Cosmological effects on the observed flux and fluence distributions of gamma-ray bursts: Are the most distant bursts in general the faintest ones?

Context. Several claims have been put forward that an essential fraction of long-duration BATSE gamma-ray bursts should lie at redshifts larger than 5. This point-of-view follows from the natural assumption that fainter objects should, on average, lie at larger redshifts. However, redshifts larger than 5 are rare for bursts observed by Swift, seemingly contradicting the BATSE estimates. Aims. The purpose of this article is to clarify this contradiction. Methods. We derive the cosmological relationships between the observed and emitted quantities, and we arrive at a prediction that can be tested on the ensembles of bursts with determined redshifts. This analysis is independent on the assumed cosmology, on the observational biases, as well as on any gamma-ray burst model. Four different samples are studied: 8 BATSE bursts with redshifts, 13 bursts with derived pseudo-redshifts, 134 Swift bursts with redshifts, and 6 Fermi bursts with redshifts. Results. The controversy can be explained by the fact that apparently fainter bursts need not, in general, lie at large redshifts. Such a behaviour is possible, when the luminosities (or emitted energies) in a sample of bursts increase more than the dimming of the observed values with redshift. In such a case dP(z)/dz > 0 can hold, where P(z) is either the peak-flux or the fluence. All four different samples of the long bursts suggest that this is really the case. This also means that the hundreds of faint, long-duration BATSE bursts need not lie at high redshifts, and that the observed redshift distribution of long Swift bursts might actually represent the actual distribution.