Gravitational waveforms, polarizations, response functions, and energy losses of triple systems in Einstein-aether theory

Gravitationally bound hierarchies containing three or more components are very common in our Universe. In this paper we study the periodic gravitational wave (GW) form, its polarizations, the response function, the Fourier transform, and the energy loss rate of a triple system through three different channels of radiation, the scalar, vector, and tensor modes, in the Einstein-aether theory of gravity. The theory violates locally the Lorentz symmetry, and yet satisfies all the theoretical and observational constraints by properly choosing its four coupling constants ci’s. In particular, in the weak-field approximations and with the recently obtained constraints of the theory, we first analyze the energy loss rate of a binary system and find that the dipole contributions from the scalar and vector modes could be of the order of O(c14)O(GNm/d)2, where c14 (≡c1+c4) is constrained to c14≲O(10-5) by current observations, and GN, m, and d are, respectively, the Newtonian constant, mass, and size of the source. On the other hand, the “strong-field” effects for a binary system of neutron stars are about 6 orders lower than that of general relativity. So, in this paper we ignore these strong-field effects and first develop the general formulas to the lowest post-Newtonian order, by taking the coupling of the aether field with matter into account. Within this approximation, we find that the scalar breather mode and the scalar longitudinal mode are all suppressed by a factor of O(c14) with respect to the transverse-traceless modes (h+ and h×), while the vectorial modes (hX and hY) are suppressed by a factor of c13≲O(10-15). Applying the general formulas to a triple system with periodic orbits, we find that the corresponding GW form, the response function, and its Fourier transform depend sensitively on the configuration of the triple system, their orientation with respect to the detectors, and the binding energies of the three compact bodies.