Charles A. Meegan ... The small reference sample of six BATSE gamma-ray bursts with known .... reference and comparison groups of gamma-ray bursts are.
The Astrophysical Journal, 584:904–910, 2003 February 20 # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
COMPARISON OF REDSHIFT-KNOWN GAMMA-RAY BURSTS WITH THE MAIN GROUPS OF BRIGHT BATSE EVENTS Igor G. Mitrofanov, Anton B. Sanin, Dmitrij S. Anfimov, and Maxim L. Litvak Space Research Institute, Profsojuznaya Str. 84/32, 117810 Moscow, Russia
Michael S. Briggs, William S. Paciesas, Geoffrey N. Pendleton, and Robert D. Preece Department of Physics, University of Alabama in Huntsville, Huntsville, AL 35899; and National Space Science and Technology Center, Huntsville, AL 35812
and Charles A. Meegan National Space Science and Technology Center and NASA Marshall Space Flight Center, Huntsville, AL 35812 Received 2001 January 10; accepted 2002 October 29
ABSTRACT The small reference sample of six BATSE gamma-ray bursts with known redshifts from optical afterglows is compared with a comparison group of the 218 brightest BATSE bursts. These two groups are shown to be consistent both with respect to the distributions of the spectral peak parameter in the observer’s frame and also with respect to the distributions of the frame-independent cosmological invariant parameter (CIP). Using the known values of the redshifts (z) for the reference sample, the rest-frame distribution of spectral parameters is built. The deredshifted distribution of the spectral parameters of the reference sample is compared with distribution of these parameters for the comparison group after deredshifting by the factor 1=ð1 þ zÞ, with z a free parameter. Requiring consistency between these two distributions produces a collective estimation of the best fitting redshifts for the comparison group z ¼ 1:8–3.6. These values can be considered as the average cosmological redshift of the sources of the brightest BATSE bursts. The most probable value of the peak energy of the spectrum in the rest frame is 920 keV. Subject heading: gamma rays: bursts
Mitrofanov et al. 1996; Dezalay et al. 1998). Some evidence has also been found that the average curves of emissivity (ACE) of dimmer bursts are stretched with respect to the ACE of bright bursts, although the results of different analyses were not entirely consistent (Norris et al. 1994; Norris 1998; Mitrofanov et al. 1996, 1999c; Stern et al. 1999). Moreover, it is not clear whether the hardness/intensity correlation and stretching/intensity anticorrelation for the same groups of bursts are simultaneously consistent with the predictions of cosmological transformation (Mitrofanov et al. 1999b). Therefore, we cannot assume that the simple cosmological model explains the observed hardness/ stretching/brightness signatures of GRBs. The source density of GRBs is now thought to be a strong function of redshift and, in this respect, the assumption of nonevolving source properties has been abandoned. The evolving density is commonly thought to be connected with the star formation rate. The hardness/intensity and stretching/intensity correlations were originally discussed using the idea that the brightness distribution resulted from the combination of a constant source density and cosmological volume effects: both correlations must now be understood in terms of GRB source density varying with redshift. It is also possible that the intrinsic properties of emission vary with redshift but, as there exists no clear evidence for this considerable complication, we will assume in this paper that the sources of GRBs do not vary with redshift in any way except spatial density. The strongest direct evidence in favor of the cosmological paradigm of gamma-ray bursts is the detection of redshifted spectral lines in the afterglowing optical counterparts of gamma-ray bursts. Among these are six BATSE events with
1. INTRODUCTION
The combined observations of the isotropy of gamma-ray bursts (GRBs) and the deviation of their log N= log P curve from a 3=2 power law produced the first observational evidence that the sources of gamma-ray bursts lie at large cosmological distances. It was thought that the observed deficit of dim events with respect to an extrapolation of a 3=2 power law results from the large cosmological distances of their emitters, which reside in the non-Euclidean periphery of the expanding universe. The bright bursts, which seem to follow a 3=2 power law, were thought to be located in the nearby flat region of cosmological space in which redshift effects are negligible, while dim events were thought to lie at cosmological distances and, hence, to incur significant redshift effects. As soon as the cosmological paradigm for gamma-ray bursts became popular, two effects were suggested as observational tests. The first effect of cosmological redshift is that the energies of spectral features of distant bursts are redshifted with respect to similar features of close events. The second effect of time dilation predicts that the light curves of distant bursts are time stretched with respect to the time profiles of close events. Based on the identification of brighter bursts with closer emitters and dimmer events with more distant sources, these two effects of redshift and time dilation were expected to cause a hardness/intensity correlation and a stretching/intensity anticorrelation for groups of BATSE bursts with different brightnesses. Indeed, the average values of hardness ratios and of the spectral peak energy Ep of F were found to correlate with brightness (Nemiroff et al. 1994; Mallozzi et al. 1995, 1998; 904
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TABLE 1 BATSE GRBs with Measured Redshifts
GRB
BATSE Trigger
64 Peak Flux Fmax 2 (photons cm s1)
970508 .......... 971214 .......... 980703 .......... 990123 .......... 990510 .......... 991216 ..........
6225 6533 6891 7343 7560 7906
1.3 2.6 2.9 17.0 11.3 91.5
measured redshifts from the fading optical counterparts (Briggs et al. 1999). These events are used below as the reference group of bursts, for which the observed physical signatures in the observer frame can be easily transformed into the physical signatures in the rest frames. The six BATSE gamma-ray bursts with measured redshifts all are long, 64 > 1:3 photons cm2 s1 bright bursts with peak fluxes Fmax (Table 1). The goal of this paper is to compare the reference sample of 6 BATSE bursts with known redshifts with a comparison group of bright BATSE bursts. This comparison will help answer four questions that are important in understanding the real properties of the cosmological emitters of GRBs. One might assume that bursts with optical afterglows constitute a separate group of transients that differ from the main sample of GRBs. We therefore ask the following question: Are gamma-ray bursts with measured redshifts from optical counterparts an unbiased subsample of the bright, long BATSE gamma-ray bursts lacking redshift measurements? We show in x 2 that the gamma-ray properties of the comparison sample are consistent with the gamma-ray properties of the reference sample of bursts with known redshifts. However, even after verifying the consistency of the observed properties (in the observer’s frame) of these two groups, one might still suspect that the intrinsic properties of the emitters are different in their rest frames. Indeed, the intrinsic physical differences between the emitters of these two groups could be compensated by differing geometrical transformations of the light curves from the rest frames into the observer’s frame. This leads to a second question: Are the intrinsic rest-frame properties of the emitters of gammaray bursts with known z consistent with the intrinsic restframe properties of the emitters of the comparison sample of bright gamma-ray bursts? We show in x 3 that the answer is also yes, the intrinsic properties of the emitters of the reference and comparison groups of gamma-ray bursts are consistent. Therefore, the deredshifted properties of bursts with known redshifts inform us of the properties of the larger sample of bright bursts lacking redshift measurements. In x 4, we compare the statistics of the comparison group in the observer’s frame with the deredshifted statistics of the reference sample in the rest frame. We then compare the two groups by deredshifting the comparison group by a freeparameter z and comparing with the rest-frame properties of the reference sample. This leads to the third question: What is the best fitting value for the average redshift of the comparison group of bursts that produces statistical consistency between the deredshifted spectral peak energies for the comparison and the reference samples of bursts? The
Observed 50 (s)
Observed 50 (keV)
z
4.3 7.6 34.0 18.1 5.4 4.2
400.3 182.0 463.6 765.5 220.3 527.0
0.835 3.418 0.966 1.600 1.619 1.020
answer to this question produces a collective estimation of the average redshift factor of the comparison group of bright bursts (see x 4). Finally, we split the comparison group into three subgroups with different brightnesses, and we ask the question: Are there any systematic trends in the best fitting redshifts for the three intensity subgroups of the comparison sample? The answer to this question will be based on the comparison of the three collective estimations of redshifts for the three intensity subgroups, as presented in x 5. The applications of these estimations to the modeling of gamma-ray bursts are discussed in x 6. Taking into account these objectives, we implement a comparison group of gamma-ray bursts as a sample of long and bright BATSE events. We also require that both DISCSC and MER data are available for each event, as these data types are needed to calculate the emission times and deconvolve the energy spectra, respectively. To define the comparison group, we made a compromise between two contradictory demands: having a large number of events for a good representation of the total sample of bursts and including the brightest events with the best counting statistics. We compromised between these goals by creating the comparison group as the sample of 218 long and bright 64 > 2:0 photons BATSE bursts with T90 > 2:0 s and Fmax 2 1 cm s . 2. CONSISTENCY BETWEEN THE REFERENCE AND COMPARISON GROUPS OF GRBs IN THE OBSERVER’S FRAME
Gamma-ray bursts are highly variable transients, and one cannot resolve brightness-dependent properties by comparing instantaneous energy spectra at selected moments of particular bursts with different brightnesses. To perform the comparison of two groups of GRBs, we have to implement a statistical summary of the spectral and temporal parameters of each group of bursts and then compare these statistics. We will use the emission time 50 and the peak energy of the emission spectrum 50 parameters to summarize each burst for statistical analysis of large samples. The emission time 50 (Mitrofanov et al. 1999a) is the total sum of all time intervals during a burst that together contribute 50% of the total fluence. The sum uses the intervals of greatest emission so that the emission time of a burst is the minimum time to emit 50% of the fluence. The spectrum of the high-intensity emission is obtained by summing the spectra of the time intervals that contribute to the emission time. From this spectrum, we obtain the peak energy 50 in the F spectrum.
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The parameters 50 and 50 are quite convenient for the statistical comparison of groups of bursts. First, they represent the time intervals of the most powerful emission, which would be similarly selected for either brighter or dimmer events. Second, the emission time intervals would be similarly selected for either stretched or nonstretched time profiles. Third, these parameters of high-power emission correspond to similar physical stages of the different bursts. And finally, these parameters are measured for the time intervals with the highest signal-to-noise ratios. For each burst in the reference and comparison groups, we sum the spectrum over the selected time intervals of emission time. We use the MER data type with the appropriate response matrix to perform the counts-to-photons deconvolution using the Band GRB function (Band et al. 1993). For each burst, we obtain the spectral parameter 50 as the peak energy of F for the spectra summed over the intervals of the emission time 50 . for The distribution of spectral parameters COMP 50 the comparison group is presented in Figure 1. It is consistent with a log normal distribution with mean value iLN ¼ 256 12 keV. The distribution is quite narhCOMP 50 row, as shown by the width factor w, defined as log w ¼ 2 , . This factor w where is a standard deviation of log COMP 50 iLN þ characterizes a ratio between the right-wing loghCOMP 50 i of this distribution, and the left-wing loghCOMP LN 50 respectively. This factor for the observed histogram of is wCOMP ¼ 3:93. The similar factor for the log norCOMP 50 ¼ 3:35. Therefore, the mal fit to this distribution is wCOMP LN ratio between the right-wing and left-wing values of COMP 50 is surprisingly small, 3–4. Previous work has used the related quantity Epeak defined in the observer’s frame and for the entire burst duration rather than 50 used herein. The narrowness of the distribution of Epeak has been previously reported (Mallozzi et al. 1995). Several authors have argued that the result is due to instrumental biases (Piran & Narayan 1996; Lloyd & Petrosian 1999), but analyses from the BATSE team argue that the effect is real (Brainerd et al. 2000). The narrowness of this distribution has important physical implications. The width of this distribution results from the convolution of the width of the intrinsic distribution of in the rest frames of the emitters with the width of the REST 50 distribution of redshifts. Since the spread of observed values is as small as 3–4, the spread of redshift factors of COMP 50 the emitters ð1 þ zÞ cannot be larger than this factor, even if the intrinsic distribution of 50 at the rest frame is a function. The analysis avoids comparing the widths of the distributions because of the uncertain width values of the reference sample. It is well known in statistics that means (first moments) converge with increasing sample size toward the true value faster than the convergence of the widths or sig-
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Fig. 1.—Histogram: Distribution of 50 for the bursts of the comparison sample. Filled circles: Values for reference sample in the rest frame. Open circles: Values for the reference sample in the observer’s frame.
mas (second moments). The reference sample of six events has the rather poorly known width factors 3:00þ1:2 0:9 and 2:34þ0:74 0:56 in the observer frame and rest frame, respectively. Therefore, the reference sample size of six, which is just adequate for the first moment estimation, seems to be of very low usefulness for comparing widths. Before using the reference sample for the collective estimation of the redshift of the comparison sample, we should test that the two samples are consistent in the observer’s frame. We use the Pearson’s 2 , Student’s t-, and Kolmogorov-Smirnov (K-S) tests to find the probability that the of the reference sample belong to six observed values REF 50 the distribution observed for the comparison sample (see Table 2). For the Pearson criteria, we calculate the consistency parameter SP as 2 P6 COMP i REF i i¼1 h50 SP ¼ ; ð1Þ DCOMP where P218 DCOMP ¼
i¼1
i COMP hCOMP i 50
2
217
ð2Þ
and estimate the probability PP that SP corresponds to the 2 distribution with 6 degrees of freedom (Table 2). We also
TABLE 2 Consistency Probabilities for the Observed Properties of the Two Samples Statistics for Comparison
PP , Pearson 2
Pt , t-test
PKS , K-S Test
50 for CS and RS in observer frame ...................................... CIP for CS and RS ................................................................ ~50 for RS in rest frame and COMP for CS observer frame ...... 50 for ~50 for RS in rest frame and ð1 þ zÞCOMP 50 CS in rest frame with fitting z.............................................
0.397 0.146 2.9 104
0.163 0.046 7.2 106
0.148 0.059 6.4 105
z ¼ 2:6 for PP ¼ 0:860
z ¼ 2:6 for Pt ¼ 1:00
z ¼ 2:6 for PKS ¼ 0:928
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use the Student’s t-test for the consistency of the means of 50 for the reference and comparison samples (Table 2). Because the F-test shows the variances of the two samples to be consistent, we use the version of the Student’s t-test that assumes equal variances (eqs. [14.2.1] and [14.2.2] of Press et al. 1992). Finally, we use the K-S test for the consistency of two distributions (Table 2). All three criteria find that the observed values of the peak energies 50 of the six bursts of the reference sample are fully consistent with the distribution of 50 for the main sample of bright bursts. Formally speaking, the consistency of spectral parameters of these two groups in the observer’s frame does not prove that the outbursting sources for these groups are identical. One cannot exclude the possibility that there are two different types of sources responsible for the bursts of the reference and comparison groups. One type of sources could emit harder gamma rays and lie at larger cosmological distance, while another type may emit softer emission and lie at closer distances. The intrinsic difference of the spectral parameters 50 for these two types of sources could be compensated by differing cosmological redshift transformations.
3. CONSISTENCY BETWEEN THE EMITTERS OF THE REFERENCE AND COMPARISON GROUPS OF GRBs IN THEIR REST FRAMES
We can test the consistency of the emitters of the different groups of gamma-ray bursts by comparing the events using parameters determined in the rest frames of the sources. We have implemented the cosmological invariant parameter (CIP) as the product of the emission time 50 and the spectral peak energy 50 , CIP ¼ 50 50 :
ð3Þ
By construction, this parameter does not depend on the cosmological transformation from the observer frame into the emitter rest frame: the factor 1=ð1 þ zÞ of time stretching is compensated by the factor ð1 þ zÞ of energy redshift. In a previous paper (Mitrofanov et al. 1999b), we implemented this concept as an average signature of large groups of bursts. In the present paper, the CIP is associated with each particular event. The CIP value is an intrinsic property of each emitter in its comoving frame. It is well known that the values of temporal parameters of GRBs are energy dependent. Because the emission time values used in this paper are obtained from a count-rate time history in an energy band fixed in the observer’s frame, the CIP is not perfectly invariant. The length of the emission time is determined mainly by the large variations of the flux rather than by spectral variations. CIP values defined using emission time values determined in an energy band fixed in the observer’s frame rather than in the comoving frame of the emitter are therefore sufficiently accurate. The comparison group has a CIP distribution that is well fitted with a log normal law (Fig. 2). The log-normal mean value of this distribution is hCIPCOMP iLN ¼ 1134þ104 96 keV s. Again, we estimate the probability that the six CIP values of the reference group belong to the distribution of CIP values for the comparison group. According to the Pearson’s 2 (see eq. [1]), Student’s t- (eq. [14.2.2] in Press et al. 1992), and Kolmogorov-Smirnov (Press et al. 1992) criteria, these
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Fig. 2.—Histogram: Distribution of the CIP for the bursts of the comparison sample. Dots: CIP values of reference sample in the rest frame and the observer’s frame.
probabilities show that the statistics of the CIP values for the reference and comparison groups are consistent (see Table 2). Therefore, in the following sections, we may use the properties of the reference sample to estimate the parameters of the comparison group.
4. COLLECTIVE ESTIMATION OF THE REDSHIFT FACTORS OF THE COMPARISON SAMPLE OF GAMMA-RAY BURSTS
Sections 2 and 3 showed that the reference and comparison groups of bursts have consistent statistical properties in the observer’s frame and that their emitters are not statistically distinguishable in the rest frames. Therefore, we may assume that these two groups correspond to the same type of gamma-ray bursts. This assumption allows us to perform a collective estimation of the average redshift factor of the bursts of the comparison group. The collective estimation is made by comparing the parameter values for the two groups using some parameter that is transformed by redshift. We use 50 rather than as the comparison parameter because our samples may have biased distributions because of the selection criterion T90 > 2 s. For each gamma-ray burst with a measured redshift, we may transform the spectral parameters 50 in the observer frame into the value ~50 for the rest frame of the emitter, ~50 ¼ 50 ð1 þ zÞ :
ð4Þ
The six values of ~50 for the reference sample are used for the collective estimation of the redshift of the comparison group of bursts. According to three commonly used criteria, the hypothesis that the rest-frame values of ~REF 50 for the reference sample for the comare consistent with the observed values of COMP 50 parison sample should be rejected with high significance, as expected (Table 2). The physical meaning is that we should
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TABLE 3 Collective z Estimates for Intensity Subgroups
Group
64 Peak Flux Fmax (photons cm2 s1)
Mean Observed Spectral Peak Energy h50 i (keV)
Mean Observed CIP (keV s)
z
1 Limits
1............ 2............ 3............
8.3–167.8 3.9–8.3 2.2–3.9
316:2þ25 23 270:9þ21 20 196:7þ16 15
1074:9þ174 150 1529:8þ227 197 893:2þ158 134
1:9þ0:9 0:7 2:4þ1:1 0:8 3:7þ1:6 1:1
1.2–2.8 1.6–3.5 2.6–5.3
reject the possibility that the sources of the comparison group have negligible redshifts. To estimate the most probable redshift for the comparison group, we calculate the deredshifted peak energies of =ð1 þ zÞ with the free redshift factor the spectra as COMP 50 ð1 þ zÞ. We use the same three criteria to estimate the best fit values of this parameter z (Table 2). Figure 3 shows these values according to the Pearson’s 2 , Student’s t-, and K-S criteria for three different levels of significance. All criteria result in similar best fit values z ¼ 2:6. The 1 limits on z are 1.8 and 3.6, according to the most conservative criterion (Table 2 and Fig. 3). Similarly, the 3 limits of z are 0.7 and 7.6.
5. COLLECTIVE ESTIMATION OF THE REDSHIFT FACTORS FOR THREE INTENSITY SUBGROUPS OF THE COMPARISON SAMPLE
We split the comparison sample of bursts into three inten64 (see Table 3). We estisity subgroups using peak flux Fmax mate the log-normal mean value of 50 for each subgroup (Table 3) and find weak evidence for a hardness-intensity correlation. This effect was first seen in a very small sample consisting of APEX bursts (Mitrofanov et al. 1992) and was confirmed with a large sample consisting of BATSE bursts (Mallozzi et al. 1995), but the reliability of the observed distributions was questioned (Lee & Petrosian 1996; Lloyd & Petrosian 1998). A correlation between hardness Epeak and fluence has been confirmed in a careful analysis by Lloyd, Petrosian, & Mallozzi (2000). The effect has been interpreted as a signature of cosmological redshift, with dimmer bursts corresponding to more distant objects with larger redshifts. We estimate the mean redshifts of each of these subgroups of bursts using the collective estimation method of the previous section (Table 3). The 1 intervals for z of these subgroups show a tendency to higher redshifts for bursts with dimmer fluxes. One might wish to compare the average z estimate for the three intensity groups with the known properties of the reference (REF) sample. The three brightest bursts agree with the z range estimated for group 1, but for the three dimmer bursts, only one has a redshift 3.418, which agrees with the interval 2.6–5.3, while GRB 970508 and GRB 980703 have z values about 3 times less than expected for group 3. However, we do not believe that one may make any conclusions about the properties of bursts by comparing the bright and dim members of the very small REF sample. We know that the spread of GRB properties is so large that much larger samples are needed before subsamples can be compared. The apparent trend easily could be due to fluctuations in the small sample drawn from the population that has a very broad distribution.
6. DISCUSSION
6.1. Consistency of Samples of Bursts with and without Optical Afterglows
Fig. 3.—Probabilities that the six rest 50 values of the reference sample and values of the comparison group belong to the same distribution the COMP 50 are shown, evaluated using Pearson’s 2 ( fine line), Student’s t- (bold line), and Kolmogorov-Smirnov (dashed line) criteria as functions of average redshift of the sources of the comparison group.
We found that the group of gamma-ray bursts with measured redshifts from optical afterglows is consistent with the main sample of long and bright BATSE gamma-ray bursts in two respects. First, consistency between the two groups is found for the directly observed spectral parameters of the spectrum summed over the time intervals of high-power emission (x 2). This shows that they belong to the same
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observational phenomenon. Second, consistency is also found for the statistics of the cosmological invariant parameters of these bursts (x 3). This test shows that the intrinsic properties of the two groups emitters are also indistinguishable. Therefore, we conclude that bursts with detected optical afterglows should be considered as a randomly selected subset of events from the general sample of long and bright events. The property of having an optical afterglow does not appear to be connected to particular values of the 50 and CIP parameters.
6.2. The Most Probable Values of Redshift for the Sources of the Comparison Sample Following the assumption of consistency between the reference and comparison samples of bursts, we have compared the statistics of the deredshifted spectral parameters of the reference sample with the observed statistics for the comparison group. We find significant inconsistency between them, as expected. We implemented a free transformation factor for deredshifting the spectral parameters of the comparison group, and we achieved the best consistency when the mean redshift value z ¼ 2:6 (x 4). The 1 limits on the mean redshift z are 1.8 and 3.6 (Fig. 3). The range 1.8–3.6 expresses the uncertainty in our knowledge of the mean rather than information about the width of the distribution of z. We use the observed width of the to constrain the interval distribution of log COMP 50 ½zmin ; zmax of the redshifts of the emitters. The ratio factors around the peak energy w between the values of COMP 50 iLN ¼ 256 12 keV are w ¼ 3:93 and 3.35 for the hCOMP 50 observed histogram and for the log normal approximation, respectively. Under the most extreme assumption that the emitters have a function intrinsic distribution of spectral peak energy in the rest frames and that the value of the peak energy is independent of z, the width of the observed distrishould be totally produced by the spread of bution of COMP 50 the redshifts of the sources. They could be distributed around z ¼ 2:6 between zmin ¼ 0:8 and zmax ¼ 6:1 for the factor w ¼ 3:93 and between zmin ¼ 1:0 and zmax ¼ 5:6 for the factor w ¼ 3:35. Accepting the mean redshift factor z ¼ 2:6 and the distribution, one has to width factor w ¼ 3:93 of log COMP 50 accept the interval [0.8–6.1] around this value as the most conservative estimation for cosmological distances of emitters of the comparison group of gamma-ray bursts. Of course, one may select another value z of mean redshift factor according to limitations of collective estimation (see Fig. 3). However, in any case, the small width log w of the determines a rather narrow interdistribution of log COMP 50 val of redshifts of emitters with zmin ¼ ð1 þ z Þ=w1=2 1 and zmax ¼ ð1 þ z Þ=w1=2 þ 1. In reality, the rest-frame distribution of spectral peak energy could have a nonzero width. Therefore, the width of the z distribution could be even narrower in the real case. The comparison group of 218 long and bright BATSE bursts could be associated with a rather narrow log-normal distribution of redshifts with the mean value z ¼ 2:6 and the width factor w ¼ 3:93, provided the intrinsic emission parameters of these sources do not vary with redshifts. The distribution of redshifts of sources with constant comoving density also has a log-normal shape with similar width but
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with a rather different mean value, z 0:9 (Krumholz, Thorsett, & Harrison 1998). Therefore, one may exclude the model with constant comoving density for the comparison group of GRBs. Another model of source density is based on the star formation rate in the cosmological space (Krumholz, Thorsett, & Harrison 1998). The best known curve for the star formation rate (Rowan-Robinson 1999) was transformed into the total number of sources at different redshifts, which could be observable now at z ¼ 0 (Lamb & Reichart 2001). The star-formation model curve has a maximum at z 2 and the redshifts of its left and 5, respec 1 and zmax right half-width points are zmax tively. Taking into account the large uncertainty of our knowledge of both curves, one may conclude that there is rather good consistency between the curve predicted from the star formation rate and our estimation for the comparison group based on the distribution of spectral peak energy. Generally speaking, the interval of z for the emitters of gamma-ray bursts could be wider than is allowed by the distribution but, in this case, one narrowness of the COMP 50 should postulate that in the rest frame, the average peak energy of emitters increases proportionally to ð1 þ zÞ with increasing z. This dependence would compensate for the redshift transformation 1=ð1 þ zÞ of the peak energy into the observer frame, and therefore, one could combine the observed narrow distribution of spectral peak energy with a very broad range of cosmological distances to the emitters. If one believes that the width of the 50 distribution derived in this paper is implausibly narrow, then the observations may be explained by abandoning the assumption of no-source evolution and supposing that a z-dependent evolution of intrinsic peak energy neatly cancels the cosmological transformation of observed peak energy. With the current small sample of GRBs with known redshifts, it is not possible to test for intrinsic evolution. Testing for intrinsic evolution will be an important task when the sample of GRBs with measured redshifts has been expanded. The collective estimations of the mean redshifts for the three intensity subsamples of the comparison sample are listed in Table 3. They are consistent with the assumption that observed burst intensity is anticorrelated with distance. The assumption that the emitters of the three subsamples are similar in their rest frames is confirmed by the comparison of their CIPs, which are consistent (Table 3). This gives additional evidence that the statistical properties of the emitters in their rest frames are similar, and the observed effect of the brightness-hardness correlation should be associated with the cosmological effect of redshift transformation. Among the bursts with known redshifts, there is clearly a lack of a flux-redshift or fluence-redshift correlation, suggesting that bursts are not standard candles if they emit isotropically (Briggs et al. 1999; Bulik 1999; Deng & Schaefer 2000; Piran, Jimenez, & Band 2000). However, afterglow observations suggest that the bursts are not isotropic emitters and, indeed, if beaming angle corrections are included, the distribution of total gamma-ray energy for the bursts with known z becomes much narrower (Frail et al. 2001). If our analysis is correct, a correlation between flux and redshift is still present in the larger burst sample, implying that the range of beaming angles is not large.
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If we accept the collective estimate of an average z 2:6 for the comparison group of bursts, we find that the restframe values of the emission time 50 are 3.6 times smaller and the emission peak energies 50 are 3.6 larger than the values observed in our frame. The rest-frame values are iLN ¼ 920 keV and hCIPiLM =½ð1 þ z Þ ð1 þ z ÞhCOMP 50 i ¼ 1:2 s. The errors of these estimations correhCOMP LN 50 spond to the errors of the average redshift factor z (see x 4). The most probable value of the peak energy of emission in the rest frame is found to be very close to the electron-positron pair rest mass of 1022 keV. We should pay attention to this fact, but we do not have any reasons from the current models of GRB emission as to why this particular energy should be preferred.
The most probable emission time of long, bright bursts in their rest frame is 1 s. The observed emission time distribution is bimodal, with a gap between the short and long modes at 0.4 s, and the most probable value for the short mode is 0.1 s (Mitrofanov et al. 1999a). Therefore, the rest-frame emission time of the long mode of GRBs is not the same as the observer-frame value of the emission time of the short mode of GRBs. Consequently, the short mode of the emission time distribution cannot be interpreted as a local population of GRBs similar to the distant sources producing the long mode. The work in the USA was supported by NASA project CGRO-98-120 of the Compton Observatory Guest Observations Program. The work in Russia was supported by RFBR grant 98-02-17380 and INTAS grant 96-0315.
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