Testing the nature of dark compact objects: a status report
Abstract
Very compact objects probe extreme gravitational fields and may be the key to understand outstanding puzzles in fundamental physics. These include the nature of dark matter, the fate of spacetime singularities, or the loss of unitarity in Hawking evaporation. The standard astrophysical description of collapsing objects tells us that massive, dark and compact objects are black holes. Any observation suggesting otherwise would be an indication of beyondthestandardmodel physics. Null results strengthen and quantify the Kerr black hole paradigm. The advent of gravitationalwave astronomy and precise measurements with very long baseline interferometry allow one to finally probe into such foundational issues. We overview the physics of exotic dark compact objects and their observational status, including the observational evidence for black holes with current and future experiments.
Keywords
Black holes Event horizon Gravitational waves Quantum gravity SingularitiesThe crushing of matter to infinite density by infinite tidal gravitation forces is a phenomenon with which one cannot live comfortably. From a purely philosophical standpoint it is difficult to believe that physical singularities are a fundamental and unavoidable feature of our universe [...] one is inclined to discard or modify that theory rather than accept the suggestion that the singularity actually occurs in nature.
Kip Thorne, Relativistic Stellar Structure and Dynamics (1966)
No testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous than the fact which it endeavors to establish.
David Hume, An Enquiry concerning Human Understanding (1748)
1 Introduction
The discovery of the electron and the known neutrality of matter led in 1904 to J. J. Thomson’s “plumpudding” atomic model. Data from new scattering experiments was soon found to be in tension with this model, which was eventually superseeded by Rutherford’s, featuring an atomic nucleus. The pointlike character of elementary particles opened up new questions. How to explain the apparent stability of the atom? How to handle the singular behavior of the electric field close to the source? What is the structure of elementary particles? Some of these questions were elucidated with quantum mechanics and quantum field theory. Invariably, the path to the answer led to the discovery of hitherto unknown phenomena and to a deeper understanding of the fundamental laws of Nature. The history of elementary particles is a timeline of the understanding of the electromagnetic (EM) interaction, and is pegged to its characteristic \(1/r^2\) behavior (which necessarily implies that other structure has to exist on small scales within any sound theory).
Arguably, the elementary particle of the gravitational interaction are black holes (BHs). Within General Relativity (GR), BHs are indivisible and the simplest macroscopic objects that one can conceive. The uniqueness results—establishing that the twoparameter Kerr family of BHs describes any vacuum, stationary and asymptotically flat, regular solution to GR—have turned BHs into somewhat of a miracle elementary particle.^{1}
Even though the first nontrivial regular, asymptotically flat, vacuum solution to the field equations describing BHs were written already in 1916 (Schwarzschild 1916; Droste 1917), several decades would elapse until such solutions became accepted and understood. The dissension between Eddington and Chandrasekhar over gravitational collapse to BHs is famous—Eddington firmly believed that nature would find its way to prevent full collapse—and it took decades for the community to overcome individual prejudices. Ironically, after that BHs quickly became the only acceptable solution. So much so, that currently an informal definition of a BH might well be “any dark, compact object with mass above roughly three solar masses.”
1.1 Black holes: kings of the cosmos?
There are various reasons why BHs were quickly adopted as the only possible dark and compact sources triggering highenergy, violent phenomena in the Universe. The BH interior is causally disconnected from the exterior by an event horizon. Unlike the classical description of atoms, the GR description of the BH exterior is selfconsistent and free of pathologies. The “inversesquare law problem”—the GR counterpart of which is the appearance of pathological curvature singularities—is swept to inside the horizon and therefore harmless for the external world. There are strong indications that classical BHs are stable against small fluctuations (Klainerman and Szeftel 2017), and attempts to produce naked singularities, starting from BH spacetimes, have failed. In addition, BHs in GR can be shown to satisfy remarkable uniqueness properties (Chruściel et al. 2012). These features promote BHs to important solutions of the field equations and ideal testbeds for new physics. But BHs are not only curious mathematical solutions to Einstein’s equations: their formation process is sound and well understood. At the classical level, there is nothing spectacular with the presence or formation of an event horizon. The equivalence principle dictates that an infalling observer crossing this region (which, by definition, is a global concept) feels nothing extraordinary: in the case of macroscopic BHs all of the local physics at the horizon is rather unremarkable. Together with observations of phenomena so powerful that could only be explained via massive compact objects, the theoretical understanding of BHs turned them into undisputed kings of the cosmos.
There is, so far, no evidence for objects other than BHs that can explain all observations. Nonetheless, given the special nature of BHs, one must question and quantify their existence. Can BHs, as envisioned in vacuum GR, hold the same surprises that the electron and the hydrogen atom did when they started to be experimentally probed? This overview will dwell on the existence of BHs, and signatures of possible alternatives. There are a number of important reasons to do so, starting from the obvious: we can do it. The landmark detection of gravitational waves (GWs) showed that we are now able to analyze and understand the details of the signal produced when two compact objects merge (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2016a, b). An increase in sensitivity of current detectors and the advent of nextgeneration interferometers on ground and in space will open the frontier of precision GW astrophysics. GWs are produced by the coherent motion of the sources as a whole: they are ideal probes of strong gravity, and play the role that EM waves did to test the Rutherford model. In parallel, novel techniques such as radio and deep infrared interferometry (Doeleman et al. 2008; Antoniadis 2013) are now providing direct images of the center of ours and others galaxies, where a dark, massive and compact object is lurking (Genzel et al. 2010; Falcke and Markoff 2013; Johannsen et al. 2016; Abuter et al. (GRAVITY Collaboration) 2018b; Akiyama et al. 2019).
The wealth of data from GW and EM observations has the potential to inform us on the following outstanding issues.
1.2 Problems on the horizon
Classically, spacetime singularities seem to be always cloaked by horizons and hence inaccessible to distant observers; this is in essence the content of the weak cosmic censorship conjecture (Penrose 1969; Wald 1997). However, there is as yet no proof that the field equations always evolve regular initial data towards regular final states.
Classically, the BH exterior is pathologyfree, but the interior is not. The Kerr family of BHs harbors singularities and closed timelike curves in its interior, and more generically it features a Cauchy horizon signaling the breakdown of predictability of the theory (Penrose 1978; Reall 2018; Dafermos 2005; Cardoso et al. 2018a). The geometry describing the interior of an astrophysical spinning BH is currently unknown. A resolution of this problem most likely requires accounting for quantum effects. It is conceivable that these quantum effects are of no consequence whatsoever on physics outside the horizon. Nevertheless, it is conceivable as well that the resolution of such inconsistency leads to new physics that resolves singularities and does away with horizons, at least in the way we understand them currently. Such possibility is not too dissimilar from what happened with the atomic model after the advent of quantum electrodynamics.
Black holes have a tremendously large entropy, which is hard to explain from microscopic states of the progenitor star. Classical results regarding for example the area (and therefore entropy) increase (Hawking 1971) and the number of microstates can be tested using GW measurements (Lai and Li 2018; Brustein et al. 2018), but assume classical matter. Indeed, semiclassical quantum effects around BHs are far from being under control. Quantum field theory on BH backgrounds leads to loss of unitarity, a selfconsistency requirement that any predictive theory ought to fulfill. The resolution of such conundrum may involve nonlocal effects changing the nearhorizon structure, or doing away with horizons completely (Giddings 1992, 2011, 2012, 2016, 2017a, b; Mazur and Mottola 2004; Mathur 2005, 2008, 2009; Barceló et al. 2016; Almheiri et al. 2013; Unruh and Wald 2017; Bianchi et al. 2018; Giddings et al. 2019).
As a matter of fact, there is no tested nor fully satisfactory theory of quantum gravity, in much the same way that one did not have a quantum theory of point charges at the beginning of the twentieth century.
GR is a purely classical theory. One expects quantum physics to become important beyond some energy scale. It is tacitly assumed that such “quantum gravity effects” are relevant only near the Planck scale: at lengths \(\ell _P\sim \sqrt{G\hbar /c^3}\sim 10^{35}\,\mathrm{m}\), the Schwarzschild radius is of the order of the Compton wavelength of the BH and the notion of a classical system is lost. However, it has been argued that, in the orders of magnitude standing between the Planck scale and those accessible by current experiments, new physics can hide. To give but one example, if gravity is fundamentally a higherdimensional interaction, then the fundamental Planck length can be substantially larger (ArkaniHamed et al. 1998; Randall and Sundrum 1999). In addition, some physics related to compact objects have a logarithmic dependence on the (reasonablydefined) Planck length (Cardoso and Pani 2017a) (as also discussed below). Curiously, some attempts to quantize the area of BHs predict sizable effects even at a classical level, resulting in precisely the same phenomenology as that discussed in the rest of this review (Bekenstein and Mukhanov 1995; Saravani et al. 2015; Foit and Kleban 2019; Cardoso et al. 2019a; Chakraborty and Lochan 2019). Thus, quantumgravity effects may be within reach.
1.3 Quantifying the evidence for black holes
Horizons are not only a rather generic prediction of GR, but their existence is in fact necessary for the consistency of the theory at the classical level. This is the root of Penrose’s (weak) Cosmic Censorship Conjecture (Penrose 1969; Wald 1997), which remains one of the most urgent open problems in fundamental physics. In this sense, the statement that there is a horizon in any spacetime harboring a singularity in its interior is such a remarkable claim, that (in an informal description of Hume’s statement above) it requires similar remarkable evidence.
It is in the nature of science that paradigms have to be constantly questioned and subjected to experimental and observational scrutiny. Most specially because if the answer turns out to be that BHs do not exist, the consequences are so extreme and profound, that it is worth all the possible burden of actually testing it. As we will argue, the question is not just whether the strongfield gravity region near compact objects is consistent with the BH geometry of GR, but rather to quantify the limits of observations in testing event horizons. This approach is common practice in other contexts. Decades of efforts in testing the pillars of GR resulted in formalisms [such as the parametrized postNewtonian approach (Will 2014)] which quantify the constraints of putative deviations from GR. For example, we know that the weak equivalence principle is valid to at least within one part in \(10^{15}\) (Bergé et al. 2018). On the other hand, no such solid framework is currently available to quantify deviations from the standard BH paradigm. In fact, as we advocate in this work, the question to be asked is not whether there is a horizon in the spacetime, but how close to it do experiments or observations go. It is important to highlight that some of the most important tests of theories or paradigms—and GR and its BH solutions are no exception—arise from entertaining the existence of alternatives. It is by allowing a large space of solutions that one can begin to exclude—with observational and experimental data—some of the alternatives, thereby producing a stronger paradigm.
1.4 The dark matter connection
Known physics all but exclude BH alternatives as explanations for the dark, massive and compact objects out there. Nonetheless, the Standard Model of fundamental interactions is not sufficient to describe the cosmos—at least on the largest scales. The nature of dark matter (DM) is one of the longeststanding puzzles in physics (Bertone and Tim 2018; Barack et al. 2018). Given that the evidence for DM is—so far—purely gravitational, further clues may well be hidden in stronggravity regions or GW signals generated by dynamical compact objects.
As an example, new fundamental fields [such as axions, axionlike particles, etc (Marsh 2016; Clifton et al. 2012)], either minimally or nonminimally coupled to gravity, are essential for cosmological models, and are able to explain all known observations concerning DM. Even the simplest possible theory of minimally coupled, massive scalar fields give rise to selfgravitating compact objects, which are dark if their interaction with Standard Model particles is weak. These are called boson stars or oscillatons, depending on whether the field if complex or real, respectively. Such dark objects have a maximum mass^{2} which is regulated by the mass of the fundamental boson itself and by possible selfinteraction terms; they form naturally through gravitational collapse and may cluster around an ultracompact configuration through “gravitational cooling” (Seidel and Suen 1994; Liebling and Palenzuela 2012; Brito et al. 2015a; Di Giovanni et al. 2018).
Furthermore, DM could be composed of entirely different fields or particles, and many of these are expect to lead to new classes of dark compact objects (Narain et al. 2006; Raidal et al. 2018; Deliyergiyev et al. 2019).
1.5 Taxonomy of compact objects: a lesson from particle physics
From a phenomenological standpoint, BHs and neutron stars could be just two “species” of a larger family of astrophysical compact objects, which might coexist with BHs rather than replacing them. These objects are theoretically predicted in extended theories of gravity but also in other scenarios in the context of GR, such as beyondtheStandardModel fundamental fields minimally coupled to gravity, or of exotic states of matter.
In this context, it is tempting to draw another parallel with particle physics. After the Thomson discovery of the electron in 1897, the zoo of elementary particles remained almost unpopulated for decades: the proton was discovered only in the 1920s, the neutron and the positron only in 1932, few years before the muon (1936). Larger and more sensitive particle accelerators had been instrumental to discover dozens of new species of elementary particles during the second half of the twentieth century, and nowadays the Standard Model of particle physics accounts for hundreds of particles, either elementary or composite. Compared to the timeline of particle physics, the discovery of BHs, neutron stars, and binary thereof is much more recent; it is therefore natural to expect that the latest advance in GW astronomy and very long baseline interferometry can unveil new species in the zoo of astrophysical compact objects. Of course, this requires an understanding of the properties of new families of hypothetical compact objects and of their signatures.
1.6 The small \(\epsilon \)limit
In addition to the above phenomenological motivations, dark compact objects are also interesting from a mathematical point of view. For instance, given the unique properties of a BH, it is interesting to study how a dark compact object approaches the “BH limit” (if the latter exists!) as its compactness increases. Continuity arguments would suggest that any deviation from a BH should vanish in this limit, but this might occur in a highly nontrivial way, as we shall discuss. The first issue in this context is how to parametrize “how close” a selfgravitating object is to a BH in a rigorous way, by introducing a “closeness” parameter \(\epsilon \), such that \(\epsilon \rightarrow 0\) corresponds to the BH limit. As we shall discuss, there are several choices for \(\epsilon \), for example the tidal deformability, the inverse of the maximum redshift in the spacetime, or a quantity related to the compactness M / R such as \(\epsilon =12M/R\), where M is the object mass in the static case and R is its radius.
In the context of DM selfgravitating objects \(\epsilon \) is expected to be of order unity. However, when quantifying the evidence for horizons or in the context of quantum corrected spacetimes, one is usually interested in the \(\epsilon \ll 1\) limit. The physics of such hypothetical objects is interesting on its own: these objects are by construction regular everywhere and causality arguments imply that all known BH physics must be recovered in the \(\epsilon \rightarrow 0\) limit. Thus, the small \(\epsilon \)limit may prove useful in the understanding of BH themselves, or to help cast a new light in old murky aspects of objects with a teleological nature. Moreover, as we will see, such limit is amenable to many analytical simplifications and describes reasonably well even finite \(\epsilon \) spacetimes. In this regard, the \(\epsilon \rightarrow 0\) limit can be compared to large spacetimedimensionality limit in Einstein field equations (Emparan et al. 2013), or even the large \({{\mathcal {N}}}\) limit in QCD (’t Hooft 1974). Here, we will focus exclusively on fourdimensional spacetimes.
2 Structure of stationary compact objects
The precise understanding of the nature of dark, massive and compact objects can follow different routes,Mumbo Jumbo is a noun and is the name of a grotesque idol said to have been worshipped by some tribes. In its figurative sense, Mumbo Jumbo is an object of senseless veneration or a meaningless ritual.
Concise Oxford English Dictionary
 i.
a pragmatic approach of testing the spacetime close to compact, dark objects, irrespective of their nature, by devising modelindependent observations that yield unambiguous answers; this often requires consistency checks and nullhypothesis tests of the Kerr metric.
 ii.
a less ambitious and more theoreticallydriven approach, which starts by constructing objects that are very compact, yet horizonless, within some framework. It proceeds to study their formation mechanisms and stability properties, and then discarding solutions which either do not form or are unstable on short timescales; finally, understand the observational imprints of the remaining objects, and how they differ from BHs.
Henceforth we shall refer to horizonless compact objects other than a neutron star as Exotic Compact Objects (ECOs). The aim of this section is to contrast the properties of BHs with those of ECOs and to find a classification for different models.
2.1 Anatomy of compact objects
2.1.1 Event horizons, trapped surfaces, apparent horizons
A BH owns its name (Herdeiro and Lemos 2018) to the fact that nothing—not even light—can escape from the region enclosed by its horizon. Since the latter is the real defining quantity of a BH, it is important to define it rigorously. In fact, there are several inequivalent concepts of horizon (Hawking and Ellis 2011; Curiel 2019). In asymptoticallyflat spacetime, a BH is the set of events from which no futurepointing null geodesic can reach future null infinity. The event horizon is the (null) boundary of this region. The event horizon is a global property of an entire spacetime: on a given spacelike slice, the event horizon cannot be computed without knowing the entire future of the slice. Strictly speaking, an event horizon does not “form” at a certain time, but it is a nonlocal property; as such, it is of limited practical use in dynamical situations.
On the other hand, in a \(3+1\) splitting of spacetime, a trapped surface is defined as a smooth closed 2surface on the slice whose futurepointing outgoing null geodesics have negative expansion (Hawking and Ellis 2011; Thornburg 2007; Wald and Iyer 1991). Roughly speaking, on a trapped surface light rays are all directed inside the trapped surface at that given time. The trapped region is the union of all trapped surfaces, and the outer boundary of the trapped region is called the apparent horizon. At variance with the event horizon, the apparent horizon is defined locally in time, but it is a property that depends on the choice of the slice. Under certain hypothesis—including the assumption that matter fields satisfy the energy conditions—the existence of a trapped surface (and hence of an apparent horizon) implies that the corresponding slice contains a BH (Hawking and Ellis 2011). The converse is instead not true: an arbitrary (spacelike) slice of a BH spacetime might not contain any apparent horizon. If an apparent horizon exists, it is necessarily contained within an event horizon, or it coincides with it. In a stationary spacetime, the event and apparent horizons always coincide at a classical level (see Bardeen 1981; York 1983; Arzano and Calcagni 2016 for possible quantum effects).
In practice, we will be dealing mostly with quasistationary solutions, when the distinction between event and apparent horizon is negligible. For the sake of brevity, we shall often refer simply to a “horizon”, having in mind the apparent horizon of a quasistationary solution. Notwithstanding, there is no direct observable associated to the horizon (Abramowicz et al. 2002; Cardoso and Pani 2017a; Nakao et al. 2019). There are signatures which can be directly associated to timelike surfaces, and whose presence would signal new physics. The absence of such signatures strengthens and quantifies the BH paradigm.
2.1.2 Quantifying the shades of dark objects: the closeness parameter \(\epsilon \)
Since we will mostly be discussing objects which look like BHs on many scales, it is useful to introduce a “closeness” parameter \(\epsilon \) that indicates how close one is to a BH spacetime. There is an infinity of possible choices for such parameter (and in fact, different choices have been made in the literature, e.g., Giddings 2014, 2016). At least in the case of spherical symmetric, Birkhoff’s theorem provides a natural choice for the closeness parameter: if the object has a surface at \(r_0\), then \(\epsilon \) is defined asAlas, I abhor informality.
That Mitchell and Webb Look, Episode 2
We should highlight that this choice of closeness parameter is made for convenience. None of the final results depend on such an arbitrary choice. In fact, there are objects—such as boson stars—without a well defined surface, since the matter fields are smooth everywhere. In such case \(r_0\) can be taken to be an effective radius beyond which the density drops sharply to zero. In some cases it is possible that the effective radius depends on the type of perturbations or on its frequency. It sometimes proves more useful, and of direct significance, to use instead the coordinate time \(\tau \) (measurable by our detectors) that a radialdirected light signal takes to travel from the light ring to the surface of the object. For spherically symmetric spacetimes, there is a onetoone correspondence with the \(\epsilon \) parameter, \(\tau =M(12\epsilon \log (4\epsilon ^2))\sim 2M\log \epsilon \), where the last step is valid when \(\epsilon \rightarrow 0\). In the rest, when convenient, we shall refer to this time scale rather than to \(r_0\).
Overall, we shall use the magnitude of \(\epsilon \) to classify different models of dark objects. A neutron star has \(\epsilon \sim {{\mathcal {O}}}(1)\) and models with such value of the closeness parameter (e.g., boson stars, stars made of DM, see below) are expected to have dynamical properties which resemble those of a stellar object rather than a BH. For example, they are characterized by observables that display \({{\mathcal {O}}}(1)\) corrections relative to the BH case and are therefore easier to distinguish. On the other hand, to test the BH paradigm in an agnostic way, or for testing the effects of quantum gravity, one often has in mind \(\epsilon \ll 1\). For instance, in certain models \(r_02M=2M\epsilon \) or the proper distance \(\sim M\sqrt{\epsilon }\) are of the order of the Planck length \(\ell _P\); in such case \(\epsilon \sim 10^{40}\) or even smaller. These models are more challenging to rule out.
Finally, in dynamical situations \(\epsilon \) might be effectively time dependent. Even when \(\epsilon \sim \ell _P/M\) at equilibrium, offequilibrium configurations might have significantly large \(\epsilon \) (see, e.g., Brustein and Medved 2018; Brustein et al. 2017a; Wang et al. 2018a, 2019b).
2.1.3 Quantifying the softness of dark objects: the curvature parameter
By comparison with the BH case, one can introduce two classes of models (Raposo et al. 2019): (i) “soft” ECOs, for which the maximum curvature is comparable to that at the horizon of the corresponding BH; and (ii) “hard” ECOs, for which the curvature is much larger. In the first class, the nearsurface geometry smoothly approaches that at the horizon in the BH limit (hence their “softness”), whereas in the second class the ECO can support large curvatures on its surface without collapsing, presumably because the underlying theory involves a new length scale, \({{\mathcal {L}}}\), such that \({{\mathcal {L}}}\ll {M}\). In these models highenergy effects drastically modify the nearsurface geometry (hence their “hardness”). An example are certain classes of wormholes (see Sect. 3).
2.1.4 Geodesic motion and associated scales
The most salient geodesic features of a compact object are depicted in Fig. 1, representing the equatorial slice of a sphericallysymmetric spacetime.
For example, assume that an experimenter far away throws (highfrequency) photons in all directions and somewhere a compact object is sitting, as in Fig. 2. Photons that have a very large impact parameter (or large angular momentum), never get close to the object. Photons with a smaller impact parameter start feeling the gravitational pull of the object and may be slightly deflected, as the ray in the figure. Below a critical impact parameter all photons “hit” the compact object. It is a curious mathematical property that the critical impact parameter corresponds to photons that circle the light ring an infinite number of orbits, before being either absorbed or scattered. Thus, the light ring is fundamental for the description of how compact objects and BHs “look” like when illuminated by accretion disks or stars, thus defining their socalled shadow, see Sect. 2.2 below.
For any regular body, the metric functions f, g are well behaved at the center, never change sign and asymptote to unity at large distances. Thus, the effective potential \(V_\mathrm{geo}\) is negative at large distances, vanishes with zero derivative at the light ring, and is positive close to the center of the object. This implies that there must be a second light ring in the spacetime, and that it is stable (Cardoso et al. 2014; Macedo et al. 2013a; Cunha et al. 2017a). Inside this region, there is stable timelike circular motion everywhere.^{3}
2.1.5 Photon spheres

Compact object: if it features an ISCO, or in other words if its surface satisfies \(r_0<6M\) (\(\epsilon <2\)). Accretion disks around compact objects of the same mass should have similar characteristics;

Ultracompact object (UCO) (Iyer et al. 1985): a compact object that features a photon sphere, \(r_0<3M\) (\(\epsilon <1/2\)). For these objects, the phenomenology related to the photon sphere might be very similar to that of a BH;

Cleanphoton sphere object (ClePhO): an ultracompact object which satisfies condition (7) and therefore has a “clean” photon sphere, \(r_0<2.038M\) (\(\epsilon \lesssim 0.019\)). The earlytime dynamics of ClePhOs is expected to be the same as that of BHs. At late times, ClePhOs should display unique signatures of their surface.
2.2 Escape trajectories and shadows
An isolated BH would appear truly as a “hole” in the sky, since we observe objects by receiving the light they either emit or reflect. The boundary of this hole, i.e., the “silhouette” of a BH, is called the shadow and is actually larger than the BH horizon and intimately related with the existence of a photon sphere.
2.3 The role of the spin
While the overall picture drawn in the previous sections is valid also for rotating objects, angular momentum introduces qualitatively new features. Spin breaks spherical symmetry, introduces frame dragging, and breaks the degeneracy between co and counterrotating orbits. We focus here on two properties related to the spin which are important for the phenomenology of ECOs, namely the existence of an ergoregion and the multipolar structure of compact spinning bodies.
2.3.1 Ergoregion
An infiniteredshift surface outside a horizon is called an ergosurface and is the boundary of the socalled ergoregion. In a stationary spacetime, this boundary is defined by the roots of \(g_{tt}=0\). Since the Killing vector \(\xi ^{\mu }=(1,0,0,0)\) becomes spacelike in the ergoregion, \(\xi ^{\mu }\xi ^{\mu }g_{\mu \nu }=g_{tt}>0\), the ergosurface is also the static limit: an observer within the ergoregion cannot stay still with respect to distant stars; the observer is forced to corotate with the spacetime due to strong framedragging effects. Owing to this property, negativeenergy (i.e., bound) states are possible within the ergoregion. This is the chief property that allows for energy and angular momentum extraction from a BH through various mechanisms, e.g., the Penrose’s process, superradiant scattering, the Blandford–Znajek mechanism, etc. (Brito et al. 2015b). An ergoregion necessarily exists in the spacetime of a stationary and axisymmetric BH and the ergosurface must lay outside the horizon or coincide with it (Brito et al. 2015b). On the other hand, a spacetime with an ergoregion but without an event horizon is linearly unstable (see Sect. 4.3).
2.3.2 Multipolar structure
In other words, the deviations of the multipole moments from their corresponding Kerr value must die sufficiently fast as the compactness of the object approaches that of a BH, or otherwise the curvature at the surface will grow and the perturbative regime breaks down (Raposo et al. 2019). The precise way in which the multipoles die depends on whether they are induced by spin or by other moments.
3 ECO taxonomy: from DM to quantum gravity
Catalogue of some proposed horizonless compact objects
Model  Formation  Stability  EM signatures  GWs 

Fluid stars  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \) 
Shapiro and Teukolsky (1983)  Iyer et al. (1985), Kokkotas and Schmidt (1999), Cardoso et al. (2014), Saida et al. (2016), Stuchlík et al. (2017), Völkel and Kokkotas (2017a, b)  Kokkotas and Schmidt (1999), Ferrari and Kokkotas (2000), Cardoso et al. (2014), Völkel and Kokkotas (2017a)  
Anisotropic stars  ✗  \(\checkmark \)  \(\checkmark \)  \(\checkmark \) 
Raposo et al. (2018), Dev and Gleiser (2003), Doneva and Yazadjiev (2012)  
Boson stars and oscillatons  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \) 
Seidel and Suen (1991, 1994), Okawa et al. (2014), Brito et al. (2016b), Liebling and Palenzuela (2012)  Gleiser and Watkins (1989), Lee and Pang (1989), Honda and Choptuik (2002), Cardoso et al. (2008a), Brito et al. (2016a), Macedo et al. (2013a)  Vincent et al. (2016), Cao et al. (2016), Shen et al. (2016)  Palenzuela et al. (2008), Kesden et al. (2005), Choptuik and Pretorius (2010), Macedo et al. (2013b), Cardoso et al. (2016b, 2017), Sennett et al. (2017), Maselli et al. (2018b)  
Gravastars  ✗  \(\checkmark \)  \(\checkmark \)  \(\sim \) 
Sakai et al. (2014), Uchikata and Yoshida (2016), Uchikata et al. (2016)  Chirenti and Rezzolla (2007), Pani et al. (2010b, c, 2009), Chirenti and Rezzolla (2016), Cardoso et al. (2016a, b, 2017), Uchikata et al. (2016), Maselli et al. (2018b), Völkel and Kokkotas (2017a, b)  
AdS bubbles  ✗  \(\checkmark \)  \(\sim \)  ✗ 
Danielsson et al. (2017)  Danielsson et al. (2017)  
Wormholes  ✗  \(\checkmark \)  \(\checkmark \)  \(\sim \) 
Gonzalez et al. (2009a, b), Bronnikov et al. (2012), Cuyubamba et al. (2018)  Nedkova et al. (2013), Ohgami and Sakai (2015), Abdujabbarov et al. (2016), Zhou et al. (2016)  
Fuzzballs  ✗  ✗  ✗  \(\sim \) 
But see Cardoso et al. (2006), Chowdhury and Mathur (2008), Eperon et al. (2016), Eperon (2017)  
Superspinars  ✗  \(\checkmark \)  ✗  \(\sim \) 
But see Patil et al. (2016)  
2–2 holes  ✗  ✗  ✗  \(\sim \) 
But see Holdom and Ren (2017)  But see Holdom and Ren (2017)  
Collapsed polymers  ✗  \(\checkmark \)  ✗  \(\sim \) 
Brustein and Medved (2019)  Brustein et al. (2017b)  
Quantum bounces/dark stars  ✗  ✗  ✗  \(\sim \) 
Barceló et al. (2017)  
Compact quantum objects\(^{*}\)  ✗  ✗  ✗  \(\checkmark \) 
Giddings et al. (2019)  
Firewalls\(^{*}\)  ✗  ✗  ✗  \(\sim \) 
3.1 A compass to navigate the ECO atlas: Buchdahl’s theorem
 1.
GR is the correct theory of gravity;
 2.
The solution is spherically symmetric;
 3.
Matter is described by a single, perfect fluid;
 4.
The fluid is either isotropic or mildly anisotropic, in the sense that the tangential pressure is smaller than the radial one, \(P_r\gtrsim P_t\);
 5.
The radial pressure and energy density are nonnegative, \(P_r\ge 0\), \(\rho \ge 0\).
 6.
The energy density decreases as one moves outwards, \(\rho '(r)<0\).
3.2 Selfgravitating fundamental fields
Selfgravitating solutions for the theory above are broadly referred to as boson stars, and can be generalized through the inclusion of nonlinear selfinteractions (Kaup 1968; Ruffini and Bonazzola 1969; Khlopov et al. 1985; Seidel and Suen 1991; Guth et al. 2015; Brito et al. 2016a; Minamitsuji 2018) (see Jetzer 1992; Schunck and Mielke 2003; Liebling and Palenzuela 2012; Macedo et al. 2013a for reviews). If the scalar is complex, there are static, sphericallysymmetric geometries, while the field itself oscillates (Kaup 1968; Ruffini and Bonazzola 1969) (for reviews, see Jetzer 1992; Schunck and Mielke 2003; Liebling and Palenzuela 2012; Macedo et al. 2013a). Analogous solutions for complex massive vector fields were also shown to exist (Brito et al. 2016a). Recently, multioscillating boson stars which are not exactly static spacetimes were constructed, and these could represent intermediate states between static boson stars which underwent violent dynamical processes (Choptuik et al. 2019). On the other hand, real scalars give rise to longterm stable oscillating geometries, but with a nontrivial timedependent stressenergy tensor, called oscillatons (Seidel and Suen 1991). Both solutions arise naturally as the endstate of gravitational collapse (Seidel and Suen 1991; Garfinkle et al. 2003; Okawa et al. 2014), and both structures share similar features.
Scalar potential and maximum mass for some scalar bosonstar models
Adapted from Cardoso et al. (2017)
Model  Potential \(V(\varPhi ^2)\)  Maximum mass \(M_{\max }/M_{\odot }\) 

\(\mu ^2\varPhi ^2\)  \(8\left( \frac{10^{11}\,\mathrm{eV}}{m_S}\right) \)  
Massive (Colpi et al. 1986)  \(\mu ^2\varPhi ^2 + \frac{\alpha }{4}\varPhi ^4\)  \(5\,\sqrt{\alpha \hbar }\left( \frac{0.1\,\mathrm{GeV}}{m_S}\right) ^2\) 
Solitonic (Friedberg et al. 1987)  \(\mu ^2\varPhi ^2\left[ 1\frac{2\varPhi ^2}{\sigma _0^2}\right] ^2\)  \(5\left[ \frac{10^{12}}{\sigma _0}\right] ^2\left( \frac{500\,\mathrm{GeV}}{m_S}\right) \) 
The simplest boson stars are moderately compact in the nonspinning case (Macedo et al. 2013a; Brito et al. 2016a; Grandclément 2017). Their mass–radius relation is shown in Fig. 5. Once spin (Grandclément 2017) or nonlinear interactions (Colpi et al. 1986; Macedo et al. 2013a; Friedberg et al. 1987) are added, boson star spacetimes can have light rings and ergoregions. The stressenergy tensor of a selfinteracting bosonic field contains anisotropies, which in principle allow to evade naturally Buchdahl’s theorem. However, there are no bosonstar solutions which evade the Buchdahl’s bound: in the static case, the most compact configuration has \(r_0\approx 2.869M\) (\(\epsilon \approx 0.44\)) (Kesden et al. 2005).
There seem to be no studies on the classification of such configurations (there are solutions known to display photon spheres, but it is unknown whether they can be as compact as ClePhOs) (Kesden et al. 2005; Grandclément 2017).
3.3 Perfect fluids
The construction of boson stars is largely facilitated by their statistics, which allow for a large number of bosons to occupy the same level. Due to Pauli’s exclusion principle, a similar construction for fermions is therefore more challenging, and approximate strategies have been devised (Ruffini and Bonazzola 1969; Shapiro and Teukolsky 1983). In most applications, such fundamental description is substituted by an effective equation of state, usually of polytropic type, which renders the corresponding Einstein equations much easier to solve (Shapiro and Teukolsky 1983).
When the stresses are assumed to be isotropic, static spheres in GR made of ordinary fluid satisfy the Buchdahl limit on their compactness, \(2M/r_0<8/9\) (Buchdahl 1959); strictly speaking, they would not qualify as a ClePhO. However, GWs couple very weakly to ordinary matter and can travel unimpeded right down to the center of stars. Close to the Buchdahl limit, the travel time is extremely large, \(\tau \sim \epsilon ^{1/2}M\), and in practice such objects would behave as ClePhOs (Pani and Ferrari 2018). In addition, polytrope stars with a light ring (sometimes referred to as ultracompact stars) always have superluminal sound speed (Saida et al. 2016). Neutron stars—the only object in our list for which there is overwhelming evidence—are not expected to have light rings nor behave as ClePhOs for currently accepted equations of state (Iyer et al. 1985). The mass–radius relation for a standard neutron star is shown in Fig. 5.
Some fermion stars, such as neutron stars, live in DMrich environments. Thus, DM can be captured by the star due to gravitational deflection and a nonvanishing crosssection for collision with the star material (Press and Spergel 1985; Gould et al. 1990; Goldman and Nussinov 1989; Bertone and Fairbairn 2008; Goldman and Nussinov 1989). The DM material eventually thermalizes with the star, producing a composite compact object. Compact solutions made of both a perfect fluid and a massive complex (Henriques et al. 1989, 1990; Lopes and Henriques 1992; Henriques and Mendes 2005; Sakamoto and Shiraishi 1998; Pisano and Tomazelli 1996) or real scalar or vector field (Brito et al. 2015a, 2016b) were built, and model the effect of bosonic DM accretion by compact stars. Complementary to these studies, accretion of fermionic DM has also been considered, by modeling the DM core with a perfect fluid and constructing a physically motivated equation of state (Leung et al. 2011, 2013; Tolos and SchaffnerBielich 2015). The compactness of such stars is similar to that of the host neutron stars, and does not seem to exceed the Buchdahl limit.
3.4 Anisotropic stars
The Buchdahl limit can be circumvented when the object is subjected to large anisotropic stresses (Andreasson 2008). These might arise in a variety of contexts: at high densities (Kippenhahn et al. 2012; Ruderman 1972; Canuto and Chitre 1974), when EM or fermionic fields play a role, or in pion condensed phase configurations in neutron stars (Sawyer and Scalapino 1973), superfluidity (Carter and Langlois 1998), solid cores (Kippenhahn et al. 2012), etc. In fact, anisotropy is common and even a simple soap bubble support anisotropic stresses (Guven and O’Murchadha 1999). Anisotropic stars were studied in GR, mostly at the level of static spherically symmetric solutions (Bowers and Liang 1974; Letelier 1980; Bayin 1982; Dev and Gleiser 2002, 2003; Mak and Harko 2003; Herrera et al. 2004; Andreasson 2008; Hillebrandt and Steinmetz 1976; Doneva and Yazadjiev 2012; Silva et al. 2015; Yagi and Yunes 2015b, c, 2016). These studies are not covariant, which precludes a full stability analysis or nonlinear evolution of such spacetimes. Progress on this front has been achieved recently (Carloni and Vernieri 2018; Isayev 2018; Raposo et al. 2018).
The compactness of very anisotropic stars may be arbitrarily close to that of a BH; compact configurations can exceed the Buchdahl limit, and some can be classified as ClePhOs. In some of these models, compact stars exist across a wide range of masses, evading one of the outstanding issues with BH mimickers, i.e., that most approach the BH compactness in a very limited range of masses, thus being unable to describe both stellarmass and supermassive BH candidates across several orders of magnitude in mass (Raposo et al. 2018). Such property of BHs in GR, visible in Fig. 5, is a consequence of the scalefree character of the vacuum field equations. It is extremely challenging to reproduce once a scale is present, as expected for material bodies. Fig. 5 summarizes the mass–radius relation for fluid stars.
3.5 Quasiblack holes
An interesting class of families of BHmimickers, the quasiblack holes, consist on extremal (charged and/or spinning) regular spacetimes. These objects can be thought of as stars, on the verge of becoming BHs (Lemos and Weinberg 2004; Lemos and Zaslavskii 2008).
3.6 Wormholes
Boson and fermion stars discussed above arise from a simple theory, with relatively simple equations of motion, and have clear dynamics. Their formation mechanism is embodied in the field equations and requires no special initial data. On the other hand, the objects listed below are, for the most part, generic constructions with a welldefined theoretical motivation, but for which the formation mechanisms are not well understood.
The above constructions show that wormholes can be constructed to have any arbitrary mass and compactness. The procedure is oblivious to the formation mechanism, it is unclear if these objects can form without carefully tuned initial conditions, nor if they are stable. Wormholes in more generic gravity theories have been constructed, some of which can potentially be traversable (Shaikh and Kar 2016; Chianese et al. 2017; Hohmann et al. 2018; Shaikh 2018; Khaybullina and Tuleganova 2019). In such theories, energy conditions might be satisfied (Kanti et al. 2011). Generically however, wormholes are linearly unstable (Gonzalez et al. 2009a, b; Bronnikov et al. 2012; Cuyubamba et al. 2018).
3.7 Dark stars
Quantum field theory around BHs or around dynamic horizonless objects gives rise to phenomena such as particle creation. Hawking evaporation of astrophysical BHs, and corresponding backreaction on the geometry is negligible (Birrell and Davies 1984). Quantum effects on collapsing horizonless geometries (and the possibility of halting collapse to BHs altogether) are less clear (Visser et al. 2009; Zeng 2017; Chen et al. 2018; Berthiere et al. 2018; Buoninfante and Mazumdar 2019; Terno 2019; Malafarina 2017). There are arguments that semiclassical effects might suffice to halt collapse and to produce dark stars, even for macroscopic configurations (Barceló et al. 2009, 2016; Kawai et al. 2013; Baccetti et al. 2017, 2018a, b; CarballoRubio 2018), but see Chen et al. (2018) for counterarguments. For certain conformal fields, it was shown that a possible endstate are precisely wormholes of the form (23). Alternative proposals, made to solve the information paradox, argue that dark stars could indeed arise, but as a “massive remnant” end state of BH evaporation (Giddings 1992; Unruh and Wald 2017).
3.8 Gravastars
Similar ideas that led to the proposal of “dark stars” were also in the genesis of a slightly different object, “gravitationalvacuum stars” or gravastars (Mazur and Mottola 2001, 2004). These are configurations supported by a negative pressure, which might arise as an hydrodynamical description of oneloop QFT effects in curved spacetime, so they 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) and because the pressure is negative [and thus violates some of the energy conditions (Mazur and Mottola 2015)]. Gravastars have been recently generalized to include antide Sitter cores, in what was termed AdS bubbles, and which may allow for holographic descriptions (Danielsson et al. 2017; Danielsson and Giri 2018). Gravastars are a very broad class of objects, and can have arbitrary compactness, depending on how one models the supporting pressure. The original gravastar model was a fivelayer construction, with an interior de Sitter core, a thin shell connecting it to a perfectfluid region, and another thinshell connecting it to the external Schwarzschild patch. A simpler construction that features all the main ingredients of the original gravastar proposal is the thinshell gravastar (Visser and Wiltshire 2004), in which a de Sitter core is connected to a Schwarzschild exterior through a thin shell of perfectfluid matter. Gravastars can also be obtained as the BHlimit of constantdensity stars, past the Buchdahl limit (Mazur and Mottola 2015; Posada and Chirenti 2019). It is interesting that such stars were found to be dynamically stable in this regime (Posada and Chirenti 2019). It has been conjectured that gravastars are a natural outcome of the inflationary universe (Wang et al. 2018b), or arising naturally within the gaugegravity duality (Danielsson et al. 2017; Danielsson and Giri 2018).
3.9 Fuzzballs and collapsed polymers
So far, quantum effects were dealt with at a semiclassical level only. A proper theory of quantum gravity needs to be able to solve some of the inherent problems in BH physics, such as the lack of unitarity in BH evaporation or the origin and nature of the huge Bekenstein–Hawking entropy \(S=k_\mathrm{B}c^3A/(4\hbar G)\) (\(k_\mathrm{B}\) is Boltzmann’s constant and A is the BH area). In other words, what is the statisticalmechanical account of BH entropy in terms of some microscopic degrees of freedom? String theory is able to provide a partial answer to some of these questions. In particular, for certain (nearly) supersymmetric BHs, the Bekenstein–Hawking entropy, as computed in the stronglycoupled supergravity description, can be reproduced in a weaklycoupled Dbrane description as the degeneracy of the relevant microstates (Strominger and Vafa 1996; Peet 1998; Das and Mathur 2000; David et al. 2002; Bena and Warner 2008).
Somewhat surprisingly, the geometric description of individual microstates seems to be regular and horizonless (Myers 1997; Mathur 2005; Bena and Warner 2008, 2013; Balasubramanian et al. 2008). This led to the “fuzzball” description of classical BH geometries, where a BH is dual to an ensemble of such microstates. In this picture, the BH geometry emerges in a coarsegrained description which “averages” over the large number of coherent superposition of microstates, producing an effective horizon at a radius where the individual microstates start to “differ appreciably” from one another (Lunin and Mathur 2002b, a). In this description, quantum gravity effects are not confined close to the BH singularity, rather the entire interior of the BH is “filled” by fluctuating geometries—hence this picture is often referred to as the “fuzzball” description of BHs.
Unfortunately, the construction of microstates corresponding to a fixed set of global charges has only been achieved in very special circumstances, either in higherdimensional or in non asymptoticallyflat spacetimes. Explicit regular, horizonless microstate geometries for asymptotically flat, fourdimensional spacetimes that could describe astrophysical bodies have not been constructed. Partly because of this, the properties of the geometries are generically unknown. These include the “softness” of the underlying microstates when interacting with GWs or light; the curvature radius or redshift of these geometries in their interior; the relevant lengthscale that indicates how far away from the Schwarzschild radius is the fuzziness relevant, etc.
A similar motivation led to the proposal of a very different BH interior in Brustein and Medved (2017) and Brustein et al. (2017a); the interior is described by an effective equation of state corresponding to a gas of highly excited strings close to the Hagedorn temperature. The behavior of such gas is similar to some polymers, and this was termed the “collapsed polymer” model for BH interiors. In both proposals, large macroscopic BHs are described by objects with a regular interior, and the classical horizon is absent. In these models, our parameter \(\epsilon \) is naturally of the order \(\sim {{\mathcal {O}}}(\ell _P/M)\in (10^{39},10^{46})\) for masses in the range \(M\in (10,10^8)M_\odot \).
3.10 “Naked singularities” and superspinars
Classical GR seems to be protected by Cosmic Censorship, in that evolutions leading to spacetime singularities also produce horizons cloaking them. Nevertheless, there is no generic proof that cosmic censorship is valid, and it is conceivable that it is a fragile, once extensions of GR are allowed. A particular impact of such violations was discussed in the context of the Kerr geometry describing spinning BHs. In GR, the angular momentum J of BHs is bounded from above by \(J\le GM^2/c\). In string theory however, such “Kerr bound” does not seem to play any fundamental role and could conceivably receive large corrections. It is thus possible that there are astrophysical objects where it is violated. Such objects were termed superspinars (Gimon and Hořava 2009), but it is part of a larger class of objects which would arise if singularities (in the classical theory of GR) would be visible. The full spacetime description of superspinars and other such similar objects is lacking: to avoid singularities and closedtimelike curves unknown quantum effects need to be invoked to create an effective surface somewhere in the spacetime. There are indications that strong GW bursts are an imprint of such objects (Harada et al. 2000), but a complete theory is necessary to understand any possible signature.
3.11 2–2 Holes and other geons
3.12 Firewalls, compact quantum objects and dirty BHs
Many 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 2017b, 2013; Giddings et al. 2019). However, the changes could also be drastic and involve “hard” structures localized close to the horizon such as firewalls and other compact quantum objects (Almheiri et al. 2013; Kaplan and Rajendran 2019; Giddings et al. 2019). Alternatively, a classical BH with modified dispersion relations for the graviton could effectively appear as having a hard surface (Zhang and Zhou 2018; Oshita and Afshordi 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 a compact horizonless object.
The zoo of compact objects is summarized in Fig. 7. In all these cases, both quantumgravity or microscopic corrections at the horizon scale select ClePhOs as wellmotivated alternatives to BHs. Despite a number of supporting arguments—some of which urgent and well founded—it is important to highlight that there is no horizonless ClePhO for which we know sufficiently well the physics at the moment.
4 Dynamics of compact objects
EM observations of compact bodies are typically performed in a context where spacetime fluctuations are irrelevant, either due to the long timescales involved or because the environment has a negligible backreaction on the body itself. For example, EM observations of accretion disks around a compact object can be interpreted using a stationary background geometry. Such geometry is a solution to the field equations describing the compact body while neglecting the accretion disk, the dynamics of which is governed by the gravitational pull of the central object and by internal forces. This approximation is adequate since the total amount of energy density around compact objects is but a small fraction of the object itself, and the induced changes in the geometry can be neglected (Barausse et al. 2014). In addition, the wavelength of EM waves of interest for Earthbased detectors is always much smaller that any lengthscale related to coherent motion of compact objects: light can be treated as a null particle following geodesics on a stationary background. Thus, the results of the previous sections suffice to discuss EM observations of compact objects, as done in Sect. 5 below.There is a crack in everything. That’s how the light gets in.
Leonard Cohen, Anthem (1992)
For GW astronomy, however, it is the spacetime fluctuations themselves that are relevant. A stationary geometry approximation would miss GW emission entirely. In addition, GWs generated by the coherent motion of sources have a wavelength of the order of the size of the system. Therefore, the geodesic approximation becomes inadequate (although it can still be used as a guide). Compact binaries are the preferred sources for GW detectors. Their GW signal is naturally divided in three stages, corresponding to the different cycles in the evolution driven by GW emission (Buonanno et al. 2007; Berti et al. 2007; Sperhake et al. 2013a): the inspiral stage, corresponding to large separations and well approximated by postNewtonian theory; the merger phase when the two objects coalesce and which can only be described accurately through numerical simulations; and finally, the ringdown phase when the merger endproduct relaxes to a stationary, equilibrium solution of the field equations (Sperhake et al. 2013a; Berti et al. 2009; Blanchet 2014). All three stages provide independent, unique tests of gravity and of compact GW sources. Overall, GWs are almost by definition attached to highly dynamical spacetimes, such as the coalescence and merger of compact objects. We turn now to that problem.
4.1 Quasinormal modes
Consider first an isolated compact object described by a stationary spacetime. Again, we start with the sphericallysymmetric case and for simplicity. Birkhoff’s theorem then implies that the exterior geometry is Schwarzschild. Focus on a small disturbance to such static spacetime, which could describe a small moving mass (a planet, a star, etc), or the latestage in the life of a coalescing binary (in which case the disturbed “isolated compact object” is to be understood as the final state of the coalescence).
The reverseengineering of the process, i.e., a reconstruction of the scattering potential V from a mode measurement was proposed in Völkel and Kokkotas (2017b, 2018), Völkel (2018). The impact of measurement error on such reconstruction is yet to be assessed.
Clearly, a perfectly reflecting surface is an idealization. In certain models, only lowfrequency waves are reflected, whereas higherfrequency waves probe the internal structure of the specific object (Saravani et al. 2015; Mathur and Turton 2014). In general, the location of the effective surface and its properties (e.g., its reflectivity) can depend on the energy scale of the process under consideration. Partial absorption is particularly important in the case of spinning objects, as discussed in Sect. 4.3.
4.2 Gravitationalwave echoes
4.2.1 Quasinormal modes, photon spheres, and echoes
For a BH, the excitation of the spacetime modes happens at the photon sphere (Davis et al. 1971, 1972; Ferrari and Mashhoon 1984). Such waves travel outwards to possible observers or down the event horizon. The structure of GW signals at late times is therefore expected to be relatively simple. This is shown in Fig. 9, for the scattering of a Gaussian pulse of axial quadrupolar modes off a BH. The pulse crosses the photon sphere, and excites its modes. The ringdown signal, a fraction of which travels to outside observers, is to a very good level described by the lowest QNMs. The other fraction of the signal generated at the photon sphere travels downwards and into the horizon. It dies off and has no effect on observables at large distances.
We end this discussion by highlighting that GW echoes are a feature of very compact ECOs, but also arise in many other contexts: classical BHs surrounded by a “hardstructure” close to the horizon (Barausse et al. 2014; Kaplan and Rajendran 2019; Ramos and Barausse 2019), or far from it (Barausse et al. 2014; Konoplya et al. 2019; Lin et al. 2019), or embedded in a theory that effectively makes the graviton see a hard wall there (Zhang and Zhou 2018; Oshita and Afshordi 2019) will respond to incoming GWs producing echoes. Finally, as we described earlier, even classical but very compact neutron or strange quark stars may be prone to exciting echoes (Ferrari and Kokkotas 2000; Raposo et al. 2018; Pani and Ferrari 2018; Mannarelli and Tonelli 2018). A simple picture of how echoes arise in a simple twobarrier system is provided in Mirbabayi (2018).
4.2.2 A blackhole representation and the transfer function
The derivation above assumes a static ECO spacetime, and a potential which vanishes at its surface. An extension of the procedure above to include both a more general potential and spin is worked out in Conklin et al. (2018). Such a “transferfunction” representation of echoes was embedded into an effectivefieldtheory scheme (Burgess et al. 2018), showing that linear “Robin” boundary conditions at \(r=r_0\) dominate at low energies. In this method the (frequency dependent) reflection coefficient and the surface location can be obtained in terms of a single lowenergy effective coupling. Recently, another model for the frequencydependent reflectivity of quantum BHs has been proposed in Oshita et al. (2019).
The previous description of echoes and of the full signal is reasonable and describes all the known numerical results. At the technical level, more sophisticated tools are required to understand the signal: the intermediatetime response is dominated by the BH QNMs, which are not part of the QNM spectrum of an ECO (Cardoso et al. 2016a; Barausse et al. 2014; Khanna and Price 2017). While this fact is easy to understand in the time domain due to causality [in terms of time needed for the perturbation to probe the boundaries (Cardoso et al. 2016a)], it is not at all obvious in the frequency domain. Indeed, the poles of the ECO Green’s functions in the complex frequency plane are different from the BH QNMs. The latetime signal is dominated by the dominant ECO poles, whereas the prompt ringdown is governed by the by the dominant QNMs of the corresponding BH spacetime.
4.2.3 A Dysonseries representation
The previous analysis showed two important aspects of the latetime behavior of very compact objects: (i) that it can be expressed in terms of the corresponding BH response if one uses a transfer function \({{\mathcal {K}}}\); (ii) that the signal after the main burst and precursor are a sequence of echoes, trapped between the object and the (exterior) peak of the potential.
Although complexlooking, Eq. (58) has a special significance, giving the amplitude of the (Fouriertransformed) nth echo of the initial burst (Correia and Cardoso 2018). When \({{\mathcal {R}}}=0\) then \(\psi =\psi _o\), the open system waveform. There are no echoes as expected. When \({{\mathcal {R}}} \ne 0\) there are additional (infinite) Dysonseries terms. The series is expected to converge, (i.e., the contribution of \(\psi _n\) becomes smaller for large n), because of two features of Eq. (58): first, if \({{\mathcal {R}}} < 1\), \({{\mathcal {R}}}^n\) contributes to damp the contribution of largen terms. Moreover, the Dyson series starts at \(k=n\). Since \(g_o\) and \(g_r\) are of the same order of magnitude, it is natural to expect that the series starting ahead (with less terms) has a smaller magnitude and contributes less to \(\psi \) than the ones preceding them. This can be verified numerically.
Finally, an important outcome of this analysis is that echoes that arise later have a smaller frequency component than the first ones: the Dyson series is basically an expansion on powers of \(V/\omega ^2\); thus by starting at \(k=n\), \(\psi _n\) skips the high frequency contribution to the series until that term. This is easily explained on physical grounds: high frequency components “leak” easily from the cavity (the cavity being formed by the ultracompact object and the potential barrier). Lower frequency components are harder to tunnel out. Thus, at late times only low frequencies are present.
Recently, this approach was extended to ECOs modeled with a multiplebarrier filter near the surface, showing that the latetime ringdown exhibits mixing of echoes (Li and Piao 2019).
4.2.4 Echo modeling
The GW signal composed of echoes is a transient signal, which captures the transition between the photosphere ringdown. GW echoes are not well described by the QNMs of the ECO, which dominate the response only at very late times. Thus, a proper understanding of the signal in the “echoing stage” requires the full understanding of the theory and ensuing dynamics of the object. Unfortunately, as we discussed, there is a plethora of proposed candidate theories and objects, with unknown properties. Thus, the GW signal is known accurately for only a handful of special setups, and under very specific assumptions on the matter content (Cardoso et al. 2016b; Price and Khanna 2017). For this reason the echo signal is very rich, and different approaches have been recently developed to model it.
A more phenomenological timedomain template, less anchored to the physics of echoes was proposed in Maselli et al. (2017b), using a superposition of sineGaussians with several free parameters. This template is very generic, but on the other hand suffers from a proliferation of parameters, which should not be in fact independent.
Note that the above two templates were directly modelled for spinning ECOs, since their underlying ingredients are very similar to the nonspinning case.
The above template depends only on two physical inputs: the reflection coefficient \({{\mathcal {R}}}\) (or \({{\mathcal {R}}}'\))—which can be in general a complex function of the frequency—and the width of the cavity d, which is directly related to the compactness of the object. For a given model of given compactness, \({{\mathcal {R}}}(\omega )\) and d are fixed and the mode does not contain other free parameters. For example, the damping factor introduced in the previous template can be written in terms of \({{\mathcal {R}}}\) and the reflection coefficient of the BH potential, \({{\mathcal {R}}}_\mathrm{BH}\) [see Eq. (41)] as \(\eta ={{\mathcal {R}}}{{\mathcal {R}}}_\mathrm{BH}\) (Testa and Pani 2018). Since \({{\mathcal {R}}}_\mathrm{BH}\) is frequency dependent so must be \(\eta \), even in the case of perfect reflectivity (\({{\mathcal {R}}}=1\)). The timedomain waveform contains all the features previously discussed for the echo signal, in particular amplitude and frequency modulation and phase inversion of each echo relative to the previous one for certain boundary conditions (Testa and Pani 2018).
Note that practically all generic modeling of echoes which do not start from a firstprinciples calculation of the GW signal assume equalspacing for the echoes. This seems certainly a good approximation for stationary geometries, but will fail for collapsing objects for example (Wang et al. 2018a, 2019b). Furthermore, if the ECO reflective properties are modeled as a multiplebarrier filter—as in certain scenarios motivated by BH area quantization (Bekenstein and Mukhanov 1995; Cardoso et al. 2019a)—mixing of echoes occurs (Li and Piao 2019).
Searches with Fourier windows. A similar but independent search technique was devised in Conklin et al. (2018), and uses the fact that echoes should pile up power at very specific frequencies (those implied by the cavity delay time) which are nearly equally spaced [cf. Eq. (30)] (but see Wang et al. 2018a, 2019b). Thus, the technique consists on producing a “combing” window in Fourier space, able to match (maximizing over extrinsic parameters) the frequencies of the cavity. The specific shape of the toothcomb was found not to be determinant, as long as it is able to capture the power in the resonant mode. An extension of this strategy is discussed in Conklin and Holdom (2019).
4.2.5 Echoes: a historical perspective
There exist in the literature examples of works where the main gist of the idea behind echoes is present, albeit only for specific examples and without the full appreciation of the role of the light ring. Already in 1978, the study of the instability of spinning horizonless compact objects (see Sect. 4.4.1) led to the understanding that the driving mechanism were the recurrent reflections of quasibound states within the ergoregion (Vilenkin 1978). Mutatis mutandis, these modes produce the echoes discussed above. Indeed, a Penrose diagram similar to that of Fig. 10 was already shown in Vilenkin (1978) (without a discussion of the GW emission slowly leaking from the potential barrier).
Probably the first example of echoes dates back to 1995, with the study of axial GWs emitted by perturbed (through Gaussian wavepackets) constantdensity compact stars (Kokkotas and Schmidt 1999; Kokkotas 1995). This was later extended in the following years to include the scattering of point particles (Tominaga et al. 1999, 2001; Andrade and Price 1999; Ferrari and Kokkotas 2000; Andrade 2001). In all these studies the GW signal shows a series of clear echoes after the main burst of radiation, which were identified as the excitation of quasitrapped modes of ultracompact stars (Chandrasekhar and Ferrari 1991). As we explained in Sect. 4.2.1, the true trappedmode behavior only sets in at much later times, and the correct description is that of echoes. The original references did not attempt to explain the pattern in the signal, but in hindsight these results fit perfectly in the description we provided above: axial modes do not couple to the fluid [nor polar modes, which couple only very weakly (Andersson et al. 1996)] and travel free to the geometrical center of the star, which is therefore the effective surface in this particular case. The time delay of the echoes in Fig. 1 of Ferrari and Kokkotas (2000) is very well described by the GW’s roundtrip time to the center, \(\tau _\mathrm{echo}\sim \frac{27\pi }{8}\epsilon ^{1/2}M\), where \(r_0=\frac{9}{4}M(1+\epsilon )\) is the radius of the star (Pani and Ferrari 2018) and \(r_0=\frac{9}{4}M\) is the Buchdahl’s limit (Buchdahl 1959).
Shortly after, but in a very different context, the overall picture of echoes would emerge in the fuzzball program. In Lunin and Mathur (2002a) and Giusto et al. (2005), the authors express the reflection coefficient of lowenergy scalars as a sum over the number of bounces at the “throat” of these geometries. The idea behind is similar to the expansion (48), and results in a series of “echoes” (Giusto et al. 2005). A quantitative calculation of the response, as well as the role of the light ring, were left undone.
In the context of wormhole physics [particularly the geometry (23)], the main features of the response of ClePhOs were identified in Damour and Solodukhin (2007). The postmerger train of echoes of the main burst was not addressed quantitatively.
Finally, in yet a different context, Barausse et al. (2014) discussed the latetime response of “dirty” BHs, modeling environmental effects (such as stars, gas etc) and showed that there are “secondary pulses” of radiation in the latetime response. These secondary pulses are just the echoes of a “mirrored” version of our original problem, where now it is the far region responsible for extra features in the effective potential, and hence the cavity is composed of the photosphere and the far region where matter is located.
4.3 The role of the spin
The previous sections dealt with static background spacetimes. Rotation introduces qualitatively new effects. For a Kerr BH, spinning with horizon angular velocity \(\varOmega \) along the azimuthal angle \(\phi \), perturbations are well understood using the Newman–Penrose formalism and a decomposition in socalled spinweighted spheroidal harmonics (Teukolsky 1972, 1973; Chandrasekhar 1983). It is still possible to reduce the problem to a PDE similar to Eq. (25), but the effective potential is frequencydependent; breaks explicitly the azimuthal symmetry, i.e., it depends also on the azimuthal number m (fluctuations depend on the azimuthal angle as \(\sim e^{im\phi }\)); is generically complex, although there exist transformations of the perturbation variables that make it real (Detweiler 1977; Maggio et al. 2019); In particular, the explicit dependence on m gives rise to a Zeeman splitting of the QNMs as functions of the spin, whereas the frequency dependence gives rise to the interesting phenomenon of superradiance whereby modes with frequency \(\omega \) are amplified when \(\omega (\omega m\varOmega )<0\). In particular, the potential is such that \(V(r\rightarrow \infty )=\omega ^2\), whereas \(V(r\rightarrow r_+)=k^2\), with \(k=\omega m\varOmega \). The relation between null geodesics and BH QNMs in the eikonal limit is more involved but conceptually similar to the static case (Yang et al. 2012).
Further features arise if the object under consideration is not a Kerr BH. In general, the vacuum region outside a spinning object is not described by the Kerr geometry. However, when \(\epsilon \rightarrow 0\) any deviation from the multipolar structure of a Kerr BH must die off sufficiently fast (Raposo et al. 2019; Glampedakis and Pappas 2018a) (see Sect. 2.3.2). Explicit examples are given in Pani (2015), Uchikata and Yoshida (2016), Uchikata et al. (2016), Yagi and Yunes (2015b, c) and Posada (2017). Therefore, if one is interested in the very small \(\epsilon \) limit, one can study a Kerrlike ECO (Cardoso et al. 2008b; Abedi et al. 2017a; Maggio et al. 2017; Nakano et al. 2017), i.e., a geometry described by the Kerr metric when \(r>r_0=r_+(1+\epsilon )\) and with some membrane with modeldependent reflective properties at \(r=r_0\). Beyond the \(\epsilon \rightarrow 0\) limit, ECOs may have arbitrary multipole moments and even break equatorial symmetry (Raposo et al. 2019; Papadopoulos and Kokkotas 2018). In such cases, it may not be possible to separate variables (Glampedakis and Pappas 2018a, b; Allahyari et al. 2019; Pappas and Glampedakis 2018) and the results below may not hold.
4.3.1 QNMs of spinning Kerrlike ECOs
4.3.2 Echoes from spinning ECOs
The transfer function of Eq. (46) can be generalized to spinning “Kerrlike” ECO subjected to boundary condition (71) near the surface. The final result reads formally the same, although \({{\mathcal {T}}}_\mathrm{BH}\) and \({{\mathcal {R}}}_\mathrm{BH}\) are defined in terms of the amplitudes of the waves scattered off the Kerr effective potential (Conklin et al. 2018; Testa et al. 2019). Echoes from Kerrlike wormholes [i.e., a spinning extension of the Damour–Solodukhin solution (Damour and Solodukhin 2007)] have been studied in Bueno et al. (2018). Phenomenological templates for echoes from Kerrlike objects were constructed in Abedi et al. (2017a), Nakano et al. (2017), Maselli et al. (2017b), Wang and Afshordi (2018) and are discussed in Sect. 4.2.4.
4.4 The stability problem
Appealing solutions are only realistic if they form and remain as longterm stable solutions of the theory. In other words, solutions have to be stable when slightly perturbed or they would not be observed (or they would not even form in the first place). There are strong indications that the exterior Kerr spacetime is stable, although a rigorous proof is still missing (Dafermos and Rodnianski 2013). On the other hand, some—and possibly most of—horizonless compact solutions are linearly or nonlinearly unstable.There is nothing stable in the world; uproar’s your only music.
John Keats, Letter to George and Thomas Keats, Jan 13 (1818)
Some studies of linearized fluctuations of ultracompact objects are given in Table 1. We will not discuss specific models, but we would like to highlight some general results.
4.4.1 The ergoregion instability
Several models of UCOs and ClePhOs are stable under radial perturbations (Iyer et al. 1985; Visser and Wiltshire 2004) (see Table 1). However, UCOs (and especially ClePhOs) can develop negativeenergy regions once spinning. In such a case, they develop a linear instability under nonradial perturbations, which is dubbed as ergoregion instability. Such instability affects any horizonless geometry with an ergoregion (Friedman 1978b; Kokkotas et al. 2004; Moschidis 2016; Cardoso et al. 2008a; Oliveira et al. 2014; Maggio et al. 2017; Vicente et al. 2018) and is deeply connected to superradiance (Brito et al. 2015b). The underlying mechanism is simple: a negativeenergy fluctuation in the ergoregion is forced to travel outwards; at large distances only positiveenergy states exist, and energy conservation implies that the initial disturbance gives rise to a positive fluctuation at infinity plus a larger (negativeenergy) fluctuation in the ergoregion. Repetition of the process leads to a cascading instability. The only way to prevent such cascade from occurring is by absorbing the negative energy states, which BHs do efficiently (and hence Kerr BHs are stable against massless fields), but perfectlyreflecting horizonless objects must then be unstable.
This instability was discovered by Friedman for ultracompact slowlyrotating stars with an ergoregion (Friedman 1978a), and later extended in Comins and Schutz (1978), Yoshida and Eriguchi (1996), Kokkotas et al. (2004). Application to Kerrlike horizonless objects started in Vilenkin (1978), whereas an analysis for gravastars, boson stars, and other objects was done in Cardoso et al. (2006, 2008a, b), Chirenti and Rezzolla (2008). More recently, Maggio et al. (2017, 2019) gave a detailed analysis of scalar, electromagnetic, and gravitational perturbations of a partiallyreflective Kerrlike ECO in the \(\epsilon \rightarrow 0\) limit.
A possible way to quench the instability is by absorbing the negativeenergy modes trapped within the ergoregion. Kerr BHs can absorb such modes efficiently and are indeed expected to be stable even if they have an ergoregion. Given its long timescales, it is possible that the instability can be efficiently quenched by some dissipation mechanism of nongravitational nature, although this effect would be modeldependent (Maggio et al. 2017, 2019). Unfortunately, the effect of viscosity in ECOs is practically unknown (Cardoso et al. 2014; Guo et al. 2018), and so are the timescales involved in putative dissipation mechanisms that might quench this instability. It is also possible that, when spinning, a partiallyabsorbing object can support quasitrapped superradiant modes with \(\omega _R<m\varOmega \), which might lead to an instability similar to that of massive bosonic fields around Kerr BHs (Brito et al. 2015b).
Finally, there are indications that instabilities of UCOs are merely the equivalent of Hawking radiation for these geometries, and that therefore there might be a smooth transition in the emission properties when approaching the BH limit (Chowdhury and Mathur 2008; Damour and Solodukhin 2007).
4.4.2 Nonlinear instabilities I: longlived modes and their backreaction
Linearized gravitational fluctuations of any nonspinning UCO are extremely longlived and decay no faster than logarithmically (Keir 2016; Cardoso et al. 2014; Eperon et al. 2016; Eperon 2017). Indeed, such perturbations can be again understood in terms of modes quasitrapped within the potential barrier shown in Fig. 8: they require a photon sphere but are absent in the BH case [hence the photon sphere is sometimes referred to as “loosely trapped” or “transversely trapping” surface (Shiromizu et al. 2017; Yoshino et al. 2017)]. For a ClePhO, these modes are very well approximated by Eqs. (30)–(31) in the static case and by their aforementioned extension in the spinning case. The long damping time of these modes has led to the conjecture that any UCO is nonlinearly unstable and may evolve through a Dyson–Chandrasekhar–Fermi type of mechanism (Keir 2016; Cardoso et al. 2014). The endstate is unknown, and most likely depends on the equation of the state of the particular UCO: some objects may fragment and evolve past the UCO region into less compact configurations, via mass ejection, whereas other UCOs may be forced into gravitational collapse to BHs.
The above mechanism is supposed to be active for any spherically symmetric UCO, and also on spinning solutions. However, it is nonlinear in nature and not well understood so far. For example, there are indications that a putative nonlinear instability would occur on very long timescales only; a model problem predicts an exponential dependence on the size of the initial perturbation (John 1981).
4.4.3 Nonlinear instabilities II: causality, hoop conjecture, and BH formation
The above argument can be made more powerful, making full use of the hoop conjecture: take two ECOs and boost them to large enough energies. Since all energy gravitates and is part of the hoop, the final object must be inside its Schwarzschild radius, hence it must be a BH [indeed, at large enough center of mass energies, the structure of the colliding objects is irrelevant (Eardley and Giddings 2002; Choptuik and Pretorius 2010; Sperhake et al. 2013b)]. It is very challenging to bypass this argument at the classical level. Nevertheless, it is important to highlight a few points: (i) Most of the arguments for ECO formation (and existence) rely directly or indirectly on unknown quantum effects associated with horizon or singularity formation (Giddings 1992, 2011, 2012, 2017b; Mazur and Mottola 2004; Mathur 2005, 2008, 2009; Barceló et al. 2016, 2017). Thus, it is very likely that horizons may form classically but that such picture is blurred by quantum effects (on unknown timescales and due to unknown dynamics);^{5} (ii) even classically, the argument does not forbid the existence of ECOs, it merely forces their interaction at high energy to result in BH formation (indeed, the same argument can be applied if the two objects are neutron stars).
4.5 Binary systems
 (i)
the way they respond to their own gravitational field—i.e., on their own multipolar structure (Krishnendu et al. 2017, 2019; Kastha et al. 2018);
 (ii)
the way they respond when acted upon by the external gravitational field of their companion—through their tidal Love numbers (TLNs) (Poisson and Will 1953);
 (iii)
on the amount of radiation that they possibly absorb, i.e., on tidal heating (Hartle 1973; Hughes 2001).
4.5.1 Multipolar structure
4.5.2 Tidal heating
A spinning BH absorbs radiation of frequency \(\omega >m\varOmega \), but amplifies radiation of smaller frequency (Brito et al. 2015b). In this respect, BHs are dissipative systems which behave just like a Newtonian viscous fluid (Damour 1982; Poisson 2009; Cardoso and Pani 2013). Dissipation gives rise to various interesting effects in a binary system—such as tidal heating (Hartle 1973; Hughes 2001), tidal acceleration, and tidal locking, as in the Earth–Moon system, where dissipation is provided by the friction of the oceans with the crust.
Thus, it might be argued that an ECO binary can be distinguished from a BH binary, because \({\dot{E}}_H=0\) for the former. However, the trapping of radiation in ClePhOs can efficiently mimic the effect of a horizon (Maselli et al. 2018b). In order for absorption to affect the orbital motion, it is necessary that the time radiation takes to reach the companion, \(T_\mathrm{rad}\), be much longer than the radiationreaction time scale due to heating, \(T_\mathrm{RR}\simeq E/{\dot{E}}_H\), where \(E\simeq \frac{1}{2} M(M\varOmega _K)^{2/3}\) is the binding energy of the binary (assuming equal masses). For BHs, \(T_\mathrm{rad}\rightarrow \infty \) because of time dilation, so that the condition \(T_\mathrm{rad}\gg T_\mathrm{RR}\) is always satisfied. For ClePhOs, \(T_\mathrm{rad}\) is of the order of the GW echo delay time, Eq. (74), and therefore increases logarithmically as \(\epsilon \rightarrow 0\). Thus, an effective tidal heating might occur even in the absence of a horizon if the object is sufficient compact. The critical value of \(\epsilon \) increases strongly as a function of the spin. For orbital radii larger than the ISCO, the condition \(T_\mathrm{rad}\gg T_\mathrm{RR}\) requires \(\epsilon \ll 10^{88}\) for \(\chi \lesssim 0.8\), and therefore even Planck corrections at the horizon scale are not sufficient to mimic tidal heating. This is not necessarily true for highly spinning objects, for example \(T_\mathrm{rad}\gg T_\mathrm{RR}\) at the ISCO requires \(\epsilon \ll 10^{16}\) for \(\chi \approx 0.9\).
4.5.3 Tidal deformability and Love numbers
Finally, the nature of the inspiralling objects is also encoded in the way they respond when acted upon by the external gravitational field of their companion—through their tidal Love numbers (TLNs) (Poisson and Will 1953). An intriguing result in classical GR is that the TLNs of BHs are zero. This result holds: (i) in the nonspinning case for weak tidal fields (Damour 1983; Binnington and Poisson 2009; Damour and Nagar 2009) and also for tidal fields of arbitrary amplitude (Gürlebeck 2015; ii) in the spinning case (Poisson 2015; Pani et al. 2015b; Landry and Poisson 2015) for weak tidal fields, at least in the axisymmetric case to second order in the spin (Pani et al. 2015b) and generically to first order in the spin (Landry and Poisson 2015). On the other hand, the TLNs of ECOs are small but finite (Pani 2015; Uchikata et al. 2016; Porto 2016; Cardoso et al. 2017; Wade et al. 2013; Giddings et al. 2019).
This is shown in Fig. 12—inspired by standard analysis to discriminate among neutronstar equations of state (Hinderer et al. 2010; Maselli et al. 2013). The figure shows the tidal deformability \(\lambda =\frac{2}{3}M^5k\) as a function of the object mass for three different toy models (gravastars, wormholes, and perfectlyreflecting Schwarzschildlike ECOs) characterized by the same Planckian scale of the correction, \(\delta \equiv r_02M={\ell }_P\approx 1.6\times 10^{33}\,\mathrm{cm}\).
To summarize, finitesize effects in the inspiral waveform provide three different nullhypothesis tests of BHs. BHs have vanishing TLNs but introduce a nonzero tidal heating (\(\psi _{\mathrm{TD}}=0\), \(\psi _{\mathrm{TH}}\ne 0\)), while ECOs have (logarithmically small) TLNs but zero tidal heating (\(\psi _{\mathrm{TD}}\ne 0\), \(\psi _{\mathrm{TH}}=0\)). In addition BHs have a very well defined set of multipole moments which depend on only two parameters (mass and angular momentum), whereas ECOs have in principle limitless possibilities. In addition, it is possible that the inspiral excites the characteristic modes of each of the objects, i.e., their QNMs. The extent to which this happens, and its impact on the inspiral stage are still to be understood (Cardoso et al. 2019b).
The TLNs were computed for boson stars (Mendes and Yang 2017; Cardoso et al. 2017; Sennett et al. 2017), very compact anisotropic fluid stars (Raposo et al. 2018), and gravastars (Uchikata et al. 2016; Cardoso et al. 2017). The TLNs of simpleminded ultracompact Schwarzschildexterior spacetimes with a stiff equation of state at the surface were computed in Cardoso et al. (2017). The TLNs of spacetimes mimicking “compact quantum objects” were recently investigated (Giddings et al. 2019).
4.5.4 Accretion and drag in inspirals around and inside DM objects
When an object moves through any medium, it will be subject to (at least) two types of drag. One is direct and caused by accretion: the accreting object grows in mass and slows down. In addition, the moving body exerts a gravitational pull on all the medium, the backreaction of which produces dynamical friction (known also as “gravitational drag”), slowing the object down. To quantify these effects, it is important to know how the medium behaves. Collisionless media cause, generically, a gravitational drag different from that of normal fluids (Macedo et al. 2013a, b). The gravitational drag caused by media which is coherent on large scales may be suppressed (Hui et al. 2017), but further work is necessary to understand this quantitatively.
Consider now a binary of two compact objects, in which one is made of DM. At large separations inspiral will be driven mostly by GW emission. However, at small distances, the dynamics will generically be dominated by accretion and gravitational drag. The phase evolution of a binary, taking gravitational radiation, accretion and drag was studied when a small BH or neutron star inspirals around and inside a massive boson star (Macedo et al. 2013a, b). These results can also be directly translated to inspirals within a DM environment (Barausse et al. 2014; Macedo et al. 2013a, b; Eda et al. 2015; Yue and Han 2018; Hannuksela et al. 2019). Full nonlinear simulations of the inspiral and merger of boson stars, oscillatons and axion stars include GW emission, drag and accretion and tidal deformations. Although considerably more difficult to systematize and perform, such studies have been undertaken recently (Bezares et al. 2017; Palenzuela et al. 2017; Bezares and Palenzuela 2018; Helfer et al. 2019; Dietrich et al. 2019; Clough et al. 2018).
4.5.5 GW emission from ECOs orbiting or within neutron stars
It is conceivable that ECOs play also a role in GW (as well as EM) emission when orbiting close to neutron stars or white dwarfs. This might arise via two different possible ways. ECOs can form via gravitational collapse of DM or unknown quantum effects, and cluster around compact stars through tidal dissipation mechanisms. Alternatively, compact stars evolving in DMrich environments may accrete a significant amount of DM in their interior: DM is captured by the star due to gravitational deflection and a nonvanishing crosssection for collision with the star material (Press and Spergel 1985; Gould et al. 1990; Goldman and Nussinov 1989; Bertone and Fairbairn 2008; Brito et al. 2015a). The DM material eventually thermalizes with the star, and accumulates inside a finitesize core (Brito et al. 2015a, 2016b; Gould et al. 1990; Goldman and Nussinov 1989).
Interaction of the core with the surrounding star may lead to characteristic EM signatures (Brito et al. 2015a, 2016b). Alternatively, a more generic imprint of such ECOs is GW emission, either via standard inspiralling processes (Maselli et al. 2017a; Horowitz and Reddy 2019) or by small oscillations of such ECOs inside neutron stars or white dwarfs (Horowitz and Reddy 2019; Ellis et al. 2018).
4.6 Formation and evolution
In the context of DM physics, the formation and existence of ECOs is very reasonable (Giudice et al. 2016). We know that DM exists, that it interacts gravitationally and that its coupling to Standard Model fields is very weak. Therefore, gravitationally bound structures made of DM particles are dark (by definition) and can potentially be compact. Examples which are well understood include boson stars, made of scalars or vectors, which constitute one notable exception to our ignorance on the formation of ECOs. These configurations can arise out of the gravitational collapse of massive scalars (or vectors). Their interaction and mergers can be studied by evolving the Einstein–Klein–Gordon (–Maxwell) system, and there is evidence that accretion of less massive boson stars makes them grow and cluster around the configuration of maximum mass. In fact, boson stars have efficient gravitational cooling mechanisms that allow them to avoid collapse to BHs and remain very compact after interactions (Seidel and Suen 1991, 1994; Brito et al. 2016b; Di Giovanni et al. 2018). Similar studies and similar conclusions hold for axion stars, where the coupling to the Maxwell field is taken into account (Widdicombe et al. 2018). The cosmological formation of such dark compact solitons, their gravitational clustering and strong interactions such as scattering and mergers was recently investigated (Amin and Mocz 2019). If DM is built out of dark fermions, then formation should parallel that of standard neutron stars, and is also a well understood process. Collisions and merger of compact boson stars (Liebling and Palenzuela 2012; Bezares et al. 2017), bosonfermion stars (Bezares and Palenzuela 2018; Bezares et al. 2019), and axion stars (Helfer et al. 2017; Clough et al. 2018) have been studied in detail.
On the other hand, although supported by sound arguments, the vast majority of the alternatives to BHs are, at best, incompletely described. Precise calculations (and often even a rigorous framework) incorporating the necessary physics are missing. Most models listed in Table 1 were built in a phenomenological way or they arise as solutions of Einstein equations coupled to exotic matter fields. For example, models of quantumcorrected objects do not include all the (supposedly large) local or nonlocal quantum effects that could prevent collapse from occurring. In the absence of a complete knowledge of the missing physics, it is unlikely that a ClePhO forms out of the merger of two ClePhOs. These objects are so compact that at merger they will be probably engulfed by a common apparent horizon. The end product is, most likely a BH as argued in Sect. 4.4.3. On the other hand, if large quantum effects do occur, they would probably act on short timescales to prevent apparent horizon formation possibly in all situations. Thus, for example quantum backreaction has been argued to lead to wormhole solutions rather than BHs (Berthiere et al. 2018). In some models, Planckscale dynamics naturally leads to abrupt changes close to the wouldbe horizon, without fine tuning (Holdom and Ren 2017). Likewise, in the presence of (exotic) matter or if GR is classically modified at the horizon scale, Birkhoff’s theorem no longer holds, and a starlike object might be a more natural outcome than a BH. However, some studies suggest that compact horizonless bodies may form naturally as the result of gravitational collapse (Beltracchi and Gondolo 2019). The generality of such result is unknown.
An important property of the vacuum field equations is their scaleinvariance, inherited by BH solutions. Thus, the scaling properties of BHs are simple: their size scales with their mass, and if a nonspinning BH of mass \(M_1\) is stable, then a BH of mass \(M_2\) is stable as well, the timescales being proportional to the mass. Such characteristic is summarized in Fig. 5. Once matter is added, this unique property is lost. Thus, it is challenging to find theories able to explain, with horizonless objects, all the observations of dark compact objects with masses ranging over more than seven orders of magnitude, although some ECO models can account for that (Raposo et al. 2018). Such “short blanket” problem is only an issue if one tries to explain away all the dark compact objects with horizonless alternatives. If particle physics is a guidance, it is well possible that nature offers us a much more diverse universe content.
5 Observational evidence for horizons
Horizons act as perfect sinks for matter and radiation. The existence of a hard or smooth surface will lead in general to clear imprints. Classically, EM waves are the traditional tool to investigate astrophysical objects. There are a handful of interesting constraints on the location of the surface of ECOs using light (Narayan et al. 1997; Narayan and Heyl 2002; McClintock et al. 2004; Broderick and Narayan 2006, 2007; Narayan and McClintock 2008; Broderick et al. 2009; Lu et al. 2017). However, testing the nature of dark, compact objects with EM observations is challenging. Some of these challenges, as we will discuss now, are tied to the incoherent nature of the EM radiation in astrophysics, and the amount of modeling and uncertainties associated to such emission. Other problems are connected to the absorption by the interstellar medium. As discussed in the previous section, testing quantum or microscopic corrections at the horizon scale with EM probes is nearly impossible. Even at the semiclassical level, Hawking radiation is extremely weak to detect and not exclusive of BH spacetimes (Paranjape and Padmanabhan 2009; Barcelo et al. 2011; Harada et al. 2019).It is well known that the Kerr solution provides the unique solution for stationary BHs in the universe. But a confirmation of the metric of the Kerr spacetime (or some aspect of it) cannot even be contemplated in the foreseeable future.
S. Chandrasekhar, The Karl Schwarzschild Lecture,
Astronomische Gesellschaft, Hamburg (September 18, 1986)
The historical detection of GWs (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2016a) opens up the exciting possibility of testing gravity in extreme regimes with unprecedented accuracy (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2016b; Yunes and Siemens 2013; Barausse et al. 2014; Berti et al. 2015; Giddings 2016; Yunes et al. 2016; Maselli et al. 2018b). GWs are generated by coherent motion of massive sources, and are therefore subjected to less modeling uncertainties (they depend on far fewer parameters) relative to EM probes. The most luminous GWs come from very dense sources, but they also interact very feebly with matter, thus providing the cleanest picture of the cosmos, complementary to that given by telescopes and particle detectors.
Henceforth we will continue using the parameter \(\epsilon \) defined by Eq. (3) to quantify the constraints that can be put on the presence/absence of a horizon. The current and projected bounds discussed below are summarized in Table 3 at the end of this section.
5.1 Tidal disruption events and EM counterparts
5.2 Equilibrium between ECOs and their environment: Sgr A*
The previous results used a large number of objects and—in addition to the caveats just pointed—assume that all are horizonless. The compact radio source Sgr A* at the center of galaxy is—due to its proximity—a good candidate to improve on the above. Sgr A* has an estimated mass \(M\sim 4\times 10^6\,M_{\odot }\), and is currently accreting at an extremely low level, with (accretion disk) luminosity \(L_\mathrm{disk}\sim 10^{36}\, \mathrm{erg\, s}^{1}\) (peaking at wavelength \(\sim 0.1\,\mathrm{mm}\)), about \(10^{9}\) times the Eddington luminosity for the central mass (Johannsen 2016a; Eckart et al. 2017). The efficiency of the accretion disk at converting gravitational energy to radiation is less than 100%, which suggests a lower bound on the accretion rate \(\dot{M} \ge L_\mathrm{disk} \sim 10^{15} \,\mathrm{g}\,\mathrm{s}^{1}\) (\(10^{24}\) in geometric units).
 i.It assumes that a thermodynamic and dynamic equilibrium must be established between the accretion disk and the central object, on relatively short timescales. However, strong lensing prevents this from happening; consider accretion disk matter, releasing isotropically (for simplicity) scattered radiation on the surface of the object. As discussed in Sect. 2.2, only a fraction \(\sim \epsilon \) is able to escape during the first interaction with the star, cf. Eq. (9). The majority of the radiation will fall back onto the surface after a time \(t_\mathrm{roundtrip} \sim 9.3 M\) given by the average of Eq. (11).^{6} Suppose one injects, instantaneously, an energy \(\delta M\) onto the object. Then, after a time \(T_a\), the energy emitted to infinity during \(N=T_a/t_\mathrm{roundtrip}\) interactions readswhere the last step is valid for \(\epsilon N\ll 1\).$$\begin{aligned} \varDelta E \sim \left[ 1  (1  \epsilon )^N\right] {\delta M} \approx \epsilon \left( \frac{T_a}{t_\mathrm{roundtrip}}\right) \delta M. \end{aligned}$$(97)We can assume \(T_a=\tau _\mathrm{Salpeter}\approx 4.5\times 10^7\,\mathrm{yr}\) and \(\dot{M}= f_\mathrm{Edd} \dot{M}_\mathrm{Edd}\), where \(\dot{M}_\mathrm{Edd}\approx 1.3 \times 10^{39}(M/M_\odot )\,\mathrm{erg/s}\) is the Eddington mass accretion rate onto a BH. Then, from Eq. (97) we getwhere we have normalized the fraction of the Eddington mass accretion rate, \(f_\mathrm{Edd}\), to its typical value for Sgr A*. Requiring this flux to be compatible with the lack of observed flux from the central spot (\({\dot{E}}\lesssim 10^{25}\)), one finds \(\epsilon \lesssim 10^{15}\).$$\begin{aligned} \dot{E}\sim 10^{25}\left( \frac{\epsilon }{10^{15}}\right) \left( \frac{f_\mathrm{Edd}}{10^{9}}\right) . \end{aligned}$$(98)Assuming \(L\sim {\dot{E}}\) and using the StefanBoltzmann law, Eq. (98) yields an estimate for the effective surface temperature of Sgr A* if the latter had a hard surface,$$\begin{aligned} T\sim 7.8\times 10^3\,\left( \frac{4\times 10^6 M_\odot }{M}\right) ^{1/2}\left( \frac{\epsilon }{10^{15}}\right) ^{1/4}\left( \frac{\delta M}{10^{7}M}\right) ^{1/4}\,\mathrm{K}. \end{aligned}$$(99)
 ii.It assumes that the central object is returning in EM radiation most of the energy that it is taking in from the disk. However, even if the object were returning all of the incoming radiation on a sufficiently short timescale, a sizable fraction of this energy could be in channels other than EM. For freelyfalling matter on a radial trajectory, its fourvelocity \(v_{(1)}^{\mu }=(E/f,\sqrt{E^2f},0,0)\). Particles at the surface of the object have \(v_{(2)}^{\mu }=(\sqrt{f},0,0,0)\). When these two collide, their CM energy reads (Banados et al. 2009),Thus, even for only moderately small \(\epsilon \), the particles are already relativistic. At these CM energies, all known particles (photons, neutrinos, gravitons, etc) should be emitted “democratically,” and in the context of DM physics, new degrees freedom can also be excited. Even without advocating new physics beyond the \(10\,\mathrm{TeV}\) scale, extrapolation of known hadronic interactions to large energies suggests that about 20% of the collision energy goes into neutrinos, whose total energy is a sizable fraction of that of the photons emitted in the process (Kelner et al. 2006). To account for these effects, we take$$\begin{aligned} E_\mathrm{CM}=m_0\sqrt{2}\sqrt{1g_{\mu \nu }v_{(1)}^\mu v_{(2)}^\mu }\sim \frac{m_0\sqrt{2E}}{\epsilon ^{1/4}}, \end{aligned}$$(100)as a reasonable conservative bound coming from this equilibrium argument.$$\begin{aligned} \epsilon \lesssim 10^{14}, \end{aligned}$$(101)
If only a fraction of the falling material interacts with the object (for example, if it is made of DM with a small interaction crosssection), then the above constraint would deteriorate even further.
 iii.
The estimate (101) was reached without a proper handling of the interaction between the putative outgoing radiation and the disk itself, and assumes spherical symmetry. Thus, there might be large systematic uncertainties associated (and which occur for any astrophysical process where incoherent motion of the radiating charges play a key role).
5.3 Bounds with shadows: Sgr A* and M87
Recent progress in very long baseline interferometry allows for direct imaging of the region close to the horizon, with the potential to provide also constraints on putative surfaces. These images are also referred to as “shadows” since they map sky luminosity to the source (typically an accretion disk), see Sect. 2.2. Two supermassive BHs have been studied, namely the Sgr A* source and the BH at the center of M87, whose imaging requires the lowest angular resolution (Doeleman et al. 2008; Doeleman 2012; Broderick et al. 2014; Goddi et al. 2016; Abuter et al. (GRAVITY Collaboration) 2018b; Amorim et al. (GRAVITY Collaboration) 2019; Johannsen et al. 2016; Akiyama et al. 2019).
In a similar spirit, tests based on stronglensing events (Nandi et al. 2018; Shaikh et al. 2019) (in fact, a variant of shadows) or quantum versions of it (Sabín 2017) have been proposed. Adding to the list of possible discriminators, GraciaLinares and Guzman (2016) studied the impact of supersonic winds blowing through BHs and boson stars. The conclusion is that, while qualitatively the stationary regime of downstream wind distribution is similar, the density may defer by almost an order of magnitude depending on the boson star configuration. At an observational level, these differences would show up presumably as friction on the compact object. However, quantitative tests based on observations are challenging to devise.
Finally, “hotspots” orbiting around supermassive objects can also provide information about nearhorizon signatures (Broderick and Loeb 2005, 2006). Recently, the first detection of these orbiting features at the ISCO of Sgr A* was reported (Abuter et al. (GRAVITY Collaboration) 2018a), implying a bound of the same order as Eq. (102).
5.4 Tests with accretion disks
Tests on the spacetime geometry can also be performed by monitoring how matter moves and radiates as it approaches the compact object. Matter close to compact objects can form an accretion disk (LyndenBell 1969; Novikov and Thorne 1973; Page and Thorne 1974), in which each element approximately moves in circular, Keplerian orbits. The disk is typically “truncated” at the ISCO (cf. Fig. 1), which represents a transition point in the physics of the accretion disk. It is in principle possible to extract the ISCO location and angular velocity—and hence infer properties of the central object such as the mass, spin, and quadrupole moment—from the EM signal emitted (mostly in the Xray band) by the accreting matter, either for a stellarmass BH or for a supermassive BH (Bambi 2017). In practice, the physics of accretion disks is very complex and extracting such properties with a good accuracy is challenging.
A promising approach is the analysis of the iron \(K\alpha \) line (Fabian et al. 1989), one of the brightest components of the Xray emission from accreting BH candidates. This line is broadened and skewed due to Doppler and (special and general) relativistic effects, which determine its characteristic shape. An analysis of this shape (assuming that the spacetime is described by the Kerr metric) provides a measurement of the BH spin and the inclination of the accretion disk (Reynolds 2014). Although limited by systematic effects (Bambi 2017), this technique has been used also to test the spacetime metric (Johannsen and Psaltis 2010, 2013; Bambi 2013; Jiang et al. 2015; Johannsen 2014; Moore and Gair 2015; Hoormann et al. 2016) and to distinguish boson stars from BHs (Cao et al. 2016; Shen et al. 2016). Another approach is the study of the thermal component of the spectrum from stellarmass BHs using the socalled continuumfitting method (Li et al. 2005; McClintock et al. 2014; Reynolds 2014), which can provide information about the ISCO location and hence the BH spin (McClintock et al. 2014). The method can be also used to test the spacetime geometry (Johannsen and Psaltis 2010; Bambi and Barausse 2011; Bambi 2012, 2014; Kong et al. 2014; Johannsen 2014; Moore and Gair 2015; Hoormann et al. 2016) but is limited by the fact that deviations from the Kerr geometry are typically degenerate with the ISCO properties, e.g., with the spin of the object (Bambi 2017; Johannsen 2016b). Finally, an independent approach is the study of the quasiperiodic oscillations observed in the Xray flux emitted by accreting compact objects (Stella and Vietri 1999; Stella et al. 1999; Abramowicz and Kluzniak 2001). The underlying mechanism is not well understood yet, but these frequencies are believed to originate in the innermost region of the accretion flow (van der Klis 2000), and they might carry information about the spacetime near compact objects. Some of the proposed models try to explain such phenomena with combinations of the orbital and epicyclic frequencies of geodesics around the object. Based on these models, constraints on boson stars have been discussed in Franchini et al. (2017).
5.5 Signatures in the massspin distribution of dark compact objects
The previous tests were based exclusively on EM measurements. There are tests which can be done either via EM or GW signals. An exciting example concerns the measurement of the spin of compact objects, which can be performed either via the aforementioned EM tests or from GW detections of binary inspirals and mergers (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2018b). This requires a large population of massive objects to have been detected and their spins estimated to some accuracy. EM or GW observations indicating statistical prevalence of slowlyspinning compact objects, across the entire mass range, indicate either a special formation channel for BHs, or could signal that such objects are in fact horizonless: the development of the ergoregion instability is expected to deplete angular momentum from spinning ClePhOs, independently of their mass, as we discussed in Sect. 4.4.1. Thus, the spinmass distribution of horizonless compact objects skews towards low spin. Although the effectiveness of such process is not fully understood, it would lead to slowlyspinning objects as a final state, see Fig. 13. On the other hand, observations of highlyspinning BH candidates can be used to constrain ECO models.
5.6 Multipole moments and tests of the nohair theorem
5.6.1 Constraints with comparablemass binaries
An estimate of the bounds on the spininduced quadrupole moment from GW detection of compactbinary inspirals was performed in Krishnendu et al. (2017, 2019) (see Fig. 14). As we discussed, this correction enters at 2PN order in the GW inspiral phase and is quadratic in the spin. Therefore, it requires relatively lowmass binaries (which perform many cycles in band before merger) and high spins. The quadrupole moment of the binary was parametrized as \({{\mathcal {M}}}_2^{(i)}=\kappa _i \chi ^2_i m_i^3\) (\(i=1,2\)), where \(\kappa _i=1\) for a Kerr BH.
In addition to projected bounds, observational bounds on parametrized corrections to the 2PN coefficient of the inspiral waveform from binary BH coalescences can be directly translated—using Eq. (82)—into a bound on (a symmetric combination of) the spininduced quadrupole moments of the binary components.^{7} This parametrized PN analysis has been recently done for various BH merger events, the combined constraint on the deviation of the 2PN coefficient reads \(\delta \varphi _2\lesssim 0.3\) at 90% confidence level (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2019). However, the component spins of these sources are compatible with zero so these constraints cannot be translated into an upper bound on the spininduced quadrupole moment in Eq. (16). They might be translated into an upper bound on the nonspin induced quadrupole moment, which is however zero in all ECO models proposed so far.
5.6.2 Projected constraints with EMRIs
Model dependent studies on the ability of EMRIs to constrain quadrupolar deviations from Kerr have been presented in Ryan (1997), Vigeland and Hughes (2010) and Moore et al. (2017).
5.7 Tidal heating
Horizons absorb incoming high frequency radiation, and serve as sinks or amplifiers for lowfrequency radiation able to tunnel in, see Sect. 4.5.2. UCOs and ClePhOs, on the other hand, are not expected to absorb any significant amount of GWs. Thus, a “nullhypothesis” test consists on using the phase of GWs to measure absorption or amplification at the surface of the objects (Maselli et al. 2018b).
Because horizon absorption is related to superradiance and the BH area theorem (Brito et al. 2015b), testing this effect is an indirect proof of the second law of BH thermodynamics. While this effect is too small to be detectable from a single event with secondgeneration detectors, a large number (\(\approx 10^4\)) of LIGOVirgo detections might support Hawking’s area theorem at \(90\%\) confidence level (Lai and Li 2018).
On the other hand, highlyspinning supermassive binaries detectable with a LISAtype GW interferometer will have a large SNR and will place stringent constraints on this effect, potentially reaching Planck scales near the horizon (Maselli et al. 2018b). This is shown in the left panel of Fig. 15, which presents the bounds on parameter \(\gamma \) defined by adding the tidalheating term in the PN phase as \(\gamma \psi _{\mathrm{TH}}^\mathrm{BH}\) [see Eq. (84)]. For a BH \(\gamma =1\), whereas \(\gamma =0\) for a perfectly reflecting ECO. Notice that the effect is linear in the spin and it would be suppressed by two further PN orders in the nonspinning case.
The effect is clearly important, but the known multiple systematics involved (e.g., due to waveform modeling and to parameter estimation in a signaldriven detector like LISA) still need to be quantified. Finally, the ability of tidal heating in constraining the closeness parameter \(\epsilon \) (or the blueshift of photons in Table 3 below) for EMRIs is yet to be understood, both because of the above systematics and also because the absence of tidal heating might be directly mapped into a bound on \(\epsilon \), since it depends mostly on the object interior rather than on the location of the surface (see, however, discussion at the end of Sect. 4.5.2).
5.8 Tidal deformability
As discussed in Sect. 4.5.3, the TLNs of a BH are identically zero, whereas those of an ECO are not. Although this correction enters at 5PN order in the waveform, the tidal deformability of an object with radius \(r_0\) is proportional to \((r_0/M)^5\), so its effect in the GW phase is magnified for less compact objects. This effect has been recently explored for bosonstar binaries, by investigating the distinguishability of binary boson stars from both binary BHs (Cardoso et al. 2017; Sennett et al. 2017; Wade et al. 2013; JohnsonMcdaniel et al. 2018) and binary neutron stars (Sennett et al. 2017). Secondgeneration GW detectors at design sensitivity should be able to distinguish bosonstars models with no selfpotential and with a quartic selfpotential (cf. Table 2) from BHs, whereas 3G (resp., LISA) is necessary to distinguish the most compact solitonic boson stars from stellarmass (resp., supermassive) BHs (Cardoso et al. 2017). As a rule of thumb, the stronger the boson selfinteraction the more compact are stable bosonstar equilibrium configurations, and hence the smaller the tidal deformability and the chances of detectability. Fits for the TLNs of various bosonstar models are provided in Sennett et al. (2017); codes to compute these quantities are publicly available (CENTRA 2019).
For ECOs inspired by Planckian corrections at the horizon scale, the TLNs scale as \(k\sim 1/\log \epsilon \) for a variety of models [see Sect. 4.5.3 and Table I in Cardoso et al. (2017)]. Due to this scaling, in these models the TLNs are only roughly 4 orders of magnitude smaller than for an ordinary neutron star. Nonetheless, measuring such small TLN is probably out of reach even with 3G and would require LISA golden binaries (Maselli et al. 2018b) (see right panel of Fig. 15). Due to the logarithmic scaling, in these models the statistical errors on \(\epsilon \) would depend exponentially on the TLNs and reaching a Planckian requires a very accurate measurement of k (Addazi et al. 2018). Nonetheless, this does not prevent to perform ECO model selection (see Fig. 12).
5.9 Resonance excitation
The contribution of the multipolar structure, tidal heating, and tidal deformability on the gravitational waveform is perturbative and produces small corrections relative to the idealized pointparticle waveform. However, there are nonperturbative effects that can be triggered during inspiral, namely the excitation of the vibration modes of the inspiralling objects. In particular, if the QNMs are of sufficiently low frequency, they can be excited during inspiral (Pani et al. 2010c; Macedo et al. 2013a, b, 2018; Cardoso et al. 2019b). This case is realized for certain models of ECOs (e.g., ultracompact gravastars and boson stars) and generically for Kerrlike ECOs in the \(\epsilon \rightarrow 0\), see Eq. (72). In addition to spacetime modes, also modeldependent fluid modes might also be excited (Yunes et al. 2016). Due to redshift effects, these will presumably play a subdominant role in the GW signal.
5.10 QNM tests
One of the simplest and most elegant tools to test the BH nature of central objects, and GR itself, is to use the uniqueness properties of the Kerr family of BHs: vacuum BHs in GR are fully specified by mass and angular momentum, and so are their vibration frequencies (Berti et al. 2006b, 2009). Thus, detection of one mode (i.e., ringing frequency and damping time) allows for an estimate of the mass and angular momentum of the object (assumed to be a GR BH). The detection of two or more modes allows to test GR and/or the BH nature of the object (Dreyer et al. 2004; Berti et al. 2006b, 2016; Berti and Cardoso 2006; Meidam et al. 2014).
Current detectors can only extract one mode for massive BH mergers, and hence one can estimate the mass and spin of the final object, assumed to be a BH (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2016b). Future detectors will be able to detect more than one mode and perform “ECO spectroscopy” (Dreyer et al. 2004; Berti et al. 2006b, 2016; Berti and Cardoso 2006; Meidam et al. 2014).
To exclude ECO models, one needs calculations of their vibration spectra. These are available for a wide class of objects, including boson stars (Berti and Cardoso 2006; Macedo et al. 2013a, b), gravastars (Pani et al. 2009; Mazur and Mottola 2015; Chirenti and Rezzolla 2016), wormholes (Konoplya and Zhidenko 2016; Nandi et al. 2017), or other quantumcorrected objects (Barceló et al. 2017; Brustein et al. 2017b). A major challenge in these tests is how to model spin effects properly, since few spinning ECO models are available and the study of their perturbations is much more involved than for Kerr BHs. In general, the postmerger signal from a distorted ECO might be qualitatively similar to that of a neutronstar merger, with several longlived modes excited (Kokkotas and Schmidt 1999) and a waveform that is more involved than a simple superposition of damped sinusoids as in the case of BH QNMs.
As discussed previously in Sect. 4.2 and in Sect. 5.12 below, all these extra features are expected to become negligible in the \(\epsilon \rightarrow 0\) limit: the prompt ringdown of an ultracompact ECOs should become indistinguishable from that of a BH in this limit, jeopardizing standard QNM tests.
5.11 Inspiralmergerringdown consistency
The full nonlinear structure of GR is encoded in the complete waveform from the inspiral and merger of compact objects. Thus, while isolated tests on separate dynamical stages are important, the ultimate test is that of consistency with the full GR prediction: is the full inspiralmergerringdown waveform compatible with that of a binary BH coalescence? Even when the SNR of a given detection is low, such tests can be performed, with some accuracy. Unfortunately, predictions for the coalescence in theories other than GR and for objects other that BHs are practically unknown. The exceptions concern evolutions of neutron stars, boson stars, composite fluid systems, and axion stars (Liebling and Palenzuela 2012; Cardoso et al. 2016b; Bezares et al. 2017, 2019; Bezares and Palenzuela 2018; Helfer et al. 2017; Widdicombe et al. 2018; Clough et al. 2018) (see Sect. 4.6), and recent progress in BH mergers in modified gravity (Okounkova et al. 2017, 2019; Hirschmann et al. 2018; Witek et al. 2019).
5.12 Tests with GW echoes
For binaries composed of ClePhOs, the GW signal generated during inspiral and merger is expected to be very similar to that by a BH binary with the same mass and spin. Indeed, the multipole moments of very compact objects approach those of Kerr when \(\epsilon \rightarrow 0\), and so do the TLNs, etc. Constraining \(\epsilon \) (or quantifying up to which point the vacuum Kerr is a description of the spacetime) is then a question of having sensitive detectors that can probe minute changes in waveforms. This would also require having sufficiently accurate waveform models to avoid systematics. However, there is a clear distinctive feature of horizonless objects: the appearance of latetime echoes in the waveforms (see Sect. 4.2). There has been some progress in modeling the echo waveform and data analysis strategies are in place to look for such latetime features; some strategies have been also implemented using real data (Abedi et al. 2017a, b; Conklin et al. 2018; Westerweck et al. 2018; Tsang et al. 2018; Nielsen et al. 2018; Lo et al. 2018; Wang et al. 2019a; Uchikata et al. 2019).
The ability to detect such signals depends on how much energy is converted from the main burst into echoes (i.e., on the relative amplitude between the first echo and the prompt ringdown signal in Fig. 9). Depending on the reflectivity of the ECO, the energy contained in the echoes can exceed that of the standard ringdown alone (Mark et al. 2017; Testa and Pani 2018), see left panel of Fig. 17. This suggests that it is possible to detect or constrain echoes even when the ringdown is marginally detectable or below threshold, as in the case of EMRIs or for comparablemass coalescences at small SNR.
Searches for echo signals in the detectors based on reliable templates can be used to find new physics, or to set very stringent constraints on several models using real data. Different groups with independent search techniques have found structure in many of the GW events, compatible with postmerger echoes (Abedi et al. 2017a, b; Ashton et al. 2016; Conklin et al. 2018; Westerweck et al. 2018). However, the statistical significance of such events has been put into question (Westerweck et al. 2018; Abedi et al. 2018). For GW150914, Abedi et al. (2017a), Ashton et al. (2016) and Conklin et al. (2018)—using independent search techniques—report evidence for the existence of postmerger echoes in the data. However, Nielsen et al. (2018) finds a lower significance and a Bayes factor indicating preference for noise over the echo hypothesis. For other GW events, there is agreement between different groups on the existence of postmerger features in the signal, found using echo waveforms. The interpretation of these features is under debate. An independent search in the LIGOVirgo Catalog GWTC1 found no statistical evidence for the presence of echoes within \(0.1\,\mathrm{s}\) of the main burst (Uchikata et al. 2019).
Any realistic search is controlled by \(\eta \) [cf. Eq. (59)] and the time delay between main burst and echoes (Abedi et al. 2017a, b; Conklin et al. 2018; Westerweck et al. 2018; Tsang et al. 2018; Nielsen et al. 2018; Lo et al. 2018). Since the SNR of the postmerger signal is controlled by \(\eta \) on a integration timescale controlled by \(\tau \), even negative searches can be used to place strong constraints on \(\epsilon \) (Westerweck et al. 2018; Nielsen et al. 2018).
Overall, in a large region of the parameter space the signal is large enough to produce effects within reach of nearfuture GW detectors, even if the corrections occur at the “Planck scale” (by which we mean \(\epsilon \sim 10^{40}\)). This is a truly remarkable prospect. As the sensitivity of GW detectors increases, the absence of echoes might be used to rule out ECO models, to set ever stringent upper bounds on the level of absorption in the object’s interior, and generically to push tests of gravity closer and closer to the horizon scale, as now routinely done for other cornerstones of GR, e.g., in tests of the equivalence principle (Will 2014; Berti et al. 2015).
5.13 Stochastic background
5.14 Motion within ECOs
In certain models, the ECO interior might be weakly interacting and a further discriminator would be the motion of test particles within the object. Among other effect, this can produce nonstandard signals in EMRIs. As discussed in Sects. 4.5.4 and 4.5.5, this motion is driven by the selfgravity of the central object, accretion, and dynamical friction. The study of geodesic motion inside a solitonic boson stars was analyzed in Kesden et al. (2005). The effects of accretion and drag were included in Macedo et al. (2013b, 2013a), Barausse et al. (2014, 2015). These effects cannot be directly translated into bounds on \(\epsilon \), but would be a smokinggun signature for the existence of structures in supermassive ultracompact objects.
6 Discussion and observational bounds
How well does the BH geometry describe the dark compact objects in our universe?
Constraints  Source  References  

\(\epsilon (\lesssim )\)  \(\frac{\nu }{\nu _\infty }(\gtrsim )\)  
1a.  \({{\mathcal {O}}}(1) \)  \({{\mathcal {O}}}(1)\)  Sgr A\(^*\) & M87  Doeleman et al. (2008), Doeleman (2012), Johannsen et al. (2016), Abuter et al. (GRAVITY Collaboration) (2018a), Akiyama et al. (2019) 
1b.  0.74  1.5  GW150914  Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) (2016a) 
2.  \({{\mathcal {O}}}(0.01)\)  \({{\mathcal {O}}}(10)\)  GW150914  Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) (2016a) 
3.  \( 10^{4.4}\)  158  All with \(M>10^{7.5}M_{\odot }\)  Lu et al. (2017) 
4.  \( 10^{14}\)  \(10^7\)  Sgr A*  Lu et al. (2017) 
5.  \(10^{40}\)  \(10^{20}\)  All with \(M<100M_{\odot }\)  Barausse et al. (2018) 
6.  \(10^{47}\)  \(10^{23}\)  GW150914  
7*.  \(e^{10^4/\zeta }\)  \(e^{5000/\zeta }\)  EMRIs  Babak et al. (2017), Barack and Cutler (2007), Pani and Maselli (2019) 
Effect and caveats  

1a.  Uses detected orbiting hotspot around Sgr A* and “shadow” of Sgr A* and M87  
Spin effects are poorly understood; systematic uncertainties not quantified  
1b.  Merger frequency of GW150914 and measurements of the masses  
Assumes merger frequency equal to Keplerian frequency at contact  
2.  Consistency of ringdown with BH signal  
Large measurement errors on the QNM frequencies. Precise bounds are model dependent  
Bounds will improve significantly with detailed searches for postmerger echoes  
3.  Lack of optical/UV transients from tidal disruption events  
Assumes: all objects are horizonless, have a hard surface, spherical symmetry, and isotropy  
4.  Uses absence of relative low luminosity from Sgr A*, compared to disk  
Spin effects and matterradiation interaction matter poorly understood; assumes spherical symmetry  
5.  Uses absence of GW stochastic background (from ergoregion instability)  
Assumes: hard surface (perfect reflection); exterior Kerr; all objects are horizonless  
6.  Uses absence of GW echoes from postmerger object  
\(90\%\) confidence level for \(\eta >0.9\), deteriorates for smaller \(\eta \). Simplified echo template, limited range of priors  
7*.  Projected EMRI constraints on the spininduced quadrupole (\(\zeta =(\delta {{\mathcal {M}}}_2/M^3)/10^{4}\)) and TLNs (\(\zeta =k_/10^{4}\))  
Assumes saturation of Eq. (16) (for \(\delta {{\mathcal {M}}}_2\)) and Eq. (95) (for k) and orderunity coefficients in those equations  
Uses PN kludge waveforms, phenomenological deviation for \({{\mathcal {M}}}_2\), and simplified parameter estimation  
Models for which \(\delta {{\mathcal {M}}}_2\sim \epsilon ^n\) or \(k\sim \epsilon ^n\) are much less constrained 
These bounds can be read in two different ways. On the one hand, they tell us how appropriate the Kerr metric is in describing some of the massive and dark objects in our universe. In other words, observations tell us that the Kerr description is compatible with observations at least down to \(r=r_+(1+\epsilon )\). Alternatively, one can view these numbers as constraints on exotic alternatives to BHs. In both cases, the constraint on \(\epsilon \) can be translated into the ratio of frequencies [or redshift, as measured by locally nonrotating observers (Bardeen et al. 1972)] of a photon as it travels from infinity down to the farthest point down to which observations are compatible with vacuum.
Most of the constraints shown in Table 3 are associated with large systematics or modelling uncertainties. From a proper understanding of astrophysical environments and their interaction with ultracompact objects, the development of a solid theoretical framework, to a proper modeling of the coalescence of such objects and data analysis to see such events, the challenges are immense. The payoff for facing these oustanding issues is to be able to quantify the statement that BHs exist in nature.
Footnotes
 1.
Quoting Subrahmanyan Chandrasekhar: “In my entire scientific life, extending over fortyfive years, the most shattering experience has been the realization that an exact solution of Einstein’s equations of general relativity provides the absolutely exact representation of untold numbers of black holes that populate the universe.” (S. Chandrasekhar, The Nora and Edward Ryerson lecture, Chicago April 22 1975).
 2.
A crucial property of BHs in GR is that—owing to the scalefree nature of vacuum Einstein’s equation—their mass is a free parameter. This is why the same Kerr metric can describe any type of BH in the universe, from stellarmass (or even possibly primordial) to supermassive. It is extremely challenging to reproduce this property with a material body, since matter fields introduce a scale.
 3.
Incidentally, this also means that the circular timelike geodesic at 6M is not really the “innermost stable circular orbit”. We use this description to keep up with the tradition in BH physics.
 4.
A similar approach is pursued to classify possible extensions of GR (Berti et al. 2015).
 5.
In this respect, a parallel can drawn with neutron stars, which can be well described within GR by a simple selfgravitating perfect fluid, but whose formation process is significantly more complex than the gravitational collapse of a perfect fluid. Incidentally, such processes involve complex microphysics and quantum effects such as those occurring in a supernova collapse. In other words, the fact that an equilibrium solution can be well described by simple matter fields does not necessarily mean that its formation is equally simple nor does it exclude more complex formation processes.
 6.
One might wonder if the trapped radiation bouncing back and forth the surface of the object might not interact with the accretion disk. As we showed in Sect. 2.2, this does not happen, as the motion of trapped photons is confined to within the photosphere.
 7.
Unfortunately, for the majority of binary BH events detected so far (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2018b), either the spin of the binary component is compatible to zero, or the event had a low signaltonoise ratio (SNR) in the early inspiral, where the PN approximation is valid. The most promising candidate for this test would be GW170729, for which the measured effective binary spin parameter is \(\chi _\mathrm{eff}\approx 0.36^{+0.21}_{0.25}\) (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2018b; Chatziioannou et al. 2019). However, for such event no parametrizedinspiral test has been performed so far (Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration) 2019). If confirmed, the recent claimed detection (Zackay et al. 2019) of a highlyspinning BH binary would be ideal to perform tests of the spininduced quadrupole moment.
Notes
Acknowledgements
We are indebted to Niayesh Afshordi, K. G. Arun, Cosimo Bambi, Carlos Barceló, Ofek Birnholtz, Silke Britzen, Ramy Brustein, Collin Capano, Raúl CarballoRubio, Ana Carvalho, Miguel Correia, Jan de Boer, Kyriakos Destounis, Valeria Ferrari, Valentino Foit, Luis Garay, Steve Giddings, Eric Gourgoulhon, Tomohiro Harada, Carlos Herdeiro, Bob Holdom, Scott Hughes, Bala Iyer, Marios Karouzos, Gaurav Khanna, Joe Keir, Matthew Kleban, Kostas Kokkotas, Pawan Kumar, Claus Laemmerzahl, José Lemos, Avi Loeb, Caio Macedo, Andrea Maselli, Samir Mathur, Emil Mottola, Kenichi Nakao, Richard Price, Sergey Solodukhin, Nami Uchikata, Chris Van den Broeck, Bert Vercnocke, Frederic Vincent, Sebastian Voelkel, Kent Yagi, and Aaron Zimmerman for providing detailed feedback, useful references, for discussions, or for suggesting corrections to an earlier version of the manuscript. VC acknowledges financial support provided under the European Union’s H2020 ERC Consolidator Grant “Matter and strongfield gravity: New frontiers in Einstein’s theory” Grant Agreement No. MaGRaTh–646597. PP acknowledges financial support provided under the European Union’s H2020 ERC, Starting Grant Agreement No. DarkGRA–757480 and support from the Amaldi Research Center funded by the MIUR program “Dipartimento di Eccellenza” (CUP: B81I18001170001). This article is based upon work from COST Action CA16104 “GWverse” supported by COST (European Cooperation in Science and Technology). This work was partially supported by the H2020MSCARISE2015 Grant No. StronGrHEP690904 and by FCT Awaken Project PTDC/MATAPL/30043/2017.
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