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 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) (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 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 Sect. 2 and also below with a different twist) or non-vacuum GR (discussed in Sect. 3.2.1). 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., 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 small-curvature 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 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. 3.2.2.
Deviations from the Kerr hypothesis
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. 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 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 (Bekenstein 1972) (see Herdeiro and Radu 2015 for a review). Nonetheless, non-standard kinetic terms (e.g. Skyrme hair; Luckock and Moss 1986), negative energies (e.g. interacting real scalar hair; Nucamendi and Salgado 2003), non-linear matter models (e.g. Yang–Mills hair; Bizon 1990) or symmetry non-inheritance 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 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 (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 (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 (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 model-dependent 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 self-gravitating lumps of such ultraligh bosons (Sanchis-Gual 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 mass-spin gaps, etc, all relevant for LISA science (cf. Sect. 4.1).
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 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 (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 semi-classical 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 would-be 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 self-gravitating 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 starsFootnote 3 arise in families of models with different classes of self-interactions 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 no-soliton theorems (Derrick 1964) 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 (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); Sanchis-Gual 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 time-dependence 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 “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 (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 first-principle 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 low-energy (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 coarse-grained 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 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 (Mazur and Mottola 2004). 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 (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 BH-limit of constant-density 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), 2-2 holes (Holdom and Ren 2017), and quasi-BHs (Lemos and Weinberg 2004; Lemos and Zaslavskii 2008) (see Carballo-Rubio 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 near-horizon 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.
Observables and tests
Inspiral-based test with SMBH binaries, IMBH binaries, and EMRIs
a. 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 (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,
$$\begin{aligned} \mathcal {M}_\ell ^{\mathrm{BH}}+\mathrm{i } \mathcal {S}_\ell ^{\mathrm{BH}} =\mathcal {M}^{\ell +1}\left( \mathrm{i } \chi \right) ^\ell \,, \end{aligned}$$
(2)
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 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 \(\overline{{{\mathcal {M}}}}_\ell \equiv \mathcal{M}_\ell /{{\mathcal {M}}}^{\ell {+}1}\) and \(\overline{{\mathcal {S}}}_\ell \equiv {{\mathcal {S}}}_\ell /{{\mathcal {M}}}^{\ell {+}1}\), the only nonvanishing moments of a Kerr BH are
$$\begin{aligned} \overline{{{\mathcal {M}}}}_{2n}^{\mathrm{BH}} = (-1)^n \chi ^{2n} \quad , \quad \overline{{{\mathcal {S}}}}_{2n{+}1}^{\mathrm{BH}} = (-1)^n \chi ^{2n{+}1} \end{aligned}$$
(3)
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.
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 (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 high-compactness limit, approaching the Kerr value (Glampedakis and Pappas 2018; Pani 2015; Raposo and Pani 2020; Raposo et al. 2019b). 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 \({{\mathcal {S}}}_2\) or the mass octopole \({{\mathcal {M}}}_3\)), or the axisymmetry (e.g. a generic mass quadrupole tensor \({{\mathcal {M}}}_{2m}\) with three independent components (\(m=0,1,2\)), as in the case of multipolar boson stars (Herdeiro et al. 2021) and of fuzzball microstate geometries (Bena and Mayerson 2020, 2021; Bianchi et al. 2020a, 2021).
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 (Barack and Cutler 2007). 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 \(M_2\) (Barack and Cutler 2007; Krishnendu et al. 2017), spin-induced octopole \(S_3\) (Krishnendu and Yelikar 2019), and current quadrupole \(S_2\) (Fransen and Mayerson 2022).
GW signals emitted by EMRIs will provide accurate measurements of the spin-induced quadrupole at the level of one part in \(10^4\) (Babak et al. 2017a; Barack and Cutler 2007), and of the equatorial symmetry breaking current quadrupole at the level of one part in \(10^2\) (Fransen and Mayerson 2022). A rather generic attempt to constrain the multipole moments of an axisymmetric and equatorially symmetric central object with EMRIs has been done in Ryan (1995, 1997a) 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. (2018, 2019); these constrain deviations from the GR expectation of the binary system without explicitly parametrizing the compact objects’ multipole structure.
c. 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 (Bernuzzi et al. 2012b; Datta and Bose 2019; Datta et al. 2020; Harms et al. 2014; Hughes 2001; Maggio et al. 2021; Taracchini et al. 2013). 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 (Datta et al. 2020; Maggio et al. 2021; Maselli et al. 2018). 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 (Datta et al. 2020; Maggio et al. 2021). Absence of a horizon can also produce resonances that can be excited during EMRIs (Cardoso et al. 2019c; Macedo et al. 2013b; Maggio et al. 2021; Pani et al. 2010).
d. 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 (Hinderer et al. 2018; Poisson and Will 2014). 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 (Hinderer et al. 2016; Maselli et al. 2012; Steinhoff et al. 2016): the tidal Love numbers (Binnington and Poisson 2009; Damour and Nagar 2009; Hinderer 2008). 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 \({\tilde{\lambda }}=\frac{2}{3} k_2 {{\mathcal {C}}}^5\), where R and \({{\mathcal {C}}}\) 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 (Abbott et al. 2018a). 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 (Binnington and Poisson 2009; Damour and Nagar 2009) and large (Gürlebeck 2015) 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 (Landry and Poisson 2015; Pani et al. 2015a, b; Poisson 2015). Very recently, this result was extended to any tidal Love number of a Kerr BH with arbitrary spin (see Charalambous et al. 2021, 2021; Chia 2021; Hui et al. 2021; Le Tiec and Casals 2021; Le Tiec et al. 2021 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 (Cardoso et al. 2017; Giddings et al. 2019; Herdeiro et al. 2020; Johnson-McDaniel et al. 2020; Maselli et al. 2018, 2019; Pani 2015; Porto 2016; Raposo et al. 2019a; Sennett et al. 2017; Uchikata et al. 2016). For nonrotating BH mimickers, featuring corrections at the horizon scale and that approach the BH compactness, the Love numbers vanish in the limit \({{\mathcal {C}}}_\text {ECO}\rightarrow {{\mathcal {C}}}_\text {BH}\), often logarithmically (Cardoso et al. 2017).
LISA will be able to measure the tidal Love numbers of BH mimickers (Maselli et al. 2018), which are otherwise unmeasurable by current and future ground based detectors (Cardoso et al. 2017). In the comparable-mass case, this measurement requires highly-spinning supermassive ECO binaries up to \(10\,\mathrm{Gpc}\). LISA may also be able to perform model selection between different families of BH mimickers (Maselli et al. 2019), although this will in general require detection of golden binaries (i.e. binaries with a very large SNR) (Addazi et al. 2019). For a large class of slowly-rotating ECOs with compactness \({{\mathcal {C}}} \lesssim 0.3\), LISA can measure the Love numbers with very good accuracy below 1% (Cardoso et al. 2017). 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 (Pacilio et al. 2020).
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 (Pani and Maselli 2019). 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 \(10^5\) (Pani and Maselli 2019).
e. 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 (Levin 2006).
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 (Zelenka et al. 2020). In some modified theories, even the geodesic orbital motion might not be integrable (Cárdenas-Avendaño et al. 2018; Lukes-Gerakopoulos et al. 2010). This is also true for some models of ECOs, such as spinning scalar boson stars and non-Kerr BHs in GR (Cunha et al. 2016). 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 (Cárdenas-Avendaño et al. 2018; Destounis et al. 2021; Gair et al. 2008; Lukes-Gerakopoulos et al. 2010).
f. 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. (2005). The effects of accretion and drag were included in Barausse et al. (2014), Barausse et al. (2015), Macedo et al. (2013a) and Macedo et al. (2013b). 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. 4). In general, they could be a smoking-gun signature for the existence of structures in supermassive ultracompact objects.
Ringdown tests
a. QNMs: Similarly to what was discussed in Sect. 2, 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 (Berti et al. 2009; Kokkotas and Schmidt 1999). 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 (Abbott et al. 2019b, 2021b). Among the entire second GW transient catalogue (Abbott et al. 2021b) 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 (Isi et al. 2019a). Overtones are particularly useful for tests of GR with equal-mass binaries (for which other angular modes can be suppressed) (Bhagwat et al. 2020; Jiménez Forteza et al. 2020; Ota and Chirenti 2020), 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 (Baibhav et al. 2018; Bhagwat et al. 2018, 2020; Brito et al. 2018; Giesler et al. 2019; Jiménez Forteza et al. 2020; Ota and Chirenti 2020).
The large SNR expected in LISA for ringdown signals of SMBH coalescences provides a unique opportunity to perform BH spectroscopy (Dreyer et al. 2004) 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 (Berti et al. 2016).
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 (Pani et al. 2009) in stars or extra degrees of freedom (e.g. scalar QNMs for boson stars; Macedo et al. 2013b), some of which—being at low frequency—could be resonantly excited during the inspiral (Cardoso et al. 2019c; Macedo et al. 2013b; Maggio et al. 2021; Pani et al. 2010); (ii) isospectrality breaking between modes that can be identified as even-parity and odd-parity in the zero-spin limit (Maggio et al. 2020). 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. (2020) by extending the BH membrane paradigm to ECOs.
b. Echoes: GW echoes (Cardoso et al. 2016a, b) in the post-merger signal of a compact binary coalescence might be a clear signature of near-horizon quantum structures (Abedi et al. 2017; Barceló et al. 2017; Cardoso et al. 2016a, b; Oshita and Afshordi 2019; Wang et al. 2020), ultracompact objects (Bueno et al. 2018; Cardoso et al. 2016a), exotic states of matter in ultracompact stars (Buoninfante and Mazumdar 2019; Ferrari and Kokkotas 2000; Pani and Ferrari 2018), and of modified theories of gravity (Buoninfante et al. 2019; Burgess et al. 2018; Delhom et al. 2019) (see Abedi et al. 2020; Cardoso and Pani 2017, 2017, 2019 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 (Ikeda et al. 2021).
If sufficiently compact, horizonless objects support quasi-bound modes trapped within their photon sphere (Cardoso et al. 2016a, b; Kokkotas 1995; Kokkotas and Schmidt 1999). 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 (Cardoso et al. 2016a, b). 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 (Maggio et al. 2020).
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 (Abedi et al. 2020; Cardoso and Pani 2017; Cardoso et al. 2016b; Oshita et al. 2020). The damping factor of subsequent echoes is related to the reflective properties of the compact object (Cardoso and Pani 2019; Maggio et al. 2019a, 2020; Price and Khanna 2017).
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 (Abedi et al. 2017; Nakano et al. 2017; Wang and Afshordi 2018); (ii) superposition of sine-Gaussians with free parameters (Maselli et al. 2017; iii) frequency-domain templatesFootnote 5 based on the physical ECO parameters (Maggio et al. 2019b; Mark et al.