Issue 
A&A
Volume 557, September 2013



Article Number  A100  
Number of page(s)  14  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/201321528  
Published online  10 September 2013 
The E_{peak} – E_{iso} relation revisited with Fermi GRBs
Resolving a longstanding debate?^{⋆}
^{1}
Université de Toulouse, UPSOMP, IRAP,
31400
Toulouse,
France
email:
vincent.heussaff@irap.omp.eu
^{2}
CNRS, IRAP, 14
avenue Édouard Belin, 31400
Toulouse,
France
^{3}
LAPP, Université de Savoie, CNRS/IN2P3,
9 chemin de Bellevue,
BP 110, 74941
AnnecyleVieux,
France
Received: 21 March 2013
Accepted: 17 July 2013
Aims. We used a sample of gammaray bursts (GRBs) detected by Fermi and Swift to reanalyze the correlation discovered by Amati et al. (2002, A&A, 390, 81) between E_{pi}, the peak energy of the prompt GRB emission, and E_{iso}, the energy released by the GRB assuming isotropic emission. This correlation has been disputed by various authors, and our aim is to assess whether it is an intrinsic GRB property or the consequence of selection effects.
Methods. We constructed a sample of Fermi GRBs with homogeneous selection criteria, and we studied their distribution in the E_{pi} – E_{iso} plane. Our sample is made of 43 GRBs with a redshift and 243 GRBs without a redshift. We show that GRBs with a redshift follow a broad E_{pi} – E_{iso} relation, while GRBs without a redshift show several outliers. We use these samples to discuss the impact of selection effects associated with GRB detection and with redshift measurement.
Results. We find that the E_{pi} – E_{iso} relation is partly due to intrinsic GRB properties and partly due to selection effects. The lower right boundary of the E_{pi} – E_{iso} relation stems from a true lack of luminous GRBs with low E_{pi}. In contrast, the upper left boundary is attributed to selection effects acting against the detection GRBs with low E_{iso} and large E_{pi} that appear to have a lower signaltonoise ratio. In addition, we demonstrate that GRBs with and without a redshift follow different distributions in the E_{pi} – E_{iso} plane. GRBs with a redshift are concentrated near the lower right boundary of the E_{pi} – E_{iso} relation. This suggests that it is easier to measure the redshift of GRBs close to the lower E_{pi} – E_{iso} boundary and that GRBs with a redshift follow the Amati relation better than the general population. In this context, we attribute the controversy about the reality of the Amati relation to the complex nature of this relation resulting from the combination of a true physical boundary and biases favoring the detection and the measurement of the redshift of GRBs located close to this boundary.
Key words: gammaray burst: general / cosmology: miscellaneous
Appendix A is available in electronic form at http://www.aanda.org
© ESO, 2013
1. Introduction
Gammaray bursts (GRBs) are powerful cosmic explosions that are visible up to high redshifts (z > 5; Lamb & Reichart 2000; Boër et al. 2006; Bloom et al. 2009; Vedrenne & Atteia 2009). GRBs have two phases: a very bright flash of highenergy radiation usually lasting several seconds (sometimes also seen in visible light), called the prompt emission, followed by a longlasting decaying afterglow, visible at longer wavelengths during several hours to months. This behavior is understood as the manifestation of a transient relativistic jet pointing towards the Earth, with the prompt emission being released inside the jet and the afterglow resulting from the interaction of the jet with the surrounding medium. The prompt emission is usually recorded with widefield gammaray imagers or spectrometers and the afterglow with followup telescopes in space and on the ground. One essential element of GRB observations is the measurement of their redshift, which is done with visible or near infrared spectrographs on large groundbased telescopes. Most often, this measurement relies on the identification of absorption lines from the host galaxy in the spectrum of the afterglow, while it is bright enough. After measuring the redshift of a GRB one can correct from cosmological effects and infer its intrinsic properties.
A surprising result obtained by Amati et al. (2002) was the discovery of a strong correlation between E_{pi}, the peak energy of the νFν spectrum of the prompt emission at the source, and E_{iso}, the energy released by a GRB assuming isotropic emission. This correlation was first observed on a sample of ten GRBs with a redshift and well measured spectral parameters (Amati et al. 2002). After this discovery, the validity of this E_{pi} – E_{iso} relation (often called Amati relation) has been widely discussed, with conflicting results (e.g. Band & Preece 2005; Ghirlanda et al. 2005; Nakar & Piran 2005; Sakamoto et al. 2006; Butler et al. 2007; Cabrera et al. 2007; Schaefer & Collazzi 2007; Butler et al. 2009; Firmani et al. 2009; Krimm et al. 2009; Butler et al. 2010; Shahmoradi & Nemiroff 2011; Collazzi et al. 2012; Kocevski 2012; Goldstein 2012). Despite these discussions, the E_{pi} – E_{iso} relation was quickly proposed as a tool to infer GRB redshifts (Atteia 2003), to constrain the physics of the prompt emission (Eichler & Levinson 2004; Rees & Mészáros 2005) and the geometry of GRB jets (Lamb et al. 2005; Toma et al. 2005), and to standardize GRBs, in much the same way as the Phillips relation was used to standardize SNe Ia (Phillips 1993). This last approach aims at constructing standard candles with GRBs and at using them to constrain the cosmological parameters (e.g. Dai et al. 2004; Friedman & Bloom 2005; Ghirlanda et al. 2006).
One difficulty of the study of the E_{pi} – E_{iso} relation is the need to measure both the peak energy of the νFν spectrum (called E_{po} in the observer’s frame, and E_{pi} in the source frame) and the redshift. The simultaneous operation of Swift and Fermi since July 2008 (Gehrels et al. 2004; Atwood et al. 2009; Meegan et al. 2009) has provided a new sample of bursts whose parameters are available, allowing exploring the E_{pi} – E_{iso} relation over a larger parameter space than it was previously possible (see also Amati 2010). This paper discusses the distribution in the plane E_{pi} – E_{iso} of 43 GRBs with a redshift and 243 GRBs without a redshift, all events having E_{po} measured with Fermi/GBM. The GRB samples are presented in Sect. 2. The distribution in the plane E_{pi} – E_{iso} is discussed in Sect. 3. The impact of selection effects on this distribution is addressed in Sects. 4 and 5, this last section addressing more specifically the selection effects associated with the measurement of redshift. The main results of our study and their interpretation are summarized in Sect. 6.
In all this paper we use a flat ΛCDM cosmology with H_{0} = 70 km s^{1} Mpc^{1} and Ω_{Λ} = 0.7, since these values allow the direct comparison of our results with those of Nava et al. (2012). Calculations of luminosity distances are done using the analytical approximation of Wickramasinghe & Ukwatta (2010) that has an accuracy better than 0.3% in the redshift range used in this paper.
2. GRB sample
We select events with well measured spectral parameters from the Fermi GRB spectral catalog of Goldstein et al. (2012; see also Paciesas et al. 2012) that provides the spectral parameters of 482 GRBs between GRB 080714086 and GRB 100709602 (about two years of data). We parametrize GRB spectra with the Band function (Band et al. 1993) that consists of two smoothly connected power laws. Following standard naming, we call α the photon spectral index of the lowenergy power law (α > −2), and β the photon spectral index of the highenergy power law (β < −2). The νFν spectrum peaks at the energy E_{peak}, near the junction of the two power laws. The selection is made with the application of the following cuts:

First, we make a selection on the duration. We consider GRBswith T_{90} between 2 and 1000 s. This criterion excludes short GRBs (T_{90} < 2 s), and very long GRBs that are superimposed on a varying background and whose E_{peak} is difficult to measure accurately.

Second, we require accurate spectral parameters. We exclude GRBs with one or more spectral parameters missing. We exclude GRBs with an error on alpha (the lowenergy index of the Band function) larger than 0.4. We exclude a few GRBs with α < −2.0, and GRBs with β > α because such values suggest a confusion between fitting parameters. We exclude GRBs with large errors on E_{po}, defined by a ratio of the 90% upper limit to the 90% lower limit larger than 3. We have less stringent constraints on beta (the highenergy index of the Band function) since we have checked that the position of GRBs in the E_{pi} – E_{iso} plane changes very little with beta. When the error on β in the catalog is lacking or larger than 1.0, we assign to β the classical value of –2.3 and we give no error. In a few cases, the high energy spectral index in the Fermi catalog is >−2, and the catalog gives the energy of a spectral break that is not E_{po}. In these cases we look for E_{po} in the GCN Circulars, and if we cannot find it, we simply remove the burst from the sample. This procedure suppresses one GRB with a redshift and ten GRBs without a redshift.
267 bursts pass the cuts, 24 with a measured redshift. This is a small number, so we extend the sample of GRBs with a redshift with the addition of 19 Fermi GRBs found in GCN circulars that respect the same selection criteria. In the end, we have two samples: 43 GRBs with a redshift, and 243 GRBs without one. GRBs in each sample are listed in Tables A.1 and A.2. The redshifts span the range [0.34–4.35], with a median value z˜ = 1.71, smaller than the median redshift of Swift GRBs (z˜ = 2.14), measured by Jakobsson et al. (2012).
Fig. 1 Fermi GRBs in the plane E_{pi} –E_{iso}. The black symbols show 43 GRBs with a redshift and the black solid lines show the corresponding best fit E_{pi} – E_{iso} relation with its two sigma limits. Black circles are 34 GRBs detected simultaneously by Fermi/GBM and Swift/BAT. Black triangles are nine GRBs detected simultaneously by Fermi/GBM and another positioning instrument (usually Fermi/LAT). The purple circles indicate GRBs whose redshift has been measured with the emission lines of the host galaxy. The green and red lines show the possible positions of GRBs without a redshift when they are placed at distances ranging from z = 0.34 (leftmost point) to z = 4.35 (rightmost point). Red lines indicate GRBs that are beyond the two sigma limit of the E_{pi} – E_{iso} relation at any redshift. The dotted blue lines show the best fit E_{pi} – E_{iso} relation of Nava et al. (2012) with its two sigma limits. The dashed blue lines show the best fit E_{pi} – E_{iso} relation of Amati (2006) with its two sigma limits. 
3. The E_{pi} – E_{iso} relation of Fermi GRBs
3.1. GRBs with a redshift
Figure 1 shows with black symbols the locations of 43 Fermi GRBs with a redshift in the E_{pi} – E_{iso} plane. These bursts show a clear correlation between E_{pi} and E_{iso}, as for the previous missions. The coefficient of correlation is 0.70, and the best fit correlation is , where E_{52} is the GRB isotropic energy in units of 10^{52} ergs (the best fit correlation is obtained by weighting each point quadratically by its error on E_{pi}). This will be the definition of the E_{pi} – E_{iso} relation in the rest of this paper.
This relation is close to the one found by Gruber et al. (2012) from a sample of Fermi GRBs (), and by Nava et al. (2012) for a larger sample including GRBs from Fermi and other missions (). If we freeze the slope to its bestfit value (0.486), we can compute the vertical standard deviation of data points around the fit: sigma = 0.34 (if we measure the standard deviation perpendicular to the best fit line, we get sigma = 0.21). The twosigma limits are shown in Fig. 1, they are defined by for the lower limit, and , for the upper limit. Figure 1 evidences a broadening of the two sigma confidence region with respect to the relation of Amati (2006) for preFermi GRBs that had a vertical sigma = 0.21. This broadening was already noted by Nava et al. (2012) and by Gruber et al. (2012), who measure sigma = 0.34 along the vertical axis, equivalent to the value found here.
3.2. GRBs without a redshift
The green and red dashed lines in Fig. 1 represent the possible locations of GRBs without a redshift in the E_{pi} – E_{iso} plane, considering that their redshift may vary between 0.34 and 4.35, the extreme values measured for GRBs with a redshift. We plot in green GRBs lying within the two sigma limits of the E_{pi} – E_{iso} relation for some redshifts, and in red the bursts that remain outside these boundaries at any redshift.
For most of the redshift range [0.34–4.35], the green and red lines are nearly parallel to the E_{pi} – E_{iso} relation, allowing assessing the position of GRBs without redshifts with respect to this relation. First, we note a clear asymmetry between GRBs above and below the best fit E_{pi} – E_{iso} relation. While GRBs with a redshift are more or less symmetric with respect to the best fit E_{pi} – E_{iso} relation (22 above and 21 below), the situation is markedly different for GRBs without a redshift. 72% of them lie entirely above the best fit E_{pi} – E_{iso} relation, 22% cross it, and 6% only lie entirely below it. This asymmetry is accompanied by a lack of events in the lower right corner of the plot. Since GRBs in this region have large E_{iso} and low E_{pi}, they should have plenty of photons and we cannot attribute their deficit to selection effects. We conclude that GRBs with a large E_{iso} and a low E_{pi} are intrinsically rare.
We finally note that 29 GRBs are located outside the two sigma limits of the E_{pi} – E_{iso} relation at all redshifts, all of them lie above the two sigma upper limit of the E_{pi} – E_{iso} relation. This is larger than the number of six GRBs expected on a purely statistical basis (assuming a gaussian shape of the distribution around the E_{pi} – E_{iso} relation ). It is difficult to infer the exact significance of this excess, because our definition of the twosigma limits is rather arbitrary and the uncertainties on E_{peak} make difficult evaluating the true number of outliers. Despite these uncertainties, it is not possible to move more than a few outliers within the twosigma limits, and we conclude safely that Fermi GRBs without a redshift include a significant fraction of outliers to the E_{pi} – E_{iso} relation.
At this point we are facing two results: on one hand GRBs with a redshift seem to follow a broad E_{pi} – E_{iso} relation, while on the other hand the overall GRB population shows a fraction of outliers of 12% in a region of the plane where GRBs are difficult to detect, suggesting a much broader distribution in the E_{pi} – E_{iso} plane. The following sections discuss how the observed E_{pi} – E_{iso} distribution is affected by selection effects. In Sect. 4 we discuss selection effects on GRB detection, while in Sect. 5 we discuss selection effects on the measurement of the redshift.
Fig. 2 Distribution of three classes of GRBs (see Fig. 1) in a photon fluence – peak photon flux plane. Several outliers to the E_{pi} – E_{iso} relation (in red) are GRBs at the limit of detection, with low photon fluence and peak photon flux. 
4. Study of the outliers
In the rest of this paper, we call outliers GRBs located above the twosigma limit of the E_{pi} – E_{iso} relation. There are 33 outliers, four GRBs with a redshift, and 29 GRBs without a redshift, they are shown in boldface in Tables A.1 and A.2. In this section we compare the properties of the 33 outliers, with those of the remaining 253 GRBs, in order to understand whether they are affected by significant selection effects. Table 1 shows the results of KolmogorovSmirnov (KS) tests comparing the duration T_{90}, spectral parameters, and brightness between the two classes. Outliers differ from GRBs that follow the E_{pi} – E_{iso} relation by their E_{po} and their brightness: on average, outliers a have larger E_{po} (a median E_{po} of 715 keV vs. 151 keV) and a smaller fluence (a median fluence of 28 × 10^{7} erg cm^{2} vs. 74 × 10^{7} erg cm^{2}). These differences are clearly visible in Fig. 3 that shows the cumulative distribution functions (CDF) of these parameters: on average outliers have an observed E_{peak} four times larger and a photon fluence five times smaller than other GRBs.
Comparison of the observed properties of 33 GRBs incompatible with E_{pi} – E_{iso} relation at all redshifts (outliers) with 253 GRBs that are within two sigma of the E_{pi} – E_{iso} relation (others).
Fig. 3 Cumulative distribution function (CDF) of 3 GRB parameters: E_{po}, photon fluence, and peak photon flux for the three classes of Fermi GRBs defined in Fig. 1. Black solid line: 43 GRBs with a redshift; green dashed line: 253 GRBs with or without a redshift that may be compatible with the E_{pi} – E_{iso} relation at some redshift; red dashdotted line: 29 GRBs without a redshift that are incompatible with the E_{pi} – E_{iso} relation at any redshift (outliers). This figure emphasizes the larger E_{po} and smaller photon fluence of outliers. 
The main consequence is that outliers are much more difficult to detect (see Band 2003, for a discussion of GRB sensitivity of different detectors). In fact, it is difficult to assess the true extension of the GRB population above the E_{pi} – E_{iso} relation because GRBs in this region have fewer photons as the direct consequence of their large E_{po}. The fraction of outliers found here is thus a lower limit of the true fraction of outliers. We conclude that, above the E_{pi} – E_{iso} relation, the observed GRB distribution is currently limited by the sensitivity of gammaray detectors.
Similar conclusions were reached by Nakar & Piran (2005), Goldstein et al. (2010), and Shahmoradi & Nemiroff (2011) based on BATSE GRBs, by Butler et al. (2007) based on Swift GRBs and by Collazzi et al. (2012) based on various GRB samples from preFermi missions, and by Kocevski (2012) based on a synthetic GRB population. Other authors have stressed the role of selection effects, however they consider that they have little impact on the E_{pi} – E_{iso} relation. Ghirlanda et al. (2008) show that the existence of an E_{po} – fluence correlation could impact the E_{pi} – E_{iso} relation of Swift GRBs and that preSwift GRBs also could be affected by unknown selection effects, and Nava et al. (2008) states that, although selection effects are present, they do not determine the spectralenergy correlations. In order to clarify this debate, we now compare the distribution of GRBs with a redshift and GRBs without a redshift in the E_{pi} – E_{iso} plane.
5. GRBs with a redshift
In order to compare GRBs with a redshift and GRBs without a redshift, we discuss the differences between the two classes in terms of their observed properties in Sect. 5.1, and in terms of their intrinsic properties in Sect. 5.2. We limit the sample of GRBs with a redshift to the bursts that have the same selection criteria as GRBs without a redshift, that is to a sample of 34 GRBs detected simultaneously by Fermi/GBM (for the spectral parameters) and Swift/BAT (for the localization). This excludes nine GRBs detected and localized by Fermi/LAT, IPN, or INTEGRAL that were outside the fieldofview of Swift/BAT at the time of the burst. Including these events in the sample of GRBs with a redshift would artificially increase the fraction of very bright GRBs, biasing the comparison between Fermi GRBs with and without a redshift, as explained in Sect. 5.3 below.
5.1. Observed properties
Table 2 compares the properties of GRBs with a redshift with the population of GRBs with no redshift. GRBs with a redshift differ from the general population by their larger brightness (fluence and peak flux). This is not surprising since GRBs that are brighter in gammarays are also expected to be brighter in visible (e.g. Wanderman & Piran 2010). The larger average fluence and peak flux of GRBs with a redshift is explained in part by a smaller distance (the median redshift of our sample is z˜ = 1.71, compared z˜ = 2.14 for Swift GRBs.) and in part by a greater ease to measure the redshift of brighter GRBs. This effect however cannot create the E_{pi} – E_{iso} relation that is an intrinsic property of GRBs. It is thus necessary to check whether the measurement of the redshift biases the intrinsic properties of GRBs, and more specifically their distribution in the E_{pi} – E_{iso} plane.
Comparison of the observed properties of 34 GRBs with a redshift and 243 GRBs without a redshift.
Fig. 4 Comparison of simulated GRB populations. Left panel. E_{pi} as a function of E_{iso}. Right panel. Slope of the E_{pi} – E_{iso} relation as a function of E_{iso}. Green: Fermi GRBs with no redshift not localized by Swift/BAT; purple: Fermi GRBs with no redshift localized by Swift/BAT; blue: Fermi GRBs with a redshift localized by Swift/BAT; black: Fermi GRBs with a redshift, irrespective of the satellite used for the localization. The crosses show the median values and the corresponding error, measured from 1000 simulated samples. The point symbols show the values measured for two actual samples: in blue 34 GRBs studied in Sect. 5.2, and in black the full sample of 43 GRBs with a redshift. 
5.2. Intrinsic properties
In order to check whether the measurement of the redshift impacts the distribution of GRBs in the E_{pi} – E_{iso} plane, we use the following procedure: from the sample of 243 GRBs without a redshift, we randomly select 34 GRBs, we assign them the redshifts of the 34 GRBs with measured redshifts, and we compute their E_{pi}, E_{iso}, and the slope of the best fit E_{pi} – E_{iso} relation. We repeat this procedure 1000 times and we compare the median values of E_{pi}, E_{iso}, and the slope of the E_{pi} – E_{iso} relation with those of GRBs with redshifts. We apply the same procedure to the subsample of 34 GRBs with a redshift, in order to understand how the redshift redistribution impacts the results. For sake of completeness, we study two additional distributions: 25 Fermi GRBs localized by Swift without a redshift, and the full sample of 43 GRBs with a redshift. The results are summarized in Fig. 4. The crosses show the median E_{iso}, E_{pi}, the median slope of the E_{pi} – E_{iso} relation and the onesigma error on these values, for the four simulated GRBs samples. The blue and black symbols show the values measured for GRBs with a redshift (excluding or not the GRBs detected by Fermi/LAT).
These simulations bring the following results:

The compatibility (within errors) of the green and purple crossesshows that the localization with Swift/BAT does not bias thedistribution of E_{pi} or E_{iso}.

The comparison of the blue and green crosses shows that GRBs with a redshift have slightly larger E_{iso} (median =9.2 × 10^{52} erg vs. 4.8 × 10^{52} erg), similar E_{pi} (median = 345 keV vs. 440 keV), and slopes of the E_{pi} – E_{iso} relation that are compatible (~0.3). The measurement of the redshift thus favors GRBs that are more energetic, with large E_{iso}.

The comparison of the simulated populations (blue and black crosses) with the actual GRBs (blue and black symbols) shows that the redshift redistribution builds simulated populations that have more or less the same E_{iso} and E_{pi} as the original population, but a significantly smaller slope of the E_{pi} – E_{iso} relation. This difference can be attributed to the fact that, unlike real GRBs, simulated GRBs can occupy the empty region in the lower right corner of the E_{pi} – E_{iso} plane, decreasing the slope of the correlation.
The differences pointed here between GRBs with a redshift and GRBs without a redshift concern average values of the parameters. They provide no information on the joint distribution in the plane E_{pi} – E_{iso} for the two groups. A look at Fig. 1 suggests different distributions of the two groups, with GRBs with a redshift gathered closer to the E_{pi} – E_{iso} boundary. To verify this hypothesis, we have divided our GRB sample (34 GRBs with a redshift plus 243 GRBs without a redshift) into four subsamples according to their distance to the E_{pi} – E_{iso} boundary (for GRBs with a redshift) or to their smallest distance to this line (for GRBs without a redshift). Since we have excluded the nine GRBs with a redshift detected by Fermi/LAT, each subsample includes eight or nine GRBs with a redshift. We have computed the fraction of GRBs with a redshift in each subsample, obtaining the following numbers with increasing distance from the boundary: 9/44 = 0.20; 9/42 = 0.21; 8/68 = 0.12; and 8/123 = 0.065. These numbers clearly show that GRBs with a redshift do not follow the general GRB distribution in the plane E_{pi} – E_{iso}, being more concentrated near the E_{pi} – E_{iso} boundary. This is illustrated in Fig. 5 that shows with different colors the groups of GRBs at different distances from the E_{pi} – E_{iso} boundary.
With this observation, we finally get a coherent picture of the origin of E_{pi} – E_{iso} relation. This correlation can be attributed to the combination of two effects: a physical boundary in the plane E_{pi} – E_{iso}, and biases favoring the detection and the measurement of redshift of GRBs located along this boundary. We have evidenced two biases acting in the same direction. First, for a given E_{iso}, GRBs far from the E_{pi} – E_{iso} boundary have fewer gammaray photons, making their detection and localization with gammaray detectors more difficult. This is confirmed with the computation of V/V_{max} as a proxy for the signaltonoise ratio^{1}. We find that the median V/V_{max} increases with the distance from the E_{pi} – E_{iso} boundary, with median (V/V_{max}) = 0.28−0.35−0.51−0.75 for the 4 groups of GRBs with a redshift defined in the previous paragraph. These numbers only concern a small fraction of GRBs but they clearly show that GRBs close to the E_{pi} – E_{iso} boundary have larger signaltonoise and are more easily detected.
Fig. 5 GRBs classification in 4 groups according to their distance to the E_{pi} – E_{iso} boundary. The parameter used to separate the 4 groups is their vertical distance (for GRBs with a redshift) or their smallest vertical distance (for GRBs without a redshift) to the black line. The black line has a slope = 0.486, assumed to be the slope of the E_{pi} – E_{iso} boundary. The position of the black line is arbitrary, it does not impact the groups as long as it stays below the GRB sample. 
The second bias is connected with the measurement of the redshift: we have found that the fraction of GRBs with a redshift increses close to the E_{pi} – E_{iso} boundary. This suggests the existence of a bias that makes easier the measurement of the redshift when a GRB is close to the E_{pi} – E_{iso} boundary. Since the capability to measure the redshift is closely linked with the optical brightness of the afterglow, we infer that there are factors enhancing the optical luminosity of GRBs close to the E_{pi} – E_{iso} boundary. The verification of this hypothesis requires a systematic study of the optical luminosity of GRB afterglows, which is beyond the scope of this paper. This bias contributes to further reduce the width of the distribution of GRBs with a redshift. As a consequence of these two biases, GRBs with a redshift appear preferentially along a narrow band above the E_{pi} – E_{iso} boundary, giving rise to correlation found by Amati et al. (2002).
5.3. Bright GRBs detected with LAT and IPN
Figure 1 shows that the 9 GRBs that were not localized by Swift (the black triangles) concentrate in the upper right of the E_{pi} – E_{iso} relation, increasing the significance of the correlation. In our interpretation, these very bright GRBs cannot have low E_{pi} because of the E_{pi} – E_{iso} boundary. Since they brightness facilitates the measurement of their redshift, they play an important role in shaping the E_{pi} – E_{iso} relation.
This is also the case of GRB 130427A (the rightmost point in Figs. 2 and 5) that is one of the few nearby energetic GRBs (like GRB 030329 detected by Hete2 at z = 0.17, Vanderspek et al. 2004). GRB 130427A is at redshift z = 0.34, the value of V/V_{max} for this GRB, V/V_{max} = 0.003, indicates that there is a 10% chance observing this GRB so close among 30 similar bursts at higher redshift. Considering that Fermi has detected more than 1100 GRBs from its launch to the end of April 2013, the detection of a nearby energetic GRB like GRB 130427A is not unexpected. Interestingly, GRB 130427A lies perfectly along the Amati relation (Amati et al. 2013). This is compatible with our interpretation that highly energetic bursts cannot have low E_{pi} because of the E_{pi} – E_{iso} boundary.
6. Discussion
The results presented in this paper are based on the distribution of GRBs in the plane E_{pi} – E_{iso}. We discuss below some selection effects that may affect this distribution. First, it is not possible to measure E_{po} outside the energy range of the detector. This restricts the range of measured E_{po} to [30–1300] keV. These bounds prevent us to include XRay Flashes (soft GRBs with E_{peak} < 30 keV) in our analysis but they do not bias the distribution of GRBs in the E_{pi} – E_{iso} plane and the conclusions of our analysis. Second, we have studied the impact of the errors on E_{po} and on the energy fluence S_{γ} given in Table A.2. If we systematically replace E_{po} with E_{po}−σ_{Epo} for all GRBs without a redshift, the number of outliers decreases from 29 to 18. Similarly, if we change the fluence S_{γ} with S_{γ}−σ_{Sγ} for all GRBs without a redshift, the number of outliers decreases from 29 to 28. Thus, we cannot get rid of the outliers, even with this rather drastic method.
Two additional effects, acting in opposite directions, can affect the observed distribution in the E_{pi} – E_{iso} plane. First, GRBs in the upper left part of the plot are more difficult to detect because they have intrinsically fewer photons than GRBs closer to the Amati relation. Since this bias acts against the detection of outliers, it leads to underestimate the true fraction of outliers, as explained in Sect. 4. On the other hand, it is possible to “create” outliers from GRBs following the Amati relation if the measurements of E_{pi} or E_{iso} are strongly biased. One possibility is the “duration bias” discussed by Kocevski & Petrosian (2013). This bias results from the inability of countlimited instruments to detect the parts of GRBs with low count rates, artificially reducing T_{90} and E_{iso} of faint GRBs. The duration bias may also affect the measurement of E_{peak} since the brightest parts of faint GRBs are often those with the largest E_{peak}. Since Table 1 shows that outliers have essentially the same duration distribution as the rest of GRBs, there is no hint in the data of a strong duration bias that could significantly alter our analysis. We have also checked the possible biases on E_{peak} with the measurement of the ratio RE = E_{peak} (at peak)/E_{peak} (average) for outliers and for other GRBs. We find that RE is broadly distributed with median(RE) = 0.74 ± 0.30 for outliers, median(RE) = 1.14 ± 0.28 for other GRBs, and median(RE) = 1.06 ± 0.28 for GRBs with a redshift. These numbers show that outliers are only marginally affected by selection effects that could modify their E_{peak}. Overall, we found no bias capable of “creating” the observed outliers from GRBs that follow the E_{pi} – E_{iso} relation.
The main conclusion of this paper is that the E_{pi} – E_{iso} relation is due to the combination of two effects: a physical limit that prevents the existence of GRBs with a large E_{iso} and a low E_{pi} (creating an E_{pi} – E_{iso} boundary ), and the sensitivity of gammaray and optical imagers that strongly favor the detection and localization of GRBs located close to the E_{pi} – E_{iso} boundary (see also Shahmoradi & Nemiroff 2011). The combination of these two effects creates a band in the E_{pi} – E_{iso} plane where GRBs are more easily detected, and a thinner band where their redshift can be measured. We believe that this interpretation explains the apparently contradictory results obtained by people who have checked the validity of E_{pi} – E_{iso} relation either using large samples of GRBs without a redshift or only using GRBs with a redshift. The first have consistently found large fractions of outliers dismissing the relation (e.g. Band & Preece 2005; Nakar & Piran 2005; Butler et al. 2007, 2010; Shahmoradi & Nemiroff 2011; Collazzi et al. 2012), while the second usually confirmed its validity (e.g. Ghirlanda et al. 2005; Sakamoto et al. 2006; Krimm et al. 2009; Nava et al. 2012). The horizontal extent of the E_{pi} – E_{iso} relation depends on the GRB luminosity function, on the evolution of the GRB rate with redshift, and on the volume of detection for each type of GRB. We have checked that the leftmost part of the E_{pi} – E_{iso} relation contains mainly faint GRBs with low E_{pi} detected at low redshift (z < 1), while the rightmost part is made of luminous GRBs with large E_{pi} that can only be found in large volumes of observation (except GRB 130427A discussed in Sect. 5.3). Finally, we note that similar studies could be performed with other relations, like the E_{pi} – L_{iso} relation (Yonetoku et al. 2004) or the E_{pi} – E_{γ} relation (Ghirlanda et al. 2004), to check their nature and their validity.
A discussion of the origin of the E_{pi} – E_{iso} boundary is beyond the scope of this paper. If it is a consequence of the dominant radiation mechanism at work in GRBs, as we expect, theoretical studies may help understanding the nature and the true location of this boundary (see for instance Eichler & Levinson 2004; Yamazaki et al. 2004; Rees & Mészáros 2005; Thompson et al. 2007; Giannios 2012; Mochkovitch et al. 2012; Beniamini & Piran 2013; Shahmoradi 2013, for explanations within the framework of different theoretical or phenomelogical models). The observation of an E_{pi} – E_{iso} relation in timeresolved spectra (e.g. Ghirlanda et al. 2010; Frontera et al. 2012; Basak & Rao 2012; Lu et al. 2012) may also provide insights into the origin of the E_{pi} – E_{iso} boundary. Finally, Since the E_{pi} – E_{iso} relation is not an intrinsic property of GRBs, it may be difficult to use it to standardize GRBs (Ghirlanda et al. 2006; Mészáros & Gehrels 2012). This might nevertheless be possible if one finds a way of identifying GRBs close to the E_{pi} – E_{iso} boundary, thanks to their temporal properties (e.g. spectral lags) or to the characteristics of their afterglows.
Online material
Appendix A: Data tables
The following tables provide the names and the spectral properties of the 286 GRBs used in this study, extracted from the Fermi GRB catalog of Goldstein et al. (2012).
43 GRBs with a redshift.
243 GRBs without a redshift.
continued.
continued.
continued.
Acknowledgments
The authors gratefully acknowledge the use of the online Fermi GRB catalog^{2}, the use of the GRB table compiled by J. Greiner (http://www.mpe.mpg.de/~jcg/grbgen.html), and the use of the table of GRB redshifts compiled by Páll Jakobsson^{3}. Y.Z. welcomes financial support from IRAP (UMR5277/CNRS/UPS).
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All Tables
Comparison of the observed properties of 33 GRBs incompatible with E_{pi} – E_{iso} relation at all redshifts (outliers) with 253 GRBs that are within two sigma of the E_{pi} – E_{iso} relation (others).
Comparison of the observed properties of 34 GRBs with a redshift and 243 GRBs without a redshift.
All Figures
Fig. 1 Fermi GRBs in the plane E_{pi} –E_{iso}. The black symbols show 43 GRBs with a redshift and the black solid lines show the corresponding best fit E_{pi} – E_{iso} relation with its two sigma limits. Black circles are 34 GRBs detected simultaneously by Fermi/GBM and Swift/BAT. Black triangles are nine GRBs detected simultaneously by Fermi/GBM and another positioning instrument (usually Fermi/LAT). The purple circles indicate GRBs whose redshift has been measured with the emission lines of the host galaxy. The green and red lines show the possible positions of GRBs without a redshift when they are placed at distances ranging from z = 0.34 (leftmost point) to z = 4.35 (rightmost point). Red lines indicate GRBs that are beyond the two sigma limit of the E_{pi} – E_{iso} relation at any redshift. The dotted blue lines show the best fit E_{pi} – E_{iso} relation of Nava et al. (2012) with its two sigma limits. The dashed blue lines show the best fit E_{pi} – E_{iso} relation of Amati (2006) with its two sigma limits. 

In the text 
Fig. 2 Distribution of three classes of GRBs (see Fig. 1) in a photon fluence – peak photon flux plane. Several outliers to the E_{pi} – E_{iso} relation (in red) are GRBs at the limit of detection, with low photon fluence and peak photon flux. 

In the text 
Fig. 3 Cumulative distribution function (CDF) of 3 GRB parameters: E_{po}, photon fluence, and peak photon flux for the three classes of Fermi GRBs defined in Fig. 1. Black solid line: 43 GRBs with a redshift; green dashed line: 253 GRBs with or without a redshift that may be compatible with the E_{pi} – E_{iso} relation at some redshift; red dashdotted line: 29 GRBs without a redshift that are incompatible with the E_{pi} – E_{iso} relation at any redshift (outliers). This figure emphasizes the larger E_{po} and smaller photon fluence of outliers. 

In the text 
Fig. 4 Comparison of simulated GRB populations. Left panel. E_{pi} as a function of E_{iso}. Right panel. Slope of the E_{pi} – E_{iso} relation as a function of E_{iso}. Green: Fermi GRBs with no redshift not localized by Swift/BAT; purple: Fermi GRBs with no redshift localized by Swift/BAT; blue: Fermi GRBs with a redshift localized by Swift/BAT; black: Fermi GRBs with a redshift, irrespective of the satellite used for the localization. The crosses show the median values and the corresponding error, measured from 1000 simulated samples. The point symbols show the values measured for two actual samples: in blue 34 GRBs studied in Sect. 5.2, and in black the full sample of 43 GRBs with a redshift. 

In the text 
Fig. 5 GRBs classification in 4 groups according to their distance to the E_{pi} – E_{iso} boundary. The parameter used to separate the 4 groups is their vertical distance (for GRBs with a redshift) or their smallest vertical distance (for GRBs without a redshift) to the black line. The black line has a slope = 0.486, assumed to be the slope of the E_{pi} – E_{iso} boundary. The position of the black line is arbitrary, it does not impact the groups as long as it stays below the GRB sample. 

In the text 
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