Black hole formation and growth with primordial non-Gaussianities 

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Linear growth of perturbations and spherical collapse model

The Universe is composed of large structures, dark matter halos, and galaxies. However, the cosmological principle predicts an uniform and isotropic distribution of matter in the Universe. If so, no structure formation can happen. Therefore, in order to form large scale structures, one needs to introduce fluctuations in the history of the Universe. The Hot Big Bang theory has no explanation for the non-uniform and isotropic distribution of matter, and it is one of the so-called problems of this theory (the two others being the flatness problem and the horizon problem). The standard description of the Universe, driven by general relativity, is expected to break down when the Universe is so dense that quantum effects may be more than important to consider.
Inflation has been considered to be a natural physical process solving Hot Big Bang theory’s problems. The first model of inflation is introduced in 1981 by Guth (1981). In addition to solve the Hot Big Bang problems, this accelerated period of our Universe allows the introduction of these quantum processes, which can produce the necessary spectrum of primordial density perturbations, that gravitational instability accentuates to produce the large structures we observe today, namely dark matter halos, clusters of galaxies and galaxies. Structure seeds or overdensities
Introduction 7 will grow with time, the overdense regions will attract their surroundings and become even more overdense. Conversely, the underdense regions will become even more underdense because matter in these regions flows away from them, leading to the formation of voids. In the following, we describe the growth of the density perturbations.
The density contrast δ(x, t) can be expressed as a function of the local density ρ(x, t), and the background density of the Universe ρ¯(t) (which is equal to the previous ρm(t)). δ(x, t) measures the deviation from a homogeneous Universe (for which δ(x, t) = 0):
The perturbative density field is predicted to be Gaussian, which is consistent with the mea-surement from the CMB (Planck Collaboration et al., 2015a). A discussion on the possible existence of non-Gaussian primordial density fluctuations at small scale, and the consequences for the assembly of dark matter and galaxies, is carried in chapter 5. The simplest initial power spectrum is the Harrison-Zeldovich spectrum (Peebles & Yu, 1970 ; Harrison, 1970 ; Zeldovich, 1972) (or scale-invariant power spectrum), defined by P (k, tinitial) ∝ kns , with the spectral index ns = 1 (“scale free”)4, is in good agreement with the CMB measurements. This initial spectrum evolves with time. We usually express the evolution with a transfer function T (k), which encodes the Universe geometry and the nature of matter (for example, the type of dark matter particles), to obtain P (k, t) = T 2(k, t) × P (k, tinitial).
To describe the evolution of perturbations, we use the ideal fluid description in the Newtonian theory. Baryons can be described by an ideal fluid, because collisions between particles are frequent, which leads to the establishment of local thermal equilibrium. (If we consider density fluctuations on characteristic length scales smaller than the Hubble length c/H, and weak gravitational field). The evolution of an ideal fluid in the Newtonian theory, is given by the 3 following equations (equation of continuity, Euler’s equation, and Poisson’s equation), which can be rewritten to take into account the expansion of the Universe. Because the Universe is expanding we move from proper distance xproper to comoving distance x, by taking xproper = a(t)x. The proper velocity is expressed as vproper = a˙(t)x + a(t)x˙ = a˙(t)x + v with a˙(t)x the Hubble velocity, and v the peculiar velocity (which describes the movement of the fluid with respect to a fundamental observer), that is used in the following 3 equations:
The density perturbations grow by self-gravity, if these perturbations stay small, we can model their growth in the linear perturbation regime (δ << 1). From the 3 equations above, we obtain the equation describing the growth of perturbations 5:
The evolution of perturbations in the density field depends on the phases of the Uni-verse. For the radiation phase, the solution of the equation of perturbation growth is δ(x, t) = A(x) ln(t) + B(x). Perturbations grow very slowly during that phase of the Universe, they will really grow during the matter-dominated phase. For the matter-dominated phase, the solution is δ(x, t) = A(x)t2/3 + B(x)t−1. The first term indicates that perturbations grow with time, with the expansion of the Universe, whereas the second term is a decaying term, which is generally not considered because it vanishes with time. Finally, today the perturbations are not growing anymore, the solution can be expressed by δ(x, t) = A(x) + B(x)e−2H×t.

Spherical collapse

As we just said, at the early stage, when perturbations are still in a linear regime (δ << 1), the overdense regions expand with the expansion of the Universe. At some point, when δ ∼ 1, the perturbations segregate from the expansion of the Universe, and over dense regions start to collapse. This phase is referred to as the turn-around, the regime becomes strongly non-linear. The growth of perturbations can not be treated anymore in the linear perturbation regime. This leads to an increase of δ, the overdense regions will attract their surroundings and become even more overdense, and will inevitably collapse under their own density due to gravity. The evolution of these overdense regions is independent of the global background evolution of the Universe; they can therefore be seen as small Universes, denser than the background density ρ¯, that collapse. To treat them we use the spherical collapse model. It assumes that the overdensity inside a sphere of radius r, is homogeneous, and describes the evolution of the radius as a function of time. The sphere is supposed composed of matter shells, which do not cross.
The newtonian equation describes the evolution of a mass shell in a spherically symmetric density perturbation:
where M is the mass within the shell, and is constant, and r is the radius of the shell. Because M is constant and so independent of t (before shell crossing), we can integrate the previous equation:
1 dr 2 GM = E (1.21) with E the specific energy of the shell. One can solve the equation for the different values of E, E = 0, E > 0 or E < 0. We focus on the last case, which corresponds to the collapse case. The motion of the shell is described by the system:
Parameters A and B can be expressed as a function of ri, ti, the density contrast δi, and the density parameter of the overdense region Ωi. Therefore the motion of the mass shell is entirely described by r and t, and the initial conditions on the radius r of the shell, and the mean overdensity enclosed in it. We can therefore compute the maximum expansion of the shell, for θ = π, which corresponds to rmax = 2A, and tmax = πB. After that, the mass shell turns around and starts to collapse, the mass shell can cross the other mass shells that were initially inside it. By the time tcol = 2tmax, all the mass shells have crossed each other many times, and have formed an extended quasi-static virialized halo. The time of virialization is the time at which the virial theorem is satisfied, so when the spherical region has collapsed to half its maximum radius, thus tcol = tvir here.
From this theory, it is possible to estimate the density contrast at which the turnaround happens, namely δ = 1.06, and at which the collapse happens, δ = 1.69. Therefore the global picture is the following, when the density contrast of a perturbation exceeds unity, it turns, and starts to collapse when it reaches δ = 1.69.

Virialization of halos

The collapse does not go to a singular point, but it is halted before reaching that stage, by what we call the virialization. Because dark matter is composed by collisionless, non or weakly interacting particles, it can not release the gravitational potential energy through radiation or shocks, therefore the virial theorem tells us that this energy is converted into a kinetic energy for the particles. Eventually the other particles will exchange with DM particles this kinetic energy, by relaxation processes, leading to a pressure supported virialized halo where finally particles will reach an equilibrium state. The overdensity at the virialization time can be derived from the theory, δ(tvir) = 178 (here we have assumed Ωm,0 = 1, otherwise we would have a weak dependence on the density parameters).

Formation of galaxies and first stars

As we have seen, baryons only represent a small fraction of the matter density in the Universe, but are present in all the structures we observe today. Therefore it is also important to treat the evolution of perturbations in the baryonic fluid, still in the Newtonian regime. Compared to the dark matter (DM) growth of perturbations, the perturbation growth in the baryonic fluid equation is slightly different, because a term OP/ρ corresponding to the pressure support is added in the Euler’s equation. The equation of perturbations growth for the baryonic fluid is now expressed by:
The differences with Eq. 1.19, is that now we have two terms coming from baryonic and dark matter gravitational potential (terms 4πGρ¯BδB, and 4πGρDM¯δDM), and a term k2c2 from the pressure gradient of the baryonic fluid. From this equation, one can define a characteristic scale, 10 1.3 Formation of galaxies and first stars the Jeans wave number kJ:
Perturbations with a physical length larger than the Jeans length (2kπ a(t) > λJ) can grow, whereas perturbation modes with a smaller physical length (2kπ a(t) < λJ) can not.
Here again, the evolution of perturbations in the density field, depends on the Universe history epochs. In the radiation-dominated phase, DM perturbations are growing logarithmically, the evolution is dictated by the term due to expansion 2aa˙δ˙ (damping term), which dominates over the gravitational potential term. However, the baryonic fluid is affected by the pressure gradient. Pressure support prevents the growth of baryonic perturbations. After the matter-radiation decoupling time (recombination), baryonic perturbations can grow, and closely follow the growth of DM perturbations.
From this, we see that baryonic perturbations follow the perturbations of the DM fluid. Without DM, the baryonic perturbations would still be in the linear regime today, making the assembly of galaxies difficult.
A characteristic mass, the Jeans mass, can be defined as the amount of baryonic mass within a sphere a radius λJ/2 (the Jeans scale length is used as a characteristic diameter of the sphere):
The Jeans mass is of order 1016 M during the radiation dominated phase, which roughly corre-spond to galaxy clusters scale. But after matter-radiation decoupling, there is no more presure support provided by photons, the baryonic gas only resists gravity by its normal gas pressure, and therefore the pressure drops significantly, the Jeans mass drops to the scale of globular cluster mass, with MJ ∼ 105 M .
In order to form galaxies, we often refer to two different stages: the assembly of mass, and the formation of stars. The assembly of mass is a long process, cold gas falls into potential well of dark matter fluctuations, increases the local density, which leads to the formation of molecular hydrogen. H2 will lead to the cooling of dense regions, then will condense and fragment. Molecular gas cloud fragmentation allows the conversion of gas into stars.
In the very early Universe, in the absence of any heavy element (metals, which are created by stellar processes), atomic and molecular hydrogen are the only coolants. At temperatures Tvir < 104 K, the cooling is done by radiative transitions of H2, which can cool the gas down to a few hundred kelvin. Contrary to H, the excitation temperature of H2 is sufficiently low (low energy levels). When Tvir > 104 K, H is able to cool the gas.
Tegmark et al. (1997) compute the necessary molecular hydrogen abundance for a halo to collapse, by computing the abundance needed to have a cooling time smaller than the Hubble time. Fig. 1.1 – Mass needed to collapse and form luminous objects at a given virialization redshift (Tegmark et al., 1997). Only clumps whose parameters (zvir,M) lie above the shaded area can collapse and form luminous objects. The dashed straight lines corresponding to Tvir = 104 K and Tvir = 103 K are shown for comparisons. The dark-shaded region is that in which no radiative cooling mechanism whatsoever could help collapse, since Tvir would be lower than the CMB temperature. The solid line corresponds to a 3-σ peak in standard CDM model.
The channel based on H+2 is efficient for 100 < z < 500. Conversely, at z < 100, H2 is mostly produced by the H− mechanism. Indeed the last reaction which represents the destruction of H− by CMB photons is not predominant, because photodetachment of H− becomes inefficient due to the decline of the cosmic background radiation.
Based on the 2 mechanisms H− and H+2, Tegmark et al. (1997) show that the H2 abundance needed for a halo to collapse is 5×10−4, which only differs slighty with the redshift of virialization (in the range 100 > zvir > 25). These results are encoded in Fig. 1.1, only halos in the non-shaded region can collapse at a given corresponding virialization redshift. In this region, the virial temperature is sufficiently high that enough H2 form for the cooling time to be smaller than the Hubble time. Halos can cool, and collapse. However, in the red shaded area, there is no radiative cooling mechanism to help the collapse, the temperature there is indeed smaller than the CMB temperature. Finally, at zvir ∼ 30, only halos more massive than 105 M are able to collapse, and to form luminous objects.
At this stage of the gas collapse, most of the hydrogen is converted into H2, but this does not increase the cooling of the gas, because the binding energy of every H2 molecule that forms (4.48 eV) is converted in thermal energy, that contributes to heat the gas.
The first generation of stars , the so-called population “PopIII”, is predicted to form in 105 M halos, often referred as “mini-halos”. With H2 cooling, primordial star-forming clouds of ∼ 1000 M collapse until a quasi-hydrostatic protostellar core of around ∼ 0.01 M forms in the inner part of these clouds (Yoshida, Omukai & Hernquist, 2008).
The question of the initial mass function of the PopIII stars is still discussed today. The number of star(s) which form in mini-halos, and the initial mass of stars, are among of the most challenging issues. This field of research has been investigated with simulations over the last decade. Several numerical works have followed protostellar formation process (Abel, Bryan & Norman, 2002 ; Yoshida, Omukai & Hernquist, 2008 ; Greif et al., 2012). On the number of stars per halos, Greif et al. (2012) ; Latif et al. (2013b), recently showed that the protostellar disks can fragment into several gas clumps, each being able to form star. The final halo could therefore host more than one single star. Small traces of metals can also lead to forming several stars in the same clump because of first dust cooling (Schneider et al., 2002 ; Omukai et al., 2005 ; Schneider et al., 2006a), therefore decreasing individual star mass.
Assuming a single star per halo, Hirano et al. (2014) derive the initial mass function of primordial stars by simulating 110 halos. They first use SPH simulations to study the formation in primordial clouds in the central part of halos, that range in Mvir = 105 − 106 M for z = 35 − 11. Radiative hydrodynamical simulations are used to follow the accretion phase of protostars. They find that PopIII star masses could range from ∼ 10 to ∼ 1000 M (this can be seen as an upper limit on the mass, because they assume that only one star forms in each halo). The mass of PopIII stars is also dictated by their radiative feedback into their surrounding gas, it can halt the accretion into the stars, and therefore regulating their growth (Bromm, 2013 ; Greif, 2015).
In the standard ΛCDM model, the first “galaxies” form after the first generation of stars. Mini-halos, which host the first PopIII stars, may indeed not be massive enough to retain the gas pushed away by the first SNe, through mechanical feedback (shock waves from SN). Potential wells may also not be deep enough to retain the gas heated by SN (thermal feedback of SN) and stellar feedback, such as photoionization by stellar radiation. This depletion of gas in mini-halos can devoid them of gas, and consequently can prevent and delay the next episode of star formation for a long time of few 107 years.
Therefore the formation of the first stars has a non-negligible impact on the Universe through different processes, which affect more than their own dark matter halos. First of all, they emit UV radiation, which can dissociate molecular hydrogen, and therefore delay star formation in neighboring halos. Second, these stars will produce and release metals in their surrounding, therefore enriching the intergalactic medium with metals. Star and BH formation will be strongly affected by the feedback from this first population of stars, we discuss the consequences of these two particular feedback processes on the formation of BHs in section 1.7.
Once gas is able to cool again, it collapses in the potential wells of halos, which have by that time, grown in mass through accretion and mergers, to 108 M . Because mini-halos are the progenitors of these massive halos, the gas is normally metal-enriched by the first population of stars, and therefore can cool even more efficiently to lower temperatures to form lower mass stars, that constitute the second generation of stars, called PopII stars.
Early metal-enrichment of the medium due to PopIII stars has been discussed in the literature (Yoshida, Bromm & Hernquist, 2004 ; Tornatore, Ferrara & Schneider, 2007 ; Greif et al., 2008), but it is generally assumed that the first galaxies are the main drivers of metal-enrichment. Radiation from the first stars, and galaxies, is also among the most commonly assumed source of radiation for the reionization of the Universe. High redshift galaxies are indeed thought to be the most important contributors of ionizing photons (Robertson et al., 2010, 2013). It is worth mentioning here, that thanks to improvements in observations, the next generation of telescopes will help us to push further our understanding of high-redshift galaxies, and their consequences on the Universe evolution. So far, we have been able to observe high redshift galaxies in the range 6 < z < 10 (Bouwens & Illingworth, 2006 ; Bouwens et al., 2015), when the Universe was less than a 1 Gyr old. James Webb Space Telescope (JWST) will open a new window on cosmic reionization, it will help us to better constrain the contribution of high redshift galaxies, in terms of the evolution of ionizing photons emitted by galaxies at z > 10, their number density, the evolution of ionized gas bubbles, and the identification of sources producing ionizing radiation. The sensitivity of JWST should ensure us to capture sources with stellar mass higher than ∼ 105 − 106 M , which is unfortunately not enough to observe the first PopIII stars (Bromm, Kudritzki & Loeb, 2001), but is still very impressive as we should observe starbursts in the first galaxies. Another source of ionizing photons is thought to be AGN, which are powered by powerful BHs. The Square Kilometre Array (SKA) will give us a better idea of the abundance of AGN at z > 6, and whether there is a faint population of AGN at such high redshift, which would favor the contribution of AGN to the reionization. In the next sections, we will focus on BHs, and their evolution within their host galaxies.

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Black holes as a key component of galaxies

In this section, we will see that BHs are a key component of galaxies. Indeed most of the local galaxies host a massive BHs, including the Milky Way and some dwarf galaxies. The discovery of luminous quasars at z > 6, 15 years ago (Fan, 2001 ; Fan et al., 2003, 2006b), showed us that massive BHs were already in place at the end of reionization epoch.

BHs and AGN

The general relativity of Einstein led us with the necessary framework, to predict theoretically the existence of BHs, immediately after 1915. The solution of Einstein’s equation derived by Schwarzschild in 1916, led us with the idea that the mass of an object can collapse to a singularity of infinite density, a black hole, from which light can not escape anymore6. The observational 6The existence of black holes has been thought/introduced earlier by Mitchell in 1784, and Laplace in 1796, but also Eddington, who predicted that “the star apparently has to […] contracting, and contracting until, I suppose, first detection of a serious candidate BH took some time from the theoretical prediction, but in 1970, the X-ray source Cygnus X-1 is observed, with a mass of more than 6 M indicating that the only possible explanation was a BH.
The first detections of supermassive BHs where through AGN observations, where the inner part of some galaxies were identified as active nuclei. We briefly remain here the main characteristics of AGN. Spectra of some galaxy nuclei present strong emission lines produced by the transitions of excited atoms. The emission lines can be broad or narrow. Broad lines correspond to high Doppler-broadening velocities of > 103 km s−1, and generally correspond to permitted lines. Narrow lines with lower velocities of 102 km s−1 are also observed, and correspond either to permitted or forbidden lines. These emission lines give us crucial information on the surroundings of BHs: broad lines are produced close to the BH, where the gas densities and velocities are high because of the potential generated by the BH, we call this region the broad line region, narrow lines are produced in more extended regions, where gas densities and velocities are lower. We call this region the narrow line region. The global picture of AGN today, is that BHs are surrounded by an accretion disk, which is itself surrounded by a small inner broad line region, around which there is a clumpy extended narrow line region. The presence of an obscuring torus, around the broad line region, is thought to hide the emission from the broad line region, depending on the axis of the line-of-sight. Broad lines in a galaxy spectrum are a diagnostic for the presence of an AGN, and they are also used to estimate the mass of BHs in AGN through the virial method. The width of the broad lines is assumed as a proxy of the Keplerian rotational velocity. Conversely, the presence of narrow emission lines does not necessarily imply the presence of an AGN, because star forming galaxies can also present narrow emission lines due to HII regions around young massive stars. Line ratios are used to distinguish between AGN and star-forming galaxies. For example, [OIII]/Hβ indicate the level of ionization and temperature, whereas ratios like [NII]/Hα give an information on the ionized zone produced by high energy photoionization. For an AGN, the level of ionization and the temperature of the emitting gas are both higher, and because the photons are more energetic, the ionized region is also expected to be larger in the case of an AGN than for a star-forming galaxies. Therefore in a line ratio diagram (BPT diagram, Baldwin, Phillips & Terlevich, 1981), high [OIII]/Hβ vs [NII]/Hα are more likely to represent an AGN.

Local galaxies

Evidence for the presence of supermassive black holes in the center of galaxies has accumulated over the last decades. Because the first observations of BHs were through AGN, we have first drawn a picture of BH that was mostly based on the most massive BHs, the most bright and accreting ones. In this section, we will see that low-mass BHs have been observed, or at least an upper limit on their mass has been estimated for some of them, with the advance of observational abilities. This has strong consequences on our vision of BH formation and evolution over cosmic time.
Massive BHs are harbored in the center of most local galaxies, some examples can be found it gets down to a few kilometers radius when gravity becomes strong enough to hold radiation and the star can at least find peace”, through not very convinced by himself “I think there should be a law of Nature to prevent the star from behaving in this absurd way.”. Finally, the first real calculation of the black hole is realized by Openheimer and Snyder in 1939, they show that an homogeneous sphere (without pressure) which gravitationally collapses, ends up its life without being able to exchange information with the rest of the Universe anymore. The term black hole is introduced later by Wheeler in 1968.
Fig. 1.2 – Relation between BH mass and the total stellar mass of local host galaxies (Reines & Volonteri, 2015). This consist of a sample of 244 broad-line AGN from which the virial BH masses are estimated through the single-epoch virial mass estimator (Reines, Greene & Geha, 2013) and shown as red points. Pink points represent 10 broad-line AGN and composite dwarf galaxies. Dark green point represents the dwarf galaxy RGG 118 with its 50,000 M BH. Light green point is Pox 52. Purple points represent 15 reverberation-mapped AGN. Blue dots represent dynamical BH mass measurements. Turquoise dots represent the S/SO galaxies with classical bulges. Orange dots the S/SO galaxies with pseudobulges. Grey lines show different BH mass- bulge mass relations.
in Kormendy & Ho (2013). For instance, galaxies NGC 1332, NGC 3091, NGC 1550, and NGC 1407, have dynamical BH mass measurement of MBH > 109 M (Kormendy & Ho, 2013). Reines & Volonteri (2015) provided a sample of local galaxies hosting BHs, at z < 0.055, that we reproduce in Fig 1.2. Blue points represent BHs in quiescent galaxies, whereas red points show AGN. On Fig 1.2, we see that indeed massive galaxies can host very massive central BHs of few 109 M . It is important to keep in mind that we have focussed so far on understanding what we were able to see until today, namely powerful BHs and massive galaxies. We have only observed the massive end of the BH and galaxy story. Understanding the BH population requires to now move on observing the low-mass end on the BH distribution, specifically the BHs that could reside in low-mass galaxies. Such observations need the support of theoretical models, such as those developed in this thesis.
That being said, observing the BH population in low-mass galaxies is not at all an easy task, for many reasons. First of all, if we simply extrapolate the BH-galaxy mass relation to low-mass galaxies, BH mass in low-mass galaxies would be lower that the ones in more massive galaxies, which makes their detections more challenging. In the hierarchical structure formation model, massive galaxies grow in mass partly because of galaxy-galaxy mergers. Low-mass galaxies, instead, do not experience as many galaxy-galaxy mergers as massive galaxies, their growth is limited compared to their massive counterparts. BH mass growth is boosted when a galaxy-galaxy merger occurs. BH in low-mass galaxies, are instead not expected to have grown much over 16 1.4 Black holes as a key component of galaxies cosmic time.
Low-mass BHs are more difficult to observe than their massive counterparts, their gravitational force is weaker, stars or gas moving around such low-mass BHs will be difficult to identify/observe.

Black holes as a key component for galaxy evolution

When gas is accreted onto a BH, there is a release of rest-mass accreted energy back to the galactic gas, which can impact the host galaxy by feedback processes (Silk & Rees, 1998). AGN feedback acts as an interaction between the energy, radiation produced by gas accretion onto the central BH, and the gas in the host galaxy. Theoretically, the energy released by the BH, can be sufficient to entirely unbind the gas of its host galaxy. If BH growth is dictated by accretion, the BH energy is expressed as EBH = ( /1 − )MBH c2, whereas the binding energy of the galaxy is expressed by Egal = Mgal σ2. The ratio between the energy released by the BH and the binding energy of the host galaxy is:
if we assume the radiative efficiency to be ∼ 0.1. If only a small fraction of the BH accretion energy was released as kinetic energy transferred to the gas, AGN feedback would be able to unbind the gas of the galaxy.

Table of contents :

1 Introduction 
1.1 Brief historical introduction
1.2 Structure formation in a homogeneous Universe
1.2.1 The homogeneous Universe
1.2.2 Linear growth of perturbations and spherical collapse model
1.3 Formation of galaxies and first stars
1.4 Black holes as a key component of galaxies
1.4.1 BHs and AGN
1.4.2 Local galaxies
1.4.3 Population of quasars at z = 6
1.5 Black holes as a key component for galaxy evolution
1.5.1 Co-evolution between BHs and their host galaxies
1.5.2 AGN feedback
1.6 Black hole growth over cosmic time
1.7 Theoretical models for black hole formation in the early Universe
1.7.1 Remnants of the first generation of stars
1.7.2 Compact stellar clusters
1.7.3 Direct collapse of gas
1.7.4 Other models
1.8 Diagnostics to distinguish between BH formation scenarios
1.9 Organization of the thesis
2 Numerical simulation 
2.1 Ramses: a numerical code with adaptive mesh refinement
2.1.1 Adaptive mesh refinement
2.1.2 Initial conditions
2.1.3 Adaptive time-stepping
2.1.4 N-body solver
2.1.5 Hydrodynamical solver
2.2 Sub-grid physics to study galaxy formation and evolution
2.2.1 Radiative cooling and photoheating by UV background
2.2.2 Star formation
2.2.3 Equation-of-state
2.2.4 SN feedback and metal enrichment
2.2.5 BH formation
2.2.6 BH accretion
2.2.7 AGN feedback
2.3 Smoothed particle hydrodynamics code Gadget
3 Pop III remnants and stellar clusters 
3.1 Introduction
3.2 Simulation set up
3.3 Seeding cosmological simulations with BH seeds
3.3.1 Selecting BH formation regions
3.3.2 Computing BH initial masses
3.3.3 BH growth and AGN feedback
3.4 The influence of star formation and metallicity on BH formation
3.5 Black hole mass function and occupation fraction
3.6 Black hole growth regulated by efficient SN feedback
3.7 Comparisons with a sample of local galaxies, and Lyman-Break Analogs
3.8 Conclusions
3.9 Perspectives
3.9.1 BH growth in the delayed cooling SN feedback simulation
3.9.2 Need for further comparisons with observations, preparing future observational missions.
4 Direct collapse model 
4.1 Introduction
4.2 Simulation set up
4.3 Method
4.4 Impact of SN feedback on metallicity and star formation
4.5 Number density of direct collapse regions in Chunky
4.6 Horizon-noAGN simulation: Can DC model explain z = 6 quasars?
4.7 Comparison between hydro. simulations and (semi-)analytical models
4.8 Conclusions
4.9 Perspectives: Applications of hybrid SAMs
5 Black hole formation and growth with primordial non-Gaussianities 
5.1 Introduction on primordial non-Gaussianities
5.1.1 Primordial bispectrum
5.1.2 Introduction of fNL parameter
5.1.3 Observational constraints, room for non-Gaussianities at small scales
5.1.4 Previous work, and the idea of running non-Gaussianities
5.2 Halo and galaxy mass functions
5.2.1 Numerical methods: from non-Gaussian N-body simulations to galaxy formation model
5.2.2 Predicted halo mass functions from theory
5.2.3 Results on halo and galaxy mass function
5.2.4 Conclusions
5.3 Reionization history of the Universe
5.3.1 Far-UV luminosity function and reionization models
5.3.2 Fraction of ionized volume of the Universe
5.3.3 Electron Thomson scattering optical depth
5.3.4 Conclusions
5.4 BH formation and growth with primordial non-Gaussianities
5.4.1 BHs formed through direct collapse
5.4.2 BHs formed from the remnants of the first generation of stars.
5.4.3 BHs in the most massive halos at z = 6
5.4.4 Conclusions
6 Conclusions


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