Search

We use Effective Field Theory techniques to derive the quadrupole-quadrupole part of the gravitational wave, obtaining a waveform in agreement with previous results found within the multipolar-post-Minkowskian method. In particular we emphasize the role of radiation-reaction terms, which affect the energy-momentum balance between source and radiation. An in-in effective action is then derived along the same principles and it is shown to provide energy and angular momentum balance equations in agreement with the corresponding fluxes carried at infinity by gravitational radiation., Comment: 21 pages, 5 figures

Joint gravitational-wave and $\gamma$-ray bursts (GRB) observations are among the best prospects for standard siren cosmology. However, the strong selection effect for the coincident GRB detection, which is possible only for sources with small inclination angles, induces a systematic uncertainty that is currently not accounted for. We show that this severe source of bias can be removed by inferring the a-priori unknown electromagnetic detection probability directly from multimessenger data. This leads at the same time to an unbiased measurement of the Hubble constant, to constrain the properties of GRB emission, and to accurately measure the viewing angle of each source. Our inference scheme is applicable to real data already in the small-statistics regime, a scenario that might become reality in the near future. Additionally, we introduce a novel likelihood approximant for GW events which treats the dependence on distance and inclination as exact., Comment: 5+6 pages, 4 figures

The Lunar Gravitational-wave Antenna (LGWA) is a proposed array of next-generation inertial sensors to monitor the response of the Moon to gravitational waves (GWs). Given the size of the Moon and the expected noise produced by the lunar seismic background, the LGWA would be able to observe GWs from about 1 mHz to 1 Hz. This would make the LGWA the missing link between space-borne detectors like LISA with peak sensitivities around a few millihertz and proposed future terrestrial detectors like Einstein Telescope or Cosmic Explorer. In this article, we provide a first comprehensive analysis of the LGWA science case including its multi-messenger aspects and lunar science with LGWA data. We also describe the scientific analyses of the Moon required to plan the LGWA mission.

We re-analyze the far zone contribution to the two-body conservative dynamics arising from interaction between radiative and longitudinal modes, the latter sourced by mass and angular momentum, which in the mass case is known as tail process. We verify the expected correspondence between two loop self-energy amplitudes and the gluing of two classical (one leading order, one at one loop) emission amplitudes. In particular we show that the factorization of the self-energy amplitude involving the angular momentum is violated when applying standard computation procedures, due to a violation of the Lorentz gauge condition commonly adopted in perturbative computations. We show however that a straightforward fix exists, as the violation corresponds to a consistent anomaly, and it can be re-absorbed by the variation of a suitable action functional., Comment: 24 pages, 3 figures. Version accepted for publication. New subsection added in Sec. II

The Einstein Telescope (ET), the European project for a third-generation gravitational-wave detector, has a reference configuration based on a triangular shape consisting of three nested detectors with 10 km arms, where in each arm there is a `xylophone' configuration made of an interferometer tuned toward high frequencies, and an interferometer tuned toward low frequencies and working at cryogenic temperature. Here, we examine the scientific perspectives under possible variations of this reference design. We perform a detailed evaluation of the science case for a single triangular geometry observatory, and we compare it with the results obtained for a network of two L-shaped detectors (either parallel or misaligned) located in Europe, considering different choices of arm-length for both the triangle and the 2L geometries. We also study how the science output changes in the absence of the low-frequency instrument, both for the triangle and the 2L configurations. We examine a broad class of simple `metrics' that quantify the science output, related to compact binary coalescences, multi-messenger astronomy and stochastic backgrounds, and we then examine the impact of different detector designs on a more specific set of scientific objectives., Comment: 197 pages, 73 figures. v2: corrections in the part on the sensitivity to stochastic backgrounds. Accepted in JCAP

A joint fit of the mass and redshift distributions of the population of Binary Black Holes detected with Gravitational-Wave observations can be used to obtain constraints on the Hubble parameter and on deviations from General Relativity in the propagation of Gravitational Waves. We first present applications of this technique to the latest catalog of Gravitational-Wave events, focusing on the comparison of different parametrizations for the source-frame mass distribution of Black Hole Binaries. We find that models with more than one feature are favourite by the data, as suggested by population studies, even when varying the cosmology. Then, we discuss perspectives for the use of this technique with third generation Gravitational-Wave detectors, exploiting the recently developed Fisher information matrix Python code GWFAST., Comment: 5 pages, 2 figures, Contribution to the ICHEP 2022 conference proceedings

We compute the contribution to the two-body scattering angle of a specific class of interactions involving the exchange of gravitational radiative degrees of freedom, including the nonlinear memory process and square of radiation reaction effects. Our computation is performed directly from the equations of motion, thus computing the overall effect of both conservative and dissipative processes. Such contributions provide in principle the last missing ingredients to compute the scattering angle at fifth post-Newtonian, at fourth post-Minkowskian order., Comment: 13 pages, 1 figure, final version published on PRD

We introduce $\texttt{GWFAST}$, a Fisher information matrix $\texttt{Python}$ code that allows easy and efficient estimation of signal-to-noise ratios and parameter measurement errors for large catalogs of resolved sources observed by networks of gravitational-wave detectors. In particular, $\texttt{GWFAST}$ includes the effects of the Earth's motion during the evolution of the signal, supports parallel computation, and relies on automatic differentiation rather than on finite differences techniques, which allows the computation of derivatives with accuracy close to machine precision. We also release the library $\texttt{WF4Py}$ implementing state-of-the-art gravitational-wave waveforms in $\texttt{Python}$. In this paper we provide a documentation of $\texttt{GWFAST}$ and $\texttt{WF4Py}$ with practical examples and tests of performance and reliability. In a companion paper we present forecasts for the detection capabilities of the second and third generation of ground-based gravitational-wave detectors, obtained with $\texttt{GWFAST}$., Comment: 18 pages, 6 figures, $\texttt{GWFAST}$ available at $\href{https://github.com/CosmoStatGW/gwfast}{\rm this\ link}$, $\texttt{WF4Py}$ available at $\href{https://github.com/CosmoStatGW/WF4Py}{\rm this\ link}$. v3 matches the version accepted for publication in ApJS. Title changed in journal

We introduce $\texttt{GWFAST}$, a novel Fisher-matrix code for gravitational-wave studies, tuned toward third-generation gravitational-wave detectors such as Einstein Telescope (ET) and Cosmic Explorer (CE). We use it to perform a comprehensive study of the capabilities of ET alone, and of a network made by ET and two CE detectors, as well as to provide forecasts for the forthcoming O4 run of the LVK collaboration. We consider binary neutron stars, binary black holes and neutron star-black hole binaries, and compute basic metrics such as the distribution of signal-to-noise ratio (SNR), the accuracy in the reconstruction of various parameters (including distance, sky localization, masses, spins and, for neutron stars, tidal deformabilities), and the redshift distribution of the detections for different thresholds in SNR and different levels of accuracy in localization and distance measurement. We examine the expected distribution and properties of `golden events', with especially large values of the SNR. We also pay special attention to the dependence of the results on astrophysical uncertainties and on various technical details (such as choice of waveforms, or the threshold in SNR), and we compare with other Fisher codes in the literature. In a companion paper we discuss the technical aspects of the code. Together with this paper, we publicly release the code $\texttt{GWFAST}$ at https://github.com/CosmoStatGW/gwfast, and the library $\texttt{WF4Py}$ implementing state-of-the-art gravitational-wave waveforms in pure $\texttt{Python}$ at https://github.com/CosmoStatGW/WF4Py., Comment: 43 + 12 pages, 24 + 5 Figures, $\texttt{GWFAST}$ available at https://github.com/CosmoStatGW/gwfast, $\texttt{WF4Py}$ available at https://github.com/CosmoStatGW/WF4Py. v3: resubmitted to APJ, minor changes in results, Appendix C added

The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of GWs can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas., Comment: Accepted in: Living Reviews in Relativity

We present the latest measurements of the Hubble parameter and of the parameter $\Xi_0$ describing modified gravitational wave propagation, obtained from the third gravitational wave transient catalog, GWTC-3, using the correlation with galaxy catalogs and information from the source-frame mass distribution of binary black holes. The latter leads to the tightest bound on $\Xi_0$ so far, i.e. $\Xi_0 = 1.2^{+0.7}_{-0.7}$ with a flat prior on $\Xi_0$, and $\Xi_0 = 1.0^{+0.4}_{-0.8}$ with a prior uniform in $\log\Xi_0$ (Max posterior and $68\%$ HDI). The measurement of $H_0$ is dominated by the single bright siren GW170817, resulting in $H_0=67^{+9}_{-6} \, \rm km \, s^{-1} \, Mpc$ when combined with the galaxy catalog., Comment: Contribution to the Gravitation session of the 56th Rencontres de Moriond 2022

Modified gravitational wave propagation is a smoking gun of modifications of gravity at cosmological scales, and can be the most promising observable for testing such theories. The observation of gravitational waves (GW) in recent years has allowed us to start probing this effect, and here we briefly review two promising ways of testing it. We will show that, already with the current network of detectors, it is possible to reach an interesting accuracy in the estimation of the $\Xi_0$ parameter (that characterizes modified gravitational wave propagation, with $\Xi_{0, {\rm GR}} = 1$) and with next generation facilities, such as the Einstein Telescope, we can get a sub-percent measurement., Comment: 4 pages, 2 figures, contribution to the 2022 Gravitation session of the 56th Rencontres de Moriond

The next generation of ground-based gravitational-wave detectors will observe coalescences of black holes and neutron stars throughout the cosmos, thousands of them with exceptional fidelity. The Science Book is the result of a 3-year effort to study the science capabilities of networks of next generation detectors. Such networks would make it possible to address unsolved problems in numerous areas of physics and astronomy, from Cosmology to Beyond the Standard Model of particle physics, and how they could provide insights into workings of strongly gravitating systems, astrophysics of compact objects and the nature of dense matter. It is inevitable that observatories of such depth and finesse will make new discoveries inaccessible to other windows of observation. In addition to laying out the rich science potential of the next generation of detectors, this report provides specific science targets in five different areas in physics and astronomy and the sensitivity requirements to accomplish those science goals. This report is the second in a six part series of reports by the GWIC 3G Subcommittee: i) Expanding the Reach of Gravitational Wave Observatories to the Edge of the Universe, ii) The Next Generation Global Gravitational Wave Observatory: The Science Book (this report), iii) 3G R&D: R&D for the Next Generation of Ground-based Gravitational Wave Detectors, iv) Gravitational Wave Data Analysis: Computing Challenges in the 3G Era, v) Future Ground-based Gravitational-wave Observatories: Synergies with Other Scientific Communities, and vi) An Exploration of Possible Governance Models for the Future Global Gravitational-Wave Observatory Network., Comment: 69 pages, 18 figures

We compute tail contributions to the conservative dynamics of a generic self-gravitating system, for every multipole order, of either electric and magnetic parity. Such contributions arise when gravitational radiation is backscattered by the static curvature generated by the source itself and reabsorbed. Tail processes are relevant for the study of compact binary dynamics starting from the fourth post-Newtonian order. Within this context, we resolve a discrepancy recently arised at fifth post-Newtonian order between Effective Field Theory and self-force results., Comment: 16 pages, 2 figures. Minor changes wrt v1, to make it conform to the version published on PRD

Modified gravitational wave (GW) propagation is a generic phenomenon in modified gravity. It affects the reconstruction of the redshift of coalescing binaries from the luminosity distance measured by GW detectors, and therefore the reconstruction of the actual masses of the component compact stars from the observed (`detector-frame') masses. We show that, thanks to the narrowness of the mass distribution of binary neutron stars, this effect can provide a clear signature of modified gravity, particularly for the redshifts explored by third generation GW detectors such as Einstein Telescope and Cosmic Explorer., Comment: v2: expanded version, to appear in Physics of the Dark Universe

It has been recently shown that quadruply lensed gravitational-wave (GW) events due to coalescing binaries can be localized to one or just a few galaxies, even in the absence of an electromagnetic counterpart. We discuss how this can be used to extract information on modified GW propagation, which is a crucial signature of modifications of gravity at cosmological scales. We show that, using quadruply lensed systems, it is possible to constrain the parameter $\Xi_0$ that characterizes modified GW propagation, without the need of imposing a prior on $H_0$. A LIGO/Virgo/Kagra network at target sensitivity might already get a significant measurement of $\Xi_0$, while a third generation GW detector such as the Einstein Telescope could reach a very interesting accuracy., Comment: 11 pages, 5 figures

We compute the effect of scattering gravitational radiation off the static background curvature, up to second order in Newton constant, known in literature as tail and tail-of-tail processes, for generic electric and magnetic multipoles. Starting from the multipole expansion of composite compact objects, and as expected due to the known electric quadrupole case, both long- and short-distance (UV) divergences are encountered. The former disappears from properly defined observables, the latter are renormalized and their associated logarithms give rise to a classical renormalization group flow. UV divergences alert for incompleteness of the multipolar description of the composite source, and are expected not to be present in a UV-complete theory, as explicitly derived in literature for the case of conservative dynamics. Logarithmic terms from tail-of-tail processes associated to generic magnetic multipoles are computed in this work for the first time., Comment: 20 pages, 1 figure. Typos corrected in v2, which is the version published in PRD

We revisit several aspects of the interaction of self-gravitating, slowly varying sources with their own emitted radiation within the context of post-Newtonian approximation to General Relativity. We discuss and clarify the choice of boundary conditions of Green's functions used to determine conservative potentials, and the interplay between the so-called near and far zones, as well as the relation between far zone ultra-violet divergences and emitted power. Both near and far zone contributions are required for the computation of the conservative dynamics. Within a field-theory approach we rederive far-zone self-energy processes, known as tail and memory effects, generalising the calculation of their divergent part to arbitrary order in the post-Newtonian expansion., Comment: 36 pages, 1 figure

We present a detailed study of the methodology for correlating `dark sirens' (compact binaries coalescences without electromagnetic counterpart) with galaxy catalogs. We propose several improvements on the current state of the art, and we apply them to the GWTC-2 catalog of LIGO/Virgo gravitational wave (GW) detections, and the GLADE galaxy catalog, performing a detailed study of several sources of systematic errors that, with the expected increase in statistics, will eventually become the dominant limitation. We provide a measurement of $H_0$ from dark sirens alone, finding as the best result $H_0=67.3^{+27.6}_{-17.9}\,\,{\rm km}\, {\rm s}^{-1}\, {\rm Mpc}^{-1}$ ($68\%$ c.l.) which is, currently, the most stringent constraint obtained using only dark sirens. Combining dark sirens with the counterpart for GW170817 we find $H_0= 72.2^{+13.9}_{-7.5} \,{\rm km}\, {\rm s}^{-1}\, {\rm Mpc}^{-1}$. We also study modified GW propagation, which is a smoking gun of dark energy and modifications of gravity at cosmological scales, and we show that current observations of dark sirens already start to provide interesting limits. From dark sirens alone, our best result for the parameter $\Xi_0$ that measures deviations from GR (with $\Xi_0=1$ in GR) is $\Xi_0=2.1^{+3.2}_{-1.2}$. We finally discuss limits on modified GW propagation under the tentative identification of the flare ZTF19abanrhr as the electromagnetic counterpart of the binary black hole coalescence GW190521, in which case our most stringent result is $\Xi_0=1.8^{+0.9}_{-0.6}$. We release the publicly available code $\tt{DarkSirensStat}$, which is available under open source license at \url{https://github.com/CosmoStatGW/DarkSirensStat}., Comment: v2: several significant technical improvements, results changed v3: minor changes, Fig 9 added. The version to appear in JCAP

A factorisation property of Feynman diagrams in the context the Effective Field Theory approach to the compact binary problem has been recently employed to efficiently determine the static sector of the potential at fifth post-Newtonian (5PN) order. We extend this procedure to the case of non-static diagrams and we use it to fix, by means of elementary algebraic manipulations, the value of more than one thousand diagrams at 5PN order, that is a substantial fraction of the diagrams needed to fully determine the dynamics at 5PN. This procedure addresses the redundancy problem that plagues the computation of the binding energy with respect to more "efficient" observables like the scattering angle, thus making the EFT approach in harmonic gauge at least as scalable as the others methods., Comment: 17 pages, 3 figures. In v3: Section 2 expanded. Published on JHEP

We apply the classical double copy to the calculation of self-energy of composite systems with multipolar coupling to gravitational field, obtaining next-to-leading order results in the gravitational coupling $G_N$ by generalizing color to kinematics replacement rules known in literature. When applied to the multipolar description of the two-body system, the self-energy diagrams studied in this work correspond to tail processes, whose physical interpretation is of radiation being emitted by the non-relativistic source, scattered by the curvature generated by the binary system and then re-absorbed by the same source. These processes contribute to the conservative two-body dynamics and the present work represents a decisive step towards the systematic use of double copy within the multipolar post-Minkowskian expansion., Comment: 20 pages, 2 figures

We provide a systematic and updated discussion of a research line carried out by our group over the last few years, in which gravity is modified at cosmological distances by the introduction of nonlocal terms, assumed to emerge at an effective level from the infrared behavior of the quantum theory. The requirement of producing a viable cosmology turns out to be very stringent and basically selects a unique model, in which the nonlocal term describes an effective mass for the conformal mode. We discuss how such a specific structure could emerge from a fundamental local theory of gravity, and we perform a detailed comparison of this model with the most recent cosmological datasets, confirming that it fits current data at the same level as $\Lambda$CDM. Most notably, the model has striking predictions in the sector of tensor perturbations, leading to a very large effect in the propagation of gravitational wave (GWs) over cosmological distances. At the redshifts relevant for the next generation of GW detectors such as Einstein Telescope, Cosmic Explorer and LISA, this leads to deviations from GR that could be as large as $80\%$, and could be verified with the detection of just a single coalescing binary with electromagnetic counterpart. This would also have potentially important consequences for the search of the counterpart since, for a given luminosity distance to the source, as inferred through the GW signal, the actual source redshift could be significantly different from that predicted by $\Lambda$CDM. At the redshifts relevant for advanced LIGO/Virgo/Kagra the effect is smaller, but still potentially observable over a few years of runs at target sensitivity., Comment: 84 pages, 22 figures

We combine different techniques to extract information about the logarithmic contributions to the two-body conservative dynamics within the post-Newtonian (PN) approximation of General Relativity. The logarithms come from the conservative part of non linear gravitational-wave tails and their iterations. Explicit, original expressions are found for conservative dynamics logarithmic tail terms up to 6PN order by adopting both traditional PN calculations and effective field theory (EFT) methods. We also determine all logarithmic terms at 7PN order, fixing a sub-leading logarithm from a tail-of-tail-of-tail process by comparison with self-force (SF) results. Moreover, we use renormalization group techniques to obtain the leading logarithmic terms to generic power $n$, appearing at $(3n+1)$PN order, and we resum the infinite series in a closed form. Half-integer PN orders enter the conservative dynamics starting at 5.5PN, but they do not generate logarithmic contributions up to next-to-next-to-leading order included. We nevertheless present their contribution at leading order in the small mass ratio limit., Comment: 17 pages, 1 figure. 2 paragraphs added to sec. IV A in v2

We discuss the first-time calculation of the static gravitational two-body potential up to fifth post-Newtonian(PN) order. The results are achieved through a manifest factorization property of the odd PN diagrams. The factorization property is illustrated also at first and third PN order., Comment: 9 pages, 5 figures, contribution to the 14th International Symposium on Radiative Corrections (RADCOR2019), 9-13 September 2019, Palais des Papes, Avignon, France

The Einstein Telescope (ET), a proposed European ground-based gravitational-wave detector of third-generation, is an evolution of second-generation detectors such as Advanced LIGO, Advanced Virgo, and KAGRA which could be operating in the mid 2030s. ET will explore the universe with gravitational waves up to cosmological distances. We discuss its main scientific objectives and its potential for discoveries in astrophysics, cosmology and fundamental physics., Comment: 51 pages, 14 figures; v3: references added, various improvements. v4: minor changes. Version published in JCAP

Recent work has shown that modified gravitational wave (GW) propagation can be a powerful probe of dark energy and modified gravity, specific to GW observations. We use the technique of Gaussian processes, that allows the reconstruction of a function from the data without assuming any parametrization, to measurements of the GW luminosity distance from simulated joint GW-GRB detections, combined with measurements of the electromagnetic luminosity distance by simulated DES data. For the GW events we consider both a second-generation LIGO/Virgo/Kagra (HVLKI) network, and a third-generation detector such as the Einstein Telescope. We find that the HVLKI network at target sensitivity, with $O(15)$ neutron star binaries with electromagnetic counterpart, could already detect deviations from GR at a level predicted by some modified gravity models, and a third-generation detector such as ET would have a remarkable discovery potential. We discuss the complementarity of the Gaussian processes technique to the $(\Xi_0,n)$ parametrization of modified GW propagation., Comment: 16 pages, 26 figures

In the context of the two-body problem in General Relativity, hereditary terms in the long range gravitational field depend on the history rather than the instantaneous state of the source at retarded time. We compute the next-to leading effects of such hereditary terms, that comprise tail and memory, on the two-body dynamics, within effective field theory methods, including both dissipative and conservative effects. The former confirm known results at 2.5 post-Newtonian order with respect to the leading order in the luminosity function; the conservative part is a new result and is an unavoidable ingredient for a derivation of the conservative two-body dynamics at fifth post-Newtonian order., Comment: 24 pages, 1 figure; footnote added at page 5

We discuss a modified gravity model which fits cosmological observations at a level statistically indistinguishable from $\Lambda$CDM and at the same time predicts very large deviations from General Relativity (GR) in the propagation of gravitational waves (GWs) across cosmological distances. The model is a variant of the RT nonlocal model proposed and developed by our group, with initial conditions set during inflation, and predicts a GW luminosity distance that, at the redshifts accessible to LISA or to a third-generation GW detector such as the Einstein Telescope (ET), can differ from that in GR by as much as $60\%$. An effect of this size could be detected with just a single standard siren with counterpart by LISA or ET. At the redshifts accessible to a LIGO/Virgo/Kagra network at target sensitivity the effect is smaller but still potentially detectable. Indeed, for the recently announced LIGO/Virgo NS-BH candidate S190814bv, the RT model predicts that, given the measured GW luminosity distance, the actual luminosity distance, and the redshift of an electromagnetic counterpart, would be smaller by as much as $7\%$ with respect to the value inferred from $\Lambda$CDM., Comment: v2: added discussion of the effect of modified GW propagation on the recent LIGO/Virgo NS-BH candidate event S190814bv

Gravitational-wave (GW) detectors can contribute to the measurement of cosmological parameters and to testing the dark-energy sector of alternatives to $\Lambda$CDM, by using standard sirens. In this paper we focus on binary neutron stars with a counterpart detected through a gamma-ray burst (GRB), both at a second-generation network made by advanced LIGO+advanced Virgo+LIGO India+Kagra, and at third-generation (3G) detectors, discussing in particular the cases of a single Einstein Telescope (ET), and of a network of ET plus two Cosmic Explorer (CE). We construct mock catalogs of standard sirens, using different scenarios for the local merger rate and for the detection of the electromagnetic counterpart. For 3G detectors we estimate the coincidences with a GRB detector with the characteristics of the proposed THESEUS mission. We discuss how these standard sirens with a GRB counterpart can improve the determination of cosmological parameters (and particularly of $H_0$) in $\Lambda$CDM, and we then study how to extract information on dark energy, considering both a non-trivial dark energy equation of state and modified GW propagation. We find that a 2G detector network can already reach, over several years of data taking, an interesting sensitivity to modified GW propagation, while a single ET detector would have a remarkable potential for discovery. We also find that, to fully exploit the potential of a ET+CE+CE network, it is necessary a much stronger program of search for electromagnetic counterparts (or else to resort to statistical methods for standard sirens), and furthermore gravitational lensing can become a limiting factor., Comment: 50 pages, 28 figures

This Astro-2020 White Paper deals with what we might learn from future gravitational wave observations about the early universe phase transitions and their energy scales, primordial black holes, Hubble parameter, dark matter and dark energy, modified theories of gravity and extra dimensions., Comment: 13 pages, 3 figures, White Paper submitted to the Astro-2020 (2020 Astronomy and Astrophysics Decadal Survey) by GWIC-3G Science Case Team (GWIC: Gravitational-Wave International Committee)

We complete the derivation of the conservative dynamics of binary systems to fourth Post-Newtonian (4PN) order in the effective field theory (EFT) approach. We present a self-contained (ambiguity-free) computation of the renormalized Lagrangian, entirely within the confines of the PN expansion. While we confirm the final results reported in the literature, we clarify several issues regarding intermediate infrared (IR) and ultraviolet (UV) divergences, as well as the renormalization procedure. First, we properly identify the IR and UV singularities using (only) dimensional regularization and the method of regions, which are the pillars of the EFT formalism. This requires a careful study of scaleless integrals in the potential region, as well as conservative contributions from radiation modes due to tail effects. As expected by consistency, the UV divergences in the near region (due to the point-particle limit) can be absorbed into two counter-terms in the worldline effective theory. The counter-terms can then be removed by field redefinitions, such that the renormalization scheme-dependence has no physical effect to 4PN order. The remaining IR poles, which are spurious in nature, are unambiguously removed by implementing the zero-bin subtraction in the EFT approach. The procedure transforms the IR singularities into UV counter-parts. As anticipated, the left-over UV poles explicitly cancel out against UV divergences in conservative terms from radiation-reaction, uniquely determining the gravitational potential. Similar artificial IR/UV poles, which are intimately linked to the split into regions, are manifest at lower orders. Starting at 4PN, both local- and nonlocal-in-time contributions from the radiation region enter in the conservative dynamics. Neither additional regulators nor ambiguity-parameters are introduced at any stage of the computations., Comment: 40 pages. 8 figures. v2: Published version

We determine the gravitational interaction between two compact bodies up to the sixth power in Newton's constant GN, in the static limit. This result is achieved within the effective field theory approach to General Relativity, and exploits a manifest factorization property of static diagrams which allows to derive static post Newtonian (PN) contributions of (2n+1)-order in terms of lower order ones. We recompute in this fashion the 1PN and 3PN static potential, and present the novel 5PN contribution., Comment: 7 pages, 3 figures. In v2: references added, published on PRL

Studies of dark energy at advanced gravitational-wave (GW) interferometers normally focus on the dark energy equation of state $w_{\rm DE}(z)$. However, modified gravity theories that predict a non-trivial dark energy equation of state generically also predict deviations from general relativity in the propagation of GWs across cosmological distances, even in theories where the speed of gravity is equal to $c$. We find that, in generic modified gravity models, the effect of modified GW propagation dominates over that of $w_{\rm DE}(z)$, making modified GW propagation a crucial observable for dark energy studies with standard sirens. We present a convenient parametrization of the effect in terms of two parameters $(\Xi_0,n)$, analogue to the $(w_0,w_a)$ parametrization of the dark energy equation of state, and we give a limit from the LIGO/Virgo measurement of $H_0$ with the neutron star binary GW170817. We then perform a Markov Chain Monte Carlo analysis to estimate the sensitivity of the Einstein Telescope (ET) to the cosmological parameters, including $(\Xi_0,n)$, both using only standard sirens, and combining them with other cosmological datasets. In particular, the Hubble parameter can be measured with an accuracy better than $1\%$ already using only standard sirens while, when combining ET with current CMB+BAO+SNe data, $\Xi_0$ can be measured to $0.8\%$ . We discuss the predictions for modified GW propagation of a specific nonlocal modification of gravity, recently developed by our group, and we show that they are within the reach of ET. Modified GW propagation also affects the GW transfer function, and therefore the tensor contribution to the ISW effect., Comment: 25 pages, 23 figures: v3: several significant improvements

In modified gravity the propagation of gravitational waves (GWs) is in general different from that in general relativity. As a result, the luminosity distance for GWs can differ from that for electromagnetic signals, and is affected both by the dark energy equation of state $w_{\rm DE}(z)$ and by a function $\delta(z)$ describing modified propagation. We show that the effect of modified propagation in general dominates over the effect of the dark energy equation of state, making it easier to distinguish a modified gravity model from $\Lambda$CDM. We illustrate this using a nonlocal modification of gravity, that has been shown to fit remarkably well CMB, SNe, BAO and structure formation data, and we discuss the prospects for distinguishing nonlocal gravity from $\Lambda$CDM with the Einstein Telescope. We find that, depending on the exact sensitivity, a few tens of standard sirens with measured redshift at $z\sim 0.4$, or a few hundreds at $1 < z < 2$, could suffice., Comment: 6 pages, 3 figures; v4: minor modifications; the version to appear in PRD

Even if the fundamental action of gravity is local, the corresponding quantum effective action, that includes the effect of quantum fluctuations, is a nonlocal object. These nonlocalities are well understood in the ultraviolet regime but much less in the infrared, where they could in principle give rise to important cosmological effects. Here we systematize and extend previous work of our group, in which it is assumed that a mass scale $\Lambda$ is dynamically generated in the infrared, giving rise to nonlocal terms in the quantum effective action of gravity. We give a detailed discussion of conceptual aspects related to nonlocal gravity and of the cosmological consequences of these models. The requirement of providing a viable cosmological evolution severely restricts the form of the nonlocal terms, and selects a model (the so-called RR model) that corresponds to a dynamical mass generation for the conformal mode. For such a model: (1) there is a FRW background evolution, where the nonlocal term acts as an effective dark energy with a phantom equation of state, providing accelerated expansion without a cosmological constant. (2) Cosmological perturbations are well behaved. (3) Implementing the model in a Boltzmann code and comparing with observations we find that the RR model fits the CMB, BAO, SNe, structure formation data and local $H_0$ measurements at a level statistically equivalent to $\Lambda$CDM. (4) Bayesian parameter estimation shows that the value of $H_0$ obtained in the RR model is higher than in $\Lambda$CDM, reducing to $2.0\sigma$ the tension with the value from local measurements. (5) The RR model provides a prediction for the sum of neutrino masses that falls within the limits set by oscillation and terrestrial experiments. (6) Gravitational waves propagate at the speed of light, complying with the limit from GW170817/GRB 170817A., Comment: 60 pages, 12 figures; v2: references added

Working within the post-Newtonian (PN) approximation to General Relativity, we use the effective field theory (EFT) framework to study the conservative dynamics of the two-body motion at fourth PN order, at fifth order in the Newton constant. This is one of the missing pieces preventing the computation of the full Lagrangian at fourth PN order using EFT methods. We exploit the analogy between diagrams in the EFT gravitational theory and 2-point functions in massless gauge theory, to address the calculation of 4-loop amplitudes by means of standard multi-loop diagrammatic techniques. For those terms which can be directly compared, our result confirms the findings of previous studies, performed using different methods., Comment: Version accepted for publication in Phys. Rev. D. Appendix C added with details of amplitude computations

We study the cosmological evolution of some nonlocal gravity models, when the initial conditions are set during a phase of primordial inflation. We examine in particular three models, the so-called RT, RR and $\Delta_4$ models, previously introduced by our group. We find that the RR and $\Delta_4$ models have a stable evolution also during inflation. The RT model has an apparent instability, but we show that, because of the smallness of the scale associated to the nonlocal term compared to the inflationary scale, this instability is innocuous and also the RT model has a viable evolution even when its initial conditions are set during a phase of primordial inflation., Comment: v3: important error corrected, which changes a conclusion (RT model viable also when initial conditions are set during inflation); v4: corrected an error in the v3 submission (abstract in the arxiv form had not been changed)

We present a comprehensive and updated comparison with cosmological observations of two non-local modifications of gravity previously introduced by our group, the so called RR and RT models. We implement the background evolution and the cosmological perturbations of the models in a modified Boltzmann code, using CLASS. We then test the non-local models against the {\em Planck} 2015 TT, TE, EE and Cosmic Microwave Background (CMB) lensing data, isotropic and anisotropic Baryonic Acoustic Oscillations (BAO) data, JLA supernovae, $H_0$ measurements and growth rate data, and we perform Bayesian parameter estimation. We then compare the RR, RT and $\Lambda$CDM models, using the Savage-Dickey method. We find that the RT model and $\Lambda$CDM perform equally well, while the performance of the RR model with respect to $\Lambda$CDM depends on whether or not we include a prior on $H_0$ based on local measurements., Comment: 42 pages, 18 figures; v2: the version to appear in JCAP

We study two nonlinear extensions of the nonlocal $R\,\Box^{-2}R$ gravity theory. We extend this theory in two different ways suggested by conformal symmetry, either replacing $\Box^{-2}$ with $(-\Box + R/6)^{-2}$, which is the operator that enters the action for a conformally-coupled scalar field, or replacing $\Box^{-2}$ with the inverse of the Paneitz operator, which is a four-derivative operator that enters in the effective action induced by the conformal anomaly. We show that the former modification gives an interesting and viable cosmological model, with a dark energy equation of state today $w_{\rm DE}\simeq -1.01$, which very closely mimics $\Lambda$CDM and evolves asymptotically into a de Sitter solution. The model based on the Paneitz operator seems instead excluded by the comparison with observations. We also review some issues about the causality of nonlocal theories, and we point out that these nonlocal models can be modified so to nicely interpolate between Starobinski inflation in the primordial universe and accelerated expansion in the recent epoch., Comment: 27 pages, 4 figures

Recent work has shown that modifications of General Relativity based on the addition to the action of a non-local term $R\,\Box^{-2}R$, or on the addition to the equations of motion of a term involving $(g_{\mu\nu}\Box^{-1} R)$, produce dynamical models of dark energy which are cosmologically viable both at the background level and at the level of cosmological perturbations. We explore a more general class of models based on the addition to the action of terms proportional to $R_{\mu\nu}\,\Box^{-2}R^{\mu\nu}$ and $C_{\mu\nu\rho\sigma}\, \Box^{-2}C^{\mu\nu\rho\sigma}$, where $C_{\mu\nu\rho\sigma}$ is the Weyl tensor. We find that the term $R_{\mu\nu}\,\Box^{-2}R^{\mu\nu}$ does not give a viable background evolution. The non-local Weyl-square term, in contrast, does not contribute to the background evolution but we find that, at the level of cosmological perturbations, it gives instabilities in the tensor sector. Thus, only non-local terms which depend just on the Ricci scalar $R$ appear to be cosmologically viable. We discuss how these results can provide a hint for the mechanism that might generate these effective non-local terms from a fundamental local theory., Comment: 25 pages, 6 figures. v2: the version to appear in PRD

We study the cosmological predictions of two recently proposed non-local modifications of General Relativity. Both models have the same number of parameters as $\Lambda$CDM, with a mass parameter $m$ replacing the cosmological constant. We implement the cosmological perturbations of the non-local models into a modification of the CLASS Boltzmann code, and we make a full comparison to CMB, BAO and supernova data. We find that the non-local models fit these datasets as well as $\Lambda$CDM. For both non-local models parameter estimation using Planck+JLA+BAO data gives a value of $H_0$ higher than in $\Lambda$CDM, and in better agreement with the values obtained from local measurements., Comment: 14 pages, 4 figures

We study the cosmological consequences of a recently proposed nonlocal modification of general relativity, obtained by adding a term $m^2R\,\Box^{-2}R$ to the Einstein-Hilbert action. The model has the same number of parameters as $\Lambda$CDM, with $m$ replacing $\Omega_{\Lambda}$, and is very predictive. At the background level, after fixing $m$ so as to reproduce the observed value of $\Omega_M$, we get a pure prediction for the equation of state of dark energy as a function of redshift, $w_{\rm DE}(z)$, with $w_{\rm DE}(0)$ in the range $[-1.165,-1.135]$ as $\Omega_M$ varies over the broad range $\Omega_M\in [0.20,0.36]$. We find that the cosmological perturbations are well-behaved, and the model fully fixes the dark energy perturbations as a function of redshift $z$ and wavenumber $k$. The nonlocal model provides a good fit to supernova data and predicts deviations from General Relativity in structure formation and in weak lensing at the level of 3-4%, therefore consistent with existing data but readily detectable by future surveys. For the logarithmic growth factor we obtain $\gamma\simeq 0.53$, to be compared with $\gamma\simeq 0.55$ in $\Lambda$CDM. For the Newtonian potential on subhorizon scales our results are well fitted by $\Psi(a;k)=[1+\mu_s a^s]\Psi_{\rm GR}(a;k)$ with a scale-independent $\mu_s\simeq 0.09$ and $s\simeq 2$, while the anisotropic stress is negligibly small., Comment: 39 pages, 20 figures

We perform a detailed study of the cosmological dynamics of a recently proposed infrared modification of the Einstein equations, based on the introduction of a non-local term constructed with $m^2g_{\mu\nu}\Box^{-1} R$, where $m$ is a mass parameter. The theory generates automatically a dynamical dark energy component, that can reproduce the observed value of the dark energy density without introducing a cosmological constant. Fixing $m$ so to reproduce the observed value $\Omega_{\rm DE}\simeq 0.68$, and writing $w(a)=w_0+(1-a) w_a$, the model provides a neat prediction for the equation of state parameters of dark energy, $w_0\simeq -1.042$ and $w_a\simeq -0.020$. We show that, because of some freedom in the definition of $\Box^{-1}$, one can extend the construction so to define a more general family of non-local models. However, in a first approximation this turns out to be equivalent to adding an explicit cosmological constant term on top of the dynamical dark energy component. This leads to an extended model with two parameters, $\Omega_{\Lambda}$ and $m$. Even in this extension the EOS parameter $w_0$ is always on the phantom side, in the range $-1.33 < w_0\leq -1$, and there is a prediction for the relation between $w_0$ and $w_a$., Comment: 30 pages, 15 figures; v2: cross-reference to 1311.3421 added

Recent work has shown that non-local modifications of the Einstein equations can have interesting cosmological consequences and can provide a dynamical origin for dark energy, consistent with existing data. At first sight these theories are plagued by ghosts. We show that these apparent ghost-like instabilities do not describe actual propagating degrees of freedom, and there is no issue of ghost-induced quantum vacuum decay., Comment: 18 pages, 3 figures. v2: cross-reference to 1311.3435 added

The energy of a compact binary system at the fifth post-Newtonian order is explicitly computed in the post-Minkowskian approximation by means of the Effective Field Theory approach. This result allows to determine, for the first time beyond the test particle limit, one coefficient of the energy expression for binary point masses on circular orbit as a function of the orbital angular frequency., Comment: 11 pages, 1 figure; in v3: Section II extended and Appendix added, final version published on PRD

In this short review we present a self-contained exposition of the effective field theory method approach to model the dynamics of gravitationally bound compact binary systems within the post-Newtonian approximation to General Relativity. Applications of this approach to the conservative sector, as well as to the radiation emission by the binary system are discussed in their salient features. Most important results are discussed in a pedagogical way, as in-depths and details can be found in the referenced papers., Comment: 37 pages, 22 figures

We derive the conservative part of the Lagrangian and the energy of a gravitationally bound two-body system at fourth post-Newtonian order, up to terms quadratic in the Newton constant. We also show that such terms are compatible with Lorentz invariance and we write an ansatz for the center-of-mass position. The remaining terms carrying higher powers of the Newton constant are currently under investigation., Comment: 24 pages, 2 figures. Typos in formulae corrected, references added, more comments in the conclusion in v2

Using a formulation of the post-Newtonian expansion in terms of Feynman graphs, we discuss how various tests of General Relativity (GR) can be translated into measurement of the three- and four-graviton vertices. In problems involving only the conservative dynamics of a system, a deviation of the three-graviton vertex from the GR prediction is equivalent, to lowest order, to the introduction of the parameter beta_{PPN} in the parametrized post-Newtonian formalism, and its strongest bound comes from lunar laser ranging, which measures it at the 0.02% level. Deviation of the three-graviton vertex from the GR prediction, however, also affects the radiative sector of the theory. We show that the timing of the Hulse-Taylor binary pulsar provides a bound on the deviation of the three-graviton vertex from the GR prediction at the 0.1% level. For coalescing binaries at interferometers we find that, because of degeneracies with other parameters in the template such as mass and spin, the effects of modified three- and four-graviton vertices is just to induce an error in the determination of these parameters and, at least in the restricted PN approximation, it is not possible to use coalescing binaries for constraining deviations of the vertices from the GR prediction., Comment: 10 pages, 5 figures; v2: an error corrected; references added