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Astron. Astrophys. 332, L57-L60 (1998)

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3. Application

Fig. 1 shows the redshift dependence of [FORMULA] 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 [FORMULA]. Qualitatively, the behavior of [FORMULA] is easily understood from Fig. 2 which shows the relative contribution of galaxies of different types to the [FORMULA] 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 [FORMULA] reflects the cessation of star formation in ellipticals in one Gyr after formation and the relatively short typical [FORMULA] merger time of 1 Gyr (PZY98). The more gradual decline of [FORMULA] 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 [FORMULA] is better described by the models with longer star-formation time scales. The lower solid line uses the same merger-time function [FORMULA] as Totani (1997). This model predicts a lower [FORMULA] merger rate.

[FIGURE] Fig. 1. Evolution of the comoving [FORMULA] merger rate density as a function of redshift, normalized to unity at z =0. The shaded area corresponds to the allowed region according to [FORMULA] with [FORMULA] in the range 1.5 to 2. The upper solid (dotted) line is for set 1b (set 1) with [FORMULA] (0)=12.16 Gyr. The dashed lines give the rates for set 2 with [FORMULA] (0)=12.16 Gyr (upper curve) and 9.92 Gyr (lower curve). The lower solid line has to be compared with the dotted line and corresponds to the merger time distribution [FORMULA] with a lower cut-off at 0.02 Gyr (like in Totani 1997).

[FIGURE] Fig. 2. Relative contribution of galaxies of different types to the [FORMULA] merger rate density. The current galactic age is [FORMULA] =12.16 Gyr and set 1 of the SF timescales is used.

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 [FORMULA] 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 [FORMULA] 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 [FORMULA] 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: [FORMULA], where [FORMULA] =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 [FORMULA] at the peak flux threshold is [FORMULA] 2.4 and 3.0 respectively. For other values of the star formation timescales and cosmological parameters [FORMULA] is found to range from 1.9 to 2.7 (with a CL [FORMULA]) for the models with a burst of star formation in elliptical galaxies (models 1b and 2b). Without an initial burst of star formation [FORMULA] ranges from 2.9 to 3.3. Similar results are found for [FORMULA] in the range 300 - 400 keV.

[FIGURE] Fig. 3. Cumulative counts of bursts [FORMULA] of the BATSE 3B catalog (histogram) and model curves computed for parameters shown in the figure (for [FORMULA]  = 12.16 Gyr)

[FIGURE] Fig. 4. Redshift [FORMULA] necessary to achieve agreement with the BATSE 3B [FORMULA] distribution. The solid line corresponds to set 1 and the dotted line is for set 1b. In both cases, the current galactic age used is [FORMULA] Gyr.

Note that for higher limiting redshift, Fig. 4 shows the existence of other possible fits with a lower CL (i.e. model 1b, [FORMULA]). The primary peak at [FORMULA] 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 [FORMULA], 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.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998