The minimal cosmological scenario predicts that bursts should occur in galaxies, and that the distance to the burst, and therefore to the galaxy, can be calculated from the intensity. Schaefer (1992) pointed out that the small error boxes of 8 bright bursts do not contain bright galaxies; if the brightest galaxy in the error box, or the detection threshold for the box, had a brightness equal to M31 (the Andromeda Galaxy), the total burst energy must have been as large as 2x1053 ergs. Fenimore et al. (1993) found that Schaefer's data were only marginally consistent with the galaxies predicted by the minimal scenario if the brightest galaxy in each error box was indeed the host galaxy. However, the brightest galaxy could also be an unrelated background galaxy. This apparent discrepancy with the minimal scenario was dubbed the "no host galaxy" problem. On the other hand, Larson and collaborators reported that their sample of error boxes, which were somewhat larger than Schaefer's, had an excess of bright galaxies, although they recognized that they could not distinguish between host and background galaxies.
D. Hartmann and I realized that a more sophisticated analysis methodology was required. Therefore we use a likelihood ratio which contrasts the hypothesis that both host and unrelated background galaxies are present with the hypothesis that all the observed galaxies are unrelated background galaxies. This ratio was developed within a Bayesian framework, but it is understandable within standard "frequentist" statistics. If this ratio is much greater than 1 then a host galaxy is clearly present in each error box, while if the ratio is much less than 1 then no host galaxy is present. Finally, if the ratio is of order unity then the data are inconclusive. By construction, this methodology accounts for the unrelated background galaxies which will be detected if the error box is searched deeply enough. We include each detected galaxy in addition to the detection limit, and we permit a more sophisticated description of the error box. This methodology demonstrates that the observations of a given error box can show conclusively that the host galaxy is present only if the expected host galaxy is on average brighter than the average brightest background galaxy, which depends on the size of the error box.
We first applied this methodology to Larson and colleagues claim that there was not a "no host galaxy" problem. In our work we use the total burst energy, corresponding to the observed energy fluence, as the standard candle. We found that the likelihood ratio for the nine fields observed by Larson and McLean 1997a is 0.25, which indicates that we are unable to determine whether host galaxies are present. On the other hand, the likelihood ratio for the four error boxes observed by Schaefer et al. 1997 with the HST is 2x 10-6 which clearly shows that the host galaxies predicted by the minimal scenario are not present.
We subsequently applied our methodology to a database of 23 IPN error boxes and 7 optical transients associated with extended or persistent emission. The optical transient fields can be treated as an error box with finite extent because there is a region around the transient in which the host galaxy would have been acceptable (galaxies have finite extent and the progenitor may have traveled away from the burst before bursting), even if the transient is localized to a fraction of an arcsecond. In this study we varied the assumed standard candle total burst energy, and calculated the odds ratio for different values. We find that a burst energy of 3x1052 ergs is ruled out, while an energy of 1-3x1053 ergs is favored. This is much larger than the energy consistent with the minimal model. Note that the energies calculated for the 4 bursts with redshifts appear to be bimodal, with 2 at ~6x1051 ergs and 2 at ~1053 ergs.
Consequently more complicated cosmological models have been suggested. The burst rate might be proportional to the star formation rate; for example, a hypernova may result from a short-lived massive star, and thus bursts will occur when and where there has been recent star formation. The universe's star formation history has recently been determined empirically, and it shows that the rate per comoving volume has plummeted since z~1.5 (Madau et al. 1996); whether the rate was constant or increasing at higher redshifts is more controversial. Using this star formation rate as the burst rate can reproduce the burst intensity distribution, with the bursts occurring at much greater distances (Wijers et al. 1997b; Totani 1997). A surprising consequence is that the portion of the intensity distribution which is a power law with an index of -3/2 results not from a uniform burst density in nearby Euclidean space but from the balance between spatial curvature and burst evolution.
Thus the minimal cosmological scenario is too simple, as was suspected on astrophysical grounds. The development of more sophisticated cosmological theories involves issues such as star formation, and consequently the study of gamma-ray bursts will be more closely integrated with cosmology and extragalactic astrophysics.