ap-6-11.dvi Acta Polytechnica Vol. 51 No. 6/2011 A Crazy Question: Can Apparently Brighter Gamma-ray Bursts Be Farther Away? A. Mészáros, J. Ř́ıpa, F. Ryde Abstract The cosmological relationships between observed and emitted quantities are determined for gamma-ray bursts (GRBs). The relationship shows that apparently fainter bursts need not, in general, lie at larger redshifts.This is possible when the luminosities (or emitted energies) in a sample of bursts increase faster than the dimming of the observed values with redshift. Four different samples of long bursts suggest that this is what really happens. Keywords: cosmology-miscellaneous, gamma-ray bursts. 1 Introduction The Burst and Transient Source Experiment (BAT- SE) instrument at the ComptonGammaRayObser- vatory detected around 2700 GRBs1, but only a few of these have directly measured redshifts from the optical afterglow (OA) observations [1, 2]. During the last couple of years the situation has improved, mainly due to the observations made by the Swift satellite2. There are alreadydozens of directly deter- mined redshifts [3]. However, this sample is only a small fraction of the thousands of detected bursts. Beside direct determination of redshifts from OAs, there are several indirectmethodswhich utilize gamma-ray data alone. In these mainly statistical studies, akeyrole is playedby theassumption that— on average — apparently fainter bursts should lie in smaller redshifts. The purpose of this paper is to show that this is not always the case. The paper is in essence a short- ened version of [4]. 2 Theoretical considerations Let the observed peak-flux or fluence P(z) of a GRB be given. This value is given by [5,6] P(z)= (1+ z)N L̃(z) 4πdl(z)2 , where it can be N = 0;1;2, and where z is the red- shift, dl(z) is the luminosity distance, and L̃(z) is either the isotropic peak-luminosity or the isotropic emitted energy. There are four possibilities [5,6]: • P(z) –fluence is in“photons/cm2”units; N =2; L̃(z) is in “photons” units, • P(z) – fluence is in “erg/cm2” units; N = 1; L̃(z) is in “erg” units, • P(z) – peak-flux is in “photons/cm2s” units; N =1; L̃(z) is in “photons/s” units, • P(z) – peak-flux is in “erg/cm2s” units; N =0; L̃(z) is in “erg/s” units. In GRB, topic P(z) is usually measured in an en- ergy interval of photons E1 < E < E2. In ad- dition, it is a good approximation that all arriving photons from this interval are detected, but none from outside. Then L̃(z) is from the energy-interval E1(1+ z) < E < E2(1+ z). It is a standard cosmological behaviour that for small z’s (z � 0.1) dl(z) ∝ z, and for large redshifts lim z→∞ dl(z) 1+ z =finite positive number for any Ho, ΩM, ΩΛ [7]. Hence, L̃(z) is a monoton- ically increasing function of the redshift along with (1 + z)2−N for fixed P(z) = P0 and for the given value of N ≤ 1. This means that z1 < z2 im- plies L̃(z1) < L̃(z2). Expressing this result in other words: more distant and brighter sources may give the same observed value of P0. Now, if a source at z2 has L̃ > L̃(z2), its observed value P ′ obs will have P ′obs > P0, i.e. it becomes apparently brighter de- spite its greater redshift than that of the source at z1. The probability of the occurrence of this kind of inversion depends on f(L̃|z), on the conditional probability density of L̃ assuming z is given, and on the spatial density of the objects. Assume now specially that L̃(z) ∝ (1+ z)q; N + q > 2; for z → ∞. Then dP(z) dz > 0; for z → ∞. 1http://www.batse.msfc.nasa.gov/batse/grb/catalog/ 2http://swift.gsfc.nasa.gov/docs/swift/swiftsc.html 45 Acta Polytechnica Vol. 51 No. 6/2011 Fig. 1: Left panel: Function Q(z) for ΩM = 0.27 and ΩΛ = 0.73. Right panel: Dependence of zturn on q for ΩM = 0.27 and ΩΛ =0.73 Fig. 2: Distribution of the fluences (left panel) and peak-fluxes (right panel) of 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 those of the GRBs in the lower left quadrant Hence, if L̃(z) increases fast enough ((N + q) > 2), then it is possible that on average dP(z)/dz > 0 can happen. This means that for small z’s P(z) always decreases as z−2, but if L̃(z) ∝ (1+z)q, (N +q) > 2 athigher redshifts, then P(z) increases as zN+p−2 for big z’s; again always. If this is the case, then the question naturally arises: Where is zturn, where dP(z)/dz = 0? Math- ematically this means that one has to search for the minimum of function Q(z)= (1+ z)N+q/dl(z) 2; i.e., when dQ(z)/dz = 0 holds. The results of numerical solutions for zturn are shown in Figure 1. 3 The samples The expectation that more distant GRBs are on av- erage apparentlybrighter for the observer canbe ver- ified on samples for which there are well-defined red- shifts, as well as measured peak-fluxes and/or flu- ences. We discuss four samples here: BATSE GRBs with known redshifts (8 long GRBs), BATSE GRBs withpseudo-redshifts (13 longGRBs), theSwift sam- ple (134 long GRBs) and the Fermi sample (6 long GRBs). For the Swift sample, the effect of inversion can easily be seen by the scatter plots of the [log fluence; z] and [log peak-flux; z] planes; Figure 2. To demon- strate the effect of inversion we marked the medians of the fluence and peak-fluxwith horizontal lines and themedians of the redshiftwithvertical dashed lines. The medians split the plotting area into four quad- rants. The GRBs in the upper right quadrants are apparentlybrighter than those in the lower left quad- rants, although their redshifts are larger. It is worth mentioning that the GRB with the greatest redshift in the sample has higher fluence than 50 % of all the items in the sample. 46 Acta Polytechnica Vol. 51 No. 6/2011 Fig. 3: Distribution of the fluences (left panel) and peak-fluxes (right panel) of GRBs with known redshifts, where Fermi GRBs are denoted by asterisks, and BATSE GRBs with determined redshifts (pseudo-redshifts) 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 those of the GRBs in the lower left quadrant Fig. 4: Distribution of the fluences (left panel) and peak-fluxes (right panel) of Swift GRBs with known redshifts. On the left panel the curves denote the values of fluences for Ẽiso = Ẽo(1+ z) q (the three constant Ẽo are in units 10 51 erg: I. 0.1; II. 1.0; III. 10.0). In the panel on the right the curves denote the values of the peak-fluxes for L̃iso = L̃o(1+ z) q (the three constant L̃o are in units 10 58ph/s: I. 0.01; II. 0.1; III. 1.0) The fluence (peak-flux) vs. redshift relationship of the Fermi and of the twoBATSE samples are sum- marized in Figure 3. To demonstrate the inversion effect — similarly to the case of the Swift sample — the medians are alsomarkedwith dashed lines. Here it is quite evident that some of the distant bursts exceed in their observed fluence and peak-fluxes the values of those with smaller redshifts. Here again the GRBs in the upper right quadrants are appar- ently brighter than those in the lower left quadrant, although their redshifts are larger. Note that the up- per rightquadrantsareevenmorepopulated than the lower right quadrants. In other words, the trend of increasing peak-flux (fluence) with redshift is really evident here. Using the special assumption L̃(z) ∝ (1 + z)q, the effect of inversion may also be illustrated dis- tinctly in the Swift sample, as follows. In Figure 4 the fluences and peak-fluxes are typified against the redshifts. Let the luminosity distances be calculated for H0 = 71km/(sMpc), ΩM = 0.27 and ΩΛ = 0.73. Then the total emitted energy Ẽiso and the peak- luminosity L̃iso can be calculated with N = 1. In the figure, the curves of the fluences and peak-fluxes are shown after substituting L̃iso = L̃o(1+ z) q and Ẽiso = Ẽo(1+z) q where L̃o and Ẽo are constants, and q =0;1,2. The inverse behaviour is quite obvious for q > 1 and roughly for z > 2. 4 Conclusions In all samples, both for the fluences and for the peak- fluxes, the “inverse” behaviour can happen. The ap- parently faintest GRBs need not also be the most 47 Acta Polytechnica Vol. 51 No. 6/2011 distant. The question of the title can really be an- swered by a clear “Yes, they can.”. It should be noted that no assumptions have been made in this paper about the models of the long GRB. In addition, no cosmological parameters needed to be specified. The results of this paper can be summarized as follows: 1. A theoretical study of the z-dependence of the observed fluences and peak-fluxes of GRBs have shownthat fainter bursts couldwell have smaller red- shifts. 2.This is really fulfilled for the four different sam- ples of long GRBs. 3. These results do not depend on the cosmologi- cal parameters and on the GRB models. 4. All this suggests that the estimations (see, for example, [8]), leading to a large fraction of BATSE bursts at z > 5, need not be correct. Acknowledgement Wewish to thankZ.Bagoly,L.G.Balázs, I.Horváth, S. Larsson, P. Mészáros, D. Szécsi and P. Veres for useful remarks. This study was supported by OTKA grant K77795, by Grant Agency of the Czech Re- public grant P209/10/0734, by Research Program MSM0021620860 of the Ministry of Education of the Czech Republic, and by the Swedish National Space Agency. References [1] Schaefer, B. E.: ApJ, 583, L67, 2003. [2] Piran, T.: Rev. Mod. Phys., 76, 1143, 2004. [3] Mészáros, P.: Rep. Progr. Phys. 69, 2259, 2006. [4] Mészáros, A., Řı́pa, J., Ryde, F.: A & A, 529, A55, 2011. [5] Mészáros, P., Mészáros, A.: ApJ, 449, 9, 1995. [6] Mészáros, A., Mészáros, P.: ApJ, 466, 29, 1996. [7] Weinberg, S.: Gravitation and Cosmology. New York : J. Wiley and Sons, 1972. [8] Lin, J.R., Zhang, S.N., Li, T.P.: ApJ,605, 819, 2004. Attila Mészáros Jakub Řı́pa Charles University Faculty of Mathematics and Physics Astronomical Institute V Holešovičkách 2, 180 00 Prague 8, Czech Republic Felix Ryde Department of Physics Royal Institute of Technology AlbaNova University Center SE-106 91 Stockholm, Sweden 48