The high-redshift star formation rate derived from GRBs: possible origin and cosmic reionization

0.26 MB·

by  F. Y. Wang

Astronomy&Astrophysicsmanuscriptno.v2 (cid:13)c ESO2014
January24,2014
The high-redshift star formation rate derived from GRBs: possible
origin and cosmic reionization
F.Y.Wang1,2,3
1 SchoolofAstronomyandSpaceScience,NanjingUniversity,Nanjing210093,China;
2 DepartmentofPhysics,TheUniversityofHongKong,PokfulamRoad,HongKong,China;
3 KeyLaboratoryofModernAstronomyandAstrophysics(NanjingUniversity),MinistryofEducation,Nanjing210093,China;
Preprintonlineversion:January24,2014
4
1
ABSTRACT
0
2
Thecollapsarmodeloflonggamma-raybursts(GRBs)indicatesthattheymaytracethestarformationhistory.SolongGRBsmaybe

n ausefultoolofmeasuringthehigh-redshiftstarformationrate(SFR).ThecollapsarmodelexplainsGRBformationviathecollapse
a ofarapidlyrotatingmassivestarwithM>30M⊙ intoablackhole,whichmayimplyadecreaseofSFRathighredshift.However,
J wefindthattheSwiftGRBsduring2005-2012arebiasedtracingtheSFR,includingafactorabout(1+z)0.5,whichisinagreement

3 withrecentresults.Aftertakingthisfactor,theSFRderivedfromGRBsdoesnot show steepdropuptoz∼9.4.Weconsider the
2 GRBsproducedbyrapidlyrotatingmetal-poorstarswithlowmassestoexplainthehigh-redshiftGRBrateexcess.Thechemically
homogeneous evolution scenario (CHES)of rapidly rotating stars with mass larger than 12M⊙ is recognized as a promising path

] towardscollapsarsinconnectionwithlongGRBs.Ourresultsindicatethatthestarsinthemassrange12M⊙<M<30M⊙ forlow
E enough metallicityZ≤0.004 withthe GRB efficiency factor 10−5 can fit the derived SFR withgood accuracy. Combining these
H twofactors, wefindthat theconversion efficiency frommassive starstoGRBsisenhanced by afactor of 10, whichmay be able
toexplaintheexcessofthehigh-redshiftGRBrate.WealsoinvestigatethecosmicreionizationhistoryusingthederivedSFR.The
.
h GRB-inferredSFRwouldbesufficienttomaintaincosmicreionizationover6<z<10andreproducetheobservedopticaldepthof
p Thomsonscatteringtothecosmicmicrowavebackground.
-
o Keywords.gamma-rayburst:general-stars:formation-reionization
r
t
s
a
1. Introduction al. 2011; Robertson & Ellis 2012; but see Elliott et al. 2012).
[
The SFR inferred from the high-redshift (z>6) GRBs seems
1 Gamma-raybursts(GRBs)arethebrightestelectromagneticex- to be too high in comparisonwith the oneobtainedfromsome
v plosions in the universe (for a recent review, see Gehrels et al. high-redshiftgalaxysurveys(Kristleretal.2009;Bouwensetal.
4 2009).Becauseoftheirveryhighluminosity,GRBscanbede-
2009).Kistleretal.(2008)foundthatthereareaboutfourtimes
6 tected out to the edge of the visible Universe (Ciardi & Loeb
asmanyGRBsatredshiftz∼4thanexpectedfromstarforma-
8 2000; Lamb & Reichart 2000; Bromm & Loeb 2002, 2006).
tion measurements. They claimed that some unknown mecha-
.5 The farthest GRB to date is GRB 090429B with a photomet- nism is leading to an enhancement about (1+z)d (d =1.5) in
1 ric redshift z=9.4 (Cucchiara et al. 2011),significantly larger
theobservedrateofhigh-redshiftGRBs.UsingmoreSwiftdata,
0 than those of the most distant quasars. This property makes
Kistleretal.(2009)foundaslightlylowervalueofenhancement
4 GRBs indispensable beacons to study the early universe, in-
about(1+z)1.2.Robertson&Ellis(2012)foundthevalueofd
1 cludingthestarformationrate(Totani1997;Wijersetal.1998;
isabout0.5bycomparingthecumulativeredshiftdistributionof
: Porciani & Madau 2001; Bromm & Loeb 2002,2006), the in-
v GRBsandSFR.Butontheotherhand,Elliottetal.(2012)found
Xi t2e0r0g7al;aMcticcQmuiendniuemt a(lI.G2M00)8()B,aarnkdanthae&mLetoaelbe2n0ri0c4h;mIennotuheiesttoarly. thatthevalueofd isaboutzerousingasmallsampleofGRBs.In
order to explainthis discrepancy,manymodels have been pro-
r (Savaglio2006;Wangetal.2012).Inaddition,GRBshavebeen
a posed. Li (2008) explained the observed discrepancy between
used as standard candles to constrain cosmological parameters
theGRBratehistoryandthestarformationratehistoryasbeing
anddarkenergy(Dai,Liang&Xu2004;Schaefer2007;Wang,
due to cosmic metallicity evolution (Langer & Norman 2006),
Qi&Dai2011).
byassumingthatlongGRBstendtooccuringalaxieswithlow
Themostpopulartheoreticalmodeloflong-durationGRBs
metallicities.Chengetal.(2010)suggestedthatthisdiscrepancy
isthecollapseofamassivestartoablackhole(Woosley1993).
could be solved if some high-redshift GRBs are produced by
ObservationsalsoshowthatGRBsareassociatedwithTypeIb/c
superconducting cosmic strings. Wang & Dai (2011) used an
supernovae (Stanek et al. 2003; Hjorth et al. 2003). So GRBs
evolvinginitialmassfunction(IMF)ofstarstoexplaintheGRB
provideacomplementarytechniqueformeasuringtheSFRhis-
redshiftdistribution.Virgilietal.(2011)discussedthepossibil-
tory(Totani1997;Wijersetal.1998;Porciani&Madau2001).
itythattheevolutionoftheGRBluminosityfunctionbreakwith
Recent studies show that Swift GRBs are not tracing the star
redshift may explain this discrepancy. Observations also show
formationhistorymeasuredbytraditionalmeansexactlybutin-
differencesinthepopulationofGRBhostgalaxiescomparedto
cludinganadditionalevolution(Le&Dermer2007;Salvaterra
expectationsforanunbiasedstar-formationtracer(Tanviretal.
&Chincarini2007;Kistleretal.2008;Yu¨kseletal.2008;Wang
2004;Fruchteretal.2006;Svenssonetal.2010).
&Dai2009;Wanderman,&Piran2010;Qinetal.2010;Caoet
1
Wang:Thehigh-redshiftSFRderivedfromGRBsandcosmicreionization
2. ThelatestGRBsample
TheexpectedredshiftdistributionofGRBsis
54 dN e (z)r˙∗(z)dVcom/dz
10 =F(z) , (1)
dz hf i 1+z
beam
53
10
where F(z) represents the ability both to detect the trigger of
1052 burstandtoobtaintheredshift,e (z)accountsforthefractionof
-1 erg s) 1051 strteaarstepdroasduccoinnsgtaGntRwBhse,nr˙∗w(ze)ciosntshiedeSrFtRhedbernisgihtyt.bTuhrsetsFw(zi)thclaunmbie-
L (iso1050 nositiessufficienttobedetectedwithinanentireredshiftrange,
so F(z)=F . GRBs that are unobservabledue to beaming are
0
1049 accounted for through hfbeami. The e (z) can be parameterized
48 as e (z)=e 0(1+z)d , where e 0 is an unknownconstant that in-
10
cludestheabsoluteconversionfromtheSFRtotheGRBratein
47 agivenGRBluminosityrange.Kistleretal.(2008)foundthein-
10
0 1 2 3 4 5 6 7 8 9 10 dexd =1.5from63SwiftGRBs.Alittlesmallervalued =1.2
z wasinferredusing119SwiftGRBs(Kistleretal.2009).Inaflat
universe,thecomovingvolumeiscalculatedby
Fig.1: Distribution of the isotropic-equivalent luminosity for
dV dD
157 long-duration Swift GRBs. Demarcated are the GRB sub- com =4p D2 com , (2)
samplesusedtoestimatetheSFR.Theshadedareaapproximates dz com dz
thedetectionthresholdofSwiftBAT.
wherethecomovingdistanceis
c z dz′
D (z)≡ . (3)
com H0Z0 W m(1+z′)3+W L
In the calculations,pwe use W m =0.27, W L =0.73 and H0=71
In this paper, we study the star formation rate history de- kms−1Mpc−1fromtheWilkinsonMicrowaveAnisotropyProbe
rived from GRBs. First we use the Swift GRB sample to test (WMAP)seven-yeardata(Komatsuetal.2011).
the evolution of GRB rate relative to SFR. If GRBs trace star WeusethelatestSwiftlong-durationGRBsampletillGRB
formation in the universe without bias, the ratio of the GRB 110403. The data is taken from Butler et al. (2007,2010) and
rate to the SFR would not be expected to vary with redshift. website1. Theisotropic-equivalentluminosityofa GRB canbe
We find that this ratio is proportional to (1+z)0.5. The index obtainedby
is smaller than the value of Kistler et al. (2009). We also de-
L =E (1+z)/T . (4)
rivethehigh-redshiftSFR usingSwift GRB samplebycorrect- iso iso 90
ingthisevolution.Then,weconsidertherapidlyrotatingmetal- ThedistributionofL for157GRBsinthesampleisshownin
iso
poorstarswith masses smaller thancriticalmass Mcri ∼30M⊙ Figure1.Weusethesameluminositycutsintheseredshiftbins
toseeiftheycanproduceGRBstoexplainthediscrepancybe- asKistleretal.(2009).Theshadedareaapproximatesthedetec-
tween high-redshift SFR and GRB rate. The collapsar model tionthresholdofSwiftBAT,whichcanbecalculatedasfollows.
indicates that stars with mass larger than 30M⊙ can produce Theluminositythresholdcan beapproximatedbya bolometric
GRBs(Woosley1993;Bissaldietal.2007;Raskinetal.2008). energyfluxlimitF =1.2×10−8ergcm−2s−1.Theluminosity
lim
ObservationalsoshowsthattheprogenitorofGRB060505hasa thresholdisthen
massabove30M (Tho¨neetal.2008).Yoon&Langer(2005)in-
⊙
vestigatedtheevolutionofrotatingsinglestarsinthemassrange L =4p D2F , (5)
lim L lim
12M <M <60M at low metallicity. They found that if the
⊙ ⊙
whereD istheluminositydistancetotheburst.
initialspinrateishighenough,thetimescaleforrotationallyin- L
In order to test the GRB rate relative to the SFR, we must
ducedmixingbecomesshorterthanthenucleartimescale.The
chooseburstswithhighluminosities,becauseonlybrightbursts
starmayevolveinaquasi-chemicallyhomogeneousway.Inpar-
can be seen at low and high-redshifts, so we choose the lumi-
ticular,forlowenoughmetallicity,thistypeofevolutioncanlead
nosity cut L >1051 erg s−1 (Yu¨ksel et al. 2008) in the red-
toretentionofsufficientangularmomentumincorestoproduce iso
shiftbin0−4.Thisremovesmanylow-redshift,low-L bursts
GRBsaccordingtothecollapsarscenario.Last,wecalculatethe iso
that could not have been seen at higher redshift. Because the
impactofthisGRB-inferredstarformationrateonthereioniza-
SFR at high redshift is poorly known (Bouwens et al. 2012;
tionhistory,includingtheopticaldepthofelectronscatteringto
Oesch et al. 2013; Coe et al. 2013), so we choose the red-
thecosmicmicrowavebackground.
shift range 0 < z < 4, in which SFR is well measured. We
Thestructureofthispaperisarrangedasfollows.Inthenext have92GRBs inthissubsample.We usetheSFRhistoryfrom
section,wecompiletheSwiftGRBsampletillGRB110403and Hopkins& Beacom(2006).We comparethe predictedandob-
test the evolution of GRB rate. The SFR derived from GRBs served cumulative GRB distributions in Figure 2. We find that
isgiveninsection3.WeshowthemodelofGRBsfromchemi- the Kolmogorov-Smirnov statistic is minimized for d = 0.5,
callyhomogeneousevolutionscenarioandtheinfluenceonhigh- which is consistent with Robertson & Ellis (2012). At the 2s
redshiftSFR derivedfromGRBs in section 4. We computethe confidencelevel,thevalueofd isintherange−0.15<d <1.6.
reionizationhistorywiththisGRB-inferredSFRinsection5.We
concludewithasummaryinsection6. 1 http://astro.berkeley.edu/∼nat/Swift/bat spec table.html
2
Wang:Thehigh-redshiftSFRderivedfromGRBsandcosmicreionization

1.0 100
GRB
SFR
-3
0.8 SFR z (1+z)
-1
<4) -3 Mpc)10
N(<z)/N( 0.6 -1 M yr 10-2
0.4 (
Hopkins & Beacom (2006)
Bouwens et al. (2009,2011)
-3 Hopkins & Beacom (2006) fit
0.2 10 SFR derived from GRBs
eff

total SFR
0.0 -4 Robertson & Ellis (2012)
0 1 2 3 4 10
0 1 2 3 4 5 6 7 8 9 10
z
redshift z
Fig.2: Cumulative distribution of 92 Swift GRBs with L >
iso Fig.3: The cosmic star formation history. The grey points are
1051erg s−1 in z=0−4 (stepwise solid line). The dashed line
taken from Hopkins & Beacom (2006), the dashed line shows
showsthe GRBrateinferredfromthe starformationhistoryof
their fitting result. The triangular points are from Bouwens et
Hopkins&Beacom(2006).ThesolidlineshowstheGRBrate
al. (2009,2011).The open circlesare taken from Robertson&
inferredfromthestarformationhistoryincluding(1+z)0.5evo-
Ellis(2012).Thefilled circlesarethe SFR derivedfromGRBs
lution.
in this work. The dotted line shows the effect SFR calculated
fromequation(19).Thethicksolidlineshowsthecombination
ofdashedanddottedlines.
OurresultissmallerthanthevaluesinKistleretal.(2009).This
maybeaccountforthedifferentGRBsample.TheGRBsample
observedbySwiftduring2005-2012isusedinthispaper.This by
biasmustbetakenintoaccountwhenonerelatestheGRBrate
totheSFR. Nth = hr˙ i A z2dz(1+z)d dVcom/dz, (7)
z1−z2 ∗ z1−z2 Zz1 1+z
3. TheSFRderivedfromGRBs wherehr˙ i istheaverageSFRdensityintheredshiftrange
∗ z1−z2
z −z . Representing the predicated numbers, Nth with the
Inthissection,weusethesamemethodasYu¨kseletal.(2008) 1 2 z1−z2
tocalculatetheSFRratefromGRBs. Becauseonlyverybright observedGRBcounts,Nzo1b−sz2,weobtaintheSFRintheredshift
burstscanbeseenfromallredshifts,weusethesameluminosity rangez1−z2,
cutsasKistler etal. (2009),as shownin Figure1.The number
countsinredshiftbinsz=4−5,5−6,6−7,7−8.5and8.5−10 hr˙ i = Nzo1b−sz2 14dzdVc1o+mz/dz(1+z)d r˙∗(z) . (8)
athreer1e0i,s4o,n2l,y1oannedG1RrBespneacmtievdelGy.RIBnt0h9e0r4e2d9shBifwtbitihnpohfo8t.o5m−et1r0ic, ∗ z1−z2 N1o−bs4 R zz12dzdVc1o+mz/dz(1+z)d
R
redshiftz∼9.4,althoughthereisalow-probabilitytailtosome- Inthecalculation,weassumethatthevalueofd isconstantatall
whatlowerredshifts(Cucchiaraetal.2011).Thebinchoiceof
redshiftrange.ThederivedSFRfromGRBsareshownasfilled
thisworkisdifferentwiththoseofRobertson&Ellis(2012).In
circlesinFigure3.Errorbarscorrespondto68%Poissonconfi-
thiswork,wechooseredshiftbinsuniforminz,andalsoensure
denceintervalsforthebinnedevents(Gehrels1986).Thehigh-
thatthenumberofGRBsineachbinsisequalorlargerthanone.
redshift SFRs obviously decrease with increasing redshifts, al-
WealsocalculatetheSFRusingbinchoiceofRobertson&Ellis
thoughanoscillationmayexist.WefindthattheSFRatz>4.48
(2012), and find that the result is a little difference compared isproportionalto(1+z)−3usingminimumc 2method,whichis
withcurrentbinchoice.TheGRBsinz=1−4actasa“control
shown as solid line in Figure 3. Because we use differentcos-
group”toconstraintheGRBtoSFRconversion,sincethisred-
mologicalparameterscomparingtoHopkins&Beacom(2006)
shiftbinhasbothgoodSFRmeasurementsandGRBcounts.We (W m=0.3,W L =0.7andH0=70kms−1 Mpc−1),SFRconver-
calculatethetheoreticallypredicatednumberofGRBsinthisbin
sion between different cosmology models must be considered.
as
Theconversionfactorforagivenredshiftrangecanbeexpressed
DW 4 r˙ (z) dV /dz as(Hopkins2004)
Nthe = D t dzF(z)e (z) ∗ com
1−4 4p Z1 hfbeami 1+z D2 (z)
r˙ (z)(cid:181) com , (9)
= A 4dz(1+z)d r˙ (z)dVcom/dz, (6) ∗ D3com(z+D z)−D3com(z−D z)
Z1 ∗ 1+z
where D is given in equation (3). At the redshift range z=
com
where A= D tDW F /4p hf i depends on the total observed 4−5,thevalueofconversionfactorsinthesetwocosmological
0 beam
time of Swift, D t, and the angular sky coverage, DW . The the- models are very similar. The relative error is less than 4%. So
oretical numberof GRBs in redshiftbin z −z can be written ourresultsareinsensitivetothechoiceofWMAP7cosmology.
1 2
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Wang:Thehigh-redshiftSFRderivedfromGRBsandcosmicreionization
The new determinationof SFR is slight smaller than the result where A = 0.06 is the normalization constant derived
Salpeter
givenbyKistleretal.(2009).Therearetworeasonsforthissitu- from,
ation.First,wederiveasmallerevolutionfactorindexd .Second,
mup
weupdatetheSwift GRBsample.Inpastthreeyears,Swift has f (M)dM=1. (11)
Z
observedmuch more GRBs with medium redshiftsthan GRBs mlow
with high redshifts. So the ratio Nobs /Nobs is smaller com-
z1−z2 1−4 Weusemlow=0.1M⊙andmup=120M⊙.Weconsiderstarswith
paredwithKistleretal.(2009).
massesbetween12M and30M .Becausethestarswithmasses
⊙ ⊙
Ishida et al. (2011) used the principal component analysis M ≥30M can produce GRBs through conventional collapse
⊙
method to measure the high-redshift SFR from the distribu- model(Woosley1993).SotherateofCHGis
tion of GRBs and found that the SFR at z∼9.4 could be up
to 0.01M⊙ yr−1 Mpc−3. Robertson & Ellis (2012) constrained R = k S (Z ,z)r˙ (z)P(x>x ) 30M⊙f (M)dM, (12)
theSFR usingGRBs byconsideringthe contributionof“dark” CHG CHG th ∗ cr Z
12M⊙
GRBs. They found that the high-redshift SFR derived from
GRBs can vary a factor of 4 using different values of d . Their wherekCHG withvalueabout10−5 istheCHGsformationeffi-
results for d =0.5 are shown as open circles in Figure 3. Our ciency,S (Zth,z)andP(x>xcr)arediscussedbelow.According
results can be marginally consistent with the open dots. Elliott toLanger&Norman(2006),thefractionalmassdensitybelong-
etal. (2012)chose43GRBsbyselectingGRBsthathavebeen ingtometallicitybelowagiventhresholdZth is
detectedbyGRONDwithin4hoursaftertheSwiftBATtrigger
andthatexhibitedanX-rayafterglow.Theyfoundthatthelinear S (Z ,z)= Gˆ[a 1+2,(Zth/Z⊙)2100.15b z], (13)
relationshipbetweenGRBrateandSFRusingthissmallsample. th G (a +2)
1
Johnsonetal.(2013)usedhigh-resolutioncosmologicalsimula-
tionstostudythehigh-redshiftSFR.Ourresultisconsistentwith where Gˆ and G are the incomplete and complete gamma func-
thatofJohnsonetal.(2013)atz≤10.Butatz≥10,theyfound tions,a 1 =−1.16is the power-lawindexin theSchechterdis-
the SFR is reduced by up to an order of magnitude due to the tribution function of galaxy stellar masses (Panter, Heavens &
molecule-dissociatingstellarradiation. Jimenez2004)andb =2istheslopeofthegalaxystellarmass-
metallicity relation (Savaglio et al. 2005; Langer & Norman
2006). We extrapolate the metallicity evolution up to redshift
z∼9.4.Observationshowsthatthismetallicityevolutioncabbe
4. GRBsfromrapidlyrotatingmetal-poorstarsand
useduptoz∼3(Kewley&Kobulnicky2007).Thisextrapola-
influenceonSFR tionhasbeenwidelyusedinliterature(Langer&Norman2006;
Li2008;Robertson& Ellis 2012).We set Z =0.004.As dis-
ThecollapsarmodelexplainsGRBformationviathecollapseof th
cussedbyYoon,Langer&Norman(2006),theGRBproduction
a rapidlyrotationmassiveironcoreintoa blackhole(Woosley
inCHESislimitedtometallicityZ ≤0.004.WithintheCHES,
1993).Thiscollapsemodelrequirestheinitialmassofthemas- th
the fraction of stars which forms a long GRB depends on the
sive stars with masses largerthan about30M (Woosley1993;
⊙ semi-convectivemixing, and the distribution functionof initial
Bissaldi et al. 2007). Yoon & Langer (2005) and Woosley &
stellar rotation velocities, D(v /v ), where v is the initial
Heger(2006)showedthatatlow metallicity,quasi-chemically- init K init
rotation velocity and v is the Keplerian velocity. We use the
homogeneousevolutionofrapidlyrotatingstarswithlowmasses K
v /v distributionfromYoon,Langer&Norman(2006)
canleadtotheformationofrapidlyrotatingheliumstarswhich init K
satisfy all the requirements of the collapsar scenario. Because l n
the rotation affects the evolution of stars significantly, espe- D(x)=BG (n )xn −1exp(−l x), (14)
ciallythroughrotationallyinducedchemicalmixing(Maeder&
Meynet 2000; Heger et al. 2000). The star remains chemically where l = 9.95, n = 2 and x ≡ v /v . The normalization
init K
homogeneousevolutionscenario(CHES). The CHESis recog- constant B = 3.2×105 is derived from ¥ D(x)dx = 1. They
0
nizedasapromisingpathtowardscollapsarsinconnectionwith found that this distribution can fit well wRith the observational
longGRBs.Yoon,Langer&Norman(2006)showedthatatlow data fromMokiem et al. (2006).Figure 4 shows the numerical
metallicity (Z ≤0.004),quasi-chemically-homogeneousevolu- valueof equation(14). In orderto produceGRBs, the valueof
tionofrapidlyrotatingstarswithmasseslargerthan12M⊙ can xshouldbelargerthan0.4(Yoon,Langer&Norman2006),so
leadtolongGRBs2.WecallthistypeofGRBsaschemicallyho- P(x>x )= ¥ D(x)dx=0.09.
cr 0.4
mogeneousGRBs(CHG)below.Ifstarsinthesamemassrange SotheexpRectednumberofCHGbetweenredshiftszandz+
havehighmetallicitiesandslowrotation,theywilldieastypeII d zcanbecalculatedas
supernovae(seeFigure3ofYoonetal.2006).Thispicturehas
alsobeenconfirmedbyobservation(Fruchteretal2006). Nexp (>L)=FD tDW z+d zdz ¥ dLF (L)R dVcom/dz,(15)
We study the rate of CHG from chemically homogeneous CHG 4p Zz ZL CHG 1+z
evolutionscenarioasfollows.Themostwidelyusedfunctional where F (L) is the luminosity function of GRBs. We use the
form for the initial mass function (IMF) is that proposed by
Schechter-functionform
Salpeter(1955):
b
1 L
F (L)= exp(−L/L ), (16)
f (M)=A M−2.35, (10) L L ⋆
Salpeter ⋆(cid:18) ⋆(cid:19)
whereb =−1.12andL =9×1052ergs−1(Wang&Dai2011).
2 Although the lower limit mass of a star with low metallicity that The integral ¥ dLF (L)⋆ equals to G (1+b , L), where G is the
caannalycosilsla.pTsheetobeGstRfiBttiinsgunpcaerartmaient.erBsuwtitlhlisshviaftluseligishtulnyimwphoenrtatnhteilnowouerr incompletegaRmLmafunction.Because1+b →L⋆0,wecanapprox-
limitmassischanged. imateG (1+b , L)→−(L)1+b /(1+b ).
L⋆ L⋆
4
Wang:Thehigh-redshiftSFRderivedfromGRBsandcosmicreionization

1.0
1.0
N=4000 fesc=0.1
0.8 N=4000 fesc=0.2
0.8
N=4000 fesc=0.3
y
babilit 0.6 QHII 0.6
pro
0.4 0.4
0.2
0.2
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30
vinit/vK redshift z
Fig.5:TheHIIfillingfactorQ asafunctionofredshiftcom-
Fig.4:Distributionoftheinitialrotationvalueofstars. HII
putedfordifferentvaluesof f .
esc
WedefineaneffectiveSFRr˙e∗ff,duetotheCHG,as fromGRBs (Kistler etal. 2013).GRBs arefoundto favorsub-
luminosity galaxies (Fynbo et al. 2003), so a larger fraction of
NCexHpG(>L) = zz+d ze r˙e∗ffdVcom(z′)/(1+z′). (17) theSFRwithinsuchhostswouldberevealedbyGRBs(Kistler
Nobs(>L) R 4e r˙ dV (z′)/(1+z′) etal.2009).WecanseethattheSFRinferredfromhigh-redshift
1−4 1 ∗ com
R GRBs can be well explained by combining equation (18) with
WeconsiderthestarformationhistoryfromHopkins&Beacom equation(19) forC∼125.TheoverallconventionallongGRB
(2006), formationefficiencyfrommassivestarsisabout<10−6(Zitouni
etal.2008;Li2008),whichissmallerthank .Thisindicates
CHG
(1+z)3.44, z<0.97, thatthesubclassofmassivestarswithlowmetallicityandchem-
r˙∗(z)(cid:181) (1+z)−0.26, 0.97<z<4.48, (18) icalhomogeneitymayproducelongGRBsmoreefficiently.

(1+z)−7.8, 4.48<z.

withr˙ (0)=0.02M yr−1Mpc−3.Forconvenience,herewefit 5. Implicationsforthecosmicreionization
∗ ⊙
the data by Nobs(> L) ∼ 60L−a with a ∼ 0.50. Substituting Determiningwhenandhowtheuniversewasreionizedbyearly
1−4 52
Eq.(15)intoEq.(17),wecanobtaintheeffectiveSFRas sources have been important questions for decades (Gunn &
Peterson 1965;Robertsonet al. 2010).Itis established thatin-
r˙e∗ff = F0D(−t(1LLt⋆−h)1b +)(b 1R1+4r˙z∗)(d 1N+obsz)(d>−1LdV)comDW4p RCHG(>Lth) tze≈rga6la.5c,tibcamseeddiounmst(rIoGnMg)Lyrea ioanbizsaotripotniomnafyrombenceoumtrpallehteyddrboy-
1−4 th gen along lines of sight to quasars at z>6 (Fan et al. 2001).
= CM yr−1Mpc−3 30M⊙f (M)dMr (z)(1+z)1+a −d +b As a measure of ionization,we follow the evolutionof the HII
⊙ Z12M⊙ ∗ 1 volume filling factor QHII = ne/nH, versus redshift, using the
2(a +b +1) SFR derivedfromGRBs (the solid line in Figure 3). The aver-
(1+z)1/2−1 age evolution of Q is found by numerical integration of the
× , (19) HII
2 rate of ionizingphotonsminusthe rate ofradiativerecombina-
!
tions(Madauetal.1999;Barkana&Loeb2001;Wyithe&Loeb
where the factor C ∼ 125 F k F1+a +b . The lumi- 2003;Yuetal.2012)
0 CHG,−5 lim,−8
n4op sDitLy(z)th2Frelismhoflodr aatgrievdesnhiflftuxz sceannsitibveitycaFlclimul.atFeodr athseLSthwi=ft dQdzHII = N˙nion −a BCnHQHII ddzt . (20)
satellite, we adopt the angular sky coverage of DW /4p ∼0.1, (cid:18) H (cid:19)
and the observation period D t ∼ 7.5 yr. In equation (19), Here,
t3h0eMi−n1t.e3g5r)a/l1.R315320MM=⊙⊙0f .(0M26).dMAlthisoupgrhopthoertrieonaarle tmoan(1y2fMac⊙−to1.r3s5−in N˙ion=(1+z)3r˙∗(z)Ng fesc/mp (21)
⊙
equation(19)maysubsumetheeffectinchangetheIMFintegral is the rate ofionizingUV photonsescapingfromthe stars into
forGRB production.The evolutionofr˙eff is shownasthe dot- theIGM,Ng isthenumberofionizingUVphotonsreleasedper
∗
tedlineinFigure3.Bouwensetal.(2009,2011)measuredhigh- baryonofthestars,(1+z)3 convertsthecomovingdensityinto
redsihftSFRusingcolor-selectedLymanbreakgalaxies(LBGs) proper density, r˙ (z) is proportional to (1+z)−3 at z > 4.48
∗
method. Their results are shown as triangular points in Figure and f is the escape fraction. The escape fraction is not well
esc
3. But LBG studies mainly probe the brightest galaxies. If the constrained.Atlowredshifts,observationsshowthattheescape
integration of UV luminosity functions down to M ≃−10, fraction from GRB hosts is about a few percent (Chen et al.
UV
the SFR inferred from LBG is consistent with that derived 2007;Fynboetal.2009).Butathighredshifts,the f islarger
esc
5
Wang:Thehigh-redshiftSFRderivedfromGRBsandcosmicreionization
Yoshidaetal. 2003).Hegeretal. (2003)andMe´sza´ros&Rees
(2010)showthatPopIIIstarscandieasGRBs. Theformation
rate of Pop III GRBs has been extensivelystudied (Campisi et
0.14
al.2011;deSouzaetal.2011).ThehighluminositiesofGRBs
make them detectable out to the edge of the visible universe
0.12
(Bromm&Loeb2002,2006;Wangetal.2012).SoGRBsmay
0.10 providetheinformationofSFRouttoz>20infuture.Theex-
WMAP9 (Hinshaw et al. 2012) trapolationofSFRtohighredshiftsmaybereasonable.Theopti-
e 0.08 caldepthisshowninFigure6.TheWMAPnine-yeardatagives
t =0.089±0.014 (Hinshaw et al. 2012), which is shown as
e
0.06
the shaded region.The combinationof Plack and WMAP data
0.04 N=4000 fesc=0.2 also givest e =0.089+−00..001124 (Plack Collaboration2013).So our
N=4000 fesc=0.3 GRB-inferredSFRcanreproducetheCMBopticaldepth.
0.02 N=4000 fesc=0.1
0.00 6. Summary
0 5 10 15 20 25 30
redshift z Using the GRB catalogs, we have constructed the cumulative
redshiftdistributionof110luminous(L >1051ergs−1)GRBs
iso
Fig.6: The optical depth t due to the scattering between the outtoredshiftz∼9.4.WefindthattheSwiftGRBsduring2005-
e
2012 are biased toward tracing the SFR, including a factor of
ionized gas and the CMB photonsis shown. The shade region
is givenby the nine-yearWMAP measurements(t =0.089± about(1+z)0.5.Correctingthisevolution,wederivethestarfor-
e
mationhistoryuptoz∼9.4usingSwiftGRBsample.Ourresults
0.014). The reionization history calculated from GRB-inferred
SFRcaneasilyreacht fromWMAPnine-yeardata. showthatnosteepdropexistsintheSFRuptoatleastz∼9.4.
e
Inordertoexplainthehigh-redshiftGRBrateexcess,theGRBs
producedbyrapidlyrotatingmetal-poorstarswithlowmassare
(Inoueetal.2005;Robertsonetal.2010).Recentestimatessug- considered. The collapsar model explains GRB formation via
gest that the clumping factor C ≈ 1−6 (Bolton & Haehnelt thecollapseofamassivestarwithM>30M⊙intoablackhole.
2007;Pawlik et al. 2009).We adoptC=3 in this paper. n is We consider that at low metallicity, quasi-chemically homoge-
H
theproperdensityofhydrogen,anda =1.63×10−13cm3 s−1 neous evolution of rapidly rotating stars with mass larger than
B
is the recombination rate for an electron temperature of about 12M⊙ can lead to the formation of GRBs. The low metallcity
104K.Becausethemassincollapsedobjectsisstillsmallathigh and rapid rotation can lead to efficiently produceGRBs in two
redshift,theIGMcontainsmostofthecosmologicalbaryons,at ways. First, rapid rotation keeps the stars chemically homoge-
meandensity neous and thus avoids the formation of a massive envelope,so
stellarcoreisfreeofspin-downduetomagneticcore-envelope
r =W r (1+z)3=4.24×10−31(1+z)3gcm−3. (22) coupling.Second,thestellarwindisweakatlowmetallicity,so
b b cr
this reduces spin-down due to stellar winds. Our fitting results
We adopt the parameters from WMAP seven-year, W bh2 = confirmthisidea.We alsocalculatethereionizationhistoryus-
0.02255±0.00054andW mh2=0.1352±0.0036(Komatsuetal. ingtheGRB-inferredSFR, andfindthatthisSFRcanmaintain
2011).The criticaldensity is r cr =1.8785×10−29 h2 g cm−3. cosmicreionizationover6<z<10andreproducetheobserved
Themeanhydrogennumberdensity, optical depth of Thomson scattering to the cosmic microwave
background.
r (1−Y)
n = b =1.905×10−7(1+z)3cm−3, (23)
H m Acknowledgements. Wethanktheanonymousrefereeforveryusefulcomments
H
andsuggestions. WethankK.S.ChengandZ.G.Daiforfruitfuldiscussion.
where Y = 0.2477±0.0029 is the helium mass fraction WeacknowledgetheuseofpublicdatafromtheSwiftdataarchive.Thisworkis
(Peimbertetal.2007).AfterthevaluesofNg and fesc aregiven, supportedbytheNationalNaturalScienceFoundationofChina(grant11103007
and11033002).
theevolutionoftheHIIvolumefillingfactorQ canbenumer-
HII
ically calculated from equation (20). In Figure 5, we show the
evolution of QHII as a function of redshift. For Ng =4000 and References
f =0.2,theIGMwascompletelyionizedatz ∼8.5.
esc rei
Barkana,R.,&Loeb,A.2001,Phys.Rep.,349,125
The cosmic microwave background (CMB) optical depth
Barkana,R.,&Loeb,A.2004,ApJ,601,64
back to redshift z can be written as the integral of nes Tdℓ, the Bissaldi,E.,Calura,F.,Matteucci,F.,Longo,F.&Barbiellini,G.2007,A&A,
electron density times the Thomsoncross section along proper 471,585
length, Bolton,J.S.&Haehnelt,M.G.,2007,MNRAS,382,325
Bouwens,R.J.,etal.2009,ApJ,705,936
z Bouwens,R.J.,etal.2011,Nature,469,504
t e(z)=Z ne(z)s T(1+z′)−1[c/H(z′)]dz′. (24) Bouwens,R.J.,Illingworth,G.D.,Oesch,P.A.,etal.2012,ApJ,752,L5
0 Bromm,V.,&Loeb,A.2002,ApJ,575,111
Bromm,V.,&Loeb,A.2006,ApJ,642,382
Soafterweobtaintheredshiftevolutionofn (z),theCMBop-
e Butler,N.R.,Bloom,J.S.,&Poznanski,D.2010,ApJ,711,495
tical depth as a function of redshift can be calculated. More Butler,N.R.,Kocevski,D.,Bloom,J.S.,&Curtis,J.L.2007,ApJ,671,656
recently, the Planck team has released the latest result on cos- Campisi,M.A.,Maio,U.,Salvaterra,R.&Ciardi,B.2011,MNRAS,416,2760
mological parameters (Plack Collaboration 2013). In the cal- Cao,X.F.,Yu,Y.W.,Cheng,K.S.&Zheng,X.P.2011,MNRAS,416,2174
culation, we extrapolate the SFR as (1+z)−3 to z ∼30. The Chen,H.W.,Prochaska,J.X.&Gnedin,N.Y.,2007,ApJ,667,L125
Cheng,K.S.,Yu,Y.,&Harko,T.2010,Phys.Rev.Lett.,104,241102
firststars,so-calledPopulationIII(PopIII)starsarepredicated
Ciardi,B.,&Loeb,A.2000,ApJ,540,687
to have formed at z > 20 in minihalos (Tegmark et al. 1997; Coe,D.,Zitrin,A.,Carrasco,M.,etal.2013,ApJ,762,32
6
Wang:Thehigh-redshiftSFRderivedfromGRBsandcosmicreionization
Cucchiara,A.etal.2011,ApJ,736,7 Yoshida,N.,Abel,T.,Hernquist,L.,Sugiyama,N.2003,ApJ,592,645
Dai,Z.G.,Liang,E.W.,&Xu,D.2004,ApJ,612,L101 Yu¨ksel,H.,Kistler,M.D.,Beacom,J.F.&Hopkins,A.M.2008,ApJ,683,L5
deSouza,R.S.,Yoshida,N.&Ioka,K.2011,A&A,533,A32 Yu,Y.W.etal.,2012,JCAP,07,023
Elliott,J.,Greiner,J.,Khochfar,S.,Schady,P.,Johnson,J.L.&Rau,A.2012, Zitouni,H.etal.2008,MNRAS,386,1597
A&A,539,A113
Fan,X.,etal.2001,AJ,122,2833
Fruchter,A.S.,etal.2006,Nature,441,463
Fynbo,J.P.U.,etal.2003,A&A,406,L63
Fynbo,J.P.U.,etal.2009,ApJS,185,526
Gehrels,N.1986,ApJ,303,336
Gehrels,N.,Ramirez-Ruiz,E.&Fox,D.B.,2009,ARA&A,47,567
Gunn,J.E.,&Peterson,B.A.,1965,ApJ,142,1633
Heger,A.,etal.2003,ApJ,591,288
Heger,A.,Langer,N.,&Woosley,S.E.2000,ApJ,528,368
Hinshaw,G.,etal.2012,arXiv:1212.5226
Hjorth,J.,etal.2003,Nature,423,847
Hopkins,A.M.2004,ApJ,615,209
Hopkins,A.M.,&Beacom,J.F.2006,ApJ,651,142
Inoue,A.K.,etal.2005,A&A,435,471
Inoue,S.,Omukai,K.,&Ciardi,B.2007,MNRAS,380,1715
Ishida,E.E.O.,deSouza,R.S.&Ferrara,A.2011,MNRAS,418,500
Johnson,J.L.,DallaVecchia,C.&Khochfa,S.2013,MNRAS,428,1857
KewleyL.&KobulnickyH.A.,2007,indeJongR.S.,ed.,IslandUniverses:
StructureandEvolutionofDiscGalaxies.Springer-Verlag,Dordrecht,p.435
Kistler,M.D.,Yu¨ksel,H.,Beacom,J.F.&Stanek,K.Z.2008,ApJ,673,L119
Kistler, M.D.,Yu¨ksel,H.,Beacom,J.F.,Hopkins,A.M.&Wyithe,J.S.B.
2009,ApJ,705,L104
Kistler,M.D.,Yu¨ksel,H.,&Hopkins,A.M.2013,arXiv:1305.1630
Komatsu,E.,etal.2011,ApJS,192,18
Lamb,D.Q.,&Reichart,D.E.2000,ApJ,536,1
Langer,N.1998,A&A,329,551
Langer,L.,&Norman,C.A.2006,ApJ,638,L63
Le,T.,&Dermer,C.D.2007,ApJ,661,394
Li,L.X.2008,MNRAS,388,1487
MacFadyen,A.I.,Woosley,S.E.,&Heger,A.2001,ApJ,550,410
Madau,P.,Haardt,F.,&Rees,M.J.1999,ApJ,514,648
Maeder,A.,&Meynet,G.2000,ARA&A,38,143
McQuinn,M.,etal.2008,MNRAS,388,1101
Me´sza´ros,P.&Ress,M.J.2010,ApJ,715,967
Mokiem,M.R.,deKoter,A.,Evans,C.J.,etal.2006,A&A,456,1131
Oesch,P.A.,Bouwens,R.J.,Illingworth,G.D.,etal.2013,arXiv:1301.6162
Panter,B.,Heavens,A.F.,&Jimenez,R.2004,MNRAS,355,764
Pawlik,A.,Schaye,J.&vanScherpenzeel,E.,2009,MNRAS,394,1812
Peimbert,M.,Luridiana,V.,&Peimbert,A.2007,ApJ,666,636
PlackCollaborationXVI.2013,submittedtoA&A
Porciani,C.&Madau,P.2001,ApJ,548,522
Qin,S.F.,Liang,E.W.,Lu,R.J.,Wei,J.Y.&Zhang,S.N.2010,MNRAS,406,
558
Raskin,C.,Scannapieco,E.,Rhoads,J.,&DellaValle,M.2008,ApJ,689,358
Robertson,B.E.&Ellis,R.S.2012,ApJ,744,95
Robertson,B.E.,Ellis,R.S.,Dunlop,J.S.,McLure,R.J.,&Stark,D.P.2010,
Nature,468,49
Salpeter,E.E.1955,ApJ,121,161
SalvaterraR.&ChincariniG.,2007,ApJ,656,L49
Savaglio,S.,2006,NewJ.Phys,8,195
Savaglio,S.,etal.2005,ApJ,635,260
Schaefer,B.E.2007,ApJ,660,16
Stanek,K.Z.,etal.2003,ApJ,591,L17
Svensson,K.M.,etal.2010,MNRAS,405,57
Tanvir,N.R.,etal.,2004,MNRAS,352,1073
Tegmark,M.,Silk,J.,Rees,M.J.,Blanchard,A.,Abel,T.,&Palla,F.1997,ApJ,
474,1
Tho¨ne,C.C.,etal.2008,ApJ,676,1151
Totani,T.1997,ApJ,486,L71
Virgili,F.J.,Zhang,B.,Nagamine,K.&Choi,J.H.2011,MNRAS,417,3025
Wanderman,D.,&Piran,T.2010,MNRAS,406,1944
Wang,F.Y.,Bromm,V.,Greif.T.H.,Stacy,A.,Dai,Z.G.,Loeb,A.andCheng,
K.S.2012,ApJ,760,27
Wang,F.Y.,&Dai,Z.G.2009,MNRAS,400,L10
Wang,F.Y.,&Dai,Z.G.2011,ApJ,727,L34
Wang,F.Y.,Qi,S.&Dai,Z.G.2011,MNRAS,415,3423
Wijers,R.A.M.J.,Bloom,J.S.,Bagla,J.S.&Natarajan,P.1998,MNRAS,
294,L13
Woosley,S.E.1993,ApJ,405,273
Woosley,S.E.&Heger,A.2006,ApJ,643,914
Wyithe,J.S.B.&Loeb,A.,2003,ApJ,586,693
Yoon,S.C.&Langer,L.,2005,A&A,443,643
Yoon,S.C.,Langer,L.&Norman,C.,2006,A&A,460,199
7
1.0
0.8
y
bilit 0.6
a
b
o
pr
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
vinit/vK