post-Newtonian Series Research Articles

Intermediate Mass-Ratio Black Hole Binaries: Applicability of Small Mass-Ratio Perturbation Theory.

The inspiral phasing of binary black holes at intermediate mass ratios (m_{2}/m_{1}∼10^{-3}) is important for gravitational wave observations, but not accessible to standard modeling techniques: The accuracy of the small mass-ratio (SMR) expansion is unknown at intermediate mass ratios, whereas numerical relativity simulations cannot reach this regime. This article assesses the accuracy of the SMR expansion by extracting the first three terms of the SMR expansion from numerical relativity data for nonspinning, quasicircular binaries. We recover the leading term predicted by SMR theory and obtain a robust prediction of the next-to-leading term. The influence of higher-order terms is bounded to be small, indicating that the SMR series truncated at next-to-leading order is quite accurate at intermediate mass ratios and even at nearly comparable mass binaries. We estimate the range of applicability for SMR and post-Newtonian series for nonspinning, quasicircular inspirals.

Spin-pálya korrekciók a kompakt kettős rendszerek mozgásának leírásában II.

A gravitációs hullámok első közvetlen detektálása után a kihívás az, hogy a detektorok segítségével minél pontosabb képet kapjunk az ezen hullámokon keresztül megfigyelhető jelenségekről, vagyis megszülessen a gravitációshullám-csillagászat. Ebben kiemelt szerepe van a kompakt objektumok által alkotott kettős rendszereknek, mivel az általuk kibocsátott hullámok széles paramétertartományban nagy pontossággal leírhatók. Ehhez azonban szükségünk van a kettős rendszer mozgásának leírására. Ezt a poszt-newtoni sorfejtés alkalmazásával tesszük meg, figyelembe véve a pálya excentricitásának és a testek forgásának hatásait. A korábbi munkák tisztázzák a szeparációs vektor nagysága változásának részleteit, azonban a kibocsátott gravitációs hullámok leírásához szükségünk van a vektor-irányokat jellemző szögmennyiségek dinamikájának leírására is. Az ehhez szükséges egyenletek kerülnek jelen cikkben származtatásra, illetve részletezésre kerül ezek megoldásának módszerei. Célkitűzés továbbá a korábbi, alapanyagként használt publikációk hibáinak feltárása, javítása.

Minimum energy and the end of the inspiral in the post-Newtonian approximation

The early inspiral phase of a compact binary coalescence is well modelled by the post-Newtonian (PN) approximation to the orbital energy and gravitational wave flux. The transition from the inspiral phase to the plunge can be defined by the minimum energy circular orbit (MECO). In the extreme mass-ratio limit, up to the highest PN order known, the PN energy equals the energy of the exact Kerr solution. However, for comparable-mass systems the MECO of the PN energy does not exist when bodies have large spins. By including the exact Kerr limit and recently published post-Newtonian terms we extract a well-defined minimum of the orbital energy beyond which the plunge or merger occurs. We study the hybrid condition for a number of cases of both black hole and neutron stars and compare to other commonly employed definitions. Our method can be used for any known order of the post-Newtonian series and enables the MECO condition to be used to define the end of the inspiral phase for highly spinning, comparable mass systems.

Parameter estimation for coalescing massive binary black holes with LISA using the full 2-post-Newtonian gravitational waveform and spin-orbit precession

Gravitational waves emitted by binary systems in the inspiral phase carry a complicated structure, consisting in a superposition of different harmonics of the orbital frequency, the amplitude of each of them taking the form of a post-Newtonian series. In addition to that, spinning binaries experience spin-orbit and spin-spin couplings which induce a precession of the orbital angular momentum and of the individual spins. With one exception, previous analyses of the measurement accuracy of gravitational wave experiments for comparable-mass binary systems have neglected either spin-precession effects or subdominant harmonics and amplitude modulations. Here we give the first explicit description of how these effects combine to improve parameter estimation. We consider supermassive black hole binaries as expected to be observed with the planned space-based interferometer LISA, and study the measurement accuracy for several astrophysically interesting parameters obtainable taking into account the full 2PN waveform for spinning bodies, as well as spin-precession effects. We find that for binaries with a total mass in the range ${10}^{5}{M}_{\ensuremath{\bigodot}}<M<{10}^{7}{M}_{\ensuremath{\bigodot}}$ at a redshift of 1, a factor $\ensuremath{\sim}1.5$ is in general gained in accuracy, with the notable exception of the determination of the individual masses in equal-mass systems, for which a factor $\ensuremath{\sim}5$ can be gained. We also find, as could be expected, that using the full waveform helps increasing the upper mass limit for detection, which can be as high as $M={10}^{8}{M}_{\ensuremath{\bigodot}}$ at a redshift of 1, as well as the redshift limit where some information can be extracted from a system, which is roughly $z\ensuremath{\gtrsim}10$ for $M\ensuremath{\le}{10}^{7}{M}_{\ensuremath{\bigodot}}$, 1.5--5 times higher than with the restricted waveform. We computed that the full waveform allows us to use supermassive black hole binaries as standard sirens up to a redshift of $z\ensuremath{\approx}1.6$, about 0.4 larger than what previous studies allowed. We found that for lower unequal-mass binary systems, the measurement accuracy is not as drastically improved as for other systems. This suggests that for these systems, adding parameters such as eccentricity or alternative gravity parameters could be achieved without much loss in the accuracy.

Inspiral, merger, and ringdown of unequal mass black hole binaries: A multipolar analysis

We study the inspiral, merger, and ringdown of unequal mass black hole binaries by analyzing a catalogue of numerical simulations for seven different values of the mass ratio (from q=M2/M1=1 to q=4). We compare numerical and post-Newtonian results by projecting the waveforms onto spin-weighted spherical harmonics, characterized by angular indices (l,m). We find that the post-Newtonian equations predict remarkably well the relation between the wave amplitude and the orbital frequency for each (l,m), and that the convergence of the post-Newtonian series to the numerical results is nonmonotonic. To leading order, the total energy emitted in the merger phase scales like η2 and the spin of the final black hole scales like η, where η=q/(1+q)2 is the symmetric mass ratio. We study the multipolar distribution of the radiation, finding that odd-l multipoles are suppressed in the equal mass limit. Higher multipoles carry a larger fraction of the total energy as q increases. We introduce and compare three different definitions for the ringdown starting time. Applying linear-estimation methods (the so-called Prony methods) to the ringdown phase, we find resolution-dependent time variations in the fitted parameters of the final black hole. By cross correlating information from different multipoles, we show that ringdown fits can be used to obtain precise estimates of the mass and spin of the final black hole, which are in remarkable agreement with energy and angular momentum balance calculations.

The speed of gravity in general relativity and theoretical interpretation of the Jovian deflection experiment

According to Einstein, the notions of geodesic, parallel transport (affine connection) and curvature of the spacetime manifold have a pure geometric origin and do not correlate with any electromagnetic concepts. At the same time, curvature is generated by matter which is not affiliated with the spacetime geometric concepts. For this reason, the fundamental constant c entering the geometric and matter sectors of the general theory of relativity has different conceptual meanings. Specifically, the letter c on the left-hand side of the Einstein equations (geometric sector) entering the Christoffel symbols and its time derivatives is the ultimate speed of gravity characterizing the upper limit on the speed of its propagation as well as the maximal rate of change of time derivatives of the metric tensor, that is gravitational field. The letter c on the right-hand side of the Einstein equations (matter sector) is the maximal speed of propagation of any other field rather than gravity. Einstein's general principle of relativity extends his principle of special relativity and equates the numerical value of the ultimate speed of gravity to that of the speed of light in the special theory of relativity but this general principle must be tested experimentally. To this end, we work out the speed of gravity parametrization of the Einstein equations (cg-parametrization) to keep track of the time-dependent effects associated with the geometric sector of general relativity and to separate them from the time-dependent effects of the matter sector. Parametrized post-Newtonian (PPN) approximation of the Einstein equations is derived in order to explain the gravitational physics of the Jovian deflection VLBI experiment conducted on 8 September 2002. The post-Newtonian series expansion in the cg-parametrized general relativity is with respect to a small parameter that is proportional to the ratio of the characteristic velocity of the bodies to the speed of propagation of the gravitational interaction cg. The Einstein equations are solved in terms of the Liénard–Wiechert tensor potentials which are used for integrating the light-ray propagation equations. An exact analytic expression for the relativistic time delay in the propagation of a radio wave from a quasar to an observer is calculated under the assumption that the light-ray deflecting bodies move with constant velocities. A post-Newtonian expansion of the time delay proves that in general relativity the time delay is affected by the speed of gravity already to the first order in 1/cg beyond the leading (static) Shapiro term. We conclude that recent measurements of the propagation of the quasar's radio signal past Jupiter are directly sensitive to the time-dependent effect from the geometric sector of general relativity which is proportional to the speed of propagation of gravity cg but not the speed of light. It provides a first confirmative measurement of the fundamental speed c of the Einstein general principle of relativity for gravitational field. A comparative analysis of our formulation with the alternative interpretations of the experiment given by other authors is provided.

Post-Newtonian approximation for isolated systems calculated by matched asymptotic expansions

Two long-standing problems with the post-Newtonian approximation for isolated slowly-moving systems in general relativity are: (i) the appearance at high post-Newtonian orders of divergent Poisson integrals, casting a doubt on the soundness of the post-Newtonian series; (ii) the domain of validity of the approximation which is limited to the near-zone of the source, and prevents one, a priori, from incorporating the condition of no-incoming radiation, to be imposed at past null infinity. In this article, we resolve the problem (i) by iterating the post-Newtonian hierarchy of equations by means of a new (Poisson-type) integral operator that is free of divergencies, and the problem (ii) by matching the post-Newtonian near-zone field to the exterior field of the source, known from previous work as a multipolar-post-Minkowskian expansion satisfying the relevant boundary conditions at infinity. As a result, we obtain an algorithm for iterating the post-Newtonian series up to any order, and we determine the terms, present in the post-Newtonian field, that are associated with the gravitational-radiation reaction onto an isolated slowly-moving matter system.

Innermost circular orbit of binary black holes at the third post-Newtonian approximation

The equations of motion of two point masses have recently been derived at the third post-Newtonian (3PN) approximation of general relativity. From that work we determine the location of the innermost circular orbit (ICO), defined by the minimum of the binary's 3PN energy as a function of the orbital frequency for circular orbits. We find that the post-Newtonian series converges well for equal masses. Spin effects appropriate to corotational black-hole binaries are included. We compare the result with a recent numerical calculation of the ICO in the case of two black holes moving on exactly circular orbits (helical symmetry). The agreement is remarkably good, indicating that the 3PN approximation is adequate to accurately locate the ICO of two black holes with comparable masses. This conclusion is reached with the post-Newtonian expansion expressed in the standard Taylor form, without using resummation techniques such as Pad\'e approximants and/or effective-one-body methods.

Gravitational waves from binary systems in circular orbits: convergence of a partially bare multipole expansion

The gravitational radiation originating from a compact binary system in circular orbit is usually expressed as an infinite sum over radiative multipole moments. In a slow-motion approximation, each multipole moment is expressed as a post-Newtonian expansion in powers of , the ratio of the orbital velocity to the speed of light. The `bare multipole truncation' of the radiation consists in keeping only the leading-order (Newtonian) term in the post-Newtonian expansion of each moment, but summing over all the multipole moments. In the case of binary systems with small mass ratios, the bare multipole series was shown in a previous paper (Simone et al 1997 Class. Quantum Grav. 14 237) to converge for all values , where e is the base of natural logarithms. (These include all physically relevant values for circular inspiral.) In this paper, we extend the analysis to a `partially bare multipole truncation' of the radiation, in which the leading-order moments are corrected with terms of relative order (first post-Newtonian, or 1PN, terms) and (1.5PN terms). We find that the partially bare multipole series also converges for all values , and that it coincides (to within 1%) with the numerically `exact' results for . Although this multipole series converges, it is an unphysical approximation, and the issue of the convergence of the true post-Newtonian series remains open. However, our analysis shows that an eventual failure of the true post-Newtonian series to converge cannot originate from summing over the Newtonian, 1PN and 1.5PN part of all the multipole moments.

Accuracy of estimating the multipole moments of a massive body from the gravitational waves of a binary inspiral

If the gravitational field of a massive, compact body is stationary, axially symmetric, and reflection symmetric across the equatorial plane, and if a much less massive compact object (such as a neutron star or a small black hole) were to orbit in a circle on the equatorial plane of the central, compact body, then the produced gravitational waves would carry the values of the central body's multipole moments. By detecting those waves and extracting from them the central body's lowest few moments, gravitational-wave detectors have the potential to test the black-hole no-hair theorem and search for exotic objects such as naked singularities and boson or soliton stars. This paper estimates how accurately we can expect to measure the central body's moments. The measurement errors are estimated using a combination of, first, the leading-order (of a post-Newtonian series) contribution of each moment to the gravitational-wave phase, second, an a priori probability distribution that constrains each moment's magnitude to a range appropriate for a compact body, and third, any relations that the multipole moments satisfy among themselves, which reduce the number of degrees of freedom for the waves (this is useful in cases when one is searching for a specific type of compact body). We find that the Earth-based LIGO detector cannot provide sufficiently precise measurements of enough multipole moments to search for exotic objects, but the space-based LISA detector can do so.