Rates of tidal disruption of stars by massive central black holes

There is strong evidence for some kind of massive dark object in the centres of many galaxy bulges. The detection of flares from tidally disrupted stars could confirm that these objects are black holes (BHs). Here we present calculations of the stellar disruption rates in detailed dynamical models of real galaxies, taking into account the refilling of the loss cone of stars on disruptable orbits by two-body relaxation and tidal forces in non-spherical galaxies. The highest disruption rates (one star per 10⁴ yr) occur in faint ( L ≲10¹⁰ L_⊙) galaxies, which have steep central density cusps. More luminous galaxies are less dense and have much longer relaxation times and more massive BHs. Dwarf stars in such galaxies are swallowed whole by the BH and hence do not emit flares; giant stars could produce flares as often as every 10⁵ yr, although the rate depends sensitively on the shape of the stellar distribution function. We discuss the possibility of detecting disruption flares in current supernova searches. The total mass of stars consumed over the lifetime of the galaxy is of the order of 10⁶ M_⊙, independent of galaxy luminosity; thus, disrupted stars may contribute significantly to the present BH mass in galaxies fainter than ∼10⁹ L_⊙.     

Gravitational waves from scattering of stellar-mass black holes in galactic nuclei

Stellar-mass black holes (BHs) are expected to segregate and form a steep density cusp around supermassive black holes (SMBHs) in galactic nuclei. We follow the evolution of a multimass system of BHs and stars by numerically integrating the Fokker–Planck energy diffusion equations for a variety of BH mass distributions. We find that the BHs 'self-segregate', and that the rarest, most massive BHs dominate the scattering rate closest to the SMBH  (≲10⁻¹ pc)  . BH–BH binaries form out of gravitational wave emission during BH encounters. We find that the expected rate of BH coalescence events detectable by Advanced LIGO is  ∼1–10² yr⁻¹  , depending on the initial mass function of stars in galactic nuclei and the mass of the most massive BHs. We find that the actual merger rate is likely ∼10 times larger than this due to the intrinsic scatter of stellar densities in many different galaxies. The BH binaries that form this way in galactic nuclei have significant eccentricities as they enter the LIGO band (90 per cent with   e > 0.9  ), and are therefore distinguishable from other binaries, which circularize before becoming detectable. We also show that eccentric mergers can be detected to larger distances and greater BH masses than circular mergers, up to  ∼700 M_⊙  . Future ground-based gravitational wave observatories will be able to constrain both the mass function of BHs and stars in galactic nuclei.     

Evolution of massive binary black holes

Since many or most galaxies have central massive black holes (BHs), mergers of galaxies can form massive binary black holes (BBHs). In this paper we study the evolution of massive BBHs in realistic galaxy models, using a generalization of techniques used to study tidal disruption rates around massive BHs. The evolution of BBHs depends on BH mass ratio and host galaxy type. BBHs with very low mass ratios (say, ≲0.001) are hardly ever formed by mergers of galaxies, because the dynamical friction time-scale is too long for the smaller BH to sink into the galactic centre within a Hubble time. BBHs with moderate mass ratios are most likely to form and survive in spherical or nearly spherical galaxies and in high-luminosity or high-dispersion galaxies; they are most likely to have merged in low-dispersion galaxies (line-of-sight velocity dispersion ≲90 km s⁻¹) or in highly flattened or triaxial galaxies.
The semimajor axes and orbital periods of surviving BBHs are generally in the range  10^-3–10 pc  and  10–10⁵ yr;  they are also larger in high-dispersion galaxies than in low-dispersion galaxies, larger in nearly spherical galaxies than in highly flattened or triaxial galaxies, and larger for BBHs with equal masses than for BBHs with unequal masses. The orbital velocities of surviving BBHs are generally in the range  10²–10⁴ km s^-1  . The methods of detecting surviving BBHs are also discussed.
If no evidence of BBHs is found in AGNs, this may be either because gas plays a major role in BBH orbital decay or because nuclear activity switches on soon after a galaxy merger, and ends before the smaller BH has had time to spiral to the centre of the galaxy.     

Impact of cosmic rays on Population III star formation

We explore the implications of a possible cosmic-ray (CR) background generated during the first supernova explosions that end the brief lives of massive Population III stars. We show that such a CR background could have significantly influenced the cooling and collapse of primordial gas clouds in minihaloes around redshifts of   z ∼ 15–20  , provided the CR flux was sufficient to yield an ionization rate greater than about 10⁻¹⁹ s⁻¹ near the centre of the minihalo. The presence of CRs with energies  ≲10⁷  eV would indirectly enhance the molecular cooling in these regions, and we estimate that the resulting lower temperatures in these minihaloes would yield a characteristic stellar mass as low as  ∼10 M_⊙  . CRs have a less-pronounced effect on the cooling and collapse of primordial gas clouds inside more massive dark matter haloes with virial masses  ≳10⁸ M_⊙  at the later stages of cosmological structure formation around   z ∼ 10–15  . In these clouds, even without CR flux the molecular abundance is already sufficient to allow cooling to the floor set by the temperature of the cosmic microwave background.     

Radiation from early black holes – I. Effects on the neutral intergalactic medium

In the pre-reionization Universe, the regions of the intergalactic medium (IGM) which are far from luminous sources are the last to undergo reionization. Until then, they should be scarcely affected by stellar radiation; instead, the X-ray emission from an early black hole (BH) population can have much larger influence. We investigate the effects of such emission, looking at a number of BH model populations (differing for the cosmological density evolution of BHs, the BH properties, and the spectral energy distribution of the BH emission). We find that BH radiation can easily heat the IGM to  10³–10⁴ K  , while achieving partial ionization. The most interesting consequence of this heating is that BHs are expected to induce a 21-cm signal (  δ T _b∼ 20–30 mK  at   z ≲ 12  ) which should be observable with forthcoming experiments (e.g. LOFAR). We also find that at   z ≲ 10  BH emission strongly increases the critical mass separating star-forming and non-star-forming haloes.     

Star clusters around recoiled black holes in the Milky Way halo

Gravitational wave emission by coalescing black holes (BHs) kicks the remnant BH with a typical velocity of hundreds of  km s⁻¹  . This velocity is sufficiently large to remove the remnant BH from a low-mass galaxy but is below the escape velocity from the Milky Way (MW) galaxy. If central BHs were common in the galactic building blocks that merged to make the MW, then numerous BHs that were kicked out of low-mass galaxies should be freely floating in the MW halo today. We use a large statistical sample of possible merger tree histories for the MW to estimate the expected number of recoiled BH remnants present in the MW halo today. We find that hundreds of BHs should remain bound to the MW halo after leaving their parent low-mass galaxies. Each BH carries a compact cluster of old stars that populated the core of its original host galaxy. Using the time-dependent Fokker–Planck equation, we find that the present-day clusters are  ≲1 pc  in size, and their central bright regions should be unresolved in most existing sky surveys. These compact systems are distinguishable from globular clusters by their internal (Keplerian) velocity dispersion greater than 100 km s⁻¹ and their high mass-to-light ratio owing to the central BH. An observational discovery of this relic population of star clusters in the MW halo would constrain the formation history of the MW and the dynamics of BH mergers in the early Universe. A similar population should exist around other galaxies and may potentially be detectable in M31 and M33.     

High-redshift galaxies, their active nuclei and central black holes

We demonstrate that the luminosity function of the recently detected population of actively star-forming galaxies at redshift z  = 3 and the B -band luminosity function of quasi-stellar objects (QSOs) at the same redshift can both be matched with the mass function of dark matter haloes predicted by standard variants of hierarchical cosmogonies for lifetimes of optically bright QSOs anywhere in the range 10⁶ to 10⁸ yr. There is a strong correlation between the lifetime and the required degree of non-linearity in the relation between black hole and halo mass. We suggest that the mass of supermassive black holes may be limited by the back-reaction of the emitted energy on the accretion flow in a self-gravitating disc. This would imply a relation of black hole to halo mass of the form M _bh ∝  v ⁵_halo ∝  M ^5/3_halo and a typical duration of the optically bright QSO phase of a few times 10⁷ yr. The high integrated mass density of black holes inferred from recent black hole mass estimates in nearby galaxies may indicate that the overall efficiency of supermassive black holes for producing blue light is smaller than previously assumed. We discuss three possible accretion modes with low optical emission efficiency: (i) accretion at far above the Eddington rate, (ii) accretion obscured by dust, and (iii) accretion below the critical rate leading to an advection-dominated accretion flow lasting for a Hubble time. We further argue that accretion with low optical efficiency might be closely related to the origin of the hard X-ray background and that the ionizing background might be progressively dominated by stars rather than QSOs at higher redshift.     

Imprints of recoiling massive black holes on the hot gas of early-type galaxies

Anisotropic gravitational radiation from a coalescing black hole (BH) binary is known to impart recoil velocities of up to  ∼1000 km s⁻¹  to the remnant BH. In this context, we study the motion of a recoiling BH inside a galaxy modelled as a Hernquist sphere, and the signature that the hole imprints on the hot gas, using N -body/smoothed particle hydrodynamics simulations. Ejection of the BH results in a sudden expansion of the gas ending with the formation of a gaseous core, similarly to what is seen for the stars. A cometary tail of particles bound to the BH is initially released along its trail. As the BH moves on a return orbit, a nearly spherical swarm of hot gaseous particles forms at every apocentre: this feature can live up to ≈10⁸ years. If the recoil velocity exceeds the sound speed initially, the BH shocks the gas in the form of a Mach cone in density near each supersonic pericentric passage. We find that the X-ray fingerprint of a recoiling BH can be detected in Chandra X-ray maps out to a distance of Virgo. For exceptionally massive BHs, the Mach cone and the wakes can be observed out to a few hundred of milliparsec. The detection of the Mach cone is of twofold importance as it can be a probe of high-velocity recoils, and an assessment of the scatter of the   M _BH− M _bulge  relation at large BH masses.     

The masses of black holes in the nuclei of spirals

We use the innermost kinematics of spirals to investigate whether these galaxies could host the massive black hole remnants that once powered the quasi-stellar object (QSO) phenomenon. Hundreds of rotation curves of early- and late-type spirals are used to place upper limits on the central black hole (BH) masses. We find that (i) in late-type spirals, the central massive dark objects (MDOs) are about 10–100 times smaller than the MDOs detected in ellipticals, and (ii) in early-type spirals, the central bodies are likely to be in the same mass range as the elliptical MDOs. As a consequence, the contribution to the QSO/active galactic nuclei (AGN) phenomenon by the BH remnants eventually hosted in spirals is negligible: ρ _BH(Sb–Im)<6×10⁴ M_⊙ Mpc⁻³ . We find several hints that the MDO mass versus bulge mass relationship is significantly steeper in spirals than in ellipticals, although the very issue of the existence of such a relation for late Hubble type objects remains open. The upper limits on the masses of the BHs resident in late-type spirals are stringent: M _BH10⁶–10⁷ M_⊙, indicating that only low-luminosity activity could possibly have occurred in these objects .     

Quantifying the coexistence of massive black holes and dense nuclear star clusters

In large spheroidal stellar systems, such as elliptical galaxies, one invariably finds a  10⁶–10⁹ M_⊙  supermassive black hole at their centre. In contrast, within dwarf elliptical galaxies one predominantly observes a  10⁵–10⁷ M_⊙  nuclear star cluster. To date, few galaxies have been found with both types of nuclei coexisting and even less have had the masses determined for both central components. Here, we identify one dozen galaxies housing nuclear star clusters and supermassive black holes whose masses have been measured. This doubles the known number of such hermaphrodite nuclei – which are expected to be fruitful sources of gravitational radiation. Over the host spheroid (stellar) mass range  10⁸–10¹¹ M_⊙  , we find that a galaxy's nucleus-to-spheroid (baryon) mass ratio is not a constant value but decreases from a few per cent to ∼0.3 per cent such that  log[( M _BH+ M _NC)/ M _sph]=−(0.39 ± 0.07) log[ M _sph/10¹⁰ M_⊙]− (2.18 ± 0.07)  . Once dry merging commences and the nuclear star clusters disappear, this ratio is expected to become a constant value.
As a byproduct of our investigation, we have found that the projected flux from resolved nuclear star clusters is well approximated with Sérsic functions having a range of indices from ∼0.5 to ∼3, the latter index describing the Milky Way's nuclear star cluster.     

The cooling of shock-compressed primordial gas

We find that at redshifts   z ≳ 10, HD  line cooling allows strongly shocked primordial gas to cool to the temperature of the cosmic microwave background (CMB). This temperature is the minimum value attainable via radiative cooling. Provided that the abundance of HD, normalized to the total number density, exceeds a critical level of  ∼10⁻⁸  , the CMB temperature floor is reached in a time which is short in comparison to the Hubble time. We estimate the characteristic masses of stars formed out of shocked primordial gas in the wake of the first supernovae, and resulting from the virialization of dark matter haloes during hierarchical structure formation to be  ∼10 M_⊙  . In addition, we show that cooling by HD enables the primordial gas in relic H  ii regions to cool to temperatures considerably lower than those reached via H₂ cooling alone. We confirm that HD cooling is unimportant in cases where the primordial gas does not go through an ionized phase, as in the formation process of the very first stars in   z ≳ 20  minihaloes of mass  ∼10⁶ M_⊙  .     

The first galaxies: assembly, cooling and the onset of turbulence

We investigate the properties of the first galaxies at   z ≳ 10  with highly resolved numerical simulations, starting from cosmological initial conditions and taking into account all relevant primordial chemistry and cooling. A first galaxy is characterized by the onset of atomic hydrogen cooling, once the virial temperature exceeds  ≃10⁴ K  , and its ability to retain photoheated gas. We follow the complex accretion and star formation history of a  ≃5 × 10⁷ M_⊙  system by means of a detailed merger tree and derive an upper limit on the number of Population III (Pop III) stars formed prior to its assembly. We investigate the thermal and chemical evolution of infalling gas and find that partial ionization at temperatures  ≳10⁴ K  catalyses the formation of  H₂  and hydrogen deuteride, allowing the gas to cool to the temperature of the cosmic microwave background. Depending on the strength of radiative and chemical feedback, primordial star formation might be dominated by intermediate-mass Pop III stars formed during the assembly of the first galaxies. Accretion on to the nascent galaxy begins with hot accretion, where gas is accreted directly from the intergalactic medium and shock heated to the virial temperature, but is quickly accompanied by a phase of cold accretion, where the gas cools in filaments before flowing into the parent halo with high velocities. The latter drives supersonic turbulence at the centre of the galaxy and could lead to very efficient chemical mixing. The onset of turbulence in the first galaxies thus likely marks the transition to Pop II star formation.     

Stellar-mass black hole binaries as ultraluminous X-ray sources

Ultraluminous X-ray sources (ULXs) with   L _x > 10³⁹ erg s⁻¹  have been discovered in great numbers in external galaxies with ROSAT , Chandra and XMM-Newton . The central question regarding this important class of sources is whether they represent an extension in the luminosity function of binary X-ray sources containing neutron stars and stellar-mass black holes (BHs), or a new class of objects, e.g. systems containing intermediate-mass BHs  (100–1000 M_⊙)  . We have carried out a theoretical study to test whether a large fraction of the ULXs, especially those in galaxies with recent star formation activity, can be explained with binary systems containing stellar-mass BHs. To this end, we have applied a unique set of binary evolution models for BH X-ray binaries, coupled to a binary population synthesis code, to model the ULXs observed in external galaxies. We find that for donor stars with initial masses  ≳10 M_⊙  the mass transfer driven by the normal nuclear evolution of the donor star is sufficient to potentially power most ULXs. This is the case during core hydrogen burning and, to an even more pronounced degree, while the donor star ascends the giant branch, although the latter phases last only ∼5 per cent of the main-sequence phase. We show that with only a modest violation of the Eddington limit, e.g. a factor of ∼10, both the numbers and properties of the majority of the ULXs can be reproduced. One of our conclusions is that if stellar-mass BH binaries account for a significant fraction of ULXs in star-forming galaxies, then the rate of formation of such systems is  ∼3 × 10⁻⁷ yr⁻¹  normalized to a core-collapse supernova rate of 0.01 yr⁻¹.     

A top-heavy stellar initial mass function in starbursts as an explanation for the high mass-to-light ratios of ultra-compact dwarf galaxies

It has been recently shown that the dynamical V -band mass-to-light ratios of compact stellar systems with masses from 10⁶ to  10⁸ M_⊙  are not consistent with the predictions from simple stellar population models. Top-heavy stellar initial mass functions (IMFs) in these so-called ultra-compact dwarf galaxies (UCDs) offer an attractive explanation for this finding, the stellar remnants and retained stellar envelopes providing the unseen mass. We therefore construct a model which quantifies by how much the IMFs of UCDs would have to deviate in the intermediate- and high-mass range from the canonical IMF in order to account for the enhanced   M / L_V   ratio of the UCDs. The deduced high-mass IMF in the UCDs depends on the age of the UCDs and the number of faint products of stellar evolution retained by them. Assuming that the IMF in the UCDs is a three-part power law equal to the canonical IMF in the low-mass range and taking 20 per cent as a plausible choice for the fraction of the remnants of high-mass stars retained by UCDs, the model suggests the exponent of the high-mass IMF to be ≈1.6 if the UCDs are  13 Gyr  old (i.e. almost as old as the Universe) or ≈1.0 if the UCDs are  7 Gyr  old, in contrast to 2.3 for the Salpeter–Massey IMF. If the IMF was as top heavy as suggested here, the stability of the UCDs might have been threatened by heavy mass loss induced by the radiation and evolution of massive stars. The central densities of UCDs must have been in the range  10⁶ to 10⁷ M_⊙ pc⁻³  when they formed with star formation rates of  10 to 100 M_⊙ yr⁻¹  .     

Occurrence of metal-free galaxies in the early Universe

The character of the first galaxies at redshifts z ≳ 10 strongly depends on their level of pre-enrichment, which is in turn determined by the rate of primordial star formation prior to their assembly. In order for the first galaxies to remain metal-free, star formation in minihaloes must be highly suppressed, most likely by H₂-dissociating Lyman–Werner (LW) radiation. We show that the build-up of such a strong LW background is hindered by two effects. First, the level of the LW background is self-regulated, being produced by the Population III (Pop III) star formation which it, in turn, suppresses. Secondly, the high opacity to LW photons which is built up in the relic H  ii regions left by the first stars acts to diminish the global LW background. Accounting for a self-regulated LW background, we estimate a lower limit for the rate of Pop III star formation in minihaloes at z ≳ 15. Further, we simulate the formation of a 'first galaxy' with virial temperature   T _vir≳ 10⁴ K  and total mass  ≳10⁸ M_⊙  at z ≳ 10, and find that complete suppression of previous Pop III star formation is unlikely, with stars of  ≳100 M_⊙  (Pop III.1) and  ≳10 M_⊙  (Pop III.2) likely forming. Finally, we discuss the implications of these results for the nature of the first galaxies, which may be observed by future missions such as the James Webb Space Telescope .     

On the possibility of observing H2 emission from primordial molecular cloud kernels

We study the prospects for observing H₂ emission during the assembly of primordial molecular cloud kernels. The primordial molecular cloud cores, which resemble those at the present epoch, can emerge around  1+ z ∼20  according to recent numerical simulations. The kernels form inside the cores, and the first stars will appear inside the kernels. A kernel typically contracts to form one of the first generation stars with an accretion rate that is as large as ∼0.01 M_⊙ yr⁻¹. This occurs owing to the primordial abundances, which result in a kernel temperature of order 1000 K, and the collapsing kernel emits H₂ line radiation at a rate ∼10³⁵ erg s⁻¹. Predominantly   J =5-3   ( v =0)  rotational emission of H₂ is expected. At redshift  1+ z ∼20  , the expected flux is ∼0.01 μJy for a single kernel. While an individual object is not observable by any facilities available in the near future, the expected assembly of primordial star clusters on subgalactic scales can result in fluxes at the sub-mJy level. This is marginally observable with ASTRO-F and ALMA. We also examine the rotational   J =2-0   ( v =0)  and vibrational   δv =1  emission lines. The former may possibly be detectable with ALMA.     

On the first generation of stars

We argue that the first stars may have spanned the conventional mass range rather than be identified with the very massive objects  (∼100–10³ M_⊙)  favoured by numerical simulations. Specifically, we find that magnetic field generation processes acting in the first protostellar systems suffice to produce fields that exceed the threshold for magneto-rotational instability (MRI) to operate, and thereby allow the MRI dynamo to generate equipartition-amplitude magnetic fields on protostellar mass scales below  ∼50 M_⊙  . Such fields allow primordial star formation to occur at essentially any metallicity by regulating angular momentum transfer, fragmentation, accretion and feedback in much the same way as occurs in conventional molecular clouds.     

Cosmic rays and the primordial gas

One of the most-outstanding problems in the gravitational collapse scenario of early structure formation is the cooling of primordial gas to allow for small-mass objects to form. As the neutral primordial gas is a poor radiator at temperatures   T ≤ 10⁴ K  , molecular hydrogen is needed for further cooling down to temperatures   T ∼ 100 K  . The formation of molecular hydrogen is catalyzed by the presence of free electrons, which could be provided by the ionization due to an early population of cosmic rays (CRs). In order to investigate this possibility, we developed a code to study the effects of ionizing CRs on the thermal and chemical evolution of primordial gas. We found that CRs can provide enough free electrons needed for the formation of molecular hydrogen, and therefore can increase the cooling ability of such primordial gas under following conditions. A dissociating photon flux with   F < 10⁻¹⁸ erg cm⁻² Hz⁻¹ s⁻¹  , initial temperature of the gas  ∼10³ K  , total gas number densities   n ≥ 1 cm⁻³  , and cosmic-ray sources with     .     

The tidal disruption rate in dense galactic cusps containing a supermassive binary black hole

We consider the problem of tidal disruption of stars in the centre of a galaxy containing a supermassive binary black hole with unequal masses. We assume that over the separation distance between the black holes, the gravitational potential is dominated by the more massive primary black hole. Also, we assume that the number density of stars is concentric with the primary black hole and has a power-law cusp. We show that the bulk of stars with a small angular-momentum component normal to the black hole binary orbit can reach a small value of total angular momentum through secular evolution in the gravitational field of the binary, and hence they can be tidally disrupted by the larger black hole. This effect is analogous to the so-called Kozai effect well known in celestial mechanics. We develop an analytical theory for the secular evolution of the stellar orbits and calculate the rate of tidal disruption. We compare our analytical theory with a simple numerical model and find very good agreement.
Our results show that for a primary black hole mass of  ∼10⁶–10⁷ M_⊙  , the black hole mass-ratio   q > 10⁻²  , cusp size ∼1 pc, the tidal disruption rate can be as large as  ∼10⁻²–1 M_⊙ yr⁻¹  . This is at least 10²–10⁴ times larger than estimated for the case of a single supermassive black hole. The duration of the phase of enhanced tidal disruption is determined by the dynamical-friction time-scale, and it is rather short: ∼10⁵ yr. The dependence of the tidal disruption rate on the mass ratio, and on the size of the cusp, is also discussed.     

Non-linear evolution of suppressed dark matter primordial power spectra

We address the degree and rapidity of generation of small-scale power over the course of structure formation in cosmologies where the primordial power spectrum is strongly suppressed beyond a given wavenumber. We first summarize the situations where one expects such suppressed power spectra and point out their diversity. We then employ an exponential cut-off, which characterizes warm dark matter (WDM) models, as a template for the shape of the cut-off and focus on damping scales ranging from 10⁶ to  10⁹  h ⁻¹ M_⊙  . Using high-resolution simulations, we show that the suppressed part of the power spectrum is quickly (re)generated and catches up with both the linear and the non-linear evolution of the unsuppressed power spectrum. From   z = 2  onwards, a power spectrum with a primordial cut-off at  10⁹  h ⁻¹ M_⊙  becomes virtually indistinguishable from an evolved cold dark matter (CDM) power spectrum. An attractor such as that described in Zaldarriaga, Scoccimarro & Hui for power spectra with different spectral indices also emerges in the case of truncated power spectra. Measurements of   z ∼ 0  non-linear power spectra at  ∼100  h ⁻¹ kpc  cannot rule out the possibility of linear power spectra damped below  ∼10⁹  h ⁻¹ M_⊙  . Therefore, WDM or scenarios with similar features should be difficult to exclude in this way.