Inspiraling compact objects with generic deformations

Self-gravitating bodies can have an arbitrarily complex shape, which implies a much richer multipolar structure than that of a black hole in general relativity. With this motivation, we study the corrections to the dynamics of a binary system due to generic, nonaxisymmetric mass quadrupole moments to leading post-Newtonian (PN) order. Utilizing the method of osculating orbits and a multiple scale analysis, we find analytic solutions to the precession and orbital dynamics of a (generically eccentric) binary in terms of the dimensionless modulus parameters ${\epsilon }_m$, corresponding to axial $m=1$ and polar $m=2$ corrections from oblateness/prolateness. The solutions to the precession dynamics are exact for $0\le {\epsilon }_2<1$, and perturbative in ${\epsilon }_1\ll 1$. We further compute the leading order corrections to the gravitational wave amplitude and phase for a quasicircular binary due to mass quadrupole effects. Making use of the stationary phase approximation and shifted uniform asymptotics (SUA), the corrections to the phase enter at relative 2PN order, while the amplitude modulations enter at $-0.5\mathrm{PN}$ order with a SUA amplitude correction at 3.25PN order, relative 2PN order to the leading order SUA correction. By investigating the dephasing due to generic quadrupole moments, we find that a phase difference $\gtrsim 0.1$ radians is achievable for ${\epsilon }_m\gtrsim {10}^{-3}$, which suggests that constraints with current and future ground-based gravitational wave detectors are possible. Our results can be implemented in parameter estimation studies to constrain generic multipolar deformations of the Kerr geometry and of neutron stars.

Figure 1

Graphical sketch of the orbital motion (blue solid line) as viewed in the fundamental reference frame. Here $\iota$ is the inclination angle, $\mathrm{\Omega }$ is the longitude of the ascending node, $\omega$ is the longitude of pericenter and $V$ is the true anomaly.

Figure 2

Top panel: comparison of the numerical evolution of $\iota$ (left), $\mathrm{\Omega }$ (middle), and $\omega$ (right) from Eqs. (49)–(51) (solid lines) to the analytic solutions (dashed lines) given in Sec. 3a4 for ${\epsilon }_2=0.9$, ${\alpha }_2=0$, ${\alpha }_1=\pi /2$. Each color represents a different value of ${\epsilon }_1$, specifically ${10}^{-3}$ (black/upper), ${10}^{-5}$ (red/middle), and ${10}^{-7}$ (cyan/lower). All of the calculations are performed with ${\iota }_0=\pi /50$. Each case is offset by a constant value from the others so that they do no overlap. Bottom panels: relative difference between the numeric and analytic solutions.

Figure 3

Top panel: comparison of the numerical evolution (solid lines) of the time variable $t_{(6)}(\tilde{u})=t(\tilde{u})/{10}^6$ (left), orbital phase ${\varphi }_{(3)}(\tilde{u})=\varphi (\tilde{u})/{10}^3$ (middle), and precession phase ${\psi }_2(\tilde{u})$ to the analytic PN expressions (dashed lines) in Eqs. (123), (124), (126), and (130), respectively. The dashed lines correspond to different values of the polar modulus ${\epsilon }_2$, specifically ${10}^{-3}$ (cyan), ${10}^{-2}$ (red), and ${10}^{-1}$ (black). The remaining parameters are held fixed at ${\epsilon }_1={10}^{-3}$, ${\alpha }_2=0$, ${\alpha }_1=\pi /2$, $\nu =1/4$, and $Q_0=M^3$. Bottom panel: dephasing in radians between the numerical evolution and analytic expressions. The dephasing in ${\psi }_2$ (bottom right) does not vary significantly for the values of ${\epsilon }_2$ considered and it is mainly due to the PN truncation and the precession-average procedure.

Figure 4

Comparison of the waveform amplitudes ${\mathcal{A}}_m$ from Eq. (135) for the spheroidal case (black), ${\epsilon }_1={\epsilon }_2={10}^{-3}$ (red, dashed), and ${\epsilon }_1={10}^{-3}$, ${\epsilon }_2={10}^{-1}$ (blue, dot-dashed). The amplitudes are all normalized such that ${\mathcal{A}}_m=1$ at the lowest frequency plotted.

Figure 5

Top panel: total GW phase difference between an oblate/prolate configuration and one with ${\epsilon }_1={\epsilon }_2={10}^{-3}$. The polar phase ${\alpha }_2=0$, while the different lines correspond to different values of ${\alpha }_1$, namely 0 (black), $\pi /4$ (red, dashed), $\pi /2$ (blue, dot-dashed), and $\pi$ (cyan, crossed). Bottom panel: the same as the top panel but with ${\epsilon }_1={10}^{-3}$ and ${\epsilon }_2={10}^{-1}$.