# New Horizons for Fundamental Physics with LISA

K. G. Arun  Chennai Mathematical Institute, Siruseri, 603103, India  Enis
Belgacem  Institute for Theoretical Physics, Utrecht University,
Princetonplein 5, 3584 CC Utrecht, The Netherlands  Robert Benkel  Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg
1, Potsdam-Golm 14476, Germany  Laura Bernard  LUTH, Observatoire de Paris,
PSL Research University, CNRS, Université de Paris, 5 place Jules Janssen,
92195 Meudon, France  Emanuele Berti  Department of Physics and Astronomy,
Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218,
USA  Gianfranco Bertone  Gravitation Astroparticle Physics Amsterdam (GRAPPA),
Institute for Theoretical Physics Amsterdam and Delta Institute for
Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands  Marc Besancon  CEA Paris-Saclay Irfu/DPhP, Bat.
141, 91191 Gif sur Yvette, France  Diego Blas  Theoretical Particle Physics
and Cosmology Group, Physics Department, King’s College London, University of
London, Strand, London WC2R 2LS, UK  Grup de Física Teòrica, Departament de
Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain  Institut
de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and
Technology,
Campus UAB, 08193 Bellaterra, Spain  Christian G. Böhmer  Department of
Mathematics, University College London, Gower Street, London, WC1E 6BT, United
Kingdom  Richard Brito  CENTRA, Departamento de Física, Instituto Superior
Técnico – IST, Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049
Lisboa, Portugal  Gianluca Calcagni  Instituto de Estructura de la Materia,
CSIC, Serrano 121, 28006 Madrid, Spain  Alejandro Cardenas-Avendaño  Programa
de Matemática, Fundación Universitaria Konrad Lorenz, 110231 Bogotá, Colombia
Illinois Center for Advanced Studies of the Universe and Department of
Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801,
USA  Department of Physics, Princeton University, Princeton, NJ, 08544, USA
Katy Clough  Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH,
UK  Marco Crisostomi  SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN
Sezione di Trieste  IFPU - Institute for Fundamental Physics of the Universe,
Via Beirut 2, 34014 Trieste, Italy  Valerio De Luca  Département de Physique
Théorique, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4,
Switzerland  Daniela Doneva  Theoretical Astrophysics, Eberhard Karls
University of Tübingen, 72076 Tübingen, Germany  INRNE – Bulgarian Academy of
Sciences, 1784 Sofia, Bulgaria  Stephanie Escoffier  Aix Marseille Univ,
CNRS/IN2P3, CPPM, Marseille, France  José María Ezquiaga  Kavli Institute for
Cosmological Physics and Enrico Fermi Institute, The University of Chicago,
Chicago, IL 60637, USA.  Pedro G. Ferreira  Astrophysics, University of
Oxford, Keble Road, Oxford OX1 3RH, UK  Pierre Fleury  Instituto de Física
Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid,
Spain  Stefano Foffa  Département de Physique Théorique, Université de Genève,
24 quai Ernest Ansermet, 1211 Genève 4, Switzerland  Gabriele Franciolini
Département de Physique Théorique, Université de Genève, 24 quai Ernest
Ansermet, 1211 Genève 4, Switzerland  Dipartimento di Fisica, Sapienza
Università di Roma & INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy
Noemi Frusciante  Instituto de Astrofisíca e Ciências do Espaço, Faculdade de
Ciências da Universidade de Lisboa, Edificio C8, Campo Grande, P-1749016,
Lisboa, Portugal  Juan García-Bellido  Instituto de Física Teórica UAM-CSIC,
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain  Carlos
Herdeiro  Departamento de Matemática da Universidade de Aveiro and CIDMA,
Campus de Santiago, 3810-183 Aveiro, Portugal  Thomas Hertog  Institute for
Theoretical Physics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
Tanja Hinderer  Institute for Theoretical Physics Utrecht University
Princetonplein 5 3584CC Utrecht, The Netherlands  Philippe Jetzer  Department
of Physics, University of Zürich, Winterthurerstrasse 190, 8057 Zürich,
Switzerland  Lucas Lombriser  Département de Physique Théorique, Université de
Genève, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland  Elisa Maggio
Dipartimento di Fisica, Sapienza Università di Roma & INFN Roma1, Piazzale
Aldo Moro 5, 00185, Roma, Italy  Michele Maggiore  Département de Physique
Théorique, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4,
Switzerland  Michele Mancarella  Département de Physique Théorique, Université
de Genève, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland  Andrea Maselli
Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy  INFN, Laboratori
Nazionali del Gran Sasso, I-67100 Assergi, Italy  Sourabh Nampalliwar
Theoretical Astrophysics, Eberhard-Karls Universität Tübingen, D-72076
Tübingen, Germany  David Nichols  Gravitation Astroparticle Physics Amsterdam
(GRAPPA), Institute for Theoretical Physics Amsterdam and Delta Institute for
Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands  Department of Physics, University of Virginia,
P.O. Box 400714, Charlottesville, VA 22904-4714, USA  Maria Okounkova  Center
for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY
10010, USA  Paolo Pani  Dipartimento di Fisica, Sapienza Università di Roma &
INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy  Vasileios Paschalidis
Departments of Astronomy and Physics, The University of Arizona, Tucson, AZ
85721, USA  Alvise Raccanelli  Dipartimento di Fisica Galileo Galilei,
Università di Padova, I-35131 Padova, Italy  INFN Sezione di Padova, I-35131
Padova, Italy  Theoretical Physics Department, CERN, 1 Esplanade des
Particules, 1211 Geneva 23,cSwitzerland  Lisa Randall  Harvard University, 17
Oxford St., Cambridge, MA, 02139, USA  Sébastien Renaux-Petel  Institut
d’Astrophysique de Paris, GReCO, UMR 7095 du CNRS et de Sorbonne Université,
98bis boulevard Arago, 75014 Paris, France  Antonio Riotto  Département de
Physique Théorique, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève
4, Switzerland  Milton Ruiz  Department of Physics, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA  Alexander Saffer  Department of
Physics, University of Virginia, Charlottesville, Virginia 22904, USA  Mairi
Sakellariadou  Theoretical Particle Physics and Cosmology Group, Physics
Department, King’s College London, University of London, Strand, London WC2R
2LS, UK  Ippocratis D. Saltas  CEICO, Institute of Physics of the Czech
Academy of Sciences, Na Slovance 2, 182 21 Praha 8, Czechia  B. S.
Sathyaprakash  Institute for Gravitation and the Cosmos, Department of
Physics, Penn State University, University Park PA 16802, USA  Department of
Astronomy and Astrophysics, Penn State University, University Park PA 16802,
USA  School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA,
United Kingdom  Lijing Shao  Kavli Institute for Astronomy and Astrophysics,
Peking University, Beijing 100871, China  Carlos F. Sopuerta  Institut de
Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193
Cerdanyola del Vallès, Spain  Institut d’Estudis Espacials de Catalunya
(IEEC), Edifici Nexus, Carrer del Gran Capità 2-4, despatx 201, 08034
Barcelona, Spain  Thomas P. Sotiriou  School of Mathematical Sciences & School
of Physics and Astronomy, University of Nottingham, University Park,
Nottingham, NG7 2RD, UK  Nikolaos Stergioulas  Department of Physics,
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece  Nicola
Tamanini  Laboratoire des 2 Infinis - Toulouse (L2IT-IN2P3), Université de
Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France  Filippo Vernizzi
Institut de physique théorique, Université Paris Saclay CEA, CNRS, 91191 Gif-
sur-Yvette, France  Helvi Witek  Department of Physics, University of Illinois
at Urbana-Champaign, Urbana, Illinois 61801, USA  Kinwah Wu  Mullard Space
Science Laboratory, University College London, Holmbury St Mary, Surrey, RH5
6NT, UK  Kent Yagi  Department of Physics, University of Virginia, P.O. Box
400714, Charlottesville, VA 22904-4714, USA  Stoytcho Yazadjiev  Theoretical
Astrophysics, Eberhard Karls University of Tübingen, 72076 Tübingen, Germany
Department of Theoretical Physics, Faculty of Physics, Sofia University, 1164
Sofia, Bulgaria  Institute of Mathematics and Informatics, Bulgarian Academy
of Sciences, Acad. G. Bonchev Street 8, 1113 Sofia, Bulgaria  Nicolás Yunes
Illinois Center for Advanced Studies of the Universe and Department of
Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801,
USA  Miguel Zilhão  Centro de Astrofísica e Gravitação (CENTRA), Departamento
de Física, Instituto Superior Técnico, Universidade de Lisboa,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Niayesh Afshordi  Perimeter Institute, Waterloo, Canada  Marie-Christine
Angonin  SYRTE, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, LNE, 61 avenue de l’Observatoire 75014 Paris, France  Vishal
Baibhav  Department of Physics and Astronomy, Johns Hopkins University, 3400
N. Charles Street, Baltimore, Maryland 21218, USA  Enrico Barausse  SISSA, Via
Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste  IFPU -
Institute for Fundamental Physics of the Universe, Sezione di Trieste  Tiago
Barreiro  Instituto de Astrofisíca e Ciências do Espaço, Faculdade de Ciências
da Universidade de Lisboa, Edificio C8, Campo Grande, P-1749016, Lisboa,
Portugal  Nicola Bartolo  Dipartimento di Fisica e Astronomia “G. Galilei",
Universitá degli studi di Padova, via Marzolo 8, I-35131, Padova, Italy
Nicola Bellomo  Theory Group, Department of Physics, University of Texas,
Austin, TX 78712, USA  Ido Ben-Dayan  Department of Physics, Ariel University,
Ariel, POB 3, 4070000, Israel  Eric A. Bergshoeff  Van Swinderen Institute for
Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG
Groningen, The Netherlands  Sebastiano Bernuzzi  Theoretisch-Physikalisches
Institut, Friedrich-Schiller-Universität Jena, 07743, Jena, Germany  Daniele
Bertacca  Dipartimento di Fisica e Astronomia “G. Galilei", Universitá degli
studi di Padova, via Marzolo 8, I-35131, Padova, Italy  Swetha Bhagwat
Dipartimento di Fisica, Sapienza Universita di Roma, Piazzale Aldo Moro 5,
00185, Roma, Italy  Béatrice Bonga  Institute for Mathematics, Astrophysics
and Particle Physics, Radboud University, 6525 AJ Nijmegen, The Netherlands
Lior M. Burko  Theiss Research, 7411 Eads Ave, La Jolla, California 92037, USA
Geoffrey Compére  Université Libre de Bruxelles, Campus Plaine CP 231, 1050
Bruxelles, Belgium  Giulia Cusin  Département de Physique Théorique and Centre
for Astroparticle Physics, Université de Genève, 24 quai E. Ansermet, CH-1211
Geneva, Switzerland  Antonio da Silva  Instituto de Astrofisíca e Ciências do
Espaço, Faculdade de Ciências da Universidade de Lisboa, Edificio C8, Campo
Grande, P-1749016, Lisboa, Portugal  Saurya Das  Department of Physics and
Astronomy, University of Lethbridge, 4401 University Drive, Lethbridge,
Alberta, Canada T1K 3M4  Claudia de Rham  Theoretical Physics, Blackett
Laboratory, Imperial College London, SW7 2AZ, UK  Kyriakos Destounis
Theoretical Astrophysics, Eberhard Karls University, 72076 Tübingen, Germany
Ema Dimastrogiovanni  Van Swinderen Institute for Particle Physics and
Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The
Netherlands  Francisco Duque  CENTRA, Departamento de Física, Instituto
Superior Técnico – IST, Universidade de Lisboa – UL, Avenida Rovisco Pais 1,
1049 Lisboa, Portugal  Richard Easther  Department of Physics, University of
Auckland, Private Bag 92019, Auckland, New Zealand  Hontas Farmer  Elmhurst
University, Department of Physics, Elmhurst, IL, USA &College of
DuPage,Science, Technology, Engineering and Math Division, Glenn Ellyn, IL,
USA  Matteo Fasiello  Instituto de Física Téorica UAM/CSIC, calle Nicolás
Cabrera 13-15, Cantoblanco, 28049, Madrid, Spain  Stanislav Fisenko
Department of Information Security, Moscow State Linguistic University ,
Moscow 119034, Russia  Kwinten Fransen  Institute for Theoretical Physics, KU
Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium  Jörg Frauendiener
Department of Mathematics and Statistics, University of Otago, Dunedin, New
Zealand  Jonathan Gair  Max Planck Institute for Gravitational Physics (Albert
Einstein Institute), Am Mühlenberg 1, Potsdam-Golm 14476, Germany  László
Árpád Gergely  Institute of Physics, University of Szeged, Tisza L. krt.
84-86, Szeged 6720, Hungary  Davide Gerosa  Dipartimento di Fisica “G.
Occhialini”, Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3,
20126 Milano, Italy  INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3,
20126 Milano, Italy  Leonardo Gualtieri  Dipartimento di Fisica, “Sapienza”
Università di Roma & Sezione INFN Roma 1, P.A. Moro 5, 00185 Roma, Italy  Wen-
Biao Han  Shanghai Astronomical Observatory, CAS, 80 Nandan Road, Shanghai,
200030, China  Aurelien Hees  SYRTE, Observatoire de Paris, Université PSL,
CNRS, Sorbonne Université, LNE, 61 avenue de l’Observatoire 75014 Paris,
France  Thomas Helfer  Department of Physics and Astronomy, Johns Hopkins
University, 3400 N. Charles Street, Baltimore, Maryland 21218, USA  Jörg
Hennig  Department of Mathematics and Statistics, University of Otago,
Dunedin, New Zealand  Alexander C. Jenkins  Department of Physics and
Astronomy, University College London, London WC1E 6BT, UK  Eric Kajfasz  Aix
Marseille Univ, CNRS/IN2P3, CPPM, IPhU, Marseille, France  Nemanja Kaloper
QMAP, Dpeartment of Physics and Astronomy, UC Davis, Davis CA 95616, USA
Vladimír Karas  Astronomical Institute, Czech Academy of Sciences, Boční II
1401, 14100 Prague, Czech Republic  Bradley J. Kavanagh  Instituto de Física
de Cantabria (IFCA, UC-CSIC), Av. de Los Castros s/n, 39005 Santander, Spain
Sergei A. Klioner  Lohrmann-Observatorium, Technische Universität Dresden,
01062 Dresden, Germany  Savvas M. Koushiappas  Department of Physics & Brown
Theoretical Physics Center, Brown University, 182 Hope St., Providence, Rhode
Island 02912, USA  Macarena Lagos  Department of Physics and Astronomy,
Columbia University, New York, NY 10027, USA  Christophe Le Poncin-Lafitte
SYRTE, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, LNE,
61 avenue de l’Observatoire 75014 Paris, France  Francisco S. N. Lobo
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da
Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016, Lisbon,
Portugal  Charalampos Markakis  School of Mathematical Sciences, Queen Mary
University of London, Mile End Road, London E1 4NS  Prado Martín-Moruno
Departamento de Física Teórica and IPARCOS, Universidad Complutense de Madrid,
E-28040 Madrid, Spain  C.J.A.P. Martins  Centro de Astrofísica da Universidade
do Porto and Instituto de Astrofísica e Ciências do Espaço, Rua das Estrelas,
4150-762 Porto, Portugal  Sabino Matarrese  Dipartimento di Fisica e
Astronomia “G. Galilei", Universitá degli studi di Padova, via Marzolo 8,
I-35131, Padova, Italy  Daniel R. Mayerson  Institute for Theoretical Physics,
KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium  José P. Mimoso
Instituto de Astrofisíca e Ciências do Espaço, Faculdade de Ciências da
Universidade de Lisboa, Edificio C8, Campo Grande, P-1749016, Lisboa, Portugal
Johannes Noller  Institute of Cosmology & Gravitation, University of
Portsmouth, Portsmouth, PO1 3FX, UK  Nelson J. Nunes  Instituto de Astrofisíca
e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa,
Edificio C8, Campo Grande, P-1749016, Lisboa, Portugal  Roberto Oliveri
CEICO, Institute of Physics of the Czech Academy of Sciences,
Na Slovance 2, 182 21 Praha 8, Czechia  Giorgio Orlando  Van Swinderen
Institute for Particle Physics and Gravity, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands  George Pappas  Department of
Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Igor Pikovski  Department of Physics, Stockholm University, SE-106 91
Stockholm, Sweden  Luigi Pilo  Dipartimento di Scienze Fisiche e Chimiche,
Università dell’Aquila, via Vetoio, 67010 L’Aquila  Jiří Podolský  Institute
of Theoretical Physics, Charles University, V Holešovičkách 2, 18000 Prague 8,
Czechia  Geraint Pratten  School of Physics and Astronomy and Institute for
Gravitational Wave Astronomy, University of Birmingham, Edgbaston, Birmingham,
B15 9TT, United Kingdom  Tomislav Prokopec  Institute for Theoretical Physics,
Princetonplein 5, 3584 CC Utrecht, The Netherlands  Hong Qi  School of Physics
and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kindom  Saeed
Rastgoo  Department of Physics and Astronomy, York University 4700 Keele
Street, Toronto, Ontario M3J 1P3, Canada  Angelo Ricciardone  Dipartimento di
Fisica e Astronomia "G. Galilei", Università degli studi di Padova, via
Marzolo 8, I-35131, Padova, Italy  Rocco Rollo  Centro Nazionale INFN di Studi
Avanzati GGI, Largo Enirico Fermi 2, I-50125 Firenze, Italy  Diego Rubiera-
Garcia  Departamento de Fśica Teórica and IPARCOS, Universidad Complutense de
Madrid, E-28040, Madrid, Spain  Olga Sergijenko  Astronomical observatory,
Taras Shevchenko National University of Kyiv, Observatorna, 3, 04053 Kiev, and
Main Astronomical Observatory of the National Academy of Sciences of Ukraine,
Zabolotnoho str., 27, Kyiv, 03680, Ukraine  Stuart Shapiro  Departments of
Physics and Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL
61801 USA  Deirdre Shoemaker  Center for Gravitational Physics, University of
Texas at Austin, Austin TX 78712  Alessandro Spallicci  Université d’Orléans -
Centre National de la Recherche Scientifique, LPC2E, 3A Avenue de la Recherche
Scientifique, 45071 Orléans, France  Oleksandr Stashko  Taras Shevchenko
National University of Kyiv, Ukraine  Leo C. Stein  Department of Physics and
Astronomy, University of Mississippi, University, Mississippi 38677, USA
Gianmassimo Tasinato  Swansea University, UK  Andrew J. Tolley  Theoretical
Physics, Blackett Laboratory, Imperial College, London, SW7 2AZ, U.K  Elias C.
Vagenas  Theoretical Physics Group, Department of Physics, Kuwait University,
P.O. Box 5969, Safat 13060, Kuwait  Stefan Vandoren  Institute for Theoretical
Physics, Princetonplein 5, 3584 CC Utrecht, The Netherlands  Daniele Vernieri
Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II” and
INFN, Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via
Cinthia, I-80126, Napoli, Italy.  Rodrigo Vicente  CENTRA, Departamento de
Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Avenida
Rovisco Pais 1, 1049 Lisboa, Portugal  Toby Wiseman  Theoretical Physics
Group, Blackett Laboratory, Imperial College, London SW7 2AZ, UK  Valery I.
Zhdanov  Astronomical observatory, Taras Shevchenko National University of
Kyiv, Observatorna, 3, 04053 Kiev, Ukraine  Miguel Zumalacárregui  Max Planck
Institute for Gravitational Physics (Albert Einstein Institute) Am Mühlenberg
1, D-14476 Potsdam-Golm, Germany

(September 3, 2024)

###### Abstract

Abstract

The Laser Interferometer Space Antenna (LISA) has the potential to reveal
wonders about the fundamental theory of nature at play in the extreme gravity
regime, where the gravitational interaction is both strong and dynamical. In
this white paper, the Fundamental Physics Working Group of the LISA Consortium
summarizes the current topics in fundamental physics where LISA observations
of GWs can be expected to provide key input. We provide the briefest of
reviews to then delineate avenues for future research directions and to
discuss connections between this working group, other working groups and the
consortium work package teams. These connections must be developed for LISA to
live up to its science potential in these areas.

###### Contents

  1. I Introduction
    1. I.1 Abbreviations
  2. II Tests of General Relativity
    1. II.1 Tests of Gravity and its fundamental principles
      1. II.1.1 The Equivalence Principle
      2. II.1.2 Lovelock’s Theorem and GR uniqueness
    2. II.2 Testing GR with compact objects
      1. II.2.1 BHs beyond GR and theories that predict deviations
      2. II.2.2 Tests with GW propagation
      3. II.2.3 Tests of GR with MBH coalescence
      4. II.2.4 Test of GR with EMRIs (non-null tests)
      5. II.2.5 Gravitational Memory and BMS symmetry
    3. II.3 Burning Questions and Needed Developments
  3. III Tests of the Nature of BHs
    1. III.1 The Kerr hypothesis
    2. III.2 Deviations from the Kerr hypothesis
      1. III.2.1 BHs in non-vacuum GR
      2. III.2.2 ECOs: deviations from (or absence of) a classical horizon
    3. III.3 Observables and tests
      1. III.3.1 Inspiral-based test with SMBH binaries, IMBH binaries, and EMRIs
      2. III.3.2 Ringdown tests
    4. III.4 Burning Questions and Needed Developments
  4. IV Dark Matter and Primordial BHs
    1. IV.1 Low mass particles $m<eV$
    2. IV.2 High mass particles $m>eV$
    3. IV.3 PBHs $m<M_{\odot}$
    4. IV.4 PBHs $m>M_{\odot}$
    5. IV.5 Exotic Compact Objects
    6. IV.6 Signatures from self-interactions
    7. IV.7 Multi-messenger signatures
    8. IV.8 Burning Questions
  5. V Tests of the $\Lambda$CDM model and dark energy
    1. V.1 Dark energy and modified gravity
      1. V.1.1 Effective field theory of dark energy
      2. V.1.2 Non-local model
      3. V.1.3 Early and interacting dark energy
      4. V.1.4 Screening
    2. V.2 Homogeneous cosmology
      1. V.2.1 Standard sirens
      2. V.2.2 Luminosity distance
      3. V.2.3 Hubble constant
    3. V.3 Large-scale structure
      1. V.3.1 Cross-correlation with large-scale structure
      2. V.3.2 Gravitational lensing of GWs
    4. V.4 Burning Questions
  6. VI Model-Independent Tests
    1. VI.1 Consistency tests of GR
      1. VI.1.1 Residual Tests
      2. VI.1.2 Inspiral-merger-ringdown (IMR) tests
      3. VI.1.3 Multipolar gravitational wave tests
    2. VI.2 Parametrized Tests
      1. VI.2.1 Inspiral tests
      2. VI.2.2 Ringdown tests
      3. VI.2.3 Mapping to non-GR Theories
    3. VI.3 Other Model-independent Tests
      1. VI.3.1 Polarization Tests
      2. VI.3.2 GW Propagation Tests
      3. VI.3.3 Stochastic Gravitational Wave Background Tests
    4. VI.4 Burning Questions and Needed Developments
  7. VII Astrophysical and Waveform Systematics
    1. VII.1 Astrophysics: Environmental effects
      1. VII.1.1 Binary and disk interactions
      2. VII.1.2 Interaction with stellar environments and hierarchical triplets
    2. VII.2 Waveform modelling and systematics for EMRIs
    3. VII.3 Waveform modelling in GR and systematics for comparable mass binaries
      1. VII.3.1 PN waveforms
      2. VII.3.2 NR waveforms
      3. VII.3.3 Phenom and EoB waveforms
    4. VII.4 Waveform modelling beyond GR and systematics for comparable mass binaries
      1. VII.4.1 NR waveforms
      2. VII.4.2 PN waveforms
      3. VII.4.3 Effects of bosonic dark matter
      4. VII.4.4 EMRI waveforms
    5. VII.5 Burning Questions and Needed Developments
  8. VIII Future Directions

##  I Introduction

Gravity is at the forefront of many of the deepest questions in fundamental
physics. These questions include the classical dynamics and quantum nature of
black holes (BHs), the matter and antimatter asymmetry of the observable
universe, the processes at play during the expansion of the universe and
during cosmological structure formation, and, of course, the intrinsic nature
of dark matter and dark energy, and perhaps of spacetime itself. These
questions are cross-generational and their answers are likely to require
cross-disciplinary explorations, instead of being the purview of a particular
sub-discipline.

The recent observations of gravitational waves (GWs) are beginning to help us
dig deeper into these questions, opening new research paths that enable the
fruitful interaction of fundamental theory and observation. GWs have the
potential to examine largely unexplored regions of the universe that are
otherwise electromagnetically obscure. Examples of these regions include the
vicinity of BH horizons, early phases in the formation of large-scale
structure, and the hot big bang. Moreover, GWs can complement electromagnetic
(EM) observations in astronomy and cosmology, thus enabling “multi-messenger”
astrophysics and creating a path toward a better understanding of our
universe.

The Laser Interferometer Space Antenna (LISA) has the potential to contribute
enormously in this quest, as this instrument is uniquely positioned to
observe, for the first time, long-wavelength GWs [56], and therefore, new
sources of GW radiation. Why is this? Because LISA can provide new information
about gravitational anomalies, perhaps related to the quantum nature of
gravity, about quantum-inspired effects in BH physics, and even about beyond
the standard model, particle physics. Examples of the information that could
be gained abound, but one concrete example is the following. LISA has the
potential to constrain (or detect) the activation of scalar or vector degrees
of freedom around black holes, which arise in certain modified gravity
theories inspired by quantum gravity. This is because these fields typically
carry energy-momentum away from BH binaries, forcing them to spiral faster
into each other than predicted in GR. This faster rate of inspiral then
modifies the time evolution of the GW frequency, and therefore, its phase,
which LISA is particularly sensitive to.

GW observations with LISA will complement the information we have gained (and
will continue to gain) with ground-based GW interferometers, such as the Laser
Interferometer Gravitational-wave Observatory (LIGO), Virgo and Kagra. This
complementarity arises because ground-based instruments operate at higher
frequencies than LISA, and therefore, they can hear GWs from a completely
different type of sources. This, in turn, implies LISA is uniquely positioned
to detect supermassive BH (SMBH) mergers, extreme mass-ratio inspirals
(EMRIs), galactic binaries, and SGWB, none of which ground-based detectors are
sensitive to. The GWs emitted by some of these sources (such as those emitted
by SMBH mergers) will be extremely loud, generating signal-to-noise ratios
(SNRs) in the thousands, which will allow us to prove fundamental physics
deeply. Other GWs will be less loud, but they will be extremely complex, as is
the case for waves generated by EMRIs; these waves will contain intricate
amplitude and phase modulations that will encode the BH geometry in which the
small compact object zooms and whirls. The observation of GWs with LISA will
also complement observations with pulsar timing arrays and EM observations of
the B-mode polarization in the cosmic microwave background (CMB), which can
inform us about fundamental physics at even lower frequencies.

The goal of this white paper is threefold. First, we aim to identify those
topics in fundamental physics beyond the current standard models of particle
physics, gravity and cosmology that are particularly relevant for the
scientific community, in order to delineate and sharpen LISA’s potential in
each of these areas. The organization of this article reflects this
identification of topics, with each Section covering one area. The range of
areas is illustrated in Figure 1. Second, within each Section we summarize the
state of the art of each of these areas, both from a theoretical and an
observational point of view. In order to keep this article both manageable and
useful, rather than presenting an extensive detailed review of each topic, we
have opted for sufficiently concise “reviews”, referring to excellent recent
articles in Living Reviews and elsewhere for more details (see e.g., Barack et
al. 109). Put differently, the philosophy behind this white paper is to strive
towards completion in terms of what are relevant branches of fundamental
physics for LISA that we think ought to be explored further, rather than
providing extensive background and technical details on each of the topics.
Third, each Section then assesses what must be done in order for LISA to live
up to its scientific potential in these areas of fundamental physics. This is
primarily where the living part of this article enters, which should be viewed
in a LISA-specific context. Finally, we bring these different strands together
and identify possible synergies in a Roadmap Section at the end.

In closing, let us note that this article is part of a series of LISA working
group (WG) articles. Some of the topics discussed here are of interest to
several WGs organized within the LISA Consortium. Strong synergies exist
between the Fundamental Physics WG, the Cosmology, the Astrophysics, and the
Waveform modelling WGs, each of which approaches the topics discussed in this
article from different, complementary angles. One of the intended goals of
this article is to promote these synergies and connect the WGs, specially the
fundamental physics one to the LISA work package groups, so that the ideas
presented here can be implemented and deployed in future LISA data analysis.

Figure 1: Schematic diagrams of topics of relevance to the Fundamental Physics
WG of the LISA Consortium. Some of these topics overlap naturally with other
WGs, such as the Waveform modelling WG, the Astrophysics WG and the Cosmology
WG.

###  I.1 Abbreviations

We here list a set of abbreviations commonly used in this review article:

  * •

BH = Black hole.

  * •

BBH = Binary black hole.

  * •

BMS = Bondi-Van der Burg-Metzner-Sachs.

  * •

CMB = Cosmic microwave background.

  * •

DM = Dark matter.

  * •

dCS = Dynamical Chern–Simons.

  * •

ECO = Exotic compact object.

  * •

EdGB = Einstein dilaton Gauss Bonnet.

  * •

EdM = Einstein dilaton Maxwell.

  * •

EEP = Einstein equivalence principle.

  * •

EM = Electromagnetic.

  * •

EMRI = Extreme mass-ratio inspiral.

  * •

EsGB = Einstein scalar Gauss–Bonnet gravity.

  * •

FLRW = Friedmann–Lemaître–Robertson–Walker.

  * •

GR = General relativity.

  * •

GW = Gravitational waves.

  * •

IMBH = Intermediate-mass black hole.

  * •

IMR = Inspiral-Merger-Ringdown.

  * •

LIGO = Laser Interferometer Gravitational-Wave Observatory.

  * •

LISA = Laser Interferometer Space Antenna.

  * •

LLI = Local Lorentz invariance.

  * •

LPI = Local position invariance.

  * •

MCMC = Markov Chain Monte Carlo.

  * •

MBH = Massive black hole.

  * •

NS = Neutron star.

  * •

QNM = Quasinormal model.

  * •

PBH = Primordial black hole.

  * •

PN = Post-Newtonian.

  * •

ppE = Parameterized post-Einsteinian.

  * •

ppN = Parametrized post-Newtonian.

  * •

SCO = Stellar compact object.

  * •

SEP = Strong equivalence principle.

  * •

SGWB = stochastic gravitational wave background.

  * •

SMBH = Supermassive black hole.

  * •

SNR = Signal-to-noise ratio.

  * •

TeVeS = Tensor-vector-scalar gravity.

  * •

WD = White dwarf.

  * •

WG = Working group.

  * •

WEP = Weak equivalence principle.

Henceforth, we use geometric units in which $G=1=c$.

##  II Tests of General Relativity

###  II.1 Tests of Gravity and its fundamental principles

GWs and some of their major sources — BHs and relativistic stars — are among
the most groundbreaking predictions of GR. It is hence rather intuitive that
GW observations should offer unique insights into the workings of gravity and
the fundamental principles that underpin GR.

####  II.1.1 The Equivalence Principle

The various versions of the equivalence principle are commonly used as a
theoretical foundation of GR and as guiding principles for tests of gravity
[1194].

The Weak Equivalence Principle (WEP) postulates that the trajectory of a
freely falling test body is independent of its structure and composition,
providing the fact that there are no external forces acting on this body such
as electromagnetism. The Einstein Equivalence Principle (EEP) goes one step
further and combines the WEP, Local Lorentz Invariance (LLI), and Local
Position Invariance (LPI). The LLI is connected with the assumption that the
outcome of any local non-gravitational test experiment is independent of the
velocity of the freely-falling frame of reference where this experiment is
performed. The LPI requires that the outcome of such an experiment is
independent of where and when it is performed. The Strong Equivalence
Principle (SEP) is equivalent to the EEP with the WEP extended to self-
gravitating bodies and the LLI and LPI to any experiment.

The EEP dictates the universality of couplings in the standard model because
of LLI and LPI. The SEP takes this even further and implies that the
gravitational coupling is fixed as well. Testing the SEP is often considered
synonymous to testing GR itself [1194]. If the SEP is fulfilled, the only
field that is a mediator of the gravitational interaction should be the
spacetime metric. Conversely, the existence of new fields mediating gravity
would generically lead to violations of the SEP (which might appear only in
specific systems, depending on how elusive the fields are).

####  II.1.2 Lovelock’s Theorem and GR uniqueness

The discussion above already suggests strongly that testing GR (classically)
amounts to looking for new fields. An independent way to reach the same
conclusion is to start from Lovelock’s theorem [803]: Einstein’s field
equations are unique, assuming that we are working in four dimensions,
diffeomorphism invariance is respected, the metric is the only field mediating
gravity, and the equations are second-order differential equations. Violating
any of these assumption can circumvent Lovelock’s theorem and lead to distinct
alternative theories of gravity. However, the vast majority of them111See
Flanagan [525], Barausse et al. [115], Pani and Sotiriou [947], Pani et al.
[955] for discussions regarding attempts to circumvent Lovelock’s theorem
without adding new fields. will share one property: they will contain one or
more additional fields (see e.g., Sotiriou 1094). Demonstrating this
mathematically might require compactifying spacetime down to 4-dimensions,
introducing additional fields to restore diffeomorphism invariance, and
performing field redefinitions.

The realization that testing the SEP and looking for deviations from GR
largely amounts to looking for new fields allows one to understand intuitively
how compact objects and GWs can be used to probe such deviations. When the new
fields have a nontrivial configuration around a BH or compact star, the latter
can be thought of as carrying a ‘charge’ (this might not be a conserved charge
associated to a gauge symmetry). Field theory intuition tells us that
accelerating charges radiate. Hence, binaries beyond GR will exhibit
additional GW polarizations. Emission in extra polarizations affects the rate
of energy loss and, in turn, the orbital dynamics. So, the pattern of emission
of conventional polarizations will also be affected. In Sect. II.2.1 we
discuss how the structure of BHs—primary sources for LISA—can be affected by
new fields, whereas in the rest of Sect. II.2 we discuss how deviations from
GR can be imprinted in different parts of the waveform or affect GW
propagation.

###  II.2 Testing GR with compact objects

####  II.2.1 BHs beyond GR and theories that predict deviations

There exist no-hair theorems for a wide range of alternative theories of
gravity [615, 147, 1097, 667, 1093, 625, 473] stating that the quiescent,
isolated BHs are indistinguishable from their GR counterpart. If the
additional fields can be excited, quasinormal mode (QNM) ringing can still be
distinct from GR [114, 868, 1132] and hence offer a possibility of test
theories that are covered by no-hair theorems. Models that manage to
circumvent no-hair theorems are expected to lead to more prominent deviations
from GR in GW signals, as they can posses additional charges and exhibit
additional interactions that can affect all parts of the waveform.

Scalar fields coupled to the Gauss–Bonnet invariant or the Pontryagin density
(dynamical Chern–Simons gravity) are known to lead to BH hair [283, 717, 1230,
1214, 1098, 1099, 167, 168, 1227, 1104, 446, 443] which can be observed with
LISA [1213, 1209, 836]. Such couplings are expected in low-energy limits of
quantum gravity (see e.g., Metsaev and Tseytlin 858, Ashtekar et al. 85,
Jackiw and Pi 687) and are part of the Horndeski class of scalar-tensor
theories [745]. A coupling to the Pontryagin density is the leading-order
parity violating term in gravity in the presence of a (pseudo) scalar [687,
45], while the linear coupling with the Gauss–Bonnet invariant is the only
term inducing hair for shift-symmetric (aka massless) scalars [1098, 1049,
443] and the leading correction for GW emission [1202]. A nonlinear coupling
with the Gauss–Bonnet invariant has recently been shown to give rise to BH
spontaneous scalarization triggered by curvature [474, 1085, 230, 1086, 813,
389, 363] or spin [464, 650, 475, 628, 187, 476]. The full set of Horndeski
theories that can give rise to BHs scalarization has been identified in
Antoniou et al. [72].

Attempts have also been made to circumvent no-hair theorems by relaxing their
assumptions. For example, superradiance can support long-lived scalar clouds
for very light scalars [79, 250] or lead to hairy BHs for complex scalars with
a time dependent phase [629, 742, 632, 623, 441, 362]; long-lived scalar
“wigs” can be formed around a Schwarzschild BH [127, 128]; configurations with
time-dependent scalar fields can be supported by some non-trivial cosmological
boundary conditions [688, 90, 183, 358, 669]; BHs immersed in an inhomogeneous
scalar field were considered in Healy et al. [619]; scalarization can be
induced by matter surrounding a BH [303, 304, 627].

BHs in Lorentz-violating theories, such as Einstein-aether theory [690, 689]
and HoVision Res. ava gravity [656, 228, 1092], will generically carry hair,
as the field that breaks Lorentz symmetry will have to be nontrivial and may
backreact on the geometry (see, however, Ramos and Barausse 1002, Adam et al.
26). In such theories, new fields can propagate superluminally or even
instantaneously [225, 202], even when all known constraints are satisfied
[1095, 488]. Indeed, BHs in Lorentz-violating theories can have a nested
structure of different horizons for different modes [487, 117] and potentially
a universal horizon that traps all signals [117, 225, 203]. GW observations
can probe this richer causal structure and yield novel constraint on Lorentz
symmetry breaking.

Hairy BHs have been studied in a variety of other scenarios and theories,
including generalized Proca theories [622, 863, 94, 620, 720, 1000, 1048] and
massive gravity theories [170, 365, 1029].

####  II.2.2 Tests with GW propagation

The propagation of GWs provides a clean test of their kinematics. In GR,
gravitons are massless spin-2 particles, while in alternative gravity
theories, they might be massive [1196, 1017, 614, 11], or even have a Lorentz-
violating structure in the dispersion relation [690, 689, 656, 228, 1092, 866,
754, 1076]. Modifications to the dispersion relation could lead to frequency-
dependent, polarization-dependent, direction-dependent propagating velocities
of GWs. Eventually, while these components travel at different velocities, the
gravitational waveforms received on the Earth are distorted, with respect to
their original chirping structure at generation [1196, 754]. Therefore, a
matched-filter analysis allowing the possibility of the modified GW
propagation in the waveform will reveal the nature of graviton kinematics (see
e.g., Abbott et al. 11) and provide stringent constraints on the mass of the
graviton and on Lorentz symmetry violations. The binary neutron star (NS)
merger GW170817 has already provided a strong, double-sided bound on the speed
of GWs to a part in $10^{15}$ [12]. However, Lorentz-violating theories have
multidimensional parameter spaces and generically exhibit additional
polarizations [1095]. The speed of these other polarizations remains virtually
unconstrained [488, 928]. A combination of multiple GW events can also be used
to simultaneously constrain a set of beyond-GR parameters [1076].

GWs travel over cosmological distances before they reach the detector. On a
Friedmann-Lemaître-Robertson-Walker (FLRW) background, we can describe GWs by
$h_{ij}$, the transverse and traceless perturbation of the spatial metric,
$g_{ij}({\bf x},t)=a^{2}(t)\left[\delta_{ij}+h_{ij}({\bf x},t)\right]$. In
Fourier space, the most general modification of the GW propagation equation
can be written as (assuming spatially flat models) [494, 122, 493]

$\ddot{h}_{ij}({\bf k},t)+\left[3H(t)+\Gamma(k,t)\right]\dot{h}_{ij}({\bf
k},t)+\left[c_{\rm T}^{2}(t)k^{2}+D(k,t)\right]h_{ij}({\bf k},t)=0\;,$ (1)

where $H\equiv\dot{a}/a$ is the Hubble rate. The parameters $c_{\rm T}$,
$\Gamma$ and $D$ respectively describe the speed of propagation of the wave,
the damping of its amplitude and additional modifications of the dispersion
relation. Scale-independent modifications, described by a $k$-independent
$\Gamma$, are discussed in Sect. V. One expects a single observation to
exclude $\Gamma\gtrsim L^{-1}$ and $D\gtrsim fL^{-1}$, where $L$ is the
distance to the source and $f$ the GW frequency, although the detailed
constraints are also controlled by the frequency dependence of these
quantities. $c_{\rm T}$, $\Gamma$ and $D$ will depend on the theory of gravity
and on the cosmological background. Hence, one can translate bounds on these
parameters into bounds on a given model. For example, the speed of GW bound
has severely constrained generalized scalar-tensor theories [657, 437, 1253,
585, 777, 386, 159] under the assumption that they account for dark energy
[800, 849, 378, 496, 1040, 100] (see also de Rham and Melville 1016). Relaxing
this assumption [533, 916, 73] can lift the constraint.

GW propagation tests can also reveal potential couplings and decays of
gravitons into other particles (e.g., Creminelli et al. 383) or oscillations
between different states (analogous to neutrino oscillations). The latter are
expected in bigravity models [614], where a massless and a massive spin-2
fields interact in a specific way to avoid ghost degrees of freedom. The
lightest tensor mode is the one that couples to matter and its speed can be
constrained as above if there is a prompt EM counterpart. Its amplitude
determines the ratio between the luminosity distance of GWs and the one of EM
radiation, which oscillates as a function of redshift. Using several
astrophysical population models for the population of massive black hole
binaries [110, 743] and performing a $\chi^{2}$ analysis, it was found that
oscillation effects can be observed for masses $m\gtrsim 2\cdot 10^{-25}\,$eV
[155]. Thus LISA will provide a $\sim 3$ order of magnitude improvement in
mass sensitivity over the current LIGO/Virgo limit, which probes $m\gtrsim
10^{-22}$eV [843], due to the larger oscillation baseline and the lower
detection frequency.

####  II.2.3 Tests of GR with MBH coalescence

##### Inspiral

LISA will observe very long inspiral phases and it will therefore allow for
high precision tests of gravity [178]. This stage can be modelled using
approximate techniques such as the low-velocity, weak-field PN expansion
[979], or the parameterized post-Einsteinian approach (ppE) [1228], that is
better suited in certain cases for alternative theories of gravity. Even
though tests of the GR nature of the waveforms can be performed without
linking to specific alternative theories of gravity within the PN or ppE
approaches, connecting the predictions of the different generalizations of
Einstein’s theory with the possible deviations in the GR expectation values of
the PN or ppE parameters is an inseparable part of testing the possible
violations of the GR fundamental symmetries [185].

Using the inspiral observations, LISA will be able to improve the constraints
on different non-GR predictions, such as the scalar dipole radiation, Lorenz
symmetry, mass of the graviton, etc., by several orders of magnitude better
compared to LIGO/Virgo [332]. Many alternative theories of gravity predict
non-zero tidal deformation of BHs [310] and that can be also tested using the
inspiral waveforms.

Even more intriguing is the possibility for multiband GW observations of BBH
mergers. The separation between the two members of a BBH determines the
frequencies of the emitted GW signal. A binary that will be in the
observational band of LISA at large separation can several years later enter
the band of ground-based detectors. Using LISA observations and assuming GR,
one can then obtain high-accuracy predictions of the time when the binary will
become observable by ground-based detectors as well as its position on the sky
[1069]. Any deviation of the former, observed by a ground based detector,
would imply a breakdown of GR. For example, joint observations of a LIGO/Virgo
and LISA of a GW150914-like event could improve constraints on BBH dipole
emission by 6 orders of magnitude [120, 1146]. The possibility of observing
the same system more than once for a prolonged period will offer the
opportunity to test completely nonlinear predictions of some generalized
scalar-tensor theories, such as dynamical BH scalarization [732].

##### Ringdown

In GR, numerical simulations have shown that the end state of a BBH merger is
a Kerr BH (e.g., Jani et al. 693, Boyle et al. 244, Healy and Lousto 618) in
agreement with the earlier analytical studies proving that the Kerr metric is
the unique stationary, asytmptotically flat, axially symmetric BH spacetime
[325, 1026, 261, 845] (see Heusler 637, Chrusciel et al. 349 for a review).
Before reaching the final Kerr state, the BH emits GWs in QNMs, which are
labeled by their overtone numbers, $n$, and angular numbers $(l,m)$ and are
determined by these numbers $(l,m,n)$ plus the mass $M$ and spin parameter $a$
of the Kerr BH [181]. Measuring several of these QNMs would allow for “BH
spectroscopy,” in which a perturbed Kerr BH is identified by the spectrum of
QNMs in its ringdown GWs [452, 477, 179]. Specifically, for a given BBH
merger, the number and amplitude of QNMs excited during the ringdown is
determined by the individual BHs and their orbital parameters prior to merger
[715]. If at least two of the QNMs can be measured from the ringdown of a BH,
then by knowing which modes were excited, one can verify that the two QNM
frequencies and two QNM damping times are both consistent with the same mass
$M$ and spin $a$ parameters of the remnant Kerr BH, within the errors of the
observation [477, 179]. For a high SNR massive BBH event measured by LISA,
this “no hair” (or “final-state”) test can be performed to verify the remnant
is consistent with a Kerr BH to high precision (e.g., Gossan et al. 592).

If the underlying gravity theory differs from GR, the ringdown of a BBH merger
will also typically differ from the ringdown in GR. Computing the QNMs in
modified theories is typically challenging, as is performing NR simulations of
BBH mergers in modified gravity theories (both of which have only been
performed in a handful of cases, e.g., Cardoso and Gualtieri 296, Blázquez-
Salcedo et al. 232, Okounkova et al. 927, Okounkova 923). This makes it
difficult to look for deviations in a particular modified theory. Parametrized
tests are instead used to look for deviations. In Tattersall et al. [1133],
deviations from linear perturbations about a Schwarzschild BH in GR were
parametrized at the level of a diffeomorphism-invariant action that
encompassed a large range of possible theories that lead to second-order
equations of motion. A more phenomenological approach to the test for
deviations from GR during ringdown is to perform a parametrized test in which
the QNM frequencies are given by the Kerr values plus small deviations, in a
“post-Kerr” expansion [582]. There are also procedures to try to combine the
deviation parameters from multiple BBH events measured by LISA [837]. With
more detailed predictions from specific theories, parametrized constraints
could be converted into constraints on a particular theory.

##### Merger

The merger phase of a BBH inspiral is arguably when the nonlinear effects of
gravity truly manifest themselves. This makes it a very challenging regime to
model, requiring full nonlinear evolutions of the field equations. A model-
independent self-consistency test, such as the IMR consistency test [564, 565]
could be used in order to avoid having to model the merger in alternative
theories of gravity. However, having theory-specific waveforms that include
the merger is essential. It can provide more stringent constraints and it is
necessary for interpreting them physically. It can also be used to
quantitatively explore deviations from GR that do not affect other parts of
the waveform — e.g. new fields that are highly excited by nonlinear effects
and decay rapidly — and guide and calibrate parametrizations.

Numerical simulations beyond GR have only been performed in a handful of
cases, most notably for scalar nonminimally coupled to curvature invariants
[168, 167, 924, 1202, 926, 927, 331, 1088, 483] and for Einstein–Maxwell-
dilaton theory [649]. Establishing whether the initial value problem is well-
posed in alternative theories is particularly challenging [959, 330, 1050,
1025, 756]. So far, known simulations have circumvented this problem by
adopted Effective Field Theory inspired treatments: either working
perturbatively in the new coupling constants [168, 167, 924, 1202, 926, 927]
or adopting a scheme similar to the Israel-Stewart treatment of viscous
relativistic hydrodynamics, in which additional fields are introduced to
render the system hyperbolic and then exponentially ‘damped’ so that the
evolution equation match the initial system asymptotically [330]. Nonlinear
evolution beyond GR is one of the major challenges in GW modelling.

####  II.2.4 Test of GR with EMRIs (non-null tests)

GW observations of EMRIs offer the opportunity to probe gravity in a mass
range which is unique to LISA. In such systems a stellar mass object orbits
around a more massive component, with typical mass ratios of the order of
$q\sim 10^{-5}-10^{-7}$, leading to a large number of accumulated cycles
before the merger, proportional to $\sim 1/q$. The signal produced by the slow
inspiral provides a detailed map of the spacetime that will be able to
pinpoint deviations from GR predictions (if any) and requires detailed
knowledge of the emitted waveform to avoid spurious systematics [983, 108].

Despite progress within GR [987], tests of gravity with EMRIs have been so far
limited to systematic calculations in modified theories, in which the
different couplings to the gravity sector require, in general, a theory-by-
theory analysis. These additional degrees of freedom, introduced as scalar,
vector or tensor modes, activate extra emission channels that modify the
binary phase evolution. These changes are expected to leave a detectable
footprint in the emitted signal and be augmented by the large number of cycles
followed by EMRIs. Calculations for non GR theories with non minimally coupled
scalar fields have mainly investigated the changes in the emitted flux within
the adiabatic approximation, for some orbital configurations (including
generic orbits) around spinning BHs [950, 232, 302, 1234, 285, 543]. These
works have shown how, depending on the magnitude of the couplings, in some
cases the accumulated GW phase can be large to produce hundreds of cycles of
difference in the binary evolution compared to GR. The projected constraints
inferred by LISA on the parameters of non-GR theories using approximate
waveforms [1234, 285], and modelling beyond the adiabatic approximation, i.e.,
taking into account self-force calculations [1248], requires further work.

Drastic simplifications in the EMRI modelling beyond GR have been recently
proven to hold for a vast class of theories, for which no-hair theorems or
separations of scales provide a decoupling of the metric and scalar
perturbations [836]. This result allows one to describe the background
spacetime as in GR, rendering all the modifications induced by the modified
theory to be universally captured by the scalar field’s charge only. It has
already been used to show that the latter can be measured with unprecedented
precision [838]. In this framework waveform modelling beyond GR can take
advantage of all the efforts devoted so far to study the evolution of scalar
charges around Kerr BHs [105, 454, 327, 458, 1183, 1182, 595, 1184, 902].

####  II.2.5 Gravitational Memory and BMS symmetry

GR predicts that the passage of a GW causes a permanent displacement in the
relative position of two inertial GW detectors. This phenomenon is known as
the displacement memory effect [806, 963, 348, 1142] (see Strominger 1106,
Compère 366 for an account of historical references). Studies on the detection
of the displacement memory effects in actual experiments have been presented
in Favata [508, 507, 509, 510]. More recently, prospects for the memory effect
detection with LISA have been discussed in Islo et al. [682] using SMBH
binaries undergoing coalescence (See also Sect. III). In addition to the
displacement memory effect, other and subdominant effects such as the spin
memory and center of mass memory effects can also take place [961, 906, 907,
527]. Generalizations of the memory effect, not necessarily motivated or
associated to a symmetry, which should in principle be measurable, have also
been discussed in Flanagan et al. [527], Compère [367], and further memory
effects with logarithmic branches have been inferred from graviton amplitude
calculations [1038, 772].

Measurement of memory effects would not only act as a test of GR [906, 907,
527, 659, 1118, 1119], but it could also potentially shed light on puzzling
infrared aspects of quantum field theories. Ideally isolated systems in GR can
be described by asymptotically flat spacetimes. At null infinity, the symmetry
group of asymptotically flat spacetimes in GR is known to be larger than the
Poincare group of symmetries. It contains the original infinite-dimensional
Bondi-Van der Burg-Metzner-Sachs (BMS) group which is a semi-direct product of
the Lorentz group and of an infinite-dimensional group of “angle dependent
translations" called supertranslations [237, 1035]. Further extensions of the
BMS group have also been proposed [126, 284]. BMS supertranslations can be
related to the gravitational memory effect in that the relative positions of
two inertial detectors before and after the passage of a GW differ by a BMS
supertranslation. In addition, perturbative quantum gravity admits remarkable
infrared identities among its scattering amplitudes, the soft theorems, that
have been demonstrated to be the Ward identities of BMS symmetries [1108,
1106]. The classical limit of the quantum soft theorems are equivalent after a
Fourier transformation to the memory effects [1108]. Measuring memory effects
therefore directly probes the infrared structure of quantum gravity
amplitudes.

The detection of GW bursts with memory with the LISA instrument could be
considered from SMBH binary mergers. Over a SMBH binary lifetime, memory
undergoes a negligible growth prior to merger (corresponding to the slow time
evolution of the binary’s inspiral), rapid accumulation of power during
coalescence, and eventual saturation to a constant value at ringdown. Islo et
al. [682] study a simulation-suite of semi-analytic models for the SMBH binary
population with a signature of a memory signal from a SMBH binary approximated
in the time domain by a step-function centered at the moment of coalescence
and assuming a simple power-law model to emulate any environmental interaction
which could influence the coalescence timing (e.g., final parsec problem
Begelman et al. 144). SMBH could stall before reaching a regime where GW
radiation can drive the binary to coalesce). Considering SMBH binaries at
$z<3$ with masses in the range $(10^{5}-10^{7})M_{\odot}$ and with mass ratios
in the range $0.25-1$, LISA prospects could be SNR > 5 events occurring 0.3 -
2.8 times per year in the most optimistic environmental interaction model, and
less than once per million years in the most pessimistic. As shown in Sesana
et al. [1070], Klein et al. [743] (and discussed in the Astrophysics WG White
Paper by Amaro-Seoane et al. 57), most massive BBHs coalescences whose
inspiral signature yields LISA SNR > 5 lie beyond $z=3$ for a 3-year LISA
lifetime, meaning that the results of this study may be interpreted as lower
limits on the number of LISA memory events.

###  II.3 Burning Questions and Needed Developments

Let us now summarize some important questions that ought to be further
investigated in the context of tests of GR, without being however exhaustive.

  * •

Even in theories where black holes deviate from Kerr, or strong field dynamics
deviates from GR, deviation can be very small once consistency and known
viability constraints are imposed or once the dependence of any new charge on
the mass or the spin are taken into account. Further work is needed to pin
down theories and scenarios that could lead to deviations that are observable
by LISA.

  * •

LISA inspiral observations will allow to put severe constraints on different
non-GR predictions. This, however, will require a more accurate and complete
development of GW waveforms including non-GR effects.

  * •

Numerical simulations with a systematic inclusion of non-GR effects for the
merger and ringdowns phase have to be performed such as to get more adequate
templates to be then used once data will be available.

  * •

Similarly more detailed studies of the EMRIs waveforms including non GR
effects have to be performed in a systematic way.

  * •

GW propagation tests are sensitive to potential couplings and decays of
gravitons into other particles. Due to the larger oscillation baseline and
lower frequency range LISA could improve substantially present LIGO/Virgo
bounds. Present studies are preliminary and further analysis could be useful.

##  III Tests of the Nature of BHs

###  III.1 The Kerr hypothesis

In vacuum GR, the Carter–Robinson [325, 1026] uniqueness theorem, with later
refinements (see Chrusciel et al. 349 for a review), establishes that the Kerr
geometry [729] is the unique physically acceptable equilibrium, asymptotically
flat BH solution. This led to the more ambitious proposal that, regardless of
the initial energy-matter content available in a gravitational collapse
scenario, the dynamically formed equilibrium BHs belong to the Kerr family (in
the absence of gauge charges) [1031]. Accordingly, (near-)equilibrium
astrophysical BH candidates are well described by the Kerr metric. This
working proposal is the Kerr hypothesis. Testing the Kerr hypothesis is an
important cornerstone of strong-gravity research in which GW science, and, in
particular, LISA, are expected to give key contributions.

Deviations from the Kerr hypothesis, that we shall refer to as non-Kerrness,
require either modified gravity (discussed in Section II and also below with a
different twist) or non-vacuum GR (discussed in subsection III.2.1). Moreover,
two approaches are, in principle, possible for studying non-Kerrness. The most
explored one is theory dependent: to consider specific choices of matter
contents or modified gravity models, compute the BH solutions (which,
generically, will be non-Kerr), and finally explore the different
phenomenology of a given model. The second one is theory agnostic: to consider
parametrized deviations from the Kerr metric, regardless of the model they
solve (if any). The latter has been fruitfully employed in studying BH
phenomenology in stationary scenarios, e.g., Johannsen and Psaltis [704],
Cardoso et al. [306], but its application in the study of dynamical properties
is more limited, given the potential lack of an underlying theory.

A substantial departure from the Kerr hypothesis is to admit deviations from,
or even the absence of, a classical horizon. This leads to hypothetical exotic
compact objects (ECOs) with a compactness comparable to that of (classical)
BHs (see Cardoso and Pani 300 for a review). One motivation for such a
dramatic scenario is that, from Penrose’s theorem [965], a classical BH
(apparent) horizon implies the existence of spacetime singularities, under
reasonable energy (and other) conditions. Thus, the absence of (or deviations
from) a classical horizon could circumvent the singularity problem. Another
motivation is that quantum corrections may be relevant at the horizon scale,
even for small-curvature supermassive objects [52, 569, 808, 809, 840, 841,
842, 844].

In this context it is also interesting that current LIGO/Virgo GW observations
(especially the recent GW190814; Abbott et al. 17 and GW190521; Abbott et al.
16, 18, respectively in the lower-mass and upper-mass gap forbidden for
standard stellar-origin BHs) do not exclude the possibility that ECOs might
co-exist along with BHs and NSs. Models of ECOs are discussed in Sect.
III.2.2.

###  III.2 Deviations from the Kerr hypothesis

####  III.2.1 BHs in non-vacuum GR

There is a large class of models wherein (covariant) matter-energy is
minimally coupled to Einstein’s gravity. These models obey the EEP (cf. Sect.
II.1.1) and fall into the realm of GR.

Including minimally coupled matter fields, with standard kinetic terms and
obeying some energy conditions (typically the dominant) can be quite
restrictive for the admissible BH solutions. In many models it prevents the
existence of non-Kerr BHs. This has been typically established by model-
specific no-hair theorems. Historically influential examples are the
Bekenstein no-scalar and no-massive-vector hair theorems [145] (see Herdeiro
and Radu 625 for a review). Nonetheless, non-standard kinetic terms (e.g.
Skyrme hair; Luckock and Moss 805), negative energies (e.g. interacting real
scalar hair; Nucamendi and Salgado 919), non-linear matter models (e.g.
Yang–Mills hair; Bizon 217) or symmetry non-inheritance between the geometry
and matter fields (e.g. synchronised bosonic hair; Herdeiro and Radu 629,
Herdeiro et al. 622) allow the existence of new families of BHs with hair222BH
“hair” refers to new macroscopic degrees of freedom that are not associated to
gauge symmetries, and cannot be computed by flux integrals at infinity., co-
existing with the vacuum Kerr solution.

The viability and relevance of any “hairy” BH model should be tested by
dynamical considerations: besides demanding well posedness of the matter
model, the non-Kerr BHs must have a dynamical formation mechanism and be
sufficiently stable to play a role in astrophysical processes. Asymptotically
flat BHs with Yang-Mills hair, for instance, are known to be perturbatively
unstable [1243] whereas BHs with Skyrme hair are perturbatively stable [638].
However, both these fields are best motivated by nuclear physics, in which
case the corresponding BH hair is, likely, astrophysically negligible (except,
possibly, for small primordial BHs (PBHs)).

Potentially astrophysically relevant hairy BHs in GR occur in the presence of
hypothetical (ultra-light) massive bosonic fields (such as the QCD axion,
axion-like particles, dark photons, etc). These ultralight fields could be a
significant component of the dark matter [80, 490, 829, 668] and are predicted
in a multitude of scenarios beyond the standard model of particle physics
[691, 490, 668, 678], including extra dimensions and string theories. They
naturally interact very weekly and in a model-dependent fashion with baryonic
matter, but their gravitational interaction is universal. The superradiant
instability of Kerr BHs [250], in the presence of (complex) ultralight bosonic
fields [482, 630], or mergers of self-gravitating lumps of such ultraligh
bosons [1045] (known as bosonic stars - see Section III.2.2) form BHs with
synchronised bosonic hair. These BHs are themselves afflicted by superradiant
instabilities [621, 552], but possibly on long timescales, even cosmologically
long [438], which can render them astrophysically relevant. Superradiance
triggered by real ultralight bosonic fields, on the other hand, leads to other
effects, such as a SGWB, continuous GW sources from isolated BHs, effects in
compact binaries, BH mass-spin gaps, etc, all relevant for LISA science (cf.
Section IV.1).

####  III.2.2 ECOs: deviations from (or absence of) a classical horizon

ECOs [577] is a generic name for a class of hypothetical dark compact objects
without a classical BH horizon that, nonetheless, can mimic the phenomenology
of BHs at the classical level. They may be described by their compactness
(i.e. the inverse of their – possibly effective – radius in units of the total
mass), reflectivity (as opposed to the perfect absorption by a classical BH
horizon), and possible extra degrees of freedom related to additional fields
[300]. Their compactness should be comparable to that of BHs; they may be
(albeit need not be) ultracompact, $i.e.$ possess bound photon orbits, such as
light rings. If they do, they could be further classified according to whether
the typical light-crossing time of the object is longer or shorter than the
instability time scale of circular null geodesics at the photon sphere, which
in turn depends on the object compactness and internal composition [300].

Several models of ECOs have been conceived in order to overcome conceptual
issues associated to BHs, such as their pathological inner structure and the
information loss paradox. Under general conditions, Penrose’s theorem [965]
implies that an apparent horizon always hides a curvature singularity wherein
Einstein’s theory breaks down. Moreover, in the semi-classical approximation,
BHs are thermodynamically unstable and have an entropy which is far in excess
of a typical stellar progenitor [616]. It has been argued that GWs may provide
smoking guns for ECOs [307, 308, 577, 121, 298, 300].

ECOs fall into two classes: some models are solutions of concrete field
theories coupled to gravity, with known dynamical properties; other models are
ad hoc proposals (to different extents) put forward to test phenomenological
responses without a complete embedding in a concrete model. In the former case
their maximum compactness is constrained by the Buchdahl’s theorem, when its
hypotheses apply [300]. In the latter case details about the dynamical
formation of the ECOs are unknown. But, as a general principle, it has been
argued that quantum effects in the near would-be horizon region could prevent
the formation of a horizon in a variety of models and theories [569, 808, 809,
846, 840, 841, 842, 844].

Amongst the first class of ECOs one of the most studied examples corresponds
to bosonic stars. These are self-gravitating solitons, composed of either
scalar [724, 1030, 700] or vector [251], massive complex fields, minimally
coupled to Einstein’s gravity – see also Herdeiro et al. [633, 624], Herdeiro
and Radu [631] for comparisons. Bosonic stars333Often called simply boson
stars in the scalar case and Proca stars in the vector case. arise in families
of models with different classes of self-interactions of the bosonic fields,
e.g., Colpi et al. [364], Schunck and Mielke [1060], Kleihaus et al. [740],
Grandclement et al. [596], Minamitsuji [864], Guerra et al. [600], Delgado et
al. [442] and different field content, e.g., Alcubierre et al. [44]; they may
also be generalized to modified gravity, e.g., Herdeiro and Radu [626].
Bosonic stars circumvent Derrick type no-soliton theorems [449] due to a
symmetry non-inheritance between matter and geometry, as the latter is
static/stationary and the former includes a harmonic time dependence (but with
a time-independent energy-momentum tensor). Some bosonic stars are dynamically
robust [793], in particular perturbatively stable, with a known formation
mechanism known as gravitational cooling [1062, 457]. These can be evolved in
binaries yielding gravitational waveforms, e.g., Liebling and Palenzuela
[793], Palenzuela et al. [938, 939], Bezares et al. [197], Sanchis-Gual et al.
[1044], that can be used – together with PN approximations [935] – as a basis
to produce waveform approximants for GW searches. Their typical GW frequency
depends crucially on the mass of the putative ultralight bosonic field and,
depending on the range of the latter, the signal can fall in the frequency
band of either LIGO/Virgo or LISA. Recently it was argued that one particular
GW event, GW190521 [16], is well mimicked by a very eccentric collision of
spinning Proca stars [277].

Bosonic stars have a cousin family of solitons in the case of real bosonic
fields, called oscillatons [1062]. They have a weak time-dependence and slow
decay, but can be very long lived, at least for spherical stars [937].
Collisions of oscillatons and the corresponding waveforms have also been
obtained, e.g., Clough et al. [357].

Bosonic stars are the prototypical example of ECOs which are not meant to
replace all BHs in the universe, but could in principle “co-exist" with them
and be exotic sources for LISA. They could also be especially interesting for
the BH seed problem at large redshift. Indeed, just like ordinary NSs, bosonic
stars have a maximum mass beyond which they are unstable against gravitational
collapse and classically form an ordinary BH. Other models that share the same
features are anisotropic stars [243, 787, 143] (see Raposo et al. 1009 for a
recent fully covariant model). Like bosonic stars, anistropic stars can evade
Buchdahl’s theorem due to their large anisotropies in the fluid.

A more ambitious first-principle model of ECO – aiming instead at replacing
the classical horizon completely – emerges in the fuzzball proposal [808, 809,
840, 841]. In the latter the classical horizon is replaced by smooth
horizonless geometries with the same mass, charges, and angular momentum as
the corresponding BH [890, 840, 162, 101, 163]. These geometries represent
some of the microstates in the low-energy (super)gravity description. For
special classes of extremal, charged, BHs [1107, 658, 822] one can precisely
count the microstates that account for the BH entropy, thus providing a
regular, horizonless, microscopic description of a classical horizon. In the
fuzzball paradigm, all properties of a BH geometry emerge in a coarse-grained
description which “averages” over the large number of coherent superposition
of microstates, or as a ‘collective behavior’ of fuzzballs [206, 207, 165,
164, 210]. Crucially, in this model quantum gravity effects are not confined
close to the BH singularity, rather the entire interior of the BH is “filled”
by fluctuating geometries, regardless of its curvature. It is worth noticing
that, while being among the most motivated models for ECOs, fuzzballs anyway
require beyond-GR physics confined at the horizon scale.

While microstate geometries emerge from a consistent low-energy truncation of
string theory, other more phenomenological models sharing similar
phenomenology have been proposed. For example, gravitational vacuum stars, or
gravastars, are dark energy stars whose interior spacetime is supported by a
negative-pressure fluid which is compensated by a thin shell of an
ultrarelativistic positive-pressure fluid. Gravastars are not endowed with an
event horizon and have a regular interior. Their model has been conceived in
order to overcome the surprisingly huge BH entropy and to provide a model for
a thermodynamically stable dark compact object [846]. The negative pressure
might arise as a hydrodynamical description of one-loop QFT effects in curved
spacetime, so gravastars do not necessarily require exotic new physics [881].
In these models, the Buchdahl limit is evaded both because the internal
effective fluid is anisotropic [328, 1009] and because the negative pressure
violates some of the energy conditions [847]. Gravastars can also be obtained
as the BH-limit of constant-density stars, past the Buchdahl limit [847, 982].
In this regime such configurations were found to be dynamically stable [982].

Other models of ECOs include: wormholes [879, 1172, 785, 410], collapsed
polymers [256, 258], nonlocal stars in the context of infinite derivative
gravity [266], dark stars [123], naked singularities and superspinars [575],
2-2 holes [653], and quasi-BHs [783, 784] (see Carballo-Rubio et al. 293,
Cardoso and Pani 300 for some reviews on ECO models). Finally, it is worth
mentioning that some of the existing proposals to solve or circumvent the
breakdown of unitarity in BH evaporation involve changes in the BH structure,
without doing away with the horizon. Some of the changes could involve “soft”
modifications of the near-horizon region, such that the object still looks
like a regular GR BH [571, 570, 572], or drastic changes in the form of “hard”
structures localized close to the horizon such as firewalls and other compact
quantum objects [52, 718, 572]. A BH surrounded by some hard structure – of
quantum origin such as firewalls, or classical matter piled up close to the
horizon – behaves for many purposes as an ECO.

Despite the wealth of models, ECOs are not without challenges. In addition to
the lack of plausible concrete formation mechanisms in many models, there are
other generic problems. One issue is that spinning compact objects with an
ergoregion but without an event horizon are prone to the ergoregion
instability when spinning sufficiently fast [538, 1219]. The endpoint of the
instability could be a slowly spinning ECO [301, 250] or dissipation within
the object could lead to a stable remnant [815, 816]. Another potential issue
is that ultracompact ECOs which are topologically trivial have not one but at
least a pair of light rings, one of which is stable, for physically reasonable
matter sources [391]. Such stable light rings have been argued to source a
spacetime instability at nonlinear level [728, 305], whose timescale or
endpoint, however, are unclear.

###  III.3 Observables and tests

####  III.3.1 Inspiral-based test with SMBH binaries, IMBH binaries, and EMRIs

##### Non-gravitational emission channels by extra fundamental fields

An obvious difference between BHs and certain models of ECOs is that the
latter could be charged under some gauge fields, as in the case of current
fuzzball microstate solutions, quasi-BHs, and potentially other models that
arise in extended theories of gravity. These fields might not be
electromagnetic and can therefore avoid current bounds on the charge of
astrophysical compact objects coming from charge neutralization and other
effects [118, 309]. In addition, their effective coupling might be suppressed,
thus evading current constraints from the absence of dipole radiation in BBHs
(see Sect. II). A detailed confrontation of given charged ECO models with
current constraints on dipolar radiation remains to be done.

##### Multipolar structure & Kerr bound

The multipole moments of a Kerr BH satisfy an elegant relation [612]444For a
generic spacetime the multipole moments of order $\ell$ are rank-$\ell$
tensors, ${\cal M}_{\ell m}$ and ${\cal S}_{\ell m}$, which reduce to scalar
quantities, ${\cal M}_{\ell}$ and ${\cal S}_{\ell}$, in the axisymmetric case,
see e.g., Bianchi et al. [208, 209] for the general definitions. ,

$\mathcal{M}_{\ell}^{\rm BH}+{\rm i}\mathcal{S}_{\ell}^{\rm
BH}=\mathcal{M}^{\ell+1}\left({\rm i}\chi\right)^{\ell}\,,$ (2)

where $\mathcal{M}_{\ell}$ ($\mathcal{S}_{\ell}$) are the Geroch–Hansen mass
(current) multipole moments [563, 612], $\mathcal{M}={\cal M}_{0}$ is the
mass, $\chi\equiv{\mathcal{J}}/{\mathcal{M}^{2}}$ the dimensionless spin, and
$\mathcal{J}=\mathcal{S}_{1}$ the angular momentum. The multipole moments of
the Kerr BH are non-trivial, but Eq. (2) implies that they are completely
determined by its mass and spin angular momentum. Thus, there is a multipolar
structure, but not multipolar freedom (unlike, say, in stars).

Furthermore, introducing the dimensionless quantities $\overline{{\cal
M}}_{\ell}\equiv{\cal M}_{\ell}/{\cal M}^{\ell{+}1}$ and $\overline{\cal
S}_{\ell}\equiv{\cal S}_{\ell}/{\cal M}^{\ell{+}1}$, the only nonvanishing
moments of a Kerr BH are

$\overline{{\cal M}}_{2n}^{\rm BH}=(-1)^{n}\chi^{2n}\quad,\quad\overline{{\cal
S}}_{2n{+}1}^{\rm BH}=(-1)^{n}\chi^{2n{+}1}$ (3)

for $n=0,1,2,...$. The fact that ${\cal M}_{\ell}=0$ (${\cal S}_{\ell}=0$)
when $\ell$ is odd (even) is a consequence of the equatorial symmetry of the
Kerr metric, whereas the fact that all multipoles with $\ell\geq 2$ are
proportional to (powers of) the spin – as well as their specific spin
dependence – is a peculiarity of the Kerr metric.

Non-Kerr compact objects (BHs or ECOs) will have, in general, a different
multipolar structure. Differences will be model dependent, but can be
considerable in some cases, e.g. for boson stars [1034] and BHs with
synchronised scalar hair [629]. For ECOs, the tower of multipole moments is,
in general, richer. The deformation of each multipole depends on the specific
ECO’s structure, and in general vanishes in the high-compactness limit,
approaching the Kerr value [944, 579, 1010, 1008]. In particular, a smoking
gun of the “non-Kerrness" of an object would be the presence of moments that
break the equatorial symmetry (e.g. the current quadrupole ${\cal S}_{2}$ or
the mass octopole ${\cal M}_{3}$), or the axisymmetry (e.g. a generic mass
quadrupole tensor ${\cal M}_{2m}$ with three independent components
($m=0,1,2$), as in the case of multipolar boson stars [635] and of fuzzball
microstate geometries [160, 208, 161, 209].

The multipolar structure of an object leaves a footprint in the GW signal
emitted during the coalescence of a binary system, modifying the PN structure
of the waveform at different orders. The lowest order contribution, entering
at 2PN order is given by the intrinsic (typically spin-induced) quadrupole
moment [107]. LISA can be able to detect deviations in the multipole moments
from supermassive binaries for comparable and unequal mass systems. So far
proposed tests of the Kerr nature have been based on constraints of the spin-
induced quadrupole $M_{2}$ [107, 760], spin-induced octopole $S_{3}$ [759],
and current quadrupole $S_{2}$ [537].

GW signals emitted by EMRIs will provide accurate measurements of the spin-
induced quadrupole at the level of one part in $10^{4}$ [107, 88], and of the
equatorial symmetry breaking current quadrupole at the level of one part in
$10^{2}$ [537]. A rather generic attempt to constrain the multipole moments of
an axisymmetric and equatorially symmetric central object with EMRIs has been
done in Ryan [1032, 1033] by mapping gauge-invariant geodesic quantities into
multipole moments in a small-orbital velocity expansion. Constraining the
radiative multipole moments of the entire binary system has been discussed in
Kastha et al. [721, 722]; these constrain deviations from the GR expectation
of the binary system without explicitly parametrizing the compact objects’
multipole structure.

##### Tidal heating

The compact objects in the binary produce a tidal field on each other which
grows as the bodies approach their final plunge and merger. If the bodies
dissipate some amount of radiation, these tides backreact on the orbit,
transferring rotational energy from their spin into the orbit. This effect is
known as tidal heating. For BHs, energy and angular momentum absorption by the
horizon is responsible for tidal heating. This effect is particularly
significant for highly spinning BHs and mostly important in the latest stages
of the inspiral. Tidal heating can contribute to thousands of radians of
accumulated orbital phase for EMRIs in the LISA band [664, 172, 1130, 613,
420, 421, 819]. If at least one binary member is an ECO instead of a BH,
dissipation is in general smaller than in the BH case or even negligible,
therefore significantly reducing the contribution of tidal heating to the GW
phase. This would allow to distinguish between BBHs and binary involving other
compact objects. For LISA binaries, constraints of the amount of dissipation
would be stronger for highly spinning objects and for binaries with large mass
ratios [834, 421, 819]. For EMRIs in the LISA band, this effect could be used
to put a very stringent upper bound on the reflectivity of ECOs, at the level
of $0.01\%$ or better [421, 819]. Absence of a horizon can also produce
resonances that can be excited during EMRIs [949, 812, 315, 819].

##### Tidal deformability

Tidal effects in compact binaries modify the dynamical evolution of the
system, accelerating the coalescence. This modifies the orbital phase, and
then in turn the GW emission [979, 645]. The imprint on the waveform is
encoded in a set of quantities which, as a first approximation, can be assumed
to be constant during the coalescence [830, 646, 1105]: the tidal Love numbers
[642, 406, 216]. These numbers can be thought of as the specific multipole
moment induced by an external tidal field, in a way akin to the electric
susceptibility in electrodynamics. The main contribution in the GW signal from
a binary is given by the quadrupolar term $k_{2}$, connected to the tidal
deformability $\lambda=\frac{2}{3}k_{2}R^{5}$, or in its dimensionless form
$\tilde{\lambda}=\frac{2}{3}k_{2}{\cal C}^{5}$, where $R$ and ${\cal C}$ are
the object radius and compactness, respectively.

The tidal Love numbers depend on the internal composition of the central
object. So far, they have been used to constrain the properties of the nuclear
equation of state through GW observations of binary NSs [6]. For a fixed
equation of state, i.e. composition, the Love numbers depend on the object
compactness only.

The tidal Love numbers of a BH in GR are precisely zero. This was shown
explicitly for Schwarzschild BHs, for both small [216, 406] and large [603]
tidal fields. The same result was shown to be valid for slowly rotating BHs up
to the second (linear) order in the spin for axisymmetric (generic) tidal
fields [957, 956, 776, 978]. Very recently, this result was extended to any
tidal Love number of a Kerr BH with arbitrary spin (see Chia 345, Le Tiec and
Casals 779, Le Tiec et al. 780, Hui et al. 670, Charalambous et al. 335, 336
for literature on this topic).

For ECOs, the tidal Love numbers are generically different from zero. In
analogy with the NS case they depend on the ECO’s structure, and may be used
to trace back the underlying properties of each model [944, 1159, 981, 310,
1064, 834, 705, 835, 1009, 573, 634]. For nonrotating BH mimickers, featuring
corrections at the horizon scale and that approach the BH compactness, the
Love numbers vanish in the limit ${\cal C}_{\textnormal{ECO}}\rightarrow{\cal
C}_{\textnormal{BH}}$, often logarithmically [310].

LISA will be able to measure the tidal Love numbers of BH mimickers [834],
which are otherwise unmeasurable by current and future ground based detectors
[310]. In the comparable-mass case, this measurement requires highly-spinning
supermassive ECO binaries up to $10\,{\rm Gpc}$. LISA may also be able to
perform model selection between different families of BH mimickers [835],
although this will in general require detection of golden binaries (i.e.
binaries with a very large SNR) [28]. For a large class of slowly-rotating
ECOs with compactness ${\cal C}\lesssim 0.3$, LISA can measure the Love
numbers with very good accuracy below 1% [310]. For (scalar) boson stars a
recent study proposed a new data analysis strategy to consistently include
several corrections (multipolar structure, tidal heating, tidal Love numbers)
in the inspiral signal from boson star binaries, improving the accuracy on the
measurement of the fundamental parameters of the theory by several orders of
magnitude compared to the case in which the effects are considered
independently [935].

Finally, EMRI observations can set even more stringent constraints, since the
measurement errors on the Love number scale as $q^{1/2}$, where $q\ll 1$ is
the mass ratio of the binary [946]. A simplistic Newtonian estimate (that
should be corroborated by a more sophisticated modelling and data analysis)
suggests that in this case the tidal Love number of the central object can be
constrained at the level of one part in $10^{5}$ [946].

##### Integrability/Chaos

One particular probe of extreme gravity that is tailor-made for EMRI signals
relates to chaos. For Hamiltonian systems, chaos refers to the non-
integrability of the equations of motion, i.e. the non-existence of a smooth
analytic function that interpolates between orbits, and has nothing to do with
a system being non-deterministic [788].

EMRIs in GR can be approximated, to zeroth-order, as geodesics of the Kerr
spacetime, and the latter has enough symmetries to guarantee that geodesics
are completely integrable and thus non-chaotic. Beyond the zeroth-order
approximation, however, other effects, such as the spin of the small compact
object, could break the integrability of the systems even within GR [1238]. In
some modified theories, even the geodesic orbital motion might not be
integrable [807, 397]. This is also true for some models of ECOs, such as
spinning scalar boson stars and non-Kerr BHs in GR [390]. In this sense, the
presence of large chaotic features in the GWs emitted by EMRIs could signal a
departure from the SEP, a violation of the Kerr hypothesis, or an
environmental effect.

Modifications to GR are expected to change the fundamental frequencies of the
orbital motion of test particles, which will be then be imprinted on the GWs
emitted by the system. A careful study of the evolution of these fundamental
frequencies will allow us to understand the importance of chaos to GR and to
the observations of GWs from EMRIs [547, 807, 397, 451].

##### Motion within ECOs

If the ECO interior is made of weakly-interacting matter, a further
discriminator of the absence of a horizon (or of a hard surface) would be the
motion of test particles within the object and its peculiar GW signal, most
notably as in the case of an EMRI moving inside a supermassive ECO. This
motion can be driven by a combination of the self-gravity of the central
object, accretion, and dynamical friction, etc. The study of geodesic motion
inside solitonic boson stars was analyzed in e.g., Kesden et al. [730]. The
effects of accretion and drag were included in Macedo et al. [811, 812],
Barausse et al. [118, 119]. These effects are model independent to a certain
extent, since they mostly depend on the density profile. For this reason they
also share some similarities with environmental tests of dark matter (see
Sect. IV). In general, they could be a smoking-gun signature for the existence
of structures in supermassive ultracompact objects.

####  III.3.2 Ringdown tests

##### QNMs

Similarly to what was discussed in Sect. II, measuring the ringdown modes in
the post-merger signal of a binary coalescence provides a clean and robust way
to test GR and the nature of the remnant. If the latter is a Kerr BH in GR,
its (infinitely countable) QNM spectrum is entirely determined only in terms
of its mass and spin. Thus, detecting several QNMs provides us with multiple
independent null-hypothesis tests, and would allow us to perform GW
spectroscopy [751, 181]. From a more theoretical perspective, the study of the
QNMs of compact objects is crucial to assess their linear stability.

The ringdown waveform originates from the perturbed remnant object, and
consists of a superposition of (complex) QNMs, whose amplitudes depend on the
binary progenitors and on the underlying theory. As previously discussed, the
fundamental QNM frequency and damping time have been measured by LIGO/Virgo
only for a few events, providing an independent measurement of the mass and
spin of the remnant which is in agreement with what inferred from the
inspiral-merger phase [10, 19]. Among the entire second GW transient catalogue
[19] the first GW event, GW150914, remains among those for which the
fundamental QNM of the remnant has been measured with the highest precision
(roughly $3\%$ and $7\%$ for the frequency and damping time, respectively).
More recently, the importance of overtones has attracted considerable
attention, especially because they allow one to start the fitting of the
ringdown signal closer to the peak of the signal, improving mass and spin
measurements [679]. Overtones are particularly useful for tests of GR with
equal-mass binaries (for which other angular modes can be suppressed) [201,
932, 703], but a detailed study for LISA remains to be done. Overall, tests of
the no-hair theorem rely also on the ability to estimate the starting time of
the ringdown when the signal is dominated by the QNMs of the remnant and on
the modelling of higher modes [96, 200, 254, 574, 201, 932, 703].

The large SNR expected in LISA for ringdown signals of SMBH coalescences
provides a unique opportunity to perform BH spectroscopy [477] and tests of
the nature of the remnant. For a single "golden merger" up to redshift $z=10$
several QNMs can be measured with unprecedented precision [184].

Besides introducing deformations in the QNM spectrum, if the remnant differs
from a Kerr BH in GR, some further clear deviations in the prompt ringdown
are: (i) possible presence of (or contamination from) other modes, e.g. fluid
modes [948] in stars or extra degrees of freedom (e.g. scalar QNMs for boson
stars; Macedo et al. 812), some of which – being at low frequency – could be
resonantly excited during the inspiral [949, 812, 315, 819]; (ii)
isospectrality breaking between modes that can be identified as even-parity
and odd-parity in the zero-spin limit [818]. This produces a characteristic
“mode doublet” in the ringdown. A generic framework to study the ringdown of a
dark compact object was recently proposed in Maggio et al. [818] by extending
the BH membrane paradigm to ECOs.

##### Echoes

GW echoes [307, 308] in the post-merger signal of a compact binary coalescence
might be a clear signature of near-horizon quantum structures [307, 308, 22,
124, 929, 1179], ultracompact objects [307, 259], exotic states of matter in
ultracompact stars [513, 945, 266], and of modified theories of gravity [269,
267, 444] (see Cardoso and Pani 298, 299, 300, Abedi et al. 23 for some recent
reviews). Detecting echoes would give us the tantalizing prospect of probing
the near-horizon structure of dark compact objects with the hope, in
particular, to shed light on putative quantum properties of BHs [673].

If sufficiently compact, horizonless objects support quasi-bound modes trapped
within their photon sphere [751, 750, 307, 308]. For ultracompact objects the
prompt ringdown is identical to that of a BH, since the signal is initially
due only to the perturbation of the photon sphere, whereas the BH horizon is
reached in infinite coordinate time [307, 308]. At late times, a modulated
train of GW echoes appears as a result of multiple reflections of the GWs
between the object interior and the photon sphere, leaking out to infinity at
each reflection. For the case of intermediate compactness, the prompt ringdown
can show some differences with the BH case due to the interference with the
first GW echoes [818].

The delay time between echoes is related to the compactness of the object
through a logarithmic dependence, which allows for tests of Planckian
corrections at the horizon scale [308, 298, 23, 930]. The damping factor of
subsequent echoes is related to the reflective properties of the compact
object [992, 816, 300, 818].

Several waveform templates for echo searches in LIGO/Virgo data have been
developed, including: (i) templates in time domain based on standard IMR
templates with additional parameters [22, 899, 1178]; (ii) superposition of
sine-Gaussians with free parameters [833]; (iii) frequency-domain templates
555Echo templates available at http://www.darkgra.org,
https://web.uniroma1.it/gmunu/.  based on the physical ECO parameters [828,
1137, 817]. The former were developed for matched-filtered searches. In
addition, unmodelled searches based on wavelets adapted from burst searches
[1151, 1152] and on Fourier windows [370, 369] have been proposed. For a
review on modelling and echo searches, see Abedi et al. [23].

LIGO/Virgo O1 and O2 events triggered some controversial claims on hints of GW
echoes detection. Independent searches found evidence for GW echoes in the
O1-O2 events [22, 370, 21]. However, a low statistical significance of such
events has been claimed [1190, 909], followed by more recent negative searches
[1152, 1160, 797]. Very recently, using a simplistic template, a dedicated
search for echoes has been performed by the LIGO/Virgo Collaboration using the
second GW transient catalogue (GWTC-2) [19], finding no evidence for echoes.
This is consistent with independent studies using physically motivated
templates, suggesting that that models with almost perfect reflectivity can be
excluded/detected with current instruments ($5\sigma$ confidence level with
SNR in the ringdown of $\approx 10$), whereas probing values of the
reflectivity smaller than $80\%$ at $3\sigma$ or more confidence level
requires SNRs of $\mathcal{O}(100)$ in the post-merger phase [1137, 817]. This
makes LISA particularly well suited for echo searches.

##### GWs as messengers from the quantum world

The high sensitivity of LISA or of other advanced detectors will also allows
to advance our understanding of quantum gravitational effects. One of the most
long-standing ideas in this context is the proposal that the area of BH
horizons is quantized in units of the Planck area $A=\alpha\ell_{p}^{2}N$,
where $N$ is an integer characterizing the BH quantum state, $\alpha$ is an
${\cal O}(1)$ dimensionless coefficient and $\ell_{p}$ is Planck’s length
[146, 883, 148]. Transitions between states occur, surprisingly, at
frequencies where ground- and space-based detectors operate. Thus, quantum BHs
may have different tidal heating properties [313, 32] or different tidal Love
numbers [257], and the coalescence of two BHs can lead to late-time echoes in
the waveforms [313, 32].

###  III.4 Burning Questions and Needed Developments

Several issues discussed in this section need further detailed studies, among
which we list the following:

  * •

A better understanding of the consequences of superradiant instabilities
triggered by e.g. real ultralight bosonic fields on the SGWB, on continuous GW
sources, and on binaries, which could be detected by LISA.

  * •

A comprehensive study of possible deviations of multipole moments for SMBH
binaries is important to have a good tool at disposal for testing the Kerr
nature of the BHs.

  * •

First-principle models of (possibly stable) ultracompact objects should be
developed to provide viable candidates for a BH mimicker.

  * •

In general for testing the different effects such as tidal heating, tidal
deformability, multipole moments, resonances, and ringdown effects, and echoes
there is need to develop quite accurate waveforms, with several parameters
describing possible deviations, and to perform accurate statistical analysis
(e.g. Bayesian model selection), especially in the case of a putative
signature of new physics in the data. Clearly, this also requires to develop
very accurate GR waveforms.

##  IV Dark Matter and Primordial BHs

The fundamental nature of dark matter (DM) is one of the major unresolved
questions in physics and cosmology. Its presence is inferred from various
observations, such as galaxy rotation curves, gravitational lensing, galaxy
cluster dynamics and CMB data (see Bertone et al. 192, Bertone and Hooper 189,
Carr et al. 318 for reviews). These observations constrain its behavior on
larger scales: DM must broadly behave as non relativistic, collisionless
matter, with an average density in galactic haloes of the order of $\sim
0.1\leavevmode\nobreak\ {\rm M_{\odot}\leavevmode\nobreak\ pc^{-3}}$
($1\leavevmode\nobreak\ \rm{GeV\leavevmode\nobreak\ cm^{-3}}$), low velocity
dispersions of order $\sim 100\leavevmode\nobreak\ {\rm km\leavevmode\nobreak\
s^{-1}}$, and it must be interacting weakly, if at all, with baryonic matter
and with itself. However, the physical properties of its constituents, in
particular their individual masses and spins, remain very poorly constrained.

Figure 2: Overview of key DM candidates [194, 191, 576]. Currently excluded
ranges of particle masses and DM fractions $f_{DM}$ are indicated in white.
Parameter regions where LISA observations may provide constraints are gray
coloured. We see that the masses of DM particle candidates remain currently
very much unconstrained and that DM particles of any allowed mass can make up
any fraction ($f_{DM}\leq 1$) of the DM. Macroscopic DM candidates such as
PBHs and ECOs are constrained by current observations but may still constitute
100% of the DM in the range $10^{-16}-10^{-11}M_{\odot}$. Approximate regions
where LISA can contribute to constraints are bracketed above and below the
plot with a brief description - see the text for more details.

This Section is concerned with models where the DM consists of or is formed
from (in the case of PBHs) some type of matter and not where the observed
effects arise due to some modification to gravity, such as e.g., Modified
Newtonian Dynamics (MOND) [1046], which is the topic of Section II. There are
two broad categories of dark matter:

  1. 1.

Particle dark matter, e.g. WIMPs or axions, where the problem of
identification lies in determining the mass, spin and fundamental interactions
of the particle (or particles).

  2. 2.

Macroscopic objects, e.g. primordial BHs (PBHs) or exotic compact objects
(ECOs), where the distributions of masses and spins, as well as the matter
from which they formed, are key aspects to be understood.

Nothing prevents DM from being a combination of several different components
drawn from either or both of these categories. Moreover, some models (e.g.
axions), are naturally hybrids, where the DM may primarily consist of unbound
particles, but also form macroscopic gravitationally bound structures such as
boson stars in overdense regions. A further aspect of DM on which observations
may shed light is in identifying possible formation mechanisms, e.g.
signatures of phase transitions, which often motivate particular DM models.

Several previous reviews have considered the potential for GWs to shed light
on the nature of dark matter [194, 191, 576]. Further, a complementary white
paper by the LISA cosmology WG assesses the prospects of GW observations of DM
from a cosmological angle. In this section we focus on the potential for LISA
specifically to identify and constrain the particle nature of dark matter and
its interactions. We summarise what is known to be possible, highlight
promising areas which require further investigation, and detail the work that
should be done prior to launch to enable us to correctly interpret the data
that we will obtain. Fig. 2 summarises the key candidates and regions which
may be probed by LISA observations.

Before we proceed we note two caveats. First, effects described in this
section may be degenerate with environmental effects described in Sect. VII
and in the waveform modling white paper. Breaking this degeneracy will be a
key factor in our ability to accurately identify DM signatures. Second, we
have in most cases been optimistic in assuming that LISA measurements will be
optimized for the searches we describe and that models can allow for
sufficiently precise predictions to proceed. Turning these caveats around,
they could both be regarded as a starting point for action items for future
work in order to ensure LISA’s science potential will be fully exploited.

We now present a number of interesting particle candidates, or rather
categories of candidates, that have been explored so far, without making the
above qualifications necessarily explicit in each section. We start by
reviewing particle candidates, separated into low mass (sub eV) and high mass
cases. This roughly divides cases where wave-like and particle-like behaviour
respectively would dominate on astrophysically significant scales. We then
turn to macroscopic candidates such as PBHs, where we separate candidates that
would clearly be of primordial origin (sub $M_{\odot}$) from those that could
arise from stellar evolution. The last two sections review the potential for
enhanced signatures from additional fundamental interactions.

###  IV.1 Low mass particles $m<eV$

In the past few years, the possibility that dark matter could be composed of
ultralight bosons, with masses ranging anywhere between $10^{-22}\,{\rm
eV}\lesssim 1$ eV, has become a popular hypothesis [518]. A non-exhaustive
list of theoretical models predicting the existence of such particles includes
the well-known QCD axion proposed to solve the strong CP problem of QCD [964,
1193], axion-like particles arising in “string axiverse” scenarios [80], dark
photons [591, 903] and “fuzzy dark matter” [668].

Quite remarkably, GW observatories such as LISA could be used to constrain or
find evidence for the existence of these particles. This stems from the fact
that rotating BHs can become unstable against the production of light bosonic
particles, if such particles exist in nature [453, 412, 467, 1201, 952, 953,
248, 1082, 880, 480, 311, 540, 468, 255], through a process known as BH
superradiance [1236, 1237, 990, 1138, 250]. The instability spins the BH down,
transferring up to a few percent of the BH’s mass and angular momentum to the
boson field, forming a long-lived bosonic “cloud” outside the horizon [80, 79,
249, 81, 629, 250, 82, 136, 482, 481, 521, 438]. Superradiance is most
effective when the boson’s Compton wavelength is comparable to the BH’s
gravitational radius [467, 1201], meaning that observations of astrophysical
BHs in the supermassive to the stellar mass range, allow us to probe bosons
with masses in the range $10^{-21}$ – $10^{-10}$ eV.

For real bosonic fields, the cloud dissipates over long timescales through the
emission of nearly-monochromatic GWs with a typical frequency $f\sim
2c^{2}m_{b}/h$, where $m_{b}$ is the mass of the field [80, 79, 1221, 81, 82,
252, 253, 136, 481, 1083]. These signals could be observable individually or
as a very strong SGWB [81, 82, 136, 252, 253, 1155, 680, 568, 418, 1112, 1244,
255, 1156], and could thus be used to constrain the existence of light bosons
in the absence of a detection. Current and future Earth-based detectors could
detect GWs emitted by bosons in the range $m_{b}\sim 10^{-14}$–$10^{-11}$ eV,
while LISA could be sensitive to bosons of mass $m_{b}\sim
10^{-19}$–$10^{-15}$ eV. It has also been proposed that LISA observations of
BBHs with total masses in the range $\sim[100,6000]\,M_{\odot}$ will provide
sufficient information to enable targeted searches for these monochromatic GW
signals with future ground-based GW detectors for masses in the range
$m_{b}\sim 10^{-14}$–$10^{-12}$ eV [905].

Furthermore, since BHs affected by the instability would spin down in the
process, accurate measurement of the spin of astrophysical BHs can be used to
strongly constrain, or find evidence for, ultralight bosons [79, 952, 953,
248, 81, 82, 136, 252, 311, 904]. Requiring that the instability acts on
timescales shorter than known astrophysical processes, such as accretion and
mergers, current measurements of SMBHs spin using continuum fitting or the
Iron K$\alpha$ (see e.g., Jovanovic 707) can probe bosons in the mass range
$m_{b}\sim 10^{-20}$–$10^{-17}$ eV, whereas similar measurements for stellar-
mass BHs probe the mass range is $m_{b}\sim 10^{-12}$–$10^{-11}$ eV. LISA is
expected to detect binaries and measure the spins of MBHs in the range $\sim
10^{2}$–$10^{7}\,M_{\odot}$ and will therefore be able to constrain the
existence of light bosons in the intermediate range $m_{b}\sim
10^{-16}$–$10^{-13}$ eV [252, 311, 255].

When the BH-boson cloud system is part of a binary, one may also expect new
phenomena that could leave imprints in the GW signal emitted by the binary
system. For example, the presence of a companion can result in resonant
transitions between energy levels of the cloud [139] or complete tidal
disruption [316], which could lead to a significant dephasing of the binary’s
GW signal. In particular, for EMRIs, resonant transitions could lead to long-
lived floating orbits [519, 1240, 140]. The presence of a cloud would also be
imprinted in the binary’s GW signal through its spin-induced multiple
moment(s) and tidal Love number(s) [139, 140, 422]. For eccentric orbits,
resonances can occur over a larger range of frequencies, extending the
possibility of detection [186]. In the case of EMRIs, dynamical friction and
the gravitational pull of the cloud may also leave sizable imprints on the GW
waveform [811, 519, 1241, 609].

Ultralight scalars can also form self-gravitating structures. These “dark
matter stars” are candidates to describe (dark) haloes in the central part of
galaxies [668]. In this context, it is crucial to understand how such
structures respond to external excitations caused by compact objects (e.g.
stars or BHs) living in these environments. Stability, proper modes,
excitation of resonances, and depletion of the scalar through radiation are
amongst the important issues that must be addressed here [66, 67]. There is
also the potential for distinctive DM density enhancements due to accretion of
DM onto BHs [669, 358, 104], which is analogous to the case of dark matter
spikes for high mass candidates (see Sect. IV.2 below). However, such density
enhancements are estimated to be several orders of magnitude weaker than those
from a superradiant build up, and concentrated near the BH horizon. They are
therefore below LISA sensitivity in the inspiral phase, but may have an impact
on merger, something which is yet to be tested using NR simulations. On the
other hand, it is also essential to know how the compact objects themselves
are affected by the surrounding dark matter. GW dephasing through dynamical
friction and scalar radiation deserves a proper and rigorous exploration [668,
66, 67]. In particular, for sources in the LISA band, it was recently found
that scalar radiation from these structures affects the gravitational waveform
at leading -6 PN order with respect to the dominant quadrupolar term [66, 67].

In summary, LISA has the potential to detect or constrain ultralight DM fields
in regions of parameter space complementary to those covered by ground-based
GW observations, although more work is needed to build waveform models that
incorporate the effects of such fields and are sufficiently accurate to be
used for data analysis purposes. It will also be important to explore in more
detail BBHs with ultralight boson clouds with full NR simulations in order to
understand how the presence of such boson clouds could be imprinted in the
late stages of BBH mergers.

###  IV.2 High mass particles $m>eV$

We now discuss what we could learn with LISA about more massive cold DM
particles. If one or both the components of a binary system are surrounded by
a dense dark matter overdensity, the binary’s inspiral will be driven by both
dynamical friction [333] and GW emission. The densities required to produce an
effect observable by LISA are typically higher than the characteristic DM
densities in most galaxies [118, 297]. There are however several mechanisms by
which the DM density can be enhanced to such higher levels, notably when a
smaller seed BH grows in mass adiabatically in a DM halo to produce a DM
“spike” [996, 589, 1161, 1036, 520]. DM spikes around SMBHs at the center of
galactic halos are prone to disruption by a number of processes, such as major
mergers and gravitational interactions with stellar cusps [857, 190]. DM
spikes around IMBHs might be more stable over cosmological timescales, and are
arguably more promising targets for indirect and GW searches [193]. The
density profile of spikes depends on the nature of DM, with warm, self-
interacting, or self-annihilating DM candidates leading to shallower profiles
with respect to cold, collisionless DM [589, 190, 1078, 610, 54].

The faster rate of inspiral in the presence of DM can be distinguished from
the slower inspiral in vacuum, thereby allowing LISA to infer the presence of
DM around a binary [811, 118, 485]. The prospects for detecting DM spikes
through their influence on the GWs emitted by IMRIs were studied in [485], in
which the dynamics of the IMRIs were treated in the leading Newtonian limit
and the DM distribution was assumed to be static throughout the inspiral.
Fisher-matrix forecasts predict that the properties of the DM spike could be
mapped precisely for the higher overdensities (comparable to the errors on the
detector-frame chirp mass) and to a few tens of a percent for lower
overdensities.

Conservation of energy means, however, that the energy dissipated through
dynamical friction must be balanced by a corresponding change in the energy of
the dark matter distribution. For a range of different DM overdensities around
different IMRIs, the energy dissipated when the DM spike is assumed to be
static turns out to be a substantial fraction of the binding energy of the
spike, rendering the assumption of staticity questionable. In these cases, it
is important to jointly evolve both the IMRIs’ orbits and their surrounding
distributions of DM [725]. When this is done, the DM distribution is found to
evolve non-trivially, significantly decreasing the amplitude of the dynamical
friction force and hence the difference between an inspiral in vacuum and in a
DM mini-spike [725]. Yet for larger DM overdensities, the amplitude of the
effect of the DM is seen to remain significant in the coupled evolution of the
IMRI and DM spike.

A Bayesian analysis of the detectability (signal-to-noise ratio),
discoverability (discrimination against in-vacuum inspiral), and measurability
(prospects for measuring the parameters) of dark matter environments shows
that inspirals in presence of dark matter can be easily discriminated against
in-vacuum inspirals. In case of detection, the DM halo’s density normalization
can be distinguished from zero at high significance, with a $95\%$ credible
interval of order 10%. The DM halo’s slope can also be measured, although the
posterior exhibits degeneracy with the chirp mass and the mass ratio.
Interestingly, unlike in-vacuum inspirals, the mass ratio can be measured even
in the Newtonian limit [373].

For EMRIs and IMRIs in vacuum, there is good evidence that gravitational
waveforms that are accurate to first post-adiabatic (1PA) order will be
sufficient to detect and perform parameter estimation [854]. Such vacuum
waveforms are under development and are expected to be ready for the beginning
of the LISA mission. By contrast, the waveform modelling for IMRIs and EMRIs
with surrounding DM spikes coupled to their evolution is only just beginning:
the calculations in Kavanagh et al. [725] assumed circular orbits and worked
at leading Newtonian order. A more complete modelling of these systems will be
necessary to produce waveforms that are suitable for describing the full range
of possible orbits and DM distributions to cover the parameter space of these
systems. The possibility of finding a simpler effective description of IMRIs
and EMRIs in DM spikes should be investigated in greater detail, in order to
decrease the size of the parameter space needed to look for signatures of cold
DM substructures. At the other end of the scale, as in the light dark matter
case, it may also be interesting to study whether there is any significant
impact of these dark matter overdensities on the merger signal using NR
simulations.

###  IV.3 PBHs $m<M_{\odot}$

PBHs might originate in the early universe from the collapse of large density
perturbations from an enhancement of the scalar curvature perturbation power
spectrum generated during inflation [685, 554, 684]. PBHs may therefore span a
wider range of masses compared to stellar BHs, which are expected to have a
mass larger than the Chandrasekhar limit, around the solar mass, from stellar
evolution.

For masses smaller than about $10^{-17}M_{\odot}$, PBHs would have evaporated
by now due to the emission of Hawking radiation, and thus do not lead to late-
time imprints [1052]. For masses larger than this lower limit, while many
observational bounds can be set on the PBH abundance in the Universe, these
still leave some open windows for PBHs to constitute the totality of DM. A
comprehensive review of the constraints on the PBH population can be found in
Carr et al. [319].

For PBHs with masses smaller than about the solar mass, the most stringent
constraints on the PBH abundance arise from evaporation limits [125] and
microlensing observations [911, 1090, 42, 50, 910, 921]. Interestingly, there
exists an open window for PBHs with masses in the range
$(10^{-16}-10^{-11})M_{\odot}$ to account for the totality of the dark matter
in the universe (see Carr et al. 319, Green and Kavanagh 598 for recent
reviews). This range of masses is however notoriously difficult to probe with
lensing observations. Indeed it was shown that both femtolensing and
microlensing searches are not able to constrain this mass window once the
extended nature of the source as well as wave optics effects are properly
considered [723, 870, 1090]. Also, bounds coming from the survival of white
dwarfs (WDs) and NSs after asteroidal mass PBHs capture were shown to be
unable to constrain the abundance in this range [870]. Finally, the GWs from
mergers of sub-solar mass PBHs, peaking at much higher frequencies compared to
the ones testable by LISA, would not lead to an observable signal. However,
LISA can search for SGWB signatures of the PBH formation in that interesting
window, thus testing the possible nature of dark matter as asteroidal-mass
PBHs, as we will describe in the following.

Curvature perturbations, responsible for the PBH production, would also lead
to the emission of a second-order induced SGWB due to the intrinsic non-linear
nature of gravity [25, 869], analysed in details in Ananda et al. [63],
Baumann et al. [138], Bugaev and Klimai [260], Saito and Yokoyama [1039],
García-Bellido et al. [555], Ando et al. [65], García-Bellido et al. [556],
Espinosa et al. [489], Kohri and Terada [748], Bartolo et al. [129, 133, 132],
Clesse et al. [355], Wang et al. [1181], De Luca et al. [427], Inomata and
Terada [676], Yuan et al. [1225], Domènech [471], Domènech et al. [472], Pi
and Sasaki [973], Chang et al. [334], Fumagalli et al. [545].666Note that even
perturbations that do not create PBHs contribute to the GWs produced, and it
is the total of all these contributions that must be considered [424]. Since
the emission mostly takes place when the length scale of the perturbation
crosses the cosmological horizon, one can relate the peak of the GW frequency
spectrum to the dominant PBH mass $M_{\text{\tiny PBH}}$ as [1039]

$f_{\star}\simeq 3\,\text{mHz}\left(\frac{M_{\text{\tiny
PBH}}}{10^{-12}M_{\odot}}\right)^{-1/2}.$ (4)

There are several current and future experiments searching for this SGWB in
various frequency ranges. In the ultra-low frequency range (around nHz), the
null detections at Pulsar Timing Array experiments like PPTA [1075], NANOGrav
[83] and EPTA [786], give rise to stringent constraints on the abundance of
GWs (although NANOGrav shows some evidence for a background, as discussed
further below). Future experiments like SKA [455] (see also Moore et al. 873)
will significantly improve the detection sensitivity. These constraints can
also be translated into a bound on the amplitude of the comoving curvature
perturbations at the corresponding scales [675, 1163]. Recently, the Earth-
Moon system has been suggested as a detector of SGWB at $\mu$Hz frequencies
[222, 223], providing a complementary probe to LISA for signals in this range.

In the intermediate frequency range (around mHz) the most relevant experiment
is LISA. LISA will be able to probe the wavenumbers of the primordial
curvature perturbation power spectrum on scales in the range
$(10^{10}-10^{15})\,{\rm Mpc}^{-1}$ [555]. Moreover, the production of PBHs
with masses around $M_{\text{\tiny PBH}}\sim{\cal
O}\left(10^{-15}-10^{-8}\right)M_{\odot}$ would be associated to GW signals
with peak frequencies which fall within the LISA sensitivity band. LISA would
be therefore able to search for DM candidates in the form of PBHs within this
range of masses [132, 555, 556]. Notice that, due to the exponential
sensitivity of the PBH abundance to the amplitude of the curvature power
spectrum, null detection of a SGWB at LISA would imply the PBH abundance in
the corresponding mass range to be negligible.

Physical SGWB spectra would typically have a white-noise ($\propto f^{3}$)
behaviour at low frequencies [489, 272], a peak corresponding to the frequency
provided in Eq. (4) and a high frequencies tail which goes like the squared
curvature power spectrum, $\Omega_{\text{\tiny GW}}(f\gg
f_{\star})\sim\mathcal{P}_{\zeta}^{2}(f)$. PBH formation from multi-field
inflationary models [940, 544], and more generally large particle production
mechanisms involving a sharp feature, lead to oscillatory power spectra that
result in a 10% oscillatory modulation in the peak of the frequency profile of
the SGWB [545, 246], thus representing a potentially observable characteristic
of these models.

Recently the NANOGrav collaboration has published a 12.5 yrs analysis of
pulsar timing data reporting strong evidence for a signal whose interpretation
in terms of a stochastic common-spectrum process is strongly preferred against
independent red-noise signals [84]. At the moment, the NANOGrav Collaboration
does not yet claim a detection of a SGWB due to the absence of evidence for
the characteristic quadrupole correlations. There are several models proposed
so far to explain the origin of this SGWB signal. One possibility is provided
by the SGWB associated to PBH formation [1166, 431, 749, 470]. The relative
PBH abundance in the mass range probed by NANOGrav would be subdominant
[1166]. However, the model proposed in De Luca et al. [431], where a flat
power spectrum of the curvature perturbation [1176, 781, 782, 204] leads to a
mass function peaked in the asteroidal mass range [432], a total abundance of
DM in the forms of PBHs allowed by observational bounds and hinted by the
lensing event candidate in the HSC data [768, 1109]. This scenario predicts a
flat SGWB spectrum which would be detectable and tested by LISA [431].

As discussed in more detail in the complementary white paper of the cosmology
WG, LISA will also be able to search for signatures of primordial non-
Gaussianity [534, 204, 494, 499, 962, 205] at small scales. Primordial non-
Gaussianity on small scales would have an impact on the shape of the SGWB,
leading to potentially detectable signatures in the spectrum of frequencies of
the observed monopole signal [271, 1162]. Possible small deformations smearing
the GWs spectrum can also arise from similar effects [469]. Non-Gaussian
signatures in the tensor three-point function would in principle be
significant, but get washed out by [132, 133] time-delay effects originating
during the propagation of the GWs in the perturbed universe, see Bartolo et
al. [129], Margalit et al. [827]. Non-Gaussian signatures may be detectable by
searching for angular anisotropies in the SGWB which, in the absence of non-
Gaussianity, are generally undetectable at LISA [134] given its angular
resolution of $\ell\lesssim 15$ [372, 97]. However, local scale-invariant non-
Gaussianity, constrained by the Planck collaboration to be $-11.1\leq
f_{\text{\tiny NL}}\leq 9.3$ at 95% C.L. [39], would correlate short and long
scales and potentially lead to an enhancement of the SGWB, see Cai et al.
[271], Unal [1162], Cai et al. [273] for details, and possible large-scale
anisotropies at detection [134]. Now if PBHs constitute a large fraction of
the DM, one expects a highly isotropic and Gaussian SGWB due to the strong
constraints by CMB observations [38] on the isocurvature modes in the DM
density fluid, associated to the non-Gaussianity [1224]. On the other hand,
the detection of a large amount of anisotropy in the signal would imply that
only a small fraction of the DM can be accounted by PBHs [134]. Moreover, the
propagation of GWs across disconnected regions of the universe may lead to
large-scale anisotropies at detection, see e.g., Alba and Maldacena [40],
Contaldi [371], Cusin et al. [394, 396], Jenkins and Sakellariadou [697],
Bartolo et al. [130, 131], Renzini and Contaldi [1015], Bertacca et al. [175].

In this range of masses, the non detection of a SGWB will rule out the
possibility of PBHs being the dark matter in the asteroidal mass range.
However, this assumes the standard scenario where GWs are produced at PBH
formation and the PBH abundance is exponentially sensitive to the size of the
fluctuations. One should investigate whether non standard scenarios would
change this conclusion, e.g. the potential role of non-Gaussianity or of other
PBH formation mechanisms.

###  IV.4 PBHs $m>M_{\odot}$

For PBHs with masses larger than about a solar mass, LISA should be able to
detect merger events of resolved sources, and unresolved signals in the form
of a SGWB. The most relevant constraints come from CMB temperature and
polarization anisotropies, which are impacted by the emission of ionizing
radiation from PBHs accreting gas in the early universe at redshift between
$z\sim(300-600)$, for which the PBH abundance is constrained to be below
$f_{\text{\tiny PBH}}(M)\lesssim(M/10M_{\odot})^{-4}$ for masses smaller than
$10^{4}M_{\odot}$ [48, 1068, 319, 598]. At larger masses, constraints coming
from CMB distortions become even more stringent [898].

Late-time constraints can be set from the comparison of emitted EM signals
from accreting PBHs with observations of galactic radio and X-ray isolated
sources [546, 825] and X-ray binaries [677], and from Dwarf Galaxy Heating
observations [804]. The totality of these bounds constrain the PBH abundance
$f_{\text{\tiny PBH}}$ (the fraction of the dark matter composed of PBHs) to
be below $f_{\text{\tiny PBH}}\lesssim 10^{-2}$ for masses larger than few
tens of $M_{\odot}$. A comparable constraint is set by the merger rates
observed by the LIGO/Virgo collaboration [8]. Indeed, for masses around
$30M_{\odot}$, a PBH abundance larger than $f_{\text{\tiny PBH}}\thickapprox
10^{-3}$ would give an expected number of events larger than the one observed.
This has been shown in Raidal et al. [1001], De Luca et al. [429], Hall et al.
[607] using the GWTC-1 data and recently in Wong et al. [1206] with the GWTC-2
catalog. Even though PBHs are expected to follow a Poisson distribution at
formation in the absence of non-Gaussianities [47, 450, 102, 877], they may
start forming clusters even before the matter-radiation equality for large
enough $f_{\text{\tiny PBH}}$ [674]. Clustering has an impact on the PBH
binary formation and subsequent disruption rates [1001, 695, 694, 1148]. While
ineffective for abundances smaller than $f_{\text{\tiny PBH}}\lesssim
10^{-2}$, it is expected to reduce the merger rate prediction for larger
abundances by an amount which is not enough to evade the LIGO/Virgo bound if
one takes into account the fraction of PBHs in dense substructures [1167, 426,
1145].Overall, we stress that a complete description of PBH clustering up to
very low redshift along with its impact on the GW signals coming from a PBH
population still represents one of the main theoretical challenges, which
deserves a thorough investigation.

Even if PBHs within this mass range can only account for a small fraction of
the dark matter in the universe, they could still both form binaries
observable by LISA and act as progenitors of SMBHs, observed at high redshift
as large as $z\gtrsim 6$, through accretion of baryonic particles during the
cosmological evolution [353, 1068]. The presence of a strong phase of
accretion through the cosmological history before the reionization epoch has
the effect of shifting the corresponding late-time mass function to larger
masses relaxing the constraints [428], even though large uncertainties are
present in modelling the accretion for massive compact objects. This is a key
challenge to overcome in order to assess the PBH contribution to merging
binaries observable by LISA. Though difficult to conclusively associate with
PBHs, the observation of events within this range of masses at high redshift
by experiments like LISA would help in understanding the physical origin of
SMBHs [1174].

Given that PBHs may potentially contribute to current GW data [536], within
this range of masses they need to be distinguished from astrophysical BHs
formed after the collapse of stars in the late universe. Various observables
could however help in discriminating the two populations. For instance, even
though PBHs are expected to form with small spins [865, 423], for PBH binaries
in the mass range observable by LISA, the baryonic mass accretion leads to
spin growth, potentially increasing their spins for the binary components at
the merger time depending on the accretion strength before the reionization
epoch [430], which is still however affected by many uncertainties. It is of
crucial importance to investigate in details the correlation between PBH
masses and spins induced by baryonic accretion as a discriminant with the
prediction for astrophysical BHs. Another possibility is to look for high
redshift events (i.e. $z\gtrsim 30$) where PBHs can contribute without any
astrophysical contamination [755, 342, 433].

Furthermore, the massive BBH merger population currently observed at
LIGO/Virgo is expected to give rise to a SGWB of unresolved sources with a
tail at low frequencies which can be detected by LISA [354, 343]. The
primordial scenario is expected to give a stronger contribution to the SGWB
with respect to the astrophysical one, due to the additional contribution
given by PBH mergers happening at higher redshift [1001]. In particular, the
PBHs merger rate is expected to increase with redshift, while the one of
stellar BHs first increases and peaks at redshift around $z\sim(1-2)$, and
then rapidly decreases [343]. Considering both observations of resolved and
unresolved signals may help to break the degeneracy within the two scenarios
[884, 434, 1180, 889].

LISA will also be able to detect the inspiral phase of binaries with masses of
at least a few tens $M_{\odot}$ with a high precision. This mass range,
falling in the large mass portion of the window observable by the LIGO/Virgo
collaboration, can be filled by a PBH population. The presence of the known
mass gap in the astrophysical BH spectrum and the recent event GW190521 [16]
make the primordial scenario investigated in Clesse and García-Bellido [352],
De Luca et al. [425], Kritos et al. [761] potentially relevant (see however
Abbott et al. 18 for different scenarios).

Finally, the detection of GWs from the coalescence of a BBH system including a