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 evaluate the redshift distribution of binary black hole (BBH), black hole–neutron star binary (BHNS), and binary neutron star (BNS) mergers, exploring the main sources of uncertainty: star formation rate (SFR) density, metallicity evolution, common envelope, mass transfer via Roche lobe overflow, natal kicks, core-collapse supernova model, and initial mass function. Among binary evolution processes, uncertainties on common envelope ejection have a major impact: the local merger rate density of BNSs varies from ∼10³ to ∼20 Gpc⁻³ yr⁻¹ if we change the common envelope efficiency parameter from α_CE = 7 to 0.5, while the local merger rates of BBHs and BHNSs vary by a factor of ∼2–3. The BBH merger rate changes by one order of magnitude, when 1σ uncertainties on metallicity evolution are taken into account. In contrast, the BNS merger rate is almost insensitive to metallicity. Hence, BNSs are the ideal test bed to put constraints on uncertain binary evolution processes, such as common envelope and natal kicks. Only models assuming values of α_CE ≳ 2 and moderately low natal kicks (depending on the ejected mass and the supernovae mechanism), result in a local BNS merger rate density within the 90 per cent credible interval inferred from the second gravitational-wave transient catalogue.

We explore hierarchical black hole (BH) mergers in nuclear star clusters (NSCs), globular clusters (GCs) and young star clusters (YSCs), accounting for both original and dynamically assembled binary BHs (BBHs). We find that the median mass of both first- and nth-generation dynamical mergers is larger in GCs and YSCs with respect to NSCs because the lighter BHs are ejected by supernova kicks from the lower mass clusters. Also, first- and nth-generation BH masses are strongly affected by the metallicity of the progenitor stars: the median mass of the primary BH of a nth-generation merger is ∼24–38 M_⊙ (∼9–15 M_⊙) in metal-poor (metal-rich) NSCs. The maximum BH mass mainly depends on the escape velocity: BHs with mass up to several thousand M_⊙ form in NSCs, while YSCs and GCs host BHs with mass up to several hundred M_⊙. Furthermore, we calculate the fraction of mergers with at least one component in the pair-instability mass gap (f_PI) and in the intermediate-mass BH regime (f_IMBH). In the fiducial model for dynamical BBHs with metallicity Z = 0.002, we find f_PI ≈ 0.05, 0.02 and 0.007 (f_IMBH ≈ 0.01, 0.002 and 0.001) in NSCs, GCs and YSCs, respectively. Both f_PI and f_IMBH drop by at least one order of magnitude at solar metallicity. Finally, we investigate the formation of GW190521 by assuming that it is either a nearly equal-mass BBH or an intermediate-mass ratio inspiral.

The gravitational-wave signal GW250114 was observed by the two LIGO detectors with a network matched-filter signal-to-noise ratio of 80. The signal was emitted by the coalescence of two black holes with near-equal masses m 1 = 33.6 − 0.8 + 1.2 M ⊙ and m 2 = 32.2 − 1.3 + 0.8 M ⊙ , and small spins χ 1 , 2 ≤ 0.26 (90% credibility) and negligible eccentricity e ≤ 0.03 . Postmerger data excluding the peak region are consistent with the dominant quadrupolar ( ℓ = | m | = 2 ) mode of a Kerr black hole and its first overtone. We constrain the modes’ frequencies to ± 30 % of the Kerr spectrum, providing a test of the remnant’s Kerr nature. We also examine Hawking’s area law, also known as the second law of black hole mechanics, which states that the total area of the black hole event horizons cannot decrease with time. A range of analyses that exclude up to five of the strongest merger cycles confirm that the remnant area is larger than the sum of the initial areas to high credibility.

The growing population of binary black holes (BBHs) observed by gravitational wave (GW) detectors is a potential Rosetta stone for understanding their formation channels. Here, we use an upgraded version of our semi-analytical codes fastcluster and cosmo|$\mathcal {R}$|ate to investigate the cosmic evolution of four different BBH populations: isolated BBHs and dynamically formed BBHs in nuclear star clusters (NSCs), globular clusters (GCs), and young star clusters (YSCs). With our approach, we can study different channels assuming the same stellar and binary input physics. We find that the merger rate density of BBHs in GCs and NSCs is barely affected by stellar metallicity (Z), while the rate of isolated BBHs changes wildly with Z. BBHs in YSCs behave in an intermediate way between isolated and GC/NSC BBHs. The local merger rate density of Nth-generation black holes (BHs), obtained by summing up hierarchical mergers in GCs, NSCs, and YSCs, ranges from ∼1 to ∼4 Gpc⁻³ yr⁻¹ and is mostly sensitive to the spin parameter. We find that the mass function of primary BHs evolves with redshift in GCs and NSCs, becoming more top-heavy at higher z. In contrast, the primary BH mass function almost does not change with redshift in YSCs and in the field. This signature of the BH mass function has relevant implications for Einstein Telescope and Cosmic Explorer. Finally, our analysis suggests that multiple channels contribute to the BBH population of the second GW transient catalogue.

On 2023 November 23, the two LIGO observatories both detected GW231123, a gravitational-wave signal consistent with the merger of two black holes with masses 13 7 − 18 + 23 M ⊙ and 10 1 − 50 + 22 M ⊙ (90% credible intervals), at a luminosity distance of 0.7–4.1 Gpc, a redshift of 0.4 0 − 0.25 + 0.27 , and with a network signal-to-noise ratio of ∼20.7. Both black holes exhibit high spins— 0.9 0 − 0.19 + 0.10 and 0.8 0 − 0.52 + 0.20 , respectively. A massive black hole remnant is supported by an independent ringdown analysis. Some properties of GW231123 are subject to large systematic uncertainties, as indicated by differences in the inferred parameters between signal models. The primary black hole lies within or above the theorized mass gap where black holes between 60–130 M _⊙ should be rare, due to pair-instability mechanisms, while the secondary spans the gap. The observation of GW231123 therefore suggests the formation of black holes from channels beyond standard stellar collapse and that intermediate-mass black holes of mass ∼200 M _⊙ form through gravitational-wave-driven mergers.