Inferring the Spatial and Energy Distribution of Gamma Ray Burst Sources. III. Anisotropic Models

We use Bayesian methods to analyze the distribution of gamma-ray burst intensities reported in the Third BATSE Catalog (3B catalog) of gamma-ray bursts, presuming the distribution of burst sources ("bursters") is isotropic. We study both phenomenological and cosmological source distribution models, using Bayes's theorem both to infer unknown parameters in the models and to compare rival models. We analyze the distribution of the time-averaged peak photon number flux, Φ, measured on both 64 ms and 1024 ms timescales, performing the analysis of data based on each timescale independently. Several of our findings differ from those of previous analyses that modeled burst detection less completely. In particular, we find that the width of the intrinsic luminosity function for bursters is unconstrained, and the luminosity function of the actually observed bursts can be extremely broad, in contrast to the findings of all previous studies. Useful constraints probably require observation of bursts significantly fainter than those visible to BATSE. We also find that the 3B peak flux data do not usefully constrain the redshifts of burst sources; useful constraints require the analysis of data beyond that in the 3B catalog (such as burst time histories) or data from brighter bursts than have been seen by BATSE (such as those observed by the Pioneer Venus Orbiter). In addition, we find that an accurate understanding of the peak flux distributions reported in the 3B almost certainly requires consideration of data on the temporal and spectral properties of bursts beyond that reported in the 3B catalog and more sophisticated modeling than has so far been attempted. We first analyze purely phenomenological power-law and broken power-law models for the distribution of observed peak fluxes. We find that the 64 ms data are adequately fitted by a single power law, but that the 1024 ms data significantly favor models with a sharp, steep break near the highest observed fluxes. At fluxes below the break, the distribution of 1024 ms fluxes is flatter than that of 64 ms fluxes. Neither data set is consistent with the power-law distribution expected from a homogeneous, Euclidean distribution of sources. Next we analyze three simple cosmological models for burst sources: standard candles with constant burst rate per comoving volume, a distribution of standard-candle sources with comoving burst rate proportional to a power law in (1 + z), and a bounded power-law burster luminosity function with constant comoving burst rate but variable power-law index and luminosity bounds. We find that the 3B data can usefully constrain the luminosity of a standard-candle cosmological population of bursts if there is no density evolution. But the 3B data allow strong density evolution and arbitrarily broad luminosity functions; consequently, they do not usefully constrain the redshifts or luminosities of cosmological burst sources. We elucidate the properties of the models responsible for these results. For sufficiently flexible models, the inferred values for parameters describing the shapes of the distributions of 64 ms and 1024 ms peak fluxes formally differ at the 68%-95% level. Because the measurements on these two timescales are not independent, it is difficult to ascertain the true significance of this discrepancy; since many bursts are common to both data sets, it is likely its significance is larger than these formal values indicate. In addition, the inferred amplitude (in bursts per year) of the distribution of 64 ms peak fluxes is about twice that of 1024 ms peak fluxes. These results strongly suggest that a complete understanding of the measured peak flux distributions requires simultaneous modeling and analysis of temporal properties of bursts. We study models that attempt to reconcile the two data sets by accounting for "peak dilution," the underestimation of the peak intensity that results from using data accumulated over a timescale exceeding the peak duration. A phenomenological model strongly correlating peak duration with peak flux is moderately successful at reconciling the data. A model that correlates peak duration with peak flux due to cosmological time dilation and relativistic beaming is less successful, but remains of interest in that it is a simple physical model illustrating how one can jointly model and analyze temporal and spectral properties of bursts with peak flux data. A more rigorous accounting for the differences between the 64 ms and 1024 ms data requires analysis of temporal and spectral information about bursts beyond that available in the 3B catalog.