The rate and luminosity function of long Gamma Ray Bursts
A. Pescalli, G. Ghirlanda, R. Salvaterra, G. Ghisellini, S. D. Vergani, F. Nappo, O. S. Salafia, A. Melandri, S. Covino, D. Götz
aa r X i v : . [ a s t r o - ph . H E ] J un Astronomy&Astrophysicsmanuscript no. PaperBAT6AA c (cid:13)
ESO 2015June 19, 2015
The rate and luminosity function of long Gamma Ray Bursts
A. Pescalli ⋆ , G. Ghirlanda , R. Salvaterra , G. Ghisellini , S. D. Vergani , F. Nappo , O. S. Salafia , A.Melandri , S. Covino , and D. Götz Universitá degli Studi dell’Insubria, via Valleggio 11, I-22100 Como, Italy INAF – Osservatorio Astronomico di Brera, via E. Bianchi 46, I-23807 Merate, Italy INAF - IASF Milano, via E. Bassini 15, I-20133 Milano, Italy GEPI, Observatoire de Paris, CNRS, Univ. Paris Diderot, 5 place Jule Janssen, F-92190 Meudon, France Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy AIM (UMR 7158 CEA/DSM-CNRS-Université Paris Diderot) Irfu/Service d’Astrophysique, Saclay, F-91191 Gif-sur-YvetteCedex, FranceJune 19, 2015
ABSTRACT
We derive, adopting a direct method, the luminosity function and the formation rate of long Gamma Ray Bursts through a complete,flux–limited, sample of
Swift bursts which has a high level of completeness in redshift z ( ∼ L = L (1 + z ) k and we derive k = .
5, consistently with recent estimates. The de-evolved luminosity function φ ( L ) of GRBs can be represented by a broken power law with slopes a = − . ± .
21 and b = − . ± .
24 below and above,respectively, a characteristic break luminosity L , b = . ± . erg/s. Under the hypothesis of luminosity evolution we find that theGRB formation rate increases with redshift up to z ∼
2, where it peaks, and then decreases in agreement with the shape of the cosmicstar formation rate. We test the direct method through numerical simulations and we show that if it is applied to incomplete (both inredshift and/or flux) GRB samples it can misleadingly result in an excess of the GRB formation rate at low redshifts.
Key words.
Gamma-ray: bursts
1. Introduction
Since the discovery of Gamma Ray Bursts (GRBs), one of themost important questions was related to their distance scale (i.e.whether galactic or cosmological) which had immediate impli-cations on their associated luminosities and energetics. Throughthe afterglow detection (Costa et al. 1997; Van Paradijs 1997)and first redshift measurements, GRBs were proven to be cos-mological sources with large isotropic equivalent luminositiesexceeding, in few cases, 10 erg s − . The pinpointing of theGRB afterglow, made available by the fast slewing of the Swift satellite (Gehrels et al. 2004), coupled with intense efforts to ac-quire early time optical spectra from ground, allowed us to mea-sure the redshifts z of GRBs with an average efficiency of 30%.Among these, GRB 090423 (with a spectroscopic z = . z = . ψ ( z ) (GRB formation rate,GRBFR hereafter), representing the number of bursts per unitcomoving volume and time as a function of redshift, and (b)their luminosity function φ ( L ) (LF hereafter), representing therelative fraction of bursts with a certain luminosity. Here, with φ ( L ) we refer to the differential luminosity function defined as dN ( L ) / dL .Recovering ψ ( z ) and φ ( L ) of GRBs allows us to test the na-ture of their progenitor (e.g. through the comparison with the ⋆ E–mail:[email protected] cosmic star formation rate) and to study the possible presence ofsub–classes of GRBs at the low end of the luminosity function(e.g. Liang et al. 2007, see also Pescalli et al. 2015). These twofunctions have been derived for the population of long GRBs(e.g. Daigne et al. 2006; Guetta & Della Valle 2007; Firmani etal. 2004; Salvaterra & Chincarini 2007; Salvaterra et al. 2009b,2012; Wanderman & Piran 2010; Yu et al. 2015; Petrosian et al.2015) through different methods and samples of bursts (§2). Forthe population of short GRBs, ψ ( z ) and φ ( L ) have been less se-curely constrained (e.g. Guetta & Piran 2005, 2006; Nakar 2006;Berger et al. 2014; D’Avanzo et al. 2015) due to the limited num-ber of bursts with measured redshifts.However, ψ ( z ) and φ ( L ) cannot be derived straightforwardlyusing all GRBs with known redshift since these samples are af-fected by observational biases. Specific methods that correct forsuch biases should be adopted. The main approaches that havebeen used so far (§2) agree on the shape of the luminosity func-tion (typically represented by a broken power law) but lead to re-markably different results on the cosmic GRB rate (particularlyat low redshifts). Independently from the method used to recoverthese two functions, most of the previous studies (see howeverSalvaterra et al. 2012) adopted either heterogeneous samples (i.e.including GRBs detected by different satellites/GRB detectorswhich have different sensitivities) and/or incomplete samples.In particular, incompleteness is induced by several effects suchas the variation of the trigger efficiency and the redshift mea-surement. Accounting for such instrumental effects is extremelydifficult in practice. Article number, page 1 of 9&Aproofs: manuscript no. PaperBAT6AA
An alternative is to work with complete samples, at the ex-pense of the number of GRBs in the sample. Salvaterra et al.(2012) (S12) defined a complete flux–limited sample of GRBs(called BAT6) detected by
Swift which, despite containing a rel-atively small number of GRBs, has a high redshift completenessand has been extensively used to test various prompt and after-glow properties of GRBs in an unbiased way (Campana et al.2012, D’Avanzo et al. 2012, Ghirlanda et al. 2012, Melandri etal. 2012, Nava et al. 2012, Covino et al. 2013, Melandri et al.2014, Vergani et al. 2014).The aim of this work is to derive φ ( L ) and ψ ( z ) through thiscomplete sample of GRBs detected by Swift . We summarise themain different approaches that have been used in the literature toderive the luminosity function and the formation rate of GRBs(§2) and present the updated sample used in this work in §3. Weadopt the C − direct method (Lynden-Bell et al. 1971) to derivethe φ ( L ) and ψ ( z ) and compare it with previous results in §4.Throughout the paper we assume a standard Λ CDM cosmolog-ical model with Ω m = . Ω Λ = . H =
70 km s − Mpc − . We use the symbol L to indicate the isotropic equivalentluminosity L iso omitting for simplicity the subscript “iso". φ ( L ) and ψ ( z ) of long GRBs The number of GRBs detectable by a given instrument above itssensitivity flux limit S can be expressed as: N ( > S ) = Ω T π Z z ( L max , S )0 Z L max L lim ( S , z ) φ ( L , z ) ψ ( z )(1 + z ) dVdz dL dz (1)where Ω and T are the instrument field of view and timeof operation, respectively, and dV / dz is the differential comov-ing volume. Here, z ( L max , S ) is the maximum redshift at which aburst with L max would still be above the instrumental flux limit S ; L lim ( S , z ) is the minimum observable luminosity as a functionof z (i.e. that corresponds to a flux above S ).If φ ( L ) and ψ ( z ) are known, it is possible to derive from Eq. 1the flux distribution of the population of GRBs observable by agiven detector, knowing its instrumental parameters. By revers-ing this argument, one can assume the functional forms of φ ( L )and ψ ( z ) (e.g. specified through a set of free parameters) and con-strain them by fitting the the model flux distribution (i.e. N ( > S ))to the observed flux distribution of a given instrument. This indi-rect method has been used to infer the luminosity function (e.g.Firmani et al. 2004; Salvaterra & Chincarini 2007; Salvaterra etal. 2009, 2012) by fitting e.g. the flux distribution of the largepopulation of GRBs detected by BATSE.The number of free parameters, if both φ ( L ) and ψ ( z ) are tobe constrained, can be large. One possibility is to assume that,based on the massive star progenitor origin of long bursts, theGRB cosmic rate traces the cosmic star formation rate, i.e. ψ ( z ) ∝ ψ ⋆ ( z ). In this way, the method allows to derive the free param-eters of φ ( L ) by fitting the result of Eq. 1 to large, statisticallysignificant, samples of observed GRBs. The assumed ψ ( z ) canbe tested by fitting the observed redshift distribution of a sampleof bursts with measured z .In the simplest scenario, the two functions φ ( L ) and ψ ( z ) areindependent. However, more realistic analyses also consideredthe possible evolution of either the luminosity function or theGRB formation rate with redshift. For example, in the case ofluminosity evolution, the burst luminosity depends on z throughthe relation L ( z ) = L (1 + z ) k (luminosity evolution scenario).Alternatively, the GRB formation rate ψ ( z ) varies with redshift with a similar analytical dependence ψ ( z ) ∝ ψ ⋆ ( z )(1 + z ) d (den-sity evolution scenario). This means that the progenitor charac-teristics evolve with z and that the ratio of the GRB formationrate to the cosmic star formation rate is not constant. Amongthe drawbacks of this method is that it relies on the assumptionof a specific functional form of φ ( L ) (and simple functions, e.g.power law, broken power law or power law with a cutoff at lowluminosities, have been adopted) and it allows to test only forevolution of the luminosity or of the rate independently.Salvaterra et al. (2012) applied the indirect method to a com-plete sample of GRBs detected by Swift (§3). They find thateither a luminosity evolution with k = . ± . d = . ± . Swift complete sample. However, they can not discriminate betweenthese two scenarios. They derive the luminosity function φ ( L )testing two analytical models: a power-law with an exponen-tial cut-off at low luminosities and a broken power-law (BPL, φ ( L ) ∝ ( L / L b ) a , b , where a and b are the slopes of the powerlaw below and above the break L b ) adopting a minimum GRBluminosity L min = erg/s. For the BPL model they found a = − . + . − . , b = − . + . − . , L b = . + . − . × erg/s and a = − . + . − . , b = . + . − . , L b = . + . − . × erg/s in the caseof luminosity and density evolution scenario, respectively.The alternative method is based on the direct derivation ofthe φ ( L ) and ψ ( z ) from observed samples of GRBs with mea-sured z and L . This method has been inherited from the studiesof the luminosity function of quasars and blazars (e.g. Chianget al. 1998, Maloney & Petrosian 1999, Singal et al. 2012,2013)and it has been applied to GRBs (Lloyd et al. 1999, Kocevski &liang 2006). Wanderman & Piran (2010) adopt a maximum like-lihood estimator to derive the discrete luminosity function andcosmic formation rate. They use the sample of ∼
100 GRBs de-tected by
Swift with measured redshift (through optical afterglowabsorption lines and photometry). Despite this sample might suf-fer from incompleteness, they derive φ ( L ) (extending from 10 erg/s up to 10 erg/s) which can be represented by a brokenpower law with a = − . b = − . L b = . erg/s. Sim-ilarly, they also derived the discrete GRB formation rate ψ ( z )which can be represented by a broken power-law as a functionof (1 + z ) with indices n = . + . − . and n = − . + . − . peaking at z = . + . − . . This rate is consistent with the SFR of Bouwenset al.(2009) for z .
3. They assume that the luminosity is independentfrom redshift.More recently, Yu et al. (2015 - Y15 hereafter) and Pet-rosian et al. (2015 - P15 hereafter) applied a statistical methodto reconstruct the discrete φ ( L ) and ψ ( z ) from a sample of Swift bursts with measured redshifts. They both find a strong lumi-nosity evolution with k ∼ .
3. Their results converge towardsa cumulative luminosity function described by a broken power-law with α = − . ± . β = − . ± . L b = . × erg/s (Y15) and α = − . β = − . L b = erg/s (P15).These indices ( α and β ) are the slopes of the cumulative lumi-nosity function which is linked to the differential one throughthe integral Φ ( L ) = N ( > L ) = R L max L φ ( L ) dL . Therefore, for aBPL luminosity function, the slopes of the differential form are( a , b ) = ( α − , β − z =
1. If compared to the SFR, this behaviour would imply arelative excess of the GRB formation rate with respect to ψ ∗ ( z ) at z ≤ the excess of GRBs at low redshifts . This result Article number, page 2 of 9. Pescalli et al.: The rate and luminosity function of long Gamma Ray Bursts is puzzling also because it is completely at odds with the find-ings of the works based on the properties of GRB host galaxies.In fact, Vergani et al. (2014), Perley et al. (2015a, 2015b, 2015c)and Kruhler et al. (2015), performed multi-wavelength and spec-troscopic studies on the properties (stellar masses, luminosities,SFR and metallicity) of GRB host galaxies of different completeGRB samples and compared them to those of the star-forminggalaxies selected by galaxy surveys. All their results clearly in-dicate that at z < Swift with measured redshifts. However,while Yu et al. (2015) work with the bolometric luminosityof GRBs, Petrosian et al. (2015) adopt the luminosity in the
Swift /BAT (15-150 keV) energy band. Y15 use all GRBs de-tected by
Swift with a measured redshift and well constrainedspectral parameters: despite their relatively large number of ob-jects ( ∼ ∼ /
250 events with measured z .Independently from the method adopted to recover φ ( L ) and ψ ( z ), one key point is the definition of the sample. S12 showedthe importance of working with complete samples of GRBs (seealso Hjorth et al. 2012). Here we start with the so called BAT6 Swift sample (S12) and extend it with additional bursts that sat-isfy its selection criteria (§3). We will then use it to derive theluminosity function and the cosmic GRB formation rate (§4).
3. BAT6 extended version
The BAT6 complete sample as defined in S12 was composed by58
Swift
GRBs with (i) favourable observing conditions for theirredshift measurement as proposed in Jackobsson et al. (2006)and (ii) a peak photon flux P ≥ . − s − (integrated overthe 15–150 keV Swift /BAT energy band). This sample, which iscomplete in flux by definition, turned out, after selection, also tobe highly complete ( ∼ z ).The study of the isotropic equivalent luminosity L of thebursts of the BAT6 sample requires the knowledge of their broadband prompt emission spectrum. Nava et al. (2012) collected the46/52 GRBs, within the BAT6, with measured E p and z . Sixbursts with measured z did not have E p measurements. One ofthe main drawbacks of the narrow/soft energy range of the BATinstrument is the difficulty to measure the peak E p of the ν F ν spectrum for several bursts it detects. Other instruments (e.g. Konus/Wind - Aptekar et al. 1995,
Fermi /GBM - Meegan et al.2009 or
Suzaku /WAM - Yamaoka et al. 2009) compensate forthis lack, thanks to their wide energy range, measuring a spec-trum extending from few keV to several MeV.Sakamoto et al. (2009) showed that for
Swift bursts with mea-sured E p there is a correlation between the slope of the spectrum α PL (when fitted with a single powerlaw model) and the peak en-ergy E p (measured by fitting a curved model). With the aim ofenlarging the sample of Nava et al. (2012), we estimated E p ofsix bursts of the BAT6, whose BAT spectrum is fitted by a sin-gle power law, through this relation (Sakamoto et al. 2009) andverified that the values obtained are consistent with those of theother bursts (we performed the K − S test finding a probabilityof ∼
70% that the two sets of peak energies originate from thesame distribution). We find that all but one GRB (i.e. 070306) have E restp = E obsp (1 + z ) consistent with the upper/lower limit re-ported in Nava et al. (2012). Therefore, we firstly extended theBAT6 sample of Nava et al. (2012) with measured z and L to50/58 bursts.Since the construction of the BAT6, other bursts satisfying itsselection criteria were detected by Swift . Moreover, some burstalready present in the original BAT6 sample were re–analysedand either their redshifts and/or their spectral properties wererevised. So our first aim was to revise the BAT6 sample. In par-ticular, the revision of 8 redshifts is here included (marked initalics in the table - their luminosity has been updated). The re-vised BAT6 sample then contains 56/58 GRBs with measured z and 54/58 with also a bolometric isotropic luminosity L . Con-sidering only the redshift, the sample is ∼
97% complete, whileif we also require the knowledge of L , the completeness level isonly slightly smaller ( ∼ Swift single power law spectrum, we estimated E p through theSakamoto et al. (2009) relation. The BAT6 extended (BAT6ext,hereafter) counts 82/99 GRBs with z ( and 81/99 with z and L .Its completeness in redshift is ∼ z , the spectral parame-ters (high and low photon indices α and β and the rest framepeak energy E restp ), the peak flux with the relative energy band,the isotropic equivalent luminosity L . The spectrum is a cut-offpower-law (CPL) if only the low energy photon index α is re-ported and a Band function if also the high energy photon index β is given. When z is not measured, we report the observed peakenergy. The luminosities reported in the table are calculated inthe [1 − ] keV rest frame energy range only for those GRBshaving both z and E p .
4. Luminosity function and GRB formation rate
In this section we will apply the C − method as originally pro-posed by Lynden-Bell et al. (1971) and applied to GRBs by e.g.Yonetoku et al. (2004,2014), Kocevski & Liang (2006), Wu etal. (2014). This method is based on the assumption that the lu-minosity is independent from the redshift. However, as discussedin Petrosian et al. (2015) a strong luminosity evolution could bepresent in the GRB population. Efron & Petrosian (1992) pro-posed a non–parametric test to estimate the degree of correlationof the luminosity with redshift induced by the flux in a flux–limited sample. This is also the case of the BAT6ext sample andthe first step will be to quantify the degree of correlation. Yu etal. (2015) and Petrosian et al. (2015) indeed found that the lumi-nosity evolves with redshift within their samples as (1 + z ) . + . − . (Y15) or (1 + z ) . ± . (P15). We applied the same method of Y15and P15 (also used in Yonetoku et al. 2004,2014) to the BAT6extsample: we define the luminosity evolution L = L (1 + z ) k (as inY15), where L is the de-evolved luminosity, and compute themodified Kendall correlation coefficient (as defined in Efron &Petrosian 1992). We find, consistently with the results of Y15and P15, k = .
5. Similar results were obtained through the in-direct method (see §2) by S12 using the BAT6 sample.We can now define the de-evolved luminosities L = L / (1 + z ) k for every GRBs and apply the Lynden–Bell C − method to Article number, page 3 of 9&Aproofs: manuscript no. PaperBAT6AA
Table 1.
BAT6ext (BAT6 extended) GRB complete sample. Columns report, in order, the redshift z , the spectral photon indices α and β , thepeak flux in units of 10 − erg cm − s − (except for those with the ⋆ which are in units of photons cm − s − ), the respective energy band, therest-frame peak energy E p and the bolometric equivalent isotropic luminosity L (calculated in the [1 − ] rest-frame energy range). For L wealso give the 1 σ error. b Bursts with missing z are reported with their observer frame E obsp for completeness, they are not used in the presentwork. c The peak energy has been estimated with the relation of Sakamoto et al. (2009). The eight GRBs (already present in the compilation ofNava et al. 2012) with an updated redshift estimate are marked in italics. In the last column we report the references, in order, for the spectralparameters and for the redshift: 1) Nava et al. (2012), 2) Covino et al. (2013), 3) Kruhler et al. (2015), 4) GCN
GRB z α [ β ] Peak flux Range E p L Ref10 − erg cm − s − (keV) (keV) ( × erg/s) ⋆ phot cm − s − . − . ± .
32 2 . ± .
17 [15 − ±
27 4 . ± .
86 1,1050401 2.9 − . − .
45] 24 . ± . − ±
117 201 ±
11 1,1050416A 0.653 − . − .
4] 5 . ± . ⋆ [15 − ± . ± .
12 1,1050525A 0.606 − . ± .
11 47 . ± . ⋆ [15 − ± . ± .
28 1,1050802 c − . ± . . ± .
35 [15 − . ± .
71 1,1050922C 2.198 − . ± .
24 45 ± − ±
118 187 ±
30 1,1060206 4.048 − . ± .
30 2 . ± .
13 [15 − ±
116 49 . ± . − . ± .
26 2 . ± . ⋆ [15 − ±
187 52 . ± . − . ± . . ± .
28 [15 − . ± . . ± .
26 1,2060614 0.125 − . . ± . ⋆ [15 − ±
45 0 . ± .
01 1,1060814 1.92 − . ± .
16 21 . ± . − ±
245 71 . ± . − − . ± .
05 13 ± −
10 000] 235 ± b − − . ± .
25 2 . ± .
23 [15 − ±
207 12 . ± . c − . ± .
08 25 ± −
10 000] 127 20 . ± . − . ± .
36 2 . ± .
17 [15 − ±
90 108 . ± . − . ± .
02 [ − . ± .
09] 120 ±
10 [20 −
10 000] 965 ±
27 109 . ± . − . ± .
13 37 . ± . − ±
485 1 . ± .
46 1,1061121 1.314 − . ± .
05 128 ±
17 [20 − ±
185 142 ±
19 1,1061222A 2.09 − . ± .
05 [ − . ± .
38] 48 ±
13 [20 −
10 000] 1091 ±
167 140 ±
38 1,1070306 c − . ± . . ± .
16 [15 − > > .
99 1,1 − . ± .
04 [ − . ± .
24] 59 ±
12 [20 −
10 000] 2349 157 . ± . − . ± .
12 41 . ± . − . ± . . ± . − . ± .
29 60 . ± . − ±
204 213 ±
73 1,1071112C c − . ± .
07 8 . ± . ⋆ [15 − . ± .
86 1,1071117 1.331 − . ± .
15 66 . ± . − ±
317 95 . ± . − . ± .
01 [ − . ± .
45] 226 ±
21 [20 − ±
43 101 . ± . − . ± .
10 33 . ± . − ±
504 96 . ± . − . ± .
25 14 . ± . − ±
34 14 . ± . c − . ± .
08 1 . ± .
13 [15 − . ± .
13 1,1 c − . ± .
63 19 . ± . − ±
17 1,3080603B 2.69 − . ± .
64 15 . ± . − ±
214 116 . ± . − . ± .
07 160 ±
33 [20 − ±
48 308 . ± . − . ± .
06 269 ±
54 [20 − ±
169 2260 ±
446 1,1080613B − − . ± .
18 47 . ± . − ± b − − . ± .
07 [ − . ± .
29] 211 ±
35 [20 − ±
223 1039 ±
173 1,1080804 2.20 − . ± .
04 7 . ± .
88 [8 −
35 000] 810 ±
45 27 . ± . − . ± .
05 4 . ± .
27 [8 −
35 000] 208 ±
11 1 . ± .
06 1,1081007 0.53 − . ± . . ± . ⋆ [25 − ±
15 0 . ± .
09 1,1081121 2.512 − . ± .
08 [ − . ± .
07] 51 . ± . −
35 000] 608 ±
42 195 . ± . − . ± .
14 2 . ± . ⋆ [15 − ±
756 28 . ± . − . ± .
01 24 . ± . −
35 000] 284 ± ± − . ± .
03 [ − . ± .
10] 17 . ± .
58 [8 −
35 000] 630 ±
31 95 ± − . ± .
01 29 . ± .
91 [8 −
35 000] 1174 ±
38 45 . ± . − . ± .
09 [ − . ± .
52] 73 . ± . − . ± . − . ± .
01 [ − . ± .
18] 91 . ± . −
35 000] 250 . ± . . ± .
18 1,1 − . ± .
08 [ − . ± .
24] 39 ± − . ± . − . ± .
37 9 . ± . − ±
164 82 . ± . − . ± .
07 2 . ± . ⋆ [100 − ±
663 96 . ± . − . ± .
06 4 . ± .
28 [8 −
35 000] 212 . ± . . ± .
25 1,1091018 0.971 − . ± .
48 4 . ± .
95 [20 − ±
26 4 . ± .
33 1,1091020 1.71 − . ± .
06 [ − . ± .
18] 18 . ± . −
35 000] 507 ±
68 32 . ± . − . ± .
05 [ − . ± .
01] 93 . ± . −
35 000] 51 . ± . . ± .
24 1,1091208B 1.063 − . ± .
04 25 . ± .
97 [8 −
35 000] 246 ±
15 17 . ± . − . ± .
07 [ − . ± .
11] 8 . ± . ⋆ [8 − . ± . . ± .
87 1,2100621A 0.542 − . ± .
13 [ − . ± .
15] 17 . ± . − ±
23 3 . ± .
34 1,1100728B 2.106 − . ± .
07 5 . ± .
35 [8 −
35 000] 404 ±
29 18 . ± . − . ± .
14 5 . ± . − ±
238 25 . ± . − . ± .
08 [ − . ± .
3] 100 ±
10 [20 − ±
50 180 . ± . − − . ± .
02 15 . ± . ⋆ [10 − ± b − − < − . + . − . ± − + − − − − . ± .
23 3 . ± . ⋆ [15 − . ± . b − − Article number, page 4 of 9. Pescalli et al.: The rate and luminosity function of long Gamma Ray Bursts
GRB z α [ β ] Peak flux Range E p L Ref10 − erg cm − s − (keV) (keV) ( × erg/s) ⋆ phot cm − s − c − . ± .
09 6 . ± . ⋆ [15 − . ± . c − − . ± .
06] 12 . ± . ⋆ [15 − . ± .
27 10,11120102A − − . ± .
03 22 . ± . ⋆ [10 − ± b − − − − . ± .
41 4 . ± . ⋆ [15 − − − − − . ± .
03 [ − . ± .
09] 16 . ± . ⋆ [10 − . ± . . ± . − . ± .
34 4 . ± . ⋆ [15 − ±
19 10 . ± . − − . + . − . ±
13 [20 −
10 000] 295 + − − − c − . ± .
08 2 . ± . ⋆ [15 − . ± .
11 19,20120802A 3.796 − . ± .
47 3 . ± . ⋆ [15 − . ± . . ± . − . ± . . ± . ⋆ [15 − . ± . . ± . − . ± .
25 4 . ± . ⋆ [10 − . ± . . ± .
27 25,26121123A 2.7 − . ± . . ± . − . ± . . ± . − − . ± .
06 4 . ± . − ± b − − c − . ± .
08 3 . ± . ⋆ [15 − . ± . − . ± .
25 3 . ± . ⋆ [15 − . ± . . ± .
65 31,32130427A 0.339 − . ± .
006 [ − . ± .
16] 6900 ±
100 [20 − . ± . . ± . c − . ± .
15 3 . ± . − ± − − . ± . ± − ± b − − − . ± .
04 [ − . ± .
03] 690 ±
30 [20 − . ± . ±
172 38,39130527A − . ± .
04 500 ±
30 [20 −
10 000] 1380 ± b − − c − . ± .
15 2 . ± . ⋆ [15 − . ± . − − . ± .
22 [ − . ± .
2] 13 . ± . ⋆ [10 − ± b − − − . ± .
21 17 . ± . − . ± . . ± . − . ± .
09 7 . ± . ⋆ [10 − . ± . b − − − . ± .
06 [ − . ± .
3] 25 ± −
10 000] 81 . ± . . ± .
45 47,48130907A 1.238 − . ± .
02 [ − . ± .
07] 220 ±
10 [20 −
10 000] 881 . ± . . ± . − . ± .
12 [ − . ± .
28] 100 ±
10 [20 −
10 000] 405 . ± . ±
11 51,52131105A 1.686 − . ± .
38 [ − . ± .
33] 20 ± −
10 000] 419 ±
102 37 . ± . − − . ± .
02 [ − . ± .
07] 49 . ± . ⋆ [10 − ± b − − − . ± .
15 19 . ± . ⋆ [15 − . ± . . ± . − − . ± .
11 [ − . ± .
35] 35 . ± . −
10 000] 214 ± b − − − . + . − . [ − . + . − . ] 47 + − [20 −
10 000] 1452 . ± . . ± . − . ± . − . ± .
1] 14 . ± . ⋆ [10 − ±
68 11 . ± . − . ± .
03 11 . ± . ⋆ [10 − ±
145 5 . ± .
47 63,64140619A − − . ± .
14 4 . ± . ⋆ [15 − . ± . b − − − − . ± .
09 2 . ± . ⋆ [15 − − − − − . ± .
54 4 . ± . −
10 000] 281 . ± . . ± . − . ± .
06 4 . ± . ⋆ [10 − . ± . . ± . Fig. 1.
Left panel:
GRB formation rate ψ ( z ) obtained with the C − method using the BAT6ext sample (black solid line). The dashed red line andthe dot-dashed cyan line are the SFR models of Hopkins & Beacom (2006) and Cole et al. (2001) shown here for reference. All the curves arenormalised to their maxima. Right panel : luminosity function φ ( L ) obtained with the C − method using the BAT6ext sample (black solid line).The best fit model describing this function is a broken power-law (dashed green line) with ( a = − . ± . b = − . ± . L b = . ± . erg/s). The orange dot-dashed line is the luminosity function obtained by S12 in the case of pure luminosity evolution. derive the cumulative luminosity function Φ ( L ) and the GRBformation rate ψ ( z ).For the ith GRB in the BAT6ext sample, de-scribed by its ( L , i , z i ), we consider the subsample J i = n j | L , j > L , i ∧ z j < z max , i o and call N i the numberof GRBs it contains. Similarly, we define the subsample J ′ i = n j | L , j > L lim , i ∧ z j < z i o and we call M i the number of GRBs it contains. L lim , i is the minimum luminosity correspond-ing to the flux limit S of the sample at the redshift z i . z max , i is themaximum redshift at which the i- th GRBs with luminosity L , i can be observed (i.e. with flux above the limit S ). L lim and z max are computed applying the K-correction because the limitingflux is computed in the observer frame Swift /BAT [15 − Article number, page 5 of 9&Aproofs: manuscript no. PaperBAT6AA
Through M i and N i we can compute the cumulative luminos-ity function Φ ( L ) and the cumulative GRB redshift distribution ζ ( z ): Φ ( L , i ) = Y j < i (1 + N j ) (2)and ζ ( z i ) = Y j < i (1 + M j ) (3)From the latter we can derive the GRB formation rate as: ψ ( z ) = d ζ ( z ) dz (1 + z ) " dV ( z ) dz − (4)where dV ( z ) / dz is the differential comoving volume. Thedifferential luminosity function φ ( L ) is obtained deriving thecumulative one Φ ( L ).The functions φ ( L ) and ψ ( z ) are shown in Fig. 1. Errors on φ ( L ) are computed propagating the errors on the cumulative oneassuming Poisson statistics. The errors on the ψ ( z ) are computedfrom the number n of GRBs within the redshift bin. We assumethat the relative error ǫ = / √ n is the same affecting ψ ( z ).
5. Results
The luminosity function φ ( L ) obtained with the BAT6ext sam-ple is shown in Fig. 1 (right panel) by the black symbols.Data are normalised to the maximum. The best fit model (greendashed line in Fig. 1 - right panel) is represented by a brokenpower law function with a = − . ± . b = − . ± . L b = . ± . erg/s (where a and b represent the slopes ofthe power law below and above L b - χ / d . o . f . = . a = − . + . − . , b = − . + . − . , L b = . + . − . × erg/s). This model is consistent with the re-sult obtained in our analysis.The GRB formation rate ψ ( z ) obtained with the BAT6ext isshown by the black symbols in Fig. 1 (left panel). The greendashed and the cyan dot-dashed lines are, respectively, the SFRof Hopkins & Beacom (2006) and Cole (2001). Data and modelsare normalised to their peak. Contrary to the results reported inP15 and Y15, the ψ ( z ) that we derive increases up to z ∼ ∼ ∼ ψ ( z ). For thisreason we also computed the ψ ( z ) using only the 56 objects ofthe revised BAT6 sample which turns out to be slightly steeperboth at low and high redshifts than the one obtained with theBAT6ext but totally consistent within the errors. We concludethat the lower completeness in redshift of the BAT6ext does notintroduce any strong bias in the ψ ( z ) and φ ( L ) obtained.
6. Monte Carlo test of the C − method We now test the C − method used to derive ψ ( z ) and φ ( L ).Through a Monte Carlo simulation (similar to e.g. Ghirlanda etal. 2015) we explore how well the method adopted in §4 canrecover the input assumptions, i.e. ψ ( z ) and LF φ ( L ). In partic-ular, we will show that if the sample used is highly incompletethe resulting GRBFR and LF can differ significantly from the in-put ones. In particular, incomplete samples (either in flux and/orredshift) may produce a misleading excess of low redshift GRBswith respect to the assumed ψ ( z ).We simulate GRBs distributed in redshift according to theGRB formation rate ψ ( z ) of Li 2008 (see also Hopkins & Bea-com 2008): ψ ( z ) = . + . z + ( z / . . (5)where ψ ( z ), in units of M ⊙ yr − Mpc − , represents the formationrate of GRBs and we assume it can extend to z ≤
10. We stressthat for the scope of the present test any other functional form of ψ ( z ) could be assumed.We adopt a luminosity function φ ( L ) as obtained by Sal-vaterra et al. 2012 from a complete sample of Swift GRBs: φ ( L ) ∝ (cid:16) LL b (cid:17) a , L ≤ L b (cid:16) LL b (cid:17) b , L > L b (6)composed by two power-laws with a break at L b . We adopted ar-bitrary parameter values: a = − . b = − .
92 and L b = × erg s − . We further assume an evolution of the luminosity propor-tional to (1 + z ) k with k = .
2. We assume this is the bolometricluminosity of the simulated bursts and compute the correspond-ing bolometric flux. Also for φ ( L ) we use this functional formbut any other function could be assumed for the scope of thepresent test.With these two assumptions we simulate a sample of GRBswith a flux limit F lim = × − erg cm − s − and we analyse itwith the C − method. Accounting for the truncation of the samplewe recover through the statistical method of Efron & Petrosian(1992) the luminosity evolution in the form (1 + z ) k , with k ∼ . L = L / (1 + z ) . and derive the GRB formation rate ψ ( z ) and theluminosity function φ ( L ) through the C − method proposed byLynden-Bell et al. (1971). The left panel of Fig. 2 shows (blacksymbols) that we recover the GRB formation rate of Eq. 5(dashed green line) that we adopted in the simulation. Similarlythe right panel of Fig. 2 shows that we also recover the lumi-nosity function that we adopted in the simulated sample (Eq. 6 -shown by the dashed green line in Fig. 2).We then tested what happens if we apply the same methodto an incomplete sample. Firstly we applied the C − method tothe same simulated sample (which is built to be complete to F lim = × − erg cm − s − ) from which we removed ran-domly a fraction of the bursts close to F lim . This new sample isclearly incomplete to F lim . The results are shown in Fig. 2 by theblue symbols. We find that the GRB formation rate ψ ( z ) is flatat low redshifts (i.e. below z =
2) showing a clear excess withrespect to the assumed function (cf blue symbols with the greendashed line in the left panel of Fig. 2). The luminosity functionis flatter than the assumed one (cf blue symbols with the greendashed line in Fig. 2 right panel).Similar results are obtained by assuming for the derivationof ψ ( z ) and φ ( L ) a flux limit which is a factor of 5 smaller than Article number, page 6 of 9. Pescalli et al.: The rate and luminosity function of long Gamma Ray Bursts
Fig. 2.
Left:
GRB formation rate (normalised to its peak) for the simulated population of GRBs with flux limit 5 × − erg cm − s − (blacksymbols). The GRB formation rate assumed in the simulation is shown by the dashed green line. The red symbols show the results obtained fromthe same sample using, for the analysis, a flux limit a factor of 5 smaller than the real one. Blue symbols are obtained by mimicking the sampleincompleteness by removing randomly some GRBs near the flux threshold adopted for the sample selection. Right panel: cumulative luminosityfunction, normalised to the first bin. The black, red and blue symbols are the same as for the left panel. The assumed luminosity function is shownby the dashed green line. that used to construct the simulated sample. This is another wayto make the sample artificially incomplete. The results are shownby the red symbols in the panels of Fig. 2. Note that in this sec-ond test the sample used is the same but it is analysed throughthe C − method assuming it is complete with respect to a fluxlimit which is smaller (a factor of 5) than the one correspondingto its real completeness (i.e. 5 × − erg cm − s − ).These simulations show that if the samples adopted arehighly incomplete in flux, one obtains an excess at the low red-shift end of the GRB formation rate and a flatter luminosity func-tion.
7. Summary and discussion
The aim of this work is to derive the luminosity function of longGRBs and their formation rate. To this aim we apply a directmethod (Lynden–Bell et al. 1971) and its specific version alreadyapplied to GRBs, e.g. Yonetoku et al. (2004, 2014), Kocevski &Liang (2006), Wu et al. (2014), P15, Y15. This is the first timethis method is applied to a well defined sample of GRBs whichis complete in flux and 82% complete in redshift.We build our sample of long GRBs starting from the BAT6complete sample (Salvaterra et al. 2012): this was composed by58 GRBs detected by the
Swift satellite satisfying the multipleobservational selection criteria of Jackobsson (2006) and havinga peak photon flux P ≥ . − s − . Here, we update theredshift measurement of 8 GRBs of the BAT6 (marked in ital-ics in Tab. 1) and accordingly revise their luminosities. Then,we update this sample to GRB 140703A ending with 99 objects.We collect their redshift measurements and spectral parametersfrom the literature (see Tab. 1). The BAT6ext sample has a red-shift completeness of ∼
82% (82/99 burst with z measured) andcounts 81/99 bursts with well determined L .We analyzed the BAT6ext sample searching for a possibleluminosity evolution induced by the flux threshold through themethod proposed by Efron & Petrosian (1992). We find that the L − z correlation introduced by the truncation due to the fluxlimit can be described as L = L (1 + z ) k with k = .
5. This resultis in agreement with what found by other authors (Yonetoku et al. 2004, Wu et al. 2012, P15, Y15). With the BAT6ext sampleafter de-evolving the luminosities for their redshift dependencewe find that: – the luminosity function φ ( L ) is a monotonic decreasingfunction well described by a broken powerlaw with slopes a = − . ± .
21 and b = − . ± .
24 below and above, re-spectively, a characteristic break luminosity L b = . ± . erg/s (green dashed line in the right panel of Fig. 1). This re-sult (shape, slopes and characteristic break) is consistent withthe luminosity function found by S12 (orange dot-dashedline in Fig. 1 - right panel). – The cosmological GRB formation rate ψ ( z ) (black solid linein the left panel of Fig. 1) increases from low redshifts tohigher values peaking at z ∼ ψ ( z ) is in contrast with theGRBFR recently found by P15 and Y15, who, applying thesame method to differently selected GRB samples, report theexistence of an excess of low redshift GRBs.However, the result that GRBs evolve in luminosity is not theprobe that GRBs had experienced a pure luminosity evolution. Infact the C − method assumes the independence between L and z and the non parametric method of Efron & Petrosian used to getthe de-evolved luminosities assigns all the evolution in the lu-minosity. For this reason we are not able to distinguish betweenan evolution in luminosity or in density (see also Salvaterra etal. 2012). The possible density evolution case requires the in-vestigation of the applicability of similar methods to the GRBsamples and will be the subject of a forthcoming work (Pescalliet al. , in preparation).Finally, intrigued by the different results with respect to Y15and P15, we performed Monte Carlo simulations in order to testthe robustness of the C − method. We showed that this methodcan correctly recover the LF φ ( L ) and the GRBFR ψ ( z ) assumedin the simulation only if the sample of GRBs it is applied to iscomplete in flux and has a high level of completeness in red-shift. Using incomplete samples or a sample incomplete in red- Article number, page 7 of 9&Aproofs: manuscript no. PaperBAT6AA shift, the resulting GRBFR and LF can be different from the as-sumed ones. Indeed, this could account for the excess of the rateof GRBs at low redshift as recently reported.
Acknowledgements.
We acknowledge the 2011 PRIN-INAF grant for financialsupport. We acknowledge the financial support of the UnivEarthS Labex programat Sorbonne Paris Cité (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02). GGthanks the Observatoire de Paris (Meudon) for support and hospitality during thecompletion of this work. We thank S. Campana and G. Tagliaferri for usefuldiscussion.
References [] Aptekar, R. L., et al. 1995, Space Sci. Rev., 71, 265[] Avni, Y., Bahcall, J. N., 1980, ApJ, 235, 694-716[] Barthelmy S. D., Sakamoto, T., Markwardt, C. B. et al. 2012, GCN, 13120[] Barthelmy S. D., Baumgartner, W. H., Cummings, J. R. et al. 2012, GCN, 13990[] Barthelmy S. D., Baumgartner, W. H., Cummings, J. R. et al. 2013, GCN, 14469[] Baumgartner, W. H., Barthelmy, S. D., Cummings, J. R. et al. 2011, GCN, 12424[] Baumgartner, W. H., Barthelmy, S. D., Cummings, J. R. et al. 2014, GCN, 16473[] Berger, E., 2014, ARA&A, 52, 43-105[] Bouwens R. J., Illingworth, G. D., Franx, M. et al. 2009, ApJ, 705, 936-961[] Campana, S., Salvaterra, R., Melandri, A. et al. 2012, MNRAS, 421, 1697-1702[] Castro–Tirado, A. J., Sánchez–Ramírez, R., Gorosabel, J. et al. 2013, GCN,14796[] Castro–Tirado, Cunniffe, R., A. J., Sánchez–Ramírez, R. et al. 2014, GCN,16505[] Chiang, J., Mukherjee, R., 1998, ApJ, 496, 752C[] Cole, S., et al. 2001, MNRAS, 326, 255[] Costa, E., Frontera, F., Heise, J. et al. 1997, Nature, 387, 783-785[] Connaughton, V., 2011, GCN, 12133[] Covino, S., Melandri, A., Salvatera, R. et al. 2013, MNRAS, 432, 1231-1244[] Cucchiara, A., Levan, A J., Fox, D. B. et al. 2011, ApJ, 736, 7[] Cucchiara, A., Levan, A. J., Cenko, S. B., 2011, GCN, 12761[] Cucchiara, A., Prochaska, J. X. et al. 2012, GCN, 12865[] Cucchiara, A., Perley, D. et al. 2013, GCN, 15144[] Cummings, J. R., Barthelmy, S. D., Baumgartner, W. H. et al. 2011, GCN, 12749[] Daigne, F., Rossi, E. M., Mochkovitch, R., 2006, MNRAS, 372, 1034-1042[] D’Avanzo, P., Salvaterra, R., Sbarufatti, B. et al. 2012, MNRAS, 425, 506-513[] D’Avanzo, P., Salvaterra, R., Bernardini, M. G. et al. 2014, MNRAS, 442, 2342-2356[] de Ugarte Postigo, A., Tanvir, N. R., Sánchez-Ramírez, R. et al. 2013, GCN,14437[] de Ugarte Postigo, A., Xu, D., Malesani, D. et al. 2013, GCN, 15187[] de Ugarte Postigo, A., Gorosabel, J., Xu, D. et al. 2014, GCN, 16310[] Efron, B., & Petrosian, V., 1992, ApJ, 399, 345[] Firmani, C., Avila–Reese, V., Ghisellini, G., Tutukov, A. V., 2004, ApJ, 611,1033-1040[] Flores, H., Covino, S., Xu, D. et al. 2013, GCN, 14491[] Flores, H., Covino, S., de Ugarte Postigo, A. et al. 2013, GCN, 14493[] Frederiks, D., 2011, GCN, 12137[] Fynbo, J. P. U., Tanvir, N. R., Jakobsson, P. et al. 2014, GCN, 16217[] Gehrels, N., Chincarini, G., Giommi, P. et al. 2004, ApJ, 611, 1005-1020[] Ghirlanda, G., Ghisellini G., Nava, L. et al. 2012, MNRAS, 442, 2553-2559[] Ghirlanda, G., Salvaterra, R., Ghisellini, G. et al. 2015, MNRAS, 448, 2514[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2011, GCN, 12135[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2012, GCN, 13412[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2013, GCN, 14487[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2013, GCN, 14575[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2013, GCN, 14720[] Golenetskii, S., Aptekar, R., Mazets, E. et al. 2013, GCN, 14808[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2013, GCN, 15145[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2013, GCN, 15203[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2013, GCN, 15413[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2013, GCN, 15452[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2014, GCN, 15853[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2014, GCN, 16134[] Golenetskii, S., Aptekar, R., Frederiks, D. et al. 2014, GCN, 16495[] Gruber, D. et al. 2012, GCN, 12874[] Guetta, D., & Piran, T., 2005, A&A, 435, 421-426[] Guetta, D., & Piran, T., 2006, A&A, 453, 823-828[] Guetta, D., & Della Valle, M., 2007, ApJ, 657, L73-L76[] Hjorth, J., Malesani, D., Jakobsson, P. et al. 2012, ApJ, 756, 187[] Hopkins, A. M. & Beacom, J. F., 2006, ApJ, 651, 142[] Jakobsson, P., Levan, A., Fynbo, J. P. U. et al. 2006, A&A, 447, 897[] Jenke, P. et al. 2014, GCN, 16220[] Jenke, P. et al. 2014, GCN, 16512[] Kocevski, D., & Liang, E., 2006, ApJ, 642, 371 [] Krimm, H. A., Barlow, B. N., Barthelmy, S. D. et al. 2012, GCN, 13634[] Kruhler, T., Malesani, D., Fynbo, J. P. U. et al. 2015, arXiv:1505.06743[] Li, L.-X., 2008, MNRAS, 388, 1487-1500[] Liang, E., Zhang, B., Virgili, F., Dai, Z. G., 2007, ApJ, 662, 1111-1118[] Lien, A. Y., Barthelmy, S. D., Baumgartner, W. H. et al. 2013, GCN, 14419[] Lloyd, N. M., & Petrosian, V., 1999, ApJ, 511, 550[] Lynden-Bell, D., 1971, MNRAS, 155, 95[] Malesani, D., Xu. D., Fynbo, J. P. U. et al. 2014, GCN, 15800[] Maloney, A. & Petrosian, V., 1999, ApJ, 518, 32-43[] McGlynn, S. et al. 2012, GCN, 13997[] Meegan, C., Lichti, G., Bhat, P. N. et al. 2009, ApJ, 702, 791-804[] Melandri, A., Sbarufatti, B., D’Avanzo, P. et al. 2012, MNRAS, 421, 1265-1272[] Melandri, A., Covino, S., Rogantini, D. et al. 2014, A&A, 565, A72[] Moskvitin, A., Burenin, R., Uklein, R. et al. 2014, GCN, 16489[] Nakar, E., Gal–Yam, A., Fox, D. B., 2006, ApJ, 650, 281-290[] Nava, L., Salvaterra, R., Ghirlanda, G. et al. 2012, MNRAS, 421, 1256-1264[] Palmer, D. M., Barthelmy, S. D., Baumgartner, W. H. et al. 2012, GCN, 12839[] Palmer, D. M., Barthelmy, S. D., Baumgartner, W. H. et al. 2012, GCN, 13536[] Palmer, D. M., Barthelmy, S. D., Baumgartner, W. H. et al. 2014, GCN, 16423[] Pelassa, V. et al. 2013, GCN, 14869[] Perley, D. A., Perley R. A., Hjorth, J. et al. 2015a, ApJ, 801, 102[] Perley, D. A., Tanvir, N. R., Hjorth, J. et al. 2015b, arXiv:1504.02479[] Perley, D. A., Kruhler, T., Schulze, S. et al. 2015c, arXiv:1504.02482[] Pescalli, A., Ghirlanda, G., Salafia, O. S., Ghisellini, G., Nappo, F., Salvaterra,R., 2015, MNRAS, 447, 1911-1921[] Petrosian, V., 1973, ApJ, 183, 359P[] Petrosian, V., Kitanidis, E., Kocevski, D., 2015, ApJ, 806, 44[] Sakamoto, T., Sato, G., Barbier, L. et al. 2009, ApJ, 693, 922-935[] Sakamoto, T., Barthelmy S. D., Baumgartner, W. H. et al. 2013, GCN, 14959[] Sakamoto, T., Barthelmy S. D., Baumgartner, W. H. et al. 2014, GCN, 15805[] Salvaterra, R. & Chincarini, G., 2007, ApJ, 656, L49-L52[] Salvaterra, R., Della Valle, M., Campana, S. et al. 2009a, Nature, 461, 1258-1260[] Salvaterra, R., Guidorzi, C., Campana, S., Chincarini, G., Tagliaferri, G., 2009b,MNRAS, 396, 299-303[] Salvaterra, R., Campana, S., Vergani, S. D. et al. 2012, ApJ, 749, 68[] Sánchez-Ramírez, R., Gorosabel, J., de Ugarte Postigo, A. et al. 2012, GCN,13723[] Schmidl, S., Nicuesa Guelbenzu, A., Klose, S. et al. 2012, GCN, 13992[] Schmidt, M., 1968, ApJ, 151, 393[] Singal, J., Petrosian, V. & Ajello, M. 2012, ApJ, 753, 45[] Singal, J., Petrosian, V. et al. 2013, ApJ, 764, 43[] Stanbro, M. et al. 2014, GCN, 16262[] Stamatikos, M. Barthelmy, S. D., Baumgartner, W. H. et al. 2012, GCN, 13559[] Tanvir, N. R., Fox, D. B., Levan, A. J. et al. 2009, Nature, 461, 1254-1257[] Tanvir, N. R., Ball, J. et al. 2012, GCN, 13532[] Tanvir, N. R., Fox, D. B., Fynbo, J. et al. 2012, GCN, 13562[] Tanvir, N. R., Levan, A. J., Matulonis, T. et al. 2013, GCN, 14567[] Tanvir, N. R., Levan, A. J., Cucchiara, A. et al. 2014, GCN, 16125[] Tello, J. C., Sánchez–Ramírez, R., Gorosabel, J. et al. 2012, GCN, 13118[] Thoene, C. C., de Ugarte Postigo, A., Gorosabel, J. et al. 2012, GCN, 13628[] Ukwatta, T. N., Barthelmy, S. D., Baumgartner, W. H. et al. 2011, GCN, 12352[] Ukwatta, T. N., Barthelmy, S. D., Baumgartner, W. H. et al. 2012, GCN, 14052[] van Paradijs, J., Groot, P. J., Galama, T. et al. 1997, Nature, 386, 6626, 686-689[] Vergani, S. D., Salvaterra, R., Japelj, J. et al. 2014, submitted to A&A,arXiv:1409.7064v1[] Wanderman, D., & Piran, T., 2010, MNRAS, 406, 1944-1958[] Wiersema, K., Flores, H., D’Elia, V. et al. 2011, GCN, 12431[] Wolter, A., Caccianiga, A., della Ceca, R., Maccacaro, T., 1994, ApJ, 433, 29-38[] Wu, S. W., Xu, D., Zhang, F. W., Wei, D. M., 2012, MNRAS, 423, 2627[] Xiong, S. et al. 2012, GCN, 12801[] Xu, D., de Ugarte Postigo, A., Malesani, D. et al. 2013, GCN, 14956[] Xu, D., Fynbo, J. P. U., Jakobsson, P. et al. 2013, GCN, 15407[] Xu, D., Malesani, D., Tanvir, N. R. et al. 2013, GCN, 15450[] Yamaoka, K. et al. 2009, PASJ, 61, S35[] Yonetoku, D., Murakami, T., Nakamura, T., Yamazaki, R., Inoue, A. K., Ioka,K., 2004, ApJ, 609, 935-951[] Yonetoku, D., Nakamura, T., Sawano, T., Takahashi, K., Toyanago, A., 2014,ApJ, 789, 65[] Younes, G. et al. 2012, GCN, 13721[] Younes, G. et al. 2013, GCN, 14545[] Yu, H., Wang, F. Y., Dai, Z. G., Cheng, K. S., 2015, ApJSS, 218, 13[] Yu, H.–F. et al. 2013, GCN, 15064[] Zaureder, A., Berger, E., 2011, GCN, 12190[] Zhang, B.–B., Bhat, N. et al. 2014, GCN, 15669
Article number, page 8 of 9. Pescalli et al.: The rate and luminosity function of long Gamma Ray Bursts
Fig. A.1.
The blue filled points represent the observed CLF Σ ( L ) whilethe orange squares represent the true CLF ˜ Σ ( L ). The solid blue line ( a = − . ± . b = − . ± . L b = . ± . erg/s) and the dashedorange line ( a = − . ± . b = − . ± . L b = . ± erg/s) arethe best fit models of the observed and true CLF respectively. Appendix A: The redshift integrated luminosityfunction
In this section we show how we derive the luminosity function ofGRBs integrated over all the redshift space using the BAT6extsample. This is not the typical luminosity function which is de-rived through indirect methods, it is free from any functionalform and only uses the 1 / V max concept, i.e. the maximum co-moving volume within which a real burst with an observed lumi-nosity could be detected by a given instrument. This is a gener-alisation of the < V / V max > method proposed by Schmidt (1968)and applied to quasars (see Avni & Bachall 1980 for an exhaus-tive description).However, the luminosity function obtained with this methodis not the canonical φ ( L ) but it is the result of the integration ofthe latter convolved with the GRB formation rate ψ ( z ) over z . Wecall this function the convolved luminosity function Σ ( L ) (CLF).In principle the luminosity function could evolve with redshift( φ ( L , z )). For this reason the shape of the CLF could be differentfrom the shape of the canonical φ ( L ). It would be simply pro-portional to φ ( L ) only if the luminosity function does not evolve,either in luminosity or in density. Indeed, it is possible to express Σ ( L ) in terms of luminosity function and GRB formation rate: Σ ( L ) = Z ∞ φ ( L , z ) ψ ( z ) dz (A.1)The advantage is that it can be obtained directly from thedata and it is extremely robust if derived through a flux limitedsample. Any evolution with redshift of the luminosity or densityis in this CFL. We use the 81/99 GRB with both z and determined L reported in Tab.1.For each GRB of the BAT6ext sample with an associated L we estimate the maximum volume V max within which the burstcould still be detected because its flux would be larger than thechosen threshold i.e. P lim = . − s − in the 15–150 keVenergy band. The observed photon flux in the Swift /BAT energyband as a function of the varying redshift is: P ( z ) = L π d ( z ) R keV keV N ( E ) dE R / (1 + z ) keV / (1 + z ) keV EN ( E ) dE (A.2) where N ( E ) is the observed photon spectrum of each GRB and d L ( z ) is the luminosity distance at redshift z . The extremes ofthe integral in the denominator correspond to the same valuesadopted to compute L . The maximum redshift z max correspondsto the redshift satisfying P ( z max ) = P lim .Considering the typical Swift /BAT field of view
Ω = . Swift T ∼ ρ i = π/ Ω T V max , i foreach GRB. We divided the observed range of luminosities in binsof equal logarithmic width ∆ and estimate Σ j ( L j ) for each bin as Σ j ( L j ) = ∆ L j X ρ i (A.3)where the sum is made over the bursts with luminosities L j − ∆ / ≤ L i ≤ L j +∆ /
2. The discrete convolved luminosity functionis shown in Fig. A.1. The normalisation is obtained also consid-ering that the bursts in the BAT6ext sample represent approxi-mately 1/3 of the total number of
Swift detected GRBs with peakflux P ≥ − s − . We verified that the BAT6ext sample (99objects) is representative in terms of peak flux distribution of thelarger population. The error bar associated to the discrete CLFare mainly related to the Poissonian error on the objects count-ing within the luminosity bin (see also e.g. Wolter et al. 1994).When computing individual rates, we have used the missionelapsed time T . However, this is the observer frame time and therate should be corrected for the cosmological time dilation. Thismeans that, at higher z , the same subset of sources should occurwith a larger frequency. Therefore, we average out the elapsedtime on the redshift interval [0 , z max ] < T > = R z max T + z dz R z max dz (A.4)and use it in the computation of Eq. A.3. As shown in Fig. A.1,the true CLF ˜ Σ ( L ) is flatter than the observed one because, ingeneral, the true elapsed time is smaller than the observed one.Moreover, the correction is more pronounced for high luminosityGRBs observable up to high redshifts.The CLF obtained with the extended BAT6ext sample isshown in Fig. A.1 by the filled symbols. The observed CFL (bluesymbols in Fig. A.1) can be adequately represented by a bro-ken power law function (solid blue line in Fig. A.1) with slopes − . ± .
14 and − . ± .
41 below and above the break lumi-nosity L b = . ± . erg/s ( χ / d . o . f . = . − . ± . − . ± .
95 below and above thebreak L b = . ± erg/s - χ / d . o . f . = . N − log S and to the redshift distribution it can beused as a constrain. The LF and the GRBFR obtained with othermethods should reproduce this CLF once convolved together andintegrated over z ..