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Maria Giovanna Dainotti Stanford University, Stanford, USA

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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.,

GRB Symposium, Nashville, April 2013

- Notwithstanding the variety of GRB’s different peculiarities, some common
- features may be identified looking at their light curves.
- A crucial breakthrough :
- 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)

Phenomenological model with SWIFT lightcurves

GRB Symposium, Nashville, April 2013

Dainotti et al. correlation

Firstly discovered in 2008 by Dainotti, Cardone, & CapozzielloMNRAS, 391, L 79D (2008)

Later uptdated by Dainotti, Willingale, Cardone, Capozziello & OstrowskiApJL, 722, L 215 (2010)

Lx(T*a) vs T*a distribution for the sample of 62 long afterglows

GRB Symposium, Nashville, April 2013

- 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 studythe low error subsamples we use the respective logarithmic errors bars to formally definethe error energy parameter

GRB Symposium, Nashville, April 2013

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 meanluminosity 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

GRB Symposium, Nashville, April 2013

The search for standard GRBs continues

Correlation coefficients ρ for for the long GRB subsamples

with the varying error parameter u

L*a vs. <L*p>45for 62 long GRBs(the σ(E) ≤ 4 subsample).

GRB Symposium, Nashville, April 2013

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)

GRB Symposium, Nashville, April 2013

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

GRB Symposium, Nashville, April 2013

ρ=-0.73 for all the distribution

b=-1.32±0.20 1σ compatiblewith 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, 135Dto check that the slope of every redshift bin is consistent with every other.BUT for a quantitative analysis we turn to …..

GRB Symposium, Nashville, April 2013

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*Xand T*aare 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!

GRB Symposium, Nashville, April 2013

- 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

GRB Symposium, Nashville, April 2013

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

GRB Symposium, Nashville, April 2013

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

GRB Symposium, Nashville, April 2013

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.

GRB Symposium, Nashville, April 2013

KL =-0.05

for 100 GRBs red line

green 53 GRBs

KT =-0.85

GRB Symposium, Nashville, April 2013

- 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

GRB Symposium, Nashville, April 2013

Log sigma (z) = Sum (1+1/m(j))

where m(j) is the number of the associated sets in the sample

For luminosity function the correction is

Not relevant

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

GRB Symposium, Nashville, April 2013

- 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 plateauYamazaki (2009)
- and the phenomenological model by Ghisellini et al. (2009).
- For a correct cosmological use we should use the right sample !!!!

GRB Symposium, Nashville, April 2013