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.

Rastall's theory is based on the nonconservation of the energy-momentum tensor. We show that, in this theory, if we introduce a two-fluid model, one component representing vacuum energy whereas the other pressureless matter (e.g. baryons plus cold dark matter), the cosmological scenario is the same as for the $\Lambda${}CDM model, both at background and linear perturbation levels, except for one aspect: now dark energy may cluster. We speculate that this can lead to a possibility of distinguishing the models at the nonlinear perturbation level.

There are evidences that the cosmic microwave background (CMB) large-angle anomalies imply a departure from statistical isotropy and hence from the standard cosmological model. We propose a $\Lambda \mathrm{CDM}$ model extension whose dark energy component preserves its nondynamical character but wields anisotropic vacuum pressure. Exact solutions for the cosmological scale factors are presented, upper bounds for the deformation parameter are evaluated and its value is estimated considering the elliptical universe proposal to solve the quadrupole anomaly. This model can be constructed from a Bianchi I cosmology with a cosmological constant from two different ways: (i) a straightforward anisotropic modification of the vacuum pressure consistently with energy-momentum conservation; (ii) a Poisson structure deformation between canonical momenta such that the dynamics remain invariant under scale factors rescalings.

We study, by means of the model proposed in Del Popolo (2009), the effect of baryon physics on the small scale problems of the CDM model. We show that, using this model, the cusp/core problem, the missing satellite problem (MSP), the Too Big to Fail (TBTF) problem, and the angular momentum catastrophe can be reconciled with observations. Concerning the cusp/core problem, the interaction among dark matter (DM) and baryonic clumps of 1% the mass of the halo, through dynamical friction (DF), is able to flatten the inner cusp of the density profiles. We moreover assume that haloes form primarily through quiescent accretion, in agreement with the spherical collapse model (SCM)-secondary infall model (SIM) prescriptions. The results of this paper follow from the two assumptions above. Concerning the MSP and TBTF problem, applying to the Via Lactea II (VL2) subhaloes a series of corrections similar to those of Brooks et al. (2013), namely applying a Zolotov et al. (2012)-like correction obtained with our model, and further correcting for the UV heating and tidal stripping, we obtain that the number of massive, luminous satellites is in agreement with the number observed in the MW. The model also produces an angular momentum distribution in agreement with observations, that is with the distribution of the angular spin parameter and angular momentum of the dwarfs studied by van den Bosch, Burkert, & Swaters (2001). In conclusion, the small scale problems of the CDM model can all be solved by introducing baryon physics.