High-redshift star formation rate up to z~8.3 derived from gamma-ray bursts and influence of background cosmology

0.29 MB·English

by  F. Y. Wang

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed6January2010 (MNLATEXstylefilev2.2)
z 8.3
High-redshift star formation rate up to ∼ derived from
gamma-ray bursts and influence of background cosmology
0 F. Y. Wang ⋆ and Z. G. Dai†
1
Department of Astronomy, Nanjing University,Nanjing 210093, P. R. China
0
2

n
6January2010
a
J

5
ABSTRACT

The high-redshift star formation rate (SFR) is difficult to measure directly even by
]
O modernapproaches.Long-durationgamma-raybursts (GRBs) can be detected to the
edge of the visible universe because of their high luminorsities. The collapsar model
C
of long gamma-ray bursts indicates that they may trace the star formation history.
.
h So long gamma-ray bursts may be a useful tool of measuring the high-redshift SFR.
p Observationsshowthat longgamma-raybursts preferto formina low-metallicityen-
- vironment.We studythe high-redshiftSFRuptoz ∼8.3consideringtheSwift GRBs
o
tracing the star formation history and the cosmic metallicity evolution in different
r
t background cosmological models including ΛCDM, quintessence, quintessence with a
s
a time-varying equationof state, and brane-worldmodel. We use latest Swift GRBs in-
[ cludingtwohighest-z GRBs,GRB 080913atz =6.7andGRB 090423atz =8.3.We
find that the SFR at z >4 shows a steep decay with a slope of ∼−5.0 in ΛCDM. In
2
the other threemodels,the high-redshiftSFR is slightlydifferentfromΛCDMmodel,
v
and also shows a steep decay.
0
4
Key words: Gamma rays: bursts – stars: formation
1
5
.
2
1
9 1 INTRODUCTION with supernovae (Stanek et al. 2003; Hjorth et al. 2003),
0 provideacomplementarytechniqueformeasuringtheSFR.
Thestarformationhistory(SFH),especiallyathigh-redshift
: GRBs at high redshfits are predicated to be observed be-
v (z >6), is important in many fields in astrophysics. Direct
cause of their high luminosities (Lamb & Reichart 2000;
i SFR measurements are quite challenging at high-redshift,
X Ciardi &Loeb 2000; Bromm &Loeb 2002, 2006; Gou etal.
particularly at the faint end of the galaxy luminosity func-
r 2004). Thefarthest GRBto dateis GRB090423 at z=8.3
tion. The star formation rate (SFR) has been comprehen-
a (Tanvir et al. 2009; Salvaterra et al. 2009). So GRBs are a
sivelyinvestigated.Hopkins&Beacom(2006)calibratedthe
promisingprobeofthestarformation history(Totani1997;
starformationhistoryouttoz>6usingtheultravioletand
Wijersetal.1998; Porciani&Madau2001;Bromm &Loeb
far-infrareddata.TheyfoundtheSFRistightlyconstrained
2002,2006; Tutukov 2003). Recent studies show that Swift
at z <1, the SFR in the redshift range of 1<z <4 is ap-
GRBsarenottracingthestarformationhistoryexactlybut
proximately a constant and shows a steep decay at z > 4
includinganadditionalevolution(Daigneetal.2006;Cen&
(Hopkins&Beacom2006).Li(2008)constrainedtheSFRup
Fang 2007; Le & Dermer 2007; Yu¨ksel & Kistler 2007; Sal-
toz=7.4byadding6newdataandfoundthesameresultas
vaterra&Chincarini2007a;Guetta&Piran2007;Kistleret
Hopkins&Beacom (2006)(Li 2008). Thehigh-redshift SFR
al. 2008; Salvaterra et al. 2008; Butler et al. 2009). Obser-
canalsobedeterminedbyobservationsofcolor-selectedLy-
vations show that GRBs prefer to form in a low-metallicity
manbreakgalaxies(LBGs)(Bouwensetal.2008;Mannucci
environment(Leetal. 2003; Staneket al.2006). Inthecol-
etal.2007;Vermaetal.2007).Otaetal.(2008)constrained
lapsar paradigm, large angular momentum is required to
theSFRusingLyαemitters(LAEs)(Otaetal.2008).InFig-
power a GRB. Because high-metallicity stars are expected
ure1,we list thedifferentobservational results. Wecan see
to have significant mass loss through winds promoting the
that different results disagree with each other even consid-
loss of angular momentum, Langer & Norman (2006) and
ering theuncertainties.
Woosley&Heger(2006)havearguedthatGRBprogenitors
Long-durationgamma-rayburststriggeredbythedeath
willhavealowmetallicity(Woosley&Heger2006;M´esz´aros
of massive stars, which have been shown to be associated
2006; Langer & Norman 2006). This has implications for
the expected redshift distribution of GRBs (Natarajan et
al. 2005; Salvaterra & Chincarini 2007a; Salvaterra et al.
⋆ [email protected]
2007b;Li2008). Charyetal.(2007) estimated alowerlimit
† [email protected]
2 F. Y. Wang and Z. G. Dai
2 THE METHOD
More recent observational studies indicated that the long
0.0 GRBhost galaxy metallicity isgenerally lower than thatof
-0.5 (1+z)-5.01 the average massive star forming galaxies (Le et al. 2003;
Stanek et al. 2006). Salvaterra & Chincarini (2007) found
-1.0 that the differential peak flux number counts obtained by
-3 Mpc) -1.5 BATSE and Swift could be well fitted using GRBs forming
-1 yr -2.0 This work iUnnldoewr-mtheetaalsliscuitmypgtaiolanxtiehsat(Stahlevaftoerrmraat&ionChoifnGcaRriBnis2f0o0ll7oaw)s.
M
( -2.5 Ho Hpokipnk &in B&e Baecoamco(m20(20060) 6 ) thecosmic starformation history and GRBsform preferen-
og LBG: Bouwens et al. (2008) tiallyinlow-metallicity galaxies, theGRBformation rateis
l
-3.0 LAE: Ota et al. (2008) given by
GRB: Yuksel et al. (2008)
-3.5 GGRRBB:: TChhias rwy oertk al. (2007) ΨGRB(z)=kGRBΣ(Zth,z)Ψ∗(z), (1)
-4.0
0.0 0.2 0.4 0.6 0.8 1.0 where kGRB is the GRB formation efficiency, Σ(Zth,z) is
log(1+z) the fraction of galaxies at redshift z with metallicity below
Zth (Langer&Norman2006) andΨ∗(z)istheobservedco-
Figure 1. The cosmic star formation history. The black circles moving SFR. The redshift distribution of observable GRBs
arefromHopkins&Beacom(2006).Thethreeopenpentaclesare is
thestarformationratesatz=4.5,z=6andz=8derivedusing dN dV/dz
Swift GRBdata. dz =F(z)/hfbeamiΣ(Zth,z)Ψ∗(z) 1+z , (2)
where F(z) represents the ability both to detect the trig-
ger of burst and to obtain theredshift (Kistler et al. 2008).
The redshifts of high-redshift GRBs are determined by a
spectral break in near infrared or infrared bands. Many
ground-based facilities could recognize the spectral break
of the SFR of 0.12±0.09 and 0.09±0.05 M⊙yr−1Mpc−3 andthenobtaintheredshift.GRONDobservedthespectral
at z = 4.5 and 6, respectively, using deep observations of break of GRB080913 between i′ and z′ bands and the red-
three z ∼5 GRBs with the Spitzer Space Telescope(Chary,
shift of GRB080913 is z=6.7 (Greiner et al. 2009). Greiner
Berger & Cowie 2007). Yu¨ksel et al.(2008) used Swift GRB
et al. (2009) show that 2m-class telescopes can identify
data to constrain the high-redshift SFR and found that no
most high-redshift GRBs. The redshift of GRB090423 is
steepdropexistsintheSFRupz∼6.5(Yu¨kseletal.2008).
determined by NIR spectroscopic observations (Tanvir et
Kistleretal.(2009)constrainedSFRuptoz ∼8usingfour-
al. 2009; Salvaterra et al. 2009). So if the luminosities of
years Swift data and found SFR to beyond z = 8 was con-
high-redshift GRBs are high enough and spacecrafts (such
sistent with LBG-basedmeasurements (Kistleretal. 2009).
as Swift and Fermi) can detect, the redshifts can be ob-
In this paper, we estimate the high-redshfit SFR us-
tained using ground-basedfacilities. hf i is thebeaming
beam
ing latest Swift long-duration GRBs, considering the GRB
factor and dV/dz is the comoving volume element per unit
formation rate tracing SFH and thecosmic metallicity evo-
redshift, given by
lution.WeusetheSFRbetweenz=1andz=4andrelate
the GRB counts in this redshift bin. The absolute conver- dV = 4πcd2LH(z). (3)
sion factor between the SFR and the GRB rate is highly dz 1+z 1+z
uncertain. But we do not need this factor in this method. The luminosity distance, d , to a source at redshift z is
L
Because weak low-redshift GRBs can not be seen at high
z 1
redshifts,soweonlyusehighluminosityGRBs.Weproceed d =c(1+z) dz′, (4)
analogouslytoYu¨kseletal.(2008).Buttherearetwodiffer- L Z H(z′)
0
ences between our method and Yu¨ksel’s method. First, we
with
consider that long GRBs prefer to form in low-metallicity
regions and trace the star formation history, but Yu¨ksel et H(z)=H0 Ωm(1+z)3+ΩΛ (5)
al. (2008) and Kistleretal. (2009) considered GRBsdonot p
in a flat ΛCDM universe. For bursts with luminosities suf-
tracestarformation historydirectly,instead implyingsome
ficient to be viewed within an entire redshift range, Kistler
kind of additional evolution. Second, we examine the influ-
etal.(2008) foundthatF(z)couldbesettoaconstant(for
enceofbackgroundcosmology.In1998,observationsontype
more details, see Kistler et al. 2008, 2009).
Iasupernovaesuggesttheacceleratinguniverse.Manymod-
Some theoretical models (Woosley & Heger 2006; see
els have been proposed to explain the accelerating expan-
M´esz´aros 2006 for a review) require that GRB progenitors
sion. Out of many particular models, we focus on four rep-
should have metallicity 6 0.1Z⊙. According to Langer &
resentativemodels:ΛCDM,quintessence,quintessencewith
Norman (2006), the fractional mass density belonging to
time-varyingequation of state, and brane-world.
metallicitybelowagiventhresholdZ is(Langer&Norman
th
The structure of this paper is as follows. In section 2,
2006)
we introduce the method. In section 3, we show our results
onhigh-redshiftSFR.Finally,section4containsconclusions Σ(Z ,z)= Γˆ[α1+2,(Zth/Z⊙)2100.15βz], (6)
and discussions. th Γ(α +2)
1
3
where Γˆ and Γ are the incomplete and complete gamma
functions, α = −1.16 is the power-law index in the
1

Schechter distribution function of galaxy stellar masses
(Panter,Heavens&Jimenez2004)andβ =2istheslopeof
1E53
the galaxy stellar mass-metallicity relation (Savaglio et al.
2005;Langer&Norman2006).WeadoptZth =0.1Z⊙ asin
Langer & Norman (2006). 1E52
GisoRtBroWspeoicbshslueormwvietndhoebsilytuymSiwisnifots1ittyil-lreGdRshBift0d90is5t2r9ibuintiFonigoufre1129.lTonhge -2-1 erg cm s 1E51
L iso
L =E /[T /(1+z)], (7)
iso iso 90
1E50
whereE istheisotropicenergyinthe1−104keVbandand
iso
T istheGRBduration.Becauseonlybrightburstscanbe
90
seen at low and high redshifts, so we choose the luminosity 1E49
cut L >3×1051 ergs s−1 (Yu¨kselet al. 2008). There are 0 1 2 3 4 5 6 7 8
iso
redshift z
four groups of GRBs defined by this L cut in z=1−4,
iso
4−5, 5−7, and 7−8.5. The GRBs in z = 1−4 act as a
Figure2.Theluminosity-redshiftdistributionof119Swift long
“control group” to base the GRB to SFR conversion, since
durationGRBsintheΛCDMmodel.Thenumbercounts inred-
thisredshiftbinhasbothgoodSFRmeasurementsandGRB
shiftbinsz=1−4, 4−5,5−7, and7−8.5are28, 6,5, and1
counts.Wecalculatethetheoretically predicatednumberof
respectively.
GRBs in thisbin as
Nthe = ∆t∆Ω 4dzF(z)Σ(Z ,z) Ψ∗(z) dV/dz
1−4 4π Z1 th hfbeami 1+z
4 dV/dz
= A Z dzΣ(Zth,z)Ψ∗(z) 1+z , (8)
1
1E53
where A=∆t∆ΩF /4πhf i depends on the total time,
0 beam
∆t,andtheangularskycoverage,∆Ω.Thetheoreticalnum-
ber in z=4−5 and 5−7 can be written by -1 s 1E52
Nzt1h−ez2 = hΨ∗iz1−z2A Zz1z2dzΣ(Zth,z)d1V+/dzz , (9) -2 L erg cm iso1E51
where hρ˙∗iz1−z2 is the average SFR density in the redshift
rangez −z .Representingthepredicatednumbers,Nthe 1E50
1 2 z1−z2
with theobserved GRBcounts,Nobs ,weobtain theSFR
z1−z2
in theredshift range z −z
1 2 1E49
0 1 2 3 4 5 6 7 8
hΨ∗iz1−z2 = NNzo11bo−−bss4z2 R14dzzz12dd1Vz+/dzdz1V+Σ/dz(zZΣth(,Zzt)hΨ,z∗)(z) . (10) Figure3.Theluminosity-redsrheidfsthidft izstributionof119Swift long
R
Below, we briefly introduce three other cosmological durationGRBsinthequintessence(w=w0).Thenumbercounts
modelsinwhichwecalculatethehigh-redshiftSFR.Wewill inredshiftbins z =1−4, 4−5, 5−7, and 7−8.5 are27, 6, 5,
restrictourattentiontoflatmodels(k=0)becausetheflat and1respectively.
geometry is strongly supported by five-years WMAP data
(Komatsu et al. 2009).
Dai & Zhu 2007). The luminosity-redshift distribution of
Wefirstconsiderthedarkenergywithaconstantequa-
GRBs in this model is shown in Figure 4.
tion of state w(z) = w , where −1 < w < −1/3. In such
0
Brane-world scenarios assume that our four-
a case this component is called “quintessence”. Confronta-
dimensional space-time is embedded into five-dimensional
tion with recent observational datasets, Wang et al. (2007)
space (Deffayet, Davli & Gabadadze 2002). Gravity in
found Ω = 0.31 and w = −0.95. We use these data in
M 0 five dimensions is governed by the usual five-dimensional
the following calculations (Wang, Dai & Zhu 2007). The
Einstein-Hilbert action. The bulk metric induces a four-
luminosity-redshift distribution of GRBs in this model is
dimensional metric on the brane. We consider the flat
shown in Figure 3.
Dvali-Gabadadze-Porrati model. The only parameter is
If we consider that the quintessence arises from an
Ω .Weusetheresult from Wanget al. (2009), Ω =0.20
evolving scalar field, it would be natural to expect that M M
(Wang,Dai&Qi2009).Theluminosity-redshiftdistribution
the equation of state should vary with time. We consider
of GRBs in this model is shown in Figure 5.
w(z)=w +w z/(1+z).WealsousetheresultsofWanget
0 1 The SFR conversion from one cosmology to another is
al. (2007), Ω = 0.30, w = −1.08 and w = 0.84 (Wang,
M 0 1 asfollows. Intheflatuniverse,thecomovingvolumeispro-
portional to comoving distance cubed, V ∝ D3, and the
c c
1 Seehttp://swift.gsfc.nasa.gov/docs/swift/archive/grb table. comoving volume between redshifts z−∆z and z+∆z is
4 F. Y. Wang and Z. G. Dai
Table 2.Thehigh-redshiftSFRsinfourmodels.

Model Redshift Starformationrate
1E53
ΛCDM 4.5±0.5 LogΨ∗=−1.287±0.30
ΛCDM 6.0±1.0 LogΨ∗=−1.750±0.30
-1 s 1E52 ΛCDM 8.0±0.5 LogΨ∗=−2.110±0.30
-2 m Quintessence 4.5±0.5 LogΨ∗=−1.296±0.30
L erg ciso1E51 QQuuiinntteesssseennccee 86..00±±01..50 LLooggΨΨ∗∗==−−21..278877±±00..3300
VarQuintessence 4.5±0.5 LogΨ∗=−1.394±0.30
1E50 VarQuintessence 6.0±1.0 LogΨ∗=−1.819±0.30
VarQuintessence 8.0±0.5 LogΨ∗=−2.214±0.30
Brane-world 4.5±0.5 LogΨ∗=−1.466±0.30
1E49 Brane-world 6.0±1.0 LogΨ∗=−1.905±0.30
0 1 2 3 4 5 6 7 8 Brane-world 8.0±0.5 LogΨ∗=−2.205±0.30
redshift z
Figure4.Theluminosity-redshiftdistributionof119Swift long 3 THE DERIVED HIGH-REDSHIFT STAR
durationGRBsinthequintessencewithatime-varyingequation FORMATION RATE
ofstate (w=w0+w1z/(1+z)). Thenumber counts inredshift
bins z = 1−4, 4−5, 5−7, and 7−8.5 are 30, 6, 5, and 1 In Figure 1, we show the measurement of the high-redshift
respectively. SFR in the ΛCDM model with ΩM = 0.27, ΩΛ = 0.73
and H = 70kms−1Mpc−1. The star formation rates are
0
LogΨ∗ = −1.287 ± 0.30, LogΨ∗ = −1.750 ± 0.30, and
LogΨ∗ =−2.110±0.30 M⊙yr−1Mpc−3 at z=4.5, 6.0, and
8.0,respectively.TakingintoaccountthePoissonconfidence
interval for four observed GRBs, we assign a statistical un-

certainty of a factor of 2 (Yu¨ksel et al. 2008). The derived
high-redshiftSFRshowsasteepdecaywithaslopeofabout
1E53 5.0(theslopeof Hopkins&Beacom (2006) isabout 7.8). If
we exclude a particular GRB or changes in redshift ranges,
thederivedSFRwillchangeinsignificantly.Thisconclusion
-2-1 m s 1E52 isconsistentwithYu¨kseletal.(2008),butthederivedhigh-
g c redshift SFR is different from that of Yu¨ksel et al. (2008).
L eriso1E51 Tinhleowm-mainetarellaicsiotnyirsegtihoants.we consider GRBs prefer to form
We show the update SFH fit of Hopkins & Beacom
1E50
(2006) at high-redshifts based on ournewGRB results. We
use a continuous form of a broken power law (Yu¨ksel et al.
1E49 2008),
0 1 2 3 4 5 6 7 8
redshift z (1+z)b η
Ψ∗(z) = Ψ∗,0(cid:26)[(1+z)a]η+(cid:20)(1+z1)b−a(cid:21)
Figure5.Theluminosity-redshiftdistributionof119Swift long
durationGRBsinthebrane-worldmodel.Thenumbercountsin (1+z)c η 1/η
+ , (12)
redshiftbinsz=1−4,4−5,5−7,and7−8.5are31,6,5,and (cid:20)(1+z1)b−a(1+z2)c−b(cid:21) (cid:27)
1respectively.
where using η ≃ −10 smoothes the power law transitions.
Our fitted result is shown by the thick gray line in Fig 1.
Here,z =1andz =4.0aretheredshiftsofthebreaks,a=
1 2
3.4,b=−0.3andc=−5.0istheslopesofthesethreepower
V (z,∆z)∝D3(z+∆z)−D3(z−∆z).Sincetheluminosity
iscproportionalcto the comovcing distance squared, L ∝ D2, laws,andthenormalization isΨ∗,0 =0.02M⊙ yr−1 Mpc−3.
c InTable2,weshowthehigh-redshiftSFRsinfourcos-
theSFRdensityforagivenredshiftrangeis(Hopkins2004)
mological models. The value of SFR in these models show
L(z) D2(z) no remarkable difference. The reasons are as follows. First,
Ψ∗(z)∝ V (z,∆z) ∝ D3(z+∆z)c−D3(z−∆z). (11) the GRB counts in redshift bins are different because the
c c c
luminosity distances in thesemodels are not equal.Second,
The Hubble functions in different dark energy models are the comoving volume of these models are different. Third,
showed in Table 1. We convert the SFR in the redshift thereis a conversion factor from one cosmological model to
range z = 1−4 in ΛCDM to other models. The derived another.The derivedSFR in thesecosmological models are
SFR can be used to calculate the high-redshift SFR in consistent with each other in quoted uncertainties. The in-
quintessence, quintessence with a time-varying equation of fluenceof backgroundcosmology can beneglected when we
state and brane-world model. usethis method to measure thehigh-redshift SFR.
5
Table1.ExpansionratesH(z)infourmodels.Weconsidertheflatuniverse.
Inthebrane-worldmodel,Ωrc =(1−Ω2m)/4.
Model CosmologicalexpansionrateH(z)).
ΛCDM H2(z)=H02 Ωm(1+z)3+1−Ωm
Quintessence H2(z)=H02 Ωm(1+(cid:2) z)3+(1−Ωm)(1+z(cid:3))3(1+w)
VarQuintessence H2(z)=H02 Ωm(1+(cid:2) z)3+(1−Ωm)(1+z)3(1+w0−w1) ex(cid:3)p(3w1z)
Braneworld H(cid:2)2(z)=H02 ( Ωm(1+z)3+Ωrc+ Ωrc)2 (cid:3)
h p p i
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