Massive black hole binaries are key targets for the space based gravitational wave interferometer LISA. Several studies have investigated how LISA observations could be used to constrain the parameters of these systems. Until recently, most of these studies have ignored the higher harmonic corrections to the waveforms. Here we analyze the effects of the higher harmonics in more detail by performing extensive Monte Carlo simulations. We pay particular attention to how the higher harmonics impact parameter correlations, and show that the additional harmonics help mitigate the impact of having two laser links fail, by allowing for an instantaneous measurement of the gravitational wave polarization with a single interferometer channel. By looking at parameter correlations we are able to explain why certain mass ratios provide dramatic improvements in certain parameter estimations, and illustrate how the improved polarization measurement improves the prospects for single interferometer operation.

We present long term evolutions of a single black hole of mass $M$ with the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system using pseudospectral methods. For our simulations we use the SGRID code where the BSSN system is implemented in its standard second order in space form. Previously we found that such simulations are quite unstable. The main goal of this paper is to present two improvements which now allow us to evolve for longer times. The first improvement is related to the boundary conditions at the excised black hole interior. We now use a gauge condition that ensures that all modes are going into the black hole, so that no boundary conditions are needed at the excision surface. The second more significant improvement has to do with our particular numerical method and involves filters based on projecting the double Fourier expansions used for the angular dependence onto spherical harmonics. With these two improvements it is now easily possible to evolve for several thousand $M$. The only remaining limitation seems to be the radiative outer boundary conditions used here. Yet this problem can be ameliorated by pushing out the location of the outer boundary, which leads to even longer run times.

In this paper, we consider the four-dimensional N = 2 supergravity theory arising from the compactification of type IIA string theory on a Calabi–Yau manifold. We analyze the supersymmetric flow equations for static, spherically symmetric, single-centered black holes. These flow equations are solved by a set of algebraic equations involving the holomorphic sections and harmonic functions. We examine black hole configurations with D0–D4–D6 charge for which the most general solution of these algebraic equations are considered. Though the black hole solution is unique for a given value of the charges, we find new phases of the black hole solutions upon varying them.

The damped harmonic gauge is important for numerical relativity computations based on the generalized harmonic formulation of Einstein's equations, and is used to reduce coordinate distortions near binary black hole mergers. However, currently there is no prescription to construct quasiequilibrium binary black hole initial data in this gauge. Instead, initial data are typically constructed using a superposition of two boosted analytic single black hole solutions as free data in the solution of the constraint equations. Then, a smooth time-dependent gauge transformation is done early in the evolution to move into the damped harmonic gauge. Using this strategy to produce initial data in damped harmonic gauge would require the solution of a single black hole in this gauge, which is not known analytically. In this work we construct a single boosted, spinning, equilibrium BH in damped harmonic coordinates as a regular time-independent coordinate transformation from Kerr-Schild coordinates. To do this, we derive and solve a set of 4 coupled, nonlinear, elliptic equations for this transformation, with appropriate boundary conditions. This solution can now be used in the construction of damped harmonic initial data for binary black holes.

The damped harmonic gauge is important for numerical relativity computations based on the generalized harmonic formulation of Einstein’s equations and is used to reduce coordinate distortions near binary black hole mergers. However, currently there is no prescription to construct quasiequilibrium binary black hole initial data in this gauge. Instead, initial data are typically constructed using a superposition of two boosted analytic single black hole solutions as free data in the solution of the constraint equations. Then, a smooth time-dependent gauge transformation is done early in the evolution to move into the damped harmonic gauge. Using this strategy to produce initial data in the damped harmonic gauge would require the solution of a single black hole in this gauge, which is not known analytically. In this work we construct a single boosted, spinning, equilibrium black hole in damped harmonic coordinates as a regular time-independent coordinate transformation from Kerr-Schild coordinates. To do this, we derive and solve a set of four coupled, nonlinear, elliptic equations for this transformation, with appropriate boundary conditions. This solution can now be used in the construction of damped harmonic initial data for binary black holes.

In this paper we consider the four dimensional N=2 supergravity theory arising from the compactification of type IIA string theory on a Calabi-Yau manifold. We analyse the supersymmetric flow equations for static, spherically symmetric, single-centered black holes. These flow equations are solved by a set of algebraic equations involving the holomorphic sections and harmonic functions. We examine black hole configurations with D0-D4-D6 charge for which the most general solution of these algebraic equations are considered. Though the black hole solution is unique for a given value of the charges, we find new phases of the black hole solutions upon varying them.

In recent series of papers, we found an arbitrary dimensional, time-evolving and spatially-inhomogeneous solutions in Einstein-Maxwell-dilaton gravity with particular couplings. Similar to the supersymmetric case the solution can be arbitrarily superposed in spite of non-trivial time-dependence, since the metric is specified by a set of harmonic functions. When each harmonic has a single point source at the center, the solution describes a spherically symmetric black hole with regular Killing horizons and the spacetime approaches asymptotically to the Friedmann-Lema\^itre-Robertson-Walker (FLRW) cosmology. We discuss in this paper that in 5-dimensions this equilibrium condition traces back to the 1st-order "Killing spinor" equation in "fake supergravity" coupled to arbitrary U(1) gauge fields and scalars. We present a 5-dimensional, asymptotically FLRW, rotating black-hole solution admitting a nontrivial "Killing spinor," which is a spinning generalization of our previous solution. We argue that the solution admits nondegenerate and rotating Killing horizons in contrast with the supersymmetric solutions. It is shown that the present pseudo-supersymmetric solution admits closed timelike curves around the central singularities. When only one harmonic is time-dependent, the solution oxidizes to 11-dimensions and realizes the dynamically intersecting M2/M2/M2-branes in a rotating Kasner universe. The Kaluza-Klein type black holes are also discussed.

We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test case we consider the evolution of a single black hole in numerical general relativity. A natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates. Instead, we consider the simpler and potentially more efficient possibility of a double Fourier expansion on the sphere for tensors in Cartesian coordinates. As usual for the double Fourier method, we employ a filter to address time-step limitations and certain stability issues. We find that a tensor filter based on spin-weighted spherical harmonics is successful, while two simplified, non-spin-weighted filters do not lead to stable evolutions. The derivatives and the filter are implemented by matrix multiplication for efficiency. A key technical point is the construction of a matrix multiplication method for the spin-weighted spherical harmonic filter. As example for the efficient parallelization of the double Fourier, spin-weighted filter method we discuss an implementation on a GPU, which achieves a speed-up of up to a factor of 20 compared to a single core CPU implementation.

In the light of recent interest in quantum gravity in de Sitter space, we investigate semiclassical aspects of four-dimensional Schwarzschild–de Sitter space-time using the method of complex paths. The standard semiclassical techniques (such as Bogoliubov coefficients and Euclidean field theory) have been useful to study quantum effects in space-times with single horizons; however, none of these approaches seem to work for Schwarzschild–de Sitter space-time or, in general, for space-times with multiple horizons. We extend the method of complex paths to space-times with multiple horizons and obtain the spectrum of particles produced in these space-times. We show that the temperature of radiation in these space-times is proportional to the effective surface gravity—the inverse harmonic sum of surface gravity of each horizon. For the Schwarzschild–de Sitter space-time, we apply the method of complex paths to three different coordinate systems—spherically symmetric, Painlevé, and Lemaître. We show that the equilibrium temperature in Schwarzschild–de Sitter space-time is the harmonic mean of cosmological and event horizon temperatures. We obtain Bogoliubov coefficients for space-times with multiple horizons by analyzing the mode functions of the quantum fields near the horizons. We propose a new definition of entropy for space-times with multiple horizons, analogous to the entropic definition for space-times with a single horizon. We define entropy for these space-times to be inversely proportional to the square of the effective surface gravity. We show that this definition of entropy for Schwarzschild–de Sitter space-time satisfies the D-bound conjecture.

We describe numerical techniques used in the construction of our 4th order evolution for the full Einstein equations, and assess the accuracy of representative solutions. The code is based on a null gauge with a quasi-spherical radial coordinate, and simulates the interaction of a single black hole with gravitational radiation. Techniques used include spherical harmonic representations, convolution spline interpolation and filtering, and an RK4 "method of lines" evolution. For sample initial data of "intermediate" size (gravitational field with 19% of the black hole mass), the code is accurate to 1 part in 10^5, until null time z=55 when the coordinate condition breaks down.Comment: Latex, 38 pages, 29 figures (360Kb compressed

The classifications of holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, SIM(n-2). Ricci-flat metrics with SIM(2) holonomy were constructed by Kerr and Goldberg, and a single four-dimensional example with a non-zero cosmological constant was exhibited by Ghanam and Thompson. Here we reduce the problem of finding the general $n$-dimensional Einstein metric of SIM(n-2) holonomy, with and without a cosmological constant, to solving a set linear generalised Laplace and Poisson equations on an (n-2)-dimensional Einstein base manifold. Explicit examples may be constructed in terms of generalised harmonic functions. A dimensional reduction of these multi-centre solutions gives new time-dependent Kaluza-Klein black holes and monopoles, including time-dependent black holes in a cosmological background whose spatial sections have non-vanishing curvature.

We present extremal stationary solutions that generalize the Israel-Wilson-Perjes class for the d+3-dimensional low-energy limit of heterotic string theory with n >= d+1 U(1) gauge fields compactified on a d-torus. A rotating axisymmetric dyonic solution is obtained using the matrix Ernst potential formulation and expressed in terms of a single d+1 X d+1-matrix harmonic function. By studying the asymptotic behaviour of the field configurations we define the physical charges of the field system. The extremality condition makes the charges to saturate the Bogomol'nyi-Prasad-Sommerfield (BPS) bound. The gyromagnetic ratios of the corresponding field configurations appear to have arbitrary values. A subclass of rotating dyonic black hole-type solutions arises when the NUT charges are set to zero. In the particular case d=1, n=6, which correspond to N=4, D=4 supergravity, the found dyon reproduces the supersymmetric dyonic solution constructed by Bergshoeff et al.

We present a time-dependent and spatially inhomogeneous solution that interpolates the extremal Reissner-Nordstr\"om (RN) black hole and the Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe with arbitrary power-law expansion. It is an exact solution of the $D$-dimensional Einstein-"Maxwell"-dilaton system, where two Abelian gauge fields couple to the dilaton with different coupling constants, and the dilaton field has a Liouville-type exponential potential. It is shown that the system satisfies the weak energy condition. The solution involves two harmonic functions on a $(D-1)$-dimensional Ricci-flat base space. In the case where the harmonics have a single-point source on the Euclidean space, we find that the spacetime describes a spherically symmetric charged black hole in the FLRW universe, which is characterized by three parameters: the steepness parameter of the dilaton potential $n_T$, the U$(1)$ charge $Q$, and the "nonextremality" $\tau $. In contrast with the extremal RN solution, the spacetime admits a nondegenerate Killing horizon unless these parameters are finely tuned. The global spacetime structures are discussed in detail.

We present long term evolutions of a single black hole of mass $M$ with the BSSN system using pseudospectral methods. For our simulations we use the SGRID code where the BSSN system is implemented in its standard second order in space form. Previously we found that such simulations are quite unstable. The main goal of this paper is to present two improvements which now allow us to evolve for longer times. The first improvement is related to the boundary conditions at the excised black hole interior. We now use a gauge condition that ensures that all modes are going into the black hole, so that no boundary conditions are needed at the excision surface. The second more significant improvement has to do with our particular numerical method and involves filters based on projecting the double Fourier expansions used for the angular dependence onto Spherical Harmonics. With these two improvements it is now easily possible to evolve for several thousand $M$. The only remaining limitation seems to be the radiative outer boundary conditions used here. Yet this problem can be ameliorated by pushing out the location of the outer boundary, which leads to even longer run-times.

We present a time-dependent and spatially inhomogeneous solution that interpolates the extremal Reissner-Nordström (RN) black hole and the Friedmann-Lemaître-Robertson-Walker (FLRW) universe with arbitrary power-law expansion. It is an exact solution of the $D$-dimensional Einstein-Maxwell-dilaton system, where two Abelian gauge fields couple to the dilaton with different coupling constants, and the dilaton field has a Liouville-type exponential potential. It is shown that the system satisfies the weak energy condition. The solution involves two harmonic functions on a ($D-1$)-dimensional Ricci-flat base space. In the case where the harmonics have a single-point source on the Euclidean space, we find that the spacetime describes a spherically symmetric charged black hole in the FLRW universe, which is characterized by three parameters: the steepness parameter of the dilaton potential $n_T$, the U(1) charge $Q$, and the nonextremality $\tau$. In contrast with the extremal RN solution, the spacetime admits a nondegenerate Killing horizon unless these parameters are finely tuned. The global spacetime structures are discussed in detail. DOI: http://dx.doi.org/10.1103/PhysRevD.81.124038 © 2010 The American Physical Society

In recent series of papers, we found an arbitrary dimensional, time-evolving, and spatially inhomogeneous solution in Einstein-Maxwell-dilaton gravity with particular couplings. Similar to the supersymmetric case, the solution can be arbitrarily superposed in spite of nontrivial time-dependence, since the metric is specified by a set of harmonic functions. When each harmonic has a single point source at the center, the solution describes a spherically symmetric black hole with regular Killing horizons and the spacetime approaches asymptotically to the Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology. We discuss in this paper that in 5 dimensions, this equilibrium condition traces back to the first-order “Killing spinor” equation in “fake supergravity” coupled to arbitrary $U(1)$ gauge fields and scalars. We present a five-dimensional, asymptotically FLRW, rotating black-hole solution admitting a nontrivial “Killing spinor,” which is a spinning generalization of our previous solution. We argue that the solution admits nondegenerate and rotating Killing horizons in contrast with the supersymmetric solutions. It is shown that the present pseudo-supersymmetric solution admits closed timelike curves around the central singularities. When only one harmonic is time-dependent, the solution oxidizes to 11 dimensions and realizes the dynamically intersecting $M2$/$M2$/$M2$-branes in a rotating Kasner universe. The Kaluza-Klein–type black holes are also discussed. DOI: http://dx.doi.org/10.1103/PhysRevD.83.024018 © 2011 American Physical Society

In light of the current (and future) gravitational wave detections, more sensitive tests of general relativity can be devised. Black hole spectroscopy has long been proposed as a way to test the no-hair theorem, that is, how closely an astrophysical black hole can be described by the Kerr geometry. We use numerical relativity simulations from the Simulating eXtreme Spacetimes project (SXS) to assess the detectability of one extra quasinormal mode in the ringdown of a binary black hole coalescence, with numbers $(\ell ,m,n)$ distinct from the fundamental quadrupolar mode (2,2,0). Our approach uses the information from the complex waveform as well as from the time derivative of the phase in two different prescriptions that allow us to estimate the point at which the ringdown is best described by a single mode or by a sum of two modes. By scaling all amplitudes to a fiducial time $t_{\mathrm{peak}}+10M$ ($t_{\mathrm{peak}}$ is the time of maximum waveform amplitude), our results for nonspinning binaries indicate that, for mass ratios of $1:1$ to approximately $5:1$, the first overtone (2,2,1) will always have a larger excitation amplitude than the fundamental modes of the other harmonics (2,1,0), (3,3,0), and (4,4,0), making it a more promising candidate for detection. Even though the (2,2,1) mode damps about 3 times faster than the fundamental higher harmonics and its frequency is very close to that of the (2,2,0) mode, its larger excitation amplitude still guarantees a more favorable scenario for detection, as we show in a preliminary Rayleigh $\mathrm{criterion}+\mathrm{Fisher}$ matrix mode resolvability analysis of a simulation with nonzero spin consistent with GW150914. In particular, for nonspinning equal-mass binaries, the ratio of the amplitude of the first overtone (2,2,1) to the fundamental mode (2,2,0) will be $\gtrsim 0.65$, whereas the corresponding ratio for the higher harmonics will be $\lesssim 0.05$. For nonspinning binaries with mass ratios larger than $5:1$, we find that the modes (2,2,1), (2,1,0), and (3,3,0) should have comparable amplitude ratios in the range 0.3–0.4. The expectation that the (2,2,1) mode should be more easily detectable than the (3,3,0) mode is confirmed with an extension of the mode resolvability analysis for nonspinning cases with larger mass ratios, keeping the mass of the final black hole compatible with GW150914.

In this New Theory a “Single Harmonic Black Hole” (SHBH) has been considered to be the Gravitational-Electromagnetic Confinement of a Single Harmonic Electromagnetic Field Configuration in which a perfect equilibrium exists between the outward directed electromagnetic radiation pressure and the inward directed Electromagnetic-Gravitational Interaction force densities. This frequency transformation is possible because of the combined Lorentz / Doppler-Effect transformation during the collapse (contraction) of the radiation when the Gravitational Electromagnetic Confinement has been formed (Implosion of Visible Light). Within the scope of this article “Single Harmonic Black Hole” (SHBH) is considered to be any kind of 3-dimensional confined Single Harmonic Electromagnetic Energy. The inner structure of a “SHBH” has been based on a 3-dimensional isotropic equilibrium within the electromagnetic field configuration. This new theory will explain how electromagnetic fields (wave packages) demonstrate inertia, mass and momentum and which forces keep the wave packages together in a way that they can be measured like particles with their own specific mass and momentum. To understand what electromagnetic inertia and the corresponding electromagnetic mass, spin and electric charge is and how the anisotropy of electromagnetic mass, spin and electric charge can be explained and how it has to be defined, a New Theory about “Electromagnetic-Gravitational Interaction” has been developed. The “New Theory” has been based on the fundamental principle of “Perfect Equilibrium within the Universe” which has already been expressed by Newton’s three equations published in 1687 in “Philosophiae Naturalis Principia Mathematica. Newton’s Equations in 3 dimensions will be published in this article in an extension into 4 dimensions. Newton’s 4-dimensional law in the 3 spatial dimensions results in an improved version of the classical Maxwell Equations and Newton’s law in the 4th dimension (time) results in the quantum mechanical Schrödinger wave equation (at non-relativistic velocities) and the relativistic Dirac equation.

Numerical simulations of binary black holes—an important predictive tool for the detection of gravitational waves—are computationally expensive, especially for binaries with high mass ratios or with rapidly spinning constituent holes. Existing codes for evolving binary black holes rely on explicit time-stepping methods, for which the time-step size is limited by the smallest spatial scale through the Courant-Friedrichs-Lewy condition. Binary inspiral typically involves spatial scales (the spatial resolution required by a small or rapidly spinning hole) which are orders of magnitude smaller than the relevant (orbital, precession, and radiation-reaction) time scales characterizing the inspiral. Therefore, in explicit evolutions of binary black holes, the time-step size is typically orders of magnitude smaller than the relevant physical time scales. Implicit time-stepping methods allow for larger time steps, and they often reduce the total computational cost (without significant loss of accuracy) for problems dominated by spatial rather than temporal error, such as for binary-black-hole inspiral in corotating coordinates. However, fully implicit methods can be difficult to implement for nonlinear evolution systems like the Einstein equations. Therefore, in this paper we explore implicit-explicit (IMEX) methods and use them for the first time to evolve black-hole spacetimes. Specifically, as a first step toward IMEX evolution of a full binary-black-hole spacetime, we develop an IMEX algorithm for the generalized harmonic formulation of the Einstein equations and use this algorithm to evolve stationary and perturbed single-black-hole spacetimes. Numerical experiments explore the stability and computational efficiency of our method.

We describe early success in the evolution of binary black hole spacetimes with a numerical code based on a generalization of harmonic coordinates. Indications are that with sufficient resolution this scheme is capable of evolving binary systems for enough time to extract information about the orbit, merger and gravitational waves emitted during the event. As an example we show results from the evolution of a binary composed of two equal mass, non-spinning black holes, through a single plunge-orbit, merger and ring down. The resultant black hole is estimated to be a Kerr black hole with angular momentum parameter a~0.70. At present, lack of resolution far from the binary prevents an accurate estimate of the energy emitted, though a rough calculation suggests on the order of 5% of the initial rest mass of the system is radiated as gravitational waves during the final orbit and ringdown.

We describe early success in the evolution of binary black-hole spacetimes with a numerical code based on a generalization of harmonic coordinates. Indications are that with sufficient resolution this scheme is capable of evolving binary systems for enough time to extract information about the orbit, merger, and gravitational waves emitted during the event. As an example we show results from the evolution of a binary composed of two equal mass, nonspinning black holes, through a single plunge orbit, merger, and ringdown. The resultant black hole is estimated to be a Kerr black hole with angular momentum parameter $a\approx 0.70$. At present, lack of resolution far from the binary prevents an accurate estimate of the energy emitted, though a rough calculation suggests on the order of 5% of the initial rest mass of the system is radiated as gravitational waves during the final orbit and ringdown.

We apply the H-FGK formalism to the study of some properties of the general class of black holes in N=2 supergravity in four dimensions that correspond to the harmonic and hyperbolic ansatze and obtain explicit extremal and non-extremal solutions for the t^3 model with and without a quantum correction. Not all solutions of the corrected model (quantum black holes), including in particular a solution with a single q_1 charge, have a regular classical limit.

Flexural thermal fluctuations in crystalline membranes affect the band structure of the carriers, which leads to an exponential density-of-states (DOS) tail beyond the unperturbed band edge. We present a theoretical description of this tail for a particular case of holes in single-layer black phosphorus, a material which exhibits an extremely anisotropic quasi-one-dimensional dispersion ($m_y/m_x\gg1$) and, as a result, an enhanced Van Hove singularity at the valence band top. The material parameters are determined by {\it ab initio} calculations and then are used for quantitative estimation of the effect of two-phonon (flexural) processes have on the charge carrier DOS. It is shown that unlike the isotropic case, the physics is determined by the phonons with wavevectors of the order of $q^*$, where $q^*$ determines the crossover between harmonic and anharmonic behavior of the flexural phonons. The spectral density of the holes in single-layer black phosphorus at finite temperatures is calculated.

Measurements of multiple harmonic modes in the gravitational wave signals from binary black hole events could provide an accurate test of general relativity, however they have never been observed before. The sub-dominant modes, other than the main (l = 2, m = 2) mode, are weak in amplitude and thus difficult to detect in a single event at the current sensitivity of the gravitational wave detectors. To recover sub-dominant modes, we propose an unmodeled method for summation of signals from a population of binary black holes. The method coherently stacks all the signal modes, effectively increasing their signal-to-noise ratio, so the amplified signal can be extracted from the noisy data. To test the method, we consider simulated numerical relativity waveforms including sub-dominant modes up to (5, 5). We inject simulated signals from a population of binary black holes into data from the first observing run of Advanced LIGO and utilize the coherent WaveBurst algorithm for signal detection and reconstruction. Using only the waveforms reconstructed by coherent WaveBurst, i.e., no a priori information about the signal model, we determine the transformation to coherently synchronize the merger and post-merger of one signal to another. We demonstrate the synchronization of the injected signals and show the efficient stacking of the (2, 2) mode and the sub-dominant modes.

A new gauge driver is introduced for the generalized harmonic (GH) representation of Einstein’s equation. This new driver allows a rather general class of gauge conditions to be implemented in a way that maintains the hyperbolicity of the combined evolution system. This driver is more stable and effective and, unlike previous drivers, allows stable evolutions using the dual-frame evolution technique. Appropriate boundary conditions for this new gauge driver are constructed, and a new boundary condition for the “gauge” components of the spacetime metric in the GH Einstein system is introduced. The stability and effectiveness of this new gauge driver are demonstrated through numerical tests, which impose a new damped-wave gauge condition on the evolutions of single black-hole spacetimes.

We present a classification of asymptotically flat, supersymmetric black hole and soliton solutions of five-dimensional minimal supergravity that admit a single axial symmetry which `commutes' with the supersymmetry. This includes the first examples of five-dimensional black hole solutions with exactly one axial Killing field that are smooth on and outside the horizon. The solutions have similar properties to the previously studied class with biaxial symmetry, in particular, they have a Gibbons-Hawking base and the harmonic functions must be of multi-centred type with the centres corresponding to the connected components of the horizon or fixed points of the axial symmetry. We find a large moduli space of black hole and soliton spacetimes with non-contractible 2-cycles and the horizon topologies are $S^3$, $S^1\times S^2$ and lens spaces $L(p,1)$.

We present a classification of asymptotically flat, supersymmetric black hole and soliton solutions of five-dimensional minimal supergravity that admit a single axial symmetry which ‘commutes’ with the supersymmetry. This includes the first examples of five-dimensional black hole solutions with exactly one axial Killing field that are smooth on and outside the horizon. The solutions have similar properties to the previously studied class with biaxial symmetry, in particular, they have a Gibbons–Hawking base and the harmonic functions must be of multi-centred type with the centres corresponding to the connected components of the horizon or fixed points of the axial symmetry. We find a large moduli space of black hole and soliton spacetimes with non-contractible 2-cycles and the horizon topologies are $$S^3$$S3 , $$S^1\times S^2$$S1×S2 and lens spaces L(p, 1).

We numerically construct asymptotically anti-de Sitter boson star solutions using a minimally coupled $\frac{D-1}{2}$-tuplet complex scalar field in $D=5,7,9,11$ dimensions. The metric admits multiple Killing vector fields in general, however the scalar fields are only invariant under a particular combination, leading to such boson star solutions possessing just a single helical Killing symmetry. These boson stars form a one parameter family of solutions, which can be parametrized by the energy density at their center. As the central energy density tends to infinity, the angular velocity, mass, and angular momentum of the boson star exhibit damped harmonic oscillations about finite central values, while the Kretschmann invariant diverges, signaling the formation of a black hole in this limit.

Gravitational waves from a binary with a single dynamically significant spin, notably including precessing black hole-neutron star (BH-NS) binaries, let us constrain that binary's properties: the two masses and the dominant black hole spin. Based on a straightforward fourier transform of $h(t)$ enabled by the corotating frame, we show the Fisher matrix for precessing binaries can be well-approximated by an extremely simple semianalytic approximation. This approximation can be easily understood as a weighted average of independent information channels, each associated with one gravitational wave harmonic. Generalizing previous studies of nonprecessing binaries to include critical symmetry-breaking precession effects required to understand plausible astrophysical sources, our ansatz can be applied to address how well gravitational wave measurements can address a wide range of astrophysical and fundamental questions. Our approach provides a simple method to assess what parameters gravitational wave detectors can measure, how well, and why.

We complete the analysis of part I in this series (Ref. \cite{Stotyn:2013yka}) by numerically constructing boson stars in 2+1 dimensional Einstein gravity with negative cosmological constant, minimally coupled to a complex scalar field. These lower dimensional boson stars have strikingly different properties than their higher dimensional counterparts, most noticeably that there exists a finite central energy density, above which an extremal BTZ black hole forms. In this limit, all of the scalar field becomes enclosed by the horizon; it does not contract to a singularity, but rather the origin remains smooth and regular and the solution represents a spinning boson star trapped inside a degenerate horizon. Additionally, whereas in higher dimensions the mass, angular momentum, and angular velocity all display damped harmonic oscillations as functions of the central energy density, in $D=3$ these quantities change monotonically up to the bound on the central energy density. Some implications for the holographic dual of these objects are discussed and it is argued that the boson star and extremal BTZ black hole phases are dual to a spontaneous symmetry breaking at zero temperature but finite energy scale.

We construct rotating boson stars and Myers-Perry black holes with scalar hair (MPBHsSH) as fully non-linear solutions of five dimensional Einstein gravity minimally coupled to a complex, massive scalar field. The MPBHsSH are, in general, regular on and outside the horizon, asymptotically flat, and possess angular momentum in a single rotation plane. They are supported by rotation and have no static limit. Such hairy BHs may be thought of as bound states of boson stars and singly spinning, vacuum MPBHs and inherit properties of both these building blocks. When the horizon area shrinks to zero, the solutions reduce to (in a single plane) rotating boson stars; but the extremal limit also yields a zero area horizon, as for singly spinning MPBHs. Similarly to the case of equal angular momenta, and in contrast to Kerr black holes with scalar hair, singly spinning MPBHsSH are disconnected from the vacuum black holes, due to a mass gap. We observe that for the general case, with two unequal angular momenta, the equilibrium condition for the existence of MPBHsSH is $w=m_1\Omega_1+m_2\Omega_2$, where $\Omega_i$ are the horizon angular velocities in the two independent rotation planes and $w,m_i$, $i=1,2$, are the scalar field's frequency and azimuthal harmonic indices.

Pulsar timing arrays (PTAs) can be used to search for very low frequency ($10^{-9}$--$10^{-7}$ Hz) gravitational waves (GWs). In this paper we present a general method for the detection and localization of single-source GWs using PTAs. We demonstrate the effectiveness of this new method for three types of signals: monochromatic waves as expected from individual supermassive binary black holes in circular orbits, GWs from eccentric binaries and GW bursts. We also test its implementation in realistic data sets that include effects such as uneven sampling and heterogeneous data spans and measurement precision. It is shown that our method, which works in the frequency domain, performs as well as published time-domain methods. In particular, we find it equivalent to the $\mathcal{F}_{e}$-statistic for monochromatic waves. We also discuss the construction of null streams -- data streams that have null response to GWs, and the prospect of using null streams as a consistency check in the case of detected GW signals. Finally, we present sensitivities to individual supermassive binary black holes in eccentric orbits. We find that a monochromatic search that is designed for circular binaries can efficiently detect eccentric binaries with both high and low eccentricities, while a harmonic summing technique provides greater sensitivities only for binaries with moderate eccentricities.

We present a detailed analysis of the expected signal-to-noise ratios of supermassive black hole binaries on eccentric orbits observed by pulsar timing arrays. We derive several analytical relations that extend the results of Peters and Mathews [Phys. Rev. D 131, 435 (1963)] to quantify the impact of eccentricity in the detection of single resolvable binaries in the pulsar timing array band. We present ready-to-use expressions to compute the increase/loss in signal-to-noise ratio of eccentric single resolvable sources whose dominant harmonic is located in the low/high frequency sensitivity regime of pulsar timing arrays. Building upon the work of Phinney (arXiv:astro-ph/0108028) and Enoki and Nagashima [Prog. Theor. Phys. 117, 241 (2007)], we present an analytical framework that enables the construction of rapid spectra for a stochastic gravitational-wave background generated by a cosmological population of eccentric sources. We confirm previous findings which indicate that, relative to a population of quasicircular binaries, the strain of a stochastic, isotropic gravitational-wave background generated by a cosmological population of eccentric binaries will be suppressed in the frequency band of pulsar timing arrays. We quantify this effect in terms of signal-to-noise ratios in a pulsar timing array.

In this New Theory a “Single Harmonic Black Hole” (SHBH) has been considered to be the Gravitational-Electromagnetic Confinement of a Single Harmonic Electromagnetic Field Configuration in which a perfect equilibrium exists between the outward directed electromagnetic radiation pressure and the inward directed Electromagnetic-Gravitational Interaction force densities. This frequency transformation is possible because of the combined Lorentz / Doppler-Effect transformation during the collapse (contraction) of the radiation when the Gravitational Electromagnetic Confinement has been formed (Implosion of Visible Light). Within the scope of this article “Single Harmonic Black Hole” (SHBH) is considered to be any kind of 3-dimensional confined Single Harmonic Electromagnetic Energy. The inner structure of a “SHBH” has been based on a 3-dimensional isotropic equilibrium within the electromagnetic field configuration. This new theory will explain how electromagnetic fields (wave packages) demonstrate inertia, mass and momentum and which forces keep the wave packages together in a way that they can be measured like particles with their own specific mass and momentum. To understand what electromagnetic inertia and the corresponding electromagnetic mass, spin and electric charge is and how the anisotropy of electromagnetic mass, spin and electric charge can be explained and how it has to be defined, a New Theory about “Electromagnetic-Gravitational Interaction” has been developed. The “New Theory” has been based on the fundamental principle of “Perfect Equilibrium within the Universe” which has already been expressed by Newton’s three equations published in 1687 in “Philosophiae Naturalis Principia Mathematica. Newton’s Equations in 3 dimensions will be published in this article in an extension into 4 dimensions. Newton’s 4-dimensional law in the 3 spatial dimensions results in an improved version of the classical Maxwell Equations and Newton’s law in the 4th dimension (time) results in the quantum mechanical Schrödinger wave equation (at non-relativistic velocities) and the relativistic Dirac equation.

A single master equation is given describing spin $s\le 2$ test fields that are gauge- and tetrad-invariant perturbations of the spinning$C$metric space-time representing a source with mass $M$ , uniformly rotating with angular momentum per unit mass $a$ , and uniformly accelerated with acceleration $A$ . This equation can be separated into its radial and angular parts. The behavior of the radial functions near the black hole (outer) horizon is studied to examine the influence of $A$ on the phenomenon of super-radiance, while the angular equation leads to modified spin-weighted spheroidal harmonic solutions generalizing those of the Kerr space-time. Finally the coupling between the spin of the perturbing field and the acceleration parameter $A$ is discussed.

In this chapter we consider perturbations and stability of higher dimensional black holes focusing on the static background case. We first review a gauge-invariant formalism for linear perturbations in a fairly generic class of (m+n)-dimensional spacetimes with a warped product metric, including black hole geometry. We classify perturbations of such a background into three types, the tensor, vector and scalar-type, according to their tensorial behavior on the n-dimensional part of the background spacetime, and for each type of perturbations, we introduce a set of manifestly gauge invariant variables. We then introduce harmonic tensors and write down the equations of motion for the expansion coefficients of the gauge invariant perturbation variables in terms of the harmonics. In particular, for the tensor-type perturbations a single master equation is obtained in the (m+n)-dimensional background, which is applicable for perturbation analysis of not only static black holes but also some class of rotating black holes as well as black-branes. For the vector and scalar type, we derive a set of decoupled master equations when the background is a (2 + n)-dimensional static black hole in the Einstein-Maxwell theory with a cosmological constant. As an application of the master equations, we review the stability analysis of higher dimensional charged static black holes with a cosmological constant. We also briefly review the recent results of a generalization of the perturbation formulae presented here and stability analysis to static black holes in generic Lovelock theory.

It has been recently proposed that quantum black holes can be described as N-graviton Bose-Einstein condensates. In this picture the quantum properties of BHs "... can be understood in terms of the single number N". However, so far, the dynamical origin of the occupational number N has not been specified. This description is alternative to the usual one, where black holes are believed to be well described geometrically even at the quantum level. In this paper we pursue the former point of view and develop a non-geometrical dynamical model of quantum black holes (BHs). In our model the occupational number N is proportional to the principal quantum number n of a Planckian harmonic oscillator. The so-called "classicalization" corresponds to the large-n limit, where the Schwarzschild horizon is recovered.

We show that nearly extremal Kerr black holes have two distinct sets of quasinormal modes, which we call zero-damping modes (ZDMs) and damped modes (DMs). The ZDMs exist for all harmonic indices $l$ and $m\ge 0$, and their frequencies cluster onto the real axis in the extremal limit. The DMs have nonzero damping for all black hole spins; they exist for all counterrotating modes ($m<0$) and for corotating modes with $0\le \mu \lesssim {\mu }_c=0.74$ (in the eikonal limit), where $\mu \equiv m/(l+1/2)$. When the two families coexist, ZDMs and DMs merge to form a single set of quasinormal modes as the black hole spin decreases. Using the effective potential for perturbations of the Kerr spacetime, we give intuitive explanations for the absence of DMs in certain areas of the spectrum and for the branching of the spectrum into ZDMs and DMs at large spins.

The standard model of cosmology is underpinned by the assumption of the statistical isotropy of the Universe. Observations of the cosmic microwave background, galaxy distributions, and supernovae, among other media, support the assumption of isotropy at scales $\gtrsim 100\mathrm{Mpc}$. The recent detections of gravitational waves from merging stellar-mass binary black holes provide a new probe of anisotropy; complementary and independent of all other probes of the matter distribution in the Universe. We present an analysis using a spherical harmonic model to determine the level of anisotropy in the first LIGO/Virgo transient catalog. We find that the ten binary black hole mergers within the first transient catalog are consistent with an isotropic distribution. We carry out a study of simulated events to assess the prospects for future probes of anisotropy. Within a single year of operation, third-generation gravitational-wave observatories will probe anisotropies with an angular scale of $\sim 36°$ at the level of $\lesssim 0.1\%$.

The standard model of cosmology is underpinned by the assumption of the statistical isotropy of the Universe. Observations of the cosmic microwave background, galaxy distributions, and supernovae, among other media, support the assumption of isotropy at scales $\gtrsim 100$\,Mpc. The recent detections of gravitational waves from merging stellar-mass binary black holes provide a new probe of anisotropy; complementary and independent of all other probes of the matter distribution in the Universe. We present an analysis using a spherical harmonic model to determine the level of anisotropy in the first LIGO/Virgo transient catalog. We find that the ten binary black hole mergers within the first transient catalog are consistent with an isotropic distribution. We carry out a study of simulated events to assess the prospects for future probes of anisotropy. Within a single year of operation, third-generation gravitational-wave observatories will probe anisotropies with an angular scale of $\sim36^\circ$ at the level of $\lesssim0.1\%$.

Observations of X‐ray emissions from binary systems have always been considered important tools to test the validity of General Relativity in strong‐field regimes. The pairs and triplets of high frequency quasi‐periodic oscillations observed in binaries containing a black hole candidate, in particular, have been proposed as a means to measure more directly the black hole properties such as its mass and spin. Numerous models have been suggested over the years to explain the QPOs and the rich phenomenology accompanying them. Many of these models rest on a number of assumptions and are at times in conflict with the most recent observations. We here propose a new, simple model in which the QPOs result from basic p‐mode oscillations of a non‐Keplerian disc of finite size. We show that within this new model all of the key properties of the QPOs: a) harmonic ratios of frequencies even as the frequencies change; b) variations in the relative strength of the frequencies with spectral energy distribution and with photon energy; c) small and systematic changes in the frequencies, can all be explained simply given a single reasonable assumption.

Among the transient black hole binary systems, 4U 1630-47 is one of the most active sources exhibiting outbursts every few hundred days, with every outburst lasting typically around hundred days. During the 2002-2004 outburst the appearance of quasi-periodic oscillations (QPOs) coincide with the onset of anomalous state. There are two distinct QPO states: namely the single QPO state with one QPO and the twin QPO state with two QPOs not related harmonically. The spectral features of this state are corroborated by a previous outburst in 1998. The evolution of the inner disc temperature and the inner disc radius suggest the possible onset of geometrically thicker, slim disc as a possible explanation of the energy spectral features. The other possibilities involving the Comptonizing cloud also exist that may explain the anomalous state. The two different QPO states exhibit different spectral features, and we provide a possible empirical physical scenario from the observation of the evolution of the spectral features.

Recently it was shown that the inclusion of higher signal harmonics in the inspiral signals of binary supermassive black holes (SMBH) leads to dramatic improvements in parameter estimation with the Laser Interferometer Space Antenna (LISA). In particular, the angular resolution becomes good enough to identify the host galaxy or galaxy cluster, in which case the redshift can be determined by electromagnetic means. The gravitational wave signal also provides the luminosity distance with high accuracy, and the relationship between this and the redshift depends sensitively on the cosmological parameters, such as the equation-of-state parameter $w=p_{ m DE}/ ho_{ m DE}$ of dark energy. With a single binary SMBH event at $z < 1$ having appropriate masses and orientation, one would be able to constrain $w$ to within a few percent. We show that, if the measured sky location is folded into the error analysis, the uncertainty on $w$ goes down by an additional factor of 2-3, leaving weak lensing as the only limiting factor in using LISA as a dark energy probe.