A Model-independent Lower Limit on the Number of Gamma-Ray Burst Hosts from Repeater Statistics

We present a general statistical analysis of Gamma Ray Bursts embedded in ahost population. If no host generates more than one observed burst, then weshow that there is a model independent lower bound on the number of hosts, $H$,of the form $H > c B^2$, where B is the number of observed bursts, and $c$ is aconstant of order one which depends on the confidence level (CL) attached tothe bound. An analysis by Tegmark et al. (1996) shows that the BATSE 3B catalogof 1122 bursts is consistent with no repeaters being present, and assuming thatthis is indeed the case, our result implies a host population with at leastH=1.2x10^6 members. Without the explicit assumption of no repeaters, a Bayesiananalysis based on the results of Tegmark et al. (1996) can be performed whichgives the weaker bound of $H>1.7\times 10^5$ at the 90% CL. In the light of thenon-detection of identifiable hosts in the small error-boxes associated withtransient counterparts to GRBs, this result gives a model independent lowerbound to the number of any rare or exotic hosts. If in fact GRBs are found tobe associated with a particular sub-class of galaxies, then an analysis alongthe lines presented here can be used to place a lower bound on the fraction ofgalaxies in this sub-class. Another possibility is to treat galaxy clusters(rather than individual galaxies) as the host population, provided that theangular size of each cluster considered is less than the resolution of thedetector. Finally, if repeaters are ever detected in a statisticallysignificant manner, this analysis can be readily adapted to find upper andlower limits on $H$.Comment: 9 pages (LaTex, aaspp4.sty); revised version includes a detailed discussion of limits which can be set using present BATSE data; to be published in ApJ Letter