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.

Dynamics and collapse of collisionless self-gravitating systems is described by the coupled collisionless Boltzmann and Poisson equations derived from f(R) gravity in the weak field approximation. Specifically, we describe a system at equilibrium by a time-independent distribution function f0(x,v) and two potentials $\Phi${}0(x) and $\Psi${}0(x) solutions of the modified Poisson and collisionless Boltzmann equations. Considering a small perturbation from the equilibrium and linearizing the field equations, it can be obtained a dispersion relation. A dispersion equation is achieved for neutral dust-particle systems where a generalized Jeans wave number is obtained. This analysis gives rise to unstable modes not present in the standard Jeans analysis (derived assuming Newtonian gravity as weak filed limit of f(R)=R). In this perspective, we discuss several self-gravitating astrophysical systems whose dynamics could be fully addressed in the framework of f(R) gravity.

Light axionic dark matter, motivated by string theory, is increasingly favored for the “no weakly interacting massive particle era”. Galaxy formation is suppressed below a Jeans scale of $\simeq {10}^8{M}_{\odot }$ by setting the axion mass to $m_B\sim {10}^{-22}\mathrm{eV}$, and the large dark cores of dwarf galaxies are explained as solitons on the de Broglie scale. This is persuasive, but detection of the inherent scalar field oscillation at the Compton frequency ${\omega }_B=(2.5\text{months}{)}^{-1}(m_B/{10}^{-22}\mathrm{eV})$ would be definitive. By evolving the coupled Schrödinger-Poisson equation for a Bose-Einstein condensate, we predict the dark matter is fully modulated by de Broglie interference, with a dense soliton core of size $\simeq 150\mathrm{pc}$, at the Galactic center. The oscillating field pressure induces general relativistic time dilation in proportion to the local dark matter density and pulsars within this dense core have detectably large timing residuals of $\simeq 400\mathrm{nsec}/(m_B/{10}^{-22}\mathrm{eV})$. This is encouraging as many new pulsars should be discovered near the Galactic center with planned radio surveys. More generally, over the whole Galaxy, differences in dark matter density between pairs of pulsars imprints a pairwise Galactocentric signature that can be distinguished from an isotropic gravitational wave background.

The cold dark-matter model successfully explains both the emergence and evolution of cosmic structures on large scales and, when we include a cosmological constant, the properties of the homogeneous and isotropic Universe. However, the cold dark-matter model faces persistent challenges on the scales of galaxies. Indeed, N-body simulations predict some galaxy properties that are at odds with the observations. These discrepancies are primarily related to the dark-matter distribution in the innermost regions of the halos of galaxies and to the dynamical properties of dwarf galaxies. They may have three different origins: (1) the baryonic physics affecting galaxy formation is still poorly understood and it is thus not properly included in the model; (2) the actual properties of dark matter differs from those of the conventional cold dark matter; (3) the theory of gravity departs from General Relativity. Solving these discrepancies is a rapidly evolving research field. We illustrate some of the solutions proposed within the cold dark-matter model, and solutions when including warm dark matter, self-interacting dark matter, axion-like particles, or fuzzy dark matter. We also illustrate some modifications of the theory of gravity: Modified Newtonian Dynamics (MOND), MOdified Gravity (MOG), and $f\left(R\right)$ gravity.

Alternative theories of gravity may serve to overcome several shortcomings of the standard cosmological model but, in their weak field limit, general relativity must be recovered so as to match the tight constraints at the Solar System scale. Therefore, testing such alternative models at scales of stellar systems could give a unique opportunity to confirm or rule them out. One of the most straightforward modifications is represented by analytical $f(R)$-gravity models that introduce a Yukawa-like modification to the Newtonian potential thus modifying the dynamics of particles. Using the geodesics equations, we have illustrated the amplitude of these modifications. First, we have integrated numerically the equations of motion showing the orbital precession of a particle around a massive object. Second, we have computed an analytic expression for the periastron advance of systems having their semimajor axis much shorter than the Yukawa-scale length. Finally, we have extended our results to the case of a binary system composed of two massive objects. Our analysis provides a powerful tool to obtain constraints on the underlying theory of gravity using current and forthcoming data sets.