Issue 
A&A
Volume 517, July 2010



Article Number  A84  
Number of page(s)  5  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201014062  
Published online  12 August 2010 
multifractality of Swift short GRBs?
F. Tamburini
Università di Padova, Dipartimento di Astronomia, vicolo dell'Osservatorio 3, 35122 Padova, Italy
Received 14 January 2010 / Accepted 10 March 2010
Abstract
Aims. We analyze and characterize the angular distribution
of selected samples of gamma ray bursts (GRBs) from BATSE and Swift
data to confirm that the division into two classes of short and
longduration GRBs also correspond to two distinct spatial populations.
Methods. The angular distribution is analyzed using multifractal
analysis and characterized by a multifractal spectrum of dimensions.
Different spectra of dimensions relate to different angular
distributions.
Results. The spectra of dimensions of short and long bursts
indicate that the two populations indeed have two different angular
distributions. Both Swift and BATSE long bursts appear to be
homogeneously distributed across the sky with a monofractal
distribution. In contrast, short GRBs follow a multifractal
distribution for both the two samples. Even if BATSE data do not enable
a unique interpretation of their angular distribution to be made
because of the instrumental selection effects that mainly favor the
detection of nearby GRBs, the results from Swift short GRBs confirm
this behavior, also when including GRBs corrected by the redshift
factor. The distributions traced by short GRBs, up to z=1,
depict a universe with a structure similar to that of a disordered
porous material with uniformly distributed heterogeneous irregular
structures, appearing more clustered than expected.
Key words: gamma rays: general  largescale structure of Universe  chaos
1 Introduction
Gamma ray bursts (GRB) are catastrophic explosions of cosmological origin that illuminate the sky once or twice a day with relatively short, intense, flashes of rays on the order of the MeV and duration that ranges from 10^{3} to about 10^{3} s. Until the launch of Swift, the most widely accepted taxonomy of GRBs had been the division between shorthard and longsoft bursts. In BATSE data, both Dezalay (1992) and Kouveliotou (1993) found a bimodal distribution in the burst duration, which was defined to be the time it takes the 90% of the flux to arrive (T_{90}) with respect to the local time of the detector. These two distinct GRB distributions are separated by a minimum located at s. This sharp division is apparently also caused by a selection effect of the instrumentation onboard the satellite. New Swift observations permit us to include in the classification scheme, as short bursts, some distant events for which apparently T_{90}>2 s with respect to Swift's proper time. The determination of the redshift allows us to correct the T_{90} for relativistic effects (Che et al. 1999; Ruffini et al. 2009). This suggests that a more solid classification of GRBs is required on the basis on a broader set of criteria, beyond the mere burst duration and the verification of the present taxonomy of long/short GRBs (Donaghy et al. 2006; Zhang et al. 2007; Bloom et al. 2008; Belczynski et al. 2008).
Long and short GRBs are understood to have been generated by different progenitors, which each contain a black hole that accretes material from either a disc or a torus with the emission of gamma rays (Ghirlanda et al. 2009). The basic model for longduration GRBs, related to the catastrophic release of energy from the collapse of massive stars (Woolsey 2001; Fryer & Kalogera 2001), has received strong support from the observations of their Xray (Gehrels et al. 2008; Nysewander et al. 2008), optical, and radio counterparts, and the association with supernova detections, of which the GRB980425/SN1998bw association was the first clear example (Galama et al. 1998; Kulkarni 1998; van Paradijs 1999). Thus, Xray flashes, which are associated to longduration GRBs, are probably produced by the highlyrelativistic jets ejected in corecollapse supernova explosions. The relativistic fireball model (Goodman 1986; Paczynski 1986, 1990; CastroTirado et al. 2001) provides a reasonable description of the observed afterglow spectrum, which is produced by the synchrotron emission of electrons accelerated in a relativistic shock with an estimated total energy budget roughly the same order of magnitude as that of supernovae Ib/c (Frail 2001). The discovery of the slowly fading Xray emission, from optical and radio afterglows of GRBs and the identification of host galaxies at cosmological distances provided additional support for their progenitors being produced shortlived massive stars at different cosmological epochs, whose detection is limited by the BATSE threshold to within distances slightly larger than (Wijiers et al. 1997; Blain & Natarajan 2000). The new data from Swift now has passed the barrier of z=8 (Salvaterra et al. 2009).
Short GRBs are understood to be produced by highly  relativistic jets ejected during different processes, such as neutron star  neutron star (NSNS) or black hole  neutron star (BHNS) binary mergers (Narayan et al. 1992; Fox et al. 2005; Metzger et al. 2008), whose averaged redshift distribution, in the Swiftera, seems to be (see e.g., Magliocchetti et al. 2003; Tanvir et al. 2005; and Ghirlanda et al. 2006). Swift and HETE2 observations have provided evidence of clear similarities with the afterglows detected in the correspondence of long GRBs, because of the detection of Xray and optical afterglows (Soderberg 2006; Grupe 2006; Berger 2007). Other similarities identified in the spectra for the initial stage of short and long GRBs and the presence, in some cases, of Xray flares infers that a common mechanism operates during the first few seconds (Barthelmyet et al. 2005; Coward, 2007). Short GRBs have been observed mostly in elliptical galaxies, but even less frequently also in nearby irregular and in starforming galaxies, confirming as progenitor the binary merging scenario (CastroTirado et al. 2002; Mêszáros 2006; Narayan et al. 2001; Belczynski et al. 2008) and the mechanism of star swapping are possible means of forming GRBs (Grindlay 2006). A similar behavior is expected for NSBH binaries (Paczynski 1991). The angular and spatial distributions of BATSE GRBs appear to be isotropic (Briggs et al. 1996), there being only a few anisotropies in the angular distribution (Mêszáros et al. 2000a,b,c). Short and long GRBs in the BATSE catalog actually show two different angular distributions (Vavrek et al. 2008; Balázs et al. 2009), but the connection to the hypothesis of the instrumental selection effect remains completely unclear.
In this paper, we characterize the angular distributions of the two classes of GRBs by performing multifractal analysis from a selected sample of BATSE and Swift observations, considering the redshift relativistic effects, and comparing the results obtained from the two catalogues. In Sect. 2, we describe the mathematical basis of our method. In Sect. 3, we estimate the fractal/multifractal dimensions by determining the distribution moments of the multifractal spectrum from the second up to the tenth order, and then draw our conclusions.
2 Fractal/multifractal analysis of the angular distribution
Being associated with galaxies, GRBs should trace the angular distribution of their host galaxies at distances slightly larger than those estimated with classical supernovae, i.e., distances on the order of a Gpc (Paczynski 1986; Usov & Chibisov 1975), across which inhomogeneity in the distribution of luminous matter should be averaged out on the Mpc scale.
Mandelbrot (1975) and Peebles (1980) used fractal geometry to describe the angular and spatial distribution of galaxies. They used LévyRayleigh random paths, i.e., infinitevariance, stable, generalized random walks in which the step lengths are described by a tailed probability distribution. Galaxies are placed at the steps of a random walk and each galaxy is randomly connected with another in its vicinity, mimicking the random motion of a fly in the air. The powerlaw constraint of the motion determines the fractal exponent of the random path and rules the distribution of the jump lengths and the direction of each jump is taken isotropically at random (Martinez 2002).
Galaxy surveys show that the distribution of the luminous matter in the universe is more complicated than that of a single fractal. The distribution of galaxies has instead multifractal properties (Pietronero 1987; Célérier & Thieberger 2001), having a tendency to cluster following the well known peculiar structures produced by patterns of voids and filaments (Kurokawa et al. 1999). The CfA survey of nearby galaxies, for example, presents the scaling properties of these strucures with a correlation dimension in the interval . The twopoint correlation function and the power spectrum analysis indicates that the distribution of galaxies on very large scales becomes homogeneous and isotropic like the Xray background emitted by active galactic nuclei (AGNs), in agreement with the Cosmological Principle (Peebles 1993).
Multifractal scaling analysis has identified aggregated structures also in some samples of galaxies with distances larger than Mpc. At scales larger than 30 Mpc, the angular distribution of luminous matter tends to homogeneity. According to Pietronero (1987), the distribution of luminous matter has multifractal properties with fractal dimensions that varies from d=1.23, for the nearby galaxies, up to the value at very high redshifts. Multifractal distributions are usually present when a structure has different fractal dimensions on different parts of the geometric support; in other words, when spatial correlations are present and change the geometrical shape of the distribution on different scales (Falconer 1990). Those distributions cannot be adequately described by a geometrical support with a single fractal dimension, but require instead a whole spectrum of dimensions.
To characterize the angular distribution of GRBs in terms of multifractals, we use the ``method of moments'', which estimates the fractality of the distribution by calculating the multifractal spectrum of generalized dimensions D_{q} in a given range (Falconer 1990; Feder 1988).
If we consider N GRBs in a box of size L divided into cells of size r, the sample is said to have a nonnull qth moment
on the scale r iff
(1) 
where n_{i} is the number of objects present in the ith cell. Higher moments reflect the emergence of structures present in the denser regions. From the scaling relation conducted on a twodimensional section, we obtain (Kurokawa et al. 1999; Kurokawa et al. 2001),
(2) 
where the coefficients D_{q} belong to the spectrum of generalized dimensions, D_{0} is the capacity dimension, D_{1} is the information dimension, and D_{2} is the correlation dimension.
To estimate the multifractality of the distribution, one has to determine each generalized dimension Dq as a function of the qth moment. One determines the dependence of each of the fractal dimensions on the moments by drawing the qD_{q} plot, which is based on the LipshitzHölder exponent (Benassi et al. 1998; Goltz 1997). In the qD_{q} plot, one defines monofractals, those simple fractal structures described by only one dimension, , and in the plot all the D_{q}s are equivalent to . In the general case of a multifractal, instead, D_{q} usually decreases for higher and higher values of the moments q: if q>q', then until converging to the asymptotic value for a distribution of infinite objects.
To characterise with high precision the multifractal distribution of a sample of objects in space, one usually does not need to calculate the generalized dimensions up to the limit dictated by the number of points in the space. For our purposes, a good estimate was obtained by taking q=10 as an upper limit.
3 Results and discussion
From Swift observations, we selected a sample of 444 GRBs that do not have ambiguous classifications, 53 of which are classified as shortGRB after correcting of T_{90} for the relativistic effects (data updated at 20090915). From BATSE 4 catalogue, instead, the total number of GRBs used in our analysis is 1843, 1447 of which are classified as longGRBs.
To verify whether the classification into the two subgroups corresponds to two distinct populations in both the sets of data collected by BATSE and Swift, we also performed a series of tests of the angular distributions in different subsamples. We randomly selected from our data a sets of either short or long GRBs, mixed together, and then tested whether GRBs might be discriminated without choosing ``a priori'' the two classes following the burst durationtime distribution. The two classes of long/short GRBs were distinguished by the fractal analysis of their angular distribution in the sky. More precisely, by progressively mixing the population of short GRBs with a growing sample of randomly chosen long GRBs, the multifractal dimension converged to that of a homogeneously distributed monofractal with dimension d=2. This simple test corroborated the true subdivision of GRBs into the two short/long populations identifiable from both the BATSE and Swift data and also when the two datasets are mixed together.
We now present and discuss the results obtained for each different class of GRBs.
Figure 1: Upper panel: qD_{q} plot of Swift long GRBs. The long bursts follow an angular homogeneous distribution. The moments D_{q} tend to decrease slowly from to with a rise of the fractal dimension after q=8. The spectrum of dimensions is that of a two dimensional Brownian motion with fractal dimension 2. Lower panel: BATSE 4 long bursts also exhibit a very similar homogeneous structure with a mean fractal dimension . 

Open with DEXTER 
3.1 Long bursts
Figure 1 clearly shows the uniform angular distribution of both the samples of Swift and BATSE 4 selected data. The sample of data from Swift shows a homogeneous distribution. In this case, the moments D_{q} tend to decrease slowly from to and the fractal dimension begins to increase after the moment q=8. In the plot, we also present some sporadic and discontinuous jumps down to that deviate from the main smoothness, a behavior that might be caused by numerical errors in the determination of certain D_{q}'s . In any case, this does not affect the interpretation of the global behavior observed in the plot. We observed a typical example of a stochastic homogeneous distribution, similar to that of a fractional Brownian motion (FBM) of dimension D=2, which is expected from distant sources that homogeneously distribute according to the Cosmological Principle. The errorbars of the D_{q}s' are calculated from the error propagation of the statistical uncertainty in the position of each GRB and from the instrumental errors. In the case of Swift data, errorbars are too small to be visible in the graph.
By analyzing BATSE data, we observed a different behavior in the interval 4<q<6, where the distribution is flat, within the experimental errors, and the angular distribution is characterized by an almost constant fractal dimension that fluctuates within the interval D_{q}= [1.82] approximately describing the angular distribution of a uniform structure. We found a similar result in both the datasets, the redshift scale being around the Gyr (as confirmed by the redshift values obtained for most of the bursts), that is, long GRBs are distributed according to the Cosmological Principle, in which the Universe is completely homogeneous and isotropic on large scales.
3.2 Short bursts
They represent a more complicated set of phenomena with many unknown properties. Figure 2 presents data for the whole sample of 53 Swift short bursts in addition to the apparentlylong bursts corrected by z and those with T_{90} already below 2 s. The sample is characterised by a spectrum of dimensions that decreases for increasing q's, the signature of a multifractal distribution.
Figure 2: qD_{q} plot of Swift short GRBs. Upper panel: spectrum of fractal dimensions of the angular distribution of the whole sample. This behavior with a maximum around q=3 clearly indicates a multifractal homogeneous distribution very close to that of a multifractional Brownian motion. Lower left panel: dimension moments of the 13 zcorrected GRBs with relatively high redshift. The structure is clearly multifractal. Lower panel: the subsample of GRBs with already T_{90}<2 s, exhibits a multifractal distribution, but with an asymptotic convergence to D_{q}=1. In this case, the structure traced by the GRBs appears to be more clustered. 

Open with DEXTER 
Short GRBs appear to be regularly angularly distributed because they follow multifractional Brownian motion (mBM) of mean index H=0.68, a distribution that begins with a clustered shape then evolves towards homogeneity on relatively larger scales. The mBM is an extension of the fractional Brownian motion in the sense that the path regularity can vary with time, as observed during the evolution of anomalous diffusion processes. The Universe traced by short GRBs exhibits a multifractal structure, which presents non null autocorrelation on lower scales and evolves to homogeneity at larger distances. This structure appears to be similar to a disordered porous material that exhibits a heterogeneous structure, or even an irregular one in a uniform sense. This type of foam contains multiple, nested natural length scales or continuously evolving scales, while moving to higher redshifts (Lim & Muniandy 2002).
The differences between the distributions of short and long bursts, as evident in the plots, is confirmed by the correlation coefficients and the pvalues obtained by comparing the results of averaging 10 sets of 53 randomly chosen (without repetition) long GRBs with respect to the total sample of short ones. The effects of the numerical problem that produced some outliers (e.g. the points at in Fig. 1, upper panel) were corrected by applying a polynomial fit to the general trend. The correlation coefficient indicates the strength and the direction of a linear relationship existing between two data records. When there is a strong correlation, the correlation coefficient is , when there is no correlation, R=0, and when an anticorrelation is present, . In general, a correlation greater than 0.8 is considered a strong correlation, whereas a correlation less than 0.5, is weak. The pvalues are instead calculated to test the hypothesis of nocorrelation, which is the probability of obtaining a correlation as large as the observed value by random chance, when the true correlation is zero. If the pvalue is less than 0.05, one rejects the null hypothesis and the correlation R is significant (Wall & Jenkins 2003). For Swift data, long and short bursts have a very weak correlation ( R = 0.2656) and a very low pvalue of p=0.2868, which rejects the nocorrelation hypothesis. From Swift data, short and long GRBs seem to follow two independent angular distributions.
The further division of short GRBs into two subclasses, one containing the short GRBs corrected for the redshift and the other containing only those GRBs that already present a T_{90} already below the 2 s, show that the two subsamples follow two mutually exclusive multifractal distributions and that the subclass of the zcorrected short GRBs has a higher fractal dimension q_{0}, close to that of a homogeneous distribution, confirming the behavior expected from a mBM (see the two lower panels in Fig. 2).
The whole sample of BATSE short GRBs exhibits a spectrum of the general dimension D_{q}, smoother than that of the whole sample of BATSE long GRBs, as reported in the upper panel of Fig. 3. The capacity dimension D_{0} and the correlation dimension D_{2} have similar values close to which are smaller than those of long GRBs. This demonstrates that short GRBs have a different multifractal and more clustered distribution than to long ones. This difference ismore evident if BATSE GRBs are divided into the two subclasses indicated by Mukherjee et al. (1998): Class II with short/faint/hard bursts and Class III with intermediate/intermediate/soft bursts (we recall that Class I correspond to long GRBs), as drawn in the lower panels of Fig. 3.
Figure 3: Upper panel: qD_{q} plot of BATSE 4 short GRBs (Class I + Class II) exhibit a fractal structure with dimensions fluctuating around the value 1.8. Lower left and right panels: qD_{q} plots of BATSE 4 short bursts, in which Class II and Class III, respectively, clearly follow a multifractal distribution. 

Open with DEXTER 
By applying the same statistical test used to quantify the distributions in Swift data, the R and p values of BATSE data are found to be R=0.0741 and p=0.7702 which have slightly different statistical behavior: almost no (anti)correlation with a high pvalue. This is a clear indication of instrumental selection effects.
In conclusion, even if R and p values of BATSE data could be ascribed to a selection effect, Swift results confirm the existence of two different angular distributions associated with the two classes of short and long GRBs. In both Swift and BATSE data sets, long bursts are homogeneously angularly distributed. In contrast, short GRBs trace a distribution that appears to differ from that expected for the clustering of luminous matter in the Universe around the redshift value . We can clearly see a multifractal distribution in a structure that should already be homogeneouslydistributed. By taking into account that the candidate progenitors of short GRBs probably migrate away from their initial positions, we should also expect an additional convolving effect on the detected structure that may tend to make the distribution more towards the total homogeneous, a result that has not been observed. This suggests either that GRBs are not good tracers of the matter distribution or that the Universe traced by GRBs appears more clustered at redshifts than expected.
AcknowledgementsThe author would like to acknowledge Massimo della Valle for the helpful discussions and suggestions.
References
 Balázs, L. G., Horváth, I., Vavrek, R., Bagoly, Z., & Mêszáros, A. 2008, AIPC, 1000, 52 [NASA ADS] [Google Scholar]
 Barthelmy, S. D., Chincarini, G., Burrows, D. N., et al. 2005, Nature, 438, 994 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Belczynski, K., Hartmann, D. H., Fryer, C. L., Holz, D. E., & O'Shea, B. 2010, ApJ, 708, 117 [NASA ADS] [CrossRef] [Google Scholar]
 Benassi, A., Cohen S., & Istas, J. 1998, Proba Lett., 39, 337 [Google Scholar]
 Berger, E. 2007, ApJ, 670, 1254 [NASA ADS] [CrossRef] [Google Scholar]
 Blain, A. W., & Natarajan, P. 2000, MNRAS, 312, L35 [NASA ADS] [CrossRef] [Google Scholar]
 Bloom, J. S., Butler, N. R., & Perley, D. A. 2008, AIPC, 1000, 11 [NASA ADS] [Google Scholar]
 Briggs, M. S., Paciesas, W. S., Pendleton, G. N., et al. 1996, ApJ, 459, 40 [NASA ADS] [CrossRef] [Google Scholar]
 CastroTirado, A. J., Sokolov, V. V., Gorosabel, J., et al. 2001, A&A, 370, 398 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 CastroTirado, A. J., Castro Cerón, J. M., Gorosabel, J., et al. 2002, A&A, 393, L55 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Célérier, M.N., & Thieberger, R. 2001, A&A, 367, 449 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Che, H., Yang, Y., Wu, M., & Li, Q. B. 1997, ApJ, 483, L25 [NASA ADS] [CrossRef] [Google Scholar]
 Coward, D. 2007, New Astron. Rev., 51, 539 [NASA ADS] [CrossRef] [Google Scholar]
 Dezalay, J.P., et al. 1992, in Gammaray bursts; Proceedings of the Workshop, Univ. of Alabama, Huntsville, 304 [Google Scholar]
 Donaghy, T. Q., Lamb, D. Q., Sakamoto, T., et al. 2006, unpublished [arXiv:astroph/0605570] [Google Scholar]
 Falconer, K. J. 1990, Fractal Geometry, mathematical foundations and applications (Brisbane (UK): Wiley & sons) [Google Scholar]
 Feder, J. 1988, Fractals (NY, USA: Plenum Press) [Google Scholar]
 Fox, D. B., Frail, D. A., Price, P. A., et al. 2005, Nature, 437, 845 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Frail, D. A., Kulkarni, S. R., Sari, R., et al. 2001, ApJ, 562, L55 [NASA ADS] [CrossRef] [Google Scholar]
 Fryer, C. L., & Kalogera, V. 2001, ApJ, 554, 548 [NASA ADS] [CrossRef] [Google Scholar]
 Galama, T. J., Vreeswijk, P. M., van Paradijs, J., et al. 1998, Nature, 395, 670 [NASA ADS] [CrossRef] [Google Scholar]
 Gehrels, N., Barthelmy, S. D., Burrows, D. N., et al. 2008, ApJ, 689, 1161 [NASA ADS] [CrossRef] [Google Scholar]
 Ghirlanda, G., Magliocchetti, M., Ghisellini, G., & Guzzo, L. 2006, MNRAS, 368, L20 [NASA ADS] [Google Scholar]
 Ghirlanda, G., Nava, L., Ghisellini, L., Celotti, A., & Firmani, C. 2009, A&A, 496, 585 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Goltz, C. 1997, Fractal and Chaotic Properties of Earthquakes (Berlin: SpringerVerlag) [Google Scholar]
 Goodman, J. 1986, ApJ, 308, L47 [NASA ADS] [CrossRef] [Google Scholar]
 Grindlay, J., Portegies Zwart, S., & McMillan, S. 2006, Nature Phys., 2, 116 [Google Scholar]
 Grupe D., Burrows, D. N., Patel, S. K., et al. 2006, ApJ, 653, 462 [NASA ADS] [CrossRef] [Google Scholar]
 Kouveliotou, C., Meegan, C. A., Fishman, G. J., et al. 1993, ApJ, 413, L101K [NASA ADS] [CrossRef] [Google Scholar]
 Kulkarni, S. R., Frail, D. A., Wieringa, M. H., et al. 1998, Nature, 395, 663 [NASA ADS] [CrossRef] [Google Scholar]
 Kurokawa, T., Morikawa, M., & Mouri, H. 1999, A&A, 344, 1 [NASA ADS] [Google Scholar]
 Kurokawa, T., Morikawa, M., & Mouri, H. 2001, A&A, 370, 358 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Lim, S. C., & Muniandy, S. V. 2002, Phys. Rev. E., 66, 021114 [NASA ADS] [CrossRef] [Google Scholar]
 Magliocchetti, M., Ghirlanda, G., & Celotti, A. 2003, MNRAS, 343, 255 [NASA ADS] [CrossRef] [Google Scholar]
 Mandelbrot, B. B. 1975, Comptes Rendues (Paris), 280A, 1551 [Google Scholar]
 Martìnez, V. J., & Saar, E. 2001, Statistics of the Galaxy Distribution (Boca Raton: Chapman and Hall/CRC Press) [Google Scholar]
 Mêszáros, A. 2006, Rep. Prog. Phys., 69, 2259 [NASA ADS] [CrossRef] [Google Scholar]
 Mêszáros, A., Bagoly, Z., Horváth, I., Balázs, L. G., & Vavrek, R., 2000a in GAMMARAY BURSTS: 5th Huntsville Symposium. AIPC, 526, 102 [Google Scholar]
 Mêszáros, A., Bagoly, Z., & Vavrek, R. 2000b, A&A, 354, 1 [NASA ADS] [Google Scholar]
 Mêszáros, A., Bagoly, Z., Horváth, I., Balázs, L. G., & Vavrek, R. 2000c, ApJ, 539, 98 [NASA ADS] [CrossRef] [Google Scholar]
 Metzger, B., Piro, A., & Quataert, E. 2009, MNRAS, 390, 781 [Google Scholar]
 Mukherjee, S., Feigelson, E. D., Babu, G. J., et al. 1998, ApJ, 508, 314 [NASA ADS] [CrossRef] [Google Scholar]
 Narayan, R., Paczynski, B., & Piran, T. 1992, ApJ, 395, L83N [NASA ADS] [CrossRef] [Google Scholar]
 Narayan, R., Piran, T., & Kumar, P. 2001, ApJ, 557, 949 [NASA ADS] [CrossRef] [Google Scholar]
 Nysewander, M., Fruchter, A. S., & Peer, A. 2009, ApJ, 701, 824 [NASA ADS] [CrossRef] [Google Scholar]
 Paczynski B. 1986, ApJ, 308, L43 [NASA ADS] [CrossRef] [Google Scholar]
 Paczynski B. 1990, ApJ, 363, 218 [NASA ADS] [CrossRef] [Google Scholar]
 Paczynski B. 1991, Acta Astron., 41, 257 [Google Scholar]
 Peebles, P. J. E. 1980, The Large Scale Structure of the Universe (Princeton University Press) [Google Scholar]
 Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton Univ. Press) [Google Scholar]
 Pietronero, L. 1987, Phys. A, 144, 257 [CrossRef] [Google Scholar]
 Ruffini, R., et al. 2009, in Proceedings of the 2008 Cefalú Conference, AIPC, 1111, 325 [Google Scholar]
 Salvaterra, R., Della Valle, M., Campana, S., et al. 2009, Nature, 461, 1258 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Soderberg, A. M., Berger, E., Kasliwal, M., et al. 2006, ApJ, 650, 261 [NASA ADS] [CrossRef] [Google Scholar]
 Tanvir, N. R., Chapman, R., Levan, A. J., & Priddey, R. S. 2005, Nature, 438, 991 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Usov V. V., & Chibisov, G. V. 1975, SVA, 19, 115 [NASA ADS] [Google Scholar]
 van Paradijs, J. 1999, Science, 286, 693 [NASA ADS] [CrossRef] [Google Scholar]
 Vavrek, R., Balázs, L. G., Mêszáros, A., Horváth, I., & Bagoly, Z. 2008, MNRAS, 391, 1741 [NASA ADS] [CrossRef] [Google Scholar]
 Wall, J. V., & Jenkins, C. R. 2003, Practical statistics for astronomers (Cambridge University Press) [Google Scholar]
 Wijiers, R. A. M. J., Bloom, J. S., Bagla, J. S., & Natarajan, P. 1997, MNRAS, 294, L13 [Google Scholar]
 Woosley, S. E. 2001, in GammaRay Bursts in the Afterglow Era, Proceedings of the International Workshop Held in Rome, Italy, 1720 October 2000, ESO ASTROPHYSICS SYMPOSIA, ed. E. Costa, F. Frontera, & J. Hjorth (SpringerVerlag), 257 [Google Scholar]
 Zhang, B., Zhang, B.B., Liang, E.W., et al. 2007, ApJ, 655, L25 [NASA ADS] [CrossRef] [Google Scholar]
All Figures
Figure 1: Upper panel: qD_{q} plot of Swift long GRBs. The long bursts follow an angular homogeneous distribution. The moments D_{q} tend to decrease slowly from to with a rise of the fractal dimension after q=8. The spectrum of dimensions is that of a two dimensional Brownian motion with fractal dimension 2. Lower panel: BATSE 4 long bursts also exhibit a very similar homogeneous structure with a mean fractal dimension . 

Open with DEXTER  
In the text 
Figure 2: qD_{q} plot of Swift short GRBs. Upper panel: spectrum of fractal dimensions of the angular distribution of the whole sample. This behavior with a maximum around q=3 clearly indicates a multifractal homogeneous distribution very close to that of a multifractional Brownian motion. Lower left panel: dimension moments of the 13 zcorrected GRBs with relatively high redshift. The structure is clearly multifractal. Lower panel: the subsample of GRBs with already T_{90}<2 s, exhibits a multifractal distribution, but with an asymptotic convergence to D_{q}=1. In this case, the structure traced by the GRBs appears to be more clustered. 

Open with DEXTER  
In the text 
Figure 3: Upper panel: qD_{q} plot of BATSE 4 short GRBs (Class I + Class II) exhibit a fractal structure with dimensions fluctuating around the value 1.8. Lower left and right panels: qD_{q} plots of BATSE 4 short bursts, in which Class II and Class III, respectively, clearly follow a multifractal distribution. 

Open with DEXTER  
In the text 
Copyright ESO 2010
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.