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 gravitational waves 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.
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1 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 crossgenerational and their answers are likely to require crossdisciplinary explorations, instead of being the purview of a particular subdiscipline.
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 largescale structure, and the hot big bang. Moreover, GWs can complement electromagnetic (EM) observations in astronomy and cosmology, thus enabling “multimessenger” 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, longwavelength GWs (AmaroSeoane et al. 2017), 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 quantuminspired 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 energymomentum 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 groundbased GW interferometers, such as the Laser Interferometer Gravitationalwave Observatory (LIGO), Virgo and KAGRA. This complementarity arises because groundbased 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 massratio inspirals (EMRIs), galactic binaries, and SGWB, none of which groundbased detectors are sensitive to. The GWs emitted by some of these sources (such as those emitted by SMBH mergers) will be extremely loud, generating signaltonoise 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 Bmode 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 Fig. 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 in Relativity and elsewhere for more details (see e.g., Barack et al. 2019). 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 LISAspecific 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.
1.1 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 = EinsteindilatonGauss–Bonnet

EdM = EinsteindilatonMaxwell.

EEP = Einstein equivalence principle

EM = Electromagnetic

EMRI = Extreme massratio inspiral

EsGB = EinsteinscalarGauss–Bonnet gravity

FLRW = Friedmann–Lemaître–Robertson–Walker

GR = General relativity

GW = Gravitational waves

IMBH = Intermediatemass black hole

IMR = Inspiralmergerringdown

LIGO = Laser Interferometer Gravitationalwave 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 = PostNewtonian

ppE = Parameterized postEinsteinian

ppN = Parametrized postNewtonian

SCO = Stellar compact object

SEP = Strong equivalence principle

SGWB = Stochastic gravitational wave background

SMBH = Supermassive black hole

SNR = Signaltonoise ratio

TeVeS = Tensorvectorscalar gravity

WD = White dwarf

WG = Working group

WEP = Weak equivalence principle
Henceforth, we use geometric units in which \(G = 1 = c\).
2 Tests of general relativity
2.1 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.
2.1.1 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 (Will 1993).
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 nongravitational test experiment is independent of the velocity of the freelyfalling 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 selfgravitating 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 (Will 1993). 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).
2.1.2 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 (Lovelock 1972): 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 secondorder 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 them^{Footnote 1} will share one property: they will contain one or more additional fields (see e.g., Sotiriou 2015b). Demonstrating this mathematically might require compactifying spacetime down to 4dimensions, 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. 2.2.1 we discuss how the structure of BHs—primary sources for LISA—can be affected by new fields, whereas in the rest of Sect. 2.2 we discuss how deviations from GR can be imprinted in different parts of the waveform or affect GW propagation.
2.2 Testing GR with compact objects
2.2.1 BHs beyond GR and theories that predict deviations
There exist nohair theorems for a wide range of alternative theories of gravity (Bekenstein 1997; Doneva and Yazadjiev 2020; Hawking 1972; Herdeiro and Radu 2015; Hui and Nicolis 2013; Sotiriou 2015a; Sotiriou and Faraoni 2012) 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 (Barausse and Sotiriou 2008; Molina et al. 2010; Tattersall and Ferreira 2018) and hence offer a possibility of test theories that are covered by nohair theorems. Models that manage to circumvent nohair 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 (Benkel et al. 2017, 2016; Campbell et al. 1992; Delgado et al. 2020b; Delsate et al. 2018; Kanti et al. 1996; Sotiriou and Zhou 2014a, b; Stein 2014; Yagi et al. 2012c; Yunes and Pretorius 2009a; Yunes and Stein 2011) which can be observed with LISA (Maselli et al. 2020a; Yagi and Stein 2016; Yagi et al. 2012b). Such couplings are expected in lowenergy limits of quantum gravity (see e.g., Ashtekar et al. 1989; Jackiw and Pi 2003; Metsaev and Tseytlin 1987) and are part of the Horndeski class of scalartensor theories (Kobayashi 2019). A coupling to the Pontryagin density is the leadingorder parity violating term in gravity in the presence of a (pseudo) scalar (Alexander and Yunes 2009; Jackiw and Pi 2003), while the linear coupling with the Gauss–Bonnet invariant is the only term inducing hair for shiftsymmetric (aka massless) scalars (Delgado et al. 2020b; Saravani and Sotiriou 2019; Sotiriou and Zhou 2014a) and the leading correction for GW emission (Witek et al. 2019). A nonlinear coupling with the Gauss–Bonnet invariant has recently been shown to give rise to BH spontaneous scalarization triggered by curvature (BlázquezSalcedo et al. 2018; Collodel et al. 2020; Cunha et al. 2019; Doneva and Yazadjiev 2018; Macedo et al. 2019; Silva et al. 2018, 2019) or spin (Berti et al. 2021; Dima et al. 2020; Doneva et al. 2020a, b; Herdeiro et al. 2021; Hod 2020). The full set of Horndeski theories that can give rise to BHs scalarization has been identified in Antoniou et al. (2018).
Attempts have also been made to circumvent nohair theorems by relaxing their assumptions. For example, superradiance can support longlived scalar clouds for very light scalars (Arvanitaki and Dubovsky 2011; Brito et al. 2015b) or lead to hairy BHs for complex scalars with a time dependent phase (Collodel et al. 2020; Delgado et al. 2019; Herdeiro et al. 2018a; Herdeiro and Radu 2014b; Herdeiro et al. 2015; Kleihaus et al. 2015); longlived scalar “wigs” can be formed around a Schwarzschild BH (Barranco et al. 2012, 2014); configurations with timedependent scalar fields can be supported by some nontrivial cosmological boundary conditions (Babichev and Charmousis 2014; Berti et al. 2013; Clough et al. 2019; Hui et al. 2019; Jacobson 1999); BHs immersed in an inhomogeneous scalar field were considered in Healy et al. (2012); scalarization can be induced by matter surrounding a BH (Cardoso et al. 2013a, b; Herdeiro et al. 2018b).
BHs in Lorentzviolating theories, such as Einsteinaether theory (Jacobson 2007; Jacobson and Mattingly 2001) and HoVision Res.ava gravity (Blas et al. 2010; Horava 2009b; Sotiriou 2011), will generically carry hair, as the field that breaks Lorentz symmetry will have to be nontrivial and may backreact on the geometry (see, however, Adam et al. 2021; Ramos and Barausse 2019). In such theories, new fields can propagate superluminally or even instantaneously (Bhattacharyya et al. 2016a; Blas and Sibiryakov 2011), even when all known constraints are satisfied (Emir Gümrükçüoǧlu et al. 2018; Sotiriou 2018). Indeed, BHs in Lorentzviolating theories can have a nested structure of different horizons for different modes (Barausse et al. 2011; Eling and Jacobson 2006) and potentially a universal horizon that traps all signals (Barausse et al. 2011; Bhattacharyya et al. 2016b; Blas and Sibiryakov 2011). 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 (Babichev et al. 2017; Heisenberg et al. 2017; Herdeiro et al. 2016; Kase et al. 2018; Minamitsuji 2017; Rahman and Sen 2019; Santos et al. 2020) and massive gravity theories (Berezhiani et al. 2012; Comelli et al. 2012; Rosen 2017).
2.2.2 Tests with GW propagation
The propagation of GWs provides a clean test of their kinematics. In GR, gravitons are massless spin2 particles, while in alternative gravity theories, they might be massive (Abbott et al. 2016b; Hassan and Rosen 2012; de Rham et al. 2011; Will 1998), or even have a Lorentzviolating structure in the dispersion relation (Blas et al. 2010; Horava 2009b; Jacobson 2007; Jacobson and Mattingly 2001; Kostelecký and Mewes 2016; Mirshekari et al. 2012; Shao 2020; Sotiriou 2011). Modifications to the dispersion relation could lead to frequencydependent, polarizationdependent, directiondependent 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 (Kostelecký and Mewes 2016; Will 1998). Therefore, a matchedfilter 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. 2016b) 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, doublesided bound on the speed of GWs to a part in \(10^{15}\) (Abbott et al. 2017e). However, Lorentzviolating theories have multidimensional parameter spaces and generically exhibit additional polarizations (Sotiriou 2018). The speed of these other polarizations remains virtually unconstrained (Emir Gümrükçüoǧlu et al. 2018; Oost et al. 2018). A combination of multiple GW events can also be used to simultaneously constrain a set of beyondGR parameters (Shao 2020).
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, \(g_{ij} (\mathbf{x},t) = a^2(t) \left[ \delta _{ij} + h_{ij} (\mathbf{x},t) \right] \). In Fourier space, the most general modification of the GW propagation equation can be written as (assuming spatially flat models) (Barausse et al. 2020; Ezquiaga 2021; Ezquiaga and GarcíaBellido 2018)
where \(H \equiv \dot{a} /a\) is the Hubble rate. The parameters \(c_{\mathrm{T}}\), \(\Gamma \) and D respectively describe the speed of propagation of the wave, the damping of its amplitude and additional modifications of the dispersion relation. Scaleindependent modifications, described by a kindependent \(\Gamma \), are discussed in Sect. 5. One expects a single observation to exclude \(\Gamma \gtrsim L^{1}\) and \(D \gtrsim f L^{1} \), 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. \(c_{\mathrm{T}}\), \(\Gamma \) 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 scalartensor theories (Ben Achour et al. 2016; Crisostomi et al. 2016; Deffayet et al. 2011; Gleyzes et al. 2015; Horndeski 1974; Langlois and Noui 2016; Zumalacárregui and GarcíaBellido 2014) under the assumption that they account for dark energy (Baker et al. 2017; Creminelli and Vernizzi 2017; Ezquiaga and Zumalacárregui 2017; Lombriser and Taylor 2016; McManus et al. 2016; Sakstein and Jain 2017) (see also de Rham and Melville 2018). Relaxing this assumption (Antoniou et al. 2021; Franchini and Sotiriou 2020; Noller et al. 2020) can lift the constraint.
GW propagation tests can also reveal potential couplings and decays of gravitons into other particles (e.g., Creminelli et al. 2020) or oscillations between different states (analogous to neutrino oscillations). The latter are expected in bigravity models (Hassan and Rosen 2012), where a massless and a massive spin2 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 (Barausse 2012; Klein et al. 2016) and performing a \(\chi ^2\) analysis, it was found that oscillation effects can be observed for masses \(m\gtrsim 2 \times 10^{25}\,\)eV (Belgacem et al. 2019c). 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 (Max et al. 2017), due to the larger oscillation baseline and the lower detection frequency.
2.2.3 Tests of GR with MBH coalescence
a. Inspiral: LISA will observe very long inspiral phases and it will therefore allow for high precision tests of gravity (Berti et al. 2005). This stage can be modelled using approximate techniques such as the lowvelocity, weakfield PN expansion (Poisson and Will 2014), or the parameterized postEinsteinian approach (ppE) (Yunes and Pretorius 2009b), 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 (Berti et al. 2018).
Using the inspiral observations, LISA will be able to improve the constraints on different nonGR predictions, such as the scalar dipole radiation, Lorenz symmetry, mass of the graviton, etc., by several orders of magnitude better compared to LIGO/Virgo (Chamberlain and Yunes 2017). Many alternative theories of gravity predict nonzero tidal deformation of BHs (Cardoso et al. 2017) 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 groundbased detectors. Using LISA observations and assuming GR, one can then obtain highaccuracy predictions of the time when the binary will become observable by groundbased detectors as well as its position on the sky (Sesana 2016). 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 GW150914like event could improve constraints on BBH dipole emission by 6 orders of magnitude (Barausse et al. 2016; Toubiana et al. 2020). 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 scalartensor theories, such as dynamical BH scalarization (Khalil et al. 2019).
b. Ringdown: In GR, numerical simulations have shown that the end state of a BBH merger is a Kerr BH (e.g., Boyle et al. 2019; Healy and Lousto 2020; Jani et al. 2016) in agreement with the earlier analytical studies proving that the Kerr metric is the unique stationary, asytmptotically flat, axially symmetric BH spacetime (Bunting 1983; Carter 1971; Mazur 1982; Robinson 1975) (see Chrusciel et al. 2012; Heusler 1996 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 (Berti et al. 2009). 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 (Berti et al. 2006; Detweiler 1980a; Dreyer et al. 2004). 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 (Kamaretsos et al. 2012). 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 (Berti et al. 2006; Dreyer et al. 2004). For a high SNR massive BBH event measured by LISA, this “no hair” (or “finalstate”) test can be performed to verify the remnant is consistent with a Kerr BH to high precision (e.g., Gossan et al. 2012).
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., BlázquezSalcedo et al. 2016; Cardoso and Gualtieri 2009; Okounkova 2020; Okounkova et al. 2020). 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. (2018), deviations from linear perturbations about a Schwarzschild BH in GR were parametrized at the level of a diffeomorphisminvariant action that encompassed a large range of possible theories that lead to secondorder 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 “postKerr” expansion (Glampedakis et al. 2017). There are also procedures to try to combine the deviation parameters from multiple BBH events measured by LISA (Maselli et al. 2020b). With more detailed predictions from specific theories, parametrized constraints could be converted into constraints on a particular theory.
c. 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 modelindependent selfconsistency test, such as the IMR consistency test (Ghosh et al. 2016a, 2017) could be used in order to avoid having to model the merger in alternative theories of gravity. However, having theoryspecific 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 (Benkel et al. 2016, 2017; Cayuso and Lehner 2020; East and Ripley 2021; Okounkova et al. 2017, 2019b, 2020; Silva et al. 2021b; Witek et al. 2019) and for Einstein–Maxwelldilaton theory (Hirschmann et al. 2018). Establishing whether the initial value problem is wellposed in alternative theories is particularly challenging (Cayuso et al. 2017; Kovács and Reall 2020; Papallo and Reall 2017; Ripley and Pretorius 2019b; Sarbach et al. 2019). So far, known simulations have circumvented this problem by adopted Effective Field Theory inspired treatments: either working perturbatively in the new coupling constants (Benkel et al. 2016, 2017; Okounkova et al. 2017, 2019b, 2020; Witek et al. 2019) or adopting a scheme similar to the IsraelStewart 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 (Cayuso et al. 2017). Nonlinear evolution beyond GR is one of the major challenges in GW modelling.
2.2.4 Test of GR with EMRIs (nonnull 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 (Barack and Pound 2019; Pound 2015a).
Despite progress within GR (Pound et al. 2020), 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 theorybytheory 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 (BlázquezSalcedo et al. 2016; Canizares et al. 2012; Cardoso et al. 2011; Fujita and Cardoso 2017; Pani et al. 2011a; Yunes et al. 2012). 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 nonGR theories using approximate waveforms (Canizares et al. 2012; Yunes et al. 2012), and modelling beyond the adiabatic approximation, i.e., taking into account selfforce calculations (Zimmerman 2015), 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 nohair theorems or separations of scales provide a decoupling of the metric and scalar perturbations (Maselli et al. 2020a). 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 (Maselli et al. 2022). 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 (Barack and Burko 2000; Castillo et al. 2018; Detweiler et al. 2003; DiazRivera et al. 2004; Gralla et al. 2015; Nasipak et al. 2019; Warburton 2015; Warburton and Barack 2010, 2011).
2.2.5 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 (Christodoulou 1991; Ludvigsen 1989; Payne 1983; Thorne 1992) (see Compère 2019a; Strominger 2018 for an account of historical references). Studies on the detection of the displacement memory effects in actual experiments have been presented in Favata (2009b, 2009a, 2010, 2011). More recently, prospects for the memory effect detection with LISA have been discussed in Islo et al. (2019) using SMBH binaries undergoing coalescence (See also Sect. 3). 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 (Flanagan et al. 2019; Nichols 2017, 2018; Pasterski et al. 2016). Generalizations of the memory effect, not necessarily motivated or associated to a symmetry, which should in principle be measurable, have also been discussed in Compère (2019b); Flanagan et al. (2019), and further memory effects with logarithmic branches have been inferred from graviton amplitude calculations (Laddha and Sen 2019; Sahoo and Sen 2019).
Measurement of memory effects would not only act as a test of GR (Flanagan et al. 2019; Hou and Zhu 2021; Nichols 2017, 2018; Tahura et al. 2021a, b), 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 infinitedimensional Bondi–Van der Burg–Metzner–Sachs (BMS) group which is a semidirect product of the Lorentz group and of an infinitedimensional group of “angle dependent translations” called supertranslations (Bondi et al. 1962; Sachs 1962). Further extensions of the BMS group have also been proposed (Barnich and Troessaert 2010; Campiglia and Laddha 2015). 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 (Strominger and Zhiboedov 2016; Strominger 2018). The classical limit of the quantum soft theorems are equivalent after a Fourier transformation to the memory effects (Strominger and Zhiboedov 2016). 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. (2019) study a simulationsuite of semianalytic models for the SMBH binary population with a signature of a memory signal from a SMBH binary approximated in the time domain by a stepfunction centered at the moment of coalescence and assuming a simple powerlaw model to emulate any environmental interaction which could influence the coalescence timing (e.g., final parsec problem Begelman et al. 1980). 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 Klein et al. (2016); Sesana et al. (2007) (and discussed in the Astrophysics WG White Paper by AmaroSeoane et al. 2022), most massive BBHs coalescences whose inspiral signature yields LISA SNR \( > 5\) lie beyond \(z = 3\) for a 3year LISA lifetime, meaning that the results of this study may be interpreted as lower limits on the number of LISA memory events.
2.3 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 nonGR predictions. This, however, will require a more accurate and complete development of GW waveforms including nonGR effects.

Numerical simulations with a systematic inclusion of nonGR 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.
3 Tests of the nature of BHs
3.1 The Kerr hypothesis
In vacuum GR, the Carter–Robinson (Carter 1971; Robinson 1975) uniqueness theorem, with later refinements (see Chrusciel et al. 2012 for a review), establishes that the Kerr geometry (Kerr 1963) is the unique physically acceptable equilibrium, asymptotically flat BH solution. This led to the more ambitious proposal that, regardless of the initial energymatter content available in a gravitational collapse scenario, the dynamically formed equilibrium BHs belong to the Kerr family (in the absence of gauge charges) (Ruffini and Wheeler 1971). 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 stronggravity 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 nonKerrness, require either modified gravity (discussed in Sect. 2 and also below with a different twist) or nonvacuum GR (discussed in Sect. 3.2.1). Moreover, two approaches are, in principle, possible for studying nonKerrness. 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 nonKerr), 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., Cardoso et al. (2014b); Johannsen and Psaltis (2011), 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 2019 for a review). One motivation for such a dramatic scenario is that, from Penrose’s theorem (Penrose 1965), 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 smallcurvature supermassive objects (Almheiri et al. 2013; Giddings 2006; Lunin and Mathur 2002a, b; Mathur 2005, 2009a, b; Mayerson 2020).
In this context it is also interesting that current LIGO/Virgo GW observations (especially the recent GW190814; Abbott et al. 2020d and GW190521; Abbott et al. 2020c, e, respectively in the lowermass and uppermass gap forbidden for standard stellarorigin BHs) do not exclude the possibility that ECOs might coexist along with BHs and NSs. Models of ECOs are discussed in Sect. 3.2.2.
3.2 Deviations from the Kerr hypothesis
3.2.1 BHs in nonvacuum GR
There is a large class of models wherein (covariant) matterenergy is minimally coupled to Einstein’s gravity. These models obey the EEP (cf. Sect. 2.1.1) 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 nonKerr BHs. This has been typically established by modelspecific nohair theorems. Historically influential examples are the Bekenstein noscalar and nomassivevector hair theorems (Bekenstein 1972) (see Herdeiro and Radu 2015 for a review). Nonetheless, nonstandard kinetic terms (e.g. Skyrme hair; Luckock and Moss 1986), negative energies (e.g. interacting real scalar hair; Nucamendi and Salgado 2003), nonlinear matter models (e.g. Yang–Mills hair; Bizon 1990) or symmetry noninheritance between the geometry and matter fields (e.g. synchronised bosonic hair; Herdeiro et al. 2016; Herdeiro and Radu 2014b) allow the existence of new families of BHs with hair, ^{Footnote 2} coexisting 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 nonKerr BHs must have a dynamical formation mechanism and be sufficiently stable to play a role in astrophysical processes. Asymptotically flat BHs with YangMills hair, for instance, are known to be perturbatively unstable (Zhou and Straumann 1991) whereas BHs with Skyrme hair are perturbatively stable (Heusler et al. 1992). 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 (ultralight) massive bosonic fields (such as the QCD axion, axionlike particles, dark photons, etc). These ultralight fields could be a significant component of the dark matter (Arvanitaki et al. 2010; Essig et al. 2013; Hui et al. 2017; Marsh 2016) and are predicted in a multitude of scenarios beyond the standard model of particle physics (Essig et al. 2013; Hui et al. 2017; Irastorza and Redondo 2018; Jaeckel and Ringwald 2010), including extra dimensions and string theories. They naturally interact very weekly and in a modeldependent fashion with baryonic matter, but their gravitational interaction is universal. The superradiant instability of Kerr BHs (Brito et al. 2015b), in the presence of (complex) ultralight bosonic fields (East and Pretorius 2017; Herdeiro and Radu 2017), or mergers of selfgravitating lumps of such ultraligh bosons (SanchisGual et al. 2020) (known as bosonic stars—see Sect. 3.2.2) form BHs with synchronised bosonic hair. These BHs are themselves afflicted by superradiant instabilities (Ganchev and Santos 2018; Herdeiro and Radu 2014a), but possibly on long timescales, even cosmologically long (Degollado et al. 2018), 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 massspin gaps, etc, all relevant for LISA science (cf. Sect. 4.1).
3.2.2 ECOs: deviations from (or absence of) a classical horizon
ECOs (Giudice et al. 2016) 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 (Cardoso and Pani 2019). 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 lightcrossing 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 (Cardoso and Pani 2019).
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 (Penrose 1965) implies that an apparent horizon always hides a curvature singularity wherein Einstein’s theory breaks down. Moreover, in the semiclassical approximation, BHs are thermodynamically unstable and have an entropy which is far in excess of a typical stellar progenitor (Hawking 1976). It has been argued that GWs may provide smoking guns for ECOs (Barausse et al. 2018; Cardoso and Pani 2017, 2019; Cardoso et al. 2016a, b; Giudice et al. 2016).
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 (Cardoso and Pani 2019). 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 wouldbe horizon region could prevent the formation of a horizon in a variety of models and theories (Giddings 2006; Lunin and Mathur 2002a, b; Mathur 2005, 2009a, b; Mayerson 2020; Mazur and Mottola 2004).
Amongst the first class of ECOs one of the most studied examples corresponds to bosonic stars. These are selfgravitating solitons, composed of either scalar (Jetzer 1992; Kaup 1968; Ruffini and Bonazzola 1969) or vector (Brito et al. 2016), massive complex fields, minimally coupled to Einstein’s gravity—see also Herdeiro et al. (2019); Herdeiro and Radu (2020); Herdeiro et al. (2017) for comparisons. Bosonic stars^{Footnote 3} arise in families of models with different classes of selfinteractions of the bosonic fields, e.g., Colpi et al. (1986); Delgado et al. (2020a); Grandclement et al. (2014); Guerra et al. (2019); Kleihaus et al. (2005); Minamitsuji (2018); Schunck and Mielke (2003) and different field content, e.g., Alcubierre et al. (2018); they may also be generalized to modified gravity, e.g., Herdeiro and Radu (2018). Bosonic stars circumvent Derrick type nosoliton theorems (Derrick 1964) due to a symmetry noninheritance between matter and geometry, as the latter is static/stationary and the former includes a harmonic time dependence (but with a timeindependent energymomentum tensor). Some bosonic stars are dynamically robust (Liebling and Palenzuela 2017), in particular perturbatively stable, with a known formation mechanism known as gravitational cooling (Di Giovanni et al. 2018; Seidel and Suen 1994). These can be evolved in binaries yielding gravitational waveforms, e.g., Bezares et al. (2017); Liebling and Palenzuela (2017); Palenzuela et al. (2008, 2017); SanchisGual et al. (2019), that can be used—together with PN approximations (Pacilio et al. 2020)—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 (Abbott et al. 2020c), is well mimicked by a very eccentric collision of spinning Proca stars (Bustillo et al. 2021).
Bosonic stars have a cousin family of solitons in the case of real bosonic fields, called oscillatons (Seidel and Suen 1994). They have a weak timedependence and slow decay, but can be very long lived, at least for spherical stars (Page 2004). Collisions of oscillatons and the corresponding waveforms have also been obtained, e.g., Clough et al. (2018).
Bosonic stars are the prototypical example of ECOs which are not meant to replace all BHs in the universe, but could in principle “coexist” 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 (Bayin 1982; Bowers and Liang 1974; Letelier 1980) (see Raposo et al. 2019a 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 firstprinciple model of ECO—aiming instead at replacing the classical horizon completely—emerges in the fuzzball proposal (Lunin and Mathur 2002a, b; Mathur 2005, 2009a). In the latter the classical horizon is replaced by smooth horizonless geometries with the same mass, charges, and angular momentum as the corresponding BH (Balasubramanian et al. 2008; Bena and Warner 2013, 2008; Mathur 2005; Myers 1997). These geometries represent some of the microstates in the lowenergy (super)gravity description. For special classes of extremal, charged, BHs (Horowitz et al. 1996; Maldacena et al. 1997; Strominger and Vafa 1996) 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 coarsegrained description which “averages” over the large number of coherent superposition of microstates, or as a ‘collective behavior’ of fuzzballs (Bena et al. 2019b, a; Bianchi et al. 2018, 2019, 2020b). 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 beyondGR physics confined at the horizon scale.
While microstate geometries emerge from a consistent lowenergy 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 negativepressure fluid which is compensated by a thin shell of an ultrarelativistic positivepressure 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 (Mazur and Mottola 2004). The negative pressure might arise as a hydrodynamical description of oneloop QFT effects in curved spacetime, so gravastars do not necessarily require exotic new physics (Mottola and Vaulin 2006). In these models, the Buchdahl limit is evaded both because the internal effective fluid is anisotropic (Cattoen et al. 2005; Raposo et al. 2019a) and because the negative pressure violates some of the energy conditions (Mazur and Mottola 2015). Gravastars can also be obtained as the BHlimit of constantdensity stars, past the Buchdahl limit (Posada and Chirenti 2019; Mazur and Mottola 2015). In this regime such configurations were found to be dynamically stable (Posada and Chirenti 2019).
Other models of ECOs include: wormholes (Damour and Solodukhin 2007; Lemos et al. 2003; Morris and Thorne 1988; Visser 1995), collapsed polymers (Brustein and Medved 2017; Brustein et al. 2017), nonlocal stars in the context of infinite derivative gravity (Buoninfante and Mazumdar 2019), dark stars (Barceló et al. 2009), naked singularities and superspinars (Gimon and Horava 2009), 22 holes (Holdom and Ren 2017), and quasiBHs (Lemos and Weinberg 2004; Lemos and Zaslavskii 2008) (see CarballoRubio et al. 2018; Cardoso and Pani 2019 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 nearhorizon region, such that the object still looks like a regular GR BH (Giddings 2013, 2017; Giddings et al. 2019), or drastic changes in the form of “hard” structures localized close to the horizon such as firewalls and other compact quantum objects (Almheiri et al. 2013; Giddings et al. 2019; Kaplan and Rajendran 2019). 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 (Friedman 1978; Yoshida and Eriguchi 1996). The endpoint of the instability could be a slowly spinning ECO (Brito et al. 2015b; Cardoso et al. 2008) or dissipation within the object could lead to a stable remnant (Maggio et al. 2017, 2019a). 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 (Cunha et al. 2017). Such stable light rings have been argued to source a spacetime instability at nonlinear level (Cardoso et al. 2014a; Keir 2016), whose timescale or endpoint, however, are unclear.
3.3 Observables and tests
3.3.1 Inspiralbased test with SMBH binaries, IMBH binaries, and EMRIs
a. Nongravitational 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, quasiBHs, 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 (Barausse et al. 2014; Cardoso et al. 2016c). In addition, their effective coupling might be suppressed, thus evading current constraints from the absence of dipole radiation in BBHs (see Sect. 2). A detailed confrontation of given charged ECO models with current constraints on dipolar radiation remains to be done.
b. Multipolar structure & Kerr bound: The multipole moments of a Kerr BH satisfy an elegant relation (Hansen 1974)^{Footnote 4},
where \(\mathcal {M}_\ell \) (\(\mathcal {S}_\ell \)) are the Geroch–Hansen mass (current) multipole moments (Geroch 1970; Hansen 1974), \(\mathcal {M}={{\mathcal {M}}}_0\) 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 nontrivial, 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 \(\overline{{{\mathcal {M}}}}_\ell \equiv \mathcal{M}_\ell /{{\mathcal {M}}}^{\ell {+}1}\) and \(\overline{{\mathcal {S}}}_\ell \equiv {{\mathcal {S}}}_\ell /{{\mathcal {M}}}^{\ell {+}1}\), the only nonvanishing moments of a Kerr BH are
for \(n=0,1,2,\ldots \). The fact that \({{\mathcal {M}}}_\ell =0\) (\({{\mathcal {S}}}_\ell =0\)) when \(\ell \) is odd (even) is a consequence of the equatorial symmetry of the Kerr metric, whereas the fact that all multipoles with \(\ell \ge 2\) are proportional to (powers of) the spin—as well as their specific spin dependence—is a peculiarity of the Kerr metric.
NonKerr 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 (Ryan 1997b) and BHs with synchronised scalar hair (Herdeiro and Radu 2014b). 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 highcompactness limit, approaching the Kerr value (Glampedakis and Pappas 2018; Pani 2015; Raposo and Pani