Cosmological effects on the observed flux and fluence distributions of
  gamma-ray bursts: Are the most distant bursts in general the faintest ones?
  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. The purpose of this article is to clarify this contradiction. 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. 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.
