High-redshift star formation rate up to z~8.3 derived from gamma-ray bursts and influence of background cosmology
aa r X i v : . [ a s t r o - ph . C O ] J a n Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 17 November 2018 (MN L A TEX style file v2.2)
High-redshift star formation rate up to z ∼ . derived fromgamma-ray bursts and influence of background cosmology F. Y. Wang ⋆ and Z. G. Dai † Department of Astronomy, Nanjing University, Nanjing 210093, P. R. China
17 November 2018
ABSTRACT
The high-redshift star formation rate (SFR) is difficult to measure directly even bymodern approaches. Long-duration gamma-ray bursts (GRBs) can be detected to theedge of the visible universe because of their high luminorsities. The collapsar modelof long gamma-ray bursts indicates that they may trace the star formation history.So long gamma-ray bursts may be a useful tool of measuring the high-redshift SFR.Observations show that long gamma-ray bursts prefer to form in a low-metallicity en-vironment. We study the high-redshift SFR up to z ∼ . Swift
GRBstracing the star formation history and the cosmic metallicity evolution in differentbackground cosmological models including ΛCDM, quintessence, quintessence with atime-varying equation of state, and brane-world model. We use latest
Swift
GRBs in-cluding two highest- z GRBs, GRB 080913 at z = 6 . z = 8 .
3. Wefind that the SFR at z > ∼ − . Key words:
Gamma rays: bursts – stars: formation
The star formation history (SFH), especially at high-redshift( z > z > z <
1, the SFR in the redshift range of 1 < z < z > z = 7 . α emitters (LAEs)(Ota et al. 2008). In Fig-ure 1, we list the different observational results. We can seethat different results disagree with each other even consid-ering the uncertainties.Long-duration gamma-ray bursts triggered by the deathof massive stars, which have been shown to be associated ⋆ [email protected] † [email protected] with supernovae (Stanek et al. 2003; Hjorth et al. 2003),provide a complementary technique for measuring the SFR.GRBs at high redshfits are predicated to be observed be-cause of their high luminosities (Lamb & Reichart 2000;Ciardi & Loeb 2000; Bromm & Loeb 2002, 2006; Gou et al.2004). The farthest GRB to date is GRB 090423 at z = 8 . Swift
GRBs are not tracing the star formation history exactly butincluding an additional evolution (Daigne et al. 2006; Cen &Fang 2007; Le & Dermer 2007; Y¨uksel & Kistler 2007; Sal-vaterra & Chincarini 2007a; Guetta & Piran 2007; Kistler etal. 2008; Salvaterra et al. 2008; Butler et al. 2009). Obser-vations show that GRBs prefer to form in a low-metallicityenvironment (Le et al. 2003; Stanek et al. 2006). In the col-lapsar paradigm, large angular momentum is required topower a GRB. Because high-metallicity stars are expectedto have significant mass loss through winds promoting theloss of angular momentum, Langer & Norman (2006) andWoosley & Heger (2006) have argued that GRB progenitorswill have a low metallicity (Woosley & Heger 2006; M´esz´aros2006; Langer & Norman 2006). This has implications forthe expected redshift distribution of GRBs (Natarajan etal. 2005; Salvaterra & Chincarini 2007a; Salvaterra et al.2007b; Li 2008). Chary et al. (2007) estimated a lower limit c (cid:13) F. Y. Wang and Z. G. Dai l og ( M y r - M p c - ) log(1+z) This work Hopkin & Beacom(2006) Hopkin & Beacom(2006) LBG: Bouwens et al. (2008) LAE: Ota et al. (2008) GRB: Y uksel et al. (2008) GRB: This work GRB: Chary et al. (2007) (1+z) -5.01
Figure 1.
The cosmic star formation history. The black circlesare from Hopkins & Beacom (2006). The three open pentacles arethe star formation rates at z = 4 . z = 6 and z = 8 derived using Swift
GRB data. of the SFR of 0 . ± .
09 and 0 . ± .
05 M ⊙ yr − Mpc − at z = 4 . z ∼ Spitzer Space Telescope (Chary,Berger & Cowie 2007). Y¨uksel et al.(2008) used
Swift
GRBdata to constrain the high-redshift SFR and found that nosteep drop exists in the SFR up z ∼ . z ∼ z = 8 was con-sistent with LBG-based measurements (Kistler et al. 2009).In this paper, we estimate the high-redshfit SFR us-ing latest Swift long-duration GRBs, considering the GRBformation rate tracing SFH and the cosmic metallicity evo-lution. We use the SFR between z = 1 and z = 4 and relatethe GRB counts in this redshift bin. The absolute conver-sion factor between the SFR and the GRB rate is highlyuncertain. But we do not need this factor in this method.Because weak low-redshift GRBs can not be seen at highredshifts, so we only use high luminosity GRBs. We proceedanalogously to Y¨uksel et al. (2008). But there are two differ-ences between our method and Y¨uksel’s method. First, weconsider that long GRBs prefer to form in low-metallicityregions and trace the star formation history, but Y¨uksel etal. (2008) and Kistler et al. (2009) considered GRBs do nottrace star formation history directly, instead implying somekind of additional evolution. Second, we examine the influ-ence of background cosmology. In 1998, observations on typeIa supernovae suggest the accelerating universe. Many mod-els have been proposed to explain the accelerating expan-sion. Out of many particular models, we focus on four rep-resentative models: ΛCDM, quintessence, quintessence withtime-varying equation of state, and brane-world.The structure of this paper is as follows. In section 2,we introduce the method. In section 3, we show our resultson high-redshift SFR. Finally, section 4 contains conclusionsand discussions. More recent observational studies indicated that the longGRB host galaxy metallicity is generally lower than that ofthe average massive star forming galaxies (Le et al. 2003;Stanek et al. 2006). Salvaterra & Chincarini (2007) foundthat the differential peak flux number counts obtained byBATSE and
Swift could be well fitted using GRBs formingin low-metallicity galaxies (Salvaterra & Chincarini 2007a).Under the assumption that the formation of GRBs followsthe cosmic star formation history and GRBs form preferen-tially in low-metallicity galaxies, the GRB formation rate isgiven byΨ
GRB ( z ) = k GRB Σ( Z th , z )Ψ ∗ ( z ) , (1)where k GRB is the GRB formation efficiency, Σ( Z th , z ) isthe fraction of galaxies at redshift z with metallicity below Z th (Langer & Norman 2006) and Ψ ∗ ( z ) is the observed co-moving SFR. The redshift distribution of observable GRBsis dNdz = F ( z ) / h f beam i Σ( Z th , z )Ψ ∗ ( z ) dV /dz z , (2)where F ( z ) represents the ability both to detect the trig-ger of burst and to obtain the redshift (Kistler et al. 2008).The redshifts of high-redshift GRBs are determined by aspectral break in near infrared or infrared bands. Manyground-based facilities could recognize the spectral breakand then obtain the redshift. GROND observed the spectralbreak of GRB080913 between i ′ and z ′ bands and the red-shift of GRB080913 is z=6.7 (Greiner et al. 2009). Greineret al. (2009) show that 2m-class telescopes can identifymost high-redshift GRBs. The redshift of GRB090423 isdetermined by NIR spectroscopic observations (Tanvir etal. 2009; Salvaterra et al. 2009). So if the luminosities ofhigh-redshift GRBs are high enough and spacecrafts (suchas Swift and Fermi) can detect, the redshifts can be ob-tained using ground-based facilities. h f beam i is the beamingfactor and d V / d z is the comoving volume element per unitredshift, given byd V d z = 4 πcd L z H ( z )1 + z . (3)The luminosity distance, d L , to a source at redshift z is d L = c (1 + z ) Z z H ( z ′ ) d z ′ , (4)with H ( z ) = H p Ω m (1 + z ) + Ω Λ (5)in a flat ΛCDM universe. For bursts with luminosities suf-ficient to be viewed within an entire redshift range, Kistleret al. (2008) found that F ( z ) could be set to a constant (formore details, see Kistler et al. 2008, 2009).Some theoretical models (Woosley & Heger 2006; seeM´esz´aros 2006 for a review) require that GRB progenitorsshould have metallicity . Z ⊙ . According to Langer &Norman (2006), the fractional mass density belonging tometallicity below a given threshold Z th is (Langer & Norman2006)Σ( Z th , z ) = ˆΓ[ α + 2 , ( Z th /Z ⊙ ) . βz ]Γ( α + 2) , (6) c (cid:13) , 000–000 where ˆΓ and Γ are the incomplete and complete gammafunctions, α = − .
16 is the power-law index in theSchechter distribution function of galaxy stellar masses(Panter, Heavens & Jimenez 2004) and β = 2 is the slope ofthe galaxy stellar mass-metallicity relation (Savaglio et al.2005; Langer & Norman 2006). We adopt Z th = 0 . Z ⊙ as inLanger & Norman (2006).We show the luminosity-redshift distribution of 119 longGRBs observed by Swift till GRB 090529 in Figure 2. Theisotropic luminosity is L iso = E iso / [ T / (1 + z )] , (7)where E iso is the isotropic energy in the 1 − keV band and T is the GRB duration. Because only bright bursts can beseen at low and high redshifts, so we choose the luminositycut L iso > × ergs s − (Y¨uksel et al. 2008). There arefour groups of GRBs defined by this L iso cut in z = 1 − −
5, 5 −
7, and 7 − .
5. The GRBs in z = 1 − N the1 − = ∆ t ∆Ω4 π Z dz F ( z ) Σ( Z th , z ) Ψ ∗ ( z ) h f beam i dV /dz z = A Z dz Σ( Z th , z ) Ψ ∗ ( z ) dV /dz z , (8)where A = ∆ t ∆Ω F / π h f beam i depends on the total time,∆ t , and the angular sky coverage, ∆Ω. The theoretical num-ber in z = 4 − − N the z − z = h Ψ ∗ i z − z A Z z z dz Σ( Z th , z ) dV /dz z , (9)where h ˙ ρ ∗ i z − z is the average SFR density in the redshiftrange z − z . Representing the predicated numbers, N the z − z with the observed GRB counts, N obs z − z , we obtain the SFRin the redshift range z − z h Ψ ∗ i z − z = N obs z − z N obs1 − R dz dV/dz z Σ( Z th , z )Ψ ∗ ( z ) R z z dz dV/dz z Σ( Z th , z ) . (10)Below, we briefly introduce three other cosmologicalmodels in which we calculate the high-redshift SFR. We willrestrict our attention to flat models ( k = 0) because the flatgeometry is strongly supported by five-years WMAP data(Komatsu et al. 2009).We first consider the dark energy with a constant equa-tion of state w ( z ) = w , where − < w < − /
3. In sucha case this component is called “quintessence”. Confronta-tion with recent observational datasets, Wang et al. (2007)found Ω M = 0 .
31 and w = − .
95. We use these data inthe following calculations (Wang, Dai & Zhu 2007). Theluminosity-redshift distribution of GRBs in this model isshown in Figure 3.If we consider that the quintessence arises from anevolving scalar field, it would be natural to expect thatthe equation of state should vary with time. We consider w ( z ) = w + w z/ (1 + z ). We also use the results of Wang etal. (2007), Ω M = 0 . w = − .
08 and w = 0 .
84 (Wang, See http://swift.gsfc.nasa.gov/docs/swift/archive/grb table . L i s o e r g c m - s - redshift z Figure 2.
The luminosity-redshift distribution of 119
Swift longduration GRBs in the ΛCDM model. The number counts in red-shift bins z = 1 −
4, 4 −
5, 5 −
7, and 7 − . L i s o e r g c m - s - redshift z Figure 3.
The luminosity-redshift distribution of 119
Swift longduration GRBs in the quintessence ( w = w ). The number countsin redshift bins z = 1 −
4, 4 −
5, 5 −
7, and 7 − . Dai & Zhu 2007). The luminosity-redshift distribution ofGRBs in this model is shown in Figure 4.Brane-world scenarios assume that our four-dimensional space-time is embedded into five-dimensionalspace (Deffayet, Davli & Gabadadze 2002). Gravity infive dimensions is governed by the usual five-dimensionalEinstein-Hilbert action. The bulk metric induces a four-dimensional metric on the brane. We consider the flatDvali-Gabadadze-Porrati model. The only parameter isΩ M . We use the result from Wang et al. (2009), Ω M = 0 . V c ∝ D c , and thecomoving volume between redshifts z − ∆ z and z + ∆ z is c (cid:13) , 000–000 F. Y. Wang and Z. G. Dai L i s o e r g c m - s - redshift z Figure 4.
The luminosity-redshift distribution of 119
Swift longduration GRBs in the quintessence with a time-varying equationof state ( w = w + w z/ (1 + z )). The number counts in redshiftbins z = 1 −
4, 4 −
5, 5 −
7, and 7 − . L i s o e r g c m - s - redshift z Figure 5.
The luminosity-redshift distribution of 119
Swift longduration GRBs in the brane-world model. The number counts inredshift bins z = 1 −
4, 4 −
5, 5 −
7, and 7 − . V c ( z, ∆ z ) ∝ D c ( z + ∆ z ) − D c ( z − ∆ z ). Since the luminosityis proportional to the comoving distance squared, L ∝ D c ,the SFR density for a given redshift range is (Hopkins 2004)Ψ ∗ ( z ) ∝ L ( z ) V c ( z, ∆ z ) ∝ D c ( z ) D c ( z + ∆ z ) − D c ( z − ∆ z ) . (11)The Hubble functions in different dark energy models areshowed in Table 1. We convert the SFR in the redshiftrange z = 1 − Table 2.
The high-redshift SFRs in four models.Model Redshift Star formation rateΛCDM 4 . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . . ± . ∗ = − . ± . In Figure 1, we show the measurement of the high-redshiftSFR in the ΛCDM model with Ω M = 0 .
27, Ω Λ = 0 . H = 70km s − Mpc − . The star formation rates areLogΨ ∗ = − . ± .
30, LogΨ ∗ = − . ± .
30, andLogΨ ∗ = − . ± .
30 M ⊙ yr − Mpc − at z = 4 .
5, 6 .
0, and8 .
0, respectively. Taking into account the Poisson confidenceinterval for four observed GRBs, we assign a statistical un-certainty of a factor of 2 (Y¨uksel et al. 2008). The derivedhigh-redshift SFR shows a steep decay with a slope of about5 . . ∗ ( z ) = Ψ ∗ , (cid:26) [(1 + z ) a ] η + (cid:20) (1 + z ) b (1 + z ) b − a (cid:21) η + (cid:20) (1 + z ) c (1 + z ) b − a (1 + z ) c − b (cid:21) η (cid:27) /η , (12)where using η ≃ −
10 smoothes the power law transitions.Our fitted result is shown by the thick gray line in Fig 1.Here, z = 1 and z = 4 . a =3 . b = − . c = − . ∗ , = 0 . M ⊙ yr − Mpc − .In Table 2, we show the high-redshift SFRs in four cos-mological models. The value of SFR in these models showno remarkable difference. The reasons are as follows. First,the GRB counts in redshift bins are different because theluminosity distances in these models are not equal. Second,the comoving volume of these models are different. Third,there is a conversion factor from one cosmological model toanother. The derived SFR in these cosmological models areconsistent with each other in quoted uncertainties. The in-fluence of background cosmology can be neglected when weuse this method to measure the high-redshift SFR. c (cid:13)000
10 smoothes the power law transitions.Our fitted result is shown by the thick gray line in Fig 1.Here, z = 1 and z = 4 . a =3 . b = − . c = − . ∗ , = 0 . M ⊙ yr − Mpc − .In Table 2, we show the high-redshift SFRs in four cos-mological models. The value of SFR in these models showno remarkable difference. The reasons are as follows. First,the GRB counts in redshift bins are different because theluminosity distances in these models are not equal. Second,the comoving volume of these models are different. Third,there is a conversion factor from one cosmological model toanother. The derived SFR in these cosmological models areconsistent with each other in quoted uncertainties. The in-fluence of background cosmology can be neglected when weuse this method to measure the high-redshift SFR. c (cid:13)000 , 000–000 Table 1.
Expansion rates H ( z ) in four models. We consider the flat universe.In the brane-world model, Ω r c = (1 − Ω m ) / H ( z )).ΛCDM H ( z ) = H (cid:2) Ω m (1 + z ) + 1 − Ω m (cid:3) Quintessence H ( z ) = H (cid:2) Ω m (1 + z ) + (1 − Ω m ) (1 + z ) w ) (cid:3) Var Quintessence H ( z ) = H (cid:2) Ω m (1 + z ) + (1 − Ω m ) (1 + z ) w − w ) exp(3 w z ) (cid:3) Braneworld H ( z ) = H h ( p Ω m (1 + z ) + Ω r c + p Ω r c ) i In this paper, we constrain the high-redshift SFR up to z ∼ . Swift
GRB data including twohighest- z GRBs, GRB 080913 at z = 6 . z = 8 . Swift
GRB data allowed the use of luminos-ity cuts to fairly compare GRBs in the full redshift range,eliminating the unknown GRB luminosity function. Second,we can calculate the high-redshift SFR by comparing thecounts of GRBs at different redshift ranges, normalized toSFR data at intermediate redshifts which have been wellconstrained by observations, eliminating the need for knowl-edge of the GRB efficiency factor. But there are two differ-ences between our method and Y¨uksel’s method. First, weconsider that long GRBs prefer to form in low-metallicityregions and trace the star formation history, but Y¨uksel etal. (2008) and Kistler et al. (2009) considered GRBs do nottrace star formation history directly, instead implying somekind of additional evolution. Second, we examine the influ-ence of background cosmology.Our results show that the SFR at z > − . − . z >
4. Our de-rived high-redshift SFR is different from Y¨uksel et al. (2008).The main reason is that we consider GRBs prefer to formin low-metallicity regions. Li (2008) derived the SFH up to z = 7 .
4, and found that the decay slope at z > . ACKNOWLEDGMENTS
We thank A. M. Hopkins and H. Y¨uksel for sharing theSFR data. This work was supported by the National Natu-ral Science Foundation of China (grants 10233010, 10221001and 10873009) and the National Basic Research Program ofChina (973 program) No. 2007CB815404. F. Y. Wang wasalso supported by Jiangsu Project Innovation for PhD Can-didates (CX07B-039z).
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