Is Angular D istribution o f GRBs random?

1. IsAngular Distribution of GRBs random? Lajos G. Balázs Konkoly Observatory, Budapest Collaborators: Zs. Bagoly (ELTE), I. Horváth (ZMNE), A. Mészáros (Ch. Univ. Prague), R. Vavrek (ESA)

2. Contents of this talk • Introduction • Mathematical considerations • formulation of the problem • preliminary studies • more sophisticated methods • Voronoi tesselation • Minimal spanning tree • Multifractal spectrum • Statistical tests • Discussion • Summary and conclusions

3. IntroductionGRB General properties GRB: energetic transient phenomena (duration < 1000 s, Eiso < 1054erg) strong evidences for cosmological origin (zmax = 8.1) physically not homogeneous population: short: T90 < 2 s, long: T90 > 2s The most comprehensive stuy 1991-2000 CGRO BATSE 2704 GRBs Recently working experiments: Swift, Agile, Fermi

4. IntroductionGRB profiles

5. IntroductionFormation of long GRBs

6. g-rays Inner Engine Relativistic Outflow Internal Shocks 106cm 1013-1015cm IntroductionOrigin of el.mag.rad. Afterglow External Shock 1016-1018cm

7. Introductionformation of short GRBs

8. IntroductionGRB and GW

9. IntroductionGRB angular distirbution

10. Mathematical considerationsformulation of the problem Cosmological distribution: large scale isotropy is expected Aitoff area conserving projection T90 > 2s T90 < 2s

11. Mathematical considerationsformulation of the problem The necessary condition ωcan be developed into series Isotropy: except except The null hypothesis (i.e. all ωkm = 0 except k=m=0) can be tested statistically

12. Mathematical considerationspreliminary studies(Balazs, L. G.; Meszaros, A.; Horvath, I., 1998, A&A., 339, 18) The relation of ωkm coffecients to the sample: Student t test was applied to test ωkm= 0 in the whole sample Results of the test Binomial tests in the subsamples

13. Mathematical considerationsmore sophisticated methods(Vavrek, R.; Balázs, L. G.; Mészáros, A.; Horváth, I.; Bagoly, Z., 2008,MNRAS, 391, 1741) Conclusion from the simple tests: short and long GRBsbehave in different ways! Definition of complete randomness: • Angular distribution independent on position i.e. P(Ω) depends only on the size ofΩandNOTon the position • Distribution in different directions independent i.e. probability of finding a GRB in Ω1 independent on finding one in Ω2 (Ω1, Ω2are NOT overlapping!)

14. Mathematical considerationsmore sophisticated methods Voronoi tesselation Cells around nearest data points Charasteristic quantities: • Cell area (A) • Perimeter (P) • Number of vertices (Nv) • Inner angle (αi) • Further combintion of these variables (e.g.): • Round factor • Modal factor • AD factor

15. Mathematical considerationsmore sophisticated methods Minimal spanning tree • Considers distances among points without loops • Sum of lengths is minimal • Distr. length and angles test randomness • Widely used in cosmology • Spherical version of MST is used

16. Mathematical considerationsmore sophisticated methods Multifractal spectrum • P(ε) probability for a point in ε area. • If P(ε) ~ εα thenα is the local fractal spectrum (α=2 for a completely random process on the plane)

17. Further statistical testsinput data and samples Most comprehensive sample of GRBs: CGRO BATSE2704 objects 5 subsamples were defined:

18. Statistical testsDefininition of test variables Voronoi tesselation • Cell area • Cell vertex • Cell chords • Inner angle • Round factor average • Round factor homegeneity • Shape factor • Modal factor • AD factor • Minimal spanning tree • Edge length mean • Edge length variance • Mean angle between edges • Multifractal spectrum • The f(α) spectrum

19. Statistical testsEstimation of the significance Assuming fully randomness 200 simulations in each subsample Obtained: simulated distribution of test variables

20. DiscussionSignificance of independent multiple tests Variables showing significant effect: differences among samples What is the probability for difference only by chance? Assuming that all the single tests were independent the probability that among ntrials at least mwill resulted significance where Particularly, giving in case of p=0.05, n=13 instead of

21. DiscussionJoint significance levels Test variables are stochastically dependent Proposition for Xk test variables (k=13 in our case): flhidden variables arenotcorrelated (m=8 in our case) Compute the Euclidean dist. from the mean of test variables:

22. DiscussionStatistical results and interpretations short1, short2, interm. samples are nonrandom long1,long2are random Swift satellite: • Long at highz (zmax=6.7) • Short at moderatez (zmax=1.8) Different progenitors anddifferent spatial samp-ling frequency

23. Discussionstatistical results and interpretetions Angular scale • Short1 12.6o • Short2 10.1o • Interm. 12.8o • Long1 7.8o • Long2 6.5o Angular distance: Sloan great wall

24. DiscussionLarge scale structures in the Universe

25. Discussionlarge scale structure of the Universe (z < 0.1)

26. Discussionlarge scale structure of the Universe (WMAP)

27. Discussionmodeling large scale structures

28. DiscussionMillenium simulation (Springel et al. 2005)

29. Discussionconstraining large scale structures ”Millenium simulation” 1010particles in 500h-1cube first structures at z=16.8100h-1 scale (Springel et al. 2005) Long GRBsmark the early stellar population Short GRBs mark the old disc population

30. Summary and conclusions • We find difference between short and long GRBs • We defined five groups (short1, short2, inter-mediate, long1, long2) • We introduced 13 test-variables (Voronoi cells, Minimal Spanning Tree, Multifractal Spectrum) • We made 200 simulation for each samples • Differences between samples in the number of test variables giving positive signal • We computed Euclidean distances from the simulated sample mean • Short1, short2, intermediate are notfully random

31. Thank you!