Dissertation Co-Evolution Of Galaxies And Black Holes Michaela Hirschmann

Dissertation Co-Evolution Of Galaxies And Black Holes Michaela Hirschmann
Dissertation
Co-Evolution Of Galaxies
And Black Holes
vorgelegt von
Michaela Hirschmann
Co-Evolution Of Galaxies
And Black Holes
Dissertation
PhD thesis
zur Erlangung der Doktorwürde
for the degree of Doctor of natural science
an der Fakultät für Physik
at the Faculty for Physics
der Ludwig-Maximilians-Universität (LMU), München
of the Ludwig-Maximilians-University (LMU) of Munich
vorgelegt von
presented by
Dipl.-Phys. Michaela Hirschmann
aus Nürnberg, Deutschland
from Nürnberg, Germany
München, 12. Mai 2011
Ludwig Maximilians University of Munich
Erster Gutachter: Prof. Dr. Andreas Burkert (USM, LMU)
First advisor
Zweiter Gutachter: Prof. Dr. Joseph Mohr (USM, LMU)
Second advisor
Tag der mündlichen Prüfung: 17.6.2011
Day of oral exam
"Zwei Dinge erfüllen das Gemüt mit immer neuer zunehmender Bewunderung und
Ehrfurcht, je öfter und anhaltender sich das Nachdenken damit beschäftigt: Der
bestirnte Himmel über mir und das moralische Prinzip in mir. "
Immanuel Kant, deutscher Philosoph, 1724-1804
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Zusammenfassung
Beobachtungen zeigen, dass die Massen schwarzer Löcher und die Eigenschaften
ihrer Heimatgalaxien (“Hostgalaxie”) deutlich korreliert sind, was als Nachweis für eine
gemeinsame Entwicklung zwischen Galaxien und ihren schwarzen Löchern angesehen
werden kann. Dies impliziert, dass schwarze Löcher zum einen das Wachstum der
Galaxien regulieren und zum anderen Galaxien die Entwicklung der schwarzen Löcher
selbst beeinflussen. Ein umfassendes Verständnis dafür fehlt allerdings bis heute. Daher
untersuche ich in dieser Arbeit verschiedene Aspekte, wie sich Galaxien und schwarze
Löcher entwickeln, z.B. die Korrelationen zwischen schwarzen Löchern und Galaxien bei
hohen Rotverschiebungen und das antihierarchische Wachstum von schwarzen Löchern. Um die Entwicklung von Galaxien und das damit zusammenhängende Wachstum
von schwarzen Löchern zu modellieren und insbesondere, um eine grosse Anzahl von
Galaxien mit schwarzen Löchern zu generieren, sind analytische oder semianalytische
Modelle (SAMs) die am besten geeigneten Methoden.
Im ersten Teil meiner Arbeit wird gezeigt wie Galaxien- und Schwarzloch-Wachstum
allein durch Verschmelzungsprozesse die Korrelation zwischen schwarzen Löchern und
Galaxien bei hohen Rotverschiebungen und die intrinsische Streuung in der Masse
des schwarzen Lochs beeinflussen. Im einfachen Fall von zufälligem Verschmelzen
von schwarzen Löchern und Galaxien nimmt die Streuung in der Schwarzlochmasse
bei gegebener Galaxienmasse mit zunehmender Anzahl von Verschmelzungsprozessen,
Zeit und Masse des schwarzen Loches ab, wie es von dem Zentralen Grenzwertsatz
in der Statistik vorhergesagt wird. Generell kann die Streuung durch σmerg (m) ≈
σini × (m + 1)−a/2 mit 0 < a < 1 abgeschätzt werden. Kosmologisches Verschmelzen
basierend auf Verschmelzungsbäumen dunkler Materie von kosmologischen N-body
Simulationen zeigt eine ähnliche Abnahme der Streuung mit a = 0.3 für m ≈ 50.
Das herausragende Fazit dieses Kapitels ist, dass die Ergebnisse, unter der Annahme
einer heutigen Streuung von 0.3 dex in der Masse des schwarzen Loches, eine Streuung von 0.6 dex bei einer Rotverschiebung von z = 3 implizieren. Dies stellt damit
ein mögliches Szenario dar, in dem schwarze Löcher bei hohen Rotverschiebungen, die
vornehmlich oberhalb bzw. unterhalb der heutigen Korrelation zwischen schwarzen
Löchern und Galaxienmassen liegen, die Folge einer grösseren Streuung bei hohen
Rotverschiebungen sein könnten. In anderen Worten, “übermassive” bzw. “untermassive” schwarze Löcher bei hohen Rotverschiebungen könnten lediglich durch statistische
Verschmelzungsprozesse erklärt werden, ohne dass das Wachstum von schwarzen Löchern und Galaxien durch gas-physikalische Prozesse verknüpft sein muss.
Trotz der Wichtigkeit von Verschmelzungsereignissen repräsentiert jedoch die Akkretion von Gas auf schwarze Löcher einen bedeutenden Beitrag zum Gesamtwachstum
von schwarzen Löchern. Dies ist insbesondere relevant, um das rätselhafte, beobachtete
antihierarchische Verhalten im Wachstum von schwarzen Löchern verstehen zu können,
was das Thema des zweiten Teils meiner Arbeit darstellt. Antihierarchisches Wachstum
von schwarzen Löchern bedeutet, dass leuchtkräftige, aktive galaktische Kerne (AGN)
bereits sehr früh im Universum vorhanden sind, wohingegen weniger leuchtkräftige
viii
AGN vornehmlich zu späteren Zeiten auftreten. Verknüpft man die Leuchtkraft einer
aktiven Galaxie mit der Schwarzlochmasse würde dies im Konflikt zu momentan favorisierten, hierarchischen Strukturmodellen stehen, in denen sich zuerst kleinskalige
Strukturen bilden, welche sich mit fortschreitender Zeit zu immer Grösseren akkumulieren. Aufgrund der Komplexität der gas-physikalischen Prozesse verwende ich für
diese Untersuchung semianalytische Modelle (SAMs), welche eine Näherung für die
wichtigsten Prozesse in Galaxienentstehung und für das Wachstum schwarzer Löcher
beinhalten. Die SAMs basieren auf einer der momentan grössten numerischen Simulationen von dunkler Materie, der Millennium-Simulation (Boxlänge: 500 Mpc, Teilchenzahl: 1010 ). Dabei scheinen Akkretion auf schwarze Löcher aufgrund von Scheibeninstabilitäten, eine “sub-Eddington” Grenze für Gasakkretion, welche von Masse und Rotverschiebung abhängt, und massive, anfängliche schwarze Löcher wichtige “Schlüssel”Prozesse darzustellen um den antihierarchischen Trend erklären zu können. Darüberhinaus, kann eine rotverschiebungsabhängige Staubverschleierung zusätzlich als wichtiges Puzzleteil für das Verständnis des antihierarchischen Verhaltens angesehen werden.
Das “best-fit” Modell dieser Arbeit, welches die oben erwähnten Prozesse beinhaltet, zeichnet sich dadurch aus, dass es die beobachtete, heutige Schwarzloch-Massenfunktion
und die heutige Galaxien-Halomassen Relation reproduzieren kann genauso wie es die
bolometrische AGN Leuchtkraftfunktion und diejenige im harten Röntgenbereich korrekt voherzusagen vermag.
Allerdings besteht ein entscheidender Nachteil von SAMs in einem höheren Grad an
analytischen Näherungen (grosser Parameterraum) als in hydrodynamischen Simulationen. Im letzten Teil meiner Arbeit vergleiche ich daher die kosmische Entwicklung
der baryonischen Komponente in Galaxien und deren Unterteilung in eine stellare Komponente, in heisses und in kaltes Gas in 48 kosmologischen “zoom” Simulationen mit
bislang unerreichter Auflösung und in SAMs, die auf derselben Verschmelzungshistorie
von dunkler Materie basieren. Die Simulationen beinhalten Kühlen durch He & H,
Photoionization, Sternentstehung und energetische Supernova Rückkopplung. Beinhalten SAMs dieselben physikalischen Prozesse wie die Simulationen, reproduzieren
sie den baryonischen Anteil in Halos besser als 20% verglichen mit Simulationen. Die
bemerkenswertesten Unterschiede zwischen den beiden Modellen sind allerdings die
folgenden: Simulationen haben eine wesentlich grössere Sternentstehungseffizienz bei
hohen Rotverschiebungen als SAMs. Ausserdem dominiert in SAMs “insitu” Sternentstehung bei allen Rotverschiebungen, wohingegen in Simulationen die Akkretion
von Sternen der dominierende Prozess bei niedrigeren Rotverschiebungen wird. Zuletzt
überschätzen die SAMs den Anteil von “heisser” relativ zu “kalter” Akkretion verglichen
mit Simulationen. Der Grund für diese Diskrepanzen besteht in vereinfachten und/oder
fehlenden physikalischen Prozessen in SAMs. Ein wichtiger, zukünftiger Aspekt könnte demnach darin bestehen, SAMs zu verbessern und auszuweiten, motiviert durch
kosmologische Simulationen, insbesondere um die Auswirkung auf die Entwicklung der
AGN Galaxien zu untersuchen.
ix
Summary
Observationally, there exist strong correlations between black hole masses and properties of the host galaxies, which may be seen as an evidence for a co-evolution between
galaxies and their residing black holes. This implies that black holes are responsible
for regulating the growth of galaxies and vice versa. However, a full understanding of
self-regulated black hole growth and how it is exactly connected to spheroid formation
of the host galaxy is still missing. In this thesis I focus on different aspects of how
galaxies and black holes are co-evolving, such as the black hole scaling relations at high
redshifts and the origin of the anti-hierarchical black hole growth. In order to model
galaxy formation and the connected black hole growth and in particular to facilitate
the construction of large, statistical samples of galaxies with black holes, a highly successful approach is provided by ’analytic’ and ’semi-analytic’ tools.
In the first part of my thesis, it is demonstrated how merger-driven growth affects the
correlations between black hole mass and host bulge mass and the evolution of the
intrinsic scatter in black hole mass. In the simple case of random merging of galaxies
and black holes, it is found that the scatter in black hole mass σ decreases with increasing merging number m, time and black hole mass as predicted by the Central-limittheorem. Generally, the scatter can be approximated by σmerg (m) ≈ σini × (m + 1)−a/2
with 0 < a < 1. Cosmological merging based on halo merger trees of cosmological
N-body simulations reveal a similar decrease of the scatter with a = 0.3 for m ≈ 50.
The striking result of this work is that assuming a present-day scatter of 0.3 dex in
black hole mass, the analysis implies a scatter of 0.6 dex at z = 3 and thus, a possible scenario, in which over- and under-massive black holes at high redshift are the
consequence only of a larger scatter in black hole mass at high redshift. This way,
an explanation for observed over- and under-massive black holes in scaling relations
at high redshift might be given solely by ’statistical’ merging without considering a
connected growth of black holes and galaxies by gas-physical processes.
However, despite of the importance of the merger events, gas accretion onto black holes
represents a significant contribution to the over-all mass growth of black holes. This
will be particularly relevant for understanding the puzzle of the observed downsizing
trend in black hole growth, which is the topic of the second part of my thesis. I use
a more complex modeling tool, such as a semi-analytic model (SAM). Downsizing or
anti-hierarchical black hole growth means that luminous active galactic nuclei (AGN)
are observationally found to be already in place very early in the Universe, whereas
moderatly luminous AGN predominantly occur at later times. Relating the luminosity directly to the black hole mass, this would be in contrast to the currently favored
hierarchical clustering scenarios, where first small structures (galaxies) are assumed to
form growing into larger ones with evolving time. Applying the semi-analytic model to
one of the currently largest dark matter simulations, the Millennium-simulation (500
Mpc box-length and 1010 particles), I find that an accretion channel onto the black
hole due to disk instabilities, a sub-Eddington limit, which is dependent on mass and
redshift, as well as a heavy black hole seeding scenario may represent important key
x
mechanisms for explaining the anti-hierarchical trend. Moreover, a redshift dependent
dust obscuration may additionally be seen as a further, fundamental “puzzle piece” in
understanding the observed downsizing. The great success of the best-fit SAM, including the outlined ingredients, can be demonstrated by a good reproduction of the
observed present-day black hole mass function and the present-day galaxy-halo mass
relation as well as the bolometric and hard X-ray luminosity function within a large
redshift range 0 < z < 5.
The primary disadvantage of semi-analytic modeling consists in a greater degree of
approximation (parameter space) than in hydrodynamical simulations. Therefore, in
the last part of my thesis, I perform a direct comparison of the cosmic evolution of
the baryon content in galaxies and its division into stars, cold and hot gas in a large
set of about 48 cosmological zoom simulations with unprecedented resolution for a
sample this large and in SAMs based on the same dark matter merging history. The
simulations include H&He cooling, Photo-ionization, star formation and thermal SN
feedback. I find that SAMs that include the same physical processes as the simulations
reproduce the total baryon fraction in halos and the fraction of cold gas plus stars in
the central galaxy to better than 20%. However, the most striking discrepancies of this
study are the following: simulations have a much larger star formation efficiency at
high redshifts than SAMs. In SAMs, in-situ star formation is always the dominating
process, whereas in simulations, accretion of stars becomes more important than in-situ
star formation at low redshift. Finally, SAMs overestimate the fraction of “hot” relative
to “cold” accretion compared to simulations. The reason for these discrepancies may
consist in simplified or missing physical processes in SAMs, which should be refined
or included. Therefore, for future models it might be an important aspect to extend
and improve SAMs - motivated by detailed high-resolution cosmological simulations,
in particular to focus on the effect on the evolution of the AGN population.
Meinen Eltern
Contents
1 Motivation for this thesis
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2 A framework for galaxy formation and black hole growth
2.1 A brief history of extragalactic astrophysics . . . . . . . . . . . . . . .
2.1.1 Towards modern cosmology . . . . . . . . . . . . . . . . . . . .
2.2 Formation and evolution of large-scale structures in the Universe . . . .
2.2.1 The homogeneous Universe . . . . . . . . . . . . . . . . . . . .
2.2.2 The mass content of the Universe . . . . . . . . . . . . . . . . .
2.2.3 The linear growth of perturbation and spherical collapse model .
2.2.4 Modeling dark matter . . . . . . . . . . . . . . . . . . . . . . .
2.3 Current picture of the joint evolution of galaxies and black holes . . . .
2.3.1 Evolution of baryonic matter . . . . . . . . . . . . . . . . . . . .
2.3.2 Observational evidence for supermassive black holes . . . . . . .
2.3.3 Current picture of co-evolving galaxies and black holes . . . . .
2.3.4 Modeling galaxy formation and black hole growth . . . . . . . .
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intrinsic scatter in black hole mass scaling relations
Black hole mass relations in the present-day universe . . . . .
Black hole mass relations at higher redshifts . . . . . . . . . .
Models for random merging . . . . . . . . . . . . . . . . . . .
3.3.1 Initial conditions . . . . . . . . . . . . . . . . . . . . .
3.3.2 Depletion scenario . . . . . . . . . . . . . . . . . . . .
3.3.3 Replenishment scenario . . . . . . . . . . . . . . . . . .
Comparison to merging in ΛCDM-Simulations . . . . . . . . .
3.4.1 Simulation setup . . . . . . . . . . . . . . . . . . . . .
3.4.2 Merger tree algorithm . . . . . . . . . . . . . . . . . .
3.4.3 Evolution of the relation between black hole and galaxy
3.4.4 Quantifying the scatter in the black hole mass relation
3.4.5 Evolution of the black hole mass function . . . . . . . .
Discussion and conclusions . . . . . . . . . . . . . . . . . . . .
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4 Origin of the anti-hierarchical growth of black holes
4.1 Motivation and observational evidence for downsizing . .
4.2 Previous studies . . . . . . . . . . . . . . . . . . . . . . .
4.3 The semi-analytic model . . . . . . . . . . . . . . . . . .
4.3.1 Merging history from the Millennium simulation .
4.3.2 Galaxy formation . . . . . . . . . . . . . . . . . .
4.3.3 Models for black hole growth in the quasar phase
4.4 Properties of nearby galaxies and black holes . . . . . . .
4.4.1 Black hole mass function . . . . . . . . . . . . . .
4.4.2 Black hole-bulge mass relation . . . . . . . . . . .
4.4.3 Galaxy-dark matter halo mass relation . . . . . .
4.5 Galaxy and black hole properties at higher redshift . . .
4.5.1 Black hole mass function . . . . . . . . . . . . . .
4.5.2 Galaxy-dark matter halo relation . . . . . . . . .
4.6 Number density evolution of AGN . . . . . . . . . . . . .
4.7 The AGN luminosity function . . . . . . . . . . . . . . .
4.7.1 Bolometric luminosities . . . . . . . . . . . . . . .
4.7.2 Hard X-ray luminosities . . . . . . . . . . . . . .
4.8 Eddington-ratio distributions . . . . . . . . . . . . . . .
4.9 Luminosity-black hole mass-relation . . . . . . . . . . . .
4.10 Discussion and conclusion . . . . . . . . . . . . . . . . .
CONTENTS
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5 Galaxy formation in semi-analytic models and zoom simulations
5.1 Different approaches for modeling galaxy formation . . . . . . . . .
5.2 Previous comparison studies . . . . . . . . . . . . . . . . . . . . . .
5.3 The simulation and merger tree construction . . . . . . . . . . . . .
5.3.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Merger trees . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 The semi-analytic model . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Redshift evolution of galaxy properties . . . . . . . . . . . . . . . .
5.5.1 Baryon fraction . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Cold gas and stars . . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Hot halo gas . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Comparison to observations . . . . . . . . . . . . . . . . . . . . . .
5.7 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . .
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6 Conclusion and Outlook
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6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.2 Next steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A Appendix
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A.1 Dark matter component . . . . . . . . . . . . . . . . . . . . . . . . . . 151
A.2 Baryonic components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
CONTENTS
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Curriculum Vitae
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List of publications
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Danksagung
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Erklärung
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xviii
CONTENTS
Chapter
1
Motivation for this thesis
It is now well established that the dark energy dominated dark matter paradigm
(ΛCDM) provides a successful basis for understanding and simulating galaxy formation. In this picture, it is now widely believed that most spheroid-dominated galaxies
host a supermassive black hole in their center at the present time. In addition, there
exist strong correlations between black hole masses and properties of the host galaxies
as the luminosity, mass and velocity dispersion of the stellar spheroidal component (e.g.
Häring & Rix, 2004). These correlations may be seen as an evidence for a co-evolution
between galaxies and their residing black holes implying that black holes are responsible
for regulating the growth of galaxies and vice versa. However, a full understanding of
self-regulated black hole growth and how it is exactly connected to spheroid formation
of the host galaxy is still missing. Therefore, in the current picture of galaxy formation
and black hole growth, there exist a number of interesting, unanswered questions, as
e.g.:
• Which trigger mechanisms are responsible for bringing gas into the host galaxy
for black hole accretion?
• How strong is the influence of black hole feedback on the host galaxy?
• How do black holes accrete their gas, i.e. how is cold gas transported to the
accretion disk?
• How do seed black holes form and how massive are they?
• What is the origin of the observed black hole mass scaling relations?
• Are black hole mass scaling relations also valid at higher redshifts or do they
evolve with cosmic time?
• What is the origin of the observed anti-hierarchical black hole growth?
2
Motivation for this thesis
Numerical, hydrodynamical simulations of isolated galaxies or galaxy mergers modeling detailed gas dynamical accretion processes in the vicinity of a black hole are
necessary in order to get a deeper understanding for exact growth mechanisms and
the connected effect of feedback on the host galaxy. However, due to a lack in current computational power for coevally resolving a large range of scales, in particular
in cosmological simulations, a further highly successful approach is provided by ’semianalytic’ galaxy formation modeling, which is computationally inexpensive compared
to hydrodynamical simulations. This facilitates the construction of samples of galaxies
orders of magnitude larger than possible with N-body simulations. This is especially
important for studying statistical properties of galaxies and black holes. However, a
major disadvantage of semi-analytical methods is the large degree of approximation
due the large parameter space, whereas hydrodynamical simulations represent a more
precise approach as gas-dynamical equations are explicitly solved. Therefore, it is of
particular importance to compare directly galaxy formation in semi-analytic models
and in cosmological ’zoom-simulations’ based on the same dark matter evolution, in
order to understand their differences, and in particular, in order to reveal limitations
of both approaches.
In this PhD thesis, I mainly focused on two statistical problems of the above outlined puzzling questions, namely the black hole mass scaling relations at higher redshifts
as well as the anti-hierarchical growth of black holes. In order to be able to generate
large samples of galaxies providing sufficiently large statistics, I worked with analytical and semi-analytical methods based on the merging histories of cosmological dark
matter simulations. The outline of my thesis is the following: In Chapter 2, I will give
a brief theoretical framework of cosmology, structure formation and the joint evolution
of galaxies and black holes, which is the basis for further understanding of the following
investigations in this work. In Chapter 3, I will concentrate on the black hole mass
scaling relations at higher redshift, in particular on explaining the observed over- and
under-massive black holes at high redshifts by studying the intrinsic scatter in black
hole mass scaling relations using analytical methods. Chapter 4 deals with the origin
of the observed anti-hierarchical (or downsizing) trend in black hole growth, which is
at first glance in disagreement with currently favored hierarchical structure formation
models. To understand, which physical processes might be causing the observed downsizing trend , I use semi-analytic models. However, in order to reveal the limitations
of semi-analytics, in Chapter 5 a detailed comparison of the evolution of the baryonic
component in semi-analytic models and in cosmological zoom simulations is performed
based on the same dark matter merging history. Finally in Chapter 6, the most striking results of this thesis will be briefly summarized and some future prospects will be
given, which might be important steps towards a better understanding of the complex
but exciting co-evolution of galaxies and the black holes residing at their centers over
cosmic times.
Chapter
2
A framework for galaxy formation and black
hole growth
2.1
A brief history of extragalactic astrophysics
Within living memory, the extraordinary beauty of the night sky was always of fascinating and impressing nature for human beings and had historically a strong impact on culture as well as on religious believes. From the ancient Greek term ’κυκλoς γαλακτ ικoς’
the name ’galaxy’ (= Milky Way) is derived for its appearance in the sky. This originates from Greek mythology, where Zeus places his son born by a mortal woman, the
infant Heracles, on Hera’s breast while she is asleep so that the baby will drink her
divine milk and will thus become immortal. But Hera wakes up while breastfeeding
and realizes an unknown baby: she pushes the baby away and a jet of milk sprays
the night sky, producing the faint band of light known as the Milky Way. With a
more academic attempt, the Greek philosopher Democritus (450-370 BC) already proposed that the Milky Way might consist of distant stars, what was finally proven by
Galileo Galilei about 2000 years later (1610 AD). The earliest recorded observation
for a distinction of stars of our Galaxy to other nebulae is made in the 10th century
by the Persian astronomer, Abd al-Rahman al-Sufi (known in the West as Azophi),
describing the Andromeda Galaxy as a "small cloud". The earliest cosmologies of the
modern era were of very speculative nature, as e.g. the ’island’ universe model of Rene
Descartes (1636 AD) or the ’New Hypothesis of the Universe’ of Thomas Wright (1750
AD), in which the sun was one of many stars which orbit about the ’Divine Center’
of the star system. At the end of the eighteenth century, William Herschel was the
first astronomer, who defined the distribution of stars in the Universe in some detail
on basis of careful consideration. By the end of the nineteenth century, a large amount
of astronomical objects was discovered that differ from stars as they were fuzzy and
not point-like. These objects were collectively referred to as ’nebulae’. E.g. Charles
Messier (1784 AD) was among the first who have catalogued bright nebulae. However,
for many years after their discovery, the nature of these nebulae was still unclear. There
4
A framework for galaxy formation and black hole growth
were two different explanations: On the one hand, nebulae were assumed to be objects
within our Milky Way, on the other hand nebulae were thought to be extragalactic objects - individual ’island universes’ like the Milky Way itself. The controversy remained
unsolved until 1925, when Edwin Hubble used distances estimated from Cepheid variables to demonstrate conclusively that some of the nebulae are indeed extragalactic,
individual galaxies comparable to our Milky Way in size and luminosity. Hubble’s
discovery was the beginning of the extragalactic astronomy. During the 1930s, highquality photographic images allowed to classify galaxies according to their different
morphological types (Ellipticals, Spirals, Irregulars).
2.1.1
Towards modern cosmology
Only four years after the discovery that the ’nebulae’ are really extragalactic objects,
Hubble made the second main discovery: he demonstrated that our Universe is expanding (the velocities of galaxies are linearly related to their distances (Hubble, 1929;
Hubble & Humason, 1931). This revolutionized the picture of our Universe profoundly.
Some years earlier, the construction of mathematical models had started to explain the
large-scale evolution of the Universe in a self-consistent way, mainly based on the general relativity of Albert Einstein and a few basic assumptions (cosmological principles).
According to solutions of the field equations, an expanding or contracting Universe
would be the natural consequence. Alexander Friedmann (1922) and George Lemaitre
(1927, Lemaître, 1927) found independently the same solution for an expanding or
static Universe. In the late 1940s George Gamow suggested that the Universe must
have been denser and perhaps also hotter at earlier times, as the Universe is expanding
leading to the Hot Big Bang model. He also realized that, as a consequence of a hotter Universe at early times, the residual heat should still be visible in the present-day
Universe as a background of thermal radiation with a few degrees of kelvin - the first
prediction for the cosmic microwave background. Therefore, the 1965 discovery of the
cosmic microwave background by Penzias & Wilson (1965) and Dicke et al. (1965) was
a great success for the Hot Big Bang model and firmly established it as the standard
model of cosmology. They found the cosmic microwave background to be isotropic and
to have a temperature of 2.7K, exactly in the range expected in the Hot Big Bang
model. However, due to several short comings of the Hot Big Bang theory, in 1981,
Alan Guth (Guth, 1981) proposed that the Universe might have experienced an early
phase of exponential expansion (’inflation’) driven by the vacuum energy of a quantum
field. This original model was later-on revised by Linde (1982) and by Albrecht &
Steinhardt (1982). In this scenario, different parts of the Universe could have been
in causal contact before inflation started, important for the principle of isotropy and
homogeneity. Another basic prediction of inflation is that the present-day Universe
should appear to have a flat geometry. Furthermore, in the 1980s, it was realized that
quantum fluctuations of a scalar field (inflaton) can generate density perturbations
close to the ones predicted by Harrison-Zel’dovich, which was found to be the only
2.2
Formation and evolution of large-scale structures in the Universe
5
possible scaling set-up in the Big Bang theory being consistent with galaxy formation
at early times. (Hawking, 1982; Guth & Pi, 1982; Bardeen et al., 1983).
In the meanwhile, a large amount of further conclusive evidences have been accumulated that the visible matter is only a small fraction and that the majority of the mass
in the Universe consists of a still unknown dark matter component. The first hint can
be traced back to 1933, when Zwicky studied the velocities of galaxies in the Coma
cluster and concluded that the total mass required for a cluster to be stable is about 400
times larger than the luminous mass in stars. However, it took more than 40 years that
the existence of dark matter was generally accepted. E.g. Ostriker et al. (1974) and
Einasto et al. (1974) claimed that massive halos around our Milky Way are required
in order to explain the motion of satellite galaxies. This hypothesis was confirmed by
measurements of spiral galaxy rotation curves (Roberts & Rots, 1973; Rubin et al.,
1978). Dark matter particles have been discussed to be neutrinos (electron neutrino in
the 1960s and 70s, e.g. Reines et al., 1980). However, in 1984, White (1984) showed
that simulations assuming a neutrino-dominated Universe (hot dark matter) result in a
stronger clustering of galaxies than is observed. Therefore, alternative models suggest
that e.g. dark matter might be a kind of weakly interacting massive particle (WIMP)
referred to as cold dark matter (Peebles, 1982; Blumenthal et al., 1982). Cold dark
matter results in the ’bottom-up’ scenario, originally proposed by Peebles, 1965, which
are nowadays favored over the top-down scenario (= hot dark matter). This means
that smaller structures collapse first and are later assembled in larger collapsing structures. Dark matter particles can decouple from radiation at earlier times than baryonic
matter so that the perturbation in density can also start to grow at earlier times. After
baryons decouple from radiation they fall into the potential wells of dark matter (Davis
et al., 1985). However, in the early 90s measurements of galaxy clustering showed that
the CDM model predicts less clustering than observed (Maddox et al., 1990; Efstathiou
et al., 1990). Thus, several alternative models have been discussed with a final model
assuming a flat geometry and adding a cosmological constant (= dark energy component) resulting in the ΛCDM model. This was verified, when it was shown by SNIa
measurements (Perlmutter et al., 1999) that the Universe is accelerating and that it is
flat (CMB measurements by de Bernardis et al., 2000).
Altogether, the ΛCDM model has now established itself firmly as the standard
paradigm for structure formation and is the basis for the evolution and formation of
galaxies.
2.2
Formation and evolution of large-scale structures
in the Universe
How large-scale structures form and evolve within the current framework of our cosmological understanding will be subject of this section, in particular with respect to a
more precise mathematical prescription than in the last Section where a short historical
6
A framework for galaxy formation and black hole growth
overview was given.
2.2.1
The homogeneous Universe
One of the fundamental assumptions in cosmology is that the Universe is on large-scales
approximately homogeneous and isotropic. Out of this basic cosmological principle and
the theory of general relativity, the space-time metric of the Universe is given by the
Robertson-Walker metric (Peacock, 1999):
!
"
dr 2
2
2
2
2
2
2
2
ds = (cdt) − a(t)
+ r (dθ + sin θdφ ) ,
(2.1)
1 − Kr 2
where r, θ and φ are spherical comoving coordinates, and t is the proper time. The
growth of the spatial expansion can be parameterized with the function a(t) := R(t)/R0 ,
which is called expansion factor, and by definition is a0 = a(t0 ) = R(t0 )/R0 ≡ 1 at
the present time. The information of the curvature of the space-time is given by the
parameter K, which can have values 1, 0, −1, corresponding to a open (hyperbolic),
flat (euclidean) or closed (spherical) geometry of the space-time. Using the RobertsonWalker metric, Einstein’s field equations simplify into the Friedmann equations:
#
$
4πG
3p
Λc2
ä
ρ+ 2 +
= −
(2.2)
a
3
c
3
# $2
ȧ
8πG
Kc2 Λc2
=
ρ− 2 +
,
(2.3)
a
3
a
3
where p represents the pressure and Λ the cosmological constant, corresponding to an
energy density of the vacuum:
ρv c2 =
Λc4
8πG
(2.4)
For the density ρ, which appears in Eq. 2.3, one distinguishes between a non-relativistic
matter component, a radiation component and a vacuum energy (cosmological constant) component. At the present time, they are denoted as ρM,0 , ρr,0 and ρΛ,0 , respectively. The Hubble parameter, that represents the expansion rate of the Universe, is
related to the scale-factor by the equation:
H(a) =
ȧ
a
(2.5)
In order to solve the above equations, the time dependence of p and ρ has to be
specified, i.e. by defining an equation of state, which depends obviously on the ratio
of energy (radiation) to matter. In the early Universe the energy is dominated by
relativistic particles and radiation; hence, the equation of state is given by p = ρ/3
and the density scales with a−4 . As the Universe expands, the energy density decays
2.2
Formation and evolution of large-scale structures in the Universe
7
Figure 2.1: Cosmological constraints in the Ωm − ΩΛ plane.
Three data
sets are shown based on supernovae, CMB and galaxy cluster measurements,
which provide independent methods to determine the cosmological parameters
(http://www.eso.org/public/images/eso0419d).
and the Universe enters a matter-dominated phase, non-relativistic matter is assumed
to be pressure-less and its energy density scales with a−3 .
It is convenient to define the today’s critical density of the Universe as
ρcr =
3H02
,
8πG
(2.6)
where H0 is the Hubble constant at the present time. Moreover, it is useful to define
the following dimensionless density parameters:
ΩM,0 =
ρM,0
Λc2
Kc2
, ΩΛ,0 =
,
Ω
=
−
r,0
ρcr
3H02
a20 H02
(2.7)
where ΩM,0 is the non-relativistic matter density parameter, ΩΛ,0 the dark energy density parameter and Ωr,0 the relativistic matter density parameter at the present time.
Thereby, whether the total density parameter Ω0 := ΩM,0 + ΩΛ,0 + Ωr,0 is smaller,
equal or larger than unity < 1, = 1, > 1 is equivalent to K = −1, 0, 1 distinguishing
between an open, flat of closed geometry. With these definitions, Eq. 2.2 becomes
8
A framework for galaxy formation and black hole growth
q0 = 1/2ΩM,0 − ΩΛ,0 with q0 := −(äa/ȧ2 )t0 (= rate, at which the Universe is accelerating) and Eq. 2.3 becomes ΩM,0 + Ωr,0 + ΩΛ,0 = 1, again at the present time.
In order to measure and to quantify today’s cosmological parameters, different
methods have been used, illustrated by Fig. 2.1 showing the ΩM,0 − ΩΛ,0 plane. The
different colored regions depict the favored parameter ranges resulting from different
measurements. Cluster of galaxies provide a powerful diagnostic for the measurement
of ΩM,0 (see red area in Fig. 2.1) as the number density of cluster at a given mass
at varying redshift is strongly dependent on ΩM,0 . Supernova of type Ia (SNIa) are
considered as excellent standard candles because of the constancy of their luminosity at
maximum brightness. Since SNIa are relatively luminous objects, they can be observed
out to high redshift, and thus, they provide a powerful tool for the determination of the
geometry and matter content of the Universe (Perlmutter et al., 1999, see green area
in Fig. 2.1). Finally, a very strong observational constraint originates from the measurement of the temperature fluctuations in the cosmic microwave backround (CMB),
most exactly with WMAP (Spergel et al., 2007; Komatsu et al., 2009). Decomposing
the temperature fluctuations in spherical harmonics, it is possible to recover the underlying fluctuation spectrum (see blue area in Fig. 2.1). Altogether, observational data
seem to converge towards a concordance model with a low-density, vacuum-dominated
Universe (ΩM,0 ∼ 0.27 ± 0.05, ΩΛ,0 ∼ 0.75 ± 0.02 and Ωr,0 ∼ 4.2 × 10−5 ± 0.02). The
Hubble constant at the present time is found to be H0 = 72 ± 5kms−1 Mpc−1 .
By putting the definitions of Eq. 2.7 into the second Friedmann equation 2.3 one
obtains the evolution of the Hubble parameter with cosmic time:
H(z) =
E(z) =
ȧ
(z) = H0 E(z), with
a
%
Ωr,0 (1 + z)4 + ΩM (1 + z)3 + (1 − Ω0 )(1 + z)2 + ΩΛ,0
(2.8)
(2.9)
Here, z is the cosmological redshift, which is directly related to the scale factor a(t):
1+z =
a(t0 )
a(te )
(2.10)
This quantity z can be understood as the Doppler shift of its emitted light due to the
expansion of the Universe:
z=
λobs − λemitted
.
λemitted
(2.11)
For small distances z ≈ v/c = d/DH is valid, where v is the velocity of the object
and DH = c/H0 represents the Hubble distance. The redshift z is directly measurable
from the spectra and is related to both the distance of the objects and the epoch of
emission of the light and thus, is often used as a time variable. Note that light emitted
2.2
Formation and evolution of large-scale structures in the Universe
9
from objects gets additionally Doppler-shifted due to their peculiar velocities besides
the global expansion of the Universe:
&
1 + vpec /c
.
(2.12)
1 + zpec =
1 − vpec /c
This means that the observed redshift of any object consists of a contribution due
to the universal expansion and one due to its peculiar velocity along the light-ofsight. Therefore, the observed redshift for the relativistic case is given by 1 + zobs =
(1 + zpec )(1 + z), and for the non-relativistic case (zpec = vpec /c) it can be calculated
by zobs = z + vpec /c · (1 + z).
2.2.2
The mass content of the Universe
Observations show that ordinary matter (baryonic component) can only account for
∼ 15 − 20% of the total matter content (CMB constraints, rotation curves of spiral
galaxies). Principally, non-baryonic particles are classified as relativistic and hot or
non-relativistic and cold dark matter particles. However, at present time, the best
agreement with observational studies is given by the cold dark matter paradigm, in
the form of Weakly Interacting Massive Particles (WIMPS), which naturally arise in
supersymmetric extensions of the standard model in particle physics. Dark matter,
which only interacts via weak and gravitational forces, is assumed to behave as a
collisionless fluid for most of the history of the Universe. As the number of particles
is assumed to be large, two-body interactions can be neglected, and the system can
be described by a distribution function in phase-space. Its evolution is given by the
collisionless Boltzmann equation (also called Vlasov equation):
df
∂f
df
dΦ df
=
+v
−
,
dt
∂t
dr
dr dv
where the potential is given by the Poisson equation:
&Φ(r, t) = 4πGρ(r, t).
2.2.3
(2.13)
(2.14)
The linear growth of perturbation and spherical collapse
model
The currently favored theories of structure formation assume that structure grows out of
primordial quantum fluctuations which get amplified during a phase of rapid expansion
(∆t ≈ 10−33 s), what is called inflation (Guth, 1981). The statistical properties of the
density field δ(x, t) can be characterized by a Fourier transformation of its two-point
correlation function, which is called power spectrum:
P (k, t) = '|δ(k, t)|2( with
'
1
ρ(x, t) − ρ
−ikx
dxδ(x,
t)e
and
δ(x,
t)
=
,
δ(k, t) =
(2π)2
ρ
(2.15)
(2.16)
10
A framework for galaxy formation and black hole growth
where ρ is the background density of the Universe. The latest measurements from the
CMB are consistent with a Gaussian scale free Harrison-Zeldovich initial fluctuation
spectrum, i.e. Pinitial (k) ∝ k n with n = 1. Perturbations in cold dark matter begin
already to grow after the radiation-matter equality (ρmatter = ρradiation ) so that by the
time of decoupling (i.e. “recombination” of H and He nuclei with electrons) dark matter density fluctuations have already grown by a factor 20 × Ωm,0 with respect to the
perturbations in the photon-baryon fluid, i.e. cold dark matter can accumulate before
baryonic matter. After recombination, baryonic matter can follow the evolution of dark
matter by collapsing into the dark matter potential wells and mass is assembled following hierarchical clustering (bottom-up scenario). The density fluctuations grow by
self-gravitation, and as long as these inhomogeneities are small, the growth of structure
can be approximated with the linear perturbation theory. Based on the Euler equation
and the continuity equation, which are given by
∂ρ
+ ∇ · (ρ · v) = 0 (Continuity : mass conservation) and (2.17)
∂t
∂v
1
+ (v · ∇) · v + ∇p = −∇Φ (Euler : momentum conservation), (2.18)
∂t
ρ
and the Poisson equation (Eq. 2.14) one can derive the growth of the perturbations in
the linear regime:
ȧ
δ̈ + 2 δ̇ = 4πGρδ
a
(2.19)
Note that the equation has two solutions, a growing mode and a decaying mode. However, the decaying solution will become negligible after some time and solely the growing
mode is the relevant solution for structure formation.
After some time the density contrast of a given perturbation becomes comparable
to unity and the linear perturbation theory will no longer represent a valid approximation. At a certain point, the perturbation will start to decouple from the background
expansion as due to the self-gravity an over-dense region will expand at a slower rate
compared to the background Universe. This will increase the density contrast and at
some point, the over-dense region will collapse under its own self-gravity, forming a
bound system. In the spherical collapse model, it can be shown that the perturbation decouples from the background expansion and collapses when it reaches a critical
overdensity of δc ≈ 1.68. However, in reality, the sphere of matter does not collapse
into one point. As a collisionless system can not dissipate its energy, the gravitational
potential energy during the collapse has to be converted into kinetic energy of the particles according to the virial theorem. Moreover, due to the collapse, a rapid change of
the gravitational potential occurs, a process called violent relaxation, which causes particle mixing in binding energy and let the particles get virialized (equilibrium state).
The over-density at virialization can be computed using the virial theorem yielding
δvir = 18π 2 ≈ 180 for Ωm,0 = 1 (resulting in M200 , r200 ). Assuming Ωm,0 += 1 and Λ += 0
2.2
Formation and evolution of large-scale structures in the Universe
11
Figure 2.2: Schematic illustration of a merger tree, depicting the evolution of dark
matter in a bottom-up scenario (Mo et al., 2010).
leads to more complex expression, namely a dependence of the virial over-density δvir
on the matter density at the time of virialization Ωm (tvir ).
2.2.4
Modeling dark matter
For modeling the structure formation in the dark matter component, current methods
consist either in using analytical techniques based on the Press-Schechter formalism
or in performing numerical N-body simulations. Within the framework of a spherical
collapse model, Press & Schechter (1974) realized that although small-scale modes have
become non-linear, large-scale modes may still follow the linear theory. Therefore, using
a filtering density function, they obtain a smoothed, linearly extrapolated density field
(with no density fluctuation smaller than of a certain size). Therefore it is possible
to determine if a given region in the space belongs to a collapsed object (when the
linearly extrapolated density becomes ≈ 1.7). Thus, the PS formalism can be used to
determine the mass function:
# $1/2
# 2 $
ρ δcrit dσ
2
δ
exp crit2 dM
(2.20)
n(M)dM = −
2
π
M σ dM
2σ
where σ 2 is the variance of the linear density field smoothed over a mass scale M. Note
that extending the PS formalism (EPS) by assuming a ellipsoidal collapse seems to
fit simulation results better. Based on PS or EPS formalisms one can derive conditional mass functions for finding progenitors of dark matter halos (using Monte-Carlotechniques) and thus, also construct merger trees for dark matter halos, which are
12
A framework for galaxy formation and black hole growth
often used as input for semi-analytical models (e.g. Kauffmann et al., 1993; Sheth &
Lemson, 1999; Somerville & Primack, 1999a). Fig. 2.2 shows a merger tree, depicting
the growth of dark matter halos as a result of mergers. Time increases from top to
bottom. A horizontal slice through the tree gives the distribution of the masses of the
progenitors at a given time (Lacey & Cole, 1993).
Even if linear theory together with the spherical collapse model provides a useful
tool for describing the growth of structure in the early universe, it can not account
completely for the non-linear processes which are occurring in hierarchical structure
formation. Therefore, a more accurate method for calculating the evolution of dark
matter is by performing numerical N-body simulations. In these simulations, the phasespace distribution function is replaced by a set of N collisionless particles which are
assumed to evolve only under self-gravity. Therefore, the following equations of motion
have to be solved for a particle i:
dri
= vi
dt
dvi
= Fi = −∇Φ|i ,
dt
(2.21)
(2.22)
where Fi is the gravitational force on the particle i determined by the gravitational
potential Φ (Equation 2.14). As in cosmological simulations the background space is
uniformly expanding with time, it is convenient to use comoving coordinates, r → x =
r/a, where a is the scale factor, and to re-write the above equations of motion 2.21
and 2.22. In order to calculate the gravitational force on a particle there exist several
algorithms (PP, Tree, PM, P3 M algorithms), with the aim to significantly reduce the
computational time. The most straight forward way to calculate the gravitational
force on a particle is given by direct summation of particle-particle interactions (PP),
but this is combined with huge computational costs. Instead, particularly the treecode
algorithm has led to substantial progress in this field by reducing the calculating time.
In the tree-algorithm, particles are arranged in a hierarchy of groups (e.g. Barnes &
Hut, 1986) and the gravitational field for each particle is calculated by a summation
over the multipole expansion of the gravitational field of these groups, where higher
order terms die off faster with increasing distance. Note that the N-body simulations I
will use in this thesis have always been performed with the treecode GADGET (Springel
et al., 2005a, see Sections 3.4.1, 4.3.1 and 5.3.1), partly combined with high-resolution
techniques. Fig. 2.3 shows a part of the density distribution of dark matter in the
Millennium simulation at redshift z = 5 (upper panel) and at z = 0 (lower panel,
Springel et al., 2005c). It illustrates nicely the growth of large-scale structures in the
Universe. Using halofinder single dark matter halos can be identified at different time
steps and this way, merger trees can also be constructed based on simulations, which
is, however, more demanding than the EPS based techniques (see 3.4.2 and 5.3.2).
2.2
Formation and evolution of large-scale structures in the Universe
13
Figure 2.3: Density distribution of dark matter in the Millennium simulation at z = 5
(upper panel) and at z = 0 (lower panel) (Springel et al., 2005c).
14
2.3
2.3.1
A framework for galaxy formation and black hole growth
Current picture of the joint evolution of galaxies
and black holes
Evolution of baryonic matter
Due to gravitational interaction, the evolution of the baryonic component is primarily
based on the structure formation of cold dark matter - as described in the last Section
(Section 2.2). However, because of additional gas-dynamical and radiative processes the
treatment of baryonic matter is much more complicated. Gas is believed to get trapped
in the potential wells of dark matter halos and subsequently cools and condenses (radiative cooling, Silk, 1977; Rees & Ostriker, 1977; Binney, 1977) in their centers. Different
cooling processes can affect the gas depending on its temperature and density. E.g.
for T > 107 K, gas is fully ionized and cools mainly through bremsstrahlung emission
from free electrons. At lower temperatures 104 K < T < 106 K, many excitation and
deexcitation processes play a role (depending strongly on the chemical composition of
the gas). However, when the temperature is smaller than T = 104 K, gas is mainly
neutral and thus, cooling gets suppressed (only heavy elements or/and molecules allow
further cooling). Cold, dense gas clouds may fragment due to self-gravity into small,
high-density cores that may then eventually form stars. As a consequence, dying stars
(SN explosions) can produce an enormous amount of energy which may heat or reheat the surrounding gas or even blow a fraction of gas out of the galaxy (winds).
Moreover, in the hierarchical evolution of galaxies, merging processes are found to play
an important role for the growth processes. Minor mergers can be seen as kind of
smooth accretion, but might be important for the size evolution of elliptical galaxies. In contrast, in major mergers, violent relaxation transforms orbital energy of the
progenitors into internal binding energy of the remnant so that any hot gas associated with the progenitors gets shock-heated during the merger accompanied by strong
star formation if the merging galaxies contain cold gas. Mergers are also influencing
the shape of the galaxies significantly and thus, they are important for reproducing
the observed different morphologies (spirals and ellipticals, e.g. Toomre & Toomre,
1972). Fig. 2.4 illustrates very schematically and simplified the relevant processes for
galaxy formation. The dashed box represents the galaxy containing a hot and cold
gas and a stellar component. Gas can cool and cold gas can form stars. Dying stars
can inject energy, mass and metals into the surrounding gas. Note that a galaxy is
not a closed box, so significant gas inflow and outflow can happen. However, besides
all these processes, it is generally accepted that galaxies contain a supermassive black
hole (SMBH) in their center, which might power active galactic nuclei (AGN), a very
important phase in the process of galaxy formation. Active galaxies (AGN) can be divided into a variety of galaxies, e.g. Quasars, Radio galaxies, Seyfert galaxies, Blazars
and BL Lacertae objects. However, common properties of AGN are that they have
a very bright and compact nuclear region often much brighter than the host galaxy,
they emit non-stellar (thus, non-thermal) continuum emission (from radio to the hard
X-ray range) and show strong emission lines. By Salpeter (1964); Zel’Dovich (1964);
2.3
Current picture of the joint evolution of galaxies and black holes
15
Figure 2.4: Schematic picture for the relevant processes in galaxy formation (Mo
et al., 2010). The dashed box represents the galaxy containing a hot and cold gas and
a stellar component as well as a black hole. Gas can cool and cold gas can form stars.
Dying stars as well as heavily accreting black holes can inject energy, mass and metals
into the surrounding gas. Note that the box is ’open’, i.e. one can have gas infall
(maybe driven by merger events or smooth gas accretion) and also gas outflows, e.g.
SN-driven winds.
Lynden-Bell (1969) it was originally proposed that the central energy source of an AGN
is gravitational in nature and thus, might be related to a SMBH. Until today, the only
reasonable explanation for AGN behavior is the paradigm that a gravitational singularity is responsible for converting infalling matter into energy (Rees, 1984). During
such active phases, black holes can influence the evolutionary processes of the galaxies
significantly as large amounts of energy are released heating up the gas component.
Therefore, in this Section I will briefly describe the interplay between the evolution of
galaxies and the growth of black holes in more detail starting with a brief review of
observational evidence for SMBHs.
2.3.2
Observational evidence for supermassive black holes
Observational evidence for black holes (see a summary in Colpi et al., 2006) - in particular, the supermassive variety - was very difficult to state, despite of the fact that
the theoretical basis was already in place immediately following the 1915 publication
of Albert Einstein’s theory of general relativity. Karl Schwarzschild’s 1916 solution of
Einstein’s field equations led to the conclusion that for a star of given mass, there exists a finite, critical radius at which light cannot escape anymore, and thus, reaches an
16
A framework for galaxy formation and black hole growth
infinite time dilation. That real stars can indeed achieve such a critical radius was later
demonstrated in a series of papers (Chandrasekhar, 1931; Landau, 1938; Oppenheimer
& Serber, 1938). The first stellar mass black hole was detected in the rapidly variable
X-ray source Cygnus X-1 (e.g. Brucato et al., 1972; Bolton, 1972), a strong confirmation of the theoretical work. In contrast, the common acceptance of supermassive
black holes (SMBHs) in galaxies - not several masses in size, but rather millions of
solar masses - was driven by an increasing amount of observational evidence. Although
many of the present-day SMBH detections are in quiescent or weakly active galaxies,
historically the SMBH paradigm evolved in the context of AGN. The first hint was
found by Carl Seyfert’s identification (early 1940s) of galaxies with ’unusual nuclei’.
Fifty years later, one of Seyfert’s original galaxy, NGC4258, was the first, for which
the existence of a SMBH was conclusively demonstrated (Miyoshi et al., 1995). The
most obvious evidence was found for M87 and Cygnus A (bright radio sources). In
both cases the presence of radio lobes and of a bright optical narrow jet suggested that
the radio emission might be due to relativistic particles ejected from the nucleus.
Although galaxies with AGN are the obvious places to look for evidence for the
existence of SMBHs, the fact that quasars were more numerous at z = 2 than today
suggests that ’dead quasar engines’ should be hiding in many nearby, quiescent galaxies. Modeling the kinematics of stars in galactic nuclei provides a proper method for
constraining the central potential and thus the mass of the central object. In particular
in the Galactic Center, kinematics can be studied on much smaller scales revealing a
black hole with a mass M• = 3 × 106M" (Genzel et al., 2000; Schödel et al., 2003; Ghez
et al., 2005). However, for cz ≈ 10, 000 km/s SMBH estimates become unfeasible.
While a way to measure black hole masses in more distant quiescent galaxies has yet
to be devised, for type-1 AGN and quasars an alternative method already exists, the
’reverberation mapping’ (Blandford & McKee, 1982; Peterson, 1993). Note that type-1
AGN refer to a orientation based distinction, and here, the observer has a direct view
onto the nucleus (in contrast to type-2 AGN, which are observed through an obscuring
torus). In the current picture of black holes, the central black hole and the accretion
disk are surrounded by a dusty torus and one distinguishes between broad-line region
(BLR) close to the black hole and a narrow-line regions (NLR) surrounding the torus.
Therefore, the difference for type-1 and type-2 AGN depends on whether the inner region, the BLR can be detected or not. The size of the BLR can be probed by the time
delay τ between variations in the ionizing continuum and in the flux of each velocity
component of the line emissions: r ∼ cτ . Meanwhile, the emission line width reflects
the velocity dispersion of the emission-line clouds in the gravitational potential which
is dominated by the black hole. Thus, the mass can be derived from the virial theorem
(M• = f rσ 2 /G, where r is the radius, f is a dimensionless parameter depending on the
kinematics, the geometry and inclination of the AGN and σ is the mean velocity dispersion). Reverberation mass measurements are consistent with other mass estimates
within a factor of a few (Bentz et al., 2009).
2.3
Current picture of the joint evolution of galaxies and black holes
2.3.3
17
Current picture of co-evolving galaxies and black holes
Providing the pure existence of the supermassive black holes in galactic centers
powering AGN, is, however, only a first step towards a complete understanding of
galaxy formation. In a second step, observational evidence was found that the black
hole masses are tightly correlated to properties of their host galaxies. Kormendy &
Richstone (1995) were the first, who noticed that SMBHs correlate with the luminosity
of the surrounding stellar component. That is the bulge of spiral galaxies or the entire
galaxy in case of ellipticals. Further detections of SMBHs have confirmed the existence
of the M• − Mbulge -correlation (e.g. Magorrian et al., 1998; Ferrarese & Merritt, 2000;
McLure & Dunlop, 2001; Marconi & Hunt, 2003; Häring & Rix, 2004). The left panel of
Fig. 2.5 shows the relation between black hole mass and bulge mass (M• ∼ 10−3 Mbulge ).
Bulge magnitudes and velocity dispersions are connected through the Faber-Jackson
Figure 2.5: Left panel: relation between black hole and bulge mass in the present-day
universe according to Magorrian et al. (1998) Right panel: relation between black hole
mass and velocity dispersion in the present-day universe according to Tremaine et al.
(2002)
relation, what directly implies that black holes are also connected with the velocity
dispersion. Indeed, observational studies found that an even tighter correlation exists
between the black hole mass and the velocity dispersion of the spheroid (Gebhardt
et al., 2000; Ferrarese & Merritt, 2000; Tremaine et al., 2002). The right panel of Fig.
γ
2.5 shows the relation: M• ∝ σve
, with γ ∼ 3.7 − 4.8. Moreover, in a very recent study,
Burkert & Tremaine (2010) showed that there exists even a tight correlation between
black hole mass and the number of globular clusters. Whether black holes are also correlated with the dark matter halo mass is subject of ongoing debate: Ferrarese (2002)
showed indirectly the existence of such a relation, however, in a very recent study of
Kormendy & Bender (2011) they claim that there is no correlation between black hole
18
A framework for galaxy formation and black hole growth
and halo mass, at least at the low mass end. Nevertheless, these observations strongly
suggest that the formation of SMBHs is tightly linked with that of the (spheroidal
component of) its host galaxy. That means the growth of black holes might not only
be understood in the general framework of galaxy formation (where the evolutionary
state of a galaxy governs black hole growth), but black holes themselves can also have
significant impact on the formation and evolution of their host galaxies.
In the following, a few basic equations, which are important for describing black hole
growth, will be given: An important approximation for an upper limit of the accretion
rate onto a black hole of mass M• is resulting from a simple spherical model with
a central source of a certain luminosity surrounded by gas. Considering the balance
between an (outwards directed) radiation pressure and the (inwards directed) force of
gravity defines a maximum luminosity, the Eddington luminosity Ledd . The gravity
force is given by:
Fgrav =
GM• mp
,
r2
(2.23)
and the force of radiation pressure can be expressed with:
Frad = σT
L
,
4πr 2 c
(2.24)
where mp is the proton mass and σT is the Thomson cross-section. With the condition
Frad = Fgrav and solving for the luminosity, one obtains (Mo et al., 2010):
#
$
4πGM• mp c
M•
46
Ledd =
≈ 1.3 × 10
erg/s
(2.25)
σT
108 M"
Above the Eddington luminosity, the source is unable to maintain steady spherical
accretion. The Eddington luminosity corresponds to a mass accretion rate, which is
the highest possible accretion rate within the simple spherical model:
Ṁedd =
Ledd
,
0c2
(2.26)
where 0 is the radiative efficiency at which the mass of accreted material is converted
into radiation. The Eddington ratio fedd is defined as the ratio of the actual luminosity
of the AGN and the Eddington luminosity (the largest possible luminosity for given
black hole mass):
fedd :=
L
Ledd
(2.27)
Observationally, bright Quasars and Seyfert galaxies are believed to be mostly connected with black holes accreting at large Eddington-ratios, whereas Radio galaxies
are mainly assumed to be linked to black holes accreting only at a small fraction of the
2.3
Current picture of the joint evolution of galaxies and black holes
19
Eddington rate.
An important growth channel for black holes is gas accretion (besides merger
events). In the picture of radiative accretion, the black hole growth rate Ṁ• (=
(1 − 0)Ṁacc , i.e. the fraction 0 of accreted mass is converted into radiation and escapes the black hole) is given by (Mo et al. (2010)):
Ṁ• =
1−0L
1 − 0 L M•
1−0
M•
=
=
fedd
,
2
0 c
0 Ledd tsalp
0
tsalp
(2.28)
where L is the Luminosity of the AGN and tsalp ∼ 4.5 × 108 yr is the Salpeter timescale.
Thus, the mass evolution of the black hole can be described by an exponential increase:
#
$
1−0
t
M• (t) = M0,• exp
fedd
(2.29)
0
tsalp
When black holes grow through gas accretion, a considerable amount of cold gas is
required in the host galaxy to be funneled into the center. Major mergers can be
possible scenarios for driving gas into the center (e.g. Di Matteo et al., 2005; Hopkins
et al., 2008c), and moreover, also bar instabilities in disks might lead to a significant
gas inflow. Another source for cold gas accretion might be provided by stellar mass
loss from evolved intermediate mass stars (Ciotti & Ostriker, 2007). One prediction of
the merger scenario is that bright AGN should preferentially be found in interacting
systems (what could not have been verified observationally at low redshifts). Moreover,
as starbursts are a further consequence of merger events, nuclear activity is often
associated with an increase in star formation. However, even if there is mainly hot gas
present in the host galaxy (in particular in massive, early-type galaxies at low redshifts),
a black hole can still accrete some gas through Bondi-accretion (Bondi, 1952). This is
a spherical accretion onto an object and occurs when the gravitational potential of the
black hole overcomes the specific thermal energy of the gas:
Ṁ• ∼ 4πrA2 ρA cs (rA )
(2.30)
where rA ∼ GM• /c2s is the accretion radius, cs is the sound speed of the hot gas and
ρA the gas density. In this phase, black holes are accreting extremely below the corresponding Eddington limit and thus, they are radiatively very inefficient.
An important consequence of the gas accretion onto the black hole is, that AGN
can release a huge amount of energy during their lifetime. The power of its energy
output may be expressed as
dE
= (0r + 0m )Ṁ• c2 ,
(2.31)
dt
where 0r and 0m is the radiative and mechanical efficiency, respectively. Comparing
the energetic feedback from an AGN to the binding energy of its host galaxy reveals
the significant impact of AGN feedback. Broadly speaking, there are three different
possible processes which can release energy from the AGN into its surrounding gas:
20
A framework for galaxy formation and black hole growth
• Radiative processes: Interaction of photons with gas particles
• Mechanical processes: Particle-particle interaction
• Energetic particles (cosmic rays)
Radiative feedback is mainly connected with high-accretion phases and one distinguishes between radiation pressure (momentum-driven winds) and radiative heating
(energy transfer). In contrast, mechanical feedback is mainly believed to occur in
low-accretion AGN in form of radio jets or lobes releasing kinetic energy. Again one
distinguishes between momentum-driven winds and energetic transfer. Through gas
heating and expelling the gas out of the host galaxy, AGN feedback has a profound
influence on the further evolution of the galaxy: cooling might be suppressed and thus,
further star formation might be quenched. Additionally, also further gas accretion onto
the black hole might be quenched leading to a self-regulation in black hole growth.
Unanswered, open questions
However, in this picture, there still exists a number of unanswered questions about
how the co-evolution of black holes and galaxies actually works. E.g. How does the
gas get into galaxies and how is it transported afterwards to the inner accretion disk
surrounding the black hole? How strong is the effect of the AGN feedback onto the host
galaxy evolution? Is the AGN phase only a once-a-lifetime event in the life of a galaxy
or is it more an intermittent process? What is the origin of the black hole mass scaling
relations and do they show a redshift evolution? What is the origin of the observed
anti-hierarchical black hole growth? In this thesis, I will especially focus on the latter
two ’mysteries’.
Currently, there exist different ways for explaining the emergence of the presentday black hole scaling relations. One possibility is that they occur due to a connected
growth of black holes and galaxies, as outlined above, as e.g. rapid cold gas accretion during gas-rich major mergers (e.g. Naab et al., 2006b; Robertson et al., 2006a;
Hopkins et al., 2008c; Johansson et al., 2009b) and/or AGN feedback for regulating
black hole growth and star formation (Granato et al., 2004; Di Matteo et al., 2005;
Croton, 2006). However, it is still not understood which of the black hole mass scaling
relations is the fundamental one. Hopkins et al. (2007c) claim that there is a “black
hole fundamental plane” relation between black hole mass, the galaxy effective radius,
the dynamical mass and the velocity dispersion. In contrast, a complete alternative
attempt for explaining the black hole mass scaling relations is purely based on statistical merging of objects without any need for a physically coupled growth of black
holes and host galaxies. Peng (2007) and Jahnke & Maccio (2010) show that for merging scenarios (either based on random or cosmological merging of galaxies and black
holes) a relation between black hole and bulge emerges automatically even if the objects are totally uncorrelated in the beginning. This is mainly a consequence of the
2.3
Current picture of the joint evolution of galaxies and black holes
21
statistical Central-Limit-Theorem. Moreover, there are claims that the relationship
between galaxy mass and black hole mass may be evolving with cosmic time due to
observations of ’over-massive’ and ’under-massive’ black holes at high redshifts (Peng
et al., 2006; Salviander et al., 2007; Woo et al., 2008), i.e. black holes which are lying
above or below the present-day relation between black hole and bulge mass and even
outside the 1-σ-range of the present-day scatter. E.g. Hopkins et al. (2007c); Croton
(2006); Robertson et al. (2006a) explain this by a shifted relation towards larger black
hole masses at high redshifts compared to the present-day relation. In their scenarios,
this is again due to gas accretion processes and thus, due to the underlying connected
growth of galaxies and black holes. However, solely considering the statistics in merging scenarios might give a complete alternative explanation, what will be the subject
of investigations in Chapter 3.
Besides, the cosmic evolution of differently luminous AGN is a further subject of
current, intense debate. Observations reveal that bright quasars start to form already
at z > 6, have an activation peak at around z ∼ 2 − 3 and decline rapidly at lower
redshifts. Here the question emerges what determines such a redshift-dependence of
the activation rate. As the number density of dark matter halos with Mhalo = 1012.5 M"
increases rapidly with time at z > 3, the increase of the quasar number density might
eventually be connected directly to the increase of massive halos. At z < 3 on the
other hand, the number density of dark matter halos keeps increasing constantly with
increasing time, implying that other processes might be necessary. If quasar activity
is mainly triggered by galaxy interactions, the decline of the quasar number density
might reflect the decline of the major merger rate. In addition to the drop of the
merger rate further physical processes as the availability of cold gas in the host galaxy
as well as the AGN feedback processes might state other plausible effects in order to
reduce the activity in bright quasars at low redshifts. Interestingly, less luminous AGN
are observationally found to peak at smaller redshift compared to bright quasars. This
indicates the ’downsizing’ or ’anti-hierarchical’ trend, i.e. it seems as if more massive
black holes have been in place already very early in the cosmic history whereas less
massive black holes seem to preferentially form only at lower redshift. At first sight,
this seems to be in clear contrast to currently favored hierarchical clustering scenarios.
Possible mechanisms, which may account for the observed downsizing trend, will be
discussed in Chapter 4.
2.3.4
Modeling galaxy formation and black hole growth
In order to model galaxy formation and black hole growth, there exist currently different
methods, which are all based on the large-scale structure formation of cold dark matter
either modeled using the analytical EPS theory or numerical N-body simulations (see
Section 2.2):
1. Analytical methods, e.g. halo occupation distribution models (HOD) (see Section
3.4.3)
22
A framework for galaxy formation and black hole growth
2. Semi-analytical models (see Section 4.3.2)
3. Hydrodynamical simulations (see Section 5.3)
In halo occupation distribution models, dark matter halos at a given mass and
redshift are populated with a number of galaxies with a certain mass/luminosity. The
dark matter halo distributions are calculated either analytically using the EPS theory
or numerically with N-body simulations. They are connected with observed luminosity/galaxy mass functions (e.g. van den Bosch et al. (2003, 2007); Mandelbaum et al.
(2006); Wechsler et al. (2006); Zheng et al. (2007); Conroy & Wechsler (2009); Guo &
White (2009); Moster et al. (2010); Zehavi et al. (2010); Behroozi et al. (2010); Wake
et al. (2011)). Formally, the connection between the luminosity function Φ(L, z) and
the dark matter halo distribution n(M, z) can be written as:
'
Φ(L, z)dL = dL Φ(L|M, z)n(M, z)dM,
(2.32)
where Φ(L|M, z)dL is the conditional luminosity function at z, which specifies the
average number of of galaxies with luminosities in the range L ± dL/2, that reside
in a halo of mass M at redshift z. The simplest assumption about the galaxy-halo
relation is that the total luminosity of a galaxy is directly proportional to its halo mass
M. This, however, results in a wrong shape of the luminosity function compared to
the observed one as at the low mass end supernova winds and at the high mass AGN
feedback become important, non-negligible processes for suppressing cooling and thus,
for quenching star formation.
However, in contrast to halo occupation models, semi-analytic models and hydrodynamical simulations represent a more thorough approach in an ’ab initio’ way for
modeling galaxy formation. In semi-analytical models (e.g. White & Frenk, 1991;
Kauffmann et al., 1993; Cole et al., 1994; Somerville & Primack, 1999b; Kauffmann
et al., 1999; Cole et al., 2000; Springel et al., 2001a; Hatton et al., 2003; Kang et al.,
2005; Baugh et al., 2005; Khochfar & Silk, 2006a; Croton, 2006; Bower et al., 2006;
De Lucia & Blaizot, 2007; Somerville et al., 2008b; Font et al., 2008; Guo & White,
2009; Weinmann et al., 2009), complicated and tightly connected physical processes
for the formation of galaxies are modeled as a set of simplified recipes and prescription
assuming a spherical halo geometry and carrying a large number of free parameters.
Within a dark matter halo, one distinguishes between three different baryonic components, cold disk gas, hot halo gas and stars assuming simple prescriptions for conversion
rates between these baryonic components. With a stellar population synthesis model,
star-formation history and metallicity can be converted into luminosity and color of the
stellar population. Dynamical friction calculations are important for the treatment of
merger processes. The free parameters in semi-analytic models are mainly set in order
to reproduce observations of the present-day Universe. However, partly they can also
be motivated by detailed hydrodynamical simulations. The main advantage of semianalytic modeling is that one can easily generate large samples of model galaxies, what
2.3
Current picture of the joint evolution of galaxies and black holes
23
is especially useful for investigating statistical properties of galaxies. Semi-analytic
models have been successful in reproducing the luminosity function, the Tully-Fisher
and Faber-Jackson relations, star formation histories, morphology, and color distributions. Thus, despite of some unresolved problems (e.g. over-prediction of the fraction
of red satellite galaxies), the overall agreement between predictions of semi-analytic
and observations is encouraging and suggest that the main physical processes are considered. However, as a number of physical processes are still poorly understood and
are thus not treated accurately from first principles, many recipes are too simplified
and thus, the semi-analytic models are far from being complete (see Mo et al., 2010).
In contrast, in cosmological hydrodynamical simulations, the evolution of the initial density field is followed explicitly by solving the gravitational and hydrodynamical
equations (see Equations 2.14, 2.17 and 2.18) numerically. This can be sampled either using a large spatial grid (Eulerian approach) or by a large number of particles
(Lagrangian approach). Note that in this work the latter approach will be used, the
smoothed particles hydrodynamics (SPH) method. As the technique is particle based,
it can follow the motion of individual mass elements (coordinates are comoving with
the fluid element) (Springel et al., 2001a). The main advantage of hydrodynamical
simulations compared to semi-analytics is that they can follow the evolution of gas
without relying on simplified approximations in a more self-consistent manner. However, simulations are still limited by numerical resolution, why some of the processes
have to be modeled on a subgrid level (e.g. Cen & Ostriker, 1993; Davé et al., 2001;
Springel & Hernquist, 2003; Maller & Bullock, 2004; Nagamine et al., 2005; Kereš
et al., 2005; Navarro et al., 2009; Schaye et al., 2010a), with recipes not so different
from semi-analytics. But it is still difficult with current computational power to have
both large statistics (large cosmological box) and a high resolution. In recent years,
many different implementations for star formation (e.g. the multi-phase model for the
interstellar medium as proposed by Springel & Hernquist, 2003), feedback from supernovae, supernova-driven winds as well as AGN feedback have been proposed (e.g. Di
Matteo et al., 2005; Cattaneo et al., 2005; Sijacki et al., 2007; Oppenheimer & Davé,
2008; Booth & Schaye, 2009; Scannapieco et al., 2009; Governato et al., 2010; Schaye
et al., 2010b; Sawala et al., 2010). For example, it has been demonstrated that star
formation can effectively be quenched after an initial burst (due to a merger event)
by AGN/SN driven outflows, producing massive red galaxies. One can conclude that
hydrodynamical simulations provide a very promising way to study galaxy formation
self-consistently in a cosmological context. However, numerical resolution is clearly
insufficient and thus, many processes are still implemented in a very crude and ad-hoc
way (see Mo et al., 2010).
As both methods, SAMs and hydrodynamical simulations, have their pros and
cons, it is of utmost importance to compare directly how galaxies form in these different approaches. It might be extremely helpful in order to reveal limitations of both
approaches and in particular to test often very simplified recipes assumed for galaxy
24
A framework for galaxy formation and black hole growth
formation processes in SAMs. The investigation of the robustness and reliability of
semi-analytic models will be the main topic in Chapter 5.
Chapter
3
The intrinsic scatter in black
hole mass scaling relations
In this Chapter, I present results on the evolution of the intrinsic scatter of
black hole masses considering different implementations of a model in which
black holes only grow via mergers. It is demonstrated how merger driven
growth affects the correlations between black hole mass and host bulge mass.
The simple case of an initially lognormal distributed scatter in black hole and
bulge masses combined with random merging within the galaxy population
results in a decreasing scatter with merging generation/number as predicted
by the central-limit theorem. In general, it is found that the decrease in scatter
σ is well approximated by σmerg (m) ≈ σini × (m + 1)−a/2 with a = 0.42 for a
range of mean number of mergers m < 50. For a large mean number of mergers
(m > 100), there is a convergence to a = 0.61. This is valid for a wide range
of different initial distributions, refill-scenarios or merger mass ratios. Growth
scenarios based on halo merger trees of cosmological N-body simulations show
a similar behavior with a scatter decrease of a = 0.30 with typical number
of mergers m < 50 consistent with random merging (best matching model:
a = 0.34). Assuming a present-day scatter of 0.3 dex in black hole mass and
a mean number of mergers not exceeding m = 50 the results imply a scatter
of 0.6 dex at z = 3 and thus, a possible scenario in which over-massive (and
under-massive) black holes at high redshift are the consequence of a larger
intrinsic scatter in black hole mass. A simple toy model connecting the growth
of black holes to the growth of haloes via mergers, neglecting any contribution
from accretion, yields a consistent M• − MBulge relation at z = 0, if the correct
initial relation is assumed. This study is published by MNRAS (Hirschmann
et al., 2010)
26
3.1
The intrinsic scatter in black hole mass scaling relations
Black hole mass relations in the present-day universe
Observationally, there exists a strong correlation between black hole masses and properties of the host galaxy, e.g. the host bulge mass, the bulge velocity dispersion and the
number of globular clusters (Häring & Rix, 2004, Ferrarese & Merritt, 2000, Gebhardt
et al., 2000, Tremaine et al., 2002, Gültekin et al., 2009, Burkert & Tremaine, 2010)
and possibly the host halo (Ferrarese, 2002) in nearby galaxies. Häring & Rix (2004)
find the relation between black hole mass M• and bulge mass Mbulge to be:
log(M• /M" ) = 8.20 + 1.12 × log(Mbulge /1011 M" ).
(3.1)
The correlation between black hole mass M• and velocity dispersion σ∗ can be written
as
log(M• /M" ) = a log(σ∗ /200kms−1 ) + b,
(3.2)
where a is the slope and b is the zero point. In the literature the values for the slope vary
between a = 3.68 and a = 4.86. For example Ferrarese & Merritt (2000) give values for
the zero point b = −2.9 and the slope a = 4.80. Gebhardt et al. (2000) claim a smaller
slope of a = 3.75, Tremaine et al. (2002) find a = 4.02 and more recently Ferrarese &
Ford (2005) estimated a slope of 4.86 and Graham (2008b) one of 3.68 for barless galaxies. Concerning the intrinsic scatter σ in black-hole mass, log M• , most studies agree
that the scatter is not larger than 0.3 dex (Gebhardt et al., 2000; Tremaine et al., 2002;
Novak et al., 2006; Graham, 2008b). Note, that with σ∗ I describe the central velocity
dispersion of a galaxy, whereas the intrinsic scatter is characterized as σ. In contrast
to Gebhardt et al. (2000) and Tremaine et al. (2002), a recent study by Gültekin et al.
(2009) obtains for Eq. 3.2 a slope a = 4.24 and an intrinsic scatter in black hole mass
of σ = 0.44 dex for a sample of 49 M• -measurements. For a subsample of early-type
galaxies they find a smaller slope as well as a smaller scatter (a = 3.96, σ = 0.31 dex)
than for the full sample. For non-elliptical galaxies the intrinsic scatter is larger with
σ = 0.53 dex. Additionally, Graham (2008a) find a reduced intrinsic scatter by removing barred galaxies from their sample (σall = 0.27 dex, σbarless = 0.17 dex). In
the following the scatter σ will always be given in dex. The existence of these tight
correlations strongly suggests a co-evolution of the black hole and the bulge component
of the host galaxy. However, the origin of these relations is uncertain and a subject
of current research (e.g. Volonteri & Natarajan, 2009; Peng, 2007; Burkert & Silk,
2001; Springel et al., 2005a; Johansson et al., 2009b). Apparently, the origin of these
relations can be connected to the gas dynamics in major galaxy mergers (Mihos &
Hernquist, 1996, Naab et al., 2006a, Robertson et al., 2006b, Hopkins et al., 2008b). In
this scenario, the black holes grow significantly in gas rich mergers of disk galaxies and
the remnants appear on the observed scaling relations. Subsequent, possibly gas poor,
major and minor merging (Khochfar & Burkert, 2003, Naab et al., 2006b, Khochfar
& Silk, 2009, Naab et al., 2009) conserves the relation (Sesana et al., 2004, Robertson
et al., 2006b, Peng, 2007, Hopkins et al., 2007a, Hopkins et al., 2007b, Springel et al.,
3.2
Black hole mass relations at higher redshifts
27
Figure 3.1: Comparison of the intrinsic scatter from observations (Tremaine et al.,
2002, Gültekin et al., 2009) and theoretical predictions (Malbon et al., 2007). From the
10-90 percentile spread in the model the corresponding theoretical scatter was calculated.
The shaded areas illustrate the error on the observationally estimated intrinsic scatter.
2005a, Johansson et al., 2009b). Moreover, AGN feedback plays an important role in
regulating the black hole growth as well as influencing the host galaxy evolution by
quenching further cooling and star formation (Di Matteo et al., 2005; Croton, 2006;
Robertson et al., 2006a).
However, a completely different approach, in order to explain the origin of the
black hole scaling relations at redshift z = 0, was made by Peng (2007) and Jahnke
& Maccio (2010). In these studies they show that the emergence of a relation can be
derived from pure statistics in merging scenarios without directly connecting the growth
of black holes and the host galaxies by gas physical processes, as e.g. AGN feedback.
They show either in random merging (Peng, 2007) or in cosmological merging scenarios
(Jahnke & Maccio, 2010) that a relation between black hole and bulge mass can be
established even if the objects are completely uncorrelated in the beginning. They
claim this is mainly a consequence of the Central-limit-theorem and the corresponding
convergence against a normal distribution independent of the initial distribution.
3.2
Black hole mass relations at higher redshifts
Observationally the black hole mass relations are well constrained only in the nearby
Universe and it is unclear, if and how they evolve with cosmic time. Several authors
28
The intrinsic scatter in black hole mass scaling relations
have found evidence that galaxies at higher redshift have a higher black hole to bulge
mass ratio M• /Mbulge than ellipticals today (McLure et al., 2006; Treu et al., 2007;
Woo et al., 2008; Walter et al., 2004; Schramm et al., 2008; Peng et al., 2006; Greene
et al., 2010; Natarajan & Treister, 2009; Salviander et al., 2007; Shields et al., 2006).
For a sample of Seyfert galaxies at moderate redshifts z < 0.1 the black holes are more
massive by ∆ log M• ∼ 0.5 dex compared to the local black hole-bulge mass relation
(Treu et al., 2007, Woo et al., 2008). Salviander et al. (2007) find at redshift z ≈ 1 an
evolution of the M• -σ∗ -relation by 0.2 dex in black hole mass. At higher redshifts of
z ∼ 2 McLure et al. (2006) observe black holes 8 times more massive than expected and
Peng et al. (2006) show that for z ≥ 2 the M• /Mbulge -ratio is 3 − 6 times larger than today. This has been confirmed by Greene et al. (2010) based on a lensed quasar sample.
Schramm et al. (2008) find evidence for an excess in M• /Mbulge at z ∼ 3 of a factor of
∼ 10. At redshifts 4 < z < 6 Shields et al. (2006) obtain for black hole masses in the
range of 8 < log(M• /M" ) < 10 a deviation from the present-day M• -Mbulge -relation
of ∆ log(M• ) ∼ 2 dex. Walter et al. (2004) report an even higher redshift object, a
quasar at z = 6, whose black hole is about 20 times more massive than expected.
Considering a present-day scatter according to Tremaine et al. (2002) (σ = 0.3) or
according to Gültekin et al. (2009) (σ = 0.31 for ellipticals) all observed black holes
at z ≥ 2 are outside the 2 − σ range of the present-day scatter. Furthermore, recent
observations (Alexander et al., 2008; Shapiro et al., 2009) show also the existence of
under-massive black holes at high redshifts. Alexander et al. (2008) find black hole
masses, which are 3 times smaller than those found in comparable massive galaxies in
the local Universe. The results in Shapiro et al. (2009) show that black hole masses
at z = 2 are an order of magnitude lower than those predicted by local scaling relations.
The most obvious explanation for the over-massive black holes at high redshifts are
possible selection effects. It is more likely to detect the most luminous and most massive black holes at high redshift than less luminous ones. However, the probability for
finding a massive black hole in the mass range 109 − 1010 M" in the observed volume at
z = 3 (e.g. Schramm et al., 2008) is extremely low as estimates of the local SMBH mass
function from SDSS (Benson et al., 2007) would predict no high mass black holes to
be found in a similar volume even assuming no evolution in the SMBH mass function.
Cosmic variance is very unlikely to be an explanation for the observed, massive black
holes. Lauer et al. (2007) point out that there is an additional bias which is due to different selection effects for high-redshift (e.g. black holes in high-z galaxies selected by
nuclear activity) and local samples (e.g. black holes in local galaxies selected by luminosity or velocity dispersion). They deduce that because of this bias M• will typically
appear to be too large in a distant sample for a given luminosity or velocity dispersion. Some authors (e.g. Croton, 2006) explain observed, over-massive black holes at
high redshift with a shifted relation towards higher black hole masses for a given bulge
mass. He uses the Millennium ΛCDM-simulation (Springel et al., 2001b) coupled with
a model of galaxy formation, where galaxy mergers are the primary drivers for black
hole and galaxy growth and explore an additional growth channel through which only
3.2
Black hole mass relations at higher redshifts
29
bulges gain mass, e.g. the disruption of stellar galactic disks in major disk mergers.
He argues, that if the bulge growth rate from such disrupted disks is not constant
with time, an evolution in the M• -Mbulge -relation can occur. Furthermore, Robertson
et al. (2006c) find from simulations of galaxy mergers, that the M• -σ∗ -relation shows
a slight redshift evolution towards higher black hole masses for a given σ∗ at higher
redshifts, but they predict no evolution for the M• -Mbulge -relation. However, Hopkins
et al. (2007c) using again simulations of major galaxy mergers show that high redshift black holes will be more massive at a fixed bulge mass than expected from the
present-day relation. They find an evolution towards lower black hole to bulge mass
ratios with cosmic time which is driven by the fact that disks (merger progenitors)
have characteristically larger gas fractions at high redshifts. It should be pointed out
that these studies (Croton, 2006; Robertson et al., 2006c; Hopkins et al., 2007c) are
consistent with each other, they are only focusing on different aspects of the evolution
of the relation. However, these methods do not provide a sufficient explanation for the
observed under-massive black hole masses at high redshifts, since these methods find
to have black holes mainly lying above the median M• -Mbulge -relation at high redshift
evolving towards the relation with decreasing redshift. However, they also predict that
the black holes go through a rapid growth phase before they end up above the relation;
these objects might be under-massive. Furthermore, if a black hole at high z is already
above the median M• -Mbulge -relation it will not end below the median relation at z=0
if only assuming merging and not gas accretion. Therefore, an alternative explanation
for the observed high and low M• /Mbulge -ratios at high redshifts could be the existence of a larger intrinsic scatter in black hole mass, even assuming no evolution of
the mean relation with cosmic time, what would be in agreement with Lauer et al.
(2007). This would be consistent with a study from Shankar et al. (2010), where they
find empirically - using the Soltan argument and quasar clustering - that the scatter
in the L-M• -relation must be large at redshifts 0.4 < z < 2.5.
In this Chapter, the question is addressed: how does the intrinsic scatter in black
hole mass evolve and change with time assuming that black holes grow only via mergers? An answer to this question is also important with respect to similarities and
differences between the observed scatter of black hole masses and predictions from
theoretical models. Malbon et al. (2007), using semi-analytic modeling, find that the
present day scatter in black hole mass decreases significantly with increasing black hole
mass. This is in contrast to observations using the full samples of e.g. Gebhardt et al.
(2000), Tremaine et al. (2002), and Gültekin et al. (2009). Here, the scatter appears
to be independent of black hole mass. In particular at the high mass end the observed
scatter is much larger than the model predictions (see Fig. 3.1). However, Gültekin
et al. (2009) demonstrate that the scatter for non-elliptical galaxies (typically at lower
masses) is larger than for elliptical galaxies. This is, at least qualitatively, in agreement
with the predictions from Malbon et al. (2007).
So far the time evolution of the scatter in black hole mass has not been investi-
30
The intrinsic scatter in black hole mass scaling relations
gated in detail. Peng (2007) deals with the evolution of the scatter, but he mainly
focuses on the aspect of how the present day M• -Mbulge -relation can form in a simple model applying random merging of galaxies. He claims that the relation develops
even if black holes and bulges are uncorrelated or wrongly correlated in the beginning.
This behavior is supposed to result from an initially exponentially decreasing SMBH
mass function where minor mergers drive the objects towards the observed correlation.
Furthermore, the scatter in black hole mass decreases with increasing merger number,
and according to his results the decrease in scatter is dominated by major mergers.
However, a quantitative study of the scatter evolution was not presented, which is the
main subject of this Chapter. A further major difference between this work and that
of Peng (2007) is that - besides Monte-Carlo generated random merging scenarios merging as it is found for dark matter haloes in large scale cosmological simulations is
included.
In the following, the evolution of the intrinsic scatter is investigated assuming:
• Simple random merging (Section 3.3.2)
• Modified random merging (Section 3.3.3)
• Merging in ΛCDM-simulations (Section 3.4)
It should be pointed out that random merging does not describe a full physical evolution
process according to currently favored structure formation models. In principle, the
model follows dry merging of galaxies and therefore is limited to high mass galaxies,
since merging at the high mass end is assumed to be almost dry, so that gas physics
and star formation do not play an important role and can be neglected in the growth
processes. However, an advantage of using a simple model such as random merging
is that separate effects on the scatter evolution can be studied, e.g. the influence
of the initial mass distribution, of the merger mass-ratio or different refill-scenarios.
Then these results are compared to merging according to currently favored structure
formation models based on dark matter ΛCDM-simulations. Since it is known that a
significant contribution to black hole growth is caused by accretion, this issue will be
discussed in Section 5.7.
3.3
3.3.1
Models for random merging
Initial conditions
In the following, two different initial distributions of bulges are described including
black holes as a starting point for random merging: a log-normal distribution and a
Schechter distribution of bulges.
3.3
Models for random merging
31
Figure 3.2: Normalized 2D-histogram of the initial distribution with the same, lognormal distributed scatter (σini = 0.6) in bulge and black hole mass. The observed
relation is shown by the black, solid line. The black dotted lines indicate the 1-σ range
of the initially applied scatter in the model.
Figure 3.3: Initial distribution based on the Schechter-fit for early-type galaxies at
z ∼ 2. An initial scatter is only applied to the black hole mass (σini = 0.6)
32
The intrinsic scatter in black hole mass scaling relations
Initial log-normal distribution
The initial log-normal distribution was constructed by taking a uniform distribution
(in the log) of bulge masses with black hole masses according to Häring & Rix (2004).
Then I applied to the bulge as well as black hole masses a log-normal distributed scatter
with a value of σ = 0.5. Having set this scatter to bulge as well as to black hole masses
results in a larger scatter in black hole and bulge masses with a value of ∼ 0.6. This
scatter is larger than the observed, present-day one specified by Tremaine et al. (2002)
and Gültekin et al. (2009). I have chosen this fiducial value a posteriori as in the most
realistic ΛCDM-simulation this results in a final observed scatter value of σ ∼ 0.32 (see
Section 3.4). However, the scatter evolution is independent of the choice of the initial
scatter value (see Section 3.3.2). I start with an initial distribution of 580, 000 bulges
with black holes. The black hole masses range from 2.0 < log(M• /M" ) < 8.0 with a
mean of 'log(M• /M" )( = 5.0 and the bulges have masses of 4.7 < log(Mbulge /M" ) <
11.1 with a mean of 'log(Mbulge /M" )( = 7.9. The resulting distribution of bulges
including black holes is depicted in the 2-D histogram in Fig. 3.2. In this plot the
number of objects Nobjects is normalized to the maximum number of objects Nmax
found in a black hole-bulge mass bin. The observed M• -Mbulge -relation according to
Häring & Rix (2004) is plotted together with the 1-σ-range of the applied scatter (solid
and dashed black lines).
Initial Schechter distribution
Observationally, it is known that the mass function of bulges follows a Schechter function (e.g. Bell et al., 2003) rather than a log-normal distribution. Therefore, the scatter
evolution of a Schechter-shaped initial distribution of bulge masses is studied. I use a
fit to the measured luminosity function (K-band magnitude) for red galaxies at redshift
z ∼ 2 according to Cirasuolo et al. (2007) to construct the initial galaxy sample,
Φ(Mk ) = 0.4 ln (10) · Φ∗ · 10−0.4∆Mk (α+1) · exp (10−0.4∆Mk ),
(3.3)
with the fitting parameters
α = −0.1
Mk∗ = −23.04
Φ(10−3 Mpc−3 ) = 0.2
where Mk are the absolute magnitudes in K-band. The luminosity function is converted
into a mass function using the mass-to-light ratios as function of the K-band magnitude
according to Cappellari et al., 2006:
#
$
M
Lk
= 1.88 ·
with
(3.4)
L
1010 · Lk,"
Mk = −2.5 log Lk + 3.28
3.3
Models for random merging
33
These mass-to-light ratios were measured for a population of ellipticals at z = 0 and for
simplicity no evolution with redshift is assumed. A consequence of this assumption is,
that I obtain quite massive galaxies at z = 2, although the stellar population was not
evolved completely at this time. Probably, the mass-to-light ratio was smaller at higher
z than the present-day value leading to smaller galaxy masses. However, for this work
it is sufficient to see the evolution of statistical properties, which are independent of the
exact choice of the mass-to-light ratio. The resulting mass distribution is scaled to a
volume of (500Mpc)3 and the evolution of ∼ 100, 000 bulges with black holes is studied,
considering only bulge masses larger than 1.6 × 108 M" (=
( log(Mbulge /M" ) > 8.2). To
keep the Schechter-distribution for the bulge masses, a log-normal scatter of σ = 0.6
was only applied to the black hole masses (see Fig. 3.3).
3.3.2
Depletion scenario
For the fiducial random merging scenario (depletion case, i.e. without refilling the
initial distribution with new galaxies), either the initial log-normal or Schechter distribution is used as described above. From the initial pool two objects are selected
randomly, are merged by adding their black hole and bulge masses and the merged
object is put back into the pool. In the next step, again, two objects are merged
randomly, but now from the new rearranged pool. This procedure is repeated iteratively until, on average, every object has had one merger, i.e. only half of the initial
objects are left over. At this point one merging generation is defined to be completed.
Then all remaining objects are considered as the initial pool for the next generation.
Therefore, after the first generation, N(1) = Nini /2 objects are remaining and after
the n-th generation the pool is reduced to N(n) = Nini /2n objects. Note that in one
generation some objects can have merged several times while others have not merged
at all. Here and in the following, if not stated otherwise, the number of mergers is
defined by counting all mergers that occur for galaxies > 104.7 M" independent of their
mass ratio.
Evolution of the black hole-bulge mass relation
If the galaxies are merged randomly from the initial log-normal or Schechter distribution in Fig. 3.2 and 3.3, an important consequence of the model is, as already pointed
out by Peng (2007), that the sample behaves according to the central-limit-theorem
(CLT). This theorem states that the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately
normally distributed. Therefore, for this case the theorem predicts that, independent
of the initial distribution, the resulting distribution always converges towards a Gaussian distribution. This trend can already be seen after only one merging generation.
In Fig. 3.4, the evolution in the black hole-bulge mass plane of the log-normal distributed sample for merger generations n = 1 − 8 is shown. Again, the black solid line
34
The intrinsic scatter in black hole mass scaling relations
Figure 3.4: Normalized 2D-histograms for the random merging generations 1 − 8 on
the basis of an initial log-normal distribution, as shown in Fig. 3.2 (depletion case).
The fit to the observed relation is illustrated by the black, solid line; the black dotted
lines show the 1 − σ range of the initially applied scatter.
Figure 3.5: Same as Fig. 3.4, but for random merging with an initial Schechterdistribution resembling z ∼ 2 red galaxies.
3.3
Models for random merging
35
shows the observed, present day, M• -Mbulge -relation with the 1-σ range of the assumed
σini = 0.6 initial scatter. The relation is conserved during all merging generations.
Note, however, that here the same initial scatter in black hole and bulge masses is
used. In addition, the overall scatter decreases significantly with increasing merger
generation. The low mass end of the distribution is depleted by merging whereas the
high mass end is populated. Moreover, there is a clear trend that the scatter decreases
more for more massive black holes and bulges than low mass systems.
The evolution of a randomly merging initial Schechter-distribution of bulges is
shown in Fig. 3.5. In this case the overall slope increases for small merger generations. For high merger generations, when the low mass end is depopulated, the slope
again becomes similar to the initial slope and the relation is shifted towards larger black
hole masses. The reason for the change in tilt as well as the shift is the different initial
scatter in black hole and bulge masses. The shift towards larger black hole masses
shows that having such initial conditions of a larger scatter in black hole masses than
in bulge masses is quite unlikely in order to explain the over-massive black holes at
high redshift. However, qualitatively the scatter evolution is similar to the previous
case. The scatter decreases with increasing merging generation and a quantitive scatter estimate is presented in the next Section 3.3.2. Note that the initially Schechter
distributed bulges evolve into a log-normal distribution for massive systems as a consequence of the CLT. Still, this might not be a bad approximation for massive galaxies
as, in principle, the Schechter distribution is a superposition of multiple Gaussians
(Blanton et al., 2003).
Quantifying the scatter in the black hole mass relation
The scatter is characterized as the σ in a log-normal-like distribution:
(x−µ)2
1
f (x) = √ · e− 2σ2
σ 2π
with x = log(M• ) and µ = 'log(M• )(.
(3.5)
Note that here the logarithm in base 10 ’log’ is used instead of the natural logarithm
’ln’. This, however, only changes the normalization. The ’log(M)’ representation is
chosen to be consistent with the observations (e.g. Tremaine et al., 2002, Gültekin
et al., 2009). To estimate the scatter σ for the black hole mass in the evolution of the
M• -Mbulge -relation (Fig. 3.4) I use the following method. For each merging generation
the bulges are divided into different mass bins. Then for each bin black hole mass histograms are constructed which resemble normal distributions which are fitted with Eq.
3.5 to derive the scatter σ. This method is consistent with the scatter determination in
observations (Gültekin et al., 2009). The fit is performed with a Lebenberg-Marquardtalgorithm which interpolates between the Gauss-Newton algorithm and the method of
gradient descent and searches iteratively for the best fit.
36
The intrinsic scatter in black hole mass scaling relations
Figure 3.6: Scatter σ vs. mean black hole mass 'log(M• /M" )( per bin for different
merging generations n in the random merging model (depletion case, initial log-normal
distribution). n = 0 is the initial distribution. The average values of σ for black hole
masses higher than 105 M" within one merging generation are shown by dotted lines.
This shows a continuous decline in scatter with merger generation.
Figure 3.7: Black hole mass 'log(M• /M" )( as a function of the mean number of
merger 'Nmerg ( for different merging generations n (depletion case, initial log-normal
distribution). For higher generations n, log'Nmerg ( correlates with 'log(M• /M" )(. The
linear fit is shown by the dashed, red line.
3.3
37
Models for random merging
Fig. 3.6 shows the scatter σ as a function of mean black hole mass 'log(M• )( per
bin for the initial log-normal distribution (n = 0) and the eight subsequent merging generations (n = 1...8) indicated by different colors. For the initial distribution
the scatter on average is σ = 0.6. However, it is not constant with mass as a lognormal distributed scatter has been applied to the bulge masses as well as the black
hole masses. Higher merger generations show a decreasing average scatter for black
hole masses larger than 105 M" (indicated by the dashed lines) as well as a decrease in
scatter for larger black hole masses within one merging generation. This indicates a
strong correlation between the scatter, the black hole mass, and the merging generation.
In Fig. 3.7 the dependence of the mean number of mergers 'Nmerg ( on the black
hole mass 'log(M• )( is shown for the 8 different merging generations indicated by the
colored lines. At the high mass end and for merger generations n ≥ 7 there is a
convergence towards a linear relation between black hole mass and mean number of
mergers. The linear relation in Fig. 3.7 (red, dashed line) is given by:
log'Nmerg ( = a · 'log(M• /M" )( + b
with a = 0.93 and b = −5.32.
(3.6)
This relation suggests that after several merging generations n, it can be predicted, how
many mergers a black hole of a certain mass must have experienced on average. E.g. a
typical supermassive black hole of 108 M" had about 100 mergers, taking into account
all mergers which an object has had during its evolution (i.e. not only mergers in the
main branch, but all progenitors in the tree since the first merging generation). Most
importantly, in Fig. 3.8 the scatter σ is plotted versus the mean number of mergers
'Nmerg ( within one bulge mass bin for the different merging generations (indicated
by different colors). The decrease in the scatter with increasing merger number is
an important consequence of the CLT. Hence, an analytic expression for the scatter
decrease can be derived from the CLT for an initial normal distribution as a function
of the merging generation n (Peng, 2007):
σmerg (n) ≈ σini · 2−n/2 ,
(3.7)
where n is the generation number and σini is the initial scatter applied to black hole
and bulge masses. The mean number of mergers m of objects within one merging
generation as a function of the generation number n can be written as
m = 2n − 1,
(3.8)
Note, that for the calculation of the mean number of mergers all merger events are
considered, which galaxies had undergone until this merging generation. With help of
Eq. 3.8, Eq. 3.7 can be rewritten to get the scatter σ as a function of the mean number
of mergers m:
σmerg (m) ≈ σini · (m + 1)−1/2 .
(3.9)
38
The intrinsic scatter in black hole mass scaling relations
Figure 3.8: Scatter log(σ) vs. mean number of mergers log('Nmerg () for different
merging generations n (depletion case, initial log-normal distribution). The black
dashed line shows the analytic solution according to the CLT for an initial normal
distribution. The blue dashed line is a fit to the scatter for 'Nmerg ( > 100, the blue one
for 'Nmerg ( < 10.
Figure 3.9: Same as in Fig. 3.8, but based on the initial Schechter-distribution at
z ∼ 2.
3.3
Models for random merging
39
This analytic expression is depicted in Fig. 3.8 by the black dashed line. Thereby
the assumption has beed made that m (= mean number of mergers per generation)
≈ 'Nmerg ( (= mean number of mergers per bulge mass bin for one generation), which
is a good approximation especially for high-mass objects. Note, that therefore it will
not be distinguished between m and 'Nmerg (. In comparison to merger results, the
analytic solution exhibits a stronger decrease in scatter. This is due to the fact that
the CLT makes predictions for the sum of independent random numbers, while, in
this study, the pool is continuously changed by removing the merging objects and by
adding the merged object. This means that the added black hole and bulge masses are
not completely independent anymore leading to a violation of the CLT principle. Note
that for merging using an unchanged pool, the scatter decrease of the distribution of
merged objects does exactly follow the CLT prediction, resulting in a stronger scatter
decrease. The red and blue dashed lines in Fig. 3.8 are a fit to the random merging data
assuming a fitting formula similar to Eq. 3.9 with the exponent a as a free parameter,
σmerg (m) ≈ σini · (m + 1)−a/2 .
(3.10)
For the mass range of convergence, ∼ m > 100, a = 0.61 ± 0.02 (blue dashed line). The
exponent a can therefore be used as a measure for the strength of the scatter decrease.
For small merger numbers (0 < m < 20) a weaker scatter decrease is obtained with a
value of a = 0.42 ± 0.02 (red dashed line). This qualitatively different behavior of the
scatter decrease depending on the merger number range will be explained in Section
3.3.2. If the initial scatter in black hole and bulge mass is varied, the same strength
of scatter decrease is obtained within the errors, i.e. a remains unchanged. This was
tested for two different initial scatter values σini = 0.40 and 0.83. Even if the value
to which the scatter converges varies, the strength of the scatter decrease is the same
(a ∼ 0.60 ± 0.02) in the limit of large m.
In Fig. 3.9 the same scatter quantification is presented for a more realistic initial
Schechter distribution. Assuming convergence for ∼ m > 50 or the region with merger
numbers between 0 < m < 10, a similar value for the exponent as for an initial lognormal distribution is obtained (a = 0.61 ± 0.02 or a = 0.42 ± 0.02, see Eq. 3.10).
This indicates that the strength of the scatter decrease is only weakly dependent on
the exact choice of the initial distribution. Again, varying the initial scatter in black
hole and bulge mass does not influence the value a in the exponent for large m.
Difference between major and minor mergers
According to Peng (2007) there is a difference for objects with only major or only minor
mergers. He claims that major mergers exhibit a stronger central-limit tendency leading
to a stronger decrease of the scatter whereas minor mergers are mainly responsible
for evolving a linear relation between bulge and black hole masses even if they are
uncorrelated in the beginning. Going beyond the qualitative estimates in Peng (2007),
here, a quantitative analysis of the scatter evolution is presented for major and minor
40
The intrinsic scatter in black hole mass scaling relations
Figure 3.10: Same as Fig. 3.4, but galaxies had undergone only major mergers in
the random merging model (depletion case).
Figure 3.11: Same as Fig. 3.4, but galaxies had undergone only minor mergers in
the random merging model (depletion case).
3.3
Models for random merging
41
mergers. The following definitions are used:
Major merger: M1 /M2 ≤ 4, M1 > M2
Minor merger: M1 /M2 ≥ 10, M1 > M2
(3.11)
(3.12)
The definition for major mergers is consistent with Peng (2007). In Figs. 3.10 and 3.11
the evolution of the black hole-bulge mass relation is shown if only major or minor
galaxy mergers are allowed, respectively, for an initial log-normal distribution. In both
cases the relation is conserved, however, by construction, for minor mergers the low
and intermediate mass range is depleted. The corresponding scatter quantification is
shown in Fig. 3.12. Indeed, there is a difference between major and minor mergers; but
in contrast to the results of Peng (2007) in the depletion case, a stronger decrease of
the scatter for minor mergers (a = 0.66) is obtained than for major mergers (a = 0.39).
However, Peng (2007) never calculated the quantitative scatter evolution, as it is done
in this study, but showed for some exemplary objects, which had undergone many
major mergers, that they seem to lie closer to the black hole-bulge mass relation than
objects with less major mergers. This might explain the different statements of Peng’s
work and this one. This result, which is found in this chapter, at first glance seems to
contradict the expectations of the CLT. However, it is important to note that here the
change of the scatter of black hole masses is considered within individual bulge mass
bins, and not over the whole population of bulge masses. During a merger generation
a bulge mass bin has a constant influx and outflux of bulges due to mergers, which
modifies the scatter behavior with respect to the CLT. Here, the scatter behavior for
major and minor mergers was investigated in the over-all distribution of bulges. I find
that when forcing galaxies to undergo only major mergers the scatter even increases,
whereas for minor mergers, again, a scatter decrease is obtained as expected from the
CLT. This shows that major mergers are a very strong constraint leading to a strong
violation of the CLT principle and causing the slower scatter decrease in smaller mass
bins. With this behavior an explanation can be deduced for the weak scatter decrease
in the small merger number range and the stronger decrease in the limit of large merger
number: the probability for having minor mergers is higher at the high mass end than
for the low mass end, where major mergers dominate.
3.3.3
Replenishment scenario
From observations as well as from simulations it is known that new galaxies form during the structure formation process. To make the simple model more realistic different
replenishment scenarios are now considered. Again, a certain initial number of objects
Nini is assumed with a log-normal or a Schechter distribution and the same iterative
random merging procedure is performed as described in Section 3.3.2. However, the
pool is now refilled after merger events with new objects from an external unchanged
reservoir.The refill-ratio Nnew /Nevent is defined to be the number of objects added from
the refill pool Nnew per number of merger events Nevent within one merging generation. The definition of one merging generation is the same as before but after the n-th
42
The intrinsic scatter in black hole mass scaling relations
Figure 3.12: Scatter evolution for the depletion case as a function of mean number of
merger for galaxies, which had either only major (dotted-dashed lines) or only minor
mergers (dashed lines).
merging generation the sample always contains more than N(n) = Nini /2n objects
depending on the refill-ratio Nnew /Nevent . At first, a refill-ratio of Nnew /Nevent = 1 is
assumed, i.e. for each merger event one new object is added randomly from the refill
pool and the total number of objects in the sample stays constant. Moreover, also
a refill-ratio of Nnew /Nevent = 1/3 is considered, i.e. one new object for every three
events, motivated by ΛCDM simulations (see Section 3.4).
For the initial log-normal distribution, a refill pool is considered identical to the
initial distribution as well as a pool of smaller mass galaxies with mean black hole
masses 'log(M• /M" )( ∼ 3.3 and the same initial scatter. For the initial Schechter
distribution, either the initial Schechter distribution itself is used as a refill-pool or
the Schechter distribution containing only bulges at the low mass end with mbulge =
1.58 × 108 − 1.58 × 1010 M" . The cases with smaller refill pools (lower galaxy masses)
allow a more realistic comparison to the ΛCDM-simulations presented in Section 3.4. In
total there are four different refill-scenarios which will be investigated in the following:
1. Refill-ratio 1:3 & initial mean (Ini 1:3 )
2. Refill-ratio 1:3 & small mean (Small 1:3 )
3. Refill-ratio 1:1 & initial mean (Ini 1:1 )
4. Refill-ratio 1:1 & small mean (Small 1:1 )
3.3
Models for random merging
43
Figure 3.13: Same as Fig. 3.4, but for random merging (using an initial log-normal
distribution) in the replenishment scenario with a refill-ratio of 1:1 and a low mass
refill pool. The black contours in the first merging generation indicate the distribution
of the unchanged refill-pool.
Figure 3.14: Same as Fig. 3.13, but for random merging (using an initial Schechter
distribution) in the replenishment scenario with a refill-ratio of 1:1 and a refill pool of
bulges with masses 108 M" < Mbulge < 1010 M" . The black contours in the first merging
generation indicate the distribution of the refill-pool.
44
The intrinsic scatter in black hole mass scaling relations
Table 3.1: Values of the fit parameter a for the scatter decrease in different random
merging models for a Schechter and a log-normal initial distribution in the limit of
small (0 < m < 20) and large merger number (100 < m).
Ini. distr.
Log-norm
high m
small m
Schechter
high m
small m
Depl.
Ini 1:3 Small 1:3 Ini 1:1 Small 1:1
0.61
0.42
0.60
0.38
0.51
0.34
0.55
0.33
0.31
0.53
0.61
0.42
0.56
0.38
0.50
0.36
0.56
0.34
0.32
0.54
Evolution of the black hole-bulge mass relation
A general feature of all replenishment models is that the scatter in the M• -Mbulge relation is again reduced with increasing merger number. However, compared to the
depletion scenario, more merger generations are needed to reduce the scatter by the
same amount. In other words, for the same merger generation the scatter decrease is
weaker as new objects with a larger initial scatter are added. For small refill pools
and large refill-ratios an interesting feature is found. Fig. 3.13 and Fig. 3.14 show the
evolution of the relation for a refill-ratio of one and the small (low mass) refill pools
(Small 1:1 ) for an initial log-normal and a Schechter distribution. The contours in the
plot of the first random merging generation illustrate the distribution of the unchanged
refill-pool. In both cases a double peak structure emerges. This is a consequence of
using a low mass refill pool and a high refill-ratio. The low-mass peak reflects the
appearance of new objects chosen from refill-distribution whereas the high mass peak
evolves through merging from the initial distribution, similar to the simple case (Section
3.3.2). This behavior will be discussed in Section 3.4 in more detail.
Quantifying the scatter in black hole relations
In contrast to the depletion scenario, the replenishment models lead to a slower decrease
of the scatter in black hole mass. The scatter quantification for the evolution of the M• Mbulge -relation in Figs. 3.13 and 3.14 (refill-ratio 1, small mean and initial log-normal
distribution or Schechter distribution) is shown in Figs. 3.15 and 3.16. The decrease
of the scatter at low merger numbers originates from merging of new objects from the
refill-pool mainly dominated by minor mergers whereas the decrease of the scatter at
high merger numbers reflects merging at the high mass end (see Section 3.3.2), mainly
dominated by major mergers. A stronger scatter decrease for small merger numbers
(a = 0.53, red dashed line) is obtained than for the large merger numbers (a = 0.31,
blue dashed line), since the probability for having minor mergers is higher at the small
merger number end.
3.3
Models for random merging
45
Figure 3.15: Same as in Fig. 3.8, but for a replenishment scenario with a refill-ratio
of 1, a refill-pool with a small mean and an initial log-normal distribution.The ’spike’
near log'Nmerg ( = 2 is due to the bimodality as discussed in the text.
Figure 3.16: Same as in Fig. 3.15, but for a replenishment scenario with a refill-ratio
of 1, a refill-pool with a small mean and an initial Schechter distribution.
46
The intrinsic scatter in black hole mass scaling relations
Table 3.2: Values of the fit parameter a for scatter decrease in different random
merging models for either only major or only minor mergers based on an initially lognormal distribution in the limit of large merger numbers (m > 100).
Major
Minor
All
Depl.
0.39
0.66
0.61
Ini 1:3 Small 1:3 Ini 1:1 Small 1:1
0.50
0.43
0.54
0.15
0.65
0.59
0.59
0.30
0.60
0.50
0.55
0.30
The influence of refill-ratio and refill-pool is analysed and the scatter evolution
resulting from the four different replenishment models is summarized in table 3.1 for
an initial log-normal and an initial Schechter distribution. The typical errors are ±0.02.
The larger the refill-ratio, the more slowly the scatter decreases. Keeping in mind that
a larger refill-ratio corresponds to a larger number of new objects added per merger
generation, this behavior can be explained as follows. New objects from the refillpool have a large initial scatter in black hole mass, in contrast to objects after several
merging generations, whose scatter has already decreased. Therefore, the more objects
are added to the sample, the less the scatter decreases with merging generation. In
addition, the lower the typical mass of objects in the refill pool, the more slowly the
scatter decreases in the limit of large merger numbers. That means that for large merger
numbers, the probability for having major mergers gets higher when using a low mass
refill-pool. However, for low merger numbers the scatter decreases more rapidly with
a low mass refill-pool. In this range, minor mergers become more likely because of the
low-mass refill-pool, leading to a stronger scatter decrease than in the major merger
dominated region (i.e. convergence region of large merger numbers). Altogether, a
small-mass refill-pool leads to a higher probability of having minor mergers in the small
merger number region than at the limit of large merger numbers, where major mergers
dominate. Furthermore, as shown in table 3.1, the choice of the initial distribution
does not change the scatter decrease a (for small as well as large m).This is expected
from the CLT as the distribution quickly evolves into a normal distribution no matter
which initial distribution is used.
Difference between major and minor mergers
Since Peng (2007) considered in his study a replenishment scenario with a refill ratio
of Nnew /Nevent = 1 using the initial distribution as the refill pool, I will also investigate
the difference between major and minor mergers for different replenishment models.
The fitted slopes for the scatter evolution for major and minor mergers are summarized
in table 3.2 with an error of about ±0.02. Note that for the case of a refill-ratio of 1 : 1
with a low-mass refill-pool, a larger error of about ±0.05 is obtained, since here the
black hole mass histograms are not fitted well by a Gaussian function anymore. As in
3.4
Comparison to merging in ΛCDM-Simulations
47
the depletion case (see Section 3.3.2) a stronger scatter decrease for galaxies undergoing
only minor mergers can be seen than for major mergers. One can argue as above that
this deviation from the CLT is most likely due to dividing the bulge masses in different
mass bins which suffer a constant influx and outflux from bulges during each merger
generation. And furthermore, forcing galaxies to undergo only major mergers is a very
strong constraint leading to a violation of the CLT.
3.4
Comparison to merging in ΛCDM-Simulations
So far, the influence of different idealized random merging models on the evolution
of the M• -Mbulge -relation and the corresponding scatter in black hole mass has been
discussed. In this Section, a more complex and astrophysically motivated model is
investigated based on merger trees from dark matter simulations following structure
formation in a ΛCDM universe.
3.4.1
Simulation setup
Here, a simulated, comoving periodic box with L = 100 Mpc box length and 5123
particles is used performed with the GADGET2 code (Springel et al., 2005a). For
this simulation a ΛCDM cosmology was assumed based on the WMAP3 (see e.g.
Spergel et al. (2003)) measurements with σ8 = 0.77, Ωm = 0.26 , ΩΛ = 0.74, and
h = H0 /(100 kms−1 ) = 0.72 (see also Moster et al. (2010) for a first analysis of this
simulation). The simulation was started at z = 43 and run until z = 0 with a fixed
comoving softening length of 2.52 h−1 kpc. Starting at an expansion factor of a = 0.06
there exist halo catalogues for 94 snapshots until z = 0 separated by ∆a = 0.01 in
time. The mass of one dark matter particle is 2 × 108 M" /h.
3.4.2
Merger tree algorithm
The merger trees of the dark matter component are constructed as follows: For every
snapshot, one has to identify the different dark matter haloes. In a first step Friendsof-Friends (FOF) groups are defined using a FOF algorithm with a linking length of
b = 0.2 (≈ 28kpc, Davis et al., 1985). In a second step, subhaloes of every FOF group
are extracted using the Subfind algorithm (Springel et al., 2001b). This halofinder
identifies over-dense regions and removes gravitationally unbound particles. This way,
the FOF group is split into a main or host halo and its satellite halos. In most cases,
90% of the total mass is located in the main halo.
The sizes and virial masses of the main halos (i.e. the most massive Subfind halos)
are determined with a spherical over-density criterion. The mininum halo mass is set
to 20 particles (4 × 109 M" /h). In the following, isolated merger trees are used which
are constructed only for the the main halos, i.e. the central objects of one FOF group
identified by Subfind. The dark matter mass of a central object is defined by the dark
matter mass within the virial radius. The over-density approximation in the spherical
48
The intrinsic scatter in black hole mass scaling relations
collapse model is used according to Bryan & Norman (1998). The algorithm to connect
the dark matter halos between the snapshots is described in detail in Maulbetsch et al.
(2007). The branches of the trees for z = 0 halos are constructed by connecting the
halos to their most massive progenitors (MMP) at previous snapshots. Thereby, halo
j with nj particles at redshift zj with the maximum probability p(i, j) is chosen to be
a MMP of halo i containing ni particles at redshift zi (where j < i and zj > zi ). The
probability p(i, j) is defined as
nov (i, j)
with
nmax (i, j)
nov = ni (zi ) ∩ nj (zj ) and
nmax (i, j) = max(ni (zi ), nj (zj ))
p(i, j) =
(3.13)
Here, nov is the number of particles found in both halos and nmax is the particle number
of the larger halo. ’Fake’ haloes are removed which exist only within one time-step and
have no connection to any branch (close to the resolution limit). The low redshift ends
of the branches are then checked for mergers. A halo j is assumed to merge into halo
i, if at least 50% of the particles of halo j are found in halo i. In case of a merger the
branches are connected. For a proper connection, the ’split’-algorithm (Genel et al.,
2009) is applied to prevent double or multiple counting of merger events within a tree.
The ’split’ algorithm was shown to produce more reliable merger rates which is also
important for this work.
3.4.3
Evolution of the relation between black hole and galaxy
mass
Two different possibilities are considered in order to populate dark matter haloes with
black holes and bulges. Either black hole masses or bulge masses are directly applied
to dark matter halos and the corresponding missing quantity is calculated using the
the black hole-bulge mass relation in Häring & Rix (2004).
Using a M• -MDM -relation
Ferrarese (2002) proposed a relation between the mass of the central black hole M•
and the mass of the dark matter halo MDM of the form
#
$1.65
MDM
M•
∼ 0.1 ·
.
(3.14)
108 M"
1012 M"
I want to point out that only the M• -Mbulge -relation is directly observable and therefore,
assumed to be more fundamental than the M• -MDM -relation. Furthermore, in a very
recent study of Kormendy & Bender (2011), they showed observationally that the black
holes seem not to correlate with dark matter halos, in particular at the low mass end.
However, the M• -MDM -relation has been supported by e.g. recent results of Booth
3.4
Comparison to merging in ΛCDM-Simulations
49
& Schaye (2009), where the authors obtain a similar relation between black hole and
dark matter halo mass using numerical simulations (GADGET III) with self-consistent
black hole growth, which are tuned to match the relations between black hole mass
and galaxy stellar properties. Furthermore there are observations from Croom et al.
(2007); Yu & Lu (2008) and White et al. (2008), which confirm the existence of a
relation between black hole and dark matter halo mass, what may justify the use of
this controversial relation in this Chapter. However, this relation is only valid for z = 0.
Since here the black holes are seeded at higher redshift, expression Eq. 3.14 is modified
in order to maintain the relation at z = 0. The same derivation is assumed as it is
described in Ferrarese (2002). Therefore, first the virial velocities vvir are assumed from
the simulations calculated by SUBFIND. This way vvir can be obtained as a function
of MDM and redshift z. Assuming that vvir ≈ vc then the relation between circular
velocity vc and velocity dispersion σc is used and the one between velocity dispersion
σc and black hole mass M• . So equation 3.14 can be rewritten:
$4.58
#
vc (MDM , z)1.19
105 M"
(3.15)
M• = 3.12 ×
200 km/s
Note, that the relation between circular velocity vc and velocity dispersion σc is derived
from observations of spiral galaxies at z = 0. Presumably at higher redshifts dissipation of baryons has a higher influence than for z = 0. This could lead to larger velocity
dispersions and therefore also to larger black hole masses. According to relation 3.15
halos can be populated with masses extracted from the dark matter simulations with
central supermassive black holes. As massive galaxies are of major interest, where gas
physics is assumed to be less important (at least at low redshifts e.g. Dekel & Birnboim, 2008; Naab et al., 2006b; Khochfar & Silk, 2009), only merger trees have been
investigated starting at z = 3 for halos more massive than 1012 M" at z = 0. At the
high redshift end of every branch starting at z = 3 the most massive progenitors of the
selected z = 0 halos are populated with black holes according to Eq. 3.15. Additionally, a log-normal distributed scatter is added to the black hole masses with σini = 0.6
(see Section 3.3). For the subsequent growth of the black holes, only dark matter halo
mergers and the corresponding mergers of their black holes are taken into account.
To allow for a comparison with the random merging cases shown previously, the
evolution of the M• - Mbulge -relation is shown in Fig. 3.17. Black holes in dark matter
haloes have been chosen according to Eq. 3.15 and then corresponding bulge masses
have been calculated by taking the median relation of Häring & Rix (2004). Once the
dark halos have been populated with these bulge masses, a scatter of sigma=0.6 has
been applied to the black hole masses for each bulge mass. Again, the growth process
is followed only via merger events. The high mass end of the relation is shifted towards
larger black hole masses as only an initial scatter to black hole masses has been applied
but not to the bulge masses. Similar to the simple models investigated before again
the scatter decreases with time. Moreover, a double peak structure can be seen at low
redshifts z < 0.4 similar to the replenishment random merging model with a low mass
50
The intrinsic scatter in black hole mass scaling relations
refill pool (Section 3.3.3). This is a consequence of the conditional mass function. Only
halos were investigated under the condition that they merged into z = 0 halos with
masses larger than 1012 M" (see e.g. Somerville et al., 2000).
Using a galaxy population model
Alternatively, to populate dark matter halos with black holes, a fitting function can
be used that relates host dark halo masses to stellar masses of galaxies to assign to
every dark matter halo mass a galaxy stellar mass. There exist many studies which
link the distribution of galaxies to that of dark matter halos (van den Bosch et al.,
2003, 2007; Mandelbaum et al., 2006; Moster et al., 2010; Guo & White, 2009). Here,
the fitting formula from Moster et al. (2010) is taken. They assume that every host
halo contains exactly one central galaxy and - as a constraint from the observed galaxy
mass function - that the stellar mass to dark matter halo mass ratio M∗ /MDM first
increases with increasing mass, reaches a maximum and then decreases again. Hence
Moster et al. (2010) adopt the following parametrization, similar to the one used in
Yang et al. (2003):
M∗ (MDM )
=2
MDM
#
M∗
MDM
$ )#
0
MDM
M1
$−β
+
#
MDM
M1
$γ *−1
(3.16)
Basically, this parametrization is set to reproduce many observations, as the galaxy
mass function or clustering. Choosing a redshift parametrization for each of the parameters in Eq. 3.16, they can predict the galaxy to dark matter mass ratio at any
redshift:
#
M∗
MDM
log M1 (z) = log M0 · (z + 1)µ
#
$
M∗
(z) =
· (z + 1)ν
M
DM z=0
0
γ(z) = γ0 · (z + 1)γ1
β(z) = β1 · z + β0
$
(3.17)
(3.18)
(3.19)
(3.20)
with log M0 = 11.88, µ = 0.019, (M∗ /MDM )z=0 = 0.0282, ν = −0.72, γ0 = 0.556,
γ1 = −0.26, β0 = 1.06 and β1 = 0.17.
To populate the galaxies with black holes, it is assumed for simplicity that all stars
are in the spheroidal component of the galaxy (M∗ ≈ Mbulge ). Using the M• -Mbulge relation (Häring & Rix, 2004), to each galaxy mass a black hole mass is applied with
the same initial scatter as already used before (σ = 0.6). If then the growth of black
holes and galaxies is taken into account through merging according to the ΛCDMsimulations, an evolution of the M• -Mbulge -relation is obtained as shown in Fig. 3.18.
Again, the same effect can be seen as already described in Section 3.4.3: a decreasing
scatter with time together with an evolving double peak structure.
3.4
Comparison to merging in ΛCDM-Simulations
51
Figure 3.17: Evolution of the M• -Mbulge -relation through merging only in ΛCDMsimulations using M• -MDM -relation.
Figure 3.18: Evolution of the M• -Mbulge -relation through merging only in ΛCDMsimulations using a galaxy population model (Moster et al., 2010).
52
The intrinsic scatter in black hole mass scaling relations
Figure 3.19: Scatter in black hole mass versus mean black hole mass 'log(M• /M" )(
at different redshifts (different colors) in the ΛCDM-simulation assuming black hole
seeding according to the M• -MDM -relation. The horizontal lines indicate the average
scatter for black holes more massive than ≈ 106 M" . The scatter for massive black
holes continuously decreases towards lower redshifts.
Figure 3.20: Relation between mean black hole mass 'log(M• /M" )( and mean number
of mergers log'Nmerg ( at different redshifts (different colors) in the ΛCDM-simulation
assuming black hole seeding according to the M• -MDM -relation. The best fit at z=0.05
is indicated by the red dashed line. For comparison the dashed black line indicates the
best fit for the random merging depletion model (Fig. 3.7).
3.4
Comparison to merging in ΛCDM-Simulations
3.4.4
53
Quantifying the scatter in the black hole mass relation
For the evolution of the M• -Mbulge -relation shown in Fig. 3.17 (the model based on
the relation between dark halo mass and and black hole mass) the same scatter quantification was performed as before for the different random merging models. Fig. 3.19
shows the evolution of the scatter in black hole mass. At the high mass end, which I
am focusing on here, the scatter decreases towards lower redshift. Furthermore - similarly to random merging - in Fig. 3.20 there is a convergence towards a linear relation
between the logarithm of the mean number of mergers and the logarithm of black hole
mass at low redshift. This relation is fitted according to
log'Nmerg ( = a · 'log(M• /M" )( + b
with a = 0.46 and b = −1.39.
(3.21)
shown by the red dashed line in Fig. 3.20. For comparison the dashed black line
illustrates the random merging case (depletion). As expected lower mass black holes
(M• < 108 M" ) experience more mergers than predicted for the simple depletion case,
whereas at the high mass end galaxies undergo less mergers than in the depletion
model.
Finally the scatter σ is plotted as a function of the mean number of mergers 'Nmerg (
in Fig. 3.21. The same qualitative behavior is found as for all random merging models.
The scatter decreases with increasing number of merger events. However, the scatter
decrease in ΛCDM-merging is weaker. The fit parameter a = 0.30 ± 0.03 is smaller
than for most of the random merging models at the limit of large merger numbers.
Note that for the case where black holes are assigned to dark matter halos using the
galaxy population model, as described in Section 3.4.3, the same qualitative scatter
behavior is obtained. The only difference is that because there are fewer objects at the
high mass end, the resulting scatter value is connected with a larger error than for the
black hole population described in Section 3.4.3. In comparison to the best-matching
random merging scenario - a refill-ratio of 1/3 and a low-mass refill-pool - ΛCDMmerging leads to a clear difference in the scatter decrease (a = 0.51 ± 0.02). However,
in the case of ΛCDM-merging there is only a maximum number of mergers of ∼ 60.
Therefore, the scatter decrease in CDM-merging is consistent with the best-matching
random merging model only in the low merger range (a = 0.34 ± 0.02), indicated by
the blue, dashed line. This shows that the scatter evolution in ΛCDM-merging can be
well approximated - quantitatively and qualitatively - by the simple model of random
merging without any structure formation model.
3.4.5
Evolution of the black hole mass function
In this Section, the evolution of the black hole mass function is investigated in the
merger-driven model to test, if merging as only growth channel provides an adequate
description for black hole formation. Fig. 3.22 shows the black hole mass function
for different redshifts (different colors) assuming an initial log-normal scatter in black
54
The intrinsic scatter in black hole mass scaling relations
Figure 3.21: Scatter log(σ) as a function of mean number of mergers log'Nmerg ( at
different redshifts in the ΛCDM-simulation assuming black hole seeding according to the
M• -MDM -relation. The red dashed line corresponds to the fit to the scatter for ΛCDM
merging. The blue dashed line shows the fit to the best-matching random merging model
(refill-ratio 1/3 and low-mass refill-pool), the black dashed line illustrates the analytic
case.
hole mass at z = 3 and a growth of black holes only via mergers. The solid lines
indicate the evolution of the black hole mass function assuming seeding according to
the M• -MDM -relation (Ferrarese, 2002) and the dashed lines show the black hole mass
function based on the galaxy population model of Moster et al. (2010). In the left
panel the local observed black hole mass function (black triangles) is shown derived
from correlations between black hole mass and bulge luminosity or stellar velocity dispersion (Marconi et al., 2004). This is in disagreement with the z = 0 model prediction
(red line), since black hole masses are underestimated. This implies that growth by
gas accretion might be an important contribution to the overall formation process of
black holes and should not be neglected. Furthermore, if we seed the black holes only
at z = 1, the disagreement with observations is - as expected - less pronounced than
for z = 3-seeding. But again, the black holes are not massive enough to reproduce the
observations. This shows that even from z = 1 till z = 0 merging only seems to be an
insufficient description for black hole growth. However, it should be pointed out most
importantly that in this Chapter it is not the primary aim to fit the present-day black
hole mass function. In fact it shall be shown how the scatter evolution in black hole
mass would be affected by merger events only. The possible influence of additional gas
physics will be discussed in the next Section.
3.4
Comparison to merging in ΛCDM-Simulations
55
Figure 3.22: Left: Evolution of the black hole mass function through ΛCDM-merging
using the M• -MDM -relation (solid lines) and using the galaxy population model (dashed
lines). The triangles correspond to the local black hole mass function according to
Marconi et al. (2004). Right: Same as in the left panel, but only for z = 0. However,
only objects are considered, that had undergone a merger event within the last 100 ×
106 yrs with a merger ratio smaller than 1 : 10. The diamonds show the observed,
present-day black hole mass function for active galaxies (Greene & Ho, 2007).
The right panel of Fig. 3.22 shows the black hole mass function for the two different
seeding mechanisms only at z = 0. However, here only galaxies have been considered
that have undergone a merger with a merger ratio smaller than 1 : 10 in the last
100×106 yrs. Therefore, they can be assumed to be in the active phase during this time.
These results deviate from observed values of the present-day black hole mass function
for a sample of active galaxies (8500 objects from SDSS DR4, Greene & Ho, 2007).
On the one hand, this could be a consequence of under-predicting the overall black
hole mass function as shown by the left panel, assuming the correct number of merger
events. Alternatively, it might imply that merger events are not frequent enough to give
an explanation for observed active galaxies and accretion is needed even without any
merger event. However, assuming that also lower mass ratios (< 1 : 100) can trigger the
activity of galaxies, a good agreement can be obtained with observations. Also raising
the time for one duty cycle would lead to a better consistence with observations. This
could be justified by the fact that within the assumed time for one duty cycle there
could exist more active objects if galaxy mergers would be considered, which happen
later in time than mergers of isolated dark matter halos (which are taken into account
in this study).
56
The intrinsic scatter in black hole mass scaling relations
3.5
Discussion and conclusions
In this Chapter, the evolution of the intrinsic scatter in black hole mass was investigated
under the assumption that the black holes only grow by mergers with other black holes
during galaxy mergers. For many different idealized random merging models with
(replenishment) and without (depletion) refilling from an external galaxy pool, the
following general results can be staten:
1. The evolution of the black hole distribution can be well described within the
framework of the central limit theorem (CLT). Independent of the initial distributions, e.g. log-normal or Schechter, of black holes the distribution always
evolves into a normal distribution after a few merger generations. All mergers
are considered independent of the mass merger ratio that galaxies with masses
> 104.7 M" experience.
2. For all random merging models a decreasing scatter σ was found with increasing
merging generation n and with increasing merger number m. As a consequence
of the mass built-up during the merger events the scatter also decreases with
increasing black hole mass. Motivated by the CLT, the scatter dependence on
the mean number of mergers can be approximated by
σmerg (m) ≈ σini · (m + 1)−a/2 .
(3.22)
Here, the exponent a is a measure of the strength of the scatter decrease. For
the different random merging models I find 0.30 < a < 0.61 for a large number
of mergers (m > 100) independent of the initial scatter applied to black hole and
bulge masses. In general, replenishment models show a weaker scatter decrease.
3. Considering either only major or only minor mergers for galaxy growth, minor
mergers are found to lead to a much stronger scatter decrease than major mergers;
hence the smaller the mass ratio of merger events, the more rapidly the scatter
decreases. This is in contrast to findings of Peng (2007) because they have not
investigated the scatter decrease in black hole explicitly and quantitatively as in
this chapter.
4. For different replenishment models, the higher the refill-ratio and the smaller
the typical mass of black holes in the refill-pool is, the more slowly the scatter
decreases.
Studying the effect of merging according to current structure formation models in
ΛCDM-simulations, a qualitatively similar behavior is found. The scatter decreases
with the number of mergers (a = 0.3), and as a consequence it also decreases with
cosmic time. This finding is quantitatively consistent with the best-matching random
merging model (refill-ratio 1/3 and low mass refill-pool) at least for the limit of the low
merger number range (a = 0.34), since in ΛCDM-merging the most massive galaxies
3.5
Discussion and conclusions
57
experience only ∼ 50 − 60 merger events. Therefore, the scatter evolution in ΛCDMmerging can be well approximated by a simple model assuming random merging of
galaxies.
From the above results, some implications can be drawn about recent observations
of high redshift black holes:
• For a simple merger driven growth of black holes it can be predicted that the
scatter in black hole mass must have been larger at higher redshift. Assuming
an initial scatter of 0.6 at z = 3 the over-massive black holes investigated by
Schramm et al. (2008), Peng et al. (2006) and McLure et al. (2006) would be
within the 2 − σ range of this large initial scatter. If these objects had on average
50 − 60 dry merger events then the present-day scatter value of ∼ 0.31 can also
be obtained for massive ellipticals (Gültekin et al., 2009). This shows that the
observations of over-massive black holes at high redshifts are consistent with a
population of galaxies that has a large scatter in black hole mass (∼ 0.6) at high
redshifts being reduced through subsequent merging.
• A further important advantage of this model is that not only over-massive black
holes at high redshifts can be explained, but it can be also accounted for the
observed under-massive black holes at z = 2 (Alexander et al., 2008; Shapiro
et al., 2009). These objects would be lying within the 2 − σ range of the scatter
at z = 2 (σ ≈ 0.5) assuming an initial scatter of 0.6 at z = 3. Here, I want
to point out, that models assuming an evolution of the M• − Mbulge relation
(from higher towards lower black hole masses with decreasing redshift) can be
an explanation for over-massive black holes at high redshifts, but not for the
observed under-massive ones.
• The scatter evolution of model predictions from GALFORM (Malbon et al., 2007)
can be explained: the decreasing scatter with increasing black hole mass (see
Fig. 3.1). This is the consequence of the merger driven growth of black holes.
More massive objects have undergone more merger events than less massive ones;
this higher merger number leads to a stronger decrease in scatter. However,
observations show a larger scatter in black hole masses, especially at the high
mass end (Tremaine et al., 2002, Gültekin et al., 2009). That might indicate
that in the model of Malbon et al. (2007) either the number of merger events
for massive objects was overestimated (unlikely) or that they started with a too
small scatter in black hole mass at high redshift.
• The finding of a decreasing scatter with increasing black hole mass as a consequence of subsequent merging are also consistent with the results of Gültekin
et al. (2009), who distinguish between two subsamples of ellipticals and nonellipticals. For ellipticals they find a smaller scatter in black hole masses than
for non-ellipticals. Referring elliptical galaxies mainly to the high-mass end and
non-elliptical ones mainly to low-mass end, these observations are also consistent
58
The intrinsic scatter in black hole mass scaling relations
with the results presented in this Chapter. Furthermore, ellipticals presumably
had more mergers.
The aim of this Chapter was to investigate the scatter evolution of supermassive
black hole for dry merger driven growth. Despite the interesting insights other important physical growth mechanisms have been neglected, the accretion of gas onto
black holes. The importance of an additional growth mechanism is also confirmed
by the comparison of the black hole mass function resulting from merging only with
observations at z = 0: the masses of black holes are under-estimated. Furthermore,
if only active galaxies are considered that have undergone a merger event in the last
100 · 106 yrs with a merger ratio smaller than 1 : 10, there is again a deviation from
observations, which could be a consequence of under-predicting the over-all black hole
mass function. Alternatively, it might indicate that merger events are not frequent
enough to be an explanation for the observed active galaxies. This implies that there
has to be additional gas accretion even if no merger event happened.
Furthermore, according to the Soltan argument (Soltan, 1982) accretion onto massive black holes is the dominant source of energy produced by quasars (L ∼ Ṁ• c2 ),
which presumably indicates that the mass in black holes is mainly generated by accretion of gas. This gas accretion might be triggered by galaxy mergers and their ability
to drive large amounts of gas to the centers of the galaxies and possibly feed a black
hole (Springel et al., 2005a; Di Matteo et al., 2005; Johansson et al., 2009b; Hopkins
et al., 2008c). It was shown in hydrodynamical galaxy-galaxy merger simulations with
self-regulated black hole growth that after the active phase of the quasar, star formation and black hole growth are quenched by gas heating through energy release by the
active quasar. Following the merger, the quasar host becomes a red and dead, massive
elliptical galaxy without any further black hole growth through gas accretion (Johansson et al., 2009b). For these galaxies the further evolution of their black holes is mainly
dominated by dry merger events. However, gas-rich mergers and cold accretion flows
seem to be the dominant growth mode for massive high redshift black holes, as the
galaxies are more gas rich (Khochfar & Silk, 2006b, 2009). The quasar activity peaks at
z ≈ 2 − 3 (e.g. Hasinger et al., 2005; Ueda, 2006). Therefore, this phase is most likely
to be responsible for the scatter in black holes at this redshift (corresponding to the
initial scatter in the models). According to Hasinger et al. (2005) the emissivity of the
high luminosity objects (log(L/L" ) ≥ 45) drops by ∼ 102 erg s−1 Mpc−3 between z ∼ 2
and z ∼ 0. Therefore, the subsequent evolution, at least for these massive black holes,
might be driven mainly by dry mergers as investigated in this Chapter. In addition,
there is theoretical evidence that for z ≤ 2 gas is more efficiently heated by accretion
shocks associated with gravitational heating in massive (MDM ≥ 1012 M" ) galaxy halos
(Khochfar & Ostriker, 2008; Dekel & Birnboim, 2008; Naab & Ostriker, 2009; Birnboim
et al., 2007), suppressing further gas cooling and star formation as well as accretion
onto the central supermassive black hole. Thus, ’red and dead’ massive spheroids evolve
starting at z ∼ 1, where again merging may dominate the further growth of these black
holes (Dekel & Birnboim, 2006; Khochfar & Silk, 2009). From disk galaxy merger sim-
3.5
Discussion and conclusions
59
ulations in Johansson et al. (2009a), it is known that over-massive black holes lying
above the black hole-bulge relation do not evolve onto the relation considering only
gas accretion driven growth. A solution for this problem could be that the evolution of
over-massive black holes onto the relation might be caused mainly by growth through
merger events leading to a scatter decrease in black hole mass as shown in this Chapter.
That would also indicate, that gas accretion may not play the most important role in
the evolution process of over-massive black holes, at least at late stages, where dry
merging is the dominating process.
Intentionally, in this Chapter the physics was kept very simple and was focused
only on the growth through merging in order to understand the influence of merging
on the evolution of the scatter in black hole mass. However, gas accretion is - even for
z < 2 - an important growth channel. The impact of gas accretion on the evolution of
the scatter in the M• − Mbulge relation is complicated and so far not easy to assess. To
get an idea for such an influence, I refer the reader to Johansson et al. (2009a). Using
merger simulations of disk galaxies with gas (3 · 109M" < Mbulge < 1010 M" ) they show
two possible scenarios: Starting with galaxies on the relation leads to an increase of the
scatter which gets even larger for higher gas fractions. However, considering initially
galaxies below and above the relation causes a decrease of the scatter. The decrease
seems to get stronger for lower gas fractions. In order to really understand the effect
of gas physics we definitely need further investigations of this process in a statistical
sense.
60
The intrinsic scatter in black hole mass scaling relations
Chapter
4
Origin of the anti-hierarchical
growth of black holes
In this Chapter, different mechanisms are presented influencing the coevolution of galaxies and black holes, which can provide an explanation of
for the observed anti-hierarchical trend of black hole growth within hierarchical clustering scenarios. Observational studies (e.g. Ueda et al., 2003; Hasinger
et al., 2005) have shown that the number densities of luminous AGN peak at
higher redshifts than the ones of less luminous AGN, implying that massive
black holes seem to have predominantly formed before lower mass black holes.
For the investigation presented here, the semi-analytic code of Somerville et al.
(2008b) is used based on the merging history of the Millennium-simulation.
The fiducial model includes a very sophisticated prescription for black hole
growth following the light curve parametrizations from a large set of merger
simulation (Hopkins et al., 2006a). It is found for the fiducial model that the
maximum number densities of AGN in different luminosity bins can match the
ones of the observations. However, the fiducial model fails to reproduce the
correct number densities of AGN in different luminosity bins at low (z < 2)
and high (z > 4) redshifts. Additional modifications, such as a sub-Eddington
limit for gas accretion dependent on mass and redshift, an additional accretion channel onto the black holes due to disk instabilities and a ’heavy’ black
hole seeding scenario seem to be key mechanisms for reproducing the observed
downsizing, i.e. the bolometric as well as the hard X-ray AGN luminosity
functions. The picture can be confirmed that disk instabilities are the main
driver for moderatly luminous Seyfert galaxies, whereas major merger events
are mostly triggering very luminous quasars. Moreover, it is shown that, when
gravitational heating is additionally assumed, the best-fit model can simultaneously predict the black hole mass function and the galaxy-halo mass relation
at z = 0.
62
4.1
Origin of the anti-hierarchical growth of black holes
Motivation and observational evidence for downsizing
As pointed out in the last Chapter, Section 5.7, besides merging of black holes, cold
gas accretion onto black holes represents a significant contribution to the over-all evolution of black holes, in particular for high-redshift and low-mass galaxies where a
large fraction of cold gas is present. During phases of strong gas accretion, it is generally accepted that black holes are powering luminous active galactic nuclei (AGN)
(Salpeter, 1964; Zel’Dovich, 1964; Lynden-Bell, 1969). Moreover, by estimating the
total energy radiated by AGN during their whole life, it can be shown that nearly all
the mass in black holes has been accumulated during periods of bright AGN activity
(Soltan, 1982). The large amounts of released energy arising from accretion onto black
holes (AGN feedback) is a very important ingredient in order to understand the joint
evolution of galaxies and black holes. On the one hand, feedback is assumed to cause
a self-regulated black hole growth and on the other hand it is thought to influence
the evolution of the host galaxy by quenching further cooling and suppressing star
formation. In order to investigate this complex scenario, several models have been
developed based on either purely analytic (Efstathiou & Rees, 1988; Haehnelt & Rees,
1993; Haiman & Loeb, 1998; Hopkins et al., 2007b) or semi-analytic approximations
(e.g. Kauffmann & Haehnelt, 2000; Volonteri et al., 2003; Granato et al., 2004; Croton,
2006; Somerville et al., 2008b). In recent years, numerical models have also become
available (Springel et al., 2005b; Hopkins et al., 2006a; Di Matteo et al., 2005; Robertson et al., 2006a; Li et al., 2007; Sijacki et al., 2007), which clearly demonstrate the
important influence of AGN feedback.
From the observational point of view with respect to AGN, one can state the following picture for black hole growth: Early surveys of quasars in the optical demonstrated
that quasars undergo significant evolution from z = 0 up to z ≈ 2 − 2.5 (Schmidt &
Green, 1983; Boyle et al., 1988; Hewett et al., 1994; Boyle et al., 2000). Beyond z = 2
the space density starts to decline (Warren et al., 1994; Schmidt et al., 1995). Studies
of Fan et al. (2000, 2001) find very bright quasars with black hole masses of the order
of 109 M" at z ≈ 6. Recent progress in detecting faint and obscured AGN was achieved
by analysing data from X-ray surveys (XMM-Newton, Chandra, ROSAT, ASCA, e.g.
Miyaji et al., 2000; La Franca et al., 2002; Cowie et al., 2003; Fiore et al., 2003; Barger
et al., 2003; Ueda et al., 2003; Hasinger et al., 2005; Barger & Cowie, 2005; Sazonov &
Revnivtsev, 2004; Nandra et al., 2005; Aird et al., 2010). Because AGN are typically
far more X-ray luminous than even actively star-forming galaxies, such deep X-ray
surveys provide the most efficient means of AGN selection and a wider range of AGN
luminosity can be probed than in the optical. This allows to study the evolution of a
wide variety of different AGN types in addition to quasars (e.g. Seyfert galaxies). All
of these studies in the hard and soft X-ray range find a strong evolution of bright AGN
with redshift: the space density of bright AGN is peaking at higher redshifts (z ≈ 2)
than that of faint AGN (z < 1). Making the simplified assumption that black holes
4.1
Motivation and observational evidence for downsizing
63
are accreting at the Eddington rate (i.e. L ∝ M• ) would imply that very massive black
holes seem to be in place already very early in the universe, whereas less massive black
holes seem to evolve predominantly at lower redshifts. This behavior is called ’downsizing’ or ’anti-hierarchical’ growth of black holes. The downsizing trend is not only
found in observations in the soft and hard X-ray bands, but also in the optical range
(Cristiani et al., 2004; Croom et al., 2004; Fan et al., 2004; Hunt et al., 2004; Richards
et al., 2006; Wolf et al., 2003) and the NIR (e.g. Matute et al., 2006). Thus, putting all
observations from different wavebands and redshift ranges together, reveals again an
anti-hierarchical trend as shown in a compilation of Hopkins et al. (2007c) (see Section
4.6). At first sight, these observations seem to be in contraction with currently favored
hierarchical structure formation models, where one would expect less massive objects
to form at early times with a subsequent hierarchical clustering of more massive objects
only at later times in the Universe. However, in our current picture, the connection of
AGN luminosity to the over-all mass assembly of black holes is pretty complex because
of the following reasons: The AGN luminosity is mainly governed by the different accretion mechanisms and phases black holes are experiencing during their life-time and
the corresponding accretion efficiencies. This means that e.g. a very massive black
hole might accrete at low Eddington-ratios leading to a small luminosity. If quasar
activity would be mainly triggered by galaxy interactions, the observed decline in the
quasar number density at z < 2 might reflect the decrease of the major merger rate
with decreasing redshifts. Also the smaller amount of cold gas, which can be accreted
onto the black hole together with AGN feedback processes might state other plausible
effects for reducing the quasar number density. For low luminous AGN, however, additional trigger mechanisms besides merger events might become important as e.g. gas
accretion onto the black hole due to stellar mass loss or disk instabilities.
Therefore, the aim of this Chapter will be to provide a better understanding for the
mechanisms triggering AGN activity and for the accretion efficiencies during the active
phases of black holes which might be causing the observed downsizing. Moreover,
I want to reproduce the AGN number density evolution in a quantitative way by
simulatenuously satisfying observational constraints of the present-day universe. For
such an investigation, simple models based only purely merging scenario, as they were
used in the previous Chapter, are certainly not sufficient anymore, and tools for a
more thorough approach for modeling the gas physics in galaxy formation are needed.
There exist several studies using semi-analytical models (’Munich’, ’Galform’ or
’Morgana’ models), where they try to give different explanations for the downsizing
trend (Fontanot et al., 2006; Malbon et al., 2007; Marulli et al., 2008; Bonoli et al.,
2009; Fanidakis et al., 2010). They will be discussed in more detail in the next Section.
However, in contrast to these previous studies, in this thesis I am using a differently
developed semi-analytic model according to Somerville et al. (2008b) (S08), which
includes a sophisticated prescription for black hole accretion in the ’bright’ quasar
mode. Following the large set of merger simulations (Springel et al., 2005b; Robertson
et al., 2006a,c; Cox et al., 2006; Hopkins et al., 2007a) typical light curve models for
64
Origin of the anti-hierarchical growth of black holes
quasars are adopted in this model. Moreover, the effect of several modifications and
extensions is discussed - based on the fiducial model - such as:
• a sub-Eddington limit for Eddington-ratios dependent on redshift and black hole
mass
• secular evolution processes triggering AGN activity, e.g. disk instabilities
• a ’heavy’ black hole seeding scenario combined with a large scatter in the accreted
gas mass onto the black hole at high redshifts.
In the course of this Chapter, first the results of previous studies (Section 4.2) are summarized with respect to the downsizing trend in black hole growth. Section 4.3 gives
a brief overview of the semi-analytic model used in this study and the different modifications and extensions for black hole growth which will be considered. In Sections
4.4 and 4.5 properties of present-day and high-redshift galaxies and their black holes
are investigated in comparison to observations. The AGN number density evolution
is studied in Section 4.6, considering the influence of the different model modifications. Section 4.7 shows a comparison of the evolution of the observed bolometric and
hard X-ray luminosity function with the model output. Moreover, the evolution of the
Eddington-ratio distributions and of the relation between black hole mass and luminosity is discussed in Sections 4.8 and 4.9. Finally, in Section 4.10, the main results
summarized and compared to previous studies.
4.2
Previous studies
In the last few years, several studies were trying to understand the observed antihierarchical trend using the ’Galform’ (Durham)- (Bower et al., 2006), the ’Munich’- (De Lucia & Blaizot, 2007) or the ’Morgana’ semi-analytic model (Monaco
& Fontanot, 2005; Fontanot et al., 2006). The first two models are applied to the
dark matter merger trees of the Millennium-simulation, whereas the latter uses merger
trees from the Pinocchio tool. All models distinguish between black hole accretion
in the bright quasar mode and the low-Eddington ratio radio-mode. However, while
in the ’Galform’ model, the quasar mode is triggered by merger events and secular
evolution processes as disk instabilities (note that the contribution to the total baryons
in black holes is dominated by disk instabilities), the ’Munich’- and the ’Morgana’model assume quasar phases to be only a consequence of merger events. The black hole
accretion rate in the ’Galform’-model is proportional to the produced stellar mass
during a starburst, whereas in the ’Munich’-model the accretion rate is dependent on
the cold gas content in the galaxy. Similarly, the ’Morgana’-model assumes black
holes growth to be proportional to a cold gas reservoir of the galaxy.
Fontanot et al. (2006) show the predictions of their ’Morgana’ model for the AGN
number density compared to luminosity functions of AGNs in the optical, soft and hard
4.2
Previous studies
65
X-ray bands. They claim that downsizing within the hierarchical ΛCDM cosmogony
can be reproduced and that this is most likely to be caused by stellar kinetic feedback
that arises in star-forming bulges leading to a removal of cold gas in small elliptical
galaxies. To obtain a good match to the amount of bright quasars they require quasartriggered galactic winds, which self-limit the accretion onto black holes.
In a theoretical study, Malbon et al. (2007) find - using the ’Galform’ semianalytic code - that the direct accretion of cold gas during starbursts is an important
growth mechanism for lower mass black holes and at high redshift. On the other hand,
the reassembly of pre-existing black hole mass into larger units via merging dominates
the growth of more massive black holes at low redshift. Therefore, they claim that
as redshift decreases progressively less massive black holes have the highest growth
rates, in agreement with downsizing. Their model output reproduces the evolution of
the optical luminosity function of quasars, however, they do not show a quantitative
comparison for the X-ray and/or bolometric AGN luminosity.
In a very recent work, Fanidakis et al. (2010), again based on the ’Galform’model, present a quantitative comparison of their model output to the observed quasar
luminosity function at different redshifts. In contrast to previous SAMs, they calculate their bolometric AGN luminosity from the accretion rates during their quasar
phases and from radio-mode, low cold accretion dominated regimes according to the
advective dominated accretion flow model (ADAFs: no radiation occurs since cooling happens through advection). At high redshift, they do not limit their accretion
to the Eddington rate, but assume super-Eddington accretion. Therefore, they claim
that super-Eddington accretion is responsible for very luminous AGN at high redshifts,
whereas the high number density of low luminous AGN at low redshift can be explained
by the luminosity derived from the ADAF model. They attribute the observed downsizing trend mainly to dust obscuration of low luminous AGN at high redshift.
Marulli et al. (2008), based on the ’Munich’-model, investigate different modifications for black hole growth in the quasar mode and compare the results directly to
the observed bolometric quasar luminosity function. For their best-fit model they find
that quasar light curve parametrizations (increasing the amount of low luminous AGN
at low z) and accretion rates dependent on mass and redshift seem to be important
(larger accretion rates at high redshift). Finally, in a study of Bonoli et al. (2009),
the follow-up work from Marulli et al. (2008), they assume that the amount of cold
gas accreted on the black hole depends linearly on redshift and is weighted with the
merger ratio. However, they are still not able to reproduce a sufficiently large amount
of high-luminous objects at high z and the one of low-luminous objects at low z. In
the course of this work, these previous results will be discussed by comparing them to
the ones presented in this Chapter.
66
Origin of the anti-hierarchical growth of black holes
4.3
The semi-analytic model
In this work, the semi-analytic code from S08 is used as the fiducial model. At this
point, a short overview of the physical recipes for galaxy formation is given and the
reader is referred to S08 for more details. Moreover, different, additional modifications
for black hole growth are presented based on the fiducial model in order to explain the
origin of the observed downsizing.
4.3.1
Merging history from the Millennium simulation
The SAM is applied to the merging history of the Millennium simulation (Springel et al.,
2005a) in order to have sufficiently large statistics. The Millennium simulation is a
cosmological N-body simulation performed with GADGET. It contains 21603 particles
with masses of 8.6 × 108 h−1 M" , has a comoving box length of 500 h−1 Mpc and
generates 64 output times from z = 127 to z = 0. As cosmological parameters they use
Ωm = 0.25, ΩΛ = 0.75, Ωb = 0.045, h = 0.73 and σ8 = 0.9. The structure identification
in each snapshot was done using first a friends-of-friends (FOF) algorithm (Davis
et al., 1985) followed by the algorithm Subfind (Springel et al., 2001a), where a
spherical overdensity criterium is chosen to identify substructures (subhalos within a
FOF group). The merger trees are constructed from the subhalos by finding a single
descendant for each subhalo at the following snapshot (see Fig. 2.3 for a visualisation
of a part of the dark matter density distribution in the Millennium simulation at z = 0
and z = 5).
4.3.2
Galaxy formation
In order to model the formation and evolution of galaxies on top of the dark matter
structure evolution, the SAM code includes physical prescriptions for gas cooling, reionization, star formation, supernova feedback, metal evolution, black hole growth and
AGN feedback. In the following, the implementations of the different mechanisms
are briefly summarized. In table 4.1, a summary of the different galaxy formation
parameters is provided.
1. Radiative cooling The rate of gas condensation via atomic cooling is computed
based on the model proposed by White & Frenk (1991). At each radius the
cooling time can be computed according to
tcool =
3/2µmp kT
.
ρg (r)Λ(T, Zh )
(4.1)
Here, T is the virial temperature, µmp is the mean molecular mass, ρg (r) is the
radial density profile of the gas and Λ(T, Zh ) is the cooling function (temperature
and metallicity dependent). The cooling time is the time required for the gas to
radiate away all its energy starting at the virial temperature. The gas density
4.3
67
The semi-analytic model
profile ρg (r) is assumed to follow an isothermal sphere: ρg (r) = mhot /(4πrvir r 2 ).
Putting this expression in Eq. 4.1 one can solve for a cooling radius rcool . Within
the cooling radius all gas can cool within the cooling time tcool . The cooling rate
for the mass within rcool is
dmcool
1
rcool 1
= mhot
.
dt
2
rvir tcool
(4.2)
Following Springel et al. (2001a) and Croton (2006), it is assumed that the cooling
time is equal to the halo dynamical time tcool = tdyn = rvir /Vvir . In order to
account for cold gas flows and hot gas accretion as found by simulations, two
different modes of accretion are distinguished: the rapid and the slow cooling
regime (= cold and hot mode cooling). In the rapid cooling regime, where the
cooling radius is larger than the virial radius rcool > rvir , the cooling rate is
equal to the gas accretion rate, governed by the mass accretion history. Slow
mode cooling occurs, whenever the cooling radius is smaller than the virial radius
rcool < rvir . Here, the cooling rate is calculated according to equation 4.2.
2. Photo-ionization Photo-ionization heating causes halos below a certain, critical
filtering mass MF to have a lower baryon fraction than the universal average. The
collapsed baryon fraction as a function of redshift and halo mass is parameterized
by the expression:
fb,coll (z, Mvir ) =
fb
,
[1 + 0.26MF (z)/Mvir ]3
(4.3)
where fb is the universal baryon fraction. The filtering mass is a function of
redshift and depends on the re-ionization history of the universe.
3. Star formation The SAM distinguishes between quiescent star formation and
merger-driven starbursts. The quiescent star formation is based on the the empirical Schmidt-Kennicutt relation (Kennicutt, 1989, 1998). The star formation
rate surface density ΣSFR is calculated according to
K
ΣSFR = AKS ΣN
gas
(4.4)
with AKS = 1.67 × 10−4 , NK = 1.4, and Σgas is the surface density of cold gas in
the disk. The normalisation is tuned for a Chabrier IMF (Chabrier, 2003). The
gas follows an exponential disk (proportional to the scale-length of the stellar
disk) and only gas above a critical surface density threshold Σcrit (= 6M" /pc2 )
is available for star formation. Computing the radius rcrit within which the gas
density exceeds the critical value the total star formation rate is
' rcrit
ṁ∗ =
ΣSFR 2πdr.
(4.5)
0
68
Origin of the anti-hierarchical growth of black holes
Star formation during starbursts is driven by merger events. The star formation
rate is assumed to be a function of the mass ratio and the combined cold gas
content of the merging galaxies, the bulge to total stellar component and burst
timescale (exponential decline). The burst continues until the cold gas reservoir
is exhausted and the burst SFR will decline exponentially. In the case that a
merger occurs, whenever a starburst from a former merger is still going on, the
new cold gas is added to the current fuel reservoir.
4. Supernova feedback Cold gas is reheated through the energy release from
supernovae explosions and the cold gas may be ejected from the galaxy by
supernova-driven winds. The heating rate of the cold gas is calculated with
ṁrh =
0SN
0
#
Vdisk
200km/s
$αrh
ṁ∗ ,
(4.6)
where 0SN
0 and αrh are free parameters. The circular velocity Vdisk is assumed to
be the maximum rotation velocity of the dark matter halo, Vmax . The heated gas
can either stay within the dark matter halo as hot gas, or, if the supernova-driven
winds are strong enough, it can be ejected from the halo into the intergalactic
medium (IGM). The fraction of reheated gas, which is ejected from the halo, is
given by
!
#
$αeject "−1
Vvir
feject (Vvir ) = 1 +
,
(4.7)
Veject
with αeject = 6 and Veject is a free parameter (≈ 100 − 150km/s). This ejected
heated gas can re-collapse onto the halo at later times and then is available for
cooling. As in Springel et al. (2001a) and De Lucia & Blaizot (2007) the rate of
reinfall of rejected gas is given by
$
#
meject
.
(4.8)
ṁreinfall = χreinfall
tdyn
Here, χreinfall is a free parameter, meject is the mass of the ejected gas outside of
the halo and tdyn = rvir /Vvir is the dynamical time of the halo.
5. Metal enrichment To track the production of metals, it is assumed that, together with a parcel of new stars dm∗ , a certain mass of metals dMZ = ydm∗
is created and instantaneously mixed with the cold gas in the disc. The yield is
assumed to be constant y = 1.5 and is treated as a free parameter. Whenever
new stars are formed, they are assumed to have the metallicity of the cold gas at
this time step. When metals are ejected from the disc due to SN-winds, either the
metals are mixed with the hot gas or ejected from the halo into the ’diffuse’ IGM
in the same proportion as reheated cold gas. Note that only metal enrichment
due to Supernovae TypeII is tracked.
4.3
The semi-analytic model
69
6. Black hole growth with AGN-driven winds and Radio-mode feedback
Black holes form at the centers of the galaxies and are thought to grow by two
channels: quasar mode and radio mode. The quasar mode is the bright mode of
black hole growth observed as optical or X-ray bright AGN radiating at a significant fraction of their Eddington limit (L ≈ (0.1 − 1)LEdd ; Vestergaard, 2004;
Kollmeier et al., 2006). Such bright AGN are believed to be fed by optically
thick, geometrically thin accretion disks (Shakura & Sunyaev, 1973). In the next
Section, the physical processes during the quasar mode and the corresponding
implementations in the code will be discussed in more detail. In contrast, AGN
activity in the radio-mode is much less dramatic. A large fraction of massive
galaxies are detected at radio wavelengths (Best et al., 2007) without showing
characteristic emission lines of classical optical or X-ray bright quasars (Kauffmann et al., 2008). Their accretion rates are believed to be a small fraction of
the Eddington rate and they are radiatively extremely inefficient. Even if AGN
spend most of their time in the radio-mode, they gain most of their mass during
the short, Eddington limited episodes of quasar phases which in the model are
assumed to be triggered by merger events. The energy being released during the
rapid growth of the black holes can drive powerful galactic scale winds. By relating the momentum of the radiative energy from the accreting black hole with the
momentum of the outflowing wind, the following expression for the mass outflow
rate can be obtained:
dMout
c
= 0wind ηrad
ṁacc ,
(4.9)
dt
vesc
where 0wind is the effective coupling efficiency, ηrad is the conversion efficiency of
matter into radiation, Mout is the ejected gas mass and vesc is the escape velocity.
In contrast to the quasar mode, the radio mode has low-Eddington ratio accretion
rates and thus, is radiatively inefficient and associated with efficient production of
radio jets that can heat gas in a quasi-hydrostatic hot halo solving the overcooling
problem of massive galaxies at low redshifts. Assuming Bondi-Hoyle accretion
combined with an isothermal cooling flow solution (Nulsen & Fabian, 2000) the
accretion rate in the radio mode can be calculated by the following formula:
!
"#
$
kT
M•
ṁradio = κradio
.
(4.10)
Λ(T, Zh )
108 M"
Thereby, κradio is a free parameter, namely the efficiency for gas accretion in the
radio mode. Note that the central black hole accretes at this rate whenever hot
halo gas is present (hot ’cooling’ mode, rcool > rvir ). The energy that effectively
couples to and heats the hot gas is given by Lheat = κheat ηrad ṁradio c2 . Assuming
that all the hot gas is at the virial temperature of the halo, the rate of the heated
gas mass is given by
Lheat
.
(4.11)
ṁheat =
2
3/4 vvir
70
Origin of the anti-hierarchical growth of black holes
The net cooling rate is then the usual cooling rate minus the heating rate from
the radio-mode.
4.3.3
Models for black hole growth in the quasar phase
The most important physical recipe for the investigation of AGN luminosities is the
prescription of how black holes grow during their ’bright’ and short quasar episodes.
Besides the fiducial implementation, three different extensions are considered based on
the fiducial calculations. They will be described in the following:
1. Fiducial model: The quasar phase is triggered by galaxy merger events with a
mass ratio of µ > 0.1. It is adopted that, whenever the two progenitor galaxies
merge, their black holes merge also instantaneously and form a new black hole
according to mass conservation. For modeling the gas accretion onto the black
hole a large set of numerical merger simulations are considered (Springel et al.,
2005b; Robertson et al., 2006a,c; Cox et al., 2006; Hopkins et al., 2006a, 2007a),
where they draw the following evolutionary picture of AGN activity and the different phases: Major merger events are leading to an infall of large amounts of
cold gas providing the fuel for the central black hole and are also triggering a
starburst event. In this first coalescence phase, the starburst dominates the luminosity (ULIRG), whereas the AGN is surrounded by a dusty torus and thus,
can be only observed in the X-ray range. The black hole is growing very rapidly
in this phase with accretion rates near the Eddington limit. Afterwards, the
black hole enters the blow-out phase, where the remaining dust and gas is expelled due to feedback processes from the gas accretion and the AGN can be
observed as a traditional quasar in the optical range. Due to lacking further
gas supply and an emerging hot halo, the quasar luminosity fades rapidly, the
accretion rate decreases and further star formation is suppressed resulting in a
’read and dead’ elliptical mainly growing by dry merger events. Moreover, they
find that AGN spend most of their time in the ’decaying’ phase, not accreting at
the Eddington-limit. To summarize, in isolated merger simulations typical light
curves for quasars can be extracted, consisting of an obscured heavy accretion
phase and a subsequent blow-out phase.
Therefore, in this model, the final black hole mass M•,final is calculated in the
beginning of each quaser phase using a parametrization according to (Hopkins
et al., 2006b):
#
$1.12
Msph
M•,final = fBH,final × 0.158
× Γ(z),
(4.12)
100M"
where Msph is the final spheroid mass after the merger, fBH,final an adjustable parameter and Γ(z) is a parameter, which describes the evolution of the black hole-
4.3
71
The semi-analytic model
Table 4.1: Summary of the galaxy formation parameters in the fiducial model according to S08.
Parameter
Description
Fiducial
value
Normalization of Kennicutt law
Power-law index in Kennicutt law
Critical surface density
1.67 × 10−4
1.4
6 M" pc−2
Burst star formation
µcrit
Critical mass ratio for burst activity
0.1
SN feedback
00SN
αrh
Veject
χre−infall
Normalization of reheating fct
Power-law slope of reheating fct
Velocity scale for ejecting gas
Time-scale for re-infall of ejected gas
1.3
2.0
120 km s−1
0.1
Chemical yield
1.5
Efficiency of conversion of rest mass to radiation
Mass of seed black hole
Scaling factor for BH mass after merger
Scaling factor for critical BH mass
0.1
100 M"
0.8
0.4
Coupling factor for AGN-driven winds
0.5
Quiescent star formation
AKS
NK
Σcrit
Chemical evolution
y
Black hole growth
ηrad
Mseed
fBH,final
fBH,crit
AGN-driven winds
0wind
Radio-mode feedback
κradio
κheat
Normalization of ’radio mode’ accretion rate 2 × 10−3
Coupling efficiency of radio jets with hot gas 1.0
72
Origin of the anti-hierarchical growth of black holes
bulge mass relation with time (Hopkins et al., 2006b). Following the merger simulations, additionally a Gaussian distributed scatter with a value of σ• = 0.3 dex
is applied to the accreted gas mass when the final black hole mass is calculated. When the black hole mass has reached this final value, the quasar mode is
switched off. During the quasar phase, the light curve models describe mainly two
different growth regimes: an Eddington-limited and a power-law decline phase of
accretion. In the first regime, the black hole accretes at the Eddington rate until
it reaches a certain critical black hole mass M•,crit :
M•,crit = fBH,crit × 1.07 (M•,final )1.1 .
(4.13)
Here, the parameter fBH,crit is set according to the merger simulations, which
determines how much of the black hole growth occurs in the Eddington-limited
versus power-law decline phase. The growth of the black hole during the first
regime can be modeled by an exponential increase of mass:
#
$
1−0
t
M•,new (t) = M• exp
fedd
,
(4.14)
0
tsalp
where 0 = 0.1 is the accretion efficiency factor and tsalp ≈ 0.45 Gyr the Salpeter
timescale. No strong observational constraints are available for 0 and if or how
it evolves with redshift. However, some observations at z = 0 indicate that
0.04 < 0 < 0.16 (Marconi et al., 2004). For simplicity in this study a constant
mean value for the accretion efficiency of 0 = 0.1 is taken at all redshifts. fedd in
Eq. 4.14 is the Eddington ratio defined by the ratio of bolometric luminosity to
the Eddington luminosity:
fedd := Lbol /Ledd .
(4.15)
The Eddington luminosity Ledd (assuming a hydrostatic equilibrium between the
inward gravitational force and the outward radiation pressure) is given by:
#
$
4πGM• mP c
M•
46
Ledd =
= 1.4 × 10
erg/s,
(4.16)
σT
108 M"
where σT is the Thomson cross section for an electron and mP the mass of a
proton. Combining eq. 4.14, 4.15 and 4.16 and the relation Lbol = 0/(1 − 0)Ṁ c2
The corresponding accretion rate in the first regime can be calculated by:
Ṁ•,I (t) = 1.26 × 1038 erg/s
1 − 0 fedd
M•,new (t).
0 c2
(4.17)
Note that in the fiducial model it is always assumed an accretion at the Eddington rate, i.e. a constant Eddington ratio of fedd = 1.
Exceeding the critical mass M•,crit in Eq. 4.13, the black hole enters the second
regime, the ’blow-out’ phase, which is described by a power-law decline in the
4.3
73
The semi-analytic model
accretion rate. Fitting the light curves in merger simulations from Hopkins et al.
(2006a) gives the following parametrization for Ṁ•,II :
Ṁ•,II (t) =
Ṁ•,peak
1 + (t/tQ )β
(4.18)
where tQ ∝ tsalp is the e-folding time, Ṁ•,peak is the peak accretion rate and β is
a parameterized function of the peak accretion rate. In the case that the initial
black hole mass is already larger than the calculated critical mass, the black hole
is not allowed to accrete at the Eddington rate at all and goes immediately into
the blow-out phase. If the initial black hole is even larger than the calculated
final mass, no quasar phase occurs at all. Note that for calculating the bolometric luminosity only the accretion rates during the quasar phases are taken into
account (thus ignoring the contribution from the radio mode accretion):
Lbol =
0
Ṁ•,QSO c2 .
1−0
(4.19)
Thereby, Ṁ•,QSO = Ṁ•,I/II is the accretion rate from regime I or II.
2. Sub-Eddington limit for the Eddington-ratios: Observational studies show
that the peak in the Eddington-ratio distributions is not constant with time,
instead, it is found to be dependent on redshift as well as on black hole mass
(Padovani, 1989; Vestergaard, 2003; Shankar et al., 2004; Kollmeier et al., 2006;
Netzer & Trakhtenbrot, 2007; Hickox et al., 2009; Schulze & Wisotzki, 2010).
Moreover, in particular at low redshifts, there seems to exist a sub-Eddington
limit for black hole accretion which is also dependent on black hole mass and
redshift (e.g. Netzer & Trakhtenbrot, 2007; Steinhardt & Elvis, 2010). In this
thesis, observations of a sample of type-1 AGNs (0 < z < 0.75) are closely
followed as shown in Netzer & Trakhtenbrot (2007). In this study they find
that the median of the Eddington-ratio distribution as well as the upper limit
of Eddington-ratios decreases with decreasing redshift and increasing black hole
mass. This can be parameterized by the following expression:
fEdd ∝ z γ(M• ) ,
(4.20)
where γ is a parameter varying with black hole mass. In Netzer & Trakhtenbrot (2007) they derive different values for γ from fitting their observations for
different black hole mass bins. Therefore, instead of assuming an accretion at
the Eddington rate below the critical black hole mass M•,crit (= first regime of
accretion) at all redshifts and for all black hole masses, a parametrization for
the sub-Eddington limits is assumed in the model motivated by the observations
74
Origin of the anti-hierarchical growth of black holes
from Netzer & Trakhtenbrot (2007):
1. M• < 3 × 108 M" :
+
fEdd (z) =
2. M• > 3 × 108 M" :
fEdd (z) =
(4.21)
1,
z>1
0.99 · z + 0.01, z < 1
+
1,
z > 1.5
0.39 · z 2.3 , z < 1.5
(4.22)
Again I want to emphasize that this parametrization is only implemented for the
first regime of black hole accretion resulting in a decreasing upper limit of the
Eddington-ratios with decreasing redshift and increasing black hole mass. In the
subsequent power-law decline phase, black holes are accreting at even smaller
Eddington-ratios. The corresponding distribution and its evolution with redshift
will be discussed in Section 4.5.
3. Additional accretion onto the black hole due to disk instabilities: So
far, AGN activity is assumed to be only triggered by major merger processes in
the model used in this study. However, different observational studies suggest
that moderately luminous AGN are typically not major-merger driven, at least
at z < 1 (Cisternas et al., 2010; Georgakakis et al., 2009; Pierce et al., 2007;
Grogin et al., 2005), as they do not find more morphological distortions for AGN
host galaxies than for inactive galaxies. Thus, the current consensus suggests
that most AGN at z < 1 − 1.5 are undergoing a ’main sequence’ secular growth,
e.g. the activity might be additionally driven by disk instabilities. Thus, in this
thesis at low redshift (z ≤ 2), a study of Efstathiou et al. (1982) is considered,
where they investigate the global stability of disk galaxies by numerical N-body
simulations. They find that the disk gets unstable if the ratio of dark matter
mass to disk mass becomes too low and give the following parameterization for
the onset of disk instabilities:
Mdisk,crit =
2
vmax
Rdisk
,
G0
(4.23)
where Mdisk,crit is the critical disk mass, above which the disk is assumed to get
unstable, vmax is the maximum circular velocity, Rdisk the exponential disk length
and 0 the stability parameter, where a slightly smaller value is used than proposed
in Efstathiou et al. (1982) (0 = 0.75). Therefore, in this model it is assumed that,
whenever the disk mass exceeds the critical disk mass, the bulge component is
enlarged by this difference in ’excess’ stellar mass ∆Mdisk = Mdisk − Mdisk,crit so
that the disk becomes stable again. Moreover, it is assume that a certain amount
4.3
75
The semi-analytic model
of cold gas (proportional to the excess disk mass) is additionally accreted onto
the black hole triggering an active phase:
M•,disknew = M• + fBH,disk × ∆Mdisk .
(4.24)
fBH,disk = 10−3 is used, motivated by the local black hole-bulge mass relation.
Assuming an Eddington-ratio of fedd = 0.01 (motivated by observations, personal
conversation with D. Alexander), the accretion rate is calculated by:
Ṁ•,diskinst = 1.26 × 1038 erg/s
1 − 0 fedd
M•,disknew
0 c2
(4.25)
and obtain the bolomotric luminosity for disk instabilities using Lbol = 0/(1 −
0)Ṁ c2 . For the total bolometric luminosity, the bolometric luminosities from
the major-merger driven quasar phase (equation 4.19) and the ones from disk
instablities are summed up:
0 ,
Ṁ•,QSO + Ṁ•,diskinst c2 .
(4.26)
Lbol =
1−0
Note that in the studies using the Munich model (Marulli et al., 2008; Bonoli
et al., 2009) no disk instabilities for calculating the AGN bolometric luminosity
are considered. However, in the Galform model disk instabilities are assumed
to destroy the whole disk, which is added completely to the bulge component and
they are believed to trigger quasar activity. In their model, disk instabilities are
found to be the major contribution for black hole growth at all redshifts (Bower
et al., 2006; Fanidakis et al., 2010).
4. ’Heavy’ seeding scenario and a larger scatter at high z: The origin of the
first massive black holes is still subject of intense debate. Currently, there exist
two favored seeding mechanisms (Haiman, 2010; Volonteri, 2010): either black
hole seeds could form out of the remnants of Pop III stars (e.g. Madau & Rees,
2001; Heger & Woosley, 2002) or during a direct core-collapse of a low-angular
momentum gas cloud (e.g. Loeb & Rasio, 1994; Koushiappas et al., 2004). In
the first case the seeds would have masses of Mseed ≈ 100 − 600M" (’light’ seeding), while in the latter case, more massive seeds between Mseed ≈ 104 − 106 M"
(’heavy’ seeding) would be expected. The detailed physical processes, in particular of the direct core-collapse, are widely unknown. Unfortunately, observational
constraints of the high-redshift universe are too weak in order to favor one of
these models. However, future observations of gravitational waves (LISA, Sesana
et al., 2005; Koushiappas & Zentner, 2006) or planned X-ray missions (WFXT:
Sivakoff et al., 2010; Gilli et al., 2010; IXO), will have technical capabilities to
detect accreting black holes at z > 6, will test these models of first black holes.
Moreover, due to an exponential growth of the black holes during accretion, it is
76
Origin of the anti-hierarchical growth of black holes
also very difficult to use the local population of massive halos to recover information about their original masses before the onset of accretion. For instance,
in theoretical studies of Volonteri et al. (2008), and Volonteri (2010), the different seeding mechanisms are investigated by following the mass assembly using
Monte-Carlo merger trees to the present time. They find that both models can
fit observational constraints at z = 0 (e.g. the black hole mass-velocity dispersion relation or the black hole mass function), when light seeds form already at
very early times (z = 20), or when heavy seeds evolve later on (z = 5 − 10). In
our fiducial model, seed black hole masses of 100M" have been assumed so far,
however, as there hardly happens seeding before z ≈ 10, now the heavy seeding
scenario is adopted with M•,seed = 105 M" . I want to stress that different seeding
mechanisms will not affect the z = 0 black hole mass population, as initial seed
masses are compensated by gas accretion growth processes. Only the black hole
distribution at high redshifts will be influenced by this modification (see Section
4.5.1).
Moreover, many observations (e.g. Walter et al., 2004; Peng et al., 2006; McLure
et al., 2006; Schramm et al., 2008) suggest that the black hole-to-bulge mass ratio was larger at higher redshifts than expected from the local black hole-bulge
mass relation what was discussed a great detail in Chapter 3. This eventually
implies that black holes were accreting more gas and thus, growing faster than
the corresponding bulges at high redshifts than at lower ones. Therefore, besides assuming an evolving black hole-bulge mass relation (see the z-dependent
Γ parameter in eq. 4.12) it is additionally adopted a larger scatter σ•,accr for
the accreted mass onto the black hole at high redshifts. This means that, when
calculating the final black hole mass M•,final , a larger Gaussian distributed scatter
is applied with a value of σ•,accr = 0.8 for z > 5. At redshift z < 5 the original
scatter value σ•,accr = 0.3 is considered. This additional assumption might also
be justified, as it was shown in the last Chapter and by Hirschmann et al. (2010)
that a large scatter in black hole mass (σ = 0.6 dex) at high redshift (z = 3) will
decrease with subsequent merging towards the observed present-day value (see
also Chapter 3).
In the course of this Chapter, the effects of the outlined modifications on the
AGN/black hole evolution are investigated. Thus, besides the fiducial model, the three
additional models are considered including the different extensions in a cumulative
way:
• FID: Fiducial model, point 1
• VE: Varying sub-Eddington limit for the Eddington-ratios fedd , points 2
• VEDI: Varying sub-Eddington limit & Disk Instabilities, points 2+3
4.4
Properties of nearby galaxies and black holes
77
• VEDIS: Best-fit model including a Varying sub-Eddington limit, Disk Instabilities
& a heavy Seeding mechanism, points 2+3+4
4.4
Properties of nearby galaxies and black holes
As shown in S08, the fiducial semi-analytic model was mainly tuned to match observational galaxy properties at z = 0. Therefore, in this Section the black hole properties
at z = 0 and the influence of the different extensions applied to the three modified
models are investigated. The Mgalaxy -MDM -relation, the black hole mass function and
the M• -Mbulge -relation will be shown, where the FID and the VEDIS models are compared to the z = 0-observations. Note that the VE and VEDI models do not result in
a stronger deviation from the FID model than the VEDIS model.
4.4.1
Black hole mass function
The present-day black hole mass function is depicted in Fig. 4.1. The blue, solid line
visualizes the FID model, whereas the black solid line together with the grey shaded
area represents observations from Shankar et al. (2004) and the black dashed line with
the open triangles corresponds to an observational study from Marconi et al. (2004).
One can find a good agreement of the FID model predictions with the observations
for black holes with masses M• < 108.5 M" . At the high mass end, however, the FID
model significantly over-predicts the number density of black holes by more than one
order of magnitude (for M• ≈ 1010 M" ). This excess of massive black holes can be seen
in many different SAM studies (e.g. Fontanot et al., 2006; Malbon et al., 2007; Marulli
et al., 2008; Fanidakis et al., 2010). However, I find that this is predominantly due to
very efficient growth of black holes during the radio-mode, which mainly contributes
to the growth of massive black holes at low redshifts (Croton, 2006, S08). When no
accretion onto the black hole during the radio-mode is assumed in the FID model, then
a much better agreement with the observations can be obtained for the whole mass
range. This implies that the radio-mode accretion is significantly over-estimated in
most of the current SAMs, as it was already shown in a recent study of Fontanot et al.
(2011), where they compared different semi-analytic models to observations. However,
in order to obtain in addition a sufficiently large suppression of gas cooling and thus, of
star formation for reproducing the luminosity function (in particular the bright end),
one does need the amount of feedback (note that the heating rate in the radio-mode
is proportional to the radio-mode accretion). This suggests that SAMs might need
an additional, alternative mechanism, in order to suppress cooling and quench star
formation in massive galaxies. One possible solution might be additionally taking into
account gravitational heating (conversion of potential energy of infalling objects into
kinetic energy of the stripped material and thermal energy of the intracluster medium),
what is mostly neglected in SAMs due to missing direct gas-dynamical modeling as in
hydrodynamical simulations. Gravitational heating may lead to similar efficiencies
78
Origin of the anti-hierarchical growth of black holes
Figure 4.1: Present-day black hole mass relation. The blue line illustrates the FID
model, the green and the red line show the FID and the VEDIS model with gravitational
heating and a reduced radio-mode accretion. Observations from Marconi et al. (2004);
Shankar et al. (2004) are illustrated by the black solid lines with the grey shaded area
and the black dashed lines with open triangles.
Figure 4.2: Present-day black hole bulge mass relation. The blue line shows the FID
model, the green and the red lines illustrate the FID and the VEDIS model including
gravitational heating and a reduced radio-mode accretion. The black line corresponds
to the observed, local relation (Häring & Rix, 2004). Dashed lines and the grey shaded
area show the scatter in models and observations, respectively.
4.4
Properties of nearby galaxies and black holes
79
as radio-mode feedback, as presented in a study of Khochfar & Ostriker (2008) and
1.1
). Following their results, a
shows an almost linear dependence on halo mass (∝ MDM
simplified approximation for an additional heating rate due to gravitational heating is
now assumed in the model, parameterized as:
$1.1
#
MDM − MDM,crit
,
(4.27)
Ṁheat,grav ∝ fgrav
MDM,crit
where MDM,crit is the critical halo mass, above which gravitational heating may occur
and fgrav is an additional free parameter (it is used MDM,crit = 1011 M" and fgrav = 1).
The green and red solid lines in Fig. 4.1 illustrate the FID and the VEDIS model
including the simplified approximation of gravitational heating together with a reduced
value for the radio mode accretion efficiency by 25% (κradio,new = 0.5 × 10−3 ). Due to
this change, both models can achieve a reasonably good agreement with the observed
black hole mass function for the whole mass range with simultaneously satisfying the
high mass end of the galaxy-dark matter halo relation (see Section 4.4.3). Moreover,
the two SAMs show almost no deviation from each other, indicating that neither the
growth of black holes by disk instabilities nor the lowered accretion rates at low redshifts
are influencing the black hole mass function significantly.
4.4.2
Black hole-bulge mass relation
The present-day relation between black hole and bulge mass is shown in Fig. 4.2. The
FID model is depicted by the blue line and the FID and VEDIS model predictions
with gravitational heating and a reduced radio-mode accretion are illustrated by the
green and red lines, respectively. The black solid line corresponds to the median of the
observed black hole-bulge mass relation according to Häring & Rix (2004), the grey
shaded area illustrates the 1−σ range of the scatter. The FID models are in reasonably
good agreement with the observations. The FID model without gravitational heating,
however, results in slightly too massive black holes at the high mass end (Mbulge >
1011.5 M" ), what is again due to the excessive growth during the radio-mode what
could have already been seen for the high-mass end of the the black hole mass function
(see Section 4.4.1). Thus, lowering the radio-mode accretion reduces the growth of
black holes and leads to a good agreement of the FID model with the observed relation
for the whole mass range (see green line in Fig. 4.2). However, comparing the FID
and the VEDIS models (both with gravitational heating and a reduced radio-mode
accretion), shows that the modifications in the VEDIS model reveal a slightly worse
match to the observed relation than the FID model. Only at the high mass end,
log(Mbulge /M" ) > 11.5, the VEDIS is still in good agreement with the FID model and
the observations, but for log(Mbulge /M" ) < 8, the black hole masses (≈ 107.5 M" ) of
the VEDIS model are found to be up to one order of magnitude larger compared to
the FID model. This is a consequence of the heavy seeding scenario in the VEDIS
model, where, by definition, no black holes smaller than 105 M" can occur. Moreover,
80
Origin of the anti-hierarchical growth of black holes
Figure 4.3: The Mgalaxy -MDM -relation at z = 0. The blue line illustrates the FID
model, the green and the red line illustrate the FID and the VEDIS model including
gravitational heating with a reduced radio-mode accretion. The black line corresponds
to the halo occupation distribution of Moster et al. (2010) at z = 0 with a 1-σ-range
shown by the grey shaded area.
for 8.5 < log(Mbulge /M" ) < 11, the VEDIS model leads only to slightly smaller black
hole masses than the FID model. This discrepancy is due to the slower growth of
black holes as it is not allowed that the black holes accrete at the Eddington-rate at
z < 1 anymore, but only at a decreasing fraction of it (sub-Eddington limit). This
means it takes a longer time to reach the final black hole mass, which correlates with
the spheroid mass in the used recipe, within one quasar accretion episode than in
the FID model. Whether the modifications assumed in this study are still reasonable
assumptions despite of the slightly worse agreement with the observed black hole-bulge
mass relation, will be discussed in Section 4.10.
4.4.3
Galaxy-dark matter halo mass relation
Fig. 4.3 shows the Mgalaxy -MDM -relation in the present-day Universe. The FID model
without gravitational heating and the FID and the VEDIS models including gravitational heating are illustrated by the blue, green and red solid lines, respectively.
The black solid line shows the halo occupation distribution function fit from Moster
et al. (2010), where the stellar mass function from the SDSS-survey is combined with
cosmological, dark matter N-body simulations. The grey shaded area depicts the corresponding 1 − σ range of the scatter. In the original FID model (blue line) can be
seen that the galaxies with masses Mgalaxy > 1013 M" are slightly over-predicted. In
4.5
Galaxy and black hole properties at higher redshift
81
S08, they suggest that this mass excess might be due to a significant underestimation
of the observed luminosities and therefore, the stellar masses of the central galaxies in
clusters (Desroches et al., 2007; Lauer et al., 2007; von der Linden et al., 2007) or due
to scattering of stars out to large radii as a consequence of a merger event (diffuse stellar component, Gonzalez et al., 2005; Zibetti et al., 2005; Murante et al., 2004, 2007).
However, including the simple approximation for an additional heating rate motivated
by the physical process of gravitational heating, leads to a significant better agreement
with the HOD model and thus, the observations even if the radio-mode accretion has
been reduced resulting in a radio-mode feedback lowered by a factor of 4. Between
the FID and the VEDIS model predictions (including gravitational heating), there is
almost no discrepancy showing that the additional modifications do not influence the
final distribution of the stellar masses significantly. Thus, one can conclude that both
models (FID and VEDIS) are in reasonably good agreement with the observations for
the whole mass range.
Note that the main results of this Chapter presented in Section 4.6 and 4.7 (the
AGN number density evolution and the luminosity function) are only marginally influenced by reducing the radio-mode accretion and adding a gravitational heating source.
However, for the following investigation, these modifications are always considered in
the FID, VE, VEDI and VEDIS models.
4.5
Galaxy and black hole properties at higher redshift
Having shown that the statistical properties of nearby black holes and galaxies can
be reproduced sufficiently well, in this Section the effect of the additional model extensions on the high-redshift evolution of galaxies and black holes is studied and the
model predictions for high-redshift objects are compared to observations. Again, only
a comparison of the FID and the VEDIS model is shown, as the VE and the VEDI
models do not lead to stronger deviations from the FID model than the VEDIS model.
4.5.1
Black hole mass function
Fig. 4.4 shows the black hole mass function at different redshift steps illustrated by
different colors (z = 0, 0.5, 1, 2, 3, 4, 5, 6). The upper panel depicts the FID model, lower
one the VEDIS model (always with gravitational heating and a reduced radio-mode
accretion rate). At redshifts z < 3, the evolution of the black holes is very similar in
the FID and the VEDIS model showing that the modifications in the VEDIS model
do not affect the black hole population at these redshifts. However, turning to higher
redshift z ≥ 3, the main difference between both models is the larger amount of black
holes more massive than M• > 106 M" . This can be explained on the one hand by
the ’heavy’ seeding scenario, on the other hand by the larger scatter in the accreted
82
Origin of the anti-hierarchical growth of black holes
Figure 4.4: Evolution of the black hole mass function for the FID and the VEDIS
models (upper and lower panel, respectively). The colored lines illustrate the SAM
results at different redshifts, the black dashed and solid lines correspond to observations
from Marconi et al. (2004) and Shankar et al. (2004). Adopting a “heavy”-seeding
scenario leads to a larger amount of black holes with M• > 106 M" at high redshift
z > 3.
4.5
Galaxy and black hole properties at higher redshift
83
Figure 4.5: The galaxy-halo mass relation for the FID (with and without gravitational
heating, green and blue lines, upper row) and the VEDIS model (red line, lower row)
at z = 1 (left column), z = 2 (middle column) and z = 4 (right column). The black
solid line illustrates the HOD fit from Moster et al. (2010), the grey shaded area show
the scatter. The thin, vertical lines depict the lower observational mass limit.
gas mass onto the black hole at z > 5 leading to larger black hole masses and a faster
growth at early redshift. Towards lower redshift, however, this effect can not be seen
anymore as the subsequent growth by gas accretion is compensating the initial black
hole masses by orders of magnitude. Compared to the study of Fanidakis et al. (2010)
this trend is even more pronounced in their model as they allow the black holes to
accrete at super-Eddington rates: e.g. at redshift z = 6, black holes with M• = 106 M"
have a number density of log Φ = −2.7 Mpc−3 dex−1 , whereas in the VEDIS model a
number density of only log Φ = −4 Mpc−3 dex−1 is obtained.
4.5.2
Galaxy-dark matter halo relation
Fig. 4.5 depicts the galaxy-dark matter halo mass relation at z = 1, z = 2 and z = 4
(left, middle and right columns, respectively). The upper row corresponds to the FID
model with and without gravitational heating (green and blue lines), while the lower
panels illustrate the VEDIS model (including gravitational heating). The black, solid
84
Origin of the anti-hierarchical growth of black holes
lines and the grey shaded areas show the halo occupation distribution (HOD) fits from
Moster et al. (2010). The black, thin vertical lines illustrate the observational mass
limit for the HOD fitting. Generally, with increasing redshift the agreement of the
model output, regarding the FID as well as the VEDIS models, to the HOD becomes
worse than what could have been seen at z = 0. Similar to the results at z = 0, there
is no difference between the FID and VEDIS models (with gravitational heating) at
high redshift showing that the additional model extensions do not influence the stellar
population significantly. At the high mass end, one can clearly see the effect of the
additional gravitational heating source. Similar to what is found at z = 0, this results
in smaller galaxy masses in the FID and the VEDIS model than in the FID model
without gravitational heating. However, this results in a slightly worse fit to but still
a sufficient agreement with the observations. Nevertheless, this may indicate that the
simplified approximation of gravitational heating might have a too strong effect at high
redshift and thus, an additional redshift dependence for calculating the gravitational
heating rate might be required in the model used here. Turning to the low mass end,
fairly good agreement of all model predictions can be obtained with the HOD at z = 1.
However, at z ≥ 2 the FID as well as the VEDIS model extensively overestimate the
amount of galaxies with masses log(MGalaxy /M" ) < 12.2. At z = 4 and for halo masses
log(MDM /M" ) ≈ 11 the models predict a larger number of galaxies by almost two
orders of magnitude. This extreme deviation is a well-known problem in most of the
currently used semi-analytic models, e.g. Fontanot et al. (2009) showed this for the
Morgana, Munich and the S08 model, and Guo et al. (2011) found the same for
the latest version of the Munich model. Moreover, the excess of low mass galaxies at
high redshift could have been also verified by cosmological simulations in a study of
Davé et al. (2011), showing that this problem is not peculiar to SAMs alone and the
corresponding larger inaccuracies in the simplified analytic models. One may speculate
that a stronger supernova feedback at high redshift might be required in order to overcome this problem (Caviglia & Somerville, in prep.). Another solution, which will be
investigated in Caviglia & Somerville (in prep.), is that the star formation efficiency is
lower in low mass galaxies at high redshift what might be due to the ability of forming
molecular hydrogen.
4.6
Number density evolution of AGN
In this Section it is investigated whether the SAMs can account for the observed downsizing behavior in black hole growth. Fig. 4.6 shows the different model predictions
for the AGN number densities versus redshift, illustrated by the open squares and the
solid lines. The AGN are divided into different bolometric luminosity classes and the
different luminosity bins are indicated by different colors:
• black: 42.5 < log(Lbol ) < 43.5
• dark blue: 43.5 < log(Lbol ) < 44.5
4.6
Number density evolution of AGN
85
Figure 4.6: Number densities of AGN versus redshift for the four different SAM
versions (FID, VE, VEDI and VEDIS). The different bolometric luminosity bins are
indicated by different colors: red: 47.5 < log(Lbol ), yellow: 46.5 < log(Lbol ) < 47.5,
green: 45.5 < log(Lbol ) < 46.5, light blue: 44.5 < log(Lbol ) < 45.5, dark blue: 43.5 <
log(Lbol ) < 44.5, black: 42.5 < log(Lbol ) < 43.5. Solid lines and open squares illustrate
the corresponding model output, dashed lines together with grey shaded areas visualize
the observational compilation from Hopkins et al. (2007c). Whereas the FID model
reveals typical downsizing problems, the best match with observations can be obtained
for the VEDIS model.
86
Origin of the anti-hierarchical growth of black holes
• light blue: 44.5 < log(Lbol ) < 45.5
• green: 45.5 < log(Lbol ) < 46.5
• yellow: 46.5 < log(Lbol ) < 47.5
• red: 47.5 < log(Lbol )
In a first step, - as the basic output from the SAMs are bolometric luminosities the SAM results are compared to the observational compilation from Hopkins et al.
(2007c) for simplicity. In this study, they convert the AGN luminosities from different
observational data sets and thus, from different wavebands (emission lines, NIR, optical,
soft and hard X-ray) and from different redshift ranges into bolometric ones. They use
a bolometric correction approximated by a double power-law:
Lbol
= c1
Lband
#
Lbol
1010 L"
$k1
+ c2
#
Lbol
1010 L"
$k2
(4.28)
with different values for (c1 , k1 , c2 , k2 ) depending on the considered waveband. Besides
the bolometric correction, they also take into account a dust correction factor, in order
to overcome the problem that obscured AGN are difficult to detect, in particular at
high redshifts and for low luminosities. Note that this ’completeness’ issue might
be the simplest explanation for the low-luminous objects peaking at lower redshifts
than high-luminous ones. In their study, they adopt a luminosity dependent observed
column density distribution (no redshift dependence) from the hard and soft X-ray
observations of Ueda et al. (2003). They also follow Ueda et al. (2003) and include
an equal fraction of Compton-thick objects with NH > 1024 cm−2 compared to that
with 1023 cm−2 < NH < 1024 cm−2 . Altogether, using these observed column density
distributions, they parameterize the observable fraction of AGNs in a certain waveband
as a power-law:
Φ(Li )
= f46
f (Lbol ) =
Φ(L[Li ])
#
Lbol
1046 erg s−1
$β
,
(4.29)
where Li is the luminosity in a certain band, Lbol is the corresponding bolometric luminosity and the parameters (f46 , β) vary for different wavebands. Even if these dust
corrections lead to flattened curves of low luminous AGN (Lbol < 1045 erg/s), they
still find a clear, anti-hierarchical behavior in the evolution of AGN luminosities as
illustrated in the four panels of Fig. 4.6 by the dashed lines together with the grey
shaded areas.
The upper left panel shows the comparison of the FID model to the observational
data. It can be seen that the number densities at the peak of each luminosity bin
are in quantitative agreement with the observations. The maximum AGN number
densities of each luminosity bin versus bolometric luminosities are directly plotted in
4.6
Number density evolution of AGN
87
Fig. 4.7 (dashed line represents the observations, solid line corresponds to the FID
model) showing that the FID model is able to produce automatically the correct order
of magnitude of AGN number densities in the different luminosity bins. However,
the observed time evolution of the peaks of the different luminosity classes cannot be
successfully predicted by the FID model, where the luminosity curves peak all at a
redshift of z ≈ 2. This discrepancy points out the “typical” anti-hierarchical behavior
within the framework of hierarchical structure formation as found by the majority of
the current SAMs (e.g. Fontanot et al., 2006; Marulli et al., 2008). In particular, at
low and high redshifts, the strongest deviations from observations are found, as follows:
1. z < 2: overprediction of AGN number densities with log(Lbol ) > 46
2. z < 2: underprediction of AGN number densities with log(Lbol ) < 46
3. z > 3: underprediction of AGN number densities with log(Lbol ) > 46
4. z > 3: overprediction of number densities with log(Lbol ) < 45
The first point suggests that at low redshifts either too massive black hole are accreting or that black holes accrete at too high rates. Assuming that activity is triggered
by merger events implies that the natural decrease in the major merger rate is not
sufficient in order to predict the observed steep decline in the AGN number densities.
However, note that in earlier version of S08 it was shown that the SAM can reproduce
the observed merger rate (Jogee et al., 2008), which is best constrained at z ≤ 1, so
that the excess in luminous quasars at low redshift is certainly not due to too many
merger events. The low number densities of moderately luminous AGN (point 2) may
indicate, however, that the AGN activity might not only be triggered by merger events,
but also by secular evolution processes as suggested by different observational studies
(Cisternas et al., 2010; Georgakakis et al., 2009; Pierce et al., 2007; Grogin et al., 2005).
The deviations at high redshifts (points 3, 4) may be a consequence of a too late formation of massive black holes as well as of an over-prediction of number densities of
less massive galaxies at high redshifts resulting in too many active galaxies with moderate luminosities. In particular the latter point, that SAMs overestimate the small
mass galaxies at high redshift, was illustrated and discussed in Fig. 4.5 in Section 4.5.2.
In order to achieve a better reproduction of the observed downsizing with the FID
model itself, in a first step the effect of solely changing several of the model parameters
has been investigated, e.g. by varying the strength of Supernova and radio-mode
feedback (as suggested in Fontanot et al., 2006). For stronger supernova feedback
(doubling the efficiency for calculating the reheated gas mass 00SN ) a decrease in the
amount of AGN can be found at all redshifts resulting in a mainly worse match to
the observations. Even if the number density of moderately luminous objects (Lbol <
1045 erg/s) at high redshifts decreases (towards the observational data), this is not
sufficient in order to achieve a reasonably good match to observations in this range.
Due to stronger supernova feedback, gas cooling and star formation is suppressed more
88
Origin of the anti-hierarchical growth of black holes
Figure 4.7: AGN number densities versus bolometric luminosities at the peak of each
luminosity bin. The green solid line shows the FID model, the red solid line the VEDIS
model and the dashed line the observational compilation. Both models can reproduce
the right order of magnitude at the peak of each luminosity bin as observed (Hopkins
et al., 2007c).
efficiently than in the FID model leading to a less efficient bulge and thus, black hole
growth at all redshifts. Increasing the strength of the radio-mode feedback leads to
a smaller amount of luminous AGN (Lbol > 1045 erg/s) between redshifts 0 < z <
4, as radio-mode feedback is very efficient in suppressing cooling and star formation
in massive halos. This again results in a worse agreement with the observational
compilation than the FID model itself. If ’halo quenching’ is assumed instead of the
radio-mode feedback, i.e. no cooling is allowed above a certain threshold halo mass
(Mhalo,thres = 1012 M" ), again, a decrease of the number of luminous AGN is obtained at
all redshifts, but no redshift dependence as observed, similar to the case of a stronger
radio-mode feedback. Moreover, in order to increase the number density of moderately
luminous objects at low redshifts, the timescale (tQ ) in the power-law decline phase of
a quasar episode has been varied. A study from Marulli et al., 2008 has shown that
additionally assuming a power-law decline growth phase in their quasar mode does
increase the number density of less luminous AGN at low redshift, resulting in a better
agreement to observations. However, within this study it was found that varying tQ
did not help to get a better agreement with observations. Therefore, one can conclude
that downsizing cannot be sufficiently reproduced by varying some of the parameters
of the FID model. Instead, the influence of the additional modifications for black hole
growth will be now studied as outlined in Section 4.3.3.
4.6
Number density evolution of AGN
89
The result for the VE model (i.e. assuming a decreasing sub-Eddington accretion
limit with decreasing redshift and increasing mass) is shown in the upper right panel
of Fig. 4.6. For bolometric luminosities larger than 1046 erg/s (green, yellow and red
curves), a mass and redshift dependent sub-Eddington limit leads to a reduced amount
of AGN and thus, to a reasonably good reproduction of the observed steep decline in
the number density of the luminous AGN at z < 2. This result can be seen as clear evidence that an upper limit for black hole accretion - decreasing with decreasing redshift
and with increasing black hole mass - seems to be necessary in order to decrease the
amount of luminous AGN at low z and to predict the observed decline in AGN number
density. This would imply that massive black holes do exist at low redshifts, but they
do not appear as luminous AGN anymore since they are less active and do not accrete
at the Eddington-rate as at earlier times. The underlying reason might be due to the
small amount of the available cold gas content in the host galaxies. One may speculate
that low cold gas densities lead to smaller viscosity so that it takes longer for the gas to
loose its angular momentum and thus, to be accreted onto the black hole. E.g. Hickox
et al. (2009) show observationally that at higher redshifts galaxies contain on average
a larger cold gas fraction than at low redshifts. Star formation and accretion onto
the black hole is quenched when a halo mass of 1012 − 1013 M" is reached. Therefore,
massive ellipticals - hosting massive black holes - contain only a small amount of cold
gas and show no quasar activity anymore.
Similar to the FID model, the VE model (see upper right panel in Fig. 4.6) still
under-predicts the number densities of AGN which are less luminous than 1046 erg/s
at z < 1.5. This failure might be due to missing trigger mechanisms for AGN activity,
since so far, a quasar phase is only driven by major merger events. Thus, the lower
left panel in Fig. 4.6 shows the AGN number density evolution for the VEDI model,
including black hole accretion due to disk instabilities at z ≤ 2. For a luminosity range
of 43 < log(Lbol ) < 45 (light and dark blue curves) it is found that accretion due to disk
instabilities increases the number density of AGN, resulting in a reasonably good match
with the observational compilation for z < 1.5. For the lowest luminosity bin, however, the amount of AGN is now over-predicted. Nevertheless, this additional accretion
mechanism seems to play an important role for triggering AGN activity of moderately
luminous objects. Compared to the Galform-model (e.g. Fanidakis et al., 2010),
however, in the model of this study disk instabilities are only important for moderately
luminous AGN whereas in their model, disk instabilities build up the major contribution for AGN number densities for all luminosities and at all redshift. Contrary, in the
Munich-model, Marulli et al. (2008) and Bonoli et al. (2009) do not account for black
hole accretion due to disk instabilities at all, but they are still slightly under-predicting
the amount of moderately luminous AGN, even in their best-fit model (like in the FID
and VE model).
Finally, the lower right panel of Fig. 4.6 shows the result for the VEDIS model.
Here, a heavy seeding scenario is additionally assumed as well as a large scatter in the
90
Origin of the anti-hierarchical growth of black holes
accreted black hole mass at z > 5 resulting in larger AGN number densities for z > 4
than for the other three models, in particular for AGN with bolometric luminosities
Lbol > 1045 erg/s. This shows clearly that black holes have to undergo at high redshifts
(z > 5) a phase of very rapid growth, even if it is observationally still unknown and
unconstrained whether and how such massive seed black holes can form out of a direct
core-collapse or whether less massive seeds have to accrete at super-Eddington ratios.
However, it is worth to point out that assuming even more massive black hole seeds
with masses of M•,seed = 106 M" results in a too small amount of moderately luminous
AGN at high redshift (z ≈ 5) in the model. Note that the studies of Marulli et al.
(2008) and Bonoli et al. (2009) assuming black hole seeds of 103 M" cannot reproduce
a large enough amount of luminous AGN at high redshift, whereas in the work of
Fanidakis et al. (2010), they obtain a sufficiently large amount of luminous AGN at
high redshift as they allow for super-Eddington accretion.
In the VEDIS model, the peak AGN number densities of the different luminosity
bins still match the observed ones what can be seen by the red, solid line in Fig. 4.7.
Moreover, in contrast to the FID model, the correct AGN number densities of high as
well as of moderately luminous objects at low redshift can now additionally be reproduced and a reasonably good match for high-luminous objects at high redshift can be
obtained. Thus, the VEDIS model is the current best-matching model for reproducing
the downsizing trend in this work. However, in this model, there still exist deviations
from observations in the lowest luminosity bin (black solid curve and light and dark
blue curves at z > 2), where the number of objects is over-predicted. This might be
due to the difficulty of detecting moderately luminous objects at high redshifts, being
a consequence of obscuration by a dusty torus around the accreting black hole, which
might eventually be modeled insufficiently for the dust correction factor applied to the
observational compilation in the study of Hopkins et al. (2007c). Another possible explanation for the over-prediction of moderately luminous AGN at high redshift might
be due to a not strong enough and/or different scalings for supernova feedback at high
redshift (see discussion in Section 4.10).
4.7
4.7.1
The AGN luminosity function
Bolometric luminosities
Despite some minor shortcomings, the success of the VEDIS can also be explicitly seen
in Fig. 4.8, where the bolometric quasar luminosity function (QLF) is plotted for different redshift bins in direct comparison to the FID model. Black symbols show the
observational compilation from Hopkins et al. (2007c), the green lines correspond to
the output of the FID model and the red lines illustrate the result from the VEDIS
model, where a reasonably good agreement is achieved with observations for the whole
redshift range. At redshift z = 5, a change in number density by about an order
4.7
The AGN luminosity function
91
Figure 4.8: Bolometric quasar luminosity functions for different redshifts. Black
symbols show the observational compilation of Hopkins et al. (2007c), the green solid
line corresponds to the output of the FID model and the red line illustrates the results
from the VEDIS model. For the VEDIS model, a reasonably good agreement with
observations can be obtained.
of magnitude is obtained due to the heavy seeding scenario and the large scatter in
the accreted black hole mass in the VEDIS model compared to the FID model. For
the redshift range between 1.5 − 3, the additional model extensions do not cause any
significant differences. However, for low redshifts, in particular at z = 0.1, the decreasing sub-Eddington limit leads to a discrepancy in AGN number densities at the
high-luminous end by 2 − 3 orders of magnitude. Moreover, at the low-luminosity end,
one can still see a change in VEDIS model by almost one order of magnitude as AGN
are assumed to be additionally triggered by disk instabilities. One can conclude, owing
to the better match to the observed QLF in the VEDIS model than in the FID model,
that the additional assumptions might be important and non-negligible ingredients for
galaxy formation and the connected black hole growth and, in particular, for the observed downsizing trend.
92
Origin of the anti-hierarchical growth of black holes
Figure 4.9: Hard X-ray AGN luminosity function for different redshifts ranges. Black
symbols show the observational compilation of Aird et al. (2010), the green solid lines
correspond to the output of the FID model and the blue dashed lines illustrates the same
results with dust correction.
4.7.2
Hard X-ray luminosities
Besides comparing the model output to the bolometric luminosities from the observational compilations of Hopkins et al. (2007c), the predicted bolometric luminosities
from the model are converted into hard X-ray luminosities in order to compare them
directly to the observed hard X-ray luminosities of a recent observational study from
Aird et al. (2010). In order to calculate the hard X-ray luminosities, we use the bolometric correction according to Marconi et al. (2004), where the hard X-ray luminosities
are approximated by the following third-degree polynomial fit:
log(LHXR /Lbol ) = −1.54 − 0.24L − 0.012L2 + 0.0015L3
(4.30)
with L = log(Lbol /L" ) − 12. These corrections are derived from template spectra,
which are truncated at λ > 1 µm in order to remove the IR bump and which, and
hence the bolometric corrections, are assumed to be independent of redshift. Moreover, a dust correction for the model luminosities is additionally assumed, as suggested
4.7
The AGN luminosity function
93
Figure 4.10: Hard X-ray AGN luminosity function for different redshifts ranges.
Black symbols show the observational compilation of Aird et al. (2010), the red solid
lines correspond to the output of the VEDIS model and the blue dashed lines illustrates
the same results with dust correction.
by different observational studies (Ueda et al., 2003; Hasinger, 2004; La Franca et al.,
2005), where they show that the fraction of obscured AGN is luminosity dependent
and decreases with increasing luminosity. However, whether there exists an additional
redshift dependence of the obscured fraction is under current, intense debate. Studies
from Ballantyne et al. (2006) and Gilli et al. (2007) propose a strong evolution of the
obscured AGN population (a relatively increasing fraction of obscured AGN with increasing redshift) to reproduce the properties of the X-ray backround, whereas Ueda
et al. (2003) and Steffen et al. (2003) do not find a clear dependence of the obscuration
on redshift. In this thesis, I follow a recent observational study from Hasinger (2008),
where they investigate a sample of X-ray selected AGN from ten independent samples.
They find that the fraction of obscured AGN increases strongly with decreasing luminosity and increasing redshift. Following Hasinger (2008), the obscured fraction of
AGN is modeled and approximated by this equation (see also Fanidakis et al. (2010)):
fobsc = −0.281(log(LHXR ) − 43.5) + 0.308(1 + z)α .
(4.31)
94
Origin of the anti-hierarchical growth of black holes
Fitting this equation to observations, as it is shown in Hasinger (2008), results in a value
for the exponent of α = 0.62. However, in this study, it is found that an even stronger
redshift dependence is needed to be consistent with the observations (α ≈ 0.8). With
calculating the obscured fraction of AGN in the hard X-ray band, the visible fraction
of AGN fvis = 1 − fobsc can be modeled, and thus the visible number density of AGN
in the hard X-ray range can be approximated by:
Φvis (LHXR ) = fvis × Φtotal (LHXR )
(4.32)
Fig. 4.9 shows the hard X-ray luminosity function for different redshift ranges. The
black symbols illustrate the observations from Aird et al. (2010), the green lines
show the predictions from the fiducial model without dust correction, whereas the
blue dashed lines correspond to the number densities of visible AGN. At low redshift
(z = 0 − 1), the fiducial model under-predicts moderately luminous AGN and overpredicts luminous AGN, as already seen for the bolometric luminosity functions in Fig.
4.8. As expected, in this redshift range, dust correction plays only a minor role and
leads to small differences at the low luminosity end. Turning to higher redshift z > 1.5,
the high luminosity end is only slightly overestimated anymore. However, at the low
luminosity end the fiducial model without dust correction extremely over-predicts the
number densities of these objects. Therefore, considering the number densities only
of the visible fraction of AGN leads to a significantly better agreement with the observations at the low mass end. This shows clearly, that the model output confirms
the observational finding of the existence of a strong redshift dependence in the obscured fraction of AGN. Besides, we can conclude that dust correction alone seems not
to a be sufficient process in order to reproduce downsizing in the FID model, as at
low redshift, the dust correction can account neither for the excess of luminous AGN
nor for the lack of moderately luminous AGN. Considering the output of the VEDIS
model, Fig. 4.10 illustrates the corresponding hard X-ray luminosities (solid, red lines)
compared to the observations (black symbols). The blue, dashed lines show the results
considering in addition the dust correction for the VEDIS model. Compared to the
FID model, the predictions of the VEDIS model lead to a significant better agreement
with the observations at redshifts z ≤ 1 as through the additional modifications the
number density of luminous AGN is reduced and the amount of moderately luminous
AGN gets increased, as it was already shown for the bolometric luminosity function
above. The results including dust obscuration show still a fairly good match with observations at this redshift range. Therefore, one can conclude that the best-fit VEDIS
model combined with a redshift and luminosity dependent dust obscuration correction
is successful in predicting the observed hard X-ray luminosity function for the whole,
observed redshift range. Note that Fanidakis et al. (2010) use the same bolometric
conversion for calculating hard X-ray luminosities and the almost the same dust obscuration correction (they use a smaller value for α). They show that they are able
to match the hard X-ray luminosity functions from an observational study of Hasinger
et al. (2005). However, their predictions are only illustrated for a comparatively small
redshift range of 0.2 < z < 1.6, where they have not applied any dust obscuration to
4.8
Eddington-ratio distributions
95
Figure 4.11: Eddington-ratio distributions for the FID, VE, VEDI and VEDIS models. Different colors correspond to different redshifts. It shows that most black holes
are not radiating at the Eddington-limit. This is in qualitative agreement with observations (Vestergaard, 2003; Kollmeier et al., 2006; Kelly et al., 2010; Schulze & Wisotzki,
2010). Moreover, with decreasing redshift, the peak of the distribution is shifted towards
smaller Eddington-ratios and the distributions are broadened.
their hard X-ray luminosities.
4.8
Eddington-ratio distributions
For a more fundamental understanding of the implications and consequences of the
downsizing behavior for galaxy evolution and black hole growth caused by the additional model modifications, in this and the following Section, the Eddington-ratio distributions and the relation between black hole mass and bolometric AGN luminosity
are investigated. The four panels in Fig. 4.11 show the Eddington-ratio distributions
during the quasar phase for the four SAMs at different redshift steps, illustrated by
different colors (z = 0, 0.5, 1, 2, 3, 4, 5, 6). In all models and at all redshift, the majority of AGN are found not to be radiating near or at the Eddington-limit as black
96
Origin of the anti-hierarchical growth of black holes
holes spend most of their time not in the first regime, accreting at the Eddington-rate
but in the second regime, accreting in the power-law decline phase of the light curves.
This is in qualitative agreement with different observational studies (Vestergaard, 2003;
Kollmeier et al., 2006; Kelly et al., 2010; Schulze & Wisotzki, 2010). For example, Kelly
et al., 2010 find that their Eddington-ratio distribution (using broad-line quasars between z = 1 − 4) peaks only at an Eddington-ratio of fedd = 0.05.
In the FID model (upper left panel), one can see clearly that the peaks of the
Eddington-ratio distributions emerge towards smaller ratios with decreasing redshift:
at z = 5 the distribution peaks at fedd ≈ 0.1, whereas at z = 0 the peak is located around fedd ≈ 10−3. This shows that with decreasing redshift an increasing
number of black holes is accreting with smaller Eddington-ratios in the power-law decline phase of the light curves. These black holes are relicts from an earlier, more
active phase with higher accretion. Moreover, the distributions are broadened with
decreasing redshift. This qualitative trend of a shifted and broadened peak towards
lower redshift with decreasing redshift is also qualitatively consistent with observational studies (see Vestergaard, 2003; Kollmeier et al., 2006; Netzer & Trakhtenbrot,
2007). In the VE model (see upper right panel), where it is additionally assume that
the black holes are not allowed anymore to accrete at the Eddington-rate in the first
regime, but instead adopt an sub-Eddington limit decreasing with redshift and increasing black hole mass, the trend of the shifted peak towards smaller Eddington-ratios
gets even amplified: by construction, there exist no larger Eddington-ratios than 0.01
at z = 0, and the distribution is shifted towards even smaller Eddington-ratios peaking
at fedd ≈ 10−4 . However, in observations this shift is much less pronounced than in
this theoretical study. This might eventually be due to observational flux limits only
reaching Eddington-limits in excess of fedd = 10−3 . Nevertheless, a stronger shift and
smaller upper accretion limit (compared to the FID model) in the Eddington-ratio
distribution, preventing especially massive quasars to accrete at the Eddington rate,
might be necessary for reproducing the small, observed amount of luminous AGN at
low redshift. Interestingly, with the Galform model (see Fanidakis et al., 2010), the
Eddington-ratio distribution at z = 0 peaks also around fedd = 10−4 in agreement with
this result even if their model is fundamentally different to the approach in this study.
Using the VEDI model (see lower left panel), at redshifts z ≤ 2, the distributions are
additionally broadened compared to the FID and the VE models, mainly towards lower
Eddington-ratios. This is due to the contribution of the black hole accretion following
disk instabilities, which results in Eddington-ratios fedd ≤ 0.01. Finally, larger seed
black hole masses in the VEDIS model (see lower right panel) lead to a much larger
amount of accreting objects near to the Eddington-limit (0.1 < fedd < 1) at z > 3. In
particular, at redshift z = 6 the number density of accreting black holes in a quasar
phase is increased by one order of magnitude (log(dN/dfedd ) ≈ −4.2) compared to the
other three models (log(dN/dfedd ) ≈ −5.2). Note that in the study of Fanidakis et al.,
2010, they obtain an even larger number density at z = 6 (log(dN/dfedd ) = −3.4) peaking at larger Eddington-ratios fedd = 0.5 (allowing additionally for super-Eddington
4.9
Luminosity-black hole mass-relation
97
accretion) than in this study.
4.9
Luminosity-black hole mass-relation
Fig. 4.12 shows the bolometric AGN luminosity versus black hole mass at z = 0 (left
panels), z = 2 (middle panels) and z = 4 (right panels) for the four different SAMs
(different rows). The different points correspond to the different objects in the sample
with bolometric luminosities higher than Lbol > 1043 erg/s. The red solid line illustrates accretion at the Eddington rate, the red dashed line depicts an Eddington-ratio
of fedd = 0.1 and the red dashed-dotted line an Eddington-ratio of fedd = 0.01.
In all models, at z = 4 (right column in Fig. 4.12) an almost linear correlation
between luminosity and black hole mass can be seen, indicating that the larger the
black hole mass the more gas is accreted by black holes resulting in larger bolometric
luminosities. Nearly all black holes at this redshift are accreting at Eddington-ratios
larger than fedd < 0.01 (see also Fig. 4.11). In the FID, VE and VEDI model black
holes with masses 104.5 M" < M• < 108 M" are active, whereas in the VEDIS model,
there exist no black holes accreting below log(M• /M" ) < 5.5 due to the heavy black
hole seeds. Moreover, in the VEDIS model, a larger amount of black holes with masses
106 M" < M• < 107.5M" is actively accreting gas than in the other three models.
Turning to redshift z = 2 (middle column Fig. 4.12), in all models, a larger number
of active black holes exists than at z = 4, however, the relation between black hole
mass and bolometric luminosity vanishes. E.g. black holes with masses M• ≈ 108 M"
can now also power moderately luminous AGN with Lbol ≈ 1043 erg/s as they are accreting clearly with Eddington-ratios below fedd < 0.01. Moreover, the probability
for black holes with M• > 107 M" to accrete at Eddington-ratios below fedd = 0.01
is even higher than to accrete at larger Eddington-ratios (consistent with Fig. 4.11).
This is due to the long power-law decline accretion phase, black holes are experiencing: massive black holes powering moderately luminous AGN are remnants of former,
high-luminous AGN. Finally at redshift z = 0 (left column Fig. 4.12), the amount of
actively accreting black holes is again clearly reduced compared to z = 2 and significant
differences between the FID, the VE and the VEDI models appear. In the FID model
(upper left panel), a few high-mass black holes (M• ≈ 109 M" ) are accreting close to
the Eddington-limit, causing the high number densities of luminous AGN at low redshifts compared to the observations seen in Fig. 4.6 (yellow and red curve). This is,
however, suppressed in the VE model, where the black holes are forced not to accrete
above Eddington-ratios of fedd = 0.01 resulting in a smaller amount of luminous AGN
at low redshift. Furthermore, in the VEDI model, there exists an increased amount
of active black holes with masses of 107 M" < M• < 108 M" due to the additional
disk instabilities triggering a quasar phase. These black holes mainly contribute to the
amount of AGN with bolometric luminosities between 1043 erg/s < Lbol < 1045 erg/s.
As Seyfert galaxies are mainly spiral galaxies with black hole masses in the range of
98
Origin of the anti-hierarchical growth of black holes
Figure 4.12: Bolometric luminosity versus black hole mass at z = 0 (left panels),
z = 1 (middle panels) and z = 2 (right panels). The points correspond to the actively
accreting black holes in the sample. The first row shows the results of the fiducial
model, the second row for the VE model, the third row for the VEDI and the lower row
corresponds to the results from the VEDIS model. The red solid line illustrates accretion
at the Eddington-rate fedd = 1, the red dashed and dotted dashed lines fractions of the
Eddington-rate fedd = 0.1 and fedd = 0.01.
4.10
Discussion and conclusion
99
107 M" < M• < 108 M" and are - compared to quasars - only moderately luminous,
disk instabilities seem indeed to provide one of the most important underlying trigger
mechanisms for their nuclear activity.
Note that in the study of Fanidakis et al. (2010), they obtain almost no evolution
of their luminosity-black hole mass relation within a redshift range 0.5 < z < 2. Black
holes above M• > 109 M" accrete always below fedd = 0.01 (ADAF regime), as in the
VE, VEDI and VEDIS models. However, in clear contrast to this work, in this redshift
range, they still have super-Eddington accretion, in particular for black hole masses
between 108 M" < M• < 109 M" (resulting in Lbol > 1046 erg/s). This is, however, in
clear contrast to several observational studies, e.g. to Steinhardt & Elvis (2010), where
they show that black holes with masses M• > 107 M" are not accreting near or at the
Eddington-limit anymore at low redshifts z = 0.2 − 0.4.
Furthermore, Kollmeier et al. (2006) investigate a sample of 407 AGN from the
AGES survey at z ∼ 0.3 − 4 with bolometric luminosities between 1045 erg/s < Lbol <
1047 erg/s. In their sample, black holes are mainly accreting between fedd ∼ 0.1 − 1.
Massive black holes (M• > 108.5M" ) are found to be active at high redshift (z = 3-4),
whereas at z < 0.5 mainly less massive objects are active (M• < 108 M" ). Such a mass
dependence, however, cannot be reproduced by the four models so far.
4.10
Discussion and conclusion
In this Chapter, different mechanisms have been presented significantly influencing the
co-evolution of galaxies and black holes, which may provide possible explanations for
the observed anti-hierarchical trend in black hole growth. For this study, the semianalytic model according to S08 has been used and three further modifications based
on this fiducial model. The semi-analytic code is applied to the merging histories of
the Millennium-simulation and contains most of the currently known physical processes
which are important for galaxy formation. Moreover, the fiducial (FID) model includes
an advanced prescription for black hole growth following the light curve models for AGN
according to isolated merger simulations. Up to a critical mass limit, black holes are
allowed to accrete at the Eddington-limit, whereas afterwards, in the power-law decline
phase, they accrete only at fractions of the Eddington-rate. Black holes spend most of
their time in the latter, low-Eddington-ratio regime. For the FID model it is found that
the peaks of the distributions of Eddington-ratios move towards smaller values and the
distributions get broadened with decreasing redshift in qualitative agreement with observational findings (Vestergaard, 2003; Kollmeier et al., 2006; Netzer & Trakhtenbrot,
2007; Kelly et al., 2010; Schulze & Wisotzki, 2010). Furthermore, the observed order
of magnitude of AGN number densities at the peak of different luminosity bins can be
reproduced. However, the FID model still reveals the characteristic anti-hierarchical
failures within the framework of a hierarchical clustering scenario: at low redshift, the
100
Origin of the anti-hierarchical growth of black holes
FID model overproduces the number densities of luminous AGN and under-predicts
the amount of moderately luminous AGN and at high redshift, this trend is reversed.
By implementing the outlined modifications, a significantly improved match to the
observations can be achieved, i.e. to the bolometric and hard X-ray AGN luminosity
function showing the observed downsizing the trend:
1. A decreasing sub-Eddington limit with increasing mass and decreasing redshift
reduces the amount of luminous AGN at low redshift (VE model).
2. Accretion onto the black hole due to disk instabilities is responsible for increasing
the amount of moderately luminous AGN at low redshift (VEDI model).
3. A heavy seeding scenario together with a large scatter in the accreted gas mass
results in a larger amount of luminous AGN at high redshift (VEDIS model).
Despite of these modifications, the observed galaxy and black hole properties of the
present-day Universe in the VEDIS model, as the galaxy-halo mass relation and as the
black hole mass function can be reproduced at z = 0. For the FID as well as for the
VEDIS model it is found that an additional approximation for gravitational heating
combined with a less efficient radio-mode accretion leads to a better agreement with
the observed black hole mass function and the galaxy-halo mass relation at the high
mass end balancing the stellar and black hole growth in a more physical way than by
assuming a more efficient radio-mode accretion and feedback alone. However, turning to the present-day black hole-bulge mass relation, the modifications in the VEDIS
model lead to a slightly worse agreement with the observations than the FID model.
This is due to the decreasing upper limit of the Eddington-ratios resulting in a slower
growth of black holes compared to the bulges and thus, in a smaller ratio of black hole
to bulge mass than expected from observations. This shows clearly that in this respect,
there is still need for improvement, even in the best-fit VEDIS model.
A sub-Eddington limit at low redshift (as suggested in observational studies by Netzer & Trakhtenbrot (2007); Steinhardt & Elvis (2010)), which is dependent on black
hole mass and redshift, leads to a strikingly good agreement with the luminous AGN
number density at low redshift as it prevents massive black hole to accrete near or at
the Eddington-rate. As one may argue that the assumption of a sub-Eddington-limit
at low redshift should be a self-consistent result from the model, a direct dependence of
the gas accretion onto the black hole on the cold gas content, which is available in the
galaxy, was tested. E.g. Hopkins et al. (2008a) have shown in simulations that allowing
dry mergers to trigger quasar activity would overproduce the observed quasar luminosity density. Thus, a simple model was assumed, where the final black hole mass in
the beginning of a quasar was linked directly to the cold gas mass M•,final = 0.005Mcold
(note that in the FID model black hole accretion is related to the spheroid component
of the merged system and thus, it is only indirectly linked to the cold gas content).
Interestingly, this results in a natural downsizing behavior of the peaks of the curves:
the peak positions evolve slightly towards lower redshifts with decreasing luminosity
4.10
Discussion and conclusion
101
of the AGN. This suggests that directly linking black hole accretion to the cold gas
fraction may represent a promising way for modeling gas accretion onto black holes in
SAMs. However, the quantitative shape of the curves, in particular the steep decline in
number densities of luminous AGN at low redshifts (Fig. 4.6), is not in agreement with
the observations so far, even if the simple approximation for gravitational heating has
been adopted. Moreover, it seems reasonable to additionally scale the final black hole
mass with the mass ratio of the two merging galaxies. In a study of Johansson et al.
(2009a) it was shown that in isolated merger simulations the growth of a black hole
during a merger event is extremely sensitive to the mass ratio (a smaller mass ratio
leads to a reduced growth of the black hole). With the additional scaling of the final
black hole mass with the mass ratio, indeed, the small number densities of luminous
AGN at z = 0 can be reproduced. However, within the redshift range of 0 < z < 3,
this model would significantly under-predict the amount of luminous AGN resulting
again in a wrong shape of the curves compared to the observations. This clearly shows
that obtaining an evolving sub-Eddington limit with redshift for the Eddington-ratios
in a self-consistent way from the FID model seems to be a pretty complex process.
Thus, either the recipe for connecting black hole accretion to the cold gas content may
have to be refined or the evolution of the cold gas content in the FID SAM may not be
correctly modeled leading to a quantitatively wrong behavior. This might be due to
incomplete prescriptions for feedback processes, in particular supernova feedback, or
due to incorrect or simplified cooling recipes in the SAM. In Section 5.5.3 of the next
Chapter, it will be shown that the cooling in SAMs compared to cosmological simulations seems to be insufficient, in particular for massive objects at high redshifts. One
possibility for refining the cooling recipe in the SAM might be to assume in general cold
gas infall and additionally to implement gravitational heating in a more self-consistent
way than it is done so far (see e.g. Khochfar & Ostriker, 2008) in order to calculate
the heated gas fraction (see also Chapter 5).
Additional gas accretion onto black holes due to disk instabilities is found to be
a non-negligible trigger mechanism for moderately luminous AGN with black hole
masses 107 M" < M• < 108 M" at low redshift. This increases the amount of AGN
with bolometric luminosities 1043 erg/s < Lbol < 1045 erg/s by a factor of 2 − 4. This is
in agreement with different observational studies (Cisternas et al., 2010; Georgakakis
et al., 2009; Pierce et al., 2007; Grogin et al., 2005), where they show that in morphologies of nearby AGN galaxies, no stronger distortions can be seen, than in the
ones of quiescent galaxies. This suggests that the nuclear activity cannot only be a
consequence of major merger events, but also by secular evolution processes as disk
instabilities. With the results in this Chapter, the current picture can be confirmed
that disk instabilities are the main trigger mechanism to power moderately luminous
Seyfert galaxies at low redshift, whereas major merger events are mostly driving the
nuclear activity in high luminous quasars at high redshift (see e.g. Hopkins & Hernquist, 2009). Note that neither the Munich nor the Galform model come to similar
conclusions (see below).
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Origin of the anti-hierarchical growth of black holes
Heavy seeding mechanisms for black holes are found to provide one, straight-forward
possibility in order to reproduce the observed amount of luminous AGN at high redshift. This is in agreement with the studies from Volonteri et al. (2008) and Volonteri
(2010) showing that either massive black hole seeds are required or less massive seeds
have to accrete at super-Eddington rates. Unfortunately, current observational constraints for such high redshift are not sufficient to favor one of these possibilities what
will be, however, possible with the next generation of planned X-ray missions (as e.g.
WFXT, IXO).
Despite the success of the final VEDIS model in reproducing the bolometric AGN
luminosity function fairly well, the amount of low luminous AGN is still over-estimated
at high redshift z > 2. However, this is a common problem of most current SAMs (see
e.g. Marulli et al., 2008; Fontanot et al., 2006). One possible explanation might be due
to dust obscuration as even 2 − 10 keV X-ray surveys will miss a significant fraction
of moderatly obscured AGN ( 25% at NH = 1023 cm−2 ) and nearly all Compton-thick
AGN (NH > 1024 cm−2 , Treister et al., 2004; Ballantyne et al., 2006). From fits to the
cosmic X-ray background, Gilli et al. (2007) predict that both moderately obscured
and Compton-thick AGN are as numerous as unobscured AGN at luminosities higher
than (log(L0.5−2keV ) > 43.5[ergs/s]), and four times as numerous as unobscured AGN
at smaller luminosities (log(L0.5−2keV ) < 43.5[ergs/s]). The obscured and Comptonthick AGN missed in deep X-ray surveys, therefore, serve not only as important probes
of SMBH/galaxy co-evolution, but likely constitute a significant fraction of the total
AGN population at all luminosities, in particular of moderately luminous AGN at high
redshift. The observational compilation of Hopkins et al. (2007c), which is used to
be compared to the model output, does take into account a dust correction factor,
where they assume a luminosity dependence of the obscured fraction (the less luminous the more obscured) and the same number of Compton-thick (NH > 1024 cm−2 )
and Compton-thin (1023 cm−2 < NH < 1024 cm−2 ) AGN. However, due to insufficient
observational constraints for obscuration of Compton-thick objects and of whether
there exists an additional redshift dependence for the obscured fraction of AGN (see
Hasinger, 2008), dust obscuration processes may be not sufficiently considered in Hopkins et al. (2007c) and thus, they may provide a possible explanation for the deviation
between the modeled bolometric luminosities of moderately luminous AGN and the
bolometric luminosities of the observational compilation at high redshift. This statement might be confirmed as the VEDIS model is in fairly good agreement with the low
luminosity end of the observed hard X-ray luminosity function at high redshift, when
a strong redshift dependence of the dust obscuration is assumed, even stronger as it is
found in Hasinger (2008). From the latter point, one can conclude that dust correction
indeed may provide a significant contribution in explaining the moderately luminous
AGN number densities peaking at low redshift, but I want to emphasize that it cannot
account for the observed downsizing completely on its own (see FID model and dust
obscuration in Fig. 4.9).
4.10
Discussion and conclusion
103
Additionally, as a relatively strong redshift dependence for the dust obscuration
is needed in order to be in agreement with observations, the over-estimation of low
luminous AGN at high redshift might be also connected to the over-estimation of low
mass galaxies at high redshift in SAMs (see Fig. 4.5). This is also found in other SAMs
(Fontanot et al., 2009; Guo et al., 2011) as well as in simulations (Davé et al., 2011)
showing that this problem is not only due to inaccuracies of SAMs. An additional or
alternative explanation might be that in the SAM, feedback processes from supernova
are not sufficiently accounted for at high redshift and the corresponding recipes might
have to be refined (Somerville & Caviglia, in prep.). Apparently, there exists currently
no trivial solution, as simply changing the corresponding parameters, which are responsible for supernova feedback, does not solve this problem.
When the results of this Chapter are compared to the findings of some previous studies (see Section 4.2), using the Munich and the Galform SAM, the following main
points can be summarized: The best-fit model based on the Munich semi-analytic
code presented in Marulli et al. (2008) and Bonoli et al. (2009) is able to reproduce
the bolometric AGN luminosity function until at redshift z = 3. This is mainly due
to the AGN light curves as well as an explicit redshift dependence of the accreted gas
mass onto the black holes during the quasar phases, which is similar to the redshift
dependence assumed for the upper limit of accretion in the model. Interestingly, even
if in their model the black hole accretion is directly connected to the cold gas mass of
the host galaxy, they still need this explicit redshift dependence for their best-fitting
model. However, contrary to this thesis, they claim that they do not need AGN activity triggered by disk instabilities, what seems to be, however, not to be consistent with
recent observations. Moreover, they fail to reproduce the AGN luminosity function
at high redshift (z > 3), as they predict too few luminous AGN and too many low
luminous AGN. At least the first point might be due to their assumption of black hole
seeds to have masses of M•,seed = 103 M" . I want to point out that their assumptions
for the parametrization of black hole accretion in their best-fit model seem to be rather
crude and ad-hoc and they do not seem to be physically motivated.
Compared to the Durham Galform model (Fanidakis et al., 2010) it should be
emphasized that the main difference to the SAM in this study is the trigger mechanism for black hole activity: in their model, the major contribution originates from
disk instabilities for all luminosities at all redshift. With their model they claim to
reproduce the AGN luminosity function at all redshift: the low-luminosity end of the
AGN luminosity function is mainly due to their ADAF model (cold gas accretion onto
the black hole in a hot halo), whereas in the mode used here, it is due to accretion in the
power-law decline regime as well as black hole accretion following disk instabilities. At
high redshift, they predict that super-Eddington accretion is the major contribution for
getting a large enough amount of luminous AGN. At this point, whether massive seed
black holes (as assumed in this study) or super-Eddington accretion describe the correct
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Origin of the anti-hierarchical growth of black holes
physical processes, the improvement of high-redshift AGN observations will be of crucial importance, both by enlarging the current high-z AGN samples and by reducing the
current uncertainty originating from incompleteness problems. They mainly attribute
the downsizing trend (that moderatly luminous objects peak at smaller redshift than
luminous objects) to dust obscuration effects and do not discuss any possible improvements for the supernova feedback and the connected reduction of low mass galaxies.
Altogether one can conclude that the model modifications presented in this Chapter
state possible attempts in order to understand the origin and the underlying ingredients
of the observed downsizing within hierarchical structure formation: at low redshift,
very massive black holes do exist and have already assembled their mass at earlier
times (mostly until z ≈ 2), but they seem to accrete only at a small fraction of the
Eddington-rate anymore and additionally, disk instabilities seem to become a more
important mechanism with decreasing redshift for triggering the activity in moderately
luminous Seyfert galaxies. In other words, downsizing does not necessarily imply that
the growth of low-mass black holes is delayed to low redshift. At high redshift, massive
black hole seeds together with a larger scatter in the accreted gas mass might be
necessary in order to allow for a sufficiently large amount of luminous AGN. However,
the remaining excess of moderately luminous AGN at high redshift might be partly
explained by a redshift dependent dust obscuration (Hasinger (2008)). Additionally, a
stronger supernova feedback required in the SAMs may also contribute to the reduction
of moderately luminous AGN allowing for further improvements of the semi-analytic
model.
Chapter
5
Galaxy formation in semi-analytic models
and zoom simulations
In this Chapter, a detailed comparison is presented between cosmological zoom
simulations and semi-analytic models (SAMs) run within merger trees extracted from the simulations. The simulations represent 48 halos with virial
masses in the range 2.4 ×1011 M" < MHalo < 3.3 ×1013 M" with unprecedented
resolution for a sample this large and covering such a broad range in halo mass.
The simulations include radiative H & He cooling, photo-ionization, star formation and thermal SN feedback. Comparisons with different SAM versions
are included. This analysis is focused on the cosmic evolution of the baryon
content in galaxies and its division into various components (stars, cold gas,
and hot gas). Also both the SAMs and simulations are compared with observational relations between halo mass and stellar mass, and between stellar
mass and star formation rate, at low and high redshift. The simulations turn
out to have much higher star formation efficiencies (by about a factor of ten)
than the SAMs, despite nominally being both normalized to the same empirical Kennicutt relation at z = 0. Therefore the cold gas is consumed much
more rapidly in the simulations and stars form much earlier. Also, simulations
show a transition between stellar mass growth that is dominated by in-situ
formation of stars to growth that is predominantly through accretion of stars
formed in external galaxies. In SAMs, the stellar growth is always dominated
by in-situ star formation, because they significantly under-predict the fraction
of mass growth from accreted stars relative to the simulations. In addition,
SAMs overestimate the fraction of “hot” relative to “cold” accretion. The reasons for these discrepancies are discussed, and several physical processes are
identified that are currently missing in the SAM, but which should be included.
This study has been submitted to MNRAS (Hirschmann et al., 2011).
106
5.1
Galaxy formation in semi-analytic models and zoom simulations
Different approaches for modeling galaxy formation
The assembly of dark matter halos, which dominate the total matter content in the
Universe, in large cosmological volumes can be followed with merger trees based on
analytic approaches, e.g. by using Monte-Carlo methods based on the extended PressSchechter formalism (EPS, Press & Schechter, 1974; Bower, 1991; Bond et al., 1991;
Somerville & Primack, 1999b; Neistein & Dekel, 2008; Zhang et al., 2008; Angulo &
White, 2010). Alternatively, as it was demonstrated in Chapter 3, the full dynamical
evolution of dark matter can be accurately followed with collisionless particles in direct
numerical simulations which are, by now, well resolved at the relevant scales (Frenk
et al., 1988; Navarro et al., 1997; Moore et al., 1999; Klypin et al., 1999; Bode & Ostriker, 2003; Springel et al., 2005a, 2008; Diemand et al., 2008). Here the identification
of dark matter halos and the construction of merger trees is considerably more demanding and various different approaches have been discussed (e.g. Davis et al., 1985;
Kauffmann et al., 1993; Ghigna et al., 2000; Springel et al., 2001a; Weller et al., 2005;
Genel et al., 2008; Fakhouri & Ma, 2008; Planelles & Quilis, 2010; Skory et al., 2010
and references therein). In particular, Springel et al. (2005a) constructed and analysed
merger trees for the Millennium simulation, which were the basic input in Chapter 4,
using the “Friends of Friends” (FOF) technique (Davis et al., 1985) to identify halos,
and Subfind (Springel et al., 2001a) to identify sub-halos (bound objects within larger
virialized dark matter halos).
Simulations of the formation and evolution of the galaxies which are believed to inhabit these dark matter halos are more demanding, theoretically as well as numerically.
Additional gas-dynamical and radiative processes, such as the formation of stars and
black holes as well as the respective feedback, have to be taken into account. To follow the evolution of galaxies two main approaches have been developed over the past
decades: Semi-analytic models (SAMs) and direct cosmological simulations. SAMs,
which have been the main tool of investigation in Chapter 4, use pre-calculated dark
matter merger trees either from EPS or direct cosmological simulations and follow the
formation of galaxies with simplified, physically and observationally motivated, analytic
recipes (White & Frenk, 1991; Kauffmann et al., 1993; Cole et al., 1994; Kauffmann,
1996; Somerville & Primack, 1999b; Kauffmann et al., 1999; Kauffmann & Haehnelt,
2000; Cole et al., 2000; Springel et al., 2001a; Hatton et al., 2003; Kang et al., 2005;
Baugh et al., 2005; Khochfar & Silk, 2006a; Croton, 2006; Bower et al., 2006; De Lucia
& Blaizot, 2007; Somerville et al., 2008b; Font et al., 2008; Guo & White, 2009; Weinmann et al., 2009). The computational costs of this approach are typically low, and the
influence of different physical mechanisms can be investigated separately in a straightforward way. Modern SAMs are quite successful at reproducing observed statistical
properties of galaxies in large cosmological volumes over a large range of galaxy masses
and redshifts (e.g. Somerville et al., 2008b; Guo et al., 2011), e.g. the downsizing
behavior in black hole growth as it was demonstrated in Chapter 4. However, disad-
5.1
Different approaches for modeling galaxy formation
107
vantages of SAMs are that the dynamics of the baryonic component (gas and stars)
and the interaction between baryonic matter and dark matter are not followed directly
and that in many cases the assumed models are simplified and use a large number of
free parameters to fit different observations simultaneously (Somerville et al., 2008b;
Benson & Bower, 2010; Bower et al., 2010).
In contrast, direct cosmological, hydrodynamical galaxy formation simulations can
follow the evolution of dark matter and gas explicitly. Even though they treat the
underlying dynamics more correctly than SAMs, the spatial and mass resolution, at
present, is not high enough to accurately simulate intermediate and low mass galaxies
in large cosmological volumes. In addition, small scale processes like e.g. the formation
of stars and black holes with the associated feedback has to be computed in a simplified
manner with sub-grid/sub-resolution models (Cen & Ostriker, 1993; Davé et al., 2001;
Springel & Hernquist, 2003; Maller & Bullock, 2004; Nagamine et al., 2005; Kereš et al.,
2005; Navarro et al., 2009; Schaye et al., 2010b), which again require the introduction
of additional free parameters. "Ab-initio" cosmological zoom simulations with proper
cosmological boundary conditions enable direct simulations of the baryonic physics of
certain regions of interest at higher resolution, either limited to small cosmological volumes (Crain et al., 2009) or, more popularly, individual halos (Navarro & Steinmetz,
1997; Governato et al., 2007; Naab et al., 2007; Brooks et al., 2009; Oser et al., 2010;
Wadepuhl & Springel, 2011; Puchwein et al., 2010; Teyssier et al., 2010b; Sawala et al.,
2010; Piontek & Steinmetz, 2011; Agertz et al., 2011). These simulations can attain
very high resolution, and provide a way to resolve galaxies of very different masses
with the appropriate resolution in each case. However they are very time consuming
and therefore not currently feasible for representative studies of large populations of
galaxies. In addition, the sub-resolution models are uncertain and it is still unclear
how sensitive various results may depend on the details of these sub-grid models or the
parameter values.
Both approaches make definite predictions for the evolution of galaxy properties
at various masses over cosmic time. Because of their greater computational efficiency,
SAMs generally include more models for physical processes than current numerical
simulations, and because of their greater flexibility, it has been possible to tune them
to obtain quite good agreement with a broad range of galaxy properties in the local
Universe. SAMs have also been shown to reproduce the statistical properties (e.g.
luminosity and stellar mass functions, star formation rates) of high redshift galaxies
(z ! 6) quite well, at least for massive galaxies (mstar " 1010 M" ; e.g. Somerville et al.
2011; Fontanot et al. 2009). In particular, in the last Chapter was shown that applying
some modifications to the model from Somerville et al. (2008b) the evolution of the
quasar luminosity function (bolometric and hard X-ray) can be reproduced reasonably
well. Therefore, one might expect SAMs to do a better job of reproducing the observed
Universe than the simulations, but one might worry that they could do so for the wrong
reasons. Because there is a great deal of uncertainty in many of the important pro-
108
Galaxy formation in semi-analytic models and zoom simulations
cesses, and most of the physical recipes contain free parameters, if one physical process
(e.g. gas cooling and accretion) is modeled inaccurately in the SAM, it is currently
possible to compensate by tuning a competing process (such as feedback). By running
SAMs within merger trees extracted from numerical simulations, in order to constrain
the evolution of the dark matter component to be the same in both cases, various
physical processes can be isolated and it can be attempted to improve the accuracy
of the semi-analytic recipes. The consequences of improving and extending the often
simplified physical prescriptions in SAMs might also be of special interest for studying
the AGN galaxy population (see Chapter 4), in particular with respect to the recipes
for gas cooling and the corresponding, possible change in the evolution of the cold
gas content being available for black hole accretion. At the same time, by comparing
the detailed predictions of the formation histories of galaxies in the simulations with
the SAM predictions, one may gain insights into the origin of existing discrepancies
between the simulations and the real Universe.
Various comparison studies between simulations and SAMs for large galaxy populations as well as individual halos have been discussed in the literature (these results
are summarized in Section 5.2), following different philosophies. For some studies only
individual physical processes, like cooling, were investigated (Lu et al., 2010; Benson
& Bower, 2010), while others focused on the evolution of individual objects, such as
a high-mass galaxy cluster (Saro et al., 2010) or a single disk galaxy (Stringer et al.,
2010). The approach in this Chapter is new in many respects. The evolution of individual halos is compared but the, up to now, largest number of 48 high-resolution
zoom simulations is used, as presented in Oser et al. (2010). The simulations cover
dark matter halos in the mass range of 2.4 × 1011 M" < MHalo < 3.3 × 1013 M" . Although a limited complement of physical processes have been taken into account in the
simulations, the more massive of these halos have been shown to represent fairly well
the evolution of observed massive galaxies (Oser et al., 2010). These simulations are
compared to results from the full Somerville et al. (2008b) SAM (as well as different
stripped down versions), which was shown to reasonably well represent present-day
galaxy properties over a wide range of masses. The SAMs are run within merger trees
extracted directly from the numerical simulations. In addition, both model predictions
are compared to observations at different redshifts and point out, where the respective
models succeed or fail either to match each other and/or the observations.
This Chapter is organized as follows: In Section 5.2 results from previous comparisons between SAMs and simulations are discussed . The hydrodynamical simulations
and the merger-tree construction method used for this study are discussed in Section
5.3 and the ingredients of the Somerville et al. (2008b) SAM is briefly reviewed in Section 5.4. The redshift evolution of the baryonic components in simulations and SAMs
is compared in Section 5.5 followed by a comparison to observations in Section 5.6.
Section 5.7 summarizes and discusses the main results of this Chapter. A resolution
study for individual halos can be found in the Appendix A.
5.2
5.2
Previous comparison studies
109
Previous comparison studies
Previous quantitative comparisons between simulations and SAMs have either focused
on whole populations of galaxies (Benson et al., 2001; Yoshida et al., 2002; Helly et al.,
2003; Cattaneo et al., 2007; Benson & Bower, 2010; Lu et al., 2010) or individual objects like a galaxy cluster (Saro et al., 2010) and a single disk galaxy (Stringer et al.,
2010). Helly et al. (2003) compared the efficiency of gas cooling for different dark
matter halos as a function of redshift between a (50Mpc/h)3 SPH (Smoothed-ParticleHydrodynamics) simulation (Hydra, Pearce et al., 2001) excluding star formation,
heating and feedback and a stripped down version – without star formation or feedback – of the Galform SAM (Cole et al., 2000). For z = 0 they find good agreement
of the cold gas mass between the SPH simulation and the SAM. At high redshifts,
however, more gas tends to cool in low-mass halos in the simulation due to the limited
numerical resolution. Still, they conclude that simulations and SAMs give consistent
results for the evolution of cooling galactic gas and confirm earlier findings of Benson
et al. (2001) and Yoshida et al. (2002).
Cattaneo et al. (2007) considered star formation and supernova feedback for a
similar comparison in a 34.19Mpc3 volume. For the SPH-simulation they used the
TreeSPH code (Dave et al., 1997) and as input for their SAM-code GalICS (Hatton
et al., 2003) the merger trees were constructed from the dark matter component of the
SPH-simulation. The SAM did not include a photo-ionising background but followed
the cooling by metals, while the simulations did include a photo-ionizing background
and assumed primordial composition of He and H. The star formation prescription in
the SAM was quite standard (star formation occurs above a certain gas surface density, according to a Kennicutt-Schmidt-like relation), including also a simple recipe for
SN-driven winds. In order to replicate AGN feedback, star formation is quenched when
the bulge component of a galaxy reaches a critical mass. For the comparison they used
two different SAM versions: one with no feedback and the ‘full’ model. In general,
they found good agreement between the simulations and the no-feedback model for the
baryonic mass functions at different redshifts and in different environments. Moreover,
simulations and the no-feedback SAM made similar predictions for the ‘hot’ and ‘cold’
mode gas accretion histories of galaxies (e.g. Kereš et al., 2005). However, at low redshifts, much less gas was left over in the simulation than in the no-feedback SAM with
both approaches over-predicting the observed baryonic mass function, in particular at
the high mass end. The full SAM, on the other hand, matched the observations due
the inclusion of supernova-driven outflows and AGN feedback, which suppresses gas
cooling in large halos. They concluded that the simulations and the no-feedback model
failed as a consequence of missing physics rather than computational inaccuracies.
Saro et al. (2010) compared the galaxy populations within a massive cluster (Mcl =
1.14 × 1015 M" ) using a high-resolution cosmological re-simulation run with Gadget2
(Dolag et al., 2009) and the SAM model of De Lucia & Blaizot (2007). They focused on
110
Galaxy formation in semi-analytic models and zoom simulations
differences between the central and the satellite galaxies considering only gas cooling
and star formation and neglecting any form of feedback. In general, they find similar
statistical properties for the galaxy populations, e.g. the stellar mass function with a
few remarkable object by object differences. The central galaxy in the simulation starts
with a more intense and shorter initial burst of star formation at high redshift and
forms fewer stars at low redshift than in the SAM. While in the SAM all stars in the
central galaxy are formed in its progenitors, in the simulations the final stellar mass is
larger than the sum of all progenitors. Satellite galaxies can lose up to 90 per cent of
their stellar mass due to tidal stripping – a process, which is, however, not included in
the De Lucia & Blaizot (2007) semi-analytic model, nor in most models discussed in
the recent literature.
Moreover, Stringer et al. (2010) presented a comparison for the evolution of a single
disk galaxy using the SPH-code Gasoline (Wadsley et al., 2004) and the semi-analytic
Galform model (Bower et al., 2006) based on the dark matter merger history of the
simulation. They find that the two techniques show a potential consistency for the
evolution of the stellar and gas components by assuming the same physics and the
same initial conditions. They try to mimic in the SAM the ‘blast wave’ SN feedback
implemented in the simulation, i.e. after a supernova explosion no cooling is allowed
in a certain volume. However, using the Galform model as described in Bower et al.
(2006) (including chemical enrichment, supernova and AGN feedback), the resulting
system is not recognisably the same as the one predicted by the simulations. At all
redshifts, the stellar mass is much larger and the hot gas fraction is much lower in the
simulation than in the SAM.
Finally, Lu et al. (2010) and Benson & Bower (2010) focus on the algorithms for
gas cooling in SAMs in great detail. Benson & Bower (2010) compare cold (rapid)
and hot (slow) accretion rates in the Galform SAM and in simulations from Kereš
et al. (2009) (50 Mpc/h, 2 × 2883 particles). They used their ‘full’ model including
feedback and metal cooling, although these processes are not included in the simulations. Moreover, they modified their SAM by adopting an updated calibration for
the transition between the rapid and slow cooling regime following the methodology of
Birnboim & Dekel (2003). They find reasonably good agreement for the hot and cold
mode accretion fraction in the SAM and the simulations and thus, they conclude that
the cold-mode physics is already adequately accounted for in SAMs. In the study of Lu
et al. (2010) five different SAMs (‘Munich’ model: Croton, 2006, ‘Kang’ model: Kang
et al., 2005, ‘Galform’ model: Cole et al., 2000, ‘GalICS’ model: Hatton et al., 2003
and the ‘Somerville’ model: Somerville & Primack, 1999b) are compared to the simulations of Kereš et al. (2009), without considering any feedback or metal enrichment
in either method. They find a significant difference between hot and cold accretion
rates: compared to the simulations, the cold mode accretion rates are lower and the
hot mode accretion are higher in the SAMs. They construct a modified cooling recipe
for the SAM to enable simultaneous hot and cold accretion, resulting in much better
5.3
The simulation and merger tree construction
111
agreement between the SAMs and the simulations.
Throughout the course of this Chapter, I will refer back to these studies and comment upon the similarities and differences with the results presented here.
5.3
5.3.1
The simulation and merger tree construction
Simulation setup
The cosmological zoom simulations presented in this Chapter are described in detail
in Oser et al. (2010) and the simulation setup is briefly reviewed here. The dark
matter halos for further refinement were selected from a dark matter only N-body simulation (Gadget-2, Springel et al., 2005a) with a comoving periodic box length of
L = 100 Mpc and 5123 particles (see also Moster et al., 2010). A ΛCDM cosmology
is assumed based on the WMAP3 measurements (see e.g. Spergel et al., 2003) with
σ8 = 0.77, Ωm = 0.26, ΩΛ = 0.74, and h = H0 /(100 kms−1 ) = 0.72. The simulation
starts at z = 43 and runs to z = 0 with a fixed comoving softening length of 2.52 h−1 kpc
and a dark matter particle mass of MDM = 2 × 108 M" /h. Starting at an expansion factor of a = 0.06 halo catalogues are constructed for 94 snapshots until z = 0 separated
by ∆a = 0.01 in time. From this simulation, 48 halos were chosen identified with the
halo finder algorithm F OF at z = 0. To construct the high-resolution initial conditions
for the re-simulations, all particles are traced back in time that are closer than 2·r200 to
the center of the halo in any snapshot and they are replaced with dark matter as well
as gas particles at higher resolution (Ωb = 0.044, ΩDM = 0.216). In the high resolution
region the dark matter particles have a mass resolution of mDM = 2.1 · 107 M" h−1 ,
which is 8 times higher than in the original simulation, and the gas particle masses are
mGas = mStar = 4.2 · 106 M" h−1 . Individual cases were run at 64 times higher mass
resolution and 4 times higher spatial resolution. The re-simulated halos cover a mass
range of two orders of magnitude (2.4 × 1011 M" < MHalo < 3.3 × 1013 M" ).
For modeling the gas component the entropy conserving formulation of SPH is used
(Gadget-2, Springel et al., 2005a). Star formation and cooling for a primordial composition of hydrogen and helium is included (Theuns et al., 1998). The cooling rates are
computed under the assumption that the gas is optically thin and in ionization equilibrium. Furthermore, the simulations include a spatially uniform redshift dependent UV
background radiation field according to Haardt & Madau (1996), where re-ionization
takes place at z ≈ 6 and the radiation field peaks at z ≈ 2 − 3.
To model star formation and SN feedback the approach of Springel & Hernquist
(2003) is used. In this model, the ISM is treated as a two-phase medium where clouds of
cold gas form from cooling of hot gas and are embedded in the hot gas phase assuming
pressure equilibrium. The hot gas is heated by supernovae and can evaporate the cold
clouds. Stars form from the cold gas whenever the local density exceeds a threshold
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.1: Visualisation of merger trees for four re-simulated halos with different
masses: upper left: Mvir = 8 × 1012 M" (M0162), upper right: Mvir = 1 × 1012 M"
(M1017), lower left: Mvir = 5 × 1011 M" (M2665), lower right: Mvir = 1 × 1011 M"
(M6782). Black circles show the dark matter halo at every time-step of the simulations.
The symbol size is proportional to the square root of the halo mass normalized to the
halo mass at z=0. The yellow stars indicate the stellar mass, the blue and red filled
circles the cold and hot gas mass within the virial radius of the dark matter halo. The
symbol sizes for the baryons scale with the square root of the masses normalized to the
maximum total baryonic mass at z=0.
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The simulation and merger tree construction
113
density (ρ > ρth = 0.205cm−3 ). The star formation rate is calculated by
dρ∗
ρc
= (1 − β)
dt
t∗
(5.1)
Here, β is the mass fraction of massive stars, which is assumed to explode as supernovae type II, ρc is the density of cold gas and t∗ = t0∗ (ρ/ρth )−1/2 is the star formation
time scale. The supernova explosions heat the surrounding gas with an energy input
of 1051 ergs. Springel & Hernquist (2003) used an idealized, isolated disk galaxy simulation to set the free parameters ρth and t0∗ , by adjusting them to obtain a match
to the observed Schmidt-Kennicutt relation. The same values of these parameters are
adopted here.
5.3.2
Merger trees
The merger trees of the dark matter component are constructed with the algorithm as
described in Section 3.4.2. The mininum halo mass is set to 20 particles (5×108 M" /h).
However, in the following, isolated merger trees are used without applying the splitalgorithm of Genel et al. (2008), as the dark matter masses in the ’split-trees’ are FOF
masses, but virial masses are needed as input for the semi-analytic model.
Note that the tree-algorithm is only applied to the dark matter particles, star or
gas particles are not separately traced back in time. They are assumed to follow the
evolution of the dark matter. Therefore, to each dark matter halo in a tree, a hot/cold
gas phase is assigned by counting hot/cold gas particles within the virial radius of the
central dark matter halo. The stellar and cold gas particles within 1/10 of the virial
radius are defined as the stellar and gas mass of the central galaxy. It is distinguished
between hot and cold gas particles by using the following definitions (code units):
log T < 0.3 log ρ + 3.2 → cold
log T > 0.3 log ρ + 3.2 → hot
(5.2)
(5.3)
The above distinction was made by looking directly at the phase diagrams of the resimulations, where it has been discriminated between the gas in the disk heated by SN
feedback and the shock heated gas in order to capture the cold star-forming gas.
Fig. 5.1 shows a visualization of four merger trees of re-simulated halos with virial
masses of 8 × 1012 M" , 1 × 1012 M" , 5 × 1011 M" and 1 × 1011 M" . The sizes of the
black circles approximate the dark matter halo masses, the yellow stars the stellar
mass within the virial radius and the blue and red filled circles the cold and hot gas
component, respectively. The symbol sizes scale with the square root of mass normalized to the final dark matter halo mass (dark matter component) and to the final
baryonic mass (star, hot and cold gas mass). One can clearly see that galaxies at high
redshift contain more cold gas, which either turns into stars or is heated towards lower
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Galaxy formation in semi-analytic models and zoom simulations
redshifts. In general, for more massive halos the fraction of cold gas and stars at z = 0
is lower.
To study the influence of numerical resolution on the evolution of the dark matter
and the baryonic components, a few halos have been simulated with a 4 × higher spatial
resolution (= 64 × higher mass resolution) than the original dark matter simulation. A
comparison of the results can be found in the Appendix. The overall mass assembly of
the main halos and the number of major mergers do not show any significant variation,
although the number of identified minor mergers increases due to the higher resolution.
Overall, one can conclude that the results are well-converged and would not change
significantly if the resolution is improved.
5.4
The semi-analytic model
The merger-trees constructed as described above are used as input for the semi-analytic
model described in Chapter 4 and in Somerville et al. (2008b) (hereafter S08). The
SAM makes use of merger trees for “isolated” halos only, and treats the evolution
of sub-structure within virialized halos using semi-analytic approximations. The ‘full’
SAM version includes photo-ionization, gas cooling, star formation, SN feedback, metal
enrichment, and black hole growth in a radio and quasar mode with corresponding
feedback. However, to provide a more meaningful comparison to the simulations, I
do not only consider the ‘full’ version, but also ‘stripped down’ models by separately
switching off AGN feedback, metal cooling, Supernova-driven winds, and ‘thermal’
Supernova feedback. The following different versions are considered:
• NF: No Feedback, primordial metallicity
• SN: thermal SN-feedback, primordial metallicity
• SNWM: thermal SN-feedback, SN-driven Winds, Metal cooling
• FULL: “full” version, including thermal SN-feedback, SN winds, metal cooling,
and AGN feedback
In contrast to Section 4.3.2 and S08 - there is a small modification for limiting the
hot gas content in the SAMs used in this Chapter. Note that in merger trees from
N-body simulations it may happen that the total mass of two merging halos at the
beginning of a merger event is larger than the mass of the merged object afterwards,
since during the merger particles can become unbound through tidal forces. Therefore,
in the SAM an upper limit is imposed on the hot halo mass of
Mhot = Mbar − Mstar,tot − Mcold − Meject .
(5.4)
Here, Meject is the mass ejected by winds, Mstar,tot and Mcold are the total star and cold
gas masses within the merged halo and Mbar is the expected baryonic fraction of the
5.5
Redshift evolution of galaxy properties
115
halo. In this way, the sum of all baryonic components in the halo is prevented from
exceeding the universal baryon fraction.
5.5
Redshift evolution of galaxy properties
In this Section, the cosmic evolution of the baryonic components of the galaxies and
halos is compared from the direct cosmological simulations to the results from the
SAMs using the dark matter merger trees constructed from the re-simulations. Here
only the evolution of the central galaxy in the main branch of the merger tree (largest
progenitor halo) is considered. The 48 halos are divided into three bins according to
their halo mass at z = 0 (every bin contains 16 halos) with 4.5 × 1012 M" < Mhalo <
4 × 1013 M" (high-mass), 1.2 × 1012 M" < Mhalo < 4.5 × 1012 M" (intermediate-mass),
and 2.4 × 1011 M" < Mhalo < 1.2 × 1012 M" (low-mass). All comparisons in this Section
make use of these bins.
Note also that a resolution study of the evolution of the baryonic component in
SAMs based on 2 × and 4 × higher resolution simulations for a high- and a low
mass halo can be found in the Appendix A. In both cases, the results based on the
simulations with different resolution are consistent.
5.5.1
Baryon fraction
For a first comparison the total baryonic mass Mbar = (Mstar + Mcold + Mhot ) is computed within the virial radius of the main halo at every redshift for the simulations and
the SAMs, respectively. In the re-simulations as well as in the SAMs, the hot gas mass
is considered within the whole halo (i.e. within the virial radius), but the stars and the
cold gas only of the central galaxy. Contributions from substructures are neglected,
as diffuse stars and cold gas and satellite stars and cold gas. The baryonic mass is
compared to the mass of available baryons within each halo, defined as fbar × Mhalo ,
where fbar = 0.169 is the cosmic baryon fraction.
Fig. 5.2 shows the average ratio of Mbar /(fbar × Mhalo ), as a function of redshift
for the three mass bins. The simulations are compared to the four SAM variants: NF,
SN, SNWM, and FULL (for details see Section 5.4). The simulations are expected to
be most directly comparable to the NF or SN SAMs, as these SAMs include the same
complement of physical processes as the simulations. Considering first the NF model,
one can see that at low redshift, the SAM overestimates the baryon fraction in high
mass halos, nearly agrees in intermediate mass halos, and slightly underestimates it
in low mass halos. At high redshift, the NF SAM overestimates the baryon fraction
at high redshift in high and intermediate mass halos, by a somewhat larger factor in
the former. Turning next to the SN SAM, one can see that the SAM predicts baryon
fractions that are everywhere higher than the simulation results, though much more so
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.2: Fraction of baryonic mass, Mbar = Mgas + M∗ , of the cosmic baryon
fraction of the halo mass, fbar × Mhalo , as function of lookback time for different semianalytic models (red: no feedback - NF; green: thermal Supernova feedback - SN; blue:
thermal Supernova feedback, SN-driven winds and metal cooling - SNWM; lila: ’full’
model including feedback from black holes - FULL) and for the SPH-re-simulation (black
lines). Upper panel: Average values for the high mass bin with 4.5 × 1012 M" < Mhalo <
3.3 × 1013 M" . Middle panel: Average values for the intermediate mass bin with 1.2 ×
1012 M" < Mhalo < 4.5 × 1012 M" . Lower panel average values for halo masses between
2.4 × 1011 M" < Mhalo < 1.2 × 1012 M" . The best match to simulations is found in
general for the NF model.
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Redshift evolution of galaxy properties
117
for the high and intermediate mass halos. This is because the “thermal” SN feedback
removes baryons from satellite galaxies, which are not counted in this census, and
deposits them in the hot gas component which is included here. In the SNWM and
FULL model, the impact of the SN-driven winds can be clearly seen, which effectively
remove baryons from the low-mass halos. AGN feedback mainly prevents hot gas from
cooling, so does not affect the total baryon fraction significantly, but will be important
for the fraction of stars and hot gas, which will be discussed later.
5.5.2
Cold gas and stars
Fig. 5.3 shows the evolution of the mass of condensed baryons (stars and cold gas,
M∗ + Mcold ) as a fraction of the total baryon mass as a function of redshift for the three
mass bins. There is fairly good agreement between the simulations and the NF model
over the whole redshift range, although the SAM is a little low at high redshift and
a bit high at low redshift, particularly in the high and intermediate mass bin. In the
SN model, the condensed baryon fraction is lowered by an almost fixed factor relative
to the NF model, and is significantly lower than the simulation results. This suggests
that the “thermal feedback” implemented in the simulation is less effective than that
included in the SAM. It may seem curious that the SNWM model results are higher
than the SN model, in fact close to the NF model in the high and intermediate mass
bin. This is because the SNWM includes metal cooling, while the SN model does not.
The enhanced cooling rates partly compensate for the removal of cold gas via the SNdriven winds. Finally, one can see that in the FULL model, the AGN feedback begins
to quench star formation in the massive halos after about z ∼ 2, while it has little
effect on the lower mass bins.
In Fig. 5.4 the evolution of the mean cold gas fraction of the central galaxy is plotted
(note that in Fig. 5.3 cold gas plus stars is plotted). The efficiency of the conversion of
cold gas to stars is clearly very different between the simulations and SAMs. In both
cases (simulations and SAMs), the final cold gas fraction is increasing with decreasing
halo mass. For the NF model, the SN and the SNWM model the cold gas fraction
varies only slightly over time and is significantly higher (about an order of magnitude
since z = 2) than for the simulations. This shows that the inclusion of SN feedback
has little impact on the gas fractions of galaxies. Only the FULL model shows a much
stronger decrease of the gas fraction with cosmic time for massive galaxies due to the
radio mode feedback. The initial cold gas fraction, at high redshifts 4 < z < 8, is
almost the same for the simulations and SAMs. With evolving cosmic time the cold
gas content decreases more rapidly in the simulations due to the more efficient conversion into stars. The cold gas in the simulations is already converted into stars at high
redshift and there is almost no more cold gas left to turn into stars at lower redshifts.
This is similar to the results found in the comparison of Cattaneo et al. (2007).
Fig. 5.5 shows the corresponding fraction of available baryons that are converted
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.3: The evolution of the fraction of the condensed baryons M∗ + Mcold of
the total baryon mass as a function of redshift for the three mass bins. The condensed
baryon fraction of all simulations is in reasonable agreement with the NF model at all
redshifts. High-mass and intermediate-mass simulations agree well with the SNWM
model. The effect of AGN feedback in the FULL model can be clearly seen in the
high-mass bin. In the low-mass bin the SN wind feedback (SNWM) makes the biggest
difference.
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Redshift evolution of galaxy properties
119
Figure 5.4: Evolution of mean cold gas fraction Mcold /(fbar × Mhalo ) of the central
galaxies. Mass bins and colors are the same as in Fig. 5.2. In all simulations the
cold gas is depleted more efficiently than in the SAMs due to the large star formation
efficiency at high redshifts. The red dotted line shows the cold gas fraction assuming
ten times higher efficiency for star formation in the NF model (see Eq. 5.5).
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.5: Comparison of the stellar baryon fraction of the central galaxy between
simulations and SAMs. The mass bins and colors are the same as in Fig. 5.2. For all
mass bins the simulations agree best with the NF SAM, but forming significantly more
stars already higher redshifts z > 1. This discrepancy can be accounted for if the star
formation efficiency parameter in the SAMs is increased by a factor 10 for the NF case
(red dashed lines). At high masses the AGN feedback (FULL) and at the low masses
the SN feedback (SN and SNWM) reduce the stellar baryon fractions in the SAMs.
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Redshift evolution of galaxy properties
121
into stars in the central galaxy M∗ /(fbar × Mhalo ), sometimes termed “baryon conversion efficiency” (Guo & White, 2009; Moster et al., 2010). In general, all simulations
predict a decreasing (high-mass) or almost constant (low-mass) conversion efficiency
with redshift, whereas most SAMs predict increasing conversion efficiencies with the
exception of high-mass galaxies in the FULL model with AGN feedback.
At low redshift z < 0.6 the conversion efficiencies agree well between the simulations
and the NF model, with higher values for lower mass galaxies. However, at high
redshifts z > 1 the conversion efficiencies are significantly higher for the simulations.
This is in contrast to the results of Cattaneo et al. (2007), where the stellar masses agree
at high redshift, but the SAM masses are larger than in simulations at low redshift.
The difference in the behavior of the SAMs and the simulations can be explained in
terms of star formation efficiency. We changed the normalization of the SK-relation in
the SAM by introducing a factor τ∗ in Eq. 4.4:
ΣSFR =
AKS NK
Σ ,
τ∗ gas
(5.5)
with τ∗ ≈ 0.1. The results of the NF model with this elevated star formation efficiency
for the stellar and cold gas mass evolution is shown in Figs. 5.4 and 5.5. At high
redshift, gas is more efficiently depleted and converted into stars, resulting in a better
agreement between this ‘high SFE’ model and the simulations. However, for the highmass and intermediate-mass bin, the ‘high SFE’ model over-predicts the stellar fraction
at low redshifts, suggesting that the Cattaneo et al. (2007) SAMs may also have had a
higher SFE, and this could explain the discrepancy between their results and the initial
results presented here. In all three mass bins, the NF model produces the most massive
stellar components. Again, the more efficient cooling due to metals in the SNWM and
the FULL model is cancelled by the effect of winds and, for massive galaxies, also by
AGN feedback, resulting in lower stellar masses than in the NF model.
Star Formation Rates
To confirm the previous findings the star formation rates are compared in Fig. 5.6.
At very high redshifts z > 4, the SFRs in the simulations are much higher than in the
SAMs. Only by assuming more efficient star formation in the NF SAM (τ∗ = 0.1) a
reasonable match to the simulations is obtained. However, at z < 1.5, the high SFE
model results in similar SFRs as the original NF model, as the larger SF efficiencies
at high redshift lead to a more rapid depletion of the cold gas. In the simulation, the
cold gas is rapidly turned into stars, resulting in lower SFRs at low redshifts compared
to SAMs because of gas depletion. Only the FULL model shows a strongly decreasing
SFR with decreasing redshift due to radio mode feedback, which becomes especially
important for low redshifts and large halo masses. This result is consistent with the
study of Saro et al. (2010), who compared their stripped-down versions of SAMs (with
no feedback) to simulations, and found find higher SFRs in the simulations for all
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.6: Evolution of the star formation rates in simulations and SAMs for three
mass bins. The mass bins and colors are the same as in Fig. 5.2. At high redshifts,
SFRs are higher in the simulations than in the SAMs leading to more rapid depletion
of the cold gas with very low SFRs at low redshifts.
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123
galaxies within a cluster (central and satellites) at high redshifts and lower SFRs at
low redshifts. In addition, Stringer et al. (2010) find a similar discrepancy for the
specific star formation rates at high redshifts (larger in simulations than in their SAM)
and good agreement for low redshifts.
To better understand this discrepancy between simulations and SAMs we take a
closer look at the respective implementations of star formation. According to Springel
& Hernquist (2003), stars in the simulations are formed locally out of cold gas with
the star formation rate density proportional to the local three-dimensional density of
gas to the power of 1.5, ρSF ∝ ρ1.5 /t∗ . The star formation timescale t∗ was set to
approximate the observed local Schmidt-Kennicutt relation (SK) for a simulation of
a smooth, isolated disk-dominated galaxy set up to resemble the Milky Way. In the
SAMs, the cold gas is assumed to settle into smooth exponential disks and stars form
according to the SK-relation, implemented in terms of surface densities.
In Fig. 5.7, the SFR surface density versus the surface density of the cold gas
is plotted for the simulated galaxies at z = 2 and z = 4 within 1/10 rvir for all
re-simulations. The black dashed line is the SK-relation assuming a Salpeter IMF
(Salpeter, 1955), as given in the original Kennicutt papers and as implemented in
the simulations following Springel & Hernquist (2003). We would naively expect the
simulations to follow this line. The red solid line shows the SK-relation for a Chabrier
IMF (Chabrier, 2003), as assumed in the SAMs. At a given gas surface density, the
SFR surface densities of the simulations lie mostly above the expected SK-relation.
The change of normalization associated with converting from Salpeter to Chabrier
cannot account for the increased star formation efficiency in the simulations. In general,
star formation in the cosmological simulations is about a factor of five more efficient
than for simulations of smooth isolated disks using the identical model (see Springel
& Hernquist, 2003). This discrepancy is a consequence of the clumpy structure of
cold gas in the cosmological simulations. In the clumps the gas can reach higher
local densities than in the idealized smooth disks that have been used by Springel &
Hernquist (2003) to calibrate the star formation timescale by matching the SK-relation.
As the implemented SK-relation is not linear, the structure of the cold gas distribution
plays an important role for the overall star formation efficiency within the galaxies (see
Teyssier et al. (2010a) for a discussion on galaxy mergers). In other words, for any
star formation model with a non-linear dependence on the local gas density (exponent
larger than unity), a more clumpy gas distribution will effectively increase the star
formation efficiency. These combined effects explain the much higher SF efficiencies at
high redshift in the simulations relative to the SAMs.
Modes of Stellar Mass Growth
In the hierarchical picture, galaxies can grow their stellar masses in two ways: 1) by
converting cold gas into stars in-situ and 2) by accreting already formed stars via
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.7: Star formation rate surface densities versus cold gas surface densities for
the simulated galaxies within 1/10 rvir . Black and blue stars correspond to different resimulations at z = 2 or z = 4, respectively. The red solid line illustrates the Kennicuttrelation implemented in the SAM assuming a Chabrier-IMF, the black dashed line the
one in the simulations (Springel & Hernquist, 2003) assuming a Salpeter-IMF.
mergers. These two modes are referred to as “in-situ” and “accreted”. The simulations
exhibit two phases of growth, with a rapid early phase at z > 2 during which stars are
formed in-situ from infalling cold gas, followed by an extended phase at z < 3 during
which the growth is primarily due to accretion of stars formed in external galaxies
(Oser et al., 2010). It is now investigated whether the SAMs show the same behavior.
Fig. 5.8 shows the fraction of cumulative in-situ over accreted stellar mass as a function
of redshift for the three different mass bins. For the SAMs the qualitative trend of a
decreasing fraction of in-situ growth is reproduced for the high mass bin. However,
the fraction of in-situ formed stars dominates over accreted stars for all models, all
masses, and at all redshifts. This is in contrast with the simulations, where accretion
dominates over in-situ formation for massive systems at low redshifts as discussed in
Oser et al. (2010).
I note several interesting trends in the in-situ to accreted fraction as vary the physics
in the SAMs. Adding thermal SN feedback increases the in-situ fraction at all redshifts
and in all mass bins. This is presumably because it suppresses star formation in lowmass satellites which are the source of accreted stars. Adding the SN-driven winds and
metal cooling further increases the in-situ fraction, again at all redshifts below z ∼ 4.
Switching on AGN feedback increases the in-situ fraction at high redshift and decreases
it at low redshift in the high mass bin (and to a lesser extent in the intermediate mass
bin). This is because the radio mode feedback shuts off cooling at late times in massive
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Redshift evolution of galaxy properties
125
Figure 5.8: Fraction of in-situ and accreted stellar mass versus redshift. The mass
bins and colors are the same as in Fig. 5.2. The bimodal behavior as seen in simulations
cannot be reproduced by the SAMs. Here, the in-situ star formation is dominating over
accretion at all redshifts.
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.9: Comparison of the mean in-situ (left column) and accreted (right column)
stellar masses. The mass bins and colors are the same as in Fig. 5.2. The in-situ stellar
masses in the SAMs agree with the ones in the simulations reasonably well, whereas in
the SAMs the accreted stellar mass is smaller than in the simulations.
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Redshift evolution of galaxy properties
127
halos, removing the supply of new gas needed to fuel ongoing in-situ star formation.
Interestingly, increasing the star formation efficiency in the NF model has almost no
effect on the in-situ to accreted fraction. This is presumably because the SF efficiency
is increased in the central and (accreted) satellite galaxies alike. However, if the SFE
were higher in high redshift galaxies than at low redshift, this would presumably increase the accreted fraction in present day galaxies. This may be part of the reason
for the higher accreted fractions in the simulations.
Fig. 5.9 shows the evolution of the cumulative mass of in-situ and accreted stars
separately for the simulations and the various SAM variants. Here one can see that the
NF SAM actually reproduces the growth of in-situ stellar masses fairly well, though
overproducing the in-situ mass at low redshift somewhat, especially in the highest mass
bin. One can speculate that gravitational heating in the simulations may prevent some
of the late cooling in the highest mass bin and leads to lower in-situ stellar mass than
the NF SAM (Naab et al., 2007; Johansson et al., 2009b; Feldmann et al., 2010). The
radio mode AGN feedback in the FULL model leads to a similar suppression of this
in-situ mass growth in the massive halos. The NF model with increased SFE gives
an even better match to the simulations at high redshift. The discrepancy arises from
the much lower accreted masses in the SAM. Here again, the SF model with high SFE
comes the closest to matching the simulation results, but it still falls short by a considerable amount.
Part of the reason for the lower predicted accreted masses in the SAMs is that the
SAMs used here only allow for cooling onto the central galaxy in the halo, effectively
assuming that the hot gas reservoir of a satellite galaxy is stripped as soon as it enters
the virial radius of the host. This is known to result in satellites that are too red and
have star formation rates that are too low compared with observations (Kimm et al.,
2009). It will also truncate their star formation, resulting in a smaller amount of stellar
mass that will eventually be accreted when they merge (Khochfar & Ostriker, 2008).
5.5.3
Hot halo gas
The evolution of the mean hot gas fraction is shown in the three panels of Fig. 5.10. In
all but the SN SAM the hot gas fraction increases with increasing halo mass, which is
qualitatively similar to the simulations. In the simulations this effect is caused by shock
heating of infalling baryonic material which becomes more efficient for massive halos
(e.g. Silk, 1977; Binney, 1977; White & Rees, 1978; Birnboim & Dekel, 2003; Birnboim
et al., 2007; Kereš et al., 2005; Khochfar & Ostriker, 2008; Kereš et al., 2009; Johansson
et al., 2009b). This trend is also seen by the SAMs, except for the SN model, where
the supernova energy input heats most of the available gas to the virial temperature
of the halos, keeping the hot gas fraction constant independent of halo mass. For the
SNWM and FULL models the supernova winds drive some of the hot gas out of the
low mass halos. The additional effect of Radio mode heating (FULL model), which
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.10: Evolution of the mass fraction of hot gas in simulations and SAMs
normalized to the available mass in baryons. The mass bins and colors are the same as
in Fig. 5.2. At low redshifts z < 1.5, simulations can be matched by the NF model. At
high redshifts the hot gas content is significanty larger for the NF model. For z > 3,
models including metal cooling provide a much better match to simulations which do
not include metal cooling.
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Redshift evolution of galaxy properties
129
Figure 5.11: Comparison of the mean accretion rates of all gas (left column), hot (intermediate column) and cold (right column) mode accretion in simulations and SAMs.
The mass bins and colors are the same as in Fig. 5.2. Compared to simulations, hot
mode accretion is over-estimated in all SAMs, whereas cold mode accretion is in general
under-estimated, particularly for SAMs without metal cooling.
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Galaxy formation in semi-analytic models and zoom simulations
prevents late cooling in massive halos and therefore leads to larger amounts of hot gas,
is apparent for the intermediate and high mass galaxies. The NF model agrees fairly
well with the simulations in all mass bins at z ! 1.5 but substantially overpredicts the
amount of hot gas at high redshift.
To understand the differences in the hot gas content at high redshift the gas accretion modes onto the central galaxies are investigated. For the simulations it is
distinguished between hot and cold accretion by considering the highest temperature a
gas particle had before it was accreted onto the galaxy, i.e. 1/10th of the virial radius.
The same definition is used to distinguish between between hot and cold gas as given
by equation 5.2 (similar to Kereš et al., 2005). The SAMs distinguishes between hot
and cold mode accretion (slow and rapid cooling) depending on whether the ratio of
the cooling radius to the virial radius rcool /rvir is larger (cold mode) or smaller (hot
mode) than unity (White & Frenk, 1991). The distinction between hot and cold mode
accretion approximately specifies whether the gas was heated to the virial temperature
of the host halo before it was accreted onto the galaxy (hot mode) or was directly
accreted without being heated (cold mode). The cold mode accretion is meant to represent the ‘cold flows’ recently discussed in the literature (Kereš et al., 2005, 2009;
Oser et al., 2010; Dekel et al., 2009). Note that for the SNWM and FULL SAMs, the
heating rates or rates of blown-out gas due to feedback processes are not subtracted
from the accretion rates.
Fig. 5.11 shows the comparison of the total (left panels), hot (middle panels) and
cold mode (right panels) gas accretion rates onto the central galaxies as a function of
redshift. For all SAMs, the total gas accretion rates onto the galaxies are significantly
higher than for the simulations. This is caused solely by higher hot mode accretion
rates from the generally larger hot gas reservoir in particular at high redshift (middle
panel in Fig. 5.11 and see Fig. 5.10). The cold mode accretion rates are much lower in
the SAMs than in the simulations, and in the SAMs without metal cooling (which are
more relevant to compare with the simulations), the cold mode is truncated at z ! 4 for
massive halos, z ! 1–1.5 for low mass halos, while it declines smoothly until z ∼ 0 in
the simulations. However, the predicted rates of cold mode accretion are much higher
in the SAMs that include metal cooling.
In contrast to the results presented here, Cattaneo et al. (2007) find a reasonably
good match for the evolution of the hot gas content as well as for the hot and cold
mode accretion rates in their SAM version without any feedback. However, they include metal cooling in their SAM, but not in their simulations. Benson & Bower (2010)
compare cold and hot mode accretion rates from SAMs to simulations, varying the supernova feedback and conditions for the rapid cooling regime according to Birnboim &
Dekel (2003). They concluded that cold-mode physics is already adequately accounted
for in SAMs — but they also used simulations with only primordial H & He cooling,
but included metal cooling in their SAMs. Lu et al. (2010) assume only H & He cooling
5.6
Comparison to observations
131
in their SAMs as well as in the simulations (like in this study) and find qualitatively
similar results to ours: a discrepancy for the hot halo gas fraction at high redshift
associated with larger hot mode and smaller cold mode accretion rates in the SAMs
than in the simulations.
This suggests that the agreement presented in Cattaneo et al. (2007) and Benson &
Bower (2010) is fortuitous, and arises because of the enhanced cold flows resulting from
the metal cooling included in the SAMs. It is unclear whether this agreement would
persist if metal cooling were included also in the simulations, but it appears that the
agreement is not nearly so good as they claim when metal cooling is omitted from both
techniques. In addition, the recipes for gas accretion in SAMs do not currently allow
co-existing cold and hot gas accretion as seen in simulations. For this, Lu et al. (2010)
proposed a new model that explicitly incorporates cold-mode accretion independent of
the hot halo gas. By fitting the hot and cold gas fraction in simulations as a function
of redshift and halo mass, and assuming accretion onto the galaxy within a free-fall
time they calculate the accretion rate of the cold component and thus, achieve a better
match of their SAMs to simulations.
5.6
Comparison to observations
In this Section, the results from the SAMs and simulations are compared to observational data and empirical constraints at different redshifts. It is focused on two key
observational constraints: the relationship between halo mass and stellar mass (the
Mgal − Mhalo -relation) and the relationship between stellar mass and star formation
rates (ṁstar − Mgal -relation).
Fig. 5.12 shows the relation of galaxy mass and dark matter halo mass for z = 0
(left panels), z = 1 (middle panels), and z = 2 (right panels). The NF and FULL SAMs
are shown, together with the simulations. One can also see the empirical constraints on
the Mgal − Mhalo -relation from Moster et al. (2010), which were obtained by asking how
halos and sub-halos in an N-body simulation must be populated in order to reproduce
the observed stellar mass functions at different redshifts (halo abundance matching).
The thin, black vertical lines illustrate the observational lower halo mass limit of the fit.
Also shown are the similar constraints from Wake et al. (2011), which are derived using
galaxy clustering data from the NEWFIRM Medium Band Survey between 1 < z < 2
(see also Wechsler et al., 2006; Zheng et al., 2007; Conroy & Wechsler, 2009; Guo &
White, 2009; Zehavi et al., 2010; Behroozi et al., 2010). Note that the fitting functions
of Moster et al. (2010) and Wake et al. (2011) are somewhat different, in particular at
the low mass end. This is not surprising, as they were derived using different methods
and from different observational data sets. At all redshifts, the simulations over-predict
the stellar masses at a given halo mass by about a factor of two for halos more massive than 1012 M" . At higher redshifts the progenitor galaxies have lower masses, and
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Galaxy formation in semi-analytic models and zoom simulations
Figure 5.12: Stellar mass versus dark matter halo mass for simulations (upper row),
NF model (middle row) and the FULL model (lower row). The left column shows the
dependence on halo mass for z = 0, the middle one for z = 1 and the right one for
z = 2. The black lines show the fit for the halo occupation distribution from Moster
et al. (2010). This thin black or green, vertical lines show the observational limit of
the fits in Moster et al. (2010) or Wake et al. (2011), respectively. The dotted-dashed
diagonal lines illustrate baryon fractions of 0.1 × fbar , 0.5 × fbar , and 1 × fbar .
5.6
Comparison to observations
133
Figure 5.13: Star formation rate versus stellar mass for simulations and two different
semi-analytic models (NF and FULL). Solid lines illustrate the observed relations at
different redshifts: z = 0, 1: Elbaz et al. (2007), z = 2: Daddi et al. (2007). Note
that following Herschel measurements the observed relation at z = 2 was re-normalized
0.3 dex downwards (see text).
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Galaxy formation in semi-analytic models and zoom simulations
deviate more from the expected distribution. At z = 2 the difference can be almost
two orders of magnitude for halos of ∼ 1011 M" , in line with the findings of the previous Section — at high redshift, gas is very efficiently converted into stars in the
simulations. Implementation of more efficient feedback from supernovae would help to
solve this problem. Indeed it has been shown that simulations that do include effective
SN feedback agree much better with expectations (see e.g. Scannapieco et al., 2009;
Sawala et al., 2010; Genel et al., 2010; Governato et al., 2010 and references therein).
In addition, it is apparent that the simulations require an additional process that can
quench star formation at late times in massive halos, such as radio mode AGN feedback.
For high mass halos, the NF model (middle row) predicts galaxy masses close to
the relation at z = 1 and z = 2 due to less efficient star formation at high redshifts
(see Fig. 5.5), but, like the simulations, over-predicts the stellar masses in low mass
halos. By z = 0 the offset is about as large as for the simulations. For the FULL SAM
there is good agreement at z = 0, which is not surprising because the model was tuned
to match the observed stellar mass function. However, the FULL model still predicts
galaxy masses that are about a factor of two to three too high for low mass halos
(log(Mhalo ) < 11.5) at high redshift. This is related to the excess of low mass galaxies
at high redshift and other connected problems discussed in Fontanot et al. (2009), and
is seen in both SAMs and hydro simulations from several groups. It is likely that these
problems are due to limitations in the current understanding or implementation of the
physics of star formation and/or SN feedback.
The relation between the star formation rates and the galaxy stellar masses Mstar
is shown in Fig. 5.13 for redshifts z = 0, z = 1, and z = 2. It is distinguished between
star-forming and non-star-forming galaxies (illustrated as crosses or open squares, respectively) using a criterion according to Franx et al. (2008): galaxies with specific
star formation rates SF R/Mstar smaller than 0.3 × t−1
hubble , are considered to be quiescent, whereas galaxies with larger specific star formation rates are assumed to be
star-forming. The black, solid lines always refer to the observed relation at the corresponding redshift (z = 0: SDSS, Elbaz et al. (2007); z = 1: GOODS, Elbaz et al.
(2007); z = 2: GOODS, Daddi et al. (2007)) for star-forming galaxies. Note that the
relation of Daddi et al. (2007) is shown at z = 2 re-normalised 0.3 dex downwards,
following the re-calibration of SFR derived from 24 micron luminosity based on recent
Herschel observations (Nordon et al., 2010).
In general, the simulations under-predict the star formation rates for star forming
galaxies at all redshifts, but most notably at z > 1, due to the high star formation
efficiencies at even higher redshifts and the resulting gas depletion (see Section 5.5.2).
Once again, implementing more effective supernova feedback would presumably suppress star formation in the small, high redshift progenitors of these galaxies and result
in higher SFR at these redshifts (see e.g. Oppenheimer & Davé, 2008; Genel et al.,
2010). The NF SAMs fit the SF sequence at z = 0, but have too many high mass
5.7
Discussion and conclusions
135
galaxies with high SFR. In the FULL model, these galaxies are quenched by radio
mode feedback, in agreement with observations (see Somerville et al., 2008b). Both
the NF and FULL SAMs SF sequences are about a factor of two too low at z ∼ 1 and
2. This also seems to be generic to many SAMs, as shown by Fontanot et al. (2009) and
others. It is interesting to note that star forming galaxies in both the NF and FULL
SAM galaxies all lie on the same SF sequence. This implies that SN feedback simply
moves galaxies along the sequence, reducing both the stellar mass and the SFR such
that the galaxies remain on the same relation, and is due to the self-regulating nature
of SN feedback in the SAMs. In contrast, AGN feedback quenches star formation and
moves massive galaxies off of the SF sequence (seen in the FULL model).
5.7
Discussion and conclusions
In this Chapter, a detailed comparison was presented between a set of 48 cosmological
hydrodynamic zoom simulations and different stripped-down versions of semi-analytic
models based on dark matter merger trees extracted from the simulations. The hydrodynamical simulations are run using the entropy-conserving formulation of SPH
with the Gadget-2 code, and include atomic cooling assuming a primordial composition of H and He, a UV background radiation field, star formation, and supernova
feedback. Results from a “no feedback” (NF) version of the SAM are presented which
contains the same physical ingredients as the simulations, together with versions that
include thermal feedback by supernovae (SN), a version that also includes large-scale
SN-driven winds and metal cooling (SNWM), and the FULL version which includes
all the previously mentioned ingredients as well as AGN feedback. With this approach
the predictions of the two methods can be compared over two orders of magnitude in
halo mass at unprecedented resolution.
The two approaches try to answer the same questions but differ in methodology.
In the simulations the full dynamical and hydrodynamical evolution of the systems
is followed by solving the equations of motion computationally. Additional physical
processes like star formation and supernova feedback are included using sub-resolution
models. The semi-analytic models are based on the computed dark matter accretion
history and approximate the gas physics, star formation, and feedback processes with
simplified recipes. The analysis is focused on the cosmic evolution of the baryon content in the central galaxies of the main branch of the merger trees and its division into
various components (stars, cold gas, and hot gas), as well as how those galaxies acquired their gas — whether through “cold” or “hot” mode accretion — and their stellar
mass (e.g. through in-situ star formation vs. accretion).
The results of the comparison are quite rich, with some surprising agreement and
some striking disagreement. First, let me note that the results of the simulations are
expected to lie somewhere in between the NF and the SN SAMs, since these SAMs
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Galaxy formation in semi-analytic models and zoom simulations
include the similar physical processes as the simulations. In most cases, the agreement
is best between the simulations and the NF model, suggesting that the SN feedback
implemented in the simulations has little effect. The NF SAMs produce very good
agreement with the simulations for the mass of cold gas plus stars at all redshifts and
for all halo masses. The SAMs slightly underestimate this “condensed baryon” fraction
at high redshift (z > 1) and overestimate it at low redshift. This indicates that the
overall cooling and accretion rates in the SAM and the simulations must be similar.
The NF SAM also produces fairly good agreement (better than ∼ 20% since z ∼ 4)
with the overall baryon fractions (i.e. hot gas plus cold gas and stars in the central
galaxy) in the simulations, here overestimating the baryon fractions at high redshift in
high and intermediate mass halos.
A striking difference is that when the evolution of the stellar and cold gas components is studied separately, the cold gas fractions are found to agree at very high
redshifts, but the gas is consumed much more rapidly in the simulations, leading to
cold gas fractions at all redshifts less than about z ∼ 3–4, and in halos of all masses,
being much lower (by up to two orders of magnitude) than in the NF SAM. Correspondingly, much higher stellar masses are found in the simulations than in the SAMs
at high redshift, although they converge to almost the same value as the NF SAMs at
z = 0. One can interpret this as an indication that the star formation efficiency is much
higher in the simulations than in the SAMs, and the reason for the convergence in the
stellar masses at low redshifts is because nearly all available gas has been consumed in
the simulations. This conjecture is supported by the finding that if the star formation
efficiency is boosted in the NF SAM by a factor of ten, excellent agreement is found
with the stellar mass fraction evolution in the simulations, and improved agreement
with the cold gas fraction evolution.
However, both the simulations and SAMs supposedly adopt the same empirical
Schmidt-Kennicutt relation between cold gas density and star formation rate. How
can the star formation efficiencies be so different? Let me note several differences
in the implementation of the star formation recipe in the simulations and SAMs. In
SAMs, the only available information about the structure of the star forming gas in
galactic disks is an estimate of the scale radius of the total baryonic component of the
disk, which comes from angular momentum conservation arguments (Mo et al., 1998;
Somerville et al., 2008a). The SAMs then make a series of assumptions — that the
gas is in a smooth, thin exponential disk with a radius that is a simple multiple of the
stellar scale radius — and apply the Kennicutt relation in terms of the predicted gas
surface density. Only gas above a critical surface density is allowed to form stars. In
contrast, the simulations provide detailed 3D predictions for the structure of the cold
gas in galaxies, and implement the SK relation in terms of 3D volume density (also
applying a threshold for SF in terms of a critical volume density). It is well known
that high redshift galaxy assembly in cosmological simulations is dominated by clumpy,
high density, cold mode accretion. Disks may be more compact than in the idealized
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Discussion and conclusions
137
case of perfect conservation of angular momentum, and are thick and clumpy. The
adopted SF recipe is super-linear in the gas volume density (i.e. the exponent in the
SK relation is larger than unity), and therefore star formation will be more efficient in
a clumpy gas distribution than in a smooth one.
The appropriate values of the SF and SN feedback efficiency parameters for the
simulations were obtained by tuning them to match the observed Kennicutt relation
for an idealized, smooth thin exponential disk, designed to resemble a Milky Way-like
galaxy at z = 0 (Springel & Hernquist, 2003). Using these same values for the parameters, the high redshift galaxies in the cosmological simulations are found to lie about a
factor of five above the Kennicutt relation with the normalization adopted by Springel
& Hernquist (2003). A second, minor issue, is that the SAMs were tuned to match
a Kennicutt relation with a normalization a factor of two lower than the one used by
Springel & Hernquist (2003), reflecting a different choice of IMF in the conversion from
observed flux to SFR.
This seems to explain the reason for the factor of ∼ 10 higher SFE in the simulations relative to the SAMs, but begs the question: is this a robust prediction of the
simulations that should be taken seriously? Are these higher star formation efficiencies
in high redshift galaxies really physical? There are several issues that are relevant
here. First, the predicted “clumpyness” of the disks is highly sensitive to the assumed
sub-grid recipes. For example, implementation of more effective SN feedback would
reduce the clumpyness of the disks at high redshift, but might increase the clumpyness
at z ≈ 2(e.g. Genel et al., 2010) in the simulations. Observed disks at high redshift are
known to be more clumpy than nearby ones (Genzel et al., 2006, 2008, 2010; Förster
Schreiber et al., 2009, 2011), but it remains highly uncertain which recipe for SN feedback will produce the “correct” degree of clumpyness and overall structure for statistical
samples of high redshift galaxies while simultaneously reproducing the properties of local spirals (Piontek & Steinmetz, 2011; Governato et al., 2009; Scannapieco et al.,
2009; Brooks et al., 2009). Moreover, recent observational studies indicate that star
formation rate densities in local galaxies as well as at high redshift correlate linearly
with the surface density of the molecular gas, with no evolution in the molecular SK
relation (Bigiel et al., 2008; Daddi et al., 2010; Genzel et al., 2010). This is also true
for galaxies with very clumpy star formation, as expected for a linear dependence with
gas density. Only interacting galaxies undergoing a significant starburst seem to show
an increased star formation efficiency (Daddi et al., 2010; Genzel et al., 2010). Neither
the SAMs nor the simulations presented here include these effects.
A second major difference between the SAMs and the simulations is in the mode
in which galaxies acquire most of their stellar mass. It is distinguished between “insitu” growth, due to stars that form out of cold gas within the galaxy in question,
and “accretion” of stars that formed in external galaxies and are accreted via mergers.
This distinction is important because it may determine the characteristic size evolu-
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Galaxy formation in semi-analytic models and zoom simulations
tion of early type galaxies (Khochfar & Silk, 2006b; Naab et al., 2009; Guo et al., 2011;
Covington et al., 2011). It has been shown previously that massive galaxies in the
simulations and semi-analytic models have an early phase of growth dominated by the
in-situ mode, and then switch over to a mode that is dominated by growth through
accretion (De Lucia et al., 2006; Khochfar & Silk, 2006b; Guo & White, 2008; Oser
et al., 2010; Feldmann et al., 2010; Zehavi et al., 2011). Although the ratio of in-situ to
accreted stars always decreases with time in the SAMs, in qualitative agreement with
the simulations, in the SAMs in-situ growth dominates over accretion in halos of all
masses at all times. Examining the absolute mass in stars formed in situ or accreted,
it is found that this is primarily because the SAMs predict much less mass growth
through accretion — the in-situ mass evolution agrees fairly well with the simulation
results. The comparison between SAMs including different physical processes gives us
further insights into the origin of the discrepancy: 1) Increasing the efficiency of SN
feedback (SN and SNWM SAMs) reduces the accreted mass because star formation is
suppressed in the low-mass satellites that eventually get accreted. 2) Including radio
mode AGN feedback (FULL SAM) reduces the in-situ growth in massive galaxies at
late times, because it shuts off the fuel supply for in-situ star formation in these objects.
Interestingly, increasing the SF efficiency in the SAM does not affect the fraction of
in-situ versus accreted mass, presumably because all galaxies are boosted equally.
In summary, it is likely that the contributions from in-situ and accreted stars are
currently incorrectly predicted in both the simulations and SAMs, for the following
reasons. In order to match observations, the simulations presented here clearly require
both a process that suppresses star formation in low mass objects at all redshifts (such
as SN-driven winds) and one that can shut off residual cooling and quench star formation in massive galaxies at late times (such as radio mode AGN feedback). The former
will reduce the accreted mass, while the latter will decrease the in-situ mass in massive
objects at late times. In addition, if the SFE is higher in high redshift galaxies than at
late times, this is likely to increase the accreted mass. On the other hand, the SAMs
presented here, like many SAMs in the literature, make the assumption that hot gas
from the halo can only be accreted onto the central galaxy. This rapidly truncates
the star formation in satellite galaxies and is likely to artificially decrease the accreted
mass fraction. In addition, the SAMs neglect gravitational heating, which is clearly
important in the simulations and should reduce the in-situ growth in massive halos at
late times. The recent study of Fontanot et al. (2011) has shown that the S08 SAM
and other similar SAMs in which radio mode feedback is used to solve the overcooling
problem over-predict the fraction of radio loud galaxies compared with observations,
and the implemented dependence of radio luminosity on stellar and halo mass is too
steep. Gravitational heating could play a similar role (Khochfar & Ostriker, 2008;
Dekel & Birnboim, 2008; Birnboim & Dekel, 2010) and thereby reduce the need for
such strong radio mode heating.
A third important result is that the cooling recipe implemented in these SAMs
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Discussion and conclusions
139
(which is widely used in many SAMs) over-predicts the overall accretion rate of gas by
a factor of 1.5 (in low mass halos) to four (in massive halos). If the accretion rates in
the simulations are accurate, this implies that the SAMs that match present day galaxy
properties are compensating for this “extra” accretion by artificially “tuning up” the
feedback. Moreover, when the gas accretion is divided into “hot mode” and “cold mode”
(Birnboim & Dekel, 2003; Kereš et al., 2005), the SAMs are found to systematically
overestimate the hot mode growth (by up to an order of magnitude) and underestimate
the cold mode. As well, in the SAMs without metal cooling, the cold mode shuts off
completely at low redshifts (the shutoff redshift depends on halo mass), while in the
simulations it declines smoothly but continues to low redshifts. This is likely to be a
result of the fact that, in the SAMs, the criterion for discriminating between hot and
cold mode accretion is based on the assumption of smooth, spherical halos, while in
simulations cold gas can stream into halos along cold, dense filaments. In addition, in
the SAMs, gas is assumed to accrete either in cold mode or hot mode, but simultaneous
cold and hot mode accretion is not allowed. In the simulations, dense, cold streams
can penetrate deep into the diffuse hot halos, allowing for both accretion modes occur
within the same halo (Kereš et al., 2005, 2009; Brooks et al., 2009). Similar results
were found in the study of Lu et al. (2010), who proposed a more accurate recipe for
treating hot and cold mode accretion in SAMs. Here, I would like to point out that
overcoming this weakness of cold and hot mode accretion in SAMs and thus, changing
the amount of cold gas present in a galaxy may automatically result in a more selfconsistent approach for modeling the downsizing in black hole growth. As shown in
Chapter 4, so far, an explicit prescription for the evolution of a sub-Eddington limit of
cold gas accretion onto the black hole is needed in order to obtain the observed decrease
in number densities of luminous AGN at z < 2 and a direct connection of black hole
accretion to the cold gas content failed in reproducing this decrease. Thus, it might be
extremely interesting to see to what extent a more physical recipe for cold and hot gas
cooling in SAMs might lead to a better agreement with the observed decrease in the
amount of luminous AGN when black hole accretion is directly connected to the cold
gas content.
Both the simulations and SAMs are compared to two key observational constraints
at z = 0, 1, and 2: the relationship between dark matter halo mass and stellar mass,
and the relationship between stellar mass and star formation rate. It is found that the
simulations predict stellar masses that are too large for their halo masses at all redshifts.
The stellar masses are too high by a factor of a few for massive halos (" 1012 M" ), and
by an order of magnitude or more for lower mass halos. The SAM results for the NF
model are qualitatively similar, although the stellar masses at high redshift are lower,
due to the lower star formation efficiencies, as already discussed. In the SAMs, the
curvature in the empirical Mgal −Mhalo relation can be achieved by including supernova
driven winds, which suppress star formation in low mass halos, and radio mode AGN
feedback, which suppresses star formation in high mass halos. The SFRs at a given
stellar mass are found to be too low in the simulations at all redshifts z ! 2, proba-
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Galaxy formation in semi-analytic models and zoom simulations
bly because of the overly efficient star formation at higher redshifts and the resulting
gas consumption. Star-forming galaxies in all SAMs were found to lie on the same
relation, with supernova feedback shifting the galaxies along the relation. The SAMs
(both FULL and NF) showed better agreement with the observed ṁstar − Mgal relation
than the simulations, but have SFRs at a given stellar mass that are about a factor of
∼ 2 lower at high redshifts.
In a final summary, I conclude that on the one hand, one can be encouraged by the
robustness of SAMs as a tool for exploring the qualitative effects of varying the physical
ingredients of galaxy formation. On the other hand, also several important areas have
been identified where the quantitative accuracy of fundamental physical recipes in the
SAMs should be improved, and several physical processes that are missing in the SAM
but which should be included. Additionally, there is a tendency to treat numerical
simulations as “truth”, but it has been shown that key predictions of these simulations
are sensitive to uncertain sub-grid recipes. Also several physical processes have been
highlighted that are neglected in the simulations studied here, but which appear to
be crucial in order to understand the properties of real galaxies. These include more
effective implementation of supernova-driven winds, chemical enrichment and metal
cooling, and a self-consistent treatment of the growth of and feedback from black
holes. Of course, these are hardly new suggestions, and considerable progress has been
made recently in all of these areas (e.g. Di Matteo et al., 2005; Cattaneo et al., 2005;
Sijacki et al., 2007; Oppenheimer & Davé, 2008; Booth & Schaye, 2009; Scannapieco
et al., 2009; Governato et al., 2010; Schaye et al., 2010b; Sawala et al., 2010; Ostriker
et al., 2010). In conclusion, I suggest that using these two complementary techniques
(SAMs and hydrodynamic simulations) together in close coordination may provide the
most powerful approach to understanding galaxy formation and evolution for the near
future.
Chapter
6
Conclusion and outlook
6.1
Summary
In our current picture of the joint evolution of galaxies and black holes, there exists a
large number of exciting and puzzling questions, which are subject of intense debate.
For example the origin of the black hole mass scaling relations, whether and how black
hole mass scaling relations evolve at high redshift, trigger mechanisms and efficiency
for black hole accretion and the origin of the observed downsizing in black hole growth.
In this thesis, I used analytical and semi-analytical methods based on the merging
history of cosmological N-body simulations in order to assess and understand some of
the unanswered questions. As, however, semi-analytic models may be criticised due to
their large degree of approximation combined with a huge parameter space compared
to hydrodynamical simulations, the limitations of semi-analytic models were tested by
a direct comparison to cosmological, hydrodynamical simulations. In summary, in this
thesis, I focused on the following topics:
• The scatter evolution in black hole mass scaling relations (Chapter 3)
• The origin of the anti-hierarchical black hole growth (Chapter 4)
• Comparison of SAMs to cosmological zoom simulations (Chapter 5)
In the following, I will briefly summarize the most striking results of the different
chapters. Chapter 3 concentrates on the influence of the merger-driven evolution of
the black hole mass scaling relations and on the evolution of the intrinsic scatter in
black hole mass. Several authors have found observational evidence that galaxies at
high redshift have significantly higher or smaller M• /Mbulge -ratios than ellipticals today (e.g. Treu et al., 2007 and Alexander et al., 2008). Some theoretical studies using
merger simulations explain these observations by an evolution of the black hole-bulge
mass relation with redshift as a consequence of the connected growth of galaxies and
black holes via gas physical processes. However, in contrast to previous work, I have
proposed a completely alternative scenario (Hirschmann et al., 2010) based on purely
142
Conclusion and Outlook
statistical merging: the existence of a larger intrinsic scatter in black hole mass at
high redshifts, even assuming no evolution of the mean relation with cosmic time and
adopting no connected black hole and host galaxy growth by gas physical processes.
In a purely merger-driven model for black hole growth (which becomes realistic for
gas-poor, massive systems at low redshifts), it is found that the scatter in black hole
mass decreases with time, the number of merger events and the black hole mass itself. This is a direct consequence of the statistical Central-limit-theorem, which might
also state an alternative explanation for establishing a relation even if objects are uncorrelated initially and if their growth is not connected via gas-physical processes as
AGN feedback (see also Peng, 2007; Jahnke & Maccio, 2010). The decrease of the
scatter in black hole mass with time and merger number m can be approximated with
σmerg (m) = σini × (m + 1)−a/2 . This is valid for a range of analytical models assuming
random merging of black holes in addition to growth scenarios based on the merging
history of ΛCDM-simulations. I have created the merger trees for a cosmological simulation (L = 100 Mpc, Np = 5123 , Gadget2), populated every newly occuring halo
with black holes and galaxies by adding a certain ’seed’ scatter and considered the
subsequent evolution solely driven by merging events. Adopting a present-day scatter of σ = 0.3 dex, the results imply a scatter of σ = 0.6 dex at z = 3. Thus, a
larger, intrinsic scatter in black hole mass is able to provide a possible scenario, which
can account simultaneously for over- and under-massive black holes at high redshifts.
Moreover, the result is also consistent with the observations of a study of Gültekin
et al. (2009), where they find a smaller scatter for ellipticals (mainly massive objects)
than for non-ellipticals (less massive objects). As a consequence of this study, I want
to emphasize that purely statistical, merging processes of galaxies and black holes seem
to be able to account for the emergence of the black hole scaling relations themselves
as well as for the observed over- and under-massive black holes without a fundamental
need for a connected growth of black holes and galaxies via gas physical processes.
However, besides the importance of merging processes, gas accretion onto black
holes does represent a significant contribution to the over-all mass assembly, which will
become especially important for explaining the puzzling question of the downsizing
trend in black hole growth (Chapter 4). The anti-hierarchical trend is revealed in many
observational studies of AGN (QSO) from different wavebands (IR, optical, soft and
hard X-ray). They find that luminous AGN peak at higher redshifts than less luminous
ones indicating that massive black holes are already in place early in the Universe
whereas less massive black holes tend to form predominantly at later times - at least
assuming that black holes are accreting at the Eddington rate (which is proportional
to the black hole mass). At first sight, this might seem to be in contradiction with the
currently favored hierarchical structure formation paradigm, where at early times low
mass objects are assumed to form, while massive objects should assemble their mass
only later-on. However, the downsizing trend is most likely caused by different accretion
mechanisms onto black holes and their corresponding accretion efficiencies. Thus, for
such an investigation, methods based on black hole growth purely due to merging
6.1
Summary
143
(as used in Chapter 3), are not sufficient anymore, as more complex gas accretion
processes onto the black holes will play a crucial role. Therefore, I used instead a semianalytic model according to Somerville et al. (2008b) and applied it for the first time
to the merger histories of one of the largest currently existing dark matter N-body
simulations, namely the Millennium simulation with a comoving box-length of 500
Mpc containing 1010 dark matter particles (this ensures good statistics). The original
SAM includes an upgraded model for black hole growth following results of SPHsimulations of merging galaxies, by distinguishing between a luminous, Eddingtonlimited period (Quasar-mode) and a low Eddington-ratio accretion, radiatively very
inefficient phase (Radio-mode). The quasar-mode is triggered by major mergers and
black hole growth during this phase is divided into an accretion phase at the Eddingtonlimit followed by a blow-out phase with fading luminosity. Using this original SAM,
the AGN number densities at the peaks of the luminosity curves can be reproduced.
However, at low and high redshifts the characteristic anti-hierarchical behavior occurs:
at low redshifts, the amount of luminous AGN is over-estimated, while moderately
luminous AGN are under-estimated. At high redshifts this trend is reversed. However,
I find that a combination of the following three modified and additional accretion
mechanisms results in a significantly better agreement with the observed downsizing,
in particular at low redshifts:
• A decreasing sub-Eddington limit with decreasing redshift and increasing black
hole mass leads to a decrease in number densities of luminous AGN at low redshift.
• Secular evolution processes triggering additional accretion onto the black holes
- disk instabilities - result in an increase of number densities of low luminous
objects at low redshift.
• Assuming a heavy seeding scenario with Mseed,• = 105 M" and a larger scatter in
accreted black hole mass at high redshift than at low ones increases the amount
of luminous AGN at high redshift.
The greatest success of the best-fit model including these modifications is that it can
reproduce the galaxy-halo mass relation as well as the black hole mass function at z = 0
and additionally its results are in fairly good agreement with the bolometric AGN luminosity function at 0 < z < 5 from the observational compilation of Hopkins et al.
(2007c). This indicates clearly that the outlined modifications might state important
mechanisms in the picture of the co-evolution of galaxies and black holes: massive black
holes can have assembled their mass until z ≈ 2, but accrete only at a small fraction
of the Eddington-rate at low redshifts so that they are hidden and do not occur as
high luminous quasars in the present-day Universe. The large amount of moderately
luminous AGN at low redshifts might be caused partly by the low-Eddington ratio accretion in the power-law decline phase, and partly by gas accretion triggered by secular
evolution processes such as disk instabilities in agreement with observational results.
Moreover, large seed black hole masses seem to be the most straight forward scenario
to explain high luminous quasars at high redshifts. Current observational constraints,
144
Conclusion and Outlook
however, are not sufficient in order to determine whether a heavy seeding scenario or
a light seeding scenario with super-Eddington accretion is the correct process. Despite
the success of the best-fit model, moderately luminous AGN at high redshifts are still
over-estimated, even in the best-fit model. Insufficient treatment of obscuration effects
in the observational compilation of Hopkins et al. (2007c) may account for this discrepancy. If namely a redshift dependent dust obscuration is additionally adopted in the
best-fit model and the bolometric luminosities are converted into hard X-ray luminosities, then the observed hard X-ray luminosity function can be reproduced reasonably
well by the best-fit model, as the dust obscuration mainly reduces the amount of moderately luminous AGN at high redshift. This shows that a redshift dependence in the
dust obscuration might provide a further ’puzzle piece’ for a complete understanding
of downsizing.
Finally, even if semi-analytic models represent the best currently available method
for investigating problems which demand large statistics, such as e.g. the downsizing
problem, a main disadvantage of semi-analytic models is that there exists a great deal
of uncertainty in many of the important processes and most of the physical recipes
contain many free parameters (compared to simulations). The extent to which this
actually matters has not yet been well assessed. Therefore, in particular in order to
be critical against the results in Chapter 4 (which were based only on semi-analytic
modeling) as well as to test their robustness and reliability, in Chapter 5, a detailed
comparison of SAMs versus cosmological zoom simulations tries to reveal the degree
of agreement in the evolution of the baryonic component (Hirschmann et al., 2011).
Various physical processes can be isolated in order to improve the accuracy of the
semi-analytic recipes and to assess the question which physical processes are missing in
the simulations. I have created the merger trees for a large sample of ≈ 48 individual,
re-simulated halos with unprecented resolution providing the basic input for the SAMs.
To make a fair comparison to the SPH-simulations, I use - besides the full SAM - three
additional different stripped-down versions of the SAMs varying the SN feedback, metal
enrichment and AGN feedback:
• No feedback, no metals (NF)
• Thermal SN-feedback, no metals (SN)
• Thermal SN-feedback with momentum-driven winds and metal evolution (SNWM)
• Additionally assuming feedback from black hole accretion (FULL)
Generally, SAMs and simulations reveal a better match at low redshift than at high
ones. The best agreement to simulations is found for the ’no feedback’ version (in
particular for the total baryon mass and the condensed baryons), indicating the weak
impact of the SN feedback in the simulations. However, a significant discrepancy occurs for the star formation efficiency at high redshift which is found to be much larger
in simulations than in the SAMs due to a clumpy cold gas structure at high redshifts
6.1
Summary
145
found in the simulations. Moreover, in the framework of how galaxies assemble their
stellar content, the bimodal behavior of the stellar evolution as it is found in the simulations by Oser et al. (2010) can not be reproduced by the SAMs. In all models, even
including AGN feedback and metal cooling, in-situ star formation is dominating over
accretion of stars at all redshifts, whereas in the simulations the fraction of accreted
stars becomes larger than the one of in-situ formed stars with evolving time. This is
due to a larger accreted stellar mass in the simulations than in the SAMs, probably a
consequence of the extremely efficient star formation in simulations at high redshifts.
Moreover, additional physical mechanisms in SAMs, as e.g. gravitational heating or
delayed strangulation, might be necessary in order to balance the accreted and in-situ
formed star fraction in the right way. Additionally, a striking discrepancy occurs for
the high redshift evolution of the hot gas content, as the SAMs predict a larger hot
gas content than simulations. This might be due to an overestimation of the heating
rates in the SAMs, while cold flows are not sufficiently accounted for. For massive
galaxies and in particular at high redshifts, accretion rates in the cold mode (= rapid
cooling) are much larger in the simulations than in the SAMs, where cold flows occur
only for very high redshifts. This points out that in the SAM a very simplified recipe
for hot and cold mode accretion is used - depending on the ratio of the cooling radius
to the virial radius, rcool /rvir . For this model, the recipes for cold and hot mode accretion should be improved significantly. A comparison of both model approaches to
observations shows that only the full SAM reveals a reasonable good agreement with
observations, whereas in simulations more physics, as e.g. a stronger supernova feedback (winds), has to be included. Here, I want to emphasize that it is very interesting
to investigate, whether and how enhanced recipes in SAMs, e.g. gas cooling, might influence the black hole growth and thus, the evolution of the connected AGN population.
In final summary, this thesis points out that in order to explain the black hole mass
scaling relations themselves and to be able to account for over- and under-massive black
holes at high redshifts, solely statistical merging processes may be sufficient neglecting
any connected growth between galaxies and black hole via gas-physical processes. However, a detailed understanding of gas accretion processes onto black holes, which states
a significant contribution to the over-all black hole growth, seem to be of particular importance and the key ingredient for explaining of the evolution of differently luminous
AGN number densities and thus, the corresponding downsizing trend. As for such an
investigation large, statistical samples of galaxy populations have to be generated with
semi-analytic models, a close coordination of these models, often including simplified
approximations for galaxy formation, with cosmological, hydrodynamical simulations
may provide one of the most powerful approaches to understand galaxy formation and
evolution in more detail and thus, to enhance the recipies used in semi-analytic models.
146
6.2
Conclusion and Outlook
Next steps
As pointed out in this work, in particular in the last Chapter, a well-known weakness
of most semi-analytic models consists in the implementation of various, over-simplified
recipes for galaxy formation, with a particular effect on the high-redshift evolution.
Compared to results of detailed numerical studies, SAMs are often neglecting important physics, as e.g., most SAMs do not incorporate a sufficient model for describing
cold flow physics yet and cannot account for the simulated bimodal behavior of the
stellar evolution. In order to compete and keep up with numerical simulations in the
future, a major task in the field of semi-analytics should be the extension and improvement of key physical ingredients by closely following the detailed knowledge of
physics and from hydrodynamical simulations, with the particular aim to reproduce
high-redshift observations. Thus, in future it might be useful to focus on more physical
implementations and examine their importance for galaxy formation. In the following
I list and describe some of the possible extensions to current models:
1. Missing cold flow physics: Self-consistent gas inflow and heating
In most of the SAMs, gas is accreted onto the halo either in the hot or in the cold
mode (= slow and rapid cooling regime) depending on the ratio of cooling to virial radius. Typically, cold mode accretion occurs for low mass halos and for higher redshifts,
hot mode accretion for high mass halos and for lower redshifts. However, simulations
show a simultaneous infall of hot and cold gas onto the halo for all redshifts with a
much larger fraction of cold infalling gas than in SAMs (assuming no metal enrichment
in both methods). The simplified recipe in the SAMs does not account for the coexistence of cold and hot halo gas and the bimodal accretion seen in simulations. This
influences strongly the evolution of galaxies, in particular at high redshifts. Thus, motivated by hydrodynamical simulations, implementing ’gravitational’ heating seems to
be a promising mechanism in order to treat gas inflow and heating in a self-consistent
way. Following Khochfar & Ostriker (Khochfar & Ostriker, 2008), the heating of the
intracluster medium (ICM) would be calculated by the net surplus of gravitational potential energy released from gas that has been stripped from infalling satellites. Note
that the simplified approximation for an additional gravitational heating source as used
in Chapter 4 can only be the start and has to be implemented in a more self-consistent
way. Gravitational heating is found to be an efficient heating source for massive dark
matter halos, where it prevents cooling, and becomes especially important at late times.
Assuming in general cold infalling gas (no shock heating) and a fraction of hot accretion, which is only due to gravitational heating, would result in an automatic, less
efficient heating at high redshifts and for low mass halos than in current SAMs. This
should result naturally in larger cold accretion fractions for objects at high redshifts.
As this new implementation might presumably mainly affect and change the high redshift evolution of the hot and cold gas content in galaxies, a direct comparison to new,
observed data of high redshift luminosity functions (z = 7 − 10, Bouwens et al. (2008))
might provide a reasonable test for the success of this new model. A further advantage
6.2
Next steps
147
of gravitational heating is - as the released energy can match the one of AGN feedback
- less AGN feedback than currently assumed might be necessary to overcome the overcooling problem and to obtain a self-consistent black hole growth (this was already
shown for the simplified approximation for gravitational heating in Chapter 4). In a
recent study of Fontanot et al. (2011) it was shown that most of the currently used
SAMs are extremely overestimating radio-mode accretion compared to observations.
Moreover, due to the change of the cold gas content, gravitational heating might also
influence the number densities of AGN, in particular at high redshifts, and thus, might
reproduce the corresponding downsizing in a more self-consistent way than in the current model.
2. Bimodal behavior of the stellar evolution
Cosmological simulations of galaxy formation appear to show a two-phase character
with a rapid early phase at z > 2 during which in-situ stars are formed within the
galaxy from infalling cold gas followed by an extended phase since z < 3 during which
ex-situ stars are primarily accreted. In the latter phase massive systems grow considerably in mass and radius by accretion of smaller satellite stellar systems formed at
quite early times (z > 3) outside of the virial radius of the forming central galaxy. In
contrast to that, in-situ star formation is found to be the dominating process in SAMs
at all redshifts. This is because in SAMs less stars are produced in smaller, accreted
galaxies. However, a well-known problem of many SAMs is that they often add excessive feedback from AGN to substitute for further important physics as gravitational
heating. Therefore, due to physically not correct feedback processes, star formation
might be prevented in small galaxies falling into larger ones. This might suggest that
implementing gravitational heating - accounting for a physically motivated heating in
galaxies - could be again a crucial ingredient in order to get an appropriate balance
between accreted and in-situ star formation, and thereby, a bimodal stellar evolution
as seen in simulations. Moreover, current SAMs are assuming instantaneous strangulation of the hot halo component of the accreted system. However, it might be more
reasonable to adopt delayed strangulation in dependence on the halo potential. Besides, as accretion of stars is closely linked to the size evolution of galaxies, which,
however, could not have been reproduced by current SAMs so far, the right fraction of
in-situ and accreted stellar mass might also account for the observed size evolution.
3. AGN feedback from X-ray emission
As AGN are known to emit X-rays through their host galaxy, a step towards a more
complex model of AGN feedback might be to take into account additional X-ray feedback from AGN. Implementing an observationally calibrated X-ray radiation field into
hydrodynamical simulations, which emanates from black holes, shows that gas is heated
out to large radii from the galactic center and thus, the effects which are reported from
’traditional’ AGN feedback are enhanced: The ’new’ feedback is found to be twice as
effective at suppressing star formation (three times less SF in the last 6 Gyr) and lowers
148
Conclusion and Outlook
the final black hole mass by 30%. Besides, less gas is accreted during major mergers
(instead accretion happens more smoothly over the following several Gyr) and the
baryonic conversion efficiency gets significantly reduced. Since one of the key strengths
of the semi-analytic approach is the ability to easily investigate the influence of varying physical recipes on galaxy properties, an interesting issue might be to study the
effect of X-ray feedback in a cosmological context by closely following the results from
hydrodynamical simulations. Due to this additional heating process a major effect will
presumably consist in the reduction of the star formation and the black hole accretion rates. Maybe, this might naturally result in a better reproduction of the observed
downsizing without assuming a priori a decreasing upper limit for the Eddington-ratios.
4. Evolution of the stellar & AGN population due to stellar mass loss
In hydrodynamical simulations (see Ciotti & Ostriker, 2007) it is found that recycled
gas from dying stars (stellar mass loss) causes important implications on galaxy formation. A typical stellar population - dependent on the adopted IMF - loses 30 − 40%
of its mass over a Hubble time influencing star formation and metal enrichment. In
addition, stellar mass loss provides an important source of fuel for the central SMBH,
even in the absence of external phenomena such as galaxy merging. As gas from dying
stars is flowing into the center, a nuclear starburst must occur coincident with an AGN
flaring. Following the simulations of Ciotti & Ostriker (2007), roughly half of the recycled gas is ejected as galactic winds, half is consumed in central starbursts and only
a small fraction of the order of 1 1% is accreted onto the central black hole inducing
AGN activity. Most SAMs account for the mass-loss from Supernovae II explosions
using the instantaneous recycling approximation so that recycled gas is available for
cooling and star formation. However, many SAMs do not model delayed stellar mass
loss from SNIa explosions and stellar winds from AGB stars and none of them do take
into account associated AGN activity for the growth of black holes. Therefore, implementing such additional scenarios would provide a very suitable method to study how
the over-all cosmological evolution of galaxies is affected, in particular the evolution of
star formation rates, metals and the number density evolution of AGN over cosmic time
compared to observations. Presumably, it would result in a significant contribution to
the AGN number density in blue, star-forming and thus less massive host galaxies.
Another interesting implication of gas recycling through star formation is that according to Martig & Bournaud (2010) the structural evolution of massive galaxies might
strongly be affected by helping the disk growth and their survival in interactions and
mergers. Taking this into account in SAMs may lead to a better reproduction of the
observed bulge-to-disk ratio of massive galaxies at low redshifts.
Finally, I want to point out that it might be very reasonable and helpful if the
improvements and changes as described above could always be accompanied by detailed comparison studies to simulations. Therefore, it would be of advantage to create
cosmological, high-resolved SPH-simulations using up-to-date simulation codes including more enhanced physics as SNIa, metal enrichment, stellar mass loss or black hole
6.2
Next steps
149
growth with feedback. Moreover, simulating large, cosmological volumes including
sub-resolution models for gas, stars and black holes, would allow to make an overall,
statistical comparison of the galaxy populations to SAMs, e.g. by investigating the
evolution of the stellar or the black hole mass function and their dependence on environment.
To summarize, a desirable aim for the future might be to modify existing models
of galaxy formation in several aspects - by improving the AGN feedback prescription
(X-ray emission), cold gas accretion and stellar mass loss - so that it incorporates the
majority of the current understanding of galaxy formation. After this implementation,
it would be of great interest to explore the extent to which such additional physical
processes would contribute to an over-all improvement of galaxy modeling by comparing to observational data, with the particular aim to achieve a deeper and more
fundamental understanding for the exciting question of how galaxies and black holes
are co-evolving, and thus, moving towards a more complete picture of the Universe we
live in.
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Conclusion and Outlook
Appendix
A
Effects of numerical resolution
A.1
Dark matter component
For one high and low mass halo with final masses of Mhalo = 3 × 1012 M" (M0501)
and of Mhalo = 8 × 1011 M" (M1646) , we have performed re-simulations with 4× the
original spatial resolution. We traced back the particles that are closer than 2 × r200
to the center of the halo in any of our snapshots and replace them with dark matter
and gas particles of higher resolution, achieving a 16× better mass resolution in the
high resolution region than in the original simulation: mDM = 2.5 × 106 M" h−1 and
mGas = mStar = 5 × 105 M" h−1 . The mass aggregation history of the dark matter
component in both halos is very similar for the two different resolution limits as shown
in the upper panel of Figs. A.1 and A.2. The red, dotted line corresponds to the
2× resolution, the black solid line to the 4× resolution. In the middle and lower
panel of Figs. A.1 and A.2 one can see that the histograms of major (> 1 : 10)
and minor mergers (< 1 : 10). The number of major mergers stays the almost the
same, whereas halos are experiencing more minor merger events (< 1 : 10) in the
higher resolution case, as more objects can be identified as substructure in the higher
resolution case. This suggests that, since galaxy formation in the SAMs is mainly
influenced by major merger events (> 1 : 10), the result from the SAMs is expected to
be mostly independent of the resolution limit of the simulation.
A.2
Baryonic components
Figs. A.3 and A.4 show explicitly the evolution of the star, cold and hot gas mass in
SAMs and simulations based on the trees of the two re-simulated individual halos for
a 2 × (dashed lines) and 4 × (solid lines) resolution. As expected from the similar
evolution of the dark matter component, for both, the low and the high mass halo
re-simulations, we find between SAMs and simulation no significant difference in the
evolution of the central galaxy/main halo. Therefore, due to computational costs, we
restrict our study to 2× re-simulations of 40 individual halos.
152
Appendix
Figure A.1: Upper panel: comparison of aggregation history (Resim M0501) of the
halo mass for two different mass resolutions (2x: red, dotted line and 4x: black, solid
line) in the SPH-re-simulations of a 3.0 ×1012 M" halo. Intermediate panel: Number of
minor mergers (> 10 : 1) as a function of lookback time for the two different resolution
limits. Lower panel: Number of major mergers (< 10 : 1) as a function of lookback
time for the two different resolution limits.
A.2
Baryonic components
153
Figure A.2: Same as Fig. A.1, but for the halo M1646 with a smaller halo mass
8.0 × 1011 M" .
154
Appendix
Figure A.3: Evolution of the baryonic components for two different mass resolutions
(2x: dotted lines, 4x: solid lines) for the halo M0501. The upper panel shows the
evolution of the stellar component, the middle panel the evolution of the cold gas fraction
and the lower panel corresponds the hot halo gas component. Black lines illustrate the
re-simulations, colored lines the different SAM versions.
A.2
Baryonic components
Figure A.4: Same as Fig. A.3, but for the halo M1646.
155
156
Appendix
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Curriculum Vitae
Curriculum Vitae
PERSONAL INFORMATION
Contact
Date of birth
Nationality
Ruemannstrasse 53, 80804 München, Germany
Tel. 0049 160 2608048
[email protected]
April 7, 1981 in Nürnberg
German
EDUCATION AND TRAINING
11/2006-pres.
PhD: research at University Observatory Munich, Munich
”Co-evolution of galaxies and black holes using analytic and
semi-analytic methods"
Supervisor: Prof. Dr. Andreas Burkert
02/2006
Diploma in Physics, Friedrich-Alexander University ErlangenNürnberg (mark: 1.0, with distinction)
02/2005-02/2006
Diploma Thesis: research at the Friedrich-Alexander University, Chair for condensed matter:
"Scattering of electrons in image potential states on the
Cu(001) surface at Co ad-atoms"
Supervisor: Prof. Dr. Thomas Fauster
10/2002-02/2005
Undergraduate and graduate studies in physics at the
Friedrich-Alexander University of Erlangen-Nürnberg
09/1987 - 06/2000 Attendance at school. Final examination: Abitur (High School
Degree, mark: 1.1), Ostendorfer Gymnasium, Neumarkt, Germany
188
AWARDS AND HONORS
07/2006:
Ohm-award of the Max-Schaldach-Stiftung for one of the best
diploma thesis during the academic year 2005/06
TALKS AT CONFERENCES AND WORKSHOPS
09/2010
08/2010
06/2010
09/2009
07/2009
12/2007
03/2006
12th Birmingham-Nottingham Extragalactic workshop, AGN:
populations, parameters and power: "Anti-hierarchical growth
of black holes in the universe"
Santa Cruz Galaxy workshop 2010: "Anti-hierarchical growth
of black holes in the universe"
Puzzles of galactic nuclei, scientific workshop at MPE: "Antihierarchical growth of black holes in the universe"
Collaborative workshop of the ΦGN-group at MPIA: "The evolution of the intrinsic scatter of massive black holes"
Physics of galactic nuclei, scientific workshop at Ringberg Castle: "The evolution of the intrinsic scatter of massive black
holes"
Start-up workshop of the ΦGN group at MPE: "Black hole
growth in a hierarchical universe"
Annual conference of the DPG: "Scattering of electrons in
image-potential states on the Cu(001) surface at Co ad-atoms",
Dresden
RESEARCH INTERESTS
Research fields:
Cosmology, structure formation, galaxy evolution, scaling relations, co-evolution of galaxies and black holes, active galactic
nuclei, comparison of semi-analytic models with simulations
Methods:
Statistical methods (central-limit-theorem), semi-analytic
models, cosmological SPH-simulations, construction of merger
trees on basis of the halofinder FOF and Subfind
List of publications
189
List of publications
REFEREED ARTICLES
• Hirschmann, M., Fauster T. "Scattering of electrons on the Cu(001) surface
by Co adatoms", Appl. Phys. A 88, 547 (2007)
• Hirschmann, M., Khochfar, S., Burkert A., Naab, T., Genel S., Somerville R.
"On the evolution of the intrinsic scatter in black hole versus galaxy mass relations", MNRAS 407, 1016-1032 (2010)
• Hirschmann, M., Naab T., Somerville R., Burkert A., Oser L. "Galaxy formation in semi-analytic models and cosmological hydrodynamic zoom simulations",
submitted to MNRAS
CONFERENCE PROCEEDINGS
• Hirschmann, M., Khochfar, S., Burkert A., Naab T. "The evolution of the intrinsic scatter of massive black holes", proceedings of the ΦGN workshop
ARTICLES IN PREPARATION
• Hirschmann, M., Somerville R., Naab T., Burkert A. "Origin of the antihierarchical growth of black holes in the universe", to be submitted
• Remus, R., Burkert A., Hirschmann M. et al. "Galaxy groups", in preparation
• Naab, T., Oser, L., Hirschmann M. et al. "Assembly of massive galaxies", in
preparation
190
Danksagung
191
Danksagung
Ich danke...
• an erster Stelle meinem Doktorvater, Prof. Dr. Andreas Burkert, für die
Möglichkeit an der Universitätssternwarte in München promovieren zu können,
für seine ständige, engagierte Unterstützung dieser Arbeit, für die zahlreichen
produktiven, wissenschaftlichen Diskussionen mit ihm genauso wie für seine kritische Anmerkungen mir mögliche Schwachstellen meiner Arbeit aufzuzeigen.
• Dr. Thorsten Naab für seinen unermüdlichen Einsatz für das Fortschreiten und Gelingen meiner Arbeit, seinen enormen Ideenreichtum, seine immerwährende Geduld (auch bei noch so “einfachen” Fragen) und seine aufbauenden
und ermunternden Worte in den manchmal auch schwierigeren Zeiten meiner
Promotion. Es war mir eine sehr grosse Freude mit ihm zu arbeiten.
• Prof. Dr. Rachel Somerville dafür, dass sie mich in die Geheimnisse der
semianalytischen Modelle eingeweiht hat, mir dabei immer mit Rat und Tat
zur Seite gestanden ist genauso wie für die zweimaligen USA Besuche an ihrem
Institut (Space Telescope Science Institute). Ohne sie wäre diese Arbeit niemals
das geworden, was sie ist.
• Dr. Sadegh Khochfar für seine engagierte Betreuung in den Fragen der Entwicklung der Streuung in der Masse des schwarzen Loches.
• der CAST & ΦGN Gruppe für interessante wissenschaftliche Diskussionen
während der “group meetings”.
• den Systemadministratoren der USM, ganz besonders Dr. Tadziu Hoffmann, der zu allen ’Unzeiten’ auftretende Probleme immer schnellstmoöglich
behoben hat.
• den Sekretärinnen der USM, dafür, dass sie mir bei anfallendem “Verwaltungskram” behilflich waren.
• und ganz besonders meinen ’Mitstreitern’, Hanna, Steffi und den ’Jungs’,
Ludwig, Simon, Michi und Matthias, die mir die Zeit meiner Promotion sehr
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’versüsst’ haben während diverser Kaffeepausen, bei mittäglichen Kochgelagen
und ganz besonders bei abendlichem Prosecco-Genuss. Die Zeit mit ihnen wird
mir unvergesslich bleiben.
• meinem guten Freund Ralf für die exzellente “psychologische” Betreuung dieser
Doktorarbeit und seine unerschöpfliche Geduld, die er mit mir hatte.
• meiner Freundin Sabine für die schönen und spassigen Zeiten, die wir in München
zusammen hatten, in denen sie mich aus den ’wissenschaftlichen Gefilden’ immer
wieder auf den Boden Realität geholt hat.
• meinem Tanzpartner Franz, der mir gezeigt hat durchs Leben und durch die
Promotion zu “tanzen”. Das Training mit ihm war der perfekte Ausgleich zu
meiner Doktorarbeit und ich werde es in Zukunft sehr vermissen.
• und nicht zuletzt meinen Eltern, mit deren ständiger Unterstützung ich es schaffen konnte, diese Doktorarbeit zu vollenden.
193
Erklärung
Ehrenwörtliche Versicherung und Erklärung
der Doktorandin
Michaela Hirschmann
Mit der Abgabe dieser Doktorarbeit versichere ich, dass ich die Arbeit selbständig
verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.
Ort, Datum
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