Gamma-ray burst as a probe for the high-z Universeextragal/2011B/rafael_2011B.pdf ·...

Gamma-ray burst as a probe for the high-z

Universe

Rafael S. de Souza1,2, Naoki Yoshida2, Kunihito Ioka3, Emille Ishida1,2, Andrea Ferrara4, Benedetta Ciardi5, Alberto-Krone Martins6

1IAG-Universidade de São Paulo2IPMU-University of Tokyo3KEK-University of Tsukuba

4SNS-University of Pisa5MPA-Garching

6SIM-Universidade de Lisboa

segunda-feira, 26 de setembro de 2011

X-RAY EVENTS

Part 1

Semi-analytical estimative of the Pop III star formation rate

Redshift distribution of Pop III Gamma-ray bursts

Radio afterglows

Upper limits from radio transients survey

Expected rate by present and future missions


LOOKING FOR ORPHAN AFTERGLOWS

Predicted number of orphans

Afterglow model

Mock sample

GAIA mission


STAR FORMATION HISTORY FROM GAMMA-RAY BURSTS

How to estimate the star formation history up to high z from current observations?

GRBs as a probe for SFH

Principal Component Analysis

Swift dataset.


FIAT LUX

The first stars in the Universe played a crucial role in the early cosmic evolution, by emitting the first light and producing the first heavy elements.

The first stars are probably responsible for reionization of the Universe

Observations of Gamma ray bursts are probably the only way to probe the death of first stars.


TWO METAL FREE POPULATIONS

Pop III.1 the first generation of metal free stars that formed from initial conditions determined cosmologically (no astrophysical feedback).

Pop III.2 zero-metallicity stars that formed from a primordial gas, but were affected by radiation from other stars. Typically are formed in initially ionized gas.


GAMMA-RAY BURST RATE

We assume that the formation rate of GRBs is proportional to the star formation rate (Totani 1997; Ishida et al. 2011). The number of observable GRBs per comoving volume per time is expressed as

A&A 533, A32 (2011)

(Mortlock et al. 2011) and z = 6.41 (Willott et al. 2003). Chandraet al. (2010) report the discovery of radio afterglow emissionfrom GRB 090423 and Frail et al. (2006) for GRB 050904.Observations of afterglows make it possible to derive the phys-ical properties of the explosion and the circumburst medium. Itis intriguing to search for these di!erent signatures in the GRBafterglows at low and high redshifts.

The purpose of the present paper is to calculate the Pop IIIGRB rate detectable by the current and future GRB missions(see also Campisi et al. 2011). We consider high-redshift GRBsof two populations following Bromm et al. (2009). Pop III.1 starsare the first-generation stars that form from initial conditionsdetermined cosmologically. Pop III.2 stars are zero-metallicitystars but formed from a primordial gas that was influenced byearlier generation of stars. Typically, Pop III.2 stars are formedin an initially ionized gas (Johnson & Bromm 2006; Yoshidaet al. 2007). The Pop III.2 stars are thought to be less massive(!40"60 M#) than Pop III.1 stars (!1000 M#) but still massiveenough for producing GRBs.

We have calculated the GRB rate for these two populationsseparately for the first time. The rest of the paper is organized asfollows. In Sect. 2, we describe a semi-analytical model to calcu-late the formation rate of primordial GRBs. In Sect. 3, we showour model predictions and calculate the detectability of Pop IIIGRBs by future satellite missions and by radio observations. InSect. 4, we discuss the results and give our concluding remarks.Throughout the paper we adopt the standard " cold dark mat-ter model with the best fit cosmological parameters from Jarosiket al. (2011) (WMAP-Yr710), #m = 0.267,#" = 0.734, andH0 = 71 km s"1 Mpc"1.

2. Gamma-ray burst rate

We assume that the formation rate of GRBs is proportional to thestar formation rate (Totani 1997; Ishida et al. 2011). The numberof observable GRBs per comoving volume per time is expressedas

$obsGRB(z) =

#obs

4!"GRB "beam$$(z)

! %

log Llim(z)p(L)d log L, (1)

where "GRB is the GRB formation e%ciency (see Sect. 2.6), "beamthe beaming factor of the burst, #obs the field of view of theexperiment,$$ the cosmic star formation rate (SFR) density, andp(L) the GRB luminosity function in X-rays to gamma rays. Theintrinsic GRB rate is given by

$GRB(z) = "GRB$$(z). (2)

The quantity Llim(z) is the minimum luminosity threshold to bedetected, which is specified for a given experiment. The non-isotropic nature of GRBs gives "beam ! 0.01"0.02 (Guetta et al.2005). Using a radio transient survey, Gal-Yam et al. (2006)place an upper limit of "beam ! 0.016. Given the average valueof jet opening angle # ! 6& (Ghirlanda et al. 2007) and "beam !5.5' 10"3, we set "beam = 0.006 as a fiducial value. The adoptedvalues of #obs are 1.4, 2, 4, and 5 for Swift, SVOM, JANUS, andEXIST, respectively (Salvaterra et al. 2008).

2.1. The number of collapsed objects

We first calculate the star formation rate (SFR) at early epochs.Assuming that stars are formed in collapsed dark matter halos,

10 http://lambda.gsfc.nasa.gov/product/map/current/

we follow a popular prescription in which the number of col-lapsed objects is calculated by the halo mass function (Hernquist& Springel 2003; Greif & Bromm 2006; Trenti & Stiavelli2009). We adopt the Sheth-Tormen mass function, fST, (Sheth& Tormen 1999) to estimate the number of dark matter halos,nST(M, z), with mass less than M per comoving volume at a givenredshift:

fST = A

"2a1

!

#1 +$$2

a1%2c

%p&%c

$exp#"a1%2

c

2$2

&, (3)

where A = 0.3222, a1 = 0.707, p = 0.3 and %c = 1.686. Themass function fST can be related to the nST(M, z) as

fST =M&m

dnST(M, z)d ln$"1 , (4)

where &m is the total mass density of the background Universe.The variance of the linear density field $(M, z) is given by

$2(M, z) =b2(z)2!2

! %

0k2P(k)W2(k,M)dk, (5)

where b(z) is the growth factor of linear perturbations normal-ized to b = 1 at the present epoch, and W(k,M) is the Fourier-space top hat filter. To calculate the power spectrum P(k), we usethe CAMB code11 for our assumed "CDM cosmology.

2.2. H2 photodissociation

The star formation e%ciency in the early Universe largely de-pends on the ability of a primordial gas to cool and condense.Hydrogen molecules (H2) are the primary coolant in a gas insmall mass “minihalos”, and are also fragile to soft ultraviolet ra-diation, and thus a ultraviolet background in the Lyman-Werner(LW) bands can easily suppress the star formation inside mini-halos. We model the dissociation e!ect by setting the minimummass for halos that are able to host Pop III stars (Yoshida et al.2003).

For the minimum halo mass capable of cooling by molecularhydrogen in the presence of a Lyman-Werner (LW) background,we adopt a fitting formula given by Machacek et al. (2001) andWise & Abel (2005), which also agrees with results from O’Shea& Norman (2008):

MH2 = exp$

fcd

0.06

% '1.25 ' 105 + 8.7 ' 105F0.47

LW,"21

(, (6)

where FLW,"21 = 4!JLW is the flux in the LW band in units of10"21 erg"1 s"1 cm"2 Hz"1, and fcd the fraction of gas that is coldand dense. We set fcd = 0.02 as a conservative estimate. Wecompute the LW flux consistently with the comoving densityin stars &$(z) via a conversion e%ciency "LW (Greif & Bromm2006):

JLW =hc

4!mH"LW&$(z)(1 + z)3. (7)

Here, "LW is the number of photons emitted in the LW bandsper stellar baryon and mH is the mass of hydrogen. The value of"LW depends on the characteristic mass of the formed primordialstars, but the variation is not very large for stars with massesgreater than ten solar masses (Schaerer 2002). We set "LW = 104

for both Pop III.1 and Pop III.2 for simplicity.

11 http://camb.info/

A32, page 2 of 9

A&A 533, A32 (2011)

(Mortlock et al. 2011) and z = 6.41 (Willott et al. 2003). Chandraet al. (2010) report the discovery of radio afterglow emissionfrom GRB 090423 and Frail et al. (2006) for GRB 050904.Observations of afterglows make it possible to derive the phys-ical properties of the explosion and the circumburst medium. Itis intriguing to search for these di!erent signatures in the GRBafterglows at low and high redshifts.

The purpose of the present paper is to calculate the Pop IIIGRB rate detectable by the current and future GRB missions(see also Campisi et al. 2011). We consider high-redshift GRBsof two populations following Bromm et al. (2009). Pop III.1 starsare the first-generation stars that form from initial conditionsdetermined cosmologically. Pop III.2 stars are zero-metallicitystars but formed from a primordial gas that was influenced byearlier generation of stars. Typically, Pop III.2 stars are formedin an initially ionized gas (Johnson & Bromm 2006; Yoshidaet al. 2007). The Pop III.2 stars are thought to be less massive(!40"60 M#) than Pop III.1 stars (!1000 M#) but still massiveenough for producing GRBs.

We have calculated the GRB rate for these two populationsseparately for the first time. The rest of the paper is organized asfollows. In Sect. 2, we describe a semi-analytical model to calcu-late the formation rate of primordial GRBs. In Sect. 3, we showour model predictions and calculate the detectability of Pop IIIGRBs by future satellite missions and by radio observations. InSect. 4, we discuss the results and give our concluding remarks.Throughout the paper we adopt the standard " cold dark mat-ter model with the best fit cosmological parameters from Jarosiket al. (2011) (WMAP-Yr710), #m = 0.267,#" = 0.734, andH0 = 71 km s"1 Mpc"1.


We assume that the formation rate of GRBs is proportional to thestar formation rate (Totani 1997; Ishida et al. 2011). The numberof observable GRBs per comoving volume per time is expressedas

$obsGRB(z) =

#obs

4!"GRB "beam$$(z)

! %

log Llim(z)p(L)d log L, (1)

where "GRB is the GRB formation e%ciency (see Sect. 2.6), "beamthe beaming factor of the burst, #obs the field of view of theexperiment,$$ the cosmic star formation rate (SFR) density, andp(L) the GRB luminosity function in X-rays to gamma rays. Theintrinsic GRB rate is given by

$GRB(z) = "GRB$$(z). (2)

The quantity Llim(z) is the minimum luminosity threshold to bedetected, which is specified for a given experiment. The non-isotropic nature of GRBs gives "beam ! 0.01"0.02 (Guetta et al.2005). Using a radio transient survey, Gal-Yam et al. (2006)place an upper limit of "beam ! 0.016. Given the average valueof jet opening angle # ! 6& (Ghirlanda et al. 2007) and "beam !5.5' 10"3, we set "beam = 0.006 as a fiducial value. The adoptedvalues of #obs are 1.4, 2, 4, and 5 for Swift, SVOM, JANUS, andEXIST, respectively (Salvaterra et al. 2008).


We first calculate the star formation rate (SFR) at early epochs.Assuming that stars are formed in collapsed dark matter halos,


we follow a popular prescription in which the number of col-lapsed objects is calculated by the halo mass function (Hernquist& Springel 2003; Greif & Bromm 2006; Trenti & Stiavelli2009). We adopt the Sheth-Tormen mass function, fST, (Sheth& Tormen 1999) to estimate the number of dark matter halos,nST(M, z), with mass less than M per comoving volume at a givenredshift:

fST = A

"2a1

!

#1 +$$2

a1%2c

%p&%c

$exp#"a1%2

c

2$2

&, (3)

where A = 0.3222, a1 = 0.707, p = 0.3 and %c = 1.686. Themass function fST can be related to the nST(M, z) as

fST =M&m

dnST(M, z)d ln$"1 , (4)

where &m is the total mass density of the background Universe.The variance of the linear density field $(M, z) is given by

$2(M, z) =b2(z)2!2

! %

0k2P(k)W2(k,M)dk, (5)

where b(z) is the growth factor of linear perturbations normal-ized to b = 1 at the present epoch, and W(k,M) is the Fourier-space top hat filter. To calculate the power spectrum P(k), we usethe CAMB code11 for our assumed "CDM cosmology.

2.2. H2 photodissociation

The star formation e%ciency in the early Universe largely de-pends on the ability of a primordial gas to cool and condense.Hydrogen molecules (H2) are the primary coolant in a gas insmall mass “minihalos”, and are also fragile to soft ultraviolet ra-diation, and thus a ultraviolet background in the Lyman-Werner(LW) bands can easily suppress the star formation inside mini-halos. We model the dissociation e!ect by setting the minimummass for halos that are able to host Pop III stars (Yoshida et al.2003).

For the minimum halo mass capable of cooling by molecularhydrogen in the presence of a Lyman-Werner (LW) background,we adopt a fitting formula given by Machacek et al. (2001) andWise & Abel (2005), which also agrees with results from O’Shea& Norman (2008):

MH2 = exp$

fcd

0.06

% '1.25 ' 105 + 8.7 ' 105F0.47

LW,"21

(, (6)

where FLW,"21 = 4!JLW is the flux in the LW band in units of10"21 erg"1 s"1 cm"2 Hz"1, and fcd the fraction of gas that is coldand dense. We set fcd = 0.02 as a conservative estimate. Wecompute the LW flux consistently with the comoving densityin stars &$(z) via a conversion e%ciency "LW (Greif & Bromm2006):

JLW =hc

4!mH"LW&$(z)(1 + z)3. (7)

Here, "LW is the number of photons emitted in the LW bandsper stellar baryon and mH is the mass of hydrogen. The value of"LW depends on the characteristic mass of the formed primordialstars, but the variation is not very large for stars with massesgreater than ten solar masses (Schaerer 2002). We set "LW = 104

for both Pop III.1 and Pop III.2 for simplicity.


A32, page 2 of 9

The intrinsic GRB rate is given by


THE NUMBER OF COLLAPSED OBJECTS

Assuming that stars are formed in collapsed dark matter haloes, we adopt the Sheth- Tormen mass function, fST, (Sheth & Tormen 1999) to estimate the number of dark matter haloes, nST(M,z), with mass less than M per comoving volume at a given redshift:

2 de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts

GRBs. They suggest that spectroscopic measurements ofmolecular and atomic absorption lines due to ambientprotostellar gas may be possible to z ! 30 and beyondwith ALMA4, EVLA5, and SKA. In the future, it willbe also promising to observe the GRB afterglows locatedby gamma-ray satellites such as Swift6, SVOM7, JANUS8

and EXIST9. Clearly, it is important to study the rate andthe detectability of Pop III GRBs at very high redshifts.

There have been already a few observations of GRBsat high redshifts. GRB 090423, at a redshift of z = 8.26(Salvaterra et al. 2009; Tanvir et al. 2009), is the objectwith the second highest redshift observed to date after thediscovered galaxy at z = 8.6 (Lehnert et al. 2010), beyondthe previous GRB 080913 at z = 6.7 (Greiner et al. 2009),GRB 050904 at z = 6.3 (Kawai et al. 2006; Totani et al.2006) and the highest redshift quasar at z = 6.41 (Willottet al. 2003). Chandra et al. (2010) reported the discoveryof radio afterglow emission from GRB 090423, and Frailet al. (2006) for GRB 050904. Observations of afterglowsmake it possible to derive physical properties of the explo-sion and the circumburst medium. It is intriguing to searchfor these di!erent signatures in the GRB afterglows at lowand high redshifts.

The purpose of the present paper is to calculate thePop III GRB rate detectable by the current and fu-ture GRB missions. We consider high-redshift GRBs oftwo populations following Bromm et al. (2009). Pop III.1stars are the first generation stars that form from ini-tial conditions determined cosmologically. Pop III.2 starsare zero-metallicity stars but formed from a primordialgas that was influenced by earlier generation of stars.Typically, Pop III.2 stars are formed in an initially ion-ized gas (Johnson & Bromm 2006; Yoshida et al. 2007).The Pop III.2 stars are thought to be less massive (! 40–60M!) than Pop III.1 stars (! 1000M!) but still mas-sive enough for producing GRBs. We calculate the GRBrate for these two populations separately for the firsttime. The rest of the paper is organized as follows. InSect. 2, we describe a semi-analytical model to calculatethe formation rate of primordial GRBs. In Sect. 3, weshow our model predictions and calculate the detectabil-ity of Pop III GRBs by future satellite missions andby radio observations. In Sect. 4, we discuss the resultsand give our concluding remarks. Throughout the paperwe adopt the standard " Cold Dark Matter model withthe best fit cosmological parameters from Jarosik et al.(2011) (WMAP-Yr710), #m = 0.267, #! = 0.734, andH0 = 71km s"1Mpc"1.

4 www.alma.nrao.edu/5 http://www.aoc.nrao.edu/evla/6 http://swift.gsfc.nasa.gov/docs/swift/swiftsc.html7 http://www.svom.fr/svom.html8 http://sms.msfc.nasa.gov/xenia/pdf/CCE2010/Burrows.pdf9 http://exist.gsfc.nasa.gov/design/



We assume that the formation rate of GRBs is propor-tional to the star formation rate (Totani 1997). The num-ber of observable GRBs per comoving volume per time isexpressed as

$obsGRB(z) =

#obs

4!"GRB "beam $#(z)

! $

Llim(z)p(L)dL, (1)

where "GRB is the GRB formation e%ciency (see section2.6), "beam is the beaming factor of the burst, #obs is thefield of view of the experiment, $# is the cosmic star for-mation rate (SFR) density and p(L) is the GRB luminos-ity function in X-rays to gamma-rays. The intrinsic GRBrate is given by

$GRB(z) = "GRB$#(z). (2)

The quantity Llim(z) is the minimum luminosity thresh-old to be detected, which is specified for a given exper-iment. The non-isotropic nature of GRBs gives "beam !0.01 " 0.02 (Guetta et al. 2005). Using a radio transientssurvey Gal-Yam et al. (2006) place an upper limit of"beam ! 0.016. We set "beam = 0.015 as a fiducial value.The adopted values of #obs are 1.4, 2, 4 and 5 for Swift,SVOM, JANUS and EXIST respectively (Salvaterra et al.2008).


We first calculate the star formation rate (SFR) at earlyepochs. Assuming that stars are formed in collapsed darkmatter haloes, we follow a popular prescription in whichthe number of collapsed objects is calculated by the halomass function (Hernquist & Springel 2003; Greif & Bromm2006; Trenti & Stiavelli 2009). We adopt the Sheth-Tormen mass function, fST, (Sheth & Tormen 1999) toestimate the number of dark matter haloes, nST(M, z),with mass less than M per comoving volume at a givenredshift:

fST = A

"

2a1

!

#

1 +

$

#2

a1$2c

%p&$c

#exp

#

"a1$2

c

2#2

&

, (3)

where A = 0.3222, a1 = 0.707, p = 0.3 and $c = 1.686.The mass function fST can be related to the nST(M, z) as

fST =M

%m

dnST(M, z)

d ln#"1, (4)

where %m is the total mass density of the backgroundUniverse. The variance of the linear density field #(M, z)is given by

#2(M, z) =b2(z)

2!2

! $

0k2P (k)W 2(k, M)dk, (5)

where b(z) is the growth factor of linear perturbationsnormalized to b = 1 at the present epoch, and W (k, M)is the Fourier-space top hat filter. To calculate the powerspectrum P (k), we use the CAMB code11 for our assumed"CDM cosmology.11 http://camb.info/










$obsGRB(z) =

#obs

4!"GRB "beam $#(z)

! $

Llim(z)p(L)dL, (1)


$GRB(z) = "GRB$#(z). (2)




fST = A

"

2a1

!

#

1 +

$

#2

a1$2c

%p&$c

#exp

#

"a1$2

c

2#2

&

, (3)


fST =M

%m

dnST(M, z)

d ln#"1, (4)


#2(M, z) =b2(z)

2!2

! $

0k2P (k)W 2(k, M)dk, (5)











$obsGRB(z) =

#obs

4!"GRB "beam $#(z)

! $

Llim(z)p(L)dL, (1)


$GRB(z) = "GRB$#(z). (2)




fST = A

"

2a1

!

#

1 +

$

#2

a1$2c

%p&$c

#exp

#

"a1$2

c

2#2

&

, (3)


fST =M

%m

dnST(M, z)

d ln#"1, (4)


#2(M, z) =b2(z)

2!2

! $

0k2P (k)W 2(k, M)dk, (5)


CAMB codehttp://camb.info/

8.3 Funcao de Press-Schechter 147

!"!#

!"#

#

#!

#!$ #!## #!#% #!#& #!#' #!#$

!"# # #! #!!

!()*!+!,-#+-

./00/(1"2# ! 3

#(1"2#(4563

789:.8(;<=(>:;8/7

789:.8(>:;8/7

$()(!

9(/(>(?(@(:(/(09(7(A(5(=(0

/9>=.8B7/C=0

0A587B/9>=.87/C=0

D:>/.8;E=0*.A7=0,

Figura 153: Variancia da flu-tuacao de massa, !M para ocenario !CDM. A linha hori-zontal tracejada e a fronteiraentre o regime linear e nao li-near das flutuacoes de densi-dade.

!"#!"

!"#$

!"#%

!"#&

!"#'

!""

!"'

" ( !" !( '"

!)*+*",*-./012]

#

!"$*.

!"!"*.

!"!'*.

!"!&*.

!*+*$3 Figura 154: Distribuicao demassas em funcao do redshiftde halos colapsados no mo-delo CDM standard. O tracohorizontal indica o densidadenumerica de galaxias com L >L! na banda V no Universoproximo.

!"#%

!"#&

!"#'

!""

!"'

!"4 !"$ !"5 !"!" !"!! !"!' !"!2 !"!& !"!(

#*6*"#%6*"7(#*6*!#*6*'#*6*&#*6*$#*6*!%

!*)*+*",*-./0128

"*-.***8

Figura 155: Distribuicao in-tegrada de massas de ha-los colapsados para diferentesredshifts.


http://camb.info

http://camb.info

SFR follows the number of collapsed objects unless for a couple of feedbacks.

Radiative Feedback, prevents the collapse

Reionization, switches from Pop III.1 to III.2

Metal Enrichment, switches from Pop III to Pop II/I


2 R. S. de Souza, N. Yoshida, K. Ioka

The purpose of this work is to calculate the Pop III GRBrate detectable by the current and future GRB missions. Weconsider high-redshift GRBs of two populations following(Bromm et al. 2009). Pop III.1 stars are the first generationstars that form from initial conditions determined cosmo-logically. Pop III.2 stars are zero-metallicity stars that formfrom a primordial gas that was influenced by earlier gen-eration stars. Typically, Pop III.2 stars are formed in aninitially ionized gas (Johnson & Bromm 2006; Yoshida et al.2007). We calculate the GRB rate for these two popula-tions separately. In Section 2, we describe a semi-analyticalmodel to calculate the formation rate of primordial GRBs.In Section 3, we show our model predictions and calcu-late the detectability of Pop III GRBs by future satellitemissions and by radio observations. In Section 4, we dis-cuss the results and give our concluding remarks. Through-out the paper we adopt the standard ! Cold Dark Mattermodel with the best fit cosmological parameters from Jarosiket al. (2011) (WMAP-Yr710), "m = 0.267, "! = 0.734, andH0 = 71km s!1Mpc!1.

2 GAMMA-RAY BURST RATE

We assume that the formation rate of GRBs is proportionalto the star formation rate. The number of observable GRBsper comoving volume per time is expressed as

#obsGRB(z) =

"obs

4!"GRB "beam #"(z)

Z #

Llim(z)

p(L)dL, (1)

where "GRB is the GRB formation e$ciency, "beam is thebeaming factor of the burst, "obs is the field of view of theexperiment, #" is the cosmic star formation rate (SFR) den-sity and p(L) is the GRB luminosity function in gamma-ray.The intrinsic GRB formation rate is given by #GRB(z) ="GRB#"(z). The quantity Llim(z) is the minimum luminos-ity threshold that is specified for a given experiment. Thenon-isotropic nature of GRBs gives "beam ! 0.02 " 0.01(Guetta et al. 2005). Using a radio transients survey Gal-Yam et al. (2006) place a upper limit of "beam ! 0.016. Weset "beam = 0.015 as a fiducial value. The adopted valuesof "obs are 1.4, 2, 4 and 5 for Swift, SVOM, JANUS andEXIST respectively.

2.1 The number of collapsed objects

We first need to calculate the star formation rate (SFR) atearly epochs. Assuming that stars are formed in collapseddark matter haloes, we follow a popular model where thenumber of collapsed objects is calculated by the halo massfunction (Hernquist & Springel 2003; Greif & Bromm 2006;Trenti & Stiavelli 2009). We adopt the Sheth-Tormen massfunction (Sheth & Tormen 1999) for the number of darkmatter haloes, nST(M, z), per unit mass per comoving vol-ume at a given redshift:

nST(M, z) = A

r

2a1

!

»

1 +

„

#2

a1$2c

«p–

$c

#exp

»

"a1$2c

2#2

–

,

(2)


where A = 0.3222, a1 = 0.707, p = 0.3 and $c = 1.686. Thevariance of the linear density field #(M, z) is given by

#2(M, z) =b2(z)

2!2

Z

#

0

k2P (k)W 2(k, M)dk, (3)

where b(z) is the growth factor of linear perturbations nor-malized to b = 1 at the present epoch, and W (k,M) is theFourier-space top hat filter. To calculate the power spectrumP (k), we used the CAMB code11 for the standard !CDMcosmology.

2.2 Radiative Feedback

The star formation e$ciency in the early universe largelydepends on the ability of a primordial gas to cool and con-dense. Hydrogen molecules (H2) are the primary coolant ina gas in small mass “minihaloes”. H2 are also fragile to softultra-violet radiation, and thus a ultra-violet background inthe Lyman-Werner (LW) bands can easily suppress star for-mation inside minihaloes. We model the e%ect by setting theminimum mass for haloes that are able to host Pop III stars(Yoshida et al. 2003).

For the minimum halo mass capable of cooling bymolecular hydrogen in the presence of a Lyman-Werner(LW) background, we adopt a fitting formula given by Wise& Abel (2005):

MMin!H2 = exp

„

fcd

0.06

«

(1.25#105+8.7#105F 0.47LW,!21), (4)

where F 0.47LW,!21 = 4!JLW is the flux in the LW band in units

of 10!21erg!1s!1cm!2Hz!1, fcd is the fraction of gas that iscold and dense. We set fcd = 0.02 as a conservative estimate.We connect the LW flux with the comoving density in stars%"(z) via a conversion e$ciency "LW (Greif & Bromm 2006):

JLW =hc

4!mH"LW%"(z)(1 + z)3. (5)

Here, "LW is the number of photons emitted in the LW bandsper stellar baryon. The value of "LW depends on the char-acteristic mass of the formed primordial stars, but the vari-ation is not very large for stars with mass greater than tensolar-masses (Schaerer 2002). We set "LW = 104 for bothPop III.1 and Pop III.2 for simplicity.

Next we calculate the stellar mass density as

%"(z) =

Z

#"(z$)

˛

˛

˛

˛

dtdz!

˛

˛

˛

˛

dz$. (6)

For a given z, the integral is performed over the maximumdistance that a LW photon can travel before it is redshiftedout of the LW bands. Haloes with virial temperature lessthan 104 Kelvin cool almost exclusively by H2 line cooling,and produce predominantly massive stars with a top heavyinitial mass function. We adopt the mass of such haloes,M(Tvir = 104K, z) as an upper limit of haloes that producePop III.1 stars. In larger haloes, the gas is ionized at viri-alization, and thus the formed stars are, according to ourdefinition, Pop III.2 stars. Namely, M(Tvir = 104K, z) is theminimum halo mass for Pop III.2 star formation.


c! 2010 RAS, MNRAS 000, 1–8





#obsGRB(z) =

"obs

4!"GRB "beam #"(z)

Z #

Llim(z)

p(L)dL, (1)




nST(M, z) = A

r

2a1

!

»

1 +

„

#2

a1$2c

«p–

$c

#exp

»

"a1$2c

2#2

–

,

(2)



#2(M, z) =b2(z)

2!2

Z

#

0

k2P (k)W 2(k, M)dk, (3)





MMin!H2 = exp

„

fcd

0.06

«

(1.25#105+8.7#105F 0.47LW,!21), (4)



JLW =hc

4!mH"LW%"(z)(1 + z)3. (5)



%"(z) =

Z

#"(z$)

˛

˛

˛

˛

dtdz!

˛

˛

˛

˛

dz$. (6)



c! 2010 RAS, MNRAS 000, 1–8

The star formation efficiency in the early Universe largely depends on the ability of a primordial gas to cool and condense. Hydrogen molecules (H2) are the primary coolant in a gas in small mass “minihaloes”. H2 are also fragile to soft ultra-violet radiation, and thus a ultra-violet background in the Lyman-Werner (LW) bands can easily suppress star formation inside minihaloes.

RADIATIVE FEEDBACK

Minimum mass able to collapse in the presence of LW background

LW flux


– 61 –

ionizing photons contribute regardless of the spatial distribution of sources; and second, the total

recombination rate is proportional to the total ionized volume, regardless of its topology. Thus,

even if two or more bubbles overlap the model remains an accurate approximation for QH II (at least

until QH II becomes nearly equal to 1). Note, however, that there still are a number of important

simplifications in the model, including the assumption of a homogeneous (though possibly time-

dependent) clumping factor, and the neglect of feedback whereby the formation of one galaxy may

suppress further galaxy formation in neighboring regions. These complications are discussed in

detail below and in §6.5 and §7.

Under these assumptions we convert equation (65), which describes individual H II regions, to

an equation which statistically describes the transition from a neutral universe to a fully ionized

one (compare Madau et al. 1999 and Haiman & Loeb 1997):

dQH II

dt=

Nion

0.76

dFcol

dt! !B

C

a3n0

HQH II , (76)

where we assumed a primordial mass fraction of hydrogen of 0.76. The solution (in analogy with

equation (67)) is

QH II(t) =! t

0

Nion

0.76

dFcol

dt!eF (t!,t)dt! , (77)

where F (t!, t) is determined by equations (68)–(71).

A simple estimate of the collapse fraction at high redshift is the mass fraction (given by

equation (31) in the Press-Schechter model) in halos above the cooling threshold, which is the

minimum mass of halos in which gas can cool e!ciently. Assuming that only atomic cooling is

e"ective during the redshift range of reionization (§3.3), the minimum mass corresponds roughly

to a halo of virial temperature Tvir = 104 K, which can be converted to a mass using equation (26).

With this prescription we derive (for Nion = 40) the reionization history shown in Figure 22 for

the case of a constant clumping factor C. The solid curves show QH II as a function of redshift for

a clumping factor C = 0 (no recombinations), C = 1, C = 10, and C = 30, in order from left to

right. Note that if C " 1 then recombinations are unimportant, but if C "> 10 then recombinations

significantly delay the reionization redshift (for a fixed star-formation history). The dashed curve

shows the collapse fraction Fcol in this model. For comparison, the vertical dotted line shows the

z = 5.8 observational lower limit (Fan et al. 2000) on the reionization redshift.

Clearly, star-forming galaxies in CDM hierarchical models are capable of ionizing the universe

at z " 6–15 with reasonable parameter choices. This has been shown by a number of theo-

retical, semi-analytic calculations (Fukugita & Kawasaki 1994; Shapiro, Giroux, & Babul 1994;

Kamionkowski, Spergel, & Sugiyama 1994; Tegmark, Silk, & Blanchard 1994; Haiman & Loeb

1997; Valageas & Silk 1999; Chiu & Ostriker 2000; Ciardi et al. 2000) as well as numerical simula-

tions (Cen & Ostriker 1993; Gnedin & Ostriker 1997; Gnedin 2000a). Similarly, if a small fraction

("< 1%) of the gas in each galaxy accretes onto a central black hole, then the resulting mini-quasars

are also able to reionize the universe, as has also been shown using semi-analytic models (Fukugita

& Kawasaki 1994; Haiman & Loeb 1998; Valageas & Silk 1999). Note that the prescription whereby

REIONIZATION– 10 –

Fig. 4.— Stages in the reionization of hydrogen in the intergalactic medium.

R. S. de Souza et al.: Populations III.1 and III.2 gamma-ray bursts


!!(z) =!!!(z")

"""""dtdz"

""""" dz". (8)

For a given z, the integral is performed over the maximum dis-tance that an LW photon can travel before it is redshifted out ofthe LW bands. The mean free path of LW photons at z = 30is #10 Mpc (physical). Photons travel over the mean free pathin #107 yr (Mackey et al. 2003). Halos with virial temperatureless than 104 Kelvin cool almost exclusively by H2 line cool-ing, and produce mostly massive stars. We adopt the mass ofsuch halos, M(Tvir = 104 K, z), as an upper limit of halos thatproduce Pop III.1 stars. In larger halos, the gas is ionized atvirialization, and thus the formed stars have, according to ourdefinition, similar properties to Pop III.2 stars. We assume thatM(Tvir = 104 K, z) is the minimum halo mass for Pop III.2 starformation.

The collapsed fraction of mass, Fcol(z), available for Pop IIIstar formation is given by

FIII.1col (z) =

1!m

! MTvir=104K

MH2

dMMnST(M, z) (9)

for Pop III.1 stars, and

FIII.2col (z) =

1!m

! $

MTvir=104K

dMMnST(M, z) (10)

for Pop III.2. Using the above criteria, the SFR of Pop III starscan be written as

!III.1! (z) = (1 % QHII(z))(1 % "(z, vwind))!m fb f!

dFIII.1col

dt(11)


!III.2! (z) = QHII(z)(1 % "(z, vwind))!m fb f!

dFIII.2col

dt(12)

for Pop III.2. Here, "(z, vwind) represents the global filling frac-tion of metals via galactic winds (see Sect. 2.4), QHII(z) the vol-ume filling fraction of ionized regions (see Sect. 2.3), and fbis the baryonic mass fraction. For the star formation e"ciency,we use the value f! = 0.001 as a conservative choice (Greif &Bromm 2006) and f! = 0.1 (Bromm & Loeb 2006) as an upperlimit. The latter choice is not strictly consistent with the assump-tion made in Eq. (6). We explore a model with f! = 0.1 simplyto show a very optimistic case.

2.3. Reionization

Inside growing H!! regions, the gas is highly ionized, and thetemperature is #104 K, so the formation of Pop III.1 stars isterminated according to our definition. The formation rate ofPop III.1 is reduced by a factor given by the volume fillingfraction of ionized regions, QHII(z). We follow Wyithe & Loeb(2003) to calculate the evolution of QHII(z) as

dQHII

dz=

Nion

0.76dFcol

dt% #B

Ca3 n0

HQHII, (13)

whose solution is

QHII(z) =! $

zdz"

dtdz"

Nion

0.76dFcol

dteF(z" ,z), (14)

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

z

QH

II!z"

Fig. 1. Reionization history calculated using our model. The blue line isour model prediction, and the dotted black line is the best fit of CAMBcode.

where

F(z", z) = %23#Bn0

H&#mH0

C[ f (z") % f (z)], (15)

and

f (z) =

#(1 + z)3 +

1 %#m

#m· (16)

Here we have assumed the primordial fraction of hydrogen of0.76. In the above equations, Nion ' N$ f! fesc is an e"ciency pa-rameter that gives the number of ionizing photons per baryon,where fesc is the fraction of ionizing photons able to escapethe host galaxy, and N$ is the time averaged number of ioniz-ing photons emitted per unit stellar mass formed. The quantityn0

H = 1.95 ( 10%7 cm%3 is the present-day comoving numberdensity of hydrogen, #B = 2.6 ( 10%13 cm3 s%1 is the hydrogenrecombination rate, and C = )n2

H*/n2H the clumping factor. We

use the average value C = 4 (see Pawlik et al. 2009 for detaileddiscussion about redshift dependence of C). We set the valuesfesc = 0.7, f! = 0.01, and N$ = 9 ( 104 as fiducial values (Greif& Bromm 2006).

In Fig. 1 we show the reionization history calculated usingour model in comparison with a fitting function that is the defaultparametrization of reionization in CAMB (Lewis et al. 2000).

2.4. Metal enrichment

We need to consider metal enrichment in the intergalacticmedium (IGM) in order to determine when the formation of pri-mordial stars is terminated (locally) and when the star formationswitches from the Pop III mode to a more conventional one.

It is thought that Pop III stars do not generate strong stel-lar winds, and thus the main contribution to the metal pollutioncomes from their supernova explosions. Madau et al. (2001) ar-gue that pregalactic outflows from the same primordial halos thatreionize the IGM could also pollute it with a substantial amountof heavy elements. To incorporate the e$ect of metal enrichmentby galactic winds, we adopt a similar prescription to Johnson(2010) and Furlanetto & Loeb (2005).

We assume that star-forming halos (“galaxies”) launch awind of metal-enriched gas at z! # 20. The metal-enriched windpropagates outward from a central galaxy with a velocity vwind,

A32, page 3 of 9



!!(z) =!!!(z")

"""""dtdz"

""""" dz". (8)



FIII.1col (z) =

1!m

! MTvir=104K

MH2

dMMnST(M, z) (9)


FIII.2col (z) =

1!m

! $

MTvir=104K

dMMnST(M, z) (10)



dFIII.1col

dt(11)



dFIII.2col

dt(12)


2.3. Reionization


dQHII

dz=

Nion

0.76dFcol

dt% #B

Ca3 n0

HQHII, (13)

whose solution is

QHII(z) =! $

zdz"

dtdz"

Nion

0.76dFcol

dteF(z" ,z), (14)

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

z

QH

II!z"


where

F(z", z) = %23#Bn0

H&#mH0

C[ f (z") % f (z)], (15)

and

f (z) =

#(1 + z)3 +

1 %#m

#m· (16)










A32, page 3 of 9



!!(z) =!!!(z")

"""""dtdz"

""""" dz". (8)



FIII.1col (z) =

1!m

! MTvir=104K

MH2

dMMnST(M, z) (9)


FIII.2col (z) =

1!m

! $

MTvir=104K

dMMnST(M, z) (10)



dFIII.1col

dt(11)



dFIII.2col

dt(12)


2.3. Reionization


dQHII

dz=

Nion

0.76dFcol

dt% #B

Ca3 n0

HQHII, (13)

whose solution is

QHII(z) =! $

zdz"

dtdz"

Nion

0.76dFcol

dteF(z" ,z), (14)

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

z

QH

II!z"


where

F(z", z) = %23#Bn0

H&#mH0

C[ f (z") % f (z)], (15)

and

f (z) =

#(1 + z)3 +

1 %#m

#m· (16)










A32, page 3 of 9


CHEMICAL ENRICHMENT4 de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

z

QHII!z"

Fig. 1. Reionization history calculated by our model, blueline in comparison with the best fit of CAMB code, dottedblack line.

F (z!, z) = !2

3

!Bn0H"

!mH0C[f(z!) ! f(z)] (13)

and

f(z) =

!

(1 + z)3 +1 ! !m

!m. (14)

where n0H = 1.95 # 10"7cm"3 is the present number

density of hydrogen comoving medium density, !B =2.6 # 10"13cm3s"1 is the hydrogen recombination rate,ne is the comoving electron density and C = $n2

H%/n2H is

the clumping factor. We use the average value C = 4 forsimplicity (see Pawlik et al. (2009) for detailed discussionabout redshift dependence of C). In figure 1 we show thereionization history estimated by our model in comparisonwith tanh fitting function that is the default parametriza-tion of reionization in CAMB code (Lewis et al. 2000).

2.4. Metal Enrichment

We need to consider the metal-enrichment in the inter-galactic medium (IGM) in order to determine when theformation of primordial stars is terminated (locally) andthe star formation switches to a more conventional mode.

Pop III stars do not generate strong stellar winds andthus the main contribution to the metal pollution comesfrom their supernova explosions. Madau et al. (2001) ar-gue that pre-galactic outflows from the same primordialhaloes that reionize the IGM could also pollute it with asubstantial amount of heavy elements.

To incorporate the e"ect of metal-enrichment by galac-tic winds, we adopt a similar prescription of Johnson(2010); Furlanetto & Loeb (2005).

We assume that star-forming haloes (“galaxies”)launch a wind of metal-enriched gas at z# & 20. The metal-enriched wind propagates outward from a central galaxy

with a velocity vwind, traveling over a comoving distanceRwind given by

Rwind =

" z

z!

vwind(1 + z!)dt

dz!dz!. (15)

Then we can express fchem, the ratio of total mass en-riched by the wind to the mass of each halo, as

fchem(M, z, vwind) =4"

3

R3wind

VH, (16)

where VH ' R3H is the comoving volume of each halo. The

halo radius RH can be approximated by

RH(M) =

#

3M

4"180#m

$1/3

. (17)

The fraction of cosmic volume enriched by the winds canbe write as

$(z, vwind) =1

#m

"

dMfbf#fchem(M, z, vwind)MnST(M, z).

(18)Although this may appear a significant over-simplification,the model, with vwind as a single parameter, indeedprovides a good insight into the impact of the metal-enrichment.

We adopt three di"erent values of vwind and examinethe e"ect of metal-enrichment quantitatively. We assumethat Pop III stars are not formed in a metal-enriched re-gion, regardless of the actual metallicity. The metallicityof the gas may not exceed the threshold necessary for thetransition to Pop II star formation (& 10"6 ! 10"3.5Z$)(Bromm & Loeb 2003; Omukai et al. 2005; Frebel et al.2007; Belczynski et al. 2010). We e"ectively assume thatthe so-called critical metallicity is very low.

Figure 2 and 3 shows the SFR history for bothPop III.1 and Pop III.2 considering three di"erent levelsof chemical enrichment, i.e. for vwind = 50, 75, 1000km/s.From figure 2, we can see that the metal enrichment hasa small influence over Pop III.1 since that Pop III.1 eraterminates early due to the reionization process, thus wejust show the lower and upper values. We compare ourmodel results with the SFRs estimated by other authorsin the literature (Bromm & Loeb 2006; Tornatore et al.2007; Trenti & Stiavelli 2009). It is important to note thatPop III formation can continue to low redshifts (z < 10)depending on the level of metal enrichment. Tornatoreet al. (2007) used cosmological simulations to show that,because of limited e#ciency of heavy element transport byoutflows, Pop III star formation could continue to formup to z = 2.5 (which matches with our predictions forvwind = 50km/s). The SFR of Tornatore et al. (2007) has

Define when the star formation switches from the Pop III to Pop II/IMain contribution comes from galactic metal enriched winds

The fraction of cosmic volume enriched by the winds can then be written as

A&A 533, A32 (2011)

Mod Op

T07

TS09

BL06

50 km/s100 km/s

0 5 10 15 20 25 30 35

10!5

10!4

0.001

0.01

z

SFR!M !y

r!1

Mpc!

3 "

Fig. 2. Pop III.1 star formation rate. Calculated for weak and strongchemical feedback models and a moderate star formation e!ciencywith f! = 0.05. The results are shown for vwind = 50 km s"1 , red line;and 100 km s"1, blue line. We also show the theoretical SFRs in the liter-ature, from Bromm & Loeb (2006) (Pop III.1+III.2), dotted black line;Trenti & Stiavelli (2009) (Pop III.2), dashed orange line; and Tornatoreet al. (2007) (Pop III.1+III.2), dot-dashed brown line. The purple lineis our optimistic model where we assume a very high star formatione!ciency, f! # 0.1, and low chemical enrichment, vwind = 50 km s"1.

traveling over a comoving distance Rwind given by

Rwind =

! z

z!vwind(1 + z$)

dtdz$

dz$. (17)

Then we can express fchem, the ratio of gas mass enriched by thewind to the total gas mass in each halo, as

fchem(M, z, vwind) =4!3

R3wind

VH, (18)

where VH % R3H is the volume of each halo. The halo radius RH

can be approximated by

RH(M) ="

3M4! & 180"m

#1/3· (19)

Equation (18) takes the self-enrichment of each halo into ac-count. The next step is to evaluate the average metallicity overcosmic scales. The fraction of cosmic volume enriched by thewinds can then be written as

#(z, vwind) =1"m

!dM fb f! fchem(M, z, vwind)MnST(M, z). (20)

Although this may appear a significant oversimplification, themodel with vwind as a single parameter indeed provides good in-sight into the impact of the metal enrichment.

We adopt three di"erent values of vwind and examine the ef-fect of metal enrichment quantitatively. We assume that Pop IIIstars are not formed in a metal-enriched region, regardless ofthe actual metallicity. Even a single pair instability supernovacan enrich the gas within a small halo to a metallicity level wellabove the critical metallicity (see e.g. Schneider et al. 2006). Wee"ectively assume that the so-called critical metallicity is verylow (Schneider et al. 2002, 2003; Bromm & Loeb 2003; Omukaiet al. 2005; Frebel et al. 2007; Belczynski et al. 2010).

Figures 2 and 3 show the star formation rate (SFR) historyfor both Pop III.1 and Pop III.2 considering three di"erent valuesof the galactic wind, vwind = 50, 75, 100 km s"1. Figure 2 showsthat the metal enrichment has little influence on Pop III.1. This is

"

""

""

""

## #

###

##

#

#

# #

##

#

# # #

###

##

###

#########

#

#

##

#

#

##

#

#

#

#

#

### # #

#

####

#### #

##

#HB2006

""

"

""

"B2008

## #

# O2008

$$

$Y2008

%

%M2007C2007

$

$

$

R2008

%%

%

W2010

&

&

&

&

B2011

Mod Op

BL06

TS09

T07100 km/s

50 km/s

75 km/s

0 5 10 15 20

10!6

10!5

10!4

0.001

0.01

0.1

1

z

SFR!M !y

r!1

Mpc!

3 "

Fig. 3. Pop III.2 star formation rate. Calculated for three di"erent chem-ical feedback models; vwind = 50 km s"1, red line; vwind = 75 km s"1,blue line; and vwind = 100 km s"1, green line. We also show the theoreti-cal SFRs in the literature, from Bromm & Loeb (2006) (Pop III.1+III.2),dotted black line; Trenti & Stiavelli (2009) (Pop III.2), dashed orangeline; and Tornatore et al. (2007) (Pop III.1+III.2), dot-dashed brownline. The purple line is our optimistic model where we assume a veryhigh star formation e!ciency, f! # 0.01, and low chemical enrichment,vwind = 50 km s"1. The light points are independent SFR determinationscompiled from the literature.

because Pop III.1 formation is terminated early due reionization.In Fig. 3 we compare the Pop III.2 SFR history with a compila-tion of independent measures from Hopkins & Beacom (2006)up to z ' 6 and from observations of color-selected Lyman breakgalaxies (Mannucci et al. 2007; Bouwens et al. 2008, 2011), Ly$Emitters (Ota et al. 2008), UV+IR measurements (Reddy et al.2008), and GRB observations (Chary et al. 2007; Yüksel et al.2008; Wang & Dai 2009) at higher z (hereafter, these will bereferred to as H2006, M2007, B2008, B2011, O2008, R2008,C2007, Y2008, and W2009, respectively). The optimistic casefor Pop III.2 is chosen to keep the SFR always below the obser-vationally determined SFR at z < 8.

We compared our model results with the SFRs estimated byother authors in the literature Bromm & Loeb (2006), Tornatoreet al. (2007), Trenti & Stiavelli (2009), and also see Naoz &Bromberg (2007). It is important to note that Pop III formationcan continue to low redshifts (z < 10) depending on the levelof metal enrichment. Tornatore et al. (2007) use cosmologicalsimulations to show that, because of limited e!ciency of heavyelement transport by outflows, Pop III star formation continuesto form down to z = 2.5 (which intriguingly matches our modelwith vwind = 50 km s"1 in Fig. 3). The SFR of Tornatore et al.(2007) has a peak value of 10"5 M( yr"1 Mpc"3 at z ' 6.

In Fig. 4 we also show the result of our model with f! =0.1"0.01 and vwind = 50 km s"1 for both Pop III.1 and Pop III.2,respectively. This model provides an “optimistic” estimate forthe detectable GRB rate for the future missions (see Sect. 3).

2.5. Luminosity function

The number of GRBs detectable by any given instrument de-pends on the instrument-specific flux sensitivity threshold andalso on the intrinsic isotropic luminosity function of GRBs.For the latter, we adopt the power-law distribution function ofWanderman & Piran (2010)

p(L) =

$%%%&%%%'

(LL!

)"0.2+0.2"0.1 L < L!,

(LL!

)"1.4+0.3"0.6 L > L!,

(21)

A32, page 4 of 9

A&A 533, A32 (2011)

Mod Op

T07

TS09

BL06

50 km/s100 km/s

0 5 10 15 20 25 30 35

10!5

10!4

0.001

0.01

z

SFR!M !y

r!1

Mpc!

3 "



Rwind =

! z

z!vwind(1 + z$)

dtdz$

dz$. (17)



R3wind

VH, (18)



RH(M) ="

3M4! & 180"m

#1/3· (19)


#(z, vwind) =1"m





"

""

""

""

## #

###

##

#

#

# #

##

#

# # #

###

##

###

#########

#

#

##

#

#

##

#

#

#

#

#

### # #

#

####

#### #

##

#HB2006

""

"

""

"B2008

## #

# O2008

$$

$Y2008

%

%M2007C2007

$

$

$

R2008

%%

%

W2010

&

&

&

&

B2011

Mod Op

BL06

TS09

T07100 km/s

50 km/s

75 km/s

0 5 10 15 20

10!6

10!5

10!4

0.001

0.01

0.1

1

z

SFR!M !y

r!1

Mpc!

3 "







p(L) =

$%%%&%%%'

(LL!

)"0.2+0.2"0.1 L < L!,

(LL!

)"1.4+0.3"0.6 L > L!,

(21)

A32, page 4 of 9


de Souza, Yoshida & Ioka: Population III.1 and III.2 Gamma-Ray Bursts 5

Mod Op

T07

TS09

BL06

50 km/s100 km/s

0 5 10 15 20 25 30 35

10!5

10!4

0.001

0.01

z

SFR!M!

yr!

1M

pc!

3 "

Fig. 2. Pop III.1 star formation rate for weak and strongchemical feedback models and a moderate star forma-tion e!ciency with f! = 0.05. The results are shown forvwind = 50 km/s (red line) and 100 km/s (blue line).We also show the theoretical SFRs in the literature, fromBromm & Loeb 2006 (Pop III.1+III.2), dotted black line,Trenti & Stiavelli 2009 (Pop III.2), dashed orange line,and Tornatore et al. 2007 (Pop III.1+III.2), dot-dashedbrown line. The purple line is our optimistic model wherewe assume a very high star formation e!ciency, f! ! 0.1,and low chemical enrichment, vwind = 50 km/s.

100 km/s

50 km/s

75 km/s

TS09

T07

BL06Mod Op

0 5 10 15 20 25 30 35

10!6

10!5

10!4

0.001

0.01

0.1

z

SFR!M!

yr!

1M

pc!

3 "

Fig. 3. Pop III.2 star formation rate for three di"erentchemical feedback models; vwind = 50 km/s (red line)vwind = 75 km/s (blue line) and vwind = 100 km/s (greenline). We also show the theoretical SFRs in the litera-ture, from Bromm & Loeb 2006 (Pop III.1+III.2), dot-ted black line, Trenti & Stiavelli 2009 (Pop III.2), dashedorange line, and Tornatore et al. 2007 (Pop III.1+III.2),dot-dashed brown line. The purple line is our optimisticmodel where we assume a very high star formation e!-ciency, f! ! 0.1, and low chemical enrichment, vwind = 50km/s.

where L! is the characteristic isotropic luminosity. We setL! ! 1053ergs/s for Pop III.1 whereas L! ! 1052ergs/sPop III.2 stars similar to ordinary GRBs (Li 2008;Wanderman & Piran 2010). Note that the Pop III.1 GRBsare assumed to be energetic with isotropic kinetic energyEiso ! 1056"57erg but long-lived T90 ! 1000 s. So that theluminosity would be moderate L! ! !! " 1056"57/1000 !

Pop III.2

Pop III.1

Optimistic case

0 5 10 15 20 25 30 35

10!5

10!4

0.001

0.01

0.1

z

SFR!M!

yr!

1M

pc!

3 "

Fig. 4. We compare the star formation rates for Pop III.1(blue dotted line) and for PopIII.2 (dashed black line), forour optimistic model with a high star formation e!ciencyf! = 0.1 and slow chemical enrichment vwind = 50km/s.

1052"53 ergs/s if !! ! 0.1 is the conversion e!ciency fromthe jet kinetic energy to gamma rays (Suwa & Ioka 2011).

Using the above relation we can predict the observableGRB rate for Swift, SVOM, JANUS and EXIST missions.For Swift, we set a bolometric energy flux limit Flim =1.2 " 10"8erg cm"2 s"1 (Li 2008). We adopt a similarlimit for SVOM (Paul et al. 2011). For JANUS, Flim !10"8erg cm"2 s"1 (Falcone et al. 2009). The luminositythreshold is then

Llim = 4" d2L Flim. (22)

Here dL is the luminosity distance for the adopted #CDMcosmology. EXIST is expected to be ! 7# 10 times moresensitive than Swift (Grindlay 2010). We set the EXISTsensitivity threshold is 10 times lower than Swift as anapproximate estimate.

2.6. Initial Mass Function and GRB FormationE!ciency

The stellar initial mass function (IMF) is critically impor-tant to determine the Pop III GRB rate. We define theGRB formation e!ciency factor per stellar mass as

#GRB = fGRB

! Mup

MGRB$(m)dm

! Mup

Mlowm$(m)dm

, (23)

where $(m) is the stellar IMF, and fGRB = 0.001 is theGRB fraction, since we expect 1 GRB every 1000 super-novae (Langer & Norman 2006). We assume that Pop IIIGRBs have a similar fraction.

We consider the following two forms of IMF. One is apower law with the standard Salpeter slope

$(m) $ m"2.35, (24)

and the other is a Gaussian IMF (Scannapieco et al. 2003;Nakamura & Umemura 2001):

$(m)m"1 dm =1%

2"%c(M)e"(m"M)2/2"c(M)2dm. (25)


Mod Op

T07

TS09

BL06

50 km/s100 km/s

0 5 10 15 20 25 30 35

10!5

10!4

0.001

0.01

z

SFR!M!

yr!

1M

pc!

3 "

Fig. 2. Pop III.1 star formation rate for weak and strongchemical feedback models and a moderate star forma-tion e!ciency with f! = 0.05. The results are shown forvwind = 50 km/s (red line) and 100 km/s (blue line).We also show the theoretical SFRs in the literature, fromBromm & Loeb 2006 (Pop III.1+III.2), dotted black line,Trenti & Stiavelli 2009 (Pop III.2), dashed orange line,and Tornatore et al. 2007 (Pop III.1+III.2), dot-dashedbrown line. The purple line is our optimistic model wherewe assume a very high star formation e!ciency, f! ! 0.1,and low chemical enrichment, vwind = 50 km/s.

100 km/s

50 km/s

75 km/s

TS09

T07

BL06Mod Op

0 5 10 15 20 25 30 35

10!6

10!5

10!4

0.001

0.01

0.1

z

SFR!M!

yr!

1M

pc!

3 "

Fig. 3. Pop III.2 star formation rate for three di"erentchemical feedback models; vwind = 50 km/s (red line)vwind = 75 km/s (blue line) and vwind = 100 km/s (greenline). We also show the theoretical SFRs in the litera-ture, from Bromm & Loeb 2006 (Pop III.1+III.2), dot-ted black line, Trenti & Stiavelli 2009 (Pop III.2), dashedorange line, and Tornatore et al. 2007 (Pop III.1+III.2),dot-dashed brown line. The purple line is our optimisticmodel where we assume a very high star formation e!-ciency, f! ! 0.1, and low chemical enrichment, vwind = 50km/s.

where L! is the characteristic isotropic luminosity. We setL! ! 1053ergs/s for Pop III.1 whereas L! ! 1052ergs/sPop III.2 stars similar to ordinary GRBs (Li 2008;Wanderman & Piran 2010). Note that the Pop III.1 GRBsare assumed to be energetic with isotropic kinetic energyEiso ! 1056"57erg but long-lived T90 ! 1000 s. So that theluminosity would be moderate L! ! !! " 1056"57/1000 !

Pop III.2

Pop III.1

Optimistic case

0 5 10 15 20 25 30 35

10!5

10!4

0.001

0.01

0.1

z

SFR!M!

yr!

1M

pc!

3 "

Fig. 4. We compare the star formation rates for Pop III.1(blue dotted line) and for PopIII.2 (dashed black line), forour optimistic model with a high star formation e!ciencyf! = 0.1 and slow chemical enrichment vwind = 50km/s.

1052"53 ergs/s if !! ! 0.1 is the conversion e!ciency fromthe jet kinetic energy to gamma rays (Suwa & Ioka 2011).

Using the above relation we can predict the observableGRB rate for Swift, SVOM, JANUS and EXIST missions.For Swift, we set a bolometric energy flux limit Flim =1.2 " 10"8erg cm"2 s"1 (Li 2008). We adopt a similarlimit for SVOM (Paul et al. 2011). For JANUS, Flim !10"8erg cm"2 s"1 (Falcone et al. 2009). The luminositythreshold is then

Llim = 4" d2L Flim. (22)

Here dL is the luminosity distance for the adopted #CDMcosmology. EXIST is expected to be ! 7# 10 times moresensitive than Swift (Grindlay 2010). We set the EXISTsensitivity threshold is 10 times lower than Swift as anapproximate estimate.

2.6. Initial Mass Function and GRB FormationE!ciency

The stellar initial mass function (IMF) is critically impor-tant to determine the Pop III GRB rate. We define theGRB formation e!ciency factor per stellar mass as

#GRB = fGRB

! Mup

MGRB$(m)dm

! Mup

Mlowm$(m)dm

, (23)

where $(m) is the stellar IMF, and fGRB = 0.001 is theGRB fraction, since we expect 1 GRB every 1000 super-novae (Langer & Norman 2006). We assume that Pop IIIGRBs have a similar fraction.

We consider the following two forms of IMF. One is apower law with the standard Salpeter slope

$(m) $ m"2.35, (24)


$(m)m"1 dm =1%

2"%c(M)e"(m"M)2/2"c(M)2dm. (25)

A&A 533, A32 (2011)

Mod Op

T07

TS09

BL06

50 km/s100 km/s

0 5 10 15 20 25 30 35

10!5

10!4

0.001

0.01

z

SFR!M !y

r!1

Mpc!

3 "



Rwind =

! z

z!vwind(1 + z$)

dtdz$

dz$. (17)



R3wind

VH, (18)



RH(M) ="

3M4! & 180"m

#1/3· (19)


#(z, vwind) =1"m





"

""

""

""

## #

###

##

#

#

# #

##

#

# # #

###

##

###

#########

#

#

##

#

#

##

#

#

#

#

#

### # #

#

####

#### #

##

#HB2006

""

"

""

"B2008

## #

# O2008

$$

$Y2008

%

%M2007C2007

$

$

$

R2008

%%

%

W2010

&

&

&

&

B2011

Mod Op

BL06

TS09

T07100 km/s

50 km/s

75 km/s

0 5 10 15 20

10!6

10!5

10!4

0.001

0.01

0.1

1

z

SFR!M !y

r!1

Mpc!

3 "







p(L) =

$%%%&%%%'

(LL!

)"0.2+0.2"0.1 L < L!,

(LL!

)"1.4+0.3"0.6 L > L!,

(21)

A32, page 4 of 9

Star formation history of Pop III stars


OBSERVED GAMMA RAY BURST RATE





#obsGRB(z) =

"obs

4!"GRB "beam #"(z)

Z #

Llim(z)

p(L)dL, (1)




nST(M, z) = A

r

2a1

!

»

1 +

„

#2

a1$2c

«p–

$c

#exp

»

"a1$2c

2#2

–

,

(2)



#2(M, z) =b2(z)

2!2

Z

#

0

k2P (k)W 2(k, M)dk, (3)





MMin!H2 = exp

„

fcd

0.06

«

(1.25#105+8.7#105F 0.47LW,!21), (4)



JLW =hc

4!mH"LW%"(z)(1 + z)3. (5)



%"(z) =

Z

#"(z$)

˛

˛

˛

˛

dtdz!

˛

˛

˛

˛

dz$. (6)



c! 2010 RAS, MNRAS 000, 1–8

ηGRB is GRB formation efficiency

ηbeam is the beaming factor of the burst

Ωobs is the field of view of the experiment

Ψ∗ is the cosmic star formation rate (SFR) density

p(L) is the GRB luminosity function

SFR


INITIAL MASS FUNCTION AND GRB EFFICIENCY

Constraints over Population III.1 and III.2 Gamma-Ray Bursts 5

the standard Salpeter slope

!(m) ! m!2.35, (20)


!(m)mdm =1"

2"#c

e!(m!M)2/2!2c dm. (21)

For the latter, we assume M = 550M" for Pop III.1 andM = 55M" for Pop III.2, with dispersion #c = (M #Mlow)/3. Mlow is the minimum mass for a given stellar type,100M" for Pop III.1 and 10M" for Pop III.2, whereas Mup

is the maximum mass for a given stellar type, 1000M" forPop III.1 and $ 100M" for Pop III.2. MGRB is the minimummass able to trigger GRBs, which we set to be 25M"(Bromm& Loeb 2006). Note that not all Pop III.1 stars will leavea blackhole behind at their deaths. In the narrow massrange of $ 140 # 260M", Pop III stars are predicted to un-dergo a pair-instability supernova (PISN) explosion (Heger& Woosley 2002).

The e!ciency factor considering Salpeter IMF are$GRB/fGRB $ 1/926M!1

" and 1/87M!1" for Pop III.1

and Pop III.2 respectively. Using the gaussian IMF,$GRB/fGRB $ 1/538M!1

" and 1/53M!1" for Pop III.1 and

Pop III.2. respectively. Thus, the GRB formation e!ciencyfor Pop III.2 can be $ 1 order of magnitude larger than PopIII.1 because of the lower mass of Pop III.2 stars.

3 REDSHIFT DISTRIBUTION OF GRBS

Over a particular time interval, "tobs, in the observer restframe, the number of observed GRBs originating betweenredshifts z and z + dz is

dNGRB

dz= #obs

GRB(z)"tobs

1 + zdVdz

, (22)

where dV/dz is the comoving volume element per unit red-shift, given by

dVdz

=4" c d2

L

(1 + z)

˛

˛

˛

˛

dtdz

˛

˛

˛

˛

. (23)

Figure 4 shows the intrinsic GRB rate. In this plot, wehave not considered observational e$ects such as beamingand instrument sensitivity. We show the GRB rate for twodi$erent IMFs discussed above. From figure 4 we can con-clude that results depend weekly of the choice of IMF.

Figure 5 shows the most optimistic case, assuming ahigh star formation e!ciency f# $ 0.1 (Bromm & Loeb2006), low chemical enrichment, vwind = 50 km/s, and aSalpeter IMF for both Pop III.1 and Pop III.2 stars.

In figures ??-??, we show the predicted observed GRBrate for Pop III.1 and III.2 detectable by Swift, SVOM,JANUS and EXIST missions for the both optimistic andconservative cases. Overall, it is more likely to observePop III.2 GRBs than Pop III.1, but the predicted ratestrongly depends on how cosmic metallicity evolves withredshift and the star formation e!ciency. The dependenceon the IMF is relatively small. Additional constraints overthese quantities should be useful to place upper limits onthe GRB observed rate and will be discussed next.

Follow-up observations of high redshift GRBs can be

done by observing their radio afterglows (Ioka & Meszaros2005; Inoue et al. 2007). We calculate the radio afterglowlight-curve for Pop III GRBs following the standard pre-scription from Sari et al. (1998); Meszaros (2006). The after-glow light-curve at the time td is given by radius rd and theLorentz factor %d of the shocked fluid at this time. They arerelated by Eiso $ 4"r3

d%2dnmpc2 and rd $ c%2

dtd. Where n isthe medium density and mp is the proton mass. The spec-trum consists of power-law segments linked by critical breakfrequencies. These are &a (the self absorption frequency),&m (the peak of injection frequency) and &c (the coolingfrequency), given by

&m ! (1 + z)1/2g(p)2'2e'1/2B E1/2

iso t!3/2d ,

&c ! (1 + z)!1/2'!3/2B n!1E!1/2

iso t!1/2d ,

&a ! (1 + z)!1'!1e '1/5

B n3/5E1/5iso ,

F",max ! (1 + z)'1/2B n1/2Eisod

!2L . (24)

Where g(p) = (p#2)/(p#1), is a function of energy spectrumindex of electrons (N(Ee)dEe ! E!p!1

e dEe), 'e and 'B arethe e!ciency factors (Meszaros 2006). There are two typesof spectra. If &m < &c, we call it the slow cooling case. Theflux at the observer, F" , is given by

F" =

8

>

>

<

>

>

:

(&a/&m)1/3(&/&a)2F",max, &a > &,(&/&m)1/3F",max, &m > & > &a,(&/&m)!(p!1)/2F",max, &c > & > &m,(&c/&m)!(p!1)/2(&/&c)

!p/2F",max, & > &c.(25)

where F",max is the observed peak flux at distance dL fromthe source.

For &m > &c, called the fast cooling case, the spectrumis

F" =

8

>

>

<

>

>

:

(&a/&c)1/3(&/&a)2F",max, &a > &,(&/&c)

1/3F",max, &c > & > &a,(&/&c)!1/2F",max, &m > & > &c,(&m/&c)

!1/2(&/&m)!p/2F",max, & > &m.(26)

Figure 8 shows the light curves for a typical GRB fromPop III.2 stars assuming an isotropic energy Eiso $ 1055ergas a lower limit since that Pop III.1 afterglows are expectedto be brighter. Similarly to the previous works we show thatit is possible to observe the light curve in radio band withALMA, LOFAR, EVLA, and ultimately by SKA.

3.1 Upper limits from radio transients survey

Since that GRBs are collimated with rather narrow openingangles, while the following afterglow could be observed overa wider angular range, afterglows are not strictly associatedwith observed prompt GRB emission. Thus, orphan after-glows are a natural prediction of GRB jets. Radio transientsources probe the high energy population of the Universeand can provide further constraints on the intrinsic rate ofGRBs. Despite the prompt emission to be highly collimated,after some time the jet starts to expand sideways and be-comes non-relativistic around the time

tNR $ 9.3 % 107

„

E1055

«1/3 „

(0.1

«2/3

n!1/3(1 + z)s, (27)

c! 2010 RAS, MNRAS 000, 1–8

Constraints over Population III.1 and III.2 Gamma-Ray Bursts 5

the standard Salpeter slope

!(m) ! m!2.35, (20)


!(m)mdm =1"

2"#c

e!(m!M)2/2!2c dm. (21)

For the latter, we assume M = 550M" for Pop III.1 andM = 55M" for Pop III.2, with dispersion #c = (M #Mlow)/3. Mlow is the minimum mass for a given stellar type,100M" for Pop III.1 and 10M" for Pop III.2, whereas Mup

is the maximum mass for a given stellar type, 1000M" forPop III.1 and $ 100M" for Pop III.2. MGRB is the minimummass able to trigger GRBs, which we set to be 25M"(Bromm& Loeb 2006). Note that not all Pop III.1 stars will leavea blackhole behind at their deaths. In the narrow massrange of $ 140 # 260M", Pop III stars are predicted to un-dergo a pair-instability supernova (PISN) explosion (Heger& Woosley 2002).

The e!ciency factor considering Salpeter IMF are$GRB/fGRB $ 1/926M!1

" and 1/87M!1" for Pop III.1

and Pop III.2 respectively. Using the gaussian IMF,$GRB/fGRB $ 1/538M!1

" and 1/53M!1" for Pop III.1 and

Pop III.2. respectively. Thus, the GRB formation e!ciencyfor Pop III.2 can be $ 1 order of magnitude larger than PopIII.1 because of the lower mass of Pop III.2 stars.

3 REDSHIFT DISTRIBUTION OF GRBS


dNGRB

dz= #obs

GRB(z)"tobs

1 + zdVdz

, (22)

where dV/dz is the comoving volume element per unit red-shift, given by

dVdz

=4" c d2

L

(1 + z)

˛

˛

˛

˛

dtdz

˛

˛

˛

˛

. (23)

Figure 4 shows the intrinsic GRB rate. In this plot, wehave not considered observational e$ects such as beamingand instrument sensitivity. We show the GRB rate for twodi$erent IMFs discussed above. From figure 4 we can con-clude that results depend weekly of the choice of IMF.

Figure 5 shows the most optimistic case, assuming ahigh star formation e!ciency f# $ 0.1 (Bromm & Loeb2006), low chemical enrichment, vwind = 50 km/s, and aSalpeter IMF for both Pop III.1 and Pop III.2 stars.

In figures ??-??, we show the predicted observed GRBrate for Pop III.1 and III.2 detectable by Swift, SVOM,JANUS and EXIST missions for the both optimistic andconservative cases. Overall, it is more likely to observePop III.2 GRBs than Pop III.1, but the predicted ratestrongly depends on how cosmic metallicity evolves withredshift and the star formation e!ciency. The dependenceon the IMF is relatively small. Additional constraints overthese quantities should be useful to place upper limits onthe GRB observed rate and will be discussed next.

Follow-up observations of high redshift GRBs can be

done by observing their radio afterglows (Ioka & Meszaros2005; Inoue et al. 2007). We calculate the radio afterglowlight-curve for Pop III GRBs following the standard pre-scription from Sari et al. (1998); Meszaros (2006). The after-glow light-curve at the time td is given by radius rd and theLorentz factor %d of the shocked fluid at this time. They arerelated by Eiso $ 4"r3

d%2dnmpc2 and rd $ c%2

dtd. Where n isthe medium density and mp is the proton mass. The spec-trum consists of power-law segments linked by critical breakfrequencies. These are &a (the self absorption frequency),&m (the peak of injection frequency) and &c (the coolingfrequency), given by

&m ! (1 + z)1/2g(p)2'2e'1/2B E1/2

iso t!3/2d ,

&c ! (1 + z)!1/2'!3/2B n!1E!1/2

iso t!1/2d ,

&a ! (1 + z)!1'!1e '1/5

B n3/5E1/5iso ,

F",max ! (1 + z)'1/2B n1/2Eisod

!2L . (24)

Where g(p) = (p#2)/(p#1), is a function of energy spectrumindex of electrons (N(Ee)dEe ! E!p!1

e dEe), 'e and 'B arethe e!ciency factors (Meszaros 2006). There are two typesof spectra. If &m < &c, we call it the slow cooling case. Theflux at the observer, F" , is given by

F" =

8

>

>

<

>

>

:

(&a/&m)1/3(&/&a)2F",max, &a > &,(&/&m)1/3F",max, &m > & > &a,(&/&m)!(p!1)/2F",max, &c > & > &m,(&c/&m)!(p!1)/2(&/&c)

!p/2F",max, & > &c.(25)

where F",max is the observed peak flux at distance dL fromthe source.


F" =

8

>

>

<

>

>

:

(&a/&c)1/3(&/&a)2F",max, &a > &,(&/&c)

1/3F",max, &c > & > &a,(&/&c)!1/2F",max, &m > & > &c,(&m/&c)

!1/2(&/&m)!p/2F",max, & > &m.(26)

Figure 8 shows the light curves for a typical GRB fromPop III.2 stars assuming an isotropic energy Eiso $ 1055ergas a lower limit since that Pop III.1 afterglows are expectedto be brighter. Similarly to the previous works we show thatit is possible to observe the light curve in radio band withALMA, LOFAR, EVLA, and ultimately by SKA.

3.1 Upper limits from radio transients survey

Since that GRBs are collimated with rather narrow openingangles, while the following afterglow could be observed overa wider angular range, afterglows are not strictly associatedwith observed prompt GRB emission. Thus, orphan after-glows are a natural prediction of GRB jets. Radio transientsources probe the high energy population of the Universeand can provide further constraints on the intrinsic rate ofGRBs. Despite the prompt emission to be highly collimated,after some time the jet starts to expand sideways and be-comes non-relativistic around the time

tNR $ 9.3 % 107

„

E1055

«1/3 „

(0.1

«2/3

n!1/3(1 + z)s, (27)

c! 2010 RAS, MNRAS 000, 1–8


Mod Op

50-100 km/sT07

TS09

BL06

0 5 10 15 20 25 3010!6

10!5

10!4

0.001

0.01

0.1

z

SFR!M!

yr!

1M

pc!

3 "

Figure 1. Pop III.1 star formation rate for three di!erent chemi-cal feedback models; vwind = 50!100 km/s and f! = 0.05, dashedblue line. We also show the theoretical SFRs in the literature,from Bromm & Loeb 2006, black line, Trenti & Stiavelli 2009,orange line, and Tornatore et al. 2007, brown line. The dashedpurple line is our optimistic model where we assume a very highstar formation e"ciency, f! " 0.1, low chemical enrichment.

100 km/s50 km/s

70 km/s

TS09

T07

Mod Op BL06

0 5 10 15 20 2510!6

10!5

10!4

0.001

0.01

0.1

z

SFR!M!

yr!

1M

pc!

3 "

Figure 2. Pop III.2 star formation rate for three di!erent chemi-cal feedback models; vwind = 50 km/s, dashed red line, vwind = 70km/s, dotted blue line, and vwind = 100 km/s, dot-dashed greenline. We also show the theoretical SFRs in the literature, fromBromm & Loeb 2006, black line, Trenti & Stiavelli 2009, orangeline, and Tornatore et al. 2007, brown line. The dashed purpleline is our optimistic model where we assume a very high starformation e"ciency, f! " 0.1, low chemical enrichment, vwind =50 km/s.

It is important to note that Pop III formation can continueto low redshifts (z < 10) depending on the level of metalenrichment. Tornatore et al. (2007) used cosmological sim-ulations to show that, because of limited e!ciency of heavyelement transport by outflows, Pop III star formation couldcontinue to form up to z = 2.5 (which matches with our pre-dictions for vwind = 50). The SFR of Tornatore et al. (2007)has a peak value of 10"5M#yr"1Mpc"3 at z ! 6 (the thinbrown solid line in figure 1).

We also show the result of our model with f! = 0.1and vwind = 50 km/s for both Pop III.1 and Pop III.2 anda comparison between both in figure 3. The model providesan “optimistic” estimate for the detectable GRB rate for thefuture missions (see Section 3).

Pop III.2

Pop III.1

5 10 15 20 25 30 3510!6

10!5

10!4

0.001

0.01

0.1

1

z

SFR!M!

yr!

1M

pc!

3 "

Figure 3. The star formation rate for our optimistic model,where we assume a very high star formation e"ciency, f! " 0.1,low chemical enrichment, vwind = 50 km/s. The dotted blue lineif for Pop III.1 and dashed black line for Pop III.2 star formationrate.

2.5 Luminosity Function

The number of bursts detectable by any given instrumentdepends on the instrument-specific flux sensitivity thresholdand also on the intrinsic isotropic luminosity function ofGRBs. For the latter, we adopt a power-law distributionfunction of Wanderman & Piran (2010)

p(L) =

8

<

:

“

LL!

”"0.2L < L!,

“

LL!

”"1.4L > L!.

, (17)

where L! is the characteristic luminosity. We set L! " 1053

for Pop III.1 whereas L! " 1052 Pop III.2 stars similar toordinary GRBs (Li 2008; Wanderman & Piran 2010).

Using the above relation we can predict the observ-able GRB rate for Swift, SVOM, JANUS and EXIST mis-sions. For Swift, we set a bolometric energy flux limitFlim = 1.2 # 10"8erg cm"2 s"1 (Li 2008). We adopt asimilar limit for SVOM (Paul et al. 2011). For JANUS,Flim " 10"8erg cm"2 s"1 (Falcone et al. 2009). The lu-minosity threshold is then

Llim = 4! d2L Flim. (18)

EXIST is expected to be " 7$10# more sensitive than Swift

(Grindlay 2010). We set the EXIST sensitivity threshold is10 times lower than Swift as an optimistic estimate.

2.6 Initial Mass Function and GRB Formation

E!ciency

The stellar initial mass function (IMF) is critically impor-tant to determine the Pop III GRB rate. We define the GRBformation e!ciency factor per stellar mass as

"GRB = fGRB

R Mup

MGRB#(m)dm

R Mup

Mlowm#(m)dm

, (19)

where #(m) is the stellar IMF, fGRB is the e!ciency of theblack hole trigger a GRBs (" 0.001), as we expect 1 GRBevery 1000 supernovae (Langer & Norman 2006).

We consider two forms of IMF. One is a power law with

c# 2010 RAS, MNRAS 000, 1–8

Salpeter

Gaussian

GRB efficiency formation rate



and the jet starts to expand sideways. Finally the shockvelocity becomes non-relativistic around the time

tNR ! 1.85"102

!

Eiso

5 " 1054

"1/3 !

!

0.1

"2/3

n!1/3(1+z) days,

(33)(Ioka & Meszaros 2005). After the time t!, the temporaldependence of the critical break frequencies should be re-placed by "c # t0, "m # t!2, "a # t!1/5 and F",max # t!1

(Sari et al. 1999). We also use the same evolution in thenon-relativistic phase for simplicity, which underestimatesthe afterglow flux after tNR.

Fig. 7 shows the light curves for a typical GRB fromPop III.2 stars assuming an isotropic kinetic energy Eiso !5 " 1054erg (in proportion to the progenitor mass) asa lower limit. Pop III.1 afterglows are expected to bebrighter. Consistently with previous works, we concludethat it is possible to observe the GRB radio afterglowswith ALMA, LOFAR, EVLA, and ultimately by SKA.

3.2. Upper limits from radio transient survey

In this section, we derive upper limits on the intrinsicGRB rate (including the o!-axis GRB) using ! 1 yeartimescale radio variability surveys. There are several ra-dio transient surveys completed so far. Bower et al. (2007)used 22 years of archival data from VLA to put an upperlimit of ! 6 deg!2 for 1-year variability transients above90 µJy, which is equivalent to ! 2.4 " 105 for all sky.Gal-Yam et al. (2006) used FIRST12 and NVSS13 radiocatalogues to place an upper limit of ! 70 radio orphanafterglows above 6 mJy in the 1.4 GHz band over the en-tire sky. This suggests less than ! 103 sources above 1 mJyon the sky, because the number of sources is expected to

be proportional to flux limit F!3/2lim (assuming Euclidian

space and no source evolution) (Gal-Yam et al. 2006).From Fig. 7, a typical GRB’s radio afterglow with

isotropic kinetic energy Eiso ! 5 " 1054 ergs stays above1 mJy over ! 102!3 days. Combining the results shownin Figs. 5 and 6, we expect ! 10 $ 104 sources (40 $ 104

events per year "102 $ 103 days) above ! 1 mJy. As aconsequence, the most optimistic case for Pop III.2 shouldbe already ruled out by the current observations of radiotransient sources, if their luminosity function follows theone assumed in the present paper. Only more conservativemodels are then viable. Radio transient surveys are not yetable to set upper limits on the Pop III.1 GRB rate. Theabove conclusion is model dependent, because the after-glow flux depends on the yet uncertain quantities such asthe isotropic energy Eiso and the ambient density n. If thecircumburst density is higher than usual, the constraintsfrom the radio transient surveys would be even severer.

In Figs. 8-9, we show the predicted observable GRBrate dNobs

GRB/dz in Eq. (26) for Pop III.1 and III.2 de-tectable by Swift, SVOM, JANUS and EXIST missions.

12 http://sundog.stsci.edu/13 http://www.cv.nrao.edu/nvss/

Pop III.2

Pop III.1

10 15 20 25 30

0.001

0.01

0.1

1

10

100

z

Intri

nsic

GRB

rate!y

r!1 "

Fig. 5. The intrinsic GRB rate dNGRB/dz, the numberof (on-axis + o!-axis) GRBs per year on the sky in Eq.(28), as a function of redshift. We set f" = 0.001 andvwind = 100km/s for this plot. For Pop III.2 with SalpeterIMF (dashed black line) and with Gaussian IMF (dottedblack line). For Pop III.1 with Salpeter IMF (dashed blueline) and with Gaussian IMF (dotted blue line).

Pop III.2

Pop III.1

Optimistic case

5 10 15 20 25 30 35 4010!4

0.01

1

100

104

z

Intri

nsic

GRB

rate!y

r!1 "

Fig. 6. The intrinsic GRB rate dNGRB/dz, the number of(on-axis + o!-axis) GRBs per year on the sky in Eq. (28),as a function of redshift for our optimistic model assuminga high star formation e"ciency, f" = 0.1, slow chemicalenrichment, vwind = 50km/s, and a Gaussian IMF, forboth Pop III.2 (dashed black line) and Pop III.1 (dottedblue line).

The shown results are still within the bounds from avail-able upper limits from the radio transient surveys. Overall,it is more likely to observe Pop III.2 GRBs than Pop III.1,but the predicted rate strongly depends on the IGM metal-licity evolution and the star formation e"ciency. The de-pendence on the IMF is relatively small.

Fig. 10 shows the GRB rate expected for EXIST obser-vations. We use the rate which is within the constraints bythe current observations of radio transients. We expect toobserve N ! 6 GRBs per year at z > 6 for Pop III.2 andN ! 0.01 per year for Pop III.1 at z > 10 at a maximum.

4. Conclusion and Discussion

There are yet no direct observations of Population IIIstars, despite much recent development in theoreticalstudies on the formation of the early generation stars.


For the latter, we assume M = 550M! for Pop III.1and M = 55M! for Pop III.2, with dispersion !c =(M ! Mlow)/3. Mlow is the minimum mass for a givenstellar type, 100M! for Pop III.1 and 10M! for Pop III.2,whereas Mup is the maximum mass for a given stellar type,1000M! for Pop III.1 and " 100M! for Pop III.2. MGRB

is the minimum mass that is able to trigger GRBs, whichwe set to be 25M! (Bromm & Loeb 2006). Note thatnot all Pop III.1 stars will leave a black hole behind attheir deaths. In the narrow mass range of " 140! 260M!

Pop III stars are predicted to undergo a pair-instability su-pernova (PISN) explosion (Heger & Woosley 2002). Thisrange of mass is excluded from the calculation of Eq. (23).

The e!ciency factor for the power-law (Salpeter) IMFis "GRB/fGRB " 1/926M"1

! and 1/87M"1! for Pop III.1

and Pop III.2 respectively. Using the gaussian IMF,"GRB/fGRB " 1/538M"1

! and 1/53M"1! for Pop III.1

and Pop III.2. respectively. Thus, the GRB formation ef-ficiency for Pop III.2 can be about an order of magnitudelarger than Pop III.1 because of the lower characteristicmass of Pop III.2 stars.

3. Redshift Distribution of GRBs


dNobsGRB

dz= #obs

GRB(z)"tobs

1 + z

dV

dz, (26)

where dV/dz is the comoving volume element per unitredshift, given by

dV

dz=

4# c d2L

(1 + z)

!

!

!

!

dt

dz

!

!

!

!

. (27)

Fig. 5 shows the intrinsic GRB rate

dNGRB

dz= #GRB(z)

"tobs

1 + z

dV

dz. (28)

In this plot, we have not considered observational e$ectssuch as beaming and instrument sensitivity. Namely, weset %obs = 4#, "beam = 1 and Llim(z) = 0 in Eq. (1). Weshow the GRB rate for our choice of two di$erent IMFs.Interestingly, Fig. 5 shows that the results depend onlyweakly on the choice of IMF.

Fig. 6 shows the most optimistic case, assuming a highstar formation e!ciency f# " 0.1, an ine!cient chemicalenrichment, vwind = 50 km/s, and a gaussian IMF for bothPop III.1 and Pop III.2 stars. We note that constraints onthese quantities will be useful to place upper limits on theGRB observed rate.

3.1. Radio Afterglows

Follow-up observations of high redshift GRBs can be doneby observing their afterglows especially in radio band(Ioka & Meszaros 2005; Inoue et al. 2007). We calculate

the radio afterglow light-curves for Pop III GRBs follow-ing the standard prescription from Sari et al. (1998, 1999);Meszaros (2006). The afterglow light-curve at the time tdis given by the shock radius rd and the Lorentz factor $d.These two quantities are related by Eiso " 4#r3

d$2dnmpc2

and rd " c$2dtd, where n is the medium density and mp

is the proton mass. The true energy is given by Etrue =%2Eiso/2, where % is the half opening angle of the shock.The spectrum consists of power-law segments linked bycritical break frequencies. These are &a (the self absorp-tion frequency), &m (the peak of injection frequency) and&c (the cooling frequency), given by

&m # (1 + z)1/2g(p)2'2e'1/2B E1/2

iso t"3/2d ,

&c # (1 + z)"1/2'"3/2B n"1E"1/2

iso t"1/2d ,

&a # (1 + z)"1'"1e '1/5

B n3/5E1/5iso ,

F!,max # (1 + z)'1/2B n1/2Eisod

"2L , (29)

where g(p) = (p! 2)/(p! 1), is a function of energy spec-trum index of electrons (N($e)d$e # $"p

e d$e, where $e isthe electron Lorentz factor), 'e and 'B are the e!ciencyfactors (Meszaros 2006). There are two types of spectra.If &m < &c, we call it the slow cooling case. The flux at theobserver, F! , is given by

F! =

"

#

#

$

#

#

%

(&a/&m)1/3(&/&a)2F!,max, &a > &,(&/&m)1/3F!,max, &m > & > &a,(&/&m)"(p"1)/2F!,max, &c > & > &m,(&c/&m)"(p"1)/2(&/&c)"p/2F!,max, & > &c.

(30)where F!,max is the observed peak flux at distance dL fromthe source.


F! =

"

#

#

$

#

#

%

(&a/&c)1/3(&/&a)2F!,max, &a > &,(&/&c)1/3F!,max, &c > & > &a,(&/&c)"1/2F!,max, &m > & > &c,(&m/&c)"1/2(&/&m)"p/2F!,max, & > &m.

(31)As the GRB jet sweeps the interstellar medium, the

Lorentz factor of the jet is decelerated. When the Lorentzfactor drops below %"1, the jet starts to expand sidewaysand becomes detectable by the o$-axis observers. Theseafterglows are not associated with the prompt GRB emis-sion. Such orphan afterglows are a natural prediction ofGRB jets. Radio transient sources probe the high energypopulation of the Universe and can provide further con-straints on the intrinsic rate of GRBs. Even if the promptemission is highly collimated, the Lorentz factor drops$d < %"1 around the time

t" " 2.14

&

Eiso

5 $ 1054

'1/3 &

%

0.1

'8/3

n"1/3(1 + z) days,

(32)

REDSHIFT DISTRIBUTION OF GRBS

A&A 533, A32 (2011)

Pop III.2

Pop III.1

10 15 20 25 30

0.001

0.01

0.1

1

10

100

z

Intr

insi

cG

RB

rate!yr!1 "

Fig. 5. The intrinsic GRB rate dNGRB/dz. The number of (on-axis + o!-axis) GRBs per year on the sky in Eq. (28), as a function of redshift.We set f! = 0.001, fGRB = 0.01 and vwind = 100 km s"1 for this plot.Salpeter IMF, dashed black line, Gaussian IMF, dotted black line, forPop III.2; and Salpeter IMF, dashed blue line, Gaussian IMF, dottedblue line, for Pop III.1.

Pop III.2

Pop III.1

Optimistic case

5 10 15 20 25 30 35 400.1

1

10

100

1000

104

105

z

Intr

insi

cG

RB

rate!yr!1 "

Fig. 6. The intrinsic GRB rate dNGRB/dz. The number of (on-axis + o!-axis) GRBs per year on the sky in Eq. (28), as a function of redshiftfor our optimistic model. We assume a high star formation e"ciency;f! = 0.1 for Pop III.1; and f! = 0.01 for Pop III.2; slow chemicalenrichment, vwind = 50 km s"1; high GRB formation e"ciency, fGRB =0.1; and a Gaussian IMF; for both Pop III.2, dashed black line; andPop III.1, dotted blue line.

3.1. Radio afterglows

Follow-up observations of high-redshift GRBs can be done byobserving their afterglows, especially in radio band (Ioka &Mészáros 2005; Inoue et al. 2007). We calculated the radio af-terglow light curves for Pop III GRBs following the standardprescription from Sari et al. (1998, 1999) and Mészáros (2006).The afterglow light curve at the time td is given by the shockradius rd and the Lorentz factor !d. These two quantities are re-lated by Eiso # 4"r3

d!2dnmpc2 and rd # c!2

dtd, where n is themedium density and mp the proton mass. The true energy is givenby Etrue = #2Eiso/2, where # is the half opening angle of theshock. The spectrum consists of power-law segments linked bycritical break frequencies. These are $a (the self absorption fre-quency), $m (the peak of injection frequency), and $c (the cooling

frequency), given by

$m $ (1 + z)1/2g(p)2%2e %1/2B E1/2

iso t"3/2d ,

$c $ (1 + z)"1/2%"3/2B n"1E"1/2

iso t"1/2d ,

$a $ (1 + z)"1%"1e %

1/5B n3/5E1/5

iso ,

F$,max $ (1 + z)%1/2B n1/2Eisod"2L , (29)

where g(p) = (p " 2)/(p " 1) is a function of energy spec-trum index of electrons (N(!e)d!e $ !"p

e d!e, where !e is theelectron Lorentz factor), and %e and %B are the e"ciency factors(Mészáros 2006). There are two types of spectra. If $m < $c, wecall it the slow cooling case. The flux at the observer, F$, is givenby

F$ =

!"""""#"""""$

($a/$m)1/3($/$a)2F$,max, $a > $,($/$m)1/3F$,max, $m > $ > $a,($/$m)"(p"1)/2F$,max, $c > $ > $m,($c/$m)"(p"1)/2($/$c)"p/2F$,max, $ > $c,

(30)

where F$,max is the observed peak flux at distance dL from thesource.

For $m > $c, called the fast cooling case, the spectrum is

F$ =

!"""""#"""""$

($a/$c)1/3($/$a)2F$,max, $a > $,($/$c)1/3F$,max, $c > $ > $a,($/$c)"1/2F$,max, $m > $ > $c,($m/$c)"1/2($/$m)"p/2F$,max, $ > $m.

(31)

As the GRB jet sweeps the interstellar medium, the Lorentz fac-tor of the jet is decelerated. When the Lorentz factor drops below#"1, the jet starts to expand sideways and becomes detectable bythe o!-axis observers. These afterglows are not associated withthe prompt GRB emission. Such orphan afterglows are a natu-ral consequence of the existence of GRB’s jets. Radio transientsources probe the high-energy population of the Universe andcan provide further constraints on the intrinsic rate of GRBs.Even if the prompt emission is highly collimated, the Lorentzfactor drops !d < #"1 around the time

t# # 2.14% Eiso

5 % 1054

&1/3 % #0.1

&8/3n"1/3(1 + z) days, (32)

and the jet starts to expand sideways. Finally the shock velocitybecomes nonrelativistic around the time

tNR # 1.85 % 102% Eiso

5 % 1054

&1/3 % #0.1

&2/3n"1/3(1 + z) days, (33)

(Ioka & Mészáros 2005). After time t#, the temporal depen-dence of the critical break frequencies should be replaced by$c $ t0, $m $ t"2, $a $ t"1/5, and F$,max $ t"1 (Sari et al. 1999).We also used the same evolution in the nonrelativistic phase forsimplicity, which underestimates the afterglow flux after tNR.

Figure 7 shows the light curves for a typical GRB fromPop III.2 stars assuming an isotropic kinetic energy Eiso #1054 erg (in proportion to the progenitor mass) as a lower limit.Pop III.1 afterglows are expected to be brighter. Consistentlywith previous works, we conclude that it is possible to observethe GRB radio afterglows with ALMA, LOFAR, EVLA, andSKA.

A32, page 6 of 9


ORPHAN AFTERGLOWS

As the GRB jet sweeps the interstellar medium, the Lorentz factor of the jet is decelerated. When the Lorentz factor drops below θ−1, the jet starts to expand sideways and becomes detectable by the off-axis observers. These afterglows are not associated with the prompt GRB emission. Even if the prompt emission is highly collimated, the Lorentz factor drops γd < θ−1 around the time

A&A 533, A32 (2011)

Pop III.2

Pop III.1

10 15 20 25 30

0.001

0.01

0.1

1

10

100

z

Intr

insi

cG

RB

rate!yr!1 "


Pop III.2

Pop III.1

Optimistic case

5 10 15 20 25 30 35 400.1

1

10

100

1000

104

105

z

Intr

insi

cG

RB

rate!yr!1 "







$m $ (1 + z)1/2g(p)2%2e %1/2B E1/2

iso t"3/2d ,

$c $ (1 + z)"1/2%"3/2B n"1E"1/2

iso t"1/2d ,

$a $ (1 + z)"1%"1e %

1/5B n3/5E1/5

iso ,

F$,max $ (1 + z)%1/2B n1/2Eisod"2L , (29)



F$ =

!"""""#"""""$


(30)



F$ =

!"""""#"""""$


(31)


t# # 2.14% Eiso

5 % 1054

&1/3 % #0.1

&8/3n"1/3(1 + z) days, (32)


tNR # 1.85 % 102% Eiso

5 % 1054

&1/3 % #0.1

&2/3n"1/3(1 + z) days, (33)



A32, page 6 of 9

and the jet starts to expand sideways. Finally the shock velocity becomes nonrelativistic around the time

A&A 533, A32 (2011)

Pop III.2

Pop III.1

10 15 20 25 30

0.001

0.01

0.1

1

10

100

zIn

trin

sic

GR

Bra

te!yr!1 "


Pop III.2

Pop III.1

Optimistic case

5 10 15 20 25 30 35 400.1

1

10

100

1000

104

105

z

Intr

insi

cG

RB

rate!yr!1 "







$m $ (1 + z)1/2g(p)2%2e %1/2B E1/2

iso t"3/2d ,

$c $ (1 + z)"1/2%"3/2B n"1E"1/2

iso t"1/2d ,

$a $ (1 + z)"1%"1e %

1/5B n3/5E1/5

iso ,

F$,max $ (1 + z)%1/2B n1/2Eisod"2L , (29)



F$ =

!"""""#"""""$


(30)



F$ =

!"""""#"""""$


(31)


t# # 2.14% Eiso

5 % 1054

&1/3 % #0.1

&8/3n"1/3(1 + z) days, (32)


tNR # 1.85 % 102% Eiso

5 % 1054

&1/3 % #0.1

&2/3n"1/3(1 + z) days, (33)



A32, page 6 of 9


Radio Afterglow light curveR. S. de Souza et al.: Populations III.1 and III.2 gamma-ray bursts

10 GHz

1.4 GHz

500 MHz

LOFAR

SKA

ALMA

EVLA

0.1 1 10 100 1000 104 10510!4

0.001

0.01

0.1

1

t !days"

F!mJy"

Fig. 7. The theoretical light curve of radio afterglow of a typicalPop III.2 GRB at z ! 10. We show the evolution of afterglow fluxF(mJy) as a function of time t (days) for typical parameters: isotropickinetic energy Eiso = 1054 erg, electron spectral index p = 2.5, plasmaparameters !e = 0.1, !B = 0.01, initial Lorentz factor "d = 200, interstel-lar medium density n = 1 cm"3, for the range of frequencies: 500 MHz(dashed brown line), 1.4 GHz (dashed red line), 10 GHz (dashed blackline), in comparison with flux sensitivity Fsen

# as a function of integra-tion time, tint(days) for SKA (dot-dashed green line), EVLA (dot-dashedorange line), LOFAR (dot-dashed blue line) and ALMA (dot-dashedpurple line).


In this section, we derive upper limits on the intrinsic GRB rate(including the o!-axis GRB) using !1 year timescale radio vari-ability surveys. There are several radio transient surveys com-pleted so far. Bower et al. (2007) used 22 years of archival datafrom VLA to put an upper limit of !6 deg"2 for 1-year variabil-ity transients above 90 µJy, which is equivalent to !2.4# 105 forthe whole sky. Gal-Yam et al. (2006) used FIRST12 and NVSS13

radio catalogs to place an upper limit of !70 radio orphan af-terglows above 6 mJy in the 1.4 GHz band over the entire sky.This suggests less than 3#104 sources above 0.3 mJy on the sky,because the number of sources is expected to be proportional toflux limit F"3/2

lim (assuming Euclidian space and no source evo-lution) (Gal-Yam et al. 2006). From Fig. 7, a typical GRB’s ra-dio afterglow with isotropic kinetic energy Eiso ! 1054 erg staysabove 0.3 mJy over !102 days.

By combining the results shown in Figs. 5 and 6, we ex-pect !30"3 # 105 sources (102"106 events per year # 102 days)above !0.3 mJy. (We integrate the event rate over redshift.) Asa consequence, the most optimistic case for Pop III.2 should al-ready be ruled out marginally by the current observations of ra-dio transient sources, if their luminosity function follows the oneassumed in the present paper. Only more conservative modelsare then viable. Radio transient surveys are not yet able to setupper limits on the Pop III.1 GRB rate. The above conclusionis model dependent, because the afterglow flux depends on thestill uncertain quantities, such as the isotropic energy Eiso andthe ambient density n. If the circumburst density is higher thanusual, the constraints from the radio transient surveys would beeven stronger. Also the GRB formation e"ciency and the beam-ing factor are not known accurately, which can a!ect both theintrinsic and observed rate more than one order of magnitude.


Pop III.1EXIST

JANUSSVOM

Swift

15 20 25 30

1"10!4

5"10!4

0.001

0.005

0.010

0.050

z

Obs

erve

dG

RB

rate!yr!1 "

Fig. 8. Predicted Pop III.1observed GRB rate. Those observed by Swift,dashed red line; SVOM, dot-dashed black line; JANUS, dotted blueline; and EXIST, green line. We adopt a GRB rate model that is con-sistent with the current upper limits from the radio transients; GaussianIMF, vwind = 50 km s"1, f$ = 0.1, fGRB = 0.1.

Pop III.2EXIST

JANUSSVOM

Swift

10 15 20 25 3010!4

0.001

0.01

0.1

1

10

z

Obs

erve

dG

RB

rate!yr!1 "

Fig. 9. Predicted Pop III.2 observed GRB rate. Those observed bySwift, dashed red line; SVOM, dot-dashed black line; JANUS, dottedblue line; and EXIST, green line; for our model with Salpeter IMF,vwind = 100 km s"1, f$ = 0.01, fGRB = 0.01.

In Figs. 8 and 9, we show the predicted observable GRBrate dNobs

GRB/dz in Eq. (26) for Pop III.1 and III.2 detectable bythe Swift, SVOM, JANUS, and EXIST missions. The resultsshown are still within the bounds of available upper limits fromthe radio transient surveys. Overall, it is more likely to observePop III.2 GRBs than Pop III.1, but the predicted rate stronglydepends on the IGM metallicity evolution, the star formation ef-ficiency and GRB formation e"ciency. The dependence on theIMF is relatively small.

Figure 10 shows the GRB rate expected for EXIST obser-vations. Because the power index of the LF is uncertain at thebright end, we added two lines to show the resulting uncertaintyin our prediction. We use the maximum rate, which is within theconstraints by the current observations of radio transients. Weexpect to observe N ! 20 GRBs per year at z > 6 for Pop III.2and N ! 0.08 per year for Pop III.1 at z > 10 with the futureEXIST satellite at a maximum. Our optimist case predicts a near-future detection of Pop III.2 GRB by Swift, and the nondetectionso far could suggest a further upper limit or di!erence betweenthe Pop III and present-day GRB spectrum.

A32, page 7 of 9


– 13 –

102 103 104 105 106 107

10−3

10−2

10−1

100

101

Flux Density (µJy)

Surfa

ce D

ensit

y (d

eg−2

)

B07

!G06

!B07.1

!B07.2

!C03

!F03

!ATATS−I

M09

!PiGSS−I

3C 286

!

MOST

ATATS−II!

Fig. 4.— Transient surface density from this and other surveys as a function of flux density.

Result from this survey is labeled 3C 286. Curves and lines indicate detected values andupper limits from a deep VLA search (B07,B07.1,B07.2; Bower et al. 2007), the comparison

of the 1.4 GHz NVSS and FIRST surveys (G06; Gal-Yam et al. 2006), from additional VLAsearches (C03 and F03; Carilli et al. 2003; Frail et al. 2003), from the first and second ATATS

papers (ATATS-I and ATATS-II; Croft et al. 2010b,a), from the first data release of PiGSS(PiGSS-I; Bower et al. 2010). from the Matsumura et al. (M09; 2009) survey, and fromthe MOST search (Bannister et al. 2010). The dashed line is proportional to S!1.5 and is

normalized to B07 estimates. Lines with arrows indicate 1! upper limits; otherwise theresults are indicative of detected transients (B07, MOST, and M09).

UPPER LIMITS FROM RADIO TRANSIENTS



10 GHz

1.4 GHz

500 MHz

LOFAR

SKA

ALMA

EVLA

0.1 1 10 100 1000 104 10510!4

0.001

0.01

0.1

1

t !days"

F!mJy"









Pop III.1EXIST

JANUSSVOM

Swift

15 20 25 30

1"10!4

5"10!4

0.001

0.005

0.010

0.050

z

Obs

erve

dG

RB

rate!yr!1 "


Pop III.2EXIST

JANUSSVOM

Swift

10 15 20 25 3010!4

0.001

0.01

0.1

1

10

z

Obs

erve

dG

RB

rate!yr!1 "





A32, page 7 of 9


10 GHz

1.4 GHz

500 MHz

LOFAR

SKA

ALMA

EVLA

0.1 1 10 100 1000 104 10510!4

0.001

0.01

0.1

1

t !days"F!mJy"









Pop III.1EXIST

JANUSSVOM

Swift

15 20 25 30

1"10!4

5"10!4

0.001

0.005

0.010

0.050

z

Obs

erve

dG

RB

rate!yr!1 "


Pop III.2EXIST

JANUSSVOM

Swift

10 15 20 25 3010!4

0.001

0.01

0.1

1

10

z

Obs

erve

dG

RB

rate!yr!1 "





A32, page 7 of 9

Theoretical predicted rate

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 adopt a GRB rate model that is consistent with the current upper limits from the radio transients; Gaussian IMF, vwind = 50 km s−1, f∗ = 0.1, fGRB = 0.1.

Predicted Pop III.2 observed GRB rate. Those observed by Swift, dashed red line; SVOM, dot-dashed black line; JANUS, dotted blue line; and EXIST, green line; for our model with Salpeter IMF, vwind =100 kms−1, f∗ =0.01, fGRB =0.01.


Upper limits for the observed GRB rate of EXIST satelliteA&A 533, A32 (2011)

Pop III.2

Pop III.1

10 15 20 25 30

0.01

0.1

1

10

z

Obs

erve

dG

RB

rate!yr!1 "

Fig. 10. Predicted maximum GRB rates observed by EXIST. Weadopted Salpeter IMF, vwind = 100 km s!1, fGRB = 0.01, f" = 0.01;for Pop III.2 GRBs, dashed black line; and Gaussian IMF, vwind =50 km s!1, f" = 0.1, fGRB = 0.1; for Pop III.1 GRBs, dotted blue line.Dotted red lines represent the same with LF’s bright end power lawindex 1.7 and 0.8.

4. Conclusion and discussion

There are still no direct observations of Population III stars, de-spite much recent development in theoretical studies on the for-mation of the early generation stars. In this paper, we follow arecent suggestion that massive Pop III stars could trigger collap-sar gamma-ray bursts. Observations of such energetic GRBs atvery high redshifts will be a unique probe of the high-redshiftUniverse. With a semi-analytical approach we estimated the starformation rate for Pop III.1 and III.2 stars including all relevantfeedback e!ects: photo-dissociation, reionization, and metal en-richment.

Using radio transient sources we are able to derive con-straints on the intrinsic rate of GRBs. We estimated the predictedGRB rate for both Pop III.1 and Pop III.2 stars, and argued thatthe latter is more likely to be observed with future experiments.We expect to observe maximum of N ! 20 GRBs per year in-tegrated over at z > 6 for Pop III.2 and N ! 0.08 per year inte-grated over at z > 10 for Pop III.1 with EXIST.

We also expect a larger number of radio afterglows thanX-ray prompt emission because the radio afterglow is long-lived,for #102 days above #0.3 mJy from Fig. 7. Combining withthe intrinsic GRB rate and constraints from radio transients, weexpect roughly #10!104 radio afterglows above "0.3 mJy al-ready on the sky. They are indeed detectable by ALMA, EVLA,LOFAR, and SKA, and have even been detected already by#1 yr-timescale variability surveys. We showed that using asemi-analytical approach combined with the current surveys,such as NVSS and FIRST, we are already able to constrain thePop III.2 GRB event rate.

Finally, it is important to note that our knowledge of the firststars and GRBs is still limited, and there are uncertainties intheir properties, most significantly in their characteristic mass.Recently, Clark et al. (2011) and Greif et al. (2011) have per-formed cosmological simulations using a sink particle techniqueto follow the evolution of a primordial protostellar accretiondisk. They find that the disk gravitationally fragments to yieldmultiple protostellar seeds. Although the final mass distributionof the formed stars is still uncertain, formation of multiple sys-tems, especially massive binary Pop III stars, would increase therate of high-redshift GRBs (e.g., Bromm & Loeb 2006; Fryer& Heger 2005, and references therein). If the GRB fraction percollapse fGRB in Eq. (23) is much larger than the current one,

say fGRB # 1, the Pop III.1 GRBs might also become detectablewith the radio telescopes (#300 afterglows above #0.3 mJy onthe sky) and the X-ray satellites (#1 event per year for EXIST)in the future.

Acknowledgements. R.S.S. thanks the Brazilian agency CNPq (200297/2010-4)for financial support. This work was supported by World Premier InternationalResearch Center Initiative (WPI Initiative), MEXT, Japan. We thank EmilleIshida, Andrea Ferrara, Kenichi Nomoto, Jarrett Johnson, and Yudai Suwa forfruitful discussion and suggestions. N.Y. acknowledges the financial support bythe Grants-in-Aid for Young Scientists (S) 20674003 by the Japan Society for thePromotion of Science. K.I. acknowledges the financial support by KAKENHI21684014, 19047004, 22244019, 22244030. We thank the anonymous refereefor their very careful reading of the paper and for several suggestions whichallowed us to improve the current work. We also thank the language editor J.Adams for his carefully revision.

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Izzard, R. G., Ramirez-Ruiz, E., & Tout, C. A. 2004, MNRAS, 348, 1215Jarosik, N., Bennett, C. L., Dunkley, J., et al. 2011, ApJS, 192, 14Johnson, J. L. 2010, MNRAS, 404, 1425Johnson, J. L., & Bromm, V. 2006, MNRAS, 366, 247Kawai, N., Kosugi, G., Aoki, K., et al. 2006, Nature, 440, 184Komissarov, S. S., & Barkov, M. V. 2010, MNRAS, 402, L25Lamb, D. Q., & Reichart, D. E. 2000, ApJ, 536, 1Langer, N., & Norman, C. A. 2006, ApJ, 638, L63Lehnert, M. D., Nesvadba, N. P. H., Cuby, J., et al. 2010, Nature, 467, 940Lewis, A., Challinor, A., & Lasenby, A. 2000, ApJ, 538, 473Li, L. 2008, MNRAS, 388, 1487Machacek, M. E., Bryan, G. L., & Abel, T. 2001, ApJ, 548, 509Mackey, J., Bromm, V., & Hernquist, L. 2003, ApJ, 586, 1Madau, P., Ferrara, A., & Rees, M. J. 2001, ApJ, 555, 92Mannucci, F., Buttery, H., Maiolino, R., Marconi, A., & Pozzetti, L. 2007, A&A,

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A32, page 8 of 9


REMARKS

We follow a recent suggestion that massive Pop III stars could trigger collapsar gamma-ray bursts. Observations of such energetic GRBs at very high redshifts will be a unique probe of the high redshift Universe.

Using a semi-analytical approach we estimate for the first time the star formation rate for Pop III.1 and III.2 stars including all relevant feedback effects: photo-dissociation, reionization and metal enrichment.

Orphan afterglows are a natural prediction of GRB jets. Using radio transient sources we are able to derive constraints on the intrinsic rate of GRBs. We estimate the predicted GRB rate for both Pop III.1 and Pop III.2 stars, and argue that the latter is more likely to be observed with future experiments. We expect to observe maximum of N < 6 GRBs per year at z > 6 for Pop III.2 and N < 0.01 per year for Pop III.1 at z > 10 with EXIST.


LOOKING FOR THE ORPHANS

One type of possible transients to be detected by Gaia are gamma-ray burst optical afterglows.

We calculate the Pop III GRB orphan afterglows rate detectable by GAIA

We make use of a star formation rate derived by a semi-analytical approach to predict the GRBs cumulative number. The orphan afterglows events are generated by a Monte-Carlo method, and realistic simulations of the Gaia observational conditions are taken into account in order to derive their observation probability expectation.


AFTERGLOW MODELde Souza, Krone-Martins & Ishida: Pop III Orphan Afterglows 3

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$

W2010

%

%

%

%

B2011

Pop III.2

0 5 10 15 2010%6

10%5

10%4

0.001

0.01

0.1

1

z

SFR!M&

yr%

1M

pc%

3 "

Fig. 1. Optimistic model for Pop III.2 star formation rateassuming a high star formation e!ciency and low chem-ical enrichment. The light points are independent SFRdeterminations from compiled from literature.

Pop III.2 GRB

5 10 15 20 25 30 351

10

100

1000

104

z

Intri

nsic

GRB

rate!y

r%1 "

Fig. 2. The intrinsic GRB rate dNGRB/dz, the numberof (on-axis + o"-axis) GRBs per year on the sky in Eq.(2), as a function of redshift for our optimistic model as-suming a high star formation e!ciency, f! = 0.01 for PopIII.2, slow chemical enrichment, vwind = 50km/s, GRBformation e!ciency, fGRB = 0.01 and a Salpeter IMF, forPop III.2 (blue line).

is able to trigger GRBs, ! 25M" (Bromm & Loeb 2006).The fGRB factor gives the fraction of stars in this rangeof mass that will produce GRBs.

The GRB formation e!ciency factor per stellar massis defined by

!GRB = fGRB

! Mup

MGRB"(m)dm

! Mup

Mlowm"(m)dm

, (3)

where "(m) is the stellar IMF. Using ! 1 year timescaleradio variability surveys de Souza et al. (2011) place up-per limits on the intrinsic GRB rate (including the o"-axisGRB). For latter we will use the values fGRB = 0.01 and!GRB/fGRB ! 1/87M#1

" as an optimistic case consistentwith their results. Figure 2 shows the upper limit for in-trinsic GRB rate constrained by radio transients surveyderived in de Souza et al. (2011).

3. Number of Observed Orphans

3.1. Afterglow Model

To calculate the afterglow light curves of Pop III GRBswe follow the standard prescription from Sari et al. (1998,1999); Meszaros (2006). The spectrum consists of power-law segments linked by critical break frequencies. Theseare #a (the self absorption frequency), #m (the peak ofinjection frequency) and #c (the cooling frequency), givenby

#m " (1 + z)1/2g(p)2$2e$1/2B E1/2

iso t#3/2d ,

#c " (1 + z)#1/2$#3/2B n#1E#1/2

iso t#1/2d ,

#a " (1 + z)#1$#1e $1/5

B n3/5E1/5iso ,

F!,max " (1 + z)$1/2B n1/2Eisod

#2L , (4)

where g(p) = (p# 2)/(p# 1), is a function of energy spec-trum index of electrons (N(%e)d%e " %#p

e d%e, where %e isthe electron Lorentz factor), $e and $B are the e!ciencyfactors (Meszaros 2006). There are two types of spectra.If #m < #c, we call it the slow cooling case. The flux at theobserver, F! , is given by

F! =

"

#

#

$

#

#

%

(#a/#m)1/3(#/#a)2F!,max, #a > #,(#/#m)1/3F!,max, #m > # > #a,(#/#m)#(p#1)/2F!,max, #c > # > #m,(#c/#m)#(p#1)/2(#/#c)#p/2F!,max, # > #c.


For #m > #c, called the fast cooling case, the spectrumis

F! =

"

#

#

$

#

#

%

(#a/#c)1/3(#/#a)2F!,max, #a > #,(#/#c)1/3F!,max, #c > # > #a,(#/#c)#1/2F!,max, #m > # > #c,(#m/#c)#1/2(#/#m)#p/2F!,max, # > #m.

(6)Initially the jet propagates as if it were spherical with

an equivalent isotropic energy of Etrue = &2jEiso/2. Even

if the prompt emission is highly collimated, the Lorentzfactor drops %d < &#1

j around the time

t" ! 2.14

&

Eiso

5 $ 1054

'1/3 &

&j

0.1

'8/3

n#1/3(1+z) days, (7)

and the jet starts to expand sideways (Ioka & Meszaros2005). Consequently, the jet becomes detectable by the o"-axis observers. These afterglows are not associated withthe prompt GRB emission.

Due to relativistic beaming, an observer located at &obs

outside the initial opening angle of the jet (&obs > &j) willobserve the afterglow emission only at t ! t". The receivedafterglow flux by an o"-axis observer in the point sourceapproximation, valid for &obs % &j , is related to that seenby an on-axis observer, by

F!(&obs, t) = '3F!/#(0, 't), (8)

de Souza, Krone-Martins & Ishida: Pop III Orphan Afterglows 3

!

!!

!

!

!!

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#

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#

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R2008

$

$

$

W2010

%

%

%

%

B2011

Pop III.2

0 5 10 15 2010%6

10%5

10%4

0.001

0.01

0.1

1

z

SFR!M&

yr%

1M

pc%

3 "


Pop III.2 GRB

5 10 15 20 25 30 351

10

100

1000

104

z

Intri

nsic

GRB

rate!y

r%1 "




!GRB = fGRB

! Mup

MGRB"(m)dm

! Mup

Mlowm"(m)dm

, (3)






#m " (1 + z)1/2g(p)2$2e$1/2B E1/2

iso t#3/2d ,

#c " (1 + z)#1/2$#3/2B n#1E#1/2

iso t#1/2d ,

#a " (1 + z)#1$#1e $1/5

B n3/5E1/5iso ,

F!,max " (1 + z)$1/2B n1/2Eisod

#2L , (4)



F! =

"

#

#

$

#

#

%




F! =

"

#

#

$

#

#

%





j around the time

t" ! 2.14

&

Eiso

5 $ 1054

'1/3 &

&j

0.1

'8/3

n#1/3(1+z) days, (7)




F!(&obs, t) = '3F!/#(0, 't), (8)


!

!!

!

!

!!

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#

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#

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$

W2010

%

%

%

%

B2011

Pop III.2

0 5 10 15 2010%6

10%5

10%4

0.001

0.01

0.1

1

z

SFR!M&

yr%

1M

pc%

3 "


Pop III.2 GRB

5 10 15 20 25 30 351

10

100

1000

104

z

Intri

nsic

GRB

rate!y

r%1 "




!GRB = fGRB

! Mup

MGRB"(m)dm

! Mup

Mlowm"(m)dm

, (3)






#m " (1 + z)1/2g(p)2$2e$1/2B E1/2

iso t#3/2d ,

#c " (1 + z)#1/2$#3/2B n#1E#1/2

iso t#1/2d ,

#a " (1 + z)#1$#1e $1/5

B n3/5E1/5iso ,

F!,max " (1 + z)$1/2B n1/2Eisod

#2L , (4)



F! =

"

#

#

$

#

#

%




F! =

"

#

#

$

#

#

%





j around the time

t" ! 2.14

&

Eiso

5 $ 1054

'1/3 &

&j

0.1

'8/3

n#1/3(1+z) days, (7)




F!(&obs, t) = '3F!/#(0, 't), (8)


!

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!

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#

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#

#

#

R2008

$

$

$

W2010

%

%

%

%

B2011

Pop III.2

0 5 10 15 2010%6

10%5

10%4

0.001

0.01

0.1

1

z

SFR!M&

yr%

1M

pc%

3 "


Pop III.2 GRB

5 10 15 20 25 30 351

10

100

1000

104

z

Intri

nsic

GRB

rate!y

r%1 "




!GRB = fGRB

! Mup

MGRB"(m)dm

! Mup

Mlowm"(m)dm

, (3)






#m " (1 + z)1/2g(p)2$2e$1/2B E1/2

iso t#3/2d ,

#c " (1 + z)#1/2$#3/2B n#1E#1/2

iso t#1/2d ,

#a " (1 + z)#1$#1e $1/5

B n3/5E1/5iso ,

F!,max " (1 + z)$1/2B n1/2Eisod

#2L , (4)



F! =

"

#

#

$

#

#

%




F! =

"

#

#

$

#

#

%





j around the time

t" ! 2.14

&

Eiso

5 $ 1054

'1/3 &

&j

0.1

'8/3

n#1/3(1+z) days, (7)




F!(&obs, t) = '3F!/#(0, 't), (8)

4 de Souza, Krone-Martins & Ishida: Pop III Orphan Afterglows

R = 20

!obs = 0.0

!obs = 0.10

!obs = 0.20

z = 3 ! j = 0.1

0.1 1 10 100 1000

0.001

0.01

0.1

1

10

t !days"

F!m

Jy"

Fig. 3. Example of afterglow light curve as a function ofobserved angle !obs. We show the evolution of afterglowflux F (mJy) as a function of time t (days) and observedangle !obs for typical parameters: isotropic kinetic energyEiso = 1054 erg, electron spectral index p = 2.5, plasmaparameters "e = 0.1, "B = 0.01, half opening angle jet !j =0.1, interstellar medium density n = 1cm!3, frequency# = 4.5 ! 1014. The horizontal dotted line is the GAIAflux limit; dashed blue line, !obs = 0; dashed red line,!obs = 0.1; dashed green line, !obs = 0.20.

where

$ " (1 # %)/(1 # % cos !obs), (9)

and % =!

1 # 1/!2. The time evolution of the Lorentzfactor in given by

!(t) =

"

#

$

#

%

!!1j

&

ttj

'!3/8t < tj

!!1j

&

ttj

'!1/2t > tj.

(10)

Figure 3 show an example of an afterglow as a function ofobserved angle for typical parameters.

3.2. Mock sample

The mock sample is generated by a Monte-Carlomethod assuming di"erent probability distribution func-tion (PDF) for each quantity as explained bellow.

3.2.1. Redshift PDF

We generate the GRB events randomly in redshift with aPDF given by Eq. (2). The probability of a given GRBappear at redshift z is

Pz(z) =dN/dz

( z0 (dN/dz)dz

(11)

Figure 4 shows the probability distribution of GRB red-shift.

5 10 150.00

0.05

0.10

0.15

z

Prob!z"

Fig. 4. Redshift PDF. Shown is the probability of a givenevent appear in a certain range of redshift.

0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

!jet

Prob!!"

Fig. 5. Half opening angle jet PDF. Shown is the proba-bility of a given GRB have a given !j .

3.2.2. Half opening angle PDF

Using an empirical opening angle estimator, Yonetokuet al. (2005) derived the opening angle PDF of GRBs.Their PDF can be fitted by a power-lay !!2 with an cuto" at $ 0.04. Their results seems also compatible withthe universal structured jet model (Perna et al. 2003). Forsimplicity, we assume a similar power-law in the range!min

j = 0.05 and !maxj = 0.5 do determine the PDF of !j ,

P!j(!) % !!2. (12)

Figure 5 shows the PDF of !j .

4. GAIA mission

In order to estimate the probability for the observationof a single event from a Pop III.2 OA by Gaia, only twoquantities play an important role: the time that the or-phan remains brighter than G=20, #t, and the coordi-nates (lgal, bgal) where the event takes place in the sky.And since those quantities are continuous distributions,it is necessary to analyze the observation probability infunction them, building Prob(#t, lgal, bgal).


R = 20

!obs = 0.0

!obs = 0.10

!obs = 0.20

z = 3 ! j = 0.1

0.1 1 10 100 1000

0.001

0.01

0.1

1

10

t !days"

F!m

Jy"


where

$ " (1 # %)/(1 # % cos !obs), (9)

and % =!


!(t) =

"

#

$

#

%

!!1j

&

ttj

'!3/8t < tj

!!1j

&

ttj

'!1/2t > tj.

(10)


3.2. Mock sample


3.2.1. Redshift PDF


Pz(z) =dN/dz

( z0 (dN/dz)dz

(11)


5 10 150.00

0.05

0.10

0.15

z

Prob!z"


0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

!jet

Prob!!"





P!j(!) % !!2. (12)


4. GAIA mission



R = 20

!obs = 0.0

!obs = 0.10

!obs = 0.20

z = 3 ! j = 0.1

0.1 1 10 100 1000

0.001

0.01

0.1

1

10

t !days"

F!m

Jy"


where

$ " (1 # %)/(1 # % cos !obs), (9)

and % =!


!(t) =

"

#

$

#

%

!!1j

&

ttj

'!3/8t < tj

!!1j

&

ttj

'!1/2t > tj.

(10)


3.2. Mock sample


3.2.1. Redshift PDF


Pz(z) =dN/dz

( z0 (dN/dz)dz

(11)


5 10 150.00

0.05

0.10

0.15

z

Prob!z"


0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

!jet

Prob!!"





P!j(!) % !!2. (12)


4. GAIA mission


We calculated the afterglow light curves for Pop III GRBs following the standard prescription from Sari et al. (1998, 1999) and Mészáros (2006). The spectrum consists of power-law segments linked by critical break frequencies. These are νa (the self absorption frequency), νm (the peak of injection frequency), and νc (the cooling



R = 20

!obs = 0.0

!obs = 0.10

!obs = 0.20

z = 3 ! j = 0.1

0.1 1 10 100 1000

0.001

0.01

0.1

1

10

t !days"

F!m

Jy"


where

$ " (1 # %)/(1 # % cos !obs), (9)

and % =!


!(t) =

"

#

$

#

%

!!1j

&

ttj

'!3/8t < tj

!!1j

&

ttj

'!1/2t > tj.

(10)


3.2. Mock sample


3.2.1. Redshift PDF


Pz(z) =dN/dz

( z0 (dN/dz)dz

(11)


5 10 150.00

0.05

0.10

0.15

z

Prob!z"


0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

!jet

Prob!!"





P!j(!) % !!2. (12)


4. GAIA mission


Example of afterglow light curve as a function of observed angle θobs. We show the evolution of afterglow flux F(mJy) as a function of time t (days) and observed angle θobs for typical parameters: isotropic kinetic energy Eiso = 1054 erg, electron spectral index p = 2.5, plasma parameters ∊e = 0.1, ∊B = 0.01, half opening angle jet θj = 0.1, interstellar medium density n = 1cm−3, frequency ν = 4.5 × 1014. The horizontal dotted line is the GAIA flux limit; dashed blue line, θobs = 0; dashed red line, θobs = 0.1; dashed green line, θobs = 0.20.


PROBABILITY DISTRIBUTION OF EACH PARAMETER


R = 20

!obs = 0.0

!obs = 0.10

!obs = 0.20

z = 3 ! j = 0.1

0.1 1 10 100 1000

0.001

0.01

0.1

1

10

t !days"

F!m

Jy"


where

$ " (1 # %)/(1 # % cos !obs), (9)

and % =!


!(t) =

"

#

$

#

%

!!1j

&

ttj

'!3/8t < tj

!!1j

&

ttj

'!1/2t > tj.

(10)


3.2. Mock sample


3.2.1. Redshift PDF


Pz(z) =dN/dz

( z0 (dN/dz)dz

(11)


5 10 150.00

0.05

0.10

0.15

z

Prob!z"


0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

!jet

Prob!!"





P!j(!) % !!2. (12)


4. GAIA mission



R = 20

!obs = 0.0

!obs = 0.10

!obs = 0.20

z = 3 ! j = 0.1

0.1 1 10 100 1000

0.001

0.01

0.1

1

10

t !days"

F!m

Jy"


where

$ " (1 # %)/(1 # % cos !obs), (9)

and % =!


!(t) =

"

#

$

#

%

!!1j

&

ttj

'!3/8t < tj

!!1j

&

ttj

'!1/2t > tj.

(10)


3.2. Mock sample


3.2.1. Redshift PDF


Pz(z) =dN/dz

( z0 (dN/dz)dz

(11)


5 10 150.00

0.05

0.10

0.15

zPr

ob!z"


0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

!jet

Prob!!"





P!j(!) % !!2. (12)


4. GAIA mission



R = 20

!obs = 0.0

!obs = 0.10

!obs = 0.20

z = 3 ! j = 0.1

0.1 1 10 100 1000

0.001

0.01

0.1

1

10

t !days"

F!m

Jy"


where

$ " (1 # %)/(1 # % cos !obs), (9)

and % =!


!(t) =

"

#

$

#

%

!!1j

&

ttj

'!3/8t < tj

!!1j

&

ttj

'!1/2t > tj.

(10)


3.2. Mock sample


3.2.1. Redshift PDF


Pz(z) =dN/dz

( z0 (dN/dz)dz

(11)


5 10 150.00

0.05

0.10

0.15

z

Prob!z"


0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

!jet

Prob!!"





P!j(!) % !!2. (12)


4. GAIA mission



R = 20

!obs = 0.0

!obs = 0.10

!obs = 0.20

z = 3 ! j = 0.1

0.1 1 10 100 1000

0.001

0.01

0.1

1

10

t !days"

F!m

Jy"


where

$ " (1 # %)/(1 # % cos !obs), (9)

and % =!


!(t) =

"

#

$

#

%

!!1j

&

ttj

'!3/8t < tj

!!1j

&

ttj

'!1/2t > tj.

(10)


3.2. Mock sample


3.2.1. Redshift PDF


Pz(z) =dN/dz

( z0 (dN/dz)dz

(11)


5 10 150.00

0.05

0.10

0.15

z

Prob!z"


0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

!jet

Prob!!"





P!j(!) % !!2. (12)


4. GAIA mission


Redshift PDF. Shown is the probability of a given event appear in a certain range of redshift.

Half opening angle jet PDF. Shown is the probability of a given GRB have a given θj.


GAIA

One of the most ambitious projects of modern Astronomy. It aims at the creation of a very precise tridimensional, dynamical and chemical census of our Galaxy, from astrometric, spectrophotometric and spectroscopic data.

Gaia satellite will perform observations of the entire sky in a continuous scanning created from the coupling rotations and precessions movements, called ‘scanning law’.

For point-sources, these observations will be unbiased and the data of all the objects under a certain limiting magnitude (G=20), will be transferred to the ground. Certainly, among all those objects, not only galactic sources will be present, but also extragalactic ones.


GAIA SCANNING LAW

In order to estimate the probability for the observation of a single event by Gaia, Prob(∆t,lgal,bgal), only two quantities play an important role: the time that the orphan remains brighter than G=20, ∆t, and the coordinates (lgal,bgal) where the event takes place in the sky.

In order to be as realistic as possible, we adopt the Gaia Data Processing and Analysis Consortiumʼs nominal implementation of it, as to derive the a transit time list comprising the instants when Gaiaʼs telescopes will be pointing at that coordinate.

!

"

#

!"#

$%&'()*(+%,-.(/

$%&'()*(+%,-.(0

1(2(34

Fig. 1.— Sketch showing how Gaia will scan thesky. The telescope axes are separated by an angleof ! = 106.5!. The satellite spins with a constantangular velocity " around it’s main axis, which iskept at an angle # = 45! from the Sun at all times.The main axis experiences slow precessional mo-tion around the Earth-to-Sun direction (!). Ro-tation around the Sun (angular velocity $) is alsotaken into account. Deatils of the dynamics aregiven in Appendix A.

ways of afterglow identification.

2. The Gaia satellite

For the purpose of our simulation, we need toknow the way Gaia will scan the sky (Lindegren2010). Gaia will carry two identical telescopes,separated by the angle of ! = 106.5! as shown inFigure 1. The satellite will make four rotatationsper day around its axis (which is perpendicular tothe direction in which both telescopes are point-ing) with the constant angular velocity ". Thedirection of the axis itself is tilted by the angle# = 45! from the direction of the Sun. The axiswill experience slow precession motion around theEarth-to-Sun direction with a period of 63 days(! in Figure 1). Gaia will have an orbit aroundL2 point and will thus experience rotation aroundthe Sun, which is shown in Figure 1 as a rotationaround x axis. Knowing ",!, #,! and $ (one yearorbital period around the Sun), we construct the

scanning law of Gaia (Appendix A).

Gaia’s two telescopes will have a field of viewof ! 0.7! " 0.7! each. The expected limitingmagnitude in broadband G magnitude (detailson the photometric system of Gaia are given inJordi et al. (2010)) is G = 20 mag. The specificsof scientific performance are given on the o"cialGaia web page3 and, for example, in Lindegren(2010). There will be a 7 " 9 astrometric CCDfield in Gaia’s focal plane. Each source will tran-sit over the 9 CCDs and will be observed by eachof them with 4.4 seconds integration time.

3. The simulation

3.1. On-axis afterglows

First we focus on on-axis afterglows, i.e. thosewith their jet cones turned in our line of sight. Insuch cases we can observe the GRB, which trig-gers a satellite and follow-up optical observations.Hence, we can base the initial parameters of oursimulations on the actually observed GRB after-glow numbers and their characteristics. Since thelaunch of Swift satellite in 2004 (Gehrels et al.2004) a large number of GRBs and their after-glows has been detected (Roaming et al. 2009).The Swift detection rate is about 100 GRBs peryear. In about half of the detected GRBs thereis no bright optical afterglow detected. Since thesatellite covers approximately 1

6 of the sky, we canestimate that there are around 300 GRBs per year,for which an optical afterglow could be detectedwith timely observations. To obtain their gen-eral properties, we used observations published inthe Gamma Ray Burst Coordinate Network Cir-culars4 (Barthelmy et al. 1995). We chose GRBsdetected between Sep 2006 and April 2009 whichhad an optical afterglow detected (and reason-ably well sampled). In general, not many opti-cal afterglows have been detected in the first fewminutes after the initial trigger (Kann et al. 2010,2011). Consequently we used measured R magni-tudes at approximately (t#t0) =0.01 day after theprompt GRB (t0 being the time of the GRB trig-ger), at which the number of detected afterglowsis larger; if an afterglow was first detected at later

3http://www.rssd.esa.int/index.php?project=GAIA&page=index

4http://gcn.gsfc.nasa.gov/

3


GAIA OBSERVATIONAL PROBABILITY


!"!#! !"$## !"$#! !"$#% !"$#&

#'##%

#'#!#

#'#(#

#'%##

!'###

!""#$%&

)*"+,-./01,23+-456%#7-8.19:;

<=:"0*1,23+->03=1=2?2,9

!

!

!

!

!

!

!

!

!!

!!!!!!!!!!!!!!

!

!

!

!

!

!

!

!

!

!

!! ! ! ! !

Fig. 6. Probability for a transient event with duration !tto be observed by GAIA. !t is the time the event staysbrighter than the GAIA limiting magnitude during the 5years nominal mission.

Given a certain coordinate in the sky, we start by com-puting the inverse Gaia scanning law. In order to be asrealistic as possible, we adopt the Gaia Data Processingand Analysis Consortium’s nominal implementation of it,as to derive the a transit time list comprising the instantswhen Gaia’s telescopes will be pointing at that coordinate.

The limiting magnitude G=20 is reached as follows.The satellite’s on board Video Processing Unit analyzesthe focal plane signal and select ”windows” of some arc-sec (the exact number depends on the focal plane’s CCDcolumn) around each detected source, and only the datainside those windows are transmitted to Earth. After themission (and during the mission for some ”problematic”cases), it will be possible to reconstruct a deeper ”image”around each source with G ! 20, where it will be possibleto reach G! 22 or even more (albeit with some contami-nation from reconstruction artifacts), but all the data forsources fainter than G=20, if those sources are more dis-tant than about 2.5 arcsec from a G! 20 source, is lost.

The detection works as follows: the satellite is contin-uously spinning, so in the Focal Plane’s reference frame,the sky moves from one side of the FP to the other,while the CCD’s charges are being synchronously trans-ferred (like in a meridian circle) to perform the integra-tion. Gaia’s focal plane is composed by several columnsof CCDs (each column produces an independent obser-vation). The two important ones for the source detectionare the Sky Mappers (2 columns) and the AstrometricFields (9 columns). When the object enters in the FOV,it will be observed by one of the SM columns (each SMcolumn ”sees” only one telescope), and it will spend 4.4seconds transiting over that CCD. Then the object will beobserved in the AF1, AF2, etc. In order to the object be

considered as a detection, it needs to be observed on oneof the SM columns and then confirmed on AF1, if bothare positive, the data around the source on every singlecolumn (SM, AFs, spectrophotometers and the radial ve-locity spectrometer) will be transferred to Earth.

So, in order to the GRB afterglow to be observed, itwill need to transit on the Gaia focal plane above G=20during at least 4.4 seconds, in order to leave the the SMcolumn and enter in the AF1 above G=20. If it manages tocross the SM column and enter in the AF1 above G=20, itis very likely that it will be tagged by the satellite’s VPUas a source and have its data sent to Earth. What willhappen after the data arrives in Earth, during the dataprocessing, is another (rather complicated) story, but atleast the raw data will be very likely transferred.

Then, we randomly select a point in time during theentire mission lifetime in order to place an event of a cer-tain !t. Using the transit time list we check if that eventwas observed, considering a window of 4.4 seconds aroundeach transit - this amount of time is the transit durationfor an object in the Gaia’s Sky Mapper CCDs,2 whichare responsible for the on-board source detection (onlythe data of sources detected on-board will be transferredto Earth). If there is a superposition between the eventduration and this time window, the event be considereddetected. This procedure is then repeated some thousandof times, as to allow a good detection probability esti-mation, which is derived by simply counting the numberof detected events over the total. Finally, the whole pro-cedure is repeated for each !t as to allow an adequatetime-sampling of the Prob(!t, lgal, bgal) distribution.

While performing these estimations, we noted thatin fact the coordinate dependency could be averagedout, since the satellite scanning law is mostly known,and that we can reasonably assume that the orphanevents can take place randomly in the sphere, allowingProb(!t, lgal, bgal) ! Prob(!t) ± !.

So, the procedure described above was repeated forseveral positions on the sphere, and the mean and thestandard deviation at each event duration were com-puted. In order to allow a good spatial sampling forthe estimation, we tessellated the celestial sphere at theHierarchical Triangular Mesh level 4 (Kunszt et al. 2001),which means that the simulation was performed at thecenter of 2048 triangles. The results, representing the be-havior of Prob(!t) ± ! can be seen in Figure 6.

Following de Souza et al. (2011) we expect between102"105 events per year. The range of possibilities comesfrom the uncertainty the e"ciency to convert gas into starsand the e"ciency to trigger GRBs (two unknown factorsfor Pop III stars.) We create a mock sample of 5 # 104

events randomly generated by Monte-Carlo method in or-der to infer the PDF of an event stays below G = 20 over!t(days). The average behavior are shown in Fig. 7. WithProb(!t) we can generate a sample with 102 " 104 events

2 For a diagram of Gaia’s focal plane, see for ex. Jordi et al.(2010).


0 5 10 15 200.00

0.05

0.10

0.15

0.20

0.25

0.30

!t !days"

Prob!!t"

Fig. 7. PDF of !t(days). Shown is the probability of anafterglow stays above the Gaia flux limit for a given timeinterval.

and test against the probability of an event to be observedby Gaia as a function the time the event stays above itsflux limit. Combining Figs. 7 and 6 we obtain the aver-age number of events observed ! 203± 80 for N = 60000events and ! 2.3 ± 0.8 for N = 60 events.

5. Conclusion and Discussion

There are yet no direct observations of Population IIIstars, despite much recent development in theoreticalstudies on the formation of the early generation stars.Following the suggestion that massive Pop III stars couldtrigger collapsar gamma-ray bursts we investigate the pos-sibility to observe their orphan afterglows. We make usefor previous results from literature to estimate the star for-mation rate for Pop III.2 stars including all relevant feed-back e"ects: photo-dissociation, reionization and metal en-richment.

Since, we expected a larger number of orphans thanon-axis GRBs, we estimate the possibility to observe suchevents during the 5 nominal operational years of the Gaiamission. We obtain the expected average number of ob-served events ! 203 ± 80 for N = 60000 events and! 2.3 ± 0.8 for N = 60 events.

However, the detection of those events among the Gaiadata will be quite a challenging task. Gaia will observemore than one billion objects all over the sky, and eachobject will be independently detected around eighty timesduring the mission, comprising a total of around 1012 as-trometric, spectrophotometric and spectroscopic observa-tions (after the detection, the observations are multiplexedin the focal plane). One can promptly imagine that find-ing the orphans events among all that data will probablybe highly non-trivial.

Finally, it is important to note that our knowledge onthe first stars and GRBs is still limited, and there are un-certainties in their properties, most importantly in theircharacteristic mass, star formation rate and e#ciency totrigger GRBs. Nonetheless, case those events are found

among Gaia data, some valuable parameters about pri-mordial stars of our Universe could be constrained.

Acknowledgements. R.S.S. thanks the Brazilian agencyFAPESP (2009/06770-2) and CNPq (200297/2010-4) forfinancial support. This work was supported by World PremierInternational Research Center Initiative (WPI Initiative),MEXT, Japan. A.K.M. thanks the Portuguese agency FCT(SFRH/BPD/74697/2010) for financial support. We alsothank the Brazilian INCT-A for providing computationalresources through the Gina machine. and the Gaia DataProcessing and Analysis Consortium Coordination Unit 2 -Simulations for providing the scanning law implementationclasses as well as the wrapper for the HTM sphere partitioningmethod.

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PDF of ∆t(days). Shown is the probability of an afterglow stays above the Gaia flux limit for a given time interval.


PARTIAL REMARKS

We estimate the possibility to observe OA events during the 5 nominal operational years of the Gaia mission. We obtain the expected average number of observed events ∼ 203 ± 80 for N = 60000 events and ∼ 2.3 ± 0.8 for N = 60 events.

However, the detection of those events among the Gaia data will be quite a challenging task. Gaia will observe more than one billion objects all over the sky, and each object will be independently detected around eighty times during the mission, comprising a total of around 1012 astrometric, spectrophotometric and spectroscopic observations. One can promptly imagine that finding the orphans events among all that data will probably be highly non-trivial.

Would be necessary apply some kind of classification method based only in the afterglow light curves to seek for Pop III OA candidates.

For a outstanding review about photometric classification methods, don’t miss the next talk of Emille Ishida A novel approach for supernova photometric classification


Given a SFR I am able to make a estimative of GRB redshift distribution if I assume that GRB follows SFR.

If this procedure is correct, I should be able in principle to recover the SFR history using GRB redshift distribution.

How to test such hypothesis against real data?


Probing cosmic star formation up to z ∼ 9.4 with gamma-ray bursts

2 E. E. O. Ishida, R. S. de Souza, A. Ferrara

peak flux number counts obtained by the Burst And Tran-sient Source Experiment (BATSE)3 and Swift4 could be wellfitted using GRBs forming in low-metallicity galaxies. Un-der the assumption that the formation of GRBs follows thecosmic star formation history and that GRBs form prefer-entially in low-metallicity galaxies, the observed GRB for-mation rate is given by

!GRB(z) = PzfGRBfb!!(z)

Z

"

Llim(z)

p(L)dL, (1)

where fGRB is GRB formation e"ciency, fb is the beam-ing factor of the burst, !! is the cosmic star formation rate(SFR), p(L) is the GRB luminosity function, Pz is the prob-ability to obtain the redshift of GRB dataset (! 0.3 for Swiftsample) and Llim(z) denotes the limit luminosity thresholdfor a given telescope.

We show detail of our hypothesis regarding each of thisterms in the following subsections.

2.1 Beaming Factor

The non-isotropic nature of GRBs gives fbeam ! 0.02" 0.01(Guetta et al. 2005). Using a radio transients survey Gal-Yam et al. (2006) place a upper limit of fbeam ! 0.016. Weset fbeam = 0.015 ± 0.005 as a fiducial value.

2.2 Initial Mass Function

The stellar initial mass function (IMF) is critically impor-tant to determine the the e"ciency of converting stars intoGRBs. We define the GRB formation e"ciency factor perstellar mass as

fGRB = "GRB

R Mup

MGRB#(m)dm

R Mup

Mlowm#(m)dm

, (2)

where #(m) is the stellar IMF, "GRB is the e"ciency of theblack hole trigger a GRBs (! 0.001), as we expect 1 GRBevery 1000 supernovae (Langer & Norman 2006).

We consider a power law IMF with the standardSalpeter slope

#(m) # m#2.35, (3)

Mlow is the minimum mass for Pop I/II stars, ! 0.1M$

whereas Mup is the maximum mass ! 100M$. MGRB is theminimum mass able to trigger GRBs, which we set to be25M$(Bromm & Loeb 2006).

2.3 Luminosity Function

The number of bursts detected by any given instrument de-pends on the instrument-specific flux sensitivity thresholdand on the poorly determined isotropic luminosity functionof GRBs. For simplicity, we assume a luminosity function ofGRBs described by a power law similar to Wanderman &

3 http://www.batse.msfc.nasa.gov/batse/4 http://swift.gsfc.nasa.gov/docs/swift/swiftsc.html

Piran (2010)

p(L) =

8

<

:

“

LL!

”#0.17L < L!,

“

LL!

”#1.44L > L!

(4)

with L! = 1052.5 . Using the above relation we can predictthe observable GRB rate for Swift mission. For Swift, we seta bolometric energy flux limit Flim = 1.2$10#8erg cm#2 s#1

(Li 2008). The luminosity threshold is then

Llim = 4$ d2L Flim. (5)

2.4 GRB Redshift Distribution

Over a particular time interval, #tobs, in the observer restframe, the number of observed GRBs originating betweenredshifts z and z + dz is

dNGRB

dz= !GRB(z)

#tobs

1 + zdVdz

, (6)

where dV/dz is the comoving volume element per unit ofredshift, given by

dVdz

=4$cd2

L

(1 + z)

˛

˛

˛

˛

dtdz

˛

˛

˛

˛

. (7)

Thus, cumulative number of GRBs up to redshift z can bewritten as

Nth(z, #tobs) =

Z z

0

dNGRB

dzdz. (8)

In the rest of this analysis, we consider the cumula-tive number of observed GRB as a function of redshift asour observable quantity. The reader might have realized bynow that we provided a physical picture of all the termscomposing equation (1), except the SFR. This was done onpurpose, since our objective is to have a model independentidea of SFR behavior with redshift, we avoid making hy-pothesis, as much as possible. Using PCA and a simulateddata set, {zi, N(zi, #tobs)}, our intention is to obtain a func-tional form for SFR(z). We present how PCA allows us todo that in the following section.

3 SWIFT DATA

We performed the calculations considering 167 GRB datawith measured redshift from Swift. The histogram showingthe number of GRBs in each redshift bin is shown in figure1. In order to be able to test di$erent redshift bins configu-rations, we used the smooth histogram function (thick, bluecurve in figure 1, hereafter HF). Both, the histogram andthe smooth function were built using a redshift bin of width#z = 1.0. The redshift bin width was chosen following theSilverman’s rule5. Using HF, we were able to generate thenumber of observed GRBs that follows the SWIFT distri-bution in various di$erent configurations of redshift bins.Although, it is important to emphasize that our observablequantity is the cumulative number of GRB as a function ofredshift. We present in figure 2 the cumulative number ofobserved GRB constructed with the real data (blue stars)

5 http://fedc.wiwi.hu-berlin.de/xplore/ebooks/html/spm/spmhtmlnode15.html

c! 2010 RAS, MNRAS 000, 1–5

Observational quantity: Cumulative number of GRBs

We propose a novel approach, based on Principal Components Analysis, to the use of Gamma-Ray Bursts (GRBs) to investigate the cosmic star formation history (SFH) up to high redshifts as suggested by the collapsar model of long GRBs.

?


measurements from the literature (section 4.2). Finally, wediscuss our results (section 5).

Throughout the paper we adopt a !CDM model withWMAP72 best fit parameters from Jarosik et al. (2011),"m = 0.267, "! = 0.734, and H0 = 71 km s!1 Mpc !1.

2 THEORETICAL GRB RATE

We assume that the formation rate of GRBs is proportionalto the star formation rate (Totani 1997). In this context, thenumber of observable GRBs per comoving volume per timeis expressed as

#GRB(z) ="obs

4!fGRBfbPz""(z)

Z #

Llim(z)

p(L)dL, (1)

where "obs is the field of view of the experiment, fGRB isthe GRB formation e$ciency, fb is the beaming factor of theburst, Pz is the fraction of GRBs with measured redshift, ""

is the cosmic SFH and p(L) is the GRB luminosity function.The quantity Llim(z) is the minimum luminosity thresholdfor a given experiment. In the following we discuss in detaileach of the terms entering equation (1).

We defined the telescope-related quantities in equation(1) based on characteristics from Swift. This lead us to"obs = 1.4 (Salvaterra et al. 2008) and Pz = 0.24 ± 0.06.3

The overall GRB rate depends on the fact that GRBsare beamed by a factor fb ! #2/2, where # is the openingangle of the jet. Using values from (Ghirlanda et al. 2007),the average value of # is ! 5±2 deg. We set fb = 0.005+0.005

!0.002

as a fiducial value. The values are in agreement with Gal-Yam et al. (2006) that, using a radio transients survey, placean upper limit of fb ! 0.016.

The stellar Initial Mass Function (IMF), $(m), is crit-ically important to determine the fraction of massive starswhich end up as long GRBs. To produce a GRB the firstcondition is that the star is massive enough to leave a blackhole remnant. Current theories indicate that the thresholdmass is MGRB = 25M$(Bromm & Loeb 2006). However,only a fraction %GRB " 10!3 (Langer & Norman 2006) ofblack holes resulting from supernova explosion actually giverise to a GRB. To be conservative, we assume the same ratefor black holes that are able to trigger GRBs and introduce asystematic error of 50% in the factor %GRB. Hence, we definethe GRB formation e$ciency factor per stellar mass as

fGRB = %GRB

R Mup

MGRB$(m)dm

R Mup

Mlowm$(m)dm

. (2)

For simplicity, we assume a “standard” Salpeter IMF,$(m) # m!2.35, with (Mlow, Mup) = (0.1M$, 100M$)(Schneider et al. 2006) and %GRB = (1.0 ± 0.5) $ 10!3.

The number of GRBs detectable by any given instru-ment depends on the instrument-specific flux sensitivitythreshold and also on the intrinsic isotropic luminosity func-tion of GRBs. For the latter, we adopt a power-law distri-

2 http://lambda.gsfc.nasa.gov/product/map/current/3 Following Wanderman & Piran (2010) we considered redshiftmeasurements obtained via absorption and photometry only. Forfurther details, please see section 4.2.

bution function of Wanderman & Piran (2010)

p(L) =

8

<

:

“

LL!

”!0.2L < L",

“

LL!

”!1.4L > L".

, (3)

where L" " 1052.5ergs/s is the characteristic isotropic GRBluminosity.

The luminosity threshold is then

Llim = 4! d2L Flim. (4)

Here dL is the luminosity distance for the adopted cosmol-ogy and Flim is the bolometric energy flux limit of ourobservational instrument. In what follows we set Flim =1.2 $ 10!8erg cm!2 s!1 (Li 2008).

Once we built equation (1), we can determine the num-ber of observed GRBs originating between redshifts z andz + dz over a time interval, %t, in the observer rest frame.This will be given by

dNGRB

dz= #GRB(z)

%t1 + z

dVdz

, (5)

where dV/dz is the comoving volume element per unit ofredshift,

dVdz

=4!cd2

L

(1 + z)

˛

˛

˛

˛

dtdz

˛

˛

˛

˛

, (6)

c is the speed of light and t is the cosmic time.Thus, the cumulative number of GRBs up to redshift z

is

N(z) =

Z z

0

dNGRB

dz%dz%. (7)

In the following, we consider N(z) as our observable.The reader might have realized by now that we have dis-cussed the physical meaning of all terms in eq. (1), exceptthe SFH. Since our aim is to build a model which is inde-pendent from the specific form of ""(z), we avoid makinghypothesis about this quantity. Instead, our intention is toderive ""(z) by using PCA and either a mock or real dataset whose data points are {zi, N(zi)} pairs.

3 PRINCIPAL COMPONENT ANALYSIS

The main goal of PCA is the dimensionality reduction ofan initial parameter space through the analysis of its inter-nal correlations. Suppose we have a model composed by Pparameters. If two of them are highly correlated, they areactually providing us the same information. This means thatit is possible to rewrite the data in a new parameter spaceconsisting of P % 1 terms, with minimum loss of informa-tion (for a complete review see Jolli&e (2002)). This new setof parameters are recognized as the principal components(PCs), or the eigenvectors of the Fisher information matrix.

The procedure to find the PCs starts from the defi-nition of the Fisher information matrix, F. We postulatethat the data set is composed by N independent obser-vations, each one characterized by a Gaussian probabilitydensity function, fi(g(xi,&&&); Gi, 'i). In our notation, xi is ameasurement of an independent variable, Gi represents themeasurement of a quantity G which depends on xi, 'i theuncertainty associated with this measures, and &&& is the pa-

c! 2010 RAS, MNRAS 000, 1–6


Probing cosmic star formation up to z ! 9.4 with gamma-ray bursts 3

rameter vector of our theoretical model. In other words, weinvestigate a specific quantity, g, which can be written as afunction of the parameters !i. In this context, the likelihoodfunction is given by L =

QNi=1 fi and the Fisher matrix is

defined as

Fkl !

fi

""2 ln L(!!!)"!k"!l

fl

. (8)

Brackets in eq. (8) represent the expectation value.We can now diagonalize F, and determine the set of

its eigenvectors/PCs, eee = {e1e1e1, e2e2e2, ..., ePePeP}, and eigenvalues,### ={#1, #2, ..., #P }. Following the standard convention, weenumerate eieiei from the largest to the smallest associatedeigenvalue. Our ability to determine the form of each PCis given by $PCi = #!1/2

i .The set eee forms a complete base of uncorrelated vectors.

This allows us to use a subspace of eee, eMeMeM = {e1e1e1, e2e2e2, ..., eMeMeM}to rewrite g as a linear combination of all the elements ineMeMeM , grec(x,%%%). The data is then used to find the appropriatevalues of the linear expansion coe!cients, %%% (for a detaileddiscussion about the reconstruction through PCA see Ishida& de Souza (2011)).

The question of how many PCs should be used in thefinal reconstruction, or how to choose the dimensionalityof eMeMeM , depends on the particular data set analyzed andour expectation towards them (for a complete review seeJolli"e (2002), Chapter 6). To provide an idea of how muchof the initial information (variance) is included in our plots,we shall order them following their cumulative percentageof total variance. A reconstruction with the first M PCsencloses a percentage of this value

tM = 100

PMi=1 #i

PPj=1 #j

. (9)

It is important to emphasize that each added PC bringsits associated uncertainty ($PCi

) into the reconstruction.So, although the best-fitted reconstruction converges to the“real” function as M increases 4, limM"P grec = g, the un-certainty associated also raises. As a consequence, the ques-tion of how many PCs turns into a matter of what percent-age of total variance we are willing to enclose.

3.1 Cosmic SFR from GRB Distribution

We shall now apply the procedure described above to theproblem of determining the cosmic SFH from GRB data.

Let us consider our data set formed by T measure-ments of the cumulative number of GRB up to redshift zi,Ndatai

= N(zi), and its corresponding uncertainty ($i). Thelikelihood is given by

L(!!!) #T

Y

i=1

exp

"

"12

„

Ndatai" N(zi,!!!)

$i

«2#

. (10)

4 This limit holds in ideal situations, where the number of datapoints are well above the number of initial parameters. If the dataset is small or approximately the same size of the initial parameterspace, one should also care for overfitting problems.

Using equation (8), the Fisher matrix components are

Fk,l =T

X

i=1

1$2

i

"N(zi,!!!)"!k!k!k

"N(zi,!!!)"!l!l!l

. (11)

The Fisher matrix determination is now a matter ofcalculating the derivatives of N(z,!!!). In order to do so wewill use the theoretical prescription shown in section 2. Al-though, as we remark before, our intention is to avoid hy-pothesis about the SFH. Aiming at model independence andsimplicity, we represent the SFH as a sum of window func-tions

&#(z,!!!) =nbinX

i=1

!i!i!ici(z), (12)

where !i are constants, nbin is the total number of redshiftbins and ci(z) is a window function which returns 1 if zi <z ! zi+1 and 0 otherwise. In our notation, zi is the lowerbound of the i " th redshift bin. Using this description, wemay write any functional form with resolution limited byour computational power (how small we keep the redshiftbins).

We are now able to analytically calculate the derivativesof N(z), resulting in

"N(z,!!!)"!k!k!k

= H(J(z) + 1 " k)

2

4

J(z)X

i=1

'k,iA(zli, zli+1)$

+'k,J(z)+1A(zlJ(z)+1, z)˜

, (13)

where H(x) is a step function which returs 0 if x < 0 and1 otherwise, J(z) corresponds to the number of integer binsup to redshift z, 'i,j is the Kronecker delta function, zli isthe lower bound of the i " th redshift bin,

A(z1, z2) !#obs$t

4(fbfGRBPz

Z z2

z1

1(1 + x)

dVdz

(x)I(x)dx,

(14)and

I(x) !

Z $

Llim(x)

p(L)dL. (15)

Using the above relations, we are able to calculate theFisher matrix and reconstruct a functional form for the SFHbased on PCA. The results provided by this procedure whenapplied to artificial and real data are shown in the nextsections.

4 RECONSTRUCTION

4.1 Mock data

In order to test the e!ciency of this procedure, we first ap-plied it to a synthetic data sample (hereafter, Mock sample).

The Mock sample is composed of TMS=70 data points,{zi, N(zi)}, each one characterizing a redshift bin of width$z = 0.1 (this corresponds to nbin = 70 in equation (12)).The fiducial model used for the SFH was a simple double-exponential function, &fid, fitted from numerical simulationby Li (2008),

log &fid(z) = a + b log (1 + z) (M%yr!1Mpc!3), (16)

c! 2010 RAS, MNRAS 000, 1–6

Aiming at model independence and simplicity, we represent the SFH as a sum of window functions

where βi are constants, nbin is the total number of redshift bins and ci(z) is a window function which returns 1 if zi < z < zi+1 and 0 otherwise.

Probing cosmic star formation up to z ! 9.4 with gamma-ray bursts 3

rameter vector of our theoretical model. In other words, weinvestigate a specific quantity, g, which can be written as afunction of the parameters !i. In this context, the likelihoodfunction is given by L =

QNi=1 fi and the Fisher matrix is

defined as

Fkl !

fi

""2 ln L(!!!)"!k"!l

fl

. (8)

Brackets in eq. (8) represent the expectation value.We can now diagonalize F, and determine the set of

its eigenvectors/PCs, eee = {e1e1e1, e2e2e2, ..., ePePeP}, and eigenvalues,### ={#1, #2, ..., #P }. Following the standard convention, weenumerate eieiei from the largest to the smallest associatedeigenvalue. Our ability to determine the form of each PCis given by $PCi = #!1/2

i .The set eee forms a complete base of uncorrelated vectors.

This allows us to use a subspace of eee, eMeMeM = {e1e1e1, e2e2e2, ..., eMeMeM}to rewrite g as a linear combination of all the elements ineMeMeM , grec(x,%%%). The data is then used to find the appropriatevalues of the linear expansion coe!cients, %%% (for a detaileddiscussion about the reconstruction through PCA see Ishida& de Souza (2011)).

The question of how many PCs should be used in thefinal reconstruction, or how to choose the dimensionalityof eMeMeM , depends on the particular data set analyzed andour expectation towards them (for a complete review seeJolli"e (2002), Chapter 6). To provide an idea of how muchof the initial information (variance) is included in our plots,we shall order them following their cumulative percentageof total variance. A reconstruction with the first M PCsencloses a percentage of this value

tM = 100

PMi=1 #i

PPj=1 #j

. (9)

It is important to emphasize that each added PC bringsits associated uncertainty ($PCi

) into the reconstruction.So, although the best-fitted reconstruction converges to the“real” function as M increases 4, limM"P grec = g, the un-certainty associated also raises. As a consequence, the ques-tion of how many PCs turns into a matter of what percent-age of total variance we are willing to enclose.

3.1 Cosmic SFR from GRB Distribution

We shall now apply the procedure described above to theproblem of determining the cosmic SFH from GRB data.

Let us consider our data set formed by T measure-ments of the cumulative number of GRB up to redshift zi,Ndatai

= N(zi), and its corresponding uncertainty ($i). Thelikelihood is given by

L(!!!) #T

Y

i=1

exp

"

"12

„

Ndatai" N(zi,!!!)

$i

«2#

. (10)

4 This limit holds in ideal situations, where the number of datapoints are well above the number of initial parameters. If the dataset is small or approximately the same size of the initial parameterspace, one should also care for overfitting problems.


Fk,l =T

X

i=1

1$2

i

"N(zi,!!!)"!k!k!k

"N(zi,!!!)"!l!l!l

. (11)

The Fisher matrix determination is now a matter ofcalculating the derivatives of N(z,!!!). In order to do so wewill use the theoretical prescription shown in section 2. Al-though, as we remark before, our intention is to avoid hy-pothesis about the SFH. Aiming at model independence andsimplicity, we represent the SFH as a sum of window func-tions

&#(z,!!!) =nbinX

i=1

!i!i!ici(z), (12)

where !i are constants, nbin is the total number of redshiftbins and ci(z) is a window function which returns 1 if zi <z ! zi+1 and 0 otherwise. In our notation, zi is the lowerbound of the i " th redshift bin. Using this description, wemay write any functional form with resolution limited byour computational power (how small we keep the redshiftbins).

We are now able to analytically calculate the derivativesof N(z), resulting in

"N(z,!!!)"!k!k!k

= H(J(z) + 1 " k)

2

4

J(z)X

i=1

'k,iA(zli, zli+1)$

+'k,J(z)+1A(zlJ(z)+1, z)˜

, (13)

where H(x) is a step function which returs 0 if x < 0 and1 otherwise, J(z) corresponds to the number of integer binsup to redshift z, 'i,j is the Kronecker delta function, zli isthe lower bound of the i " th redshift bin,

A(z1, z2) !#obs$t

4(fbfGRBPz

Z z2

z1

1(1 + x)

dVdz

(x)I(x)dx,

(14)and

I(x) !

Z $

Llim(x)

p(L)dL. (15)

Using the above relations, we are able to calculate theFisher matrix and reconstruct a functional form for the SFHbased on PCA. The results provided by this procedure whenapplied to artificial and real data are shown in the nextsections.

4 RECONSTRUCTION

4.1 Mock data

In order to test the e!ciency of this procedure, we first ap-plied it to a synthetic data sample (hereafter, Mock sample).

The Mock sample is composed of TMS=70 data points,{zi, N(zi)}, each one characterizing a redshift bin of width$z = 0.1 (this corresponds to nbin = 70 in equation (12)).The fiducial model used for the SFH was a simple double-exponential function, &fid, fitted from numerical simulationby Li (2008),

log &fid(z) = a + b log (1 + z) (M%yr!1Mpc!3), (16)

c! 2010 RAS, MNRAS 000, 1–6

The number of parameters = number of bins!!!


Principal Component Analysis (PCA)The main goal of PCA is to reduce the dimensionality of the initial

parameter space


1 PC

Mock samp le t1 ! 88.9"

0.05

0.10

0.20

0.50

1

#rec!M!yr$1Mpc$3 " t2 ! 94.4"

2 PC

t3 ! 96.4"3 PC

0.05

0.10

0.20

0.50

#rec!M!yr$1Mpc$3 " t4 ! 97.4"

4 PC

zt5 ! 98.0"5 PC

0 1 2 3 4 5 60.02

0.05

0.10

0.20

0.50

z

#rec!M!yr$1Mpc$3 " t6 ! 98.4"

6 PC

0 1 2 3 4 5 6

z

Figure 1. PCA reconstruction from Mock sample data with!z = 0.1. Reconstructions with 1 (top-left) to 6 (bottom-right)PCs, along with corresponding values of percentage of total vari-ance. The blue-thin line corresponds to our fiducial model !fid.The black-thick line is the final reconstruction for each case andthe red-dashed-thick lines corresponds to 2" confidence levels.

with (a, b) = (-1.70, 3.30) for z < 0.993; (-0.727, 0.0549) for0.993 < z < 3.80 and (2.35,-4.46) for z > 3.80.

After we generated the mock sample, the informationabout !fid(z) is discarded. Our goal from now on is to re-obtain the general functional form of equation (16) fromPCA.

The Fisher matrix was constructed using the prescrip-tion given in section 2, during an one year observation time,!tMS = 1. Our purpouse with this Mock sample is to testif the procedure works in an ideal scenario, so we did notinclude uncertainties in the parameters of equation (1).

After obtaining the PCs, we can rewrite the SFH as

!rec(z) = !fid(0) + !c +M

X

i=1

"ieieiei(z), (17)

where "i and !c are constants to be determined, M is thenumber of PCs we choose to use in the reconstruction and!fid(0) is the SFH today (for a deep discussion about therole of !c and !fid(0) in the reconstruction, see Ishida & deSouza (2011)).

The simulated data points are then used to find theappropriate values for the parameters "i as those that min-imize the expression

#2(""") !TMSX

i=1

(Ni;data " Nrec(zi,"""))2

2$2i

, (18)

where $i is the measurement uncertainty, consider equal tounity for all redshift bins.

The reconstructions obtained using 1 to 6 PCs areshown in Fig. 1. The uncertainty in the final reconstruc-tion was calculated by a quadrature sum that includes theparameters $PCi and the uncertainty in the determinationof parameters """($!i

) and !c($"c).

From this plot, we can see that the procedure does re-construct an underlying unknown function (blue-thin line),at least in an ideal scenario, with better agreement as thenumber of PCs raises. Although, we can also see that theconfidence levels (red-dashed-thick lines) also become wideras M increases. Only the reconstruction with 1 PC does notfollow this rule, since $"c

dominates the errors due to thelow freedom in trying to fit the second peak of the fiducialmodel with only 1 PC. From 2 PCs on where the fitting ismore suceptible to reproducing the second peak, and has aconsequence, the magnitude of $"c

decreases to levels bel-low those of $PCi . In such situations we can clearely see theevolution of the confidence levels due to the use of one morecomponent in each panel of figure 1.

4.2 Swift data

Given that PCA reconstruction proved to be e"ective in anideal situation, we shall now apply it to the currently avail-able Swift data set, and compare its results with completelyindependent measurements of SFH from the literature.

First of all, we need to properly choose our data set.Once only GRBs with measured redshifts can be used inour analysis, the question of how the redshift measurementswere obtained play an important role.

The GRBs redshifts are obtained in general from the op-tical afterglow spectrum using absorption lines or photome-try, or from the spectrum of the host galaxy using emissionlines. As pointed by Wanderman & Piran (2010), di"erentmethods gives di"erent redshift distributions. For redshiftsdetermined using the host’s emission lines, we do not de-tect high-redshift events, whereas absorption lines redshiftsextend over the entire range of redshifts. Furthermore, emis-sion lines are more susceptible to a selection e"ect know asthe redshift desert in the range 1.1 < z < 2.1. The distri-bution for each redshift determination method is shown infigure 2.

Being crucial for us to avoid systematics errors whicha"ect the overall redshift distribution, our data sample iscomposed by the 120 Swift GRBs with redshift measuredby absorption lines and photometry (gray region in figure 2,top panel).

The next step is to choose the appropriate redshift binwidth. We know that our final result will have better resolu-tion with higher number of bins. However, in a real situationit is not possible to make the bin width as small as we like,since we will never be able to measure non-integer numbersof GRBs. If we pick a bin width based on the available data(for example, in such a way that each bin has at least oneGRB), the bins will be too wide (# 1). In this case, the as-sumption that the SFH is constant inside the bin will nothold, leading to reconstructions with bad resolution.

In order to overcome this limitation we performed aGaussian fit to the data (black line in figure 2, top panel).Now we have a continuos probability distribution function(PDF) for dN/dz, which follows the real data distributionand allows us to set the bin width as small as we want. Wekept !z = 0.1 and use the PDF to calculate the cumulativenumber of observed GRBs in the range 0 < z < 10.0 (ornbin = 100 in equation (12)). The comparison between thereal data cumulative distribution and the one calculated viathe fitted PDF are shown in figure 2, lower panel. Our data

c! 2010 RAS, MNRAS 000, 1–6


1 PC


0.05

0.10

0.20

0.50

1

#rec!M!yr$1Mpc$3 " t2 ! 94.4"

2 PC

t3 ! 96.4"3 PC

0.05

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0.20

0.50

#rec!M!yr$1Mpc$3 " t4 ! 97.4"

4 PC

zt5 ! 98.0"5 PC

0 1 2 3 4 5 60.02

0.05

0.10

0.20

0.50

z

#rec!M!yr$1Mpc$3 " t6 ! 98.4"

6 PC

0 1 2 3 4 5 6

z






!rec(z) = !fid(0) + !c +M

X

i=1

"ieieiei(z), (17)



#2(""") !TMSX

i=1


2$2i

, (18)



) and !c($"c).




4.2 Swift data







c! 2010 RAS, MNRAS 000, 1–6

Probing cosmic star formation up to z = 9.4 with GRBs 3

encloses a percentage of this value

tM = 100

!Mi=1 !i

!Pj=1 !j

. (7)

It is important to emphasize that each added PC bringsits associated uncertainty ("PCi

) into the reconstruction.So, although the best-fitted reconstruction converges to the“real” function as M increases3, the uncertainty associatedalso raises. As a consequence, the question of how many PCsturns into a matter of what percentage of total variance weare willing to enclose.

3.1 Star formation history from GRB distribution

To specify our method of SFH reconstruction from GRBdata, let us consider a data set formed by T measure-ments of the cumulative number of GRB up to redshift zi,Ndatai

= N(zi), and its corresponding uncertainty ("i). Thelikelihood is given by

L(###) !T"

i=1

exp

#

"12

$

Ndatai"N(zi,###)

"i

%2&

. (8)


Fk,l =T'

i=1

1"2i

$N(zi,###)$#k#k#k

$N(zi,###)$#l#l#l

. (9)

The Fisher matrix determination is now a matter of calculat-ing the derivatives of N(z,###), for which we use the theoret-ical prescriptions of Sec. 2. Aiming at model independenceand simplicity, we model the SFH as

%!(z,###) =nbin'

i=1

#ici(z), (10)

where #i are constants, nbin is the total number of redshiftbins, and ci(z) is a window function which returns 1 if zi <z ! zi+1 and 0 otherwise. Using this description, we maywrite any functional form with resolution limited by ourcomputational power.

The derivatives of N(z) can be computed analytically:

$N(z,###)

$#k#k#k= H(J(z) + 1" k)

(

)

J(z)'

i=1

&k,iA(zli, zli+1)+

+&k,J(z)+1A(zlJ(z)+1, z)*

, (11)

where H(x) is a step function which returns 0 if x < 0 and1 otherwise, J(z) corresponds to the number of integer binsup to redshift z, &i,j is the Kronecker delta function, zli isthe lower bound of the i-th redshift bin, and

A(z1, z2) #!obs"t

4'fbfGRBPz $

$

+ z2

z1

1(1 + x)

dVdz

(x)

+

"

logLlim(x)

#(L)d logLdx.

(12)

3 This limit holds in ideal situations, where the number of datapoints largely exceeds the number of initial parameters. If thedata set is small or approximately the same size of the initialparameter space, one should also care for overfitting problems.

1 PC


0.05

0.10

0.20

0.50

1

#rec!M!yr$1Mpc$3 " t2 ! 94.4"

2 PC

t3 ! 96.4"3 PC

0.05

0.10

0.20

0.50

#rec!M!yr$1Mpc$3 " t4 ! 97.4"

4 PC

zt5 ! 98.0"5 PC

0 1 2 3 4 5 60.02

0.05

0.10

0.20

0.50

z

#rec!M!yr$1Mpc$3 " t6 ! 98.4"

6 PC

0 1 2 3 4 5 6

z

Figure 1. PCA reconstructions of SFH obtained from our mockdata, using 1 (top-left) to 6 (bottom-right) PCs. The blue-thinline corresponds to our fiducial model !fid. The black-thick lineis the final reconstruction for each case and the red-dashed-thicklines corresponds to 2" confidence levels. The inset shows thecumulative percentage of total variance, tM .

From these relations the Fisher matrix can be computed andthe functional form of the SFH reconstructed through PCA.

4 RECONSTRUCTION

Mock data The mock sample is composed of data pairs{zi, N(zi)}, distributed in redshift bins of width "z = 0.1,where zi represents the middle of each bin. The fiducialmodel used for the SFH is a simple double-exponential func-tion, %fid, fitted to numerical results by Li (2008),

log %fid(z) = a+ b log (1 + z), (13)

with (a, b) = ("1.70, 3.30) for z < 0.993; ("0.727, 0.0549)for 0.993 < z < 3.80 and (2.35,"4.46) for z > 3.80. We gen-erated 500 simulations, each realization with uncertainty inthe determination of N(z) set to unit and containing 65redshift bins. After we generated the mock sample, the in-formation about %fid(z) is discarded. Our goal from now onis to re-obtain the functional form of eq. (13) from PCA.

The Fisher matrix is obtained as described above, as-suming an observing time "tMS = 1 yr. As the mock samplepurpose is to test the procedure under an ideal scenario, wedid not include uncertainties in the parameters of eq. (1).Having obtained the PCs, we can rewrite the SFH as

%rec(z) = %c +M'

i=1

(ieieiei(z), (14)

where (i and %c are constants to be determined andM is thenumber of PCs we choose to use in the reconstruction. Thesimulated data points are then used to find the appropriatevalues for the parameters (i and %c as those that minimize


Principal Component Analysis (PCA)The main goal of PCA is to reduce the dimensionality of the initial

parameter space

http://www.cs.cornell.edu/courses/cs322/2008sp/images/thumb_PCA.png


1 PC


0.05

0.10

0.20

0.50

1

#rec!M!yr$1Mpc$3 " t2 ! 94.4"

2 PC

t3 ! 96.4"3 PC

0.05

0.10

0.20

0.50

#rec!M!yr$1Mpc$3 " t4 ! 97.4"

4 PC

zt5 ! 98.0"5 PC

0 1 2 3 4 5 60.02

0.05

0.10

0.20

0.50

z

#rec!M!yr$1Mpc$3 " t6 ! 98.4"

6 PC

0 1 2 3 4 5 6

z






!rec(z) = !fid(0) + !c +M

X

i=1

"ieieiei(z), (17)



#2(""") !TMSX

i=1


2$2i

, (18)



) and !c($"c).




4.2 Swift data







c! 2010 RAS, MNRAS 000, 1–6


1 PC


0.05

0.10

0.20

0.50

1

#rec!M!yr$1Mpc$3 " t2 ! 94.4"

2 PC

t3 ! 96.4"3 PC

0.05

0.10

0.20

0.50

#rec!M!yr$1Mpc$3 " t4 ! 97.4"

4 PC

zt5 ! 98.0"5 PC

0 1 2 3 4 5 60.02

0.05

0.10

0.20

0.50

z

#rec!M!yr$1Mpc$3 " t6 ! 98.4"

6 PC

0 1 2 3 4 5 6

z






!rec(z) = !fid(0) + !c +M

X

i=1

"ieieiei(z), (17)



#2(""") !TMSX

i=1


2$2i

, (18)



) and !c($"c).




4.2 Swift data







c! 2010 RAS, MNRAS 000, 1–6


encloses a percentage of this value

tM = 100

!Mi=1 !i

!Pj=1 !j

. (7)

It is important to emphasize that each added PC bringsits associated uncertainty ("PCi

) into the reconstruction.So, although the best-fitted reconstruction converges to the“real” function as M increases3, the uncertainty associatedalso raises. As a consequence, the question of how many PCsturns into a matter of what percentage of total variance weare willing to enclose.

3.1 Star formation history from GRB distribution

To specify our method of SFH reconstruction from GRBdata, let us consider a data set formed by T measure-ments of the cumulative number of GRB up to redshift zi,Ndatai

= N(zi), and its corresponding uncertainty ("i). Thelikelihood is given by

L(###) !T"

i=1

exp

#

"12

$

Ndatai"N(zi,###)

"i

%2&

. (8)


Fk,l =T'

i=1

1"2i

$N(zi,###)$#k#k#k

$N(zi,###)$#l#l#l

. (9)

The Fisher matrix determination is now a matter of calculat-ing the derivatives of N(z,###), for which we use the theoret-ical prescriptions of Sec. 2. Aiming at model independenceand simplicity, we model the SFH as

%!(z,###) =nbin'

i=1

#ici(z), (10)

where #i are constants, nbin is the total number of redshiftbins, and ci(z) is a window function which returns 1 if zi <z ! zi+1 and 0 otherwise. Using this description, we maywrite any functional form with resolution limited by ourcomputational power.

The derivatives of N(z) can be computed analytically:

$N(z,###)

$#k#k#k= H(J(z) + 1" k)

(

)

J(z)'

i=1

&k,iA(zli, zli+1)+

+&k,J(z)+1A(zlJ(z)+1, z)*

, (11)

where H(x) is a step function which returns 0 if x < 0 and1 otherwise, J(z) corresponds to the number of integer binsup to redshift z, &i,j is the Kronecker delta function, zli isthe lower bound of the i-th redshift bin, and

A(z1, z2) #!obs"t

4'fbfGRBPz $

$

+ z2

z1

1(1 + x)

dVdz

(x)

+

"

logLlim(x)

#(L)d logLdx.

(12)

3 This limit holds in ideal situations, where the number of datapoints largely exceeds the number of initial parameters. If thedata set is small or approximately the same size of the initialparameter space, one should also care for overfitting problems.

1 PC


0.05

0.10

0.20

0.50

1

#rec!M!yr$1Mpc$3 " t2 ! 94.4"

2 PC

t3 ! 96.4"3 PC

0.05

0.10

0.20

0.50

#rec!M!yr$1Mpc$3 " t4 ! 97.4"

4 PC

zt5 ! 98.0"5 PC

0 1 2 3 4 5 60.02

0.05

0.10

0.20

0.50

z

#rec!M!yr$1Mpc$3 " t6 ! 98.4"

6 PC

0 1 2 3 4 5 6

z

Figure 1. PCA reconstructions of SFH obtained from our mockdata, using 1 (top-left) to 6 (bottom-right) PCs. The blue-thinline corresponds to our fiducial model !fid. The black-thick lineis the final reconstruction for each case and the red-dashed-thicklines corresponds to 2" confidence levels. The inset shows thecumulative percentage of total variance, tM .

From these relations the Fisher matrix can be computed andthe functional form of the SFH reconstructed through PCA.

4 RECONSTRUCTION

Mock data The mock sample is composed of data pairs{zi, N(zi)}, distributed in redshift bins of width "z = 0.1,where zi represents the middle of each bin. The fiducialmodel used for the SFH is a simple double-exponential func-tion, %fid, fitted to numerical results by Li (2008),

log %fid(z) = a+ b log (1 + z), (13)

with (a, b) = ("1.70, 3.30) for z < 0.993; ("0.727, 0.0549)for 0.993 < z < 3.80 and (2.35,"4.46) for z > 3.80. We gen-erated 500 simulations, each realization with uncertainty inthe determination of N(z) set to unit and containing 65redshift bins. After we generated the mock sample, the in-formation about %fid(z) is discarded. Our goal from now onis to re-obtain the functional form of eq. (13) from PCA.

The Fisher matrix is obtained as described above, as-suming an observing time "tMS = 1 yr. As the mock samplepurpose is to test the procedure under an ideal scenario, wedid not include uncertainties in the parameters of eq. (1).Having obtained the PCs, we can rewrite the SFH as

%rec(z) = %c +M'

i=1

(ieieiei(z), (14)

where (i and %c are constants to be determined andM is thenumber of PCs we choose to use in the reconstruction. Thesimulated data points are then used to find the appropriatevalues for the parameters (i and %c as those that minimize


MOCK SAMPLE4 E. E. O. Ishida, R. S. de Souza, A. Ferrara

1 PC


0.05

0.10

0.20

0.50

1

#rec!M!yr$1Mpc$3 " t2 ! 94.4"

2 PC

t3 ! 96.4"3 PC

0.05

0.10

0.20

0.50#rec!M!yr$1Mpc$3 " t4 ! 97.4"

4 PC

zt5 ! 98.0"5 PC

0 1 2 3 4 5 60.02

0.05

0.10

0.20

0.50

z

#rec!M!yr$1Mpc$3 " t6 ! 98.4"

6 PC

0 1 2 3 4 5 6

z






!rec(z) = !fid(0) + !c +M

X

i=1

"ieieiei(z), (17)



#2(""") !TMSX

i=1


2$2i

, (18)



) and !c($"c).




4.2 Swift data







c! 2010 RAS, MNRAS 000, 1–6

PCA reconstructions of SFH obtained from our mock data, using 1 (top-left) to 6 (bottom-right) PCs. The blue-thin line corresponds to our fiducial model ρfid. The b lack- th ick l ine is the final reconstruction for each case and the red-dashed-thick lines corresponds to 2σ confidence levels. The inset shows the cumulative percentage of total variance, tM .


Swift data

Being crucial for us to avoid systematics errors which affect the overall redshift distribution, our data sample is composed by the 120 Swift GRBs with redshift measured by absorption lines and photometry


the expression

!2(""") !TMS!

i=1

(Ni;data "Nrec(zi; #c,"""))2

2$2i

, (15)

where $i = 1 for all redshift bins. The reconstructions ob-tained using 1 to 6 PCs are shown in Fig. 1. The uncertaintyin the final reconstruction was calculated by a quadraturesum that includes the parameters $PCi and the uncertaintyin the determination of parameters """($!i

) and #c($"c).From Fig. 1 we can appreciate the success of the proce-

dure in reconstructing the underlying unknown SFH in anideal scenario, with increasing agreement as the number ofPCs raises. Confidence levels also become wider as M in-creases, with the only exception of the reconstruction with1 PC. Since $"c dominates the errors due to the limited free-dom to fit the second peak of the fiducial model with only 1PC. With 2 PCs fitting the second peak becomes easier, andas a consequence, the magnitude of $"c decreases to levelsbelow those of $PCi .

Swift data After validating PCA reconstruction underideal conditions, we turn to the use of currently availableSwift data, and compare these results with independentmeasurements of SFH from the literature. First, we needto properly choose our data set. Since only GRBs with mea-sured redshifts can be used in our analysis, the question ofhow the redshift measurements were obtained must be ex-amined carefully.

GRBs redshifts are generally obtained from optical af-terglow spectra using absorption lines or photometry, orfrom the spectrum of the host galaxy using emission lines.As pointed by Wanderman & Piran (2010), di!erent meth-ods yield di!erent redshift distributions: a visual inspectionof Fig. 2 illustrates this point. Most noticeably, the GRBredshift distribution determined from their hosts lacks veryhigh-z events. Moreover, emission (and to a lesser extent, ab-sorption) lines are susceptible to a selection e!ect known asthe “redshift desert” in the range 1.1 < z < 2.1 (Fiore et al.2007; Coward 2009). Additional bias sources are preliminarydiscussed by Malesani et al. (2009). To avoid systematic er-rors a!ecting the overall redshift distribution, our data sam-ple is composed by 120 Swift GRBs with redshift determinedfrom absorption lines and photometry (gray region in Fig.2, top panel).

The next step is to choose the appropriate redshift binwidth. In principle, the quality of the reconstruction shouldincrease with the number of bins. However, as GRB are dis-crete events, if we pick a bin width based on the availabledata (for example, in such a way that each bin has at leastone GRB), the bins will be too wide (# 1). In this case,the assumption that the SFH is constant inside the bin willnot hold, leading to reconstructions with bad resolution.To overcome this limitation we performed a gaussian ker-nel fit4 to the data (black line in Fig. 2, top panel). Now wehave a continuous probability distribution function (PDF)for dN/dz, which follows the real data distribution and al-

4 A non-parametric estimate of the PDF obtained from a linearly

interpolated version of 1nh

!ni=1 k

"

x!xi

h

#

for a kernel k(x), bin

width h and a total of n bins.

absorption

photometry

fit

used sample

em ission

0 2 4 6 80

5

10

15

20

zdN!dz

!!!!!!!!!!!!

!!!!!!!

! used samplefit

0 2 4 6 80

20

40

60

80

100

z

N"z#

Figure 2. Top: Histogram showing the measured redshift distri-bution of 120 GRBs detected by Swift from 2005 to 2010, dividedby redshift measurement methods: absorption (red-dashed), pho-tometry (green-dotted) and emission (blue-dot-dashed). The grayregion corresponds to the redshift distribution of data we used(absorption + photometry). The solid black line shows the fitto the used data distribution. Bottom: The cumulative distribu-tions constructed from Swift data (gray stars) and from our fitteddistribution function (black points).

lows us to set the bin width as small as required. We kept"z = 0.1 and use the PDF to calculate the cumulative num-ber of observed GRBs in each bin. The comparison betweenthe real data cumulative distribution and the one calculatedvia the fitted PDF are shown in Fig. 2, bottom panel.

The Fisher matrix is calculated using "tSW = 6 yr ofobservation time. The parameters $i were obtained by sum-ming in quadrature the uncertainties in the quantities in-volved in eq. (1). Fig. 3 shows the first 2 PCs and the corre-sponding reconstruction using both of them, which alreadyencloses more than 97% of total variance. In the lower panel,the points correspond to completely independent measure-ments from the literature. These data points are shown onlyfor comparison purposes and have not been used in our cal-culations.

5 DISCUSSION

We have proposed the use of PCA as a powerful tool to re-construct the cosmic star formation history exploiting themeasured gamma-ray burst redshift distribution. The pro-cedure was successfully validated using synthetic data andnext applied to actual Swift data (Fig. 3).

It is important to highlight that the approach is com-pletely independent of the initial choice of the theoreticalmodel parameter vector, %%%. This has the obvious advantageof avoiding any a priori hypothesis on the SFH, #"(z). How-ever, the degeneracy between #"(z) and any other factorwe are failing to take into account cannot be removed, i.e.



PC 1

PC 2

SWIFT

0 2 4 6 8

! 0.4

! 0.2

0.0

0.2

0.4

z

e i

"

" "

"

""

"

"

"

"

""

""

"

""

"

"

"

"

"

"

"

""

"

"""""

"""

"

"

"

"

"

"

""

"

"

"

"

"

"

"

" ""

"

"

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"

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"

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!

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%

%

Thiswork

t 2 %97.2&

0 2 4 6 810!4

0.001

0.01

0.1

z

'rec!M&yr!1Mpc!3 "

Figure 3. Top: First (red solid) and second (blue dashed) PCsfrom Swift. Bottom: PCA reconstruction from Swift data using2 PCs, compared with independent SFH determinations (lightpoints, not used in our calculations). The black solid line is thePCA best-fit reconstruction using 2 PCs; the red dashed lines cor-respond to 2! confidence levels. The inset shows the cumulativepercentage of total variance, tM .

the reconstructed SFH contains also the behavior of all theagents influencing the determination of N(z) and not in-cluded in the model.

For example, Langer & Norman (2006);Woosley & Heger (2006) have argued that GRB progenitorswill have a low metallicity. Such an e!ect would be a con-sequence of the mass and angular momentum loss causedby winds in high-metallicity stars. This would prevent suchstars of becoming GRBs and consequently, change their ex-pected redshift distribution (Salvaterra & Chincarini 2007;Salvaterra et al. 2007; Li 2008). We implicitly consideredthat this and other such e!ects will span within the errorbars in our analysis.

In spite of the remaining uncertainties, which are prob-ably less severe than those a!ecting other methods aimed atrecovering the high-z tail of the SFH, there are robust indica-tions that we can gather from the analysis of our results. Thefirst is that the combination of GRB data and PCA suggestthat the level of star formation activity at z ! 9.4 could havebeen already as high as the present-day one (! 0.01M! yr"1

Mpc"3). This is a factor 3-5 times higher than deduced fromhigh-z galaxy searches through drop-out techniques, simi-larly to the trend found by Yonetoku et al. (2004). If true,this might alleviate the long-standing problem of a photon-starving reionization; it might also indicate that galaxies ac-counting for most of the star formation activity at high red-shift go undetected by even the most deep searches. Finallyit is worth noticing that a sustained high-z star formationactivity is consistent with predictions of reionization mod-els that simultaneously match a number of observable ex-perimental constraints as the Gunn-Peterson e!ect, Thom-

son free-electron optical depth, Lyman Limit Systems statis-tics etc. (Choudhury & Ferrara (2006), Bolton & Haehnelt(2007)). Given the expected large growth of GRB detec-tions from the next generation of instruments, the methodproposed here promises to become one of the most suitableand reliable tools to constrain the star formation activity inthe young Universe.

ACKNOWLEDGEMENTS

We thank K. Ioka, R. Salvaterra and N. Yoshida for useful com-

ments. E.E.O.I. thanks CAPES (1313-10-0) for financial support.

R.S.S. thanks CNPq (200297/2010-4) for financial support. AF

acknowledges support from IPMU where this research started.

This work was supported by WPI Initiative, MEXT, Japan.

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First (red solid) and second (blue dashed) PCs from Swift.

PCA reconstruction from Swift data using 2 PCs, compared with independent SFH determinations.


FINAL REMARKS

We implemented the use of PCA in the estimation of SFH based on gamma-ray burst redshift distribution

It is important to highlight that our approach is completely independent of our initial choice of β. This has an obvious advantage of avoiding hypothesis about ρ∗(z),

Given the recent dicovered GRB at z ~9.4, we are able to constraint the star formation history up to the same z, being the more distant SFR already estimated.


REFERENCES

Ishida, E. E. O., de Souza, R. S., Ferrara, A., Probing cosmic star formation with gamma-ray bursts, to appear in MNRAS, arXiv:1106.1745

de Souza, R. S., Yoshida, N. and Ioka, K., Population III.1 and III.2 Gamma-Ray Bursts: Constraints on the Event Rate for Future Radio and X-ray Surveys, A&A, 533, A32 (2011).

de Souza, R. S., Krone-Martins, A., Ferrara, A., Ciardi, B., Ishida, E., Orphan Afterglows from Pop III GRBs: Constraints on the Event Rate for Gaia Mission, in prep


HOW CAN WE SEE THE FIRST STARS?

First Stars can trigger high energetic GRBs after died.

Gamma rays are the most powerful explosions of the Universe. They tell us about extreme conditions, powerful processes, and exotic phenomena.

While supernovae explosions have luminosities similar to an entire galaxy

GRBs have luminosities equivalent to all the Universe!!!


Bower et al. (2007) used 22 years of archival data from VLA to put an upper limit of ∼ 2.4 × 105 over 1-year variability transients above 90 µJy for all sky.

Gal-Yam et al. (2006) used FIRST and NVSS radio catalogues to place an upper limit of ∼ 70 radio orphan afterglows above 6 mJy in the 1.4 GHz band over the entire sky. This suggests less than ∼ 103 sources above 1 mJy on the sky,

In our model a typical GRB’s radio afterglow with isotropic kinetic energy Eiso ∼ 5 × 1054 ergs stays above 1 mJy over ∼ 102−3 days. Combining the results shown in Figs. 5 and 6, we expect ∼ 10−104 sources above ∼ 1 mJy.

As a consequence, the most optimistic case for Pop III.2 should be already ruled out by the current observations of radio transient sources, if their luminosity function follows the one assumed in the present paper.


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