We use the effective field theory for gravitational bound states, proposed by Goldberger and Rothstein, to compute the interaction Lagrangian of a binary system at the second post-Newtonian order. Throughout the calculation, we use a metric parametrization based on a temporal Kaluza-Klein decomposition and test the claim by Kol and Smolkin that this parametrization provides important calculational advantages. We demonstrate how to use the effective field theory method efficiently in precision calculations, and we reproduce known results for the second post-Newtonian order equations of motion in harmonic gauge in a straightforward manner.
I. INTRODUCTIONIn the last two decades, significant progress has been made towards the detection of gravitational waves (GWs) via laser interferometry. Currently, the ground-based experiments LIGO [1], VIRGO [2], GEO [3], and TAMA [4] are actively searching for GWs [5]. Moreover, the proposed LISA experiment [6], due to be the first space-based GW detector, will search for GWs in a complementary frequency band to the ground-based experiments and is expected to achieve high event rates at an unprecedented signal-to-noise ratio [7].A particularly interesting source of GWs, which is expected to be detected, is the compact binary system undergoing coalescence, with neutron star (NS) and/or black hole (BH) constituents. Current experiments have yet to detect the binary inspiral signal. However, Advanced LIGO [8], an upgrade of LIGO scheduled to come online in 2014, may allow for routine detection of such events. This is due to a ∼ 10-fold increase in sensitivity over LIGO, which will in turn result in an increase of the accessible event rate by a factor ∼ 1000. Current estimates for the number of expected NS/NS, BH/BH, and BH/NS events in Advanced LIGO are roughly 10 − 100, 1 − 500, and 1 − 30 per year, respectively [9,10].All three stages of the binary coalescence, inspiral, merger, and ringdown, are potentially detectable. The inspiral phase, where the characteristic orbital velocity is v 2 ≪ 1 (in units where c = 1), can be computed analytically using an expansion in v 2 ∼ Gm/r. The merger is computed numerically [11], and there has been significant recent progress in this area [12]. The ringdown can be treated analytically using quasinormal modes [13].The perturbative calculation of the inspiral phase has been performed with a variety of methods [14,15]. Because of the phase evolution of the inspiral signal and the ability to measure the total orbital phase to ∼ 10 −3 over the LIGO bandwidth [16], these perturbation expansions must be calculated to high order. If we consider a circular orbit in the adiabatic approximation, the signal phase Φ(ω) is related to the orbital energy E(ω) and the radiated power P (ω) through the relation d 2 Φ/dω 2 ∼ (dE/dω)/P . An accuracy of ∼ 10 −3 in the cumulative orbital phase, over the LIGO bandwidth, can be achieved if the perturbation expansion is calculated to O(v 6 ) beyond Newtonian dynamics i.e., at third post-Newtonian order (3PN) [9,17]....