COSMOLOGICAL EVOLUTION AND LUMINOSITY FUNCTION EFFECTS ON NUMBER COUNTS, REDSHIFT, AND TIME DILATION OF BURSTING SOURCES

We present analytic formulae for the integral number count distributio n of cosmological bursting or steady sources valid over the entire ran ge of fluxes, including density evolution and either standard candle o r a power-law luminosity function. These are used to derive analytic f ormulae for the mean redshift, the time dilations, and the dispersion of these quantities for sources within a given flux range for Friedman n models with Ohm = 1, Lambda = 0 without K-corrections. We discuss th e extension to cases with Ohm < 1 and inclusion of K-corrections. Appl ications to the spatial distribution of cosmological gamma-ray burst s ources are discussed, both with and without an intrinsic energy stretc hing of the burst time profiles, and the implied ranges of redshift z are considered for a specific time dilation signal value. The simultan eous consideration of time dilation information and of fits of the num ber distribution versus peak flux breaks the degeneracy inherent in th e latter alone, allowing a unique determination of the density evoluti on index and the characteristic luminosity of the sources. For a repor ted time dilation signal of 2.25 and neglecting (including) energy str etching, we find that the proper density should evolve more steeply wi th redshift than comoving constant, and the redshifts of the dimmest s ources with stretching would be very large. However, the expected stat istical dispersion in the redshifts is large, especially for power-law luminosity functions, and remains compatible with that of distant qua sars. For smaller time dilation values of 1.75 and 1.35, the redshifts are more compatible with conventional ideas about galaxy formation, a nd the evolution is closer to a comoving constant or a slower evolutio n. More generally, we have considered a wide range of possible measure d time dilation ratios, and we discuss the values of the density evolu tion and the redshifts that would be expected for different values of the energy stretching.