Formation of the First Black Holes

Muhammad Latif, Dominik Schleicher

• Publisher: WSPC
• Year: 2019
• ISBN:
  9789813227965
• Topic: Naturwissenschaften
• Subtopic: Astronomie & Astrophysik

The formation of the first supermassive black holes is one of the main open questions in our understanding of high-redshift structure formation. In this book, we aim to provide a summary of state-of-the-art modern research on this topic, exploring the formation of massive black holes from a fluid-dynamical, stellar-dynamical and chemical perspective. The book thus presents a solid theoretical foundation, a comparison with current observations and future observational perspectives with upcoming missions such as the Square Kilometre Array, the European Extremely Large Telescope, the Euclid satellite as well as possible detections via gravitational waves.

Contents:

• Astrophysical Black Holes (Pedro R Capelo)
• Formation and Evolution of the Cosmic Large-Scale Structure (Björn M Schäfer)
• Thermodynamics and Chemistry of the Early Universe (Stefano Bovino & Daniele Galli)
• Formation of the First Stars (Ralf Klessen)
• Black Hole Formation via Gas-Dynamical Processes (Muhammad A Latif)
• Primordial Gas Collapse in the Presence of Radiation: Direct Collapse Black Hole or Population III Star? (Bhaskar Agarwal)
• Black Hole Formation in the First Stellar Clusters (Harley Katz)
• Evolution and Final Fates of Rapidly Accreting Supermassive Stars (Takashi Hosokawa)
• Statistical Predictions for the First Black Holes (Tilman Hartwig)
• Growth and Feedback from the First Black Holes (John H Wise)
• Super-Eddington Accretion; Flow Regimes and Conditions in High-Z Galaxies (Lucio Mayer)
• Current Observational Status (Dominik R G Schleicher)
• Probing the Formation of the Seeds of Supermassive Black Holes with Gravitational Waves (Monica Colpi)
• Prospects for Detecting the First Black Holes with the Next Generation of Telescopes (Mark Dijkstra)

Readership: Graduate students and professionals in cosmology.Black Holes;Cosmology;Early Universe0 Key Features:

• Provides the first and foremost overview on the formation mechanisms of the first supermassive black holes
• Includes a detailed treatment of the many physical processes that are relevant for the formation of the first black holes, including chemical and thermodynamical aspects as well as gas and stellar dynamics
• Provides an important outlook on future observational perspectives from both space- and ground-based missions, including the Square Kilometer Array, the European Extremely Large Telescope and the Euclid satellite, as well as the future prospects from the detection of gravitational waves

Chapter 1

Astrophysical black holes

Pedro R. Capelo

Center for Theoretical Astrophysics and Cosmology
Institute for Computational Science
University of Zurich
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
[email protected]

In this chapter, we introduce the concept of a black hole (BH) and recount the initial theoretical predictions. We then review the possible types of BHs in nature, from primordial, to stellar-mass, to supermassive BHs. Finally, we focus on the latter category and on their intricate relation with their host galaxies.

1.The concept of a black hole and the first predictions

All objects in the Universe are gravitational “holes”. Given the mass and size of a system, there is always a finite speed that needs to be overcome in order to escape its gravitational well. This is the well-known Newtonian concept of the escape velocity, which can be derived by equating the sum of the kinetic and gravitational potential energy of a particle at the surface of the object to zero, and applies to all matter — from apples to planets, from stars to galaxies. Such a simple calculation was performed by Michell (1784) — see also Laplace (1796) — when considering light particles with a finite mass and speed, to speculate on the existence of objects (so-called “dark stars”) from which light could not escape. These objects must therefore have a radius smaller than

where M_• is the mass of the object, G is the gravitational constant, c is the speed of light in vacuum, and the subscript “Schw” refers to Schwarzschild, for reasons that will become apparent below.

The key feature of these objects is not their mass nor their size, but their compactness C, defined as the ratio between the value given by Equation (1) and the actual size of the system. Planets, stars, and galaxies have values of C of the order of 10⁻⁹–10⁻⁶, whereas the known stellar compact objects — white dwarfs and neutron stars — have C ~ 10⁻⁴–10⁻¹. Only when C reaches 1 do we have a system in which gravity is strong enough that nothing can escape (assuming that c cannot be overcome).

Michell’s remarkable intellectual achievement, made using simple Newtonian physics (and assuming the corpuscular theory of light), was confirmed a little more than a century later, shortly after the publication of the theory of general relativity (GR; Einstein, 1915, 1916), when the first exact vacuum solution to Einstein’s field equations was found by Schwarzschild (1916) — see also Droste (1917); Weyl (1917) — for the case of a spherical, non-electrically charged, non-rotating system. The Schwarzschild metric can be described by (Hilbert, 1917)

where ds² is the space-time line element and the solution is written in spherical coordinates (t, r, θ, and ϕ), using the Landau–Lifshitz spacelike convention (Landau and Lifshitz, 1962; Misner et al., 1973).

Even though the right-hand side of Equation (2) diverges at both r = 0 and r_Schw, only the former is a true physical singularity (i.e. the Riemann curvature tensor is infinite only at r = 0), with the space-time being nonsingular at the so-called Schwarzschild radius.^a However, the Schwarzschild radius is of fundamental importance, as the radial coordinate of a particle travelling towards the centre changes from spacelike to timelike when crossing r_Schw, meaning that the only possible future of that particle is the singularity.^b Meanwhile, an external static observer will never observe such a boundary (or event horizon) crossing, as the observed time will be infinite (even though the proper time of the particle is finite). Moreover, any radiation sent from such particle and reaching any external observer will be infinitely redshifted. In other words, a photon sent from r_Schw would need infinite energy to reach the observer, effectively making the space-time region within the event horizon causally disconnected from the rest of the Universe. For these reasons, objects with an event horizon (i.e. with C = 1) are called BHs.

After the publication of the Schwarzschild solution, other exact solutions to Einstein’s field equations were found, in the case of electrically charged (Reissner, 1916; Nordström, 1918), rotating (Kerr, 1963), and rotating, electrically charged BHs (Newman et al., 1965). One peculiarity of BHs is that they are extremely simple. In fact, they can be described at most by three parameters: mass, spin (i.e. angular momentum), and electric charge — this is referred to as the ‘no hair’ theorem (see, e.g. Israel, 1967). Moreover, in typical astrophysical environments, it is believed that electrically charged BHs cannot exist, as any existing electric charge would be quickly cancelled by the charges in the surrounding plasma (or by spontaneous production of pairs of oppositely charged particles; see, e.g. Gibbons, 1975; Blandford and Znajek, 1977). For this reason, the most complete description for an astrophysical BH is the Kerr metric, which depends only on mass and spin.

Using the Boyer–Lindquist (Boyer and Lindquist, 1967) coordinates, one can write down the Kerr metric such that the radial coordinate of the event horizon^c is

where a ≡ c

/(

) is the BH spin and

its angular momentum. The value of |a| can vary between 0 (recovering the Schwarzschild BH) and 1 (for a maximally spinning BH^d), and its sign depends on the particle orbit we consider: 1 and −1 for corotating and counterrotating orbits (with respect to the BH angular momentum), respectively. If we take a BH with |a| = 1, the radial coordinate of the event horizon is half of that of a Schwarzschild BH (i.e. r_Kerr = r_Schw/2).

More importantly, the orbits of massive particles around Kerr BHs vary depending on the value of a. When a = −1, 0, and 1, the innermost stable circular orbit (ISCO) a massive particle can have is of a radius r_ISCO = 4.5, 3, and 0.5 × r_Schw, respectively. For r < r_ISCO, a particle can only spiral inwards (or outwards, if it has enough velocity to do so) and cannot maintain a stable circular orbit.^e

The position of the ISCO has significant consequences on how much gravitational energy can be extracted from the gas in the vicinity of the BH (via accretion processes; e.g. Shakura and Sunyaev, 1973; Blandford and Payne, 1982), as the energy lost by particles increases as the distance from the BH decreases.^f

In other words, the radiative efficiency

(i.e. how much of the rest energy of the accreting particle is released), given by

where L and

_•accr are the accretion power (or luminosity) and the mass accretion rate, respectively, depends on the BH spin (e.g. in a Novikov–Thorne disc, 0.06

0.42 for 0 ≤ a ≤ 1; Novikov and Thorne, 1973).

Indeed, astrophysical accreting BHs are believed to be normally surrounded by accretion discs (e.g. Shakura and Sunyaev, 1973), in which matter diffe...