Fig. 1 shows the redshift dependence of for several of our models. We normalize the merger rate density to unity at z =0. The gray shaded area demonstrates that the results are basically consistent with the estimates derived from the GRB source distribution as observed by BATSE (Horack et al. 1995); the computed merger rate falls roughly in the allowed region for . Qualitatively, the behavior of is easily understood from Fig. 2 which shows the relative contribution of galaxies of different types to the merger rate (for set 1). At high z it is dominated by early-type galaxies with initially a high time-averaged SFR. In set Ib (upper solid line) the sudden drop at reflects the cessation of star formation in ellipticals in one Gyr after formation and the relatively short typical merger time of 1 Gyr (PZY98). The more gradual decline of in set 1 reflects the continuous star formation in both elliptical and spiral galaxies. Comparison of sets 1 with 2 suggests that the evolution of is better described by the models with longer star-formation time scales. The lower solid line uses the same merger-time function as Totani (1997). This model predicts a lower merger rate.
The upper and lower dashed lines in Fig. 1 (set 2) demonstrate the effect of the age of galaxies on the comoving merger rate, i. e.; the influence of the cosmological parameters.
Fig. 2 gives the relative contribution to the merger-rate density for each selected subclass of galaxies. The majority of events which are potentially detectable by gravitational-wave observatories (LIGO/VIRGO) are located in early-type spiral galaxies. If GRBs originate from coalescence the dimmest bursts are expected to be hosted in elliptical galaxies.
Following the standard procedure (e.g. Horack et al. 1996) we compute the number of bursts with a peak flux greater than P. We assume that bursts are standard candles and the intrinsic luminosity doesn't evolve. The spectral form of the burst similar to that observed is adopted: , where =350 keV is a characteristic energy. For comparison, we use the observed integral brightness distribution from the BATSE 3B catalog in the energy range 50-300 keV measured at a timespan of 1024 ms (Meegan et al. 1996). Fig. 3 shows the expected brightness distributions computed for sets of parameters 1 and 1b, superimposed on the BATSE data. The curves are normalized at the peak flux threshold P =0.4 photons cm-2 s-1. Fig. 4 provides the results of a Kolmogorov-Smirnov test of the BATSE 3B catalog to the results of our computations. Only data above the peak flux of 0.4 photons cm-2 s-1 are used to avoid threshold effects. The highest confidence level (CL) is obtained for sets 1 and 1b if the limiting redshift at the peak flux threshold is 2.4 and 3.0 respectively. For other values of the star formation timescales and cosmological parameters is found to range from 1.9 to 2.7 (with a CL ) for the models with a burst of star formation in elliptical galaxies (models 1b and 2b). Without an initial burst of star formation ranges from 2.9 to 3.3. Similar results are found for in the range 300 - 400 keV.
Note that for higher limiting redshift, Fig. 4 shows the existence of other possible fits with a lower CL (i.e. model 1b, ). The primary peak at corresponds to the first change of the slope of the comoving rate density (Fig. 1). For higher redshifts, the sudden increase of the merger rate would require the same behavior of the BATSE data for consistency. As a consequence, the secondary peak at , although providing a good fit for peak flux values near the threshold, tends to depart more and more with the data for higher values of P (Fig. 3). Therefore, the redshift range related to an assumed initial burst of star formation in elliptical galaxies is likely to be beyond the actual limiting redshift for GRBs.
Finally, it has to be pointed out that the models can hardly reproduce the peak flux values of BATSE's faintest bursts (see also Totani 1997). As shown by Reichard & Mészáros (1997), this feature results from the assumption that GRBs are standard candles.
© European Southern Observatory (ESO) 1998
Online publication: March 30, 1998