Numerical simulations of accretion disks around early black holes

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Universit`

a degli Studi di Pisa

Dipartimento di fisica

Tesi di Laurea Magistrale in Fisica Curriculum di Astrofisica

Numerical simulations

of accretion disks around

early black holes

Candidato

Fabio Di Mascia

Relatore

Prof. Andrea Ferrara

Marzo 2017

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Introduction

The appearance of the first black holes is, along with the birth of the first stars, one of the most important events occurred within the first billion years of our Uni-verse. These two classes of astrophysical objects had a crucial role in the cosmic evolution, because of their ability to inject photons, energy and momentum into the surrounding interstellar or intergalactic medium, thus profoundly altering the subse-quent galaxy formation history. Massive stars also produce and disperse metals that were almost absent in the pristine gas, polluting the regions where future stars and black holes will form and leading to a transition in the star formation mode. Stars and black holes do not behave independently; instead, their formation and evolution are profoundly interwoven, because of the physical processes regulating them, which maintain efficient cross-talk between the two populations. The study of black holes formation and growth is then intrinsically linked with the study of primordial stellar evolution.

Many questions in the cosmological picture remain still open, in particular the observational evidence of the existence of black holes as massive as 109 M (named

super-massive black holes, SMBH) when the Universe was few hundred million years old. This fact is in tension with the standard theory of black holes growth, because a stellar-mass seed would require a much longer time to accumulate such a big amount of mass. Several works have been done in the last decade, greatly improving our understandings on both primordial stars evolution and black holes formation chan-nels. In particular, growing interest has been given to the Direct Collapse Black Hole (DCBH) scenario, that allows the formation of massive seeds of 105− 106 _M

if the

proper environmental conditions are satisfied. The seed of this object is supposed to grow via an accretion disk. To this regard, the standard picture of a smooth accre-tion mode has been quesaccre-tioned by recent simulaaccre-tions, that suggest that primordial accretion disks undergo fragmentation, resulting in a complicated accretion history.

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Introduction 2

Moreover, observational evidence of black holes in this mass range are still miss-ing, but with upcoming space and ground based telescopes such as JWST, ATHENA, WFIRST and SKA, the situation can finally change. Future observations represent also a great occasion to constrain the numerous theoretical models about black hole formation that have been proposed in the last years. Therefore, there is impelling necessity of predicting the observational features of a certain black hole formation mode in order to compare them directly with the future observational data.

A precise modeling of the phenomenon would require the develop of a full 3D radiation-hydrodynamic code able to follow the evolution of the gas in a proto-galactic cloud, starting from cosmological conditions and then studying the forma-tion and growth of a DCBH, predicting its observaforma-tional features. An high spatial resolution and a full radiative transfer treatment will be key points to study the self-gravitating accretion disk around the central object and its possible fragmentation along with the future evolution of gas clumps.

Codes of this kind are very complex and the work required for their implementa-tion cannot be faced during a Master Thesis. The aim of this thesis is to constitute the first step in this direction. Thus, starting from the beginning, a 2D-code, in which radiation transfer is not yet implemented, has been developed; this code will constitute the starting point to develop the final 3D code. After accurate tests of the code, we focused our attention to the study of the gravitational instability of a disk in the context of the DCBH formation.

The Thesis is structured as follow. In Chapter 1, an introduction to primordial star formation is provided. The cosmological context is described, the physics mech-anisms that lead from small matter perturbation to the birth of a star are discussed, the role of accretion disks in the growth of a star and their features are illustrated.

In Chapter 2, the problem of SMBH formation is posed and the possible channels of primordial black holes formation are reviewed. Focus is then given to the Direct Collapse Black Hole scenario, widely discussing the assumptions at the basis of this mechanism and their feasibility, with much care given to the latest analytical studies and simulations performed in this context.

In Chapter 3, we present the physical model adopted in order to study our prob-lem. We aimed to simulate the collapse of an atomic cloud under the specific physical conditions of the DCBH scenario and to follow the evolution of the disk that forms around the stellar object. We underline the approximation required to simplify the problem and to adapt it to our code.

In Chapter 4, we describe in details the 2D-Hydrodynamics code developed for the present work. Numerical tests and further expressions not explicitly reported in the text are presented in appendices.

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Introduction 3

In Chapter 5, our results are presented and critically discussed. We analyze the accretion history onto the central object dividing it intro three distinct phases. Of each stage, we illustrate the main physical properties of the cloud, focusing on the evolution of the disk during the last stage, where gravitational instability occurs, causing recurring episodes of fragmentation. In particular, we show that, from the temporal window examined, the formation of a DCBH is likely to happen despite an irregular accretion mode. We compare our results with existing literature. Finally, we discuss the validity of the assumptions previously made in the adopted physical model and we outline the future work necessary to improve our results and to confront them with observations.

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1

First stars

The first light in the Universe profoundly changed the cosmic evolution, affecting the subsequent stars and galaxy formation history in numerous ways. Primordial stars emitted an enormous amount of photons, giving start to the Reionization Epoch. They also produced and injected metals (in astrophysical context, every element heavier than helium is referred to as a metal) in the interstellar medium via supernova explosions, modifying the composition of the primordial gas, that was almost metal-free at the cosmic dawn. This feedback on the surrounding environment has also important implications for what concerns the birth and the growth of the first black holes. Understanding the formation of the first generation of stars (also called Pop III stars) and their final fate is of vital importance in determining the conditions under which the primordial black holes have formed. A unique property of primordial star formation is that its initial condition can be directly deduced from the Λ Cold Dark Matter (ΛCDM) model of cosmological structure formation. For all these reasons, we start reviewing the physics of primordial star formation, following the treatment adopted by Bromm in [1], after a brief synthesis of cosmological history from the Big Bang to the Recombination era, following the books by Loeb [2] and Bergstrom & Goobar [3].

1.1 A brief history of the Universe

”The study of cosmology is a delicate dance between what we observe but do not fully understand and what we fully understand but cannot observe” (Loeb). It is amazing considering how during the last century these two sides (what we know and what we observe) have come closer, thanks to enormous progresses both in the theoretical modeling of our Universe and in the observation capabilities. Observations have

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1.1. A brief history of the Universe 6

confirmed some theories and excluded others, but the journey is still far from being ended.

The global dynamics of our Universe can be studied using Einstein’s equations of General Relativity, that link the distribution of matter with the curvature of the space-time it occupies. The latest observational data emphasize a remarkable simplicity of our Universe. First, it satisfies the so-called cosmological principle, i.e. it is uniform and isotropic. Second, it is flat, in the sense that it has an Euclidean geometry [4], a non-trivial property because the curvature of the Universe depends on the mass it contains. Furthermore, we know since the discovery of galaxies recession by Hubble that it is expanding; given the isotropy of the Universe, this means that every observer sees itself as the center of the expansion. In order to describe this Universe, we can imagine to fill it with synchronized clocks and to define at a given time t1 the distance between two of them r1(t1). As time goes by, this distance is

stretched by the expansion and we can express this variation using a time-dependent scale-factor a(t), so that the same distance at a later time t2 can be expressed as

r2(t2) = r1(t1)[a(t2)/a(t1)]. If we consider a source at a distance r = a(t)x from us

(x is a time-independent tag, for example the present-day distance), it will move at a velocity v = dr/dt = ˙ax = ( ˙a/a)r. Defining H = ˙a/a, one obtains the Hubble law for the recession of galaxies v = Hr. It is also evident that H is a constant in space but not in time; today its value is H0 = 67.8 ± 0.6 km s−1 Mpc−1 [5].

The expansion of the Universe has an effect on the photons emitted by a luminous source too: in fact, the wavelength λem of a photon emitted at a certain time tem

will follow the expansion and will be modified, becoming at the time of its detection λobs = λem[a(tobs)/a(tem)]. The shift in wavelength ∆λ ∝ a (to which corresponds a

shift in frequency ∆ν ∝ a−1) can be parametrized introducing the redshift z defined as: 1 + z = λobs λem = a(tobs) a(tem) (1.1) The redshift is directly related with the scale factor of the expansion of the Universe: high redshifts correspond to high recession speeds of the source from us, which, in turn, correspond to an high distances and earlier emission times of the photons. Thus, the redshift provides a convenient method to express both the distance of far objects and the time in the past at which we are observing them.

In order to describe the evolution of the Universe we need to know the evolution of the scale factor a(t). This is regulated by the Friedmann equations, deduced from the Einstein’s equations combined with the cosmological principle. Solving the Friedmann equations requires to know the equation of state of the possible fluids filling the Universe, that is a relation that links the pressure P with the density ρ

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in the form P = wρc2, c being the speed of light. Three components are defined, depending on the value of w:

• matter, identified by an equation of state in which the pressure Pmat is Pmat < 1

3ρmatc

2_{, thus w <} 1 3;

• radiation, for which Prad = 1₃ρradc2, with w = 1₃;

• vacuum energy (also called cosmological constant) Λ, with Pλ = −ρΛc2 and

w = −1.

A fluid with a negative pressure may seem unusual and its introduction in the cosmo-logical picture is motivated by the following arguments. It can be demonstrated that ¨

a(t), which represents the acceleration of the expansion of the Universe, is propor-tional to P + 3ρ/c2, where P and ρ are the total pressure and density, comprehending all the fluids in the Universe. In the late 1990’s it was found observational evidence for an acceleration in the expansion rate of the Universe [6], implying w < −1/3. For this reason a fluid with negative pressure has be introduced in the cosmological scenario.

As the Universe expands, both the matter and radiation density are scaled by the volume factor a−3; however, photons also experiment a decrease in frequency by another a−1, therefore the scale laws are: ρmat∝ a−3 and ρrad ∝ a−4. These relations

imply that even if today ρmatis estimated to be larger by a factor ∼ 3300 than ρrad[5],

they were equal when 1+z ∼ 3300 and radiation dominated at earlier times. Instead, the vacuum has remained constant during the expansion and it became dominant for redshifts z < 0.3. The total contribution of these terms is usually expressed via the parameter Ω, defined by scaling ρ with a critical density ρc as Ω = ρ/ρc, where

ρc= 3H2 8πG = 9.2 × 10 −30 g cm3 H H0 2 . (1.2)

The value of Ω is crucial in order to determining the final fate of our Universe: if Ω > 1, then the Universe will at some moment end its expansion and turn around, collapsing in the so-called ”Big Crunch”; if Ω < 1, then the expansion will last forever; if Ω = 1, the Universe is in a marginally bound state. Thus, the critical density is the threshold above which the Universe will eventually collapse. In terms of the present-day contributions Ωm,0 = ρmat,0/ρc, Ωr,0 = ρrad,0/ρc and ΩΛ,0 = Λ0/3H02,

we can express Ω0 as Ω0 = Ωm,0 + Ωr,0 + ΩΛ,0; today we measure Ω0 = 1, with

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deduced from Einstein’s equations the following relation that links the evolution of the scale factor with the matter that fills the Universe:

H(t) H0 = Ωm a3 + ΩΛ+ Ωr a4 1/2 . (1.3)

This expression is particularly useful to determine the approximate behavior of a(t) in different epochs. For example, in a matter-dominated Universe a ∝ t2/3_{, for}

a radiation-dominated Universe a ∝ t1/2 _{and for a vacuum-dominated Universe}

a ∝ eHΛt_{, where H}

Λis proportional to Λ. These relations bring two interesting

impli-cations. First, we can relate the redshift with time, using the fact that a ∝ (1 + z)−1. Secondly, the expansion of a Universe dominated by the vacuum density is exponen-tial. This fact becomes very important if we think about the special properties that our Universe has, being the most simple possible Universe: homogeneous, isotropic and flat. The nature of this particular simplicity is still subject of debate. There are two general opinions: the first is that other kinds of Universe would have not been able to allow our existence; the second is that there must have been some event that determined this condition. Cosmologists have theorized that in an early period after the Big Bang the Universe was dominated by the mass density of an elevated vacuum state and experienced an exponential expansion of at least 60 e-folds: this phase is called cosmic inflation. The effect of this expansion was to erase any initial geometry curvature, making the Universe practically flat. Furthermore, we observe today regions that cannot have been in casual contact (which means that it was impossible to photons to travel from one region to another and to come back since the Big Bang) that are similar, showing a large-scale homogeneity of the Universe. The cosmic inflation gives an answer to such a high degree of order, because a rapid expansion would have enlarged homogeneous regions in the past, making us see now uniform conditions across a region far greater than our observable horizon. The vacuum state then decayed and radiation began to dominate our Universe.

In order to describe the cosmic history we can imagine to reverse the expansion we see now: then, the Universe started in a point of infinite density and temperature, a singularity named Big Bang. As the expansion began, the energy density decreased and thus the Universe started to cool while expanding. A fundamental property of the infant Universe is the fact that it can be described as a fluid composed of particles in thermal equilibrium, a condition valid only if the interactions between constituents are frequent enough. If this assumption is satisfied, then the evolution of the Universe can be described as a succession of states in thermal equilibrium, making sense the use of thermodynamical quantities, such as the temperature T (to which we associate a characteristic energy E ∼ kBT , where kB is the Boltzmann

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constant). The interaction rate per particle Γ will depend on the number density of the particles n and on their relative velocities v, that is in turn directly linked to the temperature, and on the cross-section of the process σ, as Γ = nσv. This rate has to be compared with some time to define the interactions ”frequent enough”. The characteristic time of the evolution of the Universe is obviously H−1, that represents the expansion rate of the Universe; then the condition that has to be satisfied in order to maintain thermal equilibrium is Γ H−1. Both the reaction rate and the expansion rate depend on the -decreasing- temperature of the Universe, but the reaction rate declined more rapidly, thus, as the expansion proceeded, more and more species decoupled from the thermal bath.

Therefore, the whole evolution is regulated by the characteristic energy of the Universe itself. The first time it makes sense to talk about is the Planck scale (∼ 1019 _{GeV, corresponding to t ∼ 10}−43 _{s): over this time, the theory of gravity}

and quantum physics are expected to be unified, thus the current theories have no predicting value. As the energy decreased progressively more and more fundamental forces became distinct. After the inflation, that is placed between t ∼ 10−33− 10−32

s, an unknown event named baryo-genesis or lepto-genesis generated an excess of particles over anti-particles: the origin of this asymmetry is a subject of intense study in theoretical and particle physics. When the temperature has dropped to ∼ 1 GeV (∼ 10−9 _{s), all the forces of the Standard Model were already separated}

and at this point quarks were not able to remain free in the quark-gluon plasma, becoming confined and protons and neutrons formed: this is known as the QCD-phase transition. At this point, an event that is one of the pillars of the standard cosmological model is located: the freeze-out (decoupling from the thermal bath) of an unknown specie of Cold Dark Matter. Dark matter is a type of matter different from ordinary baryonic matter and it constitutes the 82% of the total matter and the 23% of the total mass of our Universe (Ωb,0 ' 0.048 and Ωm,0 = 0.308 ± 0.012 [5]).

This matter is called dark because it does not emit or interact with light and thus it is not directly observable. However, it interacts gravitationally and there are striking evidences for its existence, for example the galaxies rotation curves: the orbital velocity of stars should decrease going far away from the center of a galaxy because most of the mass is concentrated in the center, while it remains almost flat, suggesting the presence of a diffuse component of non-visible matter surrounding the galaxy that exercises a gravitational interaction. There are several indications that this matter cannot correspond to known neutrinos and it has to be weakly interacting, with a low cross section that caused the freeze-out before every other species, neutrinos included [3, section 9.2]. Moreover, the observations about its abundance today suggested that when it went out from the thermal bath, it had

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a low random motion velocity and thus it is defined cold. The hypothesis of the existence of a vacuum density Λ and a Cold Dark Matter gave the name to the (actual) leading cosmological model, the Λ-CDM model, that is the simplest model until now developed that explains numerous observations, some of which we will talk about later on.

At T ∼ 1 MeV (t ∼ 1 s), neutrinos decoupled too. During the next few minutes, the energy decreased at the point that stable nuclei could be formed. This event, known as primordial nucleosynthesis, produced helium (∼ 24% relative abundance) and traces of deuterium and lithium. At z ∼ 3300 (t ∼ 4 × 104 yr) the Universe became matter-dominated, but matter and radiation remained coupled thanks to the efficiency of electron-photon scattering. When the Universe was 3.7 × 105 _years

old, electrons were able to be captured by hydrogen nuclei, forming the first atoms: this is the Era of Recombination (z ∼ 1100, T ∼ 3000 K). The reduction of free-electrons made the Universe transparent and photons decoupled from the thermal bath. The period with redshift 10 < z < 1100 goes under the name of Dark Ages and the first stars and galaxies are thought to be formed in this era. The light emitted by these sources was responsible for the second major phase transition of the gas in our Universe, as it dissociated atomic hydrogen, marking the start of Reionization. At z < 0.4 we entered in the Λ-dominated era, because the decreasing matter density has fallen below the dark energy component. The term Dark Ages was not chosen by chance: in fact, there are lot of uncertainties about the formation of the first structures in our Universe, an ignorance further stressed by the lack of observations in this era. We know quite well the physical conditions in which the first stars formed, because they can be inferred directly from the cosmological model, but the global picture about their properties and how they affected the Inter Galactic Medium (IGM) and following star formations are still missing pieces of this puzzle.

The most important consequence of the recombination is the fact that light was able to escape from the thermal bath and travel across the intergalactic medium, tak-ing memory of the conditions at the surface of last scattertak-ing. The cosmic microwave background CMB discovered in 1964 (figure 1.1) is the snapshot of the Universe as it was at z ∼ 1100 and maybe the most important observational result in favor of the hot Big Bang cosmological model. This radiation preserves the black-body spectrum of the thermal bath (and it is the most accurate black-body spectrum we have) at a temperature T0 = 2.73 K, that is red-shifted respect to the emitted one. From the

redshift factor (1 + z) we can deduce the CMB temperature at any moment since the recombination as T = T0(1 + z). The radiation appears almost uniform, except for a

dipole anisotropy caused by the motion of the Earth and for fluctuations of the order of 10−5. These fluctuations highlight small matter perturbations at the beginning of

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the Dark Ages, that were the seeds of the future structures formation. According with the theory of inflation, these perturbations arose from the quantum fluctuations of the primordial fluid, that were stretched during the exponential expansion.

The Λ-CDM model of which we have very briefly outlined the chronological his-tory has strong observational evidences in its favor. The first is the existence of the CMB radiation, that we have already discussed. Second, the present expansion of the Universe. Third, the predicted relative abundances of light elements (hydrogen, deuterium, helium and lithium) agree very well with the observations in space regions where there were negligible pollution by stars. It can be shown that the primordial

4_{He abundance cannot be produced in stellar interiors and thus it is very difficult}

to explain without the primordial nucleosynthesis. Furthermore, the observed deu-terium abundance constrains the baryons-to-photons ratio ηB = nγ/nB to a value of

5 − 6 × 10−10 [3, section 9.3]. Given that photons number density is dominated and experimentally determined by the CMB to be nγ ∼ 400 cm−3, this gives a limit on

the contribute of the baryonic matter ΩB to the total matter Ωm, that happens to

be ≈ 0.02. This is another reason in favor to the presence of the dark matter.

Figure 1.1: The image taken by the Planck satellite showing the Universe as it was when it became transparent 380 thousands of years after the Big Bang. Fluctuations are of the order of µK with respect to an average temperature of T0= 2.7255±0.0006 [7], resulting in a relative inhomogeneities

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1.2. The sites of primordial stellar seeds formation 12

1.2 The sites of primordial stellar seeds formation

According to the Λ-CDM model, the first structures in the Universe formed through successive mergers in a hierarchical bottom-up mode, starting from small blocks over which the bigger structures are built up. The smallest blocks are expected to form from the small perturbations of the primordial Universe. Dark matter played a crucial role in this context, because ordinary matter is influenced by the presence of the CMB radiation even after the Recombination. In fact, the diffusion of photons on small scales smoothed out the perturbation of the primordial fluid; the expansion made the smoothing scale become ∼ 100 − 1000 times the Milky Way length-scale. Then, how our galaxy could have form? The dark matter, that does not interact with CMB photons, accounts for this problem, because it could retain the shape of the primordial small scale fluctuations. Then, the first structures should have formed in regions where ordinary gas was hold in the potential well generated by the dark matter. Our current understandings suggest that these regions are dark matter mini-halos of typical masses ≈ 106 _M

[1], where the first stars are expected

to form at redshifts 20 < z < 30 (this constraint will be clear later). Such a mini-halo was formed in a region in which the gas density field was accidentally enhanced over the background matter. The self-gravity of the gas eventually amplified this perturbation, enabling the mini-halo to decoupling from the Hubble flow, making the gas turning around and collapsing. The outcome of this event was a state of approximated virial equilibrium, in which the baryonic gas is trapped in the potential well of the predominant dark matter. Using the virial theorem, we can relate the kinetic energy of the gas with the gravitational potential, expressing the average squared velocity of the gas (named the virial velocity) as

< v_vir2 > ∼ GMh < Rvir >

, (1.4)

where Mh is the total halo mass and < Rvir > the virial radius. From the theory of

gravitational instability [2,8] it can be deduced that the halo density at the virializa-tion state is ρvir ' 200ρb, where ρb(z) ' 2.5 × 10−30(1 + z)3g cm−3 is the background

density. A simple way to analytically understand the collapse and virialization pro-cess is the top-hat model described in [9], that assumes a density peak with constant density respect to the background Universe with spherical symmetry. With these assumptions, one finds for the virial radius [8]:

Rvir ' 200 pc Mh 106 _M 1/3 _{1 + z} vir 10 −1 _∆ c 200 −1/3 , (1.5)

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where ∆c= ρvir/ρb defines the overdensity of the gas when the virialization process

is complete.

During the collapse the gas heats up due to adiabatic compression or shock heat-ing. The two previous expressions allow us to express the temperature of the gas in function of the redshift, assigning to the gas a virial temperature as kBTvir ∼ mHvvir2 ,

obtaining: Tvir ' 2 × 103 K Mh 106 _M 2/3 _{1 + z} vir 20 . (1.6)

The typical gas temperature in mini-halos at the moment of first stars formation (20 < z < 30) was . 8000 K, thus atomic hydrogen cooling was inefficient. If no other cooling mechanism were available, the gas would simply remain in hydrostatic equilibrium inside the potential well generated by the DM and no further collapse and star formation would happen.

Saslaw & Zipoy [10] and Peebles & Dicke [11] were the first to realize that molec-ular hydrogen H2 could be the principal source of cooling for the low-temperature

primordial gas and that its formation should have relied on gas phase reactions. Two decades ago numerical studies investigating non-equilibrium chemistry of H2

forma-tion and destrucforma-tion and the determinaforma-tion of the cooling funcforma-tion began [12–14]. The formation of the H2 molecule is very unlikely to happen by the collisions

be-tween two H atoms, because in the early Universe there were no dust grains available to facilitate the formation of molecules. This is a consequence of the fact that the hydrogen molecule possesses an high degree of symmetry, resulting in no permanent electric dipole moment: radiative transitions can occur only via a forbidden (and thus slow) magnetic quadrupole radiation. Therefore, if two H atoms collide, forming a compound system, they rarely have the chance to radiate away the kinetic energy in excess quickly enough to make the system bound and at the end they separate again. Dust grains in the present-day gas account for this difficulty, taking the surplus of kinetic energy while the two H atoms orbit on the surface of the grain until they couple together. The formation of molecular hydrogen in the primordial Universe should rely on charged species in order to satisfy the electric dipole rule, permitting a radiative transition much faster than the magnetic quadrupole one. The two most important formation channels turn out to be the following two-step chains:

H + e− → H−+ γ (1.7)

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and

H++ H → H+₂ + γ (1.9)

H+₂ + H → H2+ H+. (1.10)

In the first chain, free electrons are the catalysts of the molecular hydrogen produc-tion and they are available at the end of recombinaproduc-tion era at z ' 1100 when the recombination typical time scale for hydrogen exceeds the Hubble time; they may also have been produced during the build-up of the first galaxies thanks to the col-lisional ionization in accretion shocks. The first chain turns out to be the dominant path [9] when both channels are active, but for high redshift z > 100, only the second path is available. This is a consequence of the fact that the ion H+₂ is more tightly bound than H− (the binding energies are respectively 2.64 eV and 0.75 eV), thus CMB photons at high redshifts are able to destroy H− but not H+₂ [9]. A rough estimate of the H2 fractional abundance fH2 in function of the virial temperature at small redshifts is deduced in [9] with a simple top-hat model and it is fH2 ∝ T

1.5 vir.

This expression, even if it is only an approximation, shows an important behavior of the gas: the deeper the potential well in which it is embedded, the higher the virial temperature, the higher the molecular hydrogen abundance and thus the following cooling, or, in other words, the gas has first to become sufficiently hot to efficiently cool later on.

In the absence of cooling, the timescales regulating the evolution of the halo are the Hubble timescale tH, that regulates the expansion rate of the Universe, and the

the dynamical timescale tdyn = (Gρ)−1/2, that is the timescale over which the collapse

would proceed if the gas loses its pressure support. Since the halo we are considering is virialized, these two timescales must be of the same order of magnitude tH∼ tdyn.

Thus, the fate of the halo and the possibility of further collapse leading to the formation of stars and galaxies relies on the cooling: the cooling time scales tcoolhas to

be shorter than the dynamical timescale, or equivalently than the Hubble timescale. This is known as the Rees-Ostriker-Silk criterion for the formation of galaxies and it was proposed in 1977 [15,16]. In order to determine the minimum halo mass that can collapse at a given redshift, we need to consider two ingredients: the relation Mh−zvir

and the Rees-Ostriker-Silk criterion. For the first relation, we need to consider the fluctuations in the primordial gas density field. Perturbations can be described by an overdensity parameter respect to the average value as δM= [(ρ − ρ)/ρ]M, where M is

the mass scale of the perturbation. Assuming a Gaussian field [2] for perturbations, they can be expressed in terms of the probability to have a certain overdensity, which is related with the standard deviation σ(M ). It is used the expression νσ-peak, such that the probability to have that peak scales as e−ν/2. Typically one focuses on

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3 − 4σ peaks, because even if higher σ peaks would collapse earlier, they would become too rare to be statistically significant for the following cosmic history. When the perturbation enters in a non-linear scale, the halo virializes. The overdensity can be expressed as a multiple of the rms of the fluctuations on the scale in question: δM= νσ(M ), that is a νσ-peak. The rms fluctuations grows approximately as:

σ(M ) ∼ σ0(M )

1 + z , (1.11)

where σ0(M ) is the present-day value. The ΛCDM model predicts σ0(M ) ' 15 for

a mini-halo with mass Mh ∼ 106 M. We can deduce the virialization redshift as

1 + zvir ∼ νσ0(M )/δM, resulting in zvir ∼ 20 − 30 for ν ∼ 3. The conditions to

be satisfied for a virialized halo to collapse are summarized in figure 1.2, where the minimum mass of Pop.III star forming regions are plotted, along with the lines that represent the physical constraints. It can be seen that a minimum halo mass of ≈ 106

M is required to collapse at redshift 20 − 30.

Figure 1.2: Minimum mass of Pop III star forming regions in function of redshift. The solid lines represent the minimum mass of DM halo for various overdensities; the dashed line represents the minimum halo mass satisfying the Rees-Ostriker-Silk criterion: halos above this line can efficiently cool allowing the gas to further collapse; the dotted line shows the minimum mass to overcome pressure support, corresponding to the cosmological Jeans condition. However, the Jeans criterion is not sufficient to enable star formation but the stronger cooling condition has to be met. For masses below 104_M

, collapse is not possible since those halos have virial temperatures below that

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1.3. Star formation 16

The framework illustrated, based on simple analytical arguments, has been com-plemented by a series of self-consistent cosmological simulations that have confirmed the general picture, adding precision and new interesting effects. In particular, sim-ulations suggest a minimum collapse mass of Mcrit ' 7 × 105 M, with only a weak

dependence on a collapse redshift. This is a consequence of the merging history of a halo: if a halo experiences a too rapid mass growth, thus virializing at high redshifts, the dynamical heating caused by DM halos merging prevents the gas to cool later; if a halo virializes later, then it has a larger threshold mass for cooling [17].

1.3 Star formation

We consider now the collapse of the central region of a mini-halo, usually a cloud of ∼ 103 M: these clouds will be the birthplaces of the future stars. Here we will

provide just a simplified review of the cloud collapse, describing the basic chemical and thermal processes that regulate its evolution. The resulting picture is a lot simpler compared to the present-day star formation, however recent progresses have started to add more physics to the simulations, a sign that this field has entered in a more mature phase.

Three stages can be identified, depending on the density of the gas. During the atomic phase (densities below 108 _cm−3_{), gas is mostly atomic, with only a small}

fraction of H2, whose formation proceeds while the density continues to increase

until it reaches a fractional abundance of n[H2]/n ∼ 10−3. The presence of molecular

hydrogen allows the gas to cool despite the compressional heating and when the density is n ∼ 104 _cm−3_{, the temperature is ∼ 200 K. At this point a transition}

named lotering state occurs, because rotational levels of H2 pass from non-LTE to

LTE populations: this changes the dependence of the cooling function from ∝ n2 to ∝ n [18]. The density and temperature at the loitering state can be used to derive a characteristic mass scale of the future Pop III star, expressed by [1]:

MJ' 500 M T 200 K 3/2 n 104 _cm−3 −1/2 . (1.12)

This value represents a pre-stellar core on the onset of gravitational runaway collapse and this mass is named Bonnor-Ebert mass. Additional coolants may prevent the gas from reaching the lotering state, for example hydrogen deuteride (HD) can act in this sense (it possesses a permanent electric dipole moment), making the temperature decrease below 200 K down to the CMB value, that sets the floor to the radiative cooling. We do not discuss here these complications.

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1.3. Star formation 17

During the lotering state, the gas evolves in a quasi-hydrostatic manner and the temperature increases due to compressional heating until the density reaches n ∼ 108

cm−3: at this point the gas undergoes a phase transition becoming fully molecular and the molecular phase begins. This happens via the three-body reactions:

H + H + H → H2+ H (1.13)

H + H + H2 → H2+ H2. (1.14)

The transition to the molecular form has two important effects on the thermal state of the gas. From one side, the increased molecules abundance results in a large increase in cooling, of a factor ∼ 103. From the other, in the H2 formation the binding energy

of 4.48 eV per molecule is released. The net thermal effect is an almost isothermal collapse at T ∼ 103 _{K. Another effect is the change in the equation of state, from an}

adiabatic index γad = 5/3 in the atomic phase to γad = 7/5 in the molecular phase,

due to the additional internal degrees of freedom of the molecules. While the density increases, progressively more lines of H2 become optically thick, complicating the

radiative transfer: at n > 1014 cm−3 all lines are optically thick and the collapse has entered in its last stage toward the proto-star.

At this point a new cooling agent comes into play, in the form of the compound H2-H2, that possesses a temporary eletric dipole moment, induced by van-der Waals

force. This triggers collision induced emission (CIE) and its reverse process, colli-sion induced absorption (CIA). However, the increasing continuum opacity quickly extinguishes the CIE cooling channel and when the density is n ∼ 1016 _cm−3 _the

temperature is ∼ 2000 K. At these high densities collisions become so frequent and energetic to dissociate molecular hydrogen, removing the binding energy of 4.48 eV from the thermal reservoir of the gas, thereby cooling it despite it is in an optically-thick regime.

The density profile of the freshly formed proto-stellar core can be approximately described by the spherically symmetric self-similarity solutions of Larson-Penston [19,20]. The LP-solutions predict a central plateux, whose size reduces as collapse proceeds, while the density increases. At high radii, the profile is close to ρ ∝ r−2 for an isothermal gas. If a polytropic equation of state in the form P ∝ ργ is valid, then the density scales as ρ ∝ r−nρ_{, where n}

ρ = 2/(2 − γ). Omukai and

Nishi [21] demonstrated the validity of this solution in the Pop III stars context, finding nρ ' 2.2, to which corresponds γ ' 1.1. The self-similarity regime is broken

once the hydrostatic inner core is formed and the proto-star begins to grow via accretion.

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1.4. Protostellar growth from an accretion disk 18

1.4 Protostellar growth from an accretion disk

The following growth of the proto-star is a crucial phase in determining the final stellar mass. The growth via accretion for the present-day star formation is a lot complex and it is not clear the geometry of the accretion process, which depends on magnetic fields, turbulence, viscosity and so on. The impact of these factors in the context of Pop III stars is still under debate and we review here the general picture. A useful estimate of the proto-stellar accretion rate, at least as order of magni-tude, can be obtained assuming that the star gains an amount of gas equal to the Jeans mass in a free-fall time:

˙ Macc∼ MJ tff ∝ c3 s ∝ T 3/2_. _(1.15)

The accretion rates in the early Universe are believed to have been much larger with respect to present-day star formation. In fact, molecular clouds have a temperature of ∼ 10 K, while regions of primordial star formation at least 200 K, that is the limit of the molecular hydrogen cooling. Even if HD cooling were active, the temperature would have been limited to cool by the CMB floor, reaching Tmin ' TCMB& 2.73(1 +

z) K, resulting in gas clouds still hotter than the present-day molecular clouds for z & 3. Therefore, typical accretion rates were even two orders of magnitude greater than in the actual star forming mode. This fact, combined with the absence of dust grains that could otherwise trigger fragmentation, suggests that the first stars could have masses of ∼ 100 M, or higher. This was the general picture until less than

ten years ago. Since then, two important effects, first suggested in analytical studies and then confirmed by improved simulations, turned out, drastically changing it.

As described by McKee and Tan [22], after the initial nearly-spherical collapse forming an hydrostatic inner core, the subsequent evolution of the star does not happen in a similar way. Matter falls with non-zero angular momentum and a ro-tationally supported disk forms around the proto-star, processing the gas from the outer regions of the cloud toward the central object. Simulations as the one per-formed by Grief et al [23] confirmed the formation of spiral structures, which are visible in all the mini-halos (figure 1.3) and they also suggested the possibility that the disk fragments in small clumps, forming a multiple stellar system. It is then clear that the growth of the proto-star is strongly dependent on the evolution of the accretion disk itself. Recent simulations as [24–27], thanks to an improved spatial resolution, were able to capture the formation of the self-gravitating accretion disk around the central object and they found that it is a general behavior that the disk undergoes a gravitational instability that leads to fragmentation.

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What happens can be understood as follows. The global gravitational stability can be investigated by a simple criterion proposed by Toomre [28], using a parameter that is Q = csΩ/(πGΣ) for a Keplerian disk; here Ω is the angular velocity and Σ

is the gas surface density. The Toomre Q-criterion says that when Q < 1, the disk undergoes gravitational instability. The physical motivation underlying the Toomre criterion can be deduced with simple arguments. For simplicity, we treat the disk as a thin disk, that is a disk in which the vertical scale height HP (the characteristic

length over which the pressure decreases by a factor 1/e) satisfies the inequality HP r, r being the radial distance from the central object. Under this assumption

the gas surface density can be related to the density as ρ ' Σ/(2HP) and the vertical

scale height with the sound speed via HP ' cs/Ω [29]. There are two forces acting on

the gas: the self-gravity, that tends to form clumps, whose pressure force is given by

π 2GΣ

2 _{(see also appendix C); the thermal pressure force, that instead tries to support}

the gas under collapse preventing fragmentation, that for simplicity we write as ρc2 s,

assuming an isothermal equation of state. The ratio of these two forces is then: Pgas Pgrav ∼ ρc 2 s π 2GΣ2 ∼ Σ 2HPc 2 s π 2GΣ2 ∼ Σ 2 Ω csc 2 s π 2GΣ2 ∼ csΩ πGΣ,

that is exactly the Toomre parameter. Even if the accurate derivation of the Q-criterion is a lot more complex, this simple reasoning expresses the essential nature of the criterion: if the self-gravity of the gas is more important than the thermal pressure, then the gas is susceptible to collapse.

In an accretion disk, gravitational torques drive mass toward the center while angular momentum is driven outward: the mass infall onto the proto-star is then determined by the efficiency of the torques in transferring gas. From the other side, the rate at which matter falls onto the disk is determined by the global collapse of the cloud hosting the pre-stellar core, given by equation (1.15). Considering for example the case of a thin disk, the accretion rate can be expressed using the formalism by Shakura & Sunyaev [30]: M˙acc ' 3πνvisΣ, where νvis represents the kinematic

viscosity and it can be dimensionally written as νvis ' αcsHP, where α indicates the

efficiency of the mechanisms that transport angular momentum. For gravitational torques a typical value is α ∼ 0.1 − 1. Recent simulations have enlightened that the accretion rates onto the disk are usually much higher than the rate at which the disk can process mass: this imbalance results in an increase of the gas surface density and thus in a reduction of the Q parameter, so that the disk becomes unstable. However, the Toomre criterion is a necessary but not sufficient condition in determining if the disk actually fragments: it suggests that the disk is susceptible to global perturbation and spiral modes, but clumps may be disrupted by the shear motion present in the

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disk, depending on the rotation of the disk itself. Thus, a stronger condition has to be met to enable fragmentation: the gas has to cool on a time scale shorter than the orbital time scale Ω−1, allowing a single portion of the disk to de-couple from the rest and to form a stable clump. This is expressed by the Gammie criterion tcool < 3Ω−1[31]. The action of cooling also reduces the effect of the thermal pressure,

further decreasing the stability of a gas region already prone to fragment.

Figure 1.3: Simulation of primordial proto-stars formation. Each square represents the density projections in a cube of side length 10 AU and each row corresponds to a different mini-halo, with time increasing from left to right. In various panels it is possible to note a disk-like structure around the growing proto-star and the formation of multiple clumps. Adopted from [23].

The radiation coming from the growing proto-star is the second important factor affecting the evolution, whose impact was first investigated analytically by McKee & Tan [32]. An upper limit to the final mass retained by a star can be naively estimated by M? ∼ M˙acctacc, where tacc is the effective accretion timescale. For

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sequence, that is the Kelvin-Helmotz time tKH. The typical values of the accretion

rate and its duration induced to think that primordial stars were a lot more massive than present-day stars, even 100 − 1000 M. What is missing in this frame is the

fact that while the proto-star grows, its surface temperature increases too and at a certain point the amount of UV photons is no more negligible.

The analytical calculation by McKee & Tan suggested that UV photons ionize the gas, forming an H-II region surrounding the central protostar. This region is bipolar and its over-pressure respect to the gas in the cloud reverses the infall in the polar directions, allowing gas to fall from the others. However, while the H-II region expands, the disk starts to photo-evaporate, until at a certain point accretion completely shuts off. This happens far before than the star has reached the main sequence, drastically limiting the duration of the accretion. They suggested an up-per limit to the stellar mass of ∼ 140 M. Following simulations confirmed these

analytical predictions, in particular Omukai et al [33] and Stacy et al [34] investi-gated the growth of the protostar focusing on the role played by the EUV photons (hν > 13.6 eV, ionizing atomic hydrogen) and FUV photons (11.2 < hν < 13.6 eV, photo-dissociating molecular hydrogen). Their results revised the upper limit for the final mass to few tens of solar masses. Figure 1.4 shows the growth of an H-II region around a central proto-star.

Figure 1.4: Protostellar radiative feedback. The three panels represent the ionization fraction of the gas in the plane x − z respectively 1500, 2000 and 3000 yr after the formation of the proto-star. White lines depict the density contours of the disk for densities ranging from 107.5 _{to 10}9

cm−3. The box size is 40.000 AU. It can be seen the formation and expansion of an H-II region hour-glass shaped. This structure gradually expands, dissipating the disk from above and below and eventually halting the accretion. Adopted from [34].

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1.5. Initial mass function of Pop. III stars 22

1.5 Initial mass function of Pop. III stars

The two effects previously described are crucial in answering two fundamental ques-tions still open about primordial stars formation: what is the binary fraction of Pop III stars? What is the initial mass function of Pop III stars? The formation of frag-ments suggests that primordial stars could easily form in binary or multiple systems and the effect of radiative feedback seems to be the most important factor in limit-ing the growth of the proto-star. Fragments may have evolved independently gainlimit-ing mass from the very same cloud of the ”central” proto-star limiting its growth, or may have finally merged within the proto-star, in which case the final mass of the star is not significantly affected by disk fragmentation. The works by [35–37] were the first to investigate this phenomenon, using cosmological simulations, noticing a common trend to binary formation in mini-halos, but simulations were not able to study the proto-stellar growth for enough time to follow the fate of the fragments. The com-mon opinion is that a fraction of about 35% of Pop III stars may have formed in binaries, that means that a single Pop III star has a probability of around 50% to have a companion. An analytical study by Latif & Schleicher [38] focused on the stability of the disk found that clumps migrate inward on a time scale shorter than the Kelvin-Helmotz time scale and therefore most of them may eventually merge with the central star, possibly leading to intermittent accretion (a phenomenon also examined by Vorobyov & Basu in various contexts, in particular in primordial star formation [39]). Latif & Schleicher also estimated that when the photo-dissociation of the disk becomes important, the accretion rate has already dropped by one or two order of magnitude respect to the first growth stage, concluding that it would be possible for a Pop III star to reach a mass of 100 M. The latest simulation by Stacy

et al [40] studied both the disk fragmentation and the impact of the proto-stellar feedback on the accretion disk, finding that numerous fragments actually form, but only 1/3 of them survive until the end of the simulation, the rest merging with each other or with the central proto-star, few fragments being ejected from the mini-halo. While the most massive star is of 20 M, most of the sinks have masses < 1 M.

This result supports a top-heavy IMF, i.e. the principal contribution to the mass of Pop III stars is contained in the most massive stars. However, the determination of the IMF of primordial stars is still in its infancy and a lot of works is still needed.

The low-mass fragments ejected by a mini-halo could have an enormous impor-tance in observations. In fact, while high-mass primordial stars have died, a star with mass < 0.8 Mcould have survived from the Dark Ages until today and it is in

prin-ciple observable. For example, from the IMF deduced by Stacy et al. [40], 106Pop III survivors should be visible in our Milky Way only. The fact that no metal-free star

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1.5. Initial mass function of Pop. III stars 23

has been detected yet has various explanations: first, the fragmentation observed in numerical simulations is somehow larger than in reality, maybe due to a pre-selection in mini-halos simulated, and the ejection fraction may be off-set too; secondly, these stars can be masked by later accretion and enrichment from interstellar materials, polluted by the following generation of stars and thus not appearing as metal-free anymore.

For the high-mass tail in the IMF, stars in the mass range 140 − 260 M are

expected to end their lives as pair-instability supernovae (PISNe) leaving no remnants behind [41,42]. These events should have leaved a footprint in the metal enrichment of metal-poor stars, showing a strong odd-even effect (low abundance ratios of odd-Z elements to even-Z elements) and the complete absence of neutron capture elements. The fact that such an evidence has not been found can be an indication that the stars generating PISNe (M > 140 M) were extremely rare, but it can also be caused

by an observational selection effect [43].

It is evident from our discussion that improving the resolution in numerical sim-ulations, the physics details and the incoming of new observational facilities as the JWST can converge toward a great increase in our knowledge about the end of the Cosmic Dark Ages and the birth of the first stars.

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2

Supermassive black

holes formation

Sky surveys in the last decade have led to observation of several quasars at redshift z > 6 [44–47], when the Universe was less than a billion years old. The term quasar (a contraction of quasi-stellar radio sources) was used in early observations referring to point-like sources (thus similar to stars), emitting in the radio waves spectrum. Progresses made thanks to infrared telescopes and the Hubble Space Telescope revealed that each quasar is at the center of a far galaxy and the enormous luminosity of these objects, that can reach values ≈ 100 times greater than the one of the Milky Way, is caused by some mechanism acting in the nucleus of the hosting galaxy. There is general consensus today that the engine powering these structures is a supermassive black hole (hereafter SMBH) with few billion solar masses, that is accreting material from a surrounding disk. The matter infalling onto the black hole releases its gravitational potential energy and, for such a compact object, the conversion mass to energy can be even more efficient than a nuclear reaction, resulting in a very powerful source.

The observational evidence of the existence of SMBHs with masses & 109 M

within 1 Gyr after the Big Bang is puzzling for several reasons, concerning both their formation and growth, and it is therefore a current topic of major research in cosmology and astrophysics.

2.1 The growth of SMBHs seeds

The presence of SMBHs in the early Universe suggests that their seeds must have formed at z ≥ 15, with masses depending on the formation mechanism in the range 10 − 105 M. They must have grown via accretion or through mergers with other

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2.1. The growth of SMBHs seeds 26

The mass growth of an accreting object (in our case a black hole) is regulated mostly by its accretion rate and lesser by the efficiency factor (the quantity of mass that is converted into energy). The accretion rate is in turn limited by the Eddington luminosity LE, that is the critical luminosity at which the radiation pressure acting

on free electrons balances the gravitational attraction toward the central object, thus halting the infall. The Eddington luminosity is a function of the black hole mass,

LE = 4πGMBHmp c σT ≈ 1.25 × 1035 M M erg s , (2.1)

where G is the gravitational Newton constant, mp the proton mass, c the speed

of light and σT the Thomson cross section (assuming that the principal source of

opacity comes from Thomson scattering). The mass growth rate can be written as ˙

MBH = λ LE

1 −

c2 , (2.2)

where λ is the ratio between the accretion luminosity and the Eddington luminosity, called the Eddington ratio. Assuming that the black hole considered is accreting at the Eddington luminosity, its growth will be exponential with an e-folding time given by: te = σT c 4πGMBH mp 1 − = tS 1 − , (2.3)

where tS is called Salpeter time and its value is tS ≈ 0.045 Gyr [48]. For a

typ-ical value ≈ 0.1, the characteristic time of BH growth is well approximated by the Salpeter time. In this picture, a stellar-mass BH seed (. 100 M) constantly

accreting at the Eddington limit would require 7 × 108 _{yr to reach the mass of a}

SMBH, a time that is barely the age of the Universe at z ' 6. Furthermore, the assumptions of constant accretion and the availability of low angular momentum gas to feed the growth for such a long period of time are weak. The study of [49] shows that supernova associated with the BH formation causes a delay of 100 Myr before the accretion becomes efficient. Thus, seeds with larger masses at z > 15 are needed to explain the presence of quasars at z > 7 in a more convincing way or a growth mechanism that can sustain a super-Eddington accretion is required.

Following the comprehensive review by Latif & Ferrara [50], we briefly describe the principal formation mechanisms of a black hole, in order to examine the possible pathways that can lead to a more massive seed and discussing the most promising ones at the current state of knowledge.

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2.2. Black hole formation mechanisms 27

2.2 Black hole formation mechanisms

The most standard way to form a BH is the collapse of a massive star in the last stage of its life, forming a so-called stellar mass BH. Primordial stars in the mass range between 40 − 140 M or above 260 M are expected to collapse into a BH of

similar masses [41,42]. If trace amounts of metals are present, all stars above 40 M

end their lives collapsing into a BH, but with mass loss increasing with metallicity. Stars of primordial composition are thus the ones that should retain most of the mass. As discussed in sections 1.4 and 1.5, the IMF of primordial stars is not yet known and the formation of stars with hundred solar masses appears to be difficult. However we can consider an upper limit, using the stellar mass distribution derived by Hirano et al [51], that simulated the evolution of one hundred stars starting from cosmological condition, including UV feedback but assuming that for each halo only a star forms, excluding the complication of multiplicity. They found stellar masses in a wide range 10 − 1000 M, suggesting that in special cases the proto-star can

grow until 1000 M. The mass distribution is represented in figure 2.1.

Figure 2.1: The stellar mass distribution of the 110 stars simulated assuming that in each mini-halo a single star forms. The different colors in the histograms represent the different paths of protostellar evolution: see [51] for details. Adopted from [51].

To sum up, BH seeds of few hundred solar masses at z = 20 − 30 can be formed, but they need to accrete mass at the Eddington rate throughout their lifetime in order to reach the desired billion of solar masses at z = 7. Furthermore, three-dimensional simulations show that the feedback from the BH reduces the accretion rate to very

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low values (10−10 M/yr), several orders of magnitude below the Eddington limit.

It is generally thought that it is impossible for a stellar mass BH to become a SMBH at z > 7 without episodes of super Eddington accretion.

For this reason, a class of black holes in the mass range 104− 106 _M

has been

postulated (there has been no observational evidence of such objects so far), indicated as intermediate massive black holes, hereafter IMBH(s). They are considered with growing consensus as the most promising seeds of the SMBH observed at z ≥ 7 and their formation in the early Universe is plausible through the two channels described below.

BHs may also have formed in dense nuclear clusters thanks to stellar dynamical processes [52,53]. A star cluster is a group of stars collectively bound by their gravitational interactions and in which star collisions are important in the dynamical evolution while rotation does not (contrary to a galaxy, that is another gravity-bound stellar system, but in which collisions are negligible in the dynamics and most of the kinetic energy is rotational energy). The dynamical evolution of a star cluster is governed by close encounters between two stars, that cause deflections and accelerations of the involved objects. This process leads to dynamical relaxation and its prolonged action guides the system toward a state of thermal equilibrium. If the relaxation time (the average time over which a star completely loses its memory about the initial conditions) becomes comparable with the core collapse time scale tcc, the core cluster can collapse [54]. Under this conditions, the stellar collisions in

its center lead to the formation of a very massive star (also called VMS, a star with a mass ≈ 1000 M), which later may collapse into a BH up to 1000 M.

The resulting BH mass depends on cluster properties and on its dynamical evo-lution. Early studies as [55,56] found that the collapse is halted by binary heating in less massive stellar clusters. Furthermore, the tcc has to be shorter than the time

required for a massive star to reach the main sequence (≈ 3 Myrs), otherwise there can be mass loss from supernova events that can stop the collapse. Later it was found that even in less massive clusters an instability called mass segregation de-velops, speeding up the collapse [57,58]. What happens is that massive stars are attracted toward the center and become trapped, making the potential well deeper and thus causing more stars to move toward the center. This mechanism strongly depends on the compactness and on the dynamical evolution of the cluster.

The relevant factors for the core collapse can be summed in the following points: • the dynamical evolution, studied via N-body simulations, investigates the im-pact and frequency of collisions: the result of a close encounter can be a devia-tion of the trajectories, a formadevia-tion or destrucdevia-tion of a binary system or even a stars merger. The works by Portegies Zwart et al. [52,59] showed that binaries

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enhance the number of stellar collisions, thus speeding up the core collapse and the formation of a single massive object. This makes possible the formation of a BH of ≈ 1000 M even in low mass systems if collisions are efficient enough.

Furthermore, the mass of the resulting VMS is about 10−3 times the cluster mass and it is mainly contributed from stars in the range of 60 − 120 M [60].

• the compactness of a cluster, represented by its half-mass radius, is a critical factor for the core collapse: for a given cluster mass, the number of collisions and the increase in mass per collision strongly depend on this property [59]. The compactness is the primal factor determining the time required by the core to collapse, which has to be shorter than the typical time scale necessary for a supernova event to occur (≈ 3 Myrs), otherwise the BH formation will not happen. Both the compactness of a cluster and the dynamical evolution are important for the mass segregation mechanism. Furthermore, the mass of a VMS increases linearly with the initial central density of a cluster [61]. • the initial mass function (IMF) and the binary fraction of the stars in the

clus-ter significantly affect the dynamical evolution of the system, which is driven by collisions, whose effects on the dynamics depend both on the kind of close encounter (binary-star or star-star) and on the masses of the stars involved. As discussed in section 1.5, latest studies suggest (with the proper caution due to limited time exploited in simulations) that the first stars may have formed in multiple systems due to fragmentation of protostellar disk triggered by gravi-tational instability, with a binary fraction of 35%. For a system with an IMF top-heavy the mass of the central VMS tends to increase, but the probability of producing a VMS is not significantly altered. Instead, adding binaries to simulations changes the evolution of the cluster, but affects the final mass of the VMS of a factor of 3. These factors are less important than the central density (compactness) and the initial mass of a cluster, that influences the final VMS by even an order of magnitude or so [61].

The mass estimate of the BH resulting from the VMS collapse is uncertain due to the complex interplay between the examined factors, thus further work is needed: simulations self-consistently modeling the formation and evolution of a nuclear stellar cluster starting from cosmological conditions and considering the N-body dynamics are still to be done. The analytical model developed by Devecchi et al. [62,63] indicates that BHs up to 1000 M can be formed at z ≥ 10. The work of Katz

et al. [61], which combines hydrodynamical cosmological simulations with N-body simulations, provides and estimate of ≈ 400 M for a VMS formed in a cluster of

104 _M

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2.3. Direct collapse black holes 30

The last channel of BH formation is the collapse of a gas cloud into a massive BH, called direct collapse black hole [64,65], hereafter DCBH. This pathway is the one on which this work is focused, thus it will be widely discussed in the following sections.

2.3 Direct collapse black holes

The direct collapse black hole scenario is one of the most promising way to solve the puzzle posed by the existence of z > 6 quasars, because it allows the formation of a BH seed of 105− 106 _M

, even two order of magnitudes higher than the typical black

hole mass expected from other mechanisms. The framework lying at the basis of this channel is the monolithic isothermal collapse of the gas of a proto-galactic cloud with mass > 107 M on a time scale of the order of 1 Myr into a single object at the center

of the DM-halo. The evolution of the stellar object fed by the infalling gas is crucial for the final mass retained by the BH: if the star evolves toward the zero-age main sequence (ZAMS), it will emit a copious amount of ionizing photons, shutting off the accretion and forming at the end of its life a black hole with a comparable mass. If instead it is able to continue its growth, it will encounter a General Relativity (GR) instability, that will induce a rapid collapse into a BH, producing virtually no ionizing photons. Thus the fresh formed black hole will be embedded into the halo under the conditions to continue accreting, passing through a feedback-regulated growth phase until it finally becomes a fully-fledged IMBH.

We then proceed to examine the physical properties of the halo, necessary to fulfill the constraints favorable to the formation of a DCBH.

2.3.1 Birthplaces of DCBHs

First of all, the formation of a DCBH requires sufficient gas reservoir to feed the growth of such a massive object. Furthermore, the gas has to be hot enough to provide the accretion rates needed for a rapid growth: in fact, the accretion rate is proportional to the gas temperature according to M˙ ∼ c3s/G ∝ T3/2. These

arguments make the primordial metal-free halos at z ≈ 15 with a virial temperature Tvir≥ 104 K and masses of > 107 M the most likely embryos of the future DCBHs.

These halos are the result of mergers and accretion of the smaller mini-halos and they have sufficient gas reservoir to form a massive object of 105_{− 10}6 _M

and a potential

well deep enough for a rapid collapse. The threshold on the temperature is motivated by the necessity of a cooling source different than molecular hydrogen cooling. Several studies [66–68] have emphasized that H2 cooling triggers fragmentation in the cloud,

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2.3. Direct collapse black holes 31

preventing the stocking of the gas reservoir into a single object. Instead, atomic lines cooling, that becomes efficient above T ≈ 104 _{K, grants a quasi-isothermal collapse}

at an accretion rate of: ˙ M∼ c3 s G ≈ 0.1 M yr −1 T 8000K 3/2 . (2.4)

Moreover, the gas cloud has to be of pristine composition for the same reasons as above. We then discuss how these requirements can be satisfied.

2.3.2 Preventing the formation of molecular hydrogen

Reminding our discussion in section 1.2 about the formation of molecular hydrogen, there are two channels through which this molecule can form, one with H− as an intermediate state and the other with H+₂. The latter path dominates at high redshifts z > 100, while in general the former is the principal process. Given that we are considering structure formation at 20 < z < 30, we focus only on the chain:

H + e−→ H−+ γ (2.5)

H + H−→ H2+ e−. (2.6)

Thus, the formation of H2 can be averted either by dissociating the H2 molecule

itself or the intermediate species H−. This can happen thanks to photons emitted by existing stars, depending on their spectra. The photo-detaching of H− requires UV photons with energy above 0.76 eV, corresponding to a star spectrum with an effective temperature of Trad = 104 K. The molecular hydrogen can be destroyed

via the two-step photo-dissociating Solomon process: photons in the energy range between 11.2 − 13.6 eV are absorbed in the Lymann-Werner bands of H2, placing the

molecule in an excited state, then radiative decay rapidly occurs, in the 10 − 15% of the cases into a vibrational continuum state, meaning that the hydrogen molecule has been dissociated. The photons fulfilling this goal are produced by stars with an effective temperature of Trad = 105 K. The two reactions contributing to the

suppression of the molecular hydrogen formation can be schematized as:

H2 + γLW → H + H (2.7)

H−+ γ0.76→ H + e−. (2.8)

The determination of the critical value of the UV flux that completely prevents molecular hydrogen from being formed, indicated with J₂₁crit (hereafter field intensity will be expressed in the usually adopted units 10−21 erg s−1 cm−2 Hz−1 sr−1), is one

Numerical simulations of accretion disks around early black holes

Universit`

a degli Studi di Pisa

Numerical simulations

of accretion disks around

early black holes

Candidato

Fabio Di Mascia

Relatore

Prof. Andrea Ferrara

Marzo 2017

Contents

Introduction

1

First stars

1.1

A brief history of the Universe

1.2

The sites of primordial stellar seeds formation

1.3

Star formation

1.4

Protostellar growth from an accretion disk

1.5

Initial mass function of Pop. III stars

2

Supermassive black

holes formation

2.1

The growth of SMBHs seeds

2.2

Black hole formation mechanisms

2.3

Direct collapse black holes

2.3.1

Birthplaces of DCBHs

2.3.2

Preventing the formation of molecular hydrogen