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.

Einstein Telescope (ET) is the European project for a gravitational-wave (GW) observatory of third-generation. In this paper we present a comprehensive discussion of its science objectives, providing state-of-the-art predictions for the capabilities of ET in both geometries currently under consideration, a single-site triangular configuration or two L-shaped detectors. We discuss the impact that ET will have on domains as broad and diverse as fundamental physics, cosmology, early Universe, astrophysics of compact objects, physics of matter in extreme conditions, and dynamics of stellar collapse. We discuss how the study of extreme astrophysical events will be enhanced by multi-messenger observations. We highlight the ET synergies with ground-based and space-borne GW observatories, including multi-band investigations of the same sources, improved parameter estimation, and complementary information on astrophysical or cosmological mechanisms obtained combining observations from different frequency bands. We present advancements in waveform modeling dedicated to third-generation observatories, along with open tools developed within the ET Collaboration for assessing the scientific potentials of different detector configurations. We finally discuss the data analysis challenges posed by third-generation observatories, which will enable access to large populations of sources and provide unprecedented precision.

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 two-point functions in massless gauge theory, to address the calculation of four-loop amplitudes by means of standard multiloop diagrammatic techniques. For those terms which can be directly compared, our result confirms the findings of previous studies, performed using different methods.

We introduce 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 LIGO-Virgo-KAGRA (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 (S/N), 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 S/N 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 S/N. 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 S/N), and we compare with other Fisher codes in the literature. In the companion paper Iacovelli et al., we discuss the technical aspects of the code. Together with this paper, we publicly release the code GWFAST, (https://github.com/CosmoStatGW/gwfast) and the library WF4Py (https://github.com/CosmoStatGW/WF4Py) implementing state-of-the-art gravitational-wave waveforms in pure Python.