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recent research results indicate that radiation from gamma - ray bursts ( grbs ) may be constrained mostly in narrow beams @xcite , which suggests that grbs may be standard candles at cosmological distances @xcite . previous studies of the grb space density distribution suggested that grb production rate may follow the observed star formation rate for small values of redshift @xcite , but may increase monotonically for very high values of redshift @xcite ; however neither the lower limit nor the upper limit of redshift for grb production could be determined reliably previously because these studies are either limited by statistics or have to rely on some empirical relationships between some statistical properties of selected samples of grbs . because both the space density distribution and beaming effect play important roles for the observed grb intensity distribution , here we make use of the batse / cgro grb intensity distribution , which is the largest sample of grbs collected so far , to determine directly the grb space density and opening angle distribution for the grb space density distribution , we assume the following model , under the cosmological parameters of @xmath2 , @xmath3 km s@xmath4 mpc@xmath4 and @xmath5 , in order to compare with previous star formation rate measurements @xcite . @xmath6    the functional form is taken from the measured star formation rate @xcite with @xmath7 and @xmath8 . our main goal in this paper is to determine @xmath9 and @xmath10 for grbs from the observed grb intensity distribution , in order to answer the question where in the universe grbs are produced predominantly .    for the grb opening angle distribution , we study three cases when the grb opening angle @xmath11 follows gaussian , exponential , or power - law distributions . for a gaussian distribution of @xmath11 , the probability that the opening angle equals to a given angle is ,    @xmath12    we assume that the total energy @xmath13 is the same for for every long grb , i.e. , they can be used as standard candles@xcite and the opening angle @xmath11 is very small , the probability that the observed fluence @xmath14 is higher than a specified value @xmath15 is then , @xmath16 @xmath17 where @xmath18 is the luminosity distance . thus the number of bursts detected is , @xmath19    similarly , for an exponential distribution with the parameter @xmath20 , @xmath21 and for a power - law distribution with the parameter @xmath22 , @xmath23    if we assume the peak luminosity of grbs is the standard candle , we only need to replace @xmath13 and @xmath15 by the standard candle luminosity @xmath24 and the peak flux @xmath25 in equations ( 3 ) , ( 4 ) , ( 5 ) and ( 6 ) , respectively . we number these slightly modified equations as ( 3)@xmath26 , ( 4)@xmath26 , ( 5)@xmath26 and ( 6)@xmath26 , respectively ; for brevity we do not list these equations explicitly . we apply equation ( 1 ) for the grb space density distribution and equations ( 4 ) , ( 4)@xmath26 , ( 5 ) , ( 5)@xmath26 , ( 6 ) and ( 6)@xmath26 for the grb opening angle distribution to fit the observed grb intensity distribution . we focus our study on long grbs ( @xmath27 ) @xcite because currently only long bursts are known to be originated at cosmological distances from the direct redshift measurements of the optical afterglows of some long grbs . all long grbs with peak flux values and detection probability above 5% included in the grb 4b+ catalog @xcite are selected , resulting in 1727 grbs as our sample . the fluence values of these grbs are taken also directly from the catalog . for the peak flux distribution , the detection efficiency correction is done directly using the detection probability of each peak flux value given in the catalog . because the fluence and peak flux does not have a good linear correlation , for each fluence range we average the detection probability of all grbs included in this fluence range and then correct for the detection efficiency of this fluence range . total energy as standard candle .    .@xmath28_dof_. @xmath24 is in units of @xmath29 _ ergs / s_. @xmath30 in units of gpc@xmath31year . all errors quoted are for 68.3% confidence . for both cases the exponential distribution of the opening angle offers the best fits ; however the residuals for the combination of peak luminosity as the standard candle and the exponential distribution is the only statistically acceptable one . [ cols="^,^,^,^,^",options="header " , ]     our fitting results are shown in @xmath32 and @xmath33 . because our fittings are done with the integral distributions , the numbers of grbs in different bins are not completely independent . to address this problem , we carried out extensive bootstrap simulations . taking the 1727 grbs as the seed distribution , we randomly generated a large numbers of peak flux or fluence distributions and fitted the sampled distributions in the same way . the best values and errors of all parameters are then calculated from these fittings ; the bootstrapped parameter values and errors are consistent with that listed in table 1 . we therefore believe the correction between the different points does not have any significant impact to our results . from the fitting results , we conclude :    * for long grbs , despite the different fitting residuals of the two standard candle models and three opening angle distributions assumed , the cosmological parameters of grbs inferred are strikingly similar , indicating the robustness of the grb cosmological model . * the power - law distribution for the grb opening angles can not fit the data and is thus rejected confidently . * the gaussian distributions for the grb opening angles fit the data only marginally , and may be rejected too . * for both standard candle models the exponential distribution for the grb opening angles provides the best fit to the data . * the combination of the peak luminosity as standard candle and the exponential distribution for the grb opening angles is the only one with statistically acceptable fit to the data ; we therefore accept this as the accepted model for long grbs . under this model the peak luminosity of a grb is @xmath34 _ ergs / s _ , the exponential constant of the opening angle distribution is @xmath35 ( radian@xmath4 ) , the lower bound to the grb redshift is @xmath36 , and the upper break redshift is @xmath37 ( no lower limit to the break is detected because not enough sufficiently faint grbs have been observed with batse ) . first we compare our study with previous studies of grb cosmological models with batse intensity distribution @xcite + @xcite . all these studies concluded that the dimmest grbs are at redshift around unity , probaly following the measured star formation rate which peaks also at redshift around unity @xcite . the main differences between our approach and previous studies are the following :    * we take the grb opening angle distribution as part of the cosmological model . this is very important , since the observed grb flux depends strongly on the opening angle , as well as the redshift . * we separate the long and short grbs as two different classes in our study . our conclusions are only applicable to long grbs . for short grbs , no conclusion can be drawn due to the limited statistical quality of data . * one simplification of our study is that we did not make the energy spectral correction to the fluence or peak flux as a function of @xmath38 ( k - correction ) ; we just take the fluence or peak flux values as listed in the catalog . technically this correction is not straightforward , because we can no longer uniquely convert the fluence or peak flux directly to redshift , because we believe that grb radiation is beamed , and the opening angle follows a certain distribution . therefore the observed fluence or peak flux is due to both the redshift and the opening angle of the grb . the good fit of the assumed simple cosmological model with only several free - parameters indicates that this correction may not impact our conclusions significantly . in fact , since the energy spectra of grbs in the batse band follows approximately a power - law with an index of @xmath392 , the k - correction is not important .    frail _ et al . _ ( 2001 ) suggested that the total energy of grbs may be a constant , based on the opening angles of several grbs determined from their afterglow light curve break . our result is not in major conflict with their suggestion , but favors the peak luminosity as the standard candle . perhaps the direction of the grb beam is not completely stable , as indicated from observations of other relativistic jets and outflows from stellar mass and supermassive black holes ; only when the beam is pointed directly at us can we observe the maximum flux from the grb . however it has been argued that for some grbs the lower limit of the total energy released is @xmath40 _ ergs _ as determined from the redshifted x - ray emission lines @xcite . however the reality of the detection of the emission lines is still under debate , and an alternative interpretation of the detected emission lines also exists @xcite . the opening angles of some grbs with afterglow detections have been determined previously from the grb afterglow data @xcite ; we call this frail - distribution . in @xmath41 we compare the frail - distribution with the opening angle distributions determined with the observed grb intensity distribution data , just as a sanity check . the power - law distribution agrees with the frail - distribution for large opening angles , but fails to predict the absence of grbs detected with opening angles smaller than 0.04 radian . no grbs with opening angles narrower than 0.04 radian was directly observed , but the recent report on the polarization of grb021206 suggests that it is possible for some grbs to have opening angles narrower than 0.01 radian @xcite . the gaussian distribution agrees with the peak of the frail - distribution distribution reasonably well ; however this model can not predict enough grbs with opening angles greater than 0.2 radian . the exponential distribution is consistent with the frail - distribution distribution for both standard candle models , but favors slightly the peak luminosity standard candle model . it was suggested that for the model in which the grbs follow the star formation rate strictly , the most possible opening angle should be around 0.1 radian @xcite . however , if the majority of faint grbs originate at higher redshift than the peak of star formation rate as our model predicts , then a narrower average opening angle is required . we therefore conclude that grb opening angles follow an exponential distribution with a mean opening angle of @xmath42 radian , and the grb space density follows a power - law with an index of @xmath393.5 between @xmath36 and @xmath43 . the general trend of the grb space density distribution determined here as a function of @xmath38 is consistent with previous studies @xcite ; however our result is statistically much more robust , and in particular @xmath9 is determined for the first time and the lower limit to @xmath10 is constrained more strictly than before . our result therefore demonstrates that grbs are no longer produced frequently in the nearby and present universe at @xmath44 , and grbs are predominantly produced in the early universe at @xmath1 . indeed most grbs with direct redshift measurements have redshifts @xmath45 @xcite , with the exceptions of grb 980425 ( @xmath38=0.0085 ) and grb 030329 ( @xmath38=0.1687 ) which may be associated with two peculiar but similar supernovae sn1998bw @xcite , and sn2003dh @xcite . it remains to be seen if these grbs belong to a subset of grbs which are different from the majority of grbs which seem to be produced at much larger redshift . it has been suggested that grbs follow star formation rate , because it is commonly believed that grbs are produced as the final gravitational collapse of massive stars @xcite . the grb space density determined here follows the measured star formation rate between @xmath0 to @xmath46 . the determination of @xmath36 suggests that the massive stars capable of producing the majority of long grbs are no longer formed frequently today . future grb experiments capable of measuring the redshift of many more grbs and thus providing direct distance measurements of grbs , such as the _ swift _ mission to be launched in late 2003 @xcite , will test this prediction firmly . the monotonic increase of the grb space density up to at least @xmath47 suggests that the formation rate of massive stars capable of producing grbs should peak at significantly higher @xmath38 than the measured star formation rate . this implies that the  dark ages " of the universe ended much earlier than previously believed at @xmath48 based on the measured space density distribution of high - redshift qsos @xcite . there are also some previous independent works showing that an essential amount of long grbs should be at @xmath49 @xcite or even up to @xmath50@xcite . a recently detected grb031203 by integral was reported to be located at @xmath51 ( formally @xmath52 ) @xcite , consistent with our model prediction . therefore grbs may be a probe of the epoch of reionization of the early universe @xcite . currently the grb intensity distribution data can not constrain the exact upper limit of @xmath10 , because not enough extremely faint grbs have been observed . future grb experiments , such as the proposed  next generation grb mission " @xcite would detect many more grbs originated in the extremely early universe , which may determine the exact ending time of the  dark ages " of the universe in order to test the prediction of some numerical simulations @xcite which suggests that the first stars may have been formed at @xmath53 . grbs may prove to be the most sensitive and ultimate probes into the early epoch of the universe when the first structures were formed . * acknowledgement : * we thank drs . zigao dai , tan lu , yongfeng huang and daming wei for valuable comments to the manuscript . this study is supported in part by the special funds for major state basic research projects ( 10233010 ) and by the national natural science foundation of china . snz also acknowledges supports by nasa s marshall space flight center and through nasa s long term space astrophysics program .
it has been known that at least some of the observed gamma - ray bursts ( grbs ) are produced at cosmological distances and the grb production rate may follow the star formation rate . 
 we model the batse detected intensity distribution of long grbs in order to determine their space density distribution and opening angle distribution . 
 our main results are : the lower and upper distance limits to the grb production are @xmath0 and @xmath1 , respectively ; the grb opening angle follows exponential distribution and the mean opening angle is about 0.03 radian ; the peak luminosity appears a better standard candle than the total energy of a grb .