Gravitational waves in scalar-tensor theory to one-and-a-half post-Newtonian order

We compute the gravitational waves generated by compact binary systems in a class of massless scalar-tensor (ST) theories to the 1.5 post-Newtonian (1.5PN) order beyond the standard quadrupole radiation in general relativity (GR). Using and adapting to ST theories the multipolar-post-Minkowskian and post-Newtonian formalisms originally defined in GR, we obtain the tail and non-linear memory terms associated with the dipole radiation in ST theory. The multipole moments and GW flux of compact binaries are derived for general orbits including the new 1.5PN contribution, and comparison is made with previous results in the literature. In the case of quasi-circular orbits, we present ready-to-use templates for the data analysis of detectors, and for the first time the scalar GW modes for comparisons with numerical relativity results.Comment: Contribution to the 2022 Gravitation session of the 56th Rencontres de Moriond. 4 page

Gravitational-wave tails of memory at 4PN order

We study a novel cubic nonlinear effect, the tails-of-memory, which consist of a combination of the tail effect (backscattering of linear gravitational waves against the curvature of spacetime generated by the source) and the memory effect (due to reradiation of gravitational waves by linear gravitational waves themselves). Our final result is consistent with a straightforward direct computation of the memory effect, but also involves many non-trivial tail-like terms.Comment: Contribution to the 2023 Gravitation session of the 57th Rencontres de Moriond. 5 pages, 1 figur

Gravitational-Wave Tails of Memory

Gravitational-wave tails are linear waves that backscatter on the curvature of space-time generated by the total mass-energy of the source. The non-linear memory effect arises from gravitational waves sourced by the stress-energy distribution of linear waves themselves. These two effects are due to quadratic multipolar interactions (mass-quadrupole and quadrupole-quadrupole) and are well known. Also known are the tails generated by tails themselves (cubic "tails-of-tails") and the tails generated by tails-of-tails or vice-versa (quartic "tails-of-tails-of-tails"). In this work, we focus on the cubic "tails-of-memory" corresponding to the mass-quadrupole-quadrupole interaction, as well as the "spin-quadrupole tails", which are due to the cubic interaction between the mass, the total angular momentum and the quadrupole. The tails-of-memory and the spin-quadrupole tails contribute to the asymptotic waveform at the fourth-post-Newtonian (4PN) order beyond quadrupolar radiation.Comment: 40 page, 1 figure, 1 table and 2 ancillary file

The Quadrupole Moment of Compact Binaries to the Fourth post-Newtonian Order: Relating the Harmonic and Radiative Metrics

Motivated by the completion of the fourth post-Newtonian (4PN) gravitational-wave generation from compact binary systems, we analyze and contrast different constructions of the metric outside an isolated system, using post-Minkowskian expansions. The metric in "harmonic" coordinates has been investigated previously, in particular to compute tails and memory effects. However, it is plagued by powers of the logarithm of the radial distance $r$ when $r\to\infty$ (with $t-r/c=$ const). As a result, the tedious computation of the "tail-of-memory" effect, which enters the gravitational-wave flux at 4PN order, is more efficiently performed in the so-called "radiative" coordinates, which admit a (Bondi-type) expansion at infinity in simple powers of $r^{-1}$, without any logarithms. Here we consider a particular construction, performed order by order in the post-Minkowskian expansion, which directly yields a metric in radiative coordinates. We relate both constructions, and prove that they are physically equivalent as soon as a relation between the "canonical" moments which parametrize the radiative metric, and those parametrizing the harmonic metric, is verified. We provide the appropriate relation for the mass quadrupole moment at 4PN order, which will be crucial when deriving the "tail-of-memory" contribution to the gravitational flux.Comment: Updated a reference: Blanchet, Faye & Larrouturou 2022 in CQ

Gravitational-Wave Phasing of Quasi-Circular Compact Binary Systems to the Fourth-and-a-Half post-Newtonian Order

The inspiral phase of gravitational waves emitted by spinless compact binary systems is derived through the fourth-and-a-half post-Newtonian (4.5PN) order beyond quadrupole radiation, and the leading amplitude mode ($\ell$, m) = (2, 2) is obtained at 4PN order. We also provide the radiated flux, as well as the phase in the stationary phase approximation. Rough numerical estimates for the contribution of each PN order are provided for typical systems observed by current and future gravitational wave detectors.Comment: 9 pages, 1 table. v2: reference of the companion paper updated. v3: post-referee version, typo corrected in Eq. (9

Gravitational Wave Flux and Quadrupole Modes from Quasi-Circular Non-Spinning Compact Binaries to the Fourth Post-Newtonian Order

This article provides the details on the technical derivation of the gravitational waveform and total gravitational-wave energy flux of non-spinning compact binary systems to the 4PN (fourth post-Newtonian) order beyond the Einstein quadrupole formula. In particular: (i) we overview the link between the radiative multipole moments measured at infinity and the source moments in the framework of dimensional regularization; (ii) we compute special corrections to the source moments due to "infrared" commutators arising at the 4PN order; (iii) we derive a "post-adiabatic" correction needed to evaluate the tail integral with 2.5PN relative precision; (iv) we discuss the relation between the binary's orbital frequency in quasi-circular orbit and the gravitational-wave frequency measured at infinity; (v) we compute the hereditary effects at the 4PN order, including those coming from the recently derived tails-of-memory; and (vi) we describe the various tests we have performed to ensure the correctness of the results. Those results are collected in an ancillary file.Comment: 32 pages, no figure. v2: reference of the companion letter updated. v3: post-referee versio

Gravitational radiation of compact binary systems in general relativity and in scalar-tensor theories

Nous améliorons les prédictions théoriques analytiques concernant les ondes gravitationnelles (OG) émises par des systèmes binaires compacts d'étoiles à neutrons ou de trous noirs pendant la phase spiralante précédant la coalescence, à la fois en relativité générale (RG) et dans une classe de théories tenseur-scalaire (TS). Nous utilisons un formalisme qui combine le développement multipolaire post-minkowskien (MPM) de la métrique dans le vide extérieur à un système de matière isolé, avec le développement post-newtonien (PN) pour des petites vitesses orbitales (v << c) et des champs gravitationnels faibles. En RG, nous calculons les modes (l,m) de la forme d'OG à l'ordre 4PN, en (v/c)^8 au-delà de l'ordre dominant, ainsi que la phase et le flux d'énergie à 4.5PN. Pour cela, nous étudions de nouveaux termes non-linéaires dans la propagation des OG. Le principal effet est le "sillage de mémoire", qui est dû aux effets combinés (i) des sillages d'OG, c'est-à-dire de la diffusion des OG sur la courbure de l'espace-temps générée par la masse totale de la binaire; et (ii) de l'effet de mémoire, dû au rayonnement gravitationnel engendré par les OG elles-mêmes. Ce calcul a nécessité l'implémentation d'une construction MPM dite "radiative" de la métrique du vide extérieur, qui élimine de la zone lointaine les logarithmes en la coordonnée radiale qui apparaissent dans la construction en coordonnées harmoniques standard. Dans les théories TS, nous adaptons le cadre PN-MPM à l'ordre 1.5PN (c'est-à-dire 2.5PN au-delà du rayonnement dipolaire dominant dans les théories TS). Nous obtenons ainsi le flux, la phase et les modes (l,m) des champs scalaires et tensoriels à l'ordre 1.5PN pour des orbites circulaires, et corrigeons une erreur dans la littérature pour les orbites générales.We improve analytical theoretical predictions for gravitational waves (GWs) emitted by compact binary systems of neutron stars and black holes during the inspiraling phase preceding merger, both in general relativity (GR) and in a class of scalar-tensor (ST) theories. We resort to a formalism that combines the multipolar post-Minkowskian (MPM) expansion of the vacuum metric exterior to an isolated matter system and the post-Newtonian (PN) expansion in small orbital velocities (v << c) and weak gravitational fields. In GR, we compute the (l,m) modes of the GW up to 4PN order, i.e. (v/c)^8 beyond leading order, as well as the phase and energy flux up to 4.5PN order. For this, we study novel nonlinear terms in the propagation of GWs. The main effect is the "tail of memory", which is due to the combined effects of (i) GW tails, i.e. the backscattering of GWs against the spacetime curvature generated by the total mass of the binary; and (ii) the memory effect, due to the reradiation of GWs by GWs themselves. This computation required implementing the so-called "radiative" MPM construction of the exterior vacuum metric, which removes the far-zone logarithms in the radial coordinate that appear in the standard construction in harmonic coordinates. In ST theories, we adapt the PN-MPM framework at 1.5PN order (i.e. 2.5PN beyond the leading dipolar radiation of ST theories). We thus obtain the flux, phase and (l,m) modes of the scalar and tensor fields at 1.5PN order for circular orbits, and correct a mistake in the literature for general orbits

Gravitational waves in scalar-tensor theory to one-and-a-half post-Newtonian order

International audienceWe compute the gravitational waves generated by compact binary systems in a class of massless scalar-tensor (ST) theories to the 1.5 post-Newtonian (1.5PN) order beyond the standard quadrupole radiation in general relativity (GR). Using and adapting to ST theories the multipolar-post-Minkowskian and post-Newtonian formalisms originally defined in GR, we obtain the tail and non-linear memory terms associated with the dipole radiation in ST theory. The multipole moments and GW flux of compact binaries are derived for general orbits including the new 1.5PN contribution, and comparison is made with previous results in the literature. In the case of quasi-circular orbits, we present ready-to-use templates for the data analysis of detectors, and for the first time the scalar GW modes for comparisons with numerical relativity results

Rayonnement gravitationnel des systèmes binaires compacts en relativité générale et dans les théories tenseur-scalaire

We improve analytical theoretical predictions for gravitational waves (GWs) emitted by compact binary systems of neutron stars and black holes during the inspiraling phase preceding merger, both in general relativity (GR) and in a class of scalar-tensor (ST) theories. We resort to a formalism that combines the multipolar post-Minkowskian (MPM) expansion of the vacuum metric exterior to an isolated matter system and the post-Newtonian (PN) expansion in small orbital velocities (v << c) and weak gravitational fields. In GR, we compute the (l,m) modes of the GW up to 4PN order, i.e. (v/c)^8 beyond leading order, as well as the phase and energy flux up to 4.5PN order. For this, we study novel nonlinear terms in the propagation of GWs. The main effect is the "tail of memory", which is due to the combined effects of (i) GW tails, i.e. the backscattering of GWs against the spacetime curvature generated by the total mass of the binary; and (ii) the memory effect, due to the reradiation of GWs by GWs themselves. This computation required implementing the so-called "radiative" MPM construction of the exterior vacuum metric, which removes the far-zone logarithms in the radial coordinate that appear in the standard construction in harmonic coordinates. In ST theories, we adapt the PN-MPM framework at 1.5PN order (i.e. 2.5PN beyond the leading dipolar radiation of ST theories). We thus obtain the flux, phase and (l,m) modes of the scalar and tensor fields at 1.5PN order for circular orbits, and correct a mistake in the literature for general orbits.Nous améliorons les prédictions théoriques analytiques concernant les ondes gravitationnelles (OG) émises par des systèmes binaires compacts d'étoiles à neutrons ou de trous noirs pendant la phase spiralante précédant la coalescence, à la fois en relativité générale (RG) et dans une classe de théories tenseur-scalaire (TS). Nous utilisons un formalisme qui combine le développement multipolaire post-minkowskien (MPM) de la métrique dans le vide extérieur à un système de matière isolé, avec le développement post-newtonien (PN) pour des petites vitesses orbitales (v << c) et des champs gravitationnels faibles. En RG, nous calculons les modes (l,m) de la forme d'OG à l'ordre 4PN, en (v/c)^8 au-delà de l'ordre dominant, ainsi que la phase et le flux d'énergie à 4.5PN. Pour cela, nous étudions de nouveaux termes non-linéaires dans la propagation des OG. Le principal effet est le "sillage de mémoire", qui est dû aux effets combinés (i) des sillages d'OG, c'est-à-dire de la diffusion des OG sur la courbure de l'espace-temps générée par la masse totale de la binaire; et (ii) de l'effet de mémoire, dû au rayonnement gravitationnel engendré par les OG elles-mêmes. Ce calcul a nécessité l'implémentation d'une construction MPM dite "radiative" de la métrique du vide extérieur, qui élimine de la zone lointaine les logarithmes en la coordonnée radiale qui apparaissent dans la construction en coordonnées harmoniques standard. Dans les théories TS, nous adaptons le cadre PN-MPM à l'ordre 1.5PN (c'est-à-dire 2.5PN au-delà du rayonnement dipolaire dominant dans les théories TS). Nous obtenons ainsi le flux, la phase et les modes (l,m) des champs scalaires et tensoriels à l'ordre 1.5PN pour des orbites circulaires, et corrigeons une erreur dans la littérature pour les orbites générales

Rayonnement gravitationnel des systèmes binaires compacts en relativité générale et dans les théories tenseur-scalaire

We improve analytical theoretical predictions for gravitational waves (GWs) emitted by compact binary systems of neutron stars and black holes during the inspiraling phase preceding merger, both in general relativity (GR) and in a class of scalar-tensor (ST) theories. We resort to a formalism that combines the multipolar post-Minkowskian (MPM) expansion of the vacuum metric exterior to an isolated matter system and the post-Newtonian (PN) expansion in small orbital velocities (v << c) and weak gravitational fields. In GR, we compute the (l,m) modes of the GW up to 4PN order, i.e. (v/c)^8 beyond leading order, as well as the phase and energy flux up to 4.5PN order. For this, we study novel nonlinear terms in the propagation of GWs. The main effect is the "tail of memory", which is due to the combined effects of (i) GW tails, i.e. the backscattering of GWs against the spacetime curvature generated by the total mass of the binary; and (ii) the memory effect, due to the reradiation of GWs by GWs themselves. This computation required implementing the so-called "radiative" MPM construction of the exterior vacuum metric, which removes the far-zone logarithms in the radial coordinate that appear in the standard construction in harmonic coordinates. In ST theories, we adapt the PN-MPM framework at 1.5PN order (i.e. 2.5PN beyond the leading dipolar radiation of ST theories). We thus obtain the flux, phase and (l,m) modes of the scalar and tensor fields at 1.5PN order for circular orbits, and correct a mistake in the literature for general orbits.Nous améliorons les prédictions théoriques analytiques concernant les ondes gravitationnelles (OG) émises par des systèmes binaires compacts d'étoiles à neutrons ou de trous noirs pendant la phase spiralante précédant la coalescence, à la fois en relativité générale (RG) et dans une classe de théories tenseur-scalaire (TS). Nous utilisons un formalisme qui combine le développement multipolaire post-minkowskien (MPM) de la métrique dans le vide extérieur à un système de matière isolé, avec le développement post-newtonien (PN) pour des petites vitesses orbitales (v << c) et des champs gravitationnels faibles. En RG, nous calculons les modes (l,m) de la forme d'OG à l'ordre 4PN, en (v/c)^8 au-delà de l'ordre dominant, ainsi que la phase et le flux d'énergie à 4.5PN. Pour cela, nous étudions de nouveaux termes non-linéaires dans la propagation des OG. Le principal effet est le "sillage de mémoire", qui est dû aux effets combinés (i) des sillages d'OG, c'est-à-dire de la diffusion des OG sur la courbure de l'espace-temps générée par la masse totale de la binaire; et (ii) de l'effet de mémoire, dû au rayonnement gravitationnel engendré par les OG elles-mêmes. Ce calcul a nécessité l'implémentation d'une construction MPM dite "radiative" de la métrique du vide extérieur, qui élimine de la zone lointaine les logarithmes en la coordonnée radiale qui apparaissent dans la construction en coordonnées harmoniques standard. Dans les théories TS, nous adaptons le cadre PN-MPM à l'ordre 1.5PN (c'est-à-dire 2.5PN au-delà du rayonnement dipolaire dominant dans les théories TS). Nous obtenons ainsi le flux, la phase et les modes (l,m) des champs scalaires et tensoriels à l'ordre 1.5PN pour des orbites circulaires, et corrigeons une erreur dans la littérature pour les orbites générales