1. Angular distribution of GRBs Lajos G. Balázs Konkoly Observatory. Budapest Collaborators: Zs. Bagoly (ELTE). I. Horváth (ZMNE). A. Mészáros (Ch. Univ. Prague). P. Veres (ELTE. ZMNE)

2. Contents of this talk  Introduction  Mathematical considerations  formulation of the problem  preliminary studies  more sophisticated methods  Statistical tests with BATSE data  Statistics with Swift GRBs  Discussion  Summary and conclusions

3. Introduction GRB General properties GRB: energetic transient phenomena (duration < 1000 s. E iso < 10 54 erg) strong evidences for cosmological origin (z max = 8.1) physically not homogeneous population: short: T90 2s The most comprehensive stuy 1991-2000 CGRO BATSE 2704 GRBs Recently working experiments: Swift. Agile. Fermi

4. Introduction GRB profiles

5. Introduction Formation of long GRBs

6. Relativistic Outflow Internal Shocks  -rays 10 13 -10 15 cm Inner Engine 10 6 cm External Shock Afterglow 10 16 -10 18 cm Introduction O rigin of el.mag.rad.

7. Introduction formation of short GRBs

8. Introduction GRB angular distirbution

9. Mathematical considerations formulation of the problem Cosmological distribution: large scale isotropy is expected Aitoff area conserving projection T 90 > 2sT 90 < 2s

10. Mathematical considerations formulation of the problem The necessary condition ω can be developed into series except Isotropy: except The null hypothesis (i.e. all ω km = 0 except k=m=0) can be tested statistically

11. Mathematical considerations preliminary studies ( Balazs. L. G.; Meszaros. A.; Horvath. I.. 1998. A&A.. 339. 18) The relation of ω km coffecients to the sample: Student t test was applied to test ω km = 0 in the whole sample Results of the testBinomial tests in the subsamples

12. Mathematical considerations more 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 GRBs behave in different ways! Definition of complete randomness:  Angular distribution independent on position i.e. P(Ω) depends only on the size of Ω and NOT on the position  Distribution in different directions independent i.e. probability of finding a GRB in Ω 1 independent on finding one in Ω 2 (Ω 1. Ω 2 are NOT overlapping!)

13. Mathematical considerations more sophisticated methods Voronoi tesselation Cells around nearest data points Charasteristic quantities: oCell area (A) oPerimeter (P) oNumber of vertices (N v ) oInner angle (α i ) oFurther combintion of these variables (e.g.):  Round factor  Modal factor  AD factor

14. Mathematical considerations more 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

15. Mathematical considerations more sophisticated methods Multifractal spectrum  P(ε) probability for a point in ε area.  If P(ε) ~ ε α then α is the local fractal spectrum (α=2 for a completely random process on the plane)

16. Further statistical tests input data and samples Most comprehensive sample of GRBs: CGRO BATSE 2704 objects 5 subsamples were defined:

17. Statistical tests Defininition 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

18. Statistical tests Estimation of the significance Assuming fully randomness 200 simulations in each subsample Obtained: simulated distribution of test variables

19. Discussion Significance 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 n trials at least m will resulted significance where Particularly. giving in case of p=0.05. n=13 instead of

20. Discussion Joint significance levels Test variables are stochastically dependent Proposition for X k test variables (k=13 in our case): f l hidden variables are not correlated (m=8 in our case) Compute the Euclidean dist. from the mean of test variables:

21. Discussion Statistical results and interpretations short1. short2. interm. samples are nonrandom long1. long2 are random Swift satellite: ―Long at high z (z max =6.7) ―Short at moderate z (z max =1.8) Different progenitors and different spatial samp- ling frequency

22. Statistics with Swift GRBs BAT sensitivity: coded mask Exposure function (slewing blocked by Earth. Moon. Sun) Generation of ran- dom samples by taking account of EF

23. Statistics with Swift GRBs (2) Tests applied ( Briggs. M.. 1993 ): Rayleigh-statistics Watson-statistics Bingham-statistics ( k eigen val. of M N )

24. 1000 MC simulations (all GRBs)

25. 1000 MC simulations (short GRBs)

26. 1000 MC simulations (long GRBs)

27. 1000 MC simulations (int. GRBs)

28. 1000 MC simulations (significance) Group N Bingham (p-value) Rayleigh (p-value) M N eigenv. λ 1 (p-value) Short 31 78.6 (0.644) 5.6 (0.409)0.387 (0.553) Interm. 46 123.8 (0.038) 10.4 (0.089) 0.441 (0.072) Long 331 831.5 (0.694) 23.2 (0.290) 0.364 (0.645) Total 408 1033.3 (0.179) 31.5 (0.183) 0.387 (0.138)

29. Sky distribution (interm.)

30. Discussion statistical results and interpretetions Angular scale (BATSE) Short1 12.6 o Short2 10.1 o Interm. 12.8 o Long1 7.8 o Long2 6.5 o Angular distance: Sloan great wall

31. Discussion modeling large scale structures

32. Discussion constraining large scale structures ”Millenium simulation” 10 10 particles in 500h -1 cube first structures at z=16.8 100h -1 scale (Springel et al. 2005) Long GRBs mark the early stellar population Short GRBs mark the old disc population

33. Summary and conclusions  We found difference between short and long GRBs  We defined five groups in BATSE (short1. short2. inter-mediate. long1. long2)  We introduced 13 test-variables (Voronoi cells. Minimal Spanning Tree. Multifractal Spectrum)  We made MC simulation for each samples  Differences between samples in the number of test variables giving positive signal  Short1, short2, intermediate are not fully random  Swift data gave some significance only for interm. population