The luminosity function and the rate of Swift's Gamma Ray Bursts
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 5 April 2018 (MN L A TEX style file v2.2)
The luminosity function and the rate of
Swift ’s Gamma RayBursts
David Wanderman and Tsvi Piran Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel. email:[email protected] email:[email protected] ABSTRACT
We invert directly the redshift - luminosity distribution of observed long
Swift
GRBsto obtain their rate and luminosity function. Our best fit rate is described by a brokenpower law that rises like (1+ z ) . +0 . − . for < z < and decrease like (1+ z ) − . +2 . − . for z > . The local rate is ρ (cid:39) . +0 . − . [ Gpc − yr − ] . The luminosity function iswell described by a broken power law with a break at L ∗ (cid:39) . ± . [ erg/sec ] andwith indices α = 0 . +0 . − . and β = 1 . +0 . − . . The recently detected GRB 090423, withredshift ≈ , fits nicely into the model’s prediction, verifying that we are allowed toextend our results to high redshifts. While there is a possible agreement with the starformation rate (SFR) for z < , the high redshift slope is shallower than the steepdecline in the SFR for < z . However we cannot rule out a GRB rate that followsone of the recent SFR models. Gamma-ray bursts (GRBs) are short and intense pulses of soft γ -rays. In this work we study theirluminosity function and their cosmic rate. These functions are essential to understand the nature ofGRBs and to determine their progenitors. They may also shed light on the still mysterious physicsof the central engine.The luminosity function and the cosmic GRB rate are observationally entangled as the ob-served rate is a convolution of the luminosity function with the cosmic rate. For this reason, almostevery work before this paper have made an a-priori assumptions on at least one of these functions.At first - having no motivation to believe otherwise - the rate was assumed to be constant. Thesimplest form for the luminosity function was a standard candle i.e. a constant luminosity. These c (cid:13) a r X i v : . [ a s t r o - ph . H E ] M a r D. Wanderman & T. Piran early studies used the measured (cid:104)
V /V max (cid:105) value to find the typical luminosity (Mao and Paczyn-ski 1992; Piran 1992; Fenimore et al. 1993; Ulmer and Wijers 1995; Ulmer et al. 1995) and corre-spondingly a maximal distance from which GRBs were observed. Later, using the flux distribution( logN − logP relation), Cohen and Piran (1995) and Loredo and Wasserman (1998) showed howrelaxing the standard candle assumption allows a corresponding relaxation of the constant rateassumption.Paczy´nski (1998) noticed that the hosts of the bursts are in star forming regions and suggestedthat GRBs follow the SFR (see also Totani 1997; Wijers et al. 1998a). The detection of a supernovaassociated with GRB980425 (Galama et al. 1998), strengthened the expectation that the GRBrate should follow the SFR. Using this proportionality, numerous studies examined the typicalluminosity assuming at first standard candles (e.g. Wijers et al. 1998a; Totani 1999) and latermore elaborate shapes of the luminosity function (e.g. Schmidt 1999, 2001b,a; Guetta et al. 2005;Firmani et al. 2004; Guetta and Piran 2005). Swift that discovers routinely GRBs afterglows detected GRBs from higher redshifts than waspreviously possible sparked renewed interest in the GRB redshift distribution (e.g. Berger et al.2005; Natarajan et al. 2005; Bromm and Loeb 2006; Jakobsson et al. 2006; Le and Dermer 2007;Y¨uksel and Kistler 2007; Salvaterra and Chincarini 2007; Liang et al. 2007; Chary et al. 2007).More and more signs appeared suggesting that the rate of GRBs does not simply follow the globalstar formation rate, as was believed earlier. Firmani et al. (2005); Le Floc’h et al. (2006); Daigneet al. (2006); Le and Dermer (2007); Guetta and Piran (2007) conclude that the GRB rate differsfrom the SFR. In particular we observe more high redshift bursts than what is expected for a ratethat follows the SFR. Y ¨uksel et al. (2008) uses the high luminosity subsample of bursts to obtainthe GRB rate without assuming any luminosity function. They find that the GRB rate at highredshifts ( z > ) is higher on than Hopkins and Beacom (2006) SFR. Alternatively luminosityevolution, has been suggested by some authors: (e.g. Lloyd-Ronning et al. 2002; Firmani et al.2004; Matsubayashi et al. 2006; Kocevski and Liang 2006). Salvaterra et al. (2008) used a sampleof long bursts with redshift and found evidence for luminosity evolution. Other papers, includingthis one, find self consistency without a luminosity evolution.We introduce here a new method for determining the rate and the luminosity, by inverting theobservations without making any assumptions on the functional form of the luminosity and ratefunctions. This direct inversion allows us to use most of the available redshift data and to obtainrobust estimation of both functions.In § c (cid:13)000
V /V max (cid:105) value to find the typical luminosity (Mao and Paczyn-ski 1992; Piran 1992; Fenimore et al. 1993; Ulmer and Wijers 1995; Ulmer et al. 1995) and corre-spondingly a maximal distance from which GRBs were observed. Later, using the flux distribution( logN − logP relation), Cohen and Piran (1995) and Loredo and Wasserman (1998) showed howrelaxing the standard candle assumption allows a corresponding relaxation of the constant rateassumption.Paczy´nski (1998) noticed that the hosts of the bursts are in star forming regions and suggestedthat GRBs follow the SFR (see also Totani 1997; Wijers et al. 1998a). The detection of a supernovaassociated with GRB980425 (Galama et al. 1998), strengthened the expectation that the GRBrate should follow the SFR. Using this proportionality, numerous studies examined the typicalluminosity assuming at first standard candles (e.g. Wijers et al. 1998a; Totani 1999) and latermore elaborate shapes of the luminosity function (e.g. Schmidt 1999, 2001b,a; Guetta et al. 2005;Firmani et al. 2004; Guetta and Piran 2005). Swift that discovers routinely GRBs afterglows detected GRBs from higher redshifts than waspreviously possible sparked renewed interest in the GRB redshift distribution (e.g. Berger et al.2005; Natarajan et al. 2005; Bromm and Loeb 2006; Jakobsson et al. 2006; Le and Dermer 2007;Y¨uksel and Kistler 2007; Salvaterra and Chincarini 2007; Liang et al. 2007; Chary et al. 2007).More and more signs appeared suggesting that the rate of GRBs does not simply follow the globalstar formation rate, as was believed earlier. Firmani et al. (2005); Le Floc’h et al. (2006); Daigneet al. (2006); Le and Dermer (2007); Guetta and Piran (2007) conclude that the GRB rate differsfrom the SFR. In particular we observe more high redshift bursts than what is expected for a ratethat follows the SFR. Y ¨uksel et al. (2008) uses the high luminosity subsample of bursts to obtainthe GRB rate without assuming any luminosity function. They find that the GRB rate at highredshifts ( z > ) is higher on than Hopkins and Beacom (2006) SFR. Alternatively luminosityevolution, has been suggested by some authors: (e.g. Lloyd-Ronning et al. 2002; Firmani et al.2004; Matsubayashi et al. 2006; Kocevski and Liang 2006). Salvaterra et al. (2008) used a sampleof long bursts with redshift and found evidence for luminosity evolution. Other papers, includingthis one, find self consistency without a luminosity evolution.We introduce here a new method for determining the rate and the luminosity, by inverting theobservations without making any assumptions on the functional form of the luminosity and ratefunctions. This direct inversion allows us to use most of the available redshift data and to obtainrobust estimation of both functions.In § c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts § § § § We consider long bursts ( t (cid:62) sec ) detected by Swift from the beginning of its operation untilburst 090726 , with a measured peak-flux and a measured redshift. Generally GRBs redshifts areobtained from the optical afterglow spectrum using absorption lines or photometry, or from thespectrum of the host galaxy using emission lines. However in the global sample we find differ-ent redshift distributions for the different detection methods, namely: Absorption, Emission andPhotometry (see Figure 1). For redshifts determined using the hosts’ emission lines, we do notdetect high-redshift events, whereas absorption lines redshifts extend over the entire range of red-shifts. Furthermore, emission lines are more susceptible to a selection effect known as the ’redshiftdesert’ in the range . < z < . , (Fiore et al. 2007, see also Coward 2008). This is in line withthe fact that we have also found that the probability to measure the redshift using emission linesstrongly depends on the gamma ray flux, favoring high fluxes . This effect is mild for absorptionlines and photometry (see appendix B).To obtain a more uniform sample we consider therefore only bursts whose redshift was mea-sured using the afterglow. For each burst we calculate the isotropic equivalent peak-luminosity: L iso (see Appendix A. for details) using the peak-flux and redshift. We use standard Λ CDM cos-mology with h = 0 . , Ω m = 0 . , Ω Λ = 0 . . We assume that the luminosity function is redshift independent. In this case the number of burstsat a given redshift and with a given luminosity is the product of the luminosity function, φ ( L ) , thatdepends only on the luminosity L and the GRB rate, R GRB ( z ) , that depends only on the redshift z . This common assumption is reasonable since a-priori there is no competing reason why the All data were taken from the
Swift information page http : //swift.gsfc.nasa.gov/docs/swift/archive/grb table/ Although a redshift determination through absorption lines is also difficult in the range . < z < . A possible explanation is that a brighter burst can be more easily localized, making follow-up possible.c (cid:13) , 000–000
D. Wanderman & T. Piran
Figure 1.
The redshift distribution for the different methods: Absorption lines, Emission lines and Photometry luminosity function should depend on the redshift. We test later the validity of this assumption andshow that it is accepted with a high statistical significance.The isotropic peak luminosity function, φ ( L ) , is defined traditionally as the fraction of GRBswith isotropic equivalent luminosities in the interval log L and log L + d log L . The rate, R GRB ( z ) ,is defined as the co-moving space density of GRBs in the interval z to z + dz . The distributiondensity: n ( L, z ) is given by: n ( L, z ) dlogLdz = φ ( L ) · R ( z ) dlogLdz , (1)where R ( z ) = R GRB ( z )(1 + z ) dV ( z ) dz , (2)is the differential co-moving rate of bursts at a redshift z, dV ( z ) /dz is the derivative of the volumeelement and the factor (1 + z ) − reflects the cosmological time dilation. We consider now a direct method to invert the observed L − z distribution and obtain the functions φ and R GRB . To do so we approximate φ ( L ) and R ( z ) as stepf unctions whose range is divided tobins with a constant value within each bin. These functions can be expressed as a sum of Heavisidefunctions. c (cid:13)000
The redshift distribution for the different methods: Absorption lines, Emission lines and Photometry luminosity function should depend on the redshift. We test later the validity of this assumption andshow that it is accepted with a high statistical significance.The isotropic peak luminosity function, φ ( L ) , is defined traditionally as the fraction of GRBswith isotropic equivalent luminosities in the interval log L and log L + d log L . The rate, R GRB ( z ) ,is defined as the co-moving space density of GRBs in the interval z to z + dz . The distributiondensity: n ( L, z ) is given by: n ( L, z ) dlogLdz = φ ( L ) · R ( z ) dlogLdz , (1)where R ( z ) = R GRB ( z )(1 + z ) dV ( z ) dz , (2)is the differential co-moving rate of bursts at a redshift z, dV ( z ) /dz is the derivative of the volumeelement and the factor (1 + z ) − reflects the cosmological time dilation. We consider now a direct method to invert the observed L − z distribution and obtain the functions φ and R GRB . To do so we approximate φ ( L ) and R ( z ) as stepf unctions whose range is divided tobins with a constant value within each bin. These functions can be expressed as a sum of Heavisidefunctions. c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts φ i ≡ φ ( L i (cid:54) L < L i +1 ) , (3) R j ≡ R ( z j (cid:54) z < z j +1 ) = 1( z j +1 − z j ) (cid:90) z j +1 z j dz R GRB ( z j )(1 + z ) dV ( z ) dz . (4)We also define the weights factors w ij ≡ (cid:90) L i +1 L i (cid:90) z j +1 z j θ ( L, z ) dlogLdz , (5)as the probability for detecting a burst with a measured redshift z and luminosity L where θ ( L, z ) ≡ θ z ( p ( L, z )) is the probability to detect and measure redshift for a burst with a luminosity L at aredshift z (see appendix B). We denote the observed number of events per bin as: N i ≡ N ( L i (cid:54) L < L i +1 ) , (6) N ,j ≡ N ( z j (cid:54) z < z j +1 ) , (7) N ij ≡ N ( L i (cid:54) L < L i +1 , z j (cid:54) z < z j +1 ) , (8) N ≡ (cid:88) ij N ij . (9)Next, we determine φ i and R j and the error estimates, using the maximum-likelihood formal-ism. We define M the (logarithmic) likelihood of the model given the observations as: M = (cid:88) ij N ij ln[ φ i R j w ij ] − N ln[ (cid:88) ij φ i R j w ij ] . (10)At the maximum all partial derivatives of M with respect to φ i and with respect to R j vanish,leading to φ i = N i (cid:80) j R j w ij (cid:80) i (cid:48) j (cid:48) φ i (cid:48) R j (cid:48) w i (cid:48) j (cid:48) N , (11)and R j = N ,j (cid:80) i φ i w ij (cid:80) i (cid:48) j (cid:48) φ i (cid:48) R j (cid:48) w i (cid:48) j (cid:48) N . (12)Notice that the second term on the right hand side of each of the equations 11 and 12 is the samenormalization factor. We have a set of non-linear equations with as many equations as variables.We solve numerically these non linear equations using successive iterations until convergence. A-priori it is not clear whether φ and R are uniquely determined and whether there is a solution at all.However, we find good convergence. We have examined a large set ( ) of initial guesses whereeach component was randomly drawn from a uniform distribution. We found a rapid convergenceto a unique solution for all initial guesses, all reaching the requested accuracy of − with lessthan 25 iterations. We thus conclude that the existence of other stable solutions is very unlikely. c (cid:13) , 000–000 D. Wanderman & T. Piran
We approximate the error as the value for which M deviate by -1 from it’s maximum: (i.e. thelikelihood is smaller by a factor e ). This reflects an σ error for a normal-distribution. − N i ln(1 + ∆ φ i φ i ) − N ln(1 + N i N ∆ φ i φ i ) , (13) − N ,j ln(1 + ∆ R j R j ) − N ln(1 + N ,j N ∆ R j R j ) . (14)The two solutions, i.e. the positive one and the negative one, give an upper and a lower boundson the error respectively. For small deviations we can approximate the error using the secondderivatives of M: ∆ φ i φ i (cid:39) √ (cid:112) N i (1 − N i /N ) , (15) ∆ R j R j (cid:39) √ (cid:112) N ,j (1 − N ,j /N ) . (16)To estimate the uncertainty induced by the specific bins choice, we preform all the analysis fora 1/2 unit redshift and Log ( L ) binning and repeat for a 1/3 unit binning (where all bins widthsare 1/3 unit, except the last two redshift bins which we cannot change because they contain too fewdata points). In the following, unless otherwise stated, we use the 1/2 unit binning for all furtheranalysis. Clearly, the results with different binning are slightly different, but they are all withineach other’s error range. When we include the uncertainty induced by the binning the error rangesbecome only slightly wider. So far, we have not assumed any functional form for the luminosity function or for the rate asour method does not require such an assumption. The ”raw” results are depicted in Figures 2 and3. Later on we will compare these step functions with models for the GRB rate that follow theSFR. However, in order to easily characterized the result we need to approximate our ”raw” stepfunctions with simple functional forms. Therefore, after obtaining the results in the form of stepfunctions we approximate them in term of broken power laws: φ ( L ) = ( LL ∗ ) − α L < L ∗ , ( LL ∗ ) − β L > L ∗ . (17) R GRB = R GRB (0) · (1 + z ) n z (cid:54) z , (1 + z ) n − n (1 + z ) n z > z , (18) c (cid:13)000
We approximate the error as the value for which M deviate by -1 from it’s maximum: (i.e. thelikelihood is smaller by a factor e ). This reflects an σ error for a normal-distribution. − N i ln(1 + ∆ φ i φ i ) − N ln(1 + N i N ∆ φ i φ i ) , (13) − N ,j ln(1 + ∆ R j R j ) − N ln(1 + N ,j N ∆ R j R j ) . (14)The two solutions, i.e. the positive one and the negative one, give an upper and a lower boundson the error respectively. For small deviations we can approximate the error using the secondderivatives of M: ∆ φ i φ i (cid:39) √ (cid:112) N i (1 − N i /N ) , (15) ∆ R j R j (cid:39) √ (cid:112) N ,j (1 − N ,j /N ) . (16)To estimate the uncertainty induced by the specific bins choice, we preform all the analysis fora 1/2 unit redshift and Log ( L ) binning and repeat for a 1/3 unit binning (where all bins widthsare 1/3 unit, except the last two redshift bins which we cannot change because they contain too fewdata points). In the following, unless otherwise stated, we use the 1/2 unit binning for all furtheranalysis. Clearly, the results with different binning are slightly different, but they are all withineach other’s error range. When we include the uncertainty induced by the binning the error rangesbecome only slightly wider. So far, we have not assumed any functional form for the luminosity function or for the rate asour method does not require such an assumption. The ”raw” results are depicted in Figures 2 and3. Later on we will compare these step functions with models for the GRB rate that follow theSFR. However, in order to easily characterized the result we need to approximate our ”raw” stepfunctions with simple functional forms. Therefore, after obtaining the results in the form of stepfunctions we approximate them in term of broken power laws: φ ( L ) = ( LL ∗ ) − α L < L ∗ , ( LL ∗ ) − β L > L ∗ . (17) R GRB = R GRB (0) · (1 + z ) n z (cid:54) z , (1 + z ) n − n (1 + z ) n z > z , (18) c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts Figure 2.
The comoving rate and the observed events number redshift distributionsUpper frame: The resulting step function and the best fit model: A broken power law with indices n = 2 . , n = − . and with a break at z = 3 . (solid line). The χ values for the models is 2.3 at 10 d.o.f. giving a rejection probability of 0.007. Lower frame: The number of detectedbursts for each redshift bin and the rates expected from the model fitted above. The two upper boxes in each column represent the statistical errorrange. We obtain the parameters of the best fit functions by minimizing the χ values.The best fit functions are shown (together with the step functions) in Figures 2 and 3. Theparameters of the best fit models are summarized in Table 1. To estimate the statistical errorsinvolved in the parameter estimates we preformed a Monte-Carlo simulation. In this simulationwe use the model with the best fit parameters to draw random sets of data (with same size asthe real sample). We then carry out the analysis on this mock data sets and obtain new best fitparameters. Repeating this process many times we obtain a scatter of points in the parametersplane around the original best fit parameters. The central and ranges for each of theparameters separately, are also shown in Table 1.The luminosity function is well described by a broken power law, with a break at L ∗ (cid:39) . ± . [ erg/sec ] and with indices α = 0 . +0 . − . and β = 1 . +0 . − . . The broken power law fit isvery good, giving χ = 0 . for 4 d.o.f. This result agrees with previous studies (e.g. Daigne et al. c (cid:13) , 000–000 D. Wanderman & T. Piran
Figure 3.
The luminosity function and the observed and predicted luminosity distributionsUpper frame: The resulting step function and the best fit model for the luminosity function: a broken power law with a break at L ∗ = 10 . , alow luminosity index α = 0 . and a high luminosity index β = 1 . . The χ value for this model is 0.63 at 4 d.o.f., giving a rejection probabilityof 0.04. Lower frame: The number of detected bursts for each luminosity bin and the rates expected from the model fitted above. The two upperboxes in each column, represents the statistical error range. / bins / binslogL ∗ . +0 . . − . − . . +0 . . − . − . α . +0 . . − . − . . +0 . . − . − . β . +0 . . − . − . . +0 . . − . − . z . +0 . . − . − . . +0 . . − . − . n . +0 . . − . − . . +0 . . − . − . n − . +2 . . − . − . − . +2 . . − . − . ρ . +0 . . − . − . . +0 . . − . − . Table 1.
Parameters results. The error ranges are 68% and 95% levels estimated using a Monte-Carlo simulations with 10000 sets, each of 101 datapoints. c (cid:13) , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts χ , rejecting such a model with high significance (98%). This contradictthe results of P´elangeon et al. (2008) who studied the HETE-2 GRBs and found a consistency witha single power law luminosity function. The rate is described as well by a broken power law for z , with a break at z = 3 . +0 . − . and indices of n = 2 . +0 . − . and n = − . +2 . − . .The local event rate is ρ (cid:39) . +0 . − . [ Gpc − yr − ] , in agreement with previous studies e.g.Schmidt (1999)[ ρ (cid:39) . ] (see however Schmidt (2001b)[ ρ (cid:39) . )]); Guetta et al. (2005)[ ρ (cid:39) . ]; Guetta and Della Valle (2007)[ ρ (cid:39) . ]; Liang et al. (2007)[ ρ (cid:39) . ]; P´elangeon et al.(2008)[ ρ (cid:38) . ]. The main factors determining the low redshift (current) event rate are the over-all GRB rate normalization, the low-redshift slope, the low end of the luminosity function slopeand most important the position of the low luminosity cutoff. This low luminosity cutoff is essen-tial for any steep enough power law to prevent its divergence. This cutoff is critical to the questionwhether the model includes the low luminosity GRB population. The cutoff used in our estimatesof ρ is L = 10 erg/sec . Before we can accept the model,we turn now to check the validity of the assumption that theluminosity function and the rate are independent. . To do so, we preform a two dimensionalKolmogorov-Smirnov test (see Fasano and Franceschini (1987), Peacock (1983), Spergel et al.(1987)). In this test the two dimensional data is compared to the modeled distribution for each of4 quadrant defined by axes crossing at each data point. The maximal difference is used to estimatethe probability that the data is drawn from the distribution implied by the model. The probabilitythat the models fits the data is displayed in the 2D K-S column in Table 4. The test give highprobability ( ), so we can accept the model and justify the underling assumption (of a redshiftindependent luminosity function).We carry out two other consistency checks. First, we compare the peak flux distribution ex-pected by our model with those observed by BATSE and by
Swift . The results are shown in Fig-ures 4 and 5. The KS-test results, 83% for BATSE bursts and 17% for the full sample of
Swift bursts (with or without redshift), indicate acceptable model. We note here, that the model is sig-nificantly rejected (KS-test < − ) when compared to BATSE peak fluxes distribution using P lim = 0 . ph/cm /sec . However, when applying the effective detection threshold calculated byBand (2002) of P lim = 0 . ph/cm /sec , the model is accepted with high significance (83%). c (cid:13) , 000–000 D. Wanderman & T. Piran
Figure 4.
Bursts count vs. peak flux for BATSE burstsCumulative bursts distribution as a function of the peak flux p . Left: logarithmic scale showing N ( < p ) Right: log linear scale showing N ( > p ) ,this is used for the KS-test, giving probability of . Figure 5.
Bursts count vs. peak flux for
Swift bursts
Cumulative bursts number distribution as a function of the peak flux p . Left: logarithmic scale showing N ( < p ) Right: log linear scale showing N ( > p ) , which is used for the KS-test, giving a probability of . Second, we compared the cumulative redshift distribution to the observed one and preformed aKS-test. Here, as well, the test gives a high probability for accepting the model ( ).To illustrate the distribution of bursts, we display the bursts in the L − z ( Luminosity − Redshif t ) plane and in a rescaled plane in which the number of detected bursts (with or withoutredshift measurement) is proportional to the area (Fig. 6). The rather uniform distribution on therescaled plane is a visual demonstration of the validity of the assumption. c (cid:13)000
Cumulative bursts number distribution as a function of the peak flux p . Left: logarithmic scale showing N ( < p ) Right: log linear scale showing N ( > p ) , which is used for the KS-test, giving a probability of . Second, we compared the cumulative redshift distribution to the observed one and preformed aKS-test. Here, as well, the test gives a high probability for accepting the model ( ).To illustrate the distribution of bursts, we display the bursts in the L − z ( Luminosity − Redshif t ) plane and in a rescaled plane in which the number of detected bursts (with or withoutredshift measurement) is proportional to the area (Fig. 6). The rather uniform distribution on therescaled plane is a visual demonstration of the validity of the assumption. c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts Figure 6.
Bursts on the L-z planeThe bursts distribution in the Luminosity-Redshift plane. Left: linear scale. Right: Axes are rescaled using R ( z ) and φ ( L ) , so that we have auniform bursts-number density. The curved lines are contour lines for equal fluxes, for the values: p = 0 . , . , , ph/sec/cm ] , from top tobottom. The p = 0 . ph/sec/cm ] line is our detection threshold. The number density is uniform in the rescaled plane as expected and it dropsto zero below the detection threshold. Swift
Sample
Our results are based only on the absorption and photometric determined redshifts of the
Swift sample. Recently, Fynbo et al. (2009) obtained emission lines redshift measurements for GRBshost galaxies that were not measured before. Most of those are at low redshifts. The growingnumber of emission lines redshifts in the range < z < raises the question whether our model- based on a sample of redshifts measured from the afterglows - is consistent. We calculate, usingour model, the expected redshift distribution of the entire bursts population and compare it to thenumber of observed bursts with a known redshift (measured using all methods). Clearly for anyrange of redshifts the number of known bursts with a given redshift range should not exceed thenumber predicted by the model. Fig 7 depicts this comparison for the cumulative number andfor the counts number in redshift bins. Our model is consistent for all redshifts (cid:38) . . Howeverthere is an excess of low redshift bursts not predicted by our model. The discrepancy arise dueto three bursts with z < . while less than one burst is predicted by the model. The redshift ofthe three bursts was determined using emission lines but no redshift was detected in this rangeusing absorption lines. These three bursts yield a rate that is significantly higher than predictedby the model. This cannot be explained by a misidentification of the host galaxies because theprobability for such an effect is too small (Cobb and Bailyn 2008). These low redshift bursts have c (cid:13) , 000–000 D. Wanderman & T. Piran
Figure 7.
Redshifts obtained by any method compared with the total number of bursts predicted by the modelUpper frames: Left: The cumulative total fraction of bursts predicted by the model (dashed curve) and the detected cumulative fraction of burstswith redshift obtained by any method (solid steps) as a function of the redshift. Right: Zoom in on the low redshift part. Lower frame: The numberdensity (per unit redshift) of predicted bursts (dashed curve) and an histogram of bursts with redshift obtained by any method (solid steps). Bothare normalized by the total number of bursts. For all frames: The number of bursts with redshifts which had an at slew, i.e. the time elapsed formthe trigger until the first optical observation is less than 300 sec (dashed dotted steps). low luminosities and they could not have been detected at a much higher redshift. We concludethat they possibly represent a low luminosity population that is different from the majority of thebursts (see a discussion in § A particularly interesting question is how many high redshift bursts are expected to be observed.Such bursts are of great interest as they may shed new light on the very early universe. Already nowGRB090423 is amongst the most distant and hence the earliest objects observed so far. Figure 8depicts the observed cumulative redshift distribution and the predicted one, for high-redshift burstsfor
Swift and for past and future missions: BATSE, EXIST (Band et al. 2008) and SVOM (Schanne c (cid:13)000
Swift and for past and future missions: BATSE, EXIST (Band et al. 2008) and SVOM (Schanne c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts Figure 8.
The cumulative redshift distribution for
Swift bursts and predictions for other missionsThe cumulative redshift distribution of
Swift observed bursts and the model prediction (solid line). The KS-test gives a probability of . Alsopresented are the predictions for some past and future missions: BATSE, SVOM and EXIST. Right frame is a zoom-in to the upper right part of theleft frame.fraction of z >
Table 2.
The high redshift fraction prediction for
Swift and several other missions z > , , , , , respectively is shown inTable 2.Detection of high redshift bursts is one of the main objectives of EXIST. The number of highredshift bursts expected to be detected by EXIST is presented in Table 3. During a five year missionEXIST will detect many high redshift ( z > ) ( ≈
30) bursts. In the best-fit model there is a goodprobability for EXIST to detect even a z > burst during mission. This is, of course, providedthat such early bursts exist and that the rate at z ≈ − can be extrapolated to such high redshifts. c (cid:13) , 000–000 D. Wanderman & T. Piran bursts per year for z > +121 − +96 − +81 − +64 − . +25 . − . . +11 . − . SVOM +12 − +9 − . . +6 . − . . +4 . − . . +1 . − . . +0 . − . Table 3.
High redshift detection rates prediction for EXIST and for SVOM. On average a redshift is obtained for only a third of the events.
The location of long GRBs in the star forming regions led to the expectation that GRB follow theSFR. We turn now to examine this hypothesis. Modeling the SFR throughout the measured redshiftrange ( < z < ) is a complicated task, involving observations in various wavelengths andvarious assumptions on observational proxies for the SFR as well as correcting due to obscuration,absorption and selection effects. In a classical work Madau (Madau et al. 1996; Madau 1998;Madau et al. 1997) considered the SFR per comoving volume vs redshift and found a rise frompresent to z ≈ − , and then a comparable decline to z ≈ . This form of the SFR vs redshift hasbecome known as the “Madau plot”. A number of developments (e.g., dust corrections, submmresults, new estimates of the SFR at low redshift) led to changes in the shape of the SFR. Followingthese developments Rowan-Robinson (1999) and Porciani and Madau (2001) suggested that theSFR rises by a factor of 10 - 20 from z = 0 to z (cid:39) . The rate at higher redshifts ( z > ) wasundecided at that time and models for the SFR at higher redshift included flat as well as rising anddeclining functions (Porciani and Madau 2001). Later, Hopkins and Beacom (2006) showed thatat high redshift ( z > ) the SFR declines. Recently, several papers estimated the SFR using thenew HST WFC3/IR camera (Bouwens et al. 2009a; Oesch et al. 2009; Bunker et al. 2009; McLureet al. 2009; Yan et al. 2009; Bouwens et al. 2009b). These works suggest a decline in the SFRfor z (cid:38) up to z ∼ − . In the following we use Bouwens et al. (2009b) as representative ofthese new high-z SFRs. Despite all the observational advances, there are still different models ofthe SFR even at low ( z < ) and intermediate ( < z < ) redshift. On one hand, the widely usedHopkins and Beacom (2006) piecewise linear fit finds a factor 10 rise in the SFR from z = 0 to z = 1 and an almost constant rate from z = 1 to z = 4 . (which follows by a steep decline athigher redshifts) .On the other hand Bouwens et al. (2009b) use data from Schiminovich (2005)and Reddy and Steidel (2009) for < z < , < z < respectively. Their data compilationcan be fairly modeled ( χ = 10 . for 9 d.o.f.) by a broken power law rising a factor of 20 from z = 0 to z = 3 . , then declining as (1 + z ) − . for . < z . Note that a rise up to z = 2 − isalso suggested by Hopkins and Beacom (2006) when fitting to a Cole (2001) form. This seemingly c (cid:13)000
The location of long GRBs in the star forming regions led to the expectation that GRB follow theSFR. We turn now to examine this hypothesis. Modeling the SFR throughout the measured redshiftrange ( < z < ) is a complicated task, involving observations in various wavelengths andvarious assumptions on observational proxies for the SFR as well as correcting due to obscuration,absorption and selection effects. In a classical work Madau (Madau et al. 1996; Madau 1998;Madau et al. 1997) considered the SFR per comoving volume vs redshift and found a rise frompresent to z ≈ − , and then a comparable decline to z ≈ . This form of the SFR vs redshift hasbecome known as the “Madau plot”. A number of developments (e.g., dust corrections, submmresults, new estimates of the SFR at low redshift) led to changes in the shape of the SFR. Followingthese developments Rowan-Robinson (1999) and Porciani and Madau (2001) suggested that theSFR rises by a factor of 10 - 20 from z = 0 to z (cid:39) . The rate at higher redshifts ( z > ) wasundecided at that time and models for the SFR at higher redshift included flat as well as rising anddeclining functions (Porciani and Madau 2001). Later, Hopkins and Beacom (2006) showed thatat high redshift ( z > ) the SFR declines. Recently, several papers estimated the SFR using thenew HST WFC3/IR camera (Bouwens et al. 2009a; Oesch et al. 2009; Bunker et al. 2009; McLureet al. 2009; Yan et al. 2009; Bouwens et al. 2009b). These works suggest a decline in the SFRfor z (cid:38) up to z ∼ − . In the following we use Bouwens et al. (2009b) as representative ofthese new high-z SFRs. Despite all the observational advances, there are still different models ofthe SFR even at low ( z < ) and intermediate ( < z < ) redshift. On one hand, the widely usedHopkins and Beacom (2006) piecewise linear fit finds a factor 10 rise in the SFR from z = 0 to z = 1 and an almost constant rate from z = 1 to z = 4 . (which follows by a steep decline athigher redshifts) .On the other hand Bouwens et al. (2009b) use data from Schiminovich (2005)and Reddy and Steidel (2009) for < z < , < z < respectively. Their data compilationcan be fairly modeled ( χ = 10 . for 9 d.o.f.) by a broken power law rising a factor of 20 from z = 0 to z = 3 . , then declining as (1 + z ) − . for . < z . Note that a rise up to z = 2 − isalso suggested by Hopkins and Beacom (2006) when fitting to a Cole (2001) form. This seemingly c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts § χ for both / and / binning. We findacceptable reduced χ values (see Table 4) for all the functions. However, a comparison of theoverall observed redshift and luminosity distributions with those predicted by models in which theGRB rate is fixed by a given SFR reveals that the 2D K-S or the K-S tests for the peak flux andthe redshift distributions show inconsistency for the first three functions (HB, PM SF2, R-R). Wefind, however, consistency for the last one (B09).Next, we optimize the luminosity function for a given SFR. We take the GRB rate as knownfollowing a model of the SFR and obtain the best fit luminosity function by solving equation 12.We now perform a 2D K-S test as well as K-S tests for the peak flux distribution and for theredshift distribution. The results of the statistical tests are shown in Table 4. Even though the fitimproves still the first three SFR models fail the KS tests. The last SFR model (B09) is, of course,consistent.We attribute the consistency of the GRB rate with the B09 SFR model but not with the firstthree SFR models to the difference between the B09 SFR and the other three models in the range < z < . While B09 keeps increasing in this range the first three are constant. This difference iscrucial and understanding the SFR at this region is critical for the question whether the GRB ratefollows the SFR or not. While there are differences at higher redshifts, the paucity of data leads towide error range in that region allowing the GRB observations to be consistent with SFRs that areconstant, decreasing or even increasing at large z.The comparison Table shows that the luminosity function parameters depends very weaklyon the GRB rate model. α and ρ changes within their 95% error range, while L ∗ and β stayalmost unchanged. This illustrates the power of the method and the robustness of the luminosityfunction results, in particular for L ∗ and β . c (cid:13) , 000–000 D. Wanderman & T. Piran
Figure 9.
GRB event rate and several star formation ratesThe results for the rate, in 1/2 unit binning. Best fit for a broken power law - heavy black solid line. Hopkins and Beacom (2006) SFR - red dashedline, Bouwens et al. (2009b) SFR - cyan solid line. SF2 of Porciani and Madau (2001) - magenta dotted line. Rowan-Robinson (1999) SFR - bluedashed dotted line. reduced χ reduced χ
2D K-S peak flux z L ∗ α β ρ (1/3 bins) (1/2 bins) K-S test K-S testthis paper 0.24 0.23 0.96 0.82 0.62 52.53 0.17 1.44 1.25R-R(1999) 0.69 0.85 · − · − · − · − · − · − · − · − · − · − B09 0.57 0.60 0.21 0.93 0.14 52.53 0.19 1.44 1.22B09 0.15 0.65 0.19 52.53 0.08 1.47 0.99Butler et al.(2009) 0.27 0.27 0.76 0.65 0.71 52.53 0.19 1.44 0.80Butler et al.(2009) 0.66 0.46 0.50 52.47 0.00 1.47 0.58Table 4.
Statistical tests for our models and for models following one of the SFRs considered (upper part) and for GRB rate from other studies(lower part). The consistent models are marked with a bold font. For each model we show the test with luminosity function from our results (firstline) and with luminosity function that best fit observations after forcing the rate to follow the SFR (second line).
Our results are in agreement with most previous works on the GRB rate. Our results agree withDaigne et al. (2006) who compared the
Swift data with Porciani and Madau (2001) SFR models c (cid:13)000
Swift data with Porciani and Madau (2001) SFR models c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts z > . the GRB rate is significantly higher than theR-R or PM SF2 SFR.A comprehensive work studying the luminosity function and the rate of long GRBs was re-cently published (Butler et al. 2009). While this work uses a different sample, adopts somewhatdifferent assumptions and uses other methods, a comparison of the results can be very useful. Wefind a good agreement for the low luminosity power law index α = 0 . +0 . − . ( ourresults : 0 . +0 . − . ) and for the break luminosity L ∗ = 10 . ± . ( our results : 10 . ± . ) . However we find differ-ent high luminosity power law index β = 2 . +2 . − . , ( our results : 1 . +0 . − . ) . The differences may beexplained by the fact that Butler et al. (2009) use a luminosity which is a time averaged whereasour luminosity is the peak luminosity. We find a nice agreement between the models for the GRBrate. Butler et al. (2009) find a rising rate for < z < - first at slope . ± . for < z < andlater at slope . ± . for < z < . This is not very different from our results, recalling that thebreak at z = 1 was not a free parameter in their model. The decline slope for z > is − . +1 . − . ,but with the big uncertainties it also matches our model. It is thus not surprising that their modelshow consistency with the statistical tests (see Table 4). Another useful comparison is with Kistleret al. (2009) who modeled the bias of the GRB rate with SFR in the range < z < and used ittogether with the high-z bursts data to estimate the high-z SFR. The high-z GRB rate they foundis roughly constant, in agreement with our results. We find that the GRB rate increases up to redshift (cid:39) and it decreases at z > ( n (cid:39) − . ), with68% confidence limits ranging from a steep decline ( n (cid:39) − . ), to a positive incline ( n (cid:39) ).The rate is compatible, of course, with a constant rate at higher redshifts. The model is consistentwith all statistical tests and thus we can accept the basic assumption that the luminosity functiondose not evolve with time. The comparison between the SFR and the GRB rate seems to be inconclusive. This arises becauseof the differences between different estimates of the SFR at the intermediate redshift range < c (cid:13) , 000–000 D. Wanderman & T. Piran z < . The rate we find is consistent with Bouwens et al. (2009b) (B09) (that follows Schiminovich2005) that describes a rising SFR from present up to z ≈ . It is inconsistent with other SFRs(Rowan-Robinson 1999; Porciani and Madau 2001; Hopkins and Beacom 2006) that suggest aconstant rate at this regime.At high redshifts the GRB data is sparse. The best fit result decreases slower than the mostrecent B09 SFR (or even the HB SFR) suggesting possibly higher GRB rate. However fast de-crease, like B09, cannot be ruled out while a flat or even slowly increasing rate are also consistent.A larger GRB rate at high redshift (compared to the SFR) can be explained within the frameworkof the massive stellar collapse model due to metallicity. Woosley and Heger (2006) suggests thatGRB rate follow the low-metallicity part of the star formation. We cautiously note however thatFynbo et al. (2009) recently found that the optical afterglow spectroscopy sample is biased againstmeasuring redshift at high metallicity environments, meaning that the result might be an artifact ofa selection effect. Another clue can be taken from Fruchter et al. (2006) who found that the GRBsdistribution within the host galaxies dose not follow the light distribution but rather some power( > ) of the light distribution, i.e. higher GRB density in the denser star forming regions. This is incontrast to the core-collapse supernovae distribution that follows the light distribution, indicatingthe GRB progenitors are different from SN progenitors and hence their rate might be different. We find an overall consistency when comparing our model and the full sample of GRBs withredshift. However, we also find three low redshift, low luminosity bursts (with emission lines red-shifts) that are not expected by the model prediction. The faintest burst in our sample GRB050724has a luminosity L = 10 . [ erg/sec ] . By applying our best fit model, we expect 0.9 bursts withluminosity L (cid:54) L , in the time span of our sample (4.5 years). Swift s weakest burst, GRB060218(Cusumano et al. 2006) with luminosity L = 10 . [ erg/sec ] , is not in our sample because ithas only emission lines redshift. Assuming that we can extrapolate the low end of our luminosityfunction we expect . · − bursts with L (cid:54) L . Even when applying the 95% level of the pa-rameter α = 0 . , we expect no more than · − bursts with L (cid:54) L . This implies that this burstrepresents a population of fainter GRBs, with much higher event rate, which cannot be directly ex-trapolated from the stronger GRB population (in agreement with e.g. Soderberg et al. 2006; Cobbet al. 2006; Guetta and Della Valle 2007; Liang et al. 2007). The emission lines redshifts sampleincludes two other low luminosity bursts: GRB051109 and GRB060505 with z = 0 . , . and c (cid:13)000
Swift data with Porciani and Madau (2001) SFR models c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts z > . the GRB rate is significantly higher than theR-R or PM SF2 SFR.A comprehensive work studying the luminosity function and the rate of long GRBs was re-cently published (Butler et al. 2009). While this work uses a different sample, adopts somewhatdifferent assumptions and uses other methods, a comparison of the results can be very useful. Wefind a good agreement for the low luminosity power law index α = 0 . +0 . − . ( ourresults : 0 . +0 . − . ) and for the break luminosity L ∗ = 10 . ± . ( our results : 10 . ± . ) . However we find differ-ent high luminosity power law index β = 2 . +2 . − . , ( our results : 1 . +0 . − . ) . The differences may beexplained by the fact that Butler et al. (2009) use a luminosity which is a time averaged whereasour luminosity is the peak luminosity. We find a nice agreement between the models for the GRBrate. Butler et al. (2009) find a rising rate for < z < - first at slope . ± . for < z < andlater at slope . ± . for < z < . This is not very different from our results, recalling that thebreak at z = 1 was not a free parameter in their model. The decline slope for z > is − . +1 . − . ,but with the big uncertainties it also matches our model. It is thus not surprising that their modelshow consistency with the statistical tests (see Table 4). Another useful comparison is with Kistleret al. (2009) who modeled the bias of the GRB rate with SFR in the range < z < and used ittogether with the high-z bursts data to estimate the high-z SFR. The high-z GRB rate they foundis roughly constant, in agreement with our results. We find that the GRB rate increases up to redshift (cid:39) and it decreases at z > ( n (cid:39) − . ), with68% confidence limits ranging from a steep decline ( n (cid:39) − . ), to a positive incline ( n (cid:39) ).The rate is compatible, of course, with a constant rate at higher redshifts. The model is consistentwith all statistical tests and thus we can accept the basic assumption that the luminosity functiondose not evolve with time. The comparison between the SFR and the GRB rate seems to be inconclusive. This arises becauseof the differences between different estimates of the SFR at the intermediate redshift range < c (cid:13) , 000–000 D. Wanderman & T. Piran z < . The rate we find is consistent with Bouwens et al. (2009b) (B09) (that follows Schiminovich2005) that describes a rising SFR from present up to z ≈ . It is inconsistent with other SFRs(Rowan-Robinson 1999; Porciani and Madau 2001; Hopkins and Beacom 2006) that suggest aconstant rate at this regime.At high redshifts the GRB data is sparse. The best fit result decreases slower than the mostrecent B09 SFR (or even the HB SFR) suggesting possibly higher GRB rate. However fast de-crease, like B09, cannot be ruled out while a flat or even slowly increasing rate are also consistent.A larger GRB rate at high redshift (compared to the SFR) can be explained within the frameworkof the massive stellar collapse model due to metallicity. Woosley and Heger (2006) suggests thatGRB rate follow the low-metallicity part of the star formation. We cautiously note however thatFynbo et al. (2009) recently found that the optical afterglow spectroscopy sample is biased againstmeasuring redshift at high metallicity environments, meaning that the result might be an artifact ofa selection effect. Another clue can be taken from Fruchter et al. (2006) who found that the GRBsdistribution within the host galaxies dose not follow the light distribution but rather some power( > ) of the light distribution, i.e. higher GRB density in the denser star forming regions. This is incontrast to the core-collapse supernovae distribution that follows the light distribution, indicatingthe GRB progenitors are different from SN progenitors and hence their rate might be different. We find an overall consistency when comparing our model and the full sample of GRBs withredshift. However, we also find three low redshift, low luminosity bursts (with emission lines red-shifts) that are not expected by the model prediction. The faintest burst in our sample GRB050724has a luminosity L = 10 . [ erg/sec ] . By applying our best fit model, we expect 0.9 bursts withluminosity L (cid:54) L , in the time span of our sample (4.5 years). Swift s weakest burst, GRB060218(Cusumano et al. 2006) with luminosity L = 10 . [ erg/sec ] , is not in our sample because ithas only emission lines redshift. Assuming that we can extrapolate the low end of our luminosityfunction we expect . · − bursts with L (cid:54) L . Even when applying the 95% level of the pa-rameter α = 0 . , we expect no more than · − bursts with L (cid:54) L . This implies that this burstrepresents a population of fainter GRBs, with much higher event rate, which cannot be directly ex-trapolated from the stronger GRB population (in agreement with e.g. Soderberg et al. 2006; Cobbet al. 2006; Guetta and Della Valle 2007; Liang et al. 2007). The emission lines redshifts sampleincludes two other low luminosity bursts: GRB051109 and GRB060505 with z = 0 . , . and c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts L = 10 . [ erg/sec ] , . [ erg/sec ] . The detection probability of such bursts, according to ourmodel is < · − , < · − respectively. These three low luminosity bursts must belong to adifferent and a distinct group and we remove them from the analysis when checking for consis-tency in § Higher redshift bursts are most interesting as they can provide clues on the very early universe.Extrapolating the rate for very high redshifts, we expect ∼ . bursts with z > and ∼ . burstswith z > (bursts with measured redshift), detected by Swift in the time span of our sample (4.5years). This is consistent with the observation of one z > burst. Future missions like EXISTcan find many more such bursts, even dozens of z > bursts (see § The local event rate found is ρ = 1 . +0 . − . [ Gpc − yr − ] , for bursts with L (cid:62) erg/sec . Withone galaxy in M pc this rate is equivalent to 1 event per galaxy per years! Taking intoconsideration the beaming factor of about 50 (see Guetta et al. 2005), the total events rate is about1 event per galaxy per · years. This should be typical to our galaxy as a recent estimationto the SFR in our galaxy (Robitaille and Whitney 2010) finds ˙ ρ = 0 . − . M (cid:12) /yr , which isequivalent to the SFR in the local universe for a galaxy in a volume of M pc . One implicationof this result will be on the possible association of GRBs with global extinctions of biologicalspices. These occurred on Earth a factor of 10 times less frequent, about once every 100 Myr.These two rates suggest that a typical Galactic GRB pointing to Earth does not cause a majorextinction event. ACKNOWLEDGMENTS
We thank Shiho Kobayashi, Ehud Nakar and Elena Rossi for fruitful discussions. The research wassupported by an ERC Advanced Research Grant, by the ISF center for High Energy Astrophysicsand by the Israel-France program in Astrophysics grant.
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APPENDIX A: THE LUMINOSITY
In this work we used only data collected by
Swift . The peak-flux of each burst, measured in
Swift ’sBAT detectors band 15keV - 150keV (Barthelmy et al. 2005). For most of the bursts we cannot es-timate the luminosity in the full γ -range (1keV - 10MeV) as the spectral shape (the Band-function)(Band et al. 1993) is poorly known. We still need however to have some quantitative measure ofthe luminosity that will be defined uniformly for all burst with a known redshift. To do so we usean average characteristic Band function, E peak = 511 keV (in source frame), α = − , β = − . (Preece et al. 2000; Porciani and Madau 2001) to estimate luminosity, using the same parametersfor all bursts, thus the estimated luminosity is proportional to the measured peak f lux (p): L iso = p πD ( z ) (1 + z ) k ( z ) C det , (A.1) D ( z ) is the proper distance at redshift z , C − det is the fraction of the total γ − ray luminosity detectedin the detectors energy band for source at redshift = 0, C det = (cid:82) MeV keV EN ( E ) dE (cid:82) E max E min N ( E ) dE , (A.2) k ( z ) is the k-correction for the given spectrum at redshift z, k ( z ) = (cid:82) E max E min N ( E ) dE (cid:82) (1+ z ) E max (1+ z ) E min N ( E ) dE , (A.3)where N ( E ) is the Band function, E min = 15 keV , E min = 150 keV . The values in this paper arethe luminosity in the range [1keV - 10MeV]. c (cid:13) , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts Figure A1.
Distributions of our fits parameters for the luminosity function ( L ∗ , α, β ) and for the rate ( z , n , n ) when choosing the γ -ray spectra(Band function) parameters randomly from their distributions. The shaded area contains the central 68% of the distribution, the dashed lines are the68% confidence range of our reported results. To study the dependence of our results on this approximation of an average Band function, wehave preformed a simulation where the spectrum of the bursts is not universal but drawn randomlyfrom the known distribution (Preece et al. 2000). Repeating the analysis in the paper 1000 times,each time calculating the luminosity of a burst using a band function with parameters randomlydrawn - independently for each burst - from the distribution. Fig A1 show the simulation resultsas an histogram for each of the parameters of the luminosity function and the rate fits. All thedistributions concentrate within the error range of the original results, demonstrating therobustness of the results and its insensitivity to the details of the distribution of spectral parameters.
APPENDIX B: THE REDSHIFT DETECTION PROBABILITY
We examine now the redshift detection probability as a function of the peak f lux . We considerhere two effects: first the probability to detect the GRB and second the probability to measure theredshift, for a given detected burst. c (cid:13) , 000–000 D. Wanderman & T. Piran
The probabilities to detect a GRB, or to measure a redshift, are a function of the burst energy,its duration and other factors, as described by (Band 2006). We restrict ourselves here only tothe dependence of these probabilities in the peak f lux , since this is the quantity we use for theanalysis in this paper.
B1 GRB Detection Probability
The simplest model often used is of a sharp threshold: No detection with p < p lim , but 100%detection of bursts with p (cid:62) p lim . For this model the value used at Swift ’s main detection band [15 − keV is p lim = 0 . ph/cm /sec . (see Guetta and Piran (2007), Gorosabel et al. (2004)).In the plot of the accumulated number of bursts as a function of log ( p ) , a linear relation appearsfor fluxes that are low or medium but above the threshold mentioned above. Although we do nottry to give a theoretical explanation for this result, we do not expect, however any strong deviationfrom that connection for lower fluxes, since our models predict a slow gradual smooth change in dN/dlog ( p ) and we can adopt this values at least as an order of magnitude estimators, to yield acontinues threshold estimation. Assuming that the deviation from linearity for fluxes below p lim isonly due to a lowered detection sensitivity, we can extrapolate the predicted number of bursts forlower fluxes and extract the detection sensitivity by comparing the number of detected bursts tothe predicted number. Figure B1 shows this: accumulated number of bursts vs. log peak flux, thelinear fit and our fit. θ γ ( p ) = (1+ c )2 + (1 − c )2 erf ( d · log ( p/p ) 0 . (cid:54) p , p < . , (B.1)with the parameters found: c = 0 . , d = 10 , log p = − . . B1.1 Redshift Measuring Probability
Redshifts are measured only for a moderate fraction of the detected bursts. A few (5-10%) are tooweak, some don’t have a clear redshift signature and some are not measured because of lack ofobservational resources.When we consider the fraction of redshift-measured GRBs to the total number of GRBs de-tected, we see a trend of increase in this fraction with the measured peak number flux of photonsin the detectors main band: keV − keV , p (see figure B2). We used a linear regression to c (cid:13)000
Redshifts are measured only for a moderate fraction of the detected bursts. A few (5-10%) are tooweak, some don’t have a clear redshift signature and some are not measured because of lack ofobservational resources.When we consider the fraction of redshift-measured GRBs to the total number of GRBs de-tected, we see a trend of increase in this fraction with the measured peak number flux of photonsin the detectors main band: keV − keV , p (see figure B2). We used a linear regression to c (cid:13)000 , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts Figure B1.
The accumulated number of bursts as function of log(peak flux) approximate the relation: θ z ( p ) /θ γ ( p ) = a · log ( p ) + b , (B.2)Where , θ γ ( p ) is the detection probability and θ z ( p ) is the probability that a redshift will be mea-sured. The parameters found in fit are: a = 0 . ± . and b = 0 . ± . . with χ = 1 . at 5 degrees of freedom, giving a rejection probability of 0.038 . We expect the fraction of numberof bursts with redshift N z ( p ) to the overall number of bursts N γ ( p ) , for any given p , to obey therelation: θ z ( p ) θ γ ( p ) = N z ( p ) N γ ( p ) . (B.3)Figure B2. depicts the fraction of bursts with a measured redshift and the linear fit, for theabsorption and photometry redshifts. Although the fit is acceptable the significant of the effect isjust one standard deviation ( σ ) - so with the current observations the option of no dependence ofredshift detection probability with peak flux ( a = 0) is still marginally consistent.Whereas the detection fraction can be fitted well with a linear model for the redshifts obtainedusing absorption lines, the case is different when considering redshifts obtained using emissionlines. Figure B3. shows the measured redshift fraction and the linear fit for the emission linesredshifts. The emission lines redshifts are obtained preferably for high flux bursts, but very few c (cid:13) , 000–000 D. Wanderman & T. Piran
Figure B2.
The fraction of bursts with a measured absorption lines and photometry redshift, as function of log(p)
Figure B3.
The fraction of bursts with a measured emission lines redshift, as function of log(p) are obtained for low and medium fluxes: for log p > and only . for log p (cid:54) . Thisfeature of the emission lines redshifts is associated with a strong bias toward lower redshifts asshown on Figure 1. These results supports our approach of selecting only the absorption lines andphotometry redshifts as it make a sample which is less biased and easier to model. c (cid:13) , 000–000 he luminosity function and the rate of Swift’s Gamma Ray Bursts Figure B4. the probability for measuring redshift by absorption lines or by photometry θ z , as a function of log(peak flux)Prob. z Prob. γ L ∗ α β z n n - - 52.54 0.17 1.32 3.13 1.85 -1.38- + 52.53 0.04 1.32 2.25 1.53 -0.00+ - 52.53 0.30 1.44 3.08 2.28 -1.06+ + 52.53 0.17 1.44 3.11 2.07 -1.36 Table B1.
Model’s best-fit parameters when taking (+) and when not taking (-) each effect into account.
The product of the above probabilities for burst detection and redshift measurement, gives thetotal probability that we detect a burst and measure its redshift θ z . Figure B4, shows the function θ z , which have the form: θ z ( p ) = ( alog ( p ) + b )( (1+ c )2 + (1 − c )2 erf ( d · log ( p/p )) 0 . (cid:54) p , p < . , (B.4)with a = 0 . , b = 0 . , c = 0 . , d = 10 , log p = − . . To check the effects of the burst detection probability and the redshift measurement probability,we repeated the process described in this paper, with all four options of taking or not takinginto account each of the modified probabilities. The best fit results for the parameters are shownin Table B1. For both corrections, we find a non-negligible effect on the results, although thedeviations induced on the parameters are in most cases within the statistical error ranges of the c (cid:13) , 000–000 D. Wanderman & T. Piran analysis. When taking both effects into account, the models give somewhat better results in thevarious statistical tests. Therefor, the function θ z ( p ) that take both effects into account is used inthis work. c (cid:13)000