Acta Polytechnica https://doi.org/10.14311/AP.2025.65.0001 Acta Polytechnica 65(1):1–8, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague THE SPATIAL DISTRIBUTION OF GRBS Zsolt Bagolya,b,∗, Lajos G. Balazsd,e, Zsuzsa Horvathc, Istvan Horvatha 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 c University of Public Service, Institute of Disaster Management, 2 Ludovika tér, H-1083 Budapest, Hungary d Eötvös University, Department of Astronomy, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary e HUN-REN Research Centre for Astronomy and Earth Sciences, Konkoly Thege Miklós Astronomical Institute, Konkoly-Thege Miklós út 15-17, H-1121 Budapest, Hungary ∗ corresponding author: bagoly.zsolt@uni-nke.hu Abstract. We analysed the different aspects of the spatial distribution of 542 Gamma-Ray Bursts with precisely determined positions and spectroscopic redshifts. The data were divided according to the origin of the redshift (afterglow or host galaxy). The yearly rate of afterglow and host-based redshift observations are different, with only a few host observations in the recent years. Since the launch of the Swift, the rate of afterglow observations fall exponentially by 50 % in 15 years, potentially affecting all planned GRB missions. We also analysed the rest-frame T90 values from the Swift BAT and FERMI GBM catalogues with the redshift data. The host- and afterglow-based points are separated in the redshift range due to observational effects, but no direct distinction could be made between the rest-frame T90 values. The correlation analysis between the GRB redshift and sky position shows that the GRB distribution could be factorised into a separate sky and radial components. Keywords: Data analysis, gamma-ray bursts. 1. Introduction Gamma-ray bursts (GRBs) are ideal candidates for probing large-scale structures due to their ability to detect up to very high redshifts. According to the Cosmological Principle, the Universe is spatially ho- mogeneous and isotropic on a large scale, and we believe that GRBs follow the distribution of baryonic matter. This allows us to use GRBs to test the bary- onic matter’s distribution in the Universe, especially the large-scale structure. GRBs are assumed as one of the most powerful and extremely bright events in the universe, which are caused by a burst of massive stars [1, 2] or the merging of binary compact objects [3]. Therefore, the link between star formation events and long duration GRBs is thought to be strong, as they are thought to originate from hypernovae originated from regions of active star formation. Observations show that GRBs are more common in the early star-forming regions of galaxies (e.g. [4–6]), in low-metallicity environments. This low metallicity affects stellar winds, allowing massive stars to retain more mass until they explode. Since GRBs can be observed at great distances, they also act as probes for investigating star formation in the early universe. The primary GRB groupings, notably the short and the long one, display diverse sky distribution, according to the early CGRO BATSE observations anisotropy investigations [7–13]. A huge GRB cluster at z ≈ 2, located in the di- rection of Hercules and Corona Borealis, has been identified in [14], where 283 GRBs with redshifts were used to study their distribution. The GRB sample was split by z based on the assumption that sky ex- posure is independent from the radial distribution. The k-th closest neighbour analysis and the boot- strap point radius technique was used on the dataset. Nearest-neighbor studies confirm the previously dis- covered massive, loose GRB cluster in the redshift range 1.6 < z ≤ 2.1, with a p = 1.6×10−4 chance [15]. The original discovery was later supported by fur- ther data and analysis as new GRBs’ redshift were observed [15, 16]. The exact nature of the structure remains unclear. [17] further investigated the k-th nearest neighbor in the GRB sample, inspired by the Hercules-Corona Borealis Great Wall. Here, the GRBs’ spatial density was approximated using the k-th Next Neighbour Statistics, rather than redshift space slices. They discovered that for k = 8, 10, 12, and 14, the analysis revealed the Giant GRB Ring, consisting of 9 GRBs with an angular major/minor diameter 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. [18] analysed the spatial point processes of GRBs with known redshifts with kernel smoothing. They concluded that the occurrence of the Giant GRB Ring is a low-probability random 1 https://doi.org/10.14311/AP.2025.65.0001 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Z. Bagoly, L. G. Balazs, Z. Horvath, I. Horvath Acta Polytechnica event, as only three ring-like patterns were revealed from 1 502 random datasets. [19] investigated the distribution of starburst galaxies from the Millennium XXL simulation at z = 0.82 distance, but the actual origin of the Giant GRB Ring is still unclear. [20] analysed the FERMI GBM data for sky distri- bution isotropy. The two-point angular correlation function was unable to identify statistical anisotropy for the both long and short GRBs due to a signifi- cant positional uncertainty. In [21], 6 289 GRBs were studied for their intrinsic features, including prompt and afterglow metrics, to determine relationships with GRB categorisation. [22] and [23] used Platinum GRB data compilation to standardise events in the Dain- otti correlation space, resulting in GRB cosmological parameter limits consistent with BAO data. Also, the Fermi GBM GRB catalogue was used by [24] to evaluate the correlation between the sky locations of GRBs and their durations, fluences, and peak fluxes. [25] added the BATSE and Swift BAT GRBs to the data. The results of the studies reveal no connection between the GRBs’ physical characteristics and their places in space. The GRB clustering might have several astrophysi- cal causes. Because of the extended inspiraling dura- tion of the pair, short GRBs are not predicted to trace star-forming activity, but the low metallicity massive stars are thought to represent a long GRB progenitor. [26] analysed a Swift GRB sub-sample composed by 58 bursts with redshifts, favourable observing condi- tions, and 1-s peak fluxes above 2.6 pg s−1 cm−2. They found that strong evolution either in luminosity or in density is required to describe the sample. [27] using the same sample found that the GRB formation rate increases with up to z ≈ 2, then drops with the star formation rate. However, [28] asserts, based on the findings of the Swift GRB Host Galaxy Legacy Survey, that long GRBs are not directly a trace of star for- mation as metallicity is the key factor the production efficiency. Using the Millennium Simulation [29], [17] showed that galaxies with a typical star creation rate compared to those with a high star formation rate likely have distinct spatial distributions. Therefore, theories regarding a wave or variation in the rate of star creation are too simplistic to fully capture the picture. A possible origin might be genuine cosmic anisotropy. For instance, [30] examined anisotropic cosmic expansions using electrodynamics and fore- casted changes in the polarisation of electromagnetic radiation in relation to these areas. In order to es- tablish scaling relations, [31] examined 570 clusters using at least X-ray, microwave, and infrared observa- tions. They found an apparent 9 % dipole-like spatial fluctuation in the local H0 at (l, b) = (280, 15) on the sky using all of the distance data that was available. Another explanation for the outcome might be a bulk flow of 900 km s−1. 2. Data selection This paper uses different redshift observation of GRBs, most of which were triggered by NASA’s Swift and/or Fermi satellites. All of them has precise spectroscopic redshifts originated from either optical afterglows or host galaxy measurements, with the corresponding investigations and locations on the celestial sphere. We used the spectroscopical redshifts of 542 GRBs up to 31st August 2022 from [16]. It has been extended with the data of [32] and [33] and the Gamma-Ray Burst Online Index (GRBOX) database. GRBOX also relies on the relevant Gamma Ray Burst Coordi- nation Network (GCN) reports – here the GCN data were also used directly. A publicly available dataset compiled by Joachim Greiner, which provides exten- sive information on nearly all GRBs observed by any instrument was also used in addition to the GRBOX and were also cross-checked with our data. Here, we used only the observations with spectro- scopic redshifts, as photometric redshifts and redshift estimates (e.g. based on Ly-alpha limits) have large redshift (radial distance) uncertainties, exceeding sev- eral hundred Mpcs. Figure 1, the galactic distribution of these 542 GRBs is shown. The colours are corresponding to the ori- gin of the redshift (afterglow or host galaxy spec- troscopy). There are 262/280 GRBs in the north- ern/southern Galactic hemisphere, and it is worth to mention that there are no visible difference between host- or afterglow-based positions. 3. Difference between host- and afterglow-based z 3.1. Redshift observations and their future Using the collected data, one can determine the yearly variances of the redshift observations. The successful spectroscopic observations will be provided either by the rapid afterglow measurements (made within the afterglow dimming timeframe) or the host galaxy’s spectral lines. The host galaxy could be observed later, and usually this is the case. Figure 2, the yearly observation rates for the afterglow- and host-based redshifts and for all spec- troscopic redshift are shown. The launch of the Swift satellite is clearly visible in the 2005’s rate. One can also observe that the rates of the afterglow- and host-based redshift observations are different: the peak in the host-based observations are dropping, only a few GRBs were observed in the last years. Without dedicated observations there’s a low chance to reverse this trend: these measurements require the largest telescopes, where the observing time is limited. Altough determined observers for ground optical follow-ups are still cruical, the new trigger sources of SVOM and Einstein Probe (beside Swift and Fermi) will hopefully expand the trigger alerts’ 2 vol. 65 no. 1/2025 The spatial distribution of GRBs 360 240 120 +90 -90 0 z from host galaxy z from afterglow Figure 1. Sky distribution of 542 GRBs with measured redshift in galactic coordinates up to 31st August 2022. The disk of the Galaxy is clearly visible, it partially prevents the optical follow-up activity. 0 10 20 30 40 50 60 1995 2000 2005 2010 2015 2020 2025 2030 2035 2040 nu mb er o f GR Bs w it h ph ot om et ri c re ds hi ft Year all z from host galaxy z from afterglow approximation TH ES EU S Figure 2. Yearly observation rates for the afterglow- and host-based redshifts and for all spectroscopic redshifts. Observation rate is clearly different for host- and/or afterglow-based redshifts. The exponential drop with a ≈ 15-year halving time means that during the THESEUS mission, only ≈ 5 redshifts are expected yearly. rate, and planned missions like Theseus will further extend it in the next 10–15 years. The drop in the number of the afterglow redshifts are clearly visible too: in ≈ 15 years, the number of observations decreased by ≈ 50 %. This lost inter- est in the GRB redshift determination could lead to a problem for all planned GRB missions, e.g. for Theseus the current estimation is ≈ 5 redshifts per year, clearly a critical situation endangering the mis- sion’s scientific output. This exponential decline in observer interest could be partially offset by rapid response/communication mechanisms: e.g. the fact that the succesful Swift XRT and UVOT afterglow detection will increase the chance of a spectroscopic redshift shows that for the success of Theseus, an on-board instrument such as IRT is central. 3.2. Selection effects in the rest-frame T90 – z distribution The short GRBs have a T90 duration of less than 2 sec- onds and have harder spectra compared to long GRBs. They are typically found in regions with low star for- mation rates, including older stellar populations and elliptical galaxies, which also support the view that 3 Z. Bagoly, L. G. Balazs, Z. Horvath, I. Horvath Acta Polytechnica Figure 3. Swift BAT data’s rest-frame T90 – z distribution of the dataset with the corresponding T90 errors (the errors in z are too small for this plot). Figure 4. Fermi GBM data’s rest-frame T90 – z distribution of the dataset with the correspondig T90 errors (the errors in z are too small for this plot). they originate from the merger of compact objects, such as neutron stars or a neutron star and a black hole, The intermediate GRBs have a T90 duration between 2 and 10 seconds, with a soft spectra. The progenitors could be potentially a mix of mechanisms from both short and long GRBs or entirely different processes. The long GRBs have a T90 duration of more than 10 seconds. They have been proposed to be associated with star-forming regions and the collapse of massive stars. The observed locations in galaxies with high rates of star formation, particularly in the arms of spiral galaxies also support this theory. Using the redshift data and the published T90 du- rations from the Swift BAT and FERMI GBM cata- logues one can plot the rest-frame T90 values with the redshifts (Figures 3–4). This method is clearly only the first approximation, considering the various obser- vational effects, e.g. the varying detector background and/or the absence of proper K-correction of a light curve with various fast spectral changes. The two satellites/detectors have different spectrum sensitivities and trigger conditions, therefore, the T90 values for the two figures cannot be directly compared (the relatively few shared observations were studied in detail by [34]). GRBs’ host galaxies have been barely seen above z > 3. The majority of short GRBs are observed by the Swift BAT and they clearly dominate the sub- second part of the plot. 4 vol. 65 no. 1/2025 The spatial distribution of GRBs -4.5 -4 -3.5 -3 -2.5 -2 5 10 15 20 25 30 35 40 z sc or e of Ma nn -W hi tn ey U te st / Wi lc ox on r an k- su m te st Spherical cone/cap radius (deg) MC 99.5% MC 99.0% all GRBs Fa ra wa y GR B Pa tc h Figure 5. 542 GRBs’ lowest Z scores and the MC 99 %/99.5 % estimate lines. It is also apparent that while the restframe T90 values cover the same, wide, range, the host- and afterglow- based observations span across different redshift ranges. As a result, the various GRB types (short, intermediate, and long) ought to be contributed in both plots for the host and the afterglow observa- tions. The figures also show diverse trigger selection effects, such as missing medium/high-z short sources for both satellites. Full consideration of these observational factors is the topic of the ongoing research. 4. The factorisation of the GRB density function Many factors, including geometrical factors, satellite activities, and optical follow-ups, affect the observa- tional probability of a GRB on the sky, and it is an important question how the environment affects the observed GRB distribution. To reconstruct the three-dimensional GRB distribu- tion, it’s important to determine if the sky distribution is independent of redshift/comoving radial distance. This involves evaluating the validity of the f (radial factor) ×g (angular factor) expansion of the density function. We examined this association earlier in [33] by calculating the radial distance distribution of the nearby GRBs inside an α-sized spherical cap surround- ing each GRB. Here, we repeat the test with the newly updated dataset. The Mann-Whitney U test is a nonparametric test used to compare two samples (here the local data within a given spherical cap and the full radial distri- bution). It is not required that the data be normally distributed. The test joins the data from both groups and rank all the values. Using these ranks, it calcu- lates the U statistic first: U1 = n1 · n2 + n1 · (n1 + 1) 2 − R1, U2 = n1 · n2 + n2 · (n2 + 1) 2 − R2, where n1 is the number of observations in group 1, n2 is the number of observations in group 2, R1 is the sum of the ranks for group 1, R2 is the sum of the ranks for group 2. The smaller value between U1 and U2 is used as the test statistic U . The Mann-Whitney U-test’s Z score is used to com- pare the data in the two samples: µU = n1 · n2 2 , σU = √ n1 · n2 · (n1 + n2 + 1) 12 , Z = U − µU σU . Probability was obtained with Monte Carlo mix- ing/randomisation between the radial (redshift) val- ues and the positional data. The Z score distribution for a whole random local dataset were determined. The same procedure was repeated for 1 000 random datasets, each for α = (5−50) degrees radius. Figure 5 shows the results for our dataset, which include the least Z score from the real data (the second smallest values are above the 99 % curve of the Monte Carlo 5 Z. Bagoly, L. G. Balazs, Z. Horvath, I. Horvath Acta Polytechnica 360 240 120 +90 -90 0 all Faraway GRB Patch Figure 6. The sky distribution of the Faraway GRB Patch. simulations). A spherical cap size of 12–14 degrees re- veals an interesting location (the Faraway GRB Patch, Figure 6), with a matching random probability of 1 %. Considering the significance of the result, it is im- portant to recall that 542 separate tests were made, all of them centred on a GRB. Therefore, there is a strong correlation between the tests because of the overlap- ping data. There are ≈ 60 separate regions on the sky for the 12–14 degree size, therefore, there should be a correction factor with this minimum value. It is important to note that [33] only included 522 GRBs; in this case, the larger dataset resulted in a greater probability, which decreased the significance. Thus, according to this analysis, we may conclude that fac- torisation is not inconsistent with the observational data. 5. The bootstrap point-radius method In [34], the point radius bootstrap method was applied to test for clustering in the redshift data. Using the results in [33], it assumed that the sky exposure is independent of z in order to use the point radius bootstrap approach. The preceding section’s results support this factorisation, therefore, we describe the present status of the investigation. The point radius bootstrap approach determines the distribution of GRBs inside a predetermined angular radius circle, which is parameterised by their A surface. Sliding the starting point k in redshift will select n consecutive GRB, in the radial distribution. All the GRBs inside the A cap will have the same sky exposure, therefore, the numbers of GRBs within and outside the redshift slice will allow us to check the uniformity of the radial distribution. We search for the greatest number of GRBs (K) for each radial beginning point k and for each GRB within the given A spherical cap. The spherical cap area of A and the slice size, n, are fixed parameters for a specific test. A Healpix partition with 49 152 positions provides a quasi-isotropic distribution of the centre of the spher- ical caps, yielding an average centre distance of ap- proximately 10 times lower than the average distance between the GRBs. A Monte-Carlo simulation is performed, repeatedly mixing radial and angular locations a thousand times. The maximum number of GRBs detected within the angular circle is taken from these 1 000 examples and compared with the actual K values. It is important to note that the distribution should be comparable for similar k and A values. The distribution of K is non-linearly dependent on the angular two-point correlation function both inside and outside the radial slice. However, the relation- ship between the spatial correlation function and the geometrical change in the angular diametre distance with the co-moving distance is quite complicated. The [34] study uses a bootstrap approach to de- termine the likelihood of receiving a number K for a given n and A, analysing the GRBs in the galactic hemispheres separately. For the northern galactic hemisphere cluster- ing occurs on three angular separation range scales, indicating considerable departures from isotropy/homogeneity. The first of these range scales occurs for n = 5 and A = 0.0628. The analysis found one occurrence (0.0628, 5) where the bursts are located within a 0.59 ≤ z ≤ 0.62 redshift range, with a likelihood of this fluctuation occurring by chance is of only 0.012. The second of these angular separation ranges is around 41 ≤ n ≤ 47, 1.6 ≤ A ≤ 1.9. Only two neigh- bouring points achieve the least significant limit in a larger (n, A) region (36 ≤ n ≤ 57, 1.57 ≤ A ≤ 1.95). A = 1.7, p = 0.038 for n = 43 and p = 0.048 for 6 vol. 65 no. 1/2025 The spatial distribution of GRBs n = 43 for A = 1.728. The redshift range of this cluster is 0.9 ≤ z ≤ 1.3. These GRBs are located in the same sky location as the Hercules-Corona Bore- alis Great Wall, however, the latter is further away (1.6 < z < 2.1) [16]. The second section is included in the third since it covers the z = 0.9–2.1 range. In the southern hemisphere, there are two possi- ble clusters. One contains the Giant GRB Ring [17] around redshift of z = 0.75–0.86, with 9 out of 19 GRBs on only 0.4396 sr in the sky. The second has 24–29 GRBs with redshifts ranging from 0.55 to 1.17–1.25, with a corresponding probability above 3 %. 6. Conclusion The yearly variances of redshift observations can be separated into two groups according to the afterglow measurements or the observation of the host galaxy’s spectral lines. The rates of afterglow- and host-based redshift observations differ, with only a few host-based GRB redshift observed in recent years. The number of afterglow redshifts has decreased exponentially, by 50 % in 15 years, potentially affecting all future GRB missions. This decrease in observer interest can be par- tially compensated by rapid response/communication mechanisms, such as successful Swift XRT and UVOT afterglow detection, which raises the chance of a spec- trosopic redshift. We plotted the rest-frame T90 values using redshift data and published T90 durations from the Swift BAT and FERMI GBM catalogues as a first approximation for the analysis of the various observational effects. The host- and afterglow-based points are separated in the redshift range due to observational effects, but no direct disctintion could be made among the rest- frame T90 values. The study also shows diverse trigger selection effects, such as missing medium/high-z short sources for both satellites. The observational likelihood of a GRB on the sky is influenced by various effects, such as satellite activity and optical follow-up factors (weather, telescope avail- ability, observer activity, etc.). To reconstruct the three-dimensional GRB distribution, it is crucial to determine if the sky distribution is independent of red- shift. The Mann-Whitney U test was used to compare the local redshift distribution within a given spherical cap and the full radial distribution. The probability was obtained with Monte Carlo mixing/randomisation between radial and positional values. The Z score dis- tribution selected only one suspicious spherical cap area with radius of 12–14 degrees with a random probability of 1 %. Because the strong correlation between the tests due to the overlapping data results in a greater probability, decreasing the significance. Thus, factorisation is not inconsistent with the ob- served data. Acknowledgements The authors thank the Hungarian TKP2021-NVA-16 and TKP2021-NKTA-64 programs for their support. References [1] S. E. Woosley. Gamma-ray bursts from stellar mass accretion disks around black holes. 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Monthly Notices of the Royal Astronomical Society 527(3):7191–7202, 2024. https://doi.org/10.1093/mnras/stad3669 8 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/10.1017/S1743921315010820 https://doi.org/10.1093/mnras/stz2754 https://doi.org/10.3847/1538-4357/ab0a86 https://doi.org/10.1093/mnras/stab3559 https://doi.org/10.1093/mnras/stac517 https://doi.org/10.3847/1538-4357/aa9708 https://doi.org/10.1093/mnras/stz921 https://doi.org/10.1088/0004-637X/749/1/68 https://doi.org/10.1051/0004-6361/201526760 https://doi.org/10.1051/0004-6361/201834179 https://doi.org/10.1111/j.1365-2966.2012.21830.x https://doi.org/10.1111/j.1365-2966.2012.21830.x https://doi.org/10.1142/S0217732312502215 https://doi.org/10.1051/0004-6361/202140296 https://doi.org/10.3390/universe8040221 https://doi.org/10.3390/universe8070342 https://doi.org/10.1093/mnras/stad3669 Acta Polytechnica 65(1):1–8, 2025 1 Introduction 2 Data selection 3 Difference between host- and afterglow-based z 3.1 Redshift observations and their future 3.2 Selection effects in the rest-frame T90 – z distribution 4 The factorisation of the GRB density function 5 The bootstrap point-radius method 6 Conclusion Acknowledgements References