Post-Newtonian expansions of extreme mass ratio inspirals of spinning bodies into Schwarzschild black holes

Author(s)

Skoupý, Viktor, Witzany, Vojtěch

Abstract

Space-based gravitational-wave detectors such as LISA are expected to detect inspirals of stellar-mass compact objects into massive black holes. Modeling such inspirals requires fully relativistic computations to achieve sufficient accuracy at leading order. However, subleading corrections such as the effects of the spin of the inspiraling compact object may potentially be treated in weak-field expansions such as the post-Newtonian (PN) approach. In this work, we calculate the PN expansion of eccentric orbits of spinning bodies around Schwarzschild black holes. Then we use the Teukolsky equation to compute the energy and angular momentum fluxes from these orbits up to the 5PN order. Some of these PN orders are exact in eccentricity, while others are expanded up to the tenth power in eccentricity. Then we use the fluxes to construct a hybrid inspiral model, where the leading part of the fluxes is calculated numerically in the fully relativistic regime, while the part linear in the small spin is analytically approximated using the PN series. We calculate LISA-relevant inspirals and respective waveforms with this model and a fully relativistic model. Through the calculation of mismatch between the waveforms from both models we conclude that the PN approximation of the linear-in-spin part of the fluxes is sufficient for lower eccentricities.

Figures

Coefficients in the PN expansion and eccentricity expansion of the linear part of the energy flux $\delta \mathcal{F}^E$ from Eq.~\eqref{eq:deltaFEn}.

Coefficients in the PN expansion and eccentricity expansion of the linear part of the energy flux $\delta \mathcal{F}^E$ from Eq.~\eqref{eq:deltaFEn}.


Coefficients in the PN expansion and eccentricity expansion of the linear part of the angular momentum flux $\delta \mathcal{F}^{J_z}$ from Eq.~\eqref{eq:deltaFJz}.

Coefficients in the PN expansion and eccentricity expansion of the linear part of the angular momentum flux $\delta \mathcal{F}^{J_z}$ from Eq.~\eqref{eq:deltaFJz}.


Relative difference between the PN expansion of the linear-in-spin part of the energy (top) and angular momentum (bottom) flux $\delta\mathcal{F}_{\text{PN}}$ and the fully relativistic value of $\delta\mathcal{F}_{\text{num}}$ for different eccentricities. The dashed lines show dependence $\order{p^{-4}}$ which should be the order of the error.

Relative difference between the PN expansion of the linear-in-spin part of the energy (top) and angular momentum (bottom) flux $\delta\mathcal{F}_{\text{PN}}$ and the fully relativistic value of $\delta\mathcal{F}_{\text{num}}$ for different eccentricities. The dashed lines show dependence $\order{p^{-4}}$ which should be the order of the error.


Adiabatic inspirals in the $p$-$e$ plane. The black line shows the separatrix $p=6+2e$. The two models are indistinguishable in this plot (see Figure \ref{fig:deltaPhi} for phase differences).

Adiabatic inspirals in the $p$-$e$ plane. The black line shows the separatrix $p=6+2e$. The two models are indistinguishable in this plot (see Figure \ref{fig:deltaPhi} for phase differences).


Absolute differences between the azimuthal (top) and radial (bottom) phases obtained from the PN model and fully relativistic model.

Absolute differences between the azimuthal (top) and radial (bottom) phases obtained from the PN model and fully relativistic model.


Waveforms of the $+$ polarization of an inspiral ending at $e=0.1$ calculated with the hybrid model (blue) and with the fully relativistic (numerical) model (yellow). The inspiral is observed from the distance of 1 Gpc at the viewing angle $\theta = \pi/3$, $\phi = \pi/4$ in the source frame.

Waveforms of the $+$ polarization of an inspiral ending at $e=0.1$ calculated with the hybrid model (blue) and with the fully relativistic (numerical) model (yellow). The inspiral is observed from the distance of 1 Gpc at the viewing angle $\theta = \pi/3$, $\phi = \pi/4$ in the source frame.


Waveforms of the $+$ polarization of an inspiral ending at $e=0.3$ calculated with the hybrid model (blue) and the fully relativistic (numerical) model (yellow). The distance and viewing angle are the same as in Figure~\ref{fig:waveform01}.

Waveforms of the $+$ polarization of an inspiral ending at $e=0.3$ calculated with the hybrid model (blue) and the fully relativistic (numerical) model (yellow). The distance and viewing angle are the same as in Figure~\ref{fig:waveform01}.


Mismatches between inspirals calculated with the hybrid model and the fully relativistic model for different final eccentricities $e_f$. The blue points show inspirals which end at $\dot{\Omega}_r/\Omega_r^2 = 10^{-2}$ while the yellow points show inspirals ending at $\dot{\Omega}_r/\Omega_r^2 = 10^{-3}$. Small differences between these cases indicate that the mismatches are almost independent of the ending criterion.

Mismatches between inspirals calculated with the hybrid model and the fully relativistic model for different final eccentricities $e_f$. The blue points show inspirals which end at $\dot{\Omega}_r/\Omega_r^2 = 10^{-2}$ while the yellow points show inspirals ending at $\dot{\Omega}_r/\Omega_r^2 = 10^{-3}$. Small differences between these cases indicate that the mismatches are almost independent of the ending criterion.


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