Acta Polytechnica https://doi.org/10.14311/AP.2025.65.0009 Acta Polytechnica 65(1):9–15, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague THE TWO-POINT CORRELATION FUNCTION OF THE GRBS Zsolt Bagolya,b a University of Public Service, Department of Natural Sciences, 2 Ludovika tér, H-1083 Budapest, Hungary b Eötvös University, Faculty of Science, Department of Physics of Complex Systems, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary correspondence: bagoly.zsolt@uni-nke.hu Abstract. We analysed the spatial distribution of 542 GRBs with measured position and spectroscopic redshift up to 31 Aug 2022. Using kernel smoothing, we determined the GRB’s Sky Exposure Function and used it in the generation of random catalogues. The spatial Two-Point Correlation Function for GRBs was determined by partitioning the data based on the origin of the redshift (afterglow or host galaxy). The resulting ξ(r) Two-Point Correlation functions remain below the 3σ noise level, suggesting no significant differences between the Two-Point Correlation functions of the random and real datasets. Keywords: Data analysis, gamma-ray bursts. 1. Introduction Gamma-ray bursts (GRBs) may be detected up to extermely high redshifts, making them excellent for studying large-scale structures. The Cosmological Principle states that the Universe is spatially homo- geneous and isotropic on the large-scale. Usually we assume that GRBs follow the distribution of baryonic matter, so GRBs can test the distribution of baryonic matter in the Universe, particularly at large scales. GRBs are assumed as one of the most powerful and extremely bright events the universe, caused by massive star bursts [1, 2] or binary compact object mergers [3]. Short GRBs are likely produced by the merger of compact objects, such as neutron stars or black holes, as proven by kilonova observations [4] and the GRB170817A event [5]. Long GRBs have been linked to collapsing huge star objects [6]. Therefore, long-duration GRBs are thought to arise from hyper- novae in active star formation zones, indicating a close link between the two. GRBs are more abundant in low-metallicity, early star-forming areas of galaxies. Low metallicity influences stellar winds, allowing large stars to maintain more mass until exploding. GRBs may be viewed from a long distance, making them useful for studying star formation in the early cosmos. The primary GRB groupings, notably the short and the long one, display distinct sky distributions, as demonstrated by the early CGRO BATSE observation of anisotropic detections [7–13]. The sky exposure is important because any association between the sky distribution and the physical parameters of the GRBs are quite interesting. There is evidence for a third intermediate group [14– 17], and the classification can be extended using dif- ferent parameters and spectral models [18]. The sky distribution of 1 669 Fermi/GBM GRBs was used by [19] for the two-point angular correla- tion tests, closest neighbour, fractal dimension, dipole and quadrupole analysis, and binomial test. The re- sults demonstrated that, with a probability of 99.98 %, short GRBs are dispersed anisotropically in the sky, but long GRBs exhibited no anisotropy. Using the two-point correlation function and closest neighbour tests, the large-scale homogeneous and isotropic distri- bution of 361 GRBs with known position and redshift was verified in [20]. [21] used conditional density and pairwise distance techniques to estimate a fractal di- mensionality1 of D = 2.55 on scales of 2–6 Gpc. [22] examined the isotropy of the sky distribution of 2 626 GRBs from the FERMI-GBM archive. For both the long and short GRBs, the applied two-point angular correlation function was unable to identify any statistical anisotropy due to the significant posi- tional uncertainty in the locations. Using prompt and afterglow parameters, [23] examined several intrinsic aspects of 6 289 GRBs and looked for relationships between the parameters and the GRBs’ categorisation. [24] tested the correlation between GRBs’ sky loca- tions, durations, fluences, and peak fluxes observed in various energy ranges using the Fermi GBM GRB observations. [25] added the BATSE and Swift BAT GRBs to the data. This study revealed no relationship between the GRBs’ physical characteristics and their sky locations. The relationship between the durations of the GRBs and redshifts/radial distances was re- cently examined by [26], who also discovered no link between them. These findings support the theory that the phys- ical characteristics of GRBs are unaffected by their location throughout the universe. Data from 542 GRBs up to 31 August 2022, were used in this analysis. The GRBs were mostly observed by NASA’s Swift and/or Fermi spacecraft, and the 1Fractal dimensionality describes the complexity and scaling properties of fractals, which are self-similar patterns exhibiting self-similarity or hierarchical structure. 9 https://doi.org/10.14311/AP.2025.65.0009 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Zsolt Bagoly Acta Polytechnica spectroscopic redshifts were obtained with a variety of redshift observations. The primary data source was In- dex (GRBOX) database [27], however, the GCN data were also directly used. Jochen Greiner’s publicly ac- cessible dataset [28] was cross-checked to the data as well. Since photometric redshifts and redshift estima- tions have significant errors that reach several hundred of Mpcs, the study exclusively employed spectroscopic redshifts only. A distinction was made according to the origin of the redshift (optical afterglows or host galaxy measurements) as the host galaxies’ distances are lower due to observational constraints. 2. The two-point correlation function A statistical technique used in cosmology to measure the large-scale features and the spatial distribution of galaxies and other sources is the two-point correlation function. It quantifies the extra likelihood of discov- ering two sources spaced apart by a specific distance when compared to a random distribution. The ξ(r) two-point correlation function, is defined as the excess probability over random of finding two sources separated by a distance r: dP = n[1 + ξ(r)]dV, (1) where dP is the probability of finding a pair of galaxies separated by r, n is the mean number density of galaxies, dV is the volume element. At a distance r, the distribution of galaxies is ran- dom if ξ(r) = 0. When ξ(r) is positive, it indicates clustering by a larger likelihood of finding galaxies separated by r than at random. If ξ(r) is negative then there is a lesser likelihood than random to locate galaxies separated by r, suggesting voids. Since the two-point correlation function is the inverse Fourier transform of the power spectrum, both forms charac- terise point distributions similarly. 3. Large-scale anomalies 3.1. Compton gamma-ray observatory observations The Compton Gamma-ray Observatory BATSE dataset [29] has been the subject of numerous stud- ies, including the first indirect observational proof for the cosmological origin of GRBs. After the first two years of BATSE observations, the observed GRBs showed an almost isotropic distribution without any concentration around the Galactic plane [30]. The results strengthened the cosmological origin model of the sources. Using angular positional measurements from the BATSE experiment, [7] found that the sky distributions of short and long GRBs are different, with a probability of 99.97 %. The anisotropic sky exposure was considered, and the effects were found to be small. However, this anisotropic sky exposure complicated the detailed statistics. The short-distance sky distribution of GRBs is anisotropic, with p = 0.00016 level [8, 31]. How- ever, due to the BATSE’s anisotropic sky exposure, the Two-Point Angular Correlation Function is not uniform too. The effects of sky exposure are small, but the non-uniformity of the sky exposure compli- cates detailed statistics, indicating that it should be specifically addressed [32]. In [10], the distribution of intermediate GRB data was analysed using a modified BATSE count-in-cell approach. The results indicate that the distribution is not isotropic with a 99.3 % confidence level. [11] em- ployed spherical harmonics to study the angular distri- bution of the intermediate group, demonstrating inher- ent anisotropy with a 97 % significance level. In [12], short bursts exhibited a greater 99.99 % significance level of anisotropy, whereas intermediate bursts had a lower 99.89 % significance level. [33] discovered the association between short GRBs and galaxies’ loca- tions in the local universe at a 99.9 % significance level. In [13], the BATSE data revealed that short GRBs depart considerably from complete randomness at a 99.90–99.98 % level, whereas the intermediate group exhibited a lesser but still significant deviation at a 98.51 % level from isotropy. 3.2. Large-scale structures using the spectroscopic redshift data In [34], a large GRB cluster at z ≈ 2 was discov- ered in the direction of Hercules and Corona Bore- alis. The study examined the spatial distribution of 283 GRBs with different redshifts. The sample was divided by z, with the assumption that sky exposure is independent of radial distribution. The dataset was analysed using the kth nearest neighbour and bootstrap point radius techniques. Nearest-neighbour investigations [35] support the existence of this huge, loose GRB cluster in the redshift range 1.6 < z ≤ 2.1, with a p = 1.6 × 10−4 likelihood. Additional data and analysis corroborated the discovery of additional GRBs with measured redshifts [36]. The structure’s actual nature is uncertain. [37] examined the kth nearest neighbour in the GRB sample, after the detection of the Hercules-Corona Borealis Great Wall. To estimate the spatial den- sity of GRBs, instead of the redshift space slices, it used kth Next Neighbour Statistics. The analysis of k = 8, 10, 12, and 14 revealed the Giant GRB Ring, consisting of 9 GRBs with an angular major/minor diametre of 43°/30° at a distance of ≈ 2 770 Mpc in the 0.78 < z < 0.86 redshift range, with a probability of 2 × 10−6 of being a random fluctuation [38]. 4. The sky exposure function The redshift-determined GRB detection probability in the sky is a multi-parameter event that relies on 10 vol. 65 no. 1/2025 The two-point correlation function of the GRBs 0 2 4 6 8 10 12 -80 -60 -40 -20 0 20 40 60 80 0 10 20 30 40 50 60 month Galactic b # of G R B s w ith m ea su re d re ds hi ft an d la tit ud e < b Figure 1. Monthly variances of the cumulative distribution of the successful spectroscopic observations in the current sample. Dedicated optical campaigns in October in the last few years are raising the orange line above the expectation – this is a clear outlier. different space instrument trigger, and detection cir- cumstances as well as the optical follow-up protocol. Satellite pointing records can be used to generate the most basic sky exposure models, however this is not very practical owing to both technological and human considerations. The information available indicates that developing a robust model for selection effects, encompassing the influence of galactic extinction and other optical observational restrictions and effects, is essentially unfeasible. In Figure 1 the yearly variances of the spectroscopic observations are shown. The cumulative number den- sity is plotted for each month against the galactic latitude. One can observe the yearly changes with the exception of October. With the exception of the afterglow’s October distri- bution, the distribution of afterglow and host galaxy measurements roughly follow the same pattern with very comparable annual modulation. The additional 10–15 observations are likely caused by optical cam- paigns in the last few years. Since the synthetic method is not practical, we must calculate the empirical Sky Exposure Function using simply the observational data. This is not in contradiction since the dataset-averaged sky detec- tion probability can be estimated, provided that the sky distribution is assumed to be independent of red- shift [20, 39]. 5. Kernel smoothing A basic non-parametric technique for estimating re- gression and probability density functions in statistics is kernel smoothing. Kernel smoothing is versatile and broadly applicable, as it avoids assuming a specific underlying distribution for the data, unlike parametric approaches. Selecting the kernel function is frequently less important than selecting the bandwidth. The Gaussian, Epanechnikov, and uniform kernels are fre- quently used kernel functions [40]. Using several methods, the optimum empirical Sky Exposure Function was rebuilt from the point distri- bution in [39]. On a random field that imitated the Swift’s exposure map, the effectiveness of the fixed and adaptive width Gaussian kernel, the Delaunay Tessellation Field Estimator [41], and the Voronoi Dia- gram Field Estimator techniques [42] were tested. The fixed-width Gaussian kernel smoothing was shown to be the best approach for both estimating the exposure map and figuring out the ideal width. Kernel-based techniques were used in [20, 43, 44] to reconstruct the empirical Sky Exposure Function of the GRBs. To get the empirical Sky Exposure Function, we smoothed the current GRB data using Gaussian kernels. The ideal kernel-smoothed Sky Ex- posure Function is shown in Figure 2. The galactic disc is visible, and the difference between the mini- mum and maximum is a factor of ≳ 5. The typical 20–40° range [39] contains the optimal kernel sizes for the usual few hundreds of GRBs, which are obvi- ously larger than the galactic optical extinction range width. As a result, kernels will provide a smoother sky exposure than the real ones, which will reduce the statistical power of detecting structures close to and below the kernel size. 6. The spatial Two-point Correlation Function The spatial Two-point Correlation Function ξ(r) was determined using the earlier estimated empirical Sky 11 Zsolt Bagoly Acta Polytechnica Figure 2. The empirical Sky Exposure Function of the GRBs reconstructed with optimal Gaussian smoothing, FWHM = 2221’. Normalised units are applied. -1 -0.5 0 0.5 1 2000 4000 6000 8000 10000 12000 MC simulations with 3σ errors all GRBs with redshift No rm al iz ed sp at ia l tw o- po in t co rr el at io n fu nc ti on Comoving distance (1/h Mpc) Figure 3. The spatial two-point correlation function for all GRBs. Exposure Function and the radial (comoving distance) distribution of the data. The simple Peebles-Hauser’s estimator [45] of the two-point correlation function ξ(r) is sometimes bi- ased and not well suitable for edge effects and survey boundaries, particularly when there is a large-scale structure present. Therefore, the Landy-Szalay esti- mator [46] was applied as the most effective technique for a lower observational probability and known par- tial vignetting: ξ(r) = DD(r) − 2DR(r) + RR(r) RR(r) , (2) where DD(r), RR(r) and DR(r) are the count of pairs between the data-data, random-random and data and random samples at separation r. The Landy-Szalay estimator minimises variance compared to other esti- mators and corrects for edge effects and the survey’s irregular geometry. For the computations, we used 10 000 random points. To generate Monte Carlo simulations, we additionally employed 100 synthetic datasets with the same marginal distributions as the original data. The Poissonian errors in the computations were found using the obtained ξ(r)’s from the random datasets. A major source of the noise at both ends of the ξ(r) arises from the lower counts, especially the RR(r) in the denomator causes big errorbars. We created here bin sizes which contain approximately an equal number of random points. Hence, by aggregating the number of events in the bins at both ends, it will slightly reduce the noise in the RR(r)−1 function. Still, at large distances, the error bars are quite large. The ξ(r) of the whole dataset is displayed in Fig- ure 3, with the mean value and the ±3σ errors. 12 vol. 65 no. 1/2025 The two-point correlation function of the GRBs MC simulations with 3σ errors -1 -0.5 0 0.5 1 2000 4000 6000 8000 10000 12000No rm al iz ed sp at ia l tw o- po in t co rr el at io n fu nc ti on GRBs with redshift from afterglow Comoving distance (1/h Mpc) Figure 4. The spatial two-point correlation function for GRBs with z from afterglows. -1 -0.5 0 0.5 1 1.5 2 0 1000 2000 3000 4000 5000 6000 7000 8000 GRBs with redshift from host galaxy No rm al iz ed sp at ia l tw o- po in t co rr el at io n fu nc ti on MC simulations with 3σ errors Comoving distance (1/h Mpc) Figure 5. The spatial two-point correlation function for GRBs with z from host galaxies. The ±3σ error lines indicate noise, and the mean of the approximately 100 synthetic datasets should be 0. For the real GRB distribution, the ξ values fall within the errors, and there is no signal at the ±3σ level. We repeated the ξ(r) calculation for the afterglow and host galaxy subsets separately. The matching ξ(r) functions are shown in Figures 4 and 5, along with the related ±3σ lines, respectively. The GRB ξ(r) values are the same within the 3σ errors in both situations. All the resulting ξ(r) two-point correlation functions are consistent with zero. Considering the observations detailed in Section 3 it can indicate that the χ2 opti- mal kernel width over-smooths the real Sky Exposure Function. Our results are similar to those of [20, 26], although we used a slightly larger number of events (n = 542 ver- sus n = 361 and 533) with a correspondingly smaller kernel size. It is important to note that a smoothed sky exposure compared to the actual sky exposure reduces statistical power, making it less effective for detecting structures near or below the kernel size. 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The Astrophysical Journal 412:64, 1993. https://doi.org/10.1086/172900 15 https://heasarc.gsfc.nasa.gov/w3browse/all/batsegrb.html https://heasarc.gsfc.nasa.gov/w3browse/all/batsegrb.html https://doi.org/10.1038/355143a0 https://doi.org/10.1007/0-306-47116-7_14 https://doi.org/10.1063/1.55312 https://doi.org/10.1038/nature04310 https://doi.org/10.1051/0004-6361/201323020 https://doi.org/10.1051/0004-6361/201424829 https://doi.org/10.1093/mnras/staa2460 https://doi.org/10.1093/mnras/stv1421 https://doi.org/10.1093/mnras/stx2550 https://doi.org/10.22323/1.233.0060 https://doi.org/10.48550/arXiv.astro-ph/0011007 https://doi.org/10.1051/0004-6361/201525736 https://doi.org/10.1093/mnras/stw559 https://doi.org/10.1086/173288 https://doi.org/10.1086/172900 Acta Polytechnica 65(1):9–15, 2025 1 Introduction 2 The two-point correlation function 3 Large-scale anomalies 3.1 Compton gamma-ray observatory observations 3.2 Large-scale structures using the spectroscopic redshift data 4 The sky exposure function 5 Kernel smoothing 6 The spatial Two-point Correlation Function Acknowledgements References