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We introduce GWFAST (https://github.com/CosmoStatGW/gwfast), a Fisher information matrix Python code that allows for 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 (GW) detectors. In particular, 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 makes possible the computation of derivatives with accuracy close to machine precision. We also release the library WF4Py (https://github.com/CosmoStatGW/WF4Py) implementing state-of-the-art GW waveforms in Python. In this paper we provide a documentation of GWFAST and WF4Py with practical examples and tests of performance and reliability. In the companion paper Iacovelli et al. we present forecasts for the detection capabilities of the second and third generation of ground-based GW detectors, obtained with GWFAST.

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.

K. G. Arun et al., 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 gravitational waves 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., E. Berti is supported by NSF Grants No. PHY-1912550 and AST-2006538, NASA ATP Grants No. 17-ATP17-0225 and 19-ATP19-0051, NSF-XSEDE Grant No. PHY-090003, and NSF Grant PHY-20043. D. Blas is supported by a ‘Ayuda Beatriz Galindo Senior’ from the Spanish ‘Ministerio de Universidades’, grant BG20/00228. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. The research leading of to these results has received funding from the Spanish Ministry of Science and Innovation (PID2020-115845GB-I00/AEI/10.13039/501100011033). K. Clough is supported by funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 693024). A. Cárdenas-Avendaño acknowledges funding from the Fundación Universitaria Konrad Lorenz (Project 5INV1) and from Will and Kacie Snellings. M. Crisostomi and E. Barausse are supported by the European Union’s H2020 ERC Consolidator Grant “GRavity from Astrophysical to Microscopic Scales” (Grant No. GRAMS-815673). P. Fleury received the support of a fellowship from “la Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/PI19/11690018. C. Herdeiro thanks the support of the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020, the projects PTDC/FIS-OUT/28407/2017, CERN/FIS-PAR/0027/2019, PTDC/FIS-AST/3041/2020 and the European Union’s Horizon 2020 research and innovation (RISE) programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. P. Pani and E. Maggio acknowledge financial support provided under the European Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480, and under the MIUR PRIN and FARE programmes (GW-NEXT, CUP: B84I20000100001), and support from the Amaldi Research Center funded by the MIUR program “Dipartimento di Eccellenza” (CUP: B81I18001170001). N.Frusciante was supported by Fundação para a Ciência e a Tecnologia (FCT) through the research grants UIDB/04434/2020, UIDP/04434/2020, PTDC/FIS-OUT/29048/2017, CERN/FIS-PAR/0037/2019, the FCT project “CosmoTests—Cosmological tests of gravity theories beyond General Relativity” with ref. number CEECIND/00017/2018 and the FCT project “BEYLA –BEYond LAmbda” with ref. number PTDC/FIS-AST/0054/2021. L.Lombriser was supported by a Swiss National Science Foundation Professorship grant (No. 170547). S.N. acknowledges support from the Alexander von Humboldt Foundation. D.N. acknowledges support from the NSF Grant No. PHY-2011784. R.B. acknowledges financial support from FCT – Fundação para a Ciência e a Tecnologia, I.P., under the Scientific Employment Stimulus - Individual Call - 2020.00470.CEECIND. V. Paschalidis acknowledges support from NSF Grant PHY-1912619 and NASA Grant 80NSSC20K1542 to the University of Arizona. B.S.S. is supported by NSF grants No. AST-2006384 and PHY-2012083. C.F.S. is supported by contracts ESP2017-90084-P and PID2019-106515GB-I00/AEI/10.13039/501100011033 (Spanish Ministry of Science and Innovation) and 2017-SGR-1469 (AGAUR, Generalitat de Catalunya). T. P. S. acknowledges partial support from the STFC Consolidated Grant No. ST/P000703/1. M. Ruiz acknowledges support from NASA Grant 80NSSC17K0070 to the University of Illinois at Urbana-Champaign. I.D. Saltas is supported by the Czech Science Foundation GAČR, Grant No. 21-16583M. N. Stergioulas is supported by the ESA Prodex grant PEA:4000132310 “LISA Stochastic Signals Analysis Pipeline”. F.V. acknowledges partial support from CNES. K.Y. acknowledges support from NSF Grant PHY-1806776, NASA Grant 80NSSC20K0523, a Sloan Foundation Research Fellowship and the Owens Family Foundation. K.Y. would like to also acknowledge support by the COST Action GWverse CA16104 and JSPS KAKENHI Grants No. JP17H06358. N. Yunes acknowledges support from NASA Grants No. NNX16AB98G, 80NSSC17M0041 and 80NSSC18K1352, NSF Award No. 1759615, and the Simons Foundation through MPS Award Number 896696. D.D. acknowledge financial support via an Emmy Noether Research Group funded by the German Research Foundation (DFG) under grant no. DO 1771/1-1.

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

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., 5 pages, 2 figures, Contribution to the ICHEP 2022 conference proceedings

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 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., 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

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., v2: expanded version, to appear in Physics of the Dark Universe

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

We apply a statistical method for constraining the Hubble parameter and modified Gravitational Wave (GW) propagation by correlating “dark sirens” (i.e. compact binary coalescences without an electromagnetic counterpart) with galaxy catalogs to the GWTC-2 catalogue and the GLADE galaxy catalogue. We propose various methodological improvements to the state of the art and discuss several sources of systematic uncertainty. We give an updated bound on H0 and the first bounds on the modified GW propagation parameter Ξ0 from dark sirens alone.

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 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 [Formula: see text] models, previously introduced by our group. We find that, during inflation, the RT model has a viable background evolution, but at the level of cosmological perturbations develops instabilities that make it nonviable. In contrast, the RR and [Formula: see text] models have a viable evolution even when their initial conditions are set during a phase of primordial inflation.