Issue 
A&A
Volume 529, May 2011



Article Number  A55  
Number of page(s)  9  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/201014918  
Published online  31 March 2011 
Cosmological effects on the observed flux and fluence distributions of gammaray bursts: Are the most distant bursts in general the faintest ones?
^{1}
Charles University, Faculty of Mathematics and Physics,
Astronomical Institute, V Holešovičkách 2, 18000 Prague 8, Czech Republic
email: meszaros@cesnet.cz, ripa@sirrah.troja.mff.cuni.cz
^{2}
Department of Physics, Royal Institute of Technology, AlbaNova
University Center, 10691
Stockholm,
Sweden
email: felix@particle.kth.se
Received:
3
May
2010
Accepted:
19
January
2011
Context. Several claims have been put forward that an essential fraction of longduration BATSE gammaray bursts should lie at redshifts larger than 5. This pointofview follows from the natural assumption that fainter objects should, on average, lie at larger redshifts. However, redshifts larger than 5 are rare for bursts observed by Swift, seemingly contradicting the BATSE estimates.
Aims. The purpose of this article is to clarify this contradiction.
Methods. We derive the cosmological relationships between the observed and emitted quantities, and we arrive at a prediction that can be tested on the ensembles of bursts with determined redshifts. This analysis is independent on the assumed cosmology, on the observational biases, as well as on any gammaray burst model. Four different samples are studied: 8 BATSE bursts with redshifts, 13 bursts with derived pseudoredshifts, 134 Swift bursts with redshifts, and 6 Fermi bursts with redshifts.
Results. The controversy can be explained by the fact that apparently fainter bursts need not, in general, lie at large redshifts. Such a behaviour is possible, when the luminosities (or emitted energies) in a sample of bursts increase more than the dimming of the observed values with redshift. In such a case dP(z)/dz > 0 can hold, where P(z) is either the peakflux or the fluence. All four different samples of the long bursts suggest that this is really the case. This also means that the hundreds of faint, longduration BATSE bursts need not lie at high redshifts, and that the observed redshift distribution of long Swift bursts might actually represent the actual distribution.
Key words: gammaray burst: general / cosmology: miscellaneous
© ESO, 2011
1. Introduction
Until the last years, the redshift distribution of gammaray bursts (GRBs) has only been poorly known. For example, the Burst And Transient Source Experiment (BATSE) instrument on Compton Gamma Ray Observatory detected around 2700 GRBs^{1}, but only a few of these have directly measured redshifts from the afterglow (AG) observations (Schaefer 2003; Piran 2004). During the last couple of years the situation has improved, mainly due to the observations made by the Swift satellite^{2}. There are already dozens of directly determined redshifts (Mészáros 2006). Nevertheless, this sample is only a small fraction of the, in total, thousands of detected bursts.
Beside the direct determination of redshifts from the AGs, there are several indirect methods, which utilize the gammaray data alone. In essence, there are two different methods which provide such determinations. The first one makes only statistical estimations; the fraction of bursts in a given redshift interval is deduced. The second one determines an actual value of the redshift for a given GRB (“pseudoredshift”).
Already at the early 1990’s, that is, far before the observation of any AG, and when even the cosmological origin was in doubt, there were estimations made in the sense of the first method (see, e.g., Paczyński 1992, and the references therein). In Mészáros & Mészáros (1996) a statistical study indicated that a fraction of GRBs should be at very large redshifts (up to z ≃ 20). In addition, there was no evidence for the termination of the increase of the numbers of GRBs for z > 5 (see also Mészáros & Mészáros 1995; Horváth et al. 1996, and Reichart & Mészáros 1997). In other words, an essential fraction (a few or a few tens of %) of GRBs should be in the redshift interval 5 < z < 20. Again using this type of estimation, Lin et al. (2004) claims that even the majority of bursts should be at these high redshifts.
The estimations of the pseudoredshifts in the sense of the second method are more recent. RamirezRuiz & Fenimore (2000), Reichart et al. (2001), Schaefer et al. (2001) and LloydRonning et al. (2002) developed a method allowing to obtain from the socalled variability the intrinsic luminosity of a GRB, and then from the measured flux its redshift. The redshifts of hundred of bursts were obtained by this method. Nevertheless, the obtained pseudoredshifts are in doubt, because there is an (1 + z) factor error in the cosmological formulas (Norris 2004; Band et al. 2004). Other authors also query these redshifts (e.g., Guidorzi et al. 2005). Norris et al. (2000) found another relation between the spectral lag and the luminosity. This method seems to be a better one (Schaefer et al. 2001; Norris 2002; Ryde et al. 2005), and led to the estimation of ~1200 burst redshifts. Remarkably, again, an essential fraction of long GRBs should have z > 5, and the redshift distribution should peak at z ~ 10. A continuation of this method (Band et al. 2004; Norris 2004), which corrected the (1 + z) error in Norris (2002), gave smaller redshifts on average, but the percentage of long GRBs at z > 5 remains open. Bagoly et al. (2003) developed a different method allowing to obtain the redshifts from the gamma spectra for 343 bright GRBs. Due to the twofold character of the estimation, the fraction of GRBs at z > 5 remains further open. No doubt has yet emerged concerning this method. Atteia (2003) also proposed a method in a similar sense for bright bursts. Other methods of such estimations also exist (Amati et al. 2002; Ghirlanda et al. 2005). These pseudoredshift estimations (for a survey see, e.g. Sect. 2.6 of Mészáros 2006) give the results that even bright GRBs should be at redshifts z ~ (1−5). For faint bursts in the BATSE data set (i.e. for GRBs with peakfluxes smaller than ≃1 photon/(cm^{2} s)) one hardly can have good pseudoredshift estimations, but on average they are expected to be at larger redshifts, using the natural assumption that these bursts are observationally fainter due to their larger distances. Hence, it is remarkable that all these pseudoredshift estimations also supports the expectation, similarly to the results of the first method, that an essential fraction of GRBs is at very high redshifts.
Contrary to these estimations for the BATSE data set, only five long bursts with z > 5 have yet been detected from direct redshift measurements from AGs using more recent satellites. In addition, the majority of measured zs are around z = 2−3, and there is a clear decreasing tendency towards the larger redshifts^{3}. In other words, the redshifts of GRBs detected by the Swift satellite do not support the BATSE redshift estimations; the redshifts of GRBs detected by the nonSwift satellites are on average even at smaller redshifts (Bagoly et al. 2006).
This can be interpreted in two essentially different ways. The first possibility is that a large fraction (a few tens of % or even the majority) of GRBs are at very high redshifts (at z > 5 or so). In such case these bursts should mainly be seen only in the gammaray band due to some observational selection (Lamb & Reichart 2000). The second possibility is that the AG data reflect the true distribution of bursts at high redshifts, and bursts at z > 5 are really very rare. In this case, however, there has to be something wrong in the estimations of redshifts from the gammaray data alone. Since observational selections for AG detections of bursts at z > 5 cannot be excluded (Lamb & Reichart 2000), the first pointofview could be also quite realistic.
The purpose of this paper is to point out some statistical properties of the GRBs, which may support the second possibility. Section 2 discusses these properties theoretically. In Sects. 3 and 4 we discuss the impact of such a behaviour on several observed burst samples. Section 5 summarizes the results of paper.
2. Theoretical considerations
2.1. The general consideration
Using the standard cosmological theory and some simple statistical considerations, we will now show that, under some conditions, apparently fainter bursts might very well be at smaller redshifts compared to the brighter ones.
As shown by Mészáros & Mészáros (1995), if there are given two photonenergies E_{1} and E_{2}, where E_{1} < E_{2}, then the flux F (in units photons/(cm^{2}s)) of the photons with energies E_{1} ≤ E ≤ E_{2} detected from a GRB having a redshift z is given by (1)where is the isotropic luminosity of a GRB (in units photons/s) between the energy range E_{1}(1 + z) ≤ E ≤ E_{2}(1 + z), and d_{l}(z) is the luminosity distance of the source. The reason of the notation , instead of the simple L(z), is that L(z) should mean the luminosity from E_{1} ≤ E ≤ E_{2} (Mészáros et al. 2006). One has d_{l}(z) = (1 + z)d_{M}(z), where d_{M}(z) is the proper motion distance of the GRB, given by the standard cosmological formula (Carroll et al. 1992), and depends on the cosmological parameters H_{0} (Hubbleconstant), Ω_{M} (the ratio of the matter density and the critical density), and (λ is the cosmological constant, c is the velocity of light). In energy units one may write and , where is a typical photon energy ensuring that the flux F_{en}(z) has the dimension erg/(cm^{2}s). in erg/s unit gives the luminosity in the interval E_{1}(1 + z) ≤ E ≤ E_{2}(1 + z). Except for an (1 + z) factor the situation is the same, when considering the fluence. Hence, in the general case, we have (2)where P(z) is either the fluence or the flux, and is either the emitted isotropic total number of photons, or the isotropic total emitted energy or the luminosity. The following values of N can be used: N = 0 if the flux is taken in energy units erg/(cm^{2}s) and is the luminosity with dimension erg/s; N = 1 if either the flux and luminosity are in photon units, or the fluence in energy units and the total energy are taken; N = 2 if the total number of photons are considered. All this means that for a given GRB – for which redshift, flux and fluence are measured – Eq. (2) defines immediately , which is then either the isotropic total emitted energy or the luminosity in the interval E_{1}(1 + z) ≤ E ≤ E_{2}(1 + z). Hence, is from the range E_{1}(1 + z) ≤ E ≤ E_{2}(1 + z) and not from E_{1} ≤ E ≤ E_{2}.
Let us have a measured value of P(z). Fixing this P(z) Eq. (2) defines a functional relationship between the redshift z and . For its transformation into the real intrinsic luminosities L(z) the beaming must be taken into account as well (Lee et al. 2000; Ryde et al. 2005; Bagoly & Veres 2009a,b; Petrosian et al. 2009; Butler et al. 2010). Additionally, we need to study the dependence of the obtained on z, and to determine the real luminosities L(z) by the Kcorrections (Mészáros 2006). It is not the aim of this paper to solve the transformation of into L(z). The purpose of this paper is to study the functional relationship among P(z), z and .
Fig. 1 Left panel: function Q(z) for Ω_{M} = 0.27 and Ω_{Λ} = 0.73. Right panel: dependence of z_{turn} on N + q for Ω_{M} = 0.27, Ω_{Λ} = 0.73. 
Using the proper motion distance d_{M}(z) Eq. (2) can be reordered as (3)The proper motion distance d_{M}(z) is bounded as z → ∞ (Weinberg 1972, Chap. 14.4.). This finiteness of the proper motion distance is fulfilled for any H_{0},Ω_{M} and Ω_{Λ}. Hence, is a monotonically increasing function of the redshift along with (1 + z)^{2−N} for the fixed P(z) = P_{0} and for the given value of N ≤ 1. It means z_{1} < z_{2} implies . Expressing this result in other words: the more distant and brighter sources may give the same observed value of P_{0}. Now, if a source at z_{2} has a , its observed value will have , i.e. it becomes apparently brighter despite its greater redshift than that of the source at z_{1}. The probability of the occurrence this kind of inversion depends on , on the conditional probability density of assuming z is given, and on the spatial density of the objects.
It is obvious from Eq. (3) that for the increasing z the is also increasing. This effect gives a bias (Lee et al. 2000; Bagoly & Veres 2009a,b; Petrosian et al. 2009; Butler et al. 2010) towards the higher values among GRBs observed above a given detection threshold. (These questions are discussed in detail mainly by Petrosian et al. 2009.) There can be also that is increasing with z due to the metallicity evolution (Wolf & Podsiadlowski 2007). There can be also an intrinsic evolution of the L(z) itself in the energy range [E_{1},E_{2}] . Hence, keeping all this in mind, can well be increasing on z, and the inverse behaviour can also occur.
2.2. A special assumption
Assume now that we have , where q is a real number, and this relation holds for any 0 < z < ∞. This means that it holds , where . Of course, this special assumption is a loss of generality, because can be really a function of z, but in general it need not have this special form. In addition, to calculate the cosmological parameters must be specified, which is a further loss of generality. Nevertheless, if this special assumption is taken, the inverse behavior may well be demonstrated.
If (N + q) > 2, then one has a highly interesting mathematical behaviour of the function P(z) (Eq. (2)). For z ≪ 0.1, P(z) decreases as z^{2}, that is, larger redshifts gives smaller fluxes or fluences as expected. However, after some z = z_{turn} this behavior must change, because as z tends to ∞, the function tends to infinity following ∝ (1 + z)^{N+q−2}. In other words, for z > z_{turn}as redshift increases, the measured P(z) will also increase. Equivalently stated, “fainter bursts are closer to us”. The possibility of this “inverse” behaviour is quite remarkable. It is important to note that the existence of a z_{turn} is exclusively determined by the value q, and the necessary and sufficient condition for it is given by the inequality (N + q) > 2. For the existence of a z_{turn} the values H_{0},Ω_{M} and Ω_{Λ} are unimportant. The value of z_{turn} can vary depending on the choice of the Ω parameters, but, however, its existence or nonexistence is unaffected.
Moreover, the value of z_{turn} itself is independent on the Hubbleconstant H_{0}. This can be seen as follows. To find z_{turn} one must simply search for the minimum of the function Q(z) = (1 + z)^{N + q}/d_{l}(z)^{2}, that is, when dQ(z)/dz = 0. But, trivially, Q(z) and have the same minimum.
The solution of the equation dQ(z)/dz = 0 must be found numerically for the general case of Omega parameters. The left panel of Fig. 1 shows the function Q(z) for Ω_{M} = 0.27 and Ω_{Λ} = 0.73. For Ω_{Λ} = 0 it can be found analytically, because d_{M}(z) is then an analytical function. For Ω_{M} = 1 and Ω_{Λ} = 0 the condition dQ(z)/dz = 0 is easily solvable. For this special case one has to search for the minimum of function (4)because here d_{M} = (2c/H_{0})/(1 − (1 + z)^{−1/2}). Then one has (5)
Fig. 2 Distribution of the fluences (left panel) and peakfluxes (right panel) of the Swift GRBs with known redshifts. The medians separate the area into four quadrants. The objects in the upper right quadrant are brighter and have larger redshifts than the that of GRBs in the lower left quadrant. 
Fig. 3 Distribution of the fluences (left panel) and peakfluxes (right panel) of the Swift GRBs with known redshifts. On the left panel the curves denote the values of fluences for (the three constant are in units 10^{51} erg: I. 0.1; II. 1.0; III. 10.0). On the right panel the curves denote the values of peakfluxes for (the three constant are in units 10^{58} ph/s: I. 0.01; II. 0.1; III. 1.0). 
3. The samples
3.1. The choice of burst samples
The frequency of the occurrence at z_{1} < z_{2}, but for their observed values P(z_{1}) < P(z_{2}), i.e. the more distant GRB is apparently brighter for the observer, can be verified on a sample for which there are welldefined redshifts, as well as measured peakfluxes and/or fluences. At a given redshift the luminosity is a stochastic variable and starting from Eq. (3) one can get the probability for , assuming that is given.
There are two different subgroups of GRBs, which can be denoted as “short” and “long”duration bursts (Norris et al. 2001; Balázs et al. 2003; Mészáros et al. 2006; Zhang et al. 2009). In addition, the existence of additional subgroups cannot be excluded (Horváth 1998, 2002; Hakkila et al. 2003; Borgonovo 2004; Varga et al. 2005; Horváth et al. 2008, 2009; Vavrek et al. 2008). The first direct redshift for a long (short) GRBs was determined in 1997 (2005) (for a detailed survey see, e.g., Piran 2004; and Mészáros 2006). The few redshifts measured for short bursts are on average small (Mészáros et al. 2009), which motivates us to concentrate on longduration bursts only in this study.
Since we try to obtain consequences of the GRBs’ redshifts also in the BATSE Catalog, we should obviously study the BATSE sample. But only a few of these bursts have directly measured redshifts from afterglow data. Therefore we will also try to obtain conclusions from a sample that uses the socalled “pseudoredshifts”, i.e. the redshifts estimated exclusively from the gamma photon measurements alone. But for them one should keep in mind that they can be uncertain. Thus, the best is to compare the BATSE samples with other samples of long GRBs. The redshifts of GRBs detected by Swift satellite – obtained from afterglow data – can well serve for this comparison. On the other hand, the redshifts of GRBs – detected by other satellites (BeppoSAX, HETE2, INTEGRAL) – are not good for our purpose, since they are strongly biased with selection effects (Lee et al. 2000; Bagoly et al. 2006; Butler et al. 2010), and represent only the brightest bursts. All this means that we will discuss four samples here: BATSE GRBs with known redshifts, BATSE GRBs with pseudoredshifts, the Swift sample and the Fermi sample. We will try to show the occurrence of the inverse behaviour, first, without the special assumptions of Sect. 2.2., and, second, using this subsection.
3.2. Swift GRBs and the inversion in this sample
In the period of 20 November 2004−30 April 2010 the Swift database gives a sample of 134 bursts with well determined redshifts from the afterglows together with BAT fluences and peakfluxes in the energy range of 15−150 keV. To abandon the short bursts only those with T_{90}/(1 + z) > 2 s were taken. They are collected in Table 1.
Swift sample with known redshifts.
Fig. 4 Left panel: vs. (1 + z) (dots). Dashed contours denote constant fluences (in units 10^{7} erg/cm^{2}): I. the maximal fluence, i.e. 1050; II. 105; III. 10.5; IV. the minimal fluence, i.e. 0.68. Right panel: vs. (1 + z) (dots). Dashed contours denote constant peakfluxes (in units ph cm^{2} s^{1}): I. the maximal peakflux, i.e. 71; II. 7.1; III. 0.71; IV. the minimal peakflux, i.e. 0.04. The objects below a curve at smaller redshifts together with those at higher redshifts and above the curve illustrate the inverse behaviour. 
continued.
BATSE sample with known redshifts.
The effect of inversion can be demonstrated by the scatter plots of the [log fluence; z] and [log peakflux; z] planes as it can be seen in Fig. 2. To demonstrate the effect of inversion we marked the medians of the fluence and peakflux with horizontal and that of the redshift with vertical dashed lines. The medians split the plotting area into four quadrants. It is easy to see that GRBs in the upper right quadrant are apparently brighter than those in the lower left one, although, their redshifts are larger. It is worth mentioning that the GRB having the greatest redshift in the sample has higher fluence than 50% of all the items in the sample.
Using the special assumption of Sect. 2.2. the effect of inversion may be illustrated in the Swift sample distinctly also as follows. In Fig. 3 the fluences and peakfluxes are typified against the redshifts. Be the luminosity distances calculated for H_{0} = 71 km/(s Mpc), Ω_{M} = 0.27 and Ω_{Λ} = 0.73. Then the total emitted energy and the peakluminosity can be calculated using Eq. (2) with N = 1. In the figure the curves of fluences and peakfluxes are shown after substituting and where and are constants, and q = 0;1;2. The inverse behaviour is quite obvious for q > 1 and roughly for z > 2.
The same effect can be similarly illustrated also in Fig. 4 showing the relation vs. (1 + z), and the relation vs. (1 + z). They were calculated again for H_{0} = 71 km/(s Mpc), Ω_{M} = 0.27 and Ω_{Λ} = 0.73. In the figure the curves of constant observable peakfluxes and fluences are also shown. These curves discriminate the bursts of lower/higher measured values. GRBs below a curve at smaller redshifts are representing the inverse behaviour with respect to those at higher redshifts and above the curve.
3.3. BATSE sample with known redshifts
There are 11 bursts, which were observed by BATSE during the period 1997−2000 and for which there are observed redshifts from the afterglow data. For one of them, GRB980329 (BATSE trigger 6665), the redshift has only an upper limit (z < 3.5), and hence will not be used here. Two cases (GRB970828 [6350] z = 0.9578 and GRB000131 [7975] z = 4.5) have determined redshifts, but they have no measured fluences and peakfluxes, hence they are also excluded. There are remaining 8 GRBs, which are collected in Table 2 (see also Bagoly et al. 2003; and Borgonovo 2004). For the definition of fluence we have chosen the fluence from the third BATSE channel between 100 and 300 keV (F_{3}). This choice is motivated by the observational fact that these fluences in the BATSE Catalog are usually well measured and correlate with other fluences (Bagoly et al. 1998; Veres et al. 2005).
Fig. 5 Distribution of the fluences (left panel) and peakfluxes (right panel) of the GRBs with known redshifts, where the Fermi GRBs are denoted by asterisks, BATSE GRBs with determined redshifts (pseudoredshifts) are denoted by dots (circles). The medians separate the area into four quadrants. The objects in the upper right quadrant are brighter and have larger redshifts than the that of GRBs in the lower left quadrant. 
BATSE sample with pseudoredshifts.
Fermi sample with known redshifts.
3.4. BATSE pseudoredshifts
In the choice of a BATSE sample with estimated pseudoredshifts one has to take care, since these redshifts are less reliable than the direct redshifts from AGs. We will use here the pseudoredshifts based on the luminositylag relation, restricted to the sample in Ryde et al. (2005), where also the spectroscopic studies support the estimations. In Table 3 we collect 13 GRBs using Table 3 of Ryde et al. (2005). We do not consider two GRBs (BATSE triggers 973 and 3648) from Ryde et al. (2005), since their estimated pseudoredshifts are ambiguous.
3.5. Fermi sample
The Fermi sample contains only 6 GRBs with known redshifts together with peakfluxes and fluences (Bissaldi & Connaughton 2009; Bissaldi 2009; van der Horst et al. 2009; Rau et al. 2009,b; Foley et al. 2010). They are collected in Table 4. The peakfluxes and fluences were measured over the energy range 50 keV−10 MeV for GRB090902B and in the range 8 keV−1 MeV for the remaining five objects.
3.6. Inversion in the BATSE and Fermi samples
The fluence (peakflux) vs. redshift relationship of the Fermi and of the two BATSE samples are summarized in Fig. 5. For demonstrating the inversion effect – similarly to the case of the Swift sample – the medians also marked with dashed lines. Here it is quite evident that some of the distant bursts exceed in their observed fluence and peakfluxes those of having smaller redshifts. Here again the GRBs in the upper right quadrants are apparently brighter than those in the lower left one, although their redshifts are larger. Note that in the upper right quadrants are even more populated than the lower right quadrants. In other words, here the trend of the increasing of peakflux (fluence) with redshift is evident, and the assumption of the Sect. 2.2. need not be used.
4. Results and discussion
It follows from the previous section that in all samples both for the fluences and for the peakfluxes the “inverse” behaviour, discussed in Sect. 2, can happen. The answer on the question of the title of this article is therefore that “this does not need to be the case”. Simply, the apparently faintest GRBs need not be also the most distant ones. This is in essence the key result of this article.
It is essential to note that in this paper no assumptions were made about the models of the long GRBs. Also the cosmological parameters did not need to be specified.
All this means that faint bursts in the BATSE Catalog simply need not be at larger redshifts, because the key “natural” assumption – apparently fainter GRBs should on average be at higher redshifts – does not hold. All this also means that the controversy about the fraction of GRBs at very high redshifts in BATSE Catalog may well be overcame: no large fraction of GRBs needs to be at very high redshifts, and the redshift distribution of long GRBs – coming from the Swift sample – may well occur also for the BATSE sample. Of course, this does not mean that no GRBs can be at, say, 8.3 < z < 20. As proposed first by Mészáros & Mészáros (1996), at these very high redshifts GRBs may well occur, but should give a minority (say 10% or so) in the BATSE Catalog similarly the Swift sample. This point of view is supported by newer studies, too (Bromm & Loeb 2006; Jakobsson et al. 2006; Gehrels et al. 2009; Butler et al. 2010).
At the end it is useful to raise again that the purpose of this paper was not to study the intrinsic evolution of the luminosities L(z) from the energy range E_{1} ≤ E ≤ E_{2}. To carry out such study, one should consider three additional reasons that may be responsible for the growth of the average value of with (1 + z): 1) Kcorrection; 2) the beaming; 3) selection biases due to the instrument’s threshold and other instrumental effects. For example, on Fig. 4 the main parts in the rightbottom sections below the curves IV – corresponding to the values below the instrumental thresholds in fluence/peakflux – are not observable. Nonetheless, even using these biased data, the theoretical considerations stated in the Sect. 2 and conclusions of the next sections are valid.
5. Conclusions
The results of this paper can be summarized as follows:

1.
the theoretical study of the zdependence of the observed fluences and peakfluxes of GRBs have shown that fainter bursts could well have smaller redshifts;

2.
this is really fulfilled for the four different samples of long GRBs;

3.
these results do not depend on the cosmological parameters and on the GRB models;

4.
all this suggests that the estimations, leading to a large fraction of BATSE bursts at z > 5, need not be correct.
Acknowledgments
We wish to thank Z. Bagoly, L. G. Balázs, I. Horváth, S. Larsson, P. Mészáros and P. Veres for useful discussions and comments on the manuscript. The useful remarks of the anonymous referee are kindly
acknowledged. This study was supported by the OTKA grant K77795, by the Grant Agency of the Czech Republic grants No. 205/08/H005, and P209/10/0734, by the project SVV 261301 of the Charles University in Prague, by the Research Program MSM0021620860 of the Ministry of Education of the Czech Republic, and by the Swedish National Space Agency.
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All Tables
All Figures
Fig. 1 Left panel: function Q(z) for Ω_{M} = 0.27 and Ω_{Λ} = 0.73. Right panel: dependence of z_{turn} on N + q for Ω_{M} = 0.27, Ω_{Λ} = 0.73. 

In the text 
Fig. 2 Distribution of the fluences (left panel) and peakfluxes (right panel) of the Swift GRBs with known redshifts. The medians separate the area into four quadrants. The objects in the upper right quadrant are brighter and have larger redshifts than the that of GRBs in the lower left quadrant. 

In the text 
Fig. 3 Distribution of the fluences (left panel) and peakfluxes (right panel) of the Swift GRBs with known redshifts. On the left panel the curves denote the values of fluences for (the three constant are in units 10^{51} erg: I. 0.1; II. 1.0; III. 10.0). On the right panel the curves denote the values of peakfluxes for (the three constant are in units 10^{58} ph/s: I. 0.01; II. 0.1; III. 1.0). 

In the text 
Fig. 4 Left panel: vs. (1 + z) (dots). Dashed contours denote constant fluences (in units 10^{7} erg/cm^{2}): I. the maximal fluence, i.e. 1050; II. 105; III. 10.5; IV. the minimal fluence, i.e. 0.68. Right panel: vs. (1 + z) (dots). Dashed contours denote constant peakfluxes (in units ph cm^{2} s^{1}): I. the maximal peakflux, i.e. 71; II. 7.1; III. 0.71; IV. the minimal peakflux, i.e. 0.04. The objects below a curve at smaller redshifts together with those at higher redshifts and above the curve illustrate the inverse behaviour. 

In the text 
Fig. 5 Distribution of the fluences (left panel) and peakfluxes (right panel) of the GRBs with known redshifts, where the Fermi GRBs are denoted by asterisks, BATSE GRBs with determined redshifts (pseudoredshifts) are denoted by dots (circles). The medians separate the area into four quadrants. The objects in the upper right quadrant are brighter and have larger redshifts than the that of GRBs in the lower left quadrant. 

In the text 
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