Detweiler's gauge-invariant redshift variable: analytic determination of the nine and nine-and-a-half post-Newtonian self-force contributions

Analytic determination of the eight-and-a-half post-Newtonian self-force contributions to the two-body gravitational interaction potential

We {\it analytically} compute, to the eight-and-a-half post-Newtonian order, and to linear order in the mass ratio, the radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies, thereby extending previous analytic results. These results are obtained by applying analytical gravitational self-force theory (for a particle in circular orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant redshift variable. We emphasize the increase in \lq\lq transcendentality" of the numbers entering the post-Newtonian expansion coefficient…

Confirming and improving post-Newtonian and effective-one-body results from self-force computations along eccentric orbits around a Schwarzschild black hole

We analytically compute, through the six-and-a-half post-Newtonian order, the second-order-in-eccentricity piece of the Detweiler-Barack-Sago gauge-invariant redshift function for a small mass in eccentric orbit around a Schwarzschild black hole. Using the first law of mechanics for eccentric orbits [A. Le Tiec, Phys. Rev. D {\bf 92}, 084021 (2015)] we transcribe our result into a correspondingly accurate knowledge of the second radial potential of the effective-one-body formalism [A. Buonanno and T. Damour, Phys. Rev. D {\bf 59}, 084006 (1999)]. We compare our newly acquired analytical inform…

New gravitational self-force analytical results for eccentric orbits around a Schwarzschild black hole

We raise the analytical knowledge of the eccentricity-expansion of the Detweiler-Barack-Sago redshift invariant in a Schwarzschild spacetime up to the 9.5th post-Newtonian order (included) for the $e^2$ and $e^4$ contributions, and up to the 4th post-Newtonian order for the higher eccentricity contributions through $e^{20}$. We convert this information into an analytical knowledge of the effective-one-body radial potentials $\bar d(u)$, $ρ(u)$ and $q(u)$ through the 9.5th post-Newtonian order. We find that our analytical results are compatible with current corresponding numerical self-force da…

Linear-in-mass-ratio contribution to spin precession and tidal invariants in Schwarzschild spacetime at very high post-Newtonian order

Using black hole perturbation theory and arbitrary-precision computer algebra, we obtain the post-Newtonian (pN) expansions of the linear-in-mass-ratio corrections to the spin-precession angle and tidal invariants for a particle in circular orbit around a Schwarzschild black hole. We extract coefficients up to 20pN order from numerical results that are calculated with an accuracy greater than 1 part in $10^{500}$. These results can be used to calibrate parameters in effective-one-body models of compact binaries, specifically the spin-orbit part of the effective Hamiltonian and the dynamically …

The First Law of Binary Black Hole Mechanics in General Relativity and Post-Newtonian Theory

First laws of black hole mechanics, or thermodynamics, come in a variety of different forms. In this paper, from a purely post-Newtonian (PN) analysis, we obtain a first law for binary systems of point masses moving along an exactly circular orbit. Our calculation is valid through 3PN order and includes, in addition, the contributions of logarithmic terms at 4PN and 5PN orders. This first law of binary point-particle mechanics is then derived from first principles in general relativity, and analogies are drawn with the single and binary black hole cases. Some consequences of the first law are …

Comparison Between Self-Force and Post-Newtonian Dynamics: Beyond Circular Orbits

The gravitational self-force (GSF) and post-Newtonian (PN) schemes are complementary approximation methods for modelling the dynamics of compact binary systems. Comparison of their results in an overlapping domain of validity provides a crucial test for both methods, and can be used to enhance their accuracy, e.g. via the determination of previously unknown PN parameters. Here, for the first time, we extend such comparisons to noncircular orbits---specifically, to a system of two nonspinning objects in a bound (eccentric) orbit. To enable the comparison we use a certain orbital-averaged quanti…

Spin-orbit precession for eccentric black hole binaries at first order in the mass ratio

We consider spin-orbit ("geodetic") precession for a compact binary in strong-field gravity. Specifically, we compute $ψ$, the ratio of the accumulated spin-precession and orbital angles over one radial period, for a spinning compact body of mass $m_1$ and spin $s_1$, with $s_1 \ll G m_1^2/c$, orbiting a non-rotating black hole. We show that $ψ$ can be computed for eccentric orbits in both the gravitational self-force and post-Newtonian frameworks, and that the results appear to be consistent. We present a post-Newtonian expansion for $ψ$ at next-to-next-to-leading order, and a Lorenz-gauge gr…

Two-body gravitational spin-orbit interaction at linear order in the mass ratio

We analytically compute, to linear order in the mass-ratio, the "geodetic" spin precession frequency of a small spinning body orbiting a large (non-spinning) body to the eight-and-a-half post-Newtonian order, thereby extending previous analytical knowledge which was limited to the third post-Newtonian level. These results are obtained applying analytical gravitational self-force theory to the first-derivative level generalization of Detweiler's gauge-invariant redshift variable. We compare our analytic results with strong-field numerical data recently obtained by S.~R.~Dolan et al. [Phys.\ Rev…

作为第九行星假说替代方案的修正牛顿动力学

Modified Newtonian Dynamics as an Alternative to the Planet Nine Hypothesis

针对新型柯伊伯带天体的轨道异常（曾被视为第九行星存在的证据），研究人员利用长期近似法证明MOND（修正牛顿动力学）可提供替代解释，其轨道长轴对齐银心、相空间团簇的预测与观测相符，提示该引力理论或可在外太阳系被观测到。

利用小型航天器搜寻第九行星：三体、后牛顿、非引力、行星及柯伊伯带效应

A search for Planet Nine with small spacecraft:Three-body, post-Newtonian, non-gravitational, planetary and Kuiper Belt effects

本文针对利用小型航天器引力探测第九行星（推测为原生黑洞）的方案，系统分析了三体、非引力、后牛顿、行星及柯伊伯带等各类效应对航天器轨道角位移的影响，明确了不同速度下的主导扰动及广义相对论效应的贡献水平。

后牛顿近似下的各向异性均匀宇宙

Anisotropic Homogeneous Cosmologies in the Post-Newtonian Approximation

在本文中，我们将探讨如何远后牛顿理论去克服困难与各向异性均匀宇宙学的牛顿近似。我们将看到，与牛顿的情况不同，后牛顿近似的宇宙学方程在9种比安基类型和奇点问题方面更符合广义相对论的精神。将旋转和剪切消失的情况分别处理。均匀的Bianchi I模型被认为是具有各向异性的无旋转宇宙学的一个例子。在牛顿近似下，如果初值问题是适定的，则需要给出所有时间的任意函数，而在后牛顿情况下则没有这种需要。对于理想流体的一般情况，只有后牛顿理论才能令人满意地描述压力的影响。这与先前一篇论文中的发现一致，该论文将后牛顿近似应用于均匀宇宙学。对于无剪切的各向异性均匀宇宙学，Heckmann和Schücking的牛顿理论被探索。与它的相对论和后牛顿对应的比较。在牛顿理论中存在着在广义相对论中没有类似物的解。后牛顿近似可能提供一条出路。

Spin-dependent two-body interactions from gravitational self-force computations

We analytically compute, through the eight-and-a-half post-Newtonian order and the fourth-order in spin, the gravitational self-force correction to Detweiler's gauge invariant redshift function for a small mass in circular orbit around a Kerr black hole. Using the first law of mechanics for black hole binaries with spin [L.~Blanchet, A.~Buonanno and A.~Le Tiec, Phys.\ Rev.\ D {\bf 87}, 024030 (2013)] we transcribe our results into a knowledge of various spin-dependent couplings, as encoded within the spinning effective-one-body model of T.~Damour and A.~Nagar [Phys.\ Rev.\ D {\bf 90}, 044018 (…

A conservative effect of the second-order gravitational self-force on quasicircular orbits in Schwarzschild spacetime

A compact object moving on a quasicircular orbit about a Schwarzschild black hole gradually spirals inward due to the dissipative action of its gravitational self-force. But in addition to driving the inspiral, the self-force has a conservative piece. Within a second-order self-force formalism, I derive a second-order generalization of Detweiler's redshift variable, which provides a gauge-invariant measure of conservative effects on quasicircular orbits. I sketch a frequency-domain numerical scheme for calculating this quantity. Once this scheme has been implemented, its results may be used to…

后牛顿理论和数值相对论的波形交汇处

Where post-Newtonian and numerical-relativity waveforms meet

我们分析了数值相对论（NR）波形，覆盖9个轨道（18个引力波周期）合并前的等质量系统与低偏心率，数值不确定性的相位为0.25弧度，幅度小于2%，这样的精度允许直接比较后牛顿（PN）波形。我们专注于一个PN近似，已被提出用于引力波数据分析，限制3.5 PN "泰勒T1“波形，并比较这些从第二个到第八个轨道，这是合并前约一个半轨道的数值波形的一部分。这对应于$Mω= 0.0455$到0.1的引力波频率范围。取决于匹配PN和NR波形的方法，在该频率范围上的累积相位不一致可以在数值不确定性内。在与替代PN近似3PN "TaylorT3“的比较中发现了类似的结果。另一方面，振幅差异约为6%，但对于所有13个比较的周期大致恒定，这表明仅需要模拟4.5个轨道以与9个轨道相同的精度匹配PN和NR波。然而，如果我们对振幅建模到2.5PN阶，则振幅不一致大致在合并前约11个周期的数值不确定性内。

旋转双星数值相对论和后牛顿波形之间的比较：轨道悬挂情况

Comparison between numerical-relativity and post-Newtonian waveforms from spinning binaries: the orbital hang-up case

我们比较的结果从数值模拟的自旋二进制的“轨道悬挂”的情况下，二进制完成至少9个轨道合并前，与后牛顿的结果使用近似泰勒T1，T4和Et。我们发现，在引力波频率达到$Mω= 0.1$之前的10个周期内，NR和2.5PN结果之间的累积相位差小于3弧度，当使用3.5PN结果时小于2.5弧度。随着黑洞自旋的增加，NR结果和限制PN结果之间的振幅不一致性增加，从等质量情况下的约6%增加到黑洞自旋为$S_i/M_i^2 = 0.85$时的12%。最后，我们的结果表明，合并波形将发挥重要作用，估计自旋从这样的inspiral波形。

Conservative second-order gravitational self-force on circular orbits and the effective one-body formalism

We consider Detweiler's redshift variable $z$ for a nonspinning mass $m_1$ in circular motion (with orbital frequency $Ω$) around a nonspinning mass $m_2$. We show how the combination of effective-one-body (EOB) theory with the first law of binary dynamics allows one to derive a simple, exact expression for the functional dependence of $z$ on the (gauge-invariant) EOB gravitational potential $u=(m_1+m_2)/R$. We then use the recently obtained high-post-Newtonian(PN)-order knowledge of the main EOB radial potential $A(u ; ν)$ [where $ν= m_1 m_2/(m_1+m_2)^2$] to decompose the second-self-force-or…

Exploiting Newton-factorized, 2PN-accurate, waveform multipoles in effective-one-body models for spin-aligned noncircularized binaries

We present a new approach to factorize and resum the post-Newtonian (PN) waveform for generic equatorial motion to be used within effective-one-body (EOB) based waveform models. The new multipolar waveform factorization improves previous prescriptions in that: (i) the generic Newtonian contribution is factored out from each multipole; (ii) the circular part is factored out and resummed using standard EOB methods and (iii) the residual, 2PN-accurate, noncircular part, and in particular the tail contribution, is additionally resummed using Padé approximants. The resulting waveform is validated i…

Accuracy of numerical relativity waveforms from binary neutron star mergers and their comparison with post-Newtonian waveforms

We present numerical relativity simulations of nine-orbit equal-mass binary neutron star covering the quasicircular late inspiral and merger. The extracted gravitational waveforms are analyzed for convergence and accuracy. Second order convergence is observed up to contact, i.e. about 3-4 cycles to merger; error estimates can be made up to this point. The uncertainties on the phase and the amplitude are dominated by truncation errors and can be minimized to 0.13 rad and less then 1%, respectively, by using several simulations and extrapolating in resolution. In the latter case finite-radius ex…

Beyond the geodesic approximation: conservative effects of the gravitational self-force in eccentric orbits around a Schwarzschild black hole

We study conservative finite-mass corrections to the motion of a particle in a bound (eccentric) strong-field orbit around a Schwarzschild black hole. We assume the particle's mass $μ$ is much smaller than the black hole mass $M$, and explore post-geodesic corrections of $O(μ/M)$. Our analysis uses numerical data from a recently developed code that outputs the Lorenz-gauge gravitational self-force (GSF) acting on the particle along the eccentric geodesic. First, we calculate the $O(μ/M)$ conservative correction to the periastron advance of the orbit, as a function of the (gauge-dependent) semi…