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)

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

Introduction GRB 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

Introduction GRB profiles

Introduction Formation of long GRBs

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

Introduction formation of short GRBs

Introduction GRB and GW

Introduction GRB angular distirbution

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

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

Mathematical considerations preliminary studies (Balazs, L. G Mathematical considerations preliminary 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

Mathematical considerations more sophisticated methods (Vavrek, R 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 Ω1 independent on finding one in Ω2 (Ω1, Ω2 are NOT overlapping!)

Mathematical considerations more 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

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

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

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

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

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

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

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

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

Discussion statistical 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

Discussion Large scale structures in the Universe

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

Discussion large scale structure of the Universe (WMAP)

Discussion modeling large scale structures

Discussion Millenium simulation (Springel et al. 2005)

Discussion constraining large scale structures ”Millenium simulation” 1010 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

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

Thank you!