Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders

Using the new methodology introduced in a recent paper [D. Bini, T. Damour, and A. Geralico, Phys. Rev. Lett. 123, 231104 (2019)], we present the details of the computation of the conservative dynamics of gravitationally interacting binary systems at the fifth post-Newtonian (5PN) level, together with its extension at the fifth-and-a-half post-Newtonian level. We present also the sixth post-Newtonian (6PN) contribution to the third-post-Minkowskian (3PM) dynamics. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, gravitational self-force, effective one-body, and Delaunay averaging. We determine the full functional structure of the 5PN Hamiltonian (which involves 95 nonzero numerical coefficients), except for two undetermined coefficients proportional to the cube of the symmetric mass ratio, and to the fifth and sixth power of the gravitational constant, $G$. We present not only the 5PN-accurate, 3PM contribution to the scattering angle but also its 6PN-accurate generalization. Both results agree with the corresponding truncations of the recent 3PM result of Bern et al. [Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019)]. We also compute the 5PN-accurate, fourth-post-Minkowskian (4PM) contribution to the scattering angle, including its nonlocal contribution, thereby offering checks for future 4PM calculations. We point out a remarkable hidden simplicity of the gauge-invariant functional relation between the radial action and the effective-one-body energy and angular momentum.

Figure 1

Matrix of the 95 nonzero numerical coefficients $C_{2k,n}^{(2l)}$ encoding the various powers of $\nu$ in the Hamiltonian (11.8). The checks indicate the coefficients determined in the present work. The only two missing coefficients are indicated by circled dots.

Figure 2

Schematic representation of the theoretical tools used to obtain the various contributions to the 5PN-accurate local Hamiltonian, adapted from Ref. [26]. These contributions are keyed, on the horizontal axis, by powers of $u=GM/r$ and squared momentum $p^2\sim p_r^2$ and, on the vertical axis, by powers of $\nu \equiv m_1{m}_2/(m_1+m_2{)}^2$. The checks indicate the coefficients determined in the present work. The question marks denote the only two missing coefficients. Note that, even if certain coefficients in Table 8 only include terms up to $O({\nu }^5)$, the identification $p^2\sim p_r^2$ done in this schematic figure lumps terms together so that $O({\nu }^6)$ terms arise in each column.