The Science of the Einstein Telescope

Author(s)

Abac, Adrian, Abramo, Raul, Albanesi, Simone, Albertini, Angelica, Agapito, Alessandro, Agathos, Michalis, Albertus, Conrado, Andersson, Nils, Andrade, Tomás, Andreoni, Igor, Angeloni, Federico, Antonelli, Marco, Antoniadis, John, Antonini, Fabio, Sedda, Manuel Arca, Artale, M. Celeste, Ascenzi, Stefano, Auclair, Pierre, Bachetti, Matteo, Badger, Charles, Banerjee, Biswajit, Barba-González, David, Barta, Dániel, Bartolo, Nicola, Bauswein, Andreas, Begnoni, Andrea, Beirnaert, Freija, Bejger, Michal, Belgacem, Enis, Bellomo, Nicola, Bernard, Laura, Bernardini, Maria Grazia, Bernuzzi, Sebastiano, Berry, Christopher P.L., Berti, Emanuele, Bertone, Gianfranco, Bettoni, Dario, Bezares, Miguel, Bhagwat, Swetha, Bisero, Sofia, Bizouard, Marie Anne, Blanco-Pillado, Jose J., Blasi, Simone, Bonino, Alice, Borghese, Alice, Borghi, Nicola, Borhanian, Ssohrab, Bortolas, Elisa, Botticella, Maria Teresa, Branchesi, Marica, Breschi, Matteo, Brito, Richard, Brocato, Enzo, Broekgaarden, Floor S., Bulik, Tomasz, Buonanno, Alessandra, Burgio, Fiorella, Burrows, Adam, Calcagni, Gianluca, Canevarolo, Sofia, Cappellaro, Enrico, Capurri, Giulia, Carbone, Carmelita, Casadio, Roberto, Cayuso, Ramiro, Cerdá-Durán, Pablo, Char, Prasanta, Chaty, Sylvain, Chiarusi, Tommaso, Chruslinska, Martyna, Cireddu, Francesco, Cole, Philippa, Colombo, Alberto, Colpi, Monica, Compère, Geoffrey, Contaldi, Carlo, Corman, Maxence, Crescimbeni, Francesco, Cristallo, Sergio, Cuoco, Elena, Cusin, Giulia, Dal Canton, Tito, Dálya, Gergely, D'Avanzo, Paolo, Davari, Nazanin, De Luca, Valerio, De Renzis, Viola, Della Valle, Massimo, Del Pozzo, Walter, De Santi, Federico, De Santis, Alessio Ludovico, Dietrich, Tim, Dimastrogiovanni, Ema, Domenech, Guillem, Doneva, Daniela, Drago, Marco, Dupletsa, Ulyana, Duval, Hannah, Dvorkin, Irina, Elias-Rosa, Nancy, Fairhurst, Stephen, Fantina, Anthea F., Fasiello, Matteo, Fays, Maxime, Fender, Rob, Fischer, Tobias, Foucart, François, Fragos, Tassos, Foffa, Stefano, Franciolini, Gabriele, Fumagalli, Jacopo, Gair, Jonathan, Gamba, Rossella, Garcia-Bellido, Juan, García-Quirós, Cecilio, Gergely, László Árpád, Ghirlanda, Giancarlo, Ghosh, Archisman, Giacomazzo, Bruno, Gittins, Fabian, Giudice, Ines Francesca, Goncharov, Boris, Gonzalez, Alejandra, Goriély, Stéphane, Graziani, Luca, Greco, Giuseppe, Gualtieri, Leonardo, Guidi, Gianluca Maria, Gupta, Ish, Haney, Maria, Hannam, Mark, Harms, Jan, Harutyunyan, Arus, Haskell, Brynmor, Haungs, Andreas, Hazra, Nandini, Hemming, Gary, Heng, Ik Siong, Hinderer, Tanja, van der Horst, Alexander, Hu, Qian, Husa, Sascha, Iacovelli, Francesco, Illuminati, Giulia, Inguglia, Gianluca, Villalba, David Izquierdo, Janquart, Justin, Janssens, Kamiel, Jenkins, Alexander C., Jones, Ian, Kacskovics, Balázs, Klessen, Ralf S., Kokkotas, Kostas, Kuan, Hao-Jui, Kumar, Sumit, Kuroyanagi, Sachiko, Laghi, Danny, Lamberts, Astrid, Lambiase, Gaetano, Larrouturou, François, Leaci, Paola, Lenzi, Michele, Levan, Andrew, Li, T.G.F., Li, Yufeng, Liang, Dicong, Limongi, Marco, Liu, Boyuan, Llanes-Estrada, Felipe J., Loffredo, Eleonora, Long, Oliver, Lope-Oter, Eva, Lukes-Gerakopoulos, Georgios, Maggio, Elisa, Maggiore, Michele, Mancarella, Michele, Mapelli, Michela, Marchant, Pablo, Margiotta, Annarita, Mariotti, Alberto, Marriott-Best, Alisha, Marsat, Sylvain, Martínez-Pinedo, Gabriel, Maselli, Andrea, Mastrogiovanni, Simone, Matos, Isabela, Melandri, Andrea, Mendes, Raissa F.P., de Souza, Josiel Mendonça Soares, Mentasti, Giorgio, Mezcua, Mar, Mösta, Philipp, Mondal, Chiranjib, Moresco, Michele, Mukherjee, Tista, Muttoni, Niccolò, Nagar, Alessandro, Narola, Harsh, Nava, Lara, Navarro Moreno, Pablo, Nelemans, Gijs, Nielsen, Alex B., Nissanke, Samaya, Obergaulinger, Martin, Oertel, Micaela, Oganesyan, Gor, Onori, Francesca, Pacilio, Costantino, Pagliaroli, Giulia, Palomba, Cristiano, Pang, Peter T.H., Pani, Paolo, Papalini, Lucia, Patricelli, Barbara, Patruno, Alessandro, Pedrotti, Alessandro, Perego, Albino, Pérez-García, Maria Angeles, Périgois, Carole, Perna, Gabriele, Péroux, Celine, Perret, J., Perrodin, Delphine, Pesci, Alessandro, Pfeiffer, Harald P., Piccinni, Ornella Juliana, Pieroni, Mauro, Piranomonte, Silvia, Pompili, Lorenzo, Porter, E.K., Porto, Rafael A., Pound, Adam, Powell, Jade, Puech, Mathieu, Pratten, Geraint, Puecher, Anna, Pujolas, Oriol, Quartin, Miguel, Raduta, Adriana R., Ramos-Buades, Antoni, Rase, Aäron, Razzano, Massimiliano, Rea, Nanda, Regimbau, Tania, Renzini, Arianna, Rettegno, Piero, Ricciardone, Angelo, Riotto, Antonio, Romero-Rodriguez, Alba, Ronchini, Samuele, Rosinska, Dorota, Rossi, Andrea, Roy, Soumen, Rubiera-Garcia, Diego, Rubio, J., Ruiz-Lapuente, Pilar, Sagun, Violetta, Sakellariadou, Mairi, Salafia, Om Sharan, Samajdar, Anuradha, Sanchis-Gual, Nicolas, Sanna, Andrea, Santoliquido, Filippo, Sathyaprakash, Bangalore, Schmidt, Patricia, Schmidt, Stefano, Schneider, Fabian R.N., Schneider, Raffaella, Sedrakian, Armen, Servant, Géraldine, Sevrin, Alexander, Shao, Lijing, Silva, Hector O., Simakachorn, Peera, Smartt, Stephen, Sotiriou, Thomas P., Spera, Mario, Stamerra, Antonio, Steer, Danièle A., Steinhoff, Jan, Stergioulas, Nikolaos, Sturani, Riccardo, Suárez, Duvier, Suresh, Jishnu, Swain, Shaun, Tagliazucchi, Matteo, Tamanini, Nicola, Tasinato, Gianmassimo, Tauris, Thomas M., Tissino, Jacopo, Tomaselli, Giovanni Maria, Toonen, Silvia, Torres-Forné, Alejandro, Turski, Cezary, Ugolini, Cristiano, Vagenas, Elias C., Valbusa Dall Armi, Lorenzo, Valenti, Elena, Valiante, Rosa, Van Den Broeck, Chris, van de Meent, Maarten, van Son, Lieke A.C., Vanvlasselaer, Miguel, Vaglio, Massimo, Varma, Vijay, Veitch, John, Vaskonen, Ville, Vergani, Susanna D., Wils, Milan, Witek, Helvi, Wong, Isaac C.F., Yazadjiev, Stoytcho, Yim, Garvin, Acernese, Fausto, Ahn, Hojae, Allocca, Annalisa, Amato, Alex, Andrés-Carcasona, Marc, Avallone, Guerino, Bachlechner, Markus, Baer, Patrick, Bagnasco, Stefano, Balbi, Gabriele, Barone, Fabrizio, Benedetti, Eugenio, Benning, Charlotte, Bini, Simone, Blázquez Salcedo, José Luis, Bozza, Valerio, Bruno, Matteo, Butz, Timo, Califano, Matteo, Calloni, Enrico, Carapella, Giovanni, Cardini, Alessandro, Subrahmanya, Shreevathsa Chalathadka, Chiadini, Francesco, Chiummo, Antonino, Cianetti, Spina, Ciani, Giacomo, Coccia, Eugenio, Contu, Andrea, Cornelissen, Robin, Cozzumbo, Andrea, Croney, Lewis, Crosta, Mariateresa, D'Agostino, Rocco, Danilishin, Stefan, D'Antonio, Sabrina, De Bolle, Jorden, Degallaix, Jérôme, De Laurentis, Mariafelicia, della Monica, Riccardo, De Marco, Francesco, de Martino, Ivan, De Rosa, Rosario, De Salvo, Riccardo, De Simone, Roberta, Detavernier, Christophe, Diaferia, Giovanni, Di Cesare, Martina, Di Fiore, Luciano, Di Giovanni, Matteo, Di Pace, Sibilla, Docherty, Jennifer, D'Urso, Domenico, Mecherfi, Oussama El, Errico, Luciano, Fabrizi, Federica, Fafone, Viviana, Fanti, Viviana, Fittipaldi, Rosalba, Fiumara, Vincenzo, Freise, Andreas, Funk, Stefan, Gaedtke, Mika, Garufi, Fabio, Gerberding, Oliver, Giangrandi, Edoardo, Giunchi, Carlo, Graham, Victoria, Granata, Massimo, Granata, Veronica, Green, Anna, Haughian, Karen, Heisenberg, Lavinia, Hennig, Margot, Hild, Stefan, Hoang, Van Long, Holland, Nathan, Iannone, Gerardo, Isleif, Katharina-Sophie, Joppe, Robert, Kim, Chang-Hee, Kim, Chunglee, Kim, Kyungmin, Korb, Erika, Korobko, Mikhail, Kranzhoff, Luise, Kuhlbusch, Tim, Lacaille, Gregoire, Lartaux-Vollard, Angélique, Lavezzi, Lia, Laycock, Paul, Lee, Sumi, Lee, Sumin, Lee, Sungho, Losurdo, Giovanni, Lucchesi, Leonardo, Lück, Harald, Macquet, Adrian, Majorana, Ettore, Mangano, Valentina, Martelli, Filippo, Martin, Iain, Martinez, Mario, Masoni, Alberto, Massaro, Luca, Melini, Daniele, Mercurio, Amata, Mereni, Lorenzo, Miller, Andrew L., Mirasola, Lorenzo, Mitchell, Alexandra, Molinari, Irene, Montani, Matteo, Mow-Lowry, Conor, Murgia, Riccardo, Murray, Peter Gordon, Muscas, Giuseppe, Naticchioni, Luca, Nela, Ardiana, Nery, Marina, Niggemann, Tom, Nippe, Niklas, Novak, Jerome, Numic, Armin, Olivieri, Marco, Orsini, Marco, Park, June Gyu, Pascucci, Daniela, Perreca, Antonio, Piergiovanni, Francesco, Pierro, Vincenzo, Pinard, Laurent, Pinto, Innocenzo, Punturo, Michele, Puppo, Paola, Quochi, Francesco, Rading, Reinhardt Omondi, Rapagnani, Piero, Ricci, Marco, Rodrigues, Davi, Romano, Rocco, Rozza, Davide, Saffarieh, Pooya, Santucci, Federica, Schramm, Steven, Schwab, Benjamin, Sequino, Valeria, Neto, Liam Shelling, Silenzi, Laura, Sintes, Alicia M., Sopuerta, Carlos F., Spencer, Andrew, Stahl, Achim, Steinlechner, Jessica, Steinlechner, Sebastian, Szabó, Róbert, Thümmler, Thomas, Tofani, Emanuele, Torniamenti, Stefano, Travaglini, Riccardo, Trozzo, Lucia, Vaccaro, M. Paola, Valentini, Michele, Ván, Peter, van Dongen, Jesse, van Heijningen, Joris, van Ranst, Zeb, Vardaro, Marco, Verdier, Patrice, Vernieri, Daniele, Wagner, Nico, Woehler, Janis, Wolf, Joachim, Zavattini, Guido, Zink, Adrian, Zmija, Andreas

Abstract

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.

Figures

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90\% upper bounds on the beyond-GR parameters $\delta\hat{\varphi}_n$ and $\delta\hat{\varphi}^{(l)}_n$ in the inspiral phase of GW150914, from $-1{\rm PN}$ to 3.5PN order. Red triangles represent bounds from Advanced LIGO observations~\cite{LIGOScientific:2019fpa}, while blue triangles show bounds from simulated GW150914-like signals injected into synthetic noise for a single triangular ET.

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An illustration of strongly lensed GWs used to disentangle polarization states with a single triangular ET. Apart from the tensor polarizations we consider the two vector polarizations, $x$ and $y$, and the longitudinal scalar polarization $l$. Posteriors are shown for the amplitudes of the vector polarizations relative to the $+$ and $\times$ amplitudes, which are denoted $A_x$ and $A_y$, and similarly for the scalar polarization, where the relative amplitude is denoted $A_l$. Green indicates lensing, blue unlensed; solid lines show true values. Without lensing, posteriors are uninformative, while lensing enables individual polarization studies.

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90\% confidence interval for the graviton mass, $A_0 = m_g^2$, as a function of luminosity distance for three GW injection sets. Triangles represent analyses with HLV, crosses and circles with a triangular ET; for the circles, luminosity distance was increased to match SNRs of LVK. The results demonstrate that ET improves bounds on $A_0$ both due to increased SNR for close-by events, and its ability to access events at larger distances despite their lower SNR.

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: Displacement memory detection

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: Spin memory detection

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: Parameter estimation

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GWs from the coalescence of equal-mass NSs in a scalar-tensor theory compared to GR. Upper panel: Plus polarization along the axis perpendicular to the orbital plane. Lower panel: GW frequency vs.\ retarded time. Simulations assume a binary total mass $m = 2.7M_\odot$ and the AH4 EoS, with $B$ denoting the scalar-Ricci coupling strength. Credits~\cite{Shibata:2013pra}.

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Comparison of the scattering angle in various PM approximations vs numerical simulations. See~\cite{Buonanno:2024vkx} for details.

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68\% constraints on $\delta k_s$, quantifying deviations from the Kerr quadrupole moment, as a function of binary total mass ($d_L=400\,{\rm Mpc}$). We assume mass ratio $q=3$, aligned spins $\chi_1=0.9$ and $\chi_2=0.8$, and inclination $\iota=\pi/3$. Blue/green triangles represent LIGO O5 and ET constraints (Fisher analysis); empty squares show Bayesian inference results. We show results for both a $10-{\rm km}$ triangular configuration and a $15-{\rm km}$ 2L configuration of the ET observatory.

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Same as in figure~\ref{fig:quadrupole_ETvsO5} but for the tidal deformability parameter $\Lambda$ as a function of the total mass of the binary ($d_L=400\,{\rm Mpc}$). In this case we assume mass ratio $q=3$, zero spins, and inclination $\iota=\pi/3$. We show results for both a $10-{\rm km}$ triangular configuration and a $15-{\rm km}$ 2L configuration of the ET observatory.

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Posterior distributions for the effective spin parameter $\chi_{\rm eff} = (m_1\chi_1 + m_2\chi_2)/(m_1 + m_2)$ and $M_B\equiv \sqrt{\lambda}/\mu^2$, with $(\mu,\lambda)$ the boson's mass and self-interaction coupling, from a mock boson-star binary signal with ET (left) and LVK O4 (right) at $d_L=400\,{\rm Mpc}$. The binary has chirp mass $\mathcal{M}=8M_\odot$, mass ratio $q=0.6$, and component (aligned) spins $(\chi_1,\chi_2)=(0.2,0.1)$. The inspiral cutoff frequency is fixed at the Roche frequency $f_{\text{RO}}=120\,{\rm Hz}$, corresponding to the binary tidal disruption.

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Left: Inverse cumulative distribution of ringdown SNRs $\rho_{\rm RD}$ for different detector configurations. The shaded band corresponds to signals with $\rho_{\rm RD}<12$. From ref.~\cite{Bhagwat:2023jwv}. Right: Forecasts for measuring the amplitude and phase of a secondary ($lmn=330$) ringdown mode in a GW190521-like event detected by ET, compared with current uncertainties and with the GR prediction. Extended from~\cite{Forteza:2022tgq}.

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BH spectroscopy with a GW150915-like event using LVKI or ET. \textit{Left:} bounds on the frequencies $f_{lmn}$ and damping times $\tau_{lmn}$. \textit{Right:} bounds on the excitation amplitudes $\mathcal{A}_{lmn}$~\cite{Gossan:2011ha,Kamaretsos:2012bs} and phases $\phi_{lmn}$.

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Posteriors of the final mass, spin, and amplitudes/phases of the 220 and 330 modes, and of the 330 GR deviations ($\delta f_{330}$ and $\delta \tau_{330}$) for a GW150914-like remnant, detected with ET in the standard 10-km triangular configuration. Here, $A_R$ is the relative mode amplitude. The 2D distributions show 68\% and 90\% credible levels, while the dashed lines in the 1D distributions mark the 90\% credible intervals. Black lines indicate the injected values.

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Fractional percentage errors on the reflectivity of compact objects for a GW150914-like event~\cite{Branchesi:2023mws}.

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Mass-radius diagram (left) and dimensionless tidal deformability (${\Lambda}$) and mass diagram (right) of a NS with a DM core. Stars containing fermionic DM (green lines) are labeled by the particle mass $\mu$ (in GeV). The ones with bosonic DM (red lines) are labeled by $\rho_0 \hbar^3$ (in $10^{-4}\,{\rm GeV}^4$). Reproduced from~\cite{Leung:2022wcf} with license from the American Physical Society.

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Predicted constraint on dark photon/baryon $U(1)_B$ coupling from ET-D (blue continuous), compared to LIGO O3 upper limit (black continuous) and current MICROSCOPE (dotted)~\cite{Berge:2017ovy} and E\"ot-Wash torsion balance experiment~\cite{Schlamminger:2007ht} (dashed) limits. Adapted from~\cite{LIGOScientific:2021ffg}.

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Maximum detectable luminosity distances for optimal scalar clouds around BHs with initial mass $M_i$ and dimensionless spin parameter $\chi_i$ for ET, assuming one year of observation and ET in the triangle configuration (results for the 2L configuration are almost indistinguishable). White contour lines indicate the values of the corresponding boson rest-energy in units of ${\rm eV}$. The dotted white lines highlight the mass and spin of a GW150914-like merger remnant. Adapted from ref.~\cite{Isi:2018pzk} and obtained using the python package gwaxion~\cite{gwaxion}. We are indebted to Max Isi for providing the code.

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ET exclusion regions for scalar boson clouds assuming a BH distance of 15 kpc, an initial spin $\chi_i=0.6$ and three different BH ages of $10^4,~10^7$ and $10^9$ years.

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Resonance frequency of the (main) Bohr transition [see \eq{bohr_resonances}], as a function of the BH mass, $M_{\rm BH}$, and~$\alpha$. See ref.~\cite{Baumann:2019ztm}.

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Left panel: relative percentage error on the scalar field mass $m_b$ for a BH binary system with mass ratio $m_2/m_1 = 1/2$, detected by ET at $d_L = 1 \, {\rm Gpc}$. Bars show the values of the innermost stable compact orbit (ISCO) and Roche frequencies. We fix the scalar cloud mass to the upper bound $M_\text{\tiny cloud} = 0.1 m_i$ for both objects ($i=1,2$). Right panel: maximum luminosity distance for which the scalar field mass can be constrained by ET with a relative percentage accuracy of $10\%$ (magenta) and $50\%$ (yellow), assuming $\alpha_1=0.2$. See ref.~\cite{DeLuca:2022xlz}.

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Left: plot of the cumulative SNR of the anisotropies as a function of the SNR of the monopole for ET (triangular) + CE ($40\, \rm km$) for five years of observation. The dashed lines represent the LVK upper bounds on the amplitude of the monopole for the three mechanisms considered (inflation, PT and SIGW). Right: plot of the monopole of the intensity (solid blue), circular polarization (solid orange) of the AGWB, and of the PLS of the intensity and circular polarization (dashed blue and orange respectively) for ET+CE in one year of observation.

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Left: plot of the intrinsic, shot noise and kinetic contributions to the dipole of the AGWB as a function of the frequency. Right: plot of the SNR of the kinetic dipole of the AGWB, obtained with the multi-frequency analysis of the anisotropies, and of the intrinsic and kinetic anisotropies of the AGWB, in cross-correlation with the galaxy survey SKAO2, as a function of the amplitude of the monopole of the AGWB. Here a network with the triangular ET configuration plus CE with the two interferometers (of 20km and 40km) placed in Hanford and Livingston for 5 yrs has been considered.

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Standard and parity-violating overlap reduction functions plotted over ground-based detector frequency range. When assuming a triangular configuration for ET, we plot the result for a pair of ET detectors (out of the three composing the triangle), and for one ET detector (out of the three composing the triangle) and a Cosmic Explorer (CE), with CE taken either in the current Hanford site or in the current Livingston sites. When assuming the 2L configuration for ET, we plot the result for this pair of L-shaped detectors. The triangular ET design is assumed in Virgo location. We denote the triangular pairing with $i$, $j$ as all pairings produce the same overlap reduction functions.

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The projected impact from correlated NN from body-waves on the search for an isotropic GWB. We present budget based on observed seimic correlations from four different seismic sites. Note that many parameters, such as depth, horizontal separation, etc., vary for the different sites; see~\cite{Janssens:2024jln} for an in-depth discussion. Figure from \cite{Janssens:2024jln}.

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Corner plot of the posterior distributions for the log-amplitude and tilt of the GWB, $\{ \log_{10}A_{\rm GW}, n_{\rm GW}\}$ for the correlation coefficient and tilt of the noise, $\{r, n_{\rm noise} \}$, when the injected signal has $\log_{10}A_{\rm GW}=-9$ and $n_{\rm GW}=2/3$, while the injected noise has $r^{\rm inj}=0.2$, $n_{\rm noise}=-8$. The shaded areas represent the 1-, 2-, and 3-$\sigma$ credible regions. The orange lines indicate the injected values for the four parameters. Figure from \cite{Caporali:2025mum}.

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As in figure~\ref{fig:corner_triangular} when the injected signal has $\log_{10}A_{\rm GW}=-11$ and $n_{\rm GW}=0$, while the injected noise has $r^{\rm inj}=0.2$, $n_{\rm noise}=-8$.

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Spectrum of the single-field slow-roll inflation as a function of frequency, together with the PLS of LISA and ET in two different configurations. The solid, dashed, dot-dashed standard inflation curves are computed by adopting $r(k_{\star}) = 0.1$, $r(k_{\star}) = 0.01$ and $r(k_{\star}) = 0.001$, respectively.

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The GW density $\Omega_{\rm GW}$ obtained in the case of a spectator axion superimposed with the sensitivity curves of GW detectors, including ET. The blue (red) dashed line corresponds to the strongly (weakly) amplified polarization. The continuous black line gives the total signal.

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Backreaction and SNR prospects as a function of the gauge coupling constant $g$ and the velocity parameter $\xi(f) = \xi_{\rm{const}}$ for two values of the coupling constant, $\lambda /f = 30/M_{\rm Pl}$. The Hubble rate on the CMB scale is assumed to be $H_{\rm{CMB}} = 2.1 \times 10^{-5}$ $M_{\rm Pl}$. The brown and red lines are the SNR contours calculated assuming 1 year of observation with the design LVK and ET + 2 CE network sensitivities, respectively. The blue and green bands represent $|\kappa -1|\leq 0.25$ and $0.1$, where we expect that the PBH overproduction bound could be relaxed in the strong backreaction regime. The light purple region represents the slow-roll parameter satisfying $\epsilon_H \leq 1$, where inflation is still ongoing. Yellow and orange regions correspond to the parameter space where the primordial curvature spectrum stays below the PBH upper limit assuming Gaussian ($\mathcal{P}_{\mathcal{R}} \leq 10^{-2}$) and $\chi^2$ ($\mathcal{P}_{\mathcal{R}} \leq 10^{-4}$) statistics. Additionally, we shade the Abelian regime in magenta (below the black dotted line) and the strong backreaction regime in gray (upper region of the $|\kappa -1|$ band), in both of which the estimation of the SNR is not reliable.

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Posterior distributions from Bayesian inference assuming a piecewise, toy model potential, with inflaton velocity $\xi_{\rm CMB}$ before pivot frequency $f_0$, and velocity $\xi_0$ after. This search is done assuming $\chi^2$ statistics. The shaded regions correspond to the $95 \%$ CL. The different colors used are: blue for ET and orange for a network consisting of ET+2CE. The red cross and line correspond to the injected values.

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GWB predicted by pre-big-bang cosmology, compared with the PLS of the considered ET configurations. Here we assume an observation time of one year and $\text{SNR} = 1$. The shaded green area is the range of the GWB allowed by a set of consistency conditions on the parameter space of the model \cite{Ben-Dayan:2024aec}. The GWB spectral shape is computed assuming $z_s=5\times 10^6$, $z_d=10^3$, $z_\sigma=2.2$ (``PBB 1'' curve, broken power law within the ET sensitivity window) and $z_s=10^8$, $z_d=10^3$, $z_\sigma=2.2$ (``PBB 2'' curve, almost flat plateau within the ET sensitivity window). Here $z_s=\eta_s/\eta_1$, $z_d=\eta_d/\eta_1$ and $z_\sigma=\eta_\sigma/\eta_1$ are ratios of conformal-time scales marking four different regimes: de Sitter evolution for $-\eta_s<\eta<-\eta_1$, radiation domination for $-\eta_1<\eta<\eta_\sigma$, matter domination for $\eta_\sigma<\eta<\eta_d$ and another radiation-domination phase for $\eta>\eta_d$.

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GW spectra from sound waves produced by first-order FOPTs at temperature $T_{\rm reh}$ \cite{Caprini:2019egz}, with the wall velocity that gives the maximal energy transfer to sound waves~\cite{Espinosa:2010hh, Gouttenoire:2022gwi}. The black curves are the power-law integrated sensitivities of ET assuming a triangular xylophone configuration with 10km arms and the 1 and 10 years observation time with ${\rm SNR}=1$.

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Regions of FOPT parameter spaces that can be probed by ET assuming triangular xylophone configuration with 10km arms for two values of $\beta/H$ (left) and $T_{\rm reh}$ (right), assuming GW from sound waves \cite{Caprini:2019egz} shown in figure~\ref{fig:pt_spectrum}. The darker regions can be probed with ET at SNR $\geq 1$ and 1 year of observation time, while the lighter regions require 10 years.

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Forecast power-law-integrated GWB sensitivity of ET, compared with signals from local Nambu-Goto strings of various tensions $G\mu$. We consider both the triangular configuration for ET with 10~km arms, and the 2L configuration with 15~km arms, misaligned as in \cite{Branchesi:2023mws}; the solid black curves corresponds to one year of observations, while the dotted black curve corresponds to 10 years. The left and right panels show the predictions for models A~\cite{Blanco-Pillado:2013qja} and B~\cite{Lorenz:2010sm} of the loop network, respectively. Both models predict that, in the triangle configuration, ET will be sensitive to $G\mu\gtrsim10^{-18}$ after one year of observations with SNR $\geq 1$.

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Spectra of GWB from global strings formed at energy scale $\eta$ are shown in solid colored lines. The black lines are the same ET-triangle sensitivity curves as shown in figure~\ref{fig:cs_stochastic_forecasts}. We used the semi-analytic calculation in~\cite{Gouttenoire:2019kij}; see also~\cite{Chang:2019mza,Chang:2021afa}.

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Expected rate of detected bursts in Einstein Telescope as a function of the string tension for models A and B. In case ET does not detect bursts from cosmic string cusps, the orange hatched region is excluded after 4 years of observations and the blue hatched region is excluded after 8 years.

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Left: $(\alpha_\text{ann}, T_\text{ann})$ parameter space which can be probed by ET, assuming a triangular xylophone configuration with 10km arms (orange), a 2L misaligned configuration with 15km arms (blue), both for an observation time of $T = 1$ year and $\text{SNR} = 1$. For comparison, we show the sensitivity region of LVK A+ (purple) with $T = 1$ year and $\text{SNR} = 1$. Contours of $h^2 \Omega_\text{peak}$ and $f_\text{peak}$ are shown by black and brown dashed lines respectively. Right: bias vs. tension parameter space with similar sensitivity regions as on the left plot. Brown dashed lines indicate the size of the bias. The gray region corresponds to a forbidden DW dominated Universe, for which $\alpha_\text{ann} \geq 1$.

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Upper bound on the amplitude of the curvature power spectrum at small scales $A$ as a function of the wavenumber $k_*$ in case of null detection of a SIGW. The green and blue regions at the top show, respectively, the exclusions from PBH constraints and from BBN/CMB constraint on the number of relativistic degrees of freedom. In red the bound from LVK is shown for comparison.

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Lower bound on the inverse time duration of the phase transition $\beta/H_*$ as a function of the temperature $T_{\rm reh}$ right after the phase transition in case of null detection of a SIGW. The green and blue regions at the bottom show the exclusions from PBH constraints and from BBN/CMB constraint on the number of relativistic degrees of freedom.

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Constraints from PBH overproduction as a function of temperature $T_\text{GW}$ and DW abundance at the peak of GW emission (adapted from \cite{Ferreira:2024eru}). Superimposed, the detectable regions with ET and LKV design as well as the exclusion from LVK O3 data. The shaded region above the solid line is excluded from the PBHs formed at $t_\text{PBH}$ or later. The dash-dotted and dotted lines refer to sub-Hubble PBHs, for representative values of $\alpha_c$, giving a reasonable sense of current systematic uncertainties. The vertical dashed lines indicate the asteroid mass range (assuming $\alpha_c=1$).

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$h^2\Omega_\text{GW}(f)$ for the amplification of vacuum fluctuations in standard single-field slow-roll inflation. The gray dashed lines corresponds to the result in GR+$\Lambda$CDM, assuming a tensor-to-scalar ratio $r=0.07$ and a scale-invariant primordial tensor spectrum, $n_T=0$. The colored lines show the result obtained with the modified cosmological discussed in the text, for fixed $f_{*}\simeq 2.5\times10^{-6}$~Hz, and $A_*=1$ (blue dashed lines), $A_*=10$ (green dot-dashed lines), $A_*=100$ (red dotted lines), and $\nu=0$ (left panels), $\nu=1$ (right panels). The colored regions refer to the BBN constraint and the projected sensitivities of various GW observatories.

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(left) Inflationary spectra induced by the intermediate matter-kination era are shown as colored lines, while the prediction from standard $\Lambda$CDM cosmology is the black line. Following the period of matter domination, the intermediate kination era starts at energy scale $E_{\rm KD}$ and lasts for $N_{\rm KD}$ e-folds of the scale-factor expansion. We assume the scale invariant tensor perturbation with inflationary energy scale $E_{\rm inf} \simeq 1.6 \times 10^{16}$ GeV. The sensitivity curves of ET assume respectively 1 and 10 years of observation with SNR $=1$, and the LISA sensitivity comes from \cite{Flauger:2020qyi}. (right) Effect of an intermediate matter era---lasting for $N_{\rm MD}$ efolds and ending at temperature $T_{\rm dec}$---on the prediction for $h^2\Omega_{\rm GW}(f) $ from local cosmic strings, shown by the blue lines \cite{Gouttenoire:2019kij,Gouttenoire:2019rtn,Ghoshal:2023sfa}.

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The ($f_{\rm RD}, f_{\rm SD}$) parameter regions accessible by ET are shown. The colored regions represent the detectable parameter spaces for three different values of $w$: red for $w=0.6$, orange for $w=0.8$, and yellow for $w=1.0$ (corresponding to kination). The gray regions indicate the parameter space excluded by indirect limits from BBN and CMB, with the darkest gray for $w=0.6$ and the lightest gray for $w=1$. We again assume the scale invariant tensor perturbation with inflationary energy scale $E_{\rm inf} \simeq 1.6 \times 10^{16}$ GeV. The parameter $f_{\rm MD}$ is chosen as a function of $f_{\rm RD}$, $f_{\rm SD}$ and $w$, as described in the text, and we denote the parameter region where $f_{\rm MD} > f_{\rm MD,max}$ with blue lines (dashed for $w=1$, dotted for $w=0.8$ and thick for $w=0.6$). The sensitivity curve of ET assumes 1 year of observation with SNR $=1$. The green area on the left is excluded, since there $f_{\rm RD}< f_{\rm BBN}$, which is not allowed.

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Dependence of the expected relative errors $\sigma$ for $l_{\rm max}=2$ (left) and $l_{\rm max}=3$ (right) harmonics. The oscillation amplitude $A_2$ and $A_3$ vary, while the other parameters are fixed as follows. For $l_{\rm max}=2$: $\Omega_0=10^{-11}$, $n_{\rm t}=0$, $\omega=10$, $A_1=0.1$, $\Phi_1=0$. For $l_{\rm max}=3$: $A_2=0.1$, $\Phi_2=0$ and $\Phi_3=0$. The black, orange and purple curves correspond to the oscillation parameters (oscillation amplitude $A_l$ and phase $\Phi_l$) for $l=1$, $l=2$, and $l=3$, respectively. From~\cite{Calcagni:2023vxg}.

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Results of the joint inference on the cosmological parameters $H_0$, $\Omega_{\rm m,0}$, employing GW+KN events detected in one year of observations by the $10$ km triangular (left panel) or the 2L-15km-$45^{\circ}$ (right panel) ET configurations, with the EM counterpart detected by the Vera Rubin Observatory. Vertical dashed lines represent the $68\%$ CI of each distribution, while the black solid lines label the fiducial values.

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Results of the joint inference on the cosmological parameters $H_0$, $\Omega_{\rm m,0}$ employing GW+GRB events detected in $5$ years of observations by the $10$ km triangular (left panel) or the 2L-15km-$45^{\circ}$ (right panel) ET configurations, with the EM counterpart detected by THESEUS. Vertical dashed lines represent the $68\%$ CI of each distribution, while the black solid lines label the fiducial values.

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Left panel: Posterior joint distribution on $H_0$ and $\Omega_{\rm m,0}$ obtained by employing BBHs as dark sirens and cross correlating the loudest GW event localization volumes with a galaxy catalog. The contours, which refer the $68\%$ and $90\%$ confidence level, are shown for two networks of 3G detectors assuming one year of continuous observations. Here ET is in a $10$ km-triangular configuration and CE1 and CE2 correspond to $40$ km and $20$ km Cosmic Explorer, respectively. Figure from \cite{Muttoni:2023prw}. Right panel: Dependence of the accuracy on $H_0$ on the GW event localization volume with a photometric (\texttt{photo-z}) or spectroscopic (\texttt{spec-z}) galaxy catalog.

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Localization capabilities for BBHs of ET in its triangular (left panel) and 2L-15km-$45^{\circ}$ (right panel) configurations. The color scale denotes the number of galaxies expected in the $90\%$ localization volume. Events marked with black dots are localized to one galaxy only.

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Number of galaxies inside the localization volume of ET golden BBH events ($\Delta\Omega<2\,\mathrm{deg^2}$) for the triangular (left) and 2L (right) configuration. The color bar represents the 90\% sky localization area. The black line and shaded region indicate the 16th and 84th percentiles, respectively.

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Effect of varying the value of $H_{0}$ and $\Omega_{M}$ on the energy density of the stochastic background from BBHs (green) and BNSs (purple). The astrophysics is fixed to the models described in \cite{Capurri:2021zli,Boco:2020pgp}. The black curve is the PLS of ET for the triangular 10 km configuration \cite{Branchesi:2023mws}.

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The functions $\delta(z)$ (left panel) and $\dgw(z)/\dem(z)$ (right panel), for the non-local gravity model proposed in \cite{Maggiore:2013mea}, for different values of a free parameter of the theory. From ref.~\cite{Belgacem:2020pdz}.

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Results of the joint inference on the cosmological parameters $H_0$, $\Omega_{\rm m,0}$, $w_0$, $w_a$, $\Xi_0$ and $n$ employing GW+KN events detected in one year of observations by the $10$ km triangular (left panel) and the 2L-15km-$45^{\circ}$ (right panel) ET configurations, together with the Vera Rubin Observatory. Vertical dashed lines represent the $68\%$ CI of each distribution, while the black solid lines label the fiducial values.

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Results of the joint inference on the cosmological parameters $H_0$, $\Omega_{\rm m,0}$, $w_0$, $w_a$, $\Xi_0$ and $n$ employing GW+GRB events detected in $5$ years of observations by the $10$ km triangular (left panel) and the 2L-15km-$45^{\circ}$ (right panel) ET configurations, together with THESEUS. Vertical dashed lines represent the $68\%$ CI of each distribution, while the black solid lines label the fiducial values.

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Left: the redshift $z_{\rm true}$ of a source, as a function of the value $z_{\rm GR}$ that would be incorrectly inferred using GR if Nature is described by a modified gravity theory with $\Xi_0\neq 1$, for different values of $\Xi_0$. Right: the effect on the distribution of the total mass of the binary from a `wrong' reconstruction using GR, assuming that the true distribution of the source-frame total mass of the binary is a (redshift-independent) Gaussian, with mean $2.66 \Msun$ and standard deviation $0.13 \Msun$. Adapted from \cite{Finke:2021eio}.

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Constraints obtained from the statistical method relying on lensed BBH events to constrain the cosmological parameters presented in~\cite{Jana:2022shb}. Left: $H_0$ vs. $\Omega_m$. Right: $w_0$ vs. $\Omega_m$. These results are obtained for a ten year observation period, assuming that a generic network of 3G detectors will have a nominal BBH detection rate of $5\times 10^5\, \mathrm{yr}^{-1}$ upto a typical redshift of $z_\text{max}\sim15$ (exact value dependent on the cosmology and population models), and that their time of arrivals will be measured precisely. The bounds obtained are competitive with other studies, and the BBH signals originate from further away, potentially carrying information from earlier periods in the Universe. Figures adapted from~\cite{Jana:2022shb}.

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Marginalised constraints combining ET KN and galaxies for $z\le0.5$, separated according to the method used. Purple: only DESI BGS galaxies; Red: DESI+KN velocities ($3\times 2$pt); Green: DESI+KN in the full $6\times 2$pt; Orange: KN distances only; Blue: $6\times 2$pt + distances; Brown: CMB. Figure from~\cite{Alfradique:2022tox}.

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Left: constraints in the $H_0 - \Omega_{\rm m}$ plane from the tomographic angular auto- and cross-correlation of resolved BBHs with a Euclid--like photometric galaxy catalog, for ET in the 2L configuration in combination with two CE detectors. Different lines correspond to fixing other cosmological parameters (baryon density $\Omega_b$, amplitude $A_s$ and spectral index $n_s$ of the primordial power spectrum), representing the case where prior knowledge is assumed on those, while still marginalising over the tracers' bias. The case of a triangular ET is not displayed since almost indistinguishable. Right: relative errors (in $\%$) on $H_0,\ \Omega_{\rm m}$ for the 2L and triangular configurations of ET. Results from~\cite{Pedrotti:2024XXX}.

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The cross-correlation between the harmonic modes of galaxies ($g$) at $z$ and BBHs ($b$) at $D_L$ leads to a cross-angular power spectrum $C^{gb}_\ell(z,D_L)$. We show the average cross-spectrum for 1000 light-cone simulations of galaxies and BBHs, for $\ell=100$ ($\sim 2^\circ$). When a shell in $z$ coincides with a shell in $D_L$, the correlation between the galaxy and BBH maps is maximal. The red, blue and black lines correspond to the Hubble diagrams $D_L(z)$ of three different cosmologies -- in the fiducial model $H_0=70.0\,\, {\rm km}\, {\rm s}^{-1}\, {\rm Mpc}^{-1}$ and $\Omega_m=0.3$. Figure taken from \cite{Ferri:2024amc}.

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Evolutionary pathways to form a GW source for isolated binary evolution. Left: the common-envelope channel and the stable mass transfer channel. Stage 1) Zero-Age-Main-Sequence (ZAMS) 2) First phase of mass transfer 3) Formation of BH or NS 4) Second phase of mass transfer 5) Formation of double compact object 6) GW merger. Right: chemically homogeneous evolution. Stage 1) ZAMS 2) Formation of binary BH 3) GW merger.

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Evolutionary pathway to form GW sources a) according to the non-interacting triple channel, b) in an interacting triple with a stellar merger in the inner binary. On the left: Stage 1) ZAMS 2) Formation of triple BH 3) ZLK cycles and orbital dissipation 4) GW merger. On the right: Stage 1) ZAMS 2) Stellar merger in the inner binary 3) Formation of binary with a rejuvenated star 4) Various phases of mass transfer 5) Formation of binary BH 6) GW merger.

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Left panel: Fraction of BBHs that merge inside the cluster for different assumptions about the binary semimajor axis sampling: assuming a Gaussian distribution peaked around the hard binary separation with $\sigma_a=0.1$ (purple straight line), or with $\sigma_a=0.3$ (blue dashed line), or following the assumption of \cite{Samsing:2017xmd}, i.e. that $a$ is distributed between $0.1-0.2$ AU according to a flat distribution. Shaded areas encompass the Poissonian error associated with the samples. The points represent data from self-consistent $N$-body and Monte Carlo simulations of star clusters. Right panel: distribution of merger generation for different cluster types. All models are performed with the \textsc{B-Pop} population synthesis tool \cite{Sedda:2021vjh}.

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\footnotesize{ The key ingredients of compact binary coalescence rates. \underline{Top:} star formation rate density (gray - total, blue - at metallicity lower than in the Small Magellanic Cloud) as a function of redshift/lookback time, spanned by observation-based $f_{\rm SFR}(Z,t)$ models from \cite{10.1093/mnras/stz2057,10.1093/mnras/stab2690}. Solid and dashed black lines show variation in the evolution of the low-metallicity Star Formation History (SFH) due to the uncertain high redshift evolution in the number density of low-mass galaxies (n$_{\rm lowM, gal}$). The arrowheads indicate the predicted range of redshifts for the peak of the star formation history in various types of environments and peak of the AGN luminosity density. \underline{Middle:} $\eta_{\rm form}$ and delay-time distribution (DTD) illustrative of merging BBH (black), BHNS (turquoise) and BNS (brown) formed in isolated binary evolution channel. Thick lines: simplified main trends found in the literature. Thin lines: $\eta_{\rm form}$ and DTD for example binary population synthesis model variations from \cite{Broekgaarden:2021efa} (left) and \cite{Boesky:2024msm} (right, normalised to the same value at peak) showing a diversity of shapes. \underline{Bottom left:} literature compilation of the local BBH $R_{\rm{merger}}$ from \cite{Mandel:2021smh}. \underline{Bottom right:} BBH $R_{\rm{merger}}$ redshift evolution examples normalised to the same local rate. Black dashed/solid lines - calculated using $f_{\rm SFR}(Z,t)$ variations corresponding to those shown as black dashed/solid lines in the top panel and the $\eta_{\rm form}$ and DTD shown as black lines in the middle panel. Purple line - example BBH $R_{\rm{merger}}$ for globular cluster channel from \cite{Ng:2020qpk}. }

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Time to detect $100$ merging binary BHs in bins of size $\Delta z=0.2$ in redshift and $\Delta \log m_1=0.1$ in primary mass, indicative of the number of measurements required to provide a constraint on the rate with an error of $10\%$ in each bin. The distribution of masses is taken to be the \textsc{Power Law+Peak} parametric model derived from GWTC-3~\cite{KAGRA:2021duu} using the median values for all parameters (which include a localized Gaussian peak at $34\Msun$), and increasing the upper cut on BH mass to $m_\mathrm{max}=250 \Msun$. This figure assumes the sensitivity curves adopted in \cite{Branchesi:2023mws} for a 2L configuration with 15 km arm-length.

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table

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NS mass measurements in binary pulsar systems, sorted by increasing mass. Names followed by (C) indicate companions to radio pulsars that are consistent with being NSs. The names in blue show systems in which the companion could also be a massive WD. The error bars indicate the 68\% confidence intervals. The histogram on the right is based on the median of the mass PDFs. Figure reproduced from V. V. Krishnan and P. Freire; for up-to-date versions, see \url{https://www3.mpifr-bonn.mpg.de/staff/pfreire/NS_masses.html}.

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Cumulative distributions of BH measured from BBHs (yellow), low-mass X-ray binaries (blue) and high-mass X-ray binaries (magenta). The distribution of prior CDFs is shown by the unfilled black bands~\cite{Fishbach:2021xqi}.

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Distribution of spin periods of 485 radio MSPs with $P<10\;{\rm ms}$. Despite intense searches, no sub-ms MSP has been discovered thus far. ET may reveal such a sub-ms MSP in a GW merger -- see text. After~\cite{Tauris:2023sxf}.

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A flowchart to test the primordial nature of a binary. Green and red refer to the condition in the box being met or violated, respectively. Adapted from ref.~\cite{Franciolini:2021xbq}. The asterisks indicate that the condition is evaluated in the scenario where PBHs are formed in a radiation-dominated Universe, and a model-dependent alternative is possible.

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Redshift posteriors for sources with $(M_{\rm total}, z, q)=(40 \Msun,30,1)$ at $\hat{\iota}=0^{\circ}, 30^{\circ}, 60^{\circ}$ and $90^{\circ}$, obtained with a waveform with (blue) and without (red) higher-order modes and for a detector network composed of two CE and one ET (see ref.~\cite{Ng:2022vbz} for more details on the assumed detector network). The solid horizontal lines show the 95\% credible intervals, whereas the dashed lines mark the injected value $z=30$. The top axis shows the optimal SNR of the two waveforms. From ref.~\cite{Ng:2022vbz}.

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Left panel: Reconstructed merger rate at $z>10$ assuming ET+CE and two populations of events: astrophysical (Pop~III) and primordial. We consider the most conservative scenario in which both populations have the same mass function, we neglect PBH accretion, and we adopt an optimistic Pop~III merger rate~\cite{Belczynski:2016ieo}. Right panel: a null observation for a primordial population with ET+CE can be translated into an upper bound on the PBH abundance. Taken from ref.~\cite{Ng:2022agi}.

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Posterior distribution of the masses $(m_1,m_2)$ and the tidal deformability $\tilde{\Lambda}$ parameter for LVK O5 (left) and ET+2CE (right) (see ref.~\cite{Crescimbeni:2024cwh} for more details on the assumed detector network). The red lines represent the injected values of a typical subsolar event. In ET+2CE case, the upper bounds on $\tilde{\Lambda}$ would exclude neutron stars or more exotic alternatives \cite{Cardoso:2019rvt}. Taken from ref.~\cite{Crescimbeni:2024cwh}.

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Left panel: posterior predictive distribution for events within the NS range based on the LVK GWTC-3 catalog \cite{Franciolini:2022tfm}. Right panel: Distribution of $\chi_\text{\rm eff}$ expected for PBH binaries as a function of the total mass $M$ depending on the accretion efficiency (see \cite{DeLuca:2020bjf,DeLuca:2023bcr} for more details). We stress that this prediction applies to PBHs born in a radiation-dominated Universe or to alternative formation scenarios for which the natal spin is negligible, and therefore represents a weaker test of the primordial nature of the binary.

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Redshift evolution of the critical halo mass threshold above which Pop~III star formation can happen (left axis) and the BH mass range in BBH mergers detectable by ET (right axis). We show $M_{\rm h, crit}$ for efficient $\rm H_{2}$- and H-cooling from \cite{Trenti:2009cj} with the thin solid and dash-dot-dotted curves. We plot $M_{\rm h, crit}$ regulated by Lyman-Werner (LW) radiation and baryon-DM streaming motion in four cases based on \cite{Schauer:2020gvx} (see also \cite{Nebrin:2023yzm}), adopting different models for the LW background intensity $J_{21}$ (in the unit of $10^{-21}\ \rm erg\ s^{-1}\ cm^{-2}\ sr^{-1}$) and streaming velocities $v_{\rm bc}$ (in the unit of the cosmic root-mean-squared velocity). The thin dashed, dotted, and long-dashed curves show the results for $v_{\rm bc}=0$, 1, and 2, respectively, under the model of $J_{21}$ from \cite{Greif:2006nr,Hartwig:2022lon}, while $J_{21}$ is boosted by a factor 10 under $v_{\rm bc}=0$ for the dash-dotted curve. We also show the halo masses corresponding to 3- and 4-$\sigma$ density peaks for the standard $\Lambda$CDM model and a fuzzy DM (FDM) model given a boson particle mass of $m_{\rm a}c^{2}=5\times 10^{-21}\ \rm eV$ with the thick solid and dotted curves. For the right axis, the inner, intermediate, and outer shaded regions show the BH mass ranges in which more than 90\%, 50\%, and 10\% of \textit{equal-mass} BBH mergers can be detected at $\rm SNR > 9$ by ET.

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Illustrative examples of the Pop III stellar IMF predicted by three high-resolution models without stellar feedback \cite{Prole:2021nym, Wollenberg2020, Stacy:2012iz} and three simulations with radiative feedback included \cite{Jaura:2022sny, Hirano:2013lba, Hirano:2015wxa}. This figure is adapted from figure~6 in \cite{Klessen2023}.

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Detection rates of Pop~III BBH mergers from isolated binary stellar evolution by ET with $\rm SNR>9$ for different models of the initial statistics of Pop~III stellar binaries ($x$ axis) and cosmic star formation histories of Pop~III stars (denoted by different markers) adapted from Fig.~4 in \cite{Santoliquido:2023wzn}. The rates predicted by other studies for isolated mergers and dynamical mergers in dense star clusters are within the range shown here.

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Stellar mergers leading to the formation of an IMBH in one of the \textsc{Dragon-II} numerical simulations \citep{ArcaSedda:2023mlv}.

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Horizon redshift for IMBH-BH mergers assuming a companion mass of $30 \Msun$ (left panel) and $100 \Msun$ (right panel). From brighter to darker colours, contours encompass regions with an SNR $> 1,~5,~10,~15,~25,~50,~100,~500,~1,000$, respectively. Calculations are performed through the \textsc{GWFish} package \citep{Dupletsa:2022scg}, assuming a PhenomD waveform approximant.

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Horizon redshift for IMBH-BH mergers assuming a companion mass of $30 \Msun$ (solid line) and $100 \Msun$ (dashed line) for a fixed SNR$=15$ and comparing a 15km 2L ET configuration with $45\deg$ inclination (black) and a 10km triangle ET configuration (red and blue for the 30 and 100 M$_\odot$ companion, respectively). The sensitivity curves come from \cite{Branchesi:2023mws}. We used the \textsc{GWFish} package \citep{Dupletsa:2022scg}, assuming a PhenomD waveform approximant, i.e. the same as Fig.~\ref{fig:horimri}.

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Characteristic strain amplitude and frequency for IMBH-IMBH binaries with different masses, assuming a luminosity distance $d_L = 1$ Gpc. The signal is overlaid to the sensitivity curves of various detectors at low-frequency (LISA), mid-range (DECIGO), and high-frequency (ET, CE, AVirgo, and ALIGO).

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The joint accuracy on luminosity distance $(\Delta d_L/d_L)$ and angular resolution $(\Delta \Omega_{90\%})$ for BNS (left) and BBH (right). Green shows results from the triangular ET detector (10-km arms) with both high- and low-frequency instruments (HFLF). Light blue indicates forecasts for LVKI O5, the most advanced 2G detector network (LIGO Hanford, LIGO Livingston, Virgo, KAGRA, and LIGO India) \citep{KAGRA:2013rdx}. The dashed red line represents a 10 deg$^2$ sky localization error. Adapted from \citep{Branchesi:2023mws}.

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Background energy density spectrum for BBHs assuming four evolution channels, isolated (Field), Young Star Cluster (YSC), Globular Cluster (GC) and Nuclear Star Cluster (NSC). Uncertainties, depicted by the shaded areas, are derived from the merger rates given in table~2 of \cite{Mapelli:2021gyv}. The black dashed curve shows the sensitivity of ET for the triangular configuration.

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Background energy density spectra for BBHs born from Pop.~III stars. Fiducial, Optimistic and Pessimistic models are the LOG\_H22, LAR\_H22 and LAR\_J19 respectively populations models from \cite{Santoliquido:2023wzn}. The residuals correspond to $\rho_{\rm thrs.} > 20$, and the grey lines correspond to the background from Pop.~I/II channels (see section~\ref{subsec:BBHchannel}).

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{\it Left panel:} Merger rate evolution for representative astrophysical models and the PBH channel. As for the astrophysical models, CE, SMT, and GC refer to common envelope mass transfer, stable mass transfer, and globular clusters, respectively. The normalization comes from the Bayesian inference on the GWTC-2 catalog in ref.~\cite{Bavera:2020uch}. The blue band reports the local merger rate bound from LVK while the dashed gray band the star formation rate (SFR). {\it Right panel:} Total SGWB (black line) coming from the various channels shown in the left panel, while we report with colored lines their contributions. We do not show the residual SGWB. Figures adapted from ref.~\cite{Bavera:2021wmw}.

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The promise of multi-messenger astrophysics. While much astronomy to date has been undertaken in the electromagnetic bands, in particular in optical light, many extreme systems in the Universe emit light across the electromagnetic spectrum and also produce additional messengers in the form of cosmic rays, neutrinos, or gravitational waves. Understanding the details of these systems and the insight they offer into central questions in astronomy, cosmology, and fundamental physics is something that can be done via multi-messenger observations.

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The physical process ongoing in the merger of a binary system of neutron stars (or a neutron star and a black hole), their multi-messenger observational signals and the scientific insight enabled from them (updated from an original concept by \cite{Metzger:2016pju}). Gravitational wave observations provide robust measurement of the component masses, and constraints on their sizes and spins. Many possible electromagnetic signals are also possible, including the detection of associated GRBs, cocoon emission, and kilonova signatures.

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The spectroscopic evolution of kilonovae. The upper (colored) lines show the X-shooter spectral sequence for AT2017gfo \cite{Pian:2017gtc,Smartt:2017fuw}, demonstrating the early blue emission with the subsequent transition to a much redder spectrum. The lower (grey/black) lines show much later time spectra of the kilonova in GRB 230307A (AT 2023vfi) obtained with {\em JWST} \cite{JWST:2023jqa}. The spectra contain numerous spectra features both early in absorption and later in emission, and these have been linked to several different $r$-process elements.

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Representation of two of our reference multi-messenger synthetic populations in a geocentric Universe. Compact binaries in the synthetic populations of C22 and C23 that are detected by ET-$\Delta$ are represented by yellow (BNS -- left semicircle) and orange (BHNS -- right semicircle) dots. The distance of each dot from the Earth (blue circle) is proportional to the redshift of the corresponding compact binary. If the simulated merger produces a relativistic jet whose prompt ($>4$ ph cm$^{-2}$ s$^{-1}$) or afterglow emission ($>0.01$ mJy, radio band) is detectable, a cyan jet is plotted centered on the dot, with its axis inclined by the actual viewing angle with respect to the line of sight to the Earth. If a kilonova is also produced, and if it is detectable ($<26$ mag, $g$ band), then a red butterfly shape is also plotted. The total number of binaries is representative of one year of ET operation.

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Schematic representation of BNS and BHNS mergers and their multi-messenger emissions.

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Distribution of ET optical KNe as a function of time assuming the BNS population from L24 (red, first row) and C22 (blue, central row), and the BHNS population from C23 (green, last row). The left column assumes the ET-$\Delta$ configuration, the right column is the 2L configuration. The plots represent the apparent AB magnitude versus post-merger time (days) for our simulated KN light curves in the $g$ (484 nm) band, restricting to ET GW-detectable sources with $\Delta\Omega_{{\rm 90}\%}<100\,\mathrm{deg}^2$. The shaded regions contain $50\%$, $90\%$, and $99\%$ of the KN light curves.

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Detection rate as a function of detection limit threshold for the KN BNS population presented in L24 and C22 (upper panels, red and blue, respectively) and for the BHNS population from C23 (lower panels, green). The left column assumes the ET-$\Delta$ configuration, and the right column is the 2L configuration. The lines indicate the KN+GW ($g$ band) detectable binaries, assuming all the BNSs (solid line from C22 upper panels and C23 lower panels), the ones with $\Delta\Omega_{{\rm 90}\%}<100\,\mathrm{deg}^2$ (dashed lines) and the ones with $\Delta\Omega_{{\rm 90}\%}<40\,\mathrm{deg}^2$ (dotted lines). The shaded region indicates the uncertainty due to the local merger rate.

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Observed spectral energy distribution peak photon energy $E_\mathrm{p,obs}$ vs photon flux for the GRB prompt BNS population presented in R22 (first row) and C22 (central row), and for the BHNS population from C23 (last row). The colored-filled regions contain 50\%, 90\%, and 99\% of the binaries, both GRB Prompt and ET detectable. The black lines contain 50\%, 90\%, and 99\% (solid, dashed, and dotted, respectively) of the ET-detectable binaries.

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Detection rate as a function of detection limit threshold for the GRB prompt BNS population presented in R22 and C22 (upper panels, red and orange, respectively) and for the BHNS population from C23 (lower panel). The left column assumes the ET delta configuration, and the right column assumes the 2L configuration. The lines indicate the GRB prompt+GW detectable binaries, assuming all the BNSs (solid lines), the ones with $\Delta\Omega_{{\rm 90}\%}<100\,\mathrm{deg}^2$ (dashed lines) and the ones with $\Delta\Omega_{{\rm 90}\%}<10\,\mathrm{deg}^2$ (dotted lines). The shaded region indicates the uncertainty at $90\%$ credible level. For C22 and C23, it indicates the uncertainty due to the local merger rate.

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Afterglow lightcurves in Radio (red), Optical (green) and X-ray (blue), for the delta (left) and 2L (right) configurations, assuming the BNS population from R22 (first row) and from C22 (central row), and the BHNS population from C23 (last row). The dashed and solid lines are the 50$\%$ and 90$\%$ containment regions of the afterglow peaks, in the respective bands. The grey lines in the background are randomly sampled optical light curves.

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Detection rate as a function of detection limit threshold for the GRB afterglow BNS population presented in R22 (first row) and C22 (central row), and for the BHNS population from C23 (last row). The left column assumes the ET delta configuration, the right column the 2L configuration. The lines indicate the GRB afterglow+GW detectable binaries in the radio (red), optical (green) and X (blue) bands, assuming all the BNSs (solid line), the ones with $\Delta\Omega_{{\rm 90}\%}<100\,\mathrm{deg}^2$ (dashed lines) and the ones with $\Delta\Omega_{{\rm 90}\%}<10\,\mathrm{deg}^2$ (dotted lines). The shaded region indicates the uncertainty at $90\%$ credible level. For C22 and C23, it indicates the uncertainty due to the local merger rate.

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Distribution in redshift of the sky localizations of BNS events (upper panel) and BHNS events (lower panel) that can power and EM counterpart, for the delta (left) and 2L (right) configurations. The line indicates the median value, the colored bands are the 50$\%$, 90$\%$ and 99$\%$ credible regions.

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Detection rates as a function of the detection limit threshold for the KN BNS population from C22 (left panel) and the BHNS population from C23 (right panel). The different colors represent various ET configurations and detector networks: blue (or green) for ET-$\Delta$, red for ET2L, gray for ET-$\Delta$+2CE, and black for ET2L+2CE. The lines correspond to KN+GW ($g$-band) detectable binaries: solid lines indicate all BNSs (or BHNSs), dashed lines represent events with $\Delta\Omega_{{\rm 90}\%}<100\mathrm{deg}^2$, and dotted lines represent events with $\Delta\Omega_{{\rm 90}\%}<40\mathrm{deg}^2$. The shaded region illustrates the uncertainty arising from the local merger rate.

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Example of redshift distribution of simulated KNe detected by Vera Rubin and WST, associated with BNS detected by ET in the 2L (top panels) and 2L + CE(40 km) (bottom panel) configurations, and having error regions of $<40$\,deg$^2$ (top left and bottom panles), and $<100$\,deg$^2$ (top right panel). The background distribution in white corresponds to the parent BNS+KN population. The green colored distribution corresponds to the KNe that are detectable with WST IFS (SNR $> 3$). KNe detectable with Vera Rubin are shown in grey. Black points refer to the y-axis scale on the right-hand side and show the fraction of detectable KN with WST with respect to those that are detectable with Vera Rubin. From Bisero et al., in preparation.

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Redshift distribution of the sky-localization uncertainty (given as 90\% credible region) for ET and ET+CE configurations. The panels show the BNS detections and the corresponding sky-localizations as a function of the redshift 15, 5, and 1 minute(s) before the merger. The blue histogram represents all detected sources, while the other colors indicate sources with sky localizations more precise than 10$^3$\,deg$^2$ (orange), 10$^2$\,deg$^2$ (green), and 10\,deg$^2$ (red). Adapted from \cite{Banerjee:2023}.

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Characteristics of most advanced EM facilities expected to be available by the late 2030s. Also available as an Excel table at \url{https://apps.et-gw.eu/tds/ql/?c=17760}. Asterisk indicates facilities and missions that are still in the project phase and have not yet been approved or adopted.

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Requirements to detect and study the EM counterparts of BNS and BHNS detected by ET, for EM facilities working in different ranges of the EM spectrum, for on-axis GRB. Also available as an Excel table at \url{https://apps.et-gw.eu/tds/ql/?c=17771}.

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Requirements to detect and study the EM counterparts of BNS and BHNS detected by ET, for EM facilities working in different ranges of the EM spectrum, for off-axis afterglow. Also available as an Excel table at \url{https://apps.et-gw.eu/tds/ql/?c=17771}.

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Requirements to detect and study the EM counterparts of BNS and BHNS detected by ET, for EM facilities working in different ranges of the EM spectrum, for KN. Also available as an Excel table at \url{https://apps.et-gw.eu/tds/ql/?c=17771}.

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The GW landscape --- a schematic representation of synergistic sources in the GW frequency-amplitude plane. The figure shows the sensitivities for ET and CE (including the current LVK detectors) at higher frequencies, and for LISA, Taiji, Tian Qin, DECIGO and LGWA at lower frequencies (sensitivities for the latter are from \cite{Colpi:2024xhw, Ajith:2024mie, Wang:2020vkg, Seto:2001qf, Kawamura:2020pcg, Li:2023umb}). Tracks (left to right) represent: an equal mass black hole binary merger (labelled SMBH) of $10^6\,\Msun$ at $z\sim 6;$ an equal-mass intermediate mass black hole merger (IMBHB) of $ 10^5 \Msun$ at $z=1.5$; a multi-band source hosting an inspiralling equal-mass stellar black hole binary (BBH) of $ 60\Msun$ at $z\sim 0.1$ merging in the ET bandwidth; an intermediate mass ratio inspiral (IMRI) hosting a stellar black of $30\,\Msun$ orbiting around a black hole of $5\times 10^3\,\Msun$ at $z=1$. The ``BBH@z=10" track denote mergers at $z=10$ of equal-mass black hole binaries forming in metal poor early galaxies, with a total mass, from bottom to top, of $20,~60,~100\, \Msun$. Squares, pentagons and diamonds refer to Galactic compact binaries detected by LISA, millions to billions of years away from merging. DLS indicates a dual line system. Each section of this chapter presents each of these sources and their scientific opportunities in detail.

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Detector horizons for the two baseline configurations of ET, alone and in conjunction with a network of two CE as described in the legend. The shaded bands, reported only for the combinations with CE, denote the range within which $50\%$ of the sources are detected, when averaging over sky location. Known classes of target sources are reported for illustration: stellar-mass black holes (SOBBH) and double neutron star systems (catalogs from ~\cite{Branchesi:2023mws}), neutron star-black hole binaries (catalogs from \cite{Iacovelli:2022bbs} based on~\cite{Broekgaarden:2021iew,Giacobbo:2018etu,Zhu:2020ffa}), Population III stars~\cite{Costa:2023xsz,Santoliquido:2023wzn}, and intermediate mass black holes (IMBH) \cite{Colpi:2024xhw}. The orange band is representative of a population of high--redshift primordial black holes (PBHs) ~\cite{Franciolini:2021tla,Ng:2022agi}. Triangles show the events detected by the LVK collaboration~\cite{KAGRA:2021vkt}.

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\small Comparison of SNR and parameter estimation error for the chirp mass, angular localization and luminosity distance for BBHs. We show the results for ET alone, for ET +1CE and for ET+2CE and, in all cases, for ET we consider both the triangle 10~km and the 2L-$45^{\circ}$ configurations. The results are obtained with the \texttt{GWfast} code~\cite{Iacovelli:2022mbg}; technical details as in \cite{Branchesi:2023mws}.

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\small As in figure~\ref{fig:cumul_SNR-Mc-Om-dL_BBH_allconf_ETCE}, for BNS.

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The cumulative number of GW detections (vertical axis) for which it will be possible to constrain NS radii below a specified threshold (horizontal axis) is shown. The lines represent different GW detector configurations: HLET, 20LA, and 40LA indicate networks with one next-generation detector, while 4020A, 20LET, and 40LET represent networks with two next-generation observatories. Figure reproduced from \cite{Gupta:2023lga}.

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Posterior probability density for $\tilde{\Lambda}$ (see \eq{deftildeLambdadiv6} for definition) for sources at 68 Mpc observed by the ET+CE network, recovered with three different models NO-PM (without post-merger phase), QU-PM (with post-merger phase and employing quasi-universal relation), and Free-PM (with post-merger phase and its parameters freed from that of the inspiral phase), in blue, orange, and green, respectively. The black dashed lines correspond to the injected values. Figure from \cite{Puecher:2022oiz}.

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Posterior probability density for $c_1$, one of the post-merger phase Lorentzian parameter for different detector networks. The dashed vertical line indicates the injected value. Figure from ~\cite{Puecher:2022oiz}.

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Localization capabilities for BBH (left panels) and BNS (right panels) of ET in its triangular and 2L configurations (first and second rows respectively), in a network with two CE. The color scale denotes the number of galaxies expected in the $90\%$ localization volume. Events marked with black dots are localized to one galaxy only.

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\small {{\bf Upper left panel}: Tracks of BH growth, for different initial {\it seed} masses assuming uninterrupted accretion at the Eddington rate and at rates twice above this limit. The different colored symbols labeled in the figure refer to $z>6$ observations of quasars by ALMA (green squares \cite{Neeleman:2019knu}) and JWST (red diamonds ~\cite{2024ApJ...966..176Y, 2024ApJ...964...90S, 2023ApJ...959...39H, Maiolino:2023bpi, 2023Natur.621...51D}), including the $z\sim 10$ massive black holes in UHZ$1$ (magenta circle \cite{Bogdan:2023ilu}), GHZ$9$ (black cross \cite{Kovacs:2024zfh}) and the candidate super-Eddington accreting black hole in GN$-$z$11$ (violet hexagon \cite{Maiolino:2023zdu}). Light seeds can explain these quasars if they accrete at a super-Eddington rate. Heavy seeds, forming at lower redshifts, do not necessarily need sustained super-Eddington accretion. There is a debate whether sustained accretion above the Eddington limit is likely in high-redshift galaxies~\cite{Pezzulli:2017ikf, 2019MNRAS.486.3892R, Trinca:2022txs, Massonneau:2022uwg, Lupi:2023oji, Greene:2024phl}. The black, blue and red tracks refer to state-of-the-art high-resolution cosmological zoom-in simulations~\cite{Lupi:2023oji, Lupi:2019jgo} where a $10^5\Msun$ black hole was implanted at $z=10$ and let evolve including accretion and feedback. {\bf Upper right panel}: Cartoon sketching the mass spectrum of black holes and the different formation paths. Arrows and labels indicate selected quasars, the Milky Way black hole, Sgr A$^*,$~\cite{Genzel:2010zy, EventHorizonTelescope:2022wkp} M$87$~\cite{EventHorizonTelescope:2019pgp}, and the post-merger mass of GW$150914$ and GW$190521$, the first and the most massive coalescence events detected to date by LVK, respectively~\cite{LIGOScientific:2016wkq, LIGOScientific:2020iuh}. Shown are the windows of exploration of ET and LISA. {\bf Lower left panel}: BH mass spectrum at $z=4$ from \cite{Trinca:2022txs} as inferred from a suite of semi-analytical models and numerical simulations to show the still large theoretical uncertainties in predicting BH evolution. Data points show the distribution of light (magenta) and heavy (red, orange) BH seeds, with the best fit for the distribution of heavy seeds in red and violet dotted lines, and the results from large-scale cosmological simulations: Illustris (dark green, cyan, and blue \cite{Sijacki:2014yfa, weinberger2017}), SIMBA (olive \cite{Dave:2019yyq}), and EAGLE (light green \cite{McAlpine:2018dua}). The observational constraints by \cite{Merloni:2008hx} and \cite{Shankar:2007zg} are shown in the grey shaded area and black dashed-dotted lines. {\bf Lower right panel}: evolution of the BH mass function from the semi-analytical model L-Galaxies \cite{Izquierdo-Villalba:2023ypb}. The figure shows how the intermediate-mass range is being progressively filled due to the growth of light seeds from population III relics (Courtesy of Izquierdo Villalba). }

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The ET and LISA cosmic horizons. The figure shows contour lines of constant signal-to-noise ratio (SNR) in the plane redshift versus total mass of the black hole binaries, as measured in the source frame. Binaries have all mass ratio 0.5. Yellow dots denote loci of BH mergers extracted from the semi-analytical model of \cite{Valiante:2020zhj}, which tracks the evolution of light and heavy seeds forming in merging halos. Light seeds form at $z\sim 20-30$ as relics of population III stars in molecular cooling dark matter halos of $10^6\,\Msun$. Heavy seeds form later ($z\sim 10-20$) in atomic cooling halos of $10^8\,\Msun$ under contrived conditions (see the review by \cite{Inayoshi:2019fun} and references therein). The bulk of the merging black holes in LISA come from the growing light seed population that ET can detect when seeds form in binaries. Red dots and blue triangles denote the quasars observed at $4 < z < 11$, from JWST \cite{2024ApJ...966..176Y, 2024ApJ...964...90S, 2023ApJ...959...39H, Maiolino:2023bpi, 2023Natur.621...51D} and ALMA\cite{Neeleman:2019knu}, respectively. Green contour lines depicting the vast population of AGN are from \cite{Shen:2010aa}.

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\small {\bf Left panel}: horizon redshift for IMRIs involving a stellar BH of $30\Msun$ as a function of the mass of the more massive primary $m_{\rm IMBH}$. Detection is set at SNR=15 using the \textsc{GWFish} tool \cite{Dupletsa:2022scg} for ET (green straight curve), LGWA (blue dashed curve), and LISA (purple dotted curve). {\bf Right panel}: cumulative mass distribution of IMRI detection versus the IMBH mass. These systems are dynamically formed in young, globular, and nuclear clusters and their overall population is shown by the filled grey histogram. The detectable sub-population of IMRIs are shown for ET, LGWA and LISA (as in the legend). Models are obtained from combined population synthesis simulations performed with the \textsc{BPop} code \cite{Sedda:2021vjh} and parameter estimation performed with \textsc{GWFish}. Adapted from Arca Sedda (in prep).

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\small {\bf Left panel:} Examples of SGWB from phase transitions (PT). {\bf Middle panel:} Examples of SGWB from cosmic strings (CS). {\bf Right panel:} Examples of SGWB from cosmic inflation. Besides LISA and ET sensitivity curves (PSD) and the corresponding PLS (see \eq{eq: PLS def} and section~\ref{sec:div9_PLSdefinition} for definitions), in blue and green lines are represented the benchmark scenarios discussed in each of the aforementioned sections.