the distance scale given by the shape of the intensity distribution. For example, by this model the faint BATSE bursts are at a redshift of z~1.
Which type of intensity (e.g., total energy, total number of photons, peak energy luminosity or peak photon luminosity) is the standard candle is of course uncertain. Because BATSE triggers on the count rate in the 50-300 keV band, and the BATSE bursts constitute the largest homogeneous database, the peak photon flux (corresponding to the peak photon luminosity) is often used. An intensity measure related to a detector's trigger is favored because the low intensity threshold is best understood. However, this is a choice based on instrumental considerations and not on physics.
Many of the proposed energy sources for cosmological bursts are the endpoints of stellar evolution, and consequently a simple assumption is that bursts occur in galaxies at a rate proportional to a galaxy's mass. Assuming a constant mass-to-light ratio for all galaxies, this implies that the rate is proportional to a galaxy's luminosity. Therefore, the minimal theory predicts that the host galaxy's luminosity function is the luminosity function for regular galaxies, weighted by one power of the luminosity.
The minimal model is based on unrealistically simplistic assumptions. The great variety of burst profiles and the large dynamic range of burst properties such as spectral hardness and time duration make the standard candle assumption suspect. Indeed, Hakkila et al. (1996) found that a standard candle cosmological model, where the peak energy flux (ergs cm-2 s-1) corresponds to the standard candle, does not fit the joint PVO-BATSE distribution (of course, a different intensity measure might correspond to the standard candle). Similarly, all known astrophysical phenomena have undergone cosmological evolution. Studies have constrained the cosmological evolution of the burst rate. While it has generally been recognized that the burst population must have undergone evolution, and that bursts at any given epoch were not standard candles, most studies fitting the burst database have adopted the minimal scenario.