Compact binary systems in scalar-tensor gravity. II. Tensor gravitational waves to second post-Newtonian order

Abstract

We derive the tensor gravitational waveform generated by a binary of nonspinning compact objects (black holes or neutron stars) in a general class of scalar-tensor theories of gravity. The waveform is accurate to second post-Newtonian order beyond the leading order quadrupole approximation. We use the direct integration of the relaxed Einstein equations formalism, appropriately adapted to scalar-tensor theories, along with previous results for the equations of motion in these theories. The self-gravity of the compact objects is treated with an approach developed by Eardley. The scalar field causes deviations from the general relativistic waveform that depend only on a small number of parameters. Among the effects of the scalar field are new hereditary terms which depend on the past history of the source. One of these, a dipole-dipole coupling, produces a zero-frequency "gravitational-wave memory" equivalent to the Christodoulou memory of general relativity. In the special case of two black holes, the waveform reduces to the general relativistic waveform. For a mixed (black hole-neutron star) system, the waveform is identical to that of Einstein's theory to first post-Newtonian order, with deviations at higher order depending only on a single parameter. The behavior in these cases matches that found for the equations of motion. PACS numbers: 04.30.Db, 04.25.Nx, 04.50.Kd I. INTRODUCTIONCompact binaries are one of the most abundant and interesting sources of gravitational waves (GWs). Systems comprising stellar-mass black holes (m BH ∼ 1-100 M ⊙ ) and/or neutron stars, which lie in the "high-frequency" GW band (1-10 3 Hz), are likely to be the first detected. Indeed, current rate estimates predict that ground-based detectors like Advanced LIGO [1] and Advanced Virgo [2] will see several to hundreds of these sources per year once they become operational (although rates are very uncertain) [3]. Higher mass binaries, containing massive or supermassive black holes, can be detected by other means. Systems with masses 10 4 -10 7 M ⊙ lie in the "low-frequency" band (10 −4 -1 Hz) and will be detected by a space-based detector like the proposed eLISA mission [4]. Heavier systems (m BH ∼ 10 8 -10 9 M ⊙ ) could be detected very soon by the timing of radio pulsars [5].Detection of the GWs is very challenging: The waves interact extremely weakly with matter, and there are many sources of noise. The separation of signal from noise is achieved by the matched filtering process, which requires the generation of extremely accurate theoretical template waveforms for the expected GW signal. For the inspiral of compact binary systems, templates are expressed in the post-Newtonian approximation to general relativity, an expansion in powers of v/c ∼ (Gm/rc 2 ) 1/2 . Each power is considered to be one-half a post-Newtonian (PN) order. Templates for nonspinning binary systems have been constructed to O((v/c) 6 ), or 3PN order [6], with partial results at 3.5PN order [7]. (For a comprehensive review, see [8].) Numerical relativity codes prov...

The Confrontation between General Relativity and Experiment

“…Mirshekari and Will [286] obtained the complete compact-binary equations of motion in general scalar-tensor theories through 2.5PN order, and obtained the energy loss rate in complete agreement with the flux result from Damour and Esposito-Farèse. Lang [241] obtained the tensor gravitational-wave signal to 2PN order.…”

“…The tensor part of the gravitational waveform has the schematic form where Contributions to the tensor waveform through 2PN order have been derived by Lang [241]. The scalar waveform is given by ϕ = ϕ 0 (1 + Ψ), where, where, ignoring terms that are constant in time, where is a unit vector directed toward the observer.…”

Extreme gravity tests with gravitational waves from compact binary coalescences: (I) inspiral–merger

“…More recently, the post-Minkowskian technique of direct integration of the relaxed Einstein equations (DIRE) [188][189][190] was used to compute the equations of motion for a system of compact objects at 2.5PN order, as well as the gravitational waveform and energy flux at 2PN (relative) order for binaries on generic orbits [191][192][193]. Sennett et al [194] specialized the results of [191][192][193] to non-spinning binary systems on quasi-circular orbits in scalar-tensor gravity at 2PN relative order 3 . Such waveforms introduce PN corrections to the mapping presented in Table 2.…”

Dipolar tidal effects in scalar-tensor theories

“…Considering the current bound on the ST parameters [25], we can determine the order of magnitude estimate of the tidal corrections to the phase, Eqs. (14) and (17). As it also requires the computation of the scalar TLNs for specific NS equations of state, which is beyond the goal of this paper, we take them as being of order ∼ 0.1, similar to the highest relativistic TLNs computed for GR.…”