New class of post-Newtonian approximants to the waveform templates of inspiralling compact binaries: Test mass in the Schwarzschild spacetime

Abstract

The standard adiabatic approximation to phasing of gravitational waves from inspiralling compact binaries uses the post-Newtonian expansions of the binding energy and gravitational wave flux both truncated at the same relative post-Newtonian order. Motivated by the eventual need to go beyond the adiabatic approximation we must view the problem as the dynamics of the binary under conservative post-Newtonian forces and gravitational radiation damping. From the viewpoint of the dynamics of the binary, the standard approximation at leading order is equivalent to retaining the 0PN and 2.5PN terms in the acceleration and neglecting the intervening 1PN and 2PN terms. A complete mathematically consistent treatment of the acceleration at leading order should include all PN terms up to 2.5PN without any gaps. These define the standard and complete non-adiabatic approximants, respectively. We propose a new and simple complete adiabatic approximant constructed from the energy and flux functions. At the leading order it uses the 2PN energy function rather than the 0PN one in the standard approximation so that in spirit it corresponds to the dynamics where there are no missing post-Newtonian terms in the acceleration. We compare the overlaps of the standard and complete adiabatic approximant templates with the exact waveform (in the adiabatic approximation) for a test particle orbiting a Schwarzschild black hole. Overlaps are computed using both the white-noise spectrum and the initial laser interferometer gravitational wave observatory (LIGO) noise spectrum. The complete adiabatic approximants lead to a remarkable improvement in the effectualness (i.e., larger overlaps with the exact signal) at lower PN (<3PN) orders. However, standard adiabatic approximants of order >=3PN are nearly as good as the complete adiabatic approximants for the construction of effectual templates. In general, faithfulness (i.e., smaller biases in the estimation of parameters) of complete approximants is also better than that of standard approximants. Standard and complete approximants beyond the adiabatic approximation are next studied using the Lagrangian models of Buonanno, Chen, and Vallisneri in the test mass limit. A limited extension of the results to the case of comparable mass binaries is provided. In this case, standard adiabatic approximants achieve an effectualness of 0.965 at order 3PN. If the comparable mass case is qualitatively similar to the test mass case then neither the improvement of the accuracy of energy function from 3PN to 4PN nor the improvement of the accuracy of flux function from 3.5PN to 4PN will result in a significant improvement in effectualness in the comparable mass case for terrestrial laser interferometric gravitational wave detectors.