Intermediate Mass Ratio Black Hole Binaries: Numerical Relativity meets Perturbation Theory Carlos O. Lousto, Hiroyuki Nakano, Yosef Zlochower, and Manuela Campanelli Center for Computational Relativity and Gravitation, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, New York 14623 (Dated: November 4, 2018) We study black-hole binaries in the intermediate-mass-ratio regime 0 . 01 < ∼ q < ∼ 0 . 1 with a new technique that makes use of nonlinear numerical trajectories and efficient perturbative evolutions to compute waveforms at large radii for the leading and nonleading modes. As a proof-of-concept, we compute waveforms for q = 1 / 10. We discuss applications of these techniques for LIGO/VIRGO data analysis and the possibility that our technique can be extended to produce accurate waveform templates from a modest number of fully-nonlinear numerical simulations. PACS numbers: 04.25.Dm, 04.25.Nx, 04.30.Db, 04.70.Bw Introduction: The dramatic breakthroughs in the numer- ical techniques to evolve black-hole binaries (BHB) [1– 3] transformed the field and made it possible to begin constructing waveform templates for use in LIGO and VIRGO gravitational wave data analysis and detection from highly-accurate, fully-nonlinear simulations. This promises to greatly aid in the detection and analysis of gravitational waves from astrophysical sources [4]. comparable-mass BHB simulations. The intermediate region of 1 / 100 < q < 1 / 10 provides a unique oppor- tunity to compare fully-nonlinear numerical and pertur- bative techniques. In this letter we introduce a novel ap- proach to deal with this intermediate-mass-ratio regime. We perform a ‘proof of principle’ comparison, to deter- mine an upper bound on the error in our technique, be- tween fully-nonlinear numerical simulations and pertur- bative ones in the borderline case of a nonspinning BHB system with mass ratio q = 1 / 10. The key element introduced in our approach is the use of the numerically generated trajectories (in the moving puncture approach [2, 3]) of a BHB during the late inspiral regime in a perturbative calculation of the waveform emitted by the binary. While LISA’s sensitivity band is well suited to observe supermassive BHBs, ground based observatories are more sensitive to BHBs with masses up to a few hundred so- lar masses. Both of these types of BHBs are most likely to have mass ratios in the range 1:10 - 1:100 [5]. This regime is hard to model with fully-nonlinear numerical simulations, which thus far have focused on the compa- rable mass regime. On the other hand, BHBs in the very small mass ratio regime can, in principle, be mod- eled accurately with perturbation theory, and significant theoretical effort was dedicated to the consistent com- putation of the self-force on a point-like source orbit- ing around Schwarzschild and Kerr black holes (see [6] for a review of new developments in the field). These perturbative techniques allow one to formally compute geodesic deviations to second order. Although secular effects still need to be quantified, these techniques are expected to provide a good approximation for mass ra- tios q = m$_{1}$/m$_{2}$ < 1 / 100. Full Numerical Simulations: In order to approxi- mate the late orbital parameters of a quasicircular inspiral we computed the momentum and puncture- position parameters using a resummed (particle limit) 3.5 PN evolution of a q = 1 / 10, non-spinning binary from r = 50 M to r = 7 . 25 M . To compute the numerical initial data, we use the puncture approach [14] along with the TwoPunctures [15] thorn. We chose the puncture mass parameters such that the system had a total ADM mass M = 1 and mass ratio q = 1 / 10. Fully-nonlinear BHB simulations with small mass- ratios are challenging and the smallest mass ratios pub- lished so far are q = 1 / 10 in the nonspinning case [7, 8] and q = 1 / 8 for highly-spinning BHBs [9]. Most fully- nonlinear numerical simulations published have been in the comparable mass regime, and consequently, most of the comparisons between NR and post-Newtonian (PN) waveforms have been in this regime (for comparisons of NR and PN waveforms from generic binaries, see [10, 11] and references therein). Similarly, phenomenological [12] and EOBNR [13] modeling of waveforms are based on We evolved this BHB data-set using the LazEv [16] implementation of the moving puncture approach [2, 3]. Our code used the Cactus toolkit [17] and the Carpet [18] mesh refinement driver to provide a ‘moving boxes’ style mesh refinement. We used a modified gauge condition suggested in [19]. Our simulation used 11 levels of re- finement (around the smaller components), with central resolutions as high as M/ 368 . 64, and 9 levels of refine- ment around the larger component. The outer bound- aries were located at 400 M and the resolution in the boundary zone was h = 2 . 7777 M for our finest resolu- tion run. The BHB performs four orbits prior to merger, arXiv:1001.2316v1 [gr-qc] 13 Jan 2010