fahmi_dry_cyc_impr.eps Acta Polytechnica Vol. 52 No. 2/2012 Angular Distribution of GRBs L. G. Balázs, A. Mészáros, I. Horváth, Z. Bagoly, P. Veres, G. Tusnády Abstract We studied the complete randomness of the angular distribution of BATSE gamma-ray bursts (GRBs). Based on their durations and peak fluxes, we divided the BATSE sample into 5 subsamples (short1, short2, intermediate, long1, long2) and studied the angular distributions separately. Weused threemethods to search for non-randomness in the subsamples: Voronoi tesselation, minimal spanning tree, and multifractal spectra. To study any non-randomness in the subsamples we defined 13 test-variables (9 from Voronoi tesselation, 3 from theminimal spanning tree and one from the multifractal spectrum). We made Monte Carlo simulations taking into account the BATSE’s sky-exposure function. We tested the randomness by introducing squaredEuclidean distances in the parameter space of the test-variables. We recognized that the short1, short2 groups deviate significantly (99.90%, 99.98%) from the fully random case in the distribution of the squared Euclidean distances but this is not true for the long samples. In the intermediate group, the squared Euclidean distances also give significant deviation (98.51%). 1 Introduction Recently, the cosmological origin of gamma-ray bursts (hereafter GRBs) has been widely ac- cepted [24,49,75]. Assuming large scale isotropy for the Universe, one would also expect the same prop- erty for GRBs. There is an increasing amount of evidence that all GRBs do not all form a physically homogeneous group [5, 27–31, 39, 50]. Hence, it is worth inves- tigating whether physically different subgroups are also different in their angular distributions. The au- thors have carried out several different tests in re- cent years [3,4,45,46] probing the intrinsic isotropy in the angular sky-distribution of the GRBs col- lected in BATSE Catalog [44]. Briefly summariz- ing the results of these studies, one may conclude: A. The long subgroup (T90 > 10s) seems to be dis- tributed isotropically; B. The intermediate subgroup (2s ≤ T90 ≤ 10s) is distributed anisotropically on the � (96–97)% significance level; C. For the short subgroup (T90 < 2s) the assumption of isotropy is re- jected only on the 92%significance level;D.The long and short subclasses, respectively, aredistributeddif- ferently on the 99.3% significance level. (For a defi- nition of the subclasses see [29–33,69].) Independently and by different tests, [43] con- firmed resultsA., B. andC., with one essential differ- ence: for the intermediate subclass a much higher — namely 99.89% — significance level of anisotropy is claimed. Again, the short subgroup is found to be “suspicious”, but only the � (85–95)% significance level is reached. The long subclass seems to be dis- tributed isotropically. [42] found a significant angular correlation on the 2◦ −5◦ scale forGRBs of T90 < 2s duration. [64] reported a correlation between the lo- cations of previously observed short bursts and the positions of galaxies in the local Universe, indicat- ing that between 10 and 25 per cent of short GRBs originate at low redshifts (z < 0.025). It is a reasonable requirement to continue these tests using more sophisticated procedures in order to see whether the angular distribution of GRBs is completely random, or whether it has some sort of regularity. This is the subject of this paper. New tests will be presented here. Mainly the clarifica- tion of the short subgroup’s behaviour is expected from these tests. In this paper, similarly to the pre- vious studies, the intrinsic randomness is tested; this means that the non-uniformsky-exposure function of BATSE instrument is eliminated. The paper is organized as follows. The three new tests are described in Section 2. This Section does not containnew results, but thisminimal surveymay be useful because the methods are not widely famil- iar. Section 3 contains the statistical tests on the data. Section 4 summarizes the results of the statis- tical tests, and Section 5 presents the main conclu- sions of the paper. The paper is based on the results published in [68]; some preliminary cosmological im- plications are also discussed by [47,48]. 2 Mathematical overview Strictly from themathematical point of view, consid- eringGRBsas a point process on the celestial sphere, the first property means that for a given Ω area on the sky the P(Ω) probability of observing a burst de- pends only on the size ofΩ, andnot on its locationon the sphere. The second property means that for the given two disjunct Ω1,Ω2 areas on the sky the joint probability is given by P(Ω1,Ω2)= P(Ω1)P(Ω2), i.e. 17 Acta Polytechnica Vol. 52 No. 2/2012 the probability of observing a burst in Ω1 is fully independent from the probability of getting one in Ω2. If both properties are fulfilled, the distribution is called completely random (for the astronomical con- text of spatial point processes, see [55]). There are several tests for checking the complete randomness of point patterns, but these procedures do not always give information for both properties simultaneously. The simplest test for isotropy is a comparison of the number counts of GRBs in disjunct areas on the sky. In the case of isotropyexpanding P(Ω) into a se- ries of spherical harmonics, all the coefficients in the series, except the 0th order, should equal zero. This property canalso be used for testing the isotropy (for testing thedipole andquadrupolemoments, see [10]). Before performing the isotropy tests the uneven ex- posure function of BATSE has to be taken into ac- count [44]. Weused threemethods, Voronoi tesselation,min- imal spanning tree and multifractal spectra to get statistical variables suitable for testing the fully ran- domness of the angular distribution of GRBs. 2.1 Voronoi tesselation (VT) The Voronoi diagram— also known as Dirichlet tes- selation or Thiessen polygons — is a fundamental structure in computational geometry and arises nat- urally in many different applications [70,62]. Gen- erally, this diagram provides a partition of a point pattern (“point field”, also “point process”), accord- ing to its spatial structure, which can be used for analyzing the underlying point process. Let us assume that there are N points (N � 1) scattered on a sphere surface with a unit radius. We say that a point field is given on the sphere. The Voronoi cell [62] of a point is the region of the sphere surface consisting of points which are closer to this givenpoint thantoanyother onesof the sphere. This cell forms a polygon on this sphere. Each such cell has its area (A) given in the steradian, perimeter (P) given by the length of boundary (one great circle of the boundary curve is also called a “chord”), number of vertices (Nv) given by an integer positive number, and by the inner angles (αi; i = 1, . . . , Nv). This method is completely non-parametric, and therefore may be sensitive for various point pattern structures in the different subclasses of GRBs. The points on a sphere may be distributed com- pletely randomly or non-randomly; the non-random distribution may have different characters (cluster- ing, filaments, etc.; for the survey of these non- random behaviors, see, e.g., [19]). The VT method is able both to detect non- randomness and also to describe its form (see [16, 17,20,34,35,59,62,63,73,74,76] for the astronomical context). 2.2 Minimal spanning tree (MST) Unlike VT, this method considers the distances (edges) among the points (vertices). Clearly, there are N(N −1)/2 distances among N points. A span- ning tree is a system of lines connecting all the pointswithout any loops. Theminimal spanning tree (MST) is a systemof connecting lines,where the sum of the lengths is minimal among all the possible con- nections between the points [40,58]. In this paper, the spherical version of MST is used following the original paper by Prim [58]. The N − 1 separate connecting lines (edges) to- gether define the minimal spanning tree. The statis- tics of the lengths and the αMST angles between the edges at the vertices can be used for testing the ran- domness of the point pattern. The MST is widely used in cosmology for studying the statistical prop- erties of galaxy samples [1,6–8,21,41]. Fig. 1: Application of Voronoi tesselation to short GRBs (Short1 sample) in the 0.65 < P256 < 2.00 peak flux range in Galactic coordinates. The peak-flux is given in dimension photon/(cm2s) 18 Acta Polytechnica Vol. 52 No. 2/2012 Fig. 2: MST for the sample in Figure 1 Fig. 3: MFR spectra of simulated (dot-dashed), Long1 (dashed), Short1 (dotted) and Short2 (three-dot-dashed) samples. Boxes represent the error of the spectrumpoints derived from Monte Carlo simulations. Note the shift of the maximum of the spectrum of the Short1 sample to- wards higher values in comparison to α =2, correspond- ing to the completely random 2D Euclidean case 2.3 Multifractal spectrum Let P(ε) denote the probability for finding a point in an area of ε radius. If P(ε) scales as εα (i.e. P(ε) ∝ εα), then α is called the local fractal dimen- sion (e.g. α =2 for a completely random process on the plane). In the case of a monofractal α is inde- pendent from the position. Amultifractal (MFR) on a point process can be defined as unification of the subsets of different (fractal) dimensions [52]. One usually denotes with f(α) the fractal dimension of the subset of points atwhich the local fractal dimen- sion is in the interval of α, α +dα. The contribution of these subsets to the whole pattern is not neces- sarily equally weighted, in practice, it depends on the relative abundances of subsets. The f(α) func- tional relationship between the fractal dimension of subsets and the corresponding local fractaldimension is called the MFR or Hausdorff spectrum. In the vicinity of the i-th point (i = 1,2, . . . , N) one canmeasure from the neighbourhood structure a local dimension αi (“Rényi dimension”). This mea- sure approximates the dimension of the embedding subset, giving the possibility to construct the MFR spectrum which characterizes the whole pattern (for more details see [52]). If themaximumof this convex spectrum is equal to the Euclidean dimension of the space, then in the classical sense the pattern is not a fractal, but the spectrum remains sensitive to the non-randomness of the point set. The concept of a multifractal can be successfully applied in astronomical problems [2,9,11–13,18,22, 25,26,36,37,53,54,60,61,65–67]. 3 Evaluation of statistical tests The three procedures outlined in Section 2 enable us to derive several stochastic quantities well suited for testing the non-randomness of the underlying point patterns. 3.1 Input data and sample definition Until now the most comprehensive all-sky survey of GRBswas done by the BATSE experiment on board theCGROsatellite in the period from1991–2000. In this period the experiment collected 2704 well jus- tified burst events, and the data is available in the Current BATSE Catalog [44]. Since there is increasing evidence ( [31] and ref- erences therein) that the GRB population is actu- 19 Acta Polytechnica Vol. 52 No. 2/2012 ally a mixture of astrophysically different phenom- ena, we divided the GRBs into three groups: short (T90 < 2s), intermediate (2s ≤ T90 ≤ 10s) and long (T90 > 10s). To avoid problems with a changing de- tection threshold, we omitted GRBs having a peak flux P256 ≤ 0.65 photonscm−2 s−1. This truncation was proposed in [56]. The burstsmay emerge at very different distances in the line of sight and itmayhap- pen that the stochastic structure of the angular dis- tribution depends on it. Therefore, we also made tests on the bursts with P256 < 2 photonscm −2 s−1 in the short and long population, separately. Table 1 defines the 5 samples to be studied here. (Due to the small number of intermediate bursts, this subsample was not divided into faint and bright parts). Table 1: Tested samples of BATSE GRBs Sample Duration Peak flux Number [s] [photonscm−2s−1] of GRBs Short1 T90 < 2 s 0.65 < P256 < 2 261 Short2 T90 < 2 s 0.65 < P256 406 Intermediate 2s ≤ T90 ≤ 10s 0.65 < P256 253 Long1 T90 > 2s 0.65 < P256 < 2 676 Long2 T90 > 10s 0.65 < P256 966 3.2 Definition of the test-variables The randomness of the point field on the sphere can be tested with respect to various criteria. Since dif- ferentnon-randombehaviorsare sensitive todifferent types of criteria of non-randomness, it is not neces- sary that all possible tests using different measures reject the assumption of randomness. In the follow- ing, we define several test-variables which are sensi- tive to different stochastic properties of the underly- ing point pattern, as proposed by [72]. Any of the quantities characterizing the Voronoi cell, i.e. area A, perimeter P , number of vertices Nv, cell chord length C, and inner angles αi can be used as test-variables, and even some combinations of these quantities. We defined the following test- variables: – Cell area A; – Cell vertex (edge) Nv; – Cell chords length C; – Inner angle αi; – Round factor (RF) average RFav =4πA/P ; – Round factor (RF) homogeneity 1− σ(RFav) RFav ; – Shape factor A/P2; – Modal factor σ(αi)/Nv; – The so-called “AD factor” defined as AD = 1 − (1 − σ(A)/〈A〉)−1, where σ(A) is the dis- persion and 〈A〉 is the average of A. To characterize the stochastic properties of a point pattern, we use three quantities obtained fromMST: – Variance of the MST edge-length σ(LMST); – Mean MST edge-length LMST ; – Mean angle between edges αMST . As to the multifractals, the only variable used here is the f(α) spectrum, which is a sensitive tool for testing the non-randomness of a point pattern. Throughout defining these variables mean (average) and variance refer to the mean and variance of the respective elements of the Voronoi foam and MST, respectively. An important problem is to study the sensitivity (discriminant power) of the different parameters to the different kind of regularity inherent in the point pattern. In the case of a fully regular mesh, e.g., A is constant and so AD = 0, σ(αi) = 0 and both in- crease towards a fully random distribution. In the case of a patchy pattern the distribution of the area of theVoronoi cells and the edgedistribution ofMST become bimodal, reflecting the average area and the edge lengthwithin and between the clusters, in com- parison to the fully random case. In a filamentary distribution, the shape of the areas becomes strongly distorted, reflecting in an increaseof themodal factor in comparison to the case of patches. [71] investigated the power of Voronoi tessela- tion and the minimal spanning tree in discriminat- ing between distributions having big and small clus- ters, full randomness and hard cores (random distri- butions, but the mutual distances of the points are constrainedby the sizeof thehardcore), respectively. They concluded that the Voronoi roundness factor did not separate small clusters and hardcore distri- butions, and the roundness factor homogeneity did not distinguish between small clusters and random distributions, nor between random and hardcore dis- tributions. MST has a very good discriminant power even in the case of hardcore distributions with close minimal interpoint distances. Since the sensitivity of the variables differs on changing the regularity properties of the underlying point patterns one may measure significant differ- ences in one parameter but not in another, even in cases when these are correlated otherwise. This is not a trivial issue. Inmost cases, one needs extended numerical simulations to study the statistical signifi- cance of the different parameters. 3.3 Estimation of significance To obtain the empirical distributions of the test- variables we made 200 simulations for each of the five samples. The number of simulated points was identical to the number for the samples defined in Section 3.1. Wegenerated the fully randomcatalogsbyMonte Carlo (MC) simulations of fully random GRB celes- tial positions and taking into account the BATSE sky-exposure function [23,44]. 20 Acta Polytechnica Vol. 52 No. 2/2012 Table 2: Calculated significance levels for the 13 test-variables and the five samples. A calculated numerical signifi- cance greater than 95% is put in bold face Name var short1 short2 interm. long1 long2 Cell area A 36.82 29.85 94.53 79.60 82.59 Cell vertex (edge) Nv 36.82 87.06 2.99 26.87 7.96 Cell chords C 47.26 52.24 18.91 84.58 54.23 Inner angle αi 96.52 21.39 87.56 37.81 63.18 RF average 4πA/P 65.17 99.98 33.83 10.95 86.07 RF homogeneity 1− σ(RFav) RFav 19.90 24.38 58.71 55.72 32.84 Shape factor A/P2 91.04 94.03 90.05 55.22 63.68 Modal factor σ(αi)/Nv 97.51 1.99 7.46 56.22 8.96 AD factor 1− ( 1− σ(A) 〈A〉 )−1 32.84 25.37 11.44 95.52 52.74 MST variance σ(LMST) 52.74 38.31 22.39 13.93 59.70 MST average LMST 97.51 7.46 89.05 56.72 8.96 MST angle αMST 85.07 14.43 36.82 73.63 60.70 MFR spectra f(α) 95.52 96.02 98.01 73.63 36.32 Squared Euclidean distance 99.90 99.98 98.51 93.03 36.81 Assuming that the point patterns obtained from the five samples defined in Table 2 are fully random, we calculated the probabilities for all the 13 test- variables selected in Section 3.2. Based on the sim- ulated distributions, we computed the level of signif- icance for all the 13 test-variables and in all defined samples. 4 Discussion of the statistical results 4.1 Evaluation of the joint significance levels We assigned to each MC simulated sample 13 values of the test variables and, consequently, a point in the 13Dparameter space. Completing 200 simulations in all of the subsampleswe get in thiswaya 13Dsample representing the joint probability distribution of the 13 test-variables. Using theEuclideandistance of the points from the sample meanwe can get a stochastic variable characterizing the deviation of the simulated points from the mean only by chance. An obvious choice would be the squared Euclidean distance. In case of a Gaussian distribution with unit vari- ances and without correlations, this would result in a χ2 distribution of 13 degrees of freedom. The test- variables in our case are correlated and have differ- ent scales. Before computing squared Euclidean dis- tances we transformed the test-variables into non- correlated ones with unit variances. Due to the strong correlation between some of the test-variables we may assume that the observed quantities can be represented with non-correlated variables of lower number. Factor analysis (FA) is a suitable way to represent the correlated observed variableswith non- correlated variables of lower number. Since our test-variables are stochastically depen- dent, following [72] we attempted to represent them by fewer non-correlated hidden variables, assuming that Xi = k∑ j=1 aij fj + si i =1, . . . , p k < p . (1) In the above equation Xi, fj , si mean the test- variables (p = 13 in our case), the hidden variables and a noise-term, respectively. Equation (1) repre- sents the basic model of FA. The covariance matrix of the Xi variables has 1 2 p(p +1) free elements. The number of free elements on the right side of Eq. (1) cannot exceed this value [38], yielding for k the fol- lowing inequality: k ≤ (2p +1− √ 8p +1)/2 (2) which gives k ≤ 8.377 in our case. Factor analysis is a common ingredient of profes- sional statistical software packages (BMDP, SAS, S- plus, SPSS1, etc). Thedefault solution for identifying 1BMDP, SAS, S-plus, SPSS are registered trademarks 21 Acta Polytechnica Vol. 52 No. 2/2012 the factor model is to perform principal component analysis (PCA). We kept as many factors as were meaningfulwith respect toEquation (2). Taking into account the constraint imposed by Equation (2) we retained 8 factors. In this waywe projected the joint distribution of the test-variables in the 13D parame- ter space into an 8D parameter space defined by the non-correlated fi hidden variables. The fj hidden variables in Equation (1) are non- correlated and have zero means and unit standard deviations. Using these variables, we defined the fol- lowing squared Euclidean distance from the sample mean: d2 = f21 + f 2 2 + . . . + f 2 k (k =8 in our case). (3) If the fj variableshad strictlyGaussiandistributions, Equation (3) would define a χ2 variable of k degrees of freedom. Sq. distance F re q u e n cy 0 5 10 15 20 25 30 35 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 short2 short1 interm. long1 long2 Fig. 4: Distribution of the squared Euclidean distances of the simulated samples from the stochastic mean of the fi hidden variables (factors) in the 8D parameter space. There are altogether 1000 simulated points. The full line marks a χ2 distribution of 8 degrees of freedom, normal- ized to the sample size. The distances of the BATSE samples are also indicated. The departures of samples “short1” and “short2” exceed all those of the simulated points. The probabilities that these deviations are non- random equal 99.9% and 99.98%, respectively 4.2 Interpretations of the statistical results Using the distribution of the squared Euclidean dis- tances, defined by Equation (3), one can get further informationonwhether aBATSEsample represented byapoint in theparameter spaceof the test-variables deviates only by chance, or whether it differs signifi- cantly from the fully random case. In all categories (short1, short2, intermediate, long1, long2) we made 200, altogether 1000, simu- lations. We calculated the d2 squared distances for all simulations and compared them with those of the BATSE samples in Table 1. Figure 4 shows a his- togramof the simulated squareddistances alongwith those of the BATSE samples. A full line represents a χ2 distribution of k = 8 degrees of freedom. Fi- gure 4 clearly shows that the departures of samples short1 and short2 exceed all those of the simulated points. The probabilities, that these deviations are non-random, equal 99.9% and 99.98%, respectively. The full randomness of the angular distribution of the long GRBs, in contrast to the regularity of the short and to some extent to the intermediate ones, points towards the differences in the angular distribution of their progenitors. The recent discov- ery of the afterglow in some short GRBs indicates that these events are associated with the old stellar population [24] accounted probably for the mergers of compact binaries, in contrast to the long bursts resulting from the collapse of verymassive stellar ob- jects in young star forming regions. The differences in progenitors also reflects the differences between the energy released by the short and long GRBs. As [33] showed, the redshift distributions of the dif- ferentGRBclasses aredifferent. Theaverage z of the short bursts is significantly smaller than that of the long ones (the average redshift of intermediate bursts is between them). Consequently, the short and long GRBs experience different sampling volumes. The sampling volume of the short bursts is much smaller and the irregularities in the distribution of the host galaxies plays a more significant role. Unfortunately, little can be said on the physical nature of the intermediate class. Statistical studies ([31] and the references therein) suggest the existence of this subgroup — at least from the purely statisti- calpoint ofview. Also anon-randomskydistribution occurs here. However, its physical origin is still fully open yet [31]. 5 Summary and conclusions We has made additional studies on the degree of randomness in the angular distribution of samples selected from the BATSE Catalog. According to the T90 durations and the P256 peak fluxes of the GRBs in the Catalog we defined five groups: short1 (T90 < 2s & 0.65 < P256 < 2), short2 (T90 < 2s & 0.65 < P256 ), intermediate (2 s ≤ T90 ≤ 10 s & 0.65 < P256), long1 (T90 > 2s & 0.65 < P256 < 2) and long2 (T90 > 10s & 0.65 < P256). To characterize the statistical properties of the point patterns given by the samples, we defined 13 test-variables based on Voronoi tesselation (VT), a minimal spanning tree (MST) and a multifrac- 22 Acta Polytechnica Vol. 52 No. 2/2012 tal spectra. For all five defined GRB samples we made 200 numerical simulations, assuming fully ran- domangular distribution and taking into account the BATSE exposure function. The numerical simula- tions enabled us to define the empirical probabilities for testing the null hypothesis, i.e. the assumption that the angular distributions of theBATSE samples are fully random. Since we performed 13 single tests simultane- ously on each subsamples, the significance obtained by calculating it separately for each test cannot be treated as a true indication for deviating from the fully random case. In fact, some of the test-variables are strongly correlated. To concentrate the infor- mation on the non-randomness experienced by the test-variables, we assumed that they can be repre- sented as a linear combination of non-correlated hid- den factors of lower number. Actually, we estimated k = 8 as the number of hidden factors. Making use of the hidden factors we computed the distribution of the squared Euclidean distances from the mean of the simulated variables. Comparing the distribu- tion of the squared Euclidean distances of the simu- lated samples with the BATSE samples we concluded that the short1, short2 groups deviate significantly (99.90%, 99.98%) from full randomness, but this is not the case for the long samples. For the intermedi- ate group, squared Euclidean distances also give sig- nificant deviation (98.51%). Acknowledgement This study was supported by OTKA grant No. K077795, by Grant Agency of the Czech Republic grant No. P209/10/0734, and by Research Program MSM0021620860 of the Ministry of Education of the Czech Republic (A.M.). References [1] Adami,C.,Mazure,A.: A&AS,134, 393, 1999. [2] Aschwanden, M. J., Parnell, C. E.: ApJ, 572, 1048, 2002. [3] Balázs, L. G., Mészáros,A., Horváth I.: A & A, 339, 1, 1998. [4] Balázs, L. G., Mészáros, A., Horváth, I., Vavrek, R.: A & A Suppl., 138, 417, 1999. [5] Balázs, L. G., Bagoly, Z., Horváth, I., Mészá- ros, A., Mészáros, P.: A & A, 401, 129, 2003. [6] Barrow, J. D., Bhavsar, S. P., Sonoda, D. H.: MNRAS, 216, 17, 1985. [7] Bhavsar, S. P., Lauer, D. A.: Examining the big bang and diffuse background radiations, In Kafatos, M. C. Kondo, Y. (eds.) Proc. IAU Symp. 168, Dordrecht : Kluwer, p. 517, 1996. [8] Bhavsar, S. P., Splinter, R. 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