Population III. 1 and III. 2 Gamma-Ray Bursts: Constraints on the Event

arXiv:1105.2395v3 [astro-ph.CO] 12 Aug 2011 rmohrsas.W mlyasm-nltclapoc tha approach semi-analytical a employ We stars). other from Pplto I;PpII tr.TePpIIGB ol esup be could GRBs III Pop The stars. III) Pop III; (Population rvdn nqepoeo h ihrdhf Universe. high-redshift the of Methods. probe unique a providing Aims. XX Xxxxx Released Japan 277-8583, 3 Chiba Kashiwa, Kashiwanoha, ervraiiyadmsl ihu Rs(rhn) hc a which of (orphans), maximum GRBs observe to without expect mostly we and variability year rg Kmsao akv21;Tm ta.2011; al. et Toma 2010; Barkov & (Komissarov erage i.e., ergs, nertdoe at over integrated vlto fteitraatcmedium. Results. intergalactic the of evolution ytecretrdotasetsace.W expect We searches. transient radio o current Interestingly, the missions. GRB by future for targets suitable 2 1 [email protected] hs oa stoi nrycudbe could energy (GRBs), isotropic bursts total gamma-ray collapsar whose produce may stars so-called the stars. of observations III) there (Pop direct populations, any III observa- stellar Population been few early yet a the not has are probe 2003; to there Palla ways Although & tional Omukai 2006). predom- 2002; al. were al. et than they Yoshida et less that (Abel was and massive Universe years inantly the million hundred of stars the few age first a the on the elements when that based heavy formed predict were studies first model Theoretical the cosmological emitting 2009). standard by producing evolution al. and et cosmic (Bromm light early played the first have in to the thought role are crucial Universe the a in stars first The Introduction 1. words. Key edopitrqet to requests offprint Send osrit nteeetrt o uuerdoadXrysurv X-ray and radio future for rate event the on Constraints E hoyCne n h rdaeUiest o dacdS Advanced for University Graduate the and Center Theory KEK nttt o h hsc n ahmtc fteUies,T Universe, the of Mathematics and Physics U the Cidade Matão 1226, for do Institute Rua São Paulo, de Universidade IAG, srnm Astrophysics & Astronomy uy2,2018 25, July eety thsbe rpsdta asv o III Pop massive that proposed been has it Recently, ecluaetetertcleetrt fgmarybursts gamma-ray of rate event theoretical the calculate We eso htPpII2GB cu oeta 0 ie oefreq more times 100 than more occur GRBs III.2 Pop that show We ecnie ohtes-aldPpII1sas(primordial) stars III.1 Pop so-called the both consider We ouainII am-a us;Rdolns X-rays lines; Radio burst; Gamma-ray III; Population & > z reso antd bv av- above magnitude of orders 2 ouain I. n I. am-a bursts: gamma-ray III.2 and III.1 Populations 0frPpII1wt XS,and EXIST, with III.1 Pop for 10 aalS eSuae-mail: Souza de S. Rafael : aucitn.article˙AA˙2011˙17242 no. manuscript < N .S eSouza de S. R. 0GB e eritgae vrat over integrated year per GRBs 20 E iso & 10 1 ∼ , < N 2 55 rotmsi oe rdcsa vn aeta sarayco already is that rate event an predicts model optimistic ur ABSTRACT .Yoshida N. , 10 − 57 0 − . o o I. Rsprya nertdoe at over integrated year per GRBs III.2 Pop for 2 osdr nooeeu yrgnrinzto n chemi and reionization hydrogen inhomogeneous considers t 10 reegtcwt h stoi nryu to up energy isotropic the with er-energetic diIsiue o dacdSuy nvriyo oy,5- Tokyo, of University Study, Advanced for Institutes odai nu ta.(07 aclt iedpnet broad-band time-dependent, calculate (2007) al. et Inoue yVLA by n hwta hycnb bevdotto out observed be can they GRBs high-redshift Mészáros that & of Ioka show afterglows 2004). 2000; radio and al. the high-redshift Loeb et study Yonetoku & the (2005) Ciardi 2004; of al. 2000; et probe Reichart Gou unique & (Lamb a Universe provide will shifts 2011). al. et Nava inas ml htteGBfrainrt osnot does rate formation GRB rela- to the luminosity decrease that energy-peak significantly imply be- observa- peak also The star spectral itself 2011). tion first the envelope al. et of the the Nagakura of tions out 2011; en- accretion Ioka break hydrogen & longlasting supergiant (Suwa can the a jet of has cause GRB star the III Pop velope, the 2011). if Ioka & Even Suwa 2010; Barkov 2010; Mészáros Rees & edtcal yAM,EL,LFR n K,while SKA, and LOFAR, EVLA, ALMA, by detectable re 4 iestai,CP05890 õPuo P Brazil SP, São Paulo, 05508-900, niversitária, CEP 3 2 1 uis(oedi,Tuua3500,Japan 305-0801, Tsukuba (Sokendai), tudies ai fegosabove afterglows radio bevtoso uheegtcGB tvr ihred- high very at GRBs energetic such of Observations http://www.skatelescope.org/ http://www.vla.nrao.edu/ http://www.lofar.org/index.htm n o I. tr pioda u ffce yradiation by affected but (primordial stars III.2 Pop and GB)fo h olpeo asv first-generation massive of collapse the from (GRBs) 1 n .Ioka K. and , 1 LOFAR, , etyta o I. Rs n hssol be should thus and GRBs, III.1 Pop than uently > z o o I. and III.2 Pop for 6 2 3 n SKA and ∼ z . J ntesywith sky the on mJy 0.3 ∼ 2(oeoue l 2004; al. et (Yonetoku 12 3 seas oa2003). Ioka also (see E < N iso > z & 0 10 . with 6

8pryear per 08 c 55 nstrained z − S 2018 ESO 57 eys ∼ Swift ergs, ∼ 1-5 cal 30 1 . 2 de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts afterglow spectra of high-redshift GRBs. They suggest 2. Gamma-ray burst rate that spectroscopic measurements of molecular and atomic We assume that the formation rate of GRBs is propor- absorption lines due to ambient protostellar gas may be tional to the star formation rate (Totani 1997; Ishida et al. possible to z 30 and beyond with ALMA4, EVLA5, and 2011). The number of observable GRBs per comoving vol- SKA. In the future,∼ it will be also promising to observe the ume per time is expressed as GRB afterglows located by gamma-ray satellites such as Swift 6 7 8 9 ∞ , SVOM , JANUS, and EXIST . Clearly, it is im- obs Ωobs portant to study the rate and the detectability of Pop III ΨGRB(z)= ηGRB ηbeam Ψ∗(z) p(L)d log L, 4π Zlog Llim(z) GRBs at very high redshifts. (1) There have been already a few observations of GRBs at where ηGRB is the GRB formation efficiency (see section high redshifts. GRB 090429B at z =9.4 (Cucchiara et al. 2.6), ηbeam the beaming factor of the burst, Ωobs the field 2011) is the current record-holding object, followed by of view of the experiment, Ψ∗ the cosmic star formation a z = 8.6 galaxy (Lehnert et al. 2010), GRB 090423 at rate (SFR) density, and p(L) the GRB luminosity function z =8.26 (Salvaterra et al. 2009; Tanvir et al. 2009), GRB in X-rays to gamma rays. The intrinsic GRB rate is given 080913 at z = 6.7 (Greiner et al. 2009), GRB 050904 by at z = 6.3 (Kawai et al. 2006; Totani et al. 2006), and ΨGRB(z)= ηGRBΨ∗(z). (2) the highest redshift quasars at z =7.085 (Mortlock et al. The quantity L (z) is the minimum luminosity threshold 2011) and z = 6.41 (Willott et al. 2003). Chandra et al. lim to be detected, which is specified for a given experiment. (2010) report the discovery of radio afterglow emission The nonisotropic nature of GRBs gives η 0.01 from GRB 090423 and Frail et al. (2006) for GRB 050904. beam 0.02 (Guetta et al. 2005). Using a radio transient∼ survey,− Observations of afterglows make it possible to derive the Gal-Yam et al. (2006) place an upper limit of η . physical properties of the explosion and the circumburst beam 0.016. Given the average value of jet opening angle θ 6◦ medium. It is intriguing to search for these different sig- (Ghirlanda et al. 2007) and η 5.5 10−3, we∼ set natures in the GRB afterglows at low and high redshifts. beam ∼ × ηbeam = 0.006 as a fiducial value. The adopted values of The purpose of the present paper is to calculate the Ωobs are 1.4, 2, 4, and 5 for Swift, SVOM, JANUS, and Pop III GRB rate detectable by the current and fu- EXIST, respectively (Salvaterra et al. 2008). ture GRB missions (see also Campisi et al. 2011). We consider high-redshift GRBs of two populations following Bromm et al. (2009). Pop III.1 stars are the first- 2.1. The number of collapsed objects generation stars that form from initial conditions deter- We first calculate the star formation rate (SFR) at mined cosmologically. Pop III.2 stars are zero-metallicity early epochs. Assuming that stars are formed in col- stars but formed from a primordial gas that was influenced lapsed dark matter halos, we follow a popular prescrip- by earlier generation of stars. Typically, Pop III.2 stars tion in which the number of collapsed objects is calculated are formed in an initially ionized gas (Johnson & Bromm by the halo mass function (Hernquist & Springel 2003; 2006; Yoshida et al. 2007). The Pop III.2 stars are thought Greif & Bromm 2006; Trenti & Stiavelli 2009). We adopt to be less massive ( 40–60M⊙) than Pop III.1 stars ( ∼ ∼ the Sheth-Tormen mass function, fST, (Sheth & Tormen 1000M⊙) but still massive enough for producing GRBs. 1999) to estimate the number of dark matter halos, We have calculated the GRB rate for these two popu- nST(M,z), with mass less than M per comoving volume lations separately for the first time. The rest of the paper at a given redshift: is organized as follows. In Sect. 2, we describe a semi- analytical model to calculate the formation rate of pri- 2 p 2 mordial GRBs. In Sect. 3, we show our model predic- 2a1 σ δc a1δc fST = A 1+ 2 exp 2 , (3) tions and calculate the detectability of Pop III GRBs r π a1δc σ − 2σ by future satellite missions and by radio observations. In where A = 0.3222,a = 0.707,p = 0.3 and δ = 1.686. Sect. 4, we discuss the results and give our concluding 1 c The mass function f can be related to the n (M,z) as remarks. Throughout the paper we adopt the standard ST ST

Λ cold dark matter model with the best ﬁt cosmologi- M dnST(M,z) 10 cal parameters from Jarosik et al. (2011) (WMAP-Yr7 ), fST = −1 , (4) ρm d ln σ −1 −1 Ωm =0.267, ΩΛ =0.734, and H0 = 71km s Mpc . where ρm is the total mass density of the background

4 Universe. The variance of the linear density ﬁeld σ(M,z) www.alma.nrao.edu/ is given by 5 http://www.aoc.nrao.edu/evla/ 6 2 ∞ http://swift.gsfc.nasa.gov/docs/swift/swiftsc.html 2 b (z) 2 2 7 σ (M,z)= k P (k)W (k,M)dk, (5) http://www.svom.fr/svom.html 2 2π Z0 8 http://sms.msfc.nasa.gov/xenia/pdf/CCE2010/Burrows.pdf 9 http://exist.gsfc.nasa.gov/design/ where b(z) is the growth factor of linear perturbations 10 http://lambda.gsfc.nasa.gov/product/map/current/ normalized to b = 1 at the present epoch, and W (k,M) de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts 3

4 is the Fourier-space top hat ﬁlter. To calculate the power assume that M(Tvir = 10 K,z) is the minimum halo mass spectrum P (k), we use the CAMB code11 for our assumed for Pop III.2 star formation. ΛCDM cosmology. The collapsed fraction of mass, Fcol(z), available for Pop III star formation is given by

MT 4 2.2. H2 Photodissociation vir=10 K III.1 1 F (z)= dMMnST(M,z) (9) col ρ Z The star formation efficiency in the early Universe largely m MH2 depends on the ability of a primordial gas to cool and con- for Pop III.1 stars, and dense. Hydrogen molecules (H2) are the primary coolant in a gas in small mass “minihalos”, and are also fragile 1 ∞ F III.2(z)= dMMn (M,z) (10) to soft ultraviolet radiation, and thus a ultraviolet back- col ST ρm ZMT 4 ground in the Lyman-Werner (LW) bands can easily sup- vir=10 K press the star formation inside minihalos. We model the for Pop III.2. Using the above criteria, the SFR of Pop III dissociation effect by setting the minimum mass for halos stars can be written as that are able to host Pop III stars (Yoshida et al. 2003). III.1 III.1 dFcol For the minimum halo mass capable of cooling by Ψ (z)=(1 Q (z))(1 ζ(z, v ))ρ f f∗ ∗ − HII − wind m b dt molecular hydrogen in the presence of a Lyman-Werner (11) (LW) background, we adopt a fitting formula given by for Pop III.1 stars, and Machacek et al. (2001) and Wise & Abel (2005), which III.2 also agrees with results from O’Shea & Norman (2008): III.2 dFcol Ψ (z)= Q (z)(1 ζ(z, v ))ρ f f∗ (12) ∗ HII − wind m b dt fcd 5 5 0.47 MH2 = exp (1.25 10 +8.7 10 FLW,−21), (6) for Pop III.2. Here, ζ(z, v ) represents the global fill- 0.06 × × wind ing fraction of metals via galactic winds (see section 2.4), where FLW,−21 = 4πJLW is the flux in the LW band in QHII(z) the volume filling fraction of ionized regions (see −21 −1 −1 −2 −1 units of 10 erg s cm Hz , and fcd the fraction of section 2.3), and fb is the baryonic mass fraction. For the gas that is cold and dense. We set fcd = 0.02 as a con- star formation efficiency, we use the value f∗ =0.001 as a servative estimate. We compute the LW flux consistently conservative choice (Greif & Bromm 2006) and f∗ = 0.1 with the comoving density in stars ρ∗(z) via a conversion (Bromm & Loeb 2006) as an upper limit. The latter choice efficiency ηLW (Greif & Bromm 2006): is not strictly consistent with the assumption made in Eq. (6). We explore a model with f∗ = 0.1 simply to show a hc 3 very optimistic case. JLW = ηLWρ∗(z)(1 + z) . (7) 4πmH

Here, ηLW is the number of photons emitted in the LW 2.3. Reionization bands per stellar baryon and mH is the mass of hydrogen. Inside growing Hii regions, the gas is highly ionized, and The value of ηLW depends on the characteristic mass of the temperature is 104 K, so the formation of Pop III.1 the formed primordial stars, but the variation is not very ∼ large for stars with masses greater than ten solar masses stars is terminated according to our definition. The for- 4 mation rate of Pop III.1 is reduced by a factor given by (Schaerer 2002). We set ηLW = 10 for both Pop III.1 and Pop III.2 for simplicity. the volume filling fraction of ionized regions, QHII(z). We Next we calculate the stellar mass density as follow Wyithe & Loeb (2003) to calculate the evolution of QHII(z) as ′ dt ′ ρ∗(z)= Ψ∗(z ) ′ dz . (8) dQHII Nion dFcol C 0 Z dz = αB n QHII, (13) dz 0.76 dt a3 H − For a given z, the integral is performed over the maxi- whose solution is mum distance that an LW photon can travel before it is ∞ ′ redshifted out of the LW bands. The mean free path of ′ dt Nion dFcol F (z ,z) QHII(z)= dz ′ e , (14) LW photons at z = 30 is 10 Mpc (physical). Photons Zz dz 0.76 dt ∼ 7 travel over the mean free path in 10 yr (Mackey et al. where 2003). Halos with virial temperature∼ less than 104 Kelvin 0 2 αBn cool almost exclusively by H2 line cooling, and produce F (z′,z)= H C[f(z′) f(z)], (15) mostly massive stars. We adopt the mass of such halos, −3 √ΩmH0 − 4 M(Tvir = 10 K,z), as an upper limit of halos that pro- and duce Pop III.1 stars. In larger halos, the gas is ionized at 1 Ωm virialization, and thus the formed stars have, according f(z)= (1 + z)3 + − . (16) r Ωm to our definition, similar properties to Pop III.2 stars. We Here we have assumed the primordial fraction of hy- 11 http://camb.info/ drogen of 0.76. In the above equations, N N f∗f is ion ≡ γ esc 4 de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts

1.0 Then we can express fchem, the ratio of gas mass enriched by the wind to the total gas mass in each halo, as 0.8 3 4π Rwind fchem(M,z,vwind)= , (18) 0.6 3 V

L H z H II

H 3

Q where VH RH is the volume of each halo. The halo radius 0.4 ∝ RH can be approximated by

1/3 0.2 3M RH(M)= . (19) 4π 180ρm 0.0 × 0 5 10 15 20 25 30 Equation (18) takes the self-enrichment of each halo into z account. The next step is to evaluate the average metal- Fig. 1. Reionization history calculated using our model. licity over cosmic scales. The fraction of cosmic volume The blue line is our model prediction, and the dotted black enriched by the winds can then be written as line is the best fit of CAMB code. 1 ζ(z, vwind)= dMfbf∗fchem(M,z,vwind)MnST(M,z). ρm Z an efficiency parameter that gives the number of ionizing (20) photons per baryon, where fesc is the fraction of ioniz- Although this may appear a significant oversimplification, ing photons able to escape the host galaxy, and Nγ is the the model with vwind as a single parameter indeed provides time averaged number of ionizing photons emitted per unit good insight into the impact of the metal enrichment. 0 −7 −3 We adopt three different values of vwind and exam- stellar mass formed. The quantity nH =1.95 10 cm is the present-day comoving number density of× hydrogen, ine the effect of metal enrichment quantitatively. We as- −13 3 −1 αB = 2.6 10 cm s is the hydrogen recombination sume that Pop III stars are not formed in a metal-enriched × 2 2 region, regardless of the actual metallicity. Even a sin- rate, and C = nH /n¯H the clumping factor. We use the average value Ch =i 4 (see Pawlik et al. (2009) for detailed gle pair instability supernova can enrich the gas within discussion about redshift dependence of C). We set the a small halo to a metallicity level well above the criti- 4 values fesc =0.7,f∗ =0.01, and Nγ =9 10 as fiducial cal metallicity (see e.g. Schneider et al. (2006)). We effec- values (Greif & Bromm 2006). × tively assume that the so-called critical metallicity is very In Fig. 1 we show the reionization history calculated low (Schneider et al. 2002, 2003; Bromm & Loeb 2003; using our model in comparison with a fitting function that Omukai et al. 2005; Frebel et al. 2007; Belczynski et al. is the default parametrization of reionization in CAMB 2010). (Lewis et al. 2000). Figures 2 and 3 show the star formation rate (SFR) history for both Pop III.1 and Pop III.2 considering three different values of the galactic wind, vwind = 50, 75, 100 2.4. Metal enrichment km/s. Figure 2 shows that the metal enrichment has lit- We need to consider metal enrichment in the intergalactic tle influence on Pop III.1. This is because Pop III.1 for- medium (IGM) in order to determine when the formation mation is terminated early due reionization. In Fig. 3 of primordial stars is terminated (locally) and when the we compare the Pop III.2 SFR history with a compila- star formation switches from the Pop III mode to a more tion of independent measures from Hopkins & Beacom (2006) up to z 6 and from observations of color- conventional one. ≈ It is thought that Pop III stars do not generate strong selected Lyman break galaxies (Mannucci et al. 2007; stellar winds, and thus the main contribution to the Bouwens et al. 2008, 2011), Lyα Emitters (Ota et al. metal pollution comes from their supernova explosions. 2008), UV+IR measurements (Reddy et al. 2008), and Madau et al. (2001) argue that pregalactic outflows from GRB observations (Chary et al. 2007; Yüksel et al. 2008; the same primordial halos that reionize the IGM could Wang & Dai 2009) at higher z (hereafter, these will be re- also pollute it with a substantial amount of heavy ele- ferred to as H2006, M2007, B2008, B2011, O2008, R2008, ments. To incorporate the effect of metal enrichment by C2007, Y2008, and W2009, respectively). The optimistic galactic winds, we adopt a similar prescription to Johnson case for Pop III.2 is chosen to keep the SFR always below (2010) and Furlanetto & Loeb (2005). the observationally determined SFR at z < 8. We assume that star-forming halos (“galaxies”) launch We compared our model results with the SFRs estimated by other authors in the literature Bromm & Loeb a wind of metal-enriched gas at z∗ 20. The metal- enriched wind propagates outward from∼ a central galaxy (2006), Tornatore et al. (2007), Trenti & Stiavelli (2009), and also see Naoz & Bromberg (2007). It is important with a velocity vwind, traveling over a comoving distance to note that Pop III formation can continue to low red- Rwind given by shifts (z < 10) depending on the level of metal enrich- z dt ment. Tornatore et al. (2007) use cosmological simula- R = v (1 + z′) dz′. (17) wind wind ′ tions to show that, because of limited efficiency of heavy Zz∗ dz de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts 5

0.01 1

ª BL06 ª § ì 0.1 BL06 æ ø ì§ L L æ ø ô

3 0.001 3 - - æ 0.01 ø æ Mod Op

Mpc Mpc æ

1 1 HB2006 ìY2008

- TS09 - ò TS09 0.001 § C2007 øW2010

yr yr ò - ò 4 ô 10 M2007 B2011

M M æ B2008 50 km/s H H 10-4 Mod Op ò O2008 100 km/s

SFR SFR ª R2008 -5 T07 -5 10 75 km/s 10 T07 50 km/s 100 km/s 10-6 0 5 10 15 20 25 30 35 0 5 10 15 20 z z Fig. 2. Pop III.1 star formation rate. Calculated for for Fig. 3. Pop III.2 star formation rate. Calculated for three weak and strong chemical feedback models and a moder- different chemical feedback models; vwind = 50 km/s, red ate star formation efficiency with f∗ = 0.05. The results line; vwind = 75 km/s, blue line; and vwind = 100 km/s, are shown for vwind = 50 km/s , red line; and 100 km/s, green line. We also show the theoretical SFRs in the liter- blue line. We also show the theoretical SFRs in the literature, from Bromm & Loeb 2006 (Pop III.1+III.2), dot- ature, from Bromm & Loeb 2006 (Pop III.1+III.2), dotted black line; Trenti & Stiavelli 2009 (Pop III.2), dashed ted black line; Trenti & Stiavelli 2009 (Pop III.2), dashed orange line; and Tornatore et al. 2007 (Pop III.1+III.2), orange line; and Tornatore et al. 2007 (Pop III.1+III.2), dot-dashed brown line. The purple line is our optimistic dot-dashed brown line. The purple line is our optimistic model where we assume a very high star formation effi- model where we assume a very high star formation efficiency, f∗ 0.01, and low chemical enrichment, v = ∼ wind ciency, f∗ 0.1, and low chemical enrichment, vwind = 50 50 km/s. The light points are independent SFR determi- km/s. ∼ nations compiled from the literature. element transport by outflows, Pop III star formation 0.1 continues to form down to z = 2.5 (which intriguingly Pop III.2 L matches our model with v = 50 km/s in Fig. 3). 3 wind - 0.01 The SFR of Tornatore et al. (2007) has a peak value of Optimistic case Mpc

− − −3 1 5 1 - 10 M⊙yr Mpc at z 6. 0.001 yr

≈

In Fig. 4 we also show the result of our model with M H -4 f∗ = 0.1 0.01 and vwind = 50 km/s for both Pop III.1 10 SFR and Pop III.2,− respectively. This model provides an “opti- Pop III.1 - mistic” estimate for the detectable GRB rate for the future 10 5 missions (see Sect. 3). 0 5 10 15 20 25 30 35 z 2.5. Luminosity function Fig. 4. Comparison of the star formation rates for Pop The number of GRBs detectable by any given instrument III.1 (blue dotted line) and for PopIII.2 (dashed black depends on the instrument-specific flux sensitivity thresh- line), for our optimistic model with a high star forma- old and also on the intrinsic isotropic luminosity function tion efficiency f∗ = 0.1 for Pop III.1, f∗ = 0.01 for Pop of GRBs. For the latter, we adopt the power-law distribu- III.2 and slow chemical enrichment vwind = 50km/s. tion function of Wanderman & Piran (2010)

+0.2 −0.2− . Using the above relation we can predict the observ- L 0 1 LL∗. − − − L∗ F = 1.2 10 8erg cm 2 s 1 (Li 2008). We adopt a lim ×  similar limit for SVOM (Paul et al. 2011). For JANUS, where L∗ is the characteristic isotropic luminosity. We set −8 −2 −1 53 52 Flim 10 erg cm s (Falcone et al. 2009). The lumi- L∗ 10 ergs/s for Pop III.1, whereas L∗ 10 ergs/s nosity∼ threshold is then for∼ Pop III.2 stars are similar to ordinary∼ GRBs (Li 2008; Wanderman & Piran 2010). The Pop III.1 GRBs 2 Llim =4π dL Flim. (22) are assumed to be energetic with isotropic kinetic energy 56−57 Eiso 10 erg but long-lived T90 1000 s, so that the Here dL is the luminosity distance for the adopted ΛCDM ∼ ∼ 56−57 luminosity would be moderate L∗ ǫγ 10 /1000 cosmology. EXIST is expected to be 7 10 times more 52−53 ∼ × ∼ ∼ − 10 ergs/s if ǫγ 0.1 is the conversion efficiency from sensitive than Swift (Grindlay 2010). We set the EXIST the jet kinetic energy∼ to gamma rays (Suwa & Ioka 2011). sensitivity threshold to ten times lower than Swift as an 6 de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts approximate estimate. For simplicity, we assume that the 3. Redshift distribution of GRBs spectral energy distribution (SED) peaks at X-to-γ ray Over a particular time interval, ∆t , in the observer rest energy (detector bandwidth) as an optimistic case. The obs frame, the number of observed GRBs originating between rate would be less if the SED is very different from Pop redshifts z and z + dz is II/I GRBs. dN obs ∆t dV GRB =Ψobs (z) obs , (26) 2.6. Initial mass function and GRB formation efficiency dz GRB 1+ z dz The stellar initial mass function (IMF) is critically impor- where dV/dz is the comoving volume element per unit tant for determining the Pop III GRB rate. We define the redshift, given by GRB formation efficiency factor per stellar mass as dV 4π c d2 dt = L . (27) Mup φ(m)dm dz (1 + z) dz MGRB ηGRB = fGRB M , (23) R up mφ(m)dm Mlow Figure 5 shows the intrinsic GRB rate R where φ(m) is the stellar IMF, and fGRB = 0.001 is the dN ∆t dV GRB =Ψ (z) obs . (28) GRB fraction, since we expect one GRB every 1000 su- dz GRB 1+ z dz pernovae (Langer & Norman 2006). We assume that Pop III GRBs have a similar fraction to the standard case. In this plot, we have not considered observational effects Izzard et al. (2004) argue that GRB formation efficiency such as beaming and instrument sensitivity; namely, we could increase by a factor of 5-7 for low-metallicity stars set Ωobs =4π, ηbeam = 1, and Llim(z)=0 in Eq. (1). We −2 ( 10 Z⊙). If most of the first stars are rotating rapidly show the GRB rate for our choice of two different IMFs. as∼ suggested by Stacy et al. (2011), we can expect that Interestingly, Fig. 5 shows that the results depend only a significant fraction of the first stars can produce GRBs. weakly on the choice of IMF. Thus given the uncertainty in the parameter for free metal Figure 6 shows the most optimistic case, assuming a high star formation efficiency f∗ = 0.1 for Pop III.1, stars, we also explore the possibility that fGRB is 10 100 higher. − × f∗ = 0.01 for Pop III.2, an inefficient chemical enrich- We consider the following two forms of IMF. One is a ment, vwind = 50 km/s, fGRB =0.1, and a Gaussian IMF power law with the standard Salpeter slope for both Pop III.1 and Pop III.2 stars. We note that constraints on these quantities will be useful for placing upper φ(m) m−2.35, (24) limits on the GRB observed rate. ∝ and the other is a Gaussian IMF (Scannapieco et al. 2003; Nakamura & Umemura 2001): 3.1. Radio afterglows

2 2 Follow-up observations of high-redshift GRBs can be done −1 1 −(m−M¯ ) /2σc(M) φ(m)m dm = e dm. (25) by observing their afterglows, especially in radio band √2πσc(M) (Ioka & Mészáros 2005; Inoue et al. 2007). We calculated For the latter, we assume M¯ = 550M⊙ for Pop III.1 the radio afterglow light curves for Pop III GRBs following and M¯ = 55M⊙ for Pop III.2, with dispersion σc = the standard prescription from Sari et al. (1998, 1999) and (M¯ M )/3. M is the minimum mass for a given stel- Mészáros (2006). The afterglow light curve at the time td − low low lar type, 100M⊙ for Pop III.1, and 10M⊙ for Pop III.2, is given by the shock radius rd and the Lorentz factor γd. 3 2 2 whereas Mup is the maximum mass for a given stellar These two quantities are related by Eiso 4πrdγdnmpc 2 ∼ type, 1000M⊙ for Pop III.1, and 100M⊙ for Pop III.2. and rd cγdtd, where n is the medium density and mp the ∼ ∼ 2 MGRB is the minimum mass that is able to trigger GRBs, proton mass. The true energy is given by Etrue = θ Eiso/2, which we set to be 25M⊙ (Bromm & Loeb 2006). Not all where θ is the half opening angle of the shock. The spec- Pop III.1 stars will leave a black hole behind at their trum consists of power-law segments linked by critical deaths. In the narrow mass range of 140 260M⊙ break frequencies. These are νa (the self absorption fre- ∼ − Pop III stars are predicted to undergo a pair-instability su- quency), νm (the peak of injection frequency), and νc (the pernova (PISN) explosion (Heger & Woosley 2002). This cooling frequency), given by range of mass is excluded from the calculation of Eq. (23). 1/2 2 2 1/2 1/2 −3/2 The efficiency factor for the power-law (Salpeter) IMF νm (1 + z) g(p) ǫeǫB Eiso td , −1 −1 ∝ is ηGRB/fGRB 1/926M⊙ and 1/87M⊙ for Pop III.1 −1/2 −3/2 −1 −1/2 −1/2 νc (1 + z) ǫ n E t , and Pop III.2,∼ respectively. Using the Gaussian IMF, ∝ B iso d −1 −1 −1 −1 1/5 3/5 1/5 η /f 1/538M and 1/53M for Pop III.1 νa (1 + z) ǫe ǫB n Eiso , GRB GRB ∼ ⊙ ⊙ ∝ and Pop III.2. respectively. Thus, the GRB formation ef- 1/2 1/2 −2 Fν,max (1 + z)ǫB n EisodL , (29) ficiency for Pop III.2 can be about an order of magnitude ∝ higher than Pop III.1 because of the lower characteristic where g(p) = (p 2)/(p 1) is a function of energy spec- mass of Pop III.2 stars. trum index of electrons− − (N(γ )dγ γ−pdγ , where γ e e ∝ e e e de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts 7 is the electron Lorentz factor), and ǫe and ǫB are the ef- timescale radio variability surveys. There are several ra- ficiency factors (Mészáros 2006). There are two types of dio transient surveys completed so far. Bower et al. (2007) spectra. If νm < νc, we call it the slow cooling case. The used 22 years of archival data from VLA to put an upper −2 flux at the observer, Fν , is given by limit of 6 deg for 1-year variability transients above ∼ 5 1/3 2 90 µJy, which is equivalent to . 2.4 10 for the whole (νa/νm) (ν/νa) Fν,max, νa > ν, ×12 13 1/3 sky. Gal-Yam et al. (2006) used FIRST and NVSS ra-  (ν/νm) Fν,max, νm >ν>νa, Fν = dio catalogs to place an upper limit of 70 radio orphan  (ν/ν )−(p−1)/2F , ν >ν>ν ,  m ν,max c m afterglows above 6 mJy in the 1.4 GHz∼ band over the en- (ν /ν )−(p−1)/2(ν/ν )−p/2F , ν>ν . c m c ν,max c tire sky. This suggests less than 3 104 sources above  (30)  0.3 mJy on the sky, because the number× of sources is ex- where Fν,max is the observed peak flux at distance dL from −3/2 the source. pected to be proportional to flux limit Flim (assuming Euclidian space and no source evolution) (Gal-Yam et al. For νm >νc, called the fast cooling case, the spectrum 2006). From Fig. 7, a typical GRB’s radio afterglow with is 54 isotropic kinetic energy Eiso 10 ergs stays above 0.3 mJy over 102 days. ∼ (ν /ν )1/3(ν/ν )2F , ν > ν, ∼ a c a ν,max a By combining the results shown in Figs. 5 and 6, we (ν/ν )1/3F , ν >ν>ν ,  c ν,max c a expect 30 3 105 sources (102 106 events per year Fν =  −1/2  (ν/νc) Fν,max, νm >ν>νc, 2 ∼ − × − −1/2 −p/2 10 days) above 0.3 mJy. (We integrate the event (νm/νc) (ν/νm) Fν,max, ν>νm. × ∼  rate over redshift.) As a consequence, the most optimistic  (31) case for Pop III.2 should already be ruled out marginally As the GRB jet sweeps the interstellar medium, the by the current observations of radio transient sources, if Lorentz factor of the jet is decelerated. When the Lorentz −1 their luminosity function follows the one assumed in the factor drops below θ , the jet starts to expand sideways present paper. Only more conservative models are then vi- and becomes detectable by the off-axis observers. These able. Radio transient surveys are not yet able to set upper afterglows are not associated with the prompt GRB emis- limits on the Pop III.1 GRB rate. The above conclusion sion. Such orphan afterglows are a natural consequence is model dependent, because the afterglow flux depends of the existence of GRB’s jets. Radio transient sources on the still uncertain quantities, such as the isotropic en- probe the high-energy population of the Universe and ergy Eiso and the ambient density n. If the circumburst can provide further constraints on the intrinsic rate of density is higher than usual, the constraints from the ra- GRBs. Even if the prompt emission is highly collimated, −1 dio transient surveys would be even stronger. Also the the Lorentz factor drops γd <θ around the time GRB formation efficiency and the beaming factor are not E 1/3 θ 8/3 known accurately, which can affect both the intrinsic and t 2.14 iso n−1/3(1 + z) days, θ ∼ 5 1054 0.1 observed rate more than one order of magnitude. × (32) In Figs. 8-9, we show the predicted observable GRB obs and the jet starts to expand sideways. Finally the shock rate dNGRB/dz in Eq. (26) for Pop III.1 and III.2 de- velocity becomes nonrelativistic around the time tectable by the Swift, SVOM, JANUS, and EXIST missions. The results shown are still within the bounds of E 1/3 θ 2/3 t 1.85 102 iso n−1/3(1+z) days, available upper limits from the radio transient surveys. NR ∼ × 5 1054 0.1 × Overall, it is more likely to observe Pop III.2 GRBs than (33) Pop III.1, but the predicted rate strongly depends on the (Ioka & Mészáros 2005). After time tθ, the temporal de- IGM metallicity evolution, the star formation efficiency pendence of the critical break frequencies should be re- and GRB formation efficiency. The dependence on the placed by ν t0,ν t−2,ν t−1/5, and F t−1 c ∝ m ∝ a ∝ ν,max ∝ IMF is relatively small. (Sari et al. 1999). We also used the same evolution in the Figure 10 shows the GRB rate expected for EXIST ob- nonrelativistic phase for simplicity, which underestimates servations. Because the power index of the LF is uncertain the afterglow flux after tNR. at the bright end, we added two lines to show the resulting Figure 7 shows the light curves for a typical GRB from uncertainty in our prediction. We use the maximum rate, Pop III.2 stars assuming an isotropic kinetic energy Eiso which is within the constraints by the current observations 54 ∼ 10 erg (in proportion to the progenitor mass) as a lower of radio transients. We expect to observe N 20 GRBs limit. Pop III.1 afterglows are expected to be brighter. per year at z > 6 for Pop III.2 and N 0.∼08 per year Consistently with previous works, we conclude that it is for Pop III.1 at z > 10 with the future∼ EXIST satellite possible to observe the GRB radio afterglows with ALMA, at a maximum. Our optimist case predicts a near-future LOFAR, EVLA, and SKA. detection of Pop III.2 GRB by Swift, and the nondetec- tion so far could suggest a further upper limit or difference 3.2. Upper limits from radio transient survey between the Pop III and present-day GRB spectrum.

In this section, we derive upper limits on the intrinsic 12 http://sundog.stsci.edu/ GRB rate (including the oﬀ-axis GRB) using 1 year 13 http://www.cv.nrao.edu/nvss/ ∼ 8 de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts

100 Pop III.2 10 GHz 10 1 L 1 - 1.4 GHz yr H 1 0.1 500 MHz

L ALMA 0.1 mJy H EVLA F 0.01 LOFAR 0.01 Intrinsic GRB rate Pop III.1 0.001 0.001 SKA

10 15 20 25 30 - 10 4 z 0.1 1 10 100 1000 104 105 t HdaysL Fig. 5. The intrinsic GRB rate dNGRB/dz. The number of (on-axis + oﬀ-axis) GRBs per year on the sky in Eq. (28), Fig. 7. The theoretical light curve of radio afterglow of a typical Pop III.2 GRB at z 10. We show the evo- as a function of redshift. We set f∗ =0.001, fGRB =0.01 ∼ and vwind = 100km/s for this plot. Salpeter IMF, dashed lution of afterglow ﬂux F (mJy) as a function of time black line, Gaussian IMF, dotted black line, for Pop III.2; t (days) for typical parameters: isotropic kinetic energy 54 and Salpeter IMF, dashed blue line, Gaussian IMF, dotted Eiso = 10 erg, electron spectral index p = 2.5, plasma blue line, for Pop III.1. parameters ǫe = 0.1, ǫB = 0.01, initial Lorentz factor −3 γd = 200, interstellar medium density n = 1 cm , for 105 the range of frequencies: 500 MHz (dashed brown line), Optimistic case 1.4 GHz (dashed red line), 10 GHz (dashed black line), in 104 sen

L comparison with ﬂux sensitivity F as a function of inte- 1 ν

- Pop III.2 yr

H gration time, t (days) for SKA (dot-dashed green line), 1000 int EVLA (dot-dashed orange line), LOFAR (dot-dashed blue 100 line) and ALMA (dot-dashed purple line). Pop III.1 10 0.050 Intrinsic GRB rate Pop III.1 1

L EXIST

1 0.010 -

0.1 yr 5 10 15 20 25 30 35 40 H 0.005 JANUS z SVOM 0.001 Fig. 6. The intrinsic GRB rate dNGRB/dz. The number - of (on-axis + off-axis) GRBs per year on the sky in Eq. 5´10 4 Swift (28), as a function of redshift for our optimistic model. We Observed GRB rate assume a high star formation efficiency; f∗ =0.1 for Pop 1´10-4 III.1; and f∗ = 0.01 for Pop III.2; slow chemical enrich- 15 20 25 30 ment, vwind = 50km/s; high GRB formation efficiency, z fGRB = 0.1; and a Gaussian IMF; for both Pop III.2, dashed black line; and Pop III.1, dotted blue line. Fig. 8. Predicted Pop III.1observed GRB rate. Those observed by Swift, dashed red line; SVOM, dot-dashed black line; JANUS, dotted blue line; and EXIST, green line. We 4. Conclusion and discussion adopt a GRB rate model that is consistent with the current upper limits from the radio transients; Gaussian IMF, There are still no direct observations of Population III vwind = 50km/s, f∗ =0.1, fGRB =0.1. stars, despite much recent development in theoretical studies on the formation of the early generation stars. In this paper, we follow a recent suggestion that massive and argued that the latter is more likely to be observed Pop III stars could trigger collapsar gamma-ray bursts. with future experiments. We expect to observe maximum Observations of such energetic GRBs at very high red- of N . 20 GRBs per year integrated over at z > 6 for Pop shifts will be a unique probe of the high-redshift Universe. III.2 and N . 0.08 per year integrated over at z > 10 for With a semi-analytical approach we estimated the star Pop III.1 with EXIST. formation rate for Pop III.1 and III.2 stars including all We also expect a larger number of radio afterglows relevant feedback effects: photo-dissociation, reionization, than X-ray prompt emission because the radio afterglow and metal enrichment. is long-lived, for 102 days above 0.3 mJy from Fig. 7. Using radio transient sources we are able to derive con- Combining with∼ the intrinsic GRB∼ rate and constraints straints on the intrinsic rate of GRBs. We estimated the from radio transients, we expect roughly 10 104 predicted GRB rate for both Pop III.1 and Pop III.2 stars, radio afterglows above & 0.3 mJy already∼ on the− sky. de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts 9

10 Bromm & Loeb 2006; Fryer & Heger 2005, and references Pop III.2 EXIST therein). If the GRB fraction per collapse fGRB in Eq. 1 L

1 (23) is much larger than the current one, say fGRB 1, - SVOM yr JANUS ∼ H the Pop III.1 GRBs might also become detectable with 0.1 Swift the radio telescopes ( 300 afterglows above 0.3 mJy on the sky) and the X-ray∼ satellites ( 1 event∼ per year 0.01 for EXIST) in the future. ∼ Observed GRB rate 0.001 Acknowledgements. R.S.S. thanks the Brazilian agency CNPq (200297/2010-4) for financial support. This work was sup- 10-4 ported by World Premier International Research Center 10 15 20 25 30 z Initiative (WPI Initiative), MEXT, Japan. We thank Emille Ishida, Andrea Ferrara, Kenichi Nomoto, Jarrett Johnson, and Fig. 9. Predicted Pop III.2 observed GRB rate. Those Yudai Suwa for fruitful discussion and suggestions. NY ac- observed by Swift, dashed red line; SVOM, dot-dashed knowledges the financial support by the Grants-in-Aid for black line; JANUS, dotted blue line; and EXIST, green Young Scientists (S) 20674003 by the Japan Society for the line; for our model with Salpeter IMF, vwind = 100km/s, Promotion of Science. KI acknowledges the financial support by KAKENHI 21684014, 19047004, 22244019, 22244030. We f∗ =0.01, fGRB =0.01. thank the anonymous referee for their very careful reading of

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