Testing the nature of dark compact objects: a status report

Testing the nature of dark compact objects: a status report

Vitor Cardoso
CENTRA, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
CERN 1 Esplanade des Particules, Geneva 23, CH-1211, Switzerland
email: vitor.cardoso@ist.utl.pt
https://centra.tecnico.ulisboa.pt/network/grit/
Paolo Pani
Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy
email: paolo.pani@uniroma1.it
https://www.roma1.infn.it/p̃ani
July 26, 2019
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 beyond-the-standard-model physics. Null results strengthen and quantify the Kerr black hole paradigm. The advent of gravitational-wave 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.

Contents
“The 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 “plum-pudding” 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 point-like 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 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 are in fact the simplest macroscopic objects that one can conceive. The uniqueness theorems – establishing that the two-parameter 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].

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? Are there any parallels that can be useful guides? 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 self-consistent and free of pathologies. The “inverse-square 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 perturbatively stable against small fluctuations, and attempts to produce naked singularities, starting from BH spacetimes, have failed. BHs are not only curious mathematical solutions to Einstein’s equations (as were they initially dismissed as), but also their formation process is sound and well understood. In fact, classically 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.

Why, then, should one question the existence of BHs? 111This is a peculiar question, being asked more than a century after the first BH solution was derived. Even though Schwarzschild and Droste wrote down the first nontrivial regular, asymptotic flat, vacuum solution to the field equations already in 1916 [2, 3], several decades would elapse until such solutions became accepted. 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. Progress in our understanding of gravitational collapse, along with breakthroughs in the understanding of singularities and horizons, contributed to change the status of BHs. 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. The irony lies in the fact that they 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.” There are a number of important reasons to do so, starting from the obvious one: 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 [4, 5]. An increase in sensitivity of current detectors and the advent of next-generation 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 [6, 7] will very soon provide direct images of the center of our galaxy, where a dark, massive and compact object is lurking [8, 9, 10, 11]. Data from such missions has the potential to inform us on the following outstanding issues:

1.1 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 [12, 13]. 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 pathology-free, 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 [14, 15, 16, 17]. 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 [18] and the number of microstates can be tested using GW measurements [19, 20], but assume classical matter. Indeed, semi-classical quantum effects around BHs are far from being under control. Quantum field theory on BH backgrounds leads to loss of unitarity, a self-consistency requirement that any predictive theory ought to fulfill. The resolution of such conundrum may involve non-local effects changing the near-horizon structure, or doing away with horizons completely [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

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 20th 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 , 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 higher-dimensional interaction, then the fundamental Planck length can be substantially larger [36, 37]. In addition, some physics related to compact objects have a logarithmic dependence on the (reasonably-defined) Planck length [38] (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 [39, 40, 41, 42, 43]. Thus, quantum-gravity effects may be within reach.

1.2 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 [12, 13], 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 strong-field 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 post-Newtonian approach [44]) 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  [45]. 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.

1.3 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 longest-standing puzzles in physics [46, 47]. Given that the evidence for DM is – so far – purely gravitational, further clues may well be hidden in strong-gravity regions or GW signals generated by dynamical compact objects.

As an example, new fundamental fields (such as axions, axion-like particles, etc [48, 49]), either minimally or non-minimally 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 self-gravitating 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 mass222A crucial property of BHs in GR is that – owing to the scale-free 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 stellar-mass (or even possibly primordial) to supermassive. It is extremely challenging to reproduce this property with a material body, since matter fields introduce a scale. which is regulated by the mass of the fundamental boson itself and by possible self-interaction terms; they form naturally through gravitational collapse and may cluster around an ultracompact configuration through “gravitational cooling” [50, 51, 52, 53].

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 [54, 55, 56].

1.4 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 co-exist 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 beyond-the-Standard-Model 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 XXth 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.

1.5 The small -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 self-gravitating object is to a BH in a rigorous way, by introducing a “closeness” parameter , such that corresponds to the BH limit. As we shall discuss, there are several choices for , for example the tidal deformability, the inverse of the maximum redshift in the spacetime, or a quantity related to the compactness such as , where is the object mass in the static case and is its radius.

In the context of DM self-gravitating objects 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 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 limit. Thus, the small -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 spacetimes. In this regard, the limit can be compared to large spacetime-dimensionality limit in Einstein field equations [57], or even the large limit in QCD [58].

2 Structure of stationary compact objects

“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

The precise understanding of the nature of dark, massive and compact objects can follow different routes,

  • a pragmatic approach of testing the spacetime close to compact, dark objects, irrespective of their nature, by devising model-independent observations that yield unambiguous answers; this often requires consistency checks and null-hypothesis tests of the Kerr metric.

  • a less ambitious and more theoretically-driven 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.

In practice, when dealing with outstanding problems where our ignorance is extreme, pursuing both approaches simultaneously is preferable. Indeed, using concrete models can sometimes be a useful guide to learn about broad, model-independent signatures. As it will become clear, one can design exotic horizonless models which mimic all observational properties of a BH with arbitrary accuracy. While the statement “BHs exist in our Universe” is fundamentally unfalsifiable, alternatives can be ruled out with a single observation, just like Popper’s black swans [59].

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

Figure 1: An equatorial slice of a very compact object, together with the most significant (from a geodesic perspective) locations. At large distances away from the central region, physics is nearly Newtonian: planets – such as the small dot on the figure – can orbit on stable orbits. The external gray area is the entire region where stable circular motion is possible. At the innermost stable circular orbit (), timelike circular motion is marginally stable, and unstable as one moves further within. High-frequency EM waves or GWs can be on circular orbit in one very special location: the light ring (). Such motion is unstable, and can also be associated with the “ringdown” excited during mergers. For horizonless objects, as one approaches the geometric center another significant region may appear: a second, stable light ring. Once rotation is turned on, regions of negative energy (“ergoregions”) are possible. The astrophysical properties of a dark compact object depends on where in this diagram its surface is located.

For simplicity, let us start with the case of a spherically symmetric object and assume that, at least its exterior, is described by GR. Static, spherically symmetric spacetimes are described (in standard coordinates with being the areal radius) by the line element

(1)

with . Birkhoff’s theorem guarantees that any vacuum, spherically-symmetric spacetime (in particular, the exterior an isolated compact, spherically-symmetric object) is described by the Schwarzschild geometry, for which

(2)

and is the total mass of the spacetime (we use geometrical units, except if otherwise stated).

2.1.1 Event horizons, trapped surfaces, apparent horizons

A BH owns its name [60] 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 [61, 62]. In asymptotically-flat spacetime, a BH is the set of events from which no future-pointing 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 splitting of spacetime, a trapped surface is defined as a smooth closed -surface on the slice whose future-pointing outgoing null geodesics have negative expansion [61, 63, 64]. 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 [61]. 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 Refs. [65, 66, 67] for possible quantum effects).

In practice, we will be dealing mostly with quasi-stationary 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 quasi-stationary solution. Notwithstanding, there is no direct observable associated to the horizon [68, 38, 69]. 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

“Alas, I abhor informality.”

That Mitchell and Webb Look, Episode 2


Since we will mostly be discussing objects which look like BHs on many scales, it is useful to introduce a “closeness” parameter 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., Refs. [70, 29]). 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 , then is defined as

(3)

We are thus guaranteed that when , a BH spacetime is recovered. For spherical objects the above definition is coordinate-independent ( is the proper equatorial circumference of the object). Furthermore, one can also define the proper distance between the surface and , , which is directly related to . Some of the observables discussed below show a dependence on , making the distinction between radial and proper distance irrelevant.

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 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 (measurable by our detectors) that a radial-directed light signal takes to travel from the light ring to the surface of the object. For spherically symmetric spacetimes, there is a one-to-one correspondence with the parameter, , where the last step is valid when . In the rest, when convenient, we shall refer to this time scale rather than to .

Overall, we shall use the magnitude of to classify different models of dark objects. A neutron star has 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 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 . For instance, in certain models or the proper distance are of the order of the Planck length ; in such case or even smaller. These models are more challenging to rule out.

Finally, in dynamical situations might be effectively time dependent. Even when at equilibrium, off-equilibrium configurations might have significantly large (see, e.g., Refs. [71, 72, 73, 74]).

2.1.3 Quantifying the softness of dark objects: the curvature parameter

In addition the closeness parameter , another important property of a dark object is its curvature scale. The horizon introduces a cut-off which limits the curvature that can be probed by an external observer. For a BH the largest curvature (as measured by the Kretschmann scalar ) occurs at the horizon and reads

(4)

For astrophysical BHs the curvature at the horizon is always rather small, and it might be large only if sub- primordial BHs exist in the universe. As a reference, the curvature at the center of an ordinary neutron star is .

By comparison with the BH case, one can introduce two classes of models [75]: (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 near-surface 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, , such that . In these models high-energy effects drastically modify the near-surface geometry (hence their “hardness”). An example are certain classes of wormholes (see Section 3).

An interesting question is whether the maximum curvature depends on . Indeed, an ECO with a surface just above the BH limit ) may always require large internal stresses in order to prevent its collapse, so that the curvature in the interior is very large, even if the exterior is exactly the Schwarzschild geometry. In other words, an ECO can be soft in the exterior but hard in the interior. Examples of this case are thin-shell gravastars and strongly anisotropic stars (see Section 3). Thus, according to this classification all ECOs might be “hard” in the limit. Likewise, the exterior of hard ECOs might be described by soft ECO solutions far from the surface, where the curvature is perturbatively close to that of a BH.

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 spherically-symmetric spacetime.

The geodesic motion of timelike or null particles in the geometry (1) can be described with the help of two conserved quantities, the specific energy and angular momentum , where a dot stands for a derivative with respect to proper time [76]. The radial motion can be computed via a normalization condition,

(5)

where for timelike or null geodesics, respectively. The null limit can be approached letting and rescaling all quantities appropriately. Circular trajectories are stable only when , and unstable for smaller radii. The surface defines the innermost stable circular orbit (ISCO), and has an important role in controlling the inner part of the accretion flow onto compact objects. It corresponds to the orbital distance at which a geometrically thin accretion disk is typically truncated [77] and it sets the highest characteristic frequency for compact emission region (“hotspots”) orbiting around accreting compact objects [78, 79].

Figure 2: A source (for example, a star) emits photons in all directions in a region of spacetime where a compact object exists (black circle). Photons with high impact parameter are weakly bent (dashed, black curve), while those with small impact parameter (short-dashed blue) are absorbed and hit the object. The separatrix corresponds to photons that travel an infinite amount of time around the light ring (solid red curve) before being scattered or absorbed. Such critical photons have an impact parameter  [76].

Another truly relativistic feature is the existence of circular null geodesics, i.e., of circular motion for high-frequency EM waves or GWs. In the Schwarzschild geometry, a circular null geodesic is possible only at  [76]. This location defines a surface called the photon sphere, or, on an equatorial slice, a light ring. The photon sphere has a number of interesting properties, and is useful to understand certain features of compact spacetimes.

For example, assume that an experimenter far away throws (high-frequency) 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 so-called shadow, see Sec. 2.2 below.

The photon sphere also has a bearing on the spacetime response to any type of high-frequency waves, and therefore describes how high-frequency GWs linger close to the horizon. At the photon sphere, . Thus, circular null geodesics are unstable: a displacement of null particles grows exponentially [80, 81]

(6)

A geodesic description anticipates that light or GWs may persist at or close to the photon sphere on timescales . Because the geodesic calculation is local, these conclusions hold irrespectively of the spacetime being vacuum all the way to the horizon or not.

For any regular body, the metric functions are well behaved at the center, never change sign and asymptote to unity at large distances. Thus, the effective potential 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 [82, 83, 84]. Inside this region, there is stable timelike circular motion everywhere333Incidentally, this also means that the circular timelike geodesic at is not really the “innermost stable circular orbit”. We use this description to keep up with the tradition in BH physics..

2.1.5 Photon spheres

An ultracompact object with surface at , with , features exactly the same geodesics and properties close to its photon sphere as BHs. From Eq. (6), we immediately realize that after a three -fold timescale, , the amplitude of the original signal is only of its original value. On these timescales one can say that the signal died away. If on such timescales the ingoing part of the signal did not have time to bounce off the surface of the object and return to the light ring, then for an external observer the relaxation is identical to that of a BH. This amounts to requiring that

(7)

Thus, the horizon plays no special role in the response of high frequency waves, nor could it: it takes an infinite (coordinate) time for a light ray to reach the horizon. The above threshold on is a natural sifter between two classes of compact, dark objects. For objects characterized by , light or GWs can make the roundtrip from the photon sphere to the object’s surface and back, before dissipation of the photon sphere modes occurs. For objects satisfying (7), the waves trapped at the photon sphere relax away by the time that the waves from the surface hit it back.

We can thus use the properties of the ISCO and photon sphere to distinguish between different classes of models:

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

  • Ultracompact object (UCO) [85]: a compact object that features a photon sphere,  (). For these objects, the phenomenology related to the photon sphere might be very similar to that of a BH;

  • Clean-photon sphere object (ClePhO): an ultracompact object which satisfies condition (7) and therefore has a “clean” photon sphere,  (). The early-time dynamics of ClePhOs is expected to be the same as that of BHs. At late times, ClePhOs should display unique signatures of their surface.

An ECO can belong to any of the above categories. There are indications that the photon sphere is a fragile concept and that it suffers radical changes in the presence of small environmental disturbances [86]. The impact of such result on the dynamics on compact objects is unknown.

2.2 Escape trajectories and shadows

Figure 3: Left: Critical escape trajectories of radiation in the Schwarzschild geometry. A locally static observer (located at ) emits photons isotropically, but those emitted within the colored conical sectors will not reach infinity. Right: Coordinate roundtrip time of photons as a function of the emission angle and for .

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. More rigorously, the shadow of a BH is the region in the observer’s sky comprising the directions of photons that asymptotically approach the event horizon when traced back in time. This property is intimately related with the existence of a photon sphere.

Indeed, according to Eq. (5), there exists a critical value of the angular momentum for a light ray to be able to escape to infinity. By requiring that a light ray emitted at a given point will not find turning points in its motion, Eq. (5) yields  [76]. Suppose that the light ray is emitted by a locally static observer at . In the local rest frame, the velocity components of the photon are [87]

(8)

where . With this, one can easily compute the escape angle, . In other words, the solid angle for escape is

(9)

where the last step is valid for . For angles larger than these, the light ray falls back and either hits the surface of the object, if there is one, or will be absorbed by the horizon. The escape angle is depicted in Fig. 3 for different emission points . The rays that are not able to escape reach a maximum coordinate distance,

(10)

This result is accurate away from , whereas for the photon approaches the photon sphere (). The coordinate time that it takes for photons that travel initially outward, but eventually turn back and hit the surface of the object, is shown in Fig. 3 as a function of the locally measured angle , and is of order for most of the angles , for . A closed form expression away from , which describes well the full range (see Fig. 3) reads

(11)

When averaging over , the coordinate roundtrip time is then , for any , where “Cat” is Catalan’s constant. Remarkably, this result is independent of in the limit.

In other words, part of the light coming from behind a UCO is “trapped” by the photon sphere. If the central object is a good absorber, an observer staring at the object sees a “hole” in the sky with radius . On the other hand, radiation emitted near the surface of the object (as for example due to an accretion flow) can escape to infinity, with an escape angle that vanishes as in the limit. This simple discussion anticipates that the shadow of a non-accreting UCO can be very similar to that of a BH, and that the accretion flow from ECOs with can also mimic that from an accreting BH [88].

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 counter-rotating 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 infinite-redshift surface outside a horizon is called an ergosurface and is the boundary of the so-called ergoregion. In a stationary spacetime, this boundary is defined by the roots of . Since the Killing vector becomes spacelike in the ergoregion, , 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 co-rotate with the spacetime due to strong frame-dragging effects. Owing to this property, negative-energy (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. [89]. 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 [89]. On the other hand, a spacetime with an ergoregion but without an event horizon is linearly unstable (see Sec. 4.3).

2.3.2 Multipolar structure

As a by-product of the BH uniqueness and no-hair theorems [90, 61] (see also [91, 92, 93]), the multipole moments of any stationary BH in isolation can be written as [94],

(12)

where () are the Geroch-Hansen mass (current) multipole moments [95, 94], the suffix “BH” refers to the Kerr metric, and

(13)

is the dimensionless spin. Equation (12) implies that () vanish when is odd (even), and that all moments with can be written only in terms of the mass and angular momentum (or, equivalently, ) of the BH. Therefore, any independent measurement of three multipole moments (e.g. the mass, the spin and the mass quadrupole ) provides a null-hypothesis test of the Kerr metric and, in turn, it might serve as a genuine strong-gravity confirmation of GR [96, 97, 98, 99, 100, 47, 101].

The vacuum region outside a spinning object is not generically described by the Kerr geometry, due to the absence of an analog to Birkhoff’s theorem in axisymmetry (for no-hair results around horizonless objects see Ref. [75, 102]). Thus, the multipole moments of an axisymmetric ECO will generically satisfy relations of the form

(14)
(15)

where and are model-dependent corrections, whose precise value can be obtained by matching the metric describing the interior of the object to that of the exterior.

For models of ECOs whose exterior is perturbatively close to Kerr, it has been conjectured that in the limit, the deviations from the Kerr multipole moments (with ) vanish as [75]

(16)
(17)

or faster, where , , , and are model-dependent numbers which satisfy certain selection and rules [75]. The coefficients and are related to the spin-induced contributions to the multipole moments and are typically of order unity or smaller, whereas the coefficients and are related to the nonspin-induced contributions. It is worth mentioning that, in all ECO models known so far, . For example, for ultracompact gravastars for any , () for odd (even) values of , and the first nonvanishing terms are  [103] and  [104].

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 [75]. The precise way in which the multipoles die depends on whether they are induced by spin or by other moments.

Note that the scaling rules (16) and (17) imply that in this case a quadrupole moment measurement will always be dominated by the spin-induced contribution, unless

(18)

For all models known so far, so obviously only the spin-induced contribution is important. Even more in general, assuming , the above upper bound is unrealistically small when , e.g. when . This will always be the case, unless some fine-tuning of the model-dependent coefficients occurs.

3 ECO taxonomy: from DM to quantum gravity

A nonexhaustive summary of possible self-gravitating compact objects is shown in Table 1. Different objects arise in different contexts. We refer the reader to specific works (e.g., Ref. [105]) for a more comprehensive review of the models.

Model Formation Stability EM signatures GWs
Fluid stars
[87] [85, 106, 82, 107, 108, 109, 110] [106, 111, 82, 109]
Anisotropic stars
[112, 113, 114] [115, 116, 117] [116, 117, 112]
Boson stars &
oscillatons [118, 50, 119, 120, 51] [121, 122, 123, 124, 125, 83] [88, 126, 127] [128, 129, 130, 131, 132, 133, 134, 135]
Gravastars
[136, 124] [137, 138, 139] [140, 141, 142, 143, 144, 145, 132, 139, 133, 135, 109, 110]
AdS bubbles
[146] [146]
Wormholes
[147, 148] [149, 150, 151, 152] [145, 133, 135]
Fuzzballs
(but see [153, 154, 155, 156]) (but see [145, 132, 157])
Superspinars
[158, 159] (but see [160]) [145, 132]
holes
(but see [161]) (but see [161]) [145, 132]
Collapsed
polymers (but see [162, 163]) [164]  [163]
Quantum bounces /
Dark stars (but see [165, 166])  [167]
Compact quantum
objects [168, 169, 70] [35]
Firewalls
[170, 132]
Table 1: Catalogue of some proposed horizonless compact objects. A tick means that the topic was addressed. With the exception of boson stars, however, most of the properties are not fully understood yet. The symbol stands for incomplete treatment. An asterisk stands for the fact that these objects are BHs, but could have phenomenology similar to the other compact objects in the list.

3.1 A compass to navigate the ECO atlas: Buchdahl’s theorem

Within GR, Buchdahl’s theorem states that, under certain assumptions, the maximum compactness of a self-gravitating object is (i.e., [171]. This result prevents the existence of ECOs with compactness arbitrarily close to that of a BH. A theorem is only as good as its assumptions; one might “turn it around” and look at the assumptions of Buchdahl’s theorem to find possible ways to evade it444A similar approach is pursued to classify possible extensions of GR [99].. More precisely, Buchdahl’s theorem assumes that [172]:

  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, ;

  5. The radial pressure and energy density are non-negative, , .

  6. The energy density decreases as one moves outwards, .

Giving up each of these assumptions (or combinations thereof) provides a way to circumvent the theorem and suggests a route to classify ECOs based on which of the underlying assumptions of Buchdahl’s theorem they violate (see Fig. 4).

Figure 4: Buchdahl’s theorem deconstructed.

3.2 Self-gravitating fundamental fields

One of the earliest and simplest known examples of a self-gravitating compact configuration is that of a (possibly complex) minimally-coupled massive scalar field , described by the action

(19)

The mass of the scalar is related to the mass parameter as , and the theory is controlled by the dimensionless coupling

(20)

where is the total mass of the bosonic configuration.

Self-gravitating solutions for the theory above are broadly referred to as boson stars, and can be generalized through the inclusion of nonlinear self-interactions [173, 174, 175, 118, 176, 125, 177] (see Refs. [178, 179, 51, 83] for reviews). If the scalar is complex, there are static, spherically-symmetric geometries, while the field itself oscillates [173, 174] (for reviews, see Refs. [178, 179, 51, 83]). Analogous solutions for complex massive vector fields were also shown to exist [125]. Recently, multi-oscillating 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 [180]. On the other hand, real scalars give rise to long-term stable oscillating geometries, but with a non-trivial time-dependent stress-energy tensor, called oscillatons [118]. Both solutions arise naturally as the end-state of gravitational collapse [118, 181, 119], and both structures share similar features.

Static boson stars form a one-parameter family of solutions governed by the value of the bosonic field at the center of the star. The mass displays a maximum above which the configuration is unstable against radial perturbations, just like ordinary stars. The maximum mass and compactness of a boson star depend strongly on the boson self-interactions. As a rule of thumb, the stronger the self-interaction the higher the maximum compactness and mass of a stable boson stars [179, 51] (see Table 2).

Model
Potential
Maximum mass
Minimal [173, 174]
Massive [182]
Solitonic [183]
Table 2: Scalar potential and maximum mass for some scalar boson-star models. In our units, the scalar field is dimensionless and the potential has dimensions of an inverse length squared. The bare mass of the scalar field is . For minimal boson stars, the scaling of the maximum mass is exact. For massive boson stars and solitonic boson stars, the scaling of the maximum mass is approximate and holds only when and when , respectively. Adapted from Ref. [133].

The simplest boson stars are moderately compact in the nonspinning case [83, 125, 184]. Their mass-radius relation is shown in Fig. 5. Once spin [184] or nonlinear interactions [182, 83, 183] are added, boson star spacetimes can have light rings and ergoregions. The stress-energy tensor of a self-interacting bosonic field contains anisotropies, which in principle allow to evade naturally Buchdahl’s theorem. However, there are no boson-star solutions which evade the Buchdahl’s bound: in the static case, the most compact configuration has ([129].

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) [129, 184].

Because of their simplicity and fundamental character, boson stars are interesting on their own. A considerable interest in their properties arose with the understanding that light scalars are predicted to occur in different scenarios, and ultralight scalars can explain the DM puzzle [185]. Indeed, dilute bosonic configurations provide an alternative model for DM halos.

Figure 5: Left: Comparison between the total mass of a boson star (complex scalar or vector fields) and an oscillaton (real scalar or vector fields), as a function of their radius . is defined as the radius containing 98% of the total mass. The procedure to find the diagram is outlined in the main text. From Ref. [120]. Right: Mass-radius diagram for nonspinning fluid stars in GR. The red dashed (blue dotted) lines are ordinary NSs (quark stars) for several representative equations of state [186, 187] (data taken from [188]); the black continuous lines are strongly-anisotropic stars [112]. Note that only the latter have photon spheres in their exterior and violate Buchdahl’s bound.

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 [174, 87]. 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 [87].

When the stresses are assumed to be isotropic, static spheres in GR made of ordinary fluid satisfy the Buchdahl limit on their compactness,  [171]; 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, , and in practice such objects would behave as ClePhOs [189]. In addition, polytrope stars with a light ring (sometimes referred to as ultra-compact stars) always have superluminal sound speed [107]. 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 [85]. The mass-radius relation for a standard neutron star is shown in Fig. 5.

Some fermion stars, such as neutron stars, live in DM-rich environments. Thus, DM can be captured by the star due to gravitational deflection and a non-vanishing cross-section for collision with the star material [190, 191, 192, 193, 192]. 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 [194, 195, 196, 197, 198, 199] or real scalar or vector field [52, 120] 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 [200, 201, 202]. 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 [203]. These might arise in a variety of contexts: at high densities [204, 205, 206], when EM or fermionic fields play a role, or in pion condensed phase configurations in neutron stars [207], superfluidity [208], solid cores [204], etc. In fact, anisotropy is common and even a simple soap bubble support anisotropic stresses [209]. Anisotropic stars were studied in GR, mostly at the level of static spherically symmetric solutions [210, 211, 212, 213, 214, 113, 215, 203, 216, 114, 115, 217, 116, 218]. 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 [219, 220, 112].

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 stellar-mass and supermassive BH candidates across several orders of magnitude in mass [112]. Such property of BHs in GR, visible in Fig. 5, is a consequence of the scale-free 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 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 well-defined theoretical motivation, but for which the formation mechanisms are not well understood.

Wormholes were originally introduced as a useful tool to teach GR, its mathematical formalism and underlying geometric description of the universe [221, 222, 223]. Wormholes connect different regions of spacetime. They are not vacuum spacetimes and require matter. The realization that wormholes can be stabilized and constructed with possibly reasonable matter has attracted a considerable attention on these objects [222, 223, 224].

Figure 6: Embedding-like diagram of a wormhole connecting two different asymptotically-flat universes. The black solid line denotes the wormhole’s throat. There are two light rings in the spacetime, one for which universe.

Different wormhole spacetimes can have very different properties. Since we are interested in understanding spacetimes that mimic BHs, consider the following two simple examples of a non-spinning geometries [222, 225, 145]. In the first example, we simply take the Schwarzschild geometry describing a mass down to a “throat” radius . At , we “glue” such spacetime to another copy of Schwarzschild. In Schwarzschild coordinates, the two metrics are identical and described by

(21)

Because Schwarzschild’s coordinates do not extend to , we use the tortoise coordinate , to describe the full spacetime, where the upper and lower sign refer to the two different universes connected at the throat. Without loss of generality we assume , so that one domain is whereas the other domain is . The surgery at the throat requires a thin shell of matter with surface density and surface pressure [226]

(22)

Although the spacetime is everywhere vacuum (except at the throat) the junction conditions force the pressure to be large when the throat is close to the Schwarzschild radius.

A similar example, this time of a non-vacuum spacetime, is the following geometry [225]

(23)

The constant is assumed to be extremely small, for example where is the Planck length. There is no event horizon at , such location is now the spacetime throat. Note that, even though such spacetime was constructed to be arbitrarily close to the Schwarzschild spacetime, the throat at is a region of large (negative) curvature, for which the Ricci and Kretschmann invariant are, respectively,

(24)

Thus, such invariants diverge at the throat in the small -limit.

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 [227, 228, 229, 230, 231]. In such theories, energy conditions might be satisfied [232].

3.6 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 back-reaction on the geometry is negligible [233]. Quantum effects on collapsing horizonless geometries (and the possibility of halting collapse to BHs altogether) are less clear [234, 235, 236, 237, 238, 239, 240]. There are arguments that semiclassical effects might suffice to halt collapse and to produce dark stars, even for macroscopic configurations [241, 242, 28, 243, 244, 245, 246], but see Ref. [236] for counter-arguments. For certain conformal fields, it was shown that a possible end-state 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 [21, 32].

3.7 Gravastars

Similar ideas that led to the proposal of “dark stars” were also in the genesis of a slightly different object, “gravitational-vacuum stars” or gravastars [247, 22]. These are configurations supported by a negative pressure, which might arise as an hydrodynamical description of one-loop QFT effects in curved spacetime, so they do not necessarily require exotic new physics [248]. In these models, the Buchdahl limit is evaded both because the internal effective fluid is anisotropic [249] and because the pressure is negative (and thus violates some of the energy conditions [250]). Gravastars have been recently generalized to include anti-de Sitter cores, in what was termed AdS bubbles, and which may allow for holographic descriptions [146, 251]. 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 five-layer construction, with an interior de Sitter core, a thin shell connecting it to a perfect-fluid region, and another thin-shell connecting it to the external Schwarzschild patch. A simpler construction that features all the main ingredients of the original gravastar proposal is the thin-shell gravastar [136], in which a de Sitter core is connected to a Schwarzschild exterior through a thin shell of perfect-fluid matter. Gravastars can also be obtained as the BH-limit of constant-density stars, past the Buchdahl limit [250, 252]. It is interesting that such stars were found to be dynamically stable in this regime [252]. It has been conjectured that gravastars are a natural outcome of the inflationary universe [253], or arising naturally within the gauge-gravity duality [146, 251].

3.8 Fuzzballs and collapsed polymers

So far, quantum effects were dealt with at a semi-classical 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 ( is Boltzmann’s constant and is the BH area). In other words, what is the statistical-mechanical 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 strongly-coupled supergravity description, can be reproduced in a weakly-coupled -brane description as the degeneracy of the relevant microstates [254, 255, 256, 257, 258].

Somewhat surprisingly, the geometric description of individual microstates seems to be regular and horizonless [259, 23, 258, 260, 261]. 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 coarse-grained 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 [262, 263]. 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 higher-dimensional or in non asymptotically-flat spacetimes. Explicit regular, horizonless microstate geometries for asymptotically flat, four-dimensional 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 Refs. [162, 72]; 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 collapse-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 is naturally of the order for masses in the range .

3.9 “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 of BHs is bounded from above by . 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 [264], 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 closed-timelike 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 [265], but a complete theory is necessary to understand any possible signature.

3.10 holes and other geons

As we remarked already, the questioning of the BH paradigm in GR comes hand in hand with the search for an improved theory of the gravitational interaction, and of possible quantum effects. A natural correction to GR would take the form of higher-curvature terms in the Lagrangian with couplings suppressed by some scale [266, 267, 268]. The study of (shell-like) matter configurations in such theories revealed the existence novel horizonless configurations, termed “2-2-holes”, which closely matches the exterior Schwarzschild solution down to about a Planck proper length of the Schwarzschild radius of the object [161]. In terms of the parameter introduced above, the theory predicts objects where  [161]. The existence and stability of proper star-like configurations was not studied. More generic theories result in a richer range of solutions, many of which are solitonic in nature and can be ultracompact (see e.g. Refs. [269, 270, 271, 272, 273]. Recently, a quantum mechanical framework to describe astrophysical, horizonless objects devoid of curvature singularities was put forward in the context of nonlocal gravity (arising from infinite derivative gravity) [274, 275, 238]. The corresponding stars can be ultracompact, although never reaching the ClePhO category.

3.11 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 near-horizon region, such that the object still looks like a regular GR BH [30, 276, 35]. However, the changes could also be drastic and involve “hard” structures localized close to the horizon such as firewalls and other compact quantum objects [31, 277, 35]. Alternatively, a classical BH with modified dispersion relations for the graviton could effectively appear as having a hard surface [278, 279]. 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 quantum-gravity or microscopic corrections at the horizon scale select ClePhOs as well-motivated 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.

Figure 7: Schematic representation of ECO models in a compactness-curvature diagram. The horizontal axis shows the compactness parameter associated to the object, which can be also mapped (in a model-dependent way) to a characteristic light-crossing timescale. The vertical axis shows the maximum curvature (as measured by the Kretschmann scalar ) of the object normalized by the corresponding quantity for a BH with the same mass . All known ECO models with have large curvature in their interior, i.e. the leftmost bottom part of the diagram is conjectured to be empty. Angular momentum tends to decrease and to increase .

4 Dynamics of compact objects

“There is a crack in everything. That’s how the light gets in.” Leonard Cohen, Anthem (1992)

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 [170]. In addition, the wavelength of EM waves of interest for Earth-based 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 Sec. 5 below.

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 [280, 281, 282]: the inspiral stage, corresponding to large separations and well approximated by post-Newtonian 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 end-product relaxes to a stationary, equilibrium solution of the field equations [282, 283, 284]. 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 spherically-symmetric 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 late-stage 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).

In the linearized regime, the geometry can be written as , where is the geometry corresponding to the stationary object, and are the small deviations induced on it by whatever is causing the dynamics. The metric fluctuations can be combined in a single master function which in vacuum is governed by a master partial differential equation of the form [285, 283]

(25)

where is a suitable coordinate. The source term contains information about the cause of the disturbance . The information about the angular dependence of the wave is encoded in the way the separation was achieved, and involves an expansion in tensor harmonics. One can generalize this procedure and consider also scalar or vector (i.e., EM) waves. These can also be reduced to a master function of the type (25), and separation is achieved with spin- harmonics for different spins of field. These angular functions are labeled by an integer . For a Schwarzschild spacetime, the effective potential is

(26)

with for scalar, vector or (axial) tensor modes. The equation does not describe completely all of the gravitational degrees of freedom. There is an another (polar) gravitational mode (in GR, there are two polarizations for GWs), also described by Eq. (25) with a slightly more complicated potential [286, 106, 283]. Note that such results apply only when there are no further degrees of freedom that couple to the GR modes [287, 288, 289, 290, 291].

The solutions to Eq. (25) depend on the source term and initial conditions, just like for any other physical system. We can gain some insight on the general properties of the system by studying the source-free equation in Fourier space. This corresponds to studying the “free” compact object when the driving force died off. As such, it gives us information on the late-time behavior of any compact object. By defining the Fourier transform through , one gets the following ODE

(27)

For a Schwarzschild spacetime, the “tortoise” coordinate is related to the original by , i.e.

(28)

such that diverges logarithmically near the horizon. In terms of , Eq. (27) is equivalent to the time-independent Schrödinger equation in one dimension and it reduces to the wave equation governing a string when . For a string of length with fixed ends, one imposes Dirichlet boundary conditions and gets an eigenvalue problem for . The boundary conditions can only be satisfied for a discrete set of normal frequencies, (). The corresponding wavefunctions are called normal modes and form a basis onto which one can expand any configuration of the system. The frequency is purely real because the associated problem is conservative.

If one is dealing with a BH spacetime, the appropriate conditions (required by causality) correspond to having waves traveling outward to spatial infinity ( as ) and inwards to the horizon ( as ) [see Fig. 8]. The effective potential displays a maximum approximately at the photon sphere, , the exact value depending on the type of perturbation and on the value of ( in the limit). Due to backscattering off the effective potential (26), the eigenvalues are not known in closed form, but they can be computed numerically [286, 106, 283]. The fundamental mode (the lowest dynamical multipole in GR) of gravitational perturbations reads [292]

(29)

Remarkably, the entire spectrum is the same for both the axial or the polar gravitational sector [286]. The frequencies are complex and are therefore called quasinormal mode (QNM) frequencies. Their imaginary component describes the decay in time of fluctuations on a timescale , and hints at the stability of the geometry. Unlike the case of a string with fixed end, we are now dealing with an open system: waves can travel to infinity or down the horizon and therefore it is physically sensible that any fluctuation damps down. The corresponding modes are QNMs, which in general do not form a complete set [106].

Figure 8: Typical effective potential for perturbations of a Schwarzschild BH (top panel) and of an horizonless compact object (bottom panel). The effective potential is peaked at approximately the photon sphere, . For BHs, QNMs are waves which are outgoing at infinity () and ingoing at the horizon (), whereas the presence of a potential well (provided either by a partly reflective surface, a centrifugal barrier at the center, or by the geometry) supports quasi-trapped, long-lived modes.

Boundary conditions play a crucial role in the structure of the QNM spectrum. If a reflective surface is placed at , where (say) Dirichlet or Neumann boundary conditions have to be imposed, the spectrum changes considerably. The QNMs in the , low-frequency limit read [293, 294, 295]

(30)
(31)

where , is a positive odd (even) integer for polar (axial) modes (or equivalently for Dirichlet (Neumann) boundary conditions), and  [296, 89, 295]. Note that the two gravitational sectors are no longer isospectral. More importantly, the perturbations have smaller frequency and are much longer lived, since a decay channel (the horizon) has disappeared. For example, for we find numerically the fundamental scalar modes

(32)
(33)

These QNMs were computed by solving the exact linearized equations numerically but agree well with Eqs. (30) and (31).

The above scaling with can be understood in terms of modes trapped between the peak of the potential (26) at and the “hard surface” at  [145, 132, 109, 297, 295] [see Fig. 8]. Low-frequency waves are almost trapped by the potential, so their wavelength scales as the size of the cavity (in tortoise coordinates), , just like the normal modes of a string. The (small) imaginary part is given by waves which tunnel through the potential and reach infinity. The tunneling probability can be computed analytically in the small-frequency regime and scales as  [296]. After a time , a wave trapped inside a box of size is reflected times, and its amplitude reduces to . Since, in this limit, we immediately obtain

(34)

This scaling agrees with exact numerical results and is valid for any and any type of perturbation.

The reverse-engineering of the process, i.e., a reconstruction of the scattering potential from a mode measurement was proposed in Ref. [110, 298, 299]. 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 low-frequency waves are reflected, whereas higher-frequency waves probe the internal structure of the specific object [40, 300]. 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 Sec. 4.3.

4.2 Gravitational-wave echoes

4.2.1 Quasinormal modes, photon spheres, and echoes

The effective potential for wave propagation reduces to that for geodesic motion () in the high-frequency, high-angular momentum (i.e., eikonal) regime. Thus, some properties of geodesic motion have a wave counterpart [80, 81]. The instability of light rays along the null circular geodesic translates into some properties of waves around objects compact enough to feature a photon sphere. A wave description needs to satisfy “quantization conditions”, which can be worked out in a WKB approximation. Since GWs are quadrupolar in nature, the lowest mode of vibration should satisfy

(35)

In addition the mode is damped, as we showed, on timescales . Overall then, the geodesic analysis predicts

(36)

This crude estimate, valid in principle only for high-frequency waves, matches well even the fundamental mode of a Schwarzschild BH, Eq. (29).

Nevertheless, QNM frequencies can be defined for any dissipative system, not only for compact objects or BHs. Thus, the association with photon spheres has limits, for instance it neglects possible coupling terms [287], nonminimal kinetic terms [301], etc. Such an analogy is nonetheless enlightening in the context of objects so compact that they have photon spheres and resemble Schwarzschild deep into the geometry, in a way that condition (7) is satisfied [145, 132, 302, 303].

For a BH, the excitation of the spacetime modes happens at the photon sphere [80]. 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.

Contrast the previous description with the dynamical response of ultracompact objects for which condition (7) is satisfied (i.e., a ClePhO) [cf. Fig. 9]. The initial description of the photon sphere modes still holds, by causality. Thus, up to timescales of the order (the roundtrip time of radiation between the photon sphere and the surface) the signal is identical to that of BHs [145, 132]. At later times, however, the pulse traveling inwards is bound to see the object and be reflected either at its surface or at its center. In fact, this pulse is semi-trapped between the object and the light ring. Upon each interaction with the light ring, a fraction exits to outside observers, giving rise to a series of echoes of ever-decreasing amplitude. From Eqs. (30)–(31), repeated reflections occur in a characteristic echo delay time [145, 132, 303] [see Fig. 10]

(37)

However, the main burst is typically generated at the photon sphere and has therefore a frequency content of the same order as the BH QNMs (29). The initial signal is of high frequency and a substantial component is able to cross the potential barrier. Thus, asymptotic observers see a series of echoes whose amplitude is getting smaller and whose frequency content is also going down. It is crucial to understand that echoes occur in a transient regime; at very late times, the signal is dominated by the lowest-damped QNMs, described by Eqs. (30)–(31).

Figure 9: Ringdown waveform for a BH (dashed black curve) compared to a ClePhO (solid red curve) with a reflective surface at with . We considered axial gravitational perturbations and a Gaussian wavepacket , (with and ) as initial condition. Note that each subsequent echo has a smaller frequency content and that the damping of subsequent echoes is much larger than the late-time QNM prediction ( with for these parameters). Data available online [292].
Figure 10: Schematic Penrose diagram of GW echoes from an ECO. Adapted from Ref. [304] (similar versions of this plot appeared in other contexts in Refs. [293, 297]).

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 “hard-structure” close to the horizon [170, 277, 305], or far from it [170, 306, 307], or embedded in a theory that effectively makes the graviton see a hard wall there [278, 279] 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 [111, 112, 189, 308].

4.2.2 A black-hole representation and the transfer function

The QNMs of a spacetime were defined as the eigenvalues of the homogeneous ordinary differential equation (27). Their role in the full solution to the homogeneous problem becomes clear once we re-write Eq. (25) in Fourier space,

(38)

where is the Fourier transformed source term . Since the potential is zero at the boundaries, two independent homogeneous solutions are

(39)

and

(40)

Note that was chosen to satisfy outgoing conditions at large distances; this is the behavior we want to impose on a system which is assumed to be isolated. On the other hand, satisfies the correct near-horizon boundary condition in the case of a BH. Define reflection and transmission coefficients,

(41)

Given the form of the ODE, the Wronskian is a constant (here ), which can be evaluated at infinity to yield . The general solution to our problem can be written as [309]

(42)

where constants. If we impose the boundary conditions appropriate for BHs, we find

(43)

This is thus the response of a BH spacetime to some source. Notice that close to the horizon the first term drops and . For detectors located far away from the source, on the other hand . It is easy to see now that QNMs correspond to poles of the propagator [283], and hence they do indeed have a significant contribution to the signal, both at infinity and near the horizon.

Consider now that instead of a BH, there is an ultracompact object. Such object has a surface at , corresponding to large negative tortoise . Then, the boundary condition (39) on the left needs to be changed to

(44)

where we assume the compact object surface to have a (possible frequency-dependent) reflectivity . We will now show that the spacetime response to an ultracompact object can be expressed in terms of the BH response and a transfer function [297] (for a previous attempt along these lines see Ref. [310]). First, notice that the ODE to be solved is exactly the same, with different conditions on one of the boundaries. We can thus still pick the two independent homogeneous solutions (39) and (40), but choose different integration constants in (42) so that the boundary condition is satisfied. Adding a homogeneous solution to (43) is allowed since it still satisfies outgoing conditions at large spatial distances. We then find at small negative

(45)

where is a constant. On the other hand, to obey the boundary condition (44), one must impose with an unknown constant . Matching outgoing and ingoing coefficients, we find

(46)

Thus, a detector at large distances sees now a signal

(47)

In other words, the signal seen by detectors is the same as the one from a BH, modified by a piece that is controlled by the reflectivity of the compact object.

Following Ref. [297], the extra term can be expanded as a geometric series

(48)

A natural interpretation emerges: a main burst of radiation is generated for example when an object crosses the light ring (where the peak of the effective potential is located). A fraction of this main burst is outward traveling and gives rise to the “prompt” response , which is equivalent to the response of a BH. However, another fraction is traveling inwards. The first term is the result of the primary reflection of at . Note the time delay factor between the first pulse and the main burst due the pulse’s extra round trip journey between the boundary the peak of the scattering potential, close to the light ring at . When the pulse reaches the potential barrier, it is partially transmitted and emerges as a contribution to the signal. The successive terms are “echoes” of this first reflection which bounces an integer number of times between the potential barrier and the compact object surface. Thus, a mathematically elegant formulation gives formal support to what was a physically intuitive picture.

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 Ref. [311]. Recently, such a “transfer-function” representation of echoes was embedded into an effective-field-theory scheme [312], showing that linear “Robin” boundary conditions at dominate at low energies. In this method the (frequency dependent) reflection coefficient and the surface location can be obtained in terms of a single low-energy effective coupling.

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 intermediate-time response is dominated by the BH QNMs, which are not part of the QNM spectrum of an ECO [145, 170, 313]. 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 [145]), 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 late-time 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 Dyson-series representation

The previous analysis showed two important aspects of the late-time behavior of very compact objects: (i) that it can be expressed in terms of the corresponding BH response if one uses a transfer function ; (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.

The response of any system with a non-trivial scattering potential and nontrivial boundary conditions includes echo-like components. To see that, let’s use a very different approach to solve (38), namely the Lippman-Schwinger integral solution used in quantum mechanics [314]. In this approach, the setup is that of flat spacetime, and the scattering potential is treated as a perturbation. In particular, the field is written as

(49)

where

(50)

is the Green’s function of the free wave operator with boundary condition (44), and is the free-wave amplitude. The formal solution of Eq. (49) is the Dyson series

(51)

which effectively works as an expansion in powers of , so we expect it to converge rapidly for high frequencies and to be a reasonable approximation also for fundamental modes. It is possible to reorganize (51) and express it as a series in powers of . We start by separating the Green’s function (50) into , with

(52)

the open system Green’s function, and

(53)

the “reflection” Green’s function. We can then write (49) as

(54)

Now, in the same way as a Dyson series is obtained, we replace the in the third integral with the entirety of the rhs of Eq. (54) evaluated at . Collecting powers of yields

(55)

If we continue this process we end up with a geometric-like series in powers of ,

(56)

with each term a Dyson series itself:

(57)

The reflectivity terms can be re-arranged as: