arXiv subjects

Kohei Inayoshi

Publications and source records attributed to Kohei Inayoshi.

At least 127 records · Page 7Linked to original sources

arXiv2021

Subaru High-z Exploration of Low-Luminosity Quasars (SHELLQs) XII. Extended [C II] Structure (Merger or Outflow) in a z = 6.72 Red Quasar

We present ALMA [C II] 158 $μ$m line and far-infrared (FIR) continuum emission observations toward HSC J120505.09$-$000027.9 (J1205$-$0000) at $z = 6.72$ with the beam size of $\sim 0''.8 \times 0''.5$ (or 4.1 kpc $\times$ 2.6 kpc), the most distant red quasar known to date. Red quasars are modestly reddened by dust, and are thought to be in rapid transition from an obscured starburst to an unobscured normal quasar, driven by powerful active galactic nucleus (AGN) feedback which blows out a cocoon of interstellar medium (ISM). The FIR continuum of J1205$-$0000 is bright, with an estimated luminosity of $L_{\rm FIR} \sim 3 \times 10^{12}~L_\odot$. The [C II] line emission is extended on scales of $r \sim 5$ kpc, greater than the FIR continuum. The line profiles at the extended regions are complex and broad (FWHM $\sim 630-780$ km s$^{-1}$). Although it is not practical to identify the nature of this extended structure, possible explanations include (i) companion/merging galaxies and (ii) massive AGN-driven outflows. For the case of (i), the companions are modestly star-forming ($\sim 10~M_\odot$ yr$^{-1}$), but are not detected by our Subaru optical observations ($y_{\rm AB,5σ} = 24.4$ mag). For the case of (ii), our lower-limit to the cold neutral outflow rate is $\sim 100~M_\odot$ yr$^{-1}$. The outflow kinetic energy and momentum are both much smaller than what predicted in energy-conserving wind models, suggesting that the AGN feedback in this quasar is not capable of completely suppressing its star formation.

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arXiv2020

Radiative feedback for supermassive star formation in a massive cloud with H2 molecules in an atomic-cooling halo

Recent three-dimensional cosmological simulations of protogalaxy formation have suggested that supermassive stars (SMSs) can form in gas clouds in which H$_2$ cooling is suppressed by dynamical heating prior to the activation of atomic cooling (Wise et al. 2019), but they stopped short of the following growth of a central protostar. Here we examine whether accretion on the protostellar core in this cloud is sufficiently rapid, in the face of the radiation feedback, to produce a SMS. We perform one-dimensional radiation-hydrodynamical simulations of the hot collapsing cloud with non-equilibrium chemical reactions directly adopting the cloud properties from Wise et al. (2019) as an initial condition. We find that the stellar Lyman-Werner (LW) radiation from the SMS dissociates H$_2$ in the inner regions of the gas flow, increasing gas temperature and thermal pressure, and temporarily stopping the accretion. However, this negative feedback ceases when the self-gravity and inward ram pressure force on larger scales push the gas inward. The central protostar is unable to expand an HII region due to the high density, and grows to a mass of $\gtrsim10^5\,{{\rm M}_\odot}$. Our results suggests the successful formation of SMSs, and resulting massive ($\sim10^5\,{{\rm M}_\odot}$) remnant black holes in the clouds, but need to be confirmed in two- or three-dimensional simulations.

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arXiv2020

Super-Eddington mass growth of intermediate-mass black holes embedded in dusty circumnuclear disks

We perform the first three-dimensional radiation hydrodynamical simulations that investigate the growth of intermediate-mass BHs (IMBHs) embedded in massive self-gravitating, dusty nuclear accretion disks. We explore the dependence of mass accretion efficiency on the gas metallicity $Z$ and mass injection at super-Eddington accretion rates from the outer galactic disk $\dot{M}_{\rm in}$, and find that the central BH can be fed at rates exceeding the Eddington rate only when the dusty disk becomes sufficiently optically thick to ionizing radiation. In this case, mass outflows from the disk owing to photoevaporation is suppressed and thus a large fraction ($\gtrsim 40\%$) of the mass injection rate can feed the central BH. The conditions are expressed as $\dot{M}_{\rm in} > 2.2\times 10^{-1}~M_\odot ~{\rm yr}^{-1} (1+Z/10^{-2}~Z_\odot)^{-1}(c_{\rm s}/10~{\rm km~s}^{-1})$, where $c_{\rm s}$ is the sound speed in the gaseous disk. With increasing numerical resolution, vigorous disk fragmentation reduces the disk surface density and dynamical heating by formed clumps makes the disk thickness higher. As a result, the photoevaorative mass-loss rate rises and thus the critical injection rate increases for fixed metallicity. This process enables super-Eddington growth of BHs until the BH mass reaches $M_{\rm BH} \sim 10^{7-8}~M_\odot$, depending on the properties of the host dark-matter halo and metal-enrichment history. In the assembly of protogalaxies, seed BHs that form in overdense regions with a mass variance of 3-4$σ$ at $z\sim 15-20$ are able to undergo short periods of their rapid growth and transits into the Eddington-limited growth phase afterwards to be supermassive BHs observed at $z>6-7$.

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arXiv2020

Pulsation-driven mass loss from massive stars behind stellar mergers in metal-poor dense clusters

The recent discovery of high-redshift (z > 6) supermassive black holes (SMBH) favors the formation of massive seed BHs in protogalaxies. One possible scenario is formation of massive stars ~ 1e3-1e4 Msun via runaway stellar collisions in a dense cluster, leaving behind massive BHs without significant mass loss. We study the pulsational instability of massive stars with the zero-age main-sequence (ZAMS) mass Mzams/Msun = 300-3000 and metallicity Z/Zsun = 0-0.1, and discuss whether or not pulsation-driven mass loss prevents massive BH formation. In the MS phase, the pulsational instability excited by the epsilon-mechanism grows in ~ 1e3 yrs. As the stellar mass and metallicity increase, the mass-loss rate increases to < 1e-3 Msun/yr. In the red super-giant (RSG) phase, the instability is excited by the kappa-mechanism operating in the hydrogen ionization zone and grows more rapidly in ~ 10 yrs. The RSG mass-loss rate is almost independent of metallicity and distributes in the range of ~ 1e-3-1e-2 Msun/yr. Conducting the stellar structure calculations including feedback due to pulsation-driven winds, we find that the stellar models of Mzams/Msun = 300-3000 can leave behind remnant BHs more massive than ~ 200-1200 Msun. We conclude that massive merger products can seed monster SMBHs observed at z > 6.

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arXiv2020

Hunting for wandering massive black holes

We investigate low-density accretion flows onto massive black holes (BHs) with masses of $\gtrsim 10^5~M_\odot$ orbiting around in the outskirts of their host galaxies, performing three-dimensional hydrodynamical simulations. Those wandering BHs are populated via ejection from the galactic nuclei through multi-body BH interactions and gravitational wave recoils associated with galaxy and BH coalescences. We find that when a wandering BH is fed with hot and diffuse plasma with density fluctuations, the mass accretion rate is limited at $\sim 10-20\%$ of the canonical Bondi-Hoyle-Littleton rate owing to a wide distribution of inflowing angular momentum. We further calculate radiation spectra from radiatively inefficient accretion flows onto the wandering BH using a semi-analytical two-temperature disk model and find that the predicted spectra have a peak at the millimeter band, where the Atacama Large Millimeter/submillimeter Array (ALMA) has the highest sensitivity and spatial resolution. Millimeter observations with ALMA and future facilities such as the next generation Very Large Array (ngVLA) will enable us to hunt for a population of wandering BHs and push the detectable mass limit down to $M_\bullet \simeq 2\times10^7~M_\odot$ for massive nearby ellipticals, e.g., M87, and $M_\bullet \simeq 10^5~M_\odot$ for the Milky Way. This radiation spectral model, combined with numerical simulations, will be applied to give physical interpretations of off-nuclear BHs detected in dwarf galaxies, which may constrain BH seed formation scenarios.

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arXiv2020

Universal transition diagram from dormant to actively accreting supermassive black holes

The vast majority of supermassive black holes (SMBHs) in the local universe exhibit levels of activity much lower than those expected from gas supplying rates onto the galactic nuclei, and only a small fraction of silent SMBHs can turn into active galactic nuclei. Revisiting observational data of very nearby SMBHs whose gravitational spheres of influence are spatially reached by the Chandra X-ray satellite, we find that the level of BH activity drastically increases from the quiescent phase when the inflow rate outside of the BH influence radius is higher than 0.1% of the Eddington accretion rate. We also show that the relation between the nuclear luminosity and gas accretion rate from the BH influence radius measured from X-ray observations is well described by the universal state transition of accreting SMBHs, as predicted by recent hydrodynamical simulations with radiative cooling and BH feedback. After the state transition, young massive stars should form naturally in the nucleus, as observed in the case of the nearest SMBH, Sagittarius A$^\ast$, which is currently quiescent but was recently active.

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arXiv2020

Hyper-Eddington accretion flows onto black holes accompanied by powerful outflows

We perform two-dimensional radiation hydrodynamical simulations of accretion flows onto black holes (BHs) at the nuclei of protogalaxies, and study the impact of mechanical and radiative feedback on rapid growth of BHs. The outflows deposit mass, momentum and energy into the surrounding medium and prevent mass accretion onto the BH, resulting in the reduction of radiative output. We find that when the BH is embedded in a dense gas core, ionizing radiation attenuated by inefficient BH feeding owing to mechanical feedback hardly affects the gas dynamics at the BH gravitational sphere of influence, from which intense inflows of neutral gas occur at rates substantially exceeding the Eddington limit without impeded by photoionization and heating. Since mechanical power of outflows driven by the rapidly accreting BH is sufficiently strong, bipolar outflows completely evacuate the surrounding gas in the polar region but mass inflows through the equatorial region maintain the BH accretion rate as high as $\sim 300-10^3~\dot{M}_{\rm Edd}$, which is reduced by one order of magnitude from those with radiative feedback alone. Furthermore, we find that the critical gas density required for rapid accretion is lower by a factor of $\sim 3$ nearly independently of BH mass, when mechanical feedback is considered. By studying the dependence on outflow model parameters (e.g., opening angle, mass-loading degree into outflows, velocity), we conclude that contrary to naive expectation, the existence of stronger outflow leads to the transition to rapid accretion phases more efficiently. Rapidly growing BHs inject mechanical power with $\sim 0.1-1\%$ of the radiative luminosity into their host galaxy scales, which is used for cosmological simulations.

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arXiv2019

The Assembly of the First Massive Black Holes

The existence of $\approx$10^9 Msun supermassive black holes (SMBHs) within the first billion year of the universe has stimulated numerous ideas for the prompt formation and rapid growth of BHs in the early universe. Here we review ways in which the seeds of massive BHs may have first assembled, how they may have subsequently grown as massive as $\approx$10^9 Msun, and how multi-messenger observations could distinguish between different SMBH assembly scenarios. We conclude the following: (1) The ultra-rare $\approx$10^9 Msun SMBHs represent only the tip of the iceberg. Early BHs likely fill a continuum from stellar-mass (approx. 10 Msun) to the super-massive ($\approx$10^9 Msun) regime, reflecting a range of initial masses and growth histories. (2) Stellar-mass BHs were likely left behind by the first generation of stars at redshifts as high as z=30, but their initial growth was typically stunted due to the shallow potential wells of their host galaxies. (3) Conditions in some larger, metal-poor galaxies soon became conducive to the rapid formation and growth of massive `seed' holes, via gas accretion and by mergers in dense stellar clusters. (4) BH masses depend on the environment (such as the number and properties of nearby radiation sources and the local baryonic streaming velocity), and on the metal enrichment and assembly history of the host galaxy. (5) Distinguishing between assembly mechanisms will be difficult, but a combination of observations by LISA (probing massive BH growth via mergers) and by deep multi-wavelength electromagnetic observations (probing growth via gas accretion) is particularly promising.

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arXiv2019

Super-Eddington growth of black holes in the early Universe: effects of disk radiation spectra

We investigate the properties of accretion flows onto a black hole (BH) with a mass of $M_{\rm BH}$ embedded in an initially uniform gas cloud with a density of $n_{\infty}$ in order to study rapid growth of BHs in the early Universe. In previous work, the conditions for super-Eddington accretion from outside the Bondi radius were studied by assuming that radiation produced at the vicinity of the central BH has a single-power-law spectrum $ν^{-α}$ at $hν\geq 13.6~{\rm eV}$ ($α\sim 1.5$). However, radiation spectra depends on the BH mass and accretion rate. Here, we perform two-dimensional multi-frequency radiation hydrodynamical simulations taking into account more realistic radiation spectra associated with the properties of nuclear accretion disks. We find that the condition for a transition to super-Eddington accretion is alleviated for a wide range of masses ($10\lesssim M_{\rm BH}/M_{\odot} \lesssim 10^6$) because photoionization for accretion disk spectra are less efficient than those for single-power-law spectra. For disk spectra, the transition to super-Eddington is more likely to occur for lower BH masses because the radiation spectra become too hard to ionize the gas. Even when accretion flows are exposed to anisotropic radiation, the effect due to radiation spectra shrinks the ionized region and likely leads to the transition to a wholly neutral accretion phase. Finally, by generalizing our simulation results, we construct a new analytical criterion required for super-Eddington accretion; $(M_{\rm BH}/10^5~M_{\odot}) (n_{\infty}/10^4~{\rm cm^{-3}}) \gtrsim 2.4~ (\langleε\rangle /100~{\rm eV})^{-5/9}$, where $\langleε\rangle$ is the mean energy of ionizing radiation from the central BH.

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arXiv2019

Titans of the Early Universe: The Prato Statement on the Origin of the First Supermassive Black Holes

In recent years, the discovery of massive quasars at z~7 has provided a striking challenge to our understanding of the origin and growth of supermassive black holes in the early Universe. Mounting observational and theoretical evidence indicates the viability of massive seeds, formed by the collapse of supermassive stars, as a progenitor model for such early, massive accreting black holes. Although considerable progress has been made in our theoretical understanding, many questions remain regarding how (and how often) such objects may form, how they live and die, and how next generation observatories may yield new insight into the origin of these primordial titans. This review focusses on our present understanding of this remarkable formation scenario, based on discussions held at the Monash Prato Centre from November 20--24, 2017, during the workshop "Titans of the Early Universe: The Origin of the First Supermassive Black Holes."

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arXiv2019

Transition of BH feeding from the quiescent regime into star-forming cold disk regime

We study the properties of rotating accretion flows onto supermassive black holes (SMBHs) using axisymmetric two-dimensional hydrodynamical simulations with radiative cooling and BH feedback. The simulations resolve the accretion dynamics of gas outside from the BH influence radius through an inner accretion disk. For lower Bondi accretion rates in units of the Eddington rate ($\dot{M}_{\rm B}\ll 10^{-3}~\dot{M}_{\rm Edd}$), the BH feeding is suppressed due to turbulent motion by several orders of magnitudes from the Bondi rate. Thus, the radiative luminosity results in as low as $\sim 10^{-10}-10^{-7}~L_{\rm Edd}$, where $L_{\rm Edd}$ is the Eddington luminosity. For higher rates of $\dot{M}_{\rm B}> 10^{-3}~\dot{M}_{\rm Edd}$, the optically-thin accreting gas cools via free-free emission and forms a geometrically-thin disk, which feeds the BH efficiently and increases the radiative luminosity to $> 10^{-3}~L_{\rm Edd}$. The transitional behavior of accreting BHs in galactic nuclei from radiatively inefficient phases to cold disk accretion naturally explains (1) the reason for the offset between the observed luminosities and theoretical predictions for nearby quiescent SMBHs, and (2) the conditions to fuel gas into the nuclear SMBH. In addition, the cold disk formed in galactic nuclei tends to be gravitationally unstable and leads to star formation when the Bondi rate is as high as $ \dot{M}_{\rm B} > 10^{-2}~M_\odot~{\rm yr}^{-1}$. This is a plausible explanation of the correlation observed between star formation rates and BH feeding rates in Seyfert galaxies.

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arXiv2019

Electromagnetic Window into the Dawn of Black Holes

Massive 10^6-10^10 Msun black holes (BHs) are ubiquitous in local galactic nuclei. They were common by the time the Universe is several Gyr old, and many of them were in place within the first 1~Gyr after the Big Bang. Their quick assembly has been attributed to mechanisms such as the rapid collapse of gas into the nuclei of early protogalaxies, accretion and mergers of stellar-mass BHs accompanying structure formation at early times, and the runaway collapse of early, ultra-dense stellar clusters. The origin of the early massive BHs remains an intriguing and long-standing unsolved puzzle in astrophysics. Here we discuss strategies for discerning between BH seeding models using electromagnetic observations. We argue that the most direct answers will be obtained through detection of BHs with masses M<10^5 Msun at redshifts z>10, where we expect them to first form. Reaching out to these redshifts and down to these masses is crucial, because BHs are expected to lose the memory of their initial assembly by the time they grow well above 10^5 Msun and are incorporated into higher-mass galaxies. The best way to detect 10^4-10^5 Msun BHs at high redshifts is by a sensitive X-ray survey. Critical constraining power is augmented by establishing the properties and the environments of their host galaxies in deep optical/IR imaging surveys. Required OIR data can be obtained with the JWST and WFIRST missions. The required X-ray flux limits (down to 10^{-19} erg/s/cm^2) are accessible only with a next-generation X-ray observatory which has both high (sub-1") angular resolution and high throughput. A combination of deep X-ray and OIR surveys will be capable of probing several generic markers of the BH seed scenarios, and resolving the long-stanging puzzle of their origin. These electromagnetic observations are also highly synergistic with the information from LISA on high-z BH mergers.

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arXiv2019

Intermediate-Mass Black Holes in Extragalactic Globular Clusters

Intermediate-mass black holes (IMBHs) have masses of about 100 to 100,000 solar masses. They remain elusive. Observing IMBHs in present-day globular clusters (GCs) would validate a formation channel for seed black holes in the early universe and inform event predictions for gravitational wave facilities. Reaching a large number of GCs per galaxy is key, as models predict that only a few percent will have retained their gravitational-wave fostering IMBHs. Related, many galaxies will need to be examined to establish a robust sample of IMBHs in GCs. These needs can be meet by using a next-generation Very Large Array (ngVLA) to search for IMBHs in the GCs of hundreds of galaxies out to a distance of 25 Mpc. These galaxies hold tens of thousands of GCs in total. We describe how to convert an ngVLA signal from a GC to an IMBH mass according to a semi-empirical accretion model. Simulations of gas flows in GCs would help to improve the robustness of the conversion. Also, self-consistent dynamical models of GCs, with stellar and binary evolution in the presence of IMBHs, would help to improve IMBH retention predictions for present-day GCs.

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arXiv2018

Stunted accretion growth of black holes by combined effect of the flow angular momentum and radiation feedback

Accretion on to seed black holes (BHs) is believed to play a crucial role in formation of supermassive BHs observed at high-redshift (z>6). Here, we investigate the combined effect of gas angular momentum and radiation feedback on the accretion flow, by performing 2D axially symmetric radiation hydrodynamics simulations that solve the flow structure across the Bondi radius and the outer part of the accretion disc simultaneously. The accreting gas with finite angular momentum forms a rotationally-supported disc inside the Bondi radius, where the accretion proceeds by the angular momentum transport due to assumed alpha-type viscosity. We find that the interplay of radiation and angular momentum significantly suppresses accretion even if the radiative feedback is weakened in an equatorial shadowing region. The accretion rate is O(alpha)\sim O(0.01-0.1) times the Bondi value, where alpha is the viscosity parameter. By developing an analytical model, we show that such a great reduction of the accretion rate persists unless the angular momentum is so small that the corresponding centrifugal radius is \lesssim 0.04 times the Bondi radius. We argue that BHs are hard to grow quickly via rapid mass accretion considering the angular momentum barrier presented in this paper.

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arXiv2018

Massive black hole and Population III galaxy formation in over-massive dark matter halos with violent merger histories

We propose the formation of massive pristine dark-matter (DM) halos with masses of $\sim 10^8~M_\odot$, due to the dynamical effects of frequent mergers in rare regions of the Universe with high baryonic streaming velocity relative to DM. Since the streaming motion prevents gas collapse into DM halos and delays prior star formation episodes, the gas remains metal-free until the halo virial temperatures $\gtrsim 2\times 10^4~{\rm K}$. The minimum cooling mass of DM halos is boosted by a factor of $\sim 10-30$ because frequent major mergers of halos further inhibit gas collapse. We use Monte Carlo merger trees to simulate the DM assembly history under a streaming velocity of twice the root-mean-square value, and estimate the number density of massive DM halos containing pristine gas as $\simeq 10^{-4}~{\rm cMpc}^{-3}$. When the gas infall begins, efficient Ly$α$ cooling drives cold streams penetrating inside the halo and feeding a central galactic disk. When one stream collides with the disk, strong shock forms a dense and hot gas cloud, where the gas never forms H$_2$ molecules due to effective collisional dissociation. As a result, a massive gas cloud forms by gravitational instability and collapses directly into a massive black hole (BH) with $M_\bullet \sim 10^5~M_\odot$. Almost simultaneously, a galaxy with $M_{\star, \rm tot}\sim 10^6~M_\odot$ composed of Population III stars forms in the nuclear region. If the typical stellar mass is as high as $\sim 100~M_\odot$, the galaxy could be detected with the James Webb Space Telescope even at $z\gtrsim 15$. These massive seed BHs would be fed by continuous gas accretion from the host galaxy, and grow to be bright quasars observed at $z\gtrsim 6$.

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arXiv2018

Low density, radiatively inefficient rotating-accretion flow onto a black hole

We study low-density axisymmetric accretion flows onto black holes (BHs) with two-dimensional hydrodynamical simulations, adopting the $α$-viscosity prescription. When the gas angular momentum is low enough to form a rotationally supported disk within the Bondi radius ($R_{\rm B}$), we find a global steady accretion solution. The solution consists of a rotational equilibrium distribution at $r\sim R_{\rm B}$, where the density follows $ρ\propto (1+R_{\rm B}/r)^{3/2}$, surrounding a geometrically thick and optically thin accretion disk at the centrifugal radius, where thermal energy generated by viscosity is transported via strong convection. Physical properties of the inner solution agree with those expected in convection-dominated accretion flows (CDAF; $ρ\propto r^{-1/2}$). In the inner CDAF solution, the gas inflow rate decreases towards the center due to convection ($\dot{M}\propto r$), and the net accretion rate (including both inflows and outflows) is strongly suppressed by several orders of magnitude from the Bondi accretion rate $\dot{M}_{\rm B}$ The net accretion rate depends on the viscous strength, following $\dot{M}/\dot{M}_{\rm B}\propto (α/0.01)^{0.6}$. This solution holds for low accretion rates of $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}< 10^{-3}$ having minimal radiation cooling, where $\dot{M}_{\rm Edd}$ is the Eddington rate. In a hot plasma at the bottom ($r<10^{-3}~R_{\rm B}$), thermal conduction would dominate the convective energy flux. Since suppression of the accretion by convection ceases, the final BH feeding rate is found to be $\dot{M}/\dot{M}_{\rm B} \sim 10^{-3}-10^{-2}$. This rate is as low as $\dot{M}/\dot{M}_{\rm Edd} \sim 10^{-7}-10^{-6}$ inferred for SgrA$^*$ and the nuclear BHs in M31 and M87, and can explain the low luminosities in these sources, without invoking any feedback mechanism.

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arXiv2018

Gravitational waves from supermassive black hole binaries in ultra-luminous infrared galaxies

Gravitational waves (GWs) in the nano-hertz band are great tools for understanding the cosmological evolution of supermassive black holes (SMBHs) in galactic nuclei. We consider SMBH binaries in high-$z$ ultra-luminous infrared galaxies (ULIRGs) as sources of a stochastic GW background (GWB). ULIRGs are likely associated with gas-rich galaxy mergers containing SMBHs that possibly occur at most once in the life of galaxies, unlike multiple dry mergers at low redshift. Adopting a well-established sample of ULIRGs, we study the properties of the GWB due to coalescing binary SMBHs in these galaxies. Since the ULIRG population peaks at $z>1.5$, the amplitude of the GWB is not affected even if BH mergers are delayed by as long as $\sim $ 10 Gyrs. Despite the rarity of the high-$z$ ULIRGs, we find a tension with the upper limits from Pulsar Timing Array (PTA) experiments. This result suggests that if a fraction $f_{\rm m,gal}$ of ULIRGs are associated with SMBH binaries, then no more than $20 f_{\rm m,gal}(λ_{\rm Edd}/0.3)^{5/3}(t_{\rm life}/30~{\rm Myr})~\%$ of the binary SMBHs in ULIRGs can merge within a Hubble time, for plausible values of the Eddington ratio of ULIRGs ($λ_{\rm Edd}$) and their lifetime ($t_{\rm life}$).

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arXiv2018

Opacity limit for supermassive protostars

We present a model for the evolution of supermassive protostars from their formation at $M_\star \simeq 0.1\,\text{M}_\odot$ until their growth to $M_\star \simeq 10^5\,\text{M}_\odot$. To calculate the initial properties of the object in the optically thick regime we follow two approaches: based on idealized thermodynamic considerations, and on a more detailed one-zone model. Both methods derive a similar value of $n_{\rm F} \simeq 2 \times 10^{17} \,\text{cm}^{-3}$ for the density of the object when opacity becomes important, i.e. the opacity limit. The subsequent evolution of the growing protostar is determined by the accretion of gas onto the object and can be described by a mass-radius relation of the form $R_\star \propto M_\star^{1/3}$ during the early stages, and of the form $R_\star \propto M_\star^{1/2}$ when internal luminosity becomes important. For the case of a supermassive protostar, this implies that the radius of the star grows from $R_\star \simeq 0.65 \,{\rm AU}$ to $R_\star \simeq 250 \,{\rm AU}$ during its evolution. Finally, we use this model to construct a sub-grid recipe for accreting sink particles in numerical simulations. A prime ingredient thereof is a physically motivated prescription for the accretion radius and the effective temperature of the growing protostar embedded inside it. From the latter, we can conclude that photo-ionization feedback can be neglected until very late in the assembly process of the supermassive object.

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