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The Direct Test of Cosmological Model for Cosmic Gamma-Ray Bursts Based on the Peak Alignment Averaging

Igor G. Mitrofanov, Maxim L. Litvak and Dmitrij A. Ushakov

arXiv:astro-ph/9707128v1 11 Jul 1997

Space Research Institute, Profsojuznaya str. 84/32, 117810 Moscow, Russia

Received

;

accepted

–2– ABSTRACT

The cosmological origin of cosmic gamma-ray bursts is tested using the method of peak alignment for the averaging of time pro?les. The test is applied to the basic cosmological model with standard sources, which postulates that di?erence between bright and dim bursts results from di?erent cosmological red-shifts of their sources. The average emissivity curve (ACEbright ) of the group of bright BATSE bursts is approximated by a simple analytical function, which takes into account the e?ect of squeezing of the time pulses with increasing energy of photons. This function is used to build the model light curve for ACEdim of dim BATSE bursts, which takes into account both the cosmological time- stretching of bursts light curves and the red-shifting of photons energies. Direct comparison between the model light curve and the ACEdim of dim bursts is performed, which is based on the estimated probabilities of di?erences between ACEs of randomly selected groups of bursts. It shows no evidence for the predicted cosmological e?ects. The 3σ upper limit of the average red- shift zdim of emitters of dim bursts is estimated to be as small, as ? 0.1 ? 0.5, which is not consistent with values ? 1 predicted by the known cosmological models of gamma-ray bursts. Subject headings: cosmology: theory-gamma rays: bursts

–3– 1. Introduction

The cosmological model of cosmic gamma-ray bursts is commonly accepted, as one of the most promising concept of the origin of gamma-ray bursts (GRBs). However, it has not been ?nally approved yet by the observational data. Two critical tests were suggested to verify the basic model with standard cosmological sources: dim bursts have to be time-stretched and red-shifted in comparison with bright events. Gamma-ray bursts are known to have very di?erent time histories, and one hardly could check the cosmological e?ects by direct comparison between particular events. These tests should be based on some averaged time-based and spectrum-based signatures, which represent the basic properties of GRBs. Several statistical tests have already been implemented to compare di?erent groups of GRBs and to resolve the predicted cosmological e?ects. In the case of time-dilation, two scienti?c groups have checked the average emissivity curves (ACEs) derived from the peak alignment averaging of bright and dim bursts, and came to opposite conclusions: time-dilation of dim bursts was seen by one group (Norris et al. 1994a, 1994b, 1996; Bonnel et al. 1996) and it was not seen by another one (Mitrofanov et al., 1992a,b, 1994, 1996). Possible reasons for disagreement were discussed (Band 1994, Mitrofanov et al. 1996), and the tentative conclusion has been drawn that the claimed dilation of dim bursts was possibly resulted from some systematic in separation of bright and dim groups of events. On the other hand, in the case of spectral red-shift, all groups involved have a consensus that bright GRBs have larger averaged spectral hardness than dim events. It was named, as the e?ect of hardness/intensity correlation (Mitrofanov et al. 1992b,c, 1994, 1996; Paciesas et al. 1993, Norris et al. 1994a,b). While this e?ect was originally seen for the average hardness ratio (de?ned as the ratio of counts at high and low energy channels), recently it was found also for the average peak energy < Ep > of νFν energy spectra (Mallozzi et

–4– al. 1995). The average peak energies < Ep > of the integral spectra of photons were found

(256) to correlate with the photons peak ?uxes Fmax at the 256 ms time scale. The e?ect of

hardness/intensity correlation may be interpreted, as a result of cosmological red- shift of dim gamma-ray bursts in respect to bright events. The corresponding cosmological red-shift factor about 1.6-2.2 (Mallozzi et al. 1995) is consistent with original cosmological models based on the interpretations of log N - log F distribution (e.g. see Emslie and Horrack 1994). Therefore, there is a discrepancy between di?erent groups about time-dilation of dim GRBs in respect to bright bursts, but, on the other hand, there is an commonly accepted agreement between them for the hardness/intensity correlation. Separate pulses of GRBs are known to squeeze with increasing energy of photons (Norris et al. 1986, Fenimore et al. 1995), and, therefore, the average curve of emissivity becomes narrower at higher energies (Mitrofanov et al. 1996). According to the cosmological model, when bright and dim bursts are detected at the same energy band in the observer frame of reference, their time pro?les were actually emitted at less hard and more hard energy ranges at comoving frames of reference, respectively. Therefore, making a comparison of bright and dim bursts from sources with small and large red-shifts, one should suppose that the intrinsic squeezing of the light curves of dim bursts due to the increase of energy of the emitted photons could partially compensate their stretching due to the cosmological timedilation. This paper provides a test of the basic cosmological models of GRBs assuming them to be standard sources. It uses the average emissivity curves for groups of bright and dim bursts and takes into account the e?ects of cosmological time- stretching in the observer frame together with the internal energy dependent squeezing of bursts light curves in the comoving frames.

–5– 2. Analytic Approximation of the Average Curve of Emissivity for Group of Bright GRBs The average curve of emissivity (ACE) of GRBs was introduced (Mitrofanov et al. 1994, 1996), as a general signature of bursts time variability. To build an ACE, all time histories of averaging bursts should be normalized by peak numbers of counts Cmax , then should be aligned at their peak bins tmax and then should be averaged at each of another bins. Comparison between the First, the Second and the Third BATSE Catalogs (Fishman et al. 1994, Meegan et al. 1994 and Meegan et al. 1995) has shown that ACE has rather stable shape: it has one peak pro?le with steep rise front and gentle back slope, and its width decreases with increasing energy of photons used for averaging (Mitrofanov et al. 1994, 1996). For the present analysis the DISCLA data were used from the large area BATSE detectors (LADs) with 1024 ms time resolution on three discriminator channels, number 1 (25-50 keV), number 2 (50-100 keV) and number 3 (100-300 keV). Two basic intensity groups of BATSE GRBs were selected from the Third BATSE Catalog (3B) (Meegan et

(1024) al. 1995): 296 bright bursts with Fmax > 1 photons cm?2 s?1 and 332 dim events with (1024) Fmax < 1 photons cm?2 s?1 . Only bursts with t90 > 1.0s were taken into account for

consideration. The group of bright bursts is used as the reference sample to ?nd the analytical approximation of ACEbright at di?erent discriminator channels (Figure 1). The function

(i) fbright (t)

=(

t0

(i) t0

(i)

+ |t ? tmax |

)aRF ,aBS

(i)

(i)

(1)

approximates ACEbright pro?les at each discriminator channel i = 1, 2, 3 with di?erent power indexes aRF at the rise front (RF) t < tmax and aBS at the back slope (BS) t > tmax ,

(i) (i)

(i)

–6– respectively. Instead of three di?erent functions (1) for each of three channels, a single function fbright (t, E) could be implemented, which approximates the shape of ACEbright at di?erent energies E, which correspond to these channels

fbright (t, E) = ( where the functions

tbright (E) )aRF (E),aBS (E) , tbright (E) + |t ? tmax |

(2)

tbright (E) = tbright · (E/173 keV )α1 aRF (E) = aRF · (E/173 keV )α2 aBS (E) = aBS · (E/173 keV )α3

(0) (0)

(0)

(3) (4) (5)

represent the change of ACEbright shape with energy. A di?erence between three observed ACEi bright pro?les (Figure 1) and the model approximation (2) could be evaluated using the function

Sbright =

i j

(ACEbright ? fbright (tj , Ei ))2 σ 2 (ACEbright )

(i,j)

(i,j)

(6)

where Ei corresponds to mean energies at three discriminator channels i = 1, 2, 3 and tj corresponds to the time bins of ACE curves from j = ?19 up to j = +19. Errors of observed ACE pro?les were estimated from the sample variance. The parameters of approximation tbright = 1.80±0.33 s, aRF = 1.31±0.13 , aBS = 1.10±0.10 , 0.28 0.12 0.09 α1 = ?0.10 ± 0.16, α2 = 0.06 ± 0.09 and α3 = 0.11 ± 0.08 were estimated from the best ?tting of all three ACEbright pro?les at channels i = 1, 2, 3. This ?tting leads to the minimum Sbright of Exp.(6), which corresponds to rather small value of the Pearson criterion: reduced χ2 = 0.66 for 108 degrees of freedom. Therefore, one might conclude that

(min) (i) (0) (0) (0)

–7– Exp. (2) gives a rather good approximation of the observed ACEbright pro?les for the basic group of bright bursts at a broad energy range from 25 up to 300 keV. On the other hand, rather small values of the reduced χ2 points out that the errors of ACEbright was probably overestimated by the sample variance algorithm, or there were some correlation between them. However, the Pearson criterion allows to determine the con?dence region for the estimated parameters of the ?tting function (2). According to Lampton et al. (1976), the con?dence region for the signi?cance level λ could be determined by the 5-dimensional contour Scountur in the 6-dimensional parameter space, which is given by the equation

Scountur = Sbright + χ2 (λ), 6 where χ2 (λ) represents the value of χ2 distribution for signi?cance λ for 6 degrees of 6

(min)

(7)

freedom. Errors ±1σ for each of six parameters, as presented above, were estimated from the condition that Exp. (6) for Sbright becomes equal to Scountur when the parameter goes up and down from the best ?tting value, while another ?ve parameters are used as free parameters for minimization. Therefore, each of these 12 points could be interpreted as ±σ deviations from the minimum point along the axes of corresponding parameter inside a 5-dimensional contour Scountur . Actually, these 12 points in the six-dimensional parameter space correspond to 12 ?tting models (2) of the ACEi bright curves. Were taken all together, they would present the ±1σ corridor of analytical approximations around the best ?tting model, which leads to Sbright = Sbright . The boundary curves of this corridor are presented at Figure 1. One might see that all these models provide rather good approximation of all three ACEbright pro?les measured at three energy discriminitors.

(i) (min)

–8– 3. Comparison between the average emissivity curves for di?erent groups of bursts Particular gamma-ray bursts are known to have very di?erent time histories and energy spectra. Therefore, ACE curves could be di?erent for particular groups of bursts randomly selected from the total data base. Groups with Nrep bursts could be de?ned, as representative samples, provided the di?erences between their ACEs would be comparable with the errors from the sample variance for each group. For smaller groups with N < Nrep a di?erence between ACE curves could be signi?cantly larger than it would be expected from the sample variance. Therefore, the comparison of ACE of di?erent groups has to take into account the actual distribution of di?erences between ACEs pro?les due to a random choice of contributing bursts. Nobody knows how large is the representative sample of time histories of GRBs, but it seems from the comparison of ACE curves for 1B, 2B and 3B databases that Nrep could be about the presently available number of bursts ? 103 (Mitrofanov et al. 1997). As it was found there, the Pearson criterion provides a rather sensitive test to measure a di?erence between ACEs for any two groups of bursts, namely groups I and II, at any discriminator channel i:

(i,j) (i,j)

(i) S(I?II)

=

j

(ACE(I) ? ACE(II) )2 σ 2 (ACE(I) ) + σ 2 (ACE(II) )

(i,j) (i,j)

(8)

The magnitude S(I?II) was used to compare ACEs pro?les for randomly selected groups of events. It was found that groups with N increasing from ? 30 up to ? 300 become more and more representative with respect to the full set. In particular, the probability distribution P300 of S(I?II) at discriminator i = 2 was obtained from 105 random choices of two groups with N=303 among the total 3B set of 638 BATSE bursts (Figure 2). This

(2)

–9– distribution does not depend signi?cantly on the intensity of selected bursts, because the main contribution into S(I?II) comes from the actual di?erence of their time histories. Thus, Exp. (8) could be used for direct comparison between ACE pro?les of groups of bright and dim bursts, and the signi?cance of a physical di?erence S between them could be estimated as the probability of obtaining S greater than S(I?II) according to the distribution P300 (S(I?II) ) provided by the Monte Carlo random choice test (Figure 2). This probability distribution will be used below to compare the analytical model based on the ACEbright of bright bursts and the actual ACEdim measured for the group of dim events.

4.

Direct cosmological Test Based on the Analytic Approximation of the Average Curve of Emissivity

The simplest test of cosmological model of GRBs could be based on the standard candle assumption, which means that everywhere at cosmological distances all sources have the same properties in their comoving frame. This basic version of the cosmological model assumes that all groups of bursts, provided would be averaged in comoving frames, should have the same ACEs. Therefore, any di?erence between ACEs of bright and dim bursts measured in the observer frame should point out on the cosmological e?ects. Let us assume that the emitters of bright and dim bursts have average red shifts zbright and zdim , respectively. If two standard sources at zbright and zdim emit bright and dim bursts with photons energy E0 and variability time scale τ0 , they would be detected in the observer frame at energies Ebright = E0 /(1 + zbright ) and Edim = E0 /(1 + zdim ) and with variability at time scales τbright = τ0 (1 + zbright ) and τdim = τ0 (1 + zdim ), respectively. The so-called stretching factor could be introduced

– 10 –

Y (zbright , zdim ) = (1 + zdim )/(1 + zbright ),

(9)

which equals to the ratio of energies of photons Ebright /Edim and/or to the ratio of variability time scales τdim /τbright at the observer frame of reference, provided they were the same in comoving frames. To test the basic cosmological model, the analytical approximation fbright (t, E) (Exp. (2)) should be transformed into the model function fdim (t, E) according to cosmological red-shifting and time-stretching transformations, which has to represent the measured ACEdim pro?les for the group of dim bursts. According to the assumption of standard candles one should postulate

t fdim (t, E) = fbright ( , E · Y ). Y Using the Exp. (2), one might represent Exp. (9), as the following

(10)

fdim (t, E; Y ) = (

Y · tbright (Y · E) )aRF (Y ·E),aBS (Y ·E), Y · tbright (Y · E) + |t ? tmax |

(11)

which could be used either as a function of one stretching parameter Y , or as a function of two red-shifts zbright and zdim . Figure 3 presents ACEdim pro?les for the basic group of 332 dim bursts from the 3B

(1024) database with Fmax < 1 photons cm?2 s?1 observed at three energy discriminators with (i) (i)

numbers i = 1, 2, 3. Expression (11) provides a trial function for the ACEdim pro?les with the factor Y , as a free parameter. To compare the model with observations, the function Sdim could be used similar to Sbright (6). Table 1 presents the best ?tting values Y ? for each of the three ACEdim pro?les ?tted separately, and one more value for the joint ?t of all

(i)

– 11 – three curves together. The errors of Y ? correspond to the range of the best ?tting values of Y for the 12 di?erent models (11) based on the initial model (2) with ±σ deviations of its six parameters (see Section 2). The values of minima Sdim

(i) (min)

for the best ?tting parameters Y ? are rather large, and

according to the Pearson criterion, the model of equation (11) does not agree with the observed ACEdim pro?les for discriminators i = 1, 3 and (1 + 2 + 3). Only in the case of discriminator i = 2 the model (11) with Y ? =0.85-0.87 formally agrees with the ACEdim pro?le. Moreover, instead of the expected stretching, all the best ?tting factors Y ? (Table 1) correspond to squeezing of ACEdim pro?les with respect to the analytic approximation of ACEbright for bright bursts (Exp. (2)). However, the classical Pearson criterion based on the χ2 -distribution could not be applied in this case, because it does not take into account the actual distribution of di?erences S(I?II) between ACEs pro?les due to a random selection of contributing events. A more accurate test of the basic cosmological model is done below, which takes into account the probability distribution of S(I?II) resulting from the random sampling of BATSE bursts (see Section 3). This test has to provide the upper limits of zdim for the basic cosmological model with standard sources, which could be deduced from the observed pro?les of ACEbright and ACEdim . According to this model, a group of bursts with ?uxes ? F corresponds to a de?nite red-shift ? z. While in the Euclidean space there is a ?ux dilution law ? R?2 , which establishes the well-known ?ux/distance relation for standard sources, the non-Euclidean dilution of ?uxes from cosmological emitters is in?uenced by the e?ects of photon energy red-shifting and light curve time- stretching. While each burst has a particular energy spectrum, the average spectral distribution could be obtained for any selected group of bursts as well as ACEs were obtained for their

(i) (i) (2)

– 12 – time histories. According to Band et al. (1993), the energy spectra of BATSE bursts φ(E) could be described by the law

φ(E) = A · ( if

E(2+α) E ? )α · e Epeak 100 keV

(12)

E < (α ? β) · φ(E) = A · ((α ? β) · if E > (α ? β) ·

Epeak (2 + α)

(13) (14)

Epeak Epeak )(α?β) · e 100 keV · (β ? α)β 100 keV (2 + α)

Epeak (2 + α)

(15)

where all energies are normalized by 100 keV. The BATSE database includes the spectral data at 2048 ms time scale, which could be used to ?nd the average spectral parameters at peak time intervals. For the group of bright BATSE bursts, the average spectral parameters at the peaks are < α >= ?0.618, < Epeak >= 329 keV and < β >= ?3.18 (Mitrofanov et al. 1997). According to the concept of standard sources, one could use the average spectra of bright bursts φ(bright) (E), as a standard distribution of photons for all emitters. Therefore, one might derive a universal relation between the observed photon ?uxes F and red-shifts z of corresponding emitters. For two basic groups of 296 bright and 332 dim bursts with

(bright) (dim) average peak ?uxes < Fmax >= 6.15±0.35 photons cm?2 s?1 and < Fmax >= 0.53±0.03

photons cm?2 s?1 , respectively, this relation corresponds to the ratio

(bright) < Fmax >

< Fmax > where

(dim)

=

E2 E1 E2 E1

φ(bright) [E(1 + zbr )]dE · R2 (zdim ) φ(bright) [E(1 + zdim )]dE · R2 (zbr )

(16)

– 13 –

R=

c · [q0 z + (1 ? q0 )(1 ? 2 (1 + z)q0 H0

1 + 2zq0 )]

(17)

is cosmological distance to a source, H0 is the Hubble constant and q0 represent the type of cosmological geometry. The geometry of the Universe with critical density is tested below with q0 = σ0 = 0.5. The peak ?ux (photons cm?2 s?1 ) was calculated in the 50-300kev energy range according to 3B Catalog database.

(bright) (dim) Using the average values < Fmax > and < Fmax > and the average spectral law

φ(bright) (E), Exp. (16) could be transformed into the relationship between two cosmological parameters: an average red-shift zdim of emitters of dim bursts and an average stretching factor Y between dim and bright bursts. Therefore, the zdim value could be implemented into the model function (11) fdim (t, E; zdim ), as a free parameter, to check the consistency between the basic cosmological model and observed ACEdim curves for dim bursts. To do this, one has to put the zdim value into the model function fdim (t, E; zdim ) and to calculate the di?erence (6) between the model and the ACEdim pro?le at the energy discriminator channel i = 2. The estimated value Sdim (zdim ) could be corresponded to the probability P300 (S(I?II) ) (Figure 2) to ?nd the di?erence S(I?II) equal to this value. The integrated probability

(2) (i)

∝

P300 (zdim ) =

Sdim (zdim )

P300 (S(I?II) )dS(I?II)

(18)

could be interpreted, as the probability that cosmological model with zdim is consistent with observed ACEdim pro?le. Changing zdim , one might create this way the probability function P300 (zdim ) (Figure 4). To take into account errors in the parameters of the basic analytical model of fbright (t, E), the main theoretical model (11) was used together with 12 additional models

– 14 – with ±σ deviations from the best ?tting parameters (3). They compose the 1σcorridor of models around the medium curve which corresponds to the best one (see Figure 4). It was found that the probability P (zdim ) decreases with increasing zdim becoming as small as the level ? 3 · 10?3 of 3σ ?uctuations at zdim = 0.07 ? 0.09 (Figure 3). Therefore, one might consider the value ? 0.1, as the 3σ upper limit for average red-shift of emitters

(1024) of the basic group of 332 dim bursts with Fmax < 1 photons cm?2 s?1 .

Two groups of bright and dim bursts are used for this estimation which been separated

(1024) by the peak ?ux Fmax = 1 photons cm?2 s?1 . In this case one has the largest possible

number of events in each sample, ? 300, with the ratio of corresponding average peak ?uxes of two samples ? 12. However, one might suspect that selected sample of ? 300 bright bursts might contain a large deal of bursts at cosmological distances, and, as such, a time dilation between bright and dim samples could be di?cult to resolve. Formally speaking, this statement is not correct: in accordance with the basic cosmological model, the increase of zbright value for the bright group results to more and more pronounced cosmological stretching of bursts from the dim sample, provided the ratio of their average peak ?uxes is ?xed. Indeed, the Exp. (16) points out that for a given ratio of

(bright) (dim) peak ?uxes < Fmax > / < Fmax > an increase of zbright from the value 0 leads to increase (bright) (dim) of stretching factor Y (zbright , zdim ). Thus, for the ratio < Fmax > / < Fmax >= 11.6

one might ?nd Y = 1.2, 1.6 and 1.8 and zdim = 0.3, 1.1 and 1.7 for zbright =0.1, 0.3 and 0.5, respectively. The found 3σ upper limit zdim ? 0.08 corresponds to the stretching factor Y ? 1.05 and zbright = 0.03. One might conclude that the basic cosmological model with ? 300 standard emitters of bright and dim bursts is consistent with the observed ACEbright and ACEdim curves, provided their red-shifts are zbright < 0.03 and zdim < 0.1, respectively.

– 15 – However, even taking into account the argument above, one could apply the proposed redshifting technique to perform a more conservative comparison between two samples of bright and dim bursts, which could be selected by a more stringent criterion based on the slope of logN ? logF distribution, and which would be truly isolated one from another by the sample of intermediate events in between.

(brightest) Let us select two samples of 102 brightest bursts with Fmax > 4.0 photons (dimmest) cm?2 s?1 and 100 dimmest events with Fmax < 0.41 photons cm?2 s?1 . The

brightest sample corresponds to the -3/2 part of the logN ? logF distribution (see 3B catalog, Meegan et al. 1995). There is about ? 400 bursts with medium peak ?uxes in between the brightest and the dimmest samples, and the ratio of the average peak

(brightest) (dimmest) ?uxes < Fmax > / < Fmax >= 49.8 is as large as possible to imply the largest

cosmological stretching between them. For the new sample of the brightest 102 bursts the analytical approximation (2) corresponds to the best ?tting parameters, which all agree quite well with the estimated ±1σ corridor with the basic sample of 296 bright bursts. The best ?tting parameters for the ACEbrightest curve are tbrightest = 1.88 s, aRF = 1.37, aBS = 1.26, α1 = ?0.25, α2 = 0.065 ? ? ? ? and α3 = 0.075. Similarly to (11), these new parameters could be used to build the trial ? function fdimmest (t, E; Y (zbrightest , zdimmest )) (11) to ?t the ACEdimmest curve of the sample of 100 dimmest bursts. The best ?tting values of Y ?? equal 1.01, 0.80 and 0.88 for ACEdimmest at the three energy discriminators with numbers i =1,2 and 3, respectively. The corresponding values of reduced χ2 are 2.95, 3.20 and 1.90, respectively. Therefore, the best ?tting stretching factors Y ?? between the samples of the dimmest and the brightest bursts do not manifest any stretching. These values are similar to the best ?tting factors between the basic samples of ? 300 bright and dim bursts, and they all are consistent with the absence of any

(i) (0) (0) (0)

– 16 – cosmological stretching. However, to ?nd the upper limit of the stretching factors between the two samples of brightest and dimmest events, one has to compare the trial model fdimmest (t, E; zdimmest ) (Exp. 11) with the ACEdimmest curve taking into account the sampling statistics of two groups. The probability distribution P100 (S(I?II) ) has to be used for the two sets of ? 100 events (see Section 3). According to Mitrofanov at al. (1997), the distribution of P100 (S(I?II) ) will have the same shape as the distribution P300 (S(I?II) ) for sets with ? 300 events. Therefore, the value of S(I?II) for the 3σ limit will be about the same. However, because of smaller statistics, for samples with ? 100 events the function (8) has denominator in ? 3 times larger than for samples with ? 300 events, and therefore, the di?erence between two ACEs pro?les allowed by 3σ limit could be in ? 1.7 times larger. Similarly to the basic case of two samples of ? 300 bursts, the new samples of ? 100 brightest and dimmest events are compared by the proposed technique, when for selected values of zdimmest the probability P100 (zdimmest ) is estimated (see (18)) to get the found di?erence between the model pro?le fdimmest (t, E; zdimmest ) and the observed ACEdimmest curve at the third energy discriminator channel. The corresponding probability function P100 (zdimmest ) is presented at the Figure 5. The 3σ upper limit of the zdimmest value is 0.46. Thus, when two samples of the brightest and the dimmest bursts with ? 100 events are compared, no signi?cant increase is found for the best ?tting stretching factors in comparison with the case of two basic samples of ? 300 bright and dim bursts. At both cases one does not see any evidence for stretching e?ect at all. Using the sampling statistics of bursts, the 3σ upper limits are estimated of zdim for ? 300 dim bursts and of zdimmest for ? 100 dimmest events, which equal ? 0.1 and ? 0.5, respectively. One could suspect that the larger upper limit in the second case results from the smaller sampling statistics of groups of ? 100 bursts, and it hardly provides more evidence for cosmological stretching in

– 17 – comparison with the basic case of groups of ? 300 events. However, formally speaking, one has to conclude that the basic cosmological models with standard candles are still allowed for gamma-ray bursts provided they correspond

(dim) to the 3σ upper limit zdim < 0.1 for the group of dim bursts with Fmax < 1.0 photons

cm?2 s?1 or to the upper limit zdimmest < 0.5 for the group of the dimmest bursts with

(dimmest) Fmax < 0.41 photons cm?2 s?1 . These limits resulted from the di?erent sampling

statistics of these groups, and further observations of bursts will allow either to decrease these limits, or to resolve the time-stretching e?ect of dim gamma-ray bursts with respect to bright ones. Two di?erent average photon spectra with power laws α = 1 and α = 2 were used for the test of the basic samples also. At the plane P300 (zdim ) versus zdim these models correspond to upper and lower lines around the main curve, which was found for the average energy spectra (Figure 6). Therefore, the shape of the photon energy spectra does not a?ect signi?cantly the upper limit of zdim . The upper limits of zdim could be estimated also for di?erent parameters of cosmological geometry. Two curves for chance probability P300 (zdim ) were derived for two di?erent sets of cosmological parameters (Figure 7): q0 = σ0 = 0.1 (open Universe) and q0 = σ0 = 1.0 (closed Universe). One might see that these cases of the Universe geometry lead to 3σ upper limits zdim ? 0.08 about the same as the case of ?at expanding Universe (q0 = σ0 = 0.1) .

5.

Discussion and Conclusions

So, the performed comparison of the ACE pro?les for groups of bright and dim bursts does not allow zdim larger than ? 0.1 ? 0.5 for the basic cosmological model with standard sources. Moreover, the ACE ? based limit of red-shift of dim bursts agrees with

– 18 – non-cosmological models of GRBs in the ?at Euclidean space. There are two well-known estimations of the red-shifts of emitters of GRBs according to cosmological models. The ?rst one is based on the average parameter < V /Vmax >= 0.33 ± 0.01 for 3B data base (Meegan et al. 1995). One should expect to have < V /Vmax >= 0.50 for homogeneous distribution of standard sources in the Euclidean space. On the other hand, the observed parameter < V /Vmax > is consistent with the non-Euclidean geometry of expanding Universe. For distant emitters of dim bursts the geometry ? based upper limit of red-shift was estimated about 0.5-2.0 (Wickramasinghe et al 1993). Taking into account the coupling between the spectral shape and the temporal pro?les of bursts, Fenimore and Bloom (1995) have obtained much larger upper limit ?2-6. Another estimation of cosmological limit of the red-shift was based on the e?ect of hardness/intensity correlation of GRBs. The average peak of νFν spectra of dim bursts was found to be much softer than the average peak of bright bursts (Mallozzi et al. 1995). The corresponding ratio between peak energies of dim and bright bursts leads to the spectra ? based upper limit of red-shift, which was estimated about ? 1. There is an agreement, at least qualitative, between geometry ? based and spectra ? based upper limits of red-shifts of distant emitters of GRBs. These estimations result to zdim ? 1 or even much larger. On the other hand, the ACE ? based upper limit of zdim ? 0.1 ? 0.5 does not agree with either the geometry ? based or the spectra ? based limits. Therefore, the basic model of GRBs with standard cosmological sources is not with all available constraints. This is the main conclusion of the present paper. Developing a cosmological model of GRBs, one should postulate some kind of z-dependent property(es) of outbursting sources which could ensure the consistency. Generally speaking, z-dependence could be attributed to di?erent properties of bursts sources, such as outbursts rate density, bursts luminosity, average energy spectra and

– 19 – average light curves. There is a reasonable consistency between geometry ? based and spectra ? based limits of red-shifts of dim bursts emitters. Therefore, one might not suggest any intrinsic z-dependence either for the outbursts rate density, or for the energy spectra of the emitted gamma-rays, because they would both lead to consistent limits of z for the model with standard sources. On the other hand, to make the agreement between them and the ACE ? based limit, one could postulate some sort of intrinsic evolution of outbursting sources which leads to intrinsic squeezing of their light curves with increasing red-shifts. There are physical conditions in local cosmological space which vary with z: the local density of matter, the local temperature of microwave background, etc., but at the present time no one knows how much these conditions could actually in?uence on bursts’ light curves, if they could at all. Obviously, a priori there is no physical reason to propose this kind of evolution, and it could be considered as a pure phenomenological speculation. In addition to ACE ? based test, the time-dilation tests should also be done with another time-based parameters of bright and dim GRBs, such as pulse width, interpulse duration, etc. Comparison of distinct time ? based signatures for di?erent intensity groups of bursts would allow to distinguish the basic e?ect of cosmological time-stretching and energy red- shifting, which should be identical for all time-energy signatures, from another e?ects resulted from z-dependent evolution, which should be di?erent for each of temporal parameters. The cosmological paradigm of GRBs could be ?nally approved at these tests, and new knowledge would be obtained about intrinsic properties of close and distant GRBs sources in the co- moving reference frames. This studies will be done elsewhere.

6.

Acknowledgments

We would like to thank members of BATSE team Drs. G.J. Fishman, W.S. Paciesas, M.S. Briggs, C.A.Meegan, R.D. Preece, G.N. Pendleton, J.J. Brainerd for fruitful

– 20 – cooperation which made possible this paper to be written. This work was supported by the RFBR 96-02-18825 grant in Russia.

– 21 –

Table 1. Best ?tting factors Y ? for ACEdim

(i)

Energy range, keV

Best ?t

Reduced χ2

P (> χ2 )

DOF

25-50 50-100 100-300 25-300

0.81±0.02 0.86±0.01 0.82±0.02 0.84±0.02

2.3 1.28 2.0 2.0

9.2 · 10?6 0.12 3.0 · 10?4 < 10?6

37 37 37 113

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A This manuscript was prepared with the AAS L TEX macros v4.0.

– 24 – Fig. 1.— At the left line three viewgraphs for ACEbright pro?les are presented for three discriminator channels number i = 1, 2, 3. The best ?tting approximation pro?les (2) are shown for each ACE. At the bottom viewgraph (i = 3) the model for i = 1 is shown (dash line) to demonstrate the energy dependence of ACE. At the right line three zooms of corresponding ACEs are presented to show the quality of approximations and the boundaries of ±σ corridor of models around the best ones. Fig. 2.— The probability distribution of S (2) values divided by number of ACE bins (38) for energy discriminator i = 2. It is provided by 105 random choices of two groups of 303 bursts among the total set of 3B database. The value corresponding to 3σ standard deviation is shown by dotted-dashed line. One sees that sampling statistics allows much larger di?erence between two sets of bursts than it could be expected from the sample variance for each of them.

(1024) Fig. 3.— ACEdim pro?les for 332 dim bursts with Fmax < 1 photons cm?2 s?1 . The best (i)

?tting models (11) of ACEs at each discriminator channel are shown by solid lines. Fig. 4.— The estimated probability P300 (zdim ) of consistency is shown between the model curve (11) based on the ACEbright pro?les and the standard cosmological model and the observed ACEdim pro?le at the second (i = 2) discriminator channel. The dashed region represents the probabilities for ±σ corridor of models around the best one. The dotteddashed line shows the probability level of standard 3σ ?uctuations. Fig. 5.— The estimated probability P100 (zdim ) of consistency is shown between the model curve (11) based on the ACEbrightest pro?les and the standard cosmological model and the observed ACEdimmest pro?le at the third (i = 3) discriminator channel. The dotted-dashed line shows the probability level of standard 3σ ?uctuations. Fig. 6.— The same probability P300 (zdim ) is shown as at the Figure 4 (solid line) together

– 25 – with two another estimations based on the energy spectra with power law: dashed and dotted lines correspond to α = 1 and α = 2, respectively. Fig. 7.— The same probability P300 (zdim ) is shown as at the Figure 4 (solid line) together with two estimations corresponded to another models of Universe: dashed and dotted lines correspond to q0 = 0.1 and q0 = 1.0, respectively.

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