1. Is Angular 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. 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 and GW

9. Introduction GRB angular distirbution

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

11. 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

12. 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

13. 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!)

14. 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

15. 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

16. 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)

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

18. 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

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

20. 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

21. 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:

22. 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

23. Discussion statistical results and interpretetions Angular scale 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

24. Discussion Large scale structures in the Universe

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

26. Discussion large scale structure of the Universe (WMAP)

27. Discussion modeling large scale structures

28. Discussion Millenium simulation (Springel et al. 2005)

29. 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

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 not fully random