© ESO, 2015

1. Introduction

The high luminosities of gamma-ray bursts (GRBs) make them ideal candidates for probing large-scale Universal structure. Gamma-ray bursts signify the presence of stellar endpoints and thus trace the location of matter in the universe. This is true whether they are long bursts (presumably originating from hypernovae), short bursts (presumably originating from compact objects), or intermediate bursts (with unknown origins that are still likely related to stellar endpoints). Assuming that the Universe is homogeneous and isotropic on a large scale implies that the large-scale distribution of GRBs should similarly be homogeneous and isotropic. The angular isotropy of GRBs has been well-studied over the past few decades (Briggs et al. 1996; Balázs et al. 1998, 1999; Mészáros et al. 2000; Magliocchetti et al. 2003; Vavrek et al. 2008). For the most part, GRBs are distributed uniformly, although some subsamples (generally believed to be those with lower luminosities and therefore thought to be cosmologically local) appear to deviate from isotropy (Balázs et al. 1998; Cline et al. 1999; Mészáros et al. 2000; Litvin et al. 2001; Magliocchetti et al. 2003; Vavrek et al. 2008). We have recently identified a surprisingly large anisotropy in the overall GRB angular distribution, suggestive of clustering, at redshift two in the constellations of Hercules and Corona Borealis. The underlying distribution of matter suggested by this cluster is large enough to question standard assumptions about the largest scale of Universal structures.

We revisit the angular and radial distributions of GRBs with known redshifts in an attempt to reexamine our previous claims suggesting the existence of this structure. As of November 2013, the redshifts of 361 GRBs have been determined¹; this represents an increase in sample size of 28% over that used in our previous analysis (283 bursts observed up until July 2012). The number of GRBs in the z = 2 redshift range, where the cluster resides, has increased from 31 bursts to 44 bursts, a 42% sample size increase that is large enough to warrant an updated analysis. We apply kth nearest neighbour analysis and the bootstrap point radius method to this database composed largely of bursts detected by NASA’s Swift experiment.

2. Nearest-neighbour statistics: the largest anisotropy is at z = 2

The larger GRB database allows us to re-examine the significance of our prior results. We employ the same statistical tests so as to retain consistency in our methodology and not introduce any potential analysis biases.

The GRB sample is subdivided by redshift z in a manner similar to our previous work (Horváth et al. 2014) so that we can base our angular studies on well-defined distance groupings. The GRB redshift uncertainties are small (many GRB redshifts are quoted to three or four significant figures), so it is possible in theory to create a large number of radial groups or bins and thus maintain a small z-dispersion in the sample. The drawback to this approach is that the 361 burst sample is still small, and angular resolution is limited based on the number of bursts in each radial group. We have subdivided the total sample into eight separate cases, containing the following numbers of radial groups: two, three, four, five, six, seven, eight, and nine. These choices allow us to examine bulk anisotropies in the GRB distribution over various distance ranges. However, binning the data limits the angular resolution we can realistically obtain within each radial bin: we are capable of finding large anisotropies. These cases are not independent of one another; each contains the same GRB sample binned differently. A choice of one radial group corresponds to the bulk angular distribution of GRBs in the plane of the sky; we do not analyse group one here, since it does not make use of the carefully measured redshifts we employed. In contrast, the choice of nine radial groups provides us with radial bins with the smallest number of bursts per bin (≈40) for which we can make reasonable, quantifiable estimates on bulk anisotropies. When choosing between 2 and 9 radial divisions, we keep the numbers of bursts in each radial group identical. The result of this approach is that we exclude GRBs with the smallest redshifts in some cases. For example, in the four group case the closest GRB (with the smallest redshift) was excluded, therefore, each of the four groups contains 90 GRBs (361 = 4 × 90 + 1).

Fig. 1 22nd (top) and 29th (bottom) neighbour distribution for the four group case (each group contains 90 GRBs), red, green, blue, and pink identifies radial groups 1–4.

We examine the angular burst distributions of each radial group by independently applying the kth nearest-neighbour statistics to the bursts in each group. As an example of our procedure, we consider the four group case again. For each radial group, we calculate the angular separations between all 90 GRBs. All neighbours of each GRB are identified and ranked as the nearest, second nearest, etc. The 90 nearest neighbour separations are collected into a first distribution, the 90 second-nearest neighbour separations are collected into a second distribution, and the process is repeated with each set of neighbours until the orthogonal set is completed with the 89th distribution being composed of the 89th nearest neighbour (farthest) separations. For each group in the four group case, these 89 nearest-neighbour distributions can be compared across the groups using a Kolmogorov-Smirnoff test. As this has been accomplished for the four group case, the same approach can be applied to the available nearest-neighbour distributions for all eight radial groupings (the two group case through the nine group case).

Each of the eight radial groupings indicates that significant anisotropies are present in one specific radial region, as defined by redshift. In other words, most of the kth nearest-neighbour distributions are not significantly different, except those that are close to one specific redshift. The bin containing the largest cluster of GRBs always comes from the redshift range 1.6 ≤ z< 2.1, as found in our previous work (Horváth et al. 2013, 2014). Figure 1 shows an example of when the sample is divided into four radial groups. In this case, each group contains 90 GRBs in the redshift ranges 2.68 ≤ z< 9.4 (group 1), 1.61 ≤ z< 2.68 (group 2), 0.85 ≤ z< 1.61 (group 3), and 0 ≤ z< 0.85 (group 4). For this example, Table 1 shows the probability that the two distributions are different. Boldface type indicates that the significance of the 31st nearest-neighbour distributions of two groups are different by more than 3σ. There are no significant differences within group 1, group 3, and group 4 distributions, but the 31st nearest-neighbour distribution in group 2 is significantly different than the distributions found in the other groups. The 31st nearest-neighbour distribution is just an example demonstrating a group 2 anisotropy; the same is also true for the 22nd, 23rd, ... 55th nearest-neighbour distributions. The GRBs clustering on small angular scales would show differences when describing close neighbour pairs, while GRBs found on opposite sides of the celestial sphere would exhibit differences when describing distant neighbour pairs. The GRBs in the 1.61 ≤ z< 2.68 redshift range have a preference for neighbours with moderately close angular separations, suggesting a large angular cluster.

3. Bootstrap point-radius method: the anisotropy represents a large GRB cluster

As demonstrated in the previous section, nearest-neighbour tests identify pairing consistent with a large, loose GRB cluster in the redshift range 1.6 <z ≤ 2.1. The significance of this cluster can also be measured using other statistical tests designed to identify clustering. Among these is the bootstrap point-radius method described in Sect. 5 of Horváth et al. (2014). The updated data set to which we apply this test contains 44 GRB in the redshift interval 1.6 <z ≤ 2.1.

Our use of the bootstrap point-radius method assumes that the sky exposure is independent of z. To carry out our analysis, we choose 44 GRBs from the observed data set and compare the sky distribution of this subsample with the sky distribution of 44 GRBs with 1.6 <z ≤ 2.1.

To study the selected bursts in two dimensions, we select random locations on the celestial sphere and find how many of the 44 points lie within a circle of predefined angular radius, for example, within 20°. We build statistics for this test by repeating the process a large number of times (i.e., 10 000). From the 10 000 Monte Carlo runs, we select the largest number of bursts found within the angular circle.

This analysis can be performed with the clustered 44 GRB positions as well as with 44 randomly chosen GRB locations from the observed data. There are some angular radii for which the maximum with the 44 GRBs with 1.6 <z ≤ 2.1 is significant. We repeat the process with 44 different randomly chosen burst positions, and we repeated the experiment 17 500 times to understand the statistical variations of this subsample. We also perform the same measurement using angular circles of different radii. The frequencies obtained are shown in Fig. 2.

Fig. 2 Results of the Monte-Carlo bootstrap point-radius method. The horizontal coordinate is the area of the circle in the sky relative to the whole sky (4π). The vertical coordinate is the logarithm of the frequency found from the 17 500 runs. Green (blue) line shows the 2σ (3σ) deviations.

Figure 2 demonstrates that the 9–18% of the sky identified for 1.6 <z ≤ 2.1 contains significantly more GRBs than similar circles at other GRB redshifts. When the area is chosen to be 0.0375 × 4π (corresponding to an angular radius of θ_max = 22.3°), 13 out of the 44 GRBs lie inside the circle. When the area is chosen to be 0.0875 × 4π (θ_max = 34.4°), 18 of 44 GRBs lie inside the circle. When the area is chosen to be 0.1875 × 4π (θ_max = 51.3°), 25 GRBs out of the 44 lie inside the circle. In this last case, only two out of the 17 500 bootstrap cases had 25 or more GRBs inside the circle. This result is, therefore, a statistically significant (p = 0.0001143) deviation, and the binomial probability for this being random is p_b = 2 × 10^-8.

The 42% increase in sample size should have led to a noticeable decrease in significance if the sample represented random sampling. However, in the radii between roughly 4° and 90°, 49 angular circles contain enough GRBs to exceed the 2σ level, compared to 28 found in our previous analysis (Horváth et al. 2014). Additionally, there are 16 angular circles containing enough GRBs to exceed the 3σ level (compared to only two in our previously published result), therefore, the evidence has strengthened that these bursts are mapping out some large-scale universal structure.

4. Sky exposure: sampling biases do not appear to be responsible for the anisotropy

Observing biases can introduce measurable angular anisotropies in a sample. However, prior results suggest that these biases are unlikely to be responsible for the observed cluster at z ≈ 2. The largest potential causes of angular biasing are:

• Sky exposure. This is a well-known bias describing favoureddetection of GRBs in some angular directions over others. Skyexposure is a function of instrumental response rather than a truesource distributional preference; some causes of anisotropic skyexposure include spacecraft pointing and a preferred orbitalplane, the avoidance of certain pointing directions such as the Sunor occultation by the Earth.

• Anisotropic measurement of GRB redshifts. GRB redshift measurements are made in the visual/infrared by ground-based telescopes, and are thus affected by observatory latitudes, seasonal weather, and Galactic extinction.

Each GRB instrument samples the sky differently, making the summed sky exposure difficult to identify for our heterogeneous GRB sample, which has been observed by many instruments since the late 1990s. However, since more than 3/4 of our sample was detected by Swift, the Swift sky exposure dominates the sampling. Thus, we assume to first order that Swift’s sky exposure is a reasonable approximation of the sky exposure of the entire burst sample. Because of its orbital characteristics, Swift (Baumgartner et al. 2013) has sampled ecliptic polar regions at slightly higher rates than ecliptic equatorial regions. Our simple model assumes that ecliptic polar regions (| β | ≥ 45°, where β is the ecliptic latitude) are sampled 1.83 times more frequently than the ecliptic equatorial region.

The location of ground-based optical and infrared telescopes measuring GRB redshifts can also lead to anisotropic observations. However, since a large number of ground-based telescopes at a variety of latitudes and longitudes have been used in GRB follow-up observations, there does not appear to be an Earth-based bias that would favour GRB afterglow measurements in some sky locations over others. Thus, our sampling model does not include a term accounting for telescope location.

Extinction due to dust from the Milky Way disk does not affect the detection of GRBs, but it does affect redshift measurements in an angularly-dependent way. Extinction removes light from extragalactic sources, making it harder to measure spectral characteristics from which redshifts can be obtained. Although the Galactic dust is strongly concentrated towards the Galactic equator, it is also very clumpy. This clumpiness makes the effect of extinction on measuring GRB redshifts very difficult to model; the details of the process depends on many variables, such as the Galactic latitude and longitude of the burst, the intrinsic luminosity and decay rate of the afterglow, the light-gathering ability of the telescope and the instrumental response of the spectrograph used, the redshift of the burst, and the observing conditions at the time of detection.

We check to see whether or not the GRB sample favours low-extinction regions by examining the distribution of visual extinctions in the directions of these 361 bursts. Extinctions are obtained from the high angular resolution DIRBE catalogue of Schlafly & Finkbeiner (2011)². The results, shown in Fig. 3, are that the sample can be modelled by a lognormal distribution centred at A_v = 0.13 mag with standard deviations σ⁺ = 0.22 and σ⁻ = 0.08. Fully 91% of the bursts in the sample have visual extinctions of A_v ≤ 0.5 mag, indicating that a characteristic of a GRB with a measured redshift is that it is not obscured by Galactic extinction.

Fig. 3 Galactic visual extinctions in the directions of the 361 GRBs in this sample. The measurement of GRB redshifts strongly favours small amounts of interstellar extinction.

It is not possible to tell if, in trying to measure GRB redshifts from afterglows, observers have avoided pointing their optical telescopes in the direction of GRBs that lie too close to the Galactic equator. This kind of a bias is possible given the potential low return on afterglow detection (especially for small- and medium-sized telescopes), but this bias cannot be modelled with the limited information provided by optical observers. Since the development of a model describing extinction and extinction-related biases on redshift measurement is very difficult, to estimate the effects of extinction on the sample we use all GRBs with measured redshifts found within 20° of the Galactic equator relative to all GRBs with measured redshifts. We find that only 3.1% of the GRB sample has had redshift measurements made within 20° of the Galactic equator.

The Galactic equatorial region is the poorest-sampled region. The ecliptic polar regions (b ≥ 20° and | β | ≥ 45° are the best-sampled regions and the ecliptic equatorial regions (b ≥ 20° and | β | < 45°) are well-sampled regions. Correcting the observations for this biasing, the expected numbers of GRBs in each group are 7.50 (best, north), 7.50 (best, south), 11.2 (good, north), 11.2 (good, south), and 5.54 (poor), whereas the actual counts are 13 (best, north), 2 (best, south), 12 (good, north), 8 (good, south), and 7 (poor). This correction results in a χ² probability of p = 0.051 that this is due to chance.

Fig. 4 Top: distribution of GRBs with measured redshift (blue). Although the distribution of all GRBs is fairly isotropic, extinction causes this sample to miss GRBs near the Galactic plane. Bottom: anisotropic distribution of GRBs near redshift z = 2 (red).

Although this probability is higher than that quoted previously (p = 0.025), the cluster density has increased relative to the rest of the z ≈ 2 sky since the last analysis. Many of the new bursts have been detected just outside the edge of the best-sampled region, in the northern well-sampled region. Unfortunately, our low-resolution angular bias correction treats all GRBs in the well-sampled region as if they are not part of the cluster. If we naively assume that the cluster comprises 17 observed bursts (13 in 50% of the northern best-sampled region and 4 in 10% of the northern well-sampled region) and recalculate the probability that this clustering is random, the probability changes to p = 1.6 × 10^-4 that exposure is responsible for the clustering. This calculation also suggests that the cluster properties might be affected slightly by exposure: the few bursts seen in regions with less exposure could represent a larger number of undetected bursts. The cluster might be shifted several degrees west of where we have previously identified it.

5. Summary and conclusion

The evidence for a possible large-scale Universal structure (Horváth et al. 2014) at a redshift of z ≈ 2 has strengthened, using a larger database of GRBs with known redshift. The new sample contains 28% more bursts than the previous sample, and 42% more bursts in the 1.6 ≤ z< 2.1 redshift range. Because the cluster has become more populated relative to the rest of the angular distribution at the same redshift, our angular tests have returned more significant results. Nearest-neighbour tests indicate that GRBs in this redshift range favour each others’ presence through moderate angular separations. The two-dimensional bootstrap point-radius method reaches the 3σ level for a number of different angular radii, indicating a large GRB cluster. Although sampling biases are present and are significant, these coupled with small number statistics, do not seem to be responsible for the observed clustering of GRBs at this redshift.

GRBs are the most luminous known form of energy release available to normal matter. As such, they are tracers for the presence of normal matter that can be detected at distances where the matter is otherwise too faint to be observed. The GRB cluster at z ≈ 2 appears to identify the presence of a larger angular structure that covers almost one-eighth of the sky. This encompasses half of the constellations of Bootes, Draco, and Lyra, and all of the constellations of Hercules and Corona Borealis. This structure has been given the popular name of the Hercules-Corona Borealis Great Wall, or Her-CrB GW.

We estimate the size of the Her-CrB GW to be about 2000–3000 Mpc across. Few limits on its radial thickness exist, other than because it appears to be confined to the 1.6 ≤ z< 2.1 redshift range. This large size makes the structure inconsistent with current inflationary Universal models because it is larger than the roughly 100 Mpc limit thought to signify the End of Greatness at which large-scale structure ceases.

However, the Her-CrB GW is not the first optical/infrared structure found to exceed the 100 Mpc size limit. Several large filamentary structures have been identified using optical and infrared redshifts of galaxies; these include the 200 Mpc-sized CfA2 Great Wall (Geller & Huchra 1989) and the 400 Mpc-sized Sloan Great Wall (Gott et al. 2005). In the ensuing years, other structures have been identified using quasars; the largest of these is the Huge Large Quasar Group (Huge-LQG) Clowes et al. (2013), which has a length of more than 1400 Mpc. Most recently, Szapudi et al. (2015) found a 440 Mpc diameter supervoid aligned with a cold spot on the cosmic microwave background.

On the other hand, many results support the cosmological principle. Yahata et al. (2005) reported that the galaxy distribution was homogeneous on scales larger than 60−70 h^-1 Mpc. Bagla et al. (2008) showed that the fractal dimension makes a rapid transition close to 3 at 40–100 Mpc scales. Sarkar et al. (2009) found the galaxy distribution to be homogeneous at length-scales greater than 70 h^-1 Mpc, and Yadav et al. (2010) estimated the homogeneity upper limit scale was close to 260 h^-1 Mpc.

As large as it appears to be, the Her-CrB GW does not necessarily have to violate the basic assumptions of the cosmological

principle (the assumptions of a homogeneous and isotropic Universe). Theoretical large-scale structure models indicate that some structures will exceed the End of Greatness on purely statistical grounds (Nadathur 2013), and this may be one such structure (albeit a very large one). Along these lines, this may not be a single structure, but a clustering of smaller adjacent and/or line-of-sight structures; the small number of bursts currently found in the cluster limits our ability to angularly resolve it. However, this becomes a semantic issue at some point, since a cluster of smaller structures is still a larger structure.

Online material

Appendix A

Acknowledgments

This research was supported by OTKA grant NN111016 and by NASA EPSCoR grant NNX13AD28A. We thank the anonymous referee for comments that greatly improved this paper.

References

All Tables

All Figures

	Fig. 1 22nd (top) and 29th (bottom) neighbour distribution for the four group case (each group contains 90 GRBs), red, green, blue, and pink identifies radial groups 1–4.
In the text

	Fig. 2 Results of the Monte-Carlo bootstrap point-radius method. The horizontal coordinate is the area of the circle in the sky relative to the whole sky (4π). The vertical coordinate is the logarithm of the frequency found from the 17 500 runs. Green (blue) line shows the 2σ (3σ) deviations.
In the text

	Fig. 3 Galactic visual extinctions in the directions of the 361 GRBs in this sample. The measurement of GRB redshifts strongly favours small amounts of interstellar extinction.
In the text

	Fig. 4 Top: distribution of GRBs with measured redshift (blue). Although the distribution of all GRBs is fairly isotropic, extinction causes this sample to miss GRBs near the Galactic plane. Bottom: anisotropic distribution of GRBs near redshift z = 2 (red).
In the text