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 distribution of cosmological bursting or steady sources valid over the entire range of fluxes, including density evolution and either standard candle or a power-law luminosity function. These are used to derive analytic formulae for the mean redshift, the time dilations, and the dispersion of these quantities for sources within a given flux range for Friedmann models with Ω = 1, Λ = 0 without K-corrections. We discuss the extension to cases with Ω < 1 and inclusion of K-corrections. Applications to the spatial distribution of cosmological gamma-ray burst sources are discussed, both with and without an intrinsic energy stretching of the burst time profiles, and the implied ranges of redshift z are considered for a specific time dilation signal value. The simultaneous consideration of time dilation information and of fits of the number distribution versus peak flux breaks the degeneracy inherent in the latter alone, allowing a unique determination of the density evolution index and the characteristic luminosity of the sources. For a reported tune dilation signal of 2.25 and neglecting (including) energy stretching, we find that the proper density should evolve more steeply with redshift than comoving constant, and the redshifts of the dimmest sources with stretching would be very large. However, the expected statistical dispersion in the redshifts is large, especially for power-law luminosity functions, and remains compatible with that of distant quasars. For smaller tune dilation values of 1.75 and 1.35, the redshifts are more compatible with conventional ideas about galaxy formation, and the evolution is closer to a comoving constant or a slower evolution. More generally, we have considered a wide range of possible measured tune dilation ratios, and we discuss the values of the density evolution and the redshifts that would be expected for different values of the energy stretching.