Quasinormal modes are eigenmodes of dissipative systems. Perturbations of classical gravitational backgrounds involving black holes or branes naturally lead to quasinormal modes. The analysis and classification of the quasinormal spectra require solving non-Hermitian eigenvalue problems for the associated linear differential equations. Within the recently developed gauge-gravity duality, these modes serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, in particular their transport coefficients, such as viscosity, conductivity and diffusion constants. In astrophysics, the detection of quasinormal modes in gravitational wave experiments would allow precise measurements of the mass and spin of black holes as well as new tests of general relativity. This review is meant as an introduction to the subject, with a focus on the recent developments in the field.

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Quasinormal modes of black holes and black branes

The following section, Management’s Discussion and Analysis of Operations, provides an overview of the consolidated financial statements of Fujitsu Limited (the “Company”) and its consolidated subsidiaries (together, the “Group”) for the year ended March 31, 2010 (fiscal 2009). Forward-looking statements in this section are based on management’s understanding and best judgment as of March 31, 2010. The impact of exchange rate movements is calculated by taking the average exchange rates in fiscal 2008 for the U.S. dollar, euro, British pound, Australian dollar, Korean won, and other currencies and applying them to foreign currencydenominated transactions in fiscal 2009.

Annual report for 2010

[ABRIDGED] We study supermassive black hole binary mergers driven by angular momentum loss to small-scale gas discs. Such binaries form after major galaxy mergers, but their fate is unclear since hardening through stellar scattering becomes very inefficient at sub-parsec distances. Gas discs may dominate binary dynamics on these scales, and promote mergers. Using numerical simulations, we investigate the evolution of the orbits of binaries embedded within geometrically thin gas discs. Our simulations directly resolve angular momentum transport within the disc, which at the radii of interest is likely dominated by disc self-gravity. We show that the binary decays at a rate which is in good agreement with analytical estimates, while the eccentricity grows. Saturation of eccentricity growth is not observed up to values e > 0.35. Accretion onto the black holes is variable, and is roughly modulated by the binary orbital frequency. Scaling our results, we analytically estimate the maximum rate of binary decay that is possible without fragmentation occuring within the surrounding gas disc, and compare that rate to an estimate of the stellar dynamical hardening rate. For binary masses in the range 10^5 < M/Msun < 10^8 we find that decay due to gas discs may dominate for separations below a ~ 0.01--0.1 pc, where the disc is optically thick. The minimum merger time scale is shorter than the Hubble time for M < 10^7 Msun. This implies that gas discs could commonly attend relatively low mass black hole mergers, and that a significant population of binaries might exist at separations of a few hundredths of a pc, where the combined decay rate is slowest. For more massive binaries, we suggest that scattering of stars formed within a fragmenting gas disc could act as a significant additional sink of binary angular momentum.

Massive black hole binary mergers within sub-pc scale gas discs

We study the efficiency and dynamics of supermassive black hole binary mergers driven by angular momentum loss to small-scale gas discs. Such binaries form after major galaxy mergers, but their fate is unclear since hardening through stellar scattering becomes very inefficient at sub-parsec distances. Gas discs may dominate binary dynamics on these scales, and promote mergers. Using numerical simulations, we investigate the evolution of the semi-major axis and eccentricity of binaries embedded within geometrically thin gas discs. Our simulations directly resolve angular momentum transport within the disc, which at the radii of interest is likely dominated by disc self-gravity. We show that the binary decays at a rate which is in good agreement with analytical estimates, while the eccentricity grows. Saturation of eccentricity growth is not observed up to values e > � 0.35. Accretion onto the black holes is variable, and is roughly modulated by the binary orbital frequency. Scaling our results, we analytically estimate the maximum rate of binary decay that is possible without fragmentation occuring within the surrounding gas disc, and compare that rate to an estimate of the stellar dynamical hardening rate. For binary masses in the range 10 5 M⊙ < � M < � 10 8 M⊙ we find that decay due to gas discs may dominate for separations below a � 10 −2 pc 0.1 pc, in the regime where the disc is optically thick. The minimum merger time scale is shorter than the Hubble time for M < � 10 7 M⊙. This implies that gas discs could commonly attend relatively low mass black hole mergers, and that a significant population of binaries might exist at separations of a few hundredths of a pc, where the combined decay rate is slowest. For more massive binaries, where this mechanism fails to act quickly enough, we suggest that scattering of stars formed within a fragmenting gas disc could act as a significant additional sink of binary angular momentum.

Massive black hole binary mergers within subparsec scale gas discs

We present a two-dimensional grid-based hydrodynamic simulation of a thin, viscous, locally isothermal corotating disk orbiting an equal-mass Newtonian binary point mass on a fixed circular orbit. We study the structure of the disk after multiple viscous times. The binary maintains a central hole in the viscously relaxed disk with radius equal to about twice the binary semimajor axis. Disk surface density within the hole is reduced by orders of magnitude relative to the density in the disk bulk. The inner truncation of the disk resembles the clearing of a gap in a protoplanetary disk. An initially circular disk becomes elliptical and then eccentric. Disturbances in the disk contain a component that is stationary in the rotating frame in which the binary is at rest; this component is a two-armed spiral density wave. We measure the distribution of the binary torque in the disk and find that the strongest positive torque is exerted inside the central low-density hole. We make connection with the linear theory of disk forcing at outer Lindblad resonances (OLRs) and find that the measured torque density distribution is consistent with forcing at the 3:2 (m = 2) OLR, well within the central hole. We also measure the time dependence of the rate at which gas accretes across the hole and find quasi-periodic structure. We discuss implications for variability and detection of active galactic nuclei containing a binary massive black hole.

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An eccentric circumbinary accretion disk and the detection of binary massive black holes

We report on the discovery of a supermassive binary black hole system in the radio galaxy 0402+379, with a projected separation between the two black holes of just 7.3 pc. This is the closest black hole pair yet found by more than 2 orders of magnitude. These results are based on recent multifrequency observations using the Very Long Baseline Array (VLBA), which reveal two compact, variable, flat-spectrum, active nuclei within the elliptical host galaxy of 0402+379. Multiepoch observations from the VLBA also provide constraints on the total mass and dynamics of the system. Low spectral resolution spectroscopy using the Hobby-Eberly Telescope indicates two velocity systems with a combined mass of the two black holes of ~1.5 × 108 M☉. The two nuclei appear stationary, while the jets emanating from the weaker of the two nuclei appear to move out and terminate in bright hot spots. The discovery of this system has implications for the number of close binary black holes that might be sources of gravitational radiation. Green Bank Telescope observations at 22 GHz to search for water masers in this interesting system are also presented.

https://iopscience.iop.org/article/10.1086/504825/pdf

A Compact Supermassive Binary Black Hole System

We report on the discovery of a supermassive binary black hole system in the radio galaxy 0402+379, with a projected separation between the two black holes of just 7.3 pc. This is the closest black hole pair yet found by more than two orders of magnitude. These results are based upon recent multi-frequency observations using the Very Long Baseline Array (VLBA) which reveal two compact, variable, flat-spectrum, active nuclei within the elliptical host galaxy of 0402+379. Multi-epoch observations from the VLBA also provide constraints on the total mass and dynamics of the system. Low spectral resolution spectroscopy using the Hobby-Eberly Telescope indicates two velocity systems with a combined mass of the two black holes of ~1.5 x 10^8 solar masses. The two nuclei appear stationary while the jets emanating from the weaker of the two nuclei appear to move out and terminate in bright hot spots. The discovery of this system has implications for the number of close binary black holes that might be sources of gravitational radiation. Green Bank Telescope observations at 22 GHz to search for water masers in this interesting system are also presented.

We have recently discovered a supermassive binary black hole system with a projected separation between the two black holes of 7.3 pc in the radio galaxy 0402+379 (Rodriguez et al. 2006). This is the most compact supermassive binary black hole pair yet imaged by more than two orders of magnitude. We present Global VLBI observations at 1.3464 GHz of this radio galaxy, taken to improve the quality of the H I data. Two absorption lines are found toward the southern jet of the source, one redshifted by 370 ± 10 km s–1 and the other blueshifted by 700 ± 10 km s–1 with respect to the systemic velocity of the source, which, along with the results obtained for the opacity distribution over the source, suggests the presence of two mass clumps rotating around the central region of the source. We propose a model consisting of a geometrically thick disk, of which we only see a couple of clumps, that reproduces the velocities measured from the H I absorption profiles. These clumps rotate in circular Keplerian orbits around an axis that crosses one of the supermassive black holes of the binary system in 0402+379. We find an upper limit for the inclination angle of the twin jets of the source to the line of sight of θ = 66°, which, according to the proposed model, implies a lower limit on the central mass of ~7 × 108 M ☉ and a lower limit for the scale height of the thick disk of ~12 pc.

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H i observations of the supermassive binary black hole system in 0402+379

Adaptive optical microscopy for neurobiology.

We have recently discovered a supermassive binary black hole system with a projected separation between the two black holes of 7.3 parsecs in the radio galaxy 0402+379. This is the most compact supermassive binary black hole pair yet imaged by more than two orders of magnitude. We present Global VLBI observations at 1.3464 GHz of this radio galaxy, taken to improve the quality of the HI data. Two absorption lines are found toward the southern jet of the source, one redshifted by 370 +/- 10 km/s and the other blueshifted by 700 +/- 10 km/s with respect to the systemic velocity of the source, which, along with the results obtained for the opacity distribution over the source, suggests the presence of two mass clumps rotating around the central region of the source. We propose a model consisting of a geometrically thick disk, of which we only see a couple of clumps, that reproduces the velocities measured from the HI absorption profiles. These clumps rotate in circular Keplerian orbits around an axis that crosses one of the supermassive black holes of the binary system in 0402+379. We find an upper limit for the inclination angle of the twin jets of the source to the line of sight of 66 degrees, which, according to the proposed model, implies a lower limit on the central mass of ~7 x 10^8 Msun and a lower limit for the scale height of the thick disk of ~12 pc .

Cristina Rodríguez

Papers

A Compact Supermassive Binary Black Hole System

A Compact Supermassive Binary Black Hole System

H i observations of the supermassive binary black hole system in 0402+379

Adaptive optical microscopy for neurobiology.

HI Observations of the Supermassive Binary Black Hole System in 0402+379