# Science with the Einstein Telescope: a comparison of different designs

Marica Branchesi, Michele Maggiore, David Alonso, Charles Badger, Biswajit Banerjee, Freija Beirnaert, Swetha Bhagwat, Guillaume Boileau, Ssohrab Borhanian, Daniel David Brown, et al.

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Marica Branchesi, Michele Maggiore, David Alonso, Charles Badger, Biswajit Banerjee, et al. Science with the Einstein Telescope: a comparison of different designs. JCAP, 2023, 07, pp.068. 10.1088/1475-7516/2023/07/068. hal-04062419

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# Science with the Einstein Telescope: a comparison of different designs

To cite this article: Marica Branchesi et al. JCAP07(2023)068

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Science with the Einstein Telescope: a
comparison of different designs

Marica Branchesi,<fnref n="1" />,<fnref n="2" />,* Michele Maggiore,³,⁴,* David Alonso,⁵

Charles Badger,⁶ Biswajit Banerjee,<fnref n="1" />,<fnref n="2" /> Freija Beirnaert,⁷

Enis Belgacem,³,⁴ Swetha Bhagwat,⁸,⁹ Guillaume Boileau,¹⁰,¹¹

Ssohrab Borhanian,¹² Daniel David Brown,¹³ Man Leong Chan,¹⁴

Giulia Cusin,¹⁵,³,⁴ Stefan L. Danilishin,¹⁶,¹⁷ Jerome Degallaix,¹⁸

Valerio De Luca,¹⁹ Arnab Dhani,²⁰ Tim Dietrich,²¹,²²

Ulyana Dupletsa,<fnref n="1" />,<fnref n="2" /> Stefano Foffa,³,⁴ Gabriele Franciolini,⁸

Andreas Freise,²³,¹⁶ Gianluca Gemme,²⁴ Boris Goncharov,<fnref n="1" />,<fnref n="2" />

Archisman Ghosh,⁷ Francesca Gulminelli,²⁵ Ish Gupta,²⁰

Pawan Kumar Gupta,¹⁶,²⁶ Jan Harms,<fnref n="1" />,<fnref n="2" /> Nandini Hazra,<fnref n="1" />,<fnref n="2" />,²⁷

Stefan Hild,¹⁶,¹⁷ Tanja Hinderer,²⁸ Ik Siong Heng,²⁹

Francesco Iacovelli,³,⁴ Justin Janquart,¹⁶,²⁶ Kamiel Janssens,¹⁰,¹¹

Alexander C. Jenkins,³⁰ Chinmay Kalaghatgi,¹⁶,²⁶,³¹

Xhesika Koroveshi,³²,³³ Tjonnie G.F. Li,³⁴,³⁵ Yufeng Li,³⁶

Eleonora Loffredo,<fnref n="1" />,<fnref n="2" /> Elisa Maggio,²² Michele Mancarella,³,⁴,³⁷,³⁸

Michela Mapelli,³⁹,⁴⁰,⁴¹ Katarina Martinovic,⁶ Andrea Maselli,<fnref n="1" />,<fnref n="2" />

Patrick Meyers,⁴² Andrew L. Miller,⁴³,¹⁶,²⁶ Chiranjib Mondal,²⁵

Niccolò Muttoni,³,⁴ Harsh Narola,¹⁶,²⁶ Micaela Oertel,⁴⁴

Gor Oganesyan,<fnref n="1" />,<fnref n="2" /> Costantino Pacilio,⁸,³⁷,³⁸ Cristiano Palomba,⁴⁵

Paolo Pani,⁸ Antonio Pasqualetti,⁴⁶ Albino Perego,⁴⁷,⁴⁸

Carole Périgois,³⁹,⁴⁰,⁴¹ Mauro Pieroni,⁴⁹,⁵⁰

Ornella Juliana Piccinni,⁵¹ Anna Puecher,¹⁶,²⁶ Paola Puppo,⁴⁵

Angelo Ricciardone,⁵²,³⁹,⁴⁰ Antonio Riotto,³,⁴ Samuele Ronchini,<fnref n="1" />,<fnref n="2" />

Mairi Sakellariadou,⁶ Anuradha Samajdar,²¹

Filippo Santoliquido,³⁹,⁴⁰,⁴¹ B.S. Sathyaprakash,²⁰,⁵³,⁵⁴

Jessica Steinlechner,¹⁶,¹⁷ Sebastian Steinlechner,¹⁶,¹⁷

Andrei Utina,¹⁶,¹⁷ Chris Van Den Broeck¹⁶,²⁶ and Teng Zhang⁹,¹⁷

<figure><img src="image_2.png" /></figure>

<footnote n="1">¹Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy</footnote>

<footnote n="2">²INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy</footnote>

<footnote>*Corresponding author.</footnote>
³Département de Physique Théorique, Université de Genève,
24 quai Ansermet, CH-1211 Genève 4, Switzerland

⁴Gravitational Wave Science Center (GWSC), Université de Genève,
CH-1211 Geneva, Switzerland

⁵Department of Physics, University of Oxford, Denys Wilkinson Building,
Keble Road, Oxford OX1 3RH, United Kingdom

⁶Theoretical Particle Physics and Cosmology Group, Physics Department,
King's College London, University of London, Strand, London WC2R 2LS, United Kingdom

⁷Department of Physics and Astronomy, Ghent University, 9000 Ghent, Belgium

⁸Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1,
Piazzale Aldo Moro 5, 00185, Roma, Italy

⁹Institute for Gravitational Wave Astronomy & School of Physics and Astronomy,
University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K.

¹⁰Universiteit Antwerpen, Prinsstraat 13, 2000 Antwerpen, Belgium

¹¹Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, ARTEMIS,
06304 Nice, France

¹²Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena,
07743, Jena, Germany

¹³ARC Centre of Excellence for Gravitational Wave Discovery, School of Physical Sciences,
University of Adelaide, 5005 Adelaide, Australia

¹⁴Department of Physics and Astronomy, The University of British Columbia,
Vancouver, BC V6T 1Z4, Canada

¹⁵Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris,
75014 Paris, France

¹⁶Nikhef — National Institute for Subatomic Physics,
Science Park 105, 1098 XG Amsterdam, The Netherlands

¹⁷Department of Gravitational Waves and Fundamental Physics,
Maastricht University, 6200 MD Maastricht, The Netherlands

¹⁸Université Lyon, Université Claude Bernard Lyon 1, CNRS,
Laboratoire des Matériaux Avancés (LMA), IP2I Lyon/IN2P3, F-69622 Villeurbanne, France

¹⁹Center for Particle Cosmology, Department of Physics and Astronomy,
University of Pennsylvania 209 S. 33rd St., Philadelphia, PA 19104, U.S.A.

²⁰Institute for Gravitation and the Cosmos, Department of Physics,
The Pennsylvania State University, University Park, PA 16802, U.S.A.

²¹Institut für Physik und Astronomie, Universität Potsdam,
Haus 28, Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany

²²Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
D-14476 Potsdam, Germany

²³Department of Physics and Astronomy, VU Amsterdam,
De Boelelaan 1081, 1081, HV, Amsterdam, The Netherlands

²⁴INFN, Sezione di Genova, I-16146 Genova, Italy

²⁵Normandie Univ, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France

²⁶Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University,
Princetonplein 1, 3584 CC Utrecht, The Netherlands

²⁷INAF — Osservatorio Astronomico d’Abruzzo, Teramo, Italy
<sup>28</sup>Institute for Theoretical Physics, Utrecht University,
Princetonplein 5, 3584 CC Utrecht, The Netherlands

<sup>29</sup>SUPA, School of Physics and Astronomy, University of Glasgow,
Glasgow G12 8QQ, United Kingdom

<sup>30</sup>Department of Physics and Astronomy, University College London,
London WC1E 6BT, United Kingdom

<sup>31</sup>Institute for High-Energy Physics, University of Amsterdam,
Science Park 904, 1098 XH Amsterdam, The Netherlands

<sup>32</sup>Institute of Technical Thermodynamics and Refrigeration: Refrigeration and Cryogenics,
Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany

<sup>33</sup>Institute of Beam Physics and Technology, Karlsruhe Institute of Technology (KIT),
76344 Eggenstein-Leopoldshafen, Germany

<sup>34</sup>Institute for Theoretical Physics, Department of Physics and Astronomy, KU Leuven,
Celestijnenlaan 200D, B-3001 Leuven, Belgium

<sup>35</sup>STADIUS, Department of Electrical Engineering (ESAT), KU Leuven,
Kasteelpark Arenberg 10, B-3001 Leuven, Belgium

<sup>36</sup>Department of Astronomy, Beijing Normal University, Beijing 100875, China

<sup>37</sup>Dipartimento di Fisica "G. Occhialini", Università degli Studi di Milano-Bicocca,
Piazza della Scienza 3, 20126 Milano, Italy

<sup>38</sup>INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy

<sup>39</sup>Physics and Astronomy Department Galileo Galilei, University of Padova,
Vicolo dell'Osservatorio 3, I-35122, Padova, Italy

<sup>40</sup>INFN — Padova, Via Marzolo 8, I-35131 Padova, Italy

<sup>41</sup>INAF — Osservatorio Astronomico di Padova,
Vicolo dell'Osservatorio 5, I-35122 Padova, Italy

<sup>42</sup>Theoretical Astrophysics Group, California Institute of Technology,
Pasadena, CA 91125, U.S.A.

<sup>43</sup>Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

<sup>44</sup>Laboratoire Univers et Théories, CNRS, Observatoire de Paris, Université PSL,
Université Paris Cité, 5 place Jules Janssen, 92195 Meudon, France

<sup>45</sup>INFN, Sezione di Roma, I-00185 Roma, Italy

<sup>46</sup>European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy

<sup>47</sup>Dipartimento di Fisica, Università di Trento, Via Sommarive 14, 38123 Trento, Italy

<sup>48</sup>INFN-TIFPA, Trento Institute for Fundamental Physics and Applications,
Via Sommarive 14, I-38123 Trento, Italy

<sup>49</sup>Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland

<sup>50</sup>Blackett Laboratory, Imperial College London,
South Kensington Campus, London, SW7 2AZ, U.K.

<sup>51</sup>Institut de Fisica d'Altes Energies (IFAE), The Barcelona Institute of Science
and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain

<sup>52</sup>Dipartimento di Fisica "E. Fermi", Università di Pisa, I-56127 Pisa, Italy

<sup>53</sup>Department of Astronomy and Astrophysics, The Pennsylvania State University,
University Park, PA 16802, U.S.A.

<sup>54</sup>School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, U.K.
E-mail: marica branchesi gssi it, michele maggiore unige ch

Received April 14, 2023

Revised June 15, 2023

Accepted June 17, 2023

Published July 28, 2023

**Abstract.** The Einstein Telescope (ET), the European project for a third-generation gravitational-wave detector, has a reference configuration based on a triangular shape consisting of three nested detectors with 10 km arms, where each detector has a ‘xylophone’ configuration made of an interferometer tuned toward high frequencies, and an interferometer tuned toward low frequencies and working at cryogenic temperature. Here, we examine the scientific perspectives under possible variations of this reference design. We perform a detailed evaluation of the science case for a single triangular geometry observatory, and we compare it with the results obtained for a network of two L-shaped detectors (either parallel or misaligned) located in Europe, considering different choices of arm-length for both the triangle and the 2L geometries. We also study how the science output changes in the absence of the low-frequency instrument, both for the triangle and the 2L configurations. We examine a broad class of simple ‘metrics’ that quantify the science output, related to compact binary coalescences, multi-messenger astronomy and stochastic backgrounds, and we then examine the impact of different detector designs on a more specific set of scientific objectives.

**Keywords:** gravitational wave detectors, gravitational waves / experiments, gravitational waves / sources

ArXiv ePrint: 2303.15923
# Contents

1 Introduction 1

2 Detector geometries and sensitivity curves 3

3 Coalescence of compact binaries 9

3.1 Binary Black Holes 11

3.1.1 Comparison between geometries 11

3.1.2 Effects of a change in the ASD 13

3.1.3 Golden events 15

3.2 Binary Neutron Stars 23

3.2.1 Comparison between geometries 23

3.2.2 Effects of a change in the ASD 24

3.2.3 Golden events 25

3.2.4 Dependence on the population model 25

3.3 ET in a network of 3G detectors 34

4 Multi-messenger astrophysics 39

4.1 BNS sky-localization and pre-merger alerts 39

4.2 Gamma-ray bursts: joint GW and high-energy detections 43

4.2.1 Prompt emission 44

4.2.2 Afterglow: survey and pointing modes 46

4.3 Kilonovae: joint GW and optical detections 48

5 Stochastic backgrounds 52

5.1 Sensitivity to isotropic stochastic backgrounds 54

5.2 Angular sensitivity 56

5.3 Astrophysical backgrounds 57

5.4 Impact of correlated magnetic, seismic and Newtonian noise 60

5.4.1 Seismic and Newtonian Noise 61

5.4.2 Magnetic noise 63

6 Impacts of detector designs on specific science cases 67

6.1 Physics near the BH horizon 67

6.1.1 Testing the GR predictions for space-time dynamics near the horizon 68

6.1.2 Searching for echoes and near-horizon structures 71

6.1.3 Constraining tidal effects and multipolar structure 73

6.2 Nuclear physics 75

6.2.1 Radius estimation from Fisher-matrix computation 76

6.2.2 Full parameter estimation results 79

6.2.3 Connected uncertainty of nuclear-physics parameters 81

6.2.4 Postmerger detectability 83

6.2.5 Conclusions: nuclear physics with ET 84

6.3 Population studies 85

6.3.1 Merger rate reconstruction 85
<table>
  <tbody>
    <tr>
      <td>6.3.2</td>
      <td>Constraints on PBHs from high-redshift mergers</td>
      <td>87</td>
    </tr>
    <tr>
      <td>6.3.3</td>
      <td>Other PBH signatures</td>
      <td>90</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Cosmology</td>
      <td>94</td>
    </tr>
    <tr>
      <td>6.4.1</td>
      <td>Hubble parameter and dark energy from joint GW/EM detections</td>
      <td>94</td>
    </tr>
    <tr>
      <td>6.4.2</td>
      <td>Hubble parameter and dark energy from BNS tidal deformability</td>
      <td>106</td>
    </tr>
    <tr>
      <td>6.4.3</td>
      <td>Hubble parameter from high-mass ratio events</td>
      <td>109</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>Cosmological stochastic backgrounds</td>
      <td>112</td>
    </tr>
    <tr>
      <td>6.5.1</td>
      <td>Cosmic strings</td>
      <td>113</td>
    </tr>
    <tr>
      <td>6.5.2</td>
      <td>First-order phase transition</td>
      <td>113</td>
    </tr>
    <tr>
      <td>6.5.3</td>
      <td>Source separation</td>
      <td>114</td>
    </tr>
    <tr>
      <td>6.6</td>
      <td>Continuous waves</td>
      <td>116</td>
    </tr>
    <tr>
      <td>6.6.1</td>
      <td>CWs from spinning neutron stars</td>
      <td>117</td>
    </tr>
    <tr>
      <td>6.6.2</td>
      <td>Transient CWs</td>
      <td>121</td>
    </tr>
    <tr>
      <td>6.6.3</td>
      <td>Search for dark matter with CWs</td>
      <td>123</td>
    </tr>
    <tr>
      <td>6.6.4</td>
      <td>Conclusions</td>
      <td>125</td>
    </tr>
    <tr>
      <td>7</td>
      <td>The role of the null stream in the triangle-2L comparison</td>
      <td>126</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Summary</td>
      <td>129</td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Comparison of different geometries</td>
      <td>130</td>
    </tr>
    <tr>
      <td>8.1.1</td>
      <td>Comparison between 15 km 2L and 10 km triangle</td>
      <td>130</td>
    </tr>
    <tr>
      <td>8.1.2</td>
      <td>Comparison between 15 km 2L and 15 km triangle</td>
      <td>134</td>
    </tr>
    <tr>
      <td>8.1.3</td>
      <td>A single L-shaped detector</td>
      <td>135</td>
    </tr>
    <tr>
      <td>8.1.4</td>
      <td>The null stream</td>
      <td>135</td>
    </tr>
    <tr>
      <td>8.1.5</td>
      <td>Further aspects of the triangle-2L comparison</td>
      <td>137</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>The role of the low-frequency sensitivity</td>
      <td>138</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Conclusions</td>
      <td>142</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Basic formalism for stochastic backgrounds</td>
      <td>144</td>
    </tr>
    <tr>
      <td>B</td>
      <td>Sensitivity to stochastic backgrounds of misaligned 2L configurations</td>
      <td>147</td>
    </tr>
    <tr>
      <td>C</td>
      <td>Tables of figures of merit for BBHs and BNSs</td>
      <td>148</td>
    </tr>
    <tr>
      <td>D</td>
      <td>Correlation between parameters for typical events</td>
      <td>153</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

Thanks to the extraordinary discoveries of the LIGO-Virgo Collaboration (LVC) in the last few years [1–7], the field of gravitational waves (GWs) is blossoming. After three observing runs the current catalog of GW detections contains about 90 binary black hole (BBH) coalescences, which in the last run have been detected on a weekly basis, as well as two binary neutron stars (BNSs) and two neutron star-black hole (NSBH) binaries [5, 6]. These discoveries are already having a significant impact on our understanding of the population properties (see, e.g. ref. [7]) and provide first results on fundamental physics and cosmology (see, e.g. ref. [8, 9]). The tremendous potential of combining multi-messenger observations including gravitational waves has been shown by GW170817, the first gravitational-wave observation from the merger of a binary neutron-star system which was detected in all the electromagnetic bands
from gamma rays (GRB 170817A), X-ray, ultraviolet-optical-near infrared (AT2017gfo), to radio [4]. This event had important implications in many fields of astrophysics, from relativistic astrophysics, nuclear physics, nucleosynthesis in the Universe, and cosmology.

However, the current infrastructures have intrinsic limitations, and for more than a decade the European community has been preparing the jump toward a ‘third-generation’ (3G) European GW detector. The Einstein Telescope (ET) [10–12] is the European observatory designed to detect gravitational-wave sources along the cosmic history up to the early Universe. The U.S. community is participating in the efforts toward 3G detectors, with the Cosmic Explorer (CE) project [13, 14]. Third-generation detectors such as ET and CE will provide an improvement in sensitivity by one order of magnitude and a significant enlargement of the bandwidth, both toward low and high frequencies, and will have extraordinary potential for discoveries in astrophysics, cosmology, and fundamental physics. Building on many previous works, the science case for ET has been summarized in [15, 16], while a recent comprehensive study of the ET capabilities can be found in [17] and, for multi-messenger observations, in [18]. For CE and for a more general discussion of 3G detectors see also [19, 20].

The current design of ET is based on several innovative concepts, with respect to the (second-generation) LIGO-Virgo detectors. In particular, ET is currently planned to be a single observatory, located 200-300 meters underground, in order to significantly reduce seismic noise; to have a triangular shape, consisting of three nested detectors, providing redundancy, the possibility of resolving the GW polarizations, and a null stream, i.e. a combination of outputs where the GW signal cancels, that can act as a veto against disturbances; and it will feature a ‘xylophone’ configuration, in which each of the three detectors actually consists of two interferometers, one tuned toward high frequencies, and therefore using high laser power, and one tuned toward low-frequency, working at cryogenic temperatures and low laser power. The first ET Conceptual Design Report dates back to 2011, and laid down the basic structure mentioned above. It has been updated in 2020, in the context of the successful proposal for including ET in the ESFRI Roadmap, the roadmap of large scientific European infrastructures.<fnref n="1" /> However, a detailed comparison of different alternatives in the design and on their impact on the science output has never been performed. The concept of ET was proposed well before the first observations of GWs. Unlike then, all the knowledge about GW sources that has accumulated in recent years makes such a study with astrophysically motivated GW source populations possible, and mandatory, today.

The aim of this work is twofold. First, we want to examine the effect of variations in the geometry, comparing a single triangle to a configuration made of two well-separated L-shaped detectors, which we will refer to as the ‘2L’ configuration, while maintaining all other innovative concepts of the ET design (underground, cryogenic, and in a xylophone configuration). Second, we want to study what the impact on the science output would be if some aspects of the design should turn out to be difficult to achieve, at least in the first stage of operations (both for a triangle and for a 2L configuration). In particular, we will study the situation in which the low-frequency (LF) instrument is not operative, and we only have the high-frequency (HF) instrument in each detector. This will allow us to quantify the impact of the LF instrument on the expected science output.

This study has been performed in the context of the activities of the Observational Science Board (OSB) of ET, which is in charge of developing the Science Cases and the

<footnote n="1">¹See http://www.et-gw.eu/index.php/relevant-et-documents for a collection of documents on the ET design study and science case.</footnote>
analysis tools relevant for ET.<fnref n="2" /> The study is structured as follows. In section 2 we will introduce and motivate the different detector geometries considered and we will give the corresponding strain sensitivities. We will then discuss the impact of different choices of geometry and design on the science that can be made with ET. The coalescence of compact binaries, such as binary black holes (BBHs) and binary neutron stars (BNSs), is a primary target of 3G detectors. The crucial properties here are the detection rates, the range and distribution in redshift of the detected events, and the accuracy in the reconstruction of the source parameters. Properties such as detection rate, range and accuracy in parameter estimation have the advantage of being very general and requiring minimal model-dependent assumptions (which are reported in the paper). Therefore, they already provide a first solid understanding of the relative advantages/disadvantages of different configurations. In section 3 we will then compare, from this point of view, the performance of the different ET configurations introduced in section 2. We will first consider ET observing as single detector (whether in its triangle or 2L configuration), discussing BBHs in section 3.1 and BNSs in section 3.2, and we will then perform in section 3.3 the same study when ET is part of a 3G detector network including one or two Cosmic Explorer detectors. The effect of the different choices of ET as single observatory on multi-messenger observations of BNSs is studied in section 4. In particular, section 4.1 discusses the sky-localization and pre-merger alerts, while sections 4.2 and 4.3 evaluate, respectively, the prospects of detecting short γ-ray bursts (GRB) and kilonovae associated with BNS mergers.

Stochastic backgrounds of GWs are another primary target of ET. The effect of different configurations on stochastic searches is studied in section 5. In particular, in section 5.1 we discuss the sensitivity to stochastic backgrounds (as expressed by the power-law integrated sensitivity, PLS) of the different configurations, while the angular resolution obtained from a multipole decomposition is discussed in section 5.2. A comparison of the different sensitivities with predictions for astrophysical backgrounds is performed in section 5.3, while the impact of correlated noise is discussed in section 5.4.

Beside the study of these rather general metrics, it is also important to investigate how these differences in detection rates, parameter estimation, or sensitivity to stochastic backgrounds, reflect on specific scientific targets of ET around which the ET Science Case is structured [15], even if this, unavoidably, may introduce some extra model-dependent assumptions. In section 6 we will then discuss the impact of the different configurations on a broad set of specific scientific targets of ET in the domains of astrophysics, fundamental physics and cosmology. Finally, in section 7 we will discuss the properties of the null stream for the triangular configuration and its potential benefits. A summary of the main results of this study is presented in section 8 and, in particular, our conclusions are discussed in section 8.3. More technical material is collected in four appendices; in particular, several tables with useful figures of merits for BBHs and BNSs are collected in appendix C. We also provide, at the end of the paper, a list of acronyms used in the text.

## 2 Detector geometries and sensitivity curves

The ET initial design included high laser power at the interferometer input, leading to high optical power in the cavity of the order of several MW, optimisation of signal recycling, frequency-dependent light squeezing, increase of the beam size, and heavier mirrors compared

<footnote n="2">²See https://www.et-gw.eu/index.php/observational-science-board for a public repository of papers produced within the ET Observational Science Board.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** A schematic picture of the different geometries considered: two widely separated L-shaped detectors (either parallel or at 45°), or a triangle made of three nested detectors.</figcaption></figure>

to 2G detectors. These specifications led to the so-called ET-B sensitivity curve [10]. This design, however, neglected the fact that high circulating power is difficult to reconcile with cryogenic test masses. This has led to a 'xylophone' concept, in which a detector is actually composed of two different interferometers, one optimized for low frequencies (LF) and one for high frequencies (HF); the LF instrument has low power (since laser power is only needed to beat down the shot noise in the high-frequency range) and cryogenic mirrors, while the HF instrument has high power and mirrors at room temperature. This has led to the ET-C sensitivity curve and, after further refinement of some noise models, to the ET-D sensitivity curve [12]. The ET-B and ET-C sensitivities must be considered as obsolete, and should not be used. The ET-D sensitivity has been the basis for all recent ET studies. Actually, we will use a further refinement of the ET-D sensitivity curve, recently developed in the context of the ET Instrument Science Board (ISB), see below.

As far as the geometry is concerned, we will study triangular configurations, in which three detectors (each one made of a LF and a HF interferometer) are nested into a triangular shape, as well as a network of two well-separated L-shaped detectors (that we will denote as the '2L' configuration), and will we examine different options for the arm length and, for the 2L configuration, different relative orientations between their arms. A schematic picture of the different geometries considered is shown in figure 1. For a single triangular configuration, currently the candidate sites are the Sos Enattos site in Sardinia, and the Meuse-Rhine three-border region across Belgium, Germany and the Netherlands. When considering a single triangle, we will locate it for definiteness in Sardinia, but there will be no significant difference in the results considering ET in the Meuse-Rhine region. When considering the 2L configuration, we will locate one detector in Sardinia and one in the Meuse-Rhine region. However, at the current level of precision, the analysis will be valid, with minor changes, for any pair of interferometers located at a comparable distance.

For two L-shaped detectors, in the coplanar limit the setting with parallel arms maximizes the sensitivity to stochastic backgrounds and the range to compact binaries, while it
is the least favorable in terms of the accuracy on angular localization and reconstruction of the distance of compact binary coalesces (or other individual sources). Conversely, the setting with arms at 45° is basically blind to stochastic backgrounds, while it maximizes the performances in terms of angular localization and distance of compact binaries, see section 5.1. When taking into account the Earth's curvature, the relative orientation between the two detectors is usually defined with reference to the great circle that connects the two detectors [21, 22]. Denoting by $\beta$ the angle describing the relative orientation of the two detectors, defined with reference to this great circle, $\beta = 0°$ corresponds to the case where the arms of the two interferometers make the same angle with respect to the great circle, while $\beta = 45°$ is when one of the two interferometers is rotated by 45° from the $\beta = 0°$ orientation. For $\beta = 45°$, the sensitivity to stochastic backgrounds is minimized, to the extent that it becomes exactly zero in the limit $f\Delta x \to 0$, where f is the GW frequency and $\Delta x$ the distance between the two detectors. In this work, we aim at studying the performance of the 2L configurations close to these limiting cases; however, setting exactly $\beta = 45°$ is not a convenient choice because it would send practically to zero the sensitivity to stochastic backgrounds; instead, as we will see, even a small deviation from $\beta = 45°$ allows us to reach an interesting sensitivity to stochastic backgrounds, without essentially affecting the performances for angular localization and parameter estimation of compact binary coalesces.

We decide to focus on two examples for the relative orientation of the interferometers arms in the 2L configurations, representatives of choices favoring, respectively, either the angular localization and distance determination of compact binaries, or the detection of stochastic backgrounds, but avoiding the limit in which the sensitivity to compact binaries is optimized at the expenses of sending to zero the sensitivity to stochastic backgrounds. We thus consider a simpler definition of the angles that uses the local North at the two detector sites as reference for the orientations. Denoting by $\alpha$ the relative angle defined in this way, and by $\beta$ the relative angle defined with reference to the great circle connecting the detectors in Sardinia and in the Netherlands, we have $\alpha \approx \beta + 2.51°$.<fnref n="3" /> We will study the cases $\alpha = 0°$ and $\alpha = 45°$, that corresponds to introducing a small offset with respect to the ‘perfectly’ parallel ($\beta = 0°$) and maximally misaligned ($\beta = 45°$) configurations. The results for compact binary coalescence (CBC) parameter estimation depends minimally on the precise value of this offset, as long as this misalignment angle is small. We have indeed checked that the results shown in section 3 are unaffected, for all practical purposes, by this small offset; e.g. the differences between the cases $\alpha = 45°$ and the truly optimal orientation $\alpha = 47.51°$ would be invisible on the scale of the plots shown in section 3, and are below the variability due to the specific sample realization of the population. In contrast, the sensitivity to stochastic backgrounds strongly depends on it, as we will discuss in appendix B. In the following, we will refer to these choices as parallel (by which we mean $\alpha = 0°$) and misaligned ($\alpha = 45°$).<fnref n="4" />

In particular, we will consider (with the definition of parallel and at 45° specified above, i.e. with respect to the local Norths at the two detector sites):

1. a triangle with 10 km arms (the current baseline ET geometry)

2. a triangle with 15 km arms

<footnote n="3">³For the Sardinia site we take (latitude 40° 31', longitude 9° 25'), while for the Meuse-Rhine site we take (latitude 50° 43' 23", longitude 5° 55' 14").</footnote>

<footnote n="4">⁴If a 2L option should be retained, it will be important to perform a more detailed study of the trade-offs between accuracy in the localization of compact binaries and sensitivity to stochastic backgrounds, as a function of the misalignment angle; such a study, however, cannot be performed abstractly but, especially when discussing misalignment at the (1–2)° level, will have to integrate constraints from the geography and the geology of the candidate sites. For instance, underground tunnels have to follow optimized paths avoiding as much as possible fractures or water springs.</footnote>
3. a 2L with 15 km arms, with parallel arms

4. a 2L with 15 km arms, with a relative orientation of 45°

5. a 2L with 20 km arms, with parallel arms

6. a 2L with 20 km arms, with a relative orientation of 45°

We will also occasionally compare with the results that can be obtained with the most advanced network of 2G detectors (LIGO Hanford, LIGO Livingston, Virgo, KAGRA and LIGO India), in order to better appreciate the improvement due to 3G detectors. We use the publicly available best sensitivities that are planned to be achieved by LIGO, Virgo and KAGRA by the end of the O5 run [23]. We will denote this network LVKI O5. LIGO India is expected to join the 2G network late in this decade. Both the Virgo and LIGO collaborations are currently discussing plans for further upgrades of the detectors to be implemented after the O5 run by developing and testing key technologies critical for the 3G detectors [24]. We will also show some results for a single L-shaped detector of 20 km (denoted as ‘1L’), that we will locate for definiteness in the Meuse-Rhine region.

It is clear that the configurations considered have different financial costs, so some configurations might be obviously better than others. The aim of this work is to provide scientific input to a broader cost-benefit analysis.<fnref n="5" />

In figure 2 we show the different noise Amplitude Spectral Densities (ASDs) used in this work to characterise the various detector configurations considered. In particular, in the left panel we show, for various arm lengths, the noise curve attainable in the full design, in which there is a xylophone configuration with an instrument tuned toward high-frequencies (HF) and one tuned toward low frequencies (LF), and the LF instrument works at cryogenic temperatures. In the right panel we show the ASDs resulting from exploiting only the HF instruments. These curves have been computed in the context of the activities of the ET Instrument Science Board (ISB).<fnref n="6" /> In particular, the line corresponding to the 10 km length in the panel labelled “HFLF cryogenic” updates the sensitivity curve known as ET-D, available at https://apps.et-gw.eu/tds/?content=3&r=14065, and shown as a reference as the black line in both panels of figure 2, and reflects more detailed data on the technology used in the ET-D design. It should be stressed that, in this phase where the design of ET is being more deeply analyzed, the sensitivity curves necessarily evolve, and will continue to evolve. The two curves shown, for the full sensitivity and for the HF instrument only, must be taken as representative curves, within other possibilities currently under study. Further intermediate sensitivities could also be studied (for instance, corresponding to a possible intermediate stage where the LF instrument is present, but operates at room temperature), and would give results in between those obtained with the HF-only and the HFLF-cryo sensitivities.

<footnote n="5">⁵An analysis of the costs of the different configurations is well beyond the scope of this study. We note, however, that in the comparison between geometries, the total linear arm length is not the only relevant parameter for estimating the relative costs. For instance, another significant parameter is the diameter of the tunnels. In an L-shaped configuration there will be only one detector per tunnel, compared to two in the triangle configuration with nested interferometers. As a result, in an L-shaped detector the diameter of the tunnels will be smaller, approximately 6.5 m, to be compared with 8 m for the triangle. Already just for what concerns the relative excavation costs, several other factors must be included for a realistic comparison, such as the fact that, in the triangle configuration, extra tunnel length is needed to displace the end towers of a detector with respect to the input towers of the other, as well as differences in the number and size of auxiliary caverns, and in the tunnels serving them. All these aspects can only be evaluated with a detailed dedicated study.</footnote>

<footnote n="6">⁶These curves are available at https://apps.et-gw.eu/tds/ql/?c=16492, see in particular the Annex files. They obtained with the help of PyGWINC package [25].</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Amplitude spectral densities (ASDs, single-sided), used in the present work for different scenarios. In both panels we show the ASD for an interferometer with 10, 15 and 20 km arms. The left panel shows the full sensitivity curves (obtained from the HF instrument and the cryogenic LF instrument, in a xylophone configuration); the right panel is the sensitivity obtained with the HF instrument only. For comparison, in both panels we also show the ASD of the 10 km ET-D interferometer (which, by definition, includes the full HFLF sensitivity). Note that the ASD are plotted as a single nested detector (each one made of a LF and a HF interferometer left plot and each one made of HF only right plot). See the text on how to take into consideration the angle between the arms and the number of interferometers for the triangular shape.</figcaption></figure>

When comparing a triangle to an L-shaped interferometer with the same instrument-noise ASD, one must take into account that the triangle is made of three nested detectors (a detector being an LF and HF interferometer pair), with an opening angle of 60°. For the triangle configuration, one must then project the GW tensor of the incoming wave onto each of these three components [see e.g. eqs. (9)–(11) of [26] for explicit expressions], and then combine the results at the level of the signal-to-noise ratio (SNR) and parameter estimation to obtain the ET capabilities. The accurate evaluations of detection horizons and parameter-estimation capabilities require the above calculation of GW-tensor projections onto the three ET components, as is done for all the results of the present paper, and in all Fisher matrix codes mentioned below. However, to have a simple and approximate rule-of-thumb for comparing the SNR of a triangle to that obtained, for the same signal, by an L-shaped detector with the same ASD, one must first of all take into account that, for each given component, the opening angle gives an extra factor $sin(60°) = \sqrt{3}/2$, see again eqs. (9)–(10) of [26]. In a first approximation, one can also assume that the three components of the triangle see the signal with the same SNR (which, however, is only approximately true). Then, since the SNR of the three components of the triangle add up in quadrature, the SNR² in a triangle is approximately larger than the SNR² in an L-shaped detector with the same ASD, by a factor $3×(\sqrt{3}/2)^2 = 9/4$. For this reason, when comparing graphically the ASD of a triangle to that of an L-shaped detector, a more fair comparison is obtained dividing the ASD of the triangle by a factor 3/2. Since some source of noise, such as Newtonian noise, are such that the SNR scales as the length of the arms, in a first approximation one could further be tempted to estimate that, for a triangle, SNR $\propto L\sqrt{3}sin(60°) = 1.5L$, while for a 2L network with the same arm-length SNR $\propto L\sqrt{2}sin(90°) \simeq 1.41L$.<fnref n="7" /> However, one must be careful not to infer

<footnote n="7">⁷In fact, not all noise relevant for 3G detectors are such that the SNR scales linearly with $L$. Examples of noise whose SNR do not scale with $L$ include coating thermal noise (since it also depends on beam size, which increases with $L$), shot noise (since one needs to adjust the arm-cavity finesse depending on $L$), controls noise</footnote>
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Detection horizons for monochromatic populations of equal-mass non-spinning binaries, for the various detector configurations considered. For comparison, we also show in each panel the curve obtained using a triangular detector with the 10 km ET-D ASD (which, by definition, includes the full HFLF sensitivity).</figcaption></figure>

from this that a 2L and a triangle configurations with similar arm lengths necessarily perform comparably. The scientific performance is determined basically by the quality of parameter estimation, which does not depend just on the SNR. As an obvious counter-example, for an ensemble of compact binary coalescence signals, and fixed ASD, 2L parallel have a higher SNR than 2L at 45°; however, as far as parameter estimation is concerned, 2L at 45° perform significantly better (as we will see explicitly for ET in section 3), exactly the contrary of what would be inferred using an SNR-only argument. Similarly, for angular localization, the baseline between the detectors is another crucial factor. A comparison between the performances of different geometries can only be obtained with a parameter-estimation study, as we will perform below, and cannot be obtained from considerations based only on the SNR.

In figure 3 we show the corresponding detection horizons for equal-mass non-spinning binaries, as a function of their source frame total mass, for the various detector configurations considered. These are defined as the maximum redshift out to which a binary with the chosen characteristics, with optimal location and inclination, can be observed by a given detector configuration with a network SNR ≥ 8. This computation is performed using the IMRPhenomHM waveform model [28], and accounting for the effect of Earth rotation, which can give a contribution in the range of masses typical of BNS systems. For comparison, we also show the results obtained for a triangular-shaped detector using the ‘older’ ET-D ASD, which reproduces the result in figure 1 of [29].

In figure 4, left panel, we show the ratios of the redshift horizons for three different geometries (triangle 15 km, 2L-15km-45° and 2L-20km-45°), all in their HFLF-cryo configurations, with respect to the 10 km triangle, also taken in its HFLF-cryo configuration. In the right panel we show the same ratios for triangle 15 km, triangle 10 km, 2L-15km-45° and 2L-20km-45°, now all in their HF-only configuration, again with respect to the horizon of the 10 km HFLF-cryo triangle. We can appreciate that, especially for total masses below

<footnote>(since *L* can change stability conditions of the opto-mechanical system, which requires adjustment of control filters) and suspension thermal noise (since one needs larger mirrors in longer detectors, which means higher suspended mass), see [27] for discussions. Therefore, the correct comparison is between detectors at fixed ASD, as discussed above, rather than at fixed *L*, and the scaling of SNR ∝ *L* is a further approximation, to be used only for a first orientation.</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Ratios of the detector horizons for the different geometries. Left panel: full HFLF-cryo sensitivities, normalized to the horizon for the HFLF-cryo sensitivity of the 10 km triangle (i.e. to the green line in the left panel of figure 3). Right: HF sensitivities (still normalized to the full HFLF-cryo sensitivity of the 10 km triangle).</figcaption></figure>

~ 10$M_{\odot}$, arm-length has a very important effect on the detection horizon: for these masses, the 15 km triangle has a range that, in terms of redshift, is larger by a factor 1.5 – 2.2 than that of the 10 km triangle, and the 2L-15km-45° is larger by a factor 1.4 – 1.8; as we see from the right panel, in this range of masses this increase in arm-length would basically compensate the loss of the LF instrument (especially for the 15 km triangle). Once again, however, it is important to recall that the detection horizon is determined just by the SNR. More important, for the science case, is the quality of parameter estimation. For instance, the 2L-15km-45°, compared to the 2L-15km-0°, has a smaller horizon reach, but better capacity to determine the parameters of the sources, as we will quantify in the next section.

## 3 Coalescence of compact binaries

In this section we study the impact of the different detector geometries and different ASDs on the detection and parameter estimation of BBHs and BNSs. Several parameter estimation codes, tuned toward 3G detectors, have been developed recently, in particular GWBENCH [19, 30], GWFISH [31], GWFAST [17, 32], TiDoFM [33, 34] and the code used in [35]. In the context of the activities of the Observational Science Board (OSB) of ET, we have performed extended cross-checks between these codes, finding very good consistency. The results presented in this section have been obtained with GWFAST which, thanks to its vectorization properties, is quite convenient for dealing with the large number of runs required, while those presented in section 4 have been obtained with GWFISH, which is also tuned toward multi-band and multi-messenger observations [18, 31]. These codes are based on the Fisher matrix formalism. The Fisher matrix formalism has well-known limitations (see e.g. [36, 37] for extensive discussion) but is currently the only computationally practical way of dealing with parameter estimation for large populations. We refer to the original papers [17, 19, 30–32] for technical details on the application of the Fisher matrix formalism to parameter estimation for 3G detectors. It should also be observed that 3G detectors will observe a large number of events with very large SNR. As we will see for instance in figure 5 below, ET, in any of its configurations, will detect $O(10^3 – 10^4)$ BBH/yr with SNR > 100. These events will largely dominate the
performance of a detector configuration for most aspects of the science case, and for large SNR the Fisher matrix approach becomes more and more reliable.

We use state-of-the-art waveforms and population models. For the waveforms, we use IMRPhenomXPHM for BBHs, which includes precessing spins and higher-order modes [38], and IMRPhenomD_NRTidalv2 [39, 40] for BNSs, which includes tidal effects. The parameters of the waveform are {$M_c, \eta, d_L, \theta, \phi, \iota, \psi, t_c, \Phi_c, \chi_{1,x}, \chi_{2,x}, \chi_{1,y}, \chi_{2,y}, \chi_{1,z}, \chi_{2,z}, \Lambda_1, \Lambda_2$}, where $M_c$ denotes the detector-frame chirp mass, $\eta$ the symmetric mass ratio, $d_L$ the luminosity distance to the source, $\theta$ and $\phi$ are the sky position coordinates (defined as $\theta = \pi/2 - \delta$ and $\phi = \alpha$, with $\alpha$ and $\delta$ right ascension and declination, respectively), $\iota$ the angle between the orbital angular momentum of the binary and the line of sight, $\psi$ the polarisation angle, $t_c$ the time of coalescence, $\Phi_c$ the phase at coalescence, $\chi_{i,c}$ the dimensionless spin of the object $i = \{1, 2\}$ along the axis $c = \{x, y, z\}$ of the coordinate system, chosen with the $z$ axis pointing along the orbital angular momentum (we use the same conventions as in section IID of [41]), and $\Lambda_i$ the dimensionless tidal deformability of the object $i$ (which is present only for systems containing a NS). Instead of $\Lambda_1, \Lambda_2$, we will actually use the two combinations $\tilde{\Lambda}$ and $\delta\tilde{\Lambda}$ defined in [42], which have the advantage that $\tilde{\Lambda}$ is the combination that enters the inspiral waveform at 5 PN, while $\delta\tilde{\Lambda}$ first enters at 6 PN. In particular,

$$ \tilde{\Lambda} = \frac{8}{13} \left[ (1 + 7\eta - 31\eta^2)(\Lambda_1 + \Lambda_2) + \sqrt{1 - 4\eta} (1 + 9\eta - 11\eta^2)(\Lambda_1 - \Lambda_2) \right]. \quad (3.1) $$

For BBHs we will perform the inference on all parameters (except, of course, $\Lambda_1, \Lambda_2$, that vanish for BHs) and, as in LVK parameter estimations, rather than $\iota$ and $\chi_{i,c}$, we will use $\theta_{JN}$ (i.e. the angle between the total angular momentum and the line of sight; note that this is the same as $\iota$ only in the absence of precession), the spin magnitudes and angles, namely $\chi_1, \chi_2, \theta_{s,1}, \theta_{s,2}, \phi_{JL}, \phi_{1,2}$.<fnref n="8" /> For BNSs, instead, we include tidal deformability but, given the small expected values of their spin magnitudes, we only consider the aligned spin components in the analysis, thus performing estimation on $\chi_{1,z}$ and $\chi_{2,z}$. The labels ‘1’ and ‘2’ always refer, respectively, to the heaviest and lightest component of the binary system. After the inversion of the Fisher matrix, we compute the sky localisation area for the events at 90% c.l., $\Delta\Omega_{90\%}$. As in [17, 18], we assume an uncorrelated 85% duty cycle in each L-shaped detector, and in each of the three instruments composing the triangle. All technical details of the inference process for the results of this section are as described in [17].

We generated the catalogs of binary neutron stars (BNSs) with the population synthesis code MOBSE [43, 44]. MOBSE evolves isolated binary stars until they possibly become binary compact objects. For the models used here, we have evolved $1.8 \times 10^8$ binary systems with 12 different metallicities, ranging from $Z = 10^{-4}$ to 0.02. Primary masses are drawn from a Kroupa initial mass function [45] between 5 and 150 $M_\odot$, while mass ratios, initial orbital periods and eccentricities follow the distributions presented in [46]. MOBSE adopts up-to-date models for stellar winds [44]. We model electron-capture, and (pulsational) pair-instability supernovae according to [47] and [48], respectively. For core-collapse supernovae, we use the rapid model by [49], which enforces a mass gap between neutron stars and black holes. The main difference with respect to [49] is that we model neutron star masses according to an uniform distribution between $m_{\text{min}} = 1.1 M_\odot$ and $m_{\text{max}} = 2.5 M_\odot$, which is a better match to current LIGO-Virgo results [7]. We draw natal kicks from a Maxwellian distribution with one-dimensional root-mean-square $\sigma = 265 \text{ km s}^{-1}$, rescaled by the mass of the compact

<footnote n="8">⁸This choice has the advantage of using quantities which are less degenerate among each other, leading to better conditioning of the Fisher matrices, thus improving the reliability of this approach.</footnote>
remnant and by the mass of the ejecta, see [50] for more details. During binary evolution we treat mass transfer, tidal evolution, gravitational-wave decay, and common envelope as described in [51]. For the common envelope, we assume an efficiency parameter $\alpha = 3$. We describe the evolution with redshift by accounting for the metallicity-dependent star formation rate across cosmic time, as described in [52]. The value for the local merger rate for this model turns out to be $R_0 \simeq 250 \text{ Gpc}^{-3} \text{yr}^{-1}$, which is consistent with the range [10, 1700] Gpc$^{-3}$yr$^{-1}$ obtained from the LVK results [7].

We generated the catalogs of binary black holes (BBHs) by mixing the isolated evolution channel (obtained as described above for BNSs) with the dynamical formation channel. We used the code FASTCLUSTER [53, 54], which evolves binary black holes in young, globular and nuclear star clusters, by accounting for their dynamical pair-up and for hierarchical mergers. In our catalogs, 55% of all BBHs come from isolated binary evolution, while 28, 15 and 2% come from dynamical formation in young, globular, and nuclear star clusters. We chose these mixing fractions because they ensure the best agreement with LIGO-Virgo black hole masses in the local Universe, see [54] for more details on the methodology. The BBH and BNS catalogs used for this study are publicly available at https://apps.eegw.eu/tds/?content=3&r=18321.

## 3.1 Binary Black Holes

In this section we present the results obtained for BBHs. The data correspond to one year of data taking. With the assumed BBH population, the total number of BBHs coalescing in one year is about $1.2 \times 10^5$. We first study the effect of varying the geometry, for detectors all with their full HFLF-cryo sensitivity, and we then discuss the effect of varying the ASD.

### 3.1.1 Comparison between geometries

In the upper-left panel of figure 5 we show the cumulative distribution of the events with a network SNR larger than a given value, while in all other panels, involving parameter estimation, we restrict to events that pass a detection threshold SNR $\ge 12$, and we show the corresponding cumulative distribution of events with an error smaller than a given value.<fnref n="9" />

<footnote n="9">⁹More precisely, when performing parameter estimation, we restrict to events that, besides passing the threshold on the SNR, also have a well-defined inversion of the Fisher matrix. In some cases, the Fisher matrix is ill-conditioned, so its inversion can lead to large errors and we then discard the corresponding event, see [17] for details. For BBHs, the fraction of ill-conditioned Fisher matrices is of order 1%, so has very little impact at this level, and is anyhow about the same percentage for different configurations, so this has essentially no impact on the comparison between configurations. For BNS, between 4% and 10% of the events are discarded, with the higher percentages appearing in the analyses using the ASD including only the HF instrument. There is also a slight dependence on the geometry especially in the HF-only case, with the 2L configurations resulting in worse conditioned matrices as compared to the triangular ones. However, the events discarded in these cases correspond preferentially to the situation when there is only one detector operating due to the duty cycle, in which case the Fisher matrix provides a poorer approximation. The ill-conditioned Fisher matrices often correspond at the population level to the events that would have anyhow large errors in parameter reconstruction (while the science output of the detectors depends mostly on the events with good reconstruction), but could also be related to events having a strong degeneracy between some parameters. In particular for BNS systems, where a higher percentage of events has ill-conditioned matrices as compared to BBH, the solution of discarding events, despite being conservative, might then leave a bias in the observed population, especially for specific science cases (as an example, events close to face-on/off are preferentially discarded due to the strong degeneracy present in this case between the inclination angle and the luminosity distance, but they are relevant e.g. for multi-messenger studies with GRBs). Another solution, adopted for the results presented in section 4, is to regularize the close-to-singular matrices according to the procedure described in [31]. This allows us to analyze the non-degenerate parameters of a larger number of the events,</footnote>
In each panel of this figure we show the results for the six geometries considered, all taken with their best ASD, i.e. in the xylophone configuration with the HF instrument and the cryogenic LF instrument (we use the label ‘HFLF-cryo’ for this ASD).

We see from the figure that even if, by itself, the performance of the 10 km triangle is extraordinary, the other geometries considered in general provide a further improvement (except for angular localization, where the 2L configurations with parallel arms have the worst performance). This is particularly evident in the reconstruction of the luminosity distance. In our population of $1.2 \times 10^5$ events (and in the specific sample drawn from it), the 10 km triangle can reach an accuracy on $d_L$ equal or better than 1% for 28 events; this raises to 202 events for 2L with 15 km arms at 45°, and 365 events for 2L with 20 km arms at 45°, which is the configuration that provides the best results (a set of tables with the explicit number of events with SNR above given thresholds, or errors on various parameters below given values, is collected in appendix C). In general (not surprisingly), the 2L configuration with the longest arm length, 20 km, and relative arm orientation of 45°, is the one that performs better for all the parameters.

We can also appreciate from figure 5 that the configurations of 2L with parallel arms performs quite poorly (comparatively) for angular localization. While this was generally expected because of the parallel arms, it is quite interesting to see that, in terms of angular resolution, even the 2L with 20 km parallel arms performs somewhat less well than a single 10 km triangle, despite the longer arm length and the possibility of performing triangulation (even if only partial, since just two detectors are involved). Again, the best results are obtained with the 2L at 45°. For example, with our choices for the population and in our sample, the 10 km triangle can detect 35 BBH/yr with angular resolution below 1 deg$^2$, to be compared with 92 BBH/yr for the 15 km 2L at 45° and 155 BBH/yr for the 20 km 2L at 45° (see table 40 in appendix C for a more extended set of examples). For the chirp mass reconstruction, the arm length sets the hierarchy among these configurations (see also table 41 in appendix C) so, again, among the configurations considered, the 10 km triangle is the least accurate. The results for the reconstruction of the symmetric mass ratio and of the spins show, instead, less pronounced differences among these configurations (note, however, the logarithmic scale, that visually flattens the differences; see again table 41 in appendix C for numerical values).

From figure 5 and tables 40, 41 we also see that the 15 km triangle has performances very similar to the 15 km 2L at 45°, except for the accuracy on the luminosity distance, where the 15 km 2L at 45° is clearly superior; for instance, the 15 km 2L at 45° would detect 202 BBH/yr with error on the luminosity distance better than 1%, to be compared with 77 for the 15 km triangle. In appendix D we examine the correlations between the various parameters for typical well-measured events and we find that, for the triangle, the error on $d_L$ is more sensitive to the error on the localization, polarization angle and orbit inclination, so that errors on these parameters have a larger effect on the marginalized error on $d_L$, compared to the 2L configuration. We also note that this large improvement is specific to the 2L configurations with arms at 45°. For instance, we see from table 40 that the number of events/yr with $d_L$ measured to better than 1% is 28 for the 10-km triangle, raising to 77 for the 15 km triangle; for the 15 km 2L with arms at 0° this number is 79, and jumps to 202 for the 15 km 2L at 45°. Therefore, the raise from 28 to 77 (or 79) events/yr can be traced to the increase in arm length, from

<footnote>at the price of introducing a dependence on the adopted regularisation threshold. For a recent discussion and comparison of these intrinsic limitations of the Fisher approach related to the inversion procedure see e.g. appendices B and C of [17]. Each method has its own benefits, and it is reassuring that both have been used in the present analysis producing consistent results.</footnote>
10 km to 15 km; the fact that the 15 km triangle and the 15 km 2L with parallel arms have a very similar performance (on this metric) means that, for a given arm length, the three nested detectors of the triangle configuration approximately compensate the large baseline of the 2L configuration with parallel arms; the further increase to order 200 events/yr is due to the better capability of the 2L configuration at 45° to measure the source location, polarization and inclination angles, and therefore disentangle them from the luminosity distance.

We can also appreciate the overall similar performance of the different geometries in the reconstruction of the intrinsic parameters (masses and spins). This is due to the fact that their reconstruction is mostly influenced by the loudness of the signal in the detector, rather than by the localization capability, and is therefore largely determined by the SNR. As we see from the SNR panel in figure 5, all configurations have similar distributions of the SNR, except for the 10 km triangle, which is disfavored, and this is reflected in the estimation of masses and spins.

Finally, it is interesting to show the joint distribution of the events with respect to $\Delta d_L/d_L$ and $\Delta\Omega_{90\%}$, since this determines the overall localization volume. To avoid a proliferation of plots, here we limit ourselves to a comparison between the 10 km triangle and 15 km 2L at 45°, which is shown in the left panel of figure 6, and between the 15 km triangle and 15 km 2L at 45°, shown in the right panel. We see that the 15 km 2L at 45° has many more events than the triangle in the region of the $(\Delta d_L/d_L, \Delta\Omega_{90\%})$ plane corresponding to an interesting three-dimensional localization, e.g., $\Delta d_L/d_L \le 1\%$ and as far as these metrics are concerned, one can conclude that, while the performance of the 10 km triangle is by itself remarkable, the 2L configuration with 15 km arms, oriented at 45°, provides a further improvement (as the 2L configuration with 20 km arms or the 15 km triangle). In contrast, the 2L with parallel arms looks quite disfavoured, because of a comparatively poor angular localization capability. The 15 km 2L at 45° and the 15 km triangle provide results quite comparable, except for the accuracy on luminosity distance, for which the 15 km 2L at 45° is clearly superior.

Finally, in figure 7 we study the performance of a single L-shaped detector with 20 km arms in the full HFLF cryo configuration, and we compare it with the 10 km triangle and with the 15 km 2L at 45°, also at their full sensitivity, as well as with LVKI O5. The most apparent feature is that, in terms of angular resolution and reconstruction of luminosity distance, a single L is clearly performing very poorly, and is even way below the LVKI O5 forecast. This shows that a single L-shaped detector, not inserted in a global network, is basically useless for those aspects of the Science Case, such as multi-messenger astronomy or cosmology, that require good estimates of sky-localization and distance of the sources. For the other parameters, however, a single L of 20 km, although it provides worse estimates, can be comparable to the 3G networks that we have considered.<fnref n="10" />

### 3.1.2 Effects of a change in the ASD

We next investigate how these results are affected by changes in ASD. In figure 8 we show, for the 10 km triangle, the cumulative distributions of the SNR and of parameter estimation (where again, for the SNR we show the cumulative distribution of events with SNR larger than a given value, while for parameter estimation we restrict to events with SNR ≥ 12 and we show the cumulative distribution of events with error smaller than a given value) for the full ASD including the HF instrument and a cryogenic LF instruments (denoted

<footnote n="10">¹⁰This, of course, assumes that spurious transient noise and glitches could be vetoed and eliminated reliably since, for a single detector, it would not be possible to reduce them by using the coincidence between independent detectors, or (as for the triangle) the null stream. In particular, the detection of unmodeled burst signals could not be reliably claimed with a single detector.</footnote>
‘HFLF cryo’), and for the ASD in which the LF instrument is completely missing and only the HF instrument is operative (denoted ‘HF only’). We also show, for comparison, our forecasts for LVKI O5. In order to be consistent with the ET results, the analysis for LVKI O5 is obtained following exactly the same Fisher-Matrix analysis used for ET. For a more precise and realistic forecast, we refer to [23] which uses a Bayesian framework close to real analysis applied to the current gravitational-wave observations.

An important message that emerges from figure 8 is that even if, at least in the first stage of operations, the LF instrument were not operational, still ET in its 10 km triangle configuration and HF-only sensitivity would provide a significant jump from 2G detectors, in terms of a number of detections, distribution of SNR and parameter reconstruction of BBHs, with the exception of angular localization (as expected, for a single infrastructure that cannot rely on triangulation). It is also worth noting that a single observatory, such as the 10-km ET triangle, at its best sensitivity, can localize a comparable number of sources with an accuracy as good as that of a network comprising five 2G detectors at full O5 sensitivity.

Note that, on a single event basis, the events observed by LVKI are almost always better localized than by ET, showing that localization benefits more from the network triangulation than from ET’s higher SNR (with fluctuations due to the assumed duty cycle). Nonetheless, the number of detections by ET is more than 10 times higher, and comprises high SNR events with low masses and/or high mass ratios. The high-SNR low-mass BBH mergers can be optimally localized accessing low frequencies and using the imprint of Earth’s rotation in the longer observed signal. The localization of high-SNR events with large mass ratio benefits from the detection of higher-order modes. This results in a similar number of optimally localized sources in the cumulative distribution up to a sky-localization of about 6 square degrees, see figure 8. The larger amount of BBHs localized with sky-localization uncertainty larger than this threshold by ET compared to LVKI network depends on the detection efficiency; ET has much higher detection efficiency, reaches higher *z* and has many more distant events where LVKI is no more able to detect sources. Going to larger and larger redshifts, on average the SNR decreases, and the ET localization worsen.

Figure 9 shows the corresponding results for the 2L configuration with 15 km arms at 45°, with the two different ASDs. For comparison, in this figure we also show both the results for the 10 km triangle in the full HFLF-cryo configuration, and the results for LVKI O5. It is interesting to observe that, even in the configuration in which the LF instrument is completely missing, for BBHs the 2L with 15 km arms at 45° is superior to the 10 km triangle with its full (HFLF cryo) sensitivity for the accuracy on the luminosity distance (particularly for the number of events with very good accuracy, say $\Delta d_L/d_L \le 1\%$, with 56 events against 28, see table 40 in appendix C) and gives basically equivalent results for angular localization, spins and orbit inclination. In contrast, for the chirp mass and symmetric mass reconstruction, the full HFLF-cryo triangle is better than the HF-only 2L-15km-45°. In the comparison with LVKI O5 we see that, even in the HF-only configuration, the 2L with 15 km arms at 45° provides a remarkable jump from 2G detectors on BBH detection number, SNR distribution and parameter reconstruction, except that the angular localization of the best-localized events would no longer be better than that of the five-detector LVKI network at O5 sensitivities (although the number of events with $\Delta\Omega_{90\%} < 10^2$ deg$^2$, relevant for multi-messenger studies, is a factor ~5 higher for the 2L with 15 km arms at 45°, compared to LVKI O5). See again tables 40 and 41 in appendix C for a compilation of number events for the different geometries and ASDs, and different cuts on the parameters.
A main message that emerges from figure 9 is that, as far as BBH parameter reconstruc-
tion is concerned, even if in a first stage of operations the whole LF instrument should not
be operative, a 2L configuration with 15 km arms at 45° would still be a very competitive 3G
instrument, in fact quite comparable to a 10 km triangle at full sensitivity (with better perfor-
mance of luminosity distance, less good performance on mass reconstruction, and equivalent
performances on all other parameters and in SNR distribution).<fnref n="11" />

Finally, another important target of 3G detectors are intermediate massive black holes (IMBHs) binaries, with masses in the range $10^2 - 10^4 M_\odot$, In this case, there are currently no observational constraints, and thus we do not use any astrophysically motivated IMBH population. Going back to figure 3, which showed the horizons for monochromatic non-spinning populations of equal-mass binaries (or figure 4, that directly gives the relative differences with respect to the baseline 10 km triangle), it can be deduced that, with the best sensitivity curve (HFLF-cryo configuration), for total masses above about $(300-400)M_\odot$, the difference between the different ET geometries is at about 10% level. For masses around $100M_\odot$ the difference is larger, and for instance for $100M_\odot$ the detection horizon ranges between $z \simeq 42$ and $z \simeq 50$ for the different geometries; however, in this redshift range we do not even expect astrophysical IMBH. Much more significant is the dependence on ASD. The right panel of figure 3 show a significant reduction in the IMBH horizon for the HF-only configuration. *This shows the importance of the low-frequency sensitivity for IMBHs.*

3.1.3 Golden events

Another important metric to assess the performance of a detector, or of a detector network, is
provided by the number and redshift distribution of ‘golden events’, i.e. events with especially
good properties in terms of SNR (e.g. BBHs with SNR ≥ 100), or error on luminosity
distance (e.g. BBHs with Δd_L/d_L ≤ 0.05) or angular localization (e.g. BBHs localized to
better than 10 deg²). Indeed, while for some aspects of the Science Case, such as population
studies, the completeness of the sample is a key element, for other aspects, such as precision
tests of General Relativity (GR) or cosmological studies, the result will be largely, if not
uniquely, determined by these ‘golden events’ (see the discussion in section 5.4 of [17] and
section VI of [19]). In figure 10 we show the redshift distribution of the BBHs detected with
SNR ≥ 100, the distribution of events with Δd_L/d_L ≤ 5%, and the distribution of events
with ΔΩ_90% ≤ 10 deg².<fnref n="12" /> In the upper row we show the results for the six detector geometries
that we are considering, all taken at full (HFLF cryo) sensitivity. The middle row focuses
on the 10 km triangle and shows how the result changes with the ASD, while the lower row
shows the same result for the 2L with 15km arms at 45° (comparing also with the reference

<footnote n="11">¹¹It should also be observed that, in most studies where the accuracy on both volume localization and masses is relevant, as for instance in joint studies of population and cosmological parameters, the bottle-neck is in the accuracy on sky localization and luminosity distance, which are always much worse than the relative accuracy on the (detector-frame) masses. As we see e.g. from figure 9, the relative error on the (detector-frame) chirp mass and on the symmetric mass ratio is typically well below $10^{-4}$ and can be as low as $10^{-6}$ for the chirp mass and $10^{-8}$ for $\eta$, while, for $d_L$, it is difficult to go below the 1% accuracy. Therefore, having a better performance on $d_L$ has a much stronger impact than having a better performance on the detector-frame masses, which in any case are very accurately measured. Note, furthermore, that at the large redshifts explored by 3G detectors, the accuracy of the reconstruction of the actual (source-frame) masses from the detector-frame masses will be determined by the accuracy of the reconstruction of the redshift.</footnote>

<footnote n="12">¹²Of course, these distributions are correlated. For instance, events with large SNR typically have small errors on $d_L$. The correlation between luminosity distance and angular resolution is shown, for BBHs, in figure 6, and the analogous result for BNS will be shown in figure 12. See figures 12 and 17 of [17] and figures 5-9 of [18] for further scatter plots of the correlations among different parameters, for the ET triangle configuration.</footnote>
triangle configuration). In all plots, we show for comparison the redshift distribution of the BBH population that we have used and, for each detector configuration, the distribution of the events detected with SNR ≥ 8 (which, on this scale, is almost indistinguishable from the whole population, at least for $z \lesssim 2$).

From the left panel in the upper row, and table 39 in appendix C, we see that, in terms of detections of events with large SNR, the 10 km triangle by itself has remarkable performances, with (in our sample realization) 2298 BBH/yr detected with SNR ≥ 100, and a redshift distribution extending up to $z \sim 5$. However, this is further improved in the other configurations; in particular, in the 2L-15km configurations, the number of BBH/yr with SNR ≥ 100 is 4933 for the misaligned configuration (and 5143 for the aligned-arms configuration), more than twice as large, and extends up to $z \approx 7$; this further raises to more than 8000 for the 2L-20km configurations. On the other hand, we already saw from figure 5 that, over the whole ensemble of detected events, the setting with parallel arms has comparatively poor performance for angular localization and accuracy on the luminosity distance; this remains true also when we restrict to golden events, as we see from the middle and the right panel in the first row of figure 10. In particular, the 2L configurations with parallel arms (whether with 15 or 20 km arm-length) are by far the less performing in terms of events with $\Delta\Omega \le 10$ deg$^2$ (followed by the 10 km triangle) while, in terms of the events with errors smaller than 5% on $d_L$, the 10 km triangle is, comparatively, the less performing, followed by the 2L configurations with parallel arms, while the 2L configurations with arms at 45° give the best results.

These results indicate that, for BBH golden events, the 2L configuration with arms at 45°, or the 15 km triangle, provide the best compromise between detecting many of them, and out to large redshift, and localizing them, further improving on the already remarkable performances of the 10 km triangle.

The middle row of figure 10 shows how the results depend on the ASD, for the 10 km triangle. From these plots we infer that, also for golden BBH events, a 10 km triangle would provide an extraordinary jump, compared to 2G detectors, even in the HF-only configuration. Indeed, by comparison, at LVKI O5, even having assumed the best-planned sensitivities for all five detectors, with our BBH population and the specific sample that we have drawn from it, we find only 4 events with SNR ≥ 100, to be compared with about 3000 for the 10 km triangle with the full HFLF-cryo ASD, and almost 800 for the 10 km triangle with the HF-only ASD, see table 40 in appendix C.

The lower row of figure 10 shows the dependence on the ASD for the 2L geometry with 15 km arms at 45° (showing, for comparison, also the result for the 10 km triangle at full sensitivity). Once again, a significant conclusion is that the 2L geometry with 15 km arms at 45°, even with the HF-only instrument, is comparable to the 10 km triangle at full sensitivity.
BBH

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Cumulative distributions of the number of detections per year, for the SNRs and for the error on the parameters, for BBH signals, for the six considered geometries, all with their best ASD, including xylophone configuration and cryogenic LF instrument.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 6.** The joint accuracy on luminosity distance and angular resolution for BBHs. Left panel: the 15 km 2L at 45° (blue) compared to the 10 km triangle (green). Right panel: the 15 km 2L at 45° (blue) compared to the 15 km triangle (red).</figcaption></figure>
BBH

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Comparison of SNR and parameter estimation error for the 10 km triangle, 15 km 2L misaligned and a single 20 km L with the HFLF-cryo ASD. By comparison, we also show the forecast for LVKI O5.</figcaption></figure>
BBH

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Comparison of SNR and parameter estimation error for the 10 km triangle with the HFLF-cryo and the HF-only ASDs. By comparison, we also show the forecast for LVKI O5.</figcaption></figure>
BBH

<figure><img src="image_11.png" /></figure>

**Figure 9.** Comparison of SNR and parameter estimation error for the 15 km 2L misaligned with the HFLF-cryo and the HF-only ASDs. For comparison, we also show the result for the 10 km triangle in the HFLF-cryo configuration, and for LVKI O5.
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Redshift distribution of BBHs detected with SNR $\ge 100$ (left column), or with relative error on the luminosity distance $\Delta d_L/d_L \le 0.05$ (central column), or with sky location $\Delta\Omega \le 10\deg^2$ (right column) for various detector geometries and sensitivity curves. The upper row shows the results for the six considered geometries, all with their best sensitivity, the central row for the 10 km triangle with the two considered ASDs, and the bottom row for the 2L with 15 km arms at 45° and the two considered ASDs (together with the 10 km triangle with the full ASD). In each panel we also show, for reference, the redshift distribution of the BBH population used (black solid line), and of the events detected in the various configurations with SNR $\ge 8$ (dotted lines).</figcaption></figure>
## 3.2 Binary Neutron Stars

We now perform the same analysis for BNSs. The corresponding results are shown in figures 11–16. We discuss again separately the dependence on geometry, on ASD, and the ‘golden events’.

### 3.2.1 Comparison between geometries

Figure 11 confirms the basic message obtained from BBHs in figure 5: the 10 km triangle has, in itself, remarkable performances and produces order-of-magnitudes improvements with respect to 2G detectors. However, a further improvement can be obtained from the other geometries that we have considered, on almost all parameters. The only exceptions are the inclination angle $i$, for which the 2L configuration with parallel arms performs, comparatively, quite poorly, both for 15 km and 20 km arm-length,<fnref n="13" /> and the luminosity distance, for which the 10 km triangle performs similarly to the 2L with 15 km parallel arms. Similarly to what we found for BBHs, also for BNSs the 2L 15 km with arms at 45° improves on the accuracy estimate of all the parameters with respect to the 10 km triangle. Comparing instead the 2L 15 km at 45° to the 15 km triangle, we see that the two perform similarly on most parameters except for luminosity distance, for which the 2L 15 km at 45° is significantly better. The 15 km triangle, however, performs better than a 15 km 2L with arms at 45° for the inclination angle $i$, while the 2L 15 km at 45° performs better for the polarization angle $\psi$. To understand better the causes of some differences between geometries, it is interesting to compare the 2L-15km with parallel arms to the triangle with the same arm length. The 2L-15km with parallel arms is significantly worse than the 15-km triangle for the reconstruction of the inclination angle $i$. This is due to the fact that, for the 2L-15km with parallel arms, only a single combination of the two polarization amplitudes is accessible. On the other hand, the 2L-15km with parallel arms is better than the 15-km triangle on the angular localization because, thanks to its long baseline, it can partially triangulate. Both the reconstruction of $i$ and the angular localization influence the reconstruction of the luminosity distance. However, we see from the panel on the luminosity distance in figure 11 that the 2L-15km with parallel arms is still significantly worse than the 15-km triangle in the reconstruction of $d_L$, so the dominant effect is due to the (comparatively) poor reconstruction of $i$.

For the parameter $\tilde{\Lambda}$ related to the tidal deformability of the two NSs, again the 10 km triangle already obtains remarkable performances, with 1040 events/yr with $\tilde{\Lambda}$ measured better than 10%, and the best events providing a measurement at a few percent level; however, all other geometries that we considered improve further on this; in particular, the 2L-15km-45° detects 2463 events/yr with $\tilde{\Lambda}$ measured better than 10%, so an improvement by a factor of 2.5, and the 15 km triangle 2783, see tables 42, 43 and 44 in appendix C.

Figure 12 shows the joint accuracy on luminosity distance and angular resolution for BNSs, comparing the 15 km 2L at 45° to the 10 km triangle (upper panel) and to the 15 km triangle (lower panel). Considering, for instance, the events with $\Delta d_L/d_L < 0.1$ and $\Delta\Omega_{90\%} \le 10^2$ deg$^2$, we see that the figure 13 shows the performance of a single L-shaped detector of 20 km for BNSs, compared to other geometries, similarly to figure 7 for BBH. We observe that, with a single L-shaped detector, no BNS will have an angular resolution $\Delta\Omega_{90\%}$ below $10^2$ deg$^2$, and (in our sample realization) only 6 BNS/yr would have $\Delta\Omega_{90\%} \le 10^2$ deg$^2$ (to be compared with 184 BNS/yr for the 10 km triangle and 559 BNS/yr for the 2L configuration

<footnote n="13">¹³The difference with respect to the angle $\theta_{JN}$ shown in the BBH case is that, for BBHs, higher-order modes and precessing spin significantly help to determine the inclination angle.</footnote>
with 15 km arms at 45°, see table 43 in appendix C). The reconstruction of distances would also be comparatively poor, with just a handful of events measured to better than 10%; so, without other 3G detectors, a single L-shaped detector, even with 20 km arms, would miss all the aspects of the Science Case, such as multi-messenger astronomy or cosmology, that require good source localization.

### 3.2.2 Effects of a change in the ASD

Figure 14 shows the result for the two ASDs of the 10 km triangle and compares them with LVKI O5. Quite interesting results emerge from the panel on the angular resolution. A five-detector network, such as LVKI O5, would of course have excellent localization capabilities for some very close events, so that it can reach a localization below $1\deg^2$ on a few events per year, which a single 10 km triangle, even with the full ASD, cannot reach. However, this will be limited to a handful of particularly close events. What is also relevant, in the panel for $\Delta\Omega_{90\%}$ of figure 14, is the number of BNS with $\Delta\Omega_{90\%}$ below about $10^2\deg^2$. In fact, such an angular localization accuracy can already be sufficient to perform multi-messenger studies with wide field of view (FoV) electromagnetic observatories, as has been demonstrated by the current follow-up of gravitational-wave signals (see e.g. [4, 55]). In this case, the single triangle is quite superior to the LVKI O5 network, by detecting a number of well-localized sources higher by a factor $\sim 3.6$ (see table 43 in appendix C for a compilation of numerical values relative to these figures, and section 4 for a detailed discussion and implications for multi-messenger studies). However, this is true only for the triangle at full sensitivity, i.e. in the HFLF-cryo configuration. The ET full sensitivity localization significantly benefits from the use of the effect of Earth's rotation on the longer signal detected, thanks to the access to lower frequencies. The benefit is much larger for closer events where the BNS signal can be followed for a longer time. In terms of the number of BNSs with angular resolution below $10^2\deg^2$, the HF-only configuration is therefore much worse than LVKI O5. For most other parameters, in contrast, a 10 km triangle provides a significant jump with respect to LVKI O5 even in the HF-only configuration (except for luminosity distance and polarization, where it is worse; this is because these quantities are more correlated with angular localization). See again table 43 in appendix C for a compilation of numerical values.

Figure 15 shows the analogous results for the 2L configurations with 15 km arms at 45°; for comparison, we also show both LVKI O5 and the 10 km triangle at full sensitivity. First of all, we see again that, at the level of full sensitivities, the configurations with 15 km arms at 45° is superior to the 10 km triangle for all parameters. We also see that, for angular localization, the HF-only configuration is much worse than LVKI O5, at least for events with $\Delta\Omega_{90\%} < 10^2\deg^2$, which are the relevant ones for multi-messenger studies, while, for all other parameters, even the HF-only configurations provides a remarkable jump compared to LVKI O5. However, contrary to the BBH case, the HF-only configuration is now very much inferior to the 10 km triangle with the full ASD, particularly for sky localization. This is due to the fact that the improvement of the low-frequency sensitivity allows a BNS to remain in the detector bandwidth for hours and even up to a day, as we already mentioned above (see also figure 2 of [17]), and then the rotation of the Earth improves the BNS localization. This is less relevant for BBHs since most of them have a total mass greater than about $\mathcal{O}(20 M_{\odot})$, and stay in the detector bandwidth for a time too short to appreciate the effect of Earth's rotation. Therefore, the low-frequency sensitivity, and therefore the LF instrument, is more important for the localization of BNSs compared to BBHs. *This shows the crucial importance of the LF instrument for BNSs.*
### 3.2.3 Golden events

Figure 16 shows the redshift distribution of ‘golden events’ for BNSs, defined here either by the condition SNR ≥ 30, or by $\Delta d_L/d_L \le 0.2$, or by $\Delta\Omega_{90\%} \le 10^2 \text{ deg}^2$. These conditions are less stringent than those used for BBHs in figure 10, reflecting the fact that BNSs are intrinsically less loud than BBHs. The upper row compares the different geometries, all taken with their best ASD, and basically confirms the picture obtained for BBH golden events, compare with the upper row of figure 10: for the SNR, the 10 km triangle gives the less good results, while the other configurations are comparable, with the 2L with 20 km parallel arms being the best; for luminosity distance and angular localization, the 2L configurations with arms at 45° give the best results and, overall, appears to provide the best compromise between having a large number of events with large SNR out to large distances, and localizing them well in volume.

The middle row quantifies the loss of performance of the 10 km triangle when the LF instrument is absent. The loss is particularly dramatic for the accuracy on luminosity distance and sky localization. *This confirms the crucial importance of the LF instrument for BNS*, as we already discussed when commenting figure 15.

From the bottom row we see that, also on this metric, the 2L configurations with arms at 45° and the HF-only ASD gives results basically identical to the 10 km triangle with full HFLF-cryo sensitivity for the SNR distribution; in contrast, we already remarked on the full ensemble of events that the LF instrument is crucial for angular localization and accuracy on luminosity distance for BNSs, and we see the same information emerging from golden events, where the HF-only configuration of the 2L configurations with arms at 45° is very much inferior to the full 10 km triangle, in contrast to what happens for BBHs, compare with figure 10. As already discussed, this is due to the fact that, with a good low-frequency sensitivity, BNSs can stay in the detector bandwidth for hours to about 1 day, with the corresponding benefit in angular localization and in all parameters correlated with it.

### 3.2.4 Dependence on the population model

Finally, it is interesting to discuss how these results depend on the model that we have used for the BNS population (and on the specific random sample extracted from it). First of all, we remark that the differences between designs that emerge from these plots, and the conclusions that we draw from them, are not driven by the tails of the distributions (that, even for a given population model, would also depend on the sample realization), but rather by their overall behavior, which in general involves hundreds of events; even for the ‘golden events’, there are several tens to several hundreds events per bin, see figure 16. To understand in full generality how the results depend on the population model used is quite difficult; the full set of runs, on all different detector configurations studied, is computationally expensive, and it is not feasible to explore a large parameter space of population models in a reasonable timeframe. Our strategy has rather been to use a state-of-the art population model [52] for BNS and NSBH (and [54] for BBHs), which are able to match the LVK rates. However, a useful check, for BNS, can be performed as follows. For the redshift distribution of the BNS rate, current models are quite sophisticated and we do not expect great variations (furthermore, we are not much interested in the absolute numbers of detections in a given configuration, but rather in the relative performances of the detectors). There is large uncertainty in the overall normalization, given by the local rate, which for BNS can be up a factor 10 higher or lower than the one that we use. However, while this will influence significantly the absolute number of detections per year, again it will not affect much the comparison between detectors, which
is our main aim. For instance, if the local rate should turn out to be a factor of 2 smaller
than the one that we use, the plots that we show will have to be understood as relative to
2 years of data, rather than 1 yr; however, the conclusions for the relative performances of
the different design would be basically unaffected (of course, the tails of rare events will be
affected for a given fixed observation time but, again, the conclusions are not really dependent
on these tails). So, for our purposes the parameter that is most significant to vary, for BNS,
is the mass distribution. As mentioned on page 10, the results for BNS presented above were
obtained with a flat mass distribution between 1.1 and 2.5 solar masses. We have repeated
the analysis using a Gaussian mass distribution, with mean $1.33M_{\odot}$ and standard deviation
$0.09M_{\odot}$, sampled independently for both masses (we then impose that the heaviest one
corresponds to $m_1$). The results, for the six considered geometries in their full HFLF-cryo
configuration, are shown in figure 17. Comparing with figure 11, we see that the results are
very consistent. There is a somewhat smaller number of overall detections (because in the
gaussian distribution events with large masses are suppressed), but the parameter estimation
is quite similar (with differences in the tails, expected because of sample variance), and the
conclusions for the comparison between detector configurations are unchanged.
BNS

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Cumulative distributions of the number of detections per year, for the SNRs and for the error on the parameters, for BNS signals, for the six considered geometries, all with their best ASD, including xylophone configuration and cryogenic LF instrument.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Figure 12.** The joint accuracy on luminosity distance and angular resolution for BNSs. Left panel: the 15 km 2L at 45° (blue) compared to the 10 km triangle (green). Right panel: the 15 km 2L at 45° (blue) compared to the 15 km triangle (red).</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Comparison of SNR and parameter estimation error for a single 20 km L-shaped detector with the HFLF-cryo ASD, compared to the 10 km triangle, a 2L with 15 km arms at 45°, and the forecast for LVKI O5.</figcaption></figure>
BNS

<figure><img src="image_16.png" /><figcaption>**Figure 14.** Comparison of SNR and parameter estimation error for the 10 km triangle with different ASDs. By comparison, we also show the forecast for LVKI O5.</figcaption></figure>
BNS

<figure><img src="image_17.png" /><figcaption>**Figure 15.** Comparison of SNR and parameter estimation error for the 2L with 15 km arms at 45°, with the two different ASDs. For comparison, we also show the 10 km triangle at full sensitivity, and the forecast for LVKI O5.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Figure 16.** Redshift distribution of BNS detected with SNR $\ge$ 30 (left column), or relative error on the luminosity distance $\Delta d_L/d_L \le 0.2$ (central column), or with sky location $\Delta\Omega \le 100\deg^2$ (right column) for various detector configurations and sensitivity curves. The upper row shows the results for the six considered geometries, all with their best sensitivity, the central row for the 10 km triangle with the two considered ASDs, and the bottom row for the 2L with 15 km arms at 45° and the two considered ASDs. In each panel, we also show, for reference, the redshift distribution of the BNS population used (black solid line) and the events detected in the various configurations with SNR $\ge 8$ (dotted lines).</figcaption></figure>
BNS gaussian

<figure><img src="image_19.png" /><figcaption>**Figure 17.** As in figure 11, for a gaussian mass distribution of the two component neutron stars.</figcaption></figure>
### 3.3 ET in a network of 3G detectors

The above analysis studied the capabilities of ET alone (whether consisting of a single-site triangle or of two L-shaped detectors on widely separated sites), comparing the results obtained with different possible geometries and sensitivity curves. We now study how the different geometries for ET (triangle or 2L) perform when they are included in a network of 3G detectors which includes Cosmic Explorer. The reference CE configuration currently consists of two L-shaped detectors (on the Earth's surface rather than underground), one with 40 km arms and one with 20 km arms [14]. We will refer to this configuration as '2CE' and we will study how different choices for the ET geometry affect the performance of a network made by ET and 2CE. Given the current uncertainties, it is also instructive to investigate what happens when ET (in a triangular or 2L geometry) is part of a network with a single CE detector, which we take to be the 40 km one ('1CE'). In order to avoid a proliferation of plots, we limit here to three options for ET: the 10 km triangle, and the 15 km 2L configurations, either with parallel arms or with arms at 45°. Therefore, we consider the following networks:

1. ET as a triangle with 10 km arms, together with 2CE (20+40 km)

2. ET as a 2L of 15 km and parallel arms, together with 2CE (20+40 km)

3. ET as a 2L of 15 km and arms at 45°, together with 2CE (20+40 km)

4. ET as a triangle with 10 km arms, together with 1CE (40 km)

5. ET as a 2L of 15 km and parallel arms, together with 1CE (40 km)

6. ET as a 2L of 15 km and arms at 45°, together with 1CE (40 km)

In particular, the last two options correspond to a global network of three L-shaped detectors, two of 15 km and with the ET ASD, and one of 40 km and the CE ASD. In all these configurations, for ET we will consider only the full sensitivity, corresponding to the ASD that we have labelled as 'HFLF-cryo' and, for CE, we use the latest publicly available official sensitivity curves.<fnref n="14" />

Figure 18 shows the SNR distribution and the errors on parameter estimation for BBHs for the six networks considered, while figure 19 shows the corresponding results for BNSs. From figure 18, to be compared with the green, blue and orange lines in figure 5 referring to ET alone, we see that, as expected, the overall performance of the network is significantly better than that of ET alone for angular localization and distance determination. As an example, the BBH/yr localized to better than 1 deg² raise from a few tens (depending on the ET geometry) to a few times 10³. On the other parameters, the gain from adding one or two CE detectors results typically in an increase by a factor of order 2 of the number of events with accuracy better than a given value of the chosen parameter; e.g., in our specific realization, for the symmetric mass ratio $\eta$, the number of events per year with $\Delta\eta \le 10^{-6}$

<footnote n="14">¹⁴The most recent PSDs of CE can be found at https://dcc.cosmicexplorer.org/CE-T2000017/public. We used the locations and orientations reported in rows C and N of table III of [30]. In particular, the 40 km detector is placed in Idaho and the 20 km one in New Mexico. These orientations are such that the relative angle between the two CE detectors, measured with respect to the great circle that joins them, is about 56.0°. When ET is taken to be in the 2L configuration, with our choice of orientations the relative angle between the Sardinia detector and the CE-Idaho is about 191.2° and that between the Sardinia detector and the CE-New Mexico one is about 237.6°. As in the discussion in footnote 4, if a 2L configuration should be retained for ET, a careful study of the optimization of the relative orientations of the ET+2CE detector network will become necessary.</footnote>
raises from 559 for ET alone in the 10 km triangle configuration, to about 922 in a network with 1CE and 1018 with 2CE; similar results hold for the other parameters.

As we see from tables 39, 40 and 41 in appendix C (factors of 2 are difficult to appreciate on the logarithmic scale of figure 18), in a network with a 40km CE, for BBHs the different geometries chosen for ET induce overall differences in the results at the level of factors ~ 1.5 or smaller. As an example, the number of BBH/yr with $d_L$ measured better than 1% raises from 2901 for the 10 km triangle+1CE, to 4301 for 2L-15km-45°+1CE, while the number of BBH/yr localized to better than 10 deg² raises from 3.0 × 10⁴ to 3.6 × 10⁴.

For BNS, the effect of the ET geometry in a network with 1 or 2CE can be seen from figure 19, and especially from tables 42, 43 and 44. For instance, the number of BNS/yr with SNR ≥ 150 raises from 87 for the 10 km triangle+1CE, to 116 for 2L-15km-45°+1CE, while those with $d_L$ measured better than 10% increases by about a factor of 2, from 4100 for the 10 km triangle+1CE to 7949 for 2L-15km-45°+1CE. Similar differences appear for the chirp mass, symmetric mass ratio and tidal deformability. For instance, the number of BNS/yr with tidal deformability measured to better than 10% raises from 2669 for the 10 km triangle+1CE to 4753 for 2L-15km-45°+1CE, and from 3847 for the 10 km triangle+2CE to 6319 for 2L-15km-45°+2CE.

So, among the three geometries considered here, the one that gives consistently the best results is, again, the 2L with 15 km arms at 45°, with typical improvement with respect to the 10 km triangle of about a factor of ~ (1.5 – 2). The configuration 2L-15km-0°, when in a network with 1CE, gives similar performances to 2L-15km-45° for masses or tidal deformability, but is worse by a factor ~ 2.5 for angular localization and ~ 4 for luminosity distance. For instance, the number of BNS/yr with $d_L$ measured better than 10% is 2079 for 2L-15km-0°+1CE and 7949 for 2L-15km-45°+1CE. In a network with 2CE the differences become smaller.

*The overall conclusion is that the difference between the performances of the ET geometries remains visible, at the level of factors ~ (1.5 – 2) for BNSs, and somewhat smaller for BBHs, also when ET is in a network with 1 or 2 CE.*
BBH

<figure><img src="image_20.png" /><figcaption>**Figure 18.** Comparison of SNR and parameter estimation error for BBHs, for the six 3G detector networks considered.</figcaption></figure>
BNS

<figure><img src="image_21.png" /><figcaption>**Figure 19.** Comparison of SNR and parameter estimation error for BNSs, for the six 3G detector networks considered.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Figure 20.** Redshift distribution of detected BBHs with SNR ≥ 100 (left column), or relative error on the luminosity distance $\Delta d_L/d_L \le 0.01$ (central column), or sky location $\Delta\Omega \le 1\deg^2$ (right column), for the six 3G the detector networks considered. In each panel, we also show, for reference, the redshift distribution of the BBH population used (black solid line), and of the events detected in the various configurations with SNR ≥ 8 (dotted lines, graphically almost indistinguishable from the full population on this scale).</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Figure 21.** Redshift distribution of detected BNS with SNR ≥ 30 (left column), or relative error on the luminosity distance $\Delta d_L/d_L \le 0.05$ (central column), or sky location $\Delta\Omega \le 5\deg^2$ (right column) for the six 3G the detector networks considered. In each panel, we also show, for reference, the redshift distribution of the BNS population used (black solid line), and of the events detected in the various configurations with SNR ≥ 8 (dotted lines).</figcaption></figure>
# 4 Multi-messenger astrophysics

In this section we study the impact of the different detector geometries and different ASDs on the multi-messenger capabilities of ET operating in synergy with electromagnetic (EM) observatories. ET will observe together with a new generation of innovative EM observatories, such as CTA [56], Athena [57], the Vera Rubin Observatory [58], JWST [59], ELT [60], SKA [61] and with the possible mission concepts UVEX [62], THESEUS [63], HERMES [64], GRINTA, ASTROGAM [65] and AMEGO [66]. These observatories will probe the formation, evolution, and physics of BNS systems in connection with kilonovae (KNe) and short gamma-ray bursts (GRBs) along the star formation and chemical evolution of the Universe. The multi-messenger studies will tremendously benefit from the larger sample of joint GW/EM detections and, in the closer Universe, from the greatly improved parameter estimates of ET with respect to the current detectors.

Taking into account the sky-localization capability of ET and the fact that the KN emission is intrinsically fainter than that of GRBs, future multi-messenger astronomy will differ at closer redshifts ($< 0.3 – 0.4$), where it will be possible to detect KN emission using the wide-field telescopes, such as the Vera Rubin Observatory, from the distant Universe where the detectable counterparts at large redshift will be almost uniquely observed in the high-energy band through the beamed emission of GRBs. While the joint GW/KN detections will enable unprecedented studies of the enrichment of the Universe with heavy elements, of nuclear physics to constrain the equation of state (EoS) of NSs and cosmology estimating the expansion rate of the Universe, the joint GW/GRBs detections will unveil the structure and properties of relativistic jets, the emission mechanism of short GRBs, the role of merger remnants such as magnetars, and will make it possible to estimate cosmological parameters and test modified gravity at the cosmological scale.

The main parameters determining the joint GW/EM detections, and thus affecting the multi-messenger performance of ET, are the ability to localize the source and the volume of the Universe up to which ET will be able to detect the source in terms of achieved redshifts. Another important aspect of third-generation GW detectors is the ability to detect BNSs before the merger and send pre-merger alerts with a good estimate of the sky localization of the source. This is critical for detecting prompt and early emissions in EM bands where all-sky monitors are not operating.

Here, we examine the sky-localization capabilities as a function of redshift and the pre-merger alert scenarios, and we evaluate the joint detections for ET observation together with examples of high-energy satellites and optical wide-FoV telescopes. Differently from the other sections, in this section, we use an SNR detection threshold of 8 taking into account that the presence of a counterpart increases the significance of the source detection.

## 4.1 BNS sky-localization and pre-merger alerts

The sky-localization capability of GW detectors has a major impact in the efficiency of the search for a counterpart determining the number of pointings of the EM instruments to cover the GW signal location, the corresponding amount of observational time to be used and the search sensitivity, and the efficiency to select and remove contaminating transients. Starting from the BNS population described in section 3, GWFISH [31] is used to evaluate the localization capability and build table 1, which gives the expected number of detections (SNR $\ge$ 8) per year with sky-localization uncertainty (given as 90% c.l.) smaller than a threshold of 10, 40, 100, 1000 deg$^2$ for the different ET configurations (we do not consider the 2L aligned
<figure><img src="image_24.png" /></figure>

**Figure 22.** Redshift distribution of the sky-localization uncertainty (given as 90% credible region) of randomly oriented BNS systems as a function of redshift for four detector geometries: the 10 km triangle (Δ10 km), the 15 km triangle (Δ15 km), the 2L with 15 km misaligned arms (2L 15 km), and the 2L with 20 km misaligned arms (2L 20 km). The plots on left and central columns show the full (HFLF cryo) sensitivity detectors. In comparison, the plots on the right columns show the performance of the detectors operating only with the HF interferometers. The absolute numbers are relative to one year of observation and assuming a duty cycle of 0.85 as described in the text.

configurations because as shown in section 3.2 they yield worse localization capabilities with
respect to the misaligned configurations). Figure 22 shows the redshift distribution of the
sky-localization uncertainty for all the detected BNS events. The distribution is given as a
function of redshift for four detector geometries; 10 km triangle, 15 km triangle, 2L with 15 km
misaligned arms (i.e. arms at a relative orientation of 45°), and 2L with 20 km misaligned
arms. The different colours show the detections localized within a certain threshold on the
sky-localization uncertainty. Comparing the plots on the left and central columns of the fig-
ure, which show the full (HFLF cryo) sensitivity detectors, it emerges that the 2L with 15 km
misaligned arms performs better than the 10 km triangle and is comparable to 15 km triangle
(in agreement with the results shown in the panel for ΔΩ90% in figure 11). Looking at the
events localized with ΔΩ90% < 10 deg² (100 deg²), the number of detection for the 2L with
15 km misaligned arms increases by factor 2 reaching z of 0.3 (1.2) with respect to the 10 km
triangle, whose well-localized detections reach z = 0.2 (0.7). The 2L with 20 km misaligned
arms increases the number of well-localized BNSs by a factor 4 with respect to the 10 km tri-
angle reaching z = 0.4 (1.6) with events localized within 10 deg² (100 deg²). Table 1 shows also
the expected number of detections per year within certain sky-localization uncertainty select-
ing events with a viewing angle, Θv,<fnref n="15" /> smaller than 15°. A fraction of these events is expected
to produce detectable high-energy emission powered by the GRB relativistic jet, assumed to
be perpendicular to the orbital plane. Also for on-axis events, the 2L with 15 km misaligned
arms performs better than the 10 km triangle and is comparable to the 15 km triangle.

<footnote n="15"><sup>15</sup>The viewing angle is defined as Θ<sub>v</sub> = ι for 0 ≤ ι ≤ 90° and Θ<sub>v</sub> = |ι - 180°| for 90° < ι < 180°.</footnote>
Full (HFLF cryo) sensitivity detectors

<table><caption>**Table 1.** Expected number of detection (SNR ≥ 8) per year with sky-localization uncertainty ΔΩ<sub>90%</sub>(deg<sup>2</sup>) smaller than the threshold indicated in the first column. While the columns 2-5 give the detections for BNS systems randomly oriented, the columns 5-9 give the detection of on-axis events, whose viewing angle is smaller than 15°. The numbers are relative to one year of observation assuming a duty cycle of 0.85 as described in the text.</caption><thead><tr><th rowspan="2">ΔΩ<sub>90%</sub>(deg<sup>2</sup>)</th><th colspan="4">All orientation BNSs</th><th colspan="4">BNSs with viewing angle Θ<sub>v</sub> &lt; 15°</th></tr><tr><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th></tr></thead><tbody><tr><td>10</td><td>11</td><td>27</td><td>24</td><td>45</td><td>0</td><td>1</td><td>2</td><td>5</td></tr><tr><td>40</td><td>78</td><td>215</td><td>162</td><td>350</td><td>8</td><td>22</td><td>20</td><td>33</td></tr><tr><td>100</td><td>280</td><td>764</td><td>644</td><td>1282</td><td>26</td><td>74</td><td>68</td><td>133</td></tr><tr><td>1000</td><td>2112</td><td>5441</td><td>7478</td><td>13482</td><td>272</td><td>632</td><td>1045</td><td>1725</td></tr></tbody></table>

HF sensitivity detectors

<table><caption>**Table 2.** Same as table 1 but considering the detectors operating with only the HF interferometers.</caption><thead><tr><th rowspan="2">ΔΩ<sub>90%</sub>(deg<sup>2</sup>)</th><th colspan="4">All orientation BNSs</th><th colspan="4">BNSs with viewing angle Θ<sub>v</sub> &lt; 15°</th></tr><tr><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th></tr></thead><tbody><tr><td>10</td><td>0</td><td>1</td><td>5</td><td>5</td><td>0</td><td>0</td><td>2</td><td>2</td></tr><tr><td>40</td><td>4</td><td>10</td><td>20</td><td>47</td><td>0</td><td>5</td><td>6</td><td>17</td></tr><tr><td>100</td><td>14</td><td>53</td><td>76</td><td>144</td><td>7</td><td>33</td><td>35</td><td>64</td></tr><tr><td>1000</td><td>145</td><td>548</td><td>1662</td><td>3378</td><td>80</td><td>336</td><td>672</td><td>1302</td></tr></tbody></table>

Losing the low-frequency interferometers, the HF GW detectors detect a significantly smaller number of well-localized events. However, as shown comparing table 1 and table 2,<fnref n="16" /> the decrease of detections of well-localized events is more severe for the triangle configurations than for 2L misaligned. The right column of figure 22 shows the redshift distribution of the sky-localization uncertainty for the HF 15 km triangle and 2L with 20 km misaligned arms. The comparison with the corresponding configurations HFLF cryo shows that a large fraction of well-localized events is already missed at small redshifts. Focusing on the on-axis events, the use of only HF interferometers decreases the number of well-localized events but in a percentage of a smaller fraction with respect to the events randomly oriented. While the HF 2L with 15 km and 20 km misaligned arms localize worse than the HFLF-cryo 10 km triangle for randomly oriented systems, for on-axis events, the HF 2L with 15 km misaligned arms localization capability is comparable to the HFLF-cryo 10 km triangle and the HF 2L with 20 km better than the HFLF-cryo 10 km triangle.

The detection of prompt/early multi-wavelength emission from BNS mergers is critical to probe the central engine of GRBs, particularly to understand the jet composition, the particle acceleration mechanism, the radiation and energy dissipation mechanisms [see e.g.

<footnote n="16">¹⁶The different numbers, comparing tables 1 and 2 with table 43 in appendix C, reside in the different SNR threshold, assumed to be 8 for multi-messenger studies and 12 for the metrics of section 3 and appendix C. The larger difference on the HF-only estimates is also related to the different approach of GWFISH (used for the tables 1 and 2) which regularizes the close-to-singular matrices [31] instead of discarding the ill-conditioned Fisher matrix as GWFAST (used for the table 43) [17, 32]. As described in the footnote 9, for the HF-only case the percentage of discarded events approaches 10%.</footnote>
<table><caption>Full (HFLF cryo) sensitivity detectors</caption><thead><tr><th rowspan="2">Configuration</th><th rowspan="2">ΔΩ<sub>90%</sub><br/>[deg<sup>2</sup>]</th><th colspan="3">All orientation BNSs</th><th colspan="3">BNSs with Θ<sub>v</sub> &lt; 15°</th></tr><tr><th>30 min</th><th>10 min</th><th>1 min</th><th>30 min</th><th>10 min</th><th>1 min</th></tr></thead><tbody><tr><td rowspan="4">Δ10km</td><td>10</td><td>0</td><td>1</td><td>5</td><td>0</td><td>0</td><td>0</td></tr><tr><td>100</td><td>10</td><td>39</td><td>113</td><td>2</td><td>8</td><td>20</td></tr><tr><td>1000</td><td>85</td><td>293</td><td>819</td><td>10</td><td>34</td><td>132</td></tr><tr><td>All detected</td><td>905</td><td>4343</td><td>23597</td><td>81</td><td>393</td><td>2312</td></tr><tr><td rowspan="4">Δ15km</td><td>10</td><td>1</td><td>5</td><td>11</td><td>0</td><td>1</td><td>1</td></tr><tr><td>100</td><td>41</td><td>109</td><td>281</td><td>6</td><td>14</td><td>36</td></tr><tr><td>1000</td><td>279</td><td>806</td><td>2007</td><td>33</td><td>102</td><td>295</td></tr><tr><td>All detected</td><td>2489</td><td>11303</td><td>48127</td><td>221</td><td>1009</td><td>4024</td></tr><tr><td rowspan="4">2L 15 km misaligned</td><td>10</td><td>0</td><td>1</td><td>8</td><td>0</td><td>0</td><td>0</td></tr><tr><td>100</td><td>20</td><td>54</td><td>169</td><td>2</td><td>7</td><td>26</td></tr><tr><td>1000</td><td>194</td><td>565</td><td>1399</td><td>23</td><td>73</td><td>199</td></tr><tr><td>All detected</td><td>2172</td><td>9598</td><td>39499</td><td>198</td><td>863</td><td>3432</td></tr><tr><td rowspan="4">2L 20 km misaligned</td><td>10</td><td>2</td><td>4</td><td>15</td><td>1</td><td>1</td><td>2</td></tr><tr><td>100</td><td>39</td><td>118</td><td>288</td><td>7</td><td>19</td><td>47</td></tr><tr><td>1000</td><td>403</td><td>1040</td><td>2427</td><td>47</td><td>128</td><td>346</td></tr><tr><td>All detected</td><td>4125</td><td>17294</td><td>56611</td><td>363</td><td>1588</td><td>4377</td></tr></tbody></table>

**Table 3.** Number of BNS mergers per year detected (SNR ≥ 8) before the merger within $z = 1.5$ for the different full (HFLF cryo) sensitivity ET configurations. Three pre-merger scenarios (30, 10, and 1 minute(s) before the merger) are shown in different columns. For each detector configuration, the rows give the number of detections with sky-localization (90% c.l.) within 10, 100, 1000 deg$^2$, and all the detected sources. The numbers of GW detections per year take into account the ET duty cycle as described in the text.

67]. Regarding the kilonova emission, it has been shown that the early phase emission is particularly sensitive to the structure of the outer sub-relativistic ejecta (see, e.g., [68]) and can give rise to early UV emission detectable by wide-field satellites such as ULTRASAT [69], and mission concepts as UVEX [62] and Dorado [70] which can benefit of pre-merger alerts. Detecting a BNS signal before the merger makes it possible to eventually detect electromagnetic emission precursors. It makes it possible also to image the sky-localization before the merger that, especially for optical observations, is extremely useful to remove contaminating transients present before the merger. Considering all the benefits of pre-merger alerts for multi-messenger astronomy, here we explore the ET capabilities to detect BNSs and give a sky-localization before the merger.
HF sensitivity detectors

<table><caption>**Table 4.** Same as table 3 but for the HF sensitivity ET.</caption><thead><tr><th rowspan="2">Configuration</th><th rowspan="2">ΔΩ<sub>90%</sub><br>[deg<sup>2</sup>]</th><th colspan="3">All orientation BNSs</th><th colspan="3">BNSs with Θ<sub>v</sub> &lt; 15°</th></tr><tr><th>30 min</th><th>10 min</th><th>1 min</th><th>30 min</th><th>10 min</th><th>1 min</th></tr></thead><tbody><tr><td rowspan="3">Δ10km</td><td>100</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>1000</td><td>0</td><td>0</td><td>4</td><td>0</td><td>0</td><td>1</td></tr><tr><td>All detected</td><td>0</td><td>3</td><td>317</td><td>0</td><td>0</td><td>26</td></tr><tr><td rowspan="3">Δ15km</td><td>100</td><td>0</td><td>0</td><td>2</td><td>0</td><td>0</td><td>0</td></tr><tr><td>1000</td><td>0</td><td>0</td><td>10</td><td>0</td><td>0</td><td>4</td></tr><tr><td>All detected</td><td>2</td><td>8</td><td>891</td><td>0</td><td>1</td><td>84</td></tr><tr><td rowspan="3">2L 15 km misaligned</td><td>100</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>1000</td><td>0</td><td>0</td><td>7</td><td>0</td><td>0</td><td>3</td></tr><tr><td>All detected</td><td>0</td><td>7</td><td>743</td><td>0</td><td>1</td><td>69</td></tr><tr><td rowspan="3">2L 20 km misaligned</td><td>100</td><td>0</td><td>0</td><td>3</td><td>0</td><td>0</td><td>0</td></tr><tr><td>1000</td><td>0</td><td>0</td><td>13</td><td>0</td><td>0</td><td>6</td></tr><tr><td>All detected</td><td>2</td><td>11</td><td>1535</td><td>0</td><td>1</td><td>146</td></tr></tbody></table>

Table 3 shows the number of BNS mergers per year detected (SNR ≥ 8) before the merger within $z = 1.5$, for the different full (HFLF cryo) sensitivity ET configurations. We consider three pre-merger scenarios; 30, 10 and 1 minute(s) before the merger. For each ET configuration, we give the number of detection with sky-localization (90% c.l.) within 10, 100, 1000 deg$^2$, and all the detected sources. We show the results for the BNSs randomly oriented of our population (columns 3-5) and select among them the BNSs with viewing angles smaller than 15° (columns 6-8). The on-axis detections are particularly important to detect the prompt/early beamed emission associated with relativistic jets, for example the very high energy prompt emission never detected so far from GRBs (see [71] for the perspectives with the Cherenkov Telescope Array [72]). *Focusing on well-localized events detected pre-merger with a sky-localization smaller than 100 deg$^2$, we highlight that the 15 km triangle turns out to perform better than the 10 km triangle and the 2L with 15 km misaligned arms, and its performance is similar to the 2L with 20 km misaligned arms. The 2L with 15 km misaligned arms performs better than the 10 km triangle. The results are similar for on-axis events.*

Table 4 shows the dramatic decrease of pre-merger alerts without low-frequency. For all the ET configurations, there are no localized pre-merger detections, except a few events with sky localization smaller than 1000 deg$^2$ per year one minute before the merger (a few < 100 deg$^2$ for 2L20km).

## 4.2 Gamma-ray bursts: joint GW and high-energy detections

Following the approach described in [18], we evaluate the expected detections of prompt and afterglow emission of short GRBs associated with BNS mergers. We explore the joint
<table><thead><tr><th>INSTRUMENT</th><th>band<br>MeV</th><th><i>F</i><sub>lim</sub><br>ph cm<sup>-2</sup> s<sup>-1</sup></th><th>FoV/4π</th><th>loc. acc.</th><th>Status</th></tr></thead><tbody><tr><td><i>Fermi</i>-GBM</td><td>0.01–25</td><td>0.5</td><td>0.75</td><td>5 deg</td><td>Operating mission</td></tr><tr><td>GECAM</td><td>0.006–5</td><td>2 × 10<sup>-8</sup>(*)</td><td>1.0</td><td>1 deg</td><td>Operating mission</td></tr><tr><td>HERMES</td><td>0.05–0.3</td><td>0.2</td><td>1.0</td><td>1 deg</td><td>Mission concept<br>Pathfinder next few yrs</td></tr><tr><td>GRINTA-TED</td><td>0.02–10</td><td>0.45</td><td>0.64</td><td>5 deg(**)</td><td>Mission concept</td></tr><tr><td>ASTROGAM</td><td>0.03–200</td><td>0.29</td><td>0.27</td><td>&lt; 1.8 deg</td><td>Mission concept</td></tr><tr><td>THESEUS-XGIS</td><td>0.002–10</td><td>3 × 10<sup>-8</sup>(*)</td><td>0.16</td><td>&lt; 15 arcmin</td><td>Mission concept<br>Possible launch 2037</td></tr></tbody></table>

<caption>**Table 5.** Instruments characteristics.</caption>

detection capabilities of the different configurations of ET operating with gamma-ray and X-ray satellites. As reference instruments, we use some examples of current and upcoming satellites and mission concepts. We analyze different observational strategies: survey mode and target pointing.

### 4.2.1 Prompt emission

Starting from our astrophysically-motivated population of BNS mergers we model the high-energy signals associated with the mergers making the BNS population able to reproduce the rate and the properties of all the short GRBs observed so far (see [18] for details). For the fraction of BNS mergers in our population that are expected to produce a jet (about 20% of the BNS see [18]), we inject both the GW and EM signals and recover them based on the detection efficiency of the GW detector and EM satellites, respectively. We define a joint detection if the signal is above the detection threshold simultaneously in both the GW and EM observatories. To properly compare the capabilities of different ET configurations and avoid differences due to the uncertainties in the EM emission, for the whole set of GW simulations we associate to each BNS population the same realization of the EM signal derived from the MCMC.

For the detection of the prompt emission, we consider the following $\gamma$-ray instruments: the Gamma-ray Burst Monitor (GBM) on board of Fermi [73], the Gravitational wave high-energy Electromagnetic Counterpart All-sky Monitor [GECAM; 74], the High Energy Rapid Modular Ensemble of Satellites [HERMES; 64], the Transient Event Detector (TED) on board of the Gamma-Ray International Transient Array Observatory (GRINTA), ASTROGAM [65], and the X/Gamma-ray Imaging Spectrometer (XGIS) on board of the Transient High-Energy Sky and Early Universe Surveyor [THESEUS; 63]. The properties of these instruments are summarized in table 5.

Table 6 shows the numbers of joint GW+$\gamma$-ray detections during one year of observation for the different $\gamma$-ray satellites operating in survey mode together with different ET full sensitivity configurations. These numbers go from ten to one hundred depending on the satellite and ET configurations. We highlight that, while some satellites maximize the number of de-

<footnote>(*) expressed in ph erg cm$^{-2}$ s$^{-1}$</footnote>

<footnote>(**) 65%(42%) of the TED/GW joint detections are detectable by the Hard X-ray Imager HXI on board of GRINTA by repointing HXI on the TED detection in 60s (5 min). HXI is expected to localize at order of 30 arcsec.</footnote>
<table><caption>Full (HFLF cryo) sensitivity detectors</caption><thead><tr><th>Instrument</th><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th></tr></thead><tbody><tr><td>Fermi-GBM</td><td>31<sup>+9</sup><sub>-9</sub></td><td>42<sup>+11</sup><sub>-13</sub></td><td>39<sup>+11</sup><sub>-9</sub></td><td>44<sup>+13</sup><sub>-11</sub></td><td>61<sup>+12</sup><sub>-11</sub>%</td><td>83<sup>+9</sup><sub>-10</sub>%</td><td>79<sup>+8</sup><sub>-11</sub>%</td><td>89<sup>+4</sup><sub>-8</sub>%</td></tr><tr><td>GECAM</td><td>61<sup>+39</sup><sub>-25</sub></td><td>89<sup>+54</sup><sub>-34</sub></td><td>81<sup>+51</sup><sub>-32</sub></td><td>96<sup>+52</sup><sub>-36</sub></td><td>51<sup>+5</sup><sub>-6</sub>%</td><td>74<sup>+5</sup><sub>-5</sub>%</td><td>70<sup>+3</sup><sub>-6</sub>%</td><td>80<sup>+4</sup><sub>-4</sub>%</td></tr><tr><td>HERMES</td><td>86<sup>+31</sup><sub>-28</sub></td><td>120<sup>+40</sup><sub>-31</sub></td><td>117<sup>+37</sup><sub>-34</sub></td><td>132<sup>+34</sup><sub>-34</sub></td><td>55<sup>+9</sup><sub>-7</sub>%</td><td>78<sup>+8</sup><sub>-7</sub>%</td><td>74<sup>+9</sup><sub>-9</sub>%</td><td>85<sup>+5</sup><sub>-6</sub>%</td></tr><tr><td>GRINTA-TED</td><td>77<sup>+31</sup><sub>-25</sub></td><td>107<sup>+31</sup><sub>-28</sub></td><td>98<sup>+31</sup><sub>-25</sub></td><td>114<sup>+34</sup><sub>-28</sub></td><td>57<sup>+10</sup><sub>-9</sub>%</td><td>79<sup>+8</sup><sub>-8</sub>%</td><td>74<sup>+9</sup><sub>-9</sub>%</td><td>85<sup>+5</sup><sub>-5</sub>%</td></tr><tr><td>ASTROGAM</td><td>18<sup>+8</sup><sub>-5</sub></td><td>24<sup>+9</sup><sub>-7</sub></td><td>24<sup>+9</sup><sub>-6</sub></td><td>27<sup>+8</sup><sub>-7</sub></td><td>59<sup>+11</sup><sub>-9</sub>%</td><td>80<sup>+8</sup><sub>-8</sub>%</td><td>77<sup>+8</sup><sub>-9</sub>%</td><td>86<sup>+6</sup><sub>-9</sub>%</td></tr><tr><td>THESEUS-XGIS</td><td>10<sup>+3</sup><sub>-3</sub></td><td>13<sup>+3</sup><sub>-3</sub></td><td>13<sup>+3</sup><sub>-3</sub></td><td>15<sup>+3</sup><sub>-4</sub></td><td>57<sup>+9</sup><sub>-10</sub>%</td><td>79<sup>+8</sup><sub>-9</sub>%</td><td>73<sup>+11</sup><sub>-7</sub>%</td><td>85<sup>+7</sup><sub>-5</sub>%</td></tr></tbody></table>

**Table 6.** Columns 2-5 give the numbers of joint GW+γ-ray detections during one year of observation for different combinations of γ-ray instruments operating in survey mode together with different ET full sensitivity configurations. The absolute numbers do not assume duty cycle for the satellites. Columns 6-9 give the fraction of detected short GRBs which will have a GW counterparts.

tections (which is useful for population studies) sacrificing the localization capability, others with a smaller number of detections give the sky-localization necessary to drive the ground-based and space-borne follow-up. This is critical for identifying the host galaxy, evaluating a redshift, and completely characterising the source and its environment. An important piece of information is given in columns 6-9, where we show what is the fraction of γ-ray bursts detected by each satellite which will have an associated GW signal. Here, we find that these fractions go from 51% – 61% for the 10 km triangle, 74% – 83% for the 15 km triangle, 70% – 79% for the 2L with 15 km misaligned arms, 80% – 89% for the 2L with 20 km misaligned arms. The ET 10 km triangle will make possible remarkable results for GRB science by detecting GW signals for a large fraction of short GRBs observed by γ-ray satellites, and it represents an impressive step forward from the LIGO, Virgo and KAGRA network, whose BNS horizon strongly limits the ability to detect events with the relativistic jet aligned with the line-of-sight (see e.g. [75, 76]). Then, the percentage of GRBs with a GW counterpart significantly increases going from the 10 km triangle to 15 km and 20 km configurations. The 15 km triangle is marginally better than the 2L with 15 km misaligned arms. Figure 23 takes as an example HERMES (full constellation) and shows the joint detections during one year. The plot shows that the most significant improvement is from the 10 km triangle to a 15 km configuration. The 15 and 20 km configurations are able to increase the number of joint detection at redshift larger than 0.9 with respect to the 10 km triangle. This is particularly important to estimate cosmological parameters or to test modified gravity at cosmological scales.

Considering the GW detectors operating with only the high-frequency instrument (see table 7), we find that the percentage of detected short GRBs with an associated GW signal significantly decreases for each configuration. The percentages become 29% – 39% for the 10 km triangle, 54% – 64% for the 15 km triangle, 49% – 60% for the 2L with 15 km misaligned arms, 66% – 76% for the 2L with 20 km misaligned arms. In terms of percentage of short GRB with a GW signal, the performance of the 15 km triangle and 2L15km HF-only detectors is comparable to the 10 km triangle full sensitivity. This is also shown in the left plot of figure 24. The right plot of figure 24 shows that 2L20km HF results to be better than the 10 km triangle full sensitivity, but again going to lower frequency significantly increase the 2L20km performance making possible to detect the GW signals associated with a large fraction (85%) of short GRBs detected by HERMES.
<table><caption>HF sensitivity detectors</caption><thead><tr><th>Instrument</th><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th></tr></thead><tbody><tr><td>Fermi-GBM</td><td>20<sup>+8</sup><sub>-7</sub></td><td>33<sup>+9</sup><sub>-9</sub></td><td>29<sup>+11</sup><sub>-9</sub></td><td>38<sup>+12</sup><sub>-10</sub></td><td>39<sup>+11</sup><sub>-8</sub>%</td><td>64<sup>+12</sup><sub>-11</sub>%</td><td>60<sup>+12</sup><sub>-11</sub>%</td><td>76<sup>+9</sup><sub>-9</sub>%</td></tr><tr><td>GECAM</td><td>35<sup>+21</sup><sub>-15</sub></td><td>62<sup>+38</sup><sub>-22</sub></td><td>58<sup>+38</sup><sub>-22</sub></td><td>77<sup>+47</sup><sub>-30</sub></td><td>29<sup>+4</sup><sub>-5</sub>%</td><td>54<sup>+4</sup><sub>-5</sub>%</td><td>49<sup>+4</sup><sub>-7</sub>%</td><td>66<sup>+4</sup><sub>-6</sub>%</td></tr><tr><td>HERMES</td><td>52<sup>+21</sup><sub>-18</sub></td><td>91<sup>+30</sup><sub>-29</sub></td><td>83<sup>+28</sup><sub>-28</sub></td><td>107<sup>+40</sup><sub>-31</sub></td><td>33<sup>+7</sup><sub>-8</sub>%</td><td>58<sup>+10</sup><sub>-8</sub>%</td><td>53<sup>+10</sup><sub>-8</sub>%</td><td>71<sup>+8</sup><sub>-8</sub>%</td></tr><tr><td>GRINTA-TED</td><td>46<sup>+22</sup><sub>-16</sub></td><td>80<sup>+31</sup><sub>-25</sub></td><td>74<sup>+28</sup><sub>-25</sub></td><td>94<sup>+33</sup><sub>-23</sub></td><td>34<sup>+9</sup><sub>-9</sub>%</td><td>61<sup>+9</sup><sub>-11</sub>%</td><td>55<sup>+9</sup><sub>-10</sub>%</td><td>72<sup>+9</sup><sub>-9</sub>%</td></tr><tr><td>ASTROGAM</td><td>12<sup>+6</sup><sub>-5</sub></td><td>19<sup>+7</sup><sub>-5</sub></td><td>18<sup>+6</sup><sub>-6</sub></td><td>23<sup>+8</sup><sub>-7</sub></td><td>37<sup>+11</sup><sub>-10</sub>%</td><td>62<sup>+9</sup><sub>-11</sub>%</td><td>57<sup>+10</sup><sub>-9</sub>%</td><td>74<sup>+9</sup><sub>-10</sub>%</td></tr><tr><td>THESEUS-XGIS</td><td>6<sup>+2</sup><sub>-2</sub></td><td>10<sup>+3</sup><sub>-3</sub></td><td>9<sup>+3</sup><sub>-3</sub></td><td>12<sup>+3</sup><sub>-3</sub></td><td>34<sup>+8</sup><sub>-9</sub>%</td><td>59<sup>+10</sup><sub>-8</sub>%</td><td>54<sup>+10</sup><sub>-9</sub>%</td><td>71<sup>+9</sup><sub>-9</sub>%</td></tr></tbody></table>

**Table 7.** Same as 6 but considering the GW detectors operating with only the HF interferometers.

<figure><img src="image_25.png" /></figure>

**Figure 23.** Histogram of the joint detection of ET and HERMES observing in survey mode during one year. The solid lines of different colors represent the different configurations of ET full sensitivity. The dashed line shows the prompt emission of short γ-ray bursts expected to be detected by HERMES during one year of observations.

<figure><img src="image_26.png" /></figure>

**Figure 24.** Histogram of the joint detection of ET and HERMES observing in survey mode during one year. The plot shows the comparison among ET full sensitivity and ET HF for Δ10km and 2L15km (left plot), and 2L20km (right plot). The dashed line shows the prompt emission of short γ-ray bursts expected to be detected by HERMES during one year of observations.

### 4.2.2 Afterglow: survey and pointing modes

This section explores the capability of detecting X-ray short GRB afterglows associated with GWs. Using the approach described in [18], we model the afterglow taking into account both
<table><caption>Full (HFLF cryo) sensitivity detectors</caption><thead><tr><th>Instrument</th><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th></tr></thead><tbody><tr><td>THESEUS-SXI survey</td><td>10<sup>+3</sup><sub>-2</sub></td><td>13<sup>+3</sup><sub>-4</sub></td><td>12<sup>+3</sup><sub>-3</sub></td><td>12<sup>+3</sup><sub>-3</sub></td></tr><tr><td>THESEUS-(SXI+XGIS) survey</td><td>21<sup>+6</sup><sub>-7</sub></td><td>21<sup>+8</sup><sub>-6</sub></td><td>20<sup>+7</sup><sub>-5</sub></td><td>21<sup>+7</sup><sub>-7</sub></td></tr></tbody></table>

<table><caption>HF sensitivity detectors</caption><thead><tr><th>Instrument</th><th>Δ10</th><th>Δ15</th><th>2L 15</th><th>2L 20</th></tr></thead><tbody><tr><td>THESEUS-SXI survey</td><td>8<sup>+2</sup><sub>-3</sub></td><td>11<sup>+2</sup><sub>-4</sub></td><td>10<sup>+2</sup><sub>-3</sub></td><td>11<sup>+2</sup><sub>-2</sub></td></tr><tr><td>THESEUS-(SXI+XGIS) survey</td><td>16<sup>+6</sup><sub>-5</sub></td><td>19<sup>+8</sup><sub>-5</sub></td><td>19<sup>+4</sup><sub>-5</sub></td><td>21<sup>+8</sup><sub>-6</sub></td></tr></tbody></table>

**Table 8.** Numbers of joint GW+X-ray detections during one year of observation for THESEUS operating in survey mode considering the instrument SXI and the combination of SXI and XGIS.

<figure><img src="image_27.png" /></figure>

**Figure 25.** Redshift distribution of joint GW+EM detections with THESEUS-(SXI+XGIS) in survey mode, where we distinguish between on-axis and off-axis cases. The ET configuration considered here is 2L 20 km HFLF cryo.

the forward shock and the high-latitude emission [77, 78] and we evaluate the joint GW/X-ray detections by considering two observational strategies; the satellite observing in survey mode or the satellite slewing to point well-localized events. As an example of performance of X-ray wide FoV satellites, we consider the Soft X-ray Imager (SXI) planned to be on board the THESEUS mission [63]. SXI is a wide field of view instrument able to cover 0.5 steradians in one observation. It has a localization accuracy of 1-2 arcmin. Since also XGIS has good sensitivity down to 2 keV in a larger FoV of 2 steradians, We also consider the combination of SXI and XGIS.

Tables 8 show the numbers of X-ray afterglow detected in survey mode by SXI and the combination of SXI and XGIS on board of THESEUS which will have an associated GW sig-
<table><caption>**Table 9.** Numbers of joint GW+X-ray detections during one year of observation for THESEUS-SXI operating in pointing mode. Only BNS localised better than 100 deg² are followed-up.</caption><thead><tr><th>Instrument</th><th>Δ10 cryo</th><th>2L 20 km cryo</th><th>Δ10 HF</th><th>2L 20 km HF</th></tr></thead><tbody><tr><td>THESEUS-SXI pointing</td><td>6<sup>+4</sup><sub>-4</sub></td><td>11<sup>+6</sup><sub>-5</sub></td><td>1<sup>+2</sup><sub>-1</sub></td><td>7<sup>+4</sup><sub>-4</sub></td></tr></tbody></table>

nal. While the top table shows the joint detection with the ET full sensitivity configurations, the bottom table shows the ones with ET HF. The number of joint detections is around 10 for SXI (20 for SXI+XGIS) per year independently of the arms lengths and the geometries, and the number remains almost the same also without accessing low-frequencies. Figure 25 shows a histogram of the joint GW+afterglow detections by THESEUS-(SXI+XGIS) in survey mode operating with full sensitivity 2L with 20 km misaligned arms as a function of redshift. The figure shows that the majority of the joint detections are within $z = 1$. This explains the reason why both arm-length and accessible frequency do not change the joint GW/X-ray afterglow detection numbers; the majority of the afterglows are detectable up to a redshift where there is no significant difference among the GW detection efficiency. The figure also shows that detecting the X-ray afterglow enables to observe events with a jet not aligned to the line of sight.

For the pointing strategy, we select events with a sky-localization uncertainty smaller than 100 deg², which is well-contained inside the SXI FoV, and we assume that SXI is able to be on target 100 s after the merger. This is a very optimistic scenario because 100 s should include the time to transmit the alert, respond to the trigger and re-point the instrument to the source position. Table 9 shows that a few detections are possible and going from the 10 km triangle to the full sensitivity 2L 20 km increases of a factor of about 2 the detections. As described in [18], there are two possible issues for the pointing strategy; being extremely fast to point to the source, and prioritising the events to be followed. No X-ray detection is possible being on source 1 hr after the merger with respect to 100 s because the X-ray afterglow emission is decaying fast. To detect a few events (see table 9) one needs to optimize the event selection among hundreds of events (see table 1) to be followed-up, to avoid to loose observational time. Selecting the sources localized better than 1000 deg², the number of joint GW+X-ray detections increases by a factor of 10 but the selection of the event to be followed becomes more complicated. As shown in table 1 the number of sources detected with sky-localization 1000 deg² are thousands. Also prioritizing the events on the basis of the viewing angle by excluding all the sources edge-on (for which the viewing angle estimate is more precise), the number of events to be followed up remains to be around several hundreds (for the 10 km triangle) to thousands (for the 2L 20 km). While the 10 km triangle gives fewer joint detections with respect to 2L 20 km full sensitivity, the trigger selection is easier because it is among a smaller number of events. The HF-only 2L with 20 km misaligned arms gives a number of joint detection comparable to the 10 km triangle.

*In summary, for the detection of the X-ray afterglow emission associated with the GW BNS signals there are no significant differences for the different configurations of ET.*

## 4.3 Kilonovae: joint GW and optical detections

This section explores the perspectives for the detection of KNe. The KN emission is not beamed as GRBs and is expected from all viewing angles. We consider as reference observatory, the Vera Rubin Observatory (VRO). VRO is the innovative 8.4-meter telescope
Full (HFLF cryo) sensitivity detectors

<table><thead><tr><th>Configuration</th><th>N<sub>GW,VRO</sub><br>Ω &lt; 20 deg<sup>2</sup></th><th>VRO<br>time</th><th>N<sub>GW,VRO</sub><br>Ω &lt; 40 deg<sup>2</sup></th><th>VRO<br>time</th><th>N<sub>GW,VRO</sub><br>Ω &lt; 100 deg<sup>2</sup></th><th>VRO<br>time</th></tr></thead><tbody><tr><td>Δ10</td><td>14 (14)</td><td>1.1% (3.3%)</td><td>36 (39)</td><td>5.1% (15%)</td><td>96</td><td>40%</td></tr><tr><td>Δ15</td><td>38 (42)</td><td>3.3% (9.8%)</td><td>84 (101)</td><td>14.2% (42%)</td><td>163</td><td>&gt; 100%</td></tr><tr><td>2L 15</td><td>28 (28)</td><td>2.2% (6.5%)</td><td>62 (77)</td><td>10.6% (31%)</td><td>189</td><td>93%</td></tr><tr><td>2L 20</td><td>55 (64)</td><td>5% (14.9%)</td><td>115 (152)</td><td>23.1% (68%)</td><td>324</td><td>&gt; 100%</td></tr></tbody></table>

**Table 10.** Numbers of expected kilonovae detected by the VRO selecting the sources detected by the different configurations of ET with sky localization smaller than 20 deg² (column 2), 40 deg² (column 4) and 100 deg² (column 6). Columns 3, 5, and 7 give the percentage of the VRO time necessary to follow up all the sources with sky-localization smaller than the above thresholds and within the VRO visibility sky area. The observational time for each event is obtained considering 600 s for each pointing and a number of the pointing corresponding to the ratio between the sky-localization uncertainty and the VRO FoV plus one pointing to be more conservative. We consider observations in two filter (g and i) the first night after the merger and the two filter observations repeated during the second night after the merger. We add 60 s to take into account filter change, slew time and overheads each night. The percentage of VRO time is evaluated assuming 2600 hours as the total amount of observational time of the VRO in one year. The expected number of detected kilonovae and percentage of VRO time considering 1800 s for each pointing (instead of 600 s) are given in parentheses. For the 100 deg² threshold, assuming for each pointing of 1800 s entails that the number of hours requested to cover the entire sky-localization is larger than the ones available in one observing night. This makes our observational strategy unfeasible, and thus we exclude this case. It is also shown that to follow all events with sky-localization < 100 deg², the required observational time becomes prohibitively large and for Δ15 and 2L 20 larger than the entire VRO observational time.

HF sensitivity detectors

<table><caption>**Table 11.** Same as 10 but for ET detectors without the low-frequency.</caption><thead><tr><th>Configuration</th><th>N<sub>GW,VRO</sub><br>Ω &lt; 20 deg<sup>2</sup></th><th>VRO<br>time</th><th>N<sub>GW,VRO</sub><br>Ω &lt; 40 deg<sup>2</sup></th><th>VRO<br>time</th><th>N<sub>GW,VRO</sub><br>Ω &lt; 100 deg<sup>2</sup></th><th>VRO<br>time</th></tr></thead><tbody><tr><td>Δ10</td><td>0 (0)</td><td>0% (0%)</td><td>2 (2)</td><td>0.3% (0.8%)</td><td>4</td><td>2%</td></tr><tr><td>Δ15</td><td>2 (2)</td><td>0.2% (0.5%)</td><td>3 (4)</td><td>0.7% (1.9%)</td><td>8</td><td>7.5%</td></tr><tr><td>2L 15</td><td>3 (4)</td><td>0.4% (1.2%)</td><td>7 (7)</td><td>1.3% (3.9%)</td><td>26</td><td>11%</td></tr><tr><td>2L 20</td><td>5 (4)</td><td>0.6% (1.6%)</td><td>15 (18)</td><td>3.1% (9.3%)</td><td>32</td><td>20.8%</td></tr></tbody></table>

characterized by a wide field (9.6 deg²) camera containing over three billion pixels of solid-state detectors [58]. Its compact shape enables to point and slew through large sky-region extremely fast. It represents the ideal instrument to observe the GW sky localizations obtaining deep observations of extremely high quality. In order to evaluate the number of possible joint optical/GW detections, we model the kilonova emission as in [79] taking the best-fit parameters obtained to reproduce the multi-filter optical observations of the kilonova AT2017gfo associated with GW170817. Our model includes cosmological and k-correction. We select all the BNS mergers detected with a sky-localization smaller than a threshold and we associate to each of them a GW17087-like kilonova taking into account the source distance and the viewing angle (our modeling starting from GW170817 reproduces the angular dependence of the expected emission). We assume three threshold on the sky-localization by selecting all the events with sky-localization < 20deg², < 40deg², and < 100deg². We
consider a Target of Opportunity strategy similar to the ones described in [80–82]; we use
observations in two filters (g and i) the first night after the merger and we repeat the two
filter observations during the second night after the merger. We consider a detection when
the kilonova emission is detected at 5σ the first and second night in both filters. Having
observations in at least two filters is important to reduce the number of contaminating tran-
sients and identify the kilonova associated with the GW signal on the basis of the color
evolution. The problem of contaminating transients is solved in real observations also using
spectroscopy able to characterize the source, but here for the configurations comparison, this
is not taken into account. Our counts take into account the visibility of the VRO.

Figure 26 shows the light-curves for GW170817-like signals associated with our popu-
lation of BNSs detected with a sky-localization smaller than 40deg² by the full (HFLF cryo)
sensitivity configurations and by the HF sensitivity configurations. The plots show the dif-
ferent numbers of events selected to be followed up for the different configurations. It is
immediately clear the significant increase in the number of targets to be followed going from
configurations with only high-frequency to the full sensitivity configurations.

Table 10 gives the number of expected kilonovae by following up all the events within the VRO visibility sky-area and with a sky-localization uncertainty smaller than 20 deg², 40 deg² and 100 deg², and the corresponding percentage of VRO time necessary for the follow-up. Enlarging the threshold on the sky-localization significantly increases the number of events to be followed-up and thus the number of joint detections but at the expense of more observational time. Also increasing the exposure from 600 s to 1800 s for each pointing tends to increase the number of detections but the percentage of time to be used becomes prohibitive. Looking at the full sensitivity detectors, *the 2L with 20 km misaligned arms is the best performing in detecting kilonovae. It enables to detect between several tens and a few hundred kilonova counterparts per year. The 15 km triangle is slightly better than the 2L with 15 km misaligned arms giving a number of detection about 30% larger.*<fnref n="17" /> *The 15 km triangle is significantly better than the 10 km triangle giving about a factor 2 larger number of detections.*

The presence of low-frequency is critical for ET operating as single observatory to detect
a large number of kilonovae counterparts. Table 11 shows the small number (a few) of
detections per year expected with the triangle-HF configurations. This number increases to
a few tens for the 2L-HF configurations.

We highlight that the absolute number per year of VRO detections is affected by the
error on the BNS local rate normalization. The astrophysical rates inferred from the ob-
servations of the first, second and third run of observations of LIGO and Virgo give a BNS
merger rate in the range 10 to 1700 Gpc⁻³yr⁻¹ [7]. Since our population corresponds to a
local rate R₀ ≈ 250 Gpc⁻³yr⁻¹, the actual numbers could be up to one order of magnitude
smaller or larger than the ones given in the tables.

<footnote n="17">¹⁷The scenario of following up all the events with sky-localization < 100 deg² case is not considered because the required observational time of VRO is prohibitive.</footnote>
<figure><img src="image_28.png" /><figcaption>(a) $\Delta$ 10 km HFLF cryo.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(c) 2L 15 km HFLF cryo.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>(e) $\Delta$ 10 km HF.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>(g) 2L 15 km HF.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>(b) $\Delta$ 15 km HFLF cryo.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>(d) 2L 20 km HFLF cryo.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>(f) $\Delta$ 15 km HF.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>(h) 2L 20 km HF.</figcaption></figure>

**Figure 26.** Light-curves for GW170817-like signals associated with BNS detected with a sky-localization smaller than 40 deg$^2$ by each of the full (HFLF cryo) sensitivity configurations (four top plot) and by each of the HF sensitivity configurations (four bottom plot). The black lines mark the (5$\sigma$ point-source depth) limiting magnitudes corresponding to 600 s of exposure for each pointing in the g band (solid black line) and the i band (dashed black line).
## 5 Stochastic backgrounds

The stochastic GW background (SGWB) is formed by the incoherent superposition of signals emitted by different GW sources in our Universe, primordial or astrophysical, at different redshifts, that we collect at our detector. Traditionally, we distinguish between an astrophysical GW background (AGWB) and a background of cosmological origin (CGWB). The latter are tensor modes produced by different processes in the early universe such as inflation, reheating, phase transitions, cosmic strings, or primordial black holes (see [83–85] for reviews). If detected, it would provide us with direct information about the very first instants of time of the evolution of our Universe. On the other hand, the AGWB is made of the superposition of all GWs emitted by different populations of astrophysical sources, from the onset of stellar activity until the present epoch. The CGWB and the AGWB are the gravitational counterparts of the Cosmic Microwave Background (CMB) [86] and of the Cosmic Infrared background (CIB) [87] in the electromagnetic context: in the same way as the CIB is a foreground for the CMB, which needs to be properly characterised and subtracted to access the CMB and its cosmological content, astrophysical backgrounds of gravitational waves represent foregrounds for primordial backgrounds of gravitational radiation. Typically it is expected that different GW sources are characterized by a different frequency profile, that can be used as a tool to disentangle them [88, 89]. As recalled in appendix A, a stochastic background is characterized by the quantity

$$ \Omega_{\text{gw}}(\hat{\mathbf{n}}, f) \equiv \frac{1}{\rho_c} \frac{d\rho_{\text{gw}}(\hat{\mathbf{n}}, f)}{d\ln f d^2\hat{\mathbf{n}}}, \qquad (5.1) $$

where $d\rho_{\text{gw}}(\hat{\mathbf{n}}, f)/d\ln f d^2\hat{\mathbf{n}}$ is the energy density per unit logarithmic frequency and unit solid angle that reaches the observer from the direction $\hat{\mathbf{n}}$ and with frequency $f$, see eqs. (A.2)–(A.4). The isotropic (normalized) energy density is defined by

$$ \Omega_{\text{gw}}(f) \equiv \int d^2\hat{\mathbf{n}} \Omega_{\text{gw}}(\hat{\mathbf{n}}, f), \qquad (5.2) $$

and gives the (normalized) energy density of the stochastic background reaching the observer from all directions. It is commonly assumed that it is possible to factorize the dependence on the frequency and that on the direction, i.e.

$$ \Omega_{\text{gw}}(\hat{\mathbf{n}}, f) = \mathcal{E}(f)\mathcal{P}(\hat{\mathbf{n}}). \qquad (5.3) $$

Furthermore, for many cosmological backgrounds (but not, for instance, for the backgrounds generated by cosmological phase transitions) the scale of variation with frequency is such that, over the bandwidth of a ground-based or a space-borne GW detector, the frequency dependence can be approximated by a simple power-law $\mathcal{E}(f) = A_{\beta} (f/f_{\text{ref}})^{\beta}$, where $A_{\beta}$ is the amplitude at an arbitrarily chosen reference frequency $f_{\text{ref}}$.

In the presence of statistical isotropy, the statistical properties of the background energy density can be characterised in terms of a frequency-dependent angular power spectrum as

$$ \langle \Omega_{\text{gw}}(\hat{\mathbf{n}}, f) \Omega_{\text{gw}}(\hat{\mathbf{n}}', f) \rangle = \sum_{\ell} C_{\ell}(f) P_{\ell}(\hat{\mathbf{n}} \cdot \hat{\mathbf{n}}'), \qquad (5.4) $$

where $P_{\ell}(\hat{\mathbf{n}} \cdot \hat{\mathbf{n}}')$ are Legendre polynomials, functions of the angular separation between the two directions $\hat{\mathbf{n}}$ and $\hat{\mathbf{n}}'$. The angular power spectrum and the frequency spectrum $\mathcal{E}(f)$
encode most of the statistical properties of a given background component. Recently, it has been shown that, similarly to CMB photons, primordial GWs are also affected by large-scale anisotropies (i.e., Sachs-Wolfe and Integrated Sachs-Wolfe effects) [90–93]. Typical CGWB components are expected to have the same level of anisotropy as the CMB: a scale-invariant spectrum $\ell(\ell+1)C_\ell \propto$ constant, and a level of anisotropy of the order of $\sim 10^{-5}$ with respect to the monopole [91]. In contrast, extra-galactic AGWBs have a scaling given by clustering, resulting in $(\ell+1)C_\ell \propto$ constant, and a higher level of anisotropy, at the level of $\sim 10^{-2}$ with respect to the monopole [94–108] (see [109] for a recent public code to compute the anisotropies of the AGWB). Interestingly, both the CGWB and the AGWB anisotropies show a correlation with CMB anisotropies which can be used both as a source discriminator and for testing systematics [110–112].

While the production of tensor modes from the early universe can typically be described in terms of a continuous emission, varying on a time scale extremely small with respect to typical observation time-scales,<fnref n="18" /> the AGWB in the frequency band of ET is expected to have a so-called popcorn-like nature, due to the discreteness of emissions in time. Indeed, in the Hz band, we expect the AGWB to be dominated by mergers of compact objects and, considering the short duration of the merger phase, events are expected to be separated in time and with a limited time overlap (some overlap is expected for BNS mergers). As a consequence, in the frequency band of ET, the angular power spectrum of the AGWB from mergers of compact binaries has an important popcorn component (due to the discrete emission in time), which adds to the clustering one (due to the discreteness of the spatial galaxy distributions) [97, 103, 104, 108]. Formally, the total angular power spectrum is given by $C_{\ell}^{\text{AGWB,tot}} = C_{\ell} + N^{\text{pop}}$, where the first term on the right-hand side is the contribution from clustering while the second component represents popcorn noise. The latter is flat in $\ell$-space (it is just an offset) and it is expected to cover the clustering contribution, see [97, 103, 104, 108]. Even if popcorn noise contains astrophysical information, it does not provide any information about the spatial distribution of sources. A possible way to overcome this problem, i.e. to extract the clustering part out of the popcorn noise threshold, is to consider cross-correlations between an SGWB map and galaxy distributions, see e.g. [108]. The study of cross-correlations, besides being an independent observable interesting on its own, provides one with a powerful detection tool, as it typically has a higher SNR than the auto-correlation, see e.g. [96–98, 108, 113] in the context of the extra-galactic astrophysical background.

Finally, kinematic anisotropies induced by a peculiar motion of the observer with respect to the rest frame of emission are expected to be quite important and potentially detectable if the corresponding monopole has a sufficiently high SNR, see e.g. [114]. Recent studies to extract the cosmic dipole using next-generation detectors have been developed both for the AGWB [114, 115] and for resolved sources [116].

We conclude this section with a remark. Because of the discrete emission in time, the AGWB in the Hz band is not irreducible: with a perfect instrument with infinite sensitivity (and with infinite computational power), all events would be detectable individually. As the sensitivity of next-generation ground-based detectors will improve more and more events will be detectable individually: in this context, it is interesting to study the so-called *residual background*, i.e. the background contribution that remains after having filtered out resolvable objects. This observable can indeed provide us with a unique way of extracting information

<footnote n="18">¹⁸For this reason most of the CGWB components represent continuous and stationary background components and they are often referred to as *irreducible* backgrounds.</footnote>
<figure><img src="image_36.png" /><figcaption>**Figure 27.** Power-law integrated sensitivity (PLS) to $\Omega_{\text{GW}}(f)$ for different ET geometries, in the full HFLF-cryo configuration. We set $\text{SNR}_{\text{th}} = 1$ and $T_{\text{obs}} = 1\text{yr}$ and we include only cross-correlations between detectors, excluding auto-correlations.</figcaption></figure>

on a faint and distant population, that cannot be detected with a catalogue approach. We will get back to this point later in this section.

## 5.1 Sensitivity to isotropic stochastic backgrounds

We now study the sensitivity, in different configurations, to a SGWB. To this goal, we compute the Power-Law integrated Sensitivity curve (PLS), whose definition is recalled in appendix A, for the triangular and L-shaped detectors, in the configurations discussed in section 2, so for the triangular configuration we consider 10 and 15-km arm, while for the L-shaped we consider 4 configurations: 15 km and 20 km arms, aligned and misaligned (by 45 deg) detectors located one in Sardinia and one in the Netherlands. We then turn to the angular sensitivity for the same ET configurations, i.e. we present how well the various configurations can reconstruct the $C_{\ell}(f)$ (i.e., the angular power spectrum), defined in eq. (5.4).

The result is shown in figure 27 for the different geometries in the full HFLF-cryo configuration, and in figure 28 for the HF-only configuration.<fnref n="19" />

<footnote n="19">¹⁹Recall from section 2 that, for the 2L cases, the alignment between the two detectors has been defined using the relative angle $\alpha$ measured with respect to the north of the Sardinia site for both detectors, rather than with respect to the great circle passing through the two sites. Therefore the $\alpha = 0^\circ$ and the $\alpha = 45^\circ$ cases shown in the figures below do not correspond to the ideal and, respectively, the worst configurations for stochastic searches, but have slight offsets from it. In terms of the angle $\alpha$, the ideally parallel and the perfectly misaligned for the configurations correspond to $\alpha = 2.51^\circ$ and $\alpha = 47.51^\circ$, respectively. This is particularly important for the misaligned case. Indeed, $\alpha = 47.51^\circ$ would correspond to essentially zero sensitivity to stochastic backgrounds, while we see from figure 27 that, for $\alpha = 45^\circ$, we already have an interesting sensitivity to stochastic backgrounds, so a small offset from the perfectly misaligned configuration is already very useful for recovering a part of the sensitivity to stochastic searches (without basically affecting the searches for compact binary coalescences, as discussed in section 2). See appendix B for more details.</footnote>
<figure><img src="image_37.png" /><figcaption>**Figure 28.** As in figure 27, for the HF-only configurations.</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>**Figure 29.** Left: ET power-law integrated sensitivity for the triangle configuration for the full HFLF-cryo configuration and the HF-only configuration, assuming $SNR_{th} = 1$ and $T_{obs} = 1$yr and including only cross-correlations between detectors. Right: the same for the 2L-15km-0° configuration.</figcaption></figure>

We observe that the triangular configurations perform better at high frequency since the overlap reduction function remains constant at high frequencies for co-located detectors rather than falling to zero. The 2L configurations do better at low frequencies, up to several tens of Hz, but only for co-aligned detectors (since this maximises the coherence for the stochastic search). The case of 45 deg 2L configurations are universally the worst.
Figure 29 compares the power-law sensitivities obtained from the HFLF-cryo and the HF-only ASDs for the 2L 15km with parallel arms (left panel) and for the 10 km triangle (right). Since $\Omega_{gw}(f)$ is quadratic in the strain, the loss of the low-frequency sensitivity has a huge impact on the sensitivity to stochastic backgrounds at low frequencies. For instance, for the 2L 15km with parallel arms, the HF-only configuration is worse than the HFLF-cryo by a factor $\sim 10^2$ at 10 Hz, and $\sim 10^3$ at 3 Hz. In contrast, the sensitivity above about 60 Hz is unaffected.

## 5.2 Angular sensitivity

In this section we study the angular resolution of different ET configurations in the context of an SGWB. We follow the formalism in [117], whom we refer the reader for details and derivations (see also [118, 119] for recent forecasts about the ET sensitivity to angular anisotropies). The sensitivity with which we can measure a given multipole of the background intensity is characterized by a quantity, $N_\ell$, such that the signal-to-noise ratio of an angular power spectrum is given by

$$ \text{SNR}^2 = \sum_{\ell} (2\ell + 1) \frac{C_{\ell}}{N_{\ell}}, \qquad (5.5) $$

where the $C_\ell$ have been defined in eq. (5.4). The $N_\ell$ can be computed from

$$ N_{\ell}^{-1} = \frac{1}{2} \sum_{ABCD} \int df \int dt \, G_{\ell}^{AB,CD}(t, f), \qquad (5.6) $$

where

$$ G_{\ell}^{AB,CD}(t, f) = \left(\frac{2\mathcal{E}(f)}{5}\right)^2 (\mathbf{N}_f^{-1})^{AB} (\mathbf{N}_f^{-1})^{CD} \frac{\sum_m \text{Re} \left(\mathcal{A}_{BC,\ell m}(t, f) \mathcal{A}_{DA,\ell m}^*(t, f)\right)}{2\ell + 1}, \qquad (5.7) $$

and

$$ \mathcal{A}_{AB,\ell m}(t, f) = \int d^2\hat{\mathbf{n}} Y_{\ell m}^*(\hat{\mathbf{n}}) \mathcal{A}_{AB}(t, f, \hat{\mathbf{n}}), \qquad (5.8) $$

is the angular multipole of the antenna pattern; we explicitly indicated its time dependence. These expressions allow one to predict the angular power spectrum of a given experiment and scan strategy. A key assumption made in obtaining this result is that the time-integrated response of the instrument can be compressed into an angular power spectrum. In practice, this will not be the case, since the scan strategy cannot be designed to average the noise over the $m$-modes and the resulting estimator will be sub-optimal. This is a necessary assumption for any analytical estimate; however, in practice, optimal estimators will have to account for inhomogeneous noise in $m$-modes.

Eq. (5.6) involves a two-dimensional integral over $t$ and $f$, where the integrand involves at least one computationally expensive spherical harmonic transform of the antenna pattern $\mathcal{A}$. This can be simplified further for experiments with a constant configuration (e.g. constant arm lengths, angular separations between arms, relative detector positions). As explained in [117], in this limit the network changes as a function of time as a rotating solid. Hence the antenna patterns at different times are related to each other through simple three-dimensional rotations (i.e. $\mathcal{A}(t, f, \hat{\mathbf{n}}) = \mathcal{A}(f, R_t \hat{\mathbf{n}})$, where $R_t$ is a time-dependent rotation matrix). Recalling that a rotation only mixes the $m$-modes for each fixed $\ell$ in $\mathcal{A}_{\ell m}$,
<figure><img src="image_39.png" /><figcaption>**Figure 30.** Noise per angular multipole at a reference frequency of 100 Hz for the four configurations under study. In the left and right panel we show the sensitivity to odd and even multipoles, respectively. All figures are obtained using the `schNell` package https://github.com/damonge/schNell.</figcaption></figure>

the function $G_ℓ$ in (5.7) does not depend on $t$ (assuming detector noise to be stationary). In this case one gets the compact expression

$$N_{\ell}^{-1} = \frac{T_{\text{obs}}}{2} \sum_{ABCD} \int df \, G_{\ell}^{AB,CD}(f), \qquad (5.9)$$

where $T_{\text{obs}}$ is the total observing time.

In figure 30 we plot the noise per multipole $N_ℓ$ at a reference frequency $f_{\text{ref}} = 100$ Hz for different detector configurations. When looking at odd multipoles, we see that the 2L configurations are much more sensitive than triangular configurations. This is due to the fact that the three nested interferometers composing a triangular detector are separated by a baseline whose size is one arm length, while the two L-shaped detectors of the network have a much larger baseline. As a consequence, the antenna pattern function of triangular configuration is almost completely even under parity, hence the instrument has a very bad sensitivity to odd multipoles, see [117]. Furthermore, as expected, aligned detectors have better sensitivity than misaligned detectors. We observe that, with the most favourable configurations, i.e. 2L aligned with 15km or 20 km length, we are potentially able to detect a dipole anisotropy at the level of $\sim 10^{-12}$ (hence $N_{\ell=1} \sim 10^{-24}$), which is the size of the kinematic dipole expected from a background component on the edge of being detected by LVC at O5 sensitivity, see also [114].<fnref n="20" />

## 5.3 Astrophysical backgrounds

Binary mergers nowadays detected by LIGO-Virgo-KAGRA are only the tip of the iceberg. In fact, a large fraction of the events at larger distances are not resolved by the detectors and are expected to constitute the dominating contribution of the astrophysical background. For compact binary coalescences (CBCs), eq. (5.1) can be rewritten in term of energy density flux as [120]

$$\Omega_{\text{gw}}(f) = \frac{1}{c\rho_c} f\mathcal{F}(f), \qquad (5.10)$$

<footnote n="20">²⁰We observe that the plateau at high-$\ell$ values is symptomatic of the fact that we have lost sensitivity to those multipoles, see [117].</footnote>
<figure><img src="image_40.png" /><figcaption>Figure 31. Background prediction for the current state-of-the-art astrophysical models, assuming a metallicity spread $\sigma_Z = 0.3$.</figcaption></figure>

where $\mathcal{F}(f)$ is defined as

$$ \mathcal{F}(f) = T^{-1} \sum_{k=1}^{N} \frac{1}{4\pi r^2} \frac{dE_{\text{gw}}^k}{df}(f), \qquad (5.11) $$

and the sum is over the $N$ individual merger contributions arriving on the detector during an observation time $T$. Rewriting the energy density, one can express the spectrum $\Omega_{\text{gw}}(f)$ in terms of frequency domain waveforms:

$$ \Omega_{\text{gw}}(f) = \frac{\pi c^2}{2G\rho_c} f T^{-1} \sum_{k=1}^{N} f^2 \tilde{h}_k^2(f), \qquad (5.12) $$

with $\tilde{h}_k^2(f)$ the sum of the squared Fourier domain GW amplitudes for the two polarization $+/-$:

$$ \tilde{h}_k^2(f) = \tilde{h}_{+,k}^2(f) + \tilde{h}_{\times,k}^2(f). \qquad (5.13) $$

Figure 31 shows predictions for the background, derived from the current state-of-the-art of population synthesis models [43, 44, 47, 48, 50, 52, 54] with a metallicity spread $\sigma_Z = 0.3$ for the three types of binary: binary neutron stars (BNSs), binary black holes (BBHs), and neutrons star-black hole systems (NSBHs). The BNS and BBH population are the ones described in section 3 and used through this paper to evaluate general metrics and several specific science cases. On these plots are also shown the PLS for the main configurations of this study, already shown in figure 27.<fnref n="21" /> The low-frequency part of the spectrum is predicted to have an evolution proportional to $f^{2/3}$, typical for the inspiral phase of compact objects binaries. This shape is common to all CBCs, therefore a sign of a specific population on the background is given by deviations from the $f^{2/3}$ behavior. For example, in figure 31 the deviation of the total background ('All', dark blue curve) from the $f^{2/3}$ shape between

<footnote n="21">²¹Here, and in figure 32, the PLS are computed including only cross-correlation between detectors, and setting $\text{SNR}_{th} = 1$ and $T_{\text{obs}} = 1$ yr, as in the plots of section 5.1.</footnote>
<figure><img src="image_41.png" /><figcaption>**Figure 32.** Location of the predicted signatures for the three main challenges for the CBC background.</figcaption></figure>

90-500Hz is due to the inclusion of BBHs from the cluster, while the deviation above 1000Hz is coming from the BNSs. Consequently, the background will be a new observation channel for population studies of compact objects.

Stellar CBC background challenges in ET can be divided into three main categories: the search of Population III stars, the study of BBH formation channels combined with the star formation history (SFH), and the residual BNSs background. All these categories are represented on figure 32 at the location of predicted background signatures, together with the ET PLS for the different geometries considered (all taken with their best ASD, see section 2).

**BBHs channels + SFH.** This part of the spectrum is where (depending on their masses and spins) individual BBHs stop emitting gravitational waves, so that the overall spectrum drops. Two main factors can imprint some signature in the range [100-1200] Hz: (1) the SFH, which is a combination of metallicity/redshift, and star formation rate/redshift relations; and (2) the formation and evolution channel of the binary (i.e. the environment of the binary formation/evolution). The detection of these signatures will be a new channel of observation for the stellar formation and evolution [121, 122]. For this specific signature the triangle configuration is preferable to any of the four 2L configurations that we have considered (15 or 20 km arm-length, aligned or at 45°), because its sensitivity to stochastic backgrounds will be better for frequencies above 200 Hz.

**Population III.** Population III stars are the first, metal-free stars that formed in the Universe [123]. The peak of the merger rate density of compact remnants left by Population III stars is expected to be at high redshift, $z \sim 8-16$ [124–128]. All of the unresolved population III merger events then combine to an SGWB in the [5–200] Hz frequency range that may deviate from the usual $f^{2/3}$ power-law behaviour [129–133]. This is due to the fact that, while population I/II stars produce an SGWB in the LIGO/Virgo frequency band during their inspiral phase, population III stars would instead show up at the merger and ringdown phases. In the frequency range where population III stars could leave their imprint, the 2L configurations with parallel arms are the best ones.
<figure><img src="image_42.png" /><figcaption>**Figure 33.** Backgrounds from all types of binaries (BBHs+BNSs+NSBHs), compared to the sensitivity of various detector configurations. The blue thick line corresponds to the total background, while the violet ones to the residuals (i.e. the background obtained after the subtraction of resolved sources) for the different configurations.</figcaption></figure>

**BNS residual mergers.** The background, as it is defined, contains only non-resolved gravitational waves. In that sense, the residual background (background obtained after subtraction of all resolved sources^22) will reveal the BNSs contributions to the background. Figure 33 shows the residual backgrounds for each type of binary and some of the configurations. As for BNSs, they are more difficult to be resolved due to their low masses, and their gravitational waves will mostly remain in the background and dominate it [121, 129].

## 5.4 Impact of correlated magnetic, seismic and Newtonian noise

Since searches for a stochastic gravitational-wave background (SGWB) [134] typically rely on cross-correlating data from two or more detectors, they are also susceptible to correlated noise. In this section we will discuss the impact of different noise sources on the ET and how they affect the different configurations, that is triangular versus 2L configuration. In section 5.4.1 we will discuss the effect of correlations in the seismic and accompanying Newtonian noise (NN), whereas section 5.4.2 focuses on correlations in magnetic field fluctuations.

In the following sections, we have used the ET-D design sensitivity in a triangular design with arm lengths of 10 km. However, the discussion depends minimally on this assumption and is a good representation for the comparison of the triangular versus 2L configurations regardless of the exact arm length and sensitivity.

At the end of the section, in table 12 the frequency regions of searches for an SGWB affected by correlated noise are summarized. We want to point out that here we only discuss the effect of fundamental noise sources, which implies that infrastructural noise sources will have to be reduced to a similar or lower level than the fundamental noise sources. This is, to some extent, discussed for magnetic noise in section 5.4.2; for all noise sources, i.e magnetic, seismic and Newtonian noise, this should be investigated further. Furthermore, we stress

<footnote>^22We assumed an SNR threshold of 12 to define a source as resolved. The resolved sources are then subtracted to obtain the residual background.</footnote>
that most of these correlated noise investigations focus on the effects of the search for an isotropic SGWB. However, as shown in a recent study on the effect of magnetic glitches due to lightning strikes [135], some correlated transient events might also affect other searches, which should be further investigated for seismic and Newtonian noise.

### 5.4.1 Seismic and Newtonian Noise

During the last decade, much research has focused on understanding the effect of seismic and Newtonian noise on the sensitivity of ET [136-140]. In a recent study the possibility of correlated seismic and Newtonian noise is investigated for the ET triangular configuration [141]. Searches for a SGWB using data from the second-generation gravitational-wave interferometric detectors, LIGO and Virgo, are considered to have no correlations in the seismic ambient field due to their large separation and the damping of seismic waves. However, in the triangular design of the ET the input and output mirrors of two interferometers are proposed to be separated by distances of only several hundred of meters. As illustrated in figure 11 of [141], there are five coupling locations between two ET interferometers, of which two are between aligned mirrors and the remaining three between mirrors rotated by 60°. Based on the ET design report [142] the horizontal separation between those mirrors is about 300-500 m for the aligned mirrors and 330-560 m for the other pairs [141].

To investigate correlations between seismic measurements over a scale of several hundreds of meters, ref. [141] analyzed data at the Earth's surface from geophones deployed near Terziet (NL) [143] and underground data from seismometers in the former Homestake mine (South Dakota, U.S.A.) [144]. The study concludes that at the surface at Terziet significant (above Gaussian noise background) seismic correlations exist between sensors with a separation of several hundreds of meters. The 50th percentile coherence is significant up to about 10 Hz, whereas the 90% coherence is significant up to 50 Hz, which is the highest recorded frequency. For 90% of the time at Homestake there is significant coherence up to 20 Hz and more than 50% even up to 40 Hz; that is the highest frequency before the response of the sensors decreases drastically. Neither the coherence nor the cross-power spectral density (CSD) were observed to have a dependence on distance (between 200m and 800m). At both locations night times are observed to be quieter, with lower CSDs. At night there are more periods when there is significant correlated noise. These studies indicate that there is a non-negligible amount of coherence during a significant amount of time (night and day), up to frequencies of 50 Hz. This leads to the conclusion that the effect of these correlated noise sources should be understood, particularly for searches for an SGWB.

Note that in [141] it is emphasized that some correlations are considered to be of anthropogenic origin. Furthermore, some of the power spectral densities (PSDs) observed at the candidate sites are lower than the CSD observed in the Homestake mine, and therefore site-specific studies should be envisioned in the future. However, the results discussed in [141] are expected to give a good indication of the order of magnitude of the possible levels of correlated seismic and Newtonian noise which might be expected in an underground environment. Homestake represents an underground environment where both ambient and anthropogenic noise sources are present. This may also be expected for ET, since its infrastructures could be expected to introduce additional anthropogenic noise, regardless of various attempts to reduce and shield them.

It is possible, based on the measurements of correlations of the seismic noise, to estimate the correlated Newtonian Noise (NN). The NN is a disturbance produced in the gravitational-wave detector by local fluctuations of the gravitational field. Two types of seismic waves
<figure><img src="image_43.png" /></figure>

**Figure 34.** The projected impact from correlated NN from body-waves $C_{NN_{BW}}$, as calculated in section III of [141], is shown in blue. The blue line represents the median value, and the associated surface is delimited by the 10% and 90% percentiles. Also curves with more optimistic assumptions (see the text) are shown. Based on figure 16 of [141]. Adapted with permission from [141], copyright (2022) by the American Physical Society.

produce NN: the first are Rayleigh waves which propagate at the surface of the Earth, and the second are body waves present in the Earth's mantle. The NN from Rayleigh waves decreases significantly with depth, so their effect on ET will be limited if one assumes an underground facility at 300 m. As shown in figure 15 of [141], the NN from Rayleigh waves significantly impacts the power-law integrated sensitivity of ET only up to about 5 Hz.<fnref n="23" /> At higher frequencies, no effect is expected.

The median contribution from correlated body waves, however, is expected to be three to five times higher than the ET sensitivity, in the range of 2 Hz to 10 Hz. While it is important to investigate the impact of noise on the sensitivity curve, correlated noise can more drastically affect some searches. The search for a SGWB typically relies on the cross-correlation of data from two detectors to decrease the Gaussian detector noise and improve the search sensitivity.

Figure 34 presents the expected impact on the search for a SGWB due to NN produced by body waves.<fnref n="24" /> It shows that the budget calculated in [141] affects the SGWB search up to at least 40 Hz (blue band), which is the highest frequency in the analysis. Near 3 Hz, this gives an effect which is larger than the 1 yr ET sensitivity curve for the SGWB (light brown line in figure 34) by a factor ~8 × 10⁶ (90% percentile), or ~6 × 10⁵ (50% percentile).

<footnote n="23">²³The power-law integrated sensitivity is defined as in [145], see also appendix A. We use an SNR of 1 after one year of data at a 100% duty cycle.</footnote>

<footnote n="24">²⁴To produce this figure, for the broadband ($\Omega_{HF-\text{only}}^{\text{PLS}}$, $\Omega_{HFLF-\text{cryo}}^{\text{PLS}}$) sensitivity to an SGWB we assumed one year of observation time (100% duty cycle). The one-year PLS curve of the A+ design for the LIGO Hanford, LIGO Livingston detectors and AdV+ design for the Virgo detector is represented by the dot-dashed curve. LIGO's A+ and Virgo's AdV+ design are the interferometer designs which are aimed to be reached during O5 [23]. This curve was obtained using the open data provided by the LVK collaborations [146] and was first presented in [147]. Note that in this figure we present the 1$\sigma$ PLS, whereas in [147] the 2$\sigma$ PLS-curve is shown.</footnote>
According to the noise estimate shown in blue in figure 34, with the triangular configuration the sensitivity of ET to an SGWB, at least below $O(50-100)$ Hz, would be of the same order of magnitude as LIGO's A+ and Virgo's AdV+ design. Even assuming that an optimistic factor of 10 NN cancellation [136] could be applied to each interferometer, i.e. a factor of 100 for the SGWB budget, the search for an SGWB would still be impacted up to $\sim 30$ Hz. When considering the night-time data with fewer anthropogenic activities, the impact decreases but stays at the same order of magnitude, i.e. a factor $\sim 10^5 - 10^6$ at 3 Hz [141].

In figure 34 we also provide the result from a more optimistic scenario (pink curve). The construction of this scenario is broken down as follows. We start from the yellow curve, which displays the affect of NN from body waves assuming a seismic spectrum identical to the Peterson low noise model [148] and 100% coherence. We then assume a NN cancellation by a factor of 3 (i.e. a factor of 9 on the stochastic budget), which is considered as a realistic target (green curve). The dark orange curve is generated from the assumption that the coherence equals to the median coherence observed in Homestake, as reported in [141]. Finally the pink curve represents the Peterson low noise model multiplied by the coherence observed at Homestake, as well as a factor of 3 NN cancellations for each detector. This pink curve represents the most optimistic scenario that one can achieve. In this case, SGWB searches will still be affected by correlated NN up to $\sim 10$ Hz.

The effect of correlated noise presented above, as well as possible additional effects from infrastructural noise, would only affect the search for an SGWB in the case of the triangular design leading to closely located interferometers. The problem of co-located correlations (of order a few hundred meters) of the seismic field should represent no problem for very distant L-shaped interferometers. However, for a triangle one can take advantage of the “null-channel” to estimate the power spectrum [149, 150], see the discussion in section 7. In the future, site-specific studies could give more detailed information. Furthermore, it would be important to estimate the order of magnitude of correlated noise expected by infrastructural noise sources caused by the ‘typical equipment’ needed for an underground, cryogenic gravitational-wave interferometer.

Finally, the effects of seismic transients such as (micro-)earthquakes should be understood. The latter would not only affect the search for an SGWB but could also be expected to affect searches for other GW signals such as CBCs, bursts or continuous waves if the noise sources are too loud. Further studies on the possible effect of correlated seismic transients would also be advisable. Such a study could be envisioned to investigate the possibility of lock-loss, similar to an earlier study for LIGO [151], or the impact on the sensitivity, similar to a study of the effect from earthquakes as part of the O3 environment investigations done at Virgo [152]. In the studies for ET one should also look into the possibility of correlations, as well as the ‘transient Newtonian Noise’, linked to the transient seismic fields and how these might introduce additional effects on the different signal searches.

### 5.4.2 Magnetic noise

One noise source commonly investigated with respect to the search for an SGWB is the Schumann resonances [153, 154]. These are standing electromagnetic waves in the cavity formed by the Earth's surface and the ionosphere, and are driven by lightning strikes all across the globe. The fundamental mode has an amplitude between $\sim 0.5$ pT and $\sim 1$ pT depending on the location and time of the day and year. The resonances are several Hz wide and have frequencies $\sim 7.8$ Hz, $\sim 14$ Hz, $\sim 21$ Hz, ...
To explain how magnetic fields couple to the interferometer, it is instructive to understand how the coupling is measured. On a regular basis, strong magnetic fields are generated in the experimental buildings of the current generation of interferometric detectors. By injecting fields many times stronger than the ambient magnetic field one tries to measure their effect on the strain ASD. We define Y and X to be the ASD of the strain and witness channel (magnetometer) respectively. We use the subscript 'inj' and 'bck' refer to times during the injection, respectively a quiet background time just before/after the injection. Then we can measure the magnetic coupling function $\kappa$ as follows [155],

$$ \kappa_{\text{measured}}(f) = \sqrt{\frac{Y_{\text{inj}}^2(f) - Y_{\text{bck}}^2(f)}{X_{\text{inj}}^2(f) - X_{\text{bck}}^2(f)}}. \qquad (5.14) $$

The coupling function describes how strong the strain data will be affected given ambient magnetic fields.

In recent work, the effect of the Schumann resonances and other forms of correlated magnetic noise on ET was investigated [156]. At low frequencies (< 30 Hz) it was shown that the magnetic coupling function has to be improved by two to four orders of magnitude compared to the second-generation detectors Advanced LIGO and Advanced Virgo in order to prevent significant coupling. Note that compared to the Advanced Virgo coupling function, a factor of 5 improvement could be expected due to the larger mirror mass of the ET [157, 158]. However, this assumes that the same actuator magnets as with Advanced Virgo will be used, which will make the control of the test mass more difficult. Furthermore, at low frequencies, the Advanced LIGO magnetic coupling is lower compared to the Advanced Virgo magnetic coupling. Therefore the electrostatic actuator design used by Advanced LIGO should be considered for ET. If ET will rather have a magnetic coupling similar to that of Advanced Virgo, the impact of correlated magnetic noise is predicted to be larger and will impact searches at a wider frequency band, up to $f \lesssim 50$ Hz. One must also stress the danger that induced eddy currents close-by to sensitive detector parts pose, so these should be minimized as much as possible.

The effect of magnetic noise is shown in figure 35, based on figure 6 of [156]. It presents the ratio of $\kappa_{\text{HF-only}}^{\text{SGWB}}$ and the magnetic coupling measured at LIGO and Virgo. The same ratio is also whon for $\kappa_{\text{HFLF-cryo}}^{\text{SGWB}}$. Here $\kappa_{\text{HF-only}}^{\text{SGWB}}$ and $\kappa_{\text{HFLF-cryo}}^{\text{SGWB}}$ are the magnetic coupling for each ET interferometer such that no effect of correlated magnetic noise is present in the search for an isotropic SGWB using one year of data with 100% duty cycle at the HF-only, respectively HFLF-cryo design sensitivity. For more details on the computation, we refer the reader to [156]. The needed improvement of the magnetic coupling such that magnetic noise is not larger than 1/10 of the ASD,<fnref n="25" /> is roughly a factor 10 less stringent than the SGWB constraint. For more information see figure 6 and the accompanying discussion of [156]. Note that in [156] was not taken into account a factor of 5 of reduction obtained by increasing the mirror mass. In the top left panel of figure 35 we applied this additional factor-of-5 reduction for Virgo's magnetic coupling below 100 Hz. At higher frequencies, the coupling function is dominated by different mechanisms. We note that in the recent LIGO-Virgo third observing run (O3), for which we use the magnetic coupling function, an additional coupling was observed [155] compared to the earlier work describing the inverse proportionality with mirror mass [158]. Therefore it is not guaranteed that this reduction could be achieved if the other coupling mechanisms are not properly understood and mitigated.

<footnote n="25">²⁵This is a typical requirement for noise sources which are considered not to be a 'fundamental' noise source, i.e. part of the calculation of the optimal sensitivity curve.</footnote>
<figure><img src="image_44.png" /></figure>

**Figure 35.** Top panels: needed improvement factor as a function of frequency for the upper limits on the ET magnetic coupling function, such that the isotropic SGWB search is unaffected. LHO and LLO refer respectively to the LIGO Hanford and LIGO Livingston observatories. The low-frequency (left panel) magnetic coupling poses a greater challenge for the operation of ET compared to the high-frequency (right panel) magnetic coupling. In both panels, the dash-dotted blue line indicates the line where no improvement is necessary. This figure is based on the two right panels of figure 6 of [156], however, we have used an additional factor of 5 in the reduction due to the larger mirror mass when going from Virgo to ET, in line with earlier work [157, 158]. This only applies to the left panel, since at high frequencies the coupling is dominated by other mechanisms. We report the upper limits for both the HF-only and HFLF-cryo sensitivity for SGWB searches (in [156] they used ET-B and ET-D). Note that, for $f > 100$ Hz, HF-only and HFLF-cryo have the same sensitivity to a SGWB, so we only present the HFLF results in the top left panel. Bottom panel: the upper limits on the maximal allowed magnetic coupling to prevent impact on the ASD and/or SGWB searches in the pessimistic scenario where magnetic infrastructural noise, as measured in the central building at Virgo, will be fully correlated between interferometers. The upper limits are provided for both HF-only and HFLF-cryo. The upper limits are compared with the mean observed magnetic coupling at the LIGO and Virgo detectors. Based on figure 7 of [156]. Adapted with permission from [156], copyright (2021) by the American Physical Society.

Up to ~20 Hz, the observed amplitude of the magnetic CSD is similar, regardless of the separation between the detectors [135]. At higher frequencies, the magnetic CSD decreases due to stronger attenuation of electromagnetic fields travelling long distances. Therefore the effect of correlations in magnetic field fluctuations up to ~30 Hz on ET is (mainly) independent of the detector configuration. Both the triangular and 2L designs will be impacted at the same order of magnitude. However, it is important to understand the distance between
the detectors and large thunderstorm regions. Regions with large thunderstorm activity are the Americas, sub-Saharan Africa and southeast Asia [159], but more research is needed to investigate the ‘local’ activity in Europe and the possible impact on gravitational-wave interferometric detectors.

To summarize the effect of fundamental magnetic noise on HFLF-cryo ET, we can state that at low frequencies (<30 Hz) an effect of up to $O(10^3)$ will be present on the ASD and up to $O(10^4)$ on the sensitivity for the SGWB. This is (mainly) independent of the triangular versus 2L configuration, due to the low attenuation of electromagnetic fields at these frequencies. Despite the dramatic effects of correlated magnetic noise, one can consider several approaches to reduce the magnetic coupling, as will be further discussed at the end of this section.

At higher frequencies, minimal effects are expected [156], however, these are more dependent on the configuration. The co-located triangular design will be more susceptible compared to two spatially separated Ls. Even more important is the location of the detector with respect to active thunderstorm regions.

The effect from individual lightning strikes is also predicted to impact the ET sensitivity [135], in case of similar magnetic coupling as at LIGO and Virgo. A glitch rate of $O(10^5)$ for correlated magnetic transients per week could be expected, with the potential to affect ET's detector sensitivity <50 Hz [135]. It might also impact background estimation for CBC and burst searches. The 2L configuration is less prone to effects coming from coherent lightning strikes compared to the triangular design.

Depending on which coupling function is used, the magnetic contamination at low frequencies (≤50 Hz) is roughly at the same level (or worse) than the NN from body waves as projected in figure 34. However, there is a crucial difference between the two scenarios: the effect of the NN can only be mitigated by noise cancellation methods. For the magnetic noise, noise cancellation is also an option [160], however one can also directly attempt to improve the magnetic coupling function. An earlier study investigated the possibility of additional magnetic shielding and considered several configurations of Helmholtz coils or a spherical shell made out of chain-link metal [161]. This study showed that a factor-of-10 reduction in the magnetic coupling function should be feasible by several different shielding approaches. These results are promising, given the ‘simple’ methods proposed. One can consider doing better by using mu metal or superconducting coils, however, this will come at a significantly larger cost. Furthermore one can seek to reduce the number of magnets (or metallic parts) attached to the lower stages of your suspensions and test masses [162]. Even better would be to move magnets and metallic structures higher up in the suspension chain, leading to the dampening of the magnetic fields by the respective pendulum stages. Finally, the induction of currents in cables of e.g. the actuator system [162] might be mitigated by using optical fibres where possible, definitely in sensitive locations. More research is needed to obtain reductions in magnetic coupling by different methods. Nevertheless, one could envision substantially decreasing the effect of magnetic fields by using a combination of the proposed methods, which might however come at a non-negligible investment cost. We would also like to point out that the study [156] showed that ET’s sensitivity curve might be affected up to ~15Hz-20Hz (top left panel of figure 6 of [156]), and some additional shielding and/or redesign of the suspensions is needed to prevent any limitations on ETs sensitivity curve.

The previous discussion assumes all effects from infrastructural noise are overcome. This infrastructural magnetic noise is typically about two orders of magnitude louder than the fundamental magnetic noise; for example, note the effects in second-generation detectors such as Virgo’s central building [156]. In the case of a very pessimistic assumption, where the
<table><thead><tr><th></th><th colspan="2">SGWB searches</th></tr></thead><tbody><tr><td>Noise type</td><td>Triangle</td><td>2L</td></tr><tr><td>Seismic ambient</td><td>Yes, &le; 4 Hz<sup>26</sup></td><td>No</td></tr><tr><td>NN ambient — Rayleigh waves</td><td>Yes, &le; 5 Hz<sup>27</sup></td><td>No</td></tr><tr><td>NN ambient — Body waves</td><td>Yes, &le; 40 Hz</td><td>No</td></tr><tr><td>Magnetic ambient</td><td>Yes, &le; 30Hz, possibly &ge; 100Hz<sup>28</sup></td><td>Yes, &le; 30Hz</td></tr></tbody></table>

**Table 12.** Frequency regions over which searches for an isotropic SGWB would be affected, considering different types of correlated noise and assuming no improvements in the magnetic coupling function with respect to Advanced LIGO and Advanced Virgo. Furthermore, in this table, we only mention fundamental noise sources and we neglect the potential effects of local (infrastructural) noise. Whereas more research is needed to address these noise sources in the case of seismic and Newtonian noise, for magnetic noise an initial study [156] showed that the correlated noise might affect the entire frequency band for which measurements of the magnetic coupling exist, i.e. between ~ 10 Hz and ~ 675 Hz. This would only affect the search for an SGWB in the case of the triangular configuration.

magnetic noise in ET is the same as in the Virgo central building and it is fully correlated between two ET interferometers, an additional one to two orders of magnitude improvement of the magnetic coupling function will be required. This would only apply to the co-located interferometers, such as the ET triangular configuration. Furthermore, this would imply that the entire frequency range over which measurements of the magnetic coupling exist (~ 10 – 675) Hz, would be affected if no improvement in the magnetic coupling is achieved with respect to Advanced LIGO and Advanced Virgo. In that scenario, the magnetic noise has the potential to limit the ET sensitivity, as shown in the bottom panel of figure 35 [156].

*This would imply that even greater investments might be needed to sufficiently reduce the magnetic coupling in the case of the triangular configuration, as compared to the 2L configuration. Further research is needed to understand to which level these infrastructural noise sources could be produced and correlated between the ET interferometers.*

# 6 Impacts of detector designs on specific science cases

We now discuss how different geometries, or different ASDs, affect specific aspects of the Science Case, examining a broad set of particularly significant questions in fundamental physics, cosmology, and astrophysics.

## 6.1 Physics near the BH horizon

A cornerstone of fundamental physics is testing the nature of compact objects and the gravitational interaction in the relativistic, highly-dynamical regime at the scale of the BH horizon. In vacuum GR, the Kerr geometry [163] is the unique physically acceptable equilibrium, asymptotically flat BH solution [164–166]. Therefore, deviations from the Kerr metric require either modified gravity or specific matter fields within GR.

<footnote n="26"><sup>26</sup>Assuming a horizontal-to-vertical and vertical-to-horizontal coupling ≲ 10<sup>-12</sup> above 4Hz. Tilt measurements and coupling was not taken into account. For details see [141].</footnote>

<footnote n="27"><sup>27</sup>Assuming an underground facility located at a depth of 300m. For details see [141].</footnote>

<footnote n="28"><sup>28</sup>This assumes a magnetic coupling function similar to LIGO Hanford. For a coupling similar to Virgo magnetic contamination is possible ≲ 50Hz and ≳ 100Hz. Furthermore, additional research is needed with respect to lightning activity in Europe and the effect might be site-dependent. For more details see [156] and [135].</footnote>