Intermediate Mass-Ratio Black Hole Binaries: Applicability of Small Mass-Ratio Perturbation Theory

The inspiral phasing of binary black holes at intermediate mass ratios ($m_2/m_1\sim {10}^{-3}$) is important for gravitational wave observations, but not accessible to standard modeling techniques: The accuracy of the small mass-ratio (SMR) expansion is unknown at intermediate mass ratios, whereas numerical relativity simulations cannot reach this regime. This article assesses the accuracy of the SMR expansion by extracting the first three terms of the SMR expansion from numerical relativity data for nonspinning, quasicircular binaries. We recover the leading term predicted by SMR theory and obtain a robust prediction of the next-to-leading term. The influence of higher-order terms is bounded to be small, indicating that the SMR series truncated at next-to-leading order is quite accurate at intermediate mass ratios and even at nearly comparable mass binaries. We estimate the range of applicability for SMR and post-Newtonian series for nonspinning, quasicircular inspirals.

Figure 1

Top: leading-order in mass-ratio contribution to the orbital phasing of the quasicircular inspiral of nonspinning black holes. Shown are the results derived here from NR simulations, as well as the SMR perturbation theory. For both curves, the 3.5PN result [14] was subtracted for clarity of plotting. Bottom: difference between SMR and the NR result. The shaded areas indicate the estimated uncertainty of the numerical calculation and ${\mathrm{\Omega }}_{\mathrm{ISCO}}$ indicates the last stable orbit for $\nu =0$.

Figure 2

Top: the three leading terms in the mass-ratio expansion of the orbital phase, as computed here. Bottom: residuals $R_{2+}$ for all 55 NR simulations indicating the combined contributions of 2PA and higher, as well as an envelope bounding these residuals in a $\nu$-independent manner.

Figure 3

Impact of the choice of expansion parameters on the 1PA and 2PA contributions to the orbital phasing. The different curves differ in whether in Eq. (3) is expanded in powers of $\nu$ or in $q$ and whether the $\varphi$ functions are written in terms of $M\mathrm{\Omega }$ or $m_1\mathrm{\Omega }$. The combination $\nu ,M\mathrm{\Omega }$ yields an exceptionally small 2PA term at low frequencies.

Figure 4

Region of applicability of different approximation techniques for nonspinning quasicircular binary black hole inspiral. The shaded regions indicate ranges within which the cumulative orbital phase error is less than $\pi /4$ and $\pi /16\mathrm{rad}$, respectively.