Semi-analytical galaxies in the multidark-universe: A perspective on the evolution of the most luminous and massive galaxies throughout cosmic history

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Semi-Analytical Galaxies

in the

MultiDark Universe

A perspective on the evolution of the most luminous and massive galaxies throughout cosmic history

A dissertation submitted in partial fulfilment of the requirements of the degree of

Doctor of Philosophy in Theoretical Physics

of the Universidad Autónoma de Madrid

by

Doris Stoppacher

advised by

Dr. F. Prada & Dr. A. Knebe

Madrid, November 2019

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Abstract

Understanding the formation and evolution of galaxies within a self-consistent cosmological context is one of the outstanding and most challenging topics of astrophysics. This dissertation is dedicated to investigating the concepts of galaxy formation with cosmology, in particular, the formation of large-scale structures and the assembly and evolution of their associated galaxies over cosmic time. This forms part of a vibrate and innovative field of research spanning all imaginable scales from The Big Bang to the Milky Way Galaxy with its many satellites.

Over the last few decades a lot of effort has been invested into the development of models which can produce statistically significant sets of galaxy properties in a computationally efficient way.

These models included the population of dark matter halos using simplified phenomenological treatments of baryonic processes and coarse-graining the properties of galaxies. As a result, relevant equation systems can be solved more efficiently – a semi-analytical model (SAM) was born.

The MultiDark-Galaxies was an ambitious project dedicated to the release of galaxy catalogues from three different SAMs: Galacticus, SAG, and SAGE; run on the MultiDark Planck 2 N-body simulation covering a cubic volume of 1h⁻¹Gpc as part of this thesis work.

To this point, the released galaxy catalogues remain one of the largest of their kind based on SAMs. We perform a comparison of the outputs of the three models and their conformity with fundamental galaxy properties derived from observations. Each model of galaxy formation has its unique recipe followed by individual calibration and tuning, therefore we highlight their differences and similarities. We demonstrate further that SAMs are an exceptional resourceful method of studying statistically significant samples of galaxy properties.

We identify modelled galaxies from Galacticus, which show similar properties as observational samples and therefore are truly comparable with luminous red galaxies (LRGs) from e.g. SDSS at z ∼ 0.1 or BOSS-CMASS at z ∼ 0.5. We extract CMASS-mock samples from Galacticus using the original photometric selection as well as alternative methods mimicking such a selection. We study these mock samples in detail and find correlations of properties related to star formation:

(specific) star formation rate, gas-phase metallicity ZCold, and cold-gas fraction MCold/M_∗, but also properties such as the halo mass M200c or the black hole mass MBH; with the large-scale structure environment e.g. filaments or knots.

We emphasise that, the bimodalities found in the properties of Galacticus’ CMASS-mocks, manifesting themselves as two distinct populations, could provide insights in the galaxy evolution in the context of the origin of the fundamental luminosity/mass-metallicity relation, merger- induced star formation, or “downsizing”. Our results may further challenge the paradigm that the large-scale environment does not influence the galaxy formation and predictions on the evolution of a galaxy inside its halo can be derived only from the halo mass and the occupation distribution, also know as the galaxy assembly bias.

We trace the progenitors of Galacticus’ LRG-samples at low redshift to z ∼ 0.5 and identify 20% of them as CMASS. We show further the full mass growth history of the progenitors of the most diverse populations, red and blue, found in the CMASS-mock of Galacticus. We find that those samples have distinct properties, cluster differently, and have been assembled at different cosmic times, most probably throughout different evolutionary paths. We will conduct further analyses in order to confirm the possible detection of a galaxy assembly bias signal in our SAM.

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Resumen

Comprender la formación y evolución de las galaxias dentro de un contexto cosmológico auto- consistente es uno de los temas más destacados de la astrofísica actual y uno de sus mayores retos. Esta tesis está dedicada a investigar los conceptos de formación de galaxias en una cos- mología distinca, y en particular a la formación de estructuras a gran escala y al ensamblaje y evolución de las galaxias asociadas a las mismas a lo largo del tiempo cósmico. Esto forma parte de un campo de investigación vibrante e innovador que abarca todas las escalas imaginables, desde el Big Bang hasta la Vía Láctea con sus numerosos satélites.

A lo largo de las últimas décadas se ha invertido mucho esfuerzo en el desarrollo de modelos capaces de producir conjuntos estadísticamente significativos de propiedades de galaxias de una manera computacionalmente eficiente. Estos modelos han incluido la población de halos de materia oscura utilizando tratamientos fenomenológicos simplificados de procesos bariónicos y procesar rudimentario las propiedades de las galaxias. Como resultado se ha obtenido una resolución más eficiente de los sistemas de ecuaciones relevantes, lo que ha supuesto el nacimiento del modelo semi-analítico (SAM).

El proyecto The MultiDark-Galaxies es un ambicioso proyecto dedicado a la obtención de catálogos de galaxias de tres SAM diferentes: Galacticus, SAG y SAGE; ejecutado en la simulación de N-cuerpos MultiDark Planck 2 cubriendo un volumen cúbico de h⁻¹Gpc como parte de este trabajo de tesis. En la actualidad, los catálogos de galaxias publicados siguen siendo algunos de los más grandes de su tipo basados en SAM. Realizamos una comparación de los resultados de los tres modelos y de su concordancia con las propiedades fundamentales de las galaxias derivadas de las observaciones. Cada modelo de formación de galaxias usa presciptiones únicas, así como calibraciones y ajustes individuales, por lo que podemos analizar sus diferencias y similitudes. Además, demostramos que los SAM son un método excepcionalmente ingenioso para estudiar muestras estadísticamente significativas de propiedades de galaxias.

Identificamos galaxias modeladas por Galacticus que muestran propiedades similares a las muestras observacionales y que son comparables con las galaxias rojas luminosas (LRG) de catálogos como SDSS en z ∼ 0.1 o BOSS-CMASS en z ∼ 0.5. Extraemos muestras simuladas de CMASS de Galacticus utilizando la selección fotométrica original, así como métodos alter- nativos que imitan dicha selección. Estudiamos estas muestras simuladas de forma detallada y encontramos correlaciones entre propiedades relacionadas con la formación estelar: tasa de for- mación de estrellas (específica), metalicidad en fase gaseosa ZColdy fracción de gas frío (MCold/ M_∗), pero también con otras propiedades como la masa de halo M200c o la masa del agujero negro MBH; con el entorno de galaxias a gran escala, por ejemplo filamentos o nudos.

Cabe resaltar que las bimodalidades encontradas en las propiedades de la muestra similada CMASS de Galacticus, que se manifiestan como dos poblaciones distintas, podrían propor- cionar información sobre la evolución de la galaxia en el contexto del origen de la relación fundamental luminosidad / masa-metalicidad, la formación estelar producida por mergers, o

“downsizing”. Nuestros resultados pueden desafiar aún más el paradigma de que el entorno a gran escala no influye en la formación de galaxias y las predicciones sobre la evolución de una galaxia dentro de su halo pueden derivarse solo de la masa de halo y la distribución de ocupación, también conocida como el sesgo de ensamblaje de galaxias.

Rastreamos los progenitores de las muestras LRG de Galacticus con un desplazamiento al rojo bajo hasta z ∼ 0.5 e identificamos el 20% de ellos como CMASS. Mostramos además el historial completo de crecimiento en masa de los progenitores de poblaciones muy diversas, rojas y azules,

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que se encuentran en la muestra similada de CMASS de Galacticus. Encontramos que esas muestras tienen propiedades distintas, se agrupan de manera diferente y se han ensamblado en diferentes tiempos cósmicos, muy propablemente siguiendo diferentes caminos evolutivos.

Llevaremos a cabo más análisis para confirmar la posible detección de una señal de sesgo de ensamblaje de galaxias en nuestro SAM.

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1 Introduction

I am fascinated! When I was very young and took a look onto the bright night sky, I was fascinated. When I learned, that all those sparkling stars, I observed, are actually suns that lived a long time ago in systems far away from us, I was fascinated. Much later, when I found out that, each tiny temperature anomaly measured in the cosmic microwave back ground radiation (CMB) [218] is about to grow towards the impossible size of a supercluster and gives rise to the most massive objects in the Universe, I was fascinated. And finally, when I researched the cosmic history and galaxy evolution for this very dissertation, I have been truly fascinated. I am fascinated by the puzzles the Universe holds for us, by the mechanisms of galaxy formation, by the formation and growth of structure, by the question where did we come from and where are we heading for, I was always fascinated.

With that fascination in my head and mind, I am going to introduce theories of galaxy evolution and computational methods of galaxy formation with the main objective of understanding what does shape galaxy properties in a cosmological context. In the first part of this introduction, a brief review of the most important paradigms of cosmology and the history of the Universe is given, before focusing on luminous red galaxies and their connection to their dark matter halos.

The second part is dedicated to the presentation of modelling techniques and introduces mainly semi-analytical models of galaxy formation and evolution. In the third and last part of this introduction, one example of modern challenges in the field of galaxy formation is presented – the assembly bias, which recently has been a topic for extensive debate.

1.1 Part I: How does the Universe work?

The Universe is all of time and space,¹ born from The Big Bang, driven by late-time acceleration [220, 231], and constrained by the fundamental laws of nature such as conservation, classical mechanics, and relativity [308]. The Universe comprises all energy and matter in their various forms as an isolated² thermodynamical system steadily pursuing the maximisation of its own entropy according to the second law of thermodynamics.³

1.1.1 A brief introduction on the history of the Universe

In the framework of modern cosmology and astrophysics, the standard model, Lambda Cold Dark Matter (ΛCDM) [41, 91, 257], describing the formation and evolution of the cosmos, is built upon two simple but important principles supported by a number of observations [139, 155, 221]:

The Universe is isotropic and homogeneous [100, 128], although we find ourselves living in a very inhomogeneous place, a galaxy. Our galaxy, the Milky Way, forms part of an even larger structure, a network of galaxies, clusters, and superclusters we call the cosmic web [34], where galaxies are aligned on rather thin filaments and concentrated towards the centre of gravity such as huge knots hosting the most massive clusters and galaxies. For an illustration of the observable Universe see Fig. 1.1.

1 ... everywhere and anywhere, every star that ever was. Where do you want to start? – Doctor Who

2 However, if the Universe is really a closed system remains controversy [117].

3 The second law of thermodynamics states that the entropy of an isolates system can never decrease over time, but reaches its maximum in the state of thermodynamical equilibrium [47]

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1 Introduction

The expansion and evolution of the Universe is determined by the Friedmann Equations [107], derived from Einstein’s field equation in the context of general relativity⁴and the under assump- tion of the cosmological principle. The solution for a flat open world model can be expressed in one single elegant equation and describes the evolution of the cosmic scale factor⁵ (a) with time and therefore the expansion of the Universe:

H²(t) =

˙a a

2

= H₀² [Ω_rad (1 + z)⁴+ Ω_m (1 + z)³+ Ω_k (1 + z)²+ Ω_Λ] (1.1) with H0 being today’s Hubble constant and the Ωs, the density parameters defined as the ratio of the observed density ρ to the critical density⁶ ρ_cr of today’s Universe. z is the redshift of the Universe and related to the scale factor as a(t) = 1/(1 + z). The density of radiation is given by Ωrad, the density of matter by Ωm, the density of curvature by Ωk, and the density of dark energy⁷ by ΩΛ, respectively:

Ω_rad = 8πG

3 H₀² ρ_rad , Ω_m= 8πG

3 H₀² ρ_m , Ω_k=−c²

H₀² k, Ω_Λ= c²

3 H₀² Λ (1.2) with G being the gravitational and Λ the cosmological constant.⁸

The density parameter given in Eq. (1.1) such as radiation or matter scale differently with redshift z. Therefore, each component of the equation dominates the expansion of the Universe at different cosmological times. Each transition, when one of the parameters started dominating the expansion, brought forward a new era in cosmic history. Currently, the energy density corresponding to the dark energy, represented by ΩΛ is dominating and causing what we call late-time acceleration. Dark energy took over the expansion of the Universe almost at the same time as the solar system formed or the first tracers of life appeared on Earth (a coincidence?!).

Before this epoch, the Universe went through a phase where collisionless dark matter together with ordinary baryonic matter, represented by Ωm, dominated its fate. Previously, baryons have not yet been cool enough to accrete into the potential wells of the cold dark matter to form galaxies. The obscure period of time, where the Universe has been transparent,⁹ but no star was born yet, is called dark ages. The only photons which could travel freely through space and time originated from the cosmic microwave background radiation (CMB), the oldest observable of the Universe at z ∼ 1100. The CMB consists of photons which could escape their coupling state and have not experienced an interaction with matter ever since. Previously to the time of recombination, where the Universe actually became transparent, the temperature

4 Spacetime tells matter how to move; matter tells spacetime how to curve – John A. Wheeler, theoretical physicist

5 The scale factor is a dimensionless parameter which represents the relative expansion of the universe and relates the comoving distances for an expanding universe with the distances at a reference time – the present.

6 The critical density is estimated to be approximately 5 atoms/m²whereas the average density of baryonic matter seems to be approximately 0.2 atoms/m²[230]. Furthermore, the contribution to the total energy budget of the Universe is with∼ 5% fairly small [221].

7 It is distinguished from ordinary matter in the sense that is has a negative pressure. It holds∼ 70% of the Universe’s total energy budget, but its true nature or origin has not been identified yet [48, 215, 237]. It further provides a general label on physics which is unknown or currently not well enough understood [8].

8 Originally introduced by Einstein in his field equation to guarantee for a static solution for the Universe and since the 1990s associated with the energy of the vacuum as the simplest explanation of dark energy. As Amendola and Tsujikawa [8] noted in their book, coincidently Aristotle first proposed an “eternal” and “incorruptible” cosmic substance in The Lambda Book (i.e. the 12^th) of his Metaphysics.

9 That means that the photons could propagate without getting scattered on or re-emitted by baryons.

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1.1 Part I: How does the Universe work?

was to high to allow atoms to recombine into their neutral states. This epoch is know as the radiation-dominated phase where the driving component of the equation of expansion around redshift z ∼ 3400 was Ωrad and the baryons and photons tightly coupled to each other in a state which is comparable to that of a hot dense plasma. With the nucleosynthesis [32] finishing the build-up of the basic elements: hydrogen, deuterium, and helium¹⁰; approximately at minute 3 of cosmic time or z ∼ 10⁸, we enter the era of the early Universe. The evolution of the Universe and its properties had already been predestinated by its phase transitions most probably a result of spontaneous symmetry breaking.¹¹ The predominance of matter over antimatter during the Baryogenesis, an example of symmetry breaking, left us with a baryonic¹²fine-tuned¹³Universe destined to bring forward life.¹⁴ Our knowledge of the physics before that is very incomplete. We only can assume that primordial density perturbations seeded the cosmic structure formation during an episode of extreme expansion called inflation¹⁵[3, 125, 267]. During the earliest epoch of the Universe, the Planck era, the laws of physics, as we understand them today, might not have even been applicable since the fundamental forces still have been unified and the Universe’s state comparable with a chaotic foam of quantised black holes,¹⁶ space and time inextricably and discontinuously, unimaginable by a human mind.¹⁷ And finally all that was initialised by The Big Bang.

The ΛCDM paradigm, described in the most straightforward way above, provides a general framework allowing galaxies to form and evolve. Thereby large-scale structure is dominated by collisionless, dissipationless cold dark matter which forms potential wells in which the baryons can accrete, cool, and subsequently form proto-galaxies. The physical mechanisms involved in this process are described by the framework of galaxy formation and evolution and includes roughly gas cooling, star formation, and feedback processes (which will be described in detail in what follows). In the next section, we will focus on the end product of those processes: Galaxies.

They are the objects on the smallest scale we discuss in this thesis. We will then subsequently move to larger scales when introducing the interaction of galaxies with their dark matter halos, their clustering dependency, and their environmental affiliation.

10We, all of us, are what happens when a primordial mixture of hydrogen and helium evolves for so long that it begins to ask where it came from. – Jill Tarter, astronomer

11It describes the spontaneous process of a symmetric state to end up asymmetrically and is a common phenomenon seen in particle physics such as the (Nobel prize winning) charge conjugation parity symmetry (CP)-violation [53].

12Physicists are made of atoms. A physicist is an attempt by an atom to understand itself. – Michio Kaku, theoretical physicist

13It seems that dimensionless physical constants such as the strength of the electromagnetic interaction between elementary charged particles, commonly known as the fine-structure constant 1/α∼ 137 [127, 261] have tightly constrained values. Small variation in them would make structure formation or nuclear fusion and therefore life, as we know it, non-existent.

14A controversial explanation for the observed fine-tuning of the Universe uses the anthropic principle which basically states that the properties of the Universe and hence fundamental constants must be compatible with the establishment of intelligent life to observe it [14, 49]. Interestingly, the fine-tuning problem as well as the anthropic principle have been debated interdisciplinary for more than 100 years (also compare to puddle thinking [2]). Richard Feynman once said that every theorist should have the following statement written on their office board: “1/137, how little we know? ” [108]. In his honour, I chose the font size of this dissertation to be 11.37pt.

15Originally proposed in the early 1980s to solve several cosmological problems such as the flatness or horizon problem, provides a casual mechanism for the origin of large-scale structures in the Universe [8].

16As described by John A. Wheeler

17Never limit yourself because of others’ limited imagination; never limit others because of your own limited imagination – Mae Jackson, astronaut

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1 Introduction

©PabloCarlosBudassi

Figure 1.1: Logarithmic scale illustration of the observable Universe.

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1.1.2 Galaxies – A Universe’s accomplishments

The above described expanding Universe luckily brought forward a diversity of objects we call galaxies. They are dynamically bound systems which consists of a variety of material such as cold and hot gas, stars, metals, black holes, and dark matter.

Fundamental properties of galaxies

Galaxies come in many different shapes and sizes. They can be divided into distinct populations such as red and blue, old and young, metal-rich and metal-poor, elliptical and disk. For this thesis work, we only need to understand the overall characteristics of the most massive luminous population, which are know to be redder, older, and can be found in the most massive structures [258]. In this section, we review briefly common terms used in the framework of galaxy formation and evolution and the most important properties of galaxies:

• Stellar masses: The stellar mass is one of the most important properties of a galaxy and can be used as a proxy for other properties as we will see in Chapter 4 of this thesis. The abundances of galaxies as a function of mass is given by the stellar mass function (SMF ). Galaxies seem to build up their mass continuously over cosmic time [183].

Thereby, their characteristic number densities follow the shape of a Schechter-function [243]. Massive galaxies formed and assembled their stars earlier than lower massive galaxies which synthesised their stellar masses later and over longer periods of time [202, 206]. In this work we are mostly interested in massive quiescent galaxies.

• Luminosities and colours: The luminosity is the total amount of electromagnetic energy emitted per unit time while its flux is the rate of energy transferred per unit surface or wavelength/frequency. The colour of an astrophysical object can be described roughly as the difference in flux/energy/magnitude measured by certain band-pass filters of a photometric system. The luminosity-colour distribution of a sample of galaxies is strongly bimodal [11] with the majority of galaxies being located in a quite narrow optical wavelength range know as the red sequence and in a slightly broader blue cloud. In the local Universe, the red sequence was found to host predominantly quiescent galaxies showing low rates of star formation and older stellar populations, whereas the blue cloud’s populations are younger and their galaxies are actively starforming. The bimodality in the luminosity-colour or colour-colour diagram provides deep insights into the evolution of a galaxy population. We will us the colour of galaxies in order to select a galaxy sample by applying photometric selection cuts Chapter 4.

• Number densities: The comoving number density of quiescent galaxies increases with time since the cosmic noon¹⁸[39], while the number of starforming galaxies has been more or less constant [206]. This implies that, with time the star formation in galaxies become less and less efficient or has be suppressed, which we call quenching. Number densities are useful to our analysis, since we will later show how to mimic a photometric selection using the same number density of objects of its observational counterpart. The quenching of star formation is further important to this thesis when discussing if ant to what degree environment shapes galaxy properties in Chapter 7.

• Scaling relations: The tight correlations of galaxy properties, also called scaling relations, have been intensively studied in the past. They help to classify galaxies according

18The peak of cosmic star formation and black hole accretion around z∼ 2 − 3 [183]

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1 Introduction

to their morphology type such as disk or elliptical, size and growth, or prominent features such as spiral arms as well as give clues about their intrinsic properties. Diverse scaling relations e.g. for dwarf galaxies revealed that the mechanisms behind the formation of these objects are actually different from the formation of normal-sized galaxies [157].

Scaling relation are important in the framework of galaxy formation, as we will see later, since they can be used to implement physical processes at the sub-grid¹⁹ level and consist of the most direct approaches to constrain baryon cycling processes. Some examples for scaling relations are:

– The cold gas fraction is the fraction of the cold gas MCold in the interstellar medium (ISM) to the total stellar mass M_∗ [12, 216]. The gas fraction decreases since cosmic noon and is assumed to tightly correlate with star formation history and the cold gas density [223].

– The mass-metallicity relation describes the relation of the metallicity or masses of metals of the stars or in the ISM to the total stellar mass [280, 304]. Its evolution suggests that galaxies of a given stellar mass M_∗ had lower cold gas-phase metallities Z_Cold at higher redshift [304] and that galaxies which form stars more rapidly have subsequently lower metallicities [168, 185]. We will come across this relation many times when presenting our results, since our adopted model of galaxy formation, introduced in Chapter 3, shows dependencies of properties related to star formation and ZCold on their large-scale environment (spoiler!).

– The fundamental plane is a combination of properties such as radius, luminosity, and velocity of galaxies based on the classic Faber-Jackson relation [101] or Kormendy relation [156] for elliptical and Tully-Fisher relation [281] for disk galaxies. Thereby the galaxies populate a fairly narrow region in the luminosity-radius-velocity plane.

– The stellar-to-halo mass relation is one of the tightest relations and provides useful information on the formation and evolution of a galaxy of given halo and stellar mass within its dark matter halo. Interestingly, the halos at intermediate masses produce most efficiently stars [22, 38, 296]. It is still poorly understood why halos with lower or higher masses form stars by orders of magnitudes less efficiently. [17].

Furthermore, this relation is crucial to our research, because we want to understand the evolution and the large-clustering of luminous red galaxies living in the most massive halos.

– Recent studies have shown that there is a fairly tight correlation between secondary galaxy properties such structural parameters or colour and the properties of dark matter, apart from the stellar mass. In what follows we will discuss the galaxy-halo connection in detail in Section 1.1.3 and introduce modelling approaches of how to parametirisise this important scaling relation in Section 1.2.2.

Luminous Red Galaxies (LRGs)

As mentioned before, LRGs are our main research object in this thesis. Here we will summarise their most important properties. They are located in the centre of dense regions as today’s cluster and supercluster and therefore hosted by the most massive dark matter halos [176]. They serve as powerful cosmological probes to study the formation of structure, the assembly of mass

19Those are the small-scale physical processes that occur at length-scales beyond resolution limits or act as a place holder for not yet understood physics.

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and cosmology itself. However, their detailed formation and evolution is still not sufficiently understood or quantified [274, 291]. To shed light on the topic one would need information about the full history of mass assembly and star formation of luminous red galaxies within a large redshift range. The Baryon Oscillation Spectroscopic Survey [hereafter BOSS, 88, 99, 244]

provides such data.

The BOSS-CMASS galaxy sample of luminous red galaxies

The Sloan Digital Sky Survey (hereafter SDSS) III, also known as BOSS, was dedicated to studying properties of the large-scale distribution of massive galaxies. The BOSS sample is divided into a low (LOWZ) and a high redshift sample (CMASS, stands for “constant mass”), respectively. The CMASS-sample was designed to target the most luminous red galaxies in order to produce a in mass uniformly distributed sample of ∼ 1.5 million LRGs. The sample exhibits a peak comoving number density n ∼ 3.4 × 10⁻⁴ h³Mpc⁻³ at z ∼ 0.5 and shows only passive evolution with little or no on-going star formation [186] along the redshift range of 0.43 < z < 0.7. The galaxies are selected by applying a set of photometric selection cuts²⁰ using (g-i) and (r-i) colours, where g, r, and i stand for the bands of the classic photometric filters of the SDSS. An important characteristic of this colour selection is that it guarantees for an extension towards the bluer colours, hence blue-cloud galaxies can enter the sample. Because the CMASS sample consists of a non- evolving population of massive galaxies, it provides an excellent “cosmic laboratory” to study galaxy formation and evolution [25, 197, 198] as well as their link to cosmology via e.g. their spatial distribution and clustering [122, 234]. BOSS LRGs were repeatedly used to determine fundamental cosmological parameters [77, 111], to probe cosmological models [4, 29, 203], and to inform the relation between the dark matter halos to their hosted galaxies [104, 121, 211]. We dedicated a great deal of our analysis to CMASS-galaxies and will show how and how successful we can select the same galaxy population from a model of galaxies formation in Chapter 4.

Galaxy Formation in a cosmological context

The amount of mass detectable by human astrophysical instrumentation in the form of baryons is only about 1/5 of the total mass budget the Universe holds [221]. The remaining 4/5 can only be measured indirectly due to the alignment and behaviour of galaxies in the gravitational field of their underlying dark matter distribution. Studying the basic properties of galaxies changed not only our entire view of the Universe, as it has been discovered that galaxies live in halos of dark matter [239], but also the fact that their properties and evolution are tightly linked their halos [291] has been an unexpected discovery of modern astrophysics. The so called, galaxy- halo connection is a fundamental and highly complex relation incooperating many physical processes, some of them possibly not even discovered yet. Exploring this relation is crucial to our understanding of galaxy formation and evolution and helps in finding an answer to the fundamental question of what shapes the observables of galaxies.

1.1.3 The galaxy-halo connection

In the framework of cosmology, the galaxy-halo connection relates multivariate distributions of properties of dark matter halos with their galaxies. Getting to the bottom of this relation would not only help in understanding the physics of galaxy formation, but also probe the properties and the distribution of the underlying dark matter [291]. The most challenging questions in this

20See the BOSS target selection web page for details on the photometric selection cutshttps://www.sdss.org/dr15/

algorithms/boss_galaxy_ts/

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1 Introduction

context is: Which properties of dark matter halos and their environment are most significantly shaping properties of galaxies or in other words How do halo properties induce star formation or quenching in their galaxies?

Early works on the galaxy-halo relation recognised that more massive galaxies and clusters should also show different clustering statistics than average galaxies, merely because their dark matter halos exhibit such a strong clustering dependency on halo mass [86, 143, 195]. The realisation of large survey missions of galaxies in the 90s of the last century made it possible to measure huge sets of galaxies simultaneously (e.g. with Two-degree Field Galaxy Redshift Survey [59] or SDSS [302]). The field of computational astrophysics grew quickly with the development of N-body simulations [58, 160] being able to resolve substructures of dark matter halos. Both disciplines together allowed for detailed studies on the galaxy-halo relation, here we summarise their most important findings:

• Scaling of stellar and halo mass: The stellar mass, M∗, of a given galaxy scales with halo mass, MHalo, as M_∗∼ MHalo2−3 for dwarf stellar masses and with M_∗∼ MHalo1/3 at the highest stellar masses [178]. The shape of the stellar-to-halo mass function (SHMF ), derived from the mismatch between the halo mass function and the galaxy stellar mass function (SMF ), provides evidence for strong feedback processes in galaxy formation which results in actively suppressing the star formation.

• Feedback: Typical feedback processes can be observed in galaxies with high and low halo masses such as induced by an active galactic nucleus (AGN) in high-mass galaxies [74, 251], or by supernovae of massive stars in low-mass galaxies [89, 136]. Both prevent the gas from cooling and limit star formation.

• Star formation efficiency: It exists a maximum efficiency in the star formation of a galaxy depending on its stellar mass, located around the mass of the Milky Way Galaxy (M∗∼ 10¹²M), and a narrow range in halo mass where galaxies most productively form stars [17, 63].

• Bias: The bias factor b represents the ratio of the clustering of galaxies in comparison to their underlying matter distribution: b² = ξ_gal/ξ_DM, ξgal and ξDM are the two-point correlation functions of galaxies and dark matter, respectively [143]. Lower mass halos have an almost constant bias, meaning that, they follow tightly their underlying dark matter distribution. Halos of higher masses are less abundant, have a larger scatter in SHMF at fixed halo mass, σlog₁₀M∗, and as a result, are stronger clustered [306]. Furthermore, their biases rise quickly with halo mass.

• Concentration is a halo property that informs about the distribution of the dark matter inside a halo. It is defined as c ≡ rhalo/r_s with rhalobeing the radius of the halo estimated by a certain over-density criterion,²¹ and rs the typical scale radius.²² The concentration is closely related to the formation time and mass assembly (see also mass-concentration relation [147, 226]) of a halo as well as the evolution of the galaxy residing in it [68, 240, 292, 293, 311]. To mentioned a few examples: halos which experienced a recent merger

21Since the dark matter halos have no strict dimensions, r_haloand subsequently the mass inside the halo need to be defined via a certain over-density criterion as e.g. the radius at which the gravitationally collapsing sphere virialises (r_vir) or when the density inside the radius reaches 200× the critical density of the Universe (r^200ρcr) [194]

22The slope of the density slope coincides with that isothermal sphere, r⁻², at rs. The density profile seems to be universal for a wide range of halo masses and variants of the curvature density parameter Ωk (see Eq. (1.1)) [207]

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are low-concentrated, while high-concentrated halos are older and had longer phases of quiescent evolution. Furthermore, more concentrated halos tend to host more massive central galaxies and are also stronger clustered. They start to host the central at lower halo masses than less concentrated halos [307].

The scatter in the stellar-to-halo mass relation

The basic shape of the SHMF provides strong evidence of feedback mechanisms driving the formation and evolution of galaxies. Feedback can be found on almost all scales and in all types of galaxies as its strength depends not only on halo mass, but also on redshift and environment.

The scatter in the SHMF , given as the variation of stellar mass at fixed halo mass, σlog₁₀M∗, is another important feature of the galaxy-halo connection in central galaxies. It basically characterises the mean value of stellar mass in bins of halo mass, indicating that a galaxies with a fixed mass can live in halos of different masses. This applies fundamental constrains on both, the galaxy and halo properties, most responsible for star formation, quenching, and/or their transformation into quiescent galaxies. The scatter σlog₁₀M∗ shows the following dependencies:

• At low masses, the scatter has a constant value but widens considerable due to the change in the slope of the SHMF (see e.g. Fig. 5 in Wechsler and Tinker [291]).

• Above the pivot mass (the halo mass which corresponds to the maximum in star formation efficiency MHalo∼ 10¹²), the scatter is monotonically increasing.

• As σlog₁₀M∗ increases, the mean halo mass at fixes stellar mass decreases

• The higher the scatter, the shallower the mean stellar mass increases with halo mass.

• If the scatter is lower, galaxies with higher stellar masses can be found in halos of lower masses.

It is important to bear in mind that, the variation of the scatter affects directly the clustering of galaxies, since with increasing σlog₁₀M∗, the halo mass that permits to host a galaxy with a certain stellar mass decreases with the scatter. In other words, an increasing scatter makes more low-mass halos to host galaxies with the same mean stellar mass as more massive halos.

Since high-mass halos cluster stronger and are also more biased, the clustering signal of samples selected by stellar mass would show deviations depending on the scatter. We explained the galaxy-halo connection and the scatter in such detail in this section, because later in Chapter 4 (spoiler!) we will show that our adopted model of galaxy formation and evolution shows not sufficient scatter in its SHMF . This has consequences on the halo occupation distribution (the probability of a halo of certain mass to host a certain amount of galaxies) and subsequently on the clustering performance of our selected galaxy sample.

Measurements on the scatter are reported in the literature as e.g. 0.18 < σlog₁₀M∗ < 0.22 [314]

at z = 0.0 using SDSS galaxies or σlog₁₀M∗= 0.18 [275] using BOSS at z = 0.5.

The clustering of galaxies

The clustering signal of galaxies, measured as the two-point auto-correlation function (hereafter 2pCF), holds fundamental information about the structure formation of the Universe. It is not only a powerful cosmological probe constraining the matter density parameter Ωm, observations also have shown that the clustering results depend strongly on luminosity, morphology, and other physical properties of a given galaxy sample [85, 90, 210]. The 2pCF is defined as the

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1 Introduction

measurement by counting and averaging the number of neighbours of each galaxy at a given scale, compared to a random distribution. This gives the typical shape of the clustering function of e.g. the baryon acoustic oscillations (BAO) measured by BOSS, where at a certain scale, the length of the sound horizon ∼ 100 h⁻¹Mpc, an excess in number density of galaxies can be found. This scale serves as a standard ruler and important cosmological constraint, since it describes the maximal distances photons could travel before decoupling from the hot, dense plasma of electrons and baryons at the time of recombination. The resulting sound waves are imprinted in the CMB power spectrum. The signal of the BAO is used for accurate distant measurements and to constrain cosmological parameters [171].

The evolution of the stellar-to-halo mass function

Does the galaxy-halo relation evolve with redshift? As we have discussed in this section, halo and galaxy properties are closely correlated. It would be interesting to know, if this relation holds at higher redshift or if a redshift-dependent factor can be derived. That information would give clues on how strongly the mass assembly of a galaxy depends on its halo mass. As mentioned before, the star formation efficiency correlates highly with mass, but weakly with redshift [17].

Behroozi et al. [16] found that the SHMF evolves only little with the peak of the relation nearly being constant until z ∼ 3. However, not enough evidence was brought forward yet to support the hypothesis that the slope of the SHMF changes significantly with halo mass and redshift.

1.1.4 The Large-Scale Structure & Environment

In the framework of galaxy formation it is vital to understand if the large and small scale environments influence the properties as well as their evolution of a given sample of galaxies [31, 95, 212, 290]. The correlation between galaxy properties and galaxy environment has been studied for almost 100 years with the result that many galaxies cannot be found in isolation, but are members of gravitationally bound larger structures. It further has been proven that, elliptical galaxies prefer cluster environments, whereas spirals the field (morphology-density relation) [94].

What is environment? In the field of galaxy formation and evolution, the term environment has been used differently depending on the context: It could be referred to as the interaction between satellites and centrals [241], but it could also be understood in terms of the “group environment” like the distance from the centre of a cluster or defined by its surface density [181, 190]. Other studies refer to environment as the location of a galaxy within the large- scale structure distribution[158, 219, 295] which can be roughly divided into four categories:

knots/nodes, filaments, sheets, and voids [126, 175]; in consequence of the hierarchical structure formation [78].

Galaxy Clusters

The large-scale distribution of galaxies in the Universe is dominated by the endeavour of galaxies to group into clusters and superclusters. Galaxy clusters are the largest gravitationally bound objects and therefore consist of excellent laboratories to study galaxy formation and evolution.

Many recent works have been presented on galaxy clusters [51, 84, 112] as well as many groups formed dedicating their research effort on their modelling and simulation: e.g. the nIFTy galaxy clusters [79], Cluster-EAGLE [9], The Three Hundred[80], or the MultiDark-Clusters [305], only to mention a few of them.

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The formation of clusters strongly depends on the applied cosmology, but also on the various complex feedback processes, regulating and shaping galaxy properties or determine their morphology as e.g. early-type galaxies can often be found in denser environments [94]. Therefore, we dedicate this section to the discussion of feedback driven environmental processes that impact cluster galaxies.

Feedback processes and properties of cluster galaxies are tightly correlated with each other, which makes them difficult to entangle. On top of that, clusters have their own characteristics regarding their formation. In the framework of halo formation, clusters grow in an inside- out manner in two growth phases, a “fast” growth phase of the central region driven by rapid accumulation and major merger events followed by a “slow” phase, where the outer regions gradually grow via moderate matter accretion [120]. As a result, the internal structure of halos contain information about their growth history [73, 292], which can then be used to track differences in the evolutional paths of cluster members.

It is important to understand, if some galaxy or halo properties determine the evolution of cluster galaxies (or the whole cluster) rather than others, and if the local and large-scale environment play a role in this scenario. For example, the most significant effect environment has on cluster galaxies is the gas stripping in outer regions of the cluster, resulting in a systematic difference in effective radius and leading to the transformation of their morphology type (e.g. from the blue cloud to the red sequence) [36, 92]. Recent works have also shown that, possibly the small scale environment influences the evolution of cluster galaxies more than the large-scale [181].

Other works studied X-ray emission from clusters and found a significant offset of the location of the central galaxy, most likely a bright cluster galaxy (BCG), with respect to the centre of the X-ray emission peak[115].

Why are galaxy cluster relevant to our discussion? The BCG consist of another sample of luminous galaxies, although their properties are similar to those of “normal” LRGs [60, 192], they enable galaxy formation studies in the visual and X-ray range of the electromagnetic spectrum (see e.g. the COSMOS survey [61, 129, 246]). The paradigm that, central galaxies are also the most massive in the system, has been proven as not being entirely correct by recent works [255].²³ Suspicious offsets of central galaxies with respect to the centre of the halo have also been found in model clusters in The Three Hundred [80] derived from semi-analytical models. Since BCGs can be found in denser environments, they would also provide a promising additional research object to the BOSS LRGs (our main research object in this thesis) and possibly help on our mission to trace the galaxy assembly bias. However, this would be beyond the scope of this dissertation, therefore we leave the studies of clusters and cluster galaxies with respect to their environment for future works.

We summarise that, baryons occupy only a small fraction of the total energy budget of the Universe and dominate at small scales such as inside a galaxies, inside a host halo as e.g. the interaction between centrals and satellites, or within groups and clusters of galaxies. In general, feedback processes such as gas cooling or AGN activity regulate the star formation in galaxies and therefore influence basic properties such as their morphological type or colour. Feedback is one of the most important and at the same time at least understood mechanism within the framework galaxy formation, since it involves a variety of processes and also is strongly linked to the host environment of a given galaxy.

23Speaking of paradigms and biases, the same leading author of this publication also wrote an interesting and very relevant article on women in physics see Skibba [254]

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1 Introduction

Quenching and feedback

The phenomenon that the star formation of a certain galaxy is very rapidly ceased is called quenching [5, 39, 72]. The existence of a tight connection between the quenching and feedback processes is widely accepted, but it seems that different mechanisms are at work for different populations of galaxies. In the low stellar mass range, feedback from massive stars typically drives the quenching [72, 82, 169]. In more massive galaxies M_∗ > 1× 10¹⁰ Mthe feedback of the central AGN is powerful enough to heat up the surrounding cold gas by injecting energy via radio jets as well as to remove the gas content in the ISM of the galaxy through outflows and winds [74]. This type of quenching is referred to as mass-quenching, mostly depending on the stellar mass and usually found in central galaxies [217].

A more complex process is the environmental-quenching where the interaction between galaxies and their surroundings, e.g. if the galaxy is a cluster of group member, quench the star formation [279]. Common phenomena thereby are: ram-pressure stripping [120, 222], strangulation [170, 201], and harassment [102, 200]; and most likely to be found in satellite galaxies.

Until now it is not entirely clear what exactly causes the quenching and to what degree it involves properties such as morphology or environment. To discuss a few examples, Wang et al. [288] confirmed a weak but significant dependency on environment, but reported that the primary driver for quenching is halo mass which regulates the star formation in both, centrals and satellites. Other works as e.g. Contini et al. [65] found that the star formation strongly depends on stellar mass, rather than environment or halo mass. A few studies have found that it is more likely for galaxy to be quenched or red, if it is hosted by more massive halos [205], but other studies have shown that there is only little dependence on either the halo mass or the distance from the central galaxy.

Summary: In this section we provided the necessary background to study luminous red galaxy by discussing their basic properties such as mass, morphology, or scaling relations. We further established a connection between the galaxies and their dark matter halos which consists of a fundamental relation. We further argued about the influence of their small- and large-scale environments and presented examples of studies from the literature derived from models and observations. In summary, it is well established that galaxies in denser environments are older, redder, metal-richer, more concentrated, more luminous, and show higher surface brightness than galaxies in voids. However, the formation of structures due to the spherical collapse and the virialisation of dark matter halos as well as the accretion of baryons into these halos and the subsequent formation of galaxies involve non-linear and highly complex processes which cannot be described analytically anymore. Therefore, modern computational techniques are necessary to investigate in detail the processes involved. In what follows, we introduce modelling techniques of galaxy formation and evolution and will provide a brief overview of cutting-edge methods and subsequently centre our discussion on semi-analytical models.

1.2 Part II: How to simulate a Universe on a computer?

In this section we introduce cosmological simulations of galaxy formation and evolution in form of a short presentation of the galaxy formation theory as well as the most common modelling techniques and their ingredients. Since the development of the first cosmological simulations in the early 80s, the field has grown ever since. Modern research in astrophysics and cosmology cannot be imagined without using computational approaches anymore. The knowledge accu- mulated over almost 40 years helps to complete our view of the Universe and provide crucial

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insights informing other field as observational cosmology or astronomy. Both domains, compu- tation and observation together, form the baseline for cosmological studies and cannot succeed one without the other. Galaxy formation depends highly on observations, because they provide the necessary constraints modellers use to calibrate or tune their models. The challenges and the ultimate objective of models are to reproduce the observed Universe as detailed and accurate as possible.

1.2.1 Galaxy formation theory

The theory of galaxy formation consists of the modelling of physical processes that affect the baryonic component distributed in their dark matter halos. It can be separated into two steps:

(i) modelling the formation and evolution of the halo population hosting the galaxies using N- body simulations or analytical methods like the Press-Schechter theory [227] and (ii) modelling the evolution of the baryonic matter distributed in their halos [70].

Overview of physical processes involved in galaxy formation and evolution

• Gravity: Provides the “skeleton” for galaxy formation and determines the shape and amplitude of the primordial power spectrum of density fluctuation, described by the initial conditions and the applied cosmology. The evolution of the power spectrum and therefore the growth of structure via merging and accretion of halos is predicted by the framework of hierarchical structure formation. The number and properties of dark matter halos existing in the simulation as well as their evolution is recorded in the so called merger trees.

• Cooling: After the skeleton of dark matter has been formed, gas will accreted into the over-dens regions and eventually cool down. Thereby, we can roughly distinguish three cooling regimes: bremsstrahlung (Tgas ∼> 10⁷ K), cooling via recombination of electrons with ions (10⁴<Tgas< 10⁷ K), and metal line cooling through collisional excitation/de- excitation of heavy elements and molecules (Tgas < 10⁴ K).

• Star formation: The gas cooled down enough to collapse into the deepest potential well usually the central region of the halo. It might become self-gravitating and collapse even more rapidly driven by its own mass rather than by the dark matter halo. Eventually, denser regions form and will reach temperatures and pressures to enable nuclear fusion and star formation. In this discussion, we describe star formation only rudimentarily, since many processes involved are still poorly understood and simulations are unable to resolve the relevant scales. Therefore, empirical sub-grid recipes, a standard procedure in large-scale cosmological simulations, are introduced to fill in for the lack of physical understanding or resolution.

• Feedback can be classified as thermal (heating the gas), kinetic (ejecting the gas due to winds), and radiative (ionising or photo-dissociating the gas). It regulates the cooling of gas and subsequently the star formation in galaxies. As we briefly raised the subject in the previous section, it can be discriminated between: supernova feedback (such as photo- heating, photo-ionisation, winds [136]) and AGN feedback (mostly in early-type galaxies and in the majority of all massive galaxies as they seem to host a central black hole [157]).

The latter is associated with high-velocity winds and cold gas ejection from the interstellar medium of the galaxy into the hot halo [131]. Both feedback processes are also treated as sub-grid implementation in current cosmological simulations.

• Mergers: In the ΛCDM paradigm large-scale structures form hierarchically via halos

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merging into more massive and more concentrated objects [55]. The merging events clearly affect the galaxies hosted by the merging halos through triggering star formation, changing their morphology types, or disrupting and accreting one another. It can be discriminated between minor and major mergers. In the standard scenario, it is assumed that during a major merger the disk of the central is destroyed and all stars migrate to a spheroidal bulge. Picturing the formation of early-type galaxies in this way is fairly simple and in practice is more complicated and therefore needs careful modelling, hence some modellers e.g. track the mass loss from satellites when moving through the dark matter halo or assume dynamical friction, the loss of momentum and kinetic energy through gravitational interaction e.g. in a system hosting central and satellite galaxies [52].

• Chemical evolution plays an important part in galaxy evolution, because metal line- cooling at intermediate temperatures and subsequent star formation is boosted by metal- rich gas. The luminosity and colour of stellar populations are also highly sensitive to metallicity, since heavy elements produce dust that reddens the galaxy in the ultra-violet and optical as well as re-radiates energy in the infrared [62].

1.2.2 Modelling techniques in galaxy formation

In this section, we introduce modelling techniques in modern computational astrophysics and cosmology. We will briefly summarise the most popular approaches and then focus on semi- analytical models as most relevant to our work.

Empirical models: Some modelling techniques are called empirical, because they map observable properties of galaxies onto properties of dark matter halos, but do not include actual modelling of physical processes. The most basic approach is called halo abundance matching (HAM). It assumes that the most massive dark matter halo also hosts the most massive galaxies, while the second most massive halo hosts the second most massive galaxy (and so forth) [141, 161, 234]. This approach is famous for its “non-parametric” implementation, meaning that there is at least only one parameter necessary to constrain the model, usually the scatter of the SHMF, σlog₁₀M∗. A more sophisticated approach is called halo occupation distribution (HOD) and describes the galaxy-halo connection as the mean number of galaxies per halo <Ngal> as a function of MHalo, passing a particular observational selection [24, 235, 312]. There are many different parameterisations available usually using a set of 5-10 parameters. The conditional luminosity/mass function (CL/MF) goes one step further and describes the full distribution of galaxy luminosity or stellar masses for a given halo mass (The luminosity or stellar mass function is “conditioned” on the halo mass) [69, 284, 298]. It further can be used to not only parameterise the galaxy-halo connection, but e.g. the relation between galaxy star formation rates and halo mass accretion rates [291].

We summarise that, empirical models have fairly simple assumptions, which do not hold when considering e.g. feedback processes we discussed in the previous section. However, empirical models as simple as they are, they are predicting the galaxy-halo connection remarkably well.

Furthermore, they are very helpful in determining the galaxy formation histories or generating mock galaxy catalogues [16, 233].

Physical models: We can roughly distinguish between three types of physical models: N- body, hydro-dynamics, and semi-analytics; although the approaches can be mixed or one build upon the other (e.g. N-body+hydro). The first step consists of choosing a cosmological model such as ΛCDM and applying initial conditions. Those can contain random perturbations in

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a homogeneous density field of the dark matter distribution, random seeds of particles in the simulation box, or mimic e.g. the over-densities in the local Universe [286].

Numerical N -body dark matter only simulations

N-body methods, also named “gravity-solvers”, establish the dark matter structure in cosmological simulation by discretely computing the forces between dark matter particles in the field of gravity. That basically means solving the Poisson’s equation. The evolution of the system is followed within a co-moving expanding cosmological frame using periodic boundaries. The expansion is derived from the Friedmann-Equations (obtained from the field equations of general relativity) or a Newtonian approximation of that, since general relativity corrections are negligible on large scales. This approach is widely used to model large-scale structure formation in a cosmological context and was used for the first time by von Hoerner [287] 1963 to describe the dynamics of stellar clusters.

N-body methods can also include baryons. Such simulations are either particle-based²⁴ such as MultiDark [147] or Euclid-Flagship [225], mesh-based²⁵ such as Millennium [265], or hybrids of both. Particle- and grid-based methods have advantages and disadvantages e.g.

particle-based codes can introduce a softening length, the length between two particle where the the gravitational interaction is suppressed, while mesh-codes are limited by the resolution of their grids. Therefore, hybrid tree-mesh codes have been developed using the particle approach on large scales and the mesh approach on small scales in order to resolve those more accurately.

A popular example for a hybrid-code is Gadget-2/3 [262].

In the context of galaxy formation, two fundamental properties are given by the underlying dark matter structure: the distribution of their masses at a certain redshift as the halo mass function and their formation histories as statistical properties also named merger trees. Within the framework of galaxy formation modelling, the general approach is to adopt the merger trees and populate their halos with galaxies.

Merger trees

The details on the formation of structure through gravitational instabilities at each time-step of the simulation is stored in the merger trees. That includes the number of dark matter halos and subhalos, the identification number of their progenitors as well as their merging time. The first step to construct a merger tree is to identify the halos and their substructures (subhalos which became bond to a larger halo) which is done by a halo-finder algorithm. That can be either done by connecting particles located near each others, the friends-of-friends method [87], or by identifying spherical overdensities in the distribution of particles. A popular method for the former is e.g. FOF [114] and for the latter e.g. AHF [154] or the 6D-phase space based code ROCKSTAR [18]. Different halo-finders tend to identify properties of halos like mass or peak circular velocity differently, which could introduce systematic errors and when populating those halos with galaxies leading to an insufficient reproduction of the real Universe. For a detail discussion on halo-finders see Knebe et al. [148], for merger trees Lee et al. [173], and for numerical methods in general Vogelsberger et al. [286].

24Also called “tree-codes”, the forces between particles are approximated by dividing the volume into cubic cells and distant cells can be treated as individual large particle. The forces between the particles are approximated via their multipole moments [13].

25Also called “mesh-codes”, particles are sampled on a discrete grid, their masses divided by the cell volume, and a local gravitational potential calculated via Fourier transformations [134].

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Hydro-cosmological simulations

The most accurately way to model galaxy formation is using numerical hydrodynamic techniques, in which the equation of gravity and fluid-dynamics are solved for particles and/or a grid of cells simultaneously. Some examples: EAGLE [242] was run on a tree-particle mesh N-body simulation, where smooth particle hydrodynamics²⁶ (SPH) was applied, whereas for Horizon- AGN [96], a particle-mesh code with an adaptive-mesh-refinement²⁷(AMR) approach was used.

Those two approaches have been combined in e.g. Arepo [263]. Furthermore, some codes like NIHAO [289] focus on the simulation of individual galaxies, while other such as Simba [83]

generate a whole galaxy population at once. For a more detailed list on simulation codes and modelling techniques see Table 2 in Vogelsberger et al. [286].

The physical processes can be most precisely modelled (within the obtained resolution limits) with “full-physics” hydro-cosmological simulations, which guarantees for most accurate predictions on the physics and is clearly the biggest advantage of this modelling technique. However, there limits are the computational exigencies such simulation bring with them and therefore hydro-codes are often restricted to studying small-scale processes or individual galaxies only.

As mentioned before, parametrisation and empirical recipes are used to describe physics beyond the resolution limits and on larger scales.

Summary: In this section we have briefly discussed the most common modelling techniques and basic concepts in the context of galaxy formation. We explained how the “skeleton” of the dark matter simulation is modelled via N-body techniques and show the necessary steps of prepare for the population of those structures with galaxies (merger trees and a halo-finders).

Subsequently, we presented hydro-cosmological simulations and some examples among them.

We complete this section with revealing why we adopted a semi-analytical modelling approach for studying galaxies in this thesis. For our study we need a large set of galaxy properties and high number densities of galaxies to guarantee for optimal clustering statistics, because we want to compare with the BOSS-CMASS sample, one of the largest observed galaxy catalogues. These requirements cannot be fulfilled with hydro-cosmological simulations due to their restriction to small scales. We cannot use the HOD approach either, because we want to make predictions on the underlying physical processes and probe the clustering of galaxy samples in relation with their intrinsic properties and environmental affiliations. In the next section, we introduce semi- analytical modelling techniques and apply what we have discussed in the general introduction of galaxy formation theory in Section 1.2.1 as well as in this section.

1.2.3 Semi-analytical models (SAMs)

Different approaches, like semi-analytical models (SAMs), of how galaxies populate dark matter halos, form and evolve, have been developed, because it is not possible to model galaxy formation on all scales simultaneously and at the same time guarantee for sufficient number densities.

SAMs are built upon N-body dark matter simulations using merger trees (information of the hierarchical formation of dark matter halos). In contrast to full-physics hydrodynamics, the SAM approach does not explicitly solve the fundamental equations, but adopts a set of simplified recipes as implementations of baryonic physics as a post-processing step. This includes phenomenological treatments of baryonic processes and coarse-graining the properties of galaxies,

26SPH is a Lagrangian method, historically most popular, where particles themselves carry the information about the fluid which is obtained via summarising over neighbouring particles closer then a smoothing length [196, 264]

27AMR is an Eulerian method of solving hydrodynamics to discretise the fluid onto grid cells and then compute advection of properties across the boundaries of the cell [258]

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Semi-analytical galaxies in the multidark-universe: A perspective on the evolution of the most luminous and massive galaxies throughout cosmic history