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