On the large-scale angular distribution of short-Gamma ray bursts
aa r X i v : . [ a s t r o - ph ] O c t On the large-scale angular distribution of short-Gamma ray bursts
A. Bernui, I. S. Ferreira, and C. A. Wuensche
Instituto Nacional de Pesquisas Espaciais, Divis˜ao de Astrof´ısica,Av. dos Astronautas 1758, CEP 12227-010 S˜ao Jos´e dos Campos, SP, Brazil [email protected]; [email protected]; [email protected]
ABSTRACT
We investigate the large-scale angular distribution of the short-Gamma ray bursts (SGRBs)from BATSE experiment, using a new coordinates-free method. The analyses performed take intoaccount the angular correlations induced by the non-uniform sky exposure during the experiment,and the uncertainty in the measured angular coordinates. Comparising the large-scale angularcorrelations from the data with those expected from simulations using the exposure functionwe find similar features. Additionally, confronting the large-angle correlations computed fromthe data with those obtained from simulated maps produced under the assumption of statisticalisotropy we found that they are incompatible at 95% confidence level. However, such differencesare restricted to the angular scales 36 ◦ − ◦ , which are likely to be due to the non-uniformsky exposure. This result strongly suggests that the set of SGRBs from BATSE are intrinsicallyisotropic. Moreover, we also investigated a possible large-angle correlation of these data with thesupergalactic plane. No evidence for such large-scale anisotropy was found. Subject headings: large-scale structure of universe — gamma rays: bursts — methods: statistical
1. Introduction
The apparent isotropy in the large-scale an-gular distribution of the Gamma ray bursts(GRBs) is a long-standing debate (Meegan et al.1992; Briggs et al. 1996; Tegmark et al. 1996;Piran & Singh 1997; Metzger et al. 1997; Bal´azs, M´esz´aros, & Horv´ath1998; M´esz´aros, Bagoly, & Vavrek 2000). Sincethe first detection with the VELA satellite (Klebesadel et al.1973) the origin of these highly energetic eventshas remained a challenge. Even if the originof GRBs turns out to be extra-galactic or cos-mological, as suggested by current data (see,e.g., Piran (2004); Zhang & M´esz´aros (2006);M´esz´aros (2002, 2006) and references therein),this does not ensure that their distribution isisotropic. Up to now, no dominant anisotropieshas been found in the angular distribution ofGRBs. However, if detected, small anisotropiceffects may reveal valuable information abouttheir origin. Additionally, the discovery of a largeangular scale pattern in the sky distribution ofGRBs may be useful to identify their sources by cross-correlating them with catalogs of cos-mic objects, e.g., early-type galaxies, hard X-raysources, etc. (Briggs et al. 1996; Tegmark et al.1996; Piran 2004; M´esz´aros 2006).The reported statistical analyses of the all-sky survey data from BATSE show that theirlarge-scale angular distribution is consistent withisotropy (Piran 2004; M´esz´aros 2006), althoughaspects like observational artifacts have not beenfully considered in some of these studies. It is wellknown that anisotropies with distinct origins man-ifest themselves on different angular scales andwith different magnitudes. In this connection, itis reasonable to consider different approaches thatcan, in principle, provide information about multi-ple types of anisotropy, imprinted as angular cor-relation signatures (ACS), that may be possiblypresent in the data.Here we apply a new coordinates-free methodto search for large-scale ACS ( ≥ ◦ ) in a subsetof the BATSE GRB data (Meegan et al. 2000),namely the Short-GRBs, and then investigate1heir significance levels through the comparisonwith a large number of Monte Carlo maps. Suchsimulated maps were produced under similar con-ditions as the catalog under analysis, that is, tak-ing into account the non-uniform sky exposure ofBATSE and the uncertainty in the coordinatesmeasurements. Furthermore, for completeness,we also compare the ACS of the catalog of GRBswith those corresponding to statistically isotropicMonte Carlo maps. Finally, we also investigatedthe possible large-scale angular correlation be-tween the set of Short-GRBs and the supergalac-tic plane, in an attempt to search for likely hostgalaxies of these events (as recently suggestedby Ghirlanda et al. (2006)).The outline of this paper is the following: insection 2 we use GRBs data from the BATSE ex-periment to determine the Short-GRBs catalog tobe investigated, and in section 3, we describe themethod employed in such studies. The data anal-yses and results are shown in section 4, and finallyin section 5 we formulate our conclusions.
2. The Short-GRBs from BATSE catalog
The physical analysis of GRBs utilizes theirtemporal and spectral properties (Fishman & Meegan1995). Despite the different light-curves observedin the spectra of GRBs, a useful parameter toclassify them is the burst duration T , defined asthe time interval during which 90% of the fluenceis measured. The current BATSE catalog 4Bc contains 2 702 events, where only 2 037 GRBshave their parameter T measured (Meegan et al.2000).At first, the T value was used to divide the setof GRBs into two different sub-classes: the Short-GRBs (SGRBs), with T < T &
10s (Kouveliotou et al. 1993;Zhang & M´esz´aros 2006; M´esz´aros 2002, 2006).However, the use of this definition of SGRBs isinstrument dependent and is susceptible to ob-servational biases (Hakkila et al. 2007b). For thisreason, one should consider in addition to the T criterium the parameter HR / (Mukherjee et al.1998) which is defined as the 100 to 300 keVfluence divided by the 25 to 100 keV fluence ofeach GRB in BATSE 4Bc (Hakkila et al. 2007a,b). http://gammaray.msfc.nasa.gov/batse/grb/catalog/current/ Thus, the appropriate definition for a catalog ofSGRBs is (Hakkila et al. 2007a,b): C = { ≤ T < .
7s and HR / > } .With this information, and using a new coordinates-free method to be described in the next section,we shall perform a detailed analysis of the large-scale ACS present in the sky distribution of theSGRBs from BATSE.
3. The 2PACF and the Sigma-Map analy-sis
Let Ω γ j ≡ Ω( θ j , φ j ; γ ) ∈ S be a spheri-cal cap region on the celestial sphere, of γ de-grees of aperture, with vertex at the j -th pixel, j = 1 , . . . , N caps , where ( θ j , φ j ) are the angularcoordinates of the center of the j -th pixel. Both,the number of spherical caps N caps and the coor-dinates of their center ( θ j , φ j ) are defined usingthe HEALPix pixelization scheme (G´orski et al.2005). The spherical caps are such that theirunion completely covers the celestial sphere S .Let C j be the catalog of cosmic objects locatedin the j -th spherical cap Ω γ j . The 2PACF of theseobjects (Chen & Hakkila 1998; Padmanabhan1993), denoted as Υ j ( γ i ; γ ), is the differencebetween the normalized frequency distributionand that expected from the number of pairs-of-objects with angular distances in the inter-val ( γ i − . δ, γ i + 0 . δ ] , i = 1 , . . . , N bins , where γ i ≡ ( i − . δ and δ ≡ γ /N bins is the bin-width.The expected frequency distribution is achievedby a large number of Monte Carlo realizations ofisotropically distributed objects in Ω γ j , contain-ing a similar number of objects as in C j (Teixeira2003; Bernui & Villela 2006). The 2PACF has theproperty that its mean, obtained by integratingover all separation angles (Chen & Hakkila 1998),is zero. A positive (negative) value of Υ j indi-cates that objects with these angular separationsare correlated (anti-correlated), while zero indi-cates no correlation.Define now the scalar function σ : Ω j
7→ ℜ + ,for j = 1 , . . . , N caps , which assigns to the j -cap, centered at ( θ j , φ j ), a real positive number σ j ≡ σ ( θ j , φ j ) ∈ ℜ + . The most natural way ofdefining a measure σ is through the variance of the2 j function, we thus define (Bernui et al. 2007) σ j ≡ N bins N bins X i =1 Υ j ( γ i ; γ ) . (1)To obtain a quantitative measure of the ACS ofthe GRBs sky map, we cover the celestial spherewith N caps spherical caps, and calculate the set ofvalues { σ j , j = 1 , ..., N caps } using eq. (1). Patch-ing together the set { σ j } in the celestial sphereaccording to a coloured scale (where, for instance, σ minimum → blue , σ maximum → red ) we obtaina sigma-map. Finally, we quantify the ACS of agiven sigma-map by calculating its angular powerspectrum. Because the sigma-map assigns a realvalue to each pixel in the celestial sphere, thatis σ = σ ( θ, φ ), one can expand it in sphericalharmonics: σ ( θ, φ ) = P ℓ, m A ℓ m Y ℓ m ( θ, φ ) wherethe set of values { S ℓ , ℓ = 0 , , , ... } , defined by S ℓ ≡ (1 / (2 ℓ + 1)) P ℓm = - ℓ | A ℓ m | , is the angularpower spectrum of the sigma-map. Because weare interested in the large-scale angular correla-tions, we shall concentrate on { S ℓ , ℓ = 1 , , ..., } .Notice that we are interested in the angular powerspectrum of the sigma-map, and not that of the ce-lestial sphere where the GRB events are located;this later case was already done by Briggs et al.(1996) and Tegmark et al. (1996). As we shallsee, the sigma-map analysis is able to reveal verysmall anisotropies, like those induced by the BAT-SEs sky exposure, despite the small burst detec-tion rate of the SGRBs.
4. Data analyses and results
In this section we shall apply the sigma-mapmethod explained in the previous section to studythe large-scale ACS present in the angular distri-bution of the 516 SGRBs listed in the catalog C . Inthe following, all the sigma-maps were calculatedusing spherical caps of γ = 90 ◦ of aperture, thatis hemispheres (smaller spherical caps have lessSGRBs in each C j , j = 1 , . . . , N caps , hence pro-duce large statistical noise in the Υ j functions).We also used N bins = 90 and N caps = 768 in theseanalyses.An important issue that deserves close inspec-tion is the presence of anisotropic ACS in the datainduced by the non-uniform sky exposure (NUSE)during the BATSE experiment (Hakkila et al. 1998), expected because some latitudes of thesky were over-observed while others were under-observed. Because there are no reports quantify-ing or tracing out the influence of the NUSE atlarge angular scales in the current BATSE catalog4Bc (see Chen & Hakkila (1998) for analyses ofthe 3B and 4B catalogs) it is interesting to usethe sigma-map method to investigate the possi-ble anisotropic angular correlations that may bepresent in the data even if their magnitudes aresmall. For this, our strategy to reveal the large-scale ACS in the data runs in three steps. First,we produce 1 000 Monte Carlo maps simulatingthe sky positions of 516 cosmic objects accord-ing to the NUSE function (Hakkila et al. 1998),then we calculate in each case the correspondingsigma-map, and finally we compute the angularpower spectrum { S ℓ , ℓ = 1 , ..., } of each of thesesigma-maps. Second, we generate 1 000 MonteCarlo maps simulating the sky positions of 516isotropically distributed cosmic objects, then wecompute for each of these Monte Carlo maps theircorresponding sigma-maps, and finally we calcu-late the angular power spectrum of these sigma-maps. Third, we calculate the sigma-maps, andtheir respective angular power spectrum, of theSGRBs listed in catalog C .In figure 1 we show two sigma-maps in galac-tic coordinates. In the top panel, we show theaverage of 100 sigma-maps, randomly chosen inbetween the 1 000 sigma-maps computed from asimilar number of Monte Carlo sky maps whichsimulate different catalogs of SGRB according tothe NUSE function. In the bottom panel we ex-hibit the sigma-map corresponding to the catalog C . In figure 2 we display a comparative analyses,taking into account isotropic and non-isotropiccases, of the angular power spectrum of the sigma-map obtained from the angular distribution of theSGRBs listed in C . In the top panel, we plot-ted the angular power spectrum S ℓ versus ℓ of thesigma-map obtained from the catalog C , togetherwith the mean of 1 000 sigma-maps computed froman equal number of statistically isotropic MonteCarlo sky maps. In the bottom panel, the plot issimilar except that the Monte Carlo sky maps haveanisotropic ACS because were produced consider-ing the NUSE function of BATSE. In both plotsthe shadowed areas correspond to 2 standard de-3iations. Besides some small differences, the an-gular power spectra corresponding to the sigma-maps computed from the SGRBs show a very sim-ilar large-scale structure when compared with themean angular power spectrum of the sigma-mapsobtained from Monte Carlos produced accordingto the NUSE function.A comparative analysis of the ACS correspond-ing to these cases, isotropic and non-isotropicdue to NUSE function, is better seen if we plot ℓ ( ℓ +1) S ℓ versus ℓ , as showed in figure 3. There wedisplay the correponding data from the SGRBs to-gether with the mean of the angular power spectraof the isotropic and non-isotropic cases, where nowthe shadowed area corresponds to 2 standard devi-ations of the isotropic case. As observed, the datahave a very similar behavior to the non-isotropiccase, and is different from the isotropic case whichshows a flat spectrum. Thus, data and simulatedisotropic maps are incompatible at 95% confidencelevel. However such differences are mainly re-stricted to the angular scales 36 ◦ − ◦ which ex-actly reproduce the imprints exhibited by the an-gular power spectrum of the non-isotropic case. Inthe absence of ACS other than those expected bythe NUSE of the BATSE experiment, this resultstrongly suggests that the SGRBs are intrinsicallyisotropic.To test the robustness of our calculationswe also performed the sigma-map analyses with N bins = 180 and N caps = 3 072, obtaining thesame result.Furthermore, we also searched for a possiblecorrelation between the SGRBs listed in C withthe supergalactic plane, where nearby galaxies ap-pear to be more concentrated. Because we donot know how many events could be originatedin these galaxies, we generate three sets of 300Monte Carlos considering in each case a differ-ent number of simulated GRBs provided by ananisotropic distribution which selects events nearthe supergalactic plane. That is, we generatedsets of maps where 33%, 50%, and 66% of theevents were produced by such anisotropic distri-bution, respectively. We then computed their cor-responding sigma-maps in order to compare themwith the sigma-map calculated from the SGRBslisted in C . To measure such a possible correlationwe computed the linear Pearson correlation coef-ficient between the sigma-map of the SGRBs and each one of the sigma-maps obtained from thesesets of Monte Carlo realizations. Notice that sucha coefficient varies from 0 (for totally uncorrelatedmaps) to 1 (for fully correlated maps). Our resultsshow that the Pearson’s coefficient is, in mean, lessthan 0 .
03 (using 300 Monte Carlos for each of thethree above mentioned cases). To realize whetherthis value is statistically significant, we computedthe Pearson’s coefficient in some illustrative cases.For example, the mean Pearson’s coefficient corre-lating one sigma-map, coming from a given set ofsigma-maps computed from the above mentioned66% anisotropic Monte Carlo maps, with the restof sigma-maps from such set is 0 . ± .
16 (theresult is similar in the other two cases). On theother hand, the mean Pearson’s coefficient corre-lating one sigma-map, chosen randomly from theset of 1 000 sigma-maps calculated from MonteCarlo maps produced according to the NUSE func-tion, with the rest of sigma-maps of this set is0 . ± .
16. Similarly, the mean Pearson’s coef-ficient correlating any sigma-map, from the setof 1 000 sigma-maps computed from Monte Carlostatistically isotropic maps, with the rest of sigma-maps of this set is 0 . ± . C with each of the1 000 sigma-maps, obtained from Monte Carlomaps generated according to the NUSE func-tion, the mean Pearson’s coefficient is 0 . ± . . ± .
09. Taken together this informa-tion suggests that there is no evidence for a large-scale angular correlation between BATSE SGRBsand simulated maps produced considering differ-ent amounts of events coming from an anisotropicdistribution that selects positions in the super-galactic plane and its surrounds. Notice that thisresult does not contradict the correlation foundby Ghirlanda et al. (2006) which is valid for smallangular distances ≤ ◦ , while the present analysisconcerns angular scales ≥ ◦ .Finally, we also tested the robustness of our re-sults under the change of the angular coordinatesof the BATSE SGRBs due to the measured er-ror boxes (Briggs et al. 1998). This was done bysorting their angular coordinates within the lim-4ts given by the error boxes ( ± . ◦ ) we found nomeasurable difference with respect to the resultspresented here.
5. Conclusions
The purpose of this study is to know the large-scale angular correlations of the set of 516 BATSESGRBs, and to discover if these correlations arecompatible with a statistically isotropic distribu-tion of events, or instead they reveal the ACS re-sulting as a consequence of the NUSE function ofBATSE experiment. To elucidate this, we needto know the angular power spectra of two sets ofsigma-maps: one set is computed from statisticallyisotropic Monte Carlo maps and the other is cal-culated from Monte Carlo maps that simulate thesky position of the events using the NUSE functionof BATSE experiment. After that, we compare thepower spectra of these sigma-maps with the angu-lar power spectrum of the sigma-map computedfrom the BATSE SGRBs data.The first thing to be noticed in the angularpower spectrum of the BATSE SGRBs, plotted infigure 2, is the absence of dominant anisotropies atthe largest angular scales 60 ◦ − ◦ ( ℓ = 1 , , ≤ ◦ ( ℓ ≥ ◦ − ◦ ( ℓ = 4 , ℓ ( ℓ + 1) S ℓ versus ℓ (see figure 3). In fact, figure 3 reveals asecond interesting thing, that is, the non-flat spec-trum of the BATSE data (represented by bullets)which clearly differs from the flat angular powerspectra showed by the statistically isotropic MonteCarlo data (the dashed line). We also observe infigure 3 that the mean of the angular power spec-tra of the sigma-maps computed from Monte Carlomaps generated according to the NUSE function(the dot-dashed line) has also a non-flat spectrumwhich is very similar to the corresponding oneobtained from BATSE SGRBs. In other words,the large-scale angular correlations of the BATSESGRBs exhibit the anisotropic imprints expectedin the data due to the NUSE of BATSE experi-ment. Other ACS are not found to be statisticallysignificant, at 95% CL. In conclusion, these resultsstrongly suggests that the SGRBs are intrinsicallyisotropic. Finally, we also studied the possible large-angle correlation between the SGRBs data andMonte Carlos with (different amounts of) sim-ulated events concentrated towards the super-galactic plane. No evidence for such large-scaleanisotropy was found in the BATSE SGRBs. Acknowledgments
We acknowledge use of the BATSE data (Meegan et al.2000). Some of the results in this paper have beenderived using the HEALPix package (G´orski et al.2005). We thank T. Villela, Z. Bagoly, A. Teix-eira, R. Tavakol, E. Berger, and B. Schaefer forinsightful comments and suggestions. We alsoacknowledge Prof. J. Hakkila for very healpful ex-changes regarding BATSE data. We are indebtedto the anonymous referee for valuable suggestionsand constructive criticisms. CAW was partiallysupported by CNPq grant 307433/2004-8. ISFand AB acknowledge CAPES and PCI/DTI-MCTfellowships, respectively.
REFERENCES
Bal´azs, L. G., M´esz´aros, A., & Horv´ath, I. 1998,A&A, 339, 1Bernui, A. & Villela, T. 2006, A&A, 445, 795Bernui, A., Mota, B., Rebou¸cas, M. J., & Tavakol,R. 2007, A&A, 464, 479Briggs, M. S., et al. et al. et al. et al. http://gammaray.msfc.nasa.gov/batse/grb/catalog/current/
M´esz´aros, A., Bagoly, Z., & Vavrek, R. 2000,A&A, 354, 1M´esz´aros, P. 2002, ARA&A, 40, 137M´esz´aros, P. 2006, Rept. Prog. Phys., 69, 2259Metzger, M., et al.
Structure formation inthe universe (Cambridge Univ. Press)Piran, T. & Singh, A. 1997, ApJ, 483, 552Piran, T. 2004, Rev. Mod. Phys., 76, 1143Tegmark, M., Hartmann D. H., Briggs M. S., &Meegan, C. A. 1996, ApJ, 468, 214Teixeira, A. F. F. 2003; physics/0312013Zhang, B. & M´esz´aros P. 2004, Int. J. Mod. Phys.A, 19, 2385
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