D      R-C-U  H, G

D      R-C-U  H, G
D
  
C F   N S   M
  R-C-U  H, G
   
D  N S
P  
M. A. P. T H. G
 : M (G)
O : J 23rd , 2009
T F   F G
R:
P. D. R S. K
P. D. M B
Zusammenfassung
Diese Arbeit befasst sich mit dem Einfluss der ersten Sterne auf die Enstehung und den Eigenschaften der ersten Galaxien. Insbesondere werden die
Auswirkungen von Turbulenz, ionisierender Strahlung, Stoßwellen und chemischer Anreicherung erläutert. Aufgrund der verbesserten Empfindlichkeit der
nächsten Generation von Teleskopen untersuchen wir die Rekombinationsstrahlung,
Bremsstrahlung und 21 cm Emission der ersten Sterne und Galaxien. Das integrierte 21 cm Signal der ersten H  Regionen könnte mit dem geplanten SKA
entschlüsselt werden, wohingegen die Rekombinationsstrahlung der ersten Sternpopulationen mithilfe von JWST gesehen werden könnte. Diese Beobachtungen sind ein wichtiger Schritt um die Strukturentstehung im frühen Universum
besser zu verstehen.
Abstract
The primary concern of this thesis is to understand the formation and properties
of the first galaxies, as well as the influence of the first stars in terms of radiative,
mechanical and chemical feedback. In particular, we elucidate the role of turbulence, ionizing radiation by massive Population III stars, mechanical feedback
by highly energetic supernovae, and chemical enrichment. In light of the next
generation of ground- and space based telescopes, we derive their observational
signature in terms of recombination radiation, bremsstrahlung and 21 cm emission. We find that the cumulative 21 cm signal of the first H  regions will likely
be observable by the planned SKA, while the recombination radiation from the
first starbursts might be observable by JWST. These probes are essential to test
the theoretical framework of the first stars and galaxies and shed some light on
this elusive period of cosmic history.
Dedicated to my family
Contents
1
Introduction
7
2
From Primordial Fluctuations to the Formation of the First Stars
9
2.1
The Background Universe . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3
First-Order Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.3.1
Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.3.2
Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3.3
Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3.4
Perturbations to the Metric . . . . . . . . . . . . . . . . . . . . . .
15
2.4
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.5
The Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.6
The Formation of the First Stars . . . . . . . . . . . . . . . . . . . . . . .
20
3
The First Galaxies: Assembly, Cooling and the Onset of Turbulence
21
3.1
Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.1.1
Initial Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.2
Refinement and Sink Particle Formation . . . . . . . . . . . . . . .
25
First Galaxy Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.2.1
Atomic Cooling Criterion . . . . . . . . . . . . . . . . . . . . . .
26
3.2.2
Cosmological Abundance . . . . . . . . . . . . . . . . . . . . . .
27
3.2.3
Assembly of Atomic Cooling Halo
. . . . . . . . . . . . . . . . .
29
3.2.4
Merger Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Cooling and Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.3.1
33
3.2
3.3
Population III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
CONTENTS
3.4
3.5
3.6
4
Population III.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.3.3
Population II . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.4.1
The Development of Turbulence: Hot versus Cold Accretion . . . .
41
3.4.2
Shocks and Fragmentation Properties . . . . . . . . . . . . . . . .
42
Massive Black Hole Growth . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.5.1
Accretion Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.5.2
Accretion Luminosity . . . . . . . . . . . . . . . . . . . . . . . .
50
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Local Radiative Feedback in the Formation of the First Protogalaxies
57
4.1
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.1.1
Cosmological Initial Conditions and Resolution . . . . . . . . . . .
59
4.1.2
Radiative Feedback . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.1.3
Sink Particle Formation . . . . . . . . . . . . . . . . . . . . . . .
65
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4.2.1
The First H II region and Lyman-Werner Bubble . . . . . . . . . .
67
4.2.2
Thermal and Chemical Evolution of the Gas . . . . . . . . . . . . .
67
4.2.3
Shielding of Molecules by Relic H II Regions . . . . . . . . . . . .
70
4.2.4
Black Hole Accretion . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.2.5
HD Cooling in Relic H II Regions . . . . . . . . . . . . . . . . . .
76
4.2.6
Star Formation in the Presence of Radiative Feedback . . . . . . .
78
Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.2
4.3
5
3.3.2
The Observational Signature of the First H II Regions
83
5.1
Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.1.1
Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.1.2
Ray-tracing Scheme . . . . . . . . . . . . . . . . . . . . . . . . .
87
5.1.3
Photoionization and Photoheating . . . . . . . . . . . . . . . . . .
89
5.1.4
Photodissociation and Photodetachment . . . . . . . . . . . . . . .
90
Observational Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5.2.1
Build-up of H II Region . . . . . . . . . . . . . . . . . . . . . . .
91
5.2.2
Recombination Radiation from Individual H II Regions . . . . . . .
93
5.2
2
CONTENTS
5.3
6
Radio Background Produced by Bremsstrahlung . . . . . . . . . .
95
5.2.4
Radio Background Produced by 21 cm Emission . . . . . . . . . .
99
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
104
The First Galaxies: Signatures of the Initial Starburst
107
6.1
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
6.1.1
The Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
6.1.2
Deriving the Observational Signature . . . . . . . . . . . . . . . .
114
Results and Implications . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
6.2.1
Evolution of Gas inside the Galaxy . . . . . . . . . . . . . . . . .
117
6.2.2
Star formation Rate Indicators . . . . . . . . . . . . . . . . . . . .
120
6.2.3
Initial Mass Function Indicators . . . . . . . . . . . . . . . . . . .
124
6.2.4
Detectability of Recombination Radiation . . . . . . . . . . . . . .
129
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
133
6.2
6.3
7
5.2.3
The First Supernova Explosions: Energetics, Feedback, and Chemical Enrichment
137
7.1
7.2
7.3
Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
7.1.1
The Supernova Progenitor . . . . . . . . . . . . . . . . . . . . . .
140
7.1.2
Energy Injection . . . . . . . . . . . . . . . . . . . . . . . . . . .
141
7.1.3
Test Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141
7.1.4
Main Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
Expansion and Cooling Properties . . . . . . . . . . . . . . . . . . . . . .
150
7.2.1
Phase I: Free Expansion . . . . . . . . . . . . . . . . . . . . . . .
150
7.2.2
Phase II: Sedov-Taylor Blast Wave . . . . . . . . . . . . . . . . . .
151
7.2.3
Phase III: Pressure-Driven Snowplow . . . . . . . . . . . . . . . .
153
7.2.4
Phase IV: Momentum-Conserving Snowplow . . . . . . . . . . . .
154
7.2.5
Summary of Expansion Properties . . . . . . . . . . . . . . . . . .
155
Feedback on Neighboring Halos . . . . . . . . . . . . . . . . . . . . . . .
156
7.3.1
Delay by Photoheating . . . . . . . . . . . . . . . . . . . . . . . .
156
7.3.2
Shock-driven Collapse . . . . . . . . . . . . . . . . . . . . . . . .
161
7.3.3
Mixing Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . .
162
7.3.4
Gravitational Fragmentation . . . . . . . . . . . . . . . . . . . . .
163
3
CONTENTS
7.4
7.5
8
9
Chemical Enrichment . . . . . . . . . . . . . .
7.4.1 Heat Conduction and Metal Cooling . .
7.4.2 Instabilities and Distribution of Metals .
Summary and Conclusions . . . . . . . . . . .
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Chemical Mixing in Smoothed Particle Hydrodynamics Simulations
8.1 Diffusion Algorithm . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Numerical Implementation . . . . . . . . . . . . . . . . .
8.1.2 Test Problems . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Application to Chemical Mixing . . . . . . . . . . . . . . . . . .
8.2.1 Chemical Mixing as Turbulent Diffusion . . . . . . . . .
8.2.2 Mixing in Supernova Remnants . . . . . . . . . . . . . .
8.3 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . .
Outlook
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173
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183
184
187
Bibliography
191
4
Publications
The results presented in this thesis have been published in the following papers:
1. The observational signature of the first H  regions, Greif, T. H., Johnson, J. L.,
Klessen, R. S., Bromm, V., 2009, MNRAS, submitted (arXiv:0905.1717)
2. The first galaxies: signatures of the initial starburst, Johnson, J. L., Greif, T. H.,
Bromm, V., Klessen, R. S., Ippolito, J., 2009, MNRAS, submitted (arXiv:0902.3263)
3. Chemical mixing in smoothed particle hydrodynamics simulations, Greif, T. H., Glover,
S. C. O., Bromm, V., Klessen, R. S., 2009, MNRAS, 392, 1381
4. The first galaxies: assembly, cooling and the onset of turbulence, Greif, T. H., Johnson,
J. L., Klessen, R. S., Bromm, V., 2008, MNRAS, 387, 1021
5. The First Supernova Explosions: Energetics, Feedback, and Chemical Enrichment,
Greif, T. H., Johnson, J. L., Bromm, V., Klessen, R. S., 2007, ApJ, 670, 1
6. Local Radiative Feedback in the Formation of the First Protogalaxies, Johnson, J. L.,
Greif, T. H., Bromm, V., 2007, ApJ, 665, 85
1
Introduction
This thesis is concerned with the formation and properties of the first stars and galaxies, their
observational signature, and their feedback on the intergalactic medium (IGM). With the introduction of the Λ cold dark matter (ΛCDM) model, the first stars, termed Population III
(Pop III), are predicted to have formed at redshifts z ∼ 20 in 105 – 106 M ’minihalos’
(Barkana & Loeb, 2001; Bromm & Larson, 2004; Fuller & Couchman, 2000; Glover, 2005;
Haiman et al., 1996b). These first bound objects formed after fluctuations in the dark matter
density became large enough to decouple from the Hubble flow and collapse (Couchman &
Rees, 1986). Understanding the origin and evolution of these perturbations has been one of
the main goals in modern cosmology, following the discovery of large-scale anisotropies in
the Cosmic Microwave Background (CMB) by the Cosmic Background Explorer (COBE),
and the analysis of these fluctuations by the Wilkinson Microwave Anisotropy Probe (WMAP)
(Komatsu et al., 2009). Theoretical work over the last few decades has revealed that these
perturbations can be attributed to quantum fluctuations in the initially extremely homogeneous fireball. Subsequently, they experienced a period of linear growth after passing
through the epochs of inflation, matter-radiation-equality, and recombination. When the
Universe was merely one per cent of its present size, previously independent modes coupled
and structure formation left the linear regime (Couchman & Rees, 1986). At this highly interesting epoch of cosmic history, numerical simulations have become invaluable. Especially
the last decade has brought a wealth of new insights, with the consensus that the first stars
were typically two orders of magnitude more massive than present-day stars (Abel et al.,
2000, 2002; Bromm et al., 1999, 2002; Yoshida et al., 2003a, 2008, 2006). Subsequent work
focused on their radiative, mechanical and chemical feedback (Abel et al., 2007; Glover &
7
1. INTRODUCTION
Brand, 2001; Greif et al., 2007; Johnson et al., 2007; Whalen et al., 2004; Wise & Abel,
2008b; Yoshida et al., 2007a), and on the formation of the first galaxies (Greif et al., 2008;
Ricotti et al., 2002a; Wise & Abel, 2007a; Yoshida, 2006). These more massive systems
formed at redshifts z ∼ 10 in & 108 M halos and relied on atomic instead of molecular hydrogen cooling. A key motivation behind this development is their potential observability by
the next generation of ground- and space-based instruments, such as the James Webb Space
Telescope (JWST) and the Square Kilometer Array (SKA). In this thesis, we investigate the
various feedback effects exerted by the first stars on the IGM, as well as their influence on
the formation of first galaxies. Furthermore, we determine the observational signature of the
first stars in terms of recombination radiation, as well as the radio background produced by
bremsstrahlung and 21 cm emission. These investigations are essential in light of the numerous missions planned for the next decade, with the goal of understanding the formation of
the first stars and galaxies at the end of the cosmic dark ages.
The organization of this thesis is as follows: In Chapter 2, we discuss the evolution of
fluctuations from inflation to the onset of non-linearity, including the formation of the first
stars. This chapter is based on Dodelson (2003), which we refer to for a more detailed
account of structure formation. In Chapter 3, we concentrate on the formation and properties of the first galaxies, followed by a treatment of radiative feedback by Pop III stars
in minihalos (Chapter 4), their observational signatures (Chapter 5), and the visibility and
characteristics of entire starbursts (Chapter 6). Subsequently, we discuss the dynamical evolution and feedback of high-redshift supernovae (Chapter 7), and introduce a new algorithm
for smoothed particle hydrodynamics (SPH) simulations for chemical mixing (Chapter 8).
Finally, in Chapter 9 we give an outlook on future work. For consistency, all quoted distances
are physical, unless noted otherwise.
8
2
From Primordial Fluctuations to the
Formation of the First Stars
Understanding the evolution of perturbations from the epoch of inflation to the formation of
the first structures is one of the most important issues in modern cosmology. After COBE
found the CMB to be isotropic over all angular scales to one part in 105 , and WMAP revealed
the detailed distribution of these inhomogeneities, theory evolved rapidly to interpret these
perturbations. In the following, we discuss first-order perturbation theory with respect to the
relevant species and derive the equations governing the evolution of fluctuations. We then
construct the dark matter power spectrum and briefly discuss the formation of the first stars.
2.1
The Background Universe
The Friedmann-Robertson-Walker (FRW) metric describes a homogeneous, isotropic, and
flat universe, with the non-negative components of the metric given by:
g00 = −1
(2.1)
gii = a2 ,
(2.2)
where a is the scale factor, and the speed of light, c, has been set to unity (we will maintain
this convention throughout this chapter). The Einstein equations couple the metric to the
9
2. FROM PRIMORDIAL FLUCTUATIONS TO THE FORMATION OF THE FIRST
STARS
total energy content of the universe, resulting in the Friedmann equation:
H2 =
8πG
ρ,
3
(2.3)
where G is the gravitational constant, H = ȧ/a is the Hubble expansion rate (the overdot
denotes derivative with respect to time, t), and ρ is the sum of all contributions to the energy
density, i.e. matter, radiation, and a cosmological constant. We obtain an additional equation
relating ρ and a by demanding that the divergence of the stress-energy tensor vanishes:
∂ρ
= −3H (ρ + P) ,
∂t
(2.4)
where P denotes the pressure. For matter, thermal pressure is negligible compared to the rest
mass, so ρm ∝ a−3 , whereas radiation pressure is important for the radiation density, yielding
ρr ∝ a−4 . The energy density and pressure of a given particle species may be derived directly
from the distribution function f :
Z
g
ρ =
E(p) f (x, p) d3 p
3
(2π)
Z
p2
g
P =
f (x, p) d3 p ,
3
3E(p)
(2π)
(2.5)
(2.6)
where g denotes the degeneracy, x the spatial coordinate (with absolute value x), E the energy, p the momentum vector (with absolute value p), and ~ has been set to unity. The
distribution function plays an important role for the evolution of perturbations in the early
Universe.
2.2
The Boltzmann Equation
A description of the interactions of particles in thermal, but not chemical equilibrium, requires the Boltzmann equation, which models the change in abundance of a certain species
due to collisions with other particles:
df
= C[ f ] ,
dt
10
(2.7)
2.3 First-Order Perturbations
where the total change of f is determined by the collisional term C[ f ]. This term is a complicated function of the abundances of all four species participating in the reaction, given
by:
Z
Z
Z
1
d3 p2
d3 p3
d3 p4
C[ f1 (p1 )] =
p
(2π)3 2E2
(2π)3 2E3
(2π)3 2E4
× (2π)4 δ3 (p1 + p2 − p3 − p4 ) δ(E1 + E2 − E3 − E4 )
× |M|2 f3 (p3 ) f4 (p4 ) − f1 (p1 ) f2 (p2 ) .
(2.8)
The first line integrates the spacial momentum of the phase-space distribution for each interacting particle, excluding the particle of interest. They should actually be over the individual
four-momenta, yet a transformation yields factors 1/2E, leaving only the spatial momenta.
The second line is relatively straightforward and applies conservation of energy and momentum based on the Dirac delta function. The third line contains the amplitude of the reaction,
denoted by |M|2 , whereas the products of the distribution functions represent the change in
f1 (p1 ) due to deviations from equilibrium.
Even for a single species the Boltzmann equation is quite formidable. A full treatment
requires the inclusion of all four involved reaction partners, which leads to a coupled set
of integro-differential equations. These may be simplified by introducing first-order perturbations to the zero-order distribution functions (i.e. Fermi-Dirac or Bose-Einstein distributions), or by simply calculating various moments of f instead of the explicit distribution.
2.3
2.3.1
First-Order Perturbations
Photons
Perhaps the most important interaction in the early universe is Compton scattering, which
couples the baryonic fluid to the photon distribution. Schematically, this reaction may be
written as
(2.9)
γ(p1 ) + e− (p2 ) ↔ γ(p3 ) + e− (p4 ) ,
11
2. FROM PRIMORDIAL FLUCTUATIONS TO THE FORMATION OF THE FIRST
STARS
where γ and e− denote photons and electrons, respectively. The photon distribution function
fr may be written as a slightly modified Bose-Einstein distribution:
!
1
p
−1,
= exp
fr
T (1 + Θ (x, p̂, t))
(2.10)
where the first-order perturbation Θ (x, p̂, t) adopts the meaning of a deviation from the background temperature T , and p̂ is the unit vector of the momentum. Since the perturbation is
small, we may rewrite the above equation:
fr '
fr(0)
∂ fr(0)
−p
Θ,
∂p
(2.11)
where fr(0) is the zero-order distribution function. Before elaborating on the collisional term,
we explicitly expand the left-hand side of equation (2.7), taking into account various partial
derivatives of fr :
d fr ∂ fr ∂ fr dxi ∂ fr dp ∂ fr d p̂i
(2.12)
=
+ i
+
+
.
dt
∂t
∂x dt
∂p dt ∂ p̂i dt
Since we are only interested in first order perturbations to fr , we neglect the last term. Expressions for dxi /dt and dp/dt require an introduction of perturbations Φ (x, t) and Ψ (x, t) to
the background metric, for which we choose conformal Newtonian gauge. In this case, Φ
corresponds to the Newtonian potential, while Ψ is the perturbation to the spatial curvature:
g00 = −1 − 2Φ
(2.13)
gii = a2 (1 + 2Ψ) .
(2.14)
After calculating the Christoffel Symbols and going through some tedious algebra, we find
!
∂ fr
∂Ψ p̂i ∂Φ
d fr ∂ fr p̂i ∂ fr
H+
.
=
+
−p
+
dt
∂t
a ∂xi
∂p
∂t
a ∂xi
(2.15)
The first two terms on the right describe the continuity and Euler equations, while the third
term dictates that photons lose energy in an expanding universe. The last two terms govern
the effects of over- and under-dense regions on the photon distribution function.
Inserting the zero-order Bose-Einstein distribution function into equation (2.15), comparing with equation (2.7), and setting the collisional term to zero (collisions are expressed
12
2.3 First-Order Perturbations
in higher-order terms of fr ), results in the well known fact that T ∝ a−1 . Inserting the first
order expression for fr into equation (2.15) yields
!
d fr
∂ fr(0) ∂Θ p̂i ∂Θ ∂Ψ p̂i ∂Φ
.
+
= −p
+
+
dt
∂p ∂t
a ∂xi
∂t
a ∂xi
(2.16)
The next step consists of explicitly computing the collisional term for Compton scattering
with equation (2.8). This includes some tedious algebra, which we do not discuss in detail.
Instead, we outline some key assumptions:
• The energy of the electron consists only of its rest-mass.
• Very little energy is transferred by Compton scattering, greatly simplifying some terms.
• Second-order terms may be neglected.
• The amplitude of the reaction is taken to be constant, i.e. |M|2 = 8πσT m2e , where σT
is the Thompson cross section and me the mass of the electron. Consequently, we
neglect a contribution from the quadrupole moment of Θ and the polarization state.
Both assumptions do not result in large dicrepancies for the purpose of determining
the matter power spectrum.
With these approximations, the final expression for the collisional term is given by
∂ fr(0)
ne σT (Θ0 − Θ + p̂vb ) ,
C[ fr ] = −p
∂p
(2.17)
where ne is the electron density, vb the bulk velocity of the baryons, and Θ0 the zeroth moR
ment of Θ, i.e. Θ0 = (1/4π) Θ dΩ. In the absence of bulk velocities, Θ is driven towards
the value for the zeroth moment. Comparing equations (2.16) and (2.17) by means of equation (2.7), and applying a Fourier transformation to Θ, Φ, Ψ, and vb , we finally obtain
Θ̇ + ikµΘ + Ψ̇ + ikµΦ = −τ̇ (Θ0 − Θ + µvb ) ,
(2.18)
where k is the wave number, an overdot represents a derivative with respect to conformal
time η (convention from now on), µ describes the angle between k and p̂, p̂vb = µvb implies
that the velocity is irrotational, and τ̇ = −ne σT a introduces the optical depth to Thompson
scattering.
13
2. FROM PRIMORDIAL FLUCTUATIONS TO THE FORMATION OF THE FIRST
STARS
2.3.2
Dark Matter
The treatment of dark matter perturbations is fairly straightforward, since collisions may be
neglected. We must therefore merely compute the left-hand side of equation (2.7). The main
difference to photons is the presence of rest-mass, leading to additional velocity terms of the
form p, p/E, and p2 /E in a modified version of equation (2.16):
!
∂ fdm
p2 p2 ∂Ψ p p̂i ∂Φ
d fdm ∂ fdm p̂i p ∂ fdm
−p
H +
=0.
=
+
+
dt
∂t
a E ∂xi
∂E
E
E ∂t
a ∂xi
(2.19)
We integrate this equation over the three-momentum to extract the zeroth moment, and ex(1 + δ (x, t)) around the zeropand the resulting dark matter particle density to ndm = n(0)
dm
(0)
order value ndm ∝ a−3 , where δ is the overdensity. This yields the first-order, Fouriertransformed equation
(2.20)
δ̇ + ikv + 3Ψ̇ = 0 ,
where the bulk dark matter velocity v is again assumed to be irrotational. This expression
shows that density perturbations are sourced by bulk velocities and changes to the spatial
curvature. A second equation may be obtained by multiplying equation (2.19) by v and
integrating over the three-momentum to extract the first moment:
v̇ + Hv + ikΦ = 0 ,
(2.21)
showing that changes in v are generated by the Hubble flow and the Newtonian potential Φ.
2.3.3
Baryons
The tight coupling of protons and electrons by Coulomb scattering forces their overdensities
and bulk velocities to common values, denoted δb and vb . This allows us to concentrate on
the collisional term for electrons, which is governed by Coulomb and Compton scattering.
Similar to the previous section, we find
δ̇b + ikvb + 3Ψ̇ = 0 ,
14
(2.22)
2.3 First-Order Perturbations
which is equivalent to equation (2.20) for dark matter density perturbations, and
v̇b + Hvb + ikΦ = τ̇
4ρr
(3iΘ1 + vb ) ,
3ρb
(2.23)
where ρr and ρb are the background energy density of photons and baryons, respectively, and
R1
Θ1 = i/2 −1 µ Θ dµ is the first moment of Θ. Changes in the bulk velocity of the baryons are
additionally sourced by the first moment of the photon distribution function. Since baryons
are tightly coupled to radiation at early times, one may omit their presence without introducing significant errors to the matter power spectrum. Nevertheless, it is instructive to write
down the governing equations, since an exact treatment requires their inclusion.
2.3.4
Perturbations to the Metric
In the last few sections, we have investigated the influence of perturbations to the metric
on various distribution functions. To obtain the full set of equations, one must investigate
how additional terms in the stress-energy tensor T µ ν , which are sourced by over- and underdensities, affect the metric itself. For this purpose we need the Einstein equations:
Gµ ν = 8πT µ ν ,
(2.24)
where Gµ ν is the Einstein tensor. Since we are mainly interested in scalar perturbations, we
concentrate on the time-time component of the stress-energy tensor. After going through
some tedious algebra, we obtain the first-order equation
G
0
0
∂Ψ
k2
2
= −6H
+ 6H Φ − 2 2 Ψ .
∂t
a
(2.25)
The right-hand side of equation (2.24) includes the contribution to the energy density from
all species, given by
G
(2.26)
T 0 0 = − (ρdm δ + ρb δb + 4ρr Θ0 ) .
2
Combining the above equations, we find
k2 Ψ + 3H Ψ̇ − HΦ = 4πGa2 (ρdm δ + ρb δb + 4ρr Θ0 ) .
15
(2.27)
2. FROM PRIMORDIAL FLUCTUATIONS TO THE FORMATION OF THE FIRST
STARS
In the absence of expansion this reduces to the well-known Poisson equation for gravity. One
may extract a separate equation for anisotropic stresses by determing the spatial components
of the Einstein tensor, yet they turn out to be negligible, implying that Ψ = −Φ. Furthermore,
the decomposition theorem shows that vector perturbations (topological defects) and tensor
perturbations (gravity waves) to the metric are not sourced by first-oder perturbations to
the time-time component of the stress-energy tensor. We therefore only discuss the linear
perturbations quoted in equations (2.13) and (2.14).
2.4
Initial Conditions
The theory of inflation predicts the exponential growth of the scale factor over a brief period
of time, caused by an almost time-invariant scalar field φ. Quantum-mechanical fluctuations
in this field oscillate with zero mean, but non-zero root-mean-square (RMS). Modes smaller
than the horizon decay ∝ η, while those larger than the horizon maintain their amplitude. As
inflation progresses, smaller and smaller modes exit the horizon, which gives the primordial
power spectrum its characteristic shape:
Pφ =
H2
.
2k3
(2.28)
Since perturbations to the spatial curvature couple directly to perturbations in the scalar field,
the primordial power spectrum of Φ is given by
PΦ p =
8πGH 2
,
9k3
(2.29)
evaluated at aH = k and where describes the rate of change of the scalar field. To quantify
deviations from scale invariance, the power spectrum is usually rescaled to
PΦ p
50π2 2 k
=
δ
9k3 H H0
!n−1
Ωm
D (a = 1)
!2
,
(2.30)
where δH is the amplitude at Horizon crossing, H0 the present Hubble rate, n the spectral
index (describes deviations from scale-invariance), Ωm the matter density today, and D the
growth function (will be defined in the next section). At the end of inflation, radiation dominates the energy content of the Universe, marking the epoch when inhomogeneities in the
16
2.5 The Power Spectrum
matter density begin to grow. In the following, we discuss how the primordial power spectrum generates these fluctuations, and construct the resulting matter power spectrum.
2.5
The Power Spectrum
For simplicity, we constuct the matter power spectrum Pδ for a universe dominated by dark
matter, setting ρdm = ρm = ρc , where ρc is the critical density of the Universe. We will
discuss the resulting minor inaccuracies at the end of this section. Using Poisson’s equation,
we relate the fluctuation power δ to the the fluctuation power of Φ when radiation no longer
dominates:
4k4 a2
Pδ =
PΦ .
(2.31)
9Ω2m H04
We then split Φ (k, a) into a transfer function T (k), which depicts the change in amplitude of
each mode through horizon-crossing and matter-radiation equality aeq , and a growth function
D(a), which describes the scale-invariant growth after these epochs, yielding
Φ (k, a) =
D(a)
9
Φ p (k)T (k)
,
10
a
(2.32)
where the factor 9/10 accounts for the fact that even the largest scales are marginally damped
through matter-radiation equality. Inserting the RMS of equation (2.32) into equation (2.31),
and using the primordial power spectrum, we obtain the matter power spectrum
Pδ =
2π2 δ2H
D1 (a)
kn 2
T
(k)
D1 (a = 1)
H0n+3
!2
.
(2.33)
To obtain the transfer function, we must follow the growth of matter perturbations through
horizon crossing and aeq , using the Boltzmann equations. Fortunately, this problem may be
decomposed: scales that cross the horizon after aeq grow ∝ a (independently of k), leading
to T (k) = 1, while sub-horizon modes are significantly damped as radiation pressure washes
out perturbations. In the following, we determine the damping term in detail.
In a first step, we neglect the influence of matter over- and under-densities on the potential, which is initially dominated by radiative perturbations. Higher moments of Θ are suppressed at early times, so we may concentrate on the first two moments of equation (2.18),
17
2. FROM PRIMORDIAL FLUCTUATIONS TO THE FORMATION OF THE FIRST
STARS
resulting in
Θ̇0 + kΘ1 = −Φ̇
(2.34)
and
k
k
Θ̇1 − Θ0 = − Φ .
(2.35)
3
3
Together with equation (2.27) this leads to a second-order differential equation for Φ:
k2
4
Φ̈ + Φ̇ + Φ = 0 .
η
3
(2.36)
which may be solved by the second spherical Bessel function
!
sin b − b cos b
,
b3
Φ = 3Φ p
(2.37)
√
where b = kη/ 3. Evidently, the overall amplitude of Φ decays ∝ η−2 once it enters the
horizon. From this we may now determine the evolution of δ using equation (2.20), resulting
in
1
3
δ̈ + δ̇ = −3Φ̈ + k2 Φ − Φ̇ .
(2.38)
η
η
With Green’s function, the solution is approximately given by
!
3kη
.
δ = 9Φ p ln
5
(2.39)
Thus, even though the gravitational potential decays, matter fluctuations still grow ∝ ln (kη).
This solution holds deep in the radiation era, yet around aeq even small perturbations in δ
begin to dominate the potential, and one must include equation (2.27) next to equations (2.20)
and (2.21):
ikv
δ0 +
= −3Φ0 ,
(2.40)
aHy
v0 +
k2 Φ =
ikΦ
v
=
,
y aHy
3y
a2 H 2 δ ,
2 (y + 1)
(2.41)
(2.42)
where a prime denotes the derivative with respect to y, and y = a/aeq . Terms ∝ ρr Θ0 as well
as H have been dropped, since aH/k 1. Combining the above, we obtain an equation for
18
2.5 The Power Spectrum
the evolution of δ once the Universe becomes matter-dominated:
δ00 +
2 + 3y 0
3
δ −
δ=0.
2y (y + 1)
2y (y + 1)
(2.43)
The solution to this equation is given by
δ = C (y + 2/3) ,
(2.44)
where C is a constant. Evidently, δ grows ∝ a after aeq , proving the inital definition of the
power spectrum useful. For the complete expression, we patch the individual solutions for
δ before and after aeq together, yielding the integration constant C. Comparing the resulting
formula for δ with equation (2.33), we finally obtain
T (k) =
2
12keq
k2
!
k
,
ln
8keq
(2.45)
p
where keq = H0 2/aeq .
In the presence of dark energy, the Boltzmann equations slightly change and the growth
factor becomes a more complicated function:
5Ωm H
D(a) =
2H0
Z
0
a
da0
,
(a0 H 0 /H0 )3
(2.46)
where the prime now simply denotes the integration variable. For higher accuracy, one must
also account for the presence of baryons and neutrinos. However, due to their small contribution to the total energy density, they only have a small effect. In the first case, the tight
coupling of baryons to photons leads to a suppression of fluctuations on small scales. This
imprints oscillations on the power spectrum, similar to the ones derived in equation (2.37).
A comparable effect is found for massive neutrinos, wish wash out perturbations due to their
ability to free-stream over large distances. Taken together, the simple analytic solution presented in equation (2.45) is accurate to ' 20 percent, which is quite good in light of the
numerous assumptions made.
19
2. FROM PRIMORDIAL FLUCTUATIONS TO THE FORMATION OF THE FIRST
STARS
2.6
The Formation of the First Stars
In the last few sections, we have derived the dark matter power spectrum that sets the overall
gravitational potential. At sufficiently late times, the baryons follow this potential, at least on
scales larger than the Jeans length. As the fluctuation power approaches unity, independent
Fourier modes couple and numerical simulations become indispensable to accurately calculate the subsequent evolution. In a more simplified spherical model, a region collapses once
a critical overdensity of δc = 1.686 is reached. At this point it free-falls to a virialized state of
high density and forms a dark matter ’halo’. The very first of these halos, termed ’minihalos’,
emerge at redshifts z . 30 with virial masses of 105 – 106 M (Tegmark et al., 1997). The
gas in these systems cools via molecular hydrogen to ' 200 K and becomes Jeans-unstable
at a hydrogen number density of nH ' 104 cm−3 (Abel et al., 2000, 2002; Bromm et al.,
1999, 2002). Under the influence of self-gravity, the central clump then further collapses to
nH ∼ 1021 cm−3 , where it becomes adiabatic and forms a protostellar seed (Yoshida et al.,
2008). The later evolution is not well understood, but analytical work has indicated that
heavy accretion from the circumstellar disk leads to a final stellar mass of ∼ 100 M (McKee
& Tan, 2008; Tan & McKee, 2004). This mass scale leads to short lifetimes, but high photon
yields (Bromm et al., 2001b; Schaerer, 2002), as well as extreme fates in the form of black
holes (BHs) or pair-instability supernovae (PISNe) (Heger et al., 2003; Heger & Woosley,
2002). A good deal of research has therefore concentrated on their feedback and influence
on the formation of the first galaxies.
20
3
The First Galaxies: Assembly, Cooling
and the Onset of Turbulence
Understanding the formation of the first stars and galaxies at the end of the cosmic dark
ages is one of the most important challenges in modern cosmology (Barkana & Loeb, 2001;
Bromm & Larson, 2004; Ciardi & Ferrara, 2005; Glover, 2005). In the standard ΛCDM
cosmology, the first stars are predicted to have formed at z . 30 in DM minihalos with virial
masses of 105 – 106 M (Abel et al., 2000, 2002; Bromm et al., 1999, 2002; Gao et al., 2007;
Nakamura & Umemura, 2001; O’Shea & Norman, 2007; Yoshida et al., 2003a, 2006). Based
on molecular hydrogen cooling, the gas can cool to ' 200 K and become Jeans-unstable once
the central clump attains ' 103 M . Such high temperatures lead to efficient accretion on
to the protostellar core and imply that the first stars were predominantly very massive, with
M∗ ∼ 100 M (Bromm & Loeb 2004; Omukai & Palla 2003; O’Shea & Norman 2007; but
see McGreer & Bryan 2008; McKee & Tan 2008; Ripamonti 2007; Tan & McKee 2004).
Due to their primordial composition, massive Pop III stars have smaller radii than their
present-day counterparts and their surface temperatures can exceed ' 105 K, resulting in high
photon yields (Bromm et al., 2001b; Schaerer, 2002; Tumlinson & Shull, 2000; Tumlinson
et al., 2003). Consequently, they exert strong feedback on the IGM via radiation in the LW
bands, which readily destroys molecular hydrogen (Glover & Brand, 2001; Greif & Bromm,
2006; Johnson et al., 2008; Machacek et al., 2001; Mesinger et al., 2006; Omukai & Yoshii,
2003; O’Shea & Norman, 2008; Ricotti et al., 2001; Wise & Abel, 2007b), and ionizing
radiation giving rise to the first H  regions (Abel et al., 2007; Ahn & Shapiro, 2007; Alvarez
21
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
et al., 2006a; Johnson et al., 2007; Kitayama et al., 2004; Susa & Umemura, 2006; Whalen
et al., 2004, 2008a; Yoshida et al., 2007a). Furthermore, a yet unknown fraction of massive,
metal-free stars is expected to end in extremely violent SN explosions (Heger et al., 2003;
Heger & Woosley, 2002), profoundly affecting the IGM in terms of dynamics and chemical
enrichment (Bromm et al., 2003; Greif et al., 2007; Kitayama & Yoshida, 2005; Machida
et al., 2005; Mackey et al., 2003; Norman et al., 2004; Salvaterra et al., 2004; Whalen et al.,
2008b; Yoshida et al., 2004). Since primordial star formation initially occurs in minihalos
that constitute the progenitors of the first galaxies, stellar feedback is expected to play an
important role during their assembly.
When and where did the first galaxies form? According to the bottom-up character of
structure formation, as described by standard Press-Schechter theory (Press & Schechter,
1974), the first & 5 × 107 M halos assembled at z & 10 via hierarchical merging. This
mass scale is set by the onset of atomic hydrogen cooling once the virial temperature exceeds ' 104 K; these objects have therefore frequently been termed ‘atomic cooling halos’.
Henceforth, we will synonymously use the term ‘first galaxy’, as these halos can retain photoheated gas and therefore might, for the first time, maintain self-regulated star formation in
a multi-phase, interstellar medium (Dijkstra et al., 2004; Mac Low & Ferrara, 1999; Madau
et al., 2001; Mori et al., 2002; Oh & Haiman, 2002; Read et al., 2006; Ricotti et al., 2002a,b,
2008; Scannapieco et al., 2002; Thacker et al., 2002; Thoul & Weinberg, 1996; Wada &
Venkatesan, 2003).
What types of stars are expected to form during the assembly of the first galaxies? Depending on the primordial initial mass function (IMF) and the strength of radiative and SNdriven feedback, the IGM could be heavily enriched with metals prior to the onset of secondgeneration star formation (Bromm et al., 2003; Greif et al., 2007; Wise & Abel, 2008b). The
existence and precise value of a critical metallicity for the transition to Pop II are still vigorously debated, and has originally been discussed in the context of fine-structure versus dust
cooling (Bromm & Loeb, 2003b; Frebel et al., 2007; Jappsen et al., 2007; Schneider et al.,
2003; Smith & Sigurdsson, 2007). In the former case, critical metallicities of Zcrit ' 10−3.5 Z
have been found (Bromm et al., 2001a; Santoro & Shull, 2006), while in the latter case uncertainties in the dust composition and gas-phase depletion lead to a range of possible values,
10−6 . Zcrit . 10−5 Z (Omukai et al., 2005; Schneider et al., 2006; Tsuribe & Omukai,
2006). Somewhat transcending this debate, recent simulations have shown that a single SN
explosion might enrich the local IGM to well above any critical metallicity (Bromm et al.,
22
2003; Greif et al., 2007; Wise & Abel, 2008b), forcing an early transition in star formation
mode (Clark et al., 2008). However, if radiative feedback was sufficiently strong, the bulk of
primordial star formation might have occurred in systems that cool via atomic hydrogen lines
(Greif & Bromm, 2006; Johnson et al., 2008; O’Shea & Norman, 2008). In light of these
uncertainties, we investigate the formation of a first galaxy in the limiting case of no feedback, allowing us to focus on the chemistry, cooling and development of turbulence during
the assembly of the atomic cooling halo.
Theoretical investigations have recently pointed towards the existence of two physically
distinct populations of metal-free stars (Johnson & Bromm, 2006; Mackey et al., 2003; Uehara & Inutsuka, 2000). Gas cooling primarily via molecular hydrogen leads to the formation of & 100 M stars, termed Pop III.1, while in regions of previous ionization hydrogen
deuteride (HD) likely enables the formation of & 10 M stars, termed Pop III.2 (Tan & McKee, 2008). This latter mode had previously been termed Pop II.5 (Greif & Bromm, 2006;
Johnson & Bromm, 2006; Mackey et al., 2003). Scenarios providing a substantial degree
of ionization include relic H  regions (Nagakura & Omukai, 2005), dense shells produced
by energetic SN explosions (Bromm et al., 2003; Machida et al., 2005; Mackey et al., 2003;
Salvaterra et al., 2004), and structure formation shocks in the virialization of the first galaxies (Greif & Bromm, 2006; Oh & Haiman, 2002). In the former case, numerical simulations
have confirmed that the gas can cool to the temperature of the CMB, reducing the Jeans-mass
by almost an order of magnitude compared to the truly primordial case (Johnson et al., 2007;
Yoshida et al., 2007b), while we here set out to investigate the effects of partial ionization
and molecule formation during the assembly of the first galaxies. One of the key questions
is therefore whether the gas can cool to the CMB limit and thus enable the formation of
Pop III.2 stars.
Another important aspect concerning the baryonic collapse of the first galaxies is the
development of turbulence. While turbulence does not seem to play a role in minihalos,
at least on scales comparable to their virial radius (Yoshida et al., 2006), turbulent motions
could become important in more massive halos, where accretion of cold gas from filaments
in combination with a softened equation of state drives strong shocks (Kereš et al., 2005;
Sancisi et al., 2008; Wise & Abel, 2007a), possibly leading to vigorous fragmentation and
the formation of the first star clusters (Clark et al., 2008). This would mark the first step towards conditions similar to present-day star formation, where supersonic turbulent velocity
fields determine the fragmentation properties of the gas. In our present study, we investi-
23
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
gate the role of turbulence by analysing the velocity field and energy content of the galaxy
during virialization. We then discuss the likely fragmentation properties of the gas and the
consequences for second-generation star formation.
The final ingredient of early galaxy formation is the co-evolution of massive black holes
(MBHs) and the surrounding stellar system, leading in extreme cases to the formation of the
highest redshift quasars observed at z & 6. The most sophisticated simulations of supermassive black hole (SMBH) growth to date have been conducted by Li et al. (2007), but their
resolution was insufficient to trace the origin of SMBHs below ' 105 M . Such SMBHs
could form via direct collapse of isothermal gas in atomic cooling halos in the presence
of a strong photo-dissociating background (Begelman et al., 2006; Bromm & Loeb, 2003a;
Koushiappas et al., 2004; Lodato & Natarajan, 2006; Spaans & Silk, 2006; Wise et al., 2008),
or via merging and accretion of gas on to the compact remnants of Pop III stars (Islam et al.,
2003; Johnson & Bromm, 2007; Madau & Rees, 2001; Pelupessy et al., 2007; Schneider
et al., 2002; Volonteri et al., 2003; Volonteri & Rees, 2005; Wise et al., 2008). Here, we
investigate the fate of BHs seeded by the remnants of Pop III stars and their prospect of
growing into SMBHs at the centres of the first galaxies. Finally, we provide an estimate for
the amplitude of ionizing and molecule-dissociating radiation emitted by the accretion disc
around the central BH.
The structure of our work is as follows. In Section 3.2, we discuss the setup of the
cosmological simulations, the implementation of multiple levels of refinement and our sink
particle algorithm. We then describe the hierarchical assembly of the galaxy (Section 3.3),
its cooling and star formation properties (Section 3.4) and the development of turbulence
(Section 3.5). In Section 3.6, we discuss the growth of the BH at the centre of the galaxy,
and in Section 3.7 we summarize our results and assess their implications.
3.1
Numerical Methodology
To investigate the formation of the first galaxies with adequate accuracy, our simulations
must resolve substructure on small scales as well as tidal torques and global gravitational
collapse on much larger scales. We capture all the relevant dynamics by performing a preliminary simulation with a coarse base resolution, but refine around the point of highest
fluctuation power. This ensures that the region containing the mass of the galaxy is well
resolved and that a given number of particles is distributed efficiently.
24
3.1 Numerical Methodology
3.1.1
Initial Setup
The simulations are carried out in a cosmological box of linear size ' 700 kpc (comoving), and are initialized at z = 99 according to a concordance ΛCDM cosmology with
matter density Ωm = 1 − ΩΛ = 0.3, baryon density Ωb = 0.04, Hubble parameter h =
H0 / 100 km s−1 Mpc−1 = 0.7, spectral index n s = 1.0, and normalization σ8 = 0.9 (Spergel
et al., 2003). Density and velocity perturbations are imprinted at recombination with a Gaussian distribution, and we apply the Zeldovich approximation to propagate the fluctuations to
z = 99, when the simulation is started. To capture the chemical evolution of the gas, we
follow the abundances of H, H+ , H− , H2 , H+2 , He, He+ , He++ , and e− , as well as the five deuterium species D, D+ , D− , HD and HD+ . We include all relevant cooling mechanisms, i.e.
H and He atomic line cooling, bremsstrahlung, inverse Compton scattering, and collisional
excitation cooling via H2 and HD (see Johnson & Bromm, 2006).
3.1.2
Refinement and Sink Particle Formation
In a preliminary run with 643 particles per species (DM and gas), we locate the formation
site of the first ' 5 × 107 M halo. This object is just massive enough to activate atomic hydrogen cooling and fulfil our prescription for a galaxy. We subsequently carry out a standard
hierarchical zoom-in procedure to achieve high mass resolution inside the region destined to
collapse into the galaxy (e.g. Gao et al., 2005; Navarro & White, 1994; Tormen et al., 1997).
We apply three consecutive levels of refinement centred on this location, such that a single
parent particle is replaced by a maximum of 512 child particles. The particle mass in the region containing the comoving volume of the galaxy with an extent of ' 200 kpc (comoving)
is ' 100 M in DM and ' 10 M in gas. This allows a baryonic resolution of ' 103 M ,
such that the ‘loitering state’ in minihalos at T ' 200 K and nH ' 104 cm−3 is marginally
resolved (Bromm et al., 2002).
To follow the growth of BHs seeded by the collapse of Pop III.1 stars in minihalos,
we apply a slightly modified version of the sink particle algorithm introduced in Johnson
& Bromm (2007). Specifically, we create sink particles once the density exceeds nH =
104 cm−3 and immediately accrete all particles within the resolution limit of the highestdensity particle, resulting in an initial sink particle mass of ' 2 × 103 M . Further accretion
is governed by the following criteria: particles must be bound (Ekin < Epot ), have a negative
25
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
divergence in velocity and fall below the Bondi radius:
rB =
µmHGMBH
,
kB T
(3.1)
where µ is the mean molecular weight, MBH the mass of the black hole, and T the temperature
of the gas at the location of the sink particle, determined by a mass-weighted average over all
accreted particles. Typical values for the initial Bondi radius are rB ' 5 pc. Here, we do not
model radiative feedback that would result from accretion, and we therefore overestimate the
Bondi radius and consequently the growth of the BH. We further assume that BH mergers
occur instantaneously once their separation falls below the resolution limit, provided that
their relative velocity is smaller than the escape velocity. Our idealized treatment provides a
robust upper limit on the growth rate of BHs during the assembly of the first galaxies, but a
more realistic calculation including the effects of feedback will be presented in future work.
3.2
First Galaxy Assembly
Among the key questions concerning the formation of the first galaxies are the degree of
complexity associated with the halo assembly process, the role of previous star formation
in minihalos, the chemical and thermal evolution of infalling gas, and the development of
turbulence. To answer these questions, we first discuss the gravitational evolution of the
DM with a merger tree that reconstructs the mass accretion history of the resulting atomic
cooling halo, and allows us to determine the maximum amount of previous star formation in
its progenitor halos.
3.2.1
Atomic Cooling Criterion
Possibly the most important characteristic distinguishing the first galaxies from their lowermass minihalo predecessors is their ability to cool via atomic hydrogen lines, which softens
the equation of state at the virial radius and allows a fraction of the potential energy to be
converted into kinetic energy. At high redshift, the virial temperature of a system with virial
mass Mvir can be expressed as
T vir
Mvir
' 10 K
5 × 107 M
4
26
!2/3
!
1+z
,
10
(3.2)
3.2 First Galaxy Assembly
such that a 5 × 107 M halo at z ' 10 is just massive enough to fulfil the atomic cooling criterion (Oh & Haiman, 2002). Related to this, they have the ability to retain gas photoheated
by hydrogen ionization (Dijkstra et al., 2004), likely allowing self-regulated star formation
for the first time. Furthermore, it is often argued that star formation in atomic cooling halos
provided the bulk of the photons for reionization due to efficient shielding from LW radiation
(Bromm & Loeb, 2003a; Choudhury et al., 2008; Greif & Bromm, 2006; Haiman & Bryan,
2006; Wise & Abel, 2008a). They may have been the key drivers of early IGM enrichment
(Madau et al., 2001), and the host systems for the formation of the first low-mass stars that
can be probed in the present-day Milky Way via stellar archaeology (Frebel et al., 2007;
Karlsson et al., 2008). Their formation thus marks an important milestone in cosmic history,
and in light of upcoming observations it is particularly important to understand the properties
of these systems.
3.2.2
Cosmological Abundance
How common are atomic cooling halos at the redshifts we consider? A frequently used tool
to estimate their abundance is the Press-Schechter mass function (Press & Schechter, 1974),
which provides an analytic expression for the number of DM halos per mass and comoving
volume. Even though the Press-Schechter mass function is inaccurate at low redshifts compared to other analytic estimates such as the Sheth-Tormen mass function (Sheth & Tormen,
2002), it provides a good fit to numerical simulations at the early times we are interested in
(Heitmann et al., 2006; Jang-Condell & Hernquist, 2001; Reed et al., 2007). In Fig. 3.1, we
show the Press-Schechter mass functions for z ' 30 and 10 (solid and dotted line, respectively), while the symbols indicate the halo distribution extracted from the simulation by the
group-finding algorithm HOP (Eisenstein & Hut, 1998). The simulation and the analytic
estimate generally agree, although there is a large scatter due to the finite box size. Better
agreement could be expected if one accounted for the fluctuation power on scales greater
than the computational box (see Yoshida et al., 2003b). At redshift z ' 10, one expects to
find roughly 10 atomic cooling halos per cubic Mpc (comoving), i.e. of the order of unity in
our computational box of length ' 700 kpc (comoving). Indeed, in our simulation a single
5 × 107 M halo forms by z ' 10.
27
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Figure 3.1: Comparison of the Press-Schechter mass function (solid and dotted line) and the
simulation results (crosses and triangles) at z ' 30 (lower set) and z ' 10 (upper set). The
simulation generally agrees with the analytic prediction, although there is a large scatter due to
the finite box size. The expected number of atomic cooling halos per cubic Mpc (comoving) at
z ' 10 is of the order of 10.
28
3.2 First Galaxy Assembly
3.2.3
Assembly of Atomic Cooling Halo
In the ΛCDM paradigm, structure formation proceeds hierarchically, with small objects collapsing first and subsequently growing via merging. This behaviour eventually leads to the
formation of halos massive enough to fulfil the atomic cooling criterion. In Fig. 3.2, we
show the DM overdensity, gas density and temperature averaged along the line of sight at
three different output times. The brightest regions in the DM distribution mark halos in virial
equilibrium, according to the commonly used criterion ρ/ρ̄ ' 178, where ρ and ρ̄ denote the
local and background density. White crosses denote the formation sites of Pop III.1 stars in
minihalos. The first star-forming minihalo at the centre of the box assembles at z ' 23 and
subsequently grows into the galaxy delineated by the insets in the right panels of Fig. 3.2
and further enlarged in Figs. 3.8, 3.9, 3.10 and 3.12. Although this structure is not yet fully
virialized and exhibits a number of sub-components, it has a common potential well and
attracts gas from the IGM towards its centre of mass, where it is accreted by the central BH
once it falls below the Bondi radius. The virial temperature of the first minihalo increases
according to equation (3.2) until it reaches ' 104 K and atomic cooling sets in, at which
point the equation of state softens and a fraction of the potential energy is converted into
kinetic energy. Star formation takes place only in the most massive minihalos, with 10 Pop
III.1 star formation sites residing in the volume that is destined to form the galaxy. This
has important consequences for the role of stellar feedback, and will be further discussed in
Section 3.4. The morphology of the galaxy can best be seen in Fig. 3.3, where we show a
three-dimensional rendering of the central 150 kpc (comoving), i.e. the same field of view as
in Fig. 3.2. The temperature is colour-coded such that the hottest regions with T ' 104 K are
displayed in bright red. Here, the true spacial structure of the galaxy becomes more apparent,
showing that its environment is organized into prominent filaments with a high amount of
substructure. In some instances, star-forming minihalos have aligned along these filaments
and will soon merge with the galaxy.
3.2.4
Merger Tree
The hierarchical assembly of the galaxy can be best described by means of a merger tree that
depicts the evolution of all progenitor halos. We construct such a merger tree by tagging all
DM particles that reside in the parent atomic cooling halo and track their location backwards
in time. If they are part of a group at a previous timestep, they are considered to reside in
29
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Figure 3.2: The DM overdensity, hydrogen number density and temperature averaged along
the line of sight within the central ' 150 kpc (comoving) at three different output times, from
z ' 23, when the first star-forming minihalo at the centre of the box collapses, to z ∼ 10, when
the first galaxy forms. White crosses denote Pop III.1 star formation sites in minihalos, and the
insets approximately delineate the boundary of the galaxy, further enlarged in Figs. 3.8, 3.9, 3.10
and 3.12. Top row: the hierarchical merging of DM halos leads to the collapse of increasingly
massive structures, with the least massive progenitors forming at the resolution limit of ' 104 M
and ultimately merging into the first galaxy with ' 5 × 107 M . The brightest regions mark halos
in virial equilibrium according to the commonly used criterion ρ/ρ̄ > 178. Although the resulting
galaxy is not yet fully virialized and is still broken up into a number of sub-components, it shares
a common potential well and the infalling gas is attracted towards its centre of mass. Middle
row: the gas generally follows the potential set by the DM, but pressure forces prevent collapse
in halos below ' 2 × 104 M (cosmological Jeans criterion). Moreover, star formation only
occurs in halos with virial masses above ' 105 M , as densities must become high enough for
molecule formation and cooling. Bottom row: the virial temperature of the first star-forming
minihalo gradually increases from ' 103 K to ' 104 K, at which point atomic cooling sets in.
30
3.2 First Galaxy Assembly
Figure 3.3: A three-dimensional rendering of the central ' 150 kpc (comoving), showing the
same field of view as in Fig. 3.2. The temperature is colour-coded such that the hottest regions
with T ' 104 K are displayed in bright red. Here, the true spacial structure of the galaxy
becomes more clear, showing that its environment is organized into prominent filaments with
a high amount of substructure. In some instances, star-forming minihalos have aligned along
these filaments and will soon merge with the galaxy.
31
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
a halo with mass equal to the sum of their individual masses. We repeat this process until
all tagged particles are no longer part of a group or the mass falls below the resolution limit.
Fig. 3.4 shows the resulting absolute and differential mass growth of the galaxy. Accretion
is fuelled by minor as well as major mergers, with the latter showing the tendency to double
the mass of the halo. In the course of ' 400 Myr, the accretion rate increases from '
5 × 10−3 M yr−1 to ' 0.5 M yr−1 , but varies significantly in between due to the highly
complex nature of bottom-up structure formation.
To illustrate the substantial degree of complexity involved, Fig. 3.5 shows the individual
paths of all progenitor halos down to the DM resolution limit of ' 104 M . The initial
widening of the tree indicates that an increasing number of minihalos collapse, while at
z ' 20 merging becomes dominant and the degree of complexity decreases again. The
timescale for the widening of the tree is ' 150 Myr, while the completion of the merging
process requires another ' 250 Myr. The total number of halos above ' 104 M that merge to
form the galaxy is ' 300. A more sophisticated analysis of the merging process is presented
in Fig. 3.6, where the individual paths and masses of all progenitor halos are shown. Each
line represents an individual halo, while the colour denotes its mass. The target halo seeding
the first galaxy is indicated by the rightmost path. Sites of Pop III.1 star formation are
denoted by star symbols, and the oval denotes the formation of the atomic cooling halo.
The history of this most basic building block of galaxy formation is highly complex, further
complicated by the formation of 10 Pop III.1 stars prior to its assembly. However, this is an
upper limit on previous star formation activity as we do not include radiative and SN-driven
feedback, which would likely reduce the net star formation rate.
The presence of DM fluctuations on mass scales below our resolution limit might imply
that Pop III star formation takes place in halos with viral mass well below ' 2 × 104 M ,
but pressure forces prevent gas from settling into these shallow potential wells (cosmological
Jeans criterion). Moreover, gas may not be able to collapse beyond the point of virialization
in . 105 M halos, as temperatures and densities do not become high enough for efficient
H2 formation. Dynamical heating by mergers counteracts cooling and thus only a fraction of
all minihalos will be able to form stars (Yoshida et al., 2003a). To ascertain the importance
of this effect, we determine the masses of all minihalos experiencing star formation. As
described in Section 3.2, the formation of a Pop III.1 star is denoted by the creation of a sink
particle once the hydrogen number density exceeds ' 104 cm−3 . We implicitly assume that
such a parcel of gas does not experience further subfragmentation. With this prescription,
32
3.3 Cooling and Star Formation
we find the following virial masses for all star-forming minihalos shown in Fig. 3.6, from top
to bottom: Mvir ' [5.8, 1.6, 7.5, 3.5, 4.3, 1.4, 3.2, 9.3, 1.4, 11.8] × 105 M . As expected, their
masses are in the range ' 105 – 106 M , emphasizing the influence of dynamical heating on
halos below ' 105 M . Interestingly, the fact that only a fraction of all minihalos forms stars
ensures a constant inflow of cold gas into existing halos, which is crucial for the growth of
the BHs at their centres.
3.3
Cooling and Star Formation
A crucial issue concerning the formation of the first galaxies is the chemical and thermal
evolution of accreted gas, which ultimately determines the mode of star formation. We here
briefly discuss radiative and SN-driven feedback exerted by the very first stars, followed
by a discussion of the chemistry and cooling properties of an atomic cooling halo and the
implications for second-generation star formation.
3.3.1
Population III.1
As shown in Section 3.3, star formation ensues in minihalos before the larger potential wells
of the first galaxies assemble. This implies that radiative and SN-driven feedback influences
star formation in other minihalos as well as second-generation star formation in the resulting
atomic cooling halo. However, recent numerical simulations have shown that local radiative
feedback via photoheating and LW radiation may not be as important as previously thought
(Ahn & Shapiro, 2007; Johnson et al., 2007; Whalen et al., 2008a), and that a global LW
background may only reduce the number of Pop III.1 stars by ∼ 50 percent (Greif & Bromm,
2006; Johnson et al., 2008). An unknown fraction of these stars end their lives as energetic
SNe and enrich the surrounding IGM to well above the critical metallicity (Bromm et al.,
2003; Greif et al., 2007; Wise & Abel, 2008b), while others collapse directly to BHs and do
not expel any metals (Heger et al., 2003; Heger & Woosley, 2002). Since the timescale for
the recollapse of enriched gas is & 100 Myr (Greif et al., 2007), and mixing is inefficient with
respect to pre-established overdensities (Cen & Riquelme, 2008), subsequent star formation
in minihalos prior to the assembly of the atomic cooling halo likely remains metal-free. It
is much more difficult, however, to predict the character of star formation inside the first
galaxies. In the following, we will first examine the consequences of pristine gas collapsing
33
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Figure 3.4: The virial mass (top panel) and accretion rate (bottom panel) of the galaxy as a
function of redshift. The growth of the underlying DM halo is fuelled by minor as well as major
mergers, with the latter showing the tendency to double the mass of the target halo. At z ' 10,
the atomic cooling criterion is fulfilled and a galaxy is born.
34
3.3 Cooling and Star Formation
Figure 3.5: The merger tree of the galaxy, illustrating its complexity as a function of time. New
branches indicate the formation of DM halos at the resolution limit of ' 104 M . The widening
of the tree increases as more and more halos collapse, until merging dominates and the degree of
complexity decreases again. The timescale for the former is ' 150 Myr, while the completion of
the merging process requires another ' 250 Myr. A total of ' 300 halos above ' 104 M merge
to form the galaxy.
35
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Figure 3.6: The full merger tree of the galaxy assembling at z ' 10. Each line represents an
individual progenitor halo and is colour-coded according to its mass. The target halo seeding the
galaxy is represented by the rightmost path, which ultimately attains ' 5 × 107 M and fulfils
the atomic cooling criterion (denoted by the red oval). Star symbols denote the formation of
Pop III.1 stars in minihalos, showing that in our specific realization 10 Pop III.1 stars form prior
to the assembly of the galaxy. Only a fraction of all minihalos form stars, as dynamical heating
via mergers partially offsets cooling. Depending on the detailed merger history, this ensures that
star-forming minihalos are supplied with cold gas, which is crucial for the growth of the BHs at
their centres.
36
3.3 Cooling and Star Formation
Figure 3.7: The phase-space distribution of gas inside the first star-forming minihalo (left-hand
panel) and the atomic cooling halo (right-hand panel). We show the temperature, electron fraction, HD fraction and H2 fraction as a function of hydrogen number density, clockwise from top
left to bottom left. Left-hand panel: in the minihalo case, adiabatic collapse drives the temperature to & 103 K and the density to nH & 1 cm−3 , where molecule formation sets in allowing
the gas to cool to ' 200 K. At this point, the central clump becomes Jeans-unstable and ultimately forms a Pop III.1 star. Right-hand panel: in the first galaxy, a second cooling channel has
emerged due to an elevated electron fraction at the virial shock, which in turn enhances molecule
formation and allows the gas to cool to the temperature of the CMB. The dashed red lines and
arrows approximately delineate the resulting Pop III.1 and Pop III.2 channels, while the solid
green lines denote the path of a representative fluid element that follows the Pop III.2 channel.
37
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
in the atomic cooling halo, and subsequently briefly address the corresponding case of preenriched gas.
3.3.2
Population III.2
The possible existence of a distinct population of metal-free stars in regions of previous ionization has attracted increasing attention (Greif & Bromm 2006; Johnson & Bromm 2006;
Mackey et al. 2003; Tan & McKee 2008; Yoshida 2006; Yoshida et al. 2007a; but see McGreer & Bryan 2008; Ripamonti 2007). According to theory, an elevated electron fraction
catalyses the formation of H2 and HD well above the level found in minihalos and enables
the gas to cool to the temperature of the CMB. This reduces the Bonnor-Ebert mass by almost an order of magnitude and likely leads to the formation of Pop III.2 stars with & 10 M
(Johnson & Bromm, 2006). Numerical simulations of star formation in relic H  regions
have largely confirmed this picture (Johnson et al., 2007; Yoshida et al., 2007b), while its
relevance during the virialization of the first galaxies has not yet been established (but see
Greif & Bromm, 2006; Johnson et al., 2008).
The chemistry of gas contracting in an atomic cooling halo is fundamentally different
from that in minihalos. The latter maintain a primordial electron fraction of ' 3 × 10−4
and form a limited amount of molecules, while the virial temperature in an atomic cooling
halo exceeds ' 104 K and the elevated electron fraction facilitates the formation of high H2
and HD abundances. This allows the gas to cool to the temperature of the CMB instead
of the canonical ' 200 K found in minihalos. In the left panel of Fig. 3.7, we show the
properties of the gas in the first star-forming minihalo at the centre of the computational
box. The primordial electron fraction remains constant until densities become high enough
for electron recombination. After adiabatic heating to & 103 K, molecule formation sets in
and the gas cools to ' 200 K, at which point the central clump becomes Jeans-unstable and
inevitably forms a Pop III.1 star. In contrast, the right panel of Fig. 3.7 shows the density
and temperature of the gas inside the atomic cooling halo. The conventional H2 cooling
channel is still visible, but a second path from low to high density has emerged, enabled by
an elevated electron fraction at the virial shock, which in turn enhances the formation of H2
and HD and allows the gas to cool to the temperature of the CMB. The dashed red lines and
black arrows in Fig. 3.7 approximately delineate both channels, showing that the electron
fraction in the Pop III.2 case is elevated by an order of magnitude to ∼ 10−3 , and the H2
38
3.3 Cooling and Star Formation
fraction rises to ' 2 × 10−3 . As already estimated in Johnson et al. (2008), the HD fraction
grows to above ' 10−6 .
To more clearly illustrate this point, we plot the path of a representative fluid element
evolving along the Pop III.2 channel (solid green lines in Fig. 3.7). Such gas indeed cools
to the CMB floor, potentially enabling the formation of Pop III.2 stars. The mass fraction
entering this channel is relatively low at the time considered here since the atomic cooling
threshold has just been surpassed, but should quickly rise as freshly accreted material is
shock-heated to ' 104 K. We cannot study any possible fragmentation, since the gas rapidly
falls to within the Bondi radius of the central BH, and is thus accreted by the sink particle.
It will be very interesting to investigate the fragmentation of the Pop III.2 mode in future,
higher-resolution simulations, in particular testing the predicted mass scale of & 10 M (e.g.
Clark et al., 2008).
What are the implications of this result? Parcels of gas that are accreted on to the galaxy
through the virial shock can cool to the temperature of the CMB and possibly become gravitationally unstable, resulting in the formation of Pop III.2 stars. Including radiative feedback
from previous star formation would only strengthen this conclusion, as the degree of ionization would be increased even further (Wise & Abel, 2008b). As long as the gas collapsing
into the first galaxies remains pristine, primordial star formation will therefore likely be dominated by intermediate-mass (Pop III.2) stars (Greif & Bromm, 2006). The crucial question,
however, is: can the gas inside the first galaxies remain metal-free?
3.3.3
Population II
In the previous sections, we have found that of order 10 Pop III.1 stars form prior to the
assembly of the atomic cooling halo. In this case it appears unlikely that all of them will
collapse into BHs without any metal-enrichment (Johnson et al., 2008). Even a single SN
from a massive Pop III star would already suffice to reach levels above the critical metallicity, at least on average (Bromm et al., 2003; Greif et al., 2007; Wise & Abel, 2008b). More
generally, at some stage in cosmic history, there must have been a transition from primordial, high-mass star formation to the ’normal’ mode that dominates today. The discovery
of extremely metal-poor stars in the Galactic halo with masses below one solar mass (Beers
& Christlieb, 2005; Christlieb et al., 2002; Frebel et al., 2005) indicates that this transition
occurs at abundances considerably smaller than the solar value. At the extreme end, these
39
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
stars have iron abundances less than 10−5 times the solar value, but show significant carbon
and oxygen enhancements, which could be due to unusual abundance patterns produced by
enrichment from BH-forming Pop III SNe (Umeda & Nomoto, 2003), or due to mass transfer from a close binary companion, whose frequency is predicted to increase with decreasing
metallicity (Lucatello et al., 2005).
Identifying the critical metallicity at which this transition occurs is subject to ongoing
research. One approach is to argue that low mass star formation becomes possible only
when atomic fine-structure line cooling from carbon and oxygen becomes effective (Bromm
et al., 2001a; Bromm & Loeb, 2003b; Frebel et al., 2007; Santoro & Shull, 2006), setting
a value for Zcrit at ' 10−3.5 Z . Another possibility, first proposed by Omukai et al. (2005),
is that low mass star formation is a result of dust-induced fragmentation occurring at high
densities, nH ' 1013 cm−3 , and thus at a very late stage in the protostellar collapse. In this
model, 10−6 . Zcrit . 10−5 Z , where much of the uncertainty in the predicted value results
from uncertainties in the dust composition and the degree of gas-phase depletion (Schneider
et al., 2002, 2006). Recent numerical simulations by Tsuribe & Omukai (2006) as well as
Clark et al. (2008) provide support for this picture. However, the existing data of metal-poor
Galactic halo stars seems to be well accommodated by the C- and O-based fine-structure
model (Frebel et al., 2007).
In the present simulation, we do not follow the metallicity evolution of the infalling
gas. Thus, we can only speculate about the properties of the resulting stellar population. It
appears reasonable to assume that some of the accreting material is still pristine and free of
metals, triggering the formation of lower-mass metal-free Pop III.2 stars. Gas that flows in at
even later times may already have experienced metal enrichment from previous Pop III SNe
in nearby minihalos. Because of the high level of turbulence within the virial radius at that
time (see Section 3.5), the incoming new material is likely to efficiently mix with the preexisting zero-metallicity gas and the era of Pop III star formation could come to an end. This
transition possibly occurs at the same time as the onset of significant degrees of turbulence in
the atomic cooling halo. We therefore speculate that some of the extremely metal-deficient
stars in the halo of the Milky Way may have formed as early as redshift z ' 10 (Clark et al.,
2008).
40
3.4 Turbulence
3.4
Turbulence
The development of turbulence in gas flowing into the central potential well of the halo
strongly influences its fragmentation behaviour and consequently its ability to form stars.
Detailed studies of the interstellar medium in the Milky Way, for instance, tell us that turbulence determines when and where star formation occurs and that it is the intricate interplay
between gravity on the one hand, and turbulence, thermal pressure and magnetic fields on
the other that sets the properties of young stars and star clusters (Ballesteros-Paredes et al.,
2007; Larson, 2003; Mac Low & Klessen, 2004). In the context of our work, we investigate
the velocity field and energy distribution that build up during the assembly of the galaxy.
As opposed to cooling flows in low-mass halos, the accretion flow on to the deep central
potential well of the atomic cooling halo considered here becomes highly turbulent within
the virial radius.
3.4.1
The Development of Turbulence: Hot versus Cold Accretion
One of the most important consequences of atomic cooling is the softening of the equation
of state below the virial radius, allowing a fraction of the potential energy to be converted
into kinetic energy (Wise & Abel, 2007a). This implies that perturbations in the gravitational
potential can generate turbulent motions on galactic scales, which are then transported to the
centre of the galaxy (e.g. Fig. 3.8). In this context it is important to investigate the accretion
of gas on to the galaxy in more detail.
In principle, there are two distinct modes of accretion. Gas accreted directly from the
IGM is heated to the virial temperature and comprises the sole channel of inflow until cooling
in filaments becomes important. This mode is termed hot accretion, and dominates in lowmass halos at high redshift. In the atomic cooling halo, the formation of the virial shock and
the concomitant heating are visible in Fig. 3.9, where we show the hydrogen number density
and temperature of the central ' 40 kpc (comoving) around the BH at the centre of the galaxy.
This case also reveals a second mode, termed cold accretion. It becomes important as soon
as filaments are massive enough to enable molecule reformation, which allows the gas to
cool and flow into the central regions of the nascent galaxy with high velocities. In Fig. 3.9,
the cold gas accreted along filaments from the left- and right-hand is clearly distinguishable
from the hot gas at the virial shock. These streams are also visible in Fig. 3.10, where we
41
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
compare the radial with the tangential velocity component, and in Fig. 3.8, where we show
the Mach number of infalling gas. Evidently, inflow velocities can be as high as 20 km s−1 ,
with Mach numbers of the order of 10.
In Fig. 3.11, we compare the energy distribution and mass fraction of cold (< 500 K)
versus hot (> 500 K) gas in radial shells for the first star-forming minihalo just before the
formation of the sink particle, and the atomic cooling halo assembling at z ' 10. The blue,
green and red lines denote the azimuthally averaged ratio of radial, tangential and thermal
to potential energy, respectively. The black lines show the sum of all three components, and
the dotted lines indicate the ratio required for perfect virialization. In the minihalo case,
the total energy is dominated by thermal energy, although its share decreases towards the
centre where cooling via molecular hydrogen becomes important. The radial kinetic energy
dominates over the tangential component down to rtan ' 5 pc, where the mass fraction of cold
gas rapidly rises and the cloud becomes rotationally supported. Efficient cooling implies that
the total energy drops below that required for perfect virialization.
In the atomic cooling halo, the total energy at rvir ' 1 kpc is dominated by bulk radial
motions. The distinction between hot and cold gas in the right panels of Fig. 3.11 shows that
a large fraction of the kinetic energy injected into the galaxy comes from cold gas accreted
along filaments, even though its mass fraction is initially small. The energy in tangential
motions begins to dominate at rtan ' 200 pc, showing that the radial energy of the cold
gas flowing in along filaments is converted into turbulent motions. This is fundamentally
different from the collapse of gas in minihalos, where the radial energy is converted into a
directed rotation along a single axis. The distinct features at r ' 350 pc are caused by a
subhalo that has not yet merged with the central clump (see also Figs. 3.2 and 3.3). Once
again, the total energy budget falls below that required for perfect virialization, as atomic
hydrogen as well as molecular cooling are able to radiate away a significant fraction of the
potential energy released. We conclude that the high energy input by cold accretion is ideally
suited to drive turbulence at the centre of the galaxy, where bulk radial inflows are converted
into turbulent motions on small scales.
3.4.2
Shocks and Fragmentation Properties
Shock fronts can arise where supersonic flows experience sudden deceleration and, if unorganized, indicate the presence of supersonic turbulence. As discussed above, cold accretion
42
3.4 Turbulence
Figure 3.8: The central ' 40 kpc (comoving) of the computational box, roughly delineated by
the insets in Fig. 3.2. Shown is the Mach number in a slice centred on the BH at the centre of the
galaxy, indicated by the filled black circle. The dashed line denotes the virial radius at a distance
of ' 1 kpc. The Mach number approaches unity at the virial shock, where gas accreted from the
IGM is heated to the virial temperature over a comparatively small distance. Inflows of cold gas
along filaments are supersonic by a factor of ' 10 and generate a high amount of turbulence at
the centre of the galaxy, where typical Mach numbers are between 1 and 5.
43
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Figure 3.9: The central ' 40 kpc (comoving) of the computational box, roughly delineated by
the insets in Fig. 3.2. Shown is the hydrogen number density (left-hand panel) and temperature
(right-hand panel) in a slice centred on the BH at the centre of the galaxy, indicated by the filled
black circle. The dashed lines denote the virial radius at a distance of ' 1 kpc. Hot accretion
dominates where gas is accreted directly from the IGM and shock-heated to ' 104 K. In contrast,
cold accretion becomes important as soon as gas cools in filaments and flows towards the centre
of the galaxy, such as the streams coming from the left- and right-hand side. They drive a prodigious amount of turbulence and create transitory density perturbations that could in principle
become Jeans-unstable. In contrast to minihalos, the initial conditions for second-generation star
formation are highly complex, with turbulent velocity fields setting the fragmentation properties
of the gas.
44
3.4 Turbulence
Figure 3.10: The central ' 40 kpc (comoving) of the computational box, roughly delineated
by the insets in Fig. 3.2. Shown is the radial (left-hand panel) and tangential velocity in the x-y
plane (right-hand panel) in a slice centred on the BH at the centre of the galaxy, indicated by the
filled black circle. The dashed lines denote the virial radius at a distance of ' 1 kpc. Streams
of cold gas from filaments, such as those coming from the left- and right-hand side, are clearly
visible and can have velocities of up to 20 km s−1 . Some even penetrate the centre and create
regions of positive radial velocities. Angular velocities are particularly high towards the centre
of the galaxy, where bulk radial inflows are converted into turbulent motions on small scales.
The presence of flows in both directions implies that these are unorganized instead of coherently
rotating, such as is the case in minihalos (see also Fig. 3.11).
45
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Figure 3.11: The energy distribution and mass fraction of cold (< 500 K) versus hot (> 500 K)
gas in radial shells for the first star-forming minihalo just before the formation of the sink particle,
and the atomic cooling halo assembling at z ' 10. The azimuthally averaged ratio of radial,
tangential and thermal to potential energy are shown as blue, green and red lines, respectively.
The black lines show the sum of all three components, and the dotted lines indicate the ratio
required for perfect virialization. In the minihalo case, the total energy is dominated by thermal
energy, although its share decreases towards the centre where cooling via molecular hydrogen
becomes important. The radial kinetic energy dominates over the tangential component down to
rtan ' 5 pc, where the mass fraction of cold gas rapidly rises and the cloud becomes rotationally
supported. In the atomic cooling halo, the total energy at rvir ' 1 kpc is dominated by bulk
radial motions. The distinction between hot and cold gas in the right panels shows that most
of the kinetic energy injected into the galaxy comes from the cold gas accreted along filaments,
even though its mass fraction is initially small. The tangential component begins to dominate
at rtan ' 200 pc, where the radial flow of cold gas along filaments is converted into turbulent
motions. The distinct features at r ' 350 pc are caused by a subhalo that has not yet merged with
the central clump (see also Figs. 3.2 and 3.3).
46
3.4 Turbulence
is a viable agent for driving turbulence, due to the prodigious amount of momentum and
kinetic energy it brings to the centre of the galaxy. In Fig. 3.12, we show the divergence
and vorticity of the velocity field during the virialization of the galaxy. A comparison with
Fig. 3.8 implies that there are indeed regions of supersonic flow that experience rapid deceleration and form shocks. In our case, two physically distinct mechanisms are responsible
for creating these shocks. The virial shock forms where the ratio of infall velocity to local
sound speed approaches unity, and is clearly visible in the left-hand panel of Fig. 3.12. The
velocity divergence is negative since the gas rapidly decelerates, while the vorticity is almost
negligible. In contrast, the unorganized multitude of shocks that form near the centre of the
galaxy are mostly caused by accretion of cold, high-velocity gas from filaments. These are
more pronounced than the virial shock and have a significantly higher angular component.
They create transitory density perturbations that could in principle become Jeans-unstable
and trigger the gravitational collapse of individual clumps.
How does the turbulence generated in the infalling material influence its fragmentation
behaviour and control subsequent star formation? From detailed observational and theoretical studies of star formation in our Milky Way we know that turbulence plays a pivotal role
in the formation of stars and star clusters. It is usually strong enough to counterbalance gravity on global scales. By the same token, however, it will usually provoke collapse locally.
Turbulence establishes a complex network of interacting shocks, where regions of high density build up at the stagnation points of convergent flows. To result in the formation of stars,
local collapse must progress to high enough densities on time scales shorter than the typical
interval between two successive shock passages. Only then can the collapsing core decouple
from the ambient flow pattern and build up a star. The accretion flow on to these objects
and consequently the final stellar mass strongly depends on the properties of the surrounding
turbulent flow.
In concert with the thermodynamic properties of the gas, leading to the cooling of highdensity material to the CMB limit (see Section 3.4), length scale and strength of the turbulence are the most important parameters governing its fragmentation behaviour and consequently the properties of star formation, such as its timescale and overall efficiency (Klessen
et al., 2000; Krumholz & McKee, 2005; Vázquez-Semadeni et al., 2003). In the atomic cooling halo discussed here, this will eventually lead to the transition to Pop II star formation.
However, a quantitative understanding of the fragmentation behaviour of the turbulent gas
would require dedicated high-resolution simulations, which is beyond the scope of this work.
47
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Figure 3.12: The central ' 40 kpc (comoving) of the computational box, roughly delineated
by the insets in Fig. 3.2. We show the divergence (left-hand panel) and z-component of the
vorticity (right-hand panel) in a slice centred on the BH at the centre of the galaxy, indicated by
the filled black circle. The dashed lines denote the virial radius at a distance of ' 1 kpc. The most
pronounced feature in the left-hand panel is the virial shock, where the ratio of infall speed to
local sound speed approaches unity and the gas decelerates over a comparatively small distance.
In contrast, the vorticity at the virial shock is almost negligible. The high velocity gradients at
the centre of the galaxy indicate the formation of a multitude of shocks where the bulk radial
flows of filaments are converted into turbulent motions on small scales.
48
3.5 Massive Black Hole Growth
3.5
Massive Black Hole Growth
Galaxy formation in general involves the co-evolution of a central black hole and the surrounding stellar system, the one influencing the other. Two crucial unsolved problems are:
what were the seeds for BH growth, and how important was this co-evolution at very high
redshifts? We here begin to address these questions. Different scenarios have been suggested to account for the seeds of BH growth (Rees, 1984): the direct collapse of gas in
atomic cooling halos in the presence of a strong photodissociating background, or stellar
remnants of massive, metal-free stars. In the following, we discuss the latter possibility and
investigate the growth of a MBH forming at the centre of the galaxy.
3.5.1
Accretion Rate
Even though studies of stellar evolution have shown that primordial stars may explode as
energetic SNe, we here assume that all Pop III.1 stars collapse directly to BHs (Heger et al.,
2003; Heger & Woosley, 2002). Their initial mass is dictated by the resolution limit to
MBH ' 2 × 103 M , with accretion on to the BH governed by the criteria discussed in
Section 3.2. Recent investigations have shown that photoheating by the progenitor star can
delay efficient accretion by reducing the central density to . 1 cm−3 (Johnson & Bromm,
2007). However, the suppression of accretion also depends on the detailed merger history
of the host halo. For example, a major merger occuring just after the formation of the BH
could transport enough cold gas to its centre to enable accretion at the Eddington rate. As
shown in Fig. 3.6, some minihalos merge shortly after forming a BH, in some cases after
only a few million years. On the other hand, if mergers are absent or incoming halos are
not sufficiently massive, accretion could be suppressed for & 100 Myr (Johnson & Bromm,
2007). In our approach to derive an upper limit on the accretion rate, we neglect the effects
of photoheating by the progenitor star such that accretion is governed solely by the supply
of cold gas brought to the centre of the halo.
A more precise modelling would also require a prescription for radiation emitted by the
BH-powered miniquasar and its feedback on the surrounding disc (Volonteri & Rees, 2006).
In a simpler approach, one can assume a given radiative efficiency , which denotes the
ratio of BH luminosity to accreted mass energy, and assume that accretion is spherically
49
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
symmetric. This leads to the Eddington accretion rate
1 MBH
,
tSalp
(3.3)
cσT
' 450 Myr .
4πGmH
(3.4)
ṀEdd =
where tSalp is the Salpeter time, defined by
tSalp =
The mass of the BH as a function of time is thus given by
!
1 − t − t0
MBH (t) = MBH (t0 ) exp
.
tSalp
(3.5)
In Fig. 3.13, we compare the mass growth of the most massive BH with the Eddingtonlimited model, using a fiducial value of = 1/10. We find that the accretion rate remains
roughly constant at ' 5 × 10−3 M yr−1 , such that the BH grows from ' 2 × 103 to ' 106 M
in the course of ' 300 Myr. Due to our neglect of radiative feedback, the BH accretes well
above the Eddington rate throughout most of its lifetime. At later times the accretion rate
stagnates, most likely caused by the high kinetic energy input at the centre via cold accretion
(see Fig. 3.11). Consequently, the fraction of unbound gas near the BH increases and its
mass growth is slowed. We conclude that Eddington accretion is possible under the most
favourable circumstances, for example, where a recent merger brings an ample supply of
cold gas to the centre of the halo, but generally radiative feedback by the progenitor star and
the disc around the BH will lead to sub-Eddington accretion rates (Johnson & Bromm, 2007;
Pelupessy et al., 2007).
3.5.2
Accretion Luminosity
The radiation generated by the BH-powered miniquasar can have numerous effects on the
formation of the first galaxies. In addition to its negative feedback on star formation via
photoheating, the emitted radiation can contribute to the LW background, as well as to the
reionization of the Universe (Kuhlen & Madau, 2005; Madau et al., 2004; Ricotti & Ostriker,
2004). In the following, we derive an upper limit on the photodissociating flux and the
number of ionizing photons emitted by a stellar remnant BH accreting at the Eddington
limit.
50
3.5 Massive Black Hole Growth
Figure 3.13: The mass (top panel) and accretion rate (bottom panel) of the central BH as a function of redshift, shown for the simulation (solid lines) and Eddington-limited accretion (dotted
lines). The accretion rate remains roughly constant at ' 5 × 10−3 M yr−1 , such that the BH
grows from ' 2 × 103 to ' 106 M in the course of ' 300 Myr. This is a strict upper limit as
radiation effects are not taken into account. Accretion is initially super-Eddington due to the high
amount of cold gas brought to the centre of the galaxy, but stagnates once turbulence becomes
important and the fraction of unbound gas increases.
51
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Photodissociating Flux
To determine the flux of LW photons, we first model the temperature profile of the surrounding accretion disc:
!1
3 GMBH ṀBH 4
T (r) =
(3.6)
,
8π σSB r3
where r is the distance from the BH, MBH its mass, ṀBH the accretion rate and σSB the
Stefan-Boltzmann constant (Pringle, 1981). For simplicity, we have taken the disc to be a
thin disc, such that each annulus radiates as a blackbody of temperature given by the above
equation. The inner-most radius of the disc is given by
rinner
!
MBH
∼ 2 km
,
M
(3.7)
corresponding to a high value for the BH spin parameter a & 0.9 (Makishima et al., 2000;
Vierdayanti et al., 2008), which we expect considering the large angular momentum of the
accreted gas (see Fig. 3.11). We integrate the flux over the surface of the disc from rinner to
router = 104 rinner , where the contributions to both the photodissociating and ionizing fluxes
are negligible. To determine the flux emitted in the LW bands, we evaluate the total emitted flux at 12.87 eV. In the upper panel of Fig. 3.14, we show the LW flux in units of
10−21 erg s−1 cm−2 Hz−1 sr−1 for the case of Eddington-limited accretion at 1 kpc distance
from the BH. We consider initial BH masses of 100 and 500 M , roughly the range expected for the direct collapse of massive Pop III stars (Heger et al., 2003; Heger & Woosley,
2002). As Fig. 3.14 shows, JLW can greatly exceed the critical value of & 10−2 required
for the suppression of star formation in minihalos, which mostly relies on efficient H2 cooling (Johnson et al., 2008; Machacek et al., 2001; O’Shea & Norman, 2008; Yoshida et al.,
2003a). Furthermore, even a modest LW flux can dissociate enough H2 such that HD formation and cooling never become important, reducing the temperature to which the gas can
cool (Yoshida et al., 2007b). The impact on primordial star formation in BH-hosting galaxies
might thus be severe even for sub-Eddington accretion rates, implying that this effect must
be taken into account in future work.
Another important issue concerns the contribution to the global LW background. We
may estimate a maximum global LW background by assuming that each atomic cooling halo
at z & 10 hosts a BH accreting at the Eddington rate. We can then find an upper limit to the
52
3.5 Massive Black Hole Growth
global LW background by summing up the contributions from BHs within a distance equal
to the maximum mean free path for a LW photon, rmax ∼ 10 Mpc at z = 15, following the
prescription in Johnson et al. (2008). This estimate yields a maximum global LW background
comparable to the LW flux from a single source, as shown in Fig. 3.14. Such a high flux
could have profound consequences for further star formation in minihalos, but in most cases
radiative feedback by the progenitor star will significantly delay accretion, such that a global
LW background fuelled by accretion on to BHs will likely be subdominant compared to a
stellar LW background (Pelupessy et al., 2007).
Ionizing Flux
The amount of ionizing radiation released by the accreting BH can be determined in analogy
to our calculation of the LW flux. An integration over the temperature profile of the accretion disc yields the total number of hydrogen-ionizing photons emitted per second, shown
in the bottom panel of Fig. 3.14. While massive Pop III stars emit of the order of 1050
hydrogen-ionizing photons per second, these stars live for only . 3 Myr (Bromm et al.,
2001b; Schaerer, 2002). However, as Fig. 3.14 shows, if Pop III relic BHs are able to accrete efficiently, they may emit 10 – 100 times this number for & 100 Myr. This enormous
flux of ionizing radiation could power H  regions with radii of the order of 10 kpc, larger
and longer-lived than the transient H  regions of individual Pop III stars (Johnson et al.,
2007; Yoshida et al., 2007a). Star formation in minihalos within the H  region could be
suppressed if accretion is continuous and drives a persistent radiative flux (Ahn & Shapiro,
2007; Whalen et al., 2008a). Due to this dramatic radiative feedback associated with high
accretion rates, though, the gas in the protogalaxy is likely to be heated and driven away
from the BH, once again resulting in sub-Eddington accretion rates (Pelupessy et al., 2007).
We have also calculated the number of He -ionizing photons emitted by the accreting
BH, and find that this is within a factor of . 2 of the number of hydrogen-ionizing photons,
owing to the high temperatures of the accretion disc. Thus, if Pop III relic BHs are able
to accrete efficiently, they may also contribute to the reionization of helium, driving He 
regions that can be as large as their H  regions (Furlanetto & Oh, 2007).
53
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
Figure 3.14: The radiation due to thermal emission from the accretion disc surrounding a Pop III
relic BH accreting at the Eddington limit, for an inital mass of 100 (solid lines) and 500 M (dotted lines). Top panel: the photodissociating flux, JLW , in units of 10−21 erg s−1 cm−2 Hz−1 sr−1 ,
at a distance of 1 kpc from the BH. Bottom panel: the number of hydrogen-ionizing photons
emitted per second from the accretion disc, in units of 1050 s−1 .
54
3.6 Summary and Conclusions
3.6
Summary and Conclusions
We have investigated the formation of the first galaxies with highly resolved numerical simulations, taking into account all relevant primordial chemistry and cooling. The first galaxies
form at redshifts z & 10 and are characterized by the onset of atomic hydrogen cooling, once
the virial temperature exceeds ' 104 K, and their ability to retain photoheated gas. We have
described the merger history of a ' 5 × 107 M system in great detail and found that in the
absence of stellar feedback 10 Pop III.1 stars form in minihalos prior to the assembly of the
galaxy. Infalling gas is partially ionized at the virial shock and forms a high amount of H2
and HD, allowing the gas to cool to the temperature of the CMB and likely form Pop III.2
stars with & 10 M . Accretion on to the galaxy proceeds initially via hot accretion, where gas
is accreted directly from the IGM and shock-heated to the virial temperature, but is quickly
accompanied by a phase of cold accretion, where the gas cools in filaments before flowing
into the parent halo with high velocities. The latter drives supersonic turbulence at the centre
of the galaxy and thus plays a key role for second-generation star formation. Finally, we
have investigated the growth of BHs seeded by the stellar remnants of Pop III.1 stars and
found that accretion at the Eddington limit might be possible under the most favourable circumstances, but in most cases radiation emitted by the progenitor star and the accretion disc
around the BH will lead to sub-Eddington accretion rates.
Depending on the strength of radiative and SN-driven feedback, some galaxies might
remain metal-free and form intermediate-mass Pop III.2 stars. The inclusion of radiative
feedback would likely increase the fraction of Pop III.2 material, as it enhances the degree
of ionization, and, consequently, the amount of molecules formed (Johnson et al., 2007;
Wise & Abel, 2008b). Observational signatures of intermediate-mass primordial stars might
include gamma-ray bursts (GRBs) for the case of a rapidly rotating progenitor (e.g. Bromm
& Loeb, 2006), or distinct abundance patterns produced by core-collapse SNe experiencing
fallback (Umeda & Nomoto, 2003). However, since the number of Pop III.1 stars formed in
minihalos prior to the assembly of the galaxy is of the order of 10, it seems much more likely
that at least one star will end in a violent SN explosion and pre-enrich the halo to supercritical
levels (Bromm et al., 2003; Greif et al., 2007; Wise & Abel, 2008b). In combination with
the onset of turbulence, metal mixing in the first galaxies will likely be highly efficient and
could lead to the formation of the first low-mass star clusters (Clark et al., 2008), in extreme
cases possibly even to metal-poor globular clusters (Bromm & Clarke, 2002). Some of the
55
3. THE FIRST GALAXIES: ASSEMBLY, COOLING AND THE ONSET OF
TURBULENCE
extremely iron-deficient, but carbon and oxygen-enhanced stars observed in the halo of the
Milky Way may thus have formed as early as redshift z ' 10.
In future work, we plan to include the effects of radiative and SN-driven feedback by
previous star formation in minihalos, as well as the distribution of metals and its concomitant
cooling. We will study the fragmentation of gas accumulating at the centre of the galaxy and
address the issue of turbulence-driven star formation in detail. The goal of making realistic
predictions for the first generation of starburst-galaxies, to be observed with the JWST, is
clearly coming within reach.
56
4
Local Radiative Feedback in the
Formation of the First Protogalaxies
The formation of the earliest galaxies plays a key role in a number of the most important
questions being addressed in cosmology today. The first galaxies are predicted to have been
the dominant sources of the radiation which reionized the universe (Ciardi et al., 2006), and
they may have hosted the majority of primordial star formation (Greif & Bromm, 2006;
Jimenez & Haiman, 2006). They are the likely sites for the formation of the most metalpoor stars that have recently been found in our Galaxy (Beers & Christlieb, 2005; Christlieb
et al., 2002; Frebel et al., 2005), and possibly for the first efficient accretion onto the stellar
black holes (see Johnson & Bromm, 2007) which may have been the seeds for the ∼ 109 M
black holes that are inferred at redshifts z & 6 (Fan et al., 2004, 2006b). Furthermore,
an understanding of the formation of the first galaxies is crucial for the interpretation of
galaxies now beginning to be observed at z & 6 (Bouwens & Illingworth, 2006; Iye et al.,
2006; Mobasher et al., 2005), as well as of the objects at redshifts z & 10 which are expected
to be detected with upcoming telescopes such as JWST (Gardner et al., 2006). Among these
systems there promise to be some of the first metal-free objects that will be observable, and
as such it is important that theoretical predictions of their properties are made.
What were the effects of the radiative feedback from the first generations of stars on the
formation of the first galaxies? It is now widely held that the first stars were likely very
massive, and therefore emitted copious amounts of radiation which profoundly affected their
surroundings (Abel et al., 2002; Bromm et al., 1999, 2002; Gao et al., 2007; Yoshida, 2006).
57
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
Recent work has demonstrated that the H  regions surrounding the first stars were able to
evacuate the primordial gas from the minihalos that hosted these objects (Abel et al., 2007;
Alvarez et al., 2006a; Kitayama et al., 2004; Whalen et al., 2004). The impact of these
H  regions on second generation star formation is complex (Ahn & Shapiro, 2007; Oh &
Haiman, 2003; Ricotti et al., 2001; Susa & Umemura, 2006). While initially the density
in these regions is suppressed and the gas within heated to & 104 K, vigorous molecule
formation can take place once the gas begins to cool and recombine after the central Pop III
star has collapsed to fom a massive black hole, leading to the possibility of the formation
of low-mass primordial stars (Johnson & Bromm, 2006, 2007; Nagakura & Omukai, 2005;
O’Shea et al., 2005; Yoshida et al., 2007a).
An additional radiative feedback effect from the first stars is the photo-dissociation of
the fragile hydrogen molecules which allow the primordial gas to cool and collapse into
minihalos, with virial temperatures . 8, 000 K (Barkana & Loeb, 2001). The effects of the
molecule-dissociating radiation from the first stars can reach far beyond their H  regions
(Ciardi et al., 2000), and thus star formation in distant minihalos may have been delayed or
quenched altogether (Haiman et al., 2000, 1997; Mackey et al., 2003). Interestingly, however,
while IGM at the epoch of the first stars becomes optically thick to LW photons only over
vast distances (Glover & Brand, 2001; Haiman et al., 2000), the high molecule fraction
that persists inside the first relic H  regions leads to a high optical depth to these photons,
potentially allowing star formation to take place in minihalos down to lower redshifts than
would otherwise be possible (Johnson & Bromm, 2007; Machacek et al., 2001, 2003; Oh &
Haiman, 2002; Ricotti et al., 2001).
In the present work, we self-consistently track the formation of, and the radiative feedback from, individual Pop III stars in the course of the formation of a primordial protogalaxy.
We compute in detail the H  regions and LW bubbles of each of these sources, and follow
the evolution of the primordial gas as it becomes incorporated into the protogalaxy. In Section 4.2, we describe our numerical methodology. Our results are presented in Section 4.3,
while we summarize our conclusions and discuss their implications in Section 4.4.
58
4.1 Methodology
4.1
4.1.1
Methodology
Cosmological Initial Conditions and Resolution
We employ the parallel version of GADGET (version 1) for our three-dimensional numerical
simulations. This code includes a tree, hierarchical gravity solver combined with the SPH
method for tracking the evolution of the gas (Springel et al., 2001). Along with H, H+ , H− ,
H2 , H+2 , He, He+ , He++ , and e− , we have included the five deuterium species D, D+ , D− , HD
and HD+ , using the same chemical network as in Johnson & Bromm (2006, 2007).
We carry out a three-dimensional cosmological simulation of high-z structure formation
which evolves both the dark matter and baryonic components, initialized according to the
ΛCDM model at z = 100. As in earlier work (Bromm et al., 2003; Johnson & Bromm,
2007), we adopt the cosmological parameters Ωm = 1 − ΩΛ = 0.3, Ωb = 0.045, h = 0.7, and
σ8 = 0.9, close to the values measured by WMAP in its first year (Spergel et al., 2003). Here
we use a periodic box with a comoving size L = 460 h−1 kpc, but unless stated explicitly,
we will always refer to physical distances. Our simulation uses a number of particles NDM =
NSPH = 1283 , where the SPH particle mass is mSPH ∼ 740 M .
We have determined the maximum density of gas that can be reliably resolved in this
simulation by carrying out a cosmological simulation from z = 100, in which we allow the
gas to cool and collapse into minihalos without including radiative effects. We then compare
the minimum resolved mass, which we take to be ∼ 64 mSPH , with the Bonnor-Ebert mass,
given by
T 3/2 −1/2
n
MBE ' 700 M
,
(4.1)
200 K
104 cm−3
where n and T are the number density and temperature of the gas, respectively. As shown
in Fig. 4.1, the gas evolves according to the canonical behavior of primordial gas collapsing
in minihalos (e.g. Bromm et al., 2002). We expect the gas in our simulations with radiative
feedback to behave similarly as it collapses to high densities, since it is the formation of, and
cooling by, molecules which will drive the collapse in both cases. Thus, we take the maximum density that we can reliably resolve to be that at which the Bonnor-Ebert mass becomes
equal to the resolution mass. As is evident in Fig. 4.1, this criterion results in a maximum
resolvable density of nres ∼ 20 cm−3 . This density is four orders of magnitude higher than
the background mean density at a redshift of z & 15, and such overdensities only occur in the
minihalos within which the first Pop III stars form (Bromm & Loeb, 2004; Yoshida, 2006).
59
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
We take it here that one Pop III star, assumed to have a mass of 100 M , will form from this
dense, collapsing primordial gas inside a minihalo, consistent with recent work which shows
that, in general, only single stars are expected to form in minihalos under these conditions
(Yoshida, 2006). Pop III stars with this mass are predicted to directly collapse to a black
hole, and therefore produce no supernova explosion (Heger et al., 2003), which allows us
to self-consistently neglect the possibility of ejection of metals into the primordial gas. We
will consider this possibility in future work. In the present work, we focus on the radiative
feedback from the first stars.
4.1.2
Radiative Feedback
In our simulations with radiative feedback, we assume that stars are formed in minihalos
which acquire densities higher than nres = 20 cm−3 . In order to account for the radiative
feedback from a star formed in a minihalo, the gas surrounding the star is first photo-heated.
We then calculate the extent of the H  region, as well as of the LW bubble around the star.
We carry out this procedure every time a star forms in the simulation. The Pop III star will
soon die, and we then let the simulation evolve once more, allowing recombination to take
place in the relic H  region, and for molecules to reform within the relic LW bubble. We
expect that this procedure will provide reliable results, as the . 3 Myr lifetime of a Pop III
star is short compared to the typical dynamical times of the gas in this simulation.
Photoionization
To account for the presence of a 100 M Pop III star in the minihalos in which the gas
collapses to a density of nres , we first photoheat and photoionize the gas within 500 pc of the
gas particle which first reaches this density for a duration of the lifetime of the star, using the
same heating and ionization rates as in Johnson & Bromm (2007). Our choice of a 500 pc
radius ensures that the entire gas within the source minihalo, with virial radius ∼ 150 pc, is
photoheated, but that we do not photoheat the dense, neutral gas in neighboring halos. Just
as in this previous work, we reproduce the basic density and velocity structure of the gas
within 500 pc of the central source that has been found in detailed one-dimensional radiation
hydrodynamics calculations (Kitayama et al., 2004; Whalen et al., 2004).
Once this density structure is in place around the point source, we employ a ray-tracing
technique to solve for the H  region that surrounds the star at the end of its life. We cast rays
60
4.1 Methodology
Figure 4.1: Determining the maximum density resolvable in our simulations. To reliably resolve
the properties of the gas in our simulation, the Bonnor-Ebert mass, similar to the Jeans mass,
must be larger than the mass in the SPH smoothing kernel. For added assurance, we take the
minimum resolvable mass to be twice the mass in the kernel. This value for the resolution mass
is shown by the dashed horizontal line. For densities higher than nres ∼ 20 cm−3 , the BonnorEbert mass may be exceeded by the resolution mass, and so we take it that we can only resolve
the properties of the gas at densities below this value. We note that the two structures emerging
at high densities are two spatially distinct halos of different mass which are undergoing collapse.
61
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
in Nray ∼ 100, 000 directions from the central source, and divide each ray into 500 segments.
Then, we add up all of the recombinations that take place over the course of the star’s lifetime
in each bin along each of the Nray rays, taking the number of recombinations to be
Nrec = αB n2mean
4π 2
t∗ r dr ,
Nray
(4.2)
where αB is the case B recombination coefficient, r is the distance of the bin from the star,
dr is length of the bin in the direction radial to the star, and t∗ is the lifetime of the star,
here taken to be 3 Myr (Schaerer, 2002). We compute nmean , the average number density of
hydrogen atoms in a bin, as
P
nH
nmean =
.
(4.3)
Npart
Here, Npart is the number of SPH particles in the bin and nH is the number density of hydrogen
of the individual SPH particles in that bin.
Next, we assume that the star radiates an equal number of photons in every direction, and
we take the total number of ionizing photons that it radiates in its lifetime to be
Nion = Qion t∗ ,
(4.4)
where we have chosen Qion , the average number of ionizing photons emitted per second
by the star, to be 1050 s−1 (Bromm et al., 2001b; Schaerer, 2002). We then add up the
recombinations in all of the bins, along each of the rays, beginning with those closest to the
star and moving outward, until the number of recombinations along a ray equals the number
of ionizing photons that are emitted along that ray. If the number of recombinations in the
bin falls below the number of atoms in the bin, then we count the number of atoms in the
bin against the number of photons as well. Doing this for each of the rays, we solve in detail
for the H  region of the star. We set the free electron fraction to unity for each of the SPH
particles that lie within the H  region. We set the temperature of the SPH particles within
the H  region, but outside of the 500 pc photo-heated region, to T = 18, 000 K, roughly the
value at the outer edge of the photo-heated region. As well, the fraction of molecules in the
H  region is set to zero, as we assume that all molecules are collisionally dissociated at the
high temperatures in the H  region.
62
4.1 Methodology
Photodissociation
To find the region in which the LW radiation from the star destroys H2 and HD molecules, the
‘LW bubble’ in our terminology, we carry out a ray-tracing procedure similar to the one used
to solve for the H  region. We use the same bins as in that procedure, but now we evaluate
the formation time of H2 molecules in each bin and compare this both to the lifetime of the
star and to the dissociation time of the molecules. For each bin, we compute the H2 formation
time as
X
nH2
/Npart ,
tform,H2 =
(4.5)
nH k1 nH+2 + k2 nH−
where nH+2 and nH− are the number densities of H+2 and H− , respectively. The sum is over
all the particles in the bin, Npart , and k1 and k2 are the rate coefficients for the following two
main reactions that produce H2 :
H + H+2 → H+ + H2 ,
H + H− → e− + H2 .
We adopt the following values for these rate coefficients (de Jong, 1972; Haiman et al.,
1996a; Karpasb) et al., 1979):
k1 = 6.4 × 10−10 cm3 s−1 ,
(4.6)
k2 = 1.3 × 10−9 cm3 s−1 .
(4.7)
The dissociation time for the molecules is obtained by finding the flux of LW photons
from a 100 M Pop III star, assumed to be a blackbody emitter with radius R∗ ' 3.9 R
and effective temperature T ∗ ' 105 K (Bromm et al., 2001b). The dissociation time for
unshielded molecules at a distance R from the star is then given by (Abel et al., 1997)
tdiss,H2
R
∼ 10 yr
1 kpc
5
!2
.
(4.8)
Next, we note that for molecules to be effectively dissociated by the LW radiation, the
dissociation time of the molecules must be shorter than both the lifetime of the star and the
formation time of the molecules. Therefore, we compare all of these timescales for each bin
63
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
along each ray and set the fraction of molecules to zero if tdiss,H2 . tform,H2 and tdiss,H2 . t∗ .
If this condition is not satisfied, then the molecule fraction is left unchanged from its value
before the formation of the star. This allows for the possibility of the effective shielding of
H2 molecules because it accounts for the build-up of H2 column density, for instance, in relic
H  regions or in collapsing minihalos where the formation time of H2 is relatively short.
We take into account the effects of self-shielding by adding up the H2 column density NH2
along the ray contributed by each bin in which the molecules are not effectively dissociated.
We then adjust the dissociation time for the molecules in shielded bins according to (Draine
& Bertoldi, 1996):
!2
!0.75
NH2
R
5
tdiss,H2 ∼ 10 yr
(4.9)
,
1 kpc
1014 cm−2
when the column density of molecules between the bin and the star is NH2 & 1014 cm−2 .
The ionic species H− and H+2 , which are reactants in the main reactions which form H2 ,
can also, in principle, be destroyed by the radiation from the star. The photo-dissociation
times for these species are given in terms of the temperature of the star T ∗ , the source of
thermal radiation in our case, and the distance from the star R, as (de Jong, 1972; Dunn,
1968; Galli & Palla, 1998)
tdiss,H+2 = 5 × 10
−2
tdiss,H− =
T ∗−1.59 exp
9.1T ∗−2.13 exp
82000
T∗
8823
T∗
!
!
R
R∗
R
R∗
!2
s,
(4.10)
!2
s.
(4.11)
For the 100 M star, we find that tdiss,H+2 ∼ 5×103 yr (R/1 kpc)2 and tdiss,H− ∼ 9×102 yr (R/1 kpc)2 .
The formation times for these species, on the other hand, are
tform,H− = nH−
dnH−
dt
!−1
dnH+2
!−1
and
t
form,H+2
=n
H+2
dt
∼ 3 × 103 yr
(4.12)
∼ 4 × 103 yr
(4.13)
for primordial gas at a temperature of T = 100 K and a density of nH = 10−2 cm−3 , typical
for gas at the outskirts of a collapsing minihalo. These formation timescales become much
64
4.1 Methodology
shorter for gas deeper inside minihalos, where the densities and temperatures are generally
higher. It is in these regions, in and around minihalos, where the presence of molecules
is most important for cooling the gas. Within these regions the photo-dissociation times
for these ionic species are less than their formation times only if they are located . 2 kpc
from the star. Thus, photo-dissociation of these species will become ineffective at distances
& 2 kpc from the star, a distance comparable to the size of the H  region of a Pop III star
(Abel et al., 2007; Alvarez et al., 2006a). Since we assume that molecules are collisionally
destroyed inside the H  region, and since the LW bubble will generally be larger than the
H  region, we ignore the photo-dissociation of H− and H+2 in our calculations.
LW photons can also be absorbed by hydrogen atoms, through the Lyman series transitions, as discussed in detail by (Haiman et al., 2000, 1997). However, this atomic absorption
will only have a significant effect on the LW flux over distances large enough that the Hubble
expansion causes many of the LW photons to redshift to wavelengths of the Lyman series
transitions. The light-crossing time for our cosmological box is much shorter than the Hubble time at the redshifts that we consider. Thus, LW photons will be negligibly redshifted as
they cross our cosmological box and we can safely neglect the minimal atomic absorption of
these photons that may take place.
It has also been found that a shell of H2 molecules may form ahead of the expanding
H  regions surrounding the first stars (see Ricotti et al., 2001). These authors find that such
shells may become optically thick to LW photons. However, Kitayama et al. (2004) have
discussed that such shells are likely short-lived, persisting for only a small fraction of the
lifetime of the star. Thus, for our calculations we neglect the possible formation of such
a shell, as we expect that the opacity to LW photons through this shell will be very small
when averaged over the lifetime of the star. Additionally, as we show in Section 4.3.1, the
regions affected by the LW feedback from a single Pop III star extend, at most, only a few
kiloparsecs beyond the H  region of such a star, which itself extends ∼ 5 kpc. If an H2 shell
forms ahead of the H  region, then the extent of the LW bubble will only be suppressed by,
at most, a factor of a few in radius.
4.1.3
Sink Particle Formation
We have carried out two simulations, one with radiative feedback and one in which the
simulation evolves without including radiative effects. For the former simulation, we allow
65
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
stars to form when the density reaches nres , and the expansion of the gas around the star due
to photo-heating suppresses the density so that our resolution limit is not violated. For the
simulation without radiative feedback we allow sink particles to form when the density of
the gas reaches nres . Since the sink particles will form only in minihalos which are expected
to form Pop III stars, we are able to track the star formation rate in the case without feedback
by tracking the formation of sink particles. We can then compare the sites, and rates, of star
formation in each of the simulations in order to elucidate the effect that radiative feedback
has on Pop III star formation.
4.2
Results
In this section we discuss the evolution of the primordial gas under the influence of the radiative feedback which arises as the first stars are formed in a region destined to subsequently
be incorporated into the first protogalaxy. Indeed, other effects will become important in
the course of the buildup of the first galaxies. Among them is the ejection of metals into
these systems by the first supernovae (Bromm et al., 2003). However, we consider the early
regime in which Pop III star formation dominates, and the effects of metals might not yet
be important. Initially only taking into account the stellar radiative feedback, and neglecting
chemical enrichment, relies on our simplifying assumption that only 100 M black holeforming Pop III stars form, which are predicted not to yield supernovae, and therefore not to
eject metals into their surroundings (Heger et al., 2003).
Although the IMF of the first generation of stars is not known with any certainty yet,
there is mounting theoretical evidence that Pop III stars were very massive, and thus it is
very likely that many of these stars ended there lives by collapsing directly to black holes,
emitting few or no metals into the IGM (Fryer et al., 2001; Heger et al., 2003). Here we
assume that all of the stars that form within our cosmological box are black hole-forming
stars which do not enrich the IGM with metals, and which therefore allow subsequent metalfree star formation to occur. Eventually, however, stars which create supernovae will form,
and the ejected metals will be incorporated into the first protogalaxies, thus drawing the
epoch of metal-free star formation to a close. In light of this, we end our simulation after
the formation of the eighth star in our box at a redshift of z ∼ 18, as we expect that at
lower redshifts the effects of the first metals ejected into the primordial gas will become
important (but see Jappsen et al., 2007). Also, at lower redshifts global LW feedback, due
66
4.2 Results
to star formation at distances far larger than our cosmological box, will become increasingly
important. That said, by tracking the formation of individual Pop III stars in our box, we are
able to find a variety of novel results concerning the local radiative feedback from the first
generations of stars.
4.2.1
The First H II region and Lyman-Werner Bubble
The first star appears in our cosmological box at a redshift of z ∼ 23. It forms inside a
minihalo with a total mass . 106 M and the gas within this halo is evaporated due to the
photo-heating from the star. The H  region that is formed around the star can be seen in
Fig. 4.2, which shows the electron fraction, H2 fraction, temperature, and density of the gas,
in projection. The H  region, which has a morphology similar to those found in previous
studies, extends out to ∼ 4 kpc from the star, also similar to results found in previous works
(Abel et al., 2007; Alvarez et al., 2006a; Yoshida et al., 2007a).
As shown in Fig. 4.2, the molecules within ∼ 5 kpc are photodissociated by the LW
radiation from the first star, and the LW bubble extends to only ∼ 1 kpc outside of the H 
region. Noting that the formation timescale for H2 in the neutral IGM at these redshifts is of
the order of ∼ 300 Myr, much longer than the lifetime of the massive stars that we consider
here, we can estimate the distance through the IGM that the LW bubble should extend, RLW ,
by evaluating the criterion for the effective dissociation of molecules at this distance from
the first star: tdiss,H2 = t∗ . Using equation (4.8) and taking the lifetime of the star to be 3 Myr
gives RLW ∼ 5 kpc, consistent with the result we find for the first star, shown in Fig. 4.2 (see
also Ferrara, 1998). Outside of this LW bubble molecules will not be dissociated effectively
by the single Pop III star, owing largely to its short lifetime. Only when continuous star
formation sets in will the LW bubbles of the first generations of stars merge and become
large enough to establish a more pervasive LW background flux (e.g. Haiman et al., 2000).
4.2.2
Thermal and Chemical Evolution of the Gas
The properties of the primordial gas within our box are strongly time-dependent, as any gradual evolution of the gas is disrupted each time a star turns on and heats the gas, ionizes atoms,
and photodissociates molecules. Certain robust patterns, however, do emerge in the course
of the evolution of the primordial gas. Fig. 4.3 shows the chemical and thermal properties of
the gas at a redshift z ∼ 18, just after the death of the eighth star in our cosmological box.
67
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
Figure 4.2: The first H  region and LW bubble. Clockwise from top-left are the temperature,
electron fraction, H2 fraction, and density, plotted in projection. While the size of our cosmological box is ∼ 27 kpc at this redshift, z ∼ 23, here we have zoomed into the inner 20 kpc, in order
to see detail around the first star. The H  region extends out to ∼ 4 kpc in radius, while the LW
bubble extends to ∼ 5 kpc, within which the molecule fraction is zero. In each panel, the lighter
shades signify higher values of the quantity plotted.
68
4.2 Results
Here, the light-shaded particles are those which have been contained within an H  region,
and so have passed through a fully ionized phase.
The ionized gas in the H  regions begins to recombine and cool once the central star
dies. The dynamical expansion of these hot regions leads to the adiabatic cooling of the
gas, as can be seen in the upper left panel of Fig. 4.3. The plot shows relic H  regions at
different evolutionary stages. The older ones are generally cooler, owing to the molecular
cooling that has had more time to lower the temperature of the gas. Indeed, the first relic
H  regions by this redshift, ∼ 70 Myr after the first star formed, have already cooled to
near the temperature of the un-ionized gas. The electron fraction of the relic H  region gas,
however, is still much higher than that of the un-ionized gas, as can be seen in the upper-right
panel of Fig. 4.3. That the cooling of the gas occurs faster than its recombination leads to the
rapid formation of molecules (Johnson & Bromm, 2006; Kang & Shapiro, 1992; Nagakura
& Omukai, 2005; Oh & Haiman, 2003). This elevated fraction of both H2 and HD molecules
in the relic H  region gas is evident in the bottom panels of Fig. 4.3.
The high abundance of molecules in relic H  regions can lead to efficient cooling of
the gas, and this has important consequences in the first protogalaxies. In particular, a high
fraction of HD in these regions could allow the gas to cool to the temperature of the CMB,
T CMB , the lowest temperature attainable by radiative cooling, and this effective cooling may
lead to the formation of lower mass metal-free stars (Johnson & Bromm, 2006; Nagakura
& Omukai, 2005; Yoshida et al., 2007b). Indeed, Fig 4.3 shows that the HD fraction can
greatly exceed the minimum value needed for efficient cooling to the CMB temperature floor
in local thermodynamic equilibrium (LTE), fHD,crit ∼ 10−8 (Johnson & Bromm, 2006).
While the LW feedback from the stars that form in our box can very effectively destroy
molecules within ∼ 5 kpc of the stars by the end of their lives, this feedback is not continuous.
Following the death of a given star, the molecules will begin to reform in the absence of LW
radiation. The time required for the formation of H2 molecules is sensitively dependent on
the ionized fraction of the gas, but the formation time can be relatively short for un-ionized
gas at high densities, as well. In relic H  regions the fraction of H2 can reach 10−4 within
∼ 1 Myr (Johnson & Bromm, 2007). In collapsing minihalos, where the molecules play a
key role in cooling the gas and allowing it to continue collapsing, the formation times are in
general longer at the densities we consider here, n . 20 cm−3 . We find that the formation
timescale for un-ionized gas collapsing in minihalos is tform,H2 ∼ 5 × 105 yr at a density
of 1 cm−3 and a temperature of 900 K, and tform,H2 ∼ 7 × 106 yr at a density of 0.1 cm−3
69
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
and a temperature of 500 K. The average time between the formation of stars in our box
is ∼ 10 Myr, and so the molecules inside sufficiently dense minihalos can often reform and
allow the gas to continue cooling and collapsing, in spite of the intermittent LW feedback
from local star forming regions.
In order to evaluate the possible effects of continuous LW feedback from sources outside
of our box, we have carried out simulations in which we include a LW background which
destroys H2 molecules at a rate given by (Abel et al., 1997)
kdiss = 1.2 × 10−12 JLW s−1 ,
(4.14)
where JLW is the flux of LW photons in units of 10−21 erg s−1 cm−2 Hz−1 sr−1 . We have
carried out simulations in which the value of JLW is taken to be zero before the formation of
the first star and 0.1, 10−2 , and 10−3 afterwards, when a LW background might be expected to
begin building up due to distant star formation. For each of these simulations, we found the
formation redshift of the second star in the box to be z2nd = 16.3, 20, and 20.5, respectively.
In our main simulation, in which we neglect a possible background LW flux, the second
star formed at z2nd = 20.6. This demonstrates that a background LW flux of JLW . 10−2
would likely have little impact on our results, while a larger LW flux would simply delay
the collapse of gas into minihalos and so lower the overall star formation rate in our box,
consistent with previous findings (e.g. Machacek et al., 2001; Mesinger et al., 2006). We
emphasize, however, that for a substantial LW background to be established, a relatively
high continuous star formation rate must be achieved, as we have shown that individual
Pop III stars can only be expected to destroy molecules within ∼ 5 kpc of their formation
sites. In the very early stages of the first star formation, when short-lived single stars are
forming in individual minihalos (e.g. Yoshida, 2006), it appears unlikely that a substantial
LW background would be established, the feedback from the sources being instead largely
local. It may be only later, when continuous star formation begins to occur in larger mass
systems that a pervasive LW background would likely be built up (Greif & Bromm, 2006;
Haiman et al., 2000).
4.2.3
Shielding of Molecules by Relic H II Regions
As star formation continues, the volume occupied by relic H  regions increases. Because
of the high molecule fraction that can develop in these regions, owing to the large electron
70
4.2 Results
Figure 4.3: The properties of the primordial gas at redshift z ∼ 18, at the end of the life of the
eighth star. The SPH particles which have experienced an ionized phase within an H  region
are colored in orange (gray), while those that have not are in black. Clockwise from the top-left,
the temperature, free electron fraction, HD fraction, and H2 fraction are plotted as functions of
gas density. The relic H  region gas cools largely by adiabatic expansion, but, importantly, also
by cooling facilitated by the high abundance of H2 and HD molecules, which arises owing to the
high electron fraction in this gas. The high electron fraction persists until the gas has collapsed
to densities of & 10 cm−3 , as can be seen in the top-right panel. The molecule fraction is highest
at low densities for gas in which the molecules have not been destroyed by LW feedback, giving
rise to the features seen at low densities in the bottom two panels.
71
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
fraction that persists for . 500 Myr (Johnson & Bromm, 2007), the increasing volume of the
IGM occupied by relic H  regions implies an increase in the optical depth to LW photons in
the vicinity of the first star formation sites. By a redshift of z ∼ 18, eight stars have formed
in our cosmological box and each has left behind a relic H  region.
As can be seen in Figs. 4.3 and 4.4, the gas inside the relic H  regions that have formed
contains an H2 fraction generally higher than the primordial abundance of 10−6 , and up to
an abundance of ∼ 10−3 in the denser regions. This elevated fraction of H2 inside the relic
H  regions leads to a high optical depth to LW photons, τLW , through the relic H  regions.
The column density through a relic H  region which recombines in the absence of LW
radiation can become of the order of NH2 ∼ 1015 cm−2 (Johnson & Bromm, 2007). Because
the molecules in the relic H  regions that we consider here are subject to LW feedback from
neighboring star formation regions, the optical depth through these regions may in general
be lower. However, the rapid rate of molecule formation in these regions, even considering
the LW feedback from local star forming regions in our box, allows the molecule fraction to
approach 10−4 as late as ∼ 100 Myr after the death of the central star. This elevated molecule
fraction combined with the growing volume-filling fraction of relic H  regions leads to an
appreciable optical depth to LW photons, which generally increases with time as more stars
form and create more relic H  regions. To quantify this effect, we calculate the average
column density of H2 molecules through a cubic region of side length l as the product of the
length l and the volume averaged number density of H2 molecules, given by
NH2
P
nH V
'l P 2 ,
V
(4.15)
where the sum is over all of the SPH particles in the volume and nH2 is the number density of
H2 at each of the SPH particles. The volume associated with each individual SPH particle,
V, is estimated as V ' mSPH /ρ, where mSPH is the mass of the SPH particle and ρ is the
mass density of the gas at that particle. The optical depth to LW photons is then computed
as (Draine & Bertoldi, 1996; Haiman et al., 2000)
τLW
!
NH2
' 0.75ln
.
1014 cm−2
(4.16)
Fig. 4.5 shows the optical depth to LW photons avergaged both over the central comoving
153 kpc h−1 of our cosmological box, in which the first star forms, and over the entire box,
72
4.2 Results
for which the comoving side length is 460 kpc h−1 . Before the formation of the first star, the
optical depth evolves largely owing to the cosmic expansion, following the relation τLW '
nH2 l ∝ (1 + z)2 , because the average H2 fraction does not change appreciably. However,
with the formation of the first star in our box at z ' 23 the optical depth begins to change
dramatically in the inner portion of the box, first falling to a value of ' 0.1 due to the LW
feedback from the first star and then steadily climbing to values & 2 as copious amounts of
molecules form inside the relic H  regions that accumulate as star formation continues.
The evolution of the optical depth averaged over the entire box is not as dramatic, as the
fraction of the volume of the whole box occupied by relic H  regions is much smaller than
the fraction of the central region that is occupied by these molecule-rich regions. However,
the optical depth averaged over the whole box, which is a better estimate of the optical depth
over cosmological distances, still rises to τLW & 1.5 across our box, an appreciable value
which will serve to impede the build-up of a cosmologically pervasive LW background.
4.2.4
Black Hole Accretion
Accretion onto Pop III relic black holes may be inefficient for some time following the formation of these objects, owing to the fact that Pop III stars photo-heat and evaporate the gas
within the minihalos which host them (Johnson & Bromm, 2007; Yoshida et al., 2007a). Indeed, accretion onto Pop III relic black holes at close to the Eddington limit can only occur
if the accreted gas has a density above ∼ 102 cm−3 , and it is only in collapsing halos that
such densities are achieved at the high redshifts at which the first stars formed (Johnson &
Bromm, 2007). By assumption, all of the stars that are formed in our simulation are black
hole-forming Pop III stars. If these black holes remain inside their host minihalos, then by
tracking the evolution of the gas within these photo-evaporated host minihalos, we can learn
when efficient accretion onto these Pop III relic black holes may occur.
The minihalo within which the first star forms at a redshift z ∼ 23 resides within the relic
H  region left by the first star. Due to the formation of a high fraction of molecules, and
to the molecular cooling that ensues, the relic H  region gas cools down to temperatures
∼ 103 K, below the virial temperature of this 106 M minihalo. The gas then re-collapses
into the minihalo, reaching a peak density of nres at a redshift z ∼ 19, or ∼ 50 Myr after
the formation of the first star. Fig. 4.6 shows the properties of the relic H  region gas as a
function of distance from the center of this minihalo, at the time when the gas has collapsed
73
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
Figure 4.4: The properties of the primordial gas at the end of the life of the eighth star in our box,
at redshift z ∼ 18. Clockwise from top-left are the temperature, electron fraction, H2 fraction, and
density of the gas, in projection. Here we show the entire cosmological box, which is ∼ 35 kpc.
Note the high electron and H2 fractions in the relic H  regions, where recombination is taking
place. The elevated H2 fraction in these regions raises the optical depth to LW photons through
them significantly, as is illustrated in Fig. 5. The temperature in the older relic H  regions is not
greatly elevated as compared to the temperature of the un-ionized gas, owing to the adiabatic and
molecular cooling that takes place in these regions. The lighter shades denote higher values of
the quantities plotted.
74
4.2 Results
Figure 4.5: The optical depth to LW photons, τLW , averaged over two volumes in our box, as a
function of redshift, z. The diamonds denote the optical depth averaged over the entire cosmological box, while the crosses denote the optical depth averaged only over a cube containing the
inner comoving 153 kpc h−1 of the box, centered in the middle of the box with a volume one
ninth that of the whole box. It is within this region that the first star forms and the star formation
rate is higher than the average star formation rate over the whole box, and this is reflected in the
higher local optical depth in this region as relic H  regions accumulate in the box. The average
optical depth through the entire box also rises, but the increase is less dramatic. The solid line
denotes the optical depth to LW photons, averaged over the whole box, that would be expected
for the case that the gas maintains the average cosmological density everywhere and that the H2
fraction does not change from the primordial value of 2 × 10−6 ; for this case, the optical depth
changes owing only to cosmic expansion. Note that the optical depth averaged over the whole
box matches well this idealized case up until the first star forms at a redshift of z ∼ 23. The
temporary drops in the optical depth occur due to LW feedback when individual stars form.
75
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
to a density of nres . We cannot, with this simulation, resolve what happens once the gas
collapses further and reaches higher densities. However, we can estimate the time it will
take for the gas at the center of the halo to reach a density of n ∼ 102 cm−3 as the free fall
time of the gas, which is tff ∼ 10 Myr. Thus, a Pop III relic black hole at the center of
this halo could be expected to begin accreting gas efficiently ∼ 60 Myr after its formation.
This is a significant delay, and could pose serious challenges to theories which predict that
efficient accretion onto Pop III relic black holes can lead to these black holes becoming the
supermassive black holes that power the quasars observed in the Sloan Digital Sky Survey at
redshifts z & 6 (Johnson & Bromm, 2007; Li et al., 2007).
4.2.5
HD Cooling in Relic H II Regions
While abundant molecules can form within relic H  regions, the LW feedback from neighboring star-forming regions can suppress the effect of this elevated fraction of molecules.
The electron fraction remains high in relic H  regions for up to ∼ 500 Myr in the general IGM, but in higher density regions where the gas is recollapsing, the electron fraction
drops much more quickly. Figs. 4.3 and 4.6 show that the electron fraction drops to a value of
. 10−4 , comparable to the electron fraction of the un-ionized gas, once the density of the relic
H  region gas becomes & 10 cm−3 . Thus, once the gas reaches these densities the ionized
fraction will become too low to catalyze the formation of a high fraction of molecules, and
of HD molecules in particular. Therefore, in order for HD to be an effective coolant of the
primordial gas in relic H  regions, the abundanct HD molecules that are formed at densities
. 10 cm−3 must not be destroyed by LW feedback from neighboring star-forming regions
before the gas collapses to high densities and forms stars. If we estimate the timescale on
which the relic H  region gas would collapse to form stars as the free-fall time of the gas,
we find that the molecules must be shielded from photodissociating radiation for at least
tff & 10 Myr in order for the high abundance of HD molecules to persist, so that the formation of so-called Pop II.5 stars might be enabled, with their hypothesized masses of ∼ 10 M
(Johnson & Bromm, 2006).
We find that the relic H  region gas that re-collapses into the minihalo in which the first
star formed, shown in Fig. 4.6, carries a high fraction of HD molecules, as LW feedback from
neighboring stars does not effectively dissociate the molecules in this relatively dense and
self-shielded gas. The HD fraction exceeds 10−7 , becoming an order of magnitude higher
76
4.2 Results
Figure 4.6: The properties of the relic H  region gas which recollapses into the minihalo which
hosted the first star. Clockwise from the top-left are the density, temperature, H2 fraction, and
free electron fraction plotted as functions of distance from the center of the minihalo at the time
when the density reaches nres , at a redshift of z ∼ 19. The temperature of the gas has dropped
to below 103 K, well below the virial temperature of the minihalo, owing to molecular cooling.
Owing to the high electron fraction that persists in this relic H  region, the molecule fraction
in this gas is higher than in the case of un-ionized primordial gas collapsing into a minihalo.
Indeed, as can be seen in Fig. 4.3, the HD fraction is roughly an order of magnitude higher at
these densities than in the case of un-ionized gas collapsing into a minihalo, which may allow
for the efficient cooling of the gas to temperatures T & T CMB and so perhaps for the formation of
metal-free stars with masses of the order of 10 M .
77
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
than its value for un-ionized primordial gas collapsing in a minihalo. Thus, for this case, HD
cooling will likely be effective at higher densities as the gas collapses further, and we expect
that a Pop II.5 star, with a mass of the order of 10 M , might form later on, if we were to
run the simulation further (Johnson & Bromm, 2006; Nagakura & Omukai, 2005; Yoshida
et al., 2007b). Had star formation taken place nearer this minihalo between the formation of
the first star and the re-collapse of the gas into the host minihalo, then the molecule fraction
would likely not be so elevated, and a higher mass metal-free star would be more likely to
form. Thus, while in our simulation it appears that the first relic H  region that forms may
give rise to Pop II.5 star formation, we emphasize that the possibility of the formation of
Pop II.5 stars in relic H  regions is very dependent on the specific LW feedback that affects
the gas in these regions.
4.2.6
Star Formation in the Presence of Radiative Feedback
To discern the effect the local radiative feedback from the first stars has on the star formation
rate, we have compared the results obtained from our simulations with and without radiative
feedback. By a redshift of z ∼ 18, a total of nine star-forming regions were identified in
our simulation without feedback, while at the same epoch eight stars had formed in our
simulation including feedback. Thus, we find that the average star formation rate at redshifts
z & 18 is diminished by a factor of perhaps only . 20 percent due to local radiative feedback,
although this result is subject to the small number statistics within our single cosmological
box. Fig. 4.7 shows the locations of the sites of star formation for both cases, plotted in
comoving coordinates against the projected density field. The orange squares denote sites
where Pop III stars could have formed in the case without feedback, while the green dots
denote sites where Pop III stars formed in the simulation including radiative feedback. Thus,
the sites where star formation is suppressed by the radiative feedback are marked by the
orange squares which are not filled by a green dot.
We point out, however, that we do not include LW feedback from stars which may have
formed outside of our box, and hence it is possible that the overall LW feedback may be
stronger than we find here. At redshift z ∼ 18, we end the simulation, but note that star
formation will likely take place at an increasing rate as the collapse fraction increases with
time. This could lead to a continuous LW background produced within our box, different
from the intermittent LW feedback produced by individual stars that occurs in the simulation
78
4.3 Summary and Discussion
down to z ∼ 18.
In addition, the limits of our resolution prohibit us from discerning the stronger shielding
of H2 molecules and I-front trapping that could occur within very dense collapsing minihalos
(Ahn & Shapiro, 2007). However, we note that these authors find that the radiative feedback
on collapsing minihalos from nearby stars generally does not greatly affect the final outcome
of the collapse, as halos which collapse in the absence of radiative feedback generally also
collapse when radiative feedback is applied (Susa & Umemura, 2006), roughly consistent
with our results in the present work. Thus, our limited resolution may not substantially impact the results that we find for the slight suppression of star formation due to local radiative
feedback, although higher resolution simulations will be necessary to more precisely study
the full impact of radiative feedback from the first stars on the first protogalaxies.
4.3
Summary and Discussion
We have performed cosmological simulations which self-consistently account for the radiative feedback from individual Pop III stars, as they form in the course of the assembly of
the first protogalaxies. We have solved in detail for the H  regions, as well as for the LW
bubbles of these stars, wherein molecule-dissociating radiation effectively destroys H2 and
HD molecules. The local radiative feedback from the first stars is complex, and we find a
variety of novel results on the evolution of the primordial gas, on the effects of the LW radiation from the first stars, on the nature of second generation star formation, and on black
hole accretion.
While the LW radiation from the first stars can, in principle, greatly suppress Pop III star
formation in the early universe, we find that a number of factors minimize the effectiveness
of this negative feedback. Firstly, the LW radiation produced locally by individual stars is
not uniform and constant, as LW feedback has been modeled in previous work (Ciardi &
Ferrara, 2005; Mesinger et al., 2006), but rather is present only during the brief lifetimes of
the individual stars that produce it. Thus, even if the molecules in collapsing minihalos and
relic H  regions are destroyed by the radiation from individual stars, they will, at the early
stages of Pop III star formation, have time to reform and continue cooling the primordial
gas in between the times of formation of local stars. Furthermore, because the LW bubbles
of individual Pop III stars extend only to RLW ∼ 5 kpc from these sources, due to the short
stellar lifetimes, the build up of a pervasive LW background would likely have to await the
79
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
Figure 4.7: The sites of star formation with and without radiative feedback, at redshift z ∼
18. The black dots show the density field in our simulation box, in projection. The orange
squares show the locations of minihalos in which Pop III star formation could take place, in our
simulation without radiative feedback. The green dots show the sites where star formation takes
place in our simulation including radiative feedback.
80
4.3 Summary and Discussion
epoch of continuous star formation, which is fundamentally different from the epoch of the
first stars, in which these sources shine for short periods within individual minihalos.
As star formation continues, the volume-filling fraction of relic H  regions increases as
well, and this, combined with the high fraction of molecules that form in these regions, leads
to an opacity to LW photons through the IGM which increases with time. This opacity can
become of the order of τLW & 2 through individual relic H  regions (Johnson & Bromm,
2007; Machacek et al., 2001, 2003; Oh & Haiman, 2002; Ricotti et al., 2001). Furthermore,
as the volume-filling fraction of relic H  regions increases with time, τLW through the general
IGM may become similarly large, and this effect will have to be considered in future work
which seeks to elucidate the effect of LW feedback on Pop III star formation.
We find that metal-free stars with masses of the order of 10 M , the postulated Pop II.5
stars (Greif & Bromm, 2006; Johnson & Bromm, 2006), might form from the moleculeenriched gas within the first relic H  regions, although we note that this may not occur in
general, due to LW feedback from neighboring star-forming regions. This susceptibility to
LW feedback is due to the fact that the high fraction of HD molecules which forms in the
electron-rich, low density regions of relic H  regions must persist until this gas has had
time to collapse to high densities and form stars. If LW feedback from neighboring starforming regions destroys the molecules after the gas has collapsed to densities & 10 cm−3 ,
then the abundance of HD molecules will not likely be elevated when the gas forms stars and
Pop II.5 star formation may be suppressed. Because this implies that the molecules must be
shielded from LW radiation for, at least, the free-fall time for gas with densities . 10 cm−3 ,
or & 10 Myr, we conclude that Pop II.5 star formation in relic H  regions may occur only
in the circumstances when local Pop III star formation is suppressed over such timescales.
However, we also point out that the shielding provided by the high H2 fraction in these relic
H  regions may help to minimize the LW feedback from neighboring star-forming regions,
and so may make Pop II.5 star formation in relic H  regions possible in many cases.
We find that the ionized primordial gas surrounding the first star formed in our simulation,
at a redshift of z ∼ 23, recombines and cools by molecular cooling to temperatures below the
virial temperature of the minihalo that hosted this first star. Thus, this relic H  region gas is
able to re-collapse to densities & 20 cm−3 within this minihalo after ∼ 50 Myr from the death
of the star. It is predicted that many Pop III stars will collapse directly to form black holes
with masses of the order of 100 M (Heger et al., 2003), and if such a black hole resides
within this host halo, then we find that it may begin accreting dense primordial gas at close
81
4. LOCAL RADIATIVE FEEDBACK IN THE FORMATION OF THE FIRST
PROTOGALAXIES
to the Eddington rate after ∼ 60 Myr from the time of its formation. This is an important
consideration to be incorporated into models of the growth of the 109 M black holes which
have been observed at redshifts & 6, as it places constraints on the amount of matter that a
given relic Pop III black hole could accrete by this redshift (Haiman & Loeb, 2001; Li et al.,
2007; Volonteri & Rees, 2006).
Finally, by comparing the star formation rates which we derive from our simulation including radiative feedback with those derived from our simulation in which feedback is left
out, we have seen that local radiative feedback from the first stars likely only diminishes the
Pop III star formation rate by a factor of, at most, a few. In our simulation, in particular, we
find that this rate is decreased by only . 20 percent, although this may be less suppression
than would be expected by the overall radiative feedback, as we did not include the possible
effects of a global LW background. Future simulations which resolve densities higher than
those reached here, and which self-consistently track the build-up of the LW background
along with the IGM opacity to LW radiation, will be necessary to more fully explore the
radiative effects of the first stars on the formation of the first galaxies. However, the goal of
understanding the formation of the first galaxies is now clearly getting within reach, and the
pace of progress is expected to be rapid.
82
5
The Observational Signature of the First
H II Regions
One of the most important questions in modern cosmology is to understand how the first stars
ended the cosmic dark ages at redshifts z . 30 (Barkana & Loeb, 2001; Bromm & Larson,
2004; Ciardi & Ferrara, 2005). Their emergence led to a fundamental transformation in
the early Universe, from its simple initial state to one of ever increasing complexity. The
emission from the hot, T eff ∼ 105 K, photospheres of Pop III stars began the reionization
of primordial hydrogen and helium in the IGM, although this process was completed only
later on, when more massive galaxies formed (Fan et al., 2006a). In addition, the supernova
explosions that ended the lives of massive Pop III stars distributed the first heavy elements
into the IGM (Bromm et al., 2003; Greif et al., 2007; Tornatore et al., 2007; Wise & Abel,
2008b). This latter process might have had a significant impact on the physics of early star
formation, as metal-enriched gas can cool more efficiently than primordial gas (Bromm &
Loeb, 2003b; Jappsen et al., 2007, 2009a; Omukai et al., 2005).
Based on numerical simulations, a general consensus has emerged that the first stars
formed in dark matter minihalos at z . 30, in isolation or at most as a small stellar multiple,
and with typical masses of M∗ ∼ 100 M (for a recent review, see Bromm et al., 2009). It
is crucial to observationally test this key prediction. However, it has become evident that
this will be very challenging. Even the exquisite near-IR (∼ nJy) sensitivity of the upcoming
JWST will not suffice to directly image such massive, single Pop III stars (Bromm et al.,
2001b; Gardner et al., 2006), unless they explode as energetic pair-instability supernovae
83
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
(Heger & Woosley, 2002; Scannapieco et al., 2005b). The direct spectroscopic detection of
recombination line emission from the H  region surrounding the Pop III star is beyond the
capability of JWST as well, although such line emission might be detectable from primordial
stellar populations inside more massive host halos (Johnson et al., 2009; Schaerer, 2002,
2003).
An alternative approach is to search for the global signature from many Pop III stars that
formed in minihalos over large cosmic volumes (Haiman & Loeb, 1997). One such probe
is the optical depth to Thomson scattering of CMB photons off free electrons along the line
of sight, determined by the five-year WMAP measurement to be τe ' 0.09 ± 0.02 (Komatsu
et al., 2009). This signal, however, is dominated by ionizing sources that must have formed
closer to the end of reionization, with only a small contribution from Pop III stars formed
in minihalos (Greif & Bromm, 2006; Schleicher et al., 2008). A second empirical signature
is the combined bremsstrahlung emission from the H  regions around those minihalos that
hosted Pop III stars. The resulting free-free radio emission leads to spectral distortions that
might be detectable in the Rayleigh-Jeans part of the CMB spectrum. Recently, the ARCADE 2 experiment has attempted to measure such a free-free contribution from the epoch
of the first stars (Kogut et al., 2006). The surprisingly strong signal found, however, cannot
originate in early Pop III stars, and in any case would overwhelm the much weaker contribution from the first stars and galaxies (Seiffert et al., 2009). The most promising detection
strategy might be to scrutinize the background from the redshifted 21 cm line of neutral hydrogen (Furlanetto et al., 2006). Once the central Pop III star has died, the relic H  region
left behind would provide a bright source of 21 cm emission (Tokutani et al., 2009). Again,
individual sources are much too weak to leave a detectable imprint, but the planned SKA
might be able to detect the cumulative signal (Furlanetto, 2006; Lazio, 2008).
We here carry out radiation hydrodynamics simulations of the evolution of H  regions
around massive Pop III stars in minihalos, giving us a detailed understanding of the properties of individual sources. We combine this with an approximate, Press-Schechter type
analysis of the cosmological number density of minihalos as a function of redshift to derive the observational signature of the first H  regions, where we specifically focus on the
free-free and 21 cm probes. In this sense, we organize our work as follows. In Section 5.2,
we describe the simulation setup and our implementation of the radiative transfer scheme
in the SPH code GADGET-2 (Springel, 2005). In Section 5.3, we discuss the properties of
the first H  regions and their observational signature in terms of recombination radiation,
84
5.1 Numerical Methodology
bremsstrahlung and 21 cm emission. Finally, in Section 5.4 we summarize our results and
assess their implications. For consistency, all quoted distances are physical, unless noted
otherwise.
5.1
Numerical Methodology
Our treatment of ionizing and photodissociating radiation emitted by massive Pop III stars
is very similar to the methodology introduced in Johnson et al. (2007) and Yoshida et al.
(2007a), with the exception that we here take the hydrodynamical response into account,
self-consistently coupled to the chemical and thermal evolution of the gas. This allows us
to model dense (D-type) as well as rarefied (R-type) ionization fronts, which is crucial for
a proper treatment of the breakout of ionizing radiation. In the following, we describe our
simulation setup, as well as the numerical implementation of the ray-tracing algorithm.
5.1.1
Simulation Setup
We perform our simulations in a cosmological box with linear size 200 kpc (comoving), and
2563 particles per species, corresponding to a particle mass of ' 17 M for dark matter and
' 3 M for gas. The simulations are initialized at z = 99 with a fluctuation power spectrum
determined by a ΛCDM cosmology with matter density Ωm = 1 − ΩΛ = 0.27, baryon density
Ωb = 0.046, Hubble parameter h = H0 /100 km s−1 Mpc−1 = 0.7, where H0 is the Hubble expansion rate today, and spectral index n s = 0.96 (Komatsu et al., 2009). We use an artificially
high fluctuation power of σ8 = 1.6 to accelerate structure formation in our relatively small
box, although the cosmological mean is given by σ8 = 0.81. We take the chemical evolution
of the gas into account by following the abundances of H, H+ , H− , H2 , H+2 , He, He+ , He++ ,
and e− , as well as the three deuterium species D, D+ , and HD. We include all relevant cooling mechanisms, i.e. H and He collisional ionization, excitation and recombination cooling,
bremsstrahlung, inverse Compton cooling, and collisional excitation cooling via H2 and HD
(Glover & Jappsen, 2007). We explicitly include H2 cooling via collisions with protons and
electrons, which is important for the chemical and thermal evolution of relic H  region gas
(Glover & Abel, 2008).
We run the simulations until the first minihalo in the box has collapsed to a density of
nH = 104 cm−3 , at which point the gas has cooled to ' 200 K and becomes Jeans-unstable
85
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
Figure 5.1: Sequential zoom-in on the first star-forming minihalo at z∗ ' 20. Shown is the
density-squared weighted average of the hydrogen density along the line of sight, just after the
formation of the Jeans-unstable clump in a ' 9.4 × 105 M minihalo. The flattening of the core
due to angular momentum conservation during the collapse is clearly visible, with the consequence that ionizing radiation from the central source breaks out anisotropically (see Fig. 5.3).
(Abel et al., 2002; Bromm et al., 2002). The first halo that fulfils this criterion collapses at
z∗ ' 20 and has a virial mass of ' 9.4×105 M and a virial radius of ' 90 pc. Highly resolved
simulations have shown that at later times, the gas condenses further under the influence of
self-gravity to nH ∼ 1021 cm−3 , where it becomes optically thick and forms a protostellar
seed (Yoshida et al., 2008). Due to its residual angular momentum, the central clump flattens
and likely evolves into an accretion disk. In our case, we find a flattened structure already
at a density of nH = 104 cm−3 (see Fig. 5.1). The high temperature of the gas then leads
to efficient accretion from the disk and it is commonly assumed that the protostar grows to
∼ 100 M within the Kelvin-Helmholtz time (Bromm & Loeb, 2004). However, we note that
under certain conditions the disk may fragment to form multiple objects of smaller masses
(Clark et al., 2008). Unfortunately, the details of the accretion phase and the concomitant
radiative feedback are poorly understood, although some analytic investigations have been
carried out (McKee & Tan, 2008; Tan & McKee, 2004). Under these circumstances it seems
best to initialize the calculation of the H  region at the onset of the initial Jeans-instability,
when the density exceeds nH = 104 cm−3 .
86
5.1 Numerical Methodology
5.1.2
Ray-tracing Scheme
The procedure used to calculate the Strömgren sphere around the star for a given time-step
∆t is similar to the ray-tracing scheme used in Johnson et al. (2007). We first designate an
individual SPH particle as the source of ionizing radiation and create a spherical grid with
typically 105 rays and 500 logarithmically spaced radial bins around the source particle. The
minimum radius is set by the smoothing length of the central particle, while the maximum
radius is chosen appropriately to encompass the entire H  region. This approach may seem
crude compared to existing methods, which use adaptive grids (e.g. HEALPix; Górski et al.,
2005), but the increased angular and radial resolution towards the center tend to mirror the
existing density profile. However, one must proceed with care if the ionization front encounters dense clumps far from the source, where the resolution may no longer be sufficient.
In a single, parallel loop, the Cartesian coordinates of all particles are converted to spherical coordinates, such that their density and chemical abundances are mapped to the bins
corresponding to their radius, zenith angle and azimuth, denoted by r, θ and φ, respectively.
The volume of each particle is approximately given by ∆V ' h3 , which transforms to ∆r = h,
∆θ = h/r and ∆φ = h/(r sin θ). If the volume element of a particle intersects with the volume
element of a bin, the particle contributes to the bin proportional to the density of the particle
squared. This dependency ensures that overdense regions are not missed if the bin size is
much larger than the smoothing length, which could occur far from the source where the
grid resolution is poor. Accidental flash-ionization of minihalos is thus avoided. Once the
above steps are complete, it is straightforward to solve the ionization front equation along
each ray:
Z rI
Ṅion
2 drI
ne n+ r2 dr ,
nn rI
=
− αB
(5.1)
dt
4π
0
where rI denotes the position of the ionization front, Ṅion the number of ionizing photons
emitted per second, αB the case B recombination coefficient, and nn , ne and n+ the number
densities of neutral particles, electrons and positively charged ions, respectively. We assume
that the recombination coefficient remains constant at its value for 104 K, which is roughly
the temperature of the H  and He  regions.
The numbers of H /He  and He  ionizing photons are given by
Ṅion
πL∗
=
4
σT eff
Z
87
∞
νmin
Bν
dν ,
hν
(5.2)
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
where h from now on denotes Planck’s constant, σ Boltzmann’s constant, and νmin the minimum frequency corresponding to the ionization threshold of H  and He . We assume that
massive Pop III stars emit a blackbody spectrum Bν (in erg s−1 cm−2 Hz−1 sr−1 ) with an effective temperature T eff = dex(4.922, 4.975, 4.999) K and luminosity L∗ = dex(5.568, 6.095,
6.574) L for a 50, 100 and 200 M star, respectively (Schaerer, 2002). This yields
Ṅion,HI/HeI = [2.80, 9.14, 26.99] × 1049 s−1
(5.3)
Ṅion,HeII = [0.72, 4.14, 15.43] × 1048 s−1 .
(5.4)
and
We do not distinguish between the H  and He  region, which is a good approximation for
massive Pop III stars (Osterbrock & Ferland, 2006). Their lifetimes are given by t∗ = 3.7, 2.7
and 2.2 Myr, respectively. We neglect the effects of stellar evolution, which might lead to a
decrease of the number of ionizing photons emitted at the end of the main sequence (Marigo
et al., 2001; Schaerer, 2002), although recent investigations have shown that rotating Pop III
stars remain on bluer evolutionary tracks and this effect might not be so strong (Vázquez
et al., 2007; Woosley & Heger, 2006; Yoon & Langer, 2005).
To obtain a discretisation of the ionization front equation, we replace the integral on the
right-hand side of equation (5.1) by a discrete sum:
Z
rI
ne n+ r2 dr '
X
0
ne,i n+,i ri2 ∆ri ,
(5.5)
i
where ∆ri is the radial extent of bin i, and the sum extends from the origin to the position
of the ionization front at the end of the current time-step ∆t. The above equation describes
the advancement of the ionization front due to an excess of ionizing photons compared to
recombinations. Similarly, the left-hand side of equation (5.1), which models the propagation
of the ionization front into neutral gas, is discretised by
nn rI2
drI
1 X
'
nn,i ri2 ∆ri ,
dt
∆t i
(5.6)
where the sum now extends from the position of the ionization front at the previous timestep to its position at the end of the current time-step. We perform the above steps separately
88
5.1 Numerical Methodology
for the H  and He  regions, since they require distinct heating and ionization rates. For
the He  region, we replace the quantities nn and n+ in equation (5.1) with nn = fHeII nH
and n+ = fHeIII nH , where fX is the number density of species X relative to nH . We adopt a
rate of αB = 1.3 × 10−12 cm3 s−1 for He  recombinations to He  (Osterbrock & Ferland,
2006). Applying the same prescription to the H  region, we find nn = ( fHI + fHeI ) nH and
n+ = ( fHII + fHeII ) nH . Similarly, we adopt a rate of αB = 2.6 × 10−13 cm3 s−1 for hydrogen
and helium recombinations to the ground state (Osterbrock & Ferland, 2006). We initialize
the calculation of the H  region at the boundary of the He  region, since hydrogen and
helium are maintained in their first ionization states by recombinations of He  to He 
(Osterbrock & Ferland, 2006). We note that the exact position of the ionization front is not
restricted to integer multiples of our pre-defined radial bins, but may instead lie anywhere
in between. For this purpose we adopt a simple linear scaling of the number of ionizations
and recombinations as a function of the relative position of the ionization front. The most
expensive step in terms of computing time is the assignment of the density and the chemical
abundances to the grid, while the ray-tracing itself requires only a negligible amount of
ressources.
5.1.3
Photoionization and Photoheating
Once the extent of the H  and He  region have been determined, the SPH particles within
these regions are assigned an additional variable that stores their distance from the source.
This information is then passed to the chemistry solver, which determines the relevant ionization and heating rates, given by
kion =
and
Γ = nn
Z
∞
νmin
F ν σν
dν
hν
νmin Fν σν 1 −
dν ,
ν
νmin
Z
(5.7)
∞
(5.8)
where Fν and σν denote the incoming specific flux and ionization cross section, respectively.
For the case of a blackbody,
L∗
Fν =
B ,
(5.9)
4 2 ν
4σT eff
r
89
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
where r is the distance from the source. The resulting rates are given by
kion,HI =
[0.45, 1.32, 3.69] × 10−6 −1
s ,
r/pc 2
(5.10)
kion,HeI =
[0.42, 1.43, 4.29] × 10−6 −1
s ,
r/pc 2
(5.11)
kion,HeII =
[0.67, 3.72, 13.57] × 10−8 −1
s ,
r/pc 2
(5.12)
[0.40, 1.28, 3.74] × 10−17
erg s−1 cm−3 ,
r/pc 2
(5.13)
[0.41, 1.57, 4.94] × 10−17
erg s−1 cm−3 ,
r/pc 2
(5.14)
[0.72, 4.46, 17.13] × 10−19
erg s−1 cm−3
r/pc 2
(5.15)
ΓHI = nHI
ΓHeI = nHeI
ΓHeII = nHeII
for a 50, 100 and 200 M Pop III star, respectively. These are taken into account every timestep, while the ray-tracing is performed only every fifth time-step. Since the hydrodynamic
time-step is generally limited to one twentieth of the sound-crossing time through the kernel,
our treatment of the coupled evolution of the ionization front and the hydrodynamic shock
is roughly correct. The computational cost of runs with and without ray-tracing are typically
within a factor of a few.
5.1.4
Photodissociation and Photodetachment
The final ingredient in our algorithm is the inclusion of molecule-dissociating radiation.
This effect turns out to be of only minor importance in the present study, but will render
our algorithm capable of addressing a general set of early Universe applications. Molecular
hydrogen is the most important coolant in low-temperature, primordial gas, but is easily
destroyed by radiation in the LW bands between 11.2 and 13.6 eV. The small residual H2
fraction in the IGM leads to a very small optical depth over cosmological distances, such
that even a small background can have a significant effect (Glover & Brand, 2001; Haiman
et al., 2000; Johnson et al., 2007). In our implementation, we do not take self-shielding
into account, which becomes important for H2 column densities & 1014 cm−2 (Draine &
90
5.2 Observational Signature
Bertoldi, 1996). Such a high column density is difficult to achieve in minihalos, and is more
likely to occur within the virial radius of the first galaxies (Oh & Haiman, 2002). However,
the onset of turbulence in the first galaxies likely leads to a reduction of self-shielding via
Doppler shifting (Greif et al., 2008; Wise & Abel, 2007a). For this reason we treat the
photodissociation of H2 in the optically thin limit, such that the dissociation rate in a volume
limited by causality to a radius r = c t∗ is given by kH2 = 1.1 × 108 FLW s−1 , where FLW is the
integral of the specific flux Fν over the LW bands, resulting in
kH2 =
[1.27, 3.38, 9.07] × 10−7 −1
s
r/pc 2
(5.16)
for a 50, 100 and 200 M Pop III star, respectively. Finally, we equate the photodissociation
rate of hydrogen deuteride to that of molecular hydrogen. For the reasons discussed in Johnson et al. (2007), we do not explicitly include photodetachment of H− and photodissociation
of H+2 .
5.2
Observational Signature
In the following, we discuss the direct observational signature of the first H  regions in
terms of recombination radiation, as well as their indirect signature in terms of a global radio
background produced by bremsstrahlung and 21 cm emission.
5.2.1
Build-up of H II Region
The build-up of the first H  regions by Pop III stars in minihalos was treated in one dimension by Kitayama et al. (2004) and Whalen et al. (2004), and in three dimensions by Alvarez
et al. (2006a), Abel et al. (2007) and Yoshida et al. (2007a). They found that recombinations
initially balanced ionizations within the virial radius of the host halo, leading to the formation of a D-type ionization front. Breakout occured after the density dropped sufficiently for
the ionization front to race ahead of the hydrodynamic shock, becoming R-type. The hydrodynamic response of the gas is self-similar, since minihalos approximately resemble singular
isothermal spheres (Alvarez et al., 2006a; Shu et al., 2002). The relevant parameters are set
by the temperature of the singular isothermal sphere and the H  region, which in our case
are T ' 200 and ' 104 K, respectively.
91
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
Figure 5.2: The hydrodynamic response of the gas to photoheating by a 100 M Pop III star after
50 kyr, 300 kyr, and 2.7 Myr (from left to right). Shown is the hydrogen density as a function
of radius for the simulation (black dots) and the analytic Shu et al. (2002) solution (green solid
line), as well as the initial density profile of a singular isothermal sphere with 200 K (red dotted
line). The functional form of the analytic solution is reproduced perpendicular to the disk, where
the ionization front breaks out after only a few 10 kyr. However, this is not the case in the plane
of the disk, where the gas remains neutral and dense until the end of the star’s lifetime.
In Fig. 5.2, we compare the density profile of the Shu et al. (2002) solution to the simulation for the case of a 100 M Pop III star. Interestingly, we find a clear deviation from
the ideal, spherically symmetric solution already during the D-type phase, which is caused
by the anisotropic collapse of the minihalo. Due to angular momentum conservation, the
gas spins up and forms a flattened, disk-like structure at a density of 104 cm−3 , which can
be seen in the right panel of Fig. 5.1, and in the left panel of Fig. 5.2, where it is evident
that the density dispersion is almost an order of magnitude within the central ' 10 pc. In
response to this anisotropy, which is further amplified by the density-squared dependence
of recombinations, the ionization front first breaks out perpendicular to the disk, where the
column density is lowest. This is visible in the left and middle panels of Fig. 5.3, as well as
in Fig. 5.2, where the Shu et al. (2002) solution is approximately reproduced perpendicular
to the disk, while the plane of the disk remains neutral and dense. Once the ionization front
becomes R-type, spherical symmetry is asymptotically restored and the H  region expands
to rHII ' 1.9, 2.7 and 3.7 kpc for the 50, 100 and 200 M Pop III star, respectively. We
find that He  ionizing photons within the He  region increase the central temperature by
a factor of ' 1.5, leaving only a small imprint on the dynamical evolution of the H  region
92
5.2 Observational Signature
(see Yoshida et al., 2007a). However, the He  λ1640 recombination line within the He 
region may be used as a distinct probe for the presence of massive Pop III stars (Bromm
et al., 2001b; Oh, 2001; Schaerer, 2002; Tumlinson et al., 2001). In the following, we use
the results obtained in this section to determine the observational signature of the first H 
regions.
5.2.2
Recombination Radiation from Individual H II Regions
The strongest direct signature of the first H  regions is likely generated by recombination
radiation within the H  region, since photons above the Lyman limit are absorbed by dense
gas in the host halo. We here concentrate on the Hα line, since Lyα photons are scattered out
of resonance by the neutral IGM, creating extended Lyα halos around high-redshift sources
(Loeb & Rybicki, 1999). In the following, we estimate the total flux of the H  region in
the Hα line and compare it to the expected sensitivity of the Mid-Infrared Instrument (MIRI)
on JWST at ∼ 10 µm wavelengths (Gardner et al., 2006). The spatial resolution is limited
by diffraction, such that a scale of ' 1 kpc at z = 20 is marginally resolved, which allows
us to approximate the H  region as a point source. Using the simulation output, the total
luminosity is given by
X mi Xρi !2
LHα = jHα
fe,i fHII,i ,
(5.17)
ρi mH
i
where jHα is the emissivity in the Hα line at 104 K (Osterbrock & Ferland, 2006), X = 0.76
is the primordial mass fraction of hydrogen, mH is the mass of the hydrogen atom, mi and
ρi are the mass and density of particle i, respectively, and the sum is over all particles in the
simulation box. From the total luminosity, we determine the observed flux with the inversesquare law
LHα
,
(5.18)
FHα =
4πD2L
where DL is the cosmological luminosity distance. In Fig. 5.4, we show the observed flux for
a 50, 100 and 200 M Pop III star as a function of time. Evidently, the H  region is brightest
before breakout, when the density in the host halo is still high, and peaks around FHα =
10−23 erg s−1 cm−2 . For a 10 σ detection with an exposure time of 100 hours, the spectrograph
on MIRI exhibits a typical limiting sensitivity of ' 10−18 erg s−1 cm−2 (Panagia, 2005),
implying that the first H  regions are five orders of magnitude too faint for a direct detection.
93
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
Figure 5.3: The H  region created by a 50, 100 and 200 M Pop III star (from top to bottom)
after 50 kyr, 300 kyr, and at the end of the star’s lifetime (from left to right). Shown is the densitysquared weighted average of the temperature along the line of sight. The spiral structure of the
central clump as well as the resulting anisotropic breakout of the ionization front are clearly
visible. For increasing stellar mass and ionizing photon output, breakout occurs earlier and is
more isotropic. Once the ionization front becomes R-type, spherical symmetry is asymptotically
restored and the H  region expands to a final radius of rHII ' 1.9, 2.7 and 3.7 kpc after 3.7, 2.7
and 2.2 Myr, respectively.
94
5.2 Observational Signature
We must therefore resort to indirect methods that rely on their cumulative signal. One such
signature is the cosmic infrared background (CIB), where the redshifted Lyα recombination
photons from z ∼ 10 – 20 might contribute at a detectable level (Kashlinsky et al., 2005;
Santos et al., 2002). Minihalos, however, are not expected to be important sources for the
CIB, as opposed to more massive dark matter halos that host the first galaxies (Greif &
Bromm, 2006). This leads us to consider the radio background as a key diagnostic of the
Pop III minihalo formation site.
5.2.3
Radio Background Produced by Bremsstrahlung
Next to recombination radiation, the first H  regions emit bremsstrahlung via thermal motions of electrons in an ionized medium. In line with our conlusions of the previous section,
the signature from an individual H  region is much too faint to be detected. However, their
cumulative radio emission might be strong enough to be detected by the upcoming SKA. We
will here further explore this possibility (for a review of earlier work, see Furlanetto et al.,
2006).
Solving the cosmological radiative transfer equation, it is straightforward to derive a
simple expression for the observed radio background Jν (in erg s−1 cm−2 Hz−1 sr−1 ):
Jν =
Z
tH,0
0
jν
c dt ,
(1 + z)3
(5.19)
where tH,0 is the present Hubble time and jν is the specific emissivity of bremsstrahlung,
given by
D E
−1/2
jν = ff n2e T/103 K
(5.20)
,
D E
where ff ' 10−39 erg s−1 cm3 Hz−1 sr−1 , n2e is the volume-averaged electron density, and
T is the temperature (Rybicki & Lightman, 1979). We universally assume T = 103 K, since
the H  region cools quite rapidly to ∼ 103 K via inverse Compton losses and adiabatic
expansion once the star has died (e.g. Greif et al., 2007; Yoshida et al., 2007a). Furthermore,
we assume jν = 0 at z < 6, since photoheating during reionization evaporates minihalos
(Dijkstra et al., 2004). This leads to:
Jν = c ff
Z
6
∞
D E
n2e dt dz ,
(1 + z)3 dz 95
(5.21)
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
Figure 5.4: The flux in the Hα line emitted by recombination radiation from ionized gas around
a 50 (dot-dashed line), 100 (dotted line) and 200 M (solid line) Pop III star, shown as a function
of time. Evidently, the H  region is brightest before breakout, when the density in the minihalo
is still high, and peaks around FHα = 10−23 erg s−1 cm−2 . For a 10 σ detection and an exposure
time of 100 hours, the limiting sensitivity of the MIRI spectrograph on JWST is approximately
10−18 erg s−1 cm−2 , indicating that the first H  regions are five orders of magnitude too faint for
a direct detection. The bumps in the 50 and 100 M cases are caused by the ionization of dense
gas near the host halo. In the 200 M case, the ionization front propagates much too rapidly for
a similar effect.
96
5.2 Observational Signature
D E
where we relate n2e to the number density of minihalos according to:
D E
dN
dz ps
.
n2e ' trec n2H,b VHII dz dt (5.22)
Here, trec = αB nH,b −1 denotes the recombination time for hydrogen atoms, αB their recombination rate for T = 103 K, nH,b the background density, Nps the number of minihalos
per comoving volume, VHII = Nion /nH,b,0 the comoving volume of an individual H  region,
which is independent of redshift, and Nion = Ṅion t∗ the total number of ionizing photons
emitted per Pop III star (see Section 5.2). In the above equation, we have implicitly assumed
that H  regions survive for a recombination time, and that all ionizing photons escape into
the IGM, which is a good approximation for massive Pop III stars in minihalos (Alvarez
et al., 2006b). We note that in the range of redshifts considered here, the recombination time
is larger than the stellar lifetime and smaller than the age of the Universe. In principle, one
must also account for the clustering of minihalos (biasing), which reduces the net volume
filling factor of H  regions (Greif & Bromm, 2006; Iliev et al., 2003; Mo & White, 1996).
However, it is extremely difficult to determine the importance of this effect, since (i) the
actual overlap depends on the relative separation of minihalos, and (ii) previous ionization
allows a nearby H  region to become larger than usual. We therefore neglect biasing, but
keep in mind that the actual signal may be somewhat lower.
In equation (5.22), the number density of minihalos is given by
Nps (z) =
Z
Mmax
nps (z, M) dM ,
(5.23)
Mmin
where nps is the well-known Press-Schechter mass function (Press & Schechter, 1974). The
minimum mass required for efficient cooling within a Hubble time may be found in Trenti &
Stiavelli (2009):
!−2
1+z
6
Mmin ' 10 M
(5.24)
,
10
while the maximum mass is set by the requirement that cooling must be dominated by molecular hydrogen, i.e. the virial temperature must not exceed T ' 104 K for atomic hydrogen
97
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
cooling, resulting in (e.g. Barkana & Loeb, 2001)
7
Mmax ' 2.5 × 10 M
1+z
10
!−3/2
.
(5.25)
We have found that our results are only marginally affected by the upper mass limit, but
depend sensitively on the lower mass limit, since most minihalos reside at the lower end of
the halo distribution function.
After combining the above equations, we obtain
Jν '
c ff Nion
Nps (z = 6) ,
αB
(5.26)
which, for an IMF consisting solely of 100 M Pop III stars, yields
Jν ' 300 mJy sr−1 .
(5.27)
The brightness temperature, T b = c2 Jν /2kB ν2 , is given by
−2
ν
.
T b ' 1 mK
100 MHz
(5.28)
In the following, we investigate whether a signal of this magnitude is observable by the
upcoming SKA.
The sensitivity of radio instruments is generally defined by the ratio of the effective collecting area Ae to the system temperature T sys . For the SKA with its proposed aperture array
configuration at low frequencies, Ae /T sys ' 5 × 103 m2 K−1 at 100 MHz 1 . In this range
the system temperature is dominated by Galactic synchrotron emission, for which a useful
approximation is given by T sky ' 180 K (ν/180 MHz)−2.6 (Furlanetto et al., 2006), resulting
in T sys ' 800 K and Ae ' 4 × 106 m2 . The minimum angular resolution for an array filling
factor of unity at 100 MHz is approximately 150 . At higher resolution, the sensitivity decreases much too rapidly for effective imaging. In Fig. 5.5, we compare the sensitivity of the
SKA for a 10 σ detection, a bandwidth of ∆νobs = 1 MHz, and an integration time of 1000 h
to the brightness temperature and specific flux expected for free-free emission. Although the
free-free signal is in principle detectable by the SKA, we have neglected biasing as well as
1
http://www.skatelescope.org
98
5.2 Observational Signature
radiative feedback in the form of a global LW background, which attenuates star formation in
minihalos (Johnson et al., 2007, 2008). Another complicating issue is the overlap with 21 cm
emission, which makes it nearly impossible to isolate the contribution from bremsstrahlung.
In consequence, we do not believe that this signal will be observable in the near future.
5.2.4
Radio Background Produced by 21 cm Emission
Perhaps the most promising observational signature comes from 21 cm emission of the relic
H  region gas once the star has died, a prospect that was already investigated by Tokutani
et al. (2009). An emission signal requires the spin temperature T S of neutral hydrogen to
be greater than the temperature of the CMB, with its relative brightness determined by T S
and the size of the H  region. The spin temperature is set by collisional coupling with
neutral hydrogen atoms, protons and electrons, as well as radiative coupling to the CMB.
Furthermore, it may be modified by the so-called Wouthuysen-Field effect, which describes
the mixing of spin states due to the absorption and re-emission of Lyα photons (Field, 1959;
Wouthuysen, 1952). The color temperature of the Lyα background is determined by the
ratio of excitations to de-excitations, which approaches the kinetic gas temperature at high
redshifts, where the optical depth to Lyα scattering is very large (Furlanetto et al., 2006).
In this case, adopting the Rayleigh-Jeans approximation and assuming T S T ∗ , where
T ∗ = hν21 /kB = 68 mK is the temperature associated with the 21 cm transition, the spin
temperature may be written as (Madau et al., 1997)
TS =
T γ + (yc + yα ) T
,
1 + yc + yα
(5.29)
where T γ is the temperature of the CMB. The collisional coupling coefficient yc is approximately given by
T∗
(nHI κHI + ne κe ) ,
yc =
(5.30)
A21 T
where A21 = 2.85×10−15 s−1 is the Einstein A-coefficient for the 21 cm transition, and κHI and
κe are the effective single-atom rate coefficients for collisions with neutral hydrogen atoms
and electrons, respectively. Good functional fits in the temperature range 100 K . T . 104 K
are given by
κHI = 10−11 T 1/2 cm3 s−1
(5.31)
99
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
Figure 5.5: The brightness temperature (top panel) and specific flux (bottom panel) of the radio
background produced by bremsstrahlung, shown as a function of observed frequency. We have
chosen a beam size of 150 to achieve the highest possible resolution and sensitivity at 100 MHz
for the currently planned configuration of the SKA. The thin dot-dashed, dotted and solid lines
correspond to an initial mass function consisting solely of 50, 100 and 200 M Pop III stars,
respectively. The thick dashed line shows the sensitivity of the SKA for a 10 σ detection, a
bandwidth of 1 MHz, and an integration time of 1000 h. Although the free-free signal is in
principle detectable by the SKA, we have here neglected biasing and radiative feedback, which
act to reduce the signal. For this reason we do not believe that the free-free signal of the first H 
regions will be observable in the near future.
100
5.2 Observational Signature
and
κe = 2 × 10−10 T 1/2 cm3 s−1 ,
(5.32)
which we have obtained from the rates quoted in Kuhlen et al. (2006). At z . 20, the electron
fraction in the IGM remains above fe = 0.1 for most of the lifetime of the relic H  region.
In this case, the collisional coupling coefficient is given by
!
!3
fe T −1/2 1 + z
yc ' 0.015
.
0.5 103 K
10
(5.33)
A derivation of the Lyα coupling coefficient yα requires radiative transfer of local as well as
global Lyα radiation, which is beyond the scope of this work. We therefore consider two
limiting cases: one in which we only activate collisional coupling, and the other in which a
strong Lyα background drives the spin temperature towards the gas temperature (i.e. yα 1
or T S = T ).
The differential brightness temperature with respect to the CMB may be derived as follows. In the Rayleigh-Jeans limit and for T S T ∗ , the monochromatic radiative transfer
equation for a ray passing through a cloud, evaluated in its comoving frame, may be written
in terms of the brightness temperature T b :
Tb = Tγ e
−τ
+
Z
τ
0
T S e−τ dτ0 ,
(5.34)
0
where the optical depth at the 21 cm line is given by
dτ =
3c2 A21 nHI
T∗
φ(ν
)
ds .
21
2
TS
32πν21
(5.35)
Here, φ(ν21 ) is the normalized line profile at the resonance frequency ν21 and ds is the distance traveled by the ray. In our case, the line profile is dominated by thermal broadening,
with a Doppler width given by
s
∆νD = ν21
2kB T
.
µmH c2
(5.36)
The amplitude of the line profile at the resonance frequency may be replaced by the Doppler
width, i.e. φ(ν21 ) = ∆νD−1 . With this definition, equation (5.34) yields the differential bright-
101
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
Figure 5.6: The observed differential brightness temperature of the relic H  region around a
100 M Pop III star, shown 20, 50 and 100 Myr after the star has turned off. We delineate the
range of possible values by showing the result for collisional coupling only (top row), as well as
perfect coupling to the Lyα background, resulting in yα 1 or T S = T (bottom row). In the first
case, the observed differential brightness temperature is of order a few 10 mK for ' 100 Myr,
while in the second case the signal is of order a few 100 mK for well over ' 100 Myr. We note
that the latter is likely more relevant at z . 20, where the observationally accessible signal is
produced (Furlanetto, 2006; Pritchard & Furlanetto, 2007).
102
5.2 Observational Signature
ness temperature δT b = T b − T γ , which becomes particularly simple for a constant spin
temperature and the fact that the relic H  regions considered here are optically thin:
δT b = (T S − T γ ) τ .
(5.37)
The observed differential brightness temperature is then simply given by δT b,obs = δT b / (1 + z).
In Fig. 5.6, we show the observed differential brightness temperature for a 100 M star
and the two limiting cases discussed above. Note that we have only taken into account
ionized gas along the line of sight. For collisional coupling only, the observed differential
brightness temperature is of order a few 10 mK for ' 100 Myr, while for perfect coupling
the signal is elevated by an order of magnitude to a few 100 mK for well over ' 100 Myr.
In reality, the expected signal lies between these extremes and is a function of redshift, since
collisional coupling becomes weaker as the background density drops, while Lyα coupling
becomes stronger as the Lyα background rises. At z . 20, where the observationally accessible signal is produced, the Lyα background is likely strong enough for the latter case to be
more important (Furlanetto, 2006; Pritchard & Furlanetto, 2007).
We may now determine the radio background produced by the integrated 21 cm emission
of relic H  regions. The differential specific flux observed at the redshifted 21 cm line from
a single relic H  region with differential brightness temperature δT b is given by
δFν =
2
2kB ν21
(1 + z)−3 ∆Ω δT b ,
c2
(5.38)
2
where ∆Ω = A/D2A denotes the solid angle subtended by the H  region, A = πrHII
its
1/3
area, rHII = 3Nion /4πnH,b
its radius, and DA the angular diameter distance. The average
differential specific flux hδFν i within a beam size ∆Ωbeam and bandwidth ∆νobs is then given
by
d2 V(z) ∆z ∆Ωbeam
,
(5.39)
hδFν i = δF Nps (z)
dz dΩ
∆νobs
where δF = δFν ∆νD / (1 + z), Nps (z) is the Press-Schechter mass function defined in equation (5.23), ∆z = ∆νobs (1 + z)2 /ν21 , and d2 V(z)/dz dΩ is the comoving volume per unit
redshift and solid angle:
2
d2 V(z) c D2A (1 + z)
=
,
(5.40)
dz dΩ
H(z)
where H(z) is the Hubble expansion rate. With the definition of the brightness temperature,
103
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
the average differential antenna temperature hδT b i is given by
hδT b i =
πc (1 + z)2 Nps (z)
2
∆νD rHII
(z) δT b (z) .
ν21
H(z)
(5.41)
Based on our argument above, we assume that the Lyα background is strong enough for
perfect coupling at all redshifts. In this case, and for T T γ , the average differential
antenna temperature becomes independent of electron fraction and temperature:
hδT b i =
9c3 A21 T ∗ Nion
(1 + z)1/2 Nps (z) ,
√
3
128πν21 H0 Ωm
(5.42)
where we have set nHI = nH,b in equation (5.35). We note that the observed frequency is
related to the redshift via νobs = ν21 / (1 + z). We have further assumed that the relic H 
region produced by each star-forming minihalo persists until the Universe is reionized (i.e.
z ' 6), which is a good approximation for perfect coupling and T T γ . Equation (5.42)
provides a robust upper limit for the collective 21 cm emission from the first relic H  regions.
In Fig. 5.7, we compare the average differential antenna temperature and specific flux
for a beam size of ∆θbeam = 150 to the sensitivity of the SKA, assuming a 10 σ detection,
a bandwidth of ∆νobs = 1 MHz, and an integration time of 1000 h. At all frequencies, the
maximum 21 cm signal from the first relic H  regions is of order 10 mK, which is well
detectable by the SKA. The effects of biasing and radiative feedback will reduce this signal,
but probably not enough to fall below the sensitivity of the SKA. Compared to free-free
emission, the 21 cm signal is typically an order of magnitude stronger, and offers the best
prospect for indirectly probing the first stars.
5.3
Summary and Conclusions
We have introduced a general-purpose radiative transfer scheme for cosmological SPH simulations that treats ionizing and photodissociating radiation from massive Pop III stars in the
early Universe. Based on this methodology, we have investigated the build-up of the first
H  regions around Pop III stars formed in minihalos, and predicted their contribution to
the extragalactic radio background via bremsstrahlung and 21 cm emission. Although recombination radiation from individual H  regions is too faint to be directly detectable even
104
5.3 Summary and Conclusions
Figure 5.7: The average differential antenna temperature (top panel) and specific flux (bottom
panel) of the radio background produced by 21 cm emission, shown as a function of observed
frequency. We have chosen a beam size of 150 to achieve the highest possible resolution and
sensitivity at 100 MHz for the currently planned configuration of the SKA. The thin dot-dashed,
dotted and solid lines correspond to an initial mass function consisting solely of 50, 100 and
200 M Pop III stars, respectively. The thick dashed line shows the sensitivity of the SKA
for a 10 σ detection, a bandwidth of 1 MHz, and an integration time of 1000 h. In all cases,
the 21 cm signal is well above the detection threshold of the SKA. The effects of biasing and
radiative feedback will reduce this signal, but probably not enough to fall below the sensitivity
of the SKA. Compared to free-free emission, the 21 cm signal is typically an order of magnitude
stronger, and offers the best prospect for indirectly probing the first stars.
105
5. THE OBSERVATIONAL SIGNATURE OF THE FIRST H II REGIONS
with JWST, their collective radio emission might be strong enough to be within reach of the
planned SKA. In particular, we have found that the integrated free-free emission results in a
maximum differential antenna temperature of ' 1 mK, while the 21 cm emission is an order
of magnitude stronger. Considering the effects of biasing and negative radiative feedback,
which would act to reduce the predicted signal, the free-free signal is likely beyond the capability of the SKA, while the 21 cm signal will most likely be observable, providing an
excellent opportunity for indirectly probing the first stars.
We note that an analysis of the angular fluctuation power spectrum will be essential to
isolate the 21 cm signal from other backgrounds (Furlanetto & Oh, 2006). Among these are
neutral minihalos, which appear in emission due to their enhanced density and temperature
(Iliev et al., 2002), or IGM gas heated by X-rays from supernovae (Oh, 2001), X-ray binaries (Glover & Brand, 2003), or the first quasars (Kuhlen et al., 2006; Madau et al., 2004). A
strong absorption signal might originate from cold, neutral gas if the Lyα background effectively couples the spin temperature to the gas temperature (Pritchard & Furlanetto, 2007). In
addition, there is the signal produced by stars (primordial or already metal-enriched) formed
in the first dwarf galaxies. All of these compete with each other, and more work is required
to understand their relative importance. One important task is to extend the simulations to
larger cosmological volumes, to measure the aggregate signal from many sources in a more
robust way.
Minihalos may not have been the dominant formation sites for primordial stars, in terms
of producing the bulk of the radiation that drove reionization, or of being the source for the
majority of the heavy elements present at high redshifts (Greif & Bromm, 2006; Schleicher
et al., 2008). Nevertheless, they are the ideal laboratory to test our current standard model
of the first stars, by providing an exceedingly simple environment for the star formation
process (Bromm et al., 2009). The next step in the hierachical build-up of structure, more
massive systems, such as the first galaxies, is already highly complex, due to the presence of
metals, turbulent velocity fields, and possibly dynamically significant magnetic fields (Greif
et al., 2008; Schleicher et al., 2009; Wise & Abel, 2007a, 2008b). It is therefore crucial to
empirically probe the minihalo environment, and the signature left in the radio background
might provide us with one of the few avenues to accomplish this in the foreseeable future.
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6
The First Galaxies: Signatures of the
Initial Starburst
The epoch of the first galaxies marked a fundamental transition in the Universe, ending the
Cosmic Dark Ages, beginning the process of reionization, and witnessing the rapid proliferation of star formation and black hole growth (Barkana & Loeb, 2001; Bromm & Larson,
2004). While the theory of primordial star formation and early galaxy formation has rapidly
developed (Barkana & Loeb, 2007; Ciardi & Ferrara, 2005; Glover, 2005), observations of
the first galaxies at redshifts z & 10 have so far been out of reach (but see Stark et al., 2007).
In the coming decade, JWST promises to provide direct observations of this critical period in
cosmic history, allowing to place new constraints on the stellar IMF at high redshift, on the
luminosity function of the first galaxies, and on the progress of the early stages of reionization (Barton et al., 2004; Gardner et al., 2006; Haiman, 2008; Ricotti et al., 2008; Windhorst
et al., 2006).
The IMF of the stellar populations which form in the first galaxies is of central importance in determining their properties and impact on early cosmic evolution. The current
theoretical consensus posits that the first stars, which formed in isolation in DM minihalos,
likely had masses of the order of 100 M (Abel et al., 2002; Bromm et al., 1999, 2002; McKee & Tan, 2008; Yoshida et al., 2008, 2006). In the first galaxies, which form in DM halos
with masses & 108 M (Greif et al., 2008; Wise & Abel, 2007a), there is no such theoretical
consensus on the IMF, as the initial conditions of the star-forming gas are uncertain (Jappsen
et al., 2009a,b). A large fraction of these first galaxies are likely to already host Pop II star
107
6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
formation, due to previous metal enrichment (Clark et al., 2008; Johnson et al., 2008; Trenti
& Stiavelli, 2009). However, clusters of primordial stars likely form in some fraction of the
first galaxies, owing to either inhomogeneities in the LW background which can suppress star
formation where galaxies are strongly clustered (Ahn et al., 2008; Dijkstra et al., 2008), or
to the direct collapse of the first stars to black holes, thereby locking up the metals produced
in their cores (Heger et al., 2003).
Due to the hard spectra of massive metal-free stars, strong nebular emission in helium
recombination lines has been suggested as an observable indicator of a population of such
stars (Bromm et al., 2001b; Oh et al., 2001; Schaerer, 2002; Tumlinson et al., 2001). In
particular, a high ratio of the luminosity emitted in He  λ1640 to that emitted in Lyα or Hα
may be a signature of a galaxy hosting massive Pop III star formation, and has already served
as the basis for searches for such galaxies (Dawson et al., 2004; Nagao et al., 2005, 2008). In
addition, high equivalent widths (EWs) of Lyα and He  λ1640 are expected to characterize
galaxies undergoing a Pop III starburst (Schaerer, 2003). While no definitive detections of
such galaxies have been achieved to date, observations of galaxies at 3 . z . 6.5 with large
Lyα EW and strong He  λ1640 emission may indicate that some galaxies host Pop III star
formation even at such relatively low redshift (Dijkstra & Wyithe, 2007; Jimenez & Haiman,
2006).
Previous analytical calculations of the recombination radiation expected from the first
galaxies have been carried out under a number of idealized assumptions, namely of a static,
uniform density field, and of the formation of a static Strömgren sphere. Taking a complementary approach, we study here the properties of the recombination radiation emitted by
the first galaxies with a focus on how the dynamical evolution of the galaxy affects the properties of this radiation. We present high-resolution cosmological radiation hydrodynamics
simulations of the production of nebular emission from a cluster of primordial stars formed
within the first galaxies. We resolve the H  and He  regions generated by the stellar cluster, thereby arriving at improved predictions for the emission properties of the first galaxies,
which will be tested by JWST.
Our paper is organized as follows. In Section 6.2, we describe our simulations and the
methods used in their analysis; in Section 6.3 our results are presented, along with the implications for both IMF and SFR indicators; in the final Section 6.4, we summarize and give
our conclusions.
108
Figure 6.1: The projected gas temperature at z = 12.7, just before the stellar cluster turns on.
Shown at left is the entire 1.4 Mpc (comoving) simulation box. The region of highest resolution
is in the center of our multi-grid simulation box; magnified at right is the central ∼ 10 kpc, where
the ∼ 108 M dark matter halo hosting the stellar cluster is located. Note that the gas is heated to
≥ 104 K when passing through the virialization shock.
109
6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
6.1
Methodology
We carry out radiation hydrodynamics simulations which track the impact of the radiation
from Pop III stellar clusters forming promptly within a dwarf galaxy at z ∼ 12.5. In this section we describe the simulations and the calculations carried out in analyzing their outcomes.
6.1.1
The Simulations
As with previous work, for our three-dimensional numerical simulations we employ the
parallel version of GADGET (version 1), which includes a tree (hierarchical) gravity solver
combined with the SPH method for tracking the evolution of gas (Springel & Hernquist,
2002; Springel et al., 2001). Along with H, H+ , H− , H2 , H+2 , He, He+ , He++ , and e− , we
have included the five deuterium species D, D+ , D− , HD and HD+ , using the same chemical
network as in Johnson & Bromm (2006, 2007).
For our simulation of the assembly of a dwarf galaxy at z ∼ 12.5, we have employed
multi-grid initial conditions which offer higher resolution in the region where the galaxy
forms (e.g. Kawata & Gibson, 2003). We initialize the simulation according to the ΛCDM
power spectrum at z = 100, adopting the cosmological parameters Ωm = 1 − ΩΛ = 0.3,
Ωb = 0.045, h = 0.7, and σ8 = 0.9, close to the values measured WMAP in its first year
(Spergel et al., 2003). Here, we use a periodic box with a comoving size L = 1h−1 Mpc for
the parent grid. Our simulations use NDM = NSPH = 1.05 × 106 particles for DM and gas,
where the SPH particle mass is mSPH ∼ 120 M in the region with the highest resolution.
For further details on the technique employed to generate our multi-grid initial conditions,
see Greif et al. (2008). The maximum gas density that we resolve is nres ∼ 103 cm−3 , while
gas at higher densities is accreted onto sink particles, as described in (Johnson et al., 2007).
We have also included the effect of a LW background radiation field, at a level of JLW =
0.04 × 10−21 erg s−1 cm−2 Hz−1 sr−1 , just as in Johnson et al. (2008).
To capture the effects of the ionizing radiation emitted by a single Pop III stellar cluster
within the dwarf galaxy, we approximate the cluster as a point source located at the center of
the most massive DM halo in our simulation box at z = 12.7. The projected gas temperature
in the entire simulation box at this redshift is shown in Fig. 6.1 (left panel); the right panel
shows the temperature in the region of the host halo at the center of the box, which has a
virial mass of 9 × 107 M , characteristic of the first galaxies. At each timestep, we find
110
6.1 Methodology
the boundaries of both the H  and He  regions generated by the stellar cluster using a
ray-tracing technique that improves our earlier implementation (Johnson et al., 2007).
The procedure used to calculate the Strömgren sphere around the stellar cluster for a
given time-step ∆t is similar to the ray-tracing scheme used in Johnson et al. (2007). We
create a spherical grid centered at the location of the cluster, consisting of ∼ 1.2 × 104 rays
and 1000 linearly spaced radial bins. We resolve the central kiloparsec around the source,
roughly the virial radius of the host halo, with 250 radial bins, while the remaining 750 bins
are linearly spaced out to ∼ 20 kpc.
In a single, parallelized loop, the Cartesian coordinates of all particles are converted to
spherical coordinates, such that their density and chemical abundances may be mapped to the
bins corresponding to their radius, azimuth and zenith, denoted by r, θ and φ, respectively. To
avoid missing dense clumps, particles contribute to bins independent of distance, but proportional to their density squared. Once this preliminary step is complete, it is straightforward
to solve the ionization front equation along each ray:
drI
nn rI2
Qion
=
− αB
dt
4π
rI
Z
ne n+ r2 dr ,
(6.1)
0
where rI denotes the position of the ionization front, Qion the number of ionizing photons
emitted per second by the stellar cluster, αB the case B recombination coefficient, and nn , ne
and n+ the number densities of nonionized particles, electrons and ionized particles, respectively. The numbers of H -, He - and He -ionizing photons are Qion = N∗ qion , where N∗
is the number of stars in the cluster (here we assume that all have the same mass) and qion is
the number of ionizing photons emitted by a single star, given by
qion
πL∗
=
4
σT eff
Z
∞
νmin
Bν
dν ,
hν
(6.2)
where σ is the Stefan-Boltzmann constant, νmin denotes the minimum frequency corresponding to the ionization thresholds of H , He  and He , and we assume that massive Pop III
stars emit a blackbody spectrum Bν (in erg s−1 cm−2 Hz−1 sr−1 ) with an effective temperature
T eff and a luminosity L∗ (e.g. Schaerer, 2002).
To obtain a discretization of the ionization front equation, we replace the integral on the
111
6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
right-hand side of equation (6.1) by a discrete sum:
Z
0
rI
ne n+ r2 dr =
X
ne,i n+,i ri2 ∆r ,
(6.3)
i
where ∆r is the radial extent of the individual bins. Similarly, the left-hand side of equation (6.1), which models the propagation of the ionization front into neutral gas, is discretized
by
drI
1 X
nn rI2
=
nn,i ri2 ∆ri ,
(6.4)
dt
∆t i
where ∆t is the current time-step and the summation is over radial bins starting with the
bin lying immediately outside of rI,old , the position of the ionization front at the end of the
previous time-step, and ending with the bin lying at the new position of the ionization front.
We perform the above steps separately for the H  and He  regions, since they require
distinct heating and ionization rates.
We have chosen the size of the bins that are used in our ray-tracing routine to roughly
match the volume of gas represented by a single SPH particle within the ∼ 1 kpc virial
radius of the halo hosting the stellar cluster, such that the boundaries of the photoionized
regions are maximally resolved while also reliably conserving ionizing photons. However,
in some cases it may occur that the mass contained in a bin is smaller than that of the SPH
particle contained within it, such that ionizing the entire SPH particle involves ionizing more
gas than is contained in the bin. In turn, this can lead to an overestimate of the number of
recombinations. While this effect is minor in our simulations, in the calculations presented
below we enforce that the total number of recombinations does not exceed the total number
of ionizing photons available.
We carry out four simulations, each with a different combination of IMF and total cluster
mass. For the IMF, we assume for simplicity, and in light of the still complete uncertainty
regarding its detailed shape, that the cluster consists either entirely of 25 M or 100 M
Pop III stars. These choices are meant to bracket the expected characteristic mass for Pop III
stars formed in the first galaxies, which depending on the cooling properties of the gas may
be Pop III.2 stars with masses of order 10 M or, possibly, Pop III.1 stars with masses
perhaps an order of magnitude higher (Greif et al. 2008; Johnson & Bromm 2006; McKee
& Tan 2008; but see Jappsen et al. 2009a). For each of these IMFs, we vary the total stellar
mass in the cluster, choosing either 2.5 × 103 M or 2.5 × 104 M for the total mass in stars.
112
6.1 Methodology
These choices correspond to ∼ 1 and ∼ 10 percent, respectively, of the cold gas available for
star formation within the central few parsecs of such a primordial dwarf galaxy (Regan &
Haehnelt, 2009; Wise et al., 2008). We calculate the ionizing flux from each of these clusters,
assuming blackbody stellar spectra at 7×104 and 105 K and bolometric luminosities of 6×104
and 106 L , for the 25 and 100 M stars, respectively, appropriate for metal-free stars on the
main sequence (Marigo et al., 2001).
For simplicity, we have chosen to keep the input stellar spectra constant in time over the
course of the simulations. Accordingly, we run the simulations only for 3 Myr, which is
roughly the hydrogen-burning timescale of a 100 M primordial star, and about half that of
a 25 M primordial star. We note that while the H -ionizing flux from 100 M primordial
stars is roughly constant over this timescale, the He -ionizing flux decreases by a factor of
∼ 2 by a stellar age of 2 Myr, and by a much larger factor near the end of hydrogen-burning
as the star evolves to the red (Marigo et al., 2001; Schaerer, 2002). We note, however, that
stellar models accounting for the effects of rotation yield less precipitous drops in the emitted
ionizing flux with time, as fast rotation, especially of low-metallicity stars, can keep the stars
on bluer evolutionary tracks (Vázquez et al., 2007; Woosley & Heger, 2006; Yoon & Langer,
2005); indeed, Pop III stars may have been fast rotators (Chiappini et al., 2008). Nonetheless,
the results that we derive pertaining to He  recombination emission from clusters of 100 M
stars may be, strictly speaking, only reliable for stellar ages . 2 Myr. An in-depth study of
the impact that stellar evolution has on the emission properties of primordial galaxies is given
in Schaerer (2002); in the present work, we take a complementary approach and instead focus
on how the emission properties are affected by the hydrodynamic evolution of the gas in the
first galaxies.
We make the related simplifying assumption that the stellar cluster forms instantaneously.
This is valid if the timescale for star formation tSF is much shorter than the lifetime of the
stars that we consider, or tSF 3 Myr. If we assume that stars form on the order of the
free-fall time tff , and take it that the star cluster forms within the central ∼ 1 pc of the halo
(Regan & Haehnelt, 2009; Wise et al., 2008), then we find tSF ∼ 5 × 105 yr, for which our
assumption is marginally valid. We note that more work is needed to accurately determine
the star formation timescale in the first galaxies, as the works cited here neglect, in particular,
the important effect of molecular cooling on the evolution of the primordial gas.
113
6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
6.1.2
Deriving the Observational Signature
The simulations described above allow us to calculate the luminosities and equivalent widths
of the recombination lines emitted from high-redshift dwarf galaxies during a primordial
starburst. A related quantity we obtain is the escape fraction of ionizing photons from such
a galaxy. Here we describe each of these calculations.
Escape Fraction of Ionizing Photons
Photons which escape the host halo from which they are emitted proceed to reionize the
IGM, where densities are generally very low, yielding long recombination times. Ionizing
photons which do not escape the host halo are, however, available to ionize dense gas which
recombines quickly, leading to appreciable emission in recombination lines. Therefore, the
luminosity of a galaxy in recombination radiation is intimately related to the escape fraction
of ionizing photons. The escape fraction of ionizing photons from the halo hosting the stellar
cluster is given by subtracting the number of recombinations Qrec per second within the virial
radius from the total number of ionizing photons emitted by the cluster:
fesc =
Qion − Qrec
,
Qion
(6.5)
again with Qion = N∗ qion . This equation is valid under the assumption that within the host
halo the number of ionizing photons which fail to escape is balanced by the number of
recombinations within the halo. This is a reasonable assumption, since the number of atoms
which become ionized within the host halo is far less than the total number of recombinations
that occur in the halo, the ionization of previously neutral gas being the only other sink for
ionizing photons within the halo. The number of recombinations is given as
Qrec =
X
i
mi ρi
αB
ρi µi mH
!2
fe fHII ,
(6.6)
where αB is the case B recombination coefficient for hydrogen, mH the mass of a hydrogen
atom, while mi , µi and ρi are the total mass, mean molecular weight, and mass density of the
ith SPH particle, respectively. For each SPH particle, fHII and fe denote the fraction of nuclei
in H  and the fraction of free electrons, respectively.
Here the summation is over all SPH particles within the virial radius of the host halo, or
114
6.1 Methodology
within distance of ∼ 1 kpc from the central stellar cluster. We calculate the escape fractions
of both H -ionizing and He -ionizing photons. These quantities are generally not equal,
and they each contribute to determining the radiative signature of the initial starbursts in the
first galaxies.
Luminosity in Recombination Lines
For each of our simulations, we compute the luminosity emitted from photoionized regions
in each of three recombination lines: Hα, Lyα, and He  λ1640. These luminosities are
calculated by again summing up the contributions from all SPH particles within the virial
radius, where virtually all of the recombination line luminosity emerges, as follows:
LHα =
X
i
LLyα =
X
i
Lλ1640 =
X
i
mi ρi
jHα
ρi µi mH
!2
mi ρi
jLyα
ρi µi mH
fe fHII ,
(6.7)
fe fHII ,
(6.8)
!2
mi ρi
jλ1640
ρi µi mH
!2
fe fHeIII ,
(6.9)
where the j are the temperature-dependent emission coefficients for the lines (Osterbrock &
Ferland, 2006), and fHeIII is the fraction of helium nuclei in He  for each SPH particle.
Given the luminosity in a recombination line over an area of the sky, we may compute
the flux in that line, as observed at z = 0 with a spectral resolution R = λ/∆λ, where λ is
the wavelength at which the emission line is observed (Oh, 2001). While Lyα photons are
scattered out of the line of sight in the IGM prior to reionization (Loeb & Rybicki, 1999), a
process which we do treat in the present calculations, Hα and He  λ1640 photons will not
suffer such severe attenuation. Assuming that the line is unresolved, the monochromatic flux
in Hα, for example, is
fHα
lHα λHα (1 + z) R
lHα
∼ 20 nJy
=
2
40
10 erg s−1
4πcDL (z)
!
1+z
10
!−1 R ,
1000
(6.10)
where lHα is the luminosity in Hα along the line of sight through the emitting galaxy, DL (z)
is the luminosity distance at redshift z (∼ 102 Gpc at z = 10), and λHα is the rest frame
115
6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
wavelength of the line, 656.3 nm. If the galaxy is spatially unresolved, appearing as a point
source, we may simply substitute LHα for lHα in equation (6.10), to compute the total flux
from the galaxy. In terms of total (integrated) line flux, we have the equivalent expression
FHα
LHα
LHα
∼ 10−20 erg s−1 cm−2
=
2
40
10 erg s−1
4πDL (z)
!
1+z
10
!−2
.
(6.11)
Recombination Line Equivalent Widths
Another observable quantity obtained from our simulations is the rest-frame EW of recombination lines. We calculate the EWs of the three recombination lines considered, following
Schaerer (2002):
LHα
0
(6.12)
WHα
=
,
Lλ,neb + Lλ,∗
0
WLyα
=
0
Wλ1640
=
LLyα
,
Lλ,neb + Lλ,∗
(6.13)
Lλ1640
,
Lλ,neb + Lλ,∗
(6.14)
where the monochromatic continuum luminosity, evaluated at the wavelength of the line, is
the sum of the nebular emission Lλ,neb and the stellar emission Lλ,∗ . The nebular continuum
luminosity is given by
c γtot
Qrec ,
(6.15)
Lλ,neb = 2
λ αB
where λ is the wavelength in question, and Qrec is again the total number of recombinations
per second in the halo. The continuous emission coefficient γtot accounts for free-free, freebound, and two-photon continuum emission, as described in Schaerer (2002). The stellar
continuum luminosity is calculated assuming a blackbody stellar spectrum and is given by
Lλ,∗
8π2 hc2 R2∗
N∗
= 5
,
λ exp (hc/λkB T eff ) − 1
(6.16)
where N∗ is the number of stars in the cluster, T eff is the effective surface temperature of a
star, and R∗ is the stellar radius.
116
6.2 Results and Implications
6.2
Results and Implications
We next discuss the observable characteristics of primordial dwarf galaxies. In particular,
we evaluate the utility of indicators for the SFR and the stellar IMF in such galaxies.
6.2.1
Evolution of Gas inside the Galaxy
With the ignition of a stellar cluster at the center of the host halo, the gas surrounding the
cluster is photoheated, raising its pressure and leading to its outward expansion. In turn, the
overall recombination rate in the host halo drops, allowing the expansion of the H  region
to continue for a constant rate of ionizing photon production. Fig. 6.2 shows the growth of
the H  region and the concomitant expansion of the gas in the center of the host halo for
the more massive 100 M cluster. The H  region breaks out of the host halo within the first
∼ 1 Myr, and after 3 Myr it extends to . 7 kpc, only slightly larger than the size of the H 
region created by a single massive Pop III star in a minihalo (Alvarez et al., 2006a).
While the gas surrounding the formation sites of the first stars in minihalos is easily
photoevacuated by a single massive Pop III star (Kitayama et al., 2004; Whalen et al., 2004),
the deeper gravitational potential well of the DM halos hosting the first galaxies allows for
the retention of gas even under intense photoheating; indeed, this is one criterion used to
define the first galaxies (Greif et al., 2008; Johnson et al., 2008; Read et al., 2006).
As shown in Fig. 6.3, a substantial portion of the gas in the galaxy, even within ∼ 100 pc
of the stellar cluster, remains neutral after 3 Myr. This gas is shielded from the ionizing
radiation, causing the ionization front to propagate outward anisotropically in the inhomogeneous cosmological density field (Abel et al., 2007; Alvarez et al., 2006a; Shapiro et al.,
2004). Even for the case of the highest ionizing flux, the fraction of ionized gas within the
central ∼ 100 pc is . 0.4, leaving the majority of the high density gas neutral. While the
photodissociating radiation from the initial stellar cluster will slow the collapse of this primordial gas (Ahn & Shapiro, 2007; Susa & Umemura, 2006; Whalen et al., 2008a), some
fraction of it will likely be converted into stars once the most massive stars in the cluster
have died out. Indeed, the shocks engendered by the supernovae that mark the end of their
lives may expedite the collapse of the gas (Greif et al., 2007; Machida et al., 2005; Mackey
et al., 2003; Sakuma & Susa, 2009; Salvaterra et al., 2004). The incomplete ionization of
the central gas confirms that the masses that we have chosen for the clusters are not overly
117
6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
Figure 6.2: The density-weighted temperature (left column), density-weighted H  fraction
(middle column), and number density (right column), each averaged along the line of sight,
of the gas surrounding the more massive 100 M star cluster, shown at three different times from
the prompt formation of the cluster: 500, 000 yr (top row), 1 Myr (middle row), and 3 Myr (bottom row). The H  region grows as the density of the gas in the center of the host halo gradually
drops in response to the intense photoheating. Note the different length scale of each column; the
density is shown only within the central region of the host halo.
118
6.2 Results and Implications
Figure 6.3: The mass fraction of enclosed gas which is photoionized, fion , as a function of the
distance from the central star cluster, after 3 Myr of photoheating. Each line corresponds to a
different choice of IMF and total mass in stars, as labeled. The ionized fraction begins to drop at
≥ 300 pc mostly due to two minihalos at ∼ 400 and ∼ 500 pc, shown in Fig. 6.2 (right panels),
which remain shielded from the ionizing radiation and are thus largely neutral. Within ∼ 100 pc,
there is a large fraction of gas which remains un-ionized at densities ≥ 100 cm−3 . This gas will
likely collapse to form stars, despite the strong radiative feedback from the central stellar cluster.
119
6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
large, as there is still some neutral gas available for subsequent star formation regardless of
the radiative feedback.
The gas that is photoionized, however, is gradually expelled from the center of the halo,
and after 3 Myr of photoheating the density of the photoionized gas within ∼ 20 pc of the
cluster drops to . 10 cm−3 for the more massive cluster of 100 M stars shown in Fig. 6.2.
For our other choices of IMF and total cluster mass the dynamical response is less dramatic,
as the ionizing flux is weaker; for example, after 3 Myr the density of the central photoionized gas is . 50 cm−3 for the less massive cluster of 25 M stars. The varying degree to
which photoheating dynamically impacts the host halo leads to important differences in the
properties of the emitted radiation.
Although the limited resolution of our simulations allows only to track the expansion of
the H  region from an initial size of & 10 pc, we expect that after roughly a sound-crossing
time of the central unresolved ∼ 10 pc, or after the first few 105 yr, the evolution of the H 
region is reliably resolved. It should thus be noted that the breakout of the H  region may
be delayed by of order this timescale compared to our simulations. We note that in the Milky
Way the expansion of the photoheated gas in an H  region may be slowed due to turbulent
pressure confinement (Mac Low et al., 2007; Xie et al., 1996), resulting in ultra-compact H 
regions persisting for & 105 yr (Wood & Churchwell, 1989), much longer than the soundcrossing timescale for such regions. As turbulence begins to play an important role in the
formation of the first galaxies (Greif et al., 2008; Wise & Abel, 2007a), the initial evolution of
H  regions therein may be similarly confined. This possibility notwithstanding, we expect
that the spatial resolution that we do achieve suffices to track changes in the luminosity
emitted in recombination lines and in the escape fraction of ionizing radiation, which we
discuss in the remainder of this section.
6.2.2
Star formation Rate Indicators
The luminosity emitted in recombination lines, such as Hα, has been found to scale remarkably well with the SFR of galaxies at low redshift (Kennicutt 1983; but see PflammAltenburg et al. 2007). The SFR obtained using such relations relies on some knowledge of
the IMF of the stars which are forming, as well as on the escape fraction of ionizing radiation. Fig. 6.4 shows our calculations of the escape fraction fesc,HII of H -ionizing photons for
each of our choices of IMF and total mass in stars, and Fig. 6.5 shows the escape fraction
120
6.2 Results and Implications
fesc,HeIII of He -ionizing photons. The corresponding luminosities emitted in Hα, Lyα, and
He  λ1640 are presented in Fig. 6.5.
As shown in Fig. 6.4, there is a clear trend toward higher H -ionizing photon escape
fractions for more massive stellar clusters, with the majority of the ionizing photons escaping from clusters with the larger total stellar mass. The escape fraction is not, however,
independent of IMF; for a given total mass in stars, the escape fraction can differ by a wide
margin. Both the variability in the escape fraction with time and the range of values that we
find are in rough agreement with other recent calculations of the escape fraction of ionizing
photons from dwarf galaxies at z & 10 (Razoumov & Sommer-Larsen, 2009; Wise & Cen,
2009). For a recent calculation of the escape fraction of ionizing photons from more massive
galaxies, see Gnedin et al. (2008).
The breakout of the H  region generated by the less massive 100 M star cluster occurs
after ∼ 1 Myr, leading to an escape fraction & 0.5 after 2 Myr. In contrast, the H  region
of the equally massive 25 M star cluster remains confined to the host halo for ≥ 3 Myr,
contributing no ionizing photons to the IGM. The progress of the initial stages of hydrogen
reionization, likely driven by star formation in the first galaxies, may thus depend on whether
these galaxies hosted massive (& 10 M ) or very massive stars (& 100 M ) (Choudhury &
Ferrara, 2007).
The evolution of the H -ionizing photon escape fraction is reflected in the evolution of the
luminosity of hydrogen recombination lines, as shown in Fig. 6.6. Comparing the panels on
the left to those on the right, the luminosity in the Lyα and Hα lines, while generally higher
for larger total mass in stars, does not scale with the total mass in stars. Indeed, owing to the
increase in the escape fraction of ionizing photons, after ∼ 1 Myr the luminosity in hydrogen
recombination lines from the clusters with greater total stellar mass drops below that of the
clusters with lower total stellar mass, for a given IMF. Overall, because of the temporal
evolution of the luminosity in a given line, there is no one-to-one relationship between the
total mass in stars and the luminosity in a given recombination line. There is thus likely
to be a relatively weak correlation between the SFR and the luminosity in the hydrogen
recombination lines emitted from the first dwarf galaxies, owing to the dynamical evolution
of the photoionized gas and the escape of ionizing radiation into the IGM.
Similar to the case of the hydrogen recombination lines, the luminosity in the He  λ1640
line is anticorrelated with the escape fraction of He -ionizing photons, shown in Fig. 6.5.
However, different from the case of the hydrogen lines, the luminosity emitted in He  λ1640
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6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
Figure 6.4: The escape fraction of hydrogen-ionizing photons, fesc,HII , from the host galaxy,
each line corresponding to a different choice of IMF and total mass in stars, as labeled. Note the
tight anticorrelation between the escape fraction plotted here and the luminosity in the hydrogen
recombination lines shown in Fig. 6, demonstrating that the vast majority of the energy emitted
in hydrogen recombination lines emanates from the dense ionized gas within the host halo, as is
shown in detail in Fig. 6.10.
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6.2 Results and Implications
Figure 6.5: The escape fraction of He -ionizing photons, fesc,HeIII , from the host galaxy, each
line labeled as in Fig. 6.4. For most cases, the negligible escape fraction leads to a tight correlation between the luminosity emitted in the He  λ1640 line and the total mass contained
in stars, in contrast to the weaker correlation for hydrogen recombination lines, as discussed in
Section 6.3.2.
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6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
line is generally much more strongly correlated with the total mass in stars, for a given
IMF. This is due to the low escape fraction of He -ionizing photons, which is essentially
zero for every case studied here, except for the case of the more massive cluster of 100 M
stars. With such a high fraction of He -ionizing photons being balanced by recombinations
within the host halo, there is a near linear relation between the total mass in stars and the
luminosity emitted in He  λ1640, making this line a potentially much more reliable SFR
indicator than hydrogen lines such as Hα. There are slight departures from linearity due to
the temperature dependence of the emission coefficient jλ1640 , which varies by a factor of ∼ 2
over the temperature range of the ionized gas in our simulations and is generally lower for
the hotter H  regions generated by the more massive stellar clusters (Osterbrock & Ferland,
2006).
6.2.3
Initial Mass Function Indicators
The luminosity emitted from a galaxy in recombination lines depends not only on the stellar
IMF, but also on the density field of the galaxy and the escape fraction of ionizing photons. Therefore, the utility of recombination line strengths as IMF indicators hinges on an
understanding of the dynamical evolution of the photoionized gas, especially for the case
of starbursts in the first dwarf galaxies, in which such dynamical effects can be most pronounced.
For the starbursts that we simulate here, the luminosity of the He  λ1640 emission line
relative to the hydrogen recombination lines can be read from Fig. 6.6, while the equivalent
widths of these lines are presented in Fig. 6.7. Comparing the top panels of Fig. 6.6 to the
bottom panels, it is evident that the ratio of the luminosity emitted in He  λ1640 to that in Hα
(or Lyα) can be very different depending on the IMF. Fig. 6.7 shows that there is a similar
distinction in the ratios of the EWs. For the 100 M star clusters the luminosity in He 
λ1640 is comparable to that in Hα, while for the 25 M star clusters the luminosity in He 
λ1640 is up to an order of magnitude lower than that in Hα. However, as the escape fraction
of H -ionizing photons increases with time for the more massive 25 M stellar cluster, the
luminosities in these two lines become comparable, revealing that there is some ambiguity
in the use of this ratio of line luminosities as an indicator of the IMF of young (. 3 Myr)
stellar clusters. Thus, in some cases dynamical effects may compromise the use of this line
ratio in distinguishing between clusters of Pop III.1 and Pop III.2 stars, with typical masses
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6.2 Results and Implications
Figure 6.6: The luminosity of the galaxy, as a function of the time from the prompt formation
of the cluster, in three recombination lines: Lyα (dot-dashed blue lines), Hα (solid red lines),
and He  λ1640 (dashed black lines). The four panels correspond to our four different choices
of IMF and total mass in stars; these are, clockwise from top-left: twenty-five 100 M stars,
two hundred fifty 100 M stars, one thousand 25 M stars, and one hundred 25 M stars. The
luminosities generally decrease with time, as the photoheating acts to decrease the density of the
ionized gas, lowering the recombination rate. Note the different evolution of the He  λ1640
luminosity as compared to that of the hydrogen recombination lines, owing to the lower escape
fraction of He -ionizing photons (see Fig. 6.5).
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6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
Figure 6.7: The rest frame equivalent widths, W 0 = W/ (1 + z), where W is the observed EW, as
a function of time, of the same three recombination lines shown in Fig. 6.6. For comparison, in
each panel we plot the observed EWs of galaxies from two different surveys: the dotted line at
400 Ådenotes the median EW of Lyα emitters detected at z = 4.5 in the LALA survey (Malhotra
& Rhoads, 2002), while the dotted line at 145 Ådenotes the average EW of the six Lyα emitters
at z ≥ 6 detected in the Subaru deep field (Nagao et al., 2007). Note that the Lyα EWs that we
compute are upper limits, as scattering in a neutral IGM has not been accounted for.
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6.2 Results and Implications
of order 100 M and 10 M , respectively.
The ratio of the observed fluxes in He  λ1640 and Hα, as calculated using equation (6.10), is displayed in Fig. 6.8. In this figure, it is clear that this line ratio is sensitive
to the IMF, although it is not a constant for each cluster. Instead, for clusters in which the
escape fraction of H I-ionizing photons increases with time dramatically, while the escape
fraction of He II-ionizing photons remains roughly constant, this line ratio varies with the
flux observed in Hα. While the ratio of the fluxes is a somewhat ambiguous IMF indicator,
the clusters with the more top-heavy IMF do consistently exhibit larger ratios of He  λ1640
to Hα. Nagao et al. (2005) present a search for He  λ1640 emission from a strong Lyα
emitter at z = 6.33, finding an upper limit for the ratio of He  λ1640 to Lyα. Assuming a
standard value of 0.07 for the ratio of luminosity in Hα to that in Lyα (Osterbrock & Ferland, 2006), we show in Fig. 6.8 the upper limit that these authors report (Dawson et al.,
2004). Although a weak upper limit, it is clear that observations with only slightly greater
sensitivity will allow to differentiate between the flux ratios predicted here for massive and
very massive Pop III IMFs.
Comparing the EW of Hα in the four panels of Fig. 6.7, it is clear that it is not strongly
dependent on the IMF or on the total mass in stars, varying by at most a factor of three
between each of the cases. While showing more variation between the four cases, the EW
of Lyα also shows considerable ambiguity as an IMF indicator, its maximum value varying
by about a factor of three between each of the four cases. This insensitivity of the Lyα EW
to the IMF arises from two effects. Firstly, the stellar continuum luminosity Lλ,∗ increases
in a similar manner as the number of ionizing photons from the massive (25 M ) IMF to
the very massive IMF (100 M ). This acts to keep the EW, roughly the ratio of the two,
relatively constant. Secondly, while the luminosity in Lyα decreases with the increasing
escape fraction of ionizing photons for the more massive clusters, the continuum luminosity
remains largely unchanged, leading to a decrease in the EW with time for these clusters. We
note that the Lyα EWs presented here are only upper limits, as we have not accounted for
scattering of Lyα photons in the IGM (Dijkstra et al., 2007).
The EW of He  λ1640 is a more definitive indicator of IMF, being higher for the clusters
of 100 M stars than for the clusters of 25 M stars, regardless of the total mass in stars or of
the age of the cluster (up to & 3 Myr). As with the utility of He  λ1640 as a SFR indicator,
this largely follows from the generally low escape fraction of He -ionizing photons.
For comparison with observed galaxies, we plot in Fig. 6.7 the two observational results:
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6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
Figure 6.8: The ratio of the integrated fluxes in He  λ1640 and Hα, Fλ1640 /FHα , as a function of
time, for each of the four clusters simulated here, as labeled. The dashed horizontal line denotes
the upper limit of this ratio for the strong Lyα emitter SDF J132440.6+273607 at z = 6.33, as
reported by Nagao et al. (2005). Similar upper limits for Lyα emitters at z = 4.5 have been
reported by Dawson et al. (2004).
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6.2 Results and Implications
the median Lyα EW of galaxies detected in the Large Area Lyman Alpha (LALA) survey,
W 0 ∼ 400 Å, and the average EW of six galaxies observed at z ≥ 6 in the Subaru deep
field, ∼ 145 Å(Nagao et al., 2007). The large LALA EWs are comparable to what we find
for primordial dwarf galaxies, although the LALA galaxies likely do not host Pop III star
formation (but see Jimenez & Haiman, 2006). The detection of an EW of the He  λ1640 line
& 10 Åwould be a stronger indication of a galaxy hosting primordial star formation, as shown
in Fig. 6.7, although none has been found as of yet. We note that observed Lyman break
galaxies at z ∼ 3 have been found to have He  λ1640 EWs of ∼ 2 Å(Shapley et al., 2003),
consistent with what is expected for Wolf-Rayet stars formed in starbursts (Brinchmann et al.,
2008; Schaerer & Vacca, 1998).
6.2.4
Detectability of Recombination Radiation
In Fig. 6.9, we present our predictions for the observable recombination line fluxes, for each
of the stellar clusters that we simulate. Fig. 6.10 shows the surface brightness in Hα as
observed on the sky, for the two more massive stellar clusters, which each have a total mass
in stars of 2.5 × 104 M . The fluxes in each plot, largely determined by our choices for the
total stellar mass, are calculated using equation (6.6). While the larger H  region generated
by the more massive stars encompasses more dense gas, creating more widely distributed
emission in Hα, as shown in Fig. 6.10, the highest flux per square arcsecond is in the central
region of the halo hosting the less massive stars. This is due again to the less dramatic
dynamical response of the gas to photoheating, leading to higher densities, and thus higher
recombination rates. Due to this effect, the highest fluxes are generated just after the birth of
a stellar cluster, as shown in Fig. 6.9, when the density of the photoionized gas is still high,
not having had time to expand in response to the concomitant heating. Indeed, Fig. 6.9 shows
that the flux in Hα from the more massive 25 M star cluster may reach & 10 × (R/1000) nJy
before the breakout of the H  region. Catching the first galaxies when still in the earliest
stages of their initial starbursts, within the first few 105 yr, is thus likely to provide one of the
best chances for observations of purely primordial stellar populations in the early Universe.
Aboard JWST, the Hα line would be observed with MIRI. Its pixel size of & 0.1 arcsec
would not resolve the brightest portions of the galaxies that we simulate, which, as shown in
Fig. 6.10, are of order 0.01 arcsec. With a resolution capability of R = 3000 the MIRI has a
sensitivity of & 200 nJy for a signal-to-noise ratio of 10, in exposures of ∼ 106 s (Panagia,
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6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
Figure 6.9: The observed fluxes, f , as a function of time, of the same three recombination
lines shown in Fig. 6.6, for the galaxy we simulate at z ∼ 12.5. The fluxes are normalized to
what would be observed with a spectroscopic resolution of R = 1000, and are computed using
equation (6.6) assuming that the galaxy appears as an unresolved point source. Note that the
flux in Lyα is an upper limit, as the present calculation does not take into account scattering in a
neutral IGM.
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6.2 Results and Implications
Figure 6.10: The flux in Hα, fHα , per square arcsecond, emitted from a primordial dwarf galaxy,
as observed on the sky at z = 0, assuming a spectroscopic resolution of R = 1000. Shown here
are the two most massive of the four stellar clusters that we simulate, one containing 25 M stars
(bottom panels), the other containing 100 M stars (top panels). From left to right, the clusters
are shown at 105 yr, 1 Myr, and 3 Myr after formation. Note that the emission is concentrated
in the densest photoionized regions, the filaments around the galaxy (compare the top panels to
Fig. 6.2) and especially the dense gas within the inner ∼ 100 pc of the galaxy. The highest total
fluxes occur at the earliest times, before the H  region has broken out; hence, the youngest stellar
clusters are the most readily observed.
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6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
2005), making it unable to detect even the brightest galaxies that we simulate, the flux in Hα
of these being . 20 nJy for R = 3000.
With a greater sensitivity of . 100 nJy (Panagia, 2005), the Near Infrared Spectrograph
(NIRSpec) operates in the wavelength range 0.7 to 5 µm, allowing it to possibly detect Lyα
out to z ∼ 40 and He  λ1640 out to z ∼ 30. However, the flux in He  λ1640 is always below
that in Hα and, hence, is too low to be detected. The Lyα line, with the highest flux of the
three recombination lines shown in Fig. 6.9, is also not directly detectable, with a flux falling
well below the ∼ 100 nJy sensitivity limit of NIRSpec. Furthermore, although the luminosity
in Lyα is always intrinsically higher than that in Hα, before reionization the observable flux
in Lyα may be dramatically decreased due to scattering in the neutral IGM (Dijkstra et al.,
2007). Because we do not account for this effect in the present calculations, the Lyα fluxes
presented here are only upper limits.
The Near Infrared Camera (NIRCam) aboard JWST, which will be used to conduct deep
surveys designed to detect the first galaxies, will be capable of detecting point source fluxes
as low as ∼ 3.5 nJy at a signal-to-noise ratio of 10 for a 105 second exposure (Gardner et al.,
2006). With a resolution of & 0.03 arcsec per pixel, the NIRCam would also not quite resolve
the galaxies that we simulate. We can evaluate the possibility that NIRCam may detect them
as point sources, however, by estimating the continuum flux of the galaxies as observed at
∼ 2 µm. As can be read from Figs. 6.6 and 6.7, the continuum flux, ∝ Lλ1640 /Wλ1640 , varies
only by a factor of a few between Lyα and He  λ1640. Thus, for simplicity we assume that
the continuum is roughly flat and calculate the specific continuum flux, as observed at z = 0,
as
Lλ1640 λ2λ1640 (1 + z)
fcont ∼
∼ 0.03 nJy ,
(6.17)
0
4πcD2L (z) Wλ1640
0
where Wλ1640
is the equivalent width in the rest frame of the galaxy, as defined in equations (6.12), (6.13) and (6.14). This flux is well below the sensitivity limit of the NIRCam,
and so we conclude that detection of the continuum radiation from the galaxies we simulate would also be undetectable. We note, however, that under favorable circumstances,
gravitationally lensed emission from a primordial galaxy undergoing an only slightly more
luminous starburst may be detectable with JWST, given that lensing can boost the flux by a
factor of order 10 (e.g. Refsdal, 1964; Stark et al., 2007).
The first dwarf galaxies could be more luminous than we find here if the efficiency of
star formation SF , defined as the fraction of the total baryonic mass in the galaxy contained
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6.3 Summary and Conclusions
in stars, is larger than what we have assumed in our simulations, where our choices for the
total mass in stars correspond to modest values of SF ∼ 10−3 – 10−4 . A larger efficiency
(SF ∼ 10−1 ), for the top-heavy IMFs considered here, would yield a cluster observable
by NIRCam (Gardner et al., 2006). However, as we have demonstrated, the much higher
ionizing flux from a ∼ 106 M cluster of massive primordial stars would induce a strong
hydrodynamic response which would lead to a rapid decline in the luminosity emitted in
recombination radiation. Thus, even if such clusters can be identified by their continuum
emission, the detection of recombination radiation, and with it information about the stellar
IMF, may be beyond the capabilities of JWST. Furthermore, the formation of such a massive
cluster of primordial stars may face an impediment due to the strong radiative feedback
within the cluster itself. Recent simulations of clustered star formation in the present-day
Universe suggest that radiative feedback influences the fragmentation behavior of the gas
and possibly lowers the overall star formation efficiency (Bate, 2009; Krumholz et al., 2007).
However, the situation is by no means clear (Dale et al., 2005, 2007).
It is possible that more massive (109 – 1010 M ) primordial galaxies form at z & 12, or
form at lower redshift, making their detection feasible. However, such more massive, and
therefore more luminous, galaxies are likely to also be more chemically evolved, and so may
already be dominated by Pop II star formation. Thus, it may be that the galaxies which host
pure Pop III starbursts, such as those we study here, will remain out of reach of even JWST,
although this critically depends on the poorly constrained process of metal enrichment in the
early Universe (Cen & Riquelme, 2008; Johnson et al., 2008; Pan & Scalo, 2007; Tornatore
et al., 2007). We emphasize, however, that the dynamical effects studied here are likely to
play a role even in the more luminous galaxies that will be detected, and are important to
take into account in evaluating observations meant to constrain the SFR or the IMF.
6.3
Summary and Conclusions
We have presented calculations of the properties of the recombination radiation emitted from
a primordial dwarf galaxy at z ∼ 12.5, during the initial stages of a starburst. Our cosmological radiation-hydrodynamical simulations allow us to track the detailed dynamical evolution
of the emitting gas in the central regions of the galaxy, and thus to study its effect on the
emerging radiation. The goal of this study has been to determine the observable signatures
of the initial starbursts in the first galaxies. In particular, we have aimed to find reliable
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6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
indicators of the star formation rate and of the stellar IMF.
Owing to the escape of H -ionizing photons into the IGM, we find only a weak correlation between the total mass in stars and the luminosity in hydrogen recombination lines.
This suggests that Lyα and Hα, despite the high luminosity in these lines, may not serve
as strong indicators of the SFR, unlike in the low-redshift Universe (Kennicutt, 1983). The
He  λ1640 line may be a more effective SFR indicator, as the luminosity in this line scales
more closely with the total mass in stars, due to the lower escape fraction of He -ionizing
photons.
We confirm that the ratio of He  λ1640 to either Lyα or Hα can be used as an IMF
indicator, although its utility is compromised in some cases by the unequal escape fractions
of H - and He -ionizing photons. The most robust IMF indicator, in terms of distinguishing
between populations of massive (& 10 M ) and very massive (& 100 M ) Pop III stars, is
the EW of He  λ1640, as it is consistently higher for the more massive stars regardless of
the total mass in stars. We note that while in principle the radiation emitted by a central
accreting BH in a primordial dwarf galaxy could introduce complications for using He 
λ1640 as IMF indicator (Tumlinson et al., 2001), recent work suggests that BH accretion
is inefficient in the early Universe (Alvarez et al., 2008; Johnson et al., 2007; Milosavljevic
et al., 2008; Milosavljević et al., 2009; Pelupessy et al., 2007). Such miniquasar activity may
thus not result in appreciable observable radiation for the first ∼ 108 yr.
In terms of the detectability of the recombination radiation from the first galaxies, we
have shown that due to the dynamical response of the gas to photoheating, a top-heavy IMF
or a high star formation efficiency can be self-defeating, leading to a decrease in the line
luminosity of the galaxy. We conclude that the detection of purely primordial dwarf galaxies
at z & 10 is likely to be beyond the capabilities of JWST, although their detection may be
just possible if the galaxies are strongly lensed. More luminous, 109 – 1010 M (total mass),
galaxies may thus be detected by JWST; however, such more massive galaxies are likely to be
already chemically enriched due to previous episodes of star formation in their progenitors.
These systems are then expected to host Pop II star formation, or a composite of Pop II
and Pop III, depending on the still poorly understood mixing of heavy elements in the first
galaxies (Karlsson et al., 2008).
Our results demonstrate how the radiation emitted from the first galaxies depends on the
hydrodynamic effects of the photoionization from clusters of massive stars. This is complementary to the results of previous studies (Schaerer, 2002, 2003), which highlight the
134
6.3 Summary and Conclusions
evolution of the emitted radiation owing to the aging of a stellar population. Clearly, both
effects must be considered in future work.
The initial starbursts of the first galaxies may constitute the formation sites of the only
metal-free stellar clusters in the Universe, since after the first several Myr supernova feedback can quickly enrich the galaxy with metals (Kitayama & Yoshida 2005; Mori et al. 2002;
but see Cen & Riquelme 2008; Tornatore et al. 2007). Also, a large fraction of the first dwarf
galaxies, with masses of order 108 M , may already form from metal-enriched gas (Johnson
et al., 2008; Omukai et al., 2008); it is an important open question what fraction of dwarf
galaxies forming at z & 10 are primordial when they begin forming stars. Future observations of those first dwarf galaxies that do host primordial star formation offer one of the few
opportunities for constraining the primordial IMF.
135
6. THE FIRST GALAXIES: SIGNATURES OF THE INITIAL STARBURST
136
7
The First Supernova Explosions:
Energetics, Feedback, and Chemical
Enrichment
One of the main goals in modern cosmology is to understand the formation of the first stars
at the end of the cosmic dark ages and how they transformed the homogeneous primordial
universe into a state of ever increasing complexity (Barkana & Loeb, 2001; Bromm & Larson, 2004; Ciardi & Ferrara, 2005; Glover, 2005; Miralda-Escudé, 2003). The first stars are
predicted to be very massive, with M∗ ∼ 100 M , and formed in DM minihalos with virial
masses of 105 – 106 M (Abel et al., 2002; Bromm et al., 1999, 2002; Gao et al., 2007;
Nakamura & Umemura, 2001; O’Shea & Norman, 2007; Yoshida et al., 2006). The first SN
explosions rapidly dispersed the heavy elements that were produced during the brief lifetime
of a Pop III star into the IGM, thus beginning the long nucleosynthetic build-up from a pure
H/He universe to one with ubiquitous metal enrichment (Bromm et al., 2003; Daigne et al.,
2006, 2004; Ferrara et al., 2000; Furlanetto & Loeb, 2003, 2005; Kitayama & Yoshida, 2005;
Madau et al., 2001; Mori et al., 2002; Norman et al., 2004; Scannapieco et al., 2002; Thacker
et al., 2002; Wada & Venkatesan, 2003; Yoshida et al., 2004).
The first SNe exerted important chemical and mechanical feedback effects on the early
universe (Ciardi & Ferrara, 2005). First, the character of star formation changed from an
early, high-mass dominated (Pop III) mode to a more normal, lower mass (Pop II) mode,
once a critical level of enrichment had been reached, the so-called critical metallicity, Zcrit .
137
7. THE FIRST SUPERNOVA EXPLOSIONS: ENERGETICS, FEEDBACK, AND
CHEMICAL ENRICHMENT
10−3.5 Z (Bromm et al., 2001a; Bromm & Loeb, 2003b; Frebel et al., 2007; Omukai, 2000;
Schneider et al., 2003, 2006; Smith & Sigurdsson, 2007). It is then crucially important to
understand the topology of early metal enrichment and when a certain region in the universe
becomes supercritical (Furlanetto & Loeb, 2005; Greif & Bromm, 2006; Mackey et al., 2003;
Matteucci & Calura, 2005; Ricotti & Ostriker, 2004; Scannapieco et al., 2003; Schneider
et al., 2002; Venkatesan, 2006). Second, the SN blast waves mechanically impacted the
halos that hosted Pop III stars by heating and subsequently evacuating the dense gas inside
them. Such a negative feedback effect, limiting the capacity for further star formation, has
previously been considered for low-mass galaxies with correspondingly shallow potential
wells (Dekel & Silk, 1986; Larson, 1974; Mac Low & Ferrara, 1999) but also for halos in the
vicinity of a SN progenitor, where the net effect is still uncertain (Cen & Riquelme, 2008). A
qualitatively different mechanical feedback effect has recently been suggested to occur in the
dense, post-shock regions of the energetic blast wave (Machida et al., 2005; Mackey et al.,
2003; Salvaterra et al., 2004) where secondary star formation might be induced by the onset
of gravitational instabilities, thus giving rise to positive feedback.
The investigation of energetic explosions in the high-redshift universe has a long and
venerable history. Most of this early work has focused on the cooling and fragmentation of
astrophysical blastwaves (Bertschinger, 1985; Vishniac et al., 1985; Wandel, 1985) and the
possibility of self-propagating galaxy formation (Carr et al., 1984; Ikeuchi, 1981; Ostriker
& Cowie, 1981). Such explosive galaxy formation models typically implied substantial distortions to the cosmic microwave background blackbody spectrum and were thus excluded
by the COBE (Fixsen et al., 1996). Many of the physical effects discussed in those seminal
papers, however, remain relevant for our current studies.
What kind of SN explosion is expected to end the life of a massive Pop III star? According to the precise progenitor mass, the SN could be either of the conventional core-collapse
type (for masses . 40 M ) or it could be a PISN for masses in the range ∼ 140 – 260 M
(Heger et al., 2003; Heger & Woosley, 2002). In this paper, we specifically consider a PISN
with a progenitor mass of 200 M and an explosion energy of Esn = 1052 ergs. A PISN
is predicted to completely disrupt the star, so that all the heavy elements produced will be
released into the IGM (Fryer et al., 2001; Heger & Woosley, 2002), resulting in extremely
high metal yields. However, our simulation would also approximately describe a hypernova
explosion, in which a rapidly rotating, massive star undergoes core collapse (Tominaga et al.,
2007; Umeda & Nomoto, 2002), as far as the energetics and overall dynamics of the blast
138
wave expansion are concerned.
Theoretically, one would expect that a PISN could only have occurred in the early universe, where Pop III stars were born massive and mass loss might have been negligible
(Baraffe et al., 2001; Kudritzki, 2002). The recent discovery of the extremely luminous,
relatively nearby SN 2006gy, which has tentatively been interpreted as a PISN (Smith et
al. 2007; but see Ofek et al. 2007), might defy this theoretical expectation. Having such a
nearby example of a PISN, in effect, would provide us with an “existence proof,” opening
up the exciting possibility of detecting the first SNe with the upcoming JWST, which will be
sensitive enough to observe a single PISN out to z ∼ 15 (Gardner et al., 2006; Mackey et al.,
2003; Scannapieco et al., 2005b; Weinmann & Lilly, 2005; Wise & Abel, 2005).
In this paper, we carry out cosmological simulations of the first SN explosions, using
the smoothed particle hydrodynamics code GADGET in its entropy-conserving formulation
(version 1.1) (Springel et al., 2001). To address the questions outlined above, it is crucial to
investigate the evolution of the SN remnant in three dimensions, starting from realistic initial
conditions and including all the relevant physics. In particular, one needs to implement a
comprehensive model for the cooling and the chemical evolution of the gas, as well as an
efficient algorithm to treat the radiative transfer around the Pop III progenitor star (Johnson et al., 2007). Our simulations greatly improve on earlier work (Bromm et al., 2003), in
that we calculate the preexplosion situation with much greater realism, using a ray-tracing
method to determine the structure and extent of the H  region. In addition, we follow the
evolution to much later times, allowing us to reach the point where the blast wave finally
stalls and effectively dissolves into the general IGM. We are thus able to analyze the expansion and cooling properties of the SN remnant in great detail, allowing us to draw robust
conclusions on the temporal behavior of the shock, its morphology, the final shock radius,
and the total swept-up mass.
The structure of our work is as follows. In Section 7.2, we describe the SN progenitor and
our method of initializing the SN explosion, followed by a test simulation to verify the accuracy of our results. Subsequently, we discuss the cosmological setup of the main simulation
and our treatment of the H  region. We then concentrate on the evolution of the SN remnant,
finding a simple analytic model that summarizes its expansion properties (Section 7.3). In
Section 7.4, we investigate the mechanical feedback of the blast wave on neighboring minihalos and elaborate on the prospect of triggering gravitational fragmentation in the dense
shell. In Section 7.5, we discuss the relevance of metal cooling, the coarse-grain dispersal of
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metals, and the general mixing efficiency. Finally, in Section 7.6 we summarize our results
and discuss important cosmological implications.
7.1
Numerical Methodology
The treatment of SN explosions in SPH simulations is a demanding problem, primarily due
to the strong discontinuities arising at the shock front. One must set up the initial conditions
with great care, and it is necessary to enforce strict timescale constraints in calculating the
subsequent evolution. In light of these challenges, we describe the SN progenitor and our
method of initializing the SN explosion, followed by a performance test to verify the correct
behavior of the shock.
7.1.1
The Supernova Progenitor
Although the IMF of Pop III is poorly constrained, numerical simulations have indicated
that stars forming in primordial halos typically attain 100 M by efficient accretion and
might even become as massive as 500 M (Bromm & Loeb, 2004; Omukai & Palla, 2003;
O’Shea & Norman, 2007). Heger & Woosley (2002) have discussed the fate of such massive
stars and found that in the range 140 – 260 M a PISN disrupts the entire progenitor, with
explosion energies ranging from 1051 to 1053 ergs and yields up to y ' 0.5.
In the present work, we aim to investigate the most representative case, assuming a stellar
mass of M∗ = 200 M . We conservatively adopt an explosion energy of Esn = 1052 ergs
and a yield of y = 0.1, to include the possibility of a less top-heavy IMF and a hypernova
explosion, but we caution that such models generally predict energy input via bipolar jets,
which may invalidate the assumption of a spherically symmetric blast wave. In light of the
uncertain progenitor, we note that the dynamics of the SN remnant are governed mainly by
the explosion energy, together with the IGM density distribution, and that the composition
and metal content of the stellar ejecta become important only at very late times when the
enriched gas recollapses (see Section 7.5.1).
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7.1 Numerical Methodology
7.1.2
Energy Injection
During the early evolution of the SN remnant, i.e., typically less than 104 yr after the explosion, the stellar ejecta are confined to a thin shell propagating at constant velocity into
the surrounding medium, while secondary shocks quickly bring the interior to a uniform
temperature (Gull, 1973). This stage in the evolution of the SN remnant is termed the freeexpansion (FE) phase, and it lasts until the swept-up mass becomes comparable to the ejecta
mass.
To reproduce these well-known initial conditions, we inject the kinetic energy of the
SN as thermal energy into the Nsn = M∗ /msph innermost particles surrounding the center
of the box, where msph ' 5 M corresponds to the SPH particle mass. This increases the
thermal energy of all particles inside rfe to Eth = Esn /Nsn , and the resulting temperature
gradient acts as a piston on the surrounding gas. The ensuing shock accelerates the material
in the vicinity of rfe and creates a thin shell of highly supersonic material, initiating the
Sedov-Taylor (ST) phase of the SN remnant. In addition, this method allows us to track the
dispersal of metals, as each particle inside rfe represents the original stellar content. Although
the so-obtained resolution is crude, we can nevertheless quantify the coarse-grain chemical
enrichment properties of the SN.
7.1.3
Test Simulation
To test our method of setting up the initial conditions, and also verify that the code reliably
calculates the shock propagation, we compare the radial profiles of a test simulation with the
well-known ST solution. For this purpose we place the SN according to the above prescriptions in a noncosmological box of length 1 kpc with 2003 gas particles and switch off gravity,
chemistry, and cooling. The particles are distributed randomly, so that the density fluctuates
slightly around the mean of nH = 0.5 cm−3 , while the temperature is set to 2 × 104 K. This
choice reflects the situation in the vicinity of a 200 M Pop III star after the surrounding
medium has been photoheated (Alvarez et al., 2006a; Johnson & Bromm, 2007; Kitayama
et al., 2004; Whalen et al., 2004), but it also reproduces the initial conditions of the main
simulation (see Fig. 7.8).
Once we initialize the simulation, the shock propagates into the surrounding medium
according to the ST solution. As the duration and time-dependent properties of the gas in
this phase are crucial for the late-time behavior of the SN remnant, we briefly review the
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relevant details.
ST Solution
For a strong shock in an adiabatic gas, the Rankine-Hugoniot jump conditions imply for the
density, velocity, pressure, and temperature directly behind the shock:
ρsh = 4ρ
3
,vsh =
ṙsh
4
3 2
,Psh =
ρṙ
4 sh
3 µmH 2
ṙ ,
,T sh =
16 kB sh
(7.1)
(7.2)
(7.3)
(7.4)
where ρ is the density of the surrounding medium, µ its mean molecular weight, mH the
mass of the hydrogen atom, and kB Boltzmann’s constant. Solving the continuity, Euler,
and energy equations, one can relate these quantities at any point behind the shock to those
directly behind the shock as a function of position. Fig. 7.1 shows the resulting self-similar,
time-independent character of the ST solution, implying that most of the swept-up mass piles
up just behind the shock and forms a dense shell, while the interior regions remain at high
temperatures.
Completing the ST solution, the absolute position of the shock as a function of time is
given by
2 !1/5
Esn tsh
(7.5)
rsh = β
,
ρ
where β = 1.17 for an adiabatic gas. Thus, given the SN explosion energy and the density of the surrounding medium, all quantities of relevance around the progenitor are welldetermined.
Test Results
Using the above results, we investigate whether the test simulation has successfully reproduced the ST solution. Fig. 7.2 compares the profiles of the test simulation after 1 Myr, when
enough time has passed to enable a relaxation to the asymptotic solution, to the analytic, but
dimensionalized, ST profiles shown in Fig. 7.1. Apart from minor deviations caused by
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7.1 Numerical Methodology
Figure 7.1: ST solution: density, velocity, pressure, and inverse temperature as a function of
position relative to the shock. Most of the mass piles up just behind the shock and forms a dense
shell, while the isobaric interior regions remain at high temperatures.
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7. THE FIRST SUPERNOVA EXPLOSIONS: ENERGETICS, FEEDBACK, AND
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initial density fluctuations and higher order shocks, in particular in regard to the velocity distribution, we find that the simulation and the ST solution are in good agreement. The slight
offset at the position of the shock is inevitable in light of kernel smoothing.
Being assured that we can reliably treat the propagation of shocks, we turn to the main
focus of this work and describe the setup for a SN explosion in the high-redshift universe.
7.1.4
Main Simulation
Initial Setup
To investigate the long-term evolution of the SN remnant, but also to correctly calculate
the structure and extent of the preceding build-up of the H  region around the progenitor,
we choose a cosmological box of length 150/h kpc (comoving), with 2003 particles per
species (DM and gas). We initialize the simulation at z = 100 deep in the linear regime,
adopting for this purpose a concordance ΛCDM cosmology with the following parameters:
matter density Ωm = 1 − ΩΛ = 0.3, baryon density Ωb = 0.04, Hubble parameter h =
H0 / 100 km s−1 Mpc−1 = 0.7, spectral index n s = 1.0, and a top-hat fluctuation power
σ8 = 0.9 (Spergel et al., 2003). Initial density and velocity perturbations are imprinted
according to a Gaussian random field and grow in proportion to the scale factor until the onset
of nonlinearity. At this point the detailed chemical evolution of the gas becomes crucial, and
we apply the same chemical network as in Johnson et al. (2007) to track the abundances
of H, H+ , H− , H2 , H+2 , He, He+ , He++ , and e− , as well as the five deuterium species D,
D+ , D− , HD, and HD+ . All relevant cooling mechanisms in the temperature range 10 –
108 K are implemented, including H and He resonance processes, bremsstrahlung, inverse
Compton (IC) scattering, and molecular cooling for H2 and HD. Metal cooling does not
become important for the entire lifetime of the SN remnant, yet we postpone a more detailed
discussion of this issue to Section 7.5. We do not take into account the emission of radiation
by the post-shock gas, which acts to create a thin layer of fully ionized material ahead of the
shock and suppresses molecule formation (Kang & Shapiro, 1992; Shapiro & Kang, 1987;
Shull & McKee, 1979), since (i) the SN remnant expands into an H  region, and (ii) we find
that molecule formation in the post-shock gas becomes important only at late times, when it
has cooled to below 104 K (see Section 7.3.4).
With these ingredients, the first star forms in a halo of Mvir ' 5 × 105 M and rvir '
100 pc at z ' 20 in the canonical fashion (Abel et al., 2002; Bromm et al., 1999, 2002). We
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7.1 Numerical Methodology
Figure 7.2: Density, velocity, pressure, and temperature of the shocked gas after 1 Myr. Black
dots represent the test simulation, while the gray lines show the dimensionalized ST solution.
Apart from deviations caused by higher order shocks and kernel smoothing, the simulation reproduces the analytic profiles relatively well.
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CHEMICAL ENRICHMENT
determine its location by identifying the first particle that reaches a density of nH = 104 cm−3 .
At this point the gas ‘loiters’ around a temperature of 200 K and typically attains a Jeans
mass of a few 103 M before further collapsing (Bromm et al., 2002; Glover, 2005). For
simplicity, we assume that such a clump forms a single star, and we find that its location is
reasonably well resolved by the minimum resolution mass, Mres ' 500 M . In Fig. 7.3, we
show the hydrogen number density in the x-y and y-z plane, centered on the formation site
of the first star. Evidently, the host halo is part of a larger overdensity that will collapse in the
near future and lead to multiple merger events. This behavior is characteristic of bottom-up
structure formation, and our simulation therefore reflects a cosmological environment typical
for these redshifts.
H  Region
The treatment of the H  region around the star is crucial for the early- and late-time behavior
of the SN remnant. The photoevaporation of the host minihalo greatly reduces the central
density and extends the energy-conserving ST phase, whereas after an intermediate stage the
enhanced pressure in the H  region leads to an earlier transition to the final, momentumconserving phase. In addition, the shock fulfills the stalling criterion (i.e. ṙsh = c s , where
c s is the sound speed of the photoheated IGM) much earlier in the H  region compared
to previously unheated gas. We have found that neglecting the presence of the H  region
around the star, which extends well into the IGM, leads to a final shock radius a factor of 2
larger, demonstrating its importance for the long-term evolution of the SN remnant.
To determine the size and structure of the H  region, we proceed analogously to Johnson
et al. (2007). In detail, we initially photoheat and photoionize a spherically symmetric region
surrounding the star up to a maximum distance of 200 pc, where we find a neighboring
minihalo. We determine the necessary heating and ionization rates by using the properties of
a 200 M Pop III star found by Bromm et al. (2001b) and Schaerer (2002). After about 2 Myr,
when the star has reached the end of its lifetime, the hydrodynamic shock has propagated to
rvir /2 and photoevaporated the central regions of the host halo. Fig. 7.8a shows the resulting
density, temperature, pressure, and velocity profiles, which display the characteristics of
the analytic Shu et al. (2002) solution (i.e. pressure equilibrium throughout the interior,
while the density [temperature] become almost constant at small [large] radii). In our case,
the average interior density drops to nH ' 0.5 cm−3 , while the central temperatures rise
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7.1 Numerical Methodology
Figure 7.3: Hydrogen number density averaged along the line of sight in a slice of 10/h kpc
(comoving) around the first star, forming in a halo of total mass Mvir ' 5 × 105 M at z ' 20.
Evidently, the host halo is part of a larger conglomeration of less massive minihalos and is subject
to the typical bottom-up evolution of structure formation.
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7. THE FIRST SUPERNOVA EXPLOSIONS: ENERGETICS, FEEDBACK, AND
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to roughly 4 × 104 K (see Fig. 7.8a). The radial profiles agree relatively well with the selfconsistent radiation-hydrodynamics simulations performed in Abel et al. (2007) and Yoshida
et al. (2007a), with slight differences most likely due to the harder spectrum and shorter
lifetime of a 200 M star compared to a 100 M star (see also Johnson & Bromm, 2007).
Since ionizing radiation escapes the minihalo after only a few thousand years (Alvarez
et al., 2006a), we determine the final size and structure of the H  region by performing the
ray-tracing algorithm introduced in Johnson et al. (2007). This routine finds the Strömgren
radius along 105 rays around the star, with each ray consisting of 200 radial bins. For this
purpose we assume that helium has the same ionization properties as hydrogen, and we do
not treat a separate He  front. Once an individual cell fulfills the Strömgren criterion, we
ionize its content and set its temperature and molecule abundances to the values at the outer
edge of the photoheated region. This method is appropriate for the propagation of ionization
fronts in the general IGM, but it does not correctly treat the photoevaporation of neighboring
minihalos, when the ionization front becomes D-type (Ahn & Shapiro, 2007). This issue
is particularly important with respect to feedback on neighboring minihalos and is further
discussed in Section 7.4.
Fig. 7.4 shows the gas temperature after the main-sequence lifetime of the star, and thus
indirectly the size and structure of the H  region. In agreement with Alvarez et al. (2006a),
we find that the H  region can be as large as 5 kpc, but efficient shielding by neighboring
minihalos may limit its size to a few 100 pc in some directions. The H  region initially
cools via IC scattering, while the pressure gradient at the boundary of the H  region leads
to a gradual adiabatic expansion on timescales comparable to the Hubble time at z ' 20
(see Figs. 7.4 and 7.9) (Johnson & Bromm, 2007). Even though molecule fractions rise to
typically xH2 ∼ 10−3 and xHD ∼ 10−7 , molecular cooling remains inefficient due to the low
densities of the photoheated gas (Johnson & Bromm, 2007). We do not treat the evolution
of a separate LW front, as molecules inside the H  region are destroyed primarily by collisional dissociation and charge transfer, while they are quickly reformed in more massive,
neighboring minihalos once the central source turns off (Johnson et al., 2007; Yoshida et al.,
2007a).
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7.1 Numerical Methodology
Figure 7.4: Temperature averaged along the line of sight in a slice of 10/h kpc (comoving)
around the star after its main-sequence lifetime of 2 Myr. Ionizing radiation has penetrated
nearby minihalos and extends up to 5 kpc around the source, heating the IGM to roughly 2 ×
104 K, while some high-density regions have effectively shielded themselves.
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7. THE FIRST SUPERNOVA EXPLOSIONS: ENERGETICS, FEEDBACK, AND
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Sink Particles
In the course of the simulation, neighboring minihalos might reach high enough densities to
form stars, and the subsequent photoheating could significantly influence the propagation of
the SN remnant. In the present work, we aim to investigate a single unperturbed SN explosion, and thus we rule out further star formation by employing the sink particle algorithm
used in Johnson et al. (2007). This routine forms sink particles once the hydrogen number density exceeds 104 cm−3 and further accretes particles inside the Bondi radius (Bondi,
1952). This procedure prevents disturbances arising from the expansion of additional ionization fronts and allows us to concentrate on the feedback caused by the SN explosion.
With these preparations in place, we reinitialize the simulation at the end of the FE phase
according to Section 7.2.2. In the following we discuss the evolution of the SN remnant until
it effectively dissolves into the IGM.
7.2
Expansion and Cooling Properties
Throughout its lifetime, the remnant goes through four evolutionary stages, each of which is
characterized by a different physical mechanism becoming dominant. Based on this chronological sequence, we discuss the expansion and cooling properties of the SN remnant with
respect to the simulation results, and we summarize the remnant’s behavior with a simple
analytic model.
7.2.1
Phase I: Free Expansion
At very early times, the SN remnant enters the FE phase and propagates nearly unhindered
into the surrounding medium. It expands with a constant velocity of v2ej = 2Esn /Mej , where
for our case Mej = M∗ . The duration of the FE phase is given by tfe = rfe /vej , or
s
tfe = rfe
Mej
,
2Esn
(7.6)
where rfe is the radius at which the swept-up mass equals the mass of the original ejecta, i.e.
3XMej
rfe =
4πmH nH
150
!1/3
,
(7.7)
7.2 Expansion and Cooling Properties
where X = 0.76 is the primordial mass fraction of hydrogen (Kitayama & Yoshida, 2005).
After tfe , the inertia of the swept-up mass becomes important, and the shock undergoes a
transition to the ST phase. Since we do not explicitly model the FE phase, we use the above
analytic arguments to obtain rfe . 20 pc and tfe . 104 yr (see Fig. 7.7).
7.2.2
Phase II: Sedov-Taylor Blast Wave
According to Section 7.2.2, the simulation begins when the shock undergoes a transition to
the ST phase. Since we have already discussed the properties of the ST solution in Section 7.2.3, we can apply these results to understand the early behavior of the SN remnant.
With the knowledge that the previous photoheating has created an average density profile of
nH ' 0.5 cm−3 in the vicinity of the progenitor (Fig. 7.8a), we find with the help of equation (7.5) that the shock approaches rvir /2 after about 105 yr. At this point it catches up with
the previously established photoheating shock, where the outlying density profile becomes
isothermal. The simulation results (Fig. 7.8b) agree with this analytic prediction, yet the profiles do not quite resemble the ST solution, partly because not enough time has passed, but
also because cooling in the dense shell becomes important, violating energy conservation.
Consequently, the ST phase ends and the remnant undergoes a second transition.
To understand the origin of this transition, we elaborate on the cooling mechanisms responsible for radiating away the thermal energy of the remnant. At the high temperatures behind the shock, these are H and He resonance processes (RPs), bremsstrahlung (BS), and IC
scattering, with the former consisting of collisional ionization, excitation, and recombination
cooling. Fig. 7.5 shows the relevant cooling rates, normalized to a hydrogen number density of unity, where the abundances of the various species follow from assuming collisional
ionization equilibrium. Evidently, IC scattering and BS are important at high temperatures,
whereas RPs become significant below 106 K. However, BS and RPs are proportional to the
density squared, while IC scattering exhibits only a linear dependence. Thus, RPs will be the
most important coolant in the dense shell, while IC scattering is dominant in the hot interior.
At z = 20, the cooling time for IC scattering is approximately 10 Myr, independent of
temperature and density, while cooling times for RPs in the dense shell are generally much
lower. To determine when radiative losses affect the energetics of the SN remnant, we must
equate the shell cooling time with the expansion time. Fig. 7.6 shows this analytic relation for
the initial conditions of the simulation, i.e. Esn = 1052 ergs and nH = 0.5 cm−3 . Confirming
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7. THE FIRST SUPERNOVA EXPLOSIONS: ENERGETICS, FEEDBACK, AND
CHEMICAL ENRICHMENT
Figure 7.5: Cooling rates as a function of temperature at z ' 20 for H and He RPs (solid line),
BS (dotted line), and IC scattering (dashed line) in collisional ionization equilibrium, with nH set
to unity. At low temperatures RPs dominate, while above 106 K IC scattering and BS become
important. However, RP’s and BS are proportional to the density squared, while IC scattering
exhibits only a linear dependence.
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7.2 Expansion and Cooling Properties
Figure 7.6: Cooling time over expansion time in the dense shell for RPs (solid line), BS (dotted
line), and IC (dashed line). The long-dashed line indicates when cooling becomes efficient,
implying that RPs are responsible for ending the ST phase after 105 yr. At z = 20, the cooling
time for IC scattering is roughly 10 Myr, independent of temperature and density.
the simulation results, this prediction yields that RPs efficiently cool the dense shell to 104 K
after about 105 yr. At this point the shocked gas separates into a hot, interior bubble with
temperatures above 106 K and a dense shell at 104 K bounded by a high-pressure gradient.
This multi-phase structure is clearly visible in Fig. 7.8b, which remains intact for . 10 Myr,
when IC scattering becomes important and cools the last remnants of the interior bubble to
the temperature of the dense shell. With energy conservation no longer valid, the ST phase
ends before the shock has relaxed to the asymptotic solution, yet we nevertheless find that
2/5
rsh ∝ tsh
fits the temporal scaling of the mass-weighted mean shock radius relatively well
(see Fig. 7.7).
7.2.3
Phase III: Pressure-Driven Snowplow
After the ST phase ends, the pressurized, interior bubble drives a dense shell, and one speaks
of a pressure-driven snowplow (PDS). To analytically describe the further evolution of the
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7. THE FIRST SUPERNOVA EXPLOSIONS: ENERGETICS, FEEDBACK, AND
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SN remnant, we assume that the entire swept-up mass Msw is confined to an infinitely thin
shell. In light of the steep density profile toward the interior, this assumption is justified,
leading to an equation of motion of the form
d (Msw vsh )
2
Pb ,
= 4πrsh
dt
(7.8)
where Pb is the pressure of the hot, interior bubble, and the external pressure has been ne−5
glected (Ostriker & McKee, 1988). Since Msw ∝ rsh in an r−2 density profile and Pb ∝ rsh
in the adiabatically expanding interior, one can solve the above equation with a power law
2/5
of the form rsh ∝ tsh
. Interestingly, this procedure yields the same temporal scaling as the
ST solution, and we therefore do not expect a change in slopes after 105 yr. This is a direct
consequence of the transition to the PDS phase once the shock approaches the isothermal
density profile in the outskirts of the halo. Fig. 7.7 confirms this result, as the simulation
2/5
shows only a slight deviation from the analytically derived tsh
slope.
Using the above model, we can further determine when the PDS phase ends. The pressure
directly behind the shock after 105 yr can be estimated with equation (7.3) as Pb /kB ' 3 ×
2/5
−5
106 K cm−3 , in agreement with the simulation (see Fig. 7.8b). With Pb ∝ rsh
and rsh ∝ tsh
,
−2
we further find Pb ∝ tsh , implying that after roughly 1 Myr the interior pressure has dropped
to Pb /kB ' 3 × 104 K cm−3 . At this point pressure equilibrium between the hot interior and
the dense shell has been established, and the shock is driven solely by its accumulated inertia.
Fig. 7.8c confirms this prediction, showing that the interior pressure has indeed dropped to
that of the dense shell. In contrast, the temperature only drops proportional to tsh and remains
high.
7.2.4
Phase IV: Momentum-Conserving Snowplow
With the pressure gradient no longer dominant, the SN remnant is driven by the accumulated
inertia of the dense shell and becomes a momentum-conserving snowplow (MCS). In analogy to the derivation performed in Section 7.3.3, the position of the shock as a function of
time can be obtained by solving equation (7.8) in the absence of a pressure term. Since the
shock has not yet propagated beyond the surrounding r−2 density profile, this yields an initial
1/2
scaling of rsh ∝ tsh
.
At later times, the shock finally leaves the host halo and encounters neighboring mini-
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7.2 Expansion and Cooling Properties
halos in the y-z plane, but underdense voids perpendicular to the y-z plane (see Fig. 7.3).
This increases its radial dispersion, while at the same time the Hubble expansion becomes
important and serves to expand the medium on which the SN remnant propagates, thus raising its physical shock velocity. We can therefore only estimate the temporal scaling of the
mass-weighted mean shock radius based on the simulation results, finding, interestingly, that
1/2
rsh maintains its tsh
scaling until it fulfills the stalling criterion (see Fig. 7.7). This occurs
after about 200 Myr, close to the Hubble time at z ' 20, when the shock velocity approaches
the local IGM sound speed and becomes indistinguishable from sound waves. Evidently,
the increased slope at late times is a fundamental difference between SNe occurring at high
1/4
redshifts and in the present-day universe, where rsh ∝ tsh
.
In the meantime, the hot interior has expanded adiabatically for roughly 10 Myr, when
IC scattering becomes important, quickly cooling the interior to temperatures of the dense
shell (see Fig. 7.8d). Even later, the high electron fraction persisting in the interior and
the dense shell leads to efficient molecule formation, and typical abundances of xH2 ∼ 10−3
and xHD ∼ 10−7 are reformed. By the end of the simulation, both phases have adiabatically
cooled to T ∼ 103 K, while densities in the dense shell have dropped to nH ∼ 10−2.5 cm−3 (see
Figs. 7.9 and 7.14). The interior is even more underdense, and we find that the SN explosion
has completely disrupted the host halo, preventing further star formation inside of it for at
least a Hubble time at z ' 20. Due to the continuous adiabatic expansion of the post-shock
gas, molecular cooling remains inefficient for the entire lifetime of the SN remnant.
7.2.5
Summary of Expansion Properties
Summarizing the expansion properties of the SN remnant, Fig. 7.7 shows the mass-weighted
mean shock radius as a function of time, together with the analytically derived power laws.
The good agreement indicates that one can quantify the evolution of the SN remnant by
means of simple physical arguments and obtain relatively accurate results.
We find a final mass-weighted mean shock radius of 2.5 kpc, which is roughly a factor
of 2 smaller than the H  region. Fig. 7.7 also shows the radial dispersion of the shock, indicating that the surrounding medium becomes highly anisotropic once the SN remnant leaves
the host halo. After about 5 Myr, the shock encounters the first neighboring minihalos in the
y-z plane and begins to stall, while it continues to propagate into the voids perpendicular to
the y-z plane. The vast majority of the shocked material resides in the general IGM, as the
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7. THE FIRST SUPERNOVA EXPLOSIONS: ENERGETICS, FEEDBACK, AND
CHEMICAL ENRICHMENT
mass-weighted mean closely traces the maximum shock radius.
To conclude this section, Fig. 7.10 shows the swept-up gas mass as a function of time
for the simulation and the analytic model, where the slopes have been determined by the
3
relation Msw ∝ ρrsh
, except at very late times when the shock structure becomes too complex
for a simple analysis. We find a final swept-up mass of 2.5 × 105 M , which is roughly a
factor of 2 smaller than the mass enclosed inside a sphere of radius 2.5 kpc, assuming the
background density at z ' 12. This further demonstrates the overall asymmetry of the shock
(see Fig. 7.14).
7.3
Feedback on Neighboring Halos
So far we have concentrated on the propagation of the SN remnant into the IGM and treated
neighboring minihalos as disturbances to the radial density profile. In the following, we
explicitly investigate the mechanical feedback of the SN on nearby minihalos and discuss
possible consequences for star formation. For this purpose we have carried out two additional
simulations, one without feedback and another with radiative feedback, which enable us to
disentangle the effects of photoheating and the SN shock.
Combining the so-obtained results, Fig. 7.11 shows the distances of all star-forming halos
from the initial SN progenitor as a function of expansion time, when they have reached the
threshold density nH = 104 cm−3 . Following our argumentation in Section 7.2.4, the halo
irreversibly collapses at this point and forms a star. The shades of the symbols in Fig. 7.11
indicate their affiliation to the different simulations (i.e. black, dark gray, and light gray
symbols represent the no-feedback, photoheating-only, and main simulation runs), while
the shapes of the symbols represent individual halos. For orientation, we show the massweighted mean shock radius at late times according to Fig. 7.7.
7.3.1
Delay by Photoheating
In the photoheating-only case, we find that the collapse of all three star-forming halos is
delayed (see Fig. 7.11). While we do not properly resolve D-type ionization fronts that
may develop within shielded minihalos, we find that ionizing radiation can penetrate deep
into their cores and suppress cooling and the accretion of gas. The distance, mass, and
maximum density of the halo at the onset of photoheating are all crucial for the extent of
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7.3 Feedback on Neighboring Halos
Figure 7.7: Evolution of the SN remnant in its entirety, from z ' 20, when the SN explodes, to
z ' 12, when the shock finally stalls. Its total lifetime is about 200 Myr, or a Hubble time at z '
20. The black dots indicate the mass-weighted mean shock radius according to the simulation,
while the dashed line shows the analytic solution. For both we find a final mass-weighted mean
shock radius of 2.5 kpc. The shaded region shows the radial dispersion of the shock, indicating
that it increases significantly once the SN remnant leaves the host halo and encounters the first
neighboring minihalos in the y-z plane. The bulk of the SN remnant propagates into the IGM,
since the mass-weighted mean closely traces the maximum shock radius.
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CHEMICAL ENRICHMENT
Figure 7.8: Density, velocity, pressure, and temperature around the SN progenitor star immediately before its death, and 105 , 106 , and 107 yr after the SN explosion. Black dots represent normal SPH particles, while gray dots represent the initial stellar ejecta. Panel (a): the hydrodynamic
shock of the H  region has approached rvir /2 at ' 50 pc after establishing pressure equilibrium
in the interior, while the density and velocity profiles assume the characteristic Shu et al. (2002)
solution of a champagne flow. The temperature of the H  region is typically 2×104 K. Panel (b):
the shock created by the SN explosion propagates into the surrounding medium according to the
ST solution, with the characteristic formation of a dense shell and a hot, interior bubble. After
105 yr, the shock approaches the previous photoheating shock with expansion velocities well in
excess of 100 km s−1 . At the same time, cooling in the dense shell by RPs becomes efficient
and the ST phase ends. The resulting sharp temperature drop between the interior bubble at
T > 106 K and the dense shell at T ' 104 K is clearly visible. Panel (c): the SN remnant has
undergone a transition to the PDS phase and is driven by the high pressure in the interior and the
momentum of the dense shell. This phase ends after about 1 Myr, when pressure equilibrium
has been established. The pressure drop in comparison with Panel (b) is clearly visible, while
temperatures in the interior remain high. Panel (d): after 10 Myr, the radial dispersion of the
shock has increased dramatically, ranging from & 200 pc where the shock encounters the first
neighboring minihalos, to . 1 kpc perpendicular to the overdensities in the y-z plane.
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7.3 Feedback on Neighboring Halos
(a) tsh = 1 Myr
(b) t sh = 10 Myr
log T
4
3
(c) tsh = 50 Myr
(d) t sh = 200 Myr
2
1
x-y plane
Boxsize: 150/h kpc (comoving)
Slice Width: 10/h kpc (comoving)
Figure 7.9: Temperature averaged along the line of sight in a slice of 10/h kpc (comoving)
around the x-y plane after 1, 10, 50, and 200 Myr. In all four panels, the H  region and SN
shock are clearly distinguishable, with the former occupying almost the entire simulation box,
while the latter is confined to the central regions. Panel (a): the SN remnant has just left the host
halo, but temperatures in the interior are still well above 104 K. Panel (b): after 10 Myr, the
asymmetry of the SN shock becomes visible, while most of the interior has cooled to well below
104 K. Panel (c): the further evolution of the shocked gas is governed by adiabatic expansion, and
its morphology develops a ‘finger-like’ structure (see also Fig. 7.13). Panel (d): after 200 Myr,
the shock velocity approaches the local sound speed and the SN remnant stalls. By this time the
post-shock regions have cooled to roughly 103 K.
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7. THE FIRST SUPERNOVA EXPLOSIONS: ENERGETICS, FEEDBACK, AND
CHEMICAL ENRICHMENT
Figure 7.10: Swept-up gas mass as a function of time for the simulation (black dots) and the
3 ,
analytic model (dashed lines), where the latter is related to the shock radius by Msw ∝ ρrsh
except at very late times, when the shock structure becomes too complex for a simple analysis.
For both we find a final swept-up mass of 2.5 × 105 M .
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7.3 Feedback on Neighboring Halos
the delay. The most massive halo in our simulation, halo (a), experiences a delay of only
25 Myr, while halos (b) and (c) are less dense at the time of photoheating and experience
delays in excess of 80 Myr. This is roughly consistent with the results of Ahn & Shapiro
(2007) who argue that radiative feedback on neighboring minihalos strongly depends on their
evolutionary stage (Mesinger et al., 2006; Susa & Umemura, 2006). For more quantitative
results, one must perform self-consistent radiation-hydrodynamics simulations that include
the effects of photodissociating radiation (Yoshida et al., 2007a).
Due to the inaccuracies mentioned above, we cannot draw robust conclusions on the
subsequent mode of primordial star formation, i.e., Pop III or Pop II.5, but we anticipate this
to be a function of the state of the collapse and the ionizing flux (Greif & Bromm, 2006;
Jappsen et al., 2007; Johnson & Bromm, 2006; Mackey et al., 2003).
7.3.2
Shock-driven Collapse
In contrast to feedback by ionizing radiation, we find that in our case the shock of the SN
remnant acts to enhance halo collapse and promote star formation. This behavior is evident from Fig. 7.11, as halos (a) and (b) collapse about 15 Myr earlier with respect to the
photoheating-only run. In both cases, the shock compresses the dense cores of each of the
halos, which serves to enhance cooling and accretion. Although our resolution is too crude
for quantitative conclusions, we find that the mechanical impact of the SN remnant mitigates
the delay caused by photoheating and leads to a slightly increased star formation rate. This
situation might be different for star-forming halos much closer to the SN explosion, where
ram pressure stripping could be sufficient to dispel a large fraction of the gas. However, in
the present simulation we find that most halos massive enough to form stars are at sufficient
distances from the SN progenitor.
We note that we neglect the possible effects of photodissociating radiation emanating
from the relic H  region, as it has been shown that recombination radiation can only produce
a relatively weak photodissociating flux, which quickly dies away as the gas in the relic H 
region recombines on timescales of the order of 1 Myr (Johnson & Bromm, 2007; Yoshida
et al., 2007a). In addition, we are able to neglect the radiation generated by the supernovashocked gas as a source of photodissociating radiation concerning neighboring minihalos,
since the shock is only moving at velocities . 20 km s−1 when it arrives at the star-forming
minihalos mentioned above. This implies a photodissociating flux of JLW . 10−5 , where
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CHEMICAL ENRICHMENT
JLW is in units of 10−21 ergs s−1 cm−2 Hz−1 sr−1 (Shull & Silk, 1979). Such a low level of
flux will not have a substantial impact on the evolution of gas in collapsing minihalos, as it
implies a timescale for photodissociation of H2 of the order of 1 Gyr, much longer than the
Hubble time at the redshifts we consider (Johnson & Bromm, 2006; Oh & Haiman, 2003).
However, a minihalo within ∼ 200 pc of the SN explosion may be subject to a significant
photodissociating flux, owing to the higher shock velocity at small distances.
7.3.3
Mixing Efficiency
An important question is whether a large fraction of metals can penetrate neighboring halos and efficiently mix with their cores, thus changing the mode of star formation from
Pop III/Pop II.5 to Pop II. Although a detailed quantitative analysis would require dedicated high-resolution simulations that are unavailable here, we can nevertheless estimate
the mixing efficiency by applying the criterion for the operation of Kelvin-Helmholtz (KH)
instabilities (Cen & Riquelme, 2008; Murray et al., 1993; Wyithe & Cen, 2007):
gDrvir
.1,
2
2πṙsh
(7.9)
where g is the gravitational acceleration at the virial radius, and D the density ratio of gas in
the halo compared to the dense shell. For the nearest star-forming halo with about 5×105 M
and rvir ' 100 pc, we find D ' 10 and ṙsh ' 20 km s−1 . The left-hand side of equation (7.9)
is thus of order 1/10, suggesting that the gas at the outer edge of the halo will be disrupted
and mix with the enriched material in the dense shell, while the core of the halo will remain
pristine and stable. This is consistent with the results of the detailed simulations conducted
by Cen & Riquelme (2008), and by the same arguments we find that halos b and c remain
pristine as well.
Once again, we note that massive minihalos very close to the SN progenitor might experience a different behavior, as expansion velocities are of the order 100 km s−1 when the SN
remnant leaves the host halo (see Fig. 7.8). In such cases, the shock could disrupt the dense
cores and trigger efficient mixing, leading to Pop II star formation once the gas recollapses.
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7.4 Chemical Enrichment
7.3.4
Gravitational Fragmentation
Various authors have suggested that cooling in the dense shell might lead to gravitational
fragmentation and trigger secondary star formation (Machida et al., 2005; Mackey et al.,
2003; Salvaterra et al., 2004). However, due to the previous photoheating, the density of
the surrounding medium is sufficiently lowered such that molecule formation is initially
inefficient. Even though we do not fully resolve the radiative shock front, we thus find that
the continuous adiabatic expansion of the dense shell renders molecular cooling unimportant
and does not trigger gravitational instabilities (see Section 7.3.4 and Fig. 7.8). We note that
the potential mixing of metals into the dense shell and the associated additional cooling might
affect these results but postpone a more detailed discussion of this issue to future work.
Another scenario for secondary star formation is at the interface of colliding SN remnants. If such encounters take place in significantly overdense regions and occur relatively
early in the evolution of individual SN remnants, densities might become high enough for
gravitational fragmentation to occur.
7.4
Chemical Enrichment
We have previously argued that mixing of enriched material with gas in existing star-forming
halos is generally inefficient (see Section 7.4.3), indicating that the dispersal of metals can
only occur via expulsion into the IGM. In Section 7.3.5, we have shown that the bulk of
the shock propagates into the voids surrounding the host halo, and we expect that chemical
enrichment proceeds via the same channel (Pieri et al., 2007). This realization is crucial, as
the detailed distribution of metals not only governs the transition to secondary star formation,
but also the properties of the first galaxies, when the enriched material recollapses into larger
potential wells.
We shed some light on these issues by determining the influence of metal cooling on
the evolution of the SN remnant and by discussing the transport of metals into the IGM.
Although our resolution is too crude for a detailed analysis, we can nevertheless discuss
the mixing efficiency in a qualitative manner and draw some preliminary conclusions on the
ultimate fate of the expelled metals.
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CHEMICAL ENRICHMENT
Figure 7.11: Collapse times and distances from the SN progenitor for all star-forming minihalos
affected by the SN shock. The shades of the symbols indicate their affiliation, i.e. black, dark
gray, and light gray symbols represent the no-feedback, photoheating-only, and main simulation
runs, respectively, while the shapes of the symbols denote the individual halos. For orientation,
the dashed line shows the mass-weighted mean shock radius at late times according to Fig. 7.7. In
our case, photoheating significantly delays star formation, while the SN shock acts to compress
gas in neighboring minihalos and slightly accelerates their collapse.
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7.4 Chemical Enrichment
Figure 7.12: Cooling time vs. expansion time in the interior. The dotted line represents cooling
by metal lines for a total metallicity of Z = Z , but consisting solely of C, O, Fe, and Si (in
equal parts), while the dashed line represents IC cooling. The long-dashed line indicates when
cooling becomes efficient, implying that metal cooling becomes momentarily important after
10 Myr, when the interior temperatures have dropped to 105 K and cooling rates peak. However,
IC scattering becomes efficient at the same time and rapidly cools the enriched regions to below
104 K, where metal cooling rates drop by a few orders of magnitude. Consequently, the presence
of metals does not effect the dynamical evolution of the SN remnant.
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CHEMICAL ENRICHMENT
Figure 7.13: Three-dimensional view of the SN remnant at z = 15, or 100 Myr after the SN
explosion, when it has reached a radius of 2 kpc. The finger-like morphology of the shock
becomes visible as the SN remnant propagates at varying speeds in different directions, caused
by an anisotropic density distribution.
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7.4 Chemical Enrichment
Figure 7.14: Hydrogen number density averaged along the line of sight in a slice of 10/h kpc
(comoving) around the x-y plane, overlaid with the distribution of all metal particles (bright
orange), after 1, 10, 50, and 200 Myr. The extent of each particle is set by the SPH smoothing
length, while the apparent mixing within neighboring minihalos is a projection effect. Panel
(a): after 1 Myr, the shock has left the host halo and enters the IGM, while the metals are still
confined well within the virial radius. Panel (b): the contours of the shock have become visible as
the SN remnant plows into the IGM 10 Myr after the SN explosion. The interior bubble expands
adiabatically into the cavities created by the shock and begins to lose its spherical symmetry.
Panel (c): after 50 Myr, the dense shell becomes distinguishable from the metal-enriched interior,
which has become substantially asymmetric and expands into the voids around the y-z plane in
the shape of an hourglass. Panel (d): when the SN remnant finally stalls after 200 Myr, the dense
shell and the metal-enriched, interior bubble have reached their maximum extent, with the former
providing a natural confinement around the interior.
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CHEMICAL ENRICHMENT
7.4.1
Heat Conduction and Metal Cooling
During the first 10 Myr, the initial stellar ejecta expand adiabatically and apparently do not
mix with the surrounding material (see Fig. 7.8). In reality, however, the high electron mean
free path behind the shock leads to heat conduction and gas from the dense shell evaporates
into the hot, interior bubble (Gull, 1973). This effect is not included in the simulation, as
it would only marginally affect the dynamics of the SN remnant, but the onset of RayleighTaylor (RT) and KH instabilities is predicted to efficiently mix the metals with primordial
material evaporated from the dense shell, reducing the metallicity of the interior by a factor
of a few to at most one order of magnitude (Madau et al., 2001). In the following, we assume
that the overall metallicity has dropped by a factor of 5, to Z = Z , in the course of the first
million years.
To determine the importance of metal cooling, we proceed analogously to Section 7.3.2
and compare the expansion and cooling timescales of the interior bubble. For this purpose
we adopt the cooling rates provided in Maio et al. (2007) for gas in collisional ionization
equilibrium enriched to Z = Z but consisting solely of C, O, Fe, and Si (in equal parts).
Although this does not represent the specific yield of a PISN or a hypernova, we here only
intend to give an approximate argument and defer a more precise treatment to future work.
Assuming the time-dependent properties of the simulation, we find that metal cooling
is only briefly important after 10 Myr (see Fig. 7.12), when the interior has adiabatically
cooled to 105 K and metal cooling is most efficient. At the same time, however, IC scattering
becomes important and temperatures quickly fall to 104 K, where cooling rates drop by a
few orders of magnitude. Due to the low densities in the interior, the onset of fine structure
cooling below 104 K also proves inefficient, and we conclude that even for initial metal yields
as high as y = 0.1, metal cooling is unimportant for the entire dynamical evolution of the SN
remnant. We emphasize that the presence of metals will become crucially important once
the enriched gas has recollapsed into a sufficiently massive halo later on, thereby reaching
high densities again.
7.4.2
Instabilities and Distribution of Metals
Due to inefficient cooling, the evolution of the metal-enriched, interior bubble is governed by
adiabatic expansion and preferentially propagates into the cavities created by the shock (see
Figs. 7.8 and 7.14). Once the shock leaves the host halo and becomes highly anisotropic, the
168
7.4 Chemical Enrichment
interior adopts the same behavior and expands into the voids surrounding the y-z plane in the
shape of an hourglass, with a maximum extent similar to the final mass-weighted mean shock
radius. This behavior is evident in Fig. 7.14, where we plot the hydrogen number density and
distribution of metal particles at various times after the SN explosion. Furthermore, since the
radius of each metal particle is determined by the smoothing length, we find that at late times
the interior becomes substantially mixed with the initial stellar ejecta.
When the shock finally stalls, the interior bubble is in pressure equilibrium with its surroundings, but it stays confined within the dense shell. To investigate the importance of RT
instabilities in this configuration, we estimate the mixing length λrt for large density contrasts
between two media according to (Madau et al., 2001):
2
λrt ' 2πgtsh
,
(7.10)
where g is the gravitational acceleration of the host halo. Estimating the distance of the dense
shell from the host halo with the final mass-weighted mean shock radius, we find that mixing
between the dense shell and the interior bubble takes place on scales . 10 pc in the course
of a few times 10 Myr. In light of the substantial extent of the interior, we conclude that
such mixing is generally inefficient and that much larger potential wells must be assembled
to recollect and mix all components of the shocked gas. Specifically, turbulence arising in
the virialization of the first galaxies could be an agent for this process (Wise & Abel, 2007a).
On the other hand, mixing could be important in the y-z plane, where, due to the progression of structure formation, material falls in along filaments toward the metal-enriched
gas near the host halo. Since this mechanism will only affect a limited volume, while most
metals are expelled into the voids (see Fig. 7.14), we again conclude that a fundamental transition in star formation from Pop III/Pop II.5 to Pop II requires the assembly of a DM halo
with a sufficiently deep potential well.
To find the minimum mass necessary to recollect the hot and underdense post-shock gas
residing at T ∼ 103 K and nH ∼ 10−2.5 cm−3 (see Section 7.3.4), we estimate by means of the
cosmological Jeans criterion that a DM halo of at least Mvir & 108 M must be assembled.
Assuming that in the course of its virialization the initial stellar ejecta mix with the entire
swept-up mass, we find a final, average metallicity of Z ' 10−2.5 Z . Although this value is
well above Zcrit , we emphasize that the final topology of metal enrichment could be highly
inhomogeneous, with pockets of highly enriched material on the one hand, but regions with
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CHEMICAL ENRICHMENT
a largely primordial composition on the other hand.
7.5
Summary and Conclusions
We have investigated the explosion of a 200 M PISN in the high-redshift universe by means
of three-dimensional, cosmological simulations, taking into account all necessary chemistry
and cooling. Using a ray-tracing algorithm to determine the size and structure of the H 
region around the progenitor star, we have followed the evolution of the SN remnant until
it effectively dissolves into the IGM, and discussed its expansion and cooling properties in
great physical detail. Specifically, we have found that a chronological sequence in its evolution, based on various physical mechanisms becoming dominant, allows the introduction
of a simple analytic model summarizing its expansion properties. The SN remnant propagates for a Hubble time at z ' 20 to a final mass-weighted mean shock radius of 2.5 kpc,
roughly half the size of the H  region, and in this process sweeps up a total gas mass of
2.5 × 105 M . We have found that its morphology becomes highly anisotropic due to encounters with filaments and neighboring minihalos in the y-z plane, but underdense voids
perpendicular to the y-z plane. Based on the high explosion energy, the host halo is entirely
evacuated, while in our case shock compression of neighboring minihalos partially offsets
the delay in star formation due to negative feedback from photoionization heating. In contrast, we do not observe gravitational fragmentation triggered by efficient cooling behind the
SN shock, which could in principle lead to secondary star formation. We have found that the
metal-enriched, interior bubble expands adiabatically into the cavities created by the shock
and preferentially propagates into the voids of the IGM with a maximum extent similar to
the final mass-weighted mean shock radius. We have estimated that RT instabilities do not
efficiently mix the dense shell with the interior, but that material falling in along filaments
could mix with metal-enriched gas near the host halo. Finally, we have concluded that a DM
halo of at least Mvir & 108 M must be assembled to recollect and mix all components of the
shocked gas.
Based on the simulation, we find that a single PISN can enrich the local IGM to a substantial degree. If energetic SNe were indeed a common fate for the first stars, they might
have deposited metals on large scales before massive galaxies formed and outflows were
suppressed by their increasingly deep potential wells. Hints on ubiquitous metal enrichment have recently been found in the low column density Lyα forest (Aguirre et al., 2005;
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7.5 Summary and Conclusions
Songaila, 2001; Songaila & Cowie, 1996) and in dwarf spheroidal satellites of the Milky
Way (Helmi et al., 2006), where the presence of a ‘bedrock’ metallicity was inferred. Based
on these observations, various authors have argued that SNe occurring in the shallow potential wells of minihalos could substantially pre-enrich the universe (Daigne et al., 2006,
2004; Greif & Bromm, 2006; Matteucci & Calura, 2005; Yoshida et al., 2004). Although
the frequency of PISNe is debated (Norman et al., 2004; Ricotti & Ostriker, 2004; Scannapieco et al., 2003; Venkatesan & Truran, 2003), we have found that the overall dynamics
of the SN remnant and the distribution of metals are largely independent of the progenitor
and are governed mainly by the explosion energy. Our simulations would therefore also approximately describe a hypernova explosion, where a rotating, massive star undergoes core
collapse (Tominaga et al., 2007; Umeda & Nomoto, 2002). Such scenarios might better explain the peculiar yields found in extremely metal-poor stars in the Galactic halo (Christlieb
et al., 2002; Frebel et al., 2005), assuming they formed out of gas enriched by a previous generation of stars (Iwamoto et al., 2005; Karlsson, 2006; Tumlinson, 2006; Tumlinson et al.,
2004; Umeda & Nomoto, 2003, 2005). However, in light of the tentative identification of
SN 2006gy as a PISN (Smith et al., 2007), we find that the plausibility of the PISN scenario
is enhanced and that it provides a viable possibility for the ultimate fate of a very massive,
metal-free star.
When and where do the first Pop II stars form? Based on the progression of structure formation, we expect that the first low-mass stars will form at the centers of the first galaxies,
where primordial material streams in along filaments and mixes with the metal-enriched gas
of the host halo. On the other hand, metals expelled into the voids are not so readily available, as the associated gas resides at too low densities and high temperatures. Because of
this diversity, secondary star formation in the virialization of the first galaxies will likely be
a highly complicated and multi-faceted process. To understand the relevant mechanisms in
detail, one must perform numerical simulations that include metal cooling as well as an efficient model for their mixing. Furthermore, one must take into account the effects of multiple
star-forming regions, including the expansion of additional H  regions and SN remnants.
In future work, we hope to shed some light on these issues by performing detailed numerical simulations incorporating all the necessary physics and evolving them to sufficiently late
times. Such investigations are key in arriving at detailed predictions for the properties of the
first galaxies to be observed with JWST.
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CHEMICAL ENRICHMENT
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8
Chemical Mixing in Smoothed Particle
Hydrodynamics Simulations
Understanding chemical enrichment and the dispersal of heavy elements in the wake of energetic SNe has become essential to a number of fields in astrophysics. The details of how
enriched material mixes with ambient gas are not only relevant for the cooling and fragmentation properties of the interstellar medium (ISM), but also manifest themselves in the
composition and dynamics of the resulting stars. As massive stars return metals to the ISM,
mixing plays a key role in the overall matter cycle in galaxies. Tracing this loop back in
time, one encounters the initial enrichment of the pristine, pure H/He gas at the end of the
cosmic dark ages, when the very first stars ended their lives in violent SN explosions and
expelled a significant fraction of their mass in metals (Greif et al., 2007; Heger et al., 2003;
Iwamoto et al., 2005; Wise & Abel, 2008b). The resulting mixing in the early Universe has
two aspects. The first concerns the enrichment inside the first galaxies, when the metals
ejected by Pop III stars recollapsed into more massive halos that cooled by atomic hydrogen lines and vigorously mixed with pristine material in the presence of a highly turbulent
medium (Greif et al., 2008; Wise & Abel, 2007a). The second relates to the enrichment of
the IGM at redshifts z > 5 (Madau et al., 2001). The resulting metallicity distribution is a
crucial ingredient in modeling the reionization history of the Universe (Furlanetto & Loeb,
2005; Yoshida et al., 2004) and in determining when cosmic star formation shifted from a
predominantly high-mass, Pop III mode, to a more normal Pop II mode (Mackey et al., 2003;
Schneider et al., 2002; Tornatore et al., 2007; Venkatesan, 2006).
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8. CHEMICAL MIXING IN SMOOTHED PARTICLE HYDRODYNAMICS
SIMULATIONS
Unfortunately, an accurate treatment of mixing, while simultaneously simulating the
larger-scale environment, is presently not feasible, as the turbulent motions responsible for
mixing typically cascade down to very small scales. A frequently encountered approach to
this problem is to assume that the products of stellar nucleosynthesis are distributed within
a fixed volume (e.g. Kobayashi et al., 2007; Norman et al., 2004; Scannapieco et al., 2005a;
Tornatore et al., 2007). Significantly better results can be achieved in grid-based simulations
by relating the mass flux between cells to the mixing efficiency, even though it remains unclear how much of this mixing is numerical instead of physical (e.g. Wise & Abel, 2008b).
Such a direct approach is difficult to implement in SPH simulations, due to their Lagrangian
nature, and instead chemical mixing has been modeled as a diffusion process (e.g. Martı́nezSerrano et al., 2008). This is somewhat more accurate since the rms displacement of a fluid
element in a homogeneously and isotropically driven turbulent velocity field can be described
by the diffusion equation, with the diffusion coefficient set by the velocity dispersion and the
typical shock travel distance (Klessen & Lin, 2003).
Prescriptions with a constant diffusion coefficient have been applied to models of the
chemical evolution of the Milky Way (Karlsson, 2005; Karlsson & Gustafsson, 2005), galaxies in a cosmological context (Martı́nez-Serrano et al., 2008), and the environment of the
first galaxies (Karlsson et al., 2008). In the present paper, we introduce an improved method
that resolves chemical mixing in space and time based on the velocity dispersion within the
SPH smoothing kernel. This yields results in agreement with more detailed investigations
of mixing in the early phases of SN remnants, without having to explicitly resolve the hydrodynamic instabilities in the post-shock gas. In future work, we plan to use our method
to investigate the long-term evolution of energetic SNe in a cosmological environment, particularly at high redshift when the Universe was enriched with the first metals. However,
since our algorithm is quite generic, we hope that it will prove a valuable tool for any SPH
simulation that attempts to follow the mixing of pollutants.
The structure of our work is as follows. In Section 8.2, we describe our model for diffusion and its numerical implementation in the cosmological simulation code GADGET (version 2) (Springel, 2005), followed by a series of idealized test simulations (Section 8.3). We
then discuss the relation between the diffusion coefficient and the velocity dispersion and
apply our prescription to the evolution of an idealized SN remnant (Section 8.4). Finally,
in Section 8.5 we summarize our results and assess their implications. For consistency, all
quoted distances are physical, unless noted otherwise.
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8.1 Diffusion Algorithm
8.1
Diffusion Algorithm
Diffusion plays an important role in a variety of astrophysical contexts. Among them are
thermal conduction (e.g. Jubelgas et al., 2004), heat transfer in shear flows (e.g. Wadsley
et al., 2008), the microscopic diffusion of particles, such as photons in stellar interiors, or the
spatial correlation of individual fluid elements in a turbulent medium (Klessen & Lin, 2003).
These processes are all described by the diffusion equation, commonly written in the form:
dc 1
= ∇ · (D∇ c) ,
dt ρ
(8.1)
where c is the concentration of a contaminant fluid per unit mass, D is the diffusion coefficient, which can be a function of space and time, and d/dt the Lagrangian derivative, or
the derivative following the motion. The diffusion coefficient has dimensions M L−1 T−1 ,
suggesting that it can be represented as the product of the local density and some typical
length-scale and velocity, such as the particle mean free path and velocity dispersion in a
microscopic picture of diffusion.
8.1.1
Numerical Implementation
In the SPH formalism, the diffusion equation can be reduced to a discrete summation over
all particles within the smoothing kernel:
dci X
=
Ki j (ci − c j ) ,
dt
j
with
Ki j =
m j 4Di D j ri j · ∇i Wi j
,
ρi ρ j (Di + D j )
ri2j
(8.2)
(8.3)
where i and j denote the particle indices, m the mass, ρ the density, Wi j the kernel and ri j ,
ri j the vector and absolute separations between particles i and j, respectively (for a more
detailed derivation, see Monaghan et al., 2005). In the above equation, the arithmetic mean
of the diffusion coefficient has been replaced by the harmonic mean, which has proven to be
more robust. Furthermore, the second derivative has been replaced by a term involving the
gradient and the particle separation, since a direct computation of the second derivative is
problematic (Monaghan, 2005).
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8. CHEMICAL MIXING IN SMOOTHED PARTICLE HYDRODYNAMICS
SIMULATIONS
The solution to equation (8.2) can either be determined explicitly, which requires an
additional constraint on the time-step to ensure numerical stability, or implicitly, which requires the solution of a coupled set of differential equations involving all ‘active’ particles
(i.e. all SPH particles that are being updated on the current time-step). The explicit approach
is extremely difficult to implement in a conservative fashion in an SPH code that allows individual particles to have different time-steps (e.g. GADGET-2). This is because in a code
of this type, neighboring particles will sometimes have different time-steps. Consequently,
the increase in concentration at particle i caused by diffusion from particle j will sometimes
be computed at a different time from the corresponding decrease in concentration at particle
j caused by diffusion to particle i. To ensure conservation, the increase and decrease must
exactly balance, but in general they will not if i and j have different time-steps. One could,
of course, avoid this problem by ensuring that all particles are synchronized before their
concentrations are updated. However, the required synchronizations would have to occur
very frequently, and so one would lose essentially all of the benefits gained by allowing the
particles to have individual time-steps. A further undesirable feature of the explicit approach
is the fact that the time-steps required to stably model the diffusion can become very small.
Consideration of equation (8.1) shows that the required time-step scales with the spatial resolution – represented in an SPH code by the smoothing length h – as
∆t ∝ h2 .
(8.4)
In comparison, the standard Courant time-step scales only linearly with h.
An implicit approach to the solution of equation (8.2) avoids some of these problems, as
it allows one to take larger time-steps without compromising numerical stability. However,
this comes at a cost: the coupled differential equations representing the diffusion must be
solved iteratively, and it is difficult to do this in a fashion that can be efficiently parallelized.
In addition, one still has to deal with the synchronization problem discussed above.
In view of the disadvantages of both standard approaches, it is interesting to explore
the viability of simpler, but more approximate approaches, such as the one presented in
this paper. To obtain our approximation, we assume that the densities and concentrations
of all active particles do not change significantly over a time interval ∆t, allowing a direct
176
8.1 Diffusion Algorithm
integration of equation (8.2):
ci (t0 + ∆t) = ci (t0 )eA∆t +
B
(1 − eA∆t ) ,
A
(8.5)
with
A=
X
Ki j
(8.6)
Ki j c j .
(8.7)
j
and
B=
X
j
For large time-steps, the concentration of particle i thus tends to the average among its neighbors, while for small time-steps it remains close to its original concentration.
We implement this approach by performing the required operations at a global synchronization point in the density routine of GADGET-2. After the new densities and smoothing lengths have been computed, we update the coefficients Ki j and subsequently use equation (8.5) to determine the new concentrations of all active SPH particles. In a final step, we
renormalize the obtained concentrations with a global factor such that the total concentration
is conserved. Since the Courant condition does not allow for significant changes in density between time-steps, and diffusion generally progresses slower than the speed of sound,
our implementation is quite generic and can be applied to a number of problems in astrophysics. We have found that the algorithm is remarkably stable even for very short diffusion
timescales, since particles tend to equilibrate their concentrations and neighboring particles
are generally active at the same time. The additional CPU consumption is minimal since
we utilize the pre-existing neighbor search. By the same token, the algorithm is easy to
implement in any SPH code and is not restricted to GADGET-2.
8.1.2
Test Problems
In this section we investigate the formal accuracy of the algorithm by performing a number
of idealized test simulations. We initialize all simulations in a periodic, uniform density
box with 1 million SPH particles and length and sound-crossing time set to unity, such that
we may conveniently quote the elapsed time in units of the sound-crossing time. We adopt
a hydrogen number density of nH = 1 cm−3 and a mean molecular weight corresponding
to that of a neutral, primordial gas. We place the particles on a grid with a very small
177
8. CHEMICAL MIXING IN SMOOTHED PARTICLE HYDRODYNAMICS
SIMULATIONS
random displacement, and in the first test problem also consider a fully random distribution
of particles, such that the density fluctuates considerably around the mean. This gives a better
handle on the performance of our algorithm under more realistic circumstances. In all cases,
we use the maximum diffusivity that is accurately modeled by our algorithm, determined by
the Courant condition (see Sections 8.2.1 and 8.3.1):
D = ρ h cs ,
(8.8)
where cs is the sound speed and ρ and h are determined by the mean density nH = 1 cm−3 ,
such that the diffusion coefficient becomes a fixed numerical value.
In the first test problem, we consider the propagation of an initial δ-distribution, to which
the analytic solution is Green’s function of the diffusion equation:
!
1
− |x − x0 |2
G x, x , t =
,
exp
(2πσ2 )3/2
2σ2
0
with variance
σ=
√
2Dt .
(8.9)
(8.10)
This configuration is reproduced by setting the concentration of the central particle to unity,
and of all others to zero. In the left-hand column of Fig. 8.1, we compare the analytic solution to the simulation results at three different output times. Early on, diffusion progresses
somewhat too rapidly, since the concentrations of neighboring particles differ substantially.
In this case, equation (8.5) effectively breaks down, but the resulting deviations remain small
and do not influence the long-term behavior, where all three curves become nearly indistinguishable. Note that the random distribution performs almost as well as the grid-based
distribution, showing that the diffusivity remains unchanged even for high density fluctuations.
The second test problem consists of two individual δ-distributions initially separated by
∆x = 1/3. The analytic solution can be obtained by a convolution of Green’s function with
the initial state of the system:
c (x, t0 + ∆t) =
Z
G x, x0 , t0 + ∆t c x0 , t0 dx0 ,
(8.11)
such that the general solution is given by the superposition of two Gaussian distributions
178
8.2 Application to Chemical Mixing
centred at x = 1/3 and 2/3. Similarly, the solution to the third test problem, consisting of a
slab of uniform concentration between x = 1/3 and 2/3, is given by
1
1/3 − x
c (x, t) = erfc √
2
4Dt
!
1
x − 2/3
c (x, t) = erfc √
2
4Dt
!
(8.12)
for x ≤ 1/2 and
(8.13)
for x > 1/2. In the middle and right-hand columns of Fig. 8.1, we compare the simulation
results to the analytic solution, showing that there are only minor deviations at early times,
similar in nature to those found in the first test problem. A slice through all three boxes
(Fig. 8.2) shows the solution in two dimensions.
Finally, in a series of resolution studies performed with 323 , 643 and 1283 particles, we
have found no correlation between the resolution and the deviation from the analytic solution.
In fact, in all cases, the deviations were comparable to those found in previous test problems.
Considering its formal simplicity, the algorithm thus performs remarkably well. Although
the diffusivity is initially slightly over-predicted for cases in which the diffusion time is
shorter than the sound-crossing time, we nevertheless find a correct long-term behavior.
8.2
Application to Chemical Mixing
In the previous section, we introduced an SPH formalism for diffusion that can be applied
to most problems that are governed by a diffusion equation. In this section, we focus on an
application that is particularly relevant to astrophysics: the mixing of chemical elements.
8.2.1
Chemical Mixing as Turbulent Diffusion
As a first step towards a model for chemical mixing, one must find a connection between
the diffusivity of a pollutant and the local physical conditions. Klessen & Lin (2003) have
provided this link by showing that the probability of finding a parcel of gas at a given location
in a homogeneously and isotropically driven turbulent velocity field can be described by the
diffusion equation, with the diffusion coefficient set by the velocity dispersion ṽ and the
179
8. CHEMICAL MIXING IN SMOOTHED PARTICLE HYDRODYNAMICS
SIMULATIONS
Figure 8.1: The propagation of a single δ-distribution (left-hand column), two δ-distributions
(middle column) and a slab of uniform concentration (right-hand column) into an otherwise pristine medium, shown for a grid-based particle distribution (solid lines) and a random particle
distribution (dashed line), compared to the analytic solution (dotted lines). The temporal evolution is depicted from top to bottom and quoted in units of the sound-crossing time. In all
three cases, the simulations reproduce the analytic solution, except at early times when the underlying assumption of constant concentration for neighboring particles is not well justified (see
Section 8.2). However, as is evident from the bottom two rows, this does not affect the long-term
behavior.
180
8.2 Application to Chemical Mixing
Figure 8.2: The propagation of a single δ-distribution (left-hand column), two δ-distributions
(middle column) and a slab of uniform concentration (right-hand column) into an otherwise
pristine medium, shown at a representative output time.
turbulent driving length l:˜
D = 2 ρ ṽ l˜ .
(8.14)
This corresponds to the classical mixing-length approach extended into the supersonic regime.
If one replaces the probability distribution with the concentration of a pollutant, this formalism can be reinterpreted to describe chemical mixing. The sole remaining task is then to
provide the parameters ṽ and l˜ to the diffusion algorithm, such that the diffusion coefficient
can become a function of space and time.
To obtain the necessary parameters, we determine the rms velocity dispersion for particle
i within its smoothing length:
ṽ2i =
2
1 X vi − v j ,
Nngb j
(8.15)
where Nngb is the number of neighbors and vi and v j are the bulk velocities of particles i and
j. Finally, we equate the turbulent driving length with the smoothing length, since this is the
minimum scale where turbulent motions can be resolved. This then yields for the diffusion
coefficient:
Di = 2 ρi ṽi hi .
(8.16)
181
8. CHEMICAL MIXING IN SMOOTHED PARTICLE HYDRODYNAMICS
SIMULATIONS
Figure 8.3: The radial velocity (left-hand column), diffusion coefficient (middle column) and
metallicity (right-hand column) as a function of distance from the SN progenitor, with the position of the forward shock denoted by the dotted line. The temporal evolution is depicted from
top to bottom and quoted in units of the sound-crossing time. The high velocity dispersion near
the main shock leads to an elevated diffusion coefficient and thus very efficient chemical mixing. Due to the dependence on velocity dispersion, mixing by hydrodynamic instabilities in the
remnant is implicitly accounted for.
182
8.2 Application to Chemical Mixing
As desired, the efficiency of chemical mixing is now governed entirely by local physical
quantities.
An important underlying assumption of this method is that the velocity field on subresolution scales corresponds to a homogeneously and isotropically driven turbulent medium,
i.e. only the magnitude of the velocity dispersion and not its three-dimensional structure is
taken into account. A further implicit assumption is that the mixing efficiency on subresolution scales is set by the corresponding value on the scale of the smoothing kernel. For
these reasons our method yields an upper limit to the amount of mixing that can occur, and
does not capture the details of a real turbulent cascade. However, since turbulent motions on
scales larger than the smoothing length are explicitly resolved, our approach should suffice
for most practical purposes.
We implement the above steps in our algorithm by performing a previous neighbor search
that finds the velocity dispersion for all active SPH particles, which is then used in equation (8.3) to obtain the coefficients Ki j . This closes the required set of equations, and in the
following subsection we use our complete model to investigate the mixing of metals in a SN
remnant.
8.2.2
Mixing in Supernova Remnants
The mixing of gas in the post-shock regions of SN remnants has previously been investigated, leading to the consensus view that secondary shocks trigger Rayleigh-Taylor and
Kelvin-Helmholtz instabilities that mix pristine gas from the dense shell with already enriched material (e.g. Chevalier et al., 1992; Gull, 1973). The aim of our algorithm is to
capture this mixing without having to explicity resolve the corresponding hydrodynamic instabilities, which cannot easily be modeled using SPH (e.g. Agertz et al., 2007; Price, 2008).
To verify its ability to do this, we perform a test simulation of an idealized SN explosion
with the same setup as in Section 8.2.2. We distribute an explosion energy of 1051 erg as
thermal energy to the Nngb nearest neighbors around the centre of the box, which results in
the formation of a shock and reproduces the ideal Sedov-Taylor solution of a blast wave in a
uniform medium (see Greif et al., 2007). Furthermore, we set the initial metallicities of the
central particles to unity.
In Fig. 8.3, we show the radial velocity, diffusion coefficient and metallicity as a function
of distance from the SN progenitor at three different output times. The high velocity disper-
183
8. CHEMICAL MIXING IN SMOOTHED PARTICLE HYDRODYNAMICS
SIMULATIONS
sion near the forward shock is responsible for the elevated diffusion coefficient, which leads
to efficient chemical mixing of pristine gas passing through the shock. As is evident from
Fig. 8.3, most parcels of gas behind the shock have nearly equilibrated their metallicities.
Our algorithm thus implicitly accounts for mixing due to the dependence of the diffusion
coefficient on the local velocity dispersion. Intriguingly, we also find metals ahead of the
forward shock. Although this initially seems unphysical, it should be remembered that our
algorithm aims to capture mixing caused by unresolved instabilities, and that in a real SN
remnant, it is these instabilities that would transport metals ahead of the mean position of
the shock. Even further out, the degree of enrichment drops roughly exponentially, since the
velocity dispersion, and hence the diffusion coefficient, tend to zero.
A second mechanism becomes important once the SN remnant has stalled: gas from
the dense shell ahead of the shock falls back on to the remnant, becomes Rayleigh-Taylor
unstable, and vigorously mixes with the interior gas. This is especially effective in a realistic
cosmological setting, where the central potential well deepens as the SN remnant expands
(Greif et al., 2007; Wise & Abel, 2008b). However, we postpone a detailed treatment of this
issue to dedicated high-resolution simulations in future work.
8.3
Summary and Conclusions
We have introduced a simple and efficient algorithm for diffusion in SPH simulations and
investigated its accuracy with a number of idealized test cases. Although the algorithm is
quite generic and can be applied to most problems involving diffusion, we have here focused
on a model for the dispersal of enriched material in supernova explosions. Adopting the
mixing length approach discussed by Klessen & Lin (2003), we link the diffusivity of the
pollutant to the local physical conditions and can thus describe the space- and time-dependent
mixing process. We have applied our prescription to an idealized SN explosion and found
that we can properly describe the mixing process without having to explicitly resolve the
hydrodynamic instabilities in the post-shock gas. Instead, this process is implicitly governed
by the dependence on the local velocity dispersion. Our method can thus be used in any SPH
simulation that attempts to follow the mixing of a pollutant but lacks the necessary resolution
on small scales. This will be relevant, in particular, for simulations that aim at simultaneously
representing the large-scale environment of the SN explosion and the fine-grained mixing.
A crucial assumption of our model is that resolved motions cascade down to subresolution
184
8.3 Summary and Conclusions
scales which then homogeneously mix the gas. For this reason our method yields an upper
limit to the mixing efficiency and the resulting degree of homogeneity, and cannot capture
the details of a real turbulent cascade. However, since turbulent motions on scales larger than
the smoothing length are explicitly resolved, this approximation should generally be valid.
Our new algorithm ties in well with the current generation of radiation-hydrodynamical
simulations of star formation, both at high redshifts and in the local Universe. The ultimate
goal is to describe the interrelated cycle of gas fragmentation and stellar feedback. To achieve
adequate realism, radiative effects have to be considered together with the mechanical and
chemical feedback due to SN explosions. The methodology developed here allows us to
include chemical feedback, suitable even for extremely large simulations. We will report on
specific applications in future studies.
185
8. CHEMICAL MIXING IN SMOOTHED PARTICLE HYDRODYNAMICS
SIMULATIONS
186
9
Outlook
Understanding the formation of the first galaxies marks the frontier of high-redshift structure
formation. It is crucial to predict their properties in order to develop the optimal search and
survey strategies for upcoming instruments such as the SKA or JWST. Whereas ab-initio
simulations of the very first stars can be carried out from first principles, and with virtually
no free parameters, one faces a much more daunting challenge with the first galaxies. Now,
the previous history of star formation has to be considered, leading to enhanced complexity
in the assembly of the first galaxies. One by one, all the complex astrophysical processes
that play a role in more recent galaxy formation appear back on the scene. Among them are
external radiation fields, comprising UV and X-ray photons, as well as local radiative feedback that may alter the star formation process on small scales. Perhaps the most important
issue, though, is metal enrichment in the wake of the first SN explosions, which fundamentally alters the cooling and fragmentation properties of the gas. Together with the onset of
turbulence, chemical mixing might be highly efficient and could lead to the formation of the
first low-mass stars and stellar clusters.
In future work, we plan to address a number of outstanding issues. Among them are
the chemical enrichment of the first galaxies and the influence of metals on neighboring star
formation sites. Another crucial question is whether the transition from Pop III to Pop II
stars is governed by atomic fine-structure or dust cooling. Theoretical work has indicated
that molecular hydrogen dominates over metal-line cooling at low densities (Jappsen et al.,
2009a,b), and that fragment masses below ∼ 1 M can only be attained via dust cooling
at high densities (Clark et al., 2008; Omukai et al., 2005). On the other hand, observational
studies seem to be in favor of the fine-structure based model (Frebel et al., 2007), even though
187
9. OUTLOOK
existing samples of extremely metal-poor stars in the Milky Way are statistically questionable (Beers & Christlieb, 2005; Christlieb et al., 2002; Frebel et al., 2005). Moreover, these
studies assume that their abundances are related to primordial star formation – a connection
that is still debated (Lucatello et al., 2005). In light of these uncertainties, it is essential
to push numerical simulation to ever lower redshifts and include additional physics in the
form of radiative feedback, metal dispersal, chemisty and cooling, and the effects of magnetic fields. We are confident that a great deal of interesting discoveries, both theoretical and
observational, await us in the rapidly growing field of early galaxy formation.
188
Acknowledgements
First of all, I would like to thank my advisors Ralf Klessen and Volker Bromm for their
dedicated support throughout these last three years. Without their help none of my work
would have been possible. I am also indepted to my co-advisor Matthias Bartelmann for assisting in various administrative issues, taking part in my doctoral committee, and providing
essential help in cosmology. I am grateful to my collaborators Jarrett Johnson, Simon Glover
and Paul Clark for their input and work concerning numerous joint projects, many of which
are part of this thesis.
Outside of physics, I would like to thank my flatmates Roy, Susanne, Angelika, Jessica
and Bianca for a wonderful stay in Heidelberg. I had lots of fun and will certainly miss the
time we spent together! Finally, I am grateful to my family for supporting me throughout
this time and providing emotional backup whenever I needed it.
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