Impact of the second-order self-forces on the dephasing of the gravitational waves from quasicircular extreme mass-ratio inspirals

The accurate calculation of the long-term phase evolution of gravitational wave (GW) forms from extreme (intermediate) mass-ratio inspirals [E(I)MRIs] is an inevitable step to extract information from this system. In order to achieve this goal, it is believed that we need to understand the gravitational self-forces. However, it has not been quantitatively demonstrated that the second-order self-forces are necessary for this purpose. In this paper, we revisit the problem to estimate the order of magnitude of the dephasing caused by the second-order self-forces on a small body in a quasicircular orbit around a Kerr black hole, based on the knowledge of the post-Newtonian (PN) approximation and invoking the energy balance argument. In particular, we focus on the averaged dissipative part of the self-force, since it gives the leading-order contribution among their various components. To avoid the possibility of the energy flux of GWs becoming negative, we propose a new simple resummation called exponential resummation, which assures the positivity of the energy flux. In order to estimate the magnitude of the yet-unknown second-order self-forces, here we point out the scaling property in the absolute value of the PN coefficients of the energy flux. Using these new tools, we evaluate the expected magnitude of dephasing. Our analysis indicates that the dephasing due to the second-order self-forces for quasicircular E(I)MRIs may be well captured by the 3 PN energy flux, once we obtain all the spin-dependent terms, except for the case with an extremely large spin of the central Kerr black hole.

Figure 1

The normalized $n\mathrm{PN}$ Taylor flux in the test particle limit, $L_{n\mathrm{PN}}^{\mathrm{T}[0]}(x,q)$, for $q=0.9$ up to $x_{\mathrm{ISCO}}(0.9)≔0.78014\dots$. The horizontal axis is the dimensionless frequency $x$ of a body. The label “Exact” means the exact numerical energy flux, calculated by Fujita and Tagoshi [55].

Figure 2

The absolute value of the contribution to the energy flux from each PN order $|C_{n,0}|$ at $x=x_{\mathrm{lr}}(q)$, with the horizontal axis being the PN order $n$. Note that the values of $|C_{n,0}|$ are accidentally small with some fixed $q$. We have safely excluded these values from the plot.

Figure 3

The dephasing due to the higher PN corrections in the test particle limit $\nu \Delta \Phi [{\mathcal{L}}_{n\mathrm{PN}}^{\exp [0]},{\mathcal{L}}_{>n\mathrm{PN}}^{\exp [0]}]$ as a function of $\nu /M$ for $q=\pm 0.9$.

Figure 4

The expected residual dephasing $\Delta {\Phi }_{\mathrm{guess}}[{\mathcal{L}}_{\mathrm{full}}^{[0]}+{\tilde{\mathcal{L}}}_{n\mathrm{PN}}^{[1]},{\tilde{\mathcal{L}}}_{>n\mathrm{PN}}^{[1]}]$ caused by the unknown part of the averaged dissipative second-order self-forces. For $n=3$, the 3.5 PN known spin-independent part of the flux at the next leading order in the mass ratio is treated as unknown.

Figure 5

The expected residual dephasing $\Delta {\Phi }_{\mathrm{guess}}[{\mathcal{L}}_{\mathrm{full}}^{[0]}+{\mathcal{L}}_{n\mathrm{PN}}^{[1]},{\mathcal{L}}_{>n\mathrm{PN}}^{[1]}]$ when we assume that the spin-dependent terms at the lower PN orders are all known.

Figure 6

The dephasing caused by the known terms up to the 3 PN order at the next leading order in the mass ratio, $\Delta {\Phi }_{\mathrm{PN}}[{\mathcal{L}}_{\mathrm{full}}^{[0]},{\tilde{\mathcal{L}}}_{3\mathrm{PN}}^{\mathrm{hyb}[1]}]$, for various dimensionless spin parameters $q$ of a Kerr black hole, based on the 3 PN hybrid flux [Eq. (16)].

Figure 7

The dephasing $\nu \Delta \Phi [{\mathcal{L}}_{\mathrm{full}}^{[0]}+{\mathcal{L}}_{2.5\mathrm{PN}}^{[0,\mathrm{H}]},{\mathcal{L}}_{>2.5\mathrm{PN}}^{[0,\mathrm{H}]}]$ due to the horizon absorption flux and the dephasing $\nu \Delta \Phi [{\mathcal{L}}_{n\mathrm{PN}}^{\exp [0]},{\mathcal{L}}_{>n\mathrm{PN}}^{\exp [0]}]$ due to the higher PN corrections, for $n=3$ and $n=3.5$, and $q=\pm 0.9$.