- 
distribution and 
-test.
We conventionally assume a flat Universe
so that 
,
with 
and 
being fractional
contributions of matter and cosmological constant term (Carroll,
Press & Turner 1992). The present value of the Hubble constant
is assumed to be 75 km/s/Mpc. The GRB were considered as
standard candles with a proper luminosity L
and have a power-law spectrum 
with a spectral
index s=1.5 (Schaefer et al. 1992). Briefly, the count rate
at the detector from a source at a redshift z is
where 
is a metric distance (see Carroll et al. 1992).
The number of events, N(>C), with an observed count rate
exceeding C is thus
The source evolution n(z) can easily be obtained using
the evolutionary Green function: 
and the dependence t(z) for a
particular cosmological model.
Fig.
shows the "Green functions" for the double compact
binary merging rate evolution assuming no collapse anisotropy.
A very strong early evolution with time is seen; however, even
for an elliptical galaxy the NS+NS merging rate is
1 per 
yrs for an age of 
yrs. The
non-monotonic character of the merging rate evolution is due to
different contributions of different type binaries (by initial
masses, separations and types of the first mass exchange); as
the detailed shape is of less importance for us now, we
postpone discussing these interesting features in a separate
paper. We should, however, note that a small decrease of
the coalescence rate observed at the age of about 4 billion years
is statistically significant and is caused by contributions
of evolutionary different types of NS+NS binaries
(they come from different ranges
of the initial mass ratios and semimajor axes). This feature can produce a
notable decrease in the - curve slope
at the corresponding 
. Thus, if 
the drop in sources production rate after 4 billion years from the beginning
corresponds (for the flat Universe), somewhat surprisingly,
just to the feature visually seen in both 2d and 3d BATSE catalog
- curves; however, it would be prematurely to take the apparent
feature seriously due to its low statistical significance.
Another possible way that could lead to arising diverse peculiarities
in the - distribution induced by the sources evolution
is connected with a possible non-monotonic
character of the star formation rate in galaxies.
For a 
spiral galaxy with constant
star formation rate our calculations give a binary NS
coalescence rate of the order of one per five thousand years only
slightly depending on the initial mass ratio spectrum and
assumptions about massive core collapse anisotropy. This rate
is close to the most ``optimistic'' estimates based on evolutionary
considerations (Lipunov et al 1987, 1995; Tutukov & Yungelson
1992; van den Heuvel 1994). In our calculations we used a
smoothed evolutionary function assuming the star formation to
occur during the first 500 million years in elliptical galaxies, and
constant in spiral galaxies. In fact, the results proved to be
only weakly sensitive to the smoothening over time provided
that it is less than 1 billlion years (the width of the wide
peak of the Green function; see Fig.).
Our theoretical models depend on three unknown parameters,
, 
, 
, as well as on s. The calculated
- were compared with the BATSE data by using Mises-Smirnov
test
(which gives essentially the same results as Kolmogorov-Smirnov
test, but uses a more smooth criterion for comparing observed and
tested distribution functions; see Bol'shev & Smirnov 1965).
Fig., ,
show two-dimensional cuts through the parameter space.
All three cuts show contour lines for confidence level according
to test higher than 90%, with the maximum level being at
in all planes.
The calculated - distributions are plotted in Fig. for
different star formation starting times .
Other parameters are fixed at the best-fit values
and 
. The most prominent
feature of the obtained theoretical - is a notable sharp
increase at low count rates due to evolutionary effects.
In terms of -test, such a turnup of the - curve would lead
to a sharp increase of the average values when weaker
sources (i.e. farther sample limits)
are taken into account (Fig.).
The curves turn out to be most
sensitive to the parameter and depend only
slightly on other parameters. We also note that a flatter GRB
spectral index s=1 would favour earlier initial star formation
and changes the - curves stronger at lower .
If the cosmological binary NS coalescences
do underly the GRB phenomenon, late epochs of the initial
star formation in galaxies ( 
) would yield
inconsistency with the already existing BATSE data.