Accuracy of the post-Newtonian approximation. II. Optimal asymptotic expansion of the energy flux for quasicircular, extreme mass-ratio inspirals into a Kerr black hole

We study the effect of black hole spin on the accuracy of the post-Newtonian approximation. We focus on the gravitational energy flux for the quasicircular, equatorial, extreme mass-ratio inspiral of a compact object into a Kerr black hole of mass $M$ and spin $J$. For a given dimensionless spin $a\equiv J/M^2$ (in geometrical units $G=c=1$), the energy flux depends only on the orbital velocity $v$ or (equivalently) on the Boyer-Lindquist orbital radius $r$. We investigate the formal region of validity of the Taylor post-Newtonian expansion of the energy flux (which is known up to order $v^8$ beyond the quadrupole formula), generalizing previous work by two of us. The error function used to determine the region of validity of the post-Newtonian expansion can have two qualitatively different kinds of behavior, and we deal with these two cases separately. We find that, at any fixed post-Newtonian order, the edge of the region of validity (as measured by $v/v_{\mathrm{ISCO}}$, where $v_{\mathrm{ISCO}}$ is the orbital velocity at the innermost stable circular orbit) is only weakly dependent on $a$. Unlike in the nonspinning case, the lack of sufficiently high-order terms does not allow us to determine if there is a convergent to divergent transition at order $v^6$. Independent of $a$, the inclusion of angular multipoles up to and including $\ell =5$ in the numerical flux is necessary to achieve the level of accuracy of the best-known ($N=8$) post-Newtonian expansion of the energy flux.

Figure 1

ISCO velocity ($v_{\mathrm{ISCO}}$, top panel) and radius ($r_{\mathrm{ISCO}}/M$, bottom panel) as a function of $a$. Here and elsewhere we use the convention that a negative spin parameter refers to counterrotating orbits.

Figure 2

Gravitational energy flux (normalized to $F_{\mathrm{Newt}}$) as a function of the normalized orbital velocity, $v/v_{\mathrm{ISCO}}$. The left panel is for corotating orbits, and the right panel is for counterrotating orbits. Different insets refer to different spin parameters $a$, as indicated. The thick black line is the numerical flux. Other linestyles refer to different PN approximations: $F^{(2)}$ (thin black), $F^{(3)}$ (long-dashed red), $F^{(4)}$ (dash-dotted green), $F^{(5)}$ (dash-dash-dotted blue), $F^{(6)}$ (dash-dot-dotted orange), $F^{(7)}$ (dotted dark green), $F^{(8)}$ (short-dashed violet).

Figure 3

Left: absolute value of the remainder of the $N=3$ PN flux, $|F-F^{(3)}|/F_{\mathrm{Newt}}$ (solid line), and the $N=4$ term $|F^{(4)}-F^{(3)}|/F_{\mathrm{Newt}}$ (dashed red line). The inset shows the modulus of their difference, Eq. (8), in a semilogarithmic scale. Right: same as the left panel, but for the $N=6$ remainder and $N=7$ term. All curves in this plot refer to the counterrotating case with spin $a=-0.5$. The lower ($v_l$, more conservative) and upper ($\overline{v}$, less conservative) edges of the region of validity are (somewhat conventionally) delimited by the vertical lines, as explained in the main text.

Figure 4

Edge of the region of validity as a function of $a$ for different PN orders, in the corotating (black, straight line) and counterrotating (red, dashed line) cases. The blue, dashed lines for $N=3$ refer to the counterrotating case, and they were obtained by an alternative method (see the discussion around Fig. 7 below).

Figure 5

Edge of the region of validity expressed in terms of the Boyer-Lindquist radius for different PN orders, in the corotating (black, straight line) and counterrotating (red, dashed line) cases. The blue, dashed lines for $N=3$ refer to the counterrotating case, and they were obtained by an alternative method (see the discussion around Fig. 7 below).

Figure 6

Edge of the region of validity as a function of $N$ for fixed values of $a$. The thick, black error bar corresponds to $a=0$, followed by $a=0.3$, 0.6, 0.9 to the right and $a=-0.1$, $-0.2$, $-0.6$, $-0.9$, $-0.99$ to the left. Note that $N=3$ is a special case. For reasons discussed in the text, we do not give error bars for $N=3$ and $a<-0.1$.

Figure 7

The solid black line shows ${\delta }^{(N)}(v)$ for $N=3$. The horizontal dashed red (dot-dashed green) lines represent ${\delta }_0$, computed from Eq. (10), and ${\delta }_0/2$. See text for discussion.

Figure 8

Relevance of multipolar components up to $\ell =2$, $\ell =3$, $\ell =4$, $\ell =5$, and summing as many $\ell$’s as necessary for the relative accuracy of the Teukolsky code to be $\mathcal{O}({10}^{-10})$ at any given velocity.