Charalampos Markakis | University of Cambridge (original) (raw)

Papers by Charalampos Markakis

Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and... more Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamilton's principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike what has been previously thought, this approach can also accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron description of ideal mag-netohydrodynamics. When Noether symmetries associated with Killing vectors or tensors are present in geodesic flows, they lead to constants of motion polynomial in the momenta. We generalize these concepts to hydrodynamic flows. Moreover, the Hamiltonian descriptions of ideal magnetohydrodynamics allow one to cast the evolution equations into a hyperbolic form useful for evolving rotating or binary compact objects with magnetic fields in numerical general relativity. In this framework, Ertel's potential vorticity theorem for baroclinic fluids arises as a special case of a conservation law valid for any Hamiltonian system. Moreover, conserved circulation laws, such as those of Kelvin, Alfvén and Bekenstein-Oron, emerge simply as special cases of the Poincaré-Cartan integral invariant of Hamiltonian systems. We use this approach to obtain an extension of Kelvin's theorem to baroclinic (nonisentropic) fluids, based on a temperature-dependent time parameter. We further extend this result to perfectly or poorly conducting baroclinic magnetoflows. Finally, in the barotropic case, such magnetoflows are shown to also be geodesic, albeit in a Finsler (rather than Riemann) space.

Stationary and axisymmetric solutions of relativistic rotating stars with strong mixed poloidal a... more Stationary and axisymmetric solutions of relativistic rotating stars with strong mixed poloidal and toroidal magnetic fields are obtained numerically. Because of the mixed components of the magnetic field, the underlying stationary and axisymmetric spacetimes are no longer circular. These configurations are computed from the full set of the Einstein-Maxwell equations, Maxwell's equations, and from first integrals and integrability conditions of the magnetohydrodynamic equilibrium equations. After a brief introduction of the formulation of the problem, we present the first results for highly deformed magnetized rotating compact stars.

Binary neutron stars in circular orbits can be modeled as helically symmetric, i.e., stationary i... more Binary neutron stars in circular orbits can be modeled as helically symmetric, i.e., stationary in a rotating frame. This symmetry gives rise to a first integral of the Euler equation, often employed for constructing equilibrium solutions via iteration. For eccentric orbits, however, the lack of helical symmetry has prevented the use of this method, and the numerical relativity community has often resorted to constructing initial data by superimposing boosted spherical stars without solving the Euler equation. The spuriously excited neutron star oscillations seen in evolutions of such data arise because such configurations lack the appropriate tidal deformations and are stationary in a linearly comoving—rather than rotating—frame. We consider eccentric configurations at apoapsis that are instantaneously stationary in a rotating frame. We extend the notion of helical symmetry to eccentric orbits, by approximating the elliptical orbit of each companion as instantaneously circular, using the ellipse's inscribed circle. The two inscribed helical symmetry vectors give rise to approximate instantaneous first integrals of the Euler equation throughout each companion. We use these integrals as the basis of a self-consistent iteration of the Einstein constraints to construct conformal thin-sandwich initial data for eccentric binaries. We find that the spurious stellar oscillations are reduced by at least an order of magnitude, compared with those found in evolutions of superposed initial data. The tidally induced oscillations, however, are physical and qualitatively similar to earlier evolutions. Finally, we show how to incorporate radial velocity due to radiation reaction in our inscribed helical symmetry vectors, which would allow one to obtain truly noneccentric initial data when our eccentricity parameter e is set to zero.

The motion of test particles in stationary axisymmetric gravitational fields is generally non-int... more The motion of test particles in stationary axisymmetric gravitational fields is generally non-integrable unless a non-trivial constant of motion, in addition to energy and angular momentum along the symmetry axis, exists. The Carter constant in Kerr–de Sitter space–time is the only example known to date. Proposed astrophysical tests of the black hole no-hair theorem have often involved integrable gravitational fields more general than the Kerr family, but the existence of such fields has been a matter of debate. To elucidate this problem, we treat its Newtonian analogue by systematically searching for non-trivial constants of motion polynomial in the momenta and obtain two theorems. First, solving a set of quadratic integrability conditions, we establish the existence and uniqueness of the family of stationary axisymmetric potentials admitting a quadratic constant. As in Kerr–de Sitter space–time, the mass moments of this class satisfy a 'no-hair' recursion relation M 2l +2 = a 2 M 2l , and the constant is Noether related to a second-order Killing–Stäckel tensor. Second, solving a new set of quartic integrability conditions, we establish non-existence of quartic constants. Remarkably, a subset of these conditions is satisfied when the mass moments obey a generalized 'no-hair' recursion relation M 2l +4 = (a 2 + b 2)M 2l +2 − a 2 b 2 M 2l. The full set of quartic integrability conditions, however, cannot be satisfied non-trivially by any stationary axisymmetric vacuum potential.

Gravitational waves from neutron-star and black-hole binaries carry valuable information on their... more Gravitational waves from neutron-star and black-hole binaries carry valuable information on their physical properties and probe physics inaccessible to the laboratory. Although development of black-hole gravitational-wave templates in the past decade has been revolutionary, the corresponding work for double neutron-star systems has lagged. Neutron stars can be well-modelled as simple barotropic fluids during the part of binary inspiral most relevant to gravitational wave astronomy, but the crucial geometric and mathematical consequences of this simplification have remained computationally unexploited. In particular, Carter and Lichnerowicz have described barotropic fluid motion via classical variational principles as conformally geodesic. Moreover, Kelvin's circulation theorem implies that initially irrotational flows remain irrotational. Applied to numerical relativ-ity, these concepts lead to novel Hamiltonian or Hamilton-Jacobi schemes for evolving relativistic fluid flows. Hamiltonian methods can conserve not only flux, but also circulation and symplecticity, and moreover do not require addition of an artificial atmosphere typically required by standard conservative methods. These properties can allow production of high-precision gravitational waveforms at low computational cost. This canonical hydrodynamics approach is applicable to a wide class of problems involving theoretical or computational fluid dynamics.

High order finite-difference or spectral methods are typically problematic in approximating a fun... more High order finite-difference or spectral methods are typically problematic in approximating a function with a jump discontinuity. Some common remedies come with a cost in accuracy near discontinuities, or in computational cost, or in complexity of implementation. However, for certain classes of problems involving piecewise analytic functions, the jump in the function and its derivatives are known or easy to compute. We show that high-order or spectral accuracy can then be recovered by simply adding to the Lagrange interpolation formula a linear combination of the jumps. Discretizations developed for smooth problems are thus easily extended to nonsmooth problems. Furthermore, in the context of one-dimensional finite-difference or pseudospectral discretizations, numerical integration and differentiation amount to matrix multiplication. We construct the matrices for such operations, in the presence of known discontinuities, by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient way to obtain solutions with moving discontinuities to evolution partial differential equations.

Information about the last stages of a binary neutron star inspiral and the final merger can be e... more Information about the last stages of a binary neutron star inspiral and the final merger can be extracted from quasiequilibrium configurations and dynamical evolutions. In this article, we construct quasiequili-brium configurations for different spins, eccentricities, mass ratios, compactnesses, and equations of state. For this purpose we employ the SGRID code, which allows us to construct such data in previously inaccessible regions of the parameter space. In particular, we consider spinning neutron stars in isolation and in binary systems; we incorporate new methods to produce highly eccentric and eccentricity-reduced data; we present the possibility of computing data for significantly unequal-mass binaries with mass ratios q ≃ 2; and we create equal-mass binaries with individual compactness up to C ≃ 0.23. As a proof of principle, we explore the dynamical evolution of three new configurations. First, we simulate a q ¼ 2.06 mass ratio which is the highest mass ratio for a binary neutron star evolved in numerical relativity to date. We find that mass transfer from the companion star sets in a few revolutions before merger and a rest mass of ∼10 −2 M ⊙ is transferred between the two stars. This amount of mass accretion corresponds to ∼10 51 ergs of accretion energy. This configuration also ejects a large amount of material during merger (∼7.6 × 10 −2 M ⊙), imparting a substantial kick to the remnant neutron star. Second, we simulate the first merger of a precessing binary neutron star. We present the dominant modes of the gravitational waves for the precessing simulation, where a clear imprint of the precession is visible in the (2,1) mode. Finally, we quantify the effect of an eccentricity-reduction procedure on the gravitational waveform. The procedure improves the waveform quality and should be employed in future precision studies. However, one also needs to reduce other errors in the waveforms, notably truncation errors, in order for the improvement due to eccentricity reduction to be effective.

Physical Review D, 2013

Using an extended set of equations of state and a multiple-group multiple-code collaborative effo... more Using an extended set of equations of state and a multiple-group multiple-code collaborative effort to generate waveforms, we improve numerical-relativity-based data-analysis estimates of the measurability of matter effects in neutron-star binaries. We vary two parameters of a parameterized piecewise-polytropic equation of state (EOS) to analyze the measurability of EOS properties, via a parameter Λ that characterizes the quadrupole deformability of an isolated neutron star. We find that, to within the accuracy of the simulations, the departure of the waveform from point-particle (or spinless double black-hole binary) inspiral increases monotonically with Λ, and changes in the EOS that did not change Λ are not measurable.

Physical Review D, 2009

As the stars approach their final plunge and merger, the gravitational wave phase accumulates mor... more As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron star compactness (the ratio of the mass of the neutron star to its radius). We estimate that realistic equations of state will lead to gravitational waveforms that are distinguishable from point particle inspirals at an effective distance (the distance to an optimally oriented and located system that would produce an equivalent waveform amplitude) of 100 Mpc or less. As Lattimer and Prakash observed, neutron-star radius is closely tied to the pressure at density not far above nuclear. Our results suggest that broadband gravitational wave observations at frequencies between 500 and 1000 Hz will constrain this pressure, and we estimate the accuracy with which it can be measured. Related first estimates of radius measurability show that the radius can be determined to an accuracy of ~1 km at 100 Mpc.

Journal of Physics: Conference Series, 2011

We report work towards a relativistic formulation for modeling strongly magnetized neutron stars,... more We report work towards a relativistic formulation for modeling strongly magnetized neutron stars, rotating or in a close circular orbit around another neutron star or black hole, under the approximations of helical symmetry and ideal MHD. The quasi-stationary evolution is governed by the frst law of thermodynamics for helically symmetric systems, which is generalized to include magnetic felds. The formulation involves an iterative scheme for solving the Einstein-Maxwell and relativistic MHD-Euler equations numerically. The resulting configurations for binary systems could be used as self-consistent initial data for studying their inspiral and merger.

Physical Review D, 2009

We report the results of a first study that uses numerical simulations to estimate the accuracy w... more We report the results of a first study that uses numerical simulations to estimate the accuracy with which one can use gravitational wave observations of double neutron-star inspiral to measure parameters of the neutron-star equation of state. The simulations use the evolution and initial-data codes of Shibata and Uryū to compute the last several orbits and the merger of neutron stars, with matter described by a parametrized equation of state. Previous work suggested the use of an effective cutoff frequency to place constraints on the equation of state. We find, however, that greater accuracy is obtained by measuring departures from the point-particle limit of the gravitational waveform produced during the late inspiral. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron-star compactness (the ratio of the mass of the neutron-star to its radius). We estimate that realistic equations of state will lead to gravitational waveforms that are distinguishable from point-particle inspirals at an effective distance (the distance to an optimally oriented and located system that would produce an equivalent waveform amplitude) of 100 Mpc or less. As Lattimer and Prakash observed, neutron-star radius is closely tied to the pressure at density not far above nuclear. Our results suggest that broadband gravitational wave observations at frequencies between 500 and 1000 Hz will constrain this pressure, and we estimate the accuracy with which it can be measured. Related first estimates of radius measurability show that the radius can be determined to an accuracy of δR˜1km at 100 Mpc.

For an equation of state in which pressure is a function only of density, the analysis of Newtoni... more For an equation of state in which pressure is a function only of density, the analysis of Newtonian stellar structure is simple in principle if the system is axisymmetric or consists of a corotating binary. It is then required only to solve two equations: one stating that the ``injection energy,'' κ, a potential, is constant throughout the stellar fluid, and the other being the integral over the stellar fluid to give the gravitational potential. An iterative solution of these equations generally diverges if κ is held fixed, but converges with other choices. To understand the mathematical reasons for this, we start the iteration from an approximation that is perturbatively different from the actual solution and, for the current study, confine ourselves to spherical symmetry. A cycle of iteration is treated as a linear ``updating'' operator, and the properties of the linear operator, especially its spectrum, determine the convergence properties. We analyze updating operators both in the finite dimensional space corresponding to a finite difference representation of the problem and in the continuum, and we find that the fixed-κ operator is self-adjoint and generally has an eigenvalue greater than unity; in the particularly important case of a polytropic equation of state with index greater than unity, we prove that there must be such an eigenvalue. For fixed central density, on the other hand, we find that the updating operator has only a single eigenvector with zero eigenvalue and is nilpotent in finite dimension, thereby giving a convergent solution.

We report work in progress towards a relativistic formulation for constructing magnetized rotatin... more We report work in progress towards a relativistic formulation for constructing magnetized rotating or binary neutron star initial data, in an ideal MHD approximation. The formulation involves a self-consistent scheme for solving the Einstein-Maxwell and MHD-Euler equations for systems with an approximate helical symmetry. Numerical codes based on this scheme are expected to model magnetars with non-axisymmetric magnetic fields as well as magnetized binary neutron star systems in quasi-equilibrium. )

For an equation of state in which pressure is a function only of density, the analysis of Newtoni... more For an equation of state in which pressure is a function only of density, the analysis of Newtonian stellar structure is simple in principle if the system is axisymmetric, or consists of a corotating binary. It is then required only to solve two equations: one stating that the "injection energy," κ, a potential, is constant throughout the stellar fluid, and the other being the integral over the stellar fluid to give the gravitational potential. An iterative solution of these equations generally diverges if κ is held fixed, but converges with other choices. We investigate the mathematical reason for this convergence/divergence by starting the iteration from an approximation that is perturbatively different from the actual solution. A cycle of iteration is then treated as a linear "updating" operator, and the properties of the linear operator, especially its spectrum, determine the convergence properties. For simplicity, we confine ourselves to spherically symmetric models in which we analyze updating operators both in the finite dimensional space corresponding to a finite difference representation of the problem, and in the continuum, and we find that the fixed-κ operator is self-adjoint and generally has an eigenvalue greater than unity; in the particularly important case of a polytropic equation of state with index greater than unity, we prove that there must be such an eigenvalue. For fixed central density, on the other hand, we find that the updating operator has only a single eigenvector, with zero eigenvalue, and is nilpotent in finite dimension, thereby giving a convergent solution.

Physical Review D, 2008

Quasi-equilibrium models of rapidly rotating triaxially deformed stars are computed in general re... more Quasi-equilibrium models of rapidly rotating triaxially deformed stars are computed in general relativistic gravity, assuming a conformally flat spatial geometry (Isenberg-Wilson-Mathews formulation) and a polytropic equation of state. Highly deformed solutions are calculated on the initial slice covered by spherical coordinate grids, centered at the source, in all angular directions up to a large truncation radius. Constant rest mass sequences are calculated from nearly axisymmetric to maximally deformed triaxial configurations. Selected parameters are to model (proto-) neutron stars; the compactness is M/R = 0.001, 0.1, 0.14, 0.2 for polytropic index n = 0.3 and M/R = 0.001, 0.1, 0.12, 0.14 for n = 0.5. We confirmed that the triaxial solutions exist for these parameters as in the case of Newtonian polytropes. However, it is also found that the triaxial sequences become shorter for higher compactness, and those may disappear at a certain large compactness for the n = 0.5 case. In the scenario of the contraction of proto-neutron stars being subject to strong viscosity and rapid cooling, it is plausible that, once the viscosity driven secular instability sets in during the contraction, the proto-neutron stars are always maximally deformed triaxial configurations, as long as the compactness and the equation of state parameters allow such triaxial sequences. Detection of gravitational waves from such sources may be used as another probe for the nuclear equation of state.

Physical Review D, 2010

Binary systems of compact objects with electromagnetic field are modeled by helically symmetric E... more Binary systems of compact objects with electromagnetic field are modeled by helically symmetric Einstein-Maxwell spacetimes with charged and magnetized perfect fluids. Previously derived thermodynamic laws for helically-symmetric perfect-fluid spacetimes are extended to include the electromagnetic fields, and electric currents and charges; the first law is written as a relation between the change in the asymptotic Noether charge dlQ\dl QdlQ and the changes in the area and electric charge of black holes, and in the vorticity, baryon rest mass, entropy, charge and magnetic flux of the magnetized fluid. Using the conservation laws of the circulation of magnetized flow found by Bekenstein and Oron for the ideal magnetohydrodynamic (MHD) fluid, and also for the flow with zero conducting current, we show that, for nearby equilibria that conserve the quantities mentioned above, the relation dlQ=0\dl Q=0dlQ=0 is satisfied. We also discuss a formulation for computing numerical solutions of magnetized binary compact objects in equilibrium with emphasis on a first integral of the ideal MHD-Euler equation.

The properties of neutron star matter above nuclear density are not precisely known. Gravitationa... more The properties of neutron star matter above nuclear density are not precisely known. Gravitational waves emitted from binary neutron stars during their late stages of inspiral and merger contain imprints of the neutron-star equation of state. Measuring departures from the point-particle limit of the late inspiral waveform allows one to measure properties of the equation of state via gravitational wave observations. This and a companion talk by J. S. Read reports a comparison of numerical waveforms from simulations of inspiraling neutron-star binaries, computed for equations of state with varying stiffness. We calculate the signal strength of the difference between waveforms for various commissioned and proposed interferometric gravitational wave detectors and show that observations at frequencies around 1 kHz will be able to measure a compactness parameter and constrain the possible neutron-star equations of state.

Physical Review D, 2009

We report the results of a first study that uses numerical simulations to estimate the accuracy w... more We report the results of a first study that uses numerical simulations to estimate the accuracy with which one can use gravitational wave observations of double neutron star inspiral to measure parameters of the neutron-star equation of state. The simulations use the evolution and initial-data codes of Shibata and Uryū to compute the last several orbits and the merger of neutron stars, with matter described by a parametrized equation of state. Previous work suggested the use of an effective cutoff frequency to place constraints on the equation of state. We find, however, that greater accuracy is obtained by measuring departures from the point-particle limit of the gravitational waveform produced during the late inspiral. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron star compactness (the ratio of the mass of the neutron star to its radius). We estimate that realistic equations of state will lead to gravitational waveforms that are distinguishable from point particle inspirals at an effective distance (the distance to an optimally oriented and located system that would produce an equivalent waveform amplitude) of 100 Mpc or less. As Lattimer and Prakash observed, neutron-star radius is closely tied to the pressure at density not far above nuclear. Our results suggest that broadband gravitational wave observations at frequencies between 500 and 1000 Hz will constrain this pressure, and we estimate the accuracy with which it can be measured. Related first estimates of radius measurability show that the radius can be determined to an accuracy of δR ∼ 1 km at 100 Mpc.

Properties of the neutron star equation of state can potentially be measured via gravitational wa... more Properties of the neutron star equation of state can potentially be measured via gravitational wave observations, by measuring departures from the point-particle limit of the waveform produced in the late inspiral of a neutron star binary. Numerical waveforms from simulations of inspiraling neutron star binaries, computed for equations of state with varying stiffness, are compared. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron star compactness. This suggests that gravitational wave observations at frequencies around ˜1 kHz will be able to measure a compactness parameter and constrain the possible neutron star equations of state.

Similar methods have been used to construct models of rapidly rotating stars, in Newtonian and re... more Similar methods have been used to construct models of rapidly rotating stars, in Newtonian and relativistic contexts. The choice of method has been based on numerical experiments, which indicate that particular methods converge quickly to a solution, while others diverge. The theory underlying these differences, however, has not been understood. In an attempt to provide a better theoretical understanding, we analytically examine the behavior of different iterative schemes near an exact solution. We find the spectrum of the linearized iteration operator and show for self-consistent field methods that iterative instability corresponds to unstable modes of this operator.

Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and... more Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamilton's principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike what has been previously thought, this approach can also accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron description of ideal mag-netohydrodynamics. When Noether symmetries associated with Killing vectors or tensors are present in geodesic flows, they lead to constants of motion polynomial in the momenta. We generalize these concepts to hydrodynamic flows. Moreover, the Hamiltonian descriptions of ideal magnetohydrodynamics allow one to cast the evolution equations into a hyperbolic form useful for evolving rotating or binary compact objects with magnetic fields in numerical general relativity. In this framework, Ertel's potential vorticity theorem for baroclinic fluids arises as a special case of a conservation law valid for any Hamiltonian system. Moreover, conserved circulation laws, such as those of Kelvin, Alfvén and Bekenstein-Oron, emerge simply as special cases of the Poincaré-Cartan integral invariant of Hamiltonian systems. We use this approach to obtain an extension of Kelvin's theorem to baroclinic (nonisentropic) fluids, based on a temperature-dependent time parameter. We further extend this result to perfectly or poorly conducting baroclinic magnetoflows. Finally, in the barotropic case, such magnetoflows are shown to also be geodesic, albeit in a Finsler (rather than Riemann) space.

Stationary and axisymmetric solutions of relativistic rotating stars with strong mixed poloidal a... more Stationary and axisymmetric solutions of relativistic rotating stars with strong mixed poloidal and toroidal magnetic fields are obtained numerically. Because of the mixed components of the magnetic field, the underlying stationary and axisymmetric spacetimes are no longer circular. These configurations are computed from the full set of the Einstein-Maxwell equations, Maxwell's equations, and from first integrals and integrability conditions of the magnetohydrodynamic equilibrium equations. After a brief introduction of the formulation of the problem, we present the first results for highly deformed magnetized rotating compact stars.

Binary neutron stars in circular orbits can be modeled as helically symmetric, i.e., stationary i... more Binary neutron stars in circular orbits can be modeled as helically symmetric, i.e., stationary in a rotating frame. This symmetry gives rise to a first integral of the Euler equation, often employed for constructing equilibrium solutions via iteration. For eccentric orbits, however, the lack of helical symmetry has prevented the use of this method, and the numerical relativity community has often resorted to constructing initial data by superimposing boosted spherical stars without solving the Euler equation. The spuriously excited neutron star oscillations seen in evolutions of such data arise because such configurations lack the appropriate tidal deformations and are stationary in a linearly comoving—rather than rotating—frame. We consider eccentric configurations at apoapsis that are instantaneously stationary in a rotating frame. We extend the notion of helical symmetry to eccentric orbits, by approximating the elliptical orbit of each companion as instantaneously circular, using the ellipse's inscribed circle. The two inscribed helical symmetry vectors give rise to approximate instantaneous first integrals of the Euler equation throughout each companion. We use these integrals as the basis of a self-consistent iteration of the Einstein constraints to construct conformal thin-sandwich initial data for eccentric binaries. We find that the spurious stellar oscillations are reduced by at least an order of magnitude, compared with those found in evolutions of superposed initial data. The tidally induced oscillations, however, are physical and qualitatively similar to earlier evolutions. Finally, we show how to incorporate radial velocity due to radiation reaction in our inscribed helical symmetry vectors, which would allow one to obtain truly noneccentric initial data when our eccentricity parameter e is set to zero.

The motion of test particles in stationary axisymmetric gravitational fields is generally non-int... more The motion of test particles in stationary axisymmetric gravitational fields is generally non-integrable unless a non-trivial constant of motion, in addition to energy and angular momentum along the symmetry axis, exists. The Carter constant in Kerr–de Sitter space–time is the only example known to date. Proposed astrophysical tests of the black hole no-hair theorem have often involved integrable gravitational fields more general than the Kerr family, but the existence of such fields has been a matter of debate. To elucidate this problem, we treat its Newtonian analogue by systematically searching for non-trivial constants of motion polynomial in the momenta and obtain two theorems. First, solving a set of quadratic integrability conditions, we establish the existence and uniqueness of the family of stationary axisymmetric potentials admitting a quadratic constant. As in Kerr–de Sitter space–time, the mass moments of this class satisfy a 'no-hair' recursion relation M 2l +2 = a 2 M 2l , and the constant is Noether related to a second-order Killing–Stäckel tensor. Second, solving a new set of quartic integrability conditions, we establish non-existence of quartic constants. Remarkably, a subset of these conditions is satisfied when the mass moments obey a generalized 'no-hair' recursion relation M 2l +4 = (a 2 + b 2)M 2l +2 − a 2 b 2 M 2l. The full set of quartic integrability conditions, however, cannot be satisfied non-trivially by any stationary axisymmetric vacuum potential.

Gravitational waves from neutron-star and black-hole binaries carry valuable information on their... more Gravitational waves from neutron-star and black-hole binaries carry valuable information on their physical properties and probe physics inaccessible to the laboratory. Although development of black-hole gravitational-wave templates in the past decade has been revolutionary, the corresponding work for double neutron-star systems has lagged. Neutron stars can be well-modelled as simple barotropic fluids during the part of binary inspiral most relevant to gravitational wave astronomy, but the crucial geometric and mathematical consequences of this simplification have remained computationally unexploited. In particular, Carter and Lichnerowicz have described barotropic fluid motion via classical variational principles as conformally geodesic. Moreover, Kelvin's circulation theorem implies that initially irrotational flows remain irrotational. Applied to numerical relativ-ity, these concepts lead to novel Hamiltonian or Hamilton-Jacobi schemes for evolving relativistic fluid flows. Hamiltonian methods can conserve not only flux, but also circulation and symplecticity, and moreover do not require addition of an artificial atmosphere typically required by standard conservative methods. These properties can allow production of high-precision gravitational waveforms at low computational cost. This canonical hydrodynamics approach is applicable to a wide class of problems involving theoretical or computational fluid dynamics.

High order finite-difference or spectral methods are typically problematic in approximating a fun... more High order finite-difference or spectral methods are typically problematic in approximating a function with a jump discontinuity. Some common remedies come with a cost in accuracy near discontinuities, or in computational cost, or in complexity of implementation. However, for certain classes of problems involving piecewise analytic functions, the jump in the function and its derivatives are known or easy to compute. We show that high-order or spectral accuracy can then be recovered by simply adding to the Lagrange interpolation formula a linear combination of the jumps. Discretizations developed for smooth problems are thus easily extended to nonsmooth problems. Furthermore, in the context of one-dimensional finite-difference or pseudospectral discretizations, numerical integration and differentiation amount to matrix multiplication. We construct the matrices for such operations, in the presence of known discontinuities, by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient way to obtain solutions with moving discontinuities to evolution partial differential equations.

Information about the last stages of a binary neutron star inspiral and the final merger can be e... more Information about the last stages of a binary neutron star inspiral and the final merger can be extracted from quasiequilibrium configurations and dynamical evolutions. In this article, we construct quasiequili-brium configurations for different spins, eccentricities, mass ratios, compactnesses, and equations of state. For this purpose we employ the SGRID code, which allows us to construct such data in previously inaccessible regions of the parameter space. In particular, we consider spinning neutron stars in isolation and in binary systems; we incorporate new methods to produce highly eccentric and eccentricity-reduced data; we present the possibility of computing data for significantly unequal-mass binaries with mass ratios q ≃ 2; and we create equal-mass binaries with individual compactness up to C ≃ 0.23. As a proof of principle, we explore the dynamical evolution of three new configurations. First, we simulate a q ¼ 2.06 mass ratio which is the highest mass ratio for a binary neutron star evolved in numerical relativity to date. We find that mass transfer from the companion star sets in a few revolutions before merger and a rest mass of ∼10 −2 M ⊙ is transferred between the two stars. This amount of mass accretion corresponds to ∼10 51 ergs of accretion energy. This configuration also ejects a large amount of material during merger (∼7.6 × 10 −2 M ⊙), imparting a substantial kick to the remnant neutron star. Second, we simulate the first merger of a precessing binary neutron star. We present the dominant modes of the gravitational waves for the precessing simulation, where a clear imprint of the precession is visible in the (2,1) mode. Finally, we quantify the effect of an eccentricity-reduction procedure on the gravitational waveform. The procedure improves the waveform quality and should be employed in future precision studies. However, one also needs to reduce other errors in the waveforms, notably truncation errors, in order for the improvement due to eccentricity reduction to be effective.

Physical Review D, 2013

Using an extended set of equations of state and a multiple-group multiple-code collaborative effo... more Using an extended set of equations of state and a multiple-group multiple-code collaborative effort to generate waveforms, we improve numerical-relativity-based data-analysis estimates of the measurability of matter effects in neutron-star binaries. We vary two parameters of a parameterized piecewise-polytropic equation of state (EOS) to analyze the measurability of EOS properties, via a parameter Λ that characterizes the quadrupole deformability of an isolated neutron star. We find that, to within the accuracy of the simulations, the departure of the waveform from point-particle (or spinless double black-hole binary) inspiral increases monotonically with Λ, and changes in the EOS that did not change Λ are not measurable.

Physical Review D, 2009

As the stars approach their final plunge and merger, the gravitational wave phase accumulates mor... more As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron star compactness (the ratio of the mass of the neutron star to its radius). We estimate that realistic equations of state will lead to gravitational waveforms that are distinguishable from point particle inspirals at an effective distance (the distance to an optimally oriented and located system that would produce an equivalent waveform amplitude) of 100 Mpc or less. As Lattimer and Prakash observed, neutron-star radius is closely tied to the pressure at density not far above nuclear. Our results suggest that broadband gravitational wave observations at frequencies between 500 and 1000 Hz will constrain this pressure, and we estimate the accuracy with which it can be measured. Related first estimates of radius measurability show that the radius can be determined to an accuracy of ~1 km at 100 Mpc.

Journal of Physics: Conference Series, 2011

We report work towards a relativistic formulation for modeling strongly magnetized neutron stars,... more We report work towards a relativistic formulation for modeling strongly magnetized neutron stars, rotating or in a close circular orbit around another neutron star or black hole, under the approximations of helical symmetry and ideal MHD. The quasi-stationary evolution is governed by the frst law of thermodynamics for helically symmetric systems, which is generalized to include magnetic felds. The formulation involves an iterative scheme for solving the Einstein-Maxwell and relativistic MHD-Euler equations numerically. The resulting configurations for binary systems could be used as self-consistent initial data for studying their inspiral and merger.

Physical Review D, 2009

We report the results of a first study that uses numerical simulations to estimate the accuracy w... more We report the results of a first study that uses numerical simulations to estimate the accuracy with which one can use gravitational wave observations of double neutron-star inspiral to measure parameters of the neutron-star equation of state. The simulations use the evolution and initial-data codes of Shibata and Uryū to compute the last several orbits and the merger of neutron stars, with matter described by a parametrized equation of state. Previous work suggested the use of an effective cutoff frequency to place constraints on the equation of state. We find, however, that greater accuracy is obtained by measuring departures from the point-particle limit of the gravitational waveform produced during the late inspiral. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron-star compactness (the ratio of the mass of the neutron-star to its radius). We estimate that realistic equations of state will lead to gravitational waveforms that are distinguishable from point-particle inspirals at an effective distance (the distance to an optimally oriented and located system that would produce an equivalent waveform amplitude) of 100 Mpc or less. As Lattimer and Prakash observed, neutron-star radius is closely tied to the pressure at density not far above nuclear. Our results suggest that broadband gravitational wave observations at frequencies between 500 and 1000 Hz will constrain this pressure, and we estimate the accuracy with which it can be measured. Related first estimates of radius measurability show that the radius can be determined to an accuracy of δR˜1km at 100 Mpc.

For an equation of state in which pressure is a function only of density, the analysis of Newtoni... more For an equation of state in which pressure is a function only of density, the analysis of Newtonian stellar structure is simple in principle if the system is axisymmetric or consists of a corotating binary. It is then required only to solve two equations: one stating that the ``injection energy,'' κ, a potential, is constant throughout the stellar fluid, and the other being the integral over the stellar fluid to give the gravitational potential. An iterative solution of these equations generally diverges if κ is held fixed, but converges with other choices. To understand the mathematical reasons for this, we start the iteration from an approximation that is perturbatively different from the actual solution and, for the current study, confine ourselves to spherical symmetry. A cycle of iteration is treated as a linear ``updating'' operator, and the properties of the linear operator, especially its spectrum, determine the convergence properties. We analyze updating operators both in the finite dimensional space corresponding to a finite difference representation of the problem and in the continuum, and we find that the fixed-κ operator is self-adjoint and generally has an eigenvalue greater than unity; in the particularly important case of a polytropic equation of state with index greater than unity, we prove that there must be such an eigenvalue. For fixed central density, on the other hand, we find that the updating operator has only a single eigenvector with zero eigenvalue and is nilpotent in finite dimension, thereby giving a convergent solution.

We report work in progress towards a relativistic formulation for constructing magnetized rotatin... more We report work in progress towards a relativistic formulation for constructing magnetized rotating or binary neutron star initial data, in an ideal MHD approximation. The formulation involves a self-consistent scheme for solving the Einstein-Maxwell and MHD-Euler equations for systems with an approximate helical symmetry. Numerical codes based on this scheme are expected to model magnetars with non-axisymmetric magnetic fields as well as magnetized binary neutron star systems in quasi-equilibrium. )

For an equation of state in which pressure is a function only of density, the analysis of Newtoni... more For an equation of state in which pressure is a function only of density, the analysis of Newtonian stellar structure is simple in principle if the system is axisymmetric, or consists of a corotating binary. It is then required only to solve two equations: one stating that the "injection energy," κ, a potential, is constant throughout the stellar fluid, and the other being the integral over the stellar fluid to give the gravitational potential. An iterative solution of these equations generally diverges if κ is held fixed, but converges with other choices. We investigate the mathematical reason for this convergence/divergence by starting the iteration from an approximation that is perturbatively different from the actual solution. A cycle of iteration is then treated as a linear "updating" operator, and the properties of the linear operator, especially its spectrum, determine the convergence properties. For simplicity, we confine ourselves to spherically symmetric models in which we analyze updating operators both in the finite dimensional space corresponding to a finite difference representation of the problem, and in the continuum, and we find that the fixed-κ operator is self-adjoint and generally has an eigenvalue greater than unity; in the particularly important case of a polytropic equation of state with index greater than unity, we prove that there must be such an eigenvalue. For fixed central density, on the other hand, we find that the updating operator has only a single eigenvector, with zero eigenvalue, and is nilpotent in finite dimension, thereby giving a convergent solution.

Physical Review D, 2008

Quasi-equilibrium models of rapidly rotating triaxially deformed stars are computed in general re... more Quasi-equilibrium models of rapidly rotating triaxially deformed stars are computed in general relativistic gravity, assuming a conformally flat spatial geometry (Isenberg-Wilson-Mathews formulation) and a polytropic equation of state. Highly deformed solutions are calculated on the initial slice covered by spherical coordinate grids, centered at the source, in all angular directions up to a large truncation radius. Constant rest mass sequences are calculated from nearly axisymmetric to maximally deformed triaxial configurations. Selected parameters are to model (proto-) neutron stars; the compactness is M/R = 0.001, 0.1, 0.14, 0.2 for polytropic index n = 0.3 and M/R = 0.001, 0.1, 0.12, 0.14 for n = 0.5. We confirmed that the triaxial solutions exist for these parameters as in the case of Newtonian polytropes. However, it is also found that the triaxial sequences become shorter for higher compactness, and those may disappear at a certain large compactness for the n = 0.5 case. In the scenario of the contraction of proto-neutron stars being subject to strong viscosity and rapid cooling, it is plausible that, once the viscosity driven secular instability sets in during the contraction, the proto-neutron stars are always maximally deformed triaxial configurations, as long as the compactness and the equation of state parameters allow such triaxial sequences. Detection of gravitational waves from such sources may be used as another probe for the nuclear equation of state.

Physical Review D, 2010

Binary systems of compact objects with electromagnetic field are modeled by helically symmetric E... more Binary systems of compact objects with electromagnetic field are modeled by helically symmetric Einstein-Maxwell spacetimes with charged and magnetized perfect fluids. Previously derived thermodynamic laws for helically-symmetric perfect-fluid spacetimes are extended to include the electromagnetic fields, and electric currents and charges; the first law is written as a relation between the change in the asymptotic Noether charge dlQ\dl QdlQ and the changes in the area and electric charge of black holes, and in the vorticity, baryon rest mass, entropy, charge and magnetic flux of the magnetized fluid. Using the conservation laws of the circulation of magnetized flow found by Bekenstein and Oron for the ideal magnetohydrodynamic (MHD) fluid, and also for the flow with zero conducting current, we show that, for nearby equilibria that conserve the quantities mentioned above, the relation dlQ=0\dl Q=0dlQ=0 is satisfied. We also discuss a formulation for computing numerical solutions of magnetized binary compact objects in equilibrium with emphasis on a first integral of the ideal MHD-Euler equation.

The properties of neutron star matter above nuclear density are not precisely known. Gravitationa... more The properties of neutron star matter above nuclear density are not precisely known. Gravitational waves emitted from binary neutron stars during their late stages of inspiral and merger contain imprints of the neutron-star equation of state. Measuring departures from the point-particle limit of the late inspiral waveform allows one to measure properties of the equation of state via gravitational wave observations. This and a companion talk by J. S. Read reports a comparison of numerical waveforms from simulations of inspiraling neutron-star binaries, computed for equations of state with varying stiffness. We calculate the signal strength of the difference between waveforms for various commissioned and proposed interferometric gravitational wave detectors and show that observations at frequencies around 1 kHz will be able to measure a compactness parameter and constrain the possible neutron-star equations of state.

Physical Review D, 2009

We report the results of a first study that uses numerical simulations to estimate the accuracy w... more We report the results of a first study that uses numerical simulations to estimate the accuracy with which one can use gravitational wave observations of double neutron star inspiral to measure parameters of the neutron-star equation of state. The simulations use the evolution and initial-data codes of Shibata and Uryū to compute the last several orbits and the merger of neutron stars, with matter described by a parametrized equation of state. Previous work suggested the use of an effective cutoff frequency to place constraints on the equation of state. We find, however, that greater accuracy is obtained by measuring departures from the point-particle limit of the gravitational waveform produced during the late inspiral. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron star compactness (the ratio of the mass of the neutron star to its radius). We estimate that realistic equations of state will lead to gravitational waveforms that are distinguishable from point particle inspirals at an effective distance (the distance to an optimally oriented and located system that would produce an equivalent waveform amplitude) of 100 Mpc or less. As Lattimer and Prakash observed, neutron-star radius is closely tied to the pressure at density not far above nuclear. Our results suggest that broadband gravitational wave observations at frequencies between 500 and 1000 Hz will constrain this pressure, and we estimate the accuracy with which it can be measured. Related first estimates of radius measurability show that the radius can be determined to an accuracy of δR ∼ 1 km at 100 Mpc.

Properties of the neutron star equation of state can potentially be measured via gravitational wa... more Properties of the neutron star equation of state can potentially be measured via gravitational wave observations, by measuring departures from the point-particle limit of the waveform produced in the late inspiral of a neutron star binary. Numerical waveforms from simulations of inspiraling neutron star binaries, computed for equations of state with varying stiffness, are compared. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron star compactness. This suggests that gravitational wave observations at frequencies around ˜1 kHz will be able to measure a compactness parameter and constrain the possible neutron star equations of state.

Similar methods have been used to construct models of rapidly rotating stars, in Newtonian and re... more Similar methods have been used to construct models of rapidly rotating stars, in Newtonian and relativistic contexts. The choice of method has been based on numerical experiments, which indicate that particular methods converge quickly to a solution, while others diverge. The theory underlying these differences, however, has not been understood. In an attempt to provide a better theoretical understanding, we analytically examine the behavior of different iterative schemes near an exact solution. We find the spectrum of the linearized iteration operator and show for self-consistent field methods that iterative instability corresponds to unstable modes of this operator.