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IsAngular Distribution of GRBs random?
Lajos G. BalázsKonkoly 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
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
IntroductionGRB profiles
Introduction Formation of long GRBs
Relativistic Outflow
InternalShocks
-rays
10101313-10-101515cmcm
InnerEngine
101066cmcm
ExternalShock
Afterglow
10101616-10-101818cmcm
IntroductionOrigin of el.mag.rad.
Introductionformation of short GRBs
IntroductionGRB and GW
IntroductionGRB angular distirbution
Mathematical considerationsformulation 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
),(),( blYbl mkkm 00 mkkivévekm except
),...,(),(),...,,,( 11 kk xxgblxxblf
Isotropy:
except
The null hypothesis (i.e. all ωkm = 0 except k=m=0) can be tested statistically
Mathematical considerationspreliminary studies(Balazs, L. G.; Meszaros, A.; Horvath, I., 1998, A&A., 339, 18)
)( knkk qp
k
nP
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
),(),(),(1
1ii
N
i
ml
mkkm blYNdblYbl
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 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 independenti.e. probability of finding a GRB in Ω1
independent on finding one in Ω2
)()(),( 2121 PPP
(Ω1, Ω2 are NOT overlapping!)
Mathematical considerationsmore sophisticated methods
Voronoi tesselationCells around nearest data points
Charasteristic quantities:o Cell area (A)o Perimeter (P)
o Number of vertices (Nv)
o Inner angle (αi)
o Further combintion of these variables (e.g.): Round factor Modal factor AD factor
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
Mathematical considerationsmore 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 testsinput data and samples
Most comprehensive sample of GRBs: CGRO BATSE 2704 objects
5 subsamples were defined:
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
Statistical testsEstimation of the significance
Assuming fully randomness 200 simulations in each subsampleObtained: simulated distribution of test variables
DiscussionSignificance of independent multiple tests
Variables showing significant effect: differences among samplesWhat is the probability for difference only by chance?
Assuming that all the single tests were independent theprobability that among n trials at least m will resulted significance
m
mk
nkn PmP )(
n
mk
nkn PmP )(
n
mk
nkn PmP )( where
knknk pp
knkn
P
)1()!(!
!
Particularly, nppP nn )1(1)1(
giving in case of p=0.05, n=13 49.0)05.01(1 13
instead of
95.005.01
95.005.01
DiscussionJoint 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:
pmpkfaX kl
m
lklk
;,,11
28
22
21
2 fffd
DiscussionStatistical 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
Discussionstatistical results and interpretetions
Angular scale• Short1 12.6o
• Short2 10.1o
• Interm. 12.8o
• Long1 7.8o
• Long2 6.5o
Angular distance:
)(1
'])'2(')'1()'1[()1(
2/1
0
2
0
spaceEuclidean
dzzzzzzH
cDd
M
M
z
A
Sloan great wall
DiscussionLarge scale structures in the Universe
Discussionlarge scale structure of the Universe (z < 0.1)
Discussionlarge scale structure of the Universe (WMAP)
Discussionmodeling large scale structures
DiscussionMillenium simulation (Springel et al. 2005)
Discussionconstraining 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 populationShort 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