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.

We apply the holographic principle in the cosmological context through the nonadditive Tsallis entropy used to describe the thermodynamic properties of nonstandard statistical systems such as the gravitational ones. Assuming the future event horizon as the infrared cutoff, we build a dark energy model free from cosmological inconsistencies, which includes standard thermodynamics and standard holographic dark energy as a limiting case. We thus describe the dynamics of Tsallis holographic dark energy in a flat Friedmann-Lemaître-Roberson-Walker background. Hence, we investigate cosmological perturbations in the linear regime on subhorizon scales. We study the growth of matter fluctuations in the case of clustering dark matter and a homogeneous dark energy component. Furthermore, we employ the most recent late-time cosmic data to test the observational viability of our theoretical scenario. We thus obtain constraints on the free parameters of the model by means of Monte Carlo numerical method. We also used Bayesian selection criteria to estimate the statistical preference for Tsallis holographic dark energy compared to the concordance $\mathrm{\Lambda }\mathrm{CDM}$ paradigm. Our results show deviations from standard holographic dark energy within the $2\sigma$ confidence level. Finally, the analysis of the dark energy equation of state indicates a quintessencelike behavior with no evidence for phantom-divide crossing at the $1\sigma$ level.

The limits of standard cosmography are here revised addressing the problem of error propagation during statistical analyses. To do so, we propose the use of Chebyshev polynomials to parameterize cosmic distances. In particular, we demonstrate that building up rational Chebyshev polynomials significantly reduces error propagations with respect to standard Taylor series. This technique provides unbiased estimations of the cosmographic parameters and performs significatively better than previous numerical approximations. To figure this out, we compare rational Chebyshev polynomials with Padé series. In addition, we theoretically evaluate the convergence radius of (1,1) Chebyshev rational polynomial and we compare it with the convergence radii of Taylor and Padé approximations. We thus focus on regions in which convergence of Chebyshev rational functions is better than standard approaches. With this recipe, as high-redshift data are employed, rational Chebyshev polynomials remain highly stable and enable one to derive highly accurate analytical approximations of Hubble’s rate in terms of the cosmographic series. Finally, we check our theoretical predictions by setting bounds on cosmographic parameters through Monte Carlo integration techniques, based on the Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic data, using the JLA supernovae sample and the most recent versions of Hubble parameter and baryon acoustic oscillation measurements. We find that cosmography with Taylor series fails to be predictive with the aforementioned data sets, while turns out to be much more stable using the Chebyshev approach.

We here propose a new model-independent technique to overcome the circularity problem affecting the use of gamma-ray bursts (GRBs) as distance indicators through the use of Ep−Eiso correlation. We calibrate the Ep−Eiso correlation and find the GRB distance moduli that can be used to constrain dark energy models. We use observational Hubble data to approximate the cosmic evolution through Bézier parametric curve obtained through the linear combination of Bernstein basis polynomials. In doing so, we build up a new data set consisting of 193 GRB distance moduli. We combine this sample with the supernova JLA data set to test the standard ΛCDM model and its wCDM extension. We place observational constraints on the cosmological parameters through Markov Chain Monte Carlo numerical technique. Moreover, we compare the theoretical scenarios by performing the Akaike and Deviance Information statistical criteria.the 2σ level, while for the wCDM model we obtain $\Omega _m=0.34^{+0.13}_{-0.15}$ and $w=-0.86^{+0.36}_{-0.38}$ at the 2σ level. Our analysis suggests that ΛCDM model is statistically favoured over the wCDM scenario. No evidence for extension of the ΛCDM model is found.

Standard sirens are the gravitational wave (GW) analog of the astronomical standard candles and can provide powerful information about the dynamics of the Universe. In this work, we simulate a catalog with 1000 standard siren events from binary neutron star mergers, within the sensitivity predicted for the third generation of the ground GW detector called the Einstein Telescope (ET). After correctly modifying the propagation of GWs as input to generate the catalog, we apply our mock dataset on scalar-tensor theories where the speed of GW propagation is equal to the speed of light. As a first application, we find new observational bounds on the running of the Planck mass, when considering appropriate values within the stability condition of the theory, and we discuss some consequences on the amplitude of the running of the Planck mass. In the second part, we combine our simulated standard sirens catalog with other geometric cosmological tests (supernovae Ia and cosmic chronometer measurements) to constrain the Hu-Sawicki $f(R)$ gravity model. We thus find new and non-null deviations from the standard $\mathrm{\Lambda }\mathrm{CDM}$ model, showing that in the future $f(R)$ gravity can be tested up to 95% confidence level. The results obtained here show that the statistical accuracy achievable by future ground-based GW observations, mainly with the ET detector (and planned detectors with a similar sensitivity), can provide strong observational bounds on modified gravity theories.