Dynamics of compact binary systems in scalar-tensor theories. II. Center-of-mass and conserved quantities to 3PN order

Abstract

The equations of motion of nonspinning compact binary systems at the third post-Newtonian (PN) order in massless scalar-tensor theories have recently been obtained. In the present paper, we complete this work by computing at 3PN order the ten integrals of motion together with the equations of motion in the center-of-mass frame. We then perform the reduction of the center-of-mass equations to quasi-circular orbits and determine the conserved energy and angular momentum for circular orbits. I. INTRODUCTIONThe detections of gravitational waves emitted by inspiralling compact binaries is a first step towards a better comprehension of gravitational physics [1]. In the future, the next generation of high precision experiments will allow a better understanding of these systems, by constraining their abundance, origin and parameters. It will also provide information about the strong-field and highly-dynamical regime of gravitational physics, by challenging the theory of general relativity (GR) in this regime.The detection and parameter estimation of gravitational wave events require a bank of highly accurate templates for the gravitational waveforms, to be match-filtered against the data. Current templates are constructed by matching the waveform at different stages of the coalescence, that are obtained using different approaches: post-Newtonian formalism for the inspiral [2] and numerical relativity for the merger and ringdown [3]. There are currently two main series of template waveforms to be used by the LIGO-Virgo collaboration, based either on a direct matching (IMR models) [4,5] or on some resummation techniques (EOB waveforms) [6,7].In order to perform tests of general relativity, one also has to construct such templates for the many alternative theories of gravity. There exist two different and complementary approaches to this problem: either theoryindependent [8] or theory-dependent [9]. Although fundamentally different, these two approaches are complementary as one needs to be able to map the constraints coming from the agnostic approach to specific theories. Here, we will focus on a specific model, namely massless scalar-tensor theory [10,11], one of the oldest, most popular and well-studied theory of gravity [12]. It consists in adding a single massless scalar field, coupled in a minimal way to gravity. One of the motivations for studying this theory, which usually arises as a low-energy limit of some string theory, relies on the fact that it may explain the accelerated expansion of the universe [13].The present work is part of a series of articles whose attempt is to construct the full gravitational and scalar waveforms in scalar-tensor theories up to 2PN order 1 [14-17]. In the companion paper [17], that will be referred to as Paper I in the following, we computed the harmonic coordinate equations of motion in scalar-tensor theories at 3PN order. In the present paper, we complete this work by determining the center-of-mass and conserved quantities to the same 3PN order in the center-of-mass frame. As the le...

Post-Newtonian dynamics and black hole thermodynamics in Einstein-scalar-Gauss-Bonnet gravity

“…The two-body Lagrangian was recently calculated at 3PN order for pure scalar-tensor theories in [55,56]. Our results extend this Lagrangian to EsGB theories: we just need to add the contribution coming from Eq.…”

“…At least in the small-α regime, the results of Ref. [56], supplemented by the Gauss-Bonnet contribution computed here [Eq. (IV.9)], yield the full EsGB Lagrangian at 3PN order.…”

Disformally coupled scalar fields and inspiralling trajectories

“…This formula is valid up to 2.5PN order. The precise form of this relation is different for a scalar-tensor theory and has very recently been computed up to 3PN order in Ref [33], but the statement that we can always recast the motion in terms of relative coordinates is not modified. So finally, in the case of a circular motion, the two unknown functions F and G are functions of time through x and v. A scalar function built out of x and v must contain only r, x · v and v by SO(3) invariance.…”

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.…”

“…Currently, the tensor waveform is known at 2PN order [14] while the scalar waveform and the energy flux are respectively known at 1.5PN and 1PN order [15]. To have a complete result at 2PN order for all quantities, one needs to use the dynamics at the higher 3PN order [16,17]. Finally, to complete these results one also has to include the finite-size effects.…”