Evolution of protostars
In order to determine the mass accretion rate, feedback and other processes that might influence the protostellar environment, we have to understand the basic principles of primordial protostellar evolution. Except for some quantitative disagreements (see e.g. Turk et al. 2011), there is broad agreement on the qualitative evolution of a Pop III protostar. Although this evolution is a complex interplay of many processes, it can be summarised by a few characteristic steps (Stahler et al. 1986; Omukai & Palla 2001).
1. At densities above n & 1019 cm 3 most of the molecular hydrogen is dissociated, the adiabatic index rises from ’ 1:1 to ’ 5=3 and the self-similar solution breaks down (Omukai & Palla 2001; Haardt et al. 2002; Omukai et al. 2005; Yoshida et al. 2006; McKee & Tan 2008; Greif et al. 2012).
2. A quasi-hydrostatic core with a mass of M ’ 0:01 M forms and a shock front develops at the protostellar surface (Omukai & Nishi 1998; Haardt et al. 2002; Greif et al. 2012; Hirano et al. 2014).
3. In the early phase, luminosity and thermal energy are mostly produced by contraction (Omukai & Palla 2001; Krumholz & McKee 2008). Consequently, temperature rises slowly and at a mass of around M ’ 5 M the optically thick outer layers expand to due radiation pressure from the luminosity of the contracting core (Stahler et al. 1986).
4. Hydrogen fusion halts the contraction at higher masses and the star reaches the zero age main sequence (ZAMS) after approximately 106 yr of protostellar evolution (Yoshida et al. 2006; Glover 2013a).
There are some special characteristics for the metal-free stellar evolution, compared to the present-day one. These characteristics of primordial protostellar evolution are summarised in the following list:
Deuterium burning does not play an important role (Omukai & Palla 2001; Schleicher et al. 2013; Glover 2013a).
High-mass stars reach the main sequence while they are still accreting (Peters et al. 2010; Klessen 2011).
Massive star formation cannot be spherically symmetric (Klessen 2011).
While Pop III protostars are large, fluﬀy objects, the final stars are smaller than their present day counterparts (Smith et al. 2011; Clark & Glover 2013).
The previously discussed evolution of primordial protostars goes along with an interdependent network of other important processes that are discussed in the following subsections.
Accretion is a crucial process in the determination of the final masses of metal-free stars. Due to the chemical composition of the primordial gas and the comparatively high temperature, accretion in the early Universe diﬀers significantly from accretion in present-day star formation. Since there are no dust grains, radiative pressure is less eﬃcient and the accretion rates are consequently higher. A first estimate for the accretion rate can be derived from the assumption that a Jeans mass is accreted in about one free-fall time:
The temperature of the star-forming gas in a primordial minihalo is higher than the 10 K of present-day star forming clouds (Omukai & Palla 2001; Bergin & Tafalla 2007; Glover 2013a). Consequently, the primordial accretion rates are much higher than local accretion rates, which are of the order of M_ ’ 10 5 M yr 1 (Glover & Abel 2008). Typical values for the accretion rate in primordial star formation range from M_ ’ 10 3 M yr 1 to M_ ’ 10 1 M yr 1 (Ripamonti et al. 2002; Hosokawa et al. 2011; Clark et al. 2011b; Hirano et al. 2014). However, these accretion rates are highly variable in time (Klessen 2011; Smith et al. 2011; Hirano et al. 2014; Stacy et al. 2013; Glover 2013a).
A main question is, when accretion is stopped and by what processes. In order to answer this question, we can distinguish two cases:
In the case of “smooth accretion”, major parts of the infalling envelope are accreted by the central object. Since protostellar radiation cannot halt this inflow, the mass accretion rate decreases rather slowly with increasing stellar mass (Machida & Doi 2013; Hirano et al. 2014).
In the case of “competitive accretion”, the gas is accreted by several protostars. Consequently, each individual protostar has to compete for gas and the reservoir might get exhausted. This scenario (also called “fragmentation induced starvation”) is discussed e.g. by Peters et al. (2010).
In reality, there is no clear distinction between these two scenarios and the accretion rate might remain high, although there are several protostars competing for infalling gas. Ultimately, accretion stops due to radiative feedback by the protostars.
As already seen in Figure 1.5, the uncertainties in the expected mass range are very large. Since the accretion rate is very high and primordial gas contains almost no metals, Pop III stars can become very massive (Omukai & Palla 2001; Clark et al. 2008). Assuming a constant accretion rate of M_ 10 3M yr 1 over the typical Kelvin-Helmholtz timescale tKH ’ 105yr yields a rough upper limit of 100M . Even if we consider continuing accretion during the main sequence lifetime, this increases the upper mass limit only by a factor of a few to 600M (Bromm & Loeb 2004). However in reality, protostellar feedback and competitive accretion limit the stellar masses to much smaller values.
Generally, radiative feedback from the accreting protostar has several eﬀects on its environment:
Radiative pressure counteracts the gravitational force and therefore reduces accretion. Radiation heats the gas and therefore stabilise it against fragmentation.
Some photons are energetic enough to photodissociate H2 .
It is still a matter of debate whether these processes actually occur and significantly influence primordial star formation.
Formation of the first stars
Although the number of ionising photons is small in the early protostellar phase, their number steeply rises with mass. Above a protostellar mass of 10 15 M , a star can eﬀectively photodissociate molecular hydrogen (Glover 2000; Glover & Brand 2003; Smith et al. 2011; Klessen 2011; Hosokawa et al. 2011;
Hirano et al. 2014). Although accretion feedback heats the gas and even removes the dominant coolant,
it is not believed to suppress fragmentation on large scales (Krumholz & McKee 2008; Peters et al. 2010;
Clark et al. 2011a,b; Greif et al. 2011; Glover 2013a). However, Smith et al. (2011) and Machida & Doi (2013) find fragmentation to be suppressed in the inner 20 AU due to accretion luminosity heating.
Even if fragmentation cannot be suppressed by accretion feedback, it is delayed by up to 1000 yr.
The radiation pressure is rather ineﬀective at early protostellar stages, but it becomes important for
protostellar masses above 50 M and can even halt further gas accretion (Omukai & Palla 2001; Haardt et al. 2002; Hosokawa et al. 2011, 2012b; Hirano et al. 2014).
Final stages elements inside the star and therefore leads to a more homogeneous evolution, the minimum mass for a rotating Pop III star to end in a pair-instability supernovae (PISN) is decreased to 65M (Chatzopoulos & Wheeler 2012; Yoon et al. 2012; Stacy & Bromm 2013). The same reasoning is valid for the upper mass limit of (pulsational) PISNe: Limongi (2017) shows that rotation and thus chemical mixing yields more massive He cores in metal-poor (10 3Z ) stars. The maximum mass for a metal-poor star to explode as a PISNe is therefore 190 M (Yoon et al. 2012).
Observational signatures of the first stars
The first stars are too far away in space and time to be directly observable (Zackrisson et al. 2011) and we rely on indirect observations to constrain their properties:
PISNe are very luminous supernovae of massive, low metallicity stars. If they exist, they can be observed out to high redshifts due to their high explosion energies and allow to probe the Universe prior to reionisation. Their (non-)detection constrains the star formation rate and the IMF of the first stars (Mackey et al. 2003; Scannapieco et al. 2005; Whalen et al. 2013; de Souza et al. 2014; Magg et al. 2016; Hartwig et al. 2017a).
We can learn about the first stars by observing their direct descendants in our galactic neigh-bourhood. This approach is called galactic archaeology, the search for stellar fossils in our own Galaxy, which provides precious information about its formation history (Frebel & Norris 2015). In Hartwig et al. (2015a) we show that already the absence of any Pop III remnant sets tight constraints on the lower mass limit of the first stars (see also Ishiyama et al. 2016; Magg et al. 2017; de Bennassuti et al. 2017).
Second generation stars are those that got enriched by exactly one previous supernova and which consequently carry the chemical fingerprint of this progenitor star. Several groups have already succeeded in determining the most likely masses of such progenitor stars (Aoki et al. 2014; Keller et al. 2014; Placco et al. 2016). In a current study, we predict the most promising host systems of these second generation stars and determine their observational signatures (Hartwig et al. 2017c).
Gravitational waves (GWs) from the compact remnants of the first stars allow indirect observations of the mass and formation eﬃciency of Pop III stars (Schneider et al. 2000; Hartwig et al. 2016a; Pacucci et al. 2017a). We will discuss this possibility in more detail in Chapter 3.
Further constraints on the Pop III IMF come from existing and upcoming observations of gamma ray bursts (Bromm & Loeb 2002; Ma et al. 2016), or the near-infrared background excess (Madau & Silk 2005).
Formation of the first supermassive black holes
In this section, we introduce existing theories of the formation and growth of the first SMBHs. We first present current observations in the local and distant Universe to motivate our theoretical framework and to highlight open questions. We discuss and compare diﬀerent formation scenarios of BH seeds and present AGN-driven winds as one potential feedback mechanism that regulates their growth.
Observations of quasars at high redshifts indicate that SMBHs of several billion solar masses were already assembled in the first billion years after the Big Bang (Fan et al. 2003, 2006a; Willott et al. 2010; Venemans et al. 2013b; De Rosa et al. 2014; Wu et al. 2015). The current record holders are a bright quasar, which hosts a SMBH with a mass of 2 109M at z = 7:085 (Mortlock et al. 2011), corresponding to 800 million years after the Big Bang, and a SMBH with 1:2 1010M at z = 6:30 (Wu et al. 2015). It is still unclear how these objects were able to acquire so much mass in this short period of time, which in turn raises questions about the formation mechanism and the involved physics.
Besides the observation of these SMBHs at high redshift, there are several observations that indicate a correlation in the local Universe between the mass of the central SMBH and large scale properties of the host galaxy, such as the stellar velocity dispersion , luminosity, or the bulge mass (Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002; Marconi & Hunt 2003; Häring & Rix 2004; Gültekin et al. 2009), see Figure 1.6. These relations imply a possible correlation between the central black hole and its host galaxy, and AGN feedback has been suggested to be responsible for regulating accretion onto the BH and star formation in the galaxy, guiding this co-evolution (see Heckman & Kauﬀmann 2011, for a review).
Formation of the first supermassive black holes
Figure 1.6: M– relation based on observations of local galaxies with dynamical measurements. The symbol indicates the method of BH mass measurement: stellar dynamical (pentagrams), gas dynamical (circles), masers (asterisks). The colour of the error ellipse indicates the Hubble type of the host galaxy. Squares are galaxies that are not included in the fit. The line is the best fit relation to the full sample. The correlation indicates a possible co-evolution between a BH and its host galaxy. Adapted from Gültekin et al. (2009).
Seed formation scenarios
We present and briefly review several scenarios to form seed BHs. We focus on the astrophysical standard scenarios to form stellar and more massive BH seeds, but we first present more exotic formation channels. So-called “primordial” BHs have formed during inflation, possibly out of large density inhomogeneities, non-linear metric perturbations, a step in the power spectrum, or cosmic string loops, which result from a scaling solution of strings formed during a phase transition in the very early Universe (Silk & Vilenkin 1984; Bramberger et al. 2015; Khlopov 2010). Overdensities of mass could collapse to primordial BHs (Hawking 1971; Carr & Hawking 1974; Carr et al. 2016), where t is the time after the Big Bang. However, the probability to form such a BH is highly improbable in a homogeneously expanding Universe, since it implies metric perturbations of order unity. If they exist, primordial BHs represent a nonrelativistic form of dark matter and their mass density has to be smaller than the dark matter density of the Universe. More stringent constraints come from primordial BHs with masses below 1014 g, which should have evaporated until today (Hawking 1975; Khlopov 2010). The (non-)detection of expected evaporated particles and their contribution to cosmic rays put constraints on the formation rate and mass spectrum of potential primordial BHs. Even the absence of observational evidence for primordial BHs is important for cosmology, since it provides unique information about the very early Universe (Khlopov 2010). Upcoming, spaceborne GW detections of BH mergers at high redshift can contribute to further constrain these models.
Pop III remnant BHs
One natural scenario are stellar mass seed black holes with masses up to a few hundred solar masses that are the remnants of Population III (Pop III) stars and then grow by mass accretion or mergers (Madau & Rees 2001; Haiman & Loeb 2001; Volonteri et al. 2003; Yoo & Miralda-Escudé 2004; Haiman 2004; Pelupessy et al. 2007; Tanaka & Haiman 2009; Whalen & Fryer 2012; Madau et al. 2014).
However, already a simple order of magnitude argument illustrates that these seed BHs require extreme conditions to be the progenitors of the most massive BHs at high redshift. Assuming accretion at the Eddington limit, the e-folding time is 45 million years. A seed BH, formed 100 Myr after the Big Bang, accreting at the Eddington limit can therefore grow by a factor of e650=45 ’ 2 106 until z = 7:085. To reach a mass of 2 109 M within this time, it is therefore necessary to start with a seed mass of about 103 M . This is larger than the mass of a typical Pop III stellar remnant (Clark et al. 2011a,b; Greif et al. 2011; Stacy et al. 2012; Latif et al. 2013a; Hirano et al. 2014; Hartwig et al. 2015b), and also constant accretion at the Eddington limit is very unlikely (Alvarez et al. 2009; Aykutalp et al. 2014; Habouzit et al. 2017). In all seed formation scenarios, BH-BH mergers or a lower radiative eﬃciency of the accretion disc could help seeds to gain mass more rapidly (Shapiro 2005; Volonteri & Rees 2006). It is still an open question, how these high gas accretion and inflow rates could be sustained during the growth of the SMBH (Johnson & Bromm 2007; Alvarez et al. 2009; Milosavljević et al. 2009b; Johnson et al. 2013b; Jeon et al. 2014) and we further address this question in Chapter 6.
Very massive stars could form through run-away stellar collisions in dense stellar clusters, which produce remnant BHs of & 1000 M (Begelman & Rees 1978; Ebisuzaki et al. 2001; Portegies Zwart et al. 2004; Belczynski et al. 2004b; Omukai et al. 2008; Devecchi & Volonteri 2009; Lupi et al. 2014; Katz et al. 2015; Mapelli 2016; Sakurai et al. 2017).
The typical masses of stellar cluster in atomic cooling halos with Tvir > 104 K are of the order 105 M and their half mass radii is about 1 pc. A large fraction of these very dense clusters undergoes core collapse before stars are able to complete stellar evolution. Runaway stellar collisions eventually lead to the formation of a very massive star, leaving behind a massive black hole remnant (Devecchi & Volonteri 2009). This scenario favours the metallicity range of about 10 5–10 3Z , because above this threshold, stellar winds eject too much gas and prevent the formation of massive stars and below this threshold gas fragmentation is not eﬃcient and we cannot form a dense cluster in the first place. Mapelli (2016) studies the metallicity dependence on the formation of BHs in dense stellar clusters. BHs can form via stellar collisions, even at solar metallicities, but the mass of the final merger product is limited to 30 M due to mass-loss by stellar winds. In the metallicity range 0:1 0:01 Z BHs with masses up to 350 M can form (Spera & Mapelli 2017). However, for these BHs with < 1000 M , formed above
0:01 Z , the probability of being ejected out of the cluster is much higher, compared to more massive BHs (Mapelli et al. 2011, 2013).
Alternatively, the central density in the core of a stellar cluster can be high enough for fast mergers of stellar mass BHs. If the binaries are hard enough and gas is funnelled to the centre in a low angular momentum flow to contract the existing stellar cluster, a BH with a final mass of MBH & 105 M can form, independent of the initial gas metallicity (Davies et al. 2011). The most promising stellar clusters for this scenario have a velocity dispersion in the range 40 100 km s 1 (Miller & Davies 2012).
Direct Collapse black holes
Another formation scenario is the direct collapse of a protogalactic gas cloud, which yields black hole seed masses of 104 106M (Rees 1984; Loeb & Rasio 1994; Bromm & Loeb 2003a; Koushiappas et al. 2004; Begelman et al. 2006; Lodato & Natarajan 2006; Volonteri 2010; Shang et al. 2010; Schleicher et al. 2010; Choi et al. 2013; Latif et al. 2013b; Regan et al. 2014a; Latif et al. 2014; Sugimura et al. 2014; Visbal et al. 2014a; Agarwal et al. 2014; Latif et al. 2015; Becerra et al. 2015; Hartwig et al. 2015c; Latif et al. 2016c; Johnson & Dijkstra 2017). To form such a massive seed, a high mass inflow rate of & 0:1 M yr 1 is required (Begelman 2010; Hosokawa et al. 2012a, 2013; Schleicher et al. 2013; Ferrara et al. 2014; Latif & Volonteri 2015). Suﬃcient conditions for such high mass inflow rates are provided in halos with Tvir > 104 K in which gas fragmentation and star formation are suppressed during the collapse (Latif et al. 2013b). To avoid fragmentation, the gas has to be metal-free and a strong radiation background has to photodissociate molecular hydrogen, which otherwise acts as a strong coolant. Strong shocks (Inayoshi & Omukai 2012) or magnetic fields (Sethi et al. 2010; Machida & Doi 2013) can also help to prevent fragmentation in primordial gas. Under these specific conditions, the gas can only cool by atomic hydrogen and collapses monolithically to form a supermassive star (SMS), which later on forms a BH seed (Shapiro 2004; Begelman 2010; Johnson et al. 2012; Hosokawa et al. 2012a, 2013; Inayoshi et al. 2014; Inayoshi & Haiman 2014; Latif et al. 2016c) or a quasi-star, which forms a stellar mass black hole that grows by swallowing its envelope (Begelman et al. 2006, 2008; Volonteri & Begelman 2010; Ball et al. 2011; Schleicher et al. 2013). The characteristic mass of direct collapse BHs is 105 M (Latif et al. 2013c; Ferrara et al. 2014), with the exact mass depending on the accretion rate, metallicity, and LW flux. We will refer to this specific type of direct collapse as ‘direct collapse scenario’ hereafter. Based on the strength of the photodissociating radiation, the cloud either monolithically collapses close to isothermality, or is able to eﬃciently cool and to fragment. The main quantity that discriminates between these two diﬀerent collapse regimes is the flux in the Lyman–Werner (LW) bands (11:2–13:6 eV). This LW radiation is emitted by the first generations of stars and it is convenient to express the flux in units of J21 = 10 21 erg s 1 cm 2 Hz 1 sr 1 (in the following, we use this convention without explicitly writing J21). The so-called critical value Jcrit sets the threshold above which a halo with Tvir > 104 K can directly collapse to a SMBH seed. Below this value, the gas is susceptible to fragmentation due to eﬃcient H2 cooling and the mass infall rates towards the centre are generally lower. However, the values for Jcrit quoted in the literature span several orders of magnitude from Jcrit = 0:5 (Agarwal et al. 2016b) to as high as Jcrit ’ 105 (Omukai 2001; Latif et al. 2015). There are several reasons for this large scatter. First of all, the value of Jcrit is highly sensitive to the spectral shape of the incident radiation field, with softer radiation fields leading to significant smaller values of Jcrit (Shang et al. 2010; Sugimura et al. 2014; Agarwal & Khochfar 2015; Latif et al. 2015). Secondly, one-zone calculations (e.g. Omukai 2001) tend to yield lower values of Jcrit than determinations made using 3D numerical simulations. This is a consequence of the fact that Jcrit depends to some extent on the details of the dynamical evolution of the gas, which are only approximately captured by one-zone calculations. This dependence on the gas dynamics also leads to Jcrit varying by a factor of a few from halo to halo (Shang et al. 2010; Latif et al. 2014). Although there seems not to be one universal value of Jcrit (Agarwal et al. 2016b), it is convenient to use this artificial threshold as a quantification of the direct collapse scenario to test the relevance of diﬀerent physical processes. Once a process significantly aﬀects the value of Jcrit, it is very likely that it plays an important role in the formation of SMSs and SMBH seeds.
We note that, besides the presented direct collapse scenario, other gas-dynamical channels to form massive BHs are possible. The high mass infall rate could also be achieved by low angular momentum material, although an additional mechanism of eﬃcient angular momentum transport is still required (Eisenstein & Loeb 1995; Koushiappas et al. 2004). Another possibility are local or global gas instabilities that could transport angular momentum outwards and gas inwards (Shlosman et al. 1989; Begelman et al. 2006). The “bar-within-bars” mechanism is triggered by global tidal torques and amplified by subsequently resulting local hydrodynamical torques. It can redistribute gas on a dynamical time scale and act over a large dynamical range from galactic scales to the centre.
Mass accretion and AGN feedback
To explain the observed SMBHs at z > 6 the BH seeds need to grow in mass by five to eight orders of magnitude in a few hundred million years. This poses two problems for the mass growth rate of BHs: first, the gas has to be transported from galactic scales to the central BH and lose its angular momentum to be finally accreted. Second, the mass growth of massive BHs is unavoidably connected to strong radiative feedback from the accretion disc. In Chapter 6, we focus on AGN-driven winds as a likely form of feedback and motivate the main observations and the basic theory in this section.
The accretion discs around SMBHs are nature’s most eﬃcient engines to convert gravitational energy of the infalling gas into radiation. Therefore, these extreme sources of energy are expected to suppress further gas accretion and to regulate the galactic gas content and star formation via ionising, thermal, and mechanical feedback. AGN feedback may also self-regulate the growth of the central SMBH and its inclusion in models of galaxy formation improves the match between the simulated and the observed galaxy luminosity function for massive galaxies (Bower et al. 2006; Croton et al. 2006; Somerville et al.
Table of contents :
1.2 Cosmological structure formation
1.2.1 Cosmological framework
1.2.2 Thermal and chemical evolution
1.2.3 Jeans analysis
1.2.4 Non-linear structure formation
1.3 Formation of the first stars
1.3.1 Primordial chemistry
1.3.2 Collapse of a primordial cloud
1.3.4 Evolution of protostars
1.3.5 Observational signatures of the first stars
1.4 Formation of the first supermassive black holes
1.4.2 Seed formation scenarios
1.4.3 Mass accretion and AGN feedback
2 Improving H2 Self-Shielding
2.2.1 Moving-mesh code AREPO
2.2.2 Initial conditions
2.2.4 H2 self-shielding
2.3.1 Determination of Jcrit
2.3.2 Differences in the H2 self-shielding
2.3.3 Impossibility of a simple correction factor
2.3.4 Effect of damping wings
2.3.5 Mass infall rate
2.4.1 Stellar spectrum
3 Gravitational waves from the remnants of the first stars
3.2.1 Self-consistent Pop III star formation
3.2.2 Binary sampling and evolution
4 Exploring the nature of the Lyman- emitter CR7
4.2 Observational constraints
4.3.1 Fiducial model
4.3.2 Models of Pop III star formation
4.3.3 Pop III remnant black hole
4.3.4 Direct collapse black hole
4.3.5 Determination of the metal tax
4.4.1 Cosmologically representative models of primordial star formation
4.4.2 Alternative scenarios of primordial star formation
4.4.3 Pop III remnant black holes
4.4.4 Direct collapse black hole
4.4.5 Mass of metal-poor gas
4.4.6 Comparison of scenarios
4.6.1 New observations of CR7
5 Statistical predictions for the first black holes
5.1 Observational constraints
5.2 Probability for stellar mass seed black holes
5.3 Probability for the direct collapse scenario
5.3.1 Atomic cooling halos
5.3.2 Pristine gas
5.3.3 Photodissociating radiation
5.3.4 Tidal and ram pressure stripping
5.3.5 Density of direct collapse seed black holes
6 2D analytical model for AGN-driven outflows in galaxy discs
6.2.1 Galaxy model
6.2.2 Quantifying the outflow
6.2.3 Shock acceleration
6.2.4 Cooling and heating
6.2.5 Mechanical luminosity and initial wind velocity
6.2.6 Momentum- to energy-driven transition
6.2.7 Validation of the model: comparison to 1D solution
6.3.1 Standard case (fiducial parameters)
6.3.2 Parameter study
6.3.3 Momentum-driven outflows for AGN luminosities above 1043 erg/s
6.3.4 Gas ejection perpendicular to the disc
6.3.5 Comparison to spherical case
6.4.1 Advantage of 2D
6.4.2 Outflow efficiency as a function of the AGN luminosity
6.4.3 Driving mechanism
6.5 Summary and conclusion
7.2 Perspectives and future projects
7.2.1 Semi-analytical model of self-regulated BH growth
7.2.2 Constraining the nature of the first stars with Galactic archaeology
7.2.3 Bayesian meta–analysis of the Pop III IMF