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

Abstract

Gravitationally bound hierarchies containing three or more components are very common in our Universe. In this paper we study periodic gravitational wave (GW) form, their polarizations, response function, its Fourier transform, and energy loss rate of a triple system through three different channels of radiation, the scalar, vector and tensor modes, in 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 (GN m/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 six orders lower than that of GR. 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, 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.

Gravitational waveforms and radiation powers of the triple system PSR J0337+1715 in modified theories of gravity

“…12, we plot the radiation power in GR for about 500 days, where the inserted image shows the details from day 12 to day 26. In ae-theory, from [22,28] we find that…”

“…where x i a is the location of the a-th body and I ≡ I kk . For more details, see [22]. From the above expressions, in addition to the usual plus and cross polarization modes, aetheory predicts three extra independent modes, h ae b , h ae X and h ae Y .…”

“…From the above expressions, in addition to the usual plus and cross polarization modes, aetheory predicts three extra independent modes, h ae b , h ae X and h ae Y . Comparing to h ae + and h ae × , these extra modes are suppressed, respectively, by a factor, c 14 10 −5 and c 13 10 −15 [22]. The longitudinal mode h ae L is proportional to the breathing mode h ae b .…”

“…and φ ij , Ψ I i and Φ II are, respectively, the gauge-invariant quantities of the tensor, vector and scalar modes defined in [22]. The quantities c V and c S denote the speeds of the vector and scalar modes, respectively.…”

“…The quantities c V and c S denote the speeds of the vector and scalar modes, respectively. Due to their presence, the polarizations of a GW have five independent modes which are given respectively by [22] h ae + ≡…”