post-Newtonian Research Articles

BIFROST: simulating compact subsystems in star clusters using a hierarchical fourth-order forward symplectic integrator code

ABSTRACT We present BIFROST, an extended version of the GPU-accelerated hierarchical fourth-order forward symplectic integrator code FROST. BIFROST (BInaries in FROST) can efficiently evolve collisional stellar systems with arbitrary binary fractions up to $f_\mathrm{bin}=100~{{\ \rm per\ cent}}$ by using secular and regularized integration for binaries, triples, multiple systems, or small clusters around black holes within the fourth-order forward integrator framework. Post-Newtonian (PN) terms up to order PN3.5 are included in the equations of motion of compact subsystems with optional three-body and spin-dependent terms. PN1.0 terms for interactions with black holes are computed everywhere in the simulation domain. The code has several merger criteria (gravitational-wave inspirals, tidal disruption events, and stellar and compact object collisions) with the addition of relativistic recoil kicks for compact object mergers. We show that for systems with N particles the scaling of the code remains good up to NGPU ∼ 40 × N/106 GPUs and that the increasing binary fractions up to 100 per cent hardly increase the code running time (less than a factor ∼1.5). We also validate the numerical accuracy of BIFROST by presenting a number of star clusters simulations the most extreme ones including a core collapse and a merger of two intermediate mass black holes with a relativistic recoil kick.

Orbital precession of stars in the Galactic Centre

ABSTRACT The region around the centre of our Galaxy is very dense of stars. The kinematics of inner moving stars in the Galaxy (the so-called S-stars) has been deeply studied by different research groups leading to the conclusion of the existence of a very compact object (Sgr A*, likely a supermassive black hole) responsible for their high speed. Here, we start from the observational evidence of orbital apsidal line precession for the S2 (also called S0-2) star to investigate on a theoretical side what level of quality in such regime of relatively strong gravitational field is reached in the orbit angular precession determination when using a direct orbital integration of the star motion subjected to an acceleration computed in the post-Newtonian (PN) scheme up to different orders. This approach, although approximated and limited to particle speed not exceeding ∼ 0.3c, allows the inclusion of various effects, like that of a possible spin of the central massive object. Our results show that the inclusion of PN terms above the standard 1PN term (the one corresponding to the classic Einstein–Schwarzschild estimate of pericenter advance) is compulsory to determine angular precession at sufficient level of accuracy for those penetrating stars that would allow to pick contemporary the value of the mass and of the spin of a rotating (Kerr-like) supermassive black hole (SMBH). We discuss how future observational data, together with a proper modelization, could allow the determination of both mass and spin of the SMBH of our Galaxy.

Axion cosmology with post-Newtonian corrections

We present first-order post-Newtonian (1PN) approximations of a general imperfect fluid and of an axion as a coherently oscillating massive scalar field, both in the cosmological context. For the axion, using the Klein transformation and Madelung transformation we derive the Schr\"odinger and Madelung hydrodynamic formulations, respectively, in exact covariant way and to 1PN order. Complete sets of equations for the 1PN formulations are derived without fixing the temporal gauge condition. We study the linear instability in cosmology and a static limit for both fluid and axion; these are presented independently of the gauge condition to 1PN order, thus are naturally gauge-invariant.

Effective-one-body Hamiltonian in scalar-tensor gravity at third post-Newtonian order

We determine the general local-in-time effective-one-body (EOB) Hamiltonian for massless Scalar-Tensor (ST) theories at third post-Newtonian (PN) order. Starting from the Lagrangian derived in [Phys. Rev. D 99, 044047 (2019)], we map it to the corresponding ordinary Hamiltonian describing the two-body interaction in ST theories at 3PN level. Using a canonical transformation, we then map this onto an EOB Hamiltonian so as to determine the ST corrections to the 3PN-accurate EOB potentials $(A,B,Q_e)$ at 3PN. We then focus on circular orbits and compare the effect of the newly computed 3PN terms, also completed with finite-size and nonlocal-in-time contributions, on predictions for the frequency at the innermost stable circular orbit. Our results will be useful to build high-accuracy waveform models in ST theory, which could be used to perform precise tests against General Relativity using gravitational wave data from coalescing compact binaries.

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.

Tidal deformations of a binary system induced by an external Kerr black hole

The dynamics of a binary system moving in the background of a black hole is affected by tidal forces. In this work, for the Kerr black hole, we derive the electric and magnetic tidal moments at quadrupole order, where the latter are computed for the first time in full generality. We make use of these moments in the scenario of a hierarchical triple system made of a Kerr black hole and an extreme-mass ratio binary system consisting of a Schwarzschild black hole and a test particle. We study how the secular dynamics of the test particle in the binary system is distorted by the presence of tidal forces from a much larger Kerr black hole. Our treatment includes strong gravitational effects beyond the post-Newtonian approximation both for the binary system and for the tidal forces since the binary system is allowed to be close to the event horizon of the Kerr black hole. We compute the shifts in the physical quantities for the secular dynamics of the test particle and show that they are gauge invariant. In particular, we apply our formalism to the innermost stable circular orbit for the test particle and to the case of the photon sphere. Our results are relevant for the astrophysical situation in which the binary system is in the vicinity of a supermassive black hole.

Modeling horizon absorption in spinning binary black holes using effective worldline theory

The mass and spin of black holes (BHs) in binary systems may change due to the infall of gravitational-wave (GW) energy down the horizons. For spinning BHs, this effect enters at 2.5 post-Newtonian (PN) order relative to the leading-order energy flux at infinity. There is currently a discrepancy in the literature in the expressions of these horizon fluxes in the test-body limit at 4PN order (relative 1.5PN order). Here, we model the horizon absorption as tidal heating in an effective worldline theory of a spinning particle equipped with tidally-induced quadrupole and octupole moments. We match the tidal response to analytic solutions of the Teukolsky equation in a scattering scenario, and obtain general formulae for the evolution of mass and spin. We then specialize to the case of aligned-spin--quasi-circular binaries, obtaining the corresponding contributions to the GW phasing through 4PN order. Importantly, we find that the number of GW cycles due to horizon fluxes with masses observed by LIGO-Virgo-KAGRA detectors is about 2-3 orders of magnitude smaller than the other contributions to the phasing at the same PN order. Furthermore, in the test-body limit, we find full agreement with results obtained earlier from BH perturbation theory, with a small mass in an equatorial circular orbit treated as a source perturbing the Kerr metric. Thus, we weigh in on one side of the previous discrepancy.

Testing the post-Newtonian expansion with GW170817

Observations of gravitational waves from compact binary mergers have enabled unique tests of general relativity in the dynamical and non-linear regimes. One of the most important such tests is constraints on the post-Newtonian (PN) corrections to the phase of the gravitational wave signal. The values of the PN coefficients can be calculated within standard general relativity, and these values are different in many alternate theories of gravity. It is clearly of great interest to constrain the deviations based on gravitational wave observations. In the majority of such tests which have been carried out, and which yield by far the most stringent constraints, it is common to vary these PN coefficients individually. While this might in principle be useful for detecting certain deviations from standard general relativity, it is a serious limitation. For example, we would expect alternate theories of gravity to generically have additional parameters. The corrections to the PN coefficients would be expected to depend on these additional non-GR parameters, whence, we expect that the various PN coefficients to be highly correlated. We present an alternate analysis here using data from the binary neutron star coalescence GW170817. Our analysis uses an appropriate linear combination of non-GR parameters that represent absolute deviations from the corresponding post-Newtonian inspiral coefficients in the TaylorF2 approximant phase. These combinations represent uncorrelated non-GR parameters which correspond to principal directions of their covariance matrix in the parameter subspace. Our results illustrate good agreement with GR. In particular, the integral non-GR phase is Psi _{{tiny int-non-GR}} = 0.0447pm 25.3000 and the deviation from GR percentile is p^{{tiny Dev-GR}}_{n}=25.85%.

Gravitational waveform and polarizationfrom binary black hole inspiral in dynamical Chern-Simons gravity: from generation to propagation

We calculate the gravitational waveform radiated from spinning black holes (BHs) binary in dynamical Chern-Simons (dCS) gravity. The equation of motion (EOM) of the spinning binary BHs is derived based on the modified Mathisson-Papapetrou-Dixon equation for the spin-aligned circular orbits. The leading-order effects induced by the dCS theory contain spin-spin interaction and monopole-quadrupole interaction, which influences both the EOM of the binary system and corresponding gravitational waveform at the second post-Newtonian (PN) order (i.e., 2PN order). After reporting the waveforms, we investigate the polarization modes of gravitational waves (GWs) in dCS theory. None of the extra modes appears in this theory up to the considered PN order. Moreover, since the time scale of the binary merger is much smaller than that of the cosmological expansion, the parity-violating effect of the dCS theory does not appear in the process of GW generation. However, during the process of GW propagation, amplitude birefringence, a typical parity-violating effect, makes plus and cross modes convert to each other, which modifies the gravitational waveform at 1.5PN order.

Relaxation-time model for the post-Newtonian Boltzmann equation

The non-equilibrium contributions to the post-Newtonian hydrodynamic equations are determined from a relaxation-time model of the post-Newtonian Boltzmann equation. The Chapman- Enskog method is used to calculate the non-equilibrium distribution function. The components of the energy–momentum tensor are found from the knowledge of the non-equilibrium and the post-Newtonian equilibrium Maxwell–Jüttner distribution functions. The linearized field equations for the mass, momentum and internal energy densities coupled with the three Poisson equations of the post-Newtonian approximation are investigated by considering a plane wave representation of the fields. The constitutive equations for the viscous stress and heat flux vector are obtained and it is shown that the transport coefficients of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential.

Gravitational spin-orbit Hamiltonian at NNNLO in the post-Newtonian framework

We present the result of the spin-orbit interaction Hamiltonian for binary systems of rotating compact objects with generic spins, up to N3LO corrections within the post-Newtonian expansion. The calculation is performed by employing the effective field theory diagrammatic approach, and it involves Feynman integrals up to three loops, evaluated within the dimensional regularization scheme. We apply canonical transformations to eliminate the non-physical divergences and spurious logarithmic behaviours of the Hamiltonian, and use the latter to derive the gauge-invariant binding energy and the scattering angle, in special kinematic regimes.

N3LO quadratic-in-spin interactions for generic compact binaries

We derive the third subleading (N3LO) corrections of the quadratic-in-spin sectors via the EFT of spinning objects in post-Newtonian (PN) gravity. These corrections consist of contributions from 4 sectors for generic compact binaries, that enter at the fifth PN order. One of these contributions is due to a new tidal interaction, that is unique to the sectors with spin, and complements the first tidal interaction that also enters at this PN order in the simple point-mass sector. The evaluation of Feynman graphs is carried out in a generic dimension via advanced multi-loop methods, and gives rise to dimensional-regularization poles in conjunction with logarithms. At these higher-spin sectors the reduction of generalized Lagrangians entails redefinitions of the position beyond linear order. We provide here the most general Lagrangians and Hamiltonians. We then specify the latter to simplified configurations, and derive the consequent gauge-invariant relations among the binding energy, angular momentum, and frequency. We end with a derivation of all the scattering angles that correspond to an extension of our Hamiltonians to the scattering problem in the simplified aligned-spins configuration, as a guide to scattering-amplitudes studies.

Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order

We obtain the total impulse in the scattering of nonspinning binaries in general relativity at fourth post-Minkowskian order, i.e., O(G^{4}), including linear, nonlinear, and hereditary radiation-reaction effects. We derive the total radiated spacetime momentum as well as the associated energy flux. The latter can be used to compute gravitational-wave observables for generic (un)bound orbits. We employ the ("in-in") Schwinger-Keldysh worldline effective field theory framework in combination with modern "multiloop" integration techniques from collider physics. The complete results are in agreement with various partial calculations in the post-Minkowskian and post-Newtonian expansion.

Priorities in gravitational waveforms for future space-borne detectors: vacuum accuracy or environment?

ABSTRACT In preparation for future space-borne gravitational-wave (GW) detectors, should the modelling effort focus on high-precision vacuum templates or on the astrophysical environment of the sources? We perform a systematic comparison of the phase contributions caused by (1) known environmental effects in both gaseous and stellar matter backgrounds, or (2) high-order post-Newtonian (PN) terms in the evolution of mHz GW sources during the inspiral stage of massive binaries. We use the accuracy of currently available analytical waveform models as a benchmark value, finding the following trends: the largest unmodelled phase contributions are likely environmental rather than PN for binaries lighter than ∼107/(1 + z)2 M⊙, where z is the redshift. Binaries heavier than ∼108/(1 + z) M⊙ do not require more accurate inspiral waveforms due to low signal-to-noise ratios (SNRs). For high-SNR sources, environmental phase contributions are relevant at low redshift, while high-order vacuum templates are required at z ≳ 4. Led by these findings, we argue that including environmental effects in waveform models should be prioritized in order to maximize the science yield of future mHz detectors.

Minimal quantum walk simulation of Dirac fermions in curved space-times

The problem of simulating through quantum walks the curved space-time propagation of Dirac fermions is revisited, taking the $$(1 + 1)$$ D case as an example. New quantum walks are introduced which encode the space-time geometry, not in the mixing operators, but in new unitary shift operators which carry out a position-dependent translation of the wave function components. These walks simulate Dirac fermions in arbitrary curved space-times and coordinates. Contrary to previously proposed general alternatives, the wave functions of the new walks have as many components as usual Dirac spinors, and not twice that number. In particular, in $$(1 + 1)$$ D space-times, only one qubit is needed at each lattice point, which makes it easier to perform quantum simulations of the Dirac dynamics on current NISQs quantum devices. Numerical simulations of the Dirac dynamics in the post Newtonian, so-called Gravitoelectromagnetism regime are presented as an illustration. The possible usefulness of the new shift operators and quantum walks in other more general contexts is also discussed.

Gravitational wave signals from finite size effects in spinning binary inspirals including parity violating constituents

We generalize the world line EFT formalism to account for parity violating finite size effects. Results are presented for potentials and radiating moments of a binary inspiral for the parity conserving sector, and agreement is found with, previous calculations. Furthermore, we generate new results in this sector, calculating the current quadrupole moment induced by finite size gravitomagnetic effects. We also present novel results for parity violating sources, which might be due to beyond standard model physics, and show that they generate GW signals with the unique signature that the current-moment appears at 0.5PN order earlier relative to the mass-moment in the PN expansion. Parity violation also induces a new type of potential, which is proportional to the S ∙ r. Finally, we present new results for the dissipative force for parity violating constituents, which leads to the curious signature of a force normal to the orbit.

Complete gravitational-waveform amplitude modes for quasicircular compact binaries to the 3.5PN order

We complete the computation of the gravitational waveform amplitude for non-spinning compact binaries to the third and a half post-Newtonian (PN) order in the quasi-circular case. This is done by performing a spin-weighted spherical harmonics decomposition of the amplitude. This computation is achieved using the post-Newtonian-Multipolar-post-Minkowskian formalism. The amplitude modes are written in a PN expanded way as well as a factorized form suitable for EOB template building. Combining this work with previous ones, we provide in the supplemental materials the expressions of the modes including all non-spinning and spinning effects to the 3.5PN order for non-precessing quasi-circular binaries.

A 3+1 formulation of the 1/c expansion of General Relativity

Expanding General Relativity in the inverse speed of light, 1/c, leads to a nonrelativistic gravitational theory that extends the Post-Newtonian expansion by the inclusion of additional strong gravitational potentials. This theory has a fully covariant formulation in the language of Newton-Cartan geometry but we revisit it here in a 3+1 formulation. The appropriate 3+1 formulation of General Relativity is one first described by Kol and Smolkin (KS), rather than the better known Arnowitt-Deser-Misner (ADM) formalism. As we review, the KS formulation is dual to the ADM formulation in that the role of tangent and co-tangent spaces get interchanged. In this 3+1 formulation the 1/c expansion can be performed in a more systematic and efficient fashion, something we use to extend the computation of the effective Lagrangian beyond what was previously achieved and to make a number of new all order observations.

Inspiral gravitational waveforms from compact binary systems in Horndeski gravity

In a subclass of Horndeski theories with the speed of gravity equivalent to that of light, we study gravitational radiation emitted during the inspiral phase of compact binary systems. We compute the waveform of scalar perturbations under a post-Newtonian expansion of energy-momentum tensors of pointlike particles that depend on a scalar field. This scalar mode not only gives rise to breathing and longitudinal polarizations of gravitational waves, but it is also responsible for scalar gravitational radiation in addition to energy loss associated with transverse and traceless tensor polarizations. We calculate the Fourier-transformed gravitational waveform of two tensor polarizations under a stationary phase approximation and show that the resulting waveform reduces to the one in a parametrized post-Einsteinian (ppE) formalism. The ppE parameters are directly related to a scalar charge in the Einstein frame, whose existence is crucial to allow the deviation from general relativity (GR). We apply our general framework to several concrete theories and show that a new theory of spontaneous scalarization with a higher-order scalar kinetic term leaves interesting deviations from GR that can be probed by the observations of gravitational waves emitted from neutron star--black hole binaries. If the scalar mass exceeds the order of typical orbital frequencies $\ensuremath{\omega}\ensuremath{\simeq}{10}^{\ensuremath{-}13}\text{ }\text{ }\mathrm{eV}$, which is the case for a recently proposed scalarized neutron star with a self-interacting potential, the gravitational waveform practically reduces to that in GR.

Collinear and triangular solutions to the coplanar and circular three-body problem in the parametrized post-Newtonian formalism

This paper investigates the coplanar and circular three-body problem in the parametrized post-Newtonian (PPN) formalism, for which we focus on a class of fully conservative theories characterized by the Eddington-Robertson parameters $\beta$ and $\gamma$. It is shown that there can still exist a collinear equilibrium configuration and a triangular one, each of which is a generalization of the post-Newtonian equilibrium configuration in general relativity. The collinear configuration can exist for arbitrary mass ratio, $\beta$, and $\gamma$. On the other hand, the PPN triangular configuration depends on the nonlinearity parameter $\beta$ but not on $\gamma$. For any value of $\beta$, the equilateral configuration is possible, if and only if three finite masses are equal or two test masses orbit around one finite mass. For general mass cases, the PPN triangle is not equilateral as in the post-Newtonian case. It is shown also that the PPN displacements from the Lagrange points in the Newtonian gravity $L_1$, $L_2$ and $L_3$ depend on $\beta$ and $\gamma$, whereas those to $L_4$ and $L_5$ rely only on $\beta$.