AN UNEXPECTEDLY LOW-REDSHIFT EXCESS OF SWIFT GAMMA-RAY BURST RATE

Gamma-ray bursts (GRBs) are some of the most violent explosions in the universe and radiate huge amounts of energy in gamma-rays over a short period of time (for reviews, see Mészáros 2006; Zhang 2007; Gehrels et al. 2009). These explosions are so bright that they can be detected at much higher redshifts than supernovae (SNe). Hitherto, the farthest GRB with a spectroscopic redshift is GRB 090423 at $z\approx 8.2$ (Salvaterra et al. 2009a; Tanvir et al. 2009). Therefore, GRBs may be useful tools to probe the early universe (Bromm & Loeb 2012; Wang et al. 2015), including dark energy (Dai et al. 2004; Schaefer 2007; Wang et al. 2011), the star formation rate (SFR; Totani 1997; Wang 2013), reionization (Totani et al. 2006; McQuinn et al. 2008), and metal enrichment (Wang et al. 2012; Castro-Tirado et al. 2013).

Theoretically, the progenitors of long GRBs with durations of ${{T}_{90}}\gt 2$ s are thought to be collapsed massive stars (Woosley 1993). Observations also show that some long GRBs are associated with the deaths of massive stars which will create core collapse SNe (Hjorth et al. 2003; Stanek et al. 2003). Therefore, GRBs can be used to investigate the SFR at high redshifts (Totani 1997; Wijers et al. 1998; Lamb & Reichart 2000; Porciani & Madau 2001; Broom & Loeb 2002; Lin et al. 2004; Wang & Dai 2009; Kistler et al. 2009; Wanderman & Piran 2010; Butler et al. 2010; Elliott et al. 2012, 2014). In order to measure SFR using GRBs, the relation between the rate of GRBs and the SFR should be known. Some studies have found that the GRB rate is consistent with SFR at about $z\lt 4.0$, but has an excess at high redshift compared with what is expected from the SFR (Le & Dermer 2007; Kistler et al. 2009). Some models have been proposed to explain the discrepancy between the SFR and GRB rate, such as the cosmic metallicity evolution (Li 2008; Qin et al. 2010), superconducting cosmic strings (Cheng et al. 2010), the evolving initial mass function of stars (Wang & Dai 2011), and the evolution of the GRB luminosity function break (Virgili et al. 2011). From the redshift distribution of GRBs and the metallicity of GRB host galaxies, Wang & Dai (2014) showed that the discrepancy between the GRB rate and the SFR can be reconciled by considering that GRBs occur in low-metallicity galaxies.

Previously in the literature, the log N–log P distribution has been used to study the luminosity function and formation rate of GRBs (Fenimore & Ramirez-Ruiz 2000; Firmani et al. 2004; Guetta et al. 2005; Guetta & Piran 2007; Salvaterra & Chincarini 2007; Salvaterra et al. 2009b; Cao et al. 2011). However, the log N–log P distribution is the production of the intrinsic luminosity function convolved with the formation rate of GRBs, and so the luminosity function and formation rate of GRBs are degenerate (Firmani et al. 2004; Guetta & Piran 2007; Howell et al. 2014).

Moreover, there are several selection effects for the observed redshift distribution of GRBs (Coward 2007), and thus the rate of GRBs. The most important effect is the observational limit of the satellite. The Swift satellite has a flux limit, which means that it cannot detect GRBs dimmer than the flux limit. Therefore, the observed data is truncated and it will be difficult to obtain the intrinsic distribution of GRBs before the selection effect is corrected. Lynden-Bell (1971) applied a novel method to study the luminosity function and density evolution from a flux-limit quasar sample, referred to as Lynden-Bell's ${{c}^{-}}$ method (Lynden-Bell 1971). Since then, this method has been widely used in other objects, such as galaxies (Kirshner et al. 1978; Loh & Spillar 1986; Merighi et al. 1986; Peterson et al. 1986), long GRBs (Llyd-Ronning et al. 2002; Yonetoku et al. 2004; Kocevski & Liang 2006; Wu et al. 2012), and short GRBs (Yonetoku et al. 2014). The basis of Lynden-Bell's ${{c}^{-}}$ method is that the distributions of luminosity L and redshift z are independent (Lynden-Bell 1971; Efron & Petrosian 1992). So before applying this method, we need to test the independence of L and z with a nonparametric test method provided by Efron & Petrosian (1992). Lynden-Bell's ${{c}^{-}}$ method is a powerful method to estimate the luminosity function and formation rate of objects in a truncated sample. For example, Yonetoku et al. (2004) used this method to derive the luminosity function and the formation rate of GRBs from 689 BATSE GRBs with pseudo redshifts. They found that the GRB formation rate increases quickly at $0\lt z\lt 1$ and remains approximately constant up to $z\sim 10$, which is consistent with the observed SFR at $z\lt 4.0$. Kocevski & Liang (2006) found that the GRB comoving rate density rises steeply at $z\lt 1.0$, flattens, and then declines at about $z\gt 3.0$. Wu et al. (2012) also studied the formation rate of 95 Swift GRBs using Lynden-Bell's ${{c}^{-}}$ method, and found that the GRB formation rate increases quickly at $0\lt z\lt 1.0$, and remains approximately constant at $1.0\lt z\lt 4.0$, but finally decreases at $z\gt 4.0$, which is consistent with the SFR (Hopkins & Beacon 2006; Yüksel et al. 2008; Wang & Dai 2009; Kistler et al. 2009).

In this paper, we study the luminosity function and formation rate of the latest Swift GRBs using Lynden-Bell's ${{c}^{-}}$ method. Lynden-Bell's ${{c}^{-}}$ method can break the degeneracy between the luminosity function and GRB formation rate. This paper is organized as follows. We introduce the data from the Swift satellite and make the K-correction in Section 2. The introduction to Lynden-Bell's ${{c}^{-}}$ method and the nonparametric τ statistical method is given in Section 3. In Section 4, we derive the luminosity function and formation rate of our GRBs. A Monte Carlo simulation is used to test our results in Section 5. Finally, in Section 6, we present our conclusions and discussion. Throughout the paper, we assume a flat Λ cold dark matter universe with ${{{\Omega }}_{m}}=0.27$ and ${{H}_{0}}=70\;{\rm km}\;{{{\rm s}}^{-1}}\;{\rm Mp}{{{\rm c}}^{-1}}$.

Swift is a multi-wavelength satellite used to observe GRBs. It has instruments designed to analyze the bursts, and X-ray and UV/optical afterglows, which can locate the positions of GRBs. We collect 127 long GRBs with well-measured spectral parameters from Fermi-GBM and Konus-Wind. These GRBs have redshift data observed by Swift. In Table 1, we list the GRB sample, including name (Column 1), redshift (Column 2), low-energy power-law index α (Column 3), high-energy power-law index β (Column 4), peak energy of the $\nu {{F}_{\nu }}$ spectrum in the observer's frame (Column 5), peak flux in a certain energy range (Column 6), energy range (Column 7), bolometric luminosity (Column 8), and references (Column 9) of GRBs.

Table 1.  List of Long GRBs used in This Paper

GRB	z	α	$\beta$ b	${{{\rm E}}_{p}}\;({\rm keV})$	Flux $({\rm erg}\;{\rm c}{{{\rm m}}^{-2}}\;{{{\rm s}}^{-1}})$	Range (keV)	$L\;({\rm erg}\;{{{\rm s}}^{-1}})$	References
050318	1.44	$-1.34_{-0.32}^{+0.32}$	⋯	$63.52_{-11.07}^{+11.07}$	$(2.2\pm 0.17)\times {{10}^{-7}}$	15–150	$4.96_{-0.38}^{+0.38}\times {{10}^{51}}$	1
050401	2.9	$-0.83_{-0.21}^{+0.21}$	$-2.37_{-0.14}^{+0.14}$	$119_{-26}^{+26}$	$(2.45\pm 0.12)\times {{10}^{-6}}$	20–2000	$2.09_{-0.10}^{+0.10}\times {{10}^{53}}$	2
050416A	0.6535	−1.0	−3.4	$15.73_{-2.42}^{+2.42}$	$5\pm 0.5$ a	15–150	$9.89_{-0.99}^{+0.99}\times {{10}^{50}}$	1
050525	0.606	$-0.99_{-0.11}^{+0.11}$	⋯	$79.08_{-3.74}^{+3.74}$	$47.7\pm 1.2$ a	15–150	$9.00_{-0.23}^{+0.23}\times {{10}^{51}}$	1
050603	2.821	$-0.79_{-0.06}^{+0.06}$	$-2.15_{-0.09}^{+0.09}$	$349_{-28}^{+28}$	$(3.2\pm 0.2)\times {{10}^{-5}}$	20–3000	$2.25_{-0.14}^{+0.14}\times {{10}^{54}}$	3
050802	1.71	$-1.6_{-0.1}^{+0.1}$	⋯	$\gt 70.59$	$(2.21\pm 3.53)\times {{10}^{-7}}$	15–150	$\gt 9.34\times {{10}^{51}}$	1
050904	6.29	$-1.15_{-0.12}^{+0.12}$	⋯	$314_{-89}^{+173}$	$(1.84\pm 0.41)\times {{10}^{-7}}$	15–5000	$9.25_{-2.06}^{+2.06}\times {{10}^{52}}$	4
050922C	2.198	$-0.83_{-0.24}^{+0.24}$	⋯	$130.8_{-36.9}^{+36.9}$	$4.5_{-1.53}^{+0.72}\times {{10}^{-6}}$	20–2000	$1.95_{-0.30}^{+0.30}\times {{10}^{53}}$	1
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051111	1.55	$-0.98_{-0.24}^{+0.25}$	⋯	$179.70_{-54.52}^{+316.76}$	$3.41_{-0.52}^{+0.66}\times {{10}^{-7}}$	15–350	$7.04_{-1.07}^{+1.36}\times {{10}^{51}}$	5
060115	3.53	$-1.0_{-0.5}^{+0.5}$	⋯	$62_{-10}^{+31}$	$0.9\pm 0.1$ a	15–150	$1.04_{-0.12}^{+0.12}\times {{10}^{52}}$	7
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060206	4.048	$-1.12_{-0.3}^{+0.3}$	⋯	$81_{-22}^{+22}$	$(2.02\pm 0.13)\times {{10}^{-7}}$	15–150	$5.29_{-0.34}^{+0.34}\times {{10}^{52}}$	1
060210	3.91	$-1.12_{-0.26}^{+0.26}$	⋯	$117_{-23}^{+23}$	$2.8\pm 0.3$ a	15–150	$5.64_{-0.60}^{+0.60}\times {{10}^{52}}$	1
060306	3.5	$-1.2_{-0.5}^{+0.5}$	⋯	$70_{-18}^{+18}$	$(4.71\pm 0.278)\times {{10}^{-7}}$	15–150	$8.85_{-0.51}^{+0.51}\times {{10}^{52}}$	1
060428B	0.348	$-0.94_{-1.30}^{+1.30}$	⋯	$21.7_{-14}^{+14}$	$0.6\pm 0.1$ a	15–150	$2.15_{-0.36}^{+0.36}\times {{10}^{49}}$	9
060614	0.125	$-1.57_{-0.14}^{+0.12}$	⋯	$302_{-85}^{+214}$	$(4.5\pm 0.7)\times {{10}^{-6}}$	20–2000	$2.33_{-0.79}^{+0.37}\times {{10}^{50}}$	10
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060708	1.92	$-0.93_{-0.43}^{+0.47}$	⋯	$87.45_{-18.94}^{+83.35}$	$1.78_{-0.33}^{+0.45}\times {{10}^{-7}}$	15–350	$5.82_{-1.08}^{+1.47}\times {{10}^{51}}$	5
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070110	2.352	$-1.15_{-0.41}^{+0.45}$	⋯	$108.33_{-16.29}^{+183.02}$	$(5.168\pm 0.831)\times {{10}^{-6}}$	15–350	$2.95_{-0.59}^{+0.87}\times {{10}^{51}}$	5
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070508	0.82	$-0.81_{-0.07}^{+0.07}$	⋯	$188_{-8}^{+8}$	$8.3_{-1.11}^{+1.03}\times {{10}^{-6}}$	20–1000	$2.96_{-0.40}^{+0.37}\times {{10}^{52}}$	16
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071003	1.605	$-0.76_{-0.07}^{+0.06}$	⋯	$780_{-70}^{+81}$	$1.22_{-0.22}^{+0.19}\times {{10}^{-5}}$	20–4000	$2.18_{-0.39}^{+0.34}\times {{10}^{53}}$	20
071010B	0.947	$-1.25_{-0.49}^{+0.74}$	$-2.65_{-0.49}^{+0.29}$	$52_{-14}^{+10}$	$8.92_{-5.99}^{+2.99}\times {{10}^{-7}}$	20–1000	$6.47_{-4.34}^{+2.17}\times {{10}^{51}}$	21
071020	2.145	$-0.65_{-0.32}^{+0.27}$	⋯	$322_{-53}^{+80}$	$6.04_{-3.88}^{+1.22}\times {{10}^{-6}}$	20–2000	$2.25_{-1.44}^{+0.45}\times {{10}^{53}}$	22
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080413B	1.1	$-1.23_{-0.25}^{+0.25}$	⋯	$78_{-16}^{+16}$	$1.4\pm 0.2$ a	15–150	$1.55_{-0.06}^{+0.06}\times {{10}^{52}}$	1
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080605	1.6398	$-0.87_{-0.12}^{+0.13}$	$-2.58_{-0.84}^{+0.31}$	$297_{-40}^{+46}$	$(1.6\pm 0.33)\times {{10}^{-5}}$	20–2000	$3.33_{-0.69}^{+0.69}\times {{10}^{53}}$	31
080607	3.036	$-0.76_{-0.06}^{+0.07}$	$-2.57_{-0.26}^{+0.18}$	$348_{-27}^{+27}$	$(2.69\pm 0.54)\times {{10}^{-5}}$	20–4000	$2.21_{-0.44}^{+0.44}\times {{10}^{54}}$	32
080721	2.602	$-0.96_{-0.07}^{+0.07}$	$-2.42_{-0.29}^{+0.29}$	$497_{-62}^{+62}$	$(2.11\pm 0.35)\times {{10}^{-5}}$	20–7000	$1.11_{-0.18}^{+0.18}\times {{10}^{54}}$	1
080804	2.2	$-0.88_{-0.1}^{+0.1}$	⋯	$315.1_{-67.4}^{+67.4}$	$(7.3\pm 0.88)\times {{10}^{-7}}$	8–35000	$2.86_{-0.34}^{+0.34}\times {{10}^{52}}$	33
080810	3.35	$-1.2_{-0.1}^{+0.1}$	−2.5	$580_{-263}^{+850}$	$1.7_{-0.2}^{+0.1}\times {{10}^{-5}}$	15–1000	$2.39_{-0.28}^{+0.14}\times {{10}^{54}}$	34
080913	6.7	$-0.82_{-0.53}^{+0.75}$	−2.5	$121_{-39}^{+232}$	$(1.4\pm 0.058)\times {{10}^{-6}}$	15–150	$1.24_{-0.18}^{+0.18}\times {{10}^{53}}$	35,36
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081118	2.58	$-0.68_{-0.09}^{+0.09}$	⋯	$98.99_{-5.01}^{+5.01}$	$(6.73\pm 0.23)\times {{10}^{-7}}$	8–35000	$3.99_{-0.14}^{+0.14}\times {{10}^{52}}$	33
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090715B	3	$-1.1_{-0.37}^{+0.37}$	⋯	$134_{-41}^{+41}$	$(9\pm 2.5)\times {{10}^{-7}}$	20–2000	$8.78_{-2.44}^{+2.44}\times {{10}^{52}}$	1
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090812	2.452	$-1.03_{-0.07}^{+0.07}$	⋯	$586_{-192}^{+192}$	$2.77\pm 0.28$ a	100–1000	$1.02_{-0.10}^{+0.10}\times {{10}^{53}}$	1
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091018	0.971	$-1.53_{-0.48}^{+0.48}$	⋯	$28_{-13}^{+13}$	$(4.32\pm 0.95)\times {{10}^{-7}}$	20–1000	$4.90_{-1.08}^{+1.08}\times {{10}^{51}}$	1
091020	1.71	$-1.20_{-0.06}^{+0.06}$	$-2.29_{-0.18}^{+0.18}$	$187_{-25}^{+25}$	$(1.88\pm 0.026)\times {{10}^{-6}}$	8–35000	$3.44_{-0.04}^{+0.04}\times {{10}^{52}}$	1
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091029	2.752	$-1.46_{-0.27}^{+0.27}$	⋯	$61.4_{-17.5}^{+17.5}$	$1.8\pm 0.1$ a	15–150	$1.40_{-0.08}^{+0.08}\times {{10}^{52}}$	39
091127	0.49	$-0.68_{-0.05}^{+0.05}$	$-2.12_{-0.01}^{+0.01}$	$59.67_{-1.81}^{+1.81}$	$(9.379\pm 0.23)\times {{10}^{-6}}$	8–35000	$7.71_{-0.19}^{+0.19}\times {{10}^{51}}$	33
091208B	1.063	$-1.29_{-0.04}^{+0.04}$	⋯	$119_{-7}^{+7}$	$(2.56\pm 0.097)\times {{10}^{-6}}$	8–35000	$1.81_{-0.06}^{+0.06}\times {{10}^{52}}$	1
100425A	1.755	$-0.53_{-1.46}^{+2.83}$	⋯	$\lt 36.02$	$4.74_{-1.97}^{+3.46}\times {{10}^{-8}}$	15–350	$\lt 1.41\times {{10}^{51}}$	40
100615A	1.398	$-1.24_{-0.06}^{+0.08}$	$-2.27_{-0.12}^{+0.11}$	$85.73_{-9.33}^{+7.82}$	$8.3\pm 0.2$ a	8–1000	$1.06_{-0.03}^{+0.03}\times {{10}^{52}}$	41
100621A	0.542	$-1.70_{-0.13}^{+0.13}$	$-2.45_{-0.15}^{+0.15}$	$95_{-15}^{+15}$	$(1.7\pm 0.13)\times {{10}^{-6}}$	20–2000	$3.24_{-0.25}^{+0.25}\times {{10}^{51}}$	1
100728A	1.567	$-0.47_{-0.15}^{+0.15}$	$-2.5_{-0.3}^{+0.2}$	$390_{-25}^{+27}$	$(4.2\pm 0.7)\times {{10}^{-6}}$	20–10000	$6.45_{-1.08}^{+1.08}\times {{10}^{52}}$	42
100728B	2.106	$-0.90_{-0.07}^{+0.07}$	⋯	$130_{-9}^{+9}$	$(5.43\pm 0.35)\times {{10}^{-7}}$	8–35000	$1.97_{-0.13}^{+0.13}\times {{10}^{52}}$	1
100814A	1.44	$-0.55_{-0.3}^{+0.3}$	⋯	$147_{-10}^{+12}$	$(7.5\pm 2.5)\times {{10}^{-7}}$	20–2000	$1.08_{-0.36}^{+0.36}\times {{10}^{52}}$	43
100816A	0.8049	$-0.31_{-0.05}^{+0.05}$	$-2.77_{-0.17}^{+0.17}$	$136.7_{-4.73}^{+4.73}$	$15.59\pm 0.25$ a	10–1000	$7.38_{-0.12}^{+0.12}\times {{10}^{51}}$	44,45
100906A	1.727	$-1.1_{-0.1}^{+0.1}$	$-2.2_{-0.3}^{+0.2}$	$180_{-40}^{+45}$	$(2.7\pm 0.3)\times {{10}^{-6}}$	20–2000	$6.90_{-0.77}^{+0.77}\times {{10}^{52}}$	46
101213A	0.414	$-1.1_{-0.07}^{+0.07}$	$-2.35_{-0.72}^{+0.29}$	$309.7_{-40.0}^{+48.9}$	$4.67\pm 0.32$ a	10–1000	$6.32_{-0.43}^{+0.43}\times {{10}^{50}}$	47
101219B	0.55	$-0.33_{-0.36}^{+0.36}$	$-2.12_{-0.12}^{+0.12}$	$70_{-8}^{+8}$	$2.0\pm 0.2$ a	10–1000	$3.81_{-0.38}^{+0.38}\times {{10}^{50}}$	48
110205A	2.22	$-1.52_{-0.14}^{+0.14}$	⋯	$222_{-74}^{+74}$	$(5.1\pm 0.7)\times {{10}^{-7}}$	20–1200	$2.65_{-0.36}^{+0.36}\times {{10}^{52}}$	1
110213A	1.46	$-1.44_{-0.05}^{+0.05}$	⋯	$98.4_{-6.9}^{+8.5}$	$17.7\pm 0.5$ a	10–1000	$2.23_{-0.06}^{+0.06}\times {{10}^{52}}$	49
110422A	1.77	$-0.53_{-0.14}^{+0.17}$	$-2.65_{-0.62}^{+0.28}$	$246_{-34}^{+37}$	$(1.2\pm 0.15)\times {{10}^{-5}}$	20–2000	$2.90_{-0.36}^{+0.36}\times {{10}^{51}}$	50
110503A	1.613	$-0.98_{-0.08}^{+0.08}$	$-2.7_{-0.3}^{+0.3}$	$219_{-19}^{+19}$	$(10\pm 1)\times {{10}^{-6}}$	20–5000	$1.89_{-0.19}^{+0.19}\times {{10}^{53}}$	1
110715A	0.82	$-1.23_{-0.08}^{+0.09}$	$-2.7_{-0.5}^{+0.2}$	$120_{-11}^{+12}$	$(1.1\pm 0.1)\times {{10}^{-5}}$	20–10000	$4.31_{-0.39}^{+0.39}\times {{10}^{52}}$	51
110731A	2.83	$-0.8_{-0.03}^{+0.03}$	$-2.98_{-0.3}^{+0.3}$	$304_{-13}^{+13}$	$20.9\pm 0.5$ a	10–1000	$3.06_{-0.07}^{+0.07}\times {{10}^{53}}$	52
110801A	1.858	$-1.7_{-0.15}^{+0.12}$	−2.5	$140_{-50}^{+1270}$	$8.91_{-1.4}^{+1.68}\times {{10}^{-8}}$	15–350	$4.43_{-0.70}^{+0.84}\times {{10}^{51}}$	53
110808A	1.348	$-1.07_{-0.11}^{+0.12}$	⋯	$4238_{-1530}^{+3270}$	$(1.1\pm 0.2)\times {{10}^{-4}}$	20–10000	$8.96_{-1.63}^{+1.63}\times {{10}^{53}}$	54
110818A	3.36	$-1.33_{-0.08}^{+0.08}$	⋯	$256.3_{-55.3}^{+55.3}$	$5.0\pm 1.4$ a	10–1000	$7.23_{-2.03}^{+2.03}\times {{10}^{52}}$	55
111008A	4.9898	$-1.36_{-0.21}^{+0.24}$	⋯	$149_{-28}^{+52}$	$(1.4\pm 0.3)\times {{10}^{-6}}$	20–2000	$4.95_{-1.06}^{+1.06}\times {{10}^{53}}$	56
111107A	2.893	$-1.38_{-0.21}^{+0.21}$	⋯	$108_{-32}^{+32}$	$2.6\pm 0.3$ a	10–1000	$1.81_{-0.21}^{+0.21}\times {{10}^{52}}$	57
111123A	3.1516	$-1.30_{-0.24}^{+0.26}$	⋯	$107.79_{-25.03}^{+125.38}$	$7.89_{-0.89}^{+1.1}\times {{10}^{-8}}$	15–350	$9.80_{-1.11}^{+1.37}\times {{10}^{51}}$	40
111228A	0.714	$-1.9_{-0.1}^{+0.1}$	$-2.7_{-0.3}^{+0.3}$	$34_{-3}^{+3}$	$27\pm 1$ a	10–1000	$6.67_{-0.25}^{+0.25}\times {{10}^{51}}$	58
120119A	1.728	$-0.98_{-0.03}^{+0.03}$	$-2.36_{-0.09}^{+0.09}$	$189.2_{-8.3}^{+8.3}$	$16.86\pm 0.39$ a	10–1000	$5.98_{-0.14}^{+0.14}\times {{10}^{52}}$	59
120326A	1.798	$-0.98_{-0.14}^{+0.14}$	$-2.53_{-0.15}^{+0.15}$	$46.45_{-3.67}^{+3.67}$	$3.1\pm 0.05$ a	10–1000	$5.91_{-0.10}^{+0.10}\times {{10}^{51}}$	60
120327A	2.813	$-1.14_{-0.28}^{+0.26}$	⋯	$106.09_{-22.76}^{+80.1}$	$3.88_{-0.52}^{+0.65}\times {{10}^{-7}}$	15–350	$3.42_{-0.46}^{+0.57}\times {{10}^{52}}$	40
120712A	4.1745	$-0.6_{-0.2}^{+0.2}$	$-1.8_{-0.2}^{+0.2}$	$124_{-26}^{+26}$	$3.5\pm 0.2$ a	10–1000	$1.35_{-0.08}^{+0.08}\times {{10}^{53}}$	61
120714B	0.3984	$-0.29_{-0.8}^{+0.96}$	⋯	$60.8_{-10.22}^{+25.92}$	$1.8_{-0.68}^{+1.13}\times {{10}^{-8}}$	15–350	$1.15_{-0.44}^{+0.72}\times {{10}^{49}}$	40
120724A	1.48	$-0.75_{-1.23}^{+2.34}$	⋯	$\lt 31.9$	$1.95_{-0.87}^{+1.58}\times {{10}^{-8}}$	15–350	$\lt 4.19\times {{10}^{50}}$	40
120802A	3.796	$-0.96_{-0.53}^{+0.60}$	⋯	$52.96_{-6.84}^{+12.58}$	$1.85_{-0.27}^{+0.34}\times {{10}^{-7}}$	15–350	$3.51_{-0.51}^{+0.65}\times {{10}^{52}}$	40
120811C	2.671	$-1.19_{-0.30}^{+0.32}$	⋯	$46.26_{-4.14}^{+4.32}$	$2.47_{-0.22}^{+0.25}\times {{10}^{-7}}$	15–350	$2.23_{-0.20}^{+0.23}\times {{10}^{52}}$	40
120907A	0.97	$-0.75_{-0.25}^{+0.25}$	⋯	$154.5_{-32.9}^{+32.9}$	$4.3\pm 0.4$ a	10–1000	$2.53_{-0.24}^{+0.24}\times {{10}^{51}}$	62
120909A	3.93	$-1.3_{-0.1}^{+0.1}$	⋯	$370_{-140}^{+140}$	$3.0\pm 0.2$ a	10–1000	$7.65_{-0.51}^{+0.51}\times {{10}^{52}}$	63
120922A	3.1	$-1.6_{-0.7}^{+0.7}$	$-2.3_{-0.1}^{+0.1}$	$37.7_{-3.5}^{+3.5}$	$3.4\pm 0.3$ a	10–1000	$3.02_{-0.27}^{+0.27}\times {{10}^{52}}$	64
121027A	1.77	$-1.49_{-0.39}^{+0.43}$	⋯	$61.75_{-13.25}^{+437.65}$	$8.34_{-1.64}^{+2.24}\times {{10}^{-8}}$	15–350	$2.92_{-0.57}^{+0.79}\times {{10}^{53}}$	40
121128A	2.2	$-0.80_{-0.12}^{+0.12}$	$-2.41_{-0.1}^{+0.1}$	$62.2_{-4.6}^{+4.6}$	$17.9\pm 0.5$ a	10–1000	$6.67_{-0.19}^{+0.19}\times {{10}^{52}}$	65
121211A	1.023	$-0.3_{-0.34}^{+0.34}$	⋯	$95.96_{-12.6}^{+12.6}$	$2.402\pm 0.202$ a	10–1000	$1.25_{-0.11}^{+0.11}\times {{10}^{51}}$	66
130215A	0.597	$-1_{-0.2}^{+0.2}$	$-1.6_{-0.03}^{+0.03}$	$155_{-63}^{+63}$	$3.5\pm 0.3$ a	10–1000	$2.16_{-0.18}^{+0.18}\times {{10}^{51}}$	67
130408A	3.758	$-0.7_{-0.15}^{+0.15}$	$-2.3_{-0.3}^{+0.3}$	$272_{-40}^{+40}$	$(5.2\pm 0.5)\times {{10}^{-6}}$	20–10000	$6.12_{-0.59}^{+0.59}\times {{10}^{53}}$	68
130420A	1.297	$-1_{-0.13}^{+0.13}$	⋯	$56_{-3}^{+3}$	$5.2\pm 0.4$ a	10–1000	$3.65_{-0.28}^{+0.28}\times {{10}^{51}}$	69
130427A	0.3399	$-0.789_{-0.003}^{+0.003}$	$-3.06_{-0.02}^{+0.02}$	$830_{-5}^{+5}$	$1052\pm 2$ a	8–1000	$1.90_{-0.00}^{+0.00}\times {{10}^{53}}$	70
130505A	2.27	$-0.31_{-0.09}^{+0.09}$	$-2.26_{-0.07}^{+0.07}$	$604_{-49}^{+49}$	$(6.9\pm 0.3)\times {{10}^{-5}}$	20–1200	$3.98_{-0.17}^{+0.17}\times {{10}^{54}}$	71
130514A	3.6	$-1.44_{-0.15}^{+0.17}$	⋯	$110_{-21}^{+42}$	$2.06_{-0.18}^{+0.23}\times {{10}^{-7}}$	15–350	$3.82_{-0.33}^{+0.43}\times {{10}^{52}}$	72
130606A	5.913	$-1.14_{-0.15}^{+0.15}$	⋯	$294_{-50}^{+90}$	$3.15_{-0.46}^{+0.56}\times {{10}^{-7}}$	15–350	$2.04_{-0.30}^{+0.36}\times {{10}^{53}}$	73
130610A	2.092	$-1_{-0.1}^{+0.1}$	⋯	$294.9_{-42.9}^{+42.9}$	$4.5\pm 0.9$ a	10–1000	$2.55_{-0.51}^{+0.51}\times {{10}^{52}}$	74
130612A	2.006	$-1.3_{-0.3}^{+0.3}$	⋯	$61.9_{-10.5}^{+10.5}$	$4.1\pm 0.2$ a	10–1000	$9.48_{-0.46}^{+0.46}\times {{10}^{51}}$	75
130701A	1.155	$-1.1_{-0.1}^{+0.1}$	⋯	$89_{-4}^{+4}$	$(4.3\pm 0.4)\times {{10}^{-6}}$	20–10000	$4.27_{-0.40}^{+0.40}\times {{10}^{52}}$	76
130831A	0.4791	$-1.51_{-0.1}^{+0.1}$	$-2.8_{-0.1}^{+0.1}$	$67_{-4}^{+4}$	$(2.6\pm 0.3)\times {{10}^{-6}}$	20–10000	$3.42_{-0.39}^{+0.39}\times {{10}^{51}}$	77
130907A	1.238	$-0.65_{-0.03}^{+0.03}$	$-2.22_{-0.05}^{+0.05}$	$390_{-16}^{+16}$	$(2.2\pm 0.1)\times {{10}^{-5}}$	20–10000	$1.82_{-0.08}^{+0.08}\times {{10}^{53}}$	78
131030A	1.295	$-0.71_{-0.12}^{+0.12}$	$-2.95_{-0.28}^{+0.28}$	$177_{-10}^{+10}$	$(10\pm 1)\times {{10}^{-6}}$	20–10000	$1.08_{-0.11}^{+0.11}\times {{10}^{53}}$	79

Note. We give the name, redshift z, spectra parameters α and β, rest frame peak energy E_p, peak flux F, energy range, bolometric luminosity L in $1-{{10}^{4}}\;{\rm keV}$, and reference of the parameters of spectrum of each GRB.

^aFor these GRBs, the peak flux is in units of ${\rm photons}\;{\rm c}{{{\rm m}}^{-2}}\;{{{\rm s}}^{-1}}$. ^bFor those GRBs with β value, the spectra are well described by the Band model. However, for the GRBs without β value, the spectra are described by a power law with an exponential cutoff model.

References. (1) Nava et al. (2012), (2) Golenetskii et al. (2005a), (3) Golenetskii et al. (2005b), (4) Sugita et al. (2009), (5) Butler et al. (2007), (6) Golenetskii et al. (2005c), (7) Barbier et al. (2006), (8) Golenetskii et al. (2006a), (9) Sakamoto et al. (2006), (10) Golenetskii et al. (2006b), (11) Stamatikos et al. (2006), (12) Golenetskii et al. (2006c), (13) Golenetskii et al. (2006d), (14) Golenetskii et al. (2006e), (15) Golenetskii et al. (2006f), (16) Golenetskii et al. (2007a), (17) Ohno et al. (2007), (18) Barbier et al. (2007), (19) Butler et al. (2010), (20) Golenetskii et al. (2007b), (21) Golenetskii et al. (2007c), (22) Golenetskii et al. (2007d), (23) Golenetskii et al. (2007e), (24) Golenetskii et al. (2007f), (25) Stamatikos et al. (2008), (26) Golenetskii et al. (2008a), (27) Golenetskii et al. (2008b), (28) Tueller et al. (2008), (29) Ohno et al. (2008), (30) Golenetskii et al. (2008c), (31) Golenetskii et al. (2008d), (32) Golenetskii et al. (2008e), (33) Nava et al. (2011), (34) Sakamoto et al. (2008), (35) Pal'Shin et al. (2008), (36) Greiner et al. (2009), (37) McBreen (2009), (38) Golenetskii et al. (2009), (39) Barthelmy et al. (2009), (40) Butler's website⁴ and Swift official website,⁵ (41) Foley & Briggs (2010), (42) Golenetskii et al. (2010a), (43) Golenetskii et al. (2010b), (44) Fitzpatrick (2010a), (45) Fitzpatrick (2010b), (46) Golenetskii et al. (2010c), (47) Gruber (2010), (48) van der Horst (2010), (49) Foley (2011), (50) Golenetskii et al. (2011a), (51) Golenetskii et al. (2011b), (52) Golenetskii et al. (2011c), (53) Sakamoto et al. (2011), (54) Golenetskii et al. (2011d), (55) Xiong (2011), (56) Golenetskii et al. (2011e), (57) Pelassa (2011), (58) Briggs & Younes (2011), (59) Gruber (2012a), (60) Collazzi (2012), (61) Gruber (2012b), (62) Younes & Barthelmy (2012), (63) Chaplin (2012), (64) Younes (2012), (65) McGlynn (2012), (66) Yu (2012), (67) Younes & Bhat (2013), (68) Golenetskii et al. (2013a), (69) Xiong & Rau (2013), (70) von Kienlin (2013), (71) Golenetskii et al. (2013b), (72) Pal'Shin et al. (2013), (73) Barthelmy et al. (2013), (74) Fitzpatrick & Pelassa (2013), (75) Fitzpatrick (2013), (76) Golenetskii et al. (2013c), (77) Golenetskii et al. (2013d), (78) Golenetskii et al. (2013e), (79) Golenetskii et al. (2013f).

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Two spectral models are used to fit the spectra of GRBs, including a power law with an exponential cutoff model (PLEXP) and the Band model (Band et al. 1993). The functional forms are as follow:

$$\begin{eqnarray}&&f(E)=A{{(\frac{E}{100\;{\rm keV}})}^{\alpha }}{{{\rm exp} }^{-\frac{(2+\alpha )E}{{{E}_{{\rm p}}}}}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&f(E)=\left\{ \begin{array}{ccccccccccccccc} \begin{array}{ccccccccccccccc} A{{(\frac{E}{100\;{\rm keV}})}^{\alpha }} \\ \times \;{\rm exp} -\frac{(2+\alpha )E}{{{E}_{{\rm p}}}} \\ \end{array} & E\lt \frac{(\alpha -\beta ){{E}_{{\rm p}}}}{2+\alpha }, \\ \begin{array}{ccccccccccccccc} A{{(\frac{E}{100\;{\rm keV}})}^{\beta }} \\ \times \;{{{\rm exp} }^{\beta -\alpha }}{{(\frac{(\alpha -\beta ){{E}_{{\rm p}}}}{(2+\alpha )100\;{\rm keV}})}^{\alpha -\beta }} \\ \end{array} & E\geqslant \frac{(\alpha -\beta ){{E}_{{\rm p}}}}{2+\alpha }, \\ \end{array} \right.\end{eqnarray} \tag{ 2 }$$

which represent a power law with an exponential cutoff model and the Band model, respectively.

Because the peak fluxes are observed in a lager range of redshifts which correspond to different range of energy bands in the rest frame of GRBs. The K-correction is required to obtain the bolometric luminosity of GRBs (Bloom et al. 2001). The bolometric luminosity of GRB is

$$\begin{eqnarray}&&L=4\pi d_{L}^{2}(z)FK,\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{{d}_{L}}(z)=\frac{c}{{{H}_{0}}}\int _{0}^{z}\frac{dz}{\sqrt{1-{{{\Omega }}_{m}}+{{{\Omega }}_{m}}{{(1+z)}^{3}}}}\end{eqnarray} \tag{ 4 }$$

is the luminosity distance at redshift z, F is the peak flux observed between between a certain energy range $({{E}_{{\rm min} }},{{E}_{{\rm max} }})$, and the K is the factor of the K-correction. If the flux F is in units of ${\rm erg}\;{\rm c}{{{\rm m}}^{-2}}\;{{{\rm s}}^{-1}}$, the parameter K is defined as

$$\begin{eqnarray}&&K=\frac{\int _{1\,{\rm keV}/(1+z)}^{{{10}^{4}}\,{\rm keV}/(1+z)}Ef(E)dE}{\int _{{{E}_{{\rm min} }}}^{{{E}_{{\rm max} }}}Ef(E)dE}.\end{eqnarray} \tag{ 5 }$$

If the flux F is in units of ${\rm photons}\;{\rm c}{{{\rm m}}^{-2}}\;{{{\rm s}}^{-1}}$, then the parameter K is defined as

$$\begin{eqnarray}&&K=\frac{\int _{1\,{\rm keV}/(1+z)}^{{{10}^{4}}\,{\rm keV}/(1+z)}Ef(E)dE}{\int _{{{E}_{{\rm min} }}}^{{{E}_{{\rm max} }}}f(E)dE},\end{eqnarray} \tag{ 6 }$$

where f(E) is the spectral model of GRBs. Then, we can obtain the bolometric luminosity L for each GRB. In Figure 1, the blue dots show the bolometric luminosity of GRBs, and the line is the observational limit of Swift. The limit is chosen as a minimum flux ${{F}_{{\rm min} }}=2.0\times {{10}^{-8}}\;{\rm erg}\;{{{\rm s}}^{-1}}\;{\rm c}{{{\rm m}}^{-2}}$, which is consistent with that of Li (2008). Therefore, the luminosity limit at redshift z is given as ${{L}_{{\rm limit}}}=4\pi d_{{\rm L}}^{2}(z){{F}_{{\rm min} }}$.

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Lynden-Bell's ${{c}^{-}}$ method is an efficient method to determinate the distribution of the luminosity and redshift of objects with a truncated data sample, including quasars (Lynden-Bell 1971; Efron & Petrosian 1992; Petrosian 1993; Maloney & Petrosian 1999), galaxies (Kirshner et al. 1978; Loh & Spillar 1986; Merighi et al. 1986; Peterson et al. 1986), and GRBs (Llyd-Ronning et al. 2002; Yonetoku et al. 2004, 2014; Wu et al. 2012). Lynden-Bell (1971) used this method to derive the luminosity function and density evolution from quasars with observational selection for the first time. This method can break the degeneracy between the luminosity function and formation rate. It is better to extract the luminosity evolution from the form of the luminosity function. If the parameters L and z in the distribution of luminosity and redshift ${\Psi }(L,z)$ are independent, then we can rewrite ${\Psi }(L,z)$ as ${\Psi }(L,z)=\psi (L)\phi (z)$ (Efron & Petrosian 1992), where $\psi (L)$, the fraction of GRBs brighter than L, is the cumulative luminosity function, and $\phi (z)$ is the redshift cumulative distribution. Unfortunately, the luminosity and redshift of GRBs are not independent (Yonetoku et al. 2004; Kocevski & Liang 2006; Wu et al. 2012), and so we should write ${\Psi }(L,z)$ as ${\Psi }(L,z)={{\psi }_{z}}(L)\phi (z)$ instead of ${\Psi }(L,z)=\psi (L)\phi (z)$, where ${{\psi }_{z}}(L)$ is the luminosity function of the GRB at redshift z. If we remove the effect of the luminosity evolution g(z), i.e., a transformation ${{L}_{0}}=L/g(z)$, then the transformed luminosity L₀ is independent of redshift. As a result, we can obtain ${\Psi }({{L}_{0}},z)=\phi (z)\psi ({{L}_{0}})$. Using the relation $L={{L}_{0}}g(z)$, we can write ${\Psi }(L,z)$ as ${\Psi }(L,z)={{\psi }_{z}}(L)\phi (z)=\psi ({{L}_{0}})\phi (z)$.Then, the luminosity function of a GRB at redshift z is ${{\psi }_{z}}(L)=\psi (L/g(z))$.

To obtain the form of the luminosity evolution g(z), we introduce a nonparametric test method proposed by Efron & Petrosian (1992) which has been widely used in previous studies (Petrosian 1993; Maloney & Petrosian 1999; Llyd-Ronning et al. 2002; Yonetoku et al. 2004, 2014; Wu et al. 2012). For the ith data in the ($L,z$) data set, we can define J_i as (Efron & Petrosian 1992)

$$\begin{eqnarray}&&{{J}_{i}}=\{j|{{L}_{j}}\geqslant {{L}_{i}},{{z}_{j}}\leqslant z_{i}^{{\rm max} }\},\end{eqnarray} \tag{ 7 }$$

where L_i is the ith GBR luminosity and $z_{i}^{{\rm max} }$ is the maximum redshift at which a GRB with luminosity L_i can be observed. This region is shown in Figure 1 as a black rectangle. The number of GRBs contained in this region is n_i. The number ${{N}_{i}}={{n}_{i}}-1$, which means taking the ith GRB out, is the same as ${{c}^{-}}$ in Lynden-Bell (1971). Similarly, $J_{i}^{\prime }$ is defined as

$$\begin{eqnarray}&&J_{i}^{\prime }=\{j|{{L}_{j}}\geqslant L_{i}^{{\rm lim} },{{z}_{j}}\lt {{z}_{i}}\},\end{eqnarray} \tag{ 8 }$$

where $L_{i}^{{\rm lim} }$ is the minimum observable luminosity at redshift z_i. This region is shown as the red rectangle in Figure 1. The number of events contained within this region is M_i.

We first consider the n_i GRBs in the black rectangle in Figure 1. The number of events that have redshift z less than or equal to z_i is defined as R_i. If L and z are independent, then R_i is uniformly distributed between 1 and n_i (Efron & Petrosian 1992). The test statistic τ is (Efron & Petrosian 1992)

$$\begin{eqnarray}&&\tau \equiv \mathop{\sum }\limits_{i}\frac{({{R}_{i}}-{{E}_{i}})}{\sqrt{{{V}_{i}}}},\end{eqnarray} \tag{ 9 }$$

where ${{E}_{i}}=\frac{1+{{n}_{i}}}{2}$, ${{V}_{i}}=\frac{n_{i}^{2}-1}{12}$ are the expected mean and the variance of R_i, respectively. If R_i is exactly uniformly distributed between 1 and n_i, then the samples of ${{R}_{i}}\leqslant {{E}_{i}}$ and ${{R}_{i}}\geqslant {{E}_{i}}$ should be nearly equal, and the test statistic τ will be nearly 0. If we choose a form of g(z) that makes test statistic $\tau =0$, then the effect of luminosity evolution can be removed by the transformation of ${{L}_{0}}=L/g(z)$.

The functional form of $g(z)={{(1+z)}^{k}}$ has been used in previous papers (Llyd-Ronning et al. 2002; Yonetoku et al. 2004, 2014; Kocevski & Liang 2006; Wu et al. 2012). We also use this form in this paper. Then, we test the independence of ${{L}_{0}}=L/g(z)$ and z by changing the value of k until the test statistic τ is zero. Figure 2 shows the value of the test statistic τ as a function of k. From this figure, we find the best fit is $k=2.43_{-0.38}^{+0.41}$ at a $1\sigma$ confidence level. Therefore, we take the luminosity evolution form of g(z) as $g(z)={{(1+z)}^{2.43}}$. A hypothesis of no luminosity evolution, k = 0, is rejected at about the $4.7\sigma$ confidence level. This value is similar to that from Yonetoku et al. (2004), whose k-value is $k=2.60_{-0.20}^{+0.15}$ and k = 0 is rejected with about $8.0\sigma$ significance from pseudo-redshift GRBs. Wu et al. (2012) also found that the value of k is $k=2.30_{-0.51}^{+0.56}$.

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After converting the observed luminosity to the non-evolving luminosity ${{L}_{0}}=L/{{(1+z)}^{2.43}}$, we can derive the local cumulative luminosity function $\psi ({{L}_{0}})$ with a nonparametric method from the following Equation (Lynden-Bell 1971; Efron & Petrosian 1992):

$$\begin{eqnarray}&&\psi ({{L}_{0i}})=\mathop{\prod }\limits_{j\lt i}(1+\frac{1}{{{N}_{j}}}),\end{eqnarray} \tag{ 10 }$$

where $j\lt i$ means that the GRB has a luminosity L_0j larger than L_0i. The cumulative number distribution $\phi (z)$ can be obtained from

$$\begin{eqnarray}&&\phi ({{z}_{i}})=\mathop{\prod }\limits_{j\lt i}(1+\frac{1}{{{M}_{j}}}),\end{eqnarray} \tag{ 11 }$$

where $j\lt i$ means that the GRB has redshift z_j less than z_i. The comoving differential form of $\phi (z)$, which represents the cosmic formation rate of GRBs $\rho (z)$, is more interesting. The formation rate of GRBs can be written as

$$\begin{eqnarray}&&\rho (z)=\frac{d\phi (z)}{dz}(1+z){{(\frac{dV(z)}{dz})}^{-1}},\end{eqnarray} \tag{ 12 }$$

where $(1+z)$ results from the cosmological time dilation and $dV(z)/dz$ is the differential comoving volume, which can be expressed as

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} \frac{dV(z)}{dz} & = & 4\pi {{(\frac{c}{{{H}_{0}}})}^{3}}{{(\int _{0}^{z}\frac{dz}{\sqrt{1-{{{\Omega }}_{{\rm m}}}+{{{\Omega }}_{{\rm m}}}{{(1+z)}^{3}}}})}^{2}} \\ {} & {} & \times \frac{1}{\sqrt{1-{{{\Omega }}_{{\rm m}}}+{{{\Omega }}_{{\rm m}}}{{(1+z)}^{3}}}}. \\ \end{array}\end{eqnarray} \tag{ 13 }$$

In this section, we present results for the luminosity function and cosmic formation rate of GRBs.

4.1. Luminosity Function

As discussed above, we obtain the form of the luminosity evolution as $g(z)={{(1+z)}^{2.43}}$ using the nonparametric τ test method. The non-evolving luminosity L₀ is defined as ${{L}_{0}}=L/g(z)$, which is shown in Figure 3. Using this new data set, the luminosity function $\psi ({{L}_{0}})$ can be derived by using Lynden Bell's ${{c}^{-}}$ method, which is shown in Figure 4. As shown in Figure 4, the luminosity function $\psi ({{L}_{0}})$ can be fitted with a broken power law after removing the redshift evolution. The forms of the luminosity function $\psi ({{L}_{0}})$ for dimmer and brighter bursts are fitted by

$$\begin{eqnarray}\psi ({{L}_{0}})\propto \left\{ \begin{array}{ccccccccccccccc} L_{0}^{-0.14\pm 0.02} & {{L}_{0}}\lt L_{0}^{b}, \\ L_{0}^{-0.70\pm 0.03} & {{L}_{0}}\gt L_{0}^{b}, \\ \end{array} \right.\end{eqnarray} \tag{ 14 }$$

where $L_{0}^{b}=1.43\times {{10}^{51}}\;{\rm erg}\;{{{\rm s}}^{-1}}$ is the break point. This result is consistent with those of previous works (Llyd-Ronning et al. 2002; Yonetoku et al. 2004; Kocevski & Liang 2006; Wu et al. 2012). It is necessary to point out that this luminosity function only represents the present distribution at z = 0 since the luminosity evolution is removed. The luminosity function ${{\psi }_{z}}(L)$ at redshift z will be ${{\psi }_{z}}(L)=\psi (L/g(z))=\psi (L/{{(1+z)}^{2.43}})$. So the break luminosity at z is $L_{z}^{b}=L_{0}^{b}{{(1+z)}^{2.43}}$.

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4.2. Formation Rate of GRBs

Figure 5 presents the cumulative GRB formation rate $\phi (z)$. According to Equation (12), in order to obtain the cosmic formation rate of GRBs, we need the differential form of the cumulative number distribution $d\phi (z)/dz$. Figure 6 shows $(1+z)d\phi (z)/dz$ as a function of redshift z. From this figure, we find that $(1+z)d\phi (z)/dz$ increases quickly at $z\lt 1$, remains approximately constant for $1\lt z\lt 4$, and then decreases sharply at $z\gt 4$ with a power-law form. However, we are more interested in the comoving density rate. From Equation (12), we can calculate the GRB formation rate $\rho (z)$, which is shown in Figure 7. In Figure 7, the blue stepwise line represents the comoving cosmic formation of GRBs as a function of redshift, and the error bar gives a $1\sigma$ confidence level. The best-fitting power laws for the different segments are

$$\begin{eqnarray}\rho (z)\propto \left\{ \begin{array}{ccccccccccccccc} {{(1+z)}^{0.04\pm 0.94}} & z\lt 1, \\ {{(1+z)}^{-0.94\pm 0.11}} & 1\lt z\lt 4, \\ {{(1+z)}^{-4.36\pm 0.48}} & z\gt 4, \\ \end{array} \right.\end{eqnarray} \tag{ 15 }$$

with a 95% confidence level. From this equation, we can derive the formation rate of GRBs in the local universe, $\rho (0)\;=7.3\pm 2.7\;{\rm Gp}{{{\rm c}}^{-3}}\;{\rm y}{{{\rm r}}^{-1}}$, which is larger than previous studies, e.g., $\rho (0)\sim 1.5\;{\rm Gp}{{{\rm c}}^{-3}}\;{\rm y}{{{\rm r}}^{-1}}$ (Schmidt 1999), $\rho (0)\sim 0.5\;{\rm Gp}{{{\rm c}}^{-3}}\;{\rm y}{{{\rm r}}^{-1}}$ (Guetta et al. 2005), and $\rho (0)\gt 0.5\;{\rm Gp}{{{\rm c}}^{-3}}\;{\rm y}{{{\rm r}}^{-1}}$ (Pélangeon et al. 2008). The main reason for this is that the GRB rate remains constant at low redshift in this paper, while it increases quickly in other studies. However, Liang et al. (2007) found that the rate of low-luminosity GRBs is $\rho (0)\sim 325\;{\rm Gp}{{{\rm c}}^{-3}}\;{\rm y}{{{\rm r}}^{-1}}$. This local rate is not corrected for the jet beaming effect.

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Obviously, the formation rate of GRBs is in contrast to previous estimates of the comoving rate density by Yonetoku et al. (2004), Kocevski & Liang (2006), and Wu et al. (2012) using the same method. Our results show that the formation rate of GRBs is almost constant at $z\lt 1.0$. However, previous results show that the formation rate increases quickly at $z\lt 1.0$ (Yonetoku et al. 2004; Kocevski & Liang 2006; Wu et al. 2012), which is consistent with SFR observations (Hopkins & Beacon 2006). However, our result is well consistent with that of Petrosian et al. (2009). Interestingly, the evolution of $(1+z)d\phi (z)/dz$ shown in Figure 6 is consistent with the behavior of $\rho (z)$ in Wu et al. (2012). We also test our program with the same GRB data from Yonetoku et al. (2004) and Wu et al. (2012), and find that our $(1+z)d\phi (z)/dz$ shows the same behavior as the $\rho (z)$ in Yonetoku et al. (2004) and Wu et al. (2012). Therefore, they might omit the $dV(z)/dz$ term in their calculations. Also, Yonetoku et al. (2004) and Wu et al. (2012) showed that the GRB rate increases as ${{(1+z)}^{6.0\pm 1.4}}$ and ${{(1+z)}^{8.24\pm 4.48}}$ at $z\lt 1$, respectively, which are much quicker than ${{(1+z)}^{2.4}}$ from Kocevski & Liang (2006).

Several comprehensive works studying the luminosity function and the rate of long GRBs have recently been published using different methods (such as Wanderman & Piran 2010; Butler et al. 2010). Assuming that the luminosity function is redshift independent, Wanderman & Piran (2010) found that the power-law index of the luminosity function is 0.22 at low luminosities, and 1.4 at high luminosities with a break of ${{10}^{52.5}}$ erg s⁻¹ using long GRBs with redshifts determined from the afterglow. The formation rate increases as ${{(1+z)}^{2.1}}$ up to $z\sim 3.0$ and decreases as ${{(1+z)}^{-1.4}}$ at $z\gt 3.0$. Butler et al. (2010) determined that the luminosity function is nearly flat $\propto {{L}^{-0.2}}$ below break ${{10}^{52.7}}$ erg s⁻¹, and declines $\propto {{L}^{-3.0}}$ using a large sample of GRBs detected by Swift. The GRB rate is similar to that of Wanderman & Piran (2010). These results are different from our results. One reason for this is the GRB sample. We use the latest GRB sample, which has redshifts observed by Swift and spectral parameters given by Fermi-GBM and Konus-Wind. The luminosity function evolution may be the most important reason. Wanderman & Piran (2010) assumed no redshift evolution for the luminosity function. In the fitting of Butler et al. (2010), no luminosity evolution is required to produce the observed number of GRBs. However, strong evolution of the luminosity function is found in the literature. The evolution of the luminosity can be parameterized as ${{(1+z)}^{1.4}}$ (Llyd-Ronning et al. 2002). Yonetoku et al. (2004) and Wu et al. (2012) found that the evolution factor is $g(z)={{(1+z)}^{2.60}}$ and $g(z)={{(1+z)}^{2.30}}$, respectively. Tan et al. (2013) found that the luminosity function of GRBs evolves with a redshift-dependent break luminosity ${{L}_{b}}=1.2\times {{10}^{51}}{{(1+z)}^{2}}\;\;{\rm erg}\;{{s}^{-1}}$, which is similar to our result. Virgili et al. (2011) found that an evolution factor ${{(1+z)}^{1.0\pm 0.2}}$ of the luminosity function can fit the BATSE and Swift data. These works suggest that it is necessary to take into account an evolution factor. Our GRB sample including those GRBs dimmer than ${{10}^{51}}\;\;{\rm erg}\;{{{\rm s}}^{-1}}$ is another important reason. For example, Kistler et al. (2008) found that the density of GRBs is much higher at $z\lt 1$ if they included GRBs dimmer than ${{10}^{51}}\;\;{\rm erg}\;{{{\rm s}}^{-1}}$. In this work, we use the Lynden-Bell ${{c}^{-}}$ method to correct the data truncated effect and consider the evolution of the GRB luminosity function. Therefore, we do not need to omit these dim GRBs.

The relation between SFR and formation rate of GRBs is attractive. We also compare our result with the observed SFR from Hopkins & Beacon (2006) in Figure 8. Obviously, it is consistent with the observed SFR at $z\gt 1.0$, but remarkably different at $z\lt 1.0$. This trend means that the formation rate of GRBs $\rho (z)$ does not trace SFR at low redshift $z\lt 1.0$. But at high redshift $z\gt 4.0$, our result is consistent with the SFR derived from GRBs (Yüksel et al. 2008; Wang & Dai 2009; Kistler et al. 2009; Wang 2013). This result is different with others in previous literatures (Llyd-Ronning et al. 2002; Yonetoku et al. 2004; Kocevski & Liang 2006; Wu et al. 2012), but it is consistent with that of Petrosian et al. (2009). There are also some previous works shown that the long GRBs may not unbiased tracers of SFR at low redshift. A strong dependence of the GRB rate on host-galaxy properties out to $z\sim 1.0$ is found by Perley et al. (2013). So use GRBs as direct tracers of the cosmic SFR is cautious at $z\lt 1.0$ (Perley et al. 2013). Vergani et al. (2014) found that the mass distribution of long GRB host galaxy is different with the expected from star-forming galaxies observed in deep survey, which suggests that long GRBs are not unbiased tracers of star formation activity at least at $z\lt 1.0$. They also found that long GRB rate can directly trace the SFR starting from $z\sim 4$ and above.

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In this section, we use Monte Carlo simulation to test our results. First, we simulate a set of data $({{L}_{0}},z)$ which follows the distribution described by Equations (14) and (15) using Monte Carlo method. Then, we transfer the luminosity L₀ to L through $L={{L}_{0}}{{(1+z)}^{k}}$, where k = 2.43. So we can get sets of pseudo data of GRB luminosity and redshift (L,z). In the simulations, we create 200 pseudo samples. Each sample contains 130 GRBs. Then we use Lynden-Bell ${{c}^{-}}$ method and nonparametric τ test method to calculate the distributions of these pseudo samples. Finally, we compare the simulated data with observed data.

Figure 9 shows the comparing results. The four panels give the luminosity–redshift distribution, luminosity function, cumulative distribution and log N–log S distribution. In panel (a), we randomly choose one pseudo sample of GRB from the 200 samples to compare with the observed data. The red dots and the blue dots represent the observed data and the simulated data, respectively. From this panel, we can see that the simulated data and the observed data have similar distributions. The other three panels (b)–(d) show the comparisons of the luminosity function, cumulative distribution and log N–log S distribution between the observed data and mock data. The red curves show the distributions of the observed data, blue curves give the distributions of all of the 200 pseudo samples of GRB data and the green curves are the mean distributions of the 200 pseudo samples. We perform the Kolmogorov–Smirnov test between observed data and the mean distributions of simulated data. The chance probabilities of the three tests are 0.49, 0.86 and 0.96, respectively. From these panels, we can also conclude that the distribution of the observed data lie in the region of those pseudo data, which means that the derived luminosity function and formation rate of GRBs are correct.

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In order to test whether long GRBs are unbiased tracers of SFR at low redshift, we simulate 200 new pseudo samples of GRBs by assuming that the GRB rate follows the SFR from Yüksel et al. (2008), i.e., $\rho (z)\propto {{(1+z)}^{3.4}}$ at $z\lt 1$, $\rho (z)\propto {{(1+z)}^{-0.3}}$ at $1\lt z\lt 4$, and $\rho (z)\propto {{(1+z)}^{-3.5}}$ at $z\gt 4$. Then, we use the same method to calculate the distributions of these pseudo data. We find that the cumulative redshift distribution of observed data is not consistent with the pseudo data, which is shown in Figure 10. The red, blue, and green curves have the same meanings as those in Figure 9. From Figure 10, we can see that part of the cumulative redshift distribution line of the observed data lies outside of the region occupied by pseudo GRB data, especially at $z\lt 1.0$. The Kolmogorov–Smirnov test between the distribution of observed data and the mean distribution of simulated data gives the chance probability of $p=6.9\times {{10}^{-12}}$. This means that long GRBs are not direct tracers of SFR at $z\lt 1.0$.

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In this paper, we use Lynden-Bell's ${{c}^{-}}$ method to study the luminosity function and formation rate of Swift long GRBs without any assumptions. First, we use a τ statistical method to separate the luminosity evolution from the stable form of the luminosity function by choosing the evolution form $g(z)={{(1+z)}^{k}}$. The most proper k is $k=2.43_{-0.38}^{+0.41}$, which gives $\tau =0$. This value is similar to those of Yonetoku et al. (2004), Wu et al. (2012), and Kocevski & Liang (2006). After correcting the luminosity evolution by ${{L}_{0}}=L/{{(1+z)}^{2.43}}$, the cumulative luminosity function $\psi ({{L}_{0}})$ and cumulative number distribution $\phi (z)$ of GRBs can be calculated, see Figures 4 and 5. The luminosity function of GRBs can be well fit with a broken power-law form as $\psi ({{L}_{0}})\propto L_{0}^{-0.14\pm 0.02}$ and $\psi ({{L}_{0}})\propto L_{0}^{-0.70\pm 0.03}$ for ${{L}_{0}}\lt L_{0}^{{\rm b}}$ and ${{L}_{0}}\gt L_{0}^{{\rm b}}$, respectively, where $L_{0}^{{\rm b}}=1.43\times {{10}^{51}}\;{\rm erg}\;{{{\rm s}}^{-1}}$ is the break point.

We also derive the formation rate of GRBs through the differential form of the cumulative number distribution $\phi (z)$. Figure 6 shows the evolution of $(1+z)\frac{d\phi (z)}{dz}$. We find that $(1+z)\frac{d\phi (z)}{dz}$ increases quickly at $z\lt 1$, then remains roughly constant at $1\lt z\lt 4$ and finally decreases rapidly at high redshift. From Equation (12), the cosmic formation rate of GRBs $\rho (z)$ is derived, which is shown in Figure 7. The best-fitting power laws for different redshift segments are $\rho (z)\propto {{(1+z)}^{0.04}}$, $\rho (z)\propto {{(1+z)}^{-0.94}}$, and $\rho (z)\propto {{(1+z)}^{-4.36}}$ for $z\lt 1.0$, $1.0\lt z\lt 4.0$, and $z\gt 4.0$, respectively. Our results show that the formation rates of GRBs are almost constant at $z\lt 1.0$. However, previous results show that the formation rate increases quickly at $z\lt 1.0$ (Yonetoku et al. 2004; Kocevski & Liang 2006; Wu et al. 2012). But Yonetoku et al. (2004) and Kocevski & Liang (2006) used the pseudo redsihfts of GRBs rather than the observed redshifts. Also, we find that the $\rho (z)$ in Wu et al. (2012) and Yonetoku et al. (2004) increase fast at $z\lt 1.0$, which has a similar behavior as $(1+z)\frac{d\phi (z)}{dz}$ shown in Figure 6. Therefore, they might omit the $dV(z)/dz$ term in their calculations. From Figure 8, it is easy to find that the GRB formation rate $\rho (z)$ is consistent with the observed SFR at $z\gt 1.0$ but entirely different at $z\lt 1.0$. This means that the formation rate of GRBs only traces SFR at $z\gt 1.0$, which is different from previous work (Yonetoku et al. 2004; Kocevski & Liang 2006; Wu et al. 2012). We find the low-redshift excess of the GRB rate for the first time.

Surprisingly, we find that the formation rate of GRBs is consistent with the SFR at $z\gt 1.0$, but shows an excess at low redshift $z\lt 1.0$ for the first time, which is different from previous works. Our results show that the formation rate of GRBs is larger than the SFR at $z\lt 1.0$. Below, we will discuss some possible reasons for this low-redshift excess.

The first reason is that the definition of long GRBs is not clear. In the classical method, long GRBs are defined by ${{T}_{90}}\gt 2$ s (Kouveliotou et al. 1993). There is no clear boundary line in this diagram to separate the long and short GRBs. Moreover, T₉₀ is an observed timescale that represents different times for GRBs at different redshifts. Meanwhile, observations of low-redshift long GRBs, such as GRB 060614 at z = 0.125 and GRB 060505 at z = 0.089, show no association with SNe (Gal-Yam et al. 2006; Gehrels et al. 2006; Fynbo et al. 2006). Therefore, more physical criteria are required to classify GRBs. Some attempts have been performed (Zhang 2006; Zhang et al. 2007, 2009; Bloom et al. 2008; Bromberg et al. 2013; Lü et al. 2014). It has been suggested that GRBs can be classified physically into Type I (compact star origin) and Type II (massive star origin; Zhang 2006; Zhang et al. 2007).

The second reason is that some selection effects have not been included in this analysis. For example, it is easier to measure the redshifts of those GRBs which are at lower redshifts, and therefore create a bias toward low-redshift GRBs. This means that we lose some high-redshift GRBs, and so the formation rate of GRBs at low redshift we calculated will be larger than the SFR. This bias can be removed by using samples with high completeness in the GRB redshift measurements. There are three such samples in the literature (Greiner et al. 2011; Hjorth et al. 2012; Salvaterra et al. 2012). This should be considered in future work.

The third reason is that there may exist a subclass of GRBs, i.e., low-luminosity GRBs (Cobb et al. 2006; Pian et al. 2006; Soderberg et al. 2006; Liang et al. 2007). The local rate of low-luminosity GRBs may be high, i.e., $\rho (0)=100-1000\;{\rm y}{{{\rm r}}^{-1}}\;{\rm Gp}{{{\rm c}}^{-3}}$ (Soderberg et al. 2006; Liang et al. 2007), much higher than high-luminosity GRBs. The progenitors of low-luminosity GRBs may be different from those of high-luminosity GRBs (Mazzali et al. 2006; Soderberg et al. 2006). Contamination from low-luminosity GRBs could lead to the low-redshift excess.

We thank the referee for helpful comments on the manuscript. We acknowledge the use of public data from the Swift data archive. We thank Bing Zhang, Yun-Wei Yu, Shi-Wei Wu, and Jin-Jun Geng for helpful discussions. This work is supported by the National Basic Research Program of China (973 Program, grant No. 2014CB845800), the National Natural Science Foundation of China (grants 11422325, 11373022, 11033002, and J1210039), the Excellent Youth Foundation of Jiangsu Province (BK20140016), and the Program for New Century Excellent Talents in University (grant No. NCET-13-0279). K.S.C. is supported by the CRF grants of the Government of the Hong Kong SAR under HUKST4/CRF/13G.