The Astrophysical Journal, 735:44 (14pp), 2011 July 1

doi:10.1088/0004-637X/735/1/44


C 2011. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

THE ROLE OF DUST IN THE EARLY UNIVERSE. I. PROTOGALAXY EVOLUTION
Daisuke Yamasawa1 , Asao Habe1 , Takashi Kozasa1 , Takaya Nozawa2 , Hiroyuki Hirashita3 ,Hideyuki Umeda4 , and Ken’ichi Nomoto2
1 Department of Cosmosciences, Graduate School of Science, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan; yamasawa@astro1.sci.hokudai.ac.jp
2 Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8583, Japan
3 Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan
4 Department of Astronomy, School of Science, University of Tokyo, Hongo, Tokyo 113-0033, Japan

Received 2010 November 29; accepted 2011 March 30; published 2011 June 14

ABSTRACTWe develop one-zone galaxy formation models in the early universe, taking into account dust formation andevolution by supernova (SN) explosions. We focus on the time evolution of dust size distribution, because H2
formation on the dust surface plays a critical role in the star formation process in the early universe. In the model,
we assume that star formation rate (SFR) is proportional to the total amount of H2 . We consistently treat (1) theformation and size evolution of dust, (2) the chemical reaction networks including H2 formation both on the surfaceof dust and in gas phase, and (3) the SFR in the model. First, we find that, because of dust destruction due to bothreverse and forward shocks driven by SNe, H2 formation is more suppressed than in situations without such dust
destruction. At the galaxy age of ∼0.8 Gyr, for galaxy models with virial mass Mvir = 109 M and formation
redshift zvir = 10, the molecular fraction is 2.5 orders of magnitude less in the model with dust destruction by bothshocks than that in the model without dust destruction. Second, we show that the H2 formation rate strongly depends
on the interstellar medium (ISM) density around SN progenitors. The SFR in higher ISM density is lower, sincedust destruction by reverse shocks is more effective in higher ISM density. We conclude that not only the amountbut also the size distribution of dust related to star formation activity strongly affects the evolution of galaxies inthe early universe.
Key words: dust, extinction – early universe – galaxies: evolution – galaxies: formation – galaxies: ISM
Online-only material: color figures

ciation by the Lyman–Werner ultraviolet (UV) background between 11.2 and 13.6 eV is important in the lower mass H2 cooling halos. Gas condensation in the lower mass H2 cooling halos
can be delayed by the Lyman–Werner background (Machacek
et al. 2001, 2003; Yoshida et al. 2003; Susa 2007; Wise &Abel 2007; O’Shea & Norman 2008). The Lyman–Werner background thus increases cooling times in the centers of such halos.
As a result, the minimum mass of a star-forming halo increases
with Lyman–Werner background intensity. The Lyman–Wernerbackground becomes less of an issue in atomic line cooling halos as Lyα cooling provides ample amounts of free electrons
for H2 cooling, and they become self-shielding to this radiation(O’Shea & Norman 2008; Susa 2008; Wise & Abel 2008; Wise
& Cen 2009).
In the later epoch, dust ejected by stars in galaxies effectively
shields the Lyman–Werner background and acts as an effective
catalyst for H2 molecule production on the dust grains. In
simulations with star formation models based on molecularhydrogen (Robertson & Kravtsov 2008; Gnedin et al. 2009),
once the gas is enriched up to Z ∼ 0.01–0.1 Z , the subsequent
star formation and enrichment of metal and dust is greatly
accelerated. Gnedin et al. (2009) show that the transition
from atomic to molecular hydrogen depends primarily on
metallicity, assuming that the dust abundance is directly related
to metallicity.
Dust plays a crucial role in star formation: (1) molecularhydrogen is produced more efficiently on dust grains than in
the gas phase, (2) dust shields dissociate UV radiation, and (3)dust allows the formation of low-mass stars in low-metallicity
environments and hence affects the initial mass function (IMF;
Omukai et al. 2005; Schneider et al. 2006; Schneider & Omukai
2010; Omukai et al. 2010).

1. INTRODUCTION
Understanding galaxy evolution in the early universe remains
one of the most important goals of modern astrophysics.
Modeling of primeval galaxy formation requires an accurate
treatment of the star formation process in low-metallicity gas(Jappsen et al. 2007; Glover & Jappsen 2007; Smith et al. 2008,2009; Jappsen et al. 2009b, 2009a). A critical challenge forachieving this goal is our poor understanding of how gas isconverted into stars under different conditions (Krumholz &McKee 2005; Robertson & Kravtsov 2008; Gnedin et al. 2009).
In particular, star formation efficiency in the primeval galaxy is
still uncertain.
The standard approach in theoretical studies of galaxy formation so far is to adopt a recipe that ties the star formation rate (SFR) to gas density both in semi-analytic models(e.g., Cole et al. 2000) and in numerical simulations (e.g.,
Springel et al. 2005). Such a recipe is based on the empirical correlations observed in local galaxies, namely, theKennicutt–Schmidt law (Kennicutt 1998). These correlations
have only been studied relatively well for nearby massive orstar bursting galaxies. However, for galaxies with low surface brightness and/or low metallicity, this empirical relation
may not be valid. Indeed, both nearby metal-poor galaxies(Bigiel et al. 2008) and high-redshift galaxies (Wolfe & Chen
2006) provide a variety of clues suggesting that gas conversion into stars in low-mass, low-metallicity galaxies is very
inefficient.
The star formation efficiency may depend on the ability toconvert a fraction of gas mass into molecular form. Molecularhydrogen is produced by chemical reactions in the gas phase in
first galaxy halos. In the reionization era, H2 molecule disso1

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

In theoretical studies on the molecular abundance in the interstellar medium (ISM), dust abundance is often scaled with themetallicity, and dust grain properties are assumed to be the sameas in the local ISM. However, the composition of dust is likely to
be different in early galaxies. The observational evidence is thatthe dust extinction curves of the broad absorption line quasars
at z > 4 are likely to be due to the Type II supernova (SN II)
dust (Maiolino et al. 2004; Gallerani et al. 2010).
Since the lifetime of an SN II progenitor is short, SN II
can be the dominant production source of dust grains in young(<1 Gyr) galaxies. Primeval SNe produced by Population III
stars (Bromm et al. 2003; Kitayama & Yoshida 2005; Whalen
et al. 2008) may contribute to dust production (Nozawa et al.
2003; Schneider et al. 2004). The winds of evolved lowmass stars contribute considerably to dust formation in nearby
galaxies, but the cosmic time is not long enough for such stars to
evolve at high redshift (z > 5), where all galaxies should haveages younger than 1 Gyr. Contribution of dust production
by low-mass stars is not dominant in such young galaxies. In
addition, dust is destroyed by SN shocks. Thus, the modelingof dust evolution in galaxies requires an accurate treatment of
production and destruction of dust grains together with starformation activities (Hirashita & Ferrara 2002).
In this paper, we investigate not only the evolution of dust
mass but also the time evolution of dust size distribution. The
dust size distribution evolves rapidly because of the destruction
by sputtering in the high-velocity shocks driven by SNe.
Collision of the expanding SN ejecta with the surrounding ISM
creates a forward shock at the interface between the ejecta andthe ISM (Nozawa et al. 2006), and a reverse shock that penetrates
into the ejecta (Bianchi & Schneider 2007; Nozawa et al. 2007;
Nath et al. 2008; Silvia et al. 2010). Since the erosion rate by
sputtering does not strongly depend on the grain size, small
grains are predominantly destroyed regardless of grain species.
Therefore, the fraction of small-size grains relatively decreases
with galaxy evolution.
We focus on the effects of molecular hydrogen abundanceon the SFR in the early stage of galaxy evolution, taking into
account molecular formation on dust, since H2 formation on
the dust surface is very effective (Hirashita & Ferrara 2002;
Cazaux & Spaans 2004). Hirashita & Ferrara (2002) show that
this effect causes an enhancement of the SFR by an orderof magnitude on a timescale of 3–5 galactic dynamical time.
However, they assumed a single dust grain size (∼0.03 μm).
We adopt more accurate analytic formulae for the formation ofmolecular hydrogen on dust grains than those in Hirashita &Ferrara (2002) by using the results of dust size distribution by
Nozawa et al. (2006, 2007).
This is the first study of galaxy evolution considering dust size
evolution for halo masses above 108 –109 in the high redshift(5 < z < 10), whose interiors we expect to be roughly selfshielded from both ionizing and Lyman–Werner UV radiation.
To clearly show the dependence of galaxy properties on dust
destruction, we use a simple one-zone galaxy model.
The paper is organized as follows: In Section 2, we describe
the dust evolution model. In Section 3, we explain our onezone galaxy model. In Section 4, we present the results. In
Section 5, we discuss the effects of the dust size evolution
on the H2 formation process and conclude by summarizing
our results. Throughout this paper, we adopt the cosmologicalparameters from the third-year Wilkinson Microwave Anisotropy
Probe results (Spergel et al. 2007): ΩΛ = 0.76, ΩM = 0.24,
Ωb = 0.04, and H0 = 73 km s−1 Mpc−1 .

2. DUST EVOLUTION MODEL2.1. Source of Dust in the Early Universe
SNe II are believed to be the dominant sources of dust at ahigh redshift of z > 5 because of the short lifetimes (<107 yr)of their massive progenitors (e.g., Dwek et al. 2007; Gallerani
et al. 2010; Gall et al. 2011). Dust formation in the ejecta ofprimordial SNe II has been investigated theoretically (Todini &Ferrara 2001; Nozawa et al. 2003; Cherchneff & Dwek 2010).
The amount and the size distributions of dust grains injected into
the ISM have been investigated by considering the destructionin SN remnants (SNRs; Bianchi & Schneider 2007; Nozawa
et al. 2007; Nath et al. 2008; Silvia et al. 2010). Although theamount of dust that forms in the ejecta is still debated (Kozasa
et al. 2009, for review), the recent observations of Cas-A SNR
revealed the presence of ∼0.07 M dust condensed in the ejecta(Barlow et al. 2010; Sibthorpe et al. 2010), which is consistent
with the dust mass predicted by the theoretical model taking
into account formation and destruction processes of dust in an
SN IIb (Nozawa et al. 2010). Valiante et al. (2009) and Dwek
& Cherchneff (2011) have proposed that the contribution fromasymptotic giant branch (AGB) stars in the high-redshift quasar
J1148+5251 cannot be neglected for the total dust budget evenat z ∼ 6. However, the size distribution of dust formed in the
mass-loss wind of AGB stars has not yet been fully studied(Ferrarotti & Gail 2006; Zhukovska et al. 2008). If SNe Ia could
occur in such an early epoch, they are unlikely to be efficient
sources of dust (Nozawa et al. 2011). Therefore, in order tofollow the evolution of dust size distribution and reveal the
resulting influence on galaxy evolution, we consider SNe II as
the source of dust in the early universe.
The basic quantity for governing the production and destruction history of dust by SNe II is the rate of SN II explosions,γSN (t), given by muSNdmΨ(t − τ (m))φ(m),(1)γSN (t) =mlSN

where Ψ(t) is the SFR at time t, φ(m) is the stellar IMF, τ (m) is
the lifetime of a star whose mass is m, and muSN and mlSN are theupper and lower mass limits of SN II progenitors, respectively. In
this paper we adopt the Salpeter IMF (φ(m) ∝ m−2.35 ; Salpeter
1955) with the stellar mass range between 0.1 M and 60 M ;we assume mlSN = 8 M and muSN = 40 M (Heger et al. 2003).
For τ (m), we adopt the model of zero-metallicity stars without
mass loss (Schaerer 2002).
In this paper, we do not consider Population III stars, forsimplicity. In the forthcoming paper, we will consider the
possible contribution of Population III stars. Population III
stars formed out of the primordial gas are considered to bemuch more massive than Population I/II stars (Yoshida et al.
2008; Bromm et al. 2009 for reviews); thus, the primordial
IMF might be biased toward a higher mass (10 M ) thanthat in the present universe. Furthermore, Population III stars
as massive as 140–260 M are predicted to end their lives as
pair-instability SNe (PISNe; Heger & Woosley 2002) and to
produce a large amount of metals and dust (Nozawa et al.
2003; Schneider et al. 2004). However, Joggerst et al. (2010a,
2010b) address the growing nucleosynthetic “forensic” evidence
that the majority of primordial stars may have been 15–40 M
objects. On the other hand, once the gas is enriched up to a
critical metallicity of Z  10−6 –10−5 Z , formation of lowmass stars is triggered, leading to the transition of the star
2

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

1
0.1

aMd [Msun]

0.01
0.001

Yamasawa et al.

where M0d,j (a, m)da is the mass of the jth dust species
produced in an SN II with radii between a and a + da as afunction of progenitor mass m, and superscript 0 indicates thecase with no destruction by a reverse shock. In Figure 1, we plota M0d (a) in the vertical axis to make clear the mass fraction ineach logarithmic bin. We can see that the grain radii range froma few Å up to a few μm and that the size spectrum of dust in
mass has a peak at a ∼ 0.1 μm.
In the course of their injection into ISM, dust grains formedin the ejecta are destroyed due to sputtering in the hot gas
between the reverse and forward shocks, which is hereafterreferred to as destruction by the reverse shock. Nozawa et al.
(2007) investigated the survival of the newly formed dust in
the shocked gas within the SNRs expanding into the uniform
ISM with hydrogen number densities of nSN = 0.1, 1.0, and10.0 cm−3 and showed that the destruction efficiency of newly
formed dust is not only sensitive to the initial size distribution but
also strongly depends on nSN . To investigate the dependence ofdestruction of dust on the ISM densities, we extend their models
to six cases of nSN = 0.03, 0.1, 0.3, 1.0, 3.0, and 10.0 cm−3 .
Figure 1 shows the IMF-averaged mass distribution of dust,
Mnd SN (a), injected into the ISM after destruction by the reverse
shock for nSN = 0.03, 0.1, 0.3, 1.0, 3.0, and 10.0 cm−3 , which
is weighted by the Salpeter IMF and is summed up over all thegrain species as in Equation (2). The mass distribution of the jthSNdust species for the case of nSN , Mnd,j(a), is the IMF-averaged
one after the destruction by the reverse shock. We can see that thechange in the dust mass distribution through processing in SNRs
becomes more (less) prominent for higher (lower) gas density;
small-size grains get deficient with increasing nSN . The dust
grains with radii below 0.01 μm are preferentially destroyed by
sputtering for nSN > 0.03 cm−3 , while dust with radii larger
than ∼1 μm is almost intact for nSN  3.0 cm−3 . As a result,the mass of dust injected into the ISM is dominated by grains
with radii above ∼0.1 μm.
The total geometrical cross section of dust per metal mass is
an important quantity for H2 formation on the grain surface ata certain metallicity level. The total geometrical cross sectionof dust injected into the ISM from an SN depends on the gas
density around the SN progenitor, nSN . In many papers, the total
geometrical cross section of dust is scaled to metal mass under
the following two assumptions: (1) the depletion factor, which
is defined as the dust mass per metal mass, is identical to that in
the Milky Way (MW); and (2) the dust size distribution, which
determines dust area per unit dust volume a 2 /a 3 , is the sameas that in the MW. However, the depletion factor and the size
distribution of the SN II dust are quite different from those in
the MW.
nSN
Figure 2 shows the depletion factor, MSN,d/mm , after thedestruction by the reverse shock as a function of ISM density
nSN
nSN , where MSN,dis the IMF-averaged total dust mass ejected
into the ISM by an SN,
 ∞nSN
MSN,d =da Mnd SN (a),
(3)

0
0.03
0.1
0.3
1.0
3.0
10

0.0001
1e-05
1e-06
1e-07
1e-08
1e-08

1e-07

1e-06

1e-05

0.0001a [cm]
Figure 1. IMF-averaged mass distributions of dust formed in the ejecta andinjected into ISM per SN II: the solid line denotes the mass distribution ofdust before the destruction through a reverse shock, a M0d (a); the dashed lines
and the dotted lines denote the mass distributions of dust at the injection inton
ISM after the destruction through a reverse shock, a Md SN (a), for the number
densities of gas around an SN progenitor, nSN = 0.03, 0.1, 0.3, 1.0, 3.0, and10.0 cm−3 , which are used to annotate the curves. The horizontal axis shows
the radius of dust in units of cm. The vertical axis is the mass distribution ofndust a Md SN (a) in units of M .
(A color version of this figure is available in the online journal.)

formation mode from massive Population III stars to low-mass
Population I/II stars, if dust is present (Omukai et al. 2005;
Schneider et al. 2006; Schneider & Omukai 2010; Omukai et al.
2010). If there is no dust, the transition of the star formation
mode is expected to occur at 10−3.5 Z (Mackey et al. 2003).
In this case, the cosmic microwave background (CMB) limits
the lower masses of stars to a few tens of M (Smith et al.
2008; Schneider & Omukai 2010). Formation history of galaxies
considering the time-dependent IMF from the top-heavy to theSalpeter-like IMF and taking into account the production anddestruction of dust by PISNe will be explored in the forthcoming
paper (D. Yamasawa et al. 2011, in preparation).
2.2. Dust Injected from SNe II into ISM
Throughout this paper, we adopt the models by Nozawa
et al. (2003, 2007) for dust formation and destruction. Nozawa
et al. (2003) investigated the dust production in the ejecta ofprimordial SNe II as well as PISNe, applying a theory of nonsteady-state nucleation and grain growth. They revealed thegrain species formed in the ejecta and their size distributions
for the unmixed and mixed elemental compositions within the
He core. In what follows, we apply the results of calculation
for the unmixed ejecta of SNe II with the progenitor mass
m = 13, 20, 25, and 30 M and the explosion energy 1051 erg,
and extrapolate the data to the mass range from 8 to 40 M .
The solid line in Figure 1 shows the IMF-averaged mass
distribution of dust formed in the ejecta of SNe II, M0d (a),which is weighted by the Salpeter IMF and is summed up over
all the grain species as  muSN
0
j mlSN dm Md,j (a, m)φ(m)00,
Md (a) =Md,j (a) =  muSNjj ml dm φ(m)0and mm is the IMF-averaged total metal mass ejected into
the ISM,
mm =

SN


i

(2)
3

  muSN
mm,i =

i

mm,i (m)φ(m)dm
mlSN.
  muSN
i ml φ(m)dm
SN(4)

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

100000

MSN,d / mm

2

Darea / mm [cm g-1]

1

0.1

10000

1000

100
0.1

0.01
0.1

1
nSN [cm ]

1

10

-3

10

nSN [cm ]

-3

n

SN
Figure 3. Ratio of total dust cross section to total metal mass, Darea/mm ,for various dust destruction models with the densities of ISM around the SN
II progenitor, nSN = 0.03, 0.1, 0.3, 1, 3, and 10 cm−3 . The dot-dashed line
0 /m , for the model without reverse shock destruction.
represents the ratio, Daream
The ejected metal mass per SN II is taken from Umeda & Nomoto (2002). The
dotted line represents the ratio in our galaxy, where we assume a dust property asa size distribution, f (a) ∝ a −3.5 (5 × 10−7 cm < a < 2.5 × 10−5 cm) (Mathis
et al. 1977; Draine & Lee 1984) and a dust-to-gas mass ratio, D = 0.00934(Pollack et al. 1994), and that the ratio of metal mass to hydrogen mass is 0.0204
and the bulk density of dust is 3.0 g cm−3 (Omukai 2000).
(A color version of this figure is available in the online journal.)

nSN
Figure 2. Depletion factor, MSN,d/mm , for various dust destruction models with the densities of ISM around the SN II progenitor, nSN =−30.03, 0.1, 0.3, 1, 3, and 10 cm . The dot-dashed line represents the deple0
tion factor, MSN,d/mm , for the model without reverse shock destruction. The
ejected metal mass per SN II is taken from Umeda & Nomoto (2002). The
dotted line represents the typical ratio in MW, where we assume the dust-to-gas
mass ratio to be D = 0.00934 (Pollack et al. 1994) and the ratio of metal mass
to hydrogen mass to be 0.0204 (Omukai 2000).
(A color version of this figure is available in the online journal.)

Here, mm,i (m) is the mass of the ith element of metal ejectedfrom an SN with progenitor m, taken from Umeda & Nomoto(2002). The depletion factor in the case of nSN = 1.0 cm−3 is afactor of six smaller than that in the MW.
Figure 3 shows the ratio of the total geometric cross sectionnSNof dust to the total metal mass, Darea/mm , after the destruction
by the reverse shock as a function of ISM density nSN . The total
geometrical cross section of dust weighted by the Salpeter IMF,
nSN
Darea
, is written as 3  ∞
nSN
SN
Darea
=
da Mnd,j(a)/a,
(5)
4ρ
j
0
j

investigated the processing of interstellar dust by sputtering in
the hot gas swept up by the SN forward shock. Adopting thedust model by Nozawa et al. (2003) as the size distributionof interstellar dust, Nozawa et al. (2006) have shown that thedestruction efficiency of dust depends on the ISM density and the
explosion energy of SNe as well as the initial size distributionof dust. It should be noted that the size distribution and thedestruction efficiency change as functions of time becauseinterstellar dust is supplied from SNe and processed in ISM
successively according to star formation activity. Thus, we must
deal with the destruction process in a way that is applicable toany dust size distribution to explore the global evolution of dust
size distribution.
In order to evaluate the destruction efficiency of interstellar
dust for any initial size distribution, here we introduce theconversion efficiency as defined below. Consider that the jth
dust species residing in the ISM, whose size distribution is givenby the number of dust grains with radii between a and a + da,
fj (a)da, is processed by sputtering in hot plasma produced
through a single passage of an SN shock. The conversion
efficiency ηj (a, a ) is defined as the number fraction of dust
grains with radii between a and a + da that are converted tograins with radii between a and a + da by sputtering through thepassage of an SN shock. The number of dust grains with radii
between a and a + da produced by the sputtering is given as
ηj (a, a )fj (a )da . Note that ηj (a, a ) = 0 for a > a . Then thechange in the number of dust grains with radii between a anda + da caused by shock processing is given by

where ρj is the bulk density of the jth dust species. The ratio,
nSN
Darea
/mm , is smaller for larger nSN because small dust grains are
efficiently destroyed by sputtering in the SNR under large nSN ;note that the surface area per dust mass is larger for smaller size
grains. Also with increasing nSN , the reverse shock becomesstronger and destroys dust by sputtering more effectively (seeNozawa et al. 2007). In addition, we plot the ratio for the casewithout reverse shock together with the typical value in the MW
nSN
for comparison. Darea/mm in the model without reverse shocknSN
is a factor of four smaller than that in the MW. Darea/mm in the−3case with nSN = 1.0 cm is 40 times smaller than that in the
MW. Thus, the rescaling of the cross section of dust by the metal
mass using the MW value results in significant overestimate forH2 formation in high-redshift galaxies.

dNj (a) =2.3. Destruction of Interstellar Dust by SN Forward Shocks
Dust grains injected into the ISM are subjected to destruction
by the blast waves (the high-velocity interstellar shocks) driven
by the ambient SNe (e.g., Jones et al. 1994). Nozawa et al. (2006)

=

∞


0

4

ηj (a, a )fj (a )da − [1 − ηj (a, a)]fj (a)da

a >a
 ∞

ηj (a, a )fj (a )da − fj (a)da.

(6)

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

sities, where the initial mass distribution of dust is set to bea Md,j (a) = 1 for clarity. As seen in the figure, small-size
grains are destroyed due to the erosion by sputtering, and moredust grains are processed for a higher ISM density.
The mass of gas swept up by the forward shock until thetruncation time ttr , Mswept , depends not only on the ISM density
but also on the initial metallicity of the gas in the ISM. As
the line cooling by heavy elements becomes more efficient fora higher gas metallicity, the forward shock is decelerated more
quickly, resulting in a smaller Mswept . By fitting Mswept calculated
for different nSN and Z, we derive the following approximation
formula,aMd [Msun]

10

1

initial
0.03
0.1
0.3
1.0
3.0
10

0.1

1e-08

1e-07

1e-06 1e-05
a [cm]

Mswept /M = 1535n−0.202[(Z/Z ) + 0.039]−0.298 ,SN(8)whose fitting accuracy is within 16% for 0.03 cm−3  nSN 
30 cm−3 and for 10−4  Z/Z  1.0.
2.4. Formulation of Dust Size Evolution

0.0001

In terms of the conversion efficiency describing the processing of dust by sputtering, here we formulate the time evolutionof the mass of jth dust grains with radii between a and a + Δa
in our model galaxies, ΔMd,j (a, t) = 4π
a 3 ρj fj (a, t)Δa, as
3

Figure 4. Dust mass, j a Md,j (a), processed by a forward shock driven by
an SN II explosion in the ambient ISM gas density of nSN = 0.03, 0.1, 0.3, 1.0,
3.0, and 10.0 cm−3 , which are used to annotate the curves, and metallicity of
Z = 10−4 Z . We assume the initial size distribution to be a Md,j (a) = 1 foreach species j. In the calculations, we use the conversion efficiency ηj (a, a )evaluated for SN explosion energy, 1051 erg, and progenitor mass, m = 20 M .
(A color version of this figure is available in the online journal.)dΔMd,j (a, t)= ΔMSN,d,j (a)γSN (t)dtMsweptγSN (t) × ΔMd,j (a, t)−
MISM (t)

 ∞
4π 3
a
da ηj (a, a )fj (a , t)ρj−
3
0
ΔMd,j (a, t)− Ψ(t),(9)
MISM (t)The corresponding change of the mass is given by
 ∞
4π 3
a ρj
dMd,j (a) =ηj (a, a )fj (a )da − Md,j (a)da,3
0(7)
where Md,j (a)da is the mass of the pre-shocked dust. The size
distribution function after the shock processing fj (a) is given
by fj (a) = fj (a) + dNj /da.
The conversion efficiency η(a, a ) and the mass of ISM gas
swept up by shock Mswept depend on the progenitor mass,
expanding energy, and type of the SN as well as the structure,
number density, and metallicity of the ISM gas. For theseparameters of SNe and ambient ISM, once Mswept and η(a, a ) for
each dust species are calculated, the time evolution of dust massand size distribution can be followed in a consistent way withthe star formation activity in galaxies as described in Section 3.
The calculations of η(a, a ) and Mswept are performed using
the method developed by Nozawa et al. (2006) as follows: theefficiency of dust destruction increases with increasing explosion energy and/or increasing nSN but is almost independent of
the progenitor mass as long as the explosion energy is the same
(Nozawa et al. 2006). We assume that SNe driving high-velocity
shock in ISM are represented by SN II with the progenitor massof 20 M and the explosion energy of 1051 erg. The ISM surrounding the SN is considered to be uniform with hydrogen
number densities nSN = 0.03, 0.1, 0.3, 1.0, 3.0, and 10 cm−3 .
By distributing dust grains with radius a uniformly in the ISM,
the conversion efficiency η(a, a ) is evaluated for each grain
species by calculating the erosion of dust by sputtering until thetruncation time ttr , which is defined as a time when the shock
velocity is decelerated below 100 km s−1 (see Nozawa et al.
2006, for the details). In the calculations, the radii of grains in
the ISM range from 0.00013 to 6.3 μm for each grain species.
In Figure 4, we present the changes in the dust size distributions due to the shock processing for different ISM den-where MISM (t) is the total mass of gas and dust, and fj (a, t)
is the size distribution function of dust species j in the ISM at
a time t. We note that (Mswept γSN (t)/MISM (t))−1 is a timescaleof sweeping the whole ISM by SNe. The IMF-averaged massof dust species j with radii between a and a + Δa injected fromSNe II into the ISM with number density nISM,SN is defined by
SN
ΔMSN,d,j (a) = Mnd,j(a)Δa. The first term on the right-handside is the injection rate of dust from SNe II. The second termis the destruction rate of interstellar dust by SN blast waves,and the third term is the rate at which the interstellar dust isincorporated into stars.
3. GALAXY EVOLUTION MODEL3.1. Dark Matter Halo and Physical State of Gas
We quantify the properties of dark matter halos, assuming a
dynamically equilibrium state. The radius of dark matter halo,
rvir , is estimated in terms of the mass of dark halo, Mvir , and theredshift of virialization, zvir , as
4 3
π r {1 + δc (zvir )} ρc0 ΩM (1 + zvir )3 = Mvir ,
3 vir

(10)

where ρc0 ≡ 3H02 /8π G is the critical density of the universe atz = 0, δc (zvir ) is the overdensity of a dark matter halo virialized
at zvir , and G is the gravitational constant.
We assume dark matter halos as singular isothermal spheres
and rotating uniform gas disks in their gravitational potentials.
Cosmological N-body simulations show that structures of dark
5

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

for H /rdisk  0.1; otherwise, we assume H /rdisk = 0.1, where1
vs = (kB T /μmH ) 2 is the isothermal sound velocity. Therefore,
the initial hydrogen density of disk, nH , is estimated as

matter halos are described well by the Navarro–Frenk–White(NFW) profile (Navarro et al. 1996). Mo et al. (1998) studied a
simple disk model in the gravitational potential of the singularisothermal sphere and a more realistic disk model in thegravitational potential of the NFW halo profile. We adopt aradius of the disk, rdisk  0.18rvir (Ferrara et al. 2000; Hirashita& Ferrara 2002), by considering the conservation of angularmomentum and assuming a typical value for the spin parameter
λ = 0.04 from the paper by Ferrara et al. (2000), who estimate
the radius of the disk as rdisk = 4.5λrvir in a modified isothermalhalo.
In our one-zone model, we need a virial temperature for the
initial gas temperature and a dynamical timescale of gas in the
disk. Gas collapsed at zvir in the dark matter halo of Mvir has avirial temperature, Tvir , defined asTvir ≡

GμmH Mvir,
3kB rvirnH =MH = Mgas − MHe = Mgas
=

vc ≡

(11)

,

(12)

(13)

Ψ(t) =

Note that the rotation timescale of the gas disk, tcir , depends
only on virialization redshift, zvir , as

11
tcir = 9.3 × 10 yr
1 + zvir

3
2



18π 2
δc (zvir )

1
2

.



2T
3Tvir

(14)

(18)

3.3. Evolution of Gas, Stars, and Metals
We calculate the time evolutions of the masses of hydrogenand helium gases, Mgas , stars, Mstar , and metal of element i,
Mm,i , in the galaxy using the following equations:dMgas (t)Mgas (t)= −Ψ(t)+ mgas γSN (t)dt
MISM (t)dMstar (t)= Ψ(t) − mejecta γSN (t)dtdMm,iMm,i (t)= −Ψ(t)+ mm,i γSN (t),(19)
dt
MISM (t)where MISM (t) = Mgas (t) + i Mm,i (t), mejecta = mgas +
i mm,i and mgas and mm,i are the gas mass of hydrogenand helium and the metal mass of element i in SN ejecta,
respectively. The mass returning to the ISM, mejecta (m), mgas (m),and mm,i (m) through an SN with progenitor mass, m, is takenfrom Umeda & Nomoto (2002) in the case of m = 13, 20, 25
and 30 M . mgas , mejecta , and mm,i are IMF-averaged mgas (m),mejecta (m), and mm,i (m), respectively. Note that metals consist
not only of heavy elements in gas phase but also those in dust
grains.

1
2

rdisk

fH2 (t)MH (t),tcir (zvir )where fH2 is the mass fraction of molecular hydrogen to the total
gas. We should note that observationally Bigiel et al. (2008) findthat H2 is converted into stars at a constant efficiency in nearby
spirals, and Gnedin et al. (2009) show that the star formation
recipe in a galaxy formation simulation in which star formation
occurs only in the molecular gas can reproduce the observationalcorrelations between SFR and the total gas density.

We must estimate the number density of the hydrogen gas,nH , because it affects both the chemical reaction rate and thecooling rate. The cooling time of halo gas is much shorter
than the Hubble timescale for the objects of interest in thispaper (Tvir  10000 K; e.g., Hutchings et al. 2002). It is widely
understood that most z ∼ 10 galaxies are not clear disk galaxies;
numerical simulations that proceed from cosmological initialconditions (z ∼ 100–200) clearly reveal that they possess highly
irregular structures whose SFRs are not easily quantifiable, thatfilamentary accretion and frequent mergers are still churning thehalo at this epoch, and that turbulent flows that arise in the centerof the halo prevent coherent disks from forming on the spatial
scales of galaxies (Johnson et al. 2008; Greif et al. 2010; Wiseet al. 2010). We assume for simplicity that a significant fractionof baryons finally collapses to a disk in the dark matter halopotential. In semi-analytic models, it is assumed that the cooled
halo gas settles into the disk (e.g., Cole et al. 2000). We make asimilar assumption in our model, but a more detailed treatment
for H2 formation, dust evolution, and star formation in the gas
disk. The radius of disk, rdisk , is determined following Hirashita& Ferrara (2002). We estimate the typical scale height, H, fromhydrostatic equilibrium (Shakura & Sunyaev 1988)√ vs
H = 2 rdisk =
vc

(17)

−1
We expect that the SFR, Ψ(t), is roughly proportional to tcir,
since a representative timescale of the dynamics of the gas disk
is tcir . We assume that

2π rdisk
.
vc

7

mH
Mvir Ωb,ΩM (mH + mHe yHe )3.2. Star Formation Law

and we also define a rotation timescale, tcir , astcir ≡

mH(mH + mHe yHe )where MHe is the initial mass of helium in the galaxy, Mgas =MH + MHe , mHe is the mass of a helium atom, and yHe is thehelium abundance. We assume yHe = 0.0972 (Omukai 2000).
Note that since massive stars ionize surrounding gas and form an
expanding H ii region (e.g., Whalen et al. 2004; Kitayama et al.
2004), we adopt gas density around the SN progenitor, nSN , asbeing different from the hydrogen gas density in our one-zone
galaxy model, nH . This is because the gas density around the SN
progenitor, nSN , is closely related to the dust destruction processby reverse shocks driven by SNe, as shown in Section 2.2.

1
2

GM virrvir(16)

The initial mass of hydrogen in the galaxy, MH , is written as

where kB is the Boltzmann constant, μ is the mean molecularweight, and mH is the mass of a hydrogen atom. The initial value
for the temperature of gas, T, is assumed to be Tvir . A circularvelocity, v c , is defined as


MH
.
2
π rdisk
2H mH

(15)
6

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

Table 1
Reaction Rates Needed to Calculate the Abundance of H2
No.

Reaction1

H + e− −→ H+ + 2e−2

H+ + e− −→ H + γ

34

H + e− −→ H− + γ
H− + H −→ H2 + e−

5

H− + H+ −→ 2H

6

H + H+ −→ H+2 + γ

789

H+2 + H −→ H2 + H+
H+2 + e− −→ 2H
H2 + H+ −→ H+2 + H

10

H2 + H −→ 3H

11
Dust

H2 + e− −→ 2H + e−
H + H + grain −→ H2 + grain

Rate(cm3 s−1 )Ref.

exp[−32.71 + 13.54 ln(T (eV))−5.739(ln(T (eV)))2 + 1.563(ln(T (eV)))3−0.2877(ln(T (eV)))4 + 3.483 × 10−2 (ln(T (eV)))5−2.632 × 10−3 (ln(T (eV)))6+1.120 × 10−4 (ln(T (eV)))7−2.039 × 10−6 (ln(T (eV)))8 ]exp[−28.61 − 0.7241(ln(T (eV)))−2.026 × 10−2 (ln(T (eV)))2−2.381 × 10−3 (ln(T (eV)))3−3.213 × 10−4 (ln(T (eV)))4−1.422 × 10−5 (ln(T (eV)))5+4.989 × 10−6 (ln(T (eV)))6+5.756 × 10−7 (ln(T (eV)))7−1.857 × 10−8 (ln(T (eV)))8−3.071 × 10−9 (ln(T (eV)))9 ]1.4 × 10−18 T 0.928 exp(−T /1.62 × 104 )4.0 × 10−9 T −0.17 (T > 300);
1.5 × 10−9 (T < 300)
5.7 × 10−6 T −1/2 + 6.3 × 10−8
−9.2 × 10−11 T 1/2 + 4.4 × 10−13 Tdex[−19.38 − 1.523 log10 T+1.118(log10 T )2 − 0.1269(log10 T )3 ]6.4 × 10−10
2.0 × 10−7 T −1/2
3.0 × 10−10 exp(−21050/T ) (T < 104 )1.5 × 10−10 exp(−14000/T ) (T > 104 )
1−a a
kH
kL
kL = 1.12 × 10−10 exp(−7.035 × 104 /T )kH = 6.5 × 10−7 T −1/2
× exp(−5.2 × 104 /T )[1 − exp(−6000/T )]a = 4.0 − 0.416 log10 (T /104 ) − 0.327(log10 (T /104 ))2
4.4 × 10−10 T 0.35 exp(−1.02 × 105 /T )See Section 3.5

1

1

1
1
1
1
1
1
2
1

1

Note. The unit of the gas temperature T is K unless otherwise stated.
References. (1) Omukai 2000; (2) Galli & Palla 1998.

ionization, recombination, and photoionization. Next, the time
evolution of the molecular fraction is written as

3.4. Chemistry and Cooling
We follow the time evolution of molecular mass fraction,
fH2 , ionization degree, x, and gas temperature, T. We define themolecular fraction of hydrogen as
2nH2
,
fH2 ≡
nH

dfH2
dt

dfH2
=
dt

+
gas

dfH2
+
dt

(20)

where nH2 and nH are the number densities of molecularhydrogen and hydrogen nuclei, respectively. The molecular
fraction is very important in our models, because it determines
the final cooling rate of low-metallicity gas. The metal-free gas
evolution with chemical reactions and cooling is studied using
the model by Tegmark et al. (1997), Hutchings et al. (2002), andHirashita & Ferrara (2002). In Table 1, we summarize chemical
reactions considered in this paper and their rate coefficients(Rn ; n = 1, . . . , 11). The equations are based on Hirashita& Ferrara (2002), but we include the effect of the dust size
distribution on H2 formation and the metal-line cooling process.
The time evolution of the ionization degree is described as

dfH2
dt

dfH2
dt

+
dust

dfH2
+
dt
UV

dest

,

(22)

star

where the terms on the right-hand side are the H2 formation
rate in the gas phase, the H2 formation rate on dust grains, thedestruction rate in the gas phase, the destruction rate by UVphotons, and the decreasing rate by star formation, respectively.
These terms are given by
dfH2
dt= 2f02 xnH (Reff,1 + Reff,2 ),
gas

dfH2
= 2Rdust DnH f0 ,dt dustdfH2= −fH2 nH (x 2 Reff,3 + f0 R10 + xR11 ),dt destdfH2= −Γ13 fH2 ,
dt UV

dx
= xf0 R1 nH − x 2 R2 nH + Γ12 f0 ,(21)
dt
where f0 = 1 − x − fH2 is the neutral fraction of hydrogen.
The terms on the right-hand side are the rates of collisional
7

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

and
dfH2
dt

= −(1 − fH2 )Ψ(t)star

1
MISM (t).

where v̄ is the mean thermal speed of hydrogen and S is the
sticking efficiency of hydrogen atoms. We assume that the gas
follows a Maxwellian distribution so that thermal speed is givenby (Krügel 2008)1

8 kB T 2
.
(31)
v̄ =
π mH

(23)

The effective formation rates of H2 including the effect of thedestruction rate of H− and H+2 are
Reff,1 ≡andR 3 R4f0 R4 + xR5 + Γ14 /nH

R 6 R7
Reff,2 ≡,f0 R7 + xR8 + Γ15 /nH

Here, we define the dust-to-gas mass ratio, D, as  ∞ 4π a 3 ρj fj (a)D≡da.
3nH mH
0
j

(24)

(25)

The reaction rate of the H2 formation on grains, Rdust , can be
estimated as  ∞  3mH v̄S  4π a 3 ρj fj (a)Rdust (a)D =da. (33)8aρj
3nH mH0jrespectively, and the destruction of H+2 due to H− collision isReff,3 ≡R 8 R9.
f0 R7 + xR8 + Γ15 /nH(26)

We will give the dust-to-gas mass ratio, D, and the production
rate of molecular hydrogen via dust surface reaction, Rdust , in
Section 3.5 and reaction rates of the photo-process, Γn (n =12, . . . , 15), in Section 3.6.
At temperature <104 K, the main coolant is molecularhydrogen in low-metallicity gas. The cooling rate for molecularhydrogen, ΛH2 , over the range 10 K  T  104 K is given by(Galli & Palla 1998)
ΛH2 (T )2= −103.0 + 97.59Tlog − 48.05Tlog
log10
nH nH2 erg cm3 s−1

We adopt S = 0.2 for T < 300 K and S = 0 for T > 300 K(Hirashita & Ferrara 2002).
3.6. Radiative Properties
We follow photo-processes in chemical reactions and heatingprocesses in thermal evolution by using the models developedby Kitayama & Ikeuchi (2000) and improved in Hirashita &Ferrara (2002). The intrinsic luminosity is assumed to be equal
to the total luminosity of OB stars whose mass is larger than
3 M (Cox 2000):
 ∞
 τm
dm
dt L(m)φ(m)Ψ(t − t ),(34)
LUV,0 (t) =3
4+10.80Tlog− 0.9032Tlog, (27)

where Tlog ≡ log10 (T /K). Glover & Abel (2008) have recently
given the H2 cooling rates, which include H − H2 collision andH2 −H2 collision pathways, while the Galli & Palla (1998) rates
include only H − H2 collisions. We assume that the lower limit
of gas temperature is the CMB temperature.
At temperature T  104 K, collisional excitation, ΛH,ce , and(less importantly) ionization of atomic hydrogen, ΛH,ci , are more
dominant cooling processes than molecular hydrogen coolingand are given by (Haiman et al. 1996)
ΛH,ce (T )1
− 1.183
T5= 7.50 × 10−19
1 exp
3
−1
ne− nH erg cm s
2
1+T

3 M

0

where L(m) is the stellar luminosity as a function of stellar
mass m. For L(m), we adopt the model of zero-metallicity stars
without mass loss in Schaerer (2002). We assume the spectrumof the incident UV radiation from stars is a power law with anindex α:ν −αIUV (ν) = I0 (νH i )
,
(35)
νH i

(28)

where ν is the frequency of photons and I0 (νH i ) is the intensity
at the ionization threshold frequency of neutral hydrogen νH i =3.3 × 1015 Hz. In this paper, we simply set α = 5 according to
Hirashita & Ferrara (2002). The normalization of the intensity
is determined by ∞LUV,0 exp(−τdisk )=IUV (ν)dν,(36)
24π rdisk
νmin

5

and
1

T2
ΛH,ci (T )− 1.578= 4.02 × 10−19 5 1 exp T5 ,3−1
ne− nH erg cm s
1+T 2

(32)

(29)

5

respectively, where T5 is the gas temperature in units of 105 K.
We consider fine-structure cooling by CI , CII , and OI , which
dominates the thermal evolution for the number density of gas
of interest in this paper (Omukai et al. 2005). The relatedparameters of transitions are given in Hollenbach & McKee(1989).

where νmin is the minimum frequency where OB stars dominate
the radiative energy of star-forming galaxies, and τdisk is the
typical dust optical depth in the disk. We assume that νmin =1015 Hz. This typical optical depth can be simply estimated as ∞τdisk = rdisk
π a 2 fj (a)da(37)

3.5. Formation of Molecular Hydrogen on Dust Grainsj

The increasing rate of molecular fraction via dust surface
reaction is estimated as
 ∞
dfH2
= 2Rdust DnH f0 =f0 fj (a)π a 2 v̄Sda,
dt dust0j0

by assuming that extinction efficiency of dust is unity in UV. We
calculate Γ12 , Γ14 , and Γ15 from Equation (A20) of Kitayama& Ikeuchi (2000) and the heating rate from Equation (A21)of Kitayama & Ikeuchi (2000). We summarize the cross section
for the photo-process in Table 2. The H2 photodissociation cross(30)
8The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

Table 2
Cross Sections for the Photoionization and Photodissociation Process, where the Frequency, ν, is in Units of Hz
No.

Reaction

Cross Section(cm2 )

ν Range
(Hz)

Ref.

12
13
14

H + γ −→ H+ + e−
H2 + γ −→ H∗2 −→ 2H
H− + γ −→ H + e−

ν > 3.3 × 1015
ν > 1.8 × 1014

1
2
3

15

H+2 + γ −→ H + H+

6.30 × 10−18 (ν/3.3 × 1015 )−3.0
see Equation (38)3.486 × 10−16 (x − 1)3/2 /x 3.11
(x ≡ ν/1.8 × 1014 )7.401 × 10−18dex(−x 2 − 0.0302x 3 − 0.0158x 4 )(x ≡ 2.762 ln(ν/2.7 × 1015 )ν > 6.4 × 1014

3

References. (1) Kitayama & Ikeuchi 2000; (2) Abel et al. 1997; (3) Tegmark et al. 1997.

section is given by Abel et al. (1997). However, if the H2 column
density becomes larger than 1014 cm−2 , self-shielding effects
become important (Draine & Bertoldi 1996). Therefore, the H2dissociation rate, Γ13 , is given by (Hirashita & Ferrara 2002)
Γ13 = (4π )1.1 × 10 IUV (3.1 × 10
nH fH2 rdisk −0.75 −1
×s ,1014 cm−2
815Table 3
The Dust Destruction Model and the Main Parameters
Model
C1m9
B1m9
A1m9 (fiducial)
A0.03m9
A0.1m9A0.3m9A3m9
A10m9
A0.1m8A0.1m10A0.1m11
A1m8
A1m10
A1m11
A10m8
A10m10
A10m11Hz)(38)

where IUV (3.1 × 1015 Hz) is in the Lyman–Werner band. We
should note that adoption of rdisk in Equation (38) gives
an extreme upper bound to the self-shielding, so we may
overestimate self-shielding to internal Lyman–Werner photons
by H2 . In this paper, we focus on the effects of dust on the
protogalaxy, so for simplicity, we set an extreme upper boundto the self-shielding.
We do not consider the Lyman–Werner background, since
we concentrate on the evolution of atomic line cooling halos(Mvir > 108 M in z < 10) in which destruction of molecularhydrogen by the Lyman–Werner background is less efficient(O’Shea & Norman 2008; Susa 2008; Wise & Abel 2008; Wise
& Cen 2009). However, 108 –109 M halos are not immune to
the Lyman–Werner background, just self-shielded at their very
centers. Not all the baryons will be protected from external
photodissociating flux, and this will affect H2 production on
dust outside the center of halo. In the lower mass halos, theLyman–Werner background may be effective to dissociate H2
molecules (Machacek et al. 2001, 2003; Yoshida et al. 2003;
Susa 2007; Wise & Abel 2007; O’Shea & Norman 2008).

Dust DestructionnSN(cm−3 )

Mvir
(M )

No destruction
Forward shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks
Forward and reverse shocks

···
1
1
0.03
0.1
0.3
3
10
0.1
0.1
0.1
1
1
1
10
10
10

109
109
109
109
109
109
109
109
108
1010
1011
108
1010
1011
108
1010
1011

4.1. The Dust Destruction
We first show the evolution of a galaxy with Mvir = 109 M
and zvir = 10 for various dust destruction models. To clarify
the dust destruction effects on galaxy evolution, we first show
the result of the model without reverse and forward shocks(model C1m9), then compare the results of the models with
only forward shocks (model B1m9) and with both forward andreverse shocks (model A1m9) to the model without both shocks(model C1m9).
In Figure 5, we show the evolution of various quantities
without dust destruction (model C1m9). The figure shows thetime evolution of the molecular fraction, fH2 ; the SFR in unitsof M yr−1 , Ψ; the stellar mass fraction, Mstar /(Mgas + Mstar );the metallicity, Z, in units of Z ; total dust cross section perunit volume, σd,−20 , in units of 10−20 cm−1 ; and the dust-to-gas
mass ratio, Dd,−2 , in units of 10−2 . The definition of Dd,−2 isconvenient for comparison with the MW value of the dust-to-gas
mass ratio. In the MW, the dust-to-gas mass ratio is 0.5 × 10−2in the diffuse ISM (Draine 2009) and 0.9 × 10−2 in molecular
clouds (Pollack et al. 1994).
The molecular fraction, fH2 , is very important, since it
determines SFR and controls galaxy evolution. The molecular
fraction reaches fH2 ∼ 1 × 10−3 around t ∼ 107 yr. This result
is robust for all models, since in this stage H2 formation in
the gas phase is dominant over that on dust grains (Tegmark
et al. 1997; Hirashita & Ferrara 2002). The gas temperature

4. RESULTS
Our fiducial model assumes Mvir = 109 M and zvir = 10
and includes the dust destruction model by both reverse shocks
and forward shocks with the ISM density around the SN
progenitor, nSN = 1 cm−3 (see Table 3 for a summary of
our models). This dark matter halo forms from a 2.5σ density
fluctuation. We stop the calculation at z = 5. It corresponds
to the galaxy age of ∼0.8 Gyr; that is, before SN II becomesa dominant source of dust grains. The initial mass of gas is
Mgas = 1.7 × 108 M and the dynamical timescale of circularmotion of the gas disk is tcir (zvir = 10) = 9.5 × 107 yr. Notethat galaxies with Mvir ∼ 109 M play a critical role in thecosmic reionization, since in the relevant redshift range forcosmological reionization, z = 6–15; most of the reionization
radiation is expected to come from galaxies with masses lessthan ∼109.5 M (Wise & Cen 2009).
9

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.
10

10

1

fH

2

fH

Ψ
Mstar/(Mgas+Mstar)ΨMstar/(Mgas+Mstar)2

1Z
σd,-20
0.1

Dd,-2

0.1

0.01

0.01

0.001

0.001

0.00010.0001

1e-05
1e+07

1e+08
yr

Z
σd,-20
Dd,-2

1e-05
1e+07

1e+09

Figure 5. Time evolution for model C1m9, in which dust destruction by thereverse and forward shocks is not considered. We show molecular fraction,
fH2 ; SFR in units of M yr−1 , Ψ; star-to-gas mass ratio, Mstar /(Mgas + Mstar );metallicity, Z, in units of Z ; total dust cross section per unit volume, σd,−20 , inunits of 10−20 cm−1 ; and dust-to-gas mass ratio, Dd,−2 , in units of 10−2 . The
horizontal axis shows the age of the galaxy in units of yr from the formation
redshift, zvir = 10.
(A color version of this figure is available in the online journal.)

1e+08
yr

1e+09

Figure 6. Same as Figure 5 but for model B1m9, in which only the dust
destruction by the forward shocks is considered.
(A color version of this figure is available in the online journal.)
10
fH

2

1

Ψ
Mstar/(Mgas+Mstar)Z
σd,-207

rapidly drops below 200 K before 10 yr. Then, the molecular
fraction rapidly increases from t ∼ 108 yr and reaches ∼0.83
at the galaxy age of ∼0.8 Gyr (z = 5). This is due to theenhancement of H2 formation on dust grains by an increaseof σd,−20 . For t  3 × 107 yr, σd,−20  1.1 × 10−4 and[dfH2 /dt]dust exceeds [dfH2 /dt]star . For σd,−20  0.001, theincrease of the molecular fraction enhances the star formation.
The cycle of the H2 formation on dust, the star formation, and thedust formation by SNe significantly accelerates galaxy evolutionby, for example, rapidly increasing the stellar mass fraction,
Mstar /(Mgas + Mstar ). At the galaxy age ∼0.8 × 109 yr, the stellar
mass fraction goes up to Mstar /(Mgas + Mstar ) ∼ 0.60. The SFR,
Ψ(t), decreases from the time when Mstar /(Mgas +Mstar ) ∼ 0.45,
since gas mass decreases significantly. The active star formation
causes the formation of dust grains and metals. At t ∼ 0.8 Gyr,the total dust cross section, σd,−20 , goes up to 2.3 and themetallicity, Z, goes up to 3.3 × 10−1 Z .
Figure 6 shows the results of the galaxy model with thedust destruction by only the forward shocks (model B1m9)
to illustrate the effects of dust destruction by forward shocks
on the galaxy evolution. In this model nSN = 1.0 cm−3 .
The dust destruction by forward shocks slightly affects thedust-to-gas mass ratio, Dd,−2 , after the galaxy age of ∼5 ×
108 yr. This is because the destruction by forward shocks
is roughly proportional to the dust-to-gas mass ratio (seeEquation (9)), and dust grains are destroyed significantly for
Dd,−2  0.1 in this case. The SFR decreases from the time
when Mstar /(Mgas + Mstar ) ∼ 0.4. At ∼0.8 Gyr, the molecular
fraction reaches fH2 ∼ 0.51 and the stellar mass fraction reaches
Mstar /(Mgas + Mstar ) ∼ 0.47.
In Figure 7, we show the results of our fiducial modelA1m9 in which we include the dust destruction by both reverseand forward shocks in the case of nSN = 1 cm−3 . The

0.1

Dd,-2

0.01

0.001

0.0001

1e-05
1e+07

1e+08
yr

1e+09

Figure 7. Same as Figure 5 but for model A1m9, in which the destruction by
reverse and forward shocks in the case of nSN = 1 cm−3 is considered.
(A color version of this figure is available in the online journal.)

molecular fraction reaches fH2 ∼ 1 × 10−3 around t ∼ 107 yr.
This is similar to C1m9 and B1m9. After t  107 yr, themolecular fraction evolution is quite different from models
C1m9 and B1m9. The molecular fraction declines slowly untilt ∼ 2 × 108 yr. After ∼2 × 108 yr, the molecular fraction
increases slowly with increase of the dust mass. This is dueto the H2 formation on the dust grains. The molecular fraction
reaches only ∼2.0 × 10−3 at t ∼ 0.8 Gyr. This is because
dust destruction by reverse shocks is very effective and hence
results in suppression of H2 formation on dust grains. On theother hand, forward shocks hardly affect the evolution of dust
size and dust mass, since the destruction of forward shocks
10The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.
1

1000

no reverse
0.03
0.1
0.3
1
3
10

fH

2

σd,-20

100

gas
dust

Ψ [ Msun / yr ]

10

0.1

UV
star

1

0.01

0.001

0.1

0.01

0.0001
1e+07
0.001

1e+09

yr

Figure 9. Time evolution of SFR in units of M yr−1 in the model with dust
destruction for various nSN = 0.03, 0.1, 0.3, 1.0, 3.0, and 10 cm−3 . The valuesof nSN are the same as those in Figure 1. The model without dust destruction
by the reverse shocks is also shown. The horizontal axis shows the age of thegalaxy in units of yr from the formation redshift, zvir = 10.
(A color version of this figure is available in the online journal.)

0.0001

1e-05
1e+07

1e+08

1e+08

1e+09

yr

Figure 8. Formation and destruction rates of H2 in model A1m9. The evolutionof the molecular formation rate in the gas phase (gas); the molecular formation
rate on dust grains (dust); the molecular destruction rate by UV photons (UV);
and the molecular decreasing rate by star formation (star), normalized to thetotal formation rate, are shown. The molecular formation on dust grains becomesdominant, compared with the other channels, after t ∼ 1.6 × 108 yr. The
molecular fraction, fH2 , and total dust cross section per unit volume in units of
10−20 cm−1 , σd,−20 , are the same as in Figure 7.
(A color version of this figure is available in the online journal.)

less than the H2 decreasing rate by star formation at this epoch.
At t ∼ 1.6 × 108 yr, the H2 formation rate on dust grainsexceeds the H2 decreasing rate by star formation. The formation on dust grains becomes the dominant process among
all of H2 formation and destruction processes. From this time
when σd,−20  5.2 × 10−5 , molecular fraction, fH2 , starts
to increase. The molecular destruction rate by UV photons
does not exceed the molecular decreasing rate by the starformation after t ∼ 4 × 107 yr. After t ∼ 4 × 107 yr,
IUV (3.1×1015 Hz) ∼ 2.0–4.8×10−20 erg s−1 cm−2 Hz−1 ster−1 ,which corresponds to J21 = 20–40 where J21 is in units of
IUV (3.1 × 1015 Hz) = J21 × 10−21 erg s−1 cm−2 Hz−1 ster−1 .
Note that J21 in this model is higher than the values, J21 ∼ 1, for
the Lyman–Werner background in the redshift of 5 < z < 10suggested in recent papers (e.g., Greif & Bromm 2006). The destruction process in the gas phase does not affect the evolutionof molecular fraction significantly after t  1 × 107 yr.

can change dust mass only for a large dust-to-gas mass ratio,
Dd,−2  10−1 . At t ∼ 0.8 Gyr, the stellar mass fraction reaches
only Mstar /(Mgas + Mstar ) ∼ 4.5 × 10−3 , which is much less thanthe model without reverse shocks shown in Figure 5 (modelC1m9) and 6 (model B1m9).
We illustrate the difference in the H2 formation rate amongmodels C1m9, B1m9, and A1m9 as follows: The H2 formation
rate depends not only on the total dust mass but also on
the dust size distribution. In models C1m9 and B1m9, thedust mass produced by an SN II without reverse shock is 
M0d,j (a)da = 0.48 M and in A1m9, the dust massjinjection into ISM through a reverse shock with nSN = 1.0 cm−3 
is j M1.0
d,j (a)da = 0.15 M . The ratio of the mean dust
area to the mean dust volume is a 2 /a 3  = 1.5 × 105 cm−1
before the reverse shock destruction. After the reverse shock
destruction, a 2 /a 3  = 4.2 × 104 cm−1 . This is a measureof dust area per dust volume and is also a measure of the H2formation rate of the dust surface. Small a 2 /a 3  leads to a low
H2 formation rate. This is the reason why model A1m9 shows asmaller H2 fraction than B1m9 and C1m9. The dust destruction
by reverse shocks changes not only the dust mass but also thegrain size distribution, and as a result drastically suppresses starformation in the galaxy.
In Figure 8, we show the evolution of the H2 formation
rate in gas phase, [dfH2 /dt]gas ; the H2 formation rate on
dust grain, [dfH2 /dt]dust ; the H2 destruction rate by UV photons, [dfH2 /dt]UV ; and the H2 decreasing rate by the star formation, [dfH2 /dt]star , normalized to the total formation rate,[dfH2 /dt]gas + [dfH2 /dt]dust , in model A1m9. At t ∼ 4 × 107 yr,
the H2 formation rate on dust grains exceeds the rate in thegas phase. However, the H2 formation rate on dust grains is4.2. The ISM Density around SN
The dependence of the time evolution of SFR on the ISM
density around an SN is presented in Figure 9. We can see thatafter t ∼ 5×107 yr higher density around SN progenitors resultsin lower molecular fraction and hence lower star formation
efficiency. The SFR, Ψ(t), is independent of nSN before t ∼5 × 107 yr, because H2 forms predominantly in the gas phase. In
the model without reverse shock destruction, the SFR increases
from ∼108 yr and saturates around 5 × 108 yr. This is because
the gas is consumed by the star formation. In model A10m9(nSN = 10 cm−3 ), SFR is suppressed until t ∼ 0.8 × 109 yr.
We should note that it is probably true that nSN < 1 cm−3
for Population III stars in the mass range of 20–40 M , since
Population III stars are massive and can photoevaporate theclouds in which they form, and ionized flows evacuate the dense
gas around the stars to well below nSN = 1 cm−3 (Whalen
et al. 2004; Kitayama et al. 2004). In this case, considerationof circumstellar densities of 10 cm−3 and greater is not relevant
to dust evolution in the SNR. In this paper, we consider
nSN > 5 cm−3 for completeness. If the stars are forming at
lower redshift and are enriched, they will have stellar winds that
also sweep away circumstellar gas to low densities.
11

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

1

0.01

Mstar/(Mgas+Mstar)

fH2

0.1

1

no reverse
0.03
0.1
0.3
1
3
10

0.001

0.00011e-05

0.00010.001
0.01
Z [Zsun]0.1

nISM,SN=0.1
nISM,SN=1
nISM,SN=10
0.1

0.01

0.001

1

1e+08

Figure 10. Molecular faction evolution, fH2 , for the model of different densities
around an SN II, nISM,SN = 0.03, 0.1, 0.3, 1.0, 3.0, and 10 cm−3 , which are thesame as in Figure 1, and the model without reverse shocks. The horizontal axis
shows the metallicity, Z, in units of Z .
(A color version of this figure is available in the online journal.)

1e+09
1e+10
Mvir [Msun]

1e+11

Figure 12. Stellar mass fraction in the galaxy, Mstar /(Mgas + Mstar ), for thevarious virial masses, Mvir , of the models with nSN = 0.1 cm−3 (solid line:
models A0.1m8, A0.1m9, A0.1m10, and A0.1m11), nISM.SN = 1.0 cm−3
(long dashed line: models A1m8, A1m9, A1m10, and A1m11), and nISM.SN =10 cm−3 (short dashed line: models A10m8, A10m9, A10m10, and A10m10).
(A color version of this figure is available in the online journal.)

1

fH2

0.1

0.01

no reverse
0.03
0.1
0.3
1
3
10

4.3. The Dark Matter Halo Mass
Finally, we show the stellar mass fraction, Mstar /(Mgas +Mstar )for Mvir = 108 , 109 , 1010 , and 1011 M at zvir = 10 in Figure 12.
We note that dark halos of virial masses, Mvir = 108 , 109 , 1010 ,and 1011 M correspond to the density fluctuation of 2.0σ ,2.5σ , 3.0σ , and 4.1σ , respectively. Mstar /(Mgas + Mstar ) is large
for large Mvir . This is explained as follows: the gas cools to the
CMB temperature ∼30 K in all Mvir after t ∼ a few × 107 yr,
although Tvir increases with Mvir , so that the final gas density
becomes higher because of smaller H /rdisk in larger Mvir (seeEquation (15)). This results in more rapid molecular formation
in a larger Mvir halo (see Equation (23)). The rapid molecularformation enhances the star formation and as a result causes thelarge stellar mass fraction. We should note that in higher zvir ,Mstar /(Mgas + Mstar ) is larger for the galaxies with Tvir  104 K,
since the rotation timescale becomes smaller in our model.

0.001

0.00011e-05

0.0001

0.001

0.01
σd,-200.1

1

Figure 11. Molecular faction evolution, fH2 , for the model of different densities
around an SN II, nISM,SN = 0.03, 0.1, 0.3, 1.0, 3.0, and 10 cm−3 , which are
the same as in Figure 1, and the model without reverse shocks. The horizontal
axis shows the total dust cross section per unit volume, σd,−20 , in units of
10−20 cm−1 .
(A color version of this figure is available in the online journal.)

5. DISCUSSION AND CONCLUSIONS
We have investigated the dust size evolution and the resulting
H2 formation on dust grains in the galaxies in the early universe(z > 5). Our dust evolution model includes the dust production
by SNe II and the effects of dust destruction by the reverse andforward shocks driven by SNe. In the galaxy model, we follow
the chemical network of H2 formation and the thermal evolutionof gas, taking into account cooling by H2 , atomic hydrogen andmetals, and heating by the stellar radiation. The SFR is assumed
to be proportional to the mass of molecular hydrogen and therotation timescale of the galactic gas disk. The evolution of thesize distribution of dust has been taken into account for the firsttime in this paper to investigate the influence on star formation
activity in galaxies.
We obtain three main results. First, we show that H2 formation
is suppressed by dust destruction, especially that resulting fromthe reverse shocks in SNRs. The resulting molecular fractionin the galaxy model with dust destruction by both reverse
shocks and forward shocks (see Figure 7) is 2.5 orders ofmagnitude less than the galaxy model without both destruction
processes (see Figure 5) and is two orders less than the galaxy
model considering dust destruction by only forward shocks (seeWe note that in a usual star formation recipe in both numericalsimulations and analytic models, the SFR is assumed to increase
with the local gas density, and our results show that the SFR isstrongly affected by nSN . We will discuss the effects of nSN on
the SFR in more detail in Section 5.
In Figure 10, we show the change of the molecular fraction,
fH2 , with metallicity, Z, for various nSN . In usual chemical
evolution models, Z is a key indicator of galaxy evolution.
However, as shown in this figure, fH2 does not solely dependon the metallicity. For Z  5 × 10−4 Z , fH2 is large in models
with small nSN . This is because Darea /Mmetal is large (small)in models with small (large) nSN for the same Z as shown in
Figure 3.
The molecular fraction is well described by the total dust
cross section per unit volume for σd,−20  0.001. In Figure 11,
we show the evolution of the molecular fraction in terms of total
dust cross section per unit volume. For σd,−20  1 × 10−3 , H2
formation on dust grains dominates fH2 evolution as shown by
the convergence of all models.
12The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.

Figure 6) for nSN = 1.0 cm−3 . Our results show that dust size
evolution has great effects on the early galaxy evolution.
The dust destruction by reverse shocks is essential in ourgalaxy model and has a more dominant role than forward shocks,
since the reverse shock destroys small dust grains earlier than
the forward shock. Dust destruction by a reverse shock becomes
more (less) prominent for a higher (lower) nSN . In early galaxy
evolution, considering dust destruction by reverse shocks isvery important to H2 formation. We note that forward shocks
affect dust size distribution in the large dust-to-gas mass ratio,
D  10−3 , as shown in Section 4.1.
Second, we show that the SFR strongly depends on the
ISM density around SN progenitors, nSN , which determines theefficiency of dust destruction by reverse shocks. As noted in
Section 3, we treat nSN differently from the mean gas density in
our one-zone galaxy model, nH , taking into account the effectof photoionization by SN progenitors. The molecular fraction
is different among the models with various nSN , even for thesame Z (see Figure 10), and it solely depends on the total dust
cross section per volume from the time when H2 formation on
dust grains becomes the most dominant of the processes of H2
formation and destruction (see Figure 11). The ratio of the total
dust cross section to total metal mass presented in Figure 3 isvery useful for a numerical simulation of galaxy formation withdust size evolution.
A higher ISM density around SN progenitors, nSN , results
in a lower SFR in the early galaxy evolution. In conventional
galaxy formation theory, SFR has been assumed as an increasing
function of gas density. Since in our one-zone model we simply
treat nSN as a parameter, it is very interesting to resolve theionized region around an SN progenitor by high-resolution
radiation hydrodynamic simulation taking into account theeffects of ionization heating by massive SN progenitors.
Finally, we show that galaxy evolution depends on Mvir
and that the stellar mass fraction, Mstar /(Mgas + Mstar ), ismonotonically increasing functions of Mvir . The halo number
density in the redshift range 5 < z < 10 is a decreasing functionof the halo virial mass, 108 M < Mvir < 1011 M . In mostpapers on the reionization, Mstar /(Mgas + Mstar ) is assumed to beconstant for various halo masses, Mvir . The difference of stellar
mass fraction with different Mvir in our results is important forgalaxy evolution in the early universe and should be consideredin the cosmic reionization process. We will study this effecttaking into account Population III star formation and the H2dissociation by the Lyman–Werner background in a forthcoming
paper, since a strong Lyman–Werner background that is effective
for dissociating molecular hydrogen, delaying gas condensation,and suppressing star formation especially in the lower mass H2
cooling halos is expected in the cosmic reionization era (D.
Yamasawa et al. 2011, in preparation).
Dust size distribution governs the extinction curve and reemitting IR spectrum (e.g., Silva et al. 1998; Granato et al. 2000;
Takeuchi et al. 2005; Li et al. 2008). A ground-based interferometric facility, the Atacama Large Millimeter Array (ALMA),5
can be used to study high-redshift galaxies, since redshifted
dust emission can be observed with ALMA. We can study dust
emission and the resulting SED of our galaxy model using our
results of the dust size distribution and test our galaxy model bycomparing it to observational data in the future.
Our galaxy model can be applied to the lower redshift (z < 5)star formation history, taking into account dust production by
5

low-mass stars and growth of dust due to accretion of metal in
ISM (Inoue 2003; Draine 2009). The accretion timescale, τacc ,depends on the metallicity and is given by τacc = τacc,0 Z /Z,
where τacc,0 ∼ 108 yr (Inoue 2003). If we assume Z ∼ 10−2 Zcorresponding τacc ∼ 1010 yr, then the accretion time is toolong to affect dust size distribution for a cosmic time 1.2 Gyr(z > 5). At a galaxy age of 1 Gyr, dust production by lowmass stars should affect the dust size evolution, because afterthat epoch dust is supplied from low-mass stars as well as SNe
II. We will include such processes to investigate low-redshift
galaxies in the future work.
Cosmological simulation of galaxy formation including ourdust formation and evolution model is needed, since it is widely
understood that most z ∼ 10 galaxies were not clear disk
galaxies (Johnson et al. 2008; Greif et al. 2010; Wise et al.
2010). We will study first galaxy formation by cosmologicalsimulation including our dust model.
We thank the anonymous referee for the very careful reading
and comments that improved this paper. The authors are gratefulto M. Fujimoto, K. Sorai, K. Omukai, A. K. Inoue, T. Takeuchi,
N. Yoshida, and B. T. Draine for helpful discussions. Numericalcomputations were carried out on NEC SX-9 at the Center for
Computational Astrophysics, CfCA, of National Astronomical
Observatory of Japan. This work was partly supported by the
Grant-in-Aid for Scientific Research of Japan Society for the
Promotion of Sciences (08091823, 18104003, 0340038). H.H.
is supported by NSC grant 99-2112-M-001-006-MY3. T.N. andK.N. are supported by the World Premier International ResearchCenter Initiative, MEXT, Japan.
REFERENCES
Abel, T., Anninos, P., Zhang, Y., & Norman, M. L. 1997, New Astron., 2, 181
Barlow, M. J., et al. 2010, A&A, 518, 138
Bianchi, S., & Schneider, R. 2007, MNRAS, 378, 973
Bigiel, F., Leroy, A., Walter, F., Brinks, E., Blok, W. J. G., Madore, B., &Thornley, M. D. 2008, AJ, 136, 2846
Bromm, V., Yoshida, N., & Hernquist, L. 2003, ApJ, 596, 135
Bromm, V., Yoshida, N., Hernquist, L., & McKee, C. F. 2009, Nature, 459, 49
Cazaux, S., & Spaans, M. 2004, ApJ, 611, 40
Cherchneff, I., & Dwek, E. 2010, ApJ, 713, 1
Cole, S., Lacey, C. G., Baugh, C. M., & Frenk, C. S. 2000, MNRAS, 319, 168
Cox, A. N. 2000, Allen’s Astrophysical Quantities (4th ed.; New York: Springer)
Draine, B. T. 2009, in ASP Conf. Ser. 414, Cosmic Dust—Near and Far, ed. Th.
Henning, E. Grün, & J. Steinacker (San Francisco, CA: ASP), 453
Draine, B. T., & Bertoldi, F. 1996, ApJ, 468, 269
Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89
Dwek, E., & Cherchneff, I. 2011, ApJ, 727, 63
Dwek, E., Galliano, F., & Jones, A. P. 2007, ApJ, 662, 927
Ferrara, A., Pettoni, M., & Sjcjekinov, Y. 2000, MNRAS, 319, 539
Ferrarotti, A. S., & Gail, H.-P. 2006, A&A, 447, 553
Gall, C., Andersen, A. C., & Hjorth, J. 2011, A&A, 528, A13
Gallerani, S., et al. 2010, arXiv:1006.4463
Galli, D., & Palla, F. 1998, A&A, 335, 403
Glover, S. C. O., & Abel, T. 2008, MNRAS, 388, 1627
Glover, S. C. O., & Jappsen, A.-K. 2007, ApJ, 666, 1
Gnedin, N. Y., Tassis, K., & Kravtsov, A. V. 2009, ApJ, 697, 55
Granato, G. L., Lacey, C. G., Silva, L., Bressan, A., Baugh, C. M., Cole, S., &Frenk, C. S. 2000, ApJ, 542, 710
Greif, T. H., & Bromm, V. 2006, MNRAS, 373, 128
Greif, T. H., Glover, S. C. O., Bromm, V., & Klessen, R. 2010, ApJ, 716, 510
Haiman, Z., Thoul, A. A., & Loeb, A. 1996, ApJ, 464, 523
Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., & Hartmann, D. H. 2003, ApJ,
591, 288
Heger, A., & Woosley, S. E. 2002, ApJ, 567, 532
Hirashita, H., & Ferrara, A. 2002, MNRAS, 337, 921
Hollenbach, D., & McKee, C. F. 1989, ApJ, 342, 306
Hutchings, R. M., Santoro, F., Thomas, P. A., & Couchman, H. M. P.
2002, MNRAS, 330, 927

http://www.almaobservatory.org/

13

The Astrophysical Journal, 735:44 (14pp), 2011 July 1

Yamasawa et al.
Omukai, K. 2000, ApJ, 534, 809
Omukai, K., Hosokawa, T., & Yoshida, N. 2010, ApJ, 722, 1793
Omukai, K., Tsuribe, T., Schneider, R., & Ferrara, A. 2005, ApJ, 626,
627
O’Shea, B. W., & Norman, M. L. 2008, ApJ, 673, 14
Pollack, L. B., Hollenbach, D., Beckwith, S., Simonelli, D. P., Roush, T., &Fong, W. 1994, ApJ, 421, 615
Robertson, B. E., & Kravtsov, A. V. 2008, ApJ, 680, 1083
Salpeter, E. E. 1955, ApJ, 121, 161
Schaerer, D. 2002, A&A, 382, 28
Schneider, R., Ferrara, A., & Salvaterra, R. 2004, MNRAS, 351, 1379
Schneider, R., & Omukai, K. 2010, MNRAS, 402, 429
Schneider, R., Omukai, K., Inoue, A., & Ferrara, A. 2006, MNRAS, 369,
1437
Shakura, N. I., & Sunyaev, R. A. 1988, Adv. Space Res., 8, 135
Sibthorpe, B., et al. 2010, ApJ, 719, 1553
Silva, L., Granato, G. L., Bressan, A., & Danese, L. 1998, ApJ, 509, 103
Silvia, D. W., Smith, B. D., & Shull, J. M. 2010, ApJ, 715, 1575
Smith, B., Sigurdsson, S., & Abel, T. 2008, MNRAS, 385, 1443
Smith, B. D., Turk, M. J., Sigurdsson, S., O’shea, B. W., & Norman, M. L.
2009, ApJ, 691, 441
Spergel, D. N., et al. 2007, ApJS, 177, 377
Springel, V., et al. 2005, Nature, 435, 629
Susa, H. 2007, ApJ, 659, 908
Susa, H. 2008, ApJ, 684, 226
Takeuchi, T., Ishii, T. T., Nozawa, T., Kozasa, T., & Hirashita, H. 2005, MNRAS,
362, 592
Tegmark, M., Silk, J., Rees, M. J., Blanchard, A., & Palla, F. 1997, ApJ, 474, 1
Todini, P., & Ferrara, A. 2001, MNRAS, 325, 726
Umeda, H., & Nomoto, K. 2002, ApJ, 565, 385
Valiante, R., Schneider, R., Bianchi, S., & Andersen, A. C. 2009, MNRAS, 397,
1661
Whalen, D., Abel, T., & Norman, M. L. 2004, ApJ, 610, 14
Whalen, D., van Veelen, B., O’shea, B. W., & Norman, M. L. 2008, ApJ, 682,
49
Wise, J. H., & Abel, T. 2007, ApJ, 671, 1559
Wise, J. H., & Abel, T. 2008, ApJ, 685, 40
Wise, J. H., & Cen, R. 2009, ApJ, 693, 984
Wise, J. H., Turk, M. J., Norman, M. L., & Abel, T. 2010, arXiv:1011.2632v2
Wolfe, A. M., & Chen, H.-W. 2006, ApJ, 652, 981
Yoshida, N., Abel, T., Hernquist, L., & Sugiyama, N. 2003, ApJ, 592, 645
Yoshida, N., Omukai, K., & Hernquist, L. 2008, Science, 321, 669
Zhukovska, S., Gail, H.-P., & Trieloff, M. 2008, A&A, 479, 453

Inoue, A. K. 2003, PASJ, 55, 901
Jappsen, A.-K., Glover, S. C. O., Klessen, R. S., & Mac Low, M.-M. 2007, ApJ,
660, 1332
Jappsen, A.-K., Klessen, R. S., Glover, S. C. O., & Mac Low, M.-M. 2009a, ApJ,
696, 1065
Jappsen, A.-K., Mac Low, M.-M., Glover, S. C. O., & Klessen, R. S. 2009b, ApJ,
694, 1161
Joggerst, C. C., Akmgren, A., Bell, J., Heger, A., Whalen, D., & Woosley, S. E.
2010a, ApJ, 709, 11
Joggerst, C. C., Akmgren, A., & Woosley, S. E. 2010b, ApJ, 723, 353
Johnson, J. L., Greif, T. H., & Bromm, V. 2008, MNRAS, 388, 26
Jones, A. P., Tielens, A. G. G. M., Hollenbach, D. J., & McKee, C. F. 1994, ApJ,
433, 797
Kennicutt, R. C. 1998, ApJ, 498, 541
Kitayama, T., & Ikeuchi, S. 2000, ApJ, 529, 615
Kitayama, T., & Yoshida, N. 2005, ApJ, 630, 675
Kitayama, T., Yoshida, N., Susa, H., & Umemura, M. 2004, ApJ, 613, 631
Kozasa, T., Nozawa, T., Tominaga, N., Umeda, H., Maeda, K., & Nomoto, K.
2009, in ASP Conf. Ser. 414, Cosmic Dust—Near and Far, ed. Th. Henning,
E. Grün, & J. Steinacker (San Francisco, CA: ASP), 43
Krügel, E. 2008, An Introduction to the Physics of Interstellar Dust (London:
Taylor and Francis)
Krumholz, M. R., & McKee, C. F. 2005, ApJ, 630, 250
Li, Y., Hopkins, P. F., Hernquist, L., Finkbeiner, D. P., Cox, T. J., Springel, V.,
Jiang, L., & Yoshida, N. 2008, ApJ, 678, 41
Machacek, M. E., Bryan, G. L., & Abel, T. 2001, ApJ, 548, 509
Machacek, M. E., Bryan, G. L., & Abel, T. 2003, MNRAS, 338, 273
Mackey, J., Bromm, V., & Hernquist, L. 2003, ApJ, 586, 1
Maiolino, R., Schneider, R., Oliva, E., Bianchi, S., Ferrara, A., Mannucci, F.,
Pedani, M., & Roca Sogorb, M. 2004, Nature, 431, 533
Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, ApJ, 217, 425
Mo, H. J., Mao, S., & White, S. D. M 1998, MNRAS, 295, 319
Nath, B. B., Laskar, T., & Shull, J. M. 2008, ApJ, 682, 1055
Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563
Nozawa, T., Kozasa, T., & Habe, A. 2006, ApJ, 648, 435
Nozawa, T., Kozasa, T., Habe, A., Dwek, E., Umeda, H., Tominaga, N., Maeda,
K., & Nomoto, K. 2007, ApJ, 666, 955
Nozawa, T., Kozasa, T., Tominaga, N., Maeda, K., Umeda, H., Nomoto, K., &Oliver, K. 2010, ApJ, 713, 356
Nozawa, T., Kozasa, T., Umeda, H., Maeda, K., & Nomoto, K. 2003, ApJ, 598,
785
Nozawa, T., Maeda, K., Kozasa, T., Tanaka, M., Nomoto, K., & Umeda, H.
2011, arXiv:1105.0973

14