Non-conformally flat initial data for binary compact objects

A new method is described for constructing initial data for a binary neutron-star (BNS) system in quasi-equilibrium circular orbit. Two formulations for non-conformally flat data, waveless (WL) and near-zone helically symmetric (NHS), are introduced; in each formulation, the Einstein-Euler system, written in 3+1 form on an asymptotically flat spacelike hypersurface, is exactly solved for all metric components, including the spatially non-conformally flat potentials, and for irrotational flow. A numerical method applicable to both formulations is explained with an emphasis on the imposition of a spatial gauge condition. Results are shown for solution sequences of irrotational BNS with matter approximated by parametrized equations of state that use a few segments of polytropic equations of state. The binding energy and total angular momentum of solution sequences computed within the conformally flat -- Isenberg-Wilson-Mathews (IWM) -- formulation are closer to those of the third post-Newtonian (3PN) two point particles up to the closest orbits, for the more compact stars, whereas sequences resulting from the WL/NHS formulations deviate from the 3PN curve even more for the sequences with larger compactness. We think it likely that this correction reflects an overestimation in the IWM formulation as well as in the 3PN formula, by $\sim 1$ cycle in the gravitational wave phase during the last several orbits. The work suggests that imposing spatial conformal flatness results in an underestimate of the quadrupole deformation of the components of binary neutron-star systems in the last few orbits prior to merger.