Using the obtained above dependence of compact binary merging rates
in the elliptical galaxy on time (Fig. 
)
and assuming the cosmological origin of GRB as products
of binary NS/BH coalescences, we can compute the theoretical
- 
curve. To do this, we need to specify the cosmological
parameters, the moment of the star formation beginning,
and the spectral power-law index of a typical GRB
(see Lipunov et al. [22] for more detail).
Taking the density of baryons in stars
(in terms of critical density to close
the Universe) [23] and varying other parameters
within limits permitted by the present theory and observations
( 
-term: 
;
the fraction of elliptical galaxies: 
;
the star formation starting redshift:
), the total compact binary merging rate
in the Universe (a constant the - curve goes at small fluxes)
is found to vary not too much from
a few 
yr 
to a few 
yr (see
Fig. ).
Figure: (Right panel) The 2-nd BATSE catalog (solid points) is fitted by
the cosmological GRB model (from Lipunov
et al. [22]). Note that the total GRB rate
in the Universe is 
per year, 3 orders of magnitude
smaller than the total binary NS merging rate.
Figure: (Left panel) - curves calculated for different
spectral power-law indices attributable to gamma- and X-ray emission
in a GRB. Values of the cosmological model parameters are
shown in the Figure.
The bar indicates the accessible range of the total GRB rates
in the entire Universe varying the parameters as discussed in the text.
This curve is consistent with observational data obtained by BATSE
(Fig. from [22]). However, the total
GRB rate in the Universe will be of order
yr . The difference in about 3 orders
can be explained either by assuming the only one merging of 1000
to yield GRB, or by the gamma-ray emission collimation into a
solid angle [22].
Lipunov, Postnov, Prokhorov (1997: Astro-ph/9703181) Totany, 1997 Astro-ph/9707051) Estimate of the redshift of GRB970228 and GRB970508 using the mean statistical properties of observed GRBs. They assume the cosmological origin of GRBs as standard-candle binary neutron star mergers.
Figure:
The 
distribution of 3B BATSE GRBs from
256-ms 1-3 (50-300 keV) channels fitted with the cosmological model
distributions in a flat 
Universe with a cosmological term
assuming gamma-ray photon power law s=-1.1. The
locations of Beppo-SAX GRBs are shown.
GRB970228 and GRB970508 are marked with asterisks.
Figure: The redshift - peak flux dependence
in the cosmologocal models assumed for different
and s=-1.1. 3B BATSE catalog data are also plotted.
Totany, 1997 Astro-ph/9707051
It is shown that the binary merger
scenario of GRBs naturally results in the rate
evolution of 
up to 
1, that has been
suggested independently from the compatibility between the number-brightness
distribution and duration-brightness correlation.
Recent progress of observation of high redshift galaxies, however,
gives more detailed information on the cosmic star formation history
(Lilly et al. 1996; Madau et al. 1996). The Canada-France Redshift
Survey (CFRS) revealed a marked evolution of 2800 Å luminosity
density, that is considered to be a star formation
indicator, as 
to 
(for 
, [24]).
The constant SFR approximation in spiral galaxies is therefore no longer
justified even at z < 1.
The redshift of GRB970508 is likely about 2, just below the upper limit that is recently determined, and the absorption system at z = 0.835 seems not to be the site of the GRB.
Figure: The number versus peak-flux distribution (upper panel) and
redshift versus peak-flux relation (lower panel).
Thick solid line is the observation by BATSE (Meegan et al. 1996).
Theoretical curves are calculated
with SFR of the galaxy evolution model (thin-solid, long- and
short-dashed lines)
and with the observational SFR-evolution model (dot-dashed line).
The line markings are the same for upper and lower panels.
The adopted values of 
, 
, 
, and 
are shown in the figure. The peak fluxes of GRB970228 and
GRB970508 are also indicated.