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arXiv:1412.3969v3 [astro-ph.HE] 7 Jan 2015
Selection effects in Gamma Ray Bursts correlations: consequences
on the ratio between GRB and star formation rates
Dainotti, M. G. 1,2,3, Del Vecchio, R. 3, Shigehiro, N. 1, Capozziello, S. 4,5,6
ABSTRACT
Gamma Ray Bursts (GRBs) visible up to very high redshift have become at-
tractive targets as potential new distance indicators. It is still not clear whether
the relations proposed so far originate from an unknown GRB physics or result
from selection effects. We investigate this issue in the case of the LX−T∗
acor-
relation (hereafter LT) between the X-ray luminosity LX(Ta) at the end of the
plateau phase, Ta, and the rest frame time T∗
a. We devise a general method to
build mock data sets starting from a GRB world model and taking into account
selection effects on both time and luminosity. This method shows how not know-
ing the efficiency function could influence the evaluation of the intrinsic slope
of any correlation and the GRB density rate. We investigate biases (small off-
sets in slope or normalization) that would occur in the LT relation as a result
of truncations, possibly present in the intrinsic distributions of LXand T∗
a. We
compare these results with the ones in Dainotti et al. (2013) showing that in both
cases the intrinsic slope of the LT correlation is ≈ −1.0. This method is general,
therefore relevant to investigate if any other GRB correlation is generated by
the biases themselves. Moreover, because the farthest GRBs and star-forming
galaxies probe the reionization epoch, we evaluate the redshift-dependent ratio
Ψ(z) = (1 + z)αof the GRB rate to star formation rate (SFR). We found a
1Astrophysical Big Bang Laboratory, Riken, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan,
maria.dainotti@riken.jp.
2Physics Department, Stanford University, Via Pueblo Mall 382, Stanford, CA, USA, E-mail: mdain-
ott@stanford.edu
3Obserwatorium Astronomiczne, Uniwersytet Jagiello´nski, ul. Orla 171, 31-501 Krak´ow, E-mails: delvec-
chioroberta@hotmail.it, dainotti@oa.uj.edu.pl, mariagiovannadainotti@yahoo.it
4Dipartimento di Fisica, Universit´adi Napoli ”Federico II”, Compl. Univ. di Monte S. Angelo, Edicio
G, Via Cinthia, I-80126 Napoli, Italy, E-mail: capozziello@na.infn.it
5INFN Sez. di Napoli, Complesso Universitario di Monte S. Angelo, Via Cinthia, Edicio N, 80126 Napoli,
Italy
6Gran Sasso Science Institute (INFN), viale F. Crispi 7, I-67100 L’Aquila, Italy
– 2 –
modest evolution −0.2≤α≤0.5 consistent with Swift GRB afterglow plateau
in the redshift range 0.99 < z < 9.4.
Subject headings: stars: gamma-ray burst: general, statistics, methods: data
analysis.
1. INTRODUCTION
GRBs are the farthest sources, seen up to redshift z= 9.46 (Cucchiara et al. 2011), and
if emitting isotropically they are also the most powerful, (with Eiso ≤1054 erg s−1), objects
in the Universe. Notwithstanding the variety of their peculiarities, some common features
may be identified by looking at their light curves. A crucial breakthrough in this area has
been the observation of GRBs by the Swift satellite, launched in 2004. With the instru-
ments on board, the Burst Alert Telescope (BAT, 15-150 keV), the X-Ray Telescope (XRT,
0.3-10 keV), and the Ultra-Violet/Optical Telescope (UVOT, 170-650 nm), Swift provides
a rapid follow-up of the afterglows in several wavelengths with better coverage than previ-
ous missions. Swift observations have revealed a more complex behavior of the light curves
afterglow (O’Brien et al. 2006; Sakamoto et al. 2007) that can be divided into two, three
and even more segments in the afterglows. The second segment, when it is flat, is called
plateau emission. A significant step forward in determining common features in the after-
glow light curves was made by fitting them with an analytical expression (Willingale et al.
2007), called hereafter W07. This provides the opportunity to look for universal features
that could provide a redshift independent measure of the distance from the GRB, as in
studies of correlations between GRB isotropic energy and peak photon energy of the νFν
spectrum, Eiso −Epeak , (Lloyd and Petrosian 1999; Amati et al. 2009), the beamed total en-
ergy Eγ−Epeak (Ghirlanda et al. 2004, 2006), the Luminosity-Variability, L-V (Norris et al.
2000; Fenimore and Ramirez-Ruiz 2000), L-Epeak (Yonetoku et al. 2004) and Luminosity- τ
lag (Schaefer 2003).
Dainotti et al. (2008, 2010), using the W07 phenomenological law for the light curves of
long GRBs, discovered a formal anti-correlation between the X-ray luminosity at the end
of the plateau LXand the rest frame plateau end-time, T∗
a=Tobs
a/(1 + z), where T∗
ais
in seconds and LXis in erg/s. The normalization and the slope parameters aand bare
constants obtained by the D’Agostini fitting method (D’Agostini 2005). Dainotti et al.
(2011a) attempted to use the LT correlation as a possible redshift estimator, but the paucity
of the data and the scatter prevents from a definite conclusion at least for a sample of
62 GRBs. In addition, a further step to better understand the role of the plateau emis-
sion has been made with the discovery of new significant correlations between LX, and
– 3 –
the mean luminosities of the prompt emission, < Lγ ,prompt >(Dainotti et al. 2011b). The
LT anticorrelation is also a useful test for theoretical models such as the accretion models,
(Cannizzo and Gehrels 2009; Cannizzo et al. 2011), the magnetar models (Dall’Osso et al.
2011; Bernardini et al. 2012a,b; Rowlinson et al. 2010, 2013, 2014), the prior emission model
(Yamazaki 2009), the unified GRB and AGN model (Nemmen et al. 2012) and the fireshell
model (Izzo et al. 2013). Moreover, Hasco¨et et al. (2014) and van Eerten (2014b) consider
both the LT and the LX-< Lγ,prompt >correlation to discriminate among several models
proposed for the origin of plateau. In Leventis et al. (2014) and in van Eerten (2014a) a
smooth energy injection through the reverse shock has been presented as a plausible expla-
nation for the origin of the LT correlation. Furthermore, also other authors were able to
reproduce and use the LT correlation to extend it in the optical band (Ghisellini et al. 2009),
to extrapolated it into correlations of the prompt emission (Sultana et al. 2012) and to use
the same methodology to build an analogous correlation in the prompt (Qi and Lu 2012).
Finally, it has been applied as a cosmological tool (Cardone et al. 2009, 2010; Dainotti et al.
2013a; Postnikov et al. 2014). Impacts of detector thresholds on cosmological standard can-
dles have also been considered (Shahmoradi and Nemiroff 2009; Petrosian and Lloyd 1998;
Petrosian et al. 1999; Petrosian 2002; Cabrera et al. 2007). However, because of large dis-
persion (Butler et al. 2010; Yu et al. 2009) and absence of good calibration none of these
correlations allow the use of GRBs as good standard candles as it has been done e.g. with
type Ia Supernovae. An important statistical technique to study selection effects for treating
data truncation in GRB correlations is the Efron and Petrosian (1992) method. Another
way to study the same problem in GRB correlations, derived modeling the high-energy
properties of GRBs, have been reported in Butler et al. (2010). In the latter paper it has
been shown that well-known examples of these correlations have common features indicative
of strong contamination by selection effects. We compare this procedure with the method
introduced by Efron & Petrosian (1992) and applied to the LT correlation (Dainotti et al.
2013b). The paper is organized as follows: section 2 introduces the relation between GRB
and SFR, section 2.1 is dedicated to the analysis of a GRB scaling relation, in particular we
consider the LT correlation as an example, but the procedure described can be adopted for
any other correlation. In section 3 we describe how to build the GRB samples, in section
4 we analyze the redshift evolution of the slope and normalization of the LT correlation.
In section 5 we study the selection effects related to simulated samples assuming different
normalization and slope values. Then, in section 6 we draw conclusions on the intrinsic slope
of the LT correlation and on the evaluation of the redshift-dependent ratio between GRB
and star formation rates.
– 4 –
2. The relation between GRB rate and the star formation rate
In order to understand the relation between GRBs and the star formation, it is often
assumed that the GRB rate (RGRB) is proportional to the SFR then the predicted distribu-
tion of the GRB redshift is compared to the observed distribution (Totani 1997; Mao and Mo
1998; Wijers et al. 1998; Porciani and Madau 2001; Natarajan et al. 2005; Jakobsson et al.
2006; Daigne and Mochkovitch 2007; Le and Dermer 2007; Coward 2007; Mao 2010). How-
ever, this relationship is not an easy task to handle, because some studies show that GRBs
do not seem to trace the star formation unbiasedly (Lloyd and Petrosian 1999). Namely, the
ratio between the RGRB and SFR, RGRB/SFR, increases with redshift (Kistler et al. 2008;
Y¨uksel and Kistler 2007) significantly. This means that GRBs are more frequent for a given
star formation rate density at earlier times. In fact, while observations consistently show
that the comoving rate density of star formation is nearly constant in the interval 1 ≤z≤4
(Hopkins and Beacom 2006), the comoving rate density of GRBs appears evolving distinctly.
In our approach we explicitly take into consideration this issue when we fit the observed
GRB rate with the model. Selection effects involved in a GRB sample are of two kinds :
the GRB detection and localization; and the redshift determination through spectroscopy
and photometry of the GRB afterglow or the host galaxy. These problems have been ob-
ject of extensive study in literature (Bloom 2003; Fiore et al. 2007; Guetta and Della Valle
2007). Moreover, the Swift trigger, is very complex and the sensitivity of the detector is
very difficult to parameterize exactly (Band 2006), but in this case not dealing with prompt
peak energy we do not have to take into consideration the double truncation present in data
(Lloyd and Petrosian 1999). In the case of plateau it is easier, since an effective luminosity
threshold appears to be present in the data which can be approximated by a 0.3−10 keV
energy flux limit Flim ≡2×1012 erg cm2s1(Dainotti et al. 2013b). The luminosity thresh-
old is then Llim = 4πD2
L(z, ΩM, H0)Flim , where DLis the luminosity distance to the burst.
Throughout the paper, we assume a flat universe with ΩM= 0.28, Ωλ= 0.72 and H0=70
km s1Mpc1. In our approach below several models are considered and then the one that
best matches the GRB rate with star formation rate has been chosen.
2.1. GRBS WORLD MODEL
We derive a model capable of reproducing the observed Swift GRB rate as a function
of redshift, luminosity and time of the plateau emission.
Rest frame time and luminosity at the end of the GRB plateau emission show strong cor-
relations as discovered by Dainotti et al. (2008) and later updated by Dainotti et al. (2010,
2011a,b, 2013a). Therefore, all these quantities must be considered in deriving reliable rates.
– 5 –
We characterize the GRB rate as a product of terms involving the redshift z of the bursts,
the isotropic equivalent luminosity release (0.3-10 keV) LXand the duration T∗
a.
Let us assume that a scaling relation exists so that the luminosity LX(Ta) for a GRB
with time scale T∗
aat redshift z is given by :
λ=α0+αττ+αζζ(1)
where we have introduced the compact notation
λ=log LX(T∗
a)
τ=log [T∗
a/(1 + z)]
ζ=log (1 + z).
(2)
and the term ζaccounts for redshift evolution. The luminosity is normalized by the
unit of 1 erg s−1and the time by the unit of 1 s, so that non dimensional quantities are
considered. All the observables in this model are computed in the rest frame, because we
are testing the role played by selection effects in the rest frame, being the LT correlation
rest frame corrected. Independently, on the physical interpretation of this relation, (in fact,
there are several models that can reproduce it as we have mentioned in the introduction) we
can nevertheless expect GRBs to follow Equation 1 with a scatter σλ. Moreover, the zero
point α0may be known only up to a given uncertainty σα. Following the approach of Butler
et al. (2010) applied for prompt correlations, we assume that λcan be approximated by
Gaussian distribution with mean λ0, expressed in Equation 1, and the variance σint as the
intrinsic scatter of the correlation. We also write the probability that a GRB with given (τ,
ζ) values has a luminosity λas follows:
Pλ(λ, τ, ζ )∝exp[−1
2[λ−(α0+αττ+αζζ)
σint
]2] (3)
with σ2
int =σ2
λ+σ2
α, with σαthe uncertainty of the α0value and σλthe uncertainty on
the luminosity value.
The approximation of a Gaussian distribution both for the luminosity and time is motivated
by the goodness of the fit which gives a probability P= 0.46 and P= 0.61 respectively,
see Fig. 1 and 2. We note that the mean, (indicated with <>)< Ta>= 3.35 (s) with a
variance σTa= 0.77 (s) and < LX>= 48.04 (erg/s) with a variance σLX= 1.37 (erg/s) are
represented respectively in Fig. 1 and 2.
– 6 –
Fig. 1.— Probability Density Distribution of T∗
a, the rest frame end time of the plateau, for GRBs
observed from 2005 January until 2014 July analyzed following the Dainotti et al. (2013a) approach with a
superimposed best fit of the Gaussian distribution.
Fig. 2.— Probability Density Distribution of LX(Ta) at the end of the plateau for GRBs observed from
2005 January until 2014 July analyzed following the Dainotti et al. (2013a) approach with a superimposed
best fit of the Gaussian distribution.
– 7 –
In order to obtain the number of GRBs with a given luminosity λ, we need to integrate
over the distributions of τand ζ. We will assume, for simplicity, that τfollows a truncated
Gaussian law.
Pτ(τ)∝(exp[−1
2(τ−τ0
στ)2]τL< τ < τU
0τ≤τLor τ ≥τU
(4)
where τLand τUindicate respectively the lower limit and upper limit of the observed τ
distribution and τ0is the mean value of this distribution. The limits of τare taken from an
updated sample of T∗
acomposed of 176 GRBs afterglows, with firm redshift determination,
from January 2005 till July 2014. The analysis follows the criteria adopted in Dainotti et al.
(2013a).
If we assume that the GRBs trace the cosmic star formation rate, we can model their
redshift distribution following Butler et al. (2010) as:
Pz(z)∝˙ρ∗(z)
1 + z
dV
dz (5)
where ˙ρ∗(z) is the comoving GRB rate density, V is the universal volume, and the factor
(1 + z) accounts for cosmic time dilatation and
dV
dz ∝r2(z)
E(z)(6)
with r(z) the comoving distance and E(z) = H(z)/H0the Hubble parameter normalized
to its present day value.
Collecting the different terms, we can finally write the true, detector-independent event N
differential rate, for z, log T∗
aand log LX, as:
dN
dλdτdz ∝Ψ(z)Pλ(λ, τ , ζ)Pτ(τ)Pz(z).(7)
We here note that we have introduced the term of the evolution in redshift, Ψ(z) =
(1 + z)α, following the approach of Lloyd & Petrosian (1999), Dermer (2007) and Robertson
& Ellis (2012). In Dermer (2007) assuming that the emission properties of GRBs do not
change with time, they find that the Swift data can only be fitted if the comoving rate
density of GRB sources exhibits positive evolution to z > 3−5. In our approach we
introduce evolution starting from z≥0.99.
– 8 –
So using the above expression for Pτ, we find that the number of GRBs with luminosity
in the range (λ,λ+dλ) and redshift between zand z+dz is:
dN
dλdz ∝Ψ(z)˙ρ∗(z)(dV /dz)
1 + zFτU− FτL
p8πσ2
τ
exp[−1
2[λ−(α0+αττ+αζζ)
pσ2
int
]2] (8)
where FτUand FτLare the error functions1of the lower and upper limit of the time
distribution. Note that Equation 8 is defined up to an overall normalization constant which
can be solved by imposing that the integral of dN/dλdz over (λ,z) gives the total number
of observed GRBs. Actually, this is not known since we do not observe all GRBs, but only
those passing a given set of selection criteria. However, we will be only interested in the
fraction of GRBs in a cell in the 2D (λ,z) space so that we do not need this quantity.
We are aware that we don’t map out the true LT relation given selection effects and the
observed LT relation. Doing this would require modeling the selection of the GRB sample
itself (using the gamma ray threshold) and also seeking to understand the tie between the
GRB flux and the afterglow LX. However, the relation between flux and LXhas been
already studied by Dainotti et al. (2013a) and reported briefly in the previous section. Here
we computed the new limit related to the updated sample, as it has been shown in the middle
panel of Fig 3.
3. SIMULATING THE GRB SAMPLES
The GRBs rate given by Equation 8 has been derived by implicitly assuming that all the
GRBs can be detected notwithstanding their observable properties. This is actually not the
case. As an example, we will consider hereafter the LT correlation although the formalism
and the method we will develop can be easily extended to whatever scaling law. For the
LT case, there are two possible selection effects. First, each detector has an efficiency which
is not the same for all the luminosities. Only GRBs with λ > λL, where λLis the lowest
detectable luminosity for a given instrument, can be detected while all the GRBs with λ
larger than a threshold luminosity λUwill be found.
Moreover, it is likely that the efficiency of the detector is not constant, but is rather a
1We remind that the usual definition of the error function is
erf (x) = 2
√πZx
0
e−t2dt. (9)
– 9 –
function of the luminosity. We will therefore introduce an efficiency function Eλ(λ) whose
functional expression is not known in advance, but can only take values in the range (0,1).
A second selection effect is related to the time duration of the GRB. Indeed, in order to
be included in the sample used to calibrate the LT correlation, the GRB afterglow has to
be measured over a sufficiently long time scale to make possible to fit the data and extract
the relevant quantities. If τis too small, as it has been shown in Dainotti et al. (2013a)
the minimum rest frame time is 14 s, few points will be available for the fit, while, on the
contrary, large τvalues will give rise to afterglow light curves which could be well sampled
by the data. Again, we can parametrize these effects introducing a second efficiency function
Eτ(τ) so that the final observable rate is the following:
dNobs
dλdz ∝dN
dλdz × Eλ(λ)Eτ(τ).(10)
We point out that our formulation, which takes into account of the efficiency functions
Eλ(λ) and Eλ(τ) in the final observed GRB rate is similar to the approach by Robertson &
Ellis (2012) in Equation 1, in which the additional factor K is presented. K is equivalent to
our Eλ(λ) and Eλ(τ).
It is worth noting that Equation 10 is actually still a simplified description. Indeed, it
is in principle possible that other selection effects take place involving observable quantities
not considered here, as for example βand the redshift. However, these parameters enter
in the determination of λso that one can (at least in a first order approximation) convert
selection cuts on them in a single efficiency function depending only on λ(for the dependence
of the flux on the redshift see left panel of Fig. 3). However, as we can see from Fig. 3 β
is constant with redshift, and there is no correlation between those two quantities, in fact
the Spearman correlation coefficient is ρ=−0.062. Nevertheless, Equation 10 provides a
Fig. 3.— Flux at the end of the plateau phase, Flux(Ta), (left panel) and the spectral index, β, (right
panel) as a function of redshift. The limiting luminosity, log LXvs 1 + zshows (middle panel) two lines, one
for the limiting flux, FSwif t,lim = 10−14 erg cm−2s−1and the other one is the most suitable for a plateau
duration of 104s, which is 2 ×10−12 erg cm−2s−1.
– 10 –
reasonably accurate description of the observable GRB rate.
In order to evaluate Equation 10 there are different quantities to determine. First, we
need to set the scaling coefficients (α0,ατ,αζ) and the intrinsic scatter σint. Second, the
mean and variance of the τdistribution (τ0,στ) has to be given. Finally, an expression for
the cosmic SFR ˙ρ∗(z) has to be assigned. None of these quantities is actually available. In
principle, one could assume a SFR law and fit for the model parameters to a large enough
GRBs sample with measured (λ,τ,ζ) values. To this end, one should know the selection
function Eλ(λ)Eτ(τ) which is not the case. Studies of how light curves would appear to a
gamma-ray detector here on Earth have been performed (Kocevski and Petrosian 2013). In
this paper the prompt emission pulses are investigated and the conclusion is that even a
perfect detector that observes over a limited energy range would not faithfully measure the
expected time dilation effects on a GRB pulse as a function of redshift.
Fig. 4.— GRBs rate density using method of Li (2008) and the observed GRBs rate density obtained by
the linear efficiency functions (upper panel), and the polynomial efficiency function (lower panel) with the
redshifts distribution of our data sample.
Nevertheless, here we study detector threshold effects on afterglow properties. Our
– 11 –
aim is to investigate how the ignorance of the efficiency function bias the estimate of the
correlation coefficients. We can therefore rely on simulated samples based on a realistic
intrinsic rate. We proceed as schematically outlined below.
(i) We assume that the available data represent reasonably well the intrinsic τdistribution
so that we can infer (τ0,στ) from the data themselves. We set τL,U =τ0±5στthus
symmetrically cutting the Gaussian distribution at its extreme ends.
(ii) Based on the shape of the cosmic SFR (Hopkins and Beacom 2006), we assume a
broken power law for the comoving GRB rate density:
˙ρ∗(z)∝
(1 + z)g0z≤z0
(1 + z)g1z0≤z≤z1
(1 + z)g2z≥z1
(11)
where the relative normalizations are set so that ˙ρ∗(z) is continuous at z0= 0.97 and
z1and (z0, z1) = (0.97,4.50), (g0, g1, g2) = (3.4,−0.3,−8.0). Moreover, besides the
equation 11, we employed other shapes of the SFR (Li 2008; Robertson and Ellis 2012;
Kistler et al. 2013) to obtain the observed GRBs rate density. The one used by (Li
2008) is:
˙ρ(z) = a+b×Log(1 + z).(12)
The a and b parameters are :
(a, b) =
(−1.70,3.30) z≤0.993
(−0.727,0.0549) 0.993 ≤z≤3.80
(2.35,−4.46) z≥3.80
(13)
Robertson & Ellis (2012) defined the SFR as:
˙ρ(z) = a+b(z/c)f
1 + (z/c)d+g, (14)
where they have a= 0.009M⊙yr−1M pc−3,b= 0.27M⊙yr−1Mpc−3,c= 3.7, d= 7.4,
and g= 10−3M⊙yr−1Mpc−3.
Instead, Kistler et al. (2013) defined the SFR as :
– 12 –
˙ρ(z) = ˙ρ0×[(1 + z)aψ + ( 1 + z
B)bψ + (1 + z
C)cψ]1
ψ,(15)
with slopes a= 3.4, b=−0.3, and c=−2.5, breaks at z1= 1 and z2= 4 corresponding
to B= (1 + z1)1−a
b∼5160 and C= (1 + z1)(b−a)
c×(1 + z2)(1−b)
c∼11.5, and ψ=−10.
Finally, we compare the fitted functions obtained with these four methods with our data
distribution. The most realible fits for our parameters is the SFR used by Li (2008),
see Fig. 4 where the best fit among linear (upper panel) and polynomial (lower panel)
ǫ(λ) functions are considered. Moreover, we adopted the constraints for the redshift
dependent ratio between SFR and GRB rate adopted by Robertson & Ellis (2012).
In this paper a modest evolution (e.g.,Ψ(z)≈(1 + z)α) with −0.2≤α≤1.5, where
the peak probability occurs for α≈0.5 is consistent with the long GRB prompt data
(P≈0.9). These values can be explained if GRBs occur primarily in low-metallicity
galaxies which are proportionally more numerous at earlier times. We note that in our
approach we assumed no evolution at low redshift for z≤0.99 consistently with the
posterior probability in Robertson & Ellis (2012) in which no evolution is possible at
the 2-σlevel. However, because a constant Ψ(z) is also ruled out (Robertson and Ellis
2012), then we fit the normalization parameters and the evolution factors obtaining
Ψ(z)≈(1 + z)−0.2for 0.993 ≤z≤3.8 and Ψ(z)≈(1 + z)0.5for z≥3.8. These
values of the evolution are compatible with Robertson et al. (2012). Regarding the
observed GRB rate we obtained that the best efficiency functions are possible both for
two polynomial and two linear as we show in Fig. 4. Table 1 and 2 show the probability
that the density rate match the afterglow plateau GRB rate assuming those efficiency
functions
(iii) For given (α0,ατ,αζ,σint) values, we divide the 2D space (λ,z) in Mcells and, for
each cell, compute the fraction of GRBs in it as:
fsim(λi, zi) = Rλi+∆λ
λi−∆λdλ Rzi+∆z
zi−∆zdz dN
dλdz
Rλmax
λmin dλ Rzmax
zmin dz dN
dλdz
(16)
where we set
(λmin, λmax ) = (42.0,52.0),(zmin, zmax) = (0,10).(17)
We find more efficient to change variable from zto ζwhen dividing the 2D space in
10 ×10 square cells.
(iv) For each given cell, we generate Nij =fsim(λi, ζj)× Nsim GRBs (with Nsim the total
number of objects to simulate) by randomly sampling (λ,ζ) within the cell boundaries
and computing τby solving Eq. 1.
– 13 –
(v) To take into account of the selection effects, for each GRB, we generate two random
numbers (uτ,uλ) uniformly sampling the range (0, 1) and only retain the GRB if
uτ≤ Eτ(τ) and uλ≤ Eλ(λ). Note that, as a consequence of this cut, the final number
Nobs of observed GRBs is smaller than the input one Nsim.
(vi) Finally, for each one of the Nobs selected GRBs, we generate new (τobs,λobs ) values
extracting from Gaussian distributions centered on the simulated (τ,λ) values and
with a 1% variance. We also associate to each GRB an error set in such a way to be
similar to what is actually obtained for GRBs having comparable (τ,λ) values.
The above procedure allows us to build simulated GRBs sample taking into account
both the intrinsic properties of any scaling relation and the selection effects induced by the
instrumental setup. Moreover, we have referred to an actual GRBs sample in order to set
both the limits on (τ,ζ,λ) and the typical measurement errors. Therefore, we can rely on
these simulated samples to investigate the impact of selection effects on the recovered slope
and intrinsic scatter of the given correlation. To this end, the last ingredient we need is a
functional expression for the efficiency functions. Since these are largely unknown, we are
forced to make some arbitrary guess. Therefore, we consider two different cases. First, we
assume that there is no selection on τ, i.e. we set Eτ= 1. Two functional expressions are
then used for Eλ, namely a power law:
Eλ(λ) =
0λ < λL
(λ−λL
λU−λL)EλλL≤λ≤λU
1λ > λU
(18)
and a fourth order polynomial, i.e. :
Eλ(λ) =
0λ < λL
E1˜
λ+E2˜
λ2+E3˜
λ3+E4˜
λ4
E1+E2+E3+E4λL≤λ≤λU
1λ > λU
(19)
with ˜
λ= (λ−λL)/(λU−λL). We try different arbitrary choices for the parameters
entering both expressions of Eλin order to investigate to which extent the results depend
on the exact choice of the efficiency function, see Fig. 5. In a second step, we abandon the
assumption Eτ= 1, to assume for it the same functional expression used for Eλ, with the
same choices for the parameters, but different upper and lower limits depending on τUand
– 14 –
Fig. 5.— The first 5 panels represent examples of the efficiency function for the linear case versus lumi-
nosities of the GRBs, λ, in our data sample, while the last 5 panels the efficiency functions for the fourth
order polynomial. The linear functions as well the polynomial ones are computed according to Eq. 18, and
Eq. 19.
– 15 –
Fig. 6.— The first 5 panels represent examples of the efficiency function for the linear case versus the
times, τof the GRBs in our data sample, while the last 5 panels the efficiency functions for the fourth order
polynomial. The linear functions as well the polynomial ones are computed according to Eq. 18, and Eq.
19.
– 16 –
τL, see Fig. 6.
4. REDSHIFT EVOLUTION ON THE NORMALIZATION AND SLOPE
PARAMETERS
Fig. 7.— ατand normalization α0using a linear function α0=−0.22x+ 52.31 (left panel) and ατ=
0.10x−1.38 (right panel).
Fig. 8.— ατand normalization α0using a polynomial function α0= 55.87 −8.13x+ 5.53x2−1.48x3+0.13x4
(left panel) and ατ=−2.35 + 2.13x−1.39x2+ 0.37x3−0.03x4(right panel).
As we have already mentioned in the previous paragraph the polynomial and the linear
model for the ǫ(λ) are unknown, then assumptions need to be made. We chose these forms,
because both normalization and slope of the LT correlation depend on the redshift either
with a polynomial or with a simple power law. Therefore, these choices for the selection
functions take into account of this redshift dependence. Namely, we consider a model redshift
– 17 –
Id λLλUEλPGRB,rate
PL1 44.34 50.86 1.25 ≤10−4
PL2 43.64 49.87 2.99 0.003
PL3 43.77 50.74 1.65 0.53
PL4 44.77 49.59 2.04 0.001
PL5 44.14 50.83 0.23 0.54
Table 1: Efficiency function parameters for the power-law Eλand no cut on τi.e. Eτ= 1.
PGRB,rate is the goodness of fit between our data and the observed GRBs rate density,
thus how the data well fit the observed GRBs density rate. To compute the probability
we compute the χ2test that performs a statistical hypothesis test in which the sampling
distribution of the test statistic is a χ2distribution when the null hypothesis is true, in order
to determine whether there is a significant difference between the expected frequencies and
the observed frequencies.
Id λLλUE1E2E3E4PGRB,rate
PoL1 44.90 49.14 0.46 0.01 0.24 0.80 0.54
PoL2 41.10 50.23 0.60 0.95 0.05 0.53 ≤10−4
PoL3 43.57 49.09 0.71 0.79 0.07 0.34 0.019
PoL4 44.37 49.52 0.51 0.03 0.46 0.78 0.15
PoL5 43.03 50.06 0.79 0.36 0.63 0.40 0.001
Table 2: Same as table 1 but for the polynomial functions. PGRB,r ate is the goodness of fit
between our data and the observed GRB rate density.
– 18 –
dependence because of the corresponding dependence of both luminosity and time. This has
been already shown in Dainotti et al. (2013a) and currently inthe middle panel of Fig. 3
for the updated data sample. To study the behavior of the redshift evolution we plot the
slope and the normalization values versus the redshift. These are obtained from the average
values for the data set divided into 5 bins, see figure 7 and into 12 bins, see figure 8. As we
can see from both figures 7 and 8 the normalization parameter α0decreases as the redshift
increases, while the slope parameter ατshows the opposite trend. Goodness of the fits is
given by the probability P= 0.79 for the data set divided in 5 bins and P= 0.87 for the
one divided into 12 bins for the linear case, while for the polynomial model P= 0.99 and
P= 0.94 for the data set divided in 5 and 12 bins respectively. These results show that both
polynomial and linear fit are possible.
5. IMPACT OF SELECTION EFFECTS
The simulated samples generated as described above are input to the same Bayesian
fitting procedure we use with real data. For each input (ατ,αζ,α0,σint) parameters, we
simulate ∼50 GRBs sample setting Nsim = 200, while the number of observed GRBs depend
on the efficiency function used. We fit these samples assuming no redshift evolution in Eq.
1, i.e. forcing αζ= 0 in the fit so that, for each simulated sample, the fitting procedure
returns both the best fit and the median and 68% confidence range of the parameters (ατ,
α0,σint). In order to investigate whether the selection effects impact the recovery of the
input scaling laws, we fit linear relations of the form:
xf=axinp +b(20)
where xinp is the input value and xfcan be either the best fit (denoted as xbf ) or
the median xf it value. When fitting the above linear relation, we use the χ2minimization
for xbf , while a weighted fit is performed for xf it with weights ωi= 1/σ2
iwhere σiis the
symmetrized 1σerror. Note that the label ihere runs over the simulations performed for
each given efficiency function.
5.1. No redshift evolution
Here, we consider input models with αζ= 0, i.e. no redshift evolution of the scaling law
(1). It is worth noting that such an assumption is actually well motivated since it has been
demonstrated in Dainotti et al. (2013a) that luminosity is almost not affected by redshift
– 19 –
Fig. 9.— Fitted vs input (ατ,α0,σint ) parameters obtained with the power law function. The first three
panels refer to the best fit values, while the other three show the median values with the 1σerror bars. Solid
red line is the best fit line while blue dashed is the no bias line when xinp =xf.
– 20 –
evolution, while time becomes to undergo redshift evolution for high redshift only. From our
point of view, however, this case allows us to directly quantify the impact of the efficiency
functions on the recovery of the scaling correlation parameters since any deviation will only
be due to the selection effects and not to any attempt of compensating the missed evolution
with z.
5.1.1. No selection on τ(Eτ= 1)
We start by considering the idealized case of no selection of τ, i.e., we force Eτ= 1,
and set the Eλparameters as listed in Tables 1 and 2 for the power-law and polynomial
expressions, respectively. As an example, figure 9 shows the results for the efficiency function,
while Table 3 summarizes the (a, b) coefficients of the linear fit between the input and
recovered quantities. The closer is ato 1, the less is the parameter biased, while b6= 0
should not be taken as evidence for bias. This result is in perfect agreement with the
intrinsic correlation slope, which is −1.07+0.09
−0.14 (Dainotti et al. 2013b), when we consider as
the best choice for the selection functions the ones that returns values of acloser to 1. If
Equation 20 is fulfilled, we can estimate the relative bias as:
∆x
x=xinp −xf
xinp
= 1 −a−b
xinp
,(21)
so that we can accept b6= 0 if xinp is much larger than b. This is indeed the case for
xinp =α0which takes typical values (∼50) much larger than the bones in Table 3.
From the proximity between solid and dashed lines, which represent respectively the best fit
line and the no bias line when xinp =xf, in the corresponding panels of Fig. 9, we see that,
for the power-law efficiency function (and no cut on τ), both the slope and the zero point
of the scaling relation are correctly recovered. The reason why is that the relative bias is
Id (a, b)τ
bf (a, b)τ
f it (a, b)α0
bf (a, b)α0
f it (a, b)σ
bf (a, b)σ
f it
∆x
x
PL1 (0.953,0.010) (0.959,0.013) (0.928,3.688) (1.000,-0.073) (0.593,0.354) (0.616,0.355) 0.004
PL2 (0.914,-0.008) (0.873,-0.024) (1.013,-0.836) (0.989,0.292) (0.689,0.299) (0.643,0.341) 0.002
PL3 (0.880,-0.052) (0.937,-0.013) (0.965,1.729) (1.008,-0.513) (0.683,0.340) (0.664,0.352) 0.003
PL4 (0.946,0.024) (0.964,0.024) (0.995,-0.076) (1.086,-4.905) (0.614,0.364) (0.585,0.380) 0.006
PL5 (0.916,-0.030) (0.962,0.004) (1.033,-2.067) (0.828,9.095) (0.716,0.333) (0.679,0.356) 0.005
Table 3: Slope aand zero point bof the fitted vs input parameters for both the best fit and median
values (labeled with subscripts bf and f it, respectively). The upperscript denotes the parameter fitted with
(τ,α0,σ) referring to (ατ,α0,σint ), respectively. ∆x
xis the bias for each efficiency function considered.
– 21 –
negligible small notwithstanding the values of the parameters setting Eλ. This is particularly
true if one relies on the median values as estimate since they are typically consistent with
the no bias line within less than 2σ.
The above results have been obtained considering a power-law Eλso that it is worth inves-
tigating whether they critically depend on this assumption. We have therefore repeated the
analysis for the polynomial Eλmodels in Table 2 obtaining the results in Table 4. A compar-
ison with the values in Table 4 shows that the (a, b) coefficients are similar so that one could
preliminarily conclude that the shape of the efficiency function does not play a major role
in the determination of the bias. Actually, although the functional expressions are different,
both the power-law and the polynomial selection functions are qualitatively similar with Eλ
increasing with λover a comparable range. Although such a behaviour is likely common to
any reasonable Eλ, we can not exclude a priori that non monotonic selection functions do
actually exist. What would the results be in such a case is not clear so that we prefer to be
cautious and conclude that the bias is roughly the same whichever monotonic Eλ(λ) func-
tion is used, but not for all the possible Eλfunctions. For non-monotonic shape of selection
function, see Stern et al. (2001), in which an assumed detection efficiency function, defined
as the ratio of the number of detected test bursts to the number of test bursts applied to
the data versus the expected peak count rate, is given by:
E(ce) = 0.70 ×[1 −exp[−(ce
ce,0
)2]]ν,(22)
where ce,0= 0.097 counts s−1cm−2and ν= 2.34 are two constants. However, quoting
from Stern et al. (2001), the best possible efficiency quality has still not yet been achieved
because in fact the detection efficiency depends on the peak count rate rather then on the
time-integrated signal.
Id (a, b)τ
bf (a, b)τ
f it (a, b)α0
bf (a, b)α0
f it (a, b)σ
bf (a, b)σ
f it
∆x
x
PoL1 (0.950,0.020) (0.928,0.011) (1.099,-5.545) (1.178,-9.935) (0.647,0.350) (0.673,0.349) 0.004
PoL2 (1.095,0.128) (1.075,0.105) (1.030,-1.662) (1.023,-1.278) (0.681,0.337) (0.625,0.372) 0.0008
PoL3 (0.984,0.063) (0.936,0.015) (0.711,15.258) (0.988,0.376) (0.741,0.289) (0.685,0.336) 0.007
PoL4 (0.870,-0.025) (0.969,0.052) (0.730,14.103) (0.734,13.872) (0.630,0.351) (0.582,0.381) 0.009
PoL5 (1.004,0.069) (0.972,0.030) (0.963,11.772) (1.002,-0.374) (0.581,0.384) (0.549,0.402) 0.18
Table 4: Same as Table 3 but for the polynomial Eλmodel.
– 22 –
5.1.2. Selection cuts on both τand λ
We now consider the case where the total selection function may be factorized as
E(τ, λ) = Eτ(τ)Eλ(λ) with both Ex(x) functions being given by power-law or fourth or-
der polynomial expressions. We consider 10 different arbitrary choices for both cases. Note
that we have to increase Nsim to 300 in order to have Nobs = 80 −100 as for the models
discussed in the previous subsection.
Table 5 gives the (a,b) coefficients for the different models considered. A comparison with
Table 3 shows that, on average, the bias on the parameters is roughly the same with the
median values giving smaller deviations and significant bias on σint only. A more detailed
analysis, however, shows that, while, in the Eτ= 1 case, biases larger than 5% are of the
order of 10%. Namely, from Table 3 and 4 we show that the relative biases, ∆x
x, both in
the linear and the polynomial case, give very small values from 0.2% to 0.9%, with the only
exception of 1 polynomial function, in which the bias is 18%, thus giving P= 10% of having
larger bias than 5%. If we consider selection cuts on both τand λwe notice in Table 5 that
number of bias whose value is greater than 5% are 4, thus increasing their probability to
occur (P∼40%). This can be qualitatively explained noting that a cut on λonly removes
the points in the luminosity axis thus possibly shifting the best fit relation, but not changing
the slope. On the contrary, removing points also along the horizontal τaxis can change
the slope ατtoo and hence also affects (α0,σint) because of the correlation among these
parameters and ατ. Similar results are obtained when both Exfunctions are modeled with
fourth order polynomials so that we will not discuss this case here. We stress that when ∆x
x
are larger than 6%, than the slope of the correlation is farther from −1.0 compared to cases
in which ∆x
x≤0.06. In fact, in the first case the slopes values range from 0.86 to 0.91, see
the functions P LT a4 and P LT a6 in Table 5. These values are not compatible in 1 σwith
the claimed intrinsic slope of the correlation, −1.07+0.09
−0.14. If we consider, instead, the lowest
∆x
x, then we obtain ranges of a= (0.94,0.99) thus showing full compatibility in 1 σwith the
intrinsic slope. In this way we have quantitatively confirmed the existence of the LX−T∗
a
correlation with the same intrinsic slope as in Dainotti et al. (2013a) if appropriate selection
functions are chosen.
6. Conclusions
Here we built a general method to evaluate selection effects for GRB correlations not
knowing a priori the efficiency function of the detector used. We have tested this method
on the LT correlation. We chose a set of GRBs and assuming Gaussian distributions for
the variables involved, for luminosity and time, and also a particular shape for the GRBs
– 23 –
rate density. We simulated a mock sample of data in order to consider the selection ef-
fects of the detectors. As we can see in paragraph 3 assuming the correct observed GRBs
rate density shape was not an easy task. In fact, we explored different methods (Li 2008;
Robertson and Ellis 2012; Kistler et al. 2013; Hopkins and Beacom 2006) that use several
SFR shapes to understand which one best matches the afterglow plateau data distribution
including the selection functions, see Fig. 4. The most reliable fits for the GRB plateau
data is the SFR used by Li (2008), while the best efficiency functions for ǫ(λ) that match
the GRB density rate can be both two polynomial and two linear, see Fig. 4. Table 1 and
2 show the probability that the density rate fits the afterglow plateau GRB rate assuming
those efficiency functions. However, we assumed there could be selection effects both for
luminosity and time. In particular, the bias is roughly the same whichever monotonic effi-
ciency function for the luminosity detection Eλis taken. From Table 3 and 4 we show that
the relative biases, ∆x
x, both in the linear and the polynomial case, give very small values
from 0.2% to 0.9%, with the only exception of 1 polynomial function, in which the bias is
18%, thus giving P= 10% of having larger bias than 5%. If we consider selection cuts on
both τand λwe notice in Table 5 that number of bias whose value is greater than 5% are
4, thus increasing the probability of having such biases (P∼40%). In addition, we studied
selection effects in the LT correlation assuming also a combination of the luminosity and time
detection efficiency functions. Different values for the parameters of the efficiency functions
in the detectors are taken into account as described in the paragraph 5. This gives distinct
fit values that inserted in Equation 20 allow to study the scattering of the correlation and its
selection effects. We have quantitatively confirmed the existence of the LX−T∗
acorrelation
with the same intrinsic slope as in Dainotti et al. (2013a) if appropriate selection functions
are chosen. In particular, when ∆x
xare larger than 6%, than the slope of the correlation is
farther from −1.0 compared to cases in which ∆x
x≤0.06. The lowest ∆x
xleads to ranges
of a= (0.94,0.99) thus showing full compatibility in 1 σwith the intrinsic slope. Finally,
the fact that the correlation is not generated by the biases themselves is a significant and
further step towards considering a set of GRBs as standard candles and their possible and
useful application as a cosmological tool.
7. Acknowledgements
This work made use of data supplied by the UK Swift Science Data Centre at the Uni-
versity of Leicester. We are particularly grateful to Cardone, V.F. for the initial contribution
to this work. M.G.D. and S.N. are grateful to the iTHES Group discussions at Riken. M.D
is grateful to the support from the JSPS Foundation, (No. 25.03786). N. S. is grateful to
JSPS (No.24.02022, No.25.03018, No.25610056, No.26287056) & MEXT(No.26105521), R.
– 24 –
D.V. is grateful to 2012/04/A/ST9/00083.
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This preprint was prepared with the AAS L
A
T
E
X macros v5.2.
– 31 –
Id (a, b)τ
bf (a, b)τ
f it (a, b)α0
bf (a, b)α0
f it (a, b)σ
bf (a, b)σ
f it
∆x
x
PLTa1 (0.842,-0.080) (0.920,-0.016) (1.099,-5.615) (1.064,-3.650) (0.615,0.363) (0.585,0.387) 0.04
PLTa2 (0.972,-0.016) (0.963,-0.027) (1.112,-5.980) (1.148,-7.746) (0.720,0.306) (0.674,0.335) 0.005
PLTa3 (0.960,-0.025) (0.974,-0.005) (0.985,0.791) (1.022,-1.228) (0.725,0.329) (0.690,0.345) 0.004
PLTa4 (0.877,-0.021) (0.933,-0.005) (1.005,-0.611) (0.959,2.196) (0.702,0.304) (0.644,0.345) 0.09
PLTa5 (0.860,-0.056) (0.901,-0.026) (0.911,4.592) (0.838,8.655) (0.687,0.338) (0.613,0.382) 0.06
PLTa6 (0.904,-0.010) (0.919,-0.003) (0.734,14.060) (0.397,32.119) (0.678,0.326) (0.680,0.330) 0.08
PLTa7 (0.915,-0.044) (0.998,0.035) (1.119,-6.577) (1.119,-6.474) (0.736,0.311) (0.722,0.327) 0.02
PLTa8 (0.967,-0.003) (0.961,-0.017) (1.027,-1.485) (1.178,-9.515) (0.731,0.287) (0.705,0.313) 0.03
PLTa9 (0.933,-0.005) (0.904,-0.031) (1.026,-1.530) (0.804,10.289) (0.641,0.357) (0.670,0.352) 0.06
PLTa10 (1.040,0.056) (0.997,0.028) (0.753,13.308) (0.822,9.345) (0.736,0.329) (0.713,0.344) 0.04
Table 5: Same as Table 3 but for the selection cuts on both (τ,λ) and power-law Exfunctions.
