Effective One-body Hamiltonian Research Articles

Third post-Newtonian effective-one-body Hamiltonian in scalar-tensor and Einstein-scalar-Gauss-Bonnet gravity

We build an effective-one-body (EOB) Hamiltonian at third post-Newtonian (3PN) order in scalar-tensor (ST) and Einstein-scalar-Gauss-Bonnet (ESGB) theories of gravity. The latter is an extension of general relativity that predicts scalar hair for black holes. We start from the known two-body Lagrangian at 3PN order, and use order-reduction methods to construct its ordinary Hamiltonian counterpart. We then reduce the conservative two-body dynamics to the (nongeodesic) motion of a test particle in an effective metric by means of canonical transformations. The resulting EOB Hamiltonian is a modification of the general relativistic Hamiltonian, and already at 3PN order, it must account for nonlocal-in-time tail contributions. We include the latter beyond circular orbits and up to sixth order in the binary's orbital eccentricity. We finally calculate the orbital frequency at the innermost stable circular orbit (ISCO) of binary black holes in the shift-symmetric ESGB model. Our work extends F.L. Juli\'e and N. Deruelle [Phys. Rev. D 95, 124054 (2017)], and it is an essential step toward the accurate modeling of gravitational waveforms beyond general relativity.

Nonlocal-in-time effective one body Hamiltonian in scalar-tensor gravity at third post-Newtonian order

We complete the nonlocal-in-time effective-one-body (EOB) formalism of conservative dynamics for massless Scalar-Tensor (ST) theories at third post-Newtonian (PN) order. The nonlocal-in-time EOB Hamiltonian is obtained by mapping the order-reduced Hamiltonian corresponding to the nonlocal-in-time Lagrangian derived in [Phys. Rev. D 99, 044047 (2019)]. To transcribe the dynamics within EOB formalism, we use a strategy of order-reduction of nonlocal dynamics to local ordinary action-angle Hamiltonian. We then map this onto the EOB Hamiltonian to determine the nonlocal-in-time ST corrections to the EOB potentials $(A,B,Q_e)$ at 3PN order.

Fourth post-Newtonian effective-one-body Hamiltonians with generic spins

In a compact binary coalescence, the spins of the compact objects can have a significant effect on the orbital motion and gravitational-wave (GW) emission. For generic spin orientations, the orbital plane precesses, leading to characteristic modulations of the GW signal. The observation of precession effects is crucial to discriminate among different binary formation scenarios, and to carry out precise tests of General Relativity. Here, we work toward an improved description of spin effects in binary inspirals, within the effective-one-body (EOB) formalism, which is commonly used to build waveform models for LIGO and Virgo data analysis. We derive EOB Hamiltonians including the complete fourth post-Newtonian (4PN) conservative dynamics, which is the current state of the art. We place no restrictions on the spin orientations or magnitudes, or on the type of compact object (e.g., black hole or neutron star), and we produce the first generic-spin EOB Hamiltonians complete at 4PN order. We consider multiple spinning EOB Hamiltonians, which are more or less direct extensions of the varieties found in previous literature, and we suggest another simplified variant. Finally, we compare the circular-orbit, aligned-spin binding-energy functions derived from the EOB Hamiltonians to numerical-relativity simulations of the late inspiral. While finding that all proposed Hamiltonians perform reasonably well, we point out some interesting differences, which could guide the selection of a simpler, and thus faster-to-evolve EOB Hamiltonian to be used in future LIGO and Virgo inference studies.

Quasicircular inspirals and plunges from nonspinning effective-one-body Hamiltonians with gravitational self-force information

The self-force program aims at accurately modeling relativistic two-body systems with a small mass ratio (SMR). In the context of the effective-one-body (EOB) framework, current results from this program can be used to determine the effective metric components at linear order in the mass ratio, resumming post-Newtonian (PN) dynamics around the test-particle limit in the process. It was shown in [Akcay et al., Phys. Rev. D 86 (2012)] that, in the original (standard) EOB gauge, the SMR contribution to the metric component $g^\text{eff}_{tt}$ exhibits a coordinate singularity at the light-ring (LR) radius. In this paper, we adopt a different gauge for the EOB dynamics and obtain a Hamiltonian that is free of poles at the LR, with complete circular-orbit information at linear order in the mass ratio and non-circular-orbit and higher-order-in-mass-ratio terms up to 3PN order. We confirm the absence of the LR-divergence in such an EOB Hamiltonian via plunging trajectories through the LR radius. Moreover, we compare the binding energies and inspiral waveforms of EOB models with SMR, PN and mixed SMR-3PN information on a quasi-circular inspiral against numerical-relativity predictions. We find good agreement between NR simulations and EOB models with SMR-3PN information for both equal and unequal mass ratios. In particular, when compared to EOB inspiral waveforms with only 3PN information, EOB Hamiltonians with SMR-3PN information improves the modeling of binary systems with small mass ratios $q \lesssim 1/3$, with a dephasing accumulated in $\sim$30 gravitational-wave (GW) cycles being of the order of few hundredths of a radian up to 4 GW cycles before merger.

High-energy gravitational scattering and the general relativistic two-body problem

A technique for translating the classical scattering function of two gravitationally interacting bodies into a corresponding (effective one-body) Hamiltonian description has been recently introduced [Phys.\ Rev.\ D {\bf 94}, 104015 (2016)]. Using this technique, we derive, for the first time, to second-order in Newton's constant (i.e. one classical loop) the Hamiltonian of two point masses having an arbitrary (possibly relativistic) relative velocity. The resulting (second post-Minkowskian) Hamiltonian is found to have a tame high-energy structure which we relate both to gravitational self-force studies of large mass-ratio binary systems, and to the ultra high-energy quantum scattering results of Amati, Ciafaloni and Veneziano. We derive several consequences of our second post-Minkowskian Hamiltonian: (i) the need to use special phase-space gauges to get a tame high-energy limit; and (ii) predictions about a (rest-mass independent) linear Regge trajectory behavior of high-angular-momenta, high-energy circular orbits. Ways of testing these predictions by dedicated numerical simulations are indicated. We finally indicate a way to connect our classical results to the quantum gravitational scattering amplitude of two particles, and we urge amplitude experts to use their novel techniques to compute the 2-loop scattering amplitude of scalar masses, from which one could deduce the third post-Minkowskian effective one-body Hamiltonian.

Spin-orbit precession along eccentric orbits for extreme mass ratio black hole binaries and its effective-one-body transcription

In this work we present an analytical gravitational self-force calculation of the spin-orbit precession along an eccentric orbit around a Schwarzschild black hole, following closely the recent prescription of Akcay, Dempsey, and Dolan. We then transcribe this quantity within the Effective-One-Body (EOB) formalism, thereby determining several new, linear-in-mass-ratio, contributions in the post-Newtonian expansion of the spin-orbit couplings entering the EOB Hamiltonian. Namely, we determine the second gyro-gravitomagnetic ratio $g_{S_*}(r,p_r,p_{\phi})$ up to order $p_r^2/r^4$ included.

Two-body problem in scalar-tensor theories as a deformation of general relativity: An effective-one-body approach

In this paper we address the two-body problem in massless Scalar-Tensor (ST) theories within an Effective-One-Body (EOB) framework. We focus on the first building block of the EOB approach, that is, mapping the conservative part of the two-body dynamics onto the geodesic motion of a test particle in an effective external metric. To this end, we first deduce the second post-Keplerian (2PK) Hamiltonian of the two-body problem from the known 2PK Lagrangian. We then build, by means of a canonical transformation a ST-deformation of the general relativistic EOB Hamiltonian which allows to incorporate the Scalar-Tensor (2PK) corrections to the currently best available General Relativity EOB results. This EOB-ST Hamiltonian defines a resummation of the dynamics that may provide information on the strong-field regime, in particular, the ISCO location and associated orbital frequency and can be compared to other, e.g. tidal, corrections.

Effective-one-body Hamiltonian with next-to-leading order spin-spin coupling

We propose a way of including the next-to-leading (NLO) order spin-spin coupling into an effective-one-body (EOB) Hamiltonian. This work extends [S. Balmelli and P. Jetzer, Phys. Rev. D 87, 124036 (2013)], which is restricted to the case of equatorial orbits and aligned spins, to general orbits with arbitrary spin orientations. This is done applying appropriate canonical phase-space transformations to the NLO spin-spin Hamiltonian in Arnowitt-Deser-Misner (ADM) coordinates, and systematically adding "effectiv" quantities at NLO to all spin-squared terms appearing in the EOB Hamiltonian. As required by consistency, the introduced quantities reduce to zero in the test- mass limit. We expose the result both in a general gauge and in a gauge-fixed form. The last is chosen such as to minimize the number of new coefficients that have to be inserted into the effective spin squared. As a result, the 25 parameters that describe the ADM NLO spin-spin dynamics get condensed into only 12 EOB terms.

Erratum: Effective one-body Hamiltonian of two spinning black holes with next-to-next-to-leading order spin-orbit coupling [Phys. Rev. D84, 084028 (2011)

Erratum: Effective one-body Hamiltonian of two spinning black holes with next-to-next-to-leading order spin-orbit coupling [Phys. Rev. D<b>84</b>, 084028 (2011)

Particle-particle and quasiparticle random phase approximations: Connections to coupled cluster theory

We establish a formal connection between the particle-particle (pp) random phase approximation (RPA) and the ladder channel of the coupled cluster doubles (CCD) equations. The relationship between RPA and CCD is best understood within a Bogoliubov quasiparticle (qp) RPA formalism. This work is a follow-up to our previous formal proof on the connection between particle-hole (ph) RPA and ring-CCD. Whereas RPA is a quasibosonic approximation, CC theory is a "correct bosonization" in the sense that the wavefunction and Hilbert space are exactly fermionic, yet the amplitude equations can be interpreted as adding different quasibosonic RPA channels together. Coupled cluster theory achieves this goal by interacting the ph (ring) and pp (ladder) diagrams via a third channel that we here call "crossed-ring" whose presence allows for full fermionic antisymmetry. Additionally, coupled cluster incorporates what we call "mosaic" terms which can be absorbed into defining a new effective one-body Hamiltonian. The inclusion of these mosaic terms seems to be quite important. The pp-RPA and qp-RPA equations are textbook material in nuclear structure physics but are largely unknown in quantum chemistry, where particle number fluctuations and Bogoliubov determinants are rarely used. We believe that the ideas and connections discussed in this paper may help design improved ways of incorporating RPA correlation into density functionals based on a CC perspective.

Effective-one-body Hamiltonian with next-to-leading order spin-spin coupling for two nonprecessing black holes with aligned spins

The canonical Arnowitt-Deser-Misner (ADM) Hamiltonian with next-to-leading order spin-spin coupling [J. Steinhoff, S. Hergt, and G. Sch\"afer, Phys. Rev. D 77, 081501 (2008); 78, 101503 (2008)] is converted into the effective-one-body (EOB) formalism of [T. Damour, P. Jaranowski, and G. Sch\"afer, Phys. Rev. D 78, 024009 (2008)] for the special case of spinning black hole binaries whose spins are aligned with the angular momentum. In particular, we propose to include the new terms by adding a dynamical term of next-to-leading order to the Kerr parameter squared entering the effective metric. The modified EOB Hamiltonian consistently reduces to the Kerr Hamiltonian as the mass ratio tends to zero; moreover, it predicts the existence of an innermost stable circular orbit. We also derive, for the general case of arbitrarily oriented spins but in the vanishing mass-ratio limit, a coordinate transformation that maps the next-to-leading order spin-spin contribution of the ADM Hamiltonian to the EOB Hamiltonian.

Extending the effective-one-body Hamiltonian of black-hole binaries to include next-to-next-to-leading spin-orbit couplings

In the effective-one-body (EOB) approach, the dynamics of two compact objects of masses ${m}_{1}$ and ${m}_{2}$ and spins ${\mathbf{S}}_{1}$ and ${\mathbf{S}}_{2}$ is mapped into the dynamics of one test particle of mass $\ensuremath{\mu}={m}_{1}{m}_{2}/({m}_{1}+{m}_{2})$ and spin ${\mathbf{S}}_{*}$ moving in a deformed Kerr metric with mass $M={m}_{1}+{m}_{2}$ and spin ${\mathbf{S}}_{\mathrm{Kerr}}$. In a previous paper, we computed an EOB Hamiltonian for spinning black-hole binaries that (i) when expanded in post-Newtonian orders, reproduces the leading-order spin-spin coupling and the leading and next-to-leading order spin-orbit couplings for any mass ratio, and (iii) reproduces all spin-orbit couplings in the test-particle limit. Here we extend this EOB Hamiltonian to include next-to-next-to-leading spin-orbit couplings for any mass ratio. We discuss two classes of EOB Hamiltonians that differ by the way the spin variables are mapped between the effective and real descriptions. We also investigate the main features of the dynamics when the motion is equatorial, such as the existence of the innermost stable circular orbit and of a peak in the orbital frequency during the plunge subsequent to the inspiral.

Effective one-body Hamiltonian of two spinning black holes with next-to-next-to-leading order spin-orbit coupling

Building on the recently computed next-to-next-to-leading order (NNLO) post-Newtonian spin-orbit Hamiltonian for spinning binaries [J. Hartung and J. Steinhoff, arXiv:1104.3079.] we improve the effective-one-body description of the dynamics of two spinning black holes by including NNLO effects in the spin-orbit interaction. The calculation that is presented extends to NNLO the next-to-leading order spin-orbit Hamiltonian computed in [T. Damour, P. Jaranowski, and G. Schaefer, Phys. Rev. D 78, 024009 (2008).]. The present effective-one-body Hamiltonian reproduces the spin-orbit coupling through NNLO in the test-particle limit case. In addition, in the case of spins parallel or antiparallel to the orbital angular momentum, when circular orbits exist, we find that the inclusion of NNLO spin-orbit terms moderates the effect of the next-to-leading order spin-orbit coupling.

Constructing effective one-body dynamics with numerical energy flux for intermediate-mass-ratio inspirals

A new scheme for computing dynamical evolutions and gravitational radiations for intermediate-mass-ratio inspirals (IMRIs) based on an effective one-body (EOB) dynamics plus Teukolsky perturbation theory is built in this paper. In the EOB framework, the dynamic essentially affects the resulted gravitational waveform for a binary compact star system. This dynamic includes two parts. One is the conservative part, which comes from effective one-body reduction. The other part is the gravitational backreaction, which contributes to the shrinking process of the inspiral of a binary compact star system. Previous works used an analytical waveform to construct this backreaction term. Since the analytical form is based on post-Newtonian expansion, the consistency of this term is always checked by numerical energy flux. Here, we directly use numerical energy flux by solving the Teukolsky equation via the frequency-domain method to construct this backreaction term. The conservative correction to the leading order terms in mass-ratio is included in the deformed-Kerr metric and the EOB Hamiltonian. We try to use this method to simulate not only quasicircular adiabatic inspiral, but also the nonadiabatic plunge phase. For several different spinning black holes, we demonstrate and compare the resulted dynamical evolutions and gravitational waveforms.

Ineffectiveness of Padé resummation techniques in post-Newtonian approximations

We test the resummation techniques used in developing Pad\'e and effective one body (EOB) waveforms for gravitational wave detection. Convergence tests show that Pad\'e approximants of the gravitational wave energy flux do not accelerate the convergence of the standard Taylor approximants even in the test mass limit, and there is no reason why Pad\'e transformations should help in estimating parameters better in data analysis. Moreover, adding a pole to the flux seems unnecessary in the construction of these Pad\'e-approximated flux formulas. Pad\'e approximants may be useful in suggesting the form of fitting formulas. We compare a 15-orbit numerical waveform of the Caltech-Cornell group to the suggested Pad\'e waveforms of Damour et al. in the equal mass, nonspinning quasicircular case. The comparison suggests that the Pad\'e waveforms do not agree better with the numerical waveform than the standard Taylor based waveforms. Based on this result, we design a simple EOB model by modifiying the Taylor-expanded EOB model of Buonanno et al., using the Taylor series of the flux with an unknown parameter at the fourth post-Newtonian order that we fit for. The 4PN parameter incorporates higher order effects of the radiation reaction. This simple EOB model generates a waveform having a phase difference of only 0.002 radians with the numerical waveform, much smaller than 0.04 radians the phase uncertainty in the numerical data itself. An EOB Hamiltonian can make use of a Pad\'e transformation in its construction, but this is the only place Pad\'e transformations seem useful.

Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling

Using a recent, novel Hamiltonian formulation of the gravitational interaction of spinning binaries, we extend the effective one body (EOB) description of the dynamics of two spinning black holes to next-to-leading order (NLO) in the spin-orbit interaction. The spin-dependent EOB Hamiltonian is constructed from four main ingredients: (i) a transformation between the ``effective'' Hamiltonian and the ``real'' one; (ii) a generalized effective Hamilton-Jacobi equation involving higher powers of the momenta; (iii) a Kerr-type effective metric (with Pad\'e-resummed coefficients) which depends on the choice of some basic ``effective spin vector'' ${\mathbf{S}}_{\mathrm{eff}}$, and which is deformed by comparable-mass effects; and (iv) an additional effective spin-orbit interaction term involving another spin vector $\mathbit{\ensuremath{\sigma}}$. As a first application of the new, NLO spin-dependent EOB Hamiltonian, we compute the binding energy of circular orbits (for parallel spins) as a function of the orbital frequency, and of the spin parameters. We also study the characteristics of the last stable circular orbit: binding energy, orbital frequency, and the corresponding dimensionless spin parameter ${\stackrel{^}{a}}_{\mathrm{LSO}}\ensuremath{\equiv}c{J}_{\mathrm{LSO}}/(G({H}_{\mathrm{LSO}}/{c}^{2}{)}^{2})$. We find that the inclusion of NLO spin-orbit terms has a significant ``moderating'' effect on the dynamical characteristics of the circular orbits for large and parallel spins.

Accurate calculation of spin-orbit coupling constants for 3d atoms and ions with effective core potentials and reduced basis sets.

We present an approximate method for obtaining spin-orbit coupling constants of 3d atoms and ions from an effective one-body Hamiltonian. The method, suggested by numerical results of the analytical theory of Blume and Watson [Proc. R. Soc. London, Sect. A 270, 127 (1962); 271, 565 (1963)], incorporates the model potential approach of Bonifacic and Huzinaga [J. Chem. Phys. 60, 167 (1974)] for representing the core-valence interactions. In combination with reduced valence basis set, the new procedure gives accurate spin-orbit constants for the 3d atoms, and slightly less accurate values for di- and tripositive 3d ions. These results suggest that this method can be an accurate economic route to crystal calculations of the spin-orbit constants within any cluster-model approach involving the Hamiltonian of Misetich and Buch [J. Chem. Phys. 41, 2524 (1964)].

Semiclassical approximations for nuclear Hamiltonians II. Spin-dependent potentials

A systematic semiclassical expansion procedure for physical quantities in nuclei, based on the Thomas-Fermi approximation to the Hartree-Fock equations and constructed in a previous work, is extended here to the realistic case where the effective one-body Hamiltonian for nucleons contains spin-dependent terms. Spin-induced corrections to the kinetic energy density and surface energy of the nucleus, and expressions for various spindependent densities inside the nucleus are computed up to fourth order in for several nuclear Hamiltonians.

Radiative muon capture in medium-heavy nuclei

Radiative muon capture in nuclei is thoroughly investigated in view of its sensitivity to the pseudoscalar coupling constant g P. In the photon energy domain of interest ( k ⩾ 60 MeV) the effective one-body hamiltonian in use is shown essentially to hold true, with minor corrections due to the muon propagator in the nuclear field and to the two-step process. The closure approximation is avoided by appropriately integrating the response function calculated in the Fermi gas model, in the energy-momentum transfer plane. Theoretical predictions in medium-heavy nuclei of the high-energy photon spectrum, of photon polarization, and of muon-spin photon angular correlation are given for various values of the pseudoscalar coupling constant.

Valence-core self-consistency in theA=17system

We develop a systematic approach to the calculation of self-consistency effects in core plus valence nucleon systems. A detailed calculation of the effective one-body Hamiltonian for A=17 in an 11h..cap omega.. model space is readily extrapolated to a 20h..cap omega.. model space which is sufficient to develop accurate tails for the single-particle wave functions. The effective one-body interaction is found using a Brillouin-Wigner type perturbation theory which eliminates folded diagrams and leads to a manifestly Hermitian interaction. A renormalized Brueckner calculation is performed for A=16 and the results are employed in a shell-model study of A=17. The self-consistent results still have a weak dependence on the initial unperturbed Hamiltonian. However, we find a single unperturbed Hamiltonian which yields a reasonable binding energy for A=16 and, except for a weak spin-orbit splitting, reasonable results for the lowest states in the A=17 system. This agreement is important for continued valence s- d shell studies.