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, 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 CGRO BATSE 2704 GRBs
Recently working experiments:
Swift, Agile, Fermi

4. Introduction GRB profiles

5. Introduction Formation of long GRBs

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

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
T90 > 2s
T90 < 2s

11. 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

12. 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

13. 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!)

14. 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

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) ~ εα 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 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:

22. 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

23. Discussion statistical results and interpretetions
Angular scale
Short o
Short o
Interm o
Long o
Long 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” 1010 particles in 500h-1 cube first structures at z= h-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

31. Thank you!