thesisDSchleicher

thesisDSchleicher
D
  
C F   N S  M
  R-C-U  H, G
   
D  N S
P  
D. P. D R. G. S
 : B (G)
O : J 9th, 2009
T  :
P   ,   
     
R:
P. D. R S. K
P. D. M B
Zusammenfassung
Das Ziel dieser Arbeit ist, das frühe Universum zwischen Rekombination und Reionisation
besser zu verstehen und neue Möglichkeiten aufzuzeigen, es zu erforschen. Dies betrifft die Sternpopulation, die Physik im frühen Universum und die Entstehung der ersten supermassereichen Schwarzen Löcher. Mit Hilfe der von WMAP 5 gemessene optische Tiefe der Reionisation wurden obere Schranken für die Stärke von primordialen
Magnetfeldern sowie die Annihilation und den Zerfall dunkler Materie und Einschränkungen für Modelle der Sternpopulation abgeleitet. Durch den Gammastrahlungs- und Neutrinohintergrund lassen sich weitere Modelleinschränkungen ableiten, die besonders für
leichte dunkle Materie signifikant sind. Ferner wurde gezeigt, dass zukünftige 21 cm
Beobachtungen wesentlich stärkere Einschränkungen an primordiale Magnetfeldstärken
liefern. Um den Ursprung der ersten supermassereichen Schwarzen Löcher und ihrer
hohen Metallizität besser zu verstehen, wurde untersucht, wie sich diese mit ALMA
und JWST zwischen Rotverschiebung 5 und 15 beobachten lassen. Hierfür wurden verschiedene Beobachtungsgrößen abgeschätzt und klassifiziert, und die Anzahl der verfügbaren Quellen abgeschätzt. Trotz großer Unsicherheiten ist zu erwarten, dass mindestens
drei Quellen in einem Raumwinkel entsprechend dem Hubble-Deep-Field gefunden werden können.
Abstract
The goal of this work is to better understand the universe between recombination and
reionization and to outline new possibilities to explore it in more detail. This concerns
the stellar population, the physics of the early universe, and the formation of the first
supermassive black holes. With the reionization optical depth from WMAP 5, I derive
upper limits for the strength of primordial magnetic fields and dark matter annihilation
/ decay, as well as constraints for stellar population models. Further constraints can be
found from the gamma-ray and neutrino background, which rule out s-wave annihilation
of light dark matter. It was shown that future 21 cm observations will constrain primordial magnetic fields even further. To improve our understanding of the origin of the first
supermassive black holes and their high metallicity, I explore how they can be observed
with ALMA and JWST between redshift 5 and 15. For this purpose, I estimated and
classified the available observables, and I provide several independent estimates for the
expected number of high-redshift black holes. In spite of large model uncertainties, one
can expect to find at least a few sources in a solid angle similar to the Hubble-Deep-Field.
Für meine Eltern
Contents
List of Figures
vii
List of Tables
xv
1
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1
1
2
5
8
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16
17
Reionization - A probe for the stellar population and the physics of the early universe
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Reionization in the early universe . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The RECFAST code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The generalized filtering mass . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Stellar feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Models for the stellar population . . . . . . . . . . . . . . . . . . . . . . . .
2.3 The effect of magnetic fields on the IGM . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Ambipolar diffusion heating . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Decaying MHD turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 The evolution of the IGM . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Implications from the dark sector . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Dark Matter annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Dark Matter decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Dark stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Constraining the parameter space with WMAP 5 . . . . . . . . . . . . . . . . . . .
19
19
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25
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29
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32
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35
2
Introduction
1.1 The timeline of cosmology . . . . . . . . . . . . . . . . . . . . .
1.1.1 From the Big Bang to cosmic recombination . . . . . . .
1.1.2 From the postrecombination epoch to cosmic reionization
1.2 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Primordial magnetic fields . . . . . . . . . . . . . . . . .
1.2.2 Dark matter decay / annihilation scenarios . . . . . . . . .
1.2.3 Implications for the first stars . . . . . . . . . . . . . . .
1.3 Probing the early universe . . . . . . . . . . . . . . . . . . . . .
iii
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CONTENTS
2.6
3
Stellar reionization with and without primordial magnetic fields . . . . . . .
35
2.5.2
Dark reionization scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Influence of primordial magnetic fields on 21 cm emission
43
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2
The evolution of the IGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.2.1
The RECFAST code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.2.2
Heating due to primordial magnetic fields . . . . . . . . . . . . . . . . . . .
48
3.2.3
The filtering mass scale and stellar feedback . . . . . . . . . . . . . . . . . .
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The 21 cm background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.3.1
The spin temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.3.2
The build-up of a Lyman-α background . . . . . . . . . . . . . . . . . . . .
54
3.3.3
21 cm fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.4
The evolution of linear perturbations in the dark ages . . . . . . . . . . . . . . . . .
57
3.5
The 21 cm power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.6
Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.3
4
2.5.1
The chemistry of the early universe and its signatures in the CMB
63
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
4.2
Imprints from primordial molecules on the CMB . . . . . . . . . . . . . . . . . . .
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4.3
Recombination and the formation of molecules in the early universe . . . . . . . . .
67
4.3.1
Hydrogen recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3.2
H2 chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.3.3
Deuterium chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
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HeH chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.4
The chemical network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
4.5
The numerical algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.6
Results from the molecular network . . . . . . . . . . . . . . . . . . . . . . . . . .
74
4.7
Effects of different species on the cosmic microwave background . . . . . . . . . . .
79
4.7.1
Molecular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.7.2
The negative hydrogen ion. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.7.3
The negative helium ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3.4
4.8
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4.7.4
Photodissociation of HeH . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.7.5
Observational relevance and results . . . . . . . . . . . . . . . . . . . . . .
83
Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
iv
CONTENTS
5
Dark stars: Implications and constraints from cosmic reionization and extragalactic
background radiation
91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.1 Main-sequence dominated models . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.2 Capture-dominated models . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Reionization constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.2 Reionization with MS-dominated dark stars . . . . . . . . . . . . . . . . . . 97
5.3.3 Reionization with CD dark stars . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Predictions for 21 cm observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.5 Cosmic constraints on massive dark matter candidates . . . . . . . . . . . . . . . . . 103
5.5.1 Gamma-ray constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5.2 Neutrino constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.5.3 Emission from dark star remnants . . . . . . . . . . . . . . . . . . . . . . . 107
5.5.4 Dependence of dark star models on the neutralino mass . . . . . . . . . . . . 109
5.6 Cosmic constraints on light dark matter . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6.1 511 keV emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6.2 Internal Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.6.3 Emission from dark star remnants . . . . . . . . . . . . . . . . . . . . . . . 115
5.7 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6
Cosmic constraints rule out s-wave annihilation of light dark matter
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Implications and discussion . . . . . . . . . . . . . . . . . . . . .
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Detecting the first quasars with ALMA and JWST
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 The high-redshift black hole population . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Constraining the relative number of active black holes . . . . . . . . . . . . . . . . .
7.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 NGC 1068: A case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Observed properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Predicted properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Uncertainties in the model assumptions . . . . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
7.6
7.7
7.8
8
7.5.1 Metallicity . . . . . . . . . . . . . . . . . . . . .
7.5.2 Variations of abundance ratios . . . . . . . . . . .
7.5.3 Internal structure . . . . . . . . . . . . . . . . . .
High-redshift observations with ALMA and JWST . . . .
7.6.1 Observables for ALMA . . . . . . . . . . . . . .
7.6.2 Observables for JWST . . . . . . . . . . . . . . .
Expected source counts and optimum observing strategies
7.7.1 The expected number of high-redshift sources . . .
7.7.2 Survey of high-J CO lines with ALMA . . . . . .
7.7.3 Continuum survey with JWST . . . . . . . . . . .
7.7.4 Uncertainties and generalizations of the model . .
7.7.5 What we will learn about the brightest quasars . .
Discussion and outlook . . . . . . . . . . . . . . . . . . .
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Discussion and outlook
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Reionization constraints . . . . . . . . . . . . . . . .
8.1.2 21 cm observations . . . . . . . . . . . . . . . . . . .
8.1.3 Constraints from the cosmic gamma-ray background
neutrino background . . . . . . . . . . . . . . . . . .
8.1.4 Signatures of primordial molecules in the CMB . . . .
8.1.5 First quasar observations with ALMA and JWST . . .
8.2 Open questions . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Physics of the early universe . . . . . . . . . . . . . .
8.2.2 Reionization and the stellar population . . . . . . . . .
8.2.3 High-redshift quasars . . . . . . . . . . . . . . . . . .
8.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 An implementation of chemistry in the RAMSES code
8.3.2 Magnetic fields . . . . . . . . . . . . . . . . . . . . .
8.3.3 Dark matter annihilation . . . . . . . . . . . . . . . .
8.3.4 The first quasars . . . . . . . . . . . . . . . . . . . .
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and the atmospheric
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A Free-free transitions involving H−
171
B Reaction rates for primordial chemistry
175
Bibliography
181
vi
List of Figures
1.1
1.2
1.3
1.4
1.5
2.1
2.2
2.3
The timeline of cosmology. See Chapter 1.1 for a detailed description. Copyright
1996 University of Cambridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
The formation of the first star, as calculated by Abel et al. [2002]. Credit: Tom Abel
(Stanford) & Ralf Kaehler (ZIB). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
An artist’s conception of a supermassive black hole and the interaction with its accretion disk. Credit: NASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
The observed 511 keV emission from the Galactic center: Evidence for dark matter
annihilation? Image: J. Knödlseder/CESR, Toulouse. . . . . . . . . . . . . . . . . .
14
An artist’s conception of the completed Atacama Large Millimeter/submillimeter
Array (ALMA). One single antenna has a size of 12 m. This telescope may observe
molecular and fine structure emission from the first high-redshift quasars. CREDIT:
ALMA/ESO/NRAO/NAOJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Evolution of different quantities as a function of redshift. a) The effective gas temperature. b) The effective ionized fraction. c) The filtering mass. d) The comoving
magnetic field strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
The reionization optical depth for Pop. III stars (model A) in the presence of primordial magnetic fields. The contour lines are equally spaced around τre = 0.87 with
∆τ = 0.017, corresponding to the different σ-errors of the measurement. Magnetic
fields higher than 20 nG are clearly excluded by the optical depth alone. The transition between stellar reionization and collisional ionization occurs at about 5 nG,
providing a more stringent limit if metal enrichment is required. . . . . . . . . . . .
35
The reionization optical depth for Pop. III stars (model B) in the presence of primordial magnetic fields. Magnetic fields higher than 20 nG are clearly excluded by the
optical depth alone. The transition between stellar reionization and collisional ionization occurs at about 5 nG. We can further exclude magnetic fields between 2 and 5
nG within 3-σ, as too high star formation efficiencies would be required to obtained
the measured optical depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
vii
LIST OF FIGURES
2.4
2.5
2.6
2.7
3.1
3.2
3.3
3.4
The reionization optical depth for a mixed population (model C) in the presence of
primordial magnetic fields. Magnetic fields higher than 20 nG are clearly excluded
by the optical depth alone. The transition between stellar reionization and collisional
ionization occurs at about 5 nG. In addition, magnetic fields between 0.7 and 5 nG
can be excluded, as they would require unreasonably high star formation efficiencies
for the measured optical depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The reionization optical depth for Pop. II stars (model D) in the presence of primordial magnetic fields. For magnetic fields lower than 5 nG, stellar reionization
dominates, but unreasonably high star formation efficiencies would be required to
reconcile the measured optical depth. For stronger magnetic fields, collisional reionization dominates, which can be excluded if metal-enrichment is imposed. Magnetic
fields larger than 20 nG can be excluded from the optical depth alone. . . . . . . . .
The reionization optical depth for Pop. III stars (model B) in the presence of dark
matter annihilation. At "σv#/mDM ∼ 10−33 cm3 /s/eV, secondary ionization of the
annihilation products starts to dominate over stellar reionization. Values higher than
3 × 10−33 cm3 /s/eV are ruled out by the WMAP 5-year data. . . . . . . . . . . . .
The reionization optical depth for Pop. III stars (model B) in the presence of dark
matter decay. For lifetimes lower than 3 × 1024 s, secondary ionization of the annihilation products starts to dominate over stellar reionization. Lifetimes lower than
3 × 1023 s are ruled out by the WMAP 5-year data. . . . . . . . . . . . . . . . . . .
Evolution of gas temperature (upper panel) and ionized fraction (lower panel) in the
medium that was not yet affected by UV feedback. The details of the models are
given in Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Evolution of the filtering mass scale for different comoving magnetic field strengths.
CMB, gas and spin temperature in the gas unaffected by reionization, for a star formation efficiency f∗ = 10−3 , in the zero field case (upper panel) and for a field
strength of 0.8 nG (lower panel). In the absence of magnetic fields, gas and CMB
are strongly coupled due to efficient Compton scattering of CMB photons. At about
redshift 200, the gas temperature decouples and evolves adiabatically, until it is reheated due to X-ray feedback. The spin temperature follows the gas temperature until
redshift ∼ 100. Collisional coupling then becomes less effective, but is replaced by
the Wouthuysen-Field coupling at z ∼ 25. In the presence of magnetic fields, additional heat goes into the gas, it thus decouples earlier and its temperature rises above
the CMB. Wouthuysen-Field coupling becomes effective only at z ∼ 15, as the high
magnetic Jeans mass delays the formation of luminous sources significantly. . . . . .
The mean evolution in the overall neutral gas, calculated for the models described
in Table 3.1. Top: The spin temperature. Middle: The mean brightness temperature fluctuation. Bottom: The frequency gradient of the mean brightness temperature
fluctuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
36
37
38
39
46
47
52
53
LIST OF FIGURES
3.5
The evolution of the 21 cm large-scale power spectrum and the growth of large-scale
fluctuations in temperature and ionization for the models given in Table 3.1. Top:
The evolution of the 21 cm power spectrum, normalized with respect to the baryonic
power spectrum at redshift 0. Middle: The evolution of the ratio gT = δT /δ. Bottom:
The evolution of the ratio gi = δxi /δ. . . . . . . . . . . . . . . . . . . . . . . . . . .
The idealized three-level hydrogen atom (ground state 1s, excited states 2s and 2p,
continuum), and the relevant transitions. . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Results for the evolution of H+2 and H− . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Results for the evolution of D+ and HeH+ . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Results for the evolution of HD+ and D− . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Results for those species which freeze-out and are not in chemical equilibrium. . . .
4.6 HeH+ photodissociation cross section obtained from detailed balance. . . . . . . . .
4.7 The HeH+ optical depth, both corrected and uncorrected for stimulated emission. The
contributions from pure rotational and ro-vibrational transitions are given separately.
4.8 The HD+ optical depth, both corrected and uncorrected for stimulated emission. The
contributions from pure rotational and ro-vibrational transitions are given separately.
4.9 The absorption optical depth due to different processes: the free-free processes of H−
and He− , the bound-free process of H− , and the photodissociation of HeH+ . Clearly,
the total optical depth is dominated by the processes involving H− . . . . . . . . . . .
4.10 The relative change in the CMB temperature due to the presence of H− . We further
plot the optical depths due to absorption, spontaneous and stimulated emission for the
bound-free process, as their overlap explains some features in the temperature change.
60
4.1
5.1
5.2
5.3
68
74
75
75
76
83
84
84
85
86
The evolution of the effective ionized fraction xeff , for reionization models with mainsequence dominated dark stars (see Table 5.1). Models MS 1 and MS 2 can be ruled
out by reionization constraints, while models MS 3 and MS 4 require a sudden increase in the star formation rate by a factor of 30 at redshift 6.5. It appears more
realistic to assume lower masses and star formation efficiencies to reconcile dark star
models with observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
The evolution of the effective ionized fraction xeff , for reionization models with capturedominated dark stars (see Table 5.2). Models CD 1a, CD 1b, CD 2a, CD 2b and CD
3 are ruled out due to reionization constraints, while the remaining models require an
artificial star burst. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
21 cm signatures of double-reionization scenarios (here MS 4 from Table5.1). Given
is the evolution after the first reionization phase, when the H gas is heated from the
previous ionization. Top: HI gas temperature, here identical to the spin temperature.
Middle: Expected mean 21 cm brightness fluctuation. Bottom: Frequency gradient
of the mean 21 cm brightness fluctuation. . . . . . . . . . . . . . . . . . . . . . . . 101
ix
LIST OF FIGURES
5.4
The predicted gamma-ray background due to direct annihilation into gamma-rays in
the presence of adiabatic contraction during the formation of dark stars, and the background measured by EGRET (squares) [Strong et al., 2004]. One finds two peaks in
the annihilation background for a given particle mass: One corresponding to annihilation at redshift zero, and one corresponding to the redshift where the enhancement
from adiabatic contraction was strongest. . . . . . . . . . . . . . . . . . . . . . . . 104
5.5
The predicted neutrino background due to direct annihilation into neutrinos in the
presence of adiabatic contraction during the formation of dark stars, and the atmospheric neutrino background [Honda et al., 2004]. One finds two peaks in the annihilation background for a given particle mass: One corresponding to annihilation
at redshift zero, and one corresponding to the redshift where the enhancement from
adiabatic contraction was strongest. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.6
The maximum gamma-ray background due to direct annihilation into gamma-rays
in the remnants of dark stars, and the background measured by EGRET [Strong
et al., 2004]. The actual contribution to the gamma-ray background is highly modeldependent (see discussion in the text). . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.7
The predicted X-ray background due to 511 keV emission for different dark matter
particle masses. Solid lines: Enhanced signal due from adiabatic contraction, dotted
lines: Conventional NFW profiles. The observed X-ray background from the HEAO
experiments (squares) [Gruber et al., 1999] and Swift/BATSE (triangles) [Ajello et al.,
2008] is shown as well. The comparison yields a lower limit of 10 MeV on the dark
matter mass for the adiabatically contracted profiles, and 7 MeV for standard NFW
halo profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.8
The predicted gamma-ray background due to bremsstrahlung emission for different
dark matter particle masses. Solid lines: Enhanced signal due from adiabatic contraction, dotted lines: Conventional NFW profiles. The lines overlap almost identically, as the main contribution comes from redshift zero, where the clumping factor is
large and dark stars are assumed not to form. The observed gamma-ray background
from the HEAO experiments (squares) [Gruber et al., 1999], Swift/BATSE (triangles) [Ajello et al., 2008], COMPTEL (crosses) [Kappadath et al., 1996] and SMM
(plusses) [Watanabe et al., 1999] is shown as well. . . . . . . . . . . . . . . . . . . . 113
5.9
The upper limit of X-ray radiation due to dark star remnants. The observed Xray background from the HEAO experiments (squares) [Gruber et al., 1999] and
Swift/BATSE (triangles) [Ajello et al., 2008] is shown as well. Only for very low
dark matter particle masses, the upper limit is somewhat higher than the observed
background. However, the actual contribution may be lower by some orders of magnitude (see discussion in the text). . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
x
LIST OF FIGURES
6.1
The predicted gamma-ray background due to internal bremsstrahlung emission for
different dark matter particle masses. In every case, we adopted a clumping factor that
yields the maximum allowed background. We compare with the observed gammaray background from COMPTEL (crosses) [Kappadath et al., 1996], SMM (plusses)
[Watanabe et al., 1999] and EGRET (squares) [Strong et al., 2004]. . . . . . . . . . . 123
6.2
The constraint on the clumping factor C0 at redshift zero due to internal bremsstrahlung
emission. For comparison, we show the weaker constraint due to 511 keV emission.
The forbidden region is shaded. We also show the upper mass limit from Galactic
center observations calculated by Beacom & Yüksel [2006] as well as the minimal
clumping factor C0 = 105 . The combination of these constraints shows that light dark
matter models with s-wave annihilation are ruled out. . . . . . . . . . . . . . . . . . 124
7.1
The co-moving number density of halos that contain black holes as a function of black
hole mass for different redshifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2
The expected contribution to the SXRB in keV cm−2 s−1 sr−1 keV−1 as a function of
the active quasar fraction fa and the spectral parameter Φ, under different assumptions
for the typical black hole luminosity. For a spectrum with the same amount of flux
in the MCD and the PL component, we expect Φ ! 0.5. Upper left: Eddington
luminosity, black holes in the mass range from 1 to 109 M' . Upper right: Eddington
luminosity, black hole masses in the range from 105 to 109 M' . Lower left: Eddington
ratio λE = 0.1, black hole masses in the range 1 − 109 M' . Lower right: Eddington
ratio λE = 0.3, black hole masses in the range 105 − 109 M' . . . . . . . . . . . . . . 133
7.3
An illustration for the inner structure of NGC 1068. It shows the central black hole
with the inner accretion disk radiating in X-rays, as well as the inner torus embedded
in a large molecular disk consisting of individual high-density clouds. The jet is
sketched as well. The torus is expexted to have a size of ∼ 3 pc, while the molecular
disk extends out to ∼ 100 pc. In the XDR model considered here, we focus on
those clouds that are unshielded by the torus and where emission is stimulated by the
impinging X-ray flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.4
A model for the X-ray chemistry in NGC 1068. The adopted flux impinging on the
cloud is 170 erg s−1 cm−2 . The adopted density is 105 cm−3 . Top: The abundances
of different species as a function of column density. Middle: The low-J CO lines as
a function of column density. Bottom: The high-J CO lines as a function of column
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.5
The X-ray chemistry in a system with X-ray flux of 1 erg s−1 cm−2 impinging on the
cloud. The adopted density is 105 cm−3 . Top: The abundances of different species as
a function of column density. Middle: The low-J CO lines as a function of column
density. Bottom: The high-J CO lines as a function of column density. . . . . . . . 139
xi
LIST OF FIGURES
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
The X-ray chemistry in a system with X-ray flux of 10 erg s−1 cm−2 impinging on the
cloud. The adopted density is 105 cm−3 . Top: The abundances of different species as
a function of column density. Middle: The low-J CO lines as a function of column
density. Bottom: The high-J CO lines as a function of column density. . . . . . . .
The observed spectrum of a starburst with 10 M' yr−1 with an age of 5 Myr located
at different redshifts. This is compared to the NIRCam sensitivities for 3σ detection
after an integration time of one day. For z ! 10, such starbursts can be detected within
a few hours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The observed spectrum of a starburst with 1 M' yr−1 with an age of 5 Myr located
at different redshifts. This is compared to the NIRCam sensitivities for 3σ detection
after an integration time of 10 days. This allows 3σ detection of such starbursts for
z ! 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The expected number of sources for a high-J CO line survey. We adopt a field of view
of (1( )2 , an active quasar fraction of 10%, as discussed in Chapter 7.3, and a redshift
interval of ∆z = 0.5, as discussed in the text. Down to z ∼ 10, a typical ALMA field
of view therefore has a high chance to find a number of quasars with black holes in
the mass range of 106 − 107 M' . Near z = 6, one may even find more massive quasars
by extending the search to larger volumes. Note that the numbers above scale linearly
with the adopted fraction of active quasars. . . . . . . . . . . . . . . . . . . . . . .
The expected number of sources for a starburst survey focusing on continuum radiation mostly from stellar light. We adopt a field of view of (2.16( )2 and an active
quasar fraction of 100%, as the relevant observables do not require quasar activity.
We explore a redshift interval of ∆z = 2. In fact, this yields a high number of sources
that are potentially accessible to JWST within the NIRCam field of view. For redshifts near z = 5, typical detection timescales are one day or less, while observations
near z = 10 require integration times between one and ten days. This is within reach
of deep surveys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The expected number of sources for a high-J CO line survey, for an evolving relation
between black hole mass and spheroid velocity dispersion (see Eq. (7.11)), for black
hole masses of at least 107 M' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional estimates for the number of sources for a high-J CO line survey. We show
extrapolations of the results by Shankar et al. [2009] and Treister et al. [2009] to high
redshift, for luminosities larger than 1044 erg s−1 . In addition, we give an estimated
based on the number of high-redshift black holes required in order to produce the
present-day 109 M' black holes. Their mass is however sensitive to model assumptions, see text and Fig. 7.13 for further discussion. . . . . . . . . . . . . . . . . . . .
The average accretion history of a black hole with 107 M' at z = 5, depending on a
coefficient p which is a function of the conversion of rest-mass energy to luminous
energy, the Eddington ratio and the duty cycle. . . . . . . . . . . . . . . . . . . . . .
xii
140
148
151
152
153
154
155
157
LIST OF FIGURES
A.1 The free-free absorption coefficient of H− for 102 , 103 , 104 and 105 GHz, as a function of temperature. Given are the fits of John [1988] and Gingerich [1961] for the
high-temperature regime, the calculation of Dalgarno & Lane [1966] based on effective range theory as well as the new calculation of this work for the low-temperature
regime up to 2000 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
xiii
LIST OF FIGURES
xiv
List of Tables
2.1
Summary of adopted stellar models. The first column gives the model name. The
second column indicates the stellar populations that contribute to reionization. The
third column gives the number of ionizing photons per baryon used in Eq (2.16). The
fourth column lists the adopted escape fractions. Model A is a highly extreme case
in which we assume all ionizing photons come from massive Pop III stars and can
escape the star-forming halo. In model B photons can escape efficiently only from
halos less massive than a corresponding virial temperature of 104 K, while for highermass halos, this fraction is reduced to 10%. In model C we assume that Pop III as well
as Pop II stars contribute to reionization, and we adopt the same escape probabilities
as in B. Model D is another extreme case which assumes that all ionizing photons
come from low-mass Pop II stars, with escape fraction 6%. . . . . . . . . . . . . . .
28
A list of models for different co-moving magnetic fields and star formation efficiencies, which are used in several figures for illustrational purposes. We give the comoving magnetic field B0 and the star formation efficiency f∗ . For illustration purposes,
all models assume a population of massive Pop. III stars. The amount of Lyman α
photons produced per stellar baryon would be larger by roughly a factor of 2 if we
were to assume Pop. II stars. Assuming that the same amount of mass goes into stars,
the coupling via the Wouthuysen-Field effect would start slightly earlier, but the delay
due to magnetic fields is still more significant. . . . . . . . . . . . . . . . . . . . .
56
4.1
4.2
4.3
4.4
Freeze-out values of free electrons, H2 and HD at z=100.
Freeze-out values of free electrons, H2 and HD at z=10. .
Abundances of D+ and HeH+ at z=100. . . . . . . . . .
Abundances of D+ and HeH+ at z=10. . . . . . . . . . .
.
.
.
.
76
77
77
77
5.1
Reionization models for MS-dominated dark stars. The parameters zPop II and zburst
give the transition redshifts to a mode of Pop. II star formation and to the sudden star
burst, while τreion is the calculated reionization optical depth and zf the redshift of
full ionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
3.1
xv
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.
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.
.
.
.
LIST OF TABLES
5.2
Reionization models for CD dark stars stars. The number of ionizing photons was
determined from the work of Yoon et al. [2008]. The parameters zPop II and zburst
give the transition redshifts to a mode of Pop. II star formation and to the sudden
star burst, while τreion is the calculated reionization optical depth and zf the redshift
of full ionization. The calculation assumes a spin-dependent scattering cross section
of 5 × 10−39 cm2 . As stellar models depend on the product of this cross section
with the threshold dark matter density, the effect of a lower scattering cross section is
equivalent to a smaller threshold density. . . . . . . . . . . . . . . . . . . . . . . . . 102
7.1
Frequency range, angular resolution θres , line sensitivity S l for a bandwith of 300 km/s
and continuum sensitivity S c for 1σ detection in one minute of integration time and
primary beam size θbeam for bands 3 − 9 of ALMA. 3 more bands might be added
in the future, band 1 around 40 GHz, band 2 around 80 GHz and band 10 around
920 GHz, which will have similar properties as the neighbouring bands. . . . . . . . 143
The main observables for ALMA, as discussed in Chapter 7.6.1. The estimated flux
ϕ for the high-J CO lines assumes that one typical resolution element of ∼ 0.01(( is
comparable to the source size, while the other fluxes correspond to the fluxes observed
for NGC 1068 if it were placed at z ∼ 8 (for lines only observable at z > 10, we
placed it at z = 12). The rest frame wavelength is given as λ. The line and continuum
sensitivities are given in Table 7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
The NIRCam sensitivity of JWST for a detection at 3σ after an integration time of
104 s. These values assume that the wide filter is used, which is appropriate for
continuum fluxes as considered here. . . . . . . . . . . . . . . . . . . . . . . . . . . 149
The main observables for JWST, as discussed in Chapter7.6.2. We estimate the flux at
z ∼ 8 as described in the text (z = 6 for Lyman α, as it is extincted at higher redshift).
The NIRCam continuum sensitivities are given in Table 7.3. The continuum fluxes
will be extincted by absorption for wavelengths smaller than 121.6(1 + z) nm, and the
same holds for the Lyman α line for redshifts before reionization is complete. As can
be seen in Figs. 7.7-7.8, this extinction does not affect the detectability of these sources.150
7.2
7.3
7.4
B.1 Collisional and radiative rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.1 Collisional and radiative rates in cgs units. . . . . . . . . . . . . . . . . . . . . . . . 177
B.1 Collisional and radiative rates in cgs units. . . . . . . . . . . . . . . . . . . . . . . . 178
xvi
Publications
The results presented in this thesis have been published in the following papers:
1. ”Effects of primordial chemistry on the cosmic microwave background”, Schleicher,
D. R. G.; Galli, D.; Palla, F.; Camenzind, M.; Klessen, R. S.; Bartelmann, M. &
Glover, S. C. O., 2008, Astronomy & Astrophysics, 490, 521
2. ”Reionization: A probe for the stellar population and the physics of the early universe”, Schleicher, D. R. G.; Banerjee, R. & Klessen, R. S., 2008, Physical Reviews
D, 78, 083005
3. “Influence of Primordial Magnetic Fields on 21 cm Emission”, Schleicher, D. R.
G.; Banerjee, R. & Klessen, R. S., 2009, The Astrophysical Journal, 692, 236S
4. ”Dark stars: Implications and constraints from cosmic reionization and extragalactic background radiation”, Schleicher, D. R. G.; Banerjee, R. & Klessen, R. S.,
2009, Physical Reviews D, 79, 043510
5. “Cosmic constraints rule out s-wave annihilation of light dark matter”, Schleicher,
D. R. G.; Glover, S. C. O.; Banerjee, R. & Klessen, R. S., 2008, Physical Reviews
D, 79, 023515
6. “Detecting the first quasars with ALMA and JWST”, Schleicher, D. R. G.; Spaans,
M. & Klessen, R. S., 2008, submitted to ApJ (arXiv:0812.3950)
LIST OF TABLES
xviii
1
Introduction
While a standard model of cosmology has been established, several open questions remain regarding the universe in particular at early times. Such questions concern, for instance, the presence of
primordial magnetic fields as well as the true nature of dark matter, but also the origin of the observed supermassive black holes at z ∼ 6 [Fan et al., 2001] and their high metallicities [Pentericci
et al., 2002]. Observational progress makes it possible to constrain some of the physics of the early
universe in more detail, like the strength of primordial magnetic fields and the effects of dark matter
annihilation / decay. I will further demonstrate that future telescopes like ALMA and JWST are capable of observing active galaxies far beyond redshift 6, as long as their black hole has at least 106 M' .
In about five years time, it is therefore possible to detect the progenitors of supermassive black holes
at high redshift and to follow the cosmic evolution of the black hole population.
I will first review the thermal history of the universe in Chapter 1.1, and discuss some of the
open questions and their consequences in Chapter 1.2. A brief overview how these questions can
be probed in more detail is given in Chapter 1.3, with a more detailed description presented in the
following chapters.
1.1 The timeline of cosmology
During the last decades, a standard model of cosmology has been established, assuming that the
universe is homogeneous and isotropic on scales larger than 100 Mpc, such that it can be described
by the Friedmann-Lemaître-Robertson-Walker metric. It originates from an early hot phase in which
radiation dominates and the light elements are formed. The composition of the universe has been
measured with high accuracy [Komatsu et al., 2008]. At z = 0, only ∼ 5% of the total mass-energy
density is due to baryons, while 23% are due to a pressureless component of dark matter and 72%
are due to a component with negative pressure that is usually referred to as dark energy, currently
consistent with a cosmological constant. Structure formation models require dark matter to be cold,
i. e. non-relativistic.
1
1. INTRODUCTION
This scenario in which the universe consists of baryons, cold dark matter (CDM) and a cosmological constant (Λ) is usually termed the ΛCDM model. This is the current standard model of cosmology.
While we know the composition of the universe, the nature of dark matter and dark energy is however
not yet understood. In this section, I will describe the timeline of cosmology as expected in the standard model, as well as some of the remaining uncertainties, in particular with respect to the nature of
dark matter and the putative presence of primordial magnetic fields. Of course, other uncertainties are
present as well, for instance regarding the nature of dark energy. The latter is however not explored
further in this work. The history of the universe, as expected in the standard model of cosmology and
discussed in the following sections, is illustrated in Fig.1.1.
1.1.1 From the Big Bang to cosmic recombination
As recently confirmed by the new WMAP 5 year data, the universe today appears close to being
spatially flat, and observations of the cosmic microwave background show that the universe has been
highly homogeneous at the epoch of recombination, over scales considerably larger than the cosmological horizon [Komatsu et al., 2008]. These observational facts, commonly referred to as the
flatness and the horizon problem, appeared hard to explain in the cosmological framework, unless an
early inflationary epoch is introduced. In this epoch, the universe is in a phase of rapid and accelerated expansion, with its size increasing by a factor of ∼ 1030 . Most inflationary models assume that
a scalar field, the inflaton field, dominates the energy budget of the universe during this epoch. It
evolves in a rather flat potential, such that its kinetic energy is considerably smaller than the potential
energy, giving rise to an equation of state with negative pressure and the accelerated expansion of the
universe.
This scenario was first introduced by Guth [1981]. It was then realized that inflation also provides
a way to generate large-scale structures in the universe from small quantum fluctuations [Hawking,
1982; Starobinskij, 1982; Guth & Pi, 1982]. A review about more recent developments is given by
Kinney [2003]. Inflationary scenarios generally predict a primordial density power spectrum which
is almost scale-invariant, which is in agreement with current data. Since the epoch of inflation must
eventually come to an end and the universe needs to be re-heated after this era of rapid expansion,
it is generally assumed that the inflaton field eventually evolves toward a minimum in its potential
where it oscillates and decays due to some couplings to ordinary matter, thus increasing the entropy
and heating the universe to extremely highly temperatures.
After reheating, the universe continues to expand and thus cools adiabatically. At temperatures of
∼ 1012 −1016 TeV, it is expected to be in the grand unification epoch, in which the weak, the strong and
the electromagnetic interaction are unified [Roos, 2003]. Such scenarios are suggested for instance
in supersymmetric extensions of the standard model [Drees, 1996]. The physics of this epoch is still
highly uncertain and subject to speculative suggestions in particle physics. It is generally assumed
that this phase is also responsible for the observed assymetry between matter and anti-matter. In order
to produce such an asymmetry, a grand unification theory generally needs to fulfill three conditions:
2
1.1 The timeline of cosmology
Figure 1.1: The timeline of cosmology. See Chapter 1.1 for a detailed description. Copyright 1996
University of Cambridge.
3
1. INTRODUCTION
1. It must contain reactions that violate baryon number conservation.
2. The theory must not be symmetric with respect to charge conjugation as well as the combination
charge conjugation and parity change.
3. The relevant processes must occur out of thermal equilibrium.
While the first condition is obvious, the second must be required as well. Otherwise, each baryonviolating reaction would be matched by equally frequent reactions that violate baryon number in the
other direction, such that the net effect is zero. Finally, a deviation from thermal equilibrium must
be required, otherwise there would be no net effect because of the inverse reactions. Since most of
the matter and anti-matter particles will annihilate during the further evolution of the universe, the
generation of a small asymmetry is sufficient to strongly suppress the abundance of anti-matter at late
times.
Once the universe cools to ∼ 100 GeV, it is in the energy range of the standard model of particle
physics [Roos, 2003]. At this scale, the electroweak symmetry is broken, and the W and Z vector
bosons that mediate the weak interaction become massive particles. Such symmetry breaking may
occur through a phase transition of first or second order. During a first order phase transition, the relevant field, here the Higgs field, tunnels through a potential barrier, while evolving smoothly from one
state to another during a second-order phase transition. Lattice computations in the framework of the
standard model of particle physics give strong evidence against a first-order phase transition [Kajantie
et al., 1996], though it remains a viable possibility if supersymmetric extensions of the standard model
are considered [e. g. Dine et al., 1992; Espinosa et al., 1993]. The electroweak baryogenesis scenario
requires a first-order phase transition [Riotto & Trodden, 1999].
At an energy scale of ∼ 150 MeV, the phase transition between low-energy hadronic physics and
QCD occurs [Kajantie & Kurki-Suonio, 1986]. At higher energies, quarks are asymptotically free and
form what is called quark matter, and at this temperature they are bound into nucleons [Bonometto &
Pantano, 1993]. Typically, a first-order phase transition takes place by bubble nucleation.
At even lower temperatures and densities, neutrinos will decouple from the plasma. This occurs at
a temperature of 3.5 MeV for µ and τ neutrinos, and at 2.3 MeV for electron-neutrinos. At energies of
∼ 0.1 MeV, the universe is in the epoch of Big Bang nucleosynthesis (BBN). In this epoch, the light
elements are formed within approximately three minutes. Big Bang nucleosynthesis was first suggested by Gamow [1946], and developed further in various works [e. g. Alpher et al., 1948; Hayashi,
1950]. The theory predicts mass abundances of ∼ 75% for hydrogen, 25% for helium, 0.01% for deuterium and trace amounts of lithium and beryllium. The generally good agreement of observed and
predicted abundances is one of the outstanding successes of the hot Big Bang theory. The observed
lithium abundance does not match these predictions very well, which may be due to further evolution
during stellar nucleosynthesis. A recent comparison of observed and predicted abundances is given
by Miele & Pisanti [2008].
As the universe cools further, it eventually reaches temperatures where the electrons and ions
combine. This is referred to as the epoch of recombination, which was predicted byPeebles [1968]
4
1.1 The timeline of cosmology
and Zeldovich et al. [1969]. While one would naively expect the recombination of hydrogen at temperatures of ∼ 104 K, the process is delayed until temperatures of ∼ 3000 K due to the large photonto-baryon ratio, which leads to a sufficient number of ionizing photons from the high-energy tail to
keep the universe ionized until z ∼ 1100. At recombination, the universe becomes transparent to the
photons that form the cosmic microwave background (CMB). The detection of this background by
Penzias & Wilson [1965] and its correct interpretation by Dicke et al. [1965] was a further milestone
of cosmology. Its spectrum appears as a perfect blackbody at a temperature of 2.725 K [Fixsen &
Mather, 2002]. Its power spectrum is now an important tool to derive for instance the cosmological
parameters, the normalization of the dark matter power spectrum and the reionization optical depth.
Ultimately, the CMB is expected to have deviations from a blackbody spectrum, for instance due to
the 21 cm line of atomic hydrogen or various molecular line. This possibility is explored in more
detail in Chapter 3 and 4.
1.1.2 From the postrecombination epoch to cosmic reionization
The universe after recombination is almost neutral, largely homogeneous and of primordial composition. Initially, the gas temperature is still coupled to the CMB temperature by Compton scattering.
This coupling becomes inefficient at z ∼ 200, and the gas temperature then evolves adiabatically and
drops below the CMB temperature. With decreasing density, the recombination timescale becomes
significantly larger than the expansion timescale, such that the universe does not become fully neutral, but rather retains a relic electron abundance of ∼ 2 × 10−4 [Peebles, 1968; Zeldovich et al., 1969;
Seager et al., 2000].
As recognized initially by Saslaw & Zipoy [1967], such an electron abundance leads to the formation of molecular hydrogen in the intergalactic medium (IGM) and the first collapsing objects. In
the IGM, H2 formation starts at z ∼ 300 via the H+2 formation channel and the abundance increases
at z ∼ 100 via the H− formation channel [Puy et al., 1993; Stancil et al., 1998; Galli & Palla, 1998].
In the first collapsing gas clouds at z ∼ 25, the H2 abundance increases further due to the higher
densities. In the absence of metal coolants, this is crucial in particular for the first minihalos with
virial temperatures below 104 K [Tegmark et al., 1997], since they cannot cool by atomic hydrogen.
A formalism to estimate the number density of halos as a function of their mass at a given redshift
has been developed by Press & Schechter [1974] and was improved by Sheth et al. [2001]. For the
current cosmological parameters, this framework predicts the first star-forming halos around z ∼ 25.
Numerical simulations indicate that gas collapse in these minihalos leads to the formation of stars with
∼ 100 M' [Abel et al., 2002; Bromm & Larson, 2004; Yoshida et al., 2006]. A result from such highresolution numerical simulations is shown in Fig. 1.2. Such stars provide strong sources of ionizing
radiation [Bromm et al., 2001; Schaerer, 2002], and are required to explain the observed reionization
optical depth [Wyithe & Loeb, 2003; Schleicher et al., 2008b]. Stars forming in primordial gas are
generally referred to as Population III stars (Pop. III), specifically Pop. III.1 if they form in neutral
gas and Pop. III.2 if they form in previously ionized gas (see below). This terminology originates
5
1. INTRODUCTION
Figure 1.2: The formation of the first star, as calculated by Abel et al. [2002]. Credit: Tom Abel (Stanford)
& Ralf Kaehler (ZIB).
from the classification of stars in the Milky Way, where metal-rich stars like the sun are called Pop. I
stars and metal-poor stars are termed Pop. II stars.
The evolution of the subsequent stellar population is more difficult to infer, since the first stars
will affect their environment by mechanical, chemical and radiative feedback. In the range of 140 −
260 M' , it is expected that they explode as a pair-instability supernova, while one expects a conventional core-collapse supernova for masses smaller than 40 M' . Inbetween these two ranges, the
star would collapse to form a black hole [Heger & Woosley, 2002]. Mechanical feedback from violent supernova explosions changes the gas distribution and thus the initial conditions for subsequent
generations of stars [Greif et al., 2007]. It also affects the ionization degree and leads to metal enrichment. While a detailed calculation of the gas fallback after a supernova explosion and the subsequent
initial conditions for star formation is still outstanding, it has been demonstrated that the correct initial conditions are indeed crucial for the further evolution of the gas [Jappsen et al., 2008]. Once a
critical metallicity is reached, arguable at ∼ 10−5 Z' or 10−3 Z' , the gas fragments and star formation
occurs according to a standard Pop. II initial mass function (IMF) [Bromm et al., 2001; Schneider
et al., 2003; Clark et al., 2008].
Radiative feedback will affect the next generation of stars in various ways. Ionization of the gas
by UV photons enhances the ability of the gas to form molecular hydrogen and HD, thus providing
an efficient cooling mechanism down to the CMB floor. This leads to smaller Jeans masses and
typically decreases the stellar mass by an order of magnitude [Yoshida et al., 2007a,b]. These are the
so-called Pop. III.2 stars. Softer UV photons in the Lyman-Werner (LW) range of 11.2 − 13.6 eV can
6
1.1 The timeline of cosmology
dissociate molecular hydrogen by the Solomon process [Stecher & Williams, 1967]. This effect was
explored a couple of years ago by Haiman et al. [2000], Glover & Brand [2001] and Machacek et al.
[2001], indicating that it may provide an efficient mechanism to suppress subsequent star formation
by destroying the main coolant. However, more recent work shows that the build-up of such a LWbackground is a self-regulated process, and the IGM may soon be optically thick to such radiation
while H2 builds up in relic HII regions [Johnson et al., 2007a, 2008]. It is thus conceivable that the
latter is only a minor correction to the total star formation rate.
In addition, the universe could be magnetized by magnetic fields generated in winds and supernova explosions from the first stars [Rees, 1987; Kandus et al., 2004]. To my knowledge, the effect
of such feedback on the subsequent generations of stars has not yet been explored. For the very first
stars, people started to consider the consequences of magnetic fields generated during protostellar
collapse (see discussion in Chapter 1.2.3 and Chapter 8.2.2), but the results are still uncertain and
controversial.
Toward z ∼ 10, halos grow in size and may harbor more massive galaxies. The so-called atomiccooling halos, those with virial temperatures above 104 K can cool via atomic hydrogen and so do
not strictly require the presence of molecular hydrogen to collapse, though simulations indicate that
this coolant is also important in these systems [Wise & Abel, 2007a; Greif et al., 2008]. They are
much more turbulent than the first minihalos and consist of a two-phase medium of both hot and cold
gas [Greif et al., 2008]. The expected stellar masses in such galaxies are currently unclear, but may
strongly depend on the metallicity [Clark et al., 2008]. It has also been suggested that these halos
could provide the seeds for the first supermassive black holes [Eisenstein & Loeb, 1995; Koushiappas
et al., 2004; Begelman et al., 2006; Spaans & Silk, 2006; Dijkstra et al., 2008]. Fig. 1.3 shows an
artist’s conception of such black holes and the interaction with their accretion disks, which regulates
the growth of the black hole [Shapiro, 2005]. However, important uncertainties remain regarding the
formation process. For instance, Lodato & Natarajan [2006] argue that the gas in such halos should
fragment if H2 cooling is efficient, and Omukai et al. [2008] and Jappsen et al. [2008] suggest that a
non-zero metallicity will lead to fragmentation.
The first stars and the first galaxies constitute the sources of light that ultimately lead to the reionization of the universe. Simulations of stellar reionization further show that the process is highly inhomogeneous and based on the growth and merging of ionized bubbles around the first stellar sources
[Gnedin, 2000; Ciardi et al., 2003; Kohler et al., 2005], and semi-analytic models have been introduced which provide an effective prescription of such processes [Shapiro & Giroux, 1987; Haiman &
Loeb, 1997a; Barkana & Loeb, 2001; Loeb & Barkana, 2001; Choudhury & Ferrara, 2005; Schleicher
et al., 2008b; Schneider et al., 2008]. X-ray feedback from high-redshift quasars may only constitute
a minor contribution, due to the constraints from the soft X-ray background [Dijkstra et al., 2004;
Salvaterra et al., 2005].
At present, reionization is constrained to end by z ∼ 6 from the spectra of high-redshift quasars
[Becker et al., 2001], which show strong Lyman α absorption features at higher redshift, as well as the
reionization optical depth due to Thomson scattering on free electrons. This boosts the polarization
7
1. INTRODUCTION
Figure 1.3: An artist’s conception of a supermassive black hole and the interaction with its accretion disk.
Credit: NASA
of the CMB on large scales, while it suppresses polarization on small scales [Zaldarriaga, 1997]. The
reionization optical depth could be derived for the first time from the WMAP 1-year data [Kogut,
2003], and has now been calculated with much higher accuracy from the five-year data to be τe =
0.087 ± 0.017.
Black holes with ∼ 109 M' have already been detected at z ∼ 6, and their accretion disks were
found to have super-solar metallicities even at such redshifts [Pentericci et al., 2002; Juarez et al.,
2009]. Observationos of their spectra provide evidence for the completion of reionization near z = 6
[Becker et al., 2001]. To explain the origin of supermassive black holes is an important goal of
cosmology and structure formation. In Chapter 7, I discuss how observations with ALMA and JWST
can detect their progenitors at higher redshift.
1.2 Open questions
In this section, I discuss some of the open questions with respect to the cosmological model described
above. This concerns the potential presence of primordial magnetic fields, as well as the true nature
of dark matter, which may reveal itself via dark matter annihilation or decay. I review both the
observational and the theoretical motivations for these scenarios. Then I discuss their impact on the
formation of the first stars, before exploring the consequences in more detail in the following chapters.
8
1.2 Open questions
1.2.1 Primordial magnetic fields
To explore the potential relevance of primordial magnetic fields in more detail, I will discuss whether
currently observed magnetic fields can be explained by astrophysical mechanisms in Chapter1.2.1.1,
and present a description of the mechanisms available in the era before cosmic recombination to
generate primordial magnetic fields in Chapter 1.2.1.2. Some of these topics are discussed in more
detail by Grasso & Rubinstein [2001].
1.2.1.1
Primordial or astrophysical?
Magnetic fields can be observed by Zeeman splitting of spectral lines, Faraday rotation measurements
of polarized electromagnetic radiation passing through an ionized medium, and from synchrotron
emission of free relativistic electrons. Comprehensive reviews of these topics are available from
Kronberg [1994] and Zweibel & Heiles [1997]. While Zeeman splitting provides a direct way of
detection, the effect is very small and only useful within our galaxy. On the other hand, synchrotron
emission and Faraday rotation measurements can trace magnetic fields at large distances, but require
an independent determination of the local electron density ne . This is sometimes possible, for instance
by studying X-ray emission in the very hot gas of galaxy clusters.
In our Galaxy, the average field strength is 3 − 4 µG, corresponding to approximate equipartition
between the magnetic field, the cosmic rays confined in the galaxy, and the small-scale turbulent motion [Kronberg, 1994]. The field is coherent over scales of a few kpc, comparable to the galactic size,
with alternating directions in the arm and interarm regions [Kronberg, 1994; Han, 2008]. For a long
time, the preferred mechanism to explain these observations was the dynamo mechanism [Zeldovich
et al., 1979; Vainshtein et al., 1980]. For the generation of large coherence lengths, the mean-field
dynamo is particularly popular, which is derived by defining a spatial average over the induction
equation on scales larger than the turbulent eddy scales [Krause & Rädler, 1980]. However, while
the mean-field dynamo would operatore on large scales, small-scale fields could be produced on a
much faster rate, leading to saturation before a significant large-scale field is built-up [Kulsrud et al.,
1997]. Also, most dynamo models predict even parity with no reversals, contrary to observations
in our Galaxy, whereas such a situation would be expected in the case of primordial fields [Grasso
& Rubinstein, 2001]. Zweibel [2006] confirms that the problem of creating a large-scale field from
astrophysical sources is still unresolved. Kulsrud & Zweibel [2008] consider some commonly raised
arguments against primordial magnetic fields, finding that these arguments are subject to considerable
uncertainties and cannot rule out the primordial case.
Strong magnetic fields have also been detected in the inter-cluster medium (ICM) of galaxy clusters [Kim et al., 1991]. Their field strength is typically described by the phenomenological equation
BICM
!
L
∼ 1.4µG
10 kpc
9
"−1/2
(h70 )−1 ,
(1.1)
1. INTRODUCTION
where L is the reversal field length and h70 the Hubble constant normalized to 70 km s−1 Mpc−1 .
Typical values of L range between 10 and 100 kpc. In the Coma cluster, a core magnetic field of
∼ 8 µG tangled at scales of about 1 kpc has been detected [Feretti et al., 1995]. While it was recently
shown that magnetic fields from outflows can lead to similar field strength in galaxy clusters [Donnert
et al., 2009], it is not yet clear if that also explains the coherence on large scales. It appears that the
results are hard to distinguish from a configuration created by primordial magnetic fields, as calculated
for instance by [Dolag et al., 1999]. As shown by Banerjee & Jedamzik [2003], primordial magnetic
fields generated during cosmological phase transitions could both explain the observed field strength
and the large-scale coherence.
Magnetic fields have also been detected in very distant objects; for instance radio quasars near z ∼
2 appear to have field strength of a few µG [Kronberg, 1994]. That by itself does not necessarily speak
for or against primordial magnetic fields, as their coherence lengths is poorly known and quasars
generally appear as highly evolved also in metallicity [Juarez et al., 2009]. However, Bernet et al.
[2008] recently showed that quasars with strong Mg+ absorption lines are unambiguously associated
with large Faraday rotation measurements. As Mg+ absorption occurs in the halos of normal galaxies
along the line of sight of quasars, they found evidence for magnetic field strengths of a few µG in
normal galaxies at z ∼ 3, with considerably less time available for a dynamo mechanism to operate.
Indeed, as noted already by Grasso & Rubinstein [2001], such field strengths can be explained by
primordial fields of ∼ 0.1 nG without requiring a dynamo to operate.
At even higher redshift, one can only give upper limits on the expected field strength. Measurements
of the small-scale CMB anisotropy yield an upper limit of 4.7 nG to the comoving field strength on a
scale of 1 Mpc [Yamazaki et al., 2006]. Constraints on the primordial field strength are also available
from the measurement of σ8 , which describes the root-mean-square of matter density fluctuations in
a comoving sphere of 8h−1 Mpc. However, they depend sensitively on the assumed power spectrum
for the magnetic field [Yamazaki et al., 2008]. Other constraints are available from Big Bang nucleosynthesis (BBN). For instance, strong magnetic fields above 4.4 × 1013 G at the epoch of BBN
significantly increase the β decay rate of neutrons, which would suppress the4 He abundance with respect to the standard case [Matese & O’Connell, 1969; O’Connell & Matese, 1969]. Strong uniform
fields would further affect the energy density of the electron gas due to the increase of its phase space
[Matese & O’Connell, 1970]. More relevant is however the contribution of the magnetic field energy
density to the total energy density, and its impact on the expansion of the universe [Greenstein, 1969].
Recent works confirm that the allowed magnetic field strength at the end of nucleosynthesis (T =
0.01 MeV) is ∼ 2 × 109 G, corresponding to ∼ 10−6 G today.
It is important to note that the constraints described here are rather weak, and magnetic fields of
such strength can have a considerable impact on the evolution of the early universe and the formation
of the first stars and galaxies. This will be described in more detail in Chapter2 and 3, where I discuss
how such constraints can be improved from measurements of the reionization optical depth and 21 cm
observations of the reionization epoch. Such observations may therefore constrain the physics in the
10
1.2 Open questions
universe after recombination. In Chapter 8, I describe work in progress that will help to tighten such
constraints even further.
1.2.1.2
Generation of primordial magnetic fields
In this section, I summarize the main mechanisms that may provide primordial magnetic fields in
the early universe, starting from mechanisms available in the pre-recombination epoch and going
backward to the QCD phase transition, the electroweak phase transition and inflation.
As pointed out by Harrison [1970], the presence of primordial vorticity before the recombination
of hydrogen can lead to the generation of magnetic fields in this epoch. This is because in this era,
Thomson scattering is much more effective for electrons than for protons. The electrons are therefore
still effectively coupled to the radiation field and behave as relativistic matter, while the ions are
already non-relativistic. The angular velocities of electrons and ions could therefore scale differently
with redshift and give rise to an electric current that generates a magnetic field. In principle, this
could lead to comoving field strenghts of up to ∼ 10 nG on scales of 1 Mpc. However, as pointed out
by Rebhan [1992], the amount of primordial vorticity is constrained by the requirement that it does
not produce too large anisotropies in the CMB. This yields the following upper limit to a present-day
field with coherence length L generated by this mechanism:
B0 (L) < 6 ×
10−18 h−2
70
!
L
1 Mpc
"−3
G,
(1.2)
where h70 is the Hubble constant normalized to 70 km s−1 Mpc−1 . As shown by Berezhiani & Dolgov
[2004], such primordial vorticity could be due to photon diffusion in the second order temperature
fluctuations. However, in this case strong fields are only obtained on small scales.
As mentioned in Chapter 1.1.1, the QCD phase transition is a first-order phase transition taking
place at temperatures of TQCD ∼ 150 MeV by bubble nucleation. Initially, these bubbles grow as
burning deflagration fronts and release heat in the form of supersonic shock fronts in an out-ofequilibrium process. When the shock fronts collide, they reheat the plasma up to TQCD and stop
further bubble growth. From this point, the transition evolves in thermal equilibrium during the socalled co-existence phase. The latent heat from these bubbles compensates for the cooling from the
expansion and keeps the temperature at TQCD , until expansion wins and the remaining quark-gluon
plasma pockets are hadronized.
According to Quashnock et al. [1989], magnetogenesis during the QCD phase transition proceeds
via the formation of an electric field behind the shock fronts of the expanding bubbles. This is a
consequence of the baryon assymetry, which is typically assumed to be present at this evolutionary
stage. Quashnock et al. [1989] then expect a typical field strength of Bl = 5 G on a scale of 100 cm.
The magnetic field on larger scales can be estimated following an approach of Hogan [1983] by
performing a proper volume average over a larger number of magnetic dipoles of size l randomly
11
1. INTRODUCTION
oriented in space. Such an average gives
! "3/2
l
B L = Bl
.
L
(1.3)
On a comoving scale of L ∼ 1 AU, one could therefore expect a field strength of 2 × 10−8 nG at
present time.
Cheng & Olinto [1994] showed that stronger fields could be produced during the coexistence
phase of the QCD phase transition. In this phase, a baryon excess builds up in front of the bubble
wall. Simulations of Kurki-Suonio [1988] indicate that this effect might enhance the baryon density
contrast by a few orders of magnitude. Even more relevant is that the thickness of this layer of
∼ 107 fm is considerably larger than the microphysical QCD length scale of ∼ 1 fm. This could
give rise to a comoving field of 10−7 nG on a scale of 1 pc. Even stronger fields can be produced
in the presence of hydrodynamic instabilities [Sigl et al., 1997]. Such instabilities can occur during
phase transitions when the transport of latent heat is dominated by the fluid flow. While it is not fully
clear if these instabilities can really develop, they appear plausible for typical parameters of the QCD
phase transition. In this case, one could obtain comoving field strengths of ∼ 1 nG on a scale of
100 kpc. Future constraints on primordial magnetic fields may therefore provide more insight in this
early epoch of the universe.
As mentioned in Chapter 1.1.1, it is not fully clear whether the electroweak phase transition is
a phase transition of first or second order. Here, I will mostly focus on the possibility of a firstorder phase transition, as also required by the electroweak baryogenesis scenario [Riotto & Trodden,
1999]. For a second-order phase transition, one expects weaker magnetic fields which are unlikely
to be astrophysically relevant [Grasso & Rubinstein, 2001]. Baym et al. [1996] suggest that strong
magnetic fields can be generated in this epoch via a dynamo mechanism. For this scenario, seed fields
are provided from random magnetic fluctuations which are always present on scales of the order
of a thermal wavelength [Grasso & Rubinstein, 2001]. Once the universe cools below the critical
temperature of Tc ∼ 100 GeV, the Higgs field locally tunnels into a phase with broken symmetry.
The tunneling gives rise to the formation of broken phase bubbles which expand and convert the false
vacuum energy into kinetic energy. For a wide range of model parameters, the expansion gives rise to
a supersonic shock wave ahead of the burning front and turbulence fully develops. On galactic scales
of ∼ 5 kpc, this gives rise to a field strength of 10−11 − 10−8 nG.
The numbers given above for the co-moving field strength assumed that the magnetic field evolves
with (1 + z)2 , such that the magnetic energy density scales as the energy density of radiation. This
is however not necessarily the case. The evolution of magnetic fields generated at the electroweak or
the QCD phase transition was calculated by Banerjee et al. [2004], finding that part of the magnetic
energy may decay in a turbulent cascade, while in the presence of non-zero helicity, energy may also
be transferred from smaller to larger scales. They find particularly promising results for magnetic
fields from the QCD phase transition, expecting co-moving field strengths in the range of 10−2 − 2 nG
12
1.2 Open questions
and typical coherence lengths of 10 kpc up to 1 Mpc. Such fields can be probed by the mechanisms
discussed in the following chapters.
Finally, the inflationary epoch provides several important ingredients for the generation of primordial magnetic fields [Turner & Widrow, 1988]:
• It naturally produces effects on very large scales, even larger than the Hubble horizon.
• It provides the dynamical means to amplify waves on long wavelengths.
• The universe is not a good conductor during the inflationary epoch, thus the ratio of magnetic
to radiation energy can increase.
• Classical fluctuations of massless fields with wavelengths smaller than the Hubble horizon can
grow super-adiabatically, with their energy density decreasing only as (1+z)2 rather than (1+z)4 .
There is however also one major obstacle for the generation of magnetic fields in this era. As shown
by Parker [1968], for a conformally flat metric like the Robertson-Walker metric, the background
gravitational field does not produce particles if the underlying theory is conformally invariant. As
the classical theory of electrodynamics is conformally invariant, Turner & Widrow [1988] proposed
three possibilities to overcome this obstacle. First, they considered to break conformal invariance
explicitly by introducing a coupling of the form RAµ Aµ or Rµν Aµ Aν , where R is the Ricci scalar, Rµν
is the Ricci tensor and Aµ the electromagnetic field. Such terms would give the photons an effective,
time-dependent mass. They showed that such a mechanism could give rise to galactic magnetic fields
without additional dynamo amplification.
Conformal invariance may also be broken by more complicated terms due to one-loop vacuum
polarization effects in curved space-time. Unfortunately, Turner & Widrow [1988] showed that their
contribution is far too small to be relevant. The third way they discussed to break conformal invariance involves a coupling of the photon to a charged field which is not conformally coupled, or
the anomalous coupling to a pseudo-scalar. This possibility was considered in more detail byRatra
[1992] and Ratra & Peebles [1995], finding that the generated comoving field strength is highly
model-dependent and ranges between 10−65 nG and 1 nG. More recent works still indicate enornous
uncertainties regarding the magnetic field strength produced during inflation, including values with
significant phenomenological consequences [Bamba et al., 2008; Campanelli et al., 2008; Campanelli,
2008].
1.2.2 Dark matter decay / annihilation scenarios
Particle physics models provide a number of different dark matter candidates. These include massive
particles of ∼ 100 GeV, like neutralinos or gravitinos that occur in supersymmetric extensions of the
standard model. Very light axions of ∼ 0.01 eV were suggested to solve the strong CP problem. Light
dark matter particles in the MeV range have received particular attention due to the observation of
511 keV emission in the Galactic center. Kaluza-Klein particles are motivated in models with spatial
13
1. INTRODUCTION
Figure 1.4: The observed 511 keV emission from the Galactic center: Evidence for dark matter annihilation? Image: J. Knödlseder/CESR, Toulouse.
14
1.2 Open questions
extra-dimensions, and even more exotic candidates are considered in the literature. For the details
of the different models, the interested reader is referred to recent reviews byBertone et al. [2004];
Bergström [2000]; Ellis [2000].
It is interesting to note that many of these models predict some mechanism for annihilation or
decay of dark matter particles, and weak interactions with baryons. Indeed, it is well-established that
a thermally-averaged dark matter annihilation cross section of ∼ 3 × 10−26 cm3 s−1 naturally leads to
the dark matter abundance observed in the universe. Conservative constraints from the Milky Way,
Andromeda (M31), and the cosmic gamma-ray background yield upper limits on the annihilation
cross section which are a few orders of magnitude larger, with the precise value depending on the
dark matter mass [Mack et al., 2008]. The experimental upper limits for scattering cross sections
between dark matter particles and baryons are 10−38 cm2 for spin-dependent scattering [Desai et al.,
2004; Angle et al., 2008] and 4 × 10−44 cm2 for spin-independent scattering [Ahmed et al., 2008].
Observations of the Galactic center in different frequencies have provided further motivation for
models that consider dark matter annihilation and/or decay. One finds an excess of GeV photons [de
Boer et al., 2005], of microwave photons [Hooper et al., 2007], of positrons [Cirelli et al., 2008] and
of MeV photons [Jean et al., 2006; Weidenspointner et al., 2006], which correlates with the Galactic
bulge instead of the disk and cannot be attributed to single sources. It is therefore controversial
whether it can be explained by conventional astrophysical sources alone or if dark matter annihilation
models are required. It is not clear whether these different phenomena are related. Hence, their interpretation is still under discussion [de Boer, 2008]. As an example, the observed 511 keV emission
from the Galactic center [Knödlseder et al., 2003] is shown in Fig. 1.4. Observations in the MeV
energy range favor models based on light dark matter. These models suggest that light dark matter
particles with masses of 1 − 100 MeV annihilate into electron-positron pairs [Boehm et al., 2004a]. In
such a scenario, the dark matter mass needs to be larger than 511 keV in order to be able to produce
electron-positron pairs through dark matter annihilation, however it should be smaller than 100 MeV,
as otherwise pion final states that produce too many gamma rays would be possible [Boehm et al.,
2004a]. Electromagnetic radiative corrections to the annihilation process require the emission of internal bremsstrahlung [Beacom et al., 2005]. It has also been proposed that such internal bremsstrahlung
emission might explain the observed gamma-ray background in the 10 − 20 MeV range [Ahn et al.,
2005; Ahn & Komatsu, 2005b] for dark matter masses of ∼ 20 MeV, though these models appear less
favorable in light of stronger upper limits on the dark matter particle mass [Beacom & Yüksel, 2006;
Sizun et al., 2006]. Supernovae data require dark matter particle masses larger than 10 MeV, although
this limit depends on assumptions made regarding the scattering cross section between dark matter
particles and neutrinos [Fayet et al., 2006].
However, while light dark matter may explain the observed 511 keV emission, they have difficulties to explain the observations at GeV energies. For this purpose, a new model of so-called eXciting
dark matter has been introduced by Finkbeiner & Weiner [2007]; Finkbeiner et al. [2008]. This model
is based on massive dark matter candidates that can produce the required emission at high energies
due to a Sommerfeld enhancement, and it assumes excited states for the dark matter particles, which
15
1. INTRODUCTION
can lead to emission of electron-positron pairs. At present, the FERMI satellite 1 searches for signals
of dark matter annihilation both in the cosmic background and nearby galaxies, and may soon provide
better constraints or even a detection of dark matter signals.
1.2.3 Implications for the first stars
Previous works often assumed that the initial conditions for the formation of the first stars are wellknown, since the initial conditions for dark matter are well understood, the gas is not yet polluted
with metals and no radiation fields are present apart from the cosmic microwave background. As a
further simplification, the dynamical generation of magnetic fields during the collapse of the protocloud is mostly neglected. However, Tan & Blackman [2004] and Silk & Langer [2006] showed that
an effective dynamo may be present in the protostellar disk, driving the magnetic field strength to
equipartition with the thermal pressure. In the case where a magnetic field strength of 1 nG is already
reached at densities of ∼ 103 cm−3 , a protostellar jet could be launched which may blow away up to
10% of the accreting matter [Machida et al., 2006]. On the other hand, simulations of Price & Bate
[2008] about present-day star formation in stellar clusters show that magnetic fields can indeed help
to suppress fragmentation and help to form more massive stars. The effect of dynamically generated
magnetic fields on the first stars is therefore not completely clear.
In the presence of primordial fields, the situation is however more complicated. Comoving field
strengths of ∼ 0.1 nG heat the IGM via ambipolar diffusion and decaying MHD turbulence, and
therefore change the initial chemical conditions [Sethi et al., 2008]. The corresponding increase in
the electron fraction enhances the formation of H2 and HD, allowing the gas to cool almost down to
the CMB floor, thus decreasing the thermal Jeans mass. On the other hand, the additional magnetic
pressure can compensate for that and give rise to a very high magnetic Jeans mass. In the IGM, one
finds
#
$3
B0
M JB ∼ 1010 M'
,
(1.4)
3 × 10−9 G
where B0 is the comoving field strength. Primordial fields can thus prevent gas collapse in small
minihalos, while in more massive halos, they may rather suppress fragmentation and support the
formation of very massive stars. As the formation of stars in the presence of primordial magnetic
fields is however not completely understood, we have explored the effect of different assumptions
regarding the stellar population in Chapter 2. In collaboration with Daniele Galli, Simon Glover,
Francesco Palla, Robi Banerjee, Raffaella Schneider and Ralf Klessen, I am currently investigating
these effects in more detail and find that indeed primordial fields tend to stabilize a massive star
formation mode.
The effect of dark matter annihilation on the first stars has been examined in more detail, but
still with significant uncertainties remaining. As the first stars are expected to form on the dark matter
cusps of the first minihalos, which are steepened further by adiabatic contraction during the formation
of the stars, the annihilation of dark matter can be a strong energy source that may power the stars
1
http://www.nasa.gov/mission pages/GLAST/main/index.html
16
1.3 Probing the early universe
[Spolyar et al., 2008]. The additional energy input leads to an increase in the stellar radius, which
may reach values up to ∼ 1000 AU, thus reducing the surface temperature to ∼ 3000 K because of
the negative stellar heat capacity. This has been explored further by Iocco [2008] and Freese et al.
[2008c], who considered the effect of scattering between baryons and dark matter particles, increasing
the dark matter abundance in the star.
Iocco et al. [2008] considered dark star masses in the range 5 ≤ M∗ ≤ 600 M' and calculated
the evolution of the pre-main-sequence phase, finding that the dark star phase where the energy input
from dark matter annihilation dominates may last for 102 − 104 yr. Freese et al. [2008a] examined
the formation process of the star in more detail, considering polytropic equilibria and additional mass
accretion until the total Jeans mass of ∼ 800 M' is reached. They find that this process lasts for
∼ 5 × 105 yr. Iocco et al. [2008], Taoso et al. [2008] and Yoon et al. [2008] have calculated the stellar
evolution for the case in which the dark matter density inside the star is enhanced by the capture
of addition WIMPs via off-scattering from stellar baryons. Iocco et al. [2008] followed the stellar
evolution until the end of He burning, Yoon et al. [2008] until the end of oxygen burning and Taoso
et al. [2008] until the end of H burning. Yoon et al. [2008] also took the effects of rotation into
account. The calculations found a potentially very long lifetime of dark stars and correspondingly a
strong increase in the number of UV photons that may contribute to reionization.
1.3 Probing the early universe
As we have seen above, many open questions remain regarding the early universe and the epoch in
which the first stars formed. The main goal of this thesis is to explore how the era between recombination and the end of reionization can be better understood. This concerns both the physics of the
early universe, such as primordial magnetic fields and dark matter annihilation / decay, the stellar
population as well as the formation of supermassive black holes and the high metallicity in their host
galaxies.
At present, we have only indirect information about the early universe, such as the observed
reionization optical depth, the Gunn-Peterson troughs in quasars beyond z = 6 and the cosmic backgrounds, which can be used to constrain models of the early universe. However, we are currently
approaching an era in which the early universe can be directly observed. This is possible for instance with 21 cm observations at high redshift with LOFAR 1 , which are sensitive to the thermal gas
properties on large scales. It may therefore observe the HII regions of the first quasars and the first
galaxies, but also detected heating effects due to dark matter annihilation and decay. This possibility
is discussed in more detail in Chapter 3.
The Planck satellite 2 , which is planned to launch in April this year (2009), will measure the CMB
with unprecedented accuracy, thus reducing the uncertainties regarding the cosmological parameters
1
2
http://www.lofar.org/
http://www.rssd.esa.int/index.php?project=planck
17
1. INTRODUCTION
Figure 1.5: An artist’s conception of the completed Atacama Large Millimeter/submillimeter Array
(ALMA). One single antenna has a size of 12 m. This telescope may observe molecular and fine structure
emission from the first high-redshift quasars. CREDIT: ALMA/ESO/NRAO/NAOJ.
as well as the reionization optical depth. This may help to tighten the constraints presented in this
work even further.
In about five years, the James Webb Space Telescope (JWST) will search for the first galaxies
during the reionization epoch. At the same time, the Atacama Large Millimeter/submillimeter Array
(ALMA) should be completed. As discussed in Chapter 7, the combination of these telescopes provides ideal conditions to explore the origin of the first quasars in the universe.
18
2
Reionization - A probe for the stellar
population and the physics of the early
universe
In this chapter, I discuss the possibility to constrain the stellar population and the physics of the early
universe with the reionization optical depth as observed by WMAP [Komatsu et al., 2008]. Particular
emphasis is on primordial magnetic fields, dark matter annihilation and decay. Stellar populations
consisting of very massive stars [Abel et al., 2002; Bromm & Larson, 2004], typical Pop. II stars
[Scalo, 1998; Kroupa, 2002; Chabrier, 2003] and dark stars powered by the annihilation of dark matter
are considered. The work presented was published by Schleicher, Banerjee, & Klessen [2008b].
The following chapters then discuss further constraints on different scenarios in more detail. In
particular, Chapter 3 examines the possibility to constrain or detect primordial magnetic fields with
21 cm observations, Chapter 5 considers further models for dark stars and the constraints from reionization and the cosmic backgrounds, and s-wave annihilation of light dark matter is ruled out in
Chapter 6.
2.1 Introduction
Over the last decades, our understanding of cosmological reionization has improved considerably.
Observations of high-redshift quasars clearly indicate that reionization must end around z ∼ 6 [Becker
et al., 2001], and the WMAP 5-year-measurement finds a reionization optical depth of τ = 0.087 ±
0.017 [Komatsu et al., 2008; Nolta et al., 2009], yielding clear evidence that reionization started significantly earlier and is thus a continuous process. Reionization through miniquasars would produce
a soft-X-ray background that is significantly higher than the observed background, and can thus be
ruled out [Dijkstra et al., 2004; Salvaterra et al., 2005]. Simulations of stellar reionization further
19
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
show that the process is highly inhomogeneous and based on the growth and merging of ionized bubbles around the first stellar sources [Gnedin & Hui, 1998; Gnedin, 2000; Ciardi et al., 2003; Mellema
et al., 2006]. Understanding the essential physical processes, it is possible to build semi-analytic
models that describe stellar reionization, which can be tested over a wide parameter space [Shapiro &
Giroux, 1987; Haiman & Loeb, 1997a; Barkana & Loeb, 2001; Loeb & Barkana, 2001; Choudhury
& Ferrara, 2005; Schneider et al., 2006]. Additional constraints on the cosmic star formation rate are
now available from gamma-ray burst observations [Yüksel et al., 2008]. The reionization framework
thus provides an increasingly reliable test for the stellar population during reionization and can be
used to constrain global physical conditions in the early universe.
Regarding the stellar population, Abel et al. [2002] and Bromm & Larson [2004] suggested that
the first stars were top-heavy with a peak in the IMF at around 100 M' . On the contrary, Clark
et al. [2008] and Omukai et al. [2008] indicate that gas especially in more massive systems can fragment because of dips in the equation of state, which may lead to the formation of a stellar cluster. It
was shown that cooling in previously ionized gas is enhanced and leads to typical stellar masses of
∼ 10 M' [Yoshida et al., 2007a,b]. It was further suggested that the presence of weak magnetic fields
is sufficient to lead to a more present-day like mode of star formation, resulting in considerably lower
masses [Silk & Langer, 2006]. Recently, a new phase of stellar evolution was suggested in which the
stars would be powered by dark matter annihilation instead of nuclear fusion [Spolyar et al., 2008].
Various follow-up works have explored such a scenarios in more detail. Studies by Iocco [2008];
Freese et al. [2008b] explored the main-sequence and pre-main-sequence phase of dark stars, and
other works calculated the effect of dark matter capture by off-scattering from baryons [Iocco et al.,
2008; Yoon et al., 2008; Freese et al., 2008a,c; Taoso et al., 2008]. While many predictions are still
model-dependent, it has often been suggested that such stars may have typical masses of 800 M' ,
giving rise to a very bright main-sequence phase, or may have much longer lifetimes due to modifications in the stellar evolution. In the end, all these suggestions must face the constraint that the stellar
population must be able to provide the correct reionization optical depth.
Complications may arise through the presence of additional physics that are typically not considered in standard reionization calculations and simulations on the first stars. Such possibilities include
the presence of primordial magnetic fields, dark matter decay and dark matter annihilation. Magnetic
fields have been observed on all scales in the universe, and recently, it has been demonstrated that
they were present already in high-redshift galaxies [Bernet et al., 2008]. They are found in the interstellar gas as well as in the intergalactic medium [Beck et al., 1996; Beck, 2001; Carilli et al., 2002],
but their origin is still unclear. There is a viable possibility that these fields have a primordial origin
[Grasso & Rubinstein, 2001; Widrow, 2002]. So far, the most stringent constraints on the strength
of these putative fields come from the measurements of the cosmic microwave background radiation
20
2.1 Introduction
(CMBR) and from big-bang nucleosynthesis (BBN) calculations. A homogeneous magnetic field
would produce temperature anisotropies in the CMBR [Zeldovich & Novikov, 1983] whose maximal
amplitudes are limited by the COBE satellite measurements which in turn limit the field strength,
as measured today, to B0 <∼ 3.5 × 10−9 G [Barrow et al., 1997]. The presence of any primordial field
would also alter the CMBR power spectrum by changing the characteristic velocities. With the sensitivity of the PLANCK satellite 1 one should be able to detect fields with present day strength of
B0 > 5 × 10−8 G [Adams et al., 1996]. So far, measurements by the WMAP satellite 2 are compatible
with the absence of primordial magnetic fields.
Strong magnetic fields in the early universe can also change the abundance of relic4 He and other
light elements during the big bang nucleosynthesis [Matese & O’Connell, 1969; Greenstein, 1969].
To comply with observational limits on light element abundances these primordial fields must not
exceed 1012 G at the time when the universe was T = 5 × 109 K which corresponds to a present day
field B0 <∼ 3 × 10−7 G [Greenstein, 1969].
Effects of primordial magnetic fields have already been considered by Kim et al. [1996], finding that density perturbations can be enhanced by the Lorentz force from tangled magnetic fields.
Both the evolution of perturbations in the presence of magnetic fields as well as their effect on the
thermodynamics via ambipolar diffusion heating and decaying MHD turbulence was considered by
Sethi & Subramanian [2005]. Recent calculations of Tashiro & Sugiyama [2006] show that the enhancement of structure due to magnetic fields is pronounced at about 5 × 106 M' , but becomes less
effective on larger mass scales and appears as a subdominant contribution on the scale of the magnetic
Jeans mass. Consequences for 21 cm observations have been explored as well [Tashiro et al., 2006;
Schleicher et al., 2009a]. In this work, we examine the consequences for reionization in more detail
and calculate the backreaction on structure formation according to the work of [Gnedin, 2000]. The
WMAP 5 year data [Nolta et al., 2009] have measured the Thomson scattering optical depth from
reionization and allow to constrain different reionization scenarios in the early universe.
The nature of dark matter is still unclear, and consequences of various particle physics like massive neutrinos or axion decay have been explored early [e. g.Doroshkevich et al., 1989; Berezhiani
et al., 1990]. Observational progress allowed to refine these studies and to explore such scenarios in the framework of ΛCDM cosmology [Chen & Kamionkowski, 2004; Hansen & Haiman,
2004; Avelino & Barbosa, 2004; Kasuya & Kawasaki, 2004; Pierpaoli, 2004; Bean et al., 2003;
Padmanabhan & Finkbeiner, 2005]. Recently, the possibility was discussed to detect such effects
using future 21 cm telescopes [Furlanetto et al., 2006b], and it was shown that the fraction of the
energy absorbed into the IGM can be calculated in detail for specific models of dark matter decay
and annihilation [Ripamonti et al., 2007a]. Indeed, secondary ionization through the decay / annihilation products can provide a way to ionize the IGM which is independent of the stellar contribution,
whereas the additional heat input increases the Jeans mass and delays the formation of the first structures and affects the chemical initial conditions [Ripamonti et al., 2007b].
1
2
http://sci.esa.int/science-e/www/area/index.cfm?fareaid=17
http://map.gsfc.nasa.gov/
21
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
In this work, we show how reionization constrains the properties of the stellar population as well
as some additional heat sources like primordial magnetic fields, dark matter annihilation and decay. In
Chapter 2.2, we present our model for stellar reionization, which considers the IGM as a two-phasemedium consisting of ionized bubbles and overall neutral gas. Based on the thermal evolution, we
self-consistently determine the minimal mass scale of halos which can collapse. In Chapter2.3, we
explain our treatment of primordial magnetic fields and show how they modify the thermal evolution.
The treatment of dark matter annihilation and decay, as well as some implications for dark stars, are
discussed in Chapter 2.4. The optical depth for different models is given in Chapter 2.5. Further
discussion and outlook is given in Chapter 2.6.
2.2 Reionization in the early universe
The thermal and ionization history of the IGM between recombination and the end of reionization is
determined by a number of different processes. At redshifts z > 300, Compton scattering of CMB
photons couples the gas temperature T to the CMB temperature Trad . At lower redshifts, this coupling
is less efficient and the gas decouples from the CMB due to adiabatic expansion. This standard
scenario can be altered if additional energy injection mechanisms are present. Additional heat input
can lead to an earlier redshift of decoupling, whereas an increase in the ionized fraction, for instance
due to secondary ionization from the decay or annihilation products of dark matter, tends to make
Compton cooling more efficient. Once structure formation sets in, the thermal and ionization history
is further influenced from X-rays produced in star forming regions and UV photons that escape from
the first galaxies. We modified the RECFAST code [Seager et al., 1999, 2000] and included all these
feedback processes as described below, modeling the IGM as a two-phase medium of ionized and
partially-ionized gas. In this picture, the fully-ionized gas refers to the gas in the HII regions of the
first luminous sources, while the partially ionized gas describes gas that is not yet affected by UV
feedback.
2.2.1 The RECFAST code
In the absence of additional energy injection mechanisms, recombination in the early universe and
freeze-out of electrons was calculated with unprecedent accuracy, following the detailed level populations of hundreds of energy levels for H, He and He+ and self-consistently calculating the radiation field [Seager et al., 2000]. They developed the RECFAST code 1 , a simplified version of the
multi-level calculations, based on an effective three-level model for the hydrogen atom. RECFAST is
capable of fully reproducing the results of the more detailed calculation [Seager et al., 1999]. Both
the detailed calculation and the RECFAST code were recently updated and improved byWong et al.
1
http://www.astro.ubc.ca/people/scott/recfast.html
22
2.2 Reionization in the early universe
[2008]. For the partially-ionized gas, we modify the equations describing the thermal and ionization
history in the following way: The equation for the temperature evolution is given as
dT
dz
=
+
4
8σT aR T rad
xe (T − T rad )
3H(z)(1 + z)me c 1 + fHe + xe
2T
2(Lheat − Lcool )
−
,
1 + z 3nkB H(z)(1 + z)
(2.1)
where Lheat is the new heating term (see Chapter 2.3.1, 2.3.2 and 2.2.3), Lcool the new cooling term
including Lyman α cooling, bremsstrahlung cooling and recombination cooling using cooling functions of Anninos et al. [1997], σT is the Thomson scattering cross section, aR the Stefan-Boltzmann
radiation constant, me the electron mass, c the speed of light, kB Boltzmann’s constant, n the total
number density, xe = ne /nH the electron fraction per hydrogen atom, H(z) is the Hubble factor and
fHe is the number ratio of He and H nuclei, which can be obtained as fHe = Y p /4(1 − Y p ) from the
mass fraction Yp of He with respect to the total baryonic mass. The evolution of the ionized fraction
of hydrogen, xp , is given as
dxp
dz
[C(z)xe xp nH αH − βH (1 − xp )e−h p νH,2s /kT ]
H(z)(1 + z)[1 + KH (ΛH + βH )nH (1 − xp )]
kionnH xp
− fion .
× [1 + KH ΛH nH (1 − xp )] −
H(z)(1 + z)
=
(2.2)
In this equation, nH is the number density of hydrogen atoms and ions, hp Planck’s constant, kion is
the collisional-ionization coefficient [Abel et al., 1997], fion describes ionization from X-rays (see
Chapter 2.2.3), and the parametrized case B recombination coefficient for atomic hydrogen αH is
given by
atb
αH = F × 10−13
cm3 s−1
(2.3)
1 + ctd
with a = 4.309, b = −0.6166, c = 0.6703, d = 0.5300 and t = T/104 K, which is a fit given
by Pequignot et al. [1991] to the coefficient of Hummer [1994]. This coefficient takes into account
that direct recombination into the ground state does not lead to a net increase of neutral hydrogen
atoms, since the photon emitted in the recombination process can ionize other hydrogen atoms in the
neighbourhood. The fudge factor F = 1.14 serves to speed up recombination and is determined from
comparison with the multilevel-code. We further introduce the clumping factor C(z) ≡ "n2e #/"ne #2
to take into account the increase in the recombination rate in structures of increased density at low
redshifts. We use the fit formula
C(z) = 27.466exp(−0.114z + 0.001328z2 )
(2.4)
obtained from simulations of Mellema et al. [2006] at redshifts z < 40 and set C(z) = 1 at higher
redshifts.
The photoionization coefficient βH is calculated from detailed balance at high redshifts as described by Seager et al. [1999, 2000]. At lower redshifts, however, radiative recombination is no
23
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
longer balanced by photoionization in the presence of additional energy injection mechanisms like
ambipolar diffusion heating. Once the ionized fraction drops below 98%, we thus calculate the photoionization coefficient directly from the photoionization cross section given bySasaki & Takahara
[1993]. The frequency νH,2p corresponds to the Lyman-α transition from the 2p state to the 1s state
of the hydrogen atom. The two-photon transition between the states 2s and 1s is close to Lymanα. Consequently we use the same frequency for both processes. Finally, ΛH = 8.22458 s−1 is the
two-photon rate for the transition 2s-1s according to Goldman [1989] and KH ≡ λ3H,2p /[8πH(z)] the
cosmological redshifting of Lyman α photons. The additional terms for Eq. (2.1) and (2.3) that are
needed to describe the effects of dark matter annihilation and decay will be discussed in Chapter2.4.
The fully-ionized component in the HII regions is described with the volume filling factor QH+
that denotes the volume fraction of ionized hydrogen bubbles. It is needed to compute the reionization
optical depth and takes UV feedback into account. It evolves as
QH+ C(z)ne,H+ αA dn ph /dz
dQH+
=
+
dz
H(z)(1 + z)
nH
(2.5)
[Shapiro & Kang, 1987; Haiman & Loeb, 1997a; Barkana & Loeb, 2001; Loeb & Barkana, 2001;
Choudhury & Ferrara, 2005; Schneider et al., 2006]. The UV photon production rate dnph /dz will be
described in Chapter 2.2.3 in more detail, ne,H+ denotes the mean electron number density in fullyionized regions, nH the neutral hydrogen density in regions that are still unaffected from UV photons
and αA = 4.2 × 10−13 [T max /104 K]−0.7 cm3 /s is the case A recombination coefficient [Osterbrock,
1989] which we have chosen here, as recombinations will preferably occur in dense regions where
the recombination photons are unlikely to escape into the IGM [Miralda-Escudé, 2003]. It is evaluated
at the temperature with Tmax = max(104 K, T ) to account for the fact that the ionized regions should
be heated at least to 104 K. If heating via ambipolar diffusion and decaying MHD turbulence increased
the gas temperature above this threshold, it is evaluated at the gas temperature instead. To compare the
models with the observational constraints from WMAP, we calculate the Thomson scattering optical
depth of free electrons, given as
nH,0 c
τT =
H0
%
0
zre
(1 + z)2
xe f f σT &
dz,
ΩΛ + Ωm (1 + z)3
(2.6)
where nH,0 is the comoving number density of ionized and neutral hydrogen, ΩΛ and Ωm the usual
cosmological density parameters, H0 the Hubble constant and zre is the redshift where reionization
starts, which we define as the point where the effective ionized fraction xe f f = QH+ + (1 − QH+ )xp
becomes larger than 6.5%. This point has been chosen after a comparison of the actual TE cross
calibration spectra with those from reionization models with fixed optical depth, but the results are
not sensitive to this choice. In summary, we adopt the following picture:
• The partially-ionized gas unaffected by UV feedback is modeled with Eqs (2.1), (2.3) that
describe the evolution of its temperature and ionization degree.
24
2.2 Reionization in the early universe
• The fully-ionized gas is described by the volume-filling factor QH+ calculated from Eq. (2.5).
Its temperature is given as the maximum of 104 K and the temperature of the partially-ionized
gas.
• Recombination both in the partially- and fully-ionized gas is enhanced by the clumping factor
given in Eq. (2.4).
• For the partially-ionized component, recombination is described using the case B recombination coefficient, appropriate for low-density gas far from the ionizing sources.
• For the fully-ionized gas, we use the case A recombination coefficient, assuming that most
recombinations take place in high-density regions near the ionizing sources.
The implementation of different feedback mechanisms is described below in more detail.
2.2.2 The generalized filtering mass
The universe becomes reionized due to stellar feedback. We assume here that the star formation rate
(SFR) is proportional to the change in the fraction of collapsed halos fcoll . As shown by Press &
Schechter [1974], fcoll is given as
'
(
δc (z)
fcoll = erfc √
,
(2.7)
2σ(mmin )
where
σ2 (m) =
%
0
∞
'
(2
dk 2
3 j1 (kR)
k
P
(k)
,
lin
kR
2π2
(2.8)
and where mmin is the minimum mass of haloes that are able to collapse at a given redshift. Here,
j1 (x) = (sin x − x cos x)/x2 is the spherical Bessel function, R is related to the halo mass Mh via
Mh = 4πρR3 /3, ρ is the mean density, δc = 1.69/D(z) the linearized density threshold for collapse
in the spherical top-hat model and D(z) the linear growth factor. In the absence of magnetic field,
mmin is determined by the filtering mass [Gnedin & Hui, 1998; Gnedin, 2000]. As discussed in
the introduction, tangled magnetic fields can potentially create more small scale structure via the
Lorentz force. The calculations of Tashiro & Sugiyama [2006] show that this effect is pronounced
in the minihalo regime. While they adopted a constant minimal collapse mass of 106 h−1 M' which
is independent of the magnetic field, we use the framework of Gnedin [2000] to take into account
the change in the mass scale of halos that can form stars. Indeed, we find that the mass scale is
changed significantly (see Fig. 2.1(d)), such that the additional small-scale structure from tangled
magnetic fields does not contribute to star formation. We introduce the magnetic Jeans mass, [Sethi
& Subramanian, 2005]
#
$3
B0
B
10
M J ∼ 10 M'
,
(2.9)
3 × 10−9 G
25
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
and the thermal Jeans mass,
M J = 2M'
!
cs
0.2 km/s
"3 #
$−1/2
n
.
103 cm−3
(2.10)
The filtering mass in the presence of magnetic fields is then given as
2/3
MF,B
3
=
a
%
0
a



da( Mg2/3(a( ) 1
!
a(
−
a
"1/2 

 ,
(2.11)
where a = 1/(1+z) is the scale factor and Mg = max(MJ , M JB). The minimum halo mass to consider is
given as mmin = max(MF,B , mcool ), where mcool denotes the minimal halo mass for which the baryons
can efficiently cool after collapse. We adopt here the fiducial value mcool = 105 M' [Greif et al.,
2008]. Similar estimates have been given by Yoshida et al. [2003], Barkana & Loeb [2001] and
Mackey et al. [2003]. For the subsequent analysis, we furthermore assume that only a certain fraction
f∗ of the collapsing halo mass turns into stars.
2.2.3 Stellar feedback
As X-ray photons have long mean free paths, they can play an important role in the ionization and
heating of the gas. Assuming that the local correlation between the SFR and the X-ray luminosity
(from 0.2− 10 keV) holds up to a renormalization factor fX [see Furlanetto et al., 2006a; Grimm et al.,
2003; Ranalli et al., 2003; Gilfanov et al., 2004; Glover & Brand, 2003], the X-ray heating function
LX is given as
!
"
SFR
LX = 3.4 × 1040 fX
erg s−1 .
(2.12)
1 M' yr−1
X-ray emission has two major sources, inverse-Compton scattering of CMB photons with relativistic
electrons accelerated in supernovae, and high-mass X-ray binaries. The former may play an increasingly important role at high redshifts, as the CMB photons are more energetic at high redshifts [Oh,
2001]. In the early universe the factor fX ∼ 0.5 if 1051 erg are released per 100 M' , corresponding
to an overall efficiency of 5% [Koyama et al., 1995]. The abundance of X-ray binaries depends on
metallicity and the stellar initial mass function, and could be especially large if very massive Pop. III
stars dominate [Glover & Brand, 2003]. In the model presented here, we adopt fX ∼ 0.5 as a generic
value and assume that the uncertainty can be ascribed to the star formation efficiency f∗ . In general,
X-rays lose their energy through three channels. A fraction fX,h goes into heating, a fraction fX,ion
into ionization, and a fraction fX,coll into excitation. These parameters should not be confused with
fion introduced in Eq. (2.3) and fcoll introduced in Eq. (2.7). We calculate them using the fit formulae
of Shull & van Steenberg [1985]. We can thus write
!
"
2LX
f∗ fX,h d fcoll /dz 1 + z
3
= 10 K fX
H(z).
(2.13)
3kB nH
0.1 0.2 0.01
10
26
2.2 Reionization in the early universe
The contribution from the X-rays to the term fion in Eq. (2.3) is then given as
fion ∼
fX,ion
LX
.
fX,h 13.6 eV H(z)(1 + z)
(2.14)
UV photons from stellar sources are in general absorbed locally, only a fraction fesc that manages
to escape from the first galaxies can contribute to the ionization of the IGM. Again, the production
of ionizing photons is associated with the star formation rate [Furlanetto et al., 2006a], yielding a
contribution
dn ph /dz
d fcoll
∼ξ
,
(2.15)
nH
dz
where
ξ = AHe f∗ fesc Nion ,
(2.16)
with AHe = 4/(4 − 3Y p ) = 1.22 and Nion is the number of ionizing photons per stellar baryon.
2.2.4 Models for the stellar population
The nature of the first stellar sources is still under discussion, and even the question whether the primordial IMF is top-heavy or closer to the locally measured IMF is not solved. While for instanceAbel
et al. [2002] and Bromm & Larson [2004] found a top-heavy IMF using adaptive-mesh refinement
(AMR) or smoothed-particle hydrodynamics (SPH) simulations, it was shown byClark et al. [2008]
that primordial and low-metallicity gas can fragment if the evolution of the gas is followed further after the formation of the first clump, due to a dip in the equation of state. Based on similar arguments,
Omukai et al. [2008] argued that even a small metallicity fraction can lead to fragmentation in the
first protogalaxies. It was further suggested that magnetic fields can have a crucial influence on the
primordial IMF [Silk & Langer, 2006]. Given these uncertainties, we discuss different models for the
stellar population. Dark stars have also been suggested as some of the first luminous sources. We will
discuss this possibility in more detail in Chapter 2.4.3.
In models A and B, we assume that reionization is solely due to metal-free massive Pop III stars.
This situation can only be examined from a theoretical point of view. Most investigations of UV
feedback from high-mass zero-metallicity stars indicate that ionizing photons can easily escape the
star-forming halo and drive large HII regions into the IGM. This suggests that the escape fraction in
high-redshift galaxies could be very high, of order unity, being much higher than in the present-day
universe [for instance Dove et al., 2000; Ciardi et al., 2002; Fujita et al., 2003]. Detailed numerical
simulations of Whalen et al. [2004] show that indeed the shock bounding HII regions of massive Pop.
III stars can easily photo-evaporate the minihalo and lead to an escape fraction of one. However, the
situation is not fully clear. There are other studies that suggest that the escape fraction could remain
small [Wood & Loeb, 2000], depending on the mass of the collapsing halo. This is why we adopt two
different models. Model A is an extreme case, in which we assume that the escape fraction is 100%
independent of the halo mass. The more realistic case is probably model B, where we distinguish
whether the virial temperature corresponding to the generalized filtering mass is smaller or larger
27
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
model
population
Nion
A
B
III
III
40,000
40,000
C
III/II
10,000
D
II
4,000
fesc
1
1 (Tvir
0.1 (Tvir
1 (Tvir
0.1 (Tvir
0.06
< 104 K)
≥ 104 K)
< 104 K)
≥ 104 K)
Table 2.1: Summary of adopted stellar models. The first column gives the model name. The second
column indicates the stellar populations that contribute to reionization. The third column gives the number
of ionizing photons per baryon used in Eq ( 2.16). The fourth column lists the adopted escape fractions.
Model A is a highly extreme case in which we assume all ionizing photons come from massive Pop III
stars and can escape the star-forming halo. In model B photons can escape efficiently only from halos
less massive than a corresponding virial temperature of 10 4 K, while for higher-mass halos, this fraction
is reduced to 10%. In model C we assume that Pop III as well as Pop II stars contribute to reionization,
and we adopt the same escape probabilities as in B. Model D is another extreme case which assumes that
all ionizing photons come from low-mass Pop II stars, with escape fraction 6%.
than 104 K. If it is smaller, we still assume that the halo is easily photoevaporated. If it is larger, we
assume that most of the mass is kept within the halo and adopt an escape fraction fesc = 0.1. The virial
/ 03/2
10
mass corresponding to 104 K is given as Mc = 5 × 107 M' 1+z
[Oh & Haiman, 2002; Greif et al.,
2008]. For model A and B, we adopt a total number of ionizing photons per baryon of Nion = 40, 000,
following Bromm et al. [2001].
The heavy elements produced by the very first stars will gradually enrich the IGM. It is very likely
that some contribution to reionization comes from low-metallicity Pop II stars as well. As these stars
are expected to have lower masses than Pop III stars, we introduce model C, which corresponds to
a stellar population of intermediate mass. It has Nion = 10, 000 and an escape fraction according to
model B, suggesting that such stars may photo-evaporate only low-mass halos.
Finally, we consider in model D the extreme case of very rapid chemical enrichment and assume
cosmic reionization is entirely driven by low-mass Pop II stars. We assume a stellar mass distribution
similar to the local IMF in the Galactic halo [Scalo, 1998; Kroupa, 2002; Chabrier, 2003] with a
metallicity of 1/20 of the solar value. This corresponds to Nion ∼ 4, 000. We furthermore adopt escape fractions that are typical for local star forming galaxies. Many upper limits and a few detections
have been found observationally [Hurwitz et al., 1997; Heckman et al., 2001; Deharveng et al., 2001;
Bland-Hawthorn & Maloney, 1999, 2001], suggesting fesc ∼ 0.06. A detection from Lyman-break
galaxies at z ∼ 3 implied a higher fraction of 10% [Steidel et al., 2001], while more recent observations place upper limits of 5 − 10% or claim detections at even lower levels [Giallongo et al., 2002;
Malkan et al., 2003; Fernández-Soto et al., 2003; Inoue et al., 2005]. As the relevant rates scale only
28
2.3 The effect of magnetic fields on the IGM
linear in fesc , it seems reasonable to adopt fesc = 0.06 for our Pop II model.
As it is unlikely that cosmic reionization occurs instantaneously with the onset of Pop III star
formation nor that chemical enrichment is so rapid that all ionizing photons come from metal-enriched
Pop II stars, we adopt the intermediate scenario C with a mixed population as our fiducial model. All
global cosmological parameters are chosen according to the WMAP 5 year data [Komatsu et al.,
2008].
2.3 The effect of magnetic fields on the IGM
As discussed by Sethi & Subramanian [2005]; Sethi et al. [2008], magnetic fields can significantly
alter the thermal evolution of the IGM via ambipolar diffusion heating and decaying MHD turbulence,
which can have a significant influence on the filtering mass scale. We explain our treatment of these
heating terms in this section.
2.3.1 Ambipolar diffusion heating
The presence of magnetic fields introduces two different contributions to the heating rate, one coming
from ambipolar diffusion and one resulting from the decay of MHD turbulence.
In the first case, the contribution can be calculated as [Sethi & Subramanian, 2005; Cowling, 1956]:
11/
12
0
ρn
11 ∇ × B
. ×B
. 111 .
Lambi =
(2.17)
16π2 γρ2b ρi
Here, ρn , ρi and ρb are the mass densities of neutral hydrogen, ionized hydrogen and all baryons. The
ion-neutral coupling coefficient for primordial gas is given as [Draine, 1980; Shang et al., 2002]
γ=
1
+
2 nH "σv#H ,H
+ 45 nHe "σv#H+ ,He
,
mH [nH + 4nHe ]
(2.18)
where nHe is the number density of He and mH the mass of the hydrogen atom. Collisions with
electrons are neglected here, as their contribution is suppressed by a factor me /mH . We adopt the zero
drift velocity momentum transfer coefficients of Pinto & Galli [2008] for collisions of H+ with H and
He , which is a good approximation in the absence of shocks. They are given by
"σv#H+ ,H = 0.649T 0.375 × 10−9 cm3 s−1 ,
2
"σv#H+ ,He = 1.424 + 7.438 × 10−6 T
3
− 6.734 × 10−9 T 2 × 10−9 cm3 s−1 .
(2.19)
(2.20)
As the power spectrum for the magnetic field is unknown and eq. (2.17) cannot be solved exactly,
we adopt a simple and intuitive approach to estimate the integral for a given average magnetic field B
with coherence length L. The heating rate can then be evaluated as
Lambi ∼
ρn
B4
.
16π2 γρ2b ρi L2
29
(2.21)
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
The coherence length L is in principle a free parameter that depends on the generation mechanism of
the magnetic field. It is constrained through the fact that tangled magnetic fields are strongly damped
by radiative viscosity in the pre-recombination universe on scales smaller than the Alfvén damping
−1 given by [Jedamzik et al., 1998; Subramanian & Barrow, 1998; Seshadri & Subramanian,
scale kmax
2001]
!
"1/4
B0 $−1 Ωm
∼ 234 Mpc
0.3
10−9 G
!
!
"
"
1/2
1/4
h
Ωb h2
×
,
0.02
0.7
−1
kmax
#
(2.22)
where B0 = B/(1 + z)2 denotes the comoving magnetic field. In fact, we expect fluctuations to
be present on all scales. As the heating term goes as L−2 , we thus adopt the minimal value L =
−1 /(1 + z).
kmax
2.3.2 Decaying MHD turbulence
For decaying MHD turbulence, we adopt the prescription ofSethi & Subramanian [2005],
Ldecay =
B0 (t)2 3α̃
[ln(1 + td /ti )]α̃ H(t)
,
8π 2 [ln(1 + td /ti ) + ln(t/ti )]α̃+1
(2.23)
where t is the cosmological time at redshift z, td is the dynamical timescale, ti the time where decay
starts, i. e. after the recombination epoch when velocity perturbations are no longer damped by
the large radiative viscosity, zi is the corresponding redshift. For a power spectrum of the magnetic
field strength with power-law index α, the parameter α̃ is given as α̃ = 2(α + 3)/(α + 5) [Olesen,
1997; Shiromizu, 1998; Christensson et al., 2001; Banerjee & Jedamzik, 2003]. In the generic case,
we expect the power spectrum of the magnetic field to have a maximum at the scale of the coherence
length, and the heat input by MHD decay should be determined from the positive slope corresponding
to larger scales [Müller & Biskamp, 2000; Christensson et al., 2001; Banerjee & Jedamzik, 2003,
2004]. We thus adopt α = 3 for the calculation. We estimate the dynamical timescale as td = L/vA ,
&
where vA = B/ 4πρb is the Alvén velocity and ρb the baryon mass density. The evolution of the
magnetic field as a function of redshift can be determined from the magnetic field energy EB = B2 /8π,
which evolves as [Sethi & Subramanian, 2005]
dE B
= −4H(t)E B − Lambi − Ldecay .
dt
(2.24)
2.3.3 The evolution of the IGM
While the redshift of reionization of course depends on the model for the stellar population, the mechanism which delays reionization in the presence of magnetic fields is always the same. We illustrate
this for our fiducal model C adopting a star formation efficiency f∗ of 1%, but point out that the general
30
2.3 The effect of magnetic fields on the IGM
(a)
(b)
(c)
(d)
Figure 2.1: Evolution of different quantities as a function of redshift. a) The effective gas temperature. b)
The effective ionized fraction. c) The filtering mass. d) The comoving magnetic field strength.
31
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
discussion is valid also for the other stellar population models. We use the cosmological parameters
of WMAP 5 [Komatsu et al., 2008]. Fig. 2.1(a) shows the thermal evolution of the IGM for different
magnetic field strengths. In the absence of magnetic fields, the gas temperature follows the temperature of radiation until z ∼ 200, where Compton scattering becomes inefficient. Afterwards, the gas
cools adiabatically until it is reheated during reionization. In the presence of magnetic fields, gas and
radiation decouple earlier due to the ambipolar diffusion heating and decaying MHD turbulence and
stays at higher temperatures, which prevents collapse in low-mass halos and thus delays reionization.
The additional heat increases the ionized fraction at early times (see Fig. 2.1(b)), while the redshift
where the IGM becomes fully ionized is delayed, due to the increased generalized filtering mass (see
Fig. 2.1(c)). Collisional ionization introduces a natural temperature plateau of the order ∼ 10000 K,
as any further temperature increase will have a backreaction on the ionized fraction of the gas, and
thus make ambipolar diffusion less efficient.
2.4 Implications from the dark sector
A large variety of different particle physics models has been suggested to explain the dark matter
content of the universe [Olive, 2008]. Many of these models predict some interactions in the dark
sector and include some form of dark matter decay and annihilation. The consequences of such
scenarios for the thermal evolution have been discussed in detail for instance byRipamonti et al.
[2007a,b]; Furlanetto et al. [2006b]. In this work, we discuss the implications of such scenarios on
reionization. As we have seen for the case of primordial magnetic fields, the additional heat input
increases the filtering mass and thus increases the minimal halo mass in which the first luminous
objects can form, making stellar reionization less effective. On the other hand, once heating through
dark matter decay or annihilation is effective, secondary ionization will likely be effective as well and
thus increase the Thomson scattering optical depth. As a consequence of dark matter annihilation
models, a new phase of stellar evolution was suggested by Spolyar et al. [2008]; Iocco [2008], in
which stars are not powered by nuclear fusion, but by annihilating dark matter within them. These
stars could reach masses of up to 103 M' [Freese et al., 2008a]. Once dark matter annihilation
becomes ineffective and such stars enter the main sequence phase, they would thus be extremely
bright sources of UV photons. The consequences of such a phase are discussed below. Further
models taking into account a dark matter capturing phase have been suggested as well [Iocco et al.,
2008; Yoon et al., 2008; Freese et al., 2008c,b]. As they are highly parameter-dependent, we study
them in a separate work [Schleicher et al., 2008a].
2.4.1 Dark Matter annihilation
The energy of the annihilation products can be deposited into heating, ionization and collisional
excitation. The latter is quickly radiated away, but may contribute to the build-up of a Lyman α
background that helps to couple the spin temperature of hydrogen to the gas temperature, which may
32
2.4 Implications from the dark sector
be relevant for 21 cm observations [Furlanetto et al., 2006b]. For given models of dark matter, it
is possible to work out the detailed absorbed fractions as a function of redshift [Ripamonti et al.,
2007a]. In this work, however, we adopt the more generic approach of Furlanetto et al. [2006b].
For definiteness, they assume that the dark matter particles annihilate to high-energy photons, which
allows to determine the energy fractions going into heat and ionization by the fitting formulae ofShull
& van Steenberg [1985]. As they show, heating by dark matter annihilation is accompanied also by
a significant increase in the electron fraction due to secondary ionization. For other models of dark
matter annihilation, the results can be rescaled by appropriate absorption efficiencies. To Eq. (2.1)
describing the temperature evolution, we thus add a term
η2 m p c2
δT
2
=−
ξX χh
δz
3 η1 kB (1 + z)H(z)
(2.25)
according to Furlanetto et al. [2006b], where η1 = 1 + fHe + xi , η2 = 1 + 4 fHe , fHe is the helium
fraction by number, χh is the fraction of energy going into heating [seeShull & van Steenberg, 1985]
and ξX the effective baryon-normalized “lifetime”, given as
!
"
ΩDM ρ0c
3 Ω DM
"σv#(1 + z)
.
(2.26)
ξX =
mDM
Ωb
Here, ΩDM and Ωb denote the cosmological parameters for dark and baryonic matter, ρ0c is the critical
density at redshift zero, mDM is the dark matter particle mass and "σv# the velocity-averaged cross
section. In the same way, we add a term to Eq. (2.3) that describes the evolution of the ionized
fraction:


 m p c2 
δxi
 ξX χi ,
= −η2 
(2.27)
δz
Eion
where Eion = 13.6 eV is the hydrogen ionization threshold and χi the fraction of the energy going into
ionization, for which we use the fitting formulae of [Shull & van Steenberg, 1985]. As the decay rate
scales with the dark matter density squared, it is most efficient at early times and may thus modify the
recombination history, which allows to place upper limits on the dark matter annihilation. As shown
by Zhang et al. [2006], the WMAP 1-year-data yield an upper limit of
#
$
−1 m DM
"σv# ≤ 2.2 × 10−29 cm3 s−1 fabs
,
(2.28)
MeV
where fabs corresponds to the energy fraction actually absorbed into the IGM and is of order 1 for
particle masses in the MeV range [Ripamonti et al., 2007a].
2.4.2 Dark Matter decay
In a similar way, dark matter decay can alter the thermal and ionization history of the universe.
The lifetimes of dark matter particles considered here will be considerably larger than the age of
the universe to ensure that the abundance of dark matter does not change significantly. Still, the
33
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
conversion of particle mass to thermal energy can have an important impact on the thermal evolution
of the universe [Ripamonti et al., 2007a; Furlanetto et al., 2006b]. In this limit, the decay rate is
constant over time and the effect at lower redshifts is more pronounced comprared to the case of dark
matter annihilation. To calculate the heating and ionization rate, the same formalism can be employed
as for dark matter annihilation, but we adopt the baryon-normalized decay rate
ξX =
ΩDM
,
Ωb t X
(2.29)
where tX is the lifetime of the dark matter particle. Based on the modified recombination and reionization histories in the presence of dark matter decay, it was shown that [Zhang et al., 2007]
fχ fabs
≤ 2.4 × 10−25 s−1 ,
tX
(2.30)
where fχ is the fraction of the particle mass-energy released through decay and fabs the fraction of
the released energy deposited into the IGM. If data of large-scale surveys are included, the constraint
can be slightly improved, yielding
fχ fabs
≤ 1.7 × 10−25 s−1 .
tX
(2.31)
2.4.3 Dark stars
Dark stars have been suggested as a possible consequence of dark matter annihilation in the early
universe [Spolyar et al., 2008; Iocco, 2008; Freese et al., 2008a]. If indeed the first stars form in
the central peaks of the dark matter distribution in the early minihaloes, they might be powered by
dark matter annihilation instead of nuclear fusion. Such stars could have up to 103 M' and surface
temperatures in the range of 3000−10000 K, i. e. significantly colder than conventional Pop. III stars.
However, this dark phase will gradually come to an end as the dark matter in the cusp annihilates,
so the star may contract and finally reach a main sequence phase. A linear stability analysis and 1D
simulations including hydrodynamics and radiation indicates the stability of such objects [Gamgami,
2007]. In this case, one can expect a similar number of photons per stellar baryon as for conventional
Pop. III stars, perhaps even higher by a factor of 2 [Bromm et al., 2001]. As it is crucial for these
objects to form on the peak of the dark matter density, we expect them to form only in low-mass halos
with a virial temperature below 104 K, as the so-called atomic cooling halos are very turbulent [Greif
et al., 2008] and are more likely to form a stellar cluster instead of a single massive star [Clark et al.,
2008]. In addition, such massive halos are more likely to have accreted material from previous metal
enrichment [Tornatore et al., 2007], which may cause fragmentation as well [Omukai et al., 2008]. In
previously ionized regions, the thermodynamics of collapse are significantly altered and simulations
generally find lower-mass stars [Yoshida et al., 2007a,b], which might be an important limitation for
dark stars as well.
34
2.5 Constraining the parameter space with WMAP 5
Figure 2.2: The reionization optical depth for Pop. III stars (model A) in the presence of primordial
magnetic fields. The contour lines are equally spaced around τ re = 0.87 with ∆τ = 0.017, corresponding
to the different σ-errors of the measurement. Magnetic fields higher than 20 nG are clearly excluded by
the optical depth alone. The transition between stellar reionization and collisional ionization occurs at
about 5 nG, providing a more stringent limit if metal enrichment is required.
2.5 Constraining the parameter space with WMAP 5
In this section, we show the results for the different stellar populations and in the context of different
additional physics.
2.5.1 Stellar reionization with and without primordial magnetic fields
We first concentrate on the effect of primordial magnetic fields and compare the results of different
models for the stellar population. We run our reionization model for a range of different star formation
efficiencies and primordial magnetic fields and obtain the corresponding optical depth. To constrain
the parameter space, some assumptions need to be made. The first is obvious: The calculated optical depth must agree with the optical depth measured by WMAP 5 at least within 3σ. In addition,
we require that the star formation efficiency may not be unreasonably high. For Pop. III stars with
∼ 100 M' forming in halos of ∼ 106 M' (dark matter), star formation efficiencies of the order 0.1%
are expected. It seems thus reasonable to reject star formation efficiencies higher by one order of
magnitude, i. e. of the order 1%. Further requirements are that reionization must be complete by
redshift 6, and that there should be a stellar contribution to reionization, so that the universe becomes
metal-enriched. Now we discuss the constraints resulting from these criteria.
35
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
Figure 2.3: The reionization optical depth for Pop. III stars (model B) in the presence of primordial
magnetic fields. Magnetic fields higher than 20 nG are clearly excluded by the optical depth alone. The
transition between stellar reionization and collisional ionization occurs at about 5 nG. We can further
exclude magnetic fields between 2 and 5 nG within 3-σ, as too high star formation efficiencies would be
required to obtained the measured optical depth.
Figure 2.4: The reionization optical depth for a mixed population (model C) in the presence of primordial
magnetic fields. Magnetic fields higher than 20 nG are clearly excluded by the optical depth alone. The
transition between stellar reionization and collisional ionization occurs at about 5 nG. In addition, magnetic fields between 0.7 and 5 nG can be excluded, as they would require unreasonably high star formation
efficiencies for the measured optical depth.
36
2.5 Constraining the parameter space with WMAP 5
Figure 2.5: The reionization optical depth for Pop. II stars (model D) in the presence of primordial
magnetic fields. For magnetic fields lower than 5 nG, stellar reionization dominates, but unreasonably
high star formation efficiencies would be required to reconcile the measured optical depth. For stronger
magnetic fields, collisional reionization dominates, which can be excluded if metal-enrichment is imposed.
Magnetic fields larger than 20 nG can be excluded from the optical depth alone.
For all stellar models, we can exclude magnetic fields larger than 20 nG, based on the optical
depth alone (see Fig. 2.2 to 2.5). a transition occurs at about 5 nG: For lower magnetic fields, reionization is still mostly due to stellar radiation, whereas for higher magnetic fields, the IGM is heated
to such high temperatures that collisional ionization becomes very efficient and increases the optical
depth. In this regime, structure formation is strongly suppressed, as the magnetic Jeans mass scales
with B30 and dominates over the thermal Jeans mass. Independently of the stellar model, such a regime
is always found between 5 and 20 nG. Therefore, structure formation is considerably impeded such
that metals do not form and the universe is not fully ionized by redshift 6. We can thus exclude magnetic field strength above 5 nG independent of the stellar model.
If we further require that the star formation efficiency must be lower than 1%, the results become model-dependent. In model A, which is an extreme case where very high escape fractions are
assumed even for very massive halos, no further constraint on the magnetic field is possible on the
3σ level. Still, even in this extreme case, one can reject fields larger than 2.5 nG on the 1σ level.
The other extreme case, model D with low escape fractions and a low number of ionizing photons
per stellar baryon, can be rejected completely, as it always requires unphysically high star formation
efficiencies.
37
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
Figure 2.6: The reionization optical depth for Pop. III stars (model B) in the presence of dark matter
annihilation. At "σv#/m DM ∼ 10−33 cm3 /s/eV, secondary ionization of the annihilation products starts to
dominate over stellar reionization. Values higher than 3 × 10 −33 cm3 /s/eV are ruled out by the WMAP
5-year data.
For more realistic cases, essentially model B and C, we find that a critical magnetic field strength
exists above which a very high star formation efficiency is needed to get into the 3σ interval around
the measured optical depth. For model B, the critical value is 2 nG, for model C, it is at 0.7 nG. As it
seems likely that a transition to less massive Pop. II stars occurs during reionization, model C might
indeed be the most realistic case and provide an upper limit of 0.7 nG.
2.5.2 Dark reionization scenarios
For the case of dark matter annihilation and decay, we will show only the results for the Pop. III
star model B. Regarding the competition between stellar reionization and secondary ionization from
high-energetic annihilation / decay products, this corresponds to a rather conservative choice if one
assumes that stellar reionization should do the main contribution. Such a stellar population could be
interpreted either as conventional Pop. III stars or as so-called dark stars in a main sequence phase.
We find that dark matter annihilation starts to become important for "σv#/mDM ∼ 10−33 cm3 /s/eV
and completely dominates the stellar contribution for higher values (see Fig.2.6). Values higher than
3×10−33 cm3 /s/eV are inconsistent with the WMAP 5-year data. The results for dark matter decay are
given in Fig. 2.7. Decay starts to become important for lifetimes below 3 × 1024 s. However, lifetimes
smaller than 3 × 1023 s are incompatible with WMAP 5. As pointed out recently, constraints on
38
2.6 Conclusions and outlook
Figure 2.7: The reionization optical depth for Pop. III stars (model B) in the presence of dark matter
decay. For lifetimes lower than 3 × 10 24 s, secondary ionization of the annihilation products starts to
dominate over stellar reionization. Lifetimes lower than 3 × 10 23 s are ruled out by the WMAP 5-year
data.
dark matter annihilation can be significantly improved taking into account the dark matter clumping
factor C(z) = "ρ2DM (z)#/"ρDM (z)#2 [Chuzhoy, 2008]. However, this quantity is highly uncertain and
may vary by six orders of magnitude [Cumberbatch et al., 2008]. In the framework of light dark
matter, we recently showed that is is constrained to be smaller than 105 at redshift zero [Schleicher
et al., 2009b], yielding an enhancement of the annihilation rate by a factor of 10 or more at redshift
20. Other dark matter models even yield a clumping factor that drops down to 1 at these redshifts,
such that our calculation altogether provides a conservative and firm upper limit [Cumberbatch et al.,
2008].
From Fig. 2.6, we can further see that very massive dark stars in a main sequence phase, which
should have higher star formation efficienes that are an order of magnitude higher than conventional
Pop. III stars, are ruled out by WMAP 5, as the models require rather low star formation efficiencies
to reproduce the Thomson scattering optical depth. Of course, this conclusion holds only if one
assumes that reionization is completely due to dark stars, which seems rather unlikely (see discussion
in Chapter 2.4.3), and we will discuss some alternatives in Chapter 2.6.
2.6 Conclusions and outlook
We have calculated the reionization optical depth for different stellar models in the presence of primordial magnetic fields as well as dark matter annihilation and decay. The results indicate which
39
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
star formation efficiencies are required in the presence of some additional heating mechanism for a
given stellar model. Considering different stellar models and primordial magnetic fields, we find the
following results:
1. Independent of the model for the stellar population, we can securely exclude primordial magnetic fields larger than 5 nG.
2. For the most realistic case with a mixed stellar population (model C), we even find an upper limit of 0.7 nG, as higher magnetic fields would require star formation efficiencies larger
than 1%, which is unrealistic. Similar results are found for model B, assuming reionization
completely due to Pop. III stars.
3. Reionization only due to population II stars (model D) is ruled out completely.
For dark matter, we found the following results:
1. Dark matter annihilation provides noticeable contributions to the reionization optical depth only
for thermally averaged mass-weighted cross sections "σv#/mDM ≥ 10−33 cm3 /s/eV.
2. Parameters "σv#/mDM ≥ 3 × 10−33 cm3 /s/eV can be ruled out by 3σ on the basis of WMAP 5
year data.
3. Dark matter decay becomes important for the reionization optical depth for lifetimes below
3 × 1024 s.
4. Dark matter lifetimes below 3 × 1023 s are ruled out by 3σ.
These results are in agreement with conservative constraints obtained from the gamma-ray background [Mack et al., 2008]. We further showed that reionization can not be due to ∼ 1000 M' dark
stars alone, as the corresponding optical depth would be significantly too high. One might wonder
whether heating from dark matter annihilation might help to significantly delay stellar reionization,
in order to reconcile this model with observations. However, as can be seen in Fig. 2.6, this is more
than compensated by the effects of secondary ionization, once that dark matter annihilation starts to
have a significant influence on the IGM. In case collider experiments like the LHC 1 or other dark
matter detection experiments 2 find evidence for a self-annihilating dark matter candidate, this can be
seen as evidence for a rapid transition towards a different mode of star formation, or a problem in our
understanding of dark stars. To reconcile dark stars with observations, the following scenarios seem
feasible:
1. Dark stars with 1000 M' may be extremely rare objects, and their actual mass scale is closer
to the mass scale for typical Pop. III stars. In such a case, reionization could not distinguish
between dark stars and conventional Pop. III stars.
1
2
http://lhc.web.cern.ch/lhc/
A list of dark matter detection experiments is given at http://cdms.physics.ucsb.edu/others/others.html.
40
2.6 Conclusions and outlook
2. The transition to lower-mass stars occurs very rapidly, such that dark stars only contribute to
the very early phase of reionization. Reasons for that might be chemical, radiative as well
as mechanical feedback (see also the discussion in Chapter 2.4.3). In fact, even a doublereionization scenario might be conceivable, in which very massive dark stars ionize the universe
at high redshifts. Due to chemical and radiative feedback, formation of such stars might be
suppressed and the universe might become neutral again, until the formation of less massive
stars becomes efficient enough to reionize the universe.
3. Dark stars do not reach a main-sequence phase, but are disrupted earlier by some non-linear
instability. Such an instability would be constrained by the fact that it neither appears in a
linear stability analysis nore in 1D simulations. Violent explosions of dark stars might even be
considered as a source for Gamma-Ray bursts.
While too definite conclusions on the existence of dark stars are not yet possible, it is at least indicated
that the possibilities mentioned above should be explored in more detailed, and a better understanding
of the properties of the dark stars. A better understanding of their evolution after the dark phase will
certainly help to better understand their possible role during reionization.
The constraints derived here are independent of other works that essentially rely on the physics
of recombination to derive upper limits on additional physics [e. g.Barrow et al., 1997; Zhang et al.,
2006, 2007]. In particular, magnetic fields can evolve dynamically and their field strength may thus
change between these epochs. It has recently been suggested that the Biermann Battery effect creates
magnetic fields in the presence of an electron pressure gradient [Xu et al., 2008]. We thus need to
probe magnetic fields at different epochs. As we will show in a separate work [Schleicher et al.,
2009a], upcoming 21 cm measurements will allow to probe the thermal history before and during
reionization in great detail, and may allow to detect primordial magnetic fields of the order of 0.1 nG.
Additional ways of probing the reionization history and the dark ages exist as well: Scattering
of CMB-photons in fine-structure lines of heavy elements may lead to a frequency-dependent CMB
power spectrum and may allow to measure metal-abundances as a function of redshift [Basu et al.,
2004]. Before reionization, molecules may form in the IGM and introduce further frequency-dependent
features in the CMB [Schleicher et al., 2008c]. Such features are likely enhanced in the presence of
either primordial magnetic fields or dark matter decay / annihilation, as the increased electron fraction
catalyses the formation of molecules, and the additional heat input leads to a departure of the level
populations from the radiation temperature.
Further improvements are expected from the upcoming measurement of Planck, that will measure
the reionization optical depth with unprecedent accuracy and thus allow to strengthen the constraints
obtained here on the stellar populations and additional physics. In addition, a more accurate determination of the cosmological parameters will remove further uncertainties in the present models of
reionization.
41
2. REIONIZATION - A PROBE FOR THE STELLAR POPULATION AND THE PHYSICS
OF THE EARLY UNIVERSE
42
3
Influence of primordial magnetic fields
on 21 cm emission
While the possibility to detect effects due to dark matter annihilation with 21 cm observations has been
discussed in quite some detail in the literature [Furlanetto et al., 2006b; Chuzhoy, 2008; Cumberbatch
et al., 2008], the signatures of primordial magnetic fields have been explored previously only for
redshift z > 30 [Tashiro et al., 2006]. This is however difficult to observe. In particular, their model
does not include the build-up of a Lyman α background by the first sources of light, which couples
the excitation temperature of atomic hydrogen to the gas temperature and can lead to a net signal
in 21 cm during the redshifts of reionization, where telescopes like LOFAR1 will observe. I will
show here why this is important to constrain primordial magnetic fields. This work was published by
Schleicher, Banerjee, & Klessen [2009a].
3.1 Introduction
Observations of the 21 cm fine structure line of atomic hydrogen have the potential to become an
important means of studying the universe at early times, during and even before the epoch of reionization. This possibility was suggested originally byPurcell & Field [1956], and significant process in
instrumentation and the development of radio telescopes has brought us close to the first observations
from radio telescopes like LOFAR. While one of the main purposes is to increase our understanding
of cosmological reionization 2 , a number of further exciting applications have been suggested in the
mean time. Loeb & Zaldarriaga [2004] demonstrated how 21 cm measurements can probe the thermal
evolution of the IGM at a much earlier time, at redshifts of z ∼ 200. Barkana & Loeb [2005a] suggested a method that separates physical and astrophysical effects and thus allows to probe the physics
of the early universe. Furlanetto et al. [2006b] showed how the effect of dark matter annihilation
1
2
http://www.lofar.org
LOFAR Science Case: http://www.lofar.org/PDF/NL-CASE-1.0.pdf
43
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
and decay would be reflected in the 21 cm line, and effects of primordial magnetic fields have been
considered by Tashiro et al. [2006].
Indeed, primordial magnetic fields can affect the early universe in various ways. The thermal
evolution is significantly altered by ambipolar diffusion heating and decaying MHD turbulence [Sethi
& Subramanian, 2005; Sethi et al., 2008; Schleicher et al., 2008b]. Kim et al. [1996] calculated the
effect of the Lorentz force on structure formation and showed that additional power is present on
small scales in the presence of primordial magnetic fields. It was thus suggested that reionization
occurs earlier in the presence of primordial magnetic fields [Sethi & Subramanian, 2005; Tashiro &
Sugiyama, 2006]. However, as pointed out by Gnedin & Hui [1998]; Gnedin [2000], the characteristic mass scale of star forming halos, the so-called filtering mass, increases signficantly when the
temperature is increased. For comoving field strengths of ∼ 1 nG, we found that the filtering mass
scale is shifted to scales where the power spectrum is essentially independent of the magnetic field
[Schleicher et al., 2008b]. Reionization is thus delayed in the presence of primordial magnetic fields.
We further found upper limits of the order 1 nG, based on the Thomson scattering optical depth measured by WMAP 5 [Komatsu et al., 2008; Nolta et al., 2009] and the requirement that reionization
ends at z ∼ 6 [Becker et al., 2001].
The presence of primordial magnetic fields can have interesting implications on first star formation as well. In the absence of magnetic fields, Abel et al. [2002] and Bromm & Larson [2004]
suggested that the first stars should be very massive, perhaps with ∼ 100 solar masses. Clark et al.
[2008] and Omukai et al. [2008] argued that in more massive and perhaps metal-enriched galaxies,
fragmentation should be more effective and lead to the formation of rather low-mass stars, due to a
stage of efficient cooling. Magnetic fields may change this picture and reduce the stellar mass by
triggering jets and outflows [Silk & Langer, 2006]. However, simulations by [Machida et al., 2006]
show that the change in mass is of the order 10%.
21 cm measurements can try to adress primordial magnetic fields in two ways: During the dark
ages of the universe, at redshifts z ∼ 200 well before the formation of the first stars, the spin temperature of hydrogen is coupled to the gas temperature via collisional de-excitation by hydrogen
atoms [Allison & Dalgarno, 1969; Zygelman, 2005] and free electrons [Smith, 1966; Furlanetto &
Furlanetto, 2007], constituting a probe at very early times. While collisional de-excitation becomes
inefficient due to the expansion of the universe, the first stars will build up a Lyman α background
that will cause a deviation of the spin temperature from the radiation temperature by the WouthuysenField effect [Wouthuysen, 1952; Field, 1958]. As primordial magnetic fields may shift the onset of
reionization, the onset of this coupling constitutes an important probe on the presence of such fields.
We adress these possibilities in the following way: In Chapter 3.2, we review our treatment of the
IGM in the presence of primordial magnetic fields. The evolution of the 21 cm background and the
44
3.2 The evolution of the IGM
role of Lyman α photons is discussed in Chapter 3.3. The evolution of linear perturbations in temperature and ionization is calculated in 3.4. The results for the power spectrum are given in Chapter 3.5,
and the results are further discussed in Chapter 3.6.
3.2 The evolution of the IGM
As indicated in the introduction, primordial magnetic fields can have a strong impact on the evolution
of the IGM before reionization. Once the first luminous objects form, their feedback must also be
taken into account. In this section, we review the basic ingredients of our treatment of the IGM
between recombination and reionization. We refer the interested reader toSchleicher et al. [2008b]
for more details.
3.2.1 The RECFAST code
Our calculation is based on a modified version of the RECFAST code1 [Seager et al., 1999, 2000] that
calculates recombination and the freeze-out of electrons. The calculation of helium recombination
was recently updated by Wong et al. [2008]. We have extended this code by including a model for
reionization in the context of primordial magnetic fields Schleicher et al. [2008b]. The equation for
the temperature evolution is given as
dT
dz
=
+
8σT aR T r4
xe
(T − T r )
3H(z)(1 + z)me c 1 + fHe + xe
2T
2(Lheat − Lcool )
−
,
1 + z 3nkB H(z)(1 + z)
(3.1)
where Lheat is the new heating term (see Chapter 3.2.2, 3.2.3 and 3.3.2), Lcool the cooling by Lyman
α emission, bremsstrahlung and recombinations, σT is the Thomson scattering cross section, aR the
Stefan-Boltzmann radiation constant, me the electron mass, c the speed of light, kB Boltzmann’s
constant, n the total number density, xe = ne /nH the electron fraction per hydrogen atom, H(z) is the
Hubble factor and fHe is the number ratio of He and H nuclei, which can be obtained as fHe = Y p /4(1−
Y p ) from the mass fraction Yp of He with respect to the total baryonic mass. Eq. (3.1) describes the
change of temperature with redshift due to Compton scattering of CMB photons, expansion of the
universe and additional heating and cooling terms that are described in the following subsections in
more detail. The evolution of the ionized fraction of hydrogen, xp , is given as
dxp
dz
1
[xe xp nH αH − βH (1 − xp )e−h p νH,2s /kB T ]
H(z)(1 + z)[1 + KH (ΛH + βH )nH (1 − xp )]
× [1 + KH ΛH nH (1 − xp )]
kion nH xp
−
− fion ,
H(z)(1 + z)
=
http://www.astro.ubc.ca/people/scott/recfast.html
45
(3.2)
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
Figure 3.1: Evolution of gas temperature (upper panel) and ionized fraction (lower panel) in the medium
that was not yet affected by UV feedback. The details of the models are given in Table 3.1.
46
3.2 The evolution of the IGM
Figure 3.2: Evolution of the filtering mass scale for different comoving magnetic field strengths.
where the ionization fraction is determined by radiative recombination, photo-ionization of excited
hydrogen atoms by CMB photons in an expanding universe and collisional ionization, as well as Xray feedback. Here, nH is the number density of hydrogen atoms and ions, hp Planck’s constant, fion
describes ionization from X-rays and UV photons (see Chapter 3.2.3), and the parametrized case B
recombination coefficient for atomic hydrogen αH is given by
αH = F × 10−13
atb
cm3 s−1
1 + ctd
(3.3)
with a = 4.309, b = −0.6166, c = 0.6703, d = 0.5300 and t = T/104 K [Pequignot et al., 1991;
Hummer, 1994] . This coefficient takes into account that direct recombination into the ground state
does not lead to a net increase of neutral hydrogen atoms, since the photon emitted in the recombination process can ionize other hydrogen atoms in the neighbourhood.
The fudge factor F = 1.14 serves to speed up recombination and is determined from comparison
with the multilevel-code. The photoionization coefficient βH is calculated from detailed balance at
high redshifts as described by Seager et al. [1999, 2000]. Once the ionized fraction drops below 98%,
it is instead calculated from the photoionization cross section given bySasaki & Takahara [1993], as
detailed balance may considerably overestimate the photoionization coefficient in case of additional
energy input at low redshift. The wavelength λH,2p corresponds to the Lyman-α transition from the
2p state to the 1s state of the hydrogen atom. The frequency for the two-photon transition between
the states 2s and 1s is close to Lyman-α and is thus approximated by νH,2s = c/λH,2p (i.e., the same
averaged wavelength is used). Finally, ΛH = 8.22458 s−1 is the two-photon rate for the transition
2s-1s according to Goldman [1989] and KH ≡ λ3H,2p /[8πH(z)] the cosmological redshifting of Lyman
47
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
α photons. To take into account UV feedback, which can not be treated as a background but creates
locally ionized bubbles, we further calculate the volume-filling factor QHII of ionized hydrogen according to the prescriptions of [Shapiro & Giroux, 1987; Haiman & Loeb, 1997a; Barkana & Loeb,
2001; Loeb & Barkana, 2001; Choudhury & Ferrara, 2005; Schneider et al., 2006]. We assume that
the temperature in the ionized medium is the maximum between 104 K and the temperature in the
overall neutral gas. The latter is only relevant if primordial magnetic fields have heated the medium
to temperatures above 104 K before it was ionized.
3.2.2 Heating due to primordial magnetic fields
The presence of magnetic fields leads to two different contributions to the heating rate, one coming
from ambipolar diffusion and one resulting from the decay of MHD turbulence. In the first case,
the contribution can be calculated as [Sethi & Subramanian, 2005; Cowling, 1956; Schleicher et al.,
2008b]:
ρn
B4
Lambi ∼
.
(3.4)
16π2 γρ2b ρi L2
Here, ρn , ρi and ρb are the mass densities of neutral hydrogen, ionized hydrogen and all baryons. The
ion-neutral coupling coefficient is calculated using the updated zero drift velocity momentum transfer
coefficients of Pinto & Galli [2008] for collisions of H+ with H and He. The coherence length L
−1 given by [Jedamzik et al., 1998;
is estimated as the inverse of the Alfvén damping wavelength kmax
Subramanian & Barrow, 1998; Seshadri & Subramanian, 2001] as
# B $−1 ! Ω "1/4
0
m
−1
kmax ∼ 234 Mpc
−9
0.3
10 G
!
"1/2 ! "1/4
2
Ωb h
h
×
,
(3.5)
0.02
0.7
which is the scale on which fluctuations in the magnetic field are damped out during recombination.
The comoving magnetic field is denoted as B0 = B/(1 + z)2 .
For decaying MHD turbulence, we adopt the prescription ofSethi & Subramanian [2005],
Ldecay =
B0 (t)2 3α̃
[ln(1 + td /ti )]α̃ H(t)
,
8π 2 [ln(1 + td /ti ) + ln(t/ti )]α̃+1
(3.6)
where t is the cosmological time at redshift z, td is the dynamical timescale, ti the time where decay
starts, i. e. after the recombination epoch when velocity perturbations are no longer damped by the
large radiative viscosity, zi is the corresponding redshift. For a power spectrum of the magnetic field
with power-law index α, implying that the magnetic field scales with the wavevector k to the power
3 + α, the parameter α̃ is given as α̃ = 2(α + 3)/(α + 5) [Olesen, 1997; Shiromizu, 1998; Christensson
et al., 2001; Banerjee & Jedamzik, 2003]. In the generic case, we expect the power spectrum of
the magnetic field to have a maximum at the scale of the coherence length, and the heat input by
48
3.2 The evolution of the IGM
MHD decay should be determined from the positive slope corresponding to larger scales [Müller &
Biskamp, 2000; Christensson et al., 2001; Banerjee & Jedamzik, 2003; Banerjee et al., 2004]. We
thus adopt α = 3 for the calculation. We estimate the dynamical timescale as td = L/vA , where
&
vA = B/ 4πρb is the Alvén velocity and ρb the baryon mass density. The evolution of the magnetic
field as a function of redshift can be determined from the magnetic field energy density EB = B2 /8π,
which evolves as [Sethi & Subramanian, 2005]
dE B
= −4H(t)E B − Lambi − Ldecay .
dt
(3.7)
For the models given in Table 3.1, the evolution of temperature and ionization in the overall neutral
medium, i. e. the gas that was not yet affected by UV photons from reionization, is given in Fig.3.1.
For comoving fields weaker than 0.02 nG, the gas temperature is closely coupled to the CMB via
Compton-scattering at redshifts z > 200. At lower redshifts, this coupling becomes inefficient and
the gas cools adiabatically during expansion, until heating due to X-ray feedback becomes important
near redshift 20. The ionized fraction drops rapidly from fully ionized at z ∼ 1100 to ∼ 2 × 10−4 , until
it increases again at low redshift due to X-ray feedback. For larger magnetic fields, the additional
heat input becomes significant and allows the gas to decouple earlier from the CMB. For comoving
fields of order 0.5 nG or more, the gas reaches a temperature plateau near 104 K. At this temperature
scale, collisional ionizations become important and the ionized fraction in the gas increases, such that
ambipolar diffusion heating becomes inefficient. The onset of X-ray feedback is delayed to the higher
filtering mass in the presence of magnetic fields, which is discussed in the next section.
3.2.3 The filtering mass scale and stellar feedback
The universe becomes reionized due to stellar feedback. We assume here that the star formation rate
(SFR) is proportional to the change in the collapsed fraction fcoll , i. e. the fraction of mass in halos
more massive than mmin . This fraction is given by the formalism of Press & Schechter [1974]. In
Schleicher et al. [2008b], we have introduced the generalized filtering mass mF,B , given as
2/3
MF,B
3
=
a
%
0
a
(
da



Mg2/3 (a( ) 1
!
a(
−
a
"1/2 

 ,
(3.8)
where a = 1/(1 + z) is the scale factor and Mg is the maximum of the thermal Jeans mass MJ and
the magnetic Jeans mass MJB , the mass scale below which magnetic pressure gradients can counteract
gravitational colapse [Sethi & Subramanian, 2005; Subramanian & Barrow, 1998]. The concept of the
filtering mass, i. e. the halo mass for which the baryonic and dark matter evolution can decouple, goes
back to Gnedin & Hui [1998], see also Gnedin [2000]. To take into account the back reaction of the
photo-heated gas on structure formation, the Jeans mass is calculated from the effective temperature
T e f f = QHII T max + (1 − QHII )T , where T max = max(104 K, T ) [Schneider et al., 2006]. The evolution
of the filtering mass for different magnetic field strengths is given in Fig.3.2.1.
49
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
When the comoving field is weaker than 0.03 nG, the filtering mass is of order 105 M' at high
redshift and increases up to 107 M' during reionization, as UV and X-ray feedback heats the gas
temperature and prevents the collapse of baryonic gas on small scales. Stronger magnetic fields provide additional heating via ambipolar diffusion and decaying MHD turbulence, such that the filtering
mass is higher from the beginning. This implies that the first luminous objects form later, as the gas
in halos below the filtering mass cannot collapse. This effect is even stronger when the comoving
field is stronger than 0.3 nG, as the magnetic Jeans mass then dominates over the thermal Jeans mass
and increases the filtering mass scale by further orders of magnitude. As discussed in Schleicher
et al. [2008b], the feedback on structure formation via the filtering mass can delay reionization significantly, which in turn allows to calculate upper limits on the magnetic field strength due to the
measured reionization optical depth.
Consequently, we define the lower limit to halo masses that can form stars as mmin = max(MF,B , 105 M' ),
where 105 M' is the minimal mass scale for which baryons can cool efficiently, as found in simulations of Greif et al. [2008]. We take into account X-ray feedback that can penetrate into the IGM
due to its long mean-free path, as well as feedback from UV photons produced in the star forming
regions. As we showed in Schleicher et al. [2008b], Population II stars can not reionize the universe
efficiently. It is thus reasonable to assume that the first luminous sources were indeed massive Pop. III
stars. While this may no longer be true once chemical and radiative feedback becomes efficient and
leads to a different mode of star formation, such a transition seems not important for the main purpose
of this work, which is to demonstrate how primordial magnetic fields shift the onset of reionization
and the epoch where a Lyman α background builds up initially. We thus adopt model B ofSchleicher
et al. [2008b], which uses an escape fraction of 100% if the virial temperature is below 104 K, and
10% for the atomic cooling halos with higher virial temperature. It further assumes that 4 × 104 UV
photons are emitted per stellar baryon during the lifetime of the star. This choice is in particular motivated by simulations of Whalen et al. [2004], that show that minihalos are easily photo-evaporated
by UV feedback from Pop. III stars, yielding escape fractions close to 100%. A new study byWise
& Abel [2008] shows that the escape fraction may be larger than 25% even for atomic cooling halos.
However, their study assumes purely primordial gas, which is unlikely in such systems, and we expect
that studies which take into account metal enrichment will find lower escape fractions, perhaps of the
order 10% as suggested here.
3.3 The 21 cm background
The 21 cm brightness temperature depends on the hyperfinestructure level populations of neutral
hydrogen, which is described by the spin temperature Tspin . In this section, we review the physical
processes that determine the spin temperature, discuss the build-up of a Lyman-α background that
can couple the spin temperature to the gas temperature at low redshift, and discuss various sources of
21 cm brightness fluctuations.
50
3.3 The 21 cm background
3.3.1 The spin temperature
The observed 21 cm brightness temperature fluctuation can then be conveniently expressed as [Furlanetto
et al., 2006a]
!
"!
"1/2
Ωb h2
0.15 1 + z
δT b = 27xH (1 + δ)
0.023 Ωm h2 10
!
"!
"
T S − T r H(z)/(1 + z)
×
mK,
(3.9)
TS
dv|| /dr||
where δ is the fractional overdensity, xH = 1 − xeff the effective neutral fraction, xeff = QH+ + (1 −
QH+ )xp the effective ionized fraction, Ωb and Ωm the cosmological density parameters for baryonic
and total mass, and dv|| /dr|| the gradient of the proper velocity along the line of sight, including the
Hubble expansion. The spin temperature Tspin is given as.
−1
T spin
=
T r−1 + xc T −1 + xα T c−1
,
1 + xc + xα
(3.10)
where Tc is the colour temperature, xc = xcHH + xeH
c the collisional coupling coefficient, given as
H n T /A T , and the
the sum of the coupling coefficient with respect to hydrogen atoms, xcHH = κ1−0
H ∗ 10 r
e n T /A T . Further, n denotes the number
coupling coefficient with respect to electrons, xeH
=
κ
H
c
1−0 e ∗ 10 r
density of hydrogen atoms, T∗ = 0.068 K the temperature corresponding to the 21 cm transition
and A10 = 2.85 × 10−15 s−1 the Einstein coefficient for radiative de-excitation. For the collisional
H , we use the recent calculation of Zygelman [2005], and
de-excitation rate through hydrogen, κ1−0
we use the new rate of Furlanetto & Furlanetto [2007] for the collisional de-excitation rate through
H . When both the spin and gas temperature are significantly larger than the spin exchange
electrons, κ1−0
temperature T se = 0.4 K, as it is usually the case, the colour temperature is given as
!
"
1 + T se /T
Tc = T
.
(3.11)
1 + T se /T spin
Spin and colour temperature thus depend on each other and need to be solved for simultaneously. The
Wouthuysen-Field coupling coefficient is given as
xα = S α
Jα
,
Jc
(3.12)
where Jc = 1.165×10−10 (1+z)/20 cm−2 s−1 Hz−1 sr−1 and Jα is the proper Lyman α photon intensity,
given as
% zmax
n:
max
(1 + z)2 c
Jα (z) =
frec (n)
dz(
2(ν(n , z( )
(3.13)
()
4π
H(z
z
n=2
[see also Barkana & Loeb, 2005a]. frec is the recycling probability of Lyman n photons [Pritchard &
Furlanetto, 2006; Hirata, 2006], H(z) the Hubble function at redshift z, c the speed of light, nmax the
highest considered Lyman n resonance, zmax the highest redshift at which star formation might occur
51
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
(a)
(b)
Figure 3.3: CMB, gas and spin temperature in the gas unaffected by reionization, for a star formation
efficiency f∗ = 10−3 , in the zero field case (upper panel) and for a field strength of 0.8 nG (lower panel).
In the absence of magnetic fields, gas and CMB are strongly coupled due to efficient Compton scattering
of CMB photons. At about redshift 200, the gas temperature decouples and evolves adiabatically, until it
is reheated due to X-ray feedback. The spin temperature follows the gas temperature until redshift ∼ 100.
Collisional coupling then becomes less effective, but is replaced by the Wouthuysen-Field coupling at
z ∼ 25. In the presence of magnetic fields, additional heat goes into the gas, it thus decouples earlier and
its temperature rises above the CMB. Wouthuysen-Field coupling becomes effective only at z ∼ 15, as the
high magnetic Jeans mass delays the formation of luminous sources significantly.
52
3.3 The 21 cm background
(a)
(b)
(c)
Figure 3.4: The mean evolution in the overall neutral gas, calculated for the models described in Table
3.1. Top: The spin temperature. Middle: The mean brightness temperature fluctuation. Bottom: The
frequency gradient of the mean brightness temperature fluctuation.
53
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
and 2(ν, z) the comoving emissivity. We adopt nmax = 30 and zmax = 40. The suppresion factor S α
can be calculated as [Chuzhoy & Shapiro, 2006; Furlanetto et al., 2006a]
#
/
01/3 $
S α = exp −0.803T −2/3 10−6 τGP
(3.14)
from the optical depth for Lyman α photons,
5
τGP ∼ 3 × 10 xHI
!
1+z
7
"3/2
.
(3.15)
Examples for the evolution of the gas, spin and CMB temperature in the zero-field case and for a
magnetic field of 0.8 nG are given in Fig. 3.3.
3.3.2 The build-up of a Lyman-α background
In the dark ages, the universe is transparent to photons between the Lyman α line and the Lyman limit.
Luminous sources can thus create a radiation background between these frequencies that is redshifted
during the expansion of the universe. When a photon is shifted into a Lyman line, it is scattered
and can couple the spin temperature to the gas temperature as described above. As shown byHirata
[2006] and Barkana & Loeb [2005a], a population of Pop. III stars can already produce a significant
amount of Lyman α radiation around redshift 25. We assume that emission is proportional to the star
formation rate, which is assumed to be proportional to the change in the collapsed fraction cf oll . It
thus depends on the evolution of the generalized filtering mass in the presence of magnetic fields. As
discussed in Furlanetto et al. [2006a] and Barkana & Loeb [2005a], the comoving emissivity is given
as
d fcoll
2(ν, z) = f∗ nb 2b (ν)
,
(3.16)
dt
where nb is the comoving baryonic number density and 2b (ν) the number of photons produces in the
frequency interval ν ± dν/2 per baryon incorporated in stars. The emissivity is approximated as a
power law 2b (ν) ∝ ναs −1 , and the parameters must be chosen according to the model for the stellar
population. As discussed above, we assume that the first stars are massive Pop. III stars, and adopt
the model of Barkana & Loeb [2005a] with Nα = 2670 photons per stellar baryon between the Lyman
α and the Lyman β line and a spectral index αs = 1.29
In addition, secondary effects from X-ray photons may lead to excitation of hydrogen atoms and
the production of further Lyman α photons. We calculate the production of Lyman α photons from
the X-ray background for consistency. The contribution from the X-rays is then given as [Chen &
Miralda-Escudé, 2004; Chuzhoy et al., 2006; Furlanetto et al., 2006a]
!
"
fX,coll f∗ d fcoll /dz
X−ray
= 0.05S α fX
∆xα
1/3 0.1 0.01
!
"3
1+z
×
.
(3.17)
10
54
3.3 The 21 cm background
We adopt fX = 0.5, and calculate fX,coll from the fit-formulae of Shull & van Steenberg [1985].
(Notation: Please recall that fcoll denotes the fraction of collapsed dark matter, while fX,coll denotes
the fraction of absorbed X-ray energy which goes into X-rays. The parameter fX provides an overall
normalization for the correlation between the star formation rate and the X-ray luminosity [Koyama
et al., 1995].) For the models described in Table 3.1, we have calculated the evolution of the spin
temperature and the mean 21 cm brightness fluctuation. From an observational point of view, it
might be more reasonable to focus on the frequency gradient of the mean 21 brightness [Schneider
et al., 2008]. As the frequency dependence of the foreground is known, such an analysis may help to
distinguish between the foreground and the actual signal. Results for the spin temperature, the mean
21 cm brightness fluctuation and its frequency gradient are shown in Fig.3.4.
At redshifts above 200, the spin temperature is coupled closely to the gas temperature due to
collisions with hydrogen atoms. At lower redshifts, the coupling becomes inefficient and the spin
temperature evolves towards the CMB temperature, until structure formation sets in and the spin
temperature is again coupled to the gas temperature via the Wouthuysen-Field effect. The point
where this happens depends on the magnetic field strength, which influences the filtering mass and
thus the mass scale of halos in which stars can form. Especially in the presence of strong magnetic
fields, this delay in structure formation also makes the production of Lyman α photons less efficient
and the departure from the CMB temperature is only weak.
This bevahiour is reflected in the evolution of the mean brightness temperature fluctuation. For
weak magnetic fields, the gas temperature is colder than the CMB because of adiabatic expansion,
so the 21 cm signal appears in absorption. At redshift 40 where the spin temperature approaches the
CMB temperature, a maximum appears as absorption goes towards zero. Comoving fields of 0.05 nG
add only little heat and essentially bring the gas and spin temperatures closer to the CMB, making
the mean brightness temperature fluctuation smaller. For stronger fields, the 21 cm signal finally
appears in emission. Due to very effective coupling and strong temperature differences at redshifts
beyond 200, a pronounced peak appears there in emission. At redshifts between 20 and 10 which
are more accessible to the next 21 cm telescopes, magnetic fields generally reduce the expected mean
fluctuation, essentially due to the delay in the build-up of a Lyman α background.
The evolution of the frequency gradient in the mean brightness temperature fluctuation essentially shows strong minima and maxima where the mean brightness temperature fluctuation changes
most significantly. In the case of weak fields, this peak is near redshift 30 when coupling via the
Wouthuysen-Field effect becomes efficient. For stronger fields, this peak is shifted towards lower
redshifts, thus providing a clear indication regarding the delay of reionization.
Scattering of Lyman α photons in the IGM does not only couple the spin temperature to the gas
temperature, but it is also a potential source of heat. Its effect on the mean temperature evolution has
been studied by a number of authors [Madau et al., 1997; Chen & Miralda-Escudé, 2004; Chuzhoy
& Shapiro, 2006; Ciardi et al., 2002]. While a detailed treatment requires to separately follow the
mean temperature of hydrogen and deuterium, a good estimate can be obtained using the approach of
55
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
Furlanetto et al. [2006a]; Furlanetto & Pritchard [2006], i. e.
"
!
0.8 xα 10
2 Lheat,Lyα
∼ 4/3
.
3 nH kB T H(z) T S α 1 + z
(3.18)
For star formation efficiencies of the order 0.1%, as they are adopted throughout most of this work,
Lyman α heating leads to a negligible temperature change of the order 1 K.
Model
B0 [nG]
1
2
3
4
5
6
7
0
0.02
0.05
0.2
0.5
0.8
0.8
f∗
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
1%
Table 3.1: A list of models for different co-moving magnetic fields and star formation efficiencies, which
are used in several figures for illustrational purposes. We give the comoving magnetic field B 0 and the
star formation efficiency f ∗ . For illustration purposes, all models assume a population of massive Pop. III
stars. The amount of Lyman α photons produced per stellar baryon would be larger by roughly a factor of
2 if we were to assume Pop. II stars. Assuming that the same amount of mass goes into stars, the coupling
via the Wouthuysen-Field effect would start slightly earlier, but the delay due to magnetic fields is still
more significant.
3.3.3 21 cm fluctuations
Fluctuations in the 21 cm brightness temperature can be caused by the relative density fluctuations δ,
relative temperature fluctuations δT and relative fluctuations in the neutral fraction δH . Fluctuations in
the Lyman α background can be an additional source of 21 cm fluctuations [Barkana & Loeb, 2005a],
but a detailed treatment of these fluctuations is beyond the scope of this work. These fluctuations are
only important in the early stage of the build-up of such a background. Once xα is significantly larger
than unity, the spin temperature is coupled closely to the gas temperature, and small fluctuations in
the background will not affect the coupling. We adopt the treatment ofLoeb & Zaldarriaga [2004] to
calculate the fractional 21 cm brightness temperature perturbation, given as
δ21 (.k) = (β + µ2 )δ + βH δH + βT δT ,
(3.19)
where µ is the cosine of the angle between the wavevector .k and the line of sight, with the expansion
coefficients [Barkana & Loeb, 2005b; Furlanetto et al., 2006a]
xc
,
(3.20)
β = 1+
xtot (1 + xtot )
56
3.4 The evolution of linear perturbations in the dark ages
xcHH − xeH
c
,
xtot (1 + xtot )
Tr
xc
d ln xc
+
.
T − T r xtot (1 + xtot ) d ln T
βH = 1 +
(3.21)
βT
(3.22)
=
The coefficient xtot is given as the sum of the collisional coupling coefficient xc and the WouthuysenField coupling coefficient xα .
3.4 The evolution of linear perturbations in the dark ages
To determine the fluctuations in the 21 cm line, we need to calculate the evolution of linear perturbations during the dark ages. We introduce the relative perturbation of the magnetic field, δB , and
the relative perturbation to the ionized fraction, δi , which is related to the relative perturbation in
the neutral fraction, δH , by δi = −δH (1 − xi )/xi . In the general case, different density modes trigger
independent temperature and ionization fluctuations. This introduces a non-trivial scale dependence
[Barkana & Loeb, 2005c; Naoz & Barkana, 2005]. However, on the large scales which might ultimately be observeable, the growing density mode dominates [Bharadwaj & Ali, 2004]. The temperature and ionization fluctuations δT and δi are thus related to the density fluctuations δ by δT = gT (z)δ
and δi = gi (z)δ with redshift-dependent coupling factors gi (z) and gT (z). The time evolution of gT
and gi is well-known in the absence of magnetic fields [Naoz & Barkana, 2005; Bharadwaj & Ali,
2004] and has also been studied for the case of dark matter annihilation and decay [Furlanetto et al.,
2006b]. We further introduce the relative fluctuation in the magnetic field, δB = gB δ. To extend the
previous analysis to the situation considered here, we must consider the effect of the new terms due
to ambipolar diffusion heating and decaying MHD turbulence in Eq. (3.1). As it has a complicated
dependence on different quantities, we will keep this discussion rather generic, so that it can also be
applied to other situations as well.
Let us consider a source term of the form
# δT $
= f (ρ, T, xi , B).
(3.23)
δz source
Here, f (ρ, T, xi , B) corresponds to the term describing ambipolar diffusion heating and decaying MHD
turbulence in Eq. (3.1). When the quantities ρ, T , xi and B are perturbed, it is straightforward to show
that this introduces further terms for the evolution of the temperature perturbation, which are of the
form
δδT
δz
2 ∂f
∂f
∂f
= T −1 δρ
+ T δT
+ δi xi
∂ρ
∂T
∂xi
3
∂f
+ δB B
− f δT .
∂B
(3.24)
These new terms describe the change in the relative temperature fluctuation, depending on the additional source term f (ρ, T, xi , B) and its derivatives with respect to gas density, temperature, ionized
57
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
fraction and magnetic field. The behaviour is dominated by the properties of the ambipolar diffusion
heating term. The time evolution of the coupling factors gT and gi is thus given as
dgT
dz
dgi
dz
gT − 2/3
xi gT T r − gi (T r − T )
+
1+z
η1 tγ T (1 + z)H(z)
2
∂
f
∂f
∂f
+ T −1 ρ
+ T gT
+ gi xi
∂ρ
∂T
∂xi
3
∂f
+ gB B
− f gT ,
∂B
αH xi nH (1 + gi + α(H gT )
gi
=
+
.
1+z
(1 + z)H(z)
=
(3.25)
(3.26)
The first term in Eq. (3.25) describes the evolution towards an adiabatic state in the absence of heating or cooling mechanisms. The second term describes the interaction with the CMB via Compton
scattering, which in general drives the gas towards an isothermal state, as the heat input per baryon
is constant. The further terms are those derived in Eq. (3.24). In Eq. (3.26), the first term holds the
relative fluctuation δi constant in the absence of recombinations, while the second term describes the
effect of hydrogen recombination, which tends to drive the corresponding coupling factor gi towards
−1. Fluctuations in density and ionization are then anticorrelated.
This system must be closed with an additional assumption for the fluctuation in the magnetic field.
The dominant mechanism for energy loss is ambipolar diffusion, which scales with ρ−2
b . It seems thus
unlikely that the magnetic field will be dissipated in regions of enhanced density. We rather assume
that the magnetic flux is approximately conserved. In the case of dynamically weak field strength, the
corresponding surface scales as ρ−2/3 , so the magnetic field scales as ρ2/3 , thus gB ∼ 2/3. Ambipolar
diffusion, the dominant new heating term, scales as ρ−2 . To relate the change in energy to the change
in temperature, it is further multiplied by 3/(2nkB ), such that in this case f ∝ ρ−3 . This behavior is
not fully compensated by the other terms, thus heating is less effective in overdense regions, and gT
tends to drop below the adiabatic value of 2/3. For the models given in Table3.1, the evolution of gT
and gi is given in Fig. 3.5.
In the zero or weak magnetic field case, the coupling factor gT evolves from zero (efficient coupling to the CMB) towards 2/3 (adiabatic), until X-ray feedback sets in. For comoving fields larger
than 0.05 nG, there is an early phase around z ∼ 500 where it evolves towards adiabatic behaviour, as
the gas recombines more efficiently in these early times and higher densities, where the factor gi is still
of order −1. Thus, ambipolar diffusion in density enhancements is more efficient and effectively heats
the gas at these redshifts. Below redshift 500, the relative fluctuation in ionization evolves towards
zero, as the gas becomes thinner and hotter. That leads to a phase in which the relative temperature
fluctuations evolves towards an isothermal state (gT = 0). At low redshifts, X-ray feedback leads to
a decrease of the coupling factor gT . Its behaviour is no longer adiabatic because of this additional
heat input. The increase in the mean ionized fraction due to secondary ionizations leads to a more
negative gi .
58
3.5 The 21 cm power spectrum
3.5 The 21 cm power spectrum
We can now calculate the 21 cm power spectrum in the presence of primordial magnetic fields from
Eq. (3.19), leading to
P21 (k, µ) = δTb2 (β( + µ2 )2 Pδδ (k),
(3.27)
where Pδδ (k) is the baryonic power spectrum and
β( = β + βT gT − βH xi gi /(1 − xi ).
For simplicity, we average over the µ dependence and define Pδδ = P0 /(1 + z)2 , such that
!
"
2 ( 1 P0 (k)
2
(2
P21 (k) = δTb β + β +
.
3
5 (1 + z)2
(3.28)
(3.29)
On the large scales considered here, the power spectrum of the 21 cm line is proportional to the
baryonic power spectrum P0 (k) for a given redshift, but the proportionality constant will evolve with
redshift and is independent of the wavenumber in the linear approximation. The ratio P21 /P0 is
shown in Fig. 3.5. It essentially reflects the behaviour that was found earlier for the mean brightness
temperature fluctuation. At high redshift, the amplitude of the 21 cm power spectrum is larger in
the presence of additional heat from ambipolar diffusion and decaying MHD turbulence. At low
redshifts, it is smaller in the presence of magnetic fields, as the high filtering mass delays the buildup of a Lyman α background and leads only to a weak coupling of the spin temperature to the gas
temperature. In addition, the point where this coupling starts is shifted towards lower redshifts. This
provides two signatures that can provide strong indications for the presence of primordial magnetic
fields.
At high redshifts z > 50, the additional heat increases the difference between gas and radiation
temperature, and the gas decouples earlier from radiation. All that tends to increase the 21 cm signal at high redshift. For redshifts z < 40, the situation is more complicated. The altered evolution
of the fluctuations and the delay in the formation of luminous objects decrease the amplitude of the
power spectrum in total, but also the onset of efficient coupling via the Wouthuysen-Field effect can
be shifted significantly.
In this context, one might wonder whether 21 cm observations can actually probe additional smallscale power, which is predicted in the context of primordial magnetic fields [Kim et al., 1996; Sethi
& Subramanian, 2005; Tashiro & Sugiyama, 2006]. However, as discussed for instance by Furlanetto
et al. [2006a], the first 21 cm telescopes will focus on rather larger scales of the order 1 Mpc. The
contribution to the power spectrum from primordial magnetic fields depends on the assumed power
spectrum for the magnetic field. Regarding the formation of additional structures, the most significant
case is a single-scale power spectrum. In this case, the contribution to the matter power spectrum can
be evaluated analytically, yielding [Kim et al., 1996; Sethi & Subramanian, 2005]
P(k) =
B40 k3 H0−4
2
(8π)3 Ω4m ρ2c kmax
59
(3.30)
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
(a)
(b)
(c)
Figure 3.5: The evolution of the 21 cm large-scale power spectrum and the growth of large-scale fluctuations in temperature and ionization for the models given in Table 3.1. Top: The evolution of the 21
cm power spectrum, normalized with respect to the baryonic power spectrum at redshift 0. Middle: The
evolution of the ratio g T = δT /δ. Bottom: The evolution of the ratio g i = δ xi /δ.
60
3.6 Discussion and outlook
for k ≤ 2kmax , where kmax is given in Eq. 3.5. For interesting field strengths of the order of 1
−1 is shifted to
nG, we thus expect modifications on scales of the order 1 kpc. For weaker fields, kmax
larger scales, but the power spectrum decreases with B40 . Detecting such changes in the matter power
spectrum is thus certainly challenging. However, the calculation of the filtering mass indicates that
the baryonic gas will not collapse in minihalos of small masses when primordial fields are present.
Star formation and HII regions from UV feedback are thus limited to more massive systems. This
might be an additional way to distinguish the case with and without magnetic fields.
3.6 Discussion and outlook
In the previous sections, we have presented a semi-analytic model describing the post-reionization
universe and reionization in the context of primordial magnetic fields, and calculated the consequences for the mean 21 cm brightness fluctuation and the large-scale power spectrum. formation
of We identify two regimes in which primordial magnetic fields can influence effects measured with
21 cm telescopes. At low redshifts, primordial magnetic fields tend to delay reionization and the
build-up of a Lyman α background, thus shifting the point where the signal is at its maximum, and
changing the amplitude of the 21 cm power spectrum. The first 21 cm telescopes like LOFAR and
others will focus mainly on the redshift of reionization and can thus probe the epoch when a significant Lyman α background builds up. As our understanding of the first stars increases due to advances
in theoretical modeling or due to better observational constraints, this may allow us to determine
whether primordial magnetic fields are needed to delay the build-up of Lyman α photons or not. As
demonstrated above, comoving field strengths of the order 1 nG can delay the build-up of a Lyman α
background by ∆z ∼ 10, which is significantly stronger than other mechanisms that might delay the
formation of luminous objects. Lyman Werner feedback is essentially self-regulated and never leads
to a significant suppression of star formation [Johnson et al., 2007b,a], X-ray feedback from miniquasars is strongly constrained from the observed soft X-ray background [Salvaterra et al., 2005],
and significant heating from dark matter decay or annihilation would be accompanied by a significant
amount of secondary ionization, resulting in a too large optical depth. On the other hand, an important issue is the question regarding the first sources of light. As shown inSchleicher et al. [2008b],
massive Pop. III stars are needed to provide the correct reionization optical depth. On the other hand,
an additional population of less massive stars with an IMF according to [Scalo, 1998; Kroupa, 2002;
Chabrier, 2003] might be present. Such a population would emit more photons between the Lyman α
and β line per stellar baryon [Leitherer et al., 1999], and could thus shift the build-up of a Lyman α
background to an earlier epoch. The onset of efficient coupling via the Wouthuysen-Field effect thus
translates into a combined constraint on the stellar population and the strength of primordial magnetic
fields.
61
3. INFLUENCE OF PRIMORDIAL MAGNETIC FIELDS ON 21 CM EMISSION
A further effect occurs at high redshifts, where the additional heat input from magnetic fields due
to ambipolar diffusion and the decay of MHD turbulence increases the 21 cm signal. As gas decouples
earlier from the radiation field, the difference between gas and radiation temperature is larger and collisions are more effective in coupling the spin temperature to the gas temperature. The determination
of the 21 cm signal from this epoch is certainly challenging, as the foreground emission corresponds
to temperatures which are higher than the expected 21 cm brightness temperature by several orders
of magnitude. However, as pointed out in other works [Di Matteo et al., 2002; Oh & Mack, 2003;
Zaldarriaga et al., 2004; Sethi, 2005; Shchekinov & Vasiliev, 2007], the foreground emission is expected to be featureless in frequency, which may allow for a sufficiently accurate subtraction. In this
context, it may also to help to focus on the frequency gradient of the mean brightness temperature,
rather than the 21 cm brightness temperature itself. Upcoming long-wavelength experiments such
as LOFAR, 21CMA (former PAST)1 , MWA2 , LWA3 and SKA4 may thus detect the additional heat
from primordial magnetic fields in the neutral gas, or otherwise set new upper limits on primordial
magnetic fields, perhaps down to B0 ∼ 0.1 nG. Like the 21 cm transition of hydrogen, rotational and
ro-vibrational transitions of primordial molecules may create interesting signatures in the CMB as
well [Schleicher et al., 2008c], which may provide a further test of the thermal evolution during the
dark ages.
1
http://web.phys.cmu.edu/ past
http://web.haystack.mit.edu/arrays/MWA/index.html
3
http://lwa.unm.edu
4
http://www.skatelescope.org
2
62
4
The chemistry of the early universe and
its signatures in the CMB
In this chapter, I explore a further possibility to observe signatures from the early universe. Similar as
the 21 cm line of atomic hydrogen, molecules in the primordial gas can lead to some net absorption in
the cosmic microwave background, which would lead to small deviations from a blackbody spectrum.
Further, scattering of CMB photons with molecules can lead to a frequency-dependent change of the
CMB power spectrum. These possibilities have been considered first by Maoli et al. [1994]. It has
been proposed that H− could lead to significant net absorption in the CMB, which would be detectable
with present technology [Black, 2006]. Here, I calculate these effects based on a self-consistent
derivation of the chemical abundances, and including the processes of spontaneous and stimulated
emission. This more detailed model shows that no observational effects can be expected for satellites
like Planck, as the absorption is strongly balanced by corresponding emission processes. This work
was published by Schleicher, Galli, Palla, Camenzind, Klessen, Bartelmann, & Glover [2008c].
4.1 Introduction
The cosmic microwave background (CMB) is one of the most powerful tools of high-precision cosmology, as it allows one to determine the cosmological parameters, the power spectrum of initial
fluctuations and various other quantities. It is thus important to have a detailed theoretical understanding of all effects that have a potential influence on CMB measurements. Following the WMAP
3 year and 5 year results [Spergel et al., 2007; Hinshaw et al., 2007; Komatsu et al., 2008] that confirmed our standard picture of cosmology, we are looking forward to the precise measurement that
will be performed with Planck1 in only a few years. The measurement of the electron scattering optical depth allows one to constrain the effective reionization redshift and yields indirect information
1
http://www.rssd.esa.int/index.php?project=planck
63
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
about an epoch that cannot yet be observed. Recalling that the cross sections of bound electrons can
be larger by orders of magnitude compared to the cross section of free electrons, the optical depth due
to molecules may provide information on the dark ages of the universe, in spite of the small molecular
abundances. In fact, the early work of Maoli et al. [1994] suggested that the molecular opacities could
smear out CMB fluctuations on the scale of the horizon, and at the same time create new secondary
fluctuations due to the interaction with the velocity fields which are present in proto-clouds in the dark
ages. This work had the intention to explain why no CMB fluctuations had been observed at that time.
Since then, there has been considerable progress both in the understanding of chemical processes
in the early universe and the molecular abundances, as well as in the interaction of molecules with
the CMB and the generation of spectral-spatial fluctuations. While recombination was originally
examined by Peebles [1968] and Zeldovich et al. [1969], and improved in several follow-up works
[Matsuda et al., 1971; Jones & Wyse, 1985; Sasaki & Takahara, 1993] based on analytic methods,
today’s computers allow a detailed treatment of the recombination process based on a reaction network that takes into account hundreds of energy levels for H, He and He+ , as in the work of Seager,
Sasselov and Scott [Seager et al., 2000]. A simplified code reproducing the results of this detailed
calculation is given by Seager, Sasselov and Scott [Seager et al., 1999]. In a recent series of papers,
Switzer & Hirata [2008a]; Hirata & Switzer [2008]; Switzer & Hirata [2008b] considered the recombination of helium in great detail. Deviations of the CMB spectrum from a pure blackbody have also
been considered in various works. Dubrovich [1975] considered the effect of hydrogen recombination lines. Rubiño-Martı́n et al. [2006, 2008] examined distortions due to helium and hydrogen lines
in more detail, and Chluba & Sunyaev [2006] examined distortions due to the two-photon process.
Wong et al. [2008] give a good overview of recent improvements and uncertainties regarding the recombination process.
Here, the main focus is on the postrecombination universe and possible imprints in the CMB from
this period. The formation of H2 during the dark ages has already been discussed bySaslaw & Zipoy
[1967]. A more detailed treatment of molecules has been performed by Puy et al. [1993], Stancil
et al. [1998] and by Galli & Palla [1998]. A useful collection of analytic formulae for estimating the
abundance of various molecules after recombination was given by Anninos et al. [1997]. Recently,
this problem was re-examined by Puy & Signore [2007], and Hirata & Padmanabhan [2006] examined H2 formation in more detail, taking into account the effects due to non-thermal photons.
Regarding the interaction of molecules with the CMB, the effects of various molecules due to their
optical depths have been considered by Dubrovich [1994]. The enhancement of spectral-spatial fluctuations due to the luminescence effect, which is well-known from stars in reflection nebulae, has
also been discussed by Dubrovich [1997]. Observational prospects for Herschel and ODIN have been
discussed by Maoli et al. [2005], and the relevance for Planck has been assessed by Dubrovich et al.
64
4.2 Imprints from primordial molecules on the CMB
[2007]. The effects discussed were based on smear-out of primary CMB fluctuations due to the molecular optical depth and the generation of secondary anisotropies due to scattering with proto-objects
in the universe. Mayer & Duschl [2005] provided a recent overview of different processes that may
contribute to the opacity in primordial gas, and derived Rosseland and Planck mean opacities.
Black [2006] recently considered the influence of the bound-free transition of the negative hydrogen ion on the CMB, and found an optical depth of more than 10−5 at 10 cm−1 , which would have
interesting implications for Planck and other CMB experiments. This contributed to our original motivation to examine this and other effects in more detail. As we will show below, however, the optical
depth is due to H− bound-free transitions is much smaller than reported by Black [2006], and must
in addition be corrected for stimulated and spontaneous emission. On the other hand, there are other
effects from H− and HeH+ that are close to observational relevance.
From a numerical point of view, the pioneering work of Anninos et al. [1997] provided a flexible and easily extendible scheme that is still widely used in state-of-the art simulations of the early
universe, and which is also adopted in the public version of the Enzo code [O’Shea et al., 2004;
Bryan et al., 1995; Norman et al., 2007]1 . This scheme is extended here to account for effects during
recombination and the evolution of primordial chemistry in the homogeneous universe. We further
provide an extended overview of the essential processes that may influence the CMB, determine
the contributions from the most relevant species, and discuss the possibilities for detection with the
Planck satellite. In Chapter 4.2, we provide a general discussion of the potential effects of primordial
chemistry on the CMB. In Chapter 4.3, we present the general picture regarding the formation of the
first molecules in the universe and give some analytic estimates for the abundances. In Chapter4.4,
we present the chemical network for our calculation, which includes some new rates for the HeH+
molecules that are given in the appendix. Section 4.5 explains the numerical algorithm for the chemical network, and Chapter 4.6 presents the model abundances as a function of redshift. In Chapter4.7,
we explain our treatment of the different species and discuss the observational implications. Further
discussion and outlook is provided in Chapter 4.8.
4.2 Imprints from primordial molecules on the CMB
The interaction with the CMB is crucial to determine the evolution and abundance of primordial
molecules, and conversely, this interaction may also leave various imprints on the CMB photons while
they travel through the dark ages. The most obvious imprint on the CMB is probably a frequencydependent change in the observed CMB spectrum I(ν) due to absorption by a species M. An upper
1
http://lca.ucsd.edu/portal/software/enzo
65
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
limit on this effect is given by
I(ν) = B(ν)e−τM (ν) ,
(4.1)
where B(ν) denotes the unaffected CMB spectrum and the optical depth τM (ν) of species M at an
observed frequency ν is given by an integration over redshift as
%
τ M (ν) =
dl σM [ν0 (1 + z)] nM
(4.2)
%
nH,0 c z f
(1 + z)2
=
fM (z)σM [ν0 (1 + z)] &
dz,
H0 0
ΩΛ + (1 + z)3 Ωm
where σM (ν) is the absorption cross section of the considered species as a function of frequency,
nM the number density of the species, nH,0 the comoving hydrogen number density, c the speed of
light, H0 Hubble’s constant, fM the fractional abundance of the species relative to hydrogen, and zf
the redshift at which it starts to form efficiently. Such an optical depth can be provided by resonant
line transitions from molecules with high dipole moments like HeH+ and HD+ , free-free processes
or photodestruction of species like H− , He− or HeH+ , which have a relatively low photodissociation
threshold. However, Eq. 4.1 gives only an upper limit because absorption can be balanced by inverse
processes (spontaneous and stimulated emission). A better estimate can be obtained by introducing
an excitation temperature Tex defined by
!
"
nu gu
Eu − El
= exp −
(4.3)
nl
gl
kT ex
where nu and nl denote the population of the upper and the lower level, gu and gl are the corresponding statistical weights and k is Boltzmann’s constant. The excitation temperature is essentially
determined by the ratio of collisional and radiative de-excitations. For molecules with non-vanishing
dipole moments, the de-excitation is dominated by radiative transitions, and the level populations
are in equilibrium with the radiation temperature. The frequency-dependent change in the radiation
temperature is given by
∆T r ∼ − (T r − T ex ) τ M (ν).
(4.4)
This expression indicates a major obstacle for the detection of primordial molecules: those species
with high dipole moments and thus high cross sections have an excitation temperature which is very
close to the temperature of the CMB, while species with low dipole moments have a very low optical
depth. Molecular resonant line transitions are thus unlikely to lead to a non-negligible net change in
the radiation temperature. For photodestruction processes, this is different because the destruction is
regulated by the CMB temperature, while the inverse formation processes are governed by the gas
temperature. Thus, photodestruction can lead to a net change in the number of CMB photons. The
same is true for free-free processes, which emit a blackbody spectrum according to the gas temperature, but absorb the spectrum of the radiation field.
66
4.3 Recombination and the formation of molecules in the early universe
In addition, as discussed by Maoli et al. [1994], the optical depth from primordial molecules may
smear out primary fluctuations in the CMB if the optical depth acts in a way that effectively scatters
the CMB photons. This is possible even in a situation where stimulated and spontaneous emission
balance the absorption of molecules, such that there is no net change in the number of photons. It
is thus somewhat complementary to the effect discussed above. However, it must be noted that it
requires spontaneous emission to be important, as stimulated emission does not change the direction
of the photons, and does not provide a mechanism for scattering. We thus emphasize the importance
to correct for stimulated emission. As shown by Basu et al. [2004] for small angular scales, such an
optical depth (corrected for stimulated emission) then leads to a change in the power spectrum given
by
∆Cl ∼ −2τCl ,
(4.5)
where the Cl ’s are the usual expansion coefficients for the observed power spectrum. As discussed by
Maoli et al. [1994], such scattering processes can in addition generate secondary anisotropies which
are proportional to the optical depth of the scattering processes. However, this effect is suppressed by
more than three orders of magnitude, as it is also proportional to the ratio of the peculiar velocity to the
speed of light. Dubrovich [1997] discussed the luminescence effect of various molecules which could
potentially amplify the generation of secondary anisotropies. Based on the new abundances found
in this work, we will give a basic estimate of this effect in Chapter 4.8. Basu [2007] considered in
addition the effects of emission from molecules like HD and LiH+ , as HD in particular is an important
coolant in cold primordial gas. Unfortunately, the effect seems to be negligible.
4.3 Recombination and the formation of molecules in the early universe
4.3.1 Hydrogen recombination
Various approaches exist to calculate the time evolution of the ionized hydrogen fraction during recombination. The traditional approach is based on solving one single ordinary differential equation
for the ionization degree xe , which can be done using the approximation of an effective three-levelatom. This was first done by Peebles [1968], and an improved version of this equation was derived
by Jones & Wyse [1985]. This treatment was based essentially on the following assumptions:
• It is sufficient to take into account only hydrogen (helium is not considered).
• Collisional processes are outweighted by radiative processes and may be ignored.
• The relative populations of the excited fine structure states are thermal.
• Recombination and photoionization rates to and from higher states are related by Saha’s formula.
• Any such recombination cascades down to settle in the first excited state.
67
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
Figure 4.1: The idealized three-level hydrogen atom (ground state 1s, excited states 2s and 2p, continuum), and the relevant transitions.
• The level populations obey n1s 0 n2s , with n1s the number density of hydrogen atoms in the
ground state and n2s the corresponding density for the 2s state.
• Each net recombination gives rise to a Lyman-α photon or two others of lower energy.
Seager et al. [2000] presented a very detailed calculation independent of these a-priori assumptions, treating helium and hydrogen as multi-level atoms with several hundred levels and evolving
one ordinary differential equation (ODE) for each level, using a self-consistent treatment of the radiation field and its interaction with matter, which effectively leads to a speed-up of recombination. To
allow the integration of their new method in other cosmological applications like CMBFAST [Seljak
& Zaldarriaga, 1996], Seager et al. [1999] produced a simplified version of this code which is capable
of reproducing the results of the full multilevel treatment. This simplified code, RECFAST1, solves
three ODEs for the ionized hydrogen fraction xp , the ionized helium fraction xHe and the gas temperature T. These ODEs are parametrized so as to reproduce the speed-up in the recombination process.
In this paper, we make use of the ODE solving for the ionized hydrogen fraction, which is given by
dxp
dz
[xe xp nH αH − βH (1 − xp )e−h p νH,2s /kT ]
H(z)(1 + z)[1 + KH (ΛH + βH )nH (1 − xp )]
× [1 + KH ΛH nH (1 − xp )].
=
(4.6)
In this equation, nH is the number density of hydrogen atoms and ions, hp Planck’s constant and the
parametrized case B recombination coefficient for atomic hydrogen αH is given by
αH = F × 10−13
1
atb
cm3 s−1
1 + ctd
http://www.astro.ubc.ca/people/scott/recfast.html
68
(4.7)
4.3 Recombination and the formation of molecules in the early universe
with a = 4.309, b = −0.6166, c = 0.6703, d = 0.5300 and t = T/104 K, which is a fit by Pequignot
et al. [1991] to the coefficient of Hummer [1994]. This coefficient takes into account that direct
recombination into the ground state does not lead to a net increase of neutral hydrogen atoms, since the
photon emitted in the recombination process can ionize other hydrogen atoms in the neighbourhood.
The fudge factor F = 1.14 serves to speed up recombination and is determined from comparison
with the multilevel-code. The photoionization coefficient βH is calculated from the recombination
coefficient as βH = αH (2πme kT/h2p )3/2 exp(−h p νH,2s /kT ). The wavelength λH,2p corresponds to the
Lyman-α transition from the 2p state to the 1s state of the hydrogen atom. The frequency for the
two-photon transition between the states 2s and 1s is close to Lyman-α and is thus approximated by
νH,2s = c/λH,2p , where c is the speed of light (i.e., the same averaged wavelength is used). Finally,
ΛH = 8.22458 s−1 is the two-photon rate for the transition 2s-1s according toGoldman [1989], H(z)
is the Hubble factor and KH ≡ λ3H,2p /[8πH(z)] the cosmological redshifting of Lyman α photons.
4.3.2 H2 chemistry
Due to the expansion of the universe, not all of the free electrons will recombine with protons. Instead,
a freeze-out occurs, as recombination becomes less efficient at lower densities. The freeze-out abundance of free electrons was fitted by Peebles [1993] as
Ω1/2
nH+
−5 0
∼ 1.2 × 10
nH
hΩb
(4.8)
for cosmologies with total mass density parameter Ω0 , baryonic density parameter Ωb and h is the
Hubble constant in units of 100 km/s/Mpc. There is no contribution from helium to the free electron fraction, as helium recombines very early. The free electron fraction leads to the formation of
molecules and ions like H− and H+2 . As discussed by Anninos & Norman [1996] and Anninos et al.
[1997], their abundances can be calculated by assuming chemical equilibrium, as their formation and
destruction timescales are much shorter than the Hubble time. This yields the approximate expressions
nH−
nH
nH+2
nH
nH+
,
nH
nH+
∼ 3 × 10−14 T 1.8
.
nH
∼ 2 × 10−9 T 0.88
(4.9)
(4.10)
The matter temperature can be calculated by assuming that at z > 200 it is the same as the radiation
temperature, owing to efficient coupling of the temperatures through the Compton scattering of CMB
photons off the residual free electrons, while at z < 200, where this coupling becomes ineffective, it
evolves as for a simple adiabatic expansion [Sunyaev & Zeldovich, 1972; Anninos & Norman, 1996].
So
T0
(1 + z)2 , z < 200,
(4.11)
T = T 0 (1 + z), z > 200, T =
1 + 200
69
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
with T 0 = 2.726 K is the CMB temperature at z = 0 [Fixsen & Mather, 2002]. At redshifts z ≥ 100,
H− is efficiently photodissociated by the CMB and H2 is mainly formed by the process H+2 + H →
H2 + H+ . Assuming that H+2 is formed most efficiently at redshift z0 = 300 without being photodissociated by the CMB and that the hydrogen mass fraction is given by fH = 0.76, one obtains for
the H2 abundance [Anninos & Norman, 1996]
fH Ω3/2
nH2
0
∼ 2 × 10−20
(1 + z0 )5.1 .
nH
hΩb
(4.12)
4.3.3 Deuterium chemistry
The HD abundance is mainly determined by the deuteration of hydrogen molecules (i. e. H2 + D+ →
HD + H+ ). It is thus crucial to have the correct abundance of D+ , which is essentially determined by
charge exchange with hydrogen atoms and ions, i.e. the processes D + H+ → D+ + H and D+ + H →
D+H+ . As will be shown below, D+ is very close to chemical equilibrium, which yields the abundance
1/2
Ω
nD+
∼ 1.2 × 10−5 exp (−43 K/T ) 0 ,
nD
hΩb
(4.13)
when expression (4.8) is used. The abundance of neutral deuterium atoms nD can be determined by
assuming that deuterium is almost fully neutral, i. e. nD ∼ Ωb fD ρc /mD , where fD is the total mass
fraction of deuterium, mD the mass of one deuterium atom, ρc = 3H02 /8πG the critical density and G
is Newton’s constant. It is clear from expression (4.13) that the abundance of D+ drops exponentially
at low temperatures. Direct deuteration of molecular hydrogen can thus only occur at redshifts where
the exponential term is still of order one, before the exponential fall-off becomes significant. We
thus evaluate the relative abundance at redshift 90. For both the formation and the destruction process
HD+H+ → H2 +D+ , we estimate the rates with the simple expressions ofGalli & Palla [1998]. Again,
we emphasize that more detailed numerical calculations should use the rates given in the appendix.
The abundance is then given at z = 90 as
!
nHD
nH
"
z=90
∼ 1.1 × 10−7 fD exp(421 K/Tz=90 )
fH Ω03/2
hΩb
.
(4.14)
As there is no efficient destruction mechanism for HD at lower redshifts, the fractional abundance
remains almost constant for z < 90.
4.3.4 HeH+ chemistry
As we will show below in more detail, chemical equilibrium is also an excellent approximation to determine the abundance of HeH+ . For the rates presented in Galli & Palla [1998], the process of stimulated radiative association of H+ and He dominates over the non-stimulated rate. With the new rates
presented in appendix B, we find that both rates roughly coincide for gas temperatures greater than
10 K. Thus, for an analytic estimate, we approximate the combined formation rate through stimulated
70
4.4 The chemical network
and non-stimulated radiative association by taking twice the rate for non-stimulated radiative association. The dominant destruction process at low redshifts is charge-exchange via HeH+ + H → He+ H+2 ,
which yields for chemical equilibrium
nHeH+
#
T $−0.24
T $
exp −
300 K
4000 K
1/2 #
$
#
−0.24
Ω
T
T $
0
−16
∼ 7.03 × 10 nH
exp −
.
hΩB 300 K
4000 K
∼ 1.76 × 10−10 nHe (nH+ /nH )
#
(4.15)
4.4 The chemical network
In our chemical network, we have included the formation paths of H2 and HD. The complete list
of rates is given in Table B, and some new rates that are relevant for HeH+ are discussed in more
detail in the appendix. H2 can be formed by two main channels, via the reactions H− + H → H2 + e
and H+2 + H → H2 + H+ . A very good compilation for the H2 chemistry was given by Yoshida
et al. [2006]. Our compilation for the H2 formation rates is similar, but we do not include all of
their three-body-processes, as they are not relevant in the low-density regime explored here. Also,
we keep those modifications for low temperatures that were originally given by Abel et al. [1997].
The ionized fraction of hydrogen is not determined by solving rate equations, but from the RECFAST
code of Seager et al. [1999]. The photodissociation rates for most of the hydrogen species are those
of Galli & Palla [1998]. For the photodissociation of H+2 , we use the rate from their standard model,
which assumes that the the levels of the molecule are populated according to LTE. Since the H+2 level
populations will be strongly coupled to the CMB at the redshifts at which H+2 photodissociation is
significant, this assumption is more reasonable than using a rate that assumes that H+2 is completely in
the ground state. However, a better understanding of this molecule would certainly be desirable. For
the molecule HD+ , we estimate the photodissociation due to reactions 22 and 23 of Table A.1 as half
of the corresponding reaction for H+2 . This is in agreement with recent isotopic helium experiments
[Pedersen, 2005; Buhr, 2007], which found a similar effect for dissociative excitation with helium
isotopes.
Photodissociation of H2 by the Solomon process [reaction 20 Stecher & Williams, 1967] is calculated following the procedure described in Glover & Jappsen [2007], with the assumption that the
rotational and vibrational levels of H2 have their LTE level populations.
In a recent work of Capitelli et al. [Capitelli et al., 2007], it was shown that the reaction rate for
the process H2 + e → H− + H is larger by several orders of magnitude than the rate given byGalli &
Palla [1998]. This is because they include the vibrational levels of molecular hydrogen in their calculation, and find that the excited vibrational levels cannot be neglected for this process. The change
in the order of magnitude, however, does not lead to a significant change in the results, as this rate is
multiplied by the rather small densities of molecular hydrogen and free electrons.
71
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
The hydrogen and helium chemistry is almost completely decoupled, as can be seen from the small
rates for the charge exchange reactions 50 and 51. Their main interaction is via the HeH+ molecule.
Owing to its relatively high contribution to the optical depth for the CMB, we have included this
molecule in our chemical network and present some new rate coefficients for it in the appendix. The
molecule HD gives a contribution to the cooling and is mainly formed through by the deuteration
reactions 30 and 32, but there are also contributions from reactions 31, 43 and 44, involving HD+ ,
D− and H− . For the formation of deuterated molecules, it is therefore important to determine the
ionized fraction of deuterium, which is given through the charge exchange reactions 27 and 28. Our
deuterium network is inspired by the compilation ofNakamura & Umemura [2002]. The main difference compared to the deuterium network of Galli & Palla [1998] is the detailed treatment of D− . We
have added reaction 44, estimating its rate from reaction 43, thus also considering the contribution
of H− . Also, reactions 27 and 28 are calculated from the revised rates of Savin [2002]. As the fit
provided by Savin [2002] for the rate of reaction 27 becomes negative for T < 2.5 K, it is set to zero
at these temperatures1 . For temperatures larger than 200 K, the revised set of deuterium rates of Galli
and Palla [Galli & Palla, 2002] are used. For lower temperatures, however, some of the new rates
show unphysical divergences. In these cases, we use the rates from Galli & Palla [1998] when the
temperature drops below 200 K.
4.5 The numerical algorithm
To determine the evolution of primordial molecules in the early universe, we have employed the
chemical network of the Enzo code [O’Shea et al., 2004; Bryan et al., 1995; Norman et al., 2007]
and extended the numerical approach developed byAnninos et al. [1997] for primordial chemistry to
determine the chemical evolution of the homogeneous universe. The main issue was the calculation of
the ionization fraction, which is determined by complex interactions between the CMB and the ground
state as well as the excited states of atomic hydrogen, and which goes beyond typical applications of
primordial chemistry. We thus included the RECFAST code of Seager et al. [1999] as a subroutine for
this calculation. For the deuterium and helium species as well as the molecules, however, we use the
first order backwards differencing (BDF) method developed by Anninos et al. [1997]. The chemical
timestep is set to 1% of the hydrodynamical timestep. The latter is given by ∆thydro = η(∆x/cs ), where
∆x is the cell size, cs is the sound speed and η is a safety factor, here taken to be 0.5. This proved
sufficient to resolve the relevant chemical timescales, and simulations performed with even smaller
timesteps gave identical results. The rate equations for the species i are given in the form
dni
= −Di ni + Ci ,
dt
1
(4.16)
Note that this does not introduce a significant error, as owing to its exponential dependence on temperature, rate 27 is
tiny at T < 2.5 K.
72
4.5 The numerical algorithm
where Di and Ci are the destruction and creation coefficient for species i, respectively, which in
general depend on the number densities of the other species and on the radiation field. Equation4.16
is discretized and the right-hand-side is evaluated at the new timestep, yielding
nnew − nold
new
+ Cinew .
= −Dnew
i n
∆t
This can be solved for nnew
i :
nnew
=
i
C new ∆t + nold
i
.
1 + Dnew ∆t
(4.17)
(4.18)
The coefficients Dnew
and Cinew are in general not known, but can be approximated using the species
i
densities from the old timestep, and those species from the new timestep which have already been
evaluated. Anninos et al. [1997] has argued that H− and H+2 can even be evaluated assuming chemical
equilibrium, since their reactions rates are much faster than those of the other species. Assuming
some species j in chemical equilibrium essentially means that ṅj = 0, yielding
n j,eq =
Cj
.
Dj
(4.19)
In our chemical model, we have some additional species which have sufficiently fast reaction rates:
HeH+ , D− and HD+ . To check the validity of this assumption, we calculate the formation of molecules
in two simulations: one using the non-equilibrium prescription (4.18) for all species, and one using
the equilibrium description (4.19) for the species with fast reaction rates. The results are presented
in Chapter 4.6 and confirm that the abundances of these species can be derived assuming chemical
equilibrium.
The BDF method is not a fully implicit numerical scheme, as several of the destruction and
creation mechanisms must be approximated using the species densities from the previous timestep.
However, we found that it is stable when the ionized fraction is provided from RECFAST. For consistency checks, we have varied the chemical timestep and explicitly ensured mass conservation for
hydrogen, helium and deuterium, yielding consistent results. Note that mass conservation, charge
conservation and positivity must be ensured as described byAnninos et al. [1997] if the ionized fraction is not provided from an independent routine.
We summarize the numerical algorithm in the following way:
• Loop over the chemical timestep tchem = 0.01thydro until the species have been evolved through
the total hydrodynamical timestep.
• Update the temperature and the abundances of H, H+ and e with RECFAST.
73
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
Species number density fraction
1e-08
H2II
HM
1e-10
1e-12
1e-14
1e-16
1e-18
1e-20
1e-22
1000
100
10
1
Redshift
Figure 4.2: Results for the evolution of H +2 and H− .
• Calculate the abundances of the atomic and ionized helium species using the non-equilibrium
prescription (4.18), and the abundance of the molecule HeH+ using (4.19).
• Calculate the abundances of H− and H+2 using (4.19), and the abundance of H2 using (4.18).
• Calculate D using (4.18), D+ , D− and HD+ using (4.19), and finally HD using (4.18).
4.6 Results from the molecular network
The formation of molecules after recombination was calculated for a ΛCDM model with Ωdm =
0.222, Ωb = 0.044, ΩΛ = 0.734, H0 = 70.9 km/s/Mpc, Y p = 0.242, [D/H] = 2.4 × 10−5 , where
Ωdm , Ωb , ΩΛ are the density parameters for dark and baryonic matter as well as dark energy, H0 is
Hubble’s constant, Yp is the mass fraction of helium with respect to the total baryonic mass and
[D/H] is the mass fraction of deuterium relative to hydrogen. These are the parameters from the
combined set of WMAP 3-year data and the data of the Sloan Digital Sky Survey (SDSS) [Spergel
et al., 2007]. We expect only marginal changes in the results for the recently published WMAP 5-year
data, as the cosmological parameters did not change significantly [Komatsu et al., 2008]. We have
performed the calculation for the HD cooling functions of Galli & Palla [1998], Flower et al. [2000]
and Lipovka et al. [2005], and find that the results are not sensitive to this choice. For our H2 cooling
function, we use that of Galli & Palla [1998], but H2 cooling of the gas is never important, and our
results would not change significantly if we were to use the revised cooling rate ofGlover & Abel
74
4.6 Results from the molecular network
Species number density fraction
1e-04
DII
HeHII
1e-06
1e-08
1e-10
1e-12
1e-14
1e-16
1e-18
1e-20
1e-22
1000
100
10
1
Redshift
Figure 4.3: Results for the evolution of D + and HeH+ .
Species number density fraction
1e-14
HDII
DM
1e-16
1e-18
1e-20
1e-22
1e-24
1e-26
1000
100
10
Redshift
Figure 4.4: Results for the evolution of HD + and D− .
75
1
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
Species number density fraction
100000
HI
HII
H2I
DI
HDI
1
1e-05
1e-10
1e-15
1000
100
10
1
Redshift
Figure 4.5: Results for those species which freeze-out and are not in chemical equilibrium.
Table 4.1: Freeze-out values of free electrons, H 2 and HD at z=100.
Cosmology / Ref.
WMAP 3 + SDSS
[Puy & Signore, 2007]
[Hirata, 2006]
Analytic approx.
[Galli & Palla, 1998]
[e /H ]
10−4
2.63 ×
∼ 10−4
2.0 × 10−4
∼ 2.4 × 10−4
76
[H2 /H]
10−7
4.23 ×
1.5 × 10−7
∼ 2 × 10−8
2.9 × 10−7
∼ 1 × 10−7
[HD/H]
2.39 × 10−10
∼ 2 × 10−10
1.6 × 10−10
8 × 10−11
4.6 Results from the molecular network
Table 4.2: Freeze-out values of free electrons, H 2 and HD at z=10.
Cosmology / Ref.
WMAP 3 + SDSS
[Puy & Signore, 2007]
[Hirata, 2006]
[Seager et al., 1999]
Analytic approx.
[Galli & Palla, 1998]
[H2 /H]
[e /H ]
10−4
1.92 ×
∼ 6 × 10−5
1.92 × 10−4
2.0 × 10−4
3.02 × 10−4
10−7
7.97 ×
1.13 × 10−6
∼ 6 × 10−7
2.9 × 10−7
1.1 × 10−6
[HD/H]
7.52 × 10−10
3.67 × 10−10
3.0 × 10−10
1.21 × 10−9
Table 4.3: Abundances of D + and HeH+ at z=100.
Cosmology / Ref.
[D+ /H]
[HeH+ /H]
WMAP 3 + SDSS
[Puy & Signore, 2007]
Analytic approx.
[Galli & Palla, 1998]
5.15 × 10−9
∼ 1.5 × 10−9
3.7 × 10−9
∼ 7.5 × 10−9
4.27 × 10−15
∼ 2.8 × 10−15
4.5 × 10−15
∼ 1.8 × 10−14
Table 4.4: Abundances of D + and HeH+ at z=10.
Cosmology / Ref.
[D+ /H]
[HeH+ /H]
WMAP 3 + SDSS
[Puy & Signore, 2007]
Analytic approx.
[Galli & Palla, 1998]
8.02 × 10−16
∼ 1 × 10−19
2.1 × 10−20
∼ 5 × 10−17
9.98 × 10−15
4.6 × 10−14
1.3 × 10−14
∼ 6.5 × 10−14
77
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
[2008]. The detailed evolution of the various species is plotted in Figs.4.6-4.6. In Fig. 4.6, we give
the results for H+2 and H− , Fig. 4.6 shows the evolution of D+ and HeH+ , and Fig. 4.7.5 the evolution of HD+ and D− . For the other species, the results are given in Fig. 4.6. For several interesting
species, we give the results at z = 100 and z = 10 in tables4.1-4.4, and compare them to the results of
Galli & Palla [1998], Puy & Signore [2007], Seager et al. [1999] and the analytical approximations of
Anninos & Norman [1996] and Chapter 4.3. As the analytic approximations do not take into account
the H− channel of H2 formation, they underestimate the abundance found in the numerical calculation
by roughly a factor of 2 at redshift 10.
The results clearly show the typical evolution of primordial chemistry as it is known from previous works. The two main channels for H2 formation, via H− and H+2 , are reflected in its cosmic
formation history, yielding a first major increase at z ∼ 300, where the relative abundance of H+2
reaches a maximum, and a second major increase at z ∼ 100, at a maximum of the H− fraction. At
redshifts below 5, there is a new rise in the abundance of H+2 . This is likely an unphysical feature
from the fit to the rate, which is not valid below 1 K. However, the real evolution at these low redshifts will in any case depart from our calculation due to reionization, metal enrichment and structure
formation, and this feature is not relevant with respect to the CMB. The evolution of the deuterium
species essentially follows the evolution of the hydrogen species. Since deuterium and hydrogen are
strongly coupled via charge-exchange reactions, they recombine at almost the same time. However,
due to the efficient charge-exchange reactions, there is no freeze-out of D+ . Instead, its abundance
drops exponentially at low redshifts. D− and HD+ peak at the same redshifts as H− and H+2 , and
the evolution of HD resembles closely the evolution of H+2 , as the dominant HD formation channel is
given by H2 + D+ → HD + H+ . The evolution of HeH+ consists of a first phase where its evolution
is determined by the effectiveness of photodissociation, and a second phase where it is determined by
charge-exchange with neutral hydrogen atoms. Due to the new formation rates presented in appendix
B, formation through stimulated and non-stimulated radiative association of He and H+ is almost
equally important and in total less effective than found in previous works [Puy & Signore, 2007; Galli
& Palla, 1998; Stancil et al., 1998].
Numerically, the free electron fraction found in this work roughly agrees with the analytic approximations of Anninos & Norman [1996] and previous results of Galli & Palla [1998], while Puy & Signore
[2007] give a somewhat lower abundance. As the electrons act as catalysts for H2 formation, this is
reflected also in the abundance of molecular hydrogen, and similarly in the abundance of HD , as it is
primarily formed by direct deuteration of molecular hydrogen. At z = 100, the H2 abundance found
by Hirata & Padmanabhan [2006] is still more than one order of magnitude below the abundance
we find here, due to the effects of non-thermal photons, while at redshift 10, this effect is much less
important and their result is only 25% below the value found here. The analytic estimate ofAnninos
et al. [1997] also underestimates the H2 abundance at z = 10 by more than a factor of 2, as it does not
take into account the H2 formation via H− . At z = 100, the abundances for D+ are of the same order
78
4.7 Effects of different species on the cosmic microwave background
of magnitude, although some differences exist, which may be due to the differences in the rates for
the charge-exchange reactions, as well as differences in the abundance of ionized hydrogen. For the
HeH+ molecule, the abundance is still comparable to previous results at redshift 100, but lower by a
factor of 5 at z = 10.
We emphasize here that the abundance of H− is similar to the one found by Black [2006]. The
peaks in the abundance at z ∼ 100 and z ∼ 1400 are reproduced, and their height agrees with the
results from our calculation when the physical number density given in his paper is converted to the
fractional abundance given in Fig. 4.6. We estimate the uncertainty from reading off the values of his
Fig. 1 to be a factor 2-3, and we point out that this is not sufficient to explain the difference in the
optical depth that we find below.
4.7 Effects of different species on the cosmic microwave background
4.7.1 Molecular lines
Due to the discreteness of the molecular lines, contributions to the optical depth arise only in narrow
redshift intervals of the order ∆z/z ∼ 10−5 , corresponding to the ratio between the thermal linewidth
to the frequency of the transition. Peculiar motions of the order 300 × (1 + z)−1/2 km s−1 may further
increase the effective linewidth by one or two orders of magnitude. This has no influence on the results presented here which depend only on the product of ∆z with the profile function that scales with
the inverse of the linewidth, but may induce additional anisotropies, as we discuss in Chapter4.8.
To evaluate Eq. 4.2 for a given frequency, we compute all the redshift intervals for which the photon
frequency lies within an absorption line of the molecule, and add up the contributions from these
frequencies. The relative importance of various molecules for the optical depth calculation can be estimated with the formulae of Dubrovich [1994]. The most promising candidate is the HeH+ molecule,
as it has a strong dipole moment and is formed from quite abundant species. Unfortunately, its destruction rate is very high as well. Another interesting molecule with a strong dipole moment is
HD+ . However, as a deuterated molecule, it has an even lower abundance. H2 D+ is not considered
because it has an even lower abundance [Galli & Palla, 1998], and H+3 has both a low abundance and
no dipole moment. The other molecule in our chemical network with a non-zero dipole moment is
HD. However, its dipole moment is eight orders of magnitude smaller than that of HeH+ , and so
even though its peak abundance is five orders of magnitude larger, its effects will still be negligible
compared to those of HeH+ . We therefore do not consider it further. In spite of its strong dipole
moment, LiH is also not considered, as it was already shown by Bougleux & Galli [1997] and Galli
& Palla [1998] that its abundance is lower by roughly 10 orders of magnitude compared to the value
assumed by Maoli et al. [1994], who discussed its potential relevance.
79
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
For HeH+ , we calculate the optical depth by using the large dataset provided by Engel et al.
[2005], which allows one to derive the line cross sections from the Einstein coefficients. The line
cross section σM,i weighted by the level population for a transition from an initial state i with vibrational quantum number v(( , rotational quantum number J(( and energy E(( , to a final state f with
vibrational quantum number v( and rotational quantum number J( is given by
!
"
1.3271 × 10−12 (2J ( + 1)c2
−E ((
exp
−
kT r
Qvr ν2
"(
'
!
hpν
A f i Φ(ν − ν f i ),
× 1 − exp −
kT r
σ M,i (ν) =
(4.20)
A f i is the Einstein coefficient, ν f i is the frequency at line centre of the transition i, Φ(ν − νf i ) =
√
1√
2 /∆ν2 ) the profile function for the line width ∆ν =
exp(−(ν
−
ν
)
2kT/m M ν f i /c, m M the
f
i
D
D
∆νD π
;
mass of the molecule and Qvr is the partition function, given by Qvr = i gi exp(−Ei /kT ) with the
degeneracies gi = 2J + 1. Again, we emphasize that the results are insensitive to the choice of the
linewidth. The total cross section σM is obtained as a sum over all σM,i . For temperatures between
500 and 10000 K, we use the fit provided by Engel et al. [2005], while
temperatures,
we
< for lower
/
0=
do linearly interpolate between the values in their Table 5. The factor 1 − exp −h p ν/kT r takes into
account the correction for stimulated emission, which is especially relevant for the pure rotational
transitions with low frequencies. As discussed in Chapter 4.2, this must be taken into account as
stimulated emission does not change the direction of the emitted photons.
For HD+ , we use the same formalism as for HeH+ , but we determine the partition function from
the accurate energy levels given byKarr & Hilico [2006]. Following Shu [1991], we use the transition
moments |D f i | given by Colbourn & Bunker [1976] to determine the Einstein coefficients for the rovibrational lines as
32π3 ν3f i
Afi =
|D f i |2 ,
(4.21)
3! p c3
where ! p is the reduced Planck constant.
The Einstein coefficients for the pure rotational transitions are calculated with the dipole moment
D0 = 0.86 Debye of Dubrovich [1994] from
A0,J→0,J−1 =
32π3 ν3f i
3! p c3
D20
J
.
2J + 1
(4.22)
4.7.2 The negative hydrogen ion.
There are two effects associated with the negative hydrogen ion than can affect the optical depth seen
by CMB photons: the bound-free process of photodetachment that has also been discussed byBlack
80
4.7 Effects of different species on the cosmic microwave background
[2006], and free-free transitions that involve an intermediate state of excited H− , i. e.
2 3∗
H + e + γ → H− → H + e.
(4.23)
While the importance of the free-free process is well-known for stellar atmospheres, there has been
little work on this process in the low-temperature regime. As the fit formulae given byJohn [1988] and
Gingerich [1961] diverge at low temperatures, we have updated previous work of Dalgarno & Lane
[1966] to calculate the free-free absorption coefficient for the low temperature regime as described
in appendix A, and we use the fit of John [1988] to the calculation of Bell & Berrington [1987] for
temperatures higher than 2000 K, where it is accurate within 1%. For the bound-free process, we
use the fit of John [1988] to the calculations of Wishart [1979]. The treatment regarding absorption,
spontaneous and stimulated emission is based on the expressions ofRuden et al. [1990], but with the
updated cross sections mentioned above. Apart from the usual optical depth due to absorption, we
introduce also effective optical depths due to stimulated and spontaneous emission. The following
expressions have to be evaluated for the redshift z, and the frequency dependence is suppressed for
simplicity. Of course, we emphasize that the observed frequencies at z = 0 must be related correctly
to the physical frequencies at higher redshifts. As usual, the contribution to absorption is given as
dτb f,abs = +nH− σb f ds,
(4.24)
where ds is the cosmological line element, σb f the cross section for the bound-free transition and
nH− the number density of H− . We further introduce the Planck spectrum BT (ν) of temperature T and
frequency ν. The contribution to the effective optical depth from stimulated emission is then given as
dτb f,stim = −(nH− )LT E σb f e−h p ν/kT ds,
(4.25)
where the LTE abundance (nH− )LT E of H− is given as
(nH− )LT E = ne nH
λ3e h p ν0 /kT
e
,
4
(4.26)
h
where λe = √2πmp kT is the thermal de Broglie wavelength and ν0 = 0.754 eV the binding energy of
e
H− . Note that the LTE abundance must be used here, as the processes of spontaneous and stimulated
emission depend on the actual density of electrons and hydrogen atoms, and in general the correction
for stimulated emission cannot be included as a factor of (1 − exp(−hν/kT )) unless the H− ion has
its LTE abundance [Ruden et al., 1990]. The effective optical depth due to spontaneous emission is
further given as
2h p ν3
dτb f,spon = −(nH− )LT E 2
σb f e−h p ν/kT ds.
(4.27)
c BT r (ν)
For the free-free effect, stimulated emission is already included in the rate coefficients aν (T ) given by
John [1988], which are normalized to the number density of neutral hydrogen atoms and the electron
81
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
pressure. The effective contributions to the optical depth from absorption and emission are then given
as
dτ f f,abs = +nH ne kT aν (T )ds,
BT g (ν)
dτ f f,em = −nH ne kT aν (T )ds
.
BT r (ν)
(4.28)
(4.29)
The absorption by free-free transitions is thus proportional to a black-body spectrum for the radiation
temperature, while the emission produces a spectrum determined by the gas temperature. In the
following, we will refer further to the optical depth from absorption, which we define as
% /
0
=
dτb f,abs + dτ f f,abs ,
(4.30)
% /
0
=
dτb f,abs + dτb f,stim + dτb f,spon + dτ f f,abs + dτ f f,em ,
(4.31)
τabs
and the effective optical depth
τe f f
which includes the corrections for emission. While the optical depth due to absorption, τabs , is essentially responsible for photon scattering and a change in the power spectrum according to Eq. (4.5),
the effective optical depth τe f f leads to a net change in the radiation flux. The resulting change in the
CMB temperature can be obtained by a linear expansion as
∆T = τe f f (ν)
BT r (ν)
= f (ν)τe f f (ν)T r ,
(∂BT r (ν)/∂T )T r
(4.32)
where we have introduced a frequency dependent correction factor f (ν), which can be evaluated to
first order as
<
= kT r
.
(4.33)
f (ν) = (1 − exp −h p ν/kT )
hν
4.7.3 The negative helium ion.
He− is to some degree similar to the H− . However, only the free-free process contributes in practice,
as any bound states autoionize on a timescale of the order of hundreds of microseconds [Holøien &
Midtdal, 1955; Brage & Fischer, 1991]. Thus, we take into account only the free-free process, which
can be treated in the same way as for H− . We approximate the corresponding free-free coefficient by
a power law proportional to ν−2 and normalize with the data given by John [1994]. Such a treatment
should be sufficient up to frequencies of 1000 GHz, and thus for the frequency range interesting for
the Planck satellite.
82
4.7 Effects of different species on the cosmic microwave background
HeHII photodissociation cross section
1e-18
HeHII photodissociation
Cross section [cm^2]
1e-19
1e-20
1e-21
1e-22
1e-23
1e-24
1e-25
1e-26
304000 305000 306000 307000 308000 309000 310000
Frequency [GHz]
Figure 4.6: HeH+ photodissociation cross section obtained from detailed balance.
4.7.4 Photodissociation of HeH+ .
We used detailed balance to determine the photodissociation cross section from inverse reaction. The
latter is essentially determined by several narrow resonances, which have been tabulated byZygelman
et al. [1998]. The photodissociation cross section is thus given as the sum over resonances i as
σ ph (ν) =
: me c2 (h p ν − E0 )
i
(h p
ν)2
Γr,i Γi /2
,
(h p ν − E0 − Er,i )2 + (Γ/2)2
(4.34)
where the parameters Er,i , Γr,i and Γi can be read off from Table 2 of Zygelman et al. [1998], and E0 is
the photodissociation threshold for HeH+ . From Dubrovich [1997], we adopt the value E0 = 1.85 eV.
The resulting photodissociation cross section is displayed in Fig.4.7.4. We consider only absorption,
as it is sufficient to rule out the contribution from this molecule with respect to Planck.
4.7.5 Observational relevance and results
As discussed in Chapter 4.2, the optical depth from resonant scattering can effect the CMB by a
change in the power spectrum according to Eq. (4.5), and it may also produce secondary anisotropies.
We neglect the latter effect for the moment, as it is suppressed by the ratio of the peculiar velocity
over the speed of light. To quantify the importance of HeH+ and HD+ for a change in the power
spectrum, we have calculated the optical depth and corrected for stimulated emission as described in
83
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
HeHII optical depth
total, uncorrected
total, corrected
rotational, corrected
ro-vibrational, corrected
Optical depth
1e-06
1e-08
1e-10
1e-12
1e-14
1e-16
10
100
Frequency [GHz]
1000
Figure 4.7: The HeH+ optical depth, both corrected and uncorrected for stimulated emission. The contributions from pure rotational and ro-vibrational transitions are given separately.
HDII optical depth
1e-07
total, uncorrected
total, corrected
rotational, corrected
ro-vibrational, corrected
1e-08
Optical depth
1e-09
1e-10
1e-11
1e-12
1e-13
1e-14
1e-15
1e-16
10
100
1000
Frequency [GHz]
Figure 4.8: The HD+ optical depth, both corrected and uncorrected for stimulated emission. The contributions from pure rotational and ro-vibrational transitions are given separately.
84
4.7 Effects of different species on the cosmic microwave background
0.01
HM free-free
HM bound-free
HeM free-free
HeHII photo
total
1e-04
Optical depth
1e-06
1e-08
1e-10
1e-12
1e-14
1e-16
1e-18
10
100
1000
Frequency [GHz]
Figure 4.9: The absorption optical depth due to different processes: the free-free processes of H − and
He− , the bound-free process of H − , and the photodissociation of HeH + . Clearly, the total optical depth is
dominated by the processes involving H − .
the previous subsection. The results are given in Figs. 4.7.5 and 4.7.5. We find that the correction for
stimulated emission is especially important for the lower frequencies of the pure rotational transitions.
As explained by Basu et al. [2004], the sensitivity is not limited by cosmic variance when power spectra at different frequencies are compared, but the limit from instrumental noise corresponds to optical
depths of 10−5 for the high-frequency bins. Thus, the signal is likely below the sensitivity of the
Planck satellite by two orders of magnitude, but reasonable upper limits on the abundance of HeH+
are feasible. From Figs. B.1.a and B.2.a in the appendix, we estimate the uncertainty in the formation
rate to be a factor of 2, while the destruction rate might be larger by up to an order of magnitude, if the
old values of Roberge and Dalgarno [Roberge & Dalgarno, 1982] are adopted. This defines the main
uncertainty in this result. Even with an improved instrumental sensitivity, very accurate foreground
subtraction would be required and may create additional noise.
Fig. 4.7.5 shows the optical depth due to absorption by several free-free and photodestruction
processes. We find that it is dominated at low frequencies by the free-free contributions of H− , and at
high frequencies by the bound-free process of H− . The optical depth from the free-free processes of
H− and He− is essentially proportional to ν−2 , at least for frequencies smaller than 1000 GHz. This
is essentially due to the characteristic frequency dependence in the absorption coefficient. Due to the
85
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
Delta T over T, Optical depth
1e-04
Delta T over T
tau_abs,bf
tau_stim,bf
tau_spont,bf
1e-06
1e-08
1e-10
1e-12
1e-14
1e-16
10
100
1000
Frequency [GHz]
Figure 4.10: The relative change in the CMB temperature due to the presence of H − . We further plot the
optical depths due to absorption, spontaneous and stimulated emission for the bound-free process, as their
overlap explains some features in the temperature change.
86
4.8 Discussion and outlook
approximation used for the He− free-free absorption coefficient, the corresponding optical depth is
somewhat overestimated for larger frequencies, but it is still only a subdominant contribution to the
total optical depth. The effect from the bound-free transition dominates, but is significantly lower than
previously reported by Black [2006], and the reason for this discrepancy is not obvious: as pointed
out in Chapter 4.6, our H− abundance agrees with his within a factor of 2 or 3, whereas the optical
depth at high frequencies is different by three orders of magnitude. Although we have been unable to
identify the reason for this disagreement, we have carefully checked our result and are confident that
it is correct.
As discussed in Chapter 4.2, photodestruction and free-free processes can change the net number
of CMB photons. The calculated change in the CMB temperature due to the free-free and bound-free
effect is given in Fig. 4.7.5, and depends on the effective optical depth defined in Eq. (4.31). The
change in the temperature due to the free-free transition is significantly lower than the absorption
optical depth in Fig. 4.7.5, as the main contribution to the absorption optical depth comes from high
redshifts z > 300, where the difference between radiation and gas temperature is very small and the
optical depth due to absorption and emission balance each other. For the bound-free transition, the
absorption is balanced to some extend by the spontaneous emission, and for frequencies larger than
1000 GHz, spontaneous emission in fact dominates, as the CMB flux at these frequencies is low by
comparison. Near 150 and 1000 GHz, the contributions from spontaneous emission and absorption
due to the bound-free process in fact become equal, making the net effect almost zero in a small frequency range.
The net change in the CMB temperature due to the free-free and bound-free processes of H− is
small because absorption and emission processes are close to equilibrium. The optical depth due to
absorption is thus considerably higher than the effective optical depth defined in Eq. (4.31), and can
lead to a change in the power spectrum according to Eq. (4.5). Finally, we note that at redshifts where
reaction 2 of Table B.1 dominates the destruction of H− , there is an uncertainty of up to an order of
magnitude in our predicted H− abundance, owing to the uncertainty in the rate of reaction 2 discussed
in detail in Glover et al. [2006]. However, this uncertainty does not appear to significantly affect the
size of our predicted signal, as the dominant contribution comes from redshifts at which reaction 2 is
unimportant.
4.8 Discussion and outlook
We have provided a detailed network for primordial chemistry and solved for the evolution in the
homogeneous universe, and we examined the various ways in which primordial species can influence
the CMB. The detailed calculation in the previous sections suggests that the H− ion is only one order
87
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
of magnitude below the detection threshold and strong upper limits on its abundance seem feasible,
even though an accurate subtraction of frequency-dependent foregrounds will be required for this
purpose. The relative deviations of the CMB from a pure blackbody have been constrained byMather
et al. [1994]; Fixsen et al. [1996] to less than 1.5 × 10−5 . Distortions are also expected due to the
two-photon process during recombination [Chluba & Sunyaev, 2006] and the helium and hydrogen
lines [Rubiño-Martı́n et al., 2006, 2008]. Finestructure transitions in heavy elements are also expect
to produce some scattering after recombination [Basu et al., 2004]. An accurate measurement of
distortions in the CMB and a precise measurement of the CMB power spectrum can thus improve
our understanding of various processes during and after recombination if an accurate foreground
subtraction is feasible. Given the tiny change in the CMB temperature found in this work, we further
conclude that detecting the change in the power spectrum caused by scattering (see Eq. (4.5)) is
the most promising way to obtain constraints on the chemistry of the dark ages by future CMB
experiments.
So far, we have only taken into account effects arising from the homogeneous universe. One
might argue that the molecular abundances could be very different if there were a protogalaxy at the
redshift of resonance. In fact, the redshift intervals for which resonance occurs are very narrow, corresponding to some 100 pc. This might be further increased by peculiar motions and local turbulence.
Núñez-López et al. [2006] demonstrated that HD in cold collapsed clouds can lead to a strong local
fluctuation of the order 10−5 . Given the small volume fraction of such clouds, we neglect their impact
in the present paper, although we will examine the effect of such inhomogeneous fluctuations in future
work.
Dubrovich [1997] suggested luminescence as an additional effect that may amplify the signal and
lead to a strong frequency dependence. This effect is well-known for stars in reflection nebulae. It
occurs at redshifts z = 300 − 100, where the rotational lines lie in the extreme Rayleigh-Jeans wing of
the CMB and the first vibrational line near its maximum. Inelastic scattering in local velocity fields
might thus provide much stronger frequency-dependent fluctuations. It is convenient to estimate the
effect with the formula given by Dubrovich [1997]:
!
"
V p Ωb
∆T r
fm
∆T r
=
,
(4.35)
−10
Tr
T r 0 10 30 km/s 0.1
where fm is the fractional abundance of the molecule, Vp the peculiar velocity which can be estimated
√
as V p = V p (0)/ 1 + z, V p (0) = 600 km/s, and the quantity (∆T r /T r )0 can be conveniently read off
from Fig. 2 of Dubrovich [1997]. For HeH+ , it yields a maximum effect of roughly ∆Tr /T r ∼ 10−11 ,
and ∆Tr /T r ∼ 10−12 for HD+ , which are clearly below the sensitivity of the Planck satellite.
As discussed by Launay et al. [1991], H2 is in general formed in excited states, but quickly decays
into the ground state. Thus, H2 formation produces additional photons that may lie within the CMB
radiation and thus produce distortions to the blackbody radiation. Prelimary estimates based on the
88
4.8 Discussion and outlook
transition between the first excited vibrational state and the ground state indicate that the effect yields
a relative change in the CMB temperature of the order 10−15 , and is thus negligible.
At the end of the dark ages, reionization will dramatically change the chemical evolution of the
intergalactic gas, and produce large regions of ionized gas. In such regions, other formation channels
for molecules could be relevant, like He+ + H → HeH+ + γ. The reaction rate of this channel is four
orders of magnitude larger than the radiative association of H+ and He. On the other hand, the influence of destruction processes, such as dissociative recombination and photodissociation, will also
be enhanced. The details of the evolution will also depend on the ratio of stellar to quasar sources
and the details of the transition from Pop III to Pop II stars. A detailed analysis of this contribution
is beyond the scope of this work, but clearly there is the possibility that this epoch could further increase the optical depths of HeH+ and H− . Other contributions may arise from heavy elements at low
redshift. Basu et al. [2004], Hernández-Monteagudo et al. [2006] and Basu [2007] suggested that one
could use the change in the power spectrum of the CMB to constrain the chemical evolution of the
low redshift universe. In fact, during the formation of metals and in early star forming regions, additional effects may occur that leave an interesting imprint in the CMB. As described byHernándezMonteagudo et al. [2007a], oxygen pumping may change the CMB temperature in metal enriched
environments in a similar way as the Wouthuysen-Field effect that is well-known from 21 cm studies
[Wouthuysen, 1952; Field, 1958], and the inhomogeneous distribution of metallicity in bubble-like
structures may influence the CMB power spectrum as described by Hernández-Monteagudo et al.
[2008]. Hernández-Monteagudo et al. [2007b] further studied the effect of resonant scattering during
reionization and recombination. In addition, star-forming regions may perturb the primordial signal
through dust and molecular emission, especially CO [Righi et al., 2008a,b]. Such a potentially rich
phenomenology will of course require a very careful analysis and a clear assessment of the different frequency dependences of various effects once the required sensitivity is reached. In the mean
time, the increasing sensitivity in instruments like Planck [Bersanelli et al., 1996], the South Pole
Telescope1 [Ruhl et al., 2004] and the Atacama Cosmology Telescope2 [Fowler, 2004] will allow at
least to set upper limits that may constrain theories involving the dark ages, reionization and recombination.
1
2
http://pole.uchicago.edu/
http://www.physics.princeton.edu/act/
89
4. THE CHEMISTRY OF THE EARLY UNIVERSE AND ITS SIGNATURES IN THE CMB
90
5
Dark stars: Implications and constraints
from cosmic reionization and
extragalactic background radiation
In this chapter, observational constraints on dark stars powered by dark matter annihilation rather than
nuclear fusion are examined in more detail. In particular, it is shown that it is difficult to reconcile
scenarios that predict ∼ 1000 M' stars or lifetimes of ∼ 100 Myr with the observed reionization
constraints. I further discuss how the dark matter densities which are enhanced by adiabatic contraction during the formation of the stars can contribute to the cosmic gamma-ray background and the
atmospheric neutrino background. The work presented here was published bySchleicher, Banerjee,
& Klessen [2008a].
5.1 Introduction
Growing astrophysical evidence suggests that dark matter in the universe is self-annihilating. X-ray
observations from the center of our Galaxy find bright 511 keV emission which cannot be attributed
to single sources [Jean et al., 2006; Weidenspointner et al., 2006], but can be well-described assuming
dark matter annihilation [Boehm et al., 2004a]. Further observations indicate also an excess of GeV
photons [de Boer et al., 2005], of microwave photons [Hooper et al., 2007], and of positrons [Cirelli
et al., 2008]. A common feature of these observations is that the emission seems isotropic and not
correlated to the Galactic disk. However, there is usually some discrepancy between the model predictions and the amount of observed radiation, which may be due to uncertainties in the dark matter
distribution, astrophysical processes and uncertainties in the model for dark matter annihilation [de
Boer, 2008].
91
5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
It is well-known that weakly-interacting massive dark matter particles may provide a natural
explanation of the observed dark matter abundance [Drees & Nojiri, 1993; Kolb & Turner, 1990].
Calculations by Ahn et al. [2005] indicated that the extragalactic gamma-ray background cannot be
explained from astrophysical sources alone, but that also a contribution from dark matter annihilation
is needed at energies between 1-20 GeV. It is currently unclear whether this is in fact the case or if
a sufficient amount of non-thermal electrons in active galactic nuclei (AGN) is available to explain
this background radiation [Inoue et al., 2008]. Future observations with the FERMI satellite 1 will
shed more light on such questions and may even distinguish between such scenarios due to specific
signatures in the anisotropic distribution of this radiation [Ando et al., 2007].
The first stars have been suggested to have high masses of the order ∼ 100 M' , thus providing
powerful ionizing sources in the early universe [Abel et al., 2002; Bromm & Larson, 2004]. The
effect of dark matter annihilation on the first stars has been explored recently in different studies.
Spolyar et al. [2008] showed that an equilibrium between cooling and energy deposition from dark
matter annihilation can always be found during the collapse of the proto-stellar cloud. This has been
explored further by Iocco [2008] and Freese et al. [2008c], who considered the effect of scattering
between baryons and dark matter particles, increasing the dark matter abundance in the star. Iocco
et al. [2008] considered dark star masses in the range 5 ≤ M∗ ≤ 600 M' and calculated the evolution
of the pre-main-sequence phase, finding that the dark star phase where the energy input from dark
matter annihilation dominates may last up to 104 yr. Freese et al. [2008a] examined the formation
process of the star in more detail, considering polytropic equilibria and additional mass accretion until
the total Jeans mass of ∼ 800 M' is reached. They find that this process lasts for ∼ 5 × 105 yr. They
suggest that dark stars are even more massive than what is typically assumed for the first stars, and
may be the progenitors for the first supermassive black holes at high redshift. Iocco et al. [2008],
Taoso et al. [2008] and Yoon et al. [2008] have calculated the stellar evolution for the case in which
the dark matter density inside the star is enhanced by the capture of addition WIMPs via off-scattering
from stellar baryons. Iocco et al. [2008] followed the stellar evolution until the end of He burning,
Yoon et al. [2008] until the end of oxygen burning and Taoso et al. [2008] until the end of H burning.
Yoon et al. [2008] also took the effects of rotation into account. The calculations found a potentially
very long lifetime of dark stars and correspondingly a strong increase in the number of UV photons
that may contribute to reionization. Dark stars in the Galactic center have been discussed byScott
et al. [2008b,a].
Such models for the stellar population in the early universe imply that the first luminous sources
produce much more ionizing photons, and reionization starts earlier than for a population of conventional Pop. III stars. In fact, we recently demonstrated that reionization based on massive Pop. III
can well reproduce the observed reionization optical depth [Schleicher et al., 2008b]. Increasing the
number of ionizing photons per stellar baryon may thus reionize the universe too early and produce a
too large reionization optical depth. This can only be avoided by introducing a transition to a stellar
population which produces less ionizing photons, such that the universe can recombine after the first
1
http://www.nasa.gov/mission pages/GLAST/science/index.html
92
5.2 The models
reionization phase. We therefore consider a double-reionization scenario in order to re-obtain the
required optical depth. We discuss such models in Chapter 5.2 and demonstrate that some models of
dark stars require considerable fine-tuning in reionization models in order to be compatible with the
reionization optical depth from the WMAP 1 5-year data [Nolta et al., 2009; Komatsu et al., 2008]
and to complete reionization at redshift z ∼ 6 [Becker et al., 2001] in Chapter 5.3. In Chapter 5.4, we
show how such scenarios can be tested via 21 cm measurements.
A further consequence of the formation of dark stars is the steepening of the density profiles in
minihalos [Freese et al., 2008a; Iocco, 2008], thus increasing the dark matter clumping factor with
respect to standard NFW models. In Chapter 5.5, we estimate the increase in the clumping factor
during the formation of dark stars and compare the calculation with our expectation for conventional
NFW profiles and heavy dark matter candidates. In Chapter 5.6, we perform similar calculations for
the light dark matter scenario. Further discussion and outlook is provided in Chapter5.7.
5.2 The models
As discussed in the introduction, various models have been suggested for dark stars. The main difference between these models comes from considering or neglecting scattering between dark matter
particles and baryons. In addition, it is not fully clear how important a phase of dark matter capture via off-scattering from baryons actually is, depending on further assumptions on the dark matter
reservoir. In the following, we will thus distinguish between main-sequence dominated models and
capture-dominated models.
5.2.1 Main-sequence dominated models
After an initial phase of equilibrium between cooling and heating from dark matter annihilation
[Spolyar et al., 2008; Iocco, 2008; Freese et al., 2008c], the dark star will contract further while
the dark matter annihilates away and the heating rate thus decreases. This duration of this adiabatic
contraction (AC) phase is currently controversial: WhileIocco et al. [2008] find it to be in the range
of (2 − 20) × 103 yr, Freese et al. [2008a] require about 106 yr. However, with a surface temperature of ∼ 6000 K, the stars are rather cold in this phase, and thus will not contribute significantly to
reionization. The uncertainty in the duration of the AC phase is therefore not crucial in this context.
If the elastic scattering cross section as well as the dark matter density around the star are sufficiently large, the star will enter a phase which is dominated by the capture of further dark matter
particles. Such a scenario will be discussed in more detail in the next subsection. Here, we assume
that the elastic scattering cross section is either too small, or that the dark matter reservoir near the star
is not sufficient to maintain the capture phase for long. Then, the star will enter the main-sequence
phase (MS), in which the luminosity is generated by nuclear burning. Stars with ∼ 1000 M' are very
bright in this phase, and emit ∼ 4 × 104 hydrogen-ionizing photons per stellar baryon during their
1
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5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
lifetime [Bromm et al., 2001; Schaerer, 2002]. We will refer to stars of such type, which have only a
short or even no phase driven by dark matter capture, as MS-dominated models.
For the case of MS-dominated models, we will focus essentially on the very massive stars suggested by Freese et al. [2008a]. For stars in the typical Pop. III mass range, it has been shown
elsewhere [e. g. Schleicher et al., 2008b] that they are consistent with reionization constraints. A star
with ∼ 800 M' forming in a dark matter halo of ∼ 106 M' corresponds to a star formation efficiency
of 1%, which we adopt for this case.
5.2.2 Capture-dominated models
For a non-zero spin-dependent scattering cross section between baryons and dark matter particles,
stars can capture additional WIMPs which may increase the dark matter density inside the star. For a
cross section of the order 5×10−39 cm2 and an environmental dark matter density of ∼ 1010 GeV cm−3 ,
this contribution becomes significant and alters the stellar evolution during the main sequence phase.
We will refer to such a scenario as a capture-dominated (CD) model. These phases have been studied
in detail by Iocco et al. [2008], Taoso et al. [2008] and Yoon et al. [2008]. They found that the number
of ionizing photons produced by such stars may be considerably increased with respect to high-mass
stars without dark matter annihilation effects, which is mostly due to a longer lifetime. In particular
for dark matter densities of (1 − 5) × 1010 GeV cm−3 , the number of produced ionizing photons
may be increased by up to two orders magnitude, while it decreases rapidly for larger dark matter
densities, and the number of ionizing photons per baryon even drops below the value for Pop. II stars
at.threshold densities of 1 × 1012 GeV cm−3 . As Yoon et al. [2008] found only a weak dependence
on stellar rotation, we will not explicitly distinguish between models with and without rotation in the
follow.
For the calculation of reionization, we will focus on some representative models ofYoon et al.
[2008] in the following. However, we point out that there are still significant uncertainties in these
models, in particular the dark matter parameters and the lifetimes of the stars. The latter should be
seen as upper limits, as they assume that a sufficient reservoir of dark matter is available in the stellar
neighborhood to allow for ongoing dark matter capture. This may however be disrupted by dynamical
processes. An apparent disagreement of dark stars in the early universe with our reionization model
may thus indicate that the stellar lifetimes are indeed smaller due to such processes.
5.3 Reionization constraints
In this section, we briefly review our reionization model and discuss reionization histories for mainsequence and capture-dominated models. These calculations implicitly assume annihilation cross
sections of the order 10−26 cm2 and dark matter particle masses of the order 100 GeV, the values
which are typically adopted in dark star models. In such models, dark matter annihilation does not
contribute to cosmic reionization [Schleicher et al., 2008b]. The chemistry in the pre-ionization era is
94
5.3 Reionization constraints
thus unchanged and well-described by previous works [Puy et al., 1993; Galli & Palla, 1998; Stancil
et al., 1998; Schleicher et al., 2008c], such that the initial conditions for star formation are unchanged.
Considering higher annihilation cross sections essentially yields an additional contribution to the
reionization optical depth, which would sharpen the constraints given below.
5.3.1 General approach
Our calculation of reionization is based on the framework developed by Schleicher et al. [2008b],
which we have implemented in the RECFAST code 1 [Seager et al., 1999, 2000]. We will review
here only those ingredients which are most relevant for this work. During reionization, the IGM
consists of a two-phase medium, i. e. a hot ionized phase and a rather cold and overally neutral phase.
The relative size of these phases is determined from the volume-filling factor QH+ of the H+ regions
[Shapiro & Giroux, 1987; Haiman & Loeb, 1997b; Barkana & Loeb, 2001; Loeb & Barkana, 2001;
Choudhury & Ferrara, 2005; Schneider et al., 2006] as a function of redshift, given by
QH+ C(z)ne,H+ αA dnph /dz
dQH+
=
+
,
dz
H(z)(1 + z)
nH
(5.1)
where C(z) = 27.466 exp(−0.114z + 0.001328z2 ) is the clumping factor [Mellema et al., 2006],
ne,H+ the number density of ionized hydrogen, αA the case A recombination coefficient [Osterbrock,
1989], H(z) the Hubble function, nH the mean neutral hydrogen density in regions unaffected by UV
feedback and dnph /dz the UV photon production rate. Our model consists of ordinary differential
equations (ODEs) for the evolution of temperature T and ionized fraction xi in the overall neutral
medium. For the application considered here, the dominant contribution to the effective ionized
fraction xe f f = QH+ +(1− QH+ )xi and the effective temperature Te f f = 104 K QH+ +T (1− QH+ ) comes
indeed from the UV feedback of the stellar population, i. e. from the hot ionized phase. According to
Gnedin & Hui [1998] and Gnedin [2000], we introduce the filtering mass scale as

! ( "1/2 
%

3 a ( 2/3 ( 
a
2/3
da M J (a ) 1 −
(5.2)
MF =
 ,
a 0
a
where a = (1 + z)−1 is the scale factor and MJ the thermal Jeans mass, given as
M J = 2M'
!
cs
0.2 km/s
"3 #
$−1/2
n
.
103 cm−3
(5.3)
Here, cs is the sound speed evaluated at temperature Te f f , in order to take into account the backreaction of heating on structure formation. In this framework, the production of UV photons can be
described as
dnph /dz
d fcoll
∼ξ
,
(5.4)
nH
dz
1
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95
5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
(a)
Figure 5.1: The evolution of the effective ionized fraction x eff , for reionization models with mainsequence dominated dark stars (see Table 5.1). Models MS 1 and MS 2 can be ruled out by reionization
constraints, while models MS 3 and MS 4 require a sudden increase in the star formation rate by a factor
of 30 at redshift 6.5. It appears more realistic to assume lower masses and star formation efficiencies to
reconcile dark star models with observations.
where ξ = AHe f∗ fesc Nion , with AHe = 4/(4 − 3Y p ) = 1.22, Nion the number of ionizing photons per
stellar baryon, f∗ is the star formation efficiency and fesc the escape fraction of UV photons from their
host galaxies. The quantity fcoll denotes the fraction of dark matter collapsed into halos, and is given
as
'
(
δc (z)
fcoll = erfc √
,
(5.5)
2σ(Mmin )
where Mmin = min(MF , 105 M' ), δc = 1.69/D(z) is the linearized density threshold for collapse in
the spherical top-hat model and σ(Mmin ) describes the power associated with the mass scale Mmin .
A relevant question in this context is also the role of Lyman-Werner (LW) feedback, which may
suppress the star formation rate in low-mass halos. The role of such feedback has been addressed
using different approaches. For instance, Machacek et al. [2001], O’Shea & Norman [2008] and Wise
& Abel [2007b] have addressed this question employing numerical simulations in a cosmological
context, assuming a constant LW-background radiation field. These simulations indicated that such
feedback can delayed star formation considerably.
More self-consistent simulations show, however, that the above calculations overestimated the
role of LW-feedback. Considering single stellar sources and neglecting self-shielding,Wise & Abel
[2008] showed that LW-feedback only marginally delays star formation in halos that already started
96
5.3 Reionization constraints
collapsing before the nearby star ignites. More detailed simulations taking into account self-shielding
show that the star formation rate may be changed by only 20% in the presence of such feedback
[Johnson et al., 2007b]. This is due to the rapid re-formation of molecular hydrogen in relic HII
regions, which leads to abundances of the order 10−4 . Such abundances effectively shield against
LW-feedback and make it ineffective [Johnson et al., 2007a]. This is the point of view adopted here,
which may translate into an uncertainty of ∼ 20% in the star formation rate. In fact, in scenarios
involving dark matter annihilation, H2 formation and self-shielding could be even further enhanced
compared to the standard case [Mapelli & Ripamonti, 2007].
The models have to reproduce the reionization optical depth given by τ = 0.087± 0.017 [Komatsu
et al., 2008] and fully ionization at z ∼ 6 [Becker et al., 2001]. In the following, we will try to construct
appropriate reionization histories for the different dark star models.
5.3.2 Reionization with MS-dominated dark stars
As shown previously [Schleicher et al., 2008b], MS-dominated dark stars with ∼ 1000 M' would significantly overproduce the reionization optical depth if this type of stars had been common throughout
the early universe. If, on the other hand, MS-dominated stars only had mass scales of ∼ 100 M' ,
comparable to conventional Pop. III stars, reionization could not discriminate between them and conventional Pop. III stars, and dark stars would be compatible with observations. Alternatively, as
explained in the introduction, a transition in the stellar population might help to alleviate the problem
for high-mass dark stars. We will explore this possibility in more detail to work out whether such a
scenario is conceiveable.
Numerical simulations by Dove et al. [2000], Ciardi et al. [2002] and Fujita et al. [2003] indicated
rather high escape fractions of order 100% for massive Pop. III stars. Wood & Loeb [2000] found
rather low escape fractions below 10%, while radiation hydrodynamics simulations byWhalen et al.
[2004] show that such stars can easily photo-evaporate the minihalo. Here we adopt the point of view
that indeed massive stars can photoevaporate small minihalos, but that the escape fraction will be
reduced to ∼ 10% in atomic cooling halos that have virial temperatures larger than 104 K. Thus, we
/ 03/2
10
set fesc = 1 if the filtering mass is below the mass scale Mc = 5 × 107 M' 1+z
that corresponds
4
to the virial temperature of 10 K [Oh & Haiman, 2002; Greif et al., 2008], and fesc = 0.1 in the
other case. To reflect the expected stellar mass of ∼ 800 M' , we choose a star formation efficiency of
f∗ ∼ 1%, an order of magnitude higher than what we expect for conventional Pop. III stars [Schleicher
et al., 2008b].
Assuming that reionization is completely due to these MS-dominated dark stars (model MS 1),
we find that the universe is fully ionized at redshift zreion = 15.5 and the reionization optical depth is
τreion ∼ 0.22, i. e. significantly larger than the WMAP 5 optical depth (see Fig.5.1). Such a model is
clearly ruled out.
To reconcile the presence of such massive dark stars with observations, one could invoke a doublereionization scenario, assuming a transition to a different mode of star formation induced by the strong
UV feedback of MS-dominated dark stars. In fact, even for conventional star formation models, it
97
5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
is discussed that such UV feedback may lead to a less massive mode of star formation [Johnson
& Bromm, 2006; Yoshida et al., 2007a,b]. In addition, chemical enrichment should facilitate such a
transition as well [Schneider et al., 2006; Clark et al., 2008; Omukai et al., 2008; Smith et al., 2008a,b;
Greif et al., 2009], although it is unclear how well metals will mix with the pristine gas. We assume
that the transition to a low-mass star formation mode with a Scalo-type IMF [Scalo, 1998] happens at
redshift 15.5, when the universe is fully ionized and UV feedback fully effective. For the subsequent
Pop. II stars, we assume a star formation efficiency of f∗ = 5 × 10−3 and Nion = 4 × 103 UV photons
per stellar baryon.
Corresponding photon escape fractions are highly uncertain. Observations ofSteidel et al. [2001]
indicate an escape fraction of 10% at z ∼ 3, while others find detections or upper limits in the range
5 − 10% [Giallongo et al., 2002; Malkan et al., 2003; Fernández-Soto et al., 2003; Inoue et al., 2005].
We adopt the generic value of 10% for simplicity, though our results do not strongly depend on this assumption. For this scenario, to which we refer as model MS 2, we find an optical depth τreion = 0.082
well within the WMAP constraint, but the universe does not get fully ionized until redshift zero. This
scenario is thus rejected based on the constraint from quasar absorption spectra [Becker et al., 2001].
To fulfill both the WMAP constraint as well as full-ionization at z ∼ 6, we need to introduce an
additional transition in our model. At redshift zburst = 6.5, we increase the star formation efficiency to
15%. This might be considered as a sudden star burst and results in full-ionization at z = 6.2. In this
case, we find τreion = 0.116, which is within the 2σ range of the WMAP data. However, we are not
aware of astrophysical models that provide a motivation for such a sudden star burst that increases the
star formation rate by a factor of 30. Based on gamma-ray burst studies,Yüksel et al. [2008] showed
that the cosmic star formation rate does not change abruptly in the redshift range between redshift
zero and zburst = 6.5. Such a sudden burst is thus at the edge of violating observation constraints.
To improve the agreement with WMAP, one can consider to shift the first transition to zPop II = 18
where full ionization is not yet reached (model MC 4), which yields an optical depth τreion = 0.086,
in good agreement with WMAP. At this redshift, 68% of the universe are already ionized, so UV
feedback might already be active and induce a transition in the stellar population. The results are
given in Fig. 5.1 and summarized in Table 5.1.
However, we find that only models MS 3 and MS 4 cannot be ruled out observationally. These
models require two severe transitions in the stellar population and cannot be considered as ”natural”.
Improved measurements of the reionization optical depth from Planck 1 will remove further uncertainties and may rule out model MS 3 as well. From a theoretical point of view, it must be checked
whether strong UV feedback can lead to the required transition to a low-mass star population, and in
addition, the plausibility of a sudden star burst near redshift 6 must be examined as well. In summary,
it seems more plausible to conclude that MS-dominated dark stars were less massive than suggested
by Freese et al. [2008c], as already hinted by Schleicher et al. [2008b].
1
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5.3 Reionization constraints
Model
zPop II
zburst
τreion
zf
MS 1
MS 2
MS 3
MS 4
15.8
15.5
18.
6.5
6.5
0.22
0.078
0.116
0.086
15.5
never
6.2
6.2
Table 5.1: Reionization models for MS-dominated dark stars. The parameters z Pop II and zburst give the
transition redshifts to a mode of Pop. II star formation and to the sudden star burst, while τ reion is the
calculated reionization optical depth and z f the redshift of full ionization.
5.3.3 Reionization with CD dark stars
For CD dark star models, the situation is complicated by the fact that the number of UV photons per
stellar baryon, Nion , is model-dependent and changes with the environmental dark matter density, ρX .
We select three representative models ofYoon et al. [2008], which assume a spin-dependent scattering
cross section of 5 × 10−39 cm2 (see Table 5.2). In general, stellar models depend on the product of this
scattering cross section with the threshold dark matter density at the stellar radius [Taoso et al., 2008;
Yoon et al., 2008]. Lower elastic scattering cross sections therefore correspond to going to smaller
threshold densities at the same elastic scattering cross section.
In the models CD 1 and 2, Nion is larger than for conventional Pop. III stars, while in the model
CD 3, it is even less than in the case of Scalo-type Pop. II stars. Such a low luminosity is unlikely
to photo-evaporate star-forming halos, and we thus adopt fesc = 10% for this case. However, such
Scalo-type Pop. II stars are ruled out as sole sources for reionization [Schleicher et al., 2008b]. As we
show in Fig. 5.2, even with a high star formation efficiency of f∗ = 1%, they never ionize the universe
completely.
In principle, one could consider the presence of other sources to ionize the universe. While dark
stars of type CD 3 may be the first stars to form, one might envision a transition to a stellar population
with the power to ionize the universe. This transition is unlikely due to UV feedback, as UV feedback
from dark stars is rather weak in this scenario. One thus has to rely on effective mixing of the produced
metallicity, or assume that the first stellar clusters in atomic cooling halos contain a sufficient number
of massive stars to reionize the universe [Clark et al., 2008].
For the other two models, Nion is significantly larger and we adopt the procedure from the previous
subsection, such that fesc depends on the filtering mass. We adopt a star formation efficiency of
f∗ = 0.1%. We examine the reionization models given in Table 5.2, which essentially follow the
philosophy of the models from the previous section. We calculate the reionization history for the case
where these dark stars are sole sources (CD 1a, CD2a) and find that the optical depth is considerably
too high. We then determine the redshift where the universe is fully ionized and assume a transition
to Pop. II stars at this redshift. In addition, to obtain full ionization at redshift 6, we assume a late
star burst as in the models MS 3 and MS 4. This approach corresponds to the models CD 1b and CD
99
5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
(a)
Figure 5.2: The evolution of the effective ionized fraction x eff , for reionization models with capturedominated dark stars (see Table 5.2). Models CD 1a, CD 1b, CD 2a, CD 2b and CD 3 are ruled out due to
reionization constraints, while the remaining models require an artificial star burst.
2b, and yields optical depth that are at least within the 2σ error of WMAP 5. In the models CD 1c
and CD 2c, we improve the agreement with WMAP by introducing the Pop. II transition at an earlier
redshift.
The results are given in Fig. 5.2. Again, it turns out that somewhat artificial models are required
to allow for an initial population of CD dark stars. The best way to reconcile these models with the
constraints from reionization might be to focus on those models that predict a parameter Nion which
is closer to the Pop. III value of 4 × 104 . This may be possible, as the transition from the models CD
1 and 2 to CD 3 is likely continuous, and an appropriate range of parameters may exist to re-concile
models with observations. This would require a ρX between 1011 GeV cm−3 and 1012 GeV cm−3 .
As mentioned earlier, the apparent violation of reionization constraints by some models depends also
on the uncertainties in the stellar lifetime. If the dark matter reservoir near the star is destroyed
earlier due to dynamical processes, the lifetime may be significantly reduced. Also, we stress that the
conclusions depend on the adopted elastic scattering cross section and the dark matter density in the
environment. The discussion here is limited to those models that have previously been worked out in
detail.
100
5.4 Predictions for 21 cm observations
(a)
(b)
(c)
Figure 5.3: 21 cm signatures of double-reionization scenarios (here MS 4 from Table 5.1). Given is the
evolution after the first reionization phase, when the H gas is heated from the previous ionization. Top:
HI gas temperature, here identical to the spin temperature. Middle: Expected mean 21 cm brightness
fluctuation. Bottom: Frequency gradient of the mean 21 cm brightness fluctuation.
101
5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
Reion. model
CD 1a
CD 1b
CD 1c
CD 2a
CD 2b
CD 2c
CD 3
ρX /1012
Nion
0.01 GeV cm−3
0.01 GeV cm−3
0.01 GeV cm−3
0.05 GeV cm−3
0.05 GeV cm−3
0.05 GeV cm−3
1 GeV cm−3
1.75 × 105
1.75 × 105
1.75 × 105
2.4 × 106
2.4 × 106
2.4 × 106
1.1 × 103
f∗
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
1%
zPop II
τreion
12.7
14.5
21.6
23
-
0.162
0.109
0.089
0.283
0.106
0.084
0.004
Table 5.2: Reionization models for CD dark stars stars. The number of ionizing photons was determined
from the work of Yoon et al. [2008]. The parameters z Pop II and zburst give the transition redshifts to a
mode of Pop. II star formation and to the sudden star burst, while τ reion is the calculated reionization
optical depth and z f the redshift of full ionization. The calculation assumes a spin-dependent scattering
cross section of 5 × 10 −39 cm2 . As stellar models depend on the product of this cross section with the
threshold dark matter density, the effect of a lower scattering cross section is equivalent to a smaller
threshold density.
5.4 Predictions for 21 cm observations
While some of the models suggested above essentially co-incide with standard reionization by mimicing the effects of conventional Pop. III stars, others may have a very distinctive signature, as they consist of a double-reionization phase, and upcoming 21 cm telescopes like LOFAR1 or SKA 2 can thus
verify or rule out such suggestions. The calculation shown in Fig.5.3 is based on the double reionization model MS 4, but clearly the models MS 3, CD 1b, CD 1c, CD 2b and CD 2c yield similar
results. In such a double-reionization scenario, the gas is heated to ∼ 104 K during the first reionization epoch. Assuming that the first reionization epoch ends at redshift zPop II , the gas temperature in
the non-ionized medium will then evolve adiabatically as
T ∼ 104 K
!
1+z
1 + zPop II
"2
.
(5.6)
In addition, the previous reionization phase will have established a radiation continuum between the
Lyman α line and the Lyman limit, where the universe is optically thin, apart from single resonances
corresponding to the Lyman series. This radiation is now redshifted into the Lyman series and may
couple the spin temperature Tspin of atomic hydrogen to the gas temperature T via the WouthuysenField effect [Wouthuysen, 1952; Field, 1958]. In fact, a small amount of Lyman α radiation suffices to
set Tspin = T [Barkana & Loeb, 2005b; Hirata, 2006; Pritchard & Furlanetto, 2006], which we assume
1
2
http://www.lofar.org/
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102
5.5 Cosmic constraints on massive dark matter candidates
here. Also, as the universe is optically thin to this radiation background, even Pop. II sources will
suffice to couple the spin temperure to the gas temperature. The mean 21 cm brightness temperature
fluctuation is then given as
δT b
!
"!
"1/2
Ωb h2
0.15 1 + z
= 27xH (1 + δ)
0.023 Ωm h2 10
!
"!
"
T S − T r H(z)/(1 + z)
×
mK,
TS
dv|| /dr||
(5.7)
where xH denotes the neutral hydrogen fraction, δ the fractional overdensity, Ωb , Ωm the cosmological density parameters for baryons and total matter, h is related to the Hubble constant H0 via
h = H0 /(100km/s/Mpc), Tr the radiation temperature and dv|| /dr|| the gradient of the proper velocity
along the line of sight, including the Hubble expansion. We further calculate the frequency gradient of
the mean 21 cm brightness temperature fluctuation to show its characteristic frequency dependence.
In Fig. 5.3, we show the evolution of the gas temperature, the mean 21 cm brightness fluctuation and
its frequency gradient for model MS 4.
As pointed out above, we expect similar results for other double-reionization models because of
the characteristic adiabatic evolution of the gas and spin temperature. The decrease of the spin temperature with increasing redshift is a unique feature that is not present in other models that like dark
matter decay [Furlanetto et al., 2006b] or ambipolar diffusion heating from primordial magnetic fields
[Sethi & Subramanian, 2005; Tashiro et al., 2006; Schleicher et al., 2009a], which may also increase
the temperature during and before reionization.
5.5 Cosmic constraints on massive dark matter candidates
In typical dark star models, it is assumed that massive dark matter candidates like neutralinos with
masses of the order 100 GeV annihilate into gamma-rays, electron-positron pairs and neutrinos
[Spolyar et al., 2008; Iocco, 2008; Freese et al., 2008b,a,c; Iocco et al., 2008; Yoon et al., 2008].
Similar to the constraint on high-redshift quasars from the X-ray background [Dijkstra et al., 2004;
Salvaterra et al., 2005; Schleicher et al., 2008d], the gamma-ray and neutrino backgrounds allow to
constrain the model for and the amount of dark matter annihilation. As detailed predictions for the
decay spectra are highly model-dependent, it is typically assumed that roughly 1/3 of the energy goes
into each annihilation channel. Constraints on such scenarios are available from the Galactic center
and the extragalactic gamma-ray and neutrino backgrouns [Ullio et al., 2002; Beacom et al., 2007;
Yüksel et al., 2007; Mack et al., 2008]. In this section, we consider how such constraints are affected
when the increase in the annihilation rate due to enhanced dark matter densities after the formation of
dark stars is taken into account.
103
5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
(a)
Figure 5.4: The predicted gamma-ray background due to direct annihilation into gamma-rays in the
presence of adiabatic contraction during the formation of dark stars, and the background measured by
EGRET (squares) [Strong et al., 2004]. One finds two peaks in the annihilation background for a given
particle mass: One corresponding to annihilation at redshift zero, and one corresponding to the redshift
where the enhancement from adiabatic contraction was strongest.
104
5.5 Cosmic constraints on massive dark matter candidates
5.5.1 Gamma-ray constraints
We adopt the formalism of Mack et al. [2008] who recently addressed the direct annihilation of massive dark matter particles into gamma-rays. The background intensity Iν is given from an integration
along the line of sight as
%
dzPν ([1 + z]ν, z)
c
Iν =
,
(5.8)
4π
H(z)(1 + z)4
where Pν (ν, z) is the (proper) volume emissivity of gamma-ray photons, which is given as
Pν = αb δ ((1 + z)ν − mDM )
mDM
keV "σv#n2DM Cγ ,
keV
(5.9)
where "σv# = 3 × 10−26 cm3 s−1 denotes the thermally-averaged annihilation cross section, αb = 1/3
is the adopted branching-ratio to gamma-rays and mDM the mass of the dark matter particle in keV. Cγ
refers to the dark matter clumping factor. This clumping factor depends on the adopted dark matter
profile and the assumptions regarding substructure in a halo [Ahn & Komatsu, 2005a; Ando, 2005;
Chuzhoy, 2008; Cumberbatch et al., 2008]. Here we use the clumping factor for a NFW dark matter
profile [Navarro et al., 1997] which has been derived by Ahn & Komatsu [2005b,a]. For z < 20, it is
given in the absence of adiabatic contraction as a power-law of the form
CDM = CDM (0)(1 + z)−β ,
(5.10)
where CDM (0) is the clumping factor at redshift zero and β determines the slope. For a NFW profile
[Navarro et al., 1997], CDM (0) ∼ 105 and β ∼ 1.8. The enhancement due to adiabatic contraction is
taken into account by defining
Cγ = CDM fenh ,
(5.11)
where the factor fenh describes the enhancement of the halo clumping factor due to adiabatic contraction (AC). We have estimated this effect based on the results of Iocco et al. [2008], comparing a
standard NFW profile with the enhanced profile that was created during dark star formation. We only
compare them down to the radius of the dark star and find an enhancement of the order ∼ 103 . For
the NFW case, the clumping factor would be essentially unchanged when including smaller radii as
well, while the AC profile is significantly steeper and the contribution from inside would dominate
the contribution to the halo clumping factor. However, as the annihilation products are trapped inside
the star, it is natural to introduce an inner cut-off at the stellar radius. In addition, we have to consider
the range of halo masses and redshifts in which dark stars may form. We assume that the halo mass
must be larger than the filtering mass to form dark stars. However, there is also an upper mass limit.
Halos with masses above
!
"3/2
10
7
(5.12)
Mc = 5 × 10 M'
1+z
correspond to virial temperatures of 104 K [Oh & Haiman, 2002] and are highly turbulent [Greif et al.,
2008]. It seems thus unlikely that stars will form on the very cusp of the dark matter distribution in
such halos, and more complex structures may arise. We thus assume that dark stars form in the mass
105
5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
(a)
Figure 5.5: The predicted neutrino background due to direct annihilation into neutrinos in the presence of
adiabatic contraction during the formation of dark stars, and the atmospheric neutrino background [ Honda
et al., 2004]. One finds two peaks in the annihilation background for a given particle mass: One corresponding to annihilation at redshift zero, and one corresponding to the redshift where the enhancement
from adiabatic contraction was strongest.
range between MF and Mc . Once Mc becomes larger than MF , dark star formation must end naturally.
In fact, it may even end before, as discussed in Chapter 5.3. To obtain the highest possible effect, we
assume that dark stars form as long as possible. We thus have
!
"
− fcoll (Mc )
.
fcoll (MF )
3 fcoll (M F )
fenh = 1 + 10
(5.13)
In Fig. 5.4, we compare the results with EGRET observations of the gamma-ray background [Strong
et al., 2004]. In the absence of adiabatic contraction, the predicted background peaks at the contribution from redshift zero [Mack et al., 2008]. We find that the enhancement of annihilation due
to adiabatic contraction produces a second peak in the predicted background which originates from
higher redshifts. In this scenario, particle masses smaller than 30 GeV can thus be ruled out.
5.5.2 Neutrino constraints
The contribution to the cosmic neutrino flux can be obtained in analogy to Eq. (5.8). As recent works
[Beacom et al., 2007; Yüksel et al., 2007], we adopt an annihilation spectrum of the form
Pν = αb δ ((1 + z)ν − mDM )
mDM
keV "σv#n2DMCneutrino ,
keV
106
(5.14)
5.5 Cosmic constraints on massive dark matter candidates
(a)
Figure 5.6: The maximum gamma-ray background due to direct annihilation into gamma-rays in the remnants of dark stars, and the background measured by EGRET [ Strong et al., 2004]. The actual contribution
to the gamma-ray background is highly model-dependent (see discussion in the text).
which is analogous to the spectrum for annihilation into gamma-rays. The branching ratio to neutrinos
is assumed to be 1/3 as well, and the annihilation comes from the same dark matter distribution, thus
yielding Cneutrino = Cγ . The atmospheric neutrino background has been calculated from different
experiments with generally good agreement [Ahrens et al., 2002; Gaisser & Honda, 2002; Honda
et al., 2004; Ashie et al., 2005; Achterberg et al., 2007]. Iocco [2008] adopted a similar atmospheric
neutrino flux for comparison with the expected neutrino flux from dark stars. We adopt here the data
provided by Honda et al. [2004] and compare them to the predicted background in Fig. 5.5. The
predicted background is always well below the observed background.
5.5.3 Emission from dark star remnants
In the previous subsections, we have included the enhancement of the halo clumping factor down
to the stellar radius, as by definition the annihilation products on smaller scales are trapped inside
the star. At the end of their lifetime, these stars may explode and the baryon density in the center
may be largely depleted. The dark matter density has certainly been significantly reduced due to
annihilations during the lifetime of the star, but it may still be enhanced compared to the usual NFW
case. A detailed calculation of this effect is strongly model-dependent. As we have seen above, the
strongest constraints are obtained for direct annihilation into gamma-rays, which is the case we pursue
here in more detail.
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5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
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So far, we assumed that dark stars form in halos between the filtering mass MF and the mass
corresponding to a virial temperature of 104 K, Mc . To obtain an upper limit, it is sufficient to
assume that in all halos above MF a dark star remnant will form at some point. Such an assumption
clearly overestimates the total contribution at low redshift. When the dark star has formed, a fraction
fcore ∼ 10−6 of the dark matter from the total halo is in the star [Freese et al., 2008a]. For the upper
limit, we assume that the total amount of dark matter in star will contribute to the X-ray background
(in fact, however, only the dark matter left over in the final remnant can contribute). In this case, we
have a proper volume emissivity
mDM
keVα511 fr fa
keV
d fcoll (MF )
× nDM fcore
,
dt
Pν = δ ((1 + z)ν − mDM )
(5.15)
where nDM is the mean proper number density of dark matter particles, mDM the particle mass in
keV and d fcoll /dt can be evaluated from Eq. (5.5). The model-dependent factor fr determines which
fraction of the dark matter in the star will be left in the remnant. We adopt fr = 1 to obtain an upper
limit. The factor fa determines the fraction of the remaining dark matter which actually annihilates,
which we set to fa = 1 as well. In Fig. 5.6, we compare the results with EGRET observations [Strong
et al., 2004]. We find that the maximum contribution is clearly above the observed background.
Whether this maximum contribution can be reached, is however uncertain and the previous work
in the literature only allows one to make rather crude estimates. For instanceIocco et al. [2008] calculate the density profile for a fiducial 100 M' protostar, finding that the density within the star roughly
scales with r−2 outside a plateau at a radius r ∼ 1011 cm. At the stellar radius of ∼ 1014 cm, the dark
matter density is still ∼ 1012 GeV cm−3 . The timescale to remove this dark matter enhancement by
annihilation is ∼ 100 Myr for 100 GeV neutralinos. We need to estimate which fraction of the dark
matter inside the star will be left at the end of its life, where the gas density is expelled by a supernova
explosion and the dark matter annihilation from this region may contribute to the cosmic gamma-ray
background.
Yoon et al. [2008] adopted a timescale of 100 Myr, the typical merger timescale at these redshifts,
as the maximum lifetime for dark stars. Depending on the scattering cross section and the environmental density, the actual dark star lifetime may be considerably shorter. Indeed, as we showed in
Chapter 5.3.3, it is difficult to reconcile lifetimes of ∼ 100 Myr with appropriate reionization scenarios. It is therefore reasonable to assume shorter timescales. In such a case, a reasonable estimate
is that ∼ 40% of the dark matter inside the star would be left at the end of its life. This would still
be enhanced compared to the standard NFW profile. In this case, the parameter fr is ∼ 40%, and
fa may be of order 1, as the annihilation timescale is comparable to the Hubble time. We note that
these numbers are highly uncertain, in particular regarding the exact evolution of dark matter density
during the lifetime of the star, the effect of a supernova explosion on the dark matter cusp as well as
the consequences of minor mergers.
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5.6 Cosmic constraints on light dark matter
There is however also a viable possibility that the dark matter distribution inside the star is significantly steeper than assumed above. In the case of dark matter capture by off-scattering from baryons,
the dark matter density inside the star follows a Gaussian shape and is highly concentrated in a small
region of r ∼ 2 × 109 cm [Griest, 1987; Iocco, 2008; Taoso et al., 2008; Yoon et al., 2008]. The
implications are not entirely clear. If capture of dark matter stops at the end of the life of the star, the
density inside the star will annihilate away quickly, and no significant contribution may come from
the remnant. If, on the other hand, dark matter capture goes on until the end of the life of the star, a
contribution to the background seems viable. In summary, this may provide a potential contribution
to the cosmic background, but its strength is still highly uncertain and should be explored further by
future work.
5.5.4 Dependence of dark star models on the neutralino mass
We conclude this section with a discussion on the constraints from cosmic backgrounds for different
neutralino masses. As dark star models in the literature mostly consider neutralinos of 100 GeV, there
are uncertainties that need to be addressed when considering different neutralino masses. For models
involving the capture of dark matter, Iocco [2008] states that the mass of the neutralino does not
change the annihilation luminosity. Taoso et al. [2008] find that variations due to different neutralino
masses are less than 5%. While these results may hold for high masses, Spergel & Press [1985]
showed that for neutralino masses below 4 GeV, they would evaporate from the star, as scattering
with baryons can upscatter them as well.
In addition, the AC phase may be modified as well, as the dark matter annihilation rate in this
phase is degenerate in the parameter "σv#/mDM . Iocco et al. [2008] find that the duration of the
AC phase may change by almost 50% if the dark matter mass is changed by a factor of 2. The
effect of different neutralino masses is therefore uncertain and should be explored in more detail. We
will however assume that the general behaviour involving adiabatic contraction in the minihalo is
still similar, such that the calculations below are approximately correct also for different neutralino
masses.
5.6 Cosmic constraints on light dark matter
Observations of 511 keV emission in the center of our Galaxy [Knödlseder et al., 2003] provide recent
motivation to models of light dark matter [Boehm et al., 2004b]. Such observational signatures can
be explained assuming dark matter annihilation, while other models still have difficulties reproducing
the observations [Boehm et al., 2004a]. The model assumes that dark matter annihilates into electronpositron pairs, which in turn annihilate into 511 keV photons. Direct annihilation of dark matter
into gamma-rays or neutrinos is assumed to be suppressed to avoid the gamma-ray constraints and
to ensure a sufficient positron production rate. It is known that electron-positron annihilation occurs
mainly via positronium-formation in our galaxy [Kinzer et al., 2001]. In addition, it was shown
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5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
[Beacom et al., 2005] that dark matter annihilation to electron-positron pairs must be accompanied
by a continuous radiation known as internal bremsstrahlung, arising from electromagnetic radiative
corrections to the dark matter annihilation process.
Motivated by these results, it was proposed that internal bremsstrahlung from dark matter annihilation may be responsible for the gamma-ray background at energies of 1-20 MeV [Ahn & Komatsu,
2005b]. Conventional astrophysical sources cannot explain the observed gamma-ray background at
these frequencies [Ahn et al., 2005]. A comparison of the observed and predicted background below
511 keV yields constraints on the dark matter particle mass [Ahn & Komatsu, 2005a]. Here we examine whether and how this scenario is affected if dark stars form in the early universe. We use a
thermally averaged cross section "σv# ∼ 3 × 10−26 cm3 s−1 to account for the observed dark matter
density [Drees & Nojiri, 1993]. This implies that "σv# is velocity-independent (S-wave annihilation).
While Boehm et al. [2004a] argue that S-wave annihilation overpredicts the flux from the galactic
center, others argue that it is still consistent [Ahn & Komatsu, 2005b,a]. The cross-section adopted
here is well-within the conservative constraints of Mack et al. [2008]. The effect of light dark matter annihilation on structure formation in the early universe has been studied in various works, e. g.
[Mapelli et al., 2006; Ripamonti et al., 2007a,b]. Constraints from upcoming 21 cm observations
have been explored by Furlanetto et al. [2006b] and Valdés et al. [2007], while constraints from background radiation have been considered by Mapelli & Ferrara [2005]. The effects of early dark matter
halos on reionization have been addressed recently byNatarajan & Schwarz [2008].
As in the previous section, we point out that significant uncertainties are present when considering
dark star models for different dark matter masses, as this question is largely unexplored. In particular,
we emphasize that no capturing phase will be present for light dark matter, as shown in the work of
Spergel & Press [1985]. Another uncertainty is the question whether to adopt self-annihilating dark
matter (i. e. Majorana particles) or particles and antiparticles of dark matter. In the calculations
below, we assume that light dark matter is self-annihilating. Otherwise, our results would be changed
by a factor of 0.5.
5.6.1 511 keV emission
The expected X-ray background from 511 keV emission is calculated from Eq. (5.8). The volume
emissivity of 511 keV photons is given as
Pν = δ ((1 + z)ν − ν511 ) 511 keVα511 "σv#n2DM C511 ,
(5.16)
where "σv# denotes the thermally-averaged annihilation cross section, α511 is the fraction producing
an electron-positron pair per dark matter annihilation process and ν511 the frequency corresponding
to 511 keV. In our galaxy, this process happens via positronium formation [Kinzer et al., 2001], and
we assume that the same is true for other galaxies. In 25% of the cases, positronium forms in a singlet
(para) state which decays to two 511 keV photons, whereas 75% form in a triplet (ortho) state which
decays into a continuum. We thus adopt α511 = 1/4 for 511 keV emission. C511 refers to the dark
110
5.6 Cosmic constraints on light dark matter
(a)
Figure 5.7: The predicted X-ray background due to 511 keV emission for different dark matter particle
masses. Solid lines: Enhanced signal due from adiabatic contraction, dotted lines: Conventional NFW
profiles. The observed X-ray background from the HEAO experiments (squares) [ Gruber et al., 1999] and
Swift/BATSE (triangles) [Ajello et al., 2008] is shown as well. The comparison yields a lower limit of 10
MeV on the dark matter mass for the adiabatically contracted profiles, and 7 MeV for standard NFW halo
profiles.
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5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
matter clumping factor, which is still highly uncertain. The main uncertainty is due to the adopted
dark matter profile and the assumptions regarding substructure in a halo [Ahn & Komatsu, 2005a;
Ando, 2005; Chuzhoy, 2008; Cumberbatch et al., 2008].
Here we use the clumping factor for a NFW dark matter profile [Navarro et al., 1997] which
has been derived by Ahn & Komatsu [2005b,a], as to our knowledge, no calculations of dark star
formation are available for other dark matter profiles. For z < 20, it is given as a power-law as
CDM = CDM (0)(1 + z)−β ,
(5.17)
where CDM (0) describes the clumping factor at redshift zero and β determines the slope. For a NFW
profile [Navarro et al., 1997], CDM (0) ∼ 105 and β ∼ 1.8. The effects of different clumping factors will
be explored in future work [Schleicher et al., 2009b]. Ahn & Komatsu [2005b,a] included contributions from all halos with masses above a minimal mass scale Mmin , which was given as the maximum
of the dark matter Jeans mass and the free-streaming mass. This approach assumes instantaneous
annihilation of the created electron-positron pairs. As pointed out byRasera & Teyssier [2006], the
assumption of instantaneuos annihilation is only valid if the dark matter halo hosts enough baryons
to provide a sufficiently high annihilation probability, postulating this to happen in halos with more
than 107 − 1010 M' , corresponding to their calculation of the filtering mass. We also calculate the
filtering mass according to the approach of Gnedin & Hui [1998]; Gnedin [2000], but obtain somewhat lower masses, with ∼ 105 M' halos at the beginning of reionization and ∼ 3 × 107 M' at the
end [Schleicher et al., 2008b]. This is also in agreement with numerical simulations of Greif et al.
[2008] which find efficient gas collapse in halos of 105 M' . The discrepancy may also be due to their
different reionization model, which assumes reionization to start at redshift 20.
We recall that the clumping factor can be considered as the product of the mean halo overdensity, the fraction of collapsed halos above a critical scale and the mean ”halo clumping factor” that
describes dark matter clumpiness within a halo. To take into account that electron-positron annihilation occurs only in halos above the filtering mass MF , we thus rescale the results of Ahn & Komatsu
[2005b,a] as
fcoll (MF )
C511 =
CDM fenh ,
(5.18)
fcoll (Mmin )
where the factor fenh is given from Eq. (5.13). As above, we assume that the halo mass must be
larger than the filtering mass, and lower than the critical mass scale Mc that corresponds to virial
temperatures of 104 K. For comparison, we will also calculate 511 keV emission with fenh = 1. We
note that the resulting background will be somewhat lower than the result ofAhn & Komatsu [2005a],
as we adopted α511 = 1/4 and include only halos above the filtering mass scale in the clumping factor.
In Fig. 5.7, we compare the results with the observed X-ray background from the HEAO-experiments
1 [Gruber et al., 1999] and SWIFT 2 /BATSE 3 observations [Ajello et al., 2008]. In the standard
1
http://heasarc.gsfc.nasa.gov/docs/heao1/heao1.html
http://heasarc.nasa.gov/docs/swift/swiftsc.html
3
http://www.batse.msfc.nasa.gov/batse/
2
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5.6 Cosmic constraints on light dark matter
(a)
Figure 5.8: The predicted gamma-ray background due to bremsstrahlung emission for different dark
matter particle masses. Solid lines: Enhanced signal due from adiabatic contraction, dotted lines:
Conventional NFW profiles. The lines overlap almost identically, as the main contribution comes from
redshift zero, where the clumping factor is large and dark stars are assumed not to form. The observed
gamma-ray background from the HEAO experiments (squares) [ Gruber et al., 1999], Swift/BATSE (triangles) [Ajello et al., 2008], COMPTEL (crosses) [Kappadath et al., 1996] and SMM (plusses) [Watanabe
et al., 1999] is shown as well.
NFW case, we find a lower limit for the dark matter particle mass of 7 MeV. For the case with
adiabatically contracted profiles due to dark star formation, we find a slightly higher lower limit of
10 MeV. This is because the enhancement is effective only for frequencies hν < 100 keV, where the
observed background is significantly larger than at 511 keV, whereAhn & Komatsu [2005a] obtained
their upper limit.
5.6.2 Internal Bremsstrahlung
The internal bremsstrahlung is calculated according to the approach ofAhn & Komatsu [2005b]. The
background intensity is given by Eq. (5.8), with a proper volume emissivity
'
(
1
4α g(ν)
2
Pν = hν"σv#Cbrems nDM
,
(5.19)
2
π ν
where α ∼ 1/137 is the finestructure constant and g(ν) is a dimensionless spectral function, defined
as

 
!
"

 s̃ 2 
s̃
1
ln 2 − 1 1 +  2   ,
(5.20)
g(ν) =
4
me
4mDM
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5. DARK STARS: IMPLICATIONS AND CONSTRAINTS FROM COSMIC
REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
(a)
Figure 5.9: The upper limit of X-ray radiation due to dark star remnants. The observed X-ray background
from the HEAO experiments (squares) [Gruber et al., 1999] and Swift/BATSE (triangles) [Ajello et al.,
2008] is shown as well. Only for very low dark matter particle masses, the upper limit is somewhat
higher than the observed background. However, the actual contribution may be lower by some orders of
magnitude (see discussion in the text).
with s̃ = 4mDM (mDM − hν). As Ahn & Komatsu [2005b] pointed out in a ’Note added in proof’,
bremsstrahlung is emitted in all dark matter halos, regardless of the baryonic content. There is thus
no need to consider any shift in the minimal mass scale, the only thing to take into account is the
enhancement of annihilation due to the AC profiles. The clumping factor Cbrems is thus given as
Cbrems = CDM fenh ,
(5.21)
where fenh is given by Eq. (5.8). In Fig. 5.8, we compare the results with the observed gammaray background from the HEAO-experiments [Gruber et al., 1999] and SWIFT/BATSE observations
[Ajello et al., 2008], as well as SMM
1
[Watanabe et al., 1999] and Comptel
2
data [Kappadath
et al., 1996]. We find that the signal is almost unchanged in the model taking into account dark star
formation. The reason is that dark stars form mainly at high redshifts, in the range where MF < Mc ,
while the dominant contribution to the background comes from redshift zero. Our results agree with
Ahn & Komatsu [2005b].
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5.7 Summary and discussion
5.6.3 Emission from dark star remnants
As in Chapter 5.5.3, we consider a scenario where dark stars explode at the end of their lifetime
and dark matter annihilation products in their remnants contribute to the cosmic background. For
simplicity, we consider 511 keV emission only, which is also mostly sensitive to modifications at
high redshift. Again, we assume that dark stars form in halos between the filtering mass MF and the
mass corresponding to a virial temperature of 104 K, Mc . In this case, the volume emissivity is given
as
Pν = δ ((1 + z)ν − ν511 ) 511 keVα511 fr fa
d fcoll (MF )
× nDM fcore
,
dt
(5.22)
where nDM is the mean proper number density of dark matter particles and d fcoll /dt can be evaluated
from Eq. (5.5). The model-dependent factor fr determines which fraction of the dark matter in the
star will be left in the remnant, we adopt fr = 1 to obtain an upper limit. The factor fa determines
the fraction of the remaining dark matter which actually annihilates, which we set to fa = 1 as
well. As in Chapter 5.6.1, α511 = 1/4 is the fraction of electron-positron annihilations per one dark
matter annihilation process, corresponding to annihilation via positronium formation. In Fig.5.9, we
compare the results with the observed X-ray background from the HEAO-experiments [Gruber et al.,
1999] and SWIFT/BATSE observations [Ajello et al., 2008].
For dark matter particle masses below 30 MeV, the upper limit found here is higher than the observed background. Again, as discussed in Chapter 5.5.4, there are significant uncertainties regarding
the question whether this high contribution can be reached, both due to uncertainties in the dark star
models, which have not been explored for light dark matter, as well as the impact of a supernova
explosion on the dark matter cusp. These possibilities should be addressed further in future work.
5.7 Summary and discussion
In this work, we have examined whether the suggestion of dark star formation in the early universe
is consistent with currently available observations. We use these observations to obtain constraints
on dark star models and dark matter properties. From considering cosmic reionization, we obtain the
following results:
• Dark stars with masses of the order 800 M' as suggested by Freese et al. [2008a] can only
be reconciled with observations if somewhat artificial double-reionization scenarios are constructed. They consist of a phase of dark star formation followed by a phase of weak Pop. II
star formation and a final star burst to reionize the universe until redshift 6.
1
2
http://heasarc.gsfc.nasa.gov/docs/heasarc/missions/solarmax.html
http://wwwgro.unh.edu/comptel/
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• The same is true for dark stars in which the number of UV photons is significantly increased
due to dark matter capture, as suggested by Iocco et al. [2008].
• It appears more reasonable to require that dark stars, if they were common, should have similar
properties as conventional Pop. III stars. For MS-dominated models, this requires that typical
dark star masses are of order 100 M' or below. For CD models it requires a dark matter density
above 1011 −1012 GeV cm−3 if a spin-dependent elastic scattering cross section of t×10−39 cm2
is assumed [Yoon et al., 2008].
• Alternatively, it may imply that the elastic scattering cross section is smaller than the current
upper limits, that the dark matter cusp is destroyed by mergers or friction with the gas or that
the star is displaced from the center of the cusp.
• A further interpretation is that dark stars are very rare. This would require some mechanism to
prevent dark star formation in most minihalos.
• However, if the double-reionization models are actually true, it would indicate that dark stars
form only at redshifts beyond 14, which makes direct observations difficult.
• We also note that 21 cm observations may either confirm or rule out double-reionization models.
We have also examined whether the formation of dark stars and the corresponding enhancement of
dark matter density in dark matter halos due to adiabatic contraction may increase the observed X-ray,
gamma-ray and neutrino background. Here we found the following results:
• For massive dark matter particles, direct annihilation into gamma-rays provides significant constraints for masses less than 30 GeV.
• For massive dark matter particles, the contribution from direct annihilation into neutrinos is
well below the observed background.
• In light dark matter scenarios, the 511 keV emission is significantly enhanced below frequencies of 100 keV in the observers restframe. For a certain range of parameters, this emission
may even form a significant contribution of the total X-ray background. In this case, we derive
a lower limit of 10 MeV for the dark matter particle mass (while we find 7 MeV for standard
NFW profiles).
• In light dark matter scenarios, the background radiation due to internal bremsstrahlung is not
affected significantly from adiabatic contraction at early times, as the main contribution comes
from low redshift.
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5.7 Summary and discussion
• Both for light and massive dark matter particles, the annihilation products in the remnants of
dark stars may provide significant contributions that may be used to constrain such models in
more detail. However, whether this contribution can be reached is highly model-dependent and
relevant questions regarding the death of dark stars has not been explored in the literature.
Future observations may provide further constraints on this exciting suggestion. Small-scale 21 cm
observations may directly probe the HII regions of the first stars and provide a further test of the
luminous sources at high redshift, and extremely bright stars might even be observed with the JamesWebb telescope, if they form sufficiently late. With this work, we would like to initiate a discussion
on observational tests and constraints on dark stars, which may tighten theoretical dark star models
and provide a new link between astronomy and particle physics.
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REIONIZATION AND EXTRAGALACTIC BACKGROUND RADIATION
118
6
Cosmic constraints rule out s-wave
annihilation of light dark matter
In this section, it is shown that light dark matter models assuming a constant thermally-averaged
annihilation cross section (s-wave annihilation) can be ruled out by a combination of cosmic and
galactic constraints. The work presented here was published by Schleicher, Glover, Banerjee, &
Klessen [2009b].
6.1 Introduction
Recent observations of the Galactic center in different frequencies have provided increasing motivation for theoretical models that consider dark matter annihilation and/or decay. One finds an excess of
GeV photons [de Boer et al., 2005], of microwave photons [Hooper et al., 2007], of positrons [Cirelli
et al., 2008] and of MeV photons [Jean et al., 2006; Weidenspointner et al., 2006], which correlates
with the Galactic bulge instead of the disk and cannot be attributed to single sources. It is therefore
controversial whether it can be explained by conventional astrophysical sources alone or if dark matter annihilation models are required. It is not clear whether these different phenomena are related.
Hence, their interpretation is still under discussion [de Boer, 2008]. For this reason, it is particularly
interesting to also consider constraints which are independent of the Galactic center observations.
Observations in the MeV energy range favor models based on light dark matter. These models
suggest that light dark matter particles with masses of 1 − 100 MeV annihilate into electron-positron
pairs [Boehm et al., 2004a]. In such a scenario, the dark matter mass needs to be larger than 511 keV
to be able to produce electron-positron pairs through dark matter annihilation. It should be smaller
than 100 MeV, as otherwise pion final states that produce too many gamma rays would be possible
[Boehm et al., 2004a]. Electromagnetic radiative corrections to the annihilation process require the
emission of internal bremsstrahlung [Beacom et al., 2005]. It has also been proposed that such internal
bremsstrahlung emission might explain the observed gamma-ray background in the 10−20 MeV range
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6. COSMIC CONSTRAINTS RULE OUT S-WAVE ANNIHILATION OF LIGHT DARK
MATTER
[Ahn et al., 2005; Ahn & Komatsu, 2005b] for dark matter masses of ∼ 20 MeV, though these models
appear less favorable in light of stronger upper limits on the dark matter particle mass [Beacom &
Yüksel, 2006; Sizun et al., 2006]. Supernovae data require dark matter particle masses larger than
10 MeV, although this limit depends on assumptions made regarding the scattering cross section
between dark matter particles and neutrinos [Fayet et al., 2006].
Calculations regarding the annihilation of light dark matter in the Milky Way show that predicted
and observed fluxes are only in agreement for p-wave annihilation models [Boehm et al., 2004a].
In addition, dark matter models can be constrained by the cosmic backgrounds [Ahn & Komatsu,
2005a; Ando, 2005; Rasera & Teyssier, 2006; Yüksel et al., 2007; Mack et al., 2008; Schleicher et al.,
2008a]. Such constraints provide a highly complementary approach based on the observed cosmic
gamma-ray background. In this letter, we show that such constraints provide a strong independent
confirmation that s-wave annihilation of light dark matter is ruled out.
6.2 Assumptions
To explain the observed abundance of dark matter in the universe requires a thermally-averaged annihilation cross-section of [Boehm et al., 2004b; Drees & Nojiri, 1993]
"σv# ∼ 3 × 10−26 cm3 s−1 .
(6.1)
In the mass range considered here, this may vary only by 10% [Ahn & Komatsu, 2005b; Boehm et al.,
2004b]. Such a cross section is in agreement with conservative constraints derived from gamma-ray
observations of the Milky Way, Andromeda (M31) and the cosmic background [Mack et al., 2008].
We adopt Eq. (6.1) in this study.
The overall intensity of annihilation radiation, however, depends sensitively on the dark matter clumping factor. This quantity, defined as C(z) = "ρ2DM (z)#/"ρDM (z)#2 , has been subject to several studies [Ahn & Komatsu, 2005a; Ando, 2005; Chuzhoy, 2008; Cumberbatch et al., 2008] and
turns out to be highly uncertain. For example, a very detailed study by Cumberbatch et al. [2008]
computed clumping factors for three commonly-adopted dark matter halo profiles: Navarro-FrenkWhite (NFW) [Navarro et al., 1997], Moore [Moore et al., 1999] and Burkert profiles [Burkert, 1995].
Depending on model assumptions, the Burkert profiles, which have a flat central core, yield clumping
factors of the order 105 − 3 × 106 , the Moore profiles, which have a steep central cusp, yield factors in
the range 107 −1011 , and intermediate NFW profiles yield factors of 106 −3×109 at redshift zero. The
large variation in values for a given profile is due to the dependence of the clumping factor on further properties like the amount of substructure and typical halo concentration parameter. There has
been a long controversy between different theoretical and observational studies, with observations
typically favoring shallow profiles and theoretical investigations favoring power-law behavior with
central slopes of the order −1 [Flores & Primack, 1994; Burkert, 1995; Fukushige & Makino, 1997,
2003; Moore et al., 1999; Salucci & Burkert, 2000; Ghigna et al., 2000; Subramanian, 2000; Taylor
& Navarro, 2001; Jing & Suto, 2000, 2002; Klypin et al., 2001; de Blok & Bosma, 2002; Ricotti,
120
6.3 Formalism
2003; Gentile et al., 2004; Tasitsiomi et al., 2004; Hoekstra et al., 2004; Broadhurst et al., 2005].
More recently, there appears some convergence towards the Einasto profile [Einasto, 1965] with a
slightly shallower slope of ∼ −0.8 [Graham et al., 2006; Springel et al., 2008a; Navarro et al., 2008].
High-resolution simulations further indicate that substructure will not provide a strong contribution
to the amount of dark matter annihilation [Springel et al., 2008b], though more realistic simulations
including baryonic physics will be needed to finally resolve this issue. As the slope of the Einasto
profile is very close to the NFW, clumping factors should be in the same range, and certainly above
the flat Burkert profile. This favors clumping factors in the range 106 − 107 , and as a firm lower
limit, we adopt 105 at redshift zero, which is the lowest value found for calculations with the Burkert
profile, and a factor of 30 below the lowest value for the NFW profile [Cumberbatch et al., 2008].
We note that some works in the literature adopt an even smaller minimal clumping factor of 2 × 104
[Ando, 2005; Yüksel et al., 2007; Mack et al., 2008], corresponding to the Kravtsov profile [Kravtsov
et al., 1998]. Such a choice would leave our conclusions unchanged.
6.3 Formalism
The gamma-ray background intensity is given as [Ahn & Komatsu, 2005b]
%
c
dzPν ([1 + z]ν, z)
Iν =
,
4π
H(z)(1 + z)4
(6.2)
where c is the speed of light, z the redshift, Pν the proper volume emissivity and H(z) the Hubble
function. The proper volume emissivity is given as
'
(
1
4α g(ν)
2
,
(6.3)
Pν = hν"σv#Cclump nDM
2
π ν
where h is Planck’s constant, Cclump is the dark matter clumping factor, nDM the number density
of dark matter particles, α ∼ 1/137 the fine-structure constant and g(ν) is a dimensionless spectral
function, defined as

 
!
"

 s̃ 2 
1
s̃
(6.4)
g(ν) =
ln 2 − 1 1 +  2   ,
4
me
4mDM
with s̃ = 4mDM (mDM − hν). The number density of dark matter particles is calculated from the
mass density ρDM = mDM nDM , which is highly constrained from WMAP observations [Komatsu
et al., 2008]. As the thermally-averaged cross section is fixed by Eq. (6.1), the predicted gamma-ray
background depends only on the assumed dark matter particle mass mDM . Between redshifts 10 and
zero, which yield the dominant contribution to the predicted gamma-ray background, the clumping
factors are well described by a power-law
Cclump = C0 (1 + z)−β
(6.5)
with β ∼ 3 and C0 being the clumping factor at redshift zero. As discussed above, the value of
C0 should be in the range 106 − 107 , and certainly above 105 . If the dark matter density in virialized
121
6. COSMIC CONSTRAINTS RULE OUT S-WAVE ANNIHILATION OF LIGHT DARK
MATTER
halos can be assumed constant, the clumping factor rises as (1+z)−3 towards lower redshift [Chuzhoy,
2008]. The evolution is however less steep if substructure in the halo is neglected [Ahn & Komatsu,
2005b,a]. Such a case would strengthen the constraint given below, as even more radiation will be
emitted at high redshift.
6.4 Results
For different dark matter masses, we calculate which clumping factor would be required to exceed
the observed gamma-ray background due to emission of internal bremsstrahlung. For illustrative
purposes, we give some examples in Fig. 6.1. In Fig. 6.2, we show the constraint for different dark
matter masses larger than 511 keV. Further limits can be calculated from 511 keV line emission. As a
conservative choice, we assume that the electron-positron-pairs annihilate via positronium formation
as observed in the Milky Way [Kinzer et al., 2001], such that only 25% of all annihilations lead to the
emission of 511 keV photons. For masses of 7 MeV, one finds a maximum permitted clumping factor
of C0 = 105 [Schleicher et al., 2008a]. As this emission always peaks at the same frequency, one can
obtain the constraint for other particle masses by requiring that C0 /m2DM = const. For dark matter
masses larger than 11 MeV, the constraint from internal bremsstrahlung is most stringent. The limit
from 511 keV emission is only important for dark matter masses below 11 MeV. For masses below
the upper limit from Galactic center observations, the constraint from 511 keV emission is strongest.
6.5 Implications and discussion
As shown by Beacom & Yüksel [2006], light dark matter scenarios that explain observations of the
Galactic center require a mass less than 3 MeV. This results from a comparison between 511 keV
emission and higher-energy gamma-rays and does not require assumptions regarding the dark matter
profile. For low masses, clumping factors C0 in the range 106 − 107 , which we would expect for
the currently-favored Einasto profiles, are excluded due to the gamma-ray background constraints.
Even if we adopted clumping factors corresponding to a flat Burket profile with little substructure,
corresponding to values C0 ≥ 105 [Cumberbatch et al., 2008], this would violate the constraints from
the gamma-ray background. This shows that the light dark matter model can no longer be maintained with a constant thermally-averaged annihilation cross section if constraints from the cosmic
background are combined with the upper mass limits from the spectral constraints, thus providing an
independent confirmation that s-wave annihilation of light dark matter can be ruled out.
These conclusions hold even if we adopted the very conservative upper mass limit ofSizun et al.
[2006], which is 7.5 MeV if the ISM in the Galactic center would be strongly ionized. This appears
unlikely and provides a very strong upper limit. In addition, more recent INTEGRAL observations
favor an even smaller emission region, which increases the inflight annihilation intensity by a factor
of 1.7 and pushes the allowed maximum positron injection energy even further down (John Beacom,
private communication).
122
6.5 Implications and discussion
(a)
Figure 6.1: The predicted gamma-ray background due to internal bremsstrahlung emission for different
dark matter particle masses. In every case, we adopted a clumping factor that yields the maximum allowed background. We compare with the observed gamma-ray background from COMPTEL (crosses)
[Kappadath et al., 1996], SMM (plusses) [Watanabe et al., 1999] and EGRET (squares) [Strong et al.,
2004].
123
6. COSMIC CONSTRAINTS RULE OUT S-WAVE ANNIHILATION OF LIGHT DARK
MATTER
(a)
Figure 6.2: The constraint on the clumping factor C 0 at redshift zero due to internal bremsstrahlung
emission. For comparison, we show the weaker constraint due to 511 keV emission. The forbidden region
is shaded. We also show the upper mass limit from Galactic center observations calculated by Beacom
& Yüksel [2006] as well as the minimal clumping factor C 0 = 105 . The combination of these constraints
shows that light dark matter models with s-wave annihilation are ruled out.
124
6.5 Implications and discussion
Still feasible are models based on p-wave annihilation, in which the annihilation cross section becomes much smaller at late times when the velocities of dark matter particles are no longer relativistic
[Bœhm & Fayet, 2004]. It is also interesting to note that the gamma-ray background at 10 − 20 MeV
might be explained by a power-law component of non-thermal electrons in active galactic nuclei
(AGN) [Inoue et al., 2008]. It is however unclear whether a sufficient number of non-thermal electrons
is actually available. We expect that the FERMI satellite 1 will shed more light on such questions,
as the anisotropic distribution of the gamma-ray background may allow one to distinguish between
astrophysical sources and dark matter annihilation [Ando et al., 2007].
We finally note that we have checked if similar constraints apply for more massive dark matter
candidates, based on the recent analysis of Mack et al. [2008] and Yüksel et al. [2007]. For such
models, however, clumping factors in the range 107 − 109 are still feasible and well within the allowed
parameter space. The upper limits derived for light dark matter do not apply here, as the annihilation
products may be different.
We finally note that if the clumpiness of dark matter was high already at early times, annihilation
of light dark matter may provide a significant contribution to the observed reionization optical depth
[Schleicher et al., 2008b; Chuzhoy, 2008; Natarajan & Schwarz, 2008]. The detailed effects of such
annihilations on the IGM have been explored by Ripamonti et al. [2007a], and consequences for
21 cm observations have been explored by Furlanetto et al. [2006b]; Chuzhoy [2008].
1
http://www.nasa.gov/mission pages/GLAST/science/index.html
125
6. COSMIC CONSTRAINTS RULE OUT S-WAVE ANNIHILATION OF LIGHT DARK
MATTER
126
7
Detecting the first quasars with ALMA
and JWST
In this chapter, it is shown how ALMA and JWST open a new observational window to detect quasars
with at least 106 M' black holes between z ∼ 5 and z ∼ 15. Both telescopes are highly complementary
in the sense that ALMA can detect molecular and fine structure lines in the mm/sub-mm regime,
while JWST will observe stellar light, the AGN continuum and UV lines. This is an extension of
a previous study by Spaans & Meijerink [2008], who calculated observables for ALMA in lowmetallicity gas in active galaxies. Here, I explore which observables can be expected for higher
metallicities, using the nearby Seyfert 2 galaxy NGC 1068 as a model AGN to calculate the expected
fluxes. In addition, I provide upper limits on the population of high-redshift quasars based on the
unresolved X-ray background, and a lower limit by considering the origin of the observed population
of supermassive black holes at z ∼ 0. This work was published by Schleicher, Spaans, & Klessen
[2008d].
7.1 Introduction
After the first detection of supermassive black holes at z ∼ 6 [Fan et al., 2001], there has been
a strong effort to find more sources at high redshift and to study their properties. Further high-z
quasi-stellar objects (QSOs) have been found in surveys like the Sloan Digital Sky Survey (SDSS)
[Fan et al., 2001; Carballo et al., 2006; Cool et al., 2006; Fan et al., 2006; Shen et al., 2007; Inada
et al., 2008], the Canada-France high-redshift quasar survey [Willott et al., 2007], the UKIDSS Large
Area Survey [Venemans, 2007] and the Subaru Telescope [Goto, 2006]. They have been studied by
follow-up observations with Chandra [Shemmer et al., 2006], with IR spectroscopy [Imanishi et al.,
2007] and mid-IR observations [Stern et al., 2007; Martı́nez-Sansigre et al., 2008], Swift observations
[Sambruna et al., 2007], VLBA observations [Momjian & Carilli, 2007], the Very Large Array as well
as the Very Long Baseline Array [Momjian et al., 2008]. Some of their host galaxies have also been
127
7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
detected in CO and submm emission [Omont et al., 1996; Carilli et al., 2002; Walter et al., 2004;
Klamer et al., 2005; Weiß et al., 2005; Maiolino et al., 2007; Walter et al., 2007; Weiß et al., 2007;
Riechers et al., 2008a,b]. Of particular relevance is the observation of APM 08279 + 5255 at z = 3.9
by Weiß et al. [2007], as its central region was significantly magnified by a gravitational lense, thus
providing a first impression of what higher-resolution telescopes like ALMA might see in the future.
The host galaxy of the quasar SDSS J114816.64 + 525150.3 at z = 6.42 has recently been observed
in C+ emission[Walter et al., 2009], finding that C+ is a very promising tracer for such high-redshift
observations.
Observational studies derived a number of relations between the mass of the central black hole
and the properties of the host galaxy. The Magorrian relation connects the velocity dispersion and
the mass of the galactic bulge with the black hole mass [Magorrian et al., 1998; Ferrarese & Merritt,
2000; Gebhardt et al., 2000; Graham et al., 2001; Merritt & Ferrarese, 2001; Tremaine et al., 2002;
Häring & Rix, 2004; Peterson, 2008; Somerville et al., 2008]. Quasars with black hole masses of
the order 109 M' are observed with supersolar metallicities even at z ∼ 6 [Pentericci et al., 2002],
which is indicated in further studies as well [Freudling et al., 2003; Maiolino et al., 2003; Dietrich
et al., 2003a,c]. In the nearby universe, correlations have been found between black hole mass and
metallicity [Warner et al., 2003; Kisaka et al., 2008], with a mild slope of 0.38 ± 0.07. This is plausible, as it is generally recognized that the overall mass of a galaxy is correlated with its metallicity
[Faber, 1973; Zaritsky et al., 1994; Jablonka et al., 1996; Trager et al., 2000]. Juarez et al. [2009]
recently examined the metallicity of high-redshift quasars, finding no evidence for an evolution of
quasar metallicity with redshift. In this paper, we will generally assume that the black hole mass is
a good indicator for the evolutionary stage of the quasar host galaxy, in particular with respect to the
Magorrian relation and the expected metallicity, but we will also discuss how the relevant observables
change in the case of lower metallicities.
It is an important goal to understand how different properties of supermassive black holes and their
host galaxies evolve with redshift. For the relation between black hole mass and velocity dispersion,
an indication for an evolution was found by Woo et al. [2006, 2008], though the uncertainties are
still large. An evolution is also indicated for the Eddington ratio λE , which is the ratio between the
bolometric black hole luminosity and the Eddington luminosity. In the local universe, Kaspi et al.
[2000] found Eddington ratios mostly between 0.01 and 0.1. Studies at higher redshift show that this
may increase up to λE ∼ 0.3, though it depends also on the black hole mass [Shankar et al., 2004;
Kollmeier et al., 2006; Shankar et al., 2009]. The number density of high redshift black holes was
recently studied by Shankar et al. [2009] and Treister et al. [2009].
In the local universe, active galactic nuclei (AGN) can be studied in much more detail and with
higher resolution. An example that has been studied particularly well, in different wavelengths and
with high resolution is the AGN NGC 1068. Its direct X-ray emission is shielded through absorption
along the line of sight, but there is evidence for reflected X-ray photons that are scattered into the
line of sight [Pounds & Vaughan, 2006]. Its molecular disk has been studied through emission from
different species, in particular molecular CO and fine-structure lines [Schinnerer et al., 2000; Galliano
128
7.1 Introduction
et al., 2003, 2005; Spinoglio et al., 2005; Davies et al., 2007b,a; Poncelet et al., 2007]. The continuum
emission and the distribution of dust has been measured in further studies [Krips et al., 2006; Mason
et al., 2006; Rhee & Larkin, 2006; Tomono et al., 2006; Howell et al., 2007; Hönig et al., 2008].
Such observations in particular allow to probe stellar and gaseous dynamics and the galactic gravitational potential [Gerssen et al., 2006; Emsellem et al., 2006; Das et al., 2007]. Observations with the
Green Bank Telescope (GBT) provide evidence for water maser activity [Greenhill & Gwinn, 1995;
Greenhill et al., 1996; Kondratko et al., 2006]. The inner radio jet can be detected with observations
at 7 mm wavelength [Cotton et al., 2008]. Recent high-resolution observations with MIDI even resolved the inner torus around the central black hole [Raban et al., 2009]. Given this amount of detail,
NGC 1068 thus provides an excellent test case for the inner structure of active galaxies.
The formation of quasars and their supermassive black holes is still one of the unresolved problems of structure formation and cosmology. The simplest scenarios assume that they have grown from
the remnants of the first stars, which are believed to be very massive [Abel et al., 2002; Bromm &
Larson, 2004; Glover, 2005], and whose black hole remnants could grow further by accretion. Such
a scenario however has problems, as the remnants of the first stars typically do not end up in the most
massive quasars at redshift z ∼ 6 [Trenti & Stiavelli, 2008], and radiative feedback from the stellar
progenitor can delay accretion as well [Johnson & Bromm, 2007; Alvarez et al., 2008; Milosavljevic
et al., 2008]. In case of Eddington accretion, seed black holes of ∼ 105 M' are required in order to
grow to the observed supermassive black holes at z ∼ 6 [Shapiro, 2005].
Recently, it has also been discussed whether the first stars in the early universe were powered
by dark matter annihilation rather than nuclear fusion [Spolyar et al., 2008; Iocco, 2008]. Such stars
could reach masses of the order 1000 M' [Freese et al., 2008a; Iocco et al., 2008] and were considered
as possible progenitors for the first supermassive black holes. The evolution of such stars on the main
sequence has been calculated by Taoso et al. [2008] and Yoon et al. [2008]. However, it was shown
that such stellar models are highly constrained by the observed reionization optical depth [Schleicher
et al., 2008b,a]. Also, we note that such seeds would still require super-Eddington accretion to grow
to the observed supermassive black holes at z ∼ 6.
Therefore, alternative scenarios have been considered that lead to the formation of massive seed
black holes by direct collapse [Eisenstein & Loeb, 1995; Koushiappas et al., 2004; Begelman et al.,
2006; Spaans & Silk, 2006; Dijkstra et al., 2008]. However, for such scenarios, the situation is also
not entirely clear. For instance, Lodato & Natarajan [2006] argue that the gas in such halos should
fragment if H2 cooling is efficient, and Omukai et al. [2008] and Jappsen et al. [2008] suggest that
a non-zero metallicity will lead to fragmentation as well. Simulations byClark et al. [2008] support
these conjectures.
The black hole population at high redshift is further constrained by observations of the soft X-ray
background. These constraints indicate that the population of high-redshift quasars was not sufficient
to reionize the universe [Dijkstra et al., 2004], and suggest an upper limit to the black hole density of
∼ 4 × 104 M' Mpc−3 [Salvaterra et al., 2005]. The latter depends in particular on the adopted spectra
and luminosity of the black hole, as well as their duty cycles, and it has been argued that larger black
129
7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
hole populations are conceivable as well [Zaroubi et al., 2007]. It is therefore important to probe the
number density of black holes and AGN activity as a function of redshift by direct observations.
In this work, we explore the possibility of searching for the first quasars in the universe with
ALMA 1 and JWST 2 . ALMA is particularly relevant in light of recent predictions on the observability
of the first miniquasars [Spaans & Meijerink, 2008] and high-redshift Lyman α galaxies [Finkelstein
et al., 2008], while it is one of the main scientific goals of JWST to find the first galaxies at high redshift. A detailed description regarding the capabilities of ALMA for high-redshift quasar observations
has been given by Maiolino [2008]. In Chapter 7.2, we introduce a model for the black hole population which is based on observed properties and correlations between black holes and their host galaxy.
In Chapter 7.3, we discuss how the number of active black holes is constrained by the unresolved Xray background. In Chapter 7.4, we analyze the observed properties of NGC 1068 and derive further
predictions for very high-J CO lines. In Chapter 7.5, the main model uncertainties with respect to
high-redshift quasars are discussed. In Chapter7.6, we derive the relevant observables for ALMA and
JWST and discuss what we can learn from follow-up observations on known high-redshift quasars.
Chapter 7.7 then provides several independent estimates for the number of detectable sources as a
function of redshift and discusses different observational strategies. Further discussion and outlook is
given in Chapter 7.8.
7.2 The high-redshift black hole population
We model the population of high-redshift black holes in the framework of the Sheth-Tormen formalism [Sheth et al., 2001], a modified version of the Press & Schechter [1974] method. Both prescriptions agree reasonably well with simulations of high-redshift structure formation [Jang-Condell &
Hernquist, 2001; Heitmann et al., 2006; Reed et al., 2007; Greif et al., 2008]. This formalism predicts the comoving number density of halos with different masses, dn/d log Mh , where Mh is the halo
mass. We calculate the virial radius rvir assuming a mean overdensity of 178 and estimate the velocity
dispersion σ as the Kepler velocity at this radius. For simple profiles such as the singular isothermal
sphere, the velocity dispersion is independent of position, while in more realistic scenarios, there
will be some scatter around this value. We adopt the correlation between black hole mass MBH and
velocity dispersion σ given by Tremaine et al. [2002] as
!
"
! "
MBH
σ
log
= α + β log
,
(7.1)
M'
σ0
with α = 8.13 ± 0.06, β = 4.02 ± 0.32 and σ0 = 200 km/s. The expected black hole population is
then given in Fig. 7.2. This shows that a significant number of potential sources is available down to
z ∼ 12, which however drops significantly towards higher redshifts.
1
2
http://www.eso.org/sci/facilities/alma/index.html
http://www.stsci.edu/jwst/overview/
130
7.2 The high-redshift black hole population
Figure 7.1: The co-moving number density of halos that contain black holes as a function of black hole
mass for different redshifts.
If black holes accrete efficiently and release the gravitational energy as radiation, the luminosity
can be estimated with the Eddington luminosity as
!
"
4 MBH
LEdd = 3.3 × 10
L' .
(7.2)
M'
On the other hand, accretion may also be limited by gas dynamics, and the Bondi solution for spherical
symmetric accretion shows that the actual accretion rate may be considerably smaller if the gas is
heated by radiative feedback or supernova explosions. As discussed byJolley & Kuncic [2008], part
of the released gravitational energy may go into mechanical feedback like jets and outflows rather than
radiation. This is consistent with measurements of the black hole mass - luminosity relation byKaspi
et al. [2000], which indicate that typical luminosities are in fact sub-Eddington. The ratio between the
total bolometric quasar luminosity and its Eddington luminosity λE , the so-called Eddington ratio, has
been discussed in a number of different works. Shankar et al. [2004] try to constrain this parameter
by matching the local and the accreted black hole mass functions and parametrize it as
!
"α
1+z
z<3
(7.3)
λE (z) = λ0
4
and λE (z) = λ0 for z ≥ 3. Their minimum χ2 analysis yields λ0 = 0.3+0.3
−0.1 and 0.3 ≤ α ≤ 3.5. The
evolution with redshift appears thus as highly uncertain, and they note that an equally good fit can be
obtained if they adopt a luminosity-dependent Eddington ratio rather than a redshift-dependent ratio.
Similar results were found by a more recent analysis [Shankar et al., 2009]. Kollmeier et al. [2006]
131
7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
find an average of λE ∼ 1/4, and their figures indicate a dependence both on redshift and on quasar
luminosity. Quasars with less massive black holes appear to have smaller Eddington ratios, and the
Eddington ratios at higher redshift appear somewhat larger. In this paper, our main interest are black
hole masses between 106 − 107 M' , which are even smaller than those considered in the literature. It
is therefore not clear which Eddington ratios should be expected, in particular for the higher redshifts.
Therefore we treat the Eddington ratio as a free parameter which needs to be determined by future
observations.
7.3 Constraining the relative number of active black holes
The population of high-redshift black holes is constrained by the unresolved soft X-ray background
(SXRB). The total SXRB in the energy range 0.5 − 2 keV is (7.53 ± 0.35) × 10−12 erg cm−2 s−1 deg−2 ,
with 94+6
−7 per cent made up of discrete X-ray sources [Moretti et al., 2003]. The unresolved fraction
of the SXRB was re-analyzed by Dijkstra et al. [2004], providing a mean and maximum intensity of
the unresolved SXRB flux of 0.35 × 10−12 and 1.23 × 10−12 erg cm−2 s−1 deg−2 , respectively. This
corresponds to specific fluxes per energy interval of the order 0.7 − 2.5 keV cm−2 s−1 sr−1 keV−1 at
1 keV, which is the upper limit for possible contributions from the early universe.
7.3.1 Assumptions
For the X-ray sources, we consider spectra of similar type as Salvaterra et al. [2005], consisting of a
soft multicolor disk (MCD) component and a power-law (PL) component at high energies [Mitsuda
et al., 1984; Miller et al., 2003; Miller & Colbert, 2004]. The MCD component peaks at frequencies
!
MBH
hν ∼ 0.095 keV
106 M'
"−1/4
(7.4)
and the spectral energy distribution (SED) scales with ν1/3 for smaller frequencies and rolls off exponentially for larger frequencies. The MCD component of high-redshift quasars is therefore redshifted
to even smaller frequencies and cannot contribute to the unresolved SXRB. For the PL SED, we
assume a scaling according to ν−α , with α ∼ 1. It is the PL component of these sources which is
redshifted into the SXRB. The spectral range in which these sources emit is not completely clear.
Here, we adopt a spectral range from 1 eV to 500 keV. The ratio r of the flux in the MCD component
to the flux in the PL component is expected to be roughly of order 1 [Mitsuda et al., 1984; Miller
et al., 2003], but we treat it as a free parameter in this analysis. A further fraction fs of the radiation
may be shielded by the inner torus. This fraction is geometry dependent, but may be of the order of
20 − 30%. The amount of luminosity that can potentially contribute to the SXRB thus scales with the
effective parameter Φ = (1 − fs )/(1 + r). For typical spectra where the amount of flux in the MCD and
the PL component is comparable, we therefore expect Φ ! 0.5. We further introduce the parameter
fa , which describes the fraction of active quasars at a given redshift.
132
7.3 Constraining the relative number of active black holes
Figure 7.2: The expected contribution to the SXRB in keV cm −2 s−1 sr−1 keV−1 as a function of the active
quasar fraction f a and the spectral parameter Φ, under different assumptions for the typical black hole
luminosity. For a spectrum with the same amount of flux in the MCD and the PL component, we expect
Φ ! 0.5. Upper left: Eddington luminosity, black holes in the mass range from 1 to 10 9 M' . Upper right:
Eddington luminosity, black hole masses in the range from 10 5 to 109 M' . Lower left: Eddington ratio
λE = 0.1, black hole masses in the range 1 − 10 9 M' . Lower right: Eddington ratio λ E = 0.3, black hole
masses in the range 10 5 − 109 M' .
133
7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
Figure 7.3: An illustration for the inner structure of NGC 1068. It shows the central black hole with the
inner accretion disk radiating in X-rays, as well as the inner torus embedded in a large molecular disk
consisting of individual high-density clouds. The jet is sketched as well. The torus is expexted to have a
size of ∼ 3 pc, while the molecular disk extends out to ∼ 100 pc. In the XDR model considered here, we
focus on those clouds that are unshielded by the torus and where emission is stimulated by the impinging
X-ray flux.
7.3.2 Formalism
The contribution to the background intensity is then given from an integration along the line of sight
as
>
?
%
c
dzPν (1 + z)ν , z
Iν =
,
(7.5)
4π
H(z)(1 + z)4
where ν is the observed frequency, H(z) the expansion rate at redshift z and Pν ((1 + z)ν, z) the mean
proper volume emissivity for photons with frequency ν(1 + z) at redshift z (in units of energy per
unit time, unit frequency and unit proper volume). To evaluate that, we first calculate the number of
photons Nν̃ of frequency ν̃ emitted at redshift z within 1 Mpc3 (comoving), which is given as
Ntot,ν̃ =
%
Mmax
Mmin
dn
fa N(ν̃, MBH )d log Mh ,
d log Mh
(7.6)
where Mmin and Mmax are the (redshift-dependent) minimum and maximum halo masses that yield
black hole masses in a reasonable range which we specify below in more detail, and N(ν̃, MBH ) is the
number of photons of frequency ν̃ emitted per second by a black hole of mass MBH . The number of
baryons Nb in 1 Mpc3 (comoving) is
ρb 1 Mpc3
Nb =
,
(7.7)
1.2m p
134
7.4 NGC 1068: A case study
where ρb is the comoving baryon density and mp the proton mass. The emissivity is then
Pν =
hν̃Ntot,ν̃ ρb (1 + z)3
.
Nb
1.2m p
(7.8)
7.3.3 Constraints
With the method described above, we calculate constraints for the fraction of active quasars af as well
as the spectral parameter Φ for four different scenarios (see Fig. 7.2). The first scenario assumes that
black holes populate the mass range from 1 to 109 M' and radiate at Eddington luminosity. This may
be the case if a sufficient number of black holes is provided as remnants from the first stars. In this
case, one finds rather tight constraints that do not allow high values for fa and Φ simultaneously. For
a spectrum with equal amount of flux in the MCD and the PL component, we expect Φ ! 0.5.
However, as the initial mass function (IMF) in the early universe is not yet understood, it is unclear
whether such stellar mass black holes are present in large numbers. As discussed in the introduction,
supermassive black holes may in fact have more massive progenitors that form through direct gas
collapse in larger halos. We assume here a lower cutoff for the black hole mass of ∼ 105 M' . That
allows for an increase in the fraction of active quasars by a factor of a few.
Finally, as discussed in Chapter 7.2, the Eddington luminosity is only an upper limit to the actual
black hole luminosity. It depends on the gas supply as well as the amount of energy going into
mechanical feedback. We therefore consider scenarios with smaller Eddington ratios as well. Our
third scenario is similar to the first, but with an Eddington ratio of 10%. This essentially leads to a
rescaling of the contribution to the X-ray background by the same factor, giving rise to a potentially
higher fraction of active black holes. Similarly, we consider a fourth scenario which is similar to the
second one, but with an Eddington ratio of 30%. Depending on the model, we find that the active
quasar fraction may vary between a few percent and ∼ 10%.
With respect to the observational search for the first quasars at high redshift, the two latter scenarios are particularly favorable, as they are consistent with a considerably larger number of active
quasars in the early universe. While it is of course unclear what will be the average Eddington ratio
at high redshift, we also note that observational studies at all redshifts have found scatter of at least
half an order of magnitude in the Eddington ratio, implying that there always will be a population of
somewhat less luminous black holes.
7.4 NGC 1068: A case study
In this section, we review known observational properties of NGC 1068, derive predictions that will
be relevant for high-redshift observations with ALMA and JWST and generalize them to objects with
different X-ray fluxes.
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
Figure 7.4: A model for the X-ray chemistry in NGC 1068. The adopted flux impinging on the cloud is
170 erg s−1 cm−2 . The adopted density is 10 5 cm−3 . Top: The abundances of different species as a function
of column density. Middle: The low-J CO lines as a function of column density. Bottom: The high-J CO
lines as a function of column density.
136
7.4 NGC 1068: A case study
7.4.1 Observed properties
As discussed in the Introduction, the nearby AGN NGC 1068, a Seyfert 2 galaxy at a distance of
14.4 Mpc, has been studied in great detail and with high resolution. As the wealth of available data
could easily give rise to a review of its own, we restrict the discussion here to only those properties
that will be relevant with respect to observations at high-redshift with ALMA and JWST.
The mass of the central black hole has been estimated from VLBI water maser observations to
be ∼ 107 M' with an X-ray luminosity of 50% Eddington [Greenhill et al., 1996]. Even though the
X-ray emission from the central black hole is shielded by an absorber on the line of sight, X-rays
that are scattered into the line of sight have been detected unambiguously [Pounds & Vaughan, 2006].
Observations of rotational CO lines and the H2 2.12 µm line indicated an inner structure consisting
of an extended molecular disk and a toroidal structure which completely absorbs the X-ray emission
along the line of sight. It is slightly tilted with respect to the disk [Galliano et al., 2003]. The H2
emission map of Galliano & Alloin [2002] shows a bright knot 70 pc in the east of the central engine,
with a size of ∼ 35 pc. The knot is also visible in CO [Galliano et al., 2003]. As they show, the
observed fluxes can be explained with the X-ray excitation model ofMaloney et al. [1996] when the
following parametrization is adopted:
• The central engine is a power-law X-ray source with spectral slope α = −0.7 and luminosity of
1044 erg s−1 in the 1 − 100 keV range.
• We observe emission from molecular clouds with a density of 105 cm−3 , a column of 1022 cm−2
at a distance of 70 pc and solar metallicity.
This region of bright emission consists therefore of a number of small clouds with densities of
105 cm−3 , which in total cover a significant part of the telescope beam. In addition, there may be
further contributions from gas at lower densities, which emits less efficiently, but fills a larger volume. For the X-ray luminosity of NGC 1068, one cannot expect CO line emission much closer to the
black hole, as the higher X-ray flux then would destroy CO very effectively, unless it is shielded by
some absorber. The black hole luminosity thus regulates the distance to the central source at which
the CO emission peaks. This situation is sketched in Fig. 7.3.2.
Further studies also measured the fluxes from other species, including the relevant fine-structure
lines. We refer here in particular to the data ofSpinoglio et al. [2005], which are used in Chapter 7.6.1
to calculate the expected fluxes from higher redshift. The situation is similar for the observed amount
of continuum emission. Here we use the data of Brauher et al. [2008].
As JWST will see the stellar light from high-redshift galaxies, the star formation rate is a highly
relevant quantity for estimates regarding the expected flux. Indeed, observations of NGC 1068 find a
starburst component from a circumnuclear ring of ∼ 3 kpc in size, with a stellar mass of ∼ 106 M' and
an age of 5 Myr [Spinoglio et al., 2005]. On scales of a few hundred parsecs, one finds star formation
rates of a few times 10 M' yr−1 kpc−2 , which can reach ∼ 100 M' yr−1 kpc−2 in the inner few ten
parsecs [Davies et al., 2007b]. This is close to the star formation rate in Eddington-limited starbursts
137
7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
as suggested by Thompson et al. [2005]. The total star formation rate of NGC 1068 therefore is
∼ 10 M' yr−1 .
7.4.2 Predicted properties
High redshift and high spatial resolution (∼ 0.01(( ) observations with ALMA will allow us to observe
redshifted high-J CO lines, J > 9 (see Chapter 7.6.1 for details). For local sources, these lines are
difficult to detect from the ground. They provide important diagnostics for the power and impact of
the accreting black hole [Spaans & Meijerink, 2008]. We use the X-ray dominated region (XDR) code
of Meijerink & Spaans [2005], an improved version of the above mentioned XDR-model ofMaloney
et al. [1996], and solve the 1D-radiation transport equation for the CO fluxes through the inferred
cloud complex. This code has been used to calculate detailed diagnostics for XDR observations
[Meijerink et al., 2007] and was successfully applied to infrared and (sub-)millimeter observations of
luminous infrared galaxies [Loenen et al., 2008].
We first calculate the chemical abundances and the expected CO line intensities for the model
used by Galliano et al. [2003] as discussed in Chapter 7.4.1. While they adopted a column density
of 1022 cm−2 , we are extending the calculation to larger columns of 1024 cm−2 , corresponding to the
typical column needed to absorb 5 keV photons. This allows us to generalize these results to systems
with larger column densities and Compton thick material. The results of the calculations are shown
in Fig. 7.4. We note that the fiducial gas density of 105 cm−3 has little impact on our results, unless
it drops to below 104.5 cm−3 . Above this limit, the XDR properties are determined by the ratio of
X-ray flux to gas density. In addition, we have checked that even if the entire disk of ∼ 100 pc was
at low densities of 103 cm−3 , the high-J CO line intensities would still be ∼ 10−5 erg s−1 cm−2 sr−1 ,
as the low-density gas extends over much larger spatial scales and thus reaches larger or comparable
column densities.
The strong X-ray flux of ∼ 170 erg s−1 cm−2 in NGC 1068 suffices to make the gas essentially
atomic and leads to high temperatures of ∼ 3000 K, as well as relatively low CO abundances of the
order 10−7 . However, the CO intensity is high, due to the strong excitation in the hot gas. For a
column of 1022 cm−2 , our results appear of the same magnitude as in the model of Galliano et al.
[2003]. For larger columns, the temperature gradually decreases, the gas becomes molecular and CO
gets more abundant, and we find intensities of the order 10−2 erg s−1 cm−2 sr−1 in the high-J CO lines.
We need to account for the fact that many systems may accrete considerably below the Eddington
limit as discussed above. Also, we want to consider quasars with somewhat smaller black hole
masses of the order 106 M' , as their number density is larger by an order of magnitude (see Fig.7.2).
Therefore, we calculate models with less X-ray flux as well. An extreme case with ∼ 1 erg s−1 cm−2 is
shown in Fig. 7.5. In this model, we find lower temperatures of ∼ 70 K, a large fraction of molecular
gas and CO abundances of the order 10−4 . While the lower temperature tends to decrease the CO line
intensities, they are still enhanced due to the larger CO abundance. Above a column of 1023 cm−2 ,
the intensities increase rather slowly as the lines become optically thick.
138
7.4 NGC 1068: A case study
Figure 7.5: The X-ray chemistry in a system with X-ray flux of 1 erg s −1 cm−2 impinging on the cloud.
The adopted density is 10 5 cm−3 . Top: The abundances of different species as a function of column
density. Middle: The low-J CO lines as a function of column density. Bottom: The high-J CO lines as a
function of column density.
139
7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
Figure 7.6: The X-ray chemistry in a system with X-ray flux of 10 erg s −1 cm−2 impinging on the cloud.
The adopted density is 10 5 cm−3 . Top: The abundances of different species as a function of column
density. Middle: The low-J CO lines as a function of column density. Bottom: The high-J CO lines as a
function of column density.
140
7.5 Uncertainties in the model assumptions
As an intermediate scenario, we consider an X-ray flux of ∼ 10 erg s−1 cm−2 (see Fig. 7.6). In this
model, the temperature is increased to ∼ 100 K. The CO abundance is initially of the order 3 × 10−6
and increases to ∼ 10−4 for larger columns. For columns less than 1022 cm−2 , the intensities are thus
reduced by about an order of magnitude compared to the previous case, while they are increased by
an order of magnitude for larger columns.
From an observational perspective, it is highly interesting that even X-ray fluxes of ∼ 1 erg s−1 cm−2
lead to significant CO line intensities. At a column of 1022 cm−2 , the high-J CO line intensity in
all models was of the order 10−4 − 10−3 erg s−1 cm−2 sr−1 , and it may increase up to values of
10−3 − 10−1 erg s−1 cm−2 sr−1 for columns of the order 1024 cm−2 .
7.5 Uncertainties in the model assumptions
In the previous section, we have considered one particular and well-studied active galaxy in quite
some detail. For observational predictions with respect to high redshifts, it is important to asses
their dependency on model assumptions and analyze the justifications of these assumptions. Here we
discuss metallicity, the metal abundance ratios and the inner structure of high-redshift AGNs. The
available X-ray flux could be considered as a similar uncertainty, but its influence has been discussed
in detail in the previous section.
7.5.1 Metallicity
For the metallicity, there is indeed a strong case that it may stay high even at high redshift. In contrast
to other typical quasar properties like the M − σ relation, the metallicity has been probed even at z ∼ 6
and was found to be supersolar at such high redshift [e. g. Pentericci et al., 2002; Freudling et al.,
2003; Maiolino et al., 2003; Dietrich et al., 2003a,c]. Recently, a systematic comparison of quasar
metallicities between redshift 4 and 6 has been performed byJuarez et al. [2009], searching for signs
of evolution within this redshift range and with respect to local quasars, without any indication for
such a scenario. This is particularly interesting, as for other quasar properties like the Eddington ratio
or the M − σ relations, indications for a deviation have been found already near z ∼ 1 [e. g.Shankar
et al., 2004; Kollmeier et al., 2006; Woo et al., 2006; Shankar et al., 2009; Woo et al., 2008]. One
might therefore expect that indeed quasars stay at metallicities larger than 0.3 Z' , in particular in the
central region of ∼ 200 pc which necessarily must be highly evolved in order to produce a massive
central black hole.
However, there might still be some evolution, in particular when systems at z > 10 are considered.
This case of low-metallicity gas has been considered previously bySpaans & Meijerink [2008]. They
find that significant CO fluxes are still reached for Z > 0.1Z' , and may even be present for smaller
metallicities, depending on the X-ray flux. In those cases where the CO flux goes down significantly,
more flux is released by H2 line emission, which is redshifted in ALMA’s frequency range for z >
141
7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
10. An evolution in metallicity is thus not a severe restriction for high-redshift quasar observations.
Instead it could be a primary result of the proposed surveys.
7.5.2 Variations of abundance ratios
Other uncertain quantities are the metal abundance ratios, which may evolve with redshift as well.
Different metal abundance ratios may change the abundances of molecules by a factor of a few, but
we do not expect a qualitative change of X-ray chemistry unless the carbon abundance exceeds the
oxygen abundance, in which case not all of the carbon is bound in CO. Instead, additional carbon
would be available that would lead to more complicated molecules and significantly change the chemistry [Herpin et al., 2002]. Still, even in this case we expect that the absorbed energy must be released
in some form and also in the mm/sub-mm regime. This is because the total heating rate is mostly
determined from absorptions by hydrogen and helium, and only to a small extend by the elemental
abundance ratios [Maloney et al., 1996; Meijerink & Spaans, 2005]. In dense gas, an equilibrium
state in which the heating rate equals the cooling rate is quickly established. The latter is dominated
by [CII] 158 µm, [OI] 63 µm and CO, with contributions from [SiII] 35 µ, [FeII] 26 µm, H2 , HCO+
and HCN [Maloney et al., 1996; Meijerink & Spaans, 2005]. While the individual contributions of
these species depend on the abundance ratios, the sum is roughly independent of them, such that the
flux in some lines will be observable. In this sense, observations with ALMA provide an opportunity
to infer the elemental abundance ratios at high redshift.
7.5.3 Internal structure
The remaining point to discuss is therefore the inner structure of high-redshift AGN. However, this
appears not as a crucial point, as our models are quite generic. We only require a central region in
which molecular clouds are irradiated by X-ray radiation. The size of this region and also its shape
may vary at different redshifts, ranging from disk-like structures as in NGC 1068 to more spherical
structures. Indeed, one could imagine that the central regions are more compact at high redshift due
to the general enhancement in density, which would lead to a larger amount of flux being produced
in a smaller region. We are largely insensitive to the geometry, since to first order only the absorbing
column density matters for the deposited X-ray energy. A column density of 1022 cm−2 is sufficient
to absorb a 1 keV photon, which is easily reached within the central 200 pc. The absorbed energy
generates the line fluxes within a certain area. Within this area, it does not matter whether the medium
is homogeneous or filled with clumps, as long as the latter fill a substantial fraction of the beam
[Spaans & Meijerink, 2008].
In this respect, the recent observation of APM 08279+5255 at z = 3.9 by Weiß et al. [2007] is
particularly encouraging, as the central region of this system has been magnified by a gravitational
lense by a factor m = 60 − 110, providing a unique opportunity to study the inner 200 pc which
otherwise could not be resolved with present telescopes. Their high-J CO and HCN 5-4 data also
indicate unambiguously that dense, 104 − 105 cm−3 gas is present. Once ALMA is available even with
142
7.6 High-redshift observations with ALMA and JWST
the initial set of 16 antennas, similar observations can be performed for the central regions of further
AGNs to provide further evidence for this scenario, while magnified objects like APM 08279+5255
can be probed down to even smaller scales.
With respect to geometrical uncertainties, we also point out that our model calculations are conservative in the sense that they assume a homogeneous medium without intrinsic velocity gradients.
In practice, the ISM will however consist of clumpy structures, in which each individual cloud has a
different velocity, thus reducing the line overlap and allowing more line flux to escape from the cloud.
In this sense, a porous medium would enhance the emissivity of optically thick lines. For a cloud
covering factor of order unity, the difference is typically a factor of two [Spaans, 1996].
7.6 High-redshift observations with ALMA and JWST
freq. [GHz]
θres [”]
S l [mJy]
S c [mJy]
θbeam [”]
84 − 116
125 − 169
163 − 211
211 − 275
275 − 373
385 − 500
602 − 720
0.034
0.023
0.018
0.014
0.011
0.008
0.005
0.42
0.45
6.8
0.54
0.96
1.6
2.1
0.06
0.07
1.3
0.14
0.25
0.86
1.3
56
48
35
27
18
12
9
Table 7.1: Frequency range, angular resolution θ res , line sensitivity S l for a bandwith of 300 km/s and
continuum sensitivity S c for 1σ detection in one minute of integration time and primary beam size θ beam
for bands 3 − 9 of ALMA. 3 more bands might be added in the future, band 1 around 40 GHz, band 2
around 80 GHz and band 10 around 920 GHz, which will have similar properties as the neighbouring
bands.
Future observations with ALMA and JWST provide new opportunities to find the first quasars at
high redshift. Their unprecedented sensitivity permits observations of less luminous sources and at
higher redshift and we can therefore find quasars with black hole masses of 106 − 107 M' , which may
be the progenitors of the known sources with ∼ 109 M' black holes. In addition, their high spatial
and spectral resolution can provide detailed information about the host galaxies of those bright highredshift quasars that we already know. In this section, we predict the main observables for ALMA
and JWST.
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
7.6.1 Observables for ALMA
In the frequency range of ALMA, it is possible to observe redshifted emission from high-J CO lines
and many fine-structure lines. A detection of at least two different lines directly provides the redshift
of the source. The main properties of ALMA are summarized in Table 7.1.
7.6.1.1
High-J CO lines
For one hour of integration time, ALMA can detect line fluxes of ∼ 0.09 mJy at 3σ within one
hour of integration time, assuming a line width of 300 km/s. This is particularly interesting with
respect to the high-J CO intensities calculated in Chapter 7.4.2. At redshift z = 8, a resolution
of ∼ 0.04” corresponds to a physical scale of ∼ 200 pc. A source of such size would fill a solid
angle of ∼ 7 × 10−15 sr. With a fiducial velocity dispersion of 300 km/s and a line intensity of
10−3 erg s−1 cm−2 sr−1 , this corresponds to a flux of ∼ 0.04 mJy, which is therefore detectable in
approximately two hours. In the calculations above, we found lower intensities only for models with
very high X-ray fluxes and small columns (see Fig.’s 7.4 to 7.6). Even such sources should however
be detectable within one day of integration time.
Observable
λ [µm]
ϕ [mJy]
Redshift
high-J CO lines
high-J HCN lines
high-J HCO+
[O I] 3 P1 →3 P2
[O III] 3 P1 →3 P0
[N II] 3 P2 →3 P1
[O I] 3 P0 →3 P1
[C II] 2 P3/2 →2 P1/2
[S III] 3 P1 →3 P0
[Si II] 2 P3/2 →2 P1/2
dust continuum
∼ 300
∼ 300
∼ 300
63.2
51.8
121.9
145.5
157.7
33.5
34.8
∼ 500
∼ 0.04
∼ 0.004
∼ 0.004
∼ 0.11
∼ 0.11
∼ 0.04
∼ 0.02
∼ 0.88
∼ 0.01
∼ 0.2
∼ 0.05
z>5
z>4
z>4
z > 5.7
z > 3.8
z > 2.5
z > 1.9
z > 1.7
z > 11.5
z > 11.1
z>0
Table 7.2: The main observables for ALMA, as discussed in Chapter 7.6.1. The estimated flux ϕ for the
high-J CO lines assumes that one typical resolution element of ∼ 0.01 (( is comparable to the source size,
while the other fluxes correspond to the fluxes observed for NGC 1068 if it were placed at z ∼ 8 (for lines
only observable at z > 10, we placed it at z = 12). The rest frame wavelength is given as λ. The line and
continuum sensitivities are given in Table 7.1.
144
7.6 High-redshift observations with ALMA and JWST
7.6.1.2
Dust emission
As discussed by Finkelstein et al. [2008], there are further contributions from dust emission which
may be relevant for ALMA as well, because the continuum sensitivity is higher by two orders of
magnitude. For NGC 1068, Brauher et al. [2008] report a flux of 85.04 Jy at 25 µm and 224 Jy at
100 µm. For z > 5, such fluxes would be redshifted into the ALMA bands. An interpolation to
500 µm yields a flux of ∼ 700 Jy, which would be redshifted into ALMA band 3 for z ∼ 5. For the
same emission at this redshift, we would therefore expect a flux of 0.06 mJy, which matches with the
band sensitivity. For objects at z = 8, we need to consider emission at ∼ 330 µm which is redshifted
into band 3, where the extrapolation yields ∼ 520 Jy. This yields an expected flux of ∼ 0.05 mJy,
which can be detected at 8σ within an hour of integration time. Such emission may be even stronger
if the dust is heated further by ongoing starbursts.
7.6.1.3
Further molecular and finestructure lines
In addition to the high-J CO lines, it seems likely that other lines can be detected with ALMA as
well. The high-J lines of HCO+ and HCN fall in ALMA’s frequency range for z > 4, and should
have ∼ 10% of the intensity of the corresponding CO lines [Meijerink et al., 2007]. For known
sources, such emission can be detected for integration times of about one day. For our fiducial AGN
NGC 1068, Spinoglio et al. [2005] measured a number of different fine-structure lines with the Long
Wavelength Spectrometer (LWS) on board the Infrared Space Observatory (ISO). With a beam size
of 80(( , this corresponds to spatial scales of ∼ 6 kpc. Of course, it is not completely straightforward to
assess what this implies for fluxes from smaller scales. However, it seems likely that this total flux is
composed from the contribution of individual clumps which may have fluxes well above the average.
To be conservative and for comparison with the estimates for CO, we rescaled them assuming source
sizes of 200 pc and a homogeneous medium. Due to the discussed inhomogeneity, the actual fluxes
could well be larger at least in the brightest knots. We also rescale them to a line width of 300 km s−1
for purposes of comparison, though both the fluxes and the sensitivity scale in the same way with the
line width and it therefore does not influence the detectability.
From the sample of Spinoglio et al. [2005], we have identified several lines that could be search for
with ALMA observations at high redshift. The fluxes are given in Table7.2). Particularly promising
is the [CII] 158 µm fine structure line, which can be detected within some minutes. For the [OI] 63 µm
line, the required integration time would be one hour. With observations of the [OI] and [CII] fine
structure lines as well as the high-J CO lines, which correspond to the main coolants in gas with
metallicities above 10−2 Z' , one has a good estimate for the total cooling rate of the gas. These
numbers should be considered as lower limits, as more flux can be contributed on larger scales.
Since at high densities in XDRs the thermal balance establishes rapidly, the total cooling flux
equals the heating flux. This allows one to infer the impinging X-ray flux, while the sum of the finestructure and CO lines divided by the heating efficiency is always a measure of the X-ray flux that
has been transformed into heat. The heating efficiency of ∼ 10% is largely independent of energy
145
7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
and the spectral shape of the flux [Dalgarno et al., 1999]. For a spatially resolved source, the linear
size of the cooling gas can be obtained from the spectroscopic redshift. The corresponding X-ray flux
on that size scale can then be converted into a luminosity. The observed profiles of the line width
may be used to derive further estimates for the black hole mass and the accretion rate [Kaspi et al.,
2000; McLure & Jarvis, 2002; Vestergaard, 2002; Warner et al., 2003; Dietrich & Hamann, 2004]. In
addition, a detection of multiple CO lines provides the density and column density distribution as a
function of distance from the central engine.
7.6.1.4
Observables in low-metallicity gas
Of course, the metallicity is also a caveat regarding some of the observables discussed here. While this
work assumes that it is close to the solar value, the situation for low-metallicity gas has been explored
previously by Spaans & Meijerink [2008]. For metallicities Z > 10−2 Z' , they find intensities in
the range 10−6 − 10−3 erg s−1 cm−2 sr−1 . For even lower metallicities, the CO lines are hard to
detect, but emission from HeH+ and H+3 becomes more prominent. Particularly interesting at very
low metallicities is also the strong emission in H2 , which will be redshifted into ALMA band 10 for
sources at z > 10 if this band is realized. While such low metallicities seem unlikely at z ∼ 8, it
may become more relevant at these higher redshift. To be conservative, we estimate the fluxes for the
low-metallicity gas based on the results ofSpaans & Meijerink [2008], assuming that they hold in the
central 200 pc. For H2 emission, this yields fluxes of 0.6 − 6 (1 + z)−3 Jy, and 0.1 − 0.01 (1 + z)−3 Jy
for CO. The H2 estimate is particularly robust, as it is insensitive to the density. CO emission could
be reduced for densities smaller than 105 cm−3 , but then a substantial fraction of this flux would be
visible in fine-structure lines, particularly [CII] 158 µm.
The choice of metallicity may be guided by other observables like the amount of dust emission, the
detection of a strong starburst by JWST as well as theoretial considerations regarding the observability
of CO lines in low-metallicity gas [Spaans & Meijerink, 2008]. Such estimates may be refined further
employing follow-up observations with other telescopes that search for the UV N4+ and C3+ lines.
Their line ratio provides an important diagnostic for the metallicity and has been used to derive the
metallicity in the known high-redshift quasars [Pentericci et al., 2002].
7.6.2 Observables for JWST
In the frequency range of JWST, it is possible to detect redshifted stellar light, the AGN continuum
and dust. While such detections will not allow to identify the object unambigously as a quasar, they
could be used to provide a source list for more detailed follow-up observations with ALMA, which
would allow us to do this identification through the shape of the CO line SED [Spaans & Meijerink,
2008]. In addition, they provide complementary information about star formation in these systems.
Quasars with unobscured broad-line regions may also be visible in further lines. In this section, we
will focus on the Near Infrared Camera (NIRCam), which will provide high sensitivity of the order
a few nJy for a 3σ detection at 0.7 − 4.44 µm wavelengths after an integration time of 104 s (see
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7.6 High-redshift observations with ALMA and JWST
Table 7.3) and a field of view of (2.16( )2 . It will consist of 40962 pixels, with an angular resolution
of 0.0317(( per pixel for the short wavelengths and 0.0648(( per pixel for the long wavelengths. For
the detection of lines, we focus on NIRSpec, which offers sensitivities of 1 × 10−18 erg s−1 cm−2 for
velocity dispersions of 111 km s−1 with S /N = 10 and integration times of 104 s at 3 µm.
7.6.2.1
Stellar light
As discussed in Chapter 7.4.1, we expect a star formation rate of ∼ 10 M' yr−1 in an object like
NGC 1068. To calculate the corresponding spectrum, we employ the evolutionary synthesis code
Starburst99 1 [Leitherer et al., 1999; Vázquez & Leitherer, 2005]. We adopt their standard IMF with
an exponent of 1.3 in the range 0.1 − 0.5 M' , and 2.3 in the range 0.5 − 100 M' and take the spectrum
after a simulation time of 5 Myr, which corresponds to the age of the circumnuclear starburst ring in
this object. We then calculate the expected flux for this starburst at different redshifts. For sources
at z > 6, we remove the flux below wavelengths of 121.6(1 + z) nm, as even a small neutral fraction
in the IGM is sufficient to lead to complete extinction. This feature may also be used to derive the
redshift of the source, provided that the spectrum is resolved with NIRSpec. The expected fluxes are
shown in Fig. 7.7 and compared to the NIRCam sensitivity. For an integration time of one day, we
find that such a starburst can be detected with 3σ at redshift z = 15. For redshifts z ! 10, even a few
hours are sufficient for detection.
For comparison, we also consider a model with a star formation rate of ∼ 1 M' yr−1 (see Fig. 7.8).
In this case, an integration time of one day is still sufficient to find a starburst at z ∼ 5. A more
extended integration time of 10 days even suffices to find such starbursts down to z ! 10. This is still
rather short compared to proposed surveys searching for light from the first galaxies, which require
integration times of 0.4 years [Windhorst et al., 2006].
7.6.2.2
The AGN continuum
For the AGN continuum, Spinoglio et al. [2005] present a model that peaks at 0.1 − 0.3 eV with a
flux of ∼ 0.01 Jy. Such radiation will be redshifted into the frequency range of JWST for z > 5. We
would expect a flux of ∼ 0.9 nJy at z = 5 and ∼ 0.3 nJy at z = 8. The first would be detectable after
an integration time of about three days, and the latter within a time of about 15 days. In practice, this
radiation component will of course overlap with the radiation from the starburst and just leads to a
mild increase in the total radiation budget. Similar to the stellar light, the flux at wavelengths smaller
than 121.6(1 + z) nm would be removed due to extinction.
7.6.2.3
Lyman α emission
Additional contributions are also present from redshifted Lyman α radiation, which are particularly
enhanced in the presence of a starburst or a supermassive black hole. Of course, such emission
1
http://www.stsci.edu/science/starburst99/
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
Figure 7.7: The observed spectrum of a starburst with 10 M ' yr−1 with an age of 5 Myr located at different
redshifts. This is compared to the NIRCam sensitivities for 3σ detection after an integration time of one
day. For z ! 10, such starbursts can be detected within a few hours.
could only be observed at redshifts where reionization is complete. For a halo mass Mh = 1011 M' ,
Dijkstra et al. [2006a,b] find Lyman α luminosities of ∼ 1042 erg s−1 . This yields a flux of 2 ×
10−18 erg s−1 cm−2 at z = 6. As they find the luminosity to scale with Mh5/3 , this contribution becomes
less significant in smaller systems.
7.6.2.4
H α and H β emission
If in addition the Mid Infrared Instrument (MIRI) is employed, one may aim at detecting the H α line
at 656.3 nm. This would allow us to derive black hole masses and accretion rates from the line width,
and is complementary to to a derivation of the X-ray luminosity with ALMA [Kaspi et al., 2000;
McLure & Jarvis, 2002; Vestergaard, 2002; Warner et al., 2003; Dietrich & Hamann, 2004]. Kaspi
et al. [2000] give light curves of the H α and H β fluxes for a sample of quasars. We have estimated
the fluxes obtained from the first five sources in their sample, assuming these objects would be shifted
to redshift z ∼ 8. We expect that the extinction of these lines would not be significant, as atomic
hydrogen in the IGM is almost completely in the ground state. From z = 8, the flux would be ∼
104 nJy, corresponding to 3 × 10−17 erg s−1 cm−2 in a frequency interval corresponding to 100 km s−1 .
This is well within the sensitivity of the MIRI spectrograph, which is 2 × 10−18 erg s−1 cm−2 for
velocity dispersions of 111 km s−1 with S /N = 10 and integration times of 104 s at 6 µm.
For the H β line at 486.1 nm, we find that it would be redshifted to ∼ 4.4 µm. Our estimates yield
fluxes of ∼ 1 × 10−18 erg s−1 cm−2 , again well within the range of NIRSpec. At higher redshift, this
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7.6 High-redshift observations with ALMA and JWST
Wavelength
Sensitivity
0.70 µm
0.90 µm
1.15 µm
1.50 µm
2.00 µm
2.77 µm
3.22 µm
3.56 µm
4.44 µm
7.24 nJy
4.62 nJy
3.70 nJy
3.41 nJy
2.83 nJy
3.81 nJy
2.75 nJy
4.08 nJy
6.18 nJy
Table 7.3: The NIRCam sensitivity of JWST for a detection at 3σ after an integration time of 10 4 s. These
values assume that the wide filter is used, which is appropriate for continuum fluxes as considered here.
will fall in the MIRI frequency range, which provides similar sensitivities.
7.6.2.5
Mg+ and C3+ emission
In addition, one may search for the Mg+ line with a rest frame wavelength of 279.8 nm and the C3+
line with 154.9 nm, which would be redshifted into the NIRSpec frequency range. Such a detection
will yield complementary information about the velocity dispersion near the black hole. As their
wavelengths are below 121.6 nm, they are not affected by extinction in the high-redshift IGM. Their
detection would provide an alternative path to deriving the redshift of the source. We find that, if the
objects from the sample of Dietrich et al. [1999] would be shifted to z ∼ 8, the expected fluxes from
the C3+ line would be of the order ∼ 1 × 10−15 erg s−1 cm−2 , and thus detectable within less than an
hour. To estimate the fluxes in the Mg+ line, we use the z = 3.4 sample of Dietrich et al. [2003b]. We
find similarly high fluxes of the order∼ 1 × 10−14 erg s−1 cm−2 .
In summary, we find that the component from the starburst will likely be the main observable for
JWST, as well as the H α, H β, Mg+ and C3+ lines in case of unobscured broad-line regions. Even the
other components alone would be detectable. At redshifts z ∼ 5, typical integration times may reach
from a few hours to a day, depending on the luminosity of the source. Near z ∼ 10, typical integration
times may vary between one and ten days. The main observables are summarized in Table7.4. We
note, however, that significant uncertainties are still present in the flux estimates, as 107 M' black
holes are currently not observed at high redshift, and the relation between the black hole mass and
the starburst component as well as different line fluxes is therefore highly uncertain.This question can
only be addressed in very detailed simulations of high-redshift quasars, which are however beyond
the scope of this work.
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
Observable
flux estimate
estimated from
stellar light
AGN continuum
L α 121.6 nm
H α 656.3 nm
H β 486.1 nm
Mg II 279.8 nm
C IV 154.9 nm
0.3 − 3 nJy
0.3 nJy
2 × 10−18 erg s−1
2 × 10−18 erg s−1
1 × 10−18 erg s−1
2 × 10−14 erg s−1
2 × 10−15 erg s−1
Davies et al. 2007
Spinoglio et al. 2005
Dijkstra et al. 2006
Kaspi et al. 2000
Kaspi et al. 2000
Dietrich et al. 1999
Dietrich et al. 2003
cm−2
cm−2
cm−2
cm−2
cm−2
Table 7.4: The main observables for JWST, as discussed in Chapter 7.6.2. We estimate the flux at z ∼ 8
as described in the text (z = 6 for Lyman α, as it is extincted at higher redshift). The NIRCam continuum
sensitivities are given in Table 7.3. The continuum fluxes will be extincted by absorption for wavelengths
smaller than 121.6(1 + z) nm, and the same holds for the Lyman α line for redshifts before reionization is
complete. As can be seen in Figs. 7.7-7.8, this extinction does not affect the detectability of these sources.
7.7 Expected source counts and optimum observing strategies
In this section, we provide several independent estimates for the number of sources at high redshift,
and discuss the corresponding observational strategies for ALMA and JWST.
7.7.1 The expected number of high-redshift sources
With the considerations above, we can attempt to estimate the number of observable sources at different redshifts. We adopt a similar formalism as Choudhury & Ferrara [2007] and count the number of
black holes N(z, MBH ) with masses larger than MBH in a redshift interval [z, z + ∆z]. This is given as
N(z, MBH ) =
×
%
% Mmax
dV
dz (
d log Mh
dz dΩ Mmin
z
dn
fa
(MBH ),
d log Mh
z+∆z
(
(7.9)
where dV dz(−1 dΩ−1 denotes the comoving volume element per unit redshift per unit solid angle,
which is given as [Peebles, 1993]
dV
dt
= D2A c .
(7.10)
dz( dΩ
dz
In this expression, DA is the angular diameter distance, c is the speed of light and dt/dz = 1/(H(z)(1 + z)),
where H(z) is the expansion rate as a function of redshift. The term dn/d log Mh describes the number
density of dark matter halos per unit mass, and the integration range in Eq.7.9 is chosen such that the
minimal halo mass corresponds to a black hole mass of MBH for our model in Chapter 7.2. The upper
150
7.7 Expected source counts and optimum observing strategies
Figure 7.8: The observed spectrum of a starburst with 1 M ' yr−1 with an age of 5 Myr located at different
redshifts. This is compared to the NIRCam sensitivities for 3σ detection after an integration time of 10
days. This allows 3σ detection of such starbursts for z ! 10.
halo mass is chosen corresponding to a black hole mass of 109 M' . The parameter fa is the fraction
of active quasars, as introduced in Chapter 7.3.
As our reference system, we have adopted the known AGN NGC 1068, which has a black hole of
∼ 107 M' and a starburst component with ∼ 10 M' yr−1 . As discussed above, even less active quasars
provide a sufficient amount of CO flux for detection, and the same holds for smaller star formation
rates. We will therefore assume that typical galaxies containing a 106 M' black hole should be
observable. To be more conservative, we will also show the corresponding numbers for more massive
black holes.
In the following, we will distinguish between two different observational strategies. The first aims
at finding high-J CO lines with ALMA, which will allow us to unambigously identify a given source
as a quasar [Spaans & Meijerink, 2008]. The second aims at finding high-redshift sources in general
and focuses mainly on continuum emission from dust or stellar light. In fact, once such sources are
found, they can be examined in more detail by other observatories like the Keck telescope or new
generations of X-ray telescopes.
7.7.2 Survey of high-J CO lines with ALMA
For the first observational strategy focusing on line observations, ALMA provides a bandwith of
8 GHz. We will refer to this scenario as a high-J CO line survey. In general, one may scan a
redshift interval of ∆z/(1 + z) = ∆ν/ν, where ∆ν refers to the difference with respect to the observed
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
Figure 7.9: The expected number of sources for a high-J CO line survey. We adopt a field of view of
(1( )2 , an active quasar fraction of 10%, as discussed in Chapter 7.3, and a redshift interval of ∆z = 0.5,
as discussed in the text. Down to z ∼ 10, a typical ALMA field of view therefore has a high chance to
find a number of quasars with black holes in the mass range of 10 6 − 107 M' . Near z = 6, one may even
find more massive quasars by extending the search to larger volumes. Note that the numbers above scale
linearly with the adopted fraction of active quasars.
frequency ν. For ν ∼ 90 GHz and z ∼ 5, this corresponds to ∆z = 0.5. This becomes even larger when
going to higher redshifts, and smaller for observations at higher frequencies, such that we slightly
underestimate the sources at high redshift. We adopt this value here as a typical mean. Based on the
results in Chapter 7.3, we assume an active quasar fraction of fa ∼ 20%. Our results can however be
directly scaled to other values. We consider a field of view of (1( )2 , even though ALMA can probably
scan an angular region of (3( )2 within a few days. The result is given in Fig. 7.9.
We find that down to z ∼ 10 there are good chances to detect several sources with black hole
masses in the range 106 − 107 M' within the ALMA beam. For an extended survey that probes an
angular field of ∼ (3( )2 , one may even find quasars with MBH > 108 M' , and at the same time find
enough sources with smaller masses to allow a statistical analysis. At redshifts beyond 10, we find
an exponential decrease in the number of high-redshift quasars, though we expect that ALMA could
still find some of them. Searching for such systems should be a high priority for future observations,
as they are a direct probe for the origin of the first supermassive black holes.
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7.7 Expected source counts and optimum observing strategies
Figure 7.10: The expected number of sources for a starburst survey focusing on continuum radiation
mostly from stellar light. We adopt a field of view of (2.16 ()2 and an active quasar fraction of 100%, as
the relevant observables do not require quasar activity. We explore a redshift interval of ∆z = 2. In fact,
this yields a high number of sources that are potentially accessible to JWST within the NIRCam field of
view. For redshifts near z = 5, typical detection timescales are one day or less, while observations near
z = 10 require integration times between one and ten days. This is within reach of deep surveys.
7.7.3 Continuum survey with JWST
For the second strategy focusing on continuum radiation, we are not restricted to active black holes
and work with fa = 1. This assumes that such galaxies have a star formation rate of at least 1 M' yr−1 .
However, our results can simply be rescaled for different star formation rates. We will refer to this
scenario as a starburst survey. For continuum observations, the bandwidth is no longer a restriction
for the redshift range, and we consider a generic interval of ∆z = 2. We adopt the NIRCam field of
view of (2.16( )2 . The result is then given in Fig. 7.10. While we cannot be sure that the detected
objects are active quasars, it is always possible to address this question with follow-up observations
involving other telescopes. In the NIRCam field of view, we expect 102 − 103 observable sources at
redshifts 5 − 10, and 10 − 100 at redshifts beyond. In addition, we expect a few very bright sources
that could be associated with 108 M' black holes within the field of view. They can be found with
integration times of a few hours, but beyond that a deep field is required to see smaller galaxies and
probe the early epoch of galaxy formation in detail.
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
Figure 7.11: The expected number of sources for a high-J CO line survey, for an evolving relation
between black hole mass and spheroid velocity dispersion (see Eq. ( 7.11)), for black hole masses of at
least 107 M' .
7.7.4 Uncertainties and generalizations of the model
After the more general considerations above, we explore the uncertainties of the previous predictions by comparing with additional independent estimates, and by considering alternative scenarios
in which the relation between black hole mass and the velocity dispersion evolves with redshift.
7.7.4.1
Evolution of the MBH − σ relation
The results discussed so far assume that our model for the black hole population holds as described
in Chapter 7.2. However, this is not necessarily true. In particular, first indications have been found
that the relation between black hole mass and bulge velocity dispersion may evolve with redshift,
and that for the same velocity dispersion, black holes may have been more massive by an amount
∆ log MBH = (3.1 ± 1.5) log(1 + z) + 0.05 ± 0.21 [Woo et al., 2006, 2008]. The error bars are large
and exploiting the 3σ range is still consistent with no evolution of even less massive black holes for
the same velocity dispersion. We will therefore adopt a general parametrization as
∆ log MBH = Cev log(1 + z)
(7.11)
and explore the effect of different coefficients Cev . At z ∼ 10, this can lead to a significant difference
in the black hole mass. We impose an additional upper limit for the black hole mass of 1% of the
total halo mass, i. e. ∼ 10% of the baryonic mass, which would already require a very efficient
accretion mechanism. To explore this effect, we adopt the black hole population model with a mass
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7.7 Expected source counts and optimum observing strategies
Figure 7.12: Additional estimates for the number of sources for a high-J CO line survey. We show extrapolations of the results by Shankar et al. [2009] and Treister et al. [2009] to high redshift, for luminosities
larger than 10 44 erg s−1 . In addition, we give an estimated based on the number of high-redshift black
holes required in order to produce the present-day 10 9 M' black holes. Their mass is however sensitive to
model assumptions, see text and Fig. 7.13 for further discussion.
range of 105 − 109 M' and a conservative Eddington ratio of λE = 0.3, though the general effects
will be similar for other models as well. As previously, we calculate the active quasar fraction from
the X-ray constraints. We take the active quasar fraction for a spectral parameter Φ ∼ 0.4 at a background contribution of 1 keV cm−2 s−1 sr−1 keV−1 . The results are given in Fig. 7.11 for black hole
masses of 107 M' for the case of a high-J CO line survey (but a similar behavior can be expected for
a starburst survey as well). Due to the normalization with respect to the X-ray background, they have
a similar order of magnitude at z ∼ 6, but the further evolution with redshift depends sensitively on
the parameter Cev . For positive Cev , as favored by recent observations, one can expect more sources
at higher redshift, while for negative Cev , the population decreases rapidly. For 107 M' black holes, it
is interesting to note that the maximum increase happens for Cev ∼ 1, as for higher values, the contribution from more massive black holes gets more significant, which effectively reduces the population
of 107 M' sources when considering the X-ray constraint. For Cev ∼ −2, the population decreases
significantly already at z ∼ 6, indicating that such a population would no longer be constrained by the
unresolved X-ray background.
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
7.7.4.2
Extrapolation of the observed population to higher redshift
The calculations above essentially provide upper limits on the population that are still in agreement
with the unresolved X-ray background. In order to quantify the uncertainty of these estimates, we
compare with further calculations based on completely different approaches. First, we extrapolate the
comoving quasar density recently derived byShankar et al. [2009] and Treister et al. [2009], focusing
on their results for luminosities larger than 1044 erg s−1 , which can be associated with a black hole
mass of 107 M' for an Eddington ratio of 10%. Treister et al. [2009] provide only numbers for
Compton-thick quasars, but give a Compton thick quasar fraction of ∼ 10%, which we adopt for this
calculation. The results are shown in Fig. 7.12. The extrapolation from Shankar et al. [2009] leads to
a steep evolution with redshift, but results in more sources at z ∼ 6 than the extrapolation ofTreister
et al. [2009]. The latter, on the other hand, is likely to overestimate the number of quasars at very
high redshift. We also note that the data of Tozzi & et al. [2006] would admit more sources with
luminosities above 1044 erg s−1 than in the model of Treister et al. [2009]. Both the extrapolations of
Shankar et al. [2009] and Treister et al. [2009] seem to agree better with negative Cev , but it appears
prelimary to draw conclusions from this coincidence, in particular as these are only extrapolations.
For a population according to the Shankar et al. [2009] extrapolation, we find reasonable chances
to detect high-redshift miniquasars in an ALMA deep field. For the extrapolation of Treister et al.
[2009], this approach would seem more difficult, but we point out that the luminosities considered
here are very conservative and ALMA should be able probe luminosities of 1044 erg s−1 , for which
the number of sources would be increased by an order of magnitude. It is not straightforward to apply
these results to JWST which has the focus on very different observables. However, one expects a
similar trend leading to deviations of the abundances by an order of magnitude.
7.7.4.3
Match with present-day black hole population
A further estimate is provided on the basis of the observed black hole population at redshift zero.
Based on the observed quasar luminosity function down to redshift z ∼ 6, Shankar et al. [2009]
calculated the average accretion history of black holes of different masses at z = 0. Their results
show that a black hole with 109 M' today had a mass of ∼ 107 M' at z = 5, comparable to the
black hole in NGC 1068. The slope of the average black hole mass of these progenitors appears as
fairly flat at that redshift, indicating that they may have high masses even at higher redshift. However,
as we will discuss below in more detail, it is clear that eventually there must be a transition in this
behavior, allowing them to build up their mass. 109 M' black holes today have a number density of
∼ 10−4 Mpc−3 [Shankar et al., 2009]. It may be even higher, as Compton-thick black holes at z = 0
are hard to detect [Treister et al., 2009]. We can therefore assume that their progenitors should have
the same comoving density at z = 5, providing a lower limit to the number of sources that we can
expect for ALMA. Based on this assumption, we have thus provided a further estimate in Fig.7.12.
To investigate the evolution of the average black hole mass at redshifts beyond the model of
Shankar et al. [2009], we calculate the average evolution towards higher redshift according toShapiro
156
7.7 Expected source counts and optimum observing strategies
Figure 7.13: The average accretion history of a black hole with 10 7 M' at z = 5, depending on a coefficient
p which is a function of the conversion of rest-mass energy to luminous energy, the Eddington ratio and
the duty cycle.
[2005] as
dMBH
MBH
=
,
dt
τgrowth
(7.12)
where
(2M /0.1) 1 1
Gyr
1 − 2M λE fduty
≡ 0.0394p Gyr,
τgrowth = 0.0394
(7.13)
where 2M = L/ Ṁ0 c2 is the efficiency of conversion of rest-mass energy to luminous energy, λE the
Eddington ratio as introduced in Chapter 7.2, and fduty the duty cycle, i. e. the fraction of quasars
active at a time. We do not use the parameter factive as introduced earlier where we determined it from
the background constraints, while the population considered now is too small to be constrained by
the background. The parameter p summarizes the complicated dependence on the latter parameters.
We explore the dependence on the mass evolution in Fig.7.13. For p > 7, the black hole mass hardly
evolves with redshift. This is certainly an extreme case which would require these black holes to be of
primordial origin. Values down to p = 2.5 require intermediate mass black holes with 104 − 105 M'
to be the progenitors of the first supermassive black holes. Such scenarios have been suggested by
Eisenstein & Loeb [1995], Koushiappas et al. [2004], Begelman et al. [2006], Spaans & Silk [2006],
Lodato & Natarajan [2006] and Dijkstra et al. [2008]. For even smaller values of p, supermassive
black holes might indeed originate from the first stars.
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
However, we note that such small values of p are difficult to achieve. For 2M ∼ 0.19, corresponding to an MHD disk model [Shapiro, 2005], we find p = 2.3 even for λE = 1 and fduty = 1.
However, observations of high-redshift Eddington ratios rather indicate values of λE ∼ 1/3 [Shankar
et al., 2004; Kollmeier et al., 2006] and fduty ∼ 0.3, which would lead to p ∼ 23. This implies that the
parameter p should be initially high to reproduce the flat slope of Shankar et al. [2009], but should
then decrease at higher redshift such that the black holes can build up their mass.
7.7.5 What we will learn about the brightest quasars
In addition to finding less massive quasars at higher redshift, instruments like ALMA and JWST can
be used to get more detailed information about those sources that we already know. At redshift z = 6,
a typical angular resolution of ∼ 0.01( corresponds to a spatial resolution of ∼ 60 pc. It is thus possible
to resolve the quasar host galaxy in high-J CO emission and dust with ALMA. As these sources are
very bright in X-ray emission, the required flux to density ratio that produces significant CO emission
is shifted to larger scales further away from the central black hole and fills a larger region in space.
With detailed diagnostics for the CO lines, it is thus possible to probe temperature and density in these
systems with spatial resolution. With a spectral resolution of up to 0.01 km s−1 , the dynamics of the
gas can be probed in detail, in particular with respect to typical infall velocities as well as winds and
jets. The observation of different lines and their relative fluxes will further provide estimates for the
metal abundance ratios in the host galaxy.
In addition, JWST will provide complementary information about stellar light, the AGN continuum and Lyman α emission. The latter could be clearly discriminated from the continuum contribution using specific narrow filters tuned to the wavelength corresponding to the redshifted Lyman α
line.
7.8 Discussion and outlook
In this paper, we have presented a model for the quasar population at high redshift, based on the
known correlations between the black hole mass and its host galaxy. We have derived constraints
on the fraction of active quasars from the SXRB and discussed the expected observables. We have
based our discussion on using NGC 1068 as a template AGN, and calculated its observability at
high redshift. Our focus lies in particular on high-J CO lines, fine-structure lines of C+ and O, dust
emission, stellar light, the AGN continuum and Lyman α radiation. Above a metallicity of 10−2 solar,
CO, C+ and O are the main coolants, and their detection can be used to derive an estimate for the
heating rate and the impinging X-ray flux. A profile of the observed line width further allows to
derive estimates for the black hole mass and probes the mass - luminosity relation at high redshift.
We expect that galaxies containing a black hole in the mass range 106 − 107 M' can be observed at
high redshift with ALMA and JWST. We predict 1 − 10 high-redshift sources that can be found via
158
7.8 Discussion and outlook
molecular lines in an ALMA field of view of (1( )2 , and 100 − 1000 sources to be found by JWST
using NIRCam and focusing on stellar emission.
We have explored the expected number of sources for ALMA and JWST with several independent
approaches. These are based on the unresolved X-ray background constraints, the data ofShankar
et al. [2009] and Treister et al. [2009] as well as the average accretion history of the supermassive
black holes at z = 0. For the case of 107 M' black holes, we expect 0.03 − 2 such black holes in a
typical field of view. These numbers would be increased by a factor of 10 for 106 M' black holes.
For an observational campaign that examines a solid angle similar to the Hubble deep field, one can
therefore expect to find sources even under unfavorable conditions. For JWST, the corresponding
estimates are 10 − 100, as the field of view is larger and continuum radiation can be searched for
in a larger redshift interval. So it is possible to use JWST for large surveys that provide sources for
follow-up observations with ALMA.
In about three years, ALMA will become operational with initially 16 antennas. This will provide
sufficient sensitivity to detect high-J CO lines, fine-structure lines and dust continuum emission from
the brightest high-redshift quasars that are already known. As these systems are large, the high
angular resolution of ALMA will provide detailed spatial information about the inner structure of
such systems, and detailed diagnostics for the CO lines allow to infer the density and temperature
distribution in the host galaxy [Meijerink et al., 2007].
Once ALMA has the full set of 50 antennas, it reaches even higher sensitivity and it becomes
possible to search for AGNs with less massive black holes. For such deep field observations, we
recommend to focus initially on modest redshifts of z ∼ 5 − 6, where we expect higher fluxes, more
sources, and where one may be more confident that the Magorrian relation is well-established. This
provides an important test to confirm the observability of high-redshift quasars, as well as important
information about the progenitors of the most massive quasars in the universe. Such studies can then
gradually be extended to higher redshifts and will provide a direct test for structure formation models
in the high-redshift universe and of the Magorrian relation at different redshifts.
JWST can aid such surveys by providing additional sources that may be detected in stellar light,
the AGN continuum or Lyman α emission. In this case, ALMA does not need to glance into the dark
sky, but can do pointed follow-ups of the known sources. This will speed up high-redshift surveys,
and provide complementary information at different frequencies. It is particularly important at very
high redshift beyond z ∼ 10, where the probability of finding sources in the ALMA field of view
decreases rapidly. The combination of these instruments can therefore truely detect the first quasars
in the universe.
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7. DETECTING THE FIRST QUASARS WITH ALMA AND JWST
160
8
Discussion and outlook
In the previous chapters, I have outlined observational constraints for the stellar population, primordial magnetic fields and dark matter models, as well as new opportunities that will help to explore the
high-redshift universe in more detail. In this chapter, I will summarize the main results, outline the
main open questions and provide an outlook for future work that will help to improve the previous
results.
8.1 Summary
In this section, I summarize the main results regarding observational constraints and future perspectives that were found in this work.
8.1.1 Reionization constraints
Reionization constraints have been considered in Chapter 2 and 5 and provide upper limits on the
existence and nature of additional physics as well as the stellar population. The main results are:
1. Pop. II stars alone cannot provide enough ionizing radiation to produce the required reionization
optical depth.
2. Constraints on the magnetic field strength result in values between 0.7 − 3 nG, depending on
the precise model for the stellar population.
−33 cm3 s−1 eV−1
3. For dark matter annihilations, the parameter combination "σv# fabs m−1
DM ≥ 3×10
can be ruled out at 3σ, where "σv# is the thermally-averaged annihilation cross section, fabs the
fraction of annihilation energy absorbed into the IGM and mDM the mass of the dark matter
particle in eV.
161
8. DISCUSSION AND OUTLOOK
−1 < 3 × 1023 s can be ruled out at 3σ,
4. For dark matter decay, the parameter combination tX fdecay
where tX is the lifetime of the dark matter particle and fdecay the fraction of the decay energy
absorbed into the IGM.
5. Dark stars with 1000 M' , as suggested by Freese et al. [2008a], are difficult to reconcile with
cosmic reionization, unless a rapid transition to a different star formation mode is assumed.
6. The same holds for dark star models which consider scattering between baryons and dark matter
particles [Iocco et al., 2008; Freese et al., 2008c; Yoon et al., 2008]. These models tend to
produce very long lifetimes of ∼ 100 Myr or more, again overproducing the amount of UV
photons.
Of course, it is important to assess the reliability of these conclusions. Points 3 and 4 are very reliable,
as their derivation assumes that reionization is due to dark matter annihilation / decay alone. Point
2 has a weak dependence on the assumptions for the stellar population, but also appears to be well
established. Point 5 appears rather firm, as well as point 6, though the latter might also be reinterpreted
as a constraint for the scattering cross section between dark matter and baryons. Regarding point 1,
one has to deal with two uncertainties: The star formation efficiency and the escape fraction of UV
photons. In this work, the latter was chosen according to the observed escape fractions in high-redshift
galaxies, while the star formation efficiency was chosen according to general considerations regarding
the Jeans mass and the mass that goes into collapse [Abel et al., 2002; Bromm & Larson, 2004]. To
alleviate this constraint, one would need to find a mechanism that increases the product of the two by
two orders of magnitude to reproduce the required optical depth. However, there is no motivation for
higher star formation efficiencies from simulations, and given that Pop. II stars produce considerably
less UV photons per stellar baryon, it seems unlikely that they can photo-evaporate the halo in the
same way as Pop. III stars.
8.1.2 21 cm observations
As shown in Chapter 3, primordial magnetic fields can be constrained further by future 21 cm observations, for instance with LOFAR or SKA. Such magnetic fields have mainly two effects on the
21 cm signal:
At redshifts z > 40, they increase the amplitude of the 21 cm power spectrum due to the additional
heat input into the gas, which is due to ambipolar diffusion heating and decaying MHD turbulence. To
measure this effect is however challenging, as the foreground emission corresponds to temperatures
which are by orders of magnitude higher. On the other hand, as pointed out by Di Matteo et al.
[2002]; Oh & Mack [2003]; Zaldarriaga et al. [2004]; Sethi [2005]; Shchekinov & Vasiliev [2007],
the foreground emission is expected to be featureless in frequency, which may allow a sufficiently
accurate subtraction.
A more relevant effect is the dynamical impact of primordial fields on the formation of galaxies.
As discussed in Chapters 2 and 3, the magnetic pressure can lead to very high Jeans masses and
162
8.1 Summary
prevent gas collapse in the first minihalos. This provides an efficient mechanism to delay reionization
and the formation of the first luminous sources. If 21 cm observations confirm that reionization started
early, such scenarios would be ruled out as well, constraining the primordial magnetic field strength
down to 0.1 nG.
Such observations also provide a relevant test for the effects of dark matter annihilation / decay
in the early universe, as shown by Furlanetto et al. [2006b]; Chuzhoy [2008]; Cumberbatch et al.
[2008]. More stringent upper limits are usually found from considerations of the cosmic gamma-ray
background, as most of the annihilation products contribute to the cosmic backgrounds rather than
being absorbed into the IGM [Cumberbatch et al., 2008]. The following subsection summarizes the
main results from these considerations.
8.1.3 Constraints from the cosmic gamma-ray background and the atmospheric neutrino background
In Chapter 5 and 6, I used the cosmic gamma-ray background and the atmospheric neutrino background to derive further constraints on dark matter models. In a similar way, as shown in Chapter7,
one can also use the unresolved X-ray background to constrain the black hole population at high
redshift. In summary, I found the following results:
1. Constraints from the unresolved X-ray background suggest that one could find up to ∼ 10
black holes of 106 M' in a field of view of (1( )2 between redshift 6 and 10, considering a
typical redshift interval of ∆z = 0.5.
2. Light dark matter models with constant thermally-averaged annihilation cross sections (s-wave)
are ruled out due to the observed gamma-ray background.
3. For heavy dark matter candidates with annihilation cross sections at the natural scale and
branching-ratio of 1/3 for gamma-rays, as typically considered in dark star models, particle
masses smaller than 30 GeV are ruled out.
4. If neutrinos are generated as annihilation products, their contribution is well below the atmospheric neutrino flux, thus providing no significant constraint on the models.
5. In the centers of primordial halos with virial temperatures smaller than 104 K, the steep cusp in
the dark matter density profile can potentially contribute to the cosmic gamma-ray background,
but this effect is very uncertain.
Again, these results are rather robust. The first conclusion depends on assumptions regarding the
black hole spectra, but holds for a reasonable range of parameters. It is particularly promising regarding the search for high-redshift quasars with ALMA and JWST, as it provides an upper limit that is
larger than other estimates for the black hole population, thus admitting a potentially large number
of sources. Other estimates taking into account the average accretion history of black holes predict
163
8. DISCUSSION AND OUTLOOK
smaller numbers, that are however still sufficient to find sources in a solid angle comparable to the
Hubble Deep Field.
The second result is of course sensitive to assumptions regarding the clumpiness of dark matter,
but they were chosen to be very conservative and can be relaxed even further. While similar results
had been obtained previously from Galactic observations [Boehm et al., 2004b], our results are less
sensitive to model assumptions regarding the details of the Milky Way, and only make use of the
observed spectral shape of the emitted radiation. Therefore, s-wave annihilation for light dark matter is now firmly ruled out. The third conclusion slightly depends on the assumption regarding the
clumpiness of dark matter at early times, but was also chosen to be conservative. It holds only for
dark matter models considered in dark star scenarios, where it is assumed that a fraction of ∼ 1/3 of
the annihilation energy goes into gamma-rays. It therefore provides a lower limit for the dark matter
particle mass in such scenarios. In more generic cases with smaller branching ratios to gamma-rays,
this constraint is relaxed considerably [Mack et al., 2008]. This is quite similar for conclusion 5,
while conclusion 4 is hardly sensitive to model assumptions.
8.1.4 Signatures of primordial molecules in the CMB
As an additional way to explore the early universe, I have considered the possibility to search for
signatures of primordial molecules in the CMB (see Chapter 4). This was in particular motivated by
a previous calculation of Black [2006] which indicated that the signatures of H− could be found with
modern technology. I therefore performed a self-consistent calculation of the chemistry in the early
universe to investigate the abundances of the first molecules, and calculated the signatures for the most
important species. The results showed that Black [2006] largely overestimated the signature from H− .
If the inverse reactions are taken into account, the net effect drops even further. It is thus considerably
below the limit where it becomes observable. Also other species like HeH+ and HD+ would need to
increase by orders of magnitude in their abundance to be detectable with the Planck satellite.
8.1.5 First quasar observations with ALMA and JWST
I examined the question how future observations may help to understand the origin of the first supermassive black holes. At z ∼ 6, black holes with 109 M' have been discovered, and their origin is still
controversial (see Chapter 7.1). Even at those redshifts, their host galaxies were found to have supersolar metallicities [Pentericci et al., 2002], and also the most recent data show no evolution of quasar
metallicity with redshift [Juarez et al., 2009]. Here, I summarize the main results regarding the detectability of their progenitors at higher redshift. This work has been inspired during a visit of Marco
Spaans in Heidelberg, who presented first estimates on the expected fluxes in high-redshift quasars
with low-metallicity gas [Spaans & Meijerink, 2008]. To examine this question in more detail, I used
the nearby Seyfert 2 galaxy NGC 1068 as a reference AGN to study the potential observables for
ALMA and JWST. I also provided several independent estimates regarding the expected number of
sources, finding that the precise number is highly uncertain, but even under unfavorable conditions,
164
8.1 Summary
a sufficient number of sources should be present in a field of view corresponding to a Hubble deep
field. The main results are the following:
1. For metallicities of at least 10−1 Z' , the gas cooling is clearly dominated by metals, and most
of the flux is expected in CO, O and C+ . Sources at z ∼ 8 can be detected with ALMA within
less than an hour, for black hole of ∼ 106 M' (not necessarily with Eddington luminosities).
2. As the main coolants can be observed, this provides a good estimate for the total heating (=cooling) rate in the dense clouds and the corresponding X-ray luminosity.
3. As shown in Table 7.2, even more molecular and fine structure lines can be detected within a
rather short integration time, providing detailed information about the gas composition.
4. JWST may provide highly complementary information by detecting stellar light, the AGN continuum and line fluxes from the broad-line region.
5. A spectral resolution of such line fluxes may provide upper limits to the black hole mass that
can be compared with the X-ray luminosity.
6. Both ALMA and JWST can detect ∼ 106 M' black holes down to redshift 15.
7. The question regarding detectability is rather insensitive to the details of the gas composition
and distribution and is mostly regulated by the absorbed amount of X-ray flux.
8. The expected number of sources is uncertain and ranges between 0.3 and 10 black holes with
106 M' within (1( )2 .
These results are very promising and show that it is possible to scan a solid angle comparable to
the Hubble Deep Field within less than a day, with high chances of finding at least three sources.
Particularly intersting is the fact that black hole masses considerably smaller than the known supermassive black holes at z ∼ 6 can be probed, thus providing the chance to understand their origin in
much more detail.
ALMA line observations provide valuable diagnostics to derive the gas temperature and density
[Meijerink & Spaans, 2005], while JWST observations yield independent information about the stellar population. A detection of the various different ALMA observables may yield further information
regarding the gas composition and metal abundance ratios. If the broad-line region is not obscured,
the velocity dispersion of the corresponding line fluxes yields an upper limit to the black hole mass.
This can be compared with the X-ray luminosity inferred from ALMA observations in order to constrain the Eddington ratio at high redshift. The velocity dispersion of the ALMA line fluxes provides
detailed information about the gas dynamics and the dynamical mass. With an upper limit from the
black hole mass, this can be used to constrain the Magorrian relation at high redshift.
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8. DISCUSSION AND OUTLOOK
8.2 Open questions
In this section, I summarize some of the main open questions regarding the early universe.
8.2.1 Physics of the early universe
At present, even if the most optimistic constraints from Chapter2 on primordial magnetic fields hold,
they hardly constrain any of the scenarios discussed in Chapter1.2.1.2, perhaps with the exception of
some inflationary scenarios, which can predict a huge range of different values. However, a comparison with the calculation of Banerjee & Jedamzik [2003] shows that the current constraints are roughly
comparable to the field strengths that could be expected from the QCD phase transition, in particular
for those models which assume high helicity. High field strengths are also predicted in case of hydrodynamic instabilities during the QCD phase transitions [Sigl et al., 1997]. Stronger observational
constraints on the magnetic field strength will therefore provide direct constraints for the physics at
the QCD phase transition.
From a theoretical point of view, it is necessary to study the impact of such magnetic fields on
the first generation of stars in more detail to better understand their role in the early universe and to
derive potential constraints. In particular, they may not only change the chemistry, but also affect the
dynamics of the gas during collapse. An outlook of the currently ongoing work regarding this topic
is given below.
The question regarding the true nature of dark matter is also unresolved, as well as its potential
annihilation and decay channels. Currently, the FERMI satellite 1 searches for signatures of dark
matter annihilation / decay in the gamma-ray background and the centers of galaxies. A positive
detection would certainly stimulate the modeling of dark matter and make it possible to specifically
explore the effects of the corresponding scenarios in the early universe. In this case, the effects of
dark matter annihilation on the first stars would need to be developed with much more detail, to take
into account the detailed dynamics of the collapse process. Again, some of the current work on this
topic is sketched below.
8.2.2 Reionization and the stellar population
Even in the absence of the primordial magnetic fields and dark matter annihilation / decay, many
questions regarding reionization and stellar populations are still open. This concerns in particular the
transition to the star formation mode in the present-day universe. A number of different feedback
mechanisms that may contribute to this transition have already been discussed in the introduction.
The main ingredient is generally assumed to be metallicity [Bromm et al., 2001; Schneider et al.,
2003; Clark et al., 2008], though it was also suggested that magnetic fields may be responsible for
this transition [Silk & Langer, 2006]. In order to understand the transition to a different star formation
mode, one needs to understand how metal enrichment proceeds in the early universe. Simulations by
1
http://www.nasa.gov/mission pages/GLAST/main/index.html
166
8.2 Open questions
Tornatore et al. [2007] show that metal enrichment is a highly inhomogeneous process, and pristine
bubbles of primordial gas may even be left near z ∼ 2. This is encouraging regarding the search
for Pop. III stars. Metal enrichment in the early universe will be explored in more detail with a new
algorithm for chemical mixing in hydrodynamic simulations [Greif et al., 2009].
This is particularly relevant for star formation in atomic cooling halos. Even in the primordial
case, they might harbor a different star formation mode than the first minihalos, as they consist of a
rather turbulent two-phase medium [Greif et al., 2008]. The gas in the hot phase has a larger ionization
degree and therefore forms coolants like H2 and HD more efficiently, such that the gas can ultimately
reach lower temperatures when it collapses. Accordingly, these will be reflected in smaller Jeans
masses. The fragmentation will be triggered further due to the presence of turbulence, as found in
simulations of Clark et al. [2008]. Of course, in the presence of metals these effects may be enhanced
even further [Schneider et al., 2003; Clark et al., 2008; Omukai et al., 2008]. This question therefore
needs to be addressed with self-consistent calculations of star formation and metal-enrichment in the
first galaxies.
As mentioned in the introduction, the effects of magnetic fields generated during protostellar
collapse are currently not well understood. Some simulations are available byMachida et al. [2006],
but with initial conditions corresponding to present-day star formation rather than the cosmological
context in the early universe. The results are thus very similar to corresponding works in present-day
star formation [Ziegler & Rüdiger, 2000; Price & Bate, 2007; Hennebelle & Fromang, 2008]. On
the other hand, Xu et al. [2008] calculated the generation of magnetic fields due to the Biermann
battery mechanism [Biermann, 1950], but could not resolve the disks in which the field strength may
be amplified by dynamo effects. To perform simulations that bridge these scales is thus one major
challenge for the future.
In addition, it is also clear that magnetic fields will be generated in the first stars. Magnetic winds
or supernova explosions will bring them in the environment and ultimately turn it into a magnetized
medium [Rees, 1987]. Small-scale turbulence from supernova explosions may contribute to the formation of magnetic fields that can affect the following stellar generations [Kandus et al., 2004]. Such
fields can affect the chemistry during collapse by ambipolar diffusion, but also the dynamics. This
form of magnetic feedback still needs to be explored in the standard framework of reionization.
8.2.3 High-redshift quasars
The main open question regarding the evolution of high-redshift quasars have already been discussed
in detail in the introduction and in Chapter 7. I will therefore just briefly summarize the main questions that need to be resolved:
• What is the origin of 109 M' black holes at z ∼ 6?
• How do such quasars build up supersolar metallicities?
• Should we expect different metallicities at higher redshift?
167
8. DISCUSSION AND OUTLOOK
• How does the Magorrian relation evolve with redshift?
As shown in Chapter 7, these questions can be addressed observationally in the era of ALMA and
JWST. I will discuss below how they can be addressed further by theoretical simulations.
8.3 Outlook
In this section, I will sketch current and future projects that will help to understand the early universe
in more detail, with the goal to arrive at more detailed predictions that can be compared with current
and future observations. Particularly promising is the launch of Planck in April this year (2009),
which will provide better constraints about cosmic reionization on a short timescale. On longer
timescales, a first set of ALMA antennas will be completed in about three years, allowing a first test
of the predictions in Chapter 7 for instance by observing known quasars at high redshifts. In five
years, ALMA should be complete and JWST should be available as well, such that one can start
searching for quasars at very high redshift.
8.3.1 An implementation of chemistry in the RAMSES code
In the framework of the DFG Priority Programme 1177 ”Witnesses of Cosmic History: Formation
and evolution of black holes, galaxies and their environment”, I will explore the observables of highredshift quasars in more detail. This requires a cosmological hydrodynamics code that can handle
star formation, metal enrichment and chemistry. For this purpose, I plan to use the RAMSES code
by Teyssier [2002], which incorporates already many of these features. An additional module for
MHD is available [Fromang et al., 2006], which may help to address questions regarding the role of
magnetic fields in the early universe. Primordial chemistry was implemented recently by Aaron Boley
at the University of Zürich, in collaboration with Romain Teyssier who is the main developer of the
RAMSES code. An extension of this network to include the effects of X-ray ionization and heating
will be straightforward. This network also provides the basic infrastructure for the implementation of
more detailed networks that include the effects of metals, likeMeijerink & Spaans [2005] or Glover &
Jappsen [2007]. While the main goal of this implementation are self-consistent simulations of quasars
and X-ray feedback in the early universe, it will also be useful for other work as sketched below.
8.3.2 Magnetic fields
To understand the effects of primordial magnetic fields in more detail, I am currently calculating
the effect of ambipolar diffusion heating on the gas chemistry during the collapse of primordial gas.
This is based on a one-zone model by Glover & Savin [2009] which includes all relevant chemical
processes down to number densities of 1014 cm−3 . Prelimary calculations indicate that even field
strengths of 0.003 nG can change the evolution of the gas temperature in this density range, and
weaker fields may become relevant at later stages. In our model, the magnetic Jeans mass is relevant
168
8.3 Outlook
only for comoving field strengths larger than 0.1 nG, while for weaker fields, the thermal Jeans mass
dominates. The calculations so far confirm that star formation can be suppressed in small halos where
the available mass is smaller than the magnetic Jeans mass. However, it is interesting that weaker
fields, which are dynamically not relevant, can still change the chemistry. This is due to the absence
of efficient coolants in primordial gas. Due to the higher temperatures, the accretion to the protostellar
cores may be more efficient, such that the stars become more massive than in the zero-field case.
To explore this effect in more detail, I am collaborating with Paul Clark and Simon Glover to
include the ambipolar diffusion heating in the GADGET code [Springel et al., 2001]. This can be done
in the case where the thermal Jeans mass dominates over the magnetic Jeans mass, such that only the
thermal effect of the fields is relevant. In this limit, it will be very interesting to compare primordial
star formation with and without primordial magnetic fields. This might be the most interesting case,
as magnetic field strengths that are dynamically relevant are already constrained by the reionization
optical depth, which may considerably improve with Planck. At a later stage, when the chemical
models in the RAMSES code are improved and contain the relevant reactions for high-density gas,
it is conceivable to perform simulations with the MHD module of RAMSES to follow the dynamical
evolution of the gas. This is also relevant to compare the effects of strong magnetic field strengths in
semi-analytic models as used in this work with detailed three-dimensional simulations.
As a side note, the methods mentioned above can also be applied to understand the effects of magnetic fields generated by the first generation of stars. I plan to extend the one-zone models mentioned
above to calculate such magnetic feedback, and to examine whether it should be pursued further in
more detailed simulations.
8.3.3 Dark matter annihilation
The effects of dark matter annihilation on the formation of dark stars have already been explored in
some detail, but important considerations are currently missing. For instance, most of the previous
works focused on the later collapse phase in which dark matter annihilation heating may match the
total cooling rate and therefore stop further collapse. However, very interesting effects can occur well
before that phase, as I currently explore in collaboration with Douglas Spolyar from the University
of Stanford. Even a modest increase of the ionization fraction in the IGM due to annihilation of dark
matter would considerably enhance the ability of the gas to form H2 and HD and thus cool down to
temperatures close to the CMB. This has a strong effect on the thermal Jeans mass and may even
trigger fragmentation well before the time a dark star is expected to form. This study will also be
performed with the one-zone model of Glover & Savin [2009].
In addition, our group plans to explore the effects on the dynamics at late times in more detail
by implementing the effects of dark matter annihilation heating in the GADGET code. As it is not
possible to follow the evolution of dark matter with the required resolution, we will rather assume an
analytic prescription for the dark matter profile that takes the adiabatic contraction into account. This
may help in particular to better understand the final mass of dark stars.
169
8. DISCUSSION AND OUTLOOK
As discussed in Chapter 5, it is important to understand when the era of dark stars in the early universe ends, and which feedback processes are responsible for this. These questions can be addressed
at least to some extent with the one-zone model of Glover & Savin [2009] by considering ionized
initial conditions and additional cooling terms due to metallicity. But also the correlation between
gas collapse and dark matter density peaks are important for the formation of dark stars. As shown
by Greif et al. [2008], gas in atomic cooling halos is much more turbulent, and cold streamers may
spash into the center. In such a situation, collapse may set in outside the central dark matter cusp, and
the formation of dark stars naturally comes to an end when halos become more massive.
8.3.4 The first quasars
In this work, I have provided estimates for the observables of high-redshift quasars with respect to
ALMA and JWST by making use of local observations of the Seyfert 2 galaxy NGC 1068 and the
X-ray dominated region code of Meijerink & Spaans [2005]. These calculations have established that
one can expect to observe black holes of ∼ 106 M' with these telescopes even at z ∼ 15. As discussed
in Chapter 7, these conclusions even hold if the gas composition is different than in the present-day
universe and for sub-solar metallicities, as the total flux essentially depends on the amount of X-ray
heating, which is not sensitive to the metallicity of the gas.
As we can expect to observe such quasars in about five years time, it is desirable to understand
in more detail what one should expect from a theoretical point of view, i. e. which metallicities
should be present at which redshift, how much star formation is going on, what is the role of X-ray
and mechanical feedback from the central black hole. For the comparison with ALMA and JWST
observations, I plan to derive how different model assumptions are reflected in different observables,
and which observables are therefore well-suited to learn about different properties of the host galaxy.
For this purpose, I want to employ detailed numerical simulations that self-consistently calculate the
effects of star formation, metal enrichment and X-ray feedback in a cosmological context.
The cosmological hydrodynamics code RAMSES [Teyssier, 2002] already provides detailed prescriptions for star formation and metal enrichment as well as a network for primordial chemistry, in
which I plan to incorporate further relevant processes as described byMeijerink & Spaans [2005] and
Glover & Jappsen [2007]. This makes it possible to self-consistently calculate the effects of X-ray
feedback, star formation and metal enrichment during the formation and evolution of high-redshift
quasars. It will be important to further establish which observables can be expected for ALMA and
JWST, and to explore a potential model dependence, which may be used to derive model parameters
from ALMA and JWST observations.
170
Appendix A
Free-free transitions involving H−
While the importance of free-free transitions is well-known in stellar atmospheres, this process is
usually not considered in colder environments. Thus, high precision calculations are only available
for stellar temperatures between 1000 and 10000 K [John, 1988; Bell & Berrington, 1987; Gingerich,
1961]. The free-free absorption coefficient kf f is normalized per hydrogen atom and per unit electron
pressure, and contributes to the optical depth by dτ = kf f nH ne kT ds. As the fits by John [1988]
and Gingerich [1961] diverge at low temperatures, thus giving an unphysical high contribution at
low redshifts, we calculate the absorption coefficient based on the formalism of Dalgarno & Lane
[1966], which takes into account only the contribution of the leading term to the transition moment.
This approximation is valid at low energies and is thus a reasonable choice for the low-temperature
regime. Assuming that the initial electron energies are described by a Maxwell distribution, the freefree coefficient k f f is given as
!
"3 !
"
% ∞
h p ν 1/2
E
−3 −5/2 2
−a 3
−1
k f f (ν) = 9.291 × 10 T
1−e
E
1+
hpν
E
0
!
"
hpν
2
× e−aE/h p ν 1 +
q0 (E)
E
3
+ q0 (E + h p ν) dE cm4 dyne−1 ,
(A.1)
where ν is the photon frequency, q0 (E) the zero-order elastic scattering cross section and a = hp ν/kT
a dimensionless parameter. While Dalgarno & Lane [1966] used effective range theory [O’Malley
et al., 1961] to expand the cross section, we use the more accurate result ofDalgarno et al. [1999] as
fitted by Pinto & Galli [2008], given by
q0 (E) =
4 × 10−15 cm2
(1 + E/(3.8eV))1.84
.
(A.2)
In the range between 1 and 107 GHz and between 0.1 and 2000 K, our results are well-fit by the
expression
2
2
k f f (ν) = 10(a1 +a2 x+a3 x )(b1 +b2 y+b3 y ) cm4 dyne−1 ,
(A.3)
171
A. FREE-FREE TRANSITIONS INVOLVING H −
Figure A.1: The free-free absorption coefficient of H − for 102 , 103 , 104 and 105 GHz, as a function of
temperature. Given are the fits of John [1988] and Gingerich [1961] for the high-temperature regime, the
calculation of Dalgarno & Lane [1966] based on effective range theory as well as the new calculation of
this work for the low-temperature regime up to 2000 K.
172
where x = log10 (ν/GHz) and y = log10 (T/K), and a1 = −3.2421, a2 = −0.502052, a3 = 0.0117164,
b1 = 4.05293, b2 = 0.169299, b3 = −0.00548517. At stellar temperatures, our results differ by about
10% from the calculation of Bell & Berrington [1987], which is based on a more detailed treatment,
whereas at lower temperatures, we expect an even higher accuracy of our result. We thus adopt the
fit of John [1988] to the calculation of Bell & Berrington [1987] for T > 2000 K and Eq. (A.3) for
lower temperatures. The different calculations are compared in Fig.A.
173
A. FREE-FREE TRANSITIONS INVOLVING H −
174
Appendix B
Reaction rates for primordial chemistry
The table on the next page summarizes the chemical model developed in Chapter4. The main differences compared to previous works like Galli & Palla [1998]; Stancil et al. [1998] have been discussed
in Chapter 4.4. This network is now also implemented in the RECFAST code and can be provided on
request.
175
Reaction
H+e → H− +γ
H− +H → H2 +e
H+H+ → H+2 +γ
H+2 +H → H2 +H+
2H+H → H2 +H
H2 +H+ → H+2 +H
H2 +e → 2H+e
H2 +H → 3H
H− +e → H +2e
H− +H → 2H+e
H− +H+ → 2H
H− +H+ → H+2 +e
H+2 +e → 2H
H+2 +H− → H +H2
H2 +e → H+H−
H− +γ → H+e
H+2 +γ → H+H+
H2 +γ → H+2 +e
H+2 +γ → 2H+ +e
H2 +γ → (H2 )∗ → 2H
D− +γ → D+e
HD+ +γ → D+H+
HD+ +γ → H+D+
HD+ +γ → H+ +D+ +e
HD+γ → HD+ +e
D+ +e → D +γ
Reaction number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Rate [cgs]
1.4 × 10−18 T 0.928 exp(−T/16200)
1.5 × 10−9 for T < 300
4.0 × 10−9 T −0.17 for T > 300
dex(−19.38 − 1.523 log10 (T ) + 1.118 log210 (T )
−0.1269 log310 (T ))
6.4 × 10−10
5 × 10−29 T −1 for T < 300
see reference
1.91 × 10−9 T 0.136 exp(−53407.1/T )
Fit to data of MSM96
see reference
see reference
1.40 × 10−7 (T/300)−0.487 exp(T/29300)
6.9 × 10−9 T −0.35 for T < 8000
9.6 × 10−7 T −0.9 for T > 8000
2.0 × 10−7 T −0.5
5 × 10−6 T −0.5 for T > 100
3.67 × 101 T −2.28 exp(− 47172
T )
1.1 × 10−1 T r2.13 exp(−8823/Tr )
1.63 × 107 exp(−32400/Tr )
2.9 × 102 T r1.56 exp(−178500/Tr )
9.0 × 101 T r1.48 exp(−335000/Tr )
1.13 × 106 T r0.369 exp(−140000/Tr )
estimated by rate 16
estimated by half of rate 17
estimated by half of rate 17
estimated by rate 19
estimated by rate 18
3.6 × 10−12 (T/300)−0.75
Table B.1: Collisional and radiative rates.
SLD98
GP98
GP98
PSS83
SKHS04b
TT02
MSM96
AAZN97
AAZN97
LSD02
GP98
GP98
GP98
AAZN97
CCDL07
GP98
GP98
GP98
GP98
GJ07
Reference
GP98
GP98
GP98
HD+H+ → D+ +H2
D+H+ → HD+ +γ
D+ +H → HD+ +γ
HD+ +e → D+H
D+e → D− + γ
D+ +D− → 2D
H+ +D− → D+H
H− +D → H+D−
D− +H → D+H−
D− +H → HD+e
D+H− → HD+e
H− +D+ → D+H
34
35
36
37
38
39
40
41
42
43
44
45
He++ +e → He+ +γ
He+ +γ → He++ + e
He+ +e →He+γ
He+γ → He+ +e
He+H+ → He+ +H
HD+ +H → HD+H+
D+ +H2 → HD+H+
HD+H → D+H2
31
32
33
46
47
48
49
50
Reaction
D+H+ → D+ +H
D+ +H → D+H+
D+H → HD+γ
D+H2 → HD+H
Reaction number
27
28
29
30
Rate [cgs]
2 × 10−10 T 0.402 exp(−37.1/T ) − 3.31 × 10−17 T 1.48
2.06 × 10−10 T 0.396 exp(−33.0/T ) + 2.03 × 10−9 T −0.332
see reference
1.69 × 10−10 exp(−4680/T + 198800/T 2 ) for T > 200
9.0 × 10−11 exp(−3876/T ) for T < 200
6.4 × 10−10
1.0 × 10−9 (0.417 + 0.846 log10 (T ) − 0.137 log210 (T ))
5.25 × 10−11 exp(−4430/T + 173900/T 2 ) for T > 200
3.2 × 10−11 exp(−3624/T ) for T < 200
1.1 × 10−9 exp(−488/T )
dex(−19.38 − 1.523 log10 (T ) + 1.118 log210 (T )
−0.1269 log310 (T ))
dex(−19.38 − 1.523 log10 (T ) + 1.118 log210 (T )
−0.1269 log310 (T ))
7.2 × 10−8 T −0.5
3.0 × 10−16 (T/300)0.95 exp(−T/9320)
1.96 × 10−7 (T/300)−0.487 exp(T/29300)
1.61 × 10−7 (T/300)−0.487 exp(T/29300)
6.4 × 10−9 (T/300)0.41
6.4 × 10−9 (T/300)0.41
1.5 × 10−9 (T/300)−0.1
estimated by rate 42
1.61 × 10−7 (T/300)−0.487 exp(T/29300)
/
0−1
3.36 × 10−10 T −0.5 (T/1000)−0.2 1 + (T/106 )0.7
see reference
see reference
see reference
4.0 × 10−37 T 4.74 for T > 10000
1.26 × 10−9 T −0.75 exp(−127500/T ) for T < 10000
Table B.1: Collisional and radiative rates in cgs units.
Cen (1992)
AAZN97
AAZN97
AAZN97
GP98
GJ07
GP98
SLD98
SLD98
LSD02
LSD02
SLD98
SLD98
SLD98
This work
LSD02
GP98
Reference
Savin (2002)
Savin (2002)
Dickinson (2005, 2008)
GP02
GP98
SLD98
GP02
GP02
GP98
GP02
He+H+2
+
→ HeH +H
He +H → HeH+ +γ
HeH+ +H → He+H+2
HeH+ +e → He+H
HeH+ +γ → He+H+
HeH+ +γ → He+ +H
+
Reaction
He+ +H → He+H+
He+H+ → HeH+ +γ, rad. ass.
He+H+ +γ → HeH+ +γ, stim. rad. ass.
Rate [cgs]
1.25 × 10−15 × (T/300) 0.25
8.0 × 10−20 (T/300)−0.24 exp(−T/4000)
3.2 × 10−20 T 1.8 /(1 + 0.1T 2.04 )
×exp(−T/4000)(1 + 2 × 10−4 T r1.1 )
3.0 × 10−10 exp(−6717/T )
4.16 × 10−16 T −0.37 exp(−T/87600)
0.69 × 10−9 (T/300)0.13 exp(−T/33100)
3.0 × 10−8 (T/300)−0.47
220Tr0.9 exp(−22740/Tr )
7.8 × 103 T r1.2 exp(−240000/Tr )
JSK95, ZSD98
GP98
SLD98
LJB95
SLD98
JSK95
GP98
Reference
ZDKL89
SLD98
In the table above, T denotes the gas temperature in Kelvin, TeV the gas temperature in eV, Tr the temperature of radiation in K.
dex(x) = 10x . Acronyms: AAZN97: Abel, Anninos, Zhang, & Norman [1997], CCDL07: Capitelli, Coppola, Diomede, & Longo
[2007], GJ07: Glover & Jappsen [2007], GP98: Galli & Palla [1998], GP02: Galli & Palla [2002], JSK95: Juřek, Špirko, & Kraemer
[1995], LJB95: Linder, Janev, & Botero [1995], LSD02: Lepp, Stancil, & Dalgarno [2002], MSM96: Martin, Schwarz, & Mandy
[1996], PSS83: Pequignot, Petitjean, & Boisson [1991], SKHS04b: Savin, Krstić, Haiman, & Stancil [2004], SLD98: Stancil, Lepp,
& Dalgarno [1998], TT02: Trevisan & Tennyson [2002], ZDKL89: Zygelman, Dalgarno, Kimura, & Lane [1989], ZSD98: Zygelman,
Stancil, & Dalgarno [1998]
54
55
56
57
58
59
Reaction number
51
52
53
Table B.1: Collisional and radiative rates in cgs units.
Acknowledgement
Mein besonderer Dank gilt meinem Betreuer Ralf Klessen für die vielen Diskussionen, sein großes Interesse
und Engagement während meiner Arbeit hier am ITA. Sehr hilfreich waren insbesondere seine zahlreichen
Anregungen und Vorschläge für interessante Projekte, die diese Arbeit ganz wesentlich bereichert haben, seine
Offenheit für neue Ideen sowie die aktive Unterstützung bei der Zusammenarbeit mit Experten außerhalb des
Instituts, ebenso wie das Angebot, meine Promotion frühzeitig abzuschliessen.
Sehr wesentlich für diese Arbeit war die Zusammenarbeit mit anderen Mitgliedern am ITA und Experten außerhalb des Instituts. Dies gilt ganz besonders für Robi Banerjee, mit dem die Arbeit an zahlreichen Projekten zu
primordialen Magnetfeldern und der Annihilation dunkler Materie sehr viel Spaßgemacht hat. Ebenso interessant war die Arbeit mit Simon Glover, der mir geholfen hat, meine Arbeit an vielen Stellen kritisch zu hinterfragen und weiter zu verbessern. Matthias Bartelmann danke ich für seinen Rat zu Fragen der Kosmologie
und vielen darüber hinausgehenden Fragestellungen, sowie seine Unterstützung im Thesis Committee und die
Bereitschaft, das Zweitgutachten für diese Arbeit zu erstellen. Ferner danke ich Max Camenzind für zahlreiche
interessante Diskussionen über Quasare und Schwarze Löcher.
Ein sehr schönes Erlebnis während meiner Doktorarbeit war die Zusammenarbeit mit Daniele Galli, Francesco
Palla und Raffaella Schneider bei zwei Besuchen in Florenz, die mich bei meiner ersten Veröffentlichung sehr
unterstützt haben und weiterhin aktiv in meine Arbeit involviert sind. Besonderer Dank gilt ferner Marco
Spaans, der bei seinem Besuch im Frühjar 2008 zahlreiche wichtige Projekte angestoßen hat, die noch weit
über meine Promotion hinausgehen, und mit dem ich bereits erfolgreich eine erste Studie zu den Beobachtungsmöglichkeiten von Quasaren bei hohen Rotverschiebungen durchführen konnte. In diesem Zusammenhang
danke ich auch Antonella Maselli und Benedetta Ciardi für die gemeinsame Arbeit am StrahlungstransportCode CRASH, der bei diesen Studien noch hilfreich sein wird.
Für wertvolle Verbesserungsvorschläge beim Schreiben dieser Arbeit danke ich Robi Banerjee, Paul Clark,
Ingo Berentzen und Ralf Klessen, sowie Svitlana Zhukovska für die zugrundeliegenden Formatvorlagen. Weiterhin danke ich Robi Banerjee, Paul Clark, Christoph Federrath, Kati Fürbach, Simon Glover, Thomas Greif,
Ralf Klessen, Milica Milosavljevic, Thomas Peters, Stefan Schmeja und Bernd Voelkl für viele spannende
Diskussionen und sehr viel Spaß beim gemeinsamen Ausgehen.
Am meisten möchte ich mich schließlich bei meinen Eltern bedanken, die mich immer unterstützen, die mir
geholfen haben, meine Interessen zu entfalten, und ohne die mein Studium und meine Promotion nicht möglich
gewesen wäre. Ebenso auch bei meiner Schwester Caro, die mich daran erinnert, dass noch andere Dinge außer
der Astronomie wichtig sind.
179
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