Determination of The intrinsic nature
of the Luminosity -time correlation in
the X-ray afterglows of GRBs
Maria Giovanna Dainotti
Stanford University, Stanford, USA
Jagellonian University, Krakow, Poland
in collaboration with
V. Petrosian, Singal,J., R. Willingale, Ostrowski M and Capozziello S.,
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Notwithstanding the variety of GRB’s different peculiarities, some common
features may be identified looking at their light curves.
A crucial breakthrough : Phenomenological model with SWIFT lightcurves
• a more complex behavior of the lightcurves, different from the broken
power-law assumed in the past (Obrien et al. 2006,Sakamoto et al. 2007)
A significant step forward in determining common features in the afterglow
• X-ray afterglow lightcurves of the full sample of Swift GRBs shows that
they may be fitted by the same analytical expression (Willingale et al. 2007)
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Dainotti et al. correlation
LX
T
*
a
Firstly discovered in 2008 by Dainotti, Cardone, & Capozziello MNRAS, 391, L 79D
(2008)
Later uptdated by Dainotti, Willingale, Cardone, Capozziello & Ostrowski
ApJL, 722, L 215 (2010)
GRB Symposium,
Nashville,
April 2013 of 62 long afterglows
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Lx(T*a) vs T*a distribution
for the
sample
Data and methodology
Sample : 77 afterglows, 66 long, 11 from IC class (short GRBs with extended
emission) detected by Swift from January 2005 up to March 2009, namely all the GRBs
with good coverage of data that obey to the Willingale et al. 2007 model with firm
redshift.
 Redshifts : from the Greiner's web page http://www.mpe.mpg.de/jcg/grb.html.
Redshift range 0.08 <z < 8.2
 Spectrum for each GRB was computed during the plateau with the Evans et al. 2010
web page http://www.swift.ac.uk/burst_analyser/
For some GRBs in the sample the error bars are so large that determination of the
observables (Lx, Ta ) is not reliable. Therefore, we study effects of excluding such cases
from the analysis (for details see Dainotti et al. 2011, ApJ 730, 135D ).
To study the low error subsamples we use the respective logarithmic errors bars to
formally define the error energy parameter
 ( E )  (   )
2
Lx
2 1/ 2
Ta
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Prompt – afterglow correlations
Dainotti et al., MNRAS, 418,2202, 2011
A search for possible physical relations between
the afterglow characteristic luminosity L*a ≡Lx(Ta)
and
the prompt emission quantities:
1.) the mean luminosity derived
as <L*p>45=Eiso/T*45
2.) <L*p>90=Eiso/T*90
3.) <L*p>Tp=Eiso/T*p
4.) the isotropic energy Eiso
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The search for standard GRBs continues
Correlation coefficients ρ for for the long
L*a vs. <L*p>45 for 62 long GRBs
(the σ(E) ≤ 4 subsample).
(L*a, <L*p>45 )
(L*a, <L*p>90)
2 1/ 2
(L*a, <L*p>Tp )
Ta
GRB Symposium, Nashville,(L*a,
April 2013
Eiso )
 ( E )  (   )
2
Lx
GRB subsamples
with the varying error parameter u
- red
- black
- green
- blue
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Conclusion I
GRBs with well fitted afterglow light curves
obey tight physical scalings, both in their afterglow
properties and in the prompt-afterglow relations.
We propose these GRBs as good candidates for
the standard Gamma Ray Burst
to be used both
- in constructing the GRB physical models and
- in cosmological applications
-
(Cardone, V.F., Capozziello, S. and Dainotti, M.G 2009, MNRAS, 400, 775C
Cardone, V.F., Dainotti, M.G., Capozziello, S., and Willingale, R. 2010, MNRAS,
408, 1181C)
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Let’s go one step back
BEFORE
• proceeding with any further application to cosmology
• or using the luminosity-time correlation as discriminant among
theoretical models for the plateau emission
We need to answer the following question:
Is what we observe a truly representation of the events or there
might be selection effect or biases?
Is the LT correlation intrinsic to GRBs, or is it only an apparent one,
induced by observational limitations and by redshift induced
correlations?
THEREFORE,
at first one should determine the true correlations among the
variables
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Division in redshift bins for the updated sample of 100 GRBs
(with firm redshift and plateau emission)
ρ=-0.73 for all the distribution
b= -1.32±0.20 1σ compatible with the previous fit
From a visual inspection it is hard to evaluate if there is a redshift induced
correlation. Therefore, we have applied the test of Dainotti et al. 2011, ApJ,
730, 135D to check that the slope of every redshift bin is consistent with every
other. BUT for a quantitative analysis we turn to …..
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The Efron and Petrosian Method
The Efron & Petrosian method (EP) (ApJ, 399, 345,1992)
• to obtain unbiased correlations, distributions, and evolution with redshift from a
data set truncated due to observational biases.
• corrects for instrumental threshold selection effect and redshift induced
correlation
• has been already successfully applied to GRBs (Lloyd,N., & Petrosian, V. ApJ, 1999)
The technique we applied
• Investigates whether the variables of the distributions, L*X and T*a are
correlated with redshift or are statistically independent.
• do we have luminosity vs. redshift evolution?
• do we have plateau duration vs. redshift evolution?
• If yes,
how to accomodate the evolution results in the analysis?
By defining new independent variables!
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How the new variables are built?
• The new variables will be uninvolved in
redshift, namely they will be not affected by
redshift evolution.
The correction will be the following
• For luminosity:
where the luminosity evolution
• For time
where the time evolution
We denote with ‘ the not evolved observables
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How to compute g(z) and f(z)?
The EP method deals with
• data subsets that can be constructed to be
independent of the truncation limit suffered
by the entire sample.
• This is done by creating 'associated sets',
which include all objects that could have been
observed given a certain limiting luminosity.
• We have to determine the limiting luminosity
for the sample
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Luminosity vs 1+z
The more appropriate Flux limit is the black dotted line Flux=
In such a way we have 90 GRBs in total, but with an
appropriate limiting flux
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A specialized version of the Kendall rank correlation
coefficient, τ
a statistic tool used to measure the association
between two measured quantities
• takes into account the associated sets and not the
whole sample
•
produces a single parameter whose value directly
rejects or accepts the hypothesis of independence.
• The values of kL and kT for which τ L,z = 0 and
τ T,z = 0 are the ones that best fit the luminosity and
time evolution respectively.
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KL =-0.05
for 100 GRBs red line
green 53 GRBs
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KT =-0.85
A new approach:
• Never applied in literature so far to find the true slope of correlation
• We consider again the method of the associated set to find the slope of
the correlation not evolved with redshift :
• The slope computed with this method is 1 σ compatible with
observational results in 1.5<b<2.0
• α=b’
For 100 GRBs
The true slope:
-1.24 <b< -0.93
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Cumulative luminosity function and density function
For luminosity function the correction is
Not relevant
Log sigma (z) = Sum (1+1/m(j))
where m(j) is the number of the
associated sets in the sample
The red points show the correction with the Efron and Petrosian (1992) method
Log Phi can be fitted with a single polinomial, the
sigma(z) with two polinomial for z<0.4 and z>0.4
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Conclusions – part II
• The correlation La-Ta exists !!!
• It can be useful as model discriminator among several
models that predict the Lx-Ta anti-correlation:
• energy injetion model from a spinning-down magnetar
at the center of the fireball Dall’ Osso et al. (2010), Xu &
Huang (2011), Rowlinson & Obrien (2011)
• Accretion model onto the central engine as the long
term powerhouse for the X-ray flux Cannizzo & Gerhels
(2009), Cannizzo et al. 2010
• Prior emission model for the X-ray plateau
(2009)
Yamazaki
• and the phenomenological model by Ghisellini et al.
(2009).
• For a correct cosmological use we should use
the right sampleGRB!!!!
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