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.

Rotating deformed neutron stars are important potential sources for ground-based gravitational wave interferometers such as LIGO, GEO600 and VIRGO. O

The favored progenitor model for short $$\gamma${}$-ray bursts (SGRBs) is the merger of two neutron stars that triggers an explosion with a burst of collimated $$\gamma${}$-rays. Following the initial prompt emission, some SGRBs exhibit a plateau phase in their X-ray light curves that indicates additional energy injection from a central engine, believed to be a rapidly rotating, highly magnetized neutron star. The collapse of this ``protomagnetar'' to a black hole is likely to be responsible for a steep decay in X-ray flux observed at the end of the plateau. In this paper, we show that these observations can be used to effectively constrain the equation of state of dense matter. In particular, we show that the known distribution of masses in binary neutron star systems, together with fits to the X-ray light curves, provides constraints that exclude the softest and stiffest plausible equations of state. We further illustrate how a future gravitational wave observation with Advanced LIGO/Virgo can place tight constraints on the equation of state, by adding into the picture a measurement of the chirp mass of the SGRB progenitor.

Recently, general relations among the quadrupole moment (Q), the moment of inertia (I) and the tidal deformability (Love number) of a neutron star were shown to exist. They are nearly independent of the nuclear matter equation of state and would be of great aid in extracting parameters from observed gravitational waves and in testing general relativity. These relations, however, do not account for strong magnetic fields. We consider this problem by studying the effect of a strong magnetic field on slowly rotating relativistic neutron stars and show that, for simple magnetic field configurations that are purely poloidal or purely toroidal, the relation between Q and I is again nearly universal. However, different magnetic field geometries lead to different I–Q relations, and, in the case of a more realistic twisted-torus magnetic field configuration, the relation depends significantly on the equation of state, losing its universality. I–Love–Q relations must thus be used with very great care, since universality is lost for stars with long spin periods, i.e. P ≳ 10 s, and strong magnetic fields, i.e. B ≳ 10¹² G.

The aim of this paper is to compare the two cases of an isolated neutron star, with a non-accreted crust, and that of an accreting neutron star, with