Our calculations show that if GRB indeed are due to binary
neutron star coalescence, they can potentially constrain the
cosmological parameters as well as the early star formation
history. Even for the simplest models we used, late epochs of
galaxy formation ( 
) do not seem to be consistent with
the observed BATSE 
- 
distribution. The best-fit
model we obtained by fitting to the 411 GRB available from the 2nd BATSE
catalogue (Meegan et al. 1994)
corresponds to 
, 
and
, with a 
agreement according
to 
-test (we also used a Kolmogorov-Smirnov test; both
criteria give qualitatively similar results but we present results
for the -test as it gives a more smooth likelihood
function). According to the best-fit, the most distant BATSE GRB
come from 
(compared to Cohen & Piran 1995 who
obtained 
; however, we made no data selection). The
results are also sensitive to the accepted mean spectral index
of the GRB (s=1.5 in our case) and favours earlier star
formation 
for flatter spectra (s=1). We also note that
the earlier epochs of the primordial star formation are not favored
by other cosmological grounds (see calculations by Cen et al. 1994).
One may also wonder how the assumption about the total 
change
the results. Obviously, we can fit the observations for a wide range of
by varying other parameters ( 
, s, 
etc.).
The dependence of - curves on was found to be rather small
(Yi 1994), and our main conclusion still holds - the - should show
a dramatic turnup at low count rates due to early evolutionary effects.
The total merging event rate predicted by our evolutionary model
is 
per year per 
,
that is 
events/yr for the entire Universe, implying
a factor of 
overproduction relative to the presently
observed BATSE GRB rate of 0.8 events per day. This could be
explained, for example, by a relativistic beaming in GRB sources (Paczynski
1994). The corresponding angle required to explain the GRB
anisotropy that high is about 
(Mao & Yi 1994). Taking this
factor into account yields the expected total GRB rate 
events per year for a limiting sensitivity lower by a
factor of 3-10 than the presently exisitng BATSE limit.
We note that use of another mean galactic
density 
in the Universe would accordingly change the overall GRB
rate 
in the Universe, but does not change the - curve shape.
Taking it less than 0.08 per cubic megaparsec would somewhat
decrease the anisotropy required ( 
).
We conclude that the crucial test of the cosmological origin of
GRBs would be observing the predicted increase of - slope at
smaller fluxes, inevitable due to early evolutionary effects.
If the cosmological origin of GRB will be confirmed, the -
and 
test could be used to independently estimating the
cosmological parameters and tracing star formation history
in galaxies.